1 Introduction↩︎

Relativistic quantum field theories of interacting fermions with four-fermion couplings provide a natural framework for the study of dynamical mass generation and spontaneous chiral symmetry breaking. A prototypical example is the Nambu–Jona-Lasinio model [1], [2], originally introduced as an effective theory of interacting nucleons and later recognized as a precursor of modern mechanisms of chiral symmetry breaking. The model has been extensively investigated using a wide range of analytical and non-perturbative techniques, including large-\(N\) expansions [3]–[5], functional renormalization group methods [6], [7], and lattice regularizations [8]. Rigorous results, however, remain comparatively scarce; notable progress includes [9], where chiral symmetry breaking is established using Reflection Positivity techniques.
From the perspective of Mathematical Physics, several rigorous frameworks have been developed to analyze interacting fermionic systems and their symmetry breaking mechanisms. In particular, lattice regularizations and Reflection Positivity play a central role in the constructive study of fermionic field theories [10], while renormalization group methods have been successfully applied to interacting fermions [11]–[14]. Rigorous results on spontaneous symmetry breaking in quantum fermionic systems have also been obtained in the operator-algebraic setting [15]. These developments provide part of the mathematical background for the present work.
The Gross–Neveu model [16] can be viewed as a lower-dimensional analogue of this framework, retaining its essential features while offering a more tractable setting for the analysis of non-perturbative phenomena. In \(1+1\) dimensions, it is renormalizable, asymptotically free, and admits an exact solution in the large-\(N\) limit, where the saddle-point approximation becomes exact. This yields a detailed analytic description of the dynamically generated mass gap and of the spectrum of bound states [16], [17], and establishes the spontaneous breaking of the discrete chiral symmetry, with the fermion condensate \(\langle \overline{\psi} \psi \rangle\) as the natural order parameter.
From a rigorous viewpoint, the Gross–Neveu model has also played an important role in the development of constructive and renormalization group techniques for fermionic quantum field theories. In particular, Kopper, Magnen and Rivasseau [18], [19] carried out a detailed renormalization group analysis, providing mathematically controlled constructions of the theory and clarifying both its ultraviolet (see also [20]–[22]) and infrared behavior.
A complementary line of research aims at establishing non-perturbative properties of interacting fermionic systems using constructive and probabilistic methods. In this context, lattice regularizations provide a natural framework, allowing one to exploit tools from statistical mechanics such as Reflection Positivity and correlation inequalities. Despite extensive analytical and numerical investigations of lattice Gross–Neveu models, a fully rigorous proof of spontaneous symmetry breaking in these systems remains elusive. The main difficulty lies in controlling the infrared behavior and the resulting Long-Range order in the presence of non-local interactions generated by integrating out the fermions, without relying on large-\(N\) techniques.
In this work, we address this problem by proving the existence of Long-Range order (LRO) in a class of lattice regularizations of the two-dimensional Euclidean Gross-Neveu model, with an even number of fermion flavors. We consider three standard lattice discretizations, originally introduced by Cohen, Elitzur and Rabinovici [23], [24], including naive fermions and staggered fermions with different interaction structures. These models correspond to formulations widely used in numerical simulations, see e.g. [25] and the more recent papers [26], [27].
Our approach is based on a Hubbard–Stratonovich transformation, which maps the fermionic theory to an effective bosonic model with an auxiliary scalar field. We prove that the resulting bosonic measures satisfy Reflection Positivity with respect to suitable lattice reflections. This structural property enables the use of chessboard estimates [28], originally developed in the context of classical spin systems and adapted here to the present fermionic setting.
The main result of the paper is the derivation of uniform upper and lower bounds on the two-point correlation function of the auxiliary field, implying Long-Range Order for the fermionic bilinear \(\langle \overline{\psi}\psi\rangle\). The proof combines Reflection Positivity with a Peierls-type argument [29], controlling configurations that interpolate between distinct minima of the effective potential and showing that they are exponentially suppressed in the number of fermion flavors. The bounds are formulated in terms of the minimizers of the effective potential and show that the long-range order parameter converges, up to controlled errors, to its mean-field prediction in the large-\(N\) regime. This provides a rigorous justification of the standard large-\(N\) picture for symmetry breaking in lattice Gross–Neveu models.
A key feature of our analysis is its robustness across different lattice discretizations. Although the models differ in their fermionic content and ultraviolet behavior, the proof applies uniformly once a Hubbard–Stratonovich representation is available. This indicates that the mechanism leading to Long-Range order is not tied to a specific formulation, but rather reflects a structural property of a broader class of fermionic systems admitting an auxiliary-field description.
More precisely, our contributions can be summarized as follows:

1. We establish Reflection Positivity for a class of Bosonized lattice Gross–Neveu models.

2. We adapt Chessboard Estimates to interacting fermionic systems with auxiliary fields.

3. We prove uniform upper and lower bounds on the bosonic two-point function with (i.e. the mass-mass fermionic correlator), implying Long-Range Order.

4. We prove uniform pointwise upper and lower bounds on the bosonic two-point function (equivalently, on the fermionic mass–mass correlator), implying Long-Range Order. Moreover, these bounds show that the bosonic two-point function is uniformly trapped between quantities determined by the minimizers of the effective potential, thereby establishing a direct quantitative connection between the rigorous analysis and the large-\(N\) (mean-field) effective-potential predictions.

.

Our results place earlier numerical and heuristic evidence on a firm mathematical footing and provide a fully rigorous and non-perturbative proof of Long-Range order in lattice versions of the Gross–Neveu model, within this class of interactions admitting an Hubbard–Stratonovich representation. More broadly, they illustrate how methods from classical Statistical Mechanics can be extended to interacting fermionic systems with auxiliary fields, and identify a mechanism of symmetry breaking that is stable across a family of lattice discretizations of independent physical interest.
The paper is organized as follows. In Section 2 we introduce the lattice models and their bosonized formulations and state the main result. In Section 3 we establish Reflection Positivity of the effective measures. In Section 4 we collect the main properties which lead to the Long-Range order, and we show how they imply the main theorem. Section 5 is devoted to the estimates of some remarkable bosonic expectations, based on Chessboard Estimates and Peierls-type arguments, which constitute a main building block for the proof of Long-Range order. Technical or minor results are collected in the appendices.

1.0.0.1 Outlook.

A natural direction for future work is the construction of infinite-volume Gibbs states associated with the symmetry-broken phases and the characterization of spontaneous symmetry breaking in terms of extremal translation-invariant measures. In particular, it would be desirable to complement the present results on Long-Range order with a direct construction of distinct infinite-volume states exhibiting a non-vanishing order parameter.

Another important problem concerns the identification of a dynamically generated mass gap through the decay properties of correlation functions. Establishing exponential decay of suitable truncated correlators would provide a direct characterization of mass generation within the present framework.

1.0.0.2 Acknowledgements.

LG and SF would like to thank Marcello Porta and Alessandro Giuliani for their supervision, insightful discussions, and critical feedback. LG is grateful to Jürg Fröhlich for references on Reflection Positivity and Phase Transitions, to Matteo Delladio for enlightening conversations on the continuum limit of lattice models, and to Amartya Harsh Singh and Alessandro Piazza for discussions on large-\(N\) limits in strongly coupled QTFs.

2 The Models↩︎

2.0.0.1 Conventions.

We work on the two-dimensional discrete torus which we represent with as its fundamental domain given by the square lattice

\[\Lambda_L \doteq \mathbb{Z}^2 \cap \Big(-\frac{L}{2},\frac{L}{2}\Big]^2,\]

with lattice spacing set equal to one and \(L\in4\mathbb{N}\). For the plaquette formulation introduced below we shall additionally assume \(L\in 8\mathbb{N}\). Lattice sites are denoted by \(x=(x_0,x_1)\).
Throughout the paper we consider three lattice realizations of the Gross–Neveu model:

1. The Naive-Fermion model (NN);

2. The Staggered-Fermion model (SN);

3. The Staggered-Fermion Plaquette model (SP).

Despite their different microscopic realizations, the three models share the same overall structure. In each case, we define a lattice Dirac operator \(\cancel{D}_{\alpha}\), introduce a quartic Gross–Neveu-like interaction, and construct the corresponding fermionic Gibbs state. We then analyze the relevant symmetries and spectral properties and derive an equivalent bosonic representation through a Hubbard–Stratonovich transformation. The arguments developed later apply to all three formulations with only minor modifications.

2.0.0.2 General Set-Up.

The Grassmann fields \((\psi,\overline{\psi})\) satisfy anti-periodic boundary conditions in both lattice directions. These boundary conditions ensure that hopping terms crossing the reflection plane enter with the sign required by the Osterwalder–Seiler Reflection Positivity construction.

The fields carry an internal index \(J\), whose meaning depends on the formulation:

\[J= \begin{cases} (s,j) & \alpha=\mathrm{NN}\\ j & \alpha=\mathrm{SN} \\ (a,j) & \alpha=\mathrm{SP} \end{cases}\]

For the naive model, \(s=1,2\) is a spin index and \(j=1,\cdots,N\) is a flavour index. For the staggered model only the flavour index is present. For the plaquette formulation, \(a\in\mathcal{I}=\{A,B,C,D\}\) labels the four internal degrees of freedom associated with a \(2\times2\) unit cell, while \(j\) again denotes the flavour index.
We shall always assume that \(N\) is even. This assumption is used only to guarantee positivity of the effective bosonic measure (see below) obtained after integrating out the fermions. Whether the restriction can be removed would require different techniques and will not be investigated here.
For the NN and SN formulations the fermionic fields are indexed by sites of \(\Lambda_L\). In the plaquette formulation it is convenient to introduce the coarse lattice

\[\widetilde{\Lambda}_L \doteq (2\mathbb{Z})^2 \cap \Big(-\frac{L}{2},\frac{L}{2}\Big]^2,\]

whose sites label the centers of the unit cells. Accordingly, we define:

\[\Lambda_L^\alpha = \begin{cases} \Lambda_L\,\,\,\,\,\,& \alpha = \mathrm{NN}\\ \Lambda_L\,\,\,\,\,\,& \alpha = \mathrm{SN} \\ \widetilde{\Lambda}_L & \alpha = \mathrm{SP} \end{cases}\]

and we denote by \(|\Lambda^{\alpha}_L|\) its cardinality, namely \(|\Lambda^{\alpha}_L|=L^2\) for \(\alpha=\)NN, SN and \(|\Lambda^{\alpha}_L|=L^2/4\) for \(\alpha=\)SP. For notational simplicity, we use a unified notation and make the dependence on the formulation explicit only when needed. In particular, we write: \[(\overline{\psi}\psi)_x \doteq \sum_J \overline{\psi}_{x,J}\psi_{x,J}\] Given a fermionic action \(S_{\alpha}(\overline{\psi},\psi)\), the expectation of an even Grassmann polynomial \(\mathcal{O}(\overline{\psi},\psi)\) is defined by: \[\langle\mathcal{O}\rangle_{\alpha} \doteq \frac{1}{Z^{\alpha}_{\Lambda_L}} \int d\overline{\psi}\,d\psi\; \mathcal{O}(\overline{\psi},\psi)\,e^{-S_{\alpha}(\overline{\psi},\psi)}, \qquad Z^{\alpha}_{\Lambda_L} \doteq \int d\overline{\psi}\,d\psi\;e^{-S_{\alpha}(\overline{\psi},\psi)},\] where \(d\overline{\psi}\,d\psi\) denotes the product of Grassmann differentials over all lattice sites and internal indices (spin, flavor, and unit-cell degrees of freedom) and \(\alpha\) indicates each possible model.

2.0.0.3 Effective bosonic measure.

In all three formulations the interaction is local and depends only on the composite field \((\overline{\psi}\psi)^2_x\). We can therefore introduce an auxiliary real field \(\phi\) through the Hubbard–Stratonovich identity

\[\label{eqn:Hub95Strat} e^{\frac{\lambda}{2N}(\overline{\psi}\psi)_x^2} = \sqrt{\frac{N}{2\pi \lambda}} \int_{-\infty}^{\infty} d\phi_x e^{-\frac{N}{2\lambda}\phi_x^2 + \phi_x(\overline{\psi}\psi)_x}.\tag{1}\]

Applying this transformation independently at every lattice site,integrating out exactly the fermionic fields and absorbing \(\phi\)-independent normalization constants into the partition function, one obtains an effective bosonic measure¹ of the form:

\[\label{eq:boson95measure} d\mu_{\alpha}(\phi) \doteq \frac{1}{Z^{\alpha}_{\Lambda_L}} \prod_{x \in \Lambda_L^{\alpha}} d\phi_x \; e^{-\frac{N}{2\lambda} \phi_x^2} \det((\cancel{D}_{\alpha})^{-}_{\Lambda_L}-M_\phi)^N.\tag{2}\]

Here \((\cancel{D}_{\alpha})^{-}_{\Lambda_L}\) denotes the appropriate lattice Dirac operator (including anti-periodic boundary conditions) used in each model while \(M_\phi\) is the multiplication operator:

\[\begin{align} (M_{\phi})_{(x,s),(y,s')}&\doteq \delta_{s,s'}\delta_{x,y}\phi_x, \quad \forall x,y\in\Lambda_L, \; s,s'\in\{1,2\} \quad \, (\alpha=\mathrm{NN});\\ (M_{\phi})_{x,y}&\doteq \delta_{x,y}\phi_x, \quad \,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall x,y\in\Lambda_L, \quad \quad \quad \quad \quad \,\,\,\,\,\,\,\,(\alpha=\mathrm{SN}); \\ (M_{\phi})_{(x,a),(y,b)}&\doteq \delta_{a,b}\delta_{x,y}\phi_x, \quad \forall x,y\in\widetilde{\Lambda}_L,\; a,b\in\mathcal{I} \quad \quad \,\,\,\,\,\, (\alpha=\mathrm{SP}). \end{align}\]

The determinant is computed by regarding \((\cancel{D}_{\alpha})^{-}_{\Lambda_L}-M_\phi\) as a square matrix (of size \(L^2\times L^2\), if \(\alpha=\mathrm{SN},\mathrm{SP}\), and \((2L)^2\times(2L)^2\), if \(\alpha=\mathrm{NN}\)).

2.0.0.4 Order parameter.

Differentiating the Hubbard–Stratonovich identity with respect to \(\phi_x\) and intergrating by parts yields

\[\label{eq:HS-2pt-common} \big\langle(\overline{\psi}\psi)_x (\overline{\psi}\psi)_y\big\rangle_{\alpha}= \Big(\frac{N}{\lambda}\Big)^2 \langle\phi_x\phi_y\rangle_{\mu_{\alpha}} -\frac{N}{\lambda}\delta_{x,y}.\tag{3}\]

Hence it is natural to identify the auxiliary field \(\phi\) as the chiral order parameter. Indeed, proving Long-Range order for the fermionic bilinear \(\overline{\psi}\psi\), namely (see [28], [30])

\[\liminf_{L \to \infty} \frac{1}{|\Lambda^{\alpha}_L|^2} \sum_{x,y \in \Lambda^{\alpha}_L} \left \langle (\overline{\psi}\psi)_x (\overline{\psi} \psi)_y \right \rangle_{\alpha} > C\]

for some \(C>0\) independent of the volume \(|\Lambda^{\alpha}_L|\), is equivalent² to proving the corresponding lower bound for the bosonic two-point function: \[\frac{1}{|\Lambda^{\alpha}_L|^2} \sum_{x, y \in \Lambda^{\alpha}_L} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} >C',\]

Therefore, throughout the paper we shall work primarily with the effective bosonic measure \(\mu_\alpha\), establishing Long-Range order for the auxiliary field \(\phi\).

2.1 Naive Fermions with Naive Interaction (NN)↩︎

The first model we consider is the lattice version of the Gross–Neveu model originally introduced in [23], formulated in terms of naive lattice fermions and therefore subject to the usual fermion–doubling phenomenon. This formulation provides the simplest discretization of the fermionic action and serves as a natural starting point for our analysis.

2.1.0.1 Spinors.

At each lattice site \(x \in \Lambda_L\), we introduce \(2N\) independent Grassmann-valued spinor fields \(\psi_{x,j}\) and \(\overline{\psi}_{x, j}\) (\(j=1, \cdots,N\)) with \(2\) entries, where \(\psi_{x,j}\) is regarded as a column vector and \(\overline{\psi}_{x,j}\) as a row vector:

\[\psi_{x,j} = \begin{pmatrix} \psi_{x,1,j} \\ \psi_{x,2,j}\end{pmatrix},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \overline{\psi}_{x,j} = \begin{pmatrix} \overline{\psi}_{x,1,j} & \overline{ \psi}_{x,2,j} \end{pmatrix}.\]

2.1.0.2 Gamma matrices.

We choose two self-adjoint matrices \(\{\gamma_\mu\}_{\mu=0,1} \subset M_2(\mathbb{C})\) as a possible irreducible representation of the \(2\)-dimensional Euclidean Clifford algebra \(\{\gamma_\mu,\gamma_\nu\} = 2\,\delta_{\mu\nu}\,\mathbbl{1}_2\). The Axial matrix \(\gamma_A\) is defined as \(\gamma_A \doteq i \gamma_0 \gamma_1\) and satisfies:

\[\gamma_A^2=\mathbbl{1}_2, \qquad \{\gamma_A,\gamma_\mu\}=0, \qquad \gamma_A^{\dagger}=\gamma_A. \label{eq:GA1}\tag{4}\]

In this paper, we adopt the following choice for the \(\gamma\)-matrices:

\[\gamma_0 = \sigma^y \equiv \left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \qquad\gamma_1 = \sigma^x\equiv \left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right),\qquad\gamma_A = \sigma^z \equiv \left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right).\]

2.1.0.3 Dirac operator.

The Lattice Dirac operator is chosen to correspond with its naive discretization, acting on \(\ell^2(\mathbb{Z}^2, \mathbb{C}^{2})\) according to:

\[\cancel{D}_{\mathrm{NN}} = \frac{1}{2} \sum_{\mu=0}^{1} \gamma_{\mu} \otimes (T_{\mu}-T_{\mu}^*),\]

where \(T_{\mu}\) is the right translation operator in the \(\mu\)-th direction, namely:

\[(T_{\mu}\psi)_x=\psi_{x+\hat{e}_{\mu}}\,\,\,\,\forall \psi\in\ell^2(\mathbb{Z}^2).\]

The Fourier representation for the Dirac operator is \[\label{eq:model95NN} (\cancel{D}_{\mathrm{NN}})_{x,y}= -i\sum_{\mu=0,1} \iint_{(-\pi,\pi]^2} \frac{d^2p}{(2\pi)^2} e^{-ip\cdot(x-y)} \gamma_{\mu} \sin(p_{\mu}).\tag{5}\]

The fermionic fields are taken to satisfy anti-periodic boundary conditions in both directions. This is implemented by using the anti-periodic extension of the lattice Dirac operator³, denoted by \((\cancel{D}_{\mathrm{NN}})_{\Lambda_L}^-\), as the kernel of the kinetic term:

\[\big((\cancel{D}_{\mathrm{NN}})_{\Lambda_L}^-\big)_{xy} = \sum_{n \in \mathbb{Z}^2} (-1)^{n_0+n_{1}} (\cancel{D}_{\mathrm{NN}})_{x+Ln, y}.\]

The proof of Lemma 1 is discussed in Appendix 6.1.

2.1.0.4 Lattice action.

The Dirac fermions are coupled through a local four-fermion interaction of Gross–Neveu type:

\[\label{NaiveA} S_{\mathrm{NN}}(\overline{\psi}, \psi) = \sum_{x, y \in \Lambda_L} \sum_{j=1}^N \overline{\psi}_{x,j} \big((\cancel{D}_{\mathrm{NN}})_{\Lambda_L}^-\big)_{x,y} \psi_{y,j} - \frac{\lambda}{2N} \sum_{x \in \Lambda_L} \bigg(\sum_{j=1}^N \overline{\psi}_{x,j} \psi_{x,j} \bigg)^2, \qquad \lambda>0,\tag{6}\]

where all spinor products have to be understood as row times column products, such as:

\[\overline{\psi}_{x,j} \psi_{x,j} = \sum_{s=1}^{2} \overline{\psi}_{x,s,j} \psi_{x,s,j}.\]

2.1.0.5 Chiral symmetry.

The action is invariant under the discrete \(\mathbb{Z}_2\)-chiral symmetry defined by:

\[\label{eq:chiral951} \begin{cases} \psi_{x,j} \to \gamma_A \psi_{x,j},\\ \overline{\psi}_{x,j} \to -\overline{\psi}_{x,j} \gamma_A, \end{cases}\tag{7}\]

while the fermion bilinear \(\overline{\psi}_{x, j}\psi_{x,j}\) obviously changes sign as one can easily infer from the first two of 4 .

2.1.0.6 Effective bosonic measure.

By applying the general Hubbard–Stratonovich construction 1 to the action 6 , and integrating out the fermionic fields, we obtain the effective bosonic measure on \(\mathbb{R}^{\Lambda_L}\):

\[d \mu_{\mathrm{NN}}(\phi) = \frac{1}{Z^{\mathrm{NN}}_{\Lambda_L}} \prod_{x \in \Lambda_L} d \phi_x e^{-\frac{N}{2\lambda}\phi_x^2} \det((\cancel{D}_{\mathrm{NN}})_{\Lambda_L}^- - M_{\phi})^N.\]

2.1.0.7 Issues, strong coupling and scaling limit.

As anticipated above, the naive lattice formulation of the Gross–Neveu model suffers from the well-known fermion–doubling problem [31], [32], implying that the number of low-energy (long-wavelength) Dirac fermions described by the free action is four, rather than one as suggested by the continuum theory. It has been pointed out that such a proliferation of fermionic degrees of freedom in free lattice theories may lead to significant difficulties once interactions among the long-wavelength modes are introduced [33]. These issues were analyzed in detail in [24], where it was argued that the interaction spoils Lorentz invariance at the quantum level. Nevertheless, at least heuristically, this model is still expected to exhibit asymptotic freedom and infrared properties analogous to those of the continuum Gross–Neveu model, up to an overall factor of four arising from the additional poles in the free propagator [24], [34].
Motivated by these difficulties, alternative lattice formulations based on staggered fermions [31] were introduced in [24]. While these models share the same continuum limit and reproduce the Gross–Neveu action, they exhibit different behaviors in the strong-coupling regime. One of these two formulations, the SP model, will be the third model under study of this paper. We first introduce a simpler staggered-fermion model, the SN model, which, despite failing to reproduce the correct continuum limit, still captures the appropriate infrared physics (see [35] and [36]) and has the simplest presentation among all three.

2.2 Staggered fermions with Naive Interaction (SN)↩︎

The second model we consider is formulated in terms of staggered fermions [31]. Compared with the naive discretization, spin degrees of freedom are encoded in site-dependent phases, while the interaction remains on-site and of Gross–Neveu type.

2.2.0.1 Staggered Fermions.

At each lattice site \(x \in \Lambda_L\), we introduce \(2N\) independent Grassmann variables \({\psi}_{x,j}\) and \(\overline{\psi}_{x,j}\), with \(j=1,\dots,N\).

2.2.0.2 Gamma matrices.

Gamma Matrices are represented by phases:

\[\Gamma_{\mu}(x) = \begin{cases} 1, \,\,\,\,&\mu=0\\ (-1)^{x_0}, &\mu=1. \end{cases}\]

The role of the Axial matrix \(\gamma_A\) is played by the multiplication operator:

\[\epsilon(x) = (-1)^{x_0+x_1}.\]

2.2.0.3 Dirac operator.

The Lattice Dirac operator in the Staggered Fermions formalism is an operator acting on \(\ell^2(\mathbb{Z}^2, \mathbb{C})\) according to:

\[\cancel{D}_{\mathrm{SN}} = \frac{1}{2} \sum_{\mu=0}^1 \Gamma_{\mu} (T_{\mu}- T_{\mu}^*).\]

By labeling the sites of \(\Lambda_L\) by \((x,a)\), with \(x\in \Lambda_L\) such that \(x_0\) is even and \(a\in\{1,2\}\), so that every \(y\in\Lambda_L\) can be written as \(\big(x_0+ \frac{1}{2}(a-1),x_1\big)\), we recover translation invariance of the Dirac operator, which can be therefore expressed by Fourier transform: \[\label{eq:model95SN} (\cancel{D}_{\mathrm{SN}})_{(x,a),(y,b)} =\iint_{(-\frac{\pi}{2},\frac{\pi}{2}]\times(-\pi,\pi]}\frac{d^2p}{(2\pi)^2} \,e^{-ip(x-y)}\, \begin{pmatrix} i\sin p_1 & \frac{1}{2}(e^{2ip_0}-1)\\ -\frac{1}{2}(e^{-2ip_0}-1) & -i\sin p_1 \end{pmatrix}_{a,b}.\tag{8}\]

Again, we impose anti-periodic boundary conditions by considering an anti-periodic extension of the above Dirac Operator:

\[\big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{xy} = \sum_{n \in \mathbb{Z}^2} (-1)^{n_0+n_1} (\cancel{D}_{\mathrm{SN}})_{x+ L n,y}.\]

The proof of Lemma 2 is discussed in Appendix 6.2.

2.2.0.4 Lattice action.

The Dirac fermions are coupled through a local four-fermion interaction of Gross–Neveu type:

\[\label{StaggeredA} S_{{\mathrm{SN}}}(\overline{\psi},\psi) = \frac{1}{2} \sum_{x,y \in \Lambda_L}\sum_{j=1}^N \overline{\psi}_{x,j} \big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{x,y} \psi_{y,j}- \frac{\lambda}{2N} \sum_{x \in \Lambda_L} \bigg(\sum_{j=1}^N \overline{\psi}_{x,j} \psi_{x,j} \bigg)^2, \qquad \lambda>0.\tag{9}\]

2.2.0.5 Chiral symmetry.

The action is invariant under the discrete \(\mathbb{Z}_2\)-chiral transformation defined by:

\[\begin{cases} \psi_{x, j} \to e^{i \frac{\pi}{2} \epsilon(x)} \psi_{x, j}\\ \overline{\psi}_{x, j} \to e^{i \frac{\pi}{2} \epsilon(x)} \overline{\psi}_{x, j}.\\ \end{cases}\]

Indeed, under this transformation, we have that

\[\begin{align} \overline{\psi}_{x,j} \big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{x,y} \psi_{y,j} \to e^{i \frac{\pi}{2} (\epsilon(x)+\epsilon(y))} \overline{\psi}_{x,j} \big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{x,y} \psi_{y,j} = \overline{\psi}_{x,j} \big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{x,y} \psi_{y,j} \end{align}\]

where we used the fact that \(\big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L}\big)_{x,y}\) is non-zero only if \(|x-y|=1\), in which case \(\epsilon(x)=-\epsilon(y)\). Conversely, under this transformation the bilinear \((\overline{\psi} \psi)_x\) changes sign: \((\overline{\psi} \psi)_x \to e^{i \pi \epsilon(x)} (\overline{\psi} \psi)_x = - (\overline{\psi} \psi)_x\).

2.2.0.6 Effective bosonic measure.

By applying the general Hubbard–Stratonovich construction 1 to the action 9 , and integrating out the fermionic fields, we obtain the effective bosonic measure on \(\mathbb{R}^{\Lambda_L}\):

\[d \mu_{\mathrm{SN}}(\phi) = \frac{1}{Z^{\mathrm{SN}}_{\Lambda_L}} \prod_{x \in \Lambda_L} d \phi_x e^{-\frac{N}{2\lambda}\phi_x^2} \det\big((\cancel{D}_{\mathrm{SN}})^-_{\Lambda_L} - M_\phi\big)^N.\]

2.3 Staggered Fermions with Plaquette Interaction (SP)↩︎

The third model, originally introduced in [24], is widely considered as the proper lattice discretization of the Gross-Neveu model, as after performing the formal continuum limit one gets exactly the usual Gross-Neveu action [24], see also Appendix 8.
The model is most naturally formulated by blocking the original lattice \(\Lambda_L\) into four-site unit cells, as illustrated in Fig. 3. This construction produces a reduced lattice \(\widetilde{\Lambda}_L = 2\mathbb{Z}^2 \cap \big(-\frac{L}{2}, \frac{L}{2}\big]^2\), such that \(\Lambda_L = \{A,B,C,D\} \times \widetilde{\Lambda}_L\), where \(a \in \{A,B,C,D\} \doteq \mathcal{I}\) labels the position within each unit cell.

The model is constructed along the same lines as the SN one described above, but with a different interaction structure. We therefore adopt the same notation, adapted to the new labeling.

2.3.0.1 Staggered fermions.

At each lattice site \((x,a) \in \widetilde{\Lambda}_L \times \mathcal{I}\), we introduce \(2N\) independent Grassmann variables \(\psi_{x,a,j}\) and \(\overline{\psi}_{x,a,j}\).

2.3.0.2 Gamma matrices.

The staggered phases are now encoded in \[\Gamma_{\mu}(a) \doteq \begin{cases} +1 & \mu=0,\\ -1 & \mu=1,\;a=A,B,\\ +1 & \mu=1,\;a=C,D. \end{cases}\] The analogue of the axial matrix is the multiplication operator \[\label{epP} \epsilon(a)= \begin{cases} +1 & a=A,D,\\ -1 & a=B,C. \end{cases}\tag{10}\]

2.3.0.3 Dirac operator.

The lattice Dirac operator acts on \(\ell^2(\widetilde{\Lambda}_L \times \mathcal{I},\mathbb{C})\) as \[\cancel{D}_{\mathrm{SP}} = \frac{1}{2}\sum_{\mu=0}^1 \Gamma_{\mu}(a)(T_\mu - T_\mu^*).\] Note that it is translationally invariant on the reduced lattice \(\widetilde{\Lambda}_L\): \[\label{eq:model95SP} (\cancel{D}_{\mathrm{SP}})_{(x,a),(y,b)} =\iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2}\frac{d^2p}{(2\pi)^2} \,e^{-ip\cdot (x-y)}\, \frac{1}{2} \begin{pmatrix} 0 &-1 + e^{2i p_1} &1 - e^{2i p_0} &0\\ 1 - e^{-2i p_1} &0 &0 &1-e^{2ip_0}\\ -1+e^{-2ip_0} &0 &0 &1 -e^{2i p_1}\\ 0 &-1+e^{-2ip_0} &-1+ e^{-2ip_1} &0 \end{pmatrix}_{a,b},\tag{11}\]

for every \(x,y\in\widetilde{\Lambda}_L\) and \(a,b\in\mathcal{I}\) (the matrix row and columns are indexed in the order \(A,B,C,D\)). Anti-periodic boundary conditions are imposed as in the previous models by defining the anti-periodic extension \[\label{eqn:51} \big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_L}\big)_{(x,a),(y,b)} = \sum_{n\in\mathbb{Z}^2} (-1)^{n_0+n_1} (\cancel{D}_{\mathrm{SP}})_{(x+Ln,a),(y,b)}, \qquad x,y\in\widetilde{\Lambda}_L, \; a,b\in\mathcal{I}.\tag{12}\]

2.3.0.4 Lattice action.

The third model is similar to the previous one but differs for the interaction, which couples degrees of freedom within the same unit cell:

\[\label{StaggeredPA} S_{\mathrm{SP}}(\overline{\psi}, \psi) = \frac{1}{2} \sum_{x,y \in \widetilde{\Lambda}_L} \sum_{a,b\in\mathcal{I}} \sum_{j=1}^N \overline{\psi}_{x,a, j} \big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_L}\big)_{(x,a),(y,b)} \psi_{y,b, j}- \frac{\lambda}{2 N} \sum_{x \in \widetilde{\Lambda}_L} \bigg(\sum_{j=1}^N \sum_{a \in \mathcal{I}}\overline{\psi}_{x,a, j} \psi_{x,a, j} \bigg)^2,\tag{13}\]

where again \(\lambda>0\).

2.3.0.5 Effective bosonic measure.

Applying the general Hubbard–Stratonovich construction 1 to the action 13 , and integrating out the fermionic fields, we obtain the effective bosonic measure on \(\mathbb{R}^{\widetilde{\Lambda}_L}\):

\[d \mu_{\mathrm{SP}}(\phi) = \frac{1}{Z^{\mathrm{SP}}_{\Lambda_L}} \prod_{x \in \widetilde{\Lambda}_L} d \phi_x e^{-\frac{N}{2\lambda}\phi_x^2} \det\big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_L} - M_\phi\big)^N.\]

This formulation makes it natural to view \(\cancel{D}\) as an operator on the reduced lattice \(\widetilde{\Lambda}_L\), endowed with four internal degrees of freedom per site. Equivalently, the SP model may be regarded as a restriction of the SN model to Hubbard–Stratonovich fields that are constant within each unit cell. This restriction, however, has significant consequences for the (formal) continuum limit (see [24], [35]) and Appendix 8.

2.4 Main result↩︎

Given the precise definitions of the models, we are ready to state our main theorem.

3 Reflection Positivity↩︎

We aim to show that the three bosonic measures \(\mu_{\alpha}(\phi)\), \(\alpha\in\{\mathrm{NN},\mathrm{SN},\mathrm{SP}\}\), constructed in the previous section, are reflection positive [28] with respect to reflections across bond planes of the underlying lattice \(\Lambda_L^\alpha\), where: \[\Lambda_L^{\mathrm{NN}}=\Lambda_L^{\mathrm{SN}}=\Lambda_L \qquad \Lambda_L^{\mathrm{SP}}=\widetilde{\Lambda}_L.\]

As a very first step towards the notion of Reflection Positivity, we need the precise definition of reflections on \(\Lambda^{\alpha}_L\), regarded as a torus, with respect to some fixed cut plane.

Any cut plane \((r,\nu)\) induces the splitting \(\Lambda^{\alpha}_L= (\Lambda^{\alpha}_L)_+\sqcup (\Lambda^{\alpha}_L)_-\), where (see Fig. 4): \[(\Lambda^{\alpha}_L)_+ \doteq \bigg\{x\in \Lambda^{\alpha}_L: \; (x_{\nu}-r)\, \text{mod}\, L \in \big(-\tfrac{L}{2}, 0\big) \bigg\}\] and \((\Lambda^{\alpha}_L)_-\doteq \Lambda^{\alpha}_L\setminus (\Lambda^{\alpha}_L)_+\). It is straightforward to check that \(\vartheta_{\nu}^L((\Lambda^{\alpha}_L)_{\pm})= (\Lambda^{\alpha}_L)_{\mp}\).
The reflection \(\vartheta_{\nu}^L\) induces an action on field configurations \(\phi\in \mathbb{R}^{\Lambda_L^\alpha}\) given by: \[(\Theta\phi)_x \doteq \phi_{\vartheta_{\nu}^L(x)}.\]

In the following, in order to avoid cluttering of the notation, we will denote \(\vartheta_{\nu}^L\) simply by \(\vartheta_{\nu}\) as reflections on the plane will not play any role in the main text. In Appendix 6 we will restore the double notation as we will need both reflections. Besides, we will possibly drop the index \(\nu\in\{0,1\}\) for the reflection, whenever its role will not be essential.
Let \(\mathcal{A}^\alpha\) be the algebra of complex polynomials on \(\mathbb{R}^{\Lambda^{\alpha}_L}\). We define the Osterwalder–Schrader reflection \(\Theta:\mathcal{A}^\alpha\to\mathcal{A}^\alpha\) by \[(\Theta F)(\phi) \doteq \overline{F(\Theta\phi)} .\]

With this definition, \(\Theta\) satisfies the following properties discussed in the following proposition.

Let \(\mathcal{A}^\alpha_\pm\subset\mathcal{A}^\alpha\) denote the subalgebra of observables depending only on \(\{\phi_x\}_{x\in(\Lambda_L^\alpha)_\pm}\). We define the pairing: \[\label{PS} \begin{align} \langle \cdot,\cdot\rangle_{\mu_{\alpha}}:\mathcal{A}^\alpha_+\times\mathcal{A}^\alpha_+&\to\mathbb{C},\\ (A,B)&\mapsto \int d\mu_\alpha(\phi)\, (\Theta A)(\phi)\, B(\phi). \end{align}\tag{14}\]

3.1 Naive Fermions with Naive Interaction↩︎

3.2 Staggered Fermions with Naive Interaction↩︎

3.3 Staggered Fermions with Plaquette Interaction↩︎

4 Long-Range Order↩︎

In this section we establish Long-Range order for the bosonic field with respect to the effective measures \(\mu_\alpha(\phi)\) and, by the Hubbard–Stratonovich identity 3 , for the fermionic bilinear \(\overline{\psi}\psi\).
In each model, the effective bosonic measure can be written in Gibbs form as:

\[d \mu_{\alpha}(\phi) = \prod_{x \in \Lambda^{\alpha}_L} d \phi_x e^{-N V^{\alpha}_{L,\lambda}(\phi)}\]

where the effective potential is given by

\[\label{EP}V^{\alpha}_{L,\lambda}(\phi) \doteq \frac{1}{2\lambda} \sum_{x\in\Lambda_L^{\alpha}}\phi_x^2 - \log\big|\det\big( (\cancel{D}_{\alpha})^-_{\Lambda_L} - M_{\phi} \big) \big|.\tag{24}\]

The strategy of the proof is dictated by the structure of the effective potential. Although \(V^{\alpha}_{L,\lambda}\) is highly non-local and does not admit a simple closed expression, Reflection Positivity implies that, among configurations constrained to lie in a given interval, its minimizers are spatially constant and can therefore be computed explicitly via Fourier transforms. The corresponding infinite-volume effective potential density:

\[\label{tdpd}v^{\alpha}_{\lambda}(\varphi) \doteq \lim_{L \to \infty} \frac{1}{|\Lambda^{\alpha}_L|} V^{\alpha}_{L, \lambda}(\phi)\Big|_{\phi= \varphi^{\Lambda^{\alpha}_L}},\tag{25}\]

has a symmetric double-well structure, with minima at \(\pm\varphi_\alpha^\star(\lambda) \neq 0\) for any \(\lambda >0\) (see Fig. 6). Long-Range order is then proved by decomposing the configuration space into regions around the two minima \(\varphi=\pm\varphi^{\star}_{\alpha}(\lambda)\), the local maximum \(\varphi=0\), and the large-field regions, and by estimating the contribution from each region separately.
The following facts about the effective potential will be repeatedly used.

The proof of Proposition 2 is discussed in Appendix 7.

4.1 Field-size decomposition↩︎

By Proposition 2 we have that the minima of the finite-volume energy density are asymptotically determined by those of \(v^{\alpha}_{\lambda}\). The latter exhibits a symmetric double-well structure (see Fig. 6), with absolute minima at \(\pm \varphi_\alpha^\star(\lambda)\). This suggests partitioning the configuration space according to the position of the field relative to the minima, the local maximum at the origin, and the large-field regime. Configurations belonging to energetically unfavorable regions, namely large fields or fields near the local maximum, are exponentially suppressed in \(N\). This suppression is quantified by comparing the energy of configurations constrained to a given region with that of optimal constant configurations, using Chessboard Estimates [28], [39], which are a remarkable corollary of Reflection Positivity, together with ?? .

The contributions which are not apriori suppressed arise from configurations where both fields lie near the minima. Configurations in which the field at different points lies around different minima can be shown to be again exponentially suppressed in \(N\), via a Peierls-type argument combined with Reflection Positivity, in the spirit of [29] (see also [40]). While this strategy applies to all our three models, the explicit estimates depend on the specific lattice realization of the Dirac operator and will be treated separately in the following subsections.
To implement this strategy, we introduce a quantitative partition of the configuration space at the level of single-site variables. Fix parameters \(C_\alpha,K_\alpha>0\) such that: \[0< C_\alpha < \varphi_\alpha^\star(\lambda) < K_\alpha.\] We then split the single-site field values into the following five regions:

1. Large positive fields: \(\phi_x\in (K_\alpha,\infty)\);

2. Large negative fields: \(\phi_x\in (-\infty,-K_\alpha)\);

3. Neighborhood of the local maximum: \(\phi_x\in (-C_\alpha,C_\alpha)\);

4. Neighborhood of the positive minimum: \(\phi_x\in [C_\alpha,K_\alpha]\);

5. Neighborhood of the negative minimum: \(\phi_x\in [-K_\alpha,-C_\alpha]\).

Accordingly, introduce the following indicator functions:

\[\begin{align} &I_x^{\pm K_\alpha}(\phi)\doteq \mathbbl{1}_{\{\pm \phi_x>K_\alpha\}}, \qquad I_x^{(-C_\alpha,C_\alpha)}(\phi)\doteq\mathbbl{1}_{\{|\phi_x|<C_\alpha\}}, \qquad I_x^{\star_{\pm}}(\phi) \doteq \mathbbl{1}_{\{C_\alpha\le \pm \phi_x\le K_\alpha\}}. \end{align}\]

The lower bound on the two-point function is obtained by isolating the contribution in which both fields lie near the same minimum, and by estimating all the remaining contributions in terms of three errors: large-field events, local-maximum events, and opposite-minimum events. Following the decomposition carried out in Appendix 9, we obtain: \[\label{eq:mainbound2} \begin{align} \inf_{\substack{x,y \in \Lambda^{\alpha}_L}} \langle \phi_x\phi_y\rangle_{\mu_\alpha} &\ge C_{\alpha}^2 - \Bigg( 2(K_{\alpha}^2+C_{\alpha}^2) \sup_{x,y\in\Lambda^{\alpha}_L} \left\langle I^{\star_+}_x I^{\star_-}_y \right\rangle_{\mu_{\alpha}} + 4C_{\alpha} (C_{\alpha}+ K_{\alpha}) \sup_{x\in\Lambda^{\alpha}_L} \left\langle I^{(-C_{\alpha},C_{\alpha})}_x \right\rangle_{\mu_{\alpha}} \\ &+ 4\big(1+ \tfrac{C_{\alpha}^2}{K_{\alpha}^2}\big)\sup_{x\in\Lambda^{\alpha}_L} \left\langle \phi_x^2 I^{+K_{\alpha}}_x \right\rangle_{\mu_{\alpha}} + 4\big(C_{\alpha}+ 2\tfrac{C_{\alpha}^2}{K_{\alpha}}+ 2K_{\alpha}\big) \sup_{x\in\Lambda^{\alpha}_L} \left\langle \phi_x^2 I^{+K_{\alpha}}_x\right\rangle_{\mu_{\alpha}}^{\frac{1}{2}} \Bigg). \end{align}\tag{26}\]

Similarly, see again Appendix 9, for the upper bound we find: \[\label{upbound} \sup_{x,y \in \Lambda^{\alpha}_L} \langle \phi_x \phi_y \rangle_{\mu_{\alpha}} \leq K_{\alpha}^2 + 2K_{\alpha} \sup_{x \in \Lambda_L^{\alpha}} \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} + 4 \sup_{x \in \Lambda_L^{\alpha}} \big \langle \phi_x^2 I_x^{+K_{\alpha}} \big \rangle_{\mu_{\alpha}}.\tag{27}\]

Thus, the proof reduces to estimating the following three expectation values: \[\big\langle \phi_x^2 I_x^{+K_\alpha}\big\rangle_{\mu_\alpha}, \qquad \big\langle I_x^{(-C_\alpha,C_\alpha)} \big\rangle_{\mu_\alpha}, \qquad \big\langle I_x^{\star_+}I_y^{\star_-}\big\rangle_{\mu_\alpha},\]

whose bounds are discussed in Proposition 3 below. The first two correspond to large-field and local-maximum events and will be controlled directly by chessboard estimates together with Proposition 2. The third one corresponds to configurations in which the field fluctuates near different minima at two distant sites and its control requires a Peierls-type argument combined with Reflection Positivity.

Assuming Proposition 3 for the moment, we first derive Theorem 1. The proof of Proposition 3 is given in Section 5.

4.2 Proof of Theorem 1↩︎

We now show how Proposition 3 implies the main theorem. By 3 we have that

\[\frac{1}{|\Lambda^{\alpha}_L|^2}\sum_{x,y\in\Lambda^{\alpha}_L} \left\langle (\overline{\psi}\psi)_x (\overline{\psi} \psi)_y \right\rangle_{\alpha}= \left(\frac{N}{\lambda}\right)^2 \frac{1}{|\Lambda^{\alpha}_L|^2} \sum_{x,y\in\Lambda^{\alpha}_L} \left\langle \phi_x \phi_y \right\rangle_{\mu_{\alpha}} - \frac{N}{\lambda} \frac{1}{|\Lambda^{\alpha}_L|}.\]

Therefore:

\[\label{eqn:32} \left(\frac{N}{\lambda}\right)^2 \min_{x,y\in\Lambda^{\alpha}_L} \langle\phi_x\phi_y\rangle_{\mu_{\alpha}}- \frac{4N}{\lambda L^2}\le \frac{1}{|\Lambda^{\alpha}_L|^2}\sum_{x,y\in\Lambda^{\alpha}_L} \left\langle (\overline{\psi}\psi)_x (\overline{\psi} \psi)_y \right\rangle_{\alpha} \le \left(\frac{N}{\lambda}\right)^2 \max_{x,y\in\Lambda^{\alpha}_L} \langle\phi_x\phi_y\rangle_{\mu_{\alpha}}.\tag{28}\]

Given any \(\varepsilon\in(0,1)\), we apply Proposition 3 with

\[C_\alpha=\sqrt{1-\frac{\varepsilon}{2}}\,\varphi_\alpha^\star(\lambda), \qquad K_\alpha=\sqrt{1+\frac{\varepsilon}{2}}\,\varphi_\alpha^\star(\lambda).\]

Using the upper and lower bound for the bosonic correlator, 27 and 26 respectively, we find that

\[\begin{align} &\sup_{x,y\in\Lambda_L^\alpha}\langle \phi_x\phi_y\rangle_{\mu_\alpha} \le \Big(1+\frac{\varepsilon}{2}\Big)\varphi_\alpha^\star(\lambda)^2+\mathcal{E}_1(N,\lambda,\varepsilon),\\ &\inf_{x,y\in\Lambda_L^\alpha}\langle \phi_x\phi_y\rangle_{\mu_\alpha} \ge \Big(1-\frac{\varepsilon}{2}\Big)\varphi_\alpha^\star(\lambda)^2-\mathcal{E}_2(N,\lambda,\varepsilon), \end{align}\]

for every \(N\ge N_1(K_{\alpha},C_{\alpha},\lambda)\) and \(L\ge L_1(K_{\alpha},C_{\alpha},\lambda)\), where \(\mathcal{E}_1\) and \(\mathcal{E}_2\) are suitable quantities, which are exponentially small in \(N\) for any \(\lambda\in(0,\lambda_0]\) and \(\varepsilon\in(0,1)\) fixed. Going back to 28 , we have that

\[\left(1-\frac{\varepsilon}{2}\right)\varphi^{\star}_{\alpha}(\lambda)^2 - \mathcal{E}_2(N,\lambda,\varepsilon) - \frac{4N}{\lambda L^2}\le \frac{1}{|\Lambda^{\alpha}_L|^2}\sum_{x,y\in\Lambda^{\alpha}_L} \left\langle (\overline{\psi}\psi)_x (\overline{\psi} \psi)_y \right\rangle_{\alpha}\le \left(1+\frac{\varepsilon}{2}\right)\varphi^{\star}_{\alpha}(\lambda)^2 + \mathcal{E}_1(N,\lambda,\varepsilon).\]

Finally we require \(N,L\) to be large enough (depending on \(\lambda\) and \(\varepsilon\)), in order to satisfy:

\[\mathcal{E}_1(N,\lambda,\varepsilon)\le \frac{\varepsilon}{2} \varphi^{\star}_{\alpha}(\lambda)^2, \qquad \mathcal{E}_2(N,\lambda,\varepsilon)\le \frac{\varepsilon}{4} \varphi^{\star}_{\alpha}(\lambda)^2, \qquad \frac{N}{L^2}\le \frac{\varepsilon}{8} \lambda\varphi^{\star}_{\alpha}(\lambda)^2,\]

so that ?? follows.

5 Proof of Proposition 3↩︎

This section is devoted to the analysis of the three main subdominant contributions to the two-point bosonic correlator \(\langle\phi_x\phi_y\rangle_{\mu_{\alpha}}\), namely the large-field, the local-maximum and the Pierles-like term. These will be shown to satisfy the bounds ?? , ?? and ?? respectively, which are the main building blocks for the proof of LRO, as discussed in Subsection 4.2. The first two estimates will be proved in Subsections 5.1 and 5.2. The third estimate, treated in Subsection 5.3, requires a separate analysis of explicit, alternating, periodic configurations, discussed in Subsection 5.4.

5.1 Large-field terms↩︎

We prove the large-field bound ?? , namely we show that configurations with \(\phi_x > K_\alpha\) are exponentially suppressed in \(N\).
The proof combines Chessboard Estimates with the fact that, for \(K_\alpha>\varphi_\alpha^\star(\lambda)\), the mean-field potential ?? is strictly larger than its minimum. Applying the Chessboard Estimate ?? to the single-site observable \(I^{+K_{\alpha}}_x(\phi)\), we obtain: \[\label{eqn:10a} \left\langle \phi_x^2 I_x^{+K_\alpha}\right\rangle_{\mu_\alpha} \le \left( \frac{ \displaystyle \int_{[K_\alpha,\infty)^{\Lambda_L^\alpha}} \Big(\prod_{x \in \Lambda_L^{\alpha}} d\phi_x\,\phi_x^2\Big) e^{-N V^{\alpha}_{L,\lambda}(\phi)} }{ \displaystyle \int_{\mathbb{R}^{\Lambda_L^\alpha}} \Big(\prod_{x \in \Lambda_L^{\alpha}} d\phi_x\Big) e^{-N V^{\alpha}_{L,\lambda}(\phi)} } \right)^{\frac{1}{|\Lambda_L^\alpha|}}.\tag{29}\]

We will proceed by looking for an upper and a lower bound for the numerator and denominator of 37 respectively.

5.1.0.1 Upper bound for the numerator of 29 .

We proceed via the following intermediate steps.

1. We first rewrite the effective potential as:

   \[V^{\alpha}_{L,\lambda}(\phi) = \frac{1}{2\lambda N}\sum_{x\in\Lambda_L^\alpha}\phi_x^2 + V^{\alpha}_{L,\tilde{\lambda}_N}(\phi), \qquad \tilde{\lambda}_N \doteq \tfrac{\lambda}{1-\frac{1}{N}}.\]

2. As a second step, we extract the infimum of \(V^{\alpha}_{L,\tilde{\lambda}_N}\) over the constrained set \(\{\phi_x \geq K_{\alpha}\, \forall x\in \Lambda^{\alpha}_L\}\) and we integrate over \(\phi\):

   \[\begin{align} &\bigg(\int_{[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} \big(\prod_{x\in \Lambda_L^{\alpha}}d\phi_x \phi_x^2 \big) \, e^{- N V^{\alpha}_{L,\lambda}(\phi)}\bigg)^{\frac{1}{|\Lambda_L^{\alpha}|}} =\\ &\bigg(\int_{[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} \big(\prod_{x\in \Lambda^{\alpha}_L}d\phi_x \phi_x^2 e^{- \frac{\phi_x^2}{2\lambda}}\big) \, e^{- N V^{\alpha}_{L,\tilde{\lambda}_N}(\phi)} \bigg)^{\frac{1}{|\Lambda_L^{\alpha}|}} \leq (\sqrt{2\pi} \lambda^{\frac{3}{2}})\, \exp \Big\{-\tfrac{N}{|\Lambda^{\alpha}_L|} \inf_{ \phi\in[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} V^{\alpha}_{L,\tilde{\lambda}_N}(\phi) \Big\}. \end{align}\]

3. By Proposition 2.[itt:1], the infimum of \(V^{\alpha}_{L, \lambda}\) is attained for constant field configurations, whose large \(L\) asymptotics can be controlled, in force of Proposition 2.[itt:2], by the infinite-volume potential density \(v^{\alpha}_{\lambda}\):

   \[\label{eqn:13s} \frac{1}{|\Lambda^{\alpha}_L|}\inf_{\phi\in[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} V^{\alpha}_{L,\tilde{\lambda}_N}(\phi) \overset{\eqref{Minpot}}{\geq} \frac{1}{|\Lambda^{\alpha}_L|}\inf_{\varphi\in[K_{\alpha},\infty)} V^{\alpha}_{L,\tilde{\lambda}_N}(\varphi) \overset{\eqref{eqn:12a}}{\geq} \inf_{\varphi\in[K_{\alpha},\infty)} v_{\tilde{\lambda}_N}^{\alpha}(\varphi) -\frac{\mathfrak{c}_0}{L}.\tag{30}\]

4. Since \(K_\alpha>\varphi_\alpha^\star(\lambda)\) and the map \(\lambda \mapsto \varphi^{\star}_{\alpha}(\lambda)\) is continuous over \((0,\infty)\) (see Proposition 2.[itt:3]), for \(N\) sufficiently large one has \(K_\alpha\ge \varphi_\alpha^\star(\tilde{\lambda}_N)\). Hence, since \(v^{\alpha}_{\tilde{\lambda}_N}\) is increasing over \([\varphi_{\alpha}^{\star}(\tilde{\lambda}_N), \infty)\) (see Proposition 2.[itt:3]), the infimum over \([K_\alpha,\infty)\) is attained at \(\varphi=K_\alpha\) and 30 yields:

   \[\frac{1}{|\Lambda^{\alpha}_L|}\inf_{\phi\in[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} V^{\alpha}_{L,\tilde{\lambda}_N}(\phi) \geq v_{\tilde{\lambda}_N}^{\alpha}(K_{\alpha}) - \tfrac{\mathfrak{c}_0}{L}.\]

All in all:

\[\label{eqn:13} \bigg(\int_{[K_{\alpha},\infty)^{\Lambda^{\alpha}_L}} \big(\prod_{x\in \Lambda_L^{\alpha}}d\phi_x \phi_x^2 \big) \, e^{- N V^{\alpha}_{L,\lambda}(\phi)}\bigg)^{\frac{1}{|\Lambda_L^{\alpha}|}} \le \left(\sqrt{2\pi} \lambda^{\frac{3}{2}}\right) \, \exp\Big\{-N \big(v^{\alpha}_{\tilde{\lambda}_N}(K_{\alpha}) - \tfrac{\mathfrak{c}_0}{L}\big)\Big\}.\tag{31}\]

5.1.0.2 Upper bound for the denominator of 29 .

The strategy for bounding from below the denominator of 29 (a.k.a. the partition function) goes through the following steps.

1. First, we restrict the integral over \(\phi\) to a neighborhood \(Q^{\alpha}_{\delta_N}\) of the homogeneous minimizer \(\varphi^{\star}_{\alpha}(\lambda)>0\), where

   \[Q^{\alpha}_{\delta_N} \doteq \big[\varphi^{\star}_{\alpha}(\lambda)-\tfrac{\delta_N}{2},\varphi^{\star}_{\alpha}(\lambda)+\tfrac{\delta_N}{2}\big]^{\Lambda^{\alpha}_L}\]

   with \(\delta_N>0\) to be specified. Since the integrand is pointwise non-negative for \(N\) even, such restriction produces a lower-bound:

   \[\label{eqn:42} \big(Z^{\alpha}_{\Lambda_L}\big)^{\frac{1}{|\Lambda^{\alpha}_{L}|}} \geq \Big[ \int_{Q^{\alpha}_{\delta_N}} \prod_{x \in \Lambda^{\alpha}_L} d \phi_x\, \;e^{-\frac{N}{2 \lambda} \phi_x^2}\, \det\big((\cancel{D}_{\alpha})^-_{\Lambda_L}-M_{\phi}\big)^N\Big]^{\frac{1}{|\Lambda^{\alpha}_{L}|}}.\tag{32}\]

2. In the second place, we perform a change of variables \(\phi_x\mapsto \varphi_\alpha^\star(\lambda)+\phi_x\) and we use the following bound (holding for \(\phi\in Q^{\alpha,0}_{\delta_N}\doteq [-\frac{\delta_N}{2},\frac{\delta_N}{2} ]^{\Lambda^{\alpha}_L}\)):

   \[e^{-\frac{N}{2 \lambda} \sum_{x} (\varphi^{\star}_{\alpha}(\lambda)+ \phi_x)^2} = e^{-\frac{N}{2 \lambda} \big(\varphi^{\star}_{\alpha}(\lambda)^2 |\Lambda^{\alpha}_L| + 2 \varphi^{\star}_{\alpha}(\lambda)\sum_{x} \phi_x + \sum_x \phi_x^2 \big)} \geq e^{-\frac{N}{2 \lambda}|\Lambda^{\alpha}_L| \big(\varphi^{\star}_{\alpha}(\lambda)^2+ 2 \varphi^{\star}_{\alpha}(\lambda) \delta_N + \delta_N^2\big)}\]

   to obtain: \[\label{eqn:44} \begin{align} &\big(Z^{\alpha}_{\Lambda_L}\big)^{\frac{1}{|\Lambda^{\alpha}_{L}|}} \ge\\ & e^{-\frac{N}{|\Lambda^{\alpha}_L|} V^{\alpha}_{L,\lambda}(\varphi_\alpha^\star(\lambda))} e^{-\frac{N}{2\lambda}\big(2\varphi_\alpha^\star(\lambda)\delta_N+\delta_N^2\big)} \bigg[ \int_{Q_{\delta_N}^{\alpha,0}} \prod_{x \in \Lambda_{L}^{\alpha}} d\phi_x\, \det\Big( \mathbbl{1}- \big((\cancel{D}_\alpha)^-_{\Lambda_L}-M_{\varphi_\alpha^\star(\lambda)}\big)^{-1} M_\phi \Big)^N\bigg]^{\frac{1}{|\Lambda^{\alpha}_{L}|}}. \end{align}\tag{33}\]

   Remark 6. Observe that the Dirac operator is invertible since a non-vanishing, constant background field opens a gap in the dispersion relations (see Appendix 7). More precisely, introducing the operator norm \(\|\,\cdot\,\|_{\mathrm{op}}\doteq \sup_{v\in\mathbb{C}^n,\|v\|=1}\|\,\cdot\,v\|_{\ell^2(X)}\), with \(X=\Lambda_L\times\{1,2\}, \Lambda_L, \widetilde{\Lambda}_L\times\{A,B,C,D\}\) for \(\alpha=\mathrm{NN},\mathrm{SN},\mathrm{SP}\) respectively, one has that

   \[\left\| \big((\cancel{D}_\alpha)^-_{\Lambda_L}- M_{\varphi^{\Lambda^{\alpha}_L}}\big)^{-1} \right\|_{\mathrm{op}}\le \frac{1}{\varphi},\]

   for any \(\varphi\ne0\). Indeed, it is readily checked (see Appendix 7.2) that the operator \((\cancel{D}_\alpha)^-_{\Lambda_L}- M_{\varphi^{\Lambda^{\alpha}_L}}\equiv (\cancel{D}_\alpha)^-_{\Lambda_L}- \varphi\mathbbl{1}\) is normal and has eigenvalues \(\Big\{\pm i \sqrt{\sin^2p_0+ \sin^2p_1} -\varphi\Big\}_{p\in(\Lambda^{\alpha}_L)^*_{--}}\), where

   \[(\Lambda^{\mathrm{NN}}_L)^*_{--} \doteq \tfrac{2\pi}{L}\big(\mathbb{Z}+ \tfrac{1}{2}\big)^2 \cap (-\pi,\pi ]^2, \qquad (\Lambda^{\mathrm{SN}}_L)^*_{--}=(\Lambda^{\mathrm{SP}}_L)^*_{--}\doteq \tfrac{2\pi}{L} \big(\mathbb{Z}+\tfrac{1}{2}\big)^2 \cap \Big(\big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big]\times(-\pi,\pi] \Big).\]

3. In order to find a lower bound for the determinant in the right–hand side of 33 , we use the following standard fact: letting \(A\) be a linear operator on a vector space \(\mathcal{V}\) of dimension \(n<\infty\), then

   \[\label{eqn:43} |\det(\mathbbl{1}_{\mathcal{V}} + A)|\ge (1-\|A\|_{\mathrm{op}})^n.\tag{34}\]

   In our case, setting \(A= \big((\cancel{D}_\alpha)^-_{\Lambda_L}-\varphi_\alpha^\star(\lambda)\big)^{-1}M_{\phi}\), with \(\phi\in Q_{\delta_N}^{\alpha,0}\), we see that, in force of Remark 6,

   \[\left\| \big((\cancel{D}_\alpha)^-_{\Lambda_L}-\varphi_\alpha^\star(\lambda)\big)^{-1}M_\phi \right\|_{\mathrm{op}} \le \left\| \big((\cancel{D}_\alpha)^-_{\Lambda_L}-\varphi_\alpha^\star(\lambda)\big)^{-1} \right\|_{\mathrm{op}} \left\| M_\phi\right\|_{\mathrm{op}} \leq \tfrac{\delta_N}{\varphi_\alpha^\star(\lambda)}.\]

   Therefore, for any choice \(0<\delta_N<\varphi^{\star}_{\alpha}(\lambda)\), we have that

   \[\big|\det\Big( \mathbbl{1}- \big((\cancel{D}_\alpha)^-_{\Lambda_L}-M_{\varphi_\alpha^\star(\lambda)}\big)^{-1} M_\phi \Big)\big|\ge \Big( 1- \tfrac{\delta_N}{\varphi^{\star}_{\alpha}(\lambda)} \Big)^{2|\Lambda_L^{\alpha}|}.\]

   Plugging the above bound in the right–hand side of 33 and performing the integral over \(Q^{\alpha,0}_{\delta_N}\), we find the following lower bound for the partition function:

   \[\label{eqn:16} (Z^{\alpha}_{\Lambda_L})^{\frac{1}{|\Lambda_L^{\alpha}|}} \ge e^{N\big[- v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda))- \tfrac{\delta_N}{\lambda}(2\varphi^{\star}_{\alpha}(\lambda)+\delta_N)+ \tfrac{1}{N}\log (2\delta_N)+ 2\log(1- \tfrac{\delta_N}{\varphi^{\star}_{\alpha}(\lambda)} ) -\tfrac{\mathfrak{c}_0}{L}\big]}.\tag{35}\]

5.1.0.3 Proof of the bound ?? .

By plugging 31 and 35 into the right–hand side of 29 , we find:

\[\label{eqn:17} \left\langle\phi_x^2 I_x^{+K_{\alpha}} \right\rangle_{\mu_{\alpha}}^{\frac{1}{|\Lambda_L^{\alpha}|}} \le e^{-N \big[v^{\alpha}_{\lambda}(K_{\alpha})-v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda)) - \mathcal{R}^{(1)}_{N,L}(\lambda)\big]} \doteq e^{-N \big[ c_1^{\alpha}(\lambda) - \mathcal{R}^{(1)}_{N,L}(\lambda)\big]},\tag{36}\]

where

\[\begin{align} \mathcal{R}^{(1)}_{N,L}(\lambda)\doteq &\big( v^{\alpha}_{\lambda}(K_{\alpha})- v^{\alpha}_{\tilde{\lambda}_N}(K_{\alpha})\big)+ \tfrac{\delta_N}{\lambda}\big(2\varphi^{\star}_{\alpha}(\lambda)+\delta_N\big)- \tfrac{1}{N}\log (2\delta_N)+\\ &-2 \log\big(1- \tfrac{\delta_N}{\varphi^{\star}_{\alpha}(\lambda)} \big)+ \tfrac{1}{2N}\big(\log(2\pi)+ 3\log\lambda\big) + \tfrac{2\mathfrak{c}_0}{{L}}. \end{align}\]

The main observation is that

\[v^{\alpha}_{\lambda}(K_{\alpha}) -v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda)) \doteq c_1^{\alpha}(\lambda)>0,\]

for every choice of \(K_{\alpha}>\varphi^{\star}_{\alpha}(\lambda)\), since the potential \(v^{\alpha}_{\lambda}\) is strictly-increasing in the interval \([\varphi^{\star}_{\alpha}(\lambda), \infty)\). Furthermore, due to the smoothness of the map \(\lambda\mapsto\varphi^{\star}_{\alpha}(\lambda)\), for every \(\varphi>0\) fixed, we have that \(v^{\alpha}_{\lambda}(K_{\alpha})- v^{\alpha}_{\tilde{\lambda}_N}(K_{\alpha})\to0\) as \(N\to\infty\). In this way, by also fixing \(\delta_N\equiv \delta_N(\lambda)\doteq \frac{\varphi^{\star}_{\alpha}(\lambda)}{N}\), we readily see that

\[\mathcal{R}^{(1)}_{N,L}(\lambda) \overset{N,L \to \infty}{\longrightarrow} 0,\]

for every \(\lambda>0\) and \(K_{\alpha}>\varphi^{\star}_{\alpha}(\lambda)\) fixed. Therefore, for \(N,L\) sufficiently large (depending on \(\lambda\) and \(K_{\alpha}\)), we can assume that \(|\mathcal{R}_{N,L}^{(1)}(\lambda)| \leq \frac{1}{2}c_1^{\alpha}(\lambda)\), yielding the bound ?? .

5.2 Local-maximum terms↩︎

In this subsection we prove the second inequality ?? , namely \[\sup_{x \in \Lambda^{\alpha}_L} \left \langle I_x^{(-C_{\alpha}, C_{\alpha})} \right \rangle_{\mu_{\alpha}} \leq e^{- \frac{N}{2} c_2^{\alpha}(\lambda)}.\] This estimate controls configurations in which the field lies in a neighborhood of a local maximum of the potential. Such configurations are energetically unfavorable and hence exponentially suppressed in \(N\). The proof relies on Chessboard Estimates, combined with the fact that, as established by Proposition 2, the potential \(v^{\alpha}_{\lambda}\) is strictly larger than its minimum whenever \(\phi_x \in (-C_{\alpha},C_{\alpha})\subsetneq (-\varphi^{\star}_{\alpha}, \varphi^{\star}_{\alpha})\). Applying the Chessboard Estimate ?? to the single-site observable, we obtain:

\[\label{eqn:10} \left \langle I_x^{(-C_{\alpha}, C_{\alpha})}\right \rangle_{\mu_{\alpha}} \leq \left(\frac{\displaystyle \int_{(-C_{\alpha},C_{\alpha})^{\Lambda^{\alpha}_L}} \big(\prod_{x \in \Lambda_L^{\alpha}}d\phi_x \big) e^{- N V^{\alpha}_{L,\lambda}(\phi)}}{ \displaystyle\int_{\mathbb{R}^{\Lambda^{\alpha}_L}} \big(\prod_{x \in \Lambda_L^{\alpha}}d\phi_x \big) e^{- N V^{\alpha}_{L,\lambda}(\phi)}}\right)^{\frac{1}{|\Lambda^{\alpha}_L|}}.\tag{37}\]

5.2.0.1 Upper bound for the numerator of 37 .

The bound of the numerator is a straightforward application of Proposition 2.

1. First, we extract the infimum of \(V^{\alpha}_{L,\lambda}\) over the constrained set \(-C_{\alpha} \leq \phi_x \leq C_{\alpha}\) and perform the integral over \(\phi\):

   \[\begin{align} \bigg(\int_{(-C_{\alpha},C_{\alpha})^{\Lambda^{\alpha}_L}} \big(\prod_{x\in \Lambda_L^{\alpha}}d\phi_x \big) \, e^{- N V^{\alpha}_{L,\lambda}(\phi)}\bigg)^{\frac{1}{|\Lambda_L^{\alpha}|}} &\leq (2C_{\alpha})\, \exp\Big\{-\tfrac{N}{|\Lambda^{\alpha}_L|} \inf_{ \phi\in (-C_{\alpha},C_{\alpha})^{\Lambda^{\alpha}_L}} V^{\alpha}_{L,\lambda}(\phi) \Big\}. \end{align}\]

2. By ?? , the infimum of \(V^{\alpha}_{L, \lambda}\) is attained over constant field configurations, whose large-\(L\) asymptotics can be controlled, due to Proposition 2.[itt:2], by the infinite-volume potential density \(v^{\alpha}_{\lambda}\):

   \[\label{eqn:13w} \frac{1}{|\Lambda^{\alpha}_L|} \inf_{\phi\in(-C_{\alpha},C_{\alpha})^{\Lambda^{\alpha}_L}} V^{\alpha}_{L,\lambda}(\phi) \overset{\eqref{Minpot}}{\geq} \frac{1}{|\Lambda^{\alpha}_L|} \inf_{\varphi\in(-C_{\alpha},C_{\alpha})} V^{\alpha}_{L,\lambda}(\varphi) \overset{\eqref{eqn:12a}}{\geq} \inf_{\varphi\in(-C_{\alpha},C_{\alpha})} v_{{\lambda}}^{\alpha}(\varphi) -\frac{\mathfrak{c}_0}{L}.\tag{38}\]

3. Since \(v^{\alpha}_{\lambda}\) is strictly decreasing on \([0, \varphi^{\star}_{\alpha}(\lambda)]\) (see Proposition 2.[itt:3]) and that \(C_{\alpha} < \varphi^{\star}_{\alpha}(\lambda)\), the right–hand side of 38 is larger than \(v^{\alpha}_{\lambda}(C_{\alpha}) - \frac{\mathfrak{c}_0}{L}\).

Therefore, we find the following upper bound for the numerator of 37 :

\[\label{eqn:16u} \begin{align} \bigg(\int_{(-C_{\alpha},C_{\alpha})^{\Lambda^{\alpha}_L}} \big(\prod_{x\in \Lambda_L^{\alpha}}d\phi_x \big) \, e^{- N V^{\alpha}_{L,\lambda}(\phi)}\bigg)^{\frac{1}{|\Lambda_L^{\alpha}|}} \leq (2C_{\alpha}) e^{- N \big[ v^{\alpha}_{\lambda}(C_{\alpha})- \tfrac{\mathfrak{c}_0}{{L}}\big]}. \end{align}\tag{39}\]

5.2.0.2 Proof of the bound ?? .

Plugging equations 31 and 39 into the right–hand side of 37 , we find: \[\begin{align} \left\langle I_x^{(-C_{\alpha},C_{\alpha})} \right\rangle_{\mu_{\alpha}}\le e^{-N\big( v^{\alpha}_{\lambda}(C_{\alpha}) - v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda))- \mathcal{R}^{(2)}_{N,L}(\lambda) \big)}, \end{align}\] where

\[\mathcal{R}^{(2)}_{N,L}(\lambda)\doteq \tfrac{\delta_N}{\lambda}\big(2\varphi^{\star}_{\alpha}(\lambda)+\delta_N\big)+ \tfrac{1}{N}\big(-\log (2\delta_N)+ \log(2C_{\alpha}) \big)- 2\log\big(1- \tfrac{\delta_N}{\varphi^{\star}_{\alpha}(\lambda)} \big) + \tfrac{2\mathfrak{c}_0}{L}.\]

We observe that

\[v^{\alpha}_{\lambda}(C_{\alpha}) -v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda)) \equiv c_2^{\alpha}(\lambda)>0,\]

for every choice of \(C_{\alpha}<\varphi^{\star}_{\alpha}(\lambda)\), since the potential \(v^{\alpha}_{\lambda}\) is strictly decreasing in the interval \([0, \varphi^{\star}_{\alpha}(\lambda))\) (see Proposition 2.[itt:3]). By fixing \(\delta_N=\frac{\varphi^{\star}_{\alpha}(\lambda)}{N}\), we see that for every \(\lambda>0\) and \(0<C_{\alpha}<\varphi^{\star}_{\alpha}(\lambda)\),

\[\mathcal{R}_{N,L}^{(2)}(\lambda)\overset{N,L\to\infty}{\longrightarrow} 0.\]

Therefore, for \(N,L\) sufficiently large (depending on \(\lambda\) and \(C_{\alpha}\)), we can assume that \(|\mathcal{R}_{N,L}^{(2)}(\lambda)| \leq \frac{1}{2}c_2^{\alpha}(\lambda)\), yielding the bound ?? .

5.3 Peierls-like terms↩︎

In this subsection we prove the bound ?? , namely \[\sup_{x,y \in \Lambda^{\alpha}_L} \left \langle I_x^{\star_+} I_y^{\star_-} \right \rangle_{\mu_{\alpha}} \leq e^{-\frac{N}{4} c_3^{\alpha}(\lambda)}.\]

This estimate controls configurations in which the field takes values near opposite minimizers of the potential at two distinct sites. Such configurations necessarily contain an interface separating regions of opposite sign, and are therefore expected to be exponentially suppressed in \(N\).
In contrast to the previous cases, this contribution cannot be handled directly by means of the Chessboard Estimates alone. Its analysis requires a more detailed control of the energy cost associated with configurations interpolating between the two minima. This is achieved via a Peierls-type argument (see e.g. [39]), combined with Reflection Positivity, which allows us to control the weight of contours separating regions with opposite sign of \(\phi\) by relating them to the energetic penalty of suitably constructed periodic configurations generated through reflections.

5.3.0.1 General Peierls’ argument.

1. For any \(x, y \in \Lambda_L^{\alpha}\), we have that

   \[\left \langle I_x^{\star_+} I_y^{\star_-} \right \rangle_{\mu_{\alpha}} \leq \left \langle I_x^+ I_y^-\right \rangle_{\mu_{\alpha}},\]

   where \(I_x^{\pm}(\phi) \doteq \mathbbl{1}_{\{\pm\phi_x > 0\}}\).

2. Since \(I_x^+ + I_x^- = 1\) almost surely, we have that

   \[\label{eqn:19} \left \langle I_x^+ I_y^-\right \rangle_{\mu_{\alpha}} = \Big\langle I_x^+ I_y^- \prod_{z \neq x,y} (I_z^+ + I_z^-)\Big\rangle_{\mu_{\alpha}} = \sum_{\substack{\zeta \in \{\pm \}^{\Lambda_L^{\alpha}}:\\ \zeta_x=+, \zeta_y=-}} \Big\langle \prod_{z\in \Lambda_L^{\alpha}} I_z^{\zeta_z} \Big\rangle_{\mu_{\alpha}}.\tag{40}\]

3. To each sign configuration \(\zeta\in \{\pm\}^{\Lambda^{\alpha}_L}\), we associate its set of Peierls’ contours, namely the closed curves in the dual lattice \((\Lambda_L^\alpha)^{\mathrm{dual}}\doteq \Lambda^{\alpha}_L+ (\frac{1}{2},\frac{1}{2})\) crossing precisely those edges \(\langle x',y'\rangle\subset \Lambda^{\alpha}_L\) for which \(\zeta_{x'}\neq \zeta_{y'}\). As usual, crossings are resolved according to the deformation rule (see [39]) shown in Fig. 7, so that the contour set is a collection of disjoint loops including the information of the values of the spins on both “sides”.
   

4. A standard Peierls argument (see [39]) then gives: \[\label{eqn:21} \left\langle I_x^{\star_+} I_y^{\star_-}\right\rangle_{\mu_\alpha} \le 2\sum_{\gamma} \Big\langle \prod_{\langle x',y'\rangle\in \mathrm{E}(\gamma)} I_{x'}^+ I_{y'}^- \Big\rangle_{\mu_\alpha},\tag{41}\] where the sum runs over all dual contours \(\gamma\) separating \(x\) and \(y\), and \(\mathrm{E}(\gamma)\) denotes the set of crossed primal edges, ordered so that \(x'\) lies on the “left hand side” of \(\gamma\) and \(y'\) on the “right hand side” (with respect to some fixed orientation of the contour \(\gamma\)).

   

5.3.0.2 Reduction to contour estimates.

The main bound for controlling the Peirls’-like contributions is provided by the following lemma.

In this way the estimate can be reduced to a simple contour-counting argument. It is readily checked that the sum over contours \(\gamma\) as in the right–hand side of 41 can be bounded as

\[\sum_{\gamma \text{ around }x} + \sum_{\gamma \text{ around }y}+ \sum_{\gamma:\; |\gamma|\ge L},\]

where “\(\gamma\) around \(x\)” means that the contour \(\gamma\) is contained in a square patch of the torus \(\Lambda^{\alpha}_L\) with side \(L\) and centered around \(x\). Therefore, assuming the bound ?? , we readily obtain that \[\label{eq:Peierls-sum} \left\langle I_x^{\star_+} I_y^{\star_-}\right\rangle_{\mu_\alpha} \le 2 \sum_{\gamma \text{ around } x} e^{-\frac{1}{4} N c_3^\alpha(\lambda)|\gamma|} + 2 \sum_{\gamma \text{ around } y} e^{-\frac{1}{4} N c_3^\alpha(\lambda)|\gamma|} + 2 \sum_{\gamma:\;|\gamma|\ge L} e^{-\frac{1}{4} N c_3^\alpha(\lambda)|\gamma|}.\tag{42}\]

Using the standard contour counting bounds: \[\#\{\gamma\text{ around }x:\;|\gamma|=k\}\le \frac{k}{2}\,3^k, \qquad \#\{\gamma:\;|\gamma|=k\}\le L^2 3^k,\] we easily find, for \(N\) and \(L\) sufficiently large: \[\sum_{\gamma\mathrm{ around }x} e^{-\frac{1}{4} N c_3^\alpha(\lambda)|\gamma|} \le \frac{1}{8} e^{-\frac{1}{4} N c_3^\alpha(\lambda)}, \qquad \sum_{\gamma:\;|\gamma|\ge L} e^{-\frac{1}{4} N c_3^\alpha(\lambda)|\gamma|} \le \frac{1}{4} e^{-\frac{1}{4} N c_3^\alpha(\lambda)}.\]

Substituting into the right–hand side 42 , we obtain the desired bound ?? .

5.3.0.3 Proof of Lemma 7.

We now prove ?? by combining Chessboard Estimates with an energetic analysis of alternating-sign periodic configurations.

1. For any contour \(\gamma\), its crossed edges can be partitioned into four classes: \[\mathrm{E}(\gamma)= \mathrm{E}_{h,e}(\gamma)\sqcup \mathrm{E}_{h,o}(\gamma)\sqcup \mathrm{E}_{v,e}(\gamma)\sqcup \mathrm{E}_{v,o}(\gamma),\]

   (here “h”/“v” stands for horizontal/vertical and “e”/“o” stands for even/odd) corresponding to the four disjoint tessellations \((\mathrm{v},\mathrm{e})\), \((\mathrm{v},\mathrm{o})\), \((\mathrm{h},\mathrm{e})\), \((\mathrm{h},\mathrm{o})\) of the torus made of \(2\times 1\) rectangles as in Fig. 8.
   

   

   By Cauchy–Schwartz’s inequality: \[\label{eqn:26} \Big\langle \prod_{\langle x,y\rangle\in \mathrm{E}(\gamma)} I_x^+ I_y^- \Big\rangle_{\mu_\alpha} = \Big\langle \prod_{\substack{a=h,v\\ b=e,o}} \prod_{\langle x,y\rangle\in \mathrm{E}_{a,b}(\gamma)} I_x^+ I_y^- \Big\rangle_{\mu_\alpha} \le \prod_{\substack{a=h,v\\ b=e,o}} \Big\langle \prod_{\langle x,y\rangle\in \mathrm{E}_{a,b}(\gamma)} I_x^+ I_y^- \Big\rangle_{\mu_\alpha}^{1/4}.\tag{43}\]

2. Applying the chessboard estimate ?? to each factor produces, up to translations, one of the four periodic patterns shown in Fig. 9. Therefore: \[\label{eq:Peierls-patterns} \Big\langle \prod_{\langle x,y\rangle\in \mathrm{E}(\gamma)} I_x^+ I_y^- \Big\rangle_{\mu_\alpha} \le \left\langle I^{(-++-)}\right\rangle_{\mu_\alpha}^{\frac{|\mathrm{E}_{h,e}(\gamma)|}{2|\Lambda_L^\alpha|}} \left\langle I^{(--++)}\right\rangle_{\mu_\alpha}^{\frac{|\mathrm{E}_{h,o}(\gamma)|}{2|\Lambda_L^\alpha|}} \left\langle I^{\left(\substack{-\\+\\+\\-}\right)}\right\rangle_{\mu_\alpha}^{\frac{|\mathrm{E}_{v,e}(\gamma)|}{2|\Lambda_L^\alpha|}} \left\langle I^{\left(\substack{-\\-\\+\\+}\right)}\right\rangle_{\mu_\alpha}^{\frac{|\mathrm{E}_{v,o}(\gamma)|}{2|\Lambda_L^\alpha|}}\tag{44}\]

   where (see also Fig. 9):

   \[\begin{align} &I^{\left(-++-\right)} \doteq \prod_{x\in \Lambda^{\alpha,h}_L} I^-_{x} I^+_{x+ e_0^{\alpha}} I^+_{x+ 2 e_0^{\alpha}} I^-_{x+ 3e_0^{\alpha}}, \qquad I^{\left(--++\right)} \doteq \prod_{x\in \Lambda^{\alpha,h}_L} I^-_{x} I^-_{x+ e_0^{\alpha}} I^+_{x+ 2 e_0^{\alpha}} I^+_{x+ 3e_0^{\alpha}}, \\ &I^{\left(\substack{-\\+\\+\\-}\right)} \doteq \prod_{x\in \Lambda^{\alpha,v}_L} I^-_{x} I^+_{x+ e_1^{\alpha}} I^+_{x+ 2e_1^{\alpha}} I^-_{x+ 3e_1^{\alpha}}, \qquad I^{\left(\substack{-\\-\\+\\+}\right)} \doteq \prod_{x\in \Lambda^{\alpha,v}_L} I^-_{x} I^-_{x+ e_1^{\alpha}} I^+_{x+ 2 e_1^{\alpha}} I^+_{x+ 3e_1^{\alpha}}, \end{align}\]

   with \(e_0^{\alpha}= 2^{\delta_{\alpha,\mathrm{SP}}}(1,0)\), \(e_1^{\alpha}= 2^{\delta_{\alpha,\mathrm{SP}}} (0,1)\), and

   \[\begin{align} \Lambda^{\alpha,h}_L&= \Lambda^{\alpha}_L \cap \left\{\begin{array}{cc} (4\mathbb{Z})\times \mathbb{Z}, & \alpha\in\{\mathrm{NN},\mathrm{SN}\} \\ (8\mathbb{Z})\times(2\mathbb{Z}), & \alpha=\mathrm{SP} \end{array}\right. \\ \Lambda^{\alpha,v}_L&= \Lambda^{\alpha}_L \cap \left\{\begin{array}{cc} \mathbb{Z}\times (4\mathbb{Z}), & \alpha\in\{\mathrm{NN},\mathrm{SN}\} \\ (2\mathbb{Z})\times(8\mathbb{Z}), & \alpha=\mathrm{SP} \end{array}. \right. \end{align}\]

   

3. By translation and rotation invariance of \(\mu_\alpha\), the four expectations coincide¹² and thus, using also that \(|\mathrm{E}_{h,e}(\gamma)|+|\mathrm{E}_{h,o}(\gamma)|+|\mathrm{E}_{v,e}(\gamma)|+|\mathrm{E}_{v,o}(\gamma)|=|\gamma|\),

   \[\label{eq:Peierls-single-pattern} \Big\langle \prod_{\langle x,y\rangle\in \mathrm{E}(\gamma)} I_x^+ I_y^- \Big\rangle_{\mu_\alpha} \le \left\langle I^{(-++-)}\right\rangle_{\mu_\alpha}^{\frac{|\gamma|}{2|\Lambda_L^\alpha|}}.\tag{45}\]

4. Following the strategy used to bound the numerator in the right–hand side of 29 , we obtain: \[\label{eq:Peierls-I-bound} \left\langle I^{(-++-)}\right\rangle_{\mu_\alpha}^{\frac{|\gamma|}{2|\Lambda^{\alpha}_L|}} \le(2 \pi \lambda)^{\frac{|\gamma|}{4}} \bigg[ \frac{1}{Z^{\alpha}_{\Lambda_L}} \exp\big\{-N \inf_{\phi \in \operatorname{supp} I^{\left(-++-\right)}} V^{\alpha}_{L,\tilde{\lambda}_N}(\phi)\big\} \bigg]^{\frac{|\gamma|}{2 |\Lambda^{\alpha}_L|}},\tag{46}\]

   with \(\tilde{\lambda}_N= \lambda\big(1-\frac{1}{N}\big)^{-1}\).

5. A further RP argument, closely analogous to Proposition 2, shows that the infimum over \(\mathrm{supp}\,I^{(-++-)}\) is attained by periodic configurations of the form: \[(\varphi_-,\varphi_+,\varphi_+,\varphi_-), \qquad \varphi_+\ge 0,\;\varphi_-\le 0,\] depending only on two real parameters \((\varphi_+,\varphi_-)\). Accordingly:

   \[\label{eqn:47} \inf_{\phi\in\mathrm{supp}\,I^{(-++-)}} V^{\alpha}_{L,\tilde{\lambda}_N}(\phi) = \inf_{\varphi_+\ge0,\;\varphi_-\le0} V^{\alpha}_{L,\tilde{\lambda}_N}(\varphi_+,\varphi_-)\tag{47}\]

   where, by abuse of notation, \(V^{\alpha}_{L,\tilde{\lambda}_N}(\varphi_+,\varphi_-)\) denotes the potential \(V^{\alpha}_{L,\tilde{\lambda}_N}(\varphi_+,\varphi_-)\) evaluated at the periodic configuration corresponding to the repetition of the pattern \((\varphi_-,\varphi_+,\varphi_+,\varphi_-)\).

   Moreover, for any \(\lambda>0\) and \(\varphi_+,\varphi_-\in\mathbb{R}\), the finite-volume potential density \(|\Lambda^{\alpha}_L|^{-1}V^{\alpha}_{L,\lambda}(\varphi_+, \varphi_-)\) is well approximated by its infinite-volume limit, to be denoted by \(w^{\alpha}_{\lambda}(\varphi_+,\varphi_-)\) (in full analogy with ?? ): \[\label{eq:Peierls-thermo} \left| \tfrac{1}{|\Lambda_L^\alpha|}V^{\alpha}_{L,\lambda}(\varphi_+,\varphi_-) - w^{\alpha}_{\lambda}(\varphi_+,\varphi_-) \right| \le \frac{\mathfrak c_1}{L}.\tag{48}\] It is possible to write down an explicit expression for \(w^{\alpha}_{\lambda}\): \[\label{eq:Peierls-w} w^{\alpha}_{\lambda}(\varphi_+,\varphi_-) = \frac{\varphi_+^2+\varphi_-^2}{4\lambda} - N_\alpha \iint_{(-\pi,\pi]^2}\frac{d^2p}{(2\pi)^2} \log \Delta^\alpha(p;\varphi_+,\varphi_-),\tag{49}\]

   where \(N_{\alpha}\) is a multiplicity factor (\(N_{\alpha}=\frac{1}{4},\frac{1}{4},\frac{1}{2}\) for \(\alpha=\mathrm{NN},\mathrm{SN},\mathrm{SP}\) respectively) and \(\Delta^\alpha(p;\varphi_+,\varphi_-)\) is an explicit function (see 52 , 57 and 64 ), related to the determinant of the Dirac operator in alternating background.

   Therefore we have that

   \[\label{eqn:56} \Big\langle I^{(-++-)}\Big\rangle_{\mu_\alpha}^{\frac{|\gamma|}{2|\Lambda^{\alpha}_L|}} \le(2 \pi \lambda)^{\frac{|\gamma|}{4}} \left[ \tfrac{1}{|\Lambda^{\alpha}_L|} \exp\Big\{-N \Big(\inf_{\varphi_+\ge0,\;\varphi_-\le0} w^{\alpha}_{\tilde{\lambda}_N}(\varphi_+,\varphi_-) -\frac{\mathfrak{c}_1}{L} \Big)\Big\}\right]^{\frac{|\gamma|}{2 |\Lambda^{\alpha}_L|}}.\tag{50}\]

6. Plugging the lower-bound on the partition function 35 in 50 , we get:

   \[\Big\langle I^{\left(-++- \right)} \Big\rangle_{\mu_{\alpha}}^{\frac{|\gamma|}{2|\Lambda^{\alpha}_L|}} \leq e^{-N \frac{|\gamma|}{2}\big(w^{\alpha}_{\tilde{\lambda}_N}(\varphi_+,\varphi_-)- v^{\alpha}_{\tilde{\lambda}_N}(\varphi_\alpha^\star(\tilde{\lambda}_N)) - \mathcal{R}^{(3)}_{N,L}(\lambda) \big) },\]

   where

   \[\begin{align} \mathcal{R}^{(3)}_{N,L}(\lambda)\doteq &\big( v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda))- v^{\alpha}_{\tilde{}\lambda_N}(\varphi^{\star}_{\alpha}(\tilde{}\lambda_N)) \big) + \tfrac{\log(2\pi\lambda)}{2N}+ \tfrac{\delta_N}{\lambda}\big(2\varphi^{\star}_{\alpha}(\lambda)+\delta_N\big)\\ &-\tfrac{1}{N}\log (2\delta_N)- 2\log\big(1- \tfrac{\delta_N}{\varphi^{\star}_{\alpha}(\lambda)} \big) + \tfrac{\mathfrak{c}_0+\mathfrak{c}_1}{L}. \end{align}\]

7. Since \(\varphi^{\star}_{\alpha}\) is a smooth function of \(\lambda\in(0,\infty)\) (see Lemma 2) and \(v^{\alpha}_{\lambda}(\varphi)\) is smooth with respect to \(\lambda\) and \(\varphi\in (0,\infty)\), we easily get that

   \[\lim_{N\to\infty} v^{\alpha}_{\tilde{}\lambda_N}(\varphi^{\star}_{\alpha}(\tilde{}\lambda_N))= v^{\alpha}_{\lambda}(\varphi^{\star}_{\alpha}(\lambda)).\]

   Using this information, we can follow a similar reasoning as before: by fixing \(\delta_N=\frac{\varphi^{\star}_{\alpha}(\lambda)}{N}\), we see that for every \(\lambda>0\):

   \[\mathcal{R}^{(3)}_{N,L}(\lambda) \overset{N,L \to \infty }{\longrightarrow} 0.\]

The crucial input, about the energetic penalty associated with the Peirels-like terms, is provided by the following lemma.

1. If \(\lambda \leq \lambda_0\), for \(N\) large enough, \(\tilde{\lambda}_N \leq \lambda_0\) as well. Therefore, we can apply Lemma 8 to obtain:

   \[\left\langle I^{\left(-++- \right)} \right\rangle_{\mu_{\alpha}}^{\frac{|\gamma|}{2|\Lambda^{\alpha}_L|}} \leq e^{-N \frac{|\gamma|}{2}\big( c_3^{\alpha}(\tilde{}\lambda_N) - \mathcal{R}^{(3)}_{N,L}(\lambda) \big) }.\]

   Also, for \(N, L\) large enough (depending on \(\lambda\)), we can assume that \(|\mathcal{R}^{(3)}_{N,L}(\lambda)| \leq \frac{1}{4} c_3^{\alpha}(\lambda)\), leading to

   \[\left\langle I^{\left(-++- \right)} \right\rangle_{\mu_{\alpha}}^{\frac{|\gamma|}{2|\Lambda^{\alpha}_L|}} \leq e^{-N \frac{|\gamma|}{2} \big(c_3^{\alpha}(\tilde{}\lambda_N) - \frac{1}{4} c_3^{\alpha}(\lambda)\big) }.\]

2. Furthermore, by the very definition (cf. Proposition 3), \(c^{\alpha}_3\) is a continuous function of \(\lambda>0\), so that \(\lim_{N\to\infty} c_3^{\alpha}(\tilde{}\lambda_N)= c_3^{\alpha}(\lambda)\). Therefore, noting also that \(c_3^{\alpha}(\lambda)>0\), for \(N\) large enough we can assume that

   \[\big|c^{\alpha}_3(\lambda)-c^{\alpha}_3(\tilde{\lambda}_N)\big| \leq \tfrac{1}{4}c^{\alpha}_3(\lambda),\]

   which yields the bound ?? .

5.4 Proof of Lemma 8↩︎

We discuss the proof for the three models, \(\alpha\in\{\mathrm{NN},\mathrm{SN},\mathrm{SP}\}\), separately. We prefer to start by discussing the SN model, which involves slightly simpler computations than the other two models.

5.4.0.1 Proof for the SN model.

By using the periodicity of the alternate field configurations \(\phi\) (consisting of only two independent variables \(\varphi_+,\varphi_-\)), see Fig. 10, one can easily compute the expression for the potential \(V^{\mathrm{SN}}_{L,\lambda}(\varphi_+,\varphi_-)\) introduced after 47 :

\[V^{\mathrm{SN}}_{L,\lambda}(\varphi_+,\varphi_-)= \frac{|\Lambda_L|}{2} \frac{\varphi_+^2+ \varphi_-^2}{2\lambda}- \sum_{p\in \Xi^*_L}\log \Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-),\]

where \(\Xi^*_L= \Big(\frac{2\pi}{L} \big( \mathbb{Z}+\frac{1}{2}\big)^2\Big) \cap \Big((-\frac{\pi}{4},\frac{\pi}{4}]\times(-\pi,\pi]\Big)\) and

\[\Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)= \det\begin{pmatrix} i \sin(p_1) - \varphi_- &-\tfrac{1}{2} &0 &\tfrac{1}{2}e^{4i p_0}\\ \tfrac{1}{2} &- i \sin(p_1) - \varphi_+ &-\tfrac{1}{2} &0\\ 0 &\tfrac{1}{2} &i \sin(p_1) - \varphi_+ &-\tfrac{1}{2}\\ -\frac{1}{2}e^{-4 i p_0} &0 &\tfrac{1}{2} &-i \sin(p_1) - \varphi_- \end{pmatrix}.\]

It follows that

\[\label{eqn:24} \begin{align} &w^{\mathrm{SN}}_{\lambda}(\varphi_+,\varphi_-)= \lim_{L\to\infty} \tfrac{1}{|\Lambda_L|} V^L_{\lambda}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+ \varphi_-^2}{4\lambda}- \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\pi,\pi]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-); \end{align}\tag{51}\]

moreover, an explicit computation leads to

\[\label{eqn:46a} \Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)= \frac{1}{4}\sin^2 (2p_0)+ \frac{1}{4}(\varphi_++\varphi_-)^2 + (\varphi_+\varphi_-)^2 + \sin^2 (p_1) \big(1+ \varphi_+^2+ \varphi_-^2 +\sin^2 (p_1) \big).\tag{52}\]

Note that due to the periodicity of \(\Delta^{\mathrm{SN}}(\,\cdot\, ;\varphi_+,\varphi_-)\) under translations of \(\frac{\pi}{2}\mathbb{Z}\times \pi\mathbb{Z}\), we can either rewrite \(w_{\lambda}^{\mathrm{SN}}\) in the form of 49 , or alternatively

\[\label{eqn:24bis} \begin{align} &w^{\mathrm{SN}}_{\lambda}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+ \varphi_-^2}{4\lambda}- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-), \end{align}\tag{53}\]

so that within the latter domain of integration, if \(\varphi_+=\varphi_-=0\), \(\Delta^{\mathrm{SN}}(\,\cdot\, ;\varphi_+,\varphi_-)\) is singular only at \(p=0\).

We are interested in finding the minimizers of \(w^{\mathrm{SN}}_{\lambda}\) and understanding their properties. By computing derivatives, we get:

\[\begin{align} \partial_{\varphi_+} w_{\lambda}^{\mathrm{SN}}(\varphi_+,\varphi_-) &= \frac{\varphi_+}{2\lambda} - 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{ \frac{1}{2}(\varphi_+ +\varphi_-) + 2 \varphi_+ \varphi_-^2 +2 \sin^2(p_1) \varphi_+}{\Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)}, \\ \partial_{\varphi_-} w^{\mathrm{SN}}_{\lambda}(\varphi_+,\varphi_-) &= \frac{\varphi_-}{2\lambda} - 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{ \frac{1}{2}(\varphi_+ +\varphi_-) + 2 \varphi_- \varphi_+^2 +2 \sin^2(p_1) \varphi_-}{\Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)}. \end{align}\]

Since we are looking for the minimizer over th constrained region \(\{\varphi_+\ge0, \varphi_-\le0\}\), looking at the zeros of \(\nabla w_{\lambda}\) is not enough. However, we observe the following:

\[\label{eqn:5} \begin{align} & (1,-1)\cdot \nabla w_{\lambda}^{\mathrm{SN}}(\varphi_+,\varphi_-)\equiv \partial_{\varphi_+}w_{\lambda}^{\mathrm{SN}}(\varphi_+,\varphi_-)- \partial_{\varphi_-}w_{\lambda}^{\mathrm{SN}}(\varphi_+,\varphi_-)=\\ & \frac{1}{2\lambda} (\varphi_+-\varphi_-) - 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{-2(\varphi_+-\varphi_-) \varphi_+ \varphi_- +2(\varphi_+-\varphi_-) \sin^2(p_1) }{\Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)}=\\ & (\varphi_+-\varphi_-) \left\{ \frac{1}{2\lambda} - 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{-2\varphi_+ \varphi_- +2\sin^2(p_1) }{\Delta^{\mathrm{SN}}(p;\varphi_+,\varphi_-)} \right\}. \end{align}\tag{54}\]

The expression in brace can be shown to be positive for \(\lambda\) small enough, say \(\lambda\le \lambda_0\) with \(\lambda_0\) a suitable constant (recall also the discussion in Remark 7). Indeed, since

\[\label{eqn:6} \Delta(p;\varphi_+,\varphi_-)\gtrsim m_{\mathrm{SN}}(\varphi_+,\varphi_-)^2 + |p|^2 \qquad \forall p\in \big(-\tfrac{\pi}{4},\tfrac{\pi}{4}\big]\times\big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big],\tag{55}\]

with \(m_{\mathrm{SN}}(\varphi_+,\varphi_-)= |\varphi_+\varphi_-|\) (the writing \(a\gtrsim b\) means that \(a\ge \mathfrak{C}b\), for some universal constant \(\mathfrak{C}>0\)), it follows that

\[\begin{align} \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \left|\frac{-2\varphi_+ \varphi_- +2\sin^2(p_1) }{\Delta^{\mathrm{SN}}(p;\phi_+,\phi_-)} \right|&\lesssim \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{|p|+m_{\mathrm{SN}}(\varphi_+,\varphi_-)}{|p|^2+m_{\mathrm{SN}}(\varphi_+,\varphi_-)^2}\\ & \lesssim \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2 \pi)^2} \frac{1}{\sqrt{|p|^2+m_{\mathrm{SN}}(\varphi_+,\varphi_-)^2}}\\ &\lesssim 1. \end{align}\]

As a consequence, for \(\lambda\) small enough the right–hand side of 54 is always non-negative on the domain of our interest, namely \(\{\varphi_+\ge0,\varphi_-\le0\}\) and so is the quantity \((1,-1)\cdot \nabla w_{\lambda}(\varphi_+,\varphi_-)\). In other words, \(w_{\lambda}(\varphi_+,\varphi_-)\) is minimized on the boundary of the region \(\{\varphi_+\ge0,\varphi_-\le0\}\). Besides, since by inspection \(w_{\lambda}(\varphi_+,\varphi_-)= w_{\lambda}(-\varphi_-,-\varphi_+)\), it suffices to look for the minima on the set \(\{\varphi_+\ge0,\varphi_-=0\}\). In order to proceed, we note the following.

1. By construction, \(w_{\lambda}^{\mathrm{SN}}(\varphi,\varphi)= \lim_{L\to\infty} \frac{1}{|\Lambda_L|} V^{\mathrm{SN}}_{L,\lambda}(\phi)\Big|_{\phi= \varphi^{\Lambda^{\alpha}_L}} \equiv v_{\lambda}^{\mathrm{SN}}(\varphi)\), with \(v^{\mathrm{SN}}_{\lambda}\) minimized by \(\varphi=\varphi^{\star}_{\mathrm{SN}}(\lambda)\) (see Proposition 2).

2. Explicitly, we have that

   \[\begin{align} & \Delta^{\mathrm{SN}}(p;\varphi,\varphi)= \tfrac{1}{4}\sin^2 (2p_0)+ \varphi^2 + \varphi^4 + \sin^2 (p_1) \big(1+ 2\varphi^2 +\sin^2 (p_1) \big),\\ &\Delta^{\mathrm{SN}}(p;\varphi,0)= \tfrac{1}{4}\sin^2 (2p_0)+ \tfrac{1}{4}\varphi^2 + \sin^2 (p_1) \big(1+ \varphi^2 +\sin^2 (p_1) \big), \end{align}\]

   from which we see that \(\Delta^{\mathrm{SN}}(p;\varphi,0) = \Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi, \tfrac{1}{\sqrt{2}}\varphi\big)- \tfrac{1}{4}\varphi^2- \tfrac{1}{4}\varphi^4\).

Therefore we have that

\[\begin{align} w_{\lambda}^{\mathrm{SN}}(\varphi,0)&= \frac{\varphi^2}{4\lambda}- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SN}}(p;\varphi,0)\\ &= \frac{\varphi^2}{4\lambda}- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\Big( \Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big) - \tfrac{1}{4}\varphi^2(1+ \varphi^2)\Big)\\ &= \frac{\varphi^2}{4\lambda}- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big) \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\left( 1 - \frac{\tfrac{1}{4}\varphi^2(1+ \varphi^2)}{\Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)} \right)\\ &= v_{\lambda}^{\mathrm{SN}}\big( \tfrac{1}{\sqrt{2}}\varphi\big)- 2\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\left( 1 - \frac{\tfrac{1}{4}\varphi^2(1+\varphi^2)}{\Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)} \right)\\ &\doteq v_{\lambda}^{\mathrm{SN}}\big( \tfrac{1}{\sqrt{2}}\varphi\big) + \delta w^{\mathrm{SN}}(\varphi). \end{align}\]

Observe that \(\Delta^{\mathrm{SN}}(p;\varphi,\varphi)\lesssim 1+ \varphi^2+\varphi^4\le (1+\varphi^2)^2\), therefore:

\[\label{eqn:30} \begin{align} \delta w^{\mathrm{SN}}(\varphi)& \equiv -2 \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\left( 1 - \frac{\tfrac{1}{4}\varphi^2(1+\varphi^2)}{\Delta^{\mathrm{SN}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)} \right) \\ &\ge - \frac{1}{4}\log\left( 1 - \mathfrak{C}_2\frac{\varphi^2(1+\varphi^2)}{\big( 1+ \varphi^2\big)^2} \right) = - \frac{1}{4}\log \left(1- \mathfrak{C}_2\frac{\varphi^2}{1+\varphi^2} \right), \end{align}\tag{56}\]

for a suitable constant \(\mathfrak{C}_2^{\mathrm{SN}}>0\). Note that the right–hand side of 56 is positive and monotone increasing for \(\varphi>0\). We deduce the following:

• if \(\varphi\in\big[0,\varphi^{\star}_{\mathrm{SN}}(\lambda)\big]\), we have that \(w_{\lambda}^{\mathrm{SN}}(\varphi,0)\ge v_{\lambda}^{\mathrm{SN}}\big(\frac{1}{\sqrt{2}}\varphi\big)\ge v_{\lambda}^{\mathrm{SN}}\big(\tfrac{1}{\sqrt{2}}\varphi_{\mathrm{SN}}^{\star}(\lambda)\big)\) (recall from Proposition 2 the form of \(v_{\lambda}^{\mathrm{SN}}\));

• if \(\varphi\in\big[\varphi^{\star}_{\mathrm{SN}}(\lambda),\infty\big)\), then

  \[w_{\lambda}^{\mathrm{SN}}(\varphi,0) \ge - \frac{1}{4} \log \left(1- \mathfrak{C}_2^{\mathrm{SN}} \tfrac{\varphi^2}{1+ \varphi^2} \right) + v_{\lambda}^{\mathrm{SN}}\big(\tfrac{1}{\sqrt{2}}\varphi\big)\ge - \frac{1}{4} \log \left(1- \mathfrak{C}_2^{\mathrm{SN}} \tfrac{\big(\varphi^{\star}_{\mathrm{SN}}(\lambda)\big)^2}{1+ \big(\varphi^{\star}_{\mathrm{SN}}(\lambda)\big)^2} \right) + v_{\lambda}^{\mathrm{SN}}\big(\varphi_{\mathrm{SN}}^{\star}(\lambda)\big).\]

We conclude that

\[\begin{align} &\min_{\varphi_+\ge0,\varphi_-\le0} w_{\lambda}^{\mathrm{SN}}(\varphi_+,\varphi_-) - v_{\lambda}^{\mathrm{SN}}\big(\varphi_{\mathrm{SN}}^{\star}(\lambda)\big) =\min_{\varphi\ge0} w_{\lambda}^{\mathrm{SN}}(\varphi,0) - v_{\lambda}^{\mathrm{SN}}\big(\varphi_{\mathrm{SN}}^{\star}(\lambda)\big) \\ &\ge \min\left\{ v_{\lambda}^{\mathrm{SN}}\big(\tfrac{1}{\sqrt{2}}\varphi_{\mathrm{SN}}^{\star}(\lambda)\big)- v_{\lambda}^{\mathrm{SN}}\big(\varphi_{\mathrm{SN}}^{\star}(\lambda)\big), \, - \frac{1}{4} \log \left(1- \mathfrak{C}_2^{\mathrm{SN}}\tfrac{\big(\varphi^{\star}_{\mathrm{SN}}(\lambda)\big)^2}{1+ \big(\varphi^{\star}_{\mathrm{SN}}(\lambda)\big)^2} \right) \right\}\equiv c_3^{\mathrm{SN}}(\lambda). \end{align}\]

5.4.0.2 Proof for the NN model.

Proceeding in analogy with the case of the SN model (see Fig. 11 for a graphical understanding of the field configuration), we find that

\[\label{eqn:46b} w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+\varphi_-^2}{4\lambda}- \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\pi,\pi]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{NN}}(p;\varphi_+,\varphi_-),\tag{57}\]

where:

\[\label{eq:myNN1} \begin{align} &\Delta^{\mathrm{NN}}(p;\varphi_+,\varphi_-)=\\ &=\det \left(\!\!\begin{array}{cccccccc} -\varphi_- & -i\sin p_1&0 &-\frac{i}{2} &0 &0 &0 &\frac{i}{2}e^{4ip_0} \\ -i\sin p_1 & -\varphi_- &\frac{i}{2}&0&0&0&-\frac{i}{2} e^{4i p_0}&0\\ 0&\frac{i}{2}&-\varphi_+ &-i\sin p_1&0&-\frac{i}{2}&0&0\\ -\frac{i}{2}&0&-i\sin p_1&-\varphi_+&\frac{i}{2}&0&0&0\\ 0&0&0&\frac{i}{2}&-\varphi_+&-i\sin p_1&0&-\frac{i}{2}\\ 0&0&-\frac{i}{2}&0&-i\sin p_1&-\varphi_+&\frac{i}{2}&0\\ 0&-\frac{i}{2} e^{-4i p_0}&0&0&0&\frac{i}{2}&-\varphi_-&-i\sin p_1\\ \frac{i}{2} e^{-4i p_0}&0&0&0&-\frac{i}{2}&0&-i\sin p_1&-\varphi_- \end{array} \!\!\right)\\ &=\frac{1}{64} \big[8+ 4\varphi_+\varphi_- + 6\varphi_+^2+ 6\varphi_-^2 + 8\varphi_+^2\varphi_-^2 - \cos(4p_0) -4\big(2+\varphi_+^2+ \varphi_-^2 \big) \cos(2p_1) + \cos(4p_1)\big]^2. \end{align}\tag{58}\]

Note that due to the periodicity of \(\Delta^{\mathrm{NN}}(\,\cdot\,;\varphi_+,\varphi_-)\) under translations of \(\frac{\pi}{2}\mathbb{Z}\times \pi\mathbb{Z}\), we can either rewrite \(w_{\lambda}^{\mathrm{NN}}\) in the form of 49 , or alternatively, by also writing \(\Delta^{\mathrm{NN}}(p;\varphi_+,\varphi_-)= \big(\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)\big)^2\),

\[w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+\varphi_-^2}{4\lambda}- 4\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-),\]

so that, on the latter domain, the integrand has only one singularity at \(p=0\), if \(\varphi_+=\varphi_-=0\). Some basic but cumbersome algebraic manipulations lead to the following rewriting:

\[\label{eq:myNN4} \begin{align} &\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)= \frac{1}{4}(\varphi_+ + \varphi_-)^2 + \varphi_+^2\varphi_-^2 + \frac{1}{4}\sin^2(2p_0) + \big(1+\varphi_+^2+ \varphi_-^2 + \sin^2 (p_1)\big) \sin^2(p_1). \end{align}\tag{59}\]

We are going to adapt the same arguments used in the proof for the SN model. As a first step, we shall show that \(\partial_{\varphi_+}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)- \partial_{\varphi_-}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)\ge0\), implying that the minimum of \(w_{\lambda}^{\mathrm{NN}}\) is attained on the set \(\{\varphi_+=0\}\cup\{\varphi_-=0\}\). Since:

\[\begin{align} &\partial_{\varphi_+}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)- \partial_{\varphi_-}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)\\ &= \frac{\varphi_+-\varphi_-}{2\lambda}- 4 \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \frac{\partial_{\varphi_+}\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)- \partial_{\varphi_-}\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)}{\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)}, \end{align}\]

and, as an explicit computation shows,

\[\label{eq:myNN2} \begin{align} \big(\partial_{\varphi_+}-\partial_{\varphi_-}\big)\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-) = 2(\varphi_+-\varphi_-) \big( -\varphi_+\varphi_- + \sin^2(p_1)\big), \end{align}\tag{60}\]

we can write:

\[\label{eq:myNN3} \begin{align} &\partial_{\varphi_+}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)- \partial_{\varphi_-}w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)= (\varphi_+-\varphi_-)\left\{\frac{1}{2\lambda}- 8 \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \frac{-\varphi_+\varphi_- + \sin^2p_1}{\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)} \right\}. \end{align}\tag{61}\]

Notice that for \(p\in (-\frac{\pi}{4},\frac{\pi}{4}]\times (-\frac{\pi}{2},\frac{\pi}{2}]\), we have that \(\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-) \gtrsim |p|^2+ m_{\mathrm{NN}}(\varphi_+,\varphi_-)^2\), with \(m_{\mathrm{NN}}(\varphi_+,\varphi_-)\doteq \sqrt{(\varphi_+ + \varphi_-)^2 + \varphi_+^2\varphi_-^2}\); furthermore \(|\varphi_+\varphi_-|\le m_{\mathrm{NN}}(\varphi_+,\varphi_-)\). It follows that the integral in the right–hand side of 61 is uniformly bounded:

\[\begin{align} \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \left|\frac{-\varphi_+\varphi_- + \sin^2p_1}{\sqrt{\Delta^{\mathrm{NN}}}(p;\varphi_+,\varphi_-)}\right|& \lesssim \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \frac{m_{\mathrm{NN}}(\varphi_+,\varphi_-)+ |p|^2}{m_{\mathrm{NN}}(\varphi_+,\varphi_-)^2+ |p|^2}\\ &\lesssim \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \left(\frac{1}{\sqrt{|p|^2 + m_{\mathrm{NN}}(\varphi_+,\varphi_-)^2}}+ 1 \right) \\ &\lesssim 1. \end{align}\]

As an immediate consequence, from 61 we have that for \(\lambda\) small enough, \((1,-1)\cdot \nabla w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)\ge0\) on the domain \(\{\varphi_+\ge0,\varphi_-\le0\}\). Using also the symmetry \(w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-)= w_{\lambda}^{\mathrm{NN}}(-\varphi_-,-\varphi_+)\), it follows that

\[\min_{\varphi_+\ge0,\varphi_-\le0} w^{\mathrm{NN}}_{\lambda}(\varphi_+,\varphi_-)= \min_{\varphi\ge0} w_{\lambda}^{\mathrm{NN}}(\varphi,0).\]

Recalling that, by construction, \(w_{\lambda}^{\mathrm{NN}}(\varphi,\varphi)= v_{\lambda}^{\mathrm{NN}}(\varphi)\); it is then natural to compare \(w_{\lambda}^{\mathrm{NN}}(\varphi,0)\) with \(w_{\lambda}^{\mathrm{NN}}\big(\frac{1}{\sqrt{2}}\varphi,\frac{1}{\sqrt{2}}\varphi\big)\), in the same way as for the SN model. From 59 , it is straightforward to check that

\[\label{eq:myNN5} \begin{align} \sqrt{\Delta^{\mathrm{NN}}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)&= \sqrt{\Delta^{\mathrm{NN}}}(p;\varphi,0)+ \tfrac{1}{4} \big(\varphi^2+\varphi^4\big). \end{align}\tag{62}\]

This allows us to rewrite:

\[\begin{align} &w_{\lambda}^{\mathrm{NN}}(\varphi,0)= \frac{\varphi^2}{4\lambda}- 4\iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \sqrt{\Delta^{\mathrm{NN}}}(p;\varphi,0)=\\ & w_{\lambda}^{\mathrm{NN}}\big(\tfrac{1}{\sqrt{2}}\varphi, \tfrac{1}{\sqrt{2}}\varphi\big) - 4\iint \frac{d^2p}{(2\pi)^2} \log \left(1- \frac{\varphi^2+\varphi^4}{4\sqrt{\Delta^{\mathrm{NN}}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)} \right) \doteq v_{\lambda}\big(\tfrac{1}{\sqrt{2}}\varphi\big)+ \delta w^{\mathrm{NN}}(\varphi). \end{align}\]

Noting that \(\sqrt{\Delta^{\mathrm{NN}}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big) \lesssim 1+ \varphi^4\), it is easy to get the following lower bound for \(\delta w^{\mathrm{NN}}\):

\[\label{eq:myNN6} \delta w^{\mathrm{NN}}(\varphi)\ge - 4 \iint_{(-\frac{\pi}{4},\frac{\pi}{4}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\left( 1- \mathfrak{C}_2^{\mathrm{NN}} \frac{\varphi^4}{1+\varphi^4}\right)= - \frac{1}{2} \log\left( 1- \mathfrak{C}_2^{\mathrm{NN}} \frac{\varphi^4}{1+\varphi^4}\right),\tag{63}\]

for a suitable constant \(\mathfrak{C}_2^{\mathrm{NN}}>0\). Note that the right–hand side of 63 is positive and monotone increasing for \(\varphi>0\). All in all:

\[w_{\lambda}^{\mathrm{NN}}(\varphi,0) \ge v_{\lambda}^{\mathrm{NN}} \big(\tfrac{1}{\sqrt{2}}\varphi\big)- \frac{1}{2} \log\left( 1- \mathfrak{C}_2^{\mathrm{NN}} \frac{\varphi^4}{1+\varphi^4}\right).\]

By replicating the same reasoning presented at the end of the previous paragraph, we find that

\[\begin{align} &\min_{\varphi_+\ge0,\varphi_-\le0} w_{\lambda}^{\mathrm{NN}}(\varphi_+,\varphi_-) - v_{\lambda}^{\mathrm{NN}}\big(\varphi_{\mathrm{NN}}^{\star}(\lambda)\big) \ge \\ & \min\left\{ v_{\lambda}^{\mathrm{NN}}\big(\tfrac{1}{\sqrt{2}}\varphi_{\mathrm{NN}}^{\star}(\lambda)\big)- v_{\lambda}^{\mathrm{NN}}\big(\varphi_{\mathrm{NN}}^{\star}(\lambda)\big), \, - \frac{1}{4} \log \left(1- \mathfrak{C}_2^{\mathrm{NN}}\frac{\big(\varphi^{\star}_{\mathrm{NN}}(\lambda)\big)^4}{1+ \big(\varphi^{\star}_{\mathrm{NN}}(\lambda)\big)^4} \right) \right\}\equiv c_3^{\mathrm{NN}}(\lambda). \end{align}\]

5.4.0.3 Proof for the SP model.

Again we follow the same ideas used for the SN model (see Fig. 12 for a graphical understanding of the field configuration). An explicit computation leads to

\[w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+\varphi_-^2}{4\lambda}- 4\iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\pi,\pi]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-),\]

where

\[\label{eq:mySP1} \begin{align} &\Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-)=\\ &\frac{1}{256}\Big\{ 16\big[\varphi_+^2+\varphi_+^4+ 2\varphi_-(\varphi_++2\varphi_+^3)+ 4\varphi_-^3(\varphi_++2\varphi_+^3)+ \varphi_-^2(1+10\varphi_+^2+12\varphi_+^4)+ \\ &+\varphi_-^4(1+12\varphi_+^2+ 16\varphi_+^4)\big] + 4\sin^2(4p_0)+ 64\sin^2p_1 \big(1+ \varphi_-^2+ \varphi_+^2+ \sin^2p_1 \big)\cdot\\ &\cdot \big[ 1+ 2\varphi_-\varphi_+ + 3\varphi_+^2 + \varphi_-^2(3+8\varphi_+^2)+ 4\sin^2p_1\big(1+\varphi_-^2+ \varphi_+^2 + \sin^2p_1\big)\big]\Big\}. \end{align}\tag{64}\]

Note that due to the periodicity of \(\Delta^{\mathrm{SP}}(\,\cdot\,;\varphi_+,\varphi_-)\) under translations of \(\frac{\pi}{4}\mathbb{Z}\times \pi\mathbb{Z}\), we can either rewrite \(w_{\lambda}^{\mathrm{SP}}\) in the form of 49 , or alternatively

\[w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-)= \frac{\varphi_+^2+\varphi_-^2}{4\lambda}- 8\iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-).\]

In this way, if \(\varphi_+=\varphi_-=0\), the integrand has only one singularity at \(p=0\). Trivial though cumbersome algebraic manipulations lead to

\[\label{eq:mySP4} \begin{align} \Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-)&= \frac{1}{16} \big[ 16\varphi_+^4\varphi_-^4 + 8\varphi_-^2\varphi_+^2(\varphi_-^2+\varphi_+^2)+ (\varphi_+^2+ \varphi_-^2)^2 + (1+2\varphi_-\varphi_+)^2(\varphi_-+\varphi_+)^2 \big]+\\ &+ \frac{1}{64}\sin^2(4p_0)+ \frac{1}{4}\sin^2p_1 \big(1+ \varphi_-^2+ \varphi_+^2 + \sin^2p_1\big)\cdot\\ &\cdot \big[ 1+ 2\varphi_-\varphi_+ + 3\varphi_+^2 + \varphi_-^2(3+8\varphi_+^2)+ 4\sin^2p_1\big(1+\varphi_-^2+ \varphi_+^2 + \sin^2p_1\big)\big]. \end{align}\tag{65}\]

Notice that since \(2\varphi_-\varphi_+ + 3\varphi_+^2 + \varphi_-^2(3+8\varphi_+^2)\ge 2\varphi_-\varphi_+ + \varphi_+^2 + \varphi_-^2\ge0\), we have that

\[\Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-)\gtrsim m_{\mathrm{SP}}(\varphi_+,\varphi_-)^2+ |p|^2 \qquad \forall p\in \big(-\tfrac{\pi}{8},\tfrac{\pi}{8}\big]\times\big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big],\]

where we have introduced

\[m_{\mathrm{SP}}(\varphi_+,\varphi_-) \doteq \sqrt{ \varphi_+^4\varphi_-^4 + \varphi_-^2\varphi_+^2(\varphi_-^2+\varphi_+^2)+ (\varphi_+^2+ \varphi_-^2)^2 }.\]

Once again our goal is to replicate the reasoning from the analysis of the SN model. An explicit computation shows that

\[\label{eq:mySP3} \begin{align} &(1,-1)\cdot\nabla w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-) = (\varphi_+-\varphi_-)\left\{\frac{1}{2\lambda}- 8 \iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \frac{\Gamma^{\mathrm{SP}}(p;\varphi_+,\varphi_-)}{\Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-)} \right\}, \end{align}\tag{66}\]

where

\[\begin{align} \Gamma^{\mathrm{SP}}(p;\varphi_+,\varphi_-) &= \frac{1}{2} \Big\{ -\varphi_+\varphi_-\big[1+3 \varphi_+^2+ \varphi_-^2(3+8\varphi_+^2) \big] +\\ &+\sin^2(p_1)\big[3-8\varphi_-^3\varphi_+ + 5\varphi_+^2 + \varphi_-^2(5+8\varphi_+^2) - \varphi_+\varphi_-(6+ 8\varphi_+^2) \big]\\ &+ 2\sin^2(p_1) \big[5+ 4(\varphi_-^2+\varphi_+^2- \varphi_-\varphi_+) + 4\sin^2(p_1) \big]\Big\}. \end{align}\]

Observe that \(|\Gamma^{\mathrm{SP}}(p;\varphi_+,\varphi_-)|\lesssim m_{\mathrm{SP}}(\varphi_+,\varphi_-)+ m_{\mathrm{SP}}(\varphi_+,\varphi_-)^2+ |p|^2\); therefore the integral in the right–hand side of 66 is uniformly bounded:

\[\begin{align} &\iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \left|\frac{\Gamma^{\mathrm{SP}}(p;\varphi_+,\varphi_-)}{\Delta^{\mathrm{SP}}(p;\varphi_+,\varphi_-)} \right| \lesssim \\ &\iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \frac{m_{\mathrm{SP}}(\varphi_+,\varphi_-)+ m_{\mathrm{SP}}(\varphi_+,\varphi_-)^2 + |p|^2}{|p|^2 + m_{\mathrm{SP}}(\varphi_+,\varphi_-)^2} \lesssim 1. \end{align}\]

As an immediate consequence, from 66 we have that for \(\lambda\) small enough, \((1,-1)\cdot \nabla w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-)\ge0\) on \(\{\varphi_+\ge0, \varphi_-\le0\}\). Using also the symmetry \(w_{\lambda}(\varphi_+,\varphi_-)= w_{\lambda}(-\varphi_-,-\varphi_+)\), it follows that

\[\min_{\varphi_+\ge0,\varphi_-\le0} w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-)= \min_{\varphi\ge0} w_{\lambda}^{\mathrm{SP}}(\varphi,0).\]

Again we aim to compare \(w_{\lambda}^{\mathrm{SP}}(\varphi,0)\) with \(w_{\lambda}^{\mathrm{SP}}\big(\frac{1}{\sqrt{2}}\varphi,\frac{1}{\sqrt{2}}\varphi\big)\), keeping in mind that \(w_{\lambda}^{\mathrm{SP}}(\varphi,\varphi)= v_{\lambda}^{\mathrm{SP}}(\varphi)\). From 65 we have that

\[\begin{align} &\Delta^{\mathrm{SP}}(p;\varphi,0)=\\ &\frac{1}{16} \big[\varphi^2+ \varphi^4\big]+ \frac{1}{64}\sin^2(4p_0)+ \frac{1}{4}\sin^2p_1 \big(1+ \varphi^2+ \sin^2p_1\big) \big[ 1+ 3\varphi^2 + 4\sin^2p_1\big(1+ \varphi^2 + \sin^2p_1\big)\big], \end{align}\]

while

\[\begin{align} \Delta^{\mathrm{SP}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)&= \frac{1}{16} \big[2\varphi^2+ 5\varphi^4 + 4\varphi^6 + \varphi^8 \big]+ \frac{1}{64}\sin^2(4p_0)+\\ &+ \frac{1}{4}\sin^2p_1 \big(1+ \varphi^2 + \sin^2p_1\big)\cdot \big[ 1+ 4\varphi^2 + 2\varphi^4+ 4\sin^2p_1\big(1+\varphi^2 + \sin^2p_1\big)\big]. \end{align}\]

Therefore we see that \(\Delta^{\mathrm{SP}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)= \Delta^{\mathrm{SP}}(p;\varphi,0)+ \tilde{\Delta}^{\mathrm{SP}}(p;\varphi)\), where

\[\label{eq:mySP5} \begin{align} \tilde{\Delta}^{\mathrm{SP}}\big(p;\varphi\big) = \tfrac{1}{16}\big(\varphi^2+ 4\varphi^4+ 4\varphi^6+ \varphi^8\big)+ \tfrac{1}{4}\sin^2p_1 \big(1+ \varphi^2 + \sin^2p_1\big)\big(\varphi^2+ 2\varphi^4\big). \end{align}\tag{67}\]

This allows us to rewrite

\[\begin{align} &w_{\lambda}^{\mathrm{SP}}(\varphi,0)=\\ &v_{\lambda}^{\mathrm{SP}}\big(\tfrac{1}{\sqrt{2}}\varphi\big)- 8\iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log \left(1- \frac{\tilde{\Delta}^{\mathrm{SP}}(p;\varphi)}{\Delta^{\mathrm{SP}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big)} \right) \doteq v_{\lambda}^{\mathrm{SP}}\big(\tfrac{1}{\sqrt{2}}\varphi\big)+ \delta w^{\mathrm{SP}}(\varphi). \end{align}\]

In order to get a lower bound for \(\delta w^{\mathrm{SP}}(\varphi)\), we observe that \(\tilde{\Delta}^{\mathrm{SP}}(p;\varphi)\gtrsim \frac{1}{16}\varphi^8\) and \(\Delta^{\mathrm{SP}}\big(p;\tfrac{1}{\sqrt{2}}\varphi,\tfrac{1}{\sqrt{2}}\varphi\big) \lesssim 1+\varphi^8\). Therefore

\[\label{eq:mySP6} \delta w^{\mathrm{SP}}(\varphi)\ge - 8 \iint_{(-\frac{\pi}{8},\frac{\pi}{8}]\times(-\frac{\pi}{2},\frac{\pi}{2}]} \frac{d^2p}{(2\pi)^2} \log\left( 1- \mathfrak{C}_2^{\mathrm{SP}} \frac{\varphi^8}{1+\varphi^8}\right)= - \frac{1}{2} \log\left( 1- \mathfrak{C}_2^{\mathrm{SP}} \frac{\varphi^8}{1+\varphi^8}\right),\tag{68}\]

for a suitable \(\mathfrak{C}_2^{\mathrm{SP}}>0\). Note that the right–hand side is positive and monotone increasing for \(\varphi>0\). All in all:

\[w_{\lambda}^{\mathrm{SP}}(\varphi,0) \ge v_{\lambda}^{\mathrm{SP}} \big(\tfrac{1}{\sqrt{2}}\varphi\big)- \frac{1}{2} \log\left( 1- \mathfrak{C}_2^{\mathrm{SP}} \frac{\varphi^8}{1+\varphi^8}\right).\]

Finally, the same argument applied to the SN and NN models yields

\[\begin{align} &\min_{\varphi_+\ge0,\varphi_-\le0} w_{\lambda}^{\mathrm{SP}}(\varphi_+,\varphi_-) - v_{\lambda}^{\mathrm{SP}}\big(\varphi_{\mathrm{SP}}^{\star}(\lambda)\big) \ge \\ & \min\left\{ v_{\lambda}^{\mathrm{SP}}\big(\tfrac{1}{\sqrt{2}}\varphi_{\mathrm{SP}}^{\star}(\lambda)\big)- v_{\lambda}^{\mathrm{SP}}\big(\varphi_{\mathrm{SP}}^{\star}(\lambda)\big), \, - \frac{1}{4} \log \left(1- \mathfrak{C}_2^{\mathrm{SP}}\frac{\big(\varphi^{\star}_{\mathrm{SP}}(\lambda)\big)^8}{1+ \big(\varphi^{\star}_{\mathrm{SP}}(\lambda)\big)^8} \right) \right\}\equiv c_3^{\mathrm{SP}}(\lambda). \end{align}\]

6 Properties of the Dirac Operator and of its anti-symmetrization↩︎

In this appendix we collect and prove several technical results concerning the Dirac operators associated with the three lattice fermion models used in the main body of the paper. These results are repeatedly employed in the proof of Reflection Positivity for the corresponding measures \(d\mu_{\alpha}(\phi)\), but are of a purely operator-theoretic nature and can therefore be isolated from the main exposition.
For each model, we analyze the algebraic and spectral properties of the associated Dirac operators, with particular emphasis on adjointness relations, symmetry under reflections, and the reality and invariance of the fermionic determinant.
In order to evoid cluttering of notation, in Subsections 6.1 and 6.2 we will use the symbol \(\cancel{D}\) to denote \(\cancel{D}_{\mathrm{NN}}\) and \(\cancel{D}_{\mathrm{SN}}\) respectively. On the other hand, we will need to distinguish between reflections on the full plane and reflections on the torus (cf. Definition 2), which will be denoted by \(\vartheta_{\nu}\) and \(\vartheta_{\nu}^L\) respectively.

6.1 Naive Dirac Operator↩︎

We now analyze the properties of the Dirac operator under reflections. To this purpose we will need the notion of the Reflection Operator.

6.2 Staggered Dirac Operator↩︎

7 Analysis of the Bosonic Potential: Proof of Proposition 2↩︎

This appendix is dedicated to the proof of the main properties, stated in Proposition 2, of the bosonic potential:

\[V^{\alpha}_{L,\lambda}(\phi) = \frac{1}{2\lambda} \sum_{x\in\Lambda_L^{\alpha}}\phi_x^2 - \log\big|\det\big( (\cancel{D}_{\alpha})^-_{\Lambda_L} - M_{\phi} \big) \big|\]

and its infinite volume limit: \(v^{\alpha}_{\lambda}(\varphi)\doteq \lim_{L\to\infty} \frac{1}{|\Lambda^{\alpha}_L|} V^{\alpha}_{\lambda}(\phi)\big|_{\phi=\varphi^{\Lambda^{\alpha}_L}}\), with \(\varphi\in\mathbb{R}\).

7.1 Proof of Item [itt:1]↩︎

Here we shall prove that the potential \(V^{\alpha}_{L,\lambda}\) is minimized by constant field configurations, namely, given any interval \(I\subseteq \mathbb{R}\),

\[\label{Minpot952} \inf_{\phi\in I^{\Lambda_L^\alpha}} V^{\alpha}_{L,\lambda}(\phi) = \inf_{\varphi\in I} V^{\alpha}_{L,\lambda}(\phi)\Big|_{\phi= \varphi^{\Lambda^{\alpha}_L}}.\tag{72}\]

The key tool is again Reflection Positivity, which implies the following lemma.

Before proving Lemma 11, let us show how it readily implies the validity of 72 . The idea is to prove that, for any given field configuration \(\phi\in I^{\Lambda^{\alpha}_L}\), there exists a uniform configuration, \(\phi=\varphi^{\Lambda^{\alpha}_L}\) for some \(\varphi\in I\), with a smaller energy. First of all observe that, since \(e^{-N V^{\alpha}_{L,\lambda}(\phi)} \geq 0\) and \(e^{-N V^{\alpha}_{L,\lambda}(\phi)} \to 0^+\) as \(\|\phi\|_{\ell^2(\Lambda^{\alpha}_L)} \to \infty\), we have that

\[\inf_{\phi\in I^{\Lambda_L^\alpha}} V^{\alpha}_{L,\lambda}(\phi)= - \frac{1}{N}\log \sup _{\phi\in I^{\Lambda_{L}^\alpha}} e^{-N V^{\alpha}_{L,\lambda}(\phi)}= - \frac{1}{N}\log \max _{\phi\in \overline{I}^{\Lambda_L^\alpha}} e^{-N V^{\alpha}_{L,\lambda}(\phi)},\]

where \(\overline{I}\) denotes the closure of the interval. If we denote by \(\tilde{\phi}\) the maximizer of \(e^{-N V^{\alpha}_{L,\lambda}}\) over \(\overline{I}^{\Lambda^{\alpha}_L}\), then by Lemma 11 we have that

\[\big(e^{-NV^{\alpha}_{L,\lambda}(\tilde{\phi})}\big)^2 \leq e^{-NV^{\alpha}_{L,\lambda}(\tilde{\phi}^+)} e^{-NV^{\alpha}_{L,\lambda}(\tilde{\phi}^-)} \leq e^{-N V^{\alpha}_{L,\lambda}(\tilde{\phi}^+)} e^{-NV^{\alpha}_{L,\lambda}(\tilde{\phi}_-)},\]

implying that \(e^{-NV^{\alpha}_{L,\lambda}(\tilde{\phi})} = e^{-N V^{\alpha}_{L,\lambda}(\tilde{\phi}^+)}\). In other words, we have shown that a configuration obtained by reflections from the minimizing one has the same energy as the latter. We can repeat this argument first for all the horizontal and then for all the vertical cutting planes, so that we end up with a constant field configuration \(\tilde{\varphi}^{\Lambda^{\alpha}_L}\), with \(\tilde{}\varphi\in \overline{I}\), which has the same energy as \(\tilde{\phi}\), thus proving 72 .

7.2 Proof of Item [itt:2]↩︎

7.2.0.1 Computation of the uniform-field potential.

Since \(\phi\) is constant, namely \(\phi= \varphi^{\Lambda^{\alpha}_L}\), the spectrum of the operator \((\cancel{D}_{\alpha})^-_{\Lambda_L}- M_{\phi}\) can be explicitly obtained using Fourier transforms. Such operation actually depends on the specific model under consideration, thus we will work out the three cases \(\alpha=\mathrm{NN},\mathrm{SN},\mathrm{SP}\) separately.

• NN model. The model has a full translational invariance by \(\mathbb{Z}^2\), thus the space of allowed momenta with anti-periodic boundary conditions is

  \[(\Lambda^{\mathrm{NN}}_L)^*_{--} \doteq \tfrac{2\pi}{L}\big(\mathbb{Z}+ \tfrac{1}{2}\big)^2 \cap (-\pi,\pi ]^2.\]

  The determinant of the finite-volume Dirac operator is computed starting from 5 :

  \[\begin{align} \det\big((\cancel{D}_{\mathrm{NN}})^{-}_{\Lambda_L}- M_{\phi}\big)\Big|_{\phi=\varphi^{\Lambda^{\alpha}_L}} &= \prod_{p \in (\Lambda^{\mathrm{NN}}_L)^*_{--}} \det\big(-i \gamma_0 \sin(p_0) - i \gamma_1 \sin(p_1) - \mathbbl{1}_2 \varphi\big)\\ &= \prod_{p \in (\Lambda^{\mathrm{NN}}_L)^*_{--}} \big(\varphi^2 +\sin^2(p_0) + \sin^2(p_1)\big), \end{align}\]

  from which:

  \[\frac{1}{|\Lambda^{\mathrm{NN}}_L|}\log\big|\det\big((\cancel{D}_{\mathrm{NN}})^{-}_{\Lambda_L}- M_\phi\big)\Big|_{\phi=\varphi^{\Lambda^{\alpha}_L}}\big| = \frac{1}{L^2}\sum_{p \in (\Lambda^{\mathrm{NN}}_L)^*_{--}} \log(\varphi^2 +\sin^2(p_0) + \sin^2(p_1)).\]

• SN model. The model has a reduced translational invariance by \((2\mathbb{Z})\times\mathbb{Z}\). It is therefore convenient to considered an enlarged \(2\times1\) fundamental cell. The resulting lattice is drawn in fig. 13.

  

  The set of allowed momenta is given by the dual of this reduced lattice:

  \[(\Lambda^{\mathrm{SN}}_L)^*_{--}= \tfrac{2\pi}{L} \big(\mathbb{Z}+\tfrac{1}{2}\big)^2 \cap \Big(\big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big]\times(-\pi,\pi] \Big).\]

  Again the determinant can be computed starting from 8 :

  \[\begin{align} \det\big((\cancel{D}_{\mathrm{SN}})^{-}_{\Lambda_L}- M_\phi\big)\Big|_{\phi=\varphi^{\Lambda^{\alpha}_L}} &= \prod_{p \in (\Lambda^{\mathrm{SN}}_L)^*_{--}} \det \begin{pmatrix} i\sin p_1 -\varphi& \frac{1}{2}(e^{2ip_0}-1)\\ -\frac{1}{2}(e^{-2ip_0}-1) & -i\sin p_1 - \varphi \end{pmatrix} \\ &= \prod_{p \in (\Lambda^{\mathrm{SN}}_L)^*_{--}} \big(\varphi^2 +\sin^2(p_0) + \sin^2(p_1)\big). \end{align}\]

  Thus: \[\frac{1}{|\Lambda^{\mathrm{SN}}_L|} \log \big|\det\big((\cancel{D}_{\mathrm{SN}})^{-}_{\Lambda_L}- M_\phi\big)\Big|_{\phi=\varphi^{\Lambda^{\alpha}_L}}\big| = \frac{1}{L^2}\sum_{p \in (\Lambda^{\mathrm{SN}}_L)^*_{--}} \log(\varphi^2 +\sin^2(p_0) + \sin^2(p_1)).\]

• SP model. The analysis is identical to the one for the model SN. Indeed, when the field \(\phi\) is constant, the models SN and SP are actually the same, as well as the set of momenta: \((\Lambda^{\mathrm{SP}}_L)^*_{--}= (\Lambda^{\mathrm{SN}}_L)^*_{--}\). The only difference comes from the fact that \(|\Lambda^{\mathrm{SP}}_L|= \frac{1}{4}|\Lambda^{\mathrm{SN}}_L|\), so that

  \[\frac{1}{|\Lambda^{\mathrm{SP}}_L|} \log \big|\det\big((\cancel{D}_{\mathrm{SP}})^{-}_{\Lambda_L}- M_\phi\big)\Big|_{\phi=\varphi^{\Lambda^{\alpha}_L}}\big| = \frac{4}{L^2}\sum_{p \in (\Lambda^{\mathrm{SP}}_L)^*_{--}} \log(\varphi^2 +\sin^2(p_0) + \sin^2(p_1)).\]

Notice that, due to the periodicity of \(\hat{f}(p;\varphi) \doteq \log\big(\varphi^2+ \sin^2(p_0)+ \sin^2(p_1)\big)\) with respect to translations of \(\pm\pi\) in both directions, it is possible to rewrite:

\[\label{eqn:37} \begin{align} \frac{1}{|\Lambda^{\alpha}_L|}V^{\alpha}_{L,\lambda}(\varphi^{\Lambda^{\alpha}_L})&= \frac{\varphi^2}{2\lambda}- \frac{1}{|\Lambda^{\alpha}_L|}\sum_{p\in(\Lambda_L^{\alpha})^*} \hat{f}(p;\varphi)= \frac{\varphi^2}{2\lambda}- \frac{C_{\alpha}}{L^2}\sum_{p\in (\Lambda^{\mathrm{NN}}_L)^*} \hat{f}(p;\varphi)\\ &= \frac{\varphi^2}{2\lambda}- \frac{4C_{\alpha}}{L^2} \sum_{p\in \frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})^2\cap (-\frac{\pi}{2},\frac{\pi}{2})^2} \hat{f}(p;\varphi), \end{align}\tag{76}\]

where \(C_{\alpha}=1,\frac{1}{2},2\) for \(\alpha=\mathrm{NN},\mathrm{SN},\mathrm{SP}\) respectively.

7.2.0.2 Convergence to the infinite-volume limit.

By standard arguments (e.g. the Dominated Convergence Theorem), we have the convergence, in the limit \(L\to\infty\), of the Riemann sum in 76 to the respective integral \(C_{\alpha}\iint_{(-\pi,\pi]^2} \frac{d^2p}{(2\pi)^2} \hat{f}(p;\varphi)\), so that

\[\label{eqn:38} \begin{align} v^{\alpha}_{\lambda}(\varphi) \equiv \lim_{L\to\infty}\frac{1}{|\Lambda^{\alpha}_L|} V^{\alpha}_{L,\lambda}(\varphi^{\Lambda^{\alpha}_L})&= \frac{\varphi^2}{2\lambda}- C_{\alpha}\iint_{(-\pi,\pi]^2} \frac{d^2p}{(2\pi)^2} \hat{f}(p;\varphi)\\ &= \frac{\varphi^2}{2\lambda}- 4C_{\alpha}\iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \hat{f}(p;\varphi). \end{align}\tag{77}\]

The representation in the second line of 77 is convenient because in the domain \((-\frac{\pi}{2},\frac{\pi}{2}]^2\), the integrand \(\hat{f}(p;\varphi)\) has only one singularity at \(p=(0,0)\), if \(\varphi=0\). Observe also that the Riemann sum in the right–hand side of 76 can be rewritten as follows:

\[\frac{1}{L^2}\sum_{p\in \frac{2\pi}{L}(\mathbb{Z}+\frac{1}{2})^2\cap (-\frac{\pi}{2},\frac{\pi}{2})^2} \hat{f}(p;\varphi)=\iint_{(-\frac{\pi}{2},\frac{\pi}{2})^2} \frac{d^2p}{(2\pi)^2} \hat{f}\big(\pi_L(p);\varphi\big),\]

where \(\pi_L: \big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big)^2\mapsto \big(-\tfrac{\pi}{2},\tfrac{\pi}{2}\big)^2 \cap \tfrac{2\pi}{L}\big(\mathbb{Z}+\tfrac{1}{2}\big)^2\) is the projection defined by the following condition (see Fig. 14):

\[\text{if}\, p\in \frac{2\pi}{L}\big([n_0,n_0+1)\times [n_1,n_1+1)\big)\text{, then}\, \pi_L(p)= \frac{2\pi}{L}\big(n_0+\frac{1}{2},n_1+\frac{1}{2}\big)\]

.

In this way:

\[\label{eqn:31} \begin{align} \left|\frac{1}{|\Lambda_L^{\alpha}|}V^{\alpha}_{L,\lambda}(\varphi^{\Lambda^{\alpha}_L})- v^{\alpha}_{\lambda}(\varphi) \right|&\le 4C_{\alpha}\iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \Big|\hat{f}\big(\pi_L(p);\varphi\big)- \hat{f}(p;\varphi)\Big|. \end{align}\tag{78}\]

In order to proceed, observe that for any \(p\in (-\frac{\pi}{2},\frac{\pi}{2})^2\) and \(\mu\in\{0,1\}\), \(|\pi_L(p)_{\mu}|\ge \frac{1}{2}|p_{\mu}|\). Note also that for any \(p\in (-\frac{\pi}{2},\frac{\pi}{2})^2\), we can write:

\[\begin{align} \Big|\hat{f}\big(\pi_L(p);\varphi\big)- \hat{f}(p;\varphi)\Big|&\le \sum_{\mu=0,1}\big|\pi_L(p)_{\mu}-p_{\mu}\big| \Bigg|\int_0^1 ds \left[\tfrac{2\sin(q_{\mu})\cos(q_{\mu})}{\varphi^2+ \sin^2(q_0)+ \sin^2(q_1)} \right]_{q= s\pi_L(p)+ (1-s)p} \Bigg|\\ &\le \sum_{\mu=0,1}\big|\pi_L(p)_{\mu}-p_{\mu}\big| \int_0^1 ds \left[\tfrac{2}{\sqrt{\sin^2(q_0)+ \sin^2(q_1)}} \right]_{q= s\pi_L(p)+ (1-s)p}\\ &\le \frac{4\pi}{L}\tfrac{2}{\sqrt{\sin^2(\frac{1}{2}p_0)+ \sin^2(\frac{1}{2}p_1)}} \le \frac{8\pi^2}{L|p|}, \end{align}\]

where we also used that for \(|t|\le\frac{\pi}{2}\), \(|\sin t|\ge \frac{2}{\pi}|t|\). Therefore from 78 we find:

\[\begin{align} &\left|\frac{1}{|\Lambda_L|} V^{\alpha}_{L,\lambda}(\varphi^{\Lambda^{\alpha}_L})- v^{\alpha}_{\lambda}(\varphi) \right|\le 8 \iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \frac{8\pi^2}{L|p|}\le \frac{32\pi^2}{L}, \end{align}\]

concluding the proof of Item [itt:2].

7.3 Proof of Item [itt:3]↩︎

7.3.0.1 Existence of non-zero minima.

It is clear by inspection that, for any \(\lambda>0\), the mean-field potential \(v^{\alpha}_{\lambda}\), as a function of \(\varphi\), is even and \(C^1\) over \(\mathbb{R}\). In order to find its minima it is convenient to analyze its derivative:

\[\frac{d v^{\alpha}_{\lambda}}{d\varphi}(\varphi)= \varphi\left( \frac{1}{\lambda}- 4 C_{\alpha} \iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \frac{1}{\varphi^2+ \sin^2p_0+ \sin^2 p_1}\right) \doteq \varphi g^{\alpha}_{\lambda}(\varphi).\]

We see that \(v^{\alpha}_{\lambda}\) has a local maximum at \(\varphi=0\). Moreover, as it is easy to check, for \(\varphi>0\) the function \(g^{\alpha}_{\lambda}\) is monotone increasing and \(\lim_{\varphi\to+\infty}g^{\alpha}_{\lambda}(\varphi)=0^+\). Besides, \(\lim_{\varphi\to 0^+}g^{\alpha}_{\lambda}(\varphi)=-\infty\), indeed:

\[\label{eqn:39} g^{\alpha}_{\lambda}(\varphi)\le \frac{1}{\lambda}- 4 C_{\alpha} \iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \frac{1}{\varphi^2+ |p|^2} \le \frac{1}{\lambda}- 2 \iint_{|p|\le 1} \frac{d^2p}{(2\pi)^2} \frac{1}{\varphi^2+ |p|^2}= \frac{1}{\lambda}- \frac{1}{2\pi}\log\big(1+ \tfrac{1}{\varphi^2} \big).\tag{79}\]

Since \(g^{\alpha}_{\lambda}\) is also obviously continuous over \((0,\infty)\), it follows that there exists a unique value \(\varphi^{\star}_{\alpha}(\lambda)\in (0,\infty)\) such that \(g^{\alpha}_{\lambda}\big(\varphi^{\star}_{\alpha}(\lambda)\big)=0\). Such point is of course an absolute minimum for \(v^{\alpha}_{\lambda}\).

7.3.0.2 Regularity of the minima.

As we just argued, the minimum \(\varphi^{\star}_{\alpha}(\lambda)\) can be interpreted as the unique positive solution to the gap equation:

\[\label{eqn:40} g^{\alpha}_{\lambda}(\varphi)=0, \qquad \varphi\in(0,\infty).\tag{80}\]

The solution of such equation can be characterized by the Implicit Function Theorem. The map \((\lambda,\varphi)\mapsto g^{\alpha}_{\lambda}(\varphi)\) is clearly \(C^{\infty}\) over \((0,\infty)^2\). Besides, since the function

\[\frac{\partial g^{\alpha}_{\lambda}}{\partial\varphi}(\varphi)= -8 C_{\alpha}\varphi\iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \frac{1}{\big(\varphi^2+ \sin^2p_0+ \sin^2 p_1\big)^2}\]

is never zero, we can apply the (global version of the) Implicit Function Theorem, which implies that the (unique) solution \(\varphi^{\star}_{\alpha}(\lambda)\) to 80 is actually \(C^{\infty}\) with respect to \(\lambda\in(0,\infty)\).

7.3.0.3 Exponential smallness of the minima.

Finally we prove ?? . Using that \(g^{\alpha}_{\lambda}\big(\varphi^{\star}_{\alpha}(\lambda)\big)=0\), from 79 we have that \(0\le \frac{1}{\lambda}- \frac{1}{2\pi}\log\big(1+ \varphi^{\star}_{\alpha}(\lambda)^{-2} \big)\), which implies that \(\varphi^{\star}_{\alpha}(\lambda) \ge e^{-\frac{\pi}{\lambda}}\). In order to provide an upper bound, we observe that

\[\begin{align} g^{\alpha}_{\lambda}(\varphi)&= \frac{1}{\lambda}- 4 C_{\alpha} \iint_{(-\frac{\pi}{2},\frac{\pi}{2}]^2} \frac{d^2p}{(2\pi)^2} \frac{1}{\varphi^2+ \sin^2p_0+ \sin^2 p_1}\\ &\ge \frac{1}{\lambda}- 8 \iint_{|p|\le \pi} \frac{d^2p}{(2\pi)^2} \frac{1}{\varphi^2+ \frac{4}{\pi^2}|p|^2}= \frac{1}{\lambda}- \frac{\pi}{2}\log\big(1+ \tfrac{2}{\varphi^2}\big), \end{align}\]

where again we used the fact that \(|\sin t|\ge \frac{2}{\pi}|t|\) for \(|t|\le\frac{\pi}{2}\). The condition \(g^{\alpha}_{\lambda}\big(\varphi^{\star}_{\alpha}(\lambda) \big)=0\) then yields, for \(\lambda\le 1\),

\[\frac{2}{\varphi^{\star}_{\alpha}(\lambda)^2}\ge e^{\frac{2}{\pi\lambda}} -1 \ge e^{\frac{2}{\pi\lambda}}\big(1- e^{-\frac{2}{\pi}} \big) \Rightarrow \varphi^{\star}_{\alpha}(\lambda)\le \sqrt{\frac{2}{1- e^{-\frac{2}{\pi}}}} e^{-\frac{1}{\pi\lambda}}.\]

8 Formal Continuum Limit of the Staggered Plaquette Model↩︎

In this appendix we derive the formal continuum limit of the Staggered Plaquette model and identify the corresponding continuum Gross–Neveu theory. We will show that:

1. after a suitable change of fermionic variables, the non-interacting lattice theory converges to one describing \(N_{\mathrm{eff}}\equiv 2N\) non-interacting Dirac fermions;

2. the plaquette interaction becomes the standard Gross–Neveu quartic interaction.

8.0.0.1 Lattice Action.

We consider a two-dimensional lattice of linear size \(L\) and mesh \(\ell>0\), \(\Lambda_{\ell,L}=\ell\mathbb{Z}^2\cap \big(-\tfrac{L}{2},\tfrac{L}{2} \big]^2\), with \(L/\ell \in 4\mathbb{N}\). We partition \(\Lambda_{\ell,L}\) into \(2\times2\) plaquettes and introduce the coarse lattice: \(\widetilde{\Lambda}_{\ell,L} = 2\ell \mathbb{Z}^2\cap\big(-\tfrac{L}{2},\tfrac{L}{2}\big]^2\) and we consider the set of internal plaquette sites \(\mathcal{I}=\{A,B,C,D\}\), so that we can identify \(\Lambda_{\ell,L}\) with \(\widetilde{\Lambda}_{\ell,L} \times \mathcal{I}\) (see Fig. 3). The action of the Staggered Plaquette model reads \(S_{\mathrm{SP}}^{\ell,L} = S_0^{\ell,L} + S_{\mathrm{int}}^{\ell,L}\), where

\[\begin{align} &S_0^{\ell,L}\doteq \ell^2 \sum_{x,y \in \widetilde{\Lambda}_{\ell,L}}\sum_{j=1}^N\sum_{a,b\in \mathcal{I}} \overline{\psi}_{x,a,j}\big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_{\ell,L}}\big)_{(x,a),(y,b)} \psi_{y,b,j},\\ &\label{eq:sp95interaction} S_{\mathrm{int}}^{\ell,L} \doteq -\frac{\lambda(2\ell)^2}{8N} \sum_{x\in\widetilde{\Lambda}_{\ell,L}} \Bigg( \sum_{a\in\mathcal{I}} \sum_{j=1}^N \overline{\psi}_{x,a,j}\psi_{x,a,j} \Bigg)^2 \end{align}\tag{81}\]

and \(\big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_{\ell,L}}\big)_{x,y} \doteq \ell^{-1} \big((\cancel{D}_{\mathrm{SP}})^-_{\Lambda_{L/\ell}}\big)_{x/\ell,y/\ell}\), with the latter defined in 12 .

8.0.0.2 Fourier Representation.

We define, for \(p\in (\widetilde{\Lambda}_{\ell,L})^*_{--} \doteq \frac{2\pi}{L} \Big(\mathbb{Z}+\tfrac12\Big)^2 \cap \Big( -\frac{\pi}{2\ell}, \frac{\pi}{2\ell} \Big)^2\),

\[\hat{\psi}^-_{p,a,j} \doteq (2\ell)^2\sum_{x\in\widetilde{\Lambda}_{\ell,L}} e^{ip\cdot x} \psi_{x,a,j}, \qquad \hat{\psi}^+_{p,a,j} \doteq \ell^2\sum_{x\in\widetilde{\Lambda}_{\ell,L}} e^{-ip\cdot x} \overline{\psi}_{x,a,j},\]

so that

\[\psi_{x,a,j} = \frac{1}{L^2} \sum_{p\in(\widetilde{\Lambda}_{\ell,L})^*_{--}} e^{-ipx} \hat{\psi}^-_{p,a,j}, \qquad \overline{\psi}_{x,a,j} = \frac{1}{L^2} \sum_{p\in(\widetilde{\Lambda}_{\ell,L})^*_{--}} e^{ipx} \hat{\psi}^+_{p,a,j}.\]

By introducing the four-component spinor:

\[\hat{\Psi}^-_{p,j} = \begin{pmatrix} \hat{\psi}^-_{p,A,j}\\ \hat{\psi}^-_{p,B,j}\\ \hat{\psi}^-_{p,C,j}\\ \hat{\psi}^-_{p,D,j} \end{pmatrix}, \qquad \hat{\Psi}^+_{p,j} = \begin{pmatrix} \hat{\psi}^+_{p,A,j}& \hat{\psi}^+_{p,B,j}& \hat{\psi}^+_{p,C,j}& \hat{\psi}^+_{p,D,j} \end{pmatrix},\]

the free action takes the form

\[S_0^{\ell,L} = \frac{1}{4L^2} \sum_{j=1}^{N} \sum_{p} \hat{\Psi}^+_{p,j} \hat{D}(p) \hat{\Psi}^-_{p,j},\]

where, letting \(\sigma_\mu(p) \doteq \frac{1-e^{-2i\ell p_\mu}}{2i\ell}\),

\[\hat{D}(p)\doteq \begin{pmatrix} 0 & -i\overline{\sigma}_1(p) & i\overline{\sigma}_0(p) & 0 \\[6pt] -i\sigma_1(p) & 0 & 0 & i\overline{\sigma}_0(p) \\[6pt] i\sigma_0(p) & 0 & 0 & i\overline{\sigma}_1(p) \\[6pt] 0 & i\sigma_0(p) & i\sigma_1(p) & 0 \end{pmatrix}.\]

8.0.0.3 Spin–Taste Basis.

To identify the proper fermionic degrees of freedom suited for the continuum limit, we perform a standard spin–taste transformation [41]. Namely we define:

\[\hat{\xi}^-_{p,\omega,s,j} \doteq \frac{1}{4} \sum_{a\in\mathcal{I}} \Gamma^{(a)}_{s,\omega} \hat{\psi}^-_{p,a,j}, \qquad \hat{\xi}^+_{p,\omega,s,j} \doteq \frac{1}{4} \sum_{a\in\mathcal{I}} \hat{\psi}^+_{p,a,j} (\Gamma^{(a)})^\dagger_{\omega,s},\]

with \(\omega\in\{\pm\}\) the taste index, \(s\in\{\uparrow,\downarrow\}\) the spin index and, letting \(e_A=(0,0), e_B=(1,0), e_C=(0,1), e_D=(1,1)\) be the normalized displacement within the unit cell (see Fig. 3),

\[\Gamma^{(a)} \equiv \left(\begin{array}{cc} \Gamma^{(a)}_{\uparrow,+} & \Gamma^{(a)}_{\uparrow,-} \\ \Gamma^{(a)}_{\downarrow,+} & \Gamma^{(a)}_{\downarrow,-} \end{array} \right) \doteq \gamma_1^{e_a\cdot e_1} \gamma_0^{e_a\cdot e_0} = \begin{cases} \mathbbl{1}_2, & a=A,\\ \sigma_x, & a=B,\\ \sigma_y, & a=C,\\ i\sigma_z, & a=D. \end{cases}\]

Alternatively, after introducing

\[\hat{\Xi}^-_{p,j} = \begin{pmatrix} \xi^-_{p,+,\uparrow,j} \\ \xi^-_{p,+,\downarrow,j} \\ \xi^-_{p,-,\uparrow,j} \\ \xi^-_{p,-,\downarrow,j} \end{pmatrix}, \qquad \hat{\Xi}^+_{p,j} = \begin{pmatrix} \xi^+_{p,+,\uparrow,j}, \xi^+_{p,+,\downarrow,j}, \xi^+_{p,-,\uparrow,j}, \xi^+_{p,-,\downarrow,j} \end{pmatrix},\]

the change of basis reads \(\hat{\Psi}^-_{p,j} = U^\dagger \hat{\Xi}^-_{p,j}\) and \(\hat{\Psi}^+_{p,j}=\hat{\Xi}^+_{p,j} U\), where

\[U = \frac{1}{\sqrt2} \begin{pmatrix} 1&0&0&i\\ 0&1&-i&0\\ 0&1&i&0\\ 1&0&0&-i \end{pmatrix}\]

is a unitary matrix. Hence:

\[S_0^{\ell,L} = \frac{1}{4L^2} \sum_{j=1}^{N} \sum_{p\in (\widetilde{\Lambda}_{\ell,L})^*_{--}} \hat{\Xi}^+_{p,j} \,\hat{S}(p)\, \hat{\Xi}^-_{p,j},\]

with \(\hat{S}(p)= U \hat{D}(p) U^\dagger\). A direct computation gives:

\[\label{eq:continuum95dirac} \hat{S}(p) = \mathbbl{1}_{\mathrm{taste}} \otimes \Big( -i p_0 \gamma_0 - i p_1 \gamma_1 \Big) + \mathcal{O}(\ell |p|^2).\tag{82}\]

Equation 82 shows that the taste sector becomes asymptotically diagonal and survives as an additional independent fermionic species. Note also that

\[\det \hat{S}(p)= \det\hat{D}(p)= \ell^{-2}\big(\sin^2(\ell p_0)+ \sin^2(\ell p_1)\big),\]

showing that \(\hat{S}(p)^{-1}\) has a pole only at \(p=0\).

Observe also that, in the spin-taste basis, the interaction term 81 becomes

\[S_{\mathrm{int}}^{\ell,L} = -\frac{\lambda(2\ell)^2}{8N} \sum_{x\in\widetilde{\Lambda}_{\ell,L}} \Big( \overline{\Xi}_x\Xi_x \Big)^2 \equiv -\frac{\lambda(2\ell)^2}{8N} \sum_{x\in\widetilde{\Lambda}_{\ell,L}} \Big(\sum_{j=1}^N\sum_{\omega=\pm} \overline{\Xi}_{x,\omega,j}\Xi_{x,\omega,j} \Big)^2.\]

8.0.0.4 Continuum Gross–Neveu Theory.

The formal limit \(\ell\to 0^+,L\to\infty\) is obtained by replacing Riemann sums by the corresponding integrals:

\[(2\ell^2)\sum_{x\in\widetilde{\Lambda}_{\ell,L}} \leadsto \iint_{\mathbb{R}^2} d^2x, \qquad L^{-2}\sum_{p\in (\widetilde{\Lambda}_{\ell,L})^*_{--}} \leadsto \iint_{\mathbb{R}^2} \frac{d^2p}{(2\pi)^2}\]

and neglecting the remainders in the right–hand side of 82 . We obtain:

\[S_{\mathrm{SP}}^{\ell,L} \leadsto S_{\mathrm{SP}} = \iint d^2x \Bigg[-\frac{1}{4}i\sum_{j=1}^N\sum_{\omega=\pm}\sum_{\mu=0,1} \overline{\Xi}_{x,\omega,j} \gamma_{\mu}\partial_{\mu} \Xi_{x,\omega,j} -\frac{\lambda}{8N} \big(\sum_{j=1}^N\sum_{\omega=\pm}\overline{\Xi}_{x,\omega,j}\Xi_{x,\omega,j}\big)^2 \Bigg].\]

The kinetic term contains an overall factor \(1/4\), which is easily removed by a trivial rescaling \(\Xi\mapsto \Xi'= \frac{1}{2}\Xi\). In this way we get:

\[S_{\mathrm{SP}} = \iint d^2x \Bigg[-\frac{1}{4}i\sum_{j=1}^N\sum_{\omega=\pm}\sum_{\mu=0,1} \overline{\Xi}'_{x,\omega,j} \gamma_{\mu}\partial_{\mu} \Xi'_{x,\omega,j} -\frac{\lambda_{\mathrm{eff}}}{2N_{\mathrm{eff}}} \big(\sum_{j=1}^N\sum_{\omega=\pm}\overline{\Xi}'_{x,\omega,j}\Xi'_{x,\omega,j}\big)^2 \Bigg],\]

with \(N_{\mathrm{eff}}=2N\) and \(\lambda_{\mathrm{eff}}=8\lambda\). We therefore conclude that the Staggered Plaquette model, after performing the formal continuum limit, reproduces the standard Gross-Neveu model with coupling \(\lambda_{\mathrm{eff}}\) and \(N_{\mathrm{eff}}\) flavors.

8.0.0.5 One-Loop Running Coupling.

The Renormalization Group equation, for the running coupling constant \(\lambda_{\mathrm{eff}}(\ell)\) at energy scale \(\ell^{-1}\), reads:

\[\ell \frac{d\lambda_{\mathrm{eff}}}{d\ell}= \beta(\lambda_{\mathrm{eff}}),\]

with \(\beta(\lambda_{\mathrm{eff}})\) the Beta Function of the Gross-Neveu model, whose one-loop expression is [42]:

\[\beta(\lambda_{\mathrm{eff}}) = \frac{1}{\pi} \left( 1-\frac{1}{N_{\mathrm{eff}}} \right) \lambda_{\mathrm{eff}}^2 + \mathcal{O}(\lambda_{\mathrm{eff}}^3).\]

The integration of the Renormalization Group equation truncated at order \(\lambda_{\mathrm{eff}}^2\), yields

\[\lambda_{\mathrm{eff}}(\ell) = \frac{\lambda_{\mathrm{eff}}(\ell_0)}{1 + \frac{1}{\pi} \left( 1-\frac{1}{N_{\mathrm{eff}}} \right) \lambda_{\mathrm{eff}}(\ell_0)^2 \log\!\big(\frac{\ell_0}{\ell}\big)}.\]

After writing \(N_{\mathrm{eff}}=2N\) and \(\lambda_{\mathrm{eff}}(\ell)=8\lambda(\ell)\), we obtain:

\[\lambda(\ell) = \frac{\lambda(\ell_0)}{1 + \frac{8}{\pi} \left( 1-\frac{1}{2N} \right) \lambda(\ell_0)^2 \log\!\big(\frac{\ell_0}{\ell}\big)}.\]

At the lowest order in \(\frac{1}{N}\), this expression coincides with ?? for \(\alpha=\mathrm{SP}\).

9 Upper and Lower-bound on the \(2\)-point function↩︎

9.1 Lower-bound of the \(2\)-point function↩︎

Since: \[1 = I_x^{+K_{\alpha}}+I_x^{-K_{\alpha}}+I_x^{(-C_{\alpha},C_{\alpha})}+I_x^{\star_+}+I_x^{\star_-},\] the two-point function can be decomposed as: \[\label{eqn:9} \langle \phi_x\phi_y\rangle_{\mu_\alpha} = \sum_{i,j \in \mathcal{R}} \big\langle \phi_x\phi_y I_x^{(i)} I_y^{(j)} \big\rangle_{\mu_\alpha},\tag{83}\] where the sum runs over the five regions \(\mathcal{R}\) listed in Items [it:large43]–[it:min-] in Section 4.1. These contributions can be naturally grouped into the three following classes.

• DT (dominant term). The two fields lie near the same minimum: \[\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{DT}} \doteq \big\langle \phi_x\phi_y \bigl( I_x^{\star_+}I_y^{\star_+} + I_x^{\star_-}I_y^{\star_-} \bigr) \big\rangle_{\mu_\alpha}.\]

• PT (Peierls’ term). The two fields lie around different minima: \[\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{PT}} \doteq \big\langle \phi_x\phi_y \bigl( I_x^{\star_+}I_y^{\star_-} + I_x^{\star_-}I_y^{\star_+} \bigr) \big\rangle_{\mu_\alpha}.\]

• EUT (energetic unfavorable term). At least one of the two fields lies in an energetically unfavorable region: \[\begin{align} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{EUT}} &\doteq \Big\langle \phi_x \phi_y \bigl(I_x^{+K_{\alpha}}+I_x^{-K_{\alpha}}+I_x^{(-C_{\alpha},C_{\alpha})}\bigr) \Big\rangle_{\mu_{\alpha}}\\ &+ \Big\langle \phi_x \phi_y \bigl( I_x^{\star_+}+ I_x^{\star_-} \bigr) \bigl(I_y^{+K_{\alpha}}+I_y^{-K_{\alpha}}+I_y^{(-C_{\alpha},C_{\alpha})}\bigr) \Big\rangle_{\mu_{\alpha}}. \end{align}\]

First, it is convenient to perform a preliminary bound of the two-point function as:

\[\label{eqn:50} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} \geq \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{DT}} - \bigg|\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{PT}}\bigg| - \bigg|\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{EUT}}\bigg|.\tag{84}\]

The bound 26 is then obtained by estimating separately the three terms in the right–hand side of 84 .

9.1.0.1 Dominant Term (DT).

We shall derive the following lower bound:

\[\label{eqn:50a} \begin{align} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{DT}} \geq C_{\alpha}^2 -\frac{4 C_{\alpha}^2}{K_{\alpha}^2} \left \langle \phi_x^2 I^{+K_{\alpha}}_x \right \rangle_{\mu_{\alpha}} - \frac{8C_{\alpha}^2}{K_{\alpha}} \left \langle \phi_x^2 I^{+K_{\alpha}}_x \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} -3C_{\alpha}^2\left \langle I^{(-C_{\alpha},C_{\alpha})}_x\right \rangle_{\mu_{\alpha}} - 2C_{\alpha}^2 \left \langle I_x^{\star_+} I_y^{\star_-} \right \rangle_{\mu_{\alpha}}, \end{align}\tag{85}\]

where notice that, due to the translation invariance of the model (see Footnote 12), averages of the form \(\langle f(\phi_x) \rangle_{\mu_{\alpha}}\) are actually constant w.r.t. \(x\in \Lambda^{\alpha}_L\).

9.1.0.2 Energetically Unfavorable Terms (EUT).

We shall prove the following upper bound: \[\label{eqn:50b} \begin{align} \bigg| \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{EUT}} \bigg| \leq 4 \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}} + 4(C_{\alpha}+2K_{\alpha}) \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} + (4K_{\alpha}C_{\alpha} + C_{\alpha}^2) \left \langle I_x^{(-C_{\alpha},C_{\alpha})} \right \rangle_{\mu_{\alpha}}. \end{align}\tag{87}\]

9.1.0.3 Peierls’ terms (PT).

Using again the invariance of \(\mu_{\alpha}\) under \(\phi\mapsto-\phi\), we get that

\[\label{eqn:50c} \bigg|\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}}^{\mathrm{PT}} \bigg| \leq 2 \big\langle |\phi_x \phi_y| I_x^{\star_+}I_y^{\star_-} \big\rangle_{\mu_\alpha} \leq 2 K_{\alpha}^2 \big\langle I_x^{\star_+}I_y^{\star_-} \big\rangle_{\mu_\alpha}.\tag{88}\]

Plugging 85 , 87 and 88 into the right–hand side of 84 , we finally get:

\[\begin{align} \inf_{x,y\in\Lambda^{\alpha}_L} \Big\langle \phi_x \phi_y \Big\rangle_{\mu_{\alpha}} &\ge C_{\alpha}^2 - \Bigg( 2(K_{\alpha}^2+C_{\alpha}^2) \sup_{x,y\in\Lambda^{\alpha}_L} \left\langle I^{\star_+}_x I^{\star_-}_y \right\rangle_{\mu_{\alpha}} + 4C_{\alpha} (C_{\alpha}+ K_{\alpha}) \sup_{x\in\Lambda^{\alpha}_L} \left\langle I^{(-C_{\alpha},C_{\alpha})}_x \right\rangle_{\mu_{\alpha}} \\ &+ 4\big(1+ \tfrac{C_{\alpha}^2}{K_{\alpha}^2}\big)\sup_{x\in\Lambda^{\alpha}_L} \left\langle \phi_x^2 I^{+K_{\alpha}}_x \right\rangle_{\mu_{\alpha}} + 4\big(C_{\alpha}+ 2\tfrac{C_{\alpha}^2}{K_{\alpha}}+ 2K_{\alpha}\big) \sup_{x\in\Lambda^{\alpha}_L} \left\langle \phi_x^2 I^{+K_{\alpha}}_x\right\rangle_{\mu_{\alpha}}^{\frac{1}{2}} \Bigg), \end{align}\]

which is nothing but 26 .

9.2 Upper bound of the \(2\)-point function↩︎

In order to perform an upper bound, we can decompose the two-point function as:

\[\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} = \left \langle \phi_x \phi_y (I_x^{(-K_{\alpha},K_{\alpha})} + I_x^{(-K_{\alpha},K_{\alpha})^c}) (I_y^{(-K_{\alpha},K_{\alpha})} + I_y^{(-K_{\alpha},K_{\alpha})^c})\right \rangle_{\mu_{\alpha}},\]

where \(I_x^{(-K_{\alpha},K_{\alpha})} \doteq \mathbbl{1}_{\{-L_{\alpha}<\phi_x <K_{\alpha}\}}\) and \(I_x^{(-K_{\alpha},K_{\alpha})^c} \doteq 1- I_x^{(-K_{\alpha},K_{\alpha})}\). Proceeding as in the previous subsection, we can decompose the correlator as:

\[\label{eq:upb} \begin{align} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} &= \left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})}\right \rangle_{\mu_{\alpha}} + 2 \left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}} + \\ &\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})^c} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}}. \end{align}\tag{89}\]

The three terms appearing in the right–hand side of 89 can be upper-bounded as follows.

\[\begin{align} \bigg|\left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})}\right \rangle_{\mu_{\alpha}}\bigg| &\leq \left \langle |\phi_x| |\phi_y| I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})}\right \rangle_{\mu_{\alpha}} \leq \left \langle I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})}\right \rangle_{\mu_{\alpha}} \leq K_{\alpha}^2;\\ \bigg|\left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}} \bigg| &\leq \left \langle |\phi_x| |\phi_y| I_x^{(-K_{\alpha},K_{\alpha})} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}} \leq K_{\alpha} \left \langle |\phi_y| I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}} \\ &= 2 K_{\alpha} \left \langle \phi_x I_x^{+K_{\alpha}}\right \rangle_{\mu_{\alpha}}\leq 2K_{\alpha}\left \langle \phi_x^2 I_x^{+K_{\alpha}}\right \rangle_{\mu_{\alpha}}^{\frac{1}{2}};\\ \bigg|\left \langle \phi_x \phi_y I_x^{(-K_{\alpha},K_{\alpha})^c} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}} \bigg| &\leq \left \langle |\phi_x| |\phi_y| I_x^{(-K_{\alpha},K_{\alpha})^c} I_y^{(-K_{\alpha},K_{\alpha})^c}\right \rangle_{\mu_{\alpha}}\\ &= 2 \left \langle \phi_x \phi_y I_x^{+K_{\alpha}} I_y^{+K_{\alpha}}\right \rangle_{\mu_{\alpha}} + 2 \left \langle |\phi_x \phi_y| I_x^{-K_{\alpha}} I_y^{+K_{\alpha}}\right \rangle_{\mu_{\alpha}} \\ &\leq 2 \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} \left \langle \phi_y^2 I_y^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} + 2 \left \langle \phi_x^2 I_x^{-K_{\alpha}}\right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} \left \langle \phi_y^2 I_y^{+K_{\alpha}}\right \rangle_{\mu_{\alpha}}^{\frac{1}{2}}\\ &= 4 \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} \left \langle \phi_y^2 I_y^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}}. \end{align}\]

Therefore, going back to 89 :

\[\sup_{x,y \in \Lambda^{\alpha}_L} \left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} \leq \sup_{x,y \in \Lambda^{\alpha}_L} \bigg|\left \langle \phi_x \phi_y \right \rangle_{\mu_{\alpha}} \bigg|\leq K_{\alpha}^2 + 2K_{\alpha}\sup_{x \in \Lambda_L^{\alpha}} \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}}^{\frac{1}{2}} + 4 \sup_{x \in \Lambda_L^{\alpha}} \left \langle \phi_x^2 I_x^{+K_{\alpha}} \right \rangle_{\mu_{\alpha}},\]

which is nothing but 27 .

References↩︎

---

1. \(d\mu_{\alpha}\) is actually a probability measure, as \(N\) is even and the determinant in the right–hand side of 2 is real, as shown in Appendix 6.
   ↩︎

2. Note that in the thermodynamic limit \(|\Lambda_L| \to \infty\), the contact term in the right–hand side of 3 is negligible in view of establishing LRO.↩︎

3. There are in general two equivalent ways to implement anti-periodic boundary conditions: either one considers anti-periodic Grassmann variables and a periodic kernel (both restricted to the finite lattice \(\Lambda_L\)) or periodic Grassmann variables and an anti-periodic kernel. In this paper we adopt the latter formulation.↩︎

4. Notice that [37] and [34] adopt different conventions for the complete elliptic integral of the first kind. Throughout [37] the argument is the parameter \(m\), whereas [34] uses the modulus \(k\). The two conventions are related by \(m=k^2\).↩︎

5. The extra normalization factor \(\ell^2\) is chosen in order for the action to scale as the volume of the system \(L^2\). Alternatively, one can think that such factor allows to reconstruct a Riemann Integral in the limit \(\ell \to 0^+\).↩︎

6. Since the Jacobian of the change of Grassmann variables cancels between numerator and denominator in the definition of the Gibbs state↩︎

7. In other words it preserves the open square lattice↩︎

8. Such a transformation leaves the multiplication operator \(M_{\phi}\) invariant, since it acts by conjugation with a diagonal multiplication operator which clearly commutes with \(M_{\phi}\) (see [38]).↩︎

9. Notice that these definitions are the correct ones only for reflections across the canonical cut plane (see Definition 1). For reflections across generic cut planes, 17 ,18 and 19 must be accompanied with the multiplication of the right-hand side by an extra sign: \[\sigma_{\nu}^L(x)\doteq (-1)^{\sum_{\mu=0,1}\frac{1}{L}(\vartheta_{\nu}(x)- \vartheta_{\nu}^L(x) )_{\mu}}.\] See also Appendix 6 for a discussion of reflections across generic cut planes.
   ↩︎

10. Explicitly: \[\theta_1(a)= \begin{cases} A \,\,\,\,\,\,\,\,\,\,&a=B\\ B &a=A\\ D &a=C\\ C &a = D\\ \end{cases},\,\,\,\,\,\,\,\,\,\,\,\, \theta_0(a)= \begin{cases} A \,\,\,\,\,\,\,\,\,\,&a=C\\ C &a=A\\ D &a=B\\ B &a= D\\ \end{cases}.\]↩︎

11. In order to prove the identity on the reflected measures, it is enough to observe that, for any function \(f\) depending on the Grassmann and real variables in \((\Lambda_L^{\alpha})_{\pm}\), the Grassmann integral of \(f\) with respect to \(d\nu_{\pm}^{\alpha}\) vanishes unless \(f\) contains all Grassmann variables associated with \((\Lambda_L^{\alpha})_{\pm}\). Hence, without loss of generality, we may restrict ourselves to observables of the form \(f = P(\phi)\prod_{x\in (\Lambda_L^{\alpha})_+}\prod_J \overline{\psi}_{x,J}\psi_{x,J}\), where \(P(\phi)\) is an integrable function depending only on \(\phi|_{(\Lambda_L^{\alpha})_+}\). For such observables, \(\Theta(f) = \Theta(P(\phi)) \prod_{x\in (\Lambda_L^{\alpha})_+}\prod_J \overline{\psi}_{\theta(x),\theta(J)}\psi_{\theta(x),\theta(J)}\), where, importantly, no sign ambiguity has arisen in this transformation. Since, by definition, \(\left\langle f \right\rangle_{\Theta(d\nu_-^{\alpha})} = \overline{ \left\langle \Theta(f) \right\rangle_{d\nu_-^{\alpha}}}\), the claimed identity immediately follows.↩︎

12. One can verify that the same conclusion holds for staggered fermions by observing the following facts. Any geometric transformation of the field \(\phi\) is implemented by conjugation with a suitable unitary operator \(U\). Since we are only interested in spectral properties, one may equivalently regard this transformation as acting on the Dirac operator, because \[\det\!\big((\cancel{D}_\alpha)^- - U M_\phi U^\ast - \lambda \mathbf{1}\big) = \det\!\big(U(\cancel{D}_\alpha)^-U^\ast - M_\phi - \lambda \mathbf{1}\big).\] Now, the \(\mathbb{Z}_2\)-holonomy background is invariant under translations and rotations, since every plaquette carries flux \(-1\). Moreover, the spectrum of a magnetic Laplacian, of which the staggered Dirac operator is an example, depends only on the holonomies of the background Gauge field (see [38]). In the proof of that result, the unitary conjugation relating two magnetic Laplacians with the same fluxes is constructed explicitly and shown to be a gauge transformation, namely multiplication by site-dependent phases. Such transformations commute with the multiplication operator \(M_\phi\), and therefore do not affect it.↩︎