1 Introduction and main results↩︎

1.1 Background and purpose↩︎

The Hubbard model is one of the simplest lattice models for interacting electrons, yet it has remained a central object in mathematical physics and condensed matter theory for many decades. It was introduced as a model for electron correlations in narrow bands [1], and has since been used as a basic framework for studying Mott physics, magnetism, and the competition between spin and charge degrees of freedom; see also the exact one-dimensional solution of Lieb and Wu [2]. Its magnetic properties have motivated several rigorous approaches. Important examples include Nagaoka-type ferromagnetism, Lieb’s spin-reflection-positivity approach to the half-filled bipartite model, and flat-band or nearly-flat-band mechanisms for itinerant ferromagnetism [3]–[6]. Rigorous analyses of Hubbard-type systems have also been developed from other viewpoints, for example by constructive renormalization group methods in the half-filled honeycomb Hubbard model [7].

Despite these developments, rigorous results for physically natural regimes remain rather limited. The difficulty is not caused by a complicated definition of the model. Rather, it comes from the delicate interplay between electron itinerancy, spin degrees of freedom, and the local Coulomb repulsion: the same hopping term which favours delocalisation also competes with the strong on-site repulsion and generates effective magnetic interactions.

In this paper we study the repulsive Hubbard model at half filling in the strong-coupling regime. The expected physical picture is simple: when \(U\gg |t|\), charge fluctuations are energetically suppressed and the remaining low-energy thermodynamics in the singly occupied sector is governed by an antiferromagnetic spin model. More precisely, virtual hopping processes of second order in \(t/U\) produce the superexchange scale \[J_0(U):=\frac{4t^2}{U}.\] Thus the natural reference model is the spin-\(\frac{1}{2}\) Heisenberg model with exchange constant \(J_0(U)\). The point of the present work is to give a quantitative mathematical formulation of this perturbative picture at the level of finite-volume canonical Gibbs states. We do not only identify the second-order effective interaction; we control the higher-order effective remainder and the charge-defect contribution with error bounds uniform in the volume and in fixed positive-field windows. In this way, the formal Hubbard-to-Heisenberg reduction is upgraded to a volume-uniform comparison of canonical thermodynamic quantities.

From the viewpoint of phase diagrams, the charge-sector estimates are also part of the significance of the result. In attractive or charge-favouring regimes, empty and doubly occupied sites may be energetically favoured, and staggered charge order can become stable; rigorous Pirogov–Sinai analyses of Borgs, Kotecký and collaborators establish such charge-ordered phases in related extended Hubbard settings [8], [9]. The present work describes the opposite large positive-\(U\) regime at half filling: charge defects are suppressed, macroscopic staggered charge order is excluded, and the remaining thermodynamics is governed by the effective spin degrees of freedom at the Heisenberg scale.

The temperature scale used here is part of the physical regime. The charge excitation scale is of order \(U\), whereas the spin interaction inside the singly occupied sector appears only at the superexchange scale \(J_0(U)=4t^2/U\). Therefore, to see the nontrivial spin thermodynamics described by the Heisenberg reference model, temperature should be measured in units of \(J_0(U)\). This is expressed by the condition \[\beta J_0(U)\ge \ell_0.\] One may try to formulate comparisons outside this scale as well, but that would require additional control of thermal regimes not addressed in the present work.

A distinctive feature of our setting is that the analysis is carried out directly in the canonical half-filled ensemble, \[\mathcal{H}_\Lambda^{\rm hf} = \ker(N_\Lambda-|\Lambda|).\] This differs from many rigorous low-temperature phase-diagram analyses of Hubbard-type or lattice-gas models, which are naturally formulated in a grand-canonical setting with the chemical potential as one of the parameters. Strong-coupling constructions of effective Hamiltonians for quantum lattice systems, including Hubbard-type tight-binding models, are a central ingredient in the quantum Pirogov–Sinai program of Datta, Fernández and Fröhlich [10], [11]. Related Pirogov–Sinai analyses, such as those of Borgs–Jedrzejewski–Kotecký and Borgs–Kotecký–Ueltschi, establish low-temperature phase diagrams and charge-ordered phases for extended Hubbard or lattice-gas type models in regimes where charge degrees of freedom are energetically favoured [8], [9], [12], [13]. In these works, effective interactions are derived and then used to analyse phase diagrams under assumptions adapted to grand-canonical or charge-order settings.

The emphasis of the present paper is different. Rather than working at the level of maximal generality, we restrict attention to the concrete repulsive Hubbard model at half filling. This allows us to keep track of the canonical constraint, the singly occupied spin sector, and the charge-defect sectors in a more explicit way. In particular, the effective Hamiltonian is used here as part of a quantitative Hubbard–Heisenberg transfer of pressure, magnetisation density, and charge-sector estimates, rather than as an input to a grand-canonical phase-diagram analysis. Possible extensions of the method to more general settings are discussed in Subsection 1.5.

In the canonical half-filled sector, the global constraint \(N_\Lambda=|\Lambda|\) prevents a naive tensor-product factorisation of local traces. Thus, when charge defects are separated from the singly occupied spin sector, one has to keep track of the half filling constraint carefully. In the proof below this appears through a restricted trace factorisation with a common outside single-occupancy projection.

The role of the staggered field differs among the results. The pressure comparison and the charge-sector estimates are uniform for \(|h|\le h_0.\) The magnetisation comparison is stated on fixed positive windows \(I\Subset(0,h_0],\) where the field is bounded away from zero. This positivity is used in the convexity argument which turns pressure comparison into magnetisation comparison. No quasi-average limit \(h\downarrow0\) is taken in this paper.

The precise model, the reference Heisenberg system, and the main results are stated in the following subsections.

1.2 Hubbard Hamiltonian at half filling↩︎

1.2.0.1 Fermions and the half-filled sector.

Fix \(d\ge1\) and let \(\Lambda_L:=\bigl(\mathbb{Z}/L\mathbb{Z}\bigr)^d\) with \(L\in2\mathbb{N}.\) Set \(\mathfrak h_{\Lambda_L}:=\ell^2(\Lambda_L)\otimes\mathbb{C}^2\) and \[\mathcal{H}_{\Lambda_L} := \bigoplus_{n=0}^{2|\Lambda_L|} \bigwedge\nolimits^{n}\mathfrak h_{\Lambda_L} .\] For \(x\in\Lambda_L\) and \(\sigma\in\{\uparrow,\downarrow\}\), let \(c_{x\sigma},c_{x\sigma}^\ast\) be the fermionic annihilation and creation operators on \(\mathcal{H}_L\), satisfying the CAR: \[\{c_{x\sigma},c_{y\tau}^\ast\} = \delta_{xy}\delta_{\sigma\tau}, \qquad \{c_{x\sigma},c_{y\tau}\}=0, \qquad \{c_{x\sigma}^\ast,c_{y\tau}^\ast\}=0.\] Define \[n_{x\sigma}:=c_{x\sigma}^\ast c_{x\sigma}, \qquad n_x:=n_{x\uparrow}+n_{x\downarrow}, \qquad N_{\Lambda_L}:=\sum_{x\in\Lambda_L} n_x .\] We work in the half-filled sector \[\mathcal{H}_{\Lambda_L}^{\rm hf} := \ker\bigl(N_{\Lambda_L}-|\Lambda_L|\bigr) = \bigwedge\nolimits^{|\Lambda_L|}\mathfrak h_{\Lambda_L} .\]

1.2.0.2 Hamiltonian.

The Hubbard Hamiltonian at half filling is given by \[H^{\rm Hub}_{\Lambda_L} := UD_{\Lambda_L}+T_{\Lambda_L},\] where \(U>0\) is the on-site Coulomb repulsion, \[D_{\Lambda_L} := \sum_{x\in\Lambda_L} n_{x\uparrow}n_{x\downarrow},\] and \(T_{\Lambda_L}\) is the nearest-neighbour hopping term. Let \(\mathscr B_{\Lambda_L}\) denote the set of unordered nearest-neighbour bonds in the torus \(\Lambda_L\), each bond being counted once. Then \[T_{\Lambda_L} := -t \sum_{\{x,y\}\in\mathscr B_{\Lambda_L}} \sum_{\sigma=\uparrow,\downarrow} \bigl( c_{x\sigma}^\ast c_{y\sigma} + c_{y\sigma}^\ast c_{x\sigma} \bigr), \qquad t\in\mathbb{R}\setminus \{0\}.\] Define the staggering \[\eta_x:=(-1)^{x_1+\cdots+x_d},\] which is well-defined on the torus \((\mathbb{Z}/L\mathbb{Z})^d\) since \(L\) is even, and set \[S_x^{(3)} := \frac{1}{2}\bigl(n_{x\uparrow}-n_{x\downarrow}\bigr), \qquad M_{\Lambda_L} := \sum_{x\in\Lambda_L}\eta_x S_x^{(3)} .\] For \(h\in\mathbb{R}\), we consider the staggered-field Hamiltonian on \(\mathcal{H}_{\Lambda_L}^{\rm hf}\), \[H^{\rm Hub}_{\Lambda_L}(h) := H^{\rm Hub}_{\Lambda_L}-hM_{\Lambda_L}.\]

1.3 Reference Heisenberg model, pressures, and quasi-averages↩︎

Throughout this subsection, for notational simplicity, we write \(\Lambda=\Lambda_L.\)

1.3.0.1 The reference Heisenberg model.

Let \[\mathcal{H}^{\rm spin}_\Lambda := \bigotimes_{x\in\Lambda}\mathbb{C}^2\] with the tensor factors ordered in any fixed way. Let \(\sigma^{(1)},\sigma^{(2)},\sigma^{(3)}\) be the Pauli matrices: \[\sigma^{(1)} = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \qquad \sigma^{(2)} = \begin{pmatrix} 0&-\mathrm{i}\,\,\\ \mathrm{i}\,\,&0 \end{pmatrix}, \qquad \sigma^{(3)} = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}.\] For each \(x\in\Lambda\) and \(a\in\{1,2,3\}\), define the local spin operator \(S_x^{(a)}\) on \(\mathcal{H}_\Lambda^{\rm spin}\), acting as \(\frac{1}{2}\sigma^{(a)}\) on the tensor factor at \(x\) and as the identity on all other tensor factors. Set \[\boldsymbol{S}_x := \bigl(S_x^{(1)},S_x^{(2)},S_x^{(3)}\bigr).\] This notation is chosen to match the corresponding fermionic spin operators introduced above. When both the fermionic Hilbert space and the reference spin Hilbert space appear in the same formula, the ambient space or the identifying map will be stated explicitly. Let \[M_\Lambda^{\rm spin} := \sum_{x\in\Lambda}\eta_x S_x^{(3)}.\] For \(J>0\), define \[H^{\rm Heis}_\Lambda(J) := J\sum_{\{x,y\}\in\mathscr B_\Lambda} \left( \boldsymbol{S}_x\cdot\boldsymbol{S}_y-\frac{1}{4} \right), \qquad \boldsymbol{S}_x\cdot\boldsymbol{S}_y := \sum_{a=1}^3 S_x^{(a)}S_y^{(a)}.\] With staggered field \(h\in\mathbb{R}\), set \[H^{\rm Heis}_\Lambda(J,h) := H^{\rm Heis}_\Lambda(J)-hM_\Lambda^{\rm spin}.\]

The Heisenberg reference model is used here as the effective spin model at the superexchange scale. Rigorous analyses of quantum spin systems and Heisenberg-type models have been developed from several viewpoints, including phase-transition and correlation methods for quantum spin systems and low-temperature spin-wave/free-energy asymptotics; see, for example, [9], [14].

1.3.0.2 Hubbard pressure and Gibbs state.

For \(\beta>0\), \(U>0\), and \(h\in\mathbb{R}\), define the finite-volume Hubbard partition function by \[\require{physics} Z_{\Lambda,\beta,U}^{\rm Hub}(h) := \Tr_{\mathcal{H}_\Lambda^{\rm hf}} e^{-\beta H_\Lambda^{\rm Hub}(h)}.\] The corresponding pressure is \[p_{\Lambda,\beta,U}^{\rm Hub}(h) := \frac{1}{\beta|\Lambda|} \log Z_{\Lambda,\beta,U}^{\rm Hub}(h).\] For \(O\in\mathcal{B}(\mathcal{H}_\Lambda^{\rm hf})\), the Hubbard Gibbs state is \[\require{physics} \omega_{\Lambda,\beta,U,h}^{\rm Hub}(O) := \frac{ \Tr_{\mathcal{H}_\Lambda^{\rm hf}} \left( Oe^{-\beta H_\Lambda^{\rm Hub}(h)} \right) }{ Z_{\Lambda,\beta,U}^{\rm Hub}(h) }.\] The finite-volume staggered magnetisation density is \[m_{\Lambda,\beta,U}^{\rm Hub}(h) := \frac{1}{|\Lambda|} \omega_{\Lambda,\beta,U,h}^{\rm Hub}(M_\Lambda).\] Since the volume is finite, \(h\mapsto p_{\Lambda,\beta,U}^{\rm Hub}(h)\) is differentiable, and \[\partial_h p_{\Lambda,\beta,U}^{\rm Hub}(h) = m_{\Lambda,\beta,U}^{\rm Hub}(h).\]

1.3.0.3 Heisenberg pressure and Gibbs state.

For \(J>0\), \(\beta>0\), and \(h\in\mathbb{R}\), define \[\require{physics} Z_{\Lambda,\beta}^{\rm Heis}(J,h) := \Tr_{\mathcal{H}_\Lambda^{\rm spin}} e^{-\beta H_\Lambda^{\rm Heis}(J,h)}.\] The corresponding pressure is \[p_{\Lambda,\beta}^{\rm Heis}(J,h) := \frac{1}{\beta|\Lambda|} \log Z_{\Lambda,\beta}^{\rm Heis}(J,h).\] For \(O\in\mathcal{B}(\mathcal{H}_\Lambda^{\rm spin})\), the Heisenberg Gibbs state is \[\require{physics} \omega_{\Lambda,\beta,J,h}^{\rm Heis}(O) := \frac{ \Tr_{\mathcal{H}_\Lambda^{\rm spin}} \left( Oe^{-\beta H_\Lambda^{\rm Heis}(J,h)} \right) }{ Z_{\Lambda,\beta}^{\rm Heis}(J,h) }.\] The finite-volume staggered magnetisation density is \[m_{\Lambda,\beta}^{\rm Heis}(J,h) := \frac{1}{|\Lambda|} \omega_{\Lambda,\beta,J,h}^{\rm Heis}(M_\Lambda^{\rm spin}).\] Again, \[\partial_h p_{\Lambda,\beta}^{\rm Heis}(J,h) = m_{\Lambda,\beta}^{\rm Heis}(J,h).\]

1.3.0.4 The effective Heisenberg reference model.

For the Hubbard model with coupling \(U\), set \[J_0(U):=\frac{4t^2}{U}.\] The Heisenberg model used as the reference model in the main comparison theorems is the nearest-neighbour antiferromagnetic Heisenberg model with coupling \(J_0(U)\) and the same staggered field \(h\): \(H_\Lambda^{\rm Heis}(J_0(U),h).\) The corresponding pressure is denoted by \[p_{\Lambda,\beta,U}^{\rm Heis}(h) := p_{\Lambda,\beta}^{\rm Heis}(J_0(U),h).\] We also write \[\omega_{\Lambda,\beta,U,h}^{\rm Heis} := \omega_{\Lambda,\beta,J_0(U),h}^{\rm Heis}\] and \[m_{\Lambda,\beta,U}^{\rm Heis}(h) := m_{\Lambda,\beta}^{\rm Heis}(J_0(U),h) = \partial_h p_{\Lambda,\beta,U}^{\rm Heis}(h).\]

1.4 Main theorems↩︎

1.4.1 Pressure and magnetisation comparison↩︎

1.4.1.1 Heisenberg-scale regime.

The comparison is made in the Heisenberg-scale temperature regime \(\beta J_0(U)\ge \ell_0.\) This means that the inverse temperature is measured on the spin-exchange scale \(J_0(U)=4t^2/U\), rather than on the original hopping scale. Thus, as \(U\to\infty\), the temperature is low enough to resolve the effective spin interaction.

Below, the pressure comparison and the charge-sector estimates are stated uniformly for \(|h|\le h_0,\) whereas the magnetisation comparison is stated on fixed field windows \(I\Subset(0,h_0].\) The condition \(\inf I>0\) is used only in the convexity step which converts pressure comparison into magnetisation comparison. No quasi-average limit \(h\downarrow0\) is taken.

The proof of Theorem 1 is given in Subsection 7.1.

The proof of Theorem 2 is given in Subsection 7.3.

To state the corresponding infinite-volume consequences in a concise form, we shall use the following standing assumption on the existence of fixed-field pressure limits.

Assume Assumption 3. For \(\sharp\in\{{\rm Hub},{\rm Heis}\}\), define \[\mathcal{D}_{\beta,U}^{\sharp} := \left\{ h\in(-h_0,h_0): p_{\beta,U}^{\sharp} \text{ is differentiable at }h \right\}.\] Since \(p_{\beta,U}^{\sharp}\) is convex, the set \((-h_0,h_0)\setminus \mathcal{D}_{\beta,U}^{\sharp}\) is at most countable. For \(h\in\mathcal{D}_{\beta,U}^{\sharp}\), set \[m_{\beta,U}^{\sharp}(h) := \partial_h p_{\beta,U}^{\sharp}(h).\] By Lemma 54, this agrees with the thermodynamic limit of the finite-volume magnetisations: \[m_{\beta,U}^{\sharp}(h) = \lim_{L\to\infty} m_{\Lambda_L,\beta,U}^{\sharp}(h), \qquad h\in\mathcal{D}_{\beta,U}^{\sharp}.\]

1.4.1.2 Positive fixed-field Hubbard magnetisation.

The fixed positive-field magnetisation comparison has a concrete consequence for the Hubbard magnetisation itself. We combine the elementary lower bound for the Heisenberg reference magnetisation with Theorem 2.

The proof of Corollary 7 is given in Appendix 7.5.

1.4.2 Charge-sector consequences↩︎

1.4.2.1 Standing assumptions and Gibbs state.

Throughout this subsubsection, fix \(\ell_0>0\). Recall the finite-volume half-filled Hubbard Gibbs state \(\omega_{\Lambda_L,\beta,U,h}^{\rm Hub}\) introduced above. All expectations below are taken with respect to this state, uniformly for \(|h|\le h_0\).

1.4.2.2 Charge observables.

Set \[q_x:=(n_x-1)^2.\] Thus \(q_x\) detects deviations from single occupancy at \(x\). We also define the staggered charge observable \[C_{\Lambda_L}^{\rm ch} := \sum_{x\in\Lambda_L}\eta_x(n_x-1).\]

The proof of Theorem 9 is given in Subsection 7.6.

1.5 Possible extensions and robustness of the comparison↩︎

The main statements above are formulated on the even torus \(\Lambda_L=(\mathbb{Z}/L\mathbb{Z})^d\) with nearest-neighbour hopping and a uniform staggered field. This choice keeps the notation simple and avoids boundary terms. The mechanism, however, is more flexible.

• Other boundary conditions and van Hove sequences. The finite-volume comparison is local in nature. Thus one expects the same thermodynamic pressure and magnetisation comparison along standard van Hove sequences of finite boxes, with periodic or open boundary conditions, provided the boundary contribution is negligible after division by the volume. The torus assumption is used here mainly to avoid carrying these boundary errors.

• More general bipartite graphs. The argument should extend to families of finite bipartite graphs with uniformly bounded degree and a compatible staggering \(\eta_x\), as long as the hopping connects the two sublattices and the relevant local estimates are uniform in the volume. In that setting the effective spin Hamiltonian is the antiferromagnetic Heisenberg model on the same graph.

• Non-uniform finite-range hopping. For hopping amplitudes \(t_{xy}\) of finite range, the second-order effective spin model should have edge-dependent antiferromagnetic couplings of order \(J_{xy}\simeq \frac{4|t_{xy}|^2}{U}.\) Thus the comparison with \(J_0(U)=4t^2/U\) is replaced by a comparison with a non-uniform effective Heisenberg model. The present paper treats the translation-invariant nearest-neighbour case in order to keep the main theorem transparent.

• Fixed positive fields beyond the uniform case. The fixed-field nature of the argument is essential, but the precise choice of a constant staggered field is not expected to be essential. A site-dependent staggered field bounded away from zero should lead to a similar field-dominated comparison, with constants depending on the lower bound of the field window.

• Zero-temperature finite-volume limits. At fixed finite volume, one may simply take \(\beta\to\infty\). Since the Hilbert space is finite-dimensional, the Gibbs state converges to the normalized projection onto the ground-state space. Hence all estimates in the main theorem which are uniform in \(\beta\) pass directly to the finite-volume zero-temperature mixed ground-state expectation. No finite-volume ground-state uniqueness is needed for this passage.

  It is plausible that, in the present positive staggered-field setting, the finite-volume ground state is in fact unique. This, however, is a separate Perron–Frobenius or reflection-positivity type input and is not used in the present comparison argument; see, for example, [4], [15], [16]. If such a uniqueness statement is established in the present setting, the mixed ground-state expectation above may be replaced by the unique ground-state expectation.

These extensions are not needed for the main results of this paper. They indicate that the comparison mechanism is not tied to the translation-invariant torus geometry, but rather to three structural features: strong repulsion at half filling, a bipartite staggered-field structure, and locality of the hopping.

1.6 Strategy and outline↩︎

The goal of the paper is to compare the canonical half-filled Hubbard model with the Heisenberg reference model on the Heisenberg scale \(\beta J_0(U)\ge \ell_0\) with \(J_0(U)=\frac{4t^2}{U}.\) The pressure comparison and the charge-sector estimates are uniform for \(|h|\le h_0.\) The magnetisation comparison is stated on fixed positive windows \(I\Subset(0,h_0],\) because the passage from pressure comparison to magnetisation comparison uses a fixed-window convexity argument. No quasi-average limit \(h\downarrow0\), reflection positivity, infrared bound, or long-range-order input is used.

1.6.0.1 Road map.

The proof has three main parts.

• First, we use the LS/SW diagonalisation scheme to conjugate the Hubbard Hamiltonian to a \(D_{\Lambda_L}\)-diagonal effective Hamiltonian, \[U_{{\rm SW},\Lambda_L}(h) H_{\Lambda_L}^{\rm Hub}(h) U_{{\rm SW},\Lambda_L}(h)^\ast = UD_{\Lambda_L}+A_{\Lambda_L}(h), \qquad [A_{\Lambda_L}(h),D_{\Lambda_L}]=0.\] The sector \(D_{\Lambda_L}=0\) is the singly occupied spin sector and is denoted by \(P\). Since the transformed Hamiltonian preserves the \(D_{\Lambda_L}\)-sectors, the \(P\)-block can be analysed separately.

• Second, we compare the effective \(P\)-block Hamiltonian with the Heisenberg reference model. The second-order term gives a Heisenberg Hamiltonian with intermediate parameters \(J(h)\) and \(h_{\rm eff}(h)\). The parameter renormalisation and the higher-order \(P\)-block remainder are controlled uniformly for \(|h|\le h_0\). This gives the pressure comparison between the effective \(P\)-block Hamiltonian and the Heisenberg reference model.

• Third, we compare the full Hubbard pressure with the \(P\)-block pressure by a soft defect estimate. Combining this estimate with the \(P\)-block comparison gives the Hubbard–Heisenberg pressure comparison for \(|h|\le h_0\). The charge-sector estimates are obtained from the same defect decomposition, together with the LS/SW dressing estimate for the normalized defect density.

The magnetisation comparison is then derived from the pressure comparison by convexity.

This proves the finite-volume pressure comparison stated in Theorem 1, the fixed-positive-field magnetisation comparison stated in Theorem 2, and the charge-sector estimates stated in Theorem 9.

1.6.0.2 Positive fixed-field magnetisation.

Since \(h\) is bounded away from zero and \(J_0(U)\to0\), the Heisenberg reference model is close to the pure staggered-field spin system at the level of pressure. This gives an explicit lower bound \[m_{\Lambda_L,\beta,U}^{\rm Heis}(h) \ge m_{\rm Heis}^{\rm lb}(U,\beta;I), \qquad h\in I,\] with \(m_{\rm Heis}^{\rm lb}(U,\beta;I)\to\frac{1}{2}\) uniformly under \(\beta J_0(U)\ge\ell_0\) as \(U\to\infty\). The Hubbard–Heisenberg magnetisation comparison then gives the corresponding positive fixed-field magnetisation bound for the Hubbard model.

1.6.0.3 Charge-sector consequences.

The same defect decomposition controls the density of charge defects \(q_x=(n_x-1)^2.\) After conjugating back to the original Hubbard Gibbs state by the LS/SW dressing estimate for the normalized defect density, we obtain \[\frac{1}{|\Lambda_L|} \sum_{x\in\Lambda_L} \omega_{\Lambda_L,\beta,U,h}^{\rm Hub}(q_x) \le \varepsilon_{\rm ch}(U,\beta;h_0).\] This implies suppression of double occupancy and excludes macroscopic staggered charge order in the normalization used here.

1.6.0.4 Organisation of the paper.

Section 2 develops the LS/SW diagonalisation and derives the \(D_{\Lambda_L}\)-diagonal effective Hamiltonian \(H_{*,\Lambda_L}(h)=UD_{\Lambda_L}+A_{\Lambda_L}(h).\) Section 3 introduces the two auxiliary deformation schemes used to identify the relevant second-order spin term and to compare it with the physical endpoint correction. The resulting second-order \(P\)-block Hamiltonian is identified with a Heisenberg-type spin Hamiltonian in Section 4.

Section 5 compares the effective \(P\)-block spin Hamiltonian with the Heisenberg reference model. This step controls both the mismatch between the renormalized parameters and the reference parameters, and the higher-order \(P\)-block remainder. Section 6 then compares the full Hubbard pressure with the \(P\)-block pressure by the soft defect estimate, and records the corresponding diagonal charge-defect bound.

The main finite-volume estimates are assembled in Section 7. First, the \(P\)-block comparison and the soft defect pressure estimate give the Hubbard–Heisenberg reference pressure comparison. Next, the magnetisation comparison is obtained from the pressure comparison on the enlarged window \(I^\sharp\) by the fixed-window convexity argument. The same section also proves the positive fixed-field magnetisation consequence, the charge-sector theorem, and the thermodynamic-limit corollaries.

The appendices contain the deferred technical estimates. Appendix 8 collects the BCH, graded-word, and quantitative LS/SW estimates, including the LS/SW dressing estimate for the normalized defect density. Appendix 9 contains the spin representation and the two-site computation underlying the Heisenberg identification. Appendix 10 proves the \(P\)-block remainder and parameter-mismatch estimates. Appendix 11 records the finite-volume pressure Lipschitz, derivative, and convexity tools used in the final assembly and in the thermodynamic-limit arguments.

Acknowledgements↩︎

This work was supported by JSPS KAKENHI Grant Number 23H01086.

Declarations↩︎

• Conflict of interest: The authors have no conflicts of interest to declare that are relevant to the content of this article.

• Data availability: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

2 \(D_\Lambda\)-diagonalisation by LS/SW and the physical output↩︎

Throughout this section, \(\Lambda=\Lambda_L\) denotes a fixed even torus, and all operators are understood on the half-filled Hilbert space \(\mathcal{H}_\Lambda^{\rm hf}\), unless explicitly stated otherwise.

The diagonalisation used below is a finite-volume strong-coupling Lie–Schwinger/Schrieffer–Wolff scheme. Such transformations go back to the work of Schrieffer and Wolff, and related \(t/U\)-expansions for the Hubbard model have been developed in several forms; see, for example, [17]–[20]. Closely related Lie–Schwinger block-diagonalisation methods have also been used in recent work on gapped quantum chains, including systems with unbounded interactions [21].

In the present paper the grading is given by the double-occupancy operator \(D_\Lambda\). The purpose of the construction is to conjugate the Hubbard Hamiltonian to a \(D_\Lambda\)-diagonal form \[UD_\Lambda+A_\Lambda(h), \qquad [A_\Lambda(h),D_\Lambda]=0,\] with estimates uniform in the volume and in the field range \(|h|\le h_0.\) The restriction to fixed positive field windows enters only later, when the pressure comparison is converted into a magnetisation comparison by convexity.

2.1 \(D\)-grading, interactions, and norms↩︎

2.1.0.1 Defect projections.

Set \[n_x:=n_{x\uparrow}+n_{x\downarrow}, \qquad q_x:=(n_x-1)^2 .\] Then \(q_x\) is the projection onto the subspace with \(n_x\in\{0,2\}\), and the family \(\{q_x\}_{x\in\Lambda}\) is commuting. On the half-filled subspace \(\mathcal{H}_\Lambda^{\rm hf}\) one has \[\label{eq:spec-qsum-2D} \sum_{x\in\Lambda}q_x=2D_\Lambda .\tag{1}\] Since we work at half filling, \(N_\Lambda=|\Lambda|\), the sector \(D_\Lambda=0\) consists precisely of the singly occupied configurations. Equivalently, \((n_{x\uparrow}+n_{x\downarrow})\psi=\psi \, (x\in\Lambda)\) for every vector \(\psi\) in this sector. We set \[\label{eq:def-PQ} P:=\mathbb{1}_{\{D_\Lambda=0\}}\restriction_{\mathcal{H}_\Lambda^{\rm hf}}, \qquad Q:=\mathbb{1}-P .\tag{2}\]

2.1.0.2 The CAR algebra, local subalgebras, and even observables.

Let \(\frak A\) denote the quasi-local CAR algebra on the infinite lattice \(\mathbb{Z}^d\).¹ For a finite set of sites \(X\), either in \(\mathbb{Z}^d\) or in a finite torus \(\Lambda\), we write \(\frak A_X\) for the CAR algebra generated by \(\{c_{x\sigma},c_{x\sigma}^\ast: x\in X,\;\sigma\in\{\uparrow,\downarrow\}\}.\) Thus, when \(X\subset\Lambda\), the symbol \(\frak A_X\) denotes the corresponding finite-volume local algebra on the torus, and \(\frak A_\Lambda\) denotes the full finite-volume CAR algebra acting on \(\mathcal{H}_\Lambda\). We work almost exclusively in finite volume; the quasi-local algebra \(\frak A\) is used only as a convenient language for locality.

Let \(\Theta\) be the fermion-parity automorphism, defined on generators by \[\Theta(c_{x\sigma})=-c_{x\sigma}, \qquad \Theta(c_{x\sigma}^\ast)=-c_{x\sigma}^\ast .\] We write \[\frak A_X^{\rm even} := \{A\in\frak A_X:\Theta(A)=A\}\] for the even local subalgebra. We shall use the elementary fact that if \(X\cap Y=\varnothing\), \(A\in\frak A_X^{\rm even}\), and \(B\in\frak A_Y\), then \([A,B]=0.\) Thus even local observables with disjoint supports commute in the usual sense.

2.1.0.3 Adjoint notation and BCH.

For \(S\in\frak A_\Lambda\), define the inner derivation \[\mathrm{ad}_S:\frak A_\Lambda\to\frak A_\Lambda, \qquad \mathrm{ad}_S(A):=[S,A].\] We also write \[\mathrm{ad}_S^{\,0}(A):=A, \qquad \mathrm{ad}_S^{\,n}(A):=\mathrm{ad}_S\bigl(\mathrm{ad}_S^{\,n-1}(A)\bigr) \quad(n\ge1).\] We repeatedly use the Baker–Campbell–Hausdorff expansion \[e^{S}Ae^{-S} = \sum_{n\ge0}\frac{1}{n!}\mathrm{ad}_S^{\,n}(A),\] whenever the series is norm convergent.

2.1.0.4 Interactions and the identification convention.

Throughout the paper, an interaction on \(\Lambda\) means a family of even local terms \[\Phi=\{\Phi_X\}_{X\subset\Lambda}, \qquad \Phi_X\in\frak A_X^{\rm even}.\] Self-adjointness or anti-self-adjointness will be specified separately when needed. We say that \(\Phi\) is finite-range if there exists \(R<\infty\) such that \(\Phi_X=0\) whenever \(\operatorname{diam}(X)>R\), where the diameter is taken with respect to the graph distance on \(\Lambda\).² The associated finite-volume operator is \[\Phi_\Lambda:=\sum_{X\subset\Lambda}\Phi_X .\] Throughout the paper we freely identify \(\Phi\) with \(\Phi_\Lambda\) when no confusion can arise. Termwise operations for interactions are collected in Appendix 8.

2.1.0.5 Interaction norm and interaction commutator.

For a finite-range interaction \(\Phi=\{\Phi_X\}_{X\subset\Lambda}\) and \(\kappa\ge0\), define \[\label{eq:spec-kappa-norm} \|\Phi\|_\kappa := \sup_{x\in\Lambda} \sum_{X\ni x} e^{\kappa |X|}\|\Phi_X\|, \qquad \|\Phi\|_0:=\|\Phi\|_{\kappa=0}.\tag{3}\] Here \(|X|\) denotes the cardinality of \(X\), and \(\|\cdot\|\) denotes the operator norm in the finite-volume representation. The notation \(\sum_{X\ni x}\) means the sum over all nonempty subsets \(X\subset\Lambda\) containing the site \(x\).

For two interactions \(A=\{A_X\}_{X\subset\Lambda}\) and \(B=\{B_Y\}_{Y\subset\Lambda}\), define the interaction \(\mathrm{ad}_A(B)=\{(\mathrm{ad}_A(B))_Z\}_{Z\subset\Lambda}\) by \[\label{eq:spec-ad-interaction-def} (\mathrm{ad}_A(B))_Z := \sum_{\substack{X,Y\subset\Lambda:\\ X\cup Y=Z}} [A_X,B_Y].\tag{4}\]

2.1.0.6 Support size of an interaction.

For an interaction \(A=\{A_X\}_{X\subset\Lambda}\), set \[\mathrm{supp}A:=\{X\subset\Lambda:\;A_X\neq0\}, \qquad s(A):=\sup\{|X|:\;X\in\mathrm{supp}A\}.\] Thus \(s(A)<\infty\) whenever \(A\) is finite-range with uniformly bounded support size.

2.1.0.7 \(D\)-grading.

For \(m\in\mathbb{N}_0\), let \[P_m := \mathbb{1}_{\{D_\Lambda=m\}}\restriction_{\mathcal{H}_\Lambda^{\rm hf}}\] be the corresponding spectral projection. We set \(P_m:=0\) if \(m\notin\operatorname{spec}(D_\Lambda)\). Then \[\sum_{m\ge0}P_m=\mathbb{1}, \qquad D_\Lambda P_m=mP_m .\] For any \(B\in\frak A_\Lambda\), define its \(D\)-graded parts by \[\label{eq:def-D-graded} B^{(k)} := \sum_{m\ge0}P_{m+k}BP_m, \qquad k\in\mathbb{Z},\tag{5}\] with the convention \(P_{m+k}=0\) if \(m+k<0\). Since the volume is finite, the sum over \(k\) is finite and \[B=\sum_{k\in\mathbb{Z}}B^{(k)}.\] Moreover, \[\label{eq:adUD-on-graded} \mathrm{ad}_{UD_{\Lambda}}(B^{(k)})=kU B^{(k)}.\tag{6}\] We write \[B^{\rm diag}:=B^{(0)}, \qquad B^{\rm off}:=\sum_{k\neq0}B^{(k)}.\] We say that \(B\) is \(D\)-diagonal if \(B=B^{\rm diag}\), equivalently \(\mathrm{ad}_{D_\Lambda}(B)=0\).

2.1.0.8 Homological inverses \(\mathcal{I}\) and \(\mathcal{I}_h\).

We use the \(D\)-grading notation introduced above. For \(B\in\frak A_\Lambda\), recall \(B^{\rm diag}:=B^{(0)},\) and \(B^{\rm off}:=\sum_{k\neq0}B^{(k)}.\)

The case \(h=0\). On the \(D\)-off-diagonal subspace, the map \(\mathrm{ad}_{UD_\Lambda}\) is inverted explicitly by \[\label{eq:def-I} \mathcal{I}(B^{\rm off}) := \sum_{k\neq0}\frac{1}{kU}B^{(k)} .\tag{7}\] Indeed, \[\label{eq:adUD-I} \mathrm{ad}_{UD_\Lambda} \bigl( \mathcal{I}(B^{\rm off}) \bigr) = B^{\rm off}.\tag{8}\] Moreover, if \(B\) is self-adjoint, then \(\mathcal{I}(B^{\rm off})^\ast = -\mathcal{I}(B^{\rm off}).\)

The case \(|h|\le h_0\). For nonzero field we use a local, \(h\)-dependent homological inverse. More precisely, let \(A\in\frak A_X\) be a local operator of \(D\)-grade \(k\neq0\). Under the standing large-\(U\) condition \(|k|U>h_0|X|,\) we define \(\mathcal{I}_h(A)\) as the local inverse of \(\mathrm{ad}_{UD_\Lambda-hM_\Lambda}\) on the \(k\)-graded piece, so that \[\label{eq:adUDh-Ih-local} \mathrm{ad}_{UD_\Lambda-hM_\Lambda} \bigl( \mathcal{I}_h(A) \bigr) = A.\tag{9}\] The concrete definition is given in Appendix 8.2 by a Neumann series on each graded local piece. It has the following properties: \(\mathcal{I}_h(A)\) is supported in the same set \(X\), has the same \(D\)-grade \(k\), and satisfies the corresponding local \(\|\cdot\|\)- and \(\|\cdot\|_\kappa\)-bounds. Moreover, if \(A\) is self-adjoint after summing over opposite grades, then the resulting \(\mathcal{I}_h\)-image is anti-self-adjoint.

At \(h=0\), this local inverse agrees with \(\mathcal{I}\). Thus one may write \(\mathcal{I}_0=\mathcal{I}.\)

Interaction-level convention. If \(B=\{B_X\}_{X\subset\Lambda}\) is a finite-range interaction, then all operations above are understood termwise in \(X\). Namely, we define \[B^{(k)} := \{(B_X)^{(k)}\}_{X\subset\Lambda}, \qquad B^{\rm off} := \{(B_X)^{\rm off}\}_{X\subset\Lambda},\] and set \[\mathcal{I}(B^{\rm off}) := \left\{ \mathcal{I}\bigl((B_X)^{\rm off}\bigr) \right\}_{X\subset\Lambda}, \qquad \mathcal{I}_h(B^{\rm off}) := \left\{ \mathcal{I}_h\bigl((B_X)^{\rm off}\bigr) \right\}_{X\subset\Lambda}.\] For \(U\) sufficiently large compared with \(h_0\) and the finite range of \(B\), this termwise definition is well-defined. In particular, \(\mathcal{I}(B^{\rm off})\) and \(\mathcal{I}_h(B^{\rm off})\) are again finite-range interactions with the same range as \(B\); see Appendix 8.2, especially Lemma 36.

2.2 Quantitative convergence of the Lie–Schwinger iteration↩︎

2.2.0.1 Convention on constants in the large-\(U\) regime.

Throughout the LS/SW analysis, constants may depend on \(d,\kappa\), on the fixed range of the hopping interaction, and on the fixed field window size \(h_0\). We fix once and for all a threshold \(U_\ast=U_\ast(d,\kappa,h_0)\) and work under the standing assumption \(U\ge U_\ast\). This threshold is chosen large enough so that the bounds for the field-dependent homological inverse \(\mathcal{I}_h\) are uniform for all \(|h|\le h_0\). After this choice, constants depending on \(h_0/U\) are absorbed into constants depending only on the fixed parameters, and we write them as \(C(d,\kappa)\), \(q(d,\kappa)\), \(\varepsilon_\ast(d,\kappa)\), and so on.

2.2.0.2 Iteration: abstract LS/SW scheme.

Fix \(U>0\) and \(|h|\le h_0\). Let \(B_0=B_{0,\Lambda}(h)\) be a self-adjoint finite-range interaction on \(\Lambda\), and set \[H^{(0)} := UD_\Lambda-hM_\Lambda+B_0 .\]

By our standing convention, each interaction is identified with its finite-volume sum whenever no confusion can arise. The norm \(\|\cdot\|_\kappa\), the \(D_\Lambda\)-grading, and the diagonal/off- diagonal decompositions are all taken at the interaction level.

For \(n\ge0\), suppose that \[H^{(n)} = UD_\Lambda-hM_\Lambda+B_n\] has been constructed, with \(B_n\) viewed as an interaction. Define \(S_n := \mathcal{I}_h\bigl((B_n)^{\rm off}\bigr),\) and set \[H^{(n+1)} := e^{S_n}H^{(n)}e^{-S_n}, \qquad B_{n+1} := H^{(n+1)}-(UD_\Lambda-hM_\Lambda).\] By the defining property of \(\mathcal{I}_h\), \[\mathrm{ad}_{S_n}(UD_\Lambda-hM_{\Lambda})=-\mathrm{ad}_{UD_{\Lambda}-hM_{\Lambda}}(S_n) = -(B_n)^{\rm off}.\] Thus the first-order \(D_\Lambda\)-off-diagonal part of \(B_n\) is cancelled at the \(n\)-th step.

If \(B_0\) is self-adjoint, then each \(B_n\) is self-adjoint. Moreover, \(S_n^\ast=-S_n,\) so every step is implemented by a unitary conjugation.

2.2.0.3 Notation for the LS/SW output.

With \(\Delta_\Lambda(h)\) as in Corollary 17, set \[\label{eq:def-A-Lambda} A_\Lambda(h) := -hM_\Lambda+T_\Lambda^{(0)}+\Delta_\Lambda(h), \qquad [A_\Lambda(h),D_\Lambda]=0.\tag{10}\] We also define the \(P\)-block Hamiltonian by \[H_{P,\Lambda}(h):=P A_\Lambda(h)P.\] The following sections analyse \(A_\Lambda(h)\), its \(P\)-block, and their comparison with the effective Heisenberg reference model.

3 Two deformations and the second-order connection↩︎

3.1 The \(\xi\)-scheme and the second-order effective term↩︎

The second-order effective spin Hamiltonian is defined using the full hopping-scale deformation \[H_\Lambda(h;\xi) := UD_\Lambda-hM_\Lambda+\xi T_\Lambda, \qquad \xi\in[0,1].\] Assume \(|h|\le h_0\), \(|\xi|\le1\), and that the hypotheses of Corollary 17 hold for the \(\xi\)-deformed Hamiltonian. Equivalently, it suffices to impose the smallness condition at \(\xi=1\). The LS/SW iteration yields a product unitary \(U_{\rm SW}(\xi)\) and a \(D_\Lambda\)-diagonal correction \(\Delta_\Lambda(h;\xi)\) such that \[U_{\rm SW}(\xi)H_\Lambda(h;\xi)U_{\rm SW}(\xi)^\ast = UD_\Lambda-hM_\Lambda+\xi T_\Lambda^{(0)} +\Delta_\Lambda(h;\xi), \qquad [\Delta_\Lambda(h;\xi),D_\Lambda]=0.\] Since \(H_\Lambda(h;0)=UD_\Lambda-hM_\Lambda\) is already \(D_\Lambda\)-diagonal, one has \(\Delta_\Lambda(h;0)=0.\) The first-order coefficient also vanishes, and we write \[\label{eq:xi-expansion-Binf} \Delta_\Lambda(h;\xi) = \sum_{n\ge2}\xi^n(\Delta_\Lambda)^{[n]}_{\xi}(h).\tag{11}\] We define the second-order effective term by \[\label{eq:def-H2} H^{(2)}_\Lambda(h) := (\Delta_\Lambda)^{[2]}_{\xi}(h) = \frac{1}{2} \frac{d^2}{d\xi^2} \Delta_\Lambda(h;\xi)\Big|_{\xi=0}.\tag{12}\] This is the term which will be identified with the antiferromagnetic Heisenberg interaction in Section 4.

For later use, we also record the endpoint remainder \[\label{eq:def-K-xi} K_\Lambda^{\xi}(h) := \Delta_\Lambda(h;\xi)\big|_{\xi=1} - H^{(2)}_\Lambda(h).\tag{13}\] Thus, at the physical endpoint, \[\Delta_\Lambda(h) = H^{(2)}_\Lambda(h)+K_\Lambda^{\xi}(h).\]

3.2 The auxiliary \(\lambda\)-deformation↩︎

We shall also use the auxiliary deformation \[H_\Lambda(h;\lambda) := UD_\Lambda-hM_\Lambda+T_\Lambda^{(0)} +\lambda (T_\Lambda)^{\rm off}, \qquad \lambda\in[0,1].\] This deformation has the same physical endpoint as the \(\xi\)-scheme, but a different reference point at zero. Indeed, \[H_\Lambda(h;\lambda)\big|_{\lambda=1} = H_\Lambda(h;\xi)\big|_{\xi=1} = UD_\Lambda-hM_\Lambda+T_\Lambda,\] whereas \[H_\Lambda(h;\lambda)\big|_{\lambda=0} = UD_\Lambda-hM_\Lambda+T_\Lambda^{(0)}, \qquad H_\Lambda(h;\xi)\big|_{\xi=0} = UD_\Lambda-hM_\Lambda.\]

Under the same smallness assumptions, the LS/SW iteration applied to \(H_\Lambda(h;\lambda)\) yields a unitary \(U_{\rm SW}(\lambda)\) and a \(D_\Lambda\)-diagonal correction \(\Delta_\Lambda(h;\lambda)\) such that \[U_{\rm SW}(\lambda)H_\Lambda(h;\lambda)U_{\rm SW}(\lambda)^\ast = UD_\Lambda-hM_\Lambda+T_\Lambda^{(0)} +\Delta_\Lambda(h;\lambda), \qquad [\Delta_\Lambda(h;\lambda),D_\Lambda]=0.\] Since the \(\lambda=0\) Hamiltonian is already \(D_\Lambda\)-diagonal, \(\Delta_\Lambda(h;0)=0.\) The first-order coefficient vanishes, and we write \[\label{eq:lambda-expansion-Binf} \Delta_\Lambda(h;\lambda) = \sum_{n\ge2}\lambda^n(\Delta_\Lambda)^{[n]}_{\lambda}(h).\tag{14}\]

The \(\lambda\)-deformation plays a specific technical role in this paper. It is not used to define the second-order effective term; that role belongs to the \(\xi\)-scheme. Instead, the \(\lambda\)-scheme is used to compare the full physical endpoint correction \[\Delta_\Lambda(h) = \Delta_\Lambda(h;\lambda)\big|_{\lambda=1}\] with the second-order coefficient \(H_\Lambda^{(2)}(h)\) defined through the \(\xi\)-scheme. This comparison is one of the main technical inputs for the control of the \(P\)-block remainder, and the required estimates are carried out in Appendix 10.

3.3 The second-order coefficient in the \(\xi\)-scheme↩︎

4 The second-order \(P\)-block and its Heisenberg form↩︎

4.1 Spin representation of the \(D_\Lambda=0\) half-filled block↩︎

4.1.0.1 Canonical identification with the spin Hilbert space.

Recall the spin Hilbert space \(\mathcal{H}^{\rm spin}_\Lambda\) introduced in Subsection 1.3. In this subsection we fix the explicit unitary identification between this spin space and the singly occupied subspace of the half-filled fermionic Hilbert space.

Fix once and for all an ordering \(\Lambda=\{x_1,\dots,x_{|\Lambda|}\}.\) We view \[\mathcal{H}^{\rm spin}_\Lambda = \bigotimes_{j=1}^{|\Lambda|}\mathbb{C}^2\] with the \(j\)-th tensor factor assigned to the site \(x_j\), and with standard basis \(|\!\uparrow\, \rangle_{x_j}\) and \(|\!\downarrow\, \rangle_{x_j}.\) On the fermionic side we work on \(\mathcal{H}^{\rm hf}_\Lambda\) and recall \(P := \mathbb{1}_{\{D_\Lambda=0\}} \restriction _{\mathcal{H}^{\rm hf}_\Lambda}.\) For \(\boldsymbol{\sigma}\in\{\uparrow,\downarrow\}^{\Lambda}\), define \[|\boldsymbol{\sigma}\rangle_{\rm f} := c_{x_1\sigma_{x_1}}^\ast \cdots c_{x_{|\Lambda|}\sigma_{x_{|\Lambda|}}}^\ast |0\rangle .\] Then \(\{ |\boldsymbol{\sigma}\rangle_{\rm f} \}_{\boldsymbol{\sigma}\in\{\uparrow,\downarrow\}^{\Lambda}}\) is an orthonormal basis of \(P\mathcal{H}^{\rm hf}_\Lambda\). We define the unitary identification \[\label{eq:def-U-spin-identification} \mathcal{U}_\Lambda: P\mathcal{H}^{\rm hf}_\Lambda \longrightarrow \mathcal{H}^{\rm spin}_\Lambda\tag{15}\] by \[\mathcal{U}_\Lambda |\boldsymbol{\sigma}\rangle_{\rm f} := \bigotimes_{j=1}^{|\Lambda|} |\sigma_{x_j}\rangle_{x_j}.\]

4.1.0.2 Spin operators and bond projector.

Let \(\sigma^{(1)},\sigma^{(2)},\sigma^{(3)}\) be the Pauli matrices. On \(\mathcal{H}^{\rm hf}_\Lambda\), define the fermionic local spin operators by \[S_x^{(a)} := \frac{1}{2} \sum_{\alpha,\beta\in\{\uparrow,\downarrow\}} c_{x\alpha}^\ast (\sigma^{(a)})_{\alpha\beta} c_{x\beta}, \qquad a\in\{1,2,3\},\] and set \(\boldsymbol{S}_x := \bigl(S_x^{(1)},S_x^{(2)},S_x^{(3)}\bigr).\)

Let \(\mathcal{U}_\Lambda: P\mathcal{H}_\Lambda^{\rm hf} \longrightarrow \mathcal{H}_\Lambda^{\rm spin}\) be the canonical identification of the singly occupied fermionic sector with the spin Hilbert space. To state the identification without ambiguity, let \(S_{x,{\rm spin}}^{(a)}\) denote the usual spin-\(\frac{1}{2}\) operator on \(\mathcal{H}_\Lambda^{\rm spin}\) acting on the tensor factor at \(x\). Then, with \(\boldsymbol{S}_{x,{\rm spin}} := \bigl( S_{x,{\rm spin}}^{(1)}, S_{x,{\rm spin}}^{(2)}, S_{x,{\rm spin}}^{(3)} \bigr),\) one has \[\label{eq:spin-identification} \mathcal{U}_\Lambda (P\boldsymbol{S}_xP) \mathcal{U}_\Lambda^\ast = \boldsymbol{S}_{x,{\rm spin}}, \qquad x\in\Lambda .\tag{16}\] After this identification is fixed, we suppress the subscript \({\rm spin}\) and write simply \(S_x^{(a)}\) and \(\boldsymbol{S}_x\) for the corresponding spin-space operators whenever the ambient Hilbert space is clear.

For a nearest-neighbour bond \(\langle x,y\rangle\), let \(P_{xy}\) be the swap operator on \(\mathcal{H}^{\rm spin}_\Lambda\), defined on the two tensor factors at \(x\) and \(y\) by \[P_{xy} \bigl( |\sigma\rangle_x\otimes|\tau\rangle_y \bigr) := |\tau\rangle_x\otimes|\sigma\rangle_y\] and extended as the identity on all other tensor factors. Define \[\label{eq:def-Bxy-spin} B_{xy} := \frac{1}{2}(\mathbb{1}-P_{xy}).\tag{17}\] The standard two-spin identities give \[\label{eq:bond-density-identities} P_{xy} = 2\boldsymbol{S}_x\cdot\boldsymbol{S}_y+\frac{1}{2}, \qquad B_{xy} = \frac{1}{4}-\boldsymbol{S}_x\cdot\boldsymbol{S}_y,\tag{18}\] where \[\boldsymbol{S}_x\cdot\boldsymbol{S}_y := \sum_{a=1}^3 S_x^{(a)}S_y^{(a)}.\] In particular, on \(\mathbb{C}^2\otimes\mathbb{C}^2\), \(B_{xy}\) is the orthogonal projection onto the singlet subspace.

4.1.0.3 Heisenberg Hamiltonian and the bond projector.

Recall the Heisenberg reference Hamiltonian with staggered field from Subsection 1.3. With \(\mathscr B_\Lambda\) denoting the set of unordered nearest-neighbour bonds in \(\Lambda\), it is given by \[H_\Lambda^{\rm Heis}(J,h) = J\sum_{\{x,y\}\in\mathscr B_\Lambda} \left( \boldsymbol{S}_x\cdot\boldsymbol{S}_y-\frac{1}{4} \right) - hM_\Lambda^{\rm spin}.\] The point used below is that the interaction term can be written in terms of the singlet bond projector \(B_{xy}\). Indeed, by 18 , \[\begin{align} H_\Lambda^{\rm Heis}(J,h) = -J\sum_{\{x,y\}\in\mathscr B_\Lambda}B_{xy} - hM_\Lambda^{\rm spin}. \label{eq:def-Heis-J} \end{align}\tag{19}\]

5 Comparison of the \(P\)-block with the Heisenberg reference model↩︎

The purpose of this section is to compare the effective Hamiltonian on the \(P\)-block with the Heisenberg reference Hamiltonian used in the pressure comparison theorem. All estimates in this section are uniform for \(|h|\le h_0.\) No positivity of the field is used here; the restriction to fixed positive field windows enters later, when pressure comparison is converted into magnetisation comparison by convexity.

Recall from 10 that the LS/SW output is \[A_\Lambda(h) = -hM_\Lambda+T_\Lambda^{(0)}+\Delta_\Lambda(h), \qquad [A_\Lambda(h),D_\Lambda]=0.\] The effective \(P\)-block Hamiltonian is \[H_{P,\Lambda}(h) := PA_\Lambda(h)P.\] Since \(PT_\Lambda^{(0)}P=0,\) we have \[\label{eq:P-block-Hamiltonian-basic} H_{P,\Lambda}(h) = P\bigl(-hM_\Lambda+\Delta_\Lambda(h)\bigr)P.\tag{20}\]

The second-order term used for the Heisenberg identification was computed through the \(\xi\)-deformation. In this section we compare the full LS/SW endpoint correction \(\Delta_\Lambda(h)\) with this second-order term, and absorb the remaining contributions into the \(P\)-block remainder. This gives a pressure comparison between the effective \(P\)-block Hamiltonian and the Heisenberg reference model.

5.1 Intermediate effective spin Hamiltonian and \(P\)-block remainder↩︎

Let \(H_\Lambda^{(2)}(h) = (\Delta_\Lambda)^{[2]}_{\xi}(h)\) be the second-order coefficient defined through the \(\xi\)-scheme. We decompose the effective \(P\)-block Hamiltonian as \[\label{eq:P-block-intermediate-decomposition} H_{P,\Lambda}(h) = P\bigl(-hM_\Lambda+H_\Lambda^{(2)}(h)\bigr)P + R_{P,\Lambda}(h),\tag{21}\] where \[\label{eq:def-P-block-remainder} R_{P,\Lambda}(h) := P\bigl(\Delta_\Lambda(h)-H_\Lambda^{(2)}(h)\bigr)P .\tag{22}\] Thus \(R_{P,\Lambda}(h)\) measures the part of the LS/SW endpoint correction which is not captured by the \(\xi\)-second-order term.

The use of \(H_\Lambda^{(2)}(h)\) here is justified by the comparison between the \(\xi\)-scheme and the auxiliary deformation. In particular, Lemma 21 identifies the explicit second-order mismatch and shows that the remaining residual is of higher order. These contributions are absorbed into the \(P\)-block remainder estimated below.

Under the spin identification \(\mathcal{U}_\Lambda: P\mathcal{H}_\Lambda^{\rm hf} \longrightarrow \mathcal{H}_\Lambda^{\rm spin},\) define the spin representative of the \(P\)-block remainder by \[\label{eq:def-spin-P-block-remainder} \mathcal{R}_{P,\Lambda}(h) := \mathcal{U}_\Lambda R_{P,\Lambda}(h)\mathcal{U}_\Lambda^\ast .\tag{23}\] By Proposition 22, \(\mathcal{U}_\Lambda P\bigl(-hM_\Lambda+H_\Lambda^{(2)}(h)\bigr)P \mathcal{U}_\Lambda^\ast = H_\Lambda^{\rm Heis}(J(h),h_{\rm eff}(h)).\) Consequently, \[\label{eq:P-block-spin-representation-with-remainder} \mathcal{U}_\Lambda H_{P,\Lambda}(h)\mathcal{U}_\Lambda^\ast = H_\Lambda^{\rm Heis}(J(h),h_{\rm eff}(h)) + \mathcal{R}_{P,\Lambda}(h).\tag{24}\]

The rest of this section compares the right-hand side of 24 with the Heisenberg reference model \(H_\Lambda^{\rm Heis}(J_0(U),h)\) with \(J_0(U)=\frac{4t^2}{U}.\) There are two contributions to control:

1. the explicit difference between the intermediate effective parameters and the reference parameters, \(H_\Lambda^{\rm Heis}(J(h),h_{\rm eff}(h)) - H_\Lambda^{\rm Heis}(J_0(U),h);\)

2. the \(P\)-block remainder \(\mathcal{R}_{P,\Lambda}(h)\).

Both are estimated uniformly for \(|h|\le h_0\). These estimates give the \(P\)-block-to-Heisenberg-reference pressure comparison used in the final assembly.

5.2 From the intermediate effective parameters to the reference model↩︎

The main theorem is stated with the Heisenberg reference Hamiltonian \(H_\Lambda^{\rm Heis}(J_0(U),h),\) Thus we compare the intermediate effective spin Hamiltonian \(H_\Lambda^{\rm Heis}(J(h),h_{\rm eff}(h))\) with the reference model \(H_\Lambda^{\rm Heis}(J_0(U),h)\).

5.3 Bound on the \(P\)-block remainder↩︎

The next estimate controls the \(P\)-block remainder uniformly on the bounded field window \(|h|\le h_0\). Its \(C^0\)-part is used for the \(P\)-block pressure comparison with the Heisenberg reference model; the \(h\)-derivative bound records the corresponding regularity of the effective \(P\)-block Hamiltonian.

Combining 24 with Proposition 24, we can write the spin representative of the effective \(P\)-block Hamiltonian as \[\label{eq:P-block-Heis-reference-plus-perturbation} \mathcal{U}_\Lambda H_{P,\Lambda}(h)\mathcal{U}_\Lambda^\ast = H_\Lambda^{\rm Heis}(J_0(U),h) + W_{P,\Lambda}(h),\tag{25}\] where \[\label{eq:def-P-block-total-perturbation} W_{P,\Lambda}(h) := G_{\Lambda,U}(h)+\mathcal{R}_{P,\Lambda}(h).\tag{26}\] Hence, by Proposition 24 and Proposition 25, \[\label{eq:P-block-total-perturbation-C1-bound} \sup_{|h|\le h_0} \|W_{P,\Lambda}(h)\|_0 + \sup_{|h|\le h_0} \|\partial_hW_{P,\Lambda}(h)\|_0 \le \varepsilon_P(U;h_0)+C(d,h_0,t)U^{-2}.\tag{27}\] In particular, the right-hand side tends to zero as \(U\to\infty\).

5.4 \(P\)-block pressure comparison with the Heisenberg reference model↩︎

Define the \(P\)-block pressure by \[\require{physics} p_{\Lambda,\beta,U}^{P}(h) := \frac{1}{\beta|\Lambda|} \log \Tr_{P\mathcal{H}_\Lambda^{\rm hf}} e^{-\beta H_{P,\Lambda}(h)}.\] Under the spin identification \(\mathcal{U}_\Lambda\), the pressure is unchanged. Therefore, by 25 , the \(P\)-block pressure is the pressure of the Heisenberg reference Hamiltonian perturbed by \(W_{P,\Lambda}(h)\).

6 Defects and comparison with the \(P\)-block↩︎

This section compares the full half-filled Hubbard model with the defect-free \(P\)-block system. The basic mechanism is the defect decomposition of the transformed Hamiltonian \[H_{*,\Lambda}(h) = UD_\Lambda+A_\Lambda(h), \qquad [A_\Lambda(h),D_\Lambda]=0,\] obtained from the LS/SW construction.

The goal is twofold. First, we prove a soft comparison between the full Hubbard pressure and the \(P\)-block pressure. Second, we derive charge-sector estimates, including suppression of double occupancy and absence of macroscopic staggered charge order.

6.1 Transformed partition function and the \(P\)-block pressure↩︎

Assume the hypotheses of Corollary 17. Recall that \[U_{\rm SW}(h)H_\Lambda^{\rm Hub}(h)U_{\rm SW}(h)^\ast = UD_\Lambda+A_\Lambda(h),\] where \(A_\Lambda(h) = -hM_\Lambda+T_\Lambda^{(0)}+\Delta_\Lambda(h).\) By unitary invariance of the trace, \[\require{physics} Z_{\Lambda,\beta,U}^{\rm Hub}(h) = \Tr_{\mathcal{H}_\Lambda^{\rm hf}} e^{-\beta(UD_\Lambda+A_\Lambda(h))}.\] The defect-free \(P\)-block partition function is \[\require{physics} Z_{\Lambda,\beta,U}^{P}(h) := \Tr_{P\mathcal{H}_\Lambda^{\rm hf}} e^{-\beta H_{P,\Lambda}(h)}, \qquad H_{P,\Lambda}(h):=P A_\Lambda(h)P.\] Equivalently, since \(P\) commutes with \(A_\Lambda(h)\), \[\require{physics} Z_{\Lambda,\beta,U}^{P}(h) = \Tr_{\mathcal{H}_\Lambda^{\rm hf}} \left( P e^{-\beta A_\Lambda(h)} \right).\] We define \[p_{\Lambda,\beta,U}^{P}(h) := \frac{1}{\beta|\Lambda|} \log Z_{\Lambda,\beta,U}^{P}(h).\]

6.2 Defect projectors and defect counting↩︎

6.2.0.1 Local defect projectors.

Define \[p_x^{(1)} := n_x-2n_{x\uparrow}n_{x\downarrow}, \qquad q_x:=\mathbb{1}-p_x^{(1)}.\] Equivalently, \[q_x=(n_x-1)^2 = \mathbb{1}-n_x+2n_{x\uparrow}n_{x\downarrow}.\] Thus \(p_x^{(1)}\) projects onto the singly occupied states at \(x\), while \(q_x\) projects onto the empty or doubly occupied states at \(x\).

For \(\mathcal{D}\subset\Lambda\), set \[Q_{\mathcal{D}} := \prod_{x\in\mathcal{D}}q_x \prod_{x\notin\mathcal{D}}p_x^{(1)}.\] Then \(\{Q_{\mathcal{D}}\}_{\mathcal{D}\subset\Lambda}\) are mutually orthogonal projections and \[\sum_{\mathcal{D}\subset\Lambda}Q_{\mathcal{D}} = \mathbb{1} \qquad \text{on }\mathcal{H}_\Lambda^{\rm hf}.\] In particular, \(Q_\varnothing=P.\)

6.2.0.2 Defect decomposition of the transformed partition function.

By Lemma 28, \[e^{-\beta UD_\Lambda} = \sum_{\mathcal{D}\subset\Lambda} e^{-\frac{\beta U}{2}|\mathcal{D}|} Q_{\mathcal{D}} \qquad \text{on }\mathcal{H}_\Lambda^{\rm hf}.\] Since \([A_\Lambda(h),D_\Lambda]=0\), we have \(e^{-\beta(UD_\Lambda+A_\Lambda(h))} = e^{-\beta UD_\Lambda}e^{-\beta A_\Lambda(h)}.\) Consequently, \[\require{physics} \label{eq:defect-decomposition-partition} Z_{\Lambda,\beta,U}^{\rm Hub}(h) = \sum_{\mathcal{D}\subset\Lambda} e^{-\frac{\beta U}{2}|\mathcal{D}|} \Tr_{\mathcal{H}_\Lambda^{\rm hf}} \left( Q_{\mathcal{D}}e^{-\beta A_\Lambda(h)} \right).\tag{28}\] The term \(\mathcal{D}=\varnothing\) is exactly the \(P\)-block partition function: \[\require{physics} \Tr_{\mathcal{H}_\Lambda^{\rm hf}} \left( Q_\varnothing e^{-\beta A_\Lambda(h)} \right) = Z_{\Lambda,\beta,U}^{P}(h).\]

6.3 Soft pressure comparison with the \(P\)-block↩︎

The defect decomposition gives a coarse but useful exponential estimate on the difference between the full Hubbard pressure and the \(P\)-block pressure.

6.4 Charge-sector consequences of the soft defect bound↩︎

We record a charge-sector consequence of the soft defect bound in the LS/SW-diagonal representation. Throughout this subsection the estimates are uniform for \(|h|\le h_0.\)

Define \[H_{*,\Lambda_L}(h):=UD_{\Lambda_L}+A_{\Lambda_L}(h)\] on \(\mathcal{H}_{\Lambda_L}^{\rm hf}\), where \(A_{\Lambda_L}(h)\) is defined in 10 . We write \[\require{physics} \omega_{\Lambda_L,\beta,U,h}^{*}(O) := \frac{ \Tr_{\mathcal{H}_{\Lambda_L}^{\rm hf}} \left( Oe^{-\beta H_{*,\Lambda_L}(h)} \right) }{ \Tr_{\mathcal{H}_{\Lambda_L}^{\rm hf}} e^{-\beta H_{*,\Lambda_L}(h)} }\] for the Gibbs state of the diagonal representative.

With \(\Gamma_U(h_0)\) as in Corollary 30, assume \(\Gamma_U(h_0)>0.\) Set \[\eta_{\rm def}^{\rm dens}(U,\beta;h_0) := \frac{4}{\beta\Gamma_U(h_0)} e^{-\beta\Gamma_U(h_0)/2}.\]

7 Proof of the main comparison theorems↩︎

This section assembles the estimates proved above. We first prove the Hubbard–Heisenberg pressure comparison on the bounded field window \(|h|\le h_0\). The magnetisation comparison is then obtained from this pressure comparison on a fixed positive field window by convexity.

7.1 Proof of the finite-volume pressure comparison↩︎

7.2 From fixed-window pressure comparison to magnetisation comparison↩︎

We next record the convexity step which converts 32 into a magnetisation comparison on \(I\).

7.3 Proof of the finite-volume fixed-field magnetisation comparison↩︎

7.4 Infinite-volume consequences↩︎

7.5 Positive fixed-field magnetisation↩︎

7.6 Proof of the charge-sector theorem↩︎

We prove Theorem 9. The only additional input needed beyond the diagonal defect-density estimate is the LS/SW dressing estimate for the normalized defect density.

For \(L\in2\mathbb{N}\), set \[\bar q_{\Lambda_L} := \frac{1}{|\Lambda_L|} \sum_{x\in\Lambda_L}q_x.\] Let \(U_{{\rm SW},\Lambda_L}(h)\) be the LS/SW product unitary from Proposition 16 and Corollary 17.

The proof of Proposition 33 is a pure LS/SW conjugation estimate for a normalized extensive observable. It is based on the summability of the generators \(S_n(h)\) and is placed in the appendix.

With \(\Gamma_U(h_0)\) as in Corollary 30, recall \(\eta_{\rm def}^{\rm dens}(U,\beta;h_0) = \frac{4}{\beta\Gamma_U(h_0)} e^{-\beta\Gamma_U(h_0)/2},\) whenever \(\Gamma_U(h_0)>0\). Define \[\varepsilon_{\rm ch}(U,\beta;h_0) := \eta_{\rm def}^{\rm dens}(U,\beta;h_0) + \rho_q(U;h_0).\]

8 BCH/LS estimates and \(D\)-graded interactions↩︎

8.1 Proofs deferred from Section 2.1↩︎

8.2 Well-definedness and bounds for \(\mathcal{I}_h\)↩︎

Throughout this subsection we fix \(h\) with \(|h|\le h_0.\)

8.3 Quantitative LS/SW iteration: deferred proofs↩︎

8.3.1 Proof of Lemma 15↩︎

Write \[H=H_0+V, \qquad H_0:=UD_\Lambda-hM_\Lambda, \qquad V:=B^{\rm diag}+B^{\rm off}.\] By the BCH expansion, \[H^+ := e^SHe^{-S} = H+\mathrm{ad}_S(H)+\sum_{n\ge2}\frac{1}{n!}\mathrm{ad}_S^{\,n}(H).\] Since \(S=\mathcal{I}_h(B^{\rm off}),\) the defining property of \(\mathcal{I}_h\) gives \(\mathrm{ad}_S(H_0) = -B^{\rm off}.\) Therefore the first-order \(D\)-off-diagonal term cancels: \(B^{\rm off}+\mathrm{ad}_S(H_0)=0.\) Thus \[H^+ = H_0 + B^{\rm diag} + \mathrm{ad}_S(B^{\rm diag}+B^{\rm off}) + R,\] where the remainder may first be written as \[R := \sum_{n\ge2}\frac{1}{n!}\mathrm{ad}_S^{\,n}(H_0+V).\]

For the estimates below, we rewrite this remainder so that \(H_0\) no longer appears. Since \(\mathrm{ad}_S(H_0)=-B^{\rm off},\) we have, for \(n\ge2\), \(\mathrm{ad}_S^{\,n}(H_0) = -\mathrm{ad}_S^{\,n-1}(B^{\rm off}).\) Hence \[R = - \sum_{m\ge1}\frac{1}{(m+1)!}\mathrm{ad}_S^{\,m}(B^{\rm off}) + \sum_{n\ge2}\frac{1}{n!} \mathrm{ad}_S^{\,n}(B^{\rm diag}+B^{\rm off}).\] In particular, \(R\) is expressed entirely in terms of \(S\), \(B^{\rm diag}\), and \(B^{\rm off}\).

Set \(B^+ := B^{\rm diag} + \mathrm{ad}_S(B^{\rm diag}+B^{\rm off}) + R.\) Then \(H^+=UD_\Lambda-hM_\Lambda+B^+.\)

BCH remainder bounds. We split \(R=R_0+R_V,\) where \[R_0 := -\sum_{m\ge1}\frac{1}{(m+1)!}\mathrm{ad}_S^{\,m}(B^{\rm off}), \qquad R_V := \sum_{n\ge2}\frac{1}{n!}\mathrm{ad}_S^{\,n}(V).\]

By Lemma 36, the map \(\mathcal{I}_h\) preserves supports. Hence \(s(S)\le s(B).\) Together with the smallness condition \(2s(B)\|S\|_\kappa\le \rho_0<1,\) this implies the hypothesis of Lemma 12 for repeated commutators with \(S\). Applying that lemma to the preceding series gives \(\|R_0\|_\kappa \le C(d,\kappa,\rho_0) \|S\|_\kappa\|B^{\rm off}\|_\kappa .\)

For \(R_V\), write \[R_V = \sum_{n\ge2}\frac{1}{n!}\mathrm{ad}_S^{\,n}(V) = \sum_{m\ge1}\frac{1}{(m+1)!} \mathrm{ad}_S^{\,m}\bigl(\mathrm{ad}_S(V)\bigr).\] By Lemma 11, \(\|\mathrm{ad}_S(V)\|_\kappa \le C(d,\kappa) \|S\|_\kappa\|V\|_\kappa .\) Applying Lemma 12 once more gives \(\|R_V\|_\kappa \le C(d,\kappa,\rho_0) \|S\|_\kappa \|\mathrm{ad}_S(V)\|_\kappa .\) Consequently, \(\|R_V\|_\kappa \le C(d,\kappa,\rho_0) \|S\|_\kappa^2 \|V\|_\kappa .\) Since \(V=B^{\rm diag}+B^{\rm off},\) we have \(\|V\|_\kappa \le \|B^{\rm diag}\|_\kappa+\|B^{\rm off}\|_\kappa.\) Combining the estimates for \(R_0\) and \(R_V\), and increasing the constant if necessary, we obtain \[\begin{align} \|R\|_\kappa \le C(d,\kappa,\rho_0) \left[ \|S\|_\kappa\|B^{\rm off}\|_\kappa + \|S\|_\kappa^2 \bigl( \|B^{\rm diag}\|_\kappa+\|B^{\rm off}\|_\kappa \bigr) \right]. \label{Inq:R95kappa} \end{align}\tag{39}\]

Extracting \(D\)-diagonal and off-diagonal parts. Since \(S\) is \(D\)-off-diagonal and \(B^{\rm diag}\) is \(D\)-diagonal, one has \(\bigl(\mathrm{ad}_S(B^{\rm diag})\bigr)^{\rm diag}=0.\) Thus \[(B^+)^{\rm off} = \mathrm{ad}_S(B^{\rm diag}) + \bigl(\mathrm{ad}_S(B^{\rm off})\bigr)^{\rm off} + R^{\rm off},\] and \[(B^+)^{\rm diag}-B^{\rm diag} = \bigl(\mathrm{ad}_S(B^{\rm off})\bigr)^{\rm diag} + R^{\rm diag}.\]

By Lemma 13, taking the \(D\)-diagonal part is contractive in \(\|\cdot\|_\kappa\), and the \(D\)-off-diagonal extraction satisfies \(\|\Phi^{\rm off}\|_\kappa \le 2\|\Phi\|_\kappa\) for every interaction \(\Phi\). Hence \[\|(B^+)^{\rm off}\|_\kappa \le \|\mathrm{ad}_S(B^{\rm diag})\|_\kappa + 2\|\mathrm{ad}_S(B^{\rm off})\|_\kappa + 2\|R\|_\kappa,\] and \[\|(B^+)^{\rm diag}-B^{\rm diag}\|_\kappa \le \|\mathrm{ad}_S(B^{\rm off})\|_\kappa + \|R\|_\kappa.\]

By Lemma 11, \[\|\mathrm{ad}_S(B^{\rm diag})\|_\kappa \le C(d,\kappa)\|S\|_\kappa\|B^{\rm diag}\|_\kappa, \qquad \|\mathrm{ad}_S(B^{\rm off})\|_\kappa \le C(d,\kappa)\|S\|_\kappa\|B^{\rm off}\|_\kappa.\] Combining these estimates and 39 , we obtain \[\|(B^+)^{\rm diag}-B^{\rm diag}\|_\kappa \le C_2(d,\kappa,\rho_0) \left[ \|S\|_\kappa\|B^{\rm off}\|_\kappa + \|S\|_\kappa^2 \bigl( \|B^{\rm diag}\|_\kappa+\|B^{\rm off}\|_\kappa \bigr) \right],\] which proves ?? .

For the off-diagonal part, the same estimates give \[\|(B^+)^{\rm off}\|_\kappa \le C(d,\kappa,\rho_0) \left[ \|S\|_\kappa\|B^{\rm diag}\|_\kappa + \|S\|_\kappa\|B^{\rm off}\|_\kappa + \|S\|_\kappa^2 \bigl( \|B^{\rm diag}\|_\kappa+\|B^{\rm off}\|_\kappa \bigr) \right].\] If \(S=0\), the desired estimate is immediate. Otherwise the smallness assumption \(2s(B)\|S\|_\kappa\le \rho_0<1\) implies \(\|S\|_\kappa\le1\), since \(s(B)\ge1\). Therefore the quadratic term is bounded by the corresponding linear term. Enlarging the constant gives \[\|(B^+)^{\rm off}\|_\kappa \le C_1(d,\kappa,\rho_0) \|S\|_\kappa \bigl( \|B^{\rm diag}\|_\kappa+\|B^{\rm off}\|_\kappa \bigr),\] which proves ?? .

Bound on the generator. Since \(S=\mathcal{I}_h(B^{\rm off}),\) Lemma 36 gives, whenever \(U>h_0s(B)\), \[\|S\|_\kappa \le \frac{2s(B)}{U-h_0s(B)} \|B^{\rm off}\|_\kappa.\] This is ?? . If in addition \(\frac{h_0}{U}s(B)\le\frac{1}{2},\) then \(U-h_0s(B)\ge \frac{U}{2},\) and hence \(\|S\|_\kappa \le \frac{4s(B)}{U}\|B^{\rm off}\|_\kappa,\) which is ?? . 0◻

8.3.2 Proof of Proposition 16↩︎

Assume \(U\ge U_\ast\). By the large-\(U\) convention and Lemma 15, there is a constant \(C_{\mathcal{I}}=C_{\mathcal{I}}(d,\kappa)\) such that, for every step of the iteration, \[\label{eq:proof-LS-Sn-bound} \|S_n\|_\kappa \le \frac{C_{\mathcal{I}}}{U} \|(B_n)^{\rm off}\|_\kappa .\tag{40}\] The constants \(C_1,C_2\) below are those in Lemma 15, with a fixed choice of \(\rho_0\in(0,1)\).

8.3.2.1 Step 1: choice of \(\varepsilon_\ast\) and bootstrap region.

We use the bootstrap conditions \[\label{eq:spec-bootstrap} \|(B_n)^{\rm off}\|_\kappa\le \varepsilon_\ast U, \qquad \|(B_n)^{\rm diag}\|_\kappa\le 2\varepsilon_\ast U .\tag{41}\] Choose \(\varepsilon_\ast>0\) sufficiently small so that \[\begin{align} &2s_\ast C_{\mathcal{I}}\varepsilon_\ast\le \rho_0, \tag{42} \\ &q:= 3C_1(d,\kappa,\rho_0)C_{\mathcal{I}}\varepsilon_\ast \le \frac{1}{2}. \tag{43} \end{align}\] Here \(s_\ast\) denotes the fixed local-size constant entering the one-step BCH estimate. Finally, enlarge \(C_\ast=C_\ast(d,\kappa)\), if necessary, and shrink \(\varepsilon_\ast\) further so that \[\label{eq:spec-eps-choice-3} C_\ast\frac{\varepsilon_\ast}{1-q^2}\le 1.\tag{44}\] This last condition will keep the diagonal part inside the bootstrap region.

8.3.2.2 Step 2: bootstrap invariance.

Assume that 41 holds at step \(n\). Then 40 gives \(2s_\ast\|S_n\|_\kappa \le 2s_\ast C_{\mathcal{I}}\varepsilon_\ast \le \rho_0,\) so Lemma 15 applies at step \(n\).

For the off-diagonal part, Lemma 15 and 40 give \[\begin{align} \|(B_{n+1})^{\rm off}\|_\kappa &\le C_1(d,\kappa,\rho_0) \|S_n\|_\kappa \bigl( \|(B_n)^{\rm diag}\|_\kappa + \|(B_n)^{\rm off}\|_\kappa \bigr) \\ &\le C_1(d,\kappa,\rho_0) \frac{C_{\mathcal{I}}}{U} \|(B_n)^{\rm off}\|_\kappa \bigl( 2\varepsilon_\ast U+\varepsilon_\ast U \bigr) \\ &= 3C_1(d,\kappa,\rho_0)C_{\mathcal{I}} \varepsilon_\ast \|(B_n)^{\rm off}\|_\kappa . \end{align}\] By the definition of \(q\) in 43 , this yields \(\|(B_{n+1})^{\rm off}\|_\kappa \le q\|(B_n)^{\rm off}\|_\kappa.\) This proves ?? and keeps \(\|(B_{n+1})^{\rm off}\|_\kappa\le \varepsilon_\ast U .\)

For the diagonal increment, Lemma 15 gives \[\begin{align} \|(B_{n+1})^{\rm diag}-(B_n)^{\rm diag}\|_\kappa \le C_2(d,\kappa,\rho_0) \Bigl[ \|S_n\|_\kappa\|(B_n)^{\rm off}\|_\kappa + \|S_n\|_\kappa^2 \bigl( \|(B_n)^{\rm diag}\|_\kappa + \|(B_n)^{\rm off}\|_\kappa \bigr) \Bigr]. \end{align}\] Using 40 and the bootstrap bounds, we obtain \[\begin{align} \|(B_{n+1})^{\rm diag}-(B_n)^{\rm diag}\|_\kappa &\le C_2(d,\kappa,\rho_0) \left[ C_{\mathcal{I}} + 3C_{\mathcal{I}}^2\varepsilon_\ast \right] \frac{\|(B_n)^{\rm off}\|_\kappa^2}{U}. \end{align}\] After enlarging \(C_\ast(d,\kappa)\), this gives \(\|(B_{n+1})^{\rm diag}-(B_n)^{\rm diag}\|_\kappa \le C_\ast \frac{\|(B_n)^{\rm off}\|_\kappa^2}{U}.\) This is ?? .

Summing the diagonal increments and using the contraction of the off-diagonal part gives \[\begin{align} \|(B_n)^{\rm diag}\|_\kappa &\le \|(B_0)^{\rm diag}\|_\kappa + C_\ast\frac{1}{U} \sum_{j=0}^{n-1} \|(B_j)^{\rm off}\|_\kappa^2 \\ &\le \varepsilon_\ast U + C_\ast \frac{\|(B_0)^{\rm off}\|_\kappa^2}{U} \sum_{j=0}^{\infty}q^{2j} \\ &\le \varepsilon_\ast U + C_\ast \frac{\varepsilon_\ast^2U}{1-q^2}. \end{align}\] By 44 , the last term is at most \(\varepsilon_\ast U\). Hence \(\|(B_n)^{\rm diag}\|_\kappa\le 2\varepsilon_\ast U.\) Thus the bootstrap region 41 is invariant, and the same \(\varepsilon_\ast\) works for all \(n\).

8.3.2.3 Step 3: convergence and the \(B_\infty\)-bound.

From ?? and 40 , we get \[\sum_{n\ge0}\|S_n\|_\kappa \le \frac{C_{\mathcal{I}}}{U} \sum_{n\ge0}\|(B_n)^{\rm off}\|_\kappa \le \frac{C_{\mathcal{I}}}{U} \frac{1}{1-q} \|(B_0)^{\rm off}\|_\kappa .\] After enlarging \(C_\ast\), this gives \[\label{eq:proof-generator-summability} \sum_{n\ge0}\|S_n\|_\kappa \le C_\ast \frac{\|(B_0)^{\rm off}\|_\kappa}{U}.\tag{45}\]

In finite volume, this summability implies the existence of the product unitary. Indeed, by the elementary bound from interaction norm to finite-volume operator norm, \(\|S_{n,\Lambda}\| \le |\Lambda|\|S_n\|_\kappa,\) and hence, for each fixed finite \(\Lambda\), \(\sum_{n\ge0}\|S_{n,\Lambda}\|<\infty.\) For notational simplicity we write \(S_n\) for the corresponding finite-volume operator in the following estimate.

Set \(U_N:=e^{S_{N-1}}\cdots e^{S_0}.\) Since each \(S_n\) is anti-self-adjoint, \(e^{S_n}\) is unitary. Moreover, \(U_{N+1}=e^{S_N}U_N,\) and therefore \(\|U_{N+1}-U_N\| = \|(e^{S_N}-\mathbb{1})U_N\| = \|e^{S_N}-\mathbb{1}\|.\) Using \(\|e^{S_N}-\mathbb{1}\| \le e^{\|S_N\|}\|S_N\|,\) we obtain, for \(M>N\), \[\|U_M-U_N\| \le \sum_{k=N}^{M-1}\|U_{k+1}-U_k\| \le \exp\left( \sum_{j\ge0}\|S_j\| \right) \sum_{k=N}^{M-1}\|S_k\|.\] Since \(\sum_{n\ge0}\|S_n\|<\infty\) in finite volume, the right-hand side tends to \(0\) as \(N\to\infty\). Thus \((U_N)_N\) is Cauchy in operator norm, and we may define \[U_{\rm SW} := \lim_{N\to\infty}U_N = \lim_{N\to\infty}e^{S_{N-1}}\cdots e^{S_0}.\] Since \(U_N\) is unitary for every \(N\), the norm limit \(U_{\rm SW}\) is also unitary.

Next, ?? and ?? imply \[\sum_{n\ge0} \|(B_{n+1})^{\rm diag}-(B_n)^{\rm diag}\|_\kappa \le C_\ast \frac{1}{U} \sum_{n\ge0} \|(B_n)^{\rm off}\|_\kappa^2 <\infty.\] Hence \((B_n)^{\rm diag}\) converges in \(\|\cdot\|_\kappa\) to a \(D_\Lambda\)-diagonal limit, which we denote by \(B_\infty(h)\). Also, by ?? , \(\|(B_n)^{\rm off}\|_\kappa \le q^n\|(B_0)^{\rm off}\|_\kappa \longrightarrow0.\) Therefore \(B_n = (B_n)^{\rm diag}+(B_n)^{\rm off} \longrightarrow B_\infty(h)\) in \(\|\cdot\|_\kappa\). For each fixed finite volume, this also implies operator-norm convergence of the corresponding finite-volume sums.

By construction, \(H^{(N)} = U_NH^{(0)}U_N^\ast = UD_\Lambda-hM_\Lambda+B_N .\) Passing to the limit in finite-volume operator norm gives \(U_{\rm SW}H^{(0)}U_{\rm SW}^{\ast} = UD_\Lambda-hM_\Lambda+B_\infty(h).\) Since each \((B_n)^{\rm diag}\) is \(D_\Lambda\)-diagonal and the diagonal subspace is closed under the \(\|\cdot\|_\kappa\)-limit, \(B_\infty(h)\) is \(D_\Lambda\)-diagonal.

Finally, summing ?? over \(n\ge0\) and using \(\|(B_n)^{\rm off}\|_\kappa \le q^n\|(B_0)^{\rm off}\|_\kappa,\) we obtain \[\|B_\infty(h)-(B_0)^{\rm diag}\|_\kappa \le C_\ast \frac{1}{U} \sum_{n\ge0} \|(B_n)^{\rm off}\|_\kappa^2 \le C_\ast \frac{\|(B_0)^{\rm off}\|_\kappa^2}{U} \sum_{n\ge0}q^{2n}.\] After enlarging \(C_\ast\), this gives \(\|B_\infty(h)-(B_0)^{\rm diag}\|_\kappa \le C_\ast \frac{\|(B_0)^{\rm off}\|_\kappa^2}{U}.\) This is ?? . 0◻

8.3.3 Proof of Corollary 17↩︎

By the choice of \(C_T(d,\kappa)\), one has \(\|(T_\Lambda)^{\rm off}\|_\kappa\le C_T(d,\kappa)|t|,\) and \(\|T_\Lambda^{(0)}\|_\kappa\le C_T(d,\kappa)|t|.\) Since \(\frac{|t|}{U}\le \varepsilon_{\rm SW}(d,\kappa) \le \frac{\varepsilon_\ast(d,\kappa)}{2C_T(d,\kappa)},\) we have \[\|(B_{0,\Lambda})^{\rm off}\|_\kappa = \|(T_\Lambda)^{\rm off}\|_\kappa \le \varepsilon_\ast U, \qquad \|(B_{0,\Lambda})^{\rm diag}\|_\kappa = \|T_\Lambda^{(0)}\|_\kappa \le \varepsilon_\ast U.\] Thus the hypotheses of Proposition 16 hold for the initial datum \[B_{0,\Lambda}=T_\Lambda, \qquad H_\Lambda^{(0)}(h) = H_\Lambda^{\rm Hub}(h) = UD_\Lambda-hM_\Lambda+T_\Lambda .\] Hence the LS/SW iteration is well-defined for all \(n\ge0\), and the bounds ?? –?? hold for the sequence generated from \(B_{0,\Lambda}=T_\Lambda\).

In particular, Proposition 16 gives a product unitary \(U_{\rm SW}(h)\) and a \(D_\Lambda\)-diagonal interaction \(B_\infty(h)\) such that \(U_{\rm SW}(h)H_\Lambda^{\rm Hub}(h)U_{\rm SW}(h)^\ast = UD_\Lambda-hM_\Lambda+B_\infty(h).\) Define \(\Delta_\Lambda(h):=B_\infty(h)-T_\Lambda^{(0)}.\) Since both \(B_\infty(h)\) and \(T_\Lambda^{(0)}\) are \(D_\Lambda\)-diagonal, one has \([\Delta_\Lambda(h),D_\Lambda]=0.\) Therefore \[U_{\rm SW}(h)H_\Lambda^{\rm Hub}(h)U_{\rm SW}(h)^\ast = UD_\Lambda-hM_\Lambda+T_\Lambda^{(0)}+\Delta_\Lambda(h).\]

Moreover, by ?? , \[\|\Delta_\Lambda(h)\|_\kappa = \|B_\infty(h)-T_\Lambda^{(0)}\|_\kappa \le C_\ast(d,\kappa) \frac{\|(T_\Lambda)^{\rm off}\|_\kappa^2}{U}.\] Using \(\|(T_\Lambda)^{\rm off}\|_\kappa\le C_T(d,\kappa)|t|,\) we obtain \(\|\Delta_\Lambda(h)\|_\kappa \le C_\ast(d,\kappa)C_T(d,\kappa)^2 \frac{t^2}{U}.\) This gives the stated estimate with \(\widetilde{C}(d,\kappa) := C_\ast(d,\kappa)C_T(d,\kappa)^2.\)

Finally, ?? gives \[\sum_{n\ge0}\|S_n(h)\|_\kappa \le C_\ast(d,\kappa) \frac{\|(T_\Lambda)^{\rm off}\|_\kappa}{U} \le C_\ast(d,\kappa)C_T(d,\kappa) \frac{|t|}{U}.\] After enlarging \(C_\ast(d,\kappa)\) once more, this becomes \(\sum_{n\ge0}\|S_n(h)\|_\kappa \le C_\ast(d,\kappa) \frac{|t|}{U}.\) This completes the proof. 0◻

8.4 Dressing of the normalized defect density↩︎

We prove Proposition 33. The only point is that the observable is normalized by the volume. This normalization turns the commutator with a short-range interaction into a volume-uniform quantity.

For a finite torus \(\Lambda_L\), set \[\bar q_{\Lambda_L} := \frac{1}{|\Lambda_L|} \sum_{x\in\Lambda_L}q_x .\]

8.5 \(h\)-derivative estimates for LS/SW increments↩︎

The estimates in this subsection are used to control the \(h\)-dependence of the \(\lambda\)-expansion remainders in Appendix 10. They are not used to prove a direct \(C^1\) comparison between the full Hubbard pressure and the \(P\)-block pressure.

8.6 Second-order computations for the \(\xi\)- and \(\lambda\)-schemes↩︎

8.6.1 Proof of Lemma 20↩︎

Set \[H_{\rm d}:=UD_\Lambda-hM_\Lambda, \qquad T^{(0)}:=T_\Lambda^{(0)}, \qquad V:=(T_\Lambda)^{\rm off}.\] Thus \[H_\Lambda(h;\xi)=H_{\rm d}+\xi T^{(0)}+\xi V, \qquad [H_{\rm d},D_\Lambda]=0.\]

Step 1. Expansion at \(\xi=0\). We use the \(\xi\)-expansion introduced in Subsection 3.1. In particular, \[B_\infty(h;\xi) = \xi T^{(0)}+\Delta_\Lambda(h;\xi), \qquad \Delta_\Lambda(h;\xi) = \sum_{m\ge2}\xi^m(\Delta_\Lambda)^{[m]}_{\xi}(h),\] and \[H_\Lambda^{(2)}(h) = (\Delta_\Lambda)^{[2]}_{\xi}(h).\] Thus it remains only to compute the \(\xi^2\)-coefficient of the diagonal correction. This coefficient is determined by the first LS/SW step, since the off-diagonal part left after that step is already of order \(\xi^2\), and later diagonal increments are at least quadratic in that remaining off-diagonal part.

Step 2. The first LS/SW step and the \(\xi^2\)-diagonal term. Let \[S^{[1]}:=\mathcal{I}_h(V), \qquad S_0(\xi):=\mathcal{I}_h\bigl((\xi T_\Lambda)^{\rm off}\bigr) = \xi S^{[1]}.\] By the defining property of \(\mathcal{I}_h\), \(\mathrm{ad}_{H_{\rm d}}(S^{[1]})=V,\) and \(\mathrm{ad}_{S^{[1]}}(H_{\rm d})=-V.\) Set \[H_0(\xi):=H_\Lambda(h;\xi), \qquad H_1(\xi):=e^{S_0(\xi)}H_0(\xi)e^{-S_0(\xi)}.\] By the BCH expansion, with the remainder controlled in \(\|\cdot\|_\kappa\) by Lemma 12 and Lemma 11, we have \[H_1(\xi) = H_0(\xi) + \mathrm{ad}_{S_0(\xi)}(H_0(\xi)) + \frac{1}{2} \mathrm{ad}_{S_0(\xi)}^2(H_0(\xi)) + O(|\xi|^3)\] in \(\|\cdot\|_\kappa\). Substituting \(H_0(\xi)=H_{\rm d}+\xi T^{(0)}+\xi V\) and \(S_0(\xi)=\xi S^{[1]},\) we obtain \[H_1(\xi) = H_{\rm d} + \xi T^{(0)} + \xi V + \xi \mathrm{ad}_{S^{[1]}}(H_{\rm d}) + \xi^2\mathrm{ad}_{S^{[1]}}(T^{(0)}) + \xi^2\mathrm{ad}_{S^{[1]}}(V) + \frac{1}{2}\xi^2\mathrm{ad}_{S^{[1]}}^2(H_{\rm d}) + O(|\xi|^3).\] The \(\xi\)-linear off-diagonal terms cancel, because \(\xi V+\xi\mathrm{ad}_{S^{[1]}}(H_{\rm d})=0.\) Moreover, \(\mathrm{ad}_{S^{[1]}}^2(H_{\rm d}) = -\mathrm{ad}_{S^{[1]}}(V).\) Hence \[H_1(\xi) = H_{\rm d} + \xi T^{(0)} + \xi^2\mathrm{ad}_{S^{[1]}}(T^{(0)}) + \frac{1}{2}\xi^2\mathrm{ad}_{S^{[1]}}(V) + O(|\xi|^3)\] in \(\|\cdot\|_\kappa\).

Since \(T^{(0)}\) is \(D_\Lambda\)-diagonal and \(S^{[1]}\) is \(D_\Lambda\)-off-diagonal, the \(D\)-grading rule gives \(\bigl(\mathrm{ad}_{S^{[1]}}(T^{(0)})\bigr)^{\rm diag}=0.\) Therefore \[\label{eq:first-step-xi2-diag} H_1(\xi)^{\rm diag} = H_{\rm d} + \xi T^{(0)} + \frac{1}{2}\xi^2 \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag} + O(|\xi|^3).\tag{48}\] Also, \[\label{eq:first-step-off-Oxi2} H_1(\xi)^{\rm off}=O(\xi^2) \qquad \text{in }\|\cdot\|_\kappa,\tag{49}\] because the \(\xi\)-linear off-diagonal term is cancelled in the first step.

Step 3. Later LS/SW steps do not change the \(\xi^2\)-diagonal coefficient. Let \(H_n(\xi)\) be the Hamiltonian after \(n\) LS/SW steps, and write \(H_n(\xi)=H_{\rm d}+B_n(\xi).\) By 49 , \(B_1(\xi)^{\rm off}=O(|\xi|^2)\) in \(\|\cdot\|_\kappa\). Applying the contraction estimate ?? from Proposition 16 from step \(1\) onward, we get \(\|B_n(\xi)^{\rm off}\|_\kappa \le C q^{n-1}|\xi|^2, \, n\ge1,\) for \(|\xi|\) sufficiently small. Here \(C\) is uniform in \(n\), \(\Lambda\), and \(|h|\le h_0\).

By the diagonal increment estimate ?? , \(\|B_{n+1}(\xi)^{\rm diag}-B_n(\xi)^{\rm diag}\|_\kappa \le C_\ast \frac{\|B_n(\xi)^{\rm off}\|_\kappa^2}{U}.\) Therefore \[\sum_{n\ge1} \|B_{n+1}(\xi)^{\rm diag}-B_n(\xi)^{\rm diag}\|_\kappa \le C|\xi|^4 \sum_{n\ge1}q^{2(n-1)} = O(|\xi|^4)\] in \(\|\cdot\|_\kappa\). Hence the total contribution of all LS/SW steps after the first one to the diagonal part is \(O(|\xi|^4)\). In particular, these later steps do not change the \(\xi^2\)-coefficient of the limiting diagonal correction.

Step 4. Identification of the second-order coefficient. By 48 , \[B_1(\xi)^{\rm diag} = \xi T^{(0)} + \frac{1}{2}\xi^2 \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag} + O(|\xi|^3)\] in \(\|\cdot \|_{\kappa}\). By Step 3, the diagonal increments from all later steps are \(O(|\xi|^4)\). Hence \[B_\infty(h;\xi) = \xi T^{(0)} + \frac{1}{2}\xi^2 \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag} + O(|\xi|^3).\] Since, by definition, \(B_\infty(h;\xi) = \xi T^{(0)}+\Delta_\Lambda(h;\xi),\) we obtain \[\Delta_\Lambda(h;\xi) = \frac{1}{2}\xi^2 \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag} + O(|\xi|^3).\] Therefore \((\Delta_\Lambda)^{[2]}_{\xi}(h) = \frac{1}{2} \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag}.\) Since \(V=(T_\Lambda)^{\rm off}\), this proves the asserted formula. 0◻

8.6.2 Proof of Lemma 21↩︎

Write \(T_\Lambda=T_\Lambda^{(0)}+V\) with \(V:=(T_\Lambda)^{\rm off},\) and set \(H_0(h):=UD_\Lambda-hM_\Lambda.\)

Second-order coefficient at the first LS/SW step. By Lemma 20, we have \((\Delta_\Lambda)^{[2]}_{\xi}(h) = \frac{1}{2} \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag}.\) For the \(\lambda\)-scheme, \(H_\Lambda(h;\lambda) = H_0(h)+T_\Lambda^{(0)}+\lambda V.\) The first generator is \(S_0(\lambda)=\lambda\mathcal{I}_h(V),\) and the defining property of \(\mathcal{I}_h\) gives \(\mathrm{ad}_{\mathcal{I}_h(V)}(H_0(h))=-V.\)

By the BCH expansion, with the remainder controlled in \(\|\cdot\|_0\) by Lemma 12 and Lemma 11, we have \[e^{S_0(\lambda)} H_\Lambda(h;\lambda) e^{-S_0(\lambda)} = H_\Lambda(h;\lambda) + \mathrm{ad}_{S_0(\lambda)}(H_\Lambda(h;\lambda)) + \frac{1}{2} \mathrm{ad}_{S_0(\lambda)}^2(H_\Lambda(h;\lambda)) + O(|\lambda|^3)\] in \(\|\cdot\|_0\). Substituting \(S_0(\lambda)=\lambda\mathcal{I}_h(V)\) and using \(\mathrm{ad}_{\mathcal{I}_h(V)}(H_0(h))=-V,\) we obtain \[\begin{align} e^{S_0(\lambda)} H_\Lambda(h;\lambda) e^{-S_0(\lambda)} &= H_0(h)+T_\Lambda^{(0)} +\lambda\,\mathrm{ad}_{\mathcal{I}_h(V)}(T_\Lambda^{(0)}) \\ &\quad +\lambda^2 \left[ \mathrm{ad}_{\mathcal{I}_h(V)}(V) + \frac{1}{2} \mathrm{ad}_{\mathcal{I}_h(V)}^2(H_0(h)) + \frac{1}{2} \mathrm{ad}_{\mathcal{I}_h(V)}^2(T_\Lambda^{(0)}) \right] + O(|\lambda|^3). \end{align}\] Since \(\mathrm{ad}_{\mathcal{I}_h(V)}^2(H_0(h)) = -\mathrm{ad}_{\mathcal{I}_h(V)}(V),\) the \(\lambda^2\)-coefficient becomes \(\frac{1}{2}\mathrm{ad}_{\mathcal{I}_h(V)}(V) + \frac{1}{2}\mathrm{ad}_{\mathcal{I}_h(V)}^2(T_\Lambda^{(0)}).\) Taking the \(D_\Lambda\)-diagonal part gives \(\frac{1}{2} \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}(V)\bigr)^{\rm diag} + \frac{1}{2} \bigl(\mathrm{ad}_{\mathcal{I}_h(V)}^2(T_\Lambda^{(0)})\bigr)^{\rm diag}.\) The first term is exactly \((\Delta_\Lambda)^{[2]}_{\xi}(h),\) and the second term is \(\mathcal{R}^{(2)}_\Lambda(h).\)

Thus, after the first LS/SW step, the \(\lambda^2\)-diagonal coefficient is \((\Delta_\Lambda)^{[2]}_{\xi}(h) + \mathcal{R}^{(2)}_\Lambda(h).\) However, unlike the \(\xi\)-scheme, the first transformed Hamiltonian still has a \(\lambda\)-linear off-diagonal term, \(\lambda\, \mathrm{ad}_{\mathcal{I}_h(V)}(T_\Lambda^{(0)}).\) The remaining contribution to the \(\lambda^2\)-diagonal coefficient comes from the subsequent LS/SW steps. We denote this total later contribution by \(\mathcal{E}^{(2)}_\Lambda(h)\). Equivalently, \[\mathcal{E}^{(2)}_\Lambda(h) = (\Delta_\Lambda)^{[2]}_{\lambda}(h) - (\Delta_\Lambda)^{[2]}_{\xi}(h) - \mathcal{R}^{(2)}_\Lambda(h).\] This gives the asserted decomposition.

Definition and estimate of \(\mathcal{E}^{(2)}_\Lambda(h)\). Let \(B_n(h;\lambda)\) be the interaction produced by the LS/SW iteration applied to \(H_\Lambda(h;\lambda) = H_0(h)+T_\Lambda^{(0)}+\lambda V .\) Thus the initial interaction is \[B_0(h;\lambda) := T_\Lambda^{(0)}+\lambda V, \qquad B_0(h;\lambda)^{\rm diag}=T_\Lambda^{(0)}, \qquad B_0(h;\lambda)^{\rm off}=\lambda V.\] After \(n\) steps we write \[H_n(h;\lambda) = H_0(h) + B_n(h;\lambda) = H_0(h) + (B_n(h;\lambda))^{\rm diag} + (B_n(h;\lambda))^{\rm off}.\] Set \[\delta_n(h;\lambda) := (B_{n+1}(h;\lambda))^{\rm diag} - (B_n(h;\lambda))^{\rm diag}.\] Since the limiting interaction \(B_\infty(h;\lambda)\) is \(D_\Lambda\)-diagonal and, by definition, \(B_\infty(h;\lambda) = T_\Lambda^{(0)}+\Delta_\Lambda(h;\lambda),\) we have \[\Delta_\Lambda(h;\lambda) = B_\infty(h;\lambda)-T_\Lambda^{(0)} = \sum_{n\ge0}\delta_n(h;\lambda),\] with convergence in the interaction norm, for \(|\lambda|\) sufficiently small.

By the analytic dependence of the LS/SW iteration on \(\lambda\), each \(\delta_n(h;\lambda)\) is analytic near \(\lambda=0\). Moreover, \(\delta_n(h;\lambda)\) has no linear term, because the diagonal increment is quadratic in the off-diagonal input and \(B_0(h;\lambda)^{\rm off}=\lambda V.\) We therefore write \[\delta_n(h;\lambda) = \sum_{m\ge2}\lambda^m\delta_n^{[m]}(h).\] By the computation of the first step above, the \(\lambda^2\)-coefficient of \(\delta_0(h;\lambda)\) is \((\Delta_\Lambda)^{[2]}_{\xi}(h) + \mathcal{R}^{(2)}_\Lambda(h).\) Hence the remaining contribution to the \(\lambda^2\)-coefficient is \[\mathcal{E}^{(2)}_\Lambda(h) := \sum_{n\ge1}\delta_n^{[2]}(h).\] Each \(\delta_n^{[2]}(h)\) is \(D_\Lambda\)-diagonal, and therefore so is \(\mathcal{E}^{(2)}_\Lambda(h)\). We now estimate this series. We use the one-step estimates with \(\kappa=0\). Choose \(r_0\in(0,1]\) so small that the LS/SW contraction estimates apply uniformly for \(|\lambda|\le r_0\). By ?? , \[\|\delta_n(h;\lambda)\|_0 \le C_\ast(d) \frac{\|(B_n(h;\lambda))^{\rm off}\|_0^2}{U}.\] For \(n\ge1\), the contraction estimate ?? gives \[\|(B_n(h;\lambda))^{\rm off}\|_0 \le q^{n-1} \|(B_1(h;\lambda))^{\rm off}\|_0 .\]

It remains to bound the first off-diagonal remainder. Put \(S:=\mathcal{I}_h(V).\) Since \(S_0(\lambda)=\lambda S\) and \(\mathrm{ad}_S(H_0(h))=-V,\) the first LS/SW step cancels the leading off-diagonal term \(\lambda V\). More precisely, after expanding by BCH and extracting the terms of order at most \(\lambda^2\), we get \[(B_1(h;\lambda))^{\rm off} = \lambda \bigl( \mathrm{ad}_S(T_\Lambda^{(0)}) \bigr) + R_1(h;\lambda),\] where \[R_1(h;\lambda) = \left[ \lambda^2 \left( \frac{1}{2}\mathrm{ad}_S(V) + \frac{1}{2}\mathrm{ad}_S^2(T_\Lambda^{(0)}) \right) + R_{\ge3}(h;\lambda) \right]^{\rm off}.\] Here \(R_{\ge3}(h;\lambda)\) denotes the part of the BCH expansion of order at least \(\lambda^3\). To see this, the \(H_0(h)\)-tail is first rewritten using \(\mathrm{ad}_S(H_0(h))=-V.\) Thus, for the \(H_0(h)\)-part, we have \(\mathrm{ad}_S^r(H_0(h)) = -\mathrm{ad}_S^{r-1}(V)\, (r\ge1),\) so no norm of \(H_0(h)\) appears in the tail estimates.

By Lemma 36 and the hopping bounds, \[\|S\|_0 = \|\mathcal{I}_h(V)\|_0 \le C(d)\frac{|t|}{U}, \qquad \|V\|_0+\|T_\Lambda^{(0)}\|_0 \le C(d)|t|.\] Using the interaction commutator bound Lemma 11, we get \[\|\mathrm{ad}_S(T_\Lambda^{(0)})\|_0 \le C(d)\frac{|t|^2}{U}, \qquad \|\mathrm{ad}_S(V)\|_0 \le C(d)\frac{|t|^2}{U}, \qquad \|\mathrm{ad}_S^2(T_\Lambda^{(0)})\|_0 \le C(d)\frac{|t|^3}{U^2}.\] After increasing the strong-coupling threshold if necessary, \(\|\mathrm{ad}_S^2(T_\Lambda^{(0)})\|_0 \le C(d)\frac{|t|^2}{U}.\)

It remains to estimate \(R_{\ge3}(h;\lambda)\). Since the terms up to order \(\lambda^2\) have already been extracted, the BCH tail is a shifted tail beginning at order \(\lambda^3\). Applying Lemma 12 in this shifted form, together with Lemma 11, gives \[\|R_{\ge3}(h;\lambda)\|_0 \le C(d)|\lambda|^3 \left( \|S\|_0^2\|V\|_0 + \|S\|_0^3\|T_\Lambda^{(0)}\|_0 \right).\] Using the bounds above and \(|\lambda|\le r_0\le1\), this implies \(\|R_{\ge3}(h;\lambda)\|_0 \le C(d)|\lambda|^2\frac{|t|^2}{U}.\) Therefore \[\|R_1(h;\lambda)\|_0 \le C(d)|\lambda|^2\frac{|t|^2}{U}\] uniformly for \(|h|\le h_0\) and \(|\lambda|\le r_0\).

Combining the leading off-diagonal term with this remainder, we obtain \[\|(B_1(h;\lambda))^{\rm off}\|_0 \le C(d)|\lambda|\frac{|t|^2}{U}.\] Consequently, for \(n\ge1\), \[\|\delta_n(h;\lambda)\|_0 \le C(d)q^{2(n-1)} |\lambda|^2 \frac{|t|^4}{U^3}.\]

We now use the analytic dependence on the auxiliary parameter. Although \(\lambda\) is real in the original deformation, we temporarily regard \(\lambda\) as a complex parameter in order to extract Taylor coefficients. The operations entering the LS/SW construction in this argument, namely the \(D\)-grading, the homological map \(\mathcal{I}_h\), commutators, and the BCH series, are complex-linear or multilinear operations on the complex Banach space of interactions. Moreover, the convergence estimates used above are norm estimates depending only on \(|\lambda|\).

Therefore, after decreasing \(r_0>0\) if necessary, the interactions \(\delta_n(h;\lambda)\) extend to Banach-valued holomorphic functions of \(\lambda\) on \(|\lambda|<r_0\), with the same bounds uniformly on \(|\lambda|\le r_0\). In this coefficient estimate we do not use self-adjointness of \(H_\Lambda(h;\lambda)\) or unitarity of the conjugations.

By Cauchy’s estimate for Banach-valued holomorphic functions, applied on the circle \(|\lambda|=r_0\), we obtain \[\|\delta_n^{[2]}(h)\|_0 \le r_0^{-2} \sup_{|\lambda|=r_0} \|\delta_n(h;\lambda)\|_0 \le C(d)q^{2(n-1)} \frac{|t|^4}{U^3}.\] Summing over \(n\ge1\), we get \[\|\mathcal{E}^{(2)}_\Lambda(h)\|_0 \le C(d) \frac{|t|^4}{U^3} \sum_{n\ge1}q^{2(n-1)} \le C_{E,0}(d)\frac{|t|^4}{U^3}.\] This proves the desired estimate for \(\mathcal{E}^{(2)}_\Lambda(h)\), uniformly in \(\Lambda\) and \(|h|\le h_0\).

Soft \(0\)-norm bound for \(\mathcal{R}^{(2)}_\Lambda(h)\). By Lemma 13, the \(D_\Lambda\)-diagonal extraction is contractive in \(\|\cdot\|_0\). Hence \[\|\mathcal{R}^{(2)}_\Lambda(h)\|_0 \le \frac{1}{2} \left\| \mathrm{ad}_{\mathcal{I}_h(V)} \left( \mathrm{ad}_{\mathcal{I}_h(V)}(T_\Lambda^{(0)}) \right) \right\|_0 .\] Applying the interaction commutator bound Lemma 11 twice, with \(\kappa=0\), gives \[\|\mathcal{R}^{(2)}_\Lambda(h)\|_0 \le C(d) \|\mathcal{I}_h(V)\|_0^2 \|T_\Lambda^{(0)}\|_0 .\] By Lemma 36, together with the large-\(U\) convention and the hopping bounds, we have uniformly in \(\Lambda\) and \(|h|\le h_0\), \(\|\mathcal{I}_h(V)\|_0 \le C(d)\frac{|t|}{U}\) and \(\|T_\Lambda^{(0)}\|_0 \le C(d)|t|.\) Therefore \(\|\mathcal{R}^{(2)}_\Lambda(h)\|_0 \le C_{R,0}(d)\frac{|t|^3}{U^2},\) after increasing \(U_\ast\) if necessary. This bound is uniform in \(\Lambda\) and \(|h|\le h_0\).

Combining the decomposition above with the estimates for \(\mathcal{R}^{(2)}_\Lambda(h)\) and \(\mathcal{E}^{(2)}_\Lambda(h)\) completes the proof. 0◻

9 Spin representation and two-site computation↩︎

9.1 Cross-bond terms vanish on the \(P\)-block↩︎

9.1.0.1 Bond hopping terms.

For a nearest-neighbour bond \(e=\langle x,y\rangle\in\mathscr B_\Lambda\), we set \[T_e := -t \sum_{\sigma\in\{\uparrow,\downarrow\}} \left( c_{x\sigma}^{*}c_{y\sigma} + c_{y\sigma}^{*}c_{x\sigma} \right).\] Thus \(T_\Lambda=\sum_{e\in\mathscr B_\Lambda}T_e.\) We use the \(D_\Lambda\)-grading convention from Subsection 2.1, and write \((T_e)^{\rm off} := \sum_{k\neq0}(T_e)^{(k)}.\)

9.2 Two-site computation↩︎

9.3 Proof of Proposition 22↩︎

10 \(P\)-block remainder estimates↩︎

This appendix contains the estimates used in Section 5. The goal is to control, on fixed positive-field windows, the difference between the effective \(P\)-block Hamiltonian and the Heisenberg reference Hamiltonian.

10.1 Parameter mismatch with the Heisenberg reference model↩︎

10.2 Uniform \(C^1\) bound on the \(P\)-block remainder↩︎

We prove Proposition 25. Recall that \[R_{P,\Lambda}(h) = P\bigl(\Delta_\Lambda(h)-H_\Lambda^{(2)}(h)\bigr)P, \qquad \mathcal{R}_{P,\Lambda}(h) = \mathcal{U}_\Lambda R_{P,\Lambda}(h) \mathcal{U}_\Lambda^\ast .\] Since \(\mathcal{U}_\Lambda\) is unitary, it suffices to estimate \(R_{P,\Lambda}(h)\).

10.2.0.1 Later-step notation in the \(\lambda\)-scheme.

For the auxiliary \(\lambda\)-deformation, write the LS/SW iterates as \[H^{(n)}(h;\lambda) = H_{\rm d}(h)+B_n(h;\lambda), \qquad H_{\rm d}(h):=UD_\Lambda-hM_\Lambda .\] We decompose \[B_n(h;\lambda) = D_n(h;\lambda)+F_n(h;\lambda), \qquad D_n(h;\lambda):=B_n(h;\lambda)^{\rm diag}, \qquad F_n(h;\lambda):=B_n(h;\lambda)^{\rm off}.\] Thus \(D_n(h;\lambda)\) is \(D_\Lambda\)-diagonal and \(F_n(h;\lambda)\) is \(D_\Lambda\)-off-diagonal. The \(n\)-th diagonal increment is \[\delta_n(h;\lambda) := D_{n+1}(h;\lambda)-D_n(h;\lambda) = B_{n+1}(h;\lambda)^{\rm diag} - B_n(h;\lambda)^{\rm diag}.\]

The next estimate is the only use of the auxiliary \(\lambda\)-expansion beyond the second-order bridge.

11 Finite-volume pressure and convexity tools↩︎

This appendix collects the finite-volume trace estimates used in Section 6. The purpose is twofold. First, we record the elementary pressure Lipschitz and Duhamel derivative bounds used to compare finite-volume pressures and their \(h\)-derivatives. Second, we prove the soft \(C^1\) defect estimate which compares the full Hubbard pressure with the \(P\)-block pressure in the fixed positive-field window.

11.1 Pressure Lipschitz and derivative formula↩︎

Let \(\mathcal{K}\) be a finite-dimensional Hilbert space. For a self-adjoint operator \(H\) on \(\mathcal{K}\), define \[\require{physics} p_\beta(H) := \frac{1}{\beta|\Lambda|} \log\Tr_{\mathcal{K}}e^{-\beta H}.\] Here \(|\Lambda|\) is the volume parameter used for normalization.

11.2 Even CAR operators and restricted trace factorisation↩︎

We record the restricted trace factorisation used in the soft defect pressure bound. Since we work in a fermionic canonical half-filled sector, we do not use an unrestricted tensor-product factorisation of half-filled traces. Instead, we insert a common outside single-occupancy projector; this freezes the outside part and reduces the half-filling constraint to the inside region.

We also use that all operators involved are even CAR operators, so that operators with disjoint supports commute.

11.3 Finite-volume pressure derivatives↩︎

Recall the finite-volume Hubbard and Heisenberg pressures \(p_{\Lambda_L,\beta,U}^{\rm Hub}(h)\) and \(p_{\Lambda_L,\beta,U}^{\rm Heis}(h)\) introduced in Subsection 1.3. Since both are finite-volume log-partition functions with \(h\) coupled linearly to the staggered magnetisation, they are convex functions of \(h\).

11.4 Positive fixed-field magnetisation↩︎

11.5 Thermodynamic derivatives and quasi-averages↩︎

The following statement explains how thermodynamic derivatives of pressures relate to limits of finite-volume magnetisations.

References↩︎

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1. For the general \(C^\ast\)-algebraic formulation of CAR algebras and quasi-local algebras, we refer to Bratteli and Robinson [22], [23].↩︎

2. Equivalently, \(\Phi_X=0\) unless \(X\) is contained in some ball of radius \(R\), up to a harmless change of \(R\).↩︎