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1 Introduction↩︎

Massless fermions provide an idealized framework for the exploration of strongly interacting quantum systems, both in condensed matter [1]–[3] and in particle physics [4]–[6]. In one spatial dimension (1D) they are characterized by a linear dispersion relation and a definite chirality, either left-moving or right-moving. The edge of a quantum Hall insulator provides a condensed-matter realization of 1D chiral fermions, with a quantized electrical conductance \(G_{\rm H}\) and thermal conductance \(G_{\rm T}\).

The zero-mass property is protected by internal symmetries which prevent the coupling of left-movers and right-movers that would gap the spectrum. Strong interactions can generate a mass by spontaneous symmetry breaking (the Anderson-Higgs mechanism [7]), but there may be an alternative: symmetric mass generation (SMG) — a process by which interactions produce mass while preserving the underlying symmetries [8]–[11].

A necessary condition for SMG to work is that the fermions are anomaly-free [12]. In the quantum Hall context this means that both \(G_{\rm H}\) and \(G_{\rm T}\) should vanish. This condition restricts the number \(N_{\rm L}\) and \(N_{\rm R}\) of left-movers and right-movers, and their integer charges \(q_n\), since \(G_{\rm H}\propto\sum_{i\in\rm{L}}q_i^2-\sum_{j\in\rm{R}}q_j^2\) and \(G_{\rm T}\propto N_{\rm L}-N_{\rm R}\).

The 3–4–5–0 model introduced by Wang and Wen [13], [14] is a test ground for SMG of 1D chiral fermions: There are two right-movers, with charges 3 and 4, and two left-movers, with charges 5 and 0, so the model is anomaly-free (\(N_{\rm L}=N_{\rm R}\) and \(3^2+4^2=5^2+0^2\), see Fig.1). The charges are co-prime, a mass term that gaps the spectrum by coupling left-movers and right-movers is prohibited by the U(1) symmetry responsible for charge conservation.

Zeng et al. [15] demonstrated the interaction-induced gap opening in the 3–4–5–0 model on a 2D lattice (2+1 dimensional space-time). The additional spatial dimension was needed to accommodate the mirror fermions that appear for any local, chirality-preserving discretization of the Hamiltonian [16]. A fine-tuning of the hopping matrix elements decouples the mirror fermions so that their system is effectively 1D.

Recently, related one-dimensional or exactly solvable approaches with local infinite-dimensional Hilbert spaces to anomaly-free chiral lattice gauge theories have also been proposed using emergent translation symmetry, symmetry disentanglers, modified Villain Hamiltonians and Euclidean models [17]–[24].

Here we demonstrate SMG on a strictly 1D lattice. By working with Stacey’s nonlocal discretization of the Hamiltonian [25] (with a tangent rather than a sine dispersion) we circumvent the fermion-doubling obstruction and enable the study of SMG in a minimally constrained setting. Our “tangent fermion” formulation builds on recent advances in quantum Monte Carlo [26] and density-matrix renormalization group (DMRG) methods [27]–[29], that preserve computational efficiency by exploiting the fact that the nonlocal Hamiltonian corresponds to a local generalized eigenvalue problem [30].

The restriction from 2D to 1D is one way in which this study goes beyond Ref.[15]. The second way is that we work around a complication of the 3–4–5–0 model, that the gapping interaction is irrelevant in the sense of the renormalization group (RG). To allow an RG scaling analysis to reliably guide the numerics, we need to make the gapping interaction relevant. Here we achieve that by introducing an additional Hubbard-type density-density interaction, which reduces the scaling dimension of the gapping interaction and makes it perturbatively relevant for Luttinger parameter \(K<2/5\). The result is a controlled setting in which the interaction responsible for SMG is the same anomaly-free 3–4–5–0 interaction, while the auxiliary Hubbard term only tunes its scaling dimension. This separates the origin of the gap from the mechanism that makes the gapping perturbation visible at accessible system sizes.

The outline of the paper is as follows. In Secs.2 and 3 we present the two key ingredients of our work: We first formulate the 3–4–5–0 model on a 1D lattice without fermion doubling and then make the 3–4–5–0 interaction RG relevant by adding a Hubbard repulsion. All of this is informed by the bosonization analysis of Sec.4. Our DMRG results are presented in Sec.5, focusing on the hallmark of symmetric mass generation: the opening of an excitation gap while keeping the ground state nondegenerate. We conclude in Sec.6.

2 3–4–5–0 tangent fermions↩︎

The 3–4–5–0 model [13], [14] on a 1D lattice (unit lattice constant) has Hamiltonian \(H=H_0+H_{3450}\), consisting of the free fermion part \[\begin{align} H_0 ={}& \sum\limits_{n>m}t_{nm}\bigl( c^\dagger_{n,3}c^{\vphantom{\dagger}}_{m,3}+c^\dagger_{n,4}c^{\vphantom{\dagger}}_{m,4}\nonumber\\ &-c^\dagger_{n,5}c^{\vphantom{\dagger}}_{m,5}-c^\dagger_{n,0}c^{\vphantom{\dagger}}_{m,0}\bigr)+ \text{H.c.},\label{H0def} \end{align}\tag{1}\] with hopping matrix elements \(t_{nm}\), and the interacting part \[\begin{align} H_{3450}={}& \sum_n\bigl(g_1 c^{\vphantom{\dagger}}_{n,3}c^\dagger_{n,4}c^\dagger_{n+1,4}c^{\vphantom{\dagger}}_{n,5}c^{\vphantom{\dagger}}_{n,0}c^{\vphantom{\dagger}}_{n+1,0} \nonumber\\ &+g_2 c^{\vphantom{\dagger}}_{n,3}c^{\vphantom{\dagger}}_{n+1,3} c^{\vphantom{\dagger}}_{n,4}c^\dagger_{n,5}c^\dagger_{n+1,5}c^{\vphantom{\dagger}}_{n,0}\bigr)+ \text{H.c.},\label{H3450def} \end{align}\tag{2}\] with coupling constants \(g_1,g_2>0\). The indices of the fermionic operators \(c_{n,\alpha}\) indicate the lattice site \(n\in\mathbb{Z}\) and the charge \(\alpha\in\{3,4,5,0\}\). (We set the electron charge \(e\) and \(\hbar\) equal to unity.)

The nearest-neighbor hopping \(t_{nm}=(t_0/2i)\delta_{n,m+1}\) produces the sine dispersion \(E(k)=t_0\sin k\), with a spurious mirror fermion at \(k=\pi\). To avoid this fermion doubling we adopt Stacey’s long-range hopping [25] \[t_{nm}=2it_0(-1)^{n-m},\label{tnmStacey}\tag{3}\] corresponding to the tangent dispersion \(E(k)=2t_0\tan(k/2)\). The highly nonlocal “all-to-all” coupling 3 becomes a local coupling if the Schrödinger equation \(H\psi=E\psi\) is reformulated as a generalized eigenvalue problem \(P\psi=EQ\psi\), with Hermitian operators \(P,Q\) that only couple nearby lattice sites [30]. One can thus work around the fermion-doubling obstruction without compromising computational efficiency [31].

Since \(\tan(k/2)\) has a positive slope in the Brillouin zone \(|k|<\pi\), the fermions with charge 3 and 4 in Eq.@eq:H3450def are right-movers, while the fermions with charge 5 and 0 are left-movers. The pair of six-fermion interaction terms \(\propto g_1,g_2\) conserve charge (at the origin of the U(1) symmetry), \[\sum_\alpha q_\alpha \ell_\alpha=0,\label{sumqell}\tag{4}\] where \(\ell_\alpha\in\mathbb{Z}\) counts how many fermion operators of charge \(\alpha\) appear in the interaction term (positive \(\ell_\alpha\) for a creation operator, negative \(\ell_\alpha\) for an annihilation operator).

Necessary conditions for an interaction-induced mass are that there are \(N_{\rm L}=N_{\rm R}\) interaction terms, with linearly independent interaction vectors \(\boldsymbol{\ell}^{(p)}\) that satisfy the null condition [32]–[35] \[\sum_{\alpha}\operatorname{sign}(v_\alpha)\ell_\alpha^{(p)}\ell_\alpha^{(p')}=0, \;\;\text{for all}\;\;p,p'.\label{vectorrule}\tag{5}\] The Hamiltonian 2 has interaction vectors \[\begin{align} &\boldsymbol{\ell}^{(1)}=(1,-2,1,2),\\ &\boldsymbol{\ell}^{(2)}=(2,1,-2,1), \end{align}\label{elldef}\tag{6}\] for \(\boldsymbol{q}=(3,4,5,0)\), so that both conditions 4 and 5 are satisfied.

We note that the charges do not uniquely follow from the interaction vectors. For example, charges 7–11–13–1 also satisfy the charge conservation rule 4 . We also note that the anomaly-free condition \[\sum_{\alpha}\operatorname{sign}(v_\alpha)q_\alpha^2=0\label{anomalyfree}\tag{7}\] is not an independent condition on the charges once the interaction vectors are given — Eq.@eq:anomalyfree follows algebraically from Eqs.@eq:sumqell and 5 .

3 SMG at weak coupling enabled by Hubbard repulsion↩︎

3.1 Renormalized scaling dimension↩︎

We recall the basics of the scaling analysis of interacting fermions [36]. Interactions can open a gap in the spectrum of an infinite 1+1 dimensional system if the scaling dimension \(D<2\). If \(D>2\) the interactions are irrelevant, the system scales to the free-fermion limit when the size tends to infinity.

In the non-interacting limit, the six-fermion interaction 2 has scaling dimension \[D_{3450}=\tfrac{1}{2}|\boldsymbol{\ell}^{(p)}|^2=5,\label{Dfree}\tag{8}\] and is therefore highly irrelevant. Strong coupling, however, may renormalize the scaling dimension, such that \(H_{3450}\) becomes relevant and opens a gap in a deeply non-perturbative regime. This is the approach taken previously in Ref.[15].

Here, we take a different route: we keep the six-fermion interaction at moderately weak coupling, where an RG scaling analysis reliably guides the numerical simulations. We add an on-site Hubbard-type density-density interaction, a four-fermion interaction that couples the density of left-movers and right-movers, \[\label{HHubbard} \begin{align} H_{\rm Hubbard}={}&U_{\rm H}\sum_n(\nu_3\delta\rho_{n,3}+\nu_4\delta\rho_{n,4})\nonumber\\ &\times(\nu_5\delta\rho_{n,5}+\nu_0\delta\rho_{n,0}),\\ \delta\rho_{n,\alpha}={}&c^\dagger_{n,\alpha}c^{\vphantom{\dagger}}_{n,\alpha}-\langle c^\dagger_{n,\alpha}c^{\vphantom{\dagger}}_{n,\alpha}\rangle, \end{align}\tag{9}\] with weight factors \(\nu_\alpha\). As the Hamiltonian \(H_{\rm Hubbard}\) causes no backscattering, it cannot open a gap. Nonetheless, it can modify the scaling dimension of \(H_{\rm 3450}\) and thereby enable a gap opening at weak coupling.

The bosonization analysis in Sec. 4 shows that if we choose the weight factors according to the interaction vectors, \((\nu_3,\nu_4,\nu_5,\nu_0)=\boldsymbol{\ell}^{(p)}\), so that the Hubbard interaction takes the form \[\label{HUdef} \begin{align} &H_{\rm Hubbard}=\sum_{n}U_n,\\ &U_n=-U^{(1)}_{\rm H}(\delta\rho_{n,3}-2\delta\rho_{n,4})(\delta\rho_{n,5}+2\delta\rho_{n,0})\nonumber\\ &\qquad-U^{(2)}_{\rm H}(2\delta\rho_{n,3}+\delta\rho_{n,4})(-2\delta\rho_{n,5}+\delta\rho_{n,0}), \end{align}\tag{10}\] that with this choice of weights, the scaling dimensions \(D^{(1)}_{3450},D^{(2)}_{3450}\) of the \(g_1\) and \(g_2\) terms in \(H_{\rm 3450}\) can be independently tuned by a pair of effective Luttinger parameters \(K_1,K_2\), according to \[\begin{align} &D^{(\alpha)}_{3450}=5K_\alpha,\;\;K_\alpha=\sqrt{\frac{2\pi t_0-CU^{(\alpha)}_{\rm H}}{2\pi t_0+CU^{(\alpha)}_{\rm H}}},\\ &C=\sqrt{\nu_3^2+\nu_4^2}\sqrt{\nu_5^2+\nu_0^2}=5. \end{align} \label{Kdef}\tag{11}\]

In the following, we set \(g_1=g_2\), \(U^{(1)}_{\rm H} =U^{(2)}_{\rm H}\equiv U_{\rm H}>0\), \(K_1=K_2\equiv K\in(0,1)\). The 3–4–5–0 interaction then becomes relevant for \(K<2/5\equiv K_c\).

3.2 Elimination of Friedel oscillations↩︎

At nonzero Fermi wave vector \(k_{\rm F}\) the electron density correlators have rapid Friedel oscillations \(\propto e^{ik_{\rm F}x}\), that reduce the effectiveness of the gap opening interactions. To eliminate these we proceed as follows.

We denote by \(N_\alpha\) the expectation value of the number of fermions of charge \(\alpha\) relative to a half-filled band (the free-fermion vacuum). The corresponding Fermi wave vector \(k_\alpha\) is \[k_\alpha = (2\pi/L) \operatorname{sign}(v_\alpha) N_\alpha.\] Friedel oscillations in the correlators are absent if \[\begin{align} &\sum_\alpha \ell^{(p)}_\alpha k_\alpha = 0\Rightarrow {\cal N}^{(p)}=0,\;\;\text{for all}\;\;p,\\ &\text{with}\;\;{\cal N}^{(p)}=\sum_\alpha \operatorname{sign}(v_\alpha) \ell^{(p)}_\alpha N_\alpha. \end{align} \label{Friedelcondition}\tag{12}\]

The Hamiltonian \(H_{3450}\) conserves charge but not particle number. We can therefore not enforce \(N_\alpha\equiv 0\) (half-filled band for each charge), to satisfy Eq.@eq:Friedelcondition . Instead, setting \({\cal N}^{(p)}\equiv 0\) for all \(p\) is permitted because \({\cal N}^{(p)}\) is conserved by \(H_{3450}\).

To see this, we note that \(N_\alpha\) can change due to the interactions by an amount \(\delta N_\alpha=\sum_p n^{(p)}\ell_\alpha^{(p)}\) with \(n^{(p)}\in\mathbb{Z}\). The corresponding change in \({\cal N}^{(p)}\) is \[\delta{\cal N}^{(p)}=\sum_{p'}n^{(p')}\sum_\alpha \operatorname{sign}(v_\alpha) \ell^{(p)}_\alpha \ell_\alpha^{(p')}=0,\] in view of the null condition 5 .

4 Bosonization analysis↩︎

4.1 Charge rotation decouples the Luttinger liquid↩︎

At first sight, the lattice model introduced in Sec. 2 appears rather complicated. Its interactions mix the four fermion flavors in a nontrivial way, and it is therefore not immediately obvious why these particular terms preserve the symmetry required for symmetric mass generation. The structure becomes more transparent after bosonization of the low-energy theory.

We begin with the noninteracting Hamiltonian 1 . The four chiral fermions are represented by chiral bosonic fields \(\varphi_{qR/L}\), where \(q\) denotes the charge and \(R/L\) the chirality (charges 3,4 right-moving, charges 5,0 left-moving). The free Hamiltonian is \[\begin{align} H_{\rm free} &= \frac{1}{4\pi}\int dx\, \bigl[ (\partial_x\varphi_{3R})^2 + (\partial_x\varphi_{4R})^2\nonumber\\ &\qquad+ (\partial_x\varphi_{5L})^2 + (\partial_x\varphi_{0L})^2 \bigr] \nonumber\\ &= \frac{1}{4\pi}\int dx\, \left[ (\partial_x\Phi_R)^2+(\partial_x\Phi_L)^2 \right],\\ \label{Hfreeboson} \end{align}\tag{13}\] where we have collected the chiral fields in vectors \[\Phi_R=(\varphi_{3R},\varphi_{4R}),\;\; \Phi_L=(\varphi_{5L},\varphi_{0L}),\;\;\Phi=(\Phi_R,\Phi_L) .\]

The free Hamiltonian is invariant under independent orthogonal rotations of the right-moving and left-moving fields, \[\widetilde{\Phi}_R=Q_R\Phi_R,\qquad \widetilde{\Phi}_L=Q_L\Phi_L .\] We will use this transformation to isolate neutral modes from charged modes. Similar changes of basis have been used, for example, in the study of conformal boundary conditions [37]–[39].

We take \[Q_R= \frac{1}{\sqrt 5} \begin{pmatrix} 1 & -2\\ 2 & 1 \end{pmatrix}, \qquad Q_L= -\frac{1}{\sqrt 5} \begin{pmatrix} 1 & 2\\ -2 & 1 \end{pmatrix}, \label{QRQL}\tag{14}\] chosen such that \[\boldsymbol{\ell}^{(p)}=\sqrt{5}\,\bigl([Q_R]_{p,1},[Q_R]_{p,2},-[Q_L]_{p,1},-[Q_L]_{p,2}\bigr),\] see Eq.@eq:elldef .

With this choice, \[\boldsymbol{\ell}^{(p)}\Phi = 2\sqrt 5\,\phi_p, \label{ellPhi}\tag{15}\] where we have introduced the nonchiral two-component bosonic fields \[\phi=\tfrac{1}{2}(\widetilde{\Phi}_R-\widetilde{\Phi}_L), \qquad \theta=\tfrac{1}{2}(\widetilde{\Phi}_R+\widetilde{\Phi}_L). \label{phitheta}\tag{16}\] The free Hamiltonian then decomposes into two decoupled Tomonaga-Luttinger liquids [40]–[42], \[H_{\rm free} = \sum_{p=1}^2 \frac{v}{2\pi} \int dx\, \left[ K_p(\partial_x\theta_p)^2 + \frac{1}{K_p}(\partial_x\phi_p)^2 \right], \label{HLL}\tag{17}\] with \(K_p=v=1\) at the free-fermion point.

The change of basis makes the charge symmetry explicit. A global \(\mathrm U(1)\) charge rotation acts on the fermions as \[c_q\mapsto e^{iq\chi}c_q ,\] and therefore shifts the bosonic fields according to \[\Phi_\alpha\mapsto \Phi_\alpha+\chi q_\alpha .\] Eq.@eq:QRQL then gives \[\phi_p\mapsto \phi_p,\;\; \theta_1\mapsto \theta_1-\sqrt 5\,\chi,\;\; \theta_2\mapsto \theta_2+2\sqrt 5\,\chi . \label{U1action}\tag{18}\] Thus the fields \(\theta\) are charged, while the fields \(\phi\) are neutral.

Vertex operators built from the charged field \(\theta\) are forbidden by charge conservation, while vertex operators built from the neutral field \(\phi\) are allowed, and can be used to introduce the 3–4–5–0 interaction. The lowest-order local vertex operators are \[V_p = \cos(\boldsymbol{\ell}^{(p)}\Phi) = \cos(2\sqrt 5\,\phi_p), \;\; p=1,2 . \label{Valpha}\tag{19}\] After re-fermionization, \(c_{x,q}\sim e^{i\varphi_q(x)}\), these are precisely the two six-fermion operators in the 3–4–5–0 interaction 2 .

The scaling dimension of an interaction \(\propto\cos \beta\phi_p\) is \(\beta^2 K_p/4\), so for \(V_p\) this is \(5K_p\). For \(K_p<2/5\) the cosine interaction is relevant and can pin the field \(\phi_p\) to a local minimum, gapping out the fermionic excitations. When both cosines are relevant, \(K_1,K_2<2/5\), both Luttinger liquids are gapped. Since the pinned fields are invariant under the charge rotation 18 , this gap opening does not break the protecting \(\mathrm U(1)\) symmetry.

4.2 Scaling dimension↩︎

We next identify the density-density interaction that reduces the scaling dimension of the operators \(V_p\). In view of Eq. 17 , a reduction of \(K_p\) is produced by the interaction \[{\cal O}_p = (\partial_x\theta_p)^2 - (\partial_x\phi_p)^2 = \partial_x\widetilde{\varphi}_{L,p}\, \partial_x\widetilde{\varphi}_{R,p} . \label{Oalpha}\tag{20}\]

Upon refermionization, \(\partial_x\varphi_q(x)\mapsto \operatorname{sign}(v_q)\delta\rho_{x,q}\), this interaction becomes the Hubbard-type density-density interaction 10 . It couples left-moving and right-moving densities, but it does not backscatter the chiral fermions and therefore does not by itself open a gap. Its role is instead to renormalize the free-fermion scaling dimension \(D_{3450}=5\) by \[D^{(p)}_{3450}=5K_p . \label{D3450K}\tag{21}\] The 3–4–5–0 interaction becomes relevant once \(K_p<2/5\), which is the criterion used to guide the numerical simulations.

4.3 Gapping without emergent ground state degeneracy↩︎

A distinctive feature of gapping by the SMG mechanism is that the ground state remains nondegenerate. Because this will play a key role in the interpretation of our numerics, we explain it in some detail.

The four components of the chiral bosonic field \(\Phi\) are defined modulo \(2\pi\), the field is compactified on the four-torus \(\mathbb{T}^4\). We define the map \({\cal A}\) from \(\mathbb{T}^4\) to \(\mathbb{T}^2\) by \[{\cal A}\Phi = A\Phi\bmod 2\pi,\;\;A=\begin{pmatrix} 1 & -2 & 1 & 2\\ 2 & 1 & -2 & 1 \end{pmatrix}.\] The two rows of \(A\) are the interaction vectors \(\boldsymbol{\ell}^{(1)}\) and \(\boldsymbol{\ell}^{(2)}\), so that \[[A\Phi]_p=\boldsymbol{\ell}^{(p)}\Phi=2\sqrt 5\,\phi_p.\]

The map \({\cal A}\) covers the whole of \(\mathbb{T}^2\) (it is surjective), because it can generate the two basis states of the two-torus: \[A (0,1,1,1)^\top=\textstyle{1\choose 0},\;\; A (1,1,1,0)^\top=\textstyle{0\choose 1}.\]

Without loss of generality [43] we may choose the sign of the couplings such that the cosine potential minima are located at \(\boldsymbol{\ell}^{(p)}\Phi=0\) mod \(2\pi\), \(p=1,2\). Its pre-image \[{\cal M}=\{x\in\mathbb{T}^4:Ax=0\bmod 2\pi\}\] is the kernel of the map \({\cal A}\).

The pinned ground state is nondegenerate if \({\cal M}\) is singly connected on \(\mathbb{T}^4\). It is a basic theorem of algebraic geometry [44] that the kernel of a surjective integer-matrix map between tori has a number of connected components equal to the greatest common divisor (gcd) of the set of maximal minors of the matrix. Since \(A\) is a \(2\times 4\) matrix, its maximal minors are the determinants of its six \(2\times 2\) submatrices. Columns 1 and 2 have gcd 5, columns 2 and 3 have gcd 3, so this already fixes the gcd of all minors at 1 and we conclude that \({\cal M}\) has only a single connected component.

An equivalent way to state this algebraic result is that the ground state is nondegenerate because the lattice of interaction vectors, \[\Lambda=\mathrm{span}_{\mathbb{Z}} \{\boldsymbol{\ell}^{(1)},\boldsymbol{\ell}^{(2)}\}\subset \mathbb{Z}^4\] is primitive [45]. The primitivity condition means that there is no allowed local vertex operator whose exponent is a nontrivial fractional linear combination of the pinned fields. In particular, \(\boldsymbol{\ell}^{(p)}/2\notin\mathbb{Z}^4\), so the half-harmonics \(\cos(\boldsymbol{\ell}^{(p)}\Phi/2)\) and \(\sin(\boldsymbol{\ell}^{(p)}\Phi/2)\) are not operators in the fermionic Hilbert space.

This is the essential difference from an ordinary symmetry-breaking sine-Gordon problem, with a non-primitive pinning potential \(\cos(2m\Phi)\). The minima \(m\Phi=0\) and \(m\Phi=\pi\) are not related by the compactification of the bosonic fields and can be distinguished by the local order parameter \(\cos(m\Phi)\). As demonstrated for the tangent fermion Luttinger liquid in Ref.[29], this produces a degenerate ground-state manifold, accompanied by spontaneous symmetry breaking.

5 Results↩︎

We consider two related signatures of symmetric mass generation: firstly in the excitation spectrum, secondly in the occupation factor. We then show numerical DMRG results on tensor networks (see App.7) that exhibit both signatures.

5.1 Excitation gap↩︎

Our goal is to distinguish three types of infinite-system spectra from finite-size data: (I) a gapless system; (II) a gapped system with broken U(1) symmetry; (III) a gapped system with preserved U(1) symmetry. The presence of a finite-size gap \(\Delta=\hbar v/L\) in a system of size \(L\) complicates the distinction. In Fig.2 we illustrate what we expect for the \(L\)-dependence of the excitation gap in each of the three cases.

Case (I) is distinguished by the \(1/L\) decay of the energy of the lowest excited state (relative to the ground state). In both cases (II) and (III) an excitation gap remains in the large-\(L\) limit; the distinguishing feature is that in case (II) the lowest excited state \(E_1\) merges with the ground state and the gap refers to the energy of the next level \(E_2\). The appearance of a degenerate ground state is the signature of the Higgs mechanism for mass generation via spontaneously broken symmetry, see Ref.[29] for a tangent fermion realization.

The DMRG results in Fig.3d) show the expected case III spectrum, indicative of SMG. The four panels isolate the two ingredients of the construction. For \(g_1=g_2=0\), the Hubbard interaction changes the Luttinger parameter but does not by itself produce the SMG gap. In contrast, for \(K>K_c\), the 3–4–5–0 interaction remains irrelevant, and the spectrum retains the finite-size behavior expected of a gapless system. Only when the 3–4–5–0 interaction is combined with a sufficiently small Luttinger parameter, \(K<K_c\), do the two lowest excitation energies separate from the gapless scaling line without an accompanying collapse of \(E_1\) onto the ground state. This is the finite-size signature expected for a symmetric, rather than symmetry-breaking, mass generation mechanism.

5.2 Occupation factor↩︎

The occupation factor \(n_\alpha(k)\), the Fourier transform of the charge-\(\alpha\) propagator \(C_\alpha(x)=\langle c_\alpha^\dagger(x)c_\alpha(0)\rangle\), has a power-law singularity at \(k=0\) for a gapless Luttinger liquid [36], \[n_\alpha(k)\propto |k|^{[K+K^{-1}]/2-1}.\] For a gapped system, instead, a smooth \(k\)-dependence follows from the exponential decay of the propagator.

In a finite system, the discreteness of \(k\) removes the singularity, but a steep rise remains in a gapless system, as can be seen in Fig.4, panels a,b,c). The steep rise is smoothed in panel d), consistent with the gap data from Fig.3.

6 Conclusion↩︎

We have studied symmetric mass generation (SMG) in the anomaly-free 3–4–5–0 model [13], [14] on a strictly one-dimensional lattice, employing fermions with a tangent dispersion (“tangent fermions”) as an alternative to lattice constructions [15] in which the unwanted mirror fermions are accommodated in an additional spatial dimension. To make the SMG mechanism visible in a weak-coupling scaling regime, we introduced a Hubbard-type density-density interaction that renormalizes the Luttinger parameter \(K\), reducing the scaling dimension. For \(K< K_c=2/5\), the gap-opening interaction becomes relevant.

The DMRG results exhibit the expected finite-size signatures of this mechanism. When either the 3–4–5–0 interaction or the sufficiently strong Hubbard renormalization is absent, the low-lying spectrum follows the behavior expected of a gapless system. When both are present, the excitation spectrum develops a gap while the ground state remains nondegenerate. The momentum occupation factor shows the corresponding smoothing of the Luttinger-liquid singularity. Taken together, these observations are consistent with interaction-induced mass generation without spontaneous breaking of the protecting \(\mathrm{U}(1)\) symmetry.

The tangent-fermion formalism realizes each chiral fermion flavor directly and independently on the lattice. This provides a useful framework in which the bosonization dictionary is particularly transparent. Indeed, this transparency was essential for identifying the Hubbard interaction in the simple form used here. We therefore expect tangent fermions to be useful more broadly in future studies of strongly interacting chiral fermions and their dynamics.

Figure 1 was suggested to us by S. Polla. AU thanks L. Lootens for helpful discussions. Data sets are available at a Zenodo repository.
Research in Leiden was supported by the Netherlands Organisation for Scientific Research (NWO/OCW), as part of Quantum Limits (project number summit.1.1016). AU was supported by BOF-GOA (Grant No.BOF23/GOA/021) and by FWO Junior Postdoctoral Fellowship (grant No.3E0.2025.0049.01). FV acknowledges funding from the UKRI (EP/Z003342/1), EOS (40007526) and IBOF (IBOF23/064).

7 DMRG calculation↩︎

7.1 MPO representation↩︎

For the DMRG calculation we need to represent the Hamiltonian as a matrix-product operator (MPO) [46], a product of matrices \(M^{(n)}\) that act only on site \(n\). The calculation is efficient if the dimension of each matrix (the MPO bond dimension \(\chi_{\rm MPO}\)) is independent of the number of lattice sites \(N\). Such a scale independent MPO is possible for short-range hoppings, and also for long-range hoppings that correspond to a local generalized eigenvalue problem [28].

7.2 Error bar estimation↩︎

We represent the eigenstates of \(H=H_0+H_{3450}\) by matrix product states (MPS, bond dimension \(\chi_{\rm MPS}\)) and carry out the tensor network DMRG algorithm [47] to variationally determine the ground state and the first few excited states. (We used the TeNPy Library [48] for these calculations.) We impose anti-periodic boundary conditions on the lattice of size \(L\), with \(L\) even to avoid the pole in the tangent dispersion at the Brillouin zone boundary.

While the matrix product representation of the Hamiltonian is exact, with small bond dimension \(\chi_{\rm MPO}\), the wave function cannot be exactly represented by a matrix product of finite bond dimension \(\chi_{\rm MPS}\). We therefore carry out the DMRG calculation at several values of \(\chi_{\rm MPS}=2^p\), increasing in powers of two, and extrapolate to estimate the \(\chi_{\rm MPS}\rightarrow\infty\) limit of the excitation energies \(E_n\).

The error bars of this extrapolation are estimated as follows, based on two empirical observations:

• Firstly, we find that the calculated \(E_n(\chi_{\rm MPS})\) decays monotonically with increasing \(\chi_{\rm MPS}\). We therefore align the upper edge of the error bar with the energy at the largest available bond dimension \(\chi_{\rm MPS}=2^{p_{\rm max}}\).

• Secondly, we find that a fit \[E_n(\chi_{\rm MPS})=E_{n}+a_{n}\,\chi_{\rm MPS}^{-\alpha_n}\label{fit}\tag{22}\] to four subsequent values of \(\chi_{\rm MPS}\) gives a monotonically increasing fit parameter \(E_n\) as larger and larger bond dimensions are included in the fit. We therefore align the lower edge of the error bar with the fit up to \(2^{p_{\rm max}}\).

Figure5 illustrates this procedure, both for the interacting case and for the case of free tangent fermions. The latter case is a test, we can compare with the known value of the excitation energy [\(E_1=4\tan(\pi/2L)\), an excitation from \(-2\tan(\pi/2L)\) to \(+2\tan(\pi/2L)\)], which is properly bracketed by the error bar.

References↩︎