Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2606.24713 · Physical Review Research

Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling

V. A. Zakharov, Atsushi Ueda, Frank Verstraete, C. W. J. Beenakker

hep-thcond-mat.str-elquant-phSubmitted Jun 23, 2026• Announced Jun 24, 2026• 0 citations• Revised Sep 2, 2026 (v2)
v1 · Jun 23, 2026v2 · Sep 2, 2026
arXivHTMLPDF

Abstract

Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter $K$) that reduces the scaling dimension of the 3-4-5-0 interaction from $5$ to $5K$, making it relevant for $K<2/5$. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.

Journal reference
Phys. Rev. Research 8, 043009 (2026)
DOI
10.1103/2v9h-25qs
Comments
11 pages, 5 figures. New appendix on a more general SMG senario. New zenodo version has the DMRG code

Related papers

Symmetric Mass Generation in the 1+1 Dimensional Chiral Fermion 3-4-5-0 Model

Meng Zeng, Zheng Zhu, Juven Wang +1 more

Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively (1+1)D. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.

arXiv:2202.12355•Physical Review Letters•May 5, 2022•44 citations

Fermi surface symmetric mass generation

Da-Chuan Lu, Meng Zeng, Juven Wang +1 more

Symmetric mass generation (SMG) is a mechanism to give gapless fermions a mass gap by nonperturbative interactions without generating any fermion bilinear condensation. The previous studies of SMG have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this paper, we generalize the concept of SMG to Fermi liquid (FL) with a finite Fermi volume and discuss how to gap out the Fermi surfaces (FSs) by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of FS SMG in both $(1+1)$-dimensional $[(1+1)\mathrm{D}]$ and $(2+1)$-dimensional FL systems when several FSs together cancel the FS anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a nonperturbative interaction mechanism is the only way to gap out the FSs. This symmetric FS reconstruction is in contrast to the conventional symmetry-breaking gapping mechanism in the FL. As a side product, our model provides a pristine one-dimensional lattice regularization for the $(1+1)\mathrm{D}$ U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction.

arXiv:2210.16304•Physical review. B./Physical review. B•May 17, 2023•20 citations

Charge Pumping Leptogenesis from the Symmetric Mass Generation Phase on a Four-Leg Ladder Lattice

Runze Chi, Meng Zeng, Tao Xiang +3 more

Quantized charge pumping in quantum Hall systems is a bulk topological property, typically manifesting on gapless boundaries as spectral flow through chiral edge modes upon flux insertion. When the boundary is anomaly-free, it can be symmetrically gapped by symmetric mass generation (SMG) interactions. However, the effects on quantized charge pumping at an interacting gapped boundary are unclear. We address this using density-matrix renormalization group simulations of a four-leg ladder quasi-(2+1)d realization of the (1+1)d 3-4-5-0 lattice chiral fermion model with six-fermion interactions at one edge. We find gapless modes near the interacting edge for both weak and strong couplings, but the edge can be in a fully gapped SMG phase at intermediate coupling. We also identify spontaneous breaking of the model's $\mathbb{Z}_2$ symmetry above a critical interaction strength. Across all three boundary phases, we find quantized charge pumping with the same pumped charge per flux quantum. The spatial distribution of the pumped charge varies between phases. Besides charge pumping, we study how a single injected fermion propagates under open boundary conditions. In the SMG phase, an interaction-induced flavor-mixing current develops at the endpoints of the gapped edge, which is absent in the weak- and strong-coupling limits. This current reveals a charge-redistribution mechanism unique to the SMG phase. Additionally, we propose an analogy between charge pumping in this model and a topological leptogenesis scenario in the (3+1)d Standard Model. The anomaly-free chiral U(1) symmetries of the 3-4-5-0 model play a role analogous to the baryon-minus-lepton symmetry of the Standard Model. In both settings, charge pumping can be associated with a continuous deformation along a classically defined but quantum-anomalous direction, while a discrete subgroup remains an exact quantum symmetry.

cond-mat.str-elCross-listedarXiv:2610.42103•Nov 2, 2026

Interacting chiral fermions on the lattice with matrix product operator norms

Jutho Haegeman, Laurens Lootens, Quinten Mortier +3 more

We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice while preserving unitarity and locality and without breaking the chiral symmetry. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. When projecting our theory on the the single-particle sector, we recover the framework of Stacey fermions, and we demonstrate that the scaling limit of the free model recovers the chiral fermion field. Technically, we make use of a matrix product operator norm to mimick the boundary of a higher dimensional topological theory. As a proof of principle, we consider a single Weyl fermion on a periodic ring with Hubbard-type nearest-neighbor interactions and construct a variational generalized DMRG code to demonstrate that the ground state for large system sizes can be determined efficiently. As our tensor network approach does not exhibit any sign problem, we can add a chemical potential and study real-time evolution.

arXiv:2405.10285•May 16, 2024•1 citation

A Lattice Non-Perturbative Definition of 1+1D Anomaly-Free Chiral Fermions and Bosons

Juven Wang, Xiao-Gang Wen

A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry $G$? In particular, we show that the 3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.

•Jul 29, 2013•1 citation

Fermionic Villain model with exact lattice chiral symmetries

Zhiyao Lu, Shu-Heng Shao

We present a fermionic lattice Hamiltonian that exactly realizes the $\mathrm{U(1)}_\mathrm{L}\times \mathrm{U(1)}_\mathrm{R}$ global symmetry and the associated chiral anomalies of a massless Dirac fermion in 1+1 dimensions. The construction couples Villain bosons to a Kitaev chain of Majorana fermions, where the bosonic fields are essential for evading Nielsen-Ninomiya-type no-go theorems. For two copies of our model, we realize the anomaly-free $\mathrm{U(1)}_{3450}$ global symmetry and construct symmetric boundary conditions. We analytically demonstrate symmetric mass generation by mapping the symmetry-preserving six-fermion interactions to fermion bilinear terms using fermionic T-duality. Finally, by gauging general anomaly-free global symmetries, we obtain a broad class of lattice chiral gauge theories. As nontrivial applications, we compute the mass spectra of the lattice Schwinger model and the 3450 gauge theory, finding agreement with the corresponding continuum results.

arXiv:2608.02728•Aug 3, 2026•0 citations
Symmetric mass generation is the interaction-induced opening of a fermion gap without spontaneous symmetry breaking. The anomaly-free 3-4-5-0 model of Wang and Wen provides a minimal one-dimensional setting for this phenomenon, but a direct lattice realization faces two obstacles: fermion doubling for local chiral discretizations and perturbative irrelevance of the six-fermion gapping interaction. We address both obstacles. First, we formulate the model on a strictly one-dimensional tangent-fermion lattice, where a nonlocal hopping produces a single chiral branch without a mirror partner while retaining an efficient tensor-network representation. Second, we add a Hubbard-type density-density interaction (Luttinger parameter KKK) that reduces the scaling dimension of the 3-4-5-0 interaction from 555 to 5K5K5K, making it relevant for K<2/5K<2/5K<2/5. Density-matrix renormalization group calculations show the opening of an excitation gap in this regime without the appearance of a degenerate ground state, the hallmark of symmetric mass generation.
Lattice regularization of chiral fermions has been a long-standing problem in physics. In this Letter, we present the density matrix renormalization group simulation of the 3-4-5-0 model of (1+1)D chiral fermions with an anomaly-free chiral U(1) symmetry, which contains two left-moving and two right-moving fermions carrying U(1) charges 3,4 and 5,0, respectively. Following the Wang-Wen chiral fermion model, we realize the chiral fermions and their mirror partners on the opposite boundaries of a thin strip of (2+1)D lattice model of multilayer Chern insulator, whose finite width implies the quantum system is effectively (1+1)D. By introducing two sets of carefully designed six-fermion local interactions to the mirror sector only, we demonstrate that the mirror fermions can be gapped out by the interaction beyond a critical strength without breaking the chiral U(1) symmetry, via the symmetric mass generation mechanism. We show that the interaction-driven gapping transition is in the Berezinskii-Kosterlitz-Thouless universality class. We determine the evolution of Luttinger parameters before the transition, which confirms that the transition happens exactly at the point when the interaction term becomes marginal. As the mirror sector is gapped after the transition, we check that the fermions in the light chiral fermion sector remain gapless, which provides the desired lattice regularization of chiral fermions.
Symmetric mass generation (SMG) is a mechanism to give gapless fermions a mass gap by nonperturbative interactions without generating any fermion bilinear condensation. The previous studies of SMG have been limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free fermion limit. In this paper, we generalize the concept of SMG to Fermi liquid (FL) with a finite Fermi volume and discuss how to gap out the Fermi surfaces (FSs) by interactions without breaking the U(1) loop group symmetry or developing topological orders. We provide examples of FS SMG in both (1+1)(1+1)(1+1)-dimensional [(1+1)D][(1+1)\mathrm{D}][(1+1)D] and (2+1)(2+1)(2+1)-dimensional FL systems when several FSs together cancel the FS anomaly. However, the U(1) loop group symmetry in these cases is still restrictive enough to rule out all possible fermion bilinear gapping terms, such that a nonperturbative interaction mechanism is the only way to gap out the FSs. This symmetric FS reconstruction is in contrast to the conventional symmetry-breaking gapping mechanism in the FL. As a side product, our model provides a pristine one-dimensional lattice regularization for the (1+1)D(1+1)\mathrm{D}(1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by utilizing a lattice translation symmetry as an emergent U(1) symmetry at low energy. This opens up the opportunity for efficient numerical simulations of chiral fermions in their own dimensions without introducing mirror fermions under the domain wall fermion construction.
Quantized charge pumping in quantum Hall systems is a bulk topological property, typically manifesting on gapless boundaries as spectral flow through chiral edge modes upon flux insertion. When the boundary is anomaly-free, it can be symmetrically gapped by symmetric mass generation (SMG) interactions. However, the effects on quantized charge pumping at an interacting gapped boundary are unclear. We address this using density-matrix renormalization group simulations of a four-leg ladder quasi-(2+1)d realization of the (1+1)d 3-4-5-0 lattice chiral fermion model with six-fermion interactions at one edge. We find gapless modes near the interacting edge for both weak and strong couplings, but the edge can be in a fully gapped SMG phase at intermediate coupling. We also identify spontaneous breaking of the model's Z2\mathbb{Z}_2Z2​ symmetry above a critical interaction strength. Across all three boundary phases, we find quantized charge pumping with the same pumped charge per flux quantum. The spatial distribution of the pumped charge varies between phases. Besides charge pumping, we study how a single injected fermion propagates under open boundary conditions. In the SMG phase, an interaction-induced flavor-mixing current develops at the endpoints of the gapped edge, which is absent in the weak- and strong-coupling limits. This current reveals a charge-redistribution mechanism unique to the SMG phase. Additionally, we propose an analogy between charge pumping in this model and a topological leptogenesis scenario in the (3+1)d Standard Model. The anomaly-free chiral U(1) symmetries of the 3-4-5-0 model play a role analogous to the baryon-minus-lepton symmetry of the Standard Model. In both settings, charge pumping can be associated with a continuous deformation along a classically defined but quantum-anomalous direction, while a discrete subgroup remains an exact quantum symmetry.
We develop a Hamiltonian formalism for simulating interacting chiral fermions on the lattice while preserving unitarity and locality and without breaking the chiral symmetry. The fermion doubling problem is circumvented by constructing a Fock space endowed with a semi-definite norm. When projecting our theory on the the single-particle sector, we recover the framework of Stacey fermions, and we demonstrate that the scaling limit of the free model recovers the chiral fermion field. Technically, we make use of a matrix product operator norm to mimick the boundary of a higher dimensional topological theory. As a proof of principle, we consider a single Weyl fermion on a periodic ring with Hubbard-type nearest-neighbor interactions and construct a variational generalized DMRG code to demonstrate that the ground state for large system sizes can be determined efficiently. As our tensor network approach does not exhibit any sign problem, we can add a chemical potential and study real-time evolution.
A non-perturbative lattice regularization of chiral fermions and bosons with anomaly-free symmetry GGG in 1+1D spacetime is proposed. More precisely, we ask whether there is a local short-range quantum Hamiltonian with a finite Hilbert space for a finite system realizing onsite symmetry GGG defined on a 1D spatial lattice with continuous time, such that its low energy physics produces a 1+1D anomaly-free chiral matter theory of symmetry GGG? In particular, we show that the 3L_LL​-5R_RR​-4L_LL​-0R_RR​ U(1) chiral fermion theory, with two left-moving fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at low energy, can be put on a 1D spatial lattice where the U(1) symmetry is realized as an onsite symmetry, if we include properly designed multi-fermion interactions with intermediate strength. In general, we propose that any 1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous time, but without suffered from Nielsen-Ninomiya theorem's fermion-doubling, if we include properly-designed interactions between matter fields. We propose how to design such interactions by looking for extra symmetries via bosonization/fermionization. We comment on the new ingredients and the differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have successful mirror-decoupling. As an additional remark, we show a topological non-perturbative proof on the equivalence relation between the 't Hooft anomaly matching conditions and the boundary fully gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) of U(1) symmetry. Our proof holds universally independent from Hamiltonian or Lagrangian/path integral formulation of quantum theory.
We present a fermionic lattice Hamiltonian that exactly realizes the U(1)L×U(1)R\mathrm{U(1)}_\mathrm{L}\times \mathrm{U(1)}_\mathrm{R}U(1)L​×U(1)R​ global symmetry and the associated chiral anomalies of a massless Dirac fermion in 1+1 dimensions. The construction couples Villain bosons to a Kitaev chain of Majorana fermions, where the bosonic fields are essential for evading Nielsen-Ninomiya-type no-go theorems. For two copies of our model, we realize the anomaly-free U(1)3450\mathrm{U(1)}_{3450}U(1)3450​ global symmetry and construct symmetric boundary conditions. We analytically demonstrate symmetric mass generation by mapping the symmetry-preserving six-fermion interactions to fermion bilinear terms using fermionic T-duality. Finally, by gauging general anomaly-free global symmetries, we obtain a broad class of lattice chiral gauge theories. As nontrivial applications, we compute the mass spectra of the lattice Schwinger model and the 3450 gauge theory, finding agreement with the corresponding continuum results.