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

Juven Wang, Xiao-Gang Wen

Published Jul 29, 2013• 1 citation
OpenAlex

Abstract

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.

Related papers

Nonperturbative regularization of ( 1+1 )-dimensional anomaly-free chiral fermions and bosons: On the equivalence of anomaly matching conditions and boundary gapping rules

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A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry $G$ in $1+1\mathrm{D}$ 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 on-site symmetry $G$ defined on a 1D spatial lattice with continuous time, such that its low-energy physics produces a $1+1\mathrm{D}$ anomaly-free chiral matter theory of symmetry $G$?'' In particular, we propose that the chiral fermion theory with chiral U(1) ${3}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{0}_{R}$ symmetry, with two left-moving fermions of charge 3 and 4, and two right-moving fermions of charge 5 and 0 at IR low energy, can emerge from a 1D UV spatial lattice with a chiral U(1) symmetry, if we include properly designed multi-fermion interactions with intermediate strength (i.e., the dimensionless coupling constant is naturally order 1). In general, we propose that any $1+1\mathrm{D}$ U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with on-site symmetry by using a quantum Hamiltonian with continuous time, but without suffering 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 anomaly-free 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. Since a lattice on-site internal symmetry can be gauged, we thus can further define a nonperturbative regularization of any anomaly-free U(1) chiral gauge theory in $1+1\mathrm{D}$. As an additional remark, we show a topological nonperturbative proof of the equivalence relation between the $G$-symmetric 't Hooft anomaly cancellation conditions and the $G$-symmetry-preserving gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multiple U(1) symmetries. We expect that our result holds universally regardless of spatial Hamiltonian or spacetime Lagrangian/path integral formulation of quantum theory. Numerical tests on our proposal are demanding tasks but highly desirable for future work.

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We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a non-perturbative definition of chiral gauge theory --- a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gravitational anomaly) by a lattice, thanks to the recently proven Poincare dual equivalence between the U(1) 't Hooft anomaly free condition and the U(1) symmetric interaction gapping rule, via a bosonization-fermionization technique.

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

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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.

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A Lattice Chiral Boson Theory in $1+1$d

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Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a $1+1$d chiral field theory as the edge of a $2+1$d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the $2+1$d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a $1+1$d spacetime lattice which carries an anomalous chiral $U(1)$ symmetry with zero chiral central charge. The lattice theory with anomalous chiral $U(1)$ symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.

arXiv:2305.03024•May 4, 2023•0 citations

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We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for $U(1)$ symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the $U(1)$ hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.

hep-thCross-listedarXiv:2601.04304•Jan 9, 2026•0 citations

1+1d Lattice Dirac Fermions from Non-Onsite Vector and Axial Symmetries

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We construct an exactly solvable Hamiltonian lattice model realizing a 1+1d Dirac fermion, with exact microscopic vector and axial vector $U(1)$ symmetries. The mixed anomaly between these is accommodated by the not-on-site action of the symmetries. Our Hilbert space is a $Z_2$-graded tensor product of local $Z_2$-graded Hilbert spaces which include infinite dimensional rotor degrees of freedom. The Hamiltonian becomes manifestly exactly solvable after a locality-preserving unitary mapping to an equivalent fermionic Villain Hilbert space. Our construction also allows an exactly solvable realization of interacting fermionic Luttinger liquids. At the free Dirac fixed point, our Hamiltonian contains irrelevant interactions, which we compute to leading order.

arXiv:2608.02722•Aug 3, 2026•0 citations
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.
A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry GGG in 1+1D1+1\mathrm{D}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 on-site symmetry GGG defined on a 1D spatial lattice with continuous time, such that its low-energy physics produces a 1+1D1+1\mathrm{D}1+1D anomaly-free chiral matter theory of symmetry GGG?'' In particular, we propose that the chiral fermion theory with chiral U(1) ${3}_{L}\text{\ensuremath{-}}{5}_{R}\text{\ensuremath{-}}{4}_{L}\text{\ensuremath{-}}{0}_{R}$ symmetry, with two left-moving fermions of charge 3 and 4, and two right-moving fermions of charge 5 and 0 at IR low energy, can emerge from a 1D UV spatial lattice with a chiral U(1) symmetry, if we include properly designed multi-fermion interactions with intermediate strength (i.e., the dimensionless coupling constant is naturally order 1). In general, we propose that any 1+1D1+1\mathrm{D}1+1D U(1)-anomaly-free chiral matter theory can be defined as a finite system on a 1D lattice with on-site symmetry by using a quantum Hamiltonian with continuous time, but without suffering 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 anomaly-free 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. Since a lattice on-site internal symmetry can be gauged, we thus can further define a nonperturbative regularization of any anomaly-free U(1) chiral gauge theory in 1+1D1+1\mathrm{D}1+1D. As an additional remark, we show a topological nonperturbative proof of the equivalence relation between the GGG-symmetric 't Hooft anomaly cancellation conditions and the GGG-symmetry-preserving gapping rules (e.g. Haldane's stability conditions for Luttinger liquid) for multiple U(1) symmetries. We expect that our result holds universally regardless of spatial Hamiltonian or spacetime Lagrangian/path integral formulation of quantum theory. Numerical tests on our proposal are demanding tasks but highly desirable for future work.
We show that the 3450 U(1) chiral fermion theory can appear as the low energy effective field theory of a 1+1D local lattice model, with an on-site U(1) symmetry and finite-range interactions. The on-site U(1) symmetry means that the U(1) symmetry can be gauged (gaugeable for both background probe and dynamical fields), which leads to a non-perturbative definition of chiral gauge theory --- a chiral fermion theory coupled to U(1) gauge theory. Our construction can be generalized to regularize any U(1)-anomaly-free 1+1D gauged chiral fermion theory with a zero chiral central charge (thus no gravitational anomaly) by a lattice, thanks to the recently proven Poincare dual equivalence between the U(1) 't Hooft anomaly free condition and the U(1) symmetric interaction gapping rule, via a bosonization-fermionization technique.
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.
A Lattice Chiral Boson Theory in 1+11+11+1d
Chiral field theories describe large classes of matter, from the edges of Quantum Hall systems to the electroweak sector of the Standard Model, but defining them on the lattice has been an ongoing challenge due to a no-go theorem precluding free local models, the potential of symmetry anomalies, and sign problems. Some approaches define a 1+11+11+1d chiral field theory as the edge of a 2+12+12+1d system and argue that the edge decouples from the bulk, but this can be difficult to verify due to finite size effects and strong interactions. On the other hand, recent work has shown how to define the 2+12+12+1d bulk theory as an exactly solvable model with zero correlation length, in which case the edge theory may be extracted exactly. We use these techniques to derive a lattice field theory on a 1+11+11+1d spacetime lattice which carries an anomalous chiral U(1)U(1)U(1) symmetry with zero chiral central charge. The lattice theory with anomalous chiral U(1)U(1)U(1) symmetry is always gapless, regardless of lattice interactions. We demonstrate the chiral anomaly by coupling to a background gauge field, develop a field theory which demonstrates the chiral behavior, and show how to assemble a chiral, anomaly-free theory where the gauge field may be taken to be dynamical.
We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for U(1)U(1)U(1) symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the U(1)U(1)U(1) hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.
We construct an exactly solvable Hamiltonian lattice model realizing a 1+1d Dirac fermion, with exact microscopic vector and axial vector U(1)U(1)U(1) symmetries. The mixed anomaly between these is accommodated by the not-on-site action of the symmetries. Our Hilbert space is a Z2Z_2Z2​-graded tensor product of local Z2Z_2Z2​-graded Hilbert spaces which include infinite dimensional rotor degrees of freedom. The Hamiltonian becomes manifestly exactly solvable after a locality-preserving unitary mapping to an equivalent fermionic Villain Hilbert space. Our construction also allows an exactly solvable realization of interacting fermionic Luttinger liquids. At the free Dirac fixed point, our Hamiltonian contains irrelevant interactions, which we compute to leading order.