arXiv:2305.03024

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

DeMarco, Michael, Ethan Lake, Xiao-Gang Wen

Published May 4, 2023• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2305.03024

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A solution to the chiral fermion problem: 1+1D U(1) chiral fermion theory realized by a truly local lattice model with U(1) on-site symmetry

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Jul 29, 20131 citation

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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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arXiv:1307.7480Physical review. B./Physical review. BJan 31, 202358 citations

arXiv:2305.03024

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

DeMarco, Michael, Ethan Lake, Xiao-Gang Wen

Published May 4, 2023• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2305.03024

Related papers

A solution to the chiral fermion problem: 1+1D U(1) chiral fermion theory realized by a truly local lattice model with U(1) on-site symmetry

Juven Wang, Xiao-Gang Wen

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.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.

Jul 16, 20180 citations

Chiral Phases on the Lattice

Michael Austin DeMarco

While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an $SU(2)$ chiral field theory, and provide a semiclassically solvable definition of Abelian $2+1$ chiral topological orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.While chiral quantum field theories (QFTs) describe a wide range of physical systems, from the standard model to topological quantum matter, the realization of chiral QFTs on a lattice has proved to be difficult due to the Nielsen-Ninomiya theorem and the possible presence of quantum anomalies. In this thesis, we use the connection between chiral phases of matter and chiral quantum field theories (QFTs) to define chiral QFTs on a lattice and allow a huge class of exotic field theories to be simulated numerically. Our work builds on the 'mirror fermion' approach to the problem of defining chiral theories on a lattice, which defines chiral field theories as the edge modes of chiral phases. We begin by reviewing the deep connections between chiral phases of matter, chiral field theories, and anomalies. We then develop numerical treatments of an $SU(2)$ chiral field theory, and provide a semiclassically solvable definition of Abelian $2+1$ chiral topological orders. This leads to an exactly solvable definition of chiral $U(1)$ SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.

arXiv:2203.01427Mar 2, 20220 citations

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.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, 20131 citation

Flavoured Lattice Schwinger Model with Chiral Anomaly

Dogukan Bakircioglu

We introduce the \emph{flavoured lattice Schwinger model}, a $(1{+}1)$ -dimensional $U(1)$ lattice gauge theory in which the fermion doubling problem is resolved by staggering a $\mathbb{Z}_{2}$ flavour degree of freedom rather than staggering chirality. Unlike the standard approaches, this construction preserves an exact axial $U(1)$ symmetry at finite lattice spacing. We derive the continuum limit, showing that the model reduces to the \emph{two-flavour} massless Schwinger model, with flavours $α\in\{0,1\}$ sharing one dynamical $U(1)$ gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge $Q_{G}^{A}$ whose continuum non-conservation $\langle dQ_{G}^{A}/dt\rangle = -(2g/π)\!\int\! dx\,\langle E(x)\rangle$ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the $χ$ flavour exposes a hidden $U_{L}(2)\times U_{R}(2)$ chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level- $1$ $SU(2)$ Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped $(2{+}1)$ D Bernevig--Hughes--Zhang topological insulator and gauging the bulk in a constant background field factorises the boundary theory into \emph{two decoupled} single-flavour Schwinger models, one on each edge, identifying the lattice factor of $2$ as one quantum of Schwinger anomaly per edge.We introduce the \emph{flavoured lattice Schwinger model}, a $(1{+}1)$-dimensional $U(1)$ lattice gauge theory in which the fermion doubling problem is resolved by staggering a $\mathbb{Z}_{2}$ flavour degree of freedom rather than staggering chirality. Unlike the standard approaches, this construction preserves an exact axial $U(1)$ symmetry at finite lattice spacing. We derive the continuum limit, showing that the model reduces to the \emph{two-flavour} massless Schwinger model, with flavours $α\in\{0,1\}$ sharing one dynamical $U(1)$ gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge $Q_{G}^{A}$ whose continuum non-conservation $\langle dQ_{G}^{A}/dt\rangle = -(2g/π)\!\int\! dx\,\langle E(x)\rangle$ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the $χ$ flavour exposes a hidden $U_{L}(2)\times U_{R}(2)$ chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level-$1$ $SU(2)$ Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped $(2{+}1)$D Bernevig--Hughes--Zhang topological insulator and gauging the bulk in a constant background field factorises the boundary theory into \emph{two decoupled} single-flavour Schwinger models, one on each edge, identifying the lattice factor of $2$ as one quantum of Schwinger anomaly per edge.

hep-latCross-listedarXiv:2604.13146Apr 15, 2026

Chiral Lattice Gauge Theories from Symmetry Disentanglers

Ryan Thorngren, John Preskill, Lukasz Fidkowski

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.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:2601.04304Jan 9, 20260 citations

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

Juven Wang, Xiao-Gang Wen

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.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.

arXiv:1307.7480Physical review. B./Physical review. BJan 31, 202358 citations

A Lattice Chiral Boson Theory in

1+1

1+1

d chiral field theory as the edge of a

2+1

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)

SU(2)

chiral field theory, and provide a semiclassically solvable definition of Abelian

2+1

chiral topological orders. This leads to an exactly solvable definition of chiral

U(1)

SPT phases with zero correlation length, which we use to extract the edge chiral field theories exactly. These zero-correlation length models are vastly more simple than previous approaches to defining chiral field theories on the lattice.

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.

We introduce the \emph{flavoured lattice Schwinger model}, a

(1{+}1)

-dimensional

U(1)

lattice gauge theory in which the fermion doubling problem is resolved by staggering a

\mathbb{Z}_{2}

flavour degree of freedom rather than staggering chirality. Unlike the standard approaches, this construction preserves an exact axial

U(1)

symmetry at finite lattice spacing. We derive the continuum limit, showing that the model reduces to the \emph{two-flavour} massless Schwinger model, with flavours

α\in\{0,1\}

sharing one dynamical

U(1)

gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge

Q_{G}^{A}

whose continuum non-conservation

\langle dQ_{G}^{A}/dt\rangle = -(2g/π)\!\int\! dx\,\langle E(x)\rangle

arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the

χ

flavour exposes a hidden

U_{L}(2)\times U_{R}(2)

chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level-

1

SU(2)

Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped

(2{+}1)

D Bernevig--Hughes--Zhang topological insulator and gauging the bulk in a constant background field factorises the boundary theory into \emph{two decoupled} single-flavour Schwinger models, one on each edge, identifying the lattice factor of

2

as one quantum of Schwinger anomaly per edge.

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.

A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral symmetry

G

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.