arXiv:2204.05328 · Physical review. B./Physical review. B

Current algebra approach to two-dimensional interacting chiral metals

Chao-Jung Lee, Michael Mulligan

Published May 25, 2023• 5 citations

Abstract

We reinterpret the chiral $U(N)$ Wess-Zumino-Witten (WZW) model at level $k>1$ in $(1+1)$ dimensions as an interacting chiral metal in two space dimensions. In this reinterpretation, spatial translations along one of the spatial dimensions in the two-dimensional chiral metal arise from a generator of the $U(N)$ symmetry of the WZW model. The WZW model at $k=1$ is equivalent to Balents and Fisher's free chiral metal [L. Balents and M. P. A. Fisher, Phys. Rev. Lett. 76, 2782 (1996)]. Here, the $U(N)$ symmetry corresponds to the IR symmetry of a chiral Fermi gas with (half of a) Fermi surface, with $N$ equal to the number of points on the Fermi surface. We argue that exactly solvable interacting generalizations occur for levels $k>1$. Importantly, these interacting chiral metals maintain the $U(N)$ symmetry of the free system. We calculate two-point correlation functions of the single-particle fermion operator, the $U(1)$ number density, and current operators in these theories for general $k$. We find that interactions ($k>1$) produce $1/N$ corrections to scaling of the single-particle fermion operator as $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and renormalize the amplitudes of the density and current two-point functions. This construction illustrates the ersatz Fermi liquid proposal of Else et al. [Phys. Rev. X 11, 021005 (2021)].

DOI: 10.1103/physrevb.107.205147

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Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d

Yichen Hu, Biao Lian

arXiv:2109.13263Physical review. B./Physical review. BMar 8, 20228 citations

Note on Wess-Zumino-Witten models and quasiuniversality in 2+1 dimensions

Adam Nahum

arXiv:1912.13468Physical review. B./Physical review. BNov 30, 202080 citations

Flavoured Lattice Schwinger Model with Chiral Anomaly

Dogukan Bakircioglu

hep-latCross-listedarXiv:2604.13146Apr 15, 2026

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

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

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

Vladimir A. Zakharov, Atsushi Ueda, Frank Verstraete +1 more

Code, data and a notebook to reproduce the figures of V. A. Zakharov, A. Ueda, F. Verstraete, C. W. J. Beenakker, "Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling".

Jun 23, 20261 citation

Dual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operators

Wu, Zhengzhi, Linhao Li

arXiv:2208.10518Aug 22, 20222 citations

arXiv:2204.05328 · Physical review. B./Physical review. B

Current algebra approach to two-dimensional interacting chiral metals

Chao-Jung Lee, Michael Mulligan

Published May 25, 2023• 5 citations

arXiv HTML PDF

Abstract

We reinterpret the chiral

U(N)

Wess-Zumino-Witten (WZW) model at level

k>1

(1+1)

dimensions as an interacting chiral metal in two space dimensions. In this reinterpretation, spatial translations along one of the spatial dimensions in the two-dimensional chiral metal arise from a generator of the

U(N)

symmetry of the WZW model. The WZW model at

k=1

is equivalent to Balents and Fisher's free chiral metal [L. Balents and M. P. A. Fisher, Phys. Rev. Lett. 76, 2782 (1996)]. Here, the

U(N)

symmetry corresponds to the IR symmetry of a chiral Fermi gas with (half of a) Fermi surface, with

N

equal to the number of points on the Fermi surface. We argue that exactly solvable interacting generalizations occur for levels

k>1

. Importantly, these interacting chiral metals maintain the

U(N)

symmetry of the free system. We calculate two-point correlation functions of the single-particle fermion operator, the

U(1)

number density, and current operators in these theories for general

k

. We find that interactions (

k>1

) produce

1/N

corrections to scaling of the single-particle fermion operator as

N\ensuremath{\rightarrow}\ensuremath{\infty}

and renormalize the amplitudes of the density and current two-point functions. This construction illustrates the ersatz Fermi liquid proposal of Else et al. [Phys. Rev. X 11, 021005 (2021)].

DOI: 10.1103/physrevb.107.205147

Related papers

Chiral Sachdev-Ye model: Integrability and chaos of anyons in 1+1d

Yichen Hu, Biao Lian

We construct and study a chiral Sachdev-Ye (SY) model consisting of $N$ chiral $\mathrm{SU}{(M)}_{1}$ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the $0+1\mathrm{d}$ quantum chaotic SY spin model into $1+1\mathrm{d}$ chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of $2+1\mathrm{d}$ gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers $N$ and $M$ , which is integrable and decomposes into a chiral $\mathrm{SU}{(M)}_{N}$ WZW model and its coset with a different speed of light. When $N=M=2$ , the model maps to a free Majorana fermion model. The second limit is the large $N$ and $M$ limit with random interactions, which is solvable to the leading $\frac{1}{NM}$ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.We construct and study a chiral Sachdev-Ye (SY) model consisting of $N$ chiral $\mathrm{SU}{(M)}_{1}$ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the $0+1\mathrm{d}$ quantum chaotic SY spin model into $1+1\mathrm{d}$ chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of $2+1\mathrm{d}$ gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers $N$ and $M$, which is integrable and decomposes into a chiral $\mathrm{SU}{(M)}_{N}$ WZW model and its coset with a different speed of light. When $N=M=2$, the model maps to a free Majorana fermion model. The second limit is the large $N$ and $M$ limit with random interactions, which is solvable to the leading $\frac{1}{NM}$ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.

arXiv:2109.13263Physical review. B./Physical review. BMar 8, 20228 citations

Note on Wess-Zumino-Witten models and quasiuniversality in 2+1 dimensions

Adam Nahum

We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}>2$ . We suggest that ${d}_{c}<3$ for all $k$ , and we compute ${d}_{c}$ in the limit of large $k$ . The flows support the conjecture that the deconfined phase transition in $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$ , controlled by a fixed point slightly outside the physical parameter space.We suggest the possibility that the two-dimensional $\mathrm{SU}{(2)}_{k}$ Wess-Zumino-Witten (WZW) theory, which has global $\mathrm{SO}(4)$ symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with $\mathrm{SO}(5)$ symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality ${d}_{c}>2$. We suggest that ${d}_{c}<3$ for all $k$, and we compute ${d}_{c}$ in the limit of large $k$. The flows support the conjecture that the deconfined phase transition in $\mathrm{SU}(2)$ magnets is a ``pseudocritical'' point with approximate $\mathrm{SO}(5)$, controlled by a fixed point slightly outside the physical parameter space.

arXiv:1912.13468Physical review. B./Physical review. BNov 30, 202080 citations

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

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

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

Vladimir A. Zakharov, Atsushi Ueda, Frank Verstraete +1 more

Jun 23, 20261 citation

Dual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operatorsDual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operators

Wu, Zhengzhi, Linhao Li

Nonlinear $σ$ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or $\text{QED}$ theory. In this paper, we propose a dimensional reduction scheme to derive the $\text{U}(1)$ gauged topological NLSM in $n$ -dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the $N_f=4$ $\text{QED}_3$ with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold $\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}$ coupled with a dynamical $\text{U(1)}$ gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the $\text{U(1)}$ monopole operators and the corresponding quantum number.Nonlinear $σ$ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or $\text{QED}$ theory. In this paper, we propose a dimensional reduction scheme to derive the $\text{U}(1)$ gauged topological NLSM in $n$-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the $N_f=4$ $\text{QED}_3$ with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold $\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}$ coupled with a dynamical $\text{U(1)}$ gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the $\text{U(1)}$ monopole operators and the corresponding quantum number.

arXiv:2208.10518Aug 22, 20222 citations

We reinterpret the chiral

U(N)

Wess-Zumino-Witten (WZW) model at level

k>1

(1+1)

U(N)

symmetry of the WZW model. The WZW model at

k=1

is equivalent to Balents and Fisher's free chiral metal [L. Balents and M. P. A. Fisher, Phys. Rev. Lett. 76, 2782 (1996)]. Here, the

U(N)

symmetry corresponds to the IR symmetry of a chiral Fermi gas with (half of a) Fermi surface, with

N

equal to the number of points on the Fermi surface. We argue that exactly solvable interacting generalizations occur for levels

k>1

. Importantly, these interacting chiral metals maintain the

U(N)

symmetry of the free system. We calculate two-point correlation functions of the single-particle fermion operator, the

U(1)

number density, and current operators in these theories for general

k

. We find that interactions (

k>1

) produce

1/N

corrections to scaling of the single-particle fermion operator as

N\ensuremath{\rightarrow}\ensuremath{\infty}

and renormalize the amplitudes of the density and current two-point functions. This construction illustrates the ersatz Fermi liquid proposal of Else et al. [Phys. Rev. X 11, 021005 (2021)].

We construct and study a chiral Sachdev-Ye (SY) model consisting of

N

chiral

\mathrm{SU}{(M)}_{1}

Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the

0+1\mathrm{d}

quantum chaotic SY spin model into

1+1\mathrm{d}

chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of

2+1\mathrm{d}

gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers

N

and

M

, which is integrable and decomposes into a chiral

\mathrm{SU}{(M)}_{N}

WZW model and its coset with a different speed of light. When

N=M=2

, the model maps to a free Majorana fermion model. The second limit is the large

N

and

M

limit with random interactions, which is solvable to the leading

\frac{1}{NM}

order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.

We suggest the possibility that the two-dimensional

\mathrm{SU}{(2)}_{k}

Wess-Zumino-Witten (WZW) theory, which has global

\mathrm{SO}(4)

symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with

\mathrm{SO}(5)

symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality

{d}_{c}>2

. We suggest that

{d}_{c}<3

for all

k

, and we compute

{d}_{c}

in the limit of large

k

. The flows support the conjecture that the deconfined phase transition in

\mathrm{SU}(2)

magnets is a ``pseudocritical'' point with approximate

\mathrm{SO}(5)

, controlled by a fixed point slightly outside the physical parameter space.

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.

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.

Dual topological nonlinear sigma models of

\text{QED}

theory by dimensional reduction and monopole operators

Nonlinear

σ

models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or

\text{QED}

theory. In this paper, we propose a dimensional reduction scheme to derive the

\text{U}(1)

gauged topological NLSM in

n

-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the

N_f=4

\text{QED}_3

with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold

\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}

coupled with a dynamical

\text{U(1)}

gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the

\text{U(1)}

monopole operators and the corresponding quantum number.