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 present a fermionic lattice Hamiltonian that exactly realizes the
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 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.
A nonperturbative lattice regularization of chiral fermions and bosons with anomaly-free chiral 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 on-site 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 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+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+1D. 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.
We consider relativistic U(1) gauge theories in
2+1 dimensions, with
Nb species of complex bosons and
Nf species of Dirac fermions at finite temperature. The quantum phase transition between the Higgs and Coulomb phases is described by a conformal field theory (CFT). At large
Nb and
Nf, but for arbitrary values of the ratio
Nb∕Nf, we present computations of various critical exponents and universal amplitudes for these CFTs. We make contact with the different spin liquids, charge liquids, and deconfined critical points of quantum magnets that these field theories describe. We compute physical observables that may be measured in experiments or numerical simulations of insulating and doped quantum magnets.
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
Z2 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∈{0,1} sharing one dynamical
U(1) gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge
QGA whose continuum non-conservation
⟨dQGA/dt⟩=−(2g/π)∫dx⟨E(x)⟩ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the
χ flavour exposes a hidden
UL(2)×UR(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.