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arXiv:2604.24872

Anomaly and symmetry-charge flow in mixed states

Ze-Min Huang, Sebastian Diehl

Published Apr 27, 2026• 0 citations
arXivHTMLPDF

Abstract

The $(1+1)$-dimensional chiral anomaly is a paradigmatic exact result in quantum field theory, traditionally formulated for zero-temperature pure states where it arises from spectral flow induced by external gauge fields and captures universal ground-state properties. In mixed states, however, the participation of many states and charge exchange with the environment invalidate this mechanism. Naive extensions yield model-dependent anomaly coefficients, calling its universality into question. Here, we resolve this problem for Abelian symmetries by deriving the anomaly from an algebraic relation between the symmetry and its flux-insertion operator. We obtain symmetry-charge flow, a mixed-state generalization of spectral flow, in which an applied field redistributes statistical weight across symmetry-resolved charge sectors. Fixed solely by symmetry, the anomaly restores universality and applies to both pure and mixed states in fermionic and bosonic systems. We substantiate these results in tight-binding fermionic models with continuous symmetry and in spin models with discrete symmetries.

DOI
10.48550/arxiv.2604.24872

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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.
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We introduce the \emph{flavoured lattice Schwinger model}, a (1+1)(1{+}1)(1+1)-dimensional U(1)U(1)U(1) lattice gauge theory in which the fermion doubling problem is resolved by staggering a Z2\mathbb{Z}_{2}Z2​ flavour degree of freedom rather than staggering chirality. Unlike the standard approaches, this construction preserves an exact axial U(1)U(1)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}α\in\{0,1\}α∈{0,1} sharing one dynamical U(1)U(1)U(1) gauge field. The central result is a well-defined, regularised, gauge-invariant lattice axial charge QGAQ_{G}^{A}QGA​ whose continuum non-conservation ⟨dQGA/dt⟩=−(2g/π) ⁣∫ ⁣dx ⟨E(x)⟩\langle dQ_{G}^{A}/dt\rangle = -(2g/π)\!\int\! dx\,\langle E(x)\rangle⟨dQGA​/dt⟩=−(2g/π)∫dx⟨E(x)⟩ arises as a direct dynamical consequence of minimal gauge coupling. A particle-hole transformation on the χχχ flavour exposes a hidden UL(2)×UR(2)U_{L}(2)\times U_{R}(2)UL​(2)×UR​(2) chiral symmetry; non-Abelian bosonisation then identifies the model with a massive abelian Schwinger sector tensored with the level-111 SU(2)SU(2)SU(2) Wess--Zumino--Witten model. Finally, we show that embedding the flavoured fermions in a ribbon-shaped (2+1)(2{+}1)(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 222 as one quantum of Schwinger anomaly per edge.