Anomaly and symmetry-charge flow in mixed states

TL;DR

This paper generalizes the concept of quantum anomalies to mixed states, showing that symmetry-charge flow can be universally defined beyond pure states, applicable to fermionic and bosonic systems.

Contribution

It derives a universal anomaly relation for mixed states using an algebraic approach, extending spectral flow concepts to non-pure states.

Findings

Anomaly coefficients become model-dependent in naive extensions for mixed states.

The derived symmetry-charge flow restores universality of anomalies in mixed states.

Validated results in fermionic and spin models with continuous and discrete symmetries.

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.