Onsiteability of Higher-Form Symmetries

TL;DR

This paper investigates the conditions under which higher-form symmetries in lattice models can be made onsite, linking onsiteability to the ability to perform higher gauging and analyzing anomalies.

Contribution

It clarifies the criteria for onsiteability of higher-form symmetries, especially in (2+1)D, by establishing an equivalence with higher gauging and anomaly conditions.

Findings

Onsiteability of 1-form symmetry in (2+1)D is characterized by a specific algebraic anomaly condition.

Any onsiteable 1-form symmetry in (2+1)D can be transformed into transversal Pauli operators via ancillas and circuits.

Necessary conditions for onsiteability in higher dimensions are derived from lattice 't Hooft anomalies.

Abstract

An internal symmetry in a lattice model is said to be onsiteable if it can be disentangled into an onsite action by introducing ancillas and conjugating with a finite-depth circuit. A standard lore holds that onsiteability is equivalent to being anomaly-free, which is indeed valid for finite 0-form symmetries in (1+1)D. However, for higher-form symmetries, these notions become inequivalent: a symmetry may be onsite while still anomalous. In this work, we clarify the conditions for onsiteability of higher-form symmetries by proposing an equivalence between onsiteability and the possibility of $higher$ gauging. For a finite 1-form symmetry in (2+1)D, we show that the symmetry is onsiteable if and only if its 't Hooft anomaly satisfies a specific algebraic condition that ensures the symmetry can be 1-gauged. We further demonstrate that onsiteable 1-form symmetry in (2+1)D can always be brought into transversal Pauli operators by ancillas and circuit conjugation. In generic dimensions, we derive necessary conditions for onsiteability using lattice 't Hooft anomaly of higher-form symmetry, and conjecture a general equivalence between onsiteability and possibility of higher gauging on lattices.