Onsiteability of Higher-Form Symmetries

TL;DR

The paper investigates when higher-form symmetries in lattice models can be realized onsite by augmenting with ancillas and applying finite-depth circuits. It proves a precise criterion for finite 1-form symmetries in (2+1)D: onsiteability is equivalent to the vanishing of the transgression $\Phi([\omega_4])$ in $H^3(BG,U(1))$ of the lattice anomaly index $[\omega_4]$ in $H^4(B^2G,U(1))$, and shows such onsiteable symmetries admit transversal Pauli realizations. Beyond (2+1)D, it introduces lattice anomaly indices valued in QCAs, such as $[\omega_3]\in H^3(B^2G,\mathbb{Q}_+)$ for (3+1)D, whose transgression $\Phi([\omega_3])\in H^2(BG,\mathbb{Q}_+)$ obstructs onsiteability, and conjectures a general criterion: a finite $p$-form symmetry in $(d+1)$D is onsiteable iff all suspended lattice-anomaly indices vanish after successive transgressions. The work provides concrete examples, including a non-onsiteable semion and an onsiteable fermionic 1-form symmetry in lattice realizations, and offers a unified lattice perspective on onsiteability and higher gauging with potential implications for fault-tolerant quantum codes and generalized symmetry classifications.

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.

Figures (4)

• Figure 1: (a): Symmetry operators are defined on a mesoscopic dual lattice $\hat{\Lambda}$. (b): The local operator $O_e$ is supported at the intersection between edges of $\Lambda$ and $\partial R$.
• Figure 2: A cycle $\hat{\gamma}$ of the dual lattice separates the disk region $R$ into half, up ($u$) and down ($d$) region.
• Figure 3: (a) By introducing a 1d ancilla and disentangler on each mesoscopic edge, the Gauss law operator $W_p$ is brought into onsite form. (b) After bringing into onsite form, each vertex has three local operators $O_j^{(g)}$ ($j=1,2,3$) that are local symmetry operators at the corners of each plaquette.
• Figure 4: For (3+1)D $G$ 1-form symmetry, the operator $\Omega(\epsilon_{01}, \epsilon_{12}, g_{012})$ is a network of 1d QCAs supported at the 2d dual lattice $\hat{\Lambda}_{\partial R}$. Each red edge in the figure carries a 1d QCA, therefore assigns a QCA index valued in $\mathbb{Q}_+$. This assignment defines a 1-cocycle $F\in Z^1(\Lambda_{\partial R}, \mathbb{Q}_+)$.