An internal symmetry in a lattice model is said to be onsiteable if it can be disentangled into an on-site 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 on site while still anomalous. In this Letter, 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. We also identify new "lattice anomalies" intrinsic to higher-form symmetry of lattice models, obstructing on-site realization of symmetry beyond the possibility of higher gauging in continuum.
We present counterexamples to the lore that symmetries that cannot be gauged or made on site are necessarily anomalous. Specifically, we construct unitary, internal symmetries of two-dimensional lattice models that cannot be consistently coupled to background or dynamical gauge fields or disentangled to a tensor product of on-site operators. These symmetries are nevertheless anomaly-free in the sense that they admit symmetric, gapped Hamiltonians with unique, invertible ground states. We show that symmetries of this kind are characterized by an index [ω]∈H^{2}(G,Q_{+}), where Q_{+} is the multiplicative group of positive rational numbers labeling one-dimensional quantum cellular automata.
We propose a Hamiltonian framework for constructing chiral gauge theories on the lattice based on symmetry disentanglers: constant-depth circuits of local unitaries that transform not-on-site symmetries into on-site ones. When chiral symmetry can be realized not-on-site and such a disentangler exists, the symmetry can be implemented in a strictly local Hamiltonian and gauged by standard lattice methods. Using lattice rotor models, we realize this idea in 1+1 and 3+1 spacetime dimensions for
U(1) symmetries with mixed 't Hooft anomalies, and show that symmetry disentanglers can be constructed when anomalies cancel. As an example, we present an exactly solvable Hamiltonian lattice model of the (1+1)-dimensional "3450" chiral gauge theory, and we argue that a related construction applies to the
U(1) hypercharge symmetry of the Standard Model fermions in 3+1 dimensions. Our results open a new route toward fully local, nonperturbative formulations of chiral gauge theories.
We clarify the lore that anomaly-free symmetries are either on-site or can be transformed into on-site symmetries. We prove that any finite, internal, anomaly-free symmetry in a 1+1D lattice Hamiltonian system can be disentangled into an on-site symmetry by introducing ancillae and applying conjugation via a finite-depth quantum circuit. We provide an explicit construction of the disentangling circuit using Gauss's law operators and emphasize the necessity of adding ancillae. Our result establishes the converse to a generalized Lieb-Schultz-Mattis theorem by demonstrating that any anomaly-free symmetry admits a trivially gapped Hamiltonian.
We study invertible phases protected by non-onsite symmetries carrying nontrivial lattice anomaly indices. While lattice anomalies are usually viewed as obstructions to symmetric short-range-entangled (SRE) states, we show that anomalous lattice symmetries can nevertheless admit invertible phases that are not SRE, and can moreover shift the set of symmetry-compatible invertible phases. In particular, we consider a fermionic
Z4F symmetry in (2+1) dimensions. For an onsite
Z4F symmetry, symmetric invertible phases have integer chiral central charge
c−∈Z, whereas a non-onsite symmetry with nontrivial lattice anomaly index obstructs all such phases despite having trivial continuum 't Hooft anomaly. We resolve this by constructing an exact exponentially quasi-local
Z4F symmetry of a
p+ip superconductor with
c−=1/2. We compute its lattice anomaly, and find that the symmetry indeed forbids
c−∈Z invertible states. The allowed invertible phases are consequently shifted to
c−∈Z+1/2. By gauging the fermion parity of the
p+ip superconductor, one obtains an Ising topological order enriched by a
Z4 symmetry. We describe the corresponding symmetry structure in terms of a fusion 2-category. We further show that the
p+ip state admits an enlarged non-onsite
U(1)f symmetry, providing an analogue of fractional quantum Hall response in an invertible phase. This non-onsite
U(1)f symmetry again forbids Chern insulators carrying
c−∈Z, while being compatible with invertible states with
c−∈Z+1/2. This result motivates a systematic study of new classes of invertible phases and spin liquids enriched by non-onsite symmetries.
Higher-form symmetry in a tensor product Hilbert space is always emergent: The symmetry generators become genuinely topological only when the Gauss law is energetically enforced at low energies. In this Letter, we present a general method for defining the 't Hooft anomaly of higher-form symmetries in lattice models built on a tensor product Hilbert space. In (2+1)D, for given Gauss-law operators realized by finite-depth circuits that generate a finite 1-form G symmetry, we construct an index representing a cohomology class in H^{4}(B^{2}G,U(1)), which characterizes the corresponding 't Hooft anomaly. This construction generalizes the Else-Nayak characterization of 0-form symmetry anomalies. More broadly, under the assumption of a specified formulation of the p-form G symmetry action and Hilbert space structure in arbitrary d spatial dimensions, we show how to characterize the 't Hooft anomaly of the symmetry action by an index valued in H^{d+2}(B^{p+1}G,U(1)).
We develop a lattice characterization of fermionic 't Hooft anomalies of finite internal symmetries in one and two spatial dimensions, formulated in terms of obstructions to symmetric short-range-entangled (SRE) states. We consider lattice systems formed by tensor product of onsite fermionic and bosonic local degrees of freedom, and finite internal symmetry given by a central extension
Z2F→Gf→Gb. We extract a hierarchy of fermionic anomaly indices for a given symmetry operator in both (1+1)D and (2+1)D. In (1+1)D, an exact lattice symmetry is characterized by a pair of cohomological data
(n2,ν3). For
Gf=Gb×Z2F, we further show that a symmetry with trivial lattice anomaly indices
(n2,ν3) is onsiteable and hence admits a symmetric SRE state, establishing that these indices faithfully detect the lattice anomaly. Comparing with continuum fermionic QFT, we find that exact lattice symmetries do not realize the additional
H1(BGb,Z2) anomaly layer in continuum QFT. In particular, for
Gb=Z2, exact lattice symmetries realize only the even
Z4 subgroup of the continuum
Z8 classification. In (2+1)D, we identify three successive anomaly layers of cohomological data
(n2,n3,ν4). We show that a nontrivial value of any layer obstructs a symmetric SRE state. For
Gf=Gb×Z2F, it also forbids a symmetric invertible state. We also construct a non-onsite
Z4F lattice symmetry in (2+1)D with nontrivial lattice anomaly index that forbids any symmetric invertible states with
c−∈Z, therefore forbids any invertible states compatible with onsite
Z4F symmetry. Our results motivate a systematic study of which continuum anomalies admit exact microscopic lattice realizations.