How do lattice anomalies of global symmetries differ from 't Hooft anomalies?

An anomaly belongs to the symmetry action, not the Hamiltonian

The paper's central move is a definition. A 't Hooft anomaly is usually introduced through a field theory: one writes down an action, then derives an obstruction to gauging the symmetry or a constraint on renormalization group flow. The authors argue that this obscures the key principle that an anomaly is a universal property of the way the symmetry acts, not a feature of a specific lattice Hamiltonian or QFT action. Their image is a stage and a play: the anomaly is the stage, and the dynamics is the particular play performed on it.

To make that principle precise, they define a lattice anomaly as a topological invariant of the symmetry action itself. The definition does not appeal to field theory or background gauge fields and is phrased in terms of the symmetry action rather than any given Hamiltonian. The controlled, mathematically precise setting of lattice systems is what allows this treatment, avoiding the technical challenges of QFT.

This reframing matters for how readers should read every later result. When the paper says two systems share a lattice anomaly, it is making a statement about an equivalence class of symmetry actions, not about a shared Hamiltonian, a shared ground state, or a shared low-energy theory.

Blend equivalence turns symmetry actions into a group

To classify anomalies, the authors need a notion of when two symmetry actions count as the same. They work with representations of a symmetry group G on the local algebra of a lattice system, called G-reps, and define blend equivalence: two G-reps are equivalent if one can be continuously deformed into the other across space, allowing the deformation to differ on the left and right halves of the lattice and to be joined in a transition region.

Stacking, meaning taking tensor products, gives the resulting set of lattice anomaly classes in d spatial dimensions the structure of an Abelian group, written as Anom_lattice^d(G). Inverses exist through a folding argument: reflecting one half of space turns a G-rep tensored with its reflection into the trivial rep. The paper restricts the blend-equivalence discussion to d ≥ 1 because the construction does not make sense in d = 0.

The authors compare this with conjugacy equivalence, an older notion in which a quantum cellular automaton V conjugates one G-rep into another. Conjugacy equivalence implies blend equivalence, but the converse is not established in general: for noncompact groups such as G = Z, one can construct noninternal symmetries that are blend equivalent to the trivial rep. The authors conjecture that blend equivalence implies conjugacy equivalence for finite groups, noting that the d = 1 case had recently been proven elsewhere.

Some lattice anomalies vanish in the infrared but remain nontrivial

The most consequential finding is a failure of one-to-one correspondence. There is a map from lattice anomalies to QFT anomalies: for a lattice system with a global symmetry, one should be able to construct a Hamiltonian whose low-energy, long-wavelength effective theory is a QFT inheriting some anomaly from the lattice symmetry. The paper gives a more precise characterization of this map.

But the map is not injective. The authors find nontrivial lattice anomalies that are infrared trivial: they admit symmetric trivial gapped ground states and map to trivial QFT anomalies at low energies. In other words, a lattice anomaly can be a genuine topological feature of the symmetry action while leaving no trace in the low-energy field theory. This contradicts a common expectation that lattice and QFT anomalies stand in one-to-one correspondence.

The result also cuts the other way for interpretation. Because an IR-trivial lattice anomaly still constrains symmetric Hamiltonians, a reader cannot conclude that an anomaly is physically irrelevant just because its low-energy limit looks anomaly-free. The paper treats these classes as interesting in their own right rather than as artifacts to be quotiented away.

A bulk-boundary correspondence built from commuting models

QFT anomalies in d dimensions are tied to symmetry-protected states in d+1 dimensions through a bulk-boundary correspondence. The paper shows an analogous correspondence for d-dimensional lattice anomalies, but the bulk must be an invertible phase of (d+1)-dimensional commuting models rather than a general gapped phase.

In the case without symmetry, the authors establish a three-way isomorphism: lattice gravitational anomalies in d dimensions, invertible phases of commuting models in d+1 dimensions, and equivalence classes of quantum cellular automata in d+1 spatial dimensions all coincide. The proof runs through a commutative diagram of homomorphisms between these groups, with the boundary algebra of the commuting model providing one map and the action of QCAs on a trivial commuting model providing another.

With symmetry, the naive extension of this three-way correspondence is flagged in the paper as incorrect. The restriction that the QCA be symmetric under an on-site G-rep limits which G-reps can appear on the boundary to those with a conjugacy inverse, all of which are internal. The authors therefore introduce conjugacy anomalies and a group of G-invertible commuting phases as the appropriate objects, and they present the corrected correspondence as a conjecture rather than a theorem.

IR-trivial anomalies still constrain MBL phases and QCAs

The paper argues that lattice anomalies, including the IR-trivial ones, have consequences in their own right. It connects them to commuting projector models, to phases of many-body localized systems, and to quantum cellular automata. These are the settings where the anomaly's constraint on symmetric Hamiltonians becomes visible without passing through a low-energy QFT.

One concrete link is the classification of invertible commuting models, which the authors show corresponds to the classification of lattice anomalies. For non-invertible commuting models the questions remain open. The paper also notes that the classification of short-range-entangled MBL phases in spatial dimension d+1 is in general isomorphic to a quotient of the QCA group by the subgroup of QCAs that can preserve an on-site LIOM algebra; the authors conjecture that this subgroup is trivial for d ≥ 1 but do not prove it.

The authors are explicit about what remains unfinished. They report substantial progress on classifying lattice anomalies and on developing theoretical tools to characterize consequences for symmetric Hamiltonians, but several statements, including the symmetric bulk-boundary correspondence, are presented as conjectures. A general definition of non-invertible symmetry on the lattice analogous to their QCA-based definition for invertible symmetries is left as a necessary first step for future work.

Where the lattice picture departs from the QFT picture

The paper's most useful contribution for a reader comparing the two frameworks is a precise statement of where they diverge. Lattice anomalies reproduce many expected properties of QFT anomalies, but the classification of lattice anomalies does not exactly correspond to the classification of QFT anomalies. The obstruction is not a technical gap to be closed later; it is the existence of nontrivial lattice anomaly classes that are IR trivial.

That changes the practical question a reader should ask. Instead of asking only whether a symmetry action descends to a nontrivial anomaly in the low-energy QFT, one should ask separately whether the lattice symmetry action is anomalous and whether that anomaly survives in the infrared. The paper's IR-trivial classes are exactly the cases where the two answers differ, and they are also the cases where the consequences for commuting models, MBL phases, and QCAs carry the weight.

The authors also caution against expecting lattice symmetric algebras to correspond one-to-one with symmetry topological field theories, because a symTFT is a QFT that does not know about the lattice. Their framework fixes the symmetry group G, whereas classifying only the local isomorphism class of the symmetric algebra can make different symmetry groups equivalent. The priority the paper establishes is therefore to keep the lattice anomaly as the primary object and treat the QFT anomaly as a possibly lossy image of it.

Sources used in this answer