Lattice Anomalies in Quantum Systems

• Lattice anomalies are defined as topological obstructions to realizing global symmetries in lattice systems using blend equivalence and topological invariants.
• They are analyzed through bulk-boundary correspondences employing quantum cellular automata and commuting projector models to link higher-dimensional phases with boundary phenomena.
• The classification uses generalized cohomology, extending traditional QFT anomaly frameworks to capture lattice-specific and fermionic symmetry structures.

Lattice anomalies are robust, topological obstructions to the realization or gauging of global symmetries in quantum many-body lattice systems, generalizing the concept of 't Hooft anomalies familiar from continuum quantum field theory (QFT). On the lattice, a symmetry anomaly is not tied to individual Hamiltonians but is a property of the symmetry action itself, classified by topological invariants associated with the system’s dimensionality, symmetry group, and the structure of locality-preserving operators such as quantum cellular automata (QCA). Lattice anomalies broadly encompass both those that have continuum (QFT) counterparts and those intrinsic to the lattice context—including anomalies that are infrared (IR) trivial, meaning they do not obstruct the existence of a trivial, symmetric, gapped ground state, yet have significant physical consequences (Tu et al., 28 Jul 2025).

1. Definition via Blend Equivalence and Topological Invariants

A lattice anomaly is formally defined using the equivalence relation of blend equivalence on local representations (G–reps) of the symmetry group $G$. Two G–reps are blend equivalent if one can interpolate between their actions on the left and right half-spaces by constructing a locally-defined symmetry that matches each action asymptotically on its respective side. The set of blend-equivalence classes forms an abelian group, which is assigned as the group of lattice anomalies for $G$ in a given spatial dimension (Tu et al., 28 Jul 2025).

The key distinction from QFT anomalies is that lattice anomalies arise from the obstruction to defining symmetry actions that are both strictly local (on-site) and global, and may persist even in settings without the usual QFT continuum structure. In fermionic systems, this framework generalizes naturally: representations act on $\mathbb{Z}_2$-graded local algebras, using the fermionic tensor product to define the stackable anomaly classes.

2. Bulk–Boundary Correspondence and Quantum Cellular Automata

A central tool for understanding lattice anomalies is the bulk–boundary correspondence, constructed using symmetric QCA and invertible commuting-projector models. Given a blend-equivalence class of G–rep on the lattice, one can always construct a $(d+1)$-dimensional symmetric QCA whose boundary algebra, with the induced action, realizes the original anomaly class. Conversely, a QCA or commuting-projector model phase with appropriate symmetry properties in $(d+1)$ dimensions defines a unique boundary anomaly in $d$ dimensions (Tu et al., 28 Jul 2025).

The precise correspondence is formulated in terms of projected or boundary algebras: the algebra of operators acting faithfully within the ground state subspace of a symmetric commuting-projector Hamiltonian. For invertible models, these boundary algebras inherit the symmetry action from the bulk QCA, giving rise to the boundary G–rep classifying the lattice anomaly.

3. Classification and Generalized Cohomology

The classification of lattice anomalies employs both algebraic and homotopical tools. The group of blend-equivalence classes can be described via a generalized cohomology theory $h^{d+1}_{\rm QCA}(BG)$, where $BG$ is the classifying space for $G$ and the coefficients come from the spectrum of QCAs. This recovers the known group cohomology invariants for finite groups in low dimensions ($H^2$, $H^3$), but encompasses a broader set of possibilities—including certain translation symmetry anomalies and fermion parity-related classes.

A crucial result is that this generalized cohomology classification provides more detailed resolution than conventional group cohomology, allowing for the identification of anomalies that do not have counterparts in the QFT infrared limit or for which the bulk-boundary correspondence breaks down in the absence of invertibility (e.g., nontrivial translation symmetry cases) (Tu et al., 28 Jul 2025).

4. Lattice vs. QFT Anomalies: Correspondence and Discrepancies

Lattice anomalies reproduce many properties familiar from QFT anomalies but fundamentally differ in several respects. Notably, there is not always a one-to-one correspondence between lattice and QFT anomalies. Some lattice anomalies are IR trivial—they admit symmetric, short-range entangled (trivial) gapped ground states and correspond to trivial QFT anomalies at low energies. Conversely, certain QFT anomalies may not have representatives that lift to the lattice setting, particularly when constraints of locality or finite onsite Hilbert space are applied (Tu et al., 28 Jul 2025).

The map from lattice anomaly classification to QFT anomalies is constructed by acting with a G–QCA on a symmetric product state and examining the resulting SPT edge anomaly; however, this map is neither injective nor surjective in general.

5. Fermionic Systems and Graded Structures

In systems with fermions, the anomaly framework is generalized to account for the graded structure of local algebras. Operators are sorted into even and odd sectors under fermion parity, and representations must respect this structure. The classification is enhanced by considering additional data such as the central extension of the symmetry group by $\mathbb{Z}_2^f$, and by analogous blend-equivalence and stacking rules in the fermionic category (Tu et al., 28 Jul 2025).

The existence and invertibility of fermionic representations (G–reps with a central $\mathbb{Z}_2^f$) can require the addition of trivial ancillas, and the anomaly classification thereby incorporates potentially richer structure, including new types of invertible and non-invertible phases.

6. Projected Algebras and Infinite Systems

For infinite systems, the algebraic formalism is refined to define projected and boundary algebras, primarily using either states on locally $C^*$-algebras or net representations with a system of compatible local projections. These approaches enable a consistent definition of the symmetry action within the ground-state manifold and thereby allow for the classification of anomalies in systems where the bulk is infinite-dimensional (Tu et al., 28 Jul 2025).

Such projected algebras are essential in establishing the structure of boundary anomalies and verifying the equivalence between the various algebraic and cohomological approaches to anomaly classification.

7. Applications, Extensions, and Open Directions

Lattice anomalies have concrete applications in analyzing commuting projector models, many-body localized (MBL) phases, and the design of symmetric quantum circuits. For example, in Abelian quantum double models, the induced automorphism of anyon excitations by the symmetry action can determine the $H^2$ anomaly invariant. The framework generalizes further to incorporate translation symmetry anomalies, higher-form (or higher-group) symmetries, and the paper of “non-invertible” phases where bulk-boundary correspondences are subtler.

The generalized cohomology perspective informs ongoing efforts to provide a complete classification of anomalies in quantum many-body systems, distinguish between those obstructions that are robust in the IR limit and those that are lattice artifacts, and connect these phenomena to anomaly inflow and SPT edge theories in the corresponding field-theoretic constructions (Tu et al., 28 Jul 2025).

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In summary, the rigorous definition and classification of lattice anomalies using blend equivalence, bulk-boundary correspondences, quantum cellular automata, and generalized cohomology forms a unifying framework that extends traditional QFT anomaly classification to the lattice setting. This approach captures both familiar and intrinsically lattice-specific anomaly phenomena, informs the construction and classification of symmetric Hamiltonians and ground states, and elucidates the interplay between microscopic symmetry realization, locality, and topological obstructions in quantum many-body systems.