Crystalline Equivalence Principle for Anomalies

Introduction

Crystalline symmetries place fundamental constraints on the phases and low-energy dynamics of quantum many-body systems. A basic example is the Lieb–Schultz–Mattis (LSM) theorem: a spin chain with translation and internal symmetries cannot host a trivial symmetric gapped state when half-odd integer spin is located per unit cell [1]. Its higher-dimensional extensions reveal a general connection between lattice symmetry actions and the necessity of non-trivial low energy spectrum [2, 3]. Subsequent developments have established such constraints involving generic crystalline symmetries, and generic distribution of projective representations at onsite Hilbert spaces [4, 5]. These development lead to various complementary ways to formulate LSM theorems for more general combination of crystalline and internal symmetries, using the method of e.g., lattice homotopy and operator algebraic approach [6, 7, 8, 9].

The interpretation of LSM constraints as symmetry anomalies has been particularly fruitful. Connections between the LSM constraints and the symmetry action on the anomalous surface of symmetry-protected-topological (SPT) phase has been discussed in Ref. [10]. The interpretation of LSM constraints in terms of continuum anomalies has been developed in Refs. [11, 12, 13, 14]. In this description, microscopic lattice translation by a lattice unit acts in the infrared as an internal symmetry, which is called an emanant symmetry [14]. The corresponding LSM constraint is then encoded in a mixed ’t Hooft anomaly between this emanant symmetry and other internal symmetries inherited from that of the UV lattice model. The dynamical consequences of LSM type constraints involving generalized symmetries, including higher-form or non-invertible symmetries, have been discussed in e.g., [15, 16, 17, 18]. Throughout these developments, a central issue is to identify precisely which information in a microscopic symmetry action produces an anomaly and what kind of dynamical consequences follow from it.

In parallel, crystalline symmetry has become an essential ingredient in the classification of topological phases. For free fermions, topological crystalline insulators and superconductors have been studied using band topology, $K$-theory, symmetry indicators, and topological quantum chemistry [19, 20, 21, 22, 23, 24]. Interacting crystalline phases admit bottom-up real-space constructions in terms of lower-dimensional building blocks and their junctions [25, 26, 27, 28]. Generalized homology and the Atiyah–Hirzebruch spectral sequence organize the consistency conditions and equivalences among such constructions [29]. Explicit constructions of crystalline phases have also tested and refined the relation between the classification of SPT phases with internal and crystalline symmetries [30, 31, 32, 33].

A unifying principle is the crystalline equivalence principle formulated by Thorngren and Else [34]. For crystalline topological phases on Euclidean space, it identifies the classification of crystalline phases with that for the same abstract symmetry group acting internally. Mathematical formulations of the crystalline equivalence principle have been developed within generalized-homology descriptions of invertible phases [35]. The equivalence principle is also supported by microscopic probes of crystalline phases such as partial rotation operations [36, 37, 38, 39].

These results motivate an analogous question for crystalline anomalies. If an infrared theory has the Lorentz invariance, a crystalline operation is expected to act in IR through a combination of a spacetime Lorentz transformation and an internal transformation, making an internal-symmetry description natural. However, a general microscopic lattice system need not possess an IR limit with Lorentz symmetry. Moreover, it remains unclear if general crystalline anomalies are described through the inflow from bulk crystalline SPT phases, or if the classification of crystalline anomalies coincides with that of crystalline SPTs in the bulk. Establishing the classification of crystalline anomalies therefore requires a direct microscopic characterization and a systematic microscopic analysis of its dynamical consequences.

There has recently been substantial progress of characterizing the ’t Hooft anomalies at microscopic lattice models [40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. In particular, microscopic anomalies involving translation have already been studied in quantum spin chains [42, 43]. These advances enable a direct microscopic characterization of crystalline anomalies in a tensor product Hilbert space, without relying on a continuum description or anomaly inflow from bulk crystalline SPT phases.

In this work, we develop such a microscopic construction for bosonic systems in $(1+1)$ and $(2+1)$ dimensions with a combination of crystalline and finite unitary internal symmetry. We obtain explicit cocycle data for combined internal and crystalline symmetry actions, identify their ambiguities to define cohomological data, and establish a dynamical consequence as an obstruction to symmetric invertible states.

Summary of results

We consider a bosonic tensor product Hilbert space

on a lattice $\Lambda$ without boundary in spatial dimension $d = 1, 2$, where each onsite Hilbert space is assumed to be isomorphic for simplicity. The internal symmetry group $G$ is finite and unitary, and its operators $U(g)$ preserve locality. A spatial symmetry group $G_{\mathrm{sp}}$ acts on the lattice and may also act on the internal symmetry group $G$ by conjugation, through a homomorphism $\phi:G_{\mathrm{sp}}\to\operatorname{Aut}(G)$. When this action is nontrivial, the internal symmetry is referred to as a modulated symmetry [50, 51].

Write the total symmetry group as

where $r$ characterizes the spatial symmetry action that reverses the orientation of the space. We denote by $U(1)_r$ the module on which $G$ acts trivially and $s\in G_{\mathrm{sp}}$ acts as $c\mapsto c^{r(s)}\in U(1)$.

First, suppose that the spatial action consists only of permutations of local Hilbert spaces,

For a given symmetry operator, we construct a microscopic anomaly index valued in

where $\pi(s)=(1,s)$. The kernel expresses the fact that the pure permutation action carries no ’t Hooft anomaly: every anomaly detected in this setting restricts trivially to $G_{\mathrm{sp}}$.

The explicit construction of the anomaly index gives the cocycles directly from symmetry operators. In one spatial dimension, the above anomaly index consists of the components

In two spatial dimensions, the index consists of the components labeled by

These anomaly indices $\{\omega^{p,q}\}$ are valued in

and obey the cocycle conditions of the double complex with the horizontal and vertical differentials $d_h,d_v$, raising the degrees of $p,q$ respectively. Possible ambiguities of the anomaly indices give the coboundary transformations again defined in the double complex. The collection of the anomaly indices $\omega=\sum_{p+q=d+2}\omega^{p,q}$ up to possible ambiguities gives a cohomology of the total complex, which is precisely given by Eq. (4). In particular, the each component of the anomaly index $\omega^{p,q}$ is a part of a single total cocycle $\omega$ and organized into a single cohomology invariant.

Second, we allow generalized spatial symmetry operators

where $U(s)$ is a locality preserving unitary, i.e., quantum cellular automata (QCA). Multiplication of crystalline symmetries may then produce an internal symmetry,

where we assume $e(s_1,s_2)\in Z(G)$. These relations define an extension

For the extensions treated here, the anomaly index takes values in

Here the orientation action on the coefficients is pulled back along $p$. When the extension splits, the above formula gives the full group cohomology of the semidirect product. We note that the restriction to the spatial subgroup generally no longer vanishes, since the spatial action can itself contribute to the anomaly data. For a non-split extension, there is generally no spatial subgroup inclusion, accordingly the formula (4) is replaced by (11).

We establish the dynamical consequence of the anomaly index by showing that a symmetric invertible state must have a trivial anomaly class. Therefore, $[\omega]\ne0$ rules out any symmetry-preserving invertible phase. This conclusion concerns the combined symmetry action. It does not require the internal symmetry action to be non-onsite; indeed, an onsite internal symmetry can participate in a mixed LSM anomaly.

Equations (4) and (11) establish a microscopic form of crystalline equivalence for the bosonic anomalies. The full index has the same cohomological description as the corresponding bosonic continuum anomaly of internal symmetry [52]. This statement is obtained purely from lattice symmetry operators without assuming specific properties of the IR theories.

Relation to previous work

Microscopic lattice anomalies Our strategy to obtain anomaly indices builds on Else–Nayak approach [40] and its two-dimensional extension [44]. It should also be compared with the microscopic formula of anomalies in (1+1)D translationally symmetric spin chains [42, 43]. For one-dimensional translation symmetry commuting with $G$, the anomaly indices are valued in $H^3(BG,U(1))$ and $H^2(BG,U(1))$, where the latter $H^2$ index labels the LSM anomaly. Our microscopic characterization provides explicit formula for each component of anomalies $\omega^{p,q}$ in both one and two spatial dimensions, generalizing the above microscopic formula for LSM anomalies. Our setup incorporates the modulated symmetry where the crystalline symmetry acts on the internal symmetry by an automorphism and admits orientation reversing spatial symmetries such as reflection.

Double complexes and spectral sequences A topological theory of LSM constraints has been developed in Ref. [27] using equivariant homology and double complexes, and discussed the role of the Lyndon–Hochschild–Serre (LHS) spectral sequence. Ref. [53] gave explicit cochain descriptions of the LHS spectral sequence and interpreted its differentials through domain-wall decoration. These developments are closely related to the real-space spectral-sequence approach to crystalline phases [29]. Ref. [54] developed the description of the LSM type theorems using the LHS spectral sequence in (1+1)D assuming the MPS representation of the ground state. More recently, Ref. [55] studied $E_{\infty}^{1,2}$-type translation anomalies and their consequences for deconfined criticality.

The use of a double complex and spectral sequence for crystalline anomalies therefore precedes the present work. Our contribution is a microscopic characterization of each cochain data of anomaly indices entering the LHS spectral sequence purely based on lattice symmetry operators, together with the proof of their obstruction to symmetric invertible states. For Eq. (10), the relevant spectral sequence is

The microscopic components $\omega^{p,q}$ are cochains before taking cohomology. The $E_2$ page organizes candidate layers, while differentials impose compatibility and identify trivializations. As described below, such differentials may trivialize some of the LSM anomalies encoded in $\omega^{p,q}$ that leads to the mechanism known as the SPT–LSM theorem.

SPT–LSM theorems and magnetic translations An SPT–LSM theorem permits a symmetric invertible ground state compatible with the seemingly nontrivial LSM anomaly index. Such phenomena were established for magnetic translations and related settings in Refs. [56, 57, 58, 59]. Their topological and spectral-sequence interpretations were discussed in Refs. [6, 53]. In this situation, a nonzero candidate LSM layer does not necessarily give a nonzero total anomaly, because a differential in the LHS spectral sequence can make that layer exact.

For instance, an example discussed in [57] makes this distinction explicit at the microscopic level. Suppose that a magnetic translation symmetry of a square lattice satisfies the algebra

and translations commute with the internal symmetry. A projective representation $\alpha\in H^2(BG,U(1))$ per unit cell defines the candidate layer $u\otimes\alpha\in E_2^{2,2}$ that corresponds to the LSM anomaly, where $u$ generates $H^2(B\mathbb{Z}^2,\mathbb{Z})$. The differential

is determined by the flux $a$ inserted at each plaquette. Using the microscopic formula of anomaly indices, we show that a symmetric invertible state requires a class $[\lambda]\in H^3(BG,U(1))$ satisfying

where $\delta_a^3$ is the slant product operation that corresponds to twisted compactification of the SPT phase with a flux insertion. Thus the obstruction to symmetric invertible states from this layer lies in $H^2(BG,U(1))/\operatorname{im}\delta_a^3$. Eq. (15) recovers the anomaly-cancellation condition underlying the dyonic SPT–LSM theorem formulated in Ref. [57].

Modulated symmetries Modulated symmetries provide a setting in which spatial and internal symmetries do not commute. LSM constraints for dipole-conserving and fractonic systems were studied in Refs. [60, 61]. Modulated SPT phases, their matrix product state descriptions, anomaly inflow, and gauging have been developed in e.g., [50, 62, 63, 64, 65, 66, 67, 68, 69]. Ref. [51] formulated stratified anomalies and derived LSM and SPT–LSM constraints for modulated symmetries. Their Sec. 3.1 also decomposes a crystalline symmetry operator into a locality-preserving internal part and a permutation of local Hilbert spaces, closely matching our starting point. Our setup includes the nontrivial automorphism action that characterizes finite modulated symmetries. Our contribution is to establish a microscopic formula each layer of anomalies for modulated symmetries, which organizes into a group cohomology of a total symmetry group where crystalline symmetries act on internal symmetries by automorphisms.

Organization of the paper

The remainder of this paper is organized as follows. Sec. 2 develops the microscopic anomaly indices in $(1+1)$ dimensions and illustrates their relation to the LSM anomaly. Sec. 3 extends the construction to $(2+1)$ dimensions. Sec. 4 considers generalized spatial symmetries that combine lattice permutations with QCAs and applies the framework to magnetic translations and SPT–LSM constraints. Technical details of this work is relegated to appendices.

Anomalies in 1+1D

In this section, we consider a 1D lattice $\Lambda$ without boundary and a tensor product Hilbert space $\mathcal{H} = \bigotimes_{x\in\Lambda} \mathcal{H}_x$, where the onsite Hilbert spaces $\mathcal{H}_x$ are all isomorphic and the system contains no fermionic degrees of freedom. A finite locality-preserving unitary symmetry is described by a representation

where $G$ is the finite internal symmetry group and $\operatorname{QCA}(\mathcal{H}) \subset\operatorname{End}(\mathcal{H})$ denotes the group of QCAs acting on $\Lambda$. Since $G$ is finite, every QCA in the image of $U$ is a finite-depth quantum circuit (FDQC).

On the other hand, a spatial symmetry is specified by a faithful group action $G_{\mathrm{sp}} \curvearrowright\Lambda$. This action induces a unitary operator $V(s)$ for each $s \in G_{\mathrm{sp}}$, defined by

Thus, the spatial symmetry considered here acts solely by permuting the local Hilbert spaces. In particular, $V(s_1s_2) = V(s_1)V(s_2)$ for every $s_1,s_2 \in G_{\mathrm{sp}}$.

Since $V(s)^{-1}U(g)V(s)$ is an internal symmetry operator for each $g \in G$ and $s \in G_{\mathrm{sp}}$, there exists a group homomorphism