Crystalline Equivalence Principle

• CEP is the principle asserting that crystalline SPT phases with spatial symmetries correspond one-to-one to TQFTs with internal symmetry via a homotopy quotient framework.
• The generalized formulation uses categorical equivalence, the Cobordism Hypothesis, and duals to rigorously classify topological phases and address anomaly detection.
• CEP has practical implications in condensed matter, underpinning classifications of modulated, fermionic, and free-fermion phases, as well as constraints like the Lieb–Schultz–Mattis theorem.

The Crystalline Equivalence Principle (CEP) is the statement that SPT phases protected by a spatial symmetry $G_{\text{sp}}$ are in one-to-one correspondence with SPT phases protected by the same symmetry group, viewed as an internal symmetry $\tilde G_{\text{sp}}$, where the orientation-reversing elements of $G_{\text{sp}}$ are mapped to anti-unitary symmetries in $\tilde G_{\text{sp}}$ (Ning et al., 19 Mar 2026). In its generalized form, CEP is an equivalence of categories between crystalline topological phases on a $G$-space $\mathcal{X}$ and TQFTs with internal symmetry encoded by the homotopy quotient $\mathcal{X}/\!/G$ (Stockall et al., 14 Aug 2025).

1. Original condensed-matter form

Thorngren–Else’s original CEP addressed crystalline SPT phases on $\mathbb{R}^d$. In the coarse-grained limit, the lattice spacing is much smaller than the unit cell, and the system can be modeled as living over the contractible space $\mathbb{R}^d$. In that setting, the original CEP says that crystalline SPT phases with spatial symmetry $G$ on $\tilde G_{\text{sp}}$0 are classified like SPTs with internal symmetry $\tilde G_{\text{sp}}$1. In the generalized framework this appears as the special case $\tilde G_{\text{sp}}$2 for a free $\tilde G_{\text{sp}}$3-action, with

$\tilde G_{\text{sp}}$4

and the corresponding corollary identifies crystalline topological phases on $\tilde G_{\text{sp}}$5 with TQFTs carrying internal $\tilde G_{\text{sp}}$6-symmetry (Stockall et al., 14 Aug 2025).

The condensed-matter meaning of this statement is that a crystalline symmetry may be treated as if it were an onsite symmetry, provided one works in the regime where the spatial background is effectively contractible. This is the form of CEP that became standard in crystalline SPT classification. A plausible implication is that the familiar “crystalline $\tilde G_{\text{sp}}$7 internal” dictionary is not a primitive statement about groups alone; it is a contractible-space shadow of a more general categorical equivalence.

2. Generalized categorical formulation

The generalized formulation fixes a symmetric monoidal $\tilde G_{\text{sp}}$8-category $\tilde G_{\text{sp}}$9 with duals. A $G_{\text{sp}}$0-space is a functor

$G_{\text{sp}}$1

and a $G_{\text{sp}}$2-category is a functor

$G_{\text{sp}}$3

A crystalline topological phase is then defined as an object of

$G_{\text{sp}}$4

that is, a $G_{\text{sp}}$5-equivariant functor $G_{\text{sp}}$6. On the internal-symmetry side, the key construction is the homotopy quotient

$G_{\text{sp}}$7

the action $G_{\text{sp}}$8-groupoid encoding both the space and the $G_{\text{sp}}$9-action (Stockall et al., 14 Aug 2025).

The main theorem gives an equivalence of categories between $\tilde G_{\text{sp}}$0-dimensional crystalline topological phases on the $\tilde G_{\text{sp}}$1-space $\tilde G_{\text{sp}}$2 valued in a $\tilde G_{\text{sp}}$3-category $\tilde G_{\text{sp}}$4, and a full subcategory of $\tilde G_{\text{sp}}$5-dimensional TQFTs with internal $\tilde G_{\text{sp}}$6-symmetry valued in $\tilde G_{\text{sp}}$7, namely those theories that intertwine the $\tilde G_{\text{sp}}$8-bundle structure on $\tilde G_{\text{sp}}$9 and on $G$0. The associated slogan is

$G$1

If the $G$2-action on $G$3 is trivial, the target can be taken to be $G$4 itself, and one has

$G$5

This formulation replaces the contractibility assumption by the homotopy quotient. The internal symmetry is therefore not merely a group action on fields; it is a family of theories parametrized by the $G$6-groupoid $G$7. The proof uses straightening/unstraightening and Lurie’s Cobordism Hypothesis, together with a conjectural technical point asserting that if $G$8 has duals and $G$9 factors through $\mathcal{X}$0-categories with duals, then the corresponding unstraightening also has duals.

3. Internal $\mathcal{X}$1-form symmetry and anomalies

In this framework, an internal symmetry of type $\mathcal{X}$2 is a $\mathcal{X}$3-form symmetry: a family of theories indexed by a space or $\mathcal{X}$4-groupoid,

$\mathcal{X}$5

Paths in $\mathcal{X}$6 describe topological defects implementing the symmetry. A nonanomalous $\mathcal{X}$7-theory with $\mathcal{X}$8-form $\mathcal{X}$9-symmetry is simply a functor

$\mathcal{X}/\!/G$0

equivalently a section of the trivial fibration

$\mathcal{X}/\!/G$1

The paper then defines anomalies by replacing this trivial bundle with a nontrivial bundle

$\mathcal{X}/\!/G$2

whose fibers are all equivalent to $\mathcal{X}/\!/G$3; an anomalous theory is a section of $\mathcal{X}/\!/G$4 (Stockall et al., 14 Aug 2025).

At the level of homotopy theory, such a bundle is classified by a map

$\mathcal{X}/\!/G$5

where $\mathcal{X}/\!/G$6 is the classifying space of autoequivalences of $\mathcal{X}/\!/G$7, denoted $\mathcal{X}/\!/G$8. The category of anomalies for $\mathcal{X}/\!/G$9-theories is equivalent to the full subcategory of

$\mathbb{R}^d$0

consisting of those $\mathbb{R}^d$1 such that $\mathbb{R}^d$2 for all $\mathbb{R}^d$3. This identifies an anomaly with a twisting of the bundle of theories over the symmetry parameter space.

Several examples sharpen this abstract description. For $\mathbb{R}^d$4 and $\mathbb{R}^d$5, one has $\mathbb{R}^d$6 with $\mathbb{R}^d$7, and anomalies are classified by

$\mathbb{R}^d$8

as for 1d projective representations. For an abelian group $\mathbb{R}^d$9, an $\mathbb{R}^d$0-form $\mathbb{R}^d$1-symmetry is equivalent to a $\mathbb{R}^d$2-form symmetry with $\mathbb{R}^d$3, so anomalies become maps $\mathbb{R}^d$4. The same construction extends from $\mathbb{R}^d$5-groupoids to general $\mathbb{R}^d$6-categories: if $\mathbb{R}^d$7 encodes a categorical symmetry, anomalies are described by

$\mathbb{R}^d$8

An anomalous theory is also a relative theory: for $\mathbb{R}^d$9, an anomalous $G$0-theory is identified with a natural transformation $G$1, and in the linear context with $G$2, with a defect between the trivial bulk theory and the anomaly bulk.

4. Matrix-product-state realizations and modulated symmetries

A constructive one-dimensional realization appears in the study of modulated symmetries, where internal symmetries act in a spatially non-uniform manner. The total symmetry group takes the form

$G$3

with $G$4 generated by translations $G$5 and/or reflections $G$6. In this setting the paper states CEP as the claim that SPT phases protected by symmetries involving spatial elements are in one-to-one correspondence with internal SPT phases protected by the same symmetries, viewed as acting internally, and shows that the classification is

$G$7

where $G$8 and $G$9 records anti-unitarity of the internalized orientation-reversing elements (Ning et al., 19 Mar 2026).

The translation case yields an explicit decomposition into strong and weak indices: $\tilde G_{\text{sp}}$00 The reflection case gives

$\tilde G_{\text{sp}}$01

These classifications are derived from symmetry constraints on injective, translation-invariant MPS tensors. The same work provides an MPS derivation of the Lyndon–Hochschild–Serre spectral sequence for $\tilde G_{\text{sp}}$02, and an explicit mapping between modulated SPT data and internal SPT cocycles. In that derivation, the strong indices arise from $\tilde G_{\text{sp}}$03 data invariant under spatial action, while the weak indices arise from $\tilde G_{\text{sp}}$04 data twisted by the spatial symmetry.

The resulting correspondence is not only classificatory. It is used to prove a Lieb–Schultz–Mattis theorem for modulated symmetries: if the local symmetry operators carry a projective representation with cocycle $\tilde G_{\text{sp}}$05 and

$\tilde G_{\text{sp}}$06

then there is no symmetric gapped short-range entangled ground state. If

$\tilde G_{\text{sp}}$07

symmetric short-range entangled phases are allowed, but any such phase must have a nontrivial bulk SPT index. The same classification is applied to non-invertible Kramers–Wannier reflection symmetries, where no gapped symmetric exponential SPT can preserve the non-invertible symmetry.

5. Fermionic versions and homotopy-theoretic formulations

The generalized categorical treatment briefly describes a fermionic version. A superspace is a space $\tilde G_{\text{sp}}$08 with a $\tilde G_{\text{sp}}$09-action encoding fermion parity. A bosonic spatial symmetry group $\tilde G_{\text{sp}}$10 may be independent of fermion parity, giving $\tilde G_{\text{sp}}$11, or may define a nontrivial extension by fermion parity, giving a supergroup $\tilde G_{\text{sp}}$12. Applying the generalized CEP to superspaces and to suitable super target categories such as $\tilde G_{\text{sp}}$13-SVect recovers a fermionic crystalline equivalence principle compatible with previous fermionic SPT classifications (Stockall et al., 14 Aug 2025).

A distinct homotopy-theoretic formulation is given for invertible fermionic phases with mixed spatial symmetries. Freed–Hopkins give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology, and the corresponding generalization introduces parametrized, equivariant symmetry types so that mixing between spatial symmetries and spin or fermion parity can be treated explicitly. In Altland–Zirnbauer classes D and A, the main theorem states

$\tilde G_{\text{sp}}$14

and

$\tilde G_{\text{sp}}$15

so that spinless crystalline phases correspond to internal spin-$\tilde G_{\text{sp}}$16 symmetry types, while spin-$\tilde G_{\text{sp}}$17 crystalline phases correspond to spinless internal symmetry types (Debray, 2021).

The proof reduces equivariant phase homology on $\tilde G_{\text{sp}}$18 to Thom spectra of virtual bundles over classifying spaces, and then uses shearing equivalences to convert crystalline data into internal symmetry data. Adams spectral sequence computations for reflections, inversions, rotations, dihedral groups, and several three-dimensional point groups reproduce classifications previously obtained by other methods. This places the fermionic CEP within a homotopy-theoretic framework that is technically distinct from the categorical generalized CEP but closely aligned in outcome.

6. Scope, failures, and terminological ambiguities

The generalized categorical theorem is explicitly stated only for topological theories: it is a theorem for TQFTs, and no analogous statement is claimed for generic gapped or gapless non-topological QFTs (Stockall et al., 14 Aug 2025). Within interacting bosonic proposals based on group cohomology, CEP is mathematically natural because

$\tilde G_{\text{sp}}$19

and, with additional internal symmetry $\tilde G_{\text{sp}}$20,

$\tilde G_{\text{sp}}$21

In that setting, crystalline and internal symmetry enter through classifying spaces and a Borel-type construction (Sheinbaum et al., 2024).

The free-fermion case is sharply different. The accepted classification uses equivariant K-theory, and the paper “Failure of the Crystalline Equivalence Principle for Weak Free Fermions” shows that CEP fails for weak free fermion crystalline phases. One obstruction is that equivariant K-theory is not Borel-type: $\tilde G_{\text{sp}}$22 A second obstruction is that internal symmetry is encoded not as $\tilde G_{\text{sp}}$23-theory of $\tilde G_{\text{sp}}$24, but through the quotient $\tilde G_{\text{sp}}$25 and the associated flavor and twist of $\tilde G_{\text{sp}}$26-theory. The paper therefore concludes that CEP is not a universal principle; its validity depends crucially on the generalized cohomology theory used, on whether one is dealing with strong or weak phases, and on the presence of translation-dependent structure. It nonetheless shows that a variant of CEP does hold for strong free fermion phases, where collapsing the torus to a point yields

$\tilde G_{\text{sp}}$27

so crystalline and internal symmetries both act only on the $\tilde G_{\text{sp}}$28-theory spectrum (Sheinbaum et al., 2024).

A common terminological ambiguity is that the acronym “CEP” is not unique across the literature. In unrelated contexts it denotes the “Classical Equivalence Principle” (Accioly et al., 2017) and the “Chameleonic Equivalence Postulate” (Zanzi, 2014). Within topological phases and TQFT, however, CEP refers to the crystalline equivalence principle and its fermionic, generalized, and homotopy-theoretic variants.