Crystalline Equivalence Principle

• Crystalline Equivalence Principle is a framework showing that spatial symmetry-protected topological phases can be reinterpreted as internal symmetry phases through cohomological classifications.
• It leverages tools like homotopy theory, spectral sequences, and matrix product state techniques to bridge interacting bosonic and fermionic phase classifications.
• The approach distinguishes between strong and weak free-fermion phases, highlighting when traditional equivalence holds and where systematic failures arise due to spatial actions.

The Crystalline Equivalence Principle (CEP) is the organizing statement that topological phases protected by spatial or crystalline symmetries can, for classification purposes, be related to phases protected by the same symmetries reinterpreted as internal symmetries. In its standard form, the principle says 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 (Ning et al., 19 Mar 2026). In later work this idea is upgraded from a classification slogan to an equivalence of categories between crystalline topological phases on a $G$-space and a category of TQFTs with internal $\mathcal{X}/\!/G$-symmetry (Stockall et al., 14 Aug 2025). The principle is central in interacting bosonic classifications, admits a fermionic version for mixed spatial symmetries, and has sharp limits: it fails for weak free-fermion phases classified by equivariant $K$-theory, although a restricted variant survives for strong free-fermion phases (Sheinbaum et al., 2024, Debray, 2021).

1. Core statement and conceptual setting

The CEP is motivated by the observation that crystalline symmetries—translations, reflections, rotations, and related point-group actions—act on space, whereas internal symmetries act on local degrees of freedom, yet in many interacting classifications the two enter through the same homotopy-theoretic machinery. In the formulation used for modulated symmetries, one starts from a symmetry group

$G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$

and the CEP asserts that modulated SPT phases with spatial symmetry $G_{\mathrm{sp}}$ and internal SPT phases obtained by reinterpreting the spatial symmetry internally are classified by the same cohomology group,

$H^2(G,U(1)_s),$

with $U(1)_s$ denoting the usual twist of coefficients when anti-unitary symmetries are present, encoded by a homomorphism $s:G\to \mathbb Z_2$ (Ning et al., 19 Mar 2026).

In the interacting bosonic setting, the principle is attractive because internal-symmetry classifications are mathematically cleaner. One later account states the motivation directly: for interacting crystalline bosonic SPTs, Thorngren and Else proposed that the classification can be expressed as

$$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$

where $G$ is a crystalline symmetry group written as an extension

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

$\mathcal{X}/\!/G$1 is the unit-cell torus, and $\mathcal{X}/\!/G$2 is the point group (Sheinbaum et al., 2024). In this Borel-style picture, the translation subgroup is encoded by the torus, the point-group action is treated equivariantly, and the resulting classification looks formally like the classification for an internal symmetry group.

This conceptual equivalence is not merely terminological. The CEP is substantive precisely when the spatial action can be absorbed into a classifying-space or quotient-space construction, so that the classification depends on the symmetry data in the same formal way as in an internal-symmetry problem. Later work makes this precise through matrix product states, Thom spectra, and straightening/unstraightening constructions, depending on the context (Ning et al., 19 Mar 2026, Debray, 2021, Stockall et al., 14 Aug 2025).

2. Interacting bosonic formulation and the standard crystalline equivalence principle

In the standard formulation recovered by the generalized categorical treatment, there is an equivalence between the category of $\mathcal{X}/\!/G$3-dimensional crystalline topological phases valued in $\mathcal{X}/\!/G$4 defined on $\mathcal{X}/\!/G$5, and the category of $\mathcal{X}/\!/G$6-dimensional topological field theories valued in $\mathcal{X}/\!/G$7 with internal $\mathcal{X}/\!/G$8-symmetry (Stockall et al., 14 Aug 2025). This is the version closest to the usual statement of CEP in the physics literature.

A more general theorem replaces $\mathcal{X}/\!/G$9 by an arbitrary $K$0-space $K$1. The theorem states an equivalence between the category of $K$2-dimensional crystalline topological phases valued in a $K$3-category $K$4 on a $K$5-space $K$6, and a full subcategory of $K$7-dimensional topological field theories valued in $K$8 with internal $K$9-symmetry, on those theories intertwining a $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$0-bundle structure on the space $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$1 and category $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$2 (Stockall et al., 14 Aug 2025). The homotopy quotient $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$3 is the action $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$4-groupoid obtained from the $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$5-action, and the key bridge is

$G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$6

The same work summarizes the conceptual content as the slogan

$G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$7

This does not mean that the two theories are literally identical. The paper explicitly warns that “the TQFTs on either side are not the same: The first kind are TQFT valued in $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$8 and the latter are TQFT valued in $G = G_{\mathrm{int}\rtimes G_{\mathrm{sp}}},$9” (Stockall et al., 14 Aug 2025). A simpler equivalence with the same target category is available when the $G_{\mathrm{sp}}$0-action on $G_{\mathrm{sp}}$1 is trivial: in that case crystalline phases on $G_{\mathrm{sp}}$2 are equivalent to TQFTs valued in $G_{\mathrm{sp}}$3 with internal $G_{\mathrm{sp}}$4-symmetry.

This formulation clarifies why the CEP is strongest in interacting, Borel-type settings. The relevant constructions identify equivariant families of TQFTs with TQFTs over a homotopy quotient, thereby replacing explicit spatial action by internal symmetry data attached to the quotient space (Stockall et al., 14 Aug 2025).

3. One-dimensional modulated symmetries and matrix product state realization

A recent matrix product state treatment extends the CEP to modulated symmetries, defined as internal symmetries that act in a spatially non-uniform manner (Ning et al., 19 Mar 2026). In this setting the total symmetry takes the semidirect-product form

$G_{\mathrm{sp}}$5

where the spatial symmetry acts nontrivially on the internal symmetry through a homomorphism

$G_{\mathrm{sp}}$6

For translation $G_{\mathrm{sp}}$7, the local symmetry operators satisfy

$G_{\mathrm{sp}}$8

and for reflection $G_{\mathrm{sp}}$9,

$H^2(G,U(1)_s),$0

In one dimension the maximal spatial symmetry group considered is

$H^2(G,U(1)_s),$1

Using injective MPS for gapped 1D ground states, the paper derives push-through equations for symmetry action on the MPS tensor $H^2(G,U(1)_s),$2. For translation-modulated symmetry,

$H^2(G,U(1)_s),$3

and translation invariance reduces this to

$H^2(G,U(1)_s),$4

The virtual operators $H^2(G,U(1)_s),$5 form a projective representation,

$H^2(G,U(1)_s),$6

with $H^2(G,U(1)_s),$7 a $H^2(G,U(1)_s),$8-cocycle. The strong index is the invariant part of $H^2(G,U(1)_s),$9 under the spatial action. For translation,

$U(1)_s$0

while for reflection,

$U(1)_s$1

Besides strong indices, the classification contains weak indices from $U(1)_s$2. For translation they are classified by

$U(1)_s$3

and for reflection by

$U(1)_s$4

The resulting 1D MPS classification is

$U(1)_s$5

for translation, and

$U(1)_s$6

for reflection; the paper emphasizes that this matches the CEP prediction

$U(1)_s$7

(Ning et al., 19 Mar 2026).

The same analysis gives a physical interpretation of the two types of index. The strong index corresponds to projective boundary symmetry action and is visible on open chains and in entanglement-spectrum degeneracy. The weak index is a symmetry charge assigned to unit cells or domain walls and can be detected by varying system size, inserting translation defects, or examining momentum shifts in the presence of defects (Ning et al., 19 Mar 2026). This provides an explicit realization of CEP in terms of virtual symmetry action rather than an abstract cohomological slogan.

4. Spectral-sequence structure, Lieb–Schultz–Mattis constraints, and anomalies

The matrix product state derivation also produces a concrete form of the Lyndon–Hochschild–Serre spectral sequence for

$U(1)_s$8

The virtual symmetry data satisfy four consistency conditions: an internal cocycle condition, compatibility with $U(1)_s$9 action, compatibility of sequential spatial actions, and a spatial cocycle condition. These are identified as exactly the LHS spectral sequence constraints, and the paper constructs an explicit $s:G\to \mathbb Z_2$0-cocycle on $s:G\to \mathbb Z_2$1 from the internal cocycle, the spatial action phases, and the spatial cocycle (Ning et al., 19 Mar 2026). In this sense the CEP is not only a classification coincidence; it is realized by an explicit MPS-to-cohomology map.

One application is a modulated Lieb–Schultz–Mattis-type anomaly. If the local symmetry operators form a projective representation

$s:G\to \mathbb Z_2$2

then the MPS consistency condition becomes

$s:G\to \mathbb Z_2$3

If

$s:G\to \mathbb Z_2$4

then no symmetric short-range-entangled ground state exists: the system must either break symmetry or be gapless. If $s:G\to \mathbb Z_2$5 is compatible but nontrivial, the ground state can exist, but it must realize a nontrivial SPT with nontrivial entanglement and projective edge modes; this is the paper’s SPT-LSM constraint (Ning et al., 19 Mar 2026).

The generalized categorical formulation develops a parallel anomaly theory for crystalline phases. A $s:G\to \mathbb Z_2$6-theory with nonanomalous $s:G\to \mathbb Z_2$7-form $s:G\to \mathbb Z_2$8-groupoid symmetry is a functor into $s:G\to \mathbb Z_2$9, equivalently a section of the trivial fibration $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$0 (Stockall et al., 14 Aug 2025). In the anomalous case, the trivial bundle is replaced by a nontrivial bundle with fiber $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$1: $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$2 The paper defines an anomalous $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$3-theory as a section of $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$4, and proves that the category of anomalies is the full subcategory of

$$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$5

on those $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$6 such that $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$7 for all $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$8 (Stockall et al., 14 Aug 2025).

This anomaly framework is then interpreted through relative theories. A theory with anomaly $$H^{d+2}(BG;\mathbb{Z}) = H^{d+2}_P(T^d;\mathbb{Z}),$$9 is a natural transformation $G$0, so an anomalous theory is a boundary theory and the anomaly is canceled by a bulk in one higher dimension (Stockall et al., 14 Aug 2025). The same formalism is stated to extend from groupoid symmetry to categorical symmetry, including non-invertible symmetry.

5. Fermionic crystalline equivalence principle

A fermionic version of the principle is proved in a homotopy-theoretic framework based on the Freed–Hopkins ansatz for invertible phases (Debray, 2021). There the basic input is a symmetry type $G$1 or $G$2, together with a spatial action of a group $G$3 on space. The paper generalizes the ansatz to situations where the symmetry type varies over space and mixes nontrivially with the $G$4-action, leading to the generalized phase homology

$G$5

with $G$6, and the classification statement

$G$7

The mixed symmetry is encoded by an extension

$G$8

together with a faithful spatial representation

$G$9

In the fermionic crystalline case of interest, the mixing is determined by the Stiefel–Whitney class

$\mathcal{X}/\!/G$00

which defines a central extension

$\mathcal{X}/\!/G$01

The paper distinguishes two local systems of symmetry types on $\mathcal{X}/\!/G$02: $\mathcal{X}/\!/G$03, the spinless crystalline local system, and $\mathcal{X}/\!/G$04, the spin-$\mathcal{X}/\!/G$05 crystalline local system. The main theorem gives natural isomorphisms

$\mathcal{X}/\!/G$06

$\mathcal{X}/\!/G$07

In words, a spinless crystalline phase is equivalent to an internal spin-$\mathcal{X}/\!/G$08 phase, and a spin-$\mathcal{X}/\!/G$09 crystalline phase is equivalent to a spinless internal phase (Debray, 2021). This exchange is the paper’s fermionic crystalline equivalence principle.

The proof proceeds in two steps. First, the equivariant phase homology is reduced to homotopy classes of maps out of a Thom spectrum

$\mathcal{X}/\!/G$10

Second, a shearing argument identifies the mixed Thom spectrum with

$\mathcal{X}/\!/G$11

for a suitable rank-zero virtual bundle $\mathcal{X}/\!/G$12, thereby converting the mixed spatial problem into an ordinary internal symmetry Thom spectrum problem (Debray, 2021). Computationally, this reduction allows the use of the Adams spectral sequence and related tools to calculate explicit classifications for reflections, inversions, rotations, dihedral symmetries, and several three-dimensional point groups.

6. Limits, failures, and terminological boundaries

The strongest limitation of the CEP is exhibited by weak free-fermion phases. A recent topological analysis argues that the principle, while useful and often correct for interacting crystalline SPT phases, does not extend to all free-fermion crystalline phases (Sheinbaum et al., 2024). The weak free-fermion classification is expressed in equivariant $\mathcal{X}/\!/G$13-theory,

$\mathcal{X}/\!/G$14

and this theory distinguishes sharply between crystal symmetry, which acts on the torus and on the $\mathcal{X}/\!/G$15-theory equivariantly, and internal symmetry, which determines the flavor of $\mathcal{X}/\!/G$16-theory used.

The failure is structural rather than technical. The paper highlights two mismatches: $\mathcal{X}/\!/G$17 and

$\mathcal{X}/\!/G$18

For spatial symmetries there is only a relation via the Atiyah–Segal completion theorem, not equality; for internal symmetries with antiunitary structure there is no known Atiyah–Segal-type completion theorem that would repair the mismatch (Sheinbaum et al., 2024). Since equivariant $\mathcal{X}/\!/G$19-theory retains essential information about how the group acts, the classification cannot in general erase the distinction between spatial and internal symmetry in the way required by the standard CEP.

The same paper therefore draws a sharp distinction between weak and strong crystalline free-fermion phases. Weak phases depend on the full lattice or torus structure and are precisely the phases for which CEP fails. Strong phases survive when the torus is collapsed to a point; for them a restricted CEP-like statement does hold, because the symmetry no longer acts on geometry in an essential way and only affects the flavor of the $\mathcal{X}/\!/G$20-theory (Sheinbaum et al., 2024). This restricted result is explicitly said not to require Borel-type cohomology.

The generalized categorical theorem has its own caveats. It depends on a conjecture asserting that if a monoidal functor into $\mathcal{X}/\!/G$21-categories factors through categories with duals, then the unstraightened category still has duals (Stockall et al., 14 Aug 2025). It is also stated to be a theorem for crystalline topological phases, not for arbitrary non-topological theories with spatial symmetry (Stockall et al., 14 Aug 2025).

Finally, the term Crystalline Equivalence Principle should not be confused with the gravitational equivalence principle literature. The paper “Terrestrial vs. spaceborne, quantum vs. classical tests of the equivalence principle” (Hohensee et al., 2013) does not discuss the “Crystalline Equivalence Principle” by name, and the paper “Composite system in noncommutative space and the equivalence principle” (Gnatenko, 2014) addresses recovery of the weak equivalence principle in noncommutative space through the condition

$\mathcal{X}/\!/G$22

rather than SPT classification. The shared phrase “equivalence principle” therefore spans distinct research programs, and within condensed-matter and topological-field-theory usage the CEP refers specifically to the crystalline-to-internal correspondence described above.