Subsystem Symmetry Fractionalization

• Subsystem symmetry fractionalization is a phenomenon in topological phases where symmetry generators act on lower-dimensional submanifolds, leading to a fractionalization of global relations rather than individual charges.
• The algebraic framework partitions relations into non-fractionalizable local ones and fractionalizable global ones, with homomorphisms mapping these global relations to the anyon fusion group.
• In two-dimensional models, this mechanism enforces restricted mobility of anyons—yielding lineons or immobile fractons—and introduces novel classification and anomaly phenomena in subsystem SET phases.

Subsystem symmetry fractionalization is a form of symmetry enrichment in topological quantum matter in which the relevant symmetry generators live on rigid lower-dimensional submanifolds, such as lines or fractals, rather than on the entire system. In two-dimensional topological phases, the phenomenon differs sharply from conventional global symmetry fractionalization: isolated anyons do not carry fractional charge under any single subsystem generator, but they can fractionalize a global relation among macroscopically many generators whose product is the identity in the microscopic algebra. In the framework developed for two-dimensional topological order with subsystem symmetry, the central result is that only such global relations can fractionalize, while local relations are non-fractionalizable; moreover, fractionalized anyons must have restricted mobility when the symmetry is enforced, becoming lineons for line symmetries and pointlike immobile excitations for fractal symmetries (Stephen et al., 2022).

1. Definition and basic mechanism

A subsystem symmetry in two dimensions “live[s]” on rigid sub-manifolds, for example lines or fractals, rather than on the whole system. Concretely, one has a group $G$ generated by operators $U(s)$, each acting nontrivially only on spins lying on a fixed line, fractal, or other lower-dimensional set. These generators obey relations, some “local,” involving finitely many generators, and some “global,” involving macroscopically many generators whose number grows with system size (Stephen et al., 2022).

The defining distinction from ordinary symmetry fractionalization is that individual subsystem generators act linearly on an anyon. There is no fractional charge under a single line. Instead, the relevant structure is a product of many subsystem generators that equals identity in the microscopic algebra,

$\prod_{g\in R} U_g \;=\; I.$

For an anyon state $|a\rangle$, that same relation can act as

$$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$

so that the relation itself is fractionalized (Stephen et al., 2022).

This mechanism is qualitatively different from the standard global case, where one usually writes

$U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$

with $\omega$ valued in abelian anyons, and an anyon picks up a phase under each symmetry element. For subsystem symmetries, the rigid geometric support of generators obstructs that familiar pattern in many cases. The nontrivial possibility instead lies in extensive relations among generators, such as the product of all row and column flips on a square lattice, which is trivial only macroscopically but can carry a nontrivial total charge when evaluated on an anyon sector (Stephen et al., 2022).

A common misconception is therefore that subsystem symmetry fractionalization should simply be the analogue of global projective symmetry action applied line by line. The two-dimensional analysis rules this out: the fractionalized object is not a single generator but a global relation among many generators (Stephen et al., 2022).

2. Algebraic formulation

The algebraic framework writes the subsystem symmetry group as

$G \;=\; F(\mathcal S)\,/\,\mathfrak R,$

where $F(\mathcal S)$ is the free group on generators $\mathcal S$ and $U(s)$0 is the normal subgroup generated by all relations. The relations are partitioned into “non-fractionalizable” $U(s)$1, consisting of local relations, and “fractionalizable” $U(s)$2, consisting of global ones. The abelian group of fractionalizable relations is

$U(s)$3

If the anyon fusion group is $U(s)$4, then fractionalization classes are specified by a group homomorphism

$U(s)$5

assigning to each fractionalizable relation $U(s)$6 an anyon $U(s)$7 such that the localized product of symmetry operators $U(s)$8 around a region $U(s)$9 is equivalent to braiding $\prod_{g\in R} U_g \;=\; I.$0 around $\prod_{g\in R} U_g \;=\; I.$1 (Stephen et al., 2022).

The same structure can be expressed in cocycle language. For global symmetries one has localized operators $\prod_{g\in R} U_g \;=\; I.$2 satisfying

$\prod_{g\in R} U_g \;=\; I.$3

with

$\prod_{g\in R} U_g \;=\; I.$4

In the subsystem case, however, only those $\prod_{g\in R} U_g \;=\; I.$5 arising from $\prod_{g\in R} U_g \;=\; I.$6 and trivial on $\prod_{g\in R} U_g \;=\; I.$7 are realizable. The localizations of local generators are unambiguous, so there are no further coboundary ambiguities of the usual kind (Stephen et al., 2022).

A concise consequence is that fractionalization classes of a subsystem symmetry group $\prod_{g\in R} U_g \;=\; I.$8 acting on a topological order with abelian anyons $\prod_{g\in R} U_g \;=\; I.$9 are classified by homomorphisms

$|a\rangle$0

This is the classification statement specific to the mechanism based on global relations (Stephen et al., 2022).

3. Constraints, no-go structure, and restricted mobility

The two-dimensional theory is organized by a strong constraint: subsystem symmetry fractionalization is not possible in many cases due to the additional rigid geometric structure of the symmetries. In particular, all local relations are non-fractionalizable because their boundary support is too sparse to braid an anyon string (Stephen et al., 2022).

Only global relations among subsystem symmetry generators can fractionalize. These are relations involving an extensive number of generators, for instance the product of all row and column operators on a finite torus or all preserved generators in a fractal-symmetry construction. Locally the individual generators may commute and square to identity, but the relation is trivial only at the macroscopic level (Stephen et al., 2022).

A second necessary property is symmetry-protected restricted mobility. Fractionalized anyons must have restricted mobility when the symmetry is enforced, such that they are confined to a single line for line symmetries or to a single point for fractal symmetries. In the line-symmetry examples this gives lineons; in the fractal-symmetry examples the isolated $|a\rangle$1 anyon is fully immobile, a “symmetry-protected fracton” (Stephen et al., 2022).

This restricted mobility is not an auxiliary feature. The stated result is that it is a necessary signature of subsystem symmetry fractionalization in two dimensions, since it enforces emergent conservation laws on subsystems and allows fractional charges under global relations (Stephen et al., 2022). A plausible implication is that mobility constraints are part of the data distinguishing subsystem-symmetry-enriched topological phases from ordinary SET phases.

The later field-theoretic treatment sharpens the same distinction by embedding subsystem symmetry into higher-form symmetry. There, subsystem symmetry fractionalization again arises from global relations among rigid line or plane generators, and truncating such a relation to a finite region produces a topological line or surface operator at the boundary. That work presents this as a qualitatively new mechanism, different from the usual projective representations of point-group symmetries, and develops anomalous examples as well (Hsin et al., 2024).

4. Exactly solvable two-dimensional models

The canonical two-dimensional example is a $|a\rangle$2 toric-code “line” SSET. Its degrees of freedom are qubits on sites $|a\rangle$3 and edges $|a\rangle$4 of a square lattice, with subsystem symmetry generators

$|a\rangle$5

These obey $|a\rangle$6 and commute linewise, together with the global relation

$|a\rangle$7

The Hamiltonian is

$|a\rangle$8

It is unitarily, but not symmetrically, equivalent to the usual toric code and supports $|a\rangle$9 anyons (Stephen et al., 2022).

In this model, the anyon $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$0 splits into two lineons, $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$1 and $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$2, moving only horizontally or vertically. Truncated line symmetry creates $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$3-pairs at its ends, and the product of all truncated lines $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$4 is exactly a closed $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$5-loop $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$6. Hence

$$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$7

A more refined statement is that $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$8 in row $$\Bigl(\prod_{g\in R} U_g\Bigr)\,\lvert a\rangle\;=\; e^{2\pi i\sum_{g\in R}\alpha_g(a)}\,\lvert a\rangle\,,\qquad\sum_{g\in R}\alpha_g(a)\neq0\ (\mathrm{mod}\,1)\,,$$9 has charge $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$0 under $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$1 and $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$2 under all other generators, so the total charge under the relation $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$3 is $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$4 (Stephen et al., 2022).

Variants of the same construction realize

$U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$5

by modifying local Hamiltonian terms to attach symmetry charges of suitable anyon type. Section III C contains a model $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$6 with $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$7 (Stephen et al., 2022).

A second family uses three-direction line symmetries with generators $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$8, $U(g_1)U(g_2)=\omega(g_1,g_2)U(g_1g_2)$9, and $\omega$0 along horizontal, vertical, and one diagonal direction. The fractionalizable relations are

$\omega$1

In this case $\omega$2, and the fractionalization maps $\omega$3 can be chosen independently, subject to one overall product constraint, by decorating certain Hamiltonian terms. The $\omega$4 anyons split into three lineons moving along the three directions (Stephen et al., 2022).

A third class involves fractal symmetries. The model $\omega$5 is obtained by gauging one sublattice of a fractal-symmetry cluster state. The symmetry generators take the form

$\omega$6

where $\omega$7 is a reversible cellular automaton rule. Truncated fractal symmetry operators produce pairs of $\omega$8 anyons along the boundary of the truncated fractal. One can define preserved generators $\omega$9 so that only the global relation

$G \;=\; F(\mathcal S)\,/\,\mathfrak R,$0

is fractionalizable. The isolated $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$1 anyon is fully immobile, and one computes

$G \;=\; F(\mathcal S)\,/\,\mathfrak R,$2

by counting that $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$3 anticommutes with an odd number of $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$4 (Stephen et al., 2022).

5. Foliated field theory, higher-form embedding, and anomalous classes

A later extension formulates subsystem symmetry fractionalization through the principle of embedding subsystem symmetry into higher-form symmetry. In that language, a subsystem symmetry is a symmetry whose generators live on rigid lower-dimensional submanifolds but are not fully topological. A subsystem $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$5-form symmetry has generators on rigid codimension-$G \;=\; F(\mathcal S)\,/\,\mathfrak R,$6 subspaces (Hsin et al., 2024).

The continuum framework introduces foliation one-forms $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$7 satisfying

$G \;=\; F(\mathcal S)\,/\,\mathfrak R,$8

and foliated gauge fields $G \;=\; F(\mathcal S)\,/\,\mathfrak R,$9 obeying

$F(\mathcal S)$0

In $F(\mathcal S)$1 dimensions, a subsystem one-form symmetry acting on lines of constant $F(\mathcal S)$2 and constant $F(\mathcal S)$3 is described by

$F(\mathcal S)$4

with $F(\mathcal S)$5. The embedding into the ordinary one-form symmetry is

$F(\mathcal S)$6

typically with $F(\mathcal S)$7 (Hsin et al., 2024).

A simple continuum theory is the $F(\mathcal S)$8D $F(\mathcal S)$9 gauge theory

$\mathcal S$0

Within this formulation, the distinction between trivial and nontrivial subsystem fractionalization is again encoded in global relations. For the quotient relation

$\mathcal S$1

nontrivial fractionalization is characterized by an anyon $\mathcal S$2 whose Wilson loop appears at the boundary of the truncated product. Equivalently,

$\mathcal S$3

and the resulting anomaly lives in

$\mathcal S$4

The same work states that in $\mathcal S$5D, planar subsystem symmetry fractionalization on particles is captured by classes in

$\mathcal S$6

and by the bulk SPT action

$\mathcal S$7

with $\mathcal S$8 (Hsin et al., 2024).

The field-theoretic analysis also introduces anomalous subsystem symmetry fractionalization. One anomalous example is a $\mathcal S$9D boundary of a $U(s)$00D SSPT with action

$U(s)$01

supporting a modified toric-code gauge theory with linear symmetry whose truncated global relation creates a semion line. Another example is a $U(s)$02D mixed anomaly obtained by embedding a subsystem one-form symmetry and a two-form symmetry in a $U(s)$03D SPT with

$U(s)$04

These examples show that some fractionalization patterns cannot be realized in a purely lower-dimensional system (Hsin et al., 2024).

6. Relation to fracton order and subsystem-enriched phases

The broader symmetry-fracton literature studies a related but distinct setting: three-dimensional fracton phases with planar subsystem symmetry. In that context, subsystem-symmetry fractionalization means that emergent excitations such as fractons and lineons transform projectively under an un-gauged factor $U(s)$05. Concretely, if $U(s)$06 is the $U(s)$07 action restricted to a plane $U(s)$08, then at a subsystem-symmetry-invariant location one may have

$U(s)$09

with $U(s)$10 a $U(s)$11-cocycle. The corresponding classification is described by a group cohomology

$U(s)$12

where $U(s)$13 is the abelian fusion group of the emergent excitations (You et al., 2018).

A central construction is partial gauging of one factor in

$U(s)$14

After gauging only a $U(s)$15 subsystem symmetry factor, one obtains a symmetry-enriched X-cube model in which lineon flux excitations carry half-charge under the remaining $U(s)$16 subsystem symmetry. The endpoint of an explicit flux-string operator transforms projectively, with

$U(s)$17

The same framework also produces $U(s)$18-enriched fracton order by decorating subsystem domain-frame lines with valence-tube-solid structure whose ends carry Kramers doublets; after gauging, the lineons satisfy

$U(s)$19

(You et al., 2018).

The continuum description uses a higher-rank mutual Chern-Simons theory with two symmetric, traceless $U(s)$20 rank-2 gauge fields $U(s)$21 and $U(s)$22, gauge transformations

$U(s)$23

and topological response

$U(s)$24

Varying $U(s)$25 gives twisted Gauss laws that bind flux in one sector to charges in the other,

$U(s)$26

This formulation naturally captures both the gapless boundary modes of the ungauged subsystem SPT and the statistical interactions of the gauged fracton order (You et al., 2018).

The relation between the two-dimensional global-relation mechanism and the three-dimensional fracton cohomology mechanism is not an identity. The former is formulated in terms of fractionalizable global relations $U(s)$27 and homomorphisms $U(s)$28, whereas the latter uses projective actions on subsystem-invariant locations and $U(s)$29 (Stephen et al., 2022). A plausible implication is that “subsystem symmetry fractionalization” names a family of enrichment phenomena whose precise algebraic description depends on both dimensionality and the geometry of symmetry support.

7. Classification significance and open directions

The two-dimensional classification by

$U(s)$30

identifies global relations as the basic carriers of fractionalization data. This result is presented as a new mechanism that should enrich the classification of subsystem-symmetry-enriched topological phases in all dimensions (Stephen et al., 2022).

The later foliated-field-theory perspective broadens that claim by stating that the framework extends the range of subsystem symmetry fractionalization through new examples derived from embedding subsystem symmetry into higher-form symmetry, and by exhibiting both continuum and lattice realizations, including anomalous ones (Hsin et al., 2024). In this language, obstructions can arise from nontrivial Postnikov classes, described as an “$U(s)$31 obstruction,” while the interplay of $U(s)$32-form, $U(s)$33-form, and subsystem symmetry can form higher-group structures (Hsin et al., 2024).

Several research directions are stated directly. One is that the importance of global relations in fractionalization should hold significance for the classification of phases with subsystem symmetries in all dimensions (Stephen et al., 2022). Another is the possibility of novel criticalities at SSET-breaking transitions and possible anomalies, meaning patterns not realizable strictly in two dimensions but only at boundaries of three-dimensional systems (Stephen et al., 2022). The anomalous semion and mixed-anomaly constructions in the foliated framework provide concrete realizations of those questions (Hsin et al., 2024).

Taken together, the current picture is technically sharp on three points. First, subsystem symmetry fractionalization is not the straightforward subsystem analogue of global projective symmetry action. Second, in two-dimensional topological order with rigid subsystem symmetry, the realizable fractionalization data are carried by global relations and classified by $U(s)$34. Third, restricted mobility and anomaly phenomena are not peripheral but intrinsic to the structure of subsystem-symmetry-enriched topological phases (Stephen et al., 2022).