---
title: Subsystem Symmetry Fractionalization
url: https://www.emergentmind.com/topics/subsystem-symmetry-fractionalization
type: topic
---

# Subsystem Symmetry Fractionalization

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 [2203.13244].

## 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 [2203.13244].

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 [2203.13244].

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 [2203.13244].

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 [2203.13244].

## 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 $\mathfrak R$ is the normal subgroup generated by all relations. The relations are partitioned into “non-fractionalizable” $\mathfrak R_{nf}$, consisting of local relations, and “fractionalizable” $\mathfrak R_f$, consisting of global ones. The abelian group of fractionalizable relations is
\[
\mathfrak R_f \;=\;\mathfrak R/\mathfrak R_{nf}\,,\quad \mathfrak R_f\cong\mbox{(some } \mathbb Z_2^k\mbox{)}.
\]
If the anyon fusion group is $\mathcal A$, then fractionalization classes are specified by a group homomorphism
\[
\phi:\mathfrak R_f\;\to\;\mathcal A,
\]
assigning to each fractionalizable relation $r$ an anyon $\phi(r)$ such that the localized product of symmetry operators $V_R(r)$ around a region $R$ is equivalent to braiding $\phi(r)$ around $R$ [2203.13244].

The same structure can be expressed in cocycle language. For global symmetries one has localized operators $V_R(g)$ satisfying
\[
V_R(g_1)\,V_R(g_2)\;=\;\omega(g_1,g_2)\;V_R(g_1g_2),
\]
with
\[
\omega(g_1,g_2)\,\omega(g_1g_2,g_3)=\omega(g_1,g_2g_3)\,\omega(g_2,g_3).
\]
In the subsystem case, however, only those $\omega$ arising from $\phi(r)$ and trivial on $\mathfrak R_{nf}$ are realizable. The localizations of local generators are unambiguous, so there are no further coboundary ambiguities of the usual kind [2203.13244].

A concise consequence is that fractionalization classes of a subsystem symmetry group $G$ acting on a topological order with abelian anyons $\mathcal A$ are classified by homomorphisms
\[
\phi:\mathfrak R_f\;\to\;\mathcal A.
\]
This is the classification statement specific to the mechanism based on global relations [2203.13244].

## 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 [2203.13244].

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 [2203.13244].

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 $e$ anyon is fully immobile, a “symmetry-protected fracton” [2203.13244].

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 [2203.13244]. 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 [2403.09098].

## 4. Exactly solvable two-dimensional models

The canonical two-dimensional example is a $\mathbb Z_2$ toric-code “line” SSET. Its degrees of freedom are qubits on sites $v$ and edges $e$ of a square lattice, with subsystem symmetry generators
\[
U(x_j)=\prod_i X_{(i,j)},\qquad U(y_i)=\prod_j X_{(i,j)}.
\]
These obey $x_j^2=y_i^2=1$ and commute linewise, together with the global relation
\[
r_{\rm all}=\Bigl(\prod_j x_j\Bigr)\Bigl(\prod_i y_i\Bigr)^{-1}=1.
\]
The Hamiltonian is
\[
H_\ell=-\sum_v\bigl(X_v\,Z_{v\pm\hat x/2}\bigr)\;-\;\sum_v\bigl(X_v\,Z_{v\pm\hat y/2}\bigr)\;-\;\sum_p\Bigl(\prod_{e\in p}X_e\Bigr)\Bigl(\prod_{v\in p}Z_v\Bigr).
\]
It is unitarily, but not symmetrically, equivalent to the usual toric code and supports $e,m,\epsilon$ anyons [2203.13244].

In this model, the anyon $e$ splits into two lineons, $e^x$ and $e^y$, moving only horizontally or vertically. Truncated line symmetry creates $m$-pairs at its ends, and the product of all truncated lines $V(r_{\rm all})$ is exactly a closed $m$-loop $S^m_{\partial R}$. Hence
\[
\phi(r_{\rm all})=m.
\]
A more refined statement is that $e^x$ in row $j$ has charge $-1$ under $U(x_j)$ and $+1$ under all other generators, so the total charge under the relation $\prod x_j\prod y_i=1$ is $-1$ [2203.13244].

Variants of the same construction realize
\[
\phi(r_{\rm all})=e
\qquad\text{or}\qquad
\phi(r_{\rm all})=\epsilon
\]
by modifying local Hamiltonian terms to attach symmetry charges of suitable anyon type. Section III C contains a model $H_f$ with $\phi(r_{\rm all})=\epsilon$ [2203.13244].

A second family uses three-direction line symmetries with generators $x_j$, $y_i$, and $d_k$ along horizontal, vertical, and one diagonal direction. The fractionalizable relations are
\[
r_{a,b}=\Bigl(\prod_{j\equiv a\,(2)}x_j\Bigr)\Bigl(\prod_{i\equiv b\,(2)}y_i\Bigr)\Bigl(\prod_{k\equiv a+b+1\,(2)}d_k\Bigr)=1,\quad a,b\in\{0,1\}.
\]
In this case $\mathfrak R_f\cong \mathbb Z_2^3$, and the fractionalization maps $\phi(r_{a,b})$ can be chosen independently, subject to one overall product constraint, by decorating certain Hamiltonian terms. The $e$ anyons split into three lineons moving along the three directions [2203.13244].

A third class involves fractal symmetries. The model $H_F$ is obtained by gauging one sublattice of a fractal-symmetry cluster state. The symmetry generators take the form
\[
U(q)=\prod_v X_v^{q_v},\qquad q_{j+1}=f\,q_j,
\]
where $f$ is a reversible cellular automaton rule. Truncated fractal symmetry operators produce pairs of $m$ anyons along the boundary of the truncated fractal. One can define preserved generators $p_k=f_kf_{k+1}$ so that only the global relation
\[
r_{\rm all}=\prod_k p_k=1
\]
is fractionalizable. The isolated $e$ anyon is fully immobile, and one computes
\[
\phi(r_{\rm all})=m
\]
by counting that $e$ anticommutes with an odd number of $p_k$ [2203.13244].

## 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 $q$-form symmetry has generators on rigid codimension-$q$ subspaces [2403.09098].

The continuum framework introduces foliation one-forms $e^k$ satisfying
\[
e^k\wedge e^k=0,
\]
and foliated gauge fields $B_n^k$ obeying
\[
B_n^k\wedge e^k=0,\qquad B_n^k\mapsto B_n^k+d\lambda_{n-1}^k,\qquad \lambda_{n-1}^k\wedge e^k=0.
\]
In $2+1$ dimensions, a subsystem one-form symmetry acting on lines of constant $x$ and constant $y$ is described by
\[
B_2^x=A^x\wedge dx,\qquad B_2^y=A^y\wedge dy,
\]
with $B_2^k\wedge dx^k=0$. The embedding into the ordinary one-form symmetry is
\[
B_2=\sum_{k=x,y}v^k B_2^k,
\]
typically with $v^k=1$ [2403.09098].

A simple continuum theory is the $2+1$D $\mathbb Z_N$ gauge theory
\[
S \;=\;\frac{N}{2\pi}\int a\wedge db \;+\;\frac{N}{2\pi}\int b\wedge B_2
\;=\;\frac{N}{2\pi}\int \Bigl(a\wedge db + b\wedge\sum_k B_2^k\Bigr).
\]
Within this formulation, the distinction between trivial and nontrivial subsystem fractionalization is again encoded in global relations. For the quotient relation
\[
\prod_i Q_i^x=\prod_j Q_j^y,
\]
nontrivial fractionalization is characterized by an anyon $a\in\mathcal A$ whose Wilson loop appears at the boundary of the truncated product. Equivalently,
\[
B_2=B_2^x+B_2^y
\quad\Longrightarrow\quad
\text{nontrivial fractionalization }\Leftrightarrow \theta([B_2^x],[B_2^y])\neq0,
\]
and the resulting anomaly lives in
\[
\int B_2^x\wedge B_2^y.
\]
The same work states that in $3+1$D, planar subsystem symmetry fractionalization on particles is captured by classes in
\[
H^3(B^2G,\mathcal A^{(2)})
\]
and by the bulk SPT action
\[
S_{\rm bulk}=\frac{2\pi}{N}\int B_2\wedge dB_2
\]
with $B_2=\sum_k B_2^k$ [2403.09098].

The field-theoretic analysis also introduces anomalous subsystem symmetry fractionalization. One anomalous example is a $2+1$D boundary of a $3+1$D SSPT with action
\[
\pi\int B_2^xB_2^yB_2^z,
\]
supporting a modified toric-code gauge theory with linear symmetry whose truncated global relation creates a semion line. Another example is a $3+1$D mixed anomaly obtained by embedding a subsystem one-form symmetry and a two-form symmetry in a $4+1$D SPT with
\[
S=\tfrac{2\pi}{2\pi}\int B_2\wedge B_3.
\]
These examples show that some fractionalization patterns cannot be realized in a purely lower-dimensional system [2403.09098].

## 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 $H\subset G_{\rm sub}$. Concretely, if $U_h(p)$ is the $H$ action restricted to a plane $p$, then at a subsystem-symmetry-invariant location one may have
\[
U_{h_1}(p)\,U_{h_2}(p)=\omega(h_1,h_2)\,U_{h_1h_2}(p)
\]
with $\omega(h_1,h_2)=\pm1$ a $2$-cocycle. The corresponding classification is described by a group cohomology
\[
H^2_{\rm sub}(H,A),
\]
where $A$ is the abelian fusion group of the emergent excitations [1805.09800].

A central construction is partial gauging of one factor in
\[
G_{\rm sub}=\mathbb Z_2\times H_{\rm sub}.
\]
After gauging only a $\mathbb Z_2$ subsystem symmetry factor, one obtains a symmetry-enriched X-cube model in which lineon flux excitations carry half-charge under the remaining $\mathbb Z_2$ subsystem symmetry. The endpoint of an explicit flux-string operator transforms projectively, with
\[
(U_b)^2=-1.
\]
The same framework also produces $\mathcal T$-enriched fracton order by decorating subsystem domain-frame lines with valence-tube-solid structure whose ends carry Kramers doublets; after gauging, the lineons satisfy
\[
\mathcal T^2=-1
\]
[1805.09800].

The continuum description uses a higher-rank mutual Chern-Simons theory with two symmetric, traceless $U(1)$ rank-2 gauge fields $A_{ij}^1$ and $A_{ij}^2$, gauge transformations
\[
A^I_{ij}\mapsto A^I_{ij}+\partial_i\partial_j\alpha^I,\qquad
A_0^I\mapsto A_0^I+\partial_t\alpha^I,
\]
and topological response
\[
\mathcal{L}_{\rm CS}
= -\frac{1}{4\pi}\,\epsilon^{ijk}\Bigl[A^1_{jk}\,\partial_tA^2_{\,i\ell}
+A^2_{jk}\,\partial_tA^1_{\,i\ell}\Bigr]
-A_0^I\rho^I + A_{ij}^I J_{ij}^I.
\]
Varying $A_0^I$ gives twisted Gauss laws that bind flux in one sector to charges in the other,
\[
\tfrac1{2\pi}\,\epsilon^{ijk}\partial_iA^2_{jk}=\rho^1,\qquad
\tfrac1{2\pi}\,\epsilon^{ijk}\partial_iA^1_{jk}=\rho^2.
\]
This formulation naturally captures both the gapless boundary modes of the ungauged subsystem SPT and the statistical interactions of the gauged fracton order [1805.09800].

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 $\mathfrak R_f$ and homomorphisms $\phi:\mathfrak R_f\to\mathcal A$, whereas the latter uses projective actions on subsystem-invariant locations and $H^2_{\rm sub}(H,A)$ [2203.13244]. 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
\[
\phi:\mathfrak R_f\to\mathcal A
\]
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 [2203.13244].

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 [2403.09098]. In this language, obstructions can arise from nontrivial Postnikov classes, described as an “$H^3$ obstruction,” while the interplay of $0$-form, $1$-form, and subsystem symmetry can form higher-group structures [2403.09098].

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 [2203.13244]. 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 [2203.13244]. The anomalous semion and mixed-anomaly constructions in the foliated framework provide concrete realizations of those questions [2403.09098].

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 $\phi:\mathfrak R_f\to\mathcal A$. Third, restricted mobility and anomaly phenomena are not peripheral but intrinsic to the structure of subsystem-symmetry-enriched topological phases [2203.13244].

Source: https://www.emergentmind.com/topics/subsystem-symmetry-fractionalization