Symmetry-Enriched Criticality

• Symmetry-enriched criticality is defined as the refinement of conventional critical universality by incorporating symmetry actions on local and nonlocal operators, leading to distinct gapless states.
• Key insights include the role of symmetry fluxes, twisted sector charges, and conformal boundary conditions in distinguishing critical points sharing the same central charge and local exponents.
• Applications span gapless SPT phases, disorder-induced criticality, and non-Hermitian and noninvertible systems, illustrating how symmetry alters dynamic and topological properties.

Symmetry-enriched criticality denotes critical points or critical phases for which bulk universality data are not sufficient once symmetry is enforced: two gapless states may share the same central charge, the same local critical exponents, or the same long-wavelength field theory, yet remain sharply distinct because symmetry acts differently on local operators, nonlocal or twisted sectors, boundaries, interfaces, or generalized defects (Verresen et al., 2019, Yu et al., 2021). In current usage, the term spans several related settings: gapless analogues of SPT distinctions in $1+1$ dimensions, symmetry-sensitive boundary and interface classifications, disorder- and Floquet-enriched infinite-randomness fixed points, dynamical and measurement-induced critical phenomena, and extensions involving non-Hermitian, noninvertible, and higher-form symmetries (Duque et al., 2020, Chou et al., 11 Sep 2025, Li et al., 2024, Moy, 21 Jul 2025).

1. Core definitions and organizing principles

A standard formulation of symmetry-enriched criticality in $1+1$ dimensions treats a critical theory as a $G$-enriched CFT, meaning that the symmetry action on the low-energy theory carries discrete information that cannot be removed without leaving the universality class or breaking symmetry (Verresen et al., 2019). In this formulation, the basic nonlocal object is a symmetry flux for $g\in G$,

$$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$

whose endpoint $\mathcal O_n^g$ is chosen so that

$$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$

has the slowest possible decay. In gapless systems one typically has

$$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle \sim \frac{1}{|n-m|^{2\Delta_g}},$$

whereas long-range order is the special case $\Delta_g=0$ (Verresen et al., 2019). The symmetry charge of such a dominant nonlocal operator, or equivalently of the corresponding twisted-sector ground state, is then a topological invariant of the critical point (Verresen et al., 2019).

This perspective already shows why symmetry enrichment is not reducible to ordinary bulk scaling data. The same bare CFT can split into several symmetry-enriched versions, distinguished by the symmetry properties of local primaries, disorder operators, or twisted sectors rather than by different values of $c$ or different local exponents (Verresen et al., 2019). A complementary formulation emphasizes that not all symmetry-enriched critical points exhibit protected degenerate edge modes; conformal boundary conditions can be the more generic universal fingerprint. In the generalized Ising and three-state Potts chains studied in this framework, critical points with the same bulk Ising or Potts CFT are nevertheless distinct because they realize different conformal boundary conditions once symmetry is enforced (Yu et al., 2021).

A broader implication, explicit in the coupled spin-ladder construction, is that the same compact-boson universality class can support several distinct gapless phases with the same unbroken symmetries, distinguished only by how microscopic symmetry is embedded into the infrared fields (Mondal et al., 2023). This suggests that symmetry-enriched criticality is best understood as a refinement of universality: the critical fixed point is specified not only by its local field content, but also by symmetry action on local and nonlocal sectors.

2. Boundary conditions, edge modes, and one-dimensional realizations

Boundary data provide some of the clearest concrete realizations of symmetry-enriched criticality. In generalized Ising and three-state Potts chains, the critical point between an SPT phase and a symmetry-breaking phase lies in the same bulk universality class as the ordinary Ising or Potts critical point, but realizes a different conformal boundary condition (Yu et al., 2021). For the Ising-family example, the ordinary transverse-field Ising chain realizes free boundary conditions, whereas the cluster-Ising critical point realizes fixed boundary conditions. For the Potts-family example, ordinary disorder–FM and disorder–not-$1+1$0 transitions realize free Potts boundary conditions, while the RSPT-related transitions realize the dual-mixed boundary condition (Yu et al., 2021). This is a direct demonstration that boundary CFT data can refine bulk universality.

The spin-ladder triangular model sharpens this point by exhibiting a critical line between a spontaneous symmetry breaking phase and an SPT phase, with one endpoint in symmetry-enriched Ashkin–Teller universality and the remaining critical line in symmetry-enriched Ising universality (Wang et al., 2023). Under open boundary conditions, both classes support symmetry-protected degenerate edge modes. The finite-size splitting of these boundary sectors is not uniform along the critical line: at the SEATU endpoint it closes as

$1+1$1

at the special SEIU point $1+1$2 it closes as

$1+1$3

and at other SEIU points it behaves as

$1+1$4

The coexistence of algebraic and exponential edge splittings within the same bulk symmetry-enriched Ising universality shows that boundary-sector data can contain structure not fixed by the bulk CFT alone (Wang et al., 2023).

A still more extensive example is the coupled XXZ spin ladder, which realizes five distinct gapless symmetry-enriched phases,

$1+1$5

all with the same unbroken symmetries and the same long-wavelength compact-boson description, yet not mutually connected without a phase transition or intermediate phases (Mondal et al., 2023). Their distinction is encoded in the symmetry action on the infrared bosons and in the behavior of charged local and nonlocal operators. One phase, $1+1$6, is a gapless topological phase with nonzero string order and protected edge modes under open boundary conditions (Mondal et al., 2023). Here the enrichment is not a change in local bulk criticality; it is a change in symmetry realization, operator content, and boundary topology at fixed $1+1$7.

3. Interfaces, anomalies, and universal constraints

Interfaces supply a more robust diagnostic than edges in situations where protected boundary zero modes are absent. A general interface criterion states that if there exists a $1+1$8-symmetric invertible interface between two $1+1$9d critical theories, then the two theories must have the same symmetry-charge assignments for all operators, local and nonlocal (Prembabu et al., 29 Dec 2025). Consequently, whenever two symmetry-enriched realizations of the same CFT differ in local operator charges or in twisted-sector charges, any symmetry-preserving spatial interface between them must flow to a non-invertible, degenerate, or factorizing defect rather than to a transparent invertible interface (Prembabu et al., 29 Dec 2025). In the Ising examples, interfaces between distinct $G$0-enriched versions of the Ising CFT realize $G$1d symmetry-breaking interface phases, and the finite-size splittings of the corresponding interface sectors scale as

$G$2

when the leading allowed defect operator has scaling dimension $G$3 (Prembabu et al., 29 Dec 2025).

A complementary boundary-CFT viewpoint yields a symmetry-enriched $G$4-theorem. When a $G$5d rational CFT can interpolate, by symmetry-preserving relevant deformations, between distinct trivially gapped phases or SPT phases protected by a global symmetry $G$6, the strip Hilbert space between the corresponding conformal boundary conditions carries a projective $G$7-action. If the difference class $G$8 is nontrivial, then every energy eigenspace is degenerate, in particular

$G$9

for the ground state degeneracy on the interval (Cordova et al., 2022). Modular invariance and boundary CFT then imply a strictly positive lower bound on admissible central charge for such SPT transitions (Cordova et al., 2022). This makes symmetry enrichment a quantitative constraint on critical CFT data, not merely a boundary decoration.

Self-duality gives an anomaly-based version of the same idea. For lattice models self-dual under a transformation connecting distinct SPT phases, symmetry-twisted boundary conditions produce exact degeneracies, implying that the self-dual point cannot realize a unique gapped symmetric ground state under periodic boundary conditions (Li et al., 2022). The paper formulates this as a mixed ’t Hooft anomaly involving the ordinary symmetry and the duality symmetry at criticality, and extends the argument beyond ordinary $g\in G$0-form symmetries to higher-form and subsystem symmetries (Li et al., 2022). This places symmetry-enriched criticality in direct continuity with anomaly-protected criticality and deconfined criticality.

4. Dynamical, measurement-induced, and disordered forms of enrichment

The term is also used in dynamical and stochastic settings, although the mechanism of enrichment is then different. In the three-dimensional clock model with emergent $g\in G$1 symmetry, the critical point carries two divergent spatial scales,

$g\in G$2

and the nonequilibrium relaxation dynamics correspondingly require two time-like scales. The proposed scaling form is

$g\in G$3

with the ordinary magnetization governed by the usual $g\in G$4D XY exponent $g\in G$5, while the anisotropy-sensitive angular order parameter $g\in G$6 relaxes first with $g\in G$7 and then with a distinct late-time exponent $g\in G$8 (Shu et al., 2023). Here emergent symmetry enriches dynamic universality rather than boundary topology.

A more explicitly information-theoretic usage appears in dynamical quantum phase transitions of the quenched Lipkin–Meshkov–Glick model. The asymmetry monotone

$g\in G$9

and its time average

$$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$0

detect the same dynamical critical region as the conventional dynamical order parameter $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$1, while peaks in time-averaged asymmetry coincide with peaks in time-averaged entropy production (Nascimento et al., 31 Jan 2026). In this sense the enrichment is dynamical and informational rather than topological: symmetry does more than label phases, because asymmetry quantifies symmetry-sector coherence generated during critical dynamics (Nascimento et al., 31 Jan 2026).

Measurement-induced criticality shows that such enrichment can also be fragile. In monitored $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$2-symmetric random circuits, symmetry-preserving measurements can support symmetry-enriched measurement-induced transitions, but explicit symmetry-breaking measurements are a relevant perturbation (Russotto et al., 9 Jul 2026). Measuring in a rotated basis

$$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$3

produces a finite charge correlation length

$$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$4

so any nonzero symmetry breaking ultimately drives the transition to the same universality class as a generic monitored circuit with no symmetry constraint (Russotto et al., 9 Jul 2026). The paper therefore gives a sharp negative result: a symmetry-enriched monitored critical point is not stable to explicit symmetry-breaking monitoring in the class studied.

Disorder provides a different route to enriched criticality. In random quantum spin chains, strong-randomness critical points can have local bulk properties indistinguishable from conventional random-singlet or infinite-randomness fixed points, yet remain topologically distinct because of protected edge modes, altered boundary criticality, and symmetry charges of nonlocal operators (Duque et al., 2020). The random Ising$$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$5 critical point and the symmetry-enriched random-singlet phase are the disordered analogues of gapless topological phases; their local bulk exponents agree with conventional infinite-randomness criticality, while boundary and nonlocal observables do not (Duque et al., 2020). In a more general categorical setting, a one-dimensional disorder ensemble with exact symmetry category $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$6 and only average realization of the nontrivial $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$7-graded sectors of $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$8 is anomaly-free if and only if the Drinfeld center $$\mathcal S_n^g=\cdots U_{n-2}^g U_{n-1}^g\,\mathcal O_n^g,$$9 admits a magnetic Lagrangian algebra invariant under the permutation action of $\mathcal O_n^g$0 on anyons (Li et al., 9 Feb 2026). When that average anomaly is present, a single disorder realization is long-range entangled with probability one in the thermodynamic limit, and the paper argues that Griffiths-type or infinite-randomness behavior should be expected (Li et al., 9 Feb 2026).

5. Non-Hermitian, noninvertible, and generalized-symmetry extensions

Non-Hermitian systems furnish a distinct notion of symmetry-enriched criticality. In $\mathcal O_n^g$1-symmetric non-Hermitian SSH chains, the same non-unitary $\mathcal O_n^g$2 CFT appears on both trivial and topological critical lines, but only some critical points carry robust edge modes (Chou et al., 11 Sep 2025). The topological degeneracy is encoded directly in the complex entanglement entropy,

$\mathcal O_n^g$3

so that

$\mathcal O_n^g$4

counts the winding sector $\mathcal O_n^g$5 (Chou et al., 11 Sep 2025). This enriches non-unitary criticality in a way unavailable in Hermitian systems: the same bulk non-unitary universality class splits into topologically distinct $\mathcal O_n^g$6-protected sectors, and the topological distinction is visible in the imaginary part of the entropy itself (Chou et al., 11 Sep 2025).

Noninvertible symmetry provides another extension. In one-dimensional critical chains with $\mathcal O_n^g$7 symmetry, the paper identifies noninvertible symmetry-enriched quantum critical points separating distinct noninvertible SPT phases or separating a noninvertible SPT from a noninvertible-symmetry-breaking phase (Li et al., 2024). The low-energy theories are ordinary Ising or compact-boson CFTs, but twisted-sector $\mathcal O_n^g$8-charges and open-chain degeneracies distinguish the resulting critical points. Finite-entanglement scaling gives $\mathcal O_n^g$9 for the Ising-type transitions and $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$0 for the compact-boson transition, while the topological distinctions survive in twisted-boundary sectors and open-chain edge structure (Li et al., 2024). This is a direct generalization of group-symmetry-enriched criticality to categorical symmetry.

A higher-dimensional generalization replaces ordinary order parameters by generalized symmetry breaking. In $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$1 dimensions, $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$2 gauge theory coupled to adjoint Majorana fermions yields continuous transitions between phases with different breaking patterns of a magnetic $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$3 one-form symmetry (Moy, 21 Jul 2025). For even $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$4 and sufficiently large odd $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$5, the massless theory separates a phase where the $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$6 one-form symmetry is completely broken from one where it is broken only to $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$7, producing $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$8 topological order (Moy, 21 Jul 2025). In a second class of examples, enforcing a non-invertible analogue of time-reversal symmetry $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle$$9 and $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle \sim \frac{1}{|n-m|^{2\Delta_g}},$$0 flavor symmetry produces a transition between a topologically ordered phase and a phase that spontaneously breaks the non-invertible time-reversal symmetry (Moy, 21 Jul 2025). Here generalized symmetry enrichment means that the critical theory is constrained simultaneously by ordinary, higher-form, and non-invertible symmetries.

6. Scope, fragility, and open problems

The literature does not use “symmetry-enriched criticality” in a single narrowly fixed sense. In the clean $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle \sim \frac{1}{|n-m|^{2\Delta_g}},$$1d CFT setting it usually refers to gapless analogues of SPT distinctions, often encoded in nonlocal operator charges, conformal boundary conditions, edge sectors, or interface anomalies (Verresen et al., 2019, Yu et al., 2021, Prembabu et al., 29 Dec 2025). In nonequilibrium work it can instead mean that symmetry enriches critical dynamics by introducing new dynamical exponents or by organizing asymmetry generation during a dynamical phase transition (Shu et al., 2023, Nascimento et al., 31 Jan 2026). A plausible implication is that the phrase now denotes a family of related phenomena rather than a single classification scheme.

Several common simplifications are explicitly contradicted by the cited work. Symmetry-enriched criticality is not equivalent to protected edge zero modes: conformal boundary conditions and non-invertible interfaces can be the more generic universal signatures (Yu et al., 2021, Prembabu et al., 29 Dec 2025). It is also not automatically robust: explicit symmetry-breaking measurements destabilize the monitored-circuit version, and the asymptotic critical behavior flows to the non-symmetric universality class (Russotto et al., 9 Jul 2026). Nor is it always a purely bulk-CFT notion: the spin-ladder triangular model shows that boundary-sector gap scaling can exhibit a finer structure than bulk symmetry-enriched universality labels alone would suggest (Wang et al., 2023).

Many results remain model-specific or numerical. The asymmetry-based diagnosis of dynamical criticality is demonstrated in the quenched Lipkin–Meshkov–Glick model and is presented as a numerical correspondence rather than a universal theorem (Nascimento et al., 31 Jan 2026). The $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle \sim \frac{1}{|n-m|^{2\Delta_g}},$$2-symmetric non-Hermitian constructions are established in one-dimensional free-fermion models, and a broader classification of topologically distinct non-unitary universality classes is left open (Chou et al., 11 Sep 2025). The noninvertible $$\langle \mathcal S_m^{g\dagger}\mathcal S_n^g\rangle \sim \frac{1}{|n-m|^{2\Delta_g}},$$3 examples are one-dimensional and rely heavily on Kennedy–Tasaki duality (Li et al., 2024). The average categorical symmetry framework provides a sharp anomaly criterion but not a full classification of disorder-controlled universality classes (Li et al., 9 Feb 2026).

Open directions are correspondingly broad. Explicitly proposed extensions include higher dimensions and interacting many-body versions of non-Hermitian symmetry-enriched criticality (Chou et al., 11 Sep 2025); mixed-state, finite-temperature, open-system, and alternative-asymmetry versions of dynamical criticality (Nascimento et al., 31 Jan 2026); other symmetries, deterministic dynamics, non-Abelian cases, and higher dimensions in monitored circuits (Russotto et al., 9 Jul 2026); systematic classifications of topologically distinct non-unitary or noninvertible universality classes (Chou et al., 11 Sep 2025, Li et al., 2024); and numerical frameworks for disordered boundary criticality in higher dimensions, where strong-disorder real-space RG has already been proposed as a route toward symmetry-enriched boundary criticality and gapless SPT edges (Tenkila et al., 2024). The cumulative lesson is that symmetry-enriched criticality is now a broad research program: it asks how symmetry, topology, and criticality continue to constrain one another once the bulk is gapless, and it does so across equilibrium, nonequilibrium, disordered, non-Hermitian, and generalized-symmetry regimes.