Order Parameter Fractionalization

• Order Parameter Fractionalization is a phenomenon where the typical local bosonic order parameter is replaced by an emergent composite field carrying fractional quantum numbers, gauge charge, or nonlocal characteristics.
• The framework factorizes conventional order parameters into emergent constituents with inherent gauge redundancies, explaining transitions in systems like Kondo–Kitaev models, disordered PDWs, and doped Mott insulators.
• This conceptual shift provides critical insights into unconventional superconductivity, topological order, and symmetry fractionalization, facilitating new diagnostics and methods in strongly correlated systems.

Searching arXiv for recent and relevant papers on order parameter fractionalization and closely related uses of the term. Order parameter fractionalization denotes a set of related phenomena in which the object governing ordering is not a conventional local bosonic bilinear, but an emergent composite carrying fractional quantum numbers, gauge charge, or projective symmetry quantum numbers. In the literature considered here, the term covers several distinct but connected usages: a spinorial charge-$e$, $S=\tfrac12$ condensate in Kondo–Kitaev systems, a disorder-driven decomposition of pair-density-wave order into a uniform charge-$2e$ superconductor and a destroyed charge-density component, nonlocal Wilson-loop order parameters in doped Mott insulators, and symmetry-fractionalized anyons whose condensation necessarily produces ordinary symmetry breaking (Komijani et al., 2018, Tsvelik et al., 2021, May-Mann et al., 24 Sep 2025, Schuler et al., 2022). A common thread is that the observable ordered phase is controlled by fractionalized constituents together with an emergent gauge structure, so that the physically measurable order parameter is often composite, nonlocal, or secondary.

1. Conventional order parameters and their fractionalized counterparts

In the Landau paradigm, broken symmetry is characterized by a bosonic order parameter, and in Yang’s formulation of off-diagonal long-range order the relevant operator is likewise bosonic, built from an even number of fermion fields and carrying integer spin and even charge. One formulation of order fractionalization explicitly relaxes that assumption: the order parameter may transform in a fundamental spinorial representation, carry half-integer quantum numbers, and appear through long-time factorization of dynamical quantities rather than through a purely equal-time local expectation value (Komijani et al., 2018).

In that formulation, the central object is the single-particle irreducible self-energy. Conventional broken symmetry yields an effectively local and instantaneous self-energy, $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$, where $M_{\alpha\beta}$ is a bosonic order parameter. By contrast, fractionalized order is associated with a fermionic pole and asymptotic factorization,

$$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$

with $V$ a spinorial order parameter. This construction extends off-diagonal long-range order into the time domain and ties ordering to a long-lived composite fermion or “dark fermion” (Komijani et al., 2018).

A second, complementary usage appears in systems with intertwined order. There the primary order parameter can itself be re-expressed as a product of emergent constituents. For pair-density waves, for example,

$$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$

with $\Delta_0$ a uniform charge-$2e$ superconducting field and $S=\tfrac12$0 a charge-density-wave field. This decomposition introduces a local $S=\tfrac12$1 redundancy,

$S=\tfrac12$2

so the physical PDW remains the gauge-invariant composite, while the factors acquire separate low-energy significance only in the disordered regime (May-Mann et al., 24 Sep 2025).

A third formulation appears in doped Mott systems, where the relevant “order parameters” are not local bosonic fields at all, but Wilson loops of emergent compact gauge fields. In the phase-string approach to the $S=\tfrac12$3–$S=\tfrac12$4 model, the underdoped regime is organized by two Wilson loops $S=\tfrac12$5 and $S=\tfrac12$6, which diagnose confinement and deconfinement of holons and spinons and thereby determine whether the system is antiferromagnetic, superconducting, or Bose insulating (Ye et al., 2011). In this sense, order parameter fractionalization means that conventional antiferromagnetic and superconducting orders are composites of spinon or holon condensation together with a specific confinement pattern.

2. Fractionalization as factorization, gauge redundancy, and composite order

Across these settings, a recurring structural motif is the factorization of a physical operator into emergent fields together with a gauge redundancy. In the pair-density-wave problem, the clean order parameter

$S=\tfrac12$7

has no uniform charge-$S=\tfrac12$8 component, but it admits composite bilinears

$S=\tfrac12$9

corresponding respectively to a charge-density wave at $2e$0 and a uniform charge-$2e$1 superconducting field (May-Mann et al., 24 Sep 2025). The fractionalization ansatz $2e$2 reorganizes these intertwined orders into emergent pieces with a local $2e$3 gauge symmetry, making it possible for disorder to select different fates for the superconducting and charge sectors.

In impurity and Kondo-lattice realizations, factorization occurs at the level of composite fermions. In the single-channel Kondo problem,

$2e$4

while in the channel-asymmetric two-channel Kondo model,

$2e$5

The spinorial amplitude $2e$6 or $2e$7 carries the symmetry quantum numbers that would ordinarily be attributed to a conventional order parameter, whereas $2e$8 is an emergent fermion. The transformation

$2e$9

shows the associated gauge redundancy directly (Komijani et al., 2018).

In the $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$0–$$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$1 model, the same logic is formulated in gauge-theoretic terms. After the phase-string transformation, holons and spinons see each other as $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$2-flux sources through compact $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$3 gauge fields $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$4 and $$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$5. The order of the antiferromagnetic and superconducting phases is then encoded in the confinement properties of these gauge sectors. The superconducting phase has holon condensation and logarithmic confinement of spinons,

$$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$6

while the antiferromagnetic phase has spinon condensation and logarithmic confinement of holons,

$$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$7

The general transport composition rule,

$$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$8

in the isotropic case, makes electron transport itself a composite of spinon and holon transport (Ye et al., 2011).

3. Kondo–Kitaev and spin–charge realizations

A direct realization of fractionalized order appears in the Kitaev–Kondo model built from conduction electrons coupled to a gapless Yao–Lee spin liquid. There the Kondo interaction can be rewritten in terms of the composite boson

$$\Sigma_{\alpha\beta}(2,1)\approx M_{\alpha\beta}(1)\delta(2-1)$$9

where $M_{\alpha\beta}$0 is a three-component Majorana spinon. The corresponding Hubbard–Stratonovich field $M_{\alpha\beta}$1 is a spinor order parameter carrying charge $M_{\alpha\beta}$2 and spin $M_{\alpha\beta}$3. The ordered phase develops off-diagonal long-range order only in a gauge-invariant form,

$M_{\alpha\beta}$4

where $M_{\alpha\beta}$5 is a $M_{\alpha\beta}$6 gauge string; in an axial gauge without visons, this correlator factorizes at long distance, and electrons can coherently tunnel arbitrarily long distances through the spin liquid (Tsvelik et al., 2021).

The same theme is developed in a solvable three-dimensional Kondo lattice on the hyper-octagonal lattice. That model exhibits an instability at arbitrarily small Kondo coupling because the charge-$M_{\alpha\beta}$7 pairing susceptibility diverges logarithmically,

$M_{\alpha\beta}$8

The resulting ground state is a pair density wave with a fractionalized charge-$M_{\alpha\beta}$9, $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$0 order parameter formed between electrons and Majorana fermions. The corresponding superconductivity is odd in frequency,

$$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$1

and the ordered phase coexists with a neutral or quasi-neutral Majorana Fermi surface (Coleman et al., 2022).

An exactly solvable spin–charge ladder provides a discrete-symmetry counterpart. In that model the basic composite bosons are

$$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$2

built from spin-chain Majoranas $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$3 and charge-wire Majoranas $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$4. Their combinations

$$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$5

transform oddly under both spin and charge $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$6 parities, so a nonzero $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$7 simultaneously breaks both symmetries. In spin variables the $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$8 are nonlocal string operators, and the exact solution reveals $$\Sigma_{\alpha\beta}(2,1)\sim \bar V_{\alpha}(2)\,V_{\beta}(1), \qquad \Sigma_{\alpha\beta}(\omega)\sim \frac{\bar V_\alpha V_\beta}{\omega},$$9 order fractionalization with dual symmetry breaking, intertwined order parameters, a correlated topological superconductor TSC$V$0, gapped $V$1 Kondo flux excitations, and odd-frequency pairing induced by Majorana spinons (Miao et al., 2024).

The same family of ideas has a well-defined breakdown mechanism. In the half-filled CPT model, asymptotic analysis in the small-$V$2 and large-$V$3 limits identifies a quantum critical point $V$4 between a superconductor and a Kondo insulator. Thermal gauge fluctuations, measured by the vison gap, control the finite-temperature destruction of the fractionalized superconducting phase, and at large $V$5 the Kondo insulator undergoes orbital decoupling into a decoupled Kitaev orbital liquid (Panigrahi et al., 2024).

4. Disorder, intertwined order, and selective survival of ordered components

In disordered pair-density-wave superconductors, order parameter fractionalization takes a different form. A unidirectional Larkin–Ovchinnikov PDW is described by

$V$6

with composite CDW and charge-$V$7 orders $V$8 and $V$9. Disorder couples as a random field to $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$0, and “the charge order of the clean PDW is inevitably lost (via Imry-Ma).” The paper then studies a strongly inhomogeneous regime of dilute PDW puddles in a metallic background, with hierarchy

$$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$1

where $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$2 is puddle size, $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$3 the interpuddle separation, and $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$4 the metallic coherence length (May-Mann et al., 24 Sep 2025).

For puddle $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$5, the PDW gap is written

$$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$6

with $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$7 the total phase and $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$8 the relative phase. Disorder pins $$\Delta_{\pm \mathbf{Q}}(\mathbf{r})=\Delta_0(\mathbf{r})\,\rho_{\pm \mathbf{Q}}(\mathbf{r}),$$9 locally through

$\Delta_0$0

leaving $\Delta_0$1 free. After integrating out the metallic background, the effective Hamiltonian becomes

$\Delta_0$2

with a local $\Delta_0$3 gauge symmetry $\Delta_0$4. In the dilute limit the leading Josephson couplings are secretly unfrustrated; after an explicit $\Delta_0$5 gauge transformation they reduce to a ferromagnetic XY model (May-Mann et al., 24 Sep 2025).

The resulting low-temperature state is macroscopically equivalent to a uniform charge-$\Delta_0$6 $\Delta_0$7-wave superconductor, even though each puddle retains a local PDW modulation. The fractionalization statement is that the PDW behaves as though it had split into a uniform charge-$\Delta_0$8 superconducting field $\Delta_0$9, which orders, and a CDW field $2e$0, which is pinned locally but loses long-range order. This stands in explicit contrast to the vestigial charge-$2e$1 superconductivity proposed for weakly disordered PDWs (May-Mann et al., 24 Sep 2025).

A related but distinct defect-based mechanism appears in spontaneous integer quantum Hall systems generated by noncoplanar magnetic order. There the order parameter manifold has

$2e$2

so the ordered state supports magnetic $2e$3 vortices. The electron problem in a slowly varying spin texture acquires a non-Abelian gauge field

$2e$4

and the Hall response to the vortex flux yields a bound charge

$2e$5

so odd-vorticity defects carry half-odd-integer charge in a bulk integer Hall phase (Muniz et al., 2011). Here the topological defect of a conventional order parameter acquires fractional quantum numbers.

5. Symmetry fractionalization, anyon condensation, and emergent criticality

A large branch of the literature uses “fractionalization” to denote the projective action of symmetry on anyons in a topologically ordered phase. When the symmetry does not permute anyon types, local symmetry actions around an anyon define projective representations classified by

$2e$6

where $2e$7 is the Abelian anyon group. When symmetry permutes anyon types, the classification generalizes to twisted group cohomology, and the data are encoded in the fusion and associativity of extrinsic twist defects (Tarantino et al., 2015, Chen, 2016). In this setting, conventional symmetry quantum numbers—charge, spin, crystal momentum, reflection parity—are carried fractionally by anyons rather than by local quasiparticles.

Reflection-enriched $2e$8 topological order provides an explicit anomaly criterion. Electric and magnetic quasiparticles can carry fractional reflection quantum numbers characterized by cocycles $2e$9. For even $S=\tfrac12$00 there are four possible fractionalization patterns, and anomalies occur if and only if both electric and magnetic quasiparticles possess nontrivial fractional reflection quantum numbers. The anomaly can be cancelled by a $S=\tfrac12$01D $S=\tfrac12$02 symmetry-protected topological phase living on the mirror plane inside an otherwise trivial $S=\tfrac12$03D bulk (Lake, 2016).

For square-lattice $S=\tfrac12$04 topological order with space-group symmetry, the algebraic classification gives 2080 symmetry classes consistent with the fusion rules. In the family of generalized toric-code models studied explicitly, exactly 487 symmetry classes are realized; with the more restrictive symmetry action in which space-group operations act trivially in the internal Hilbert space of each spin, exactly 82 symmetry classes are realized. The same work gives a single model realizing all $S=\tfrac12$05 fractionalization types allowed for a single anyon species as Hamiltonian parameters are varied (Song et al., 2014).

When symmetry-fractionalized anyons condense, ordinary symmetry breaking must follow. In the toric-code Ising model perturbed by nearest- and next-nearest-neighbor Ising couplings, the magnetic $S=\tfrac12$06 anyons carry a nontrivial fractionalization class under the combined global Ising and square-lattice symmetries. A general theorem then implies that a continuous condensation transition of $S=\tfrac12$07 cannot preserve the full symmetry. Numerically and field-theoretically, the model exhibits two symmetry-breaking patterns and a line of emergent $S=\tfrac12$08D XY$S=\tfrac12$09 transitions ending at a fine-tuned $S=\tfrac12$10 critical point; the star indicates that the physical local order parameter is composite in terms of the fractionalized critical field (Schuler et al., 2022).

The same work makes the distinction quantitative through anomalous dimensions. At the fine-tuned $S=\tfrac12$11 point, the physical spin correlation factorizes into a product of two Ising correlators, implying

$S=\tfrac12$12

with numerical estimate $S=\tfrac12$13. Along the XY$S=\tfrac12$14 line, the correlation-length exponent is that of the $S=\tfrac12$15D XY universality class,

$S=\tfrac12$16

but the measured anomalous dimension of the physical order parameter is much larger than the conventional XY value, $S=\tfrac12$17, reflecting its composite origin (Schuler et al., 2022).

6. Diagnostics, nonlocal probes, and conceptual boundaries

Because the primary ordered objects are often gauge charged or nonlocal, diagnostics of order parameter fractionalization are frequently nonlocal as well. In the $S=\tfrac12$18–$S=\tfrac12$19 model, the Wilson loops $S=\tfrac12$20 and $S=\tfrac12$21 form “a complete set of order parameters determining the phase diagram in the underdoped regime.” Their large-loop behavior distinguishes antiferromagnetic, superconducting, and Bose-insulating phases through the confinement or deconfinement of spinons and holons (Ye et al., 2011).

Holography provides a further extension of the order-parameter concept. For finite-density states in large-$S=\tfrac12$22 gauge theories, the electric flux through a bulk surface $S=\tfrac12$23,

$S=\tfrac12$24

acts as a charged-sector order parameter refining entanglement entropy. In fully fractionalized phases the flux obeys a volume law,

$S=\tfrac12$25

while in deconfined cohesive phases it scales between a boundary and a volume law, and in confined cohesive phases it vanishes. This construction distinguishes charge hidden behind a horizon from charge carried by bulk matter and defines a nonlocal order parameter for charge fractionalization (Hartnoll et al., 2012).

A related boundary of the concept is the distinction between order fractionalization and its breakdown. In the CPT model, the vison gap determines the thermal stability of the fractionalized superconducting phase, and its collapse accompanies the destruction of the $S=\tfrac12$26 Higgs structure (Panigrahi et al., 2024). In the disorder-driven PDW problem, fractionalization does not produce a topologically ordered phase but rather a selective survival of the superconducting component under random fields (May-Mann et al., 24 Sep 2025). These examples show that order parameter fractionalization is not a single universality class but a structural principle: conventional order may be reorganized into emergent constituents with different gauge, symmetry, and disorder responses.

This suggests a broad taxonomy. In one class, the order parameter itself is a spinorial or gauge-charged condensate, as in Kondo–Kitaev systems (Komijani et al., 2018, Tsvelik et al., 2021, Coleman et al., 2022). In a second, a conventional order parameter is decomposed into emergent sectors with distinct fates under perturbations, as in disordered PDWs (May-Mann et al., 24 Sep 2025). In a third, the relevant “order parameters” are nonlocal Wilson loops or symmetry-twist data, as in doped Mott systems and symmetry-enriched topological orders (Ye et al., 2011, Tarantino et al., 2015). In a fourth, topological defects of a conventional ordered state acquire fractional quantum numbers, as in spontaneous integer Hall systems (Muniz et al., 2011). What unifies these usages is the same departure from the Landau expectation that the ordering field should be a local, gauge-invariant boson built directly from microscopic degrees of freedom.