Fractional Topological Invariants

• Fractional topological invariants are quantities that can take fractional or discretely fractional values, defined via Berry phases, winding numbers, and twisted boundary conditions in gapped quantum systems.
• They are used to characterize multimerized lattice models, fractional quantum Hall states, and crystalline topological phases, providing insights beyond conventional integer invariants.
• Various computational approaches—including Green’s-function integrals, partial symmetry actions, and many-body twist techniques—offer practical methods for probing fractionalized phases in condensed matter.

Fractional topological invariants are topological quantities associated with gapped quantum systems, topologically ordered phases, or symmetry-enriched states, for which the invariant either takes fractional or discretely fractional values—such as $\gamma \equiv 2\pi n/Q \pmod{2\pi}$, $\nu\in\mathbb Q$, or $\theta=\pi/t^2$—or remains integer-valued while characterizing a fractionalized phase through Green’s-function or many-body data (Hatsugai et al., 2010, Basa et al., 2022, Ye et al., 2017, Gurarie et al., 2013). In condensed-matter usage, these constructions appear in fractional quantum Hall and fractional Chern insulating states, multimerized lattice models, projective or open-system band problems, and crystalline or higher-order topological phases. They are defined through Berry phases, winding numbers, Green’s-function integrals, twisted-boundary-condition Chern numbers, partial-symmetry expectation values, or topological response terms.

1. Forms of fractionality

The literature treats several distinct mechanisms by which topological invariants become “fractional.” In multimerized gapped systems, the invariant is a Berry phase quantized in units of $2\pi/Q$, with $Q=d+1$ in $d$ spatial dimensions; in this setting the invariant is a $Z_Q$ order parameter for dimerization, trimerization, or tetramerization. In interacting many-body systems with a $p$-fold ground-state degeneracy, one defines averaged quantities such as $\bar C({\phi})=C({\phi})/p$ and $\bar \Gamma=\Gamma/p$, which are rational and remain stable as long as the gap and degeneracy are preserved. In projectively realized or open-system band problems, the winding number itself can be valued in $\nu\in\mathbb Q$0, either because the Clifford algebra is centrally extended or because the steady-state Bloch map closes only after multiple Brillouin-zone cycles (Hatsugai et al., 2010, Wang et al., 2013, Basa et al., 2022, Wu et al., 4 Mar 2026).

A second distinction is between invariants of the electron Green’s function and invariants of the fractionalized phase itself. In fractional quantum Hall and fractional Chern insulating systems, the Green’s-function winding can remain integer-valued even though the physical Hall conductance or many-body Chern number is fractional. By contrast, quasiparticle, crystalline, and higher-order constructions often attach the fractional value directly to the invariant: examples include the quasiparticle Chern number $\nu\in\mathbb Q$1, the fractional corner anomaly $\nu\in\mathbb Q$2, and the fractional axion angle $\nu\in\mathbb Q$3 (Grusdt et al., 2015, Peterson et al., 2020, Ye et al., 2017).

2. Green’s-function invariants in fractional Hall and Chern phases

For a two-dimensional interacting system with Matsubara Green’s function $\nu\in\mathbb Q$4, a natural generalization of the noninteracting Chern number is the integer

$\nu\in\mathbb Q$5

In the noninteracting limit $\nu\in\mathbb Q$6, this reduces exactly to the sum of Chern numbers of the filled bands and hence to the integer Hall conductance $\nu\in\mathbb Q$7. For interacting fractional quantum Hall states, the invariant is computed through the bulk–boundary relation

$\nu\in\mathbb Q$8

For Abelian states described by a symmetric nondegenerate $\nu\in\mathbb Q$9-matrix, one finds

$\theta=\pi/t^2$0

Thus a Laughlin state with $\theta=\pi/t^2$1 has $\theta=\pi/t^2$2. For Read–Rezayi $\theta=\pi/t^2$3 states, the edge Green’s function gives $\theta=\pi/t^2$4, independently of $\theta=\pi/t^2$5; in particular, the Moore–Read state at $\theta=\pi/t^2$6 has $\theta=\pi/t^2$7, while the anti-Pfaffian has $\theta=\pi/t^2$8 (Gurarie et al., 2013).

In fractional Chern insulators, the relevant Green’s-function quantity is the Ishikawa–Matsuyama invariant

$\theta=\pi/t^2$9

together with the Luttinger decomposition

$2\pi/Q$0

In the fractional Chern insulating phase of the fermionic Harper–Hofstadter–Hubbard model, exact diagonalization shows a clear violation of Luttinger’s theorem: the many-body Chern number is fractional, while $2\pi/Q$1 remains integer. The fractional nature of the many-body Chern number is encoded in the Středa response of the Luttinger integral, whereas the integer invariant $2\pi/Q$2 arises from the Středa response of the Luttinger count (Markov et al., 17 Mar 2026).

3. Berry-phase and rational-winding constructions

For gapped electronic systems with a $2\pi/Q$3-multimer structure, Hatsugai and Maruyama define adiabatic $2\pi/Q$4 invariants by quantized Berry phases. Writing the Berry phase along a closed loop $2\pi/Q$5 in twist-angle space as

$2\pi/Q$6

the global $2\pi/Q$7 equivalence and the vanishing of the sum of the $2\pi/Q$8 loops imply

$2\pi/Q$9

For $Q=d+1$0, $Q=d+1$1, this gives the familiar $Q=d+1$2 invariant of polyacetylene; for $Q=d+1$3, $Q=d+1$4, the Kagome trimer phases carry $Q=d+1$5; for $Q=d+1$6, $Q=d+1$7, the Pyrochlore tetramer phases carry $Q=d+1$8. In this framework the invariant is a quantum $Q=d+1$9-multimer order parameter for topological phase transitions by multimerization (Hatsugai et al., 2010).

A distinct route to rational invariants is the “fractional twist” of the Kitaev chain. There the Pauli matrices in the BdG Hamiltonian are replaced by rational operator powers,

$d$0

which closes the algebra only after adding a central generator proportional to the identity. The real-space winding number is then evaluated through Prodan’s formula

$d$1

Numerically, $d$2 remains $d$3 for $d$4, passes through exactly $d$5 at the bulk-gap closing $d$6, and for $d$7 takes a dense set of rational values in $d$8. The resulting phase is described as pseudo-metallic, and the lattice construction is interpreted as the analogue of projective Dirac operators with rational indices (Basa et al., 2022).

Open systems provide a third mechanism. In a periodic SSH chain governed by the Gorini–Kossakowski–Sudarshan–Lindblad master equation, a multi-valued gain matrix $d$9 with periodicity $Z_Q$0 produces a purified Bloch loop that does not close over the naive interval $Z_Q$1. The Berry phase over one fundamental zone is then

$Z_Q$2

while integer quantization is restored over the extended interval:

$Z_Q$3

In this setting, the fractional invariant no longer possesses quantized topology in the conventional sense, but multi-period re-quantization survives on the extended Brillouin zone (Wu et al., 4 Mar 2026).

4. Many-body wavefunction and quasiparticle formulations

A fully many-body formulation uses twisted boundary conditions on a torus. For an interacting insulator with many-body ground state $Z_Q$4 and, when present, a $Z_Q$5-fold ground-state degeneracy, the $Z_Q$6 Berry curvature in the $Z_Q$7 plane defines

$Z_Q$8

In the fractional case one uses the average $Z_Q$9. For $p$0, this reproduces the Niu–Thouless–Wu formula,

$p$1

which is a rational multiple of $p$2 with denominator $p$3. In odd dimensions, one similarly defines a Berry phase $p$4 and, for $p$5, the magneto-electric angle

$p$6

which in the fractional case is defined mod $p$7. These formulas are valid in the presence of arbitrary many-body interaction and disorder (Wang et al., 2013).

A complementary formulation attaches the invariant to an elementary excitation rather than to the ground state. In a Laughlin-type fractional quantum Hall system, a mobile impurity can bind to a single quasiparticle and form a “topological polaron.” Since the quasiparticle carries charge $p$8, the composite experiences a reduced magnetic field $p$9. The Chern number of the quasiparticle band is then

$\bar C({\phi})=C({\phi})/p$0

Operationally, the invariant is extracted by combining Ramsey interference with Bloch oscillations, measuring the Zak phase $\bar C({\phi})=C({\phi})/p$1, and integrating its winding over one period:

$\bar C({\phi})=C({\phi})/p$2

This gives a direct probe of fractional charge in the bulk of fractional quantum Hall systems (Grusdt et al., 2015).

5. Crystalline and higher-order fractional invariants

In rotationally symmetric higher-order topological phases, a bulk-band invariant can be defined from boundary-localized fractional density rather than from isolated in-gap corner modes. For a finite crystal with both $\bar C({\phi})=C({\phi})/p$3 and $\bar C({\phi})=C({\phi})/p$4 cuts, a crystal cut only along $\bar C({\phi})=C({\phi})/p$5, and a crystal cut only along $\bar C({\phi})=C({\phi})/p$6, the fractional corner anomaly is

$\bar C({\phi})=C({\phi})/p$7

In the zero-correlation-length limit this reduces to

$\bar C({\phi})=C({\phi})/p$8

By rotation symmetry, $\bar C({\phi})=C({\phi})/p$9 is quantized in units of $\bar \Gamma=\Gamma/p$0. In microwave-resonator realizations, this invariant was measured directly: the $\bar \Gamma=\Gamma/p$1-symmetric metamaterial yielded $\bar \Gamma=\Gamma/p$2, and the $\bar \Gamma=\Gamma/p$3-symmetric Kagome metamaterial yielded $\bar \Gamma=\Gamma/p$4 or $\bar \Gamma=\Gamma/p$5, depending on the band (Peterson et al., 2020).

For fractional Chern insulators and continuum fractional quantum Hall states with rotational symmetry, partial rotations centered at high-symmetry points produce a more refined set of crystalline invariants. If $\bar \Gamma=\Gamma/p$6 rotates only the degrees of freedom inside a large disk $\bar \Gamma=\Gamma/p$7 and may include a compensating $\bar \Gamma=\Gamma/p$8 charge rotation, then

$\bar \Gamma=\Gamma/p$9

The reduced invariant $\nu\in\mathbb Q$00 is obtained from $\nu\in\mathbb Q$01 modulo the relabelings allowed by symmetry fractionalization. For the topological orders considered in projected parton wave functions, the Hall conductivity, filling fraction, and partial rotation invariants fully characterize the crystalline invariants of the system. In the continuum limit, the same construction yields invariants of fractional quantum Hall states protected by spatial rotational symmetry (Kobayashi et al., 2024).

A broader many-body crystalline framework combines several such probes. In interacting lattice systems with $\nu\in\mathbb Q$02 charge conservation and crystalline symmetry, one uses flux insertion, momentum polarization, and partial rotation expectation values to define fractional Chern numbers, translation invariants, and rotation invariants. The residue

$\nu\in\mathbb Q$03

is interpreted as the fractional angular momentum localized at the rotation center, while defect responses such as fractional disclination charge or fractional crystalline polarization encode the same symmetry-enriched topological data (Manjunath et al., 11 Apr 2026).

6. Fractional response theories and broader mathematical usage

In $\nu\in\mathbb Q$04 dimensions, fractional topological insulators are characterized by a fractional axion angle. If the smallest electric charge is $\nu\in\mathbb Q$05, then magnetic flux is quantized in units of $\nu\in\mathbb Q$06, the periodicity of the $\nu\in\mathbb Q$07 term becomes

$\nu\in\mathbb Q$08

and time-reversal symmetry restricts

$\nu\in\mathbb Q$09

Under stacking, these phases separate into two topologically distinct classes: type-I for odd $\nu\in\mathbb Q$10, which can be written as a conventional topological insulator stacked with a fractionalized gapped fermionic state of vanishing $\nu\in\mathbb Q$11, and type-II for even $\nu\in\mathbb Q$12, which cannot be realized through such stacking. This classification is formulated through a fractional $\nu\in\mathbb Q$13-duality of $\nu\in\mathbb Q$14 with fractionally charged excitations (Ye et al., 2017).

Outside condensed-matter usage, the phrase “fractional topological invariant” also appears in topology and geometric analysis. In mapping-class and contact-topological settings, the fractional Dehn twist coefficient

$\nu\in\mathbb Q$15

measures the boundary rotation required to pass from a relative isotopy class $\nu\in\mathbb Q$16 to its Thurston representative; it is estimated via properly embedded arcs and detects phenomena such as right-veering, stabilization bounds, and overtwistedness criteria (Kazez et al., 2012).

In stable homotopy theory, the $\nu\in\mathbb Q$17-invariant is a tertiary index-theoretic and homotopy-theoretic invariant of framed manifolds. Its target is a quotient of modular-form power series by divided congruences,

$\nu\in\mathbb Q$18

and its coefficients are typically rational rather than integral. In this sense, the “fractional” aspect is carried by denominators coming from divided congruences and Bernoulli-number phenomena rather than by condensed-matter fractionalization (Bunke et al., 2008).

7. Limitations, nonuniqueness, and interpretive issues

A recurring limitation is that a single invariant is rarely a complete label of a fractional phase. For fractional quantum Hall states, $\nu\in\mathbb Q$19 is the integer winding number of the single-particle Green’s function and, in the interacting case, no longer measures the physical Hall conductance, which is rational; instead, it encodes the total scaling dimension of the electron operator at the edge. It can distinguish Pfaffian from anti-Pfaffian, or Laughlin $\nu\in\mathbb Q$20 from integer $\nu\in\mathbb Q$21, but different states may share the same $\nu\in\mathbb Q$22 (Gurarie et al., 2013).

A second issue is the mismatch between single-particle and many-body topology. In fractional Chern insulators, the integer Ishikawa–Matsuyama invariant $\nu\in\mathbb Q$23 is inherited from the underlying band topology, whereas the fractional many-body Hall response is carried by the Luttinger integral and by zeros of the Green’s function. This separates the topology of the electron propagator from the topology of the fractionalized phase itself (Markov et al., 17 Mar 2026).

A third issue concerns the parameter space over which the invariant is defined. In open SSH chains with multi-valued dissipation, the fractional winding over one Brillouin zone is not conventionally quantized, while integer quantization is recovered only after extending the zone to the true periodicity. This suggests that the distinction between a genuinely fractional invariant and an integer invariant defined on a larger parameter space is model-dependent (Wu et al., 4 Mar 2026).

Higher-order and crystalline settings add a complementary caution. Boundary-localized in-gap modes are not always spectrally isolated, and their energies need not be protected; the fractional corner anomaly was introduced precisely because a bulk-band indicator is needed even when corner modes overlap bulk or edge bands. In that sense, fractional topological invariants are often most robust when formulated through Berry phases, Green’s functions, twisted boundary conditions, or partial symmetry actions rather than through the energies of a small subset of boundary states (Peterson et al., 2020).