Maximally Fractional Defects in Topological Systems

• Maximally fractional defects are topological constructs exhibiting extreme fractionalization of charges or quantum numbers, defined by their sensitivity to both symmetry group representations and internal RCFT data.
• They appear across diverse systems such as higher-order topological insulators, fractional quantum Hall bilayers, orbifold CFTs, and generalized XY models, each with distinct measurable signatures like fractional charge values up to 3/4 e.
• Experimental and theoretical approaches—ranging from mode density measurements and defect-bound state analysis to information-theoretic methods—provide compelling evidence for these intricate fractionalization phenomena.

Searching arXiv for the specified papers and topic to ground the article in current arXiv records. In the cited literature, “maximally fractional defects” does not denote a single universal object. The phrase is used for several classes of defects in which fractionalization is pushed to an extremal form allowed by symmetry, topology, or ramification data: disclinations in higher-order topological crystalline insulators (HOTIs) that trap the largest nontrivial charge fraction modulo $e$; twist defects and genons in fractional quantum Hall (FQH) systems whose species labels, fusion spaces, and braiding are intrinsically defect-sensitive; non-universal orbifold defects that simultaneously encode permutation-group and seed-RCFT data; fully-ramified Gukov–Witten surface defects realized by fractional branes; and related fractional-vortex or zero-mode phenomena in graphene and generalized $XY$ models (Peterson et al., 2020, Teo et al., 2013, Liu et al., 2017, Ghasemi, 14 Feb 2026, Ashok et al., 2020, Obispo et al., 2014, Badal et al., 2016).

1. Terminological scope and defect-theoretic meaning

In symmetric-product orbifold CFTs, topological defects in $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$ are divided into two broad classes: universal $\mathrm{Rep}(S_N)$ defects and maximally fractional defects. Universal defects are labeled by an irreducible $S_N$-representation $R$ and depend only on the group $S_N$, while maximally fractional defects depend both on an $S_N$-representation $R$ and a topological defect $K$ in the seed RCFT $XY$0. In that setting, “maximally fractional” therefore means non-universal and explicitly sensitive to both permutation-group data and internal RCFT data (Ghasemi, 14 Feb 2026).

In abelian bosonic bilayer $XY$1 FQH states, the same phrase is used for twofold twist defects $XY$2 implementing the ungauged layer-exchange symmetry $XY$3. Their species label $XY$4 determines a fractional charge

$XY$5

and the review of these defects identifies “maximal fractionalization” through several quantities at once: the number of species $XY$6, the largest fractional charges up to $XY$7, maximal spin-fractionalization, and the largest ground-state degeneracies $XY$8 for $XY$9 defects (Teo et al., 2013).

In HOTIs, by contrast, maximal fractionality is defined directly in terms of defect-bound electric charge modulo $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$0. Because $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$1 is defined modulo $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$2, the largest nontrivial fraction of the electron charge that can be trapped is

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$3

for an elementary disclination in an $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$4-fold lattice (Peterson et al., 2020).

In the fractional-brane realization of surface defects in $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$5 SYM, “maximally fractional” is used synonymously with “fully-ramified” Gukov–Witten defects. The distinguished case is $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$6, with one fractional D3-brane of each type, so that $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$7 is broken to $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$8 (Ashok et al., 2020).

A common misconception is therefore avoided by keeping the usage context-specific: the phrase does not pick out a single invariant across all subfields, but rather an extremal defect construction within each theory.

2. HOTI disclinations and maximally fractional charge

For a disclination defect in a $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$9-symmetric insulator, the charge bound to the defect is

$\mathrm{Rep}(S_N)$0

where $\mathrm{Rep}(S_N)$1 is the Frank angle, $\mathrm{Rep}(S_N)$2 is the Burgers vector, $\mathrm{Rep}(S_N)$3 is a second-order Wannier indicator, and $\mathrm{Rep}(S_N)$4 is the quantized bulk polarization. In a purely $\mathrm{Rep}(S_N)$5-symmetric HOTI with vanishing dipole moment $\mathrm{Rep}(S_N)$6 and nontrivial corner Wannier index, this reduces for type-I disclinations to

$\mathrm{Rep}(S_N)$7

For the simplest HOTI model studied experimentally, $\mathrm{Rep}(S_N)$8, so that

$\mathrm{Rep}(S_N)$9

Equivalently, for $S_N$0,

$S_N$1

Because $S_N$2 is defined modulo $S_N$3, the largest nontrivial fraction is $S_N$4. In $S_N$5 symmetry, the maximal nonzero fractions are $S_N$6, $S_N$7, and $S_N$8; a $S_N$9 disclination traps $R$0, while a $R$1 disclination traps $R$2. The “most fractional” defect is therefore the $R$3 wedge in a $R$4 lattice, carrying $R$5 (Peterson et al., 2020).

Peterson et al. realized the relevant $R$6 HOTI on a printed-circuit board of Rogers RT/duroid 5880. Each site is a half-wavelength copper microstrip resonator with $R$7 and $R$8. Strong bonds use $R$9 between adjacent unit cells and weak bonds use $S_N$0 within each cell, realizing the prototypical quadrupole model. Four resonators per cell give $S_N$1 symmetry in the bulk; at edges and corners the model exhibits $S_N$2 edge charge and $S_N$3 corner charge at quarter-filling. Disclinations are introduced by a cut-and-glue procedure: removing one $S_N$4 wedge produces a $S_N$5 global board with $S_N$6 and $S_N$7, while inserting one extra wedge produces a $S_N$8 board with $S_N$9 and $S_N$0 (Peterson et al., 2020).

The experimental observable is mode density. A vector network analyzer probes each resonator’s reflection $S_N$1; absorptance

$S_N$2

is converted to local density of states

$S_N$3

After normalizing $S_N$4 per resonator, integrating over a given bulk-band window gives the filled-band mode density, interpreted as fractional charge density. The central cell shows $S_N$5 in the singly degenerate bands for $S_N$6, and $S_N$7 for $S_N$8 (Peterson et al., 2020).

The same work also connects trapped fractional charge to defect-localized topological states. In the pristine disclination, the anomalous mode density resides entirely in the defective core cell, which has an odd number of sites, so there is no genuine in-gap eigenstate at the core. After locally deforming the central cell into a trivial cluster and creating an interior boundary, the trapped fractional charge splits onto surrounding intact cells, and the interior corners host $S_N$9 topological bound modes in the bandgap: three for $R$0, five for $R$1. Further symmetry breaking gaps out all but one bound mode, leaving a single robust midgap state. The cited interpretation is that disclination charge provides a genuine bulk probe of crystalline topology even when edge or corner spectra show no obvious gapless features.

3. Twist defects, species labels, and genons in fractional quantum Hall systems

An abelian bilayer FQH state is described by the $R$2-matrix

$R$3

and is invariant under the layer-exchange symmetry $R$4. The quasiparticle lattice is

$R$5

A twofold defect $R$6 is a semiclassical point-like object implementing the ungauged $R$7 symmetry on anyon labels. Since $R$8 is fixed by $R$9, each defect carries fractional charge

$K$0

Fusion with an anyon $K$1 shifts the species label by $K$2, and the fundamental fusion rules are

$K$3

Since there are $K$4 abelian anyon channels in $K$5, the quantum dimension is

$K$6

The exchange spin is

$K$7

and the corresponding $K$8-rotation phase is

$K$9

For $XY$00 defects on the sphere, the ground-state degeneracy is

$XY$01

The torus-with-branch-cut construction preserves the congruence subgroup $XY$02, generated by $XY$03 and $XY$04, rather than the full modular group. The phrase “maximally fractional” is justified there by the simultaneous growth of species count $XY$05, fractional charge resolution up to $XY$06, spin-fractionalization, and defect Hilbert-space dimension (Teo et al., 2013).

A closely related construction appears in the lattice FQH realization of genons. Before defects, the system consists of two layers $XY$07 of square-lattice sites $XY$08 with uniform flux $XY$09 per plaquette and long-range Kapit–Mueller hopping,

$XY$10

with two exactly flat Chern-$XY$11 bands at zero energy, one per layer. Introducing $XY$12 twist defects in pairs via $XY$13 straight branch cuts flips the layer index whenever a particle hops across a cut. On a torus with $XY$14 cuts, the low-energy manifold acquires $XY$15 midgap states localized at the defect cores. A local defect potential

$XY$16

restores a new flat lowest band of dimension

$XY$17

with the dispersion of the new lowest $XY$18 bands suppressed by a factor $XY$19 while leaving their wave-function subspace essentially unchanged (Liu et al., 2017).

Each twist pair acts like a wormhole connecting the two layers, so $XY$20 pairs raise the genus to

$XY$21

For $XY$22-body correlated bosons in the $XY$23 Read–Rezayi sequence at filling $XY$24, the expected topological ground-state degeneracies are

$XY$25

Equivalently,

$XY$26

The defect quantum dimension is defined by

$XY$27

which yields

$XY$28

The many-body spectra, twisted-boundary spectral flow, and particle entanglement spectra together provide the cited “proof-of-concept” evidence that wormhole-like twist defects in lattice FQH models are the predicted genons (Liu et al., 2017).

4. Maximally fractional defects in symmetric-product orbifold CFTs

For the symmetric-product orbifold

$XY$29

universal defects are labeled by an irreducible $XY$30-representation $XY$31 and can be written as

$XY$32

These implement the non-invertible $XY$33 symmetry of the orbifold. Maximally fractional defects are defined by choosing both an $XY$34-representation $XY$35 and a topological defect $XY$36 in the seed RCFT $XY$37, giving

$XY$38

When $XY$39 is a diagonal RCFT and $XY$40 is the Verlinde line labeled by $XY$41, the $XY$42-cycle seed defect is

$XY$43

and one recovers the familiar maximally fractional defect $XY$44 (Ghasemi, 14 Feb 2026).

The defect relative entropy between two maximally fractional defects $XY$45 and $XY$46 is defined by

$XY$47

and computed by the replica trick

$XY$48

In the IR limit, the leading vacuum block dominates, and the result factorizes into an $XY$49-character part and a seed-RCFT part. The final expression is

$XY$50

The two probability distributions are

$XY$51

and, in the diagonal case,

$XY$52

In the general rational case one uses $XY$53 built from $XY$54 and defect-interface coefficients (Ghasemi, 14 Feb 2026).

The information-theoretic interpretation is explicit. For universal defects, only the permutation-group data contributes. For maximally fractional defects, both permutation and modular data enter and together define the relevant probability distributions. The maximally fractional defect therefore behaves exactly like the product measure $XY$55, and the relative entropy splits additively into group-theoretic and seed-theoretic KL divergences.

5. Fractional zero modes, valley number, and string-connected vortices

In graphene with a topological defect modeled by an Aharonov–Bohm-like pseudomagnetic flux, the low-energy Hamiltonian is

$XY$56

with pseudomagnetic potential

$XY$57

Because the Hamiltonian commutes with $XY$58, one defines the valley-number operator

$XY$59

Only the zero-energy states contribute a nonzero $XY$60-number to the vacuum expectation value. The vacuum valley number is

$XY$61

where $XY$62 is the fractional part. Hence $XY$63 vanishes when $XY$64, while

$XY$65

For generic values of $XY$66, the induced valley number is irrational. The same mechanism gives an analogous induced spin polarization,

$XY$67

when the axial gauge field is coupled to physical spin instead of valley (Obispo et al., 2014).

A different realization of fractional defects appears in the generalized $XY$68-well $XY$69 model. The discrete energy is

$XY$70

For sequences with

$XY$71

the asymptotic ground states exhibit exactly $XY$72 discrete vortices of charge $XY$73. The $XY$74-limit of

$XY$75

is

$XY$76

where

$XY$77

and the surface term

$XY$78

measures the total anisotropic length of the jump set, i.e. the string defects. Since the lifted field $XY$79 has integer winding $XY$80, the original field $XY$81 has degree $XY$82, and the cited result is that the only nonzero fractional charges are $XY$83, which are maximal in absolute value for a single defect. Minimizers pair $XY$84 and $XY$85 vortices and connect them by shortest $XY$86-strings; for a single dipole at separation $XY$87,

$XY$88

This model therefore exhibits a defect theory in which maximal single-defect fractionality is inseparable from string tension and anisotropic network optimization (Badal et al., 2016).

6. Fully ramified surface defects from fractional branes

In Type IIB on $XY$89, orbifolding two complex directions $XY$90 by $XY$91,

$XY$92

and introducing $XY$93 fractional D3-branes of types $XY$94 with one-dimensional Chan–Paton factors $XY$95 transforming as

$XY$96

realizes the maximally fractional, or fully-ramified, Gukov–Witten surface defect. Each brane extends along

$XY$97

and because there is one brane of each type, the $XY$98 gauge group is $XY$99, embedded in $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$00. The defect corresponds to the partition

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$01

(Ashok et al., 2020).

In the $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$02-th twisted sector there are $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$03 NS/NS scalars $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$04 and $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$05 R/R scalars $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$06. Defining the singlet and doublet combinations and then the $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$07-charged linear combinations

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$08

one obtains precisely the Gukov–Witten singularity data. The gauge-field monodromy and Higgs-field singularity are

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$09

and

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$10

The R/R scalars generate a $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$11 $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$12-term

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$13

so the full parameter set is

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$14

Equivalently,

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$15

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$16

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$17

The $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$18 real closed-string moduli thereby map one-to-one onto the defect parameters (Ashok et al., 2020).

The corresponding low-energy world-volume theory is a coupled $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$19–$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$20 system with a $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$21 chain-saw quiver gauge theory of gauge group $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$22. Its twisted effective superpotential is

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$23

with

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$24

The same data can be encoded by the vortex partition function

$\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$25

In this setting, “maximally fractional” refers to the fully ramified defect for which the fractional-brane construction resolves the entire $\mathrm{Sym}^N(M)=M^{\otimes N}/S_N$26 stack into one brane of each type, and the defect parameters are explicit linear combinations of twisted-sector closed-string vacuum expectation values.

The surveyed works therefore treat maximally fractional defects as privileged probes of hidden structure. In HOTIs, disclination charge depends only on bulk invariants and defect geometry and functions as a bulk probe of crystalline topology. In lattice and bilayer FQH systems, defect number and species control degeneracy, braiding, and effective genus. In symmetric orbifolds, maximally fractional defects are exactly those for which both permutation and modular data enter the defect relative entropy. In the fully ramified surface-defect construction, the maximal defect is the one with the finest fractional-brane resolution and a one-to-one map from twisted closed-string moduli to defect couplings.