Fractionally Charged Excitations

• Fractionally charged excitations are quantum states with charges that are genuine fractions of an electron, emerging from strong correlations and topological order.
• Experimental detection leverages methods like shot noise, local electrometry, and interferometry to measure fractional charges such as e/3, e/4, and e/m.
• Theoretical frameworks including Laughlin states and composite fermion constructions illuminate the physics of fractionalization and its applications in quantum computing.

A fractionally charged excitation is a localized or extended quantum excitation whose electric charge, as measured relative to the vacuum or ground state, is a genuine rational or, in certain cases, irrational fraction of the electron charge $e$. Such excitations are fundamentally distinct from composite objects built from electrons or holes and are typically associated with strong correlation, topological order, and the breakdown of the canonical integer charge quantization in the low-energy spectrum. Fractionalization of charge constitutes one of the central signatures of quantum many-body effects in low-dimensional condensed matter platforms, most notably the fractional quantum Hall states, but also in Chern insulators, correlated insulators at commensurate filling, 1D and higher-dimensional lattice models, and in engineered or emergent topological defects.

1. Formal Definitions and Fundamental Mechanisms

Fractional charge arises when the spectrum of low-energy excitations above a gapped ground state supports objects whose local charge quantization is decoupled from that of bare electrons. Two principal mechanisms underlie this:

• Topological order and anyonic statistics: In systems with nontrivial braiding (anyonic) statistics—characterized by emergent gauge structures and robust ground-state degeneracies on nontrivial manifolds—fractionally charged quasiparticles appear, such as Laughlin quasiholes in FQH states with $q = e/m$ at $\nu = 1/m$, Ising non-Abelian charge $e/4$ in the Moore-Read state at $\nu=5/2$, and $e/3$ in $1/3$ Laughlin states (Bisognin et al., 2019, Venkatachalam et al., 2010, Balram et al., 2014).
• Domain wall and symmetry-related fractionalization: In certain 1D and lattice models, fractional excitations are domain walls between degenerate, translation-symmetry-related ground states, with the fractional charge $Q = \pm e/(mq)$ set by block size and filling fraction (Wikberg, 2012).

Topological order enforces constraints—e.g., via the Lieb-Schultz-Mattis-Oshikawa-Hastings theorem—that at fractional filling and in translation-invariant insulators, only two possibilities exist: either gapless neutral excitations or gapped systems with fractionally charged anyons and ground-state degeneracy (e.g., "quantum charge liquids") (Musser et al., 7 Aug 2024).

2. Paradigmatic Examples Across Physical Platforms

The canonical realization is the fractional quantum Hall effect (FQHE), where strongly interacting electrons in two-dimensional Landau levels at rational filling ($\nu=p/q$) exhibit:

• Laughlin states ($\nu=1/m$): Quasiparticle/​quasihole excitations with charge $e/m$ and Abelian braiding. The local measurement of charge via shot noise and point contact transport gives direct access to $q$ (Bisognin et al., 2019).
• Hierarchical/Jain sequences and non-Abelian states ($\nu=5/2,\,7/3$): Quasiparticles with $e/4,\,e/3$ observed via SET electrometry and interference (Venkatachalam et al., 2010). The non-Abelian Moore–Read (Pfaffian) state at $\nu=5/2$ hosts $e/4$ charges, crucial for topological quantum computation (Venkatachalam et al., 2010).

Fractionalization appears in fractional Chern insulators (FCIs): topological flat bands realized in lattice models, e.g., flux-lattice Bose–Hubbard Hamiltonians and moiré systems (1804.02002, Gonçalves et al., 5 Jun 2025). Interacting bosons/fermions at fractional band filling stabilize FCIs with fractionally charged (e.g. $e/2,\,e/3$) localized and dispersive excitations. In moiré twisted-bilayer TMDs, spinless and spinful $e/3$ quasi-electrons and quasi-holes are resolved, with their energetic structure and magnetic-field response quantitatively modeled via exact diagonalization (Gonçalves et al., 5 Jun 2025).

One-dimensional models (e.g., Su-Schrieffer-Heeger (SSH), soliton modes in superfluids) exhibit fractional charge/spin domain walls with $Q = \varphi/(2\pi)$ in soliton textures or $e/q$ in lattice block structures (Ye et al., 2015, Wikberg, 2012). In 1D fermionic superfluids, phase-twisted order parameters give rise to fractional spin and Majorana zero modes with half-integer occupation.

Generalizations to lattice systems of arbitrary dimension (e.g., generalized Laughlin liquids on $D$-dimensional hypercubic lattices) yield deconfined excitations with charge $e/m^D$ (Chern et al., 2011).

3. Experimental Detection and Quantification

Experimental detection and quantification of fractional charge are realized through various approaches:

• Shot noise and finite-frequency noise: Measurements of current fluctuations at point contacts, both in the low-frequency and microwave regime, yield the effective charge $q$ via Fano factor analysis and threshold frequency $f_c = q V_{\mathrm{dc}}/h$ for photon emission (Bisognin et al., 2019, Kapfer et al., 2018). Microwave photon detection enables direct, calibration-free extraction of $q$ and probes the exclusion statistics of the quasiparticles (Bisognin et al., 2019).
• Local electrometry and charge sensing SETs: In FQH states, localized $e/4$ or $e/3$ charge addition leads to discrete jumps in local potential, directly mapping to the local charge (Venkatachalam et al., 2010).
• Cold atom quantum gas microscopy and flux insertion: In optical lattice FCIs, single-site addressing and local potential/flux insertion protocols lead to quantized changes in local density integrating to fractional values, as confirmed by quantum gas microscope imaging (1804.02002).
• Resonant inelastic light scattering (RILS): Optical probe of spinful fractionally charged skyrmions at $\nu=1/3$, resolving fractional charge via energetically separated collective modes (Balram et al., 2014).
• Interferometry and quantum tomography: Electronic Mach–Zehnder, Hong–Ou–Mandel, and Hanbury–Brown–Twiss interferometers reconstruct the single-particle correlation function and wavefunction of fractionally charged states (levitons, half-levitons) (Moskalets, 2016).
• Collider searches for fractionally charged particles (FCPs): At the particle-physics scale, minimal extensions of the SM with FCPs (e.g., $Q=e/6$ particles) are probed via low-$dE/dx$ tracks, jets$+E_T$ signatures, and specialized detectors such as milliQan, directly testing the quantization structure of electric charge (Koren et al., 22 Jul 2025).

4. Theoretical Frameworks and Generalization

A diversity of theoretical constructs underpin fractionally charged excitations:

• Wavefunction-based approaches: Laughlin and Moore–Read trial states, composite fermion (CF) construction $$\Psi_{CF} = \mathcal{P}_{LLL} \prod_{i<j}(z_i-z_j)^{2p}\Phi_{\nu^*}$$, parton constructions for FCIs and quantum charge liquids (Venkatachalam et al., 2010, Musser et al., 7 Aug 2024).
• Topological path fusion: The fusion of winding paths around flux tubes in multiply connected geometries encodes the splitting of the electron charge into rational/irrational fractions. The continued fraction and $K$-matrix formalism capture both the Laughlin and Jain sequences, as well as irrational charge series, and universally connect to 3D knot topology (Si, 2020).
• Lattice gauge theory and anyon condensation: General structure of topological order, emergent anyon species, and minimal ground-state degeneracy (GSD) characterize "quantum charge liquids" as opposed to charge-ordered insulators. For a fermionic QCL at filling $\nu=p/q$ and even $q$, the minimal GSD is $4q^2$, realized via Abelian $\mathbb{Z}_{2q}$ gauge theory with fractional charge $p/q$ anyons (Musser et al., 7 Aug 2024).
• Correlation function analysis: For driven Fermi seas, the excess first-order correlation function $G^{(1)}$ encodes the distinction between pure charge and electron–hole admixtures. Half-levitons, excited by half-integer flux Lorentzian pulses, yield factorizable $g_{HL}$ (single-particle with $e/2$ charge) and $g_{eh}$ (electron–hole pairs) terms, producing a zero-energy, fractionally charged excitation (Moskalets, 2016).

5. Classification, Quantum Numbers, and Statistics of Fractional Excitations

The properties of fractionally charged excitations include:

• Charge: Quantized as $e^*$, typically $e/m$ in Laughlin-type states, with generalizations to $e/(mq)$ in 1D/block models (Wikberg, 2012) and $e/4,\,e/3$ or more exotic values (e.g., $2/(3+\sqrt 2)$) in knot-fusion chains (Si, 2020).
• Statistics: Excitations may be Abelian or non-Abelian anyons, distinguished by their topological spin and braiding. The quantum dimension $d$ captures the ground-state degeneracy scaling with anyon number ($d=1$ for Abelian, $d=\sqrt2$ for Majorana zero modes) (Ye et al., 2015).
• Spin structure: Fractionally charged skyrmions combine charge $e/3$ with $S^z=1$ topological spin structure in FQH ferromagnets, exhibiting a nontrivial interplay between charge and SU(2) texture (Balram et al., 2014). In twisted-bilayer FCIs, excitations may be spinless or spinful, with spin-flip (spinful) excitations energetically penalized by the Zeeman effect (Gonçalves et al., 5 Jun 2025).
• Ground-state degeneracy: Topological order mandates robust GSD linked to filling and underlying symmetry: GSD = $q$ (1D lattice), $q^2$ (2D Laughlin), $4q^2$ (fermionic QCL with even $q$) (Musser et al., 7 Aug 2024).

A table summarizing principal fractional charges in representative systems:

System/Class	Principal Fractional Charge(s)	Topological Order/Statistics
Laughlin $\nu=1/m$ (FQHE/FCI)	$e/m$	Abelian anyons, $d=1$
Moore-Read $\nu=5/2$	$e/4$	Non-Abelian (Ising), $d=\sqrt{2}$
1D domain wall (convex/nonconvex)	$e/q$, $e/(mq)$	Abelian (SSH case), fusion structure
Lattice FQH/FCI and QCLs	$e/q$ or $p/q$	Abelian ($\mathbb{Z}_q$/$\mathbb{Z}_{2q}$ gauge)
Knot-fusion path networks	$p/(2kp+1)$, $2/(3+\sqrt{2})$, …	Abelian/irrational, Chern–Simons/Knots
Skyrmion in FQHE	$e/3$, $S=1$	Composite texture, Abelian/statistical
Integer QH with $Z_2$ spin vortex	$e/2$	$\mathbb{Z}_2$-protected, Berry phase

6. Extensions, Exotic Cases, and Future Directions

Fractionalization persists and generalizes beyond classic 2D electron gases:

• Higher dimensions: Explicit lattice constructs yield deconfined $e/m^D$ charges in D-Dimensions, and topological path-fusion unifies FQHE, irrational charge sequences, and knot-theoretic structures in 3D (Chern et al., 2011, Si, 2020).
• Even denominator and irrational charges: Specialized braiding or fusion of multi-flux clusters generates even-denominator and irrational $e^*$, extending the paradigm beyond Laughlin/Jain series (Si, 2020).
• Fractionalization in quantum charge liquids: Insulating, translation-invariant phases at fractional filling that evade charge ordering, with robust anyonic excitations distinguished by emergent gauge structure and large GSD, are predicted for moiré and other lattice platforms (Musser et al., 7 Aug 2024).
• Driven Fermi sea and engineered electronic excitations: Minimal excitation pulses (levitons and half-levitons) in mesoscopic electron quantum optics realize single- or fractionally charged states with controlled time/energy structure, enabling new interferometric probes and quantum information schemes (Moskalets, 2016, Kapfer et al., 2018).
• Beyond the Standard Model particle physics: Massive, stable or metastable fractionally charged particles as probes of global gauge structure and one-form symmetries, with LHC and dedicated detector constraints on $Q=e/6,\,e/3,\,2e/3$ species (Koren et al., 22 Jul 2025).

Open research directions include non-Abelian fusion and statistics in lattice and 3D topological systems, extension of experimental detection to non-Abelian and irrational-charge anyons, quantum information encoding schemes utilizing engineered fractionalization, and the design of platforms where translation-invariant topological order coexists with fractional charge.

7. Conceptual Clarifications and Theoretical Implications

Contrary to the notion that fractional charge only appears in FQHE, strong correlation and emergent gauge structure enable fractionalization in a much broader class of systems—including FCIs, quantum charge liquids, topological insulators with spin texture, and generalized lattice models (Muniz et al., 2011, Musser et al., 7 Aug 2024). Importantly, not all "fractionalization" is topological: in coupled-channel IQHE, interaction-induced eigenmodes fractionalize injected charge between channels, yet these excitations lack the robustness and statistical structure of FQHE anyons (Inoue et al., 2013). Moreover, within finite systems, fractional quantum numbers may coexist with exact integer quantization at the global level, with the localized fraction being compensated by a vanishing uniform background (Ye et al., 2015).

Fractionally charged excitations thus serve as powerful probes of emergent quantum order, dualities between lattice and continuum, and the interplay between symmetry, topology, and correlation in quantum matter.