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Topological Spin Derivation of Generalized Exclusion Statistics and Fano Factor in Fractional Quantum Hall States

Rowan Brad Quni-Gudzinas

Published Sep 4, 2026• 0 citations
Publisher

Abstract

The generalized exclusion statistics (GES) parameter $g$ characterizes the fractionalization of anyonic excitations in fractional quantum Hall (FQH) states. While $g$ is phenomenologically linked to the filling factor $\nu$ in Laughlin states, a universal derivation from the underlying topological field theory for arbitrary Abelian and non-Abelian states remains an open theoretical target. Furthermore, whether this topologically derived $g$ can quantitatively predict the shot-noise Fano factor $F$ measured in quantum point contact (QPC) tunneling experiments has not been rigorously established across general FQH phases. Here, we derive $g$ directly from the topological spin $\theta = e^{2\pi i h}$, where $h$ is the conformal dimension of the corresponding conformal field theory (CFT) primary field. For Abelian anyons, $g = 2h$, while non-Abelian anyons yield a matrix form. Using the Keldysh formalism for the QPC tunneling Hamiltonian, we show this topologically derived $g$ dictates the tunneling density of states exponent and predicts the Fano factor $F = g \nu$ in the weak backscattering regime. For Laughlin states at $\nu=1/m$, this yields $F=\nu$, matching fractional charge $e^*=\nu e$. For non-Abelian states, the derived $g$ matrix predicts a modified Fano factor revealing topological spin contributions to nonequilibrium noise. ## 1. Introduction The fractional quantum Hall (FQH) effect provides a physical realization of topological phases of matter hosting anyonic excitations. These excitations obey generalized exclusion statistics (GES), a framework introduced to generalize the Pauli exclusion principle to particles with fractional braiding statistics. In a quantum point contact (QPC) geometry, the tunneling of these anyons generates shot noise, characterized by the Fano factor $F$ (the ratio of noise power to twice the Poissonian current). While the connection between GES and the filling factor $\nu$ is well-established for Laughlin states, a universal deriv

DOI
10.5281/zenodo.22306738

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The generalized exclusion statistics (GES) parameter ggg characterizes the fractionalization of anyonic excitations in fractional quantum Hall (FQH) states. While ggg is phenomenologically linked to the filling factor ν\nuν in Laughlin states, a universal derivation from the underlying topological field theory for arbitrary Abelian and non-Abelian states remains an open theoretical target. Furthermore, whether this topologically derived ggg can quantitatively predict the shot-noise Fano factor FFF measured in quantum point contact (QPC) tunneling experiments has not been rigorously established across general FQH phases. Here, we derive ggg directly from the topological spin θ=e2πih\theta = e^{2\pi i h}θ=e2πih, where hhh is the conformal dimension of the corresponding conformal field theory (CFT) primary field. For Abelian anyons, g=2hg = 2hg=2h, while non-Abelian anyons yield a matrix form. Using the Keldysh formalism for the QPC tunneling Hamiltonian, we show this topologically derived ggg dictates the tunneling density of states exponent and predicts the Fano factor F=gνF = g \nuF=gν in the weak backscattering regime. For Laughlin states at ν=1/m\nu=1/mν=1/m, this yields F=νF=\nuF=ν, matching fractional charge e∗=νee^*=\nu ee∗=νe. For non-Abelian states, the derived ggg matrix predicts a modified Fano factor revealing topological spin contributions to nonequilibrium noise. ## 1. Introduction The fractional quantum Hall (FQH) effect provides a physical realization of topological phases of matter hosting anyonic excitations. These excitations obey generalized exclusion statistics (GES), a framework introduced to generalize the Pauli exclusion principle to particles with fractional braiding statistics. In a quantum point contact (QPC) geometry, the tunneling of these anyons generates shot noise, characterized by the Fano factor FFF (the ratio of noise power to twice the Poissonian current). While the connection between GES and the filling factor ν\nuν is well-established for Laughlin states, a universal deriv
The quantum Hall effect has been the prototypical example for many emerging fields of Physics, including the study of topological phases of matter, topological order, and topological insulators. The objects of interest in this thesis are anyons, which are elementary excitations in fractional quantum Hall (FQH) fluids. Anyons are quasiparticles with fractional charge and fractional statistics. The latter prop- erty makes it a subject of immense interest due to application in fault-tolerant quantum computing. Here, we employ a perspective that anyons are the appro- priate degree of freedom to describe the relevant physics of FQH systems. This is because when a topological phase is assumed and the energy gap goes to infinity, anyons (“quasiholes”) are the only possible elementary excitations, since they are not energetically punished by the correspondng model Hamiltonian. With respect to a realistic Hamiltonian, however, this choice of degree of freedom depends on the exact effective interaction in the system. For example, while the Laughlin state is the model state at filling factor n + 1/3 (where n is some integer), when the effective electron-electron interaction is more long-ranged, the quasiholes of a Gaffnian state may be excited and become the better degree of freedom to describe physics in the same filling factor. This perspective results in a surprising observa- tion that a Laughlin quasihole may fractionalize under certain conditions, leading to a quasiparticle of charge e/6 that can potentially be detected in experiments. Another important aspect of anyons is their intrinsic spin and the corresponding spin-statistics relation, which must account for the fact that FQH anyons are not point particles, but objects with finite sizes and intricate internal structures. Our investigation leads to a rigorous derivation of the general spin-statistics relation for anyons in Abelian FQH phases. Furthermore, we show it is possible to fully capture the important features of the statistics by a conformal Hilbert space (CHS) description. Our analysis provides the necessary tools to study such geometric effects, which we use to investigate how disorder in a realistic system may affect different experimental schemes for measuring anyon statistics.
We investigate the response to modular transformations and the fractional statistics of Abelian multicomponent fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using the anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the KKK matrix, further strengthens the correspondence between the two-dimensional (2D) CFTs and (2+1)D(2+1)\mathrm{D}(2+1)D topological orders for multicomponent FQH states.
Master's thesis in Theoretical Physics. The official university title is "Fractional Quantum Hall Effect: Current State and Perspectives"; this record uses the author's preferred, more descriptive title. This thesis develops a unified description of the fractional quantum Hall effect (FQHE) and topological order through three complementary frameworks: microscopic trial many-body wavefunctions, the algebraic theory of anyons and unitary modular tensor categories, and Chern-Simons effective field theory. The first part covers the physical foundations—Landau quantization, the TKNN invariant, Laughlin's wavefunctions, the hierarchy and composite fermion theories, and some non-Abelian states like the Moore-Read Pfaffian state. The second part develops the algebraic theory of anyons, from the braid group to fusion rules, quantum dimensions, topological spins, and the F- and R-matrices that characterize a unitary modular tensor category. The third part constructs the Chern-Simons effective field theory, deriving fractional charge, statistics, and ground-state degeneracy, unifying Abelian states via the K-matrix formalism, and then generalizing to non-Abelian states through 𝑈(1)ₘ × 𝑆𝑈(𝑁)ₖ Chern-Simons theory. The central result is that Chern–Simons theory provides a field-theoretic realization of the topological order introduced microscopically and algebraically in the earlier chapters, closing the circle between the three descriptions and providing a unified framework for the FQHE and topological order and anyon's physics.
Quasihole excitations in fractional quantum Hall (FQH) systems exhibit fractional statistics and fractional spin, but how the spin-statistics relation emerges from many-body physics remains poorly understood. Here we prove a spin-statistics relation using only FQH wave functions, on both the sphere and disk geometry. In particular, the proof on the disk generalizes to all quasiholes in realistic systems, which have a finite size and could be deformed into arbitrary shapes. Different components of the quasihole spins are linked to different conformal Hilbert spaces (CHSs), which are null spaces of model Hamiltonians that host the respective FQH ground states and quasihole states. Understanding how the intrinsic spin of the quasiholes is linked to different CHSs is crucial for the generalized spin-statistics relation that takes into account the effect of metric deformation. In terms of the experimental relevance, this enables us to study the effect of deformation and disorder that introduces an additional source of Berry curvature, an aspect of anyon braiding that has been largely neglected in previous literature.
Even-denominator fractional quantum Hall (FQH) states can be viewed as topological superconductors of composite fermions, supporting a charged chiral mode and ∣Ccf∣|\mathcal{C}_{cf}|∣Ccf​∣ neutral Majorana modes set by the Chern number Ccf\mathcal{C}_{cf}Ccf​. Despite ongoing efforts, distinguishing the many competing paired phases remains an open problem. In this work, we propose a unified theory of charge transport across a quantum point contact (QPC) for general paired FQH states described by an so(N)1×u(1)so(N)_1 \times u(1)so(N)1​×u(1) conformal field theory. We derive the boundary effective action for an arbitrary number of Majorana fermions N=∣Ccf∣N=|\mathcal{C}_{cf}|N=∣Ccf​∣ and develop a non-perturbative instanton approximation to describe tunneling processes. We establish a weak-strong duality relating strong quasiparticle tunneling to weak electron tunneling. We calculate the scaling dimensions of the tunneling operators and demonstrate that while the weak-coupling fixed point is generally unstable, the strong-coupling fixed point is stable for physically relevant filling fractions and number of Majorana fermions. These transport exponents provide a distinct experimental fingerprint to identify the topological phases of even-denominator FQH states.
Nearly fifty years after its experimental debut, the quantum hall effect has produced perhaps more unique descriptions than any other subject of condensed matter physics. From topological band theory to non-commutative geometry, topological and conformal field theories, random matrix models and more, each approach captures a subset of the essential features, and compliments the others to produce a beautiful mosaic description. In this thesis, we revisit the fractional quantum hall (FQH) effect from its oldest vantage point, that of the trial wavefunctions and the parent Hamiltonians which stabilize them. Laughlin's observation of the importance of the theory of holomorphic functions to describing incompressible states in a partially filled Landau level has produced a coherent paradigm for realizing FQH parent Hamiltonians as frustration-free exactly-solvable one-dimensional models through dimensional reduction. Trial wavefunctions are then classified in terms of a small number of product states which generate all other states in the orbital-occupation expansion of the full state by a sequence of angular-momentum-preserving squeezing operations. These unentangled states are called the DNA of the FQH state, and they can be described in terms of simple local rules called generalized Pauli principles (GPP) of the form "at most r electrons can occupy k consecutive orbitals". It was only recently shown that an analysis of this form can be carried out in second-quantization for states with Landau level mixing, such as the Jain composite-fermion and parton series. In this case, the GPPs are generalized once further to entangled Pauli principles (EPP) for which the DNA may be entangled with respect to the extra Landau level index. In the first part of this thesis, we prove a theorem that allows for a rigorous computation of EPPs given a frustration free k-body parent Hamiltonian by examining only k-particle zero-modes, and give an example of its utility by demonstrating the exact-solvability of a correlated pair-hopping model that can be derived as a special limit of a FQH parent Hamiltonian. In the second part, taking advantage of the second quantized EPP analysis from the continuum, we derive a class of bosonic lattice models for states in multiple Landau levels which, in the presence of a contact interaction, are shown to stabilize non-Abelian ground states. Furthermore, we show how the EPP can be used to compute the degeneracies of quasihole excitations, and therefore the topological order through the bulk-boundary correspondence, which we verify with simple numerics. To the best of our knowledge, this work represents the first exponentially-local exactly-solvable fractional Chern insulator model for non-Abelian states with only two-body interactions.