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Fundamental Dual-Phase Theory of Identical Particles: The Existence Conditions for Non-Abelian Anyons in 2D Topology (Series II)

Jian-Hua Wang

Published Oct 3, 2025• 0 citations
Publisher

Abstract

This work uncovers an essential dual-phase structure of identical particles, revealing a fundamental limitation in existing quantum statistical theories. For a considerable period, it was commonly assumed that an interchange of two particles would be accompanied by a fixed phase factor. Here, the author discovers that the exchange process inherently involves two distinct phases: a relative phase α and an exchange phase β, rigorously related through α = β-1 = e±iθ. Specifically, α = β = 1 characterizes bosons, while α = β = -1 describes fermions. This finding necessitates a radical change in handling identical particles. Within this new framework, clockwise exchange yields α = eiθ and β = e-iθ, while counter-clockwise exchange gives α = e-iθ and β = eiθ. Crucially, both scenarios, despite their topological differences, are physically equivalent, governed by the universal constraint αβ = 1 for all identical particles. This framework not only fundamentally changes the conceptual basis of quantum statistics but also provides critical theoretical constraints on the existence conditions of non-Abelian anyons in two-dimensional topological systems.

DOI
10.20944/preprints202510.0229.v1

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This work uncovers an essential dual-phase structure of identical particles, revealing a fundamental limitation in existing quantum statistical theories. For a considerable period, it was commonly assumed that an interchange of two particles would be accompanied by a fixed phase factor. Here, the author discovers that the exchange process inherently involves two distinct phases: a relative phase α and an exchange phase β, rigorously related through α = β-1 = e±iθ. Specifically, α = β = 1 characterizes bosons, while α = β = -1 describes fermions. This finding necessitates a radical change in handling identical particles. Within this new framework, clockwise exchange yields α = eiθ and β = e-iθ, while counter-clockwise exchange gives α = e-iθ and β = eiθ. Crucially, both scenarios, despite their topological differences, are physically equivalent, governed by the universal constraint αβ = 1 for all identical particles. This framework not only fundamentally changes the conceptual basis of quantum statistics but also provides critical theoretical constraints on the existence conditions of non-Abelian anyons in two-dimensional topological systems.
As opposed to classical mechanics, quantum mechanical particles can be truly identical and lead to new and interesting phenomena. Identical particles can be of different types, determined by their exchange symmetry, which in turn gives rise to statistical repulsion. The exchange symmetry is given by a representation of the exchange group; the fundamental group of the configuration space of identical particles. In three dimensions the exchange group is the permutation group and there are only two types of identical particles; bosons and fermions. While any number of bosons can be at the same place or in the same state, fermions repel each other. In two dimensions the exchange group is the braid group and essentially any exchange symmetry is allowed, such particles are called anyons. Abelian anyons are described by abelian representations of the exchange group and can be seen as giving a continuous interpolation between bosons and fermions. Non-abelian anyons are much more complex and their statistical repulsion is yet largely unexplored. We use the framework of modular tensor categories to show how the statistical repulsion of non-abelian anyons depends on the exchange symmetry. The Fibonacci anyon model is studied, for which explicit results are obtained. We also show how Fibonacci anyons can be used to implement topological quantum computation, providing topologically stable quantum information encoded in the state of non-abelian anyons that can be manipulated via the non-abelian exchange symmetry.
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons is not simply and automatically equal to the phase factor one can observe in an interference experiment that involves physically exchanging two such particles. The observable phase contains, in general, single-particle geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. By extending the analysis to (non-abelian) anyons it is argued that, similarly, there are single-anyon geometric and dynamical contributions in addition to purely topological unitary transformations that accompany physical exchanges of anyons. Work remains to be done in order to demonstrate---if it is still true---that those additional contributions to the gates in anyonic topological quantum computers do not destroy the inherent robustness of the ideal gates. This negative result is described most clearly in terms of the Berry matrix.
When leaving classical physics and entering the realm of quantum physics, there are many new concepts being introduced. One of the most fundamental ideas in quantum mechanics is that particles no longer have exact known positions, but instead expected values and prob- abilities. This leads to the phenomena of truly identical particles, since they no longer can be distinguished simply by their positions. An important property differentiating different kinds of particles is how a system behaves when two such identical particles are exchanged. Historically, this divided particles into bosons and fermions, corresponding to symmetry and antisymmetry under an exchange. However, in two dimensions a new type of particle appears. These particles are called anyons, and behave differently when particles are exchanged. Anyons can be further divided into abelian and non-abelian anyons, of which this thesis will focus on the latter. The ex- changes can then be represented by the fundamental group of the configuration space of the particles, and in two dimensions this fundamental group is the braid group. Using rotors from a Clifford algebra and studying excitations of Majorana fermions, this thesis will show a way to calculate the exchange matrices of non-abelian anyons, and their corresponding eigenvalues. Furthermore, suggestions on a generalization of this framework along with areas where it can be applied are given.
The conventional framework for quantum statistics is built upon gauge theory, where particle exchanges generate path-dependent phases. However, the apparent consistency of this approach masks a deeper question: is gauge invariance truly sufficient to satisfy the physical requirement of indistinguishability? We demonstrate that gauge transformations, while preserving probabilities in a formal sense, are inadequate to capture the full constraints of identical particles, thereby allowing for unphysical statistical outcomes. This critical limitation necessitates a reconstruction of the theory by strictly enforcing indistinguishability as the foundational principle, thus moving beyond the conventional topological paradigm. This shift yields a radically simplified framework in which the statistical phase emerges as a path-independent quantity, α=e±iθ \alpha = e^{\pm i\theta} α=e±iθ, unifying bosons, fermions, and anyons within a single consistent description. Building upon the operator-based formalism of Series I and the dual-phase theory of Series II, we further present an exact and computationally tractable approach for solving N-anyon systems.
The phase factor (−1)2s(-1)^{2s}(−1)2s that features in the exchange symmetry for identical spin-sss fermions or bosons contributes to but is not identical to the phase factor one may observe when one actually exchanges two such particles. The observable phase contains {\em single-particle} geometric and dynamical phases as well, induced by both spin and spatial exchange transformations. Extending the analysis to (abelian) anyons by incorporating the Aharonov-Bohm effect shows that the topological phase two anyons pick up under an interchange is at bottom a single-particle effect, in contrast to the fermion/boson exchange phase factor (−1)2s(-1)^{2s}(−1)2s, which is a two-particle effect.
"Anyonic quantum statistics is an exotic phenomenon of identical particles in quantum mechanics. When particles are confined in two spatial dimensions, exchanges of identical particles can induce phase factors in the wave function of Abelian anyons or matrix valued transformations of the wave function of non-Abelian anyons. As a result, systems of anyons may have richer properties than those of bosons and fermions in three spatial dimensions. There is strong theoretical support for the existence of anyons in some engineered two-dimensional systems such as 2D electron gases, strongly correlated spin lattices and as edge modes of nanowires. In the future, anyons could be used in topological quantum computation to perform highly efficient information processing with very small error rates. The phases of matter described by dynamically interacting anyons have recently been studied in chains where anyons interact via Heisenberg-type exchange interactions. In this thesis, new kind of anyonic interactions are studied, induced purely by braiding during free evolution. Such interactions are of topological origin, and the information about the interactions is stored non-locally. A quantum walk model is used to study the effects of these braiding interactions on the dynamical behaviour of anyons. The anyonic quantum walk is a quasi-one-dimensional generalization of the discrete-time quantum walk which allows the simulation of anyonic dynamics analytically. The moving anyon and a chain of stationary anyons interact via braiding statistics, and the behaviour of the anyon is studied in three cases. Striking differences are found between particles with conventional boson or fermion statistics and non-Abelian anyonic statistics. The random walk is a dynamical model that describes the motion of a particle on a lattice. In physics, it is used to describe Brownian motion of fluids and gases. In such systems, energy transport is diffusive, and the order of the system approaches a highly mixed state without any information content. The quantum version of the random walk, the quantum walk, has nonintuitive properties. Generally, the information content in a unitarily evolving system is fixed, which leads to unexpected transport phenomena. An initially localized particle does not propagate diffusively, but escapes the starting point with ballistic speed. Most of our results for non-Abelian anyons use the Ising model anyon which is most likely to be measured in experiments. First, when each anyonic site is occupied by one Ising anyon, the propagation of the anyon becomes diffusive. More precisely, the variance of the spatial probability distribution of the particle depends linearly on the number of time steps. This is in stark contrast to bosons, fermions and Abelian anyons which propagate ballistically, and the variance depends linearly on the square of the number of time steps. The essential reason for this slowdown is that the non-Abelian anyons possess an extra degree of freedom called the fusion Hilbert space. This space can be viewed as an environment for the normal degrees of freedom of the particle, inducing decoherence in the quantum walk. This thesis opens the line for studies of this novel kind of decoherence mechanism in quantum walks. In the second case the study is extended to more general anyon models by losing some of the information about the history of the evolution. In this case the system is subject to decoherence, and the total system evolution is not unitary. The behaviour of the particle in this model is found to be diffusive for all the non-Abelian anyon models studied, while Abelian anyons behave ballistically also in this model. One peculiarity of quantum systems is their behaviour under disorder. Quantum mechanical particles moving in random local potentials are known to freeze and not move at all. Such a phenomenon is known as Anderson localization. Studies have shown that Anderson localization happens also in quantum walks with random spatial fluctuations in the coin parameters. In the third case, the transport properties of anyons are studied under topological randomness, allowing the occupations of anyons change between experiments. Bosons and fermions would propagate ballistically under such randomness, and Abelian anyons are shown to localize. The results show that non-Abelian anyons behave diffusively at short time scales, and it is argued that they do so in the long time limit as well. In all cases, non-Abelian anyons are shown to have very different dynamical properties than bosons, fermions and Abelian anyons. A possible simulation of the anyonic quantum walk in Fractional Quantum Hall systems is also discussed.