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Physical review. B, Condensed matter

Multisheet configuration space and fractional quantum statistics

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

Published May 15, 1992• 0 citations
Publisher

Abstract

We show that if one uses a multisheet configuration space for a system of identical particles, then fractional quantum statistics in two dimensions can be geometrically characterized as a topological quantum number for going from one sheet to the next. Furthermore, the spectrum of such a q-sheet system is shown to consist of the spectra of all species of anyons obeying different fractional statistics \ensuremath{\theta}=2\ensuremath{\pi}p/q (p=0,1, . . .,q-1) with the same denominator q. A lattice computation has been done to verify this. Various aspects of the multisheet formalism are discussed.

DOI
10.1103/physrevb.45.11161

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We show that if one uses a multisheet configuration space for a system of identical particles, then fractional quantum statistics in two dimensions can be geometrically characterized as a topological quantum number for going from one sheet to the next. Furthermore, the spectrum of such a q-sheet system is shown to consist of the spectra of all species of anyons obeying different fractional statistics \ensuremath{\theta}=2\ensuremath{\pi}p/q (p=0,1, . . .,q-1) with the same denominator q. A lattice computation has been done to verify this. Various aspects of the multisheet formalism are discussed.
Anyons and fractional statistics are by now well established in two-dimensional systems. In one dimension, fractional statistics has been established so far only through Haldane's fractional exclusion principle, but not via a fractional phase the wave function acquires as particles are interchanged. At first sight, the topology of the configuration space appears to preclude such phases in one dimension. Here we argue that the crossings of one-dimensional anyons are always unidirectional, which makes it possible to assign phases consistently and hence to introduce a statistical angle $\ensuremath{\theta}$. The fractional statistics then manifests itself in fractional spacings of the single-particle momenta of the anyons when periodic boundary conditions are imposed. These spacings are given by $\ensuremath{\Delta}p=2\ensuremath{\pi}\ensuremath{\hbar}/L(|\ensuremath{\theta}|/\ensuremath{\pi}+\text{non-negative}\text{ }\text{integer})$ for a system of length LLL. This condition is the analog of the quantization of relative angular momenta according to ${l}_{\text{z}}=\ensuremath{\hbar}(\ensuremath{-}\ensuremath{\theta}/\ensuremath{\pi}+2\ifmmode\times\else\texttimes\fi{}\text{integer})$ for two-dimensional anyons.
We review the theory of statistics of identical systems in terms of the properties of the classical configuration space. Special emphasis is placed on the influence of the connectivity of this space on quantum statistics. The theory shows that exotic statistical systems, such as those obeying fractional statistics, can occur in all dimensions. Particle statistics in one dimension involves novel features particular to this dimension and these are briefly described. The proof of a topological spin-statistics theorem which avoids field theory or relativity is also outlined.
The standard topological approach to indistinguishable particles formulates exchange statistics by using the fundamental group to analyze the connectedness of the configuration space. Although successful in two and more dimensions, this approach gives only trivial or near-trivial exchange statistics in one dimension because two-body coincidences are excluded from configuration space. Instead, we include these path-ambiguous singular points and consider configuration space as an orbifold. This orbifold topological approach allows unified analysis of exchange statistics in any dimension and predicts possibilities for anyons in one-dimensional systems, including non-Abelian anyons obeying alternate strand groups. These results clarify the nontopological origin of fractional statistics in one-dimensional anyon models.
The quantum-mechanical description of assemblies of particles whose motion is confined to two (or one) spatial dimensions offers many possibilities that are distinct from bosons and fermions. We call such particles anyons. The simplest anyons are parameterized by an angular phase parameter θ. θ = 0, π correspond to bosons and fermions, respectively; at intermediate values, we say that we have fractional statistics. In two dimensions, θ describes the phase acquired by the wave function as two anyons wind around one another counterclockwise. It generates a shift in the allowed values for the relative angular momentum. Composites of localized electric charge and magnetic flux associated with an abelian U(1) gauge group realize this behavior. More complex charge-flux constructions can involve nonabelian and product groups acting on a spectrum of allowed charges and fluxes, giving rise to nonabelian and mutual statistics. Interchanges of nonabelian anyons implement unitary transformations of the wave function within an emergent space of internal states. Anyons of all kinds are described by quantum field theories that include Chern–Simons terms. The crossings of one-dimensional anyons on a ring are unidirectional, such that a fractional phase θ acquired upon interchange gives rise to fractional shifts in the relative momenta between the anyons. The quasiparticle excitations of fractional quantum Hall states have long been predicted to include anyons. Recently, the anyon behavior predicted for quasiparticles in the ν = 1/3 fractional quantum Hall state has been observed in both scattering and interferometric experiments. Excitations within designed systems, notably including superconducting circuits, can exhibit anyon behavior. Such systems are being developed for possible use in quantum information processing.
General kinematical restrictions on quantum-theories of N identical anyons have been established by using braid groups. For an orientable compact surface (without boundary) of genus g, the statistical parameter θ is a rational multiple of π, θ=πp/v (p and v mutually prime integers), and the number of components of the wavefunction is (a multiple of) vg. The particle-number is N=rv+1−g (r integer) for spinless (S=0) and N=rv for spinning (S=θ/2π) anyons. The restrictions for spinning anyons are consistent with results for fractional quantum Hall systems, nonlinear O(3) field theory, and Chern-Simons theory. The multi-component structure, which appears for g≥1, reflects an internal collective degree of freedom. A comparison with the multi-component wavefunctions needed to describe systems with fractionally charged quasi-particles yields consistency relations between charge and statistics. Non-orientable surfaces do not allow fractional statistics.
Abstract The book presents the wide range of topics in two-dimensional physics of quantum Hall systems, especially fractional quantum Hall states. It starts with the fundamental problems of quantum statistics in two dimensions and the corresponding braid group formalism. The braid group formalism of anyons (previously known) is developed for composite fermions. The effective formalism used in many-body quantum Hall theories — the Chern–Simons theory is also presented. The Chern–Simons theory of anyons (particles obeying fractional statistics) and composite fermions (related to Hall systems) is given, in detail. Numerical studies, which play the important role in quantum Hall analyses, are presented for spherical systems (Haldane sphere). The composite fermion theory is tested in numerical studies. The concept of the hierarchy of condensed states of composite fermion excitations is introduced (in analogy to the Haldane hierarchy). The hierarchies of odd-denominator states and even-denominator states are presented. The BCS paired Hall state is also discussed. An introduction to multi-component quantum Hall systems and spin quantum Hall systems is sketched.