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arXiv:cond-mat/9507143 · Journal of the Physical Society of Japan

Higher-Spin Generalization of Fractional Exclusion Statistics

Takahiro Fukui, Norio Kawakami, Sung-Kil Yang

Published Jun 15, 1996• 4 citations
arXivHTMLPDF

Abstract

We study fractional exclusion statistics for quantum systems with SU(2) symmetry (arbitrary spin $S$), by generalizing the thermodynamic equations with squeezed strings proposed by Ha and Haldane. The bare hole distributions as well as the statistical interaction defined by an infinite-dimensional matrix specify the universality class. It is shown that the system is described by the level-$2S$ WZW model and has a close relationship to non-abelian fractional quantum Hall states. As a low-energy effective theory, the sector of {\it massless} Z$_{2S}$ parafermions is extracted, whose statistical interaction is given by a finite-dimensional matrix.

DOI
10.1143/jpsj.65.1617

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We study fractional exclusion statistics for quantum systems with SU(2) symmetry (arbitrary spin SSS), by generalizing the thermodynamic equations with squeezed strings proposed by Ha and Haldane. The bare hole distributions as well as the statistical interaction defined by an infinite-dimensional matrix specify the universality class. It is shown that the system is described by the level-2S2S2S WZW model and has a close relationship to non-abelian fractional quantum Hall states. As a low-energy effective theory, the sector of {\it massless} Z2S_{2S}2S​ parafermions is extracted, whose statistical interaction is given by a finite-dimensional matrix.
The idea of fractional exclusion statistics proposed by Haldane is applied to systems with internal degrees of freedom, and its thermodynamics is examined. In the case of one dimension, various calculated bulk quantities show that the critical behavior of such systems can be described by c=1 conformal field theories and conformal dimensions are completely characterized by statistical interactions. It is also found that statistical interactions have an intimate relationship with a topological order matrix in Chern-Simons theory for the fractional quantum Hall effect.
Haldane's fractional exclusion statistics (FES) describes a generalized Pauli exclusion statistics, which can be regarded as an emergent quantum statistics induced by the intrinsic dynamical interaction. A non-mutual FES has been identified at the quantum criticality of the one-dimensional (1D) and 2D interacting Bose Gas [Nat. Sci. Rev. 9, nwac027 (2022)]. It is naturally asked if such a non-mutual FES can be induced by the spin-spin interaction in the antiferromagnetic spin-1/2 XXX chain? In this article, we first represent the Bethe ansatz equations of spin strings in terms of the FES equations of different species. Then we show that the 1D spin XXX chain remarkably possesses the non-mutual FES in the critical region. We observe that the equation of state in terms of the FES gives rises to full statistical properties of the model at quantum criticality, which are in good agreement with the results obtained from the thermodynamic Bethe ansatz (TBA) equations of the model. From the non-mutual FES, we also precisely determine the quantum scaling functions, which further agree well with the previous TBA results [Phys. Rev. B 96, 220401(R) (2017)]. Finally, we also build up an exact mapping between the scaling functions of the Lieb-Liniger model and the spin Heisenberg spin chain at quantum criticality. Our method provides deep insights into the critical phase of matter from quantum FES point of view.
Haldane's fractional exclusion statistics (FES) describe generalized Pauli exclusion statistics, which can be regarded as emergent quantum statistics induced by the intrinsic dynamical interaction. Nonmutual FES have been identified at the quantum criticality of the one-dimensional (1D) and 2D interacting Bose gas [Nat. Sci. Rev. 9, nwac027 (2022)]. It is naturally asked if such nonmutual FES can be induced by the spin-spin interaction in the antiferromagnetic spin-1/2 XXXXXXXXX chain. In this article, we first represent the Bethe ansatz equations of spin strings in terms of the FES equations of different species. Then we show that the 1D spin XXXXXXXXX chain remarkably possesses nonmutual FES in the critical region. We observe that the equation of state in terms of the FES gives rises to full statistical properties of the model at quantum criticality, which are in good agreement with the results obtained from the thermodynamic Bethe ansatz (TBA) equations of the model. From the nonmutual FES, we also precisely determine the quantum scaling functions, which further agree well with the previous TBA results [Phys. Rev. B 96, 220401(R) (2017)]. Finally, we also establish an exact mapping between the scaling functions of the Lieb-Liniger model and the spin Heisenberg spin chain at quantum criticality. Our method provides deep insights into the critical phase of matter from a quantum FES point of view.
We determine the exclusion statistics properties of the fundamental edge quasiparticles over a specific $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ non-Abelian quantum Hall state known as the q=2q\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2q=2 Pfaffian state. The fundamental excitations are the edge electrons of charge $\ensuremath{-}e$ and the edge quasiholes of charge +e/4+e/4+e/4. We explicitly determine thermodynamic distribution functions and establish a duality which generalizes the duality for fractional exclusion statistics as described by F. D. M. Haldane.
We report on a study of virial expansions for interacting electrons in the lowest Landau level of a two-dimensional electron gas. For hard-core-model interactions, we derive analytic results valid at low temperatures and filling factors smaller than 1/31/31/3, and comment on their relationship with virial expansions for exclusion statistics models. In the high-temperature limit the virial coefficients reduce to those for a noninteracting electron system. For the first five virial coefficients, the crossover between low- and high-temperature limits has been studied numerically by using partition functions obtained from small system exact diagonalization calculations. Our results show that the exclusion statistics description of fractional Hall thermodynamics breaks down when the temperature exceeds a small fraction of the gap temperature.
We construct thermodynamics of the one-dimensional supersymmetric {\it t-J} model with the $ 1/\sin^2$ interaction and hopping. The thermodynamics is described exactly in terms of free spinons and holons obeying Haldane's fractional exclusion statistics at all temperatures. Moreover, at low temperatures the semionic spinons and holons decouple resulting in the spin-charge separation in thermodynamic properties. We obtain explicit results for the spin and charge susceptibilities and specific heat, and interpret them in terms of the fractional exclusion statistics. Extension to the multi-component {\it t-J} model shows that the excitations obey either fractional statistics for g-ons with partial polarization of components, or the parafermionic one without polarization.