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Physical Review Letters

von Neumann Entropy Associated with the Haldane Exclusion Statistics

A. K. Rajagopal

Published Feb 13, 1995• 71 citations
Publisher

Abstract

We obtain the von Neumann entropy per state of the Haldane exclusion statistics with parameter $g$ in terms of the mean occupation number $\overline{n}$ as ${S}_{H}=\ensuremath{-}\overline{n}{w\mathrm{ln}w\ensuremath{-}(1+w)\mathrm{ln}(1+w)}$, where $w=\frac{(1\ensuremath{-}\overline{n}g)}{\overline{n}}$. This reduces correctly to the well known expressions in the limiting cases of Bose ($g=0$) and Fermi ($g=1$) statistics. We have derived the second and third order fluctuations in the occupation numbers for arbitrary $g$. An elegant general duality relationship between the $w$ factor associated with the particle and that associated with the hole at the reciprocal $g$ is deduced along with the attendant relationship between the two respective entropies.

DOI
10.1103/physrevlett.74.1048

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We obtain the von Neumann entropy per state of the Haldane exclusion statistics with parameter ggg in terms of the mean occupation number n‾\overline{n}n as SH=\ensuremath−n‾wlnw\ensuremath−(1+w)ln(1+w){S}_{H}=\ensuremath{-}\overline{n}{w\mathrm{ln}w\ensuremath{-}(1+w)\mathrm{ln}(1+w)}SH​=\ensuremath−nwlnw\ensuremath−(1+w)ln(1+w), where w=(1\ensuremath−n‾g)n‾w=\frac{(1\ensuremath{-}\overline{n}g)}{\overline{n}}w=n(1\ensuremath−ng)​. This reduces correctly to the well known expressions in the limiting cases of Bose (g=0g=0g=0) and Fermi (g=1g=1g=1) statistics. We have derived the second and third order fluctuations in the occupation numbers for arbitrary ggg. An elegant general duality relationship between the www factor associated with the particle and that associated with the hole at the reciprocal ggg is deduced along with the attendant relationship between the two respective entropies.
I study basic properties of a Haldane liquid with mutual exclusion statistics (MES), which is a liquid of quasiparticles interacting only through the MES that are assumed to be parameters as usually done in the definition of Haldane statistics. By this approximation for the statistics, I obtain the following: The ground state of the system has multiple pseudo--Fermi surfaces for all species of quasiparticles when T=0,T=0,T=0, where the pseudo--Fermi surface for each species exhibits the particle-hole asymmetry. I define the dressed energies so as to satisfy the Sutherland-Wu (SW) functional equations that are the defining relations for the MES, from which the elementary excitations near the ground state are given. And finally, I show that the SW equations have an unusual type of duality such as spin-charge duality in the system.
This paper is a revision of the combinatorics of fractional exclusion statistics (FES). More specifically, the following exact statement of the generalized Pauli principle is derived: for an NNN-particles system exhibiting FES of extended parameter g=q/rg=q/rg=q/r (qqq and rrr are co-prime integers such that 0<q≤r0 < q \leq r0<q≤r), we found that the allowed occupation number of a state is smaller than or equal to r−q+1r-q+1r−q+1 and \emph{not} to 1/g1/g1/g whenever q≠1q\neq 1q=1 and, moreover, the global occupancy shape (merely represented by a partition of NNN) is admissible if the number of states occupied by at least two particles is less than or equal to (N−1)/r(N-1)/r(N−1)/r (N≡1(modr)N \equiv 1 \pmod rN≡1(modr)). These counting rules allow distinguishing infinitely many families of FES systems depending on the parameter ggg and the size NNN. As an application of the main result, we study the probability distributions of occupancy configurations. For instance, the number of occupied states is found to be a hypergeometric random variable. Closed-form expressions for the expectation values and variances in the thermodynamic limit are presented. By way of comparison, we obtain parallel results regarding the Gentile intermediate statistics and demonstrate subtle similarities and contrasts with respect to FES.
A new distribution for systems of particles obeying statistical exclusion of correlated states is presented following the Haldane's state counting. It relies upon a conjecture to deal with the multiple exclusion that takes place when the states available to single particles are spatially correlated and it can be simultaneously excluded by more than one particle. The Haldane's statistics [F. D. M. Haldane, Phys. Rev. Lett. 67, 937 (1991)] and Wu's distribution [Y.-S. Wu, Phys. Rev. Lett. 52, 2103 (1984)] are recovered in the limit of non-correlated states (constant statistical exclusion) of the multiple exclusion statistics. In addition, the exclusion spectrum function G(n)G(n)G(n) is introduced to account for the dependence of the statistical exclusion on the occupation-number nnn. Results of thermodynamics and state occupation are shown for ideal lattice gases of linear particles of size kkk (kkk-mers) where multiple exclusion occurs. Remarkable agreement is found with simulations from k=2k=2k=2 to 101010 where multiple exclusion dominates as kkk increases.