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Physica Scripta

Interpolative Quantum Statistics: Towards a Statistical Mechanics of Laughlin Liquids

R Ramanathan

Published Jul 1, 1999• 0 citations
Publisher

Abstract

It is pointed out that Haldane's 'statistical interaction' is implicit in the generalised, interpolative quantum statistics developed in a different context and this relationship has been further clarified by the recent work of Wu and Rajagopal. The interpolative quantum statistics, therefore, also applies to the thermodynamics of the elementary excitations in pure Laughlin liquids.

DOI
10.1238/physica.regular.060a00015

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It is pointed out that Haldane's 'statistical interaction' is implicit in the generalised, interpolative quantum statistics developed in a different context and this relationship has been further clarified by the recent work of Wu and Rajagopal. The interpolative quantum statistics, therefore, also applies to the thermodynamics of the elementary excitations in pure Laughlin liquids.
A review of the interpolative quantum statistics is presented. It is shown, by imposing a positivity condition on the distribution function, that this statistics subsumes Gentile's intermediate statistics and is far richer than Gentile's statistics, apart from being a genuine quantum statistics. It is also capable of accounting for the possible very small violations of statistics in physical processes. It is finally pointed out that this statistics offers an elegant and useful parametrization of Haldane's 'statistical interaction', while retaining all the interesting features of the latter. Some of the statistical mechanics that these distribution functions lead to are also examined.
We show that the thermodynamic Bethe ansatz equations for one-dimensional integrable many-body systems can be reinterpreted in such a way that they only code the statistical interactions, in the sense of Haldane, between particles of identical or different momenta. Thus, the thermodynamic properties of these systems can be characterized by the generalized ideal gases recently proposed by one of us. For example, the Yang-Yang δδδ-function gas is a gas with specific statistical interactions between particles of different momenta, while the Calogero-Sutherland system provides a model for an ideal gas of particles with a fractional statistics.
The elementary excitations of a fractional quantum Hall liquid are quasiparticles or quasiholes that are neither bosons nor fermions, but are so-called anyons. Here we study impurity particles immersed in a quantum Hall liquid that bind to the quasiholes via repulsive interactions with the liquid. We show that the angular momentum of an impurity is given by the multiple of a fractional "quantum" of angular momentum, and can directly be observed from the impurity density. In a system with several impurities bound to quasiholes, their total angular momentum interpolates between the values for free fermions and for free bosons. This interpolation is characterized by the fractional statistical parameter of the anyons, which is typically defined via their braiding behavior.
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.
Bethe ansatz and bosonization procedures are used to describe the thermodynamics of the strong-coupled Hubbard chain in the spin-incoherent Luttinger liquid (LL) regime: $J(\ensuremath{\equiv}4{t}^{2}/U)\ensuremath{\ll}{k}_{B}T\ensuremath{\ll}{E}_{F}$, where ttt is the hopping amplitude, $U(\ensuremath{\gg}t)$ is the repulsive on-site Coulomb interaction, and ${k}_{B}T({E}_{F}\ensuremath{\sim}t)$ is the thermal (Fermi) energy. We introduce a fractional Landau LL approach, whose $U=\ensuremath{\infty}$ fixed point is exactly mapped onto an ideal gas with two species obeying the Haldane-Wu exclusion fractional statistics. This phenomenological approach sheds light on the behavior of several thermodynamic properties in the spin-incoherent LL regime: specific heat, charge compressibility, magnetic susceptibility, and Drude weight. In fact, besides the hopping (mass) renormalization, the fractional Landau LL parameters, due to quasiparticle interaction, are determined and relationships with velocities of holons and spinons are unveiled. The specific heat thus obtained is in very good agreement with previous density matrix renormalization group (DMRG) simulations of the $t\text{\ensuremath{-}}J$ model in the spin-incoherent regime. A phase diagram is provided and two thermodynamic paths to access this regime clarifies both the numerical and analytical procedures. Further, we show that the high-TTT limit of the fractional Landau LL entropy and chemical potential exhibit the expected results of the $t\text{\ensuremath{-}}J$ model, under the condition $U\ensuremath{\gg}{k}_{B}T$. Lastly, finite-temperature Lanczos simulations of the single-particle distribution function confirm the characteristics of the spin-incoherent regime and the high-TTT limit observed in previous DMRG studies.