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arXiv:cond-mat/9504112 · Physical review. B, Condensed matter

Mutual-exclusion statistics in exactly solvable models in one and higher dimensions at low temperatures

Yasuhiro Hatsugai, Mahito Kohmoto, Tohru Koma, Yong-Shi Wu

Published Aug 15, 1996• 32 citations
arXivHTMLPDF

Abstract

We study statistical characterization of the many-body states in exactly solvable models with internal degrees of freedom. The models under consideration include the isotropic and anisotropic Heisenberg spin chains, the Hubbard chain, and a model in higher dimensions which exhibits the Mott metal-insulator transition. It is shown that the ground states of these systems are all described by that of a generalized ideal gas of particles (called exclusons) which have mutual-exclusion statistics, either between different rapidities or between different species. For the Bethe ansatz solvable models, the low-temperature properties are well described by the excluson description if the degeneracies due to string solutions with complex rapidities are taken into account correctly. For the Hubbard chain with strong but finite coupling, charge-spin separation is shown for thermodynamics at low temperatures. Moreover, we present an exactly solvable model in arbitrary dimensions which, in addition to giving a perspective view of spin-charge separation, constitutes an explicit example of mutual-exclusion statistics in more than two dimensions. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.54.5358

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We study statistical characterization of the many-body states in exactly solvable models with internal degrees of freedom. The models under consideration include the isotropic and anisotropic Heisenberg spin chains, the Hubbard chain, and a model in higher dimensions which exhibits the Mott metal-insulator transition. It is shown that the ground states of these systems are all described by that of a generalized ideal gas of particles (called exclusons) which have mutual-exclusion statistics, either between different rapidities or between different species. For the Bethe ansatz solvable models, the low-temperature properties are well described by the excluson description if the degeneracies due to string solutions with complex rapidities are taken into account correctly. For the Hubbard chain with strong but finite coupling, charge-spin separation is shown for thermodynamics at low temperatures. Moreover, we present an exactly solvable model in arbitrary dimensions which, in addition to giving a perspective view of spin-charge separation, constitutes an explicit example of mutual-exclusion statistics in more than two dimensions. \textcopyright{} 1996 The American Physical Society.
We study statistical characterization of the many-body states in the exactly solvable model with internal degree of freedom in more than one dimension. The model exhibits the Mott metal-insulator transition. It is shown that the ground state is described by that of a generalized ideal gas of particles (called exclusons) which have mutual exclusion statistics between different species. In addition to giving a perspective view of spin-charge separation, the model constitutes an explicit example of mutual exclusion statistics in more than two dimensions.
We show that the Hubbard model with infinite-range Coulomb coupling is equivalent to an ideal gas of three species of particles obeying fractional exclusion statistics. A full appreciation of this mapping requires an extension of the pertinent formalism. This very simple, but rather peculiar model is exactly solvable in any dimension and exhibits a Mott metal-insulator transition, whose universality class is shown to be that of a free spinless Fermi gas. A modified version of the Luttinger theorem is shown to apply in any dimension. We also characterize the metallic and insulating phases by obtaining the electronic band structure as well as the interacting density of states. The fractional statistics manifests itself on the amplitudes of several thermodynamic quantities and, in particular, the Pauli spin susceptibility is subdominant in all metallic phases, and a Curie-type of response appears.
In this article we review recent developments in the one-dimensional Bethe ansatz solvable multi-chain quantum models. The algebraic version of the Bethe ansatz (the quantum inverse scattering method) permits us to construct new families of integrable Hamiltonians using simple generalizations of the well known constructions of the single-chain model. First we consider the easiest example (`basic' model) of this class of models: the antiferromagnetic two-chain spin-1/2 model with the nearest-neighbour and next-nearest-neighbour spin-frustrating interactions (zigzag chain). We show how the algebra of the quantum inverse scattering method works for this model, and what are the important features of the Hamiltonian (which reveal the topological properties of two dimensions together with the one-dimensional properties). We consider the solution of the Bethe ansatz for the ground state (in particular, commensurate-incommensurate quantum phase transitions present due to competing spin-frustrating interactions are discussed) and construct the thermal Bethe ansatz (in the form of the `quantum transfer matrix') for this model. Then possible generalizations of the basic model are considered: an inclusion of a magnetic anisotropy, higher-spin representations (including the important case of a quantum ferrimagnet), the multi-chain case, internal degrees of freedom of particles at each site, etc. We observe the similarities and differences between this class of models and related exactly solvable models: other groups of multi-chain lattice models, quantum field theory models and magnetic impurity (Kondo-like) models. Finally, the behaviour of non-integrable (less constrained) multi-chain quantum models is discussed.
Abstract This book focuses on the theory of quantum matter, strongly interacting systems of quantum many–particle physics, particularly on their study using exactly solvable and quantum integrable models with Bethe ansatz methods. Part 1 explores the fundamental methods of statistical physics and quantum many–particle physics required for an understanding of quantum matter. It also presents a selection of the most important model systems to describe quantum matter ranging from the Hubbard model of condensed matter physics to the Rabi model of quantum optics. The remaining five parts of the book examines appropriate special cases of these models with respect to their exact solutions using Bethe ansatz methods for the ground state, finite–size, and finite temperature properties. They also demonstrate the quantum integrability of an exemplary model, the Heisenberg quantum spin chain, within the framework of the quantum inverse scattering method and through the algebraic Bethe ansatz. Further models, whose Bethe ansatz solutions are derived and examined, include the Bose and Fermi gases in one dimension, the one–dimensional Hubbard model, the Kondo model, and the quantum Tavis–Cummings model, the latter a model descendent from the Rabi model.
Chapter 1. Exact and explicit results are derived for the thermodynamic properties (isochores, isotherms, isobars, response functions, speed of sound) of a quantum gas in dimensions [special characters omitted] ≥ 1 and with fractional exclusion statistics 0 ≤ g ≤ 1 connecting bosons (g = 0) and fermions (g = 1). In [special characters omitted] = 1 the results are equivalent to those of the Calogero-Sutherland model, a gas with long-range two-body interaction. Emphasis is given to the crossover between boson-like and fermion-like features, caused by aspects of the statistical interaction that mimic long-range attraction and short-range repulsion. A phase transition along the isobar occurs at a nonzero temperature in all dimensions. The T-dependence of the speed of sound is in simple relation to isochores and isobars. The effects of soft container walls are accounted for rigorously for the case of a pure power-law potential. Chapter 2. The exact thermodynamics (isochores, isotherms, isobars, response functions, speed of sound) is worked out for a statistically interacting quantum gas in [special characters omitted] dimensions. The results in [special characters omitted] = 1 are those of the thermodynamic Bethe ansatz for the Nonlinear Schrödinger model, a gas with repulsive two-body contact potential. In all dimensions the ideal boson and fermion gases are recovered in the weak-coupling and strong-coupling limits, respectively. For all nonzero couplings ideal fermion gas behavior emerges for [special characters omitted] >> 1 and, in the limit [special characters omitted] → ∞, a phase transition occurs at T > 0. Significant deviations from ideal quantum gas behavior are found for intermediate coupling and finite [special characters omitted]. Chapter 3. Methodology previously developed in the framework of the coordinate Bethe ansatz applied to integrable quantum gas models is employed to calculate some ground-state properties and elementary excitations for quantum gas models in [special characters omitted] = 1 dimensions with statistical interactions that are not equivalent to dynamical interactions. The focus in this comparative study is on modifications of the Calogero-Sutherland and Nonlinear Schrödinger models. Statistical interactions cannot be chosen arbitrarily. They must satisfy certain physicality conditions, which will be discussed in some detail.
The ground state and low-temperature properties of the Hubbard model with infinite-range attractive interaction in the presence of a magnetic field are considered. This model is exactly solvable in any dimension and is mapped onto an ideal gas of three species of exclusons. The system is found to behave as a noninteracting Fermi gas whenever the correlation function $C(\mathbf{k})=⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩\ensuremath{-}⟨{n}_{\mathbf{k}\ensuremath{\uparrow}}⟩⟨{n}_{\mathbf{k}\ensuremath{\downarrow}}⟩$ is negligible, otherwise it displays nontrivial correlated behavior which manifests as fractional statistical effects. The corresponding effective Hamiltonians and free energies governing the distinct regimes are presented. In one dimension, our results are analyzed in light of those derived using the on-site attractive Hubbard chain.