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Modern Physics Letters B

Generalization of Tomonaga–Luttinger Liquids and the Calogero–Sutherland Model: Microscopic Models for the Haldane Liquids

Kazumoto Iguchi

Published Jul 10, 1998• 1 citation
Publisher

Abstract

We discuss the quantum many-body system interacting with a separately symmetric two-body potential in higher dimensions as a generalization of the Calogero–Sutherland model in one dimension. This system exhibits the properties of a Haldane liquid such as Haldane–Wu (i.e., fractional exclusion) statistics, broken particle–hole symmetry and the existence of pseudo-Fermi surface as a generalization of the cencept of Tomonaga–Luttinger liquids in one dimension.

DOI
10.1142/s0217984998000718

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We discuss the quantum many-body system interacting with a separately symmetric two-body potential in higher dimensions as a generalization of the Calogero–Sutherland model in one dimension. This system exhibits the properties of a Haldane liquid such as Haldane–Wu (i.e., fractional exclusion) statistics, broken particle–hole symmetry and the existence of pseudo-Fermi surface as a generalization of the cencept of Tomonaga–Luttinger liquids in one dimension.
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We study in this paper the properties of a many body system of fermions obeying exclusion-statistics (Haldane liquid) where the origin of exclusion statistics is coming from an interaction-induced displacement field ak\mathbf{a}_{\mathbf{k}}ak​ introduced by Sutherland (Haldane-Sutherland liquid). In particular we show how the Luttinger Theorem becomes compatible with exclusion statistics as a result of momentum conservation and adiabaticity. As a result, the low energy properties of Haldane-Sutherland liquids are Fermi/Luttinger liquid-like.
Applying boundary conformal field theory we study the low-energy critical behavior of the Calogero-Sutherland model of BCNBC_NBCN​-type. The universality class of the model is found to be a {\it chiral} Tomonaga-Luttinger liquid. Various correlation exponents depending on the interaction strength are obtained.
Universal scaling relations for dissipative Tomonaga-Luttinger (TL) liquids with SU(N\mathrm{SU}(NSU(N) spin symmetry are obtained for both fermions and bosons, by using asymptotic Bethe-ansatz solutions and conformal field theory (CFT) in one-dimensional non-Hermitian quantum many-body systems with SU(N\mathrm{SU}(NSU(N) symmetry. We uncover that the spectrum of dissipative TL liquids with SU(N\mathrm{SU}(NSU(N) spin symmetry is described by the sum of one charge mode characterized by a complex generalization of c=1c=1c=1 U(1) Gaussian CFT, and $N\ensuremath{-}1$ spin modes characterized by level-1SU(N1\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N1SU(N) Kac-Moody algebra with the conformal anomaly $c=N\ensuremath{-}1$, and thereby dissipation only affects the charge mode as a result of spin-charge separation in one-dimensional non-Hermitian quantum systems. The derivation is based on a complex generalization of Haldane's ideal-gas description, which is implemented by the SU(N\mathrm{SU}(NSU(N) Calogero-Sutherland model with inverse-square long-range interactions.
In the present chapter, we discuss the Sutherland model for particles without internal degrees of freedom such as spin. We call this model the singlecomponent Sutherland model. For two particles, the eigenenergies and eigenstates of the Sutherland model (1.3) have been obtained explicitly in the previous chapter. The most striking feature of the Sutherland model is that one can derive not only the energy spectrum but also the dynamics for the many-particle case with an exact account of interaction effects. Thus, the Sutherland model provides an ideal framework to study a one-dimensional quantum liquid in detail. In Section 2.1, we derive the eigenenergies of eigenstates. In Section 2.2, we present different but equivalent physical pictures for the energy spectrum. Namely, the energy spectrum is naturally regarded as that of interacting bosons or fermions. The same spectrum can also be interpreted as that of free particles obeying nontrivial quantum statistics, i.e., free anyons in one dimension. The exclusion statistics proposed by Haldane will be explained on this occasion. Spectrum and statistics of elementary excitations are derived in Section 2.3. In Section 2.4, we discuss thermodynamic properties, which can be rewritten as those of free anyons. In Section 2.5, we identify the eigenfunctions with Jack symmetric polynomials, and discuss their basic properties. In Section 2.6, we consider dynamical correlation functions such as Green's functions and the density correlation function. These quantities are derived with the use of Jack polynomials, and are naturally interpreted in terms of elementary excitations with fractional charge. The Sutherland model is the simplest model to realize the Tomonaga–Luttinger liquid.