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arXiv:cond-mat/9402011 · Progress of Theoretical Physics

Critical Properties of Quantum Many-Body Systems with 1/r2 Interaction

N. Kawakami

Published Feb 1, 1994• 21 citations
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Abstract

We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.

DOI
10.1143/ptp/91.2.189

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We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.
We review recent results obtained for a class of one-dimensional quantum models with 1/r2 long-range interaction. Based on the asymptotic Bethe-ansatz solution and conformal field theory, we study critical properties of the continuum boson model, the SU(ν) spin chain, the OSp(ν, 1) supersymmetric t-J model and a new hierarchy of models related to the fractional quantum Hall effect. We further investigate the class of 1/r2 models with harmonic confinement by means of a newly proposed method of the renormalized-harmonic oscillator solution.
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Asymptotic Bethe-ansatz solutions are obtained for one-dimensional quantum systems with a 1r2\frac{1}{{r}^{2}}r21​ long-range interaction by generalizing Sutherland's method to multicomponent quantum systems. We obtain the solutions to the supersymmetric $t\ensuremath{-}J$ model of Kuramoto and Yokoyama, the $\mathrm{SU}(\ensuremath{\nu})$ Haldane-Shastry model, and the multicomponent $t\ensuremath{-}J$ model. The excitation spectrum as well as bulk quantities are computed analytically. Conformal properties of low-energy excitations are discussed in connection with Luttinger liquid theory.
A novel hierarchy of the one-dimensional SU(N) electron models with 1/r2{\mathit{r}}^{2}r2 interaction is proposed and solved by the asymptotic Bethe ansatz both for the continuum and lattice cases. The construction of the hierarchy is closely related to that for the fractional quantum Hall effect (FQHE) of the filling factor ${\ensuremath{\nu}}_{\mathit{c}}$=1/[p1{\mathit{p}}_{1}p1​-1/(p2{\mathit{p}}_{2}p2​-...-1/pN{\mathit{p}}_{\mathit{N}}pN​)...]. Under the chiral constraint the model describes the essential properties of the edge states for the FQHE with the above filling fraction. Furthermore the matrix deduced from the excitation spectrum characterizes the topological order of the FQHE state.
A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU(\ensuremath{\nu}) symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU(\ensuremath{\nu}) Lie algebra.