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arXiv:cond-mat/9510012 · Physical Review Letters

Random Matrix Model and the Calogero-Sutherland Model: A Novel Current-Density Mapping

Nobuhiko Taniguchi, B. Sriram Shastry, B. L. Altshuler

Published Nov 13, 1995• 28 citations
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Abstract

We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths $\ensuremath{\lambda}=\frac{1}{2}, 1, \mathrm{and} 2$, we explore the local current-density mapping applicable to arbitrary $\ensuremath{\lambda}$ including irrational values, which results from the novel structure of the Calogero-Sutherland model. We find an interesting and novel relationship between equal time current and density correlations for any coupling, which exist in addition to the usual Ward identities for this class of systems.

DOI
10.1103/physrevlett.75.3724

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arXiv:cond-mat/9705279•May 27, 1997•0 citations
We investigate the relation between the invariant correlators of random matrix theory and correlators of the integrable one-dimensional systems. Starting from the relation between correlators for the coupling strengths \ensuremathλ=12,1,and2\ensuremath{\lambda}=\frac{1}{2}, 1, \mathrm{and} 2\ensuremathλ=21​,1,and2, we explore the local current-density mapping applicable to arbitrary \ensuremathλ\ensuremath{\lambda}\ensuremathλ including irrational values, which results from the novel structure of the Calogero-Sutherland model. We find an interesting and novel relationship between equal time current and density correlations for any coupling, which exist in addition to the usual Ward identities for this class of systems.
By utilizing the symmetric property known as the Ward-Takahashi identity in disordered systems, we explore the novel symmetry relations which hold in one-dimensional systems with inverse square interaction (the Calogero-Sutherland model). The identities emerge totally from the algebraic structure of the model. They show that the dynamical correlators are connected with one another, involving the higher-order integrals of motion. We obtain the result for the coupling strengths \ensuremath{\lambda}=1/2, 1, and 2, and conjecture that a similar relation may hold for arbitrary rational \ensuremath{\lambda}.
The Calogero-Sutherland model represents a paradigmatic example of an integrable quantum system with applications ranging from cold atoms to random matrix theory. Combining sum rules with the Monte Carlo technique, we introduce a stochastic method that allows one to compute the dynamic structure factor and obtain an exact description of excitations beyond the conventional Luttinger liquid regime. We explore a broad range of interaction regimes, including weak interactions, where a Bogoliubov-type spectrum emerges, the Tonks-Girardeau regime, where excitations resemble those of an ideal Fermi gas, and strong interactions, where umklapp scattering leads to a Brillouin zone structure, typical of a crystal. Additionally, we discuss the connection between the hydrodynamic description of one-dimensional quantum gases, liquids, and solids with the Calogero-Sutherland wave function. The model's universality extends beyond atoms in waveguides, with implications for disordered systems and random matrix theory.
Abstract We study a class of random matrices arising from the Lax matrix structure of classical integrable systems, particularly the Calogero family of models. Our focus is the density of eigenvalues for these random matrices. The problem can be mapped to analyzing the density of eigenvalues for generalized versions of conventional random matrix ensembles, including a modified form of the log-gas. The mapping comes from the underlying integrable structure of these models. Such deep connection is confirmed by extensive Monte-Carlo simulations. Thereby we move forward not only in terms of understanding such a class of random matrices arising from integrable many-body systems but also by providing a building block for the generalized hydrodynamic description of integrable systems.
We obtain the exact ground state for the Calogero-Sutherland problem in arbitrary dimensions. In the special case of two dimensions, we show that the problem is connected to the random matrix problem for complex matrices, provided the strength of the inverse-square interaction g = 2. In the thermodynamic limit, we obtain the ground state energy and the pair-correlation function and show that in this case there is no long-range order. Calogero-Sutherland type models are a class of exactly solvable models. Starting from the inception [1], till today, these models continue to be of interest, especially due to their exact solvability [2]. Moreover, these models are related to (1 + 1)-dimensional conformal field theory, random matrices and a host of other things [3]. Some variants of the models have attracted attention in very recent past [3–5]. The original model of Calogero [1] described N particles in one dimension, interacting through an inverse square law and a two-body harmonic interaction. The latter interaction was added to the effect of complete discretization of the spectrum. This model gave in to an analytic treatment yielding an exact solution. Sutherland [6] considered a variant of this problem where the harmonic interaction was replaced by a harmonic well containing the N particles. It was shown that one can exactly solve for the ground state of the system. The norm of the ground state wave-function was shown to be of a form that coincides with the joint probability density function of the eigenvalues of random matrices from an orthogonal, unitary or symplectic Gaussian ensemble, provided, the
For a one-dimensional model in which the two-body interactions are long-range and strong, the system almost crystallizes. The harmonic modes of such a lattice were used by Krivnov and Ovchinnikov to compute the ground-state wave function and the dynamical density-density correlations. We review this method, and apply it to the Calogero-Sutherland model, whose density-density correlation functions are exactly known for certain values of the coupling constant. We show numerically that the correlations obtained are quite accurate even if the coupling is not very large. Such comparisons have been made earlier by Forrester. The lattice method is considerably simpler than the ones used to derive the exact results, and yields expressions for the correlations- which are easily plotted. The equal-time correlations can be expanded in inverse powers of coupling; we show that the two leading order terms agree with the exact results which are known for integer values of the coupling. The strength-dependent power law fall-off is typical of a Luttinger liquid.In a general one-dimensional model where the two-body interaction decreases as a power of the relative distance, we argue, following Schulz, that at zero temperature the system behaves as a Luttinger liquid if the power exceeds 1, and as a Wigner crystal if it is less than 1.PACS Nos.: 63.20-e, 71.10Pm
For a one-dimensional model in which the two-body interactions are long-range and strong, the system almost crystallizes. The harmonic modes of such a lattice can be used to compute the ground state wave function and the dynamical density-density correlations. We use this method to calculate the density corrrelations in the Calogero-Sutherland model. We show numerically that the correlations obtained are quite accurate even if the coupling is not very large. Our method is considerably simpler than the ones used to derive the exact results, and yields expressions for the correlations which are easily plotted. Our equal-time correlations can be expanded in powers of the inverse coupling; we show that the two leading order terms agree with the exact results which are known for integer values of the coupling.