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arXiv:hep-th/9404192 · Physical review. B, Condensed matter

Density-correlation functions in Calogero-Sutherland models

Joseph A. Minahan, Alexios P. Polychronakos

Published Aug 1, 1994• 69 citations
arXivHTMLPDF

Abstract

Using arguments from two-dimensional Yang-Mills theory and the collective coordinate formulation of the Calogero-Sutherland model, we conjecture the dynamical density-correlation function for coupling l and 1/l, where l is an integer. We present overwhelming evidence that the conjecture is indeed correct.
DOI
10.1103/physrevb.50.4236

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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.
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
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.