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arXiv:cond-mat/0207301 · Journal of Physics A Mathematical and General

Correlation functions of the BC Calogero Sutherland model

Shinsuke M. Nishigaki, Dimitri M. Gangardt, Alex Kamenev

Published Mar 13, 2003• 18 citations
arXivHTMLPDF

Abstract

The BC-type Calogero–Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica σ-model formalism suitable to treat all chiral RMEs. By exploiting ‘generalized colour–flavour transformation’ we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c = 1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.

DOI
10.1088/0305-4470/36/12/316

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The BC-type Calogero–Sutherland model (CSM) is an integrable extension of the ordinary A-type CSM that possesses a reflection symmetry point. The BC-CSM is related to the chiral classes of random matrix ensembles (RMEs) in exactly the same way as the A-CSM is related to the Dyson classes. We first develop the fermionic replica σ-model formalism suitable to treat all chiral RMEs. By exploiting ‘generalized colour–flavour transformation’ we then extend the method to find the exact asymptotics of the BC-CSM density profile. Consistency of our result with the c = 1 Gaussian conformal field theory description is verified. The emerging Friedel oscillations structure and sum rules are discussed in details. We also compute the distribution of the particle nearest to the reflection point.
At coupling strengths \ensuremath{\lambda}=1/2, 1, or 2, the Calogero-Sutherland model (CSM) is related to Brownian motion in a Wigner-Dyson random matrix ensemble with orthogonal, unitary, or symplectic symmetry. Using this relation in conjunction with superanalytic techniques developed in mesoscopic conductor physics, we derive an exact integral representation for the CSM two-particle Green's function in the thermodynamic limit. Simple closed expressions for the single-particle Green's functions are extracted by separation of points. For the advanced part, where a particle is added to the ground state and later removed, a sum of two contributions is found: the expected one, with just one particle excitation present, plus an extra term arising from fractionalization of the single particle into a number of elementary particle and hole excitations.
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.
We study correlation functions of the Calogero-Sutherland model in the whole range of the interaction parameter. Using the replica method we obtain analytical expressions for the long-distance asymptotics of the one-body density matrix in addition to the previously derived asymptotics of the pair-distribution function [D.M. Gangardt and A. Kamenev, Nucl. Phys. B 610, 578 (2001)]. The leading analytic and nonanalytic terms in the short-distance expansion of the one-body density matrix are discussed. Numerical results for these correlation functions are obtained using Monte Carlo techniques for all distances. The momentum distribution and static structure factor are calculated. The potential and kinetic energies are obtained using the Hellmann-Feynman theorem. Perfect agreement is found between the analytical expressions and numerical data. These results allow for the description of physical regimes of the Calogero-Sutherland model. The zero temperature phase diagram is found to be of a crossover type and includes quasicondensation, quasicrystallization and quasisupersolid regimes.
Spin generalization of the relativistic Calogero - Sutherland model is constructed by using the affine Hecke algebra and shown to possess the quantum affine symmetry . The spinless model is exactly diagonalized by means of the Macdonald symmetric polynomials. The dynamical density - density correlation function, as well as the one-particle Green function, are evaluated exactly. We also investigate the finite-size scaling of the model and show that the low-energy behaviour is described by the C = 1 Gaussian theory with a new selection rule. The results indicate that the excitations obey the fractional exclusion statistics and also exhibit the Tomonaga - Luttinger liquid behaviour.
Using the collective field technique, we give the description of the spin Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates. JSPS fellow
The effective extended conformal field theory with symmetry [Formula: see text] that describes the thermodynamic limit of the Calogero–Sutherland (CS) model is considered. The dynamic structure factor of the chiral component in the repulsive regime is determined and compared with the corresponding one of the free bosonic theory, given that both share isomorphic Hilbert spaces but differ in the time evolution of quantum states. In either case, a sharp response function peaked at only one resonant frequency is found, and the physical implications of this outcome are addressed. Furthermore, a detailed comparison between this result and the corresponding one obtained in the first quantized formulation of the CS model is provided. Complete agreement in the parameter region in which both results overlap is found. This outcome provides further support to the equivalence between the first and second quantized treatments of the CS model, in which the computational advantages from integrability in the first formulation are paralleled by those from the algebraic structure of the second.