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arXiv:2507.17397

Dynamic correlations in Calogero-Sutherland model

G. Lleopart Motis, Dimitri M. Gangardt, Michael Pustilnik, G. E. Astrakharchik

Published Jul 23, 2025• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2507.17397

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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.
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In this paper, we demonstrate a recently developed technique which addresses the problem of obtaining nonuniversal prefactors of the correlation functions of one-dimensional (1D) systems at zero temperature. Our approach combines the effective field theory description of generic 1D quantum liquids with the finite-size scaling of form factors (matrix elements) which are obtained using microscopic techniques developed in the context of integrable models. We thus establish exact analytic forms for the prefactors of the long-distance behavior of equal-time correlation functions as well as prefactors of singularities of dynamic response functions. In this paper, our focus is on three specific integrable models: the Calogero-Sutherland, Lieb-Liniger, and XXZXXZXXZ models.
Recent ground-breaking experiments have realized strongly interacting quantum degenerate Fermi gas in a cold atomic system with tunable interactions. This has provided a table-top system which is extremely hydrodynamic in nature. This experimental realization helps us to investigate several aspects such as the interplay between nonlinearity, dissipation and dispersion. We find, for instance, that the dynamics in such a system shows near perfect agreement with a hydrodynamic theory. In collaboration with the group of John Thomas at Duke we interpreted studies of collision of two strongly interacting Fermi gases that led to shock waves which are a hallmark of nonlinear physics. Due to reasons such as the nature of interactions, higher dimensionality, these cold atomic systems are non-integrable and moreover the underlying field theory construction is mostly phenomenological in nature. On the other hand there are certain one-dimensional systems which are not only integrable but also facilitate more formal and rigorous ways of deriving the corresponding integrable field theories. One such family of models is the family of Calogero models (and their generalizations). They provide an extraordinary insight into the field of strongly correlated systems and hydrodynamics. We study the collective field theory of such models and address aspects of nonlinear physics such as Spin-Charge Interaction, Emptiness Formation Probability, Solitons etc; We derive a two-component nonlinear, nonlocal, integrable field theory. We also show that the Calogero family which is integrable even in an external harmonic trap (usually unavoidable in cold atom setups) is relatively short ranged thereby qualifying as a toy model for cold atom experiments. Transport in certain strongly correlated systems (impurity models) was studied using few low-dimensional techniques such as a 1/N diagrammatic expansion, Slave Boson Mean Field Theory and the Bethe Ansatz. A mesoscopic setup such as parallel quantum dots forms an ideal platform for such an investigation and comparison between different low-dimensional techniques. We studied transport, correlations and nature of the ground state of double quantum dots. We probed several non-perturbative aspects of this double-impurity model. For example, we showed that the RKKY interaction in closely spaced dots can be non-ferromagnetic due to its non-pertubative nature. This study helped us to point some discrepancies between different methods (such as the Numerical Renormalization Group). We give possible reasons for these discrepancies.
Ultracold bosonic and fermionic quantum gases confined to quasi-one-dimensional (1D) geometry are promising candidates for probing fundamental concepts of Luttinger liquid (LL) physics. They can also be exploited for devising applications in quantum information processing and precision measurements. Here, we focus on 1D dipolar Bose gases, where evidence of super-strong coupling behavior has been demonstrated by analyzing the low-energy static and dynamical structures of the fluid at zero temperature by a combined reptation quantum Monte Carlo (RQMC) and bosonization approach. Fingerprints of LL behavior emerge in the whole crossover from the already strongly interacting Tonks–Girardeau at low density to a dipolar density wave regime at high density. We have also shown that a LL framework can be effectively set up and utilized to describe this strongly correlated crossover physics in the case of confined 1D geometries after using the results for the homogeneous system in LL hydrodynamic equations within a local density approximation. This leads to the prediction of observable quantities such as the frequencies of the collective modes of the trapped dipolar gas under the more realistic conditions that could be found in ongoing experiments. The present paper provides a description of the theoretical framework in which the above results have been worked out, making available all the detailed derivations of the hydrodynamic Luttinger equations for the inhomogeneous trapped gas and of the correlation functions for the homogeneous system.
For many years, the Luttinger liquid theory has served as a useful paradigm for the description of one-dimensional (1D) quantum fluids in the limit of low energies. This theory is based on a linearization of the dispersion relation of the particles constituting the fluid. Recent progress in understanding 1D quantum fluids beyond the low-energy limit is reviewed, where the nonlinearity of the dispersion relation becomes essential. The novel methods which have been developed to tackle such systems combine phenomenology built on the ideas of the Fermi-edge singularity and the Fermi-liquid theory, perturbation theory in the interaction strength, and new ways of treating finite-size properties of integrable models. These methods can be applied to a wide variety of 1D fluids, from 1D spin liquids to electrons in quantum wires to cold atoms confined by 1D traps. Existing results for various dynamic correlation functions are reviewed, in particular, the dynamic structure factor and the spectral function. Moreover, it is shown how a dispersion nonlinearity leads to finite particle lifetimes and its impact on the transport properties of 1D systems at finite temperatures is discussed. The conventional Luttinger liquid theory is a special limit of the new theory, and the relation between the two is explained.