Calogero-Sutherland Model Research Articles

Effect of magnetic flux on the critical behavior of a system with long-range hopping

We study the effect of a magnetic flux in a 1D disordered wire with long range hopping. It is shown that this model is at the metal-insulator transition (MIT) for all disorder values and the spectral correlations are given by critical statistics. In the weak disorder regime a smooth transition between orthogonal and unitary symmetry is observed as the flux strength increases. By contrast, in the strong disorder regime the spectral correlations are almost flux independent. It is also conjectured that the two level correlation function for arbitrary flux is given by the dynamical density-density correlations of the Calogero-Sutherland (CS) model at finite temperature. Finally we describe the classical dynamics of the model and its relevance to quantum chaos.

Second Quantization of the Elliptic Calogero-Sutherland Model

We construct a quantum field theory model of anyons on a circle and at finite temperature. We find an anyon Hamiltonian providing a second quantization of the elliptic Calogero-Sutherland model. This allows us to prove a remarkable identity which is a starting point for an algorithm to construct eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland Hamiltonian.

Long-range interactions in the quantum many-body problem in one dimension: ground state.

We investigate the ground state properties of a family of N-body systems in one dimension, trapped in a polynomial potential and having long-range two-body interaction in addition to the inverse square potential studied in the Calogero-Sutherland model (CSM). We show that for such a Hamiltonian, the ground state energy is similar to that of free fermions in a harmonic well with a displacement that depends on the number of particles and depth of the well. We obtain the ground state wave function and using random matrix results, study the particle density and pair correlation function (PCF). We observe that the particles are arranged in bands. Due to the presence of long-range interaction, the PCF shows a departure from the CSM.

On the quasiparticle description ofc= 1 CFTs

We show that the description of $c=1$ Conformal Field Theory in terms of quasiparticles satisfying fractional statistics can be obtained from the sine-Gordon model with a chemical potential $A$, in the limit where $A \gg M$. These quasiparticles are related to the excitations of the Calogero-Sutherland (CS) model. We provide a direct calculation of their 2-particle S-matrix using Korepin's method. We also reconsider the computation of the CS S-matrix in terms of particles with fractional charge.

Calogero–Sutherland model and bulk-boundary correlations in conformal field theory

We show that, in any conformal field theory, the weights of all bulk primary fields that couple to Nφ2,1 fields on the boundary are given by the spectrum of an N-particle Calogero–Sutherland model. The corresponding correlation function is simply related to the N-particle wave function. Applications are discussed to the minimal models and the non-unitary O(n) model.

Sea-boson theory of Landau-Fermi liquids, Luttinger liquids and Wigner crystals

It is shown how Luttinger liquids may be studied using sea-bosons. The main advantage of the sea-boson method is its ability to provide information about short-wavelength physics in addition to the asymptotics and is naturally generalisable to more than one dimension. In this article, we solve the Luttinger model and the Calogero-Sutherland model, the latter in the weak-coupling limit. The anomalous exponent we obtain in the former case is identical to the one obtained by Mattis and Lieb. We also apply this method to solve the two-dimensional analog of the Luttinger model and show that the system is a Landau Fermi liquid. Then we solve the model of spinless fermions in one-dimension with long-range (gauge) interactions and map the Wigner crystal phase of the system.

Applications of Ideas from Random Matrix Theory to Step Distributions on “Misoriented” Surfaces

Arising as a fluctuation phenomenon, the equilibrium distribution of meandering steps with mean separation $<\ell>$ on a "tilted" surface can be fruitfully analyzed using results from RMT. The set of step configurations in 2D can be mapped onto the world lines of spinless fermions in 1+1D using the Calogero-Sutherland model. The strength of the ("instantaneous", inverse-square) elastic repulsion between steps, in dimensionless form, is $\beta(\beta-2)/4$. The distribution of spacings $s< \ell>$ between neighboring steps (analogous to the normalized spacings of energy levels) is well described by a {\it "generalized" Wigner surmise}: $p_{\beta}(0,s) \approx a s^{\beta}\exp(-b s^2)$. The value of $\beta$ is taken to best fit the data; typically $2 \le \beta \le 10$. The procedure is superior to conventional Gaussian and mean-field approaches, and progress is being made on formal justification. Furthermore, the theoretically simpler step-step distribution function can be measured and analyzed based on exact results. Formal results and applications to experiments on metals and semiconductors are summarized, along with open questions. (conference abstract)

Some results on the eigenfunctions of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra D4

We express the Hamiltonian of the quantum trigonometric Calogero–Sutherland model related to the Lie algebra D4 in terms of a set of Weyl-invariant variables, namely, the characters of the fundamental representations of the Lie algebra. This parametrization allows us to solve for the energy eigenfunctions of the theory and to study properties of the system of orthogonal polynomials associated with them such as recurrence relations and generating functions.

Finite-temperature structure factor in the Haldane–Shastry spin chain

The Haldane-Shastry spin chain can be mapped to the infinite coupling limit of the SU(2) spin Calogero-Sutherland model. We use the $\mathfrak{gl}_2$ Jack polynomials' technology to compute the form factors of the spin operator on the multi-spinon spectrum. The spin structure factor is obtained through a form factor expansion. The expansion is proven to converge in the small momentum limit. Numerics based on two- and four-spinons contributions give an approximate result for the infinite temperature static and dynamic spin structure factor.

Unified algebraic approach to few- and many-body correlated systems

The present paper is an extended version of another paper [Phys. Rev. B 59, R2490 (1999)], where we have established the equivalence of the Calogero-Sutherland model to decoupled oscillators. Here, we first employ the same approach for finding the eigenstates of a large class of Hamiltonians, dealing with correlated systems. A number of few- and many-body interacting models are studied and the relationship between their respective Hilbert spaces, with that of oscillators, is found. This connection is then used to obtain the spectrum generating algebras for these systems and make an algebraic statement about correlated systems. The procedure to generate solvable interacting models is outlined. We then point out the inadequacies of the present technique and make use of a method for solving linear differential equations to diagonalize the Sutherland model and establish a precise connection between this correlated system's wave functions, with those of the free particles on a circle. In the process, we obtain an expression for the Jack polynomials. In two dimensions, we analyze the Hamiltonian having Laughlin wave function as the ground state and point out the natural emergence of the underlying linear ${W}_{1+\ensuremath{\infty}}$ symmetry in this approach.

Critical statistics in quantum chaos and Calogero-Sutherland model at finite temperature.

We investigate the spectral properties of a generalized Gaussian orthogonal ensemble capable of describing critical statistics. The joint distribution of eigenvalues of this model is expressed as the diagonal element of the density matrix of a gas of particles governed by the Calogero-Sutherland (CS) Hamiltonian. Taking advantage of the correspondence between CS particles and eigenvalues, and utilizing a recently conjectured expression by Kravtsov and Tsvelik for the finite temperature density-density correlations of the CS model, we show that the number variance of our random matrix model is asymptotically linear with a slope depending on the parameters of the model. Such linear behavior is a signature of critical statistics. This random matrix model may be relevant for the description of spectral correlations of complex quantum systems with a self-similar or fractal Poincaré section of its classical counterpart. This is shown in detail for two examples: the anisotropic Kepler problem and a kicked particle in a well potential. In both cases the number variance and the Delta(3) statistic are accurately described by our analytical results.

Multiple algebraisations of an elliptic Calogero–Sutherland model

Recently, Gómez-Ullate et al. [Phys. Lett. B 511, 112 (2001)] have studied an N-particle quantum problem with elliptic-function potentials. They have shown that the Hamiltonian operator preserves a finite dimensional space of functions and as such is quasi-exactly solvable (QES). In this article we show that other types of invariant function spaces exist, which are in close relation to the algebraic properties of the elliptic functions. Accordingly, series of new algebraic eigenfunctions can be constructed.

Correlation measures of the Calogero–Sutherland model at T=0

We are exploiting the results of the exact solution for the ground state of the one-dimensional spinless quantum gas of Fermions with μ/ x ij 2 particle–particle interaction, and analyze particle-number fluctuations for μ=− 1 4 , 0, and 2 . These are suppressed by repulsive interaction ( μ>0), enhanced by attraction ( μ<0), and may therefore measure the kind and strength of correlation. Other recently proposed purely quantum-kinematical measures of the correlation strength arise from the small-separation behavior of the pair density or from the non-idempotency of the momentum distribution and its large-momenta behavior.

Correlation functions of the BC Calogero Sutherland model

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-CS Mi sr elated 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 develo pt he 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 = 1G aussian 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.

A perturbative approach to the quantum elliptic Calogero–Sutherland model

We solve perturbatively the quantum elliptic Calogero–Sutherland model in the regime in which the quotient between the real and imaginary semiperiods of the Weierstrass P function is small.

Asymptotic Bethe ansatz for the Sutherland–Römer model with open boundary conditions

By constructing the reflection spin-Dunkl operators, the integrable Sutherland–Römer model (SRM) with open boundary condition is established, which describes a one-dimensional, two-component, quantum many-particle system in which like particles interact with a pair potential g( g+1)/sinh 2( r), while unlike particles interact with a pair potential − g( g+1)/cosh 2( r). By solving the Schrödinger equation and using the properties of the hypergeometric functions and gamma functions, the two-particle scattering matrix and the reflection matrix are obtained in the framework of the asymptotic Bethe ansatz method. The Bethe ansatz equations of the system are obtained. The Hamiltonians of SRM with some other open boundary conditions are expressed explicitly. Our method can be generalized, as a example, to the boundary Calogero–Sutherland model which is also constructed by the reflection spin-Dunkl operators.

Exact solution of a two-component Calogero-Sutherland model with molecular field

By constructing the spin-Dunkl operators, a two-component Calogero-Sutherland model with molecular field is established. The integrability of the system is proved by constructing the conserved quantities. With the use of the asymptotic Bethe ansatz method, the exact solution of the model is derived and the thermodynamics of the rational limit is discussed.

The Asymptotic Dynamics of two-dimensional (anti-)de Sitter Gravity

We show that the asymptotic dynamics of two-dimensional de Sitter or anti-de Sitter Jackiw-Teitelboim (JT) gravity is described by a generalized two-particle Calogero-Sutherland model. This correspondence is established by formulating the JT model of (A)dS gravity in two dimensions as a topological gauge theory, which reduces to a nonlinear 0+1-dimensional sigma model on the boundary of (A)dS space. The appearance of cyclic coordinates allows then a further reduction to the Calogero-Sutherland quantum mechanical model.

Duality and quasiparticles in the Calogero-Sutherland model: Some exact results

The quantum-mechanical many-body system with the potential proportional to the pairwise inverse-square distance possesses a strong-weak coupling duality. Based on this duality, particle and/or quasiparticle states are described as SU(1,1) coherent states. The constructed quasiparticle states are of hierarchical nature.

COHERENT STATES AND GEOMETRIC PHASES IN THE CALOGERO–SUTHERLAND MODEL

Exact coherent states in the Calogero–Sutherland models (of time-dependent parameters) which describe identical harmonic oscillators interacting through inverse-square potentials are constructed, in terms of the classical solutions of a harmonic oscillator. For quasi-periodic coherent states of the time-periodic systems, geometric phases are evaluated. For the AN-1 Calogero–Sutherland model, the phase is calculated for a general coherent state. The phases for other models are also considered.