Loop Groups, Anyons and the Calogero–Sutherland Model

The positive energy representations of the loop group of U(1) are used to construct a boson-anyon correspondence. We compute all the correlation functions of our anyon fields and study an anyonic W-algebra of unbounded operators with a common dense domain. This algebra contains an operator with peculiar exchange relations with the anyon fields. This operator can be interpreted as a second quantized Calogero–Sutherland (CS) Hamiltonian and may be used to solve the CS model. In particular, we inductively construct all eigenfunctions of the CS model from anyon correlation functions, for all particle numbers and positive couplings. Received: 12 May 1998 / Accepted: 4 August 1998     

On the Eigenstates of the Elliptic Calogero–Moser Model

It is known that the trigonometric Calogero–Sutherland model is obtained by the trigonometric limit (–1) of the elliptic Calogero–Moser model, where (1, ) is a basic period of the elliptic function. We show that for all square-integrable eigenstates and eigenvalues of the Hamiltonian of the Calogero–Sutherland model, if exp(2–1) is small enough then there exist square-integrable eigenstates and eigenvalues of the Hamiltonian of the elliptic Calogero–Moser model which converge to the ones of the Calogero–Sutherland model for the 2-particle and the coupling constant l is positive integer cases and the 3-particle and l=1 case. In other words, we justify the regular perturbation with respect to the parameter exp(2–1). With some assumptions, we show analogous results for N-particle and l is positive integer cases.     

Source Identity and Kernel Functions for Elliptic Calogero–Sutherland Type Systems

Kernel functions related to quantum many-body systems of Calogero–Sutherland type are discussed, in particular for the elliptic case. The main result is an elliptic generalization of an identity due to Sen that is a source for many such kernel functions. Applications are given, including simple exact eigenfunctions and corresponding eigenvalues of Chalykh–Feigin–Veselov–Sergeev-type deformations of the elliptic Calogero–Sutherland model for special parameter values.     

Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

A new generalization of the Jack polynomials that incorporates fermionic variables is presented. These Jack superpolynomials are constructed as those eigenfunctions of the supersymmetric extension of the trigonometric Calogero–Moser–Sutherland (CMS) model that decomposes triangularly in terms of the symmetric monomial superfunctions. Many explicit examples are displayed. Furthermore, various new results have been obtained for the supersymmetric version of the CMS models: the Lax formulation, the construction of the Dunkl operators and the explicit expressions for the conserved charges. The reformulation of the models in terms of the exchange-operator formalism is a crucial aspect of our analysis.     

AN-Type Dunkl Operators and New Spin Calogero–Sutherland Models

A new family of A _N -type Dunkl operators preserving a polynomial subspace of finite dimension is constructed. Using a general quadratic combination of these operators and the usual Dunkl operators, several new families of exactly and quasi-exactly solvable quantum spin Calogero–Sutherland models are obtained. These include, in particular, three families of quasi-exactly solvable elliptic spin Hamiltonians. Received: 17 February 2001 / Accepted: 8 March 2001     

Conformal blocks in Virasoro and W theories: Duality and the Calogero–Sutherland model

We study the properties of the conformal blocks of the conformal field theories with Virasoro or W-extended symmetry. When the conformal blocks contain only second-order degenerate fields, the conformal blocks obey second order differential equations and they can be interpreted as ground-state wave functions of a trigonometric Calogero–Sutherland Hamiltonian with non-trivial braiding properties. A generalized duality property relates the two types of second order degenerate fields. By studying this duality we found that the excited states of the Calogero–Sutherland Hamiltonian are characterized by two partitions, or in the case of _WAk−1${\mathrm{WA}}_{k-1}$ theories by k   partitions. By extending the conformal field theories under consideration by a u(1)$u(1)$ field, we find that we can put in correspondence the states in the Hilbert state of the extended CFT with the excited non-polynomial eigenstates of the Calogero–Sutherland Hamiltonian. When the action of the Calogero–Sutherland integrals of motion is translated on the Hilbert space, they become identical to the integrals of motion recently discovered by Alba, Fateev, Litvinov and Tarnopolsky in Liouville theory in the context of the AGT conjecture. Upon bosonization, these integrals of motion can be expressed as a sum of two, or in general k, bosonic Calogero–Sutherland Hamiltonian coupled by an interaction term with a triangular structure. For special values of the coupling constant, the conformal blocks can be expressed in terms of Jack polynomials with pairing properties, and they give electron wave functions for special Fractional Quantum Hall states.     

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 $\text{P}$ function is small.     

Some Connections Between the Classical Calogero–Moser Model and the Log-Gas

Journal of Statistical Physics - In this work we discuss connections between a one-dimensional system of N particles interacting with a repulsive inverse square potential and confined in a harmonic...     

Calogero–Sutherland system with two types interacting spins

The dynamics of low-temperature (T = 5 K) photoluminescence spectra of Si/Si_1-x Ge _x /Si heterostructures (x = 0.045) under the influence of a stream of nonequilibrium phonons (heat pulses) propagating in the structure is investigated. The rapid evaporation of the electron–hole liquid in the quantum well of the structure is observed as the liquid is heated by nonequilibrium phonons. It is established that an increase in the exciton-gas density in the quantum well is caused by the evaporation of the electron–hole liquid and by an increase in the rate of exciton capture by the quantum well. It is shown that the interaction with nonequilibrium phonons results in the dissociation of bound-exciton complexes in the Si layers, which is accompanied by an increase in the exciton concentration and lifetime.     

A New Model in the Calogero–Ruijsenaars Family

Hamiltonian reduction is used to project a trivially integrable system on the Heisenberg double of SU(n, n), to obtain a system of Ruijsenaars type on a suitable quotient space. This system possesses BC _n symmetry and is shown to be equivalent to the standard three-parameter BC _n hyperbolic Sutherland model in the cotangent bundle limit.     

On the eigenfunctions of hyperbolic quantum Calogero–Moser–Sutherland systems in a Morse potential

Letters in Mathematical Physics - We zoom in on special instances of recently found eigenfunctions for the hyperbolic quantum Calogero–Moser–Sutherland n-particle system in a Morse...     

Exact solutions of a new elliptic Calogero–Sutherland model

A quantum Hamiltonian describing N particles on a line interacting pairwise via an elliptic function potential in the presence of an external field is introduced. For a discrete set of values of the strength of the external potential, it is shown that a finite number of eigenfunctions and eigenvalues of the model can be exactly computed in an algebraic way.     

Calogero–Moser Systems and Hitchin Systems

We exhibit the elliptic Calogero–Moser system as a Hitchin system of G-principal Higgs pairs. The group G, though naturally associated to any root system, is not semi-simple. We then interpret the Lax pairs with spectral parameter of d'Hoker and Phong [dP1] and Bordner, Corrigan and Sasaki [BCS1] in terms of equivariant embeddings of the Hitchin system of G into that of GL(N). Received: 8 May 2000 / Accepted: 2 July 2001     

Quantum Spin Liquid Phase in the Shastry–Sutherland Model Detected by an Improved Level Spectroscopic Method

We study the spin-1/2 two-dimensional Shastry–Sutherland spin model by exact diagonalization of clusters with periodic boundary conditions, developing an improved level spectroscopic technique using energy gaps between states with different quantum numbers. The crossing points of some of the relative(composite) gaps have much weaker finite-size drifts than the normally used gaps defined only with respect to the ground state, thus allowing precise determination of quantum critical points even wit...     

Trigonometric and Elliptic Ruijsenaars–Schneider Systems on the Complex Projective Space

We present a direct construction of compact real forms of the trigonometric and elliptic \({n}\)-particle Ruijsenaars–Schneider systems whose completed center-of-mass phase space is the complex projective space \({{\mathbb{CP}}^{n-1}}\) with the Fubini–Study symplectic structure. These systems are labeled by an integer \({p\in\{1,\ldots,n-1\}}\) relative prime to \({n}\) and a coupling parameter \({y}\) varying in a certain punctured interval around \({p\pi/n}\). Our work extends Ruijsenaars’s pioneering study of compactifications that imposed the restriction \({0 < y < \pi/n}\), and also builds on an earlier derivation of more general compact trigonometric systems by Hamiltonian reduction.     

Integrability of Calogero–Coulomb problems

In this short review we describe the integrability properties of the Calogero-type perturbations of one- and two-center Coulomb problems and of the Stark–Coulomb problem. We present the explicit expressions of their constants of motion and show that these systems admit partial separation of variables.     

New Elliptic Solutions of the Yang–Baxter Equation

We consider two finite index endomorphisms \({\rho}\), \({\sigma}\) of any AFD factor M. We characterize the condition for there being a sequence \({\{ u_n\}}\) of unitaries of the factor M with \({\mathrm{Ad}u_n \circ \rho \to \sigma}\). The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of AFD factors, obtained by Kawahiashi–Sutherland–Takesaki and Masuda–Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.     

New Formulas for the Eigenfunctions of the Two-Particle Difference Calogero–Moser System

We present a new proof of the integrability of the DDPT-I equation. The DDPT-I equation represents a functional-difference deformation of the well-known Darboux–Pöschl–Teller equation. The proof is based on some formula for special Casorati determinants established in the paper. This formula provides some new representation for the DDPT-I potentials and for the general solution for the DDPT-I equation. It allows also a very easy computation of the action of the difference KdV flow on the DDPT-I initial data. In other words we obtain the new formulas for the eigenfunctions of the Hamiltonians of the two-particle difference BC ₁ Calogero–Moser system also known as quantum relativistic Calogero–Moser, (QRCM), system.