Alexios P. Polychronakos

Lattice integrable systems of Haldane-Shastry type.

We present a new lattice integrable system in one dimension of the Haldane-Shastry type. It consists of spins positioned at the static equilibrium positions of particles in a corresponding classical Calogero system and interacting through an exchange term with strength inversely proportional to the square of their distance. We achieve this by viewing the Haldane-Shastry system as a high-interaction limit of the Sutherland system of particles with internal degrees of freedom and identifying the same limit in a corresponding Calogero system. The commuting integrals of motion of this system are found using the exchange operator formalism.

Integrable systems for particles with internal degrees of freedom

Abstract We show that a class of models for particles with internal degrees of freedom are integrable. These systems are basically generalizations of the models of Calogero and Sutherland. The proofs of integrability are based on a recently developed exchange operator formalism. This allows us to construct the invariants of the motion for these systems. We also calculate the wave-functions for the Calogero-like models in the ferromagnetic and antiferromagnetic regimes.

Equivalence of two-dimensional QCD and the c = 1 matrix model

Abstract We consider two-dimensional QCD with the spatial dimension compactified to a circle. We show that the states in the theory consist of interacting strings that wind around the circle and derive the Hamiltonian for this theory in the large N limit, complete with interactions. Mapping the winding states into momentum states, we express this Hamiltonian in terms of a continuous field. For a U( N ) gauge with a background source of Wilson loops, we recover the collective field Hamiltonian found by Das and Jevicki for the c = 1 matrix model, except the spatial coordinate is on a circle. We then proceed to show that two-dimensional QCD with a U( N ) gauge group can be reduced to a one-dimensional unitary matrix model and is hence equivalent to a theory of N free nonrelativistic fermions on a circle. A similar result is true for the group SU( N ), but the fermions must be modded out by the center of mass coordinate.

Generalized Statistics in One Dimension

An exposition of the different definitions and approaches to quantum statistics is given, with emphasis in one-dimensional situations. Permutation statistics, scattering statistics and exclusion statistics are analyzed. The Calogero model, matrix model and spin chain models constitute specific realizations.

Exact spectrum of SU(n) spin chain with inverse-square exchange

The spectrum and partition function of a nonuniform spin chain with long-range interactions are derived. The model consists of SU(n) spins positioned at the equilibrium positions of a classical Calogero model and interacting with an external magnetic field and through mutual exchange terms with strength inversely proportional to the square of their distance. The energy levels are found to be equidistant and to have a high degree of degeneracy, with several SU(n) multiplets grouping into the same energy eigenspace. The partition function takes the form of a q-deformed polynomial. This leads to a description of the system in terms of an effective parafermionic hamiltonian, and to a classification of the states in terms of “modules” consisting of base-n string of integers.

Interacting fermion systems from two-dimensional QCD

Abstract We consider two-dimensional U ( N ) QCD on the cylinder with a time-like Wilson line in an arbitrary representation. We show that the theory is equivalent to N fermions with internal degrees of freedom, which interact among themselves with a generalized Sutherland-type interaction. By evaluating the expectation value of the Wilson line in the original theory we explicitly find the spectrum and degeneracies of these particle systems.

WAVES AND SOLITONS IN THE CONTINUUM LIMIT OF THE CALOGERO-SUTHERLAND MODEL

We examine a collection of classical particles interacting with inverse-square two-body potentials in the thermodynamic limit of finite particle density. We find explicit large-amplitude density waves and soliton solutions for the motion of the system. Waves can be constructed as coherent states of either solitons or phonons (small-amplitude waves). Therefore, either solitons or phonons can be considered as the fundamental excitations. The generic wave is shown to correspond to a two-band state in the quantum description of the system, while the limiting cases of solitons and phonons correspond to particle and hole excitations.

Classical solutions for two-dimensional QCD on the sphere

Abstract We consider U( N ) and SU( N ) gauge theory on the sphere. We express the problem in terms of a matrix element of N free fermions on a circle. This allows us to find an alternative way to show Wittens result that the partition function is a sum over classical saddle points. We then show how the phase transition of Douglas and Kazakov occurs from this point of view. By generalizing the work of Douglas and Kazakov, we find other “stringy” solutions for the U( N ) case in the large- N limit. Each solution is described by a net U(1) charge. We derive a relation for the maximum charge for a given area and we also describe the critical behaviour for these new solutions. Finally, we describe solutions for lattice SU( N ) which are in a sense dual to the continuum U( N ) solutions.

On the electromagnetic interactions of anyons

Abstract Using the appropriate representation of the Poincare group and a definition of minimal coupling, we discuss some aspects of the electromagnetic interactions of charged anyons. In a nonrelativistic expansion, we derive a Schrodinger-type equation for the anyon wave function which includes spin-magnetic field and spin-orbit couplings. In particular, the gyromagnetic ratio for charged anyons is shown to be 2; this last result is essentially a reflection of the fact that the spin is parallel to the momentum in (2 + 1) dimensions.

Density-correlation functions in Calogero-Sutherland models

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 [ital l] and 1/[ital l], where [ital l] is an integer. We present overwhelming evidence that the conjecture is indeed correct.