We construct a non-chiral conformal field theory (CFT) on the torus that accommodates a second quantization of the elliptic Calogero-Sutherland (eCS) model. We show that the CFT operator that provides this second quantization defines, at the same time, a quantum version of a soliton equation called the non-chiral intermediate long-wave (ncILW) equation. We also show that this CFT operator is a second quantization of a generalized eCS model which can describe arbitrary numbers of four different kinds of particles; we propose that these particles can be identified with solitons of the quantum ncILW equation.

We study Hartree-Fock theory at half-filling for the 3D anisotropic Hubbard model on a cubic lattice with hopping parameter t in the x- and y-directions and a possibly different hopping parameter t(z) in the z-direction; this model interpolates between the 2D and 3D Hubbard models corresponding to the limiting cases t(z )= 0 and t(z )= t, respectively. We first derive all-order asymptotic expansions for the density of states. Using these expansions and units such that t = 1, we analyze how the Neel temperature and the antiferromagnetic mean field depend on the coupling parameter, U, and on the hopping parameter t(z). We derive asymptotic formulas valid in the weak coupling regime, and we study in particular the transition from the three-dimensional to the two-dimensional model as t(z )-> 0. It is found that the asymptotic formulas are qualitatively different for t(z )= 0 (the two-dimensional case) and t(z )> 0 (the case of nonzero hopping in the z-direction). Our results show that certain universality features of the three-dimensional Hubbard model are lost in the limit t(z )-> 0 in which the three-dimensional model reduces to the two-dimensional model.

We present a generalization of the Mehler kernel, providing the exact analytic time evolution of the Calogero model that describes bosons or fermions propagating on the real line with inverse-square two-body interactions and harmonic confinement. The key to our result is a simple relation between the exact propagator of the Calogero model and a Baker-Akhiezer function, together with explicit combinatorial formulas for the latter function.

We discuss elliptic quantum Calogero-Moser-Sutherland models, including their relativistic generalizations due to Ruijsenaars and van Diejen, and the relations of these models to classes of special functions developed and explored in recent and on-going research efforts. We give an introduction to these models and corresponding special functions aimed at non-experts. We also describe a few open problems in the field.

We study the algebraic structure of electron density operators in gapless Weyl fermion systems in d=3,5,7, » spatial dimensions and in topological insulators (without any protecting symmetry) in d=4,6,8, » spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket - a generalization of commutator for more than two operators - of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the W1+∞ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a c-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry.

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Backlund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin-Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.

We study analytic solutions to the Bardeen-Cooper-Schrieffer (BCS) gap equation for isotropic superconductors with finite-range interaction potentials over the full range of temperatures from absolute zero to the superconducting critical temperature 0≤T≤Tc. Using these solutions Δ(ϵ,T), we provide a proof of the universality of the temperature dependence of the BCS gap ratio at the Fermi level Δ(ϵ=0,T)/Tc. Moreover, by examining the behavior of this ratio as a function of energy ϵ, we find that nonuniversal features emerge away from the Fermi level, and these features take the form of a temperature-independent multiplicative factor F(ϵ), which is equal to Δ(ϵ,T)/Δ(ϵ=0,T) up to exponentially small corrections, i.e., the error terms vanish like e-1/λ in the weak-coupling limit λ→0. We discuss the model-dependent features of both F(ϵ) and Tc, and we illustrate their behavior focusing on several concrete examples of physically relevant finite-range potentials. Comparing these cases for fixed coupling constants, we highlight the importance of the functional form of the interaction potential in determining the size of the critical temperature and provide guidelines for choosing potentials which lead to higher values of Tc. We also propose experimental signatures which could be used to probe the energy dependence of the gap and potentially shed light on the underlying mechanisms giving rise to superconductivity.

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product. 

Kajihara obtained in 2004 a remarkable transformation formula connecting multiple basic hypergeometric series associated with A-type root systems of different ranks. By specialisations of his formula, we deduce kernel identities for deformed Macdonald-Ruijsenaars (MR) and Noumi-Sano (NS) operators. The deformed MR operators were introduced by Sergeev and Veselov in the first order case and by Feigin and Silantyev in the higher order cases. As applications of our kernel identities, we prove that all of these operators pairwise commute and are simultaneously diagonalised by the super-Macdonald polynomials. We also provide an explicit description of the algebra generated by the deformed MR and/or NS operators by a Harish-Chandra type isomorphism and show that the deformed MR (NS) operators can be viewed as restrictions of inverse limits of ordinary MR (NS) operators.