International Journal of Modern Physics B

In this paper, we introduce the cyclic basic hypergeometric series _{p + 1}Φ_p with q → ω where ω^N = 1. This is a terminating series with N terms, whose summand has period N. We show how the Fourier transform of the weights of the integrable chiral Potts model are related to the ₂Φ₁, which is summable. We show that ₃Φ₂ satisfies certain transformation formulae. We then show that the Saalschützian ₄Φ₃ series is summable at argument z = ω. This then gives the simplest proof of the star-triangle relation in the chiral Potts model. Finally, we let N → ∞, where the star-triangle equation becomes a two-sided identity for the hypergeometric series.

We find raising and lowering operators distinguishing the degenerate states for the Hamiltonian at x = ± 1 for spin 1 that was given by Happer et al.^1,2 to interpret the curious degeneracies of the Zeeman effect for condensed vapor of ⁸⁷Rb. The operators obey Yangian commutation relations. We show that the curious degeneracies seem to verify the Yangian algebraic structure for quantum tensor space and are consistent with the representation theory of Y(sl(2)).

Here we review a method for constructing exact eigenvalues and eigenfunctions of a many-particle quantum system, which is obtained by adding some nonhermitian but PT invariant (i.e., combined parity and time reversal invariant) interaction to the Calogero model. It is shown that such extended Calogero model leads to a real spectrum obeying generalised exclusion statistics. It is also found that the corresponding exchange statistics parameter differs from the exclusion statistics parameter and exhibits a 'reflection symmetry' provided the strength of the PT invariant interaction exceeds a critical value.

We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O(n = 1) model we find the number A_V(2m + 1;3), which 3-enumerates vertically symmetric alternating sign matrices.

We use Monte Carlo, transfer-matrix and finite-size scaling methods to investigate two-dimensional O(n) models with n > 2, in particular the case n = 3 which includes the classical Heisenberg model. Depending on the type of interaction and the lattice structure, two different types of phase transitions are present. One type resembles the hard-hexagon transition and occurs in the loop representation of the honeycomb O(n) model. The other type is a first-order transition which occurs for spin-spin interactions that are strongly nonlinear in the neighbor-spin products. When the nonlinearity is decreased, the first-order line ends in a critical point. The existence of the first-order line is in agreement with mean-field theory as well as with high- and low-temperature approximations.

We review an algebraic method for constructing degenerate eigenvectors of the transfer matrix of the eight-vertex Cyclic Solid-on-Solid lattice model (8V CSOS model), where the degeneracy increases exponentially with respect to the system size. We consider the elliptic quantum group E_τ,η(sl₂) at the discrete coupling constants: 2N η = m₁ + im₂τ, where N, m₁ and m₂ are integers. Then we show that degenerate eigenvectors of the transfer matrix of the six-vertex model at roots of unity in the sector S^Z ≡ 0 (mod N) are derived from those of the 8V CSOS model, through the trigonometric limit. They are associated with the complete N strings. From the result we see that the dimension of a given degenerate eigenspace in the sector S^Z ≡ 0 (mod N) of the six-vertex model at Nth roots of unity is given by , where is the maximal value of the total spin operator S^Z in the degenerate eigenspace.

By varying (x,y,z) within a manifold ℳ, the positive (negative)-energy eigenvectors of the 2 × 2 Hamiltonian H = xσ_x + yσ_y + zσ_z (where σ_x,y,z are the Pauli matrices) form a U(1) fiber bundle. For certain ℳ the bundle has non-trivial topology. For example when ℳ = S² the associated bundle has non-zero Chern number indicating that it is topologically non-trivial at the highest level. In this paper we construct a simple 2 × 2 Hamiltonian whose eigen-vector bundle exhibits a more subtle topological non-triviality when ℳ is a closed three-manifold. This non-trivial topology is characterized by non-zero Chern-Simons invariant.

We compare two different methods of computing form factors. One is the well established procedure of solving the form factor consistency equations and the other is to represent the field content as well as the particle creation operators in terms of fermionic Fock operators. We compute the corresponding matrix elements for the complex free fermion and the Federbush model. The matrix elements only satisfy the form factor consistency equations involving anyonic factors of local commutativity when the corresponding operators are local. We carry out the ultraviolet limit, analyze the momentum space cluster properties and demonstrate how the Federbush model can be obtained from the SU(3)₃-homogeneous sine-Gordon model. We propose a new class of Lagrangians which constitute a generalization of the Federbush model in a Lie algebraic fashion. For these models we evaluate the associated scattering matrices from first principles, which can alternatively also be obtained in a certain limit of the homogeneous sine-Gordon models.

Deformation quantization (of a commutative algebra) is based on the introduction of a new associative product, expressed as a formal series, . In the case of the algebra of functions on a symplectic space the first term in the perturbation is often identified with the antisymmetric Poisson bracket. There is a wide-spread belief that every associative *-product is equivalent to one for which C₁(f,g) is antisymmetric and that, in particular, every abelian deformation is trivial. This paper shows that this is far from being the case and illustrates the existence of abelian deformations by physical examples.

Recent studies of the topological properties of a general class of lattice Dirac operators are reported. This is based on a specific algebraic realization of the Ginsparg-Wilson relation in the form γ₅ (γ₅D) + (γ₅D)γ₅ = 2a^2k+1(γ₅D)^2k+2 where k stands for a non-negative integer. The choice k = 0 corresponds to the commonly discussed Ginsparg-Wilson relation and thus to the overlap operator. It is shown that local chiral anomaly and the instanton-related index of all these operators are identical. The locality of all these Dirac operators for vanishing gauge fields is proved on the basis of explicit construction, but the locality with dynamical gauge fields has not been fully established yet.

The spectral flows of the hermitian Wilson-Dirac operator for a continuous family of abelian gauge fields connecting different topological sectors are shown to have a characteristic structure leading to the lattice index theorem. The index of the overlap Dirac operator is shown to coincide with the topological charge for a wide class of gauge field configurations. It is also argued that in two dimensions the eigenvalue spectra for some special but nontrivial background gauge fields can be described by a set of universal polynomials and the index can be found exactly.

This paper reviews some recent progress on dimer and spanning tree enumerations. We use the Kasteleyn formulation to enumerate close-packed dimers on a simple-quartic net embedded on non-orientable surfaces, and obtain solutions in the form of double products. For spanning trees the enumeration is carried out by evaluating the eigenvalues of the Laplacian matrix associated with the lattice, a procedure which holds in any spatial dimension. In two dimensions a bijection due to Temperley relates spanning tree and dimer configurations on two related lattices. We use this bijection to enumerate dimers on a net with a vacancy on the boundary. It is found that the occurrence of a vacancy induces a correction to the enumeration, where N is the linear size of the lattice, and changes the central charge from c = 1 to -2.

We study the geometry of a knot invariant defined in terms of the quantum dilogarithm function. We show that a hyperbolic structure naturally arises in the classical limit of the invariant; the completeness conditions can also be identified with the saddle point equations by studying a (1,1)-tangle.

The "Laughlin" picture of the Fractional Quantum Hall effect can be derived using the "exotic" model based on the two-fold centrally-extended planar Galilei group. When coupled to a planar magnetic field of critical strength determined by the extension parameters, the system becomes singular, and "Faddeev-Jackiw" reduction yields the "Chern-Simons" mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law.

We give a brief summary of our recent works on the three-state chiral clock model. In these works, we use improved effective field theories with clusters being strips, infinite in the chiral direction and finite in the non-chiral direction. Hence, effective-field transfer matrix methods can be employed in these studies. The effective fields are determined by the Gibbs-Bogoliubov free energy variational principle, leading to Weiss or Bethe approximations in different studies respectively. By systematic improvement of these approximations, i.e. widening the strips, these studies point to the conclusion that there is no Lifshitz point existing at finite non-zero chirality.

We show how growth by agglomeration can be described by means of algebraic or differential equations which determine the evolution of probabilities of various local configurations. The minimal fluctuation condition is used to define vitrification. Our methods have been successfully used for the description of glass formation.

After a short summary on the elliptic quantum group and the elliptic algebra , we present a free field representation of the Drinfeld currents and the vertex operators (VO's) in the level k. We especially demonstrate a construction of the higher spin type I VO's by fusing the spin 1/2 type I VO's and fix a rule of attaching the screening current S(z) associated with the q-deformed ℤ_k-parafermion theory. As a result we get a free field representation of the higher spin type I VO's which commutation relation by the fused Boltzmann weight coefficients is manifest.

We compute the survival probability of an initial state, with an energy in a certain window, by means of random matrix theory. We determine its probability distribution and show that is is universal, i.e. characterised only by the symmetry class of the hamiltonian and independent of the initial state.

We extend a recent construction for an integrable model describing Josephson tunneling between identical BCS systems to the case where the BCS systems have different single particle energy levels. The exact solution of this generalized model is obtained through the Bethe ansatz.

I will review the finite density algorithm for lattice QCD based on finite chemical potential and summarize the associated difficulties. I will propose a canonical ensemble approach which projects out the finite baryon number sector from the fermion determinant. For this algorithm to work, it requires an efficient method for calculating the fermion determinant and a Monte Carlo algorithm which accommodates unbiased estimate of the probability. I shall report on the progress made along this direction with the Padé–Z₂ estimator of the determinant and its implementation in the newly developed Noisy Monte Carlo algorithm.

Recent results on the short-time behaviors of a few models possessing a common feature of long-ranged interaction will be summarized. For the disorder initial state, the initial order increase is observed for each model in a heat-bath at the critical temperature. The dynamic exponents are calculated. For arbitrary initial order and environment temperature, universal characteristic functions are introduced in order to generalize the scaling relations. Remarkable consistence between the theoretic renormalization group results and the simulations are found in the long-range regime.

We describe generalized D = 11 Poincaré and conformal supersymmetries. The corresponding generalization of twistor and supertwistor framework is outlined with OSp(1|64) superspinors describing BPS preons. The BPS states as composed out of n = 32 - k preons are introduced, and basic ideas concerning BPS preon dynamics is presented. The lecture is based on results obtained by J.A. de Azcarraga, I. Bandos, J.M. Izquierdo and the author.¹

We have performed a Random Matrix Theory (RMT) analysis of the quantum four state chiral Potts chain for different sizes of the quantum chain up to eight sites, and for different unfolding methods. Our analysis shows that one generically has a Gaussian Orthogonal Ensemble statistics for the unfolded spectrum instead of the GUE statistics one could expect. Furthermore a change from the generic GOE distribution to a Poisson distribution occurs when the hamiltonian becomes integrable. Therefore, the RMT analysis can be seen as a detector of "higher genus integrability".

We obtain an explicit expression for the defining relation of the deformed W_N algebra, . Using this expression we can show that, in the q→1 limit, with reduces to the -version of the Lepowsky–Wilson's -algebra of level k, . In other words with can be considered as a q-deformation of .

In the appendix given by H. Awata, S. Odake and J. Shiraishi, we present an interesting relation between and ζ-function regularization.

In order to achieve in configuration space a dimensional reduction from dimension two to dimension one, the lowest Landau level (LLL) projection, also called the Peierls substitution, is not sufficient. One has also, once in the LLL, to take the vanishing magnetic field limit.

An exactly solvable hard-core Bose–Hubbard model, which is equivalent to a mean-filed plus nearest-level pairing theory, for a description of well-deformed nuclei is used and applied to the actinide region. Binding energies and pairing excitation energies of ^226–234Th, ^230–240U, and ^236–243Pu isotopes are calculated and compared with the corresponding experimental values.

We study the algebra , the basis of the Hilbert space ℋ_n in terms of θ functions of the positions of n solitons. Then we embed the Heisenberg group as the quantum operator factors in the representation of the transfer matrices of various integrable models. Finally we generalize our result to the generic θ case.

We briefly review recent progress on calculating susceptibilities in planar Ising models.

In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.

In this note we give an introduction to the topic of Quantum Control, explaining what its objectives are, and describing some of its limitations.

In this talk, we shall briefly review some results on the strongly correlated electron systems, derived recently by applying Lieb's spin-reflection-positivity method. To explain the basic ideas of this method to a wide audience, we emphasize the important role played by Marshall's rule in studying the many-body systems.

An algebraic treatment of the eigenstates of the (A_{N - 1}-)Calogero model is presented, which provides an algebraic construction of the nonsymmetric orthogonal eigenvectors, symmetrization, antisymmetrization and calculation of square norms in a unified way.

The energy eigenvalues of the superintegrable chiral Potts model are determined by the zeros of special polynomials which define finite representations of Onsager's algebra. The polynomials determining the low-sector eigenvalues have been given by Baxter in 1988. In the ℤ₃-case they satisfy 4-term recursion relations and so cannot form orthogonal sequences. However, we show that they are closely related to Jacobi polynomials and satisfy a special "partial orthogonality" with respect to a Jacobi weight function.

On the basis of Bethe ansatz solution of two-component bosons with SU(2) symmetry and δ-function interaction in one dimension, we study the thermodynamics of the system at finite temperature by using the strategy of thermodynamic Bethe ansatz (TBA). It is shown that the ground state is an isospin "ferromagnetic" state by the method of TBA, and at high temperature the magnetic property is dominated by Curie's law. We obtain the exact result of specific heat and entropy in strong coupling limit which scales like T at low temperature. While in weak coupling limit, it is found there is still no Bose-Einstein Condensation (BEC) in such 1D system.

A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applications of this method to untwisted and twisted quantum affine superalgebras.

Free field and twisted parafermionic representations of twisted current algebra are obtained. The corresponding twisted Sugawara energy-momentum tensor is given in terms of three (β,γ) pairs and two scalar fields and also in terms of twisted parafermionic currents and one scalar field. Two screening currents of the first kind are presented in terms of the free fields.