Table of contents

We obtain the diagonal reflection matrices for a recently introduced family of dilute lattice models in which the model can be viewed as an Ising model in a magnetic field. We calculate the surface free energy from the crossing-unitarity relation and thus directly obtain the critical magnetic surface exponent for L odd and surface specific heat exponent for L even in each of the various regimes. For L = 3 in the appropriate regime we obtain the Ising exponent , which is the first determination of this exponent without the use of scaling relations.

A new correlated electron model is presented which is derived using the quantum inverse scattering method and is thus integrable in one dimension on a periodic lattice. The R-matrix is constructed from a representation of the Temperley - Lieb algebra. The associated Hamiltonian describes correlated hopping, electron pair hopping and a generalized spin interaction.

We consider a quantum system with N degrees of freedom which is classically chaotic. When N is large, and both and the quantum energy uncertainty are small, quantum chaos theory can be used to demonstrate the following results: (i) given a generic observable A, the infinite time average of the quantum expectation value is independent of all aspects of the initial state other than the total energy, and equal to an appropriate thermal average of A; (ii) the time variations of are too small to represent thermal fluctuations; (iii) however, the time variations of can be consistently interpreted as thermal fluctuations, even though these same time variations would be called quantum fluctuations when N is small.

The spin-1 Ising model with bilinear and biquadratic exchange interactions and single-ion crystal field is solved on the Bethe lattice using exact recursion equations. The general procedure of investigation of critical properties is discussed and a full set of phase diagrams are constructed for both positive and negative biquadratic couplings. A comparison is made with the results of other approximation schemes.

We study the phase diagram of the extended Hubbard model in the atomic limit. At zero temperature, the phase diagram decomposes into six regions: three with homogeneous phases (characterized by particle densities , 1 and 2 and staggered charge density ) and three with staggered phases (characterized by the densities , 1 and and staggered densities , 1, and ). Here we use Pirogov - Sinai theory to analyse the details of the phase diagram of this model at low temperatures. In particular, we show that for any sufficiently low non-zero temperature the three staggered regions merge into one staggered region S, without any phase transitions (analytic free energy and staggered order parameter ) within S.

The generalization ability of an extremely dilute feedback neural network with multi-state neurons is studied by means of a deterministic noiseless parallel dynamics. The overlap with any one of a macroscopic number of binary, full activity, concepts is determined when the network is trained with examples of variable activity according to a Hebbian learning algorithm that favours stable symmetric mixture states. Explicit results about the phase diagram and the generalization error are obtained for a network with three-state neurons which remain inactive below a threshold . It is shown that the generalization ability can be considerably enhanced either by training the network with low-activity examples or by means of a moderate increase in .

We develop further a recent dynamical replica theory to describe the dynamics of the Sherrington - Kirkpatrick spin-glass in terms of closed evolution equations for macroscopic order parameters. We show how microscopic memory effects can be included in the formalism through the introduction of a dynamic order parameter function: the joint spin-field distribution. The resulting formalism describes very accurately the relaxation phenomena observed in numerical simulations, including the typical overall slowing down of the flow that was missed by the previous simple two-parameter theory. The advanced dynamical replica theory is either exact or a very good approximation.

Within a real-space renormalization group (RG) which preserves two-site correlation functions, we study, on the square lattice, the criticality of the bond-diluted Z(q) ferromagnetic model. We generalize the `break-collapse method' which simplifies greatly the exact calculation of arbitrary Z(q) two-terminal clusters (commonly appearing in RG approaches) mainly for a large value of q. We reproduce, in the pure case, several known exact results. The structure of the phase diagrams, for all the values of q, is obtained with a good precision. The massless spin - wave-like phase, which evolves into the Kosterlitz - Touless phase for , occurs around (in agreement with the well known exact result). The structure of the phase diagrams in the diluted case is qualitatively similar to that obtained from the pure model. The massless spin - wave-like phase resists in an interval of concentration p which increases for the large values of q.

We study complex-temperature singularities of the Ising model on the triangular and honeycomb lattices. We first discuss the complex-T phases and their boundaries. From exact results, we determine the complex-T singularities in the specific heat and magnetization. For the triangular lattice we discuss the implications of the divergence of the magnetization at the point (where ) and extend a previous study by Guttmann of the susceptibility at this point with the use of differential approximants. For the honeycomb lattice, from an analysis of low-temperature series expansions, we have found evidence that the uniform and staggered susceptibilities and both have divergent singularities at , and our numerical values for the exponents are consistent with the hypothesis that the exact values are . The critical amplitudes at this singularity were calculated. Using our exact results for and together with numerical values for from series analyses, we find that the exponent relation is violated at z = -1 on the honeycomb lattice; the right-hand side is consistent with being equal to 4 rather than 2. The connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed.

We investigate the spin- Heisenberg star introduced by Richter and Voigt. The model is defined by ; , . In extension to the work of Richter and Voigt we consider a more general describing the properties of the spins surrounding the central spin . The Heisenberg star may be considered as an essential structure element of a lattice with frustration (namely a spin embedded in a magnetic matrix ) or, alternatively, as a magnetic system with a perturbation by an extra spin. We present some general features of the eigenvalues, the eigenfunctions as well as the spin correlation of the model. For being a linear chain, a square lattice or a Lieb - Mattis-type system we present the ground-state properties of the model in dependence on the frustration parameter . Furthermore, the thermodynamic properties are calculated for being a Lieb - Mattis antiferromagnet.

We characterize the steady state of a driven diffusive lattice gas in which each site holds several particles, and the dynamics is activated and asymmetric. Using a quantum Hamiltonian formalism, we show that for arbitrary transition rates the model has product invariant measure. In the steady state, a pairwise balance condition is shown to hold. Configurations and leading respectively into and out of a given configuration are matched in pairs so that the flux of transitions from to is equal to the flux from to . Pairwise balance is more general than the condition of detailed balance and holds in the non-equilibrium steady state of a number of stochastic models.

The correct Hamiltonian for an extended Hubbard model with quantum group symmetry as introduced by Montorsi and Rasetti is derived for a D-dimensional lattice. It is shown that the superconducting holds as a true quantum symmetry only for D = 1 and that terms of higher order in the fermionic operators are needed in addition to phonons. A discussion of quantum symmetries in general is given in a formalism that should be readily accessible to non-Hopf algebraists.

The determination of the continuous symmetries of differential equations follows a well known algorithm, and is reduced to solution of a set of linear equations; this is based on considering infinitesimal generators of the symmetries, so that the method does not extend to discrete symmetries. In this paper, we present a method to determine discrete symmetries in a certain class by means of a linear system, although this is considerably more difficult to solve than the one connected with continuous symmetries. We also consider the inverse (and simpler) problem of determining the most general equation admitting a given discrete symmetry. In the last part, we consider a number of examples, dealing in particular with symmetries of relevance to physics.

We find the invariant measure for two new types of S matrices relevant for chaotic scattering from a cavity in a waveguide. The S matrices considered can be written as a matrix of blocks, each of rank N, in which the two diagonal blocks are identical and the two off-diagonal blocks are identical. The S matrices are unitary; in addition, they may be symmetric because of time-reversal symmetry. The invariant measure, with and without the condition of symmetry, is given explicitly in terms of the invariant measures for the well known circular unitary and orthogonal ensembles. Some implications are drawn for the resulting statistical distribution of the transmission coefficient through a chaotic cavity.

For both the Schrödinger equation in quantum mechanics and the Riccati-type equation satisfied by the superpotential in supersymmetric quantum mechanics, we explicitly show that there exists an Ermakov-type functional invariant with respect to the space variable. An energy-like interpretation is suggested for this invariant.

Even and odd phase coherent states associated with the Hermetian phase operator theory are introduced in terms of the creation operation of the phase quanta defined in a finite-dimensional phase state space. Some mathematical and physical properties of these quantum states are studied in some detail. It is shown that the even phase coherent states together with the odd ones build an overcomplete Hilbert space. Even and odd coherent-state formalism of the Pegg - Barnett phase operator is given in terms of the projection operator in the even and odd phase coherent-state space. The number - phase uncertainty relation is investigated for these quantum states. It is shown that even and odd phase coherent states are not minimum uncertainty and intelligent states for the number and phase operators.

Ermakov systems of arbitrary order and dimension are constructed. These inherit an underlying linear structure based on that recently established for the classical Ermakov system. As an application, alignment of a (2 + 1)-dimensional Ermakov and integrable Ernst system is shown to produce a novel integrable hybrid of a (2 + 1)-dimensional sinh - Gordon system and of a conventional Ermakov system.

The modified Born - Oppenheimer equation arising from Berry's phase is studied extensively in two physical systems. One is the spin- neutral particle in the spherically symmetric magnetic field, the other is the electron in the cylindrically symmetric magnetic field. The results show the significant topological effects in simple quantum systems.

A new quasiclassical formula for scars is obtained by using the Fredholm method. We show that it can be expanded into a formula obtained earlier by Agam and Fishman. The derivation is simple and direct. It is also more rigorous and more general than that of Agam and Fishman. It also clarifies the remarkable process of resurgence, relating the high-order terms based on long orbits to the Weyl term whose origin is the zero length orbits.