Table of contents

The zero-temperature critical behaviours of the classical Heisenberg and XY ferromagnets in the uniaxial random field are studied in the arbitrary spatial dimension. Exact results are obtained for the magnetization, transversal to the random field direction, at the Gaussian and bimodal distributions of the random field. For the Gaussian distribution the critical behaviour in strong random fields is independent of the spatial dimension. The transversal magnetization, , where is the distribution width. For the bimodal distribution of the random field, the transversal magnetization obeys the law , where is a critical field, and H is the random field amplitude. The same critical behaviour is expected for related systems, for example random antiferromagnets in the uniform field.

The continuous one-dimensional Burridge - Knopoff model is generalized by introducing plastic creep in addition to rigid sliding. The resulting equations, for an order parameter (sliding rate) and a control parameter (driving force), exhibit a velocity-strengthening and a velocity-softening instability. In the former regime, reminiscent of self-organized criticality in continuum systems, anomalous diffusion is described by a nonlinear diffusion equation. The latter regime, characteristic of deterministic chaos, is described by a time-dependent Ginzburg - Landau equation. Implications of the model with respect to earthquake predictability are discussed.

The vectorial extension of the Ribaucour transformation for the Lamé equations of orthogonal conjugate nets in multidimensions is given. We show that the composition of two vectorial Ribaucour transformations with appropriate transformation data is again a vectorial Ribaucour transformation, from which follows the permutability of the vectorial Ribaucour transformations. Finally, as an example we apply the vectorial Ribaucour transformation to the Cartesian background.

For the dynamical glassy transition in the p-spin mean-field spin-glass model a thermodynamic description is given. The often considered marginal states are not the relevant ones for this purpose. This leads us to consider a cooling experiment on exponential timescales, where lower states are accessed. The very slow configurational modes are at quasi-equilibrium at an effective temperature. A system-independent law is derived that expresses their contribution to the specific heat. -scaling in the ageing regime of two-time quantities is explained.

The associated Hamiltonian for an su(3) spin chain combining and representations is calculated. The ansatz equations for this chain are obtained and solved in the thermodynamic limit, and the ground state and excitations are described. Thus, relations between the number of roots and the number of holes in each level have been found. The excited states are characterized by means of these quantum numbers. Finally, the exact S-matrix for a state with two holes is found.

We give a definition of quantum states which behave classically based on minimal number of physically reasonable requirements. We prove that an infinite unique class of states exists which satisfies this definition and we show that every state from this class may be generated in the unique way departing from some corresponding strictly quantum state. We discuss some implications of the obtained results.

We present and analyse low-temperature series and complex-temperature partition function zeros for the q-state Potts model with q = 4 on the honeycomb lattice and q = 3,4 on the triangular lattice. A discussion is given on how the locations of the singularities obtained from the series analysis correlate with the complex-temperature phase boundary. Extending our earlier work, we include a similar discussion for the Potts model with q = 3 on the honeycomb lattice and with q = 3,4 on the kagomé lattice.

We examine the effects of introducing a wall or edge into a directed percolation process. Scaling ansatze are presented for the density and survival probability of a cluster in these geometries, and we make the connection to surface critical phenomena and field theory. The results of previous numerical work for a wall can thus be interpreted in terms of surface exponents satisfying scaling relations generalizing those for ordinary directed percolation. New exponents for edge directed percolation are also introduced. They are calculated in mean-field theory and measured numerically in 2 + 1 dimensions.

We continue our analysis of the physics of quantum lattice gas automata (QLGA). Previous work has been restricted to periodic or infinite lattices; simulation of more realistic physical situations requires finite sizes and nonperiodic boundary conditions. Furthermore, envisioning a QLGA as a nanoscale computer architecture motivates consideration of inhomogeneities in the `substrate'; this translates into inhomogeneities in the local evolution rules. Concentrating on the one-particle sector of the model, we determine the various boundary conditions and rule inhomogeneities which are consistent with unitary global evolution. We analyse the reflection of plane waves from boundaries, simulate wavepacket refraction across inhomogeneities, and conclude by discussing the extension of these results to multiple particles.

The SO(4) invariance of the transfer matrix for the one-dimensional Hubbard model is clarified from the viewpoint of the quantum inverse scattering method. We demonstrate the SO(4) symmetry by means of the fermionic L-operator and the fermionic R-matrix, which satisfy the graded Yang-Baxter relation. The transformation law of the fermionic L-operator under the SO(4) rotation is identified with a kind of gauge transformation, which determines the corresponding transformation of the fermionic creation and annihilation operators under the SO(4) rotation. The transfer matrix is confirmed to be invariant under the SO(4) rotation, which ensures the SO(4) invariance of the conserved currents including the Hamiltonian. Furthermore, we show that the representation of the higher conserved currents in terms of the Clifford algebra gives manifestly SO(4) invariant forms.

The dynamics of the (1 + 1)-dimensional crystalline surface with long-range interactions is investigated using the renormalization group (RG) technique. The system in question displays a roughening transition which does not occur for systems with short-range interactions. The linear macromobility continuously decreases to zero as temperature decreases to the critical temperature in contrast to the usual two-dimensional roughening transition with a universal jump in the mobility at the transition point. The nonlinear mobility is also derived using the RG recursion relations. Two different RG schemes are employed and their differences are addressed.

Dynamic behaviour of Aharonov-Bohm-type electron interference in the presence of a nonclassical electromagnetic field is investigated. The visibility of the time-averaged interference pattern is discussed for SU(1,1) coherent state (CS) and a comparison with other states is made. It is shown that the dynamic behaviour of the electron interference exhibits collapse and revival (CR) phenomenon for SU(1,1) CS. It is also shown that CR phenomenon of electron interference is closely related to the fluctuation of a nonclassical electromagnetic field.

In the present paper we study the Faddeev-Popov path-integral quantization of electrodynamics in an inhomogeneous dielectric medium. We quantize all polarizations of the photons and introduce the corresponding ghost fields. Using the heat kernel technique, we express the heat kernel coefficients in terms of the dielectricity and calculate the ultraviolet divergent terms in the effective action. No cancellation between ghosts and `non-physical' degrees of freedom of the photon is observed.

The density of states and the Hall conductivity of a two-dimensional electron gas in a uniform magnetic field and in the presence of a -impurity are exactly calculated using elementary field theoretic techniques. The impurity creates one localized state per Landau level, but the Hall conductivity is unaffected. Our treatment is explicitly gauge invariant, and can be easily adapted to other problems involving zero-range potentials.

We construct a Hopf algebra cocycle for the Yangian double ), conjugating Drinfeld's co-product to the usual one. To do this, we factorize the twist between two `opposite' versions of Drinfeld's co-product, introduced in an earlier work, using the decomposition of the algebra in its negative and non-negative modes sub-algebras.

It is sometimes desirable to produce for a nonlinear system of ODEs a new representation of simpler structural form, but it is well known that this goal may imply an increase in the dimension of the system. This is what happens if in this new representation the vector field has a lower degree of nonlinearity or a smaller number of nonlinear contributions. Until now both issues have been treated separately, rather unsystematically and, in some cases, at the expense of an excessive increase in the number of dimensions. We unify here the treatment of both issues in a common algebraic framework. This allows us to proceed algorithmically in terms of simple matrix operations.

Rational solutions for the Painlevé IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinants of the Hermite polynomials, or by a special case of the Schur polynomials.

Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck scale, one of the many problems one has to face in this enterprise is to find the discrete protoforms of the building blocks of continuum physics and mathematics. A core concept is the notion of dimension. In the following we develop such a notion for irregular structures such as (large) graphs and networks and derive a number of its properties. Among other things we show its stability under a wide class of perturbations which is important if one has ` dimensional phase transitions' in mind. Furthermore we systematically construct graphs with almost arbitrary ` fractal dimension' which may be of some use in the context of ` dimensional renormalization' or statistical mechanics on irregular sets.

A relativistic spin operator for Dirac particles is identified and it is shown that a coupling of spin to angular velocity arises in the relativistic case, just as Mashhoon had speculated, and Hehl and Ni had demonstrated, in the non-relativistic case.

The algorithmic method introduced by Fokas and Ablowitz to investigate the transformation properties of Painlevé equations is used to obtain one-to-one correspondence between the Painlevé IV, V and VI equations and the second-order second-degree equations of Painlevé type.

The Landau-Lifshitz equation for a spin chain with an easy plane in the case of spin non-flip is solved by the method of inverse scattering transform. To avoid complexity caused by the Riemann surface of the usual spectral parameter, a particular parameter k is introduced. After performing a gauge transformation corresponding to , the resulting Lax pair is independent of particular solutions in this limit. An inverse scattering transform is then developed in terms of k. A system of linear equations is derived in the reflectionless case. An expression of the gauge transformation and hence expressions of multi-soliton solutions are found explicitly by using the Binet-Cauchy formula. As an example, an explicit expression of the 1-soliton is given in terms of elementary functions of x and t.

A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The two-point function is found from the requirement that it transforms covariantly under this realization. The result is in agreement with explicit lattice calculations of the (1 + 1)-dimensional Ising model and the d-dimensional spherical model. A hard core is found which is not present in the continuum. For a semi-infinite lattice, profiles are also obtained.