Table of contents

Having developed a hybrid genetic algorithm, we have studied low-lying excited states of the ±J Heisenberg model in two (d = 2) and three (d = 3) dimensions. We have found evidence of the occurrence of the Parisi states in d = 3 but not in d = 2. That is, in L^d lattices, there exist metastable states with a finite excitation energy of ΔE ∼ O(J) for L → ∞, and energy barriers ΔW between the ground state and those metastable states are ΔW ∼ O(JL^θ) with θ > 0 in d = 3 but with θ < 0 in d = 2. This finding favours the replica-symmetry-breaking or the trivial–nontrivial scenario of the SG phase over the droplet scenario.

We consider random walks on the surface of the sphere S_n−1(n ⩾ 2) of the n-dimensional Euclidean space E_n, in short a hypersphere. By solving the diffusion equation in S_n−1 we show that the usual law ⟨r²⟩ ∝ t valid in E_n−1 should be replaced in S_n−1 by the generic law ⟨cos θ⟩ ∝ exp(−t/τ), where θ denotes the angular displacement of the walker. More generally one has ⟨C^n/2−1_L(cos θ)⟩ ∝ exp(−t/τ(L, n)) where C^n/2−1_L is a Gegenbauer polynomial. Conjectures concerning random walks on a fractal inscribed in S_n−1 are given tentatively.

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24-vertex model which is known in the literature as the fully packed double loop model (FPL²). In the particular case in which the fugacities of the polygons are the same, the model admits an exact solution. The solution is obtained using coordinate Bethe ansatz and provides a closed expression for the free energy. In particular, we find the free energy of the four-colouring model and the double Hamiltonian walk and recover the known entropy of the Ice model. When both fugacities are set equal to 2 the model undergoes an infinite-order phase transition.

A new stochastic cellular automaton (CA) model of traffic flow, which includes slow-to-start effects and a driver's perspective, is proposed by extending the Burgers CA and the Nagel–Schreckenberg CA model. The flow–density relation of this model shows multiple metastable branches near the transition density from free to congested traffic, which form a wide scattering area in the fundamental diagram. The stability of these branches and their velocity distributions are explicitly studied by numerical simulations.

We examine the behaviour of non-interacting Fermi gases at low temperature. If there is a confining potential present the thermodynamic behaviour is altered from the familiar results for the unconfined gas. The role of de Haas–van Alphen type oscillations that are a consequence of the confining potential is considered. Attention is concentrated on the behaviour of the chemical potential and the specific heat. Results are compared and contrasted with those for an unconfined and a totally confined gas.

A quasi-periodic packing of interpenetrating copies of , most of them only partially occupied, can be defined in terms of the strip projection method for any icosahedral cluster . We show that in the case when the coordinates of the vectors of belong to the quadratic field the dimension of the superspace can be reduced, namely can be re-defined as a multi-component model set by using a six-dimensional superspace.

We present a dynamical analysis of a classical billiard chain—a channel with parallel semi-circular walls, which can serve as a model for a bent optical fibre. An interesting feature of this model is the fact that the phase space separates into two disjoint invariant components corresponding to the left and right uni-directional motions. Dynamics is decomposed into the jump map, a Poincaré map between the two ends of a basic cell, and the time function, travelling time across a basic cell of a point on a surface of section. The jump map has a mixed phase space where the relative sizes of the regular and chaotic components depend on the width of the channel. For a suitable value of this parameter, we can have almost fully chaotic phase space. We have studied numerically the Lyapunov exponents, time auto-correlation functions and diffusion of particles along the chain. As a result of the singularity of the time function, we obtain marginally normal diffusion after we subtract the average drift. The last result is also supported by some analytical arguments.

We develop a matrix model to describe bilayered quantum Hall fluids for a series of filling factors. Considering two coupling layers, and starting from a corresponding action, we construct its vacuum configuration at ν = q_iK⁻¹_ijq_j, where K_ij is a 2 × 2 matrix and q_i is a vector. Our model allows us to reproduce several well-known wavefunctions. We show that the wavefunction Ψ_(m,m,n) constructed years ago by Yoshioka, MacDonald and Girvin for the fractional quantum Hall effect at filling factor and in particular Ψ_(3,3,1) at filling can be obtained from our vacuum configuration. The unpolarized Halperin wavefunction and especially that for the fractional quantum Hall state at filling factor can also be recovered from our approach. Generalization to more than two layers is straightforward.

We present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group USp(4). Three different families of coherent states are obtained, corresponding to the subgroups U(1) × U(1), U(2) and SU(2) × SU(2). The symplectic structure on the manifold of coherent states is obtained, and canonical coordinates are used to express the classical limit of quantum observables. One of the families is seen to provide a trivial classical limit.

In this paper, by means of the discrete zero curvature representation, nonisospectral negative Volterra flows and mixed Volterra flows are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for the two nonisospectral flows and obtain the formulae of the corresponding conserved densities and associated fluxes. Integrable time discretizations for several isospectral equations of the two flows are also presented.

A derivation of Belavkin's stochastic Schrödinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schrödinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence.

We present an example of quantum computational tasks whose performance is enhanced if we distribute quantum information using quantum cloning. Furthermore, we give achievable efficiencies for probabilistically cloning the quantum states used in implemented tasks for which cloning provides some enhancement in performance.

In this paper, we introduce a new family of photon-added as well as photon-depleted q-deformed coherent states related to the inverse q-boson operators. These states are constructed via the generalized inverse q-boson operator actions on a newly introduced family of q-deformed coherent states (Quesne C 2002 J. Phys. A: Math. Gen.35 9213) which are defined by slightly modifying the maths-type q-deformed coherent states. The quantum statistical properties of these photon-added and photon-depleted states, such as quadrature squeezing and photon-counting statistics, are discussed analytically and numerically in the context of both conventional (nondeformed) and deformed quantum optics.

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.

We show that general 3n-j(n > 2) symbols of the first and second kinds for the group SU(2) can be reformulated in terms of binomial coefficients. The proof is based on the graphical technique established by Yutsis et al and through a definition of a reduced 6-j symbol. The resulting 3n-j symbols thereby take a combinatorial form which is simply the product of two factors. The one is an integer or polynomial which is the single sum over the products of reduced 6-j symbols. They are in the form of summing over the products of binomial coefficients. The other is a multiplication of all the triangle relations appearing in the symbols, which can also be rewritten using binomial coefficients. The new formulation indicates that the intrinsic structure for the general recoupling coefficients is much nicer and simpler, which might serve as a bridge for study with other fields. Along with our newly developed algorithms, this also provides a basis for a direct, exact and efficient calculation or tabulation of all the 3n-j symbols of the SU(2) group for all the range of quantum angular momentum arguments. As an illustration, we present the results for the 12-j symbols of the first kind.

This paper is devoted to the solution of the bi-fractional differential equation for real 0 < α ⩽ 1, β > 0 and λ ≠ 0, with the initial conditions Here (^CD^α_tu)(t, x) is the partial derivative coinciding with the Caputo fractional derivative for 0 < α < 1 and with the usual derivative for α = 1, while (^LD^β_xu)(t, x)) is the Liouville partial fractional derivative (^LD^β_tu)(t, x)) of order β > 0. The Laplace and Fourier transforms are applied to solve the above problem in closed form. The fundamental solution of these problems is established and its moments are calculated. The special case α = 1/2 and β = 1 is presented, and its application is given to obtain the Dirac-type decomposition for the ordinary diffusion equation.

In standard Poincare and anti de Sitter SO(2, 3) invariant theories, antiparticles are related to negative energy solutions of covariant equations while independent positive energy unitary irreducible representations (UIRs) of the symmetry group are used for describing both a particle and its antiparticle. Such an approach cannot be applied in de Sitter SO(1, 4) invariant theory. We argue that it would be more natural to require that (*) one UIR should describe a particle and its antiparticle simultaneously. This would automatically explain the existence of antiparticles and show that a particle and its antiparticle are different states of the same object. If (*) is adopted then among the above groups only the SO(1, 4) one can be a candidate for constructing elementary particle theory. It is shown that UIRs of the SO(1, 4) group can be interpreted in the framework of (*) and cannot be interpreted in the standard way. By quantizing such UIRs and requiring that the energy should be positive in the Poincare approximation, we conclude that (i) elementary particles can be only fermions. It is also shown that (ii) C invariance is not exact even in the free massive theory and (iii) elementary particles cannot be neutral. This gives a natural explanation of the fact that all observed neutral states are bosons.

An angular momentum (2j + 1)-dimensional Hilbert space H is considered. Symplectic transformations S on the tensor product of N of these spaces H ⊗ ⋅⋅⋅ ⊗ H are studied for the case when the 2j + 1 is a power of a prime (Galois case). The corresponding operators are calculated numerically. The formalism is applied to quantum coding. A simple repetition code based on the space H_A spanned by the direct products of N angular momentum states with the same m, has distance 1. It is shown that a code based on the symplectically transformed space SH_A has (in general) larger distance. An example with three qutrits is discussed in detail.