Table of contents

We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at Δ = -½, along with the ground-state wavefunction of the corresponding O(n) loop model at n = 1. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some cases the largest component itself, are conjectured to be directly related to the total number of alternating-sign matrices and plane partitions in certain symmetry classes.

Recent investigations by Bender and Boettcher and by Mezincescu have argued that the discrete spectrum of the non-Hermitian potential V(x) = -ix³ should be real. We give further evidence for this through a novel formulation which transforms the general one-dimensional Schrodinger equation (with complex potential) into a fourth-order linear differential equation for |Ψ(x)|². This permits the application of the eigenvalue moment method, developed by Handy, Bessis and coworkers, yielding rapidly converging lower and upper bounds to the low-lying discrete state energies. We adapt this formalism to the pure imaginary cubic potential, generating tight bounds for the first five discrete state energy levels.

We apply the transfer-matrix density-matrix renormalization group (TMRG) to a stochastic model, the Domany-Kinzel cellular automaton, which exhibits a non-equilibrium phase transition in the directed percolation universality class. Estimates for the stochastic time evolution, phase boundaries and critical exponents can be obtained with high precision. This is possible using only modest numerical effort since the thermodynamic limit can be taken analytically in our approach. We also point out further advantages of the TMRG over other numerical approaches, such as classical DMRG or Monte Carlo simulations.

We discuss a new numerical method for the determination of excited states of a quantum system using a generalization of the Feynman-Kac formula. The method relies on introducing an ensemble of non-interacting identical systems with a fermionic statistics imposed on the systems as a whole, and on determining the ground state of this fermionic ensemble by taking the long-time limit of the Euclidean kernel. Due to the exclusion principle, the ground state of an n-system ensemble is realized by the set of individual systems occupying successively the n lowest states, all of which can therefore be sampled in this way. To demonstrate how the method works, we consider a one-dimensional oscillator and a chain of harmonically coupled particles.

We investigate an XY spin-glass model in which both spins and interactions (or couplings) evolve in time, but with widely separated time-scales. For large times this model can be solved using replica theory, requiring two levels of replicas, one level for the spins and one for the couplings. We define the relevant order parameters, and derive a phase diagram in the replica-symmetric approximation, which exhibits two distinct spin-glass phases. The first phase is characterized by freezing of the spins only, whereas in the second phase both spins and couplings are frozen. A detailed stability analysis also leads to two distinct corresponding de Almeida-Thouless lines, each marking continuous replica-symmetry breaking. Numerical simulations support our theoretical study.

We study the phase-ordering dynamics of the O(n) model with a conserved order parameter for systems with topological defects. We present results from both cell dynamical simulations and predictions of a Gaussian auxiliary field (GAF) approximation for the XY (n = 2) model in two and three dimensions, and the Heisenberg (n = 3) model in three dimensions. We describe the results for the structure factor S(q) and growth law L(t) from simulations. The growth laws obtained are consistent with theoretical predictions based on energy-scaling arguments. The structure factor shows good dynamical scaling using a length extracted from its first moment. The simulations are compared with the theoretical predictions of the GAF for the scaling functions. Our results show that the GAF gives a good qualitative description of most features of the structure factor. However, it overestimates the amplitude of the Porod tail in the large-q limit. Moreover, for small q, the structure factor exhibits a q²-behaviour instead of the expected (generalized) Yeung result of q⁴.

We show that the recent application of non-extensive statistical mechanics to fully developed turbulence (FDT) satisfies Novikov's inequality only in the near-Gaussian limit and exhibits compatibility with various other formulations of FDT. We define relations between the non-extensivity parameter q and the corresponding parameters of the lognormal, multifractal and random-β model.

We prove several results concerning the numbers of n-edge self-avoiding polygons and walks in the lattice Z^d which had previously been conjectured on the basis of numerical results. If the number of n-edge self-avoiding polygons (walks) with k contacts is p_n(k) (c_n(k)) then we prove that κ₀ ≡ lim_n→∞n^-1 log p_n(k) = lim_n→∞n^-1 log c_n(k) exists for all fixed k and is independent of k. For polygons in Z², we prove that there exist two positive functions B₁ and B₂, independent of n but depending on k, such that B₁n^kp_n(0) ⩽ p_n(k) ⩽ B₂n^kp_n(0) for fixed k and n large. Also, provided the limit exists, we prove that 0 < lim_n→∞ ⟨k⟩_n/n < 1.

In addition, we consider the number of polygons with a density of contacts, i.e. k = αn, and show that the corresponding connective constant, κ(α), exists and is a concave function of α. For d = 2, we prove that lim_α→0⁺ κ(α) = κ₀ and the right derivative of κ(α) at α = 0 is infinite.

High-dimensional energy landscapes of complex systems often exhibit a very complicated structure, with many local minima separated by a multitude of barriers of various heights. For the analysis of the dynamics on such landscapes, simplified models based on combining many microstates to form physically relevant macrostates are of great advantage. In particular, knowledge of the relative sizes of minimum and transition regions is crucial. As an example, we analyse transitions in low-energy regions belonging to a simple model of the crystalline compound MgF₂. We find that the minimum regions, i.e. the states associated with only one particular minimum, extend to energies far above the saddle points, and we show that the size of the transition regions is small compared with the minimum regions.

The self-diffusion coefficient of a dilute gas composed of infinitely thin hard needles was studied by classical trajectory calculations in relation to the moment of inertia of the needle, I. The calculated self-diffusion coefficient was compared with the value obtained by the independent collision approximation (ICA) in which each collision is assumed to be uncorrelated. Both the self-diffusion coefficient and the collision frequency increase with decreasing moment of inertia of the needle as expected by the ICA. The increasing rate of the self-diffusion coefficient was proportional to I ^-0.83 at I→0, that is larger than I ^-1/2 by the ICA. However, the ICA gives a larger self-diffusion coefficient at large moment of inertia; the correlation between the impulses during the chattering collisions changes its sign from positive to negative with decreasing moment of inertia. The thermal rotational motion at the initial configuration reduces the effect of the collision-induced rotational motion that leads to the positive correlation. The rapid thermal rotational motion at small moment of inertia makes the chattering collisions a kind of reciprocating motion in the orientation of the needles.

In the classification of Hietarinta, three triangular 4×4 R-matrices lead, via the FRT formalism, to matrix bialgebras which are not deformations of the trivial one. In this paper, we find the bialgebras which are in duality with these three exotic matrix bialgebras. We note that the L-T duality of FRT is not sufficient for the construction of the bialgebras in duality. We find also the quantum planes corresponding to these bialgebras both by the Wess-Zumino R-matrix method and by Manin's method.

The quantum-electrodynamic binding energiesB are determined perturbatively to order (nα)² for single macroscopic bodies (quasi-continua mimicking atomic solids) having the dispersive dielectric function ε(ω)≃{1 + 4πnαΩ²/(Ω²-(ω²-i0)²}, as if each atom were an oscillator of frequency Ω, and n the number density of atoms (pairwise separations ρ). The familiar divergences all persist although they are modified by dispersion (finite rather than infinite Ω); they must be controlled instead by imposing the condition ρ>λ~(minimum lattice spacing) <<c/Ω. QED gives identically the same B = -(nα)²(1/2)∫_λ^∞dρ ρ²f(ρ)g(ρ) as one obtains from the properly retarded attraction -α²f(ρ) between atoms, with g(ρ) a correlation function defined purely by the geometry of the body. The first three terms of the Taylor series for g are determined, respectively, by volume V, surface area S and any sharp edges. To order (nα)², but not beyond, the results for solid bodies lead directly to those for cavities of the same shape and size in otherwise unbounded material.

Unlike the attraction between disjoint bodies, B for any single finite body (typical linear dimensions a>>c/Ω) is dominated by components proportional, respectively, to (nα)²ℏΩ×{-V/λ³, + S/λ², -a/λ (if there are edges) and ±log (c/2Ωλ)}. These always tend to induce collapse rather than expansion. The pure Casimir components are of order (nα)²ℏc/a, and (like the logarithmic terms) sometimes positive, which makes them impossible to understand if the dominant terms are disregarded. The B are found in closed form for spheres, spherical shells and cubes, up to corrections vanishing with λ. For unit length of an infinitely long right circular cylinder of radius a, the standard V- and S-proportional terms are corrected only by -(nα)²(π²ℏΩ/128a)log (c/2Ωλ); the pure Casimir component, which would be proportional to (nα)²ℏc/a², vanishes through apparently accidental cancellations peculiar to order (nα)².

The exceptional superalgebra D(2,1;α) has been classified as a candidate conformal supersymmetry algebra in two dimensions. We propose an alternative interpretation of it as an extended BFV-BRST quantization superalgebra in 2D (D(2,1;1)≃osp(2,2|2)). A superfield realization is presented wherein the standard extended phase space coordinates can be identified. The physical states are studied via the cohomology of the BRST operator. Finally we reverse engineer a classical action corresponding to the algebraic model we have constructed, and identify the Lagrangian equations of motion.

We obtain three new solvable, real, shape-invariant potentials starting from the harmonic oscillator, Pöschl-Teller I and Pöschl-Teller II potentials on the half-axis and extending their domain to the full line, while taking special care to regularize the inverse-square singularity at the origin. The regularization procedure gives rise to a delta-function behaviour at the origin. Our new systems possess underlying nonlinear potential algebras, which can also be used to determine their spectra analytically.

We study the stationary problem of a charged Dirac particle in (2+1) dimensions in the presence of a uniform magnetic field B and a singular magnetic tube of flux Φ = 2πκ/e. The rotational invariance of this configuration implies that the subspaces of definite angular momentum l + 1/2 are invariant under the action of the Hamiltonian H. We show that for κ-l⩾1 or κ-l⩽0 the restriction of H to these subspaces, H_l, is essentially self-adjoint, while for 0<κ-l<1 H_l admits a one-parameter family of self-adjoint extensions (SAEs). In the latter case, the functions in the domain of H_l are singular (but square integrable) at the origin, their behaviour being dictated by the value of the parameter γ that identifies the SAE. We also determine the spectrum of the Hamiltonian as a function of κ and γ, as well as its closure.

An exact single-level resonance formula for the survival probability S(t) in the full time interval, that depends only on the resonance energy _r and the decay width Γ_r and fulfils time-reversal invariance, is used to discuss the non-exponential contributions to decay. At short times the formula behaves as S(t)≈1-ct^1/2 with c a constant, whereas at long times it behaves as S(t)≈dt^-3, d being a constant. With the time expressed in lifetime units, the onset of non-exponential decay is given at short times by τ_S≈4/[π(R² + R + 1/4)] and at long times by τ_L≈5.41 ln (R) + 12.25, where R = _r /Γ_r. The predictions of the formula are compared with numerical examples and some experimental results searching for non-exponential contributions to decay.

It is shown that two (1 + 1)-dimensional (2D) free Abelian and self-interacting non-Abelian gauge theories (without any interaction with matter fields) belong to a new class of topological field theories (TFTs). These new theories capture together some of the key features of Witten and Schwarz types of TFT because they are endowed with symmetries that are reminiscent of the Schwarz-type theories but their Lagrangian density has the appearance of the Witten-type theories. The topological invariants for these theories are computed on a 2D compact manifold and their recursion relations are obtained. These new theories are shown to provide a class of tractable field theoretical models for the Hodge theory in two dimensions of flat (Minkowski) spacetime where there are no propagating degrees of freedom associated with the 2D gauge boson.

We consider the classical elliptic Calogero-Moser model. A set of canonical separated variables for this model has been constructed in Kuznetsov et al. However, the generating function of the separating canonical transform is known only for two- and three-particle cases. We construct this generating function for the next A₃ case as the limit of the conjectured form of the quantum separating operator. We show explicitly that this generating function gives a canonical transform from the set of original variables to the separated ones.

Through an ℏ-expansion of the confined Calogero model with spin exchange interactions, we extract a generating function for the involutive conserved charges of the Frahm-Polychronakos spin chain. The resulting conservation laws possess the spin chain Yangian symmetry, although they are not expressible in terms of these Yangians.

The more usual problems discussed in classical mechanics in elementary textbooks are the non-dissipative ones in which there is a Hamiltonian representing the energy as a constant of motion. The translation of this type of problem to quantum mechanics is very well known. Conversely, there are very simple classical mechanical problems that involve dissipation, but whose translation to quantum mechanics on the basis of a corresponding Hamiltonian is sometimes misinterpreted, since the underlying classical formalism involves non-canonical transformations that lead to non-unitary transformations of the quantum mechanical wavefunctions. In this paper we shall discuss the problem of a beam of particles of given momentum incident from the left on a shutter that is opened at time t = 0. The solution has the well known properties of diffraction in time, but we will analyse it quantum mechanically both when dissipation is and is not present. The objective is to get a better understanding of the effect of dissipation in the quantum mechanical picture and to estimate the influence of the above mentioned non-unitary transformations on this particular problem.

In their paper Doplicher, Fredenhagen and Roberts (DFR) proposed a simple model of a particle in quantum spacetime. We give a new formulation of the model and propose some small changes and additions which improve the physical interpretation. In particular, we show that the internal degrees of freedom e and m of the particle represent external forces acting on the particle. To obtain this result we follow a constructive approach. The model is formulated as a covariance system. It has projective representations in which not only the spacetime coordinates but also the conjugated momenta are two-by-two noncommuting. These momenta are of the form P_µ-(b/c)A_µ, where b is the charge of the particle. The electric and magnetic fields obtained from the vector potential A_µ coincide with the variables e and m postulated by DFR. Similarly, the spacetime position operators are of the form Q_µ-(al²/ℏc)Ω_µ, where a is a generalized charge and l a fundamental length, and with vector potentials Ω_µ which are in some sense dual w.r.t. the A_µ.

A Poisson superpair is a pair of Poisson superalgebra structures on a supercommutative associative algebra, whose any linear combination is also a Poisson superalgebra structure. In this paper, we first construct certain linear and quadratic Poisson superpairs over a semi-finitely-filtered polarized ₂-graded associative algebra. Then we give a construction of certain Hamiltonian superpairs in the formal variational calculus over any finite-dimensional ₂-graded associative algebra with a supersymmetric nondegenerate associative bilinear form. Our constructions are based on the Adler mapping in a general sense. Our results in this paper can be viewed as noncommutative generalizations of the Adler–Gel'fand–Dikii Hamiltonian pair.

Equation (10) contains a typographic error: the factor r_l in the sum should be omitted, so the correct equation is:

Note that for r_l ≠ 1, the two steps, equations (3) and (4), cannot be combined in this way. The numerical and analytical treatment were based on the correct expression.

Another error concerns equation (11), in which the last sum should run from k = 1 to N:

Notice furthermore that the generalization error, equation (6), was evaluated with respect to an isotropic input distribution, whereas for the training we assume structured inputs, cf. equation (8). Taking into account the same data distribution for generalization yields epsilon_g as a function of , and (see, e.g., Marangi C, Biehl M and Solla S A 1995 Europhys. Lett.30 117). However, the simpler expression (6) is just as suitable to quantify the success of learning.

The address details of Fl Stancu are incomplete in the printed edition of the journal. Her main address should read as University of Liège, Institute of Physics B5, Sart Tilman, B-4000 Liège 1, Belgium, and her second address as ECT*, Strada delle Tabarelle, 286, I-38050 Villazzano, Trento, Italy. Her main address was omitted from the paper edition. This error has already been corrected in the online edition.