Table of contents

Stochastic Loewner evolution (SLE_κ) has been introduced as a description of the continuum limit of cluster boundaries in two-dimensional critical systems. We show that the problem of N radial SLEs in the unit disc is equivalent to Dyson's Brownian motion on the boundary of the disc, with parameter β = 4/κ. As a result, various equilibrium critical models give realizations of circular ensembles with β different from the classical values of 1, 2 and 4 which correspond to symmetry classes of random U(N) matrices. Some of the bulk critical exponents are related to the spectrum of the associated Calogero–Sutherland Hamiltonian. The main result is also checked against the predictions of conformal field theory.

Distillable entanglement (E_d) is one of the acceptable measures of entanglement of mixed states. On the basis of discrimination through local operation and classical communication, this letter gives E_d for two classes of orthogonal multipartite maximally entangled states.

There is ongoing interest in the kinetic energy functional T_s[ρ] in density functional theory. The present study lies in this area and concerns the Pauli potential V_P[ρ]. A differential equation is obtained here for V_P(x) in one dimension for a general two-level system. Also, as a specific example, such a functional of ρ(x), the ground-state Fermion density, is given for the case of N Fermions which are harmonically confined.

A constant-time solution of the continuous global optimization problem (GOP) is obtained by using an ensemble algorithm. We show that under certain assumptions, the solution can be guaranteed by mapping the GOP onto a discrete unsorted search problem, whereupon Brüschweiler's ensemble search algorithm is applied. For adequate sensitivities of the measurement technique, the query complexity of the ensemble search algorithm depends linearly on the size of the function's domain. Advantages and limitations of an eventual NMR implementation are discussed.

Two-point topological charge correlation functions of several types of geometric singularity in Gaussian random fields are calculated explicitly, using a general scheme: zeros of n-dimensional random vectors, signed by the sign of their Jacobian determinant; critical points (gradient zeros) of real scalars in two dimensions signed by the Hessian; and umbilic points of real scalars in two dimensions, signed by their index. The functions in each case depend on the underlying spatial correlation function of the field. These topological charge correlation functions are found to obey the first Stillinger-Lovett sum rule for ionic fluids.

The exact finite-size scaling properties of clusters in compact directed percolation on a square lattice are derived. The results are implicit in previous work on the enumeration of staircase polygons, but their explicit form has not been presented as such before. The analysis provides important insights into the nature of this type of percolation transition.

We study free massive fermionic ghosts, in the presence of an extended line of impurities, relying on the Lagrangian formalism. We propose two distinct defect interactions, respectively, of relevant and marginal nature. The corresponding scattering theories reveal the occurrence of resonances and instabilities in the former case and the presence of poles with imaginary residues in the latter. Correlation functions of the thermal and disorder operators are computed exactly, exploiting the bulk form factors and the matrix elements relative to the defect operator. In the marginal situation, the one-point function of the disorder operator displays a critical exponent continuously varying with the interaction strength.

The number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory. In this paper, solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by Monte Carlo simulations using Wang–Landau sampling method for integers up to 8000. It is shown that lim_n→∞ ln(p₃(n))/n^3/4 = 1.79 ± 0.01, where p₃(n) is the number of solid partitions of the integer n. This result strongly suggests that the MacMahon conjecture for solid partitions, though not exact, could still give the correct leading asymptotic behaviour.

We enumerate self-avoiding walks and polygons, counted by perimeter, on the quasiperiodic rhombic Penrose and Ammann-Beenker tilings, thereby considerably extending previous results. In contrast to similar problems on regular lattices, these numbers depend on the chosen initial vertex. We compare different ways of counting and demonstrate that suitable averaging improves convergence to the asymptotic regime. This leads to improved estimates for critical points and exponents, which support the conjecture that self-avoiding walks on quasiperiodic tilings belong to the same universality class as self-avoiding walks on the square lattice. For polygons, the obtained enumeration data do not allow us to draw decisive conclusions about the exponent.

We have studied the two-dimensional ±J Ising model on the square lattice with a view to determine whether a spin glass can exist at small finite temperatures. By mapping the Ising model onto an ensemble of non-interacting lattice fermions, we have shown that the critical temperature for the spin glass phase transition appears to vanish. This result applies for all concentrations of negative bonds where the ground state is a spin glass.

We analyse a method to determine the short-time exponent θ related to the critical initial slip in stochastic lattice models. In this method it suffices to start with an uncorrelated state with a vanishing order parameter instead of departing, as is usually done, from an initial state with a nonvanishing order parameter. The exponent θ is calculated by the time correlation of the order parameter. This method, deduced previously for up–down symmetry models, is extended here to include models with other symmetries. We also consider the extension to cover models with absorbing states.

Period-3 oscillations of pendulum are investigated using the method developed in our previous paper [1]. Values of the driving force within very narrow ranges may give rise to this kind of motion. Because of the extreme sensitivity of the equation to the force strength and initial conditions, some features of the system can hardly be depicted, either numerically or experimentally. However, by analytically obtaining a map of states it is possible to detect the underlying structure of the system of solutions. The theory predicts the existence of unstable periodic solutions. Also, it predicts stable period-3 solutions around the top position of pendulum. Trajectories obtained by numerically integrating the pendulum equation in a phase-locked condition agree with our diagrams.

Upper bounds for the number () of Casimir operators of perfect Lie algebras with nontrivial Levi decomposition are obtained, and in particular the existence of nontrivial invariants is proved. It is shown that for high-ranked representations R the Casimir operators of the semidirect sum →⊕_R(deg R)L₁ of a semisimple Lie algebra and an Abelian Lie algebra (deg R)L₁ of dimension equal to the degree of R are completely determined by the representation R, which also allows the analysis of the invariants of subalgebras which extend to operators of the total algebra. In particular, for the adjoint representation of a semisimple Lie algebra the Casimir operators of →⊕_ad()(dim )L₁ can be explicitly constructed from the Casimir operators of the Levi part .

The system of N particles that are non-fermions interacting with non-positive pair potentials is considered. Each system is represented by a graph with points for particles and links for existence of bound states in the corresponding particle pair. A general sufficient condition for stability (the stability is defined as the existence of a bound state with the energy below all dissociation thresholds) is formulated as a theorem: any connected graph represents a stable system. The theorem also shows that the system may dissociate only into those clusters that contain full connected components of the graph. The applications to stability of nuclei are discussed.

We show in this paper that a particular family of Askey–Wilson polynomials can be interpreted directly in the light of q-deformed su_q(2) algebras. This approach allows us to correct previous results concerning the q-Legendre functions investigated in Granovskii and Zhedanov (1993 J. Phys. A: Math. Gen.26 4331). We also establish the orthonormalization and the special cases q → 1 (classical limit) and q → ∞ (asymptote). We conclude that these q-Legendre functions differ significantly from their classical counterparts only when q is in the vicinity of the unitary circle, where the singular points of the absolute value of these q-functions undergo a series of bifurcations.

In this paper, symmetry reductions for a cubic nonlinear Schrödinger (NLS) equation to complex ordinary differential equations are presented. These are obtained by means of Lie's method of infinitesimal transformation groups. It is shown that ten types of subgroups of the symmetry group lead, via symmetry reduction, to ordinary differential equations. These equations are solved and the similarity solutions are obtained.

Partial reversibility of quantum operations (or quantum channels) is considered from the information-theoretical point of view. The necessary and sufficient condition for quantum operations to be partially reversible is shown. The condition can be expressed in terms of information-theoretical quantities (von Neumann entropy and Ψ-information). The quantum information-theoretical meanings of the condition are discussed. The results are compared with those obtained for completely reversible quantum operations. The Ψ-information is calculated for the quantum depolarizing channel of a qubit and the linear dissipative channel of a single-mode bosonic system.

This paper investigates finite-dimensional -symmetric Hamiltonians. In doing so, it clarifies some of the claims made in earlier papers on -symmetric quantum mechanics. In particular, it is shown here that there are two ways to extend real symmetric Hamiltonians into the complex domain: (i) the usual approach is to generalize such Hamiltonians to include complex Hermitian Hamiltonians. (ii) Alternatively, one can generalize real symmetric Hamiltonians to include complex -symmetric Hamiltonians. In the first approach, the spectrum remains real, while in the second approach the spectrum remains real if the symmetry is not broken. Both generalizations give a consistent theory of quantum mechanics, but if D > 2, a D-dimensional Hermitian matrix Hamiltonian has more arbitrary parameters than a D-dimensional -symmetric matrix Hamiltonian.

A neutral particle with general spin and magnetic moment moving in an arbitrarily varying magnetic field is studied. The time evolution operator for the Schrödinger equation can be obtained if one can find a unit vector that satisfies the equation obeyed by the mean of the spin operator. There exist at least 2s + 1 cyclic solutions in any time interval. Some particular time interval may exist in which all solutions are cyclic. The nonadiabatic geometric phase for cyclic solutions generally contains extra terms in addition to the familiar one that is proportional to the solid angle subtended by the closed trace of the spin vector.

We show with the help of examples that discrete wavelets can be a useful tool in perturbation theory of finite-dimensional quantum Hamilton systems.

Operator-Schmidt decompositions of the quantum Fourier transform on ^N₁ ⊗ ^N₂ are computed for all N₁, N₂ ≥ 2. The decomposition is shown to be completely degenerate when N₁ is a factor of N₂ and when N₁ > N₂. The first known special case, N₁ = N₂ = 2ⁿ, was computed by Nielsen in his study of the communication cost of computing the quantum Fourier transform of a collection of qubits equally distributed between two parties (M A Nielsen 1998 PhD Thesis University of New Mexico ch 6 Preprint quant-ph/0011036). More generally, the special case N₁ = 2^n₁ ⩽ 2^n₂ = N₂ was computed by Nielsen et al in their study of strength measures of quantum operations (M A Nielsen et al 2002 Preprint quant-ph/0208077 (2003 Phys. Rev. A at press)). Given the Schmidt decompositions presented here, it follows that in all cases the bipartite communication cost of exact computation of the quantum Fourier transform is maximal.

We provide a systematic study on the possibility of supersymmetry (SUSY) for one-dimensional quantum mechanical systems consisting of a pair of lines or intervals [−l, l] each having a point singularity. We consider the most general singularities and walls (boundaries) at x = ±l admitted quantum mechanically, using a U(2) family of parameters to specify one singularity and similarly a U(1) family of parameters to specify one wall. With these parameter freedoms, we find that for a certain subfamily the line systems acquire an N = 1 SUSY which can be enhanced to N = 4 if the parameters are further tuned, and that these SUSY are generically broken except for a special case. The interval systems, on the other hand, can accommodate N = 2 or N = 4 SUSY, broken or unbroken, and exhibit a rich variety of (degenerate) spectra. Our SUSY systems include the familiar SUSY systems with the Dirac δ(x)-potential, and hence are extensions of the known SUSY quantum mechanics to those with general point singularities and walls. The self-adjointness of the supercharge in relation to the self-adjointness of the Hamiltonian is also discussed.

We derive a local, gauge-invariant action for the SU(N) nonlinear σ-model in 2 + 1 dimensions. In this setting, the model is defined in terms of a self-interacting pseudo-vector field θ_μ, with values in the Lie algebra of the group SU(N). Thanks to a non-trivially realized gauge invariance, the model has the correct number of physical degrees of freedom: only one polarization of θ_μ, like in the case of the familiar Yang–Mills theory in 2 + 1 dimensions. Moreover, since θ_μ is a pseudo-vector, the physical content corresponds to one massless pseudo-scalar field in the Lie algebra of SU(N), as in the standard representation of the model. We show that the dynamics of the physical polarization corresponds to that of the SU(N) nonlinear σ-model in the standard representation, and also construct the corresponding BRST-invariant gauge-fixed action.

Casimir energy is a nonlocal effect; its magnitude cannot be deduced from heat-kernel expansions, even those including the integrated boundary terms. On the other hand, it is known that the divergent terms in the regularized (but not yet renormalized) total vacuum energy are associated with the heat-kernel coefficients. Here a recent study of the relations among the eigenvalue density, the heat kernel and the integral kernel of the operator e^−t√H is exploited to characterize this association completely. Various previously isolated observations about the structure of the regularized energy emerge naturally. For over 20 years controversies have persisted stemming from the fact that certain (presumably physically meaningful) terms in the renormalized vacuum energy density in the interior of a cavity become singular at the boundary and correlate to certain divergent terms in the regularized total energy. The point of view of the present paper promises to help resolve these issues.

We propose a general formalism to compute exact correlation functions for Cardy's boundary states. Using the free-field construction of boundary states and applying the Coulomb-gas technique, it is shown that charge neutrality conditions pick up particular linear combinations of conformal blocks. As an example we study the critical Ising model with free and fixed boundary conditions, and demonstrate that conventional results are reproduced. This formalism thus directly associates algebraically constructed boundary states with correlation functions which are in principle observable or numerically calculable.

The authors of this paper have noticed that they made a mistake concerning the stability of the fixed points where the molecule is located horizontally. This results in figures 9 and 10 being incorrect. For full details and corrected figures, please see pdf.

This issue was initially put live on 3 June (for one day only) with incorrect page numbering (6283--6568). The page numbers are now correct (with effect from 5 June)and correspond to the print issue. We apologize for any inconvenience this may have caused.