Table of contents

We present an interesting reformulation of a collection of dilaton gravity models in two spacetime dimensions into a field theory of two decoupled Liouville fields in flat space, in the presence of a Maxwell gauge field. An effective action is also obtained, encoding the dynamics of the dilaton field and the single gravitational degree of freedom in a decoupled regime. This effective action represents an interesting starting point for future work, including the canonical quantization of these classes of nontrivial models of gravity-coupled matter systems.

We study numerically the statistics of curves which form the boundaries of toppling wave clusters in the deterministic Bak, Tang and Wiesenfeld sandpile model and stochastic Manna model on a square lattice. We consider the Abelian version of each model. Multiple tests show that the boundary of toppling wave clusters in both deterministic and stochastic models can be described by SLE_κ curves with diffusivity κ = 2.

We consider the O(n) loop model on tetravalent maps and show how to rephrase it into a model of bipartite maps without loops. This follows from a combinatorial decomposition that consists in cutting the O(n) model configurations along their loops so that each elementary piece is a map that may have arbitrary even face degrees. In the induced statistics, these maps are drawn according to a Boltzmann distribution whose parameters (the face weights) are determined by a fixed point condition. In particular, we show that the dense and dilute critical points of the O(n) model correspond to bipartite maps with large faces (i.e. whose degree distribution has a fat tail). The re-expression of the fixed point condition in terms of linear integral equations allows us to explore the phase diagram of the model. In particular, we determine this phase diagram exactly for the simplest version of the model where the loops are 'rigid'. Several generalizations of the model are discussed.

We obtain long series expansions for the bulk, surface and corner free energies for several two-dimensional statistical models, by combining Enting's finite lattice method (FLM) with exact transfer matrix enumerations. The models encompass all integrable curves of the Q-state Potts model on the square and triangular lattices, including the antiferromagnetic transition curves and the Ising model (Q = 2) at temperature T, as well as a fully packed O(n) type loop model on the square lattice. The expansions are around the trivial fixed points at infinite Q, n or 1/T. By using a carefully chosen expansion parameter, q ≪ 1, all expansions turn out to be of the form $\prod _{k=1}^\infty (1-q^k)^{\alpha _k + k \beta _k}$, where the coefficients α_k and β_k are periodic functions of k. Thanks to this periodicity property, we can conjecture the form of the expansions to all orders (except in a few cases where the periodicity is too large). These expressions are then valid for all 0 ⩽ q < 1. We analyse in detail the q → 1⁻ limit in which the models become critical. In this limit the divergence of the corner free energy defines a universal term which can be compared with the conformal field theory (CFT) predictions of Cardy and Peschel. This allows us to deduce the asymptotic expressions for the correlation length in several cases. Finally we work out the FLM formulae for the case where some of the system's boundaries are endowed with particular (non-free) boundary conditions. We apply this in particular to the square-lattice Potts model with Jacobsen–Saleur boundary conditions, conjecturing the expansions of the surface and corner free energies to arbitrary order for any integer value of the boundary interaction parameter r. These results are in turn compared with CFT predictions.

We propose a useful method to analyze one-dimensional (1D)-cellular automata traffic models based on the distribution of cluster size in the system. By applying this method, we reproduce the exact solution of the totally asymmetric exclusion process and the zero range process (ZRP). Moreover, we confirm that a certain kind of slow-to-start model can be interpreted as the ZRP and also obtain the exact solution of it. Finally, we extend the Fukui–Ishibashi model and obtain promising expressions of the flux.

The two-point correlation function of chaotic systems with spin 1/2 is evaluated using periodic orbits. The spectral form factor for all times thus becomes accessible. Equivalence with the predictions of random matrix theory for the Gaussian symplectic ensemble is demonstrated. A duality between the underlying generating functions of the orthogonal and symplectic symmetry classes is semiclassically established.

In this paper, we consider spinor Bose–Einstein condensates with spin f = 5 and f = 6 in the presence and absence of an external magnetic field at the mean field level. We calculate all of the so-called inert states of these systems. Inert states are a very unique class of stationary states because they remain stationary while Hamiltonian parameters change. Their existence comes from Michel's theorem. For illustration of symmetry properties of the inert states we use a method that allows for the classification of the systems as a polyhedron with 2f vertices proposed by Barnett et al (2006 Phys. Rev. Lett.97 180412).

A new class of special functions of three real variables, based on the alternating subgroup of the permutation group S₃, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the E-functions of C₃ is shown. Continuous interpolation of the three-dimensional data is studied and exemplified.

A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r, t) at position r and time t is completely determined by its previous values at all other locations r' and retarded times t' ⩽ t, provided that the function vanishes at infinity and has continuous second partial derivatives. Functions of this kind occur in many areas of physics and it seems somewhat surprising that they are constrained in this way.

The Kontsevich–Penner model, an Airy matrix model with a logarithmic potential, may be derived from a simple Gaussian two-matrix model through a duality. In this dual version, the Fourier transforms of the n-point correlation functions can be computed in the closed form. Using Virasoro constraints, we find that in addition to the parameters t_n, which appear in the Korteweg–de Vries hierarchies, one needs to introduce half-integer indices t_n/2. The free energy as a function of those parameters may be obtained from these Virasoro constraints. The large N limit follows from the solution to an integral equation. This leads to explicit computations for a number of topological invariants.

In the paper, we construct a hierarchy of integrable Hamiltonian systems which describe the variation of n-wave envelopes in a nonlinear dielectric medium. The exact solutions for some special Hamiltonians are given in terms of elliptic functions of the first kind. These solutions can be used to study such physical phenomena as the Kerr effect or parametric conversion.

For the case of the bipolaron, it has been proved recently that for U ⩾ 53.2α, where U is the repulsion parameter of the electrons and α is the coupling constant of the polaron, no binding occurs. We show that actually for U ⩾ 52.1α, there is no binding. Furthermore, we obtain optimized results for small and large values of α: more specifically, we prove that for each ε > 0, there is an α₁ and an α₂, such that if 0 < α ⩽ α₁, a condition for no-binding becomes U ⩾ (40.4 + ε)α, and if α ⩾ α₂, it is U ⩾ (38.7 + ε)α. We show that α₁ can be computed with any desired accuracy, whereas we are merely able to prove the existence of such an α₂.

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms of the τ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.

We construct a multi-component interacting hard-core anyons model with SU(N) invariance and obtain the exact solutions of the systems by using the nested algebraic Bethe ansatz method with anyonic grading. We find that the grading parameters enter the Bethe ansatz equations as the gauge potentials, and the magnetic flux-like effects are different in the different rapidity sectors. Our results cover the interacting multi-component bosonic or multi-component fermionic models with SU(N) symmetry. In the system, the effective interactions among each component can be tuned by the grading parameters which do not break the integrability, while the integrability of conventional multi-component bosons or fermions requires that all the interactions are equal. The results will be useful to theories and experiments of the cold atomic systems.

Saari conjectured that the N-body motion with a constant configurational measure is a motion with fixed shape. Here, the configurational measure μ is a scale-invariant product of the moment of inertia I = ∑_km_k|q_k|² and the potential function U = ∑_{i < j}m_im_j/|q_i − q_j|^α, α > 0. Namely, μ = I^α/2U. We will show that this conjecture is true for a planar equal-mass three-body problem under the strong force potential ∑_{i < j}1/|q_i − q_j|².

Partition functions often become τ-functions of integrable hierarchies, if they are considered dependent on infinite sets of parameters called time variables. The Hurwitz partition functions Z = ∑_Rd^{2 − k}_Rχ_R(t⁽¹⁾)...χ_R(t^(k))exp (∑_nξ_nC_R(n)) depend on two types of such time variables, t and ξ. KP/Toda integrability in t requires that k ⩽ 2 and also that C_R(n) are selected in a rather special way, in particular the naive cut-and-join operators are not allowed for n > 2. Integrability in ξ further restricts the choice of C_R(n), forbidding, for example, the free cumulants. It also requires that k ⩽ 1. The quasi-classical integrability (the WDVV equations) is naturally present in ξ variables, but also requires a careful definition of the generating function.

We show that 2ⁿ discrete coherent states of an n-qubit system, generated by application of the discrete displacement operators to a symmetric fiducial state, have isotropic fluctuations, with n/2 ⩽ 〈ΔS²〉 ⩽ n, in a specific tangent plane, which in general is not orthogonal to the mean spin direction. This allows us to use them as reference states to define a discrete squeezing for non-symmetric n-qubit states. Examples of states with reduced fluctuations, obtained after application of XOR gates to correlate (partially entangle) qubits, are analyzed.

We consider properties of quantum channels with the use of unified entropies. Extremal unravelings of quantum channel with respect to these entropies are examined. The concept of map entropy is extended in terms of the unified entropies. The map (q, s)-entropy is naturally defined as the unified (q, s)-entropy of a rescaled dynamical matrix of given quantum channel. Inequalities of Fannes type are obtained for introduced entropies in terms of both the trace and Frobenius norms of difference between corresponding dynamical matrices. Additivity properties of introduced map entropies are discussed. The known inequality of Lindblad with the entropy exchange is generalized to many of the unified entropies. For the tensor product of a pair of quantum channels, we derive a two-sided estimate on the output entropy of a maximally entangled input state.

We investigate the Zel'dovich effect in the context of ultracold, harmonically trapped quantum gases. We suggest that currently available experimental techniques in cold-atom research offer an exciting opportunity for a direct observation of the Zel'dovich effect without the difficulties imposed by conventional condensed matter and nuclear physics studies. We also demonstrate an interesting scaling symmetry in the level rearrangements which has heretofore gone unnoticed.

A methodology based on generalized Sturmian functions is put forward to solve two- and three-body scattering problems. It uses a spectral method which allows for the inclusion of the correct asymptotic behavior when solving the associated driven Schrödinger equation. For the two-body case, we demonstrate the equivalence between the exterior complex scaling (ECS) and the Sturmian approaches and illustrate the latter by using Hulthén Sturmian functions. Contrary to the ECS approach, no artificial cut-off of the potential is required in the Surmian approach. For the three-body scattering problem, the theoretical framework is presented in hyperspherical coordinates and a set of hyperspherical generalized Sturmian functions possessing outgoing asymptotic behavior is introduced. The Sturmian procedure is a direct generalization of the method discussed for the two-body problem; thus, the comparison with the ECS method is similar. For both the two- and three-body cases, Sturmian bases are efficient as they possess the correct outgoing behavior, diagonalize part of the potentials involved and are essentially localized in the region where the unsolved interaction is not negligible. Moreover, with the Sturmian basis, the operator (H − E) is represented by a diagonal matrix whose elements are simply the Sturmian eigenvalues.

Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O(N⁷) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O(N⁵) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.

A brane picture in type IIA superstring for the Yang monopole is reconsidered. It makes use of D2 and D4-branes wrapped on cycles in the K3 surface. When the model was first presented, some problems concerning the charges of the monopoles arose. In this paper, they are shown to be cured by the model itself. Surprisingly, the incompatibility between the multi-charge configuration and the spherical symmetry of the Yang monopole is seen in the brane description as the emergence of the enhançon shell and the fuzzy geometry. This consistency is deep and surprising, and is the point that triggered this work. It nontrivially relates a purely geometrical problem in ordinary spacetime with the emergence of noncommutative geometries. Besides, this paper includes an extended model for SO(4)-monopoles and a T-dual model in type IIB superstring.

I construct solutions to the heterotic supergravity BPS equations on products of Minkowski space with a non-symmetric coset. All of the bosonic fields are homogeneous and non-vanishing, the dilaton being a linear function on the non-compact part of spacetime.