Volume 54, Issue 10, October 2013
Index of content:

We provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the LogSobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information—a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped freefermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrixproduct operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.
 ARTICLES


Partial Differential Equations

Reduction of weakly nonlinear parabolic partial differential equations
View Description Hide DescriptionIt is known that the SwiftHohenberg equation can be reduced to the GinzburgLandau equation (amplitude equation) by means of the singular perturbation method. This means that if ɛ > 0 is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations is proposed. An amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution. In particular, near the periodic solution, the error estimate of solutions holds uniformly in t > 0.

Ground state solutions for nonperiodic Dirac equation with superquadratic nonlinearity
View Description Hide DescriptionThis paper is concerned with the following nonperiodic Dirac equation in , where M(x) is a vector potential. Under weak superquadratic condition on the nonlinearity, we establish the existence of ground state solutions of Nehari type by using the generalized Nehari manifold method developed recently by Szulkin and Weth for above problem.

Energy decay result in a Timoshenkotype system of thermoelasticity of type III with distributive delay
View Description Hide DescriptionIn this paper, we consider a onedimensional linear thermoelastic system of Timoshenko type with distributive delay, where the heat conduction is given by Green and Naghdi theories. We establish the stability of the system for the case of equal and nonequal speeds of wave propagation.

Representation Theory and Algebraic Methods

Stokes matrices for the quantum cohomologies of a class of orbifold projective lines
View Description Hide DescriptionWe prove the Dubrovin's conjecture for the Stokes matrices for the quantum cohomology of orbifold projective lines. The conjecture states that the Stokes matrix of the first structure connection of the Frobenius manifold constructed from the GromovWitten theory coincides with the Euler matrix of a full exceptional collection of the bounded derived category of the coherent sheaves. Our proof is based on the homological mirror symmetry, primitive forms of affine cusp polynomials, and the PicardLefschetz theory.

Classical rmatrices via semidualisation
View Description Hide DescriptionWe study the interplay between double cross sum decompositions of a given Lie algebra and classical rmatrices for its semidual. For a class of Lie algebras which can be obtained by a process of generalised complexification we derive an expression for classical rmatrices of the semidual Lie bialgebra in terms of the data which determines the decomposition of the original Lie algebra. Applied to the local isometry Lie algebras arising in threedimensional gravity, decomposition, and semidualisation yields the main class of nontrivial rmatrices for the Euclidean and Poincaré group in three dimensions. In addition, the construction links the rmatrices with the Bianchi classification of threedimensional real Lie algebras.

ManyBody and Condensed Matter Physics

Observed asymptotic differences in energies of stable and minimal point configurations on and the role of defects
View Description Hide DescriptionConfigurations of N points on the twosphere that are stable with respect to the Riesz senergy have a structure that is largely hexagonal. These stable configurations differ from the configurations with the lowest reported Npoint senergy in the location and structure of defects within this hexagonal structure. These differences in energy between the stable and minimal configuration suggest that energy scale at which defects play a role. This work uses numerical experiments to report this difference as a function of N, allowing us to infer the energy scale at which defects play a role. This work is presented in the context of established estimates for the minimal Npoint energy, and in particular we identify terms in these estimates that likely reflect defect structure.

Quantum Mechanics

On quantization of nondispersive wave packets
View Description Hide DescriptionNondispersive wave packets are widely used in optics and acoustics. We found it interesting that such packets could be also a subject of quantum field theory. Canonical commutation relations for the nondispersive wave packets are constructed.

New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some twodimensional systems
View Description Hide DescriptionNew ladder operators are constructed for a rational extension of the harmonic oscillator associated with type III Hermite exceptional orthogonal polynomials and characterized by an even integer m. The eigenstates of the Hamiltonian separate into m + 1 infinitedimensional unitary irreducible representations of the corresponding polynomial Heisenberg algebra. These ladder operators are used to construct a higherorder integral of motion for two superintegrable twodimensional systems separable in cartesian coordinates. The polynomial algebras of such systems provide for the first time an algebraic derivation of the whole spectrum through their finitedimensional unitary irreducible representations.

The velocity operator in quantum mechanics in noncommutative space
View Description Hide DescriptionWe tested the consequences of noncommutative (NC from now on) coordinates x k , k = 1, 2, 3 in the framework of quantum mechanics. We restricted ourselves to 3D rotationally invariant NC configuration spaces with dynamics specified by the Hamiltonian , where is an analogue of kinetic energy and denotes an arbitrary rotationally invariant potential. We introduced the velocity operator by ( being the position operator), which is a NC generalization of the usual gradient operator (multiplied by −i). We found that the NC velocity operators possess various general, independent of potential, properties: (1) uncertainty relations indicate an existence of a natural kinetic energy cutoff, (2) commutation relations , which is nontrivial in the NC case, (3) relation between and that indicates the existence of maximal velocity and confirms the kinetic energy cutoff, (4) all these results sum up in canonical (general, not depending on a particular form of the central potential) commutation relations of Euclidean group E(4) = SO(4)▷T(4), (5) Heisenberg equation for the velocity operator, relating acceleration to derivatives of the potential.

On the solvability of the quantum Rabi model and its 2photon and twomode generalizations
View Description Hide DescriptionWe study the solvability of the timeindependent matrix Schrödinger differential equations of the quantum Rabi model and its 2photon and twomode generalizations in Bargmann Hilbert spaces of entire functions. We show that the Rabi model and its 2photon and twomode analogs are quasiexactly solvable. We derive the exact, closedform expressions for the energies and the allowed model parameters for all the three cases in the solvable subspaces. Up to a normalization factor, the eigenfunctions for these models are given by polynomials whose roots are determined by systems of algebraic equations.

Simple bound state energy spectra solutions of Dirac equation for Hulthén and Eckart potentials
View Description Hide DescriptionWithin the framework of simple similarity transformation method, we present simple Schrödingerlike equation for Dirac equation with Hulthén and Eckart Lorentz vectors potentials. Using the asymptotic iteration method we have managed to obtain the explicit analytical solution of the bound state energy spectra as well as the eigenfunctions for both potentials. Discussion related to the eigenfunction for both cases is also considered. We have shown that the wave function is sensitive for both potentials as well as the parameter δ.

Nonpolynomial extensions of solvable potentials à la AbrahamMoses
View Description Hide DescriptionAbrahamMoses transformations, besides Darboux transformations, are wellknown procedures to generate extensions of solvable potentials in onedimensional quantum mechanics. Here we present the explicit forms of infinitely many seed solutions for adding eigenstates at arbitrary real energy through the AbrahamMoses transformations for typical solvable potentials, e.g., the radial oscillator, the DarbouxPöschlTeller, and some others. These seed solutions are simple generalisations of the virtual state wavefunctions , which are obtained from the eigenfunctions by discrete symmetries of the potentials. The virtual state wavefunctions have been an essential ingredient for constructing multiindexed Laguerre and Jacobi polynomials through multiple DarbouxCrum transformations. In contrast to the Darboux transformations, the virtual state wavefunctions generate nonpolynomial extensions of solvable potentials through the AbrahamMoses transformations.

Quantum Information and Computation

Rapid mixing implies exponential decay of correlations
View Description Hide DescriptionWe provide an analysis of the correlation properties of spin and fermionic systems on a lattice evolving according to open system dynamics generated by a local primitive Liouvillian. We show that if the Liouvillian has a spectral gap which is independent of the system size, then the correlations between local observables decay exponentially as a function of the distance between their supports. We prove, furthermore, that if the LogSobolev constant is independent of the system size, then the system satisfies clustering of correlations in the mutual information—a much more stringent form of correlation decay. As a consequence, in the latter case we get an area law (with logarithmic corrections) for the mutual information. As a further corollary, we obtain a stability theorem for local distant perturbations. We also demonstrate that gapped freefermionic systems exhibit clustering of correlations in the covariance and in the mutual information. We conclude with a discussion of the implications of these results for the classical simulation of open quantum systems with matrixproduct operators and the robust dissipative preparation of topologically ordered states of lattice spin systems.

Remarks on Kim's strong subadditivity matrix inequality: Extensions and equality conditions^{a)}
View Description Hide DescriptionWe describe recent work of Kim [J. Math. Phys.53, 122204 (2012)] to show that operator convex functions associated with quasientropies can be used to prove a large class of new matrix inequalities in the tripartite and bipartite setting by taking a judiciously chosen partial trace over all but one of the spaces. We give some additional examples in both settings. Furthermore, we observe that the equality conditions for all the new inequalities are essentially the same as those for strong subadditivity.

Relativistic Quantum Mechanics, Quantum Field Theory, Quantum Gravity, and String Theory

Modularity, quaternionKähler spaces, and mirror symmetry
View Description Hide DescriptionWe provide an explicit twistorial construction of quaternionKähler manifolds obtained by deformation of cmap spaces and carrying an isometric action of the modular group . The deformation is not assumed to preserve any continuous isometry and therefore this construction presents a general framework for describing NS5brane instanton effects in string compactifications with N = 2 supersymmetry. In this context the modular invariant parametrization of twistor lines found in this work yields the complete nonperturbative mirror map between type IIA and type IIB physical fields.

General Relativity and Gravitation

Probing pure Lovelock gravity by Nariai and BertottiRobinson solutions
View Description Hide DescriptionThe product spacetimes of constant curvature describe in Einstein gravity, which is linear in Riemann curvature, Nariai metric which is a solution of Λvacuum when curvatures are equal, k 1 = k 2, while it is BertottiRobinson metric describing uniform electric field when curvatures are equal and opposite, k 1 = −k 2. We probe pure Lovelock gravity by these simple product spacetimes and prove that the same characterization of these solutions is indeed true in general for pure Lovelock gravitational equation of order N in d = 2N + 2 dimension. We also consider these solutions for the conventional setting of EinsteinGaussBonnet gravity.

A characterization of causal automorphisms on twodimensional Minkowski spacetime
View Description Hide DescriptionIt is shown that causal automorphisms on twodimensional Minkowski spacetime can be characterized by the invariance of the wave equations.

Killing symmetries in spaces with Λ
View Description Hide DescriptionAll Killing symmetries in complex spaces with Λ in terms of the PlebańskiRobinsonFinley coordinate system are found. All metrics with Λ admitting a null Killing vector are explicitly given. It is shown that the problem of nonnull Killing vector reduces to looking for solution of the BoyerFinleyPlebański (Toda field) equation.

Covariant differential identities and conservation laws in metrictorsion theories of gravitation. II. Manifestly generally covariant theories
View Description Hide DescriptionThe present paper continues the work of Lompay and Petrov [J. Math. Phys.54, 062504 (2013)] where manifestly covariant differential identities and conserved quantities in generally covariant metrictorsion theories of gravity of the most general type have been constructed. Here, we study these theories presented more concretely, setting that their Lagrangians are manifestly generally covariant scalars: algebraic functions of contractions of tensor functions and their covariant derivatives. It is assumed that Lagrangians depend on metric tensor g, curvature tensor R, torsion tensor T and its first and second covariant derivatives, besides, on an arbitrary set of other tensor (matter) fields and their first and second covariant derivatives: . Thus, both the standard minimal coupling with the RiemannCartan geometry and nonminimal coupling with the curvature and torsion tensors are considered. The studies and results are as follow: (a) A physical interpretation of the Noether and Klein identities is examined. It was found that they are the basis for constructing equations of balance of energymomentum tensors of various types (canonical, metrical, and Belinfante symmetrized). The equations of balance are presented. (b) Using the generalized equations of balance, new (generalized) manifestly generally covariant expressions for canonical energymomentum and spin tensors of the matter fields are constructed. In the cases, when the matter Lagrangian contains both the higher derivatives and nonminimal coupling with curvature and torsion, such generalizations are nontrivial. (c) The Belinfante procedure is generalized for an arbitrary RiemannCartan space. (d) A more convenient in applications generalized expression for the canonical superpotential is obtained. (e) A total system of equations for the gravitational fields and matter sources are presented in the form more naturally generalizing the EinsteinCartan equations with matter. This result, being a one of the more important results itself, is to be also a basis for constructing physically sensible conservation laws and their applications.

Flagdipole spinor fields in ESK gravities
View Description Hide DescriptionWe consider the RiemannCartan geometry as a basis for the EinsteinSciamaKibble theory coupled to spinor fields: we focus on f(R) and conformal gravities, regarding the flagdipole spinor fields, type(4) spinor fields under the Lounesto classification. We study such theories in specific cases given, for instance, by cosmological scenarios: we find that in such background the Dirac equation admits solutions that are not Dirac spinor fields, but in fact the aforementioned flagdipoles ones. These solutions are important from a theoretical perspective, as they evince that spinor fields are not necessarily determined by their dynamics, but also a discussion on their structural (algebraic) properties must be carried off. Furthermore, the phenomenological point of view is shown to be also relevant, since for isotropic Universes they circumvent the question whether spinor fields do undergo the Cosmological Principle.
