Poisson–Lie T-Duality¶for Quasitriangular Lie Bialgebras

We introduce a new 2-parameter family of sigma models exhibiting Poisson–Lie T-duality on a quasitriangular Poisson–Lie group G. The models contain previously known models as well as a new 1-parameter line of models having the novel feature that the Lagrangian takes the simple form , where the generalised metric E is constant (not dependent on the field u as in previous models). We characterise these models in terms of a global conserved G-invariance. The models on G=SU ₂ and its dual G ^* are computed explicitly. The general theory of Poisson–Lie T-duality is also extended, notably the reduction of the Hamiltonian formulation to constant loops as integrable motion on the group manifold. The approach also points in principle to the extension of T-duality in the Hamiltonian formulation to group factorisations D=G⋈M, where the subgroups need not be dual or connected to the Drinfeld double. Received: 22 August 1999 / Accepted: 4 February 2000     

General relations among observables in neutral-current phenomena mediated by one or twoZ bosons

Ifμ-e universality is assumed, there are 17 neutral-current parameters of current experimental interest, including the parity-violating nuclear force sector. We deduce the general relations among these parameters implied by gauge models. In single-Z boson models, there are 10 relations, while two-boson models lead to 4 relations. Ifμ-e universality is abandoned, the number of parameters increases to 31, while the number of relations becomes 21 in single-boson models and 12 in two-boson models. We derive all these relations.     

LRS Bianchi Type II Bulk Viscous Fluid Universe with?Decaying Vacuum Energy Density Λ

The present study deals with spatially homogeneous and locally rotationally symmetric (LRS) Bianchi type II cosmological models with bulk viscous fluid distribution of matter and decaying vacuum energy density Λ. To get the deterministic models of the universe, we assume that the expansion (θ) in the model is proportional to the shear (σ). This leads to condition R=mS ⁿ , where R and S are metric potentials, m and n are constants. We have obtained two types of models of the universe for two different values of n. The vacuum energy density Λ for both models is found to be a decreasing function of time and it approaches a small positive value at late time which is supported by recent results from the observations of (SN Ia). Some physical and geometric behaviour of these models are also discussed.     

Bianchi Type V Universe Models with Specific Expansion Anisotropy

We investigate Bianchi type V cosmological models for perfect fluid source with time varying cosmological term Λ. We examine the possibility of cosmological models assuming the expansion anisotropy (the ratio σ/θ of the shear scalar σ to the volume expansion θ) to be a function of average scale factor R. The resulting models begin with initial anisotropy and approach isotropy at late times. Our models present an initial epoch with decelerating expansion followed by late time acceleration consistent with observations.     

Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas

Many two-dimensional spin models can be transformed into Coulomb-gas systems in which charges interact via logarithmic potentials. For some models, such as the eight-vertex model and the Ashkin-Teller model, the Coulomb-gas representation has added significantly to the insight in the phase transitions. For other models, notably theXY model and the clock models, the equivalence has been instrumental for almost our entire understanding of the critical behavior. Recently it was shown that theq-state Potts model and then-vector model are equivalent to a Coulomb gas with an asymmetry between positive and negative charges. Fieldlike operators in these spin models transform noninteger charges and magnetic monopoles. With the aid of exactly solved models the Coulombgas representation allows analytic calculation of some critical indices.     

New Spin Generalisation for Long Range Interaction Models

We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N) or sp(N). We find the symmetry algebras of these new models: the half-loop algebra based on so(N) or sp(N) for the Calogero models and the Yangian of so(N) or sp(N) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant.Dedicated to my PhD supervisor and friend D. Arnaudon.     

Nonexistence of H Theorem for Some Lattice Boltzmann Models

In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal models, and D2Q16 and D3Q40 thermal (energy-conserving) models. The proof does not require the equilibria to be polynomials.     

Effective actions and large-N limits

The saddlepoint action for a large-N theory is given as an effective action for composite operators. This effective action is computed explicitly forO(N) models and as a series in large-N invariants for matrix models. In the latter case, the use of the first term of the series is found to give good numerical agreement with the exact solutions of the solvable models.     

Investigation of quasi-realistic heterotic string models with reduced Higgs spectrum

Quasi-realistic heterotic-string models in the free fermionic formulation typically contain an anomalous U(1) that leads to supersymmetry breaking. Supersymmetry is restored by imposing F- and D-flatness on the vacuum. A three generation free fermionic standard-like model which did not admit stringent F- and D-flat directions has been presented (Cleaver et al. in Phys. Rev. D 78, 046009, 2008) and it was argued that all the moduli in that model were fixed. The particular property of that model was the reduction of the untwisted Higgs spectrum by a combination of symmetric and asymmetric internal dimension boundary conditions with respect to the internal fermions associated with the compactified dimensions. This reduction occurred without the need or presence of flat directions. In this paper we extend the analysis of free fermionic models with reduced Higgs spectrum to models with the same internal space but with modified gauge groups: SO(10) or flipped SU(5) subgroup. We show that all the models studied in this paper do admit stringent flat directions. Currently, the only examples of models that do not admit stringent flat directions are the standard-like models of Cleaver et al. (Phys. Rev. D 78, 046009, 2008). We comment on the relationship between flat directions and reduced Higgs in free fermionic models as well as the possible connection between moduli stabilization and model phenomenology.     

Time Asymmetry and Chaos in General Relativity

In this work the late-time evolution of Bianchi type VIII models is discussed. These cosmological models exhibit a chaotic behaviour towards the initial singularity and our investigations show that towards the future, far from the initial singularity, these models have a non-chaotic evolution, even in the case of vacuum and without inflation. These space-time solutions turn out to exhibit a particular time asymmetry. On the other hand, investigations of the late-time behaviour of type VIII models by another author have the result that chaos continues for ever in the far future and that these solutions have a time symmetric behaviour: this result was obtained using the approximation methods of Belinski, Khalatnikov and Lifshitz (BKL) and we try to find out a possible reason explaining why the different approaches lead to distinct outcomes. It will be shown that, at a heuristic level, the BKL method gives a valid approximation of the late-time evolution of type VIII models, agreeing with the result of our investigations.     

Model for infrared and Raman studies of molecular rotations in liquids and gases

Experimental infrared and Raman data for molecular rotations in dense phases often lie in between the results predicted by theJ- andM-diffusion models of Gordon. In this paper, we present a theory which is similar in its basic approach to Gordon’s extended diffusion models (EDM) but in which the restrictions of theJ andM limits are removed. The outcome is a scheme which allows one to describe situations which fall between the two extreme pictures of theJ andM models. Application of this scheme to experiments is discussed.     

Discrete symmetry groups of vertex models in statistical mechanics

We analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations. They can be seen as generated by involutions corresponding respectively to two kinds of transformations onq×q matrices: the inversion of theq×q matrix and an (involutive) permutation of the entries of the matrix. We show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics. This approach enables one to analyze two-dimensionalq ⁴-state vertex models as simply as three-dimensional vertex models, or higher-dimensional vertex models. Various examples of birational symmetries of vertex models are analyzed. A particular emphasis is devoted to a three-dimensional vertex model, the 64-state cubic vertex model, which exhibits a polynomial growth of the complexity of the calculations. A subcase of this general model is seen to yield integrable recursion relations. We also concentrate on a specific two-dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes Yang-Baxter integrable. It is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces, or higher-dimensional algebraic varieties.     

$\mathcal{PT}$ Symmetry in Statistical Mechanics and the Sign Problem

Generalized PT\mathcal{PT} symmetry provides crucial insight into the sign problem for two classes of models. In the case of quantum statistical models at non-zero chemical potential, the free energy density is directly related to the ground state energy of a non-Hermitian, but generalized PT\mathcal{PT}-symmetric Hamiltonian. There is a corresponding class of PT\mathcal{PT}-symmetric classical statistical mechanics models with non-Hermitian transfer matrices. We discuss a class of Z(N) spin models with explicit PT\mathcal{PT} symmetry and also the ANNNI model, which has a hidden PT\mathcal{PT} symmetry. For both quantum and classical models, the class of models with generalized PT\mathcal{PT} symmetry is precisely the class where the complex weight problem can be reduced to real weights, i.e., a sign problem. The spatial two-point functions of such models can exhibit three different behaviors: exponential decay, oscillatory decay, and periodic behavior. The latter two regions are associated with PT\mathcal{PT} symmetry breaking, where a Hamiltonian or transfer matrix has complex conjugate pairs of eigenvalues. The transition to a spatially modulated phase is associated with PT\mathcal{PT} symmetry breaking of the ground state, and is generically a first-order transition. In the region where PT\mathcal{PT} symmetry is unbroken, the sign problem can always be solved in principle using the equivalence to a Hermitian theory in this region. The ANNNI model provides an example of a model with PT\mathcal{PT} symmetry which can be simulated for all parameter values, including cases where PT\mathcal{PT} symmetry is broken.     

The parallel complexity of growth models

This paper investigates the parallel complexity of several nonequilibrium growth models.Invasion percolation, Eden growth, ballistic deposition, andsolid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale asO(log² N), whereN is the system size, and the number of processors required scales as a polynomial inN. The algorithms are based on fast parallel procedures for finding minimum-weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such asdiffusion-limited aggregation, for which fast parallel algorithms probably do not exist.     

Limiting Shapes for Deterministic Centrally Seeded Growth Models

We study the rotor router model and two deterministic sandpile models. For the rotor router model in ℤ ^d , Levine and Peres proved that the limiting shape of the growth cluster is a sphere. For the other two models, only bounds in dimension 2 are known. A unified approach for these models with a new parameter h (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension d≥1. For the rotor router model, the limiting shape is a sphere for all values of h. For one of the sandpile models, and h=2d−2 (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit h→−∞. Finally, we prove that the rotor router shape contains a diamond.     

Energy distribution in <Emphasis Type="Italic">f</Emphasis> (<Emphasis Type="Italic">R</Emphasis>) gravity

The well-known energy problem is discussed in f (R) theory of gravity. We use the generalized Landau–Lifshitz energy–momentum complex in the framework of metric f (R) gravity to evaluate the energy density of plane symmetric solutions for some general f (R) models. In particular, this quantity is found for some popular choices of f (R) models. The constant scalar curvature condition and the stability condition for these models are also discussed. Further, we investigate the energy distribution of cosmic string spacetime.     

Super-WZNW with reductions to supersymmetric and fermionic integrable models

A systematic construction for an action describing a class of supersymmetric integrable models as well as for pure fermionic theories is discussed in terms of the gauged WZNW model associated to half integer graded affine Kac–Moody algebras. Explicit examples of the N=1,2 super-sinh(sine)-Gordon models are discussed in detail. Pure fermionic theories arises for cosets sl(p,1)/sl(p)u(1) when a maximal kernel condition is fulfilled. The integrability condition for such models is discussed and it is shown that the simplest example when p=2 leads to the constrained Bukhvostov–Lipatov, Thirring, scalar massive and pseudo-scalar massless Gross–Neveu models.     

Bulk Viscous Bianchi Type V Cosmological Models with Decaying Cosmological Term Λ

We present bulk viscous Bianchi type V cosmological models with time-dependent cosmological term Λ. Exact solutions of Einstein field equations have been obtained by assuming shear scalar σ proportional to volume expansion θ. The coefficient of bulk viscosity is taken to be power function of energy density ρ or volume expansion θ. In these models cosmological term Λ come out to be negative. It is found that models obtained are expanding, shearing and non-rotating. They do not approach isotropy for large values of time t. Some observational parameters for the model have also been discussed.     

A study of the GaAs/AlAs(001) superlattice spectrum within the abrupt- and smooth-boundary models

The electron states and miniband spectra of (GaAs) _m (AlAs) _n (001) superlattices related to the Г- and X _z -valleys of conduction bands of bulk GaAs and AlAs crystals are considered within the framework of simplified abrupt- and smooth-heteroboundary models. The models are constructed on the basis of calculations made by the pseudopotential method. It is shown that the proposed models describe the results of the corresponding exact calculations in the approximation of the abrupt and smooth boundaries reasonably well. It is found that the difference between the smooth-boundary and abrupt-boundary models becomes pronounced in the case of superlattices with thin layers, where this difference results in a qualitatively different dispersion of lower minibands.     

Polarization of the cosmic microwave background radiation

In re-ionized models, the measurement of polarization of CMBR can be a good criterion to narrow down the parameter space for cosmological models. A Vishniac-type effect in second order polarization over arc minute scales has been calculated. It has been shown that while the effect is very small (∼10⁻² μK) for CDM models, it can be significant (∼0.3μK) for some isocurvature models.