Exact variational methods and cluster-variation approximations

It is shown that for classical,d-dimensional lattice models with finite-range interactions the translation-invariant equilibrium states have the property that their mean entropy is completely determined by their restriction to a subset of the lattice that is infinite in (d–1) dimensions and has a width equal to the range of the interaction in the dth dimension. This property is used to show proper convergence toward the exact result for a hierarchy of approximations of the cluster-variation method that uses one-dimensionally increasing basis clusters in a two-dimensional lattice.     

The Phase Transition of the Quantum Ising Model is Sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called ‘random-parity’ representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.     

String Geometry and the Noncommutative Torus

We construct a new gauge theory on a pair of d-dimensional noncommutative tori. The latter comes from an intimate relationship between the noncommutative geometry associated with a lattice vertex operator algebra ? and the noncommutative torus. We show that the tachyon algebra of ? is naturally isomorphic to a class of twisted modules representing quantum deformations of the algebra of functions on the torus. We construct the corresponding real spectral triples and determine their Morita equivalence classes using string duality arguments. These constructions yield simple proofs of the O(d,d;ℤ) Morita equivalences between d-dimensional noncommutative tori and give a natural physical interpretation of them in terms of the target space duality group of toroidally compactified string theory. We classify the automorphisms of the twisted modules and construct the most general gauge theory which is invariant under the automorphism group. We compute bosonic and fermionic actions associated with these gauge theories and show that they are explicitly duality-symmetric. The duality-invariant gauge theory is manifestly covariant but contains highly non-local interactions. We show that it also admits a new sort of particle-antiparticle duality which enables the construction of instanton field configurations in any dimension. The duality non-symmetric on-shell projection of the field theory is shown to coincide with the standard non-abelian Yang–Mills gauge theory minimally coupled to massive Dirac fermion fields. Received: 26 October 1998/ Accepted: 9 April 1999     

Lyapunov Bounds for Lattice Gauge Dynamics

We study the classical Hamiltonian dynamics of the Kogut–Susskind model for lattice gauge theories on a finite box in a d-dimensional integer lattice. The coupling constant for the plaquette interaction is denoted λ². When the gauge group is a real or a complex subgroup of a unitary matrix group U(N), N≥ 1, we show that the maximal Lyapunov exponent is bounded by , uniformly in the size of the lattice, the energy of the system as well as the order, N, of the gauge group. Received: 20 December 1997 / Accepted: 21 July 1998     

Transfer tensor method in statistical mechanics

We generalize the transfer matrix method to get exact solutions of d-dimensional classical spin lattice models in statistical mechanics.     

A link between quantum and classical Potts models

We study ground states of quantum Potts models. We construct ground states of certaind-dimensional quantum models as Gibbs measures of ad-dimensional classical spin system. Our results imply that various phenomena of classical spin systems can also be found in quantum ground states.     

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

Hypercube stacking: A potts-spin model for surface growth

We present a deposition and evaporation model for surface growth under a solid-on-solid constraint. We generalize the Ising-spin representation of a two-dimensional surface by Blöte and Hilhorst to ad-dimensional surface of a (d+1)-dimensional hypercubic lattice. The allowed surface configurations correspond to the (degenerate) ground states of a chirald-state Potts model. We describe a vectorized multisite-coding implementation for the corresponding kinetic Potts-spin model ford=2 andd=3. For thed=2 equilibrium surface our simulation results show excellent agreement with an exact analysis.     

Approximating the number of monomer-dimer coverings of a lattice

We study the problem of counting the number of coverings of ad-dimensional rectangular lattice by a specified number of monomers and dimers. This problem arises in several models in statistical physics, and has been widely studied. A classical technique due to Fisher, Kasteleyn, and Temperley solves the problem exactly in two dimensions when the number of monomers is zero (the dimer covering problem), but is not applicable in higher dimensions or in the presence of monomers. This paper presents the first provably polynomial-time approximation algorithms for computing the number of coverings with any specified number of monomers ind-dimensional rectangular lattices with periodic boundaries, for any fixed dimensiond, and in two-dimensional lattices with fixed boundaries. The algorithms are based on Monte Carlo simulation of a suitable Markov chain, and, in constrast to most Monte Carlo algorithms in statistical physics, have rigorously derived performance guarantees that do not rely on any assumptions. The method generalizes to counting coverings of any finite vertex-transitive graph, a class which includes most natural finite lattices with periodic boundary conditions.     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

Renormalization group equations in lattice gauge theories

New recursion equations for renormalization group transformations of the Migdal-Kadanoff type are obtained for gauge systems including fermion variables on a d-dimensional Euclidean space-time lattice. It is shown that in the weak gauge coupling region these equations have β-functions similar to those of continuum field theories in the case of U(1), SU(2) gauge groups (QED, QCD). On the other hand in the strong-coupling limit there is an infrared attractive fixed point corresponding to a color-confining effective system in both groups. A possible entire trajectory of the non-Abelian system is briefly conjectured.     

On Equilibrium Distribution of a Reversible Growth Model

We study a probabilistic model of interacting spins indexed by elements of a finite subset of the d-dimensional integer lattice, d≥1. Conditions of time reversibility are examined. It is shown that the model equilibrium distribution converges to a limit distribution as the indexing set expands to the whole lattice. The occupied site percolation problem is solved for the limit distribution. Two models with similar dynamics are also discussed.     

LRO in Lattice Systems of Linear Classical and Quantum Oscillators. Strong Nearest-Neighbor Pair Quadratic Interaction

For systems of one-component interacting oscillators on the d-dimensional lattice, d>1, whose potential energy besides a large nearest-neighbour (n-n) ferromagnetic translation-invariant quadratic term contains small non-nearest-neighbour translation invariant term, an existence of a ferromagnetic long-range order for two valued lattice spins, equal to a sign of oscillator variables, is established for sufficiently large magnitude g of the n-n interaction with the help of the Peierls type contour bound. The Ruelle superstability bound is used for a derivation of the contour bound.     

Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics in flat space

The solution of the euclidean Schrödinger equation in an arbitrary fermion sector of supersymmetric quantum mechanics in flat d-dimensional space is represented in terms of a markovian stochastic process extending the known stochastic process of the zero- or d-fermion sector.     

Quantum theory of a massless spinning particle

The classical and quantum mechanics of a free massless point-like fermion is presented. The action is invariant under local 1-dimensional reparametrizations and supersymmetry, as well as a large set of rigid symmetries, including conformal and chiral symmetries. An infinite set of rigid symmetries is derived starting from the chiral invariance, but only a finite subset does not vanish on shell. The BRST-transformations corresponding to thed=1 local symmetries are constructed and the quantization is performed. Finally, the BRST-cohomology is investigated and the conditions producing the physical states are derived.     

A mean spherical model with coulomb interactions. II. Correlations at a free surface

Studies of the mean spherical model with Coulomb interactions are continued, by considering a system on ad-dimensional lattice which is periodic ind–1 dimensions and has a free surface in the remaining dimension. It is shown explicitly that correlations along the free surface decay asy ^–d ind dimensions and show that the surface properties of this model are those expected for a charged system in its plasma phase.     

Magnetic system under a fluctuating field

We study nonequilibrium steady states, phase transitions and critical phenomena in a d-dimensional lattice model which represents a magnetic system under the action of a field fluctuating very rapidly with time. This induces competing kinetics which produces a sort of (dynamical) frustration which might occur also in some natural disordered systems. The exact solution for d = 1, partial exact results for d ≥ 2, and a comparison with some related models are reported.     

Density of states in a disordered solid,Borel-Padé analysis

Wegner's gauge-invariant model of the localization problem is treated on ad-dimensional lattice. Borel-Padé methods are used to analyze the perturbation series for the averaged one-particle Green function. Results for the density of states are obtained.     

Direct estimates on intersection probabilities of random walks

A study is made of intersection properties of independent random walks ind-dimensional lattice spaceZ ^d . A simple method is developed which makes it possible to estimate intersection probalities of two random walks with killing ratem directly. It is expected that the method can be generalized and extended to other issues.     

The complete N = 2, d = 6 supergravity with matter and yang-mills couplings

We derive all the quartic fermion terms in the action, and all the cubic fermion tterms in the transformation rules of the N = 2, d = 6 supergravity plus matter-coupled Yang-Mills system constructed by the authors in an earlier paper. We also show how compactification to the 4-dimensional Minkowski spacetime is always automatically realized even with fermionic condensates, based on the argument of scale covariance by Witten.