Site-bond percolation problems

We study a percolation process in which both sites and bonds are randomly blocked, independent of each other. In the Bethe lattice, the exact solution for the percolation threshold is found to be a hyperbola in thex-p plane, wherex andp are the respective probabilities of each site and bond being unblocked. Percolation threshold for a square and a simple cubic lattice is obtained by computer simulation. We also present a result obtained by a real-space renormalization group technique for the square lattice.     

Lattice simulations for light nuclei: Chiral effective field theory at leading order

We discuss lattice simulations of light nuclei at leading order in the chiral effective field theory. Using lattice pion fields and auxiliary fields, we include the physics of instantaneous one-pion exchange and the leading-order S-wave contact interactions. We also consider higher-derivative contact interactions which adjust the S-wave scattering amplitude at higher momenta. By construction our lattice path integral is positive definite in the limit of exact Wigner SU(4) symmetry for any even number of nucleons. This SU(4) positivity and the approximate SU(4) symmetry of the low-energy interactions play an important role in suppressing sign and phase oscillations in Monte Carlo simulations. We assess the computational scaling of the lattice algorithm for light nuclei with up to eight nucleons and analyze in detail calculations of the deuteron, triton, and helium-4.     

The influence of fixed spins on the thermodynamic behavior of an Ising ferromagnet

We study the thermodynamic behavior of a ferromagnetic Ising system on a Bethe lattice in the presence of given boundary conditions. More specifically, we study the interface of the system when the spins on half of the surface are fixed opposite to the spins on the other half. We find an interface width that remains finite in the whole range (0,T _c ), a feature due to the special topology of the Bethe lattice. We also study the case where the spin on a certain lattice site belonging to a domain is fixed in a direction opposite to the domain magnetization at all temperaturesT _c . We obtain the influence of that spin on the local magnetization, and we find that the fixed spin nucleates a local domain that extends over a distance of only a few lattice sites from it at all temperaturesT _c .     

The role of the modular pairs in the category of complete orthomodular lattice

We study the modular pairs of a complete orthomodular lattice i.e. a CROC. We propose the concept ofm-morphism as a mapping which preserves the lattice structure, the orthogonality and the property to be a modular pair. We give a characterization of them-morphisms in the case of the complex Hilbert space to justify this concept.     

Magnetic and non-magnetic ground states of the Kondo lattice

We use the variational method to investigate the ground state phase diagram of the Kondo lattice Hamiltonian for arbitraryJ/W, and conduction electron concentrationn _c (J is the Kondo coupling andW the bandwidth). We are particularly interested in the question under which circumstances the globally singlet (collective Kondo) Fermi liquid type ground state becomes unstable against magnetic ordering. For the collective Kondo singlet we use the lattice generalization of Yosida's wavefunction which implies the existence of a large Fermi volume, in accordance with Luttinger's theorem. Using the Gutzwiller approximation, we derive closed-form results for the ground state energy at arbitraryJ/W andn _c, and for the Kondo gap atn _c=1. We introduce simple trial states to describe ferromagnetic, antiferromagnetic, and spiral ordering in the small-J (RKKY) regime, and Nagaoka type ferromagnetism at largeJ/W. We study three particular cases: a band with a constant density of states, and the (tight binding) linear chain, and square lattice periodic Kondo models. We find that the lattice enhancement of the Kondo effect, which is described in our theory of the Fermi liquid state, pushes the RKKY-to-nonmagnetic phase boundary to much smaller values ofJ/W than it was previously thought. In our study of the square lattice case, we also find a region of itinerant, Nagaoka-type ferromagnetism at largeJ/W forn _c 1/3.     

Compactly Generated de Morgan Lattices, Basic Algebras and Effect Algebras

We prove that a de Morgan lattice is compactly generated if and only if its order topology is compatible with a uniformity on L generated by some separating function family on L. Moreover, if L is complete then L is (o)-topological. Further, if a basic algebra L (hence lattice with sectional antitone involutions) is compactly generated then L is atomic. Thus all non-atomic Boolean algebras as well as non-atomic lattice effect algebras (including non-atomic MV-algebras and orthomodular lattices) are not compactly generated.     

The zero-point energy in the molecular hydrogen crystal

We propose a modified Einstein approximation to describe zero-point energy vibrations in a quantum crystal. Our aim was to develop a computationally cheap tool suitable for lattice structure optimisation. As in the classical Einstein model the representative atom vibrates in an effective potential due to the surrounding atoms of the crystal; the atoms however are not strictly placed at the positions corresponding to the crystal potential energy minima but their positions are described by the quantum mechanical density distributions. The effective potential computed that way is suitable for the application in solid para-hydrogen in contrast to the normal (unmodified) Einstein approximation. We compute the cohesive energy of the para-hydrogen crystal and perform lattice structure optimisation. The hexagonal closed packed is more stable than the fcc closed packed lattice and the lattice constants obtained are in very good agreement with the experimental values.     

Exactly Solvable Model of Reactions on a Random Catalytic Chain

In this paper we study a catalytically-activated A+A0 reaction taking place on a one-dimensional regular lattice which is brought in contact with a reservoir of A particles. The A particles have a hard-core and undergo continuous exchanges with the reservoir, adsorbing onto the lattice or desorbing back to the reservoir. Some lattice sites possess special, catalytic properties, which induce an immediate reaction between two neighboring A particles as soon as at least one of them lands onto a catalytic site. We consider three situations for the spatial placement of the catalytic sites: regular, annealed random, and quenched random. For all these cases we derive exact results for the partition function, and the disorder-averaged pressure per lattice site. We also present exact asymptotic results for the particles' mean density and the system's compressibility. The model studied here furnishes another example of a 1D Ising-type system with random multisite interactions which admits an exact solution.     

Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund transformations for rational solutions of the Painlevé VI equation. These transformations act on a family of Painlevé VI tau functions. They are obtained from reducing the Hirota bilinear equations that describe the relation between certain points in the 3 component polynomial KP Grassmannian. In this way we obtain transformations that act on the root lattice of A⁵. We also show that this A⁵ root lattice can be related to the F₄⁽¹⁾ root lattice. We thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized by the elements of this F₄⁽¹⁾ root lattice.     

Random walks on the Bethe lattice

We obtain random walk statistics for a nearest-neighbor (Pólya) walk on a Bethe lattice (infinite Cayley tree) of coordination numberz, and show how a random walk problem for a particular inhomogeneous Bethe lattice may be solved exactly. We question the common assertion that the Bethe lattice is an infinite-dimensional system.Supported in part by the U.S. Department of Energy.     

Effects of three-body atomic interaction and optical lattice on solitons in quasi-one-dimensional Bose-Einstein condensate

We make use of a coordinate-free approach to implement Vakhitov-Kolokolov criterion for stability analysis in order to study the effects of three-body atomic recombination and lattice potential on the matter-wave bright solitons formed in Bose-Einstein condensates. We analytically demonstrate that (i) the critical number of atoms in a stable BEC soliton is just half the number of atoms in a marginally stable Townes-like soliton and (ii) an additive optical lattice potential further reduces this number by a factor of √1 − bg ₃ with g ₃ the coupling constant of the lattice potential and b = 0.7301.      

Temperature and spatial diffusion of atoms cooled in a 3D lin ⊥lin bright optical lattice

We present a detailed experimental study of a three-dimensional lin⊥lin bright optical lattice. Measurements of the atomic temperature and spatial diffusion coefficients are reported for different angles between the lattice beams, i.e. for different lattice constants. The experimental findings are interpreted with the help of numerical simulations. In particular we show, both experimentally and theoretically, that the temperature is independent of the lattice constant. Received 5 July 2001 and Received in final form 13 August 2001     

Critical exponents of Manhattan Hamiltonian walks in two dimensions,from Potts andO(n) models

We consider a set of Hamiltonian circuits filling a Manhattan lattice, i.e., a square lattice with alternating traffic regulation. We show that the generating function (with fugacityz) of this set is identical to the critical partition function of aq-state Potts model on an unoriented square lattice withq ^1/2 =z. The set of critical exponents governing correlations of Hamiltonian circuits is derived using a Coulomb gas technique. These exponents are also found to be those of an O(n) vector model in the low-temperature phase withn =q ^1/2 =z. The critical exponents in the limitz = 0 are then those of spanning trees (q= 0) and of dense polymers (n=0,T < T_c), corresponding to a conformal theory with central chargeC = –2. This shows that the Manhattan orientation and the Hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of Hamiltonian walks.     

Lattice Uniformities on Effect Algebras

Let L be a lattice ordered effect algebra. We prove that the lattice uniformities on L which make uniformly continuous the operations − and + of L are uniquely determined by their system of neighborhoods of 0 and form a distributive lattice. Moreover we prove that every such uniformity is generated by a family of weakly subadditive [0,+∞]-valued functions on L.     

The correlation between strain and electronic structure in monolayer thick films

We explote and empirical relationship between strain (and lattice constant) and the valence electronic structure for mercury, bromine and iron overlayers. These overlayers have a range of lattice constants. For the Hg overlayers, all share a common cubic crystallography. We generally observe that the smaller the overlayer lattice constant, the greater the energy separation betweend-bands (Hg and Fe) orp-bands (Br). These results have important implications in relating electronic structure to fundamental properties such as magnetism. In addition, the film thickness limits for pseudomorphic growth calculated from the bulk properties are consistent with the experimental studies of Hg pseudomorphic growth on Ag (100) at 90 K.     

Q¯ modes in the Quark-Gluon Plasma

We study the evolution of heavy quarkonium states with temperature in a Quark-Gluon Plasma (QGP) by evaluating an in-medium Qˉ T-matrix within a reduced Bethe-Salpeter equation in S- and P-wave channels. The interaction kernel is extracted from finite-temperature QCD lattice calculations of the singlet free energy of a Qˉ pair. Quarkonium bound states are found to gradually move across the Qˉ threshold after which they rapidly dissolve in the hot system. We calculate Euclidean-time correlation functions and compare to results from lattice QCD. We also study finite-width effects in the heavy-quark propagators.     

Provable First-Order Transitions for Nonlinear Vector and Gauge Models with Continuous Symmetries

We consider various sufficiently nonlinear vector models of ferromagnets, of nematic liquid crystals and of nonlinear lattice gauge theories with continuous symmetries. We show, employing the method of Reflection Positivity and Chessboard Estimates, that they all exhibit first-order transitions in the temperature, when the nonlinearity parameter is large enough. The results hold in dimension 2 or more for the ferromagnetic models and the RP^N–1 liquid crystal models and in dimension 3 or more for the lattice gauge models. In the two-dimensional case our results clarify and solve a recent controversy about the possibility of such transitions. For lattice gauge models our methods provide the first proof of a first-order transition in a model with a continuous gauge symmetry.Acknowledgement We thank in particular E. Domany and A. Schwimmer who suggested to us to consider lattice gauge models, and also L. Chayes, D. v.d. Marel, A. Messager, K. Netocný, S. Romano and A. Sokal for stimulating discussions and/or correspondence. S.S. acknowledges the financial support of the RFFI grant 03-01-00444.     

Topological order in an exactly solvable 3D spin model

We study a 3D generalization of the toric code model introduced recently by Chamon. This is an exactly solvable spin model with six-qubit nearest-neighbor interactions on an FCC lattice whose ground space exhibits topological quantum order. The elementary excitations of this model which we call monopoles can be geometrically described as the corners of rectangular-shaped membranes. We prove that the creation of an isolated monopole separated from other monopoles by a distance R requires an operator acting on Ω(R²) qubits. Composite particles that consist of two monopoles (dipoles) and four monopoles (quadrupoles) can be described as end-points of strings. The peculiar feature of the model is that dipole-type strings are rigid, that is, such strings must be aligned with face-diagonals of the lattice. For periodic boundary conditions the ground space can encode 4g qubits where g is the greatest common divisor of the lattice dimensions. We describe a complete set of logical operators acting on the encoded qubits in terms of closed strings and closed membranes.     

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.     

The deconfining phase transition in lattice quantum chromodynamics

We present a large-scale Monte Carlo calculation of the deconfining phase transition temperature in lattice quantum chromodynamics without fermions. Using the Wilson action, the transition temperature as a function of the lattice couplingg is consistent with scaling behavior dictated by the perturbativeα function for 6/g²>6.15. Speaker at the conference; on leave from CRIP, Budapest.