Decay of Correlations in Random-Cluster Models

We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that (here, d is the number of dimensions). Specifically, if throughout the subcritical phase, then for some α > 0. This implies the exponential decay of the two-point correlation function of subcritical Potts models, subject to a hypothesis of (at least) polynomial decay of this function. Similar results are known already for percolation and Ising models, and for Potts models when the number q of available states is sufficiently large; indeed the hypothesis of polynomial decay has been proved rigorously for these cases. In two dimensions, the hypothesis that is weaker than requiring that the susceptibility be finite, i.e., that the two-point function be summable. The principal new technique is a form of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, and Kesten. For the current application, this leads to an analysis of a first-passage problem for random-cluster models, and a proof that the associated time constant is strictly positive if and only if the tail of R decays exponentially. Received: 25 September 1996 / Accepted: 21 February 1997     

Haldane gap in the S = 1 Haldane-Shastry model

We present evidence that the S = 1 Haldane- Shastry model has a gap in the energy spectrum, extending to the case of a model with long-range interactions results previously obtained by Haldane and, numerically, by other authors for the case of short-range interactions. We studied the groundstate and first excited state using a modified Lanczos algorithm and obtained the low temperature behavior diagonalizing exactly small chains. The correlation functions decay exponentially with distance and the low-T susceptibility decays exponentially to zero. The gap is larger than for the Heisenberg model and the correlation functions decay faster indicating a smaller correlation length.     

High temperature expansions and dynamical systems

We study continuous and discrete spin systems on a lattice with random interactions of finite range. In particular for sufficiently large interactions we prove no spectral gap property in the high temperature region. Moreover we show that in two dimensions, if the temperature is sufficiently high and the probability of interaction to be large is small enough, we have an almost sure stretched exponential upper bound for the decay to equilibrium.Supported by EEC grant SC1-CT92-784.     

Ground States in Relatively Bounded Quantum Perturbations of Classical Lattice Systems

We consider ground states in relatively bounded quantum perturbations of classical lattice models. We prove general results about such perturbations (existence of the spectral gap, exponential decay of truncated correlations, analyticity of the ground state), and also prove that in particular the AKLT model belongs to this class if viewed on a large enough length scale. This immediately implies a general perturbation theory about this model. On leave from Institute for Information Transmission Problems, Moscow The author is an Irish Research Council for Science, Engineering and Technology Postdoctoral Fellow     

Dynamical Localization in Disordered Quantum Spin Systems

We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.     

On the exponential decay of correlation functions

We use the properties of subharmonic functions to prove the following results, First, for any lattice system with finite-range forces there is a gap in the spectrum of the transfer matrix, which persists in the thermodynamic limit, if the fugacityz lies in a regionE of the complex plane that contains the origin and is free of zeros of the grand partition function (with periodic boundary conditions) as the thermodynamic limit is approached. Secondly, if the transfer matrix is symmetric (for example, with nearest and next-nearest neighbor interactions in two dimensions), and if infinite-volume Ursell functions exist that are independent of the order in which the various sides of the periodicity box tend to infinity, then these Ursell functions decay exponentially with distance for all positivez inE. (For the Ising ferromagnet with two-body interactions, exponential decay holds forz inE even if the range of interaction is not restricted to one lattice spacing). Thirdly, if the interaction potential decays moreslowly than any decaying exponential, then so do all the infinite-volume Ursell functions, for almost all sufficiently small fugacities in the case of general lattice systems, and for all real magnetic fields in the case of Ising ferromagnets.     

Large deviation principle for spatial density of local observables from Gibbs distributions

We establish the large deviation principle characterising, in the thermodynamic limit, the exponential decay rates for the probabilities of macroscopic fluctuations of spatial densities generated by local observables from Gibbs lattice systems with absolutely summable interactions.     

Uniform andL 2 convergence in one dimensional stochastic Ising models

We study the rate of convergence to equilibrium of one dimensional stochastic Ising models with finite range interactions. We donot assume that the interactions are ferromagnetic or that the flip rates are attractive. The infinitesimal generators of these processes all have gaps between zero and the rest of their spectra. We prove that if one of these processes is observed by means of local observables, then the convergence is seen to be exponentially fast with an exponent that is any number less than the spectral gap. Moreover this exponential convergence is uniform in the initial configuration.The authors were partially supported by N.S.F. Grants DMS-8609944 and DMS-8611487, respectively. Also, the second author acknowledges supports from DAAL 03-86-K-0171     

Fractons in proteins: can they lead to anomalously decaying time autocorrelations?

Motivated by recent studies on the fractal nature of folded proteins, we analyze the time-dependent autocorrelation function < x (-->)(t).x(-->)(0) >of the distance between two points on a thermally vibrating fractal. Using fractons, the vibrational excitations of a fractal, we show that for both strongly underdamped and overdamped vibrations this correlation function decays anomalously, displaying a crossover from a nearly stretched exponential decay at short times to a slow algebraic decay at long times. Relationship to single molecule experiments is discussed.     

Asymptotic behavior of correlation functions for electric potential and field fluctuations in a classical one- component plasma

The correlations of the electric potential fluctuations in a classical one-component plasma are studied for large distances between the observation points. The two-point correlation function for these fluctuations is known to decay slowly for large distances, even if exponential clustering holds for the charge correlation functions. In this paper the asymptotic behavior of the generalk-point electric potential correlation functions is analyzed. Each of these correlation functions can be split into a reducible part, which is given by a sum of products of lower-order correlation functions, and a remaining irreducible part. It is shown, on the basis of an exponential clustering hypothesis for the charge correlation functions, that for allk3 the irreducible parts of the electric potential correlation functions decay faster than any inverse power of the distance, if one or more of the observation points move far away from the others. Hence, the two-point electric potential correlation function is the only one with a slow algebraic decay. The same statement holds for the correlation functions of the electric field fluctuations.     

CP violation and the absence of second class currents in charmless B decays

The absence of second class currents together with the assumption of the factorization for non-leptonic B decays provide new constraints on CP observables in the decay . The kinematics of this decay does not allow for interference between the oppositely charged resonances in the Dalitz plot as in . Nonetheless, under the assumption of factorization, the two-body time-dependent isospin analysis leads to a more robust extraction of the angle than in the isospin-pentagon analysis. The absence of second class currents might lead to enhanced direct CP violation and/or allows for a test of some assumptions made in the analysis in other decays like , and . The effects from non-factorizable contributions on the determination of are estimated by means of a numerical study. Received: 3 July 2001 / Revised version: 19 September 2001 Published online: 21 November 2001     

A two-dimensional isotropic quantum antiferromagnet with unique disordered ground state

We continue the study of valence-bond solid antiferromagnetic quantum Hamiltonians. These Hamiltonians are invariant under rotations in spin space. We prove that a particular two-dimensional model from this class (the spin-3/2 model on the hexagonal lattice) has a unique ground state in the infinite-volume limit and hence no Néel order. Moreover, all truncated correlation functions decay exponentially in this ground state. We also characterize all the finite-volume ground states of these models (in every dimension), and prove that the two-point correlation function of the spin-2 square lattice model with periodic boundary conditions has exponential decay.     

Valence bond ground states in isotropic quantum antiferromagnets

Haldane predicted that the isotropic quantum Heisenberg spin chain is in a massive phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.Supported in part by N.S.F. Grant PHY-80-19754. Fellow of the A.P. Sloan Foundation and the Canadian Institute for Advanced ResearchN.S.F. Post-doctoral FellowSupported in part by N.S.F. Grant PHY-85-15288-A01     

Long-range inverse two-spin correlations in one-dimensional Potts lattices

The inverse two-spin correlation function of a one-dimensional three-state Potts lattice with constant nearest-neighbor interactions in a uniform external field is derived exactly. It is shown that the external field induces long-range correlations. The inverse two-spin correlation function decays in a monotonie exponential fashion for a ferromagnetic lattice, while it decays in an oscillatory exponential fashion for an antiferromagnetic lattice. With no external field the inverse two-spin correlation function has a finite range equal to that of the interactions.     

Self-Similarity,Operators and Dynamics

We construct a large class of infinite self-similar (fractal, hierarchical or substitution) graphs and show, under a certain strong symmetry assumption, that the spectrum of the Laplacian can be described in terms of iterations of an associated rational function (so-called 'spectral decimation'). We prove that the spectrum consists of the Julia set of the rational function and a (possibly empty) set of isolated eigenvalues which accumulate to the Julia set. In order to obtain our results, we start with investigation of abstract spectral self-similarity of operators.     

Probing anomalous <Emphasis Type="Italic">Wtb</Emphasis> couplings in top pair decays

We investigate several quantities, defined in the decays of top quark pairs, which can be used to explore non-standard Wtb interactions. Two new angular asymmetries are introduced in the leptonic decay of top (anti)quarks. Both are very sensitive to anomalous Wtb couplings, and their measurement allows for a precise determination of the W helicity fractions. We also examine other angular and energy asymmetries, the W helicity fractions and their ratios, as well as spin correlation asymmetries, analysing their dependence on anomalous Wtb couplings and identifing the quantities which are most sensitive to them. It is explicitly shown that spin correlation asymmetries are less sensitive to new interactions in the decay of the top quark; therefore, when combined with the measurement of other observables, they can be used to determine the tt̄ spin correlation even in the presence of anomalous Wtb couplings. We finally discuss some asymmetries which can be used to test CP violation in tt̄ production and complex phases in the effective Wtb vertex.     

Radiative corrections to semileptonic decays of charged hyperons

We have studied the radiative corrections to the lepton energy spectrum in semileptonic hyperon decays. The calculation is performed relativistically for the baryons as well as for the leptons, under the assumption of the effective current-current interaction of the V-?A type for the baryonic part. We obtain the explicit formula of radiative corrections to the lepton energy spectrum which we can exactly evaluate in case of charged hyperon decays. Numerical values of the radiative corrections to the decays rate and the shape of the lepton energy spectrum are also given for some decay modes. It is shown that the spectral shape is little affected by the radiative corrections.     

Abundance of translation invariant pure states on quantum spin chains

We construct a set of translation invariant pure states of a quantum spin chain, which is w -dense in the set of all translation invariant states of the chain. Each of the approximating states has exponential decay of correlations, and is the unique ground state of a finite range Hamiltonian with a spectral gap above the ground state energy.     

Critical Points Inside the Gaps of Ground State Laminations for Some Models in Statistical Mechanics

We consider models of interacting particles situated in the points of a discrete set Λ. The state of each particle is determined by a real variable. The particles are interacting with each other and we are interested in ground states and other critical points of the energy (metastable states). Under the assumption that the set Λ and the interaction are symmetric under the action of a group G—which satisfies some mild assumptions—, that the interaction is ferromagnetic, as well as periodic under addition of integers, and that it decays with the distance fast enough, it was shown in a previous paper that there are many ground states that satisfy an order property called self-conforming or Birkhoff. Under some slightly stronger assumptions all ground states satisfy this order property. Under the assumption that the interaction decays fast enough with the distance, we show that either the ground states form a one dimensional family or that there are other Birkhoff critical points which are not ground states, but lying inside the gaps left by ground states. This alternative happens if and only if a Peierls–Nabarro barrier vanishes. The main tool we use is a renormalized energy. In the particular case that the set Λ is a one dimensional lattice and that the interaction is just nearest neighbor, our result establishes Mather’s criterion for the existence of invariant circles in twist mappings in terms of the vanishing of the Peierls–Nabarro barrier. The work of RdlL was supported by NSF grants. The work of EV was supported by GNAMPA and MIUR Variational Methods and Nonlinear Differential Equations.     

Application of the real space dynamic renormalization group method to the one-dimensional kinetic ising model

We apply the recently developed real space dynamic renormalization group method to the one-dimensional kinetic Ising model. We show how one can develop block spin methods that lead to recursion relations for the space and time dependent correlation functions that correspond to the observables for this system. We point out the importance of carefully choosing the appropriate parameters governing the behavior of individual blocks of spins and the necessity of worrying about the high temperature properties of the temperature recursion relations if one is to obtain the proper exponential decay of correlation functions at large distances away from the critical point at zero temperature. We systematically investigate the accuracy of our approximate recursion relations for various correlation functions by checking them against the known exact results. Our simple methods work surprisingly well over a wide range of temperatures, wavenumbers and frequencies.