Metastates in disordered mean-field models: Random field and hopfield models

We rigorously investigate the size dependence of disordered mean-field models with finite local spin space in terms of metastates. Thereby we provide an illustration of the framework of metastates for systems of randomly competing Gibbs measures. In particular we consider the thermodynamic limit of the empirical metastate , whereμ _n (η) is the Gibbs measure in the finite volume {1,…,n} and the frozen disorder variableη is fixed. We treat explicitly the Hopfield model with finitely many patterns and the Curie-Weiss random field Ising model. In both examples in the phase transition regime the empirical metastate is dispersed for largeN. Moreover, it does not converge for a.e.η, but rather in distribution, for whose limits we given explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.     

Analogues of Non-Gibbsianness in Joint Measures of Disordered Mean Field Models

It is known that the joint measures on the product of spin-space and disorder space are very often non-Gibbsian measures, for lattice systems with quenched disorder, at low temperature. Are there reflections of this non-Gibbsianness in the corresponding mean-field models? We study the continuity properties of the conditional probabilities in finite volume of the following mean field models: (a) joint measures of random field Ising, (b) joint measures of dilute Ising, (c) decimation of ferromagnetic Ising. The conditional probabilities are functions of the empirical mean of the conditionings; so we look at the large volume behavior of these functions to discover non-trivial limiting objects. For (a) we find (1) discontinuous dependence for almost any realization and (2) dependence of the conditional probabilities on the phase. In contrast to that we see continuous behavior for (b) and (c), for almost any realization. This is in complete analogy to the behavior of the corresponding lattice models in high dimensions. It shows that non-Gibbsian behavior which seems a genuine lattice phenomenon can be partially understood already on the level of mean-field models.     

Hierarchical interfaces in random media II: The Gibbs measures

We continue the analysis of hierarchical interfaces in random media started in earlier work. We show that from the estimates on the renormalized random variables established in that work, it follows that these models possess unique Gibbs states describing mostly flat interfaces in dimensionD > 3, if the disorder is weak and the temperature low enough. In the course of the proof we also present very explicit formulas for expectations of local observables.     

Metastates in the Hopfield Model in the Replica Symmetric Regime

We study the finite dimensional marginals of the Gibbs measure in the Hopfield model at low temperature when the number of patterns, M, is proportional to the volume with a sufficiently small proportionality constant > 0. It is shown that even when a single pattern is selected (by a magnetic field or by conditioning), the marginals do not converge almost surely, but only in law. The corresponding limiting law is constructed explicitly. We fit our result in the recently proposed language of metastates which we discuss some length. As a byproduct, in a certain regime of the parameters and (the inverse temperature), we also give a simple proof of Talagrands recent result that the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky can be rigorously justified.     

Stochastic Symmetry-Breaking in a Gaussian Hopfield Model

We study a two-pattern Hopfield model with Gaussian disorder. We find that there are infinitely many pure states at low temperatures in this model, and that the metastate is supported on an infinity of symmetric pairs of pure states. The origin of this phenomenon is the random breaking of a rotation symmetry of the distribution of the disorder variables.     

Magnetic Field Dependent Tunneling of Atoms and Molecules in Non-Magnetic Disordered Solids

The low-temperature properties of disordered solids, such as glasses or crystals with certain substitutional defects are governed by atomic tunneling systems. Until recently it was believed that the dielectric properties of insulating materials devoid of magnetic impurities should not—or only very weakly—depend on external magnetic fields. In contrast, new experiments on glasses and crystalline defect systems show a pronounced magnetic field dependence of the dielectric properties of such materials at ultra-low temperatures. In particular, the low-frequency dielectric susceptibility and the amplitude of polarization echoes appear to be strongly affected by magnetic fields. These very surprising findings clearly indicate that atomic tunneling systems can couple to magnetic fields. We summarize the available data and discuss the possible origin of these intriguing phenomena.     

Spherical Model in a Random Field

We investigate the properties of the Gibbs states and thermodynamic observables of the spherical model in a random field. We show that on the low-temperature critical line the magnetization of the model is not a self-averaging observable, but it self-averages conditionally. We also show that an arbitrarily weak homogeneous boundary field dominates over fluctuations of the random field once the model transits into a ferromagnetic phase. As a result, a homogeneous boundary field restores the conventional self-averaging of thermodynamic observables, like the magnetization and the susceptibility. We also investigate the effective field created at the sites of the lattice by the random field, and show that at the critical temperature of the spherical model the effective field undergoes a transition into a phase with long-range correlations ∼r ^4−d .     

Analyticity of density of states in a gauge-invariant model for disordered electronic systems

Then-orbital gauge-invariant model of disordered electronic systems proposed by Wegner is studied in the regime of dominant diagonal disorder. Analyticity of the density of states is established in two cases: (a) when the number of orbitals is small, (b) when the number of orbitals is large and the energy is in the expected extended states region.     

The State(s) of Replica Symmetry Breaking: Mean Field Theories vs. Short-Ranged Spin Glasses

We prove the impossibility of recent attempts to decouple the Replica Symmetry Breaking (RSB) picture for finite-dimensional spin glasses from the existence of many thermodynamic (i.e., infinite-volume) pure states while preserving another signature RSB feature—space filling relative domain walls between different finite-volume states. Thus revisions of the notion of pure states cannot shield the RSB picture from the internal contradictions that rule out its physical correctness in finite dimensions at low temperature in large finite volume.     

Analyticity in Hubbard Models

The Hubbard model describes a lattice system of quantum particles with local (on-site) interactions. Its free energy is analytic when t is small, or t ²/U is small; here, is the inverse temperature, U the on-site repulsion, and t the hopping coefficient. For more general models with Hamiltonian H=V+T where V involves local terms only, the free energy is analytic when T is small, irrespective of V. There exists a unique Gibbs state showing exponential decay of spatial correlations. These properties are rigorously established in this paper.     

Localization and interaction in two dimensions: The disordered Tomonaga model

The effect of electron-electron interactions on the localization behavior of two-dimensional disordered systems is investigated in the framework of a generalized version of Tomonaga's model. The main result is a closed expression for the dynamical conductivity which allows to predict the critical electron concentration below which localization sets in as a function of microscopic parameters only. Experimental verification is possible, although difficult.     

On phase separation in the spherical model of a ferromagnet: Quasiaverage approach

The spherical model of a ferromagnet is investigated in the framework of the generalized quasiaverage approach where an external field positive in one half of a square lattice and negative in the other half is used. It is shown that in addition to the well-known critical point, a second one can be produced by the field. Although the main asymptotic of the free energy is analytic at this point, the next-to-leading asymptotic possesses a singularity here, as well as at the point where the free energy per site is nonanalytic. An order parameter of the model also has singularities at both critical points. The magnetization profile is studied at different scales. It is shown that (in an appropriate regime), below the new critical temperature the magnetization profile freezes, that is, becomes temperature independent.     

An Ultimate Frustration in Classical Lattice-Gas Models

A classical lattice-gas model is called frustrated if not all of its interactions can attain their minima simultaneously. The antiferromagnetic Ising model on the triangular lattice is a standard example.^{(1, 29)} However, in all such models known so far, one could always find nonfrustrated interactions having the same ground-state configurations. Here we constructed a family of classical lattice-gas models with finite-range, translation-invariant, frustrated interactions and with unique ground-state measures which are not unique ground-state measures of any finite-range, translation-invariant, nonfrustrated interactions.Our ground-state configurations are two-dimensional analogs of one-dimensional, most homogeneous,⁽¹³⁾ nonperiodic ground-state configurations of infinite-range, convex, repulsive interactions in models with devil's staircases.Our models are microscopic (toy) models of quasicrystals which cannot be stabilized by matching rules alone; competing interactions are necessary.     

“Golden Oldie”: The Spatially Homogeneous Cosmological Models

Schücking's program of calculating all spatially homogeneous dust-filled spacetimes is presented.     

Relaxation to Stationary Nonequilibrium States in Stochastic Ginzburg–Landau Models

We propose an approach to investigate properties of the time relaxation to stationary nonequilibrium states of correlation functions of stochastic Ginzburg–Landau models with noise (temperature of the reservoirs in contact with the system) changing in space. The formalism relates the stochastic expectations to correlation functions of an imaginary time field theory, and it allows us to study the nonlinear dynamics in terms of a field theory given by a perturbation of a Gaussian measure related to the (easier) linear dynamical problem. To show the usefulness of the formalism, we argue that a perturbative analysis within the integral representation is enough to give us the time relaxation rates of the correlations in some situations.     

Random-bond Ising chain in a transverse magnetic field: A finite-size scaling analysis

We investigate the zero-temperature quantum phase transition of the randombond Ising chain in a transverse magnetic field. Its critical properties are identical to those of the McCoy-Wu model, which is a classical Ising model in two dimensions with layered disorder. The latter is studied via Monte Carlo simulations and transfer matrix calculations and the critical exponents are determined with a finite-size scaling analysis. The magnetization and susceptibility obey conventional rather than activated scaling. We observe that the order parameter and correlation function probability distribution show a nontrivial scaling near the critical point, which implies a hierarchy of critical exponents associated with the critical behavior of the generalized correlation lengths.     

Long time tails in stationary random media II: Applications

In a previous paper we developed a mode-coupling theory to describe the long time properties of diffusion in stationary, statistically homogeneous, random media. Here the general theory is applied to deterministic and stochastic Lorentz models and several hopping models. The mode-coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse-grained diffusion coefficient and a coarse-grained free volume. For one-dimensional models these amplitudes can be evaluated, and the mode-coupling theory is shown to agree with exact solutions obtained for these models. For higher-dimensional Lorentz models the formal theory yields expressions which are difficult to evaluate. For these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers.Work supported by grant No. CHE 77-16308 from the National Science Foundation and by a Nato Travel Grant.     

Entropy Production in Non-Markovian Collision Models: Information Backflow vs. System-Environment Correlations

We investigate the irreversible entropy production of a qubit in contact with an environment modelled by a microscopic collision model in both Markovian and non-Markovian regimes. Our main goal is to contribute to the discussions on the relationship between non-Markovian dynamics and negative entropy production rates. We employ two different types of collision models that do or do not keep the correlations established between the system and the incoming environmental particle, while both of them pertain to their non-Markovian nature through information backflow from the environment to the system. We observe that as the former model, where the correlations between the system and environment are preserved, gives rise to negative entropy production rates in the transient dynamics, the latter one always maintains positive rates, even though the convergence to the steady-state value is slower as compared to the corresponding Markovian dynamics. Our results suggest that the mechanism underpinning the negative entropy production rates is not solely non-Markovianity through information backflow, but rather the contribution to it through established system-environment correlations.     

A mean-field,effective medium theory of random magnetic alloys. II. The random heisenberg model

A classical Heisen berg model is analysed. The interaction is of the RKKY type and only between sites randomly occupied by magnetic atoms. The possible phases are described at various temperatures and concentration of magnetic atoms. The procedure is realistic and not the ‘exactly’ solvable kind studied by earlier workers.