Griffiths Inequalities Research Articles

Correlation Inequalities for Schrödinger Operators

The purpose of the present paper is to analyze correlation structures of the ground states of the Schrodinger operator. We construct Griffiths inequalities for the ground state expectations by applying operator-theoretic correlation inequalities. As an example of such an application, we study the ground state properties of Schrodinger operators.

Spectral properties of the Ruelle operator on the Walters class over compact spaces

Recently the Ruelle–Perron–Frobenius theorem was proved for Hölder potentials defined on the symbolic space , where (the alphabet) M is any compact metric space. In this paper, we extend this theorem to the Walters space , in similar general alphabets. We also describe in detail an abstract procedure to obtain the Fréchet analyticity of the Ruelle operator under quite general conditions and we apply this result to prove the analytic dependence of this operator on both Walters and Hölder spaces. The analyticity of the pressure functional on Hölder spaces is established. An exponential decay of the correlations is shown when the Ruelle operator has the spectral gap property.A new (and natural) family of Walters potentials (on a finite alphabet derived from the Ising model) which do not have an exponential decay of the correlations is presented. Because of the lack of exponential decay, for such potentials there is an absence of the spectral gap for the Ruelle operator. The key idea in proving the lack of exponential decay of the correlations is the Griffiths–Kelly–Sherman inequalities.

Quantum Griffiths Inequalities

We present a general framework of Griffiths inequalities for quantum systems. Our approach is based on operator inequalities associated with self-dual cones and provides a consistent viewpoint of the Griffiths inequality. As examples, we discuss the quantum Ising model, quantum rotor model, Bose--Hubbard model, Hubbard model, and spin-1/2 quantum Heisenberg model. We present a model-independent structure that governs the correlation inequalities.

Critical exponents of the quark-gluon bags model with a critical endpoint

The critical indices α', β, γ', and δ of the Quark Gluon Bags with Surface Tension Model that has a critical endpoint are calculated and compared with the exponents of other models. These indices are expressed in terms of the most general parameters of the model. Despite the usual expectations the found critical indices do not depend on the Fisher exponent τ and on the parameter [symbol: see text] which relates the mean bag surface to its volume. The scaling relations for the obtained critical exponents are verified, and it is demonstrated that for the standard definition of the index α' the Fisher and the Griffiths scaling inequalities are not fulfilled in general case, whereas the Liberman scaling inequality is always obeyed. This is not surprising for the phase diagram with the asymmetric properties of pure phases, but the present model also provides us with the first and explicit example that the specially defined index α'(s) does not recover the scaling relations as well. Therefore, here we suggest the physically motivated definition of the index α' = α'(c) and demonstrate that such a definition recovers the Fisher scaling inequality, while it is shown that the Griffiths inequality should be generalized for the phase diagram with the asymmetric properties. The critical exponents of several systems that belong to different universality classes are successfully described by the parameters of the present model, and hence its equation of state can be used for a variety of practical applications.

Griffith–Kelly–Sherman Correlation Inequalities for Generalized Potts Model

Inspired by the work of D.G.Kelly and S.Sherman on general Griffiths inequalities on correlations in Ising ferromagnets, we formulate and prove Griffith–Kelly–Sherman-type inequalities for the ferromagnetic Potts model with a general number q of local states. We take as local state space for the q-state Potts model the set Fc = { − l, − l + 1, ⋯ ,l − 1,l},where \(l=\frac{q-1}{2}\). The important properties of Fc for what follows are that |Fc| = q and Fc = − Fc.

Griffiths Inequalities for Ising Spin Glasses on the Nishimori Line

The Griffiths inequalities for Ising spin glasses are proved on the Nishimori line with various bond randomness which includes Gaussian and $\pm J$ bond randomness. The proof for Ising systems with Gaussian bond randomness has already been carried out by Morita et al, which uses not only the gauge theory but also the properties of the Gaussian distribution, so that it cannot be directly applied to the systems with other bond randomness. The present proof essentially uses only the gauge theory, so that it does not depend on the detail properties of the probability distribution of random interactions. Thus, the results obtained from the inequalities for Ising systems with Gaussian bond randomness do also hold for those with various bond randomness, especially with $\pm J$ bond randomness.

On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems

The existence of an unbounded order parameter (magnetization) is established for a broad class of lattice Gibbs (equilibrium) systems of linear oscillators interacting via a strong pair nearest-neighbor polynomial potential and other many-body potentials. The considered systems are characterized by a general polynomial short-range interaction potential energy that generates Gibbs averages satisfying two generalized GKS inequalities.

Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States

Quantum models on the hyper-cubic d-dimensional lattice of spin- 1 particles interacting with linear oscillators are shown to have three ferromagnetic ground state order parameters. Two order parameters coincide with the magnetization in the first and third directions and the third one is a magnetization in a continuous oscillator variable. The proofs use a generalized Peierls argument and two Griffiths inequalities. The class of spin- oscillator Hamiltonians considered manifest maximal ordering in their ground states. The models have relevance for hydrogen-bond ferroelectrics. The simplest of these is proven to have a unique Gibbsian ground state.

Inequalities for the Local Energy of Random Ising Models

We derive a rigorous lower bound on the average local energy for the Ising model with quenched randomness. The result is that the lower bound is given by the average local energy calculated in the absence of all interactions other than the one under consideration. The only condition for this statement to hold is that the distribution function of the random interaction under consideration is symmetric. All other interactions can be arbitrarily distributed including non-random cases. A non-trivial fact is that any introduction of other interactions to the isolated case always leads to an increase of the average local energy, which is opposite to ferromagnetic systems where the Griffiths inequality holds. Another inequality is proved for asymmetrically distributed interactions. The probability for the thermal average of the local energy to be lower than that for the isolated case takes a maximum value on the Nishimori line as a function of the temperature. In this sense the system is most stable on the Nishimori line.

Griffiths Inequalities in the Nishimori Line

The Griffiths inequalities for Ising spin-glass models with Gaussian randomness of non-vanishing mean are proved using the properties of the Gaussian distribution and the gauge symmetry of the system. These inequalities imply that the correlation functions are non-negative and monotonic along the Nishimori line in the phase diagram. From this result, the existence of the thermodynamic limit for the correlation functions and the free energy is proved under free and fixed boundary conditions. Relations between the location of multi-critical points are also derived for different lattices.

Griffiths inequalities for the Gaussian spin glass

The Griffiths inequalities for Ising spin-glass models with Gaussian randomness of non-vanishing mean are proved using properties of the Gaussian distribution and gauge symmetry of the system. These inequalities imply that correlation functions are non-negative and monotonic along the Nishimori line in the phase diagram. From this result, the existence of thermodynamic limit for correlation functions and pressure is proved under free and fixed boundary conditions. Relations between the location of multicritical points are also derived for different lattices.

Right and wrong near critical endpoints

The implications of convexity (or thermodynamic stability) for isothermal density (or composition) phase diagrams near critical endpoints are investigated. In particular, Schreinemakers’ rules for the geometry of three coexisting phases in a space of densities are refined and extended physically to apply at and in the vicinity of a critical endpoint. Seemingly plausible phase diagrams are presented which violate the extended rules (and, in most cases, the Second Law of Thermodynamics). The requirements of convexity, supplemented for Ising-related systems by the Griffiths–Kelly–Sherman inequalities, also restrict the signs of important expansion coefficients of thermodynamic scaling functions for criticality and provide bounds on universal critical amplitude ratios.

Index of authors and papers to volume 206

The partition function of a double layer classical nearest neighbor XY-model is expressed in terms of three coupled Coulomb gases. Renormalization-group equations, based on this representation, are derived and used in combination with a Griffiths inequality, to discuss the phase diagram for arbitrary inter-layer coupling.

Inequalities and many phase transitions in ferromagnetic systems

A general construction of ferromagnetic systems with many phase transitions is given. It is based on two new results: an extension of one of the GKS inequalities to not necessarily ferromagnetic interactions, and a uniqueness of the Gibbs state theorem for perturbations of some simple systemsat all temperatures.

On the positivity of correlations in nonequilibrium spin systems

We consider Ising spin systems, equivalently lattice gases evolving under discrete- or continuous-time Markov processes, i.e., “stochastic cellular automata” or “interacting particle systems.” We show that for certain spin-flip probabilities or rates and suitable initial states the expectation values of products of spin variables taken at equal or different times are nonnegative; they satisfy the same inequalities as the equal-time correlations of ferromagnetic systems in equilibrium (first Griffiths inequality). Extensions of FKG inequalities to time-displaced correlations are also discussed.

Griffiths inequalities for noninteractingN-vector (classical heisenberg) models and applications to interacting systems

An order relation for tensors is defined. With this ordering it is shown that in noninteractingN-vector models 〈σAσB〉−〈σA〉〈σB〉 is positive. Applications to interacting models include a proof for the alignment of spins and the subadditivity of the free energy.

On the group structure, GKS and FKG inequalities for Ising models

The FKG inequality is applied to a lattice of states of the Ising model which are related to subgroups of the group of spin variables. The resulting correlation inequalities ar exactly the GKS inequalities.

Symmetries and correlation inequalities for classicaln-vector models

We describe a new class of single spin measures on then-dimensional sphereS r n of radiusr (n ⩽ 4) for which Lebowitz-type [J. Lebowitz,J. Stat. Phys.16:463 (1977)] inequalities hold. This is achieved by an appropriate parametrization ofS r n . The above class includes the uniform measures onxs e ℝRn ∶ρ ⩽ ¦x¦ ⩽r for any 0 ⩽νp ⩽ r. The second topic of this paper is an abstract formulation of the first Griffiths inequality [R. B. Griffiths,J. Math. Phys.8:478 (1967)] and the underlying symmetry property.

Renormalization group studies of phase diagrams of Ising and XY-model films of variable thickness

We compare the critical behavior of spin- 1 2 Ising model (ordinary phase transition in two dimensions) and classical XY-model (topological phase transition in two dimensions) films, with two flat surfaces and nearest neighbor couplings K S between surface spins and K B between all others, as a function of film thickness. We carry out a real space Migdal-style renormalization in two stages. In the bulk stage the film is first renormalized towards a double layer, with the renormalized parameters as inter- and intra-layer couplings. Then the double layer is renormalized with those couplings as initial parameters. From the RG-equations for the bulk stage we find a tricritical point, not only for the Ising model (in which case it is well known) but also for the XY-model. It signals the existence of distinct surface and bulk transitions for sufficiently large values of K S K B . For the Ising model the complete program can be carried out and the phase diagram for films of arbitrary thickness is constructed. For the double layer XY-model a sufficiently complex Migdal-style renormalization appears to be unfeasible, presumably due to the possible presence of strings between the layers. Therefore, in an alternative approach, two representations for the partition function of the double layer XY-model are given. The system can be described in terms of its topological excitations, i.e., vortices on each layer and strings, either closed or terminating in vortices, between them. The system is also written as three coupled Coulomb gases. Based on this representation a renormalization group is found, and used, together with a Griffiths inequality for the correlation functions, to obtain information on the phase diagram. If there is a single phase transition, there is one phase where the correlations exhibit power law behavior and another where they fall off exponentially. The transition temperature increases monotonously with interlayer coupling to twice its value in two dimensions, but the nature of the phase transition for any finite inter-layer coupling appears to be different from that at zero coupling. We suggest that this is associated with the behavior of the internal strings. These results should be relevant for the renormalization of the film with isotropic bulk couplings as well as for layered systems of finite thickness, with different inter- and intra-layer couplings.

Critical behavior of the double layer classical XY-model

The partition function of a double layer classical nearest neighbor XY-model is expressed in terms of three coupled Coulomb gases. Renormalization-group equations, based on this representation, are derived and used in combination with a Griffiths inequality, to discuss the phase diagram for arbitrary inter-layer coupling.