Ising model in the high density limit

It is proved that the free energy per spin in the thermodynamic limit of an Ising model on a lattice with coordination numberz approaches the classical Curie-Weiss free energy in the limitz→∞. The infinite spacial dimension limit of nearest neighbour lattice models is a special case of this result.     

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.     

Mixing properties in lattice systems

We study mixing or spatial cluster properties and some of their consequences in classical lattice systems, in particular complete regularity and the weaker notion of strong mixing. Introducing the notion of reflection positivity as a generalization ofT-positivity of [1], we construct a generalized transfer matrixP and relate complete regularity to a spectral gap inP. It is shown that all reflection invariant Ising systems with n.n. and ferromagnetic n.n.n. interaction satisfy reflection positivity. For Ising ferromagnets with reflection positivity, exponential decay of the truncated 2-point function implies complete regularity. In particular, the 2-dimensional spin-1/2 Ising model is completely regular, except at the critical point. This complements a result of [2] that strong mixing fails at the critical point of this model and in this case verifies the suggestion of Jona-Lasinio [3] that critical behaviour should be linked with failure of strong mixing. We then show that strong mixing imposes severe restrictions on the possible form of limits of block spins. Strong mixing in each direction allows onlyindependent Gaussians as non-zero limit if the 2-point function exists; strong mixing in a single direction only will allow infinitely divisible distributions.     

Decay of correlations in surface models

A convergent low-temperature expansion for a variety of models of twodimensional surfaces is presented. It yields existence of the thermodynamic limit for the pressure and correlation functions as well as analyticity inz =e ^– In addition, the estimates give exponential decay of truncated correlations, which proves the existence of a gap in the spectrum of the transfer matrix below the ground state eigenvalue. Two particular examples included in the general framework are the solid-on-solid and discrete Gaussian models.Supported in part by the National Science Foundation under grant No. PHY 79-16812.     

Mathematical properties of position-space renormalization-group transformations

Properties of position-space or cell-type renormalization-group transformations from an Ising model object system onto an Ising model image system, of the type introduced by Niemeijer, van Leeuwen, and Kadanoff, are studied in the thermodynamic limit of an infinite lattice. In the case of a KadanofF transformation with finitep, we prove that if the magnetic field in the object system is sufficiently large (i.e., the lattice-gas activity is sufficiently small), the transformation leads to a well-defined set of image interactions with finite norm, in the thermodynamic limit, and these interactions are analytic functions of the object interactions. Under the same conditions the image interactions decay exponentially rapidly with the geometrical size of the clusters with which they are associated if the object interactions are suitably short-ranged. We also present compelling evidence (not, however, a completely rigorous proof) that under other conditions both the finite- and infinite-p (majority rule) transformations exhibit peculiarities, suggesting either that the image interactions are undefined (i.e., the transformation does not possess a thermodynamic limit) or that they fail to be smooth functions of the object interactions. These peculiarities are associated (in terms of their mathematical origin) with phase transitions in the object system governed not by the object interactions themselves, but by a modified set of interactions.Supported in part by NSF Grant No. DMR 76-23071.     

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.     

On aC*-algebra approach to phase transition in the two-dimensional Ising model

We investigate the state on theC*-algebra of Pauli spins on a one-dimensional lattice (infinitely extended in both directions) which gives rise to the thermodynamic limit of the Gibbs ensemble in the two-dimensional Ising model (with nearest neighbour interaction). It is shown that the representation of the Pauli spin algebra associated with the state is factorial above and at the known critical temperature, while it has a two-dimensional center below the critical temperature. As a technical tool, we derive a general criterion for a state of the Pauli spin algebra corresponding to a Fock state of the Fermion algebra to be primary. We also show that restrictions of two quasifree states of the Fermion algebra to its even part are equivalent if and only if the projection operatorsE ₁ andE ₂ (on the direct sum of two copies of the basic Hilbert space) satisfy the following two conditions: (1)E ₁ ?E ₂ is in the Hilbert-Schmidt class, (2)E ₁ ∧ (1 ?E ₂) has an even dimension, where the even-oddness of dimE ₁ ∧ (1 ?E ₂) is called ?₂-index ofE ₁ andE ₂ and is continuous inE ₁ andE ₂ relative to the norm topology.     

Constant coupling approximation for antiferromagnets with mixed ferro- and antiferromagnetic interactions

The constant-coupling approximation is extended to an antiferromagnetic spin-$\text{1}\text{2}$ system with two distinct anisotropic exchange interactions. The thermodynamic properties such as the transition temperature, magnetization, susceptibility and specific heat are discussed for three special cases: (i) ferro- and antiferromagnetic Ising interactions, (ii) isotropic ferro- and antiferromagnetic Heisenberg interactions; and (iii) isotropic ferromagnetic Heisenberg interactions and antiferromagnetic Ising interactions, allowing in each case for two different nearest-neighbour coupling constants. Numerical calculations have been performed for a layer structure with z = 6 intraplanar and z' = 6 interplanar nearest neighbours and the results are compared with those obtained in other approximations. Applying the theory to FeCl₂, the exchange constants are evaluated. It is shown that the calculated magnitudes of the interactions strongly depend upon the exchange-interaction model assumed.     

Critical dynamics of the three-dimensional Ising model: A Monte Carlo study

Extensive Monte Carlo simulations have been performed to analyze the dynamical behavior of the three-dimensional Ising model with local dynamics. We have studied the equilibrium correlation functions and the power spectral densities of odd and even observables. The exponential relaxation times have been calculated in the asymptotic one-exponential time region. We find that the critical exponentz=2.09 ±0.02 characterizes the algebraic divergence with lattice size for all observables. The influence of scaling corrections has been analyzed. We have determined integrated relaxation times as well. Their dynamical exponentz _int agrees withz for correlations of the magnetization and its absolute value, but it is different for energy correlations. We have applied a scaling method to analyze the behavior of the correlation functions. This method verifies excellent scaling behavior and yields a dynamical exponentz _scal which perfectly agrees withz.     

The φ4 lattice field theory as an asymptotic expansion about the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins, an asymptotic expansion of expectations about the Ising limit is established in inverse powers of the bare coupling constant λ. In the thermodynamic limit (N → ∞), the expansion is expected to be valid in the noncritical region of the Ising system.     

On a soluble model of an antiferromagnetic chain with Dzyaloshinsky interactions. II

Time-dependent properties of the one-dimensional XY model with Dzyaloshinsky interactions in the presence of a magnetic field in the z direction are investigated. Explicit expressions are derived for the time correlation functions of the z components of two spins and the time auto-correlation functions of M^z (= the z component of the magnetization). The ergodic behaviour of M^z in the thermodynamic limit is discussed in some detail. Furthermore an exact expression is derived for the time dependence of the average of M^z in a typical nonequilibrium situation. Finally the frequency-dependent susceptibility is evaluated.     

The Potts Model Built on Sand

We consider theq = 4 Potts model on the square lattice with an additional nonlocal interaction. That interaction arises from the choice of the reference measure taken to be the uniform measure on the recurrent configurations for the abelian sandpile model. In that reference measure some correlation functions have a power-law decay. We investigate the low-temperature phase diagram and we prove the existence of a single stable phase with exponential decay of correlations. For all boundary conditions the density of 4 in the infinite volume limit goes to one as the temperature tends to zero.     

Thermodynamic properties of the φ4 lattice field theory near the Ising limit

For a d-dimensional φ⁴ lattice field theory consisting of N spins with nearest-neighbor interactions, the partition function is transformed for large bare coupling constant λ into an Ising-like system with additional neighbor interactions. For d = 2 a mean field approximation is then used to estimate the difference in critical temperature between the lattice φ⁴ field theory and its Ising limit (λ = ∞). Expansions are obtained for the susceptibility and specific heat. The critical exponents are shown to be identical to the Ising exponents.     

Fluctuation susceptibility relations for classical spin systems

We prove identities between integrated Ursell functions and derivatives of the pressure in the thermodynamic limit, for multicomponent classical spin systems which obey the Lee-Yang theorem and some form of Gaussian domination, when the susceptibility is finite (T>T _c). Following Refs. 3 and 4, we view the moment generating function of the magnetization as the inverse of an infinitely divisible characteristic function. Fluctuation susceptibility relations of all orders then follow by bounding the corresponding cumulants, taken in zero external field. High-order cumulants are bounded in terms of the susceptibility using Gaussian and Simon's inequalities for short-range interactions.     

Ising models on the pentagon lattice

We solve inhomogeneous Ising models on the pentagon lattice using the transfer matrix formalism. As two special cases we study the ferromagnetic and the fully frustrated antiferromagnetic model on this lattice. The ferromagnet shows a phase transition at someT _c>0 with the usual Ising behaviour. In the frustrated case no transition occurs at any temperature due to frustration. Frustration also causes a nonvanishing rest entropy. We also calculate the spin-spin-correlation for large distance in both cases. In the ferromagnetic model we thus get the magnetization and the expected algebraic (exponential) decay of the correlations at (above)T _c. The correlations of the frustrated model decay exponentially for all temperatures, includingT=0, indicating that evenT=0 belongs to the disordered high temperature phase. Superimposed to the exponential decay the correlation shows an interesting oscillatory behaviour with temperature dependent wave number, i.e. an incommensurate structure.Work performed within the research program of the Sonderforschungsbereich 125, Aachen-Jülich-Köln     

Partition function zeros and the three-dimensional Ising spin glass

We have computed the exact partition function of the 3D Ising spin glass on lattices of effective size 3×3×L_z, 4×4×L_z, and 5×5×L_z forL _z up to 9, and several random bond configurations. Studying the distribution of zeros of the associated partition functions, we find further evidence that these systems have a singularity in the thermodynamic limit.     

Correlations in inhomogeneous Ising models

Using general methods developed in a previous treatment we study correlations in inhomogeneous Ising models on a square lattice. The nearest neighbour couplings can vary both in strength and sign such that the coupling distribution is translationally invariant in diagonal direction. We calculate correlations parallel to the layering in the diagonally layered model with periodv=2, the so-called “general square lattice” model (GS). If the model has a finite critical temperature,T _c>0, we have a spontaneous magnetization belowT _c vanishing atT _c with the Ising exponent β=1/8. AtT _c correlations decay algebraically with critical exponnet η=1/4 and exponentially forT>T _c. In the frustrated case we have oscillatory behaviour superposed on the exponential decay where the wavevector of the oscillations changes at some “disorder temperature”T _D(>T _c) from commensurate to temperature-dependent in commensurate periods. If the critical temperature vanishes,T _c=0 we always have exponential decay at finite temperatures, while atT=T _c=0 we encounter either long-range order or algebraic decay with critical index η=1/2, i.e.T=0 is thus a critical point.     

Generalized transformation for decorated spin models

The paper discusses the transformation of decorated Ising models into an effective undecorated spin model, using the most general Hamiltonian for interacting Ising models including a long range and high order interactions. The inverse of a Vandermonde matrix with equidistant nodes [−s,s] is used to obtain an analytical expression of the transformation. This kind of transformation is very useful to obtain the partition function of decorated systems. The method presented by Fisher is also extended, in order to obtain the correlation functions of the decorated Ising models transforming into an effective undecorated Ising model. We apply this transformation to a particular mixed spin-(1/2, 1) and (1/2, 2) square lattice with only nearest site interaction. This model could be transformed into an effective uniform spin-S square lattice with nearest and next-nearest interaction, furthermore the effective Hamiltonian also includes combinations of three-body and four-body interactions; in particular we considered spin 1 and 2.     

On mean field equations for spin glasses

In this paper we study rigorously the random Ising model on a Cayley tree in the limit of infinite coordination numberz 8. An iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice. In this way we obtain local properties of the model in the (thermodynamic) limit of an infinite number of shells. When the coupling constants are independent Gaussian random variables the SK expressions emerge as stable fixed points of our scheme and provide a valid local mean-field theory of spin glasses in which negative local entropy (at low temperatures) while perfectly possible mathematically may still perhaps be physically undesirable. Finally we examine the TAP equations and show that if the average over bond disorder and the limitz 8 are actually performed, one recovers our iterative scheme and hence the SK equations also in the thermodynamic limit.On leave from Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia.