Zeros of partition functions via correlation inequalities

We show analyticity of the pressure for some classical ferromagnetic systems in the region ¦Im ¦ < Re of the external field. The proof, via correlation inequalities, is simpler than existing proofs for the Lee and Yang region {Re 0} and applies, without any approximation procedure, to more general continuous spin variables, e.g., distributed as , where _2n is an arbitrary real number and the other parameters are positive. It also applies directly to plane rotators in the region ¦Im ¦ ¦Re ¦ (Euclidean norms), but the proof will be given in a subsequent article, together with new inequalities between truncated correlation functions.     

Zeros of the Potts model partition function on Sierpinski graphs

We calculate zeros of the q-state Potts model partition function on m  ?th-iterate Sierpinski graphs, _Sm$S_m$, in the variable q and in a temperature-like variable, y  . We infer some conjectured asymptotic properties of the loci of zeros in the limit m→∞$m\to \infty$ and relate these to thermodynamic properties of the q  -state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, _S∞$S_{\infty }$.     

Zeros of the finite-size scaling region partition function for a model with a wetting transition

We derive a finite-size scaling representation for the partition function for an Onsager-Temperley string model with a wetting transition, and analyze the zeros of this partition function in the complex scaled coupling parameter of relevance. The system models the one-dimensional interface between two phases in a rectangular two-dimensional region (x, y) ²,–L yL,oxN. The two phases are at coexistence. The string or interface has a surface tension 2KkT per unit length and an extra Boltzmann weighta per unit length if it touches the surfaces aty=±L. There is a critical valuea _c=1/2K and fora>a _c the string is confined to one of the surfaces, while fora a _c the string moves roughly in the rectangular region. The finite-size scaling parameters are =a _c ² N/L ² and =L(a–a _c)/a _c ² . We find that for || large, the zeros of the scaled partition function lie close to the lines arg()=±/4 with re()>0. We discuss the motion of all the zeros as changes by both analytic and numerical arguments.     

New inequalities for Ising ferromagnets

It is shown that for Ising ferromagnets which obey the Lee-Yang theorem the Ursell functions or cumulants of the magnetization variable at nonzero external field satisfy series of inequalities. Several relations connecting Ursell functions with nonzero and zero field are derived.     

The Gaussian inequality for multicomponent rotators

The Gaussian inequality is proven for multicomponent rotators with negative correlations between two spin components. In the case of one-component systems, the Gaussian inequality is shown to be a consequence of Lebowitz' inequality. For multicomponent models, the Gaussian inequality implies that the decay rate of the truncated correlation (or Schwinger) functions is dominated by that of the two-point function. Applied to field theory, these inequalities give information on the absence of bound states in the (₁ ² + _{1 2})² model.     

Correlation inequalities and the Kosterlitz-Thouless transition for anisotropic rotators

I give a proof of the Kosterlitz-Thouless transition for sufficiently anisotropic (J _zJ _x ^–1 =J _zJ _y ^–1 <2q ^–(JKT)^–J) two-dimensionalN-component rotators (N 3). The method is based on Wells' inequality and is related to mean field Gaussian inequalities.     

Gaussian fluctuations for the magnetization of Lee-Yang ferromagnets at zero external field

We consider the fluctuations of the block spin magnetization normalized by the square root of the considered number of spins in a block for Lee-Yang ferromagnets. It is established that the fluctuations are Gaussian when ^d at zero external field whenever the susceptibility is finite (i.e., above the critical temperature) and converges to the second derivative of the pressure at zero field. The validity of this fluctuation-dissipation condition is known to hold for a large class of Lee-Yang models, including, for instance, classical Heisenberg ferromagnets.     

Correlation inequalities and uniqueness of the equilibrium state for the plane rotator ferromagnetic model

We derive new inequalities for the plane rotator ferromagnetic model and use them to obtain the following results:

1. If the model is isotropic, the derivability of the free energy as function of the magnetic fieldh implies the existence of a unique translation invariant Gibbs state and if furthermoreh=0 all Gibbs states are invariant by rotation of the spins.
2. If the model is anisotropic the above assertion holds forh non-zero.
3. If the model is anisotropic then there are at most two extremal translation invariant Gibbs states for almost all values of the anisotropy parameter.

     

The partition function zeros of the anisotropic Ising model with multisite interactions on a zigzag ladder

It is shown that the spin- anisotropic Ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin- Axial-Next-Nearest-Neighbor Ising (ANNNI) model with multisite interactions. The partition function zeros of the ANNNI model with multisite interactions are investigated. A comprehensive analysis of the partition function zeros of the ANNNI model with and without three-site interactions on a zigzag ladder is done using the transfer matrix method. Analytical equations for the distribution of the partition function zeros in the complex magnetic field (Yang-Lee zeros) and temperature (Fisher zeros) planes are derived. The Yang-Lee and Fisher zeros distributions are studied numerically for a variety of values of the model parameters. The densities of the Yang-Lee and Fisher zeros are studied and the corresponding edge singularity exponents are calculated. It is shown that the introduction of three-site interaction terms in the ANNNI model leads to a simpler distribution of the partition function zeros. For example, the Yang-Lee zeros tend to a circular distribution when increasing by modulus the three-site interactions term coefficient. It is found that the Yang-Lee and Fisher edge singularity exponents are universal and equal to each other, .     

Restrictions on the phase diagrams for a large class of multisite interaction spin systems

Through the proof of two very general theorems involving Ising spin systems with multisite interactions, specific regions of the complexh plane, whereh is the external magnetic field, are shown to be free of zeros of the partition function. Hence in these regions the partition function is analytic and phase transitions are absent. As an example: for systems with ferromagnetic multisite interactions involving even numbers of sites, no phase transition occurs outside of an interval centered on the origin of the realh axis and of the form (–C(T),C(T)), whereT is the temperature. For T0,C(T)0 and phase transitions can occur only ath=0.     

One-particle subspaces in the stochasticXY model

We study the spectrum of the generatorH of the Glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1/. We construct two so-called one-particle subspacesH _± forH and describe the spectrum of the generator in these subspaces. As a consequence we find time asymptotics of the correlations for the Glauber dynamics.     

Zeros of the partition function using theorems of Ruelle

Theorems of Ruelle which provide a technique for finding regions of the relevant complex planes free of zeros of the partition function are used to study certain Ising spin systems. Of particular interest is the antiferromagnetic triangle lattice system with h≠0 and systems having three-body interactions.     

Partition function zeros for aperiodic systems

The study of zeros of partition functions, initiated by Yang and Lee, provides an important qualitative and quantitative tool in the study of critical phenomena. This has frequently been used for periodic as well as hierarchical lattices. Here, we consider magnetic field and temperature zeros of Ising model partition functions on several aperiodic structures. In 1D, we analyze aperiodic chains obtained from substitution rules, the most prominent example being the Fibonacci chain. In 2D, we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices.     

The Ising model on a closed Cayley tree: Zeros of the partition function

It is shown that the zeros of the partition function of an Ising model defined on the closed Cayley tree have a particularly simple structure in the complex interface interaction plane.     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

GKS inequalities for arbitrary spin ising ferromagnets

Elementary proofs of the first and second Griffiths-Kelly-Sherman (GKS) inequalities are given for higher-spin Ising systems with a Hamiltonian containing only a quadratic form in the spin variables and integer powers of single spin variables. These proofs are obtained using Gaussian random variables. A slight generalization of previous results has been obtained in that the coefficients of the even powers of the spin variables are allowed to be negative.Work supported by NSF Grant GP-36564X.     

Gaussian bounds for correlations in lattice spin systems

In quite generalN-component ferromagnetic spin systems, it is proved that an arbitrary correlation function is bounded by the corresponding correlation function of a Gaussian model. The bound is useful for the analysis of high-temperature behavior of the system. Similar bounds for truncated correlation functions are also obtained for a class of single-component spin systems.     

Mean-field bounds and correlation inequalities

I prove a new correlation inequality for a class ofN-component classical ferromagnets (1N4). This inequality implies that the correlation functions decay exponentially and the spontaneous magnetization is zero, above the mean-field critical temperature.     

Spectral anomalies of focused hollow Gaussian beams at the geometrical focal plane

On the basis of diffraction integral and the expansion of the hard-aperture function into a finite series of complex Gaussian functions, an approximate expression for spatially fully coherent polychromatic hollow Gaussian beams passing through aperture lens is obtained. Detailed numerical results indicate that remarkable spectral changes always occurs near the points where the field amplitude has zero value. The effects of truncation parameter, Fresnel number and the beam order on spectral shifts and spectral switches are investigated numerically.     

Properties of the Zeros of the Sum of two Polynomials

Some properties—including relations having a Diophantine character—of the zeros of the sum of two polynomials are reported.