Griffiths' singularities in diluted ising models on the Cayley tree

The Griffiths singularities are fully exhibited for a class of diluted ferromagnetic Ising models defined on the Cayley tree (Bethe lattice). For the deterministic model the Lee-Yang circle theorem is explicitly proven for the magnetization at the origin and it is shown that, in the thermodynamic limit, the Lee-Yang singularities become dense in the entire unit circle for the whole ferromagnetic phase. Smoothness (infinite differentiability) of the quenched magnetizationm at the origin with respect to the external magnetic field is also proven for convenient choices of temperature and disorder. From our analysis we also conclude that the existence of metastable states is impossible for the random models under consideration.     

Some remarks on the location of zeroes of the partition function for lattice systems

We use techniques which generalize the Lee-Yang circle theorem to investigate the distribution of zeroes of the partition function for various classes of classical lattice systems.     

A general Lee-Yang theorem for one-component and multicomponent ferromagnets

We show that any measure on ⁿ possessing the Lee-Yang property retains that property when multiplied by a ferromagnetic pair interaction. Newman's Lee-Yang theorem for one-component ferromagnets with general single-spin measure is an immediate consequence. We also prove an analogous result for two-component ferromagnets. ForN-component ferromagnets (N 3), we prove a Lee-Yang theorem when the interaction is sufficiently anisotropic.Research supported in part by NSF grant PHY 78-25390 A01Research supported in part by NSF grant PHY 78-23952     

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.     

Infinitely divisible distribution functions of class L and the Lee-Yang theorem

It is shown that the free-energy density of a large class of ferromagnets satisfying the Lee-Yang property is to be connected with the limit characteristic function of a suitably renormalized sum of independent and non-identically distributed random variables. Using the canonical representation formulae of such characteristic functions, various chains of inequalities are derived for the Ursell functions.     

An application of the GHS inequalities to show the absence of phase transition for Ising spin systems

We show that the GHS inequalities can be used instead of the Lee-Yang circle theorem to prove that there is no phase transition for the ν-dimensional Ising model in the presence of a (non-zero) external field.     

On the nature of the Lee-Yang measure for Ising ferromagnets

For the two- and three-dimensional nearest neighbors Ising model in the presence of a magnetic field, we study numerically asymptotic properties of the set of orthogonal polynomials associated with the Lee-Yang measure. This provides an insight into the nature of this measure near its end points, on the Lee-Yang circle. We introduce a smoothness index which analyzes the structure of the measure. Its value is found to be equal to 2 within 10^–3 for all the models tested in two and three dimensions, at any temperatures. The results strongly suggest the absence of any singular part (continuous or pure point) in the measure, even in dimension 3. We also confirm, using a different method, known results on the behavior of the measure near its end points.Research Assistant of the Belgian National Fund for Scientific Research.On leave of absence from CEN-Saclay France     

Scaling relations for logarithmic corrections

Multiplicative logarithmic corrections to scaling are frequently encountered in the critical behavior of certain statistical-mechanical systems. Here, a Lee-Yang zero approach is used to systematically analyze the exponents of such logarithms and to propose scaling relations between them. These proposed relations are then confronted with a variety of results from the literature.     

Rigorous results for general ising ferromagnets

Several new results are given concerning the Lee-Yang theorem, the GHS inequality, and spin-1/2 approximations for general Ising ferromagnets, and the extension of these results to vector spin models is discussed.Research supported in part by National Science Foundation Grant MPS 74-04870 A01.     

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 functions of descendants in the scaling Lee-Yang model

Correlation functions of the composite field \(T\bar T\) in the scaling Lee-Yang model are studied. Using the analytic expression for form factors of this operator recently proposed by Delfino and Niccoli hep-th/0407142 [1], we show numerically that the constraints on the \(T\bar T\) expectation values obtained by Zamolodchikov hep-th/0401146     

On the uniqueness of the equilibrium state for plane rotators

We study the classical statistical mechanics of the plane rotator, and show that there is a unique translation invariant equilibrium state in zero external field, if there is no spontaneous magnetization. Moreover, this state is then extremal in the equilibrium states. In particular there is a unique phase for the two dimensional rotator, and a unique phase for the three dimensional rotator above the critical temperature. It is also shown that in a sufficiently large external field the Lee-Yang theorem implies uniqueness of the equilibrium state.     

New rigorous theorems on Ising ferromagnets

The Lee-Yang theorem was extended to some correlation functions of Ising ferromagnets. Every “fugacity” zero of the spin function (partition function or correlation functions) was proved to be non-degenerate in the completely connected Ising ferromagnet, except for infinite temperature.     

Construction of modular branching functions from Bethe's equations in the 3-state Potts chain

We use the single-particle excitation energies and the completeness rules of the 3-state antiferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of theD ₄ representation ofZ ₄ parafermions which complements the bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented.     

Phase Transitions with Four-Spin Interactions

Using an extended Lee-Yang theorem and GKS correlation inequalities, we prove, for a class of ferromagnetic multi-spin interactions, that they will have a phase transition (and spontaneous magnetization) if, and only if, the external field h = 0 (and the temperature is low enough). We also show the absence of phase transitions for some nonferromagnetic interactions. The FKG inequalities are shown to hold for a larger class of multi-spin interactions.     

General theory of lee-yang zeros in models with first-order phase transitions

We present a general, rigorous theory of Lee-Yang zeros for models with first-order phase transitions that admit convergent contour expansions. We derive formulas for the positions and the density of the zeros. In particular, we show that, for models without symmetry, the curves on which the zeros lie are generically not circles, and can have topologically nontrivial features, such as bifurcation. Our results are illustrated in three models in a complex field: the low-temperature Ising and Blume-Capel models, and the q-state Potts model for large q.     

L e e-Yang circl e analysis of e+ e− and $p\ov erlin e{p}$ g en eraliz ed multiplicity distribution

We study the evolution of Lee-Yang zeros structure of generalized multiplicity distribution (GMD) in high energy collision. Starting our study with electron-positron e ⁺ e ⁻ scattering data, we extend the study by Chan and Chew (Z. Phys. C 55:503, 1992) on TASSO and AMY multiplicity data for , 22, 34.8, 43.6 and 57 GeV to the ones from DELPHI and OPAL Collaboration for , 133, 161, 172, 183 and 189 GeV. We compare the results with the Lee-Yang structure for proton-antiproton at , 546 and 900 GeV from UA5 Collaboration. Our preliminary result shows that there is indeed a change in the shape and size of the Lee-Yang zeros with increasing energy, accompanied by the development of the so-called “ear”-like structure in the Lee-Yang plot. We expect that the development of this “ear”-like structure is related to the “shoulder” structure in the multiplicity data, which further indicates an ongoing phase transition from soft to semihard scattering. We also extend our prediction to LHC’s  TeV. Insert your abstract here.     

Classification of phase transitions in small systems

We present a classification scheme for phase transitions in finite systems like atomic and molecular clusters based on the Lee-Yang zeros in the complex temperature plane. In the limit of infinite particle numbers the scheme reduces to the Ehrenfest definition of phase transitions and gives the right critical indices. We apply this classification scheme to Bose-Einstein condensates in a harmonic trap as an example of a higher order phase transition in a finite system and to small Ar clusters.