Lee-Yang Theorem Research Articles

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.

Critical Behavior of Transport in Lattice and Continuum Percolation Models

It has been observed that the critical exponents of transport in the continuum, such as in the Swiss cheese and random checkerboard models, can exhibit nonuniversal behavior, with values different than the lattice case. Nevertheless, it is shown here that the transport exponents for both lattice and continuum percolation models satisfy the standard scaling relations for phase transitions in statistical mechanics. The results are established through a direct, analytic correspondence between transport coefficients for two component random media and the magnetization of the Ising model, which is based on the observation we made previously that both problems share the Lee-Yang property.

Hierarchical ferromagnetic vector spin model possessing the Lee-Yang property. Thermodynamic limit at the critical point and above

The hierarchical ferromagneticN-dimensional vector spin model as a sequence of probability measures onRN is considered. The starting element of this sequence is chosen to belong to the Lee-Yang class of measures that is defined in the paper and includes most known examples (ϕ4 measures, Gaussian measures, and so on). For this model, we probe two thermodynamic limit theorems. One of them is just the classical central limit theorem for weakly dependent random vectors. It describes the convergence of classically normed sums of spins when temperature is sufficiently high. The other theorem describes the convergence of “more than normally” normed sums that holds for some fixed temperature. It corresponds to the strong dependence of spins, which appears at the critical point of the model.

Some new results on Yang-Lee zeros of the Ising model partition function

We prove that for the Ising model on a lattice of dimensionality d ≥ 2, the zeros of the partition function Z in the complex μ plane (where μ = e −2 βH ) lie on the unit circle lhaiμlhai = 1 for a wider range of K nn′ = βJ nn′ than the range K nn′ ≥ 0 assumed in the premise of the Yang-Lee circle theorem. This range includes complex temperatures, and we show that it is lattice-dependent. Our results thus complement the Yang-Lee theorem, which applies for any d and any lattice if J nn′ ≥ 0. For the case of uniform couplings K nn′ = K, we show that these zeros lie on the unit circle lhaiμlhai = 1 not just for the Yang-Lee range 0 ≤ u ≤ 1, but also for (i) − u c,sq ≤ u ≤ 0 on the square lattice, and (ii) − u c,l ≤ u ≤ 0 on the triangular lattice, where u = z 2 = e −4 K , u c,sq = 3 − 2 3 2 , and u c,t = 1 3 . For the honeycomb, 3 × 12 2, and 4 × 8 2 lattices we prove an exact symmetry of the reduced partition functions, Z r( z, − μ) = Z r(− z, μ). This proves that the zeros of Z for these lattices lie on lhaiμlhai = 1 for −1 ≤ z ≤ 0 as well as the Yang-Lee range 0 ≤ z ≤ 1. Finally, we report some new results on the patterns of zeros for values of u or z outside these ranges.

Moderate Deviations of Dependent Random Variables Related to CLT

This paper consists of three-parts. In the first-part, we find a common condition-the $C^2$ regularity--both for CLT and for moderate deviations. We show that this condition is verified in two important situations: the Lee-Yang theorem case and the FKG system case. In the second part, we apply the previous results to the additive functionals of a Markov process. By means of Feynman-Kac formula and Kasto's analytic perturbation theory, we show that the Lee-Yang theorem holds under the assumption that 1 is an isolated, simple and the only eigenvalue with modulus 1 of the operator $P_1$ acting on an appropriate Banach space $(b\mathscr{E}, C_b(E), L^2 \cdots)$. The last part is devoted to some applications to statistical mechanical systems, where the $C^2$-regularity becomes a property of the pressure functionals and the two situations presented above become exactly the Lee-Tang theorem case and the FKG system case. We shall discuss in detail the ferromagnetic model and give some general remarks on some other models.

Generalized circle theorem on zeros of partition function at asymmetric first order transitions.

We present a generalized circle theorem which includes the Lee-Yang theorem for symmetric transitions as a special case. It is found that zeros of the partition function can be written in terms of discontinuities in the derivatives of the free energy. For asymmetric transitions, the locus of the zeros is tangent to the unit circle if the partition functions of the two phases are added up with unequal prefactors. This conclusion is substantiated by explicit calculation of zeros of the partition function for the Blume-Capel model near and at the triple line at low temperatures.

Epsilon -entropy and critical distortion of random fields

The link between epsilon -entropy for binary random fields and the Lee-Yang theorem in statistical mechanics makes it possible to prove the existence of positive critical distortion for most binary random fields with pair interactions and some with many body interactions. Lower bounds for critical distortion of random fields on two-dimensional (2-D) lattices have been obtained. In particular, the new lower bounds for some 2-D Ising models improve upon previously known bounds. >

D-component rotators as the classical limit of quantum SO(D) vector models

The author considers a class of spin systems whose single-site configuration space is an orbit of a representation of a compact Lie group G. For these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions. If a certain group-theoretic condition is satisfied, then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the 'corresponding' classical one. This condition is shown to be satisfied, in particular, for the D-component rotators when D is odd. The result could be useful for the extension of the Lee-Yang theorem to such models.

The Lee-Yang property for some isotropic spin models

The Lee-Yang property for some isotropic spin models

Random vectors with the Lee-Yang property

Random vectors with the Lee-Yang property

On the DLR equation for the ( λ: φ4: + b : φ2: + μφ, μ ≠ 0) 2 euclidean quantum field theory: The uniqueness theorem

The Dobrushin-Lanford-Ruelle (DLR) equation is studied in a certain space of measures in the case of 2-dimensional λ : φ 4: + b : φ 2: + μφ, μ ≠ 0, euclidean quantum field theory. The uniqueness theorem stating that for any μ ≠ 0 there exists unique, translationally invariant 8-regular solution of the corresponding DLR equation is proved. The theorem is proved by combining the Lee-Yang theorem with the general van Hove type theorem for infinite volume free energy densities.

Hierarchical model of a vector ferromagnet · self-similar block-spin distributions and the Lee-Yang theorem

For a translationally invariant model of a vector ferromagnet the approximating model possessing hierarchical structure is constructed. The equation for seld-similar distributions of block-spins in the hierarchical model is investigated in the framework of suitable function theory methods. In particular, the generalised Lee-Yang theorem is proved for arbitrary block-spin dimensionalities.

Yang-Lee zeros of the Ising model on the closed Cayley tree

The Yang-Lee zeros of the partition function of the ferro-, antiferro- and of the partially antiferromagnetic anisotropic Ising models defined on the closed symmetric Cayley tree are studied. The applicability of the Yang-Lee theorem to the antiferromagnetic systems is shown to be a consequence of the invariance of the unit circle under the Bethe-Peierls map. The relationship as well as the distinction between the set of zeros and the Julia set is established. The fractal dimension of the Julia set is shown to be equal to one in the low temperature phase and to be a decreasing function of the temperature in the paramagnetic phase of the three systems.

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.

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>Tc). 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.

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.

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.

Location of zeros in the complex temperature plane: absence of Lee-Yang theorem

In the Yang-Lee theory (1952) the occurrence of a phase transition at real (physical) values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane. As shown by Lee and Yang, these zeros fall on lines for a wide class of ferromagnetic models. In extensions of the Yang-Lee theory to the complex temperature plane, it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas. Though this happens to be true for the isotropic Ising model, the authors point out that there is in general no basis for this assumption. They discuss the location of the zeros of the partition function of the anisotropic Ising model in the complex temperature plane and show that they indeed always fall in areas.

Hierarchical vector model of a ferromagnet in the method of collective variables. The Lee-Yang theorem

Hierarchical vector model of a ferromagnet in the method of collective variables. The Lee-Yang theorem

Structure and motion of the Lee–Yang zeros

For an Ising model satisfying the Lee–Yang condition, the zeros of the partition function Z and those of the associated functions ZA in the space of imaginary magnetic fields at all lattice sites are determined by a single analytic hypersurface. The sense of motion of the zeros of Z as the interactions are varied can be related to the positions of the zeros of the ZA . Contrary to a plausible conjecture, it is not true that all of the zeros of Z in a uniform field tend towards the point ẑ=1 in the complex fugacity plane as the temperature is lowered, but it is possible that the first zero (that nearest to ẑ=1) has a monotone motion. Various simplicity and intertwining properties of the zeros of Z and ZA which generalize earlier results are proved by a new argument which makes direct use of the Lee–Yang property.