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     

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     

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.     

Remarks on the relation between the Lee-Yang circle theorem and the correlation inequalities

We investigate the relation between the Lee-Yang circle theorem and the correlation inequalities. These results are general and independent of models. General properties of the partition functions which belong to the Lee-Yang class are given.     

Fractal dimensions and homeomorphic conjugacies

We investigate the behavior of the spectrum of singularities associated with the invariant measure of some dynamical systems under nonsmooth coordinate changes. When the homeomorphic conjugacy is not Lipschitz continuous, we discuss how its singularities can affect the whole set of generalized fractal dimensions. We give applications to homeomorphisms that conjugate critical circle maps with irrational (golden mean) winding numbers. We present numerical studies corroborating the theoretical predictions.     

Singularities in Gravitational Collapse with Radial Pressure

We analyze spherical dust collapse with non-vanishing radial pressure, II, and vanishing tangential stresses. Considering a barotropic equation of state, II = , we obtain an analytical solution in closed form—which is exact for = –1, 0, and approximate otherwise—near the center of symmetry (where the curvature singularity forms). We study the formation, visibility, and curvature strength of singularities in the resulting spacetime. We find that visible, Tipler strong singularities can develop from generic initial data. Radial pressure alters the spectrum of possible endstates for collapse, increasing the parameter space region that contains no visible singularities, but cannot by itself prevent the formation of visible singularities for sufficiently low values of the energy density. Known results from pressureless dust are recovered in the = 0 limit.     

Percolation systems away from the critical point

This article reviews some effects of disorder in percolation systems away from the critical density p _c. For densities below p _c, the statistics of large clusters defines the animals problem. Its relation to the directed animals problem and the Lee-Yang edge singularity problem is described. Rare compact clusters give rise to Griffiths singularities in the free energy of diluted ferromagnets, and lead to a very slow relaxation of magnetization. In biased diffusion on percolation clusters, trapping in dead-end branches leads to asymptotic drift velocity becoming zero for strong bias, and very slow relaxation of velocity near the critical bias field.     

p + ip-wave pairing symmetry at type-II van Hove singularities

Based on the random phase approximation calculation in two-orbital honeycomb lattice model, we investigate the pairing symmetry of Ni-based transition-metal trichalcogenides by electron doping access to type-II van Hove singularities (vHs). We find that chiral even-parity d + id-wave (E_g) state is suppressed by odd-parity p + ip-wave (E_u) state when electron doping approaches the type-II vHs. The type-II vHs peak in density of states (DOS) enables to strengthen the ferromagnetic fluctuation, which is responsible for triplet pairing. The competition between antiferromagnetic and ferromagnetic fluctuation results in pairing phase transition from singlet to triplet pairing. The Ni-based transitionmetal trichalcogenides provide a promising platform to unconventional superconductor emerging from electronic DOS.     

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.     

Theory of monomer-dimer systems

We investigate the general monomer-dimer partition function,P(x), which is a polynomial in the monomer activity,x, with coefficients depending on the dimer activities. Our main result is thatP(x) has its zeros on the imaginary axis when the dimer activities are nonnegative. Therefore, no monomer-dimer system can have a phase transition as a function of monomer density except, possibly, when the monomer density is minimal (i.e.x=0). Elaborating on this theme we prove the existence and analyticity of correlation functions (away fromx=0) in the thermodynamic limit. Among other things we obtain bounds on the compressibility and derive a new variable in which to make an expansion of the free energy that converges down to the minimal monomer density. We also relate the monomer-dimer problem to the Heisenberg and Ising models of a magnet and derive Christoffell-Darboux formulas for the monomer-dimer and Ising model partition functions. This casts the Ising model in a new light and provides an alternative proof of the Lee-Yang circle theorem. We also derive joint complex analyticity domains in the monomer and dimer activities. Our considerations are independent of geometry and hence are valid for any dimensionality.Work supported by National Science Foundation Grant GP-26526.     

Lifshitz tails and long-time decay in random systems with arbitrary disorder

In random systems, the density of states of various linear problems, such as phonons, tight-binding electrons, or diffusion in a medium with traps, exhibits an exponentially small Liftshitz tail at band edges. When the distribution of the appropriate random variables (atomic masses, site energies, trap depths) has a delta function at its lower (upper) bound, the Lifshitz singularities are pure exponentials. We study in a quantitative way how these singularities are affected by a universal logarithmic correction for continuous distributions starting with a power law. We derive an asymptotic expansion of the Lifshitz tail to all orders in this logarithmic variable. For distributions starting with an essential singularity, the exponent of the Lifshitz singularity itself is modified. These results are obtained in the example of harmonic chains with random masses. It is argued that analogous results hoid in higher dimensions. Their implications for other models, such as the long-time decay in trapping problems, are also discussed.     

Exchange bias in nearly perpendicularly coupled ferromagnetic/ferromagnetic system

Exchange bias phenomena appear not only in ferromagnetic/antiferromagnetic systems but also in ferromagnetic/ferromagnetic systems in which two layers are nearly perpendicularly coupled. We investigated the origin of the symmetry-breaking mechanism and the relationship between the exchange bias and the system's energy parameters. We compared the results of computational Monte Carlo simulations with those of theoretical model calculation. We found that the exchange bias exhibited nonlinear behaviors, including sign reversal and singularities. These complicated behaviors were caused by two distinct magnetization processes depending on the interlayer coupling strength. The exchange bias reached a maximum at the transition between the two magnetization processes.     

Tunneling singularities in the open Hubbard chain

We study singularities in the I–V characteristics for sequential tunneling from resonant localized levels (e.g. a quantum dot) into a one-dimensional electron system described by a Hubbard model. Boundary conformal field theory together with the exact solution of the Hubbard model subject to boundary fields allow to compute the exponents describing the singularity arising when the energy of the local level is tuned through the Fermi energy of the wire as a function of electron density and magnetic field. For boundary potentials with bound states a sequence of such singularities can be observed.     

String theory and cosmological singularities

In general relativity space-like or null singularities are common: they imply that ‘time’ can have a beginning or end. Well-known examples are singularities inside black holes and initial or final singularities in expanding or contracting universes. In recent times, string theory is providing new perspectives of such singularities which may lead to an understanding of these in the standard framework of time evolution in quantum mechanics. In this article, we describe some of these approaches.      

Griffiths singularities in unbinding of strongly disordered polymers

Griffiths singularities occurring in the unbinding of strongly disordered heteropolymers are studied. A model with two randomly distributed binding energies, -1 and -v, is introduced and studied analytically by analyzing the Lee-Yang zeros of the partition sum. It is demonstrated that in the limit v--> infinity the model exhibits a Griffiths type singularity at a temperature T(G)=O(1) corresponding to melting of long homogeneous domains of the low binding energy. For finite v>1 the model is expected to exhibit an additional, unbinding, transition at a high temperature T(M)=O(v).     

飞秒激光在锥形镀膜探针传输中相位奇异的研究

利用数值模拟的方法对飞秒激光脉冲中光波振荡一个周期内不同时刻的光场进行了计算、比较, 获得了输出端平面内各空间点处光场的最大值及达到最大值时对应的时间, 得到了探针近场的振幅和相位分布. 在研究振幅分布的过程中发现, 在探针输出端的有些区域振幅始终为零, 即在光波振荡的一个周期内这些区域始终为相位奇异点; 通过分析相位分布得到了相位奇异随时间的演化及相位奇异的特点.     

On the genericity of spacetime singularities

We consider here the genericity aspects of spacetime singularities that occur in cosmology and in gravitational collapse. The singularity theorems (that predict the occurrence of singularities in general relativity) allow the singularities of gravitational collapse to be either visible to external observers or covered by an event horizon of gravity. It is shown that the visible singularities that develop as final states of spherical collapse are generic. Some consequences of this fact are discussed.      

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.     

LETTER: Quantum Singularity of Quasiregular Spacetimes

Some of the mildest singularities in classical general relativity are shown to be singular quantum mechanically as well. A class of the mild, topological singularities known as quasiregular singularities remains singular when probed by quantum wave packets. These static spacetimes possessing dislocations and disclinations are quantum-mechanically singular since the spatial portion of the wave operator is not essentially self-adjoint and thus the evolution of a test quantum wave packet is not uniquely determined by the initial wave function.