The Yang-Lee edge singularity in spherical models

The density of Yang-Lee zeros in the thermodynamic limit is discussed for ferromagnetic spherical models of general dimensionalities and arbitrary range of interaction. In all cases the zeros lie on the imaginary axis in the complex magnetic field planeH=H+iH with a density (H) that exhibits a square root singularity(H) (H-H ₀), with=1/2, as the edge of the gap atH=H ₀(T) is approached forT>T _c. WhenTT _c one hasH ₀(T)(TT _c ) with critical exponent=+.Supported by the National Science Foundation in part through the Materials Science Center at Cornell University.     

Analyticity properties and power law estimates of functions in percolation theory

We consider percolation on the sites of a graphG, e.g., a regulard-dimensional lattice. All sites ofG are occupied (vacant) with probabilityp (respectively,q=1–p), independently of each other.W denotes the cluster of occupied sites containing a fixed site (which will usually be taken to be the origin) andW the cardinality ofW. The percolation probability is the probability that #W=, i.e.,(p)=P _p{# W=}. Some critical values ofp,p _H andp _T, are defined, respectively, as the smallest value ofp for which(p)> 0, and for which the expectation of #W is infinite. Formally,p _H=inf {p(p)>0} andp _T=inf{p E _p{#W}=}. We show for fairly general graphsGthat ifp _T, thenP _P{#W n} decreases exponentially inn. For the special casesG =G ₀= the simple quadratic lattice andG ₁= the graph which corresponds to bond-percolation on ², we obtain upper and lower bounds for(p) of the formC¦p¦-P _H¦, and bounds forEp{#W} of the formC¦p–p _H¦^–. We also investigate smoothness properties of (p)=E _p{number of clusters per site} =E _p {(#W)^–1; (#W) 1}. This function was introduced by Sykes and Essam, who assumed that (·) has exactly one singularity, namely, atp=p _H. For the graphsG ₀ andG ₁, (i.e., site or bond percolation on ²) we show that (p) is analytic atp p _H and has two continuous derivatives atp=p _H. The emphasis is on rigorous proofs.Research supported by the NSF through a grant to Cornell University.     

Critical thermodynamics of dysprosium ethyl sulphate,a pure dipolar Ising ferromagnet

Using a SQUID magnetometer the magnetization of the Ising ferromagnet dysprosium ethyl sulphate has been measured in the temperature range 0.015–3K and in external fields 0.004–30 Oe. The results atT0.4 K (precritical regime) confirm the pure dipolar nature of the magnetic interactions. Particular interest has been paid to the immediate neighbourhood of the critical pointT _c =118.17 mK andH=0 where the zero-field susceptibility(T) aboveT _c , the spontaneous magnetizationM _s (T), and the magnetization atT _c ,M _c (H), can be described by mean field laws modified by logarithmic singularities. The critical amplitudes, B, andD of, M _s andM _c , respectively, satisfy the relation (2/3B ²)^1/3=D. All these results agree with predictions of (i) the Larkin-Khmel'nitskii theory and (ii) exact solutions of the renormalization group equations for 3-dimensional dipolar Ising ferromagnets.A project of the Sonderforschungsbereich 65 Festkörperspektroskopie Darmstadt-Frankfurt, financed by special funds of the Deutsche Forschungsgemeinschaft     

Note on the interfacial tension of phase-separated polymer solutions

Earlier theoretical calculations of the interfacial tension of phase-separated polymer solutions as a function of the degree of polymerizationN and the temperatureT, based partly on the mean-field approximation, had led toN ^–1/4(1–T/T _c )^3/2 for fixedN1 andT approaching the critical solution temperatureT _c It is here remarked that the scaling procedure of de Gennes then modifies this toN ^–0.37(1–T/T _c )^1.26, which is in close accord with the experimentalN ^–0.44(1–T/T _c )^1.26. The simplest mean-field picture yieldsN ^–1/2(1–T/T _c )^3/2.     

Weak localization and fluctuation electron-electron interaction in disordered indium films

The effects of perpendicular magnetic fieldsH on the temperature dependence of the resistanceR(T) (T<20 K), the superconducting transition temperatureT _c and the fluctuation superconductivity (T«T _c ) have been investigated on high-resistance (R _>1 kOhm) indium films prepared by simultaneous condensation of the metal and hydrogen onto a substrate cooled to 5 K. AtT«T _c the resistance slightly decreased with rising temperature, in agreement with the weak localization and electron-electron interaction (EEI) theories. Processing of theR(T, H) curves in terms of the theories permitted the electron phase relaxation time (T) to be determined and the time _so of electron spin relaxation due to spin-orbital interaction (SOI) to be estimated. In the temperature rangeT–T _c /k the effect of the fluctuation EEI on conductivity was investigated. The contribution of the fluctuation EEI is shown to be given by a sum of two corrections: the Cooper and the Maki-Thompson ones. In the rangeT–T _c /k the influence of magnetic field was investigated on the transition from two-to zerodimensional fluctuations of the order parameter resulting from temperature increase in the rangeT>T _c . Increasing field was found to cause enlargement of the zero-dimensional fluctuation critical region and lowering of the fluctuation dimensionality alteration temperature, leading eventually to complete two-dimensional fluctuation suppression. The experimental dependencesT _c (H) experienced, in addition to orbital effects of the magnetic field, depairing effect of the field on electron spins. Thus the _so magnitude could also be estimated and proved roughly equal to _so determined from the analysis ofR(T, H)     

Nonuniversal critical phenomena along the lambda line of4He

General relations between the critical specific heat above and below the lambda transition of⁴He are derived. The specific heat aboveT is calculated in three dimensions up to two-loop order. The relations can be used to determine the effective renormalized static couplings of modelF without integrating the renormalization-group flow equations. The effect of different normalizations of the couplings is discussed. The theory is applicable both to asymptotic universal and to nonasymptotic nonuniversal properties well away fromT (P) and provides part of the basis for quantitative predictions of other nonuniversal critical phenomena along the lambda line.     

Scaling of the surface tension of phase-separated polymer solutions

The tension of the interface between the equilibrium phases of a phase-separated polymer solution is obtained in the simplest mean-field approximation from the functional equation for the composition profile of the interface. For temperaturesT near the critical solution temperatureT _c, i.e., for Flory parameter near _c, and for high degrees of polymerizationN, the profile and tension scale with=N ^1/2(– _c), just as do the compositions of the coexisting phases in mean-field approximation. The surface tension in the asymptotic limitN, _c at fixedx, is found to be given bya ²/kT _c (2c'/c)^1/2 N ^-5/4(x), wherea is the lattice spacing of an underlying lattice (or, roughly, the length of a monomer),c andc are the vertical and total coordination numbers of the lattice, and(x) is a scaling function, known for allx, with the asymptotic behavior asx0 and asx. The latter implies that becomes independent ofN asN at fixedT nearT _c; the former implies that becomes proportional toN ^–1/2(1–T/T _c)^3/2 asTT _c at fixedN1, as found previously.     

Cabibbo angle

An expression is derived for the Cabibbo angle from a consideration of theM ² matrices ofd, s andu, c quarks. Withm _u=m _d0.3 GeV and expressions derived earlier form _s andm _c, the Cabibbo angle is found to be 1342, an excellent result.     

Critical current density of a YBa2Cu3O7 film: Comparison between experiment and collective pinning theory

With the vibrating reed technique we have measured the critical current densityJ _c of a Y 123 200 nm thin film as a function of magnetic field perpendicular to the CuO₂ planes (O T<B8 T) and temperature (20KT60 K). At fieldsB<0.1 TJ _c is magnetic field independent and decreases at higher fields. A comparison with theory indicates that a crossover from a single pinning to a small bundle collective pinning regime may explain the observed behaviour. According to our estimate the main pinning centers are weak point pins due to oxygen vacancies. From the temperature dependence ofJ _c atB0 we obtain a temperature dependence of the thermodynamical critical fieldB _c (1–T/T _c )² forT20 K which agrees with the anomalous temperature dependence ofB _c2 observed recently in highly anisotropic high temperature superconductors.     

Corrections to cluster-radius scaling for branched polymers and percolation

We report analyses of series enumerations for the mean radius of gyration for isotropic and directed lattice animals and for percolation clusters, in two and three dimensions. We allow for the leading correction to the scaling behaviour and obtain estimates of the leading correction-to-scaling exponent . We find -0.640±0.004 and =0.87±0.07 for isotropic animals in 2d, and =0.64±0.06 in 3d. For directed lattice animals we argue that the leading correction has= or= ; we also estimate =0.82±0.01 and 0.69 ±0.01 ind=2, 3 respectively. For percolation clusters at and abovep _c, we find (p _c) =0.58±0.06 and (p>p _c)=0.84±0.09 in 2d, and (p _c)=0.42±0.11 and (p>p _c)=0.41 ±0.09 in 3d.     

Dynamics of Ising spin glasses far below the lower critical dimension: The one-dimensional case and small clusters

The Glauber model is studied for symmetric distributions (±J and gaussian) of the nearest-neighbour interactionJ, including a magnetic field. For small clusters of spins (closed rings ofN bonds, withN7) the complex magnetic susceptibility () and the time-dependent remanent magnetizationm(t) are found exactly for given bond configurations {J _ij } by diagonalization of the Liouville operator; apart from the ±J model, the average over {J _ij } must be done numerically by simple random sampling Monte Carlo. Nevertheless our accuracy is much better than corresponding dynamic Monte Carlo simulations, even if one considers the extrapolation toN.We analyze the results along the lines of corresponding experimental work, studying the frequency-dependence of the peak in (), theT lnt-scaling ofm(t) at low temperaturesT, and the decomposition of () into a spectrum of relaxation times. Many results are strikingly similar to experimental data for systems such as the Holmium-Borate spinglass or the superparamagnet Eu_0.05Sr_0.95S, for instance. Frequency-dependent critical fieldsH _c () in theH-T plane are also identified but do not have the familiar Almeida-Thouless shape, however.Dedicated to B. Mühlschlegel on the occasion of his 60th birthday     

Probabilistic formulation of classical mechanics

Starting axiomatically with a system of finite degrees of freedom whose logic _c is an atomic Boolean -algebra, we prove the existence of phase space _c, as a separable metric space, and a natural (weak) topology on the set of statesI (all the probability measures on _c) such that _c, the subspace of pure statesP, the set of atoms of _c and the spaceP( _c) of all the atomic measures on _c, are all homeomorphic. The only physically accessible states are the points of _c. This probabilistic formulation is shown to be reducible to a purely deterministic theory.     

The superconducting transition temperatures of the thorium-hydrides

The contributions to the quantity ² F() coming from the different vibration modes are calculated for the compounds Th₄H₁₅ and Th H₂. The electronic quantities have been evaluated in a cluster approximation while simple models are used for the phonons. The high transition temperature of Th₄H₁₅ is explained by the large contribution of the hydrogen modes to the McMillan, which overcompensates the deterioration of the Th-lattice in comparison to the Th-metal. In Th H₂ the hydrogen contribution turns out to be insufficient for a highT _c . An important property of these substances is the significant admixture off-states at the Fermi level.     

Specific heat of Eu x Sr1−x Te

In this paper, we report on measurements of the specific heatC of single-crystalline Eu _x Sr_1–x Te at temperatures between 60 mK and 15 K and in magnetic fields up to 6 T. Pure antiferromagnetic EuTe shows unusual critical behavior in the vicinity of the Néel temperatureT _N=9.8 K with a positive critical exponent instead of the 3d-Heisenberg exponent =–0.12. Possible reasons for this discrepancy between theory and experiment include magnetic anisotropy effects due to magnetic dipole-dipole interactions, which may give rise to a cross-over of the critical behavior very close toT _N. This anisotropy is also seen in the specific heat below 1 K where an exponential decay ofC is observed, and in the dependence of the magnetic susceptibility on the direction of the applied field. With increasing dilution of EuTe with nonmagnetic Sr, the critical behavior changes: becomes negative and decreases continuously towards –1 atxx _c. This concentration dependence of was previously observed in the diluted ferromagnetic system Eu _x Sr_1–x S. Our data thus support that the apparent change in the critical behavior depends on the degree of disorder. Samples with concentrationx lower than the critical concentrationx _c reveal spin-glass behavior in the specific heat. In addition, the dependence ofT _N on magnetic fields is discussed. The data yield a normalized magnetic phase boundaryB _c(T)/B_c(T=0) vs.T _N(B)/T_N(B=0) which is independent of concentration.     

Almost periodic Jacobi matrices associated with Julia sets for polynomials

Let be the Jacobi matrix associated with polynomialT(z) of degreeN2. The spectrum of is the Julia set associated withT(z) which in many cases is a Cantor set. Let ⁽¹⁾ denote the result of omitting the first row and column ofJ. Then it is shown that the spectrum of ⁽¹⁾ may be purely discrete.It is also shown that forT(z)= ^NC_N(z/) for > , whereC _N is a Chebychev polynomial the coefficients of and ⁽¹⁾ are limit periodic extending the work of Bellissard, Bessis, and Moussa (Phys. Rev. Lett.49, 701–704 (1982)).Supported in part by N.S.F. grant DMS-8401609Supported in part by N.S.F. grant MCS-8203325     

Möbius function for the set of acyclic directed backbone graphs

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let _uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b _uv withbb, by(b,b)=(–1) ^c-c , wherec andc are the respective cycle ranks ofb andb. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.     

Scaling behavior of permeability and conductivity anisotropy near the percolation threshold

We use the finite-size scaling method to estimate the critical exponent that characterizes the scaling behavior of conductivity and permeability anisotropy near the percolation thresholdp _c . Here is defined by the scaling lawk _l /k _t –1(p–p _c ), wherek _t andk _t are the conductivity or permeability of the system in the direction of the macroscopic potential gradient and perpendicular to this direction, respectively. The results are (d=2)0.819±0.011 and (d=3)0.518±0.001. We interpret these results in terms of the structure of percolation clusters and their chemical distance. We also compare our results with the predictions of a scaling theory for due to Straley, and propose that (d=2)=t- _B , wheret is the critical exponent of the conductivity or permeability of the system, and _B is the critical exponent of the backbone of percolation clusters.     

Scaling function for energy-density correlations: Dispersion theory andε-expansion forT>T c

A dispersion representation for the static energy-density correlation function ² (q) ²(–q) _c =C(q,T)=A+Bt ^–h(z ²), wherez=q , t=(T—T)_c/T _c and is the correlation length, is discussed.h(z ²) is calculated to order ² in the zero-field critical region (T>T _c) for the standard isotropicn-component ⁴Ginzburg-Landau-Wilson model. Utilizing a procedure similar to that introduced by Bray for the two-point correlation function, the-expansion results are used in conjunction with an approximant for the spectral functionF(z/2) Imh(—z ²) based on the asymptotically exact short-distance expansion resulth ^–1(z ²)z ^/v[D ₀+D ₁ z ^{–(1 —)/v} +D ₂ z ^–1/v ] to predict quantitatively the full momentum dependence ofC(q,T) forT>T _c. In contrast to the two-point correlation function,C(q,T) is found to be a monotonic function as the critical temperature is approached at fixedq (forT>T _c).     

Self-consistent calculation of statics and dynamics of the roughening transition

Statics and dynamics of the modified kinetic discrete Gaussian model are treated selfconsistently using a Gaussian probability assumption. A non-trivial roughening temperatureT _R is found in exactly two dimensions only. The free energyF, the correlation length and the interface roughness h ² are found to behave—lnFlnh ²(T _R –T)^–1 for temperaturesT approachingT _R from below. The linear relaxation rate of the order parameter is found to be proportional to ^–2. As a model for crystal growth, the growth rate depends linearly upon the chemical potential difference aboveT _R , shows a metastable regime belowT _R with a spinodal limit of metastability _c , beyond which oscillatory growth starts. The critical behavior of _c is found to be ln _c –(T _R –T)^–1+O(ln (T _R –T)).     

Note on the quantum displacement of the critical point

The quantum corrections to the law of corresponding states are studied by calculating the critical pressure, temperature, and density to first order in Planck's constanth on an exactly soluble model. The ratio of the critical parameters to the corresponding classical values are found to be (p _c/p _c ⁰)^1/2=_c/_c ⁰ = T_c/T_c ⁰ = 1–0.67, with=h _c ^1/3(mkT _c)^–1/2. The critical ratio is independent ofh to first order. The results are compared with critical data for noble gases and hydrogen isotopes.