“Almost” mean-field ising model: An algebraic approach

We study the thermodynamic limit of the algebraic dynamics for an almost mean-field Ising model, which is a slight generalization of the Ising model in the mean-field approximation. We prove that there exists a family of relevant states on which the algebraic dynamics ^t can be defined. This ^t defines a group of automorphisms of the algebra obtained by completing the standard spin algebra with respect to the quasiuniform topology defined by our states.     

“Statistical” symmetry with applications to phase transitions

Hermann proposed that mesomorphic media should be classified by assigning certain statistical symmetry groups to each possible partially ordered array. Two translational groups introduced were called superordinate and subordinate. We find that the average density in such a partially ordered medium has the superordinate symmetry ₁, while the pair correlation function has the subordinate symmetry ₂. A complete listing is made of all compatible combinations of ₁ and ₂ in two and three dimensions. This leads to more possible symmetries than Hermann obtained, e.g., also to nonstoichiometric crystals. The order parameter space for the systems is found to be the quotient space ₁/₂. In most cases it is identical to the order parameter space of low-dimensionalXY spin systems. The Landau free energy is expanded as functional of the two-particle correlation functionK; the translation group is found to be ₁×₂. A Landau mean-field theory can then be carried out by expanding the system free energy into a series of invariants of the active irreducible representations ofK and mapping the free energy onto that for anXY planar spin system. We predict novel critical behavior for transitions between mesomorphic phases and go nogo selection rules for continuous transitions. We give the structure factors for X-ray scattering so changes in all such phase transitions are observable. The statistical symmetry groups, which describe point and translational symmetries of the mesophases, are classified. Proposals are made to include quasi-long-range or topological order in the classification scheme.This work supported in part by National Science Foundation (Division of International Programs), the PSC-BHE—Faculty Research Award CUNY and Deutsche Forschungsgemeinschaft.     

Correlations in Ising ferromagnets. III

An inequality relating binary correlation functions for an Ising model with purely ferromagnetic interactions is derived by elementary arguments and used to show that such a ferromagnet cannot exhibit a spontaneous magnetization at temperatures above the mean-field approximation to the Curie or critical point. (As a consequence, the corresponding lattice gas cannot undergo a first order phase transition in density (condensation) above this temperature.) The mean-field susceptibility in zero magnetic field at high temperatures is shown to be an upper bound for the exact result.Research supported in part by the National Science Foundation. Alfred P. Sloan research fellow.     

Colored noise driven systems with inertia

We present a novel approximation scheme, termed unified colored noise approximation (UCNA), for colored Gaussian noise driven nonlinear systems with inertia. This approximation allows one to evaluate static (stationary distributions, moments) as well as dynamical quantities (correlation functions) for small-to-moderate-to-large values of the correlation time. The approximation replaces a three-dimensional Markovian process by a reduced, two-dimensional Markovian dynamics with new drift and diffusion coefficients. For a harmonic potential the stationary moments are reproduced exactly. Most importantly, we present a criterion involving the noise strengthD, the friction strength and the noise color , which describes the region of validity of UCNA in the parameter space given by (D, , ). At small -values we contrast the UCNA with the well-known small approximation. In order to have a comparison onanalytical grounds, we test the static and dynamical predictions of UCNA versus the well-known analytical results obtained from a three-dimensional Ornstein-Uhlenbeck process.     

Metastates in Disordered Mean-Field Models II: The Superstates

We continue to investigate the size dependence of disordered mean-field models with finite local spin space in more detail, illustrating the concept of superstates as recently proposed by Bovier and Gayrard. We discuss various notions of convergence for the behavior of the paths (t_[tN]()) _{t(0, 1]} in the thermodynamic limit N. Here _n () is the Gibbs measure in the finite volume {1,..., n} and is the disorder variable. In particular we prove refined convergence statements in our concrete examples, the Hopfield model with finitely many patterns (having continuous paths) and the Curie–Weiss random-field Ising model (having singular paths).     

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.     

Absence of asymptotic-time symmetry breaking in zero-bias spin-boson model with partial relaxation

The symmetric spin-boson model without external field is treated for any type of coupling to the boson bath and any initial bath density matrix. With initially fully aligned spin (_z (0)= =1), the proof is given that a partial relaxation (_z (+) t1<) implies that there is no asymptotic-time (up-and-down) symmetry breaking (i.e. that _z (+)=0). For the problem of a particle (interacting with free bosons) in a symmetric double well without spatial symmetry breaking before the infinite time limit, this means that att + the particle distribution becomes symmetric (irrespective of the full initial asymmetry) unless the particle fully remains (att + ) in Ihe starting well.     

Approximate ground state in a model with spontaneous symmetry breakdown

The spontaneous symmetry breakdown is treated by means of a variational approach. Use is made of coherent states of Glauber and of pairing states of BCS-type as the translationally invariant vacuum states for the discussion of the real scalar field ⁴ withm ₀ ² 0. The first type of trial states reproduces the usual approach to spontaneous symmetry breakdown (-) in the tree approximation (which is possible only form ₀ ² <0), while the second type of trial states offers the possibility of spontaneous symmetry breakdown even form ₀ ² =0.     

Implementation of comparative probability by normal states. Infinite dimensional case

Let be an infinite dimensional Hilbert space and () the set of all (orthogonal) projections on . A comparative probability on () is a linear preorder on () such thatOP1,1O and such that ifP┴R,Q┴R, thenPQP+RQ+R for allP, Q, R in (). We give a sufficient and necessary condition for to be implemented in a canonical way by a normal state onB(), the bounded linear operators on .     

Disordered ground states for classical discrete-state problems in one dimension

It is known that one-dimensional lattice problems with a discrete, finite set of states per site generically have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (I) inversion symmetry (or both). We show that such constraints give rise to the possibility ofdisordered GSs over a finite fraction of the coupling-parameter space—that is, without invoking any nongeneric fine tuning of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of statesk at each site and the ranger of the interaction. The Ising (k=2) case is the least prone to disorder:I symmetry allows for disordered GSs (without fine tuning) only forr5, whileS symmetry never gives rise to disordered GSs.     

On the hardening in modulated structure

Hardening in modulated structure is evaluated using the periodic approximation. The critical shear stress increment due to the periodic structure is calculated in the constant line energy approximation. The results are applicable to any periodic structure (concentration waves must be neither homophase nor symmetric) exerting on the dislocation local glide forces with an amplitude smaller than ( denotes the line energy of corresponding straight dislocation directed along the concentration variations with the wave vector). In the zero approximation, the critical forceb is then simply the glide force on the straight dislocation averaged along its length in its most hardened position.     

Nonsymmetric Kaluza-Klein and Jordan-Thiry theory in a general non-Abelian case

This paper is devoted to an (n+4)-dimensional unification of NGT (nonsymmetric gravitation theory) and Yang-Mills theory in a Jordan-Thiry manner. We find interference effects between gravitational and Yang-Mills fields which appear to be due to the skew-symmetric part of the metric on the (n+4)-dimensional manifold (nonsymmetrically metrized principal fiber bundle). Our unification, called the nonsymmetric-non-Abelian Jordan-Thiry theory, becomes classical if the skew-symmetric part of the metric is zero. We find the Yang-Mills field Lagrangian up to the second order of approximation inh =g – . We also deal with the Lagrangian for the scalar field (connected to the gravitational constant). We consider the spin content of the theory and a relationship between the cosmological constant and the coupling constant between the skewon field and the gauge field in the first order of approximation. We show how to derive a dielectric model of a confinement from interference effects in these theories. We underline some similarities between the nonsymmetric Jordan-Thiry Lagrangian in the flat space limit and the soliton bag model Lagrangian.     

Anisotropic low-temperature heat transport in YBa2Cu3O6.9 single crystal

We report measurements of the low-temperature thermal conductivity of YBa₂Cu₃O_7– (0.1) single crystals (T _c =84 K) both parallel ( ^{a, b} ) and perpendicular ( ^c ) to the CuO₂ planes. Whereas ^c (T) is found to be identical, within experimental resolution, with the phonon contribution _ph (T), ^{a, b} (T) contains an additional term linear in temperature,AT. We ascribeAT to the contribution of unpaired electronic carriers residing in the chain layers. Measurements performed in external magnetic fieldsB8 T support this interpretation. Our observations can be explained by an internal multilayer (IML) model in which it is assumed that strong superconductivity is generated within the CuO₂ layers and weak superconductivity is induced in the chain layers by the proximity effect. The fit of the experimental results to the IML model reveals that approximately 15% of the electronic carriers remain unpaired in YBa₂Cu₃O₇ belowT=1 K.     

The crystal structure of copper dipyridine dibromide Cu(C5H5N)2Br2

The X-ray structural analysis of copper dipyridine dibromide was carried out. The substance is monoclinic, its space group isP2₁/n, having the lattice constantsa=8.30 kXU,b=17.72 kXU,c=4.04 kXU,=96°,Z=2. The localization of the heavy atoms was carried out by means of the projections of the Patterson functionsP(u, v) andP(v, w), which provided the bases for determining the signs of the majority of structure factors. The projections of the electron density(x, y) and(y, z) were calculated on the basis of these data. The positions of the atoms were refined three times by methods of differential syntheses and by geometrical analysis. The structure of CuPy₂Br₂ is very close to that of CuPy₂Cl₂ [1], [2], differing from it primarily in the orientation of the symmetry elements with respect to the crystallographic axes. In both structures the copper atom is octahedrally coordinated with four halogen and two nitrogen atoms. The nitrogen atoms and two bromine atoms are bound to the copper covalently in the structure of the bromine derivative, as follows from the length of the bonds (Cu-N 1.99 kXU, Cu-Br, 2.46 kXU), while the remaining two bromine atoms are bound to the copper by weaker bonds and mediate the chain formation of molecules in the direction of thec axis.

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Cu(C₅H₅N)₂Br₂

. , 2₁/ ==8,30 kX,b=17,72 kX,=4,04 kX,=96°,z=2. (, v) P(v, w), . (, ) (, z). . CuPy₂Br₂ u₂1₂, . . , (-N 1,99 kX, u-r 2,46 kX), .

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The authors thank M. Serátor for supplying the crystal samples and for initiating the work, the management of the Institute of Technical Physics, Czechoslovak Academy of Sciences, in Prague and especially Dr. A. Línek for making it possible to calculate some of the Fourier maps on the special computer Elika and finally A. lechtová for carefully calculating the Fourier maps and structure factors.

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The first stage of this work was reported on at the 3rd State Conference on Inorganic Chemistry, held in Bratislava from June 29th to July 4th, 1959.     

More inequalities for critical exponents

A variety of rigorous inequalities for critical exponents is proved. Most notable is the low-temperature Josephson inequalitydv +2 2–. Others are 1 1 +v, 1 1 , 1,d 1 + 1/ (for d),dv, ₃ + (for d), ₄ , and _{2m 2m+2} (form 2). The hypotheses vary; all inequalities are true for the spin-1/2 Ising model with nearest-neighbor ferromagnetic pair interactions.NSF Predoctoral Fellow (1976–1979). Research supported in part by NSF Grant PHY 78-23952.     

On some frequent but controversial statements concerning the Einstein-Podolsky-Rosen correlations

Quite often the compatibility of the EPR correlations with the relativity theory has been questioned; it has been stated that the first in time of two correlated measurements instantaneously collapses the other subsystem; it has been suggested that a causal asymmetry is built into the Feynman propagator. However, the EPR transition amplitude, as derived from the S matrix, is Lorentz andCPT invariant; the correlation formula is symmetric in the two measurements irrespective of their time ordering, so that the link of the correlations is the Feynman zigzag, and that causality isCPT invariant at the microlevel; finally, although the Feynman propagator has theP andCT symmetries, no causal asymmetry follows from that. As for Stapp's views concerning process and becoming, and his Whiteheadean concept of an advancing front, I object that they belong to factlike macrophysics, and are refuted at the microlevel by the EPR phenomenology, which displays direct Fokker-like space-time connections. The reason for this is a radical one. The very blending of a space-time picture and of a probability calculus is a paradox. The only adequate paradigm is one denying objectivity to space-time—but this, of course, is also required by the complementary of the x and the k pictures, which only look compatible at the macrolevel. Therefore, the classical objectivity must yield in favor of intersubjectivity. Only the macroscopic preparing and measuring devices have factlike objectivity; the transition of the quantal system takes place beyond both thex and thek 4-spaces. Then, the intrinsic symmetries between retarded and advanced waves, and statistical prediction and retrodiction, entails that the future has no less (but no more) existence than the past. It is the future that is significant in creative process, the elementary forms of which should be termed precognition or psychokinesis—respectively symmetric to the factlike taboos that we can neither know into the future nor act into the past. It is gratifying that Robert Jahn, at the Engineering School of Princeton University, is conducting (after others) conclusive experiments demonstrating low level psychokinesis—a phenomenon implied by the very symmetry of the negentropy-information transition. So, what pierces the veil of maya is the (rare) occurrence of paranormal phenomena. The essential severance between act and potentia is not a spacelike advancing front, but the out of and the into factlike space-time. Finally, I do not feel that an adequate understanding of the EPR phenomenology requires going beyond the present status of relativistic quantum mechanics. Rather, I believe that the potentialities of this formalism have not yet been fully exploited.     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.     

Nonequilibrium phase transitions in stochastic lattice systems: Influence of the hopping rates

Two-dimensional lattice-gas models with attractive interactions and particle-conserving hopping dynamics under the influence of a very large external electric field E along a principal axis are studied in the case of different ratios between the jump rates in the field direction and perpendicular to it using different transition probabilities. We investigate the dependence of the non-equilibrium steady-state properties on the transition mechanism. We find self-similarity with respect to (T, ) and a coexistence curve critical exponent which, for small, seems independent of. There is some evidence that this exponent might be halfway between the equilibrium mean field and Onsager's values. A crossover toward mean-field behavior for large seems also identified.Partially supported by the US-Spanish Cooperative Research Program, Grant CCB-8402025.     

Investigation of nematic order by coherent neutron scattering

The mathematical relationship between the orientational order parameters and the coherent neutron scattering cross section for a nematic liquid crystal is given. For deuterated para-azoxyanisole the single-molecule part of the cross section is evaluated within the meanfield approximation and combined with experimental results to give information about molecular orientational order in terms of P ₂, P ₄ and P ₆. Both P ₂ and P ₄ are found necessary for describing the molecular order. Discrepancies between experimental and theoretical results are interpreted as possibly reflecting the inadequacy of the meanfield theory of Maier and Saupe.     

On the magnetic susceptibility of Pd1−x Ag x in the range 0.5≦x≦1

The weak variation of the magnetic bulk susceptibility of Pd_1–x Ag _x with temperature T and silver mole fractionx within 0.5x1 has been investigated in the range 5KT400K. Experimental evidence can be given for an intersection point of the susceptibility isotherms (T=const,x) atx=0.55. The observed dependence of on T andx is interpreted by means of a semiphenomenological alloy susceptibility function (T,x).