Infrared bounds,phase transitions and continuous symmetry breaking

We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (·) ₃ ² quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k ^–2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).Research supported by USNSF under grants GP-38048 and MPS-74-13252A. Sloan Fellow; also in the Department of Physics     

Stochastic phase spaces and master Liouville spaces in statistical mechanics

The concept of probability space is generalized to that of stochastic probability space. This enables the introduction of representations of quantum mechanics on stochastic phase spaces. The resulting formulation of quantum statistical mechanics in terms of -distribution functions bears a remarkable resemblance to its classical counterpart. Furthermore, both classical and quantum statistical mechanics can be formulated in one and the same master Liouville space overL ²(). A joint derivation of a classical and quantum Boltzman equation provides an illustration of the practical uses of these formalisms.Supported in part by an NRC grant.     

Electronic phase separation in La-Cuprates

The mechanisms driving the phase separation in high-T _c La cuprates is studied by magnetic susceptibility and electrical resistivity measurements. The experiments are performed under different thermal treatment. Three temperature ranges can be resolved characteristic for different physical processes: At 150T170K a space limited diffusion of superparamagnetic clusters (single hole clusters) occurs leading to the formation of a small but rather tight percolative subphase which at highT _c values becomes superconducting. Above 180 K the cluster motion becomes unrestricted in space and a more extended subphase is built up, however, with lowerT _c values. Above 230 K oxygen diffusion starts leading to an unexpected destruction of the conducting/superconducting phase.     

Effects of pressure and of thermal history on the structural phase transition of KCN

The cubic-noncubic structural phase transition of a KCN single crystal is studied as a function of thermal history and of hydrostatic pressure up to 2 kbar. The standard sequence of phases is cubic-monoclinic-orthorhombic on cooling and orthorhombic-cubic on heating. The monoclinic phase is also absent in first cooling runs at low pressures (p250 bar). The width of the monoclinic field inp, T phase diagram is increased in second cooling runs. The effects are discussed in terms of a martensitic transformation behavior and random strain fields.     

Spatial transport phase transitions in systems with high polariton densities

The behaviour of high intense laser beams in matter is examined with respect to a possible pattern formation. The latter is thought to be a collective phenomenon. We establish a microscopic transport theory for anharmonically interacting polaritons, which is valid even for regions far from thermal equilibrium. It is shown that then there must necessarily be phase transitions in the space distribution of polaritons as a function of the laser power. Numerical calculations are performed for an ionic crystal. They yield two critical intensitiesI _c1 andI _c2, where qualitative alterations in the spatial distribution are to be expected. We ascribe the transport phase transitions atI _c1 andI _c2 to the formation of large and small scale filaments respectively. The phenomenon appears to be very similar to the Bénard problem.     

Statistical thermodynamics of the formation of a new phase in the boiling up of volatile liquids

The fluctuation growth of a macroscopic bubble containing a vapor in a moderately superheated or tensile-stressed volatile liquid is treated as the two-dimensional diffusion of a nucleus of a new phase in the space of variables made up of its volumev and the pressure of the vapor in itp. The shape of the free energy surface of the system liquid plus bubble with vapor in the plane (V; p) in the neighborhood of the labile equilibrium of the system is examined, and a two-dimensional nucleus distribution function given with respect to its variables is derived. Close to the pass in the surface a nondiagonal diffusion tensor in the space (V, p) is also calculated. A two-dimensional stationary equation of the kinetics of the formation of a new phase of Kramers type is solved, and an expression is derived for the probability of homogeneous nucleation for an arbitrary viscosity and volatility of a liquid far from its critical point. Various limiting cases are examined.     

Secondary order parameters in the structural phase transitions

In the Landau type phase transitions the primary order parameter is related with irreducible representation which subduces the high symmetry space groupG _o to the low symmetry space subgroupG andG _o G. We propose to define the secondary order parameters as irreducible representations which subduceG _o to the subgroupsH _l , whereH _l are simultaneously supergroups forG, i.e.G _o H _l G. From this statement general conclusions are drawn. In particular, for some phase transitions with elargement of the elementary cell, in addition to the superlattice reflections from the primary order parameter, the superlattice reflections from the secondary order parameters can arise in another non-equivalent high symmetry point of the reciprocal space.     

Characterisations of phase transitions in Ising spin systems

We construct aC*-algebraic formalism designed to provide a framework for the characterisation of phase transitions in a class of Ising spin systems: this class is large enough to include the rectangular lattice models, of arbitrary finite dimensionality, with nearest neighbour interactions. Using an extension of Onsager's transfer matrix formalism, we express properties of a Gibbs state of a system in terms of a contractive linear transformation, ₀, of a certain Hilbert space, the properties of ₀ being governed by the temperature as well as the interactions in the system. We obtain conditions on ₀ under which the system exhibits a phase transition characterised by (A) a thermodynamical singularity, (B) a change in symmetry, associated with theG-ergodic decomposition of Gibbs states, (C) a divergence of a correlation length (appropriately defined) at the critical point, and (D) scaling laws in the critical region. Applying our formalism to the rectangular two-dimensional Ising model with nearest neighbour interactions, we show that its phase transition possesses the properties (B) and (C), as well as (A).     

Star-polarization: A natural link between phase space representation and operator representation of quantum mechanics

The method of *-polarization connects phase space mechanics to the usual operator formulation of quantum theory. A *-polarization is a linear submanifold of the space of C functions on phase space. Elements of a *-polarization are in direct correspondence with the Schroedinger wave functions and this correspondence induces the Weyl correspondence between classical observables and operators. All generalized Moyal algebras admit *-polarizations and a general method is thus available for translating *-quantization into operator language.     

Area in phase space as determiner of transition probability: Bohr-Sommerfeld bands,Wigner ripples,and Fresnel zones

We consider an oscillator subjected to a sudden change in equilibrium position or in effective spring constant, or both—to a squeeze in the language of quantum optics. We analyze the probability of transition from a given initial state to a final state, in its dependence on final-state quantum number. We make use of five sources of insight: Bohr-Sommerfeld quantization via bands in phase space, area of overlap between before-squeeze band and after-squeeze band, interference in phase space, Wigner function as quantum update of B-S band and near-zone Fresnel diffraction as mockup Wigner function.     

A heuristic theory of phase transitions

LetZ be a suitable Banach space of interactions for a lattice spin system. Ifn+1 thermodynamic phases coexist for ₀ Z, it is shown that a manifold of codimensionn of coexistence of (at least)n+1 phases passes through ₀. There are alson+1 manifolds of codimensionn–1 of coexistence of (at least)n phases; these have a common boundary along the manifold of coexistence ofn+1 phases. And so on for coexistence of fewer phases. This theorem is proved under a technical condition (R) which says that the pressure is a differentiable function of the interaction at ₀ when restricted to some codimensionn affine subspace ofZ. The condition (R) has not been checked in any specific instance, and it is possible that our theorem is useless or vacuous. We believe however that the method of proof is physically correct and constitutes at least a heuristic proof of the Gibbs phase rule.     

On a quantum algebraic approach to a generalized phase space

We approach the relationship between classical and quantum theories in a new way, which allows both to be expressed in the same mathematical language, in terms of a matrix algebra in a phase space. This makes clear not only the similarities of the two theories, but also certain essential differences, and lays a foundation for understanding their relationship. We use the Wigner-Moyal transformation as a change of representation in phase space, and we avoid the problem of negative probabilities by regarding the solutions of our equations as constants of the motion, rather than as statistical weight factors. We show a close relationship of our work to that of Prigogine and his group. We bring in a new nonnegative probability function, and we propose extensions of the theory to cover thermodynamic processes involving entropy changes, as well as the usual reversible processes.     

The broken supersymmetry phase of a self-avoiding random walk

We consider a weakly self-avoiding random walk on a hierarchical lattice ind=4 dimensions. We show that for choices of the killing ratea less than the critical valuea _cthe dominant walks fill space, which corresponds to a spontaneously broken supersymmetry phase. We identify the asymptotic density to which walks fill space, (a), to be a supersymmetric order parameter for this transition. We prove that (a)(a _c–a) (–log(a _c–a))^1/2 asaa _c, which is mean-field behavior with logarithmic corrections, as expected for a system in its upper critical dimension.Research partially supported by NSF Grants DMS 91-2096 and DMS 91-96161.     

Signal distortion noise in volume phase holograms

Signal distortion noise in volume phase holograms is analysed for the case in which the object beam consists of two plane waves. It is shown that, for a given (on-Bragg) signal diffraction efficiency, the magnitude of the signal distortion noise is inversely proportional to the square of crystal thickness. Furthermore, as a function of the angular position of the reconstruction beam, the signal distortion noise has two peaks: one at the Bragg angle and the other at a deviation ₀ from the Bragg angle, where ₀ is a function of relative angles between the components of the object beam and the reference beam. The on-Bragg peak of the signal distortion noise is experimentally located for volume phase holograms recorded in Fe-doped LiNbO₃. The measured angular position is found to be in accord with the calculated value.     

Quantum evolution and classical flow in complex phase space

For a class of holomorphic perturbations of the harmonic oscillator inn degrees of freedom a local solution of the time-dependent Schrödinger equation in the Bargmann representation is constructed which pointwise propagates, to leading order in , along the classical trajectories in complex phase space.     

Low-temperature phase transitions in GdS,GdSe and GdTe

In the Gd monochalcogenides up to three different structural distortions have been detected in the magnetically ordered regime. Our experiments indicate that all three phases show type II antiferromagnetic order but with different orientations of the magnetic moments: A truly trigonal phase withm _k k=[1/2, 1/2, 1/2], a monoclinic phase withabc, =/2+, and the spins oriented along [110], and a pseudo-rhombohedrally compressed monoclinic phase with the spins in the (111) planes. The spin flips can be accounted for by a model including anisotropic exchange interactions.     

Scattering theory,cross sections,and the lattice of propositions

The asymptotic conditions for the nonrelativistic quantum scattering of a particle by a center of force are derived in terms of a metric on the space of states on a complete orthocomplemented lattice. The flux of particles scattered into a coneC per unit incident flux, averaged over all displacements of the center of force at right angles to the axis of the incident beam, is expressed in terms of the differential cross sectiond/d when the motion is classical, and in terms of the scattering amplitudef when the motion is quantum mechanical. This enables the usual identificationd/d=|f|² to be made.     

Structural phase transition in elpasolite-like (NH4)2KWO3F3

A new chemical compound, (NH₄)₂KWO₃F₃, was synthesized. The Rietveld-refined crystal structure was found to be cubic at room temperature and to belong to the elpasolite family (space group ). The heat capacity and unit cell parameters were studied within a broad temperature range. A second-order phase transition was found to occur at 235.4 K and to be well described in terms of phenomenological theory. Hydrostatic pressure broadens the temperature interval of stability of the cubic phase (dT₀/dp = −10.8 K GPa⁻¹). A possible model of structural ordering based on a comparison of the entropy parameters and electron density distribution in oxygen and fluorine atoms is discussed.     

Spherical models with a Gates-Penrose-type phase transition

Gates and Penrose have given criteria under which classical gases with weak long-range interactions fail to be described by the van der Waals equation with Maxwell's rule. Unfortunately, examples of equations of state for such systems have not yet been produced. This paper examines the Gates-Penrose class of interactions-i.e.,U (r)=q(r)+(r), in the limit0, where the Fourier transform (p) has a minimum at a nonzero value ofp-for the spherical model on a one-dimensional lattice. Free energy and magnetization isotherms are computed; it is seen that there is a phase transition, but that the zero-field spontaneous magnetization is always zero (a parahelicoidal phase). However, the pair-correlation function may exhibit either long-range order or the appearance of oscillation.     

Long-time correlations in a model of structural phase transitions with infinite range interaction

The long-time value of the displacement-displacement correlation function is calculated within the ⁴-lattice model used to simulate structural phase transitions. Its temperature dependence and dispersion is discussed for infinite range interaction. Special attent is devoted to the natural explanation of a central peak in the scattering function using nonvanishing long-time correlations.