A Monte Carlo simulation study of associated liquid crystals

We have performed a Monte Carlo simulation study of a system of ellipsoidal particles with donor—acceptor sites modelling complementary hydrogen-bonding groups in real molecules. We have considered elongated Gay—Berne particles with terminal interaction sites allowing particles to associate and form dimers. The changes in the phase transitions and in the molecular organization and the interplay between orientational ordering and dimer formation are discussed. Particle flip and dimer moves have been used to increase the convergency rate of the Monte Carlo (MC) Markov chain.     

Phase Transitions in Dynamical Random Graphs

We study a large-time limit of a Markov process whose states are finite graphs. The number of the vertices is described by a supercritical branching process, and the dynamics of edges is determined by the rates of appending and deleting. We find a phase transition in our model similar to the one in the random graph model G _n,p . We derive a formula for the line of critical parameters which separates two different phases: one is where the size of the largest component is proportional to the size of the entire graph, and another one, where the size of the largest component is at most logarithmic with respect to the size of the entire graph. In the supercritical phase we find the asymptotics for the size of the largest component.     

Phase transitions in “small” systems

Traditionally, phase transitions are defined in the thermodynamic limit only. We discuss how phase transitions of first order (with phase separation and surface tension), continuous transitions and (multi)-critical points can be seen and classified for small systems. “Small” systems are systems where the linear dimension is of the characteristic range of the interaction between the particles; i.e. also astrophysical systems are “small” in this sense. Boltzmann defines the entropy as the logarithm of the area of the surface in the mechanical N-body phase space at total energy E. The topology of S(E,N) or more precisely, of the curvature determinant allows the classification of phase transitions without taking the thermodynamic limit. Micro-canonical thermo-statistics and phase transitions will be discussed here for a system coupled by short range forces in another situation where entropy is not extensive. The first calculation of the entire entropy surface S(E,N) for the diluted Potts model (ordinary (q=3)-Potts model plus vacancies) on a square lattice is shown. The regions in {E,N} where D>0 correspond to pure phases, ordered resp. disordered, and D<0 represent transitions of first order with phase separation and “surface tension”. These regions are bordered by a line with D=0. A line of continuous transitions starts at the critical point of the ordinary (q=3)-Potts model and runs down to a branching point P_m. Along this line vanishes in the direction of the eigenvector of D with the largest eigen-value . It characterizes a maximum of the largest eigenvalue . This corresponds to a critical line where the transition is continuous and the surface tension disappears. Here the neighboring phases are indistinguishable. The region where two or more lines with D=0 cross is the region of the (multi)-critical point. The micro-canonical ensemble allows to put these phenomena entirely on the level of mechanics. Received 18 October 1999 and received in final form 17 November 1999     

Antiferromagnetic order and phase transitions in GdS as studied with X-ray resonance-exchange scattering

We report on X-ray resonance exchange and neutron scattering of metallic GdS. At the L_II and L _III absorption edges of Gd, resonance enhancements of more than two orders of magnitude over the non-resonant magnetic scattering are observed. Polarisation analysis proves that these enhancements are due to dipolar transitions from the 2p to the 5d states. The branching ratio between the L_II and L _III edges of 2.5 suggests a polarisation of the 5d electrons in the ground state. The antiferromagnetic order is of type II in the fcc lattice. Single crystal diffraction of hot neutrons suggests that the spin direction lies within the (111) planes with a value for the sublattice magnetisation of 6.51(3) . The critical exponent for the sublattice magnetisation has a value of β = 0.38(2) in agreement with a pure Heisenberg model. Above T _N, a sharp component persists in the critical diffuse scattering. Lattice distortions give indications for two additional low-temperature phase transitions at about 49 K and 32 K. We argue that these transitions are not connected to spin reorientations and discuss the possible influence of fourth-order exchange interactions. Received 19 November 1999 and Received in final form 12 December 2000     

Long Paths and Cycles in Dynamical Graphs

We study the large-time dynamics of a Markov process whose states are finite directed graphs. The number of the vertices is described by a supercritical branching process, and the edges follow a certain mean-field dynamics determined by the rates of appending and deleting. We find sufficient conditions under which asymptotically a.s. the order of the largest component is proportional to the order of the graph. A lower bound for the length of the longest directed path in the graph is provided as well. We derive an explicit formula for the limit as time goes to infinity, of the expected number of cycles of a given finite length. Finally, we study the phase diagram.     

Phase transitions in random Potts systems and the community detection problem: spin-glass type and dynamic perspectives

Phase transitions in spin-glass type systems and, more recently, in related computational problems have gained broad interest in disparate arenas. In the current work, we focus on the “community detection” problem when cast in terms of a general Potts spin-glass type problem. As such, our results apply to rather broad Potts spin-glass type systems. Community detection describes the general problem of partitioning a complex system involving many elements into optimally decoupled “communities” of such elements. We report on phase transitions between solvable and unsolvable regimes. A solvable region may further split into “easy” and “hard” phases. Spin-glass type phase transitions appear at both low and high temperatures (or noise). Low-temperature transitions correspond to an “order by disorder” type effect wherein fluctuations render the system ordered or solvable. Separate transitions appear at higher temperatures into a disordered (or an unsolvable) phase. Different sorts of randomness lead to disparate behaviors. We illustrate the spin glass character of both transitions and report on memory effects. We further relate Potts type spin systems to mechanical analogs and suggest how chaotic-type behavior in general thermodynamic systems can indeed naturally arise in hard computational problems and spin glasses. The correspondence between the two types of transitions (spin glass and dynamic) is likely to extend across a larger spectrum of spin-glass type systems and hard computational problems. We briefly discuss potential implications of these transitions in complex many-body physical systems.     

Structural phase transition induced by the Jahn-Teller effect. Antiferrodistortive ordering

We study the phase transitions induced by the Jahn-Teller effect ofE-doublet ions in a cubic crystal with antiferrodistortive interactions. AnS=1 pseudospin model is constructed which takes the three lowest vibronic levels of the Jahn-Teller complexes into account. We find a second-order phase transition to a tetragonal phase with two inequivalent sublattices. The transitions between the vibronic levels give rise to bands of collective vibronic excitations with strongly temperature-dependent frequencies. The nature of the various modes is analyzed in detail. We also study the coupling to the elastic displacement field of the crystal. For a sufficiently large coupling constant, this coupling stabilizes a different low-temperature tetragonal phase with two equivalent sublattices. In a certain region of coupling constants, a transition occurs between the two tetragonal phases by second-order transitions to an intermediate phase of lower symmetry. The influence of the coupling on the dynamic behaviour is discussed.Supported by Schweizerischer Nationalfonds     

Structural phase transition induced by the Jahn-Teller effect. Antiferrodistortive ordering

We study the phase transitions induced by the Jahn-Teller effect ofE-doublet ions in a cubic crystal with antiferrodistortive interactions. AnS=1 pseudospin model is constructed which takes the three lowest vibronic levels of the Jahn-Teller complexes into account. We find a second-order phase transition to a tetragonal phase with two inequivalent sublattices. The transitions between the vibronic levels give rise to bands of collective vibronic excitations with strongly temperature-dependent frequencies. The nature of the various modes is analyzed in detail. We also study the coupling to the elastic displacement field of the crystal. For a sufficiently large coupling constant, this coupling stabilizes a different low-temperature tetragonal phase with two equivalent sublattices. In a certain region of coupling constants, a transition occurs between the two tetragonal phases by second-order transitions to an intermediate phase of lower symmetry. The influence of the coupling on the dynamic behaviour is discussed.     

Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

The quantum compass chain in a transverse magnetic field

We study the magnetic behaviors of a spin-1/2 quantum compass chain (QCC) in a transverse magnetic field, by means of the analytical spinless fermion approach and numerical Lanczos method. In the absence of the magnetic field, the phase diagram is divided into four gapped regions. To determine what happens by applying a transverse magnetic field, using the spinless fermion approach, critical fields are obtained as a function of exchanges. Our analytical results show, the field-induced effects depend on in which one of the four regions the system is. In two regions of the phase diagram, the Ising-type phase transition happens in a finite field. In another region, we have identified two quantum phase transitions (QPT)s in the ground state magnetic phase diagram. These quantum phase transitions belong to the universality class of the commensurate-incommensurate phase transition. We also present a detailed numerical analysis of the low energy spectrum and the ground state magnetic phase diagram. In particular, we show that the intermediate state (h _c1 < h < h _c2) is gapful, describing the spin-flop phase.     

Generalized Farey Trees,Transfer Operators and Phase Transitions

We consider a family of Markov maps on the unit interval, interpolating between the tent map and the Farey map. The latter is not uniformly expanding. Each map being composed of two fractional linear transformations, the family generalizes many particular properties which for the case of the Farey map have been successfully exploited in number theory. We analyze the dynamics through the spectral properties of the generalized transfer operator. Application of the thermodynamic formalism to the family reveals first and second order phase transitions and unusual properties like positivity of the interaction function.     

The donnachie-landshoff pomeron and gauge invariance

We discuss a model for heavy mesons where the light quark (u or d) moves in the colour-electric field from a heavy quark (c or b) placed in the center of a bag. We calculate energy spectra for pionic and photonic transitions from excited states. The transition amplitudes and the branching ratios between electromagnetic and pionic transitions compare favorably with the limited amount of known experimental data. Received: 19 January 1999 / Published online: 18 June 1999     

Thermal expansion anisotropy of β-TlInS2 crystals in the course of phase transitions

Basing on the thermal expansion studies performed for β-TlInS₂ crystals in the course of their phase transitions, we have found that the crystals aged for one year are composed from at least two polytypes, which undergo phase transitions at different temperatures. The detailed analysis of anisotropy of the thermal expansion demonstrates that all of the diagonal components of the thermal expansion tensor associated with the eigensystem have almost the same modules but differ by their signs. This leads to appearance of an elliptical conical surface of zero thermal expansion. The orientation of this surface does not depend on the temperature in the temperature interval under study. We have also found that the thermal expansion coefficients along the crystallographic a and b axes are close to zero.     

Phase transition and high pressure behavior of Zirconium and Niobium carbides

We have predicted the phase transition pressure (P _T )and high pressure behavior of Zirconium and Niobium carbide (ZrC, NbC). The high pressure structural phase transitions in ZrC and NbC has been studied by using a two body inter-ionic potential model, which includes the Coulomb screening effect, due to the semi-metallic nature of these compounds. These transition metal carbides have been found to undergo NaCl (B1) to CsCl (B2)-type structural phase transition, at high pressure like other binary systems. We predict such structural transformation in ZrC and NbC at a pressure of 98GPa and 85GPa respectively. We have also predicted second order elastic constant and bulk modulus. The present theoretical work has been compared with the corresponding experimental data and prediction of LAPW and GGA and LDA theories.      

Fracture and second-order phase transitions

Using the global fiber bundle model as a tractable scheme of progressive fracture in heterogeneous materials, we define the branching ratio in avalanches as a suitable order parameter to clarify the order of the phase transition occurring at the collapse of the system. The model is analyzed using a probabilistic approach suited to smooth fluctuations. The branching ratio shows a behavior analogous to the magnetization in known magnetic systems with second-order phase transitions. We obtain a universal critical exponent beta approximately = 0.5 independent of the probability distribution used to assign the strengths of individual fibers.     

Uniform uniaxial anisotropic systems with coupled vector order parameters

Phase transitions in magnets, described by two coupled, m-component vector order parameters, having uniform uniaxial anisotropies are studied. Using a phenomenological model, it is shown that when both order parameters are anisotropic, phase transitions are always second order, in either the uniaxial or the (m???1)-isotropic phase. This is contrary to the isotropic case of two coupled order parameters, for which phase transitions are fluctuation-induced first order. The transitions are still continuous into the m-isotropic phase even when the only anisotropic order parameter is the one with the lowest mean-field critical temperature. New discontinuous transitions still occur in either the uniaxial or the (m???1)-isotropic phase, when the only anisotropic order parameter has the highest mean-field critical temperature.     

Phase Transitions and Topology Changes in Configuration Space

The relation between thermodynamic phase transitions in classical systems and topological changes in their configuration space is discussed for two physical models and contains the first exact analytic computation of a topologic invariant (the Euler characteristic) of certain submanifolds in the configuration space of two physical models. The models are the mean-field XY model and the one-dimensional XY model with nearest-neighbor interactions. The former model undergoes a second-order phase transition at a finite critical temperature while the latter has no phase transitions. The computation of this topologic invariant is performed within the framework of Morse theory. In both models topology changes in configuration space are present as the potential energy is varied; however, in the mean-field model there is a particularly strong topology change, corresponding to a big jump in the Euler characteristic, connected with the phase transition, which is absent in the one-dimensional model with no phase transition. The comparison between the two models has two major consequences: (i) it lends new and strong support to a recently proposed topological approach to the study of phase transitions; (ii) it allows us to conjecture which particular topology changes could entail a phase transition in general. We also discuss a simplified illustrative model of the topology changes connected to phase transitions using of two-dimensional surfaces, and a possible direct connection between topological invariants and thermodynamic quantities.     

Ground-state properties of the Falicov-Kimball model in one and two dimensions

A new numerical method is used to study the ground-state properties of the spinless Falicov-Kimball model in one and two dimensions. The resultant solutions are used to examine the phase diagram of the model as well as possibilities for valence and metal-insulator transitions. In one dimension a comprehensive phase diagram of the model is presented. On the base of this phase diagram, the complete picture of valence and metal-insulator transitions is discussed. In two dimensions the structure of ground-state configurations is described for intermediate interactions between f and d electrons. In this region the phase separation and metal-insulator transitions are found at low f-electron concentrations. It is shown that valence transitions exhibit a staircase structure. Received 20 October 2000     

Interference in Phase Space

A central problem in quantum mechanics is the calculation of the overlap, that is, the scalar product between two quantum states. In the semiclassical limit (Bohr's correspondence principle) we visualize this quantity as the area of overlap between two bands in phase space. In the case of more than one overlap the contributing amplitudes have to be combined with a phase difference again determined by an area in phase space. In this sense the familiar double-slit interference experiment is generalized to an interference in phase space. We derive this concept by the WKB approximation, illustrate it by the example of Franck-Condon transitions in diatomic molecules, and compare it with and contrast it to Wigner's concept of pseudo-probabilities in phase space.     

Overview of phases and phase transitions

We provide an overview of some modern developments in the theory of phases and phase transitions in classical and quantum systems. We show the link between non-ergodicity and fidelity in quantum systems and discuss topological phase transitions. We show that the quantum phase transitions are associated with qualitative changes in some properties of the quantum wavefunctions across the phase transition. We discuss the topological phase transition associated with p-wave superconductor since it is a topic of wide interest because of the possible observation of Majorana fermions.