Interaction Parameter Space Research Articles

Parameter symmetries of quantum many-body systems

We analyze the occurrence of dynamically equivalent Hamiltonians in the parameter space of general many-body interactions for quantum systems, particularly those that conserve the total number of particles. As an illustration of the general framework, the appearance of parameter symmetries in the interacting boson model-1 and their absence in the Ginocchio SO(8) fermionic model are discussed.

Ferroelectric phase transitions in systems of axially symmetric dipolar particles in elongational flow

Effects of steady elongational flow on orientational phase transitions in systems of uniaxial, elongated particles with dipole moments are considered. The Onsager model is used with mean-field dipole–dipole and excluded-volume quadrupole interactions, combined with quadrupole effects introduced by elongational flow. Self-consistent, equilibrium orientation distribution of dipole particles, equilibrium thermodynamic properties and phase transitions are controlled by three dimensionless parameters related to dipole–dipole interactions, excluded-volume effects and intensity of the flow field. Dipole and axial order parameters, phase equilibria, and phase instability points are discussed in the space of dipole and quadrupole interaction parameters. Apolar-to-polar phase transitions, as well as transitions between individual apolar phases are affected significantly by coupling between excluded-volume effects and elongational flow. Three ranges of flow intensity inducing different phase behaviour are discussed. In a the weak flow range, a single polar phase and two different apolar phases are predicted. Phase diagram shows a triple point and three equilibrium lines for different pairs of phases. In the medium flow range, the triple point disappears and phase diagram reduces to two phases, apolar and polar, with a single line of coexistence. The first-order apolar-to-polar phase transition appears at relatively low values of the quadrupole interaction parameter, while at higher values of the parameter, second-order transition takes place. The first-order ferroelectric transition, as well as the coexistence of apolar and polar phases, disappear at a critical flow intensity separating the range of medium and strong flow. Within the strong flow range, the ferroelectric transition is second-order in the entire range of quadrupole interactions. Dipole–dipole interactions, critical for the ferroelectric transition, are significantly reduced by flow potential, especially at stronger excluded-volume interactions. A critical flow intensity has been found at which ferroelectric transition is stabilised solely by the elongational flow. Above the critical flow intensity, the system exhibits single-phase behaviour with stable polar order. No elongational flow is capable to induce ferroelectric phase transition in systems with dipole interaction parameter lower than unity.

A generalization of the three-state binary gas–liquid lattice models and analysis of their critical lines

The concept of a global phase diagram – a diagram in the space of fluid interaction parameters in which regions of similar types of phase behavior are demarcated – has proved very valuable in both theoretical and experimental studies of fluid mixtures. This type of work sheds light on the essential transition mechanism involved in the change of character of the phase diagram, such as multi-critical points and double points, and other phenomena of great practical importance in supercritical fluid extraction processes. The first step in this process is to understand the changes in critical line structure which underlies these differences. In this work, we study and generalize following compressible binary gas–liquid models; the three-state lattice gas, the van der Waals equation for binary mixtures, the Tompa model for polymer solutions, and the modified version of Tompa model for polymer mixture solutions. Then, the critical lines of these models are displayed in the density–density, P, T and T, x planes. The crossing point, a point at which two critical lines meet and exchange branches, is also calculated analytically. The P, T diagrams are discussed according to Scott and van Konynenburg phase diagram classifications.

Exact solutions in aS=12quantum spin ladder model with pair and quartic interactions

The effects of multispin interactions upon quantum spin ladders are investigated theoretically. A two-leg $S=\frac{1}{2}$ quantum spin ladder is taken to have intrarung (inter-rung) quadratic (biquadratic) Heisenberg exchange-type interactions and is placed in a uniform magnetic field. Due to the commutability properties of the Hamiltonian, the magnetic canonical partition function of the model is exactly calculable by a method of rung transfer matrix. Exact solutions are then derived for thermodynamic quantities like free and internal energies, magnetization, and magnetic susceptibilities (initial and nonlinear). The zero-temperature results comprise exact solutions in the complete interaction parameter space for the ground-state energy and degeneracy, zero-field spin gap, loci of quantum phase transitions, and the critical magnetic field.

Cluster variation approach to the Ising square lattice with two- and four-spin interactions

The phase diagram of the Ising square ferromagnet with nearest-neighbor, next-nearest-neighbor, and four-spin (plaquette) interactions is investigated using the cluster variation method in the square approximation. A complete analysis in temperature is performed for the ferromagnetic, superantiferromagnetic, and frustrated region of the ground state. Particular attention is devoted to the role of the four-spin interaction. The most remarkable effect is the occurrence, in several regions of the interaction parameter space, of first-order transitions and sometimes reentrance phenomena. Critical and tricritical points are almost analytically determined.

Toulouse points and non-Fermi-liquid states in the mixed-valence regime of the generalized Anderson model.

We study the mixed valence regime of a generalized Anderson impurity model using the bosonization approach. This single impurity problem is defined by the $U=\infty$ Anderson model with an additional density-density interaction, as well as an explicit exchange interaction, between the impurity and conduction electrons. We find three points in the interaction parameter space at which all the correlation functions can be calculated explicitly. These points represent the mixed valence counterparts of the ususal Toulouse point for the Kondo problem, and are appropriately named the Toulouse points of the mixed valence problem. Two of the Toulouse points exhibit the strong coupling, Fermi liquid behavior. The third one shows spin-charge separation; here, the spin-spin correlation functions are Fermi-liquid-like, the charge-charge correlation functions and the single particle Green function have non-Fermi-liquid behaviors, and a pairing correlation function is enhanced compared to the Fermi liquid case. This third Toulouse point describes the novel intermediate mixed valence phase we have previously identified. In deriving these results, we emphasize the importance of keeping track of the anticommutation relation between the fermion fields when the bosonization method is applied to quantum impurity problems.

Tetrahis(dimethylamino)ethylene-C60: Multicomponent superexchange and Mott ferromagnetism.

An interacting, quasi-one-dimensional, orbitally degenerate hopping model is introduced, motivated by the structure of tetrahis(dimethylamino)ethylene (TDAE)-${\mathrm{C}}_{60}$. The ground state is a Mott insulator for average ionization ${\mathrm{C}}_{60}^{\mathrm{\ensuremath{-}}}$, with several possible magnetically ordered phases. The effective Hamiltonian is derived via a multicomponent superexchange transformation. Orbital degeneracy is represented as an additional ``isospin'' degree of freedom. We explore the phase diagram of the one-dimensional model in interaction parameter space, and identify various special points: (i) an SU(4) antiferromagnet (solved by Sutherland and approximated by a large-N mean-field theory), (ii) a spin-ferromagnet/isospin-antiferromagnet, and (iii) an isospin-ferromagnet/spin-antiferromagnet. The exact ground states for (ii) and (iii) are proven using Marshall's theorem. Away from these points we determine the phase diagram in the classical limit. Three-dimensional ordering for case (ii) is found using interchain mean-field theory. We find two transition temperatures: ${\mathit{T}}_{\mathit{C}}$\ensuremath{\propto} \ensuremath{\surd}${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$${\mathit{J}}_{\mathrm{\ensuremath{\parallel}}}$ for the ferromagnetic spin order and ${\mathit{T}}_{\mathit{N}}$\ensuremath{\propto}${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ for the antiferromagnetic isospin ordering, where ${\mathit{J}}_{\mathrm{\ensuremath{\parallel}}}$, ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$ are the intrachain and interchain couplings, respectively. We discuss the relevance of our theory to recent experiments on TDAE-${\mathrm{C}}_{60}$.

Magnetism in Two-Band Systems with Electron Correlation

Magnetic phase diagram for the two-band Hubbard model in one dimension is explored. In the parameter space of direct and exchange interactions, the insulating ferromagnetism with an orbital superstructure transforms into metallic ferromagnetism as the strength of Hund's coupling is increased. When the orbital degeneracy is lifted, the ferromagnetic state is destabilized except in the region where metallic ferromagnetism appears

Successive reentrances and phase transitions in exactly solved dilute centered square Ising lattices.

We solve exactly several periodically dilute centered square Ising lattices by transforming the systems into eight-vertex models. We find that for a given set of interactions, there may be five transitions with decreasing temperature with two reentrant paramagnetic phases. These phases extend to infinity in the space of interaction parameters. Moreover, two additional reentrant phases are found, each in a limited region of phase space.

Analysis of the critical lines for a binary gas-liquid mixture using a modified lattice-gas models

The concept of a global phase diagram — a diagram in the space of fluid interaction parameters in which regions of similar types of phase behavior are demarcated — has proved very valuable in both theoretical and experimental studies of fluid mixtures. This type of work sheds light on the essential transition mechanisms involved in the change of character of the phase diagram, such as multi-critical points and double points, and other phenomena of great practical importance in supercritical fluid extraction processes. The first step in this process is to understand the changes in critical-line structure which underlies these differences. It is shown in this paper that the so called lattice gas equation-of-state, when modified for chain-like molecules, is extremely sensitive to small modifications in the chain length parameter. The cause of this effect is traced by analyzing the complete algebraic solution of the polynomial representing the critical lines.

Re-entrant phase transitions of the Blume-Emery-Griffiths model

We have investigated the spin-1 Ising model on the simple cubic lattice with bilinear, biquadratic interaction and anisotropic energy (BEG model). We have been specially interested in the case of antiferro biquadratic interaction, because the interaction will cause the competition with bilinear interaction and anisotropy. A two-sublattice ordering, so called the staggered quadrupole (SQ) phase, occurs as long as biquadratic interaction is negative large enough. We have obtained a full phase diagram in the hole interaction parameter space

Exact phase diagram of a generalized Kagomé Ising lattice: Reentrance and disorder lines.

We solve exactly a generalized Kagom\'e Ising lattice with three kinds of interactions by transforming the system into a 16-vertex model that satisfies the free-fermion condition. The phase diagram shows several interesting phenomena due to the frustration generated by the competing interactions: successive transitions with reentrance, partial disorder, and disorder lines. The reentrance region in the space of interaction parameters is found to be infinite, unlike previous exactly solved models. Two disorder solutions with interesting behavior are found.

Periodic structures in a 2D lattice with competing interactions and a ϕ4substrate

The authors study a 2D triagonal lattice with competing first- and second-neighbour harmonic interactions and a 4 on-site potential. At zero temperature they determine the classical ground-state configurations of low periodicity. The phase diagram in the interaction parameter space is significantly different from that calculated for the piecewise quadratic on-site potential. The main difference is in the phonon stability regions of the different periodicities.

Analysis of Nonadditive Multiway Classifications

Abstract This article considers the problems of testing additivity and estimating σ2 in unreplicated multiway classifications. To model nonadditivity and jointly estimate σ2, the interaction parameter space must be restricted; otherwise the model is saturated. The parameterization we use is a multiway extension of the two-way multiplicative interaction model of Mandel (1971) and Johnson and Graybill (1972a). For example, in a three-way classification, we model interaction as θijk = λδ 1i δ2j δ3k . This structure is a special case of the k-mode principal components model, which has received considerable attention in the psychometric literature (Kapteyn, Neudecker, and Wansbeek 1986). We construct an exact test of λ = 0 and propose an estimator of σ2 that can be used when interaction has been detected. Our test is an approximation to the likelihood ratio test (LRT) of Ho : λ = 0. The proposed test has essentially the same power as the LRT but is easier to compute, and the exact null distribution of the test...

Bifurcation sets of the self-consistent flow in generalized SU(2) models.

We analyze the possible bifurcations of the stationary points of the self-consistent flow on the Bloch sphere. We propose a generalized SU(2) model Hamiltonian for which, by means of a geometrically simple classification, we find the topologically invariant regions of the mean field flow in the space of interaction parameters. The borders of these regions are the bifurcation sets corresponding to general nonthermodynamic phase transitions. When these separatrices are crossed, the mean field flow undergoes a qualitative change. We also examine the consequences of the catastrophical configurations on the exact dynamics in quasispin space.

Multicritical properties of uniaxial Heisenberg antiferromagnets.

The phase diagrams of spin-1 Heisenberg antiferromagnets with ferromagnetic next-nearest-neighbor (NNN) interactions, single-site uniaxial anisotropy, and a magnetic field along the easy axis are studied. If we treat uniaxial anisotropy exactly and exchange interactions in the mean-field approximation, novel multicritical behavior is obtained at the coexistence of paramagnetic, spin-flop, and antiferromagnetic phases. New higher-order critical points are found which combine bicritical, tricritical, critical-end-point, and triple-point properties. Occurring at well-defined values of anisotropy and NNN interactions, the multicritical points merge at a new multicritical point in the four-dimensional parameter space of temperature, magnetic field, NNN interactions, and anisotropy. Appropriate phase diagrams as well as topology of the various multicritical points involved are discussed.

Statics and dynamics of incommensurate-commensurate phase transitions in A2BX4 type crystals

A unified treatment has been constructed which gives a perspective view of incommensurate(IC)-commensurate (C) phase transition schemes in A2BX4 type crystals. We have shown that a microscopic treatment based on 'Extended Antiferroelectric Interaction Ising' model can explain most of the observed features of IC-C phase transitions. Within this model, each substance is characterized by a point in a two-dimensional interaction parameter space. The unified view has been visualized by constructing a distribution map of the representative points of the hitherto observed materials in this interaction parameter space. The kinetic process of IC-C phase transition in real crystals is discussed in terms of the dislocation loop in soliton lattices.

A Unified View of Incommensurate-Commensurate Phase Transitions in A2BX4 Type Crystals

A unified treatment has been constructed which gives a perspective view of incommensurate(IC)-commensurate(C) phase transition schemes in A 2 BX 4 type crystals. We have shown that a microscopic treatment based on `Extended Antiferroelectric Interaction Ising' model can explain most of the observed features of IC-C phase transitions. The wide variety of the lock-in phases is attributed to the fact that there are three different ground states in the extended antiferroelectric interaction system, and that each substance passes through various transient lock-in phases in the process of reaching one of those ground states. Within this model, each substance is characterized by a point in a two-dimensional interaction parameter space. The unified view has been visualized by constructing a distribution map of the representative points of the hitherto observed materials in this interaction parameter space.

Multiple phase coexistence and the scaling transformation

It is shown that a simple extension of the finite size scaling method in the theory of critical phenomena can yield a sequence of approximants to any point subset of the interaction parameter space where multiple phase coexistence is possible. The method is illustrated by an application to the three-, four- and five-state Potts models, and even in its lowest order of approximation is able to distinguish the difference in the zero-field critical behaviour between the five-state and the three- and four-state models.

Exact preservation of the free energy in a modified Migdal-Kadanoff approximation

The simple, but ad hoc, prescription for bond shifting in the Migdal-Kadanoff approximation is replaced by a systematic calculational procedure. In this method, the free energy is required to be preserved exactly by means of matching series expansions. The resulting renormalization group is characterized by a small parameter, and can be extended by working to higher order. A significant quantitative improvement over the standard Migdal-Kadanoff method is obtained with little increase in calculational effort. This approach is ultimately limited, however, by being restricted to a one-dimensional interaction parameter space.