Free field construction for the eight-vertex model: representation for form factors

The free field realization of the eight-vertex model is extended to form factors. It is achieved by constructing off-diagonal with respect to the ground state sectors matrix elements of the Λ operator which establishes a relation between corner transfer matrices of the eight-vertex model and of the SOS model. As an example, the two-particle form factor of the σ^z operator is evaluated explicitly.     

On the Solutions of the Yang-Baxter Equations with General Inhomogeneous Eight-Vertex R-Matrix: Relations with Zamolodchikov’s Tetrahedral Algebra

We present most general one-parametric solutions of the Yang-Baxter equations (YBE) for one spectral parameter dependent R _ij (u)-matrices of the six- and eight-vertex models, where the only constraint is the particle number conservation by mod(2). A complete classification of the solutions is performed. We have obtained also two spectral parameter dependent particular solutions R _ij (u,v) of YBE. The application of the non-homogeneous solutions to construction of Zamolodchikov’s tetrahedral algebra is discussed.     

Disorder-induced quantum-Griffiths-like features from a non-conventional renormalization group analysis near four dimensions

We investigate the role played by symmetry conserving quenched disorder on quantum criticality of a variety of d-dimensional systems with a continuous symmetry order parameter. We employ a non-standard procedure which combines a preliminary reduction to an effective classical random problem and a successive conventional renormalization group treatment. Solving the effective flow equations to first order in ε=4−d and then restoring the original coupling parameters, for d<4 we find a quantum critical point scenario exhibiting unusual features, which remind us of some predictions of the quantum Griffiths phase model.     

Row transfer matrix spectra of cyclic solid-on-solid lattice models

The eigenvalue spectra of cyclic solid-on-solid (CSOS) row transfer matrices are studied. An equivalence is established between the inversion identity and the Bethe ansatz functional equations and these equations are solved in the thermodynamic limit by a Wiener-Hopf perturbation technique for the bands of leading excitations. TheL-state CSOS model, with crossing parameter=s/L, possesses a 2(L – s)-fold degenerate largest eigenvalue corresponding to the 2(L – s) coexisting phases. The expressions for the largest eigenvalue and free energy coincide with those of the eight-vertex model. The string excitations for 2s < L and 2s > L admit different classifications and are treated separately. The correlation length is calculated in both regimes, yielding the critical exponentv=L/2s, in agreement with the scaling relations.     

Symmetry relations in the sixteen-vertex model

We explore the consequences of the existence of a group of linear symmetry transformations for the sixteen-vertex model. An arbitrary sixteen-vertex model is shown tobe equivalent to a standard model, described in terms of ten parameters having an invariant significance. The general eight-vertex model is obtained when four of the invariant parameters vanish. Two further subciasses of the sixteen-vertex model are found to be equivalent to the eight-vertex model. The invariant conditions leading to the solved cases (i.e the general six-vertex, the free-fermion and the symmetric eight-vertex models) are given.     

Spontaneous symmetry breakings in Z2 gauge theories for doped quantum dimer and eight-vertex models

Behavior of doped fermions in Z₂ gauge theories for the quantum dimer and eight-vertex models is studied. Fermions carry charge and spin degrees of freedom. In the confinement phase of the Z₂ gauge theories, these internal symmetries are spontaneously broken and a superconducting or Neél state appears. On the other hand in the deconfinement-topologically ordered state, all symmetries are respected. From the view point of the quantum dimer and eight-vertex models, this result indicates interplay of the phase structure of the doped fermions and background configuration of the dimer or the eight-vertex groundstate. At the quantum phase transitions in these systems, structure of the doped fermions groundstate and also that of the background dimer or eight-vertex groundstate both change. Translational symmetry breaking induces a superconducting or antiferromagnetic state of the doped fermions.     

De Gennes parameter limit for the occurrence of a single vortex in a square mesoscopic superconductor

The time dependent Ginzburg–Landau equations (TDGLE) are used to study the properties of a mesoscopic superconducting square surrounded by different metallic materials. The properties of the metallic environment are taken into account by De Gennes boundary conditions, via the extrapolation length b. The external magnetic field is applied perpendicularly to the square surface. The TDGLE are used upon taking the magnetic field and the order parameter invariant along z-direction. It is determined the b-limit for the occurrence of a single vortex in a mesoscopic square of area d², d varies discretely from 4ξ(0) to 10ξ(0). We can show a logarithmical dependence of the sample size as a function on b parameter.     

Effective-field theory on the kinetic Ising model

As an analytical method, the effective-field theory (EFT) is used to study the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field. The effective-field equations of motion of the average magnetization are given for the square lattice (Z=4) and the simple cubic lattice (Z=6), respectively. The dynamic order parameter, the hysteresis loop area and the dynamic correlation are calculated. In the field amplitude h₀/ZJ-temperature T/ZJ plane, the phase boundary separating the dynamic ordered and the disordered phase has been drawn, and the dynamical tricritical point has been observed. We also make the compare results of EFT with that given by using the mean field theory (MFT).     

Drinfeld Twist and Symmetric Bethe Vectors of Open XYZ Chain with Non-Diagonal Boundary Terms

With the help of the Drinfeld twist or factorizing F-matrix for the eight-vertex solid-on-solid(SOS) model,we find that in the F-basis provided by the twist the two sets of pseudo-particle creation operators simultaneously take completely symmetric and polarization free form.This allows us to obtain the explicit and completely symmetric expressions of the two sets of Bethe states of the model.     

Interpretation of the symmetry group of the homogeneous 16-vertex model in terms of Lorentz similarity transformations

A new analysis is made of the symmetry group of the general homogeneous 16-vertex model on a square lattice, i.e. the group of transformations in the parameter space of the model leaving invariant its partition function. The set of 16 vertex weights is decomposed in such a way that the ensuing matrix P of 16 composite parameters transforms according to the group of Lorentz similarity transformations. Equivalence classes of models can be characterized by a suitably chosen ‘normal’ matrix P⁽ⁿ⁾, depending on 10 parameters, four having the significance of principle values, and the remaining six (two angles and two 3-dimensional unit vectors) determining a Lorentz transformation. The analysis is applied to the general eight-vertex model as well as to its soluble subclasses, the symmetric eight-vertex model, the general six-vertex model and the free fermion model.     

The Six and Eight-Vertex Models Revisited

Elliott Lieb's ice-type models opened up the whole field of solvable models in statistical mechanics. Here we discuss the “commuting transfer matrix” T,Qequations for these models, writing them in a more explicit and transparent notation that we believe offers new insights. The approach manifests the relationship between the six-vertex and chiral Potts models, and between the eight-vertex and Kashiwara–Miwa models.     

Correlations in the XY model and screw dislocations in the solid-on-solid model

An exact relation is proven between the two-point correlation function of the classical XY model and the free energy of a step associated with two screw dislocations on a crystal surface described by a solid-on-solid model. This result suggests a way to obtain the XY critical exponent η_c from Monte Carlo studies of the solid-on-solid model.     

Embedded elliptic curves and the Yang-Baxter equations

The completely $\text{Z}$_n symmetric S-matrix defined by Belavin is shown to satisfy the Yang-Baxter equations. In the projective space of Boltzmann weights, the curves on which there exist commuting transfer matrices are shown to be embedded elliptic curves. Explicit polynomial equations for these curves are given. For n=2 these results reduce to the results of Baxter for the symmetric eight-vertex model.     

Lattice models in statistical mechanics for N = 2 superconformal field theory

We study exactly solvable cyclic solid-on-solid models extended to arbitrary higher degrees of the fusion construction. A detailed calculation of their local state probabilities via the corner transfer matrix method is presented. The order parameters thus obtained include new characters of the conformal field theories. We discuss a series of lattice analogues of the N = 2 superconformal minimal models included in this family of the cyclic solid-on-solid models.     

Topological order and conformal quantum critical points

We discuss a certain class of two-dimensional quantum systems which exhibit conventional order and topological order, as well as quantum critical points separating these phases. All of the ground-state equal-time correlators of these theories are equal to correlation functions of a local two-dimensional classical model. The critical points therefore exhibit a time-independent form of conformal invariance. These theories characterize the universality classes of two-dimensional quantum dimer models and of quantum generalizations of the eight-vertex model, as well as and non-abelian gauge theories. The conformal quantum critical points are relatives of the Lifshitz points of three-dimensional anisotropic classical systems such as smectic liquid crystals. In particular, the ground-state wave functional of these quantum Lifshitz points is just the statistical (Gibbs) weight of the ordinary two-dimensional free boson, the two-dimensional Gaussian model. The full phase diagram for the quantum eight-vertex model exhibits quantum critical lines with continuously varying critical exponents separating phases with long-range order from a deconfined topologically ordered liquid phase. We show how similar ideas also apply to a well-known field theory with non-Abelian symmetry, the strong-coupling limit of 2+1-dimensional Yang–Mills gauge theory with a Chern–Simons term. The ground state of this theory is relevant for recent theories of topological quantum computation.     

Analytic Approximations for the Velocity of Field-Driven Ising Interfaces

We present analytic approximations for the field, temperature, and orientation dependences of the interface velocity in a two-dimensional kinetic Ising model in a nonzero field. The model, which has nonconserved order parameter, is useful for ferromagnets, ferroelectrics, and other systems undergoing order–disorder phase transformations driven by a bulk free-energy difference. The solid-on-solid (SOS) approximation for the microscopic surface structure is used to estimate mean spin-class populations, from which the mean interface velocity can be obtained for any specific single-spin-flip dynamic. This linear-response approximation remains accurate for higher temperatures than the single-step and polynuclear growth models, while it reduces to these in the appropriate low-temperature limits. The equilibrium SOS approximation is generalized by mean-field arguments to obtain field-dependent spin-class populations for moving interfaces, and thereby a nonlinear-response approximation for the velocity. The analytic results for the interface velocity and the spin-class populations are compared with Monte Carlo simulations. Excellent agreement is found in a wide range of field, temperature, and interface orientation.     

Difference equations in spin chains with a boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the eight-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary S-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal-Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.     

Deposition of particles under gravity in a kinetic lattice model

We study the growth through particle deposition of the surface of a discrete two-dimensional system, in which the motion of particles is affected by infinite gravity and the Kob-Andersen kinetic rule. Computer simulation results are found to be consistent with previous results in literature, showing that this particular case belongs to the same universality class as Ballistic Deposition, the Eden model, one step solid-on-solid (SOS) deposition and Kardar-Parisi-Zhang (KPZ) characterized by scaling exponents α=0.5, β=1/3=0.33 and z=α/β=1.5.     

Stability and instability in a class of car following model on a closed loop

A velocity-matching car following model is modified to represent the motion of n vehicles travelling on a closed loop. Each vehicle is given a preferred velocity profile, which it attempts to achieve while also attempting to maintain a zero relative velocity between itself and the vehicle in front. The crucial distinctive of the looped model, as opposed to ‘non-looped’ models, is that the last vehicle in the stream is itself being followed by the lead (first) vehicle. The model gives rise to a system of n coupled time delay differential equations which are solved approximately (using a Taylor series expansion in time delay) and numerically using a fourth-order Runge-Kutta routine.The stability of the model is considered and an analytic form of the stable region in parameter space is found in the limit as n approaches infinity.