International Journal of Modern Physics B

The early history of exactly soluble models in statistical mechanics and the quantum-mechanical many-body problem, especially in the 1960's, is briefly reviewed.

In this talk, we give a brief overview of several aspects of the theory of the chiral Potts model, including higher-genus solutions of the star–triangle and tetrahedron equations, cyclic representations of affine quantum groups, basic hypergeometric functions at root of unity, and possible applications.

The homogeneous three-layer Zamolodchikov model is equivalent to a four-state model on the checkerboard lattice which closely resembles the four-state critical Potts model, but with some of its Boltzmann weights negated. Here we show that it satisfies a "star-to-reverse-star" (or simply star-star) relation, even though we know of no star-triangle relation for this model. For any nearest-neighbour checkerboard model, we show that this star-star relation is sufficient to ensure that the decimated model (where half the spins have been summed over) satisfies a "twisted" Yang-Baxter relation. This ensures that the transfer matrices of the original model commute in pairs, which is an adequate condition for "solvability".

This paper gives a self-contained exposition of the basic structure of quantum link invariants as state summations for a vacuum-vacuum scattering amplitude. Models of Vaughan Jones are expressed in this context. A simple proof is given that an important subset of these invariants are built from Vassiliev invariants of finite type.

A star-star transformation is introduced for the Potts model as a generalization of the Onsager star-triangle relation. By considering a special limit of the star-star relation, we deduce a dual transformation relation for the Potts model on the 3-12 lattice. We also obtain numerical bounds on the critical point of the Potts model on the kagomé lattice, a long unsolved problem.

Recent work on the finite temperature correlators of the Ising chain in a transverse field is reviewed. The crossover phase diagram in the coupling-constant and temperature plane is discussed, and various exact results and conjectures for static and dynamic correlators are presented.

We summarize some recent connections between a class of nonlinear integral equations related to the thermodynamic Bethe Ansatz and a class of linear integral equations related to the Painlevé equations.

We discuss an interrelation between quantum integrable models and classical soliton equations with discretized time. It appeared that spectral characteristics of quantum integrable systems may be obtained from entirely classical set up. namely, the eigenvalues of the quantum transfer matrix and the scattering S-matrix itself are identified with a certain τ-functions of the discrete Liouville equation. The Bethe ansatz equations are obtained as dynamics of zeros. For comparison we also present the Bethe ansatz equations for elliptic solutions of the classical discrete Sine-Gordon equation. The paper is based on the recent study of classical integrable structures in quantum integrable systems.¹

The kagomé Ising antiferromagnet is highly frustrated with its pair correlation decaying exponentially at large distance for all temperatures including absolute zero. Hence, the spin system does not support long-range orderings and is devoid of any phase transition. One proves, via local star-triangle and decoration-decimation transformations, that correlations in the kagomé Ising antiferromagnet at arbitrary temperatures can be represented as linear combinations of correlations in the honeycomb Ising ferromagnet at high temperatures (disordered region). Existent knowledge of all honeycomb Ising correlations upon a select (spatially compact) 10-site cluster is thus sufficient to determine all present kagomé Ising correlations upon an associated 9-site cluster. Examples of resulting exact solutions for pair and multisite correlations in the kagomé Ising antiferromagnet are presented at all temperatures. Applications include joint configuration probabilities, thermodynamic response functions such as the specific heat and the initial perpendicular susceptibility, and the inelastic neutron scattering function.

The exact form factors of the Heisenberg models XXX and XXZ have been recently computed through the quantum affine symmetry of XXZ model in the thermodynamic limit. We use them to derive an exact formula for the contribution of two spinons to the dynamical correlation function of XXX model at zero temperature.

We briefly review some common diffusion-limited reactions with emphasis on results for two-species reactions with anisotropic hopping. Our review also covers single-species reactions. The scope is that of providing reference and general discussion rather than details of methods and results. Recent exact results for a two-species model with anisotropic hopping and with 'sticky' interaction of like particles, obtained by a novel method which allows exact solution of certain single-species and two-species reactions, are discussed.

We present a solution for a chain-like cluster model, when the elements of the cluster are Ising spins interacting via a short-range potential. The solution is obtained in the limit where no spin agglomeration or percolation type processes are allowed. We also determine the magnetic susceptibility of the model.

It has been established that the infinite-state Potts model in d dimensions generates restricted partitions of integers in d-1 dimensions, the latter a well-known intractable problem in number theory for d>3. Here we consider the d=4 problem. We consider a Potts model on an L × M × N × P hypercubic lattice whose partition function G_LMNP(t) generates restricted solid partitions on an L × M × N lattice with each part no greater than P. Closed-form expressions are obtained for G_222P(t) and we evaluated its zeroes in the complex t plane for different values of P. On the basis of our numerical results we conjecture that all zeroes of the enumeration generating function G_LMNP(t) lie on the unit circle |t|=1 in the limit that any of the indices L, M, N, P becomes infinite.

This paper is concerned with a curious gap in a string of exactly solvable models, a gap that is suggestively related to a completely integrable nonlinear PDE in d=2 known as Liouville's equation. This PDE emerges in a limit N→∞ from the equilibrium statistical mechanics of classical point particles with gravitational interactions (SMGI) in dimension d=2 which, accordingly, is an exactly solvable continuum model in this limit. Interestingly, in d=1 and all d>2, the SMGI can be, and partly has been, exactly evaluated for all N≤∞. This entitles one to suspect that the SMGI for d=2 is likewise exactly solvable for N>∞, but currently this is an unproven hypothesis. If this conjecture can be answered in the affirmative, spin-offs in various areas associated with Liouville's equation, such as vortex gases, superfluidity, random matrices, and string theory can be expected.

We review recent results for the universal part of the free energy of fully-finite two-dimensional regions at criticality. Their effect on the total free energy is considered. Including non-universal edge free energy terms leads to various Casimir instabilities toward elongation, sharp corners on the boundary, and other behavior. Universal features of certain matrix elements of the transfer matrix and corner transfer matrix are also obtained.

The supersymmetric tJ model in one dimension is investigated by means of the quantum transfer matrix (QTM). The free energy of the quantum system is given by just the largest eigenvalue of the QTM whereas the next-largest ones determine the correlation lengths at finite temperature. The first step of the analysis consists of an embedding of the QTM into a family of commuting matrices. In the second step the eigenvalues equations are derived by an algebraic Bethe ansatz. Finally, the eigenvalue equations are transformed into a set of nonlinear integral equations. Some numerical results are presented. The specific heat for instance shows the typical Luttinger liquid behavior at low temperatures.

We present exact analytical results for a critical field and correlations functions for N site ring and arbitrary size two dimensional coupled chains in a two component Hubbard model with a strong magnetic field and exactly at half-filling. The critical field near the Nagaoka instability with one spin and two spin flips is driven by Coulomb interaction, lattice geometry, dimensionality and crucially depends on the number of atomic sites N. Unsaturated ferromagnetism, reminiscent of Nagaoka-like behavior, and spin liquid state are examined in the structure of off-diagonal spin correlation at large U.

We develop a coarse-graining procedure for two-dimensional models of fluctuating loops by mapping them to interface models. The result is an effective field theory for the scaling limit of loop models, which is found to be a Liouville theory with imaginary couplings. This field theory is completely specified by geometry and conformal invariance alone, and it leads to exact results for the critical exponents and the conformal charge of loop models. A physical interpretation of the Dotsenko-Fateev screening charge is found.

The integrable state function of delta interaction on an infinite line is constrained to a ring of finite circumference. The state function remains consistent with the Bethe ansatz, but in a multiply connected state space. The constants of integration of the integrable state function are roots of a transcendental equation determined by the topological properties of this multiply connected space.

A model ternary polymer mixture is considered with bi- and -functional monomers and a solvent embedded on a honeycomb lattice. The model is shown to be equivalent to an eight-vertex model on the same lattice. Using this equivalence which further maps the problem into an Ising model in a non-zero magnetic field, an exact analysis of the model is carried out. The two phase coexistence surface in the temperature- composition space is determined exactly and connectivity properties of the model are studied.

We review the method of constructing solvable models in three dimensions, by starting from two-dimensional solvable models. The solvable three-dimensional models thus constructed do possess positive Boltzmann weights. These are multilayer two-dimensional systems with interactions in the third direction which can be interpreted as nearest-neighbour interactions. The set of conditions corresponding to the general 3D multilayer extension of solvable 2D models is derived.

The shape of the boundary of a finite-region of a grid has a strong effect on the local entropy and local statistics (frequencies of local patterns) of a random dimer configuration. For large regions these quantities appear to be locally constant almost everywhere, but for many regions the local statistics are subject to macroscopic variation. The local frequencies associated with individual bonds can be determined in closed form for important special cases, and their large-scale non-constancy is probably related to the existence of a two-parameter family of extremal Gibbs measures for the models in question.

We propose a generalization of Bailey's lemma, useful for proving q-series identities. As an application, generalizations of Euler's identity, the Rogers–Ramanujan identities, and the Andrews–Gordon identities are derived. This generalized Bailey lemma also allows one to derive the branching functions of higher-level cosets.

The integrable Heisenberg quantum chain with certain non-diagonal boundary fields is the generator of a Markov process known as asymmetric exclusion process with open boundary conditions. This is a driven lattice gas where particles hop randomly along a one-dimensional chain and are injected and absorbed at the boundaries. This model has been suggested in 1968 by MacDonald, Gibbs and Pipkin as a model for the kinetics of protein synthesis on nucleic acid templates. The exact solution of the steady state of the system (corresponding to the exact ground state of the Heisenberg chain) which was obtained recently is shown to be in qualitative agreement with experimental data. The exact solution supports some of the original conclusions drawn from a mean field treatment by MacDonald et al. but gives deeper insight into one important aspect.

We present two independent approaches for computing the thermodynamics for classical particles interacting via the Moser-Calogero potential Combining the results we conjecture the form of equation of state or, what is equivalent, the asymptotics of the Jacobian between volume elements corresponding to two symplectic structures on the phase space.

We show that free-fermion conditions on vertex models amount to degeneration conditions for the inversion relations and the symmetries of the Yang-Baxter equations. We also show how solutions of the tetrahedron equations enjoy a similar property, and analyze the phenomenon.