Yasuhiro Akutsu

Exactly Solvable Models in Statistical Mechanics

Recent studies on exactly solvable models in statistical mechanics are reviewed. A brief summary of the quantum inverse scattering method is given to emphasize the soliton theoretic aspect of the theory. Introducing a class of lattice models called the IRF models, it is shown that there exists an infinite number of exactly solvable models in 2-dimen-sional statistical mechanics. Significances both in physics and mathematics are discussed.

Exactly solvable models and knot theory

Abstract Presented is a review on theory of exactly solvable models in statistical mechanics and its application to knot theory. The Yang-Baxter relation, a sufficient condition for the solvability of models, is introduced for scattering matrices in (1 + 1)-dimensional field theory and for Boltzmann weights of vertex models and IRF models in two-dimensional statistical mechanics. A systematic study of solutions of the Yang-Baxter relation shows that there exists at least an infinite number of two-dimensional exactly solvable models in classical statistical mechanics. The result implies that each universality class has at least one exactly solvable model. A novel connection between physics and mathematics is exposed. Namely, a general theory to derive link polynomials, topological invariants for knots and links, from the exactly solvable models is presented. It is emphasized that the Yang-Baxter relation is a key to relate various new developments in recent theoretical physics.

Stochastic Korteweg-de Vries Equation with and without Damping

Introducing the Korteweg-de Vries equation with a damping term and an external force, the behaviors of solitons under the Gaussian white noise is studied. In the non-damping case, a height and a width of the soliton are proportional to t -3/2 and t 3/2 respectively, for large t . The asymptotic form of the multi-soliton is also discussed. In the damping case, it is found that a width of the soliton is much narrow while a height is much small as compared with the non-damping case. A new idea for the analysis of inhomogeneous soliton equation is suggested.

Exactly Solvable Models and New Link Polynomials.I.N-State Vertex Models

Presented is a general method to construct representations of the braid group, a basic object in the knot theory, from the Boltzmann weights of the exactly solvable models in statistical mechanics at criticality. The method is applied to a class of N -state vertex models ( N =2, 3 and 4) to have explicit braid group representations. Furthermore, the Markov traces are introduced and a sequence of new link polynomials, topological invariants for knots and links, is constructed.

Knot invariants and the critical statistical systems

A new invariant polynomial for knots and links is constructed from a solvable vertex model describing a critical statistical system. Various implications and the possible generalizations are discussed in connection with the recent development in the study of critical phenomena in two dimensions.

A New Approach to Quantum Spin Chains at Finite Temperature

Presented is a new approach to the study of one dimensional quantum spin systems at finite temperature. The approach is based upon a general structure of solvable models; the Yang-Baxter equation and the Bethe ansatz equation, and in addition the evaluation of finite size corrections. Provided that the ground state is constructed by the Bethe ansatz method, exact low temperature expansions of physical quantities are given without further assumption.

Yang-Baxter Relations for Spin Models and Fermion Models

A systematic method to derive the Yang-Baxter relations for the fermion models which are equivalent to the solvable spin models is presented. The method is applied to prove the Yang-Baxter relation and the commutability of the transfer matrices for the one-dimensional Hubbard model.

Exactly Solvable Models and New Link Polynomials. II. Link Polynomials for Closed 3-Braids

Using a generalized Alexander-Conway relation derived from a three-state exactly solvable model in statistical mechanics, new invariant polynomials for knots and links are explicitly evaluated. It is shown that the invariant polynomials for closed 3-braids are obtained recursively. It is also shown that the invariant polynomials are more powerful than the Jones polynomials.

INVARIANTS OF COLORED LINKS

We define a new hierarchy of isotopy invariants of colored oriented links through oriented tangle diagrams. We prove the colored braid relation and the Markov trace property explicitly.

Statistical Mechanics of Biomembrane Phase Transition. I. Excluded Volume Effects of Lipid Chains in Their Conformation Change

The main phase transition of a membrane is attributed to conformation changes of the hydrocarbon chains constituting the lipid bilayers. The excluded volume effect, which is. crucial for cooperative nature of the conformation changes, is taken into consideration by a modified dimer model. Statistical mechanics of the model is investigated rigorously by means of the transfer matrix formalism without introducing any extra simplification such as the continuum approximation. The model is found to exhibit a new type of second order phase transition which does not contain critical fluctuation: Specific heat is zero for T < T c and is bounded by a finite value for T ≧ T c . A variational evaluation of the maximum eigen-value of the transfer matrix is given for the purpose of quantitative (though approximate) analysis of the phase transition.