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Progress of Theoretical Physics

New Factorized S-Matrix and Its Application to Exactly Solvable q-State Model. I

Kiyoshi Sogo, Yuko Akutsu, Takayuki Abe

Published Sep 1, 1983• 97 citations
Publisher

Abstract

Presented is a new q-state factorized S-matrix which possesses the CPT invariances, crossing symmetry and the charge conservation symmetry. The result is a generalization of Zamolodchikov and Fateev’s case (q = 3) to arbitrary q case. Another solution of the factorization equations is given and asymmetric generalization of the q-state model is discussed.

DOI
10.1143/ptp.70.730

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Presented is a new q-state factorized S-matrix which possesses the CPT invariances, crossing symmetry and the charge conservation symmetry. The result is a generalization of Zamolodchikov and Fateev’s case (q = 3) to arbitrary q case. Another solution of the factorization equations is given and asymmetric generalization of the q-state model is discussed.
Multi-state generalization, which is applicable to arbitrary IRF model, is presented. The generalization is made in such a way that each spin is replaced by an S N -symmetric multiplet. If the original IRF model is exactly solvable satisfying the star-triangle equation, so is the generalized one. A simple symmetry-breaking procedure is also presented. In the special case N =2, another kind of generalization is proposed and an exactly solvable 4-state model with S 2 -symmetry between the highest spins is newly constructed.
The Schrodinger equation i Psi =(A+Bt) Psi with constant Hermitian matrices A and B is studied. In the form where B is diagonalized, this equation is a generalization of the Landau-Zener problem to an arbitrary number of crossing energy levels. An approach-the independent crossing approximation-leading to a partial understanding of the general case in terms of the two-level problem is introduced. It is found that certain S-matrix elements (and thus the corresponding transition probabilities) for the general problem are exactly given by formulae of unexpected simplicity, suggesting that some kind of general analytic solution might exist. The full S-matrix is calculated for a solvable special case of the problem. The solution of another special case, previously discovered for systems with three states, is generalized to any number of states. The asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1/t.
In the case of two-component models, all possible solutions for factorization equations of S-matrix are obtained. These solutions give a classification of solvable vertex models. It is found that Yang-Baxter relation is more general than the factorization equation. The six vertex model by Yang, Yang and Sutherland is derived not from the factorization equation but from Yang-Baxter relation. Thus it is concluded that Yang-Baxter relation lies at the basis of all the known solvable two component models. Furthermore, spin Hamiltonians corresponding to our vertex models are presented in explicit forms.
We give a systematic construction of new quasi-exactly solvable systems via Bethe ansatz and supersymmetric quantum mechanics (SUSYQM). Methods based on the intertwining of supercharges have been extensively used in the literature for exactly solvable systems. We generalize the state-deleting (Krein-Adler) supersymmetric transformations to quasi-exactly exactly solvable (QES) systems building on the Bethe ansatz method and related Bethe roots. This enables us to construct superpartners for a wide class of known QES systems classified previously through a hidden sl(2)sl(2)sl(2) algebra. We present our constructions of factorizations and intertwining relations related to 1st-order SUSYQM and the n=1n=1n=1 state for 10 nonequivalent types, denoted I,...,X. In order to have a unified treatment we rely on their ODE standard form as this is also the appropriate setting to obtain the Bethe ansatz equations which constrain the polynomial solutions. This setting also allows one to deal with systems with nnn states in a unified manner, using analysis based on the Bethe ansatz equations to build the supersymmetric transformations in terms of the Bethe ansatz roots. We derive the Schrödinger potentials for the n=1n=1n=1 superpartners of the 10 QES cases and give closed-form solutions for the spectra and wavefunctions of the corresponding QES SUSYQM systems. Furthermore, we present numerical results for higher excited states up to the n=10n=10n=10 level. The results obtained may have wider applicability as our framework is built on ODEs with polynomial coefficients.
We solve the SU(N)-invariant Yang-Baxter equations imposing only the unitarity condition. The usual S matrices should satisfy the crossing symmetry which originates from the CPT invariance of relativistic quantum-field theory. In this paper, we consider nonrelativistic SU(N)-invariant factorizable S matrices by relaxing the crossing symmetry and making the amplitudes for creating and annihilating new particles vanish and find that these S matrices are exactly the same as those of the multicomponent Calogero-Sutherland model, the quantum-mechanical model with the hyperbolic potential between particles and antiparticles. This particular solution is of interest since it cannot be obtained as a nonrelativistic limit of any known relativistic solutions of the SU(N)-invariant Yang-Baxter equations. \textcopyright{} 1996 The American Physical Society.