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Progress of Theoretical Physics

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

Kiyoshi Sogo, Yuko Akutsu, T. Abe

Published Sep 1, 1983• 106 citations
Publisher

Abstract

The partition function of the q-state vertex model is derived by the use of the inversion method. The connection of the vertex model to the solid on solid model is established by the generalized Wu-Kadanoff-Wegner transformation. The critical behavior of the model is discussed in relation with the roughening transition.
DOI
10.1143/ptp.70.739

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Journal of the Physical Society of Japan•Jul 15, 1989•16 citations
These are introductory lectures on application of the free field representation (bosonization) techniques to the solid-on-\S olid (SOS) and eight-vertex models.We start from the very beginnings, including the physical badcground of lattice models and some basic information on quantum integrability.After definitions of the eight-vertex and SOS models, we describe their relation known as the vertex-face correspondence.Then, skipping the Bethe ansatz solution, wc turn to the problem of calculation of correlation functions by mcaiis of the fit.cficld rcprcscntation.We explain, how the vertex-face correspondence works on the level of vertex operators and bosonization, making it possible to express the correlation functions of the eight-vertex model in terms of the free field representation aimed to describe the SOS model.1 Eight-vertex model arid commuting traxisfer Ilatl ⋅\cdot⋅ ice.\'Since we shall speak about some models of physical importance.let us formulate first the general physical framework.Consider a classical system of interacting particles.Let CCC be a state of this system, which can run some (generally infinite) set of admissible states.Let E(C)E(C)E(C) be the energy of the state C. defined by usual IIamiltonian mechanics.Suppose that the system weakly interacts with a thermal bath of the temperature TTT .The most fundamental postulate of statistical mechanics, the Gibbs law, says that the probability of the state CCC is given by w(C)=1Ze−B(C)/Tw(C)= \frac{1}{Z}e^{-B(C)/T}w(C)=Z1​e−B(C)/T .(1.1) lt is easy to understand, what is the proportionality coefficient 1/Z1/Z1/Z .As the total probability for thc system to be in any state is unity.we have for the partition functions of the system Z=∑ce−F(C)/7Z= \sum_{c}e^{-F_{}(C)/\mathit{7}}Z=∑c​e−F​(C)/7 .(1.2) Since the space of configurations can be continu$ \mathit{0}\prime 18,$ †,11e\sum\dagger,11\mathrm{e}_{\backslash }\mathrm{s}\mathrm{u}\mathrm{m}†,11e\​sum may turn out to be an itιtegra↓↓ion\mathrm{i}\mathrm{t}\iota \mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\downarrow\downarrow \mathrm{i}o\mathrm{n}itιtegra↓↓ion , but we shall not consider this general case in these lectures.The partition function encodes the most fundamental observable thermodynamic function of the model, the free eneryy:F=E−TS=−Tlog⁡F=E-TS=-T\logF=E−TS=−Tlog Z.(1.3)Here EEE is the total energy of the system, E=∑E(C)w(c,)=T∼dZ(T)dT,Z(T)=−T2dFdTT=F−TdFdTE= \sum E(C)w(c,)=\frac{T^{\sim}d}{Z(T)dT},Z(T)=-T^{2}\frac{dF}{dTT}=F-T\frac{dF}{dT}E=∑E(C)w(c,)=Z(T)dTT∼d​,Z(T)=−T2dTTdF​=F−TdTdF​ , while S=−∑w(c,)log⁡w(C)=−dΓ′dTS=-\sum w(c,)\log w(C)=-\frac{d\Gamma'}{dT}S=−∑w(c,)logw(C)=−dTdΓ′​is the entropy.The first equality in (1.3) is the thermodynamic definition of the free energy, which, in principle, was established before Gibbs, while the second equality provides its statistical interpretation.Let fi⟨Cf_{i}\langle Cfi​⟨C ) be some functions of the state of the system.The correlation functions are generic expectation values ⟨f1…fr′)=∑c∣ι)(C)fι(C)…fn(C)\langle f_{1}\ldots f_{r}')=\sum_{c}|\iota)(C)f\iota(C)\ldots f_{n}(C)⟨f1​…fr′​)=∑c​∣ι)(C)fι(C)…fn​(C) .
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.
Background of the problem The spin 1/21/21/2 XXZ model for Δ<−1\Delta <-1Δ<−1 The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The XXXXXXXXX limit q→−1q\rightarrow-1q→−1 Discussions List of formulas.
For a solvable N -state vertex model in the low-temperature antiferroelectric phase, exact per-site free-energy f is calculated as a function of the order-parameter p (`vertical polarization'), based on the quantum inverse scattering method. The Bethe-ansatz integral equation is solved as an expansion with respect to p , which leads to the Gruber-Mullins-Pokrovsky-Talapov-type expansion, f ( p )= f (0)+ a ·| p |+ b ·| p | 3 + O (| p | 4 ). It is found that the coefficients a and b are identical to those of the six-vertex model ( N =2 case). This fact gives support for the universal Gaussian curvature jump for the N -state vertex model interpreted as a model of interface roughening.