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arXiv:cond-mat/9710106 · Journal of the Physical Society of Japan

Test of Guttmann and Enting's Conjecture in the Eight-Vertex Model

Hiroshi Tsukahara, Takeo Inami

Published Apr 15, 1998• 4 citations
arXivHTMLPDF

Abstract

We investigate the analyticity property of the partially resummed series expansion(PRSE) of the partition function for the eight-vertex model. Developing a graphical technique, we have obtained a first few terms of the PRSE and found that these terms have a pole only at one point in the complex plane of the coupling constant. This result supports the conjecture proposed by Guttmann and Enting concerning the ``solvability'' in statistical mechanical lattice models.

DOI
10.1143/jpsj.67.1067

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We investigate the analyticity property of the partially resummed series expansion(PRSE) of the partition function for the eight-vertex model. Developing a graphical technique, we have obtained a first few terms of the PRSE and found that these terms have a pole only at one point in the complex plane of the coupling constant. This result supports the conjecture proposed by Guttmann and Enting concerning the ``solvability'' in statistical mechanical lattice models.
We investigate the analyticity property of the partially resummed series expansion(PRSE) of the partition function for the eight-vertex model. Developing a graphical technique, we have obtained a first few terms of the PRSE and found that these terms have a pole only at one point in the complex plane of the coupling constant. This result supports the conjecture proposed by Guttmann and Enting concerning the “solvability ” in statistical mechanical lattice models. Recently Guttmann and Enting introduced a new point of view into the study of the solvability of lattice models in the statistical mechanics, including models for combinatorial problems [1]. Their approach is based on a study of the connection between the solvability of lattice models by use of the inversion relation and the analyticity properties of their associated series expansion, which is called the partially resummed series expansion(PRSE). For exactly solvable models, the inversion relation for the partition function was derived using the star-triangle relation [2,3]. Subsequently, it was shown to hold true for the partition
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) .
Abstract Any planar set of intersecting straight lines forms a four-coordinated graph, or ‘lattice’, provided no three lines intersect at a point. For any such lattice an eight-vertex model can be constructed. Provided the interactions satisfy certain constraints (which are in general temperature-dependent), the model can be solved exactly in the thermodynamic limit, its local properties at a particular site being those of a related square lattice. A particular case is a solvable model on the Kagomé lattice. It is shown that this model includes as special cases many of the models in statistical mechanics that have been solved exactly, notably the square, triangular and honeycomb Ising models, and the square eight-vertex model. Some remarkable equivalences between correlations on different lattices are also established.