Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Test of Guttmann and Enting’s conjecture in the eight-vertex model

Hiroshi Tsukahara, Takeo Inami

Published Jan 1, 1997• 5 citations
OpenAlex

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. 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

Related papers

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

Hiroshi Tsukahara, Takeo Inami

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.

arXiv:cond-mat/9710106•Journal of the Physical Society of Japan•Apr 15, 1998•4 citations

A distant descendant of the six-vertex model

Vladimir V. Bazhanov, С. М. Сергеев

In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.

arXiv:2310.08427•Nuclear Physics B•May 8, 2024•3 citations

Lectures on the eight-vertex model and bosonization(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Michael Lashkevich

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 $C$ be a state of this system, which can run some (generally infinite) set of admissible states.Let $E(C)$ 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 $T$ .The most fundamental postulate of statistical mechanics, the Gibbs law, says that the probability of the state $C$ is given by $w(C)= \frac{1}{Z}e^{-B(C)/T}$ .(1.1) lt is easy to understand, what is the proportionality coefficient $1/Z$ .As the total probability for thc system to be in any state is unity.we have for the partition functions of the system $Z= \sum_{c}e^{-F_{}(C)/\mathit{7}}$ .(1.2) Since the space of configurations can be continu$ \mathit{0}\prime 18,$ $\dagger,11\mathrm{e}_{\backslash }\mathrm{s}\mathrm{u}\mathrm{m}$ may turn out to be an $\mathrm{i}\mathrm{t}\iota \mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\downarrow\downarrow \mathrm{i}o\mathrm{n}$ , 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:$F=E-TS=-T\log$ Z.(1.3)Here $E$ is the total energy of the system, $E= \sum E(C)w(c,)=\frac{T^{\sim}d}{Z(T)dT},Z(T)=-T^{2}\frac{dF}{dTT}=F-T\frac{dF}{dT}$ , while $S=-\sum w(c,)\log w(C)=-\frac{d\Gamma'}{dT}$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 $f_{i}\langle C$ ) be some functions of the state of the system.The correlation functions are generic expectation values $\langle f_{1}\ldots f_{r}')=\sum_{c}|\iota)(C)f\iota(C)\ldots f_{n}(C)$ .

数理解析研究所講究録•Apr 1, 2006•0 citations

Solvable eight-vertex model on an arbitrary planar lattice

R. J. Baxter

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.

Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences•May 2, 1978•301 citations

SOLVABLE EIGHT-VERTEX MODEL ON AN ARBITRARY PLANAR LATTICE

R. J. Baxter

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 Kagome 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.

Advanced series in mathematical physics•Mar 1, 1990•0 citations

Partition function of the eight-vertex model with domain wall boundary condition

Wen‐Li Yang, Yao-Zhong Zhang

We derive the recursive relations of the partition function for the eight-vertex model on an N×N square lattice with domain wall boundary condition. Solving the recursive relations, we obtain the explicit expression of the domain wall partition function of the model. In the trigonometric/rational limit, our results recover the corresponding ones for the six-vertex model.

arXiv:0903.3089•Journal of Mathematical Physics•Aug 1, 2009•14 citations
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
In this paper we present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local “spin variables” can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an equivalent “dual” formulation of the model, where the spins take continuous real values on the circle. From algebraic point of view this model is closely related to the to the 6-vertex model. It is connected with the construction of an intertwiner for two infinite-dimensional representations of the quantum affine algebra Uq(slˆ(2)) without the highest and lowest weights. The partition function of the model in the large lattice limit is calculated by the inversion relation method. Amazingly, it coincides with the partition function of the off-critical 8-vertex free-fermion model.
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.
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 Kagome 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.