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数理解析研究所講究録

Form factors, correlation functions and vertex operators in the eight-vertex model at reflectionless points(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Yas-Hiro Quano

Published Apr 1, 2006• 0 citations
OpenAlex

Abstract

The eight-vertex model at the reflectionless points is considered on the basis of Smirnov's axiomatic approach.Integral formulae for form factors of the eight-vertex model can be obtained in terms of those of the eight-vertex SOS model, by using vertex-face transbrma- tion.The resulting formulae have very simple forms at the reflectionless points, and suggest us the free field representation of type II vertex operators in the eight-verterc model.

Related papers

Form factors, correlation functions and vertex operators in the eight-vertex model at reflectionless points

Yas-Hiro Quano

The eight-vertex model at the reflectionless points is considered on the basis of Smirnov's axiomatic approach. Integral formulae for form factors of the eight-vertex model can be obtained in terms of those of the eight-vertex SOS model, by using vertex-face transformation. The resulting formulae have very simple forms at the reflectionless points, and suggests us the explicit expressions of the type II vertex operators of the eight-vertex model.

arXiv:hep-th/0410084•Oct 8, 2004•3 citations

Form factors, correlation functions and vertex operators in the eight-vertex model at reflectionless points(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

泰宏 桑野

数理解析研究所講究録•Apr 1, 2006•0 citations

Form factors and vertex operators in the XYZ antiferromagnet

Yas-Hiro Quano

This article summarizes my talk on form factors in the reflectionless eight-vertex model, given in the conference 'Elliptic Integrable Systems', held as a part of RIMS Project, Department of Mathematics, Kyoto University, November 8-11, 2004.It is shown that the form factors of σ z in the eight-vertex model at the so-called reflectionless points can be expressed in terms of the sum of theta functions without any integrals.

•Jan 1, 2005•0 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

A Vertex Operator Approach for Form Factors of Belavin's (Z/nZ)-Symmetric Model and Its Application

Yas-Hiro Quano

A vertex operator approach for form factors of Belavin's (Z/nZ)-symmetric model is constructed on the basis of bosonization of vertex operators in the A (1) n-1 model and vertex-face transformation.As simple application for n = 2, we obtain expressions for 2m-point form factors related to the σ z and σ x operators in the eight-vertex model.

arXiv:1010.4382•Symmetry Integrability and Geometry Methods and Applications•Jan 15, 2011•0 citations

Form factors and vertex operators in the eight-vertex model at reflectionless points

Yas-Hiro Quano

素粒子論研究•Jan 20, 2006•0 citations
The eight-vertex model at the reflectionless points is considered on the basis of Smirnov's axiomatic approach.Integral formulae for form factors of the eight-vertex model can be obtained in terms of those of the eight-vertex SOS model, by using vertex-face transbrma- tion.The resulting formulae have very simple forms at the reflectionless points, and suggest us the free field representation of type II vertex operators in the eight-verterc 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) .