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数理解析研究所講究録

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

Michael Lashkevich

Published Apr 1, 2006• 0 citations
OpenAlex

Abstract

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 $\langle f_{1}\ldots f_{r}')=\sum_{c}|\iota)(C)f\iota(C)\ldots f_{n}(C)$ .

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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 )

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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 continu0′18, \mathit{0}\prime 18,0′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) .
This chapter introduces the six-vertex model defined on a two-dimensional square lattice. It describes the model in detail, since it gives an important prototype of many solvable lattice models defined on two-dimensional lattices. The eight-vertex model, which generalizes the six-vertex model directly, may be considered as the most important exactly solvable model in statistical mechanics. The chapter explores some features of the six-vertex model defined on a square lattice. It reviews a method for diagonalizing the transfer matrix, which is called the coordinate Bethe ansatz, and provides the expressions of the free energy per site in the ferroelectric, the antiferroelectric and the disordered phases, respectively. The chapter discusses some mathematical theories associated with integrable models such as the braid group and the quantum groups. It outlines the Ising model, the Potts models and the chiral Potts model and then the eight-vertex model and the interaction-round-a-face models.
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
Abstract Vertex models more general than the ice model are possible and often have physical applications. The square lattice admits the general sixteen-vertex model of which the special cases, the eight- and the six-vertex model, are the most relevant and physically interesting, in particular through their connection to the one-dimensional integrable quantum mechanical models and the Bethe ansatz. This chapter introduces power- ful tools to examine vertex models, including the R- and L-matrices to encode the Boltzmann vertex weights and the monodromy and transfer matrices, which encode the integrability of the vertex models (i.e. that transfer matrices of different spectral parameters commute). This integrability is ultimately expressed in the Yang–Baxter relations.
This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School ``Integrability in Atomic and Condensed Matter Physics'' in the summer of 2018. The responsibility for the selection of the material is partially with the organisers, Jean-Sebastien Caux, Nikolai Kitanine, Andreas Klümper and Robert Konik. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. My lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. I was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. I was further asked to explain the graphical language of vertex models and to introduce the concepts of the Trotter decomposition and the quantum transfer matrix. This was basically the contents of the first four lectures presented at the school. In the remaining two lectures I started filling these notions with life by deriving an integral representation of the free energy per lattice site for the Heisenberg-Ising chain (alias XXZ model) using techniques based on non-linear integral equations. Up to small corrections the following sections 1-6 display the six lectures almost literally. The only major change is that the example of the XXZ chain has been moved from section 5 to 2. During the school it was not really necessary to introduce the model, since other speakers had explained it before. But for these notes I thought it might be useful to introduce the main example rather early. I also supplemented each lecture with a comment section which contains additional references and material of the type that was discussed informally with the participants.