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数理解析研究所講究録

Quantum group symmetry and lattice correlation functions(Algebraic Analysis and Number Theory)

Michio Jimbo

Published Sep 1, 1992• 0 citations
OpenAlex

Abstract

Quantum groups play a role of deformed symmetry of inte- grable models in lattice statistics or quantum field theory.We give a survey of how they arise in the context of the one dimensional XXZ spin chain.We then outline the recent works concerning the structure of the space of states of this model and its spin correlation functions (Joint work with Brian

Related papers

Quantum Group Symmetry and Lattice Correlation Functions

Michio Jimbo

Quantum groups play a role of deformed symmetry of integable models in lattice statistics or quantum field theory. We give a survey of how they arise in the context of the one dimensional XXZ spin chain. We then outline the recent works concerning the structure of the space of states of this model and its spin correlation functions.

Chinese Journal of Physics•Dec 1, 1992•2 citations

Correlation Functions of Quantum Integrable Models: The XXZ Spin Chain

Jean Michel Maillet

A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed. It is based on the resolution of the quantum inverse scattering problem. The application of this method to theXXZ Heisenberg spin- $$\tfrac{1}{2}$$ in a magnetic field leads to multiple integral representations of its n-point correlation functions. For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra $${\mathcal{U}_q}(\widehat s{l_2})$$ together with the corner transfer matrix approach (massive regime). We also discuss further problems, related in particular to long distance asymptotics of these correlation functions.

Birkhäuser Boston eBooks•Jan 1, 2002•0 citations

Quantum group symmetry and lattice correlation functions(Algebraic Analysis and Number Theory)

道夫 神保

数理解析研究所講究録•Sep 1, 1992•0 citations

Local Operators in Integrable Models I

Michio Jimbo, Tetsuji Miwa, Fedor Aleksandrovich Smirnov

ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.

Mathematical surveys and monographs•Jul 2, 2021•2 citations

Algebraic analysis of solvable lattice models

道夫 神保, 哲二 三輪

Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ 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 $XXX$ limit $q\rightarrow-1$ Discussions List of formulas.

American Mathematical Society eBooks•Jan 1, 1999•1 citation

Correlation functions of quantum integrable systems

Jean Michel Maillet

A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed.It is based on the resolution of the so-called quantum inverse scattering problem, namely, the reconstruction of local operators in terms of the (operator) entries of the quantum monodromy matrix obeying a Yang-Baxter algebra.The application of this method to the XXZ Heisenberg spin-1 2 chain in a magnetic field leads to multiple integral representations of its n-point correlation functions.For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q ( ŝl 2 ) together with the corner transfer matrix approach (massive regime).

Non-perturbative Quantum Effects•Sep 25, 2000•1 citation
Quantum groups play a role of deformed symmetry of inte- grable models in lattice statistics or quantum field theory.We give a survey of how they arise in the context of the one dimensional XXZ spin chain.We then outline the recent works concerning the structure of the space of states of this model and its spin correlation functions (Joint work with Brian
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed. It is based on the resolution of the quantum inverse scattering problem. The application of this method to theXXZ Heisenberg spin- 12\tfrac{1}{2}21​ in a magnetic field leads to multiple integral representations of its n-point correlation functions. For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra Uq(s^l2){\mathcal{U}_q}(\widehat s{l_2})Uq​(sl2​) together with the corner transfer matrix approach (massive regime). We also discuss further problems, related in particular to long distance asymptotics of these correlation functions.
ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.
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.
A general approach to the computation of exact correlation functions of lattice quantum integrable models proposed recently is reviewed.It is based on the resolution of the so-called quantum inverse scattering problem, namely, the reconstruction of local operators in terms of the (operator) entries of the quantum monodromy matrix obeying a Yang-Baxter algebra.The application of this method to the XXZ Heisenberg spin-1 2 chain in a magnetic field leads to multiple integral representations of its n-point correlation functions.For zero magnetic field, it gives a proof, in both the massless and massive (anti-ferromagnetic) regimes, of the formulas obtained from the q-deformed KZ equations (massless regime) and the representation theory of the quantum affine algebra U q ( ŝl 2 ) together with the corner transfer matrix approach (massive regime).