Keywords:
Integral equations.
;
Operator theory.
;
Quantum field theory.
;
Electronic books.
Description / Table of Contents:
Integrable 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.Going through the book, readers will find themselves at the forefront of this rapidly developing research field.
Type of Medium:
Online Resource
Pages:
1 online resource (208 pages)
Edition:
1st ed.
ISBN:
9781470465766
Series Statement:
Mathematical Surveys and Monographs ; v.256
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=6661094
DDC:
530.13
Language:
English
Note:
Intro -- Introduction -- Chapter 1. Formulation of the Problem -- 1.1. Six-vertex Model -- 1.2. Using Tensor Notation -- Disordered phase -- Ordered phase -- 1.3. The Main Object of Our Study -- 1.4. Spectral Parameter and R-Matrix -- 1.5. Six-vertex Model on a Plane -- 1.6. XXZ Anti-ferromagnet at Finite Temperature -- 1.7. Density Matrix and Entanglement von Neumann Entropy -- 1.8. Our Strategy in Volume I -- Chapter 2. Spectral Problem in Matsubara Direction and Quantum Groups -- 2.1. Algebraic Bethe Ansatz -- 2.2. Algebra _{ }(̂ ₂) -- 2.2.1. General Definitions -- 2.2.2. Algebra _{ }( ₂) -- 2.2.3. Algebra _{ }(̂ ₂) -- 2.3. Bazhanov-Lukyanov-Zamolodchikov Construction -- 2.3.1. q-Oscillator Representation of _{ } ⁺ -- 2.3.2. Intertwiner for ^{±}_{ ₁}⊗ ^{±}_{ ₂} -- 2.3.3. Triangular Structures of ⁺⊗ ⁻ -- 2.3.4. Triangular Structure of ⁽¹⁾_{ }⊗ ^{±}_{ } -- 2.4. Higher Spins in Matsubara Space -- 2.4.1. Summary -- 2.5. Q-Operators -- 2.6. Destri-deVega Equation -- 2.6.1. General Procedure -- 2.6.2. Finite temperature case -- Chapter 3. Fermions -- 3.1. Intertwiner and Quasi-intertwiner for Fused Modules -- 3.1.1. Intertwiner for Representations of the Same Kind -- 3.1.2. Quasi-intertwiner for Operators of Different Kind -- 3.2. Operators ( , ) and ( )( , ) -- 3.2.1. Adjoint Action of R Matrices -- 3.2.2. Definition and Reduction Properties -- 3.2.3. Commutation Relations -- 3.2.4. Analytic Properties -- 3.3. Annihilation Operators -- 3.4. Creation Operators -- 3.4.1. Operator *( ) -- 3.4.2. Commutation Relations with , ̄ -- 3.5. Fermionic Creation Operators -- 3.6. Homogeneous Versus Inhomogeneous Cases: Russian Doll Construction -- 3.7. Commutation Relations Between Creation and Annihilation Operators -- 3.8. Summary -- Chapter 4. Main Theorem -- 4.1. Fermionic Basis and Difference Equations -- 4.2. Deformed Abelian Differentials.
,
4.3. Main Theorem -- 4.4. Completeness in Homogeneous Case -- 4.4.1. Linear Independence -- 4.4.2. Operators ̄*, ̄*, ̄* -- 4.4.3. Basis -- 4.5. Summary -- Chapter 5. Applications and Generalisations -- 5.1. Function ( , | ) via Integral Equation -- 5.2. Main Theorem and Inverse Problem -- 5.2.1. General Idea -- 5.2.2. Matsubara Data -- 5.2.3. Making Equations -- 5.2.4. Examples -- 5.3. The Case =0 -- 5.3.1. General Remarks -- 5.3.2. Reduction to the Quotient Space -- 5.3.3. The Case =0. -- 5.3.4. Computation of the Function . -- 5.3.5. Entanglement Entropy -- 5.3.6. Invariant Operators -- 5.4. XXX Case -- 5.5. Remarks on XYZ Case -- 5.5.1. Another Way of Presenting the XXZ Results -- 5.5.2. XYZ Model and Sklyanin Algebra -- 5.5.3. Trace -- 5.5.4. Formula for Correlation Functions -- 5.5.5. Discussion -- Appendix A. Quasi-classical Limit and Algebraic Geometry -- A.1. Algebraic Interpretation of Quantum Results -- A.2. Canonical Differential in the Classical Case -- A.3. Riemann Surfaces -- A.4. Affine Jacobi Variety -- A.5. Classical Interpretation of Fermionic Basis -- Notation -- Bibliography -- Index.
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