It is unlikely that today there is a specialist in theoretical physics who has not heard anything about the algebraic Bethe ansatz. Over the past few years, this method has been actively used in quantum statistical physics models, condensed matter physics, gauge field theories, and string theory. This book presents the state-of-the-art research in the field of algebraic Bethe ansatz. Along with the results that have already become classic, the book also contains the results obtained in recent years. The reader will get acquainted with the solution of the spectral problem and more complex problems that are solved using this method. Various methods for calculating scalar products and form factors are described in detail. Special attention is paid to applying the algebraic Bethe ansatz to the calculation of the correlation functions of quantum integrable models. The book also elaborates on multiple integral representations for correlation functions and examples of calculating the long-distance asymptotics of correlations. This text is intended for advanced undergraduate and postgraduate students, and specialists interested in the mathematical methods of studying physical systems that allow them to obtain exact results.
Author(s): Nikita Slavnov
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 398
City: Singapore
Contents
Preface
About the Author
Acknowledgment
1. Quantum integrable systems
1.1 XXX Heisenberg chain
1.2 Construction of integrable systems
1.3 Yang–Baxter equation
1.4 Constructing the monodromy matrix
1.5 Hamiltonian of the XXX spin chain
1.6 Examples of monodromy matrices
1.6.1 Inhomogeneous XXX chain
1.6.2 Quantum nonlinear Schr¨odinger equation
1.6.3 Trigonometric R-matrix
1.7 Hilbert space
1.8 Commutation relations
1.8.1 Lattice QNLSE model
1.8.2 Quantum determinant
2. Algebraic Bethe ansatz
2.1 Actions of the monodromy matrix entries on vectors
2.1.1 Shorthand notation
2.1.2 Action of operators A and D on Bethe vectors
2.1.3 Eigenvectors and Bethe equations
2.1.4 Example of on-shell Bethe vector
2.1.5 Compatibility of commutation relations
2.2 Twisted monodromy matrix
2.3 Bethe equations
2.3.1 Coinciding roots
2.4 Spurious solutions in spin chains
2.4.1 Spurious solution of twisted Bethe equations
2.4.2 On the completeness of the twisted on-shell Bethe vectors
2.5 Dual Bethe vectors
2.6 Generalized L-operator
3. Quantum inverse problem
3.1 Cyclic permutations in the monodromy matrix
3.2 Quantum inverse problem
4. Composite model
4.1 Summation over partitions
4.2 Definition of composite model
4.3 Multi-composite model
4.3.1 Spin chains
4.3.2 One-dimensional Bose gas
4.4 Special on-shell Bethe vector
5. Scalar products of off-shell Bethe vectors
5.1 Scalar product of off-shell Bethe vectors
5.1.1 Action of C(v) on Bethe vectors
5.1.2 Generalized model and scalar products
5.1.3 Highest coefficient of the scalar products
5.2 Determinant representation for the highest coefficient
5.2.1 Properties of the determinant representation
5.2.2 Summation identity
5.3 Different representations for the scalar product
6. Scalar products with on-shell Bethe vectors
6.1 Identities for determinants
6.2 Determinant formula for the scalar product
6.3 Orthogonality of the eigenvectors
6.4 Norm of the twisted on-shell vector
6.4.1 Basis of twisted on-shell Bethe vectors
6.5 Particular cases of scalar products
7. Alternative methods to compute scalar products
7.1 Scalar product and multiple action of transfer matrix
7.1.1 Scalar product and completely indirect action
7.1.2 Completely indirect action
7.1.3 Multiple action of transfer matrices
7.2 System of linear equations for scalar products
7.2.1 Cauchy matrix
7.2.2 Transformation of the system of linear equations
8. Form factors of the monodromy matrix elements
8.1 Form factors of the off-diagonal elements
8.2 Form factors of the diagonal elements
8.3 Universal form factor
8.3.1 Determinant representations
8.4 Relationships between universal form factors
9. Form factors of local operators
9.1 Form factors of local operators in the spin chains
9.1.1 Magnetization
9.2 Form factors of local operators in the one-dimensional Bose gas
9.2.1 Field form factors
9.3 Calculation of generating function
9.4 One more determinant representation
10. Thermodynamic limit
10.1 Bethe equations in the one-dimensional Bose gas
10.2 Ground state in the thermodynamic limit
10.2.1 Ground state density
10.2.2 Ground state counting function
10.3 Excited states
10.3.1 Counting function of excited states
10.3.2 Shift function
10.3.3 Energy and momentum of the excited states
10.4 Chemical potential
10.5 Dressed charge and shift function
10.6 Thermodynamic limit of the XXZ chain
10.6.1 Zero magnetic field
10.6.2 Dressed charge at the Fermi boundary
10.7 Examples of calculating the thermodynamic limit
10.7.1 Thermodynamic limit of sums and products
10.7.2 Definition of Fredholm determinant
10.7.3 Norm of on-shell Bethe vectors in the thermodynamic limit
10.7.4 Form factors in the thermodynamic limit
11. Multiple integral representations for correlation functions
11.1 Two-point correlation functions via the algebraic Bethe ansatz
11.2 Two-point functions in the XXZ chain
11.3 Generating function
11.4 Multiple integral representation for generating function
11.4.1 Replacement of sums with integrals
11.4.2 Sum over partitions of inhomogeneities
11.4.3 Sum over partitions of Bethe parameters
11.4.3.1 Transformation of Lm(N)
11.4.3.2 Ratio of scalar products
11.4.3.3 Two particular cases
11.5 Generating function via form factor expansion
11.5.1 Sums over solutions of twisted Bethe equations
11.5.2 Poles of the integrand
11.5.3 Calculating the integral
11.5.4 Spurious solutions
11.6 Correlation function of free fermions
11.7 Emptiness formation probability
11.7.1 Emptiness formation probability at Δ = 1/2
12. Asymptotics of correlation functions via form factor expansion
12.1 General scheme of the form factor approach
12.2 Generating function
12.3 Special form factor
12.3.1 Smooth part
12.3.1.1 Analytical properties of the scalar product
12.3.1.2 Thermodynamic limit of the smooth part
12.3.2 Thermodynamic limit of the discrete part
12.3.2.1 Discrete part for impenetrable bosons
12.3.2.2 Discrete part in the general case
12.4 Form factor with particles and holes
12.4.1 Smooth part
12.4.2 Discrete part
12.4.3 Excitations away from the Fermi boundaries
12.4.4 Critical states and critical form factors
12.4.5 Discrete part of critical form factor amplitude
12.5 Summation of critical form factors
12.6 Long-distance asymptotics of the correlation function
12.6.1 The case ℓ = 0
12.6.2 The case ℓ = 0
12.7 Correlation function of fields
12.8 Large-time asymptotics
12.8.1 Correlation function of fields
12.8.2 Density correlation function
Appendix A The ψ-function and the Barnes G-function
A.1 The ψ-function
A.2 The Barnes G-function
Appendix B Finite-size corrections to the excitation energy
B.1 Corrections to the shift function
B.2 Excitation energy
Appendix C Identities for Fredholm determinants
Appendix D Integrals with Vandermonde determinant
Bibliography
Index
