Finite size effects in correlated electron models: exact results

Andrei A. Zvyagin1860945031, 9781860945038, 9781860946868

The book presents exact results for one-dimensional models (including quantum spin models) of strongly correlated electrons in a comprehensive and concise manner. It incorporates important results related to magnetic and hybridization impurities in electron hosts and contains exact original results for disordered ensembles of impurities in interacting systems. These models describe a number of real low-dimensional electron systems that are widely used in nanophysics and microelectronics. An important method of modern theoretical and mathematical physics — the Bethe’s Ansatz (BA) — is introduced to readers. This book presents different forms of the BA for periodic and open quantum chains. Other forms dealt with are the co-ordinate BA, thermodynamic BA, nested BA, algebraic BA, and thermal BA. The book also contains a compact description of other theoretical methods such as scaling, conformal field theory, Abelian and non-Abelian bosonizations. The book is suitable for use as a textbook by graduate students in non-perturbative methods of low-dimensional quantum many-body theory. It will also be a useful source of reference for qualified physicists, as well as non-experts in low-dimensional physics, as it explores material necessary for further studies in the fields of exactly solvable quantum models and low-dimensional correlated electron systems.

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Table of contents :
Contents……Page 12
Preface……Page 8
1.1 Why is the Topic of the Book Worthwhile Studying?……Page 14
1.2 Thermodynamics……Page 18
1.3 Statistical Mechanics: Simple Models……Page 20
1.4 Mermin–Hohenberg Theorem……Page 30
2. Quantum Spin-1/2 Chain with the Nearest-Neighbour Couplings……Page 38
2.1 One-Dimensional Spin Hamiltonian……Page 39
2.2 Ising Chain……Page 40
2.3 Isotropic XY Ring……Page 43
2.4 Ising Chain in a Transverse Magnetic Field……Page 48
2.5 Dimerized XY Chain……Page 51
3.1 Bethe Ansatz……Page 62
3.2 Simple Solutions of the Bethe Ansatz Equations: Strings……Page 68
3.3 Thermodynamic Bethe Ansatz……Page 70
3.4 The Ground State Behaviour……Page 75
3.5 Magnetic Field Behaviour in the Ground State: Wiener–Hopf Method……Page 79
4. Correlated Electron Chains: Co-ordinate Bethe Ansatz……Page 86
4.1 Hubbard Chain……Page 87
4.2 t-J Chain……Page 110
5.1 The Algebraic Bethe Ansatz for a Spin-1/2 Chain……Page 128
5.2 SU(2)-Symmetric Spin-S Chain……Page 137
5.3 The Algebraic Bethe Ansatz for Correlated Electron Models……Page 150
6.1 Open Boundaries. XY and Ising Chains……Page 162
6.2 Open Boundaries: Co-ordinate Bethe Ansatz for the Heisenberg–Ising Chain……Page 166
6.3 Open Boundaries. The Algebraic Bethe Ansatz……Page 177
6.4 Open Hubbard Chain……Page 184
6.5 Open Supersymmetric t-J Chain……Page 194
7.1 Impurities in XY Chains……Page 202
7.2 Impurities in Spin Chains: Bethe Ansatz……Page 213
7.3 Impurity in Correlated Electron Chains……Page 232
8.1 Impurities’ Bands……Page 256
8.2 Disodered Ensembles of Impurities in Correlated Chains. “Quantum Transfer Matrix” Approach……Page 275
9.1 Finite Size Corrections for Quantum Spin Chains……Page 298
9.2 Finite Size Corrections for Correlated Electron Chains……Page 307
9.3 Elements of Conformal Field Theory……Page 317
9.4 Asymptotics of Correlation Functions……Page 329
9.5 Persistent Currents in Correlated Electron Rings……Page 336
10.1 Scaling Analysis……Page 348
10.2 Bosonization……Page 352
Bibliography……Page 368
Index……Page 378