Preface ......................................................... v
Acknowledgements ............................................... ix
Note to the Readers ............................................ xi
Chapter 1: Classical Statistical Mechanics
Introduction .................................................... 1
The Takahashi Nearest-Neighbor Gas .............................. 2
The Kac-Baker Mode ............................................. l7
The Coulomb Gas and the Statistical Theory of Energy Levels .... 12
Bibliography ................................................... 22
Reprinted papers
A Simple Method for Treating the Statistical Mechanics of
One-Dimensional Substances ..................................... 25
H. Takahashi Proc. Phys.-Math. Soc. Japan 24, 60 (1942)
Sur l'lntegrale de Configuration pour les Systemes de
Particules a Une Dimension ..................................... 28
L. van Hove Physica 16, 137 (1950)
Exact Partition Functions for some One-Dimensional Models
via the Isobaric Ensemble ...................................... 35
M. Bishop and M.A. Boonstra Amer. J. Phys. 51, 564 (1983)
Molecular Distribution Functions in a One-Dimensional Fluid .... 38
Z.W. Salsburg, R.W. Zwanzig and J.G. Kirkwood J. Chem.
Phys. 21, 1098 (1953)
Certain General Order-Disorder Models in the Limit of
Long-Range Interactions ........................................ 48
G.A. Baker, Jr. Phys. Rev. 126, 2071 (1962)
On the van der Waals Theory of the Vapor-Liquid
Equilibrium, I. Discussion of a One-Dimensional Model .......... 56
M. Kac, G. E. Uhlenbeck and P. C. Hemmer J. Math. Phys.
4, 216 (1963)
On the van der Waals Theory of the Vapor-Liquid
Equilibrium, III. Discussion of the Critical Region ............ 69
P.C. Hemmer, M. Kac and G.E. Uhlenbeck J. Math. Phys. 5,
60 (1964)
Exact Statistical Mechanics of a One-Dimensional System with
Coulomb Forces ................................................. 84
A. Lenard J. Math. Phys. 2, 682 (1961)
One-Dimensional Plasma Model at Thermodynamic Equilibrium ...... 96
O.C. Eldridge and M. Feix Phys. Fluids 5, 1076 (1962)
Statistical Mechanics of a One-Dimensional Coulomb System
with a Uniform Charge Background .............................. 101
R.J. Baxter Proc. Camb. Phil. Soc. 59, 779 (1963)
Chapter 2: Spectrum of Disordered and/or Anharmonic Chains of
Oscillators
Introduction .................................................. 111
Dyson's Method ................................................ 112
Other Rigorous Solutions ...................................... 116
Special Theorems .............................................. 118
Some Recent Work .............................................. 120
Anharmonic Chains — Classical and Quantum ..................... 121
Anharmonicity and Randomness .................................. 124
Bibliography .................................................. 125
Reprinted papers
The Dynamics of a Disordered Linear Chain ..................... 129
F.J. Dyson Phys. Rev. 92, 1331 (1953)
Disordered One-Dimensional Crystals ........................... 137
H. Schmidt Phys. Rev. 105, 425 (1957)
Special Frequencies in the Vibrational Spectra of
Disordered Chains ............................................. 154
H. Matsuda Progr. Theor. Phys. {Kyoto) 31, 161 (1964)
Some Exact Results for the Vibrational Spectrum of a
Disordered Chain .............................................. 156
R.R. Borland Proc. Phys. Soc. (London) 83, 1027 (1964)
Structure of the Spectra of Disordered Systems, I.
Fundamental Theorems .......................................... 162
J. Hori and H. Matsuda Progr. Theor. Phys. (Kyoto)
32, 183 (1964)
The Vibrational Spectrum of a Disordered Linear Syst .......... 169
R.L. Agacy Proc. Phys. Soc. (London) 83, 591 (1964)
Vibrations of Glass-Like Disordered Chains .................... 175
P. Dean Proc. Phys. Soc. (London) 84, 727 (1964)
Vibrations of a Chain with Nonlinear Interaction .............. 193
M. Toda J. Phys. Soc. Japan 22, 431 (1967a)
Wave Propagation in Anharmonic Lattices ....................... 199
M. Toda J. Phys. Soc. Japan 23, 501 (1967b)
A Soliton and Two Solitons in an Exponential Lattice
and Related Equations ......................................... 205
M. Toda and M. Wadati J. Phys. Soc. Japan 34, 18 (1973)
Integrals of the Toda Lattice ................................. 213
M. Henon Phys. Rev. B9, 1921 (1974)
The Toda Lattice, II. Existence of Integrals .................. 216
H. Flaschka Phys. Rev. B9, 1924 (1974)
Solution of a Three-Body Problem in One Dimension ............. 218
F. Calogero J. Math. Phys. 10, 2191 (1969)
Solution of the One-Dimensional N-Body Problems with
Quadratic and/or Inversely Quadratic Pair Potentials .......... 224
F. Calogero J. Math. Phys. 12, 419 (1971)
A Brief History of the Quantum Soliton with New Results on
the Quantization of the Toda Lattice .......................... 242
B. Sutherland Rocky Mountain J. Math. 8, 413 (1978)
Chapter 3: Electron Energy Bands in Ordered and Disordered
"Crystals"
General Considerations about Periodic Structures .............. 259
Special Periodic Structures ................................... 262
Slight Deviations from Periodicity ............................ 263
Large Deviations from Periodicity ............................. 264
Localizations of Eigenstates .................................. 267
A Nonlinear Eigenvalue Problem ................................ 268
Miscellania ................................................... 270
Bibliography .................................................. 272
Reprinted papers
The Approximate Solution of One-Dimensional Wave Equations .... 277
С. Eckart Revs. Mod. Phys. 20, 399 (1948)
Quantum Mechanics of Electrons in Crystal Lattices ............ 296
R. de L. Kronig and W.G. Penney Proc. Roy. Soc. (London)
A130, 499 (1931)
New Soluble Energy Band Problem ............................... 311
F.L. Scarf Phys. Rev. 112, 1137 (1958)
Energy Bands and Wave Functions in Periodic Potentials ........ 315
H.M. James Phys. Rev. 76, 1602 (1949)
Wave Propagation in One-Dimensional Structures ................ 324
J.M. Luttinger Philips Research Rep. 6, 303 (1951)
Analytic Properties of Bloch Waves and Wannier Functions ...... 332
W. Kohn Phys. Rev. 115, 809 (1959)
A Theory of the Electrical Breakdown of Solid Dielectrics ..... 345
С. Zener Proc. Roy. Soc. (London) 145, 523 (1934)
Electron Levels in a One-Dimensional Random Lattice ........... 352
H.L. Frisch and S.P. Lloyd Phys. Rev. 120, 1175 (1960)
Existence of Energy Gaps in One-Dimensional Liquids ........... 367
R.E. Borland Proc. Phys. Soc. (London) 78, 926 (1961)
The Electronic Structure of a One-Dimensional Random Alloy .... 373
R.L. Agacy and R.E. Borland Proc. Phys. Soc.
(London) 84, 1017 (1964)
Existence of Energy Gaps in the Spectrum of a
One-Dimensional Atomic Chain .................................. 383
L. Dworin Phys. Rev. 138, A1121 (1965)
Saxon-Hutner Theorem for One-Dimensional General Alloys ....... 389
B.Y. Tong and S.Y. Tong Phys. Rev. 180, 739 (1969)
Short-Range versus Long-Range Order in a Model Binary Alloy ... 394
M. Plischke and D.С. Mattis Phys. Rev. Lett. 27, 42
(1971)
The Statistics of One-Dimensional Resistances ................. 398
P.D. Kirkman and J.B. Pendry J. Phys. C: Solid State
Phys. 17, 4327 (1984)
Exact Solution of a Nonlinear Eigenvalue Problem .............. 416
F.J. Roraeiras and G. Rowlands Phys. Rev. A33, 3499
(1986)
Chapter 4: The Many-Fermion Problem
Introduction .................................................. 419
Tomonaga's Model .............................................. 421
Exactly Soluble Model ......................................... 423
To Fill or Not To Fill? ....................................... 424
Is Perturbation Theory Valid? ................................. 425
Fermionization ................................................ 426
The Hubbard Model ............................................. 430
Some Theorems and Exact Results ............................... 431
Fractional Charge ............................................. 433
Field Theory .................................................. 434
General ....................................................... 436
Bibliography .................................................. 437
Reprinted papers
Remarks on Bloch's Method of Sound Waves Applied to
Many-Fermion Problems ......................................... 441
S. Tomonoga Progr. Theoret. Phys. {Kyoto) 5, 544 (1950)
Exact Solution of a Many-Fermion System and its Associated
Boson Field ................................................... 467
D.C. Mattis and E.H. Lieb J. Math. Phys. 6, 304 (1965)
Tomonaga's Model and the Threshold Singularity of X-Ray
Spectra of Metals ............................................. 476
K.D. Schotte and U. Schotte Phys. Rev. 182, 479 (1969)
New Wave-Operator Identity Applied to the Study of
Persistent Currents in ID ..................................... 480
D.С. Mattis J. Math. Phys. 15, 609 (1974)
Conductivity of One-Dimensional Interacting Fermions .......... 484
D.С. Mattis Phys. Rev. Lett. 32, 714 (1974)
Backward Scattering in the One-Dimensional Electron Gas ....... 488
A. Luther and V.J. Emery Phys. Rev. Lett. 33, 589 (1974)
Calculation of Critical Exponents in Two Dimensions from
Quantum Field Theory in One Dimension ......................... 492
A. Luther and I. Peschel Phys. Rev. B12, 3908 (1975)
Generalization of the Landau Liquid Concept: Example of
the Luttinger Liquids ......................................... 502
J. Carmelo and A.A. Ovchinnikov J. Phys.: Cond.
Matter 3, 757 (1991)
Structure and Solution of the Massive Thirring Model .......... 511
H. Bergknoff and H. B. Thacker Phys. Rev. D19, 3666 (1979)
One-Dimensional Backward-Scattering Fermion Model with
Built-in Cutoff ............................................... 527
S. Rudin Phys. Rev. B28, 4825 (1983)
Strange Solutions to Field Theories in One
Spatial Dimension ............................................. 531
D.С. Mattis and B. Sutherland J. Math. Phys. 22, 1692 (1981)
Some Exact Results for the Many-Body Problem in One
Dimension with Repulsive Delta-Function Interaction ........... 535
C.N. Yang Phys. Rev. Lett. 19, 1312 (1967)
S Matrix for the One-Dimensional N-Body Problem with
Repulsive or Attractive δ-Function Interaction .......... 539
С.N. Yang Phys. Rev. 168, 1920 (1968)
Absence of Mott Transition in an Exact Solution of the
Short-Range, One-Band Model in One Dimension .................. 543
E.H. Lieb and F.Y. Wu Phys. Rev. Lett. 20, 1445 (1968)
Theory of Ferromagnetism and the Ordering of Electronic
Energy Levels ................................................. 547
E.H. Lieb and D.C. Mattis Phys. Rev. 125, 164 (1962)
Magnetic Properties of Some Itinerant-Electron Systems
at T > 0 ...................................................... 556
M. Aizenman and E.H. Lieb Phys. Rev. Lett. 65, 1470
(1990)
One-Dimensional Plasma as an Example of a Wigner Solid ........ 560
B. Sutherland Phys. Rev. Lett. 35, 185 (1975)
Soliton Excitations in Polyacetylene .......................... 564
W.P. Su, J.R. Schrieffer and A.J. Heeger Phys. Rev.
B22, 2099 (1980)
Soliton with Fermion Number 1/2 in Condensed Matter and
Relativistic Field Theories ................................... 577
R. Jackiw and J. R. Schrieffer Nucl. Phys. B190 [FS3],
253 (1981)
Chapter 5: The Bose Gas
The Hard Core Gas ............................................. 592
Point Interactions: Bethe Ansatz Solution in the Continuum
and Numerical Solution on a Lattice ........................... 596
Bose Gas with Attractive Forces ............................... 597
Repulsive Forces .............................................. 598
The Quantum Nonlinear Schrödinger Equation .................... 598
The Yang-Yang Thermodynamic Formalism ......................... 599
Bibliography .................................................. 601
Reprinted papers
Exact Analysis of an Interacting Bose Gas, I. The General
Solution and the Ground State ................................. 605
E.H. Lieb and W. Liniger Phys. Rev. 130, 1605 (1963)
Exact Analysis of an Interacting Bose Gas, II. The Excitation
Spectrum ...................................................... 617
E.H. Lieb Phys. Rev. 130, 1616 (1963)
Momentum Distribution in the Ground State of the
One-Dimensional System of Impenetrable Bosons ................. 626
A. Lenard J. Math. Phys. 5, 930 (1964)
One-Dimensional Impenetrable Bosons in Thermal
Equilibrium ................................................... 640
A. Lenard J. Math. Phys. 7, 1268 (1966)
Study of Exactly Soluble One-Dimensional ЛГ-Body Problems ..... 645
J.B. McGuire J. Math. Phys. 5, 622 (1964)
Quantum Critical Phenomena in One-Dimensional Bose
Systems ....................................................... 660
G.G. Batrouni, R.T. Scalettar and G.T. Zimanyi Phys.
Rev. Lett. 65, 1765 (1990)
Thermodynamics of a One-Dimensional System of Bosons with
Repulsive Delta-Function Interaction .......................... 664
C.N. Yang and С.Р. Yang J. Math. Phys. 10, 1115 (1969)
Chapter 6: Magnetism
Origins of the Magnetic Force ................................. 673
Algebra of Spin Operators ..................................... 674
The Heisenberg Model .......................................... 675
Antiferromagnetic s = 1/2 Heisenberg Chain .................... 677
Anisotropic Exchange .......................................... 682
Three Exactly Soluble Models .................................. 684
Integer-Spin Antiferromagnetism ............................... 685
Miscellania ................................................... 687
Bibliography .................................................. 689
Reprinted papers
On the Theory of Metals, I. Eigenvalues and Eigenfunctions of
a Linear Chain of Atoms (English Translation) ................. 693
H. Bethe Zeitsf. Physik 71, 205 (1931)
Linear Magnetic Chains with Anisotropic Coupling .............. 721
J.C. Bonner and M. E. Fisher Phys. Rev. 135, A640
(1964)
Magnetization Curve at Zero Temperature for the
Antiferromagnetic Heisenberg Linear Chain ..................... 740
R.B. Griffiths Phys. Rev. 133, A768 (1964)
What is the Spin of a Spin Wave? .............................. 748
L.D. Faddeev and L.A. Takhtajan Phys. Lett. 85A, 375
(1981)
A Survey of Results for the 1-D Heisenberg Magnets ............ 751
J.D. Johnson J. Appl. Phys. 52, 1991 (1981)
Two Soluble Models of an Antiferromagnetic Chain .............. 753
E. Lieb, T. Schultz and D. Mattis Ann. Phys. 16, 407
(1961)
Nonlinear Field Theory of Large-Spin Heisenberg
Antiferromagnets: Semiclassically Quantized Solitons of
the One-Dimensional Easy-Axis Neel State ...................... 813
F.D.M. Haldane Phys. Rev. Lett. 50, 1153 (1983)
Comment on "Ground State Properties of a Spin-1
Antiferromagnetic Chain" ...................................... 817
D.С. Mattis Phys. Rev. B31, 4698 (1985)
Rigorous Results on Valence-Bond Ground States in
Antiferromagnets .............................................. 819
I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki Phys.
Rev. Lett 59, 799 (1987)
Quantum Spin Chains and the Haldane Gap ....................... 823
I. Affleck J. Phys.: Cond. Matter 1, 3047 (1989)
Chapter 7: Time-Dependent Phenomena and the Approach to
Equilibrium
Diffusion ..................................................... 849
The Lack of Thermalization .................................... 850
Conclusions on FPU Paradox .................................... 851
A Historical Note ............................................. 852
Bibliography .................................................. 853
Reprinted papers
Studies of Nonlinear Problems. I .............................. 857
E. Fermi, J. Pasta, S. Ulam (with M. Tsingou)
Los Alamos National Lab. Report LA-1940, May 1955
(unpublished)
The Fermi-Pasta-Ulam Problem: Paradox Turns Discovery ......... 877
J. Ford Phys. Rep. 213, 272 (1992)
Statistical Mechanics of Assemblies of Coupled Oscillators .... 916
G.W. Ford, M. Kac and P. Mazur J. Math. Phys. 6, 504
(1965)
Korteweg-deVries Equation and Generalizations, VI. Methods
for Exact Solution ............................................ 928
C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura
Comm. Pure Appl. Math. 27, 97 (1974)
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