The Bethe Ansatz
Table of contents
Introduction
i
What these pages are about
i.a
Notes to the reader
i.n
Section and equation labelling
i.n.l
Format
i.n.f
Copyright and license; citing these notes
i.n.c
Contextual colors
i.n.x
Acknowledgements
i.n.ack
Zoology of models solvable by the Bethe Ansatz
i.z
Spin chains
i.z.sc
Heisenberg chain
i.z.sc.h
Anisotropic Heisenberg chain
i.z.sc.xxz
Fermions on a lattice
i.z.fl
Hubbard
i.z.fl.hu
Impurity models
i.z.im
The Kondo (s-d exchange) model
i.z.im.k
The Anderson model
i.z.im.ai
Continuum models
i.z.cm
Lieb-Liniger
i.z.cm.l
Gaudin-Yang
i.z.cm.gy
Two-component Bose gas
i.z.cm.t
Yang permutation
i.z.cm.y
Gaudin magnets and their generalizations
i.z.g
The Richardson model
i.z.g.r
Gaudin magnets
i.z.g.g
Chronology of exactly solvable models
i.c
Models and eigenstates: the Coordinate Bethe Ansatz
c
The Lieb-Liniger gas
c.l
Interacting particles in one dimension
c.l.i
The Lieb-Liniger model
c.l.l
Bogoliubov theory
c.l.b
Solving the Schrödinger equation
c.l.s
Two particles
c.l.s.2
Many particles
c.l.s.N
Quantization and the Bethe equations
c.l.q
Two particles
c.l.q.2
Many particles
c.l.q.N
Properties of solutions to the Bethe equations
c.l.p
Norms of Bethe eigenstates: Gaudin's conjecture
c.l.n
The Heisenberg spin-\(1/2\) chain
c.h
Definitions
c.h.d
Solving the Schrödinger equation
c.h.s
Parametrization in terms of rapidities
c.h.r
Eigenstates of the XXX antiferromagnet
c.h.e
Ground state of the XXX antiferromagnet
c.h.e.gs
States with real rapidities
c.h.e.rr
String states
c.h.e.s
Supplement: Completeness of the Bethe Ansatz for the \(M = 2\) case of the \(XXX\) antiferromagnet
c.h.e.M2
Further spin chains
c.sc
The axial Heisenberg antiferromagnet (XXZ with \(\Delta > 1\))
The planar Heisenberg chain (XXZ with \(-1 < \Delta < 1\))
String state classification
c.sc.p.s
Further continuum models
c.cm
The attractive Lieb-Liniger gas
c.cm.al
Bethe equations for strings
c.cm.al.BE
The ground state
c.cm.al.gs
Excited states
c.cm.al.e
Fermions on a lattice
c.fl
Impurity models
c.im
Gaudin magnets
c.gm
Ground states: identification, properties and excitations
g
Lieb-Liniger
g.l
The thermodynamic limit
g.l.tl
The ground state and the Lieb equation
g.l.Le
Excitations in the vicinity of the ground state
g.l.e
Particle-like excitations (Type I)
g.l.e.I
Hole-like excitations (Type II)
g.l.e.II
Heisenberg
g.h
Going to infinite size
g.h.tl
The ground state of the infinite isotropic antiferromagnet
g.h.gs
Excitations at zero field: spinons
g.h.e
Further spin chains
g.sc
The planar Heisenberg chain
g.sc.p
The ground state of the planar antiferromagnet
g.sc.p.gs
Excitations at zero field: XXZ spinons
g.sc.p.e
Equilibrium at finite \(T\): the Thermodynamic Bethe Ansatz
e
Lieb-Liniger
e.l
Yang-Yang formalism; Gibbs free energy
e.l.YY
Thermodynamic properties
e.l.tp
The \(c \rightarrow \infty\) (Tonks-Girardeau) limit
e.l.TG
The \(c \rightarrow 0^+\) limit
e.l.c0
Heisenberg
e.h
Equations for the thermal equilibrium state
e.h.eq
Factorized form of TBA equations
e.h.f
Limiting cases
e.h.l
High-temperature limit
e.h.l.hT
Low-temperature limit
e.h.l.lT
The Algebraic Bethe Ansatz
a
General considerations and the Yang-Baxter equation
a.g
Constructing \(R\) matrices
a.R
Diagonal \(R\) matrices
a.R.d
The simplest nondiagonal form for an \(R\) matrix
a.R.s
Operator algebra
a.R.s.o
Eigenstates of the transfer matrix
a.R.s.t
Dual states
a.R.s.d
General comments
a.R.s.gc
Finding an explicit \(R\)-matrix of the simplest form
a.R.se
Constructing integrable models
a.cim
Finding an explicit model associated to the simplest \(R\)-matrix
a.cim.m
ABA for the isotropic \(S=1/2\) antiferromagnet (\(XXX\) model)
a.h
Another example: the trigonometric \(R\)-matrix and the anisotropic \(S=1/2\) antiferromagnet (\(XXZ\) model)
a.xxz
Scalar products: Slavnov's formula
a.S
Matrix elements of physical operators
a.me
Solution of the quantum inverse problem
a.me.sqip
Determinant representation for matrix elements
a.me.d
Dynamical correlation functions
d
Basics
d.b
Schrödinger, Heisenberg and Interaction pictures
d.b.p
Fermi's Golden Rule
d.b.F
Correlators; Lehmann representation
d.b.c
Detailed balance
d.b.db
Sum rules
d.sr
Integrated intensity
d.sr.ii
The f-sumrule
d.sr.f
Lieb-Liniger
d.sr.f.l
\(XXZ\) chain
d.sr.f.xxz
Lieb-Liniger
d.l
Spin-\(1/2\) chains
d.shc
Integrability out of equilibrium
o
The Quench Action
o.qa
Formalism
o.qa.f
Interaction quench in Lieb-Liniger
o.qa.il
Anisotropy quench in \(XXZ\)
o.qa.aa
Literature
l
Books
l.b
b-Gaudin
b-Gaudin
b-KBI
b-KBI
Articles
l.a
1925
l.a.1925
1925.Ising.ZP.31
1925.Ising.ZP.31
1928
l.a.1928
1928.Heisenberg.ZP.49
1928.Heisenberg.ZP.49
1930
l.a.1930
1930.Bloch.ZP.61
1930.Bloch.ZP.61
1931
l.a.1931
1931.Bethe.ZP.71
1931.Bethe.ZP.71
1938
l.a.1938
1938.Hulthen.AMAF.26A
1938.Hulthen.AMAF.26A
1938.Sauter.AP.425
1938.Sauter.AP.425
1952
l.a.1952
1952.Anderson.PR.86
1952.Anderson.PR.86
1958
l.a.1958
1958.Orbach.PR.112
1958.Orbach.PR.112
1962
l.a.1962
1962.desCloizeaux.PR.128
1962.desCloizeaux.PR.128
1963
l.a.1963
1963.Lieb.PR.130.1
1963.Lieb.PR.130.1
1963.Lieb.PR.130.2
1963.Lieb.PR.130.2
1964
l.a.1964
1964.Griffiths.PR.133
1964.Griffiths.PR.133
1964.McGuire.JMP.5
1964.McGuire.JMP.5
1966
l.a.1966
1966.desCloizeaux.JMP.7.1
1966.desCloizeaux.JMP.7.1
1966.Yang.PR.150.1
1966.Yang.PR.150.1
1966.Yang.PR.150.2
1966.Yang.PR.150.2
1966.Yang.PR.151
1966.Yang.PR.151
1967
l.a.1967
1967.Gaudin.PLA.24
1967.Gaudin.PLA.24
1967.Yang.PRL.19
1967.Yang.PRL.19
1968
l.a.1968
1968.Lieb.PRL.20
1968.Lieb.PRL.20
1969
l.a.1969
1969.Yang.JMP.10
1969.Yang.JMP.10
1971
l.a.1971
1971.Gaudin.JMP.12.I
1971.Gaudin.JMP.12.I
1971.Gaudin.JMP.12.II
1971.Gaudin.JMP.12.II
1971.Gaudin.PRL.26
1971.Gaudin.PRL.26
1971.Lai.PRL.26
1971.Lai.PRL.26
1971.Takahashi.PLA.36
1971.Takahashi.PLA.36
1971.Takahashi.PTP.46
1971.Takahashi.PTP.46
1972
l.a.1972
1972.Baxter.AP.70.1
1972.Baxter.AP.70.1
1972.Baxter.AP.70.2
1972.Baxter.AP.70.2
1972.Johnson.PLA.38
1972.Johnson.PLA.38
1972.Johnson.PRA.6
1972.Johnson.PRA.6
1972.Takahashi.PLA.41
1972.Takahashi.PLA.41
1972.Takahashi.PTP.48
1972.Takahashi.PTP.48
1981
l.a.1981
1981.Faddeev.PLA.85
1981.Faddeev.PLA.85
1981.Gaudin.PRD.23
1981.Gaudin.PRD.23
1982
l.a.1982
1982.Babujian.PLA.90
1982.Babujian.PLA.90
1982.Korepin.CMP.86
1982.Korepin.CMP.86
1988
l.a.1988
1988.Sklyanin.JPA.21
1988.Sklyanin.JPA.21
1989
l.a.1989
1989.Slavnov.TMP.79
1989.Slavnov.TMP.79
1990
l.a.1990
1990.Slavnov.TMP.82
1990.Slavnov.TMP.82
1992
l.a.1992
1992.Essler.JPA.25
1992.Essler.JPA.25
1999
l.a.1999
1999.Kitanine.NPB.554
1999.Kitanine.NPB.554
2007
l.a.2007
2007.Pereira.JSTAT.P08022
2007.Pereira.JSTAT.P08022
2007.Hagemans.JPA.40
2007.Hagemans.JPA.40
2008
l.a.2008
2008.Caux.JSTAT.P08006
2008.Caux.JSTAT.P08006
Models and eigenstates: the Coordinate Bethe Ansatz
Impurity models
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[c]
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Author: Jean-Sébastien Caux
Created: 2024-01-18 Thu 14:24