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\htmlonly{Heisenberg Model Bibliography }
\htmlonly{}
\line{\hfill\tt Date printed: \today}
\bigskip
\texonly{\centerline{{\bf The Heisenberg Model - a Bibliography}}}
\htmlonly{# The Heisenberg Model - a Bibliography

}
\bigskip
\no This document started out as a list of references for a
series of four lectures on the Heisenberg model given by Tom
Kennedy and Bruno Nachtergaele at the Erwin Schr\"odinger Institute in
Vienna as part of the workshop on the Hubbard and Heisenberg Models,
August 27 - September~9, 1995. We would like to thank the ESI for the
opportunity to give these lectures and for their kind hospitality
during our visit there.
\no This document consists of two parts. The first is essentially
a terse summary of the topics discussed in the original lectures
with references to some of the relevant literature. In the html version
each citation is a link to the corresponding item in the
reference list. The second part is the list of references itself.
In addition to the standard reference to the published book or
article, we tried to include links to electronic versions of the
abstract and/or paper in electronic archives where available, as
well as a link to the Mathematical Reviews item when available.
\no Regular updates to this
\htmlonly{web page will be made.}
\texonly{document will be made and will be available at the URL: }
\texonly{\hfill \break }
\texonly{\no {\tt http://www2.math.arizona.edu/\~{ }tgk/qs.html }.}
\texonly{\hfill \break }
This document is also available as a
\htmlonly{}
\htmlonly{ TeX file .}
\texonly { TeX file.}
We hope that with time this document will evolve into a fairly
comprehensive bibliography for mathematically rigorous references on
quantum spin systems. References to relevant non-rigorous and
experimental work will also be included without aiming at any degree
of completeness. All readers are kindly invited to send us
corrections, comments, suggestions and references to be considered
for future editions, preferably by email.
\no {\bf Mirrors:}
\htmlonly{}
\htmlonly{The homepage }
\htmlonly{of this document is in Tucson, Arizona, USA.}
\htmlonly{There is a }
\htmlonly{}
\htmlonly{European mirror in Leuven, Belgium.}
\htmlonly{Anyone who would like to set up another mirror should contact us,}
\htmlonly{and we will happily include a link to it.}
\no ``Abstract(lanl...)'' refers to the
\texonly{physics preprint archive xxx at Los Alamos. }
\texonly{The home page URL is \hfill \break}
\texonly{\no {\tt http://xxx.lanl.gov/} \hfill \break}
\htmlonly{ }
\htmlonly{ physics preprint archive xxx at Los Alamos.}
Clicking on ``Abstract (lanl ...)'' will bring up the abstract of
the paper. Further options on the abstract page allow you to download the
paper itself.
\no ``Abstract(Texas ...)'' and ``Paper(Texas ...)'' refer to the
\texonly{mathematical physics archive mp\_arc at the University of Texas }
\texonly{Mathematics Department. The home page is \hfill \break}
\texonly{\no URL {\tt http://www.ma.utexas.edu/mp\_arc/mp\_arc-home.html}.}
\texonly{\hfill \break}
\htmlonly{

}
\htmlonly{mathematical physics archive mp_arc }
\htmlonly{at the University of Texas Mathematics Department.}
Clicking on ``Abstract (Texas ...)'' will bring up the abstract of
the paper. Clicking on ``Paper (Texas ...)'' will bring up the paper itself.
This is typically a TeX file, so you won't want to read it on line, but
you can download it with your browser.
\no ``Math Reviews'' refers to the Mathematical Reviews (on-line).
These links are, very unfortunately, only accessible to users
of accounts on computer hosts at institutions with a subscription to MathSciNet
of the American Mathematical Society.
For more information check out the MathSciNet home page at URL
\texonly{\hfill \break}
\texonly{\no {\tt http://e-math.ams.org:80/msnhtml/mathscimain.html}.}
\htmlonly{}
\htmlonly{MathSciNet}
\texonly{\no Tom Kennedy (email: {\tt tgk@math.arizona.edu}),}
\texonly{Department of Mathematics,}
\texonly{University of Arizona, Tucson, AZ 85721-0001, USA.}
\htmlonly{

}
\htmlonly{

}
\htmlonly{Tom Kennedy

}
\htmlonly{email: ` tgk@math.arizona.edu `

}
\htmlonly{Department of Mathematics

}
\htmlonly{University of Arizona

}
\htmlonly{Tucson, AZ 85721, USA

}
\texonly{\no Bruno Nachtergaele (email: {\tt bxn@math.ucdavis.edu}),}
\texonly{Department of Mathematics, University of California, Davis}
\texonly{Davis, CA 95616-8633, USA}
\htmlonly{

}
\htmlonly{Bruno Nachtergaele

}
\htmlonly{email: ` bxn@math.ucdavis.edu `

}
\htmlonly{Department of Mathematics

}
\htmlonly{University of California, Davis

}
\htmlonly{Davis, CA 95616-8633, USA

}
\texonly{\medskip}
\texonly{\no Copyright \copyright\ 1996 by Tom Kennedy and Bruno Nachtergaele}
\htmlonly{

}
\htmlonly{Copyright © 1996 by Tom Kennedy and Bruno Nachtergaele}
\bigskip
\hrule
\bigskip
\bigskip
\texonly{\centerline{{\bf Lecture 1 : Introduction to the Heisenberg Model}}}
\htmlonly{ # Lecture 1 : Introduction to the Heisenberg Model

}
\bigskip
\t Where it comes from
\no Anderson showed that
in the half-filled band case with large U, the ground state of the
Hubbard model is given (to second order in perturbation theory)
by the ground state of the antiferromagnetic Heisenberg model
\cite{And59}. See also \cite{Matt87}.
It is also possible to get ferromagnetism from Hubbard type models,
but this is much more subtle. See \cite{Mie91}, \cite{MT93}, \cite{Tas95a},
\cite{Tas95b}.
\t The infinite volume model (general quantum spin systems)
The infinite volume models are most conveniently defined by the
Heisenberg dynamics of quasi-local observables of the infinite volume
system. See \cite{BR81} or \cite{Sim93}.
A recent improvement on the theorem giving the existence of the infinite
volume dynamics is in \cite{Mat93}.
\t Equilibrium states
\no A derivation of the Energy-Entropy Balance inequalities (EEB) from the
variational principle (minimization of free energy) is given in
\cite{FV78}. Equivalence between EEB inequalities and the KMS condition
was shown in \cite{AS77} and \cite{FV77}.
\t High temperature
\no At high temperatures there is only one equilibrium state and a finite
correlation length. This can be proved by standard polymer expansions
\cite{XXX}, or by using KMS techniques, see, e.g., \cite{Mat94}.
\t Ferro- versus Antiferromagnetic ground states
\no Quantum fluctuations imply that the exact ground state energy in
general is not computible.
Minimizing the energy per bond is not a local calculation
because it is not known what local density matrices (describing
the state at the two sites of the bond) extend to
translation invariant states of the infinite system.
Insisting on minimizing the energy at a particular bond with disregard
of the other bonds generically implies raising the energy at other
bonds. In other words, the two sets of density matrices describing the
state at the bonds
\texonly{$\{x, x+1\}$}
\htmlonly{{x,x+1}}
and
\texonly{$\{x+1, x+2\}$}
\htmlonly{{x+1,x+2}}
, are not independent.
See \cite{Wer90} for a more detailed description of this problem.
\t Long Range Order
\no For a detailed discussion of the relation between LRO and symmetry
breaking for the Heisenberg model see \cite{KomTas93}.
\t The Mermin-Wagner-Hohenberg theorem
\no The term ``Mermin-Wagner-Hohenberg theorem'' refers to a
series of theorems showing absence of continuous symmetry breaking
at finite temperature in one- and two-dimensional models with
not too long range interactions. The proofs all use a basic idea
due to Mermin, Wagner, and Hohenberg \cite{MW66}, \cite{Hoh67}.
Important more recent and more general versions are in \cite{FP81}.
There is a techically simpler and even more general Mermin-Wagner argument
based on the EEB inequalities, which, e.g., can also be used to
prove absence of breaking of translation invariance in one-dimensional
models with long range interactions. See \cite{FVV84}.
\t Low energy spectrum
\no There is a theorem, often called ``Goldstone Theorem'',
that says that in a system with a continuous symmetry
and a gap the symmetry cannot be broken \cite{LPW81}.
\t Free energy as the temperature goes to zero
\no Best lower bound (so far) on the pressure of spin 1/2 Heisenberg ferromagnet
is in \cite{Tot93}.
\t Pitfalls of finite system calculations
\no The number of ground states for a finite volume system need not
equal the number of infinite volume ground states. For example, in
the Heisenberg model there are infinitely many ground states in two
or more dimensions \cite{DLS78}, \cite{KLS88a}, but the ground state
for a finite system is unique \cite{LM62}, \cite{LSM61}.
\no The spin 1 chain with open boundary conditions provides an example
where the number of finite volume ground states is greater than the
number of infinite volume ground states \cite{Ken90}.
\no How the non-unique infinite volume ground states can be obtained
as thermodynamic limits of finite volume {\it excited} states
is discussed in \cite{KomTas94}.
\bigskip
\bigskip
\texonly{\centerline{{\bf Lecture 2 : Low temperature and the existence of LRO}}}
\htmlonly{# Lecture 2 : Low temperature and the existence of LRO

}
\bigskip
\t Infrared bound
\no The proof of the infrared bound for the quantum case and
the proof of long range order is in \cite{DLS78}.
The proof of LRO for the anisotropic AF Heisenberg model in two or
more dimensions using reflection positivity is in \cite{FL78}.
The proof of LRO in the ground state of the antiferromagnet in two
dimensions is in \cite{JF86}.
An improvement on the original sum rule argument \cite{DLS78} for the
existence of LRO can be found in \cite{KLS88a}.
\no The proof of LRO in the 2d XY model's ground state appeared
simultaneously in the two references \cite{KLS88b} and \cite{KK88}.
The second reference also considered the full range of anisotropic
(XXZ) models.
\no Reflection positivity cannot hold in full generality for the
ferromagnet as was shown in \cite{Spe85}.
\t Kirwood-Thomas equation for the ground state
\no The Kirkwood-Thomas approach to the ground state appeared in
\cite{KirTho83}. For a simpler and more general treatment see \cite{Ken95}.
\no A generalization of the KT method to higher spin was developed in \cite{Mat90a}
and used to prove uniqueness of the ground state in \cite{Mat90b}.
\t The dressing transformation
\no The dressing transformation for quantum spin systems was introduced in
\cite{Alb89}. For a unitary version of the dressing transformation see
\cite{Alb90}.
The dressing transformation may be used in conjunction with
a path space expansion \cite{AD95}.
\t Path space expansions (Trotter product, Feynman-Kac, Duhamel)
\no This general approach has a long history. Early references in the
context of quantum spin systems are \cite{Gin69}, \cite{TY83}, and \cite{TY84}.
Recent expositions can be found in \cite{KenTas92a} (see Section 4), \cite{BKU95},
and \cite{DFF95}.
\t Applications of expansion methods
\no Existence of a gap in highly anisotropic Heisenberg AF via the dressing
transformation is in \cite{Alb89}.
Existence of a gap in wide class of models via a path space space expansion
is in section 4 of \cite{KenTas92a}.
\no LRO order in the XXZ ferromagnet can be proved in two or more dimensions
if the zz coupling is stronger than the xx=yy couplings by a resummed
expansion \cite{Ken85}.
\no Existence of continuous spectrum just above the gap via
path-space expansions was shown in \cite{Pok93}.
\no Ornstein-Zernike decay of a two point function in the Ising model
in a strong transverse field is proved in \cite{Ken91}.
\t Adiabatic approximation
\no The adiabatic approximation for quantum spin dynamics
is studied in \cite{Alb95}.
\bigskip
\bigskip
\texonly{\centerline{{\bf Lecture 3 : Exact solutions and Feynman-Kac representations}}}
\htmlonly{# Lecture 3 : Exact solutions and Feynman-Kac representations

}
\bigskip
\t Affleck-Haldane theory of the Heisenberg chain
Haldane initiated the study of the general structure of the ground state
phase diagram of quantum spin chains and predicted
the existence of a massive phase for integer spin antiferromagnetic
chains \cite{Hal83a}, \cite{Hal83b}. For a review see \cite{AH87}.
\t The spin 1 chain
$$H= \sum_x \, J_1 S_x \cdot S_{x+1} + J_2 (S_x \cdot S_{x+1})^2 $$
Let $\theta$ be the angle of the point $(J_1,J_2)$, so
$\tan \theta = J_2/J_1$.
\no $\theta=\pi/4$ and $\theta= -3 \pi/4$ are Sutherland models with
SU(3) symmetry \cite{Sut75}.
\no The model $\theta =-\pi/4$ (and $3\pi/4$) \cite{Tak82}, \cite{Bab82}.
\no $\tan \theta = 1/3$ is the VBS chain introduced by
Affleck, Kennedy, Lieb, and Tasaki \cite{AKLT87}, \cite{AKLT88}, thus providing
the first rigorous example of the Haldane phase.
\no $\theta= -\pi /4$ is equivalent to a Potts model, has a finite correlation
length, a gap and two dimerized ground states \cite{BB89}, \cite{Klu90},
\cite{AN94}.
\no $\theta=0$ is the usual Heisenberg model. It is not solved and there are
no rigorous results, but there are very good numerics \cite{WH93}.
\t Infinite dimensional symmetries
\no One may hope that for special models that posses infinite dimensional
symmetries (e.g., the spin 1/2 XXZ chain) exact eigenstates can be
constructed using the representation theory of those infinite
dimensional symmetry algebras. For a review of the status of theis
project see \cite{JM95}.
For a discussion of some of the mathematical problems with this
approach see \cite{FNW96}.
\t Valence bond solid states
\no The VBS construction of \cite{AKLT87} and \cite{AKLT88} was generalized
in \cite{FNW89} and \cite{FNW92a}, where a convenient transfer matrix
formalism was introduced. The states produced by the general
construction are called ``Finitely Correlated States''. A subclass,
the so-called purely generated finitely correlated states, are ground
states of models with finite range interactions. The latter were
later publicized under the name ``Matrix Product States'' \cite{KSZ92}.
For further properties of finitely correlated states and the corresponding
generalized VBS models see \cite{FNW91}, \cite{FNW92b}, \cite{FNW92c},
\cite{FNW92d}, and \cite{FNW94}.
\no Lower bounds for arbitrary generalized VBS models with a finite number
of ground states are obtained in \cite{Nac96}.
\t Feynman-Kac representations
\no A stochastic geometric analysis of special antiferromagnetic spin S
chains is found in \cite{AN94}.
Stochastic representations of a general class of ferro- and
antiferromagnetic interactions for spin S systems are
described in \cite{Nac94}.
\no Lower bounds on the pressure of the ferromagnet, using a Feynman-Kac
type representation, were obtained in \cite{CS91}.
Better bounds are given in \cite{Tot93}. Some older works where
similar representations were used are \cite{Tho80} and \cite{Gin68}.
\bigskip
\bigskip
\texonly{\centerline{{\bf Lecture 4 : Recent developments and open problems}}}
\htmlonly{# Lecture 4 : Recent developments and open problems

}
\bigskip
\t Interfaces
\no Interfaces in one dimensional ferromagnet (kinks) are described
and studied in \cite{GW95}, \cite{ASW95}, \cite{AKS95}, and \cite{KN96a}.
In \cite{KN96b} existence of diagonal interfaces in twodimensional XXZ
ferromagnets is proved and their excitations are studied.
\t Quantum Pirogov-Sinai theory
\no The general theory is in \cite{BKU95} and \cite{DFF95}.
An earlier paper which considered a particular model is \cite{AD95}.
Extensions of the theories and applications to particular models are
in \cite{DFFR96} and \cite{FR96}.
\t Quantum spin liquids
\no This usually refers to a model in which the Hamiltonian has the usual
SU(2) symmetry and the symmetries of the lattice, but the ground state
is unique, correlations decay exponentially and there is a gap in the
spectrum. (Sometimes this is called an incompressible quantum
spin liquid, and a compressible quantum spin liquid refers to a system with
only power law decay of the correlations and no gap.)
In one dimension, the VBS models provide rigorous examples.
There is a two dimensional VBS example, but it has
spin 3/2 and a complicated Hamiltonian \cite{KLT88}.
(The existence of a gap in this model is open.)
\no An interesting problem is to find a rigorous example of a two
dimensional quantum spin liquid with spin 1/2.
\t Dimensional crossover
\no This mostly refers to crossover from onedimensional to twodimensional
behavior in systems that consist of a plane of weakly coupled chains as the
coupling between chains is made stronger. Some physics references
are \cite{AGS94} and \cite{AffHal96}.
\t Localization
\no The exponential decay of correlations in the ground state of various
models with random interactions is in \cite{CKP91} and \cite{AKN92}.
\no These papers are concerned with the ground state of a random system.
An interesting problem is to study how the introduction of randomness
in a quantum spin system can change continuous spectrum into pure point
spectrum.
\t Haldane phase
\no A review article from the perspective of physics is \cite{Aff89}.
\no A rigorous example of a spin 1 Hamiltonian that is in the Haldane phase
(the VBS model) is in \cite{AKLT87} and \cite{AKLT88}.
\no The hidden-order parameter for the Haldane phase was introduced in
\cite{NR89}. See also \cite{GA89}.
\no Rigorous examples of Hamiltonians which are in the Haldane phase and
are not VBS type models (but unfortunately are not translation
invariant) may be found in \cite{KenTas92a} and \cite{KenTas92b}.
\no A unitary tranformation which makes the off diagonal matrix elements
nonegative for a large class of spin-1 Hamiltonians is given in \cite{Ken94}.
\no Results for half-integer spin: In addition to \cite{AN94}, see \cite{AL86}
and \cite{Kol85}.
\t Spin ladders
\no A recent review of the physics is \cite{DR96}.
\bigskip
\bigskip
\hrule
\bigskip
\bigskip
\texonly{\centerline{{\bf The References}}}
\htmlonly{# The References

}
\bigskip
\baselineskip=12pt
\ref{AD95} C. Albanese and N. Datta,
Quantum criticality, Mott transition and
sign problem for a model of lattice fermions.
\cmp {\bf 167}, 571 (1995).
\ref{Aff89} I. Affleck,
Quantum spin chains and the Haldane gap.
\jpc {\bf 1}, 3047 (1989).
\ref{AffHal96} I. Affleck , B.I. Halperin,
On a Renormalization Group Approach to Dimensional Crossover,
\arclanl cond-mat/9603078
\ref{AGS94} I. Affleck, M.P. Gelfand, and R.R.P. Singh,
A Plane of Weakly Coupled Heisenberg Chains: Theoretical Arguments and
Numerical Calculations,
\jpa {\bf 27}, 7313 (1994), erratum {\bf 28} 1787 (1995),
\arclanl cond-mat/9408062
\ref{AH87} I. Affleck, F. D. M. Haldane,
Critical Theory of Quantum Spin Chains,
\prb 36, 5291--5300 (1987).
\arcmr 89a:82011
\ref{AKLT87} I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki,
Rigorous results on valence-bond ground states in antiferromagnets.
\prl {\bf 59}, 799-802 (1987).
\ref{AKLT88} I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki,
Valence-bond ground states in isotropic quantum antiferromagnets.
\cmp {\bf 115}, 477-528 (1988).
\arcmr 89d:82025
\ref{AKN92} M. Aizenman, A. Klein, C. Newman,
Percolation methods for disordered quantum Ising models,
in proceedings of 1992 Prague conference on
``Phase transitions: Mathematics, Physics, Biology, ...'' (R. Kotecky, ed.)
World Scientific.
\ref{AN94} M. Aizenman and B. Nachtergaele,
Geometric aspects of quantum spin states.
\cmp {\bf 164}, 17--63 (1994),
\arclanl cond-mat/9310009
\ref{AKS95} G. Albertini, V.E. Korepin, A. Schadschneider
XXZ model as an effective Hamiltonian for generalized Hubbard models
with broken $\eta$-symmetry,
\jpa, {\bf 25}, L303--L309 (1995).
\arclanl cond-mat/9411051
\ref{AL86} I. Affleck, E. H. Lieb,
A proof of part of Haldane's conjecture on spin chains.
\lmp {\bf 12}, 57 (1986).
\arcmr 87h:82023
\ref{Alb89} C. Albanese,
On the spectrum of the Heisenberg Hamiltonian.
\jsp {\bf 55}, 297 (1989).
\ref{Alb90} C. Albanese,
Unitary dressing transformations and exponential
decay below threshold for quantum spin systems.
\cmp {\bf 134}, 1 (1990),
\arcmr 92b:82015a
,
\cmp {\bf 134}, 237 (1990).
\arcmr 92b:82015b
\ref{Alb95} C. Albanese,
The adiabatic approximation for quantum spin systems with a spectral gap.
preprint.
\ref{And59} P. W. Anderson,
New approach to the theory of superexchange interactions,
\pr {\bf 115}, 2--13 (1959).
\ref{AS77} H. Araki and G.L. Sewell,
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\end
-------------------- 96-252 --------------------
Parreira J. R., Bolina O., Perez J. F.
Energy Gap in Heisenberg Antiferromagnetic Half-Integer Spin Chains
with Long-Range Interactions
(10K, TeX)
ABSTRACT. We show that there is no gap in the excitation spectrum of
antiferromagnetic chains with half-integer spins and long-range
interactions provided the exchange function has a sufficiently
rapid decay
______________________________________________________________________
David.E.Evans & Yasuyuki Kawahigashi, Quantum Symmetries on Operator
Algebras, Oxford University Press, to appear shortly.