# The Heisenberg Model - a Bibliography

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 Schrödinger 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.

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.

Regular updates to this web page will be made. This document is also available as a 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.

Mirrors: The homepage of this document is in Tucson, Arizona, USA. There is a European mirror in Leuven, Belgium, and a Far East mirror in Tokyo, Japan. Anyone who would like to set up another mirror should contact us, and we will happily include a link to it.

Abstract(lanl...)'' refers to the 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.

Abstract(Texas ...)'' and Paper(Texas ...)'' refer to the
mathematical physics archive mp_arc
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.

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 MathSciNet

Tom Kennedy
email: tgk@math.arizona.edu
Department of Mathematics
University of Arizona
Tucson, AZ 85721, USA

Bruno Nachtergaele
email: bxn@math.ucdavis.edu
Department of Mathematics
University of California, Davis
Davis, CA 95616-8633, USA

# Lecture 1 : Introduction to the Heisenberg Model

## 1. Where it comes from

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 [And59] . See also [Matt87] . It is also possible to get ferromagnetism from Hubbard type models, but this is much more subtle. See [Mie91] , [MT93] , [Tas95a] , [Tas95b] .

## 2. 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 [BR81] or [Sim93] . A recent improvement on the theorem giving the existence of the infinite volume dynamics is in [Mat93] .

## 3. Equilibrium states

A derivation of the Energy-Entropy Balance inequalities (EEB) from the variational principle (minimization of free energy) is given in [FV78] . Equivalence between EEB inequalities and the KMS condition was shown in [AS77] and [FV77] .

## 4. High temperature

At high temperatures there is only one equilibrium state and a finite correlation length. This can be proved by standard polymer expansions [XXX] , or by using KMS techniques, see, e.g., [Mat94] .

## 5. Ferro- versus Antiferromagnetic ground states

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 {x,x+1} and {x+1,x+2} , are not independent. See [Wer90] for a more detailed description of this problem.

## 6. Long Range Order

For a detailed discussion of the relation between LRO and symmetry breaking for the Heisenberg model see [KomTas93] .

## 7. The Mermin-Wagner-Hohenberg theorem

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 [MW66] , [Hoh67] . Important more recent and more general versions are in [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 [FVV84] .

## 8. Low energy spectrum

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 [LPW81] .

## 9. Free energy as the temperature goes to zero

Best lower bound (so far) on the pressure of spin 1/2 Heisenberg ferromagnet is in [Tot93] .

## 10. Pitfalls of finite system calculations

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 [DLS78] , [KLS88a] , but the ground state for a finite system is unique [LM62] , [LSM61] .

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 [Ken90] .

How the non-unique infinite volume ground states can be obtained as thermodynamic limits of finite volume excited states is discussed in [KomTas94] .

# Lecture 2 : Low temperature and the existence of LRO

## 11. Infrared bound

The proof of the infrared bound for the quantum case and the proof of long range order is in [DLS78] . The proof of LRO for the anisotropic AF Heisenberg model in two or more dimensions using reflection positivity is in [FL78] . The proof of LRO in the ground state of the antiferromagnet in two dimensions is in [JF86] . An improvement on the original sum rule argument [DLS78] for the existence of LRO can be found in [KLS88a] .

The proof of LRO in the 2d XY model's ground state appeared simultaneously in the two references [KLS88b] and [KK88] . The second reference also considered the full range of anisotropic (XXZ) models.

Reflection positivity cannot hold in full generality for the ferromagnet as was shown in [Spe85] .

## 12. Kirwood-Thomas equation for the ground state

The Kirkwood-Thomas approach to the ground state appeared in [KirTho83] . For a simpler and more general treatment see [Ken95] .

A generalization of the KT method to higher spin was developed in [Mat90a] and used to prove uniqueness of the ground state in [Mat90b] .

## 13. The dressing transformation

The dressing transformation for quantum spin systems was introduced in [Alb89] . For a unitary version of the dressing transformation see [Alb90] . The dressing transformation may be used in conjunction with a path space expansion [AD95] .

## 14. Path space expansions (Trotter product, Feynman-Kac, Duhamel)

This general approach has a long history. Early references in the context of quantum spin systems are [Gin69] , [TY83] , and [TY84] . Recent expositions can be found in [KenTas92a] (see Section 4), [BKU95] , and [DFF95] .

## 15. Applications of expansion methods

Existence of a gap in highly anisotropic Heisenberg AF via the dressing transformation is in [Alb89] . Existence of a gap in wide class of models via a path space space expansion is in section 4 of [KenTas92a] .

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 [Ken85] .

Existence of continuous spectrum just above the gap via path-space expansions was shown in [Pok93] .

Ornstein-Zernike decay of a two point function in the Ising model in a strong transverse field is proved in [Ken91] .

The adiabatic approximation for quantum spin dynamics is studied in [Alb95] .

# Lecture 3 : Exact solutions and Feynman-Kac representations

## 17. 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 [Hal83a] , [Hal83b] . For a review see [AH87] .

## 18. 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 $\tan \theta = J_2/J_1$.

$\theta=\pi/4$ and $\theta= -3 \pi/4$ are Sutherland models with SU(3) symmetry [Sut75] .

The model $\theta =-\pi/4$ (and $3\pi/4$) [Tak82] , [Bab82] .

$\tan \theta = 1/3$ is the VBS chain introduced by Affleck, Kennedy, Lieb, and Tasaki [AKLT87] , [AKLT88] , thus providing the first rigorous example of the Haldane phase.

$\theta= -\pi /4$ is equivalent to a Potts model, has a finite correlation length, a gap and two dimerized ground states [BB89] , [Klu90] , [AN94] .

$\theta=0$ is the usual Heisenberg model. It is not solved and there are no rigorous results, but there are very good numerics [WH93] .

## 19. Infinite dimensional symmetries

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 [JM95] . For a discussion of some of the mathematical problems with this approach see [FNW96] .

## 20. Valence bond solid states

The VBS construction of [AKLT87] and [AKLT88] was generalized in [FNW89] and [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'' [KSZ92] . For further properties of finitely correlated states and the corresponding generalized VBS models see [FNW91] , [FNW92b] , [FNW92c] , [FNW92d] , and [FNW94] .

Lower bounds for arbitrary generalized VBS models with a finite number of ground states are obtained in [Nac96] .

## 21. Feynman-Kac representations

A stochastic geometric analysis of special antiferromagnetic spin S chains is found in [AN94] . Stochastic representations of a general class of ferro- and antiferromagnetic interactions for spin S systems are described in [Nac94] .

Lower bounds on the pressure of the ferromagnet, using a Feynman-Kac type representation, were obtained in [CS91] . Better bounds are given in [Tot93] . Some older works where similar representations were used are [Tho80] and [Gin68] .

# Lecture 4 : Recent developments and open problems

## 22. Interfaces

Interfaces in one dimensional ferromagnet (kinks) are described and studied in [GW95] , [ASW95] , [AKS95] , and [KN96a] . In [KN96b] existence of diagonal interfaces in twodimensional XXZ ferromagnets is proved and their excitations are studied.

## 23. Quantum Pirogov-Sinai theory

The general theory is in [BKU95] and [DFF95] . An earlier paper which considered a particular model is [AD95] . Extensions of the theories and applications to particular models are in [DFFR96] and [FR96] .

## 24. Quantum spin liquids

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 [KLT88] . (The existence of a gap in this model is open.)

An interesting problem is to find a rigorous example of a two dimensional quantum spin liquid with spin 1/2.

## 25. Dimensional crossover

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 [AGS94] and [AffHal96] .

## 26. Localization

The exponential decay of correlations in the ground state of various models with random interactions is in [CKP91] and [AKN92] .

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.

## 27. Haldane phase

A review article from the perspective of physics is [Aff89] .

A rigorous example of a spin 1 Hamiltonian that is in the Haldane phase (the VBS model) is in [AKLT87] and [AKLT88] .

The hidden-order parameter for the Haldane phase was introduced in [NR89] . See also [GA89] .

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 [KenTas92a] and [KenTas92b] .

A unitary tranformation which makes the off diagonal matrix elements nonegative for a large class of spin-1 Hamiltonians is given in [Ken94] .

Results for half-integer spin: In addition to [AN94] , see [AL86] and [Kol85] .

A recent review of the physics is [DR96] .

# The References

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[AffHal96] I. Affleck , B.I. Halperin, On a Renormalization Group Approach to Dimensional Crossover, Abstract (lanl cond-mat/9603078)

[AGS94] I. Affleck, M.P. Gelfand, and R.R.P. Singh, A Plane of Weakly Coupled Heisenberg Chains: Theoretical Arguments and Numerical Calculations, J. Phys. A: Math. Gen. 27 , 7313 (1994), erratum 28 1787 (1995), Abstract (lanl cond-mat/9408062)

[AH87] I. Affleck, F. D. M. Haldane, Critical Theory of Quantum Spin Chains, Phys. Rev. B 36, 5291--5300 (1987). Math Reviews 89a:82011

[AKLT87] I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59 , 799-802 (1987).

[AKLT88] I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki, Valence-bond ground states in isotropic quantum antiferromagnets. Commun. Math. Phys. 115 , 477-528 (1988). Math Reviews 89d:82025

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[Ken91] T. Kennedy, Ornstein-Zernike decay in the ground state of the quantum Ising model in a transverse magnetic field. Commun. Math. Phys. 137 , 599 (1991). Math Reviews 92i:82013

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[Ken95] T. Kennedy, unpublished notes available by email request.

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[KLS88b] T. Kennedy, E. H. Lieb, B. S. Shastry, The XY model has long-range order for all spins and all dimensions greater than one. Phys. Rev. Lett. 61 , 2582 (1988).

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[KN96a] T. Koma, B. Nachtergaele, The spectral gap of the ferromagnetic XXZ chain, to appear in Lett. Math. Phys. . Abstract (lanl cond-mat/9512120)

[KN96b] T. Koma, B. Nachtergaele, Low Energy Excitations Above Interface Ground States of Quantum Heisenberg Ferromagnets, in preparation

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[Mat90a] T. Matsui, A link between quantum and classical Potts models. J. Stat. Phys. 59 , 781 (1990). Math Reviews 91f:82021

[Mat90b] T. Matsui, Uniqueness of the translationally invariant ground state in quantum spin systems. Commun. Math. Phys. 126 , 453 (1990).

[Mat93] T. Matsui, Markov Semigroups on UHF Algebras, Rev. Math. Phys. 5 , 587 (1993). Math Reviews 94j:82049

[Mat94] T. Matsui, Purification and Uniqueness of Quantum Gibbs States, Commun. Math. Phys. 162 , 321--332 (1994). Math Reviews 95d:82012

[Matt87] D. C. Mattis, The Theory of Magnetism, vols. I and II . Springer-Verlag, New York 1987.

[Mie91] A. Mielke, Ferromagnetism in the Hubbard model on line graphs and further considerations, J. Phys. A: Math. Gen. 24 , L73, 3311 (1991). Math Reviews 92c:82023

[MT93] A. Mielke and H. Tasaki, Ferromagnetism in the Hubbard model. Examples from models with degenerate single-electron ground states, Commun. Math. Phys. 158 , 341--371 (1993). Math Reviews 94k:82116 , Abstract (lanl cond-mat/9305026 )

[MW66] D. Mermin and H. Wagner, Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models, Phys. Rev. Lett. 17 , 1133--1136 (1966)

[Nac94] B. Nachtergaele, Quasi-state decompositions for quantum spin systems in Probability Theory and Mathematical Statistics (Proceedings of the 6th Vilnius Conference) , B. Grigelionis et al. (Eds), VSP/TEV, Utrecht-Tokyo-Vilnius, 1994, pp 565-590, Abstract (lanl cond-mat/9312012 )

[Nac96] B. Nachtergaele, The spectral gap of some quantum spin chains with discrete symmetry breaking, Commun. Math. Phys. 175 , 565--606 (1996), Abstract (lanl cond-mat/9410110)

[NR89] M. den Nijs, K. Rommelse, Preroughning transitions in crystal surfaces and valence-bond phases in quantum spin chains. Phys. Rev. B 40 , 4709 (1989).

[Pok93] M. Pokorny, Continuous spectrum in the ground state of two spin 1/2 models in the infinite volume limit, J. Stat. Phys. 72 , 381--403 (1993) Math Reviews 94e:82013

[Sim93] B. Simon, The Statistical Mechanics of Lattice Gases . Princeton University Press, Princeton, N.J. 1993. Math Reviews 95a:82001

[Spe85] E. R. Speer, Failure of reflection positivity in the quantum Heisenberg ferromagnet. Lett. Math. Phys. 10 , 41 (1985). Math Reviews 86j:82016

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