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LONG RANGE ORDER FOR THE QUANTUM HEISENBERG MODEL

Contributed by: Elliott H. Lieb (Princeton Univ.), Jan. 27, 1999.

Abstract   The Peierls argument (as modified by Griffiths and by Dobrushin) showed that long range order (LRO) exists in the Ising model for low temperature, T, in dimension D=2 or greater. For many years the analogous statement for the Heisenbeg model, in which the spin symmetry is continuous rather than discrete, was an open problem. It was finally proved for the classical model (for D >2) in [1]. The quantum case was solved in [2] for T>0, but only for the antiferromagnet and for large enough spin. (Note: classically there is no mathematical difference between the ferro- and antiferromagnets, but there is a huge difference quantum-mechanically. In particular, reflection positivity, which holds for the antiferromagnet, fails for the ferromagnet.) In [3] LRO was proved for the ground state of the 2D antiferromagnet for spin 1 or greater and in [4] this was extended to all spins greater than 0 for dimensions D>2. Several open problems seem to be annoyingly difficult, but their solution would probably add some new techniques to the theory.



A. Prove LRO for the quantum Heisenberg ferromagnet for D>2 at T>0. (It is easy to see that LRO exists in the ground state for all D>0.)

B. Prove LRO for spin 1/2 in 2D in the ground state.

C. All the proofs mentioned above use infrared bounds, which so far require strict translation invariance. In contrast the Peierls argument is robust and requires only lower bounds on the exchange energies. Find a robust proof for the cases of continuous symmetry mentioned above, i.e, one that does not need perfect translation invariance.

References:
  1. J. Froehlich, B. Simon and T. Spencer, Commun. Math. Phys. 50, 79-95 (1976).

  2. F. J. Dyson, E. H. Lieb and B. Simon, J. Stat. Phys. 18, 335-383 (1978).

  3. E. Jordao Neves and J. Fernendo Perez, Phys. Lett. 114A, 331-333 (1986).

  4. T. Kennedy, E. H. Lieb and S. Shastry, J. Stat. Phys. 53, 1019-1030 (1988).

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lieb@math.princeton.edu (contributor), aizenman@princeton.edu (editor)