nLab Heisenberg model

Idea

In solid state physics

In solid state physics, the Heisenberg model is spin chain-model for magnetic quantum materials.

The Heisenberg model is akin to the Ising model, but where the latter only has internal symmetry-group $\mathbb{Z}/2$ (spin flips along one axis), the Heisenberg model has the full Spin(3) $\simeq$ SU(2)-group of internal symmetries acting on spin representations located at each of the “sites” of the model.

There are generalizations of the model with $SU(N)$-internal symmetry for any $N \in \mathbb{N}$.

In SYM theory and AdS/CFT

As an abstract model, the Heisenberg spin chain has a curious re-incarnation as describing the dynamics of BMN operators in D=4 N=4 super Yang-Mills theory under the dilatation operator in the large-N limit of the AdS/CFT correspondence. See here for more.

Properties

Symmetry protected topological phase

The Heisenberg model has a curious internal-symmetry protected topological phase, observed in Haldane 1983 and hence known as the Haldane phase.

Related entries

• spin chain

• Ising model

References

General

The original articles:

• Werner Heisenberg: Zur Theorie des Ferromagnetismus, Zeitschrift für Physik 49 (1928) 619–636 (1928) [doi:10.1007/BF01328601]

• Hans Bethe: Zur Theorie der Metalle, Zeitschrift für Physik 71 (1931) 205 [doi:10.1007/BF01341708]

Extension to the AKLT model:

• Ian Affleck, Tom Kennedy, Elliott H. Lieb, Hal Tasaki: Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59 (1987) 799 [doi:10.1103/PhysRevLett.59.799]

Observation of the “Haldan gap phase” in antiferromagnetic integer Heisenberg spin chains (and relation to the SO(3) sigma-model):

• F. Duncan M. Haldane: Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State, Phys. Rev. Lett. 50 (1983) 1153 [doi:10.1103/PhysRevLett.50.1153]

• Duncan Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model, Physics Letters A 93 9 (1983) 464-468 [doi:10.1016/0375-9601(83)90631-X]

• Duncan Haldane: Ground State Properties of Antiferromagnetic Chains with Unrestricted Spin: Integer Spin Chains as Realisations of the $O(3)$ Non-Linear Sigma Model [arXiv:1612.00076]

• T. Jolicoeur, Olivier Golinelli: Physics of integer-spin antiferromagnetic chains: Haldane gaps and edge states, Comptes Rendus Chimie 22 (2019) 445-451 [doi:10.1016/j.crci.2019.05.005, arXiv:1906.12207]

See also:

• Wikipedia, Quantum Heisenberg model

SPTs

On symmetry protected topological phases of the Heisenberg model:

• Duncan Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model, Physics Letters A 93 9 (1983) 464-468 [doi:10.1016/0375-9601(83)90631-X]

• Pierre Fromholz, Philippe Lecheminant, Symmetry-protected topological phases in the $SU(N)$ Heisenberg spin chain: a Majorana-fermion approach, Phys. Rev. B 102 094410 (2020) [arXiv:2006.13553, doi:10.1103/PhysRevB.102.094410]