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

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$]$