Thursday, March 7, 2013

How does the spin-statistics theorem apply in condensed matter?

The spin-statistics theorem is an important result in quantum field theory. It shows that particles with integer spin must be bosons and particles with half-integer spin must be fermions.

Confusion then arises in the quantum many-body theory of condensed matter because there are theories [and materials!] which involve quasi-particles which appear to violate this theorem. Here are some examples:
  • Spinless fermions. These arise in one-dimensional models. For example, the transverse field Ising model.
  • Schwinger bosons which are spinors (bosonic spinons). These arise in Sp(N) representations of frustrated quantum antiferromagnets. They were introduced by Read and Sachdev.
  • Anyons. In two dimensions one can have quasi-particles which obey neither bose nor Fermi statistics.
How is this possible? Like a lot of inconsistencies, the answer is to look at the underlying assumptions required to prove the theorem. These include assuming:
  1. The theory has a Lorentz invariant Lagrangian.
  2. The vacuum is Lorentz invariant.
  3. The particle is a localized excitation. Microscopically, it is not attached to a string or domain wall.
  4. The particle is propagating, meaning that it has a finite, not infinite, mass.
  5. The particle is a real excitation, meaning that states containing this particle have a positive definite norm.
I think three dimensions [and a non-interacting, i.e. quadratic Hamiltonian] may be other assumptions.

In condensed matter, one or more of the above assumptions may not hold. For example, 
  • inclusion of a discrete lattice breaks Galilean invariance
  • spontaneous symmetry breaking  
  • topological order can lead to non-local excitations
  • in one dimension spinless fermions may be non-local [e.g. associated with a domain wall]
Thanks to Ben Powell for reminding me of this question.

1 comment:

  1. There is some work by Duck and Sudarshan which draws on some earlier work by Sudarshan and which is summarized in a pedagogical manner in Am. J Phys, 66, 4, 284 (1998), that deals with the restrictions that are required for the theorem to be valid. It would seem that Lorentz invariance isn't a requirement and most of the other possible restrictions you mention are probably not required either.