Thursday, November 24, 2016

The many scales of emergence in the Haldane spin chain

The spin-1 antiferromagnetic Heisenberg chain provides a nice example of emergence in a quantum many-body system. Specifically, there are three distinct phenomena that emerge that were difficult to anticipate: the energy gap conjectured by Haldane, topological order, and the edge excitations with spin-1/2.

An interesting question is whether anyone could have ever predicted these from just knowing the atomic and crystal structure of a specific material. I suspect Laughlin and Pines would say no.

To understand the emergent properties one needs to derive effective Hamiltonians at several different length and energy scales. I have tried to capture this in the diagram below. In the vertical direction, the length scales get longer and the energy scales get smaller.

It is interesting that one can get the Haldane gap from the non-linear sigma model. However, it coarse grains too much and won't give the topological order or the edge excitations.

It seems to me that the profundity of the emergence that occurs at the different strata (length scales) is different. At the lower levels, the emergence is perhaps more "straightforward" and less surprising or less singular (in the sense of Berry).

Aside. I spend too much time making this figure in PowerPoint. Any suggestions on a quick and easy way to make such figures?

Any comments on the diagram would be appreciated.


  1. Pedantic, but, I think, important point.

    There is no topological order in the Heisenberg chain (or 1d in general). The Haldane state is a symmetry protect topological (SPT) phase.

    SPT states (while very interesting and exotic) are much less exotic than topological order (TO). For example, TO is characterised by long-range entanglement whereas SPT is not.

    1. Hi Ben,

      Thanks for the expert comment. Perhaps I should put "string order".
      The definition of topological order you are using is a more recent (and more precise) one.

      For a long time people did call this string order "topological order". For example,

  2. Also, something I find interesting and don't really understand fully the implications of is the role of symmetry at different levels of emergence.

    In the Heisenberg model any of three symmetries protect the SPT state. Specifically these symmetries are: inversion, time reversal and diherdral (i.e. rotations by pi).

    But in fermionic models there are also charge fluctuations. These mean that time reversal and dihedral symmetries are not longer protective, and only inversion protects the SPT state.

    This was mostly explained by Frank Pollmann et al. But Henry Nourse recently reported some nice numerical evidence in three-leg Hubbard ladders (arxiv:1606.04297).

    So what I don't understand is what this says about the validity of the Heisenberg model. For example, it seems like you'd get important differences between spin and fermionic models in time reversal symmetric systems with broken inversion symmetry.