# Condensed Matter Seminar Series

## Title: Lieb-Schultz-Mattis in Higher Dimensions

### Matthew Hastings

#### Los Alamos National Lab

##### Thursday September 11, 4:00 pm, Room 298, Physics Building

Abstract:
In 1961, Lieb, Schultz, and Mattis showed the absence of a gap in a
class of one-dimensional spin chains: chains with half-integer spin per
unit cell and SU(2)-invariant short-range interactions. This
basic result has guided research on spins chains ever since. For
example, the discovery of the Haldane gap in chains with integer spin
was surprising as it indicated a fundamental difference between integer
and half-integer spins.

Since then, there has been much work searching for
higher dimensional extensions of this result, in particular due to
possible connections to high-temperature superconductivity. The
clearest statement of the basic physical reasons to expect such an
extension are due to Misguich et. al, who argued that any such system
would either have long-range spin order, and hence have gapless spin
wave excitations, or else would have a class of topological excitations
with vanishing gap. Thus, showing this result in higher
dimensions would connect directly to recent ideas on topological order
in quantum systems. I will sketch my proof of this result,
emphasizing connections to these basic physical ideas, and discuss more
recent results on the stability of topological order. In the
process, I will derive various results about locality of correlation
functions in these systems.

*Host: Harold Baranger*