Abstract
Grégoire Misguich (Institut de Physique Théorique, CEA-Scalay, France)
Topological order in Mott insulators and dimer models
A few years ago, Hastings [1] extended the famous one-dimensional (1D)
Lieb-Shultz-Mattis theorem [2]
to arbitrary dimensions. In the first part of this talk, we will explain
that this result
essentially implies that a gapped Mott insulator cannot be disordered at
T=0.
Instead, it must either be "conventionnaly" ordered, or "topologically"
ordered.
In the second part of the talk, we will briefly review a (standard)
theoretical framework (so-called "large N" [3]) which can
be used to describe some topologically ordered states of Mott insulators.
These states, called short-ranged resonating valence-bond (RVB) spin
liquids can be realized as the ground states of some
frustrated quantum antiferromagnets. To highlight their (Z_2) topological
properties, we will present a
connection to Kitaev's toric code [4].
The last part of the talk will be about a few recent results about quantum
dimer models (QDM) [5].
These models are toy models which are simpler to analyze than frustrated
quantum
antiferromagnets, but they can capture some important properties of short
range resonating valence-bond (RVB) spin liquids [6].
We will in particular discuss the nature of vortex excitations in the
topological phase of the triangular lattice QDM [6,7] and
their relation to magnetic excitations in Kitaev's toric code.
[1] M. Hasintgs, 69, 104431 (2004). See also Oshikawa, Phys. Rev. Lett.
84, 1535 (2000)
and B. Nachtergaele and R. Sims, Com. Math. Phys. 276, 437 (2007)
[2] E. H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. 16, 407
(1961)
[3] N. Read and S. Sachdev, Phys Rev. Lett 66, 1773 (1991)
[4] A. Kitaev, Ann. Phys., 303, 2 (2003)
[5] D. Rokhsar and S. Kivelson, Phys. Rev. Lett. (1988)
[6] R. Moessner and S. Sondhi, Phys. Rev. Lett. (2001)
[7] G. Misguich and Mila, Phys. Rev. B 77, 134421 (2008)