Fractionalization from Crystallography

A standard tenet of condensed matter physics is that when a crystal
with a partially filled energy band is forced to insulate, interesting
physics must arise.  As band insulators appear only when the filling --
the number of electrons per unit cell and spin projection -- is an
integer, at fractional filling an insulating phase that preserves all
symmetries is a Mott insulator, i.e. it is either gapless or, if
gapped, displays fractionalized excitations and topological order.
Remarkably, the little-studied inverse question -- whether a 'trivial'
band insulator is always possible at integer filling -- has a rich
answer involving basic ideas of crystallography. In my talk, I will
show that lattice symmetries may forbid a band insulator even at
certain integer fillings, if the crystal is *non-symmorphic* -- a
property of the majority of three-dimensional crystal structures. In
these cases, one may infer the existence of topological order and the
presence of fractionalized excitations, if the ground state is gapped
and fully symmetric -- in other words, these are Mott insulators, but
with fully filled bands! This is demonstrated using a non-perturbative
flux threading argument, and has immediate applications to quantum spin
systems and bosonic insulators in addition to electronic band
structures in the absence of spin-orbit interactions [1]. Along the
way, we will naturally be led to systematic methods for constructing
symmetric gapped ground states for interacting boson and spin models
on generic symmorphic lattices [2, 3].

References: 
1. S.A. Parameswaran, A. M. Turner, D.P. Arovas & A. Vishwanath,
   Nature Physics 9, 299 (2013) 
2. I. Kimchi, S.A. Parameswaran, A. M. Turner, F. Wang &
   A. Vishwanath, arXiv:1207.0498, in press, PNAS. 
3. S.A. Parameswaran, I. Kimchi, A. M. Turner, D. M. Stamper-Kurn &
   A. Vishwanath, Phys. Rev. Lett. 110, 125301.