Sufficient Conditions for Topological Order in Insulators

– We prove the existence of low energy excitations in insulating systems at general filling factor under certain conditions, and discuss in which cases these may be identified as topological excitations. In the specific case of half-filling this proof provides a significantly shortened proof of the recent higher dimensional Lieb-Schultz-Mattis theorem. The classic 1961 result of Lieb, Schultz, and Mattis (LSM) [1], proving the existence of an excitation within energy ∼ 1/L of the ground state for certain one dimensional spin chains, has had a large effect on the field. While it was then proven by Affleck and Lieb [2] that one dimensional systems either have gapless localized excitations or a local symmetry breaking, it has long been suspected that in higher dimensions there is a more interesting possibility of topological order [3]. One way to understand topological order is based on flux insertion. We give the physical argument here, and then discuss the difficulties in this argument which give rise to the need for the more careful argument of this paper. We consider a higher-dimensional system which is periodic in one direction. A spin-1/2 system can be mapped to a hard-core boson system on a lattice, with the presence or absence of a particle denoting spin up or down. If the particle system is superfluid, there is long range order in the x and y components of the spin in the original system, implying the existence of low energy excitations. On the other hand, if the particle system is insulating, it should be possible to insert 2π of gauge flux in the hopping of particles across a given line cutting the system, returning the Hamiltonian to the original one, but, for non-integer filling fraction, taking the system to an excited state which is very close in energy to the ground state. Using adiabatic flux insertion, this was suggested as a way to prove a higher dimensional LSM theorem [4]. The two possibilities would thus seem to be a superfluid system (or other system which resists flux insertion) which has low energy excitations, or an insulating system which has topological order. In either case, there is a state close in energy to the ground state. However, there is a serious problem with this argument. The definition of adiabatic flux insertion depends on the existence of a gap (but does not require anything about the magnitude of a gap); however, in a spin system with no disorder, any gap in the spectrum at zero flux must close at some non-zero value of the flux, thus making it impossible even to define an adiabatic flux insertion [5]. In a fractional quantum Hall system with disorder, there is a related problem. In a 1/3 quantum Hall state, the gap between the three approximately