arXiv:0812.2278v1 [cond-mat.str-el] 12 Dec 2008A TOPOLOGICAL PHASE IN A QUANTUM GRAVITY MODELSOLVAY CONFERENCE TALK, OCTOBER 2008MICHAEL H. FREEDMANIntroductionThe concept of a topological phase may be traced to the interpretation [] of theinteger quantum Hall effect in terms of (what topologists call) a Chern class (orK-theory class) over the Brillion zone (momentum torus) and to Wilczek’s realization[] that in(2+1)-dimensions, anyon statistics was a possibility to be consideredon an equal footing with the more familiar fermionic and bosonic statistics.The discovery of the fractional quantum Hall effect, Laughlin’s wave function,charge fractionalization, and Halperin’s realization [] that the whole package mustnecessarily include anyonic statiatics, had by the mid 1980’s presented us with arather firm “existence proof” of topological phases. Unlike in mathematics, “proof”preceded definition.I would claim that two decades later we do not have a suitably general definitionof what a topological phase is, or more importantly, any robust understanding ofhow to enter one even in the world of mathematical models. The latter is, of course,the more important issue and the main subject of this note. But a good definitioncan sharpen our thinking and a poor definition can misdirect us. I will not attempta final answer here but merely comment on the strengths and weaknesses of possibledefinitions and argue for some flexibility. In particular, I describe a rather simpleclass of “quantum gravity” models which are neither lattice nor field theoretic butappear to contain strong candidates for topological phases.What is a topological phase?The easiest answer is that a topological phaseis a system whose effective low energy theory is governed by a Chern-Simons La-grangian. This answer is extremely efficient but limiting as it overlooks Dijkgraaf-Witten finite group [] TQFTs which can be very interesting even in the absenceof a Chern-Simons term (in this case a twist classβ∈H4(BF;U(1))) and possiblyother, as yet unknown, topological structures. This definition would be a bit likedefining a group to be a set of matrices with certain properties; the definition istoo limiting since there are non-matrix groups.Similarly, any definition which contains phrases such as “spin-charge separation,”“fractional charge,” “point-like excitations,” and “string-operator” presume toomuch: that electrons carry the relevant microscopic degrees of freedom or that thesystem is quasi(2+1)-dimensional, and may even unnecessarily exclude novel statesof electrons confined in two dimensions. I prefer a spectral definition (but will alsocriticize it!).Definition 0.1.LetHbe a Hilbert space with local degrees of freedom. A Hamil-tonianH∶ H → His said to describe atopologial phaseif:(1)Hhas degenerate ground state1
2MICHAEL H. FREEDMAN(2)Hhas a gap to the first excited state(3) (1) and (2) are “stable” with respect to any sum of local perturbations.