A Topological Phase In A Quantum Gravity Model - A TOPOLOGICAL PHASE IN A QUANTUM GRAVITY MODEL arXiv:0812.2278v1[cond-mat.str-el 12 Dec 2008 SOLVAY

A Topological Phase In A Quantum Gravity Model - A...

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arXiv:0812.2278v1 [cond-mat.str-el] 12 Dec 2008 A TOPOLOGICAL PHASE IN A QUANTUM GRAVITY MODEL SOLVAY CONFERENCE TALK, OCTOBER 2008 MICHAEL H. FREEDMAN Introduction The concept of a topological phase may be traced to the interpretation [[1]] of the integer quantum Hall effect in terms of (what topologists call) a Chern class (or K - theory class) over the Brillion zone (momentum torus) and to Wilczek’s realization [[2]] that in ( 2 + 1 ) -dimensions, anyon statistics was a possibility to be considered on 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 [[3]] that the whole package must necessarily include anyonic statiatics, had by the mid 1980’s presented us with a rather 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 definition of what a topological phase is, or more importantly, any robust understanding of how 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 definition can sharpen our thinking and a poor definition can misdirect us. I will not attempt a final answer here but merely comment on the strengths and weaknesses of possible definitions and argue for some flexibility. In particular, I describe a rather simple class of “quantum gravity” models which are neither lattice nor field theoretic but appear to contain strong candidates for topological phases. What is a topological phase? The easiest answer is that a topological phase is 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 [[4]] TQFTs which can be very interesting even in the absence of a Chern-Simons term (in this case a twist class β H 4 ( BF ; U ( 1 )) ) and possibly other, as yet unknown, topological structures. This definition would be a bit like defining a group to be a set of matrices with certain properties; the definition is too 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 too much: that electrons carry the relevant microscopic degrees of freedom or that the system is quasi ( 2 + 1 ) -dimensional, and may even unnecessarily exclude novel states of electrons confined in two dimensions. I prefer a spectral definition (but will also criticize it!). Definition 0.1. Let H be a Hilbert space with local degrees of freedom. A Hamil- tonian H ∶ H → H is said to describe a topologial phase if: (1) H has degenerate ground state 1
2 MICHAEL H. FREEDMAN (2) H has a gap to the first excited state (3) (1) and (2) are “stable” with respect to any sum of local perturbations.

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