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A class of $$P,T$$-invariant topological phases of interacting electrons. (English) Zbl 1057.81053
Summary: We describe a class of parity- and time-reversal-invariant topological states of matter which can arise in correlated electron systems in $$2+1$$-dimensions. These states are characterized by particle-like excitations exhibiting exotic braiding statistics. $$P$$ and $$T$$ invariance are maintained by a ‘doubling’ of the low-energy degrees of freedom which occurs naturally without doubling the underlying microscopic degrees of freedom. The simplest examples have been the subject of considerable interest as proposed mechanisms for high-$$T_c$$ superconductivity. One is the ‘doubled’ version of the chiral spin liquid. The chiral spin liquid gives rise to anyon superconductivity at finite doping and the corresponding field theory is $$U(1)$$ Chern-Simons theory at coupling constant $$m=2$$. The ‘doubled’ theory is two copies of this theory, one with $$m=2$$ the other with $$m=-2$$. The second example corresponds to $$Z_2$$ gauge theory, which describes a scenario for spin-charge separation. Our main concern, with an eye towards applications to quantum computation, are richer models which support non-Abelian statistics. All of these models, richer or poorer, lie in a tightly organized discrete family indexed by the Baraha numbers, $$2\cos(\pi/(k+2))$$, for positive integer $$k$$. The physical inference is that a material manifesting the $$Z_2$$ gauge theory or a doubled chiral spin liquid might be easily altered to one capable of universal quantum computation. These phases of matter have a field-theoretic description in terms of gauge theories which, in their infrared limits, are topological field theories. We motivate these gauge theories using a parton model or slave-fermion construction and show how they can be solved exactly. The structure of the resulting Hilbert spaces can be understood in purely combinatorial terms. The highly constrained nature of this combinatorial construction, phrased in the language of the topology of curves on surfaces, lays the groundwork for a strategy for constructing microscopic lattice models which give rise to these phases.

##### MSC:
 81T45 Topological field theories in quantum mechanics 81V70 Many-body theory; quantum Hall effect 82D10 Statistical mechanics of plasmas 82B10 Quantum equilibrium statistical mechanics (general) 81P68 Quantum computation
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