Exact entanglement renormalization for string-net models

Physical Review B - Condensed Matter and Materials Physics

Volumn 79, Issue 19, 2009, Pages

  König, Robert ^a   Reichardt, Ben W ^b   Vidal, Guifré ^c  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

^c UNIVERSITY OF QUEENSLAND   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 67649452290     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.79.195123     Document Type: Article

References (34)

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21. That is, linear dependencies between amplitudes of locally differing configurations.
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24. This is essential to avoid increasing the number of degrees of freedom. Our RG transformation fits nicely into the general theoretical framework of Ref. but its derivation is independent of Ref. and motivated by topological properties of string nets.
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29. Indeed, if |Ψ G is a ground state of HG then by Lemma 2 |Ψ G = Zp† | Ψ′ G′ and thus Zp |Ψ G = | Ψ′ G′ for some ground state | Ψ′ G′ of H G′. Conversely, if | Ψ′ G′ is a ground state of H G′ then let |Ψ G = Zp† | Ψ′ G′ so | Ψ′ G = Zp |Ψ G. Equation 7 implies that Zp† B q′ = Bq Qv Bp Zp† so |Ψ G = Bq Qv Bp |Ψ G. Thus |Ψ G is fixed by all plaquette and vertex operators, so is a ground state of HG. Alternatively, we could have used the fact that HG and H G′ have the same ground-space degeneracy.
30. The degeneracy is a function of the genus and the fusion rules of the tensor category. It can be computed by considering, as in Ref., a minimal set of inequivalent string-net configurations. For example, for the Fibonacci model, which has one nontrivial string label, the ground space on the torus is four dimensional. This corresponds to the different configurations of strings along the two fundamental 1 cycles.
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