Topological order in a three-dimensional toric code at finite temperature

Physical Review B - Condensed Matter and Materials Physics

Volumn 78, Issue 15, 2008, Pages

  Castelnovo, Claudio ^a   Chamon, Claudio ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b Boston University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 55449127517     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.78.155120     Document Type: Article

References (28)

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14. Note that this decomposition is visible only at finite temperature, where the full Hamiltonian enters the calculations for the von Neumann entropy of the system via the density matrix. At zero temperature, the two contributions do remain distinct, but they cannot be told apart as there is no explicit dependence on λA and λB in the GS wave function.
15. Our conclusions, from the topological entropy, are in disagreement with the ones obtained by Z. Nussinov and G. Ortiz in Ref.. While the authors discuss both phase transitions in the model, at T=0 and at finite temperature, they argue that only the former has a topological nature, and they indeed conclude that topological order is fragile at finite temperature. As explained in Sec. 4, this discrepancy is due to the fact that the authors consider winding loop operators as (nonlocal) order parameters, which vanish intrinsically at any finite temperature and cannot be used (at least in a naive way) to investigate the robustness of topological order to thermal fluctuations.
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26. The familiar reader may have noticed that the construction of ZJ tot is based on the well-known duality between the 3D Ising model and the Z2 Ising gauge theory in three dimensions, discussed, for example, in Refs..
27. Note that the classical model at finite T that obtains by setting λA =0 is nothing but a classical Z2 gauge theory in three dimensions. Therefore, our results show that the topological entropy of this classical system behaves as a proper (nonlocal) order parameter that captures its finite-temperature phase transition.
28. We thank John Cardy for pointing us in the direction of this replica trick to handle the delta function terms in Eq. 15.