Ising order parameter and topological phase transitions: Toric code in uniform field. (arXiv:1907.06219v1 [quant-ph])

Quantum Ising model in a transverse field is of the simplest quantum many
body systems used for studying universal properties of quantum phase
transitions. Interestingly, it has been shown that such phase transitions can
be mapped to topological phase transitions in Toric code models by a dual
transformation [M. H. Zarei, Phys. Rev. B 96, 165146 (2017)]. Therefore, one
can expect that well-known properties of the transverse Ising model are used
for characterizing topological phase transition in Toric code model. In this
paper, we consider the magnetization of Ising model and show while it is a
local order parameter, it is mapped to a non-local order parameter in the
topological side. Consequently, we introduce a string order parameter for
topological phase transitions in Toric code models defined on different
lattices with different dimensions in presence of a uniform magnetic field.
Since such string order parameter is dual of the Ising order parameter with
well-known properties, it leads to a topological (non local) characterization
of phase transition in the Toric code model in uniform field. Our results show
that although topological phase transitions do not follow a symmetry-breaking
mechanism, it might be still possible to use concepts of symmetry-breaking
phase transitions for topological ones by finding suitable mappings.

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