Interplay of topological order and spin glassiness in the toric code under random magnetic fields

We analyze the toric code model in the presence of quenched disorder, which is introduced via different types of random magnetic fields. In general, close to a quantum phas e transition between a spin polarized phase and a topologically ordered one, we find that increasing the amou nt of disorder favors the topological phase. For some realizations of disorder, topological order can be rob ust against arbitrarily strong magnetic fields. In the case of the toric code in a random ±h field, we show that the system exhibits a quantum phase transi tion to a spin glass phase in an appropriate dual variables descript ion. The survival of topological order in the spin glass phase is directly related to the percolation properti es of the rigid lattice in the Edwards-Anderson bimodal spin glass model. According to recent numerical results for this model [Phys. Rev. B 82, 214401 (2010)], it is likely that the rigid lattice does not percolate and, as a res ult, a new intermediate quantum phase appears in the random-field toric code. In this intermediate quantum phase , topological order coexists with spin glassiness.