Topological multicritical point in the phase diagram of the toric code model and three-dimensional lattice gauge Higgs model

Abstract

We construct a mapping between the two-dimensional toric code model in external magnetic fields, h z and h x , and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional Z 2 gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large ͑up to 60 3 ͒ lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the Z 2 gauge system. An effective field theory for this type of multicritical point ͑if one actually exists͒ is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

Phase diagram of the toric code model in a parallel magnetic field

“…Using highaccuracy results for the transverse-field Ising model 15 , we immediately establish the critical value of the h z -field, h (c) z (h x = 0)=0.328474(3). When h x is finite but weak, TCM undergoes the same Ising-type continuous transition driven by large h z -field 9 . Though in finite fields the predominantly charge-condensed and predominantly vortex condensed phases can be continuously transformed into each other, along the self-duality line (h x = h z ) the symmetry between charges and vortices can be broken in a first-order transition over a certain range of fields h z (=h x ).…”

“…The extrapolation of intersection points for two subsequent sizes is shown in Fig. 16 using separate topologically protected ground states from magnetically ordered phases A and B characterized by condensation of charges and vortices, respectively 9,14 . Along the blue dashed line in Fig.…”

“…In Ref. 9 the authors used an approximate mapping of TCM onto the anisotropic Z 2 gauge Higgs model, and computed the phase diagram of the corresponding 3D classical model in the isotropic limit, extending previous studies of the Z 2 gauge Higgs model, see Ref. 13, to larger system sizes.…”

Relaxation dynamics of the toric code in contact with a thermal reservoir: Finite-size scaling in a low-temperature regime

“…Under local perturbations to H TC , such topologically nontrivial error strings only occur at order L in perturbation theory; consequently both the splitting of the ground state degeneracy and transitions between ground states are suppressed exponentially, and thus this ground state subspace is "topologically protected" from such unitary perturbations [34][35][36][37][38][39][40][41] . The toric code can thus act as a selfcorrecting quantum memory at zero temperature.…”

Quantum memories at finite temperature

“…The topologically ordered phase is then preserved in the thermodynamic limit, and any given degree of suppression can be efficiently realized. The explicit example of the toric code Hamiltonian perturbed by magnetic fields has been well studied by Trebst et al (2007), Vidal, Dusuel, and Schmidt (2009), Tupitsyn et al (2010), and Dusuel et al (2011.…”