arXiv:1201.6409v1 [cond-mat.stat-mech] 31 Jan 2012 Phase diagram of the toric code model in magnetic ﬁeld Fengcheng Wu, 1 Youjin Deng, 1,2, ∗ and Nikolay Prokof’ev 2, † 1 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China 2 Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA We study the phase diagram of the toric code model in magnetic ﬁeld which has three dis- tinct phases: topological phase, charge-condensed phase and vortex-condensed phase. An implicit local order parameter that characterizes the transition between the topological phase and charge- condensed phase is sampled in the continuous-time Monte Carlo simulation, and the corresponding second-order transition line is obtained by ﬁnite-size scaling analysis of this order parameter. The symmetry breaking between charges and vortices along the ﬁrst-order transition line is also observed, and our numerical result h (c) x = h (c) z =0.418(2) provides the most accurate estimate of the end point of the ﬁrst-order transition line, compared to h (c) x = h (c) z =0.48(2) obtained in [Phys. Rev. B 79, 033109 (2009)]. PACS numbers: 05.30.Rt, 02.70.Ss, 03.65.Vf, 05.30.Pr I. INTRODUCTION Topological phases are attracting a lot of attention for a variety of reasons from precise determination of physical constants to unusual properties of quasiparti- cles and promise for quantum information and computa- tion applications. The search for such phases has been mainly focused on electron systems, such as quantum Hall states and topological insulators 1,2 . However, topo- logical phases can also exist in magnetic systems and the toric code model (TCM) 3 based on spin-1/2 degrees of freedom provides a key example. The ground state of TCM in zero external ﬁeld is topologically ordered and characterized by two types of excitations (charges and vortices) with mutual anyonic statistics. To be useful for quantum computing, the topological character of TCM should be robust against local magnetic ﬁeld perturba- tions. Thus one of the most important questions to ask is what are the critical values of magnetic ﬁelds which cause condensation of charges or vortices, i.e. break the topological order. The purpose of this paper is to per- form an accurate study of phase transition lines in TCM subject to external magnetic ﬁeld. As TCM involves four-spin interactions, the continuous-time Monte Carlo approach of Ref. 4 has to be modiﬁed appropriately and supplemented with additional updates. Since the topological phase is characterized by non-local correlations, it is non-trivial to construct a quantity which distinguishes between the topological and condensed phases. However, an implicit local order parameter can still be deﬁned to characterize the transition between the topological phase and charge-condensed phase. The corresponding second-order transition line is then obtained from the standard ﬁnite-size scaling analysis of this order param- eter. Along the self-duality line, there is a symmetry between charges and vortices which is, however, broken in a certain range of parameters. It is observed in the Monte Carlo simulation as the ﬁrst-order phase transition between the charge- and vortex-condensed phases. Similar studies of TCM were performed in the past using other methods 5,6 . In Ref. 5 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. 7 , to larger system sizes. In Ref. 6 the problem was addressed by several perturbative (high- order) treatments. The parameter space of the model considered in this paper is the same as in Ref. 6 . Not surprisingly, we ﬁnd the same topology of the phase dia- gram, but due to larger system sizes investigated in this work and accurate ﬁnite-size scaling, our numerical re- sult h (c) x = h (c) z =0.418(2) provides a better estimate of the end point of the ﬁrst-order transition line, compared to h (c) x = h (c) z =0.48(2) calculated in Ref. 6 . II. MODEL The toric code model is deﬁned on a square lattice with spins located on lattice edges. The Hamiltonian (in zero ﬁeld) is based on four-spin interactions: H TCM = −J s  s A s − J p  p B p , (1) where A s =  j∈s σ x j and B p =  j∈p σ z j (σ α j are the Pauli matrices). Here subscripts s and p refer to sites and plaquettes of the square lattice, see Fig. 1, respectively, and address all spins surrounding the corresponding site or plaquette. Since all A s and B p commute with each other, the ground state manifold is known exactly. It cor- responds to eigenstates of all A s and B p with maximal eigenvalues +1. The ground-state degeneracy depends on boundary conditions, and on a torus there are four de- generate ground states. Elementary excitations, a charge