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arXiv:0807.0487 · Physical Review B

Low-energy effective theory of the toric code model in a parallel magnetic field

Julien Vidal, Sébastien Dusuel, Kai Phillip Schmidt

Published Jan 27, 2009• 222 citations
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Abstract

We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong- and weak-field limits, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line, giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.

DOI
10.1103/physrevb.79.033109

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We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong- and weak-field limits, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line, giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.
Ground-state phase diagram of the toric code model in a parallel magnetic field has three distinct phases: topological, charge-condensed, and vortex-condensed states. To study it we consider an implicit local order parameter characterizing the transition between the topological and charge-condensed phases, and sample it using continuous-time Monte Carlo simulations. The corresponding second-order transition line is obtained by finite-size scaling analysis of this order parameter. Symmetry breaking between charges and vortices along the first-order transition line is also observed, and our numerical result shows that the end point of the first-order transition line is located at hx(c)=hz(c)=0.418(2){h}_{x}^{(c)}={h}_{z}^{(c)}=0.418(2)hx(c)​=hz(c)​=0.418(2).
We construct a mapping between the two-dimensional toric code model in external magnetic fields, hz{h}_{z}hz​ and hx{h}_{x}hx​, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional Z2{\mathbb{Z}}_{2}Z2​ 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 603{60}^{3}603) 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 Z2{\mathbb{Z}}_{2}Z2​ gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.
We study the robustness of topologically ordered phases under perturbation by considering the specific examples of the toric code and Kitaev's honeycomb models.More precisely, by means of high-order series expansions (pCUT) combined with a variational method (iPEPS) we analyze transitions between the topological and polarized phases of the toric code in a uniform magnetic field.We first describe the phase diagram of the perturbed toric code, and develop a quasi-particle picture of its elementary anyonic excitations.The effective model of interacting quasi-particles we derive allows us to analyze spectral properties of the low-energy physics of the perturbed toric code and unveil the presence of bound states.In a second step, we compute dynamical correlation functions (spectral densities) that are relevant for potential scattering experiments.Finally, we draw a connection between Kitaev's honeycomb model in a vortex superlattice and graphene in a strongly-modulated magnetic field, and present analytical results on metal-insulator transitions in these systems.
We investigate the quantum robustness of the topological order in the toric code on the honeycomb lattice in the presence of a uniform parallel field. For a field in z z -direction, the low-energy physics is in the flux-free sector and can be mapped to the transverse-field Ising model on the honeycomb lattice. One finds a second-order quantum phase transition in the 3D Ising ^\star ⋆ universality class for both signs of the field. The same is true for a postive field in x x -direction where an analogue mapping in the charge-free sector yields a ferromagnetic transverse-field Ising model on the triangular lattice and the phase transition is still 3D Ising ^\star ⋆ . In contrast, for negative x x -field, the charge-free sector is mapped to the highly frustrated antiferromagnetic transverse-field Ising model on the triangular lattice which is known to host a quantum phase transition in the 3D XY ^\star ⋆ universality class. Further, the charge-free sector does not always contain the low-energy physics for negative x x -fields and a first-order phase transition to the polarized phase in the charge-full sector takes place at larger negative field values. We quantify the location of this transition by comparing quantum Monte Carlo simulations and high-field series expansions. The full extension of the topological phase in the presence of x x - and z z -fields is determined by perturbative linked-cluster expansions using a full graph decomposition. Extrapolating the high-order series of the charge and the flux gap allows to estimate critical exponents of the gap closing. This analysis indicates that the topological order breaks down by critical lines of 3D Ising ^\star ⋆ and 3D XY ^\star ⋆ type with interesting potential multi-critical crossing points. All findings for the toric code on the honeycomb lattice can be transferred exactly to the toric code on the triangular lattice.
We investigate the quantum phase diagram of the KKK-layer Ising toric code corresponding to KKK layers of two-dimensional toric codes coupled by Ising interactions. While for small Ising interactions the system displays Z2K{\mathbb{Z}}_{2}^{K}Z2K​ topological order originating from the toric codes in each layer, the system shows Z2{\mathbb{Z}}_{2}Z2​ topological order in the high-Ising limit. The latter is demonstrated for general KKK by deriving an effective low-energy model in KthK\mathrm{th}Kth-order degenerate perturbation theory, which is given as an effective anisotropic single-layer toric code in terms of collective pseudospins 1/2 referring to the two ground states of isolated Ising chain segments. For the specific cases K=3K=3K=3 and K=4K=4K=4 we apply high-order series expansions to determine the gap series in the low- and high-Ising limit. Extrapolation of the elementary energy gaps gives convincing evidence that the ground-state phase diagram consists of a single quantum critical point and our findings suggest a quantum phase transition in the 3D Ising* universality class for both KKK separating both types of topological order, which is consistent with former findings for the bilayer Ising toric code.
We investigate the stability of the topological phase of the toric code model in the presence of a uniform magnetic field by means of variational and high-order series expansion approaches. We find that when this perturbation is strong enough, the system undergoes a topological phase transition whose first- or second-order nature depends on the field orientation. When this transition is of second order, it is in the Ising universality class except for a special line on which the critical exponent driving the closure of the gap varies continuously, unveiling a new topological universality class.