Numerical study of a transition betweenZ2topologically ordered phases

Abstract

Distinguishing different topologically ordered phases and characterizing phase transitions between them is a difficult task due to the absence of local order parameters. In this paper, we use a combination of analytical and numerical approaches to distinguish two such phases and characterize a phase transition between them. The "toric code" and "double semion" models are simple lattice models exhibiting Z2 topological order. Although both models express the same topological ground state degeneracies and entanglement entropies, they are distinct phases of matter because their emergent quasi-particles obey different statistics. For a 1D model, we tune a phase transition between these two phases and obtain an exact solution to the entire phase diagram, finding a second-order Ising×Ising transition. We then use exact diagonalization to study the 2D case and find indications of a first-order transition. We show that the quasi-particle statistics provides a robust indicator of the distinct topological orders throughout the whole phase diagram.

Entanglement phases as holographic duals of anyon condensates

“…Finally, let us turn towards the direct Toric Code -Double Semion transition, previously only studied with exact diagonalization and on quasi-1D systems [42], whose nature is yet to be resolved. As one would assume that interactions generally give rise to condensation of excitations, one expects that an interpolation between the two models would typically drive the Toric Code through some condensation transition, either into a trivial or a more complex phase [such as the D(Z 4 ) model], and from there through another condensation-driven transition to the Double Semion model, and a direct transition would at least require some fine-tuning of interactions.…”

Quantum error correction with the semion code

“…Any phase picked up by the algorithm yields a valid string operator. Once the value for the class representative i  is fixed, the algorithm assigns values to the rest of the configurations in the same configuration class by making use of equation (23). Afterwards, a configuration, belonging to a different configuration class, where the values have not yet been fixed, is chosen and the same procedure is repeated until F  has been fixed for all possible configurations.…”

“…It can be thought of as a simple two-dimensional lattice gauge theory with gauge group G 2  = . In D=2 spatial dimensions, there is another lattice gauge theory with the same gauge group but different topological properties: the double semion (DS) model [16][17][18][22][23][24][25]. Although the Kitaev and the DS models are lattice gauge theories sharing the same gauge group, G 2  = , the braiding properties of their quasiparticle excitations are radically different.…”

Tensor-network approach to phase transitions in string-net models

“…The stringnet Hamiltonian allows to describe all doubled achiral topological phases. Thus, it has been the starting point for many studies concerning phase transitions [3][4][5][6][7][8][9][10][11][12][13][14] . Nevertheless, in the absence of local order parameter, the nature of these transitions remains an open question in many cases since one cannot use the Landau's theory of symmetry breaking.…”