Quantum robustness and phase transitions of the 3D Toric Code in a field

Abstract

We study the robustness of 3D intrinsic topogical order under external perturbations by investigating the paradigmatic microscopic model, the 3D toric code in an external magnetic field. Exact dualities as well as variational calculations reveal a ground-state phase diagram with first and second-order quantum phase transitions. The variational approach can be applied without further approximations only for certain field directions. In the general field case, an approximative scheme based on an expansion of the variational energy in orders of the variational parameters is developed. For the breakdown of the 3D intrinsic topological order, it is found that the (im-)mobility of the quasiparticle excitations is crucial in contrast to their fractional statistics.

Quantum robustness of fracton phases

“…In the past, this has been exploited successfully for two-and three-dimensional topological codes in a magnetic field in Refs. [63] and [8], respectively.…”

“…Elementary excitations in two dimensions are so-called anyons [4,5] having a generalized particle statistics distinct from conventional bosons and fermions. Although anyonic point particles are forbidden in three spatial dimensions according to the spinstatistics theorem, the concept of fractional statistics can be generalized to spatially extended objects like membrane excitations in 3D [6][7][8].…”

“…Apart from understanding the role of thermal fluctuations on fracton codes, it is important and interesting to investigate the breakdown of fracton topological order at zero temperature. For the more conventional topological models like the toric code, color codes, or stringnet models very rich phase diagrams with quantum critical behavior have been found when adding perturbations like an external magnetic field [8,[53][54][55][56][57][58][59][60][61][62][63][64][65]. This is widely unexplored for three-dimensional fracton order, which is the main motivation for this work.…”

Competing topological orders in three dimensions

“…Furthermore, in three-dimensional topologically-ordered systems point-like anyonic excitations are excluded but nontrivial statistics can be found for extended objects such as membranes. However, in 3D, one must distinguish between two main categories of topologically-ordered long-range entangled ground states [16,17] depending on whether their degeneracy is finite [18][19][20][21] or sub-extensive with the system size [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. The topological order for systems with sub-extensive ground-state degeneracy is called fracton order.…”

Ising order parameter and topological phase transitions: Toric code in a uniform magnetic field

“…Of the most interesting topological systems are topological quantum codes such as Toric code (TC) which has been introduced in context of quantum error correction [23]. It has been shown that nonlocal nature of such quantum codes leads to a natural robustness against local perturbations [24][25][26]. Beside the above practical applications, topological quantum codes can also be used as toy models for studying topological order because of their simplicity where topological phase transition out of a topological code state has attracted much attention [27][28][29][30][31][32].…”