Citation: Tupitsyn, I. S., et al. "Topological multicritical point in the Toric Code and 3D gauge Higgs Models." arXiv preprint arXiv:0804.3175 (2008).
Web: https://arxiv.org/abs/0804.3175
Tags: Physical, Toric-code, Phase-transition
In this lovely paper, the authors use theoretical and numerical tools to analyze the phase diagram of the toric code, in a phase space with two parameters. One parameter condenses m-type anyons (a magnetic field) and the second parameter condenses e-type anyons. These condensations are second-order phase transitions. The e-condensed phase and m-condensed phase are both topologically trivial, but if you tune between them in some parts of the paramter space you get a first-order phase transition between them. In that way the phase diagram looks sort of like the phase diagram of water.
The TC-trivial and TC-trivial boundaries in the phase diagram correspond to the physical gapped boundaries corresponding to condensing e/m particles. The relationship between boundaries in the phase diagram to boundaries on the lattice is a sort of topological Wick rotation, discussed at length in Kong-Zheng's work:
> Kong, Liang, and Hao Zheng. "A mathematical theory of gapless edges of 2d topological orders. Part I." Journal of High Energy Physics 2020.2 (2020): 1-62.
> Kong, Liang, and Hao Zheng. "A mathematical theory of gapless edges of 2d topological orders. Part II." Nuclear Physics B 966 (2021): 115384.
There were earlier studies of the phase diagram of the toric code in the presence of just a magnetic field, which was a trend-setting paper for understanding anyon condensation:
> Trebst, Simon, et al. "Breakdown of a topological phase: Quantum phase transition in a loop gas model with tension." Physical review letters 98.7 (2007): 070602.
The authors highlight the previous works have tried to understand the phase diagram of the toric code and have reached incorrect conclusions, since their numerics did not use techniques which allowed for very accurate data.