2014/08/31 by Pramod Padmanabhan, Juan Pablo Ibieta Jimenez, Juan Pablo Ibieta-Jimenez +3
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #Gauge theory #Lattice (music) #Operator algebra #Quantum #Quantum affine algebra #Quantum algebra #Quantum algorithm #Quantum many-body systems #Tensor product #Toric code #Transfer matrix #cond-mat.str-el #hep-th
paper · pdf · doi:10.1016/j.aop.2016.05.014
published as Annals Phys. 372 (2016) 238-259 · 21 pages, 5 figures, A new section has been added which shows how the double semion model can be obtained from the transfer matrix formalism. This section also shows how the twisted quantum double model of arXiv:1211.3695 [cond-mat.str-el] can be obtained from the transfer matrix formalism. Algebra of operators added, main results unchanged
arxiv created 2015/12/10 · openalex publication_date 2016/05/28 · openalex created_date 2016/06/24 · arxiv updated 2016/06/30 · openalex updated_date 2026/08/05
Quantum double models, such as the toric code, can be constructed from transfer matrices of lattice gauge theories with discrete gauge groups and parametrized by the center of the gauge group algebra and its dual. For general choices of these parameters the transfer matrix contains operators acting on links which can also be thought of as perturbations to the quantum double model driving it out of its topological phase and destroying the exact solvability of the quantum double model. We modify these transfer matrices with perturbations and extract exactly solvable models which remain in a quantum phase, thus nullifying the effect of the perturbation. The algebra of the modified vertex and plaquette operators now obey a deformed version of the quantum double algebra. The Abelian cases are shown to be in the quantum double phase whereas the non-Abelian phases are shown to be in a modified phase of the corresponding quantum double phase. These are illustrated with the groups ℤn and S3. The quantum phases are determined by studying the excitations of these systems namely their fusion rules and the statistics. We then go further to construct a transfer matrix which contains the other ℤ2 phase namely the double semion phase. More generally for other discrete groups these transfer matrices contain the twisted quantum double models. These transfer matrices can be thought of as being obtained by introducing extra parameters into the transfer matrix of lattice gauge theories. These parameters are central elements belonging to the tensor products of the algebra and its dual and are associated to vertices and volumes of the three dimensional lattice. As in the case of the lattice gauge theories we construct the operators creating the excitations in this case and study their braiding and fusion properties.