Topological quantum Lego Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states

Imprint of topological degeneracy in quasi-one-dimensional fractional quantum Hall states Some of the most beautiful discoveries of the last couple of years were made in the field of topological phases of matter. The simpler, non-interacting phases are included in the periodic table of topological insulators, and are distinguished by the appearance of the edge states protected by the bulk gap and the symmetry group of the underlying phase. While there are a lot of open problems about the non-interacting phases, I think it is fair to say that the most obvious and pressing concerns about them are resolved. The full classification of all the possible topological systems with all the possible kinds of symmetries might not yet exist; however, if one asks the right question, available tools guarantee finding the answer with a reasonable amount of effort. In particular, it is always easy to construct a topological model with a certain symmetry, and study all of its properties. The situation is much less satisfactory, when it comes to the interacting and fractionalized systems. Historically, the first approach to these was to use trial wave functions, instrumental for the description of the fractional quantum Hall effect. This approach involves essentially guessing the correct solution from the start, and it requires therefore a great deal of ingenuity. Further, the trial wave functions do not always provide a recipe for constructing a Hamiltonian that produces them as a ground state. Often one can guess the kind of the Hamiltonian that would result in a desired behavior, and verify whether the guess is correct by using exact diagonalization or more advanced numerical techniques. An alternative approach is to construct exactly solvable models where all the terms in the Hamiltonian commute. These models were used to construct the most complicated topological phases, and in my personal understanding their limitation is that they rely on a very complicated and fine-tuned Hamiltonian in order to achieve what is needed. This means that perhaps the easiest way to construct topological phases through the exactly solvable models approach is to use a full-fledged quantum computer. The work of Weizmann and Harvard team of researchers, highlighted above, uses the technique known as the 'wire construction', developed for fractional quantum Hall effect [1], and further