I read some material in this forum and realize that entanglement entropy does not correspond to long range entanglement. Then what quantity can be used to characterize the topological order in 1+1D topological superconductor that can be obtained numerically?


For bosonic systems, there is no topological order in 1+1D. For fermionic systems, the only topological order in 1+1D is the p-wave state, that has Majorana zero mode at the chain end.

  • $\begingroup$ Thank you for your reply. I know that in Kitaev model there is Majorana zero model at the chain end, However, in this model, U(1) symmetry is explicitly broken. I am trying to search for models beyond mean field theory level that show topological order in one dimension by density-matrix renormalization group method, for example, Hubbard-like model with various extra terms. I am wondering what kind of quantities, which are numerically accessible, can be used to characterize topological order in 1+1D. $\endgroup$ – fangniuwawa Oct 2 '13 at 9:57
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    $\begingroup$ If you bosonize the 1+1D fermion model, the fermion parity conservation becomes a $Z_2$ symmetry in the bosonize model. If the bosonize model spontaneously break such a $Z_2$ symmetry, then the 1+1D fermion model is in the topologically ordered phase. $\endgroup$ – Xiao-Gang Wen Oct 3 '13 at 1:24

For simple systems, when a simple Bogoliubov-deGennes Hamiltonian is sufficient, you can calculate the band structure with periodic boundary condition. Then you calculate the band structure imposing open boundary conditions. The topological aspects usually show themselves as zero energy band crossing.

The previous method is particularly efficient when you do not need to consider the self-consistency condition for superconductivity, and/or without impurities. Adding these two effects... well I do not know other numerical method than the previous one, sorry.

I'm a bit under rush. Please ask for further precisions if you need some.


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