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How does the ground state energy of the system change when we braid two anyons? Can the braiding of anyons be simulated with a computational method such as the density matrix renormalization group, and if so, could someone please give me some references? I've tried looking around, but I haven't been able to find anything.

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I am specifically referring to the following slides, specifically around slide 56 or so, where he talks about DMRG and the quantum Hall Hamiltonian:

http://www.phys.virginia.edu/Announcements/Seminars/Slides/S2678.pdf

If, as Slaviks states below, that the ground state doesn't change, how can we learn about braiding statistics from DMRG?

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    $\begingroup$ The ground state energy does not change - anyon picture requres highly degenerate ground state. Search for Laughlin wave function for insights. $\endgroup$
    – Slaviks
    Commented Apr 13, 2015 at 19:32
  • $\begingroup$ @Slaviks I've edited my question after considering your question--how then, in the document that I've cited above, can we look at braiding statistics with DMRG if the ground state doesn't change? $\endgroup$
    – Joshuah Heath
    Commented Apr 13, 2015 at 20:18
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    $\begingroup$ @Bronzeclocksofbenin The ground state does change, the energy does not. Personally I would be surprised if you can extract braiding statistics - a global property of the ground state manifold - from DMRG or its time-dependent generalisations, which allow efficient access mainly to local observables. The notes you link seem to be about extracting topological information from the Schmidt coefficients, which are the only non-local properties that "come for free". Almost all other non-local observables are extremely hard to compute. Perhaps there is a clever way to do this, though. $\endgroup$
    – Mark Mitchison
    Commented Apr 13, 2015 at 22:33

2 Answers 2

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In these modern DMRG algorithms for topological phases, braiding statistics is rarely computed directly. The reason is that it is not clear how to trap a particular anyon in the bulk, and to get braiding statistics requires a careful calculation of adiabatic non-Abelian Berry phase which is often very computationally demanding. Instead, one calculates modular transformations on cylinders (to be more precise, the Dehn twist) by measuring the so-called momentum polarization of the Schmit states. It contains a lot of useful information about the topological order. The latter part of the slides explained these ideas.

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Yes tensor network algorithms have been developed to describe braiding of anyons, Abelian and non-Abelian. The networks are constructed from tensors that explicitly conserve topological charge and the braiding, fusion, and recoupling data are taken as input to the algorithms. This reference: https://arxiv.org/abs/1311.0967 describes how to use the Time Evolved Block Decimation (TEBD) algorithm with anyonic Matrix Product States to compute ground states and dynamics of braiding anyons. This reference: https://arxiv.org/abs/1505.00100 shows how to compute ground states of braiding anyons using DMRG.

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