2
$\begingroup$

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.

EDIT:

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?

$\endgroup$
  • 2
    $\begingroup$ The ground state energy does not change - anyon picture requres highly degenerate ground state. Search for Laughlin wave function for insights. $\endgroup$ – Slaviks Apr 13 '15 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 Apr 13 '15 at 20:18
  • 2
    $\begingroup$ @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 Apr 13 '15 at 22:33
2
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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: http://arxiv.org/pdf/1311.0967.pdf 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: http://arxiv.org/pdf/1505.00100v1.pdf shows how to compute ground states of braiding anyons using DMRG.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.