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nLab > Latest Changes: braid group statistics

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2012

    stub for braid group statistics (again, for the moment mainly in order to record a reference)

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2021

    I am adding missing cross-links anyon – quantum Hall effect – quantum computation

    In the process, I have added this reference:

    • Ville Lahtinen, Jiannis K. Pachos, A Short Introduction to Topological Quantum Computation, SciPost Phys. 3, 021 (2017) (arXiv:1705.04103)

    diff, v5, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 14th 2021

    Have been expanding the list of references a little: more of the original articles, and more reviews.

    On the actual experimental confirmation of anyons, it seems the first robust result is from last year(!):

    diff, v8, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 17th 2021

    am now adding references on the other kind of anyon excitations, not in quantum Hall liquids but in topological superconductors. (Will move this into a separate page, to be !include-ed here.)

    So far I have:

    via Majorana zero modes:

    Original proposal:

    • N. Read, Dmitry Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect, Phys. Rev. B61:10267, 2000 (arXiv:cond-mat/9906453)

    Review:

    via Majorana zero modes restricted to edges of topological insulators:

    • Biao Lian, Xiao-Qi Sun, Abolhassan Vaezi, Xiao-Liang Qi, and Shou-Cheng Zhang, Topological quantum computation based on chiral Majorana fermions, PNAS October 23, 2018 115 (43) 10938-10942; first published October 8, 2018 (doi:10.1073/pnas.1810003115)

    diff, v11, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 25th 2021

    added pointer to:

    diff, v17, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2022

    added references (here) on “anyonic braiding in momentum space”

    (had collected these in April already, for this MO reply, but back then the nLab was down)

    will give this list its own stand-alone entry to be !include-ed here, since the same list should also go at semi-metal

    diff, v20, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMay 28th 2022
    • (edited May 28th 2022)

    added (here) a commented list of references on “defect anyons”.

    Will put this list into a separate page now and re-!include it here.

    diff, v22, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2022
    • (edited May 29th 2022)

    I have re-worked and substantially expanded the Idea-section (here).

    This could (and should) be expanded much further, of course.

    For the moment the point I tried to bring out is that the literature is a little undecided about two somewhat different conceptualization of what counts as an anyon: one being associated with the term “quasiparticle”, the other really being solitonic defects.

    Even on just this point one could and should expand further. But so much for now.

    diff, v23, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 30th 2022

    have added pointer (here) to vortex anyons with bound Majorana zero modes, and included the graphics from Fig. 1 here.

    diff, v24, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2022
    • (edited May 31st 2022)

    pointer to this book had been missing:

    diff, v25, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2022

    added this pointer:

    diff, v26, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 1st 2022

    I have further worked on the Idea section (now called “Anyon braiding”, here*) further trying to bring out the issue as in #8 by adding/point to more of the original references.

    diff, v27, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022

    started adding a section As a “fictitious” Aharonov-Bohm effect (here) on the model due to

    diff, v33, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022
    • (edited Jun 4th 2022)

    have further worked on the Idea-section (here), now highlighting that one may recognize in the literature two different mathematical conceptualizations of anyon statistics, which the entry now refers to as anyonic quanta and as anyonic defects, respectively

    diff, v36, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022
    • (edited Jun 4th 2022)

    Finally found authors admitting this here:

    If θ∈/12ℤ\theta \in\!\!\!\!\!/ \frac{1}{2}\mathbb{Z} the Hilbert space of anyon wave functions must be chosen to be a space of multi-valued functions with half-monodromies given by the phase factors exp(2πiθ)exp(2 \pi \mathrm{i} \theta). Such wave functions can be viewed as single-valued functions on the universal cover M˜ n\widetilde M_n of M nM_n [[the configuration space of points]].

    Namely, p. 20 in:

    diff, v38, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 4th 2022

    of course that’s essentially already the statement of

    albeit not quite as explicitly

    diff, v39, current

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeJun 11th 2022

    above my definition of “adiabatic defect braiding” (here), I have added more and more authorative reference which agree with this (or would, if they thought of stating definitions).

    diff, v46, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 14th 2022

    I have added (here) a further reference/quote on anyons appearing as defects

    diff, v48, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2022

    added (here) brief pointer to and a figure from

    diff, v49, current

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 21st 2022

    added pointer to:

    diff, v51, current

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeNov 22nd 2022

    added pointer to the recent:

    diff, v53, current

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 5th 2022
    • (edited Dec 5th 2022)

    added pointer to this article:

    diff, v54, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2024

    added pointer to today’s

    diff, v57, current

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2024

    added pointer to:

    diff, v59, current

1 to 24 of 24

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