Topology vs. Anderson localization: non-perturbative solutions in one dimension

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Altland, Alexander ; Bagrets, Dmitry ; Kamenev, Alex (2014)
  • Related identifiers: doi: 10.1103/PhysRevB.91.085429
  • Subject: Condensed Matter - Mesoscale and Nanoscale Physics | Condensed Matter - Disordered Systems and Neural Networks

We present an analytic theory of quantum criticality in quasi one-dimensional topological Anderson insulators. We describe these systems in terms of two parameters $(g,\chi)$ representing localization and topological properties, respectively. Certain critical values of $\chi$ (half-integer for $\Bbb{Z}$ classes, or zero for $\Bbb{Z}_2$ classes) define phase boundaries between distinct topological sectors. Upon increasing system size, the two parameters exhibit flow similar to the celebrated two parameter flow of the integer quantum Hall insulator. However, unlike the quantum Hall system, an exact analytical description of the entire phase diagram can be given in terms of the transfer-matrix solution of corresponding supersymmetric non-linear sigma-models. In $\Bbb{Z}_2$ classes we uncover a hidden supersymmetry, present at the quantum critical point.
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    55 references, page 1 of 6

    3. Our goal is to calculate transfer matrix element between two neighboring grains e S(Q( );Q~( )), where ; = denote two parts of the manifold, and and in the o -diagonal terms we kept a term / ~2. We rst evaluate ~ integral in the Gaussian approximation near ~ = 0:

    1 1 2g cosh2 y

    2 sinh2 y 4.

    4 sin y1 sinh y0 sin 2 (ei 3.

    1 We here consider a topological superconductor as a thermal insulator, i.e. our usage of the term `insulator' encompasses both conventional insulators, and superconductors.

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    4 As a matter of principle, the momentum space description can be maintained at the expense of extending the unit cell from atomistic scales, a, to the system size, L (a disordered system is periodic in its own size.) However, the price to be payed is that the topological information is now encoded in the con guration dependent structure of O((L=a)d) bands. While the multi-band framework may still be accessible by numerical means, it is less suited for analytical theory building.

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