Topics for self-study


1D localization

Our aim now is to verify that Anderson localization works in one-dimensional systems.

Simulate the Anderson model of a ribbon of appropriate and large width \(W\) as a function of length \(L\).

Anderson model is just the simpest tight binding model on a square lattice with random onsite potential.

Tune your model in the clean limit such that it has a relatively large number of modes (at least 3). Then calculate conductance as a function of \(L\) at a finite disorder, while keeping \(W\) constant.

The weak disorder regime should look ohmic or classical i.e \(g \sim N_{ch}\lambda_{MFP}/L\). Here \(\lambda_{MFP}\) is the mean free path, and \(N_{ch}\) is the number of channels.

First, verify that when \(g \gtrsim 1\) you observe the classical behavior and evaluate the mean free path.

Verify that the scaling also holds for different disorder strengths and different widths.

Examine the plot for larger \(L\), but this time plot \(\textrm{ln}(g)\) to verify that at large \(L\) the conductance \(g\) goes as \(g \sim \exp(-L/\xi)\). Try to guess how \(\xi\) is related to \(\lambda_{MFP}\) by comparing the numbers you get from the plot in this part and the previous.

Check what happens when you reduce the disorder? Is there sign of a insulator- metal transition at lower disorder?

Griffiths phase

A disordered Kitaev chain has a peculiar property. Close to the transition point it can have infinite density of states even despite it is insulating.

Calculate the energies of all the states in a finite Kitaev chain with disorder. You’ll need to get the Hamiltonian of the chain by using syst.hamiltonian_submatrix method, and diagonalize it (check the very beginning of the course if you don’t remember how to diagonalize matrices).

Do so for many disorder realizations, and build a histograph of the density of states for different values of average \(m\) and of disorder strengh around the critical point \(m=0\).

If all goes well, you should observe different behaviors: the density of states in a finite region around \(m=0\) has a weak power law divergence, that eventually turns into an actual gap. Check out this paper for details:

  • Griffiths effects and quantum critical points in dirty superconductors without spin-rotation invariance: One-dimensional examples (arXiv:cond-mat/0011200) Olexei Motrunich, Kedar Damle, David A. Huse

Now share your results:

Review assignment

Theory of the topological Anderson insulator (arXiv:0908.0881)

C. W. Groth, M. Wimmer, A. R. Akhmerov, J. Tworzydło, C. W. J. Beenakker

We present an effective medium theory that explains the disorder-induced transition into a phase of quantized conductance, discovered in computer simulations of HgTe quantum wells. It is the combination of a random potential and quadratic corrections proportional to p^2 sigma_z to the Dirac Hamiltonian that can drive an ordinary band insulator into a topological insulator (having an inverted band gap). We calculate the location of the phase boundary at weak disorder and show that it corresponds to the crossing of a band edge rather than a mobility edge. Our mechanism for the formation of a topological Anderson insulator is generic, and would apply as well to three-dimensional semiconductors with strong spin-orbit coupling.

Hint: The topological Anderson insulator.

Demonstration of one-parameter scaling at the Dirac point in graphene (arXiv:0705.0886)

J. H. Bardarson, J. Tworzydło, P. W. Brouwer, C. W. J. Beenakker

We numerically calculate the conductivity \(\sigma\) of an undoped graphene sheet (size \(L\)) in the limit of vanishingly small lattice constant. We demonstrate one-parameter scaling for random impurity scattering and determine the scaling function \(\beta(\sigma)=d\ln\sigma/d\ln L\). Contrary to a recent prediction, the scaling flow has no fixed point (\(\beta>0\)) for conductivities up to and beyond the symplectic metal-insulator transition. Instead, the data supports an alternative scaling flow for which the conductivity at the Dirac point increases logarithmically with sample size in the absence of intervalley scattering – without reaching a scale-invariant limit.

Hint: One-parameter scaling in graphene.

Topological delocalization of two-dimensional massless Dirac fermions (arXiv:0705.1607)

Kentaro Nomura, Mikito Koshino, Shinsei Ryu

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which e.g., underlies the theoretical description of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing \(g\). We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of states of the massless Dirac Hamiltonian can be localized.

Hint: Scaling with Dirac fermions.

Topology, Delocalization via Average Symmetry and the Symplectic Anderson Transition (arXiv:1208.3442)

Liang Fu, C. L. Kane

A field theory of the Anderson transition in two dimensional disordered systems with spin-orbit interactions and time-reversal symmetry is developed, in which the proliferation of vortex-like topological defects is essential for localization. The sign of vortex fugacity determines the \(Z_2\) topological class of the localized phase. There are two distinct, but equivalent transitions between the metallic phase and the two insulating phases. The critical conductivity and correlation length exponent of these transitions are computed in a \(N=1-\epsilon\) expansion in the number of replicas, where for small \(\epsilon\) the critical points are perturbatively connected to the Kosterlitz Thouless critical point. Delocalized states, which arise at the surface of weak topological insulators and topological crystalline insulators, occur because vortex proliferation is forbidden due to the presence of symmetries that are violated by disorder, but are restored by disorder averaging.

Hint: The average symmetry and weak transitions.

Topology vs. Anderson localization: non-perturbative solutions in one dimension (arXiv:1411.5992)

Alexander Altland, Dmitry Bagrets, Alex Kamenev

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.

Hint: A technical paper about localization in 1D, but you don’t need to follow the calculations.

Bonus: Find your own paper to review!

Do you know of another paper that fits into the topics of this week, and you think is good? Then you can get bonus points by reviewing that paper instead!