As usual, start by grabbing the notebooks of this week (w9_disorder
). They are once again over here.
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?
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
http://arxiv.org/abs/cond-mat/0011200
Olexei Motrunich, Kedar Damle, David A. Huse
MoocSelfAssessment description
In the live version of the course, you would need to share your solution and grade yourself.
Discussion entitled 'Disorder' is available in the online version of the course.
Theory of the topological Anderson insulator
http://arxiv.org/abs/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
http://arxiv.org/abs/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
http://arxiv.org/abs/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
http://arxiv.org/abs/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
http://arxiv.org/abs/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!
Read one of the above papers and see how it is
related to the current topic.
In the live version of the course, you would need to write a summary which is then assessed by your peers.
Do you have questions about what you read? Would you like to suggest other papers? Tell us:
Discussion entitled 'Disorder' is available in the online version of the course.