Abstract:
While Anderson localisation is largely well-understood, its description has
traditionally been rather cumbersome. A recently-developed theory --
Localisation Landscape Theory (LLT) -- has unparalleled strengths and
advantages, both computational and conceptual, over alternative methods. To
begin with, we demonstrate that the localisation length cannot be conveniently
computed starting directly from the exact eigenstates, thus motivating the need
for the LLT approach. Then, we confirm that the Hamiltonian with the effective
potential of LLT has very similar low energy eigenstates to that with the
physical potential, justifying the crucial role the effective potential plays
in our new method. We proceed to use LLT to calculate the localisation length
for very low-energy, maximally localised eigenstates, as defined by the
length-scale of exponential decay of the eigenstates, (manually) testing our
findings against exact diagonalisation. We then describe several mechanisms by
which the eigenstates spread out at higher energies where the
tunnelling-in-the-effective-potential picture breaks down, and explicitly
demonstrate that our method is no longer applicable in this regime. We place
our computational scheme in context by explaining the connection to the more
general problem of multidimensional tunnelling and discussing the
approximations involved. Our method of calculating the localisation length can
be applied to (nearly) arbitrary disordered, continuous potentials at very low
energies.
