Sophie S. Shamailov, Dylan J. Brown, Thomas A. Haase, Maarten D. Hoogerland
SciPost Phys. Core 4, 017 (2021) ·
published 9 June 2021
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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demonstrate that our method is no longer applicable in this regime.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.
SciPost Phys. 4, 018 (2018) ·
published 31 March 2018
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Superconducting Josephson vortices have direct analogues in ultracold-atom
physics as solitary-wave excitations of two-component superfluid Bose gases
with linear coupling. Here we numerically extend the zero-velocity Josephson
vortex solutions of the coupled Gross-Pitaevskii equations to non-zero
velocities, thus obtaining the full dispersion relation. The inertial mass of
the Josephson vortex obtained from the dispersion relation depends on the
strength of linear coupling and has a simple pole divergence at a critical
value where it changes sign while assuming large absolute values. Additional
low-velocity quasiparticles with negative inertial mass emerge at finite
momentum that are reminiscent of a dark soliton in one component with
counter-flow in the other. In the limit of small linear coupling we compare the
Josephson vortex solutions to sine-Gordon solitons and show that the
correspondence between them is asymptotic, but significant differences appear
at finite values of the coupling constant. Finally, for unequal and non-zero
self- and cross-component nonlinearities, we find a new solitary-wave
excitation branch. In its presence, both dark solitons and Josephson vortices
are dynamically stable while the new excitations are unstable.