28 research outputs found
Topological delocalization in the completely disordered two-dimensional quantum walk
We investigate numerically and theoretically the effect of spatial disorder
on two-dimensional split-step discrete-time quantum walks with two internal
"coin" states. Spatial disorder can lead to Anderson localization, inhibiting
the spread of quantum walks, putting them at a disadvantage against their
diffusively spreading classical counterparts. We find that spatial disorder of
the most general type, i.e., position-dependent Haar random coin operators,
does not lead to Anderson localization but to a diffusive spread instead. This
is a delocalization, which happens because disorder places the quantum walk to
a critical point between different anomalous Floquet-Anderson insulating
topological phases. We base this explanation on the relationship of this
general quantum walk to a simpler case more studied in the literature and for
which disorder-induced delocalization of a topological origin has been
observed. We review topological delocalization for the simpler quantum walk,
using time evolution of the wave functions and level spacing statistics. We
apply scattering theory to two-dimensional quantum walks and thus calculate the
topological invariants of disordered quantum walks, substantiating the
topological interpretation of the delocalization and finding signatures of the
delocalization in the finite-size scaling of transmission. We show criticality
of the Haar random quantum walk by calculating the critical exponent in
three different ways and find 0.52 as in the integer quantum
Hall effect. Our results showcase how theoretical ideas and numerical tools
from solid-state physics can help us understand spatially random quantum walks.Comment: 18 pages, 18 figures. Similar to the published version. Comments are
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Correlated metallic two-particle bound states in Wannier--Stark flatbands
Tight-binding single-particle models on simple Bravais lattices in space
dimension , when exposed to commensurate DC fields, result in the
complete absence of transport due to the formation of Wannier--Stark flatbands
[Phys. Rev. Res. , 013174 (2021)]. The single-particle states
localize in a factorial manner, i.e., faster than exponential. Here, we
introduce interaction among two such particles that partially lifts the
localization and results in metallic two-particle bound states that propagate
in the directions perpendicular to the DC field. We demonstrate this effect
using a square lattice with Hubbard interaction. We apply perturbation theory
in the regime of interaction strength hopping strength
field strength , and obtain estimates for the group velocity of
the bound states in the direction perpendicular to the field. The two-particle
group velocity scales as . We calculate the dependence
of the exponent on the DC field direction and on the dominant
two-particle configurations related to the choices of unperturbed flatbands.
Numerical simulations confirm our predictions from the perturbative analysis.Comment: 11 pages, 7 figures. Comments are welcom
Intermediate super-exponential localization with Aubry-Andr\'e chains
We demonstrate the existence of an intermediate super-exponential
localization regime for eigenstates of the Aubry-Andr\'e chain. In this regime,
the eigenstates localize factorially similarly to the eigenstates of the
Wannier-Stark ladder. The super-exponential decay emerges on intermediate
length scales for large values of the -- the
quasi-period of the Aubry-Andr\'e potential. This intermediate localization is
present both in the metallic and insulating phases of the system. In the
insulating phase, the super-exponential localization is periodically
interrupted by weaker decaying tails to form the conventional asymptotic
exponential decay predicted for the Aubry-Andr\'e model. In the metallic phase,
the super-exponential localization happens for states with energies away from
the center of the spectrum and is followed by a super-exponential growth into
the next peak of the extended eigenstate. By adjusting the parameters it is
possible to arbitrarily extend the validity of the super-exponential
localization. A similar intermediate super-exponential localization regime is
demonstrated in quasiperiodic discrete-time unitary maps.Comment: 9 pages, 9 figures. Comments are welcom