41 research outputs found
Critical states and anomalous mobility edges in two-dimensional diagonal quasicrystals
We study the single-particle properties of two-dimensional quasicrystals
where the underlying geometry of the tight-binding lattice is crystalline but
the on-site potential is quasicrystalline. We will focus on the 2D generalised
Aubry-Andr\'e model which has a varying form to its quasiperiodic potential,
through a deformation parameter and varied irrational periods of cosine terms,
which allows a continuous family of on-site quasicrystalline models to be
studied. We show that the 2D generalised Aubry-Andr\'e model exhibits
single-particle mobility edges between extended and localised states and a
localisation transition in a similar manner to the prior studied
one-dimensional limit. However, we find that such models in two dimensions are
dominated across large parameter regions by critical states. The presence of
critical states results in anomalous mobility edges between both extended and
critical and localised and critical states in the single-particle spectrum,
even when there is no mobility edge between extended and localised states
present. Due to this, these models exhibit anomalous diffusion of initially
localised states across the majority of parameter regions, including deep in
the normally localised regime. The presence of critical states in large
parameter regimes and throughout the spectrum will have consequences for the
many-body properties of quasicrystals, including the formation of the Bose
glass and the potential to host a many-body localised phase.Comment: 11 pages, 13 figures, comments welcom
Driven Topological Systems in the Classical Limit
Periodically-driven quantum systems can exhibit topologically non-trivial
behaviour, even when their quasi-energy bands have zero Chern numbers. Much
work has been conducted on non-interacting quantum-mechanical models where this
kind of behaviour is present. However, the inclusion of interactions in
out-of-equilibrium quantum systems can prove to be quite challenging. On the
other hand, the classical counterpart of hard-core interactions can be
simulated efficiently via constrained random walks. The non-interacting model
proposed by Rudner et al. [Phys. Rev. X 3, 031005 (2013)], has a special point
for which the system is equivalent to a classical random walk. We consider the
classical counterpart of this model, which is exact at a special point even
when hard-core interactions are present, and show how these quantitatively
affect the edge currents in a strip geometry. We find that the interacting
classical system is well described by a mean-field theory. Using this we
simulate the dynamics of the classical system, which show that the interactions
play the role of Markovian, or time dependent disorder. By comparing the
evolution of classical and quantum edge currents in small lattices, we find
regimes where the classical limit considered gives good insight into the
quantum problem.Comment: 15 pages, 15 figures, new content on the quantum mode
Mean-field phases of an ultracold gas in a quasicrystalline potential
The recent experimental advancement to realise ultracold gases scattering off
an eight-fold optical potential [Phys. Rev. Lett. 122, 110404 (2019)] heralds
the beginning of a new technique to study the properties of quasicrystalline
structures. Quasicrystals possess long-range order but are not periodic, and
are still little studied in comparison to their periodic counterparts. Here, we
consider an ultracold bosonic gas in an eight-fold symmetric lattice and assume
a toy model where the atoms occupy the ground states of the local minima of the
potential. The ground state phases of the system are studied, with particular
interest in the local nature of the phases. The usual Mott-insulator, density
wave, and supersolid phases of the standard and extended Bose-Hubbard model are
observed. For non-zero long-range interactions, we find that density wave
states can spontaneously break the eight-fold symmetry, and may even possess no
rotational symmetry. We find the local variation in the number of nearest
neighbours to play a vital role in the phase transitions, local structure, and
global symmetries of the ground states. This variation in the number of nearest
neighbours is not a unique property of the considered eight-fold lattice, and
we expect our results to be generalisable to any quasicrystalline potential
where there are only small position dependent variations in the site energy,
tunnelling and interactions.Comment: 10 pages, 10 figures, accepted to PR
Staggered Ground States in an Optical Lattice
Non-standard Bose-Hubbard models can exhibit rich ground state phase
diagrams, even when considering the one-dimensional limit. Using a
self-consistent Gutzwiller diagonalisation approach, we study the mean-field
ground state properties of a long-range interacting atomic gas in a
one-dimensional optical lattice. We first confirm that the inclusion of
long-range two-body interactions to the standard Bose-Hubbard model introduces
density wave and supersolid phases. However, the introduction of pair and
density-dependent tunnelling can result in new phases with two-site periodic
density, single-particle transport and two-body transport order parameters.
These staggered phases are potentially a mean-field signature of the known
novel twisted superfluids found via a DMRG approach [PRA \textbf{94}, 011603(R)
(2016)]. We also observe other unconventional phases, which are characterised
by sign staggered order parameters between adjacent lattice sites.Comment: 11 pages, 7 figures, comments welcom
Critical states and anomalous mobility edges in two-dimensional diagonal quasicrystals
We study the single-particle properties of two-dimensional quasicrystals where the underlying geometry of the tight-binding lattice is crystalline but the on-site potential is quasicrystalline. We will focus on the two-dimensional (2D) generalized Aubry-André model which has a varying form to its quasiperiodic potential, through a deformation parameter and varied irrational periods of cosine terms, which allows a continuous family of on-site quasicrystalline models to be studied. We show that the 2D generalized Aubry-André model exhibits many single-particle mobility edges which we confirm for finite systems and supports critical states across large parameter regions. Critical states are neither fully localized nor extended. We observe that diagonal quasicrystalline models can support many energy intervals of critical states in the spectrum while stabilizing both localized and extended states in other energy intervals; we refer to these as anomalous mobility edges. We show that critical states are present independent of system size through a scaling analysis of the inverse participation ratio and that they are present in spectra that also contain extended and localized states, confirming that at least one anomalous mobility edge is present. Due to this, these models exhibit anomalous diffusion of initially localized states across the majority of parameter regions, including deep in the normally localized regime. The presence of critical states in large parameter regimes and throughout the spectrum will have consequences for the many-body properties of quasicrystals, including the formation of the Bose glass and the potential to host a many-body localized phase
Exact edge, bulk and bound states of finite topological systems
Finite topologically non-trivial systems are often characterised by the
presence of bound states at their physical edges. These topological edge modes
can be distinguished from usual Shockley waves energetically, as their energies
remain finite and in-gap. On a clean 1D or reducible 2D model, in either the
commensurate or semi-infinite case, the edge modes can be obtained
analytically, as shown in [PRL 71, 3697 (1993)] and [PRA 89, 023619 (2014)]. We
put forward a method for obtaining the spectrum and wave functions of
topological edge modes for arbitrary finite lattices, including the
incommensurate case. A small number of parameters are easily determined
numerically, with the form of the eigenstates remaining fully analytical. We
also obtain the bulk modes in the finite system analytically and their
eigenenergies, which lie within the infinite-size limit continuum. Our method
is general and can be easily applied to obtain the properties of
non-topological models and/or extended to include impurities. As an example, we
consider the case of an impurity located next to one edge of a 1D system,
equivalent to a softened boundary in a separable 2D model. We show that a
localised impurity can have a drastic effect on the edge modes of the system.
Using the periodic Harper and Hofstadter models to illustrate our method, we
find that, on increasing the impurity strength, edge states can enter or exit
the continuum, and a trivial Shockley state bound to the impurity may appear.
The fate of the topological edge modes in the presence of impurities can be
addressed by quenching the impurity strength. We find that at certain critical
impurity strengths, the transition probability for a particle initially
prepared in an edge mode to decay into the bulk exhibits discontinuities that
mark the entry and exit points of edge modes from and into the bulk spectrum.Comment: 13 pages, 6 figure
Linked and knotted synthetic magnetic fields
We show that the realisation of synthetic magnetic fields via light-matter
coupling in the Lambda-scheme implements a natural geometrical construction of
magnetic fields, namely as the pullback of the area element of the sphere to
Euclidean space via certain maps. For suitable maps, this construction
generates linked and knotted magnetic fields, and the synthetic realisation
amounts to the identification of the map with the ratio of two Rabi frequencies
which represent the coupling of the internal energy levels of an ultracold
atom. We consider examples of maps which can be physically realised in terms of
Rabi frequencies and which lead to linked and knotted synthetic magnetic fields
acting on the neutral atomic gas. We also show that the ground state of the
Bose-Einstein condensate may inherit topological properties of the synthetic
gauge field, with linked and knotted vortex lines appearing in some cases.Comment: 8 pages, 4 figures, supplementary videos attached. Comments welcom
Topological models in rotationally symmetric quasicrystals
We investigate the physics of quasicrystalline models in the presence of a
uniform magnetic field, focusing on the presence and construction of
topological states. This is done by using the Hofstadter model but with the
sites and couplings denoted by the vertex model of the quasicrystal, giving the
Hofstadter vertex model. We specifically consider two-dimensional quasicrystals
made from tilings of two tiles with incommensurate areas, focusing on the
five-fold Penrose and the eight-fold Ammann-Beenker tilings. This introduces
two competing scales; the uniform magnetic field and the incommensurate scale
of the cells of the tiling. Due to these competing scales the periodicity of
the Hofstadter butterfly is destroyed. We observe the presence of topological
edge states on the boundary of the system via the Bott index that exhibit two
way transport along the edge. For the eight-fold tiling we also observe
internal edge-like states with non-zero Bott index, which exhibit two way
transport along this internal edge. The presence of these internal edge states
is a new characteristic of quasicrystalline models in magnetic fields. We then
move on to considering interacting systems. This is challenging, in part
because exact diagonalization on a few tens of sites is not expected to be
enough to accurately capture the physics of the quasicrystalline system, and in
part because it is not clear how to construct topological flatbands having a
large number of states. We show that these problems can be circumvented by
building the models analytically, and in this way we construct models with
Laughlin type fractional quantum Hall ground states.Comment: 13 pages, 16 figures, published PR
Synthetic mean-field interactions in photonic lattices
Photonic lattices are usually considered to be limited by their lack of
methods to include interactions. We address this issue by introducing
mean-field interactions through optical components which are external to the
photonic lattice. The proposed technique to realise mean-field interacting
photonic lattices relies on a Suzuki-Trotter decomposition of the unitary
evolution for the full Hamiltonian. The technique realises the dynamics in an
analogous way to that of a step-wise numerical implementation of quantum
dynamics, in the spirit of digital quantum simulation. It is a very versatile
technique which allows for the emulation of interactions that do not only
depend on inter-particle separations or do not decay with particle separation.
We detail the proposed experimental scheme and consider two examples of
interacting phenomena, self-trapping and the decay of Bloch oscillations, that
are observable with the proposed technique.Comment: 7 pages, 5 figures, comments welcom