175 research outputs found
The quantum group, Harper equation and the structure of Bloch eigenstates on a honeycomb lattice
The tight-binding model of quantum particles on a honeycomb lattice is
investigated in the presence of homogeneous magnetic field. Provided the
magnetic flux per unit hexagon is rational of the elementary flux, the
one-particle Hamiltonian is expressed in terms of the generators of the quantum
group . Employing the functional representation of the quantum group
the Harper equation is rewritten as a systems of two coupled
functional equations in the complex plane. For the special values of
quasi-momentum the entangled system admits solutions in terms of polynomials.
The system is shown to exhibit certain symmetry allowing to resolve the
entanglement, and basic single equation determining the eigenvalues and
eigenstates (polynomials) is obtained. Equations specifying locations of the
roots of polynomials in the complex plane are found. Employing numerical
analysis the roots of polynomials corresponding to different eigenstates are
solved out and the diagrams exhibiting the ordered structure of one-particle
eigenstates are depicted.Comment: 11 pages, 4 figure
Synthetic Gauge Fields for Vibrational Excitations of Trapped ions
The vibrations of a collection of ions in a microtrap array can be described
in terms of hopping phonons. We show theoretically that the vibrational
couplings may be tailored by using a gradient of the microtrap frequencies,
together with a periodic driving of the trapping potential. These ingredients
allow us to induce effective gauge fields on the vibrational excitations, such
that phonons mimic the behavior of charged particles in a magnetic field. In
particular, microtrap arrays are ideally suited to realize the famous
Aharonov-Bohm effect, and observe the paradigmatic edge states typical from
quantum-Hall samples and topological insulators.Comment: replaced with published versio
The role of a form of vector potential - normalization of the antisymmetric gauge
Results obtained for the antisymmetric gauge A=[Hy,-Hx]/2 by Brown and Zak
are compared with those based on pure group-theoretical considerations and
corresponding to the Landau gauge A=[0,Hx]. Imposing the periodic boundary
conditions one has to be very careful since the first gauge leads to a factor
system which is not normalized. A period N introduced in Brown's and Zak's
papers should be considered as a magnetic one, whereas the crystal period is in
fact 2N. The `normalization' procedure proposed here shows the equivalence of
Brown's, Zak's, and other approaches. It also indicates the importance of the
concept of magnetic cells. Moreover, it is shown that factor systems (of
projective representations and central extensions) are gauge-dependent, whereas
a commutator of two magnetic translations is gauge-independent. This result
indicates that a form of the vector potential (a gauge) is also important in
physical investigations.Comment: RevTEX, 9 pages, to be published in J. Math. Phy
Predicted signatures of p-wave superfluid phases and Majorana zero modes of fermionic atoms in RF absorption
We study the superfluid phases of quasi-2D atomic Fermi gases interacting via
a p-wave Feshbach resonance. We calculate the absorption spectra of these
phases under a hyperfine transition, for both non-rotating and rotating
superfluids. We show that one can identify the different phases of the p-wave
superfluid from the absorption spectrum. The absorption spectrum shows clear
signatures of the existence of Majorana zero modes at the cores of vortices of
the weakly-pairing phase
Adiabatic continuity between Hofstadter and Chern insulator states
We show that the topologically nontrivial bands of Chern insulators are
adiabatic cousins of the Landau bands of Hofstadter lattices. We demonstrate
adiabatic connection also between several familiar fractional quantum Hall
states on Hofstadter lattices and the fractional Chern insulator states in
partially filled Chern bands, which implies that they are in fact different
manifestations of the same phase. This adiabatic path provides a way of
generating many more fractional Chern insulator states and helps clarify that
nonuniformity in the distribution of the Berry curvature is responsible for
weakening or altogether destroying fractional topological states
Hofstadter butterfly for a finite correlated system
We investigate a finite two-dimensional system in the presence of external
magnetic field. We discuss how the energy spectrum depends on the system size,
boundary conditions and Coulomb repulsion. On one hand, using these results we
present the field dependence of the transport properties of a nanosystem. In
particular, we demonstrate that these properties depend on whether the system
consists of even or odd number of sites. On the other hand, on the basis of
exact results obtained for a finite system we investigate whether the
Hofstadter butterfly is robust against strong electronic correlations. We show
that for sufficiently strong Coulomb repulsion the Hubbard gap decreases when
the magnetic field increases.Comment: 7 pages, 5 figures, revte
Majorana Fermions in superconducting 1D systems having periodic, quasiperiodic, and disordered potentials
We present a unified study of the effect of periodic, quasiperiodic and
disordered potentials on topological phases that are characterized by Majorana
end modes in 1D p-wave superconducting systems. We define a topological
invariant derived from the equations of motion for Majorana modes and, as our
first application, employ it to characterize the phase diagram for simple
periodic structures. Our general result is a relation between the topological
invariant and the normal state localization length. This link allows us to
leverage the considerable literature on localization physics and obtain the
topological phase diagrams and their salient features for quasiperiodic and
disordered systems for the entire region of parameter space.Comment: 5 pages, 2 figure
Hofstadter Problem on the Honeycomb and Triangular Lattices: Bethe Ansatz Solution
We consider Bloch electrons on the honeycomb lattice under a uniform magnetic
field with flux per cell. It is shown that the problem factorizes
to two triangular lattices. Treating magnetic translations as Heisenberg-Weyl
group and by the use of its irreducible representation on the space of theta
functions, we find a nested set of Bethe equations, which determine the
eigenstates and energy spectrum. The Bethe equations have simple form which
allows to consider them further in the limit by the technique
of Thermodynamic Bethe Ansatz and analyze Hofstadter problem for the irrational
flux.Comment: 7 pages, 2 figures, Revte
Moire Butterflies
The Hofstadter butterfly spectral patterns of lattice electrons in an
external magnetic field yield some of the most beguiling images in physics.
Here we explore the magneto-electronic spectra of systems with moire spatial
patterns, concentrating on the case of twisted bilayer graphene. Because
long-period spatial patterns are accurately formed at small twist angles,
fractal butterfly spectra and associated magneto-transport and
magneto-mechanical anomalies emerge at accessible magnetic field strengths
The longitudinal conductance of mesoscopic Hall samples with arbitrary disorder and periodic modulations
We use the Kubo-Landauer formalism to compute the longitudinal (two-terminal)
conductance of a two dimensional electron system placed in a strong
perpendicular magnetic field, and subjected to periodic modulations and/or
disorder potentials. The scattering problem is recast as a set of
inhomogeneous, coupled linear equations, allowing us to find the transmission
probabilities from a finite-size system computation; the results are exact for
non-interacting electrons. Our method fully accounts for the effects of the
disorder and the periodic modulation, irrespective of their relative strength,
as long as Landau level mixing is negligible. In particular, we focus on the
interplay between the effects of the periodic modulation and those of the
disorder. This appears to be the relevant regime to understand recent
experiments [S. Melinte {\em et al}, Phys. Rev. Lett. {\bf 92}, 036802 (2004)],
and our numerical results are in qualitative agreement with these experimental
results. The numerical techniques we develop can be generalized
straightforwardly to many-terminal geometries, as well as other multi-channel
scattering problems.Comment: 13 pages, 11 figure
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