27 research outputs found
Fidelity susceptibility of one-dimensional models with twisted boundary conditions
Recently it has been shown that the fidelity of the ground state of a quantum
many-body system can be used to detect its quantum critical points (QCPs). If g
denotes the parameter in the Hamiltonian with respect to which the fidelity is
computed, we find that for one-dimensional models with large but finite size,
the fidelity susceptibility \chi_F can detect a QCP provided that the
correlation length exponent satisfies \nu < 2. We then show that \chi_F can be
used to locate a QCP even if \nu \ge 2 if we introduce boundary conditions
labeled by a twist angle N\theta, where N is the system size. If the QCP lies
at g = 0, we find that if N is kept constant, \chi_F has a scaling form given
by \chi_F \sim \theta^{-2/\nu} f(g/\theta^{1/\nu}) if \theta \ll 2\pi/N. We
illustrate this both in a tight-binding model of fermions with a spatially
varying chemical potential with amplitude h and period 2q in which \nu = q, and
in a XY spin-1/2 chain in which \nu = 2. Finally we show that when q is very
large, the model has two additional QCPs at h = \pm 2 which cannot be detected
by studying the energy spectrum but are clearly detected by \chi_F. The peak
value and width of \chi_F seem to scale as non-trivial powers of q at these
QCPs. We argue that these QCPs mark a transition between extended and localized
states at the Fermi energy.Comment: 12 pages, 10 figures; made some changes in response to referees; this
is the published versio
Majorana edge modes in the Kitaev model
We study the Majorana modes, both equilibrium and Floquet, which can appear
at the edges of the Kitaev model on the honeycomb lattice. We first present the
analytical solutions known for the equilibrium Majorana edge modes for both
zigzag and armchair edges of a semi-infinite Kitaev model and chart the
parameter regimes of the model in which they appear. We then examine how edge
modes can be generated if the Kitaev coupling on the bonds perpendicular to the
edge is varied periodically in time as periodic -function kicks. We
derive a general condition for the appearance and disappearance of the Floquet
edge modes as a function of the drive frequency for a generic -dimensional
integrable system. We confirm this general condition for the Kitaev model with
a finite width by mapping it to a one-dimensional model. Our numerical and
analytical study of this problem shows that Floquet Majorana modes can appear
on some edges in the kicked system even when the corresponding equilibrium
Hamiltonian has no Majorana mode solutions on those edges. We support our
analytical studies by numerics for finite sized system which show that periodic
kicks can generate modes at the edges and the corners of the lattice.Comment: 13 pages, 10 figures; made some minor changes and added several
reference
From fractional boundary charges to quantized Hall conductance
We study the fractional boundary charges (FBCs) occurring in nanowires in the
presence of periodically modulated chemical potentials and connect them to the
FBCs occurring in a two-dimensional electron gas in the presence of a
perpendicular magnetic field in the integer quantum Hall effect (QHE) regime.
First, we show that in nanowires the FBCs take fractional values and change
linearly as a function of phase offset of the modulated chemical potential.
This linear slope takes quantized values determined by the period of the
modulation and depends only on the number of the filled bands. Next, we
establish a mapping from the one-dimensional system to the QHE setup, where we
again focus on the properties of the FBCs. By considering a cylinder topology
with an external flux similar to the Laughlin construction, we find that the
slope of the FBCs as function of flux is linear and assumes universal quantized
values, also in the presence of arbitrary disorder. We establish that the
quantized slopes give rise to the quantization of the Hall conductance.
Importantly, the approach via FBCs is valid for arbitrary flux values and
disorder. The slope of the FBCs plays the role of a topological invariant for
clean and disordered QHE systems. Our predictions for the FBCs can be tested
experimentally in nanowires and in Corbino disk geometries in the integer QHE
regime.Comment: 14 pages, 12 figure
Floquet Majorana and Para-Fermions in Driven Rashba Nanowires
We study a periodically driven nanowire with Rashba-like conduction and
valence bands in the presence of a magnetic field. We identify topological
regimes in which the system hosts zero-energy Majorana fermions. We further
investigate the effect of strong electron-electron interactions that give rise
to parafermion zero energy modes hosted at the nanowire ends. The first setup
we consider allows for topological phases by applying only static magnetic
fields without the need of superconductivity. The second setup involves both
superconductivity and time-dependent magnetic fields and allows one to generate
topological phases without fine-tuning of the chemical potential. Promising
candidate materials are graphene nanoribbons due to their intrinsic
particle-hole symmetry.Comment: 8 pages, 5 figure
Transport signatures of topological phases in double nanowires probed by spin-polarized STM
We study a double-nanowire setup proximity coupled to an -wave
superconductor and search for the bulk signatures of the topological phase
transition that can be observed experimentally, for example, with an STM tip.
Three bulk quantities, namely, the charge, the spin polarization, and the
pairing amplitude of intrawire superconductivity are studied in this work. The
spin polarization and the pairing amplitude flip sign as the system undergoes a
phase transition from the trivial to the topological phase. In order to
identify promising ways to observe bulk signatures of the phase transition in
transport experiments, we compute the spin current flowing between a local
spin-polarized probe, such as an STM tip, and the double-nanowire system in the
Keldysh formalism. We find that the spin current contains information about the
sign flip of the bulk spin polarization and can be used to determine the
topological phase transition point.Comment: 12 pages, 7 figure
Majorana Fermions in superconducting wires: effects of long-range hopping, broken time-reversal symmetry and potential landscapes
We present a comprehensive study of two of the most experimentally relevant
extensions of Kitaev's spinless model of a 1D p-wave superconductor: those
involving (i) longer range hopping and superconductivity and (ii) inhomogeneous
potentials. We commence with a pedagogical review of the spinless model and, as
a means of characterizing topological phases exhibited by the systems studied
here, we introduce bulk topological invariants as well as those derived from an
explicit consideration of boundary modes. In time-reversal invariant systems,
we find that the longer range hopping leads to topological phases characterized
by multiple Majorana modes. In particular, we investigate a spin model, which
respects a duality and maps to a fermionic model with multiple Majorana modes;
we highlight the connection between these topological phases and the broken
symmetry phases in the original spin model. In the presence of time-reversal
symmetry breaking terms, we show that the topological phase diagram is
characterized by an extended gapless regime. For the case of inhomogeneous
potentials, we explore phase diagrams of periodic, quasiperiodic, and
disordered systems. We present a detailed mapping between normal state
localization properties of such systems and the topological phases of the
corresponding superconducting systems. This powerful tool allows us to leverage
the analyses of Hofstadter's butterfly and the vast literature on Anderson
localization to the question of Majorana modes in superconducting quasiperiodic
and disordered systems, respectively. We briefly touch upon the synergistic
effects that can be expected in cases where long-range hopping and disorder are
both present.Comment: 30 pages, 13 figure
Floquet generation of Majorana end modes and topological invariants
We show how Majorana end modes can be generated in a one-dimensional system
by varying some of the parameters in the Hamiltonian periodically in time. The
specific model we consider is a chain containing spinless electrons with a
nearest-neighbor hopping amplitude, a p-wave superconducting term and a
chemical potential; this is equivalent to a spin-1/2 chain with anisotropic XY
couplings between nearest neighbors and a magnetic field applied in the
z-direction. We show that varying the chemical potential (or magnetic field)
periodically in time can produce Majorana modes at the ends of a long chain. We
discuss two kinds of periodic driving, periodic delta-function kicks and a
simple harmonic variation with time. We discuss some distinctive features of
the end modes such as the inverse participation ratio of their wave functions
and their Floquet eigenvalues which are always equal to +/- 1 for time-reversal
symmetric systems. For the case of periodic delta-function kicks, we use the
effective Hamiltonian of a system with periodic boundary conditions to define
two topological invariants. The first invariant is a well-known winding number
while the second invariant has not appeared in the literature before. The
second invariant is more powerful in that it always correctly predicts the
numbers of end modes with Floquet eigenvalues equal to +1 and -1, while the
first invariant does not. We find that the number of end modes can become very
large as the driving frequency decreases. We show that periodic delta-function
kicks in the hopping and superconducting terms can also produce end modes.
Finally, we study the effect of electron-phonon interactions (which are
relevant at finite temperatures) and a random noise in the chemical potential
on the Majorana modes.Comment: 15 pages, 11 figures; added more numerical and analytical results
about second topological invariant, and a discussion of effects of
electron-phonon interactions and noise on Majorana end mode
Majorana Kramers pairs in Rashba double nanowires with interactions and disorder
We analyze the effects of electron-electron interactions and disorder on a
Rashba double-nanowire setup coupled to an s-wave superconductor, which has
been recently proposed as a versatile platform to generate Kramers pairs of
Majorana bound states in the absence of magnetic fields. We identify the regime
of parameters for which these Kramers pairs are stable against interaction and
disorder effects. We use bosonization, perturbative renormalization group, and
replica techniques to derive the flow equations for various parameters of the
model and evaluate the corresponding phase diagram with topological and
disorder-dominated phases. We confirm aforementioned results by considering a
more microscopic approach which starts from the tunneling Hamiltonian between
the three-dimensional s-wave superconductor and the nanowires. We find again
that the interaction drives the system into the topological phase and, as the
strength of the source term coming from the tunneling Hamiltonian increases,
strong electron-electron interactions are required to reach the topological
phase.Comment: 24 pages, 12 figures, published versio