[Show abstract][Hide abstract] ABSTRACT: Topological phases of matter that depend for their existence on interactions
are fundamentally interesting and potentially useful as platforms for future
quantum computers. Despite the multitude of theoretical proposals the only
interaction-enabled topological phase experimentally observed is the fractional
quantum Hall liquid. To help identify other systems that can give rise to such
phases we present in this work a detailed study of the effect of interactions
on Majorana zero modes bound to vortices in a superconducting surface of a 3D
topological insulator. This system is of interest because, as was recently
pointed out, it can be tuned into the regime of strong interactions. We start
with a 0D system suggesting an experimental realization of the
interaction-induced $\mathbb{Z}_8$ ground state periodicity previously
discussed by Fidkowski and Kitaev. We argue that the periodicity is
experimentally observable using a tunnel probe. We then focus on
interaction-enabled crystalline topological phases that can be built with the
Majoranas in a vortex lattice in higher dimensions. In 1D we identify an
interesting exactly solvable model which is related to a previously discussed
one that exhibits an interaction-enabled topological phase. We study these
models using analytical techniques, exact numerical diagonalization (ED) and
density matrix renormalization group (DMRG). Our results confirm the existence
of the interaction-enabled topological phase and clarify the nature of the
quantum phase transition that leads to it. We finish with a discussion of
models in dimensions 2 and 3 that produce similar interaction-enabled
topological phases.
Physical Review B 06/2015; 92(7). DOI:10.1103/PhysRevB.92.075438 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The Hubbard chain and spinless fermion chain are paradigms of strongly
correlated systems, very well understood using Bethe ansatz, Density Matrix
Renormalization Group (DMRG) and field theory/renormalization group (RG)
methods. They have been applied to one-dimensional materials and have provided
important insights for understanding higher dimensional cases. Recently, a new
interacting fermion model has been introduced, with possible applications to
topological materials. It has a single Majorana fermion operator on each
lattice site and interactions with the shortest possible range that involve 4
sites. We present a thorough analysis of the phase diagram of this model in one
dimension using field theory/RG and DMRG methods. It includes a gapped
supersymmetric region and a novel gapless phase with coexisting Luttinger
liquid and Ising degrees of freedom. In addition to a first order transition,
three critical points occur: tricritical Ising, Lifshitz and a novel
generalization of the commensurate-incommensurate transition. We also survey
various gapped phases of the system that arise when the translation symmetry is
broken by dimerization and find both trivial and topological phases with 0, 1
and 2 Majorana zero modes bound to the edges of the chain with open boundary
conditions.
[Show abstract][Hide abstract] ABSTRACT: We show that a strongly interacting chain of Majorana fermions exhibits a
supersymmetric quantum critical point corresponding to the $c={7\over 10}$
tricritical Ising model, which separates a critical phase in the Ising
universality class from a supersymmetric massive phase. We verify our
predictions with numerical density-matrix-renormalization-group computations
and determine the consequences for tunnelling experiments.
[Show abstract][Hide abstract] ABSTRACT: The effect of surface disorder on electronic systems is particularly
interesting for topological phases with surface and edge states. Using exact
diagonalization, it has been demonstrated that the surface states of a 3D
topological insulator survive strong surface disorder, and simply get pushed to
a clean part of the bulk. Here we explore a new method which analytically
eliminates the clean bulk, and reduces a $D$-dimensional problem to a
Hamiltonian-diagonalization problem within the $(D-1)$-dimensional disordered
surface. This dramatic reduction in complexity allows the analysis of
significantly bigger systems than is possible with exact diagonalization. We
use our method to analyze a 2D topological spin-Hall insulator with
non-magnetic and magnetic edge impurities, and we calculate the probability
density (or local density of states) of the zero-energy eigenstates as a
function of edge-parallel momentum and layer index. Our analysis reveals that
the system size needed to reach behavior in the thermodynamic limit increases
with disorder. We also compute the edge conductance as a function of disorder
strength, and chart a lower bound for the length scale marking the crossover to
the thermodynamic limit.
Physical Review B 03/2015; 92(7). DOI:10.1103/PhysRevB.92.075110 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Systems of strongly interacting particles, fermions or bosons, can give rise
to topological phases that are not acessible to non-interacting systems. Many
such interaction-enabled topological phases have been discussed theoretically
but few experimental realizations exists. Here we propose a new platform for
interacting topological phases of fermions with time reversal symmetry $\bar T$
(such that $\bar T^2=1$) that can be realized in vortex lattices in the surface
state of a topological insulator. The constituent particles are Majorana
fermions bound to vortices and antivortices of such a lattice. We explain how
the $\bar T$ symmetry arises and discuss ways in which interactions can be
experimentally tuned and detected. We show how these features can be exploited
to realize a class of interaction-enabled crystalline topological phases that
have no analog in weakly interacting systems.
[Show abstract][Hide abstract] ABSTRACT: Interesting phases of quantum matter often arise when the constituent
particles -- electrons in solids -- interact strongly. Such strongly
interacting systems are however quite rare and occur only in extreme
environments of low spatial dimension, low temperatures or intense magnetic
fields. Here we introduce a new system in which the fundamental electrons
interact only weakly but the low energy effective theory is described by
strongly interacting Majorana fermions. The system consists of an Abrikosov
vortex lattice in the surface of a strong topological insulator and is
accessible experimentally using presently available technology. The simplest
interactions between the Majorana degrees of freedom exhibit an unusual
nonlocal structure that involves four distinct Majorana sites. We formulate
simple lattice models with this type of interaction and find exact solutions in
certain physically relevant one- and two-dimensional geometries. In other cases
we show how our construction allows for the experimental realization of
interesting spin models previously only theoretically contemplated.
Physical Review B 11/2014; 91(16). DOI:10.1103/PhysRevB.91.165402 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Electronic states associated with a chain of magnetic adatoms on the surface
of an ordinary s- wave superconductor have been shown theoretically to form a
one dimensional topological phase with unpaired Majorana fermions bound to its
ends. In a simple 1D effective model the system exhibits an interesting
self-organization property: the pitch of the spiral formed by the adatom
magnetic moments tends to adjust itself so that electronically the chain
remains in the topological phase whenever such a state is physically
accessible. Here we examine the physics underlying this self-organization
property in the framework of a more realistic 2D model of a superconducting
surface coupled to a 1D chain of magnetic adatoms. Treating both the
superconducting order and the magnetic moments selfconsistently we find that
the system retains its self-organization property, even if the topological
phase extends over a somewhat smaller portion of the phase diagram compared to
the 1D model. We also study the effect of imperfections and find that, once
established, the topological phase survives moderate levels of disorder.
Physical Review B 06/2014; 90(8). DOI:10.1103/PhysRevB.90.085124 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is generally thought that adiabatic exchange of two identical particles is
impossible in one spatial dimension. Here we describe a simple protocol that
permits adiabatic exchange of two Majorana fermions in a one-dimensional
topological superconductor wire. The exchange relies on the concept of
``Majorana shuttle'' whereby a $\pi$ domain wall in the superconducting order
parameter which hosts a pair of ancillary Majoranas delivers one zero mode
across the wire while the other one tunnels in the opposite direction. The
method requires some tuning of parameters and does not, therefore, enjoy the
full topological protection. The resulting exchange statistics, however,
remains non-Abelian for a wide range of parameters that characterize the
exchange.
[Show abstract][Hide abstract] ABSTRACT: Most physical systems known to date tend to resist entering the topological phase and must be fine-tuned to reach that phase. Here, we introduce a system in which a key dynamical parameter adjusts itself in response to the changing external conditions so that the ground state naturally favors the topological phase. The system consists of a quantum wire formed of individual magnetic atoms placed on the surface of an ordinary s-wave superconductor. It realizes the Kitaev paradigm of topological superconductivity when the wave vector characterizing the emergent spin helix dynamically self-tunes to support the topological phase. We call this phenomenon a self-organized topological state.
[Show abstract][Hide abstract] ABSTRACT: It has been suggested recently, based on subtle field-theoretical considerations, that the electromagnetic response of Weyl semimetals and the closely related Weyl insulators can be characterized by an axion term θE·B with space and time dependent axion angle θ(r,t). Here we construct a minimal lattice model of the Weyl medium and study its electromagnetic response by a combination of analytical and numerical techniques. We confirm the existence of the anomalous Hall effect expected on the basis of the field theory treatment. We find, contrary to the latter, that chiral magnetic effect (that is, ground state charge current induced by the applied magnetic field) is absent in both the semimetal and the insulator phase. We elucidate the reasons for this discrepancy.
[Show abstract][Hide abstract] ABSTRACT: A surface of a strong topological insulator (STI) is characterized by an
odd number of linearly dispersing gapless electronic surface states. It
is well known that such a surface cannot be described by an effective
two-dimensional lattice model (without breaking the time-reversal
symmetry), which often hampers theoretical efforts to quantitatively
understand some of the properties of such surfaces, including the effect
of strong disorder, interactions and various symmetry-breaking
instabilities. Here we describe a lattice model that can be used to
describe a pair of STI surfaces and has an odd number of Dirac fermion
states with wavefunctions localized on each surface. The Hamiltonian
consists of two planar tight-binding models with spin-orbit coupling,
representing the two surfaces, weakly coupled to each other by terms
that remove the redundant Dirac points from the low-energy spectrum. The
utility of this model is illustrated by studying the magnetic and
exciton instabilities of the STI surface state driven by short-range
repulsive interactions.
[Show abstract][Hide abstract] ABSTRACT: Two-dimensional topological insulators (2D TIs) have been proposed as platforms for many intriguing applications, ranging from spintronics to topological quantum information processing. Realizing this potential will likely be facilitated by the discovery of new, easily manufactured materials in this class. With this goal in mind, we introduce a new framework for engineering a 2D TI by hybridizing graphene with impurity bands arising from heavy adatoms possessing partially filled d shells, in particular, osmium and iridium. First-principles calculations predict that the gaps generated by this means exceed 0.2 eV over a broad range of adatom coverage; moreover, tuning of the Fermi level is not required to enter the TI state. The mechanism at work is expected to be rather general and may open the door to designing new TI phases in many materials.
[Show abstract][Hide abstract] ABSTRACT: A surface of a strong topological insulator (STI) is characterized by an odd
number of linearly dispersing gapless electronic surface states. It is well
known that such a surface cannot be described by an effective two-dimensional
lattice model (without breaking the time-reversal symmetry), which often
hampers theoretical efforts to quantitatively understand some of the properties
of such surfaces, including the effect of strong disorder, interactions and
various symmetry-breaking instabilities. Here we formulate a lattice model that
can be used to describe a {\em pair} of STI surfaces and has an odd number of
Dirac fermion states with wavefunctions localized on each surface. The
Hamiltonian consists of two planar tight-binding models with spin-orbit
coupling, representing the two surfaces, weakly coupled by terms that remove
the extra Dirac points from the low-energy spectrum. We illustrate the utility
of this model by studying the magnetic and exciton instabilities of the STI
surface state driven by short-range repulsive interactions and show that this
leads to results that are consistent with calculations based on the continuum
model as well as three-dimensional lattice models. We expect the model
introduced in this work to be widely applicable to studies of surface phenomena
in STIs.
[Show abstract][Hide abstract] ABSTRACT: It has been shown previously that a finite-length topological insulator
nanowire, proximity-coupled to an ordinary bulk s-wave superconductor and
subject to a longitudinal applied magnetic field, realizes a one-dimensional
topological superconductor with an unpaired Majorana fermion (MF) localized at
each end of the nanowire. Here, we study the stability of these MFs with
respect to various perturbations that are likely to occur in a physical
realization of the proposed device. We show that the unpaired Majorana fermions
persist in this system for any value of the chemical potential inside the bulk
band gap of order 300 meV in Bi$_2$Se$_3$ by computing the Majorana number.
From this calculation, we also show that the unpaired Majorana fermions persist
when the magnetic flux through the nanowire cross-section deviates
significantly from half flux quantum. Lastly, we demonstrate that the unpaired
Majorana fermions persist in strongly disordered wires with fluctuations in the
on-site potential ranging in magnitude up to several times the size of the bulk
band gap. These results suggest this solid-state system should exhibit unpaired
Majorana fermions under accessible conditions likely important for experimental
study or future applications.
[Show abstract][Hide abstract] ABSTRACT: We show that a three dimensional topological insulator doped with magnetic
impurities in the bulk can have a regime where the surface is magnetically
ordered but the bulk is not. This is in contrast to conventional materials
where bulk ordered phases are typically more robust than surface ordered
phases. The difference originates from the topologically protected gapless
surface states characteristic of topological insulators. We study the problem
using a mean field approach in two concrete models that give the same
qualitative result, with some interesting differences. Our findings could help
explain recent experimental results showing the emergence of a spectral gap in
the surface state of Bi2Se3 doped with Mn or Fe atoms, but with no measurable
bulk magnetism.
[Show abstract][Hide abstract] ABSTRACT: The surface of a topological insulator hosts a very special form of a
quasi-two dimensional metallic system when it is embedded in a topologically
trivial medium like the vacuum. The electronic properties of this unusual 2D
metal are distinct in many aspects from both the conventional two-dimensional
electron gas systems in quantum well heterostructures as well as those of a
single layer graphene. In this paper, we study one of these distinct features
i.e., the response of the electronic spins to an applied magnetic field
perpendicular to the surface. We find an unusual behaviour of the spin
magnetization and susceptibility as a function of both the magnetic field and
the chemical potential for a generic topological surface. We propose that this
behavior could be studied by the recently developed experimental technique
called \beta NMR which is highly sensitive to the surface electron spins. We
explain how this technique could be used to probe for spontaneous magnetic
ordering caused by magnetic dopants or interactions discussed in the recent
literature.
[Show abstract][Hide abstract] ABSTRACT: We construct a simple model for electrons in a three-dimensional crystal where a combination of short-range hopping and spin-orbit coupling results in nearly flat bands characterized by a nontrivial Z2 topological index. The flat band is separated from other bands by a band gap Δ that is much larger than the bandwidth W. When the flat band is partially filled we show that the system remains nonmagnetic for a significant range of repulsive interactions. In this regime we conjecture that the true many-body ground state may become a three-dimensional fractional topological insulator.
[Show abstract][Hide abstract] ABSTRACT: A finite-length topological-insulator nanowire, proximity-coupled to an
ordinary bulk s-wave superconductor and subject to a longitudinal
applied magnetic field, is shown to realize a one-dimensional
topological superconductor with unpaired Majorana fermions localized at
both ends. This situation occurs under a wide range of conditions and
constitutes an easily accessible physical realization of the elusive
Majorana particle in a solid-state system.
[Show abstract][Hide abstract] ABSTRACT: The 2007 discovery of quantized conductance in HgTe quantum wells delivered
the field of topological insulators (TIs) its first experimental confirmation.
While many three-dimensional TIs have since been identified, HgTe remains the
only known two-dimensional system in this class. Difficulty fabricating HgTe
quantum wells has, moreover, hampered their widespread use. With the goal of
breaking this logjam we provide a blueprint for stabilizing a robust TI state
in a more readily available two-dimensional material---graphene. Using symmetry
arguments, density functional theory, and tight-binding simulations, we predict
that graphene endowed with certain heavy adatoms realizes a TI with substantial
band gap. For indium and thallium, our most promising adatom candidates, a
modest 6% coverage produces an estimated gap near 80K and 240K, respectively,
which should be detectable in transport or spectroscopic measurements.
Engineering such a robust topological phase in graphene could pave the way for
a new generation of devices for spintronics, ultra-low-dissipation electronics
and quantum information processing.
Physical Review X 04/2011; 1(2). DOI:10.1103/PhysRevX.1.021001 · 9.04 Impact Factor