[Show abstract][Hide abstract] ABSTRACT: In a Dirac semimetal, the conduction and valence bands contact only at
discrete (Dirac) points in the Brillouin zone (BZ) and disperse linearly in all
directions around these critical points. Including spin, the low energy
effective theory around each critical point is a four band Dirac Hamiltonian.
In two dimensions (2D), this situation is realized in graphene without
spin-orbit coupling. 3D Dirac points are predicted to exist at the phase
transition between a topological and a normal insulator in the presence of
inversion symmetry. Here we show that 3D Dirac points can also be protected by
crystallographic symmetries in particular space-groups and enumerate the
criteria necessary to identify these groups. This reveals the possibility of 3D
analogs to graphene. We provide a systematic approach for identifying such
materials and present ab initio calculations of metastable \beta-cristobalite
BiO_2 which exhibits Dirac points at the three symmetry related X points of the
BZ.
[Show abstract][Hide abstract] ABSTRACT: We formulate a theory of non-Abelian fractional quantum Hall states by
considering an anisotropic system consisting of coupled, interacting one
dimensional wires. We show that Abelian bosonization provides a simple
framework for characterizing the Moore Read state, as well as the more general
Read Rezayi sequence. This coupled wire construction provides a solvable
Hamiltonian formulated in terms of electronic degrees of freedom, and provides
a direct route to characterizing the quasiparticles and edge states in terms of
conformal field theory. This construction leads to a simple interpretation of
the coset construction of conformal field theory, which is a powerful method
for describing non Abelian states. In the present context, the coset
construction arises when the original chiral modes are fractionalized into
coset sectors, and the different sectors acquire energy gaps due to coupling in
"different directions". The coupled wire construction can also can be used to
describe anisotropic lattice systems, and provides a starting point for models
of fractional and non-Abelian Chern insulators. This paper also includes an
extended introduction to the coupled wire construction for Abelian quantum Hall
states, which was introduced earlier.
Physical Review B 11/2011; 89(8). DOI:10.1103/PhysRevB.89.085101 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The topological insulating phase results from inversion of the band gap due to spin-orbit coupling at an odd number of time-reversal symmetric points. In Bi2Se3, this inversion occurs at the Gamma point. For bulk Bi2Se3, we have analyzed the effect of arbitrary strain on the Gamma point band gap using density functional theory. By computing the band structure both with and without spin-orbit interactions, we consider the effects of strain on the gap via Coulombic interaction and spin-orbit interaction separately. While compressive strain acts to decrease the Coulombic gap, it also increases the strength of the spin-orbit interaction, increasing the inverted gap. Comparison with Bi2Te3 supports the conclusion that effects on both Coulombic and spin-orbit interactions are critical to understanding the behavior of topological insulators under strain, and we propose that the topological insulating phase can be effectively manipulated by inducing strain through chemical substitution.
[Show abstract][Hide abstract] ABSTRACT: The three-dimensional topological insulator (originally called "topological
insulators") is the first example in nature of a topologically ordered
electronic phase existing in three dimensions that cannot be reduced to
multiple copies of quantum-Hall-like states. Their topological order can be
realized at room temperatures without magnetic fields and they can be turned
into magnets and exotic superconductors leading to world-wide interest and
activity in topological insulators. One of the major challenges in going from
quantum Hall-like 2D states to 3D topological insulators is to develop new
experimental approaches/methods to precisely probe this novel form of
topological-order since the standard tools and settings that work for IQH-state
also work for QSH states. The method to probe 2D topological-order is
exclusively with charge transport, which either measures quantized transverse
conductance plateaus in IQH systems or longitudinal conductance in quantum spin
Hall (QSH) systems. In a 3D topological insulator, the boundary itself supports
a two dimensional electron gas (2DEG) and transport is not (Z$_2$)
topologically quantized. In this paper, we review the birth of momentum- and
spin-resolved spectroscopy as a new experimental approach and as a directly
boundary sensitive method to study and prove topological-order in
three-dimensions via the direct measurements of the topological invariants
{$\nu_o$} that are associated with the Z$_2$ topology of the spin-orbit band
structure and opposite parity band inversions, which led to the experimental
discovery of the first 3D topological insulators. We also discuss how
spectroscopic methods are leading to the identification of spin-orbit
superconductors that may work as Majorana platforms and can be used to identify
topological superconductors - yet another class of new state of matter.
[Show abstract][Hide abstract] ABSTRACT: We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.
[Show abstract][Hide abstract] ABSTRACT: Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, Bi2Te3 and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.
Physics World 11/2010; 82(4). DOI:10.1103/RevModPhys.82.3045 · 0.24 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We develop a unified framework to classify topological defects in insulators
and superconductors described by spatially modulated Bloch and Bogoliubov de
Gennes Hamiltonians. We consider Hamiltonians H(k,r) that vary slowly with
adiabatic parameters r surrounding the defect and belong to any of the ten
symmetry classes defined by time reversal symmetry and particle-hole symmetry.
The topological classes for such defects are identified, and explicit formulas
for the topological invariants are presented. We introduce a generalization of
the bulk-boundary correspondence that relates the topological classes to defect
Hamiltonians to the presence of protected gapless modes at the defect. Many
examples of line and point defects in three dimensional systems will be
discussed. These can host one dimensional chiral Dirac fermions, helical Dirac
fermions, chiral Majorana fermions and helical Majorana fermions, as well as
zero dimensional chiral and Majorana zero modes. This approach can also be used
to classify temporal pumping cycles, such as the Thouless charge pump, as well
as a fermion parity pump, which is related to the Ising non-Abelian statistics
of defects that support Majorana zero modes.
[Show abstract][Hide abstract] ABSTRACT: We develop a unified framework to classify topological defects in insulators and superconductors described by spatially modulated Bloch and Bogoliubov de Gennes Hamiltonians. We consider a Hamiltonian H(k,r) that varies slowly with adiabatic parameters r away from the defect. Band theories are grouped into ten classes according to the presence or absence of anti-unitary symmetries, time reversal 2ˆ=±1 and/or particle-hole 2ˆ=±1. Both send k-k and rr. Stable classification of topological band theories are characterized by a unified set of integral formulae for all the symmetry classes in any dimensions. Examples that fall into this framework include edge and surface states along an interface, 1D chiral, helical and Majorana modes along a line defect, bound charge and Majorana zero mode at a point defect. This approach also applies to time dependent phenomena, such as the Thouless charge pumb, the Z2 spin pumb and the exchange statistics of Majorana bound states in three dimensions.
[Show abstract][Hide abstract] ABSTRACT: We show that three dimensional superconductors, described within a Bogoliubov
de Gennes framework can have zero energy bound states associated with pointlike
topological defects. The Majorana fermions associated with these modes have
non-Abelian exchange statistics, despite the fact that the braid group is
trivial in three dimensions. This can occur because the defects are associated
with an orientation that can undergo topologically nontrivial rotations. A new
feature of three dimensional systems is that there are "braidless" operations
in which it is possible to manipulate the groundstate associated with a set of
defects without moving or measuring them. To illustrate these effects we
analyze specific architectures involving topological insulators and
superconductors.
[Show abstract][Hide abstract] ABSTRACT: A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry based classification of condensed matter. Exotic spin transport phenomena such as the dissipationless quantum spin Hall effect have been speculated to originate from a novel topological order whose identification requires a spin sensitive measurement. Using Spin-resolved-ARPES, we probe the spin degrees of freedom and demonstrate that topological quantum numbers are uniquely determined from spin-texture Berry Phase imaging measurements. Applying this method to pure antimony (Sb) and Bi-Sb, we identify the origin of its novel Topological Order and the negative value of the mirror Chern number. These results taken together constitute the first observation of surface electrons collectively carrying a topological Berry's phase and definite mirror Chern chirality in pure Antimony (Sb) which are the key electronic properties for realizing topological quantum computing via the interface Majorana fermion framework. This paper contains the details of the above mentioned previously reported (Science \textbf{323}, 919 (2009)) results. Comment: 6 Figures, 10 Pages, RevTex Format, Detailed version of Hsieh et.al., SCIENCE 323, 919 (2009)
[Show abstract][Hide abstract] ABSTRACT: We propose two experiments to probe the Majorana fermion edge states that
occur at a junction between a superconductor and a magnet deposited on the
surface of a topological insulator. Combining two Majorana fermions into a
single Dirac fermion on a magnetic domain wall allows the neutral Majorana
fermions to be probed with charge transport. We will discuss a novel
interferometer for Majorana fermions, which probes their Z_2 phase. This setup
also allows the transmission of neutral Majorana fermions through a point
contact to be measured. We introduce a point contact formed by a
superconducting junction and show that its transmission can be controlled by
the phase difference across the junction. We discuss the feasibility of these
experiments using the recently discovered topological insulator Bi_2 Se_3.
[Show abstract][Hide abstract] ABSTRACT: We study a quantum point contact in a quantum spin Hall insulator. It has recently been shown that the Luttinger liquid theory of such a structure maps to the theory of a weak link in a Luttinger liquid with spin with Luttinger liquid parameters g_\rho = 1/g_\sigma = g < 1. We show that for 1/2<g<1, the pinch-off of the point contact as a function of gate voltage is controlled by a novel quantum critical point, related to a nontrivial intermediate fixed point found previously in the Luttinger liquid model. We predict that the dependence of the conductance on gate voltage and temperature near the pinch-off transition collapses onto a universal curve described by a crossover scaling function. We compute the conductance, the critical exponents and the scaling function in solvable limits, which include g=1-\epsilon, g=1/2+\epsilon and g=1/\sqrt{3}. These results, along with a general scaling analysis provide an overall picture of the critical behavior as a function of g. In addition, we analyze the structure of the four terminal conductance of the point contact in the weak tunneling and weak backscattering limits. We find that different components of the conductance can have different temperature dependence. We identify a skew conductance G_{XY}, which we predict vanishes as T^\gamma with \gamma\ge 2. This behavior is a direct consequence of the unique edge state structure of the quantum spin Hall insulator. Finally, we show that for g<1/2 the presence of spin non conserving spin orbit interactions leads to a novel time reversal symmetry breaking insulating phase. In this phase, the transport is carried by spinless chargons and chargeless spinons. These lead to nontrivial correlations in the low frequency shot noise. Implications for experiments on HgCdTe quantum well structures will be discussed. Comment: 23 pages, 11 figures
[Show abstract][Hide abstract] ABSTRACT: We study junctions between superconductors mediated by the edge states of a
quantum spin Hall insulator. We show that such junctions exhibit a fractional
Josephson effect, in which the current phase relation has a 4\pi, rather than a
2\pi periodicity. This effect is a consequence of the conservation of fermion
parity - the number of electrons modulo 2 - in a superconducting junction, and
is closely related to the Z_2 topological structure of the quantum spin Hall
insulator. Inelastic processes, which violate the conservation of fermion
parity, lead to telegraph noise in the equilibrium supercurrent. We predict
that the low frequency noise due these processes diverges exponentially with
temperature T as T -> 0. Possible experiments on HgCdTe quantum wells will be
discussed.
[Show abstract][Hide abstract] ABSTRACT: A topologically ordered material is characterized by a rare quantum organization of electrons that evades the conventional spontaneously broken symmetry-based classification of condensed matter. Exotic spin-transport phenomena, such as the dissipationless quantum spin Hall effect, have been speculated to originate from a topological order whose identification requires a spin-sensitive measurement, which does not exist to this date in any system. Using Mott polarimetry, we probed the spin degrees of freedom and demonstrated that topological quantum numbers are completely determined from spin texture-imaging measurements. Applying this method to Sb and Bi(1-x)Sb(x), we identified the origin of its topological order and unusual chiral properties. These results taken together constitute the first observation of surface electrons collectively carrying a topological quantum Berry's phase and definite spin chirality, which are the key electronic properties component for realizing topological quantum computing bits with intrinsic spin Hall-like topological phenomena.
[Show abstract][Hide abstract] ABSTRACT: A topologically ordered material is characterized by a rare quantum
organization of electrons that evades the conventional spontaneously
broken symmetry based classification of condensed matter. Exotic spin
transport phenomena such as the dissipationless quantum spin Hall effect
have been speculated to originate from a novel topological order whose
identification requires a spin sensitive measurement, which does not
exist to this date in any system (neither in Hg(Cd)Te quantum wells nor
in the topological insulator BiSb). Using Mott polarimetry, we probe the
spin degrees of freedom of these quantum spin Hall states and
demonstrate that topological quantum numbers are uniquely determined
from spin texture imaging measurements. Applying this method to the
Bi{1-x}Sb{x} series, we identify the origin of its novel order and
unusual chiral properties. These results taken together constitute the
first observation of surface electrons collectively carrying a
geometrical quantum (Berry's) phase and definite chirality (mirror Chern
number, n_M =-1), which are the key electronic properties for realizing
topological computing bits with intrinsic spin Hall-like topological
phenomena. Our spin-resolved results not only provides the first clear
proof of a topological insulating state in nature but also demonstrate
the utility of spin-resolved ARPES technique in measuring the quantum
spin Hall phases of matter.
[Show abstract][Hide abstract] ABSTRACT: Graphene is an extended two-dimensional (2D) sheet of carbon atoms bound by strong nearest neighbor sp2 bonds and ordered in the form of a 2D honeycomb lattice. Stacked sheets of graphene occur naturally in the mineral graphite, an extremely well studied form of carbon that historically has attracted attention because of its unique electronic and structural properties. The chapter discusses the physical properties of single-layer graphene and the practical experimental methods for isolating macroscopic single- and few-layer samples. The chapter presents a class of remarkable quantum electronic transport phenomena in ultrathin one- and few-layer samples at room temperature; the related effects are quickly suppressed when the graphene layers are stacked to form a three-dimensional (3D) material. It is possible that some of these phenomena will be harnessed to develop a new family of submicron electronic devices based on their 2D physics. The chapter presents a relatively simple model for the electronic structure.
Contemporary Concepts of Condensed Matter Science 12/2008; 3. DOI:10.1016/S1572-0934(08)00007-3
[Show abstract][Hide abstract] ABSTRACT: Experiment has now proved the existence of the predicted
three-dimensional 'topological insulator' in the semiconducting alloy
Bi1-xSbx.
[Show abstract][Hide abstract] ABSTRACT: We study the electronic surface states of the semiconducting alloy bismuth antimony (Bi1−xSbx). Using a phenomenological tight-binding model, we show that the Fermi surface for the 111 surface states encloses an odd number of time-reversal-invariant momenta (TRIM) in the surface Brillouin zone. This confirms that the alloy is a strong topological insulator in the (1;111) Z2 topological class. We go on to develop general arguments which show that spatial symmetries lead to additional topological structure of the bulk energy bands, and impose further constraints on the surface band structure. Inversion-symmetric band structures are characterized by eight Z2 “parity invariants,” which include the four Z2 invariants defined by time-reversal symmetry. The extra invariants determine the “surface fermion parity,” which specifies which surface TRIM are enclosed by an odd number of electron or hole pockets. We provide a simple proof of this result, which provides a direct link between the surface-state structure and the parity eigenvalues characterizing the bulk. Using this result, we make specific predictions for the surface-state structure for several faces of Bi1−xSbx. We next show that mirror-invariant band structures are characterized by an integer “mirror Chern number” nM, which further constrains the surface states. We show that the sign of nM in the topological insulator phase of Bi1−xSbx is related to a previously unexplored Z2 parameter in the L point k⋅p theory of pure bismuth, which we refer to as the “mirror chirality” η. The value of η predicted by the tight-binding model for bismuth disagrees with the value predicted by a more fundamental pseudopotential calculation. This explains a subtle disagreement between our tight-binding surface-state calculation and previous first-principles calculations of the surface states of bismuth. This suggests that the tight-binding parameters in the Liu-Allen model of bismuth need to be reconsidered. Implications for existing and future angle-resolved photoemission spectroscopy (ARPES) experiments and spin-polarized ARPES experiments will be discussed.
[Show abstract][Hide abstract] ABSTRACT: We study the proximity effect between an s-wave superconductor and the surface states of a strong topological insulator. The resulting two-dimensional state resembles a spinless px+ipy superconductor, but does not break time reversal symmetry. This state supports Majorana bound states at vortices. We show that linear junctions between superconductors mediated by the topological insulator form a nonchiral one-dimensional wire for Majorana fermions, and that circuits formed from these junctions provide a method for creating, manipulating, and fusing Majorana bound states.