bibliography.yml

- abstract: Motivated by recent developments in the field of one-dimensional topological
    superconductors, we investigate the topological properties of s-matrix of generic
    superconducting junctions where dimension should not play any role. We argue that
    for a finite junction the s-matrix is always topologically trivial. We resolve
    an apparent contradiction with the previous results by taking into account the
    low-energy resonant poles of s-matrix. Thus no common topological transition occur
    in a finite junction. We reveal a transition of a different kind that concerns
    the configuration of the resonant poles.
  authors:
  - D. I. Pikulin
  - Yuli V. Nazarov
  doi: 10.1134/S0021364011210090
  id: '1103.0780'
  journal: Jetp Lett.
  journal_url: http://dx.doi.org/10.1134/s0021364011210090
  on_arxiv: 2011-03-03
  page: '693'
  published: 2012-01-01
  title: Topological properties of superconducting junctions
  vol: '94'
- abstract: The phase-dependent bound states (Andreev levels) of a Josephson junction
    can cross at the Fermi level, if the superconducting ground state switches between
    even and odd fermion parity. The level crossing is topologically protected, in
    the absence of time-reversal and spin-rotation symmetry, irrespective of whether
    the superconductor itself is topologically trivial or not. We develop a statistical
    theory of these topological transitions in an N-mode quantum-dot Josephson junction,
    by associating the Andreev level crossings with the real eigenvalues of a random
    non-Hermitian matrix. The number of topological transitions in a 2pi phase interval
    scales as sqrt(N) and their spacing distribution is a hybrid of the Wigner and
    Poisson distributions of random-matrix theory.
  authors:
  - C. W. J. Beenakker
  - J. M. Edge
  - J. P. Dahlhaus
  - D. I. Pikulin
  - Shuo Mi
  - M. Wimmer
  doi: 10.1103/PhysRevLett.111.037001
  id: '1305.2924'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.111.037001
  on_arxiv: 2013-05-13
  page: '037001'
  published: 2013-07-17
  title: Wigner-Poisson statistics of topological transitions in a Josephson junction
  vol: '111'
- abstract: Semiconducting nanowires in proximity to superconductors are promising
    experimental systems for Majorana fermions, which may ultimately be used as building
    blocks for topological quantum computers. A serious challenge in the experimental
    realization of the Majorana fermions is the supression of topological superconductivity
    by disorder. We show that Majorana fermions protected by a robust topological
    gap can occur at the ends of a chain of quantum dots connected by s-wave superconductors.
    In the appropriate parameter regime, we establish that the quantum dot/superconductor
    system is equivalent to a 1D Kitaev chain, which can be tuned to be in a robust
    topological phase with Majorana end modes even in the case where the quantum dots
    and superconductors are both strongly disordered. Such a spin-orbit coupled quantum
    dot - s-wave superconductor array provides an ideal experimental platform for
    the observation of non-Abelian Majorana modes.
  authors:
  - Jay D. Sau
  - S. Das Sarma
  doi: 10.1038/ncomms1966
  id: '1111.6600'
  journal: Nat Commun
  journal_url: http://dx.doi.org/10.1038/ncomms1966
  on_arxiv: 2011-11-28
  page: '964'
  published: 2012-01-01
  title: How to realize a robust practical Majorana chain in a quantum dot-superconductor
    linear array
  vol: '3'
- abstract: A single two-dimensinoal Dirac fermion state has been recently observed
    on the surface of topological insulator Bi$_2$Te$_3$ by angle-resolved photoemission
    spectroscopy (ARPES). We study the surface band structure using $k \cdot p$ theory
    and find an unconventional hexagonal warping term, which is the counterpart of
    cubic Dresselhaus spin-orbit coupling in rhombohedral structures. We show that
    this hexagonal warping term naturally explains the observed hexagonal snow-flake
    Fermi surface. The strength of hexagonal warping is characterized by a single
    parameter, which is extracted from the size of the Fermi surface. We predict a
    number of testable signatures of hexagonal warping in spectroscopy experiments
    on Bi$_2$Te$_3$. We also explore the possibility of a spin-density wave due to
    strong nesting of the Fermi surface.
  authors:
  - Liang Fu
  doi: 10.1103/PhysRevLett.103.266801
  id: '0908.1418'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.103.266801
  on_arxiv: 2009-08-10
  page: '266801'
  published: 2009-12-21
  title: Hexagonal Warping Effects in the Surface States of Topological Insulator
    Bi$_2$Te$_3$
  vol: '103'
- abstract: We report on electronic transport measurements of dual-gated nano-devices
    of the low-carrier density topological insulator Bi1.5Sb0.5Te1.7Se1.3. In all
    devices the upper and lower surface states are independently tunable to the Dirac
    point by the top and bottom gate electrodes. In thin devices, electric fields
    are found to penetrate through the bulk, indicating finite capacitive coupling
    between the surface states. A charging model allows us to use the penetrating
    electric field as a measurement of the inter-surface capacitance $C_{TI}$ and
    the surface state energy-density relationship $\mu$(n), which is found to be consistent
    with independent ARPES measurements. At high magnetic fields, increased field
    penetration through the surface states is observed, strongly suggestive of the
    opening of a surface state band gap due to broken time-reversal symmetry.
  authors:
  - Valla Fatemi
  - Benjamin Hunt
  - Hadar Steinberg
  - Stephen L. Eltinge
  - Fahad Mahmood
  - Nicholas P. Butch
  - Kenji Watanabe
  - Takashi Taniguchi
  - Nuh Gedik
  - Ray Ashoori
  - Pablo Jarillo-Herrero
  doi: 10.1103/PhysRevLett.113.206801
  id: '1410.0655'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.113.206801
  on_arxiv: 2014-10-02
  page: '206801'
  published: 2014-11-14
  title: Electrostatic Coupling between Two Surfaces of a Topological Insulator Nanodevice
  vol: '113'
- abstract: Existence of the magnetic monopole is compatible with the fundamental
    laws of nature, however, this illusive particle has yet to be detected experimentally.
    In this work, we show that an electric charge near the topological surface state
    induces an image magnetic monopole charge due to the topological magneto-electric
    effect. The magnetic field generated by the image magnetic monopole can be experimentally
    measured, and the inverse square law of the field dependence can be determined
    quantitatively. We propose that this effect can be used to experimentally realize
    a gas of quantum particles carrying fractional statistics, consisting of the bound
    states of the electric charge and the image magnetic monopole charge.
  authors:
  - Xiao-Liang Qi
  - Rundong Li
  - Jiadong Zang
  - Shou-Cheng Zhang
  doi: 10.1126/science.1167747
  id: '0811.1303'
  journal: Science
  journal_url: http://dx.doi.org/10.1126/science.1167747
  on_arxiv: 2008-11-08
  page: '1184'
  published: 2009-02-27
  title: Seeing the magnetic monopole through the mirror of topological surface states
  vol: '323'
- abstract: The role of disorder in the field of three-dimensional time reversal invariant
    topological insulators has become an active field of research recently. However,
    the computation of Z2 invariants for large, disordered systems still poses a considerable
    challenge. In this paper we apply and extend a recently proposed method based
    on the scattering matrix approach, which allows the study of large systems at
    reasonable computational effort with few-channel leads. By computing the Z2 invariant
    directly for the disordered topological Anderson insulator, we unambiguously identify
    the topological nature of this phase without resorting to its connection with
    the clean case. We are able to efficiently compute the Z2 phase diagram in the
    mass-disorder plane. The topological phase boundaries are found to be well described
    by the self consistent Born approximation, both for vanishing and finite chemical
    potential.
  authors:
  - Björn Sbierski
  - Piet W. Brouwer
  doi: 10.1103/PhysRevB.89.155311
  id: '1401.7461'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.89.155311
  on_arxiv: 2014-01-29
  page: '155311'
  published: 2014-04-15
  title: Z2 phase diagram of three-dimensional disordered topological insulator via
    scattering matrix approach
  vol: '89'
- abstract: Many promising building blocks of future electronic technology - including
    non-stoichiometric compounds, strongly correlated oxides, and strained or patterned
    films - are inhomogeneous on the nanometer length scale. Exploiting the inhomogeneity
    of such materials to design next-generation nanodevices requires a band structure
    probe with nanoscale spatial resolution. To address this demand, we report the
    first simultaneous observation and quantitative reconciliation of two candidate
    probes - Landau level spectroscopy and quasiparticle interference imaging - which
    we employ here to reconstruct the multi-component surface state band structure
    of the topological semimetal antimony(Sb). We thus establish the technique of
    band structure tunneling microscopy (BSTM), whose unique advantages include nanoscale
    access to non-rigid band structure deformation, empty state dispersion, and magnetic
    field dependent states. We use BSTM to elucidate the relationship between bulk
    conductivity and surface state robustness in topological materials, and to quantify
    essential metrics for spintronics applications.
  authors:
  - Anjan Soumyanarayanan
  - Michael M. Yee
  - Yang He
  - Hsin Lin
  - Dillon R. Gardner
  - Arun Bansil
  - Young S. Lee
  - Jennifer E. Hoffman
  id: '1311.1758'
  on_arxiv: 2013-11-07
  title: Imaging the Nanoscale Band Structure of Topological Sb
- abstract: A direct signature of electron transport at the metallic surface of a
    topological insulator is the Aharonov-Bohm oscillation observed in a recent study
    of Bi_2Se_3 nanowires [Peng et al., Nature Mater. 9, 225 (2010)] where conductance
    was found to oscillate as a function of magnetic flux $\phi$ through the wire,
    with a period of one flux quantum $\phi_0=h/e$ and maximum conductance at zero
    flux. This seemingly agrees neither with diffusive theory, which would predict
    a period of half a flux quantum, nor with ballistic theory, which in the simplest
    form predicts a period of $\phi_0$ but a minimum at zero flux due to a nontrivial
    Berry phase in topological insulators. We show how h/e and h/2e flux oscillations
    of the conductance depend on doping and disorder strength, provide a possible
    explanation for the experiments, and discuss further experiments that could verify
    the theory.
  authors:
  - J. H. Bardarson
  - P. W. Brouwer
  - J. E. Moore
  doi: 10.1103/PhysRevLett.105.156803
  id: '1005.3762'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.105.156803
  on_arxiv: 2010-05-20
  page: '156803'
  published: 2010-10-07
  title: Aharonov-Bohm oscillations in disordered topological insulator nanowires
  vol: '105'
- abstract: We present a simple prescription to flatten isolated Bloch bands with
    non-zero Chern number. We first show that approximate flattening of bands with
    non-zero Chern number is possible by tuning ratios of nearest-neighbor and next-nearest
    neighbor hoppings in the Haldane model and, similarly, in the chiral-pi-flux square
    lattice model. Then we show that perfect flattening can be attained with further
    range hoppings that decrease exponentially with distance. Finally, we add interactions
    to the model and present exact diagonalization results for a small system at 1/3
    filling that support (i) the existence of a spectral gap, (ii) that the ground
    state is a topological state, and (iii) that the Hall conductance is quantized.
  authors:
  - Titus Neupert
  - Luiz Santos
  - Claudio Chamon
  - Christopher Mudry
  doi: 10.1103/PhysRevLett.106.236804
  id: '1012.4723'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.106.236804
  on_arxiv: 2010-12-21
  page: '236804'
  published: 2011-06-06
  title: Fractional quantum Hall states at zero magnetic field
  vol: '106'
- abstract: The search for topologically non-trivial states of matter has become an
    important goal for condensed matter physics. Here, we give a theoretical introduction
    to the quantum anomalous Hall (QAH) effect based on magnetic topological insulators
    in two-dimension (2D) and three-dimension (3D). In 2D topological insulators,
    magnetic order breaks the symmetry between the counter-propagating helical edge
    states, and as a result, the quantum spin Hall effect can evolve into the QAH
    effect. In 3D, magnetic order opens up a gap for the topological surface states,
    and chiral edge state has been predicted to exist on the magnetic domain walls.
    We present the phase diagram in thin films of a magnetic topological insulator
    and review the basic mechanism of ferromagnetic order in magnetically doped topological
    insulators. We also review the recent experimental observation of the QAH effect.
    We discuss more recent theoretical work on the coexistence of the helical and
    chiral edge states, multi-channel chiral edge states, the theory of the plateau
    transition, and the thickness dependence in the QAH effect.
  authors:
  - Jing Wang
  - Biao Lian
  - Shou-Cheng Zhang
  doi: 10.1088/0031-8949/2015/T164/014003
  id: '1409.6715'
  journal: Phys. Scr.
  journal_url: http://dx.doi.org/10.1088/0031-8949/2015/t164/014003
  on_arxiv: 2014-09-23
  page: '014003'
  published: 2015-12-01
  title: Quantum anomalous Hall effect in magnetic topological insulators
  vol: T164
- abstract: 'Motivated by new capabilities to realise artificial gauge fields in ultracold
    atomic systems, and by their potential to access correlated topological phases
    in lattice systems, we present a new strategy for designing topologically non-trivial
    band structures. Our approach is simple and direct: it amounts to considering
    tight-binding models directly in reciprocal space. These models naturally cause
    atoms to experience highly uniform magnetic flux density and lead to topological
    bands with very narrow dispersion, without fine-tuning of parameters. Further,
    our construction immediately yields instances of optical Chern lattices, as well
    as band structures of higher Chern number, |C|>1.'
  authors:
  - N. R. Cooper
  - R. Moessner
  doi: 10.1103/PhysRevLett.109.215302
  id: '1208.4579'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.109.215302
  on_arxiv: 2012-08-22
  page: '215302'
  published: 2012-11-20
  title: Designing Topological Bands in Reciprocal Space
  vol: '109'
- abstract: When electrons are confined in two dimensions and subjected to strong
    magnetic fields, the Coulomb interactions between them become dominant and can
    lead to novel states of matter such as fractional quantum Hall liquids. In these
    liquids electrons linked to magnetic flux quanta form complex composite quasipartices,
    which are manifested in the quantization of the Hall conductivity as rational
    fractions of the conductance quantum. The recent experimental discovery of an
    anomalous integer quantum Hall effect in graphene has opened up a new avenue in
    the study of correlated 2D electronic systems, in which the interacting electron
    wavefunctions are those of massless chiral fermions. However, due to the prevailing
    disorder, graphene has thus far exhibited only weak signatures of correlated electron
    phenomena, despite concerted experimental efforts and intense theoretical interest.
    Here, we report the observation of the fractional quantum Hall effect in ultraclean
    suspended graphene, supporting the existence of strongly correlated electron states
    in the presence of a magnetic field. In addition, at low carrier density graphene
    becomes an insulator with an energy gap tunable by magnetic field. These newly
    discovered quantum states offer the opportunity to study a new state of matter
    of strongly correlated Dirac fermions in the presence of large magnetic fields.
  authors:
  - Kirill I. Bolotin
  - Fereshte Ghahari
  - Michael D. Shulman
  - Horst L. Stormer
  - Philip Kim
  doi: 10.1038/nature08582
  id: '0910.2763'
  journal: Nature
  journal_url: http://dx.doi.org/10.1038/nature08582
  on_arxiv: 2009-10-15
  page: '196'
  published: 2009-11-01
  title: Observation of the Fractional Quantum Hall Effect in Graphene
  vol: '462'
- abstract: Non-Abelian anyons--particles whose exchange noncommutatively transforms
    a system's quantum state--are widely sought for the exotic fundamental physics
    they harbor as well as for quantum computing applications. There now exist numerous
    blueprints for stabilizing the simplest type of non-Abelian anyon, defects binding
    Majorana modes, by judiciously interfacing widely available materials. Following
    this line of attack, we introduce a device fabricated from conventional fractional
    quantum Hall states and s-wave superconductors that supports exotic non-Abelian
    anyons that bind `parafermions', which can be viewed as fractionalized Majorana
    fermions. We show that these modes can be experimentally identified (and distinguished
    from Majoranas) using Josephson measurements. We also provide a practical recipe
    for braiding parafermions and show that they give rise to non-Abelian statistics.
    Interestingly, braiding in our setup produces a richer set of topologically protected
    qubit operations when compared to the Majorana case. As a byproduct, we establish
    a new, experimentally realistic Majorana platform in weakly spin-orbit-coupled
    materials such as GaAs.
  authors:
  - David J. Clarke
  - Jason Alicea
  - Kirill Shtengel
  doi: 10.1038/ncomms2340
  id: '1204.5479'
  journal: Nat Commun
  journal_url: http://dx.doi.org/10.1038/ncomms2340
  on_arxiv: 2012-04-24
  page: '1348'
  published: 2013-12-01
  title: Exotic non-Abelian anyons from conventional fractional quantum Hall states
  vol: '4'
- abstract: We present a comprehensive and self-contained simplified review of the
    quantum computing scheme of Phys. Rev. Lett. 98, 190504 (2007), which features
    a 2-D nearest neighbor coupled lattice of qubits, a threshold error rate approaching
    1%, natural asymmetric and adjustable strength error correction and low overhead
    arbitrarily long-range logical gates. These features make it by far the best and
    most practical quantum computing scheme devised to date. We restrict the discussion
    to direct manipulation of the surface code using the stabilizer formalism, both
    of which we also briefly review, to make the scheme accessible to a broad audience.
  authors:
  - Austin G. Fowler
  - Ashley M. Stephens
  - Peter Groszkowski
  doi: 10.1103/PhysRevA.80.052312
  id: '0803.0272'
  journal: Phys. Rev. A
  journal_url: http://dx.doi.org/10.1103/physreva.80.052312
  on_arxiv: 2008-03-03
  page: '052312'
  published: 2009-11-11
  title: High threshold universal quantum computation on the surface code
  vol: '80'
- abstract: We consider an annular superconductor-insulator-superconductor Josephson-junction,
    with the insulator being a double layer of electron and holes at Abelian fractional
    quantum Hall states of identical fillings. When the two superconductors gap out
    the edge modes, the system has a topological ground state degeneracy in the thermodynamic
    limit akin to the fractional quantum Hall degeneracy on a torus. In the quasi-one-dimensional
    limit, where the width of the insulator becomes small, the ground state energies
    are split. We discuss several implications of the topological degeneracy that
    survive the crossover to the quasi-one-dimensional limit. In particular, the Josephson
    effect shows a $2\pi d$-periodicity, where $d$ is the ground state degeneracy
    in the 2 dimensional limit. We find that at special values of the relative phase
    between the two superconductors there are protected crossing points in which the
    degeneracy is not completely lifted. These features occur also if the insulator
    is a time-reversal-invariant fractional topological insulator. We describe the
    latter using a construction based on coupled wires. Furthermore, when the superconductors
    are replaced by systems with an appropriate magnetic order that gap the edges
    via a spin-flipping backscattering, the Josephson effect is replaced by a spin
    Josephson effect.
  authors:
  - Eran Sagi
  - Yuval Oreg
  - Ady Stern
  - Bertrand I. Halperin
  doi: 10.1103/PhysRevB.91.245144
  id: '1502.01665'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.91.245144
  on_arxiv: 2015-02-05
  page: '245144'
  published: 2015-06-19
  title: Imprint of topological degeneracy in quasi-one-dimensional fractional quantum
    Hall states
  vol: '91'
- abstract: We report the discovery of Weyl semimetal NbAs featuring topological Fermi
    arc surface states.
  authors:
  - Su-Yang Xu
  - Nasser Alidoust
  - Ilya Belopolski
  - Chenglong Zhang
  - Guang Bian
  - Tay-Rong Chang
  - Hao Zheng
  - Vladimir Strokov
  - Daniel S. Sanchez
  - Guoqing Chang
  - Zhujun Yuan
  - Daixiang Mou
  - Yun Wu
  - Lunan Huang
  - Chi-Cheng Lee
  - Shin-Ming Huang
  - BaoKai Wang
  - Arun Bansil
  - Horng-Tay Jeng
  - Titus Neupert
  - Adam Kaminski
  - Hsin Lin
  - Shuang Jia
  - M. Zahid Hasan
  doi: 10.1038/nphys3437
  id: '1504.01350'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys3437
  on_arxiv: 2015-04-06
  page: '748'
  published: 2015-09-01
  title: Discovery of a Weyl Fermion semimetal state in NbAs
  vol: '11'
- abstract: 'A topological insulator is characterized by a dichotomy between the interior
    and the edge of a finite system: While the bulk has a non-zero energy gap, the
    edges are forced to sustain excitations traversing these gaps. Originally proposed
    for electrons governed by quantum mechanics, it has remained an important open
    question if the same physics can be observed for systems obeying Newton''s equations
    of motion. Here, we report on measurements that characterize the collective behavior
    of mechanical oscillators exhibiting the phenomenology of the quantum spin hall
    effect. The phononic edge modes are shown to be helical and we demonstrate their
    topological protection via the stability against imperfections. Our results open
    the door to the design of topological acoustic meta-materials that can capitalize
    on the stability of the surfaces phonons as reliable wave guides.'
  authors:
  - Roman Süsstrunk
  - Sebastian D. Huber
  doi: 10.1126/science.aab0239
  id: '1503.06808'
  journal: Science
  journal_url: http://dx.doi.org/10.1126/science.aab0239
  on_arxiv: 2015-03-23
  page: '47'
  published: 2015-07-03
  title: Observation of phononic helical edge states in a mechanical 'topological
    insulator'
  vol: '349'
- abstract: We study the effect of electrostatic disorder on the conductivity of a
    three-dimensional antiferromagnetic insulator (a stack of quantum anomalous Hall
    layers with staggered magnetization). The phase diagram contains regions where
    the increase of disorder first causes the appearance of surface conduction (via
    a topological phase transition), followed by the appearance of bulk conduction
    (via a metal-insulator transition). The conducting surface states are stabilized
    by an effective time-reversal symmetry that is broken locally by the disorder
    but restored on long length scales. A simple self-consistent Born approximation
    reliably locates the boundaries of this socalled "statistical" topological phase.
  authors:
  - P. Baireuther
  - J. M. Edge
  - I. C. Fulga
  - C. W. J. Beenakker
  - J. Tworzydło
  doi: 10.1103/PhysRevB.89.035410
  id: '1309.5846'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.89.035410
  on_arxiv: 2013-09-23
  page: '035410'
  published: 2014-01-09
  title: Quantum phase transitions of a disordered antiferromagnetic topological insulator
  vol: '89'
- abstract: Topological band structures in electronic systems like topological insulators
    and semimetals give rise to highly unusual physical properties. Analogous topological
    effects have also been discussed in bosonic systems, but the novel phenomena typically
    occur only when the system is excited by finite-frequency probes. A mapping recently
    proposed by Kane and Lubensky [Nat. Phys. 10, 39 (2014)], however, establishes
    a closer correspondence. It relates the zero-frequency excitations of mechanical
    systems to topological zero modes of fermions that appear at the edges of an otherwise
    gapped system. Here we generalize the mapping to systems with an intrinsically
    gapless bulk. In particular, we construct mechanical counterparts of topological
    semimetals. The resulting gapless bulk modes are physically distinct from the
    usual acoustic Goldstone phonons, and appear even in the absence of continuous
    translation invariance. Moreover, the zero-frequency phonon modes feature adjustable
    momenta and are topologically protected as long as the lattice coordination is
    unchanged. Such protected soft modes with tunable wavevector may be useful in
    designing mechanical structures with fault-tolerant properties.
  authors:
  - Hoi Chun Po
  - Yasaman Bahri
  - Ashvin Vishwanath
  doi: 10.1103/PhysRevB.93.205158
  id: '1410.1320'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.93.205158
  on_arxiv: 2014-10-06
  page: '205158'
  published: 2016-05-31
  title: Phonon analogue of topological nodal semimetals
  vol: '93'
- abstract: 'We show that time-reversal invariant superconductors in d=2 (d=3) dimensions
    can support topologically stable Fermi points (lines), characterized by an integer
    topological charge. Combining this with the momentum space symmetries present,
    we prove analogs of the fermion doubling theorem: for d=2 lattice models admitting
    a spin X electron-hole structure, the number of Fermi points is a multiple of
    four, while for d=3, Fermi lines come in pairs. We show two implications of our
    findings for topological superconductors in d=3: first, we relate the bulk topological
    invariant to a topological number for the surface Fermi points in the form of
    an index theorem. Second, we show that the existence of topologically stable Fermi
    lines results in extended gapless regions in a generic topological superconductor
    phase diagram.'
  authors:
  - B. Béri
  doi: 10.1103/PhysRevB.81.134515
  id: '0909.5680'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.81.134515
  on_arxiv: 2009-09-30
  page: '134515'
  published: 2010-04-14
  title: Topologically stable gapless phases of time-reversal invariant superconductors
  vol: '81'
- abstract: Frames, or lattices consisting of mass points connected by rigid bonds
    or central force springs, are important model constructs that have applications
    in such diverse fields as structural engineering, architecture, and materials
    science. The difference between the number of bonds and the number of degrees
    of freedom of these lattices determines the number of their zero-frequency "floppy
    modes". When these are balanced, the system is on the verge of mechanical instability
    and is termed isostatic. It has recently been shown that certain extended isostatic
    lattices exhibit floppy modes localized at their boundary. These boundary modes
    are insensitive to local perturbations, and appear to have a topological origin,
    reminiscent of the protected electronic boundary modes that occur in the quantum
    Hall effect and in topological insulators. In this paper we establish the connection
    between the topological mechanical modes and the topological band theory of electronic
    systems, and we predict the existence of new topological bulk mechanical phases
    with distinct boundary modes. We introduce model systems in one and two dimensions
    that exemplify this phenomenon.
  authors:
  - C. L. Kane
  - T. C. Lubensky
  doi: 10.1038/nphys2835
  id: '1308.0554'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys2835
  on_arxiv: 2013-08-02
  page: '39'
  published: 2014-01-01
  title: Topological Boundary Modes in Isostatic Lattices
  vol: '10'
- abstract: Networks of rigid bars connected by joints, termed linkages, provide a
    minimal framework to design robotic arms and mechanical metamaterials built out
    of folding components. Here, we investigate a chain-like linkage that, according
    to linear elasticity, behaves like a topological mechanical insulator whose zero-energy
    modes are localized at the edge. Simple experiments we performed using prototypes
    of the chain vividly illustrate how the soft motion, initially localized at the
    edge, can in fact propagate unobstructed all the way to the opposite end. We demonstrate
    using real prototypes, simulations and analytical models that the chain is a mechanical
    conductor, whose carriers are nonlinear solitary waves, not captured within linear
    elasticity. Indeed, the linkage prototype can be regarded as the simplest example
    of a topological metamaterial whose protected mechanical excitations are solitons,
    moving domain walls between distinct topological mechanical phases. More practically,
    we have built a topologically protected mechanism that can perform basic tasks
    such as transporting a mechanical state from one location to another. Our work
    paves the way towards adopting the principle of topological robustness in the
    design of robots assembled from activated linkages as well as in the fabrication
    of complex molecular nanostructures.
  authors:
  - Bryan Gin-ge Chen
  - Nitin Upadhyaya
  - Vincenzo Vitelli
  doi: 10.1073/pnas.1405969111
  id: '1404.2263'
  journal: Proceedings of the National Academy of Sciences
  journal_url: http://dx.doi.org/10.1073/pnas.1405969111
  on_arxiv: 2014-04-08
  page: '13004'
  published: 2014-09-09
  title: Nonlinear conduction via solitons in a topological mechanical insulator
  vol: '111'
- abstract: 'Mechanical metamaterials are artificial structures with unusual properties,
    such as negative Poisson ratio, bistability or tunable vibrational properties,
    that originate in the geometry of their unit cell. At the heart of such unusual
    behaviour is often a soft mode: a motion that does not significantly stretch or
    compress the links between constituent elements. When activated by motors or external
    fields, soft modes become the building blocks of robots and smart materials. Here,
    we demonstrate the existence of topological soft modes that can be positioned
    at desired locations in a metamaterial while being robust against a wide range
    of structural deformations or changes in material parameters. These protected
    modes, localized at dislocations, are the mechanical analogue of topological states
    bound to defects in electronic systems. We create physical realizations of the
    topological modes in prototypes of kagome lattices built out of rigid triangular
    plates. We show mathematically that they originate from the interplay between
    two Berry phases: the Burgers vector of the dislocation and the topological polarization
    of the lattice. Our work paves the way towards engineering topologically protected
    nano-mechanical structures for molecular robotics or information storage and read-out.'
  authors:
  - Jayson Paulose
  - Bryan Gin-ge Chen
  - Vincenzo Vitelli
  doi: 10.1038/nphys3185
  id: '1406.3323'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys3185
  on_arxiv: 2014-06-12
  page: '153'
  published: 2015-02-01
  title: Topological modes bound to dislocations in mechanical metamaterials
  vol: '11'
- abstract: States of self-stress, tensions and compressions of structural elements
    that result in zero net forces, play an important role in determining the load-bearing
    ability of structures ranging from bridges to metamaterials with tunable mechanical
    properties. We exploit a class of recently introduced states of self-stress analogous
    to topological quantum states to sculpt localized buckling regions in the interior
    of periodic cellular metamaterials. Although the topological states of self stress
    arise in the linear response of an idealized mechanical frame of harmonic springs
    connected by freely-hinged joints, they leave a distinct signature in the nonlinear
    buckling behaviour of a cellular material built out of elastic beams with rigid
    joints. The salient feature of these localized buckling regions is that they are
    indistinguishable from their surroundings as far as material parameters or connectivity
    of their constituent elements are concerned. Furthermore, they are robust against
    a wide range of structural perturbations. We demonstrate the effectiveness of
    this topological design through analytical and numerical calculations as well
    as buckling experiments performed on two- and three-dimensional metamaterials
    built out of stacked kagome lattices.
  authors:
  - Jayson Paulose
  - Anne S. Meeussen
  - Vincenzo Vitelli
  doi: 10.1073/pnas.1502939112
  id: '1502.03396'
  journal: Proc Natl Acad Sci USA
  journal_url: http://dx.doi.org/10.1073/pnas.1502939112
  on_arxiv: 2015-02-11
  page: '7639'
  published: 2015-06-23
  title: Selective buckling via states of self-stress in topological metamaterials
  vol: '112'
- abstract: Topological insulators (TIs) are a novel class of materials with nontrivial
    surface or edge states. Time-reversal symmetry (TRS) protected TIs are characterized
    by the Z2 topological invariant and their helical property becomes lost in an
    applied magnetic field. Currently there exist extensive efforts searching for
    TIs that are protected by symmetries other than TRS. Here we show, a topological
    phase characterized by a spin Chern topological invariant is realized in an inverted
    electron-hole bilayer engineered from indium arsenide-gallium antimonide (InAs/GaSb)
    semiconductors which retains robust helical edges under a strong magnetic field.
    Wide conductance plateaus of 2e2/h value are observed; they persist to 12T applied
    in-plane magnetic field without evidence for transition to a trivial insulator.
    In a perpendicular magnetic field up to 8T, there exists no signature to the bulk
    gap closing. While the Fermi energy remains inside the bulk gap, the longitudinal
    conductance increases from 2e2/h in strong magnetic fields suggesting a trend
    towards chiral edge transport. Our findings are first evidences for a quantum
    spin Hall (QSH) insulator protected by a spin Chern invariant. These results demonstrate
    that InAs/GaSb bilayers are a novel system for engineering the robust helical
    edge channels much needed for spintronics and for creating and manipulating Majorana
    particles in solid state.
  authors:
  - Lingjie Du
  - Ivan Knez
  - Gerard Sullivan
  - Rui-Rui Du
  doi: 10.1103/PhysRevLett.114.096802
  id: '1306.1925'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.114.096802
  on_arxiv: 2013-06-08
  page: 096802
  published: 2015-03-04
  title: Observation of Quantum Spin Hall States in InAs/GaSb Bilayers under Broken
    Time-Reversal Symmetry
  vol: '114'
- abstract: We propose and analyze inter-edge tunneling in a quantum spin Hall corner
    junction as a means to probe the helical nature of the edge states. We show that
    electron-electron interactions in the one-dimensional helical edge states result
    in Luttinger parameters for spin and charge that are intertwined, and thus rather
    different than those for a quantum wire with spin rotation invariance. Consequently,
    we find that the four-terminal conductance in a corner junction has a distinctive
    form that could be used as evidence for the helical nature of the edge states.
  authors:
  - Chang-Yu Hou
  - Eun-Ah Kim
  - Claudio Chamon
  doi: 10.1103/PhysRevLett.102.076602
  id: '0808.1723'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.102.076602
  on_arxiv: 2008-08-13
  page: '076602'
  published: 2009-02-17
  title: Corner Junction as a Probe of Helical Edge States
  vol: '102'
- 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.
  authors:
  - Conan Weeks
  - Jun Hu
  - Jason Alicea
  - Marcel Franz
  - Ruqian Wu
  id: '1104.3282'
  on_arxiv: 2011-04-17
  title: Engineering a robust quantum spin Hall state in graphene via adatom deposition
- abstract: Topological insulators are a newly discovered phase of matter characterized
    by a gapped bulk surrounded by novel conducting boundary states. Since their theoretical
    discovery, these materials have encouraged intense efforts to study their properties
    and capabilities. Among the most striking results of this activity are proposals
    to engineer a new variety of superconductor at the surfaces of topological insulators.
    These topological superconductors would be capable of supporting localized Majorana
    fermions, particles whose braiding properties have been proposed as the basis
    of a fault-tolerant quantum computer. Despite the clear theoretical motivation,
    a conclusive realization of topological superconductivity remains an outstanding
    experimental goal. Here we present measurements of superconductivity induced in
    two-dimensional HgTe/HgCdTe quantum wells, a material which becomes a quantum
    spin Hall insulator when the well width exceeds d_{C}=6.3 nm. In wells that are
    7.5 nm wide, we find that supercurrents are confined to the one-dimensional sample
    edges as the bulk density is depleted. However, when the well width is decreased
    to 4.5 nm the edge supercurrents cannot be distinguished from those in the bulk.
    These results provide evidence for superconductivity induced in the helical edges
    of the quantum spin Hall effect, a promising step toward the demonstration of
    one-dimensional topological superconductivity. Our results also provide a direct
    measurement of the widths of these edge channels, which range from 180 nm to 408
    nm.
  authors:
  - Sean Hart
  - Hechen Ren
  - Timo Wagner
  - Philipp Leubner
  - Mathias Mühlbauer
  - Christoph Brüne
  - Hartmut Buhmann
  - Laurens W. Molenkamp
  - Amir Yacoby
  doi: 10.1038/nphys3036
  id: '1312.2559'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys3036
  on_arxiv: 2013-12-09
  page: '638'
  published: 2014-09-01
  title: Induced Superconductivity in the Quantum Spin Hall Edge
  vol: '10'
- abstract: We study the influence of electron puddles created by doping of a 2D topological
    insulator on its helical edge conductance. A single puddle is modeled by a quantum
    dot tunnel-coupled to the helical edge. It may lead to significant inelastic backscattering
    within the edge because of the long electron dwelling time in the dot. We find
    the resulting correction to the perfect edge conductance. Generalizing to multiple
    puddles, we assess the dependence of the helical edge resistance on temperature
    and doping level, and compare it with recent experimental data.
  authors:
  - Jukka I. Väyrynen
  - Moshe Goldstein
  - Leonid I. Glazman
  doi: 10.1103/PhysRevLett.110.216402
  id: '1303.1766'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.110.216402
  on_arxiv: 2013-03-07
  page: '216402'
  published: 2013-05-21
  title: Helical edge resistance introduced by charge puddles
  vol: '110'
- abstract: Certain band insulators allow for the adiabatic pumping of quantized charge
    or spin for special time-dependences of the Hamiltonian. These "topological pumps"
    are closely related to two dimensional topological insulating phases of matter
    upon rolling the insulator up to a cylinder and threading it with a time dependent
    flux. In this article we extend the classification of topological pumps to the
    Wigner Dyson and chiral classes, coupled to multi-channel leads. The topological
    index distinguishing different topological classes is formulated in terms of the
    scattering matrix of the system. We argue that similar to topologically non-trivial
    insulators, topological pumps are characterized by the appearance of protected
    gapless end states during the course of a pumping cycle. We show that this property
    allows for the pumping of quantized charge or spin in the weak coupling limit.
    Our results may also be applied to two dimensional topological insulators, where
    they give a physically transparent interpretation of the topologically non-trivial
    phases in terms of scattering matrices.
  authors:
  - Dganit Meidan
  - Tobias Micklitz
  - Piet W. Brouwer
  doi: 10.1103/PhysRevB.84.195410
  id: '1107.2215'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.84.195410
  on_arxiv: 2011-07-12
  page: '195410'
  published: 2011-11-02
  title: Topological classification of adiabatic processes
  vol: '84'
- 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.
  authors:
  - Liang Fu
  - C. L. Kane
  doi: 10.1103/PhysRevB.79.161408
  id: '0804.4469'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.79.161408
  on_arxiv: 2008-04-28
  page: '161408'
  published: 2009-04-28
  title: Josephson Current and Noise at a Superconductor-Quantum Spin Hall Insulator-Superconductor
    Junction
  vol: '79'
- 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 p_x+ip_y 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 non chiral 1 dimensional wire for Majorana fermions, and that circuits
    formed from these junctions provide a method for creating, manipulating and fusing
    Majorana bound states.
  authors:
  - Liang Fu
  - C. L. Kane
  doi: 10.1103/PhysRevLett.100.096407
  id: '0707.1692'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.100.096407
  on_arxiv: 2007-07-11
  page: 096407
  published: 2008-03-06
  title: Superconducting proximity effect and Majorana fermions at the surface of
    a topological insulator
  vol: '100'
- abstract: Topological insulators are characterized by an insulating bulk with a
    finite band gap and conducting edge or surface states, where charge carriers are
    protected against backscattering. These states give rise to the quantum spin Hall
    effect without an external magnetic field, where electrons with opposite spins
    have opposite momentum at a given edge. The surface energy spectrum of a threedimensional
    topological insulator is made up by an odd number of Dirac cones with the spin
    locked to the momentum. The long-sought yet elusive Majorana fermion is predicted
    to arise from a combination of a superconductor and a topological insulator. An
    essential step in the hunt for this emergent particle is the unequivocal observation
    of supercurrent in a topological phase. Here, we present the first measurement
    of a Josephson supercurrent through a topological insulator. Direct evidence for
    Josephson supercurrents in superconductor (Nb) - topological insulator (Bi2Te3)
    - superconductor e-beam fabricated junctions is provided by the observation of
    clear Shapiro steps under microwave irradiation, and a Fraunhofer-type dependence
    of the critical current on magnetic field. The dependence of the critical current
    on temperature and length shows that the junctions are in the ballistic limit.
    Shubnikov-de Haas oscillations in magnetic fields up to 30 T reveal a topologically
    non-trivial two-dimensional surface state. We argue that the ballistic Josephson
    current is hosted by this surface state despite the fact that the normal state
    transport is dominated by diffusive bulk conductivity. The lateral Nb-Bi2Te3-Nb
    junctions hence provide prospects for the realization of devices supporting Majorana
    fermions.
  authors:
  - M. Veldhorst
  - M. Snelder
  - M. Hoek
  - T. Gang
  - X. L. Wang
  - V. K. Guduru
  - U. Zeitler
  - W. G. v. d. Wiel
  - A. A. Golubov
  - H. Hilgenkamp
  - A. Brinkman
  doi: 10.1038/nmat3255
  id: '1112.3527'
  journal: Nature Mater
  journal_url: http://dx.doi.org/10.1038/nmat3255
  on_arxiv: 2011-12-15
  page: '417'
  published: 2012-05-01
  title: Josephson supercurrent through a topological insulator surface state
  vol: '11'
- abstract: We prove a topological criterion for the existence of zero-energy Majorana
    bound-state on a disclination, a rotation symmetry breaking point defect, in 4-fold
    symmetric topological crystalline superconductors (TCS). We first establish a
    complete topological classification of TCS using the Chern invariant and three
    integral rotation invariants. By analytically and numerically studying disclinations,
    we algebraically deduce a Z_2-index that identifies the parity of the number of
    Majorana zero-modes at a disclination. Surprisingly, we also find weakly-protected
    Majorana fermions bound at the corners of superconductors with trivial Chern and
    weak invariants.
  authors:
  - Jeffrey C. Y. Teo
  - Taylor L. Hughes
  doi: 10.1103/PhysRevLett.111.047006
  id: '1208.6303'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.111.047006
  on_arxiv: 2012-08-30
  page: '047006'
  published: 2013-07-26
  title: Majorana Fermions and Disclinations in Topological Crystalline Superconductors
  vol: '111'
- abstract: Three-dimensional topological insulators are classified into "strong"
    (STI) and "weak" (WTI) according to the nature of their surface states. While
    the surface states of the STI are topologically protected from localization, this
    does not hold for the WTI. In this work we show that the surface states of the
    WTI are actually protected from any random perturbation that does not break time-reversal
    symmetry, and does not close the bulk energy gap. Consequently, the conductivity
    of metallic surfaces in the clean system remains finite even in the presence of
    strong disorder of this type. In the weak disorder limit the surfaces are found
    to be perfect metals, and strong surface disorder only acts to push the metallic
    surfaces inwards. We find that the WTI differs from the STI primarily in its anisotropy,
    and that the anisotropy is not a sign of its weakness but rather of its richness.
  authors:
  - Zohar Ringel
  - Yaacov E. Kraus
  - Ady Stern
  doi: 10.1103/PhysRevB.86.045102
  id: '1105.4351'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.86.045102
  on_arxiv: 2011-05-22
  page: '045102'
  published: 2012-07-02
  title: The strong side of weak topological insulators
  vol: '86'
- abstract: 'The unrelated discoveries of quasicrystals and topological insulators
    have in turn challenged prevailing paradigms in condensed-matter physics. We find
    a surprising connection between quasicrystals and topological phases of matter:
    (i) quasicrystals exhibit nontrivial topological properties and (ii) these properties
    are attributed to dimensions higher than that of the quasicrystal. Specifically,
    we show, both theoretically and experimentally, that one-dimensional quasicrystals
    are assigned two-dimensional Chern numbers and, respectively, exhibit topologically
    protected boundary states equivalent to the edge states of a two-dimensional quantum
    Hall system.We harness the topological nature of these states to adiabatically
    pump light across the quasicrystal. We generalize our results to higher-dimensional
    systems and other topological indices. Hence, quasicrystals offer a new platform
    for the study of topological phases while their topology may better explain their
    surface properties.'
  authors:
  - Yaacov E. Kraus
  - Yoav Lahini
  - Zohar Ringel
  - Mor Verbin
  - Oded Zilberberg
  doi: 10.1103/PhysRevLett.109.106402
  id: '1109.5983'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.109.106402
  on_arxiv: 2011-09-27
  page: '106402'
  published: 2012-09-04
  title: Topological States and Adiabatic Pumping in Quasicrystals
  vol: '109'
- abstract: Electron edge states in graphene in the Quantum Hall effect regime can
    carry both charge and spin. We show that spin splitting of the zeroth Landau level
    gives rise to counterpropagating modes with opposite spin polarization. These
    chiral spin modes lead to a rich variety of spin current states, depending on
    the spin flip rate. A method to control the latter locally is proposed. We estimate
    Zeeman spin splitting enhanced by exchange, and obtain a spin gap of a few hundred
    Kelvin.
  authors:
  - D. A. Abanin
  - P. A. Lee
  - L. S. Levitov
  doi: 10.1103/PhysRevLett.96.176803
  id: cond-mat/0602645
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.96.176803
  on_arxiv: 2006-02-27
  page: '176803'
  published: 2006-05-03
  title: Spin Filtered Edge States and Quantum Hall Effect in Graphene
  vol: '96'
- abstract: In a graphene Landau level (LL), strong Coulomb interactions and the fourfold
    spin/valley degeneracy lead to an approximate SU(4) isospin symmetry. At partial
    filling, exchange interactions can spontaneously break this symmetry, manifesting
    as additional integer quantum Hall plateaus outside the normal sequence. Here
    we report the observation of a large number of these quantum Hall isospin ferromagnetic
    (QHIFM) states, which we classify according to their real spin structure using
    temperature-dependent tilted field magnetotransport. The large measured activation
    gaps confirm the Coulomb origin of the broken symmetry states, but the order is
    strongly dependent on LL index. In the high energy LLs, the Zeeman effect is the
    dominant aligning field, leading to real spin ferromagnets with Skyrmionic excitations
    at half filling, whereas in the `relativistic' zero energy LL, lattice scale anisotropies
    drive the system to a spin unpolarized state, likely a charge- or spin-density
    wave.
  authors:
  - Andrea F. Young
  - Cory R. Dean
  - Lei Wang
  - Hechen Ren
  - Paul Cadden-Zimansky
  - Kenji Watanabe
  - Takashi Taniguchi
  - James Hone
  - Kenneth L. Shepard
  - Philip Kim
  doi: 10.1038/nphys2307
  id: '1201.4167'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys2307
  on_arxiv: 2012-01-19
  page: '550'
  published: 2012-07-01
  title: Spin and valley quantum Hall ferromagnetism in graphene
  vol: '8'
- abstract: We have determined the finite temperature coherence length of edge states
    in the Integer Quantum Hall Effect (IQHE) regime. This was realized by measuring
    the visibility of electronic Mach-Zehnder interferometers of different sizes,
    at filling factor 2. The visibility shows an exponential decay with the temperature.
    The characteristic temperature scale is found inversely proportional to the length
    of the interferometer arm, allowing to define a coherence length $\l_\phi$. The
    variations of $\l_\phi$ with magnetic field are the same for all samples, with
    a maximum located at the upper end of the quantum hall plateau. Our results provide
    the first accurate determination of $\l_\phi$ in the quantum Hall regime.
  authors:
  - P. Roulleau
  - F. Portier
  - P. Roche
  - A. Cavanna
  - G. Faini
  - U. Gennser
  - D. Mailly
  doi: 10.1103/PhysRevLett.100.126802
  id: '0710.2806'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.100.126802
  on_arxiv: 2007-10-15
  page: '126802'
  published: 2008-03-25
  title: Direct measurement of the coherence length of edge states in the Integer
    Quantum Hall Regime
  vol: '100'
- abstract: We investigate the effects that a tilting of the magnetic field from the
    parallel direction has on the states of a 1D Majorana nanowire. Particularly,
    we focus on the conditions for the existence of Majorana zero modes, uncovering
    an analytical relation (the sine rule) between the field orientation relative
    to the wire, its magnitude and the superconducting parameter of the material.
    The study is then extended to junctions of nanowires, treated as magnetically
    inhomogeneous straight nanowires composed of two homogeneous arms. It is shown
    that their spectrum can be explained in terms of the spectra of two independent
    arms. Finally, we investigate how the localization of the Majorana mode is transferred
    from the magnetic interface at the corner of the junction to the end of the nanowire
    when increasing the arm length.
  authors:
  - Javier Osca
  - Daniel Ruiz
  - Llorenç Serra
  doi: 10.1103/PhysRevB.89.245405
  id: '1403.4464'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.89.245405
  on_arxiv: 2014-03-18
  page: '245405'
  published: 2014-06-05
  title: Effects of tilting the magnetic field in 1D Majorana nanowires
  vol: '89'
- abstract: We investigate Josephson currents in mesoscopic rings with a weak link
    which are in or near a topological superconducting phase. As a paradigmatic example,
    we consider the Kitaev model of a spinless p-wave superconductor in one dimension,
    emphasizing how this model emerges from more realistic settings based on semiconductor
    nanowires. We show that the flux periodicity of the Josephson current provides
    signatures of the topological phase transition and the emergence of Majorana fermions
    situated on both sides of the weak link even when fermion parity is not a good
    quantum number. In large rings, the Majorana fermions hybridize only across the
    weak link. In this case, the Josephson current is h/e periodic in the flux threading
    the loop when fermion parity is a good quantum number but reverts to the more
    conventional h/2e periodicity in the presence of fermion-parity changing relaxation
    processes. In mesoscopic rings, the Majorana fermions also hybridize through their
    overlap in the interior of the superconducting ring. We find that in the topological
    superconducting phase, this gives rise to an h/e-periodic contribution even when
    fermion parity is not conserved and that this contribution exhibits a peak near
    the topological phase transition. This signature of the topological phase transition
    is robust to the effects of disorder. As a byproduct, we find that close to the
    topological phase transition, disorder drives the system deeper into the topological
    phase. This is in stark contrast to the known behavior far from the phase transition,
    where disorder tends to suppress the topological phase.
  authors:
  - Falko Pientka
  - Alessandro Romito
  - Mathias Duckheim
  - Yuval Oreg
  - Felix von Oppen
  doi: 10.1088/1367-2630/15/2/025001
  id: '1210.3237'
  journal: New J. Phys.
  journal_url: http://dx.doi.org/10.1088/1367-2630/15/2/025001
  on_arxiv: 2012-10-11
  page: '025001'
  published: 2013-02-01
  title: Signatures of topological phase transitions in mesoscopic superconducting
    rings
  vol: '15'
- abstract: Majorana fermions are particles identical to their own antiparticles.
    They have been theoretically predicted to exist in topological superconductors.
    We report electrical measurements on InSb nanowires contacted with one normal
    (Au) and one superconducting electrode (NbTiN). Gate voltages vary electron density
    and define a tunnel barrier between normal and superconducting contacts. In the
    presence of magnetic fields of order 100 mT we observe bound, mid-gap states at
    zero bias voltage. These bound states remain fixed to zero bias even when magnetic
    fields and gate voltages are changed over considerable ranges. Our observations
    support the hypothesis of Majorana fermions in nanowires coupled to superconductors.
  authors:
  - V. Mourik
  - K. Zuo
  - S. M. Frolov
  - S. R. Plissard
  - E. P. A. M. Bakkers
  - L. P. Kouwenhoven
  doi: 10.1126/science.1222360
  id: '1204.2792'
  journal: Science
  journal_url: http://dx.doi.org/10.1126/science.1222360
  on_arxiv: 2012-04-12
  page: '1003'
  published: 2012-05-25
  title: Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire
    devices
  vol: '336'
- abstract: We calculate the conductance of a ballistic point contact to a superconducting
    wire, produced by the s-wave proximity effect in a semiconductor with spin-orbit
    coupling in a parallel magnetic field. The conductance G as a function of contact
    width or Fermi energy shows plateaus at half-integer multiples of 4e^2/h if the
    superconductor is in a topologically nontrivial phase. In contrast, the plateaus
    are at the usual integer multiples in the topologically trivial phase. Disorder
    destroys all plateaus except the first, which remains precisely quantized, consistent
    with previous results for a tunnel contact. The advantage of a ballistic contact
    over a tunnel contact as a probe of the topological phase is the strongly reduced
    sensitivity to finite voltage or temperature.
  authors:
  - M. Wimmer
  - A. R. Akhmerov
  - J. P. Dahlhaus
  - C. W. J. Beenakker
  doi: 10.1088/1367-2630/13/5/053016
  id: '1101.5795'
  journal: New J. Phys.
  journal_url: http://dx.doi.org/10.1088/1367-2630/13/5/053016
  on_arxiv: 2011-01-30
  page: '053016'
  published: 2011-05-12
  title: Quantum point contact as a probe of a topological superconductor
  vol: '13'
- abstract: Topological quantum computation provides an elegant way around decoherence,
    as one encodes quantum information in a non-local fashion that the environment
    finds difficult to corrupt. Here we establish that one of the key operations---braiding
    of non-Abelian anyons---can be implemented in one-dimensional semiconductor wire
    networks. Previous work [Lutchyn et al., arXiv:1002.4033 and Oreg et al., arXiv:1003.1145]
    provided a recipe for driving semiconducting wires into a topological phase supporting
    long-sought particles known as Majorana fermions that can store topologically
    protected quantum information. Majorana fermions in this setting can be transported,
    created, and fused by applying locally tunable gates to the wire. More importantly,
    we show that networks of such wires allow braiding of Majorana fermions and that
    they exhibit non-Abelian statistics like vortices in a p+ip superconductor. We
    propose experimental setups that enable the Majorana fusion rules to be probed,
    along with networks that allow for efficient exchange of arbitrary numbers of
    Majorana fermions. This work paves a new path forward in topological quantum computation
    that benefits from physical transparency and experimental realism.
  authors:
  - Jason Alicea
  - Yuval Oreg
  - Gil Refael
  - Felix von Oppen
  - Matthew P. A. Fisher
  doi: 10.1038/nphys1915
  id: '1006.4395'
  journal: Nature Phys
  journal_url: http://dx.doi.org/10.1038/nphys1915
  on_arxiv: 2010-06-23
  page: '412'
  published: 2011-05-01
  title: Non-Abelian statistics and topological quantum information processing in
    1D wire networks
  vol: '7'
- abstract: We study multiband semiconducting nanowires proximity-coupled with an
    s-wave superconductor. We show that when odd number of subbands are occupied the
    system realizes non-trivial topological state supporting Majorana modes localized
    at the ends. We study the topological quantum phase transition in this system
    and analytically calculate the phase diagram as a function of the chemical potential
    and magnetic field. Our key finding is that multiband occupancy not only lifts
    the stringent constraint of one-dimensionality but also allows to have higher
    carrier density in the nanowire and as such multisubband nanowires are better-suited
    for observing the Majorana particle. We study the robustness of the topological
    phase by including the effects of the short- and long-range disorder. We show
    that in the limit of strong interband mixing there is an optimal regime in the
    phase diagram ("sweet spot") where the topological state is to a large extent
    insensitive to the presence of disorder.
  authors:
  - Roman M. Lutchyn
  - Tudor Stanescu
  - S. Das Sarma
  doi: 10.1103/PhysRevLett.106.127001
  id: '1008.0629'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.106.127001
  on_arxiv: 2010-08-03
  page: '127001'
  published: 2011-03-21
  title: Search for Majorana fermions in multiband semiconducting nanowires
  vol: '106'
- abstract: We introduce a strong-disorder renormalization group (RG) approach suitable
    for investigating the quasiparticle excitations of disordered superconductors
    in which the quasiparticle spin is not conserved. We analyze one-dimensional models
    with this RG and with elementary transfer matrix methods. We find that such models
    with broken spin rotation invariance {\it generically} lie in one of two topologically
    distinct localized phases. Close enough to the critical point separating the two
    phases, the system has a power-law divergent low-energy density of states (with
    a non-universal continuously varying power-law) in either phase, due to quantum
    Griffiths singularities. This critical point belongs to the same infinite-disorder
    universality class as the one dimensional particle-hole symmetric Anderson localization
    problem, while the Griffiths phases in the vicinity of the transition are controlled
    by lines of strong (but not infinite) disorder fixed points terminating in the
    critical point.
  authors:
  - Olexei Motrunich
  - Kedar Damle
  - David A. Huse
  doi: 10.1103/PhysRevB.63.224204
  id: cond-mat/0011200
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.63.224204
  on_arxiv: 2000-11-11
  page: '224204'
  published: 2001-05-22
  title: 'Griffiths effects and quantum critical points in dirty superconductors without
    spin-rotation invariance: One-dimensional examples'
  vol: '63'
- abstract: We present an effective medium theory that explains the disorder-induced
    transition into a phase of quantized conductance, discovered in computer simulations
    of HgTe quantum wells. It is the combination of a random potential and quadratic
    corrections proportional to p^2 sigma_z to the Dirac Hamiltonian that can drive
    an ordinary band insulator into a topological insulator (having an inverted band
    gap). We calculate the location of the phase boundary at weak disorder and show
    that it corresponds to the crossing of a band edge rather than a mobility edge.
    Our mechanism for the formation of a topological Anderson insulator is generic,
    and would apply as well to three-dimensional semiconductors with strong spin-orbit
    coupling.
  authors:
  - C. W. Groth
  - M. Wimmer
  - A. R. Akhmerov
  - J. Tworzydło
  - C. W. J. Beenakker
  doi: 10.1103/PhysRevLett.103.196805
  id: '0908.0881'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.103.196805
  on_arxiv: 2009-08-06
  page: '196805'
  published: 2009-11-06
  title: Theory of the topological Anderson insulator
  vol: '103'
- abstract: We numerically calculate the conductivity $\sigma$ of an undoped graphene
    sheet (size $L$) in the limit of vanishingly small lattice constant. We demonstrate
    one-parameter scaling for random impurity scattering and determine the scaling
    function $\beta(\sigma)=d\ln\sigma/d\ln L$. Contrary to a recent prediction, the
    scaling flow has no fixed point ($\beta>0$) for conductivities up to and beyond
    the symplectic metal-insulator transition. Instead, the data supports an alternative
    scaling flow for which the conductivity at the Dirac point increases logarithmically
    with sample size in the absence of intervalley scattering -- without reaching
    a scale-invariant limit.
  authors:
  - J. H. Bardarson
  - J. Tworzydło
  - P. W. Brouwer
  - C. W. J. Beenakker
  doi: 10.1103/PhysRevLett.99.106801
  id: '0705.0886'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.99.106801
  on_arxiv: 2007-05-07
  page: '106801'
  published: 2007-09-05
  title: Demonstration of one-parameter scaling at the Dirac point in graphene
  vol: '99'
- abstract: The beta function of a two-dimensional massless Dirac Hamiltonian subject
    to a random scalar potential, which e.g., underlies the theoretical description
    of graphene, is computed numerically. Although it belongs to, from a symmetry
    standpoint, the two-dimensional symplectic class, the beta function monotonically
    increases with decreasing $g$. We also provide an argument based on the spectral
    flows under twisting boundary conditions, which shows that none of states of the
    massless Dirac Hamiltonian can be localized.
  authors:
  - Kentaro Nomura
  - Mikito Koshino
  - Shinsei Ryu
  doi: 10.1103/PhysRevLett.99.146806
  id: '0705.1607'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.99.146806
  on_arxiv: 2007-05-11
  page: '146806'
  published: 2007-10-05
  title: Topological delocalization of two-dimensional massless Dirac fermions
  vol: '99'
- abstract: A field theory of the Anderson transition in two dimensional disordered
    systems with spin-orbit interactions and time-reversal symmetry is developed,
    in which the proliferation of vortex-like topological defects is essential for
    localization. The sign of vortex fugacity determines the $Z_2$ topological class
    of the localized phase. There are two distinct, but equivalent transitions between
    the metallic phase and the two insulating phases. The critical conductivity and
    correlation length exponent of these transitions are computed in a $N=1-\epsilon$
    expansion in the number of replicas, where for small $\epsilon$ the critical points
    are perturbatively connected to the Kosterlitz Thouless critical point. Delocalized
    states, which arise at the surface of weak topological insulators and topological
    crystalline insulators, occur because vortex proliferation is forbidden due to
    the presence of symmetries that are violated by disorder, but are restored by
    disorder averaging.
  authors:
  - Liang Fu
  - C. L. Kane
  doi: 10.1103/PhysRevLett.109.246605
  id: '1208.3442'
  journal: Phys. Rev. Lett.
  journal_url: http://dx.doi.org/10.1103/physrevlett.109.246605
  on_arxiv: 2012-08-16
  page: '246605'
  published: 2012-12-14
  title: Topology, Delocalization via Average Symmetry and the Symplectic Anderson
    Transition
  vol: '109'
- abstract: We present an analytic theory of quantum criticality in quasi one-dimensional
    topological Anderson insulators. We describe these systems in terms of two parameters
    $(g,\chi)$ representing localization and topological properties, respectively.
    Certain critical values of $\chi$ (half-integer for $\Bbb{Z}$ classes, or zero
    for $\Bbb{Z}_2$ classes) define phase boundaries between distinct topological
    sectors. Upon increasing system size, the two parameters exhibit flow similar
    to the celebrated two parameter flow of the integer quantum Hall insulator. However,
    unlike the quantum Hall system, an exact analytical description of the entire
    phase diagram can be given in terms of the transfer-matrix solution of corresponding
    supersymmetric non-linear sigma-models. In $\Bbb{Z}_2$ classes we uncover a hidden
    supersymmetry, present at the quantum critical point.
  authors:
  - Alexander Altland
  - Dmitry Bagrets
  - Alex Kamenev
  doi: 10.1103/PhysRevB.91.085429
  id: '1411.5992'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.91.085429
  on_arxiv: 2014-11-21
  page: 085429
  published: 2015-02-27
  title: 'Topology vs. Anderson localization: non-perturbative solutions in one dimension'
  vol: '91'
- abstract: The search for topologically non-trivial states of matter has become an
    important goal for condensed matter physics. Recently, a new class of topological
    insulators has been proposed. These topological insulators have an insulating
    gap in the bulk, but have topologically protected edge states due to the time
    reversal symmetry. In two dimensions the helical edge states give rise to the
    quantum spin Hall (QSH) effect, in the absence of any external magnetic field.
    Here we review a recent theory which predicts that the QSH state can be realized
    in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum
    well, the band structure changes from a normal to an "inverted" type at a critical
    thickness $d_c$. We present an analytical solution of the helical edge states
    and explicitly demonstrate their topological stability. We also review the recent
    experimental observation of the QSH state in HgTe/(Hg,Cd)Te quantum wells. We
    review both the fabrication of the sample and the experimental setup. For thin
    quantum wells with well width $d_{QW}< 6.3$ nm, the insulating regime shows the
    conventional behavior of vanishingly small conductance at low temperature. However,
    for thicker quantum wells ($d_{QW}> 6.3$ nm), the nominally insulating regime
    shows a plateau of residual conductance close to $2e^2/h$. The residual conductance
    is independent of the sample width, indicating that it is caused by edge states.
    Furthermore, the residual conductance is destroyed by a small external magnetic
    field. The quantum phase transition at the critical thickness, $d_c= 6.3$ nm,
    is also independently determined from the occurrence of a magnetic field induced
    insulator to metal transition.
  authors:
  - Markus Koenig
  - Hartmut Buhmann
  - Laurens W. Molenkamp
  - Taylor L. Hughes
  - Chao-Xing Liu
  - Xiao-Liang Qi
  - Shou-Cheng Zhang
  doi: 10.1143/JPSJ.77.031007
  id: '0801.0901'
  journal: J. Phys. Soc. Jpn.
  journal_url: http://dx.doi.org/10.1143/jpsj.77.031007
  on_arxiv: 2008-01-07
  page: '031007'
  published: 2008-03-15
  title: 'The Quantum Spin Hall Effect: Theory and Experiment'
  vol: '77'
- abstract: Topological insulators are electronic materials that have a bulk band
    gap like an ordinary insulator, but have protected conducting states on their
    edge or surface. The 2D topological insulator is a quantum spin Hall insulator,
    which is a close cousin of the integer quantum Hall state. A 3D topological insulator
    supports novel spin polarized 2D Dirac fermions on its surface. In this Colloquium
    article we will review the theoretical foundation for these electronic states
    and describe recent experiments in which their signatures have been observed.
    We will describe transport experiments on HgCdTe quantum wells that demonstrate
    the existence of the edge states predicted for the quantum spin Hall insulator.
    We will then discuss experiments on Bi_{1-x}Sb_x, Bi_2 Se_3, Bi_2 Te_3 and Sb_2
    Te_3 that establish these materials as 3D topological insulators and directly
    probe the topology of their surface states. We will then describe exotic states
    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. We will close by discussing prospects
    for observing these exotic states, a well as other potential device applications
    of topological insulators.
  authors:
  - M. Z. Hasan
  - C. L. Kane
  doi: 10.1103/RevModPhys.82.3045
  id: '1002.3895'
  journal: Rev. Mod. Phys.
  journal_url: http://dx.doi.org/10.1103/revmodphys.82.3045
  on_arxiv: 2010-02-20
  page: '3045'
  published: 2010-11-08
  title: Topological Insulators
  vol: '82'
- abstract: Topological insulators are new states of quantum matter which can not
    be adiabatically connected to conventional insulators and semiconductors. They
    are characterized by a full insulating gap in the bulk and gapless edge or surface
    states which are protected by time-reversal symmetry. These topological materials
    have been theoretically predicted and experimentally observed in a variety of
    systems, including HgTe quantum wells, BiSb alloys, and Bi$_2$Te$_3$ and Bi$_2$Se$_3$
    crystals. We review theoretical models, materials properties and experimental
    results on two-dimensional and three-dimensional topological insulators, and discuss
    both the topological band theory and the topological field theory. Topological
    superconductors have a full pairing gap in the bulk and gapless surface states
    consisting of Majorana fermions. We review the theory of topological superconductors
    in close analogy to the theory of topological insulators.
  authors:
  - Xiao-Liang Qi
  - Shou-Cheng Zhang
  doi: 10.1103/RevModPhys.83.1057
  id: '1008.2026'
  journal: Rev. Mod. Phys.
  journal_url: http://dx.doi.org/10.1103/revmodphys.83.1057
  on_arxiv: 2010-08-12
  page: '1057'
  published: 2011-10-14
  title: Topological insulators and superconductors
  vol: '83'
- abstract: 'This is a colloquium-style introduction to the midgap excitations in
    superconductors known as Majorana fermions. These elusive particles, equal to
    their own antiparticle, may or may not exist in Nature as elementary building
    blocks, but in condensed matter they can be constructed out of electron and hole
    excitations. What is needed is a superconductor to hide the charge difference,
    and a topological (Berry) phase to eliminate the energy difference from zero-point
    motion. A pair of widely separated Majorana fermions, bound to magnetic or electrostatic
    defects, has non-Abelian exchange statistics. A qubit encoded in this Majorana
    pair is expected to have an unusually long coherence time. We discuss strategies
    to detect Majorana fermions in a topological superconductor, as well as possible
    applications in a quantum computer. The status of the experimental search is reviewed.
    Contents: I. What Are They? (Their origin in particle physics; Their emergence
    in superconductors; Their potential for quantum computing) II. How to Make Them
    (Shockley mechanism; Chiral p-wave superconductors; Topological insulators; Semiconductor
    heterostructures) III. How to Detect Them (Half-integer conductance quantization;
    Nonlocal tunneling; 4\pi-periodic Josephson effect; Thermal metal-insulator transition)
    IV. How to Use Them (Topological qubits; Read out; Braiding) V. Outlook on the
    Experimental Progress [scheduled for vol. 4 of Annual Review of Condensed Matter
    Physics]'
  authors:
  - C. W. J. Beenakker
  doi: 10.1146/annurev-conmatphys-030212-184337
  id: '1112.1950'
  journal: Annu. Rev. Condens. Matter Phys.
  journal_url: http://dx.doi.org/10.1146/annurev-conmatphys-030212-184337
  on_arxiv: 2011-12-08
  page: '113'
  published: 2013-04-01
  title: Search for Majorana fermions in superconductors
  vol: '4'
- abstract: 'The 1937 theoretical discovery of Majorana fermions--whose defining property
    is that they are their own anti-particles--has since impacted diverse problems
    ranging from neutrino physics and dark matter searches to the fractional quantum
    Hall effect and superconductivity. Despite this long history the unambiguous observation
    of Majorana fermions nevertheless remains an outstanding goal. This review article
    highlights recent advances in the condensed matter search for Majorana that have
    led many in the field to believe that this quest may soon bear fruit. We begin
    by introducing in some detail exotic `topological'' one- and two-dimensional superconductors
    that support Majorana fermions at their boundaries and at vortices. We then turn
    to one of the key insights that arose during the past few years; namely, that
    it is possible to `engineer'' such exotic superconductors in the laboratory by
    forming appropriate heterostructures with ordinary s-wave superconductors. Numerous
    proposals of this type are discussed, based on diverse materials such as topological
    insulators, conventional semiconductors, ferromagnetic metals, and many others.
    The all-important question of how one experimentally detects Majorana fermions
    in these setups is then addressed. We focus on three classes of measurements that
    provide smoking-gun Majorana signatures: tunneling, Josephson effects, and interferometry.
    Finally, we discuss the most remarkable properties of condensed matter Majorana
    fermions--the non-Abelian exchange statistics that they generate and their associated
    potential for quantum computation.'
  authors:
  - Jason Alicea
  doi: 10.1088/0034-4885/75/7/076501
  id: '1202.1293'
  journal: Rep. Prog. Phys.
  journal_url: http://dx.doi.org/10.1088/0034-4885/75/7/076501
  on_arxiv: 2012-02-06
  page: '076501'
  published: 2012-07-01
  title: New directions in the pursuit of Majorana fermions in solid state systems
  vol: '75'
- abstract: This short review article provides a pedagogical introduction to the rapidly
    growing research field of Majorana fermions in topological superconductors. We
    first discuss in some details the simplest "toy model" in which Majoranas appear,
    namely a one-dimensional tight-binding representation of a p-wave superconductor,
    introduced more than ten years ago by Kitaev. We then give a general introduction
    to the remarkable properties of Majorana fermions in condensed matter systems,
    such as their intrinsically non-local nature and exotic exchange statistics, and
    explain why these quasiparticles are suspected to be especially well suited for
    low-decoherence quantum information processing. We also discuss the experimentally
    promising (and perhaps already successfully realized) possibility of creating
    topological superconductors using semiconductors with strong spin-orbit coupling,
    proximity-coupled to standard s-wave superconductors and exposed to a magnetic
    field. The goal is to provide an introduction to the subject for experimentalists
    or theorists who are new to the field, focusing on the aspects which are most
    important for understanding the basic physics. The text should be accessible for
    readers with a basic understanding of quantum mechanics and second quantization,
    and does not require knowledge of quantum field theory or topological states of
    matter.
  authors:
  - Martin Leijnse
  - Karsten Flensberg
  doi: 10.1088/0268-1242/27/12/124003
  id: '1206.1736'
  journal: Semicond. Sci. Technol.
  journal_url: http://dx.doi.org/10.1088/0268-1242/27/12/124003
  on_arxiv: 2012-06-08
  page: '124003'
  published: 2012-12-05
  title: Introduction to topological superconductivity and Majorana fermions
  vol: '27'
- abstract: I. Introduction (What is new in RMT, Superconducting quasiparticles, Experimental
    platforms) II. Topological superconductivity (Kitaev chain, Majorana operators,
    Majorana zero-modes, Phase transition beyond mean-field) III. Fundamental symmetries
    (Particle-hole symmetry, Majorana representation, Time-reversal and chiral symmetry)
    IV. Hamiltonian ensembles (The ten-fold way, Midgap spectral peak, Energy level
    repulsion) V. Scattering matrix ensembles (Fundamental symmetries, Chaotic scattering,
    Circular ensembles, Topological quantum numbers) VI. Electrical conduction (Majorana
    nanowire, Counting Majorana zero-modes, Conductance distribution, Weak antilocalization,
    Andreev resonances, Shot noise of Majorana edge modes) VII. Thermal conduction
    (Topological phase transitions, Super-universality, Heat transport by Majorana
    edge modes, Thermopower and time-delay matrix, Andreev billiard with chiral symmetry)
    VIII. Josephson junctions (Fermion parity switches, 4{\pi}-periodic Josephson
    effect, Discrete vortices) IX. Conclusion
  authors:
  - C. W. J. Beenakker
  doi: 10.1103/RevModPhys.87.1037
  id: '1407.2131'
  journal: Rev. Mod. Phys.
  journal_url: http://dx.doi.org/10.1103/revmodphys.87.1037
  on_arxiv: 2014-07-08
  page: '1037'
  published: 2015-09-03
  title: Random-matrix theory of Majorana fermions and topological superconductors
  vol: '87'
- abstract: Topological quantum computation has recently emerged as one of the most
    exciting approaches to constructing a fault-tolerant quantum computer. The proposal
    relies on the existence of topological states of matter whose quasiparticle excitations
    are neither bosons nor fermions, but are particles known as {\it Non-Abelian anyons},
    meaning that they obey {\it non-Abelian braiding statistics}. Quantum information
    is stored in states with multiple quasiparticles, which have a topological degeneracy.
    The unitary gate operations which are necessary for quantum computation are carried
    out by braiding quasiparticles, and then measuring the multi-quasiparticle states.
    The fault-tolerance of a topological quantum computer arises from the non-local
    encoding of the states of the quasiparticles, which makes them immune to errors
    caused by local perturbations. To date, the only such topological states thought
    to have been found in nature are fractional quantum Hall states, most prominently
    the \nu=5/2 state, although several other prospective candidates have been proposed
    in systems as disparate as ultra-cold atoms in optical lattices and thin film
    superconductors. In this review article, we describe current research in this
    field, focusing on the general theoretical concepts of non-Abelian statistics
    as it relates to topological quantum computation, on understanding non-Abelian
    quantum Hall states, on proposed experiments to detect non-Abelian anyons, and
    on proposed architectures for a topological quantum computer. We address both
    the mathematical underpinnings of topological quantum computation and the physics
    of the subject using the \nu=5/2 fractional quantum Hall state as the archetype
    of a non-Abelian topological state enabling fault-tolerant quantum computation.
  authors:
  - Chetan Nayak
  - Steven H. Simon
  - Ady Stern
  - Michael Freedman
  - Sankar Das Sarma
  doi: 10.1103/RevModPhys.80.1083
  id: '0707.1889'
  journal: Rev. Mod. Phys.
  journal_url: http://dx.doi.org/10.1103/revmodphys.80.1083
  on_arxiv: 2007-07-12
  page: '1083'
  published: 2008-09-12
  title: Non-Abelian Anyons and Topological Quantum Computation
  vol: '80'
- abstract: The dichotomy between fermions and bosons is at the root of many physical
    phenomena, from metallic conduction of electricity to super-fluidity, and from
    the periodic table to coherent propagation of light. The dichotomy originates
    from the symmetry of the quantum mechanical wave function to the interchange of
    two identical particles. In systems that are confined to two spatial dimensions
    particles that are neither fermions nor bosons, coined "anyons", may exist. The
    fractional quantum Hall effect offers an experimental system where this possibility
    is realized. In this paper we present the concept of anyons, we explain why the
    observation of the fractional quantum Hall effect almost forces the notion of
    anyons upon us, and we review several possible ways for a direct observation of
    the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian
    anyons, motivating their existence from the point of view of trial wave functions,
    giving a simple exposition of their relation to conformal field theories, and
    reviewing several proposals for their direct observation.
  authors:
  - Ady Stern
  doi: 10.1016/j.aop.2007.10.008
  id: '0711.4697'
  journal: Annals of Physics
  journal_url: http://dx.doi.org/10.1016/j.aop.2007.10.008
  on_arxiv: 2007-11-29
  page: '204'
  published: 2008-01-01
  title: Anyons and the quantum Hall effect - a pedagogical review
  vol: '323'
- abstract: Contribution to the 44th IFF Spring School held at the Forschungszentrum
    J\"ulich in 2013 on "Quantum Information Processing". The notes include a pedagogic
    (but incomplete) introduction to Majorana fermions; especially paying attention
    to the usefulness for quantum information purposes. It contains an introduction
    to topological aspects of superconductors, braiding, physical implementations,
    and measurement of the quantum information using superconducting qubits.
  authors:
  - Fabian Hassler
  id: '1404.0897'
  on_arxiv: 2014-04-03
  title: Majorana Qubits
- abstract: Topological insulators represent unique phases of matter with insulating
    bulk and conducting edge or surface states, immune to small perturbations such
    as backscattering due to disorder. This stems from their peculiar band structure,
    which provides topological protections. While conventional tools (pressure, doping
    etc.) to modify the band structure are available, time periodic perturbations
    can provide tunability by adding time as an extra dimension enhanced to the problem.
    In this short review, we outline the recent research on topological insulators
    in non equilibrium situations. Firstly, we introduce briefly the Floquet formalism
    that allows to describe steady states of the electronic system with an effective
    time-independent Hamiltonian. Secondly, we summarize recent theoretical work on
    how light irradiation drives semi-metallic graphene or a trivial semiconducting
    system into a topological phase. Finally, we show how photons can be used to probe
    topological edge or surface states.
  authors:
  - Jérôme Cayssol
  - Balázs Dóra
  - Ferenc Simon
  - Roderich Moessner
  doi: 10.1002/pssr.201206451
  id: '1211.5623'
  journal: Phys. Status Solidi RRL
  journal_url: http://dx.doi.org/10.1002/pssr.201206451
  on_arxiv: 2012-11-23
  page: '101'
  published: 2013-02-01
  title: Floquet topological insulators
  vol: '7'
- abstract: In this review, we discuss recent progress in the explorations of topological
    materials beyond topological insulators; specifically, we focus on topological
    crystalline insulators and bulk topological superconductors. The basic concepts,
    model Hamiltonians, and novel electronic properties of these new topological materials
    are explained. The key role of symmetries that underlie their topological properties
    is elucidated. Key issues in their materials realizations are also discussed.
  authors:
  - Yoichi Ando
  - Liang Fu
  doi: 10.1146/annurev-conmatphys-031214-014501
  id: '1501.00531'
  journal: Annu. Rev. Condens. Matter Phys.
  journal_url: http://dx.doi.org/10.1146/annurev-conmatphys-031214-014501
  on_arxiv: 2015-01-03
  page: '361'
  published: 2015-03-01
  title: 'Topological Crystalline Insulators and Topological Superconductors: From
    Concepts to Materials'
  vol: '6'
- abstract: Topological properties of physical systems can lead to robust behaviors
    that are insensitive to microscopic details. Such topologically robust phenomena
    are not limited to static systems but can also appear in driven quantum systems.
    In this paper, we show that the Floquet operators of periodically driven systems
    can be divided into topologically distinct (homotopy) classes, and give a simple
    physical interpretation of this classification in terms of the spectra of Floquet
    operators. Using this picture, we provide an intuitive understanding of the well-known
    phenomenon of quantized adiabatic pumping. Systems whose Floquet operators belong
    to the trivial class simulate the dynamics generated by time-independent Hamiltonians,
    which can be topologically classified according to the schemes developed for static
    systems. We demonstrate these principles through an example of a periodically
    driven two--dimensional hexagonal lattice model which exhibits several topological
    phases. Remarkably, one of these phases supports chiral edge modes even though
    the bulk is topologically trivial.
  authors:
  - Takuya Kitagawa
  - Erez Berg
  - Mark Rudner
  - Eugene Demler
  doi: 10.1103/PhysRevB.82.235114
  id: '1010.6126'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.82.235114
  on_arxiv: 2010-10-29
  page: '235114'
  published: 2010-12-10
  title: Topological characterization of periodically-driven quantum systems
  vol: '82'
- abstract: Recently, several authors have investigated topological phenomena in periodically-driven
    systems of non-interacting particles. These phenomena are identified through analogies
    between the Floquet spectra of driven systems and the band structures of static
    Hamiltonians. Intriguingly, these works have revealed that the topological characterization
    of driven systems is richer than that of static systems. In particular, in driven
    systems in two dimensions (2D), robust chiral edge states can appear even though
    the Chern numbers of all the bulk Floquet bands are zero. Here we elucidate the
    crucial distinctions between static and driven 2D systems, and construct a new
    topological invariant that yields the correct edge state structure in the driven
    case. We provide formulations in both the time and frequency domains, which afford
    additional insight into the origins of the "anomalous" spectra which arise in
    driven systems. Possible realizations of these phenomena in solid state and cold
    atomic systems are discussed.
  authors:
  - Mark S. Rudner
  - Netanel H. Lindner
  - Erez Berg
  - Michael Levin
  id: '1212.3324'
  on_arxiv: 2012-12-13
  title: Anomalous edge states and the bulk-edge correspondence for periodically-driven
    two dimensional systems
- abstract: Topological crystalline insulators are new states of matter in which the
    topological nature of electronic structures arises from crystal symmetries. Here
    we predict the first material realization of topological crystalline insulator
    in the semiconductor SnTe, by identifying its nonzero topological index. We predict
    that as a manifestation of this nontrivial topology, SnTe has metallic surface
    states with an even number of Dirac cones on high-symmetry crystal surfaces such
    as {001}, {110} and {111}. These surface states form a new type of high-mobility
    chiral electron gas, which is robust against disorder and topologically protected
    by reflection symmetry of the crystal with respect to {110} mirror plane. Breaking
    this mirror symmetry via elastic strain engineering or applying an in-plane magnetic
    field can open up a continuously tunable band gap on the surface, which may lead
    to wide-ranging applications in thermoelectrics, infrared detection, and tunable
    electronics. Closely related semiconductors PbTe and PbSe also become topological
    crystalline insulators after band inversion by pressure, strain and alloying.
  authors:
  - Timothy H. Hsieh
  - Hsin Lin
  - Junwei Liu
  - Wenhui Duan
  - Arun Bansil
  - Liang Fu
  doi: 10.1038/ncomms1969
  id: '1202.1003'
  journal: Nat Commun
  journal_url: http://dx.doi.org/10.1038/ncomms1969
  on_arxiv: 2012-02-05
  page: '982'
  published: 2012-01-01
  title: Topological Crystalline Insulators in the SnTe Material Class
  vol: '3'
- abstract: We define a class of insulators with gapless surface states protected
    from localization due to the statistical properties of a disordered ensemble,
    namely due to the ensemble's invariance under a certain symmetry. We show that
    these insulators are topological, and are protected by a $\mathbb{Z}_2$ invariant.
    Finally, we prove that every topological insulator gives rise to an infinite number
    of classes of statistical topological insulators in higher dimensions. Our conclusions
    are confirmed by numerical simulations.
  authors:
  - I. C. Fulga
  - B. van Heck
  - J. M. Edge
  - A. R. Akhmerov
  doi: 10.1103/PhysRevB.89.155424
  id: '1212.6191'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.89.155424
  on_arxiv: 2012-12-26
  page: '155424'
  published: 2014-04-21
  title: Statistical Topological Insulators
  vol: '89'
- 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.
  authors:
  - Jeffrey C. Y. Teo
  - C. L. Kane
  doi: 10.1103/PhysRevB.82.115120
  id: '1006.0690'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.82.115120
  on_arxiv: 2010-06-03
  page: '115120'
  published: 2010-09-22
  title: Topological Defects and Gapless Modes in Insulators and Superconductors
  vol: '82'
- abstract: We discover novel topological pumps in the Josephson effects for superconductors.
    The phase difference, which is odd under the chiral symmetry defined by the product
    of time-reversal and particle-hole symmetries, acts as an anomalous adiabatic
    parameter. These pumping cycles are different from those in the "periodic table",
    and are characterized by $Z\times Z$ or $Z_2\times Z_2$ strong invariants. We
    determine the general classifications in class AIII, and those in class DIII with
    a single anomalous parameter. For the $Z_2\times Z_2$ topological pump in class
    DIII, one $Z_2$ invariant describes the coincidence of fermion parity and spin
    pumps whereas the other one reflects the non-Abelian statistics of Majorana Kramers
    pairs, leading to three distinct fractional Josephson effects.
  authors:
  - Fan Zhang
  - C. L. Kane
  doi: 10.1103/PhysRevB.90.020501
  id: '1310.5281'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.90.020501
  on_arxiv: 2013-10-20
  page: '020501'
  published: 2014-07-08
  title: Anomalous Topological Pumps and Fractional Josephson Effects
  vol: '90'
- abstract: We apply ideas from $C^*$-algebra to the study of disordered topological
    insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian,
    describing band projected coordinate matrices. By considering topological obstructions
    to approximating these matrices by exactly commuting matrices, we are able to
    compute invariants quantifying different topological phases. We generalize previous
    two dimensional results to higher dimensions; we give a general expression for
    the topological invariants for arbitrary dimension and several symmetry classes,
    including chiral symmetry classes, and we present a detailed $K$-theory treatment
    of this expression for time reversal invariant three dimensional systems. We can
    use these results to show non-existence of localized Wannier functions for these
    systems. We use this approach to calculate the index for time-reversal invariant
    systems with spin-orbit scattering in three dimensions, on sizes up to $12^3$,
    averaging over a large number of samples. The results show an interesting separation
    between the localization transition and the point at which the average index (which
    can be viewed as an "order parameter" for the topological insulator) begins to
    fluctuate from sample too sample, implying the existence of an unsuspected quantum
    phase transition separating two different delocalized phases in this system. One
    of the particular advantages of the $C^*$-algebraic technique that we present
    is that it is significantly faster in practice than other methods of computing
    the index, allowing the study of larger systems. In this paper, we present a detailed
    discussion of numerical implementation of our method.
  authors:
  - M. B. Hastings
  - T. A. Loring
  doi: 10.1016/j.aop.2010.12.013
  id: '1012.1019'
  journal: Annals of Physics
  journal_url: http://dx.doi.org/10.1016/j.aop.2010.12.013
  on_arxiv: 2010-12-05
  page: '1699'
  published: 2011-07-01
  title: 'Topological Insulators and C^*-Algebras: Theory and Numerical Practice'
  vol: '326'
- abstract: The topological invariant of a topological insulator (or superconductor)
    is given by the number of symmetry-protected edge states present at the Fermi
    level. Despite this fact, established expressions for the topological invariant
    require knowledge of all states below the Fermi energy. Here, we propose a way
    to calculate the topological invariant employing solely its scattering matrix
    at the Fermi level without knowledge of the full spectrum. Since the approach
    based on scattering matrices requires much less information than the Hamiltonian-based
    approaches (surface versus bulk), it is numerically more efficient. In particular,
    is better-suited for studying disordered systems. Moreover, it directly connects
    the topological invariant to transport properties potentially providing a new
    way to probe topological phases.
  authors:
  - I. C. Fulga
  - F. Hassler
  - A. R. Akhmerov
  doi: 10.1103/PhysRevB.85.165409
  id: '1106.6351'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.85.165409
  on_arxiv: 2011-06-30
  page: '165409'
  published: 2012-04-05
  title: Scattering theory of topological insulators and superconductors
  vol: '85'
- abstract: A number of recent articles have reported the existence of topologically
    non-trivial states and associated end states in one-dimensional incommensurate
    lattice models that would usually only be expected in higher dimensions. Using
    an explicit construction, we here argue that the end states have precisely the
    same origin as their counterparts in commensurate models and that incommensurability
    does not in fact provide a meaningful connection to the topological classification
    of systems in higher dimensions. In particular, we show that it is possible to
    smoothly interpolate between states with commensurate and incommensurate modulation
    parameters without closing the band gap and without states crossing the band gap.
  authors:
  - Kevin A. Madsen
  - Emil J. Bergholtz
  - Piet W. Brouwer
  doi: 10.1103/PhysRevB.88.125118
  id: '1307.2577'
  journal: Phys. Rev. B
  journal_url: http://dx.doi.org/10.1103/physrevb.88.125118
  on_arxiv: 2013-07-09
  page: '125118'
  published: 2013-09-11
  title: Topological equivalence of crystal and quasicrystal band structures
  vol: '88'
- abstract: The quasi-bound states of a superconducting quantum dot that is weakly
    coupled to a normal metal appear as resonances in the Andreev reflection probability,
    measured via the differential conductance. We study the evolution of these Andreev
    resonances when an external parameter (such as magnetic field or gate voltage)
    is varied, using a random-matrix model for the $N\times N$ scattering matrix.
    We contrast the two ensembles with broken time-reversal symmetry, in the presence
    or absence of spin-rotation symmetry (class C or D). The poles of the scattering
    matrix in the complex plane, encoding the center and width of the resonance, are
    repelled from the imaginary axis in class C. In class D, in contrast, a number
    $\propto\sqrt{N}$ of the poles has zero real part. The corresponding Andreev resonances
    are pinned to the middle of the gap and produce a zero-bias conductance peak that
    does not split over a range of parameter values (Y-shaped profile), unlike the
    usual conductance peaks that merge and then immediately split (X-shaped profile).
  authors:
  - Shuo Mi
  - D. I. Pikulin
  - M. Marciani
  - C. W. J. Beenakker
  doi: 10.1134/S1063776114120176
  id: '1405.6896'
  journal: J. Exp. Theor. Phys.
  journal_url: http://dx.doi.org/10.1134/s1063776114120176
  on_arxiv: 2014-05-27
  page: '1018'
  published: 2014-12-01
  title: X-shaped and Y-shaped Andreev resonance profiles in a superconducting quantum
    dot
  vol: '119'