Time Reversal Invariant Ζ 2 Topological Insulator

Size: px
Start display at page:

Download "Time Reversal Invariant Ζ 2 Topological Insulator"

Transcription

1 Time Reversal Invariant Ζ Topological Insulator D Bloch Hamiltonians subject to the T constraint 1 ( ) ΘH Θ = H( ) with Θ = 1 are classified by a Ζ topological invariant (ν =,1) Understand via Bul-Boundary correspondence : dge States for <<π/a ν= : Conventional Insulator ν=1 : Topological Insulator Kramers degenerate at time reversal invariant momenta * = * + G *= *=π/a *= *=π/a ven number of bands crossing Fermi energy Odd number of bands crossing Fermi energy

2 3D Topological Insulators There are 4 surface Dirac Points due to Kramers degeneracy y Λ 4 Λ 3 x Λ 1 Λ D Dirac Point Surface Brillouin Zone ν = : Wea Topological Insulator =Λ a =Λ b OR =Λ a =Λ b How do the Dirac points connect? Determined by 4 bul Z topological invariants ν ; (ν 1 ν ν 3 ) y Related to layered D QSHI ; (ν 1 ν ν 3 ) ~ Miller indices Fermi surface encloses even number of Dirac points x ν = 1 : Strong Topological Insulator Fermi circle encloses odd number of Dirac points Topological Metal : 1/4 graphene Berry s phase π Robust to disorder: impossible to localize y x F

3 Bi 1-x Sb x Theory: Predict Bi 1-x Sb x is a topological insulator by exploiting inversion symmetry of pure Bi, Sb (Fu,Kane PRL 7) xperiment: ARPS (Hsieh et al. Nature 8) Bi 1-x Sb x is a Strong Topological Insulator ν ;(ν 1,ν,ν 3 ) = 1;(111) 5 surface state bands cross F between Γ and M Bi Se 3 ARPS xperiment : Y. Xia et al., Nature Phys. (9). Band Theory : H. Zhang et. al, Nature Phys. (9). ν;(ν1,ν,ν3) = 1;() : Band inversion at Γ nergy gap: ~.3 ev : A room temperature topological insulator Control F on surface by exposing to NO Simple surface state structure : Similar to graphene, except only a single Dirac point F

4 Unique Properties of Topological Insulator Surface States Half an ordinary DG ; ¼ Graphene Spin polarized Fermi surface Charge Current ~ Spin Density Spin Current ~ Charge Density F π Berry s phase Robust to disorder Wea Antilocalization Impossible to localize, Klein paradox xotic States when broen symmetry leads to surface energy gap: Quantum Hall state, topological magnetoelectric effect Fu, Kane 7; Qi, Hughes, Zhang 8, ssin, Moore, Vanderbilt 9 Superconducting state Fu, Kane 8

5 Orbital QH : Surface Quantum Hall ffect = Landau Level for Dirac fermions. Fractional IQH σ xy e = n + h 1 e σ xy = h e σ xy = h B ν=1 chiral edge state Anomalous QH : Induce a surface gap by depositing magnetic material H = ψ ( iv σ µ + Mσz ) ψ e e F Mass due to xchange field e σ xy = sgn( M ) h + h M M TI h gap = M Chiral dge State at Domain Wall : M M

6 Topological Superconductors, Majorana Fermions and Topological Quantum Computation 1. Bogoliubov de Gennes Theory. Majorana bound states, Kitaev model 3. Topological superconductor 4. Periodic Table of topological insulators and superconductors 5. Topological quantum computation 6. Proximity effect devices

7 BCS Theory of Superconductivity mean field theory : * Ψ ΨΨ Ψ Ψ Ψ ΨΨ = ΨΨ ( ) H H BdG Ψ Ψ = Ψ Ψ Bogoliubov de Gennes Hamiltonian H BdG H = * H Intrinsic anti-unitary particle hole symmetry 1 ΞHBdGΞ = Ξ =+1 Particle hole redundancy H BdG ϕ =Ξϕ = Ξ ϕ= τϕ* x = τ x same state 1 = 1 ΞH Ξ = 1 ( ) H ( ) Bloch - BdG Hamiltonians satisfy BdG BdG Topological classification problem similar to time reversal symmetry

8 1D Ζ Topological Superconductor : ν =,1 Bul-Boundary correspondence : Discrete end state spectrum ND (Kitaev, ) ν= trivial ν=1 topological - Γ Γ =Γ = Zero mode Γ =Γ = = Majorana fermion bound state Majorana Fermion : Particle = Antiparticle = Real part of a Dirac fermion : =Ψ+Ψ ; Ψ= + i 1 1 i( ) ; i = Ψ Ψ Ψ = { } 1 = 1 i, = δ i j ij Half a state Two Majorana fermions define a single two level system occupied H iε ε = 1 = ΨΨ ε empty

9 Kitaev Model for 1D p wave superconductor H µ N= tcc ( + c c) µ cc+ ( cc + c c) i i i+ 1 i+ 1 i i i i i+ 1 i+ 1 i c = ( c c ) HBdG ( ) c H ( ) = τ (tcos µ ) + τ sin = d( ) τ BdG z x µ >t : Strong pairing phase trivial superconductor d x t d() d z + µ <t : Wea pairing phase topological superconductor d x d() d z Similar to SSH model, except different symmetry : ( d, d, d x y z) = ( d x, d y, d z)

10 Majorana Chain c + i i 1i i µ cc iµ i i 1i i ( i i+ 1+ i+ 1 i) ( 1 ii+ 1 i1i+ 1) ( cc i i+ 1 ci+ 1ci) i ( 1 ii+ 1 i1 i+ 1) t c c c c it + + H = i t + t 1 1i i i 1i + 1 i For =t : nearest neighbor Majorana chain t = µ, t = t 1 1i t 1 t i t 1 >t trivial SC t 1 <t topological SC Unpaired Majorana Fermion at end

11 D Ζ topological superconductor (broen T symmetry) Bul-Boundary correspondence: n = # Chiral Majorana Fermion edge states = xamples Spinless p x +ip y superconductor (n=1) Chiral triplet p wave superconductor (eg Sr RuO 4 ) (n=) T-SC m µ Read Green model : H= cc + ( ( ) cc + cc..) ( ) = ( x + iy) H ( ) = τ t cos + cos µ + τ sin + τ sin = d( ) τ Lattice BdG model : BdG ( z x y ) ( x x y y ) µ >4t : Strong pairing phase trivial superconductor d y d x d() d z Chern number d x µ <4t : Wea pairing phase topological superconductor d y d() d z Chern number 1

12 Majorana zero mode at a vortex Φ= h p e Hole in a topological superconductor threaded by flux Boundary condition on fermion wavefunction p 1 ( ) ( 1) + () ψ L = L A π m ψ ( x) e iq x ; q = ( m+ 1+ p) ψ m L p even p odd π v ε ~ L q q zero mode Without the hole : Caroli, de Gennes, Matricon theory ( 64) ε ~ F

13 Majorana Fermions and Topological Quantum Computing The degenerate states associated with Majorana zero modes define a topologically protected quantum memory (Kitaev 3) Majorana separated bound states = 1 fermion - degenerate states (full/empty) = 1 qubit N separated Majoranas = N qubits Quantum Information is stored non locally - Immune from local decoherence Braiding performs unitary operations Non-Abelian statistics Ψ= + i 1 Measure ( ) 11 / Interchange rule (Ivanov 3) i j j i t Braid These operations, however, are not sufficient to mae a universal quantum computer Create 134

14 Potential condensed matter hosts for Majorana bound states Quasiparticles in fractional Quantum Hall effect at ν=5/ Moore Read 91 Unconventional superconductors - Sr RuO 4 Das Sarma, Naya, Tewari 6 - Fermionic atoms near feshbach resonance Gurarie 5 Proximity ffect Devices using ordinary s wave superconductors - Topological Insulator devices Fu, Kane 8 - Semiconductor/Magnet devices Sau, Lutchyn, Tewari, Das Sarma 9, Lee 9, among others Current Status : Not Observed

15 H Superconducting Proximity ffect = ψ ( iv σ µψ ) + S ψψ + ψψ * S proximity induced superconductivity at surface s wave superconductor Topological insulator Half an ordinary superconductor Similar to spinless p x +ip y superconductor, except : - Does not violate time reversal symmetry - s-wave singlet superconductivity - Required minus sign is provided by π Berry s phase due to Dirac Point Nontrivial ground state supports Majorana fermion bound states at vortices - Dirac point

16 Majorana Bound States on Topological Insulators 1. h/e vortex in D superconducting state SC TI h/e Quasiparticle Bound state at = Majorana Fermion Half a State = =. Superconductor-magnet interface at edge of D QSHI M S.C. QSHI m< m = S M m> gap = m Domain wall bound state

17 1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface M SC x TI = : Half a 1D chiral Dirac fermion. S-TI-S Josephson Junction H = i v F x SC φ SC φ = π φ π TI Gapless non-chiral Majorana fermion for phase difference φ = π H = i + i v ( ) F L xl R xr cos( φ/ ) L R

18 Manipulation of Majorana Fermions Control phases of S-TI-S Junctions Tri-Junction : A storage register for Majoranas φ φ 1 + Majorana present Create A pair of Majorana bound states can be created from the vacuum in a well defined state >. Braid A single Majorana can be moved between junctions. Allows braiding of multiple Majoranas Measure Fuse a pair of Majoranas. States,1> distinguished by presence of quasiparticle. supercurrent across line junction 1 φ π φ π φ π

19 Another route to the D p+ip superconductor Semiconductor - Magnet - Superconductor structure Superconductor Semiconductor Magnetic Insulator Single Fermi circle with Berry phase π ε Topological superconductor with Majorana edge states and Majorana bound states at vortices. Sau, Lutchyn, Tewari, Das Sarma 9 Rashba split DG bands F Zeeman splitting Variants : - use applied magnetic field to lift Kramers degeneracy (Alicea 1) - Use 1D quantum wire (eg InAs ). A route to 1D p wave superconductor with Majorana end states. (Oreg, von Oppen, Alicea, Fisher 1 Challenge : requres very low electron density high purity.

20 Periodic Table of Topological Insulators and Superconductors Anti-Unitary Symmetries : - Time Reversal : - Particle - Hole : Unitary (chiral) symmetry : ΘH Θ =+ Θ =± 1 ( ) H( ) ; 1 ΞH Ξ = Ξ =± 1 ( ) H( ) ; 1 1 ΠH( ) Π = H( ) ; Π ΘΞ 8 antiunitary symmetry classes Altland- Zirnbauer Random Matrix Classes Complex K-theory Real K-theory Kitaev, 8 Schnyder, Ryu, Furusai, Ludwig 8 Bott Periodicity d d+8

Topological Insulators

Topological Insulators Topological Insulators Aira Furusai (Condensed Matter Theory Lab.) = topological insulators (3d and 2d) Outline Introduction: band theory Example of topological insulators: integer quantum Hall effect

More information

Multichannel Majorana Wires

Multichannel Majorana Wires Multichannel Majorana Wires Piet Brouwer Frascati, 2014 Dahlem Center for Complex Quantum Systems Physics Department Inanc Adagideli Freie Universität Berlin Mathias Duckheim Dganit Meidan Graham Kells

More information

Quantum dots and Majorana Fermions Karsten Flensberg

Quantum dots and Majorana Fermions Karsten Flensberg Quantum dots and Majorana Fermions Karsten Flensberg Center for Quantum Devices University of Copenhagen Collaborator: Martin Leijnse and R. Egger M. Kjærgaard K. Wölms Outline: - Introduction to Majorana

More information

A Short Introduction to Topological Superconductors

A Short Introduction to Topological Superconductors A Short Introduction to Topological Superconductors --- A Glimpse of Topological Phases of Matter Jyong-Hao Chen Condensed Matter Theory, PSI & Institute for Theoretical Physics, ETHZ Dec. 09, 2015 @ Superconductivity

More information

Lecture notes on topological insulators

Lecture notes on topological insulators Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan Dated: May 8, 07 I. D p-wave SUPERCONDUCTOR Here we study p-wave SC in D

More information

Topological invariants for 1-dimensional superconductors

Topological invariants for 1-dimensional superconductors Topological invariants for 1-dimensional superconductors Eddy Ardonne Jan Budich 1306.4459 1308.soon SPORE 13 2013-07-31 Intro: Transverse field Ising model H TFI = L 1 i=0 hσ z i + σ x i σ x i+1 σ s:

More information

Crystalline Symmetry and Topology. YITP, Kyoto University Masatoshi Sato

Crystalline Symmetry and Topology. YITP, Kyoto University Masatoshi Sato Crystalline Symmetry and Topology YITP, Kyoto University Masatoshi Sato In collaboration with Ken Shiozaki (YITP) Kiyonori Gomi (Shinshu University) Nobuyuki Okuma (YITP) Ai Yamakage (Nagoya University)

More information

Manipulation of Majorana fermions via single charge control

Manipulation of Majorana fermions via single charge control Manipulation of Majorana fermions via single charge control Karsten Flensberg Niels Bohr Institute University of Copenhagen Superconducting hybrids: from conventional to exotic, Villard de Lans, France,

More information

Surface Majorana Fermions in Topological Superconductors. ISSP, Univ. of Tokyo. Nagoya University Masatoshi Sato

Surface Majorana Fermions in Topological Superconductors. ISSP, Univ. of Tokyo. Nagoya University Masatoshi Sato Surface Majorana Fermions in Topological Superconductors ISSP, Univ. of Tokyo Nagoya University Masatoshi Sato Kyoto Tokyo Nagoya In collaboration with Satoshi Fujimoto (Kyoto University) Yoshiro Takahashi

More information

Symmetric Surfaces of Topological Superconductor

Symmetric Surfaces of Topological Superconductor Symmetric Surfaces of Topological Superconductor Sharmistha Sahoo Zhao Zhang Jeffrey Teo Outline Introduction Brief description of time reversal symmetric topological superconductor. Coupled wire model

More information

Exotic Phenomena in Topological Insulators and Superconductors

Exotic Phenomena in Topological Insulators and Superconductors SPICE Workshop on Spin Dynamics in the Dirac System Schloss Waldthausen, Mainz, 6 June 2017 Exotic Phenomena in Topological Insulators and Superconductors Yoichi Ando Physics Institute II, University of

More information

Classification theory of topological insulators with Clifford algebras and its application to interacting fermions. Takahiro Morimoto.

Classification theory of topological insulators with Clifford algebras and its application to interacting fermions. Takahiro Morimoto. QMath13, 10 th October 2016 Classification theory of topological insulators with Clifford algebras and its application to interacting fermions Takahiro Morimoto UC Berkeley Collaborators Akira Furusaki

More information

Topological Superconductivity and Superfluidity

Topological Superconductivity and Superfluidity Topological Superconductivity and Superfluidity SLAC-PUB-13926 Xiao-Liang Qi, Taylor L. Hughes, Srinivas Raghu and Shou-Cheng Zhang Department of Physics, McCullough Building, Stanford University, Stanford,

More information

Topological Defects inside a Topological Band Insulator

Topological Defects inside a Topological Band Insulator Topological Defects inside a Topological Band Insulator Ashvin Vishwanath UC Berkeley Refs: Ran, Zhang A.V., Nature Physics 5, 289 (2009). Hosur, Ryu, AV arxiv: 0908.2691 Part 1: Outline A toy model of

More information

Modern Topics in Solid-State Theory: Topological insulators and superconductors

Modern Topics in Solid-State Theory: Topological insulators and superconductors Modern Topics in Solid-State Theory: Topological insulators and superconductors Andreas P. Schnyder Max-Planck-Institut für Festkörperforschung, Stuttgart Universität Stuttgart January 2016 Lecture Four:

More information

Effective Field Theories of Topological Insulators

Effective Field Theories of Topological Insulators Effective Field Theories of Topological Insulators Eduardo Fradkin University of Illinois at Urbana-Champaign Workshop on Field Theoretic Computer Simulations for Particle Physics and Condensed Matter

More information

Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona

Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether

More information

Introductory lecture on topological insulators. Reza Asgari

Introductory lecture on topological insulators. Reza Asgari Introductory lecture on topological insulators Reza Asgari Workshop on graphene and topological insulators, IPM. 19-20 Oct. 2011 Outlines -Introduction New phases of materials, Insulators -Theory quantum

More information

Disordered topological insulators with time-reversal symmetry: Z 2 invariants

Disordered topological insulators with time-reversal symmetry: Z 2 invariants Keio Topo. Science (2016/11/18) Disordered topological insulators with time-reversal symmetry: Z 2 invariants Hosho Katsura Department of Physics, UTokyo Collaborators: Yutaka Akagi (UTokyo) Tohru Koma

More information

Field Theory Description of Topological States of Matter

Field Theory Description of Topological States of Matter Field Theory Description of Topological States of Matter Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter Quantum Hall effect: bulk and edge Effective field

More information

Topological nonsymmorphic crystalline superconductors

Topological nonsymmorphic crystalline superconductors UIUC, 10/26/2015 Topological nonsymmorphic crystalline superconductors Chaoxing Liu Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA Chao-Xing Liu, Rui-Xing

More information

MAJORANAFERMIONS IN CONDENSED MATTER PHYSICS

MAJORANAFERMIONS IN CONDENSED MATTER PHYSICS MAJORANAFERMIONS IN CONDENSED MATTER PHYSICS A. J. Leggett University of Illinois at Urbana Champaign based in part on joint work with Yiruo Lin Memorial meeting for Nobel Laureate Professor Abdus Salam

More information

Majorana Fermions and Topological Quantum Information Processing. Liang Jiang Yale University & IIIS. QIP 2013, Beijing

Majorana Fermions and Topological Quantum Information Processing. Liang Jiang Yale University & IIIS. QIP 2013, Beijing Majorana Fermions and Topological Quantum Information Processing Liang Jiang Yale University & IIIS QIP 2013, Beijing 2013.1.21 Conventional Quantum Systems Local degrees of freedom E.g., spins, photons,

More information

arxiv: v1 [cond-mat.mes-hall] 5 Jan 2015

arxiv: v1 [cond-mat.mes-hall] 5 Jan 2015 Equivalence of topological mirror and chiral superconductivity in one dimension Eugene Dumitrescu 1, Girish Sharma 1, Jay D. Sau 2, and Sumanta Tewari 1 1 Department of Physics and Astronomy, Clemson University,

More information

Majorana single-charge transistor. Reinhold Egger Institut für Theoretische Physik

Majorana single-charge transistor. Reinhold Egger Institut für Theoretische Physik Majorana single-charge transistor Reinhold Egger Institut für Theoretische Physik Overview Coulomb charging effects on quantum transport through Majorana nanowires: Two-terminal device: Majorana singlecharge

More information

Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators

Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Topological Electromagnetic and Thermal Responses of Time-Reversal Invariant Superconductors and Chiral-Symmetric band insulators Satoshi Fujimoto Dept. Phys., Kyoto University Collaborator: Ken Shiozaki

More information

arxiv: v2 [cond-mat.mes-hall] 29 Oct 2013

arxiv: v2 [cond-mat.mes-hall] 29 Oct 2013 Topological invariant for generic 1D time reversal symmetric superconductors in class DIII Jan Carl Budich, Eddy Ardonne Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden Dated:

More information

Topological insulator (TI)

Topological insulator (TI) Topological insulator (TI) Haldane model: QHE without Landau level Quantized spin Hall effect: 2D topological insulators: Kane-Mele model for graphene HgTe quantum well InAs/GaSb quantum well 3D topological

More information

M. Zahid Hasan Joseph Henry Laboratories of Physics Department of Physics, Princeton University KITP (2007, 2008)

M. Zahid Hasan Joseph Henry Laboratories of Physics Department of Physics, Princeton University KITP (2007, 2008) Experimental Discovery of Topological Insulators Bi-Sb alloys, Bi 2 Se 3, Sb 2 Te 3 and Bi 2 Te 3 Observation of Quantum-Hall-like effects Without magnetic field 1 st European Workshop on Topological Insulators

More information

Single particle Green s functions and interacting topological insulators

Single particle Green s functions and interacting topological insulators 1 Single particle Green s functions and interacting topological insulators Victor Gurarie Nordita, Jan 2011 Topological insulators are free fermion systems characterized by topological invariants. 2 In

More information

Topological insulators. Pavel Buividovich (Regensburg)

Topological insulators. Pavel Buividovich (Regensburg) Topological insulators Pavel Buividovich (Regensburg) Hall effect Classical treatment Dissipative motion for point-like particles (Drude theory) Steady motion Classical Hall effect Cyclotron frequency

More information

TOPOLOGY IN CONDENSED MATTER SYSTEMS: MAJORANA MODES AND WEYL SEMIMETALS. Jan 23, 2012, University of Illinois, Urbana-Chamapaign

TOPOLOGY IN CONDENSED MATTER SYSTEMS: MAJORANA MODES AND WEYL SEMIMETALS. Jan 23, 2012, University of Illinois, Urbana-Chamapaign TOPOLOGY IN CONDENSED MATTER SYSTEMS: MAJORANA MODES AND WEYL SEMIMETALS Pavan Hosur UC Berkeley Jan 23, 2012, University of Illinois, Urbana-Chamapaign Acknowledgements Advisor: Ashvin Vishwanath UC Berkeley

More information

Topological Physics in Band Insulators II

Topological Physics in Band Insulators II Topological Physics in Band Insulators II Gene Mele University of Pennsylvania Topological Insulators in Two and Three Dimensions The canonical list of electric forms of matter is actually incomplete Conductor

More information

Dirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators. Nagoya University Masatoshi Sato

Dirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators. Nagoya University Masatoshi Sato Dirac-Fermion-Induced Parity Mixing in Superconducting Topological Insulators Nagoya University Masatoshi Sato In collaboration with Yukio Tanaka (Nagoya University) Keiji Yada (Nagoya University) Ai Yamakage

More information

Topological Insulators and Superconductors

Topological Insulators and Superconductors Topological Insulators and Superconductors Lecture #1: Topology and Band Theory Lecture #: Topological Insulators in and 3 dimensions Lecture #3: Topological Superconductors, Majorana Fermions an Topological

More information

The Quantum Spin Hall Effect

The Quantum Spin Hall Effect The Quantum Spin Hall Effect Shou-Cheng Zhang Stanford University with Andrei Bernevig, Taylor Hughes Science, 314,1757 2006 Molenamp et al, Science, 318, 766 2007 XL Qi, T. Hughes, SCZ preprint The quantum

More information

Topological Insulators

Topological Insulators Topological Insulators A new state of matter with three dimensional topological electronic order L. Andrew Wray Lawrence Berkeley National Lab Princeton University Surface States (Topological Order in

More information

Field Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti)

Field Theory Description of Topological States of Matter. Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Field Theory Description of Topological States of Matter Andrea Cappelli INFN, Florence (w. E. Randellini, J. Sisti) Topological States of Matter System with bulk gap but non-trivial at energies below

More information

Symmetry Protected Topological Phases of Matter

Symmetry Protected Topological Phases of Matter Symmetry Protected Topological Phases of Matter T. Senthil (MIT) Review: T. Senthil, Annual Reviews of Condensed Matter Physics, 2015 Topological insulators 1.0 Free electron band theory: distinct insulating

More information

Topological minigap in quasi-one-dimensional spin-orbit-coupled semiconductor Majorana wires

Topological minigap in quasi-one-dimensional spin-orbit-coupled semiconductor Majorana wires Topological minigap in quasi-one-dimensional spin-orbit-coupled semiconductor Majorana wires Sumanta Tewari 1, T. D. Stanescu 2, Jay D. Sau 3, and S. Das Sarma 4 1 Department of Physics and Astronomy,

More information

Many-body Characterization of Particle-Conserving Topological Superfluids

Many-body Characterization of Particle-Conserving Topological Superfluids Many-body Characterization of Particle-Conserving Topological Superfluids Gerardo Ortiz Department of Physics - Indiana University INT-15-1 - March 2015 Collaborators: Jorge Dukelsky: CSIC - Madrid Emilio

More information

Topological Kondo Insulator SmB 6. Tetsuya Takimoto

Topological Kondo Insulator SmB 6. Tetsuya Takimoto Topological Kondo Insulator SmB 6 J. Phys. Soc. Jpn. 80 123720, (2011). Tetsuya Takimoto Department of Physics, Hanyang University Collaborator: Ki-Hoon Lee (POSTECH) Content 1. Introduction of SmB 6 in-gap

More information

KITP miniprogram, Dec. 11, 2008

KITP miniprogram, Dec. 11, 2008 1. Magnetoelectric polarizability in 3D insulators and experiments! 2. Topological insulators with interactions (3. Critical Majorana fermion chain at the QSH edge) KITP miniprogram, Dec. 11, 2008 Joel

More information

Topological Photonics with Heavy-Photon Bands

Topological Photonics with Heavy-Photon Bands Topological Photonics with Heavy-Photon Bands Vassilios Yannopapas Dept. of Physics, National Technical University of Athens (NTUA) Quantum simulations and many-body physics with light, 4-11/6/2016, Hania,

More information

Detecting and using Majorana fermions in superconductors

Detecting and using Majorana fermions in superconductors Detecting and using Majorana fermions in superconductors Anton Akhmerov with Carlo Beenakker, Jan Dahlhaus, Fabian Hassler, and Michael Wimmer New J. Phys. 13, 053016 (2011) and arxiv:1105.0315 Superconductor

More information

Topological states of matter in correlated electron systems

Topological states of matter in correlated electron systems Seminar @ Tsinghua, Dec.5/2012 Topological states of matter in correlated electron systems Qiang-Hua Wang National Lab of Solid State Microstructures, Nanjing University, Nanjing 210093, China Collaborators:Dunghai

More information

What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU

What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU What is a topological insulator? Ming-Che Chang Dept of Physics, NTNU A mini course on topology extrinsic curvature K vs intrinsic (Gaussian) curvature G K 0 G 0 G>0 G=0 K 0 G=0 G

More information

Hartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany

Hartmut Buhmann. Physikalisches Institut, EP3 Universität Würzburg Germany Hartmut Buhmann Physikalisches Institut, EP3 Universität Würzburg Germany Part I and II Insulators and Topological Insulators HgTe crystal structure Part III quantum wells Two-Dimensional TI Quantum Spin

More information

arxiv: v1 [cond-mat.supr-con] 17 Dec 2009

arxiv: v1 [cond-mat.supr-con] 17 Dec 2009 Odd-Parity Topological Superconductors: Theory and Application to Cu x Bi Se 3 Liang Fu and Erez Berg Department of Physics, Harvard University, Cambridge, MA 0138 arxiv:091.394v1 [cond-mat.supr-con] 17

More information

Reducing and increasing dimensionality of topological insulators

Reducing and increasing dimensionality of topological insulators Reducing and increasing dimensionality of topological insulators Anton Akhmerov with Bernard van Heck, Cosma Fulga, Fabian Hassler, and Jonathan Edge PRB 85, 165409 (2012), PRB 89, 155424 (2014). ESI,

More information

Organizing Principles for Understanding Matter

Organizing Principles for Understanding Matter Organizing Principles for Understanding Matter Symmetry Conceptual simplification Conservation laws Distinguish phases of matter by pattern of broken symmetries Topology Properties insensitive to smooth

More information

Topological Insulators and Superconductors. Tokyo 2010 Shoucheng Zhang, Stanford University

Topological Insulators and Superconductors. Tokyo 2010 Shoucheng Zhang, Stanford University Topological Insulators and Superconductors Tokyo 2010 Shoucheng Zhang, Stanford University Colloborators Stanford group: Xiaoliang Qi, Andrei Bernevig, Congjun Wu, Chaoxing Liu, Taylor Hughes, Sri Raghu,

More information

arxiv: v2 [cond-mat.supr-con] 13 Aug 2010

arxiv: v2 [cond-mat.supr-con] 13 Aug 2010 Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures Roman M. Lutchyn, Jay D. Sau, and S. Das Sarma Joint Quantum Institute and Condensed Matter Theory

More information

Topological Insulators in 3D and Bosonization

Topological Insulators in 3D and Bosonization Topological Insulators in 3D and Bosonization Andrea Cappelli, INFN Florence (w. E. Randellini, J. Sisti) Outline Topological states of matter: bulk and edge Fermions and bosons on the (1+1)-dimensional

More information

Bell-like non-locality from Majorana end-states

Bell-like non-locality from Majorana end-states Bell-like non-locality from Majorana end-states Alessandro Romito with Yuval Gefen (WIS) 07.09.2016, Daejeon, Workshop on Anderson Localiation in Topological Insulators Outline Topological superconductors

More information

Composite Dirac liquids

Composite Dirac liquids Composite Dirac liquids Composite Fermi liquid non-interacting 3D TI surface Interactions Composite Dirac liquid ~ Jason Alicea, Caltech David Mross, Andrew Essin, & JA, Physical Review X 5, 011011 (2015)

More information

Introduction to topological insulators. Jennifer Cano

Introduction to topological insulators. Jennifer Cano Introduction to topological insulators Jennifer Cano Adapted from Charlie Kane s Windsor Lectures: http://www.physics.upenn.edu/~kane/ Review article: Hasan & Kane Rev. Mod. Phys. 2010 What is an insulator?

More information

Topological insulators

Topological insulators Oddelek za fiziko Seminar 1 b 1. letnik, II. stopnja Topological insulators Author: Žiga Kos Supervisor: prof. dr. Dragan Mihailović Ljubljana, June 24, 2013 Abstract In the seminar, the basic ideas behind

More information

Lecture III: Topological phases

Lecture III: Topological phases Lecture III: Topological phases Ann Arbor, 11 August 2010 Joel Moore University of California, Berkeley, and Lawrence Berkeley National Laboratory Thanks Berkeley students: Andrew Essin Roger Mong Vasudha

More information

Defects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014

Defects in topologically ordered states. Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 Defects in topologically ordered states Xiao-Liang Qi Stanford University Mag Lab, Tallahassee, 01/09/2014 References Maissam Barkeshli & XLQ, PRX, 2, 031013 (2012) Maissam Barkeshli, Chaoming Jian, XLQ,

More information

Unconventional pairing in three-dimensional topological insulators with warped surface state Andrey Vasenko

Unconventional pairing in three-dimensional topological insulators with warped surface state Andrey Vasenko Unconventional pairing in three-dimensional topological insulators with warped surface state Andrey Vasenko Moscow Institute of Electronics and Mathematics, Higher School of Economics Collaborators Alexander

More information

Topological Physics in Band Insulators. Gene Mele Department of Physics University of Pennsylvania

Topological Physics in Band Insulators. Gene Mele Department of Physics University of Pennsylvania Topological Physics in Band Insulators Gene Mele Department of Physics University of Pennsylvania A Brief History of Topological Insulators What they are How they were discovered Why they are important

More information

Topological insulator with time-reversal symmetry

Topological insulator with time-reversal symmetry Phys620.nb 101 7 Topological insulator with time-reversal symmetry Q: Can we get a topological insulator that preserves the time-reversal symmetry? A: Yes, with the help of the spin degree of freedom.

More information

Vortex States in a Non-Abelian Magnetic Field

Vortex States in a Non-Abelian Magnetic Field Vortex States in a Non-Abelian Magnetic Field Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University SESAPS November 10, 2016 Acknowledgments Collin Broholm IQM

More information

Majorana Fermions in Superconducting Chains

Majorana Fermions in Superconducting Chains 16 th December 2015 Majorana Fermions in Superconducting Chains Matilda Peruzzo Fermions (I) Quantum many-body theory: Fermions Bosons Fermions (II) Properties Pauli exclusion principle Fermions (II)

More information

Topological Phases of Matter Out of Equilibrium

Topological Phases of Matter Out of Equilibrium Topological Phases of Matter Out of Equilibrium Nigel Cooper T.C.M. Group, Cavendish Laboratory, University of Cambridge Solvay Workshop on Quantum Simulation ULB, Brussels, 18 February 2019 Max McGinley

More information

Graphite, graphene and relativistic electrons

Graphite, graphene and relativistic electrons Graphite, graphene and relativistic electrons Introduction Physics of E. graphene Y. Andrei Experiments Rutgers University Transport electric field effect Quantum Hall Effect chiral fermions STM Dirac

More information

arxiv: v2 [cond-mat.mes-hall] 16 Nov 2012

arxiv: v2 [cond-mat.mes-hall] 16 Nov 2012 TOPICAL REVIEW arxiv:1206.1736v2 [cond-mat.mes-hall] 16 Nov 2012 Introduction to topological superconductivity and Majorana fermions 1. Introduction Martin Leijnse and Karsten Flensberg Center for Quantum

More information

arxiv: v1 [cond-mat.mes-hall] 16 Feb 2013

arxiv: v1 [cond-mat.mes-hall] 16 Feb 2013 Proposal for Manipulation of Majorana Fermions in Nano-Patterned Semiconductor-Superconductor Heterostructure arxiv:1302.3947v1 [cond-mat.mes-hall] 16 Feb 2013 Abstract Long-Hua Wu,, Qi-Feng Liang, Zhi

More information

Quantum Computing: the Majorana Fermion Solution. By: Ryan Sinclair. Physics 642 4/28/2016

Quantum Computing: the Majorana Fermion Solution. By: Ryan Sinclair. Physics 642 4/28/2016 Quantum Computing: the Majorana Fermion Solution By: Ryan Sinclair Physics 642 4/28/2016 Quantum Computation: The Majorana Fermion Solution Since the introduction of the Torpedo Data Computer during World

More information

Recent developments in topological materials

Recent developments in topological materials Recent developments in topological materials NHMFL Winter School January 6, 2014 Joel Moore University of California, Berkeley, and Lawrence Berkeley National Laboratory Berkeley students: Andrew Essin,

More information

Spin Hall and quantum spin Hall effects. Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST

Spin Hall and quantum spin Hall effects. Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST YKIS2007 (Kyoto) Nov.16, 2007 Spin Hall and quantum spin Hall effects Shuichi Murakami Department of Physics, Tokyo Institute of Technology PRESTO, JST Introduction Spin Hall effect spin Hall effect in

More information

Basics of topological insulator

Basics of topological insulator 011/11/18 @ NTU Basics of topological insulator Ming-Che Chang Dept of Physics, NTNU A brief history of insulators Band insulator (Wilson, Bloch) Mott insulator Anderson insulator Quantum Hall insulator

More information

Topological Kondo Insulators!

Topological Kondo Insulators! Topological Kondo Insulators! Maxim Dzero, University of Maryland Collaborators: Kai Sun, University of Maryland Victor Galitski, University of Maryland Piers Coleman, Rutgers University Main idea Kondo

More information

Universal phase transitions in Topological lattice models

Universal phase transitions in Topological lattice models Universal phase transitions in Topological lattice models F. J. Burnell Collaborators: J. Slingerland S. H. Simon September 2, 2010 Overview Matter: classified by orders Symmetry Breaking (Ferromagnet)

More information

POEM: Physics of Emergent Materials

POEM: Physics of Emergent Materials POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Reference: Bernevig Topological Insulators and Topological Superconductors Tutorials:

More information

Wiring Topological Phases

Wiring Topological Phases 1 Wiring Topological Phases Quantum Condensed Matter Journal Club Adhip Agarwala Department of Physics Indian Institute of Science adhip@physics.iisc.ernet.in February 4, 2016 So you are interested in

More information

POEM: Physics of Emergent Materials

POEM: Physics of Emergent Materials POEM: Physics of Emergent Materials Nandini Trivedi L1: Spin Orbit Coupling L2: Topology and Topological Insulators Tutorials: May 24, 25 (2017) Scope of Lectures and Anchor Points: 1.Spin-Orbit Interaction

More information

5 Topological insulator with time-reversal symmetry

5 Topological insulator with time-reversal symmetry Phys62.nb 63 5 Topological insulator with time-reversal symmetry It is impossible to have quantum Hall effect without breaking the time-reversal symmetry. xy xy. If we want xy to be invariant under, xy

More information

Symmetry Protected Topological Insulators and Semimetals

Symmetry Protected Topological Insulators and Semimetals Symmetry Protected Topological Insulators and Semimetals I. Introduction : Many examples of topological band phenomena II. Recent developments : - Line node semimetal Kim, Wieder, Kane, Rappe, PRL 115,

More information

Topological Properties of Quantum States of Condensed Matter: some recent surprises.

Topological Properties of Quantum States of Condensed Matter: some recent surprises. Topological Properties of Quantum States of Condensed Matter: some recent surprises. F. D. M. Haldane Princeton University and Instituut Lorentz 1. Berry phases, zero-field Hall effect, and one-way light

More information

Magnets, 1D quantum system, and quantum Phase transitions

Magnets, 1D quantum system, and quantum Phase transitions 134 Phys620.nb 10 Magnets, 1D quantum system, and quantum Phase transitions In 1D, fermions can be mapped into bosons, and vice versa. 10.1. magnetization and frustrated magnets (in any dimensions) Consider

More information

arxiv: v2 [cond-mat.mes-hall] 31 Mar 2016

arxiv: v2 [cond-mat.mes-hall] 31 Mar 2016 Journal of the Physical Society of Japan LETTERS Entanglement Chern Number of the ane Mele Model with Ferromagnetism Hiromu Araki, Toshikaze ariyado,, Takahiro Fukui 3, and Yasuhiro Hatsugai, Graduate

More information

Topological Kondo effect in Majorana devices. Reinhold Egger Institut für Theoretische Physik

Topological Kondo effect in Majorana devices. Reinhold Egger Institut für Theoretische Physik Topological Kondo effect in Maorana devices Reinhold Egger Institut für Theoretische Physik Overview Coulomb charging effects on quantum transport in a Maorana device: Topological Kondo effect with stable

More information

Topological Physics in Band Insulators. Gene Mele DRL 2N17a

Topological Physics in Band Insulators. Gene Mele DRL 2N17a Topological Physics in Band Insulators Gene Mele DRL 2N17a Electronic States of Matter Benjamin Franklin (University of Pennsylvania) That the Electrical Fire freely removes from Place to Place in and

More information

arxiv: v1 [cond-mat.mes-hall] 29 Jul 2010

arxiv: v1 [cond-mat.mes-hall] 29 Jul 2010 Discovery of several large families of Topological Insulator classes with backscattering-suppressed spin-polarized single-dirac-cone on the surface arxiv:1007.5111v1 [cond-mat.mes-hall] 29 Jul 2010 Su-Yang

More information

Majorana bound states in spatially inhomogeneous nanowires

Majorana bound states in spatially inhomogeneous nanowires Master Thesis Majorana bound states in spatially inhomogeneous nanowires Author: Johan Ekström Supervisor: Assoc. Prof. Martin Leijnse Division of Solid State Physics Faculty of Engineering November 2016

More information

Transport through interacting Majorana devices. Reinhold Egger Institut für Theoretische Physik

Transport through interacting Majorana devices. Reinhold Egger Institut für Theoretische Physik Transport through interacting Maorana devices Reinhold Egger Institut für Theoretische Physik Overview Coulomb charging effects on quantum transport through Maorana nanowires: Two-terminal device: Maorana

More information

arxiv: v3 [cond-mat.supr-con] 4 Apr 2017

arxiv: v3 [cond-mat.supr-con] 4 Apr 2017 Topological superconductors: a review Masatoshi Sato 1, and Yoichi Ando 2, 1 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 2 Physics Institute II, University of Cologne,

More information

Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots

Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots Energy Spectrum and Broken spin-surface locking in Topological Insulator quantum dots A. Kundu 1 1 Heinrich-Heine Universität Düsseldorf, Germany The Capri Spring School on Transport in Nanostructures

More information

Kitaev honeycomb lattice model: from A to B and beyond

Kitaev honeycomb lattice model: from A to B and beyond Kitaev honeycomb lattice model: from A to B and beyond Jiri Vala Department of Mathematical Physics National University of Ireland at Maynooth Postdoc: PhD students: Collaborators: Graham Kells Ahmet Bolukbasi

More information

Superconductivities of doped Weyl semimetals

Superconductivities of doped Weyl semimetals UIUC, Feb 5 th (2013) Superconductivities of doped Weyl semimetals Phys. Rev. B. 86, 214514 (2012) - Editors suggestion Gil Young Cho UC Berkeley Jens H Bardarson Yuan-Ming Lu Joel E Moore Plan. Part 1

More information

Symmetry, Topology and Phases of Matter

Symmetry, Topology and Phases of Matter Symmetry, Topology and Phases of Matter E E k=λ a k=λ b k=λ a k=λ b Topological Phases of Matter Many examples of topological band phenomena States adiabatically connected to independent electrons: - Quantum

More information

Topological Bandstructures for Ultracold Atoms

Topological Bandstructures for Ultracold Atoms Topological Bandstructures for Ultracold Atoms Nigel Cooper Cavendish Laboratory, University of Cambridge New quantum states of matter in and out of equilibrium GGI, Florence, 12 April 2012 NRC, PRL 106,

More information

Berry s phase in Hall Effects and Topological Insulators

Berry s phase in Hall Effects and Topological Insulators Lecture 6 Berry s phase in Hall Effects and Topological Insulators Given the analogs between Berry s phase and vector potentials, it is not surprising that Berry s phase can be important in the Hall effect.

More information

π-junctions in the Kitaev chain

π-junctions in the Kitaev chain π-junctions in the Kitaev chain Master Thesis in Theoretical Physics 60 credits Author: Nikolaos Palaiodimopoulos Supervisor: Prof. Thors Hans Hansson Co-supervisor: Dr. Eddy Ardonne February 7, 2016 2

More information

Lecture notes on topological insulators

Lecture notes on topological insulators Lecture notes on topological insulators Ming-Che Chang Department of Physics, National Taiwan Normal University, Taipei, Taiwan (Dated: November 1, 18) Contents I. D Topological insulator 1 A. General

More information

Chiral Majorana fermion from quantum anomalous Hall plateau transition

Chiral Majorana fermion from quantum anomalous Hall plateau transition Chiral Majorana fermion from quantum anomalous Hall plateau transition Phys. Rev. B, 2015 王靖复旦大学物理系 wjingphys@fudan.edu.cn Science, 2017 1 Acknowledgements Stanford Biao Lian Quan Zhou Xiao-Liang Qi Shou-Cheng

More information

Majorana-type quasiparticles in nanoscopic systems

Majorana-type quasiparticles in nanoscopic systems Kraków, 20 IV 2015 Majorana-type quasiparticles in nanoscopic systems Tadeusz Domański / UMCS, Lublin / Kraków, 20 IV 2015 Majorana-type quasiparticles in nanoscopic systems Tadeusz Domański / UMCS, Lublin

More information

v. Tε n k =ε n k T r T = r, T v T = r, I v I = I r I = v. Iε n k =ε n k Berry curvature: Symmetry Consideration n k = n k

v. Tε n k =ε n k T r T = r, T v T = r, I v I = I r I = v. Iε n k =ε n k Berry curvature: Symmetry Consideration n k = n k Berry curvature: Symmetry Consideration Time reversal (i.e. motion reversal) 1 1 T r T = r, T v T = v. Tε n k =ε n k n k = n k Inversion Symmetry: 1 1 I r I = r, I v I = v. Iε n k =ε n k n k = n k θ

More information