Topological Superconductors, Majorana Fermions and Topological Quantum Computation
0. … from last time: The surface of a topological insulator
1. Bogoliubov de Gennes Theory
2. Majorana bound states, Kitaev model 3. Topological superconductor
4. Periodic Table of topological insulators and superconductors
5. Topological quantum computation
6. Proximity effect devices
Unique Properties of Topological Insulator Surface States
“Half” an ordinary 2DEG ; ¼ Graphene EF Spin polarized Fermi surface
• Charge Current ~ Spin Density • Spin Current ~ Charge Density
Berry’s phase
• Robust to disorder • Weak Antilocalization • Impossible to localize
Exotic States when broken symmetry leads to surface energy gap:
• Quantum Hall state, topological magnetoelectric effect • Superconducting state Even more exotic states if surface is gapped without breaking symmetry • Requires intrinsic topological order like non-Abelian FQHE Surface Quantum Hall Effect Orbital QHE : E=0 Landau Level for Dirac fermions. “Fractional” IQHE
2 2 e xy 1 2h B 2 0 e 1 xy n -1 h 2 2 -2 e n=1 chiral edge state xy 2h
Anomalous QHE : Induce a surface gap by depositing magnetic material
† 2 2 Hi0 ( - v - DM z ) e e - Mass due to Exchange field 2h 2h M↑ M↓ e2 xyDsgn( M ) EF 2h TI Egap = 2|DM| Chiral Edge State at Domain Wall : DM ↔ -DM Topological Magnetoelectric Effect Qi, Hughes, Zhang ’08; Essin, Moore, Vanderbilt ‘09
Consider a solid cylinder of TI with a magnetically gapped surface
M 2 e 1 J xy E n E M h 2 J Magnetoelectric Polarizability topological “q term” 2 DL EB E e 1 ME n e2 h 2 q 2 h
TR sym. : q 02 or mod The fractional part of the magnetoelectric polarizability is determined by the bulk, and independent of the surface (provided there is a gap) Analogous to the electric polarization, P, in 1D.
DL formula “uncertainty quantum” e d=1 : Polarization P PE Tr[]A e (extra end electron) 2 BZ 2 d=3 : Magnetoelectric e 2 eh2 / (extra surface EB 2 Tr[]AAAAAd poliarizability 43 h BZ quantum Hall layer) BCS Theory of Superconductivity
mean field theory : ††††* D
H D 1 † Bogoliubov de Gennes 0 HH H BdG † Hamiltonian BdG 2 * k D-H0
Intrinsic anti-unitary particle – hole symmetry
-1 01 HH - x * x BdG BdG 10 2 1 E E Particle – hole redundancy k = k † same state --EEEE
-1 Bloch - BdG Hamiltonians satisfy HHBdG()()kk - BdG - Topological classification problem similar to time reversal symmetry 1D 2 Topological Superconductor : n = 0,1 (Kitaev, 2000)
Bulk-Boundary correspondence : Discrete end state spectrum END
n=0 “trivial” n=1 “topological” E D D Zero mode E E E=0 † 0 0 EE00 -E † -EE Majorana fermion -D -D bound state
† Majorana Fermion : Particle = Antiparticle
† ; i 2 1 Real part of a Dirac fermion : 1 1 2 i -ii( - †† ) ; - ,2 2 1 2 i j ij “Half a state”
Two Majorana fermions define a single two level system occupied † Hi2 0 1 2 0 empty 0 Kitaev Model for 1D p wave superconductor
††††† H- N tcc()()i i1 cc i 1 i - cc i i D cc i i 1 cc i 1 i i - t c † k D ck c- k H BdG () k † k c-k
HBdG( k ) z (2 t cos k - ) x D sin k d ( k ) dx d(k) ||>2t : Strong pairing phase d trivial superconductor z
dx d(k) ||<2t : Weak pairing phase
topological superconductor dz
Similar to SSH model, except different symmetry : (,,)(,,)d d d - d - d d x y zkk x y z - Majorana Chain
H2 i t t cii12 i i 1 1i 2 i 2 2 i 1 i 1 i † ci c i 2 i 12 i i †† For D=t : nearest neighbor Majorana chain t ci c i1 c i 1 c i 2 it 1 i 2 i 1 - 2 i 1 i 1 †† t12, t 2 t Dci c i1 c i 1 c i 2 i D 1 i 2 i 1 2 i 1 i 1
t1 t2 1i 2i t1>t2 trivial SC
t1 Unpaired Majorana Fermion at end 2D topological superconductor (broken T symmetry) Bulk-Boundary correspondence: n = # Chiral Majorana Fermion edge states E T-SC k Examples † k • Spinless px+ipy superconductor (n=1) -kk • Chiral triplet p wave superconductor (eg Sr2RuO4) (n=2) k 2 † D()k Dk ik Read Green model : H - ck c k D()..k c k c- k c c 0 xy k 2m Lattice BdG model : HBdG(kd ) z 2 t cos k x cos k y - D x sin k x y sin k y ( k ) dx d(k) ||>4t : Strong pairing phase d trivial superconductor z Chern number 0 dy dx d(k) ||<4t : Weak pairing phase dz Chern number 1 topological superconductor dy Majorana zero mode at a vortex Boundary condition on fermion wavefunction h 0 p1 p A (L )- ( 1) (0) 2e L (x ) eiqm x ; q 2 m 1 p m L Hole in a topological superconductor threaded by flux p even p odd E E 2v D ~ L q q zero mode D2 Without the hole : Caroli, de Gennes, Matricon theory (’64) D ~ EF Periodic Table of Topological Insulators and Superconductors Anti-Unitary Symmetries : - Time Reversal : HH(kk ) -1 ( - ) ; 2 1 -1 2 - Particle - Hole : HH(kk ) - ( - ) ; 1 -1 Unitary (chiral) symmetry : HH()()kk - ; 8 antiunitary symmetry classes Complex K-theory Altland- Zirnbauer Random Matrix Real Classes K-theory Kitaev, 2008 Schnyder, Ryu, Furusaki, Ludwig 2008 Bott Periodicity d→d+8 Majorana Fermions and Topological Quantum Computing (Kitaev ’03) The degenerate states associated with Majorana zero modes define a topologically protected quantum memory • 2 Majorana separated bound states = 1 fermion 12 i - 2 degenerate states (full/empty) = 1 qubit • 2N separated Majoranas = N qubits • Quantum Information is stored non locally - Immune from local decoherence Braiding performs unitary operations Measure 012 0 34 1 12 1 34 / 2 Non-Abelian statistics Interchange rule (Ivanov 03) t ij Braid ji- These operations, however, are not sufficient to make a universal quantum computer Create 0012 34 Potential condensed matter hosts for topological superconductivity • Quasiparticles in fractional Quantum Hall effect at n=5/2 Moore Read ‘91 • Unconventional superconductors - Sr2RuO4 Das Sarma, Nayak, Tewari ‘06 - Fermionic atoms near feshbach resonance Gurarie ‘05 - CuxBi2Se3 ? • Proximity Effect Devices using ordinary superconductors - Topological Insulator devices Fu, Kane ‘08 - 2D Semiconductor/Magnet devices Sau, Lutchyn, Tewari, Das Sarma ’09, Lee ’09 - 1D Semiconductor devices: eg In As quantum wires Oreg, von Oppen, Alicea, Fisher ’10 Lutchyn, Sau, Das Sarma ‘10 Expt : Maurik et al. (Kouwenhoven) ‘12 - 1D Ferromagnetic atomic chains on superconductors Expt : Nadj-Perg et al. (Yazdani) ‘14 Topological Superconductors Spinless p-wave superconductor: -k †† i ck c- k D e() k x ik y k Ordinary Superconductor : -k↓ †† i ckk c - D e (s-wave, singlet pairing) k↑ Surface of topological insulator -k ← Dirac point c†† cD ei kk- surface ↓ ↑ (s-wave, singlet pairing) → Half an ordinary superconductor k Nontrivial ground state supports Majorana fermions at vortices Majorana Bound States on Topological Insulators 1. h/2e vortex in 2D superconducting state E † E h/2e D † 0 00 SC -D † TI -EE Quasiparticle Bound state at E=0 Majorana Fermion 0 “Half a State” 2. Superconductor-magnet interface at edge of 2D QSHI m | DSM | - | D | M S.C. m>0 Egap =2|m| QSHI m<0 Domain wall bound state 0 1D Majorana Fermions on Topological Insulators 1. 1D Chiral Majorana mode at superconductor-magnet interface E M SC kx TI † kk - : “Half” a 1D chiral Dirac fermion Hi -vF x 2. S-TI-S Josephson Junction f 0 f f SC SC TI Gapless non-chiral Majorana fermion for phase difference f H - ivF L x L - R x R i Dcos( f / 2) L R Another route to the 2D p+ip superconductor Sau, Lutchyn, Tewari, Semiconductor - Magnet - Superconductor structure Das Sarma ‘09 E Superconductor Rashba split 2DEG bands Semiconductor EF Magnetic Insulator k • Single Fermi circle with Berry phase - Zeeman splitting • Topological superconductor with Majorana edge states and Majorana bound states at vortices. • Variants : - use applied magnetic field to lift Kramers degeneracy (Alicea ’10) - Use 1D quantum wire (eg InSb ). A route to 1D p wave superconductor with Majorana end states. (Oreg, von Oppen, Alicea, Fisher ’10 Experiments on Proximity Devices Superconducting Proximity Effect: Use ordinary superconductors and topological materials to engineer topological superconductivity Superconductor - Topological Insulator Devices Expt: Hart, … Yacoby ‘14 (HgTe); Theory: Fu, Kane ‘07, ’08 Pribiag, … Kouwenhoven ‘14 (InAs/GaSb) Majorana Boundary Mode magnetic s-wave insulator superconductor QSHI SC SC B Quantum Spin Hall Insulator “Two slit” interference pattern in a S-TI-S Josephson Junction Demonstrates edge superconductivity Superconductor - Semiconductor Nanowire Devices tunneling conductance Theory: Lutchyn, Sau , Das Sarma ‘10 B Oreg, Refael, von Oppen ’10 InSb nanowire Expt: Mourik, …Kouwenhoven ‘12 SC Majorana end mode Ferromagnetic Atomic Chains STM Spectroscopy on Superconductors E=0 Nadj-Perg, …, Yazdani ’14 (Fe on Pb) STM Majorana topography end mode Fractional Josephson Effect Kitaev ’01 Kwon, Sengupta, Yakovenko ’04 Fu, Kane ’08 1 2 o 1112 34 e 3 4 0012 34 • 4 perioidicity of E(f) protected by local conservation of fermion parity. • AC Josephson effect with half the usual frequency: f = eV/h A Z2 Interferometer for Majorana Fermions A signature for neutral Majorana fermions probed with charge transport N even ci† - 12 2 ci12e e h 1 N 2e N odd • Chiral electrons on magnetic domain wall split into a pair of chiral Majorana fermions 2 -2 • “Z2 Aharonov Bohm phase” converts an e h electron into a hole 1 G N Akhmerov, Nilsson, Beenakker, PRL ’09 Fu and Kane, PRL ‘09 Symmetry Respecting Gapped (SRG) State of a TI Surface Without interactions such as state is impossible. With interactions a gapped state with intrinsic topological order is possible. Laughlin argument: = h/e Thread flux h/e through a slab : e/2 e/2 TI e/2 TI e/2 2 SRG M↑ xy = e /2h e/2 -e/2 SRG State must support charge e/2 excitations Split a chiral Dirac fermion : n = 1 n = 1/2 Much recent progress : c = 1 c = 1/2 Metlitski, Kane, Fisher arXiv: 1306:3286 Bonderson, Nayak, Qi arXiv: 1306: 3230 M↑ Chen, Fidkowski, Vishwanath arXiv: 1306: 3250 SRG Wang, Potter, Senthil arXiv: 1306: 3223 M↓ Mross, Essen, Alicea arXiv: 1410.4201 Metlitski, Vishwanath arXiv:1505.05142 SRG State must have non-Abelian topological order Wang, Senthil arXiv:1505.05141