Topological Kondo effect in Majorana devices Reinhold Egger Institut für Theoretische Physik Overview Coulomb charging effects on quantum transport in a Majorana device: „Topological Kondo effect“ with stable non-Fermi liquid behavior Beri & Cooper, PRL 2012 With interactions in the leads: new unstable fixed point Altland & Egger, PRL 2013 Zazunov, Altland & Egger, New J. Phys. 2014 ‚Majorana quantum impurity spin‘ dynamics near strong coupling Altland, Beri, Egger & Tsvelik, PRL 2014 Non-Fermi liquid manifold: coupling to bulk superconductor Eriksson, Mora, Zazunov & Egger, PRL 2014 Majorana bound states Majorana fermions Beenakker, Ann. Rev. Con. Mat. Phys. 2013 Alicea, Rep. Prog. Phys. 2012 Leijnse & Flensberg, Semicond. Sci. Tech. 2012 + j j γ i , γ j = 2δ ij statistics { γ =γ Non-Abelian exchange Two Majoranas = nonlocal fermion d = γ 1 + iγ 2 + 2 Occupation of single Majorana ill-defined: γ γ = γ d + d = 0,1 Count state of Majorana pair } =1 Realizable (for example) as end states of spinless 1D p-wave superconductor (Kitaev chain) Recipe: Proximity coupling of 1D helical wire to s-wave superconductor For long wires: Majorana bound states are zero energy modes Experimental Majorana signatures Mourik et al., Science 2012 InSb nanowires expected to host Majoranas due to interplay of • strong Rashba spin orbit field • magnetic Zeeman field • proximity-induced pairing Oreg, Refael & von Oppen, PRL 2010 Lutchyn, Sau & Das Sarma, PRL 2010 Transport signature of Majoranas: Zero-bias conductance peak due to resonant Andreev reflection Bolech & Demler, PRL 2007 Law, Lee & Ng, PRL 2009 Flensberg, PRB 2010 See also: Rokhinson et al., Nat. Phys. 2012; Deng et al., Nano Lett. 2012; Das et al., Nat. Phys. 2012; Churchill et al., PRB 2013 Zero-bias conductance peak Mourik et al., Science 2012 Possible explanations: Majorana state (most likely!) Disorder-induced peak Smooth confinement Kondo effect Bagrets & Altland, PRL 2012 Kells, Meidan & Brouwer, PRB 2012 Lee et al., PRL 2012 Suppose that Majorana mode is realized… Quantum transport features beyond zero-bias anomaly peak? Coulomb interaction effects? Simplest case: Majorana single charge transistor ‚Overhanging‘ helical wire parts serve as normal-conducting leads Nanowire part coupled to superconductor hosts pair of Majorana bound states Include charging energy of this ‚dot‘ γL γ R Majorana single charge transistor Hützen et al., PRL 2012 Floating superconducting ‚dot‘ contains two Majorana bound states tunnel-coupled to normal-conducting leads Charging energy finite Consider universal regime: Long superconducting wire: Direct tunnel coupling between left and right Majorana modes is assumed negligible No quasi-particle excitations: Proximity-induced gap is largest energy scale of interest Hamiltonian: charging term Majorana pair: nonlocal fermion d = γ L + iγ R Condensate gives another zero mode Cooper pair number Nc, conjugate phase ϕ Dot Hamiltonian (gate parameter ng) H island = EC (2 N c + d d − ng ) + 2 Majorana fermions couple to Cooper pairs through the charging energy Tunneling Normal-conducting leads: effectively spinless helical wire Applied bias voltage V = chemical potential difference Tunneling of electrons from lead to dot: Project electron operator in superconducting wire part to Majorana sector Spin structure of Majorana state encoded in tunneling matrix elements Flensberg, PRB 2010 Tunneling Hamiltonian Source (drain) couples to left (right) Majorana only: Ht = + t c ∑ j j η j + h.c. j = L,R η j = (d ± e − iφ d + ) 2 respects current conservation 2 Hybridizations: Γj ~ ν t j + + Normal tunneling ~ c d , d c Either destroy or create nonlocal d fermion Condensate not involved Anomalous tunneling ~ c + e − iφ d + , deiφ c Create (destroy) both lead and d fermion & split (add) a Cooper pair Absence of even-odd effect Without Majorana states: Even-odd effect With Majoranas: no even-odd effect! Tuning wire parameters into the topological phase removes even-odd effect (a) E ! Δ 2N-3 2N-1 N 2N+1 !Δ 2N-4 2N-2 2N 2N+2 (b) 2N-3 2N-4 2N+1 2N-1 2N-2 2N 2N+2 picture from: Fu, PRL 2010 Noninteracting case: Resonant Andreev reflection Bolech & Demler, PRL 2007 Law, Lee & Ng, PRL 2009 Ec=0 Majorana spectral function − Im Gγ j (ε ) = ret Γj ε 2 + Γ j2 T=0 differential conductance: 2e 2 G (V ) = Currents IL and IR fluctuate independently, superconductor is effectively grounded Perfect Andreev reflection via Majorana state 1 h 1 + (eV Γ )2 Zero-energy Majorana bound state leaks into lead Strong blockade: Electron teleportation Fu, PRL 2010 Peak conductance for half-integer ng Strong charging energy then allows only two degenerate charge configurations Model maps to spinless resonant tunneling model 2 Linear conductance (T=0): G = e / h Interpretation: Electron teleportation due to nonlocality of d fermion Topological Kondo effect Beri & Cooper, PRL 2012 Altland & Egger, PRL 2013 Beri, PRL 2013 Altland, Beri, Egger & Tsvelik, PRL 2014 Zazunov, Altland & Egger, NJP 2014 Now N>1 helical wires: M Majorana states tunnelcoupled to helical Luttinger liquid wires with g≤1 Strong charging energy, with nearly integer ng: unique equilibrium charge state on the island 2N-1-fold ground state degeneracy due to Majorana states (taking into account parity constraint) Need N>1 for interesting effect! „Klein-Majorana fusion“ Abelian bosonization of lead fermions Klein factors are needed to ensure anticommutation relations between different leads Klein factors can be represented by additional Majorana fermion for each lead Combine Klein-Majorana and ‚true‘ Majorana fermion at each contact to build auxiliary fermions, fj All occupation numbers fj+fj are conserved and can be gauged away purely bosonic problem remains… Charging effects: dipole confinement High energy scales > EC : charging effects irrelevant Electron tunneling amplitudes from lead j to dot renormalize independently upwards −1+ 1 2g t j (E ) ~ E RG flow towards resonant Andreev reflection fixed point For E < EC: charging induces ‚confinement‘ In- and out-tunneling events are bound to ‚dipoles‘ with coupling λ j≠ k : entanglement of different leads Dipole coupling describes amplitude for ‚cotunneling‘ from lead j to lead k −3+ 1 ‚Bare‘ value (1) t j (EC ) tk (EC ) λ jk = ~ EC g large for small EC EC RG equations in dipole phase Energy scales below EC: effective phase action 2 g dω S= ω Φ j (ω ) − ∑ λ jk ∫ dτ cos(Φ j − Φ k ) ∑ ∫ 2π j 2π j ≠k Lead DoS One-loop RG equations M dλ jk = − g −1 − 1 λ jk + ν ∑ λ jmλmk dl m ≠ ( j ,k ) ( ) suppression by Luttinger liquid tunneling DoS enhancement by dipole fusion processes RG-unstable intermediate fixed point with isotropic −1 couplings (for M>2 leads) g −1 * λ j ≠k = λ = ν M −2 RG flow RG flow towards strong coupling for λ(1) > λ* Always happens for moderate charging energy Flow towards isotropic couplings: anisotropies are RG irrelevant Perturbative RG fails below Kondo temperature TK ≈ EC e − λ* λ(1 ) Topological Kondo effect Beri & Cooper, PRL 2012 Refermionize for g=1: M ∞ H = −i ∫ dx ∑ψ +j ∂ xψ j + iλ ∑ψ +j (0 )S jkψ k (0 ) −∞ j =1 j ≠k Majorana bilinears S jk = iγ jγ k ‚Reality‘ condition: SO(M) symmetry [instead of SU(2)] nonlocal realization of ‚quantum impurity spin‘ Nonlocality ensures stability of Kondo fixed point Majorana basis ψ ( x ) = µ ( x ) + iξ ( x ) SO2(M) Kondo model for leads: H = −i ∫ dxµ T ∂ x µ + iλµ T (0 )Sˆµ (0 ) + [µ ↔ ξ ] Minimal case: M=3 Majorana states SU(2) representation of „quantum impurity spin“ i S j = ε jklγ k γ l [S1 , S2 ] = iS3 4 Spin S=1/2 operator, nonlocally realized in terms of Majorana states can be represented by Pauli matrices Exchange coupling of this spin-1/2 to two SO(3) lead currents → multichannel Kondo effect Transport properties near unitary limit Temperature & voltages < TK: Dual instanton version of action applies near strong coupling limit Nonequilibrium Keldysh formulation Linear conductance tensor ∂I j 2e 2 T 2 y − 2 1 1 − δ jk − G jk = e = ∂µk h TK M 1 Non-integer scaling dimension y = 2 g 1 − M > 1 implies non-Fermi liquid behavior even for g=1 completely isotropic multi-terminal junction Correlated Andreev reflection Diagonal conductance at T=0 exceeds resonant tunneling („teleportation“) value but stays below resonant Andreev reflection limit 2e 2 1 2e 2 e2 < G jj < G jj = 1 − ⇒ h M h h Interpretation: Correlated Andreev reflection Remove one lead: change of scaling dimensions and conductance Non-Fermi liquid power-law corrections at finite T Fano factor Zazunov et al., NJP 2014 Backscattering correction to current near unitary 2 y −2 limit for ∑ µ j = 0 j 1 µk e δI j = − ∑ δ jk − µk TK M Shot noise: S~jk (ω → 0) = dt eiωt ( I j (t )I k (0) − I j I k ∫ 2 ge ~ S jk = − k 2 1 1 µl ∑l δ jl − M δ kl − M T K ) 2 y −2 µl universal Fano factor, but different value than for SU(N) Kondo effect Sela et al. PRL 2006; Mora et al., PRB 2009 Majorana spin dynamics Altland, Beri, Egger & Tsvelik, PRL 2014 Overscreened multi-channel Kondo fixed point: massively entangled effective impurity degree remains at strong coupling: „Majorana spin“ Probe and manipulate by coupling of Majoranas H Z = ∑ h jk S jk jk ‚Zeeman fields‘ h jk = − hkj : overlap of Majorana wavefunctions within same nanowire Couple to S jk = iγ jγ k Majorana spin near strong coupling Bosonized form of Majorana spin at Kondo fixed point: S jk = iγ jγ k cos[Θ j (0 ) − Θk (0 )] Dual boson fields Θ j (x ) describe ‚charge‘ (not ‚phase‘) in respective lead 2 Scaling dimension y Z = 1 − → RG relevant M Zeeman field ultimately destroys Kondo fixed point & breaks emergent time reversal symmetry Perturbative treatment possible for Th < T < TK M /2 h12 dominant 1-2 Zeeman coupling: Th = T T K K Crossover SO(M)→SO(M-2) Lowering T below Th → crossover to another Kondo model with SO(M-2) (Fermi liquid for M<5) Zeeman coupling h12 flows to strong coupling → γ 1 , γ 2 disappear from low-energy sector Same scenario follows from Bethe ansatz solution Altland, Beri, Egger & Tsvelik, JPA 2014 Observable in conductance & in thermodynamic properties SO(M)→SO(M-2): conductance scaling for single Zeeman component h12 ≠ 0 consider G jj ( j ≠ 1,2 ) (diagonal element of conductance tensor) Multi-point correlations Majorana spin has nontrivial multi-point correlations at Kondo fixed point, e.g. for M=3 (absent for SU(N) case!) Tτ S j (τ 1 )Sk (τ 2 )Sl (τ 3 ) ε jkl ~ 1/ 3 TK (τ 12τ 13τ 23 ) Observable consequences for time-dependent ‚Zeeman‘ field B j = ε jkl hkl with B (t ) = (B1 cos(ω1t ), B2 cos(ω2t ),0 ) Time-dependent gate voltage modulation of tunnel couplings Measurement of ‚magnetization‘ by known read-out methods Nonlinear frequency mixing S3 (t ) ~ B1B2 cos[(ω1 ± ω2 )t ] Oscillatory transverse spin correlations (for B2=0) cos(ω1t ) S2 (t )S3 (0 ) ~ B1 (ω1t )2 / 3 Adding Josephson coupling: Non Fermi liquid manifold Eriksson, Mora, Zazunov & Egger, PRL 2014 H island = EC (2 N c + nˆ − ng ) − E J cos ϕ 2 Yet another bulk superconductor: Topological Cooper pair box Effectively harmonic oscillator for EJ >> EC with Josephson plasma oscillation frequency: Ω = 8E J EC Low energy theory Tracing over phase fluctuations gives two coupling mechanisms: Resonant Andreev reflection processes H A = ∑ t jγ j (ψ +j (0) − ψ j (0) ) j Kondo exchange coupling, but of SO1(M) type ( )( ) H K = ∑ λ jk ψ +j (0 )+ ψ j (0 ) ψ k+ (0 )+ψ k (0 ) γ j γ k j≠k Interplay of resonant Andreev reflection and Kondo screening for Γ < TK Quantum Brownian Motion picture Abelian bosonization now yields (M=3) H A + H K ∝ − ∑ Γj sin Φ j − TK ∑ cos Φ j cos Φ k j Simple cubic lattice j ≠k bcc lattice Quantum Brownian motion Leading irrelevant operator (LIO): tunneling transitions connecting nearest neighbors Scaling dimension of LIO from n.n. distance d y LIO d2 = 2π 2 Yi & Kane, PRB 1998 Pinned phase field configurations correspond to Kondo fixed point, but unitarily rotated by resonant Andreev reflection corrections Stable non-Fermi liquid manifold as long as LIO stays irrelevant, i.e. for y LIO > 1 Scaling dimension of LIO M-dimensional manifold of non-Fermi liquid states spanned by parameters δ j = Γj TK Scaling dimension of LIO 1 M 2 δj y = min 2, ∑ 1 − arcsin 2 j =1 π 2(M − 1) ) Stable manifold corresponds to y>1 For y<1: standard resonant Andreev reflection scenario applies For y>1: non-Fermi liquid power laws appear in temperature dependence of conductance tensor Conclusions Coulomb charging effects on quantum transport in a Majorana device: „Topological Kondo effect“ with stable non-Fermi liquid behavior Beri & Cooper, PRL 2014 With interactions in the leads: new unstable fixed point Altland & Egger, PRL 2013 Zazunov, Altland & Egger, New J. Phys. 2014 ‚Majorana quantum impurity spin‘ dynamics near strong coupling Altland, Beri, Egger & Tsvelik, PRL 2014 Non-Fermi liquid manifold: coupling to bulk superconductor Eriksson, Mora, Zazunov & Egger, PRL 2014 THANK YOU FOR YOUR ATTENTION

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