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

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1 Topological Kondo effect in Maorana devices Reinhold Egger Institut für Theoretische Physik

2 Overview Coulomb charging effects on quantum transport in a Maorana device: Topological Kondo effect with stable non-fermi liquid behavior Beri & Cooper, PRL 01 With interactions in the leads: new unstable fixed point Altland & Egger, PRL 013 Zazunov, Altland & Egger, New J. Phys. 014 Maorana quantum impurity spin dynamics near strong coupling Altland, Beri, Egger & Tsvelik, PRL 014 Non-Fermi liquid manifold: coupling to bulk superconductor Eriksson, Mora, Zazunov & Egger, PRL 014

3 Maorana bound states (MBSs) Maorana fermions Non-Abelian exchange statistics + = γ Two MBS yield one (nonlocal) fermion Beenakker, Ann. Rev. Con. Mat. Phys. 013 Alicea, Rep. Prog. Phys. 01 Leinse & Flensberg, Semicond. Sci. Tech Occupation of single MBS is ill-defined: γ γ = γ = 1 Count state of MBS pair d + d = 0,1 Realizable (for example) as end states of spinless 1D p-wave superconductor (Kitaev chain) Recipe: Proximity couple 1D helical wire to s-wave superconductor For long wires: MBSs are zero energy modes γ { γ } i, γ = δ i d = γ + i 1 γ

4 Experimental Maorana signatures InAs or InSb nanowires expected to host Maoranas due to interplay of strong Rashba spin orbit field magnetic Zeeman field proximity-induced pairing Oreg, Refael & von Oppen, PRL 010 Lutchyn, Sau & Das Sarma, PRL 010 Mourik et al., Science 01 Transport signature of Maoranas: Zero-bias conductance peak due to resonant Andreev reflection Bolech & Demler, PRL 007 Law, Lee & Ng, PRL 009 Flensberg, PRB 010 see also: Rokhinson et al., Nat. Phys. 01; Deng et al., Nano Lett. 01; Das et al., Nat. Phys. 01; Churchill et al., PRB 013; Nad-Perge et al., Science 014

5 Zero-bias conductance peak Mourik et al., Science 01 Possible explanations: Maorana state (most likely) Disorder-induced peak Bagrets & Altland, PRL 01 Smooth confinement Kells, Meidan & Brouwer, PRB 01 Kondo effect Lee et al., PRL 01

6 Suppose that Maorana mode is realized Quantum transport features beyond zero-bias anomaly peak? Coulomb interaction effects? Simplest case: Maorana single charge transistor Overhanging helical wire parts serve as normal-conducting leads Nanowire part coupled to superconductor hosts pair of Maorana bound states Include charging energy of this dot γ L γ R

7 Maorana single charge transistor Floating superconducting dot contains two Maorana bound states tunnel-coupled to normal-conducting leads Charging energy finite Consider universal regime: Long superconducting wire: Direct tunnel coupling between left and right Maorana modes is assumed negligible No quasi-particle excitations: Proximity-induced gap is largest energy scale of interest Hützen et al., PRL 01

8 Hamiltonian: charging term Fu, PRL 010 Maorana pair: nonlocal fermion = γ + iγ Condensate gives another zero mode Cooper pair number N c, conugate phase ϕ Dot Hamiltonian (gate parameter n g ) d L R H island = E C ( + N + d d n ) c g Maorana fermions couple to Cooper pairs through the charging energy

9 Tunneling Normal-conducting leads: effectively spinless helical wire Applied bias voltage V = chemical potential difference Tunneling of electrons from lead to dot: Proect electron operator in superconducting wire part to Maorana sector Spin structure of Maorana state encoded in tunneling matrix elements Flensberg, PRB 010

10 Tunneling Hamiltonian Source (drain) couples to left (right) Maorana only: H = t + t η + = L, R respects charge conservation Hybridizations: Normal tunneling c Γ ~ c ~ ν t + d, d h. c. Either destroy or create nonlocal d fermion Condensate not involved + iφ + iφ Anomalous tunneling ~ c e d, de c Create (destroy) both lead and d fermion & split (add) a Cooper pair + c η ( iφ + d ± e ) = d

11 Absence of even-odd effect Without MBSs: Even-odd effect With MBSs: no even-odd effect! Tuning wire parameters into the topological phase removes even-odd effect (a) Δ! N-3 N-1 N+1 N-4 N- N N+ E N! Δ (b) N-4 N-3 N-1 N+1 N- N N+ Fu, PRL 010

12 Noninteracting case: Resonant Andreev reflection E c =0 Maorana spectral function ImG T=0 differential conductance: G( V ) Currents I L and I R fluctuate independently, superconductor is effectively grounded Perfect Andreev reflection via MBS Zero-energy MBS leaks into lead Bolech & Demler, PRL 007 Law, Lee & Ng, PRL 009 ret γ = Γ ( ε ) = e h ε 1+ + Γ 1 ( ev Γ)

13 Strong blockade: Electron teleportation Fu, PRL 010 Peak conductance for half-integer n g Strong charging energy then allows only two degenerate charge configurations Model maps to spinless resonant tunneling model Linear conductance (T=0): G = e / h Interpretation: Electron teleportation due to nonlocality of d fermion

14 Topological Kondo effect Beri & Cooper, PRL 01 Altland & Egger, PRL 013 Beri, PRL 013 Altland, Beri, Egger & Tsvelik, PRL 014 Zazunov, Altland & Egger, NJP 014 Now N>1 helical wires: M Maorana states tunnelcoupled to helical Luttinger liquid wires with g 1 Strong charging energy, with nearly integer ng: unique equilibrium charge state on the island N-1 -fold ground state degeneracy due to Maorana states (taking into account parity constraint) Need N>1 for interesting effect!

15 Klein-Maorana fusion Abelian bosonization of lead fermions Klein factors are needed to ensure anticommutation relations between different leads Klein factors can be represented by additional Maorana fermion for each lead Combine Klein-Maorana and true Maorana fermion at each contact to build auxiliary fermions, f All occupation numbers f + f are conserved and can be gauged away purely bosonic problem remains

16 Charging effects: dipole confinement High energy scales : charging effects irrelevant Electron tunneling amplitudes from lead to dot renormalize independently upwards RG flow towards resonant Andreev reflection fixed point For E < E C ( ) g E ~ E : charging induces confinement In- and out-tunneling events are bound to dipoles with coupling : entanglement of different leads λ k > E C t Dipole coupling describes amplitude for cotunneling from lead to lead k Bare value t ( EC ) tk ( EC ) 1 (1) 3+ g λ k = ~ EC E large for small E C C 1+ 1

17 RG equations in dipole phase Energy scales below E C : effective phase action g dω S = ω Φ ω λ k dτ cos Φ Φ π π One-loop RG equations dλ k = g dl ( ) ( ) k ( 1 1) λ k + ν λ mλmk suppression by Luttinger liquid tunneling DoS enhancement by dipole fusion processes RG-unstable intermediate fixed point with isotropic couplings (for M> leads) 1 * g 1 λ k = λ = ν M k M m ( Lead DoS, k )

18 RG flow (1) * RG flow towards strong coupling for λ > λ Always happens for moderate charging energy Flow towards isotropic couplings: anisotropies are RG irrelevant Perturbative RG fails below Kondo temperature T K E C e λ * λ ( 1)

19 Topological Kondo effect Refermionize for g=1: Maorana bilinears Reality condition: SO(M) symmetry [instead of SU()] nonlocal realization of quantum impurity spin Nonlocality ensures stability of Kondo fixed point Maorana basis H H = i M + xψ + iλ = 1 k dx ψ ψ x = µ x + iξ x SO (M) Kondo model S k = iγ γ ( ) ( ) ( ) = T T i dxµ µ + iλµ ˆ 0 x k ψ + ( 0) S ψ ( 0) k k for leads: ( 0) Sµ ( ) + [ µ ξ ]

20 Minimal case: M=3 Maorana states SU() representation of quantum impurity spin i S = ε klγ kγ l 4 [ S 1, S ] = is 3 Spin S=1/ operator, nonlocally realized in terms of Maorana states can be represented by Pauli matrices Exchange coupling (= dipole coupling) of this spin-1/ to two SO(3) lead currents multichannel Kondo effect

21 Transport properties near unitary limit Temperature & voltages < T K : Dual instanton version of action applies near strong coupling limit Nonequilibrium Keldysh formulation Linear conductance tensor Non-integer scaling dimension implies non-fermi liquid behavior even for g=1 completely isotropic multi-terminal unction = = M T T h e I e G k y K k k 1 1 δ µ > = M g y

22 Correlated Andreev reflection Diagonal conductance at T=0 exceeds resonant tunneling ( teleportation ) value but stays below resonant Andreev reflection limit G = e h 1 1 M Interpretation: Correlated Andreev reflection e h Remove one lead: change of scaling dimensions and conductance Non-Fermi liquid power-law corrections at finite T e h < G <

23 Fano factor Zazunov et al., NJP 014 Backscattering correction to current near unitary limit for µ = 0 y e µ k 1 δ = δ I k µ k k T M K Shot noise: ~ ω k S i t ( ω 0) = dt e I ( t) I ( 0) ( I I ) k k y ~ ge 1 1 µ l S k = δ l δkl µ l l M M TK universal Fano factor, but different value than for SU(N) Kondo effect Sela et al. PRL 006; Mora et al., PRB 009

24 Maorana spin dynamics Altland, Beri, Egger & Tsvelik, PRL 014 Overscreened multi-channel Kondo fixed point: massively entangled effective impurity degree remains at strong coupling: Maorana spin Probe and manipulate by coupling of MBSs H = h Zeeman fields h k = h k describe overlap of MBS wavefunctions within same nanowire Zeeman fields couple to S = iγ γ Z k k k k S k

25 Maorana spin near strong coupling Bosonized form of Maorana spin at Kondo fixed point: S = iγ γ cos Θ 0 Θ 0 Dual boson fields Θ x describe charge (not phase ) in respective lead Scaling dimension = 1 RG relevant Zeeman field ultimately destroys Kondo fixed point & breaks emergent time reversal symmetry Perturbative treatment possible for k ( ) k y Z M [ ( ) ( )] k T < T < dominant 1- Zeeman coupling: h T K T h h 1 = T K M / T K

26 Crossover SO(M) SO(M-) Lowering T below T h crossover to another Kondo model with SO(M-) (Fermi liquid for M<5) Zeeman coupling h 1 flows to strong coupling γ 1,γ disappear from low-energy sector Same scenario follows from Bethe ansatz solution Altland, Beri, Egger & Tsvelik, JPA 014 Observable in conductance & in thermodynamic properties

27 SO(M) SO(M-): conductance scaling for single Zeeman component h 1 0 consider G ( 1, ) (diagonal element of conductance tensor)

28 Multi-point correlations Maorana spin has nontrivial multi-point correlations at Kondo fixed point, e.g. for M=3 (absent for SU(N) case) T τ S ( τ ) S ( τ ) S ( τ ) 1 k l 3 ~ Observable consequences for time-dependent Zeeman field B ε h with B = ω ω kl kl t B1 cos 1t, B cos t,0 Time-dependent gate voltage modulation of tunnel couplings Measurement of magnetization by known read-out methods Nonlinear frequency mixing T K ( τ τ τ ) 1/ 3 Oscillatory transverse spin correlations (for B =0) 1 = ( ) ( ( ) ( ) ) S ( t) S ( 0) 3 ~ ε B kl 13 3 ( t) ~ B B [( ω ± ω ) t] S cos ( ω1t ) ( ω t) / 3 cos 1

29 Adding Josephson coupling: Non Fermi liquid manifold Eriksson, Mora, Zazunov & Egger, PRL 014 H island = E C ( ) N + nˆ n E cosϕ c g J with another bulk superconductor: Topological Cooper pair box Effectively harmonic oscillator for E J >> E C with Josephson plasma oscillation frequency Ω = 8E E J C

30 Low energy theory Tracing over phase fluctuations gives two coupling mechanisms: Resonant Andreev reflection processes H Kondo exchange coupling, but of SO 1 (M) type H K = = A t ( + ( ) ( )) + ψ 0 + ψ 0 ψ ( 0) ψ ( ) ( ) λ + 0 k k γ ( + ψ 0) ψ (0)) ( Interplay of resonant Andreev reflection and Kondo screening for k Γ < T K k γ γ k

31 Quantum Brownian Motion picture Abelian bosonization now yields (M=3) H A + H K Γ sin Φ TK k cosφ cos Φ k Simple cubic lattice bcc lattice

32 Quantum Brownian motion Leading irrelevant operator (LIO): tunneling transitions connecting nearest neighbors Scaling dimension of LIO from n.n. distance d d y LIO = Yi & Kane, PRB 1998 π 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 > 1 y LIO

33 Scaling dimension of LIO M-dimensional manifold of non-fermi liquid states spanned by parameters Γ δ = TK Scaling dimension of LIO M 1 y = min, 1 arcsin = π δ ( ) M 1 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

34 Conclusions Coulomb charging effects on quantum transport in a Maorana device: Topological Kondo effect with stable non-fermi liquid behavior Beri & Cooper, PRL 014 With interactions in the leads: new unstable fixed point Altland & Egger, PRL 013 Zazunov, Altland & Egger, New J. Phys. 014 Maorana quantum impurity spin dynamics near strong coupling Altland, Beri, Egger & Tsvelik, PRL 014 Non-Fermi liquid manifold: coupling to bulk superconductor Eriksson, Mora, Zazunov & Egger, PRL 014 THANK YOU FOR YOUR ATTENTION