Transport through interacting Majorana devices. Reinhold Egger Institut für Theoretische Physik
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1 Transport through interacting Maorana devices Reinhold Egger Institut für Theoretische Physik
2 Overview Coulomb charging effects on quantum transport through Maorana nanowires: Two-terminal device: Maorana single-charge transistor Zazunov, Levy Yeyati & Egger, PRB 84, (2011) Hützen, Zazunov, Braunecker, Levy Yeyati & Egger, PRL 109, (2012) Multi-terminal device: Topological Kondo effect with stable non-fermi liquid behavior With interactions in the leads: novel unstable fixed point Altland & Egger, PRL 110, (2013) Zazunov, Altland & Egger, NJP 16, (2014) Maorana spin dynamics near strong coupling Altland, Beri, Egger & Tsvelik, arxiv: Non-Fermi liquid manifold: coupling to bulk superconductor, (not discussed here) Eriksson, Mora, Zazunov & Egger, arxiv:
3 Maorana bound states Maorana fermions Non-Abelian exchange statistics Two Maoranas = nonlocal fermion Occupation of single Maorana ill-defined: Count state of Maorana pair Beenakker, Ann. Rev. Con. Mat. Phys Alicea, Rep. Prog. Phys Leinse & Flensberg, Semicond. Sci. Tech γ = γ { γ } i, γ = 2δ i d + d d = γ 1 + iγ 2 + γ γ = γ 2 = 1 = 0,1 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: Maorana bound states are zero energy modes
4 Experimental Maorana signatures 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 2010 Lutchyn, Sau & Das Sarma, PRL 2010 Mourik et al., Science 2012 Transport signature of Maoranas: 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
5 Zero-bias conductance peak Mourik et al., Science 2012 Possible explanations: Maorana state (most likely!) Disorder-induced peak Bagrets & Altland, PRL 2012 Smooth confinement Kells, Meidan & Brouwer, PRB 2012 Kondo effect Lee et al., PRL 2012
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 2012
8 Hamiltonian: charging term 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 c = E c ( + N + d d n ) 2 2 c g Maorana fermions couple to Cooper pairs through the charging energy
9 Tunneling Normal-conducting leads: noninteracting fermions (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 2010
10 Tunneling Hamiltonian Source (drain) couples to left (right) Maorana only: H respects current conservation Hybridizations: Normal tunneling t c + t η + = L, R = Γ ~ h. c. ρ t L / R ~ 0 L / R c + d, d 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 2 η ( iφ + d ± e ) 2 = d
11 Absence of even-odd effect Without Maorana states: Even-odd effect With Maoranas: no even-odd effect! Tuning wire parameters into the topological phase removes even-odd effect (a) Δ! 2N-3 2N-1 2N+1 2N-4 2N-2 2N 2N+2 E N! Δ (b) 2N-4 2N-3 2N-1 2N+1 2N-2 2N 2N+2 picture from: Fu, PRL 2010
12 Maorana Meir-Wingreen formula Exact expression for interacting Maorana dot eγ ε ( ε µ ) ret I L, R = = d F ImGη ( ε ) h Lead Fermi distribution encoded in + Proof uses η η = 1 Differential conductance: Here: symmetric case Γ L = Γ R F G = di dv I = I L I R = Γ 2 ( ) ( ) ε = tanh ε ( ) 2 2T
13 Noninteracting case: Resonant Andreev reflection E c =0 Maorana spectral function ImG T=0 differential conductance: G( V ) Bolech & Demler, PRL 2007 Law, Lee & Ng, PRL 2009 ( ε ) = 2 2 Currents I L and I R fluctuate independently, superconductor is effectively grounded Perfect Andreev reflection via Maorana state Zero-energy Maorana bound state leaks into lead ret γ = 2e h 2 ε 1+ Γ + Γ 1 ( ev Γ) 2
14 Strong blockade: Electron teleportation Fu, PRL 2010 Peak conductance for half-integer n g 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
15 Crossover from resonant Andreev reflection to electron teleportation Keldysh approach yields full action in phase representation Zazunov, Levy Yeyati & Egger, PRB 2011 Practically useful in weak Coulomb blockade regime: interaction corrections to conductance Full crossover from three other methods: Hützen et al., PRL 2012 Master equation for T>Γ: include sequential and all cotunneling processes (incl. local and crossed Andreev reflection) Equation of motion approach for peak conductance Zero bandwidth model for leads: exact solution
16 Coulomb oscillations Master equation T = 2Γ Valley conductance dominated by elastic cotunneling
17 Peak conductance: from resonant Andreev reflection to teleportation T=0
18 Finite bias sidepeaks n g = 1/ 2 n g = 1 Master equation T = 2Γ
19 Finite bias sidepeaks ev = 4nE c µ L,R resonant with two (almost) degenerate higher order charge states: additional sequential tunneling contributions On resonance: sidepeaks at Requires change of Cooper pair number - only possible through anomalous tunneling: without Maoranas no side-peaks Similar sidepeaks away from resonance
20 Multi-terminal case Zazunov, Altland & Egger, NJP 2014 Altland & Egger, PRL 2013 Beri & Cooper, PRL 2012; Beri, PRL 2013 Altland, Beri, Egger & Tsvelik, arxiv: & arxiv: Now N>1 helical wires: M Maoranas modes, tunnel-coupled to helical Luttinger liquid wires, g 1 Bosonization of leads: Klein-Maorana fusion Klein factors additional Maorana fermion for each lead Combine Klein-Maorana and true Maorana at each contact to build auxiliary fermion f All occupation numbers f + f are conserved and can be gauged away: purely bosonic problem remains!
21 Charging effects: dipole confinement High energy scales : charging effects irrelevant Electron tunneling amplitudes from lead to dot renormalize independently upwards 1+ 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 teleportation 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 2
22 RG equations in dipole phase Energy scales below E c : effective phase action S = [ Φ] λ ( Φ Φ ) k τ cos k S Lutt d k One-loop RG equations dλ k = g dl ( 1 1) λ k + ν λ mλmk suppression by Luttinger tunneling DoS enhancement by dipole fusion processes RG-unstable intermediate fixed point with isotropic couplings (for M>2 leads) 1 * g 1 λ k = λ = ν M 2 M m ( Lead DoS, k )
23 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)
24 Topological Kondo effect Refermionize for g=1: H = i dx Maorana bilinears M + Ψ xψ + iλ = 1 k Ψ Reality condition: SO(M) symmetry [instead of SU(2)] nonlocal realization of quantum impurity spin Nonlocality ensures stability of Kondo fixed point Real Maorana basis Ψ x = µ + i SO 2 (M) Kondo model H S k = iγ γ k + ( ) ξ ( 0) S Ψ ( 0) k k for leads: = T T i dxµ µ + iλµ ˆ 0 x Beri & Cooper, PRL 2012 ( 0) Sµ ( ) + [ µ ξ ]
25 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 2 2 y 2 I 2e = = T 1 G k e 1 δ T k µ k h K M 1 Non-integer scaling dimension y = 2 g 1 > M implies non-fermi liquid behavior even for g=1 completely isotropic multi-terminal unction 1
26 Fano factor Zazunov et al., NJP 2014 Backscattering correction to current near unitary limit for µ = 0 2 y 2 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 2 2 y 2 ~ 2ge 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 2006; Mora et al., PRB 2009
27 Maorana spin dynamics Altland, Beri, Egger & Tsvelik, arxiv: Overscreened multi-channel Kondo fixed point: massively entangled effective impurity degree remains at strong coupling: Maorana spin Probe and manipulate by coupling of Maoranas H = h Zeeman fields h : overlap of Maorana k = h k wavefunctions within same nanowire Couple to S = iγ γ k k Z k k S k
28 Maorana spin near strong coupling Bosonized form of Maorana spin at Kondo fixed point: = γ γ S cos 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 i ( ) k 2 y Z M [ Θ ( 0) Θ ( 0) ] k T < T < dominant 1-2 Zeeman coupling: h T K T h h 12 = T K M / 2 T K
29 Crossover SO(M) SO(M-2) Lowering T below T h crossover to another Kondo model with SO(M-2) (Fermi liquid for M<5) Zeeman coupling h 12 flows to strong coupling γ,γ 1 2 disappear from low-energy sector Same scenario follows from Bethe ansatz solution Altland, Beri, Egger & Tsvelik, arxiv: Observable in conductance & in thermodynamic properties
30 SO(M) SO(M-2): conductance scaling for single Zeeman component h 12 0 consider G ( 1,2 ) (diagonal element of conductance tensor)
31 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 2 l 3 ~ Observable consequences for time-dependent Zeeman field B ε h with B t = B ω t, B cos ω,0 kl kl 1 cos 1 2 2t 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 2 =0) 12 ε kl s kl ( t) ~ B B [( ω ± ω ) t] s3 1 2 cos 1 2 = ε = ( ) ( ( ) ( ) ) s 2 ( t) s ( 0) 3 ~ B 1 ( ω1t ) ( ω t) 2 / 3 cos 1 S kl
32 Conclusions Coulomb charging effects on quantum transport through Maorana nanowires: Two-terminal device: Maorana single-charge transistor Zazunov, Levy Yeyati & Egger, PRB 84, (2011) Hützen, Zazunov, Braunecker, Levy Yeyati & Egger, PRL 109, (2012) Multi-terminal device: Topological Kondo effect with stable non-fermi liquid behavior With electron-electron interactions in the leads: novel unstable fixed point Altland & Egger, PRL 110, (2013) Zazunov, Altland & Egger, NJP 16, (2014) Dynamics of impurity spin in strong coupling regime Altland, Beri, Egger & Tsvelik, arxiv: Non-Fermi liquid manifold: coupling to bulk superconductor (not discussed here) Eriksson, Mora, Zazunov & Egger, arxiv: , THANK YOU FOR THE ATTENTION!
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