Topological Superconductivity with Magnetic Atoms

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1 Topological Superconductivity with Magnetic Atoms Yang Peng Falko Pientka Yuval Vinkler-Aviv Felix von Oppen L.I. Glazman Yale University Phys. Rev. B 88, (2013), Phys. Rev. B 89, (R) (2014), Phys. Rev. Lett. 114, (2015), arxiv: SPICE Workshop, August 2015

2 Topological Superconductivity with Magnetic Atoms Yang Peng Falko Pientka Felix von Oppen Yuval Vinkler-Aviv L.I. Glazman Yale University SPICE Workshop, August 2015

3 Outline Motivation: predictions & observations for engineered Majoranas Electronic structure of a magnetic atom in a superconductor A chain of embedded magnetic atoms: 1D band in the dilute ( Shiba chain ) and dense ( wire ) limits Induced superconductivity in the 1D band Majoranas at the chain ends and 1D bulk DOS

4 Engineered Majorana states: SO wire A quantum wire with SO interaction in proximity with s-wave superconductor placed in a uniform magnetic field these states map onto 1D p-wave superconductor gap in the host s-wave superconductor

5 Engineered Majorana states: B-helix A quantum wire without SO SO interaction in placed proximity in a helical with s-wave magnetic superconductor field placed in a uniform magnetic field Is equivalent to SO wire in a uniform B field (by gauge transf.) forms a 1D p-wave superconductor B. Braunecker et al., (2010), Choy et al., (2011), Kjaergaard + et al., PRB (2012), Ends of the proximity-induced 1D p-wave superconducting wire carry Majorana states

6 Engineered Majoranas: magnetic atoms Helical B-field Analogy with a proximitized nanowire in a helical field (low energies, ) Klinovaja et al., PRL; Braunecker&Simon, PRL ; Vaziveh&Franz, PRL (2013)

7 RKKY vs. AF; other interactions Superconductivity is irrelevant for RKKY at Menzel et al, PRL 108, (2012) (a mild constraint) N.Yao et al PRL 2014 Magnetic order in a 1D chain of magnetic impurities Other interactions: single-ion anisotropy (weak for L=0 impurities); Dzyaloshinskii-Morya interaction (picks chirality; affects pitch, angle?) long history of studies (, R.J. Elliott, PRB1961,, Yosida, Theory of Magnetism, Springer, 1996)

8 Experiment (S. Nadj-Perge et al, Princeton) The atomic chain is ferromagnetic, not helical Fe-Fe approx 4.1 Pb-Pb approx 3.5 Fe:Pb The induced (p-wave?) gap is narrow (if any) The end (Majorana?) states are strongly localized ( )

9 Experiment (S. Nadj-Perge et al, Princeton) The atomic chain is ferromagnetic, not helical Fe-Fe approx 4.1 Pb-Pb approx 3.5 Fe:Pb The induced (p-wave?) gap is narrow (if any) The end (Majorana?) states are strongly localized ( ) A 1D chain of Yu-Shiba-Rusinov states ( helical Shiba chain ), any energy F. Pientka et al., PRL 2015

10 The perceived puzzle The induced (p-wave?) gap is narrow (if any) The end (Majorana?) states are strongly localized ( ) arxiv: v1 JCCM-October 2014: Commentary on S. Nadj-Perge et al by P. A. Lee:

11 Magnetic Atom in a Normal-Metal Host Anderson impurity model PW Anderson, PR 124, 41 (1961) no hybridization : energy of the first electron is ; the addition energy for the second electron is

12 Magnetic Atom in a Normal-Metal Host Levels broadened by hybridization Magnetic moment formation resonance MAGNETIC We are interested in

13 Magnetic Atom in a Superconductor, Levels broadened by hybridization resonance upon increasing at no bulk states to hybridize with a discrete d-level state within the superconducting gap at Fermi sea a discrete state remains within the gap (cross-over to a Shiba state: a bound, sub-gap state of a quasiparticle)

14 Shallow Shiba States, Exchange integral Density of states Identical to potential in 1D

15 Sub-gap (Shiba) state: repulsion from the qp band Density of states Slow spatial decay

16 Weight of the d-level in the sub-gap state [Yu-Shiba-Rusinov ( ) limit] at at at any Shiba, Progr Theor Phys 50 (1973)

17 Isolated Shiba from BdG equations is a 4-component vector in spin and particle-hole space Solve for Fourier transform Inverse Fourier transform yields an eigenvalue (4x4 matrix) equation; eigen- values, vectors:

18 Shiba bands from BdG equations z-oriented magnetic moment polarization, do Fourier transf. generalize for N moments of arbitrary Pretend to solve, do inverse Fourier, get a 4Nx4N matrix equation All states are encoded here, including the entire 3D particle-hole continuum, J E is an energy-dependent matrix

19 the deep-shibas projection Shiba states well separated from continuum

20 Long-range hopping Slow spatial decay

21 the projected Hamiltonian 2Nx2N matrix Includes long-range hopping: spins of available Shiba states and odd in i-j, manifestly p-wave pairing: spins of a Cooper pair in the host supercond

22 Spin helix bulk SO & estimates SO angle for the part of SO interaction violating inversion symmetry (equiv. by gauge transf., Jian Li et al, PRB 2014) Band width of Shiba states at Characteristic velocity p-wave gap p-wave coherence length, independent of

23 Shiba Chain: Solving for the Spectrum solvable by discrete Fourier transformation Includes long-range hopping: and odd in i-j, manifestly p-wave pairing:

24 Phase diagram from rigorous solution S S

25 Topological index Topological phase, B winds around origin Top. index is 0

26 Majoranas in a Shiba chain Majorana wave function at S v S

27 Shiba chain Majoranas: a bigger picture This is a distant and tiny tail, that follows an exponential decay v

28 Shiba band is narrow : Majoranas require fine-tuning 1 mev scale Fine-tuning on mev scale In experiment, the Fe atoms are closely spaced (~0.4 nm)

29 Magnetic ion in a host metal: Anderson impurity 1 ev scale Fe: 3d shell

30 1D chain of impurities: spin-polarized d-band 1 ev scale not-so-fine-tuning on 1 ev scale Spin-polarized, itinerant electrons + SO interaction in the host

31 Estimates of 1D band parameters at Reduces the hopping attempt frequency! Shiba, Progr Theor Phys 50 (1973) Pb host Characteristic velocity p-wave gap at at W~Γ, matches the Shiba limit Fe dominates over the indirect hopping at Fe p-wave coherence length, independent of

32 Details of real calculation - 1 Hartree-Fock for a chain of Anderson impurities in a superconducting host, 8Nx8N matrix equation (superconductor + d-levels) Solve for and exclude d-levels degrees of freedom, reduce to 4Nx4N matrix equation (for the host component only)

33 Details of real calculation - 2 Project on single spin direction for each site, 2Nx2N matrix equation (spin-polarized d-levels) Discrete Fourier transform, 2x2 matrix eigenvalue equation non-local bare-wire spectrum Excitations E(k), DOS, Majorana states spectrum renormalization induced by hybridization the induced p-wave gap

34 Phase Diagram and Excitations

35 Density of States and Majoranas end state (artificially broadened) Van Hove singularities coming from flat parts of the excitations band short, independent of Estimates of Γ, W, E d +U: from numerical simulations in: Jian Li et al, PRB (2014); estimate of p : from STM, ibid.

36 Resolving the Spectrum SACLAY

37 Peng, Pientka, Vinkler, LG, von Oppen, arxiv June 2015 Resolving the Spectrum

38 Conclusions short, independent of A better way to study spectrum: superconducting STM tip

CLASSIFICATION OF TOPOLOGICAL INSULATORS AND SUPERCONDUCTORS, RESPONSES AND QUANTUM ANOMALIES

CLASSIFICATION OF TOPOLOGICAL INSULATORS AND SUPERCONDUCTORS, RESPONSES AND QUANTUM ANOMALIES ANDREAS W.W. LUDWIG (UC-Santa Barbara) work done in collaboration with: Shinsei Ryu (UC-Berkeley) Andreas Schnyder