Yasu Asano (Hokkaido Univ.) Novel Quantum States in Condensed Matter - PowerPoint PPT Presentation

Flat-band Andreev bound states and Odd-frequency pairs Yasu Asano (Hokkaido Univ.) Novel Quantum States in Condensed Matter at Yukawa Institute of Theoretical Physics 16 November, 2017, MEXT of Japan Core-to-core by JSPS

Outline Summary Flat-band Andreev bound states in a nodal SC Paramagnetic response of a superconducting disk Conductance in a NS hybrid Relation to Majorana physics Tunable - junction with a QAHI ϕ

Unconventional Superconductors Andreev bound states with flat dispersion at a clean surface x=0 Nodal! (out of the ten-fold symmetry classes) Sign change is necessary to be nontrivial k y - + - γ k x + - + - + f Fermi surface s p d xy x d xy f p x

Topological characterization M. Sato. et. al, PRB(2011) Dimensional reduction ky Fix ky and consider 1D BZ s ± = ∆ ( ± k x , k y ) s + s − - + | ∆ ( ± k x , k y ) |  � ∆ ( k ) ξ H = = ∆ ( k ) τ 1 + ξτ 3 ∆ ( k ) − ξ Z π dk x Tr[ τ 2 H ( k x ) − 1 ∂ k x H ( k x )] = 1 − s + s − w 1D ( k y ) = i s + 2 π 2 − π

What happen on the flat ZESs under the potential disorder? How the flat ZESs affect observable values? Clean Potential disorder Theory Experiment (Translational symmetry) High degeneracy High symmetry d xy f p x w 1D = 1 or − 1

Zero-bias conductance in a NS junction Classical Ohm’s law would be expected G − 1 NS = R NS = R B + R N R N →∞ G NS = 0 lim

Quasiclassical Usadel equation in N Quantum Ohm’s law In a singlet superconductor In a triplet superconductor Tanaka et. al. PRB (2004) G Q = 2 e 2 h with ✏ = 0 R N →∞ G NS = 0 lim Im( θ ) = 0 R N →∞ G NS = 4 e 2 θ = i ( x/L + 1) β 0 h | N ZES | lim β 0 =2 G Q R N N ZES

an invariant in differential equation Flat surface Andreev bound states Topological classification of nodal SC Dimensional reduction Chiral symmetry of Hamiltonian topological invariant d xy f p x H BdG = ( ξ + V )ˆ τ 3 + ∆ ˆ τ 1 w 1D = 1 or − 1 { H BdG , − ˆ τ 2 } + = 0 X N ZES = w 1D ( k y ) eigenvalue of − ˆ λ = 1 or − 1 τ 2 k y N ZES = N + − N −

In mathematics, Atiyah-Singer Index theorem connects topology and analysis topological invariant The number of ZES X N ZES = w 1D = N + − N − k y belong to λ = ± 1

linear combination of Sato et.al., PRB (2011) and (2) non ZES: chirality one-by-one Ikegaya, YA, PRB (2015) chiral symmetry { ˆ τ 2 } + = 0 H BdG , − ˆ eigenvalue of − ˆ τ 2 λ = 1 or − 1 (1) ZES: eigenstate of − ˆ τ 2 λ = 1 λ = − 1 χ E 6 =0 = a + χ + + a � χ � , | a + | = | a � |

robust E=0 robust E=0 fragile ZESs at a clean surface f px dxy E=0 d xy f p x translational sym. λ: Fragile 1 −1 disorder different chirality N ZES =0 N + =1 N − =1 disorder same chirality N ZES =2 Robust! N + =2 N − =0 disorder Robust! different chirality N + =2 N − =1 N ZES =1

In physics, describes the number of zero-energy states that penetrate into dirty normal metal and Quantization of Conductance minimum form resonant transmission channels Atiyah-Singer index X N ZES = w 1D = N + − N − k y R N →∞ G NS = 4 e 2 h | N ZES | lim

Classification in bulk |Z| dirty surface ZESs at a |Z| clean surface ZESs at a (if TRS is preserved) W(k) Schnyder et.al. (2008) Z Topo # necessary not necessary symmetry Translational (to be nontrivial) nodal full gap Pair potential Real SC Ikegaya, Suzuki, Tanaka, YA, PRB 94, 054512 (2016) X | W ( k ) | k | N ZES |

Conductance minimum is quantized at Atiyah-Singer index Cooper pairs? Spin X Parity X Frequency In SC triplet p-wave (odd) even In dirty N triplet s-wave (even) odd to satisfy a requirement of Fermi-Dirac statistics degenerate ZESs spin-triplet SC

Symmetry Classification singlet Spin Orbital s, d (even-parity) triplet p, f (odd-parity) Spin-flip potential mix spin-singlet and spin-triplet Fourier trans. f σ , σ 0 ( r � r 0 ) = � h ψ σ ( r ) ψ σ 0 ( r 0 ) i f σ , σ 0 ( p ) spin X orbital = - 1 Fermi-Dirac statistics Surface & interface mix even- and odd-parity

Odd-freq. Pairs General definition of pairing function Fourier trans. Topological surfaces (ZES) generate odd-freq. pairs f σ , σ 0 ( r � r 0 , τ � τ 0 ) = � h T τ ψ σ ( r, τ ) ψ σ 0 ( r 0 τ 0 ) i f σ , σ 0 ( p, ω n ) spin X orbital X frequency = - 1

Paramagnetic response of a small superconductor Diamagnetic Paramagnetic Odd-frequency pairs are paramagnetic! YA, Golubov, Fominov, Tanaka, PRL 107 , 087001 (2011)

Small unconventional superconductors are paramagnetic due to odd-freq. pair at their surface Solve Eilenberger and Maxwell Eqs. simultaneousely Pair potential and vector potential are determined self-consistently on 2D disks We consider… Suzuki and YA, PRB 89 , 184508 (2014)

Paramagnetic response of a singlet d-wave SC subdominant component odd-freq. paramagnetic χ ( r ) = [ H ( r ) − H ex ] / [4 π H ex ] R =3 ξ 0 λ L =3 ξ 0 H ex =0 . 001 H c 2

Paramagnetic response of a triplet p-wave SC odd-freq. paramagnetic subdominant component

Susceptibility v.s. Temperature d-wave p-wave Crossover to paramagnetic phase at low temperature odd-freq. pairs are paramagnetic energetically localize near E=0 Tp

Crossover temperature v.s. Size of disk In larger discs, relative area of ‘surface’ becomes smaller odd-freq. pairs are confined at surface within ξ 0

Any difference between p and d? Yes! in the presence of surface roughness

Effects of surface roughness Suzu ki and Asano, PRB 91, 214510 (2015) Odd-w p-wave Odd-w s-wave N ZES 6 = 0 N ZES = 0

Susceptibility v.s. Temperature under surface roughness

Relating papers on d-wave SC Higash itani, JPSJ 66 , 2556 (1997) Fogelstrom, Rainer, and Sauls, PRL 79 , 281 (1997) Barash, Kalenkov, and Kurkijarvi, PRB 62 , 6665 (2000) Zare, Dahm, and Schophl, PRL 104 , 237001 (2010) Vorontsov, PRL 102 , 177001 (2009). Hakans son, Lofwander and Fogelstrom, Nat. Phys. 11 , 755 (2015). Suzuki and YA, PRB 89 , 184508 (2014) Suzuki and YA, PRB 91 , 214510 (2015) Suzuki and YA, PRB 94 , 155302 (2016) Our papers on d, p, chiral-d, chiral-p, chiral-f energetics of flat-band ZESs odd-frequency pairs

Trouble! A spin-triplet p-wave superconductor has never been discovered yet! Why don’t we make it? Sure! Why not! Ikegaya, Kobayashi, YA, in preparation N ZES 6 = 0

What we have done spin-triplet p-wave Necessary conditions? single-band BdG Hamiltonian must belong to the class BDI specify realistic models A sufficient condition for N ZES 6 = 0

Dresselhaus [110] + in-plane Zeeman Solutions Alicea, PRB 81, 125381 (2010) You, Oh, Vedral, PRB 87, 054501 (2013) 2D helical p-wave + in plane Zeeman Mizushima, Sato, Machida, PRL 109, 165031 (2012) Wong, Oriz, Law, Lee, PRB 88, 060504 (2014) Majorana! N ZES = Majorana number

Majorana SCs SCs with ? N ZES 6 = 0

Tunable -junction with a QAHI Sakurai, Ikegaya, and YA, arXiv:1709.02338. ϕ

Josephson Junction -junction 0-junction SIS SXS SFS J ( θ ) ∝ ∂ θ E ( θ ) J 0 0 0 J J J -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 θ / π θ / π θ / π E 0 0 0 E E E -1.0 0.0 1.0 -1.0 0.0 1.0 -1.0 0.0 1.0 θ / π θ / π θ / π θ / π θ / π θ / π ϕ -junction π

-junction Reynoso,et. al., PRL 101, 107001 (2008). Current at zero phase difference value built-in …. Buzdin, PRL 101, 107005 (2008) 
 Dolcini, et. al., PRB 92,035428 (2015) Tanaka, et. al., PRL 103, 107002 (2009). Campagnano, et. al., J. Phys.27, 2053012015). Zazunov, et. al., PRL 103, 147004 (2009). Dell’Anna, et. al, PRB 75, 085305 (2007). Heim, et. al., J. Phys. 25, 215701 (2013). TRS + Inversion Breaking YA et. al, PRB 2007 F F F S S Yokoyama, Eto, Nazarov, PRB 89, 195407 (2014). ϕ J = J 0 sin( θ − ϕ ) = J sin θ cos ϕ − J 0 cos θ sin ϕ J ∝ ( M 1 × M 2 · M 3 ) cos θ + J 0 sin( θ ) ϕ

Quantum Anomalous Hall Insulator Zeeman Spin-orbit ˆ H Q ( r ) =( ε r − m z )ˆ σ 3 + i λ∂ x ˆ σ 2 − i λ∂ y ˆ σ 1

Current-phase relationship (CPR) ϕ ϕ θ / π

Andreev reflections ① ① ② ② ③ ③ ④ ④ e ik e L e − i θ R e − ik h L e i θ L = e i θ e i ( k e − k h ) L − ϕ k e = k h = 0 k e = − k 1 , k h = k 1

Magnetic mirror reflection symmetry Zeeman random H = ˇ ˇ H L + ˇ H R + ˇ H Q H L,R ( − θ L,R ) = ˇ ˇ H ∗ L,R ( θ L,R ) ˇ Q = ˇ 0 or π E ( θ ) = E ( − θ ) H ∗ H Q ˇ Q 6 = ˇ E ( θ ) 6 = E ( � θ ) ϕ -junction H ∗ H Q ˆ + V ( x, y ) − V y ˆ H Q ( r ) =( ε r − m z )ˆ σ 3 + i λ∂ x ˆ σ 2 − i λ∂ y ˆ σ 1 σ 2 ˆ Q ( r ) =( ε r − m z )ˆ σ 3 + i λ∂ x ˆ σ 2 + i λ∂ y ˆ + V ( x, − y ) H ∗ σ 1 + V y ˆ σ 2 This sign can be changed by y → − y

Impurity potential Changing width Zeeman field Impurities Junction shape ϕ -junction

Summary Conductance minimum and index theorem Paramagnetic response of a small superconductor Flat-band Andreev bound states in a nodal SC Flat-band ZESs = Majorana Flat-band ZESs = odd-frequency Cooper pairs odd-frequency pair Andreev bound state Majorana BS