Features of edge states and domain walls in chiral superconductors - PowerPoint PPT Presentation

Features of edge states and domain walls in chiral superconductors Manfred Sigrist NQS2017, YITP Kyoto E k k

Chiral superconductors Chiral superconductors - candidates tetragonal URu 2 Si 2 Sr 2 RuO 4 crystal structure odd-parity k = ∆ 0 ( k x ± ik y ) ∆ ~ chiral p -wave even-parity k = ∆ 0 ( k x ± ik y ) k z ∆ ~ chiral d -wave µ SR L z = ± 1 polar Kerr effect polar Kerr effect

Chiral superconductors Chiral superconductors - candidates hexagonal SrPtAs UPt 3 crystal structure Pt U odd-parity k = ∆ 0 ( k x ± ik y ) 2 k z ∆ ~ chiral f -wave even-parity k = ∆ 0 ( k x ± ik y ) 2 ∆ ~ chiral d -wave µ SR µ SR L z = ± 2 polar Kerr effect

Content focus on Sr 2 RuO 4 as a chiral p -wave SC edge states and edge currents in a chiral p -wave SC chiral domains many other collaborators: Y. Imai, K. Wakabayashi, A. Furusaki, M. Matsumoto, C. Honerkamp, M. Fischer, T.M. Rice, J. Goryo, W. Huang, ... Sarah Etter Adrien Bouhon former doctor students at ETH Zurich

Sr 2 RuO 4 possible odd-parity spin-triplet states Maeno et al 1994 ✓ ◆ 0 k x ± ik y ˆ Ψ ( ~ k ) = layered crystal 0 k x ± ik y structure quasi-2D metal A -phase chiral phase pair wave function ✓ − k x + ik y ◆ 0 ✓ Ψ ↑↑ Ψ ( ~ ˆ k ) = ◆ Ψ ↑↓ 0 k x + ik y ˆ Ψ = Ψ ↓↑ Ψ ↓↓ B -phase helical phase

Sr 2 RuO 4 - chiral p-wave superconductor Maeno et al 1994 ✓ ◆ 0 k x ± ik y ˆ Ψ ( ~ k ) = layered crystal 0 k x ± ik y structure quasi-2D metal A -phase chiral phase identifaction • intrinsic magnetism ✓ − k x + ik y ◆ 0 • inplane spin polarizable Ψ ( ~ ˆ k ) = 0 k x + ik y • multi-component • polar Kerr effect B -phase helical phase • phase-sensitive SQUID

Sr 2 RuO 4 - chiral p-wave superconductor Maeno et al 1994 ke ± i θ k ✓ ◆ 0 ˆ Ψ ( ~ k ) = layered crystal ke ± i θ k 0 structure quasi-2D metal A -phase chiral phase identifaction phase winding around the FS • intrinsic magnetism gap function • inplane spin polarizable nodeless gap ∆ k = | ∆ 0 | e + i θ k • multi-component 2-fold degenerate • polar Kerr effect ∆ k = | ∆ 0 | e − i θ k chiral domains • phase-sensitive SQUID chiral - broken time reversal symmetry

intrinsic magnetism in Sr 2 RuO 4 ? random local magnetism edge state currents ''edge currents'' around scanning probes at mesoscopic discs inhomogeneities & defects disc SC N R ≈ 5 µ m µ SR - zero-field relaxtion SC-N T > T c T < T c Luke et al (1998) scanning Hall probe muon-spin depolarization intrinsic magnetism Mackenzie group (2014)

Edge currents

Sr 2 RuO 4 - edge state spectrum edge states for the chiral p-wave state scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space) “Andreev reflection” hole Bohr-Sommerfeld quantization I 1 k k surface θ p · d s + φ k 1 + φ k 2 = 2 π n ~ k 1 k 2 phase shifts at turning points ∆ k = | ∆ 0 | e i θ k for | E | ⌧ | ∆ 0 | φ k 1 + φ k 2 = π + θ k 2 − θ k 1 electron | p | ≈ E E = 0 θ k 2 − θ k 1 = π v F specular scattering

Sr 2 RuO 4 - edge state spectrum edge states for the chiral p-wave state scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space) “Andreev reflection” hole Bohr-Sommerfeld quantization k k k k surface θ E = E k k = ∆ 0 sin θ = ∆ 0 k 1 k 2 k F 2 θ = π − ( θ ~ k 1 ) k 2 − θ ~ ∆ k = | ∆ 0 | e i θ k electron phase shifts specular scattering

Sr 2 RuO 4 - bulk and edge spectrum edge states for the chiral p-wave state scattering of quasiparticles at the surface solution of Bogolyubov-de Gennes equations subgap bound states (close orbits in particle-hole space) “Andreev reflection” E Bohr-Sommerfeld quantization continuum k k + ∆ 0 E = E k k = ∆ 0 sin θ = ∆ 0 − k F k F k k + k F − ∆ 0 2 θ = π − ( θ ~ k 1 ) k 2 − θ ~ continuum phase shifts note: θ ~ k 2 − θ ~ k 1 = ± π E = 0 result of topological property

Sr 2 RuO 4 - bulk and edge spectrum order parameter deformation E continuum ∆ ~ k = η x k x + η y k y + ∆ 0 − k F | η y | occupied | ∆ 0 | k k + k F − ∆ 0 | η x | continuum x driving currents hole c h fields surface spontaneous a charge B z r current B z ∼ 10 G g e screening electron currents λ

Topology and edge currents Are edge currents a unique topological property? lattice version of chiral p-wave superconductor (tight-binding): k = − 2 t (cos k x a + cos k y a ) + .... − ✏ F ⇠ ~ k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ +1-winding k y zeros of ∆ ~ k -1-winding k x Chern numbers N = +1 1 st Brillouin zone

Topology and edge currents Are edge currents a unique topological property? lattice version of chiral p-wave superconductor (tight-binding): k = − 2 t (cos k x a + cos k y a ) + .... − ✏ F ⇠ ~ k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ +1-winding k y zeros of ∆ ~ k -1-winding k x Chern numbers = +1 − 4 × 1 N 2 = − 1 × 1 = − 1 1 st Brillouin zone

Topology and edge currents k y k y N = +1 N = − 1 k x k x E E k k k k

Topology and edge currents k y k y N = +1 N = − 1 k x k x quasiparticle velocity dE k k v q = 1 E E ~ dk k k k k k v q > 0 v q < 0

Topology and edge currents k y k y charge current N = +1 N = − 1 k x k x hole c h surface k k a r g e electron E E backward backward k k k k opposite chirality, but identical current direction

Topology - thermal Hall effect “Spontaneous” Righi-Leduc effect Chern number T 2 chiral QP edge states κ xy = π k 2 B T temperature ∆ kBT ) 12 ~ N + O ( e − gradient heat current Read & Green; Qin, Niu & Shi; Sumiyoshi & Fujimoto; … T 1 chiral QP edge states Lishitz-transition property of the N = +1 ( topological transition) 12 ~ κ xy (subgap) quasiparticles π k 2 B T T ⌧ T c N = − 1 µ

Topology - thermal Hall effect “Spontaneous” Righi-Leduc effect Chern number T 2 chiral QP edge states κ xy = π k 2 B T temperature ∆ kBT ) 12 ~ N + O ( e − gradient heat current Read & Green; Qin, Niu & Shi; Sumiyoshi & Fujimoto; … T 1 chiral QP edge states Lishitz-transition property of the N = +1 ( topological transition) 12 ~ κ xy (subgap) quasiparticles π k 2 B T T ⌧ T c N = − 1 µ v QP > 0 v QP < 0 analog to ν =1 Quantum Hall effect

Topology and edge currents - a further twist particle-hole symmetry n = (1 , 0) ~ y k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ E x Q = ⇡ ∆ ~ Q = − ∆ ~ ~ a (1 , 1) k + ~ k k k k y ~ ~ k x k 2 k 1

Topology and edge currents - a further twist particle-hole symmetry n = (1 , 0) ~ y k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ E x Q = ⇡ ∆ ~ Q = − ∆ ~ ~ a (1 , 1) k + ~ k k k k y n = (1 , 1) ~ y ~ k 1 E x k x ~ k 2 k k

Topology and edge currents - a further twist particle-hole symmetry n = (1 , 0) ~ y k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ E x Q = ⇡ ∆ ~ Q = − ∆ ~ ~ a (1 , 1) k + ~ k k k k y n = (1 , 1) ~ y ~ k 1 E x k x ~ k 2 k k

Topology and edge currents - a further twist particle-hole symmetry n = (1 , 0) ~ y k = ∆ 0 (sin k x a + i sin k y a ) ∆ ~ E x Q = ⇡ ∆ ~ Q = − ∆ ~ ~ a (1 , 1) k + ~ k k k k y n = (1 , 1) ~ y ~ k 1 E x ~ k x Q ~ k 2 new zero-energy momenta k k θ ~ k 2 − θ ~ k 1 = ± π E = 0 no change in Chern number

Topology and edge currents - a further twist n = (1 , 1) ~ E E k k k k current reversal k y k y ~ k 1 ~ k 1 ~ k x k x Q ~ ~ k 2 k 2

Topology and edge currents - a further twist n = (1 , 1) ~ E E k k k k current reversal y y x x A. Bouhon non-circular supercurrent circular supercurrent

Ginzburg-Landau approach - chiral p-wave order parameter: d ( k ) = ˆ z η · k = ˆ z ( η x k x + η y k y ) Z a | η | 2 + b 1 | η | 4 + b 2 ( η ∗ 2 x η 2 y + η 2 x η ∗ 2 y ) + b 3 | η x | 2 | η y | 2 � F = dV + K 1 ( | Π x η x | 2 + | Π y η y | 2 ) + K 2 ( | Π x η y | 2 + | Π y η x | 2 ) + [ K 3 ( Π x η x ) ∗ ( Π y η y ) + K 4 ( Π x η y ) ∗ ( Π y η x ) + c.c. ] Π = ~ i r + 2 e + K 5 | Π z η | 2 + ( r × A ) / 8 π and c A

Ginzburg-Landau approach - chiral p-wave order parameter: d ( k ) = ˆ z η · k = ˆ z ( η x k x + η y k y ) Z a | η | 2 + b 1 | η | 4 + b 2 ( η ∗ 2 x η 2 y + η 2 x η ∗ 2 y ) + b 3 | η x | 2 | η y | 2 � F = dV + K 1 ( | Π x η x | 2 + | Π y η y | 2 ) + K 2 ( | Π x η y | 2 + | Π y η x | 2 ) + [ K 3 ( Π x η x ) ∗ ( Π y η y ) + K 4 ( Π x η y ) ∗ ( Π y η x ) + c.c. ] Π = ~ i r + 2 e + K 5 | Π z η | 2 + ( r × A ) / 8 π and c A length scales for amplitude modulations for the two order parameter components

Ginzburg-Landau approach - chiral p-wave order parameter: d ( k ) = ˆ z η · k = ˆ z ( η x k x + η y k y ) Z a | η | 2 + b 1 | η | 4 + b 2 ( η ∗ 2 x η 2 y + η 2 x η ∗ 2 y ) + b 3 | η x | 2 | η y | 2 � F = dV + K 1 ( | Π x η x | 2 + | Π y η y | 2 ) + K 2 ( | Π x η y | 2 + | Π y η x | 2 ) + [ K 3 ( Π x η x ) ∗ ( Π y η y ) + K 4 ( Π x η y ) ∗ ( Π y η x ) + c.c. ] Π = ~ i r + 2 e + K 5 | Π z η | 2 + ( r × A ) / 8 π and c A edge currents for chiral phase