Rashba vs Kohn-Luttinger: evolution of p-wave superconductivity in magnetized two-dimensional Fermi gas subject to spin-orbit interaction

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1 Rashba vs Kohn-Luttinger: evolution of p-wave superconductivity in magnetized two-dimensional Fermi gas subject to spin-orbit interaction Oleg Starykh, University of Utah with Dima Pesin, Ethan Lake, Caleb Webb PHYSICAL REVIEW B 93, (2016) Nordita, Multi-component and strongly-correlated superconductors, July 15, 2016

2 E.I. Rashba W. Kohn J. Luttinger under grads! Ethan Lake Caleb Webb

3 Condensed matter physics in 21 century: the age of spin-orbit spintronics topological insulators, Majorana fermions Kitaev s non-abelian spin liquid/toric code The key issue: spin-orbit + e-e interaction

4 Outline Spin-orbit + interaction: surprises in 1d Kohn-Luttinger (KL) mechanism in 2d magnetized gas KL in the presence of spin-orbit!!

5 Important: fully broken spin symmetry Eigenvalues Toy d=1 problem B=0: spin-orbit can be gauged away Eigenstates χ + and χ : orthogonal at the same k but not at the same energy spinors k < 0: clock-wise rotation of spins µ 1 2 k > 0: counterclock-wise rotation of spins

6 E-e interaction adds a new process: Cooper scattering (inter-band Josephson coupling) Cooper channel: spin non-conserving inter-subband pair tunneling possible due to Spin-Orbit only

7 SDW instability Easy limit: E F >> gµb >> αk F Free charge: Interacting spin: K c < 1 + Cooper process K s > 1 relevant! Strong-coupling limit: minimal Thus θ s is frozen, hence φ s fluctuates wildly - remember [φ,θ]=iδ(x-y). 2k F component of spin operators: but Ising order in the spin sector!

8 Spin chain with uniform DM term (J x R J x L) h z γ γ D x Rotate right (left) current by γ ( γ) (J x R J x L) Backscattering is modified

9 Magnetic field can now be absorbed Transverse to field (t) components oscillate So that The final Hamiltonian Cooper term

10 Phase diagram of Heisenberg+uniform DM chain e = J z J D 2 2J 2 Lesson:! ordered phases below this line are stabilized by spin-orbit+zeeman field + e-e interaction Gangadharaiah, Sun, OS, PRB 2008! Garate, Affleck 2010! Povarov et al, PRL 2011! Karimi, Affleck 2011! Smirnov et al 2015! Wen Jin, OS unpublished

11 Outline Spin-orbit + interaction: surprises in 1d Kohn-Luttinger (KL) mechanism in 2d magnetized gas KL in the presence of spin-orbit!!

12 Kohn-Luttinger mechanism: Superconductivity from repulsion NEW MECHANISM FOR SUPERCONDUCTIVITY* %. Kohn University of California, San Diego, La Jolla, California and J. M. Luttinger Columbia University, New York, New York (Received 16 August 1965) Attraction from Friedel oscillations U eff U eff (r) cos(2k F r) (2k F r) 3 r

13 More recent history

14 Hamiltonian Diagrams to U 2 order: k k k k k k k k U eff = + + k k k k k k k k k k k k + + p + k k k k k k p

15 Hamiltonian Diagrams to U 2 order: H I = Un " n # k k k k k k k k triplet channel U eff = + + k k k k k k k k k k k k + + p + k k k k k k p

16 k k U eff = k k p + k k p k k k k

17 Two dimensions: one-sided singularity s U2D e U 2 2kf 2 0 Re 1 q k q k k q apple 2k f always k F U e 2D =0 k

18 Zeeman field produces superconductivity k k k k k U eff = k k p p + k k k q Spin-up electrons experience effective attraction mediated by spin-down electrons! q Spin-down electrons remain in the normal state q apple 2k f,"! One-sided superconductivity 2k f,#

19 Outline Spin-orbit + interaction: surprises in 1d Kohn-Luttinger (KL) mechanism in 2d magnetized gas KL in the presence of spin-orbit!!

20 Basic solid state: spin-orbit interaction is unavoidable 2DEG E H SOC = α R (k x σ y k y σ x ) General set-up: arbitrary H but Zeeman energy >> Spin-orbit energy z y H quantization axis along the magnetic field 2DEG x

21

22 Expectations

23 1. Switch to band basis (spin is not conserved) E(k) = k2 2m 0 ~ H ~ + R (k x y k y x ) H k y H k y z y H 2DEG x k x +Q Q k x Q = m R sin chirality = ±1

24 2. Project the interaction into band basis keep U 2,U 2 R drop U 3,U 2 R Kohn Luttinger term

25 2. Project the interaction into band basis (via Schrieffer-Wolff transformation) Π < 0 KL process O(U 2 ) R 1 > 0 Repulsion O(Uα 2 R ) λ =1 λ =2 J<0 R 2 > 0 Josephson exchange Repuslion O(Uα 2 R ) O(Uα 2 R )

26 Mean-field order parameter dispersion Two solutions of self-consistent equations

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28 Decoupled phase has lower energy Coupled phase suffers exponentially from intra-band repulsion R1,2

29 Pairing symmetry coupled chirality of the order parameter is opposite to that of the band decoupled chirality of the order parameter matches that of the band

30 Experimental probe: angle-sensitive specific heat z y H from the node of the order parameter 2DEG x

31 Unexpected bonus: finite-momentum pairing c.f. D. F. Agterberg and R. P. Kaur, 2007 Can be seen from 2-particle problem:

32 Conclusions d=1: spin-orbit stabilized phases d=2: spin-orbit only relieves degeneracy

33 Previous studies: no mag. field g eff 0.02 u 2 ν 2 2D g eff u 2 ν2d Θ j z = j z =4 j z = Θ Luyang Wang, Oskar Vafek Physica C 497, pp (2014)