SOC Fermi Gas in 1D Optical Lattice —Exotic pairing states and Topological properties 中科院物理研究所 胡海平 Collaborators : Chen Cheng, Yucheng Wang, Hong-Gang Luo,Shu Chen 08/02/2015 Aug. 2, 2015 HHP 1

Outlook: Introduction to 1D SOC Bosonization study Experimental realization, Single particle physics Bosonization study Half-Filling: FFLO-BCS transition Topological Superfluid and MFs p-wave Superfluid, MFs and Number conservation Phase diagram Edge states Conclusions Aug. 2, 2015 HHP 2

Why study SOC? Spintronics Topoloical insulators 2D topological insulators:QSHE 3D topological insulators protected by time reversal symmetry Topological Superconductors Intrinsic topological superconductors Effective p-wave superonductors: s-wave +SOC+Zeemann field. Aug. 2, 2015 HHP 3

Non-abelian gauge field The ground states of have m-fold degeneracy, one can obtain a non-Abelian adiabatic gauge potential-Berry's connection which is generically a matrix. Hui Zhai, Rep. Prog. Phys. 78 (2015) 026001 Hui Zhai, International Journal of Modern Physics B, 2012 Aug. 2, 2015 HHP 4

Model and experimental setups Lin Y-J, Jiménez-García K and Spielman I B 2011 Nature 471,83 Wang P, Yu Z-Q, Fu Z, Miao J, Huang L, Chai S, Zhai H and Zhang J 2012 Phys. Rev. Lett. 109 095301 Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S and Zwierlein M W 2012 Phys. Rev. Lett. 109 095302 For K40 Hui Zhai, Rep. Prog. Phys. 78 (2015) 026001 Hui Zhai, International Journal of Modern Physics B, 2012 Aug. 2, 2015 HHP 5

Single-particle Physics Exotic pairings : FFLO, BCS , FFLO -BCS Coexistence Lattice Model: Topological Superfluid States: p-wave nature after including pairings induced by proximity effects: (a) α= 0, h=0.8 (b) α= 1, h=0 (c) α= 0.4, h=1 Aug. 2, 2015 HHP 6

Bosonization of chiral modes Crossing the Fermi energy Zeeman term: Interaction term: a=0,2 label the modes at k=0 and |k|=2k0 H+: single gapless phonon mode corresponding to fluctuatioons of total charge H-: gapped by cosine terms, 2 phases seperated by a critical point as the two cosine terms will compete (1)Trivial: interaction dominates , i.e., Luther-Emery (spin-gapped ) phase (2) Topological: Zeemann term dominates and is strongly fluctuating, single fermion excitation is gapless Aug. 2, 2015 HHP 7

Exotic pairings: FFLO or BCS? (a) Fixing magnetic field, increasing SOC strength: FFLO → FFLO-BCS → BCS (b) Fixing SOC, increasing magnetic field: BCS → FFLO-BCS → FFLO Order parameter: s-wave pairing peak: p-wave pairing peak: Magnetization: Aug. 2, 2015 HHP 8

Exotic pairings: FFLO or BCS? (a) Fixing magnetic field, increasing SOC strength: FFLO → FFLO-BCS → BCS (b) Fixing SOC, increasing magnetic field: BCS → FFLO-BCS → FFLO Aug. 2, 2015 HHP 9

Introduction to Majorana Fermions(MFs) 1. What is a majorana fermion? A majorana fermion is a fermion which is its own anti-particle 2. Theoretical prediction in condensed matter systems p-wave SC chain, 2D p+ip vortex core SOC+s-wave+Zeemann,… 3. Experimental progress Recent experimental signature for observing MFs in SC wires L.P.Kouwenhoven, Science, 336,1003(2012) Still on debate Aug. 2, 2015 HHP

Kitaev model and parity (a) Degeneracy: 2-fold belong to different parity space. Label as and (b) Parity: (c) How about number conservation systems? Aug. 2, 2015 HHP

Degeneracy and Number conservations Degeneracy? No! Must contain at least two TSF region which will be naturally realized in harmonic traps. For Particle Number Even: Odd: Excitation energy Thermodynamical limit: Trivial phase: TSFs: Aug. 2, 2015 HHP

Phase diagram at filling 1/4 (a) FFLO & MP & BCS (noraml pairing states) (b) LE phase (c) TSF (d) Metal phase : Here (b)(c)(d) are charge gapless (a)(b)(c) are superconducting phase with Aug. 2, 2015 HHP

Filling-1/4: Phase transitions (a) Order parameter The parings in momentum space exhibit zero-peak and two sharp shoulders around two Fermi points. (b) Gap closings (single particle gap & EB) (c)Scaling behaviour:of charge gap: Normal SC states & LE phase: Finite in TL LE & TSC phase: nearly 0 Normal gas: Linear to zero in TL Aug. 2, 2015 HHP

Edge states (a) Transverse magnetization: (b)Edge states Normal SC states A.N.D. LE phase: Mainly in bulk with modulation ½ period of Normal gas TSC phase: fluctuation mainly at the end Normal gas: Friedel Osillations in the bulk Aug. 2, 2015 HHP

Summary For filling ½, SOC induces a series of phase transitions between different pairing states : FFLO, FFLO-BCS and BCS pairings states. The order parameters including magnetization, s-wave pairing peak, p-wave pairing peak. For filling ¼, TSF states exist in a large parameter region which are characterized by its gapless charge excitation gap , and edge states. i.e., the TSF states in particle conserved system is a gapless topological state! Different phase transitions are accompanied by slope discontinuity of order parameter (2nd?). Aug. 2, 2015 HHP 16

Thank you! Aug. 2, 2015 HHP