1. 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
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2. 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
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3. 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.
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4. 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)
Hui Zhai, International Journal of Modern Physics B, 2012
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5. 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 Phys. Rev. Lett
Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S and Zwierlein M W Phys. Rev. Lett
For K40
Hui Zhai, Rep. Prog. Phys. 78 (2015)
Hui Zhai, International Journal of Modern Physics B, 2012
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6. 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= (b) α= 1, h=0 (c) α= 0.4, h=1
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7. 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
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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
Order parameter:
s-wave pairing peak:
p-wave pairing peak:
Magnetization:
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9. 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
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10. 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
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11. Kitaev model and parity
(a) Degeneracy: 2-fold
belong to different parity space.
Label as and
(b) Parity:
(c) How about number conservation systems?
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12. 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:
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13. 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
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14. 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
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15. 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
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16. 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?).
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17. Thank you!
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