Effects of spin-orbit coupling on the BKT transition and the vortex- - PowerPoint PPT Presentation

Effects of spin-orbit coupling on the BKT transition and the vortex- antivortex structure in 2D Fermi Gases Carlos A. R. Sa de Melo Georgia Institute of Technology QMath13 Mathematical Results in Quantum Physics Atlanta: October 10 th , 2016 1

Main References for Talk (UNPUBLISHED) 2 2

Acknowledgements Li Han Ian Spielman Jeroen Devreese Jacques Tempere 3

Outline 1) Introduction to 2D Fermi gases. 2) Creation of artificial spin-orbit coupling (SOC). 3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. 4) The BKT transition and the vortex-antivortex structure. 5) Conclusions 5) Conclusions 4

Conclusions in words • Ultra-cold fermions in the presence of spin-orbit and Zeeman fields are special systems that allow for the study of exciting new phases of matter, such as topological superfluids, with a high degree of accuracy. • Topological quantum phase transitions emerge as function of Zeeman fields and binding energy for fixed spin-orbit coupling. 5

Conclusions in words • The critical temperature of the BKT transition as a function of pair binding energy is affected by the presence of spin-orbit effects and Zeeman fields. While the Zeeman field tends to reduce the critical temperature, SOC tends to stabilize it by introducing a triplet component in the superfluid order parameter. • In the presence of a generic SOC the sound velocity in the superfluid state is anisotropic and becomes a sensitive probe of the proximity to topological quantum phase transitions. The vortex and antivortex shapes are also affected by the SOC and acquire a corresponding anisotropy. 6

Conclusions in Pictures Change in topology TRANSITION FROM GAPLESS TO GAPPED SUPERFLUID 7

BKT transition and vortex-antivortex structure Rashba ERD 8

Outline 1) Introduction to 2D Fermi gases. 1) Introduction to 2D Fermi gases. 2) Creation of artificial spin-orbit coupling (SOC). 3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. 4) The BKT transition and the vortex-antivortex structure. 5) Conclusions 9

Condensed Matter meets Atomic Physics In optical lattices many types of atoms Neutral atoms (bosons or fermions) can be loaded like bosonic, Sodium-23, Potassium-39, Rubidium-87, or Cesium-133; and fermionic Lithium-6, Potassium-40, Strontium-87, etc… In real crystals electrons or holes (absence of electrons) may be responsible for many “electronic” phases of condensed matter physics, such as metallic, insulating, superconducting, ferromagnetic, anti- Electrons of holes (fermions only) ferromagnetic, etc… 10

How atoms are trapped? ( ) ( ) r d E = − • V t t , , • Atom-laser interaction r r r r ( ) ( ) d E r = − α ω t • Induced dipole , moment. ( ) ( ) 2 r E r = − α ω V t t , ( )[ , ] • Trapping potential 11

Atoms in optical lattices ( ) ( ) ( ) E = − α ω < > V t 2 [ , ] r r r r r r r r 1 ( ) ( ) ( ) 2 = − α ω E V [ ] r r r r r r r r 2 12

How optical lattices are created? 13

Single plane excitations Vortex-antivortex pairs BKT transition: Physics of 2D XY model 14

Critical Temperature Pairing Temperature Bose Liquid 0.125 Fermi Liquid BCS-Bose Superfluidity in 2D 15

2D Fermi gases with increasing attractive interactions, but no SOC. 16

Outline 1) Introduction to 2D Fermi gases. 2) Creation of artificial spin-orbit coupling (SOC). 2) Creation of artificial spin-orbit coupling (SOC). 3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. 4) The BKT Transition and the vortex-antivortex structure. 5) Conclusions 17

Raman process and spin-orbit coupling k k − δ Ω 2 ( )   + R   m 2 2 2   k k Ω + δ 2 ( ) −   R   m 2 2 2   18

SU(2) rotation to new spin basis: σ x � σ z ; σ z � σ y ; σ y � σ x k + Ω δ k k 2 2   + − −   i k R R     x m m 2 2 2     k δ + Ω k k 2 2    −  − i R k R   x  m m  2 2 2     19

k + Ω δ k k 2 2 spin-orbit   + − −   i k R R     x m m 2 2 2 detuning     k δ + Ω k k 2 2    −  − i R k R Raman   x  m m  2 2 2     coupling 20

Hamiltonian with spin-orbit Hamiltonia n with spin - orbit k + k + = ε − H c c h c c ( ) ( )   ks ks s s ks ks ' ' k s k s , , 21

Parallel and perpendicular fields k k = h h ( ) ( ) z || k k k = − h h ih ( ) ( ) ( ) ⊥ x y k k k ε − − h h ( ) ( ) ( )   ⊥ H k = || ( )   k k k − ε + h h 0 * ( ) ( ) ( )   ⊥ ||   22

Hamiltonian in terms of k -dependent magnetic fields Hamiltonia n Matrix H k k 1 k k k = ε − − − h h h ( ) ( ) ( ) ( ) ( ) x x y y z z σ σ σ 0 Momentum Space Two - Level System in a momentum dependent magnetic field [ ] h k k k k = h h h ( ) ( ), ( ), ( ) x y z 23

Eigenvalues k k k ε = ε − h ( ) ( ) ( ) eff ⇑ k k k ε = ε + h ( ) ( ) ( ) eff ⇓ 2 k k 2 k k 2 = + + h h h h ( ) ( ) ( ) ( ) x y z eff 24

Rashba Spin-Orbit Coupling 25

Equal-Rashba-Dresselhaus (ERD) Spin-Orbit Coupling 26

Energy Dispersions in the ERD case Simpler case : k ε = k m 2 ( ) /( 2 ) k = h ( ) 0 x k = h vk ( ) y x k = h ( ) 0 z k k ε = ε − vk ( ) ( ) x ⇑ k k ε = ε + vk ( ) ( ) x ⇓ 27

Energy Dispersions and Fermi Surfaces k ε = ± k m vk 2 ( ) /( 2 ) α x k / x k k / x k F F 28

Momentum Distribution (Parity) k = h ( ) 0 x k h k ( ) y = x 0 . 71 ε k F F k h ( ) = z 0 . 05 ε F 29

Outline 1) Introduction to 2D Fermi gases. 2) Creation of artificial spin-orbit coupling (SOC). 3) Quantum phases and topological quantum phase 3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. transitions of 2D Fermi gases with SOC. 4) The BKT Transition and the vortex-antivortex structure. 5) Conclusions 30

Bring Interactions Back (real space) Kinetic Energy Spin-orbit and Zeeman Contact Interaction 31

Bring Interactions Back (momentum space) + q q ∆ = − = ∆ = − = g b g b * ( 0 ) and ( 0 ) 0 0 32

Bring interactions back: Hamiltonian in initial spin basis ψ ψ + + ψ ψ ↑ ↓ k k − k ↓ − k ↑ + ψ ↑ k + ψ ↓ k ψ − k ↑ ψ − k ↓ ~ k k k = ε − µ − K sh ( ) ( ) ( ) s z 33

Bring interactions back: Hamiltonian in the generalized helicity basis + + Φ k Φ k Φ k Φ k − − ⇑ ⇓ ⇑ ⇓ + Φ k ⇑ + Φ k ⇓ Φ k − ⇑ Φ k − ⇓ 34

Order Parameter: Singlet & Triplet η z η y η x k k = = h vk h h ( ) ( ) ERD ⊥ x z z k 2 = + k h vk h = 2 h vk x h ( ) ( ) ( 0 , , ) x z z eff eff 35

Excitation Spectrum Can be zero 36

Excitation Spectrum Making singlet and triplet sectors explicit k k k k ↔ E ↔ E E E ( ) ( ) ( ) ( ) − + 2 1 singlet sector 37

Excitation Spectrum (ERD) US-1 US-2 i-US-0 d-US-0 = 0 38

Lifshitz transition Change in topology 39

Topological invariant (charge) in 2D 40

Vortices and Anti-vortices of m(k) z ε = h z ε = / 0 . 2 h / 1 . 5 F ε = E F / 1 . 0 b F − US − 0 US 1

For T = 0 phase diagram need chemical potential and order parameter δ Ω ∂ Ω Order 0 = 0 = Number = − N 0 0 Parameter + ∂ µ δ ∆ Equation + 0 Equation 42

T = 0 Phase Diagram in 2D 43

Momentum distributions in 2D ↑ ↓ 44

Thermodynamic signatures of topological transitions 45

T = 0 Thermodynamic Properties in 2D 46

Outline 1) Introduction to 2D Fermi gases. 2) Creation of artificial spin-orbit coupling (SOC). 3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. 4) The BKT transition and the vortex-antivortex structure. 4) The BKT transition and the vortex-antivortex structure. 5) Conclusions 47

Hamiltonian in Real Space Kinetic Energy Spin-orbit and Zeeman Contact Interaction 48

Effective Action at finite T 49

Effective Action at finite T 50

BKT Transition Temperature 51

Beyond the Clogston Limit 52

Full Finite Phase Diagram 53

Anisotropic speed of sound 54

Vortex-Antivortex Structure RASHBA ERD 55