Effects of spin-orbit coupling on the BKT transition and the vortexantivortex structure in D Fermi Gases Carlos A. R. Sa de Melo Georgia Institute of Technology QMath13 Mathematical Results in Quantum Physics Atlanta: October 10 th, 016 1
Main References for Tal UNPUBLISHED
Acnowledgements Li Han Ian Spielman Jacques Tempere Jeroen Devreese 3
Outline 1 Introduction to D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and topological quantum phase transitions of D 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 D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and topological quantum phase transitions of D 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 can be loaded lie bosonic, Sodium-3, Potassium-39, Rubidium-87, or Cesium-133; and fermionic Lithium-6, Potassium-40, Strontium-87, etc Neutral atoms bosons or fermions 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, antiferromagnetic, etc Electrons of holes fermions only 10
How atoms are trapped? Atom-laser interaction r, t = d E r t V, Induced dipole moment. d = α ω E r,t Trapping potential V r, t = α ω[ E r, t ] 11
Atoms in optical lattices V V r = α ω < [ E r, t 1 r = α ω [ E r ] ] > 1
How optical lattices are created? 13
Single plane excitations Vortex-antivortex pairs BKT transition: Physics of D XY model 14
Critical Temperature Pairing Temperature Bose Liquid 0.15 Fermi Liquid BCS-Bose Superfluidity in D 15
D Fermi gases with increasing attractive interactions, but no SOC. 16
Outline 1 Introduction to D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and topological quantum phase transitions of D Fermi gases with SOC. 4 The BKT Transition and the vortex-antivortex structure. 5 Conclusions 17
Raman process and spin-orbit coupling R m Ω + δ Ω + R m δ 18
SU rotation to new spin basis: σ x σ z ; σ z σ y ; σ y σ x Ω + Ω + + m m i m i m R x R x R R δ δ 19
spin-orbit detuning Raman coupling Ω + Ω + + m m i m i m R x R x R R δ δ 0
Hamiltonian with spin-orbit Hamiltonian with spin - orbit H = ε c + c s s, s, s h s' s c + s' c s 1
+ = = = * 0 H h h h h ih h h h h y x z ε ε Parallel and perpendicular fields
Hamiltonian in terms of -dependent magnetic fields z z y y x x h h h σ σ σ 1 H Hamiltonian Matrix 0 = ε [ ],, momentum dependent magnetic field in a Momentum Space Two - LevelSystem h z y x h h h = 3
Eigenvalues eff eff eff z y x h h h h h h + + = + = = ε ε ε ε 4
Rashba Spin-Orbit Coupling 5
Equal-Rashba-Dresselhaus ERD Spin-Orbit Coupling 6
Energy Dispersions in the ERD case / m = ε x x z x y x v v h v h h + = = = = = 0 0 Simpler case : ε ε ε ε 7
Energy Dispersions and Fermi Surfaces ε α = /m ± v x x / F / x F 8
h h Momentum Distribution Parity x y = 0 ε F hz ε F = = 0.71 0.05 x F 9
Outline 1 Introduction to D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and and topological quantum quantum phase phase transitions of D Fermi gases with SOC. 4 The BKT Transition and the vortex-antivortex structure. 5 Conclusions 30
Bring Interactions Bac real space Kinetic Energy Spin-orbit and Zeeman Contact Interaction 31
Bring Interactions Bac momentum space * 0 = g b q = 0 and 0 = g b + q = 0 3
Bring interactions bac: Hamiltonian in initial spin basis + ψ + ψ ψ ψ ψ + ψ + ψ ψ ~ K s = ε µ sh z 33
Bring interactions bac: Hamiltonian in the generalized helicity basis Φ + + Φ Φ Φ Φ Φ + + Φ Φ 34
Order Parameter: Singlet & Triplet η z η y h = = v h x z h z η x ERD h = eff 0, v, h x z v h eff x z h = + 35
Excitation Spectrum Can be zero 36
Excitation Spectrum Maing singlet and triplet sectors explicit E E E1 E+ singlet sector 37
Excitation Spectrum ERD US- US-1 i-us-0 d-us-0 = 0 38
Lifshitz transition Change in topology 39
Topological invariant charge in D 40
Vortices and Anti-vortices of m h z / ε US F = 0. 0 Eb / ε =1.0 F h z / ε US F = 1.5 1
For T = 0 phase diagram need chemical potential and order parameter δ δ Ω 0 = 0 0 Order Parameter Equation N + = Ω 0 = 0 µ + Number Equation 4
T = 0 Phase Diagram in D 43
Momentum distributions in D 44
Thermodynamic signatures of topological transitions 45
T = 0 Thermodynamic Properties in D 46
Outline 1 Introduction to D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and topological quantum phase transitions of D Fermi gases with SOC. 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 5
Full Finite Phase Diagram 53
Anisotropic speed of sound 54
Vortex-Antivortex Structure RASHBA ERD 55
Outline 1 Introduction to D Fermi gases. Creation of artificial spin-orbit coupling SOC. 3 Quantum phases and topological quantum phase transitions of D Fermi gases with SOC. 4 The BKT transition and the vortex-antivortex structure. 5 Conclusions 5 Conclusions 56
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. 57
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. 58
Conclusions in Pictures Change in topology TRANSITION FROM GAPLESS TO GAPPED SUPERFLUID 59
BKT transition and vortex-antivortex structure Rashba ERD 60
THE END 61