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 10th, 2016 1

Main References for Talk

2

(UNPUBLISHED)

2

Acknowledgements

Jeroen Devreese Li Han

3

Jacques Tempere Ian Spielman

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

• f 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 7

TRANSITION FROM GAPLESS TO GAPPED SUPERFLUID

BKT transition and vortex-antivortex structure

8

Rashba ERD

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

1) Introduction to 2D Fermi gases. 9

Condensed Matter meets Atomic Physics

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- ferromagnetic, etc…

Neutral atoms (bosons or fermions) Electrons of holes (fermions only)

In optical lattices many types of atoms can be loaded like bosonic, Sodium-23, Potassium-39, Rubidium-87, or Cesium-133; and fermionic Lithium-6, Potassium-40, Strontium-87, etc…

10

How atoms are trapped?

• Atom-laser interaction
• Induced dipole

moment.

• Trapping potential

( ) ( )

t , r E d ω α − =

( ) ( )

t t V , , r r r r E d r

• −

=

( ) ( ) 2

] , )[ ( , t t V r E r ω α − =

11

Atoms in optical lattices

( ) ( ) ( ) ( ) ( ) ( ) 2

2

] [ 2 1 ] , [ r r r r r r r r r r r r r r r r E E ω α ω α − = > < − = V t V

12

How optical lattices are created?

13

Single plane excitations

Vortex-antivortex pairs BKT transition: Physics of 2D XY model

14

Critical Temperature

BCS-Bose Superfluidity in 2D 0.125

Bose Liquid Fermi Liquid Pairing Temperature 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). 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

2) Creation of artificial spin-orbit coupling (SOC). 17

Raman process and spin-orbit coupling

            − + Ω Ω + − 2 2 ) ( 2 2 2 2 ) (

2 2

δ δ m m

R R

k k k k

18

SU(2) rotation to new spin basis: σx σz; σz σy ; σy σx

            Ω − +       −       − − Ω + + 2 2 2 2 2 2

2 2 2 2

m k k m k i k m k i m k

R x R x R R

k k δ δ

19

spin-orbit detuning Raman coupling

            Ω − +       −       − − Ω + + 2 2 2 2 2 2

2 2 2 2

m k k m k i k m k i m k

R x R x R R

k k δ δ

20

Hamiltonian with spin-orbit

ks ks s k s s ks ks s k

c c h c c H

+ +

 

− =

' , ' ,

) ( ) (

• rbit
• spin

n with Hamiltonia k k ε

21

        + − − − = − = =

⊥ ⊥ ⊥

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (

|| * || ||

k k k k k k k H k k k k k h h h h ih h h h h

y x z

ε ε

Parallel and perpendicular fields

22

Hamiltonian in terms of k-dependent magnetic fields

z z y y x x

h h h σ k σ k σ k 1 k k H ) ( ) ( ) ( ) ( ) ( Matrix n Hamiltonia − − − = ε

[ ]

) ( ), ( ), ( ) ( field magnetic dependent momentum a in System Level

• Two

Space Momentum k k k k h

z y x

h h h =

23

Eigenvalues

2 2 2 eff eff eff

) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( k k k k k k k k k k

z y x

h h h h h h + + = + = − =

⇓ ⇑

ε ε ε ε

24

Rashba Spin-Orbit Coupling

25

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

26

Energy Dispersions in the ERD case

) 2 /( ) (

2

m k = k ε

x x z x y x

vk vk h vk h h + = − = = = =

⇓ ⇑

) ( ) ( ) ( ) ( ) ( ) ( ) ( : case Simpler k k k k k k k ε ε ε ε

27

x

vk m k ± = ) 2 /( ) (

2

k

α

ε

F x k

k /

F x k

k /

Energy Dispersions and Fermi Surfaces

28

Momentum Distribution (Parity)

05 . ) ( 71 . ) ( ) ( = = =

F z F x F y x

h k k h h ε ε k k k

29

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

3) Quantum phases and topological quantum phase transitions of 2D Fermi gases with SOC. 30

Bring Interactions Back (real space)

Kinetic Energy Contact Interaction Spin-orbit and Zeeman

31

Bring Interactions Back (momentum space)

) ( and ) (

*

= − = ∆ = − = ∆

+ q

q b g b g

32

Bring interactions back: Hamiltonian in initial spin basis

↑ k

ψ

↓ k

ψ

+ ↓ −k

ψ

+ ↑ k

ψ

+ ↓ k

ψ

+ ↑ −k

ψ

↑ −k

ψ

↓ −k

ψ

) ( ) ( ) ( ~ k k k

z s

sh K − − = µ ε

33

Bring interactions back: Hamiltonian in the generalized helicity basis

⇑

Φk

⇓

Φk

+ ⇑ −

Φ k

+ ⇓ −

Φ k

+

⇑

Φ

k

+

⇓

Φ

k

⇑ −

Φ k

⇓ −

Φ k

34

Order Parameter: Singlet & Triplet

ERD ηz ηx ηy

z z x

h h vk h = =

⊥

) ( ) ( k k ) , , ( ) (

eff z x h

vk h = k

2 2 eff

) (

z x

h vk h + = k

35

Excitation Spectrum

Can be zero 36

Excitation Spectrum

singlet sector Making singlet and triplet sectors explicit

37

) ( ) (

2

k k

−

↔ E E ) ( ) (

1

k k

+

↔ E E

Excitation Spectrum (ERD)

US-2 US-1 d-US-0 i-US-0 = 0

38

Lifshitz transition

Change in topology 39

Topological invariant (charge) in 2D

40

Vortices and Anti-vortices of m(k)

1 5 . 1 / − = US h

F z ε

. 1 / =

F b

E ε 2 . / − = US h

F z ε

For T = 0 phase diagram need chemical potential and order parameter

0 =

∆ Ω δ δ

0 =

∂ Ω ∂ − =

+ +

µ N

Order Parameter Equation Number 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. 5) Conclusions

4) The BKT transition and the vortex-antivortex structure. 47

Hamiltonian in Real Space

Kinetic Energy Contact Interaction Spin-orbit and Zeeman

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

55

RASHBA ERD

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 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

• f 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 59

TRANSITION FROM GAPLESS TO GAPPED SUPERFLUID

BKT transition and vortex-antivortex structure

60

Rashba ERD

THE END

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