effects of spin-orbit coupling on the bkt transition and ... · • the critical temperature of the...
TRANSCRIPT
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, 20161
Main References for Talk
2
(UNPUBLISHED)
2
Acknowledgements
Jeroen Devreese
Li Han
3Jacques 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) Conclusions5) 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
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, rEd ωα−=
( ) ( )ttV ,, rrrrEdr •−=
( ) ( ) 2],)[(, ttV rEr ωα−=
11
Atoms in optical lattices
( ) ( ) ( )( ) ( ) ( ) 2
2
][2
1
],[
rrrrrrrr
rrrrrrrr
E
E
ωα
ωα
−=
><−=
V
tV
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+−
22
)(
2
222
)(
2
2
δ
δ
m
mR
R
kk
kk
18
SU(2) rotation to new spin basis:
σx σ
z; σ
z σ
y; σ
y σ
x
Ω−+
−
−−Ω++
222
22222
22
m
kk
m
ki
km
ki
m
k
Rx
R
xRR
k
k
δ
δ
19
spin-orbit
detuning
Ramancoupling
Ω−+
−
−−Ω++
222
22222
22
m
kk
m
ki
km
ki
m
k
Rx
R
xRR
k
k
δ
δ
20
Hamiltonian with spin-orbit
kskssk
sskskssk
cchccH ++ −= ',
',
)()(
orbit-spinn with Hamiltonia
kkε
21
+−−−
=
−=
=
⊥
⊥
⊥
)()()(
)()()()(
)()()(
)()(
||*
||0
||
kkk
kkkkH
kkk
kk
hh
hh
ihhh
hh
yx
z
εε
Parallel and perpendicular fields
22
Hamiltonian in terms of
k-dependent magnetic fields
zzyyxx hhh σkσkσk1kkH )()()()()(
Matrixn Hamiltonia
0 −−−= ε
[ ])(),(),()(
field magneticdependent momentum ain
System Level-Two Space Momentum
kkkkh zyx hhh=
23
Eigenvalues
222
eff
eff
eff
)()()()(
)()()(
)()()(
kkkk
kkk
kkk
zyx hhhh
h
h
++=
+=
−=
⇓
⇑
εεεε
24
Rashba Spin-Orbit Coupling
25
Equal-Rashba-Dresselhaus (ERD)
Spin-Orbit Coupling
26
Energy Dispersions in the ERD case
)2/()( 2 mk=kε
x
x
z
xy
x
vk
vk
h
vkh
h
+=
−==
==
⇓
⇑
)()(
)()(
0)(
)(
0)(
:caseSimpler
kk
kk
k
k
k
εεεε
27
xvkmk ±= )2/()( 2kαε
Fx kk /Fx kk /
Energy Dispersions and Fermi Surfaces
28
Momentum Distribution (Parity)
05.0)(
71.0)(
0)(
=
=
=
F
z
F
x
F
y
x
h
k
kh
h
ε
εk
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)
)0(and)0( *00 =−=∆=−=∆ + qq bgbg
32
Bring interactions back:
Hamiltonian in initial spin basis
↑kψ ↓k
ψ +↓−k
ψ+↑k
ψ+↓k
ψ
+↑−k
ψ
↑−kψ
↓−kψ
)()()(~
kkk zs shK −−= µε33
Bring interactions back: Hamiltonian
in the generalized helicity basis
⇑Φ
k ⇓Φ
k+⇑−Φ
k+
⇓−Φk
+⇑
Φk
+⇓
Φk
⇑−Φk
⇓−Φk
34
Order Parameter: Singlet & Triplet
ERD
ηz
ηx
ηy
zzx hhvkh ==⊥ )()( kk
),,0()(eff zx hvkh =k 22
eff )( zx hvkh +=k35
Excitation Spectrum
Can bezero
36
Excitation Spectrum
singlet sector
Making singlet and triplet sectors explicit
37
)()(2 kk −↔ EE )()(1 kk +↔ EE
Excitation Spectrum (ERD)
US-2 US-1
d-US-0i-US-0
= 038
Lifshitz transition
Change in topology
39
Topological invariant (charge) in 2D
40
Vortices and Anti-vortices of m(k)
1
5.1/
−=
US
h Fz ε0.1/ =FbE ε0
2.0/
−=
US
h Fz ε
For T = 0 phase diagram need chemical
potential and order parameter
00
0 =∆Ω
δδ
00 =∂Ω∂−=
++ µ
NOrder ParameterEquation
NumberEquation
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) Conclusions5) 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
59
TRANSITION FROM
GAPLESS TO GAPPED SUPERFLUID
BKT transition and
vortex-antivortex structure
60
Rashba ERD
THE END
61