Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical...
Transcript of Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical...
![Page 1: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/1.jpg)
Superfluid state in the multi-component fermionic optical lattice systems
Tokyo Tech. A. Koga
Y. Okanami et al. Phys. Rev. A 89, 053622 (2014).
![Page 2: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/2.jpg)
Superfluid state in the multi-component fermionic optical lattice systems
CollaboratorsTokyo Tech. Y. Okanami
N. TakemoriFribourg P. Werner
Y. Okanami et al. Phys. Rev. A 89, 053622 (2014).
![Page 3: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/3.jpg)
A. KOGA
Ultracold fermions
Optical lattices (40K, 6Li, 171Yb, 173Yb)
Number of fermions
Confining potential
Onsite interactions
I. Bloch, Nature Physics 1, 23 (2005) Controllable
Fermi surface
Superfluid state
Mott insulating state
M. Köhl et al., Phys. Rev. Lett. 94, 080403 (2005)
J. K. Chin et al., Nature 443, 961 (2006)
R. Jördens et al., Nature 455, 204 (2008)
![Page 4: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/4.jpg)
A. KOGA
Spin degrees of freedom
Potassium 40K (F=9/2, 7/2)I=4L=0S=1/2
Lithium 6Li (F=3/2, 1/2)I=1L=0S=1/2
Magnetic field (G)
tenfold
F=9/2
Two-component systems
Three components
![Page 5: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/5.jpg)
A. KOGA
Superfluid state in 40K (F=9/2)
Two component fermionsBCS-BEC crossover
Cooper pair
Attractive interaction
Feshbach resonance
C. A. Regal, et al., Phys. Rev. Lett. 92, 040403 (2004).
Two-component systems
BCSBEC
![Page 6: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/6.jpg)
A. KOGA
Three-component fermions
Lithium 6Li : S=1/2, I=1 |F,mF> = |1/2, 1/2>, |1/2, -1/2>, |3/2, -3/2>
Ottenstein et al., Phys. Rev. Lett. 101, 203202 (2008)J. H. Huckans et al., Phys. Rev. Lett. 102, 165302 (2009)
Magnetic field (G)
Scattering length
(a0)SU(3)
![Page 7: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/7.jpg)
A. KOGA
Two-component systems (DMFT)
BCS-BEC crossoverCooper pairs
A. Garg et al., PRB 72, 023517 (2005)AK and P. Werner, PRA 84, 023638 (2011)
BCS-BEC crossover (T≠0)
2nd orderBCS
BEC
Attractive case only
![Page 8: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/8.jpg)
A. KOGA
Three-component fermions (half filling)
Color superconductivityU
U’ U’
Metal
SF
PM
K. Inaba and S. Suga, Phys. Rev. Lett. 108, 255301 (2012)
IPTSFA(T=0)
U’/t
U/U’=0.1
Pair potential
Superfluid
Repulsiveinteraction
![Page 9: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/9.jpg)
A. KOGA
Questions
Superfluid state2 component system vs. 3 component system
4 components ?
U
U’ U’
2 comp. U
U’U
U’ U’
U
U’ U’
T=0
Attractive Repulsive
![Page 10: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/10.jpg)
A. KOGA
Contents
Superfluid state in multicomponent systems
BCS-BEC crossover ?Paramagnetic phase diagram ?
2comp. U
U’U
U’ U’
U
U’ U’
Okanami et al., Phys. Rev. A 89, 053622 (2014)
BCSBEC
![Page 11: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/11.jpg)
A. KOGA
Contents
Four component systems
U
U
U’U’
S-wave superconductivity?orbital degrees of freedom?condensed matter?
S-wave superfluid in degenerate Hubbard model ?
Orbital 1Orbital 2
U U’
![Page 12: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/12.jpg)
Model and Methods
![Page 13: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/13.jpg)
A. KOGA
Three component fermions
Hubbard model
Pair potentialU
U’ U’
t
U U’ U’
K. Inaba and S. Suga, Phys. Rev. Lett. 108, 255301 (2012)
FrustrationNo Density Wave & Magnetic order
![Page 14: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/14.jpg)
A. KOGA
Simple static mean-field theory
BCS theory
U’ no role for Superfluid
Static mean-field → Dynamical mean-field
U
U’ U’
![Page 15: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/15.jpg)
A. KOGA
Impurity system
Effective Bath
Dynamical mean-field theoryPruschke, Jarrell, Freericks, Adv. Phys. 44, 187 (1995)Georges, Kotliar, Krauth, Rozenberg, Rev. Mod. Phys. 68, 13 (1996) Kotliar & Vollhardt, Physics Today 53, (2004)
Lattice system Σ(ω), G(ω)
Self-consistently
Local particle correlations
^^
![Page 16: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/16.jpg)
A. KOGA
Impurity solver
Continuous-Time QMC method (Nambu)Strong coupling approach
Treat SF directly
P. Werner, et al., Phys. Rev. Lett. 97, 076405 (2006).E. Gull, et al., Rev. Mod. Phys. 83, 349 (2011).
![Page 17: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/17.jpg)
Three-component fermions
2 comp. U
U’
?
![Page 18: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/18.jpg)
A. KOGA
Three component fermions
Hubbard model
U
U’ U’
K. Inaba and S. Suga, Phys. Rev. Lett. 108, 255301 (2012)
2 comp. U
U’
![Page 19: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/19.jpg)
A. KOGA
Stability of superfluid state (T/D=0.03)
Pair potential
Y. Okanami et al., Phys. Rev. A 89, 053622 (2014)
2 comp. U
U’
U
U’ U’
![Page 20: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/20.jpg)
A. KOGA
Phase diagram
T=0.03repulsiveattractive
Pair Mott state
2 comp. & 3 comp.
U
U’ U’
Y. Okanami et al., Phys. Rev. A 89, 053622 (2014)
![Page 21: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/21.jpg)
A. KOGA
Double occupancy (Para)
T/D=0.03
2 comp. U
U’
U
U’ U’
1st order phase transitions
Pair Mott state color Mott state
![Page 22: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/22.jpg)
A. KOGA
Phase diagram (para)
T/D=0.03
CSM
PM
K. Inaba et al., Phys. Rev. A 82, 051602 (2010).
-4 -2 0 2 40
1
2
3
4
5T=0 SFA
Crossover
![Page 23: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/23.jpg)
A. KOGA
Comparison (T/D=0.03)
Stability of superfluid state
Superfluidstable along the phase boundary between PM and metal
-4 -2 0 2 40
1
2
3
4
5
T=0 SFA
Superfluid
?
K. Inaba et al., Phys. Rev. A 82, 051602 (2010).
PM
MetalCSM
Superfluid
![Page 24: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/24.jpg)
BCS-BEC crossover
![Page 25: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/25.jpg)
A. KOGA
Attractive Hubbard model
BCS-BEC crossoverBCS weak couplingBEC strong coupling
2nd order phase transitions
2comp
Nature of BCS & BEC ?
1st order
2nd order
3 components
![Page 26: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/26.jpg)
A. KOGA
’delta_beta_67.txt’ u 1:2:(-$6)
-20 -15 -10 -5 0 0
0.5
1
1.5
2
2.5
3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Pair potential (T/D=0.03)
Density plot
BCS
BEC
U/D
U’/D
BCS(1,2)
BEC(-1,0.9)
UU’ U’
![Page 27: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/27.jpg)
A. KOGA
Phase transitions
BEC regionU/D=-1.0, U’/D=0.9
BCS regionU/D=1.0, U’/D=2.0
2nd order 1st order
![Page 28: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/28.jpg)
A. KOGA
Density of states
BCS regionU/D=1.0, U’/D=2.0
BEC regionU/D=-1.0, U’/D=0.9
2nd order 1st order
T>Tc
T<Tc
T=Tc
UU’ U’
![Page 29: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/29.jpg)
A. KOGA
Effect of repulsive interaction
U/D=1.0, U’/D=2.0
Δ=F(0)
Δ’ U
U’ U’
![Page 30: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/30.jpg)
A. KOGA
Summary (3 components)
Superfluid state in three component fermionsRepulsive interactionCompetition PM & Metal
1st order
2nd order
Dynamical correlations important!
![Page 31: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/31.jpg)
Four component fermions
U
U
U’
![Page 32: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/32.jpg)
A. KOGA
Four component fermions
Degenerate Hubbard model (Mn, V, Ti, …)U
U
U’
Hund coupling J=0
Momoi & Kubo, Phys. Rev. B 58, 567 (1998)
U,U’ : independentEffective Hamiltonian U=U’
CeTIn5 (T=Co, Ir, Rh)Takimoto et al., J. Phys. 14, 369 (2002)
Orbital 1Orbital 2
tij U U’
![Page 33: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/33.jpg)
A. KOGA
Dynamical mean-field theory
Green functionsAssumption only intra-orbital pairs
Impurity solversStrong-coupling expansion CTQMC method
U
U
U’
![Page 34: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/34.jpg)
A. KOGA
Phase diagram (para)
Degenerate Hubbard modelHalf filling
0 1 2 3 4 50
1
2
3
4
5
U
U'
Z=0.2 Z=0.4 Z=0.6 Z=0.8 Z=0.9Metal
Mott insulator
Superconductivity?Mott transitions?
U’>U
AK, Imai & Kawakami, Phys. Rev. B 66, 165107 (2002)
D=1.0
U U’
![Page 35: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/35.jpg)
A. KOGA
S-wave superconductivity (T=0.01D)
Pair potential
2 comp.
U
U’
Repulsive systems s-wave superconductivity
U U’
![Page 36: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/36.jpg)
A. KOGA
Mott transitions (paramagnetic state)
Renormalization factors (T=0.01D)
2 comp.
U
U’
Mott transitions
![Page 37: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/37.jpg)
A. KOGA
Phase diagram
Degenerate Hubbard model
S-wave SCrepulsive interacting case
Between Metal & Mott states
0 1 2 3 4 50
1
2
3
4
5
U
U'
Z=0.2 Z=0.4 Z=0.6 Z=0.8 Z=0.9
T=0 (para)
?
![Page 38: Superfluid state in the multi-component fermionic optical ...A. KOGA Ultracold fermions `Optical lattices (40K, 6Li, 171Yb, 173Yb) 9Number of fermions 9Confining potential 9Onsite](https://reader035.fdocuments.us/reader035/viewer/2022071104/5fddacdc603e116149222696/html5/thumbnails/38.jpg)
A. KOGA
Summary
Superfluid state in multicomponent fermions
3 comp.2comp. Vs 3comp.Repulsive interaction
Competition PM & Metal
4 comp. (2 orbitals)Repulsion-induced SF
Dynamical correlations important!
Okanami et al., Phys. Rev. A 89, 053622 (2014).