Download - Pure Fulde-Ferrel-Larkin-Ovchinnikov state in optical lattices of off-diagonal confinement
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Pure Fulde-Ferrel-Larkin-Ovchinnikov state in optical lattices of off-diagonal confinement
高先龙
2011.8.5 兰州
Collaborators: Reza Asgari, 汪泾泾,陈阿海
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金华
八月 五月
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框架• Intro: 1D system of FFLO phase
• Confinement: Diagonal confinement versus Off-diagonal confinement
• Results: Pure FFLO state
• Conclusions
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1D system of FFLO phase
Introduction
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实验物理学家
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理论物理工作者
2 2
121 1 1
( )2
fN N N
D i ji i ji
H g x xm x
external potential
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Why 1D: Non-Fermi liquid
1D
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BCS: Δ(r) const∝FF: Δ(r) exp(iq r)∝ ⋅LO: Δ(r) cos(q r)∝ ⋅
Hunt for the Elusive FFLO State
Attractive Fermi systems, spin polarization and superfluidity are enemies
Conventional: a partially polarized Fermi gas undergoes macroscopic phaseseparation into a polarized normal region and an unpolarized superfluid region
FFLO state: Unconventional superfluid state when , in which fermion pairs with nonzero momentum form a spatially modulated inhomogeneous superfluid phase,
[Fulde & Ferrell (1964); Larkin & Ovchinnikov (1964)]
FFkk
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1D Exotic phase:FFLO
Bosonization: Yang Phys. Rev. B 63, 140511 (2001); Zhao & Liu PRA (2008)
Bethe-ansatz: Orso, PRL98 (2007); Hu, Liu, Drummond, PRL98 (2007); Guan, Batchelor, Lee, Bortz, PRB (2007)
DMRG: E. Feiguin and F. Heidrich-Meisner, PRB (2007); M. Tezuka and M. Ueda, PRL (2008); M. Rizzi, et al, PRB (2008)
QMC: M. Casula, D. M. Ceperley, and E. J. Mueller, PRA (2008)
DFT: Gao Xianlong & Reza Asgari, PRA (2008)
Related: mass imbalanced Fermi Hubbard model, B Wang, et al., PRA (2009) SJ Gu, PRB (200?); Cazalilla and Giamarchi, PRL (2005)
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Why FFLO in cold atom?
Condensed matter systems: FFLO physics is obscured by impurities, orbital effects, and spin-orbit coupling.
Ultracold atomic systems: the interaction, lattice, and polarization can be chosen at will.
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Characterization of the FFLO phase
• Pairing at finite k; nonzero pairing momentum,
q0= kF ≠ 0
• oscillating pairing function, F~cos(kFx).
oscillations in order parameter Δ(r)
• Fulde-Ferrell vs Larkin-Ovchinnikov
• Translational & rotational invariance broken
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Suggestions for the experimental observationof the FFLO state
• Image density profiles of : search for oscillations,
absorption imaging; phase-contrast imaging technique
• RF-spectroscopy: Kinunnen et al. PRL 96, 110403 (2006)
• Rapid-sweep-method, time-of-flight: peaks at finite velocities.
• Noise correlations:
• density of states: RF spectroscopy
Greiner et al. PRL 94, 110401 (2005)Altman et al. Phys. Rev. A 70, 013603 (2004) Luescher et al. PRA (2009)Yang, PRB (2001)
nn ,
kkkkkk nnnnn ,
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Inhomogeneous FFLO state in 1D
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The 1D attractive Hubbard model: Phase diagram
Bethe ansatz, Phases:
I. Empty lattice
II. (n < 1, p = 1): Fully polarized
III. (n = 1, p = 1): Fully polarized
IV. (n < 1, p < 1): Less than half-filled,
partially polarized: FFLO
V. (n < 1, p = 0): no polarization, fully paired
Essler’s Book, The One-Dimensional Hubbard Model, 2005
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Diagonal confinement versus Off-diagonal confinement
Confinement:
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DC: 1D-Pairing at finite Q & Spatial decay
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DC: Power-law decay of correlations, spatial oscillations
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The asymmetric Hubbard model
“BCS”
“FFLO”
( cf. B. Wang et al., PRA79, 2009 )
1 component gas
Spin-independent hopping
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The asymmetric Hubbard model
0.0 0.1 0.2 0.3 0.4
10-6
10-5
10-4
10-3
10-2
x/L
x/L
|(x)|
superconducting correlations
‘incommensurate’ densities
unequal hoppings: the model is no longer integrable, hence use DMRG
‘commensurate’ densities
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The attractive Gaudin model
Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007).
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The attractive Gaudin model
Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007) ; XW Guan, PRA
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Predictions from field theory and LDA
The attractive Gaudin model: in a trap
Partially-polarized phase associated with FFLO state Yang Phys. Rev. B 63, 140511 (2001);
Bethe-Ansatz + local density approximation:
Two-phase structures: centre partially polarized; edge either fully paired or fully polarized.
Orso, PRL 98, 070402 (2007)Hu, Liu & Drummond, PRL 98, 070403 (2007)Guan, Batchelor, Lee & Bortz, PRB 76, 085120 (2007).
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Predictions from BA and LDA
The attractive Gaudin model: in a trap
Mean field theory vs. exact solution
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The attractive Gaudin model: in a trap
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The attractive Gaudin model: in a trap
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FFLO---Experimental Results 6Li
Liao et al., Nature 467, 567 (2010)
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FFLO---Experimental Results 6Li
No unambiguous demonstration for FFLO state is obtained in cold atomic systems until now!
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Liao et al., Nature 467, 567(2010)
一维系统
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Phases induced by external potential
M. Rigol et al., PRL (2003) ; G. Xianlong et al., PRL (2007)
U< 0 U >0
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Pure state possible? through different designing
harmonic trapping
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Results
Pure FFLO state
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Predictions from Bethe-ansatz based DFT: N=36
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Predictions from Bethe-ansatz based DFT: N=36
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Predictions from Bethe-ansatz based DFT: N=36
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Critical FFLO state in a 1D attractive Fermi gas
Pure FFLO state occurs only at the critical polarization!
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The effect of disorder on the 1D attractive Fermi gas
Wang Jingjing, Gao Xianlong, JPB (2011)
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speckle intensity
the spatial (auto)correlation
FFLO-BCS phase could change to FFLO-N phase while increasing disorder
The effect of disorder on the 1D attractive Fermi gas
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Off-diagonal confinement
harmonic trapping
t=0t=0
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Phase diagram in DC system
M.P.A. Fisher et al.,PRB 40,546 (1989)
Phase Diagram
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The model
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Phase diagram
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Particle-hole symmetry
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Pairing correlations
均匀体系
非均匀体系
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N=80
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N=70
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Spin-spin correlations
detectable in a non-destructive way via spatially resolved quantum polarization spectroscopy.
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Spin-spin correlations.
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Conclusions
• We show that the off-diagonal confinement supports a region of pure FFLO state, thus provides an ideal system to detect the FFLO state in 1D systems.
• deviates from linear relations
• Magnetic structure factor shows a kink related to finite FFLO momentum
Note for helpful ALPS (Algorithms and Libraries for Physics Simulations) http://alps.comp-phys.org/
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Team
感谢: NSFC 的支持
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Thanks for your attention