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

金华

八月 五月

框架• Intro: 1D system of FFLO phase

• Confinement: Diagonal confinement versus Off-diagonal confinement

• Results: Pure FFLO state

• Conclusions

1D system of FFLO phase

Introduction

实验物理学家

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

Why 1D: Non-Fermi liquid

1D

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

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)

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.

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

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 ,

Inhomogeneous FFLO state in 1D

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

Diagonal confinement versus Off-diagonal confinement

Confinement:

DC: 1D-Pairing at finite Q & Spatial decay

DC: Power-law decay of correlations, spatial oscillations

The asymmetric Hubbard model

“BCS”

“FFLO”

( cf. B. Wang et al., PRA79, 2009 )

1 component gas

Spin-independent hopping

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

The attractive Gaudin model

Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007).

The attractive Gaudin model

Yang Phys. Rev. B 63, 140511 (2001); Orso, Phys. Rev. Lett. 98, 070402 (2007) ； XW Guan, PRA

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).

Predictions from BA and LDA

The attractive Gaudin model: in a trap

Mean field theory vs. exact solution

The attractive Gaudin model: in a trap

The attractive Gaudin model: in a trap

FFLO---Experimental Results 6Li

Liao et al., Nature 467, 567 (2010)

FFLO---Experimental Results 6Li

No unambiguous demonstration for FFLO state is obtained in cold atomic systems until now!

Liao et al., Nature 467, 567(2010)

一维系统

Phases induced by external potential

M. Rigol et al., PRL (2003) ; G. Xianlong et al., PRL (2007)

U< ０ U ＞０

Pure state possible? through different designing

harmonic trapping

Results

Pure FFLO state

Predictions from Bethe-ansatz based DFT: N=36

Predictions from Bethe-ansatz based DFT: N=36

Predictions from Bethe-ansatz based DFT: N=36

Critical FFLO state in a 1D attractive Fermi gas

Pure FFLO state occurs only at the critical polarization!

The effect of disorder on the 1D attractive Fermi gas

Wang Jingjing, Gao Xianlong, JPB (2011)

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

Off-diagonal confinement

harmonic trapping

t=0t=0

Phase diagram in DC system

M.P.A. Fisher et al.,PRB 40,546 (1989)

Phase Diagram

The model

Phase diagram

Particle-hole symmetry

Pairing correlations

均匀体系

非均匀体系

N=80

N=70

Spin-spin correlations

detectable in a non-destructive way via spatially resolved quantum polarization spectroscopy.

Spin-spin correlations.

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/

Team

感谢： NSFC 的支持

Thanks for your attention