Pure Fulde-Ferrel-Larkin-Ovchinnikov state in optical lattices of off-diagonal confinement2011.8.5 Collaborators: Reza Asgari,

Intro: 1D system of FFLO phase

Confinement: Diagonal confinement versus Off-diagonal confinement

Results: Pure FFLO state

Conclusions

1D system of FFLO phase Introduction

Why 1D: Non-Fermi liquid1D

BCS: (r) const FF: (r) exp(iqr) LO: (r) cos(qr)Hunt for the Elusive FFLO StateAttractive 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)]

1D Exotic phase:FFLOBosonization: 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 (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)

Inhomogeneous FFLO state in 1D

The 1D attractive Hubbard model: Phase diagramBethe ansatz, Phases:I. Empty latticeII. (n < 1, p = 1): Fully polarized III. (n = 1, p = 1): Fully polarizedIV. (n < 1, p < 1): Less than half-filled,partially polarized: FFLOV. (n < 1, p = 0): no polarization, fully paired

Esslers 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 BCSFFLO( cf. B. Wang et al., PRA79, 2009 )1 component gasSpin-independent hopping

The asymmetric Hubbard model superconducting correlationsincommensurate densities unequal hoppings: the model is no longer integrable, hence use DMRG

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 LDAThe 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 LDAThe 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 6LiLiao et al., Nature 467, 567 (2010)

FFLO---Experimental Results 6LiNo unambiguous demonstration for FFLO state is obtained in cold atomic systems until now!

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

Phases induced by external potentialM. Rigol et al., PRL (2003) ; G. Xianlong et al., PRL (2007)U

Pure state possible? through different designing harmonic trapping

ResultsPure 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 intensitythe spatial (auto)correlationFFLO-BCS phase could change to FFLO-N phase while increasing disorderThe effect of disorder on the 1D attractive Fermi gas

Off-diagonal confinementharmonic trappingt=0t=0

Phase diagram in DC systemM.P.A. Fisher et al.,PRB 40,546 (1989)Phase Diagram

The model

Phase diagram

Particle-hole symmetry

Pairing correlations

N=80

N=70

Spin-spin correlationsdetectable in a non-destructive way via spatially resolved quantum polarization spectroscopy.

Spin-spin correlations.

ConclusionsWe 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 NSFC

Thanks for your attention

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