Pure FuldeFerrelLarkinOvchinnikov state in optical lattices of offdiagonal

Pure Fulde-Ferrel-Larkin-Ovchinnikov state in optical lattices of off-diagonal confinement 高先龙 Collaborators: Reza Asgari, 汪泾泾，陈阿海 2011. 8. 5 兰州

框架 • Intro: 1 D system of FFLO phase • Confinement: Diagonal confinement versus Off-diagonal confinement • Results: Pure FFLO state • Conclusions

Introduction 1 D system of FFLO phase

Why 1 D: Non-Fermi liquid 1 D

Hunt for the Elusive FFLO State BCS: Δ(r) ∝ const FF: Δ(r) ∝ exp(iq⋅r) LO: Δ(r) ∝ cos(q⋅r) Attractive Fermi systems, spin polarization and superfluidity are enemies Conventional: a partially polarized Fermi gas undergoes macroscopic phase separation 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)]

1 D Exotic phase: FFLO Bosonization: Yang Phys. Rev. B 63, 140511 (2001); Zhao & Liu PRA (2008) Bethe-ansatz: Orso, PRL 98 (2007); Hu, Liu, Drummond, PRL 98 (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, q 0= k. F ≠ 0 • oscillating pairing function, F~cos( k. Fx). oscillations in order parameter Δ(r) • Fulde-Ferrell vs Larkin-Ovchinnikov • Translational & rotational invariance broken

Suggestions for the experimental observation of 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 1 D

The 1 D 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

Confinement: Diagonal confinement versus Off-diagonal confinement

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

DC: Power-law decay of correlations, spatial oscillations

The asymmetric Hubbard model Spin-independent hopping “FFLO” “BCS” ( cf. B. Wang et al. , PRA 79, 2009 ) 1 component gas

The asymmetric Hubbard model unequal hoppings: the model is no longer integrable, hence use DMRG superconducting correlations ‘commensurate’ densities ‘incommensurate’ 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

The attractive Gaudin model: in a trap Partially-polarized phase associated with FFLO state Yang Phys. Rev. B 63, 140511 (2001); Predictions from field theory and LDA 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).

The attractive Gaudin model: in a trap Predictions from BA and LDA Mean field theory vs. exact solution

The attractive Gaudin model: in a trap

The attractive Gaudin model: in a trap

FFLO---Experimental Results 6 Li Liao et al. , Nature 467, 567 (2010)

FFLO---Experimental Results 6 Li No unambiguous demonstration for FFLO state is obtained in cold atomic systems until now!

一维系统 Liao et al. , Nature 467, 567(2010)

Phases induced by external potential U<０ U＞０ M. Rigol et al. , PRL (2003) ; G. Xianlong et al. , PRL (2007)

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 1 D attractive Fermi gas Pure FFLO state occurs only at the critical polarization!

The effect of disorder on the 1 D attractive Fermi gas Wang Jingjing, Gao Xianlong, JPB (2011)

The effect of disorder on the 1 D attractive Fermi gas speckle intensity the spatial (auto)correlation FFLO-BCS phase could change to FFLO-N phase while increasing disorder

Off-diagonal confinement harmonic trapping t=0

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

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 1 D 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/

Thanks for your attention