Figure 1

(Color online) Correlation exponents of the polarized attractive Hubbard model. (a) Dominant two-particle correlation exponents ${\Delta }_{\xi }\left(k\right)$ in the strong-coupling limit $(\mid U\mid ⪢t)$ as a function of polarization $p$ for different densities $n$. Singlet superfluid fluctuations (SS) at momentum $\mid k_{F,\uparrow }-k_{F,\downarrow }\mid$ are dominant, while $2k_{F\downarrow }$ charge density waves (CDW) are the subleading fluctuations. Note that the exponents ${\Delta }_{\mathrm{SS}}$ differ by $1∕2$ for $p=0$ and $p\to 0^{+}$, as predicted in Ref. 27. (b) Evolution of the Green’s function correlation exponents with polarization for different densities in the strong-coupling limit. The SS exponents are shown again for comparison. The Green’s function of the majority particles $\left({\Delta }_{\mathrm{G}\uparrow }\right)$ always decays more slowly than the minority one $\left({\Delta }_{\mathrm{G}\downarrow }\right)$. (c) Phase diagram for an infinite number of weakly coupled chains as a function of density and polarization for various interactions $U$. In the FFLO phase, SS correlations are dominant, while in the Fermi liquid phase, the asymptotic decay of the Green’s functions is slower than that of any two-particle correlator. For finite $U$, only the boundaries determined by the majority particles $({\Delta }_{\mathrm{SS}}={\Delta }_{\mathrm{G}\uparrow })$ are shown. SS denotes the conventional balanced $s$-wave superfluid for all $n$ at $p=0$.Reuse & Permissions

Figure 2

(Color online) Noise correlations in the attractive 1D Hubbard model for several spin polarizations $p$ at density $n=3∕4$. In the unpolarized case [row (a)], one identifies dominating $s$-wave superfluid correlations in $G_{\uparrow \downarrow }$, (a,ii), along the antidiagonal, indicating the presence of pairs with zero total momentum. In a population-imbalanced system [rows (b)–(f)], the FFLO state is clearly visible in $G_{\uparrow \downarrow }$ [column (ii)] and also $G$ [column (i)] as strongly peaked signals that shift apart as the polarization increases. The separation $Q$ [indicated in (f,ii)] between the peaks is equal to the total momentum of the pairs. Finite-size-scaling extrapolations of $L{G}_{\uparrow \downarrow }$ taken along $q^{\prime }=-q-k_{F\uparrow }+k_{F\downarrow }$, indicated by the arrows shown in column (iii), illustrate the growth of the signal with increasing system size. The momentum distributions [column (iv)] show a clear difference between the gapped $(p=0)$ and the gapless $(p>0)$ phases, but do not contain direct evidence for the presence of an FFLO state.Reuse & Permissions

Figure 3

(Color online) Off-diagonal noise correlations $G_{\uparrow \downarrow }$ in an unpolarized system with $L=64$ sites at filling $n=3∕4$ for different interactions $U$. The Fermi surfaces (nine-tile pattern), visible for small interactions, become weaker and finally disappear for strong couplings. SS correlations develop around opposite Fermi points and start to spread out along the whole antidiagonal with increasing attraction.Reuse & Permissions

Figure 4

(Color online) Local densities for systems with $L=64$ sites in the strong-coupling limit $U∕t=-10$ at filling $n=1∕2$ and polarization $p=1∕2$ for various depths of the trap $V$ [row (a)]. Even for small confinement, some majority particles are pushed away from the center and are arranged in fully polarized wings. In the central area of the trap (shaded area), an FFLO state at an effective filling $n_{\mathrm{eff}}$ and polarization $p_{\mathrm{eff}}$ is formed, as can be deduced from the characteristic signal in the off-diagonal noise correlator $G_{\uparrow \downarrow }$ [row (b)] and in the total shot noise $G$ [row (c)].Reuse & Permissions