Nonlocal Quantum Fluctuations and Fermionic Superfluidity in the Imbalanced Attractive Hubbard Model

We study fermionic superfluidity in strongly anisotropic optical lattices with attractive interactions utilizing the cluster dynamical mean-field theory method, and focusing in particular on the role of nonlocal quantum fluctuations. We show that nonlocal quantum fluctuations impact the BCS superfluid transition dramatically. Moreover, we show that exotic superfluid states with a delicate order parameter structure, such as the Fulde-Ferrell-Larkin-Ovchinnikov phase driven by spin population imbalance, can emerge even in the presence of such strong fluctuations.

Figure 1

(a) A schematic of the system geometry. The optical lattice consists of one-dimensional chains which are coupled to form an anisotropic cubic lattice. In our cluster DMFT scheme, the 1D chain is treated as a single cluster with periodic boundaries. (b) The BCS and FFLO phase transitions as a function of the interchain hopping $t_{\perp }$ both in the cluster DMFT model (with nonlocal quantum fluctuations) and the real-space DMFT model (excluding nonlocal quantum fluctuations). The critical temperature of the BCS state is given by the continuous and dashed blue line for the cluster $[c]$ and single-site [s] models, respectively. Similarly, the FFLO critical temperature is given by the green line for the cluster $[c]$ model and by the dashed green line for the single-site $[s]$ model. As a point of contrast, the static mean-field prediction of the BCS critical temperature is $T_{c,\mathrm{MF}}=0.52$ in the corresponding parameter range, while the FFLO critical temperature would be on the order of $0.5T_{c,\mathrm{MF}}$ [12]. In each case the system is at half-filling, and the FFLO transition is found by varying the spin polarization of the system while maintaining the total filling fraction constant. For $t_{\perp }\ge 0.15$, we perform the calculations in a cluster of $N_c=36$ lattice sites, whereas for $t_{\perp }=0.1$ a cluster size of $N_c=42$ is required for convergence.

Figure 2

The dependence of the self-energy on the dimensionality. In each panel we plot the normal spin-$\uparrow$ and anomalous Nambu components, ${\mathrm{\Sigma }}_{\uparrow ,ij}(i{\omega }_n)$ and $S_{ij}(i{\omega }_n)$, of the self-energy between sites $i$ and $j$ for the lowest Matsubara frequency. Here, the system is at a temperature of $T=0.08$ and at half-filling with zero spin polarization. At (a) $t_{\perp }=0.3$ and (b) $t_{\perp }=0.2$ this corresponds to the BCS state, while at (c) $t_{\perp }=0.1$ the system is in the normal phase in which case the anomalous self-energy is identically zero and thus not plotted. The nonlocal quantum fluctuations grow substantially as the interchain hopping $t_{\perp }$ decreases. The oscillating structure of the self-energy in $i-j$ is predominantly a single particle signature and corresponds to the Fermi momentum of the system. Note that the BCS order parameter is chosen real and therefore the BCS anomalous self-energy is also real.

Figure 3

The FFLO state at $t_{\perp }=0.25$. Here, the temperature is $T=0.05$ and the polarization $P=0.035$. (a) The density difference $N_{\uparrow }-N_{\downarrow }$ and the order parameter $\mathrm{\Delta }$ as a function of the cluster site $i$. In our results, the translation invariance of the system is spontaneously broken and in a particular simulation in the FFLO regime; e.g., the minima of the order parameter and density may fall on any given lattice site. (b) The absolute value of the anomalous part of the self-energy, $S_{ij}(i{\omega }_n)$, plotted as a function of the cluster site $i$ and the distance $i-j$ to the site $j$ for the lowest Matsubara frequency.