Fulde-Ferrell-Larkin-Ovchinnikov state in the dimensional crossover between one- and three-dimensional lattices

We present a full phase diagram for the one-dimensional (1D) to three-dimensional (3D) crossover of the Fulde-Ferrell–Larkin-Ovchinnikov (FFLO) state in an attractive Hubbard model of 3D-coupled chains in a harmonic trap. We employ real-space dynamical mean-field theory which describes full local quantum fluctuations beyond the usual mean-field and local density approximation. We find strong dimensionality effects on the shell structure undergoing a crossover between distinctive quasi-1D and quasi-3D regimes. We predict an optimal regime for the FFLO state that is considerably extended to intermediate interchain couplings and polarizations, directly realizable with ultracold atomic gases. We find that the 1D-like FFLO feature is vulnerable to thermal fluctuations, while the FFLO state of mixed 1D-3D character can be stabilized at a higher temperature.

Figure 1

Phase diagram of the 3D coupled-chain Hubbard model. The particle densities $n_{\uparrow ,\downarrow }$ and the order parameter $\Delta$ are calculated as a function of the interchain coupling and the polarization at zero temperature. (a) The oscillation amplitude ${\Delta }_0$ of the order parameter and the density difference $n_{\uparrow }-n_{\downarrow }$ at the trap center. The phases at the trap center can be divided into three: the fully paired superfluid (SF) ($n_{\uparrow }=n_{\downarrow },\Delta \ne 0$), FFLO (oscillating $n$ and $\Delta$), and normal ($\Delta =0$) phases. (b) Phases I–III associated with the shell structures in the trap, explained in the text.

Figure 2

Evolution of the FFLO oscillations in the 1D-3D crossover. The profiles of the particle densities $n_{\uparrow ,\downarrow }$ and the order parameter $\Delta$ along the chain sites $i$ are presented for the two regimes of interchain couplings (a) $t_{\perp }=0.2$ (quasi-1D) and (b) $t_{\perp }=0.5$ (quasi-3D). In the quasi-1D regime, for small polarization $P=0.04$, the FFLO-type oscillations at the center are surrounded by the fully paired shoulders. The region of oscillating $\Delta$ develops from the center, expands toward the edges as $P$ increases, and then emerges over the whole area for $P=0.13$. On the contrary, in the quasi-3D regime, the oscillations are initially at the partially polarized intermediate regions and spread toward the center as $P$ increases. At finite interchain couplings, the far edges of the trap are always polarized. In (c), the evolution of the oscillation envelope of $\Delta$ is shown with increasing $P$ at $t_{\perp }=0.2$ in the quasi-1D regime. Uniform oscillations occur along the trap at an intermediate polarization.

Figure 3

Polarized superfluid phase at finite temperature. For interchain coupling $t_{\perp }=0.3$, the profiles at temperature $T=0.05$ are compared with those at $T=0$ for the particle densities $n_{\uparrow ,\downarrow }$ and the order parameter $\Delta$ along the chain sites $i$. At polarization (a) $P\simeq 0.09$, corresponding to phase II, the structure of the FFLO core surrounded by fully paired shoulders found at $T=0$ is completely changed into the polarized superfluid phase with a uniform order parameter at $T=0.05$. The order parameter oscillations at the edges are still observed at $T=0.05$. In contrast, at a higher polarization (b) $P\simeq 0.14$, corresponding to phase III, similar FFLO characteristics are identified at both temperatures $T=0.05$ and $T=0$.