Finite-temperature quantum Monte Carlo study of the one-dimensional polarized Fermi gas

Quantum Monte Carlo (QMC) techniques are used to provide an approximation-free investigation of the phases of the one-dimensional attractive Hubbard Hamiltonian in the presence of population imbalance. The temperature at which the “Fulde-Ferrell-Larkin-Ovchinnikov” (FFLO) phase is destroyed by thermal fluctuations is determined as a function of the polarization. It is shown that the presence of a confining potential does not dramatically alter the FFLO regime and that recent experiments on trapped atomic gases likely lie just within the stable temperature range.

Figure 1

The effect of temperature on the pair momentum distribution. A peak at nonzero momentum is a signature of the FFLO state. As $\beta$ decreases, the peak disappears at a crossover value ${\beta }_c=1/T_c$. In this case, ${\beta }_c\approx 6$. The error bars are of the order of the symbol size.

Figure 2

Pair Green function calculated at low temperatures where the system is in the FFLO state ($\beta =32$ and $\beta =16$) and at the temperature at which FFLO disappears ($\beta =6$). Note that the oscillations disappear at the higher temperature.

Figure 3

Double occupancy versus $\beta$ for three polarizations exhibits a very sharp drop for $\beta <1$, indicating that pairs are being broken near $\beta ~1/\left|U\right|$.

Figure 4

(Color online). Phase diagram in the polarization-temperature plane. (a) The system at $N=L$ (half filling); (b) the system at $N=L/2$ (quarter filling). At $P=0$, the system is in the BCS state. The regions above the open (red) squares and to the left of the vertical (blue) lines are unexplored. Up to the open squares the system is in the FFLO phase. Phase boundaries represent crossover behavior not phase transitions. In the FFLO phase, $n_{\mathrm{pair}}(k)$ peaks at $k\ne 0$; in the PPP phase, the peak is at $k=0$.

Figure 5

Majority ($N_1$) and minority ($N_2$) density histograms at the FFLO-PPP boundary for two system sizes. The larger system size offers more grid points and, therefore, a finer resolution of the density fluctuations. A single peak is seen for each population, indicating the absence of phase separation.

Figure 6

Histograms of local densities for $L=30$, $\beta =16t$, and $U=-3.5t$. Left peaks correspond to the minority species and right ones to the majority. The peaks move smoothly as ${\mu }_1$ and ${\mu }_2$ are tuned to take the system across the boundary between PPP and FFLO. No double-peak structure is observed, indicating the absence of phase separation.

Figure 7

(Color online). Pair momentum distribution in a trapped system for several polarizations. The FFLO peak moves to higher momentum values as $P$ increases, as in the uniform system. The majority population is fixed $N_1=39$; $T/T_F=0.008$ in the balanced case $N_1=N_2=39$.

Figure 8

Density profiles of the two species for the low temperature and low polarization case. Very narrow fully paired regions are seen at the edges of the cloud. $N_1=39$ and $N_2=35$.

Figure 9

Density profile differences for different polarizations. The oscillations are standing waves showing the pair-rich (minima) and pair-depleted (maxima) regions in the confined system. $T/T_F=0.008$ for the balanced case with $N=78$ particles.

Figure 10

Trapped system at finite temperature and low polarization. Panels (a) and (b) show density profiles at $T/T_F=0.016$ and $T/T_F=0.1$, respectively. (c) Density profile difference for different temperatures.

Figure 11

Pair momentum distribution of the system at $P=0.05$ (Fig. 10) and increasing temperature. The FFLO peak disappears at $T/T_F=0.016$.

Figure 12

Trapped system at low temperature and high polarization. $N_1=39$, $N_2=11$, $L=120$, $U/t=-10$, $V_t=0.0007$. $T_F$ is calculated for the total balanced population of $N=50$. Large fully polarized regions are seen at the boundaries of the system.

Figure 13

Same trapped system as in Fig. 12 at higher temperature. $P=0.56$, $L=120$, $U/t=-10$, $V_t=0.0007t$, $N_1=39$, $N_2=11$. Panels (a) and (b) show the density profiles and differences for two temperature. (c) Pair momentum distribution.