Spin polarized nonrelativistic fermions in $1+1$ dimensions

We study by Monte Carlo methods the thermodynamics of a spin polarized gas of nonrelativistic fermions in $1+1$ dimensions. The main result of this work is that our action suffers no significant sign problem for any spin polarization in the region relevant for dilute degenerate Fermi gases. This lack of sign problem allows us to study attractive spin polarized fermions nonperturbatively at spin polarizations not previously explored. For some parameters values we verify results previously obtained by methods which include an uncontrolled step like complex Langevin and/or analytical continuation from imaginary chemical potential. For others, larger values of the polarization, we deviate from these previous results.

Figure 1

Continuum time limit for the density (left) and spin (right) at $\beta \mu =2.0$ and $\beta h=2.0$. The spatial lattice spacing is fixed to $\mathrm{\Delta }x=1$ and the number of sites to $N_x=61$. The extrapolation is done with a quadratic correction for the exponentiated form and with a linear correction for the other case.

Figure 2

Left: Continuum limit in space for the density at $\beta \mu =2.0$ and $\beta h=2.0$. The spatial lattice size is fixed to $L=61$. The extrapolation is done with a quadratic correction for the continuum dispersion form and with a quadratic plus quartic correction for the other case. Right: Thermodynamic limit for the density for $\beta \mu =2.0$ and $\beta h=2.0$.

Figure 3

Results for the shifted integration contour $A_{xt}\to A_{xt}+i\delta$ as a function of $\delta$. On the left we have the density and in the right panel we plot the value of the phase average $⟨\cos (\mathrm{Im}S)⟩$. The horizontal line in the density plot corresponds to a common fit to all data points.

Figure 4

Density (left) and spin (right) normalized by the nonpolarized density of the free Fermi gas $n_0(\beta ,\mu )$. The continuous lines show the results obtained in [10] by an analytic continuation from imaginary spin chemical potential. This method is expected to work well for $\beta h\le \pi$, the region indicated by solid lines, but it quickly becomes unreliable as we move to higher polarizations, as we can see by comparing our results with the dotted lines.

Figure 5

Our results for the density compared with results from analytical continuation from imaginary polarization [10] and complex Langevin simulations [18]. The parameters for these simulations are $G=1/\sqrt{8}$, $\beta =8$, and $\beta h=2.0$. The right panel indicates the difference between our results and the other methods. The error-bars indicate the statistical uncertainties of our results since we did not have information about the error bars on the results from the other approaches.