Thermodynamics of balanced and slightly spin-imbalanced Fermi gases at unitarity

In this article we present a Monte Carlo calculation of the critical temperature and other thermodynamic quantities for the unitary Fermi gas with a population imbalance (unequal number of fermions in the two spin components). We describe an improved worm-type algorithm that is less prone to autocorrelations than the previously available methods and show how this algorithm can be applied to simulate the unitary Fermi gas in presence of a small imbalance. Our data indicate that the critical temperature remains almost constant for small imbalances $h=\Delta \mu /{\varepsilon }_F\lesssim 0.2$. We obtain the continuum result $T_c=0.171(5){\varepsilon }_F$ in units of Fermi energy and derive a lower bound on the deviation of the critical temperature from the balanced limit, $T_c(h)-T_c(0)>-0.5{\varepsilon }_F{h}^2$. Using an additional assumption a tighter lower bound can be obtained. We also calculate the energy per particle and the chemical potential in the balanced and imbalanced cases.

Figure 1

The relative difference $[g(L_i,L_j)-\mathrm{g̃}(L_i,L_j)]/g(L_i,L_j)$ as a function of $c$ (in lattice units) for several values of $L_i$ and $L_j$ ranging between $10$ and $16$.

Figure 2

A typical fit of the rescaled correlation function $R(L,T)$ according to Eq. (10). Here data were taken at four different lattices sizes and temperatures. For this fit, ${\mathrm{\chi }}^2/\mathrm{d}.\mathrm{o}.\mathrm{f}=1.4$. The value for $c$ was found to be $-1.4(5)$. All quantities are given in lattice units.

Figure 3

The first $100\hspace{0.5em}000$ numerical measurements of the interaction energy (a measurement takes place every $100$ MC steps) with the worm setup (left) and the diagonal setup (right). Strong autocorrelations are visible in the worm setup.

Figure 4

The error analysis of blocked data for the interaction energy with the worm setup (left) and the diagonal setup (right). The blocked error is much higher in the worm setup and continues increasing even for large block sizes.

Figure 5

The error analysis of the interaction energy. Red circles correspond to the pure diagonal setup and blue squares to a combination of diagonal and modified worm updates with equal probabilities for each kind of update.

Figure 6

Schematic plot of the average sign near the critical point as a function of imbalance. The shaded area covers the range of values the sign can take at different values of lattice size and chemical potential. The lower boundary of this area is the “worst-case” curve of the sign, corresponding to lowest densities and largest lattice sizes used.

Figure 7

The critical temperature versus filling factor for different values of the chemical potential. The continuum limit corresponds to $\nu \to 0$. The linear extrapolation (solid line) of the seven data points at lowest filling factors (filled circles) yields a continuum value of $T_c/{\varepsilon }_F=0.173(6)$. The dashed line corresponds to a quadratic fit through all data points.

Figure 8

The nonuniversal constant $c$ (left) and the first two coefficients in the expansion of the universal scaling function $f(x)$ (right) in lattice units versus filling factor [see Eq. (10) with $f(0)=f_0$ and $f^{\prime }(0)=f_1{T}_c$].

Figure 9

The energy per particle (left) and the chemical potential (right) versus filling factor at the critical point. The linear fits were performed for data at filling factors ${\nu }^{1/3}<0.75$. Filled circles indicate data included in the fit and empty circles stand for data excluded from the fit (see text for discussion).

Figure 10

Relation between the chemical potential difference and the relative density difference at $T_c$.

Figure 11

Projection of the data onto the $({\nu }^{1/3}$-$T_c)$ plane. Red circles denote the balanced data and blue triangles data at nonzero imbalance. The line corresponds to the constant fit (33). Dashed lines denote the error margins.

Figure 12

The continuum limit of the critical temperature as a function of imbalance. The solid line is the value obtained from the constant fit (32); the shaded area corresponds to one standard deviation. The dashed line is the lower bound obtained from fit (29) and the dot-dashed line is the tighter lower bound obtained from fit (30).

Figure 13

Three-dimensional plot of the critical temperature versus filling factor and imbalance. The surface corresponds to the constant fit (33).

Figure 14

Projection of the data for the energy per particle onto the $({\nu }^{1/3}$-$E)$ plane. Red circles denote the balanced data and blue triangles data at nonzero imbalance. For comparison, the fit at zero imbalance is shown. The points at nonzero imbalance tend to lie above the balanced fit line.

Figure 15

Projection of the data for the average chemical potential onto the $({\nu }^{1/3}$-$\mu )$ plane. Red circles denote the balanced data and blue triangles data at nonzero imbalance. The solid line corresponds to the constant fit and the dashed lines indicate the error margins.