Quantum Monte Carlo calculation of the zero-temperature phase diagram of the two-component fermionic hard-core gas in two dimensions

Motivated by potential realizations in cold-atom or cold-molecule systems, we have performed quantum Monte Carlo simulations of two-component gases of fermions in two dimensions with hard-core interactions. We have determined the gross features of the zero-temperature phase diagram by investigating the relative stabilities of paramagnetic and ferromagnetic fluids and crystals. We have also examined the effect of including a pairwise, long-range $r^{-3}$ potential between the particles. Our most important conclusion is that there is no region of stability for a ferromagnetic fluid phase, even if the long-range interaction is present. We also present results for the pair-correlation function, static structure factor, and momentum density of two-dimensional hard-core fluids.

Figure 1

Two-body Jastrow factors $\exp [u(r)]$ for $N=50$ paramagnetic hard-core particles with $D=0.5$ a.u.

Figure 2

DMC energy $E$ against the reciprocal of the square of the system size $N^{-2}$ for (a) $D=0.5$ a.u., (b) $D=0.63$ a.u., (c) $D=0.75$ a.u., (d) $D=0.88$ a.u., and (e) $D=1$ a.u. The largest system size considered for ferromagnetic crystals is $N=49$ in each case, apart from $D=1$, where the largest size considered is $N=100$. The largest system size considered for the antiferromagnetic crystal is $N=36$. The largest system size considered for the paramagnetic fluid (square cell) is $N=50$, 58, 50, 58, and 98 for $D=0.5$, 0.63, 0.75, 0.88, and 1, respectively. The largest system size considered for the ferromagnetic fluid is $N=49$ in each case.

Figure 3

Optimal Gaussian exponent $C$ for the ferromagnetic crystal phase against the reciprocal of system size $N^{-1}$.

Figure 4

DMC energy $E$ against hard-core diameter $D$. Note that, when $D=0$, the energies of the paramagnetic and ferromagnetic fluids are 0.5 and 1 a.u. per particle, respectively.

Figure 5

DMC energy of the different phases relative to the energy of the paramagnetic fluid.

Figure 6

PCFs $g(r)$ of hard-core systems at (a) $D=0.5$ a.u., (b) $D=0.75$ a.u., (c) $D=0.88$ a.u., and (d) $D=1$ a.u. for different system sizes $N$. The results were all obtained with VMC except where indicated otherwise.

Figure 7

VMC structure factor of the paramagnetic fluid at $D=0.63$ a.u. for a range of system sizes $N$.

Figure 8

VMC and DMC structure factors of the paramagnetic fluid at $D=0.88$ a.u. with $N=18$ particles.

Figure 9

VMC structure factors of paramagnetic fluids. For $D=0$, the analytic result for a paramagnetic free-particle gas is given.

Figure 10

VMC structure factors of ferromagnetic fluids and crystals. The $D=0$ result is the analytic structure factor for a ferromagnetic free-particle gas.

Figure 11

VMC momentum density of the paramagnetic fluid at $D=0.88$ a.u. at different system sizes $N$.

Figure 12

VMC momentum densities of paramagnetic fluids at different hard-core diameters $D$.

Figure 13

VMC momentum densities of ferromagnetic fluids at different hard-core diameters $D$.

Figure 14

Renormalization factor $Z$ against $D$, from fits to the momentum density.

Figure 15

VMC momentum densities of ferromagnetic crystals.

Figure 16

Change in energy resulting from the inclusion of a $\Lambda r^{-3}$ interaction for a paramagnetic fluid of $N=18$ hard-core particles of diameter $D=0.88$ a.u.

Figure 17

VMC-evaluated expectation value of ${\sum }_{i>j}{r}_{\mathit{ij}}^{-3}$ with respect to the nonreoptimized VMC wave function.

Figure 18

Energy of the ferromagnetic fluid and crystal relative to the energy of the paramagnetic fluid against the strength of the $r^{-3}$ interaction for hard-core particles of diameter $D=0.88$ a.u.