Equation of state of two-dimensional ${}^3$He at zero temperature

We have performed a quantum Monte Carlo study of a two-dimensional bulk sample of interacting $1/2$-spin structureless fermions, a model of ${}^3$He adsorbed on a variety of preplated graphite substrates. We have computed the equation of state and the polarization energy using both the standard fixed-node approximate technique with a simple backflow trial function and a formally exact methodology, relying on bosonic imaginary-time correlation functions of operators suitably chosen in order to extract fermionic energies. As the density increases, the fixed-node approximation predicts a transition to an itinerant ferromagnetic fluid, whereas the unbiased methodology indicates that the paramagnetic fluid is the stable phase until crystallization takes place. We find that two-dimensional ${}^3$He at zero temperature crystallizes from the paramagnetic fluid at a density of 0.061 Å${}^{-2}$ with a narrow coexistence region of 0.002 Å${}^{-2}$. Remarkably, the spin susceptibility turns out in very good agreement with experiments.

Figure 1

Upper panel: Imaginary time correlation functions, ${\mathcal{C}}_F\left(\tau \right)$, corresponding to the two different choices of the operator in (13). Lower panel: Reconstructed spectral functions $s_F\left(\omega \right)$ obtained with the GIFT method.

Figure 2

Equation of state of ${}^3$He in two dimensions. Solid line (broken across the coexistence region): liquid and solid ${}^3$He; dashed line: mass-3 boson fluid; dotted line: liquid ${}^3$He, after Ref. 16. The latter is only reliable at low densities.

Figure 3

Bose-Fermi gap in the energy per particle, ${\Delta }_{BF}/N$, as a function of the spin polarization, $\zeta$, evaluated via PW-FN, BF-FN, and BF-FC. Upper panel, $\rho =0.020$ Å${}^{-2}$ and $N=18$; middle panel, $\rho =0.045$ and $N=18$; lower panel, $\rho =0.070$ Å${}^{-2}$ and $N=26$. The statistical uncertainties are below the symbols size.

Figure 4

FC versus FN equation of state. Thick solid line (broken across the coexistence region): paramagnetic liquid and solid ${}^3$He (FC); dashed line: paramagnetic liquid (FN); dotted line: ferromagnetic liquid (FN); the dashed and dotted lines terminate at the FN freezing density; thin solid line: energy of the solid, down to the FN melting density. For each density, the energy is relative to the energy $E_B$ of the mass-3 boson fluid.

Figure 5

Fixed-node results for the polarization energy $E\left(\zeta \right)-E_{\mathrm{fit}}\left(0\right)$ relative to the Fermi energy $E_F$ at $\rho =0.020$, 0.045, 0.050, 0.055, 0.060, 0.065, 0.070 Å${}^{-2}$ from top to bottom (respectively, open triangles, open squares, filled squares, open diamonds, filled diamonds, open circles, filled circles). The function $E_{\mathrm{fit}}\left(\zeta \right)$ is a quadratic polynomial in ${\zeta }^2$ fitted to the simulation data; the solid line is the density-independent result for noninteracting particles.

Figure 6

FC results for the polarization energy $E\left(\zeta \right)-E_{\mathrm{fit}}\left(0\right)$ relative to the Fermi energy $E_F$ at $\rho =0.020$, 0.045, 0.050, 0.055, 0.060, 0.065, 0.070 Å${}^{-2}$ in order of decreasing dispersion (respectively, open triangles, open squares, filled squares, open diamonds, filled diamonds, open circles, filled circles). The function $E_{\mathrm{fit}}\left(\zeta \right)$ is a quadratic polynomial in ${\zeta }^2$ fitted to the simulation data; the solid line is the density-independent result for noninteracting particles.

Figure 7

Enhancement of the spin susceptibility as a function of the density: (filled circles) as measured in the second layer of ${}^{3}He$ on graphite[2]; (open circles) as calculated assuming a quadratic dispersion over the whole polarization range in Fig. 6. The corresponding Fixed-node result from Fig. 5 would diverge at $\rho \simeq 0.050$ Å${}^{-2}$.