Quantum phase transition with a simple variational ansatz

We study the zero-temperature quantum phase transition between liquid and hcp solid ${}^4\mathrm{He}$. We use the variational method with a simple yet exchange-symmetric and fully explicit wave function. It is found that the optimized wave function undergoes spontaneous symmetry breaking and describes the quantum solidification of helium at 22 atm. The explicit form of the wave function allows us to consider various contributions to the phase transition. We find that the employed wave function is an excellent candidate for describing both a first-order quantum phase transition and the ground state of a Bose solid.

Figure 1

Energy per particle obtained with the two-parameter symmetric wave function ${\psi }_{\text{snj}}^{\{b,\gamma \}}$ given by Eqs. (1, 2, 3), as a function of variational parameters $b$ and $\gamma$, for three different densities. Parameters are shown in terms of $\sigma =2.556$ Å. Simulation used $N_{\mathrm{p}}=N_{\text{s}}=900$ particles. (a) Density $\rho =22.2$ ${\mathrm{nm}}^{-3}$ displays a single minimum, at $\gamma =0$, which corresponds to a liquid phase; (b) intermediate density $\rho =25.8$ ${\mathrm{nm}}^{-3}$ is in the liquid-solid coexistence region, and has two separate local minima; (c) density $\rho =29.3$ ${\mathrm{nm}}^{-3}$ displays only one minimum, corresponding to the solid phase. Contours are separated by 0.4 K.

Figure 2

Optimized parameters of the two-parameter symmetric wave function given by Eqs. (1, 2, 3), as a function of density. Parameters are shown in units of $\sigma =2.556$Å. Left vertical axis corresponds to the value of the site-localization parameter $\gamma$, while the right axis shows parameter $b$.

Figure 3

Gibbs free energy, per particle, of the optimized state of the two-parameter symmetric wave function given by Eqs. (1, 2, 3). Arrow indicates the location of the phase transition. The inset shows the free energies with subtracted linear fit to the solid phase of the 180-particle system.

Figure 4

Optimized density as a function of pressure. Blue lines show density for the two-parameter symmetric wave function ${\psi }_{\text{snj}}$ for varying number of particles, $N=180,448,\text{and}900$. The size effects in the transition location are below the statistical error, which is about 1 atm. Purple line shows density for five-parameter symmetric wave function with improved pair-correlation factor (9), with transition at 22 bars. Green line shows density for the traditional, nonsymmetric Nosanow-Jastrow wave function of Eq. (5), with transition at 5 bars. Red cross marks show experimental melting and freezing densities [50]. The piecewise appearance of data comes from different equations of state $\rho \left(P\right)$ selected by the varying pressure-optimized variational parameters; size of the steps is thus indicative of the statistical error.