Iterative backflow renormalization procedure for many-body ground-state wave functions of strongly interacting normal Fermi liquids

We show how a ground-state trial wave function of a Fermi liquid can be systematically improved by introducing a sequence of renormalized coordinates through an iterative backflow transformation. We apply this scheme to calculate the ground-state energy of liquid ${}^3\mathrm{He}$ in two dimensions at freezing density using variational and fixed-node diffusion Monte Carlo. Compared with exact transient estimate results for systems with a small number of particles, we find that variance extrapolations provide accurate results for the true ground state together with stringent lower bounds. For larger systems these bounds can in turn be used to quantify the systematic bias of fixed-node calculations. These wave functions are size consistent and the scaling of their computational complexity with the number of particles is the same as for standard backflow wave functions.

Figure 1

Optimized potentials of the trial function ${\Psi }_T^{\left(4\right)}$ for $N=26,\zeta =0$. Lines are broken where the pair distribution functions of the relevant (quasi)coordinates become negligibly small, $g\left(r\right)\lesssim {10}^{-3}$ (see Fig. 2).

Figure 2

Pair correlation functions calculated in a VMC simulation with the ${\mathrm{BF}}^{\left(4\right)}$ trial function using the bare coordinates $\left\{{\mathbf{r}}_i\right\}$ (solid line) and the renormalized backflow coordinates $\left\{{\mathbf{q}}_i^{\left(0\right)}\right\},\left\{{\mathbf{q}}_i^{\left(2\right)}\right\}$, and $\left\{{\mathbf{q}}_i^{\left(4\right)}\right\}$ (dotted, dash-dotted, and dashed lines, respectively). The statistical noise reaches its maximum value, ∼0.003, at the highest peak.

Figure 3

VMC and DMC energies per particle of $N=26$ unpolarized ${}^3\mathrm{He}$ atoms in two dimensions as a function of the variance ${\sigma }^2/N$. Each point corresponds to a different trial function [PW and ${\mathrm{BF}}^{\left(\alpha \right)}$ with $\alpha =0$ to 4, from higher to lower variance]. The TE energy has been subtracted. Their dependence is nearly linear, and their extrapolations to zero variance, the entries ${\mathrm{VMC}}_{\mathrm{ext}}$ and ${\mathrm{DMC}}_{\mathrm{ext}}$ in Table 1, are very close to the exact result. We further show the energy lower bound $(E_T-\sqrt{{\sigma }^2})/N$, whose extrapolation to zero variance, the entry ${\mathrm{LB}}_{\mathrm{ext}}$ in Table 1, is also very close to the exact result. The dashed line is a rough estimate of the first excited state (the difference between the two slowest exponential decay constants in the fermionic signal of the TE procedure [3]). It shows that the conditions for the validity of the energy lower bound (see text) are met by the iterated backflow trial functions. The dotted line is the alternate estimate $1/\left(2AN\right)$ of Eq. (11) for the validity of the lower bound.

Figure 4

Same as Fig. 3, for $N=29$ spin-polarized ${}^3\mathrm{He}$ atoms. Both the estimate of the first excited state, ∼0.6 K, and the value of $1/\left(2AN\right)$, 0.772 K, are off scale.

Figure 5

Same as Fig. 3, for $N=58$ unpolarized ${}^3\mathrm{He}$ atoms. Due to the lack of TE results, we take a reference energy $E_0/N$ halfway between ${\mathrm{VMC}}_{\mathrm{ext}}$ and ${\mathrm{LB}}_{\mathrm{ext}}$. Also, we do not have an estimate of the first excited state. Comparison of the dotted line here with that in Fig. 3 shows that the range of $E_T-E_0$ where the lower bound is expected to be valid shrinks as $1/N$.