Quantum Monte Carlo study of the metal-to-insulator transition on a honeycomb lattice with $1/r$ interactions

Describing correlated electron systems near phase transitions has been a major challenge in computational condensed-matter physics. In this paper, we apply highly accurate fixed-node quantum Monte Carlo techniques, which directly work with many-body wave functions and simulate electron correlations, to investigate the metal-to-insulator transition of a correlated hydrogen lattice. By calculating spin and charge properties, and analyzing the low-energy Hilbert space, we identify the transition point and identify order parameters that can be used to detect the transition. Our results provide a benchmark for density functional theories seeking to treat correlated electron systems.

Figure 1

Energy vs 1/(number of atoms). The lattice constant is fixed as $a=2.95$ Å and the state is Néel. The energy starts to converge linearly at lattice cell size $4\times 4$ (32 atoms).

Figure 2

DFT energy vs lattice constant. Here the vertical axis is the energy difference between the Néel state and spin-unpolarized state. Lines correspond to different hybridization. The inset plot shows the symmetry-breaking point [where the antiferromagnetic (AFM) functional produces the Néel state] as a function of hybridization.

Figure 3

Shifted FN-RMC energy vs lattice constant. The y axis is the shifted energy with respect to the averaged value corresponding to the specific lattice constant. The blue dots correspond to the spin-unpolarized states' wave functions; green dots correspond to Néel states. For clarity, we have drawn regions around the trial functions of the same spin state.

Figure 4

Order parameters computed with a PBE trial function. Green, blue, and red colors represent the results calculated with VMC, DMC, and RMC, respectively. Gray dots are extrapolated values with Eq. (6). All statistical uncertainties are much smaller than the symbols.

Figure 5

Order parameters vs lattice constant. Heat maps are colored by energy; blue represents low energy and red represents high energy. The curve on top of the heat map depicts the fitted minimum-energy order parameters.

Figure 6

Finite-size effects on the order parameters as a function of lattice constant.