Auxiliary field quantum Monte Carlo for multiband Hubbard models: Controlling the sign and phase problems to capture Hund's physics

In the study of strongly correlated, many-electron systems, the Hubbard-Kanamori (HK) model has emerged as one of the prototypes for transition metal oxide physics. The model is multiband in nature and contains Hund's coupling terms, which have pronounced effects on metal-insulator transitions, high-temperature superconductivity, and other physical properties. In the following, we present a complete theoretical framework for treating the HK model using the ground-state auxiliary field quantum Monte Carlo (AFQMC) method and analyze its performance on few-band models whose parameters approximate those observed in ruthenates, rhodates, and other materials exhibiting Hund's physics. Unlike previous studies, the constrained path and phaseless approximations are used to respectively control the sign and phase problems, which enables high-accuracy modeling of the HK model's ground-state properties within parameter regimes of experimental interest. We demonstrate that, after careful consideration of the Hubbard-Stratonovich transformations and trial wave functions employed, relative errors in the energy of less than $1\%$ can routinely be achieved for moderate to large values of the Hund's coupling constant. Crucially, our methodology also accurately predicts magnetic ordering and phase transitions. The results presented open the door to more predictive modeling of Hund's physics within a wide range of strongly correlated materials using AFQMC.

Figure 1

AFQMC ground-state energy vs the density-density parameter $U$ for the two-band, $5\times 1$ HK model using the charge decomposition and FE trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1,U_1=0, U_2=0$, and $J=0$ with $N_{\uparrow }=N_{\downarrow }=6$. Relative errors, $\mathrm{\Delta }E$, taken with respect to ED results are plotted in the inset for clarity.

Figure 2

AFQMC ground-state energy vs the Hund's coupling parameter $J$ for the two-band, $5\times 1$ HK model using the charge decomposition and FE/RHF trial wave functions (WF). Here, all of the other Hamiltonian parameters are held fixed at $t=1,U=0,U_1=0$, and $U_2=0$ with $N_{\uparrow }=N_{\downarrow }=6$. Relative errors, $\mathrm{\Delta }E$, taken with respect to ED results are plotted in the inset for clarity.

Figure 3

Comparison of phaseless and constrained path AFQMC energy errors as a function of $J$ for a two-band, $5\times 1$ HK model. Open circles denote parameters at which the constrained path approximation was employed, while closed circles denote parameters at which the phaseless approximation was employed. Here, we set $N_{\uparrow }=N_{\downarrow }=6, t=1,U=0,U_1=0$, and $U_2=0$. FE trial wave functions were used for both the initial and trial wave functions, and 560 walkers were employed in each calculation.

Figure 4

Schematic of our three-band model on a $4\times 4$ lattice. At each site, there is one atom with three bands, one of which is lower in energy by $\mathrm{\Delta }$ than the other two degenerate bands. The top right box illustrates a situation in which AFM order is present between adjacent lattice sites.

Figure 5

AFQMC ground-state energy as a function of the band gap magnitude, $\mathrm{\Delta }$, for the three-band, $2\times 2$ HK model using the charge decomposition and GHF trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1,U=6,U_1=U-2J,U_2=U-3J$, and $J=0.15U$ with $N_{\uparrow }=N_{\downarrow }=8$. Relative errors, $\mathrm{\Delta }E$, taken with respect to ED results are plotted in the inset for clarity.

Figure 6

AFQMC ground-state energy vs $U$ for the three-band, $2\times 2$ HK model using the charge decomposition and GHF trial wave functions. Here, all of the other Hamiltonian parameters are held fixed at $t=1,\mathrm{\Delta }=0.8,U_1=U-2J,U_2=U-3J$, and $J=0.15U$ with $N_{\uparrow }=N_{\downarrow }=8$. Relative errors, $\mathrm{\Delta }E$, are taken with respect to ED results are plotted in the inset for clarity.

Figure 7

AFQMC ground-state energy vs band gap magnitude, $\mathrm{\Delta }$, for the three-band, $4\times 4$ HK model using the charge decomposition. GHF trial wave functions with both FM order and AFM order are used. QMC predicted a phase transition to occur at around $\mathrm{\Delta }=1.15$, as illustrated by the orange dotted line. Hartree-Fock predicted a phase transition to occur at $\mathrm{\Delta }=0.5$, as illustrated by the green dotted line. Here, all of the other Hamiltonian parameters are held fixed at $t=1,U=6,U_1=U-2J,U_2=U-3J$, and $J=0.15U$ with $N_{\uparrow }=N_{\downarrow }=32$.