Study of the triangular-lattice Hubbard model with constrained-path quantum Monte Carlo

Abstract

We benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and $U$ values and show that symmetry-adapted trial wave functions are essential for quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state symmetry yield energy deviations $\lesssim 1\%$ from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large $U$ regimes necessitates symmetry-projected trials to reach comparable accuracy, where both free-electron and symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.

Figures (7)

• Figure 1: Triangular-lattice clusters (left) and corresponding one-electron spectra (eigenvalues of $\hat{H}_1$) grouped by $D_2$ irreps (right). The black outline shows a $4 \times 4$ region with periodic boundaries across the dashed lines. Note that while the symmetry groups of the XC $4 \times 4$ and YC $4 \times 4$ lattices are $D_4$ and $D_8$, respectively, one can label the orbitals using irreps of the $D_2$ subgroup (see App. \ref{['app:xc4_symm']} for details).
• Figure 2: (Top panels) Ground state energies per site obtained from CPMC and DMRG on YC$n$ cylinders with widths $n \equiv N_y = 4, 6, 12$ at quarter-filling ($\nu = 1/2$) and $U = 4, 8, 12$. The insets highlight the three largest $N_x$ points, which exhibit a clear linear dependence on $N_x^{-1}$. Colored labels indicate the extrapolated infinite-cylinder energies $\varepsilon_{\infty}$ obtained from data of the corresponding color. The quoted uncertainties include both statistical and systematic contributions from the fitting procedure, detailed in App. \ref{['app:cpmc_extrap']}. (Bottom panels) Relative error between CPMC and DMRG energies for the YC4 and YC6 cylinders, which remain below $1\%$.
• Figure 3: GHF ground state solutions on XC$n$ cylinders at half-filling as a function of $U$.
• Figure 4: Ground state energies per site obtained from CPMC and ED for the XC $4 \times 4$ lattice at half-filling ($\nu = 1$). The inset depicts the relative error between CPMC and ED energies. Employing the symmetry-projected ($K$, SG, $S^2$)-GHF trial yields improved energies compared to the o-FE trial.
• Figure 5: Relative error between CPMC and ED energies for $U = 4, 8, 12$ from o-FE and a variety of symmetry-projected trials. The relative errors are systematically reduced as symmetry projectors are successively applied onto the GHF state.