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arXiv:2603.14808 · Physical review. B./Physical review. B

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

Shu Fay Ung, Ankit Mahajan, David R. Reichman

cond-mat.str-elquant-phSubmitted Mar 16, 2026• Announced Mar 17, 2026• 1 citation
v1 · Mar 16, 2026v2 · Jun 12, 2026v3 · Jun 26, 2026v4 · Jul 12, 2026
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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.

DOI
10.1103/j85h-z664

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We benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and UUU 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 ≲1%\lesssim 1\%≲1% from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large UUU 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.
We consider symmetry-projected Hartree-Fock trial wave functions in constrained-path Monte Carlo (CPMC) calculations. Previous CPMC calculations have mostly employed Hartree-Fock (HF) trial wave functions, restricted or unrestricted. The symmetry-projected HF approach results in a hierarchy of wave functions with increasing quality: the more symmetries that are broken and restored in a self-consistent manner, the higher the quality of the trial wave function. This hierarchy is approximately maintained in CPMC calculations: the accuracy in the energy increases and the statistical variance decreases when further symmetries are broken and restored. Significant improvement is achieved in CPMC with the best symmetry-projected trial wave functions over those from simple HF. We analyze and quantify the behavior using the two-dimensional repulsive Hubbard model as an example. In the sign-problem-free region, where CPMC can be made exact but a constraint is deliberately imposed here, spin-projected wave functions remove the constraint bias. Away from half filling, spatial symmetry restoration in addition to that of the spin leads to highly accurate results from CPMC. Since the computational cost of symmetry-projected HF trial wave functions in CPMC can be made to scale algebraically with system size, this provides a potentially general approach for accurate calculations in many-fermion systems.
We present results from a constrained path Monte Carlo (CPMC) study of a modified two-dimensional Hubbard model. We include more general forms of the band structure and electron interaction in order to examine their effects on ground-state properties, such as electron pairing correlations. Both next-nearest neighbor hopping, t', and third-nearest neighbor hopping, t″, are introduced in the Hamiltonian. A nearest neighbor interaction of strength V is also included. We carry out CPMC calculations on system sizes up to 16 x 16, at various electron fillings, to investigate the ground state of the model for different values of these parameters. For realistic systems, these calculations indicate that Hubbard-like models are not capable of showing enhanced pairing. The modified Hamiltonian also presents an opportunity to more closely examine the accuracy and robustness of the CPMC algorithm. Results of further benchmark calculations involving CPMC are presented. These benchmarks confirm that CPMC is able to show enhanced pairing in systems where such behavior is known to exist.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we have studied the ground-state energies and the spin and hole densities of a 12×312 \times 312×3 Hubbard model with open boundary conditions and 6 holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate UUU regimes. For U/t≥6U/t \geq 6U/t≥6 we find a ground-state with stripes.
Using both the density-matrix renormalization group method and the constrained-path quantum Monte Carlo method, we studied the ground-state energies and the spin and hole densities of a $12\ifmmode\times\else\texttimes\fi{}3$ Hubbard model with open boundary conditions and six holes doped away from half-filling. Results obtained with these two methods agree well in the small and intermediate U regimes. For U/t> 6U/t>~6U/t> 6 we find a ground-state with charge inhomogeneities consistent with stripes.
We simulated the three-band Hubbard model using the constrained-path Monte Carlo (CPMC) method in search for a possible superconducting ground state. The CPMC is a ground-state method that is free of the exponential scaling of computing time with system size. We calculated the binding energy of a pair of holes for systems up to $6\ifmmode\times\else\texttimes\fi{}4$ unit cells. We also studied the pairing correlation functions versus distance for both the ddd-wave and extended sss-wave channels in systems up to $6\ifmmode\times\else\texttimes\fi{}6$. We found that holes bind for a wide range of parameters and that the binding increased as the system size is increased. However, the pairing correlation functions decay quickly with distance. For the extended sss channel, we found that as the Coulomb interaction Ud{U}_{d}Ud​ on the Cu sites is increased, the long-range part of the correlation functions is suppressed and fluctuates around zero. For the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ channel, we found that the correlations decay rapidly with distance towards a small positive value. However, this value becomes smaller as the interaction Ud{U}_{d}Ud​ or the system size is increased.