Quantum Monte Carlo (QMC) methods are useful for studies of strongly correlated materials because they are many body in nature and use the physical Hamiltonian. Typical calculations assume as a starting point a wave function constructed from single-particle orbitals obtained from one-body methods, e.g., density functional theory. However, mean-field-derived wave functions can sometimes lead to systematic QMC biases if the mean-field result poorly describes the true ground state. Here, we study the accuracy and flexibility of QMC trial wave functions using variational and fixed-node diffusion QMC estimates of the total spin density and lattice distortion of antiferromagnetic iron oxide (FeO) in the ground state
B1 crystal structure. We found that for relatively simple wave functions the predicted lattice distortion was controlled by the choice of single-particle orbitals used to construct the wave function, rather than by subsequent wave function optimization techniques within QMC. By optimizing the orbitals with QMC, we then demonstrate starting-point independence of the trial wave function with respect to the method by which the orbitals were constructed by demonstrating convergence of the energy, spin density, and predicted lattice distortion for two qualitatively different sets of orbitals. The results suggest that orbital optimization is a promising method for accurate many-body calculations of strongly correlated condensed phases.
Electronic structure quantum Monte Carlo Quantum Monte Carlo (QMC) is an advanced simulation methodology for studies of manybody quantum systems. The QMC approaches combine analytical insights with stochastic computational techniques for efficient solution of several classes of important many-body problems such as the stationary Schrödinger equation. QMC methods of various flavors have been applied to a great variety of systems spanning continuous and lattice quantum models, molecular and condensed systems, BEC-BCS ultracold condensates, nuclei, etc. In this review, we focus on the electronic structure QMC, i.e., methods relevant for systems described by the electron-ion Hamiltonians. Some of the key QMC achievements include direct treatment of electron correlation, accuracy in predicting energy differences and favorable scaling in the system size. Calculations of atoms, molecules, clusters and solids have demonstrated QMC applicability to real systems with hundreds of electrons while providing 90-95% of the correlation energy and energy differences typically within a few percent of experiments. Advances in accuracy beyond these limits are hampered by the so-called fixed-node approximation which is used to circumvent the notorious fermion sign problem. Many-body nodes of fermion states and their properties have therefore become one of the important topics for further progress in predictive power and efficiency of QMC calculations. Some of our recent results on the wave function nodes and related nodal domain topologies will be briefly reviewed. This includes analysis of few-electron systems and descriptions of exact and approximate nodes using transformations and projections of the highly-dimensional nodal hypersurfaces into the 3D space. Studies of fermion nodes offer new insights into topological properties of eigenstates such as explicit demonstrations that generic fermionic ground states exhibit the minimal number of two nodal domains. Recently proposed trial wave functions based on Pfaffians with pairing orbitals are presented and their nodal properties are tested in calculations of first row atoms and molecules. Finally, backflow "dressed" coordinates are introduced as another possibility for capturing correlation effects and for decreasing the fixed-node bias.
Describing correlated electron systems has been a major challenge in computational condensed-matter physics. Quantum Monte Carlo, a powerful computational tool for the study of correlated systems, solves electron correlation problems explicitly. It has been taken as a benchmark method for understanding the correlated systems. Instead of making approximations to Hamiltonian, QMC methods work with the wave functions, and the computational cost scales well with the system size. With the development of parallel computing, QMC calculations on large systems are becoming more and more feasible. We have investigated two correlated systems with highly accurate fixed node QMC techniques. The first system is a correlated hydrogen model system near the metal to insulator transition. We have successfully identified the transition point by calculating spin and charge properties and analyzing the low energy Hilbert space. The second one is a strongly correlated Fe/O system. Calculations on the Fe atoms, O atoms, and FeO molecules are conducted with multiple highly accurate many-body techniques. The source of errors has been disentangled by comparing the results of the many body techniques with the experimental results. For the Fe and O atoms, the calculated properties coincide well with previous experimental results. For the basis-based techniques, the performance is mainly limited by the basis set. The calculated equilibrium bond length, excitation energy and vibrational frequency of the FeO molecules are also in close agreement with the known values from previous experiments.
We develop a formalism and present an algorithm for optimization of the trial wave function used in fixed-node diffusion quantum Monte Carlo (DMC) methods. The formalism is based on the DMC mixed estimator of the ground-state probability density. We take advantage of a basic property of the walker configuration distribution generated in a DMC calculation, to (i) project out a multideterminant expansion of the fixed-node ground-state wave function and (ii) to define a cost function that relates the fixed-node ground-state and the noninteracting trial wave functions. We show that (a) locally smoothing out the kink of the fixed-node ground-state wave function at the node generates a new trial wave function with better nodal structure and (b) we argue that the noise in the fixed-node wave function resulting from finite sampling plays a beneficial role, allowing the nodes to adjust toward the ones of the exact many-body ground state in a simulated annealing-like process. Based on these principles, we propose a method to improve both single determinant and multideterminant expansions of the trial wave function. The method can be generalized to other wave-function forms such as pfaffians. We test the method in a model system where benchmark configuration-interaction calculations can be performed and most components of the Hamiltonian are evaluated analytically. Comparing the DMC calculations with the exact solutions, we find that the trial wave function is systematically improved. The overlap of the optimized trial wave function and the exact ground state converges to 100% even starting from wave functions orthogonal to the exact ground state. Similarly, the DMC total energy and density converges to the exact solutions for the model. In the optimization process we find an optimal noninteracting nodal potential of density-functional-like form whose existence was predicted in a previous publication [Phys. Rev. B 77, 245110 (2008)]. Tests of the method are extended to a model system with a conventional Coulomb interaction where we show we can obtain the exact Kohn-Sham effective potential from the DMC data.
This article describes the variational and fixed-node diffusion quantum Monte Carlo methods and how they may be used to calculate the properties of many-electron systems. These stochastic wave-function-based approaches provide a very direct treatment of quantum many-body effects and serve as benchmarks against which other techniques may be compared. They complement the less demanding density-functional approach by providing more accurate results and a deeper understanding of the physics of electronic correlation in real materials. The algorithms are intrinsically parallel, and currently available high-performance computers allow applications to systems containing a thousand or more electrons. With these tools one can study complicated problems such as the properties of surfaces and defects, while including electron correlation effects with high precision. The authors provide a pedagogical overview of the techniques and describe a selection of applications to ground and excited states of solids and clusters.
We investigate the ground-state properties of two strongly-correlated systems: the two-dimensional
t−J model that can be used to study the high-temperature superconducting phase in the doped antiferromagnet, and the frustrated N\'eel antiferromagnet described by the
J1−J2 Heisenberg model on the square lattice, which is widely considered as the prototype model for spin frustration. In this thesis, we apply to these two systems state-of-the-art quantum Monte Carlo techniques, including the variational and the Green's function Monte Carlo with the fixed-node approximation. Few Lanczos steps are used to systematically improve the accuracy of the trial wave functions. By introducing a suitable regularization scheme for the variational calculations with few Lanczos steps, stable and controllable simulations can be performed up to very large cluster sizes with very good accuracy. In the two-dimensional
t−J model at
J/t=0.4, we show that the accuracy of the Gutzwiller-projected variational state (containing
dx2−y2 pairing) can be improved much by few Lanczos steps; in addition, the fixed-node Monte Carlo with these systematically improvable trial wave functions gives results that are comparable with the best accurate DMRG ones. Our main outcome is that the ground state is homogeneous and no evidence of stripes is detected around the doping
δ=1/8. Indeed, our best approximation to the ground state does not show any tendency towards charge inhomogeneity. Furthermore, our results show that a uniform state containing superconductivity and antiferromagnetism is stabilized at low hole doping, i.e.,
δ≲0.1. In the
J1−J2 Heisenberg model on the square lattice, we use the projected mean-field state that is built from Abrikosov fermions having a
Z2 gauge structure and four Dirac points in the spinon spectrum. No spin or dimer order is found in the strongly frustrated regime and our calculations imply that a spin liquid phase may faithfully represent the exact ground state around
J2/J1=0.5. The few Lanczos step technique is used to systematically improve the accuracy of the variational states both for the ground state and for few relevant low-energy excitations. This procedure allows us to estimate, in a valuable and systemic way, the spin gaps within thermodynamical limit and to show a solid evidence of an unconventional gapless excitation spectrum in the strongly frustrated regime, i.e.,
J2/J1≃0.5. In particular, we found gapless triplet excitations at momenta
(π,0) and
(0,π), which are compatible with the presence of four Dirac points at momenta
(±2π,±2π) in the spinon spectrum.