Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. Here we show that combining both, and introducing a diagrammatic quantum Monte Carlo method, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply it to the quintessential problem of solid state, the uniform electron gas, which is at the heart of the density functional theory success in describing real materials, yet it has not been adequately solved for over 90 years. Our method allows us to calculate numerically exact momentum and frequency resolved spin and charge response functions. This method can be applied to a number of moderately interacting electron systems, including models of realistic metallic and semiconducting solids.
Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. Here we show that combining both, and introducing a diagrammatic quantum Monte Carlo method, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply it to the quintessential problem of solid state, the uniform electron gas, which is at the heart of the density functional theory success in describing real materials, yet it has not been adequately solved for over 90 years. Our method allows us to calculate numerically exact momentum and frequency resolved spin and charge response functions. This method can be applied to a number of moderately interacting electron systems, including models of realistic metallic and semiconducting solids.
Two of the most influential ideas developed by Richard Feynman are the Feynman diagram technique and his variational approach. The former provides a powerful tool to construct a systematic expansion for a generic interacting system, while the latter allows optimization of a perturbation theory using a variational principle. Here we show that combining a variational approach with a new diagrammatic quantum Monte Carlo method, both based on the Feynman's original ideas, results in a powerful and accurate solver to the generic solid state problem, in which a macroscopic number of electrons interact by the long range Coulomb repulsion. We apply the solver to the quintessential problem of solid state, the uniform electron gas (UEG), which is at the heart of the density functional theory (DFT) success in describing real materials, yet it has not been adequately solved for over 90 years. While some wave-function properties, like the ground state energy, have been very accurately calculated by the diffusion Monte Carlo method (DMC), the static and dynamic response functions, which are directly accessed by the experiment, and are crucial for development of non-local and non-adiabatic density functionals, remain poorly understood. Our method allows us to calculate the momentum-frequency resolved spin response functions for the first time, and to improve on the precision of the charge response function. The accuracy of both response functions is sufficiently high, so as to uncover previously missed fine structure in these responses. This method can be straightforwardly applied to a large number of moderately interacting electron systems in the thermodynamic limit, including realistic models of metallic and semiconducting solids.
This thesis is devoted to the development of numerical methods for correlated electrons in realistic systems based on perturbation theories using Feynman diagrams. Electron correlations in realistic systems give rise to a wide range of interesting physical properties. Diagrammatic expansions and its low-order approximations are key building blocks of theoretical and numerical methods for studying correlated electrons. Recent developments of the diagrammatic quantum Monte Carlo (DiagMC) methods enable efficient stochastic evaluations of diagrams and does not suffer from numerical sign problem which worsens with increasing system size. It is therefore a promising direction to apply DiagMC techniques to realistic electron systems in the pursuit of ab initio calculations of correlated materials. This thesis starts by reviewing basic concepts and definitions of realistic electron systems, the perturbation theory based on Feynman diagrams, Monte Carlo evaluations of the series, as well as self-consistent diagrammatic methods. Low-order diagrammatic methods are applied to realistic calculations and benchmarked against large collections of state-of-the-art many-body methods. The thesis then proposes a diagrammatic Monte Carlo solver for realistic molecules and impurities based on a recursive evaluation of diagrams, which is tested on realistic molecules and impurities with varying temperature, system size, and interaction strength. We observe that this method is ideal for problems with many orbitals with moderate correlations, but may suffer from series divergence at strong correlations. Next, we propose the inchworm Monte Carlo algorithm for interaction expansion, which overcomes limitations of divergent diagrammatic series by replacing the standard perturbation series with a sequence of incremental series expansions. A prototype implementation is presented for system with on-site Hubbard interactions. Finally, we discuss technical aspects of numerical representations of the Green's function, which has significant impact on the efficiency of diagrammatic methods. We review several common numerical representations, and introduce our development of the sparse sampling method, which provides compact representations of the Green's function in both imaginary time and frequency as well as fast transformation formula between the two.
We calculate the single-particle excitation spectrum and the Landau liquid parameters for the archetypal model of solids, the three-dimensional uniform electron gas, with the variational diagrammatic Monte Carlo method, which gives numerically controlled results without systematic error. In the metallic range of density, we establish benchmark values for the wave-function renormalization factor Z, the effective mass [Formula: see text], and the Landau parameters [Formula: see text] and [Formula: see text] with unprecedented accuracy, and we resolve the long-standing puzzle of non-monotonic dependence of mass on density. We also exclude the possibility that experimentally measured large reduction of bandwidth in Na metal can originate from the charge and spin fluctuations contained in the model of the uniform electron gas.
Quantum Monte Carlo (QMC) is one of the most promising methods for solving quantum many-body problems. QMC combines known analytical properties of wave functions, results from other methods, and stochastic techniques, into a powerful tool for investigation of the electronic structure of real materials. Furthermore, the method has a favorable scaling ( O[N3] in the number of electrons), perfect scalability on parallel architectures, and can be applied to a wide range of systems. Our work has focused both on expanding the capabilities of QMC for calculation of quantities beyond energies, such as interatomic forces, and on application of the method to challenging problems in nanostructure materials research. For the calculation of interatomic forces we have implemented a finite difference correlated sampling method. The correlated sampling enables us to avoid the problem of statistical noise in evaluating the energy differences and allows us to obtain forces with an average 1% error. We have analyzed the effect of the method's main approximations on the estimate of the forces, both by theoretical arguments and by performing a variety of tests on small systems. Furthermore, we have used this newly developed method for estimating equilibrium geometries in a problem where other methods have failed. QMC enabled us to treat the electron correlation effects with high accuracy and to provide accurate predictions in a number of interesting applications. In particular, for carbon rings of C4N+2 stoichiometry we have resolved a long-standing problem regarding the competition between the aromaticity and second order Jahn-Teller effects. We have found that the dominant mechanism at small and intermediate sizes, which stabilizes the bond angle and bond length alternated geometries, is the secondorder Jahn-Teller effect while the aromatic isomer is always found to be a transition state. These high accuracy calculations which have involved multi-reference wave functions, extensive variational optimizations, and the use of massively parallel platforms demonstrate clearly the unique capabilities of the QMC approach.
We develop a Monte Carlo scheme for sampling series of Feynman diagrams for the proper self-energy which are self-consistently expressed in terms of renormalized particle propagators. This approach is used to solve the problem of a single spin-down fermion resonantly interacting with the Fermi gas of spin-up particles. Though the original series based on bare propagators are sign-alternating and divergent one can still determine the answer behind them by using two strategies (separately or together): (i) using proper series re-summation techniques, and (ii) introducing renormalized propagators which are defined in terms of the simulated proper self-energy, i.e. making the entire scheme self-consistent. Our solution is important for understanding the phase diagram and properties of the BCS-BEC crossover in the strongly imbalanced regime. On the technical side, we develop a generic sign-problem tolerant method for exact numerical solution of polaron-type models, and, possibly, of the interacting many-body Hamiltonians.
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.