"Quantum Monte Carlo (QMC) is one of the most powerful methods for solving many body problems in Physics and Chemistry. MC sampling allows one to measure physical properties systems with very complicated wave functions that incorporate correlation between the electrons. Additionally, QMC has O(N3) scaling verses exponential scaling for methods such as Conguration Interaction and “exact diagonalization"", and it’s perfect parallelism allows one to speed up computation with little effort on large parallel computers. Our work has focused on two systems. The first is a 1D ionic Hubbard model with a complex phase diagram that we find to exhibit a ferroeletric \bond ordered"" phase. A new many-body berry's phase formulation of the electronic polarization and localization is used in conjunction with QMC to analyze whether a bond order phase exists between the customary band and Mott insulating phases. The bond order and bond order correlation functions are measured to illustrate the stability of this intermediate state, and phase swapping between two degenerate states of bond order (§B) is observed. We propose that the Mott state it is unstable to bond ordering for any degree of ionicity. The second system we have studied is a set of idealized (harmonic) and realistic quantum dots. Two idealized dots, spherical and squashed, quasi-2D structures, are analyzed. In the first QMC is contrasted with a large set of commonly used electronic structure methods including MSFT, KLI, LSDA, and CCSD, and in the second we scrutinize whether SDW states truly exist as predicted by LSDA. In conjunction with the Computational Electronics Group at the Beckman Institute, we have performed QMC calculations on realistically modeled quantum dots in the effective mass approximation. The potentials of these dots are determined using the FEM method to solve Poisson's equation and the effective mass is allowed to spatially vary in our QMC simulations. We find that in arrays of QD's, LSDA may be insufficiently accurate to determine the proper spin states of the electrons. Additionally, we illustrate that QMC has a bright future ahead of it in modeling such devices using arbitrary functions represented on grids using splines."
"Quantum Monte Carlo (QMC) is one of the most powerful methods for solving many body problems in Physics and Chemistry. MC sampling allows one to measure physical properties systems with very complicated wave functions that incorporate correlation between the electrons. Additionally, QMC has O(N3) scaling verses exponential scaling for methods such as Conguration Interaction and “exact diagonalization"", and it’s perfect parallelism allows one to speed up computation with little effort on large parallel computers. Our work has focused on two systems. The first is a 1D ionic Hubbard model with a complex phase diagram that we find to exhibit a ferroeletric \bond ordered"" phase. A new many-body berry's phase formulation of the electronic polarization and localization is used in conjunction with QMC to analyze whether a bond order phase exists between the customary band and Mott insulating phases. The bond order and bond order correlation functions are measured to illustrate the stability of this intermediate state, and phase swapping between two degenerate states of bond order (§B) is observed. We propose that the Mott state it is unstable to bond ordering for any degree of ionicity. The second system we have studied is a set of idealized (harmonic) and realistic quantum dots. Two idealized dots, spherical and squashed, quasi-2D structures, are analyzed. In the first QMC is contrasted with a large set of commonly used electronic structure methods including MSFT, KLI, LSDA, and CCSD, and in the second we scrutinize whether SDW states truly exist as predicted by LSDA. In conjunction with the Computational Electronics Group at the Beckman Institute, we have performed QMC calculations on realistically modeled quantum dots in the effective mass approximation. The potentials of these dots are determined using the FEM method to solve Poisson's equation and the effective mass is allowed to spatially vary in our QMC simulations. We find that in arrays of QD's, LSDA may be insufficiently accurate to determine the proper spin states of the electrons. Additionally, we illustrate that QMC has a bright future ahead of it in modeling such devices using arbitrary functions represented on grids using splines."
The second system we have studied is a set of idealized (harmonic) and realistic quantum dots. Two idealized dots, spherical and squashed, quasi-2D structures, are analyzed. In the first QMC is contrasted with a large set of commonly used electronic structure methods including MSFT, KLI, LSDA, and CCSD, and in the second we scrutinize whether SDW states truly exist as predicted by LSDA. In conjunction with the Computational Electronics Group at the Beckman Institute, we have performed QMC calculations on realistically modelled quantum dots in the effective mass approximation. The potentials of these dots are determined using the FEM method to solve Poisson's equation and the effective mass is allowed to spatially vary in our QMC simulations. We find that in arrays of QD's, LSDA may be insufficiently accurate to determine the proper spin states of the electrons. Additionally, we illustrate that QMC has a bright future ahead of it in modelling such devices using arbitrary functions represented on grids using splines.
Quantum Monte Carlo methods are used to study a quantum phase transition in a one-dimensional Hubbard model with on-site interaction U and a staggered ionic potential $(\ensuremath{\Delta}).$ Using recently formulated methods, the electronic polarization and localization are determined directly from the correlated ground state wave function and compared to results of previous work using exact diagonalization and Hartree-Fock. We find a transition from a symmetric band insulator to a broken-symmetry bond ordered (BO) phase as the ratio of $U/\ensuremath{\Delta}$ is increased, with a metallic point at the transition. Since it is known that at $\ensuremath{\Delta}=0$ the usual Hubbard model is a Mott insulator (MI) with no long-range order, we have searched for a second transition to this state by (i) increasing U at fixed $\ensuremath{\Delta}$ and (ii) decreasing $\ensuremath{\Delta}$ at fixed U. We find no transition from the BO to MI state, and we propose that the symmetric MI state exists only at $\ensuremath{\Delta}\ensuremath{\equiv}0.$
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.
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.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.
In this thesis we apply recently developed, as well as sophisticated quantum Monte Carlo methods to numerically investigate models of strongly correlated electron systems on honeycomb structures. The latter are of particular interest owing to their unique properties when simulating electrons on them, like the relativistic dispersion, strong quantum fluctuations and their resistance against instabilities. This work covers several projects including the advancement of the weak-coupling continuous time quantum Monte Carlo and its application to zero temperature and phonons, quantum phase transitions of valence bond solids in spin-1/2 Heisenberg systems using projector quantum Monte Carlo in the valence bond basis, and the magnetic field induced transition to a canted antiferromagnet of the Hubbard model on the honeycomb lattice. The emphasis lies on two projects investigating the phase diagram of the SU(2) and the SU(N)-symmetric Hubbard model on the hexagonal lattice. At sufficiently low temperatures, condensed-matter systems tend to develop order. An exception are quantum spin-liquids, where fluctuations prevent a transition to an ordered state down to the lowest temperatures. Previously elusive in experimentally relevant microscopic two-dimensional models, we show by means of large-scale quantum Monte Carlo simulations of the SU(2) Hubbard model on the honeycomb lattice, that a quantum spin-liquid emerges between the state described by massless Dirac fermions and an antiferromagnetically ordered Mott insulator. This unexpected quantum-disordered state is found to be a short-range resonating valence bond liquid, akin to the one proposed for high temperature superconductors. Inspired by the rich phase diagrams of SU(N) models we study the SU(N)-symmetric Hubbard Heisenberg quantum antiferromagnet on the honeycomb lattice to investigate the reliability of 1/N corrections to large-N results by means of numerically exact QMC simulations. We study the melting of phases as correlations increase with decreasing N and determine whether the quantum spin liquid found in the SU(2) Hubbard model at intermediate coupling is a specific feature, or also exists in the unconstrained t-J model and higher symmetries.