problem, due to the Fermion sign problem. While still scaling exponentially, full configuration-interaction Monte Carlo (FCI-QMC) [1] mitigates some of the exponential variance by allowing annihilation of noise ‐ whenever two walkers arrive at the same configuration with opposite signs, they are removed from the simulation. While FCI-QMC has been quite successful for quantum chemistry problems, it has had less success with strongly correlated systems; for example, it is unstable for even small systems sizes when applied to the one of the simplest many-body systems: the Fermi-Polaron. In this talk, we explore a series of algorithmic improvements which significantly increase its eectiveness. These improvements include modifying fixed node to work in this method, developing a partial node approach which allows for an extrapolation to the exact answer and developing a variant of release node. Additionally, we find a way to work directly in the thermodynamic limit for some subset of important systems. We apply these new approaches to the Fermi-Polaron finding that it accurately reproduces known analytic and numerical results going beyond them in certain cases [2].
problem, due to the Fermion sign problem. While still scaling exponentially, full configuration-interaction Monte Carlo (FCI-QMC) [1] mitigates some of the exponential variance by allowing annihilation of noise ‐ whenever two walkers arrive at the same configuration with opposite signs, they are removed from the simulation. While FCI-QMC has been quite successful for quantum chemistry problems, it has had less success with strongly correlated systems; for example, it is unstable for even small systems sizes when applied to the one of the simplest many-body systems: the Fermi-Polaron. In this talk, we explore a series of algorithmic improvements which significantly increase its eectiveness. These improvements include modifying fixed node to work in this method, developing a partial node approach which allows for an extrapolation to the exact answer and developing a variant of release node. Additionally, we find a way to work directly in the thermodynamic limit for some subset of important systems. We apply these new approaches to the Fermi-Polaron finding that it accurately reproduces known analytic and numerical results going beyond them in certain cases [2].
Finding the ground state of a fermionic Hamiltonian using quantum Monte Carlo (QMC) is a very difficult problem, due to the Fermi sign problem. While still scaling exponentially, full configuration-interaction Monte Carlo (FCIQMC) mitigates some of the exponential variance by allowing annihilation of noise---whenever two walkers arrive at the same configuration with opposite signs, they are removed from the simulation. While FCIQMC has been quite successful for quantum chemistry problems, its application to problems in condensed systems has been limited. In this paper, we apply the FCIQMC algorithm to the Fermi polaron problem, which provides an ideal test bed for improving the algorithm, although since we restrict the number of many-body excitations, our algorithm is more precisely a QMC implementation of the configuration-interaction method (CIQMC). In its simplest form, CIQMC is unstable for even a fairly small system sizes. However, with a series of algorithmic improvements, we are able to significantly increase its effectiveness. We modify fixed node QMC to work in these systems, introduce a well chosen importance sampled trial wave function, a partial node approximation, and a variant of release node. Finally, we develop a way to perform CIQMC directly in the thermodynamic limit.
This thesis provides an investigation of the application of the full configuration interaction quantum Monte Carlo (FCIQMC) algorithm to ultracold atom problems. This includes working on the algorithm, understanding the sign problem, and applying it to the study of strongly-correlated lattice polaron physics. The thesis conceptually consists of two parts. In the first, we discuss the FCIQMC algorithm, which is a modern variant of projector Monte Carlo and can be used for calculating the ground states or low-lying excited states of quantum Hamiltonians. We present our implementation of the algorithm in the Julia programming language and discuss the details of the algorithm and data structures employed. Next, we turn to a treatment of the FCIQMC sign problem, which arises when the number of walkers employed by FCIQMC is insufficient to capture the sign structure of the ground state eigenvector of the Hamiltonian. We present a new scalar model that accurately reconstructs the dynamic behaviour of FCIQMC and shows us that as the number of walkers increases, the walker population switches from a sign-incoherent regime to a sign-coherent one in way that is reminiscent of a continuous phase transition. In the second part, we focus on the application of FCIQMC to the Bose-Hubbard model (BHM) with an impurity near quantum criticality. We use FCIQMC to guide and validate a diagrammatic theory based on the quantum Gutzwiller method, which describes the polaron by treating interactions between the impurity and the elementary excitations in the BHM. This theory shows that at strong interactions, the polaron reflects the state of the host system and as such, polarons could be used as quantum sensors in experiments. Lastly, we show numerical evidence that the polaron energy --- the energy shift caused by introducing an impurity to the BHM --- is scale invariant near the phase transition and that it exhibits critical phenomena not yet explained by theory. Furthermore, we show that the polaron energy is linked to the impurity-boson density-density correlation function and that it exhibits a flattening near the phase transition. This flattening indicates that in the thermodynamic limit, the length scales associated with the polaron diverge.
We present a study of fixed- and partial-node approximations in Slater determinant basis sets, using full configuration interaction quantum Monte Carlo (FCIQMC) to perform sampling. Walker annihilation in the FCIQMC method allows partial-node simulations to be performed, relaxing the nodal constraint to converge to the FCI solution. This is applied to ab initio molecular systems, using symmetry-projected Jastrow mean-field wave functions for complete active space (CAS) problems. Convergence and the sign problem within the partial-node approximation are studied, which is shown to eventually be limited in its use due to the large walker populations required. However, the fixed-node approximation results in an accurate and practical method. We apply these approaches to various molecular systems and active spaces, including ferrocene and acenes. This also provides a test of symmetry-projected Jastrow mean-field wave functions in variational Monte Carlo for a new set of problems. For trans -polyacetylene molecules and acenes, we find that the time to perform a constant number of fixed-node FCIQMC iterations scales as and, respectively, resulting in an efficient method for CAS-based problems that can be applied accurately to large active spaces.
The study of strongly correlated quantum many-body systems is an outstanding challenge. Highly accurate results are needed for the understanding of practical and fundamental problems in condensed-matter physics, high energy physics, material science, quantum chemistry and so on. Our familiar mean-field or perturbative methods tend to be ineffective. Numerical simulations provide a promising approach for studying such systems. The fundamental difficulty of numerical simulation is that the dimension of the Hilbert space needed to describe interacting systems increases exponentially with the system size. Quantum Monte Carlo (QMC) methods are one of the best approaches to tackle the problem of enormous Hilbert space. They have been highly successful for boson systems and unfrustrated spin models. For systems with fermions, the exchange symmetry in general causes the infamous sign problem, making the statistical noise in the computed results grow exponentially with the system size. This hinders our understanding of interesting physics such as high-temperature superconductivity, metal-insulator phase transition. In this thesis, we present a variety of new developments in the auxiliary-field quantum Monte Carlo (AFQMC) methods, including the incorporation of symmetry in both the trial wave function and the projector, developing the constraint release method, using the force-bias to drastically improve the efficiency in Metropolis framework, identifying and solving the infinite variance problem, and sampling Hartree-Fock-Bogoliubov wave function. With these developments, some of the most challenging many-electron problems are now under control. We obtain an exact numerical solution of two-dimensional strongly interacting Fermi atomic gas, determine the ground state properties of the 2D Fermi gas with Rashba spin-orbit coupling, provide benchmark results for the ground state of the two-dimensional Hubbard model, and establish that the Hubbard model has a stripe order in the underdoped region.
By performing a stochastic dynamic in a space of Slater determinants, the full configuration interaction quantum Monte Carlo (FCIQMC) method has been able to obtain energies which are essentially free from systematic error to the basis set correlation energy, within small and systematically improvable error bars. However, the weakly exponential scaling with basis size makes converging the energy with respect to basis set costly and in larger systems, impossible. To ameliorate these basis set issues, here we use perturbation theory to couple the FCIQMC wavefunction to an explicitly correlated strongly orthogonal basis of geminals, following the [2](R12) approach of Valeev et al. The required one- and two-particle density matrices are computed on-the-fly during the FCIQMC dynamic, using a sampling procedure which incurs relatively little additional computation expense. The F12 energy corrections are shown to converge rapidly as a function of sampling, both in imaginary time and number of walkers. Our pilot calculations on the binding curve for the carbon dimer, which exhibits strong correlation effects as well as substantial basis set dependence, demonstrate that the accuracy of the FCIQMC-F12 method surpasses that of all previous FCIQMC calculations, and that the F12 correction improves results equivalent to increasing the quality of the one-electron basis by two cardinal numbers.
The full configuration interaction quantum Monte Carlo (FCIQMC) method, as well as its "initiator" extension (i-FCIQMC), is used to tackle the complex electronic structure of the carbon dimer across the entire dissociation reaction coordinate, as a prototypical example of a strongly correlated molecular system. Various basis sets of increasing size up to the large cc-pVQZ are used, spanning a fully accessible N-electron basis of over 10(12) Slater determinants, and the accuracy of the method is demonstrated in each basis set. Convergence to the FCI limit is achieved in the largest basis with only O[10(7)] walkers within random errorbars of a few tenths of a millihartree across the binding curve, and extensive comparisons to FCI, CCSD(T), MRCI, and CEEIS results are made where possible. A detailed exposition of the convergence properties of the FCIQMC methods is provided, considering convergence with elapsed imaginary time, number of walkers and size of the basis. Various symmetries which can be incorporated into the stochastic dynamic, beyond the standard abelian point group symmetry and spin polarisation are also described. These can have significant benefit to the computational effort of the calculations, as well as the ability to converge to various excited states. The results presented demonstrate a new benchmark accuracy in basis-set energies for systems of this size, significantly improving on previous state of the art estimates.