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Full configuration interaction quantum Monte Carlo and its application to the lattice Bose polaron at quantum criticality

Matija Čufar

Published Jan 1, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.82921/mro-79

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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.
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.
Full configuration interaction quantum Monte Carlo for accurate ab initio\textit{ab initio}ab initio nuclear structure calculations: algorithms and calculation details
Full configuration interaction quantum Monte Carlo (FCIQMC) is a stochastic many-body solver that has been widely applied to electronic, molecular, and condensed-matter systems. In this work we apply FCIQMC to ab initio\textit{ab initio}ab initio nuclear structure calculations using interactions derived from chiral effective field theory. We describe the algorithm in detail, including imaginary-time propagation, excitation generation, estimator choices, the initiator approximation with adaptive shift correction, and reduced-density-matrix (RDM) sampling. Benchmark calculations in small model spaces, where deterministic full configuration interaction (FCI) results are available, validate the stochastic calculation of energies, radii, and RDM-based pure estimators. For large model spaces, we analyze the residual finite-walker bias through systematic walker-number convergence and infinite-walker extrapolations. We also demonstrate that FCIQMC can be extended beyond ground-state calculations by computing the low-lying spectrum of 6^66Li.
To validate the ability of full configuration interaction quantum Monte Carlo (FCIQMC) for studying the 2D Hubbard model near half-filling regime, the ground state energies of a 4 × 4 square lattice system with various interaction strengths are calculated. It is found that the calculated results are in good agreement with those obtained by exact diagonalization (i.e., the exact values for a given basis set) when the population of psi particles (psips) is higher than the critical population required to correctly sample the ground state wave function. In addition, the variations of the average computational time per 20 Monte Carlo cycles with the coupling strength and the number of processors are also analyzed. The calculated results show that the computational efficiency of an FCIQMC calculation is mainly affected by the total population of psips and the communication between processors. These results can provide useful references for understanding the FCIQMC algorithm, studying the ground state properties of the 2D Hubbard model for the larger system size by the FCIQMC method and using a computational budget as effectively as possible.
There has been recent literature discussion on the origin and severity of the `sign problem' in full configuration interaction quantum Monte Carlo (FCIQMC) and its `initiator' adaptation (i-FCIQMC), methods of interest and potential because they allow for exact (FCI) ground-state solutions to be obtained often at a much reduced computational cost. In this study we aim to use a simple order parameter, describing the `sign structure' of the stochastic wavefunction representation, to empirically characterise the fundamentally different collective behaviour of the walker population in both methods.
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].
We investigate the sign problem for full configuration interaction quantum Monte Carlo (FCIQMC), a stochastic algorithm for finding the ground-state solution of the Schr\"odinger equation with substantially reduced computational cost compared with exact diagonalization. We find kkk-space Hubbard models for which the solution is yielded with storage that grows sublinearly in the size of the many-body Hilbert space, in spite of using a wave function that is simply a linear combination of states. The FCIQMC algorithm is able to find this sublinear scaling regime without bias and with only a choice of the Hamiltonian basis. By means of a demonstration we solve for the energy of a 70-site half-filled system (with a space of 1038{10}^{38}1038 determinants) in 250 core hours, substantially quicker than the $\ensuremath{\sim}{10}^{36}$ core hours that would be required by exact diagonalization. This is the largest space that has been sampled in an unbiased fashion. The challenge for the recently developed FCIQMC method is made clear: Expand the sublinear scaling regime while retaining exact-on-average accuracy. We comment upon the relationship between this and the scaling law previously observed in the initiator adaptation (i-FCIQMC). We argue that our results change the landscape for the development of FCIQMC and related methods.