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Bulletin of the American Physical Society

Numerical investigations of the sign problem in full configuration interaction quantum Monte Carlo

James J. Shepherd, James S. Spencer, Gustavo E. Scuseria

Published Mar 4, 2015• 0 citations
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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.

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This paper deals with the sign problem in full configuration-interaction quantum Monte Carlo. After putting this article on the arxiv, it was pointed out to us that our argument applies to a number of model Hamiltonians we considered numerically but not to the most generic case. Please see arxiv: 1209.3044, where we have a corrected comprehensive discussion of the necessary and sufficient conditions for when the sign problem for a given Hamiltonian (and basis) differ between first and second quantization.

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Emergence of Critical Phenomena in Full Configuration Interaction Quantum Monte Carlo

James J. Shepherd, Lauretta R. Schwarz, Robert E. Thomas +3 more

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.

arXiv:1209.4023•Sep 18, 2012•0 citations
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.
This paper deals with the sign problem in full configuration-interaction quantum Monte Carlo. After putting this article on the arxiv, it was pointed out to us that our argument applies to a number of model Hamiltonians we considered numerically but not to the most generic case. Please see arxiv: 1209.3044, where we have a corrected comprehensive discussion of the necessary and sufficient conditions for when the sign problem for a given Hamiltonian (and basis) differ between first and second quantization.
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.