Stochastic wavefunction approach to Hubbard-like models

This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ( $p-d$ ) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\textit{ab-initio}$ systems, including a fully periodic $\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band ($p-d$) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the $p-d$ Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as $\textit{ab-initio}$ systems, including a fully periodic $\textit{ab-initio}$ graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.

Oct 1, 20172 citations

A generating functional approach to the Hubbard model

Yu. A. Izyumov, N. I. Chaschin, D. S. Alexeev +1 more

arXiv:cond-mat/0504723The European Physical Journal BMay 1, 200515 citations

Quantum Monte Carlo simulations of Hubbard type models

Fakher F. Assaad, R. Preuss, A. Muramatsu +1 more

Journal of Low Temperature PhysicsApr 1, 19944 citations

Phaseless quantum Monte-Carlo approach to strongly correlated superconductors with stochastic Hartree–Fock–Bogoliubov wavefunctions

Olivier Juillet, Alexandre Leprévost, J. Bonnard +1 more

Abstract The so-called phaseless quantum Monte-Carlo method currently offers one of the best performing theoretical framework to investigate interacting Fermi systems. It allows to extract an approximate ground-state wavefunction by averaging independent-particle states undergoing a Brownian motion in imaginary-time. Here, we extend the approach to a random walk in the space of Hartree–Fock–Bogoliubov (HFB) vacua that are better suited for superconducting or superfluid systems. Well-controlled statistical errors are ensured by constraining stochastic paths with the help of a trial wavefunction. It also guides the dynamics and takes the form of a linear combination of HFB ansätze. Estimates for the observables are reconstructed through an extension of Wick’s theorem to matrix elements between HFB product states. The usual combinatory complexity associated to the application of this theorem for four- and more- body operators is bypassed with a compact expression in terms of Pfaffians. The limiting case of a stochastic motion within Slater determinants but guided with HFB trial wavefunctions is also considered. Finally, exploratory results for the spin-polarized Hubbard model in the attractive regime are presented.Abstract The so-called phaseless quantum Monte-Carlo method currently offers one of the best performing theoretical framework to investigate interacting Fermi systems. It allows to extract an approximate ground-state wavefunction by averaging independent-particle states undergoing a Brownian motion in imaginary-time. Here, we extend the approach to a random walk in the space of Hartree–Fock–Bogoliubov (HFB) vacua that are better suited for superconducting or superfluid systems. Well-controlled statistical errors are ensured by constraining stochastic paths with the help of a trial wavefunction. It also guides the dynamics and takes the form of a linear combination of HFB ansätze. Estimates for the observables are reconstructed through an extension of Wick’s theorem to matrix elements between HFB product states. The usual combinatory complexity associated to the application of this theorem for four- and more- body operators is bypassed with a compact expression in terms of Pfaffians. The limiting case of a stochastic motion within Slater determinants but guided with HFB trial wavefunctions is also considered. Finally, exploratory results for the spin-polarized Hubbard model in the attractive regime are presented.

arXiv:1610.08022Journal of Physics A Mathematical and TheoreticalFeb 24, 20174 citations

p-d

) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the

p-d

Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as

\textit{ab-initio}

systems, including a fully periodic

\textit{ab-initio}

graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.