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arXiv:cond-mat/0009447 · Physical review. B, Condensed matter

Efficient calculation of imaginary-time-displaced correlation functions in the projector auxiliary-field quantum Monte Carlo algorithm

M. Feldbacher, Fakher F. Assaad

Published Jan 29, 2001• 38 citations
arXivHTMLPDF

Abstract

The calculation of imaginary-time-displaced correlation functions with the auxiliary-field projector quantum Monte Carlo algorithm provides valuable insight (such as spin and charge gaps) into the model under consideration. One of the authors and M. Imada proposed a numerically stable method to compute those quantities [J. Phys. Soc. Jpn. 65, 189 (1996)]. Although precise, this method is expensive in CPU time. Here we present an alternative approach which is an order of magnitude quicker, just as precise, and very simple to implement. The method is based on the observation that for a given auxiliary field the equal-time Green-function matrix G is a projector: ${G}^{2}=G.$

DOI
10.1103/physrevb.63.073105

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The calculation of imaginary-time-displaced correlation functions with the auxiliary-field projector quantum Monte Carlo algorithm provides valuable insight (such as spin and charge gaps) into the model under consideration. One of the authors and M. Imada proposed a numerically stable method to compute those quantities [J. Phys. Soc. Jpn. 65, 189 (1996)]. Although precise, this method is expensive in CPU time. Here we present an alternative approach which is an order of magnitude quicker, just as precise, and very simple to implement. The method is based on the observation that for a given auxiliary field the equal-time Green-function matrix G is a projector: G2=G.{G}^{2}=G.G2=G.
Quantum Monte Carlo simulations provide a powerful tool to compute static properties of quantum many-particles models and realistic systems. A vast array of ground-state energies and properties defined by equal-time correlation functions can be computed with high accuracy. On the other hand, the extensions needed to study time-displaced correlation functions, which provide access to excited states, have been challenging, especially for fermionic strongly correlated systems. The authors present a methodology that allows a high-accuracy calculation of the dynamical Green functions of many-fermions systems in the framework of the auxiliary-field quantum Monte Carlo method. They extract the the charge gap of the repulsive two-dimensional Hubbard model at half-filling as a function of the interaction strength and propose strategies for applying their methodology to more realistic systems.
We present a numerically stable Quantum Monte Carlo algorithm to calculate zero-temperature imaginary-time Green functions $ G(\vec{r}, τ) $ for Hubbard type models. We illustrate the efficiency of the algorithm by calculating the on-site Green function $ G(\vec{r}=0, τ) $ on 4×44 \times 44×4 to 12×1212 \times 1212×12 lattices for the two-dimensional half-filled repulsive Hubbard model at U/t=4U/t = 4U/t=4. By fitting the tail of $ G(\vec{r}=0, τ) $ at long imaginary time to the form e−τΔce^{-τΔ_c}e−τΔc​, we obtain a precise estimate of the charge gap: Δc=0.67±0.02Δ_c = 0.67 \pm 0.02Δc​=0.67±0.02 in units of the hopping matrix element. We argue that the algorithm provides a powerful tool to study the metal-insulator transition from the insulator side.
Quantum Monte Carlo (QMC) algorithms have long relied on Jastrow factors to incorporate dynamic correlation into trial wave functions. While Jastrow-type wave functions have been widely employed in real-space algorithms, they have seen limited use in second-quantized QMC methods, particularly in projection methods that involve a stochastic evolution of the wave function in imaginary time. Here we propose a scheme for generating Jastrow-type correlated trial wave functions for auxiliary-field QMC methods. The method is based on decoupling the two-body Jastrow into one-body projectors coupled to auxiliary fields, which then operate on a single determinant to produce a multideterminant trial wave function. We demonstrate that intelligent sampling of the most significant determinants in this expansion can produce compact trial wave functions that reduce errors in the calculated energies. Our technique may be readily generalized to accommodate a wide range of two-body Jastrow factors and applied to a variety of model and chemical systems.
We evaluate imaginary time density-density correlation functions for a two-dimensional homogeneous electron gas using the phaseless auxiliary field quantum Monte Carlo method. We show that such methodology, once equipped with suitable numerical stabilization techniques necessary to deal with exponentials, products and inversions of large matrices, gives access to the calculation of imaginary time correlation functions for medium-sized systems; we present simulations of a number up to 42 correlated fermions in the continuum, using up to 300 plane waves as basis set elements. We discuss the numerical stabilization techniques and the computational complexity of the methodology. We perform the inverse Laplace transform of the obtained density-density correlation functions, assessing the ability of the phaseless auxiliary field quantum Monte Carlo method to evaluate dynamical properties of many-fermion systems.
Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.