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arXiv:cond-mat/9508113 · Journal of the Physical Society of Japan

Stable Quantum Monte Carlo Algorithm forT=0Calculation of Imaginary Time Green Functions

Fakher F. Assaad, Masatoshi Imada

Published Jan 15, 1996• 44 citations
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Abstract

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 $4 \times 4$ to $12 \times 12$ lattices for the two-dimensional half-filled repulsive Hubbard model at $U/t = 4$. By fitting the tail of $ G(\vec{r}=0, τ) $ at long imaginary time to the form $e^{-τΔ_c}$, we obtain a precise estimate of the charge gap: $Δ_c = 0.67 \pm 0.02$ 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.

DOI
10.1143/jpsj.65.189

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We present a numerically stable Quantum Monte Carlo algorithm to calculate zero-temperature imaginary-time Green functions G(r⃗,τ)G(\vec{r}, τ)G(r,τ) for Hubbard type models. We illustrate the efficiency of the algorithm by calculating the on-site Green function G(r⃗=0,τ)G(\vec{r}=0, τ)G(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(r⃗=0,τ)G(\vec{r}=0, τ)G(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.
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 use quantum Monte Carlo methods to determine T=0T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0T=0 Green functions, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega})$, on lattices up to $16\ifmmode\times\else\texttimes\fi{}16$ for the 2D Hubbard model at U/t=4U/t\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4U/t=4. For chemical potentials $\ensuremath{\mu}$ within the Hubbard gap $|\ensuremath{\mu}|<{\ensuremath{\mu}}_{c}$ and at long distances $\stackrel{\ensuremath{\rightarrow}}{r}$, $G(\stackrel{\ensuremath{\rightarrow}}{r},\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\mu})\ensuremath{\sim}{e}^{\ensuremath{-}|\stackrel{\ensuremath{\rightarrow}}{r}|/{\ensuremath{\xi}}_{l}}$ with critical behavior ${\ensuremath{\xi}}_{l}\ensuremath{\sim}|\ensuremath{\mu}\ensuremath{-}{\ensuremath{\mu}}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}$, $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.26\ifmmode\pm\else\textpm\fi{}0.05$. This result stands in agreement with the assumption of hyperscaling with correlation exponent $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/4$ and dynamical exponent z=4z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}4z=4. In contrast, the generic band insulator as well as the metal-insulator transition in the 1D Hubbard model are characterized by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/2$ and z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2.
We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.
We present an algorithm for the computation of unbiased Green functions and selfenergies for quantum lattice models, free from systematic errors and valid in the thermodynamic limit. The method combines direct lattice simulations using the Blankenbecler-Scalapino-Sugar quantum Monte Carlo (BSS-QMC) approach with controlled multigrid extrapolation techniques. We show that the half-filled Hubbard model is insulating at low temperatures even in the weak- coupling regime; the previously claimed Mott transition at intermediate coupling does not exist.
We present a numerical solution of the infinite-dimensional Hubbard model at finite temperature in the paramagnetic phase. The problem reduces to a single-impurity Anderson model supplemented by a self-consistency condition. Using Monte Carlo methods and complete enumeration we determine the imaginary-time Green's function, the density of doubly occupied sites, and the compressibility close to half filling. All three quantities present direct evidence for a Mott insulating phase above a critical value of U.