Improved estimators for the self-energy and vertex function in hybridization-expansion continuous-time quantum Monte Carlo simulations

Abstract

We propose efficient measurement procedures for the self-energy and vertex function of the Anderson impurity model within the hybridization expansion continuous-time quantum Monte Carlo algorithm. The method is based on the measurement of higher-order correlation functions related to the quantities being sought through the equation of motion, a technique previously introduced in the NRG context. For the case of interactions of density-density type, the additional correlators can be obtained at essentially no additional computational cost. In combination with a recently introduced method for filtering the Monte Carlo noise using a representation in terms of orthogonal polynomials, we obtain data with unprecedented accuracy. This leads to an enhanced stability in analytical continuations of the self-energy or in two-particle based theories such as the dual fermion approach. As an illustration of the method we reexamine the previously reported spin-freezing and high-spin to low-spin transitions in a two-orbital model with density-density interactions. In both cases, the vertex function undergoes significant changes, which suggests significant corrections to the dynamical mean-field solutions in dual fermion calculations.

Towards ab initio Calculations with the Dynamical Vertex Approximation

“…Furthermore, CT-QMC algorithms naturally operate in the imaginary time basis, requiring an extra Fourier transform to Matsubara frequencies. Intrinsic statistical uncertainties in the high-frequency region of the two-particle quantities 100,101) combined with the previously mentioned computational requirements limit us hence to small frequency box sizes of the two-particle Green's func- Fig. 2.…”

Effect of local electron-electron correlation in hydrogen-like impurities in Ge

“…The Dyson equation gives the difference between Green's functions obtained from different Monte Carlo simulations, preventing the error from canceling. Indeed, the error in the self-energy is proportional to the absolute error in the Monte Carlo simulation, 42 becoming much larger than the actual data even at relatively low frequencies. Moreover, one cannot accurately determine other quantities that are sensitive to the Green's function and self-energy at all frequencies (such as the occupation of the orbitals).…”

“…Boehnke et al 43 showed that, by measuring the Green's functions in an orthogonal Legendre basis (limited to a relatively small number of polynomials), one can filter out the Monte Carlo noise without losing any accuracy in the computation of the Green's functions and self-energies. Hafermann et al 42 derived an expression for the self-energy involving a quotient of Green's functions rather than a difference. In this formulation, the error in the self-energy is proportional to the relative Monte Carlo error, leading to greatly reduced error into high frequencies.…”

Improved Green’s function measurement for hybridization expansion quantum Monte Carlo

“…The statistical noise in the selfenergy (and the two-particle Green's function) can be significantly suppressed when improved estimators based on the equation of motion are used [18]. Besides that, filtering out the stochastic noise using orthogonal polynomial representation can be used to further improve the results [19].…”

“…Recently, an efficient estimator for the selfenergy in CT-HYB was proposed in [18]. The method is based on the observation that the selfenergy can be expressed from the equation of motion for the Green's function as…”