Symmetric improved estimators for continuous-time quantum Monte Carlo

J. Kaufmann, P. Gunacker, A. Kowalski, G. Sangiovanni, and K. Held

Phys. Rev. B 100, 075119 – Published 8 August 2019

Abstract

We derive equations of motion for Green's functions of the multiorbital Anderson impurity model by differentiating symmetrically with respect to all time arguments. The resulting equations relate the one- and two-particle Green's function to correlators of up to six particles at four times. As an application we consider continuous-time quantum Monte Carlo simulations in the hybridization expansion, which hitherto suffered from notoriously high noise levels at large Matsubara frequencies. Employing the derived symmetric improved estimators overcomes this problem.

Received 5 June 2019

DOI:https://doi.org/10.1103/PhysRevB.100.075119

Physics Subject Headings (PhySH)

Quantum field theory (low energy)

Anderson impurity model Quantum Monte Carlo

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

J. Kaufmann^1,2,*, P. Gunacker^1,*, A. Kowalski³, G. Sangiovanni⁴, and K. Held¹

¹Institute for Solid State Physics, TU Wien, 1040 Vienna, Austria
²Institute for Theoretical Solid State Physics, IFW Dresden, 01069 Dresden, Germany
³Institut für Theoretische Physik und Astrophysik, Universität Würzburg, 97074 Würzburg, Germany
⁴Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Universität Würzburg, 97074 Würzburg, Germany

^*These authors contributed equally to this work.

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Issue

Vol. 100, Iss. 7 — 15 August 2019

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Images

Figure 1
Feynman-diagrammatic visualization of the symmetric improved estimator Eq. (13) for the one-particle Green's function. The part of the diagram that is drawn in green will be computed by CT-QMC in the following. Solid lines are full one-particle Green's functions $G$ , dashed lines are noninteracting Green's functions $G$ , and dots are $U$ matrices. The three-particle Green's function is represented by the hexagon. Note that the Hartree term $\propto \sum_{j} U_{[a j] [a j]} n_{j}$ is excluded.
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Figure 2
Heuristic drawing of the Feynman-diagrammatic decomposition of the connected part of the two-particle Green's function, as obtained by symmetric improved estimators. The estimators that are later computed by CT-QMC are drawn in green. Solid lines represent interacting one-particle Green's functions $G$ , dashed lines are noninteracting Green's functions $G$ , and dots are $U$ matrices. The terms involving $ϑ$ are to be understood as products. Note that Hartree-like terms and products thereof are not shown, in order to make the picture more concise.
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Figure 3
Symbolic representation of a “replacement” move applied to a worm operator that is at equal time with two other operators. Operators are represented as symbols (flavors) on the imaginary time axis before (top) and after (bottom) the move, with filled symbols representing creators and little vertical tags at the symbols' top and bottom signifying operators connected to hybridization events. Changes due to the move are marked in orange.
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Figure 4
Upper panel: Comparison of the self-energy (for the AIM specified in Sec. 5), as calculated from the directly measured one-particle Green's function (direct) vs the result obtained with improved (impr.) and symmetrically improved (sym. impr.) estimators. Lower panel: logplot of the absolute difference to exact diagonalization data.
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Figure 5
Reducible vertex $F_{↑ ↑ ↓ ↓}^{ν ν^{'} ω}$ . Upper row: Bosonic frequency $ν_{1} - ν_{2} = ω = 0$ . Lower row: $ν_{1} - ν_{2} = ω = 10 \times 2 π / β$ . First column: $F$ as calculated with symmetric improved estimators. Second column: Difference of the symmetric improved to the exact result. Third column: Difference of the conventional calculation (directly measured two-particle Green's function) to the exact result.
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Figure 6
Three-leg kernel function $K_{↑ ↑ ↓ ↓}^{(2), p h, ν (ν - ω)}$ , calculated by Eq. (31) (left), its difference to the exact result (middle), and the difference of the conventional direct calculation to the exact result (right).
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