Bold-line diagrammatic Monte Carlo method: General formulation and application to expansion around the noncrossing approximation

We present a general framework for performing “bold-line” diagrammatic Monte Carlo calculations using an analytical partial resummation as a starting point for a stochastic summation of all diagrams. As a stringent test case we assess the accuracy of the method by solving the equations of single-site dynamical mean-field theory, using the noncrossing approximation as a starting point. We establish the validity of the starting approximations and show that the bold method provides a very accurate treatment of the Mott-insulating phase.

Figure 1

(Color online) Typical diagrams arising in the hybridization expansion of the partition function of the Anderson impurity model: the four local states are described by wavy lines $\left(|0⟩,|\uparrow \downarrow ⟩\right)$ and light red (dark blue) dashed lines $\left(|\uparrow ⟩,|\downarrow ⟩\right)$. Light red (dark blue) solid lines denote hybridization functions, and empty (filled) circles denote local annihilation (creation) operators. (a) NCA diagram. (b) Bold propagator (double dashed line) which resums NCA diagrams including (a). (c) Diagram with crossing lines. (d) Diagram in expansion which resums diagrams including $c$. In BoldOCA, all of diagram (c) is contained in the single bold line (b).

Figure 2

(Color online) Probability $p$ for a diagram of $Z$ to contain $k$ spin-up hybridization lines, computed for Anderson impurity model with semicircular density of states and hybridization function fixed by dynamical mean-field self-consistency condition with parameters $\eta =0.15$, $n=1$, and $\beta t=10$ at $U$ values indicated. Full symbols: BoldNCA. Empty symbols: BoldOCA.

Figure 3

(Color online) Upper panel: expectation (over $C_W$) of sign as a function of interaction $U$ at carrier concentration $n=1$ per site and temperatures indicated. Lower panel: expectation value of sign as a function of carrier concentration $1-x$ at $U=6t$ and temperatures indicated.

Figure 4

(Color online) The Green’s function of Anderson impurity model with semicircular density of states and hybridization function fixed by a DMFT self-consistency condition, calculated using NCA, OCA, CT-HYB, and BoldNCA (BoldOCA would be indistinguishable), $\beta t=10$, starting from converged and accurate (Ref. 20) CT-HYB results at interactions and dopings indicated. Upper panel: Fermi-liquid case $U/t=1$. Lower panel: Correlated metal $U/t=4$.

Figure 5

(Color online) The Green’s function of Anderson impurity model with semicircular density of states and hybridization function fixed by a DMFT self-consistency condition, calculated using NCA, OCA, CT-HYB, and BoldNCA (BoldOCA would be indistinguishable), $\beta t=10$, starting from converged and accurate (Ref. 20) CT-HYB results at interactions and dopings indicated. Upper panel: Doped Mott insulator at filling $n=0.95$ and interaction $U/t=6$. Lower panel: Mott insulator, $U/t=6$, $n=1$.

Figure 6

(Color online) Upper panel: analytical continuation of the Green’s functions obtained from different solvers from DMFT of half-filled Hubbard model at $U=6t$ and $\beta t=7$. Lower panel: difference $\Delta G\left(\tau \right)$ between $G\left(\tau \right)$ obtained from CT-HYB, BoldNCA, and BoldOCA measurements and $G\left(\tau \right)$ reconstructed from the analytically continued $A\left(\omega \right)$.