Continuous-time quantum Monte Carlo calculation of multiorbital vertex asymptotics

We derive the equations for calculating the high-frequency asymptotics of the local two-particle vertex function for a multiorbital impurity model. These relate the asymptotics for a general local interaction to equal-time two-particle Green's functions, which we sample using continuous-time quantum Monte Carlo simulations with a worm algorithm. As specific examples we study the single-orbital Hubbard model and the three $t_{2g}$ orbitals of ${\mathrm{SrVO}}_3$ within dynamical mean-field theory (DMFT). We demonstrate how the knowledge of the high-frequency asymptotics reduces the statistical uncertainties of the vertex and further eliminates finite-box-size effects. The proposed method benefits the calculation of nonlocal susceptibilities in DMFT and diagrammatic extensions of DMFT.

Figure 1

Decomposition of the two-particle Green's function into disconnected parts and a connected part.

Figure 2

Diagram of the bare local interaction $U$. The bare vertex does not contain any Green's function, and the (amputated) legs drawn in gray indicate the direction of the incoming/outgoing particles and their spin-orbital flavor $a,b,c,d$ (the Matsubara frequencies are suppressed for simplicity).

Figure 3

Vertex diagrams that depend on only one bosonic frequency, in $ph$ channel (top left), $\overline{ph}$ channel (right), and $pp$ channel (bottom left). Frequencies are given in the channel-specific notation (see Appendix pp1).

Figure 4

Vertex diagrams that depend on one bosonic and one fermionic frequency, in $ph$ channel (top left), $\overline{ph}$ channel (right), and $pp$ channel (bottom left). Frequencies are given in the channel-specific notation (see Appendix pp1).

Figure 5

Local full vertex $F$ for the half-filled Hubbard model in DMFT for $U=2D$ and $\beta =8/D$. Upper row: $F_d$ in the the density channel. Lower row: $F_m$ in the magnetic channel. First column: $F$ extracted from an improved-estimator CT-QMC measurement with full frequency dependence. Second column: $F$ from QMC, combined with asymptotics according to Eq. (40) with $l=10$. Third column: $F$ obtained with ED for comparison. We use the particle-hole frequency representation (see Appendix pp1) and fix ${\omega }_{ph}=15\frac{2\pi }{\beta }$.

Figure 6

Difference of ED vertex and asymptotic vertex in density (top) and magnetic channel (bottom) for three different bosonic frequencies ${\omega }_{ph}=m\frac{2\pi }{\beta }$ (columns). The fermionic Matsubara frequencies on the $x$ and $y$ axis are shifted by $m$ as indicated.

Figure 7

Local susceptibilities ${\chi }_{\mathrm{loc}}\left(i{\omega }_n\right)$ in density (top) and magnetic channel (bottom) for the Hubbard model in DMFT at $U=2D$ and $\beta =8/D$. The bosonic Matsubara frequencies are ${\omega }_n=n\frac{2\pi }{\beta }$. We compare the direct calculation via Eq. (22) (solid line) to that using the summation Eq. (41) over fermionic Matsubara frequencies without vertex asymptotics in a small box (x) to that using vertex asymptotics and hence a large box (+). Inset: Zoom-in showing the box effect and noise reduction.

Figure 8

Matrix element of the full vertex $F_{d,1111}^{\nu {\nu }^{\prime }{\omega }_{15}}$ (upper row) and $F_{m,1111}^{\nu {\nu }^{\prime }{\omega }_{15}}$ (lower row) for four times at the same $t_{2g}$ orbital. Left column: $F$ extracted from an improved-estimator CT-QMC measurement with full frequency dependence. Right column: $F$, combined with asymptotics according to Eq. (40) with $l=15$. To remove the constant background, $F_d$ was shifted by $U_d=U$ and $F_m$ by $U_m=-U$.

Figure 9

Difference of $F_{d,1111}^{\nu {\nu }^{\prime }{\omega }_{15}}$ (top) and $F_{m,1111}^{\nu {\nu }^{\prime }{\omega }_{15}}$ (bottom) to their respective purely asymptotic version for different bosonic frequencies ${\omega }_m=\frac{2\pi }{\beta }m$.

Figure 10

Local susceptibility ${\chi }_{d/m,1122}^{\mathrm{loc}}$ of ${\mathrm{SrVO}}_3$ between two different $t_{2g}$ orbitals in density (top) and magnetic channel (bottom). Inset: Zoom-in.

Figure 11

The $ph, \overline{ph}$, and $pp$ parts of the asymptotic vertex $F_{\uparrow \downarrow }^{\mathrm{asympt},\nu {\nu }^{\prime }{\omega }_{10}}-U$ in $ph$ notation at $U=2$ and $\beta =8$.

Figure 12

$F_{\uparrow \downarrow }^{\mathrm{asympt},\nu {\nu }^{\prime }{\omega }_{10}}$ (left) and $R_{\uparrow \downarrow }^{\nu {\nu }^{\prime }{\omega }_{10}}$ (right) in $ph$ notation at $U=2$ and $\beta =8$.