Diagrammatic Monte Carlo approach for diagrammatic extensions of dynamical mean-field theory: Convergence analysis of the dual fermion technique

The dual fermion approach provides a formally exact prescription for calculating properties of a correlated electron system in terms of a diagrammatic expansion around dynamical mean-field theory (DMFT). Most practical implementations, however, neglect higher-order interaction vertices beyond two-particle scattering in the dual effective action and further truncate the diagrammatic expansion in the two-particle scattering vertex to a leading-order or ladder-type approximation. In this work, we compute the dual fermion expansion for the two-dimensional Hubbard model including all diagram topologies with two-particle interactions to high orders by means of a stochastic diagrammatic Monte Carlo algorithm. We benchmark the obtained self-energy against numerically exact diagrammatic determinant Monte Carlo simulations to systematically assess convergence of the dual fermion series and the validity of these approximations. We observe that, from high temperatures down to the vicinity of the DMFT Néel transition, the dual fermion series converges very quickly to the exact solution in the whole range of Hubbard interactions considered ($4\le U/t\le 12$), implying that contributions from higher-order vertices are small. As the temperature is lowered further, we observe slower series convergence, convergence to incorrect solutions, and ultimately divergence. This happens in a regime where magnetic correlations become significant. We find, however, that the self-consistent particle-hole ladder approximation yields reasonable and often even highly accurate results in this regime.

Figure 1

The CWO update creates a pair of worms with excess 4-momentum $\delta$ on a random vertex corner. The converse DWO move removes a pair of worms in the same location.

Figure 2

The MWL/MWV/MWH updates move a worm to a neighboring location and update momentum and frequency indices of the touched propagator or vertex accordingly. The pictures show the moving of $\mathcal{I}$. Moving $\mathcal{M}$ has the same effect except that the sign of the 4-momentum $\delta =(\mathrm{\Delta },\mathbf{\delta })$ is flipped.

Figure 3

The RWL update reconnects the lines adjacent to the worms. In this example, the new excess 4-momentum is ${\delta }^{\prime }=\delta +k-k^{\prime }$.

Figure 4

The AIV update replaces the worms by a new interaction vertex $\gamma$. If $\sigma \ne {\sigma }^{\prime }$, we may alternatively create a vertex with different spin configuration by connecting the $(\sigma ,k)$ line to the lower right and the $({\sigma }^{\prime },k^{\prime }+\delta )$ line to the upper right corner. The converse move is called RIV.

Figure 5

Dual (top) and lattice-fermion (bottom) self-energy at the lowest Matsubara frequency along the high-symmetry lines of the Brillouin zone for moderate interaction $U=4$ and temperature $T=0.5$ at half-filling $n=1$. The imaginary part of $\tilde{\mathrm{\Sigma }}$ has been multiplied by a factor of two for better visibility. Different colors correspond to diagram order cutoffs $N_{*}=$ 1 (DMFT), 2, 4, 6. Grey crosses indicate results from the one-shot ($\times$) and self-consistent ($+$) ladder approximation, circles the DDMC benchmark. Real (imaginary) parts are shown with solid (dashed) lines. Stochastic errors on DDMC and DiagMC@DF data are displayed as vertical bars, but often indiscernible because they are smaller than the line width.

Figure 6

Dual (top) and lattice-fermion (bottom) self-energy at the lowest Matsubara frequency along the high-symmetry lines of the Brillouin zone at half-filling and $T=0.5$ for stronger interactions $U=8$ (left) and $U=12$ (right). For better visibility, the imaginary parts of the dual self-energies have been magnified by factors of 2 and 10, respectively. See Fig. 5 for an explanation of the different lines and symbols.

Figure 7

Lattice self-energy for $U=12$, $T=0.5$, $n=1$ at the high-symmetry points $\mathrm{\Gamma }(0,0)$, $\mathrm{X}(\pi ,0)$, and $\mathrm{M}(\pi ,\pi )$ plotted vs the Matsubara frequency ${\omega }_n$. Shown are the sixth-order DiagMC@DF result (yellow filled symbols) and the DDMC benchmark (open black symbols) as well as the momentum-independent DMFT data (blue lines). Symbols referring to real and imaginary parts are connected by solid and dashed lines, respectively.

Figure 8

Lattice self-energies at half-filling and lower temperatures. The parameters are $U=4$, $T=0.2$ (top), $U=8$, $T=0.3$ (center), and $U=12$, $T=0.25$ (bottom). See Fig. 5 for an explanation of the different lines and symbols.

Figure 9

The dual self-energy series at half-filling and $U=12$, $T=0.167$ appears strongly divergent. For better visibility, the imaginary part is magnified by a factor of 10. Real (imaginary) parts are shown with solid (dashed) lines. Stochastic errors are displayed as vertical bars, but often indiscernible because they are smaller than the line width.

Figure 10

Overview of our DiagMC@DF simulations at half-filling (filled symbols). Red diamonds mark parameters where the DF series appears to diverge. Green circles (yellow squares) indicate rapid (slow) series convergence, red crosses convergence to incorrect results. For comparison, the lines show DMFT Néel temperatures from Ref. [37] and temperatures where nearest-neighbor spin correlations $C_s=-4\langle S_i^z{S}_{i+1}^z\rangle$ or the correlation length $\xi$ reach a given value (see main text for details).

Figure 11

Double occupancy of the half-filled model at $U=8$ and temperatures $T=0.5,1$. Shown are DF series convergence with diagram order $N_{*}$ and finite-size extrapolated DDMC results. Diagram order $N_{*}=1$ corresponds to DMFT.

Figure 12

Lattice self-energy at 24% hole doping for $U=4$ and $T=0.5$. Grey bands indicate the estimate from conventional (weak-coupling) DiagMC calculations. Different colors represent DiagMC@DF data with varying order cutoff. Real and imaginary parts are indicated by solid and dashed lines, respectively. Stochastic errors are displayed as vertical bars, but often indiscernible because they are smaller than the line width.

Figure 13

Lattice self-energy at 24% hole doping for $U=4$, $T=0.2$. (Top) Momentum dependence of $\mathrm{\Sigma }\left({\omega }_0\right)$, see Fig. 12 for an explanation of the different lines. (Bottom) Convergence at the high-symmetry points $\mathrm{\Gamma }$, $X$, $M$ (blue, orange, green) with diagram order $N_{*}$ in the DF series (squares connected by solid lines) and in the weak-coupling expansion sampled by conventional DiagMC (triangles, dashed lines).

Figure 14

Lattice self-energy at 20% hole doping for $U=12$, $T=0.5$. Different colors represent DiagMC@DF data with varying order cutoff. Real and imaginary parts are indicated by solid and dashed lines, respectively. Stochastic errors are displayed as vertical bars, but often indiscernible because they are smaller than the line width.

Figure 15

Lattice self-energy at 20% hole doping for $U=8$, $T=0.25$. (Top) Momentum dependence, see Fig. 14 for an explanation of the different lines. (Bottom) Convergence with diagram order $N_{*}$ at the high-symmetry points.

Figure 16

Double occupancy vs order cutoff $N_{*}$ at $U=12$, $T=0.5$ for three different values of the doping $\delta =1-n$. Points at $N_{*}=1$ correspond to DMFT. Triangles indicate cluster-size extrapolated DCA results from Ref. [12], the circle our DDMC benchmark calculation at half-filling. Error estimates are indicated by black vertical lines.

Figure 17

Modified lattice self-energy (green lines) calculated according to Eq. (19) compared to the original self-energy [Eq. (7), orange lines] and DDMC benchmarks for three sets of parameters at half-filling. In all cases the cutoff order is $N_{*}=6$. Self-energy data from different orders and ladder approximations for the same parameters can be found in Fig. 8 (top) and Fig. 6, respectively. Stochastic errors are displayed as vertical bars on all data, but often indiscernible because they are smaller than the line width.

Figure 18

(Top and center) Lattice self-energy computed from different hybridization functions for the half-filled system at $U=8$ and $T=0.3$. Shown is the dependence on cutoff order $N_{*}$ at the high-symmetry points $\mathrm{\Gamma }$ (blue), $X$ (orange), and $M$ (green). Points connected by solid (dashed) lines correspond to an expansion around DMFT (self-consistent DF ladder approximation). Both show a divergence in the imaginary part at the $X$ point. For comparison, the corresponding DDMC and sc-ladder results are indicated, too. (Bottom) DMFT and sc-ladder hybridization functions that have been used as inputs for the two different DiagMC@DF calculations.

Figure 19

Lattice self-energy at the $M$ point for $U=4$, $T=0.5$, and half-filling. Black and red squares are DCA and DiagMC@DF data, respectively, digitized from Fig. 12 of Ref. [46]. Blue and green symbols represent our DiagMC@DF and DDMC data. DF (DDMC) data are plotted as a function of cutoff order $N_{*}$ (linear system size $L$). Reference [46] does not provide error estimates, but their plot symbols have a radius $\sim 0.003$ on the $y$-axis scale. Blue crosses connected by dashed lines indicate the effect of reducing the momentum basis cutoff from $n_{\text{max}}=8$ to 3.

Figure 20

Diagram topologies of first- and second-order contributions to the dual self-energy. In- and outgoing corners of interaction vertices are marked with open and filled circles, respectively. In the symmetrized theory (cf. Appendix), primed and nonprimed diagrams are considered equivalent and only one variant is treated explicitly, whereas the unsymmetrized theory sums all shown diagrams.