Convergence acceleration and stabilization of dynamical mean-field theory calculations

The convergence to the self-consistency in the dynamical mean-field theory (DMFT) calculations for models of correlated electron systems can be significantly accelerated by using an appropriate mixing of hybridization functions which are used as the input to the impurity solver. It is shown that the techniques and the past experience with the mixing of input charge densities in the density-functional theory calculations are also effective in DMFT. As an example, the increase in the computational requirements near the Mott metal-insulator transition in the Hubbard model due to critical slowing down can be strongly reduced by using the modified Broyden’s method to numerically solve the nonlinear self-consistency equation. Speed-up factors as high as three were observed in practical calculations even for this relatively well-behaved problem. Furthermore, the convergence can be achieved in difficult cases where simple linear mixing is either not effective or even leads to divergence. Unstable and metastable solutions can also be obtained. We also determine the linear response of the system with respect to the variations in the hybridization function, which is related to the propagation of the information between the different energy scales during the iteration.

Figure 1

(Color online) The DMFT loop using the numerical renormalization group as the impurity solver. The Broyden solver is incorporated in the loop as a correction step which modifies the input hybridization function in order to accelerate the convergence to the self-consistency. The new elements in the loop are shown in gray (red online).

Figure 2

(Color online) Comparison of the convergence of the impurity spectral function $A\left(\omega \right)$ in a calculation for the Hubbard model defined on the Bethe lattice in the paramagnetic phase at half-filling. We compare the simple mixing (here $\alpha =1$, i.e., the output from the previous iteration is used directly as the input for the new iteration, so this is actually direct iteration rather than mixing) and Broyden’s mixing. The inset shows the converged density of states.

Figure 3

The Broyden corrections for consecutive iterations; the curves are offset for clarity. The parameters are as in Fig. 2.

Figure 4

(Color online) Comparison of the convergence as a function of the interaction strength $U/W$. The convergence is defined to occur when two consecutive solutions for the density of states differ by not more than $\lambda ={10}^{-6}$ (integrated absolute value of the difference). The vertical dashed line corresponds to the point of the Mott metal-insulator transition at $U_{\text{c}}/W=1.46$.

Figure 5

(Color online) Majority-spin spectral functions for the half-filled Hubbard model in a magnetic field. The arrows show the direction of the increasing magnetic field.

Figure 6

(Color online) The response function $R_E\left(\omega \right)$ for the Hubbard model for the excitation energy $E/W=0.2$ (graphically represented as the vertical dashed line). Model parameters are the same as in Fig. 2.

Figure 7

(Color online) Evolution of the paramagnetic solution for the half-filled Hubbard model after switching off the Broyden mixing. (a) Initial paramagnetic spectral function. (b) Resulting antiferromagnetic spin-dependent spectral functions. (c) Magnetization and (d) convergence as a function of the number of iterations. (e) Difference between the spectral functions at iteration 9 and the initial spectral function indicating the progressive breaking of the spin symmetry.

Figure 8

Evolution of the solution for the Hubbard model at half-filling in an external magnetic field after switching off the Broyden mixing. (a) Magnetization as a function of the number of iterations, (b) ratio between the differences of consecutive hybridization functions, which provides an estimate for the dominant eigenvalue $\lambda \sim -3.3$ for $B/W=0.02$.