Submatrix updates for the continuous-time auxiliary-field algorithm

We present a submatrix update algorithm for the continuous-time auxiliary field method that allows the simulation of large lattice and impurity problems. The algorithm takes optimal advantage of modern CPU architectures by consistently using matrix instead of vector operations, resulting in a speedup of a factor of $\approx$8 and thereby allowing access to larger systems and lower temperature. We illustrate the power of our algorithm at the example of a cluster dynamical mean field simulation of the Néel transition in the three-dimensional Hubbard model, where we show momentum dependent self-energies for clusters with up to 100 sites.

Figure 1

Illustration of update formulas. (a) “rank-one” updates of Ref. 31, accessing $O(m^2)$ data points for $O(m^2)$ operations and performing one update. (b) submatrix updates, accessing $O(m^2)$ values but performing $O(m^{2}k)$ operations, for $k$ updates.

Figure 2

Time per update (in arbitrary units) for submatrix and rank-one CT-AUX updates. Open circles: rank-one updates. Filled diamonds, triangles, squares, and left triangles: submatrix updates for $k_{\max }=32,64,128,$ and $256$. Dashed line: ideal $O(k^2)$ scaling, arbitrary prefactor.

Figure 3

Updates per time as a function of $k_{\max }.$ Sixteen-site cluster $U/t=8$ for temperatures indicated.

Figure 4

Time to solution as a function of the number of CPU’s, for a $16$-site cluster impurity problem. Squares (black online): $U/t=8$, $\beta t=10$ ($\langle k\rangle =550$), half filling. Circles (red online): $\beta t=20$ ($\langle k\rangle =1100$). The dashed lines show the ideal scaling.

Figure 5

Extrapolation of the cluster energy as a function of cluster size at $U/t=8$, $T/t=0.5$. Dashed line: DMFT results. Circles (black online) denote DCA results from clusters with size $18$, $26,$ $36,$ $56,$ $64,$ $84,$ and $100$. Solid line (green online): least squares fit. Triangle (blue online): extrapolated result. The error bar denotes the fitting error; statistical (Monte Carlo) errors are smaller than symbol size.

Figure 6

Real and imaginary parts of the lowest Matsubara frequency of the interpolated DCA cluster self-energy $\Sigma (k,i{\omega }_0)$ of a 3D Hubbard model above the Néel temperature,[44] for $U/t=8$, $T/t=1$ (left panel), $T/t=0.5$ (middle panel), and $T/t=0.35$ (right panel), at half filling. The lines denote DMFT results (horizontal straight lines) and results for clusters of size 18, 84, and 100. The interpolation follows a path along the high-symmetry points $\Gamma =(0,0,0)$, $X=(\pi ,0,0)$, $M=(\pi ,\pi ,0)$, and $R=(\pi ,\pi ,\pi )$.

Figure 7

Real and imaginary parts of the frequency dependence of the interpolated DCA cluster self-energy $\Sigma (k,i{\omega }_0)$ at selected $k$ points $(0,0,0)$, $(\pi ,0,0)$, and $(\pi /2,\pi /2,\pi /2)$. 3D Hubbard model above the Néel temperature[44] for $U/t=8$, $T/t=0.5$ (upper row) and $T/t=0.35$ (lower row) at half filling. The lines denote DMFT results (horizontal straight lines) and results for clusters of size 18, 84, and 100.