Reply to “Comment on ‘Cluster methods for strongly correlated electron systems’ ”

We reply to the Comment by Aryanpour, Maier, and Jarrell [Phys. Rev. B 71, 037101 (2005)] on our paper [Phys. Rev. B 65, 155112 (2002)]. We demonstrate, using general arguments and explicit examples, that whenever the correlation length is finite, local observables converge exponentially fast in the cluster size $L_c$ within cellular dynamical mean field theory. This is a faster rate of convergence than the $1∕L_c^2$ behavior of the dynamical cluster approximation, thus refuting the central assertion of their Comment.

Figure 1

$\Delta \chi =({\chi }_x-\chi )∕\chi$ as a function of $x$, for $\beta$, the inverse temperature, equal to 4, $L_c=38$, and $t=\mu =1$. The continuous line is the relative error of the DCA prediction on $\chi$. The dotted line is a guide for the eye.

Figure 2

$\Delta {\chi }_{\mathit{\text{bulk}}}$ (circles, solid line) and $\Delta {\chi }_{fa}$ (squares, dotted line) as a function of $L_c$. Inset: $\Delta {\chi }_{\mathit{\text{bulk}}}{L}_c^2$ (circles, solid line) and $\Delta {\chi }_{fa}{L}_c^2$ (squares, dotted line) as a function of $L_c$.

Figure 3

$\Delta {\chi }_{\mathit{\text{bulk}}}$ as a function of $L_c$ in a logarithmic scale (circles) compared to its DCA counterpart (squares).