Ergodicity of the hybridization-expansion Monte Carlo algorithm for broken-symmetry states

With the success of dynamical mean-field theories, solvers for quantum-impurity problems have become an important tool for the numerical study of strongly correlated systems. Continuous-time quantum Monte Carlo sampling of the expansion in powers of the hybridization between the “impurity” and the bath provides a powerful solver when interactions are strong. Here we show that the usual updates that add or remove a pair of creation-annihilation operators are rigorously not ergodic for several classes of broken symmetries that involve spatial components. We show that updates with larger numbers of simultaneous updates of pairs of creation-annihilation operators remedy this problem. As an example, we apply the four-operator updates that are necessary for ergodicity to the case of $d$-wave superconductivity in plaquette dynamical mean-field theory for the one-band Hubbard model. While the results are qualitatively similar to those previously published, they are quantitatively better than previous ones, being closer to those obtained by other approaches.

Figure 1

Diagrams contributing to the weight of a second-order configuration, cf. Eq. (8). The bold black circle represents the trace with the impurity operators, connected in all different ways by the hybridization function.

Figure 2

$d$-wave superconducting order parameter $\Phi$ as a function of doping $\delta$, for the low temperature $T=1/100$, with and without four-operator updates (circles and squares, respectively). The value of the interaction $U=9.0$ is larger than $U_{\mathrm{MIT}}$.

Figure 3

$d$-wave superconducting order parameter $\Phi$ as a function of temperature $T$ for $U=9.0$ and $\delta =0.04$, with and without four-operator updates (circles and squares, respectively).

Figure 4

Revised temperature versus doping phase diagram of the two-dimensional Hubbard model within plaquette CDMFT for $U=6.2$. The only modification compared with Refs. [19, 27] is for the superconducting region delineated by $T_c^d$ (blue/light gray area). With two-operator updates, superconductivity occurs below the dotted blue (light gray) line. With the four-operator updates, superconductivity extends to the end of the blue (light gray) area. For completeness, we describe the rest of the phase diagram. The first-order transition (red/dark gray area) terminating at the critical endpoint $({\delta }_p,T_p)$ (circle) separates a correlated metal from a pseudogap metal. $T_{{\sigma }_c}\left(\delta \right)$ is the temperature where ${\sigma }_c\left(\mu \right)$ has an inflection point. It follows $T^{*}$ and $T_{\mathrm{WL}}$, i.e., the dynamic and thermodynamic supercritical crossovers determined by the inflection in the local density of states $A(\omega =0,T)$ and in the charge compressibility $\kappa \left(\mu \right)$, respectively [28]. The pseudogap scale can be identified also as inflection points in the local spin susceptibility ${\chi }_0\left(T\right)$, $T_{{\chi }_0}$. $T_{{\rho }_c,\min }$ is the temperature where ${\rho }_c\left(T\right)$ has a minimum. It scales with the temperature where $A(\omega =0,T)$ [${\chi }_0\left(T\right)$] peaks, $T_{A,\max }$ [$T_{{\chi }_0,\max }$].