Diagrammatic determinantal quantum Monte Carlo methods: Projective schemes and applications to the Hubbard-Holstein model

We extend the weak-coupling diagrammatic determinantal algorithm to projective schemes as well as to the inclusion of phonon degrees of freedom. The projective approach provides a very efficient algorithm to access zero-temperature properties. To implement phonons, we integrate them out in favor of a retarded density-density interaction and simulate the resulting purely electronic action with the weak-coupling diagrammatic determinantal algorithm. Both extensions are tested within the dynamical mean-field approximation for the Hubbard and Hubbard-Holstein models.

Figure 1

(Color online) Green’s function $G\left(\tau \right)$ for the Anderson model of Eq. (20). (a) Particle-hole symmetric point, $\mu =0$. To avoid the vanishing of the weight at odd values of $n$, we have used $\delta =0.1$ for this simulation. (b) Away from particle-hole symmetry, $\mu =-0.5$. For this higher temperature, we provide a comparison with the Hirsch-Fye algorithm with Trotter step $\Delta \tau t=0.1$. Note that in both cases we have used a $32\times 32$ square lattice to generate the noninteracting Green’s function $G^0\left(\tau \right)$. $⟨n⟩$ corresponds to the average order expansion parameter.

Figure 2

(Color online) DMFT calculation of the double occupancy for the half filled Hubbard model as a function of (i) temperature $1∕\beta$ and (ii) projection parameter $1∕\Theta$. In the projective approach, observables are computed in the imaginary time range $[\Theta ∕4,3\Theta ∕4]$. This rather large measurement interval explains the slight discrepancy with the data of Ref. 5 for the projective code and at finite values of $\Theta$. However, extrapolation to $\Theta \to \infty$, where the measurement range becomes irrelevant, yields results consistent with Ref. 5.

Figure 3

(Color online) (a) Double occupancy, (b) local charge susceptibility, and (c) local spin susceptibility for the Hubbard-Holstein model in the DMFT approximation.

Figure 4

(Color online) The normalized histogram of the local density $\rho$ of the Hubbard-Holstein model at half filling with $U∕t=1$, ${\omega }_0=0.2t$, and $\beta t=40$, for several values of the electron-phonon coupling $g$.