Continuous momentum dependence in the dynamical cluster approximation

The dynamical cluster approximation (DCA) is a quantum cluster extension to the single-site dynamical mean-field theory that incorporates spatially nonlocal dynamic correlations systematically and nonperturbatively. The ${\mathrm{DCA}}^{+}$ algorithm addresses the cluster shape dependence of the DCA and improves the convergence with cluster size by introducing a lattice self-energy with continuous momentum dependence. However, we show that the ${\mathrm{DCA}}^{+}$ algorithm is plagued by a fundamental problem when its self-consistency equations are formulated using the bare Green's function of the cluster. This problem is most severe in the strongly correlated regime at low doping, where the ${\mathrm{DCA}}^{+}$ self-energy becomes overly metallic and local, and persists to cluster sizes where the standard DCA has long converged. In view of the failure of the ${\mathrm{DCA}}^{+}$ algorithm, we propose to complement DCA simulations with a post-interpolation procedure for single-particle and two-particle correlation functions to preserve continuous momentum dependence and the associated benefits in the DCA. We demonstrate the effectiveness of this practical approach with results for the half-filled and hole-doped two-dimensional Hubbard model.

Figure 1

Interpolated DCA results for the momentum dependence of the imaginary part of the self-energy at the lowest Matsubara frequency for the half-filled Hubbard model at $U=7t$ and temperatures $T=0.8t$ (left) and $0.2t$ (right), respectively. The curves of the three different $N_c=256$ clusters and the square $N_c=288$ cluster match nearly perfectly, verifying convergence and cluster shape independence of the results.

Figure 2

Comparison of the momentum dependence of the imaginary part of the ${\mathrm{DCA}}^{+}$ lattice self-energy at the lowest Matsubara frequency for various cluster sizes with the interpolated $N_c=256A$ DCA result. The set of parameters is the same as in Fig. 1: $U=7t, \langle n\rangle =1$, and $T=0.8t$ (left) and $T=0.2t$ (right), respectively.

Figure 3

(Top) Momentum dependence of the imaginary part of the self-energy at the lowest Matsubara frequency for $U=7t, \langle n\rangle =1, N_c=256A$, and $T=0.8t$ (left) and $T=0.2t$ (right), respectively. Performing only one ${\mathrm{DCA}}^{+}$ iteration (green stars) or one DCA-interp-latt iteration (orange circles) on the converged DCA results (blue squares) yields a far more local self-energy at $T=0.2t$. (Bottom) Decay of the real space self-energy at the lowest Matsubara frequency for the same set of parameters. The DCA cluster self-energy (blue squares) is, by construction, periodic on the cluster. The interpolated (orange circles) and the deconvoluted (green stars) DCA cluster self-energy have the full momentum resolution of the lattice and are not periodic on the cluster.

Figure 4

Local bare cluster Green's function (top) and local hybridization function (bottom) in imaginary time for $U=7t, \langle n\rangle =1, N_c=256A$, and $T=0.8t$ (left) and $T=0.2t$ (right), respectively. The solid blue lines correspond to the DCA results. The dashed orange lines represent results where the Green's function is coarse-grained with the interpolated DCA cluster self-energy. In the dash-dotted green results, the Green's function is coarse-grained with the deconvoluted DCA cluster self-energy, and the cluster-exclusion step is done with the coarse-grained average of the latter.

Figure 5

Average QMC sign ${\sigma }_{\text{QMC}}$ vs temperature $T$ for the 10% hole-doped Hubbard model with $U=4t$ and cluster size $N_c=24$.

Figure 6

DCA and ${\mathrm{DCA}}^{+}$ results for the momentum dependence of the imaginary part of the self-energy at the lowest Matsubara frequency for the 10% hole-doped Hubbard model with $U=4t, N_c=24$, and temperatures $T=0.5t$ (left) and $T=0.02t$ (right), respectively.

Figure 7

Momentum dependence of the piecewise constant (left) and the interpolated and symmetrized (right) DCA self-energy at the lowest Matsubara frequency for the half-filled Hubbard model at $U=7t$ and $T=0.2t$ for various $\left[\right[2L,0],[0,2L\left]\right]$ clusters.

Figure 8

Momentum dependence of the piecewise constant (left) and the interpolated and symmetrized (right) DCA self-energy at the lowest Matsubara frequency for the half-filled Hubbard model at $U=7t$ and $T=0.2t$ for two different 16-site clusters.

Figure 9

Momentum dependence of the piecewise constant (left) and the interpolated and symmetrized (right) DCA self-energy at the lowest Matsubara frequency for the 10% hole-doped Hubbard model at $U=4t$ and $T=0.02t$ for several similar sized clusters.

Figure 10

Superconducting transition temperature $T_c$ vs cluster size $N_c$ for the 10% hole-doped Hubbard model with $U=4t$. The two data points at $N_c=14$ and 16 correspond to the 14A and 14B clusters and the 16A and 16B clusters, respectively.