Accessing thermodynamics from dynamical cluster-embedding approaches

Dynamical quantum-cluster approaches, such as different cluster extensions of the dynamical mean-field theory (cluster DMFT) or the variational cluster approximation (VCA), combined with efficient cluster solvers, such as the quantum Monte Carlo (QMC) method, provide controlled approximations of the single-particle Green’s function for lattice models of strongly correlated electrons. To access the thermodynamics, however, a thermodynamical potential is needed. We present an efficient numerical algorithm to compute the grand potential within cluster-embedding approaches that are based on novel continuous-time QMC schemes. It is shown that the numerically exact cluster grand potential can be obtained from a quantum Wang-Landau technique to reweight the coefficients in the expansion of the partition function. The lattice contributions to the grand potential are computed by a proper infinite summation over Matsubara frequencies. A proof of principle is given by applying the VCA to antiferromagnetic (short-range) order in the two-dimensional Hubbard model at finite temperatures.

Figure 1

(Color online) Left: Histogram $p\left(k\right)$ of the order $k$ in the weak-coupling expansion. Calculation for a $2\times 2$ Hubbard cluster at half filling, $\beta t=5$ and $U/t=4$. Right: Histogram with Wang-Landau reweighting, $\tilde{p}\left(k\right)$, choosing $k_c=40$.

Figure 2

(Color online) The grand potential per site as a function of the chemical potential $\mu$ for small Hubbard clusters at $\beta t=10$ and $U/t=4$. The CT-QMC results are shown as dots. The statistical error is smaller than the symbol size. Full ED results are shown as lines for comparison.

Figure 3

(Color online) The on-site Green’s function for a $2\times 2$ cluster at $\beta t=5$. From bottom to top, different lines corresponds to different interaction strengths varying from $U/t=0.5$ to $U/t=4.0$ in steps of 0.5. The Wang-Landau factor $g\left(k\right)$ was determined at $U/t=4$, and the Green’s function for other interaction strengths are calculated from Eq. (21).

Figure 4

(Color online) The internal energy per site determined from a $4\times 4$ embedded cluster at different temperatures. From bottom to top, the interaction strength increases from $U/t=1.0$ to 4.0. Statistical error bars are smaller than the symbol size and hardly visible. Our results are in good agreement with the QMC solution on an $8\times 8$ cluster obtained in Ref. 32 at $U/t=4.0$.

Figure 5

(Color online) The grand potential $\Omega$ per site from Eq. (25) and the cluster grand potential ${\Omega }^{\prime }$ per site as functions of temperature for two different interaction strengths.

Figure 6

(Color online) Entropy (per site) as calculated from Eq. (32) for the same parameters as for Figs. 4, 5.

Figure 7

(Color online) The comparison of the entropy from Eq. (32) and the numerical differentiation of the the lattice grand potential, $S=-\partial \Omega /\partial T$, at $U/t=1$ for different temperatures.

Figure 8

(Color online) Lattice grand potential $\Omega$ as a function of the variational parameter $h$ as obtained by embedding a $4\times 4$ cluster [(a) and (b)] and a $2\times 2$ cluster (c). Results for different interaction strengths $U/t$ as indicated in the figure. Note that the difference $\Omega \left(h\right)-\Omega \left(0\right)$ is plotted. (a) $\beta t=5.0$, $N_c=4\times 4$, (b) $\beta t=2.0$, $N_c=4\times 4$, and (c) $\beta t=5.0$, $N_c=2\times 2$. Symbols with statistical errors represent the numerical data. Lines are obtained by a spline interpolation and serve as a guide to the eyes only in panels (a) and (b). Lines in panel (c) are from full ED calculations for comparison.