Variational Cluster Approach to Correlated Electron Systems in Low Dimensions

A self-energy-functional approach is applied to construct cluster approximations for correlated lattice models. It turns out that the cluster-perturbation theory [D. Sénéchal et al., Phys. Rev. Lett. 84, 522 (2000)] and the cellular dynamical mean-field theory [G. Kotliar et al., Phys. Rev. Lett. 87, 186401 (2001)] are limiting cases of a more general cluster method. The results for the one-dimensional Hubbard model are discussed with regard to boundary conditions, bath degrees of freedom, and cluster size.

Figure 1

Reference systems considered within various cluster approximations for the $D=2$ Hubbard model. Filled circles: On-site interaction. Solid lines: Intracluster hopping. Dashed lines: Intercluster hopping. Open circles: Additional $n_{\mathrm{b}}$ uncorrelated bath sites. Big circles: Bath with $n_{\mathrm{b}}=\infty$. Common to all methods is (i) the numerical solution of the system of decoupled clusters and (ii) the subsequent coupling of the clusters via an RPA-like or Dyson equation. Bath parameters are determined self-consistently. The self-energy-functional theory (SFT) comprises the extreme limits CPT ($n_{\mathrm{b}}=0$) and C-DMFT ($n_{\mathrm{b}}=\infty$).

Figure 2

$E_0\equiv \Omega +\mu ⟨N⟩$ vs $t^{\prime }$ as obtained by evaluating the self-energy functional $\Omega =\Omega [\mathbf{\Sigma }(t^{\prime })]$. Original system $H$: $D=1$ Hubbard model with nearest neighbor (NN) hopping $t=1$ and $U=8$ for $T=0$ and $\mu =U/2$ (half filling). Reference system $H^{\prime }$: Set of decoupled Hubbard chains with $N_{\mathrm{c}}$ sites each; $N_{\mathrm{c}}=2,4,6,8,10$ as indicated. Variational parameter: NN hopping $t^{\prime }$ of $H^{\prime }$. The inset shows $\Delta E_0=E_0-E_{0,\mathrm{m}\mathrm{i}\mathrm{n}}$ vs $t^{\prime }$.

Figure 3

$E_0$ as in Fig. 2 but vs different variational parameters: NN hopping $t_{\mathrm{b}}$ to uncorrelated bath sites (left), hopping $t_r$ between the edge sites (right). Dashed lines: $E_0$ without parameter optimization. Arrows: $E_0$ calculated for an isolated cluster. Exact result for $E_0$ from Ref. 20.