Extended variational cluster approximation for correlated systems

The variational cluster approximation (VCA) proposed by M. Potthoff et al. [Phys. Rev. Lett. 91, 206402 (2003)] is extended to electron or spin systems with nonlocal interactions. By introducing more than one source field in the action and employing the Legendre transformation, we derive a generalized self-energy functional with stationary properties. Applying this functional to a proper reference system, we construct the extended VCA (EVCA). In the limit of continuous degrees of freedom for the reference system, EVCA can recover the cluster extension of the extended dynamical mean-field theory (EDMFT). For a system with correlated hopping, the EVCA recovers the cluster extension of the dynamical mean-field theory for correlated hopping. Using a discrete reference system composed of decoupled three-site single impurities, we test the theory for the extended Hubbard model. Quantitatively good results as compared with EDMFT are obtained. We also propose VCA (EVCA) based on clusters with periodic boundary conditions. It has the (extended) dynamical cluster approximation as the continuous limit. A number of related issues are discussed.

Figure 1

Structure of a three-site unit cell of the reference system Hamiltonian (66). Filled dot denotes impurity site, open dot and square denote fermion and boson bath site, respectively.

Figure 2

${\Omega }_f$ and ${\Omega }_b$ in the vicinity of the metallic stationary point at $T=0.01$, $U=1.0$, and $V_0=0.5$. They are shifted by their stationary values ${\Omega }_f^{*}$ and ${\Omega }_b^{*}$, respectively, and magnified by $\beta =100$. $V^{*},{\xi }^{*}$, and ${\gamma }^{*}$ denote the corresponding values at the stationary point.

Figure 3

(a) The density-density Green’s function at the stationary point for $T=0.01$, $U=1.0$, and $V_0=0.5$; (b) the discrepancies between certain quantities, which should be zero in the continuous limit of bath; see text. $\alpha =1∕2$.

Figure 4

Comparison of the three quantities related to the electron Green’s function at the stationary point for $T=0.01$, $U=1.0$, and $V_0=0.5$.

Figure 5

(a) Double occupancy as a function of $V_0$ for $T=0.01$, $U=1.0$. The vertical dashed line marks the boundary $V_{\mathit{max}}\approx 0.72$, above which the symmetry-unbroken three-site EVCA solution does not exist. (b) The density-density Green’s function $\Pi \left(i{\nu }_n\right)$ for several $V_0$ values. From top to bottom, $V_0=0.7$, 0.5, 0.3, and 0.0.