Figure 1

(a) The phase diagram of the paramagnetic half-filled Hubbard model within plaquette CDMFT. Inset: The histogram of the two insulating states. It shows the probability for a given cluster eigenstate among the 16 eigenstates of the half filled plaquette. The singlet plaquette ground state has the highest probability. (b) For comparison, the corresponding phase diagram of the single-site DMFT (using the same 2D density of states) is shown. The coexistence region is shown as the shaded region. The dashed line marks the crossover above the critical point. The crossover line was determined by the condition that the imaginary part of the self-energy at few lowest Matsubara frequencies is flat at the crossover value of $U$. For easier comparison, the $x$ axis is rescaled and the reduced value of $U_r=\frac{U-U_{\mathrm{MIT}}}{U_{\mathrm{MIT}}}$ is used. The critical value of $U$ is $U_{\mathrm{MIT}}=6.05t$ in the cluster case and $U_{\mathrm{MIT}}=9.35t$ in the single-site case. Pentagons in panel (a) mark the points in phase diagram for which we present the local spectral functions in Fig. 2.Reuse & Permissions

Figure 2

The local spectral function for four representative values of $U/t\mathrm{s}$ and temperature $T=0.01t$ marked by pentagons in Fig. 1. (a) For $U$ below $U_{c1}$ the system is in Fermi liquid regime with rather large coherence temperature. (b) In the coexistence region, the insulating solution has a small but finite gap ($\sim 0.2t$). (c) The metallic solution in the same region is strongly incoherent and the value at zero frequency decreases due to the finite scattering rate [see self-energy in Fig. 3a]. (d) For $U$ above $U_{c2}$, the Mott gap steadily increases with $U$.Reuse & Permissions

Figure 3

Top: The imaginary part of the cluster self-energies for the same parameters as in Fig. 2. Because of particle-hole symmetry, the ($\pi$, $\pi$) and (0, 0) cluster self-energies have the same imaginary part and we show only one of them. Below the MIT shown here in (a), the momentum dependence of the self-energy is rather weak and the cluster solution is very similar to the single-site DMFT solution. In the coexistence region, the metallic solution shown here in (c) is strongly incoherent especially in the ($\pi$, 0) orbital. For the insulating solutions in (b) and (d), the ($\pi$, 0) scattering rate diverges which opens the gap in the spectra. bottom: (e) $\mathrm{Re}{\Sigma }_K(0^{-})-\mu$ as a function of $U$. Because of particle-hole symmetry, $\mathrm{Re}{\Sigma }_{K=(\pi ,0)}-\mu$ vanishes.Reuse & Permissions

Figure 4

The quasiparticle residue $Z$ vs $U/t$ for different orbitals in CDMFT. Below the transition point, the (0, 0) and ($\pi$, $\pi$) orbitals have essentially the same $Z$ as single-site DMFT (dotted line) while the quasiparticles are more renormalized in the ($\pi$, 0) orbital.Reuse & Permissions