Metal-insulator transition in cluster dynamical mean field theory: Intersite correlation, cluster size, interaction strength, and the location of the transition line

To gain insight into the physics of the metal-insulator transition and the effectiveness of cluster dynamical mean field theory (DMFT) we have used 1, 2-, and 4-site dynamical mean field theory to solve a polaron model of electrons coupled to a classical phonon field. The metal to polaronic insulator phase boundary is determined along with electron spectral functions and cluster correlation functions. Pronounced cluster size effects occur in the intermediate coupling region in which the cluster calculation leads to a gap and the single-site approximation does not. Differences in spectra (in particular a sharper band edge in the cluster calculation) persist in the strong-coupling regime. A partial density of states is defined encoding a generalized nesting property of the band structure; variations in this density of states account for differences between the dynamical cluster approximation and the cellular-DMFT implementations of cluster DMFT, as well as for differences in behavior between the single-band models appropriate for cuprates and the multiband models appropriate for manganites. A pole or strong resonance in the self-energy is associated with insulating states; the momentum dependence of the pole is found to distinguish between Slater-like and Mott-like mechanisms for metal-insulator transition. Implications for the theoretical treatment of doped manganites are discussed.

Figure 1

(Color online) Partition of the Brillouin zone. (a) 2-site DCA on square lattice. (b) 4-site DCA on square lattice. (c) 2-site DCA on cubic lattice.

Figure 2

(Color online) The PDOS for 2-site and 4-site DCA partitionings on the square lattice with nearest-neighbor hopping. The total bandwidth is 12.

Figure 3

(Color online) Partial DOS for $3D$, $2S$, and $e_g$ bands.

Figure 4

(Color online) Results from single-site DMFT calculation for two-dimensional square lattice with nearest-neighbor hopping $t=1.5$ (bandwidth 12), polaronic coupling [Eq. (20)], $U=3,4,5$, and $T=0.1$: (a) spectral functions; (b) real part of self-energies.

Figure 5

(Color online) Results from 2-site DCA calculation for two-dimensional square lattice with nearest-neighbor hopping $t=1.5$ (bandwidth 12), polaronic coupling [Eq. (20)], $U=4$, and $T=0.1$. (a) Real part of self-energies for $S$ and $P$ sectors. (b) Partial spectral functions for $S$ and $P$ sectors.

Figure 6

(Color online) Results from 4-site DCA calculation for two-dimensional square lattice with nearest-neighbor hopping $t=1.5$ (bandwidth 12), polaronic coupling [Eq. (20)], $U=4$, and $T=0.1$. (a) Real part of self energies for $S$ $P$ and $D$ sectors. (b) Partial spectral functions for $S$ $P$ and $D$ sectors.

Figure 7

(Color online) Spectral functions calculated by 2-site and 4-site DCA approximations for the two-dimensional square lattice with nearest-neighbor hopping $t=1.5$, polaronic coupling [Eq. (20)], (a) $U=3$ and $T=0.2$, and (b) $U=4$ and $T=0.1$.

Figure 8

(Color online) Spectral functions from S-DMFT and 2-site DCA calculation for the two-dimensional square lattice with nearest-neighbor hopping $t=1.5$, polaronic coupling (a) $U=5$, and (b) $U=10$, $T=0.1$, and $N=1$. For $U=10$ only the upper Hubbard band is shown.

Figure 9

(Color online) Spectral functions (left panel) and real part of self-energy (right panel) from 2-site DCA calculation for the two-dimensional square lattice with nearest-neighbor hopping $t=1.5$ (bandwidth 12), polaronic coupling $U=4$, and $T=0.1–0.3$.

Figure 10

(Color online) Self-energy (left panel) and spectral function (right panel) calculated from single-site (solid line) and 2-site DCA (dashed line) approximations for the square lattice with nearest-neighbor hopping $t=1.5$, polaronic coupling $U=5$, $T=0.1$ at $N=0.2$.

Figure 11

(Color online) Spectral functions calculated by single-site DMFT, 2-site, and 4-site DCA approximations at $U=4$, $T=0.1$, (a) $N=0.5$ and (b) $N=0.4$.

Figure 12

(Color online) Self-energies calculated by (a) 2-site and (b) 4-site DCA approximations for the two-dimensional square lattice with bandwidth 12 at $U=4$, $T=0.1$, $N=0.5$.

Figure 13

(Color online) Partial DOS for 2-site DCA and Cellular-DMFT on the square lattice with nearest-neighbor hopping $t=1.5$ (bandwidth 12). $U_c$ for 1-site DMFT, 2-site cellular DMFT, and 2-site DCA approximations are roughly 5, 4, and 3, respectively.

Figure 14

(Color online) 2-site cellular-DMFT and DCA spectral functions calculated for the square lattice with nearest-neighbor hopping $t=1.5$ at half-filling polaronic coupling $U=4$, $T=0.2$. The DCA result is more insulating (having smaller density of state around the Fermi energy) than cellular-DMFT.

Figure 15

(Color online) Spectral functions for the coupling close to the gap opening $U_c$. (a) For two degenerate $S$ band with bandwidth 12, the $U_c^{\text{1-site}}\sim 4$ (solid) and $U_c^{2\text{-DCA}}\sim 3$ (dashed). (b) For $e_g$ band with bandwidth 12, the $U_c^{1\text{-site}}\sim 6$ (solid) and $U_c^{2\text{-DCA}}\sim 5$ (dashed). Those curves are calculated at $T=0.1$.