Dynamical mean-field theory of the Anderson-Hubbard model with local and nonlocal disorder in tensor formulation

To explore correlated electrons in the presence of local and nonlocal disorder, the Blackman-Esterling-Berk method for averaging over off-diagonal disorder is implemented into dynamical mean-field theory using tensor notation. The impurity model combining disorder and correlations is solved using the recently developed fork tensor-product state solver, which allows one to calculate the single particle spectral functions on the real-frequency axis. In the absence of off-diagonal hopping, we establish exact bounds of the spectral function of the noninteracting Bethe lattice with coordination number $Z$. In the presence of interaction, the Mott insulating paramagnetic phase of the one-band Hubbard model is computed at zero temperature in alloys with site- and off-diagonal disorder. When the Hubbard $U$ parameter is increased, transitions from an alloy band insulator through a correlated metal into a Mott insulating phase are found to take place.

Figure 1

Noninteracting case: Comparison of spectral functions for different dimensionless hopping parameters ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}$. The parameters are ${\underline{\mathrm{U}}}^{\mathsf{A}}={\underline{\mathrm{U}}}^{\mathsf{B}}=0, {\underline{\mathrm{v}}}^{\mathsf{A}}={\underline{\mathrm{v}}}^{\mathsf{B}}=0, c^{\mathsf{A}}=0.1=1-c^{\mathsf{B}}$, and ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{A}}={\underline{\mathrm{T}}}^{\mathsf{B}\mathsf{B}}=1$. The solid lines represent the component spectral functions ${\underline{\mathrm{A}}}^{\alpha }\left(z\right)=c^{\alpha }\mathbb{E}(A_i\left(z\right)|i\mapsto \alpha )$, where $\mathsf{A}$ is red and $\mathsf{B}$ is blue; the dotted yellow line shows the average spectral function $\mathbb{E}\left(A\left(z\right)\right)={\underline{\mathrm{A}}}^{\mathsf{A}}\left(z\right)+{\underline{\mathrm{A}}}^{\mathsf{B}}\left(z\right)$. The thin vertical lines show the maximal spectral bounds given by the Gershgorin circle theorem [59].

Figure 2

Comparison of spectral functions for different values of the dimensionless hopping parameter ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}$ with ${\underline{\mathrm{U}}}^{\mathsf{A}}={\underline{\mathrm{U}}}^{\mathsf{B}}=3D, {\underline{\mathrm{v}}}^{\mathsf{A}}={\underline{\mathrm{v}}}^{\mathsf{B}}=-1.5D, c^{\mathsf{A}}=0.1=1-c^{\mathsf{B}}$, and ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{A}}={\underline{\mathrm{T}}}^{\mathsf{B}\mathsf{B}}=1$ at $T=0$. For ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}=0$ a shift $\eta =0.12$ had to be used; the other panels were calculated for $\eta =0.08$.

Figure 3

Comparison of spectral functions for different values of the dimensionless hopping parameter ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}$ with ${\underline{\mathrm{U}}}^{\mathsf{A}}=0, {\underline{\mathrm{U}}}^{\mathsf{B}}=3D, {\underline{\mathrm{v}}}^{\mathsf{A}}=0, {\underline{\mathrm{v}}}^{\mathsf{B}}=-1.5D, c^{\mathsf{A}}=0.1=1-c^{\mathsf{B}}$, and ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{A}}={\underline{\mathrm{T}}}^{\mathsf{B}\mathsf{B}}=1$ at $T=0$.

Figure 4

Quasiparticle weight $Z$ corresponding to ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}=5.0$ in Figs. 2 and 3 for parameters ${\underline{\mathrm{v}}}^{\mathsf{A}}=-{\underline{\mathrm{U}}}^{\mathsf{A}}/2, {\underline{\mathrm{v}}}^{\mathsf{B}}=-{\underline{\mathrm{U}}}^{\mathsf{B}}/2, {\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{A}}={\underline{\mathrm{T}}}^{\mathsf{B}\mathsf{B}}=1$, and ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}=5.0$ at $T=0$, calculated for a shift $\eta =0.12$.

Figure 5

Comparison of the spectral functions for ${\underline{\mathrm{U}}}^{\mathsf{A}}={\underline{\mathrm{U}}}^{\mathsf{B}}=U, -({\underline{\mathrm{v}}}^{\mathsf{A}}+U/2)={\underline{\mathrm{v}}}^{\mathsf{B}}+U/2=1.5D, c^{\mathsf{A}}=0.5=c^{\mathsf{B}}, {\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{A}}={\underline{\mathrm{T}}}^{\mathsf{B}\mathsf{B}}=1$, and $T=0$ for different values of ${\underline{\mathrm{T}}}^{\mathsf{A}\mathsf{B}}$ and $U$.