Two-particle correlations and the metal-insulator transition: Iterated perturbation theory revisited

Recent advances in many-body physics have made it possible to study correlated electron systems at the two-particle level. In dynamical mean-field theory (DMFT), it has been shown that the metal-insulator phase diagram is closely related to the eigenstructure of the susceptibility. So far, this situation has been studied using accurate but numerically expensive solvers. Here, the iterated perturbation theory (IPT) approximation is used instead. Its simplicity makes it possible to obtain analytical results for the two-particle vertex and the DMFT Jacobian. The limited computational cost also enables a detailed comparison of analytical expressions for the response functions to results obtained using finite differences. At the same time, the approximate nature of IPT precludes an interpretation of the metal-insulator transition in terms of a Landau free energy functional.

Figure 1

Phase diagram of IPT for the Bethe lattice. The position of the critical point at $U_c\approx 2.46$ and $T_c\approx 4.69\times {10}^{-2}$, i.e., ${\beta }_c\approx 21.3$, is taken from Ref. [24]. $U_{c1}$ and $U_{c2}$ denote the boundaries of the hysteresis region; the parameters of Figs. 4 and 5 are also marked.

Figure 2

Iterated perturbation theory self-energy and vertex.

Figure 3

$d\mathrm{\Sigma }/dU$ evaluated analytical via Eq. (23) (green crosses) and numerical via the finite difference $\mathrm{\Delta }U=0.001$ (orange pluses). The DMFT response is enhanced compared to the impurity response with $G_0$ held constant (blue circles) due to the self-consistent feedback.

Figure 4

Calculations at $\beta =35, U=1.5$ (left) and $U=3$ (right). (a), (b) Converged solution and normalized difference vector of $G_0, \mathrm{\Sigma }$, and $G$. (c), (d) The self-energy converges exponentially in the forward iteration, $\mathrm{\Delta }\mathrm{\Sigma }\left(i{\nu }_n\right)\propto {\lambda }_J^{\text{iteration}}$, at all frequencies. (e), (f) The first eigenvalues of the Jacobian, sorted by their absolute value. The value of ${\lambda }_J$ found in (c) and (d) corresponds to the leading eigenvalue here.

Figure 5

Results at $\beta =20$, which is just above the hysteresis region. (a) Eigenvalues of the Jacobian. A thick colored line is used to highlight the leading eigenvalue, i.e., the one with the largest absolute value. For small $U$, the leading eigenvalue is negative, while all eigenvalues eventually turn positive in the atomic limit. The leading eigenvalue reaches a maximum just below unity close to $U_c$ (vertical line; value from Strand et al. [24]). (b) The imaginary-time Green's functions and self-energy evaluated at $\tau =\beta /2$. They change rapidly in the region where ${\lambda }_J\approx +1$. (c) The evolution of $\mathrm{\Sigma }$ in the Legendre basis shows that not all properties are monotonic functions of $U$.

Figure 6

The Green's function in the Matsubara and Legendre representation at $T=1/20$, just above $T_c$ and close to $U_c$. The shape of the curves at low frequency changes rapidly with $U$.

Figure 7

The IPT vertex is given by the difference between a Toeplitz matrix (left) and a Hankel matrix (right), and a prefactor $3U^2/{\beta }^2$.