Comparison of computer-algebra strong-coupling perturbation theory and dynamical mean-field theory for the Mott-Hubbard insulator in high dimensions

We present a large-scale combinatorial-diagrammatic computation of high-order contributions to the strong-coupling Kato-Takahashi perturbation series for the Hubbard model in high dimensions. The ground-state energy of the Mott-insulating phase is determined exactly up to the 15th order in $1/U$. The perturbation expansion is extrapolated to infinite order and the critical behavior is determined using the Domb-Sykes method. We compare the perturbative results with two dynamical mean-field theory (DMFT) calculations using a quantum Monte Carlo method and a density-matrix renormalization group method as impurity solvers. The comparison demonstrates the excellent agreement and accuracy of both extrapolated strong-coupling perturbation theory and quantum Monte Carlo based DMFT, even close to the critical coupling where the Mott insulator becomes unstable.

Figure 1

Absolute differences (13) between the SCPT and DMFT ground-state energies as functions of $U$ for the orders $m=9$ (circle), $m=11$ (square), and $m=15$ (diamond). Open and solid symbols correspond to DMRG-DMFT and QMC-DMFT, respectively. Vertical lines mark the critical coupling $U_c$ deduced from DMRG-DMFT (dotted-dashed line) and QMC-DMFT (dashed line) studies. Other lines are guides for the eye. The inset shows the same data on a logarithmic scale.

Figure 2

SCPT ground-state energy $E_m(U=4.8)$ as a function of $x=\frac{2}{m+1}$ (open circles) and $x=\frac{2}{m-1}$ (open squares). Lines represent least-square quadratic fits (14) of these data using all points ($m\le 15$, black or red dashed lines) and all but the leftmost two points ($m\le 11$, blue or green solid lines). The solid circle and square correspond to the QMC-DMFT and DMRG-DMFT results for $U=4.8$, respectively. The inset shows an expanded view of the box in the lower left corner.

Figure 3

Domb-Sykes plot of the ratio $R_n$ as a function of $x=\frac{2}{m+1}$ (open circles) and $x=\frac{2}{m-1}$ (open squares). Lines represent quadratic least-square fits (19) of these data using all points ($m\le 15$, black or red dashed lines) and all but the leftmost two points ($m\le 11$, blue or green solid lines).