Evaluation of high-order terms for the Hubbard model in the strong-coupling limit

The ground-state energy of the Hubbard model on a Bethe lattice with infinite connectivity at half filling is calculated for the insulating phase. Using Kohn's transformation to derive an effective Hamiltonian for the strong-coupling limit, the resulting class of diagrams is determined. We develop an algorithm for an algebraic evaluation of the contributions of high-order terms and check it by applying it to the Falicov-Kimball model that is exactly solvable. For the Hubbard model, the ground-state energy is exactly calculated up to order $t^{12}/U^{11}$. The results of the strong-coupling expansion deviate from numerical calculations as quantum Monte Carlo (or density-matrix renormalization group) by less than $0.13\%$ ($0.32\%$ respectively) for $U>4.76$.

Figure 1

Correspondence between sequences of hops on the lattice (left) and Butcher trees (right): Shown are the sequences in first ($n=2$) and third order ($n=3$) and the related Butcher trees. The first hop defines the root of the tree (encircled).

Figure 2

Butcher trees up to the order $i=11$. The order expressed by the number of sites is $i=2n-3$. The encircled sites denote the root of the trees. Note that for $n=7$, 29 Butcher trees are skipped (ellipsis symbol).

Figure 3

The intermediate states for the two processes and all sequences contributing to the ground-state energy in third order in $\frac{1}{U}$. The arrows correspond to the application of $T$. The symbol $◯\uparrow \downarrow$ denotes a doubly occupied lattice site; the symbols $◯$ and $◯·$ denote a hole and a singly occupied lattice site, respectively. In the right column, the contribution of each sequence to the ground-state energy is indicated. They sum up to $-\frac{1}{2U^3}$.

Figure 4

QMC and DDMRG (Ref. 10) results for the ground-state energy $E$ (top) and the double occupancy $D$ (bottom) in comparison with PT; see Eqs. (19) and (20).

Figure 5

Construction of ePT: PT values for $U_c\left(s\right)=\sqrt{a_{2s+2}/a_{2s}}$ (squares) are extrapolated to $1/s\to 0$ using a quadratic least-squares fit (solid line). Evaluations at smaller $1/s$ (circles) define ePT coefficients to all orders; see Ref. 17.

Figure 6

Flowcharts of the procedures $\mathtt{hopping}$ and $\mathtt{hopping}\_\mathtt{fwd}$.