Systematic mapping of the Hubbard model to the generalized $t\text{−}J$ model

The generalized $t\text{−}J$ model conserving the number of double occupancies is constructed from the Hubbard model at and in the vicinity of half-filling at strong coupling. The construction is realized by a self-similar continuous unitary transformation. The flow equation is closed by a truncation scheme based on the spatial range of processes. We analyze the conditions under which the $t\text{−}J$ model can be set up and we find that it can only be defined for sufficiently large interaction. There, the parameters of the effective model are determined.

Figure 1

(Color online) Schematic phase diagram of the Hubbard model at half-filling in higher dimensions $d>2$. The tick labels at the $x$ axis are indicative only. The bare band width is denoted by $W$, the Hubbard repulsion by $U$, the temperature by $T$; PI stands for paramagnetic insulator, PM for paramagnetic metal, and AFI for antiferromagnetic insulator. The dashed line indicates the transition or the crossover from insulator to metal.

Figure 2

The operator from Eq. (8) in the two-dimensional square lattice. Closed sites are involved in the hopping process. The operators are shown at the sites on which they act.

Figure 3

Closed circles indicate the two clusters defining the truncation rule of the NN truncation. Only terms fitting on these clusters are kept.

Figure 4

(Color online) Results for the Heisenberg coupling $J_1$ from the minimal and the NN truncation compared to perturbation theory. All curves are plotted relative to $J_1^{\left(2\right)}=4t^2∕U$. The numbers in the legend in front of Padé or $D\log P$ indicate the degree of the polynomials in $x=t^2∕U^2$ in the numerator and denominator of the Padé approximant. Recall $W=8t$.

Figure 5

Various clusters defining the operators to be kept in the CUT. Only one of various clusters that are related by the point group symmetry of the two-dimensional square lattice is shown.

Figure 6

(Color online) Evolution of the residual off-diagonality in the double-plaquette calculation (v) for various values of $W∕U$.

Figure 7

Distribution of coefficients as function of $W∕U$ according to the spatial extension $d$ of the respective terms in the double-plaquette calculation (v). The sum of the squares of the coefficients is plotted relative to the second-order Heisenberg exchange $J_1^{\left(2\right)}=4t^2∕U$.

Figure 8

(Color online) Apparent charge gap as calculated with Lanczos diagonalization. Gray lines show the minimal eigenvalues, black lines the result of a linear extrapolation.

Figure 9

(Color online) Couplings of two-spin operators relative to $J_1^{\left(2\right)}$ as function of $W∕U$.

Figure 10

(Color online) Couplings of four-spin operators relative to $J_1^{\left(2\right)}$ as function of $W∕U$.

Figure 11

(Color online) Effective hopping coefficients between (a) NN sites, (b) NNN sites, (c) spin-dependent NNN hopping, (d) 3NN sites, and (e) spin-dependent 3NN hopping.

Figure 12

(Color online) Hubbard $U$ (a) repulsion and NN interaction $V$ (b) and NNN interaction $V^{\prime }$ (c) between holes or double occupancies.

Figure 13

Further coefficients of interactions of two DO that are of second order in $t∕U$. For explanation, see the text.