Low-energy theory of the $t-t^{\prime }-t^{″}-U$ Hubbard model at half-filling: Interaction strengths in cuprate superconductors and an effective spin-only description of ${\text{La}}_2{\text{CuO}}_4$

Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, $t^{\prime }$ and $t^{″}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on ${\text{La}}_2{\text{CuO}}_4$. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of $U$, $t$, $t^{\prime }$, and $t^{″}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for ${\text{La}}_2{\text{CuO}}_4$. The ring exchange terms are found essential, in particular to determine the relative sign of $t^{\prime }$ and $t^{″}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\simeq 8t$ consistent with Mott insulating behavior.

Figure 1

(Color online) Hopping processes and resulting spin interactions. (a) shows the different hopping processes characterized by parameters $t$, $t^{\prime }$, and $t^{″}$ in the Hubbard model considered in this paper. $t$, $t^{\prime }$, and $t^{″}$ are hopping parameters between first-, second-, and third-nearest neighbors, respectively. (b) At half-filling, and when $t^{\prime }=t^{″}=0$, canonical perturbation theory leads to order $t^4/U^3$ to an effective spin-1/2 Hamiltonian, $H_s^{\left(4\right)}$, characterized by first- $\left(J_1\right)$, second- $\left(J_2\right)$, and third- $\left(J_3\right)$ nearest-neighbor exchange interactions as well as a four-spin ring (cyclic) exchange interaction with coupling strength $J_c$. (c) In a large $S$ expansion, $H_s^{\left(4\right)}$ can be recast to order $1/S$ as an effective spin Hamiltonian which only involves bilinear (pairwise) spin-spin exchange interactions between first- $\left(J_1^{\text{eff}}\right)$, second- $\left(J_2^{\text{eff}}\right)$, and third- $\left(J_3^{\text{eff}}\right)$ nearest neighbors. To order $1/S$, the effect of the ring exchange of strength $J_c$ merely renormalizes the first- and second-nearest-neighbor exchange as discussed in Ref. 6. (d) illustrates an example of a four hops (ring exchange) electronic process that involves $t^{\prime }$ and $t^{″}$ and contributes to order $t^2{t}^{\prime }{t}^{″}/U^3$ to the spin Hamiltonian $H_s^{\left(4\right)}$. (e) shows that the ring exchange term originating from the hopping path illustrated in (d) introduces in the $1/S$ approximation of $H_s^{\text{eff}}$ a fourth-nearest-neighbor exchange, $J_4^{\text{eff}}$ as well as renormalize the $J_1^{\text{eff}}$, $J_2^{\text{eff}}$, and $J_3^{\text{eff}}$ of (c). (f) illustrates all the additional $J_{n\ge 4}^{\text{eff}}$ exchanges beyond those shown in (c) and which are generated by all the hopping processes involving an allowed combination of $t$, $t^{\prime }$, and $t^{″}$ to order $1/U^3$.

Figure 2

(Color online) Magnon energy in ${\text{La}}_2{\text{CuO}}_4$, as a function of wave vector $\mathbf{k}$, across the Brillouin zone at a temperature of 10 K. The experimental data (red squares) and fit (full line) are from Ref. 6. The fitting parameters $t$ and $U$ are given in Eq. (36).

Figure 3

(Color online) Magnon energy in ${\text{La}}_2{\text{CuO}}_4$, as a function of wave vector $\mathbf{k}$, across the Brillouin zone at a temperature of 10 K. The experimental data (red squares) are from Ref. 6. The fit is made using the parameters given by Eq. (38).

Figure 4

(Color online) $\mathbf{k}$ dependence of ${\Delta }_{\mathbf{k}}$. Data are shown over the structural Brillouin zone of the square lattice. As can be seen, the dispersion has the symmetry of the zone corresponding to the antiferromagnetic order.

Figure 5

(Color online) Goodness of fit parameter ${\chi }^2$ as a function $t^{″}/t$ and $t^{\prime }/t$ with the terms with odd powers of $t^{\prime }/t$ and $t^{″}/t$ (a) excluded and (b) included.

Figure 6

Label of the different sites involved in the many-spin terms induced by $t^{\prime }$ and $t^{″}$ to order $1/U^3$.

Figure 7

(Color online) Evolution of ${⟨{\Delta }_k⟩}_k$ as a function of the number of points in the Brillouin zone.

Figure 8

(Color online) Evolution of $L\times {⟨{\Delta }_k⟩}_k$ as a function of the number of points in the Brillouin zone.