Unified one-band Hubbard model for magnetic and electronic spectra of the parent compounds of cuprate superconductors

Using low-energy projection of the one-band $t$-$t^{\prime }$-$t^{\prime \prime }$ Hubbard model we derive an effective spin Hamiltonian and its spin-wave expansion to order $1/S$. We fit the spin-wave dispersion of several parent compounds to the high-temperature superconducting cuprates La${}_2$CuO${}_4$, Sr${}_2$CuO${}_2$Cl${}_2$, and Bi${}_2$Sr${}_2$YCu${}_2$O${}_8$. Our accurate quantitative determination of the one-band Hubbard model parameters allows prediction and comparison to experimental results. Among those we discuss the two-magnon Raman peak line shape, the K-edge resonant inelastic x-ray scattering 500-meV peak, and the high-energy kink in the angle-resolved photoemission spectroscopy quasiparticle dispersion, also known as the waterfall feature.

Figure 1

Top: Example of an exchange process. contributing to the matrix element $\langle \begin{array}{cc}\uparrow & \downarrow \\ \downarrow & \uparrow \end{array}\left|T_{-1}{T}_0{T}_0{T}_1\right|\begin{array}{cc}\downarrow & \uparrow \\ \uparrow & \downarrow \end{array}\rangle$. Plain (open) points are DOs (holes). Bottom: CuO${}_2$ planes from the perovskite structure. Examples of exchange loops from the effective spin Hamiltonian of Eq. (3) are indicated with first-, second-, and third-nearest-neighbor hoppings $t$, $t^{\prime }$, and $t^{\prime \prime }$.

Figure 2

(a) Fits of the measured SW dispersions (solid lines) and the 1JHei SW dispersion (dotted lines) with $J_{\mathrm{NN}}={\omega }_{(\pi /2,\pi /2)}/2Z_c$. From top to bottom: BSYCO ($J_{\mathrm{NN}}=0.15$ eV), SCOC (Ref. 6) ($J_{\mathrm{NN}}=0.12$ eV), and LCO (Ref. 7) ($J_{\mathrm{NN}}=0.14$ eV). (b)–(d) Same as in (a) but subtracted from the 1JHei SW dispersion to enhance details. Dashed lines are the SW dispersions obtained from the Hubbard model with only NN hopping $t$. Data points from experiments are folded onto the equivalent high-symmetry axes of the first Brillouin zone.

Figure 3

Summary of the fitting results. (a)–(c) Goodness (${\chi }^2$) of the $(t/U,t^{\prime }/t)$ fit for fixed $(U,t^{\prime \prime }/t)$. Solid (blue) and dashed (green) lines are best-fit solutions as functions of $U$ for $t^{\prime }{t}^{\prime \prime }<0$ and $t^{\prime }{t}^{\prime \prime }>0$, respectively. (d)–(f) ${\chi }^2$ along the best-fit lines defined in (a)–(c). (g)–(i) $(t/U,t^{\prime }/t,t^{\prime \prime }/t)$ solutions along the best-fit lines defined in (a)–(c). (j)–(l) Staggered magnetization, DO density, and $\mathbf{k}$-averaged quantum $1/S$ renormalization $Z_c$ along the $t^{\prime }{t}^{\prime \prime }<0$ best-fit lines defined in (a)–(d). Lines: Solid blue BSYCO, dashed red SCOC, and dotted black LCO. Shaded areas indicate $\mathbf{k}$ variation of $Z_c$.

Figure 4

(a), (b) $S^{zz}(\mathbf{k},\omega )$ for 1JHei with $J_{\mathrm{NN}}={\omega }_{(\pi ,0)}/2Z_c$ and the 1bHub with the SCOC parameters. (c), (d) $S^{zz}$ energy line shapes at the ZB points for 1JHei (dashed red line) and 1bHub (solid blue line). (e), (f) Corresponding 2M DOS. (g), (h) DOS profiles at $(0,0)$ and $(\pi ,0)$. (i), (j) Single-magnon wave vectors contributing to the DOS peaks at $(0,0)$ and $(\pi ,0)$ in 1JHei and (k), (l) in the SCOC 1bHub.