Numerical investigation of spin excitations in a doped spin chain

We study the doping evolution of spin excitations in a one-dimensional (1D) Hubbard model and its downfolded spin Hamiltonians, by using exact diagonalization combined with cluster perturbation theory. In all models we observe hardening (softening) of spin excitations upon electron (hole) doping, which are reminiscent of recent experiments on two-dimensional (2D) cuprate materials. We also find that the three-site and even higher-order terms are crucial for the low-energy effective spin models to reproduce the magnetic spectra of doped Hubbard systems at a quantitative level. To interpret the numerical results, we further employ a strong coupling slave-boson mean-field theory. The mean-field theory provides an intuitive understanding of the overall compact support of dynamic spin structure factors, including the shift of zero-energy modes and change of spin excitation bandwidth with doping. Our results can serve as predictive benchmarks for future inelastic x-ray or neutron scattering experiments on doped 1D antiferromagnetic Mott insulators.

Figure 1

$S(q,\omega )$ calculated using ED+CPT for various doping levels (left to right: number of electrons $n=1$, 0.875, 0.75, and 0.625) on a 16-site 1D system with particle-hole symmetry ($t^{\prime }=0$). Upper panels: Hubbard model with $U=8t$; middle panels: $t-J-3s$ model with $J=0.5t$; bottom panels: $t-J$ model with $J=0.5t$. A Lorentzian broadening of $\delta =0.15t$ is adopted in these calculations. Note that because of the particle-hole symmetry only the hole-doped spectrum is shown.

Figure 2

$S(q,\omega )$ calculated using the slave-boson mean-field approach for the $t-J$ model with $J=0.5t, t^{\prime }=0$, at various doping levels (left to right: number of electrons $n=1$, 0.875, 0.75, and 0.625). A Lorentzian broadening of $\delta =0.05t$ is adopted in these calculations. Note that because of the particle-hole symmetry only the hole-doped spectrum is shown.

Figure 3

$S(q,\omega )$ calculated using ED+CPT for various doping levels (left to right: number of electrons $n=1.25$, 1.125, 0.875, and 0.75) on a 16-site 1D system without particle-hole symmetry ($t^{\prime }=-0.3t$). Upper panels: Hubbard model with $U=8t$; middle panels: $t-J-3s$ model with $J=0.5t$; bottom panels: $t-J$ model with $J=0.5t$. A Lorentzian broadening of $\delta =0.15t$ is adopted in these calculations.

Figure 4

$S(q,\omega )$ calculated using the slave-boson mean-field approach for the $t-J$ model with $J=0.5t, t^{\prime }=-0.3t$, at various doping levels (left to right: number of electrons $n=1.25$, 1.125, 0.875, and 0.75). A Lorentzian broadening of $\delta =0.05t$ is adopted in these calculations.

Figure 5

The characteristic spin excitation energy ${\omega }_c$ (see text) as a function of doping $x$, obtained from ED+CPT results for the Hubbard model (solid circles), $t-J-3s$ model (open squares), and $t-J$ model (open triangles) with different values of $t^{\prime }$. Negative (positive) $x$ corresponds to hole (electron) doping. The gray curve denotes the corresponding mean-field fermionic $\xi \left(k\right)$ bandwidth. The inset of (c) illustrates the definition of ${\omega }_c$ and ${\omega }_m$ as explained in the text.

Figure 6

(a)–(c) $S(q,\omega )$ calculated using the slave-boson mean-field approach for the $t-J$ model with $J=0.5t, t^{\prime }=-0.3t$, at the doping level $n=1.25$, 1.0 and 0.75, respectively. (d)–(f) Mean-field fermionic spinon band ${\xi }_k$ Eq. (13) for the same doping levels. The shaded area of the fermionic band ${\xi }_k$ is occupied by particles (spinons), while the gray dashed lines denote the Fermi energy $E_F$ and momenta $k_F$. The vectors $q_1$ and $q_2$ denote the allowed momenta for zero-energy particle-hole excitations (see text). The colored (red, green, or blue) arrows in (d)–(f) indicate the particle-hole excitations that contribute to the excitations at particular ($q,\omega$) points denoted by the corresponding colored dots in (a)–(c).