Dynamic structure factor in impurity-doped spin chains

The effects of impurities in spin-1/2 Heisenberg chains have recently experienced a renewed interest due to experimental realizations in solid state systems and ultracold gases. Nonmagnetic impurities lead to effectively isolated finite chain segments with a discrete spectrum and characteristic correlations, which have a distinct effect on the dynamic structure factor. Using bosonization and the numerical density matrix renormalization group, we provide detailed quantitative predictions for the momentum- and energy-resolved structure factor in doped systems. Due to the impurities, spectral weight is shifted away from the antiferromagnetic wave vector $k=\pi$ into regions which normally have no spectral weight in the thermodynamic limit. The effect can be quantitatively described in terms of scaling functions, which are derived from a recurrence relation based on bosonization.

Figure 1

(a) The dynamical structure factor $S_L({\omega }_m,k)$ at $L=100$ as a function of $k$ near $\pi$ from bosonization compared to numerical DMRG calculations for ${\omega }_m=8\frac{\pi v}{L}\approx 0.31J$ and $9\frac{\pi v}{L}\approx 0.348J$. The $L\to \infty$ behavior $S_{\infty }$ from Eq. (3) and the averaged signal ${\overline{S}}_p$ with doping level $p$ at the same energies are also shown where vertical lines mark $v|k-\pi |\approx {\omega }_m$. (b) Schematic two-spinon continuum. (c) Bosonization error over finite-size deviation $\mathrm{\Delta }S$ in Eq. (14) for $m=4$ and different $k$ using $K=0.8$ as a function of $1/L$.

Figure 2

The rescaled signal ${\left(\frac{\omega }{v}\right)}^{2-2K}{S}_{\infty }$ and ${\left(\frac{\omega }{v}\right)}^{3-2K}{S}_{\mathrm{imp}}$ as functions of the scaling variable $vq/\omega$ for $K=0.7$ and $K=0.9$. Inset: Relative impurity contribution $\omega {\widehat{S}}_{\mathrm{imp}}\left(\omega \right)/v{\widehat{S}}_{\infty }\left(\omega \right)$ from the $k$-integrated signal as a function of $K$.