Figure 1

Imaginary-time susceptibility ${\chi }_{\text{QMC}}\left(q,\tau \right)$ at $q=\pi /64$ on 128 sites, for two temperatures $T$, fitted to $\chi \left(q,\tau \right)$ from Eq. (9) (lines) in three ways, namely, ${\gamma }_{q,\text{QMC}}$ optimized (solid), ${\gamma }_B$ taken from Ref. 20 (dashed), and $\gamma$ forced to zero (dotted). The index “$i$” on the $y$ axis refers to $\chi \left(q,\tau \right)$ from Eq. (9) for the lines in panels (a) and (b) as well as to QMC for the symbols in panel (a). Panel (a) Global behavior of ${\chi }_{\text{QMC}}\left(q,\tau \right)/{\chi }_{\text{QMC}}\left(q,0\right)$ for $T/J=0.1$ (QMC, squares) and 0.25 (QMC, circles). In this panel the three fits (lines) are indistinguishable on the scale of the plot. Panel (b) error $2\sigma$ of ${\chi }_{\text{QMC}}\left(q,\tau \right)$ for each $\tau$ evaluated (error bars) and difference $\chi \left(q,\tau \right)-{\chi }_{\text{QMC}}\left(q,\tau \right)$ between QMC and the three fits (lines). $2\sigma$ for the QMC data is $O\left({10}^{-7}\right)$. Plots corresponding to $T/J=0.1$ have been shifted by $2\times {10}^{-6}$.Reuse & Permissions

Figure 2

Momentum dependence of the renormalized Luttinger parameter $K_q$, spinon velocity $v_q$, and scattering rate ${\gamma }_{q,\text{QMC}}/{\gamma }_B$ for the first nonzero six momenta on a 128 site system for two temperatures $T/J=0.1$ (white symbols) and 0.25 (black symbols). Note that ${\gamma }_{q,\text{QMC}}/{\gamma }_B$ for $T=0.1$ has been scaled by 2 to fit into the plot.Reuse & Permissions

Figure 3

${\omega }_n{\sigma }_{\text{QMC}}\left(q,{\omega }_n\right)$ from QMC for the first three nonzero Matsubara frequencies ${\omega }_n=2\pi nT$ and wave vectors $q=n\pi /64$, with $n=1$, 2, and 3 as compared to ${\omega }_n^2\chi \left(q,{\omega }_n\right)/q^2$ using Eq. (9) with $\gamma =0$ (dashed) and $\gamma ={\gamma }_{q,\text{QMC}}$ (solid) on a 128 site system for (a) $T/J=0.1$ and (b) 0.25. (See text regarding statistical error.)Reuse & Permissions