Coexistence of diffusive and ballistic transport in integrable quantum lattice models

We investigate the high-temperature dynamical conductivity $\sigma \left(\omega \right)$ in two one-dimensional integrable quantum lattice models: the anisotropic XXZ spin chain and the Hubbard chain. The emphasis is on the metallic regime of both models, where besides the ballistic component, the regular part of conductivity might reveal a diffusivelike transport. To resolve the low-frequency dynamics, we upgrade the microcanonical Lanczos method enabling studies of finite-size systems with up to $L\le 32$ sites for the XXZ spin model with the frequency resolution $\delta \omega \sim {10}^{-3}J$. Results for the XXZ chain reveal a fine structure of $\sigma \left(\omega \right)$ spectra, which originates from the discontinuous variation of the stiffness, previously found at commensurate values of the anisotropy parameter $\mathrm{\Delta }$. Still, we do not find clear evidence for a diffusive component, at least not for commensurate values of $\mathrm{\Delta }$, particularly for $\mathrm{\Delta }=0.5$, as well as for $\mathrm{\Delta }\to 0$. Similar is the conclusion for the Hubbard model away from half-filling, where the spectra reveal more universal behavior.

Figure 1

High-$T$ spin conductivity $\tilde{\sigma }\left(\omega \right)=T{\sigma }_{\mathrm{reg}}\left(\omega \right)$ in the XXZ spin chain for the anisotropy range $\mathrm{\Delta }\le 1$ ($\mathrm{\Delta }=0,0.025,\cdots ,1.0$), as calculated by (a) ED for $L=20$ sites and (b) MCLM for $L=32$ sites. Each consecutive curve is given an offset for better visibility. (c) and (d) Heat maps of results from (a) and (b), respectively. For clarity, the latter results are normalized by a maximum of ${\tilde{\sigma }}_{\mathrm{reg}}\left(\omega \right)$. Guidelines on both panels mark the positions of the low-frequency peaks obtained for $L=20$.

Figure 2

Integrated and normalized dynamical spin conductivity $\tilde{I}\left(\omega \right)=I\left(\omega \right)/I_{\infty }$ in a log-$\omega$ scale within the XXZ spin chain for selected $\mathrm{\Delta }=0.2,0.6,1.0$, as calculated for different sizes, i.e., $L=16,20$ using full ED, and $L=24,28,32$ using MCLM.

Figure 3

Spin conductivity $\tilde{\sigma }\left(\omega \right)$ obtained for different $L=20–32$ for two commensurate $\mathrm{\Delta }=cos(\pi /m)$: (a) $m=3$ ($\mathrm{\Delta }=0.5$), where the dotted line is the result of tDMRG [25], and (b) $m=4, \mathrm{\Delta }=1/\sqrt{2}\simeq 0.707$. Inset shows ${\tilde{\sigma }}_{\mathrm{reg}}\left(\omega \right)$ (using logarithmic scale) obtained from ED for selected values of $\mathrm{\Delta }$ together with a guideline $\sim {\omega }^2$.

Figure 4

Integrated spin conductivity $I\left(\omega \right)$ for the whole range of $\mathrm{\Delta }=0.1–1.0$, as obtained via MCLM on $L=32$ XXZ spin chain. For $\mathrm{\Delta }=1$ we show also the fit for superdiffusive $I\left(\omega \right)\propto {\omega }^{2/3}$.

Figure 5

Detailed analysis of ${\tilde{\sigma }}_{\mathrm{reg}}\left(\omega \right)$ spectra in the vicinity of (a) ${\mathrm{\Delta }}_4=1/\sqrt{2}$ and (b) ${\mathrm{\Delta }}_2=0$ anisotropy, as calculated for $L=20$ sites by ED. The DPT result is also marked in (b).

Figure 6

(a) Integrated and normalized regular part of the optical conductivity $I_{\mathrm{reg}}\left(\omega \right)$ obtained for $L=16–28$ sites via the DPT calculation for $\mathrm{\Delta }\to 0$. The inset shows rescaled data $I_{\mathrm{reg}}(\omega /{\omega }_p)$, where ${\omega }_p$ is the position of maximum. (b) The low-frequency part of $I_{\mathrm{reg}}(\omega \ll \mathrm{\Delta })$ fitted by $I_{\mathrm{reg}}\propto {\omega }^{b+1}$ (fits are shown as dashed lines). (c) Exponential decay of $1-I_{\mathrm{reg}}\left(\omega \right)$. (d) Finite-size scaling of the exponent $b$ from (b), where the dashed line shows fits linear in $1/L$ and the continuous guideline goes through results for $L=24$ and $L=28$. In (d) we present also the position of peak ${\omega }_p$ and median ${\omega }_m$ together with the linear fits.

Figure 7

(a) Charge conductivity ${\tilde{\sigma }}_c\left(\omega \right)$ for the 1D Hubbard model at quarter-filling $\overline{n}=1/2$ and zero magnetization $\overline{m}=0$, obtained via MCLM for a chain of $L=20$ sites and $M_L=5\times {10}^3$ steps. (a) The scan for the whole interaction $0<U/t<2$ range. (b) Spectra for selected moderate/large $U/t=1,2,4,8$.