Finite-temperature Drude weight within the anisotropic Heisenberg chain

Finite-temperature Drude weight (spin stiffness) $D\left(T\right)$ is evaluated within the anisotropic spin-$1/2$ Heisenberg model on a chain using the exact diagonalization for small systems. It is shown that odd-side chains allow for more reliable scaling and results, in particular, if one takes into account corrections due to low-frequency finite-size anomalies. At high $T$ and zero magnetization $D$ is shown to scale to zero approaching the isotropic point $\Delta =1$. On the other hand, for $\Delta >2$ at all magnetizations $D$ is nearly exhausted with the overlap with the conserved energy current. Results for the $T$ variation $D\left(T\right)$ are also presented.

Figure 1

High-$T$ Drude weight $C=TD$ vs $1/L$ for zero magnetization $s=0$ and different $\Delta$ as obtained for systems with odd $L=5\text{--}19$.

Figure 2

Regular part of dynamical conductivity ${\sigma }_{\mathrm{reg}}\left(\omega \right)$ and the integrated one $I\left(\omega \right)$ (inset) for $s=0$: (a) $\Delta =0.5$ and (b) $\Delta =0.86$ for different sizes $L=13\text{--}21$.

Figure 3

Integrated dynamical conductivity $I\left(\omega \right)$ for $\Delta =0.25,s=0$, and various sizes $L$. The inset focuses on the low-$\omega$ regime.

Figure 4

High-$T$ Drude weight $C=TD$ vs $\Delta$ at magnetization $s=0$ obtained using the ED and finite-size scaling of $D\left(L\right)$ (full curve), adding the low-$\omega$ correction (dashed curve), and within the analytical TBA (Refs. 15 and13) (dotted curve).

Figure 5

High-$T$ Drude weight $C$ vs magnetization $s$ within the Ising-type regime $\Delta =1.7,3$ obtained for fixed $L=19$, with the $1/L\to 0$ extrapolation for $s=0,1/4,1/3$ (dots), and the analytical approximation $C_3$ [Eq. (11)] for $\Delta =3$.

Figure 6

Normalized Drude weight $D^{*}$ within the plane $\Delta ,s$ as calculated in systems with fixed size $L=19$.

Figure 7

Finite-$T$ Drude weight $D\left(T\right)$ for $\Delta <1$ at magnetization $s=0$ as calculated numerically using ED with $L=15\text{--}21$ and finite-size scaling (full line with dots), extrapolating the high-$T$ numerical result, i.e., $D=C/T$ (thin lines), and thermodynamic Bethe ansatz result (dots) from Ref. 13.