Thermal transport of the $XXZ$ chain in a magnetic field

We study the heat conduction of the spin-$1∕2$ $XXZ$ chain in finite magnetic fields where magnetothermal effects arise. Due to the integrability of this model, all transport coefficients diverge, signaled by finite Drude weights. Using exact diagonalization and mean-field theory, we analyze the temperature and field dependence of the thermal Drude weight for various exchange anisotropies under the condition of zero magnetization-current flow. First, we find a strong magnetic field dependence of the Drude weight, including a suppression of its magnitude with increasing field strength and a nonmonotonic field-dependence of the peak position. Second, for small exchange anisotropies and magnetic fields in the massless as well as in the fully polarized regime, the mean-field approach is in excellent agreement with the exact diagonalization data. Third, at the field-induced quantum critical line between the para- and ferromagnetic region, we propose a universal low-temperature behavior of the thermal Drude weight.

Figure 1

Thermal Drude weight $K_{\mathrm{th}}(h,T)$ of the $XXZ$ chain for $\Delta =0.1$: comparison of mean-field theory (MF) and exact diagonalization (ED). The thermal Drude weight $K_{\mathrm{th}}(h,T)$ is shown for $h=0,0.5,1.1,1.5$. Increasing field is indicated by the arrow. Thick lines denote results from the Hartree-Fock approximation; thin lines: ED for $N=18$. Deviations at low temperatures for $h=0$ and $h=0.5$ are due to finite-size effects in the ED results.

Figure 2

Thermal Drude weight $K_{\mathrm{th}}(h,T)$ of the Heisenberg chain $(\Delta =1)$. Panel (a) $K_{\mathrm{th}}(h,T)$ computed from Eq. (14) using $D_{\mathrm{s}}=D_{\mathrm{s}}^{\mathit{I}}$ (solid lines) and $D_{\mathrm{s}}=D_{\mathrm{s}}^{\mathit{II}}$ (dashed lines), both for $h=0.5$ and $N=12,14,16,18,20$. Arrows indicate increasing system size. The dotted line is the result for $h=0$ and $N\to \infty$ from Ref. 16. Inset: comparison of $D_{\mathrm{s}}^{\mathit{I}}(h,T)$ [solid lines] and $D_{\mathrm{s}}^{\mathit{II}}(h,T)$ [dashed lines]; $h=0.5$, $\Delta =1$. Panel (b) $K_{\mathrm{th}}(h,T)$ for $h=1.5,2,2.5,3$ and $N=20$ [thick lines; $D_{\mathrm{s}}(h,T)=D_{\mathrm{s}}^{\mathit{I}}(h,T)$]. The curve for $N=18$, $h=1.5$ is included (thin solid line).

Figure 3

Main panel: field dependence of the thermal Drude weight $K_{\mathrm{th}}(h,T)$ for $\Delta =0.5,1,2$ and $T∕J=0.5$ (ED for $N=18$ sites). Inset: field dependence of $D_{\mathrm{th}}(h,T)$ for the same parameter sets as in the main panel.

Figure 4

Thermal Drude weight $K_{\mathrm{th}}(h,T)$ at the critical field $h_{c2}=1+\Delta$ for $\Delta =0,0.1,0.5,1,2$. For $\Delta \ne 0$, we show numerical results for $N=18$ sites, while the curve for the free fermion case (thin solid line) is valid in the thermodynamic limit.

Figure 5

Comparison of the spin Drude weight $D_{\mathrm{s}}^{\mathit{I}}(h,T)$ (thick lines) and its lower bound $D_{\mathrm{sub}}$ (thin lines; see text for further details). The figure shows results for $\Delta =1$, $N=20$, and $h=0.5,2,2.5$.