Ballistic transport in integrable quantum lattice models with degenerate spectra

We study ballistic transport in integrable quantum lattice models, i.e., in the spin XXZ and Hubbard chains, close to the noninteracting limit. It is by now well established that the stiffnesses of spin and charge currents reveal, at high temperatures, a discontinuous reduction (jump) when the interaction is introduced. We show that the jumps are related to the large degeneracy of the parent noninteracting models and, more generally, can appear in other integrable models with macroscopic degeneracies. These degeneracies are properly captured by the degenerate perturbation calculations which may be performed for large systems. We find that the discontinuities and the quasilocality of the conserved current in this limit can be traced back to the nonlocal character of an effective interaction. From the latter observation we identify a class of observables which show discontinuities.

Figure 1

XXZ model at half filling $n_{\mathrm{av}}=1/2$. (a) Spin stiffness $D_s$ from the perturbative approach [right-hand side of Eq. (4)] compared with (canonical) results from ED of the Hamiltonian [left-hand side of Eq. (4)] for $\mathrm{\Delta }=0.01$. (b) Stiffness from the perturbative approach for the operators $E_2$ and $F_3$ [see Eq. (9)]. In order to filter out the finite-size effects arising from $L$ dependence of the norm $\langle AA\rangle /L$, in (a) and (b) we show the renormalized stiffnesses, $D_A\times {\langle AA\rangle }_{L\to \infty }/{\langle AA\rangle }_L$. Results for $\mathrm{\Delta }=0$ ($D_A^0=1/8$) and analytical Mazur lower-bound/GHD results from Refs. [14, 18] are also shown. (c) Dynamical spin conductivity ${\sigma }_s\left(\omega \right)$ and integrated conductivity $I_s\left(\omega \right)$ obtained via MCLM on $L=28$ sites, shown in the inset and in the main panel, respectively, where $\omega$ is rescaled by $\mathrm{\Delta }$ in the latter case. The horizontal line shows $D_s$ from (a).

Figure 2

Hubbard model. (a) Charge stiffness for $n_{\mathrm{av}}=1/2$ from ED, $\langle \widetilde{J_c}\widetilde{J_c}\rangle /L$ (lines with points), and from the perturbative approach, $\langle {\overline{J_c}}^{\parallel }{\overline{J_c}}^{\parallel }\rangle /L$ (horizontal lines). (b) Charge stiffness $D_c\left(n_{\mathrm{av}}\right)$ for fixed $m_{\mathrm{av}}\simeq 0$ and (c) spin stiffness $D_s\left(m_{\mathrm{av}}\right)$ for fixed $n_{\mathrm{av}}\simeq 1$obtained from the perturbation approach. In (b) and (c) we show also $L=14$ results for $D_{c,s}^0$ at $U=0$.

Figure 3

Results for the XXZ chain with grand-canonical averaging over the Hilbert space. (a) Spin stiffness $D_s$ for $\mathrm{\Delta }=0.5$ and $\mathrm{\Delta }=0.51$. (b) Relative difference ${\text{Err}}_A$ between the ED results and the perturbative approach [see Eq. (B4)]. We show results for $\mathrm{\Delta }\to 0.5$ for the spin current $A=F_1=J_s$ and for $A=F_3$; see Eq. (9) in the main text for the definition of operators. For clarity, the results for $F_3$ are multiplied by a factor of 10. For comparison we show also the results for the spin current when $\mathrm{\Delta }\to 0$.

Figure 4

The same as in Fig. 3, but for charge current ($A=J_c$) in the quarter-filled Hubbard chain.