Microscopic Origin of Ideal Conductivity in Integrable Quantum Models

Nonergodic dynamical systems display anomalous transport properties. Prominent examples are integrable quantum systems, whose exceptional properties are diverging dc conductivities. In this Letter, we explain the microscopic origin of ideal conductivity by resorting to the thermodynamic particle content of a system. Using group-theoretic arguments we rigorously resolve the long-standing controversy regarding the nature of spin and charge Drude weights in the absence of chemical potentials. In addition, by employing a hydrodynamic description, we devise an efficient computational method to calculate exact Drude weights from the stationary currents generated in an inhomogeneous quench from bipartitioned initial states. We exemplify the method on the anisotropic Heisenberg model at finite temperatures for the entire range of anisotropies, accessing regimes that are out of reach with other approaches. Quite remarkably, spin Drude weight and asymptotic spin current rates reveal a completely discontinuous (fractal) dependence on the anisotropy parameter.

Figure 1

Particle content of the $XXZ$ Heisenberg model for $|\mathrm{\Delta }|\ge 1$ (top) and $|\mathrm{\Delta }|<1$ (bottom). While the former consist of infinitely many bound magnons with densities ${\rho }_a(u)$, $a\in {\mathbb{Z}}_{\ge 1}$, the latter reduces to $N_p$ particles ($n^{\prime }=N_p-2$) whose number depends discontinuously on $\mathrm{\Delta }$. Morphology of the graphs reflects how the particles effectively scatter among each other. Black and white end nodes label a distinguished pair that forms a doublet with an effective magnetic moment (orange arrow) being the only particles in the spectrum which transform nontrivially under the spin-reversal operation.

Figure 2

Partitioning protocol: the initial state is prepared in two nearly identical grand canonical equilibria ${\widehat{\varrho }}_{L,R}\propto \exp (-{\beta }_{L,R}\widehat{H}+h_{L,R}{\widehat{S}}^z)$, representing a $q$-charged ($q=s$, $e$) domain wall of size $\delta q$, with the corresponding chemical potential drop $\delta {\mu }_q={\mu }_{q,L}-{\mu }_{q,R}$ (where ${\mu }_e=\beta$, ${\mu }_s=h$). The initial defect expands in an inhomogeneous state localized within the light cone $v_{\max }^L<\zeta <v_{\max }^R$. In the $t\to \infty$ limit, the state along each ray $\zeta =x/t$ relaxes in a quasistationary state, which is uniquely characterized by particle distributions ${\rho }_a(u,\zeta )$, for $a=1,\dots ,N_p$. Drude weight ${\mathcal{D}}^{(q)}$ is proportional to the increment of the total current rate ${\lim }_{t\to \infty }{\mathcal{J}}^{(q)}(t;\delta {\mu }_q)/t$ in the limit $\delta {\mu }_q\to 0$.

Figure 3

Asymptotic spin (blue) and energy (red) current rates ${\mathcal{R}}^{(q)}={\lim }_{t\to \infty }{\mathcal{J}}^{(q)}(t)/t={\int }_{-v_{\max }^L}^{v_{\max }^R}\mathrm{d}\zeta j^{(q)}(\zeta ;\delta \beta ,\delta h)$, emerging by joining two equilibrium states with chemical potentials $h_{L,R}=\pm 1$ and inverse temperatures ${\beta }_{L,R}=1$, 3. Considering the sequence $\mathrm{\Delta }=\cos [\pi /(3+1/{\nu }_2)]$ for ${\nu }_2=2,3,\dots ,20,10^3$ (the points at ${\nu }_2=10^3$ are obtained by linear extrapolation of other ${\nu }_2$ points), we demonstrate that ${\lim }_{{\nu }_2\to \infty }{\mathcal{D}}^{(s)}\ne {\mathcal{D}}^{(s)}[\gamma =(\pi /3)]$ (open circle). The same holds in general when approaching a value of $\gamma$ parametrized by $l-1$ integers ${\nu }_i$ as the ${\nu }_l\to \infty$ limit of the order-$l$ sequence of ${\nu }_i$. This indicates that spin current is a nowhere-continuous function of $\mathrm{\Delta }$ within $|\mathrm{\Delta }|<1$ [cf. Eq. (5) for the exact analytic high-temperature results]. Unlike ${\mathcal{R}}^{(s)}$, the thermal current rate ${\mathcal{R}}^{(e)}$ depends continuously on $\mathrm{\Delta }$, as shown for ${\nu }_1=3$ (open square).

Figure 4

Rescaled spin Drude weight ${\tilde{\mathcal{D}}}^{(s)}=(16/\beta ){\mathcal{D}}^{(s)}$ obtained from Eq. (7) for various temperatures at $\mathrm{\Delta }=\cos [\pi /(3+1/{\nu }_2)]$, for ${\nu }_2=\{2,3,\dots ,12,10^3\}$ (${\nu }_2=10^3$ are obtained by linear extrapolation of other ${\nu }_2$-points), and for $\mathrm{\Delta }=\cos (\pi /3)=0.5$ (open circles). For a dense set of commensurate anisotropies $\mathrm{\Delta }=\cos (\pi m/\ell )$, ${\mathcal{D}}^{(s)}$ is found to be a discontinuous function of $\mathrm{\Delta }$ at arbitrary finite temperature (see explanation in the caption of Fig. 3).