Diffusive Heat Waves in Random Conformal Field Theory

We propose and study a conformal field theory (CFT) model with random position-dependent velocity that, as we argue, naturally emerges as an effective description of heat transport in one-dimensional quantum many-body systems with certain static random impurities. We present exact analytical results that elucidate how purely ballistic heat waves in standard CFT can acquire normal and anomalous diffusive contributions due to our impurities. Our results include impurity-averaged Green’s functions describing the time evolution of the energy density and the heat current, and an explicit formula for the thermal conductivity that, in addition to a universal Drude peak, has a nontrivial real regular contribution that depends on details of the impurities.

Figure 1

Illustration of a velocity $v(x)$ effectively describing a lattice system with impurities varying on a mesoscopic scale $a_{\mathrm{imp}}$ much larger than the lattice spacing $a$. For the $XXZ$ spin chain described in the main text, $x=ia$, and the color and size of the dots indicate the magnitude of the couplings $J_i$.

Figure 2

$\mathrm{Re}{\kappa }_{\mathrm{th}}^{\mathrm{reg}}(\omega )$ in (13) for Examples (a)–(d) in Table 1. In all plots, ${\omega }_0=v/{\mathrm{\Gamma }}_0$, ${\mathrm{\Gamma }}_0/a_0=0.6$ is fixed, and the parameter varied is $\beta {\omega }_0$ equal to 1.8 (blue solid line), 1.2 (red dotted), and 0.6 (yellow dashed).