Superdiffusive transport in chaotic quantum systems with nodal interactions

Abstract

We introduce a class of interacting fermionic quantum models in $d$ dimensions with nodal interactions that exhibit superdiffusive transport. We establish non-perturbatively that the nodal structure of the interactions gives rise to long-lived quasiparticle excitations that result in a diverging diffusion constant, even though the system is fully chaotic. Using a Boltzmann equation approach, we find that the charge mode acquires an anomalous dispersion relation at long wavelength $ω(q) \sim q^{z} $ with dynamical exponent $z={\rm min}[(2n+d)/2n,2]$, where $n$ is the order of the nodal point in momentum space. We verify our predictions in one dimensional systems using tensor-network techniques.

Figures (2)

• Figure 1: Superdiffusion in chaotic nodal chains. (a) Distribution of many-body level spacing $s$ in the middle half of the spectrum of nodal interaction model with $\tilde{\phi}_k=1+e^{i(k+\pi/2)}$. In this figure, we choose momentum $k=0$, particle number $N=(L+1)/2$ and interaction strength $W=4$. The $r$ ratio PhysRevB.75.155111 is consistent with Wigner-Dyson GOE. There is clear level repulsion in this model, ruling out integrability. (b) Low-energy spectrum (real part) of normal modes of the linearized Boltzmann equation for nodal interactions with $\tilde{\phi}_k = 1+e^{i(k+\pi/2)}$, exhibiting a node of order $n=1$ at $k_0 = \pi/2$. We choose the interactions to be noisy in time ($\gamma(k) = \beta(k,k') = 1$) to break energy conservation and focus on charge transport. We find two gapless modes: a ballistic mode corresponding to $n_{k_0}$ (which is exactly conserved by the interactions), and a hydrodynamic charge mode with dispersion relation $\omega \sim q^{3/2}$ with $q$ the momentum, indicating superdiffusive transport with dynamical exponent $z=3/2$. (c) Structure factor $C(x,t)\equiv\langle n(x,t)n(0,0)\rangle$ in same setup as (b), obtained from the Boltzmann equation. The main component of the structure factor follows the scaling relation $C(x,t)=t^{-2/3}f(xt^{-2/3})$, consistent with the dynamical exponent $z=3/2$. The decreasing ballistic component originates from modes near the node $k_0$.
• Figure 2: Tensor-network simulations.(a) NESS current under boundary driving. When the Hamiltonian possesses node ($a=1$), the scaling of NESS current and system size satisfies $j\sim L^{-1/2}$ at large $L$ limit. In the case $a=0.5$, the interactions do not exhibit a node, and the scaling of NESS current and system size satisfies $j\sim L^{-1}$ at large $L$ limit consistent with diffusive transport. (b) Time dependence of the charge transfer $\Delta s$ starting from a domain wall initial state. (c) The time-dependent exponent $\alpha(t)$ calculated as the numerical logarithmic derivative $\mathrm{d} \log \Delta s/\mathrm{d} \log t$. Fig. (b) and (c) demonstrate that in the model with nodal interaction, $\Delta s \propto t^{2/3}$, corresponding to a dynamical exponent $z = 3/2$. In contrast, with non-nodal interaction, $\Delta s \propto t^{1/2}$, indicating a dynamical exponent $z = 2$. For the NESS current simulations (a), we chose $\Gamma=1$, $\mu=0.02$, and a maximum bond dimension $\chi = 180$. In the charge transfer simulations (b) and (c), we chose $\nu = 0.01$ and a maximum bond dimension $\chi = 200$.