Quantum Critical Dynamics Induced by Topological Zero Modes

Abstract

We investigate the low-frequency ac transport in the Su-Schrieffer-Heeger (SSH) chain with chiral disorder near the topological delocalization transition. Our key finding is that the formation of hybridized pairs of topological domain wall zero modes leads to the anomalous logarithmic scaling of the ac conductivity $σ(ω) \sim \log ω$ at criticality, and $σ(ω) \sim ω^{2 δ} \log ^2 ω$ away from it. Using the combination of real-space renormalization group analysis and qualitative hybridization arguments, we demonstrate that the form of the scaling of ac conductivity at criticality stems directly from the stretched-exponential ($ψ(x) \sim e^{-s \sqrt{x}}~\,$) spatial decay of zero-mode wavefunctions at the critical point.

Figures (4)

• Figure 1: Hopping disorder in the SSH chain creates isolated rare regions characterized by extremely small effective tunneling probability $t_{\rm eff}$. Two zero modes appearing on both sides of the link $t_{\rm eff}$ hybridize away from zero energy with energy splitting $\omega \sim t_{\rm eff} \sim e^{-\sqrt[\alpha]{d/\xi}}$, where $\xi$ encodes the typical size of individual zero modes and both $\alpha=1$ and $\alpha=2$ are possible in this model. The transition amplitude between these states in a time-dependent electric field is $|\braket{\psi_-| \hat{x}| \psi_+}| \sim d \sim \log^\alpha \omega$, implying an insulator for $\alpha = 1$ with ac conductivity following a modified Mott-Berezinskiy law \ref{['soffcrit']}, and, for $\alpha=2$, a glassy metal characterized by activated dynamical scaling \ref{['scrit']}.
• Figure 2: $a)\,$Dynamical conductivity in the SSH chain with chiral box disorder at half filling for different values of the parameter $\delta$, which encodes the distance to the topological phase transition. The numerical data is fitted to $\sigma(\omega) = c_1 \omega^{2 |\delta_{\rm fit}|} \log^2 (c_2/\omega)$, and $|\delta_{\rm fit}|= 0.78$, $0.78$, $1.23$, $1.24$ indicate a good agreement with the actual values of dimerization $\delta$, confirming our expression for ac conductivity in this regime \ref{['soffcrit']}. The crossover to the critical $\sigma(\omega) \sim \log \omega$ regime occurs beyond a certain $\delta$-dependent frequency indicated by the dotted lines. Note that ac conductivity is enhanced for $\delta>0$, corresponding to the topological phase in the model. $b)$ Schematic representation of the crossover between different types of states found in the SSH model with bond disorder. When $\delta \neq 0$, the closest to $E = 0$ states are the hybridized exponential modes. Immediately above it lie the stretched-exponential double-peak wavefunctions arising from the surviving critical regions. At even higher energies, regular exponentially localized states are found. $c)$ Same as $a)$ for the critical value $\delta = 0$ and different disorder strengths $s$. The curves exhibit good data collapse after rescaling both the frequency and the conductivity by $s^2$. The unscaled curves are shown in the inset. Importantly, the $\log \omega$ law holds at all disorder strengths, consistent with \ref{['sigcrit']}. The simulations were performed for chains of $10^3$ sites and $10^6$ disorder configurations.
• Figure 3: Dipole transition amplitude between states separated by $\omega$ with disorder strength $s^2 = 0.3$, fit to $\log^2 \omega$ (black line) and $\log \omega$ (red dashed line). The best data fit to $\log^2 \omega$ is evidence for stretched exponential behavior of the zero-energy wavefunction at criticality, justifying the result \ref{['logsq']}. Inset: coefficient $c_s$ as a function of disorder strength $s$, extracted from fitting $\log^2 \omega$. The predicted relation $c_s = c_{\rm crit}s^{-2}$ in \ref{['logsq']} is established with $c_{\rm crit} \simeq 0.13$. The calculation used $10^3$ sites at criticality ($t=t'=1$), chiral box disorder, and averaging over $10^4$ configurations.
• Figure S4: $a)$ Two zero modes ($\psi_{\rm L}$ and $\psi_{\rm R}$) appear on both sides of the topological region created by the bond-dilution disorder in the SSH chain. $b)$ If the segment is sufficiently long, the zero modes hybridize into sub-gap bonding, and anti-bonding states $\psi_{\pm}$. Due to the finite gap, only the optical transitions between such states (indicated with a purple arrow) will contribute to the low-frequency ac conductivity.