Superdiffusion and anomalous fluctuations in chiral integrable dynamics

TL;DR

This work investigates how breaking time-reversal symmetry affects superdiffusive magnetization transport in integrable SU(2) spin ladders. Using thermodynamic Bethe ansatz and full counting statistics, the authors find robust $z=3/2$ superdiffusion at infinite temperature even when TRS is broken, but the magnetization fluctuation statistics deviate from the KPZ universality class. Remarkably, the spin current obeys a fluctuation symmetry despite broken TRS and space reflection, due to a $C_2$ ladder symmetry. An integrable chiral quantum circuit with the same symmetries reproduces the same transport behavior without energy conservation, suggesting that TRS breaking is not sufficient to induce KPZ universality and that additional structural features are required for KPZ in quantum many-body systems.

Abstract

Symmetries strongly influence transport properties of quantum many-body systems, and can lead to deviations from the generic case of diffusion. In this work, we study the impact of time-reversal symmetry breaking on the transport and its universal aspects in integrable chiral spin ladders. We observe that the infinite-temperature spin transport is superdiffusive with a dynamical critical exponent z = 3/2 matching the one of the Kardar-Parisi-Zhang (KPZ) universality class, which also lacks the time reversal symmetry. However, we find that fluctuations of the net magnetization transfer deviate from the KPZ predictions. Moreover, the full probability distribution of the associated spin current obeys fluctuation symmetry despite broken time-reversal and space-reflection symmetries. To further investigate the role of conserved quantities, we introduce an integrable quantum circuit that shares the essential symmetries with the chiral ladder, and which exhibits analogous dynamical behaviour in the absence of energy conservation. Our work shows that time-reversal symmetry breaking is compatible with superdiffusion, but insufficient to stabilize the KPZ universality in integrable systems. This suggests that additional fundamental features are missing in order to identify the emergence of such dynamics in quantum matter.

Figures (6)

• Figure 1: SU(2)-invariant spin-1/2 systems considered in this work: (a) Triangular ladder parametrized by $\eta\in[0,1]$, with nearest-neighbour exchange $J_1=1-\eta$, next-nearest-neighbour exchange $J_2=\eta/2$, and uniform three-spin chiral interaction $J_\chi=\sqrt{\eta(1-\eta)}$. (b) Brickwork circuit constructed from the composition of four elementary gates $\check{R}(\tau)=(\mathds{1}+{\rm i}\tau P)/(1+{\rm i}\tau)$, some of which are deformed by the parameter $\delta=-\sqrt{\eta/(1-\eta)}$. Here, $P$ is the permutation of two neighbouring spins $1/2$. The composite four-site unitary gate $U_{[4j-1,4j+2]}$, acting in sites $4j\!-\!1,4j,4j\!+\!1,4j\!+\!2$, is encircled in red. Up to $O(\tau^3)$ corrections, it is equivalent to the exponential of the four-site Hamiltonian density of the chiral spin ladder shown in panel (a). The continuous- and discrete-time dynamics are equivalent in the limit $\tau\to0$.
• Figure 2: Second moments and cumulants of the magnetization and energy transfer. Panel (a) shows the second moment of the net magnetization transfer through the middle of the system, $\mu_2(t)=\langle\left[S_{\rm left}^z(t)-S_{\rm left}^z(0)\right]^2\rangle$. The orange and dark red solid lines correspond to $\mu_2(t)$ computed at finite chemical potential $\mu=0.5$, for $\eta=0$ and $\eta=0.5$ respectively. The black and green dashed lines correspond to $\sim D_{\rm sw}t$, with the hydrodynamic predictions for the Drude self-weights $D_{\rm sw}$ reported in Section \ref{['sec:drude-ballistic']}. Panel (b) shows the second cumulant of the net magnetization transfer through the middle of the system, $\kappa_2(t)$, for different values of the model's parameter $\eta$, at zero chemical potential ($\mu=0$). We observe the asymptotic scaling $\kappa_2(t)\sim t^{2/3}$, implying the dynamical exponent $z=3/2$. Panel (c) shows the second cumulant $\kappa^{E}_2(t)$ of the net energy transfer through the middle of the system, at infinite temperature and for different values of $\eta$. Irrespectively of the model's parameter $\eta$, we find $\kappa^{E}_2(t) \sim t$, indicating ballistic energy transport.
• Figure 3: Higher-order fluctuations corresponding to the magnetization transfer. Plotted is the excess kurtosis $\gamma_4^\mathrm{QGF}(t)$ of the magnetization transfer for the chiral ladder with $L=128$ sites and $\eta=1/2$. Various bond dimensions $\chi$ are used. The dashed lines correspond to the predictions within the KPZ universality class (either Baik-Rains or Tracy-Widom), and to Gaussian fluctuations. The inset shows $\gamma_4^\mathrm{QGF}(t)$ as a function of $1/t$ for $\chi=1024$. Extrapolating to $t\to \infty$ gives a negative and non-zero value of the excess kurtosis, suggesting a non-Gaussian nature of the fluctuations.
• Figure 4: Spin transport and higher-order fluctuations in the chiral circuit. Panel (a) shows the second moment $\mu_{2}(t)$ for $\delta=1$ and $2$. The dashed line indicating the power-law scaling $\propto t^{2/3}$ serves as a guide to the eye. Panel (b) shows the excess kurtosis $\gamma_4^\mathrm{(QGF)}$ for the same values of $\delta$. Dashed lines show theoretical predictions for the standard KPZ (TW and BR) distributions, as well as for the Gaussian fluctuations. The inset highlights the $1/t$ infinite-time extrapolation. Results were computed using the QGF method with maximum bond dimension $\chi=1024$, for a system of size $L=256$, and circuit step $\tau=1$.
• Figure 5: Hamiltonian dynamics: Convergence test for the second and fourth moments of the magnetization transfer for different bond dimensions. The other parameters are $L=128,\ dt=0.1, \eta=0.5$.