Anomalous Dynamical Scaling at Topological Quantum Criticality

TL;DR

The paper shows that topological edge modes at quantum critical points induce anomalous dynamical scaling beyond the conventional Kibble–Zurek framework. Through analyses of interacting Ising-type chains and free-fermion α-chains, it reveals a universal boundary scaling distinct from bulk KZ behavior, tied to edge-state topology and conformal boundary conditions. The authors further demonstrate analogous anomalous scaling in two-dimensional Chern criticality and establish robustness to disorder, indicating a general topology-driven mechanism for driven critical dynamics. Together, these results broaden the understanding of quantum critical dynamics and provide a unified scaling picture for boundary observables at topological QCPs.

Abstract

We study the nonequilibrium driven dynamics at topologically nontrivial quantum critical points (QCPs),and find that topological edge modes at criticality give rise to anomalous universal dynamical scaling behavior. By analyzing the driven dynamics of bulk and boundary order parameters at topologically distinct Ising QCPs, we demonstrate that, while the bulk dynamics remain indistinguishable and follow standard Kibble Zurek (KZ) scaling, the anomalous boundary dynamics is unique to topological criticality, and its explanation goes beyond the traditional KZ mechanism. To elucidate the unified origin of this anomaly, we further study the dynamics of defect production at topologically distinct QCPs in free-fermion models and demonstrate similar anomalous universal scaling exclusive to topological criticality. These findings establish the existence of anomalous dynamical scaling arising from the interplay between topology and driven dynamics, challenging standard paradigms of quantum critical dynamics.

Figures (11)

• Figure 1: (a) Schematic phase diagram of the spin model in Eq. (\ref{['E1']}) (upper panel) and its fermionic dual (lower panel). In the spin models, the control parameter $\lambda$ tunes the system across the topologically nontrivial or trivial QCP $\lambda_c$, separating the FM phase from the cluster-SPT phase or the PM phase. In the fermion model, the control parameter $\lambda$ tunes the system across the topologically nontrivial or trivial QCP $\lambda_c$, separating the topological superconductor (TSC) phase with winding number $\omega = 1$ from the TSC with $\omega = 2$ or topologically trivial with $\omega = 0$. The control parameter $\lambda$ is linearly ramped from the FM phase (TSC with $\omega = 1$) toward the QCP at $\lambda_c$ (blue dashed arrow). (b) Relevant time scales for driven dynamics near criticality. The correlation time diverges as the system approaches the QCP (black solid curve), while the blue dashed line denotes the time distance $|\lambda - \lambda_c|/R$ to the QCP for different quench rates.
• Figure 2: The boundary magnetization $m_s$ at the end of the quench as a function of the quench rate $R$ for (a) the CI chain and (b) the TFI chain of different system sizes. The dynamical scaling of $m_{s}$ at QCPs exhibits power law scalings with distinctive exponents close to $1$ and $1/4$, respectively (solid lines). In contrast, the bulk magnetization $m_b$ in both (c) the CI chain and (d) the TFI chain exhibit the same power-law behavior, with an exponent close to 1/16 (solid lines).
• Figure 3: The anomalous dynamical scaling behaviors of the edge excitation density $n_{\text{ex}}$ when the system is driven to topologically nontrivial QCPs (a-1)–(a-2) for the clean system and (b-1)–(b-2) for the disordered system. (a-1) The dependence of edge excitation density $n_{\text{ex}}$ on the quench rate $R$ for different values of $\epsilon_i$. In the slow-quench regime, $n_{\text{ex}}$ follows a power-law scaling with an exponent close to 1.5. In the fast-quench regime, $n_{\text{ex}}$ saturates to a value independent of the quench rate. (a-2) The saturation value $n^s_{\text{ex}}$ and the critical quench rate $R^c$ both exhibit power-law scaling with the dimensionless distance $\epsilon_i$, with exponents close to 2 and 4/3, respectively. (a-3) The excitations $n_{\text{ex}}$ generated by quenching to the topologically trivial QCP as a function of the quench rate $R$ for different values of $\epsilon_i$ in the clean system. (b-1)-(b-2) Same as (a-1)-(a-2) but for the presence of symmetry-preserving disorder. The anomalous dynamical power-law scaling remains robust and exhibits the same critical exponents within error bars. (b-3) Across the critical point of the Creutz ladder model, the topology-induced anomalous power-law behavior of defect production reported in Ref. Barankov2008PRL is destroyed by symmetry-preserving disorder. Error bars denote standard deviations. The size of the clean system is $L=1200$, while that of the disordered system is $L=400$. Average is performed over $600$ independent disorder realizations.
• Figure S1: In equilibrium, the local magnetization $m_{b,s}$ as a function of the dimensionless distance $\epsilon=|\lambda-\lambda_c|/\lambda_c$ to the QCP $\lambda_c$. The black solid lines provide the guidance for an eye for the expected scaling (slope) on the log-log plot. The corresponding slopes approach the values of (a) $1$, (b) $1/2$, (c) $1/8$, and (d) $1/8$, respectively.
• Figure S2: The rescaled magnetization $m_a(R)L^{\Delta_a}$ as a function of $RL^r$ for different $L$. (a)-(d) correspond to panels (a)-(d) in Fig. 2 of the main text. The plots collapse onto a single scaling function, consistent with the dynamical scaling hypothesis (Eq. ( \ref{['eq:sres1']})) and the following scaling exponents: (a) $\Delta_s = 2, r = 2$, (b) $\Delta_s = \frac{1}{2}, r = 2$, (c) $\Delta_s = \frac{1}{8}, r = 2$, and (d) $\Delta_s = \frac{1}{8}, r = 2$. All plots are displayed on log-log scales.