Anomalous resonance in Weyl semimetals: A holographic study of non-linear effects

TL;DR

The paper addresses anomalous resonance of currents in a CFT with a holographic 't Hooft anomaly under a magnetic field, and whether nonlinear back-reaction can sustain time-translation symmetry–breaking oscillations after an electric quench. It employs fully back-reacted five-dimensional Einstein-Maxwell-Chern-Simons-Dilaton holography with a magnetic background and scalar deformations, evolving time-dependent bulk gauge fields via a characteristic formulation to track the boundary current. The key finding is that nonlinear effects generally increase the decay rate at finite final temperature $T_{final}$, but for sufficiently small $T_{final}$ and large Chern-Simons coupling $\lambda$, the current oscillations can have an arbitrarily long lifetime, with late-time dynamics well captured by the lowest quasi-normal mode; this yields an approximate holographic time-crystal in a strongly coupled system. The work provides non-perturbative insights into anomalous transport in Weyl semimetals and clarifies the role of back-reaction and temperature in sustaining long-lived, driven-like coherent modes, offering potential experimental signatures and guiding future explorations of energy injections and scalar deformations.

Abstract

We consider hairy, magnetic black brane solutions to Einstein-Maxwell-Chern-Simons-Dilaton theory with full back-reaction of the scalar and gauge fields and compute the time evolution with time dependent sources. The dual field theory's 't Hooft anomaly leads to long-lived current oscillations in the low temperature limit long after the electric pulse which created these modes has been switched off. We find that, generally, non-linear effects increase the decay rates of long-lived modes by a temperature dependent amount. Nonetheless, our results suggest that the life-time of time-translation symmetry breaking states can be made arbitrarily large if the temperature after the quench is sufficiently small. Experimentally this might be realizable in Weyl semimetals.

Figures (5)

• Figure 1: The $x$ component of the current $J=J^2_x$ as a function of time $t\Lambda$, driven out of equilibrium by an electric quench, starting from a magnetic black brane with scalar hair, $B=2.52 \Lambda^2$, and initial temperature $T=0.549 \Lambda$. The scalar-potential-parameter $\sigma$ was set to $1$, such that the potential is bounded. The Chern-Simons coupling $\lambda$ is set to 3 here. The mass squared of the (bulk-) scalar field is $-3$, corresponding to a relevant deformation of the boundary CFT. We compare the numerically computed results including the full back-reaction of the scalar, the magnetic field and the current (blue solid curve) with the linearized approximation (red dotted curve). We linearize around the geometry at late times. From $t\Lambda =0$ to $t\Lambda \approx 2$ non-linear effects dominate, however, the quasi-normal modes determine the behavior of the current oscillations already from $t\Lambda \approx 2$ onward.
• Figure 2: On the left we show the temperature $T/\Lambda$ as a function of time. The parameters regarding the potential parameter $\sigma$, the Chern-Simons coupling $\lambda$, the initial temperature, the magnetic background field, the electric quench and the mass of the scalar field are the same as for the evolution in Fig. \ref{['fig:current_b=1.2']}. On the right we show the difference between the exact and linearized decay rates $\delta \gamma$.
• Figure 3: The linearized and exact decay rates of current fluctuations as a function of time $t\sqrt{B}$. The left side shows results for $\lambda=1.5$, while the right side is for $\lambda=2.5$. The green curve corresponds to the exact decay rates extracted from fitting the numerical results with back-reaction for the amplitude of the current oscillations to $e^{-\gamma t}$. The blue and orange line correspond to the linearized decay rates extracted from quasi-normal modes for the initial and final temperature (initial and final mass), respectively. The exact decay rates asymptote towards the linearized decay rates at late times, while this by itself is expected, it is surprising how well the linearized results approximate the exact results, given the size of the amplitude of the current oscillations (on the left, at, e.g., $t\sqrt{B}=300$, we find that the amplitude $J$ of the current oscillations $J^2_x$ divided by $8\pi G_5$ is $J\approx0.85 B^{3/2}$, on the right, at $t\sqrt{B}=300$, we find $J\approx0.074 B^{3/2}$).
• Figure 4: The late time temperature, in units of $\sqrt{B}$, after the electric pulse injected energy, as a function of the amplitude $E$ of the electric pulse in units of $B$. We show the results for various sizes of the Chern-Simons coupling $\lambda$. As the limit $\lambda \to \infty$ corresponds to the probe limit, in which the differential equation for $a^2_x$ and the field equations for the metric decouple, the temperature increase tends to zero for large $\lambda$. Note that the upper index in $a^2_x$ refers to the flavor index of the gauge field, not a squared term.
• Figure 5: On the left we show the normalized difference between the exact and linearized decay rates (blue curve) compared with the amplitude squared of the current oscillations, normalized by $J_{max}^2$, the maximum of the current's amplitude squared (red curve) after the electric pulse. Both curves agree well. On the right we show the relative difference between exact and linearized decay rates $(\gamma_{\text{lin}}-\gamma)/\gamma_{\text{lin}}$ compared with the amplitude of the current oscillations squared in units of the magnetic field $B$. As can be seen there, the relative difference in decay rates is of the same order Here $\lambda =1.5$ and the initial temperature is the same as in Fig. \ref{['fig:temp']}. In contrast to $\delta \gamma =\gamma_{\text{qnm}}-\gamma$, which we defined for Fig. \ref{['fig:temperatureAnddelta']}, we compare here the exact decay rate $\gamma(t)$ with the linearized decay rate at late times $\gamma_{\text{lin}}=\lim_{t\to \infty}\gamma_{\text{qnm}}$.