Nonequilibrium steady states in driven holographic Weyl semi-metals

Abstract

Three-dimensional Weyl materials provide a controlled setting for exploring Floquet dynamics in open quantum systems, including nonequilibrium steady states (NESS). Motivated by the desire for a strongly-coupled description, we employ holography to analyze the formation and stability of a NESS in a Weyl semi-metal induced by an external circularly polarized electric field. A time-periodic steady-state solution is constructed and its stability is determined from the spectrum of out-of-equilibrium quasinormal modes (Floquet exponents). A stable region in the drive parameter space is identified; beyond a critical curve, the Floquet exponents enter the upper half of the complex plane, leading to a superharmonic response. At sufficiently strong driving, chaotic time evolution emerges in the fully nonlinear initial-boundary value problem. The anomaly-induced response of the NESS to an external magnetic field is also computed, and the resulting behavior is related to the previously proposed chiral pumping effect.

Figures (5)

• Figure 1: The response of $A_3(z=1)$ to the driving parameters $(P,\omega_\textrm{D})$. Top row ($\alpha=1$, $b=0$): Monotonic behavior. Bottom row ($\alpha=10$, $b=1$) Non-monotonic behavior with level crossings. In contrast to free fermions ebihara2016chiral, we do not make assumptions on the largeness of $\omega_\textrm{D}$.
• Figure 2: Left: Phase diagram showing the stability/instability transition of the NESS with $B=0$. The beige region indicates that the dominant Floquet exponent $\omega_{\max}$ has entered the upper half of the complex plane. Right: Cross-sections showing $\operatorname{Im}(\omega_{\max})$ as a function of driving frequency $\omega_\textrm{D}$ for fixed amplitude $P$.
• Figure 3: From left to right: Time evolution of the vector response corresponding to the points labeled from bottom to top in Fig. \ref{['fig:phase']}. The source is turned on at time $T=10$. The source $S_1$ is labeled in blue and the vector response $\langle J_V^1\rangle_\textrm{ren}$ is labeled in red. The vector response is evidently commensurate with the source.
• Figure 4: From left to right: Time evolution of the axial response corresponding to the points labeled from bottom to top in Fig. \ref{['fig:phase']}. The source is turned on at time $T=10$. The horizon-value of the gauge field $A_3(z=1)$ is labeled in blue and the axial current $\langle J_A^3\rangle_\textrm{ren}$ is labeled in red. Despite not corresponding to a quantum observable, $A_3(z=1)$ provides an effective probe of the nonequilibrium stability of the NESS. The overlaid green dots indicate the temporal position of the nodes of the vector source $S_1$. Unlike the vector response, the axial response is clearly superharmonic.
• Figure 5: Demonstration of the chiral pumping effect at vanishing chemical potential ($\mu=0$). The response of the charge density and axial current to an external magnetic field $\vec{B}=B \hat{e}_3$ is shown.