Holographic study of shear viscosity and butterfly velocity for magnetic field-driven quantum criticality

Abstract

We investigate the shear viscosity and butterfly velocity of a magnetic field-induced quantum phase transition in five dimensional Einstein-Maxwell-Chern-Simons theory, which is holographically dual to a class of strongly coupled quantum field theories with chiral anomalies. Our analysis reveals that the ratio of longitudinal shear viscosity to entropy density $η_\parallel/s$ exhibits a pronounced non-monotonic dependence on temperature $T$ when the magnetic field $B$ is slightly below the critical value $B_c$ of the quantum phase transition. In particular, it can develop a distinct minimum at an intermediate temperature. This contrasts sharply with the monotonic temperature scaling observed at and above $B_c$, where $η_\parallel/s$ follows the scaling $T^{2/3}$ at $B=B_c$ and transitions to $T^2$ for $B>B_c$ as $T\to0$. The non-vanishing of $η_\parallel/s$ for $B<B_c$ in the zero temperature limit suggests that it could serve as a good order parameter of the quantum phase transition. We also find that all butterfly velocities change dramatically near the quantum phase transition, and thus their derivatives with respect to $B$ can be independently used to detect the quantum critical point.

Figures (8)

• Figure 1: The entropy density $s$ as a function of temperature $T$ in different phases. There is a QPT at the critical magnetic field at $B_c\approx 0.332$. We consider $k=2/\sqrt{3}$ and fix the chemical potential $\mu=1$.
• Figure 2: The ratio $\eta_{||}/s$ as a function of magnetic field at different temperatures. The blue point marks the maximum of each curve. The dashed horizontal line represents the KSS bound, while the dashed vertical line marks the location of QCP. We choose $k=2/\sqrt{3}$ and $\mu=1$.
• Figure 3: Left panel: The maximum $(B_0,(\eta_\parallel/s)_0)$ (blue points in Figure \ref{['fig:etaB']}) as a functions of temperature. Right panel: The ratio $\eta_{||}/s$ as a function temperature for different $B>B_c$. The plots are generated for $k=2/\sqrt{3}$ and $\mu=1$.
• Figure 4: Left panel: The ratio $\eta_{||}/s$ as a function of temperature for selected values of $B$. Right panel: $\eta_\parallel/s$ versus temperature at $B=B_c$ (top-right) and $B=0.33$ (bottom-right). The dashed horizontal line represents the KSS value $\eta/s=\frac{1}{4\pi}$. The plots are generated for $k=2/\sqrt{3}$ and $\mu=1$.
• Figure 5: Temperature dependence of $\eta_\parallel/s$ at $B_c$. The red solid curve is obtained by fitting low temperature data (denoted by blue dots). We have set $k=2/\sqrt{3}$ and $\mu=1$.