Odd Viscosity in the Quantum Critical Region of a Holographic Weyl Semimetal

We study odd viscosity in a holographic model of a Weyl semimetal. The model is characterized by a quantum phase transition from a topological semimetal to a trivial semimetal state. Since the model is axisymmetric in three spatial dimensions there are two independent odd viscosities. Both odd viscosity coefficients are nonvanishing in the quantum critical region and nonzero only due to the mixed axial gravitational anomaly. It is therefore a novel example in which the mixed axial gravitational anomaly gives rise to a transport coefficient at first order in derivatives at finite temperature. In the quantum critical region, the physics of viscosities as well as conductivities is governed by the quantum critical point.

Figure 1

The schematic picture for the holographic Weyl semimetal model (1) in the coupling constant ($M/b$)–temperature ($T$) plane. At zero temperature a topological quantum phase transition occurs at the critical value $(M/b{)}_c$. At finite temperature the dashed line is a smooth crossover and in the quantum critical regime the physics is governed by the quantum critical behavior.

Figure 2

Odd viscosity ${\eta }_{H_{\parallel }}$ as a function of $M/b$ at different temperatures.

Figure 3

Odd viscosity ${\eta }_{H_{\perp }}$ as a function of $M/b$ at different low temperatures.

Figure 4

The temperature scaling exponents ${\gamma }_i$ with $i\in \{1,\dots ,6\}$ for viscosities ${\eta }_{\parallel }$, ${\eta }_{H_{\parallel }}$, and ${\eta }_{H_{\perp }}$ and for electric conductivities ${\sigma }_{\parallel }$, ${\sigma }_{\perp }$, and ${\sigma }_{\mathrm{AHE}}$ at the critical value $M/b=0.744$ for low temperatures. The dashed lines in the picture denote the analytic values of the scaling exponents from the scaling analysis.