Chern-Simons diffusion rate across different phase transitions

We investigate how the dimensionless ratio given by the Chern-Simons diffusion rate ${\mathrm{\Gamma }}_{\mathrm{CS}}$ divided by the product of the entropy density $s$ and temperature $T$ behaves across different kinds of phase transitions in the class of bottom-up nonconformal Einstein-dilaton holographic models originally proposed by Gubser and Nellore. By tuning the dilaton potential, one is able to holographically mimic a first order, a second order, or a crossover transition. In a first order phase transition, ${\mathrm{\Gamma }}_{\mathrm{CS}}/sT$ jumps at the critical temperature (as previously found in the holographic literature), while in a second order phase transition it develops an infinite slope. On the other hand, in a crossover, ${\mathrm{\Gamma }}_{\mathrm{CS}}/sT$ behaves smoothly, although displaying a fast variation around the pseudo-critical temperature. In all the cases, ${\mathrm{\Gamma }}_{\mathrm{CS}}/sT$ increases with decreasing $T$. The behavior of the Chern-Simons diffusion rate across different phase transitions is expected to play a relevant role for the chiral magnetic effect around the QCD critical end point, which is a second order phase transition point connecting a crossover band to a line of first order phase transition. Our findings in the present work add to the literature the first predictions for the Chern-Simons diffusion rate across second order and crossover transitions in strongly coupled nonconformal, non-Abelian gauge theories.

Figure 1

Thermodynamics of the Einstein-dilaton model with a first order phase transition. (a) Black hole temperature normalized by the critical temperature $T_c$ as a function of the radial position of the black hole horizon ${\varphi }_H$. The dashed line lies at 1, while the minimum temperature for the existence of black hole solutions at the bottom of the curve is $T_{\min }\approx 0.9T_c$. In the inset, we show the range of ${\varphi }_H$ for which the black hole solutions are unstable, displaying negative pressure (within this region, the thermal gas solution corresponds to the dominant saddle point of the model). (b) Speed of sound squared. (c) From top to bottom at high $T$: ${\kappa }^{2}s/T^3$ (solid black curve), ${\kappa }^2\epsilon /T^4$ (dot-dashed red curve), ${\kappa }^{2}p/T^4$ (dashed blue curve), and ${\kappa }^{2}I/T^4$ (long-dashed green curve).

Figure 2

Ratio of the Chern-Simons diffusion rate divided by the product $sT$ normalized by its value at large $T$ in the Einstein-dilaton model with a first order phase transition for different axion-dilaton coupling functions.

Figure 3

Thermodynamics of the Einstein-dilaton model with a second order phase transition. (a) Black hole temperature normalized by the critical temperature $T_c$ as a function of the radial position of the black hole horizon ${\varphi }_H$. At the critical temperature the curve has a stationary point of inflection. (b) Speed of sound squared: it is never negative, but it vanishes at $T=T_c$. (c) From top to bottom at high $T$: ${\kappa }^{2}s/T^3$ (solid black curve), ${\kappa }^2\epsilon /T^4$ (dot-dashed red curve), ${\kappa }^{2}p/T^4$ (dashed blue curve), and ${\kappa }^{2}I/T^4$ (long-dashed green curve).

Figure 4

Ratio of the Chern-Simons diffusion rate divided by the product $sT$ normalized by its value at large $T$ in the Einstein-dilaton model with a second order phase transition for different axion-dilaton coupling functions.

Figure 5

Thermodynamics of the Einstein-dilaton model with a crossover. (a) Black hole temperature normalized by the critical temperature $T_c$ as a function of the radial position of the black hole horizon ${\varphi }_H$. (b) Speed of sound squared. (c) From top to bottom at high $T$: ${\kappa }^{2}s/T^3$ (solid black curve), ${\kappa }^2\epsilon /T^4$ (dot-dashed red curve), ${\kappa }^{2}p/T^4$ (dashed blue curve), and ${\kappa }^{2}I/T^4$ (long-dashed green curve).

Figure 6

Ratio of the Chern-Simons diffusion rate divided by the product $sT$ normalized by its value at large $T$ in the Einstein-dilaton model with a crossover for different axion-dilaton coupling functions.