A New Phase Transition Related to the Black Hole's Topological Charge

The topological charge $\epsilon$ of AdS black hole is introduced in Ref.[1,2], where a complete thermodynamic first law is obtained. In this paper, we investigate a new phase transition related to the topological charge in Einstein-Maxwell theory. Firstly, we derive the explicit solutions corresponding to the divergence of specific heat $C_{\epsilon}$ and determine the phase transition critical point. Secondly, the $T-r$ curve and $T-S$ curve are investigated and they exhibit an interesting van der Waals system's behavior. Critical physical quantities are also obtained which are consistent with those derived from the specific heat analysis. Thirdly, a van der Waals system's swallow tail behavior is observed when $\epsilon>\epsilon_{c}$ in the $F-T$ graph. What's more, the analytic phase transition coexistence lines are obtained by using the Maxwell equal area law and free energy analysis, the results of which are consistent with each other.Comment: 11 pages, 5 figure

Attractive Interaction between Vortex and Anti-vortex in Holographic Superfluid

Annihilation process of a pair of vortices in holographic superfluid is numerically simulated. The process is found to consist of two stages which are amazingly separated by vortex size $2r$. The separation distance $\delta(t)$ between vortex and anti-vortex as a function of time is well fitted by $\alpha (t_{0}-t)^{n}$, where the scaling exponent $n=1/2$ for $\delta (t)>2r$, and $n=2/5$ for $\delta(t)<2r$. Then the approaching velocity and acceleration as functions of time and as functions of separation distance are obtained. Thus the attractive force between vortex and anti-vortex is derived as $f(\delta)\propto 1/\delta^{3}$ for the first stage, and $f(\delta)\propto 1/\delta^{4}$ for the second stage. In the end, we explained why the annihilation rate of vortices in turbulent superfluid system obeys the two-body decay law when the vortex density is low.Comment: 14 pages, 5 figure

Revisiting van der Waals like behavior of f(R) AdS black holes via the two point correlation function

Van der Waals like behavior of $f(R)$ AdS black holes is revisited via two point correlation function, which is dual to the geodesic length in the bulk. The equation of motion constrained by the boundary condition is solved numerically and both the effect of boundary region size and $f(R)$ gravity are probed. Moreover, an analogous specific heat related to $\delta L$ is introduced. It is shown that the $T-\delta L$ graphs of $f(R)$ AdS black holes exhibit reverse van der Waals like behavior just as the $T-S$ graphs do. Free energy analysis is carried out to determine the first order phase transition temperature $T_*$ and the unstable branch in $T-\delta L$ curve is removed by a bar $T=T_*$. It is shown that the first order phase transition temperature is the same at least to the order of $10^{-10}$ for different choices of the parameter $b$ although the values of free energy vary with $b$. Our result further supports the former finding that charged $f(R)$ AdS black holes behave much like RN-AdS black holes. We also check the analogous equal area law numerically and find that the relative errors for both the cases $\theta_0=0.1$ and $\theta_0=0.2$ are small enough. The fitting functions between $\log\mid T -T_c\mid$ and $\log\mid\delta L-\delta L_c\mid$ for both cases are also obtained. It is shown that the slope is around 3, implying that the critical exponent is about $2/3$. This result is in accordance with those in former literatures of specific heat related to the thermal entropy or entanglement entropy.Comment: Revised version. Match the published version. 14pages,5figure