Thermodynamics, phase transitions and Ruppeiner geometry for   Einstein-dilaton Lifshitz black holes in the presence of Maxwell and   Born-Infeld electrodynamics
  In this paper, we first obtain the ($n+1$)-dimensional dilaton-Lifshitz black hole (BH) solutions in the presence of Born-Infeld (BI) electrodynamics. We find that there are two different solutions for $z=n+1$ and $z\neq n+1$ cases ($z$ is dynamical critical exponent). We show that the thermodynamics first law is satisfied for both cases. Then, we turn to study different phase transitions (PTs) for our BHs. We start with study of Hawking-Page PT for both linearly and BI charged BHs. After that, we discuss the PTs inside the BHs. We present the improved Davies quantities and prove that the PT points shown by them coincide with Ruppeiner ones. We show that the zero temperature PTs are transitions on radiance properties of BHs by using Landau-Lifshitz theory. Next, we turn to study Ruppeiner geometry of linearly and BI charged BHs. For linearly charged case, we show that there are no PT at finite temperature for the case $z\geq 2$. For $z<2$, it is found that the number of finite temperature PT points depends on the value of BH charge and is not more than two. When we have two finite temperature PT points, there are no thermally stable BH between these two points and we have discontinues small/large BH PTs. As expected, for small BHs, we observe finite magnitude for Ruppeiner invariant which shows the finite correlation between possible BH molecules while for large BHs, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz BHs for different values of $z$. We observe that small BHs are thermally unstable in some situations. Also, the behavior of correlation between possible BH molecules for large BHs is the same as linearly charged case. In both linearly and BI charged cases, for some choices of parameters, the BH systems behave like a Van der Waals gas near transition point. 
M. Kord Zangeneh, A. Dehyadegari, M. R. Mehdizadeh, B. Wang and A.   Sheykhi
hep-th