Quantum dynamics in sine-square deformed conformal field theory: Quench from uniform to non-uniform CFTs

In this work, motivated by the sine-square deformation (SSD) for (1+1)-dimensional quantum critical systems, we study the non-equilibrium quantum dynamics of a conformal field theory (CFT) with SSD, which was recently proposed to have continuous energy spectrum and continuous Virasoro algebra. In particular, we study the time evolution of entanglement entropy after a quantum quench from a uniform CFT, which is defined on a finite space of length $L$, to a sine-square deformed CFT. We find there is a crossover time $t^{\ast}$ that divides the entanglement evolution into two interesting regions. For $t\ll t^{\ast}$, the entanglement entropy does not evolve in time; for $t\gg t^{\ast}$, the entanglement entropy grows as $S_A(t)\simeq \frac{c}{3}\log t$, which is independent of the lengths of the subsystem and the total system. This $\log t$ growth with no revival indicates that a sine-square deformed CFT effectively has an infinite length, in agreement with previous studies based on the energy spectrum analysis. Furthermore, we study the quench dynamics for a CFT with M$\ddot{\text{o}}$bius deformation, which interpolates between a uniform CFT and a sine-square deformed CFT. The entanglement entropy oscillates in time with period $L_{\text{eff}}=L\cosh(2\theta)$, with $\theta=0$ corresponding to the uniform case and $\theta\to \infty$ corresponding to the SSD limit. Our field theory calculation is confirmed by a numerical study on a (1+1)-d critical fermion chain.Comment: are welcome; 10 pages, 4 figures; v2: refs added; v3: refs added; A physical interpretation of t* is added; v4: published version (selected as Editors' Suggestion

Phase transitions, geometrothermodynamics and critical exponents of black holes with conformal anomaly

We investigate the phase transitions of black holes with conformal anomaly in canonical ensemble from different perspectives. Some interesting and novel phase transition phenomena have been discovered. Firstly, we discuss the behavior of the specific heat and the inverse of the isothermal compressibility. It is shown that there are striking differences in Hawking temperature and phase structure between black holes with conformal anomaly and those without it. In the case with conformal anomaly, there exists local minimum temperature corresponding to the phase transition point. Phase transitions take place not only from an unstable large black hole to a locally stable medium black hole but also from an unstable medium black hole to a locally stable small black hole. Secondly, we probe in details the dependence of phase transitions on the choice of parameters. The results show that black holes with conformal anomaly have much richer phase structure than those without it. There would be two, only one or no phase transition points depending on the parameters we have chosen. The corresponding parameter region are derived both numerically and graphically. Thirdly, geometrothermodynamics are built up to examine the phase structure we have discovered. It is shown that Legendre invariant thermodynamic scalar curvature diverges exactly where the specific heat diverges. Furthermore, critical behaviors are investigated by calculating the relevant critical exponents. It is proved that these critical exponents satisfy the thermodynamic scaling laws, leading to the conclusion that critical exponents and the scaling laws can reserve even when we consider conformal anomaly.Comment: some new references adde

Non-extended phase space thermodynamics of Lovelock AdS black holes in grand canonical ensemble

Recently, extended phase space thermodynamics of Lovelock AdS black holes has been of great interest. To provide insight from a different perspective and gain a unified phase transition picture, non-extended phase space thermodynamics of $(n+1)$-dimensional charged topological Lovelock AdS black holes is investigated detailedly in the grand canonical ensemble. Specifically, the specific heat at constant electric potential is calculated and phase transition in the grand canonical ensemble is discussed. To probe the impact of the various parameters, we utilize the control variate method and solve the phase transition condition equation numerically for the case $k=1,-1$. There are two critical points for the case $n=6,k=1$ while there is only one for other cases. For $k=0$, there exists no phase transition point. To figure out the nature of phase transition in the grand canonical ensemble, we carry out an analytic check of the analog form of Ehrenfest equations proposed by Banerjee et al. It is shown that Lovelock AdS black holes in the grand canonical ensemble undergo a second order phase transition. To examine the phase structure in the grand canonical ensemble, we utilize the thermodynamic geometry method and calculate both the Weinhold metric and Ruppeiner metric. It is shown that for both analytic and graphical results that the divergence structure of the Ruppeiner scalar curvature coincides with that of the specific heat. Our research provides one more example that Ruppeiner metric serves as a wonderful tool to probe the phase structures of black holes

P-V Criticality of Topological Black Holes in Lovelock-Born-Infeld Gravity

To understand the effect of third order Lovelock gravity, $P-V$ criticality of topological AdS black holes in Lovelock-Born-Infeld gravity is investigated. The thermodynamics is further explored with some more extensions and details than the former literature. A detailed analysis of the limit case $\beta\rightarrow\infty$ is performed for the seven-dimensional black holes. It is shown that for the spherical topology, $P-V$ criticality exists for both the uncharged and charged cases. Our results demonstrate again that the charge is not the indispensable condition of $P-V$ criticality. It may be attributed to the effect of higher derivative terms of curvature because similar phenomenon was also found for Gauss-Bonnet black holes. For $k=0$, there would be no $P-V$ criticality. Interesting findings occur in the case $k=-1$, in which positive solutions of critical points are found for both the uncharged and charged cases. However, the $P-v$ diagram is quite strange. To check whether these findings are physical, we give the analysis on the non-negative definiteness condition of entropy. It is shown that for any nontrivial value of $\alpha$, the entropy is always positive for any specific volume $v$. Since no $P-V$ criticality exists for $k=-1$ in Einstein gravity and Gauss-Bonnet gravity, we can relate our findings with the peculiar property of third order Lovelock gravity. The entropy in third order Lovelock gravity consists of extra terms which is absent in the Gauss-Bonnet black holes, which makes the critical points satisfy the constraint of non-negative definiteness condition of entropy. We also check the Gibbs free energy graph and the "swallow tail" behavior can be observed. Moreover, the effect of nonlinear electrodynamics is also included in our research.Comment: 13 pages, 7 figure