Time Evolution after Double Trace Deformation

In this paper, we consider double trace deformation to single CFT${}_2$, and study time evolution after the deformation. The double trace deformation we consider is nonlocal: composed of two local operators placed at separate points. We study two types of local operators: one is usual local operator in CFT, and the other is HKLL bulk local operator, which is still operator in CFT but has properties as bulk local operator. We compute null energy and averaged null energy in the bulk in both types of deformations. We confirmed that, with the suitable choice of couplings, averaged null energies are negative. This implies causal structure is modified in the bulk, from classical background. We then calculate time evolution of entanglement entropy and entanglement Renyi entropy after double trace deformation. We find both quantities are found to show peculiar shockwave-like time evolution.Comment: 16 pages, 12 figures, references added, typos correcte

Butterflies from Information Metric

We study time evolution of distance between thermal states excited by local operators, with different external couplings. We find that growth of the distance implies growth of commutators of operators, signifying the local excitations are scrambled. We confirm this growth of distance by holographic computation, by evaluating volume of codimension 1 extremal volume surface. We find that the distance increases exponentially as $e^{\frac{2\pi t}{\beta}}$. Our result implies that, in chaotic system, trajectories of excited thermal states exhibit high sensitivity to perturbation to the Hamiltonian, and the distance between them will be significant at the scrambling time. We also confirm the decay of two point function of holographic Wilson loops on thermofield double state.Comment: 10 pages, 3 figures, reference added, minor modification

Chiral phase transition in the linear sigma model within Hartree factorization in the Tsallis nonextensive statistics

We studied chiral phase transition in the linear sigma model within the Tsallis nonextensive statistics in the case of small deviation from the Boltzmann-Gibbs (BG) statistics. The statistics has two parameters: the temperature $T$ and the entropic parameter $q$. The normalized $q$-expectation value and the physical temperature \Tph were employed in this study. The normalized $q$-expectation value was expanded as a series of the value $(1-q)$, where the absolute value $|1-q|$ is the measure of the deviation from the BG statistics. We applied the Hartree factorization and the free particle approximation, and obtained the equations for the condensate, the sigma mass, and the pion mass. The physical temperature dependences of these quantities were obtained numerically. We found following facts. The condensate at $q$ is smaller than that at $q'$ for $q>q'$. The sigma mass at $q$ is lighter than that at $q'$ for $q>q'$ at low physical temperature, and the sigma mass at $q$ is heavier than that at $q'$ for $q>q'$ at high physical temperature. The pion mass at $q$ is heavier than that at $q'$ for $q>q'$. The difference between the pion masses at different values of $q$ is small for \Tph \le 200 MeV. That is to say, the condensate and the sigma mass are affected by the Tsallis nonextensive statistics of small $|1-q|$, and the pion mass is also affected by the statistics of small $|1-q|$ except for \Tph \le 200 MeV.Comment: 9 pages, 6 figure