Split degenerate states and stable p+ip phases from holography

In this paper, we investigate the p+$i$p superfluid phases in the complex vector field holographic p-wave model. We find that in the probe limit, the p+$i$p phase and the p-wave phase are equally stable, hence the p and $i$p orders can be mixed with an arbitrary ratio to form more general p+$\lambda i$p phases, which are also equally stable with the p-wave and p+$i$p phases. As a result, the system possesses a degenerate thermal state in the superfluid region. We further study the case with considering the back reaction on the metric, and find that the degenerate ground states will be separated into p-wave and p+$i$p phases, and the p-wave phase is more stable. Finally, due to the different critical temperature of the zeroth order phase transitions from p-wave and p+$i$p phases to the normal phase, there is a temperature region where the p+$i$p phase exists but the p-wave phase doesn't. In this region we find the stable p+$i$p phase for the first time.Comment: 16 pages, 5 figures; typos correcte

Characterization of four-qubit states via Bell inequalities

A set of Bell inequalities classifying the quantum entanglement of four-qubit states is presented. These inequalities involve only two measurement settings per observer and can characterize fully separable, bi-separable and tri-separable quantum states. In addition, a quadratic inequality of the Bell operators for four-qubit systems is derived

{μ-6,6′-Dimeth­oxy-2,2′-[ethane-1,2-diylbis(nitrilo­methyl­idyne)]diphenolato-1κ4 O 6,O 1,O 1′,O 6′:2κ4 O 1,N,N′,O 1′}(methanol-1κO)(perchlorato-1κO)copper(II)sodium(I)

The mol­ecule of the title compound, [CuNa(C18H18N2O4)(ClO4)(CH3OH)], is almost planar with mean deviation of 5.2 (8)°. The coordination environment of the CuII ion is distorted square-planar and it is N2O2-chelated by the 6,6′-dimeth­oxy-2,2′-[ethane-1,2-diylbis(nitrilo­methyl­idyne)]diphenolate Schiff base ligand. The Na atom is chelated by the four O atoms of the Schiff base ligand, and coordinated by a methanol mol­ecule and a perchlorate anion. The six-coordinate Na atom adopts a distorted octa­hedral coordination geometry. The O atoms of the perchlorate anion are disordered over two sites with site-occupancy factors of 0.697 (5) and 0.303 (5). O—H⋯O hydrogen bonding occurs

Thermodynamics of phase transition in higher dimensional Reissner-Nordstr\"{o}m-de Sitter black hole

It is well known that there are black hole and the cosmological horizons for the Reissner-Nordstr\"{o}m-de Sitter spacetime. Although the thermodynamic quantities on the horizons are not irrelevant, they satisfy the laws of black hole thermodynamics respectively. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in $(n+2)$-dimensional Reissner-Nordstr\"{o}m-de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed, moreover, the critical temperature, critical pressure and critical volume are obtained. We carry out an analytical check of Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the spacetime undergoes a second order phase transition at the critical point. This result is consistent with the nature of liquid--gas phase transition at the critical point, hence deepening the understanding of the analogy of charged dS spacetime and liquid--gas systems.Comment: 18 pages, 7 figures, 1 tabl

Phase transition of the higher dimensional charged Gauss-Bonnet black hole in de Sitter spacetime

We study the phase transition of charged Gauss-Bonnet-de Sitter (GB-dS) black hole. For black holes in de Sitter spacetime, there is not only black hole horizon, but also the cosmological horizon. The thermodynamic quantities on the both horizons satisfy the first law of the black hole thermodynamics, respectively; moreover, there are additional connections between them. Using the effective temperature approach, we obtained the effective thermodynamic quantities of charged GB-dS black hole. According to Ehrenfest classification, we calculate some response functions and plot their figures, from which one can see that the spacetime undergoes a second-order phase transition at the critical point. It is shown that the critical values of effective temperature and pressure decrease with the increase of the value of GB parameter $\alpha$.Comment: 9 pages, 16 figure

Clapeyron equation and phase equilibrium properties in higher dimensional charged topological dilaton AdS black holes with a nonlinear source

Using Maxwell's equal area law, we discuss the phase transition of higher dimensional charged topological dilaton AdS black holes with a nonlinear source. The coexisting region of the two phases is found and we depict the coexistence region in $P-v$ diagrams. The two-phase equilibrium curves in $P-T$ diagrams are plotted, and we take the first order approximation of volume $v$ in the calculation. To better compare with a general thermodynamic system, the Clapeyron equation is derived for higher dimensional charged topological black hole with a nonlinear source. The latent heat of isothermal phase transition is investigated. We also study the effect of the parameters of the black hole on the region of two-phases coexistence. The results show that the black hole may go through a small-large phase transition similar to those of usual non-gravity thermodynamic systems.Comment: 21 pages,25 figures. published version in EPJC. arXiv admin note: substantial text overlap with arXiv:1411.7202; text overlap with arXiv:1506.01786, arXiv:hep-th/0605042 by other author