Fundamental Plane of Black Hole Activity in Quiescent Regime

A correlation among the radio luminosity ($L_{\rm R}$), X-ray luminosity ($L_{\rm X}$), and black hole mass ($M_{\rm BH}$) in active galactic nuclei (AGNs) and black hole binaries is known to exist and is called the "Fundamental Plane" of black hole activity. Yuan & Cui (2005) predicts that the radio/X-ray correlation index, $\xi_{\rm X}$, changes from $\xi_{\rm X}\approx 0.6$ to $\xi_{\rm X}\approx 1.2-1.3$ when $L_{\rm X}/L_{\rm Edd}$ decreases below a critical value $\sim 10^{-6}$. While many works favor such a change, there are also several works claiming the opposite. In this paper, we gather from literature a largest quiescent AGN (defined as $L_{\rm X}/L_{\rm Edd} < 10^{-6}$) sample to date, consisting of $75$ sources. We find that these quiescent AGNs follow a $\xi_{\rm X}\approx 1.23$ radio/X-ray relationship, in excellent agreement with the Yuan \& Cui prediction. The reason for the discrepancy between the present result and some previous works is that their samples contain not only quiescent sources but also "normal" ones (i.e., $L_{\rm X}/L_{\rm Edd} > 10^{-6}$). In this case, the quiescent sources will mix up with those normal ones in $L_{\rm R}$ and $L_{\rm X}$. The value of $\xi_{\rm X}$ will then be between $0.6$ and $\sim1.3$, with the exact value being determined by the sample composition, i.e., the fraction of the quiescent and normal sources. Based on this result, we propose that a more physical way to study the Fundamental Plane is to replace $L_{\rm R}$ and $L_{\rm X}$ with $L_{\rm R}/L_{\rm Edd}$ and $L_{\rm X}/L_{\rm Edd}$, respectively.Comment: 11 pages, 7 figures, accepted for publication in The Astrophysical Journa

The tensor renormalization group study of the general spin-S Blume-Capel model

We focus on the special situation of $D=2J$ of the general spin-S Blume-Capel model on the square lattice. Under the infinitesimal external magnetic field, the phase transition behaviors due to the thermal fluctuations are discussed by the newly developed tensor renormalization group method. For the case of the integer spin-S, the system will undergo $S$ first-order phase transitions with the successive symmetry breaking with the magnetization $M=S,S-1,...0$. For the half-integer spin-S, there are similar $S-1/2$ first order phase transition with $M=S,S-1,...1/2$ stepwise structure, in addition, there is a continuous phase transition due to the spin-flip $Z_2$ symmetry breaking. In the low temperature regions, all first-order phase transitions are accompanied by the successive disappearance of the optional spin-component pairs($s,-s$), furthermore, the critical temperature for the nth first-order phase transition is the same, independent of the value of the spin-S. In the absence of the magnetic field, the visualization parameter characterizing the intrinsic degeneracy of the different phases clearly demonstrates the phase transition process.Comment: 6 pages, 7 figure

Quantum Synchronizable Codes From Quadratic Residue Codes and Their Supercodes

Quantum synchronizable codes are quantum error-correcting codes designed to correct the effects of both quantum noise and block synchronization errors. While it is known that quantum synchronizable codes can be constructed from cyclic codes that satisfy special properties, only a few classes of cyclic codes have been proved to give promising quantum synchronizable codes. In this paper, using quadratic residue codes and their supercodes, we give a simple construction for quantum synchronizable codes whose synchronization capabilities attain the upper bound. The method is applicable to cyclic codes of prime length