The fragmentation of spin-orbit coupled spin-1 Bose gas with a weak interaction in external harmonic trap is explored by both exact diagonalization and mean-field theory. This fragmentation tendency, which originates from the total angular momentum c
onservation, is affected obviously by the spin-orbit coupling strength and the spin-dependent interaction. Strong spin-orbit interaction raises the inverse participation ratio, which describes the number of significantly occupied single-particle states. As the spin-dependent interaction changes from anti-ferromagnetic to ferromagnetic, the peak values in the inverse participation ratio become lower. Without the confinement of the appointed total angular momentum, the condensate chooses a zero or finite total angular momentum ground state, which is determined by both the interaction and the spin-orbit coupling strength.
We analytically and numerically investigate the ground state of the spin-orbit coupled spin-1 Bose-Einstein condensates in an external parabolic potential. When the spin-orbit coupling strength $kappa$ is comparable with that of the trapping potentia
l, the density distribution centers of different components of the spinor condensate deviate evidently from the trap center in the plane wave and stripe phases. When $kappagg1$, the magnitude of this deviation decreases as $kappa$ is getting larger and larger. Correspondingly, periphery half-skyrmions textures arise. This deviation can be reflected by the non-uniform magnetic moment in the $z$ direction, $mathcal{F}_z$. With the manipulation of the external trap, the local magnitude of $mathcal{F}_z$ can be increased evidently. This kind of increase of $mathcal{F}_z$ is also observed in the square vortex lattice phase of the condensate.
We investigate the fractionalized Skyrmion excitations induced by spin-orbit coupling in rotating and rapidly quenched spin-1 Bose-Einstein condensates. Our results show that the fractionalized Skyrmion excitation depends on the combination of spin-o
rbit coupling and rotation, and it originates from a dipole structure of spin which is always embedded in three vortices constructed by each condensate component respectively. When spin-orbit coupling is larger than a critical value, the fractionalized Skyrmions encircle the center with one or several circles to form a radial lattice, which occurs even in the strong ferromagnetic/antiferromagnetic condensates. We can use both the spin-orbit coupling and the rotation to adjust the radial lattice. The realization and the detection of the fractionalized Skyrmions are compatible with current experimental technology.
Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by J=GM^2) is considered. It is shown that consistent boundary conditions exist, for which the asymptotic symmetry generato
rs form one copy of the Virasoro algebra with central charge c_L=12J / hbar. This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature T_L=1/2pi and conjugate energy given by the zero mode generator, L_0, of the Virasoro algebra. Assuming unitarity, the Cardy formula then gives a microscopic entropy S_{micro}=2pi J / hbar for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy S_{macro}=Area / 4hbar G. The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge c_L=12J / hbar, and in particular that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with c_L sim 2 times 10^{79}.
Three dimensional Einstein gravity with negative cosmological constant -1/ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for g
eneric mu due to negative-energy massive gravitons. However at the critical value muell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3ell /G).
Three dimensional topologically massive gravity (TMG) with a negative cosmological constant -ell^{-2} and positive Newton constant G admits an AdS_3 vacuum solution for any value of the graviton mass mu. These are all known to be perturbatively unsta
ble except at the recently explored chiral point muell=1. However we show herein that for every value of muell< 3 there are two other (potentially stable) vacuum solutions given by SL(2,R)x U(1)-invariant warped AdS_3 geometries, with a timelike or spacelike U(1) isometry. Critical behavior occurs at muell=3, where the warping transitions from a stretching to a squashing, and there are a pair of warped solutions with a null U(1) isometry. For muell>3, there are known warped black hole solutions which are asymptotic to warped AdS_3. We show that these black holes are discrete quotients of warped AdS_3 just as BTZ black holes are discrete quotients of ordinary AdS_3. Moreover new solutions of this type, relevant to any theory with warped AdS_3 solutions, are exhibited. Finally we note that the black hole thermodynamics is consistent with the hypothesis that, for muell>3, the warped AdS_3 ground state of TMG is holographically dual to a 2D boundary CFT with central charges c_R={15(muell)^2+81over Gmu((muell)^2+27)} and c_L={12 muell^2over G((muell)^2+27)}.
In a recent paper (arXiv: 0801.4566) it was shown that all global energy eigenstates of asymptotically $AdS_3$ chiral gravity have non-negative energy at the linearized level. This result was questioned (arXiv: 0803.3998) by Carlip, Deser, Waldron an
d Wise (CDWW), who work on the Poincare patch. They exhibit a linearized solution of chiral gravity and claim that it has negative energy and is smooth at the boundary. We show that the solution of CDWW is smooth only on that part of the boundary of $AdS_3$ included in the Poincare patch. Extended to global $AdS_3$, it is divergent at the boundary point not included in the Poincare patch. Hence it is consistent with the results of (arXiv: 0801.4566).
