Abstract:
In the chiral title compound, C38H28O4P2, the intramolecular P...P separation is 3.671 (2) and the dihedral angle between the two benzene rings in the biphenyl unit is 77.9 (2)°.

Abstract:
The title potential antibacterial compound, C14H12N2O4·H2O, is a Schiff base which has an intramolecular O—H...N hydrogen bond and crystallizes with one uncoordinated water molecule, which links three symmetry-related molecules through two O—H...O and one N—H...O hydrogen bond. In the crystal structure, further intermolecular O—H...O hydrogen bonds link symmetry-related molecules, forming layers parallel to the bc plane.

Abstract:
We construct in the K matrix formalism concrete examples of symmetry enriched topological phases, namely intrinsically topological phases with global symmetries. We focus on the Abelian and non-chiral topological phases and demonstrate by our examples how the interplay between the global symmetry and the fusion algebra of the anyons of a topologically ordered system determines the existence of gapless edge modes protected by the symmetry and that a (quasi)-group structure can be defined among these phases. Our examples include phases that display charge fractionalization and more exotic non-local anyon exchange under global symmetry that correspond to general group extensions of the global symmetry group.

Abstract:
Casini et al raise the issue that the entanglement entropy in gauge theories is ambiguous because its definition depends on the choice of the boundary between two regions.; even a small change in the boundary could annihilate the otherwise finite topological entanglement entropy between two regions. In this article, we first show that the topological entanglement entropy in the Kitaev model which is not a true gauge theory, is free of ambiguity. Then, we give a physical interpretation, from the perspectives of what can be measured in an experiement, to the purported ambiguity of true gauge theories, where the topological entanglement arises as redundancy in counting the degrees of freedom along the boundary separating two regions. We generalize these discussions to non-Abelian gauge theories.

Abstract:
In this paper we initiate the study of holographic quantum liquids in 1+1 dimensions. Since the Landau Fermi liquid theory breaks down in 1+1 dimensions, it is of interest to see what holographic methods have to say about similar models. For theories with a gapless branch, the Luttinger conjecture states that there is an effective description of the physics in terms of a Luttinger liquid which is specified by two parameters. The theory we consider is the defect CFT arising due to a probe D3 brane in the AdS Schwarzschild planar black hole background. We turn on a fundamental string density on the worldvolume. Unlike higher dimensional defects, a persistent dissipationless zero sound mode is found. The thermodynamic aspects of these models are considered carefully and certain subtleties with boundary terms are explained which are unique to 1+1 dimensions. Spectral functions of bosonic and fermionic fluctuations are also considered and quasinormal modes are analysed. A prescription is given to compute spectral functions when there is mixing due to the worldvolume gauge field. We comment on the Luttinger conjecture in the light of our findings.

Abstract:
In this paper we would like to demonstrate how the known rules of anyon condensation motivated physically proposed by Bais \textit{et al} can be recovered by the mathematics of twist-free commutative separable Frobenius algebra (CSFA). In some simple cases, those physical rules are also sufficient conditions defining a twist-free CSFA. This allows us to make use of the generalized $ADE$ classification of CSFA's and modular invariants to classify anyon condensation, and thus characterizing all gapped domain walls and gapped boundaries of a large class of topological orders. In fact, this classification is equivalent to the classification we proposed in Ref.1.

Abstract:
We relate the ground state degeneracy (GSD) of a non-Abelian topological phase on a surface with boundaries to the anyon condensates that break the topological phase to a trivial phase. Specifically, we propose that gapped boundary conditions of the surface are in one-to-one correspondence to the sets of condensates, each being able to completely break the phase, and we substantiate this by examples. The GSD resulting from a particular boundary condition coincides with the number of confined topological sectors due to the corresponding condensation. These lead to a generalization of the Laughlin-Wu-Tao (LWT) charge-pumping argument for Abelian fractional quantum Hall states (FQHS) to encompass non-Abelian topological phases, in the sense that an anyon loop of a confined anyon winding a non-trivial cycle can pump a condensate from one boundary to another. Such generalized pumping may find applications in quantum control of anyons, eventually realizing topological quantum computation.

Abstract:
We study the Levin-Wen string-net model with a $Z_N$ type fusion algebra. Solutions of the local constraints of this model correspond to $Z_N$ gauge theory and double Chern-simons theories with quantum groups. For the first time, we explicitly construct a spin-$(N-1)/2$ model with $Z_N$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a $Z_N$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a $Z_N$ gauge field living on the links of the triangular lattice. The ungauged $Z_N$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group $H^3(Z_N,U(1))$ of $Z_N$. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantusm group algebra.