Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries

We propose a general approach to construct symmetry protected topological (SPT) states i.e the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state of a spin-1 model, a SO(3) SPT state of a boson system with spin-1 bosons and spinless bosons, and a SU(2) SPT state of a spin-1/2 boson system. By applying the "spin gauge field" which directly couples to the spin density and spin current of $S^z$ components, we also calculate the quantum spin Hall conductance in each SPT state. The projective ground states can be further studied numerically in the future by variational Monte Carlo etc.Comment: 7+ pages, accepted by Phys. Rev.

Weak Decays of Doubly-Heavy Tetraquarks bcˉqqˉ{b\bar c}{q\bar q}

We study the weak decays of exotic tetraquark states ${b\bar c}{q\bar q}$ with two heavy quarks. Under the SU(3) symmetry for light quarks, these tetraquarks can be classified into an octet plus a singlet: $3\bigotimes\bar 3=1\bigoplus8$. We will concentrate on the octet tetraquarks with $J^{P}=0^{+}$, and study their weak decays, both semileptonic and nonleptonic. Hadron-level effective Hamiltonian is constructed according to the irreducible representations of the SU(3) group. Expanding the Hamiltonian, we obtain the decay amplitudes parameterized in terms of a few irreducible quantities. Based on these amplitudes, relations for decay widths are derived, which can be tested in future. We also give a list of golden channels that can be used to look for these states at various colliders.Comment: 14 pages,3 figure