Possibility of S=1 spin liquids with fermionic spinons on triangular lattices

In this paper we generalize the fermionic representation for $S=1/2$ spins to arbitrary spins. Within a mean field theory we obtain several spin liquid states for spin $S=1$ antiferromagnets on triangular lattices, including gapless f-wave spin liquid and topologically nontrivial $p_x+ip_y$ spin liquid. After considering different competing orders, we construct a phase diagram for the $J_1$-$J_3$-$K$ model. The application to recently discovered material $\mathrm{NiGa_2S_4}$ is discussed.Comment: 5 pages, 3 figure

Fermionic theory for quantum antiferromagnets with spin S > 1/2

The fermion representation for S = 1/2 spins is generalized to spins with arbitrary magnitudes. The symmetry properties of the representation is analyzed where we find that the particle-hole symmetry in the spinon Hilbert space of S =1/2 fermion representation is absent for S > 1/2. As a result, different path integral representations and mean field theories can be formulated for spin models. In particular, we construct a Lagrangian with restored particle-hole symmetry, and apply the corresponding mean field theory to one dimensional (1D) S = 1 and S = 3/2 antiferromagnetic Heisenberg models, with results that agree with Haldane's conjecture. For a S = 1 open chain, we show that Majorana fermion edge states exist in our mean field theory. The generalization to spins with arbitrary magnitude S is discussed. Our approach can be applied to higher dimensional spin systems. As an example, we study the geometrically frustrated S = 1 AFM on triangular lattice. Two spin liquids with different pairing symmetries are discussed: the gapped px + ipy-wave spin liquid and the gapless f-wave spin liquid. We compare our mean field result with the experiment on NiGa2S4, which remains disordered at low temperature and was proposed to be in a spin liquid state. Our fermionic mean field theory provide a framework to study S > 1/2 spin liquids with fermionic spinon excitations.Comment: 16 pages, 4 figure

Maximum Estrada Index of Bicyclic Graphs

Let $G$ be a simple graph of order $n$, let $\lambda_1(G),\lambda_2(G),...,\lambda_n(G)$ be the eigenvalues of the adjacency matrix of $G$. The Esrada index of $G$ is defined as $EE(G)=\sum_{i=1}^{n}e^{\lambda_i(G)}$. In this paper we determine the unique graph with maximum Estrada index among bicyclic graphs with fixed order