Valence Bond Phases in $S=1/2$ Kane-Mele-Heisenberg Model

The phase diagram of Kane-Mele-Heisenberg (KMH) model in classical limit~\cite{zare}, contains disordered regions in the coupling space, as the result of to competition among different terms in the Hamiltonian, leading to frustration in finding a unique ground state. In this work we explore the nature of these phase in the quantum limit, for a $S=1/2$. Employing exact diagonalization (ED) in $S_z$ and nearest neighbor valence bond (NNVB) bases, bond and plaquette valence bond mean field theories, We show that the disordered regions are divided into ordered quantum states in the form of plaquette valence bond crystal(PVBC) and staggered dimerized (SD) phases.

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