Rotational symmetry breaking toward a string-valence bond solid phase in frustrated J1 - J2 transverse field Ising model

We study the effect of quantum fluctuations by means of a transverse magnetic field (G) on the highly degenerate ground state of antiferromagnetic J(1) - J(2) Ising model on the square lattice, at the limit J(2)/J(1) = 0.5. We show that harmonic quantum fluctuations based on single spin flips can not lift such degeneracy, however an-harmonic quantum fluctuations based on multi spin cluster flip excitations lift the degeneracy toward a unique ground state with string-valence bond solid (VBS) nature. A cluster operator formalism has been implemented to incorporate an-harmonic quantum fluctuations. We show that cluster-type excitations of the model lead not only to lower the excitation energy compared with a single-spin flip but also to lift the extensive degeneracy in favor of a string-VBS state, which breaks lattice rotational symmetry with only two fold degeneracy. The tendency toward the broken symmetry state is justified by numerical exact diagonalization. Moreover, we introduce a map to find the relation between the present model on the checkerboard and square lattices.