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

Highlights: • Harmonic quantum fluctuations can not lift the large degeneracy of the ground state. • An-harmonic quantum fluctuations lift the degeneracy toward a unique string-VBS phase. • String-VBS phase break rotational symmetry with only two-fold degeneracy. • The string-VBS phase can be obtained from plaquette-VBS phase of the checkerboard lattice. - Abstract: We study the effect of quantum fluctuations by means of a transverse magnetic field (Γ) 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.