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We investigate the ground state nature of the transverse field Ising model onthe $J_1-J_2$ square lattice at the highly frustrated point $J_2/J_1=0.5$. Atzero field, the model has an exponentially large degenerate classical groundstate, which can be affected by quantum fluctuations for non-zero field towarda unique quantum ground state. We consider two types of quantum fluctuations,harmonic ones by using linear spin wave theory (LSWT) with single-spin flipexcitations above a long range magnetically ordered background and anharmonicfluctuations, by employing a cluster-operator approach (COA) with multi-spincluster type fluctuations above a non-magnetic cluster ordered background. Ourfindings reveal that the harmonic fluctuations of LSWT fail to lift theextensive degeneracy as well as signaling a violation of the Hellmann-Feynmantheorem. However, the string-type anharmonic fluctuations of COA are able tolift the degeneracy toward a string-valence bond solid (VBS) state, which isobtained from an effective theory consistent with the Hellmann-Feynman theoremas well. Our results are further confirmed by implementing numerical treetensor network simulation. The emergent non-magnetic string-VBS phase is gappedand breaks lattice rotational symmetry with only two-fold degeneracy, whichbears a continuous quantum phase transition at $\Gamma/J_1 \cong 0.50$ to thequantum paramagnet phase of high fields. The critical behavior is characterizedby $\nu \cong 1.0$ and $\gamma \cong 0.33$ exponents.

Emergence of string-valence bond solid state in the frustrated $J_1-J_2$ transverse field Ising model on the square lattice