We study the effect of quantum fluctuations by means of a transverse magnetic field (
Γ) on the antiferromagnetic
J1−J2 Ising model on the checkerboard lattice, the two dimensional version of pyrochlore lattice. A plaquette operator formalism has been implemented to bosonise the model, in which a single boson is associated to each eigenstate of a plaquette and the inter-plaquette interactions define an effective Hamiltonian. The excitations of a plaquette would represent an-harmonic fluctuations of the model, which 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 plaquette ordered state, which breaks lattice rotational symmetry for
J2>J1 (the collinear phase). The bosonic excitation gap vanishes at the critical boundary to N\'{e}el (
J2J1) phases, respectively. At the homogeneous coupling (
J1=J2) a (canted) resonating plaquette state (RPS) is established from an-harmonic fluctuations, which lasts for low fields,
Γ/J1≲0.4, followed by a crossover to the fully polarized phase. We will show that our results predict an order-by-disorder as a results of transverse field at the critical point (
J2=0.5J1) of the antiferromagnetic
J1−J2 Ising square lattice.
The ferromagnetic-phase stability limit and magnetization of a quenched bond-mixed spin-1/2 transverse Ising model in an anisotropic simple square lattice are studied within the framework of an effective-field theory considering both competing and noncompeting interactions. In the bond-disordered case, where bonds
J1 and
J2 occur with probabilities 1-p and p, respectively, critical curves are obtained for various \ensuremath{\alpha}'s (\ensuremath{\alpha}\ensuremath{\equiv}
J1/
J2), and the critical bond concentrations
pc at which the ferromagnetic phase breaks down are determined. As \ensuremath{\alpha}\ensuremath{\rightarrow}0, the mixed model reduces to the diluted Ising model and the results obtained are compared to those available in the literature. The effect of the transverse field (for different \ensuremath{\alpha}'s) in the thermal behavior of the magnetization as a function of the concentration is also discussed.
The one-dimensional (1D)
J1−J2 quantum spin model can be viewed as a strong-coupling analogue of the Schrieffer-Su-Heeger model with two inequivalent alternating Ising couplings along the wire, associated to the physics of resonating valence bonds. Similar to the quantum Ising model, which differently presents a long-range N\' eel ordered phase, this model also maps onto a p-wave superconducting wire which shows a topological phase transition with the emergence of low-energy Majorana fermions. We show how signatures of the topological phase transition for the p-wave superconducting wire, i.e. a half Skyrmion, are revealed through local (short-range) spin observables and their derivatives related to the capacitance of the pairing fermion model. Then, we present an edge correspondence through the edge spin susceptibility in the
J1−J2 model revealing that the topological phase transition is a metal of Majorana fermions. We justify that the spin magnetization at an edge at very small transverse magnetic field is a good marker of the topological invariant and of Majorana zero modes. We identify a correspondence between the quantum information of resonating valence bonds and the charge fluctuations in a p-wave superconductor through our method "the bipartite fluctuations". Physical properties of this 1D model are in fact robust when including additional interactions, which is optimistic for practical applications e.g. in quantum circuits.