The one-dimensional (1D) $J_1-J_2$ quantum spin model can be viewed as a strong-coupling analog 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 Néel 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 $J_1-J_2$ 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.

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• Capacitance
• Edge states
• Geometric & topological phases
• Magnetic phase transitions
• Magnetic susceptibility
• Majorana fermions
• Quantum correlations in quantum information
• Quantum field theory
• Topological superconductors
• Density matrix renormalization group