Topological Signatures of Magnetic Phase Transitions with Majorana Fermions through Local Observables and Quantum Information

TL;DR

The paper connects magnetic quantum phase transitions in the 1D $J_1$-$J_2$ spin chain to topological properties of a p-wave Kitaev-like wire by leveraging local spin observables and quantum information probes. Using a Majorana/bond-fermion mapping and Bogoliubov–de Gennes formalism, it links edge magnetization, half-Skyrmions on the Bloch sphere, and a topological invariant $C$ to the presence of Majorana zero modes, and shows that short-range spin correlators and edge susceptibilities encode the transition through logarithmic singularities. The work further presents a concrete realization scheme via a wire of resonating Bloch spheres, develops the AKLT-based generalizations, and demonstrates robustness to perturbations with numerical evidence from DMRG. It also highlights a correspondence between resonating valence bonds and bipartite charge fluctuations in a $p$-wave superconductor, suggesting practical routes for quantum-circuit implementations of Majorana-based physics.

Abstract

The one-dimensional (1D) $J_1-J_2$ 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 $J_1-J_2$ 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.

Figures (6)

• Figure 1: Spin magnetization at the edge obtained with DMRG (400 sites) DatasLink as a function of $h_1$ and $J_1$ with $J_2=1$. Comparison between the edge magnetization in blue when $h_1\rightarrow 0$ and $\langle \sigma_{2m}^y \sigma_{2m+1}^y\rangle$ correlation function in orange in Fig. \ref{['Clarifynumbers']} (with $\langle \sigma_{2m}^y \sigma_{2m+1}^y\rangle\rightarrow - \langle \sigma_{2m}^y \sigma_{2m+1}^y\rangle$).
• Figure 2: Observables from Eqs. (\ref{['spinobservables']}), (\ref{['observables']}) and DMRG DatasLink.
• Figure 3: Bulk-edge correspondence: Edge spin susceptibility with $h=h_1$ from DMRG which diverges logarithmically when $J_1=J_2$ (see also Eq. (\ref{['chi1']})). Logarithmic divergence of the derivative associated to the short-range spin correlation function(s) which also agrees with Eq. (\ref{['transition']}).
• Figure 4: (Top) $\langle \sigma_{2m-1}^x\sigma_{2m}^z\sigma_{2m+1}^z\sigma_{2m+2}^x\rangle$ (solid lines) and $\langle \sigma_{2m}^y\sigma_{2m+1}^y\rangle$ (dashed lines) from DMRG when including an additional term $J_{1}^z$. A link to the DMRG data can be found in Ref. [33] of the Letter. (Bottom) Logarithmic divergences of the derivatives associated to short-range spin correlation functions.
• Figure 5: (Top) $\langle \sigma_{2m-1}^x\sigma_{2m}^z\sigma_{2m+1}^z\sigma_{2m+2}^x\rangle$ (solid lines) and $\langle \sigma_{2m}^y\sigma_{2m+1}^y\rangle$ (dashed lines) from DMRG when including an additional term $J_{2}^z$. A link to the DMRG data can be found in Ref. [33] of the Letter. (Bottom) Logarithmic divergences of the derivatives associated to short-range spin correlation functions.