Many-body wave function and edge magnetization of an open $p+is$ superconducting chain

TL;DR

This work addresses the explicit construction of the many-body BCS ground-state wave function for a one-dimensional spin-1/2 $p+is$ superconductor under open boundary conditions, focusing on special parameter sets where exact results are attainable. By mapping the spin-1/2 $p$-wave sector to two decoupled Kitaev chains and treating the $s$-wave component perturbatively, the authors derive a degenerate-ground-state structure and analytic edge-magnetization results; they then elevate this to an exact solution via a Bogoliubov transformation and recursive wave-function construction, yielding explicit expressions for the ground state and edge magnetization that depend on the ratio $\frac{Δ_s}{\sqrt{Δ_s^2+Δ_p^2}}$. The key finding is that edge magnetization is localized at the chain ends with magnitudes $\langle S_N^z\rangle= -\frac{1}{4}\left(1+\frac{Δ_s}{\sqrt{Δ_s^2+Δ_p^2}}\right)$ and $\langle S_1^z\rangle= +\frac{1}{4}\left(1+\frac{Δ_s}{\sqrt{Δ_s^2+Δ_p^2}}\right)$ in the exact solution, reducing to the previously reported $\pm\frac{1}{4}$ in the small-$Δ_s$ limit. This wave-function–level treatment provides a deeper, quantitative understanding of edge phenomena and topological features in open $p+is$ chains beyond standard Majorana-based arguments, with implications for realizing and probing edge states in quasi-1D superconductors and proximity-coupled systems.

Abstract

Although BCS wave function for superconductors under periodic boundary conditions are well-established, obtaining an explicit form of the many-body BCS wave function under open boundary condition is usually a nontrivial problem. In this work, we construct the exact BCS ground state wave function of a one-dimensional spin-1/2 superconductor with $p+ is$ pairing symmetry under open boundary conditions for special sets of parameters. The spin magnetization on the edges are calculated explicitly using the obtained wave function. Approximate expression of the wave function is also discussed based on degenerate perturbation theory when the $s$-wave component is much smaller than the $p$-wave one, which provides more intuitive understanding for the system. Our work is useful for obtaining deeper understandings of open $p+ is$ superconducting chains on a wave function level.

Figures (5)

• Figure 1: Numerical results for edge magnetization at site $1$ as a function of $\Delta_s/\Delta_p$ shown by the red dots, calculated on an open chain with $N=20$ sites. Numerical values exhibits excellent agreement with the analytical expression $\frac{1}{4}(1+\frac{\Delta_s/\Delta_p}{\sqrt{1+(\Delta_s/\Delta_p)^2}})$ plotted by the black curve.
• Figure 2: Schematic plot of the pairing structure for the spin-1/2 $p$-wave superconducting chain at special parameters as two decoupled spinless Kitaev chains. The upper chain corresponds a copy of Kitaev superconducting chain with $\Delta=t$, described by the Hamiltonian $\hat{H}_{p,+}$ in Eq. (\ref{['decompose']}), while the lower chain corresponds to $\Delta=-t$, described by the Hamiltonian $\hat{H}_{p,-}$.
• Figure 3: Schematic plot of the pairing structure for the 1D spin-1/2 $p_z+is$ superconducting model at special parameters. The interactions in the $p$-wave and $s$-wave pairing Hamiltonians are represented by the horizontal black lines and vertical red lines, respectively, in the figure.
• Figure 4: Schematic plot of the pairing structure for the 1D spin-1/2 $p+s$ superconducting model. The $s$-wave pairing, indicated by red lines, is nonlocal, in contrast to the $p+is$ case.
• Figure 5: Numerical results for the magnetization $\langle \tilde{G}_{p+is} | S_m^z | \tilde{G}_{p+is} \rangle$ as a function of site $m$ at $\Delta_p = \Delta_s$ on an open $p+is$ superconducting chain with $N = 20$ sites. The magnetization in the bulk vanishes, while its value at the edge sites $1$ and $N$ agree well with the analytical results, which are $\pm \frac{1}{4}\left(1+\frac{1}{\sqrt{2}}\right)\approx \pm 0.427$.