Nontopological Majorana Zero Modes in Inhomogeneous Spin Ladders

We show that the coupling of homogeneous Heisenberg spin-$1/2$ ladders in different phases leads to the formation of interfacial zero energy Majorana bound states. Unlike Majorana bound states at the interfaces of topological quantum wires, these states are void of topological protection and generally susceptible to local perturbations of the host spin system. However, a key message of our Letter is that, in practice, they show a high degree of resilience over wide parameter ranges which may make them interesting candidates for applications.

Figure 1

Phase diagram for the Hamiltonian (1) obtained from SU(2) DMRG simulations of a $100\times 2$ site ladder with bond dimension $\chi =1500$ states. The red (blue) phase boundary shows the critical line with Majorana fermion mass $m_t=0$ ($m_s=0$). Inset figures show schematic representations of singlet (blue) and triplet (red) bond order within each phase, and the corresponding signs of $m_t$ and $m_s$.

Figure 2

Bond formation patterns when parameters $(J_{\times },J_{\perp })$ are varied (in green) in order to form a RS–${\mathrm{VBS}}_{-}$–RS ladder. (a) For sudden parameter changes at interfaces. (b) For smooth parameter changes, Jackiw-Rebbi zero modes emerge when the singlet mass, $m_s$, changes sign (lowest sketch).

Figure 3

Finite size scaling of low-energy SU(2) DMRG eigenstates in RS–${\mathrm{VBS}}_{-}$–RS ladders of total length $L$. Blue and red indicate singlet and triplet states ($S=0/1$) (the ground state is not shown). We use $J=1$, with the other couplings varied as $J_{\perp }(x)=\frac{4}{3}[1-w(x)]$ and $J_{\times }(x)=-\frac{4}{3}w(x)$, with $w(x)=f(x-x_{+})-f(x-x_{-})$, $f(x)=[1+\exp (x/\delta x){]}^{-1}$, $L_c\equiv x_{+}-x_{-}$, and $x_{\pm }=(L\pm L_c)/2$. The width $\delta x$ controls the interface smoothness. The ground state degeneracy develops quickly with increasing $\delta x$ and is only marginally affected by the length $L_c$ of the ${\mathrm{VBS}}_{-}$ region (see Sec. IV of [12]).