Majorana states in inhomogeneous spin ladders

We propose an inhomogeneous open spin ladder, related to the Kitaev honeycomb model, which can be tuned between topological and nontopological phases. In extension of Lieb's theorem, we show numerically that the ground state of the spin ladder is either vortex free or vortex full. We study the robustness of Majorana end states (MES) which emerge at the boundary between sections in different topological phases and show that while the MES in the homogeneous ladder are destroyed by single-body perturbations, in the presence of inhomogeneities at least two-body perturbations are required to destabilize MES. Furthermore, we prove that $x$, $y$, or $z$ inhomogeneous magnetic fields are not able to destroy the topological degeneracy. Finally, we present a trijunction setup where MES can be braided. A network of such spin ladders provides thus a promising platform for realization and manipulation of MES.

Figure 1

Inhomogeneous spin ladder. Each site contains a quantum spin-$\frac{1}{2}$ which interacts with its nearest-neighbor spins via bond-dependent Ising interactions $J_x,J_y,J_z$. The $x$, $y$, and $z$ bonds are indicated by red, green, and blue lines, respectively, the $A$ ($B$) sublattice site by black (white) dots, and the n$\mathit{th}$ unit cell composed of four spins by the black dashed square. In contrast to the standard honeycomb model, the $z$ couplings are inhomogeneous, that is, site-dependent $J_z\to J_{z_{ij}}$.

Figure 2

Inhomogeneous spin ladder with different topological sections. Shown are two topological sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_3$ (thin $z$ bonds $J_{z_1}$ and $J_{z_3}$) separated by a nontopological section ${\mathcal{S}}_2$ (thick $z$ bonds $J_{z_2}$). For the corresponding $J_{x,y,z}$ values, see main text. The wave functions of the four MES ${\gamma }_{1,...,4}$ are mainly localized at the phase boundaries and, for $J_x>J_y$, on the lower sites of the ladder as indicated by the large dots.

Figure 3

Inhomogeneous spin ladder as defined in Fig. (2). (a) MES wave functions $\varphi \left(i\right)$ (corresponding to ${\gamma }_{1,...,4}$) as functions of site $i$. The curves for ${\gamma }_{2,3,4}$ are shifted vertically for clarity. The order used for the site labeling of the spin ladder is shown in (b). The circles represent the wave function weight of ${\gamma }_1$ (proportional to the area enclosed by the circle) at the corresponding site. For both plots we have all $u_{ij}=+1$ (vortex-free), $N=60$, $J_x=1.0$, $J_y=-0.4$, $J_{z_1}=J_{z_3}=0.2$ in ${\mathcal{S}}_{1,3}$, and $J_{z_2}=3$ in ${\mathcal{S}}_2$. Section ${\mathcal{S}}_2$ starts at unit cell $n=41$ and ends at $n=79$.

Figure 4

Excitation energy $\Delta E_V$ of a single vortex as a function of its position $p$ (square plaquette) on the ladder for different $N$. The values of $J_{x,y}$ and $J_{z_{ij}}$ are given in the main text. The junction between sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is at $p=2N/3$ and between ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ at $p=4N/3$. Note the slight increase of $\Delta E_V$ (as compared to the bulk) for $N=45$ and 120 when the vortex is located at the end of the ladder.

Figure 5

Network of spin ladders. (a) Majoranas are exchanged through the trijunctions, which are shown in detail in (b). The connection between the three spin ladders is given by Ising couplings $J_x$ (red dashed lines) and $J_y$ (green dashed lines). Braiding is performed by varying $J_{x,y}$ (see main text). MES ${\gamma }_{1,...,4}$ (large dots) are localized at the left and right ends of the ladder and at the junction between topological (thin $z$ links) and nontopological (thick $z$ links) sections.

Figure 6

Inhomogeneous Kitaev spin ladder. This spin ladder possesses two topological sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_3$ (thin $z$ links with couplings $J_{z_1}$ and $J_{z_3}$ which we choose to be equal, i.e, $J_{z_1}=J_{z_3}=J_z$) separated by a nontopological section ${\mathcal{S}}_2$ (thick $z$ links with couplings $J_{z_2}=J_{z^{\prime }}$). The main components of the four MES wavefunctions ${\gamma }_{1,...,4}$ lie on the lower sites for $J_x>J_y$ and are represented by large dots.

Figure 7

Energy $\Delta E_V$ of a single vortex as a function of its position $p$ on the ladder. We recall that a vortex can be placed at $2N-1$ different positions on a ladder with $N$ unit cells. The five different curves correspond to $N=9,15,30,60,150$. We see a clear difference between the vortex energy in the bulk and near the boundaries: Boundary effects increase the energy of a vortex lying near to one end of the ladder. It is also worth pointing out that the vortex energy converges quickly (with $N$) to its thermodynamic limit value. We see that the vortex energy is positive for each curve irrespective of the vortex's position. This plot thus supports our claim that the ground state is vortex free. The values of the different couplings are (a) $J_x=1.0$, $J_y=-0.5$, $J_z=0.3$ in ${\mathcal{S}}_{1,3}$ and $J_{z^{\prime }}=0.3$ in ${\mathcal{S}}_2$. (b) $J_x=1.0$, $J_y=-0.65$, $J_z=4.3$ in ${\mathcal{S}}_{1,3}$ and $J_{z^{\prime }}=4.3$ in ${\mathcal{S}}_2$. The curves for $N=15,30,60,150$ are shifted vertically by $0.005,0.01,0.015$, and $0.02$, respectively, for clarity.

Figure 8

(a),(b) Energy $\Delta E_V$ of a single vortex as a function of its position $p$ on a ladder with $N=120$, $J_x=1.0$, $J_y=-0.37$, $J_z=0.25$ in ${\mathcal{S}}_{1,3}$, while $J_{z^{\prime }}=4$ in ${\mathcal{S}}_2$ for (a) and $J_x=1.0$, $J_y=-0.37$, $J_z=0.25$ in ${\mathcal{S}}_{1,3}$ and $J_{z^{\prime }}=0.25$ in ${\mathcal{S}}_2$ for (b). The junction between sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is at $p=2N/3$ and between ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ is at $p=4N/3$.

Figure 9

(a),(b) Energy $\Delta E_{2V}$ of two vortices as a function of the position $p$ of the second vortex. The first vortex lies on the $p=1$ square plaquette. The $J_{x,y,z,z^{\prime }}$ parameters are chosen respectively as in Fig. 8. The vortex-vortex interaction is attractive and rapidly decaying as a function of distance between the two vortices. Indeed, already for $p=5$ the energy of the two vortices is roughly $0.04$ [$0.03$ (energy of the vortex at the boundary $p=1$) plus $0.01$ (energy of the vortex in the bulk)]. However, as mentioned in the main text, the attraction is never strong enough to favor the creation of vortices and the energy of the two vortices is always positive. The junction between sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is at $p=2N/3$ and between ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ is at $p=4N/3$.

Figure 10

Energy $E_{VF}$ of the vortex-full sector as a function of $N$ with $J_x=1.0$, $J_y=-0.55$, $J_z=0.25$ in ${\mathcal{S}}_{1,3}$, while $J_{z^{\prime }}=4$ in ${\mathcal{S}}_2$. As expected, the energy of the vortex-full sector is always positive and grows linearly with $N$. The slope of the of the straight line can be interpreted as an average vortex energy. This plot indicates again that the vortex-vortex interaction does not favor the creation of vortices and the ground state is vortex-free. The junction between sections ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ is at $p=2N/3$ and between ${\mathcal{S}}_2$ and ${\mathcal{S}}_3$ is at $p=4N/3$.

Figure 11

Pictorial representation of the ground states in the limit $J_x\ne 0$ and $J_{y,z}=0$ in sections ${\mathcal{S}}_{1,3}$ and $J_z\ne 0$ and $J_{x,y}=0$ in ${\mathcal{S}}_2$. Here $\Uparrow$ and $\Downarrow$ are a pictorial representations of ${\tau }^x$ eigenstates.

Figure 12

Energy eigenvalues ${\epsilon }_k$ of $\mathcal{H}$ in Eq. (D4) for $xx$-$yy$ chains with $J_x=0.4$, $J_y=1.0$, and $2N=20$ for (a) and $2N=100$ for (b). There are $2N$ zero-energy modes in addition to the two expected MES. The presence of these additional zero modes can be understood by mapping Hamiltonian $H_{xx-yy}$ to an array of $\pi$ junctions [see Eq. (D3)].

Figure 13

$-\langle {\Psi }_n\left|{\sigma }_1^x{\sigma }_{4N}^x\right|{\Psi }_n\rangle$ as a function of $N$, with all $u_{ij}=+1$, $J_x=1.0$, $J_y=-0.4$, and $J_{z_1}=J_{z_2}=J_{z_3}=0.2$ for (a), and $J_{z_1}=J_{z_2}=J_{z_3}=2$ for (b). We make use of the projection protocol of Ref. 29 in order to determine if the physical ground state of the vortex-free sector has even ($n=0$) or odd ($n=1$) parity.

Figure 14

Plot of correlator $-\langle {\Psi }_n\left|{\sigma }_1^x{\sigma }_{4N}^x\right|{\Psi }_n\rangle$ as a function of position of a single vortex, $p$, for $N=50$, $J_x=1.0$, $J_y=-0.4$, and $J_{z_1}=J_{z_2}=J_{z_3}=0.2$ in (a) and $J_{z_1}=J_{z_3}=0.2$, $J_{z_2}=2$ in (b). The junctions between sections ${\mathcal{S}}_{1,3}$ and ${\mathcal{S}}_2$ are at plaquettes $p=41,71$. We used the projection protocol of Ref. 29 to determine if the physical ground state of the corresponding single-vortex sector has even ($n=0$) or odd ($n=1$) parity. Note that the physical ground states we consider have a fixed parity $i{\gamma }_2{\gamma }_3=+1$ and oscillating parity $i{\gamma }_1{\gamma }_4$.