Exact zero modes in interacting Majorana X- and Y-junctions

TL;DR

This work shows that exact zero modes emerge in junctions of interacting Kitaev chains with two, three, and four arms, driven by incommensurate short-range correlations and tunable through a junction coupling $\alpha$. By mapping the interacting Kitaev model to spin chains via the Jordan-Wigner transformation, the authors reveal parity-paired in-gap states whose level crossings signal EZMs, including a symmetry-protected center Majorana in the Y-junction and tunable center modes in the X-junction. The results highlight a universal mechanism where outer-edge Majorana interactions produce parity pairs, while center-end degrees of freedom control cross-pair splittings, with potential implications for junction-based Majorana manipulation and braiding. The study also demonstrates the robustness of EZMs against certain disorder and emphasizes the distinct roles of odd vs even arm junctions in stabilizing center modes.

Abstract

We report the emergence of exact zero modes in junctions of two, three and four short interacting Majorana wires, equivalent to a chain with an impurity bond, Y- and X- junctions respectively. These exact zero modes are due to incommensurate short-range correlations induced by the interacting Majorana fermions, and they appear as unavoided level crossings between in-gap states upon continuously tuning the interaction. In a junction of only two chains we report exact zero modes and parity switching as soon as the coupling between the chains across a junction is positive. Remarkably, for junctions with multiple chains the in-gap states group up into sets of parity pairs -- pairs of states with opposite parity and similar energies. We demonstrate that the formation of these parity pairs are always due to the interaction of the outer edges of the junction. The behavior within each pair can be efficiently described by two coupled chains. In the Y-junction, we detect four in-gap states (two parity pairs) that show exact zero modes not only within each pair but also between them. This is attributed to an additional Majorana fermion localized at the center of junction that is protected by symmetry. Therefore, coupling between the Majorana fermions at the outer edges of the junction is mediated by that in the center. We argue that this is a generic feature of junctions with an odd number of arms. In the X-junction we detect eight in-gap states (four parity pairs) that are the result of two Majorana degrees of freedom localized at the center of the junction. However, we demonstrate that, by contrast to the Y-junction, the appearance of Majorana fermions at the center of the X-junction is not protected and the interaction across the junction can be tuned to the point where there are only Majorana fermions localized at the four outer edges of the junction, forming four in-gap states.

Figures (16)

• Figure 1: Sketches of the junction geometries addressed in this paper; two coupled Majorana chains (left), Y-junction (middle), and X-junction (right). Dark green dotted lines denote the coupling between the ends of the chains that is described by $\mathcal{H}_\mathrm{coupling}(\alpha)$. In red we mark the Majorana degrees of freedom $\gamma_{i,k}$ in arm $i$, with $1 \leq k \leq N_i$. And in dark blue the corresponding spin-1/2 $\sigma_{i,k}$, with $1 \leq k \leq \tilde{N}_i = N_i/2$.
• Figure 2: Numerical results for the energy spectrum of two chains of $N=20$ Majorana fermions each, coupled by Eq.\ref{['eq: 2-chain coupling']}. We fix the four-Majorana coupling strength at $g=0.2$. All data are centered around the average of the two lowest energy levels. (a) Four lowest energy levels as a function of the coupling strength $\alpha$. The magnetic field $h=0.3$ is fixed. When coupling the two chains ($\alpha > 0$), two states with opposite parity (light blue crosses and dark blue dots) form the in-gap spectrum while the other two become part of the bulk excitations (grey lines). (b) In-gap spectrum as a function of the magnetic field $h$ for a small coupling strength $\alpha=0.015$. Exact zero modes appear as soon as $\alpha>0$. For small $\alpha$ the intervals where negative parity sector forms the ground-state are much shorter compared to that of positive one. Inset: first crossings between the parity sectors zoomed in. (c) Same as in (b) but for $\alpha=0.6$. Interval in which the ground state has a negative parity is larger compared to $\alpha=0.015$. (d) Same as in (b) but for negative coupling $\alpha=-1$. We observe no parity switching and no exact zero modes. Inset: Final avoided crossing zoomed in.
• Figure 3: Spatial profile two chains containing $N=20$ Majorana fermions each, coupled by four different strengths $\alpha$. Data is provided for the two in-gap states. Four-Majorana interaction strength is fixed at $g=0.2$. Blue circles and orange diamonds depict the strength of the magnetic field $h$. (a) Spatial profile for the ground state. Left side indicates the first chain while the right side the second chain. Amplitude of local density of states for $\alpha=0.015$ in the middle of the chain is slightly smaller than at the edges. As expected, the local density of states in the center vanishes when approaching the single interacting Kitaev chain limit, i.e. $\alpha \rightarrow 1$. For negative coupling $\alpha=-1$ the Majoranas in the center is also absent. (b) Same as in (a) but for the excited state. Local density of states is only shown for the first arm. Note that the profile is visually identical to that of the ground state.
• Figure 4: Energy spectrum of three interacting Kitaev chains, each containing $N=14$ Majorana fermions, coupled as defined by Eq. \ref{['eq: 3-chain coupling full']} in a Y-junction geometry for a fixed four-Majorana interaction strength $g=0.2$. Dots and crosses denote the positive and negative parity sectors respectively. (a) Eight lowest energies, centered around the average of the four in-gap states, as a function of the coupling strength $\alpha$ and a fixed magnetic field $h=0.45$. In a Y-junction four states make up the in-gap spectrum and group up into two parity pairs (red and blue curves), while the other four become bulk excitations (grey lines). (b) In-gap spectrum as a function of $h$ at $\alpha=1$, centered around the average of the lowest four levels. Note that there are two levels marked with red and two different levels with blue that are barely distinguishable on this scale; the oscillations within each pair is analyzed in (c). Parity pairs show multiple crossings as a fincvtion of $h$. For reference, we show level crossings in a single interacting Kitaev chain of $N=14$ Majorana fermions (grey curve). (c) Same data as in (b) but centered around the average of each parity pair. For comparison, we show two chains of $N=14$ Majorana fermions each coupled with a strength $\alpha=1.25$. (d)-(f) In-gap spectrum for $\alpha=-1$ plotted with the same convention as in (a)-(c). While we still see two parity pairs (d) and crossing between them (e), the crossings within each pair are now avoided (f).
• Figure 5: Spatial profile of Majorana fermions in the Y-junction for two coupling strenghts $\alpha$. Data is presented for the ground state -- for other in-gap states we refer to Appendix \ref{['Ap: Y-junction LDOS in-gap states']} -- and for three different magnetic fields $h$ (blue circles, orange diamonds and green squares). The four-Majorana interaction strength is fixed at $g=0.2$. (a) Local density of states for all three chains for $\alpha=1$. Third chain is greyed out for visual clarity. Each chain has a clear Majorana zero mode localized at its edge and another mode appears at the center of the junction with its amplitude distributed equally among the three arms. The amplitude and spatial distribution varies with the magnetic field. Asymmetry in the energy spectrum with respect to the magnetic field (see Fig.\ref{['fig: Y-junction EZM']}(b)) is also present here. (b) Same as in (a) but for $\alpha=-1$. Note, the spectrum appears the same as in (a) but mirrored around $h=0$.