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Interacting Majorana modes at surfaces of noncentrosymmetric superconductors

机译：非中心对称超导体表面的相互作用Majorana模

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Noncentrosymmetric superconductors with line nodes are expected to possess topologically protected Hat zero-energy bands of surface states, which can be described as Majorana modes. We here investigate their fate if residual interactions beyond BCS theory are included. For a minimal square-lattice model with a plaquette interaction, we find stringlike integrals of motion that form Clifford algebras and lead to exact degeneracies. These degeneracies strongly depend on whether the numbers of sites in the x and y directions are even or odd and are robust against disorder in the interactions. We show that the mapping of the Majorana model onto two decoupled spin compass models [Y. Kamiya el al., Phys. Rev. B 98, 161409(R) (2018)] and extra spectator degrees of freedom only works for open boundary conditions. The mapping shows that the three-leg and four-leg Majorana ladders are integrable, while systems of larger width are not. In addition, the mapping maximally reduces the effort for exact diagonalization, which is utilized to obtain the gap above the ground states. We find that this gap remains open if one dimension is kept constant and even, while the other is sent to infinity, at least if that dimension is odd. Moreover, we compare the topological properties of the interacting Majorana model to those of the toric-code model. The Majorana model has long-range entangled ground states that differ by Z_2 fluxes through the system on a torus. The ground states exhibit string condensation similar to the toric code but the topological order is not robust. While the spectrum is gapped—due to spontaneous symmetry breaking inherited from the compass models—states with different values of the Z_2 fluxes end up in the ground-state sector in the thermodynamic limit. Hence, the gap does not protect these fluxes against weak perturbations.
机译：具有线节点的非中心对称超导体有望具有表面状态的拓扑保护的Hat零能带，可以将其描述为Majorana模式。如果包括BCS理论之外的剩余相互作用，我们将在这里研究它们的命运。对于具有板球相互作用的最小方格模型，我们发现形成Clifford代数并导致精确简并的运动的线状积分。这些简并性在很大程度上取决于在x和y方向上的位点数是偶数还是奇数，并且对于交互作用的无序性具有鲁棒性。我们表明，马约拉纳模型到两个解耦自旋罗盘模型的映射[Y. Kamiya el。，物理学。 Rev. B 98，161409（R）（2018）]和额外的观众自由度仅适用于开放边界条件。该映射显示三腿和四腿的Majorana梯子是可集成的，而较大宽度的系统则不可。另外，该映射最大程度地减少了精确对角化的工作量，该工作用于获得基态以上的间隙。我们发现，如果一个维保持恒定不变，而另一个维被发送到无穷大，则至少在该维为奇数时，该间隙仍然保持打开状态。此外，我们将交互的Majorana模型与复曲面代码模型的拓扑特性进行了比较。 Majorana模型具有长距离纠缠的基态，这些基态在圆环上通过系统的Z_2通量有所不同。基态表现出类似于复曲面编码的字符串缩合，但拓扑顺序不稳健。由于从罗盘模型继承而来的自然对称性破裂，光谱出现了缺口，具有Z_2通量值不同的状态最终处于热力学极限的基态扇区。因此，该间隙不能保护这些通量免受微扰。

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