Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
Interacting Majorana modes at surfaces of noncentrosymmetric superconductors
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DOI：
10.1103/physrevb.101.024519
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发表时间：
2019-11
影响因子：
3.7
通讯作者：
J. E. Rückert;Gergő Roósz;C. Timm
中科院分区：
文献类型：
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作者：
J. E. Rückert;Gergő Roósz;C. Timm
具有线节点的非中心对称超导体被认为具有拓扑保护的表面态平坦零能带，这可以被描述为Majorana模式。我们在这里调查他们的命运，如果剩余的相互作用超出BCS理论。对于一个最小的正方形格子模型与格元相互作用，我们发现弦状积分的运动，形成Clifford代数，并导致确切的退化。这些简并性强烈地依赖于在x和y方向上的位点数目是偶数还是奇数，并且对相互作用中的无序是鲁棒的。我们证明了马约拉纳模型到两个解耦自旋罗盘模型的映射[Kamiya等人，Phys. Rev. B 98，161409（2018）]和额外的观众自由度仅适用于开放边界条件。映射表明，三腿和四腿的马约拉纳阶梯是可积的，而更大的宽度的系统是不可积的。此外，映射最大限度地减少了精确对角化的努力，这是用来获得差距以上的基态。我们发现，如果一个维度保持不变且为偶数，而另一个维度被发送到无穷大，至少如果该维度为奇数，则该间隙保持开放。此外，我们比较的拓扑性质的相互作用的马约拉纳模型的环面码模型。Majorana模型具有长程纠缠基态，通过环面上的系统的通量相差$\mathbb{Z}_2$。基态表现出类似于环面代码的弦凝聚，但拓扑顺序是不稳健的。虽然光谱是有间隙的-由于自发对称性破缺继承自指南针模型-具有不同$\mathbb{Z}_2$通量值的状态最终在热力学极限的基态扇区。因此，差距不能保护这些通量免受弱扰动。
Noncentrosymmetric superconductors with line nodes are expected to possess topologically protected flat 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 string-like 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 [Kamiya et al., Phys. Rev. B 98, 161409 (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 $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ fluxes end up in the ground-state sector in the thermodynamic limit. Hence, the gap does not protect these fluxes against weak perturbations.