Most theoretical studies of topological superconductors and Majorana-based quantum computation rely on a mean-field approach to describe superconductivity. A potential problem with this approach is that real superconductors are described by number-conserving Hamiltonians with long-range interactions, so their topological properties may not be correctly captured by mean-field models that violate number conservation and have short-range interactions. To resolve this issue, reliable results on number-conserving models of superconductivity are essential. As a first step in this direction, we use rigorous methods to study a number-conserving toy model of a topological superconducting wire. We prove that this model exhibits many of the desired properties of the mean-field models, including a finite energy gap in a sector of fixed total particle number, the existence of long-range Majorana-like correlations between the ends of an open wire, and a change in the ground state fermion parity for periodic vs antiperiodic boundary conditions. These results show that many of the remarkable properties of mean-field models of topological superconductivity persist in more realistic models with number-conserving dynamics.

中文翻译：

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带有粒子数守恒的拓扑超导的严格结果。

拓扑超导体和基于Majorana的量子计算的大多数理论研究都依赖于均值场方法来描述超导性。这种方法的潜在问题是，真正的超导体是由具有长距离相互作用的守恒哈密顿量来描述的，因此它们的拓扑性质可能无法通过违反数守恒并具有短程相互作用的均值场模型来正确捕获。为了解决这个问题，在超导数守恒模型上可靠的结果至关重要。作为朝这个方向迈出的第一步，我们使用严格的方法来研究拓扑超导线的保数玩具模型。我们证明该模型展示了均场模型的许多理想特性，包括固定总粒子数扇区中的有限能隙，导线末端之间存在远距离的类似Majorana的相关性，以及周期性和反周期性边界条件下基态费米子平价的变化。这些结果表明，拓扑超导平均场模型的许多显着特性在具有数守恒动力学的更实际的模型中仍然存在。