蜂窝格点上Dirac理论凝聚相中的近零模
Near zero modes in condensate phases of the Dirac theory on the honeycomb lattice
摘要：
我们研究了蜂窝晶格上的一些费米凝聚相，以确定这些相中的拓扑缺陷（涡旋和边缘）是否可以支持零能量的束缚态。我们认为，拓扑零模式绑定到旋涡和边缘不仅连接，但实际上应该是\emph{identified}。最近，已经表明，在精确半填充的蜂窝晶格的狄拉克费米子近似下，最简单的s波超导态支持涡旋核心内的零模式（P. Ghaemi和F. Wilczek，2007年）。我们发现，在连续狄拉克理论的零模式是不唯一的，既不是这个阶段，也不是半填充。此外，我们还找到了涡旋束缚零模的\emph{exact}波函数，以及我们讨论的相位的完整边缘态谱。在我们研究的所有相位中的零模都具有偶数简并度，因此任何马约拉纳模对都简单地等价于一个普通的费米子。因此，与$p_x+i p_y$超导体中的束缚态零模相反，这里的涡旋\emph{not}表现出非阿贝尔交换统计。在纯狄拉克理论中，零模似乎受到理论的有效低能对称性的拓扑保护，但在原始蜂窝晶格模型中，这些零模被分裂，通过有效低能对称性的显式破缺。
英文摘要：
We investigate a number of fermionic condensate phases on the honeycomb lattice, to determine whether topological defects (vortices and edges) in these phases can support bound states with zero energy. We argue that topological zero modes bound to vortices and at edges are not only connected, but should in fact be \emph{identified}. Recently, it has been shown that the simplest s-wave superconducting state for the Dirac fermion approximation of the honeycomb lattice at precisely half filling, supports zero modes inside the cores of vortices (P. Ghaemi and F. Wilczek, 2007). We find that within the continuum Dirac theory the zero modes are not unique neither to this phase, nor to half filling. In addition, we find the \emph{exact} wavefunctions for vortex bound zero modes, as well as the complete edge state spectrum of the phases we discuss. The zero modes in all the phases we examine have even-numbered degeneracy, and as such pairs of any Majorana modes are simply equivalent to one ordinary fermion. As a result, contrary to bound state zero modes in $p_x+i p_y$ superconductors, vortices here do \emph{not} exhibit non-Abelian exchange statistics. The zero modes in the pure Dirac theory are seemingly topologically protected by the effective low energy symmetry of the theory, yet on the original honeycomb lattice model these zero modes are split, by explicit breaking of the effective low energy symmetry.
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