The fidelity between two infinitesimally close states or the fidelity susceptibility of a system are known to detect quantum phase transitions. Here we show that the k-space fidelity between two states far from each other and taken deep inside (bulk) of two phase s, generically vanishes at the k-points where there are gapless points in the energy spectrum that give origin to the lines (edges) separating the phases in the phase diagram. We consider a general case of two-band models and present a sufficient condition for the existence of gapless points, given there are pairs of parameter points for which the fidelity between the corresponding states is zero. By presenting an explicit counter-example, we showed that the sufficient condition is not necessary. Further, we showed that, unless the set of parameter points is suitably constrained, the existence of gapless points generically imply the accompanied pairs of parameter points with vanishing fidelity. Also, we showed the connection between the vanishing fidelity and gapless points on a number of concrete examples (topological triplet superconductor, topological insulator, 1d Kitaev model of spinless fermions, BCS superconductor, Ising model in a transverse field, graphene and Haldane Chern insulator), as well as for the more general case of Dirac-like Hamiltonians. We also briefly discuss the relation between the vanishing fidelity and gapless points at finite temperatures.
Correspondence between maximally entangled states in discrete and Gaussian regimes
利用量子纯化方法研究了离散变量量子态与连续变量（特别是高斯态）态之间的一般对应原理。在以前的工作中，我们已经研究了高斯最大混合态（GMMS）和它们的纯化（称为高斯最大纠缠态（GMESs））之间的信息论对应关系[Phys. Lett. A { \bf 380}，3607（2016）]。通过计算纠缠态之间的量子保真度，比较了$N\times N$维最大纠缠态和我们以前提出的GMES。利用这些结果，我们可以很自然地得出结论：我们的GMES更适合于高斯量子信息中的\emph{maximally entangled}态的概念，因此它可能比高斯区域中的双模压缩真空（TMSV）态更适用于量子信息任务.
