Consider a generic quantum spin chain that can be mapped to free quadratic fermions via Jordan-Wigner (JW) transformation. In the presence of arbitrary boundary magnetic fields, this Hamiltonian is no longer a quadratic Hamiltonian after JW transformation. Using ancillary sites and enlarging the Hamiltonian we first introduce a bigger quadratic Hamiltonian. Then we diagonalize this enlarged Hamiltonian in its most generic form and show that all the states are degenerate because of the presence of a zero mode. The eigenstates of the original spin chain with boundary magnetic fields can be derived after appropriate projection. We study in-depth the properties of the eigenstates of the enlarged Hamiltonian. In particular, we find: 1) the eigenstates in configuration bases, 2) calculate all the correlation functions, 3) find the reduced density matrices, 4) calculate the entanglement entropy. We show that the generic eigenstate of the enlarged Hamiltonian (including the eigenstates of the original spin chain) breaks the parity number symmetry and consequently one needs to take care of some technicalities regarding the calculation of the reduced density matrix and entanglement entropy. Interestingly we show that the entanglement structure of these eigenstates is quite universal and independent of the Hamiltonian. We support our results by applying them to a couple of examples.
Parity Broken Chiral Spin Dynamics in Ba$_3$NbFe$_3$Si$_2$O$_{14}$
用非极化和极化非弹性中子散射研究了镧系镓酸盐Ba $_3$ NbFe $_3$ Si $_2$ O $_{14}$对映体纯晶体中手性螺旋调制120 $^{\circ}$磁序的自旋波激发.获得了手性基态的动力学指纹，通过（i）对结构手性负责的谱重不对称性和（ii）在整个能谱上观察到的自旋相关性的全手性来奇异化。自旋波元激发的内在手征性质的宏观时间反演对称性破缺的情况下示出。
