一维晶格上的Poincaré晶体
Poincaré crystal on the one-dimensional lattice
摘要：
本文发展了一维Bravais晶格上具有离散Poincaré对称性粒子的量子理论。我们回顾了最近发现的离散洛伦兹对称性，这是唯一的洛伦兹对称性共存的离散空间平移对称的布拉维晶格。离散的洛伦兹变换和时空平移形成离散的庞加莱群，在量子理论中用幺正算符表示。我们找到了表象存在的条件，这些条件用准动量和准能量的同余关系表示。然后，我们建立了洛伦兹不变的多体理论的不可区分的粒子表示的酉算子和Floquet哈密顿量的领域运营商。一些典型的哈密顿量包括长程跳跃，其随着位置之间的距离增加而波动。我们计算了格点理论的格林函数。格林函数不为零的时空点显示出晶格结构。在传播过程中，粒子保持在单个或几个位置上，以保持洛伦兹对称性。
英文摘要：
In this paper, we develop the quantum theory of particles that has discrete Poincaré symmetry on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry, which is the unique Lorentz symmetry that coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and spacetime translations form the discrete Poincaré group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green's functions of the lattice theory. The spacetime points where the Green's function is nonzero display a lattice structure. During the propagation, the particles stay localized on a single or a few sites to preserve the Lorentz symmetry.
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