International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science, Tsukuba 305-0044, Japan.
Department of Physics, University of California, Berkeley, California 94720, USA.
Phys Rev Lett. 2018 Jun 15;120(24):247202. doi: 10.1103/PhysRevLett.120.247202.
We show that the Z_{N} Berry phase (Berry phase quantized into 2π/N) provides a useful tool to characterize symmetry protected topological phases with correlation that can be directly computed through numerics of a relatively small system size. The Z_{N} Berry phase is defined in a N-1-dimensional parameter space of local gauge twists, which we call the "synthetic Brillouin zone," and an appropriate choice of an integration path consistent with the symmetry of the system ensures exact quantization of the Berry phase. We demonstrate the usefulness of the Z_{N} Berry phase by studying two 1D models of bosons, SU(3) and SU(4) Affleck-Kennedy-Lieb-Tasaki models, where topological phase transitions are captured by Z_{3} and Z_{4} Berry phases, respectively. We find that the exact quantization of the Z_{N} Berry phase at the topological transitions arises from a gapless band structure (e.g., Dirac cones or nodal lines) in the synthetic Brillouin zone.
我们表明，Z_{N} 贝里相位（贝里相位量化为 2π/N）为具有相关性的对称保护拓扑相提供了一种有用的工具，这种相关性可以通过相对较小系统尺寸的数值计算直接计算。Z_{N} 贝里相位在局部规范扭曲的 N-1 维参数空间中定义，我们称之为“合成布里渊区”，并且与系统对称性一致的适当积分路径选择确保了贝里相位的精确量化。我们通过研究两个一维玻色子模型，SU(3)和 SU(4) Affleck-Kennedy-Lieb-Tasaki 模型，展示了 Z_{N} 贝里相位的有用性，其中拓扑相变分别由 Z_{3}和 Z_{4} 贝里相位捕获。我们发现，在拓扑相变处 Z_{N} 贝里相位的精确量化源于合成布里渊区中无带隙的能带结构（例如，狄拉克锥或节线）。
