We present the geodesic nature and quantization of geometric shift vector in quantum systems,with the parameter space defined by the Bloch momentum,using the Wilson loop approach.Our analysis extends to include bosoni...We present the geodesic nature and quantization of geometric shift vector in quantum systems,with the parameter space defined by the Bloch momentum,using the Wilson loop approach.Our analysis extends to include bosonic phonon drag shift vectors with non-vertical transitions.We demonstrate that the gauge invariant shift vector can be quantized as integer values,analogous to the Euler characteristic based on the Gauss-Bonnet theorem for a manifold with a smooth boundary.We reveal intricate relationships among geometric quantities such as the shift vector,Berry curvature,and quantum metric.Our findings demonstrate that the loop integral of the shift vector in the quantized interband formula contributes to the non-quantized component of the trace of conductivity in the circular photogalvanic effect.The Wilson loop method facilitates first-principles calculations,providing insights in the geometric underpinnings of these interband gauge invariant quantities and shedding light on their nonlinear optical manifestations in real materials.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.12522411,12304049,and 12474240 for H.W.,92265203 and 12488101 for K.C.)the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant Nos.XDB28000000 and XDB0460000 for K.C.)+1 种基金the Innovation Program for Quantum Science and Technology(Grant No.2024ZD0300104 for K.C.)the Fundamental Research Funds for the Central Universities(for H.W.)。
文摘We present the geodesic nature and quantization of geometric shift vector in quantum systems,with the parameter space defined by the Bloch momentum,using the Wilson loop approach.Our analysis extends to include bosonic phonon drag shift vectors with non-vertical transitions.We demonstrate that the gauge invariant shift vector can be quantized as integer values,analogous to the Euler characteristic based on the Gauss-Bonnet theorem for a manifold with a smooth boundary.We reveal intricate relationships among geometric quantities such as the shift vector,Berry curvature,and quantum metric.Our findings demonstrate that the loop integral of the shift vector in the quantized interband formula contributes to the non-quantized component of the trace of conductivity in the circular photogalvanic effect.The Wilson loop method facilitates first-principles calculations,providing insights in the geometric underpinnings of these interband gauge invariant quantities and shedding light on their nonlinear optical manifestations in real materials.
