Fig. 5: Geometric phases obtained from measured Berry curvatures.
From: Observation of Berry curvature in non-Hermitian system from far-field radiation
A Schematic of an individual valley (blue shading) around the \({{{{\mathcal{K}}}}}_{1}\) point (blue dot) in the reciprocal lattice. The integral on Berry curvature over the individual valley gives the geometric (Berry) phases. Considering that the Berry curvatures congregate around the \({{{{\mathcal{K}}}}}_{1}\) point when δr is relatively small, we perform the integral on a circular region (black circle) to simplify the calculation. Gray shading: the first BZ; purple dot: the Γ point; blue dot: the \({{{{\mathcal{K}}}}}_{1}\) point. B, C The geometric phases for TEA,B modes. Blue solid line: theoretical bulk Berry phase γn;t of the two-level model according to Eq. (4); red circles: numerical bulk Berry phase γn obtained from the integral of bulk Berry curvatures Bn calculated in Fig. 4; yellow triangles: geometric phase \({\gamma }_{n;t}^{r}\) obtained from measured radiation Berry curvature \({B}_{n;t}^{r}\) shown in Fig. 4. When δr = 0, the theoretical Berry phases are exactly  ±π for TEA,B modes owing to the existence of DP, corresponding to quantized valley-Chern numbers of  ±1/2. When δr gradually increases, all the geometric phases gradually deviate from the quantized  ±π. Moreover, when δr becomes relatively large, the theoretical γn;t and measured \({\gamma }_{n;t}^{r}\) slightly deviate from the numerical γn, due to the impact of TEC mode.
