Fig. 1
Different ways of encircling multiple expectational points (EPs). a Illustration of Riemann surface associated with the square root function associated with an archetypal 2 × 2 non-Hermitian Hamiltonian. A loop that encircles the EP (also known as the branch point) starting from the state s1 will map it onto s2 and vice versa. In the complex plane projection, this is represented by adding a branch cut as shown by the red line. The gray arrow in the projection plane indicates the encircling direction. b A scenario that exhibits three EPs. In this case, loops can encircle the same EPs in different ways as illustrated by the two loops (solid/dashed lines) that enclose EP1,3 starting from the same point (gray dot)
