Table 1 Transformation of interband Berry connection rmn, shift vector Rmn, group velocity difference Δmn, and Berry curvature Ωmn under space inversion I and time reversal T symmetry operation.

Rmn(k) is odd under I and even under T in moment space. Ωmn(k) is even under I and odd under T in moment space. These transformation rules govern the coupling between ferroelectric polarization and nonlinear SC and CC photocurrent: JSCy,(Py)=JSCy,(Py), JSCy,(Py)=JSCy,(Py), JCCx,(Py)=JCCx,(Py), and JCCx,(Py)=JCCx,(Py).

QuantitySymmetry operationGauge dependency
Space inversion (I)Time reversal (T)
rmn(k)rmn( −k)rmn*(k)Yes
Rmn(k)Rmn( −k)Rmn( −k)No
Δmn(k)Δmn( −k)Δmn( −k)No
Ωmn(k)Ωmn( −k)Ωmn( −k)No