  Nonlinear optical responses to external electromagnetic field, characterized
by second and higher order susceptibilities, play crucial roles in nonlinear
optical devices and novel optoelectronics. Herein we present a quantum
nonlinear ferroic optical Hall effect (QNFOHE) in multiferroics - a Hall-like
direct photocurrent originated from the second order current response to
intense electromagnetic field whose direction can be controlled by both
internal ferroic orders and external light polarization. QNFOHE consists of two
types of nonlinear photocurrent responses - shift current and circular
photocurrent under linearly and circularly polarized light irradiation
dominated by topological shift vector and Berry curvature, respectively. We
elucidate the microscopic mechanism of QNFOHE in a representative class of 2D
multiferroic materials using group theoretical analyses and first-principles
electronic structure theory. The complex interplay of crystalline, permutation,
gauge, and time reversal symmetries as well as inherent causality governs the
symmetry properties of shift current and circular photocurrent. QNFOHE combined
with rich ferroic degrees of freedom in multiferroic materials will open up a
variety of new avenues for realizing tunable and configurable nonlinear
optoelectronics etc.
