  We examine thermal Green's functions of fermionic operators in quantum field
theories with gravity duals. The calculations are performed on the gravity side
using ingoing Eddington-Finkelstein coordinates. We find that at negative
imaginary Matsubara frequencies and special values of the wavenumber, there are
multiple solutions to the bulk equations of motion that are ingoing at the
horizon and thus the boundary Green's function is not uniquely defined. At
these points in Fourier space a line of poles and a line of zeros of the
correlator intersect. We analyze these `pole-skipping' points in
three-dimensional asymptotically anti-de Sitter spacetimes where exact Green's
functions are known. We then generalize the procedure to higher-dimensional
spacetimes. We also discuss the special case of a fermion with half-integer
mass in the BTZ background. We discuss the implications and possible
generalizations of the results.
