LAPACK routine ZGEEV - gives wrong eigenvalues - fortran90

Question

I wrote the following program in fortran that uses a lapack subroutine called ZGEEV. The idea was to see how the eigenvalues of the matrix change as k goes from real to complex. Analytically, the answers should be 2 and 0, whether k is complex or not. But I obtain a plot that shows a lot variation.
Especially for real k, the plot looks like this -
Here is the code i wrote -
program main
implicit none
!**********************************************
complex(8) :: k,mat(2,2)
complex(8) :: eigenvals(2)
real(8), parameter :: kmax = 2.d0
real(8), parameter :: dk = 1.d-1
real(8) :: kr,ki
!**********************************************
kr=-kmax
do while (kr.le.kmax)
ki= -1.d-3
do while (ki.le.1.d-3)
k=cmplx(kr,ki)
call init_mat(k,mat)
call diagonalize(mat,eigenvals)
print*, real(k), real(eigenvals(2)),aimag(eigenvals(2))
ki=ki+1.d-4
end do
kr=kr+dk
end do
end program main
subroutine init_mat(k,mat)
implicit none
complex(8),intent(in) :: k
complex(8),intent(out):: mat(2,2)
complex(8),parameter :: di=(0.d0,1.d0)
complex(8),parameter :: d1=(1.d0,0.d0)
!**********************************************
mat(1,1) = d1
mat(1,2) = exp(di*k)
mat(2,1) = exp(di*k)
mat(2,2) = d1
return
end subroutine init_mat
subroutine diagonalize(mat,eigenvals)
implicit none
complex(8),intent(in) :: mat(2,2)
complex(8),intent(out):: eigenvals(2)
complex(8) :: vl(2,2),vr(2,2)
complex(8),allocatable:: work(:)
integer(4) :: lwork
complex(8) :: rwork(4)
complex(8) :: mat2(2,2)
integer(4) :: info
!**********************************************
mat2(:,:) = mat(:,:)
allocate(work(6))
call zgeev('N', 'N', 2, mat2, 2, eigenvals, vl, 2, vr, 2, work, -1, rwork, info)
lwork = work(1)
deallocate(work)
allocate(work(lwork))
call zgeev('V', 'V', 2, mat2, 2, eigenvals, vl, 2, vr, 2, work, lwork, rwork, info)
if (info.ne.0) print*, info
stop 'diagonalize failed'
end subroutine diagonalize
Any lazy theorizing as to the causes of this aberration is welcome in the comments!
PS: i wrote up a similar code in python and there the eigenvalues are two constant lines at y=2 and y=0.

Answer 1

in subroutine init_mat(k,mat)
mat(1,2) = exp(di*k)
and
mat(2,1) = exp(di*k)
But one of them, e.g., mat(2,1) should = exp(-di*k)
Although your math project calls for a matrix with e^ik and e^-ik on the off-diagonals, the code shown instead is creating a matrix with e^ik on both off-diagonals. The matrix actually coded has complex eigenvalues, so the subroutines for finding eigenvalues may be working correctly and the input as shown has a mis-specification.
So what are the eigenvalues of [[1, e^ik], [e^ik, 1]]?
Well, the trace is still 2, so the eigenvalues sum to 2.
And the determinant is 1-e^(2ik), so the product is complex.
This suggests that the eigenvalues of the matrix actually input are complex conjugates that sum to 2. By inspection, the eigenvalues seem to be 1 +/- e^ik