I am working with a circular lattice of N oscillators, where each oscillator have an interaction with its adjacent oscillators. After tracing out n from N of the oscillators I ended up with the following reduced density operator, where A is complex symmetric, A∗ is the complex conjugate(not to be confused with Hermitian Conjugate), B is Hermitian x=(xn+1,…,xN) (the position coordinates of the oscillators)
ρout(x,x′)=exp(xTAx+x′TA∗x′+xTBx′)
The eigenvalues of this operator is proportional to the determinant of B. If one of the eigenvalues of B is zero, then the entanglement entropy is also zero. Is there any physical interpretation to a zero eigenvalue of B ?