They state in their book on page 792 the following:

It can be proved, however, that if $A$ and $B$ are operators linear in the $u(R)$ and $P(R)$ of a harmonic crystal, then: $$\langle e^A e^B \rangle = \exp((1/2)\langle A^2+2AB+B^2 \rangle)$$

They give as a reference the following paper by Mermin:

http://www.phys.ufl.edu/~maslov/phz6426/Mermin_JMathPhys_7_1038.pdf

But I don't see how to use this paper to prove the identity in the book.

Can anyone help with this?

P.S

$\langle \cdot \rangle$ denotes averaging.