# What is *diagonal* long range order?

I have seen this question about off-diagonal long range order in superfluids.

What’s the difference and the significance between long range diagonal and off-diagonal long range order?

While off-diagonal long-range order is defined as $$n^{(1)}(r,r')=\langle \psi^\dagger (r)\psi (r')\rangle$$, diagonal long-range order is the special case where $$r = r'$$. Qualitatively, one can see that when $$r=r'$$ the argument of the expectation value is $$|\psi(r)|^2$$, i.e. the probability density of finding the particle in space. Hence, diagonal long-range order $$\leftrightarrow$$ density profile of the system.