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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?

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Found the answer. It means density modulation.

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.

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