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In the context of condensed matter physics, particularly phase transitions of transition metal compounds, I often encounter charge ordering (CO) and orbital ordering (OO). For me, the terms look similar because orbitals represent electron distributions thus, the way I see it, OO includes CO. But it seems that CO and OO are distinguished concepts. For example, in this paper, the authors wrote both terms.

Charge ordering and orbital ordering in ....

What is the difference between CO and OO?

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Typically, different orbitals have different properties under the point group symmetries of the lattice, and so if there is orbital ordering, there must also be charge ordering, and vice versa. This could be thought of in the context of a Landau theory of two fields that break the same symmetry.

$$F= r_1 \phi_1^2 + r_2 \phi_2^2 + u_1\phi_1^4 + u_2\phi_2^4 +\lambda \phi_1 \phi_2,$$ where e.g $\phi_1$ is orbital and $\phi_2$ is charge.

Clearly if one of the fields orders then the $\lambda$ term ensures that the other orders as well. There remains the question of what is driving the ordering: Are the orbital degrees of freedom the main player and the charge just goes along, or is it the other way around (in this example that will be determined by having one of the $r$'s negative and the other positive.).

This is clearly not a very sharply defined question. In fact, being able to define a unique 'mechanism' relies on having small coupling constants (in this example $\lambda$) between these degrees of freedom. Nevertheless this can have physical (and experimentally accessible) consequences. See for example arXiv:1312.6085 and arXiv:1306.4377 in the context of iron based superconductors. There may be better examples but this is what came to mind.

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