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I've seen the picture of nematic phase in liquid crystals like this one https://saylordotorg.github.io/text_general-chemistry-principles-patterns-and-applications-v1.0/s15-08-liquid-crystals.html, which is quite intuitive.

However, I find it confusing when considering it with crystal solids. It's said that in the nematic phase of superconductors, the rotational symmetry is broken but the translation symmetry is preserved. What does this mean? What breaks the symmetry?

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It's not a nematic phase of the underlying atomic lattice, but rather of the superconducting order parameter $\psi$, which is a complex scalar for $s$-wave superconductors but e.g. an element of $\mathbb C^5$ for $d$-wave superconductors. In the latter case, one can have translational symmetry ($\psi$ takes the same value everywhere) without rotational symmetry.

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