How do crystals break translational and rotational invariance? If I arbitrary displace a crystal, it is still the same and its energy is still the same. So how then is translational symmetry broken? Same applies for rotation.
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I am also a learner, but I want to lay foundation for people to come and make corrections and offer you the best answer:
Suppose you move a gas molecule in a homogeneous gas from one point to another, the state of the gas molecule does not change. The homogeneous gas in this case is said to have translational symmetry. However, if you move an electron in a crystal from one point to another, the state functions describing the electron at the two points cannot be said to the same. The crystal in this case is said to break translational symmetry spontaneously. In Bloch's theorem, you can take the advantage of the periodicity of the Crytal lattice to translate the wave vector:
From this one-dimensional translation, we say that $\Psi_1(x)\neq\Psi_2(x)$ breaks translational symmetry. Using the periodicity of the lattice constant(L), we can write $\Psi_1(x)=\Psi_2(x+L)$.
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1$\begingroup$ Welcome to PSE! Instead of using images, you can type in formulas using MathJax. You can find a tutorial here: math.meta.stackexchange.com/questions/5020/… $\endgroup$ Commented Sep 18, 2022 at 13:15
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The translational and rotational symmetries of a crystal are discrete- a cubic crystal appears the same if you rotate it about a face by a right angle, not by a smaller angle.
The same way, the crystal remains the same if it is translated by certain distances along certain directions only- these are the lattice translations. Both symmetries are discrete, which is lower than the continuous symmetry where you could translate/rotate in any manner and have the crystal look the same.
The energy need not remain the same if you arbitrarily rotate it. Consider a magnetic crystal in a magnetic field. Because magnetisation is anisotropic, rotation would change the energy of the crystal.
