How to understand crystal momentum, its relation to translational symmetry, Noether theorem, and to symmetry breaking/Landau-Ginzburg theory?

Question

This question continues from my another question How to understand critical points of the Brillouin zone, (in)direct bands of transition-metal dichalcogenides?, and is related to Is crystal momentum really momentum?.

My understanding of crystal momentum is that it is the usual momentum modulo something, that is why it takes values only in the Brillouin zone. Meanwhile crystal momentum takes discrete values, which is for other reasons. We are talking about phonons which are the quantum version of vibrations, so the momentum (as well as the momentum modulo sth), energy, etc. of phonons are discrete.

We consider crystal momentum instead of momentum, because the translational symmetry is broken and therefore the momentum $\mu$ is not conserved. But since the crystal is still symmetric with respect to translations of $p*a, \forall p\in \mathbb{Z}$ where $a$ is a constant length, ($\mu \mod \frac{2\pi}a$) is conserved (as said below to be a non-strict application of Noether theorem (I am not familiar with this theorem)), we define crystal momentum for convenience, as ($\mu \mod \frac{2\pi}a$).

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The discussion of symmetry here is similar to that in Xiao-Liang Qi and Shou-Cheng Zhang, 2010, Topological insulators and superconductors.

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My questions are

1. Why translational symmetry is the prerequisite of momentum conservation?
2. When we talk about translational symmetry, what space is being discussed? For example, when we talk about crystals, the potential function $V(x)$ doesn't have translational symmetry (except for certain translations). For me, the crystal is 'embedded' in $\mathbb{R}^3$, which certainly has translational symmetry. So the space being discussed must involve potential or Hamiltonian function or a scalar field, etc.? In other words, we are actually talking about the symmetry of a field in the space?
3. Are the discussion of translational symmetry here similar to (or part of) the discussion of symmetry breaking and effective field theory (I am not familiar with these concepts either) in Qi's paper? If yes, why? (Since the paper says the theory is a universal description of quantum states except for quantum Hall..., the answer seems to be yes.)

Answer 1

To understand why translational symmetry implies momentum conservation at an intuitive level, think about the converse: if translational symmetry is broken, namely if the potential varies in space, then that means there is an applied force. But a system under an applied force does not have a conserved momentum.

This idea is formalized in Noether's theorem which says if a system has a symmetry, then there is a corresponding conserved quantity. For translational invariance this is momentum conservation. For time translation invariance it is energy conservation. For rotational invariance it is angular momentum conservation.

For the crystal case, you are right that it is embedded in R3, but when the crystal forms, it breaks that symmetry. Think about when the material is in the liquid state above its melting point; there it DOES have the full R3 symmetry. However, when the crystal forms, that symmetry is "spontaneously broken" because the crystal chooses to bond along some directions and not others, and the solid crystal now has a symmetry group that is a subset of the full R3 symmetry. You can still apply Noether's theorem to this smaller set of symmetries, but now the conserved quantity is the crystal momentum and not the real momentum.

The Ginzburg-Landau effective field theory is a way to think about all continuous spontaneous symmetry breaking phase transitions in general. Essentially it posits that these transitions can generally be described by expanding the free energy of the system in terms of a local order parameter that measures the degree of symmetry breaking. This order parameter is equal to zero in the symmetric phase but becomes nonzero in the symmetry broken phase. For the crystal solid formation transition, that order parameter would be the Fourier components of the charge density at the reciprocal lattice vectors. Another classic example is the magnetization for a ferromagnetic transition.

The quantum hall state doesn't fit in here because its an example of topological order where there is no local order parameter.