Is there a conservation law associated with C4 symmetry?

• csmallw
In summary, continuous rotational symmetry implies conservation of angular momentum and continuous translational symmetry implies conservation of linear momentum. For discrete translational symmetry, conservation of crystal momentum is only up to a reciprocal lattice vector. While there is no direct conservation law associated with discrete rotational symmetry, it can be used to predict macroscopic properties of crystals such as ferroelectric polarization. These predictions can be stated in terms of conservation laws, such as the conservation of a ferroelectric moment perpendicular to a C4 axis. However, this does not have a direct connection to superconductivity.
csmallw
I know--because of Noether's theorem--that continuous rotational symmetry implies conservation of angular momentum, and that continuous translational symmetry implies conservation of linear momentum. It also turns out that the discrete translational symmetry exhibited by a Bravais lattice implies conservation of crystal momentum.

Are there conservation laws associated with discrete rotational symmetries (like C4 symmetry)?

I was originally thinking about Cooper pairs in a superconductor when this question came to mind. If there is no conservation law associated with discrete rotational symmetry, then it seems to me that the concept of the orbital angular momentum of a Cooper pair ("s-wave," "d-wave," etc.) might be kind of meaningless.

Note that the conservation of crystal momentum is only up to a reciprocal lattice vector.
Even with full rotational symmetry a scattering like d+d->s+g would be in accordance with angular momentum conservation.
But you are right that in principle it would be more precise to use group labels from C4 (or whatever applies to the crystals under consideration) instead of labels from continuous rotation groups.

Discrete rotation symmetries can be used to predict certain macroscopic properties of crystals.

For example, ferroelectric polarization can occur only parallel to a C4 axis (or C2 or C3 for that matter).

Consequently, crystals with two non-collinear rotation axes cannot be ferroelectric.

If you want to state this in the form of a conservation law, then the ferroelectic moment perpendicular to the axis is conserved -at zero.

This has no direct connection to superconductivity that I am aware of.

1. What is C4 symmetry?

C4 symmetry refers to a type of symmetry in which an object or system can be rotated by 90 degrees and still appear unchanged. This means that there are four distinct orientations in which the object or system can exist without any visible differences.

2. Is there a conservation law associated with C4 symmetry?

Yes, there is a conservation law associated with C4 symmetry. This law states that certain physical quantities, such as energy or momentum, remain constant when the system undergoes a transformation with C4 symmetry. This means that these quantities are conserved and do not change even when the system is rotated by 90 degrees.

3. How does C4 symmetry relate to conservation laws?

C4 symmetry is closely related to conservation laws because it describes a type of symmetry that preserves certain physical quantities within a system. This means that the system is constrained to behave in a certain way in order to maintain this symmetry and conserve these quantities.

4. Can C4 symmetry be broken?

Yes, C4 symmetry can be broken in certain systems. This can occur if external forces or interactions cause the system to deviate from its symmetrical state, leading to a violation of the associated conservation law. However, in many cases, C4 symmetry is a fundamental property of the system and cannot be broken.

5. What are some examples of systems with C4 symmetry?

Some examples of systems with C4 symmetry include crystals, molecules, and certain physical systems such as a spinning top. These systems exhibit four-fold rotational symmetry and have associated conservation laws, such as the conservation of angular momentum, due to this symmetry.

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