# Is non-temperature related Symmetry Breaking possible?

In physics. Symmetry Breaking is related to temperature like in electroweak unification at high energies. Does symmetry breaking always involve temperature? Can you give examples of symmetry breaking (related to broken symmetry and stuff) in low temperature daily process that doesn't involve temperature?

If symmetry breaking always involve temperature. Then what term one must use if one want something similar to symmetry breaking but not involving temperature?

• Off the top of my head how about pressure, magnetization, concentration changes that lead to phase changes? Commented Jul 1 at 1:50

Temperature is not needed to define symmetry breaking. Symmetry breaking simply refers to the loss of some symmetry during the evolution of a system. Given that a system is in a state $$A$$ that is symmetric, meaning it is invariant under some operation $$S$$ (such as rotation, translation, etc.), $$SA=A$$. If the system ends up in a state B that is not symmetric ($$SB\neq B$$), then symmetry breaking has occurred.

Example: Using your finger you mantain a pencil completely vertical over a table, so that only the tip of the pencil is in contact with the surface of the table. This is state $$A$$ that is invariant under rotations. State $$A$$ is unstable, and as soon as you remove your finger, the pencil falls and rests in horizontal position on the table. This is state $$B$$ that is not rotationally invariant.

In this example, there is symmetry breaking because there is a multiplicity of ground states (all possible horizontal orientations of the pencil over the table). In this case, the system evolves towards the ground state simply following the equations of motion once the constraint (the finger) has been removed. In the context of phase transitions, the ground state is typically obtained by reducing the temperature, but this is not always the case, as shown in this example.

I recommend reading Wikipedia: Symmetry breaking for other examples of symmetry breaking that do not involve temperature.

In physics. Symmetry Breaking is related to temperature [...]

Symmetry breaking in classical physics
More correct way of saying this is that in (classical) physics symmetry breaking is related to a thermodynamic state/equilibrium. Without relation to thermodynamics, there is nothing unusual about symmetry breaking - if we have a symmetric potential with several minima and put an object in one of these minima, we break the symmetry.

However, in thermodynamics/statistical physics we postulate that the system will reach equilibrium configuration, where the probability of a state is determined solely by its energy (there are some caveats related to other integrals of motion, but these are usually excluded in the first chapter of a thermodynamics/stat physics textbook.) Then the symmetric states have the same probabilities, i.e., the probability distribution should be symmetric.

This is not what we observe in practice - a ferromagnet has a well-defined magnetization direction, even though all the directions are equivalent. This has to do with violation of the basic thermodynamic assumptions - one can say that the system either hasn't reached equilibrium or that it evolves too slowly to equate time averaging with statistical averaging (ergodicity). Hence the seeming violation of thermodynamic laws (which should not apply), which we call symmetry breaking.

Symmetry breaking in Quantum mechanics
I stressed that the above argument applies to classical systems, since in the quantum case we can have probability distributions of different nature, even without resorting to thermodynamics. E.g., a two-well potential has a symmetric ground state, but if we start with a wave packet in one well, the symmetry is broken and it may take a very long time for the packet to tunnel into the other well. Without invoking thermodynamics the wave function might become symmetric for an instant, but this symmetry is not preserved - rather we will observe back and forth oscillations between the two minima, because our initial state was not a ground state, but a superposition of states. Now, if we demanded that the system be in the ground state, we expect it to be symmetric - but here we get back to the thermodynamics, which is the reason why we expect a quantum system to be in the ground state (at zero temperature).

Quantum phase transitions
Finally, changes of symmetry may happen at zero temperature, controlled by parameters other than temperature - e.g., a disordered state may become an ordered one or the other way around:

Here the system remains truly thermodynamic, but the symmetry of the ground state changes. I am not sure whether anyone uses symmetry breaking language to describe such situations, but this is probably the closest that one can get to symmetry breaking without temperature.

• I asked all these because electroweak symmetry breaking involves high temperature. Is it possible monopoles could be more symmetric state of electromagnetism that occurs at low temperature? What theory involves something like this?
– Jtl
Commented Jul 1 at 13:24
• @Jtl I am rather ignorant about HEP, so I have no idea in what sense the temperature is high? Doesn't it simply means that it is not zero? Commented Jul 1 at 13:28
• I read "During this era, only the electromagnetic and nuclear weak forces are still combined. The temperature of the universe at this stage is more than 1015 K, and there are no ordinary particles yet, just photons and pure energy.".
– Jtl
Commented Jul 1 at 14:14