# Understanding Z2 Symmetry in Models

• touqra
In summary, Z2 symmetry is a symmetry that allows for the same equations of motion when the field is taken to -\phi. It is usually necessary for models with physical applications, such as the Ising model or \mathbb{Z}_2 topological quantum field theory.
touqra
Can someone explain what's Z2 symmetry ? Is it necessary to have it in a model, even SM ?

Are you talking about $\mathbb{Z}_2$?

Z2 is usually a symmetry like something goes to - something.

So, for example, I can write this lagrangian:

$$\mathcal{L} = \frac{1}{2}\partial_{\mu} \phi \partial^{\mu} \phi+\lambda\phi^4$$

The Z2 symmetry is manifest---that is I can always take $$\phi$$ to $$-\phi$$ and get the same lagrangian back.

As far as necessarily needing it for anything, I don't know, but I don't suspect there's anything particularly deep about it.

Even simple, $\mathbb{Z}_2$ refers to a group, namely that of additon mod 2. It only has two elements, the identity and element a, which satisfies a^2 = 1.

You can also view this as the numbers 0 and 1 with "group multiplication" identified with addition modulo 2. See for yourself:
0+0 = 0
0+1 = 1+ 0 = 1
1+1 = 0 (mod 2)

In the context of physics the $\mathbb{Z}_2$ symmetry usually refers to the fact that we are dealing with some system which, among a lot more stuff, contains an invariance with respect to some $\mathbb{Z}_2$ operation. A simple case is the example given by BenTheMan. Other contexts include the Ising model, $\mathbb{Z}_2$ topological quantum field theory, orbifolds, etc.

BenTheMan said:
Z2 is usually a symmetry like something goes to - something.

So, for example, I can write this lagrangian:

$$\mathcal{L} = \frac{1}{2}\partial_{\mu} \phi \partial^{\mu} \phi+\lambda\phi^4$$

The Z2 symmetry is manifest---that is I can always take $$\phi$$ to $$-\phi$$ and get the same lagrangian back.

As far as necessarily needing it for anything, I don't know, but I don't suspect there's anything particularly deep about it.

What does it mean to have a $$-\phi$$ field ?

touqra said:
What does it mean to have a $$-\phi$$ field ?

Hmm. I don't know. In the context of the example I gave you, it doesn't mean anything because of the symmetry. $$\phi$$ and $$-\phi$$ give the same lagrangian, so the equations of motion are unchanged---that is, the physics is exactly the same, so (in some sense) there IS no meaning to $$-\phi$$.

In a classical model $-\phi(x)$ stands for the amplitude of the field. The amplitude can be all sorts of things, most prominent example being the displacement of an atom with respect to some mean lattice.

In that case a postive $\phi(x)$ is a displacement of the field in the positive direction at the point x, and a negative $\phi(x)$ is one in the negative direction. (think of it as a rubber sheet stretched out, with bumps here and there).

The fact that we have a symmetry means that the different states $\phi(x)$ and $-\phi(x)$ carry the same energy.

$\mathbb{Z}_2$ is useful in higher dimensional theories to project out unwanted zero modes for the photon.

You can see this quite nicely in the Kaluza-Klein lagrangian where the metric is parameterized on an $S^1$ compactification:

$$ds^2=\phi^{-1/3}(g_{\mu\nu}+A_\mu A_\nu\phi)dx^\mu dx^\nu+\phi^{2/3}A_\mu dx^\mu dy+\phi^{2/3}dy^2$$

The $\mathbb{Z}_2$ orbifolding is obtained after identifying y with -y.

The invariance of the interval $ds^2$ under the symmetry determines the transformation properties of all the fields:

$$g_{\mu\nu}(y)=g_{\mu\nu}(-y)$$

$$A_{\mu}(y)=-A_{\mu}(-y)$$

$$\phi(y)=\phi(-y)$$

Since the field $A_\mu$ is odd under this symmetry it cannot have a zero mode, eg when you write the fields out like:

$$A_\mu(\vec{x},y)=\sum_{n=-\infty}^\infty A(\vec{x})e^{iny/r}$$

:)

## 1. What is Z2 symmetry?

Z2 symmetry, also known as discrete or binary symmetry, is a type of symmetry that involves only two possible values or states. In physics, it is often used to describe systems that exhibit a property where the system remains unchanged after a certain transformation, such as a reflection or a rotation by 180 degrees.

## 2. How is Z2 symmetry relevant in models?

Z2 symmetry is relevant in models because it can simplify and constrain the behavior of a system. By imposing Z2 symmetry, certain interactions or terms in the model can be forbidden, leading to a more elegant and predictive model. Z2 symmetry is commonly used in particle physics, condensed matter physics, and other fields of physics.

## 3. What are some examples of models that utilize Z2 symmetry?

One example is the Ising model, which describes the behavior of ferromagnetic materials and uses Z2 symmetry to constrain the orientation of spins. Another example is the Higgs mechanism in particle physics, where the Higgs field is assumed to have Z2 symmetry. This symmetry breaking leads to the spontaneous generation of the masses of particles.

## 4. How is Z2 symmetry broken?

Z2 symmetry can be broken in two ways: explicitly or spontaneously. Explicit breaking occurs when the symmetry is not present in the model or when certain terms are added that violate the symmetry. Spontaneous breaking occurs when the symmetry is present, but the system chooses a particular state that is not symmetric. This can lead to the appearance of new phenomena, such as mass generation in the Higgs mechanism.

## 5. What are the implications of understanding Z2 symmetry in models?

Understanding Z2 symmetry in models can provide insights into the behavior and properties of physical systems. It allows for the simplification and prediction of complex systems, and can also lead to the discovery of new phenomena. Additionally, understanding Z2 symmetry can help bridge the gap between theoretical models and experimental observations, leading to a better understanding of the natural world.

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