Introduction to Spontaneous Symmetry Breaking in Ising Model
Symmetry Breaking is an important topic appearing in different fields of Physics.
It has become "widely" famous when it has been proposed as an explanation for a common phenomenon like Ferromagnetism.
Let's think about a Ising Model, in first instance without specifying the number of dimensions, and so it's possible to formalize it this way
A N particles system with 2 spin values possible
Then let's consider the Hamiltonian of the System
Standard Hamiltonian taking into account both
the endogenous interaction that's a pairwise nearest neighbour interaction $ \sigma_{i} \sigma_{j} $
the exogenous interaction that's single spin interaction with an external field
Now let's consider just the endogenous interaction, assuming that the exogenous one is negligible compared to the first one
This assumption transforms the Hamiltonian this way
Approximated Hamiltonian under the assumption that the exogenous term is negligible compared to the endogenous one
Let's now define a Transformation called Spin Flip Transformation this way
Spin Flip Transformation just invertes the value of each spin in the System
Now it's time for a couple of important observation
Observation 1:
Approximated Hamiltonian is invariant to Spin Flip Transformation
The Energy Value computed using the approximated Hamiltonian is the same for the Normal System and the Spin Flipped System
The reason for this invariance stays in the exclusive presence of the pairwise interaction term: flipping the spin value doesn't change the spin product value.
Observation 2:
Initial Hamiltonian is not invariant to Spin Flip Transformation
The reason for the absence of invariance stays in the fact that in addition to the pairwise interaction term (that's invariant to spin flip transformation) there's a spin-external field interaction term
that's not invariant to spin flip.
Let's now consider an observable for the system like the system magnetization defined as follows
The System Magnetization is just the mean magnetization given by the orientation of every spin in the system
Let's now observe that the system magnetization presents odd symmetry with respect to the spin flip transformation
It means that the average system magnetization (defined as the average of all the possible system magnetization values for a given energy value) under the assumption that the external field is neglectible, thus to use the Approximated Hamiltonian, is expected to vanish.
This dynamic is indeed observed for
finite systems (no Thermodynamic Limit)
in the Thermodynamic Limit for $ D = 1 $ Monodimensional Systems
in the Thermodynamic Limit for $ D \ge 2 $ only if the Temperature of the System is high enough
It is indeed known that $ D \ge 2 $ Ising Systems present a Paramagnetic Phase and a Ferromagnetic Phase.
In the Paramagnetic Phase the abovementioned Symmetry holds and thus the measured value of the system magnetization is 0 like it is expected to be (according to the model).
In the Ferromagnetic Phase the measured value for the system magnetization in known to be all but 0 and this implies that the Symmetry has been broken.
According to the Model, the case for no Symmetry is the one where the Assumption that allowed us to use the Approximated Hamiltonian is not holding anymore and it implies necessarily that some sort of external field is acting in a non negligible way.
The abovementioned observations tell us that some sort of external field (or something acting as an external field) is always present
It's very interesting to observe that this sort of external field is not just an unerasable element of real world experiments, because of the observation that in every Monodimensional System the Symmetry is never broken and thus they present just the Paramagnetic Phase.
This should suggest that there is a correlation of some kind between this sort of external field and the number of dimensions of the system.
It could be shown that this correlation is due to the presence of Long Range Interactions that are responsible for this sort of external field.
They don't appear in $ D = 1 $ Systems but appear in $ D \ge 2 $ Systems.
