# The role of symmetry

We can also see that the symmetry of the Hamiltonian of the system plays an important role for the possibility of phase transitions, and also in determining the lower critical dimension of the system. In particular the fact that for the Ising model phase transitions can occur for ${\displaystyle d>1}$ relies also on the fact that the Hamiltonian of the system has a discrete symmetry, namely its symmetry group is ${\displaystyle \mathbb {Z} _{2}}$. This means that the boundary of magnetic domains has finite "thickness" (one lattice unit, since a boundary in this case simply separates two adjacent antiparallel spins).

However, if the degrees of freedom have a continuous symmetry things start getting different. For example, the Hamiltonian of the Heisenberg model (see A slightly trickier system: the Heisenberg model) is:

${\displaystyle {\mathcal {H}}=-J\sum _{\left\langle i,j\right\rangle }{\vec {S}}_{i}\cdot {\vec {S}}_{j}-{\vec {H}}\cdot \sum _{i}{\vec {S}}_{i}}$
and in the case ${\displaystyle {\vec {H}}=0}$ the system has a rotational invariance. In fact, if ${\displaystyle R\in O(3)}$ then we easily see that:
${\displaystyle {\mathcal {H}}\left(\lbrace R{\vec {S}}_{i}\rbrace ,H=0\right)={\mathcal {H}}\left(\lbrace {\vec {S}}_{i}\rbrace ,H=0\right)}$
(of course if ${\displaystyle {\vec {H}}\neq 0}$ then the Hamiltonian will be invariant only under ${\displaystyle O(2)}$ transformations). In this case since the system has a wider symmetry its entropy will be much larger[1], and so we need more energy if we want long range order in the Heisenberg model at ${\displaystyle T>0}$ (in other words, the dimension of the domain wall will be comparable to the size of the system).

1. Very intuitively, since there are "more directions" for the spins, there will be much more possible configurations for the system.