conditions. we are interested on the continuous limit that correspond to matrices of infinite size. We will consider various size of the matrices expecting a stabilization of the values of observables like the energy density, increasing the matrix size. In order to find same possible phase transitions will be used the specific heat which is a measure of the dispersion of the energy. The phase transitions are registered as peaks of the specific heat, increasing the matrices size.

8-dim spectral action 
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In this section will be computed a spectral action starting from a non-commutative spectral triple. The feature of this particular triple is the choice of a 4-dimension Harmonic Dirac operator. The idea behind this construction [@8-dim] is to relate the Dirac operator with the oscillator Hamiltonian operator. Roughly speaking, we look at the Dirac operator as a generalization of the Laplace operator so we have $\mathcal{D}^2\approx H$. Considering the spectrum $\omega(2n+1)$ $n \in N$ of the one-dimensional harmonic oscillator Hamiltonian $H$, can be deduced that $H^{-1}$ is a non-commutative infinitesimal of order one. The non-commutative dimension of a spectral triple, equipped with the 4D Harmonic Dirac operator $ \mathcal{D}^2_4= H_4$, is fixed by the non-commutative order of the 