Large deviations in mean-field quantum spin systems

Continuous fields (or bundles) of $C^{\ast }$-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^{\ast }$-bundle generated by increasing symmetric tensor powers of the complex $\mathrm{\ell }\times \mathrm{\ell }$ matrices $M_{\mathrm{\ell }}(\mathbb{C})$, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.