Thermodynamics relies on the possibility to describe systems composed of a large number of constituents in terms of few macroscopic variables. Its foundations are rooted into the paradigm of statistical mechanics, where thermal properties originate from averaging procedures which smoothen out local details. While undoubtedly successful, elegant and formally correct, this approach carries over an operational problem, namely determining the precision at which such variables are inferred, when technical/practical limitations restrict our capabilities to local probing. Here we introduce the local quantum thermal susceptibility, a quantifier for the best achievable accuracy for temperature estimation via local measurements. Our method relies on basic concepts of quantum estimation theory, providing an operative strategy to address the local thermal response of arbitrary quantum systems at equilibrium. At low temperatures, it highlights the local distinguishability of the ground state from the excited sub-manifolds, thus providing a method to locate quantum phase transitions.

We study the distribution of the Schmidt coefficients of the reduced density matrix of a quantum system in a pure state. By applying general methods of statistical mechanics, we introduce a fictitious temperature and a partition function and translate the problem in terms of the distribution of the eigenvalues of random matrices. We investigate the appearance of two phase transitions, one at a positive temperature, associated with very entangled states, and one at a negative temperature, signaling the appearance of a significant factorization in the many-body wave function. We also focus on the presence of metastable states (related to two-dimensional quantum gravity) and study the finite size corrections to the saddle point solution

We exactly diagonalize the finite-size XY model with periodic boundary conditions and analytically determine the ground state energy. We show that there are two types of fermions: singles and pairs, whose dispersion relations have a completely arbitrary gauge-dependent sign. It follows that the ground state is determined by a competition between the vacuum states (with a suitable gauge) of two parity sectors. We finally exhibit some points in finite size systems that forerun criticality. They are associated to single Bogoliubov fermions and to the level crossings between physical and unphysical states. In the thermodynamic limit they approach the ground state and build up singularities at logarithmic rates.