Mutual entropy and thermal area law in C*-algebraic quantum lattice systems

We present a general definition of quantum mutual entropy for infinitely extended quantum spin and fermion lattice systems. Using this, we establish a thermal area law in these infinitely extended quantum systems. The proof is based on the local thermodynamical stability (LTS), a variational principle in terms of the conditional free energy. Our thermal area law in quasi-local C*-systems applies to general interactions with well-defined surface energies. We also examine the quantum mutual entropy between the left- and right-sided infinite regions of one-dimensional lattice systems. For general translation-invariant finite-range interactions on such systems, the thermal equilibrium state at any temperature exhibits a finite mutual entropy between these infinite disjoint regions. This further implies that the infinitely large quantum entanglement characteristic of critical ground states in one-dimensional systems is drastically destroyed by even a small positive temperature.