Operator Reflection Positivity Inequalities and their Applications to Interacting Quantum Rotors

In the Reflection Positivity theory and its application to statistical mechanical systems, certain matrix inequalities play a central role. The Dyson-Lieb-Simon [1] and Kennedy—Lieb—Shastry [2] inequalities constitute prominent examples. In this paper we extend the KLS inequality to the case where matrices are replaced by certain operators. As an application, we prove the occurrence of the long-range order in the ground state of two-dimensional quantum rotors.