13 Quantum Cluster Methods: CPT and CDMFT David S´ en´ echal D´ epartement de physique Universit ´ e de Sherbrooke, Qu´ ebec, Canada Contents 1 Introduction 2 1.1 The Hubbard model and Green functions ..................... 2 1.2 Clusters ...................................... 6 1.3 Cluster Perturbation Theory ........................... 7 2 Periodic systems 8 2.1 Cluster kinematics ................................ 8 2.2 The CPT Green function and periodization .................... 11 3 The exact diagonalization method 15 4 Cellular Dynamical Mean-Field Theory 16 4.1 The dynamical mean field ............................. 16 4.2 The equivalent Anderson impurity model .................... 18 4.3 The self-consistency condition .......................... 20 4.4 Inhomogeneous systems ............................. 22 5 Applications 23 5.1 The Mott transition ................................ 23 5.2 Superconductivity ................................. 25 6 Extended interactions 28 E. Pavarini, E. Koch, and P. Coleman (eds.) Many-Body Physics: From Kondo to Hubbard Modeling and Simulation Vol. 5 Forschungszentrum J ¨ ulich, 2015, ISBN 978-3-95806-074-6 http://www.cond-mat.de/events/correl15