Theory of Strongly Correlated Systems

This book provides an analytically tractable framework for exploring strongly correlated quantum systems beyond conventional paradigms such as Landau’s Fermi liquid theory. This book offers a pedagogical and self-contained introduction to SYK physics, emphasizing its role in describing strange metals —systems with linear-in-temperature resistivity, Planckian dissipation, and quasiparticle breakdown. Beginning with an invitation to the SYK paradigm, the text develops both Majorana and complex fermion variants and presents modern many-body tools including large-N formulations, Schwinger–Dyson and Kadanoff–Baym equations, Matsubara and Keldysh formalisms, and nonperturbative contour deformations. These methods are applied to study equilibrium thermodynamics, quantum chaos, quench dynamics, and transport phenomena. Extensive appendices provide mathematical background and Mathematica® implementations for critical exponents, conductivity, and transport across temperature regimes. Bridging analytical depth with computational practice, this book also highlights the deep connections between the SYK model, black hole thermodynamics, and holographic duality, illustrating how a simple fermionic model offers insight into both condensed matter systems and quantum gravity. It serves as an essential resource for advanced students and researchers in strongly correlated and holographic quantum matter.