Many-Body Green's Functions for Time-Dependent Problems
Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.
Contents
Part I. Normal Phase; 1. Introduction; 2. The Schrödinger and Heisenberg Representations; 3. Splitting the Hamiltonian: Heisenburg and Interaction Pictures; 4. Time-dependent Quantum and Ensemble Averages: Initial Preparation of the System; 5. Quantum Averages over the Ground State and Gell-Mann-Low Theorem; 6. The Contour Idea for Time-dependent Averages: Forward and Backward Branches; 7. Closed Time Path Green's Functions; 8. Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane; 9. Perturbation Theory: Wick's Theorem for Strings of Operators Ordered Along a Contour; 10. Non-equilibrium Diagrammatics: Feynman Rules; 11. Non-equilibrium Dyson Equations; 12. Kubo-Martin-Schwinger Boundary Conditions; 13. Converting Contour-time to Real-time Arguments; 14. Langreth Rules: Convolutions and Products; 15. The Kadanoff-Baym Equations; 16. The T-matrix Approximation in the Normal Phase; 17. Contour Diagrammatic Structure in Terms of Functional Derivatives; 18. Beyond Linear-response Theory; 19. Time-dependent Hartree-Fock Approximation and Mean-field Decoupling; 20. Miscellany and Addenda to Part I; 21. Time-dependent Version of the BCS Hamiltonian: Gor'kov Equations for the Normal and Anomalous Single-particle Green's Functions; 22. The Hamiltonian in the Nambu Representation and Role of the Hartree-Fock-BCS Self-energy; 23. Contour-ordered Green's Functions in the Nambu Representation; 24. The T-matrix Approximation in the Superfluid Phase; 25. Derivation of the Time-dependent Bogoliubov-deGennes Equations; 26. A Brief Excursus to the BCS-BEC Crossover; 27. Analytic Continuation from the Imaginary to the Real Time Axis; 28. Derivation of the Time-dependent Gross-Pitaevskii Equation for Composite Bosons in the BEC Limit of the BCS-BEC Crossover; 29. Derivation of the Time-dependent Ginzburg-Landau (TDGL) Equation for Cooper pairs in the BCS Limit of the BCS-BEC Crossover; 30. Real-frequency Green's Functions from the Kadanoff-Baym Equations in the Equilibrium Case; 31. Miscellany and Addenda to Part II; 32. An Overview on Applications: Yesterday, Today, and Tomorrow; 33. Driven Open Quantum Systems; 34. Extension to Superfluid Fermi systems; 35. Connection between the Schwinger-Keldysh Closed-contour Approach and the Lindblad Master Equation; 36. State-of-the-art Numerical Methods; 37. Miscellany and Addenda to Part III.
Review 1
'Professor Strinati provides us with a pleasant and expert overview of the various standard techniques used in handling the many-body problem, mainly in fermionic systems. The book covers not only the classical Green's functions methods appropriate for equilibrium situations, but also the remarkable extension to out of equilibrium situations with the Keldysh formalism. It also addresses the famous Kadanoff and Baym approach. As a physical example for application Professor Strinati has chosen the fascinating phenomenon of the BEC-BCS crossover in ultracold fermionic atoms, a field in which he has made very important contributions. I highly recommend this remarkable book.' Roland Combescot, Ecole Normale Supérieure, Paris
Review 2
'Echoing the pedagogical style of Enrico Fermi, Professor Strinati guides the reader step-by-step through the nonequilibrium Green's function formalism. The text stands out for its mathematical completeness, offering rigorous derivations that expose theoretical details frequently omitted elsewhere. Bridging this foundational clarity with modern applications in ultracold gases, it serves as a valuable tutorial for students and an essential reference for researchers.' Hui Zhai, Institute for Advanced Study, Tsinghua University
Review 3
'Since Feynman introduced diagrammatic methods in quantum field theory, the techniques have undergone extensive development and now underpin a wide range of applications. These methods have required even further expansion to meet the challenges of describing nonequilibrium phenomena and nonlinear processes. This book is devoted to a key breakthrough in this field – the Keldysh formalism, or nonequilibrium Green's functions. Professor Giancarlo Calvanese Strinati, a world-renowned expert in condensed-matter theory, first presents the foundations and subtleties of the method and then applies it to a broad set of modern problems—from high-temperature superconductivity and ultracold gases to other strongly correlated systems, mesoscopic physics, and nanoscale electronics. I am confident that this monograph will help new generations of researchers master modern diagrammatic techniques, appreciate their elegance, and use them to drive future advances in understanding complex quantum systems.' Andrey Varlamov, CNR-SPIN, Rome
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Description
Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.
Contents
Part I. Normal Phase; 1. Introduction; 2. The Schrödinger and Heisenberg Representations; 3. Splitting the Hamiltonian: Heisenburg and Interaction Pictures; 4. Time-dependent Quantum and Ensemble Averages: Initial Preparation of the System; 5. Quantum Averages over the Ground State and Gell-Mann-Low Theorem; 6. The Contour Idea for Time-dependent Averages: Forward and Backward Branches; 7. Closed Time Path Green's Functions; 8. Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane; 9. Perturbation Theory: Wick's Theorem for Strings of Operators Ordered Along a Contour; 10. Non-equilibrium Diagrammatics: Feynman Rules; 11. Non-equilibrium Dyson Equations; 12. Kubo-Martin-Schwinger Boundary Conditions; 13. Converting Contour-time to Real-time Arguments; 14. Langreth Rules: Convolutions and Products; 15. The Kadanoff-Baym Equations; 16. The T-matrix Approximation in the Normal Phase; 17. Contour Diagrammatic Structure in Terms of Functional Derivatives; 18. Beyond Linear-response Theory; 19. Time-dependent Hartree-Fock Approximation and Mean-field Decoupling; 20. Miscellany and Addenda to Part I; 21. Time-dependent Version of the BCS Hamiltonian: Gor'kov Equations for the Normal and Anomalous Single-particle Green's Functions; 22. The Hamiltonian in the Nambu Representation and Role of the Hartree-Fock-BCS Self-energy; 23. Contour-ordered Green's Functions in the Nambu Representation; 24. The T-matrix Approximation in the Superfluid Phase; 25. Derivation of the Time-dependent Bogoliubov-deGennes Equations; 26. A Brief Excursus to the BCS-BEC Crossover; 27. Analytic Continuation from the Imaginary to the Real Time Axis; 28. Derivation of the Time-dependent Gross-Pitaevskii Equation for Composite Bosons in the BEC Limit of the BCS-BEC Crossover; 29. Derivation of the Time-dependent Ginzburg-Landau (TDGL) Equation for Cooper pairs in the BCS Limit of the BCS-BEC Crossover; 30. Real-frequency Green's Functions from the Kadanoff-Baym Equations in the Equilibrium Case; 31. Miscellany and Addenda to Part II; 32. An Overview on Applications: Yesterday, Today, and Tomorrow; 33. Driven Open Quantum Systems; 34. Extension to Superfluid Fermi systems; 35. Connection between the Schwinger-Keldysh Closed-contour Approach and the Lindblad Master Equation; 36. State-of-the-art Numerical Methods; 37. Miscellany and Addenda to Part III.
Review 1
'Professor Strinati provides us with a pleasant and expert overview of the various standard techniques used in handling the many-body problem, mainly in fermionic systems. The book covers not only the classical Green's functions methods appropriate for equilibrium situations, but also the remarkable extension to out of equilibrium situations with the Keldysh formalism. It also addresses the famous Kadanoff and Baym approach. As a physical example for application Professor Strinati has chosen the fascinating phenomenon of the BEC-BCS crossover in ultracold fermionic atoms, a field in which he has made very important contributions. I highly recommend this remarkable book.' Roland Combescot, Ecole Normale Supérieure, Paris
Review 2
'Echoing the pedagogical style of Enrico Fermi, Professor Strinati guides the reader step-by-step through the nonequilibrium Green's function formalism. The text stands out for its mathematical completeness, offering rigorous derivations that expose theoretical details frequently omitted elsewhere. Bridging this foundational clarity with modern applications in ultracold gases, it serves as a valuable tutorial for students and an essential reference for researchers.' Hui Zhai, Institute for Advanced Study, Tsinghua University
Review 3
'Since Feynman introduced diagrammatic methods in quantum field theory, the techniques have undergone extensive development and now underpin a wide range of applications. These methods have required even further expansion to meet the challenges of describing nonequilibrium phenomena and nonlinear processes. This book is devoted to a key breakthrough in this field – the Keldysh formalism, or nonequilibrium Green's functions. Professor Giancarlo Calvanese Strinati, a world-renowned expert in condensed-matter theory, first presents the foundations and subtleties of the method and then applies it to a broad set of modern problems—from high-temperature superconductivity and ultracold gases to other strongly correlated systems, mesoscopic physics, and nanoscale electronics. I am confident that this monograph will help new generations of researchers master modern diagrammatic techniques, appreciate their elegance, and use them to drive future advances in understanding complex quantum systems.' Andrey Varlamov, CNR-SPIN, Rome
