Fifteen Eighty Four

Purpose of the book:

This book provides an advanced and detailed, yet pedagogical, account of the theoretical formal-
ism that describes quantum many-body systems departing from equilibrium under quite general
conditions. It deals specifically with the contour Green’s functions formalism, which is a general
and versatile framework that can be applied to finite and extended quantum many-body systems,
whether in equilibrium or in non-equilibrium, either at zero or finite temperature. The formalism
was originally introduced by Schwinger and subsequently extended by Keldysh, while an indepen-
dent related approach was put forward by Baym and Kadanoff.

Focus of the book:

This book focuses on the broad task of providing the readers with a practical working knowl-
edge on the way to use the tools of the contour many-body Green’s functions for time-dependent
problems in all of its details. Its main scope is to highlight the universality and versatility of the
contour Schwinger-Keldysh formalism to a whole class of physical phenomena. To this end, it
provides a self-contained introduction to the topic together with a considerable amount of detailed
derivations, which make the text accessible to graduate students with minimal training in Green’s
functions methods. At the same time, the book possesses a distinct degree of originality and
contains material which is not commonly found in other books or review articles on the subject.

Relevance of time-dependent phenomena in quantum many-body systems:

Many-body Green’s functions methods aim at reducing the detailed information contained in
the wave functions, by addressing directly dynamical quantities that are closely related to exper-
iments. This motivation is also borne out by more conventional Green’s functions methods, like
the zero-temperature method to describe systems weakly perturbed out of equilibrium at low tem-
peratures, or the Matsubara method for systems at equilibrium at any temperatures. The contour
Green’s functions formalism encompasses and generalizes both these more conventional methods,
by maintaining the same diagrammatic formal structure of the zero-temperature method, and ob-
taining explicit expressions of diagrammatic terms via a Wick’s theorem similar to that of the
Matsubara method.

Thus far, there is only a limited number of books available dealing with the non-equilibrium contour Green’s functions approach. This is because (i) while the zero-temperature and Matsubara methods often lead to analytic or semi-analytic results for which only limited numerical efforts may be sufficient, implementing the Schwinger-Keldysh contour Green’s functions formalism unavoid-ably requires considerable numerical efforts to begin with, and (ii) only quite recently experiments have developed the capability of following the time evolution of transient and/or metastable out-of-equilibrium dynamics of condensed-matter systems (as well as of ultra-cold gases). This book is meant to provide a bridge between this kind of experiments and the formal structure of the Schwinger-Keldysh contour Green’s functions formalism.

Technological vs theoretical aspects:

On the technological side, the interest in ultrafast excitation and relaxation phenomena has been
prompted by the needs of nano-electronics and the attainability of ultrashort coherent light sources
(generated, for instance, from electron storage rings), with pulses at the level of the femtosecond
(10⁻¹⁵s) and even sub-femtosecond in ultrafast pump-probe spectroscopies. The importance of
this new branch of physics was recognized by the 2023 Nobel Prize in Physics awarded to Pierre
Agostini, Ferenc Krausz, and Anne L’Huillier “for experimental methods that generate attosecond
pulses (10⁻¹⁸s) of light for the study of electron dynamics in matter.”

On the theoretical side, fundamental questions arise about the behavior of matter at these
ultrashort time scales, for which the structural dynamics and the formation of many-particle cor-
relations acquire special significance. All of this should then be described in a non-equilibrium
fashion, without any a-priori assumption on the statistical distribution of particles out of equilib-
rium and on the separation of time scales, as well as on the weakness of external perturbations and
of spatial inhomogeneities. These are precisely the tasks which the contour Green’s functions (or
Schwinger-Keldysh) formalism is optimally suited for.

Contents of the book:

The book is divided intro three Parts. The main introductory aspects of the Schwinger-Keldysh
formalism are presented in Part I, where bosons and fermions are dealt with on equal footing
in the “normal” phase (in the sense that no condensate is considered). Part II deals with the
“superfluid” phase for fermions, which are also allowed to undergo the BCS-BEC crossover. In
this way, emphasis will be given to the extension of the contour Schwinger-Keldysh formalism
to superconductors driven out of equilibrium and, more generally, to fermionic superfluids (like
ultra-cold Fermi gases) with their vastly different characteristic time scale compared to electron
systems. Part III is devoted to the discussion of a number of specific applications of the non-
equilibrium Schwinger-Keldysh formalism to various topics, including driven open quantum systems
(also in the superfluid phase and with emphasis on dissipation), as well as the relation with the
Lindblad equation whose role is acquiring increasing importance in the literature of non-equilibrium
processes. In addition, Part III covers at a qualitative level foreseen applications of the Schwinger-
Keldysh formalism to various topics of current and increasing experimental interest, like pump
and probe photoemission, metastable photo-induced superconductivity, and the dynamics induced
by quenches and ramps in closed quantum systems. State-of-the-art numerical methods are also
discussed in Part III.

Style of the book:

Throughout, the presentation style of the book proceeds at an even pace while providing all
necessary details, without “sweeping under the rug” the theoretical subtleties encountered along
the way but rather addressing them in full depth, with the intention of making the book a future
reference manual for practitioners and even specialists in the field. The many formal derivations
presented in the book are thus fully detailed, self-contained and rigorous, but still reader-friendly.

Target audience:

The disciplines for which the book will be of interest to are mainly Condensed Matter Physics,
Quantum Physics and Technologies, and ultra-cold Fermi gases undergoing the BCS-BEC crossover.
The book is also meant to be a pedagogical text for independent reading on time-dependent
problems for quantum many-body systems with a special perspective on superconductors, in such
a way that even advanced researchers could consult it at their needs. In addition, the book provides a ready-to-use manual, which instructors could easily use for their courses on this topic. The level of reader will be at the graduate and post-graduate level. Accordingly, the book may serve as a text for a one- or two-semester graduate course, as well as a useful reference for a wide audience of physicists dealing with physical problems for which use of the contour Schwinger-Keldysh formalism is relevant.

Title: Many-Body Green’s Functions for Time-Dependent Problems

Author: Giancarlo Calvanese Strinati

ISBN: 9781009411547