Non-Equilibrium Quantum Spin Transport Theory Based on Schwinger-Keldysh Formalism

We extend our study of the large- $N$ expansion of general non-equilibrium many-body systems with matrix degrees of freedom $M$ , and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields $M_\pm$ -- defined as the values of $M$ on the forward and backward segments of the closed time contour -- for their linear combinations $M_{\textrm{cl}}$ and $M_{\textrm{qu}}$ , known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface $Σ$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts $Σ^\pm$ corresponding to the forward and backward segments of the closed time contour, and $Σ^\wedge$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface $Σ$ of the dual string theory undergoes a very different natural decomposition: $Σ$ consists of a "classical" part $Σ^{\textrm{cl}}$ , and a "quantum embellishment" part $Σ^{\textrm{qu}}$ . We show that both parts of $Σ$ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each $Σ$ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.We extend our study of the large-$N$ expansion of general non-equilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields $M_\pm$ -- defined as the values of $M$ on the forward and backward segments of the closed time contour -- for their linear combinations $M_{\textrm{cl}}$ and $M_{\textrm{qu}}$, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface $Σ$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts $Σ^\pm$ corresponding to the forward and backward segments of the closed time contour, and $Σ^\wedge$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface $Σ$ of the dual string theory undergoes a very different natural decomposition: $Σ$ consists of a "classical" part $Σ^{\textrm{cl}}$, and a "quantum embellishment" part $Σ^{\textrm{qu}}$. We show that both parts of $Σ$ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each $Σ$ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

arXiv:2010.10671Oct 20, 20202 citations

Quantum transport: the Non-Equilibrium Green's Function formalism

WORLD SCIENTIFIC eBooksOct 5, 20160 citations

Effective equilibrium theory of nonequilibrium quantum transport

Prasenjit Dutt, Jens Koch, Jong E. Han +1 more

arXiv:1004.5591Annals of PhysicsJul 22, 201125 citations

A UNIFIED FORMALISM OF THERMAL QUANTUM FIELD THEORY

H. Chu, Hiroomi Umezawa

We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.We present a comprehensive review of the most fundamental and practical aspects of thermo-field dynamics (TFD), including some of the most recent developments in the field. To make TFD fully consistent, some suitable changes in the structure of the thermal doublets and the Bogoliubov transformation matrices have been made. A close comparison between TFD and the Schwinger-Keldysh closed time path formalism (SKF) is presented. We find that TFD and SKF are in many ways the same in form; in particular, the two approaches are identical in stationary situations. However, TFD and SKF are quite different in time-dependent nonequilibrium situations. The main source of this difference is that the time evolution of the density matrix itself is ignored in SKF while in TFD it is replaced by a time-dependent Bogoliubov transformation. In this sense TFD is a better candidate for time-dependent quantum field theory. Even in equilibrium situations, TFD has some remarkable advantages over the Matsubara approach and SKF, the most notable being the Feynman diagram recipes, which we will present. We will show that the calculations of two-point functions are simplified, instead of being complicated, by the matrix nature of the formalism. We will present some explicit calculations using TFD, including space-time inhomogeneous situations and the vacuum polarization in equilibrium relativistic QED.

International Journal of Modern Physics AJun 10, 199457 citations

Nonequilibrium Quantum Transport Theory of Spinful and Topological Systems

F. A. Buot

WORLD SCIENTIFIC eBooksJul 21, 20220 citations

We extend our study of the large-

N

expansion of general non-equilibrium many-body systems with matrix degrees of freedom

M

, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields

M_\pm

-- defined as the values of

M

on the forward and backward segments of the closed time contour -- for their linear combinations

M_{\textrm{cl}}

and

M_{\textrm{qu}}

, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface

Σ

in the dual string theory expansion was found to exhibit a triple decomposition into the parts

Σ^\pm

corresponding to the forward and backward segments of the closed time contour, and

Σ^\wedge

which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface

Σ

of the dual string theory undergoes a very different natural decomposition:

Σ

consists of a "classical" part

Σ^{\textrm{cl}}

, and a "quantum embellishment" part

Σ^{\textrm{qu}}

. We show that both parts of

Σ

carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each

Σ

into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.