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

Duality and bosonization in Schwinger–Keldysh formulation

R E Gamboa Saraví, Carlos Naón, F Schaposnik

We present a path-integral bosonization approach for systems out of equilibrium based on a duality transformation of the original Dirac fermion theory combined with the Schwinger–Keldysh time closed contour technique, to handle the non-equilibrium situation. The duality approach to bosonization that we present is valid for D ≥ 2 space–time dimensions leading for D = 2 to exact results. In this last case we present the bosonization rules for fermion currents, calculate current–current correlation functions and establish the connection between the fermionic and bosonic distribution functions in a generic, non-equilibrium situation.We present a path-integral bosonization approach for systems out of equilibrium based on a duality transformation of the original Dirac fermion theory combined with the Schwinger–Keldysh time closed contour technique, to handle the non-equilibrium situation. The duality approach to bosonization that we present is valid for D ≥ 2 space–time dimensions leading for D = 2 to exact results. In this last case we present the bosonization rules for fermion currents, calculate current–current correlation functions and establish the connection between the fermionic and bosonic distribution functions in a generic, non-equilibrium situation.

arXiv:1406.4578Journal of Statistical Mechanics Theory and ExperimentSep 30, 20142 citations

Dynamical equations for time-ordered Green’s functions: from the Keldysh time-loop contour to equilibrium at finite and zero temperature

H. Ness, L. K. Dash

We study the dynamical equation of the time-ordered Green's function at finite temperature. We show that the time-ordered Green's function obeys a conventional Dyson equation only at equilibrium and in the limit of zero temperature. In all other cases, i.e. finite temperature at equilibrium or non-equilibrium, the time-ordered Green's function obeys instead a modified Dyson equation. The derivation of this result is obtained from the general formalism of the non-equilibrium Green's functions on the Keldysh time-loop contour. At equilibrium, our result is fully consistent with the Matsubara temperature Green's function formalism and also justifies rigorously the correction terms introduced in an ad hoc way with Hedin and Lundqvist. Our results show that one should use the appropriate dynamical equation for the time-ordered Green's function when working beyond the equilibrium zero-temperature limit.We study the dynamical equation of the time-ordered Green's function at finite temperature. We show that the time-ordered Green's function obeys a conventional Dyson equation only at equilibrium and in the limit of zero temperature. In all other cases, i.e. finite temperature at equilibrium or non-equilibrium, the time-ordered Green's function obeys instead a modified Dyson equation. The derivation of this result is obtained from the general formalism of the non-equilibrium Green's functions on the Keldysh time-loop contour. At equilibrium, our result is fully consistent with the Matsubara temperature Green's function formalism and also justifies rigorously the correction terms introduced in an ad hoc way with Hedin and Lundqvist. Our results show that one should use the appropriate dynamical equation for the time-ordered Green's function when working beyond the equilibrium zero-temperature limit.

arXiv:1211.2602Journal of Physics Condensed MatterNov 20, 20121 citation

Introduction to the Keldysh Formalism

Robert van Leeuwen, Nils Erik Dahlen, Gianluca Stefanucci +2 more

Lecture notes in physicsJan 1, 200672 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.