From the Keldysh Formalism to Non-equilibrium Dynamical Mean-Field Theory

Kouki Nakata

Martin Eckstein

Martin Eckstein

Antonius Dorda

Filippo Bovo

In this work we examine the dynamics of one-dimensional systems. We present a continuum approach to the construction of the non-equilibrium Keldysh functional integral and derive a relation between the classical and quantum fields of the Keldysh formalism and the occupation of a state. Using the Keldysh formalism, we give a self-contained presentation of bosonization and refermionisation of fermionic and bosonic interacting particles. We derive the first two non-linear corrections to linear bosonization due to the fermionic spectrum curvature using the functional integral formalism. Using the previous results, we study the thermal dynamics of one-dimensional systems. We consider both the phonons, coming from bosonization, and fermionic quasiparticles, coming from refermionisation, in a single theory. Having them in a single theory is justified by a scale separation between them. Studying the dynamical structure factor we find three regimes. From higher to lower energies we have a ballistic and a collisional regimes of fermionic quasiparticles and a hydrodynamical regime of non-linear phonons. The phase field of the non-linear bosons satisfies the Kardar-Parisi-Zhang equation, thereby linking the dynamics of one-dimensional quantum systems to the universality class of surface growth. The time-scale separating the phononic and fermionic quasiparticle regimes is very long for up-to-date experiments, meaning that the ballistic regime of fermionic quasiparticles is the only one likely to be observed. Since this approach has a limited range of validity for weakly interacting bosons, we derive their dynamical structure factor using a semiclassical approach.

Christian Käding, Mario Pitschmann

We provide a description of interacting quantum fields in terms of density matrices for any occupation numbers in Fock space in a momentum basis. As a simple example, we focus on a real scalar field interacting with another real scalar field, and present a practicable formalism for directly computing the density matrix elements of the combined scalar–scalar system. For deriving the main formula, we use techniques from non-equilibrium quantum field theory like thermo-field dynamics and the Schwinger–Keldysh formalism. Our results allow for studies of particle creation/annihilation processes at finite times and other non-equilibrium processes, including those found in the theory of open quantum systems.