Strongly correlated electron systems are among the biggest challenges of modern condensed matter physics due to the exponential growth of the many-body Hilbert space. In particular, nonequilibrium Green’s function (NEGF) methods provide a powerful framework for describing the real-time dynamics of interacting quantum systems. However, the numerical treatment of this framework is often computationally expensive, even for limited time grids and moderate lattice sizes.

In this thesis, we first investigate the use of tensor network methods for the compression of NEGFs, with particular attention devoted to tensor cross interpolation (TCI). Several tensor network topologies are benchmarked for the compression of Hubbard-type Green’s functions, including standard matrix product states( MPS) and tree tensor networks (TTN).
We analyze how the network topology influences the required bond dimension, computational cost, and achievable compression accuracy. Our results demonstrate that TTN topologies offer significant improvements for Green’s functions under external DC fields or when momentum dependency is directly encoded in the network structure. Specifically, we show that the required bond dimensions are systematically lower than or equal to those of linear structures.

In the second part, we extend this analysis to the electron-phonon coupled Holstein model, using interleaved Quantics Tensor Trains (QTTs) to solve diagrammatic selfconsistent equations. We first present the equilibrium temperature-filling phase diagram for charge density waves (CDW). We then generalize the formalism to compute both CDW and pairing (SC) susceptibilities in out-of-equilibrium systems.