Progress of Non-Equilibrium Green's Functions 9 (PNGF9)

Generalized Keldysh formalism for nonequilibrium correlation functions and its quantics-tensor-train implementation

26 Aug 2026, 09:35

20m

Auditorium (Paul Scherrer Institut)

Contributed Talk

Speaker

Dr Ken Inayoshi (Department of Physics, Saitama University)

Description

Recent developments in time-resolved Raman scattering [1] and resonant inelastic X-ray scattering [2,3] have enabled the observation of the time evolution of various low-energy excitations, such as charge, spin, and phonon excitations, in materials. From a theoretical perspective, the spectra of these excitations can be computed from the nonequilibrium two-particle correlation functions. To compute the time evolution of these correlation functions, one can straightforwardly solve the nonequilibrium Bethe–Salpeter equation, but this incurs prohibitive computational and memory costs. Therefore, conventional approaches often neglect vertex corrections (the bubble approximation) [4], which is inconsistent with the Baym–Kadanoff conserving approximation.

To overcome this difficulty, we propose an approach to calculate nonequilibrium correlation functions, naturally including vertex corrections, using the generalized Keldysh formalism [5] for a Hamiltonian that includes a virtual external probe field [6]. In particular, we introduce a Bethe–Salpeter-like integral equation for correlation functions and develop an efficient solver of this equation with quantics tensor trains (QTT) [7–11].

In this talk, we present this formalism together with its QTT implementation, and apply it to the fluctuation dynamics of nonequilibrium antiferromagnetic states in the Hubbard model. We find that the dynamics differs qualitatively depending on whether vertex corrections are included, and that, as the system approaches a nonthermal critical point, the maximum value and the decay time of the fluctuations increase [12].

[1] J.-A. Yang et al., Sci. Rep. 7, 40876 (2017).
[2] M. Dean et al., Nature Mat. 15, 601 (2016).
[3] M. Mitrano and Y. Wang, Commun. Phys. 3, 184 (2020).
[4] P. Werner, M. Eckstein and N. Tsuji, Phys. Rev. B 108, 245157 (2023).
[5] E. Canovi, P. Werner and M. Eckstein, Phys. Rev. Lett. 113, 265702 (2014).
[6] O. P. Matveev, A. M. Shvaika, and J. K. Freericks, Phys. Rev. B 113, 195148 (2026).
[7] H. Shinaoka et al., Phys. Rev. X 13, 021015 (2023).
[8] M. Murray, H. Shinaoka, and P. Werner, Phys. Rev. B 109, 165135 (2024).
[9] M. Środa, K. Inayoshi, H. Shinaoka, and P. Werner, Phys. Rev. Lett. 135, 226501 (2025).
[10] K. Inayoshi, M. Środa, A. Kauch, P. Werner, and H. Shinaoka, SciPost Phys. 20, 077 (2026).
[11] M. Środa, K. Inayoshi, M. Schüler, H. Shinaoka, and P. Werner, Phys. Rev. B 113, 165113 (2026).
[12] K. Inayoshi, H. Shinaoka, and Y. Murakami, arXiv:2607.11055.