Causality in the Fermi problem and the Magnus expansion

C. Compagno, C. M. Palma, Roberto Passante +1 more

W. R. Salzman

General and greatly simplified methods are presented for calculating terms in the exponent in Magnus and Magnus-like expansions to first order in a ‘‘small’’perturbation and to infinite order in the overall Hamiltonian in the Schrödinger representation. These techniques are applied to four simple but important models and it is shown that in each case the Magnus exponent diverges for some range of the parameters of the model. This result casts serious doubt on the utility of the Magnus expansion in the Schrödinger representation. Several of the problems are also treated in the interaction representation giving results which converge throughout the useful range of parameters. There is no evidence that a similar question exists in this representation.

Laraib-Ul- Nissa, Muhammad Abdullah, Waqar Yousaf

We investigate the nonequilibrium relaxation dynamics of the one-dimensional (1D) Fermi-Hubbard model subjected to abrupt, global interaction quenches. Specifically, we benchmark the convergence properties, structural accuracy, and algorithmic breakdown of the Magnus expansion against numerically exact results obtained via full Exact Diagonalization (ED) on small, periodic lattice clusters. By tracking the real-time evolution of local observables, double occupancy (doublon density), and many-body state fidelity metrics, we map out the validity bounds of the low-order Magnus series across weak, moderate, and strong interaction regimes. Our findings demonstrate that while the Magnus expansion provides an exceptionally accurate description of short-time coherent dynamics, rapid phase matching, and initial prethermalization tendencies, its convergence is fundamentally bottlenecked at longer timescales. This breakdown is driven by the rapid growth of multi-particle entanglement, non-local operator spreading via nested commutators, and the emergence of severe state-space fragmentation inherent to dense, strongly interacting many-body spectra.

G. Compagno, G. Massimo Palma, Roberto Passante +1 more

Eugene Stephane Mananga, Thibault Charpentier

In this paper, we revisit, from different perspectives, the applications of Floquet–Magnus expansion (FME) and Fer expansion (FE) in NMR and physics. We use these expansion schemes for the calculation of effective Hamiltonians and propagators. We found that the FME is related to the FE. While both approaches are powerful methods, there are subtle differences between them. To understand the differences, we performed explicit calculation on various problems, namely, Bloch–Siegert shift, Cross Polarization (CP) and Continuous Waves (CW). We put the work in a prospect that is related to spin physics in order to make the calculation more general. As the propagator from the FME takes the form of the evolution operator, which removes the constraint of a stroboscopic observation, we appreciated the effects of time-evolution under Hamiltonians with different orders separately.

Amin Faraji Astaneh, Parsa Kafashi

We investigate Krylov spread complexity for states evolving under time-dependent Hamiltonians. For periodically driven systems, we formulate the problem within Floquet theory and show how the Magnus expansion provides a systematic approximation when the Floquet Hamiltonian is not available in closed form. We then extend this framework beyond periodic driving and demonstrate that, in addition to the globally truncated Magnus expansion, a piecewise Magnus expansion provides a reliable method when the global expansion loses convergence or accuracy. Our results provide practical tools for analyzing complexity growth in a broad class of time-dependent quantum systems.