Analytical solution of a free-fermion chain with time-dependent ramps

¹ Institute of Theoretical and Computational Physics, Graz University of Technology - Petersgasse 16, A-8010 Graz, Austria
² Institute of Physics, University of Graz - Universitätsplatz 5, A-8010 Graz, Austria
³ Department of Theoretical Physics, Institute of Physics, Budapest University of Technology and Economics Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁴ HUN-REN-BME Quantum Error Correcting Codes and Non-equilibrium Phases Research Group, Budapest University of Technology and Economics - Műegyetem rkp. 3., H-1111 Budapest, Hungary
⁵ Laboratoire de Physique de l’École Normale Superieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité, F-75005 Paris, France

Received: 22 October 2025
Accepted: 7 January 2026

Abstract

We provide an exact analytical solution of the single-particle Schrödinger equation for a chain of non-interacting fermions subject to a time-dependent linear potential, with its slope varied as an arbitrary function of time. The resulting dynamics exhibit self-similar behavior, with a structure reminiscent of the domain wall melting problem, albeit characterized by a nontrivial time-dependent length scale and phase. Building on this solution, we derive hydrodynamic predictions for the evolution of particle density, current, and entanglement entropy along the chain. In the special case of a sudden quench, the system develops a breathing interface region, which may be interpreted as a realization of Wannier-Stark localization, as previously suggested on the basis of hydrodynamic arguments.