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

TL;DR

We solve the single-particle Schrödinger equation for a one-dimensional lattice of non-interacting fermions subject to a time-dependent linear potential with slope $ξ(t)$, providing an exact construction of the mode dynamics. The solution uses an ansatz involving Bessel functions with a time-dependent width $ℓ(t)$ and phase $θ(t)$, whose evolution is fixed by the ODEs $\dot{ℓ}=\sinθ$ and $\dot{θ}=\cosθ/ℓ-1/ξ$, plus a dynamical phase $φ(t)=\int_0^t ds\,1/ξ(s)$; the many-body two-point function then reads $\tilde{C}_{i,j}(t)=e^{i(i-j)θ(t)}C_{i,j}(ℓ(t))$. Building on this, we derive hydrodynamic predictions for density, current, and entanglement, and we examine the quench limit where the interface exhibits breathing modes linked to Wannier–Stark localization. The results connect to domain-wall melting physics and provide exact benchmarks for driven non-interacting systems, with implications for quantum control and transport in inhomogeneous lattices.

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.

Figures (6)

• Figure 1: Illustration of the linear potential $V_j(t) = j/\xi(t)$ in \ref{['eq:H']}, with time-dependent slope. For $t<0$, the system features a correlated interface region $|j| \leq \xi_0$. During the driving the interface remains self-similar, with its extent given by $\ell(t)$, as obtained in the main text as a function of $\xi(t)$. Site occupations are represented using a color scale, with intensity increasing from lighter to darker shades.
• Figure 2: Convergence to the adiabatic limit: Length $\ell(t)$ (top) and phase $\theta(t)$ (bottom) for the driving $\xi(t)/\xi_0 = R + (1-R) \exp(-\frac{t}{\xi_0\delta})$, which interpolates from the initial slope $\xi_0$ to the final value $\xi_1 = R \xi_0$ ($R=0.25$ in the plots) over a timescale $\xi_0\delta$. For $\delta \gg 1$, the system approaches the adiabatic regime, with $\ell(t) \simeq \xi(t)$ and $\theta(t) \simeq 0$, up to oscillatory corrections given in eq. \ref{['eq:non-ad']}.
• Figure 3: (a) -- Length $\ell(t)$ and (b) -- phase $\theta(t)$ shown as functions of the rescaled time $t/\xi_0$ for the driving protocol $\xi(t) = \xi_0\bigl(1-\frac{1}{3}\sin \frac{t}{\xi_0}\bigr)$. (c) -- Corresponding Fermi points $p_\pm(x,t)$ in \ref{['eq:fermi-pts']} shown as functions of the rescaled position $x/\xi_0$ at three different times: $t_0=0$, $t_1=3\pi\xi_0/4$, and $t_2=2\pi \xi_0$ (marked with symbols in panels (a)–(b)). In the phase-space picture of panel (c), note that $\ell(t)$ controls the size of the correlated region occupied by the quasiparticles, while $\theta(t)$ determines the momentum boost imparted by the drive (marked with horizontal dashed lines).
• Figure 4: Particle density (a), current (b), and entanglement entropy (c) under the driving protocol of Fig. \ref{['fig:fermi_pts']}, shown as functions of the rescaled position $x/\xi_0$ at three times: $t_0=0$, $t_1=3\pi\xi_0/4$, and $t_2=2\pi \xi_0$. Solid lines correspond to the exact results from eqs. \ref{['eq:exact-density']} and \ref{['eq:ent-def']}, while thin dashed lines show the hydrodynamic predictions from eqs. \ref{['eq:dens']}–\ref{['eq:curr']} and \ref{['eq:hydro-ent']}. Symbols show numerical data from exact diagonalization of the two-point correlation \ref{['eq:2pt-corr']} of a finite chain with $\xi_0=10$ and $N=40$, followed by Trotter decomposition of the unitary evolution.
• Figure 5: Convergence to the quench limit. -- Length $\ell(t)$ (top) and phase $\theta(t)$ (bottom) for the driving protocol $\xi(t)/\xi_0 = R+(1-R) \exp(-\frac{t}{\xi_0\delta})$, $R=\xi_1/\xi_0=0.25$, also shown in Fig. \ref{['fig:ad-lim']}. For $\delta \ll 1$, they converge to the results given in eqs. \ref{['eq:l-quench']}–\ref{['eq:th-quench']}, shown by solid lines.