Exact Mobility Edges in a Disorder-Free Dimerized Stark Lattice with Effective Unbounded Hopping

TL;DR

This paper introduces a disorder-free 1D Hamiltonian with an exact mobility edge, derived from a dimerized Stark lattice with unbounded hopping, and demonstrates its robustness and experimental feasibility.

Contribution

It presents an analytically solvable model with an exact mobility edge in a disorder-free setting, evading traditional no-go theorems through unbounded and staggered hopping.

Findings

Identified a sharp mobility edge where energy equals inter-cell hopping strength.

Discovered coexistence of extended states with localized branches, including a Wannier-Stark ladder.

Showed the mobility edge's robustness against experimental imperfections.

Abstract

We propose a disorder-free one-dimensional single-particle Hamiltonian hosting an exact mobility edge (ME), placing the system outside the assumptions of no-go theorems regarding unbounded potentials. By applying a linear Stark potential selectively to one sublattice of a dimerized chain, we generate an effective Hamiltonian with unbounded, staggered hopping amplitudes. The unbounded nature of the hopping places the model outside the scope of the Simon-Spencer theorem, while the staggered scaling allows it to evade broader constraints on Jacobi matrices. We analytically derive the bulk spectrum in reciprocal space, identifying a sharp ME where the energy magnitude equals the inter-cell hopping strength. This edge separates a continuum of extended states from two distinct localized branches: a standard unbounded Wannier-Stark ladder and an anomalous bounded branch accumulating at the ME. The existence of extended states is supported by finite-size scaling of the inverse participation ratio up to system sizes $L \sim 10^9$. Furthermore, we propose an experimental realization using photonic frequency synthetic dimensions. Our numerical results indicate that the ME is robust against potential experimental imperfections, including frequency detuning errors and photon loss, establishing a practical path for observing MEs in disorder-free systems.