The ground state of a free-fermionic chain with inhomogeneous hoppings at half-filling can be mapped into the Dirac vacuum on a static curved space-time, which presents exactly homogeneous occupations due to particle-hole symmetry. Yet, far from half-filling, we observe density modulations and depletion effects. The system can be described by a 1D Schrödinger equation on a different static space-time, with an effective potential, which accounts for the depleted regions. We provide a semiclassical expression for the single-particle modes and the density profiles associated to different hopping patterns and filling fractions. Moreover, we show that the depletion effects can be compensated for all filling fractions by adding a chemical potential proportional to the hoppings. Interestingly, we can obtain exactly the same density profiles on a homogeneous chain if we introduce a chemical potential, which is inverse to the hopping intensities, even though the ground state is different from the original one.

• Received 28 September 2022
• Accepted 29 November 2022
• Corrected 4 January 2023

DOI:https://doi.org/10.1103/PhysRevB.106.224204