Local fermion density in inhomogeneous free-fermion chains: A discrete WKB approach, by Martín Zapata, Federico Finkel, Artemio González-López

SciPost Physics Home Authoring Refereeing Submit a manuscript About Local fermion density in inhomogeneous free-fermion chains: A discrete WKB approach Martín Zapata, Federico Finkel, Artemio González-López SciPost Phys. 20, 078 (2026) · published 10 March 2026 doi: 10.21468/SciPostPhys.20.3.078 pdf BiBTeX RIS Submissions/Reports Abstract We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.20.3.078TI - Local fermion density in inhomogeneous free-fermion chains: A discrete WKB approachPY - 2026/03/10UR - https://scipost.org/SciPostPhys.20.3.078JF - SciPost PhysicsJA - SciPost Phys.VL - 20IS - 3SP - 078A1 - Zapata, MartínAU - Finkel, FedericoAU - González-López, ArtemioAB - We introduce a novel analytical approach for studying free-fermion (XX) chains with smoothly varying, site-dependent hoppings and magnetic fields. Building on a discrete WKB-like approximation applied directly to the recurrence relation for the single-particle eigenfunctions, we derive a closed-form expression for the local fermion density profile as a function of the Fermi energy, which is valid for arbitrary fillings, hopping amplitudes and magnetic fields. This formula reproduces the depletion and saturation effects observed in previous studies of inhomogeneous free-fermion chains, and provides a theoretical framework to understand entanglement entropy suppression in these models. We demonstrate the accuracy of our asymptotic formula in several chains with different hopping and magnetic field profiles. Our findings are thus the first step towards an analytical treatment of entanglement in free-fermion chains beyond the reach of conventional field-theoretic techniques.ER - × @Article{10.21468/SciPostPhys.20.3.078, title={{Local fermion density in inhomogeneous free-fermion chains: A discrete WKB approach}}, author={Martín Zapata and Federico Finkel and Artemio González-López}, journal={SciPost Phys.}, volume={20}, pages={078}, year={2026}, publisher={SciPost}, doi={10.21468/SciPostPhys.20.3.078}, url={https://scipost.org/10.21468/SciPostPhys.20.3.078},} Ontology / Topics See full Ontology or Topics database. Exactly solvable models Free fermions Heisenberg spin chains Jordan-Wigner fermions/transformation Orthogonal polynomials Authors / Affiliations: mappings to Contributors and Organizations See all Organizations. 1 Martín Zapata, 2 Federico Finkel, 2 Artemio González-López 1 Max-Planck-Institut für Quantenoptik / Max Planck Institute of Quantum Optics [MPQ] 2 Universidad Complutense de Madrid / Complutense University of Madrid Funders for the research work leading to this publication Ministerio de Ciencia, Innovación y Universidades / Ministry of Science, Innovation and Universities Universidad Complutense de Madrid / Complutense University of Madrid