Local fermion density in inhomogeneous free-fermion chains: a discrete WKB approach
arXiv:2511.16473 · doi:10.21468/SciPostPhys.20.3.078
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.
Revised version: 36 pages, 14 figures, supplementary material linked, two new references and a remark added
References in corpus (12)
- Tools for quantum simulation with ultracold atoms in optical lattices
- Universal parity effects in the entanglement entropy of XX chains with open boundary conditions
- Entanglement over the rainbow
- From conformal to volume-law for the entanglement entropy in exponentially deformed critical spin 1/2 chains
- More on the rainbow chain: entanglement, space-time geometry and thermal states
- Inhomogeneous XX spin chains and quasi-exactly solvable models
- Entanglement entropy of inhomogeneous XX spin chains with algebraic interactions
- Engineering entanglement Hamiltonians with strongly interacting cold atoms in optical traps
- Entanglement Hamiltonian and orthogonal polynomials
- Depletion in fermionic chains with inhomogeneous hoppings
- Fermionic logarithmic negativity in the Krawtchouk chain
- Rainbow chains and numerical renormalisation group for accurate chiral conformal spectra