Discrete WKB approach to density profiles in inhomogeneous two-band free-fermion chains

Abstract

We extend the discrete WKB method to one-dimensional bipartite tight-binding chains of free fermions with slowly varying hopping amplitudes and on-site potentials. We focus on the inhomogeneous Rice-Mele model, suggesting possible experimental realizations thereof in optical superlattices. We derive explicit semiclassical expressions for the single-particle wave functions and density of states, as well as the sublattice-resolved local particle densities and filling fraction at arbitrary Fermi energy. In particular, we show that the WKB density profiles agree with those obtained from the local-density approximation. We analyze the spatial dependence of the local band edges and its consequences for the global spectral structure, and clarify the role played by the phase-space topology in determining the Bohr-Sommerfeld quantization conditions. Finally, we illustrate our results with two representative examples: a homogeneous Rice-Mele chain subject to a harmonic trap and a fully inhomogeneous chain containing a domain wall.

The paper addresses bulk particle densities and single-particle states in one-dimensional bipartite free-fermion chains whose hoppings and on-site potentials vary slowly compared with the lattice spacing. It focuses on the inhomogeneous Rice–Mele model at arbitrary Fermi energy, including spatially varying dimerization, staggered potential, and scalar confinement. A two-component discrete WKB expansion is applied directly to the lattice eigenvalue equations. It yields oscillatory wave functions with envelopes proportional to $|v_F|^{-1/2}$, evanescent solutions, and semiclassical densities of states. Integrating the sublattice-resolved spectral densities gives explicit density profiles, with the sublattice imbalance expressed through incomplete elliptic integrals. These leading-order results coincide with the local-density approximation. Local band edges determine metallic, half-filled insulating, empty, and fully occupied regions. The treatment also formulates Bohr–Sommerfeld actions using $-x\,dp$ to accommodate momentum winding on cylindrical phase space. Numerical comparisons use chains of $1000$ cells, examining harmonic confinement and a smoothly varying domain wall. For constant local half-gap $E_{\min}$ and trap amplitude $\mu_0$, the bulk global gap exists when $\mu_0<2E_{\min}$ and has width $2E_{\min}-\mu_0$. The domain-wall example additionally supports a Jackiw–Rebbi mode with width scaling as $a\sqrt{L}$. The analysis assumes slow variation and neglects exponentially small interwell tunneling. Turning-point connection formulas, band crossings, coalescing turning points, and localized edge or defect states require separate treatment. The domain-wall mode is analyzed through a separate continuum approximation; its energy-shift estimate omits other lattice corrections of the same order.