Bound states of quasiparticles with quartic dispersion in an external potential: WKB approach

TL;DR

The paper develops a semiclassical WKB framework for bound states of quasiparticles with quartic dispersion in one dimension, requiring fourth-order differential equations and higher-order Airy functions. By deriving the hyperasymptotics of the fourth-order Airy functions $Ai_4$ and $\tilde{Ai}_4$, the authors obtain accurate connection formulas and a generalized Bohr-Sommerfeld quantization condition that includes nonperturbative in $\hbar$ corrections. They validate the approach on harmonic and double-quartic potentials, in both momentum- and coordinate-space representations, and provide explicit quantization relations that improve low-lying energy estimates. The results have relevance for Dirac-materials like bilayer graphene, where higher-order kinetic terms are present, and they open pathways to extensions to $E(p)\sim p^{2n}$ and higher dimensions. Overall, the work highlights the necessity of hyperasymptotic analysis in matching WKB solutions for higher-order dispersions and offers a practical framework for predicting bound-state spectra in soft-dispersion systems.

Abstract

The Wentzel-Kramers-Brillouin semiclassical method is formulated for quasiparticles with quartic-in-momentum dispersion which presents the simplest case of a soft energy-momentum dispersion. It is shown that matching wave functions in the classically forbidden and allowed regions requires the consideration of higher-order Airy-type functions. The asymptotics of these functions are found by using the method of steepest descents and contain additional exponentially suppressed contributions known as hyperasymptotics. These hyperasymptotics are crucially important for the correct matching of wave functions in vicinity of turning points for higher-order differential equations. A quantization condition for bound state energies is obtained, which generalizes the standard Bohr-Sommerfeld quantization condition for particles with quadratic energy-momentum dispersion and contains non-perturbative in $\hbar$ correction. This non-perturbative correction, usually associated with tunneling effects or the presence of complex turning points, occurs even for the harmonic potential with quartic dispersion where complex turning points and tunneling are absent. The quantization condition is used to find bound state energies in the case of quadratic and quartic potentials.

Figures (4)

• Figure 1: Schematics of the bound state problem with potential $V(x)$ (blue solid line), energy $E$ (orange dashed line), turning points $x_a$ and $x_b$, and classically forbidden (I, III) and classically allowed (II) regions.
• Figure 2: The integration paths in the complex plane $t$ defining generalized Airy functions.
• Figure 3: The plots of the functions $Ai_4(x)$ (in blue) and $\widetilde{Ai}_4(x)$ (in red) on the real axis.
• Figure 4: Sectors in complex plane $t$ with $\text{Re}\,t^5 >0$ (shaded regions) and extrema of $t+\frac{t^5}{5}$, denoted by small circles, in the classically forbidden region $z>0$ (left panel), and extrema of $t-\frac{t^5}{5}$ in the classically allowed region $z<0$ (right panel). Paths of steepest descent passing through the extrema are shown in blue and paths of strongest ascent in red.