Excitons in Large Disordered Boron-Nitride Layer using Linear-Scaling Bethe-Salpeter Simulations

Abstract: We introduce a real-space, linear-scaling Bethe-Salpeter framework that enables excitonic spectroscopy in large and possibly disordered boron-nitride-derived systems. Thanks to the use of a sublattice-resolved perturbative decoupling that maps localized electron-hole pairs onto a sparse tight-binding model, we implement the Kernel Polynomial Method to compute absorption spectra with O(N) cost. To illustrate the capabilities of our method, we apply it to Anderson-disordered monolayer hexagonal boron nitride with up to $10^{5}$ orbitals. The method reveals a disorder-induced asymmetric broadening of bright excitons, a crossover from quadratic to linear redshift of the main absorption peak, and Anderson localization of the exciton center of mass. This approach extends excitonic calculations beyond the reach of conventional ab initio Green's function methods (GW approximation and Bethe-Salpeter equation), opening optical spectroscopy to large-scale, disordered, moiré, quasicrystalline, and structurally complex quantum materials.

• The paper introduces a real-space, linear-scaling Bethe-Salpeter method to simulate excitonic spectroscopy in disordered BN monolayers.
• It demonstrates that disorder induces asymmetric spectral broadening, systematic redshifts, and a quadratic-to-linear transition in excitonic peak behavior.
• The study reveals pronounced center-of-mass localization of exciton wavefunctions while preserving a compact electron-hole pair and strong binding energy.

Linear-Scaling Bethe-Salpeter Simulations of Excitons in Disordered Boron Nitride Layers

Introduction

This work presents a real-space, linear-scaling computational methodology for excitonic spectroscopy in atomistically-large, structurally disordered two-dimensional (2D) systems, with a focus on hexagonal boron nitride (hBN) monolayers. Standard approaches to exciton physics based on many-body perturbation theory, specifically the GW-BSE framework, face severe computational bottlenecks when translational symmetry is broken, as in the case of disordered, moiré, or large-unit-cell materials. The methodology developed addresses this by mapping the Bethe-Salpeter equation (BSE) to a sparse real-space Hamiltonian and employing the Kernel Polynomial Method (KPM) for linear-scaling evaluation of optical properties. The approach enables direct access to the effects of static disorder on excitonic spectra, center-of-mass localization, and the emergence of new optical features in supercell calculations with up to $10^5$ orbitals.

Methodology

The formalism begins with a $p_z$-projected tight-binding Hamiltonian for hBN, separable by sublattice, and introduces Anderson-type diagonal disorder. Using perturbative decoupling, the electronic structure is mapped to effective single-particle electron and hole Hamiltonians. The direct Coulomb interaction is modeled via the Rytova-Keldysh potential, enabling efficient evaluation of the BSE Hamiltonian in a localized basis of electron-hole pairs. A real-space cutoff, $R_{\mathrm{cut}}$, is used to truncate pairs with large separation, ensuring linear scaling with the number of atoms in the unit cell. The optical absorption spectrum is calculated as the imaginary part of the dielectric function, avoiding explicit diagonalization via KPM.

The direct exchange term is neglected, justified by its minimal contribution compared to the direct interaction in large-gap materials. The approach is explicitly tailored to systems lacking translational invariance, allowing simulation of atomistically-resolved excitons and optical spectra under strong disorder or aperiodicity.

Optical Absorption under Disorder

The simulated absorption spectra as a function of disorder strength $W_0$ reveal several key features:

• The main excitonic absorption peak exhibits a pronounced asymmetric broadening and systematic redshift with increasing $W_0$.
• For low disorder, the redshift of the absorption peak scales quadratically with $W_0$, transitioning to a linear regime at higher disorder.
• The excess full width at half maximum (FWHM) increases as $W_0^2 A$ with $A \approx 1.1$ eV$^{-1}$, up to $W_0 \sim 0.5$ eV, closely resembling exciton-phonon linewidth broadening phenomenology.

  Figure 1: Calculated optical response $p_z$0 for increasing Anderson disorder strength, showing peak broadening and redshift; the inset highlights the evolution and fits of the main peak energy vs. disorder strength.

Exciton Density of States and Oscillator Strength Redistribution

Beyond absorption spectra, the exciton density of states (DOS) and its interplay with oscillator strength are analyzed:

• In the pristine case, bright excitons correspond to discrete eigenvalues at the band edge; optical activity is restricted to transitions at $p_z$1 due to selection rules.
• In the presence of moderate disorder ($p_z$2 eV), the DOS broadens substantially and optically active states are redistributed over a continuum, reflecting strong breaking of translational invariance and energetic mixing across former Brillouin zone sectors.
• The optical response becomes dominated by a disorder-activated continuum of states, in contrast to the sharply peaked spectrum of the pristine system.

  Figure 2: Exciton DOS and absorption spectra for pristine (blue) and disordered ($p_z$3 eV, red) hBN monolayer in large supercells, illustrating the transition from discrete to continuum features.

Exciton Wavefunction Localization

A central result pertains to the spatial localization properties of excitonic states under disorder:

• Direct evaluation of the exciton wavefunctions in real space exposes Anderson localization of the center-of-mass coordinate, with the participation ratio (PR) and its scaling with $p_z$4 quantifying the loss of exciton mobility.
• Despite pronounced localization of the center of mass, the relative electron-hole coordinate remains compact and nearly invariant with increasing disorder, preserving a strong binding energy ($p_z$5 eV).
• The compactness, $p_z$6, does not increase with disorder, indicating that the exciton remains tightly bound at the atomic scale even as its overall spatial extent collapses.
• Quantitatively, for $p_z$7 eV, $p_z$8 decreases as $p_z$9, consistent with Anderson localization theory; the threshold is set by the finite size of the supercell.

  Figure 3: (a) Electron-hole projected density maps show increasing center-of-mass localization with disorder; (b) Relative coordinate probability density remains compact; (c) Disorder dependence of localization $R_{\mathrm{cut}}$0 and compactness $R_{\mathrm{cut}}$1 for the lowest excitonic states.

Implications and Outlook

The methodology enables ab initio-informed studies of excitons in systems beyond the reach of conventional GW-BSE approaches, providing atomistic insight into disorder-driven exciton physics. The principal implications include:

• Spectral fingerprints of disorder are shown to go beyond mere peak broadening, involving activation of dark continua and qualitative reshaping of absorption lines.
• The decoupling of center-of-mass localization and electron-hole pair compactness establishes that strongly localized excitons remain robust against static potential fluctuations in terms of binding energy and radiative character.
• The quadratic-to-linear crossover in the excitonic peak redshift with disorder is a robust numerical result, offering a direct experimental observable for distinguishing weak and strong disorder regimes.
• The technique opens pathways to address new physical phenomena in moiré superlattices, quasicrystals, functionalized heterostructures, and electrically or magnetically tuned exciton systems that lack lattice periodicity (2606.30585).

The approach is also extensible to exciton diffusion simulations and could be adapted to study time-dependent or non-equilibrium phenomena in disordered 2D materials. The results suggest that, in experimentally relevant hBN monolayers, exciton localization is controlled by static disorder rather than phonons, with significant ramifications for device engineering and fundamental research.

Conclusion

This contribution establishes a linear-scaling Bethe-Salpeter framework that unlocks statistical and microscopic analysis of excitons in structurally complex, disordered, and large-unit-cell materials. By capturing both the spectral and real-space signatures of disorder, it offers a robust computational platform for probing optical and transport phenomena where excitonic effects in non-crystalline environments play a pivotal role. The work enables rigorous evaluation of the interplay between electron-electron interactions and disorder, a regime that is increasingly relevant for next-generation quantum materials and optoelectronic device architectures.