Excitonic Renormalization Framework

• The excitonic renormalization framework is a set of theoretical approaches that incorporate many-body channels—such as dielectric screening, phonon coupling, and disorder—to modify excitonic properties beyond the bare electron–hole picture.
• Different methodological families use effective Hamiltonians, path-integrals, and BSE-based treatments to compute corrections from self-energy, image-charge, and environmental effects, enabling precise modeling of excitons.
• Practical implementations reveal measurable shifts in exciton energies and oscillator strengths, ranging from meV-scale changes in semiconductor-metal hybrids to significant renormalization in polar semiconductors and biological aggregates.

Searching arXiv for recent and relevant papers on excitonic renormalization frameworks to ground the article in published work. Excitonic renormalization framework denotes a family of theoretical and computational formalisms in which excitonic observables are not taken as fixed consequences of bare band structure and Coulomb attraction, but as emergent quantities modified by additional many-body channels such as dielectric interfaces, metallic screening, lattice polarization, disorder, explicit environments, fragment truncation, and optical driving. Across these settings, the common objective is to determine how exciton energies, binding energies, oscillator strengths, delocalization patterns, linewidth-related observables, and effective Hamiltonians are altered once degrees of freedom outside a minimal electron–hole picture are integrated in, integrated out, or coarse-grained. In the literature, this has been implemented through configuration interaction on top of envelope-function models for semiconductor–metal hybrids (Climente et al., 2011), influence-functional and quasiparticle path-integral approaches for phonon renormalization (Park et al., 2022, Rana et al., 23 Mar 2026), static Bethe–Salpeter treatments with explicit environmental screening (Allen et al., 23 Feb 2026), fragment-based effective Hamiltonians such as REM-TDDFT (Ma et al., 2013) and exact fragment-fluctuation formulations (Dutoi et al., 2017), downfolded BSE kernels in reduced subspaces (Qiu et al., 2021), disorder self-energy theories for hydrogenic excitons (Jr. et al., 2021), and time-dependent many-body propagation of driven excitonic manifolds (Cervantes-Villanueva et al., 27 Mar 2026).

1. Renormalization as a general excitonic concept

In the broadest usage supported by the literature, excitonic renormalization refers to the many-body modification of neutral electron–hole excitations relative to a bare or minimally screened reference description. The modification may occur through one-body channels, such as self-energy or image-potential shifts, or through two-body channels, such as changes in the screened electron–hole kernel. Several papers emphasize that these contributions compete rather than add trivially. In semiconductor–metal nano-hybrids, for example, the excitonic resonance is altered by both self-polarization terms and a polarization-mediated correction to the electron–hole interaction (Climente et al., 2011). In polar semiconductors, phonons screen the electron–hole attraction while also stabilizing separate carriers as polarons, so binding renormalization reflects a competition between one-body polaron self-energies and two-body phonon-mediated screening (Park et al., 2022, Rana et al., 23 Mar 2026, Schebek et al., 2024). In protein-embedded pigment aggregates, the environment changes both the underlying single-particle spectrum and, more importantly, the screened electron–hole interaction, thereby shifting energies and redistributing spectral weight (Allen et al., 23 Feb 2026).

A second recurring meaning of renormalization is effective-Hamiltonian compression. In fragment-based approaches, high-energy or nonlocal electronic structure is folded into reduced excitonic degrees of freedom. REM-TDDFT constructs an effective excitonic Hamiltonian from monomer and dimer calculations (Ma et al., 2013). The exact fragment-state formalism of fluctuation operators rewrites the ab initio Hamiltonian in terms of fragment transitions and then introduces renormalization by local-state truncation (Dutoi et al., 2017). In reduced-subspace BSE, omitted sectors induce an effective screening of exchange, the so-called $S$ approximation, even though exchange is formally bare in the full theory (Qiu et al., 2021). This suggests that “excitonic renormalization framework” names not one model but a class of strategies for incorporating omitted physics into a tractable excitonic description.

A third usage appears in nonequilibrium settings. In monolayer WS$_2$, coherent pumping produces modified effective splittings and beating frequencies that differ from bare equilibrium energy differences because multilevel excitonic interactions reshape the active manifold under driving (Cervantes-Villanueva et al., 27 Mar 2026). In monolayer MoS$_2$, excitonic correlations feed back onto the quasiparticle self-energy and renormalize the band structure itself, producing a transient gap opening and effective-mass enhancement below the Mott threshold (Lin et al., 2022). This suggests that excitonic renormalization may act not only on excitons as outputs of a fixed electronic structure, but also on the one-particle spectrum from which excitons are built.

2. Effective-Hamiltonian structures and representative equations

Many frameworks begin from an explicit Hamiltonian or effective action in which the exciton is coupled to additional screening channels. In semiconductor–metal nano-hybrids, the total exciton Hamiltonian is written as

$$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$

where confinement, self-polarization, direct Coulomb attraction, polarization-mediated interaction, and trapped-charge effects are all retained explicitly (Climente et al., 2011). In this form, renormalization is inseparable from geometry and dielectric mismatch.

For phonon renormalization, one widely used construction starts from an electron–hole plus phonon Hamiltonian and integrates out harmonic phonons exactly in imaginary time. In the quasiparticle path-integral treatment,

$$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$

and the phonons generate a nonlocal effective interaction with kernel

$$\mathcal H_{\mathrm{eff},\tau}^{ij} = - \sigma_{ij} \frac{\alpha_{ij}\omega^2\sqrt{\hbar}}{\beta \sqrt{8m_{ij}\omega}} \int_0^{\beta\hbar} d\tau' \, \frac{e^{-\omega |\tau-\tau'|}}{|\mathbf x_{i,\tau}-\mathbf x_{j,\tau'}|},$$

where the sign structure distinguishes attractive self-interactions from repulsive induced electron–hole terms (Park et al., 2022). A later first-principles influence-functional approach generalizes this picture with branch-resolved couplings obtained from GW and DFPT, again integrating out phonons exactly at the Gaussian level (Rana et al., 23 Mar 2026).

In BSE-based frameworks, renormalization is often phrased as a change in the effective electron–hole kernel. For PSII-RC, the static screened interaction is

$W(r,r',\omega=0)=\epsilon^{-1}(r,r')\,v(r-r'),$

and the practical scheme replaces the exact nonlocal $W$ by a fitted translationally invariant kernel $v_W(r-r')$, with reciprocal-space inverse dielectric

$\epsilon^{-1}(k)=1+\frac{v_W(k)}{v(k)}.$

This is then used in a static BSE-like TDHF@$_2$0 formulation beyond the Tamm–Dancoff approximation (Allen et al., 23 Feb 2026). In low-dimensional materials treated on a reduced Hilbert space, the effective Hamiltonian after Löwdin partitioning becomes

$_2$1

and exchange inside the retained space is replaced by an effectively screened exchange kernel induced by the eliminated sector (Qiu et al., 2021).

Fragment-based excitonic renormalization uses a different formal language. REM-TDDFT defines a model space spanned by single-block excitations,

$_2$2

then builds effective monomer and dimer contributions and solves

$_2$3

for excitation energies (Ma et al., 2013). The exact fragment-fluctuation formalism goes further, rewriting the ab initio Hamiltonian in terms of operators $_2$4 that fluctuate fragments between internally correlated states, with an exact expansion up to four-fragment terms because the electronic Hamiltonian is two-body (Dutoi et al., 2017). The later Hermitian XR series constructs overlap and Hamiltonian operators as expansions in interfragment orbital overlap, tending toward a Hermitian projected Hamiltonian while retaining modularity (Bauer et al., 2024).

3. Screening channels and physical mechanisms

A central encyclopedic distinction is between the different channels that renormalize excitons. The data support at least six recurring mechanisms.

Dielectric and metallic polarization modifies both one-particle energies and two-particle attraction. In CdS-based nano-hybrids, dielectric mismatch and metallic interfaces induce image charges that shift electron and hole confinement energies through self-polarization and alter the electron–hole attraction through a polarization-mediated correction (Climente et al., 2011). Geometry determines whether this is weak, as in a neutral matchstick tip, or strong, as in a core-shell geometry that surrounds the radial confinement region.

Phonon-mediated screening weakens exciton binding through retarded lattice polarization. In the Fröhlich path-integral model, phonons induce attractive self-interactions for individual carriers and a repulsive retarded interaction between electron and hole, thereby screening the exciton (Park et al., 2022). The first-principles influence-functional formulation sharpens this by showing that optical phonons, especially LO modes, dominate exciton binding renormalization, whereas acoustic and TO deformation couplings can strongly affect single-carrier polaron energies but largely cancel out of the exciton binding itself (Rana et al., 23 Mar 2026). A BSE-based first-principles treatment reaches a closely related conclusion: vibrational screening in ZnS, MgO, and GaN is dictated by long-range Fröhlich coupling involving polar LO phonons, while remaining vibrational degrees of freedom are negligible (Schebek et al., 2024).

Explicit environments renormalize excitons through anisotropic screening and spectral rehybridization. In the PSII reaction center, embedding the chlorin hexamer in a $_2$5 Å protein environment changes the effective electron–hole interaction at low $_2$6, shifts $_2$7-region excitations, redistributes oscillator strengths, and alters participation ratios and pigment character (Allen et al., 23 Feb 2026). The same paper argues that for sufficiently large systems atomistic details in the screened interaction self-average, so collective $_2$8-dependent polarization becomes the key object.

Disorder generates a state-dependent self-energy for excitons. In a 3D hydrogenic model with short-range disorder, large-scale disorder fluctuations broaden excitons, but short-scale fluctuations of order the Bohr radius produce a systematic down-shift of exciton levels that exceeds the broadening parametrically (Jr. et al., 2021). Because different $_2$9 states shift by different amounts, disorder effectively renormalizes the excitonic Rydberg scale and lifts accidental hydrogenic degeneracies, notably producing an S–P splitting analogous in spirit, though not in sign, to the Lamb shift.

Subspace truncation and omitted screening sectors can renormalize excitonic kernels even when the full-theory object is formally unscreened. In reduced-subspace BSE, the direct interaction is physically screened by the environment, but exchange remains formally bare. However, once the Hilbert space is partitioned into retained and omitted sectors, the omitted transitions induce an effective screening of exchange inside the retained space (Qiu et al., 2021). This is especially relevant for singlet–triplet splitting and convergence with respect to empty states.

Optical driving and nonequilibrium correlations modify effective excitonic splittings and couplings dynamically. In time-dependent GW-BSE/HSEX for WS$_2$0, coherent pumping creates excitonic superpositions whose oscillation energies can differ from equilibrium excitonic splittings because intermediate excitons such as $_2$1 alter the effective coupling dynamics (Cervantes-Villanueva et al., 27 Mar 2026). In pumped MoS$_2$2, bound excitonic correlations below the Mott threshold renormalize the quasiparticle self-energy, causing a gap opening of about $_2$3 meV and a strong effective-mass enhancement (Lin et al., 2022).

4. Methodological families

The literature in the data block divides naturally into several methodological families.

Family	Core object	Representative papers
Electrostatic CI / envelope-function	Renormalized exciton Hamiltonian with image and polarization terms	(Climente et al., 2011)
Path-integral phonon frameworks	Imaginary-time influence functional after integrating out phonons	(Park et al., 2022, Rana et al., 23 Mar 2026)
BSE with explicit environmental or phonon screening	Screened electron–hole kernel modified by environment or lattice	(Allen et al., 23 Feb 2026, Qiu et al., 2021, Schebek et al., 2024)
Fragment-based effective Hamiltonians	Excitonic Hamiltonian built from monomer/dimer or fragment fluctuation data	(Ma et al., 2013, Dutoi et al., 2017, Bauer et al., 2024)
Disorder or lattice-model renormalization	Self-energy or PRM-renormalized susceptibility in model Hamiltonians	(Jr. et al., 2021, Phan et al., 2011, Ohki et al., 2019)
Nonequilibrium many-body propagation	Time-dependent density matrix or spectral function under driving	(Cervantes-Villanueva et al., 27 Mar 2026, Lin et al., 2022)

These families differ in what is integrated out and what is solved explicitly. Electrostatic and BSE-based approaches usually keep the exciton basis explicit and renormalize kernels. Path-integral approaches integrate out phonons exactly within harmonic models and sample the remaining retarded two-particle problem. Fragment methods integrate out local high-energy electronic complexity into reusable subsystem tensors. Disorder and lattice-model PRM methods compute renormalized susceptibilities or effective quasiparticle dispersions directly in model space. Time-dependent methods propagate the many-body density matrix or transient spectral function rather than solving a static eigenvalue problem.

A plausible implication is that “framework” should be understood structurally rather than tied to a single formalism: it denotes a procedure for incorporating environmental, phononic, geometric, or truncation effects into an excitonic description.

5. Representative systems and benchmark behaviors

Specific systems clarify how different frameworks manifest.

In semiconductor–metal nano-hybrids, neutral matchstick structures show only meV-scale metal-induced changes because the exciton remains centered in the rod and the tip-localized image well is too weak and localized to bind carriers (Climente et al., 2011). Charged matchsticks behave differently: trapped charges in the metallic tip produce strong red-shifts and spatially indirect excitons, with the hole localizing near the metal and the electron pushed away. Neutral core-shell structures lie between these extremes, showing tens-of-meV red-shifts because the shell perturbs the dominant radial confinement channel throughout the nanorod (Climente et al., 2011).

In polar semiconductors, phonon renormalization scales with material polarity. In the first-principles BSE correction for MgO, ZnS, and GaN, the lowest excitonic shifts are about $_2$4, $_2$5, and $_2$6 meV, respectively, and higher absorption peaks can shift by up to $_2$7 meV (Schebek et al., 2024). In the nonperturbative influence-functional treatment, the renormalized zero-temperature binding energies are $_2$8 meV for MgO, $_2$9 meV for CdS, $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$0 meV for AgCl, and $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$1 meV for CsPbBr$$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$2, all substantially smaller than static Wannier estimates and generally closer to experiment (Rana et al., 23 Mar 2026). The two phonon papers together support a consistent picture in which lattice screening reduces exciton binding by a few tens to hundreds of meV depending on coupling strength and dielectric contrast (Park et al., 2022, Rana et al., 23 Mar 2026, Schebek et al., 2024).

In biological nanostructures, the PSII reaction center provides a many-body environment-renormalization benchmark. Embedding the pigment core in an explicit quantum-mechanical protein environment shifts the bright $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$3-like excitation from $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$4 to $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$5 eV and reduces its participation ratio from $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$6 to $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$7, indicating environment-induced localization in transition space (Allen et al., 23 Feb 2026). Other low-lying states shift differently and acquire altered pigment character, demonstrating that environmental renormalization is polarization dependent rather than a rigid site-energy shift.

In molecular aggregates, REM-TDDFT typically keeps low-lying excitation-energy errors below $$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$8 eV relative to full TDDFT when paired with a long-range corrected functional such as LC-BLYP (Ma et al., 2013). The Hermitian XR series on Be$$H_X^{NH} = H_{e,h}^{SNR} + H_{e,h}^{pol} + H_C^{SNR} + H_C^{pol} + V_{e,h}^{imp},$$9 shows a different benchmark logic: the zeroth-order method is comparable to CCSD(T), the first-order method to FCI, and the second-order method agrees with FCI well up the inner repulsive wall, indicating rapid convergence of the overlap expansion (Bauer et al., 2024).

In low-dimensional semiconductors, the effect of environment on singlet–triplet splitting differs from its effect on exciton binding. For h-BN encapsulated by semiconducting slabs or in an h-BN/graphene superlattice, the singlet–triplet splitting changes only by a few meV, even though environmental screening can strongly modify direct-term physics in general (Qiu et al., 2021). This reflects the short-range nature of exchange relative to the more environment-sensitive direct interaction.

In disordered 3D excitons, the renormalization is primarily spectral. The disorder-induced real shift can exceed broadening parametrically, and the state-dependent coefficients $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$0 produce both effective Rydberg renormalization and S–P splitting (Jr. et al., 2021).

6. Ordered phases, collective regimes, and driven excitonic manifolds

Several papers extend excitonic renormalization beyond single-exciton energies into collective or ordered regimes.

In the 2D extended Falicov–Kimball model, the projector-based renormalization method separates Hartree terms from fluctuation terms, renormalizes the band structure and spontaneous hybridization field, and then computes the dynamical electron–hole susceptibility (Phan et al., 2011). The resulting picture distinguishes a BCS-like excitonic insulator on the semimetal side from a BEC-like excitonic insulator on the semiconducting side. Above the transition temperature on the semiconductor side, the imaginary part of the susceptibility shows a $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$1 excitonic resonance, interpreted as a precursor or “excitonic halo” regime (Phan et al., 2011).

In tilted Dirac systems, the instability is controlled by chemical-potential shift and in-plane field rather than by interaction strength alone. The RG treatment renormalizes the Dirac velocities while the ladder approximation identifies an even-parity, spin-transverse, intervalley excitonic instability with criterion $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$2 in a linearized eigenvalue equation (Ohki et al., 2019). The NMR relaxation rate $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$3 is then linked to excitonic spin fluctuations over an extended parameter region. This is a renormalization framework in the sense that RG-improved quasiparticles feed a ladder susceptibility whose divergence diagnoses spontaneous mass generation.

In microcavity exciton-polariton systems, the projector-based renormalization framework of the exciton polariton model treats Coulomb interaction and electron–hole/photon coupling on an equal footing and finds a crossover from excitonic-insulator character to polariton and photonic condensed states as excitation density increases at large detuning (Phan et al., 2015). A plausible implication is that here renormalization concerns not only the excitonic subsystem but also its hybridization with a bosonic light mode, producing a renormalized quasiparticle band structure and characteristic luminescence signatures.

In ultrafast driven 2D materials, the excitonic manifold itself becomes time dependent. The time-dependent HSEX equation of motion

$$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$4

propagates pump-induced coherences in WS$$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$5, while effective few-level equations show that inter-exciton coherence depends on population imbalance and Rabi couplings (Cervantes-Villanueva et al., 27 Mar 2026). This suggests that in nonequilibrium settings renormalization should include coherent activation of couplings, not only screened interaction changes.

7. Approximations, controversies, and recurrent limitations

The data repeatedly stress that each framework isolates some renormalization channels while neglecting others. This is a major source of apparent disagreement across subfields.

A common misconception is that “screening” is a single scalar shift. Several papers explicitly reject this simplification. In semiconductor–metal hybrids, the net excitonic shift is a competition between self-energy/image-charge terms and polarization-mediated modification of electron–hole attraction (Climente et al., 2011). In phonon problems, self-trapping and dynamical screening must both be included to define binding properly (Park et al., 2022, Rana et al., 23 Mar 2026). In low-dimensional BSE, environment strongly screens the direct term but does not equivalently screen exchange in the full theory (Qiu et al., 2021). This suggests that excitonic renormalization should not be reduced to a single “screening correction.”

Another recurring issue is the distinction between formal many-body exactness within a model and physical completeness for real materials. Path-integral phonon methods are nonperturbative relative to the assumed harmonic Fröhlich-type model, but remain approximate because of effective-mass bands, limited phonon branches, and neglected anharmonicity (Park et al., 2022, Rana et al., 23 Mar 2026). Static BSE with fitted $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$6 is exact only within its static-screening and self-averaging assumptions (Allen et al., 23 Feb 2026). REM-TDDFT is accurate when low-lying excitations are local or weakly delocalized but becomes less reliable for higher excitations or stronger interfragment charge transfer (Ma et al., 2013). Disorder theories assume weak short-range disorder and hydrogenic excitons (Jr. et al., 2021). Time-dependent HSEX with frozen screening captures coherent dressing but not full nonequilibrium $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$7 (Cervantes-Villanueva et al., 27 Mar 2026).

A further controversy concerns the role of the environment in fine structure. The low-dimensional BSE subspace study finds that singlet–triplet splitting is largely unaffected by the external dielectric environment for most quasi-two-dimensional materials (Qiu et al., 2021). This may appear at odds with the strong environment-induced changes in optical spectra reported elsewhere, but the distinction follows from the direct-versus-exchange decomposition: binding energies and optical gaps can shift strongly while short-range exchange-dominated splittings shift only weakly.

The data on monolayer WSe$$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$8 present a separate caution. Only a supplementary fragment is available, so a full exciton–phonon renormalization formalism cannot be reconstructed without inventing missing equations. The supplied text therefore limits itself to DFT convergence, $$\hat{\mathcal H}=\hat{\mathcal H}_{\mathrm{eh}}+\hat{\mathcal H}_{\mathrm{ph}}+\hat{\mathcal H}_{\mathrm{int}},$$9 band stretching, and the inferred GW–BSE–phonon workflow, rather than explicit exciton–phonon equations (Mishra et al., 2018). This is a methodological reminder that “excitonic renormalization framework” often spans incomplete or hybrid workflows in practice.

Overall, the literature suggests that no single universal excitonic renormalization framework exists. Instead, the term designates a technically diverse but conceptually unified program: identify the external or integrated-out degrees of freedom that modify neutral excitations, construct the corresponding effective one- and two-particle corrections, and solve or sample the resulting renormalized excitonic problem at the level appropriate to geometry, material class, and target observable [(Climente et al., 2011); (Park et al., 2022); (Allen et al., 23 Feb 2026); (Ma et al., 2013); (Dutoi et al., 2017); (Qiu et al., 2021); (Jr. et al., 2021); (Rana et al., 23 Mar 2026); (Schebek et al., 2024)].