Non-Phononic Vibrational Density of States

• Non-phononic vibrational DoS is the spectrum of vibrational modes in disordered materials that diverge from the Debye model, incorporating quasi-localized and diffusive excitations.
• The topic employs methods like imposed-wave analysis, finite-size scaling, and random pinning to effectively separate phononic and non-phononic contributions.
• Key insights include varied low-frequency scaling laws (e.g., ω^4 and ω^2) with implications for understanding boson peaks, marginal stability, and phonon–defect hybridization.

Searching arXiv for recent and foundational papers on non-phononic vibrational density of states, boson peak, and related low-frequency anomalies. I’m unable to invoke the arXiv search tool directly in this environment, so I will ground the article strictly in the supplied arXiv records and cite them accordingly. Non-phononic vibrational density of states (DoS) denotes the part of the vibrational spectrum that is not exhausted by ordinary acoustic phonons and their Debye counting. In disordered solids, this topic arises because the measured vibrational density of states deviates from the Debye form even at relatively low frequencies, and those deviations can originate from physically distinct mechanisms: disorder-renormalized phonons, genuinely non-phononic quasi-localized or diffusive excitations, and, in liquids, overdamped localized motions rather than propagating waves. A central modern theme is therefore not merely the existence of non-Debye behavior, but the separation of phononic and non-phononic contributions, their scaling laws, and their relation to the boson peak, marginal stability, frozen-in stress, and phonon–defect hybridization (Lerner et al., 6 May 2026).

1. Debye baseline and the meaning of “non-phononic”

The reference point is the Debye spectrum generated by long-wavelength elastic waves. In three dimensions, the Debye vibrational density of states is written as

$${\cal D}_{\rm D}(\omega)=A_{\rm D}\omega^2, \qquad A_{\rm D}=\frac{V}{6\pi^2N}\left(\frac{2}{c_{\rm s}^3}+\frac{1}{c_\ell^3}\right),$$

so the reduced spectrum ${\cal D}(\omega)/\omega^2$ is flat when only continuum-elastic phonons are present. Within this usage, a rise of the reduced spectrum above the Debye plateau is a non-Debye anomaly, and the boson peak is the maximum of that reduced spectrum (Lerner et al., 6 May 2026).

In this framework, “phononic” and “non-phononic” are not synonymous with “extended” and “localized” in any simple way. Phonons are the continuum-elastic excitations enforced by translational invariance, with linear dispersion in the long-wavelength limit. Non-phononic modes are instead excitations that are not ordinary elastic plane waves. In glasses, these are often quasi-localized or localized low-frequency modes; in jamming-related mean-field sectors they can be extended anomalous modes; in liquids they may be overdamped local motions rather than oscillatory waves (Kapteijns et al., 2018).

A recurrent source of confusion is that the same low-frequency exponent can have different physical meanings in different settings. In three-dimensional solids, $D(\omega)\sim \omega^2$ may be Debye phonons, but in marginally stable amorphous packings the same $\omega^2$ scaling can instead describe a non-phononic anomalous sector with a pressure-dependent prefactor. Conversely, an $\omega^4$ contribution may describe quasi-localized glassy modes, but in other treatments an $\omega^4$ excess can also arise from disorder-renormalized phonons or continuum-elastic scattering corrections (Charbonneau et al., 2015).

2. Competing theoretical decompositions

One influential decomposition writes the measured vibrational density of states as the sum of a phononic part, obtained from disorder-renormalized acoustic dispersion, and a non-phononic part associated with genuinely disorder-induced excitations. In that picture,

$${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$

with ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$ for the low-frequency non-phononic tail in glasses. The phononic part itself is not Debye once disorder softens the dispersion. Using

$$\omega(k)\simeq ck-c\Upsilon\xi^2k^3,\qquad \Upsilon=\frac{1}{96},$$

one obtains

$${\cal D}_{\rm ph}(\omega)\simeq A_{\rm D}\omega^2+A_{\rm w}\omega^4+A_6\omega^6+{\cal O}(\omega^8),$$

so the onset of excess modes is controlled by the competition between two distinct ${\cal D}(\omega)/\omega^2$0 amplitudes, ${\cal D}(\omega)/\omega^2$1 and ${\cal D}(\omega)/\omega^2$2 (Lerner et al., 6 May 2026).

Other frameworks do not begin from such an additive split. In the effective-field-theory treatment of amorphous materials, disorder broadens the acoustic spectrum through momentum diffusion, and the non-phononic sector is identified with disorder-broadened, non-ballistic excitations (“diffusons”) generated by the same Green’s function that describes propagating phonons at lower ${\cal D}(\omega)/\omega^2$3. There the boson peak appears as a crossover from propagons to diffusons rather than as a separately inserted family of modes (Baggioli et al., 2019). A closely related continuum theory with power-law correlated elastic disorder likewise retains a Debye baseline but predicts a disorder-induced excess above it, including a three-dimensional low-frequency excess ${\cal D}(\omega)/\omega^2$4 and a logarithmic correction in the reduced excess DoS near the boson peak (Cui et al., 2020).

A third decomposition, developed within generalized heterogeneous-elasticity theory, distinguishes two non-phononic sectors. Type-I excitations are random-matrix-like modes associated with marginal stability and the boson-peak sector, with ${\cal D}(\omega)/\omega^2$5 in small marginal systems. Type-II excitations arise from local non-irrotational oscillations associated with frozen-in stress; in that theory their low-frequency DoS scales as

${\cal D}(\omega)/\omega^2$6

with ${\cal D}(\omega)/\omega^2$7 controlled by the statistics of small local stresses and by the tapering of the interaction potential near cutoff (Schirmacher et al., 2023).

3. Reported scaling laws across systems

The literature does not support a single universal low-frequency exponent for all non-phononic spectra. Instead, different systems and different separation procedures yield distinct laws.

System or regime	Reported low-frequency law	Physical interpretation
Three-dimensional glasses	${\cal D}(\omega)/\omega^2$8	Gapless quasi-localized non-phononic tail
Two-dimensional glasses with random pinning	${\cal D}(\omega)/\omega^2$9 or $D(\omega)\sim \omega^2$0	Bare non-phononic spectrum after phonon suppression
Two-dimensional glasses below $D(\omega)\sim \omega^2$1	$D(\omega)\sim \omega^2$2	Alternative quasi-localized scaling claim
Soft-sphere packings above jamming	$D(\omega)\sim \omega^2$3	Mean-field anomalous non-phononic sector
Liquids, overdamped regime	$D(\omega)\sim \omega^2$4	Overdamped localized non-propagating motions
GHET type-II sector	$D(\omega)\sim \omega^2$5	Frozen-stress-controlled non-phononic modes

The quartic law has especially broad support in structural glasses. Direct measurements in two, three, and four dimensions found a universal non-phononic law $D(\omega)\sim \omega^2$6, with dimensionality affecting the prefactor and localization rather than the exponent. In that work, the associated modes were soft quasilocalized vibrational modes with a disordered core and power-law elastic tails, and the cumulative spectrum obeyed $D(\omega)\sim \omega^2$7 below a cutoff scale $D(\omega)\sim \omega^2$8 (Kapteijns et al., 2018).

Two-dimensional systems remain the most controversial. One line of work argues that once phonons are suppressed by random pinning, the intrinsic non-phononic vDoS is again $D(\omega)\sim \omega^2$9, with exponentially decaying mode profiles and no far-field elastic tail (Shiraishi et al., 2023). Another finds that for systems larger than about 100 particles, the cumulative density of states below the first transverse sound mode scales as $\omega^2$0, implying $\omega^2$1, together with a distinctive $\omega^2$2 finite-size scaling (Wang et al., 2022). This discrepancy remains an active point of debate rather than a settled matter.

Near jamming, a different anomalous sector is reported. Simulations of soft-sphere packings in dimensions $\omega^2$3 through $\omega^2$4 find a low-frequency $\omega^2$5 sector over the accessible window, consistent with mean-field predictions and distinct from Debye scaling in $\omega^2$6. The relevant crossover frequency obeys $\omega^2$7, and the true Debye crossover $\omega^2$8 is either below numerical resolution or absent near jamming (Charbonneau et al., 2015).

Liquids add a further distinction. One analysis argues that the low-frequency DoS is Debye-like, $\omega^2$9, for propagating collective modes, while an $\omega^4$0-linear sector appears only for overdamped, localized, non-propagating motions in the regime $\omega^4$1 (Brazhkin, 2024). Another derives the same linear law from purely imaginary instantaneous normal modes, obtaining

$\omega^4$2

so that $\omega^4$3 at low frequency as a direct spectral fingerprint of relaxational, non-phononic dynamics (Zaccone et al., 2021).

4. Spatial structure and microscopic origin

In many glass models, the low-frequency non-phononic modes are quasi-localized. Their characteristic structure is a localized or compact core, often of linear size of order ten particle diameters, dressed by an elastic far field. In the dimensional comparison of two-, three-, and four-dimensional glasses, the far-field displacement decays as $\omega^4$4, which implies much weaker localization in two dimensions than in three or four. This difference explains the strong system-size dependence of the participation ratio and prefactor in two dimensions, compared with the much weaker finite-size effects in higher dimensions (Kapteijns et al., 2018).

Random pinning changes that morphology. In pinned two-dimensional Kob–Andersen glasses, the participation ratios of low-frequency modes become extremely small, and the radial decay profile of a representative low-frequency mode is exponential rather than $\omega^4$5. The reported conclusion is that pinning removes phonon hybridization and reveals truly localized non-phononic modes without far-field contributions (Shiraishi et al., 2023).

A more specific microscopic picture has been developed for the boson-peak regime. A cage-relative filtering procedure applied to boson-peak eigenmodes reveals localized defects hidden inside hybridized phonon–defect states. These defects are described as anisotropic, compact, and predominantly pure-shear objects. In two dimensions their shape distribution peaks near $\omega^4$6, inconsistent with both isotropic disks and one-dimensional strings, while in three dimensions their total deviatoric strain distribution peaks near $\omega^4$7, i.e. close to pure shear. Their density $\omega^4$8 tracks the boson-peak excess $\omega^4$9 across the systems studied (Mahajan et al., 2024).

Other theories locate the microscopic origin in frozen-in stress rather than in quasi-localized defects alone. In GHET, the additional non-phononic modes arise from local non-irrotational oscillations associated with the stress field. Their low-frequency spectrum is linked to the small-$\omega^4$0 behavior of the local stress distribution, and the resulting states obey GOE level statistics rather than Poisson statistics, leading that theory to classify both type-I and type-II low-frequency non-phononic excitations as extended rather than localized (Schirmacher et al., 2023). This suggests that “non-phononic” does not imply a single mode morphology.

Network and gel systems provide another microscopic route. In a patchy-particle gel, the low-frequency excess grows as the network becomes more chain-like, and analysis of representative motifs and linear chains attributes the excess to strong translation–rotation coupling in chain segments, especially transverse strand motions softened by coherent particle rotations (Rovigatti et al., 2011).

5. Methods of identification and separation

The central methodological problem is that in disordered systems a plane wave is generally not an exact Hessian eigenmode, while localized or quasi-localized modes may hybridize with phonons. Modern work therefore relies on several complementary separation procedures rather than on a single diagnostic.

One strategy begins from the actual phonon dispersion. In disordered solids, the imposed-wave method applies a wave-like force field,

$\omega^4$1

computes the linear response $\omega^4$2, and assigns a frequency through a Rayleigh quotient. This yields disorder-averaged longitudinal and transverse dispersions, from which the phononic contribution $\omega^4$3 can be constructed without adjustable parameters once the elastic data are known (Lerner et al., 6 May 2026).

A second strategy isolates non-phononic modes by finite-size spectral separation. In small glasses, phonon bands move upward, opening a low-frequency window dominated by non-phononic modes. In that regime, the cumulative spectrum can be fitted to $\omega^4$4 to extract the quartic tail. Participation ratios are then used to distinguish extended phonon bands from quasi-localized excitations; in glasses the latter scale roughly as $\omega^4$5, whereas in two-dimensional analyses based on sub-$\omega^4$6 windows the cumulative DoS and its $\omega^4$7 scaling play the main role because participation-ratio thresholds become unreliable (Lerner et al., 6 May 2026).

A third route suppresses phonons directly. Random pinning breaks translational invariance, removes the two zero-frequency global translations, and strongly suppresses the low-frequency Goldstone sector. In the pinned two-dimensional glass study, this exposure of the localized sector was the methodological reason the cumulative vDoS could be read as $\omega^4$8 with much weaker contamination from phonon hybridization (Shiraishi et al., 2023).

Boson-peak studies now also use local non-affinity filters. The cage-relative displacement

$\omega^4$9

suppresses locally affine phononic motion and retains localized defect content inside hybridized boson-peak modes. Particles in the top $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$0 of $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$1 are then clustered by Voronoi adjacency to define defects, making it possible to estimate defect densities even when the original eigenmodes look extended (Mahajan et al., 2024).

In liquids, the methodological distinction is different. Current-current correlation functions and inelastic scattering peak positions are taken to probe propagating phonon-like excitations and hence a quadratic low-frequency DoS, whereas instantaneous normal mode analyses can mix real and imaginary frequencies and thereby count overdamped, non-vibrational local motions as if they were ordinary vibrations. This difference is central to the disagreement over whether a measured linear low-frequency DoS in a liquid is intrinsic, overdamped, or partly an analysis artifact (Brazhkin, 2024).

6. Boson peak, disorder dependence, and unresolved controversies

The boson peak is the most prominent context in which non-phononic DoS is discussed, but current work does not support a single-origin explanation. In the additive decomposition based on measured phonon dispersion, the boson-peak excess in realistic glasses is a mixed object. The non-phononic fraction up to $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$2 may be quantified by

$${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$3

and representative values reported for glasses are $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$4, $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$5, $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$6, $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$7, and approximately $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$8 for a hyperquenched BIPL glass. These values indicate a sizable but not exclusive non-phononic contribution, and directly challenge purely phononic identifications of the boson peak with a shifted Van Hove-like feature (Lerner et al., 6 May 2026).

A more defect-centered interpretation argues that localized boson-peak defects are the microscopic origin of the excess. In that picture, the boson peak is not merely a softened-phonon effect, because the density of extracted compact quadrupolar defects correlates with $${\cal D}(\omega)\approx {\cal D}_{\rm ph}(\omega)+{\cal D}_{\rm G}(\omega),$$9 across two- and three-dimensional systems. At the same time, those defects are not generally visible as clean isolated eigenmodes because phonons at ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$0 strongly hybridize with them (Mahajan et al., 2024).

Jamming-related work advances a different claim. In soft-sphere packings above jamming, the anomalous ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$1 sector is interpreted as a universal mean-field signature of marginal stability, while quasilocalization is treated as a lower-dimensional correction rather than as the origin of the excess DoS itself. In that view, the boson peak is “purely mean-field in nature,” and any ultimate Debye regime is pushed to extremely low or inaccessible frequencies (Charbonneau et al., 2015).

Continuum-elastic theories add further interpretations. The unified theory of amorphous spectra identifies the boson peak with the crossover from propagons to diffusons, driven mainly by transverse disorder broadening (Baggioli et al., 2019). The power-law-correlated-disorder theory instead attributes the boson peak and the associated excess DoS to long-ranged elastic disorder that produces logarithmically enhanced attenuation and a logarithmic correction in the reduced excess DoS near the boson peak (Cui et al., 2020).

The status of the quartic law is itself controversial. One line of work treats ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$2 as universal for non-phononic glassy modes across dimensions when phonons are cleanly separated (Kapteijns et al., 2018). Another argues that the low-frequency exponent is non-universal, especially for stress-controlled type-II modes in small stable systems, and depends on cutoff tapering and on the small-stress statistics induced by the pair potential. In that picture, the frequently observed ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$3 law may result from the common ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$4 tapering procedure, while other choices yield ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$5 or ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$6 (Schirmacher et al., 5 Sep 2025).

The resulting picture is therefore plural rather than singular. Non-phononic vibrational density of states is a family of phenomena: gapless quasi-localized quartic tails in many glasses, anomalous ${\cal D}_{\rm G}(\omega)\simeq A_{\rm g}\omega^4$7 sectors near jamming, overdamped linear spectra in liquids, frozen-stress-controlled spectra with non-universal exponents, and boson-peak defects embedded in hybridized modes. The common denominator is not a unique exponent, but the failure of ordinary acoustic phonons to account for the full low-frequency spectrum of disordered matter (Schirmacher et al., 2023).