Boson Peak in Disordered Materials

• Boson Peak is a low-frequency excess in the vibrational density of states in disordered solids, indicating nonphononic vibrations and elastic heterogeneity.
• Experimental findings show the peak appears in the THz/meV range with dispersionless behavior and strong correlation with structural and elastic properties.
• Recent theories attribute the Boson Peak to mechanisms such as quasi-localized vibrations, marginal stability, and local inversion-symmetry breaking.

The boson peak (BP) is a low-energy excess in the vibrational density of states (vDOS) over the Debye contribution, conventionally identified as a maximum of the reduced vDOS $g(\omega)/\omega^{d-1}$, or $g(\omega)/\omega^2$ in three dimensions. It is a hallmark of glasses and amorphous materials, typically appears in the THz or meV range, and is linked to anomalous low-temperature heat capacity, thermal conductivity, sound attenuation, and nonaffine elasticity. Despite decades of work, its microscopic origin remains debated; recent literature has associated the BP with quasi-localized nonphononic vibrations, local inversion-symmetry breaking, marginal stability near isostaticity, elastic heterogeneity, dispersionless bands in $S(q,\omega)$, and one-dimensional string-like excitations called stringlets (Jiang et al., 2023, Moriel et al., 2023, Mizuno et al., 27 Jan 2026).

1. Definition and spectral characterization

In an isotropic elastic continuum, the Debye prediction is $g_{\rm Debye}(\omega)=A_D\,\omega^2$ in three dimensions, and more generally $g_D(\omega)\propto\omega^{d-1}$ in spatial dimension $d$. The BP is the excess above this baseline. A standard definition uses the reduced vDOS,

$$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$

whose maximum defines $\omega_{\rm BP}$ in 3D; in 2D, the analogous reduced quantity is $g(\omega)/\omega$ (Moriel et al., 2023, Tømterud et al., 2022).

A useful decomposition separates phononic and nonphononic sectors:

$g(\omega)=A_D\,\omega^2+g_G(\omega),$

where $g(\omega)/\omega^2$0 is the nonphononic vDOS. In this representation, the nonphononic contribution itself can display both a low-frequency power-law tail and a peak at $g(\omega)/\omega^2$1, which coincides up to small shifts with $g(\omega)/\omega^2$2. Recent work emphasized that the BP is not only a feature of the ratio $g(\omega)/\omega^2$3 but also an intrinsic feature of the nonphononic spectrum (Moriel et al., 2023).

The reduced vDOS is not the only route to identifying the BP. In scattering experiments, it also appears through the dynamical structure factor $g(\omega)/\omega^2$4, in Raman and far-infrared observables, in heat-capacity anomalies such as the hump in $g(\omega)/\omega^2$5, and in low-frequency dielectric response. This multiplicity of definitions is part of the reason the phenomenon supports several partially overlapping theoretical interpretations (Kabeya et al., 2017, Li et al., 2023, Mahajan et al., 8 Sep 2025).

2. Experimental manifestations across materials and dimensions

The BP has been observed in oxide, metallic, polymeric, colloidal, granular, and atomically thin systems. In vitreous glucose, integrated terahertz-band spectroscopy found a pronounced peak in $g(\omega)/\omega^2$6; at $g(\omega)/\omega^2$7 the BP appears at $g(\omega)/\omega^2$8 with full-width-at-half-maximum $g(\omega)/\omega^2$9, and as temperature rises toward $S(q,\omega)$0 the BP shifts to lower $S(q,\omega)$1 and broadens (Kabeya et al., 2017).

In Zr$S(q,\omega)$2Cu$S(q,\omega)$3Al$S(q,\omega)$4 metallic glasses, inelastic neutron scattering identified a pronounced ridge of intensity at $S(q,\omega)$5, virtually independent of $S(q,\omega)$6 over $S(q,\omega)$7–$S(q,\omega)$8; Raman scattering showed a peak at $S(q,\omega)$9 and a heat-capacity fit yielded $g_{\rm Debye}(\omega)=A_D\,\omega^2$0 (Li et al., 2023). In femtosecond MeV ultrafast electron diffraction on a metallic glass film, collective atomic oscillations were observed in both reciprocal and real space within the BP frequency range of $g_{\rm Debye}(\omega)=A_D\,\omega^2$1–$g_{\rm Debye}(\omega)=A_D\,\omega^2$2, and the extracted wave velocity $g_{\rm Debye}(\omega)=A_D\,\omega^2$3 affirmed the transverse acoustic wave nature of the oscillations (Tian et al., 2021).

Surface and confinement effects can also shift the BP. In Al/Al$g_{\rm Debye}(\omega)=A_D\,\omega^2$4O$g_{\rm Debye}(\omega)=A_D\,\omega^2$5 core-shell nanoparticles with a $g_{\rm Debye}(\omega)=A_D\,\omega^2$6 amorphous alumina shell, neutron spectroscopy found an intense low-energy peak at $g_{\rm Debye}(\omega)=A_D\,\omega^2$7; its intensity scales inversely with particle size and oxide fraction, indicating a surface origin, and molecular dynamics showed that the frequency is softened compared with a hypothetical bulk amorphous Al$g_{\rm Debye}(\omega)=A_D\,\omega^2$8O$g_{\rm Debye}(\omega)=A_D\,\omega^2$9 (Cortie et al., 2019). In amorphous bilayer SiO$g_D(\omega)\propto\omega^{d-1}$0/Ru(0001), helium atom scattering provided the first experimental evidence of the BP in a 2D material: a dispersionless BP appears at $g_D(\omega)\propto\omega^{d-1}$1, corresponding to $g_D(\omega)\propto\omega^{d-1}$2, with reasonable evidence for a double excitation at $g_D(\omega)\propto\omega^{d-1}$3 (Tømterud et al., 2022).

Model systems have been equally important. In hard-sphere colloidal glasses, the experimental determination of the vibrational density of states showed that the emergence of the BP and high values of the specific heat are directly related and are specific to the glass; a very defected crystal with disorder only slightly smaller than for the glass had significantly smaller low-frequency DOS and specific heat (Zargar et al., 2014). In 2D disordered granular crystals, the BP was observed in a system where the disorder is due to the force network and the spatial heterogeneity of elasticity (Zhang et al., 2015).

3. Momentum-space structure and real-space correlations

A major development is the shift from viewing the BP purely as a scalar excess in $g_D(\omega)\propto\omega^{d-1}$4 to treating it as a feature with a definite $g_D(\omega)\propto\omega^{d-1}$5-space structure. In metallic glasses, the BP energy was found to be largely dispersionless, while the BP intensity scales with the static structure factor. Specifically, in Zr$g_D(\omega)\propto\omega^{d-1}$6Cu$g_D(\omega)\propto\omega^{d-1}$7Al$g_D(\omega)\propto\omega^{d-1}$8, $g_D(\omega)\propto\omega^{d-1}$9 tracks $d$0 one-to-one, and only shear-strain fluctuations among several tested local structural descriptors produced both a dispersionless peak at $d$1 and $d$2 (Li et al., 2023).

This dispersionless-band picture was developed further in simulations of both network- and packing-type glasses. Two routes were formulated for extracting the vDOS from $d$3: a high-$d$4 route, analogous to velocity-autocorrelation approaches, and a low-$d$5 route that resolves contributions from different wavenumber sectors. The key result is that the BP in the vDOS emerges as the spectral consequence of a dispersionless excitation band in $d$6; integrating the high-$d$7 sector of the transverse response yields a nonphononic contribution $d$8 that peaks at $d$9 (Mizuno et al., 27 Jan 2026).

Real-space measurements support the same conclusion. Ultrafast electron diffraction showed that the oscillation frequency has reciprocal dependence on interatomic pair distances, $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$0, and the corresponding dispersion $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$1 is consistent with the transverse acoustic branch in Zr-based metallic glasses (Tian et al., 2021). A related real-space solution of the inhomogeneous elastic wave equation found a BP-associated flat dispersion relation, localized vibration in exponential decay in a soft spot, and a BP frequency that depends on the fluctuation length of the shear modulus, with $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$2 in the reported scaling form (Jiang, 2024).

These results collectively support a representation of the BP as a nonphononic, nearly $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$3-independent band that coexists with acoustic branches. This suggests that the BP is not exhausted by a purely frequency-space excess; it also reflects the wavenumber-resolved organization of vibrational excitations in disordered solids (Li et al., 2023, Mizuno et al., 27 Jan 2026).

4. Microscopic theories and competing mechanisms

One influential framework identifies the BP with interacting quasi-localized nonphononic vibrations. In this picture, the very low-frequency nonphononic sector follows

$$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$4

and as frequency increases these excitations hybridize through random elastic couplings and internal stresses, piling up around a characteristic frequency $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$5 and forming the BP excess. The same framework explains why the peak frequency and magnitude vary mildly with thermal history, while the $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$6 tail can vary strongly, and it finds that modes near $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$7 are collective hybrids of many quasi-localized excitations (Moriel et al., 2023).

Another line of work ties the BP to local symmetry. Numerical lattice dynamics on random-network glasses and randomly depleted FCC crystals introduced a local inversion-symmetry order parameter $$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$8 and showed that both the BP and the nonaffine softening of the material display a strong positive correlation with this quantity, whereas the standard bond-orientational order parameter is a poor correlator. In these models, the boson-peak frequency scales linearly with coordination,

$$R(\omega)\equiv \frac{g(\omega)}{g_D(\omega)}=\frac{g(\omega)}{A_D\,\omega^2},$$9

and the shear modulus obeys $\omega_{\rm BP}$0 (Milkus et al., 2016).

A third framework emphasizes isostaticity and marginal stability. In silica glass, counting bond-stretching and bond-bending constraints confirms isostaticity at the tetrahedral composition. Simulations then distinguish three limits: an isostatic network with $\omega_{\rm BP}$1 finite, an overconstrained unstressed network where the BP shifts to a finite frequency, and the physical pre-stressed glass in which frustration pushes soft modes downward and produces quasi-localized vibrations with $\omega_{\rm BP}$2 below a crossover frequency. In this description, the BP in the full glass reflects near-isostatic constraints and marginal stability, and appears as a broad, nearly wavenumber-independent band in the dynamical structure factor (Mizuno et al., 28 Aug 2025).

Heterogeneous elasticity theory, in coherent-potential-approximation form, takes a different route. It models the shear modulus as a fluctuating field $\omega_{\rm BP}$3 and introduces a coarse-graining wavenumber $\omega_{\rm BP}$4 together with a variance parameter for shear-modulus fluctuations. A strong correlation was reported between $\omega_{\rm BP}$5 and the first sharp diffraction peak (FSDP) wavenumber, leading to the two-step interpretation that the FSDP determines the unit size of elastic-modulus heterogeneity, while the magnitude of the modulus fluctuation determines the BP frequency and intensity (Kyotani et al., 10 Jan 2025).

Stringlet-based theories treat the relevant excess modes as one-dimensional vibrating strings with an exponential size distribution. A recent quantitative model generalized a framework originally proposed by Lund and provided an analytical prediction for $\omega_{\rm BP}$6 in 2D and 3D amorphous systems in the low-temperature regime well below the glass transition temperature, with no free parameters and quantitative agreement with prior simulation observations. In a closely related formulation, the low-temperature predictions are

$\omega_{\rm BP}$7

and the theory naturally reproduces the softening of the BP upon heating, the scaling with the shear modulus in the glass state, and strong damping above $\omega_{\rm BP}$8 that generates a large low-frequency contribution to the 3D vDOS (Jiang et al., 2023, Jiang et al., 2023).

A more agnostic synthesis is the flat-mode perspective, which proposes that the BP may universally originate from a dispersionless, optic-like excitation at $\omega_{\rm BP}$9. This perspective is explicitly framed as remaining agnostic about the microscopic origin of the flat mode itself (Mahajan et al., 8 Sep 2025).

5. Structural control parameters, thermal history, and dimensionality

Several structural parameters recur across otherwise different BP theories. Coordination is central in random-network, defective-crystal, jamming, and covalent-network descriptions. Near isostaticity, the distance from the Maxwell threshold controls the onset of excess modes; in random-network and depleted-FCC models, decreasing $g(\omega)/\omega$0 causes the Debye window to shrink and the BP to grow and shift down (Milkus et al., 2016). In silica, the statement that $g(\omega)/\omega$1 as the network approaches isostaticity is part of the proposed universality across network and packing glasses (Mizuno et al., 28 Aug 2025).

Medium-range order also enters explicitly. In the HET/CPA description, the FSDP sets the largest structural correlation in the glass and is used to determine the unit size of elastic-modulus heterogeneity, while the modulus variance determines the BP frequency and intensity (Kyotani et al., 10 Jan 2025). In glucose, an optic-like flat L-mode associated with medium-range-order clusters of size $g(\omega)/\omega$2 was reported to hybridize with the transverse acoustic branch, increasing the local vDOS and contributing to the BP (Kabeya et al., 2017).

Thermal history produces a characteristic contrast between peak and tail. In the quasi-localized-vibration picture, annealing changes the BP peak frequency and magnitude only mildly, while the amplitude of the $g(\omega)/\omega$3 tail can be suppressed strongly, even by an order of magnitude in reanalyzed boron-oxide data (Moriel et al., 2023). In glucose, increasing temperature shifts the BP to lower frequency and broadens it (Kabeya et al., 2017). In the stringlet theory, the BP softens on heating because both the transverse sound speed decreases and the characteristic stringlet length increases, while above $g(\omega)/\omega$4 strong damping produces a large low-frequency tail (Jiang et al., 2023, Jiang et al., 2023).

Dimensionality changes the appropriate reduced spectrum. In 2D one examines $g(\omega)/\omega$5 rather than $g(\omega)/\omega$6, and the observation of a dispersionless BP in 2D silica shows that the anomaly survives in a strictly two-dimensional network (Tømterud et al., 2022). Confinement and free surfaces can shift the characteristic scale further, as shown by the softened BP of ultrathin alumina glass compared with hypothetical bulk amorphous alumina (Cortie et al., 2019).

6. Controversies, misconceptions, and current synthesis

A persistent misconception is that the BP is merely a broadened and shifted van Hove singularity. Studies of disordered granular crystals argued instead for an interplay of mesoscopic screening and microscopic elasticity disorder, causing broadening and attenuation of the first and second van Hove singularities rather than a trivial shift of a crystalline singularity (Zhang et al., 2015). Related work on defective crystals showed that bond-orientational order alone does not predict the BP, and colloidal measurements found that even very defected crystals do not reproduce the large low-frequency excess and specific-heat anomaly of the glass (Milkus et al., 2016, Zargar et al., 2014).

A second controversy concerns localization versus collectivity. The quasi-localized-vibration literature emphasizes localized cores of $g(\omega)/\omega$7 atoms with elastic tails and subsequent hybridization of many such excitations near the BP (Moriel et al., 2023). By contrast, a Green’s-function analysis that dispersed the DOS onto each degree of freedom reported that $g(\omega)/\omega$8 out of $g(\omega)/\omega$9 degrees of freedom, or $g(\omega)=A_D\,\omega^2+g_G(\omega),$0, contribute to the BP in a 2D glass, challenging the view that the BP is contributed by only a minority of particles and highlighting a global and collective origin (Jiang, 14 Jan 2026).

A third dispute is whether a universal phenomenology exists at all. Random-matrix theory proposed a definition of the BP based on universal eigenvector statistics rather than solely on excess over Debye scaling, and identified a large class of random matrices with generalized translational invariance whose central-band eigenvectors share the same statistics. In sparse positive-definite ensembles, the characteristic frequency scales as $g(\omega)=A_D\,\omega^2+g_G(\omega),$1, connecting the BP to coordination and isostaticity from yet another angle (Manning et al., 2013).

The present literature therefore supports a restricted consensus rather than a single microscopic theory. Across many approaches, the BP is associated with excess nonphononic vibrations, strong transverse character, disorder-sensitive hybridization, and a nearly dispersionless spectral feature in $g(\omega)=A_D\,\omega^2+g_G(\omega),$2. What remains unsettled is whether the most fundamental microscopic driver is local inversion-symmetry breaking, near-isostatic marginal stability, heterogeneous elasticity, stringlet excitations, medium-range cluster resonances, or some common flat-mode phenomenology that these frameworks describe in different languages (Li et al., 2023, Mizuno et al., 28 Aug 2025, Jiang et al., 2023, Mahajan et al., 8 Sep 2025).