Quartic Scaling Law

• Quartic scaling law is a family of fourth-order power dependencies observed in varied fields, from acoustic attenuation in silica to double-scaling in tensor models.
• Applications include direct f^4 fitting in sub-THz spectroscopy, quartic conductivity in T-odd transport, and O(N^4) complexity reduction in quantum chemistry.
• Its practical significance lies in unifying diverse scaling behaviors, offering a benchmark for experimental validations and computational efficiency across disciplines.

“Quartic scaling law” does not denote a single formal theorem across contemporary arXiv literature. The expression is used for several distinct technical structures: a direct fourth-power dependence of an observable, as in the athermal attenuation law $\alpha_{bp}(f)=Af^4$ in vitreous silica (Wang et al., 2023); a nonlinear conductivity that is generally a quartic function of $\sigma_{xx}$ in time-reversal-odd transport (Huang et al., 2023); the double-scaling limit of a rank-$D$ quartic tensor model (Dartois et al., 2013); a homogeneous Casimir-based scaling law for generalized quartic oscillators (Klink et al., 2020); and an $\mathcal O(N^4)$ complexity reduction for periodic MP2 in plane waves (Schäfer et al., 2016). A qualified use also appears in quadratically parameterized SGD, where the proven scaling law is not formally “quartic,” even though the source condition and loss geometry are quartic in the underlying parameters (Ding et al., 13 Feb 2025). More recently, a universal factorization $\chi/2\pi=\eta E_4/h$ has been proposed for Kerr-type interactions generated by quartic nonlinearities projected onto normal modes (Liu et al., 18 Apr 2026).

1. Semantic scope and neighboring terminology

The term is therefore best understood as a family resemblance rather than a single concept. In different fields, the quartic element may reside in the measured observable, in a polynomial dependence on a control parameter, in the interaction term of a Hamiltonian, in a homogeneous scaling rule for invariants, or in algorithmic complexity.

Domain	Quartic quantity or law
Acoustic spectroscopy	$\alpha_{bp}(f)=Af^4$
$\mathcal T$-odd nonlinear transport	$\chi_{yxx}$ quartic in $\sigma_{xx}$
Tensor models	double scaling of a quartically interacting model
Quartic oscillators	$E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$
Periodic MP2	$\sigma_{xx}$0 runtime
Quadratically parameterized SGD	quartic objective and source condition, but no named quartic law
Kerr-type interactions	$\sigma_{xx}$1

A recurrent source of confusion is the proximity of “quartic scaling” to “quarter-power scaling.” The latter is a plant-allometric concept involving exponents that are multiples of $\sigma_{xx}$2, especially $\sigma_{xx}$3, and in the hydraulic derivation of flow similarity it arises from $\sigma_{xx}$4 and $\sigma_{xx}$5. It is therefore a distinct usage: the exponent is not $\sigma_{xx}$6, but a multiple of $\sigma_{xx}$7 (Price et al., 2015).

2. Fourth-power attenuation below the boson peak

A particularly literal quartic scaling law appears in sub-THz acoustic spectroscopy of vitreous silica. There the central quantity is the longitudinal-acoustic attenuation coefficient, and the paper formulates the boson-peak-related contribution as

$\sigma_{xx}$8

The fitted coefficient is reported as

$\sigma_{xx}$9

and is temperature independent between $D$0 K and $D$1 K. The crucial spectral window is the sub-THz region below the boson peak, approximately $D$2–$D$3 GHz, with the quartic athermal term beginning to emerge above about $D$4 GHz (Wang et al., 2023).

The experimental basis is a combination of narrow-band superlattice detection and broadband phase-sensitive THz acoustic spectroscopy. The latter directly accesses both attenuation and constant-frequency phase velocity and yields the broadband fit

$D$5

with $D$6 in $D$7 and $D$8 in THz. When the measured velocity dispersion is included and attenuation is re-expressed versus wavevector, the result becomes slightly subquartic,

$D$9

The quartic law is thus sharpest when attenuation is treated as a constant-frequency quantity.

This usage is also notable for what it does not claim. The observed $\mathcal O(N^4)$0 law is presented as evidence for an athermal boson-peak-related attenuation channel, but not as unique proof of a single microscopic mechanism. The paper emphasizes near-quantitative agreement with the soft-potential-model prediction

$\mathcal O(N^4)$1

while also stating that fluctuating-elasticity theories can account for similar damping and dispersion phenomenology. The quartic scaling law here is therefore an empirical law for a specific observable, with ongoing interpretive competition at the microscopic level.

3. Quartic conductivity scaling in time-reversal-odd transport

In nonlinear transport theory, the quartic object is not a frequency dependence but a conductivity dependence. The current is expanded as

$\mathcal O(N^4)$2

and the focus is the $\mathcal O(N^4)$3-odd second-order conductivity, especially $\mathcal O(N^4)$4. The paper’s central claim is that $\mathcal O(N^4)$5 is generally a quartic function of the longitudinal conductivity $\mathcal O(N^4)$6, with the highest allowed scaling power being fourth order (Huang et al., 2023).

In the experimentally common two-source case, the conductivity-form law contains constant, linear, quadratic, cubic, and quartic terms. The cubic and quartic pieces explicitly include

$\mathcal O(N^4)$7

and

$\mathcal O(N^4)$8

This is not merely a curve-fitting statement. The derivation uses the semiclassical Boltzmann equation with conventional scattering, side-jump-related scattering, and skew scattering, together with Matthiessen’s rule

$\mathcal O(N^4)$9

A central conceptual point is that the $\chi/2\pi=\eta E_4/h$0-independent term is not purely intrinsic. The constant contribution is

$\chi/2\pi=\eta E_4/h$1

so extrinsic mechanisms also enter the zeroth-order term. The paper names these zeroth-order extrinsic contributions and shows in a four-band Dirac model,

$\chi/2\pi=\eta E_4/h$2

that they can be comparable to or even larger than the intrinsic part, and can even reverse the sign of the total $\chi/2\pi=\eta E_4/h$3 contribution.

The quartic structure is also conditional. In the single-scattering-source limit, the law collapses to

$\chi/2\pi=\eta E_4/h$4

Hence cubic and quartic terms diagnose the coexistence of at least two scattering sources and require skew scattering. Here the phrase “quartic scaling law” refers to the full polynomial structure of nonlinear transport, not to a single isolated $\chi/2\pi=\eta E_4/h$5 term.

4. Quartic interaction and double scaling in tensor models

In random tensor theory, the quartic aspect lies in the interaction itself. The model is a rank-$\chi/2\pi=\eta E_4/h$6 complex tensor theory with quartic melonic invariants, normalized by $\chi/2\pi=\eta E_4/h$7, which the paper states is the unique large-$\chi/2\pi=\eta E_4/h$8 scaling leading to a nontrivial large-$\chi/2\pi=\eta E_4/h$9 limit. The leading large-$\alpha_{bp}(f)=Af^4$0 sector is melonic, with critical point

$\alpha_{bp}(f)=Af^4$1

and square-root susceptibility exponent

$\alpha_{bp}(f)=Af^4$2

In this setting, “quartic scaling law” is more precisely the double-scaling law of a quartically interacting tensor model (Dartois et al., 2013).

The distinctive contribution of the paper is the identification of subleading families through pruning and reduction. For $\alpha_{bp}(f)=Af^4$3, the dominant subleading reduced graphs are the cherry trees. They saturate the maximal edge bound, maximize the number of critical multicolored edges, and avoid genus penalties. Their counting is governed by Catalan numbers,

$\alpha_{bp}(f)=Af^4$4

The double-scaling variable is

$\alpha_{bp}(f)=Af^4$5

and the resummed two-point function in the melonic-plus-cherry approximation is

$\alpha_{bp}(f)=Af^4$6

This defines a controlled double-scaling limit whose singularity is again square-root in the scaled variable.

The paper’s comparison with matrix models is integral to the meaning of the law. In contrast with quartic matrix models, the tensor-model double scaling is described as stable, and $\alpha_{bp}(f)=Af^4$7 appears as an unexpected upper critical dimension. The law is therefore not “quartic” because the critical exponent equals $\alpha_{bp}(f)=Af^4$8, but because the interacting theory being scaled is quartic from the outset.

5. Quartic Hamiltonians in quantum and wave systems

A more algebraic notion of quartic scaling law appears in generalized symmetric quartic oscillators. The potential

$\alpha_{bp}(f)=Af^4$9

is rewritten through generators of a step-3 nilpotent Lie algebra, the “quartic group,” via

$\mathcal T$0

The spectrum depends only on the Casimir invariants

$\mathcal T$1

Under the unitary scaling operator $\mathcal T$2, the energy obeys

$\mathcal T$3

with general solution

$\mathcal T$4

Here the quartic scaling law is exact and group-theoretic: the quartic potential induces a homogeneous scaling structure in the Casimir variables rather than an empirical fourth-power fit (Klink et al., 2020).

A related but broader proposal appears in Kerr-type nonlinear quantum platforms. Starting from a quartic interaction in a finite normal-mode basis,

$\mathcal T$5

the rotating-wave approximation yields the factorization

$\mathcal T$6

The same structure is extended to self-Kerr,

$\mathcal T$7

This formulation treats self-Kerr, cross-Kerr, and cross-phase modulation as observable projections of an intrinsic quartic energy scale (Liu et al., 18 Apr 2026).

The factorization is stated under explicit assumptions: quartic expansion about a stable operating point, finite normal-mode truncation, weak-to-intermediate nonlinearity, validity of the rotating-wave approximation, and Markovian dissipation. The worked quarton example uses

$\mathcal T$8

to predict

$\mathcal T$9

with reported uncertainty

$\chi_{yxx}$0

against a measured

$\chi_{yxx}$1

a deviation of $\chi_{yxx}$2. The paper further states that the dominant uncertainty comes from Josephson-energy extraction rather than the geometric projection factor. In this usage, “quartic scaling law” means factorization of Kerr rates into an intrinsic quartic scale and a dimensionless projection coefficient.

6. Complexity reduction and qualified usages

In electronic-structure theory, quartic scaling is computational rather than physical. For periodic MP2 in a plane-wave basis, the conventional bottleneck arises from explicit summation over all virtual pairs and scales as

$\chi_{yxx}$3

which becomes the familiar quintic $\chi_{yxx}$4 behavior when $\chi_{yxx}$5, $\chi_{yxx}$6, and $\chi_{yxx}$7 all grow linearly with system size. The Laplace-transformed reformulation eliminates all summations over virtual orbitals by contracting them into transformed states, after which the dominant steps scale as

$\chi_{yxx}$8

Under $\chi_{yxx}$9, both become quartic, yielding an exact $\sigma_{xx}$0 algorithm. The paper supplements this with internal basis-set extrapolation using

$\sigma_{xx}$1

which improves effective accuracy-cost tradeoffs without changing formal scaling. In a LiH $\sigma_{xx}$2 supercell benchmark, runtime at $\sigma_{xx}$3 cores is reported as $\sigma_{xx}$4 of the $\sigma_{xx}$5-core time, close to the ideal $\sigma_{xx}$6 (Schäfer et al., 2016).

A more restrictive use appears in learning theory. The paper on stochastic gradient descent in quadratically parameterized linear regression is explicitly relevant only in a qualified sense: it does not prove a named or formal quartic scaling law. The model is linear in features but quadratic in parameters,

$\sigma_{xx}$7

and the quartic structure appears in the source condition

$\sigma_{xx}$8

in the prediction-error decomposition

$\sigma_{xx}$9

and in the resulting quartic population geometry. The actual scaling laws proved in the paper concern excess risk as a function of sample size $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$0, model size $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$1, and effective dimension $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$2; the rates are powers of $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$3 and $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$4, not a formally quartic exponent. This is therefore a case where quartic structure is real but terminologically indirect (Ding et al., 13 Feb 2025).

Taken together, these usages show why the phrase resists a single encyclopedia definition. Sometimes it denotes a literal fourth-power law for an observable; sometimes a quartic polynomial dependence on a transport coefficient; sometimes the scaling limit of a quartically interacting field theory; sometimes a homogeneous rule tied to Casimir invariants or mode-projection of a quartic Hamiltonian; and sometimes an $E_n(t^3\beta_3,t^4c)=t^2E_n(\beta_3,c)$5 algorithmic complexity class. The stable common core is not one equation, but the repeated appearance of fourth-order structure as the organizing element of asymptotics, interaction geometry, or computation.