Intrinsic Anomalous Roughening

• Intrinsic anomalous roughening is a kinetic scaling regime where the global roughness exponent differs from local and spectral exponents, resulting in nonstationary short-scale fluctuations.
• The methodology employs analyses of structure factors and local width scaling to decisively separate genuine intrinsic anomalies from transient finite-time or finite-size effects.
• Models and experiments—from Ising interfaces to thin-film electrodeposition—demonstrate distinct exponent relations that refine the scaling taxonomy and provide practical diagnostic tools.

Searching arXiv for the cited works on intrinsic anomalous roughening and closely related scaling frameworks. arXiv search: "Intrinsic anomalous roughening Ramasco Lopez Rodriguez scaling KPZ CV model" Intrinsic anomalous roughening is a kinetic-scaling regime in which the global roughness exponent that governs system-size dependence differs from the local and spectral exponents, so that short-scale fluctuations remain nonstationary even after the correlation length exceeds the observation scale. Its diagnostic form is $w(l,t)\sim t^\kappa l^{\alpha_{\mathrm{loc}}}$, with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$, while the spectral and local exponents coincide, $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$. This separates it from ordinary Family–Vicsek scaling, from super-roughening, and from faceted anomalous scaling, and it also separates genuine anomalies from finite-time or finite-size crossovers that only mimic them (Rodriguez-Fernandez et al., 2022).

1. Scaling structure and taxonomy

The standard global width is

$W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$

with $W\sim t^\beta$ in the growth regime and $\beta=\alpha/z$. In ordinary dynamic scaling, the same roughness exponent controls global, local, and spectral observables. A local width or height-difference measure obeys

$w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$

so that, for $l\ll t^{1/z}$, the prefactor becomes time independent. In the anomalous framework, the structure factor

$S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$

has a large-$u$ asymptote $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$0, implying $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$1 at sufficiently large $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$2. Intrinsic anomalous roughening is the subclass with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$3, so that the local sector is governed by a distinct exponent from the global one (Rodriguez-Fernandez et al., 2022).

The standard classification used in anomalous roughening studies distinguishes four main cases (Torres et al., 2012):

Class	Exponent relations	Signature
Family–Vicsek	$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$4	Single self-affine scaling
Intrinsic anomalous	$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$5	$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$6
Super-rough	$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$7	Global exponent exceeds unity
Faceted anomalous	$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$8	Piecewise-linear faceted morphology

This taxonomy is not merely terminological. It determines which observable is decisive. Width measurements alone can hide intrinsic anomalies, because $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$9 depends only on $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$0 and $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$1, whereas the high-$\alpha_{\mathrm{loc}}=\alpha_s<\alpha$2 structure factor and the small-$\alpha_{\mathrm{loc}}=\alpha_s<\alpha$3 local width reveal $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$4 and $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$5. In intrinsic anomalous roughening, the decisive asymptotic statement is that short-scale fluctuations never become stationary in the same manner as under Family–Vicsek scaling.

2. Diagnostics and the distinction between intrinsic and apparent anomalies

The most stringent distinction is between a genuine intrinsic anomaly and an apparent anomaly generated by long-lived corrections. The decisive tests are asymptotic. A fixed-$\alpha_{\mathrm{loc}}=\alpha_s<\alpha$6 local roughness must retain a power-law prefactor $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$7 at the largest accessible times, $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$8 must remain different from $\alpha_{\mathrm{loc}}=\alpha_s<\alpha$9, and scaling collapses must require an explicit anomalous prefactor. If, instead, the effective anomaly decays away, the system is not intrinsically anomalous (Assis et al., 2015).

The Clarke–Vvedensky thin-film study provides a canonical counterexample. For mesoscopically thick films without a step-edge barrier, the global roughness scales as

$W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$0

with $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$1 and $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$2, while the local roughness is described by

$W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$3

with $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$4, both consistent with the Villain–Lai–Das Sarma class (Assis et al., 2015). At very low temperatures, $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$5, small-box local roughness displays splitting and effective exponents $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$6 between $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$7 and $W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$8, which imitates intrinsic anomalous roughening. However, the same study shows that the small-box roughness behaves as

$W(L,t)\sim L^\alpha f\!\left(\frac{t}{L^z}\right),$9

with $W\sim t^\beta$0, so the time-dependent correction vanishes and the apparent anomaly shrinks toward $W\sim t^\beta$1 (Assis et al., 2015).

That case has become a methodological benchmark because it shows why exponent splitting at modest times is insufficient. In the same parameter range, the correlation length follows

$W\sim t^\beta$2

and successful collapses of both global and local roughness recover normal VLDS forms once the corrections are handled properly. The conclusion is unambiguous: under $W\sim t^\beta$3, $W\sim t^\beta$4, thickness up to $W\sim t^\beta$5 monolayers, and no step-edge barrier, the anomaly is apparent rather than intrinsic (Assis et al., 2015).

3. Mechanisms that generate intrinsic anomalous roughening

One robust route is quenched disorder coupled to avalanche dynamics. In the ferromagnetic thin-film magnetic-domain-wall model of Buceta and Muraca, the global width obeys Family–Vicsek scaling with $W\sim t^\beta$6, $W\sim t^\beta$7, and $W\sim t^\beta$8, but the local and spectral exponents are both approximately $W\sim t^\beta$9: $\beta=\alpha/z$0. The second-order height-difference correlation acquires the anomalous prefactor $\beta=\alpha/z$1, and the local exponents decrease with correlation order, $\beta=\alpha/z$2, $\beta=\alpha/z$3, $\beta=\alpha/z$4, and $\beta=\alpha/z$5, showing multi-affinity in addition to intrinsic anomaly (Torres et al., 2012). In that model, quenched pinning and avalanche-like motion generate nonstationary local observables even though the global width has a standard collapse.

A second route is time dependence in the correlation-generating mechanism itself. In competitive growth models where the correlated process occurs with a rate

$\beta=\alpha/z$6

the anomalous exponents are determined directly from the normal exponents of the correlated component and the aggregation-mechanism exponent $\beta=\alpha/z$7. The framework yields

$\beta=\alpha/z$8

and

$\beta=\alpha/z$9

Here the anomaly is generated by the progressive slowing of correlation propagation, so local slopes continue to increase even though the underlying correlated dynamics remains in a known universality class (Reis, 2013).

A third route is a time-dependent noise amplitude within an otherwise linear diffusion-dominated equation. In Cu$w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$0O electrodeposition, the coarse-grained dynamics is governed by the Mullins–Herring equation

$w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$1

with

$w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$2

For $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$3, the local roughness satisfies

$w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$4

so $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$5 yields $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$6. On $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$7-Si(100), the measured $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$8 and $w(l,t)\sim l^\alpha g\!\left(\frac{t}{l^z}\right),$9 with $l\ll t^{1/z}$0 identify intrinsic anomalous roughening, while growth on Ni/$l\ll t^{1/z}$1-Si keeps $l\ll t^{1/z}$2 approximately constant and remains normal (Brandt et al., 2015). In that system the substrate controls the grain-size distribution, step-edge barriers, and hence the effective coarse-grained noise amplitude.

4. Representative realizations

Intrinsic anomalous roughening has been reported in systems with distinct microscopic rules, disorder structures, and morphological constraints. The common feature is exponent splitting between the global and local or spectral sectors, but the numerical values and even the asymptotic fate can differ substantially.

System	Representative exponents	Scaling character
Ferromagnetic thin-film MDW (Torres et al., 2012)	$l\ll t^{1/z}$3	Intrinsic anomalous, multi-affine
Ising interfaces in 2D (Rodriguez-Fernandez et al., 2022)	$l\ll t^{1/z}$4	Intrinsic anomalous throughout time evolution for $l\ll t^{1/z}$5
Coffee-ring fronts, $l\ll t^{1/z}$6 (Barreales et al., 2022)	$l\ll t^{1/z}$7	Intrinsic anomalous pre-pinning
Coffee-ring fronts, $l\ll t^{1/z}$8 (Barreales et al., 2022)	$l\ll t^{1/z}$9	Intrinsic anomalous in strong-crossover regime
2D cross section of 3D Ising (Dashti-Naserabadi et al., 2019)	$S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$0	Super-rough globally, intrinsic-anomalous locally

The Ising-interface model revisited in two dimensions is particularly important because it challenged the expectation that quenched disorder or deterministic instability is necessary. In that system, the interface is the upper envelope of the cluster of $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$1 spins connected to the bottom boundary, which makes the height definition intrinsically nonlocal. The global roughness exponent remains $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$2, but the large-$S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$3 spectrum gives $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$4, and the anomaly persists throughout the full time evolution for system sizes up to $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$5. The same work reported two distinct dynamic regimes, $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$6 at early times and $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$7 at late times, neither of which matches a known roughening universality class asymptotically (Rodriguez-Fernandez et al., 2022).

The coffee-ring front model provides a morphologically unstable realization. For ballistic aggregation of patchy colloids, $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$8 produces a discontinuous pinning–depinning transition and intrinsic anomalous roughening before pinning, while $S(k,t)\sim k^{-(2\alpha+d)}\,f_S(k^z t)$9 remains in a moving phase but shows strong crossover from the $u$0 behavior. In both cases the hallmark is an upward shift of $u$1 and $u$2 at small scales, with $u$3 at high $u$4 (Barreales et al., 2022). The underlying mechanism is a morphological instability caused by the finite interaction range and alignment rule of the patchy-binding dynamics.

The equilibrium 3D Ising model offers a different geometry. Its 2D cross section at $u$5 has a globally super-rough width, $u$6 with $u$7, yet the local width and structure factor obey $u$8 and $u$9 with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$00. The study therefore describes the cross section as super-rough globally and intrinsically anomalous in its local properties (Dashti-Naserabadi et al., 2019).

5. Neighboring anomalous classes and class boundaries

Intrinsic anomalous roughening is best understood in relation to the anomalous classes it is not. Temporally correlated KPZ growth in one dimension provides a clean contrast. When the temporal correlation exponent exceeds a threshold, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$01 in ballistic deposition and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$02 in the stabilized discretized KPZ equation, the surface develops macroscopic facets and the spectral exponent becomes larger than one. For $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$03, the reported values are $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$04, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$05, and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$06, which identifies a faceted anomalous phase rather than an intrinsic one (Alés et al., 2019).

The same distinction appears in anharmonic elastic interfaces with temporally correlated noise. In $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$07, for any anharmonicity degree $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$08, the threshold is $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$09. Above it, the exact theory gives $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$10 and

$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$11

with faceted morphology and anomalous amplitude scaling. For $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$12, the model reduces to Edwards–Wilkinson behavior with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$13 and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$14, while for $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$15 anomalous roughening is ruled out because the Laplacian elasticity remains asymptotically dominant (Alés et al., 2021).

A linear benchmark is the Edwards–Wilkinson equation with spatiotemporally correlated noise. In $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$16, the exact exponents are

$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$17

and the slope field becomes rough when $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$18. Exactly at that threshold the height field becomes super-rough, with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$19 and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$20. The anomalous branch is therefore super-rough rather than intrinsic, and the condition for anomaly is identical to the condition for slope roughening (Alés et al., 2019).

The anharmonic Larkin model gives a closely related faceted case. In $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$21 and any finite $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$22, it exhibits $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$23 and a local exponent consistent with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$24. The unusual hierarchy $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$25 invalidates single-exponent scaling for two-point correlations and produces steady-state piecewise linear facets, again placing the model outside the intrinsic subclass (Purrello et al., 2018). These neighboring cases clarify why the inequalities among $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$26, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$27, and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$28 matter as much as the mere presence of anomalous scaling.

6. Contemporary extensions, scale dependence, and unresolved issues

Recent work has extended the concept beyond classical stochastic growth. In the easy-axis XXZ chain, the roughness is the variance of a block charge, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$29. For quenches from non-fluctuating product states, the early-time behavior is diffusive,

$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$30

but the stationary block fluctuations are sub-extensive,

$\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$31

with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$32. The reported easy-axis values are $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$33, while $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$34, so the Family–Vicsek identity $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$35 fails. The mechanism is local relaxation to squeezed generalized Gibbs ensembles with vanishing static susceptibility $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$36 and sub-extensive charge fluctuations, rather than to canonical GGEs (Cecile et al., 2023). This is a genuine broadening of anomalous roughening into integrable quantum dynamics.

A distinct contemporary development comes from nonconserved critical dynamics of the two-dimensional Ising model. When the order-parameter field is analyzed as a height field, an ordered quench follows Family–Vicsek scaling with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$37 and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$38, whereas a disordered quench shows an overgrowth regime classified through Ramasco–López–Rodríguez spectral scaling as intrinsic anomalous roughening, with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$39, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$40, and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$41 in TDGL or $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$42 in Glauber dynamics. The anomalous sector grows with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$43 until it reaches a cutoff $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$44, after which the system relaxes toward equilibrium. The corresponding integral GL model shifts the roughness exponent by one and exhibits faceted anomalous roughening with $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$45 and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$46 for the disordered quench (Pino et al., 11 Jul 2025). This suggests a broader operational use of the term in critical-dynamics settings, where the anomaly can be transient and controlled by instability saturation.

Two further issues recur across the literature. First, scaling can fail at the largest accessible scales even when smaller systems show convincing anomalous collapses. In the Ising-interface study, the $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$47 data weaken width saturation and make the long-time structure factor effectively time independent, suggesting a fixed length scale rather than continued roughening (Rodriguez-Fernandez et al., 2022). Second, geometry can change the measured exponents. In forced radial imbibition, the growth exponent increases linearly with the flow rate while the roughness exponent decreases with it, and the roughening dynamics differ markedly from one-dimensional planar imbibition. At high flow rates the data are consistent with intrinsic-type anomalous roughening, while low flow rates are super-rough; the study therefore argues that the “universality class” is not universal under geometric change (Chen et al., 2015).

Taken together, these results define intrinsic anomalous roughening less as a single microscopic mechanism than as a precise scaling condition: asymptotic exponent splitting between global and local or spectral sectors, with nonstationary short-scale fluctuations that cannot be absorbed into ordinary Family–Vicsek scaling. The strongest evidence still comes from joint analysis of $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$48, $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$49, and $\kappa=(\alpha-\alpha_{\mathrm{loc}})/z>0$50, together with explicit tests that the anomaly survives the removal of finite-time corrections.