Size-Dependent Critical States

• Size-dependent critical states are finite-size phenomena where system geometry and perturbations define non-universal critical behavior.
• They arise in various domains, from dislocation dynamics to non-Hermitian localization, illustrating power-law relaxation and explicit scaling cutoffs.
• These states highlight the interplay between structure and size, offering insights for materials science, quantum systems, and biological assemblies.

Size-dependent critical states are states or regimes in which critical or critical-like behavior is controlled by finite system size, geometry, or a perturbation whose amplitude itself scales with size. In the cited literature, the term covers several nonidentical phenomena: sub-threshold dislocation ensembles with power-law relaxation cut off at a size-controlled time, quasiperiodic non-Hermitian eigenstates that display critical localization only at finite size and cross over to Anderson localization as $L\to\infty$, and finite systems whose critical amplitudes or pseudo-critical points drift with $L$ according to explicit scaling laws [(Ispánovity, 2011); (Liang et al., 23 Sep 2025); (Turban, 2023); (Monthus, 2016)]. A recurring feature is that finite size is not merely a correction to thermodynamic criticality; it is part of the mechanism that defines the observed state.

1. Conceptual scope and distinctions

In the cited literature, size-dependent criticality does not denote a single universality class. One usage concerns systems that remain critical only over a finite size window. In quasiperiodic non-Hermitian lattices, size-dependent critical states are defined as eigenstates with multifractal or critical signatures at finite size that cross over to ordinary Anderson localization in the thermodynamic limit (Liang et al., 23 Sep 2025). A second usage concerns systems whose relaxation is scale free below a threshold, but whose power-law regime is cut off by a time scale controlled only by the number of degrees of freedom, as in discrete dislocation systems (Ispánovity, 2011). A third usage concerns finite-size scaling at a bulk critical point in the presence of a perturbation $\Delta=A/L^\omega$, where the perturbation can be irrelevant, marginal, or relevant depending on the comparison of $\omega$ with the RG eigenvalue $y_\Delta$ (Turban, 2023).

These usages differ from conventional critical states that remain scale invariant in the thermodynamic sense. In the non-Hermitian localization setting, the distinction is explicit: extended and critical states satisfy $\gamma=0$, localized states satisfy $\gamma>0$, and size-dependent critical states are those for which finite-size diagnostics mimic criticality even though the asymptotic Lyapunov-exponent analysis points to localization (Liang et al., 23 Sep 2025). In the many-body-localization setting, the emphasis shifts from an asymptotic critical phase to finite-size pseudo-critical points of individual eigenstates, defined so that a given eigenstate is “truly critical” at its own control parameter value (Monthus, 2016). This suggests that “size-dependent critical state” is best understood as a family of finite-size critical phenomena rather than a single model-specific object.

A persistent source of confusion is the relation between size dependence and universality. Several cited works explicitly argue against simple universality. In dislocation systems, the scaling exponents depend on external stress and on the initial correlation structure (Ispánovity, 2011). In thin-film delamination, the size effect depends crucially on crack shape (Zaiser, 2011). In monometallic clusters, the critical size for collective rearrangement is material dependent (Rossi et al., 2017). The presence of scale-free behavior therefore does not imply that the associated exponents or crossover sizes are universal across histories, geometries, or materials.

2. Collective dislocation dynamics and sub-threshold criticality

A central realization in dislocation plasticity is that small-sample plasticity can be interpreted through collective finite-size scaling rather than through a change of microscopic dislocation mechanisms. Compressed micropillars display a size-dependent yield stress and intermittent plastic strain bursts, and these two signatures are presented as manifestations of fundamental collective phenomena governing dislocation dynamics on all scales; agreement is reported between 2D and 3D dislocation-dynamics simulations and experiments on compressed micropillars (Beato et al., 2011).

The most explicit finite-size critical-state construction in this area is the two-dimensional discrete dislocation model studied below a threshold stress $\tau_{\text{th}}$. The model consists of parallel edge dislocations moving on parallel glide planes with overdamped dynamics,

$$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$

where $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$ (Ispánovity, 2011). In yielding tests, the average plastic response follows a power law at low stress and then breaks down smoothly around $L$0. The threshold is interpreted not as a sharply defined single critical point but as a transition regime.

Below $L$1, relaxation is power law rather than exponential. The symmetric and antisymmetric parts of the velocity distribution scale as

$L$2

implying

$L$3

The combination $L$4 is identified as the Andrade exponent. For a random initial state driven at $L$5, the reported values are $L$6, $L$7, $L$8, so that $L$9 (Ispánovity, 2011). The relaxation law is thus derived from a scaling form of the full velocity statistics rather than fitted independently.

The decisive finite-size signature is the cutoff time

$\Delta=A/L^\omega$0

which depends only on system size and is independent of external stress (Ispánovity, 2011). Larger systems sustain the scale-free regime longer. Together with the dependence of the exponents on external stress and initial correlations, this leads to the conclusion that the system is in a critical state throughout the studied sub-threshold regime rather than only at a single yield point. A plausible implication is that intermittent strain bursts and size-dependent flow stress can be viewed as different observables of the same collective finite-size criticality.

3. Localization, non-Hermitian skin physics, and finite-size critical eigenstates

In non-Hermitian quasiperiodic systems, size-dependent critical states are defined as states that look critical only at finite size. Their finite-size signatures include wavefunction profile, multifractal behavior, fractal dimension $\Delta=A/L^\omega$1, participation-related diagnostics, entanglement entropy, and spectral structure, but in the thermodynamic limit they cross over to Anderson localization rather than remaining genuinely critical (Liang et al., 23 Sep 2025). The physical mechanism is the interplay between local non-reciprocal domain walls and the non-Hermitian skin effect. The domain wall hinders directional accumulation, while non-reciprocity drives amplitude toward one boundary. The competition produces a finite-size window with critical-like structure.

The asymptotic diagnostic is the Lyapunov exponent $\Delta=A/L^\omega$2. In the cited formulation, extended and critical states have $\Delta=A/L^\omega$3, localized states have $\Delta=A/L^\omega$4, and the localization length is $\Delta=A/L^\omega$5 (Liang et al., 23 Sep 2025). The analysis organizes the spectrum through a mobility ring in the complex-energy plane: states inside the ring have $\Delta=A/L^\omega$6, states outside have $\Delta=A/L^\omega$7. This is precisely why the work treats size-dependent critical localization as distinct from conventional critical phases.

A complementary experimental line establishes exact quantum critical states in a programmable quasiperiodic mosaic model implemented on a superconducting quantum processor with up to 56 qubits (Huang et al., 26 Feb 2025). In the short-range, uniform-potential limit $\Delta=A/L^\omega$8 and $\Delta=A/L^\omega$9, the model exhibits a localized-to-critical transition at $\omega$0. The localized phase for $\omega$1 has localization length

$\omega$2

while the critical phase occurs for $\omega$3. The rigorous criteria for critical states are the incommensurately distributed zeros in the dominating hopping couplings $\omega$4 and the delocalized nature in the regime $\omega$5. Weak long-range couplings preserve the critical state, whereas strong long-range couplings remove the quasiperiodic zeros and drive the system into an extended phase; experimentally, the breakdown is reported around $\omega$6. At the exactly solvable line $\omega$7, the anomalous mobility edges are $\omega$8, with critical states for $\omega$9 and localized states for $y_\Delta$0 (Huang et al., 26 Feb 2025).

Related non-Hermitian transitions show different size laws. In a Hermitian Anderson lattice with a single non-Hermitian coupling impurity, clean scale-free localized states satisfy $y_\Delta$1, and disorder converts them into Anderson-localized states with a size-dependent critical disorder $y_\Delta$2 as $y_\Delta$3. Numerically, $y_\Delta$4 with $y_\Delta$5 to $y_\Delta$6 (Yılmaz et al., 2024). In a Hatano–Nelson chain with one impurity, the impurity acts as an effective boundary, driving a transition from non-skin to skin states at a critical impurity strength

$y_\Delta$7

and in the non-reciprocal SSH model

$y_\Delta$8

so the critical value increases exponentially with system size (Liu et al., 2021). These examples show that size dependence may appear as a finite-size critical window, as an algebraically drifting disorder threshold, or as an exponentially growing impurity threshold.

4. Finite-size scaling formalisms and pseudo-critical points

A general scaling framework for homogeneous size-dependent perturbations starts from

$y_\Delta$9

with RG eigenvalue $\gamma=0$0 (Turban, 2023). At the bulk critical point, the expectation value of a local observable $\gamma=0$1 with scaling dimension $\gamma=0$2 obeys

$\gamma=0$3

The perturbation is irrelevant for $\gamma=0$4, marginal for $\gamma=0$5, and relevant for $\gamma=0$6. In the irrelevant regime, only subleading corrections appear. In the marginal regime, the exponent remains $\gamma=0$7 but the finite-size amplitude becomes a universal function of $\gamma=0$8. In the relevant regime, the finite-size decay exponent changes to $\gamma=0$9, and depending on the sign of the perturbation, the observable may exhibit either algebraic decay or an essential singularity (Turban, 2023).

Exactly solvable examples make the classification concrete. In the one-dimensional Ising chain in a transverse field with $\gamma>0$0, the surface magnetization at $\gamma>0$1 satisfies $\gamma>0$2 in the unperturbed case, while the marginal case $\gamma>0$3 yields

$\gamma>0$4

and the relevant regime $\gamma>0$5 changes the decay to $\gamma>0$6 for $\gamma>0$7 (Turban, 2023). The fully connected Ising model provides an analogous threshold at $\gamma>0$8, again separating perturbative corrections, amplitude renormalization, and exponent change.

A different finite-size formalism appears at the many-body-localization transition, where a pseudo-critical point is assigned to each individual eigenstate $\gamma>0$9 rather than to a disorder sample or an averaged observable (Monthus, 2016). The defining criterion is that the hybridization ratio with the most dangerous resonant partner becomes order one. The general scaling form is

$\tau_{\text{th}}$0

In the toy MBL model discussed in the cited work, the eigenstate scaling is governed by $\tau_{\text{th}}$1, while sample-to-sample fluctuations at nonzero energy density scale as $\tau_{\text{th}}$2, corresponding to $\tau_{\text{th}}$3 (Monthus, 2016). The conceptual point is that standard disorder averages mix localized, delocalized, and genuinely critical states, whereas evaluating each eigenstate at its own pseudo-critical point isolates a “truly critical” finite-size state.

5. Geometry, defects, and material-dependent crossover scales

Several cited systems show that size-dependent criticality is inseparable from geometry. In thin-film delamination, the shear failure strength depends strongly on crack shape (Zaiser, 2011). Circular cracks exhibit a crossover between a nucleation-controlled, size-dependent regime at small crack radius and a propagation-controlled, size-independent regime at large radius. By contrast, transversely spanning cracks show a logarithmic system-size dependence of the failure stress for all crack lengths studied. The interpretation is in terms of extreme-value statistics: nucleation-controlled failure behaves as a weakest-link problem, with failure-stress distributions well fitted by a Gumbel distribution.

In monometallic nanoparticles, the dominant rearrangement mechanism changes at a material-dependent critical size (Rossi et al., 2017). Lipscomb’s Diamond-Square-Diamond mechanism, a single-step screw-dislocation-like motion of the whole cluster, occurs readily in 55-atom clusters and remains accessible only below a material-specific limit. The reported crossover sizes are 147 atoms for Au, about 309 atoms for Pt, about 561 atoms for Ni, Ag, and Pd, and above 561 atoms for Cu. Beyond these sizes, the system rearranges through layer-by-layer dislocations, surface peeling, defect formation, and low-symmetry defected motifs. The paper relates the critical size to the ratio between bulk modulus and atomic cohesive energy.

In magnetic nanostructures, size and symmetry regulate topological phase transitions (Wang et al., 18 May 2026). Nanodisks exhibit the sequence

$\tau_{\text{th}}$4

as the diameter increases. The cited size windows are $\tau_{\text{th}}$5 nm for the ferromagnetic state, about $\tau_{\text{th}}$6–$\tau_{\text{th}}$7 nm for skyrmion–skyrmionium interconversion, $\tau_{\text{th}}$8–$\tau_{\text{th}}$9 nm for energetically favorable skyrmions, and $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$0–$$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$1 nm for skyrmionium and multi-state behavior. Squares and rectangles suppress this topological complexity through corner-induced demagnetization and reduced symmetry, while a perpendicular magnetic field drives skyrmionium-to-skyrmion collapse at $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$2–$$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$3 T (Wang et al., 18 May 2026).

Critical finite-size behavior can also be shape dependent even in canonical lattice models. For the critical two-dimensional Ising model on a triangular lattice with free boundaries, the free-energy density admits bulk, surface, corner, and higher-order terms, and the leading corner free-energy contribution agrees with the Cardy–Peschel conformal-field-theory prediction and is universal (Wu et al., 2012). However, higher-order logarithmic corrections are present for rhombus, trapezoid, and hexagon geometries but absent for triangle and rectangle geometries, and the corner contributions to internal energy and specific heat are not universal. The cited work therefore distinguishes universal and nonuniversal parts of critical finite-size structure.

In elastomer fracture, the critical state for crack growth is controlled by a finite fracture process zone with intrinsic length

$$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$4

where $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$5 is the fracture energy and $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$6 is the critical local energy density (Lee et al., 2024). The strongest size effects occur when $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$7, with larger cracks lowering the rupture stretch and smaller cracks or specimens raising it. This usage of “critical” concerns the onset of crack advance rather than an equilibrium critical phase, but it preserves the same structural point: the threshold state is governed by a competition between a macroscopic size $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$8 and an intrinsic length $$\dot{x}_i = s_i \left[ \sum_{j=1;\ j\ne i}^{N} s_j \tau_\text{ind}(\mathbf r_i - \mathbf r_j) + \tau_\text{ext}(\mathbf r_i) \right], \qquad \dot{y}_i=0,$$9.

6. Biological and nonequilibrium realizations

Protein native states provide an example of finite-size criticality inferred from equilibrium fluctuation ensembles. Analysis of 7678 NMR-determined protein structure ensembles shows that the orientational correlation $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$0 has a strong short-range maximum near $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$1, crosses zero at a correlation length $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$2, becomes negative, and returns toward zero at larger separations (Tang et al., 2016). The key scaling result is

$$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$3

so the correlation length grows proportionally with protein size. A susceptibility

$$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$4

peaks at a size-dependent shape factor

$$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$5

and finite-size scaling is reported to be consistent with $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$6, $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$7, and $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$8 (Tang et al., 2016). The interpretation is that proteins near their native state behave like finite-size critical systems: stable enough to remain folded, but with correlations that span the whole molecule.

Aggregation with fragmentation yields a different kind of size-dependent critical state, the super-cluster state (Brilliantov et al., 2021). For rates $$\tau_\text{ind}(\mathbf r)=\cos(\varphi)\cos(2\varphi)r^{-1}$$9 and $L$00, the super-cluster state occurs for

$L$01

It is non-extensive: the total number of clusters scales sublinearly,

$L$02

so the cluster density vanishes in the thermodynamic limit. In the exactly analyzed case $L$03, the crossover time obeys

$L$04

the cluster-size distribution scales as

$L$05

and the total cluster number scales as

$L$06

corresponding to $L$07, $L$08, $L$09, and $L$10 (Brilliantov et al., 2021). The cited work emphasizes that conventional van Kampen expansion fails because fluctuations eventually dominate the deterministic term.

Taken together, these examples clarify two general points. First, size-dependent critical states are not always finite-size approximations to a thermodynamic critical phase; in some systems they disappear asymptotically, while in others they remain the appropriate description of finite functional objects such as proteins or mesoscopic dislocation ensembles. Second, the control variable is often geometric or structural rather than thermal: crack shape, domain-wall arrangement, baryonic hard-core volume, cluster size, or protein compactness can play the role ordinarily assigned to temperature or field. This suggests that the modern literature on size-dependent critical states is best read as an extension of criticality into finite, heterogeneous, and history-dependent systems rather than as a simple modification of standard bulk phase-transition theory.