Griffiths Phase in Disordered Systems

• Griffiths phase is a disorder-induced regime characterized by rare, locally ordered regions amid a globally disordered background, leading to nonanalytic behavior over a finite parameter interval.
• It exhibits anomalous susceptibility, broad distributions of observables, and slow relaxation dynamics, often following algebraic decay as seen in various magnetic and quantum systems.
• Analytical and numerical methods like rare-region analysis, cluster frameworks, and large-deviation Monte Carlo simulations offer critical insights into diagnosing Griffiths-phase behavior across diverse materials.

Searching arXiv for recent and foundational papers on Griffiths phases to ground the article in current literature. A Griffiths phase is a disorder-induced extended regime adjoining a phase transition in which the system is globally disordered, yet contains exponentially rare regions that are locally ordered or locally active. In classical disordered magnets, the canonical temperature window is $T_c(p)<T<T_f\equiv T_c(1)$, where the diluted system has no bulk magnetization but rare locally ordered regions already exist; in absorbing-state dynamics, the analogous interval is $\lambda_0<\lambda<\lambda_c$, where the bulk is inactive but rare active domains remain long lived (Münster et al., 2024, Cota et al., 2018). Its defining signatures are nonanalytic response over a finite parameter interval rather than at a single point, anomalous susceptibility, broad distributions of observables, and slow relaxation that is often algebraic with continuously varying exponents (Moretti et al., 2013, Ódor et al., 2010).

1. Rare regions and the canonical mechanism

The standard Griffiths construction is a rare-region argument. In disordered magnets, the disorder is quenched, so some spatial regions are atypically well connected or weakly diluted and therefore behave as if they were ordered even though the full sample is not. In the two-dimensional bond-diluted Ising model, this is the interval $T_c(p)<T<T_f$, where the bulk remains paramagnetic but exponentially rare regions are locally ordered (Münster et al., 2024). In absorbing-state models, the same logic produces an extended interval $\lambda_0<\lambda<\lambda_c$ where rare active domains survive for anomalously long times (Cota et al., 2018).

The basic asymptotic mechanism is the competition between exponentially small probability and exponentially large lifetime. In hierarchical modular brain-network models, the lifetime of a rare active region of size $\zeta$ is written as

$$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$

while the probability of such a region decays exponentially with $\zeta$. Saddle-point evaluation then yields

$\rho(t)\sim t^{-\theta},$

with an exponent that varies continuously with the control parameter (Moretti et al., 2013). The same structure appears in modular network SIS dynamics, where loosely coupled modules act as effective rare regions and produce an extended interval of control parameters with continuously changing dynamical exponents (Cota et al., 2018).

This rare-region construction also clarifies why Griffiths phases are extended critical-like regimes rather than ordinary critical points. In a clean system, scale invariance is restricted to one tuned point. In a Griffiths phase, scale-free behavior arises generically from a broad spectrum of rare-region sizes and lifetimes. A plausible implication is that Griffiths phenomena are better viewed as a family of disorder-controlled rare-event regimes than as a single universality class.

2. Response functions, scaling forms, and diagnostic frameworks

In magnetic materials, the most widely used experimental signature is the anomalous inverse susceptibility. Instead of a purely Curie-Weiss form,

$\chi=\frac{C}{T-\theta_{CW}},$

the inverse susceptibility develops a downward deviation above the bulk ordering temperature because ferromagnetic clusters are more easily polarized than a homogeneous paramagnet (Ghorai et al., 2020, Rathi et al., 2016). A common phenomenological form is

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$

where $\lambda_0<\lambda<\lambda_c$0 measures the Griffiths singularity and $\lambda_0<\lambda<\lambda_c$1 is the disorder-renormalized lower scale entering the fit (Rathi et al., 2016, Bhoi et al., 2011). In this language, the Griffiths window is $\lambda_0<\lambda<\lambda_c$2, with $\lambda_0<\lambda<\lambda_c$3 marking recovery of ordinary Curie-Weiss behavior (Rathi et al., 2016, Ghorai et al., 2020).

A more microscopic description was developed in Bray’s cluster framework. In Sr$\lambda_0<\lambda<\lambda_c$4IrO$\lambda_0<\lambda<\lambda_c$5, the inverse-eigenvalue distribution of the susceptibility matrix is taken as

$\lambda_0<\lambda<\lambda_c$6

and the average susceptibility as

$\lambda_0<\lambda<\lambda_c$7

In that system, Bray analysis yielded $\lambda_0<\lambda<\lambda_c$8, whereas conventional modified-Arrott analysis gave $\lambda_0<\lambda<\lambda_c$9, $T_c(p)<T<T_f$0, and $T_c(p)<T<T_f$1, which the authors regarded as unrealistic because the transition is not from a clean paramagnet to a clean ferromagnet but from a ferromagnetic Griffiths regime to long-range order (Rathi et al., 2016).

The modern large-deviation perspective makes the same point in distributional form. For the bond-diluted two-dimensional Ising model, Bray’s prediction for the susceptibility distribution in the Griffiths phase is

$T_c(p)<T<T_f$2

with $T_c(p)<T<T_f$3 vanishing as $T_c(p)<T<T_f$4. A large-deviations Monte Carlo study sampled this tail down to probabilities of order $T_c(p)<T<T_f$5, showing directly that the Griffiths phase is encoded not only in averaged susceptibilities but in the heavy right tail of the full disorder distribution (Münster et al., 2024).

These methods also delimit a recurrent methodological issue: conventional critical-scaling procedures can fail in the Griffiths regime because cluster contributions contaminate asymptotic critical fits. This is explicit in Sr$T_c(p)<T<T_f$6IrO$T_c(p)<T<T_f$7, where Bray-model analysis was proposed as a more appropriate tool than standard Arrott plots for Griffiths ferromagnets (Rathi et al., 2016).

3. Classical magnetic realizations

Hole-doped and mixed-valence oxides supply some of the clearest classical realizations. In La$T_c(p)<T<T_f$8Pb$T_c(p)<T<T_f$9MnO$\lambda_0<\lambda<\lambda_c$0, all samples with $\lambda_0<\lambda<\lambda_c$1 remain single-phase rhombohedral, so the anomalous magnetism is not attributable to impurity phases. The Griffiths phase is identified from the downturn of $\lambda_0<\lambda<\lambda_c$2 above the bulk Curie temperature, and its strength decreases with increasing Pb content and increasing magnetic field; $\lambda_0<\lambda<\lambda_c$3 is largest for $\lambda_0<\lambda<\lambda_c$4 and smallest for $\lambda_0<\lambda<\lambda_c$5 (Ghorai et al., 2020). The same study found critical exponents $\lambda_0<\lambda<\lambda_c$6, $\lambda_0<\lambda<\lambda_c$7, and $\lambda_0<\lambda<\lambda_c$8, close to mean-field values, and interpreted this not as evidence for a clean mean-field magnet but as a consequence of long-range interactions among ferromagnetic clusters formed in the Griffiths regime (Ghorai et al., 2020).

La$\lambda_0<\lambda<\lambda_c$9Sr$\zeta$0Mn$\zeta$1Co$\zeta$2O$\zeta$3 sharpens a different aspect of the phenomenon. There, the Griffiths phase appears over broad composition ranges and is attributed to quenched Co–O–Mn antiferromagnetic bonds embedded in a ferromagnetic background. The inverse susceptibility obeys

$\zeta$4

with representative values $\zeta$5 for $\zeta$6 and $\zeta$7 for $\zeta$8 (Bhoi et al., 2011). Crucially, this system shows that a Griffiths phase can exist entirely in a metallic paramagnetic state for $\zeta$9 and $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$0, separating Griffiths physics from the metal-insulator-percolation narrative often emphasized in manganites (Bhoi et al., 2011).

Layered Sr$$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$1IrO$$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$2 illustrates how strong anisotropy and spin-orbit coupling modify the phenomenology. The Griffiths phase occupies

$$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$3

with $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$4, $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$5, and $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$6 at $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$7 Oe; $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$8 decreases to $$\tau(\zeta)\simeq t_0 \exp\!\big[A(\lambda)\zeta\big],$$9 at $\zeta$0 kOe, confirming field suppression (Rathi et al., 2016). The same work argued that the apparent critical exponents extracted by conventional Arrott analysis were distorted precisely because the free energy is nonanalytic in the Griffiths regime (Rathi et al., 2016).

Antiferromagnets require different diagnostics. In DyBaCo$\zeta$1O$\zeta$2, in-field susceptibility yielded only $\zeta$3 K for one specimen and missed the Griffiths regime entirely in another, whereas thermoremanent magnetization revealed $\zeta$4 K in both high- and low-symmetry samples (Kumar et al., 2016). The remanent relaxation within the Griffiths regime followed the Heisenberg form

$\zeta$5

rather than a simple exponential, and $\zeta$6 remained essentially invariant under changes in oxygen non-stoichiometry even though $\zeta$7 shifted (Kumar et al., 2016). This establishes an important experimental point: in antiferromagnets, a missing $\zeta$8 downturn does not exclude a Griffiths phase.

4. Quantum Griffiths phases and correlated-electron systems

In quantum systems, the same rare-region logic is transferred to $\zeta$9 criticality, where the hallmark is ultraslow dynamics and a diverging effective dynamical exponent. In Ni$\rho(t)\sim t^{-\theta},$0V$\rho(t)\sim t^{-\theta},$1, the ferromagnetic phase itself was argued to host a quantum Griffiths regime near the disorder-driven critical concentration $\rho(t)\sim t^{-\theta},$2. On the paramagnetic side, the anomalous field dependence is $\rho(t)\sim t^{-\theta},$3; on the ferromagnetic side it becomes

$\rho(t)\sim t^{-\theta},$4

with a nonuniversal exponent $\rho(t)\sim t^{-\theta},$5 that is strongly $\rho(t)\sim t^{-\theta},$6-dependent and nearly symmetric about $\rho(t)\sim t^{-\theta},$7 (Wang et al., 2016). Muon spin rotation further showed inhomogeneous magnetic order and fluctuating clusters, supporting the interpretation that long-range order coexists with disconnected rare regions inside the ordered phase (Wang et al., 2016).

URu$\rho(t)\sim t^{-\theta},$8Si$\rho(t)\sim t^{-\theta},$9 represents a more controversial application. One interpretation of its hidden-order regime is a Griffiths phase produced by the collapse of long-range antiferromagnetism into weakly coupled magnetic clusters. The reported thermodynamic singularities are

$\chi=\frac{C}{T-\theta_{CW}},$0

with $\chi=\frac{C}{T-\theta_{CW}},$1 close to $\chi=\frac{C}{T-\theta_{CW}},$2, together with frequency-dispersive AC susceptibility and a unidirectional anisotropy in rotating-field resistivity (Liu et al., 2018). The same paper presents this as an alternative to hidden-order parameter scenarios, but its own discussion also leaves room for a more cautious reading as a Griffiths-phase-like cluster regime rather than a definitive replacement of the hidden-order concept (Liu et al., 2018).

CeRhSn extends the same language to a structurally disordered Kondo-lattice metal. There, a classical Griffiths phase is proposed below $\chi=\frac{C}{T-\theta_{CW}},$3 K and a quantum Griffiths phase below $\chi=\frac{C}{T-\theta_{CW}},$4 K, supported by $\chi=\frac{C}{T-\theta_{CW}},$5, $\chi=\frac{C}{T-\theta_{CW}},$6, and $\chi=\frac{C}{T-\theta_{CW}},$7, together with STM evidence for nanometer-scale structural inhomogeneity (Ślebarski et al., 4 Jun 2025). The same work locates a Griffiths-fit scale $\chi=\frac{C}{T-\theta_{CW}},$8 K and $\chi=\frac{C}{T-\theta_{CW}},$9 from inverse susceptibility (Ślebarski et al., 4 Jun 2025).

A distinct quantum realization appears at the superconductor-insulator transition of mirror-symmetric twisted trilayer graphene. Under out-of-plane magnetic field, the transition is broadened into a quantum Griffiths phase with activated scaling,

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$0

and an effective exponent obeying

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$1

with $$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$2 mT and $$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$3 (Mahapatra et al., 14 Jul 2025). Under in-plane field, the broad Griffiths regime collapses into a nearly single quantum critical point, making field orientation an explicit control parameter for disorder-dominated quantum criticality (Mahapatra et al., 14 Jul 2025).

5. Nonequilibrium systems, networks, and temporal generalizations

Absorbing-state dynamics provides the cleanest nonequilibrium setting. In finite-dimensional hierarchical modular brain-network models, rare active regions embedded in an inactive background produce algebraic decay

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$4

over a finite interval of spreading rates, and the dynamic susceptibility

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$5

diverges throughout that interval rather than only at a single critical point (Moretti et al., 2013). This was used to argue that hierarchical modular structure can “stretch” criticality into a Griffiths phase and thereby enlarge the dynamic range (Moretti et al., 2013).

Later work loosened the structural requirements. On highly modular but non-hierarchical networks, loosely coupled modules themselves act as rare regions, producing extended intervals of nonuniversal power-law relaxation and avalanche-size distributions

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$6

with $$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$7 in the reported SIS simulations (Cota et al., 2018). This explicitly relaxes the earlier conjecture that hierarchy is required. At the same time, Griffiths phases weaken or disappear when module-size heterogeneity becomes too broad, because a few giant modules dominate the dynamics (Cota et al., 2018).

The role of topology was already isolated in the contact process on complex networks. There, purely topological quenched disorder can generate Griffiths phases when the network has finite topological dimension, whereas effectively infinite-dimensional small-world regimes do not generically support the same behavior (Ódor et al., 2010). Hierarchical modular networks with small-world edges refine this further: small-worldness, modularity, and localized spectral modes are not sufficient by themselves; what matters is an exponentially decreasing inter-moduli connectivity probability across hierarchy levels (Li, 2016).

Temporal disorder generates an exact counterpart. In temporal Griffiths phases, rare absorbing time intervals in a globally active system play the role that rare spatial regions play in ordinary Griffiths phases. The defining size dependence becomes

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$8

while at criticality

$$\chi^{-1}(T)\propto (T-T_C^R)^{1-\lambda}, \qquad 0<\lambda<1,$$9

This behavior was established for absorbing-state systems with temporal disorder in $\lambda_0<\lambda<\lambda_c$00, and explicitly formulated as a space-time-reversed counterpart of ordinary Griffiths phases (Vazquez et al., 2011).

A cold-atom realization appears in facilitated Rydberg gases. In the high-temperature limit, motion restores a homogeneous absorbing-state transition. In the frozen low-temperature limit, however, facilitation is constrained to a random network resembling an Erdős-Rényi graph, and the absorbing-state transition is replaced by an extended Griffiths phase accurately described by SIS dynamics on that network, including blockade corrections (Brady et al., 2023).

6. Conceptual boundaries, controversies, and scope

Several recurring issues delimit the concept. First, a Griffiths phase is not synonymous with any broad crossover or any clustered state. In the two-dimensional Potts model with long-range correlated disorder, Monte Carlo work found algebraically divergent susceptibility across a finite temperature interval and systematic violation of hyperscaling through cancellation of leading disorder-fluctuation terms (Chatelain, 2013). A later stability analysis showed that this interval cannot be reduced to a simple spreading of local transition temperatures governed only by disorder fluctuations, and that its width is controlled by disorder strength; however, the author still noted that finite accessible sizes preclude an absolute thermodynamic-limit proof (Chatelain, 2014).

Second, anomalous critical exponents extracted by standard methods need not be intrinsic universality exponents. This is explicit in Sr$\lambda_0<\lambda<\lambda_c$01IrO$\lambda_0<\lambda<\lambda_c$02, where Bray analysis and modified-Arrott analysis yielded sharply different $\lambda_0<\lambda<\lambda_c$03 values because the latter was contaminated by Griffiths nonanalyticity (Rathi et al., 2016). It is also explicit in La$\lambda_0<\lambda<\lambda_c$04Pb$\lambda_0<\lambda<\lambda_c$05MnO$\lambda_0<\lambda<\lambda_c$06, where mean-field-like exponents were interpreted not as evidence against disorder, but as evidence that long-range interactions among ferromagnetic clusters dominate the onset of bulk order (Ghorai et al., 2020).

Third, “Griffiths-like” language can exceed the strict statistical-mechanical meaning of the term. The proposed “cellular Griffiths-like phase” in liquid-liquid phase separation is explicitly framed as an analogy: spatially distributed protein-rich droplets are treated as rare regions, and slow heterogeneous dynamics are emphasized, but the work does not derive Griffiths singularities from a quenched-disorder partition function or compute rare-region probability laws in the conventional sense (Squillante et al., 2024). The distinction is important because it separates rigorous Griffiths phenomenology from heuristic borrowing of the term.

Taken together, these examples suggest that “Griffiths phase” is best reserved for disorder-dominated regimes in which rare regions generate extended nonanalytic behavior, broad distributions, and nonuniversal slow dynamics, while adjacent phrases such as “Griffiths-like” are more appropriate when only the phenomenological analogy is established.