Harris CriterionEdit
The Harris criterion is a foundational result in the theory of critical phenomena that tells us when the inevitable imperfections of real materials—random impurities, vacancies, and other quenched disorder—will change the way a system behaves as it approaches a phase transition. Named after A. B. Harris, who introduced it in the mid-1970s, the criterion provides a simple, practical rule of thumb for assessing whether disorder is relevant to the universal properties of a system near its critical point. In essence, it helps physicists decide when the clean (idealized) model and its associated universality class will survive the presence of imperfections, and when disorder will push the system into a different universality class.
The practical appeal of the Harris criterion lies in its reliance on a small set of quantities that characterize the clean system: the space dimension d and the correlation length exponent nu, which governs how the correlation length diverges as the system nears the critical point critical point of a phase transition. In a condensed form, the criterion states that disorder is relevant if the product d × nu is less than 2, and irrelevant if that product exceeds 2; the borderline case d × nu = 2 is considered marginal. Since nu is a property of the pure system (the one without disorder), the criterion can be applied with relatively little model complexity to predict whether introducing randomness will alter the critical behavior critical exponent correlation length.
Background and formulation
Disorder in condensed matter and statistical mechanics comes in many flavors, but quenched disorder is a particularly important case. It refers to randomness that is fixed in time, such as lattice vacancies, dopant atoms, or random bond strengths that do not fluctuate during the observation of the system. Near a phase transition, where long-range correlations emerge and the system becomes scale-invariant, even small amounts of quenched disorder can have outsized effects. The Harris criterion codifies when those effects are expected to appear in the universal signatures of the transition, such as the scaling of the order parameter, susceptibility, and specific heat.
The argument rests on finite-size scaling and the idea that different regions of a disordered sample sample the disorder landscape. If fluctuations in the local critical temperature due to impurities are large on the scale of the correlation length, then disorder does not simply wash out or average away; instead, it changes the way fluctuations accumulate across scales. When d × nu < 2, those local fluctuations are sufficiently potent to modify the critical behavior, leading to a new universality class. When d × nu > 2, the clean system’s critical behavior remains effectively intact in the presence of weak disorder. The marginal case d × nu = 2 is subtle and often exhibits logarithmic corrections that depend on the details of the disorder and the model in question. These ideas are embedded in the broader framework of the renormalization group, where the relevance or irrelevance of perturbations determines whether a given fixed point governs the long-distance physics renormalization group finite-size scaling.
Mathematical expression and implications
- Disorder is relevant if d × nu < 2.
- Disorder is irrelevant if d × nu > 2.
- Disorder is marginal if d × nu = 2, often accompanied by logarithmic corrections.
In the language of critical exponents, alpha, the exponent describing the specific heat, is linked to nu by the hyperscaling relation alpha = 2 − d × nu for many systems. Thus, a positive alpha (a divergent specific heat in the clean system) aligns with the intuitive statement that disorder can be relevant. Conversely, if alpha < 0 (a non-diverging specific heat in the clean system), disorder is typically irrelevant. These connections help translate the Harris criterion into concrete predictions for experiments and simulations specific heat critical exponent.
The criterion has proven robust across a variety of classical systems, particularly in spin models. For example, in the Ising model family, the criterion suggests that in three dimensions the product 3 × nu of the clean system is below 2, pointing toward a disorder-relevant scenario in many parameter regimes, with numerical studies often finding a shift to a new universality class under quenched randomness such as random bonds or random sites. In two dimensions, nu = 1 for the clean Ising model, giving d × nu = 2, the marginal case where disorder can induce logarithmic corrections to scaling rather than a wholesale change in exponents. These behaviors have been explored in detail in Ising models and related systems, with caveats arising from the specifics of the disorder and interactions Ising model random-bond Ising model quenched disorder.
Quantum phase transitions, which occur at zero temperature as a function of some nonthermal control parameter, invite a quantum generalization of the Harris criterion. The effective dimension becomes d + z, where z is the dynamical critical exponent, and the same logic applies to determine whether disorder is relevant in the quantum case. This connection to quantum criticality highlights the broader reach of the criterion beyond purely classical systems quantum phase transition dynamical critical exponent.
Applications and examples
- Ising-like magnets with impurities: In many three-dimensional magnetic systems with quenched randomness, the Harris criterion signals a tendency for disorder to alter the universality class, and a corresponding shift in critical exponents is observed in simulations and experiments on diluted or randomly bonded magnets Ising model random-bond Ising model.
- Two-dimensional systems: The borderline case d × nu = 2 in the clean 2D Ising model leads to logarithmic corrections when disorder is present, a pattern seen in numerous simulations and analytic studies of two-dimensional Ising model with random bonds or site dilution.
- Random-field systems: In the random-field Ising model and related setups, quenched disorder tends to have a strong impact on critical behavior, and the Harris criterion helps frame when those effects are expected to be decisive for the universality class. See discussions of the random-field Ising model in the literature.
- Percolation and porous media: Problems with connectivity and disorder in real materials also intersect with the Harris perspective, as the presence of randomness can influence the scaling of cluster sizes and the approach to percolation thresholds percolation theory.
- Extensions to aperiodic and correlated disorder: In systems where disorder is not purely random but follows aperiodic or correlated patterns, the Luck criterion provides a related, more refined tool for assessing relevance. The Harris criterion and its extensions continue to guide analyses in these contexts Luck criterion.
In many of these contexts, the key takeaway is that the mere presence of impurities is not a guarantee of altered critical behavior. The decisive ingredient is how those impurities scale with the growing correlation length as a system nears its phase transition, which the Harris criterion encodes in a compact inequality linking d and nu critical exponent correlation length.
Extensions, limitations, and controversies
- Quantum generalization: As mentioned, the extension to quantum phase transitions uses the effective dimension d + z, so the criterion reads (d + z) × nu < 2 for relevance, with nu and z defined in the quantum setting. This extension provides a practical rule of thumb for disordered quantum magnets and related systems quantum phase transition dynamical critical exponent.
- Correlated and long-range disorder: The original Harris criterion presumes short-range, uncorrelated randomness. When disorder is long-range correlated or exhibits power-law correlations, the simple d × nu criterion can fail to capture the true relevance of disorder, and more sophisticated analyses are needed. In these situations, theoretical work and simulations often appeal to generalized criteria or nonperturbative approaches renormalization group.
- Rare-region effects and Griffiths phenomena: In some disordered systems, rare fluctuations create regions that locally appear critical and influence observable scaling over broad parameter ranges. These Griffiths phases can complicate the interpretation of Harris-type predictions and can mask or alter the way disorder affects critical exponents in finite samples Griffiths phase.
- Experimental challenges and nonuniversal behavior: Real materials are messy, with competing interactions, finite-size effects, and measurement constraints. While the Harris criterion provides a valuable guide, experimental determinations of whether disorder changes universality often display a mix of consistent shifts, small corrections, and in some cases apparent preservation of clean-system exponents within error bars. Critics of overreliance on a single criterion emphasize the need to complement it with comprehensive numerical and experimental analyses critical point finite-size scaling.
- Pragmatic perspective: From a practical, engineering-minded standpoint, the Harris criterion offers a clear, testable expectation that helps guide material design and interpretation of data. It supports a disciplined approach to predicting when impurities will qualitatively change critical behavior and when they can be safely neglected, a philosophy that aligns with a preference for robust, falsifiable criteria in complex systems.