Renormalisation group theory was developed in the 1940s in the context of quantum field theory and used later in various fields such as cosmology, quantum mechanics and condensed matter physics. Renormalisation group theory provides an efficient tool to describe continuous phase transitions, in which the correlation length, the distance over which fluctuations in the microscopic degrees of freedom are correlated to each other, becomes infinite and the system is scale-invariant.
The basic idea of renormalisation group applied to crackling noise is to start from a microscopic description of the system and iteratively shrink, or coarse-grain the system in such a way that microscopic degrees of freedom are removed but the large-scale properties are preserved. As the system is coarse-grained and the model parameters are renormalised, the system can be seen as flowing in the space of model parameters. In this space, which has as many dimensions as there are parameters in the model, fixed points can exist, at which coarse-graining yields the same model and the system is self-similar.
The process of iterative shrinking (coarse-graining) of the system can be visualised in figure3.2 in the example of the RFIM model of equation3.1.
The model is seen as the red line on the figure. Depending on the starting values of the disorder R, the system flows towards either snapping or popping.
AtRc, the model flows towards a fixed point S∗, where avalanches occur on a broad range of scales and are power-law distributed.
Power-law distributions result naturally from the coarse-graining process if one assumes that the distribution that is considered remains the same upon coarse-graining. The distribution of event sizes is typically of the form D(S)∼S−τFs, whereτ is a universal exponent independent of the microscopic details of the model and Fs is a function controlling the large scale cutoff to the power-law behaviour. The further the parameters of a model are from their critical value corresponding to S∗ in figure 3.2, the shorter the cutoff to the pure power-law behaviour.FS is model-dependent and not universal, and will depend on details of the system.
Multiple models starting with different microscopic descriptions can converge towards the same fixed point, meaning that they share large scale properties which are independent of the microscopics of the particular systems
3.2 An experimentalist’s guide to renormalisation group theory
under consideration. These models share critical exponents and are said to be part of the same universality class.
Universality is a very powerful concept, which in principle allows the large-scale statistics of any system within its universality class to be predicted by using insight obtained from a single system within the universality class.
Different systems that are expected to have very different physics at small scale can share a universality class. For example, the distribution of slip sizes in slowly compressed microcrystals has been shown to be part of the same universality class as the size distribution of avalanches in soft magnets [73].
Universality is useful, not only as an abstract theoretical tool but also for applications such as non-destructive stress testing. The Barkhausen noise of slowly compressed crystals depends on the applied stress, which allows the failure stress to be extracted in a non-destructive way from the noise spectrum. The same idea can also be applied to the monitoring of structures and mine shafts. In these applications, universality means that the details of the materials or rocks that are being monitored do not matter as long as they belong to the expected universality class.
Figure 3.2: Visualisation of renormalisation group flow in the space of model parameters in the context of the RFIM. The model flows towards a fixed pointS∗in which the model is self-similar. From [63].
Although systems within a universality class share critical exponents, the cut-off functions differ from model to model. In principle, this allows different models belonging to the same universality class to be distinguished.
Experimental data obtained with different values of some relevant parameter
can be plotted in terms of the dependence of the cutoff function on that parameter. For the RFIM for example, the cutoff function for the distribution of event sizes D(S) takes the form D(S) ∼ S−τFs(Sσ R−RR c) where σ is related to the fractal dimension of the avalanches. D(S)/S−τ can then be plotted as a function of Sσ(R−Rc)/R in order to collapse curves of different avalanches at different values of R. This is not easy however. On the theoretical side, it requires some knowledge of the form of the scaling function. On the experimental side, the data needs to be acquired over a range large enough to see the effect of the cutoff function experimentally, which requires repeated measurements for different values of relevant parameters of the model (here R). Measuring power-law exponents is already not easy as measurements have to be performed over several decades and enough data needs to be gathered to be statistically relevant. Measuring exponents over a range of scales wide enough to see the cutoffs to the studied probability distributions is experimentally highly non-trivial.
Figure 3.3:Scaling collapse of avalanche sizes in the RFIM wherer= (R−Rc)/R.
Dintrefers to the distribution of avalanche sizes integrated over the entire external field hysteresis loop in a soft magnet and ¯τis the corresponding power-law exponent.
σis another universal exponent related to the fractal dimension of the avalanches.
The RFIM model discussed above is only one of many models showing crackling behaviour. One class of models called elastic interface models has been very successful in describing a variety of systems such as soft magnets, slip faults, contact lines of fluids on rough surfaces and will be discussed in the next section.
3.3 Avalanches in elastic systems in a disor-dered medium
Elastic models have proven very powerful in describing the statics and dynamics of various systems, from domain walls in ferroelectric and ferro-magnetic materials [67,69, 71] to Abrikosov lattices [94], wetting fronts [95]
3.3 Avalanches in elastic systems in a disordered medium
and crack propagation [96]. Elastic models provide a general description of interface statics and dynamics by focusing on the boundary itself and including only a few key ingredients rather than the detailed microscale physics of the system. The interface is modelled as an elastic boundary of dimensiond, withm transverse components along the direction of move-ment of the interface and living in aD-dimensional space, withD=d+m. Elastic models can describe a moving interface such as a ferroelectric or ferromagnetic domain wall as well as periodic structures such as Abrikosov vortex lattices in type-II superconductors. Here, the case of a d-dimensional interface living inD=d+ 1 is discussed. The interface is described using a displacement functionu(z, t) wherez is the position along the interface and x is the transverse coordinate, describing the direction of propagation.
Figure 3.4: Description of the coordinate system describing the interface. The interface is shown in blue and is propagating along the xdirection. The function u(z, t) describes the interface position.
u(z) is a single-valued function, meaning that this description does not take overhangs or nucleation sites into account. In the case of real interfaces, where these overhangs do exist, they can be interpolated out. The main ingredients affecting the interface shape are the external force driving the overall movement of the interface, the interface elasticity which tends to favour a flat line configuration (or plane or hyperplane depending on the interface dimensionality), and the disorder, which pins the interface and promotes meandering. The system displays glassy physics as the energy landscape exhibits many local minima and the interface wanders through consecutive metastable states. The competition between these three a priori simple ingredients leads to the rich physics of avalanches and scale-invariance of event size and energy distributions. A simple description of the dynamics of the interface is the quenched Edwards-Wilkinson equation
η∂tu(z, t) =f(z, t) +F(u(z, t), z) +c∇2u(z, t) +µ(z, t) (3.2) whereη is a microscale friction coefficient,f(z, t) describes the applied external force, andF(u(z, t), x) is a random force which mimics the effect of the disorder.c∇2u(z, t) describes a short-range interface elasticity with
modulus c, while µ(z, t) describes thermal noise. It is assumed that the disorder is uncorrelated along the interface direction, whereas along the transverse coordinates, the disorder is usually described as being correlated in one of two ways, each belonging to a distinct universality class [97]. In the so-called random-bond case illustrated in figure3.5(a,c), it is assumed that the disorder affects the phases on both sides of the interface symmetrically.
In the case of uniaxial ferroelectrics, random-bond disorder affects the depth of both energy wells corresponding to the two stable polarisation states in the same way. The pinning potential in random-bond disorder is correlated on a short range and only impurities located close to the interface contribute to its pinning. In the random-field case shown in figure3.5(b,d) however, both sides of the interface are affected asymmetrically and one polarisation orientation is favoured over another. In random-field disorder, all the impurities within the region corresponding to the phase favoured by the external force contribute and the pinning potential has long-ranges correlations.
Figure 3.5:(a,b) Schematic of how impurities affect the pinning in the random-bond and random-field cases respectively. The defects contributing to the pinning are shown in red. After [71]. (c,d) In the case of a ferroelectric, the disorder affects the depths of the free energy minima as defined by the Ginzburg-Landau-Devonshire (GLD) theory, where the free energyF =12aP2+14bP4+16cP6 is minimised in terms of the polarisation order parameter P. The energy wells are affected symmetrically in the random-bond case and asymmetrically in the random-field case.
At a temperature of 0 and below a critical external forcefc, the interface is pinned with no observable motion. Atf >> fc, the interface velocity is linear with the driving force and the defects act as a viscous drag. Closer to the critical force, however, wheref > fc, the interface is depinned and the overall velocity exhibits a power-law increase v∼(f−fc)β, whereβ is a universal exponent. Crossing throughfc, the system undergoes a second-order dynamic phase transition with the velocity as an second-order parameter and
3.3 Avalanches in elastic systems in a disordered medium
the external field as a control parameter. Atf =fc, motion of the interface occurs in a wide range of event sizes and with a power-law distribution P(S) ∼S−τdepfs(S/Sc), where fs is a cutoff function of the sizes, which decays sharply asS≥Sc and is a constant forS < Sc.Sc is a characteristic event size which depends on the external force, the dimensionality of the interface and its static roughness. In this regime, avalanches are spatially uncorrelated and no aftershocks are observed, as opposed to earthquakes, where following large events, the faults rearrange in a series of smaller earthquakes. At low temperature and belowfc, the interface is in the so-called creep regime, where the motion of the boundary is thermally activated.
This regime has been experimentally demonstrated by magneto-optic Kerr imaging in ferromagnetic Pt/Co/Pt films [98] and by piezoresponse force microscopy in ferroelectric materials [99–101]. This regime is particularly rich and interesting.
The overall interface velocity in the creep regime follows a stretched exponential behaviour v ∼exp(−βUc(ffc)µ) where β = 1/kbT, Uc is the characteristic height of the barriers the interface needs to cross and µ is an exponent dependent on the type of disorder and the dimensionality of the interface. The motion typically happens in two steps. In the first step, a portion of the interface of typical lengthloptmoves through thermal activation, triggering a rearrangement of the interface on a larger scale, through a fast avalanche process in the second step, as shown in figure3.6.
Figure 3.6:In the creep regime, a thermal nucleus of typical size loptmoves first, triggering a larger avalanche. After [67]
As a consequence of the glassy physics describing the interface, the optimal thermal nucleus size lopt ∼ f−2−ζ1 is inversely proportional to the external force f and ζ describes the size scaling of the domain wall roughness [67]. The size distribution of these events also follows a power-law P(Seve) ∼ Seve−τfs(Seve/Sc) where the cutoff size increases as the driving force decreases. This means that most events are smaller thanloptsince the probability of events larger thanSc is cut off byfs. It is therefore the few larger events that drive the overall motion of the interface according to the
creep law.
Figure 3.7:In the creep regime, avalanches are predicted to cluster in space and time, with aftershocks as seen in earthquakes, while in the depinning regime where f > fc, the avalanches are uncorrelated and no aftershocks are predicted. From [102].
Another interesting feature is the spatial localisation of events in the creep regime recently identified in theoretical studies [102]. In d= 1 and in the presence of random-bond disorder, the avalanche events in the creep regime tend to cluster together in space and time, similar to aftershocks in earthquakes, while these correlations are absent in the depinning regime. The distribution of cluster sizes follows a power-law with a crossover of exponents at S =Sc. BelowSc, the size exponent in the random bond case is that of equilibrium with τeq = 0.8, whereas aboveSc, it follows the depinning characteristic exponent of τdep= 1.11.
Figure 3.8:The distribution of clusters of event sizes in the creep regime is predicted to show a crossover from the 1D equilibrium exponentτeq= 0.8 to a larger exponent τdep= 1.11 compatible with the 1D depinning exponent of 1.11. From [102].