Rigidity theory was used first in glasses by Phillips (1979) to understand a general experimental observation: best network glass formers have an average coordination (< r > ) equal to < r > = 2.4. Here, best network glass formers means that the cooling speed is minimal in order to form the glass (Phillips, 1979). This special < r > = 2.4 coordination is obtained via chemical doping.

Why this magical number? The answer lies in a topological phase transition between a flexible to a rigid network (Thorpe, 1983). By considering each bond between any of the N atoms as a mechanical constraint, one can ask the question of how many bonds are needed to make the system rigid. If the number of constraints is bigger than the dimension of the configurational space, given by 3N, then the system is rigid. When the constraints are equal to 3N, the network is isostatic and has very special properties (Selvanathan et al., 2000; Wang et al., 2001). Otherwise is flexible.

When N_c is the number of constraints of a 3 dimensional system (3D), then a fraction (3N–N_c)/3N of the 3N configurational coordinates are cyclic, since the energy of the system does not depend on such variables (Thorpe, 1983). Thus, f is also the fraction of vibrational modes with zero frequency (f), called floppy modes.

Such fraction can be found using a mean-field approximation known as the Maxwell counting (Thorpe, 1983). This counting goes as follows: each of the r bonds in a site is shared by two sites. There are r/2 constraints due to distance fixing between neighbors. Angular forces also give constraints for directional bonding, and in 3D there are (2r − 3) constraints to give, f=3N−Nc3N=1−∑r r∕2+(2r−3)xr3=2−56r, when f  = 0 the network passes from a floppy network to a rigid one. In 3D, the mean-field approach predicts the transition at the critical value ⟨r_c⟩ = 2.4 if all angular constraints are present.

Notice that rigidity percolation theory was made for zero temperature (Thorpe, 1983). However, the original idea of Phillips’s RT was to explain the minimal cooling speeds to form glasses of a liquid supercooled melt (Phillips, 1979). Why a T = 0 theory works for the melt? This question is still in the process of being answered, and is a very active field of research. However, some preliminary conclusions are available, depending upon whether T is bigger or lower than T_g, where T_g is the glass transition temperature.

All glasses present an excess of low-frequency vibrational modes (LVFMs) relative to the Debye crystal model, like the Boson peak (BP) or floppy modes (FM). They are in the THz range of frequencies (Buchenau et al., 1984; Elliot, 1990; Hehlen et al., 2000; Binder and Kob, 2005). Crystals can also present a Boson peak under pressure, like in SiO₂ (Nakayama, 2002), where the rigid SiO₄ tetrahedra units are connected by flexible bonds (Trachenko et al., 2004). In glasses, the nature of the Boson peak is still debated, while floppy modes are well understood.

During the last years, it has become clear that the Boson peak can also be explained as a consequence of rigidity. To understand this, suppose that we have an over-constrained network in which N_c > 3N, as shown in Figure 1. If bonds are removed at random, for example, by dilution, eventually N_c = 3N. At this point, the lattice becomes isostatic, see Figure 1. If we further dilute the lattice, eventually N_c < 3N and 3N–N_c zero frequency modes will appear. But what happens when we dilute the lattice starting from N_c < 3N?.

This question can be answered first by studying random bond dilution in periodic lattices (Flores-Ruiz and Naumis, 2011), then a Boson peak appears at a frequency Ω_BP. It can appear at most at one-third of the Debye frequency (Flores-Ruiz and Naumis, 2011) ω_D, while for example in almost all glasses (Rufflé et al., 2008) Ω_BP = 0.1ω_D. In addition, the position of the peak scales as (Flores-Ruiz and Naumis, 2011) <r>−rc, where < r > is the coordination of the network and r_c is the critical coordination of the isostatic lattice. For disordered lattices, one can extend this approach by using perturbation theory or by performing numerical simulations (Flores-Ruiz et al., 2010).

Furthermore, the peak has an almost transverse mode nature and can be related with a Van Hove singularity occurring in the glass (Flores-Ruiz and Naumis, 2013). Recent evidence suggests that the Boson Peak in glasses is equivalent to the transverse acoustic van Hove singularity in crystals (Chumakov et al., 2011), supporting the analytical and computer model (Flores-Ruiz and Naumis, 2013). The shift of the Boson peak due to pressure effects (Monaco et al., 2006) can also be understood as an increase in the number of contacts (Flores-Ruiz and Naumis, 2013).

Thus, a new picture emerges by the process of bond dilution. The Boson peak is due to a diluted connectivity in an over-constrained network. As < r > goes to a critical r_c, an isostatic lattice is reached. The Boson peaks reach a zero frequency and if dilution is performed again, floppy modes are obtained. Both low-frequency modes anomalies are due to a reduced connectivity of the network.

For example, in SiO₂ under pressure a blue shift of the LFVM anomalies is observed (Trachenko et al., 2004). Such effect can be understood as a departure from isostaticity by an increased number of atomic contacts due to pressure. Below a pressure of 3GPa, constraint counting in SiO₂ reveals an isostatic network due to the connection of rigid unit modes (RUM) made from SiO₄ tetrahedra with flexible bonds. Pressure leads to the formation of defects with an increased coordination number (Trachenko et al., 2004).

Notice that in real glasses, floppy modes, and the Boson peak can coexist since regions with different elastic properties can coexist (Bhosle et al., 2011).

In the next section, we will discuss the importance of low-frequency modes for the glass transition.

Do low-frequency modes anomalies influence glass transition? There are several arguments against this. For example, within the energy landscape picture (Debenedetti and Stillinger, 2000), if a glass relax into a floppy mode channel reaction coordinate, eventually it will reach a new basin in which the floppy coordinates are different. Observe that floppy modes are important for fast relaxation. Long-time scale relaxation (α) does not retain memory of the almost instantaneous configuration of a floppy mode. However, energy landscape basins are in fact very similar (Flores-Ruiz and Naumis, 2012). Even in protein folding, an analysis of the landscape leads to the conclusion that the similarity of the minima is behind the remarkable phenomena of relaxation along soft coordinates (Pontiggia et al., 2007). For protein G, slow modes display a very mild dependence on the trajectory duration in the landscape. This originates from a striking self-similarity of the free-energy landscape embodied by the consistency of the principal directions of the local minima (Pontiggia et al., 2007), where the system dwells for several nanoseconds, and of the virtual jumps connecting them. Incidentally, proteins also have a boson peak (Ciliberti et al., 2006). Recently, a careful study of the short-time behavior distribution of normal modes on different meta-basins in Lennard-Jones binary glasses leads to the same conclusion (Flores-Ruiz and Naumis, 2012). In fact, such a remarkable result is behind another puzzling question, why short-time dynamics provides information about long-time dynamics in glasses (Dyre et al., 2006).

At the same time, there is a clear indication that low-frequency modes are paramount to the thermodynamical stability of a system. At this point, we will use an opposite approach to that used by most people. Instead of looking at melt cooling, here we assume that a glass was already formed, and has a temperature T lower than T_g. Then the glass is heated, until reach T_g.

Mechanical rigidity provides a clue on the thermodynamical stability of a system that contains a delicate balance between dimensionality and low-frequency vibrational modes. In particular, stability can be tested by looking at long-range correlations of the quadratic displacement u²(r_j) of atom j at position r_j, u(ri)−u(rj)2∕2=u2(T)−u(ri)⋅u(ri+Rij). where R_ij is the vector R_ij = r_j – r_i that joins atom i with j, and the bracket ⟨⟩ denotes average at temperature T and over all atoms. For a laboratory timescale, a glass is in a metastable state. Thus, we can suppose that the glass can be represented as an harmonic Hamiltonian in thermal equilibrium. Under this approximation, it is easy to prove that, (5)u2(T)≈3kBTm∫0∞ ρ(ω)ω2dω. where < m > is the average mass. An excess of LFVM is enhanced by the 1/ω² inside the integral in equation (5). LFVMs lead to an increased ⟨u²(T)⟩of the glass when compared with the crystal. Dimensionality (d) is also important for the crystal and glass. For crystals, the Debye model indicates that ρ(ω) = Cω^d–1. Only for d = 3 the integral is convergent for non-zero T.

Summarizing, the stability of a solid is mainly contained in the factor ρ(ω)/ω². Any excess of low-frequency modes will decrease the stability of the system.

Let us now further develop the previous arguments to show how LFVM can determine T_g. At low T, the system is trapped in an energy landscape basin (Debenedetti and Stillinger, 2000). As the system is heated, it can visit new basins. Following density functional theory, we look for transition states between basins, using the ansatz that atoms travel the least motion path between adjacent minima (Hall and Wolynes, 1987). The viscosity η(T) can be estimated as (Hall and Wolynes, 1987), (6)ln(η(T)∕η0)≈3R024u2(T), where η₀ is the typical value of the melt viscosity, R₀ is a constant related with the range of the inter-atomic potential. The mean quadratic displacement ⟨u²(T)⟩should be taken as measured by Mossbäuer scattering, since ⟨u²(T)⟩ will diverge for long-time scales. Since T_g corresponds to the temperature where η₀ ≈10¹³ Poise, from equation (6) we get, (7)u2(Tg)R02≈34(13−y)ln10≈0.32513−y, where y is an exponent defined by η = 10^y Poise. For typical fluids (Hall and Wolynes, 1987) y ≈ 3. Also, for typical inter-atomic potentials (Hall and Wolynes, 1987) 0.3σ < R₀ < 0.5σ (where σ is the atomic size). Thus, we obtain the following bounds, (8)0.0080<u2(Tg)σ2<0.0081, The maximal quadratic displacement allowed for the system to be solid is determined by, (9)u2(Tg)<0.9σ. Since σ is of the order of the inter-atomic distance, it turns out that we just obtained for glasses the well known Lindemann criterion, i.e., for crystals melting occurs when the mean atomic displacement u2(T) is around 10% of the atomic spacing a (Tabor, 1996). In fact, this criterion is valid for many glasses (Buchenau et al., 1984; Buchenau and Zorn, 1992). At this point, the fundamental role of low-frequency modes to determine T_g and T_m is clear, since we can separate the problem in two time scales.

Using equation (5) and that u2(Tg)≈0.01σ, T_g is given by, (10)Tg≈0.0013ρ2∕3kBm∫0∞ ρ(ω)ω2dω.−1 The previous analysis can be combined with RT to estimate T_g as a function of chemical doping (Naumis, 2006a,b). Using the simple model of ρ(ω) described in equation (2) to feed equations (5) and (10), we get that (11)Tg(r)≈Tg(<r>=2.0)∕1−γ(r−2). where γ a constant determined by ρ_R(ω) (the density of states when f  = 0) and ω₀, (12)γ≡ω02∫0∞ ρR(ω)ω2dω−1−1. This functional form deduced here for T_g as a function of ⟨r⟩ has been observed experimentally by many groups (Tatsumisago et al., 1990; Sreeram et al., 1991), and is called the empirically modified Gibbs-DiMarzio law (Naumis and Kerner, 1998; Kerner and Naumis, 2000).

Here, we discuss recent progress in two fields: strain and thermal relaxation in glasses. In glasses, quenching produces defects with some residual internal stress. Such stress relaxation is known to be described experimentally by stretched exponentials (Phillips, 2006; Welch et al., 2013), (13)I(t)≈exp[−(t∕τ)β] The relaxation parameters τ and β have been usually fitted from experimental data. Not so much work has been done on the theoretical side to understand a fundamental issue: how to get these parameters from a theory. Eventually, by considering defects as traps and a diffusion model, stretched relaxation theory (Phillips, 2006) (SER) was able to predict that in d dimensions β = d/(d + 2), giving the magic number β = 3/5 for d = 3. Later on, it was recognized that long-range forces can have an important effect in certain kind of glasses (Naumis and Phillips, 2012a,b), and the previous formula is modified by an effective dimensionality β = d*/(d* + 2), with d* = fd, and f  = 1/2. This produces another magic number (Naumis and Phillips, 2012a,b), β = 3/7.

In real glasses, the accurate predictions of SER theory have been confirmed in a beautiful experiment by using a big homogeneous glass plate (Welch et al., 2013). Surprisingly, the same magic number appears in a study of twentieth century citations, involving 25 million papers (Wallace et al., 2009; Naumis and Phillips, 2012b). If papers are ranked by citations, then the ranking does not follow a power law. Instead, a stretched exponential is followed (Wallace et al., 2009). Before 1960, the number of citations has an exponent β = 3/5. After 1960, β = 3/7. The transition can be explained by the jet age, which increased the connectivity of the network (Naumis and Phillips, 2012b). The similarity with glasses is not coincidental, it is just the result of combining relaxation and topology, leading to a new exciting field. In fact, stretched exponentials lead to the modified beta rank law (Naumis and Cocho, 2008), which has many applications in biology, social sciences, soft-matter, physics, etc., i.e., when network topology matters (Naumis and Cocho, 2007).

Finally, we will discuss how thermal relaxation is modified by the rigid or flexible nature of the system. This is important for glass formation ability, since, for example, a fast relaxation will enhance crystallization. As an example, it has been observed that for metallic glasses, T_g is related with the chemical composition through the thermal conductivity (Louzguine-Luzgin et al., 2008). Also, the inclusion of group IV impurities leads to an enhanced glass formation ability (Reyes-Retana and Naumis, 2014, 2015).

The thermal conductivity for low temperatures of diluted flexible systems has been studied under the harmonic approximation, revealing also anomalies (Romero-Arias et al., 2009). For higher temperatures, the problem requires the study of non-linear Hamiltonians, which is a topic that even in one dimensional periodic case has many open questions (Campbell et al., 2004).

In fact, the study of thermal relaxation started with the well known Fermi–Pasta–Ulam (FPU) problem (Fermi et al., 1965; Campbell et al., 2004). They tried to solve the paradox that from a pure theoretical point of view, harmonic solids can never reach energy equipartition since there is no sharing of energy between normal modes. Fermi–Pasta–Ulam proposed a simple model to understand the problem. The model is a linear chain with equal masses and non-linear springs (Fermi et al., 1965; Campbell et al., 2004). The dynamical equations were solved using one of the first computers. The results were difficult to understand at that time, since equipartition did not appear as expected. After years of research, it is now clear that relaxation depends upon low-frequency vibrational modes (Ford, 1961; Reigada et al., 2001; Ponno, 2005), due to their quasi-resonant nature (Ford, 1961; Cerón et al., 2005; Ponno, 2005). As a result, they share energy in an efficient way (Ponno, 2005) when compared with high-frequency modes.

The FPU model can be modified to include rigidity (Romero-Arias and Naumis, 2008) by adding second-neighbor non-linear bonds in the original FPU model (Romero-Arias and Naumis, 2008). Constraints bonds are added at random or in a periodic way.

The relaxation from an initial temperature T to zero temperature can be studied in the following two steps (Reigada et al., 2002): a thermal bath is applied until thermal equilibrium is reached, then the bath is retired and a damping term is added at both ends of the chains. For long times, relaxation is always slower when the number of LFVMs is reduced. The relaxation of high-frequency modes requires a transference of energy to LFVMs (Reigada et al., 2001). Such a phenomenon is akin to turbulence (Ponno, 2005), in which energy is injected at large scales, and transferred via a cascade of self-similar eddies to a small scale, where energy is finally dissipated.

The study of relaxation in two and three dimensional systems using non-linear terms is still an open field since even for the original FPU model the work is scarce.

One of the less studied problems is how rigidity theory enters inside the electronic properties picture. At first sight, all vibrational modes affects electron scattering, and thus their effects are hidden inside the thermal noise. However, after the discovery of graphene and other pure bi-dimensional materials (Geim and Novoselov, 2007), there is a growing interest in the subject. Graphene, which is a 2D crystal with a one-atom thickness, can be considered as a flexible membrane. In such case, floppy modes are called flexural modes, since they correspond to vibrations in a perpendicular direction to the graphene’s plane (Geim and Novoselov, 2007). Note that there are predictions concerning the possibility of finding amorphous graphene, which is proposed to be as a simple realization of Zachariasen’s glass (Kumar et al., 2012).

Let us consider a simple example of the effects of floppy modes. For simplicity, we focus on a system described with a one-orbital nearest-neighbor tight-binding Hamiltonian, as happens in graphene (Geim and Novoselov, 2007), (14)H=−∑x′,n tn′cx′†cx′+δn′+H.c., where x′ runs over all sites of the distorted lattice due to a floppy mode. Here, cx′† is the creation operator for an electron at site x′ and cx′+δn′ is the annihilation operator at site x′+δn′. The vectors δn′ point in the direction of each nearest-neighbor for a given atom. The nearest-neighbor hopping parameters tn′ are usually modified by deformations, since they depend on the inter-atomic distances. For carbon, they fulfill an exponential decay (Oliva-Leyva and Naumis, 2013), tn′=texp[−β(|δn′|∕a−1)], where t is the equilibrium hopping parameter.

Suppose the system is distorted by a floppy mode. Since bond lengths are not modified, the parameters tn′ are invariant (although in many systems, angular effects can induce changes depending of the orbital overlaps. For graphene, this effect is negligible for π orbitals). Under our approximation, the Hamiltonian matrix remains the same for floppy mode deformations. Thus, the eigenvalues are invariant under such distortion, and the electronic density of states also remains invariant. However, the electron’s speeds are modified. To see this in a simple way, consider a periodic system with an elastic deformation field u(x). The position of the atoms in the undeformed (x) and deformed lattices (x′) are related by, (15)x′=x+u(x) with an associated strain tensor, (16)∈¯ij=(∂iuj(x)+∂jui(x))∕2 that here we assume that is uniform, i.e., without a spatial dependence. Replacing the creation/annihilation operators with their Fourier expansions, the Hamiltonian in momentum space is (Oliva-Leyva and Naumis, 2013) (17)H=−∑k,ntn′e−ik⋅(I¯+ϵ¯)⋅δnck†ck+H.c., and therefore, the dispersion relation is, (18)E(k)=±∑ntn′e−ik⋅(I¯+ϵ¯)⋅δn. It is easy to see that the spectrum is the same as in the undeformed system, as expected from the eigenvalues of the Hamiltonian matrix. However, the dispersion relationship is distorted through a deformation of the reciprocal space. For strained or rippled graphene, this effect produces a deformation of the Dirac cone (Oliva-Leyva and Naumis, 2013).

Floppy modes in graphene are not only important for electron scattering. When boundaries or impurities appear, a fraction of the electronic modes have zero energy (with respect to the Fermi level). Such states are known as Dirac modes in graphene (Barrios-Vargas and Naumis, 2013), but appears in other bipartite lattices as confined states. For example, they have been observed in quasiperiodic systems (Naumis et al., 1994) and in random binary alloys (Naumis et al., 2002). Such modes can be described in terms of a mechanical instability when isostaticity appears (Kane and Lubensky, 2014). They are similar to the protected electronic boundary modes that occur in the quantum Hall effect and in topological insulators (Kane and Lubensky, 2014). Recently, Kane and Lubensky (2014) established the connection between the topological mechanical modes and the topological band theory of electronic systems, predicting new topological bulk mechanical phases.

These topological “floppy modes” are, in fact, behind the remarkable metal-insulator transition in doped graphene, which transforms graphene in a narrow-gap semiconductor with an enormous technological potential (Naumis, 2007; Barrios-Vargas and Naumis, 2011). They are also fundamental in graphene’s magnetic properties (Barrios-Vargas and Naumis, 2013).

Rigidity theory allows one to understand, via chemical modification, how low-frequency modes anomalies play an important role in glass transition. The Boson peak can also be included in the theory by considering bond dilution of over-constrained networks. When relaxation is combined with topology, stretched exponentials with magical β exponents are obtained. The ideas of rigidity can also be extended to study the electronic properties of solids, providing bridges between glasses and new topics in material science like graphene and quantum topological phases.

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.