# Quasi-critical fluctuations: a novel state of matter?

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## Abstract

Quasi-critical fluctuations occur close to critical points or close to continuous phase transitions. In three-dimensional systems, precision tuning is required to access the fluctuation regime. Lowering the dimensionality enhances the parameter space for quasi-critical fluctuations considerably. This enables one to make use of novel properties emerging in fluctuating systems, such as giant susceptibilities, Casimir forces or novel quasi-particle interactions. Examples are discussed ranging from simple metal–adsorbate systems to unconventional superconductivity in iron-based superconductors.

## Keywords

Phase transitions Fluctuations Low-dimensional systems Charge density wave High-*T*

_{c}superconductors

## Introduction

*T*

_{c}. The Landau theory of phase transitions allows a very succinct phenomenological description of such phase transitions. It uses an expansion of the Free Energy of the system in terms of powers of the order parameter:

*h*is an external field.

In field-free space, the expression contains only even powers of the order parameter, since the sign of the order parameter is irrelevant for the free energy. Consider for instance the order parameter of a ferromagnetic material, i.e. the local magnetisation. Obviously, the free energy depends only on the magnitude, not on the direction of the magnetisation. In the presence of an external field, however, the term given in square brackets in Eq. (1) has to be added to the expansion. It is linear in the order parameter, and hence changes sign, as either the order parameter or the external field are reversed.

Now we concentrate on the phase transition at \( T_{\text{c}} \) in the absence of an external field. At \( T_{\text{c}} \) both the first and the second derivative with respect to \( m \) of the free energy are zero for \( m = 0 \). As a consequence, the order parameter is allowed to fluctuate around zero. Thus, the system is in a quite exceptional state, it exhibits critical fluctuations. This implies that the local order parameter varies in time and space from point to point. However it does so in a very peculiar manner, since the fluctuations are correlated. As \( T \) approaches \( T_{\text{c}} \) the correlation length diverges. If the phase with \( m = 0 \) is denoted as phase 1 and the phase with \( m \ne 0 \) as phase 2, then a diverging correlation length means that there are domains of either phase in the system which momentarily span the whole system. Moreover, at every moment, phase 2 is nucleating within phase 1 and vice versa. Thus, at a given time, phase 2 domains of every size are found in phase 1 and phase 1 domains of every size in phase 2. Consequently at \( T_{\text{c}} \) the correlation length \( \zeta \) is the only characteristic length scale in the system. With \( \zeta \) diverging at \( T_{\text{c}} \), the system is said to be scale free. The proliferation of phase boundaries in this state causes strong light scattering, the so-called critical opalescence.

## Novel properties

Thus, a fluctuating system is exceptionally sensitive against external perturbations. Consequently a material which can be kept in such a fluctuating state lends itself to applications in switching and sensing devices. The problem is of course that normally a precise fine-tuning of the thermodynamic parameters is required to keep the system close to the critical condition.

A further interesting aspect of fluctuations is the Casimir effect. If, for instance, point-, line-, or planar defects are immersed into the fluctuating system, the boundary conditions at the defects enforce a change of the fluctuation spectrum. Accordingly, the energy density between the defects is altered which results in an attractive or repulsive interaction between the defects depending on the boundary conditions (Hertlein et al. 2008). This is a generalised Casimir effect, the analogue to the interaction resulting from a modification of the vacuum fluctuation spectrum between to dielectric bodies.

## Low-dimensional systems

*N*with exchange interaction

*J*< 0, so that the groundstate is ferromagnetic. Flipping one of the spins in the chain changes the free energy

*F*:

*N*positions the spin is flipped. Obviously, for

*T*> 0 the free energy is always lowered, provided that

*N*is large enough. Hence the correlation length in such a system with discrete symmetry and solely nn interactions diverges only for \( T \to 0 \) K. Similarly, in a 2D system with continuous symmetry, e.g. a Heisenberg spin array, the correlation length will also diverge only for \( T \to 0 \) K (Mermin and Wagner 1966). Accordingly, fluctuations will prevail in such materials down to 0 K. The broad range where fluctuations are expected to dominate the behaviour of quasi-1D systems is also illustrated by the calculations of Anderson and co-workers (Lee et al. 1973) who show that in their system, the correlation length diverges only at about 20 % of the critical temperature \( T_{\text{c}} \). The bottom line is that quasi-critical fluctuations are difficult to establish in a 3D material, but by engineering low-D materials it is possible to considerably enlarge the parameter space for such a fluctuating state and eventually harness the exotic properties associated with it for practical applications.

^{18 }m

^{−2}). Halogen adsorption lifts the (1 × 2)-missing-row reconstruction of the clean surface and at room temperature the Br forms a long-range-ordered c(2 × 2) structure as shown in Fig. 3 (Blum et al. 2002). This is a very common structure, because the quasi-hexagonal one packing the repulsive energy between the adatoms is minimised. Since the Pt(110) surface features close-packed atom rows with a nn distance of 0.277 nm, with the rows being separated by a lattice constant, i.e. 0.392 nm, the surface is strongly anisotropic. Actually, 1D electronic surface states are present and thus the surface may be considered as quasi-1D. Upon heating, a continuous disordering transition takes place.

*T*(2 × 1) structure an extra energy cost has to be spent on the inter-adsorbate repulsion and the distortion of the substrate. This extra energy, however, is over-compensated by an increased bonding strength of the Br to the substrate, as the latter offers more favourable bonding sites on the CDW maxima. Rising the temperature increases both, the lattice entropy and the electronic entropy in the substrate until the combined PLD/CDW ‘melts’. In other words, the periodic lattice and charge density modulation is increasingly blurred by thermal excitation of phonons and by excitation of electrons across the Peierls gap. As the PLD/CDW order parameter is thermally suppressed, the energy balance tilts in favour of the c(2 × 2) structure, since in the latter the inter-adsorbate repulsion is minimised. To substantiate this idea we investigate the individual free energy contributions in the system. The inter-adsorbate repulsion can be represented in an Ising-type model. To each of the adsorption sites \( i \) is assigned an occupation number \( s_{i} = \pm {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2} \), depending on whether it is occupied (+) or not (−). Since the Br–Br distance along the close-packed row direction (the [1 −1 0] direction) is the same in both structures, the difference in repulsive energies arises solely from the difference in the occupation of nn sites in the [001] direction. Accordingly, in a (minimal) 1D model the repulsive energy can be represented by a term \( \sum\limits_{i} {Js_{i} s_{i + 1} } \), where \( i \) counts the adsorption sites in [001] direction. For the c(2 × 2) structure, the occupation numbers change sign from place to place, and therefore the contribution to the free energy is negative. In the (2 × 1) structure in contrast, all \( s_{i} \) are positive and a positive contribution to the free energy is obtained. The substrate contribution is provided by the usual Landau expansion of the free energy in terms of powers of the order parameter, here the amplitude of the PLD/CDW. Finally we need to represent the adsorbate coupling to the substrate CDW. To this end we introduce a coupling term \( \sum\limits_{i} {gm\left( {s_{i} + s_{i + 1} } \right)} \). For \( g < 0 \) and \( m \ne 0 \) , this term lowers the total free energy, if the adsorbate forms a (2 × 1) phase, but it is zero in the c(2 × 2) phase and also, if \( m = 0 \), i.e. if the substrate PLD/CDW is suppressed. Thus, one obtains in total (ignoring a constant contribution \( F_{0} \) to the free energy):

The resulting free energy surface as a function of temperature and order parameter is discussed in detail in ref (Cordin et al. 2012). Obviously, on the flat substrate \( m = 0 \) and owing to the first term the free energy is minimised, if \( s_{i} = - s_{i + 1} \). This is the signature of the c(2 × 2) structure. If the substrate is buckled \( \left( {m \ne 0} \right) \), then the coupling term becomes effective. For strong enough coupling, it will outweigh the first term and thus favour the (2 × 1) structure. Note that the second term has the form of an external field contribution to the free energy (compare Eq. 1). Thus, it stabilises a finite value of the order parameter even for \( T > T_{\text{c}} \). Actually, \( T_{\text{c}} \) could even be negative implying that the PLD/CDW is unstable at all temperatures on the clean surface. Nevertheless, on the adsorbate covered surface an order parameter \( m \ne 0 \) could persist up to some finite temperature due to the stabilisation of the PLD/CDW as the adsorbate locks into the CDW fluctuations. The adsorbate freezes them into a (more or less) static PLD/CDW phase in a bootstrap type mechanism. The presence of the coupling term in Eq. (4) renders the phase transition weakly first order. The stability of the system then depends on the barrier in the free-energy surface at a given temperature. Actually, in the present system there are still c(2 × 2) ⟷ (2 × 1) fluctuations present at \( T = 50 \) K, albeit on a time scale of several seconds.

As pointed out before, such a type of phase transition is extremely rare in adsorbate systems. Obviously it requires an instability of the substrate to CDW fluctuations. For Pt, this is not too surprising, as a strong Kohn anomaly has been observed even in the bulk (Tsunoda 2011). But is there any significance beyond this system? The answer is yes.

## Application to iron-based superconductors

Very recently a discussion arose about the origin of surface structures observed on the cleavage planes of 122 Fe-based superconductors AFe_{2}As_{2}, where 122 refers to the chemical composition and A is an earth alkali metal (Ba, Sr, Ca) (Hoffman 2011). On these cleavage planes c(2 × 2) and (2 × 1) structures where observed, with the former prevailing after room temperature cleavage of BaFe_{2}As_{2} and SrFe_{2}As_{2}, while the latter was found after low-temperature cleavage. On CaFe_{2}As_{2} the (2 × 1) structure was stable also after room temperature cleavage. Long-range ordered domains were often found to co-exist with disordered areas. The results were similar for the undoped parent compounds and the doped superconductors, with a possible dependence of the relative stability of the two structures on doping.

The interpretation of the observed structures was subject to controversy. While some groups favoured an explanation in terms of earth alkaline metal adsorbate superstructures (Boyer et al. 2008; Yin et al. 2009; Hsieh et al. 2008; Massee et al. 2009; Zhang et al. 2010), other groups proposed reconstructions of the arsenic top-layer (Niestemski et al. 2009; Nascimento et al. 2009; Li et al. 2010). Here we take side with the adsorbate superstructure interpretation. This is motivated by (i) the striking similarity of STM topographs obtained on Br/Pt(110) as compared to results from AFe_{2}As_{2} cleavage planes [see Figs. 6a, 2h in ref (Niestemski et al. 2009)], (ii) the similar relative stability of the long-range ordered structures as a function of temperature and composition and (iii) similar structures in the Fermi surface mapping obtained by angle-resolved photoemission spectroscopy (ARPES).

Argument (i) is perhaps somewhat phenomenological, but it would be surprising, if a reconstruction of a bare surface would produce not only a similar contrast, but also identical domain structures of coexisting adsorbate phases with a coverage of 0.5 ML.

Argument (ii) deserves a more detailed consideration. The existence of two different long-range ordered structures depending solely on temperature is not trivial to explain, neither in the bare-surface, nor in the adsorbate-on-surface model. In both cases, the substrate has to co-operate in the phase transition delivering a substantial entropy contribution in the high-temperature structure. As pointed out above, the entropy difference could result from the ‘melting’ of a CDW in the substrate. This possibility has in fact been considered by Niestemski et al. (2009), but was discarded, because they did not observe a contrast inversion in the STM image, as the bias was reversed. This conclusion, however, is not justified, if the CDW is decorated by an adsorbate. Moreover, on a bare surface the melting of a CDW above the critical temperature should result in a (1 × 1) structure rather than a c(2 × 2) as it is found in the present case. Further support for the adsorbate-on surface hypothesis derives from theoretical work by Gao et al. (2010). Their ab initio DFT calculations showed the surface with 0.5 ML earth-alkaline metal coverage to be the energetically preferred one. As to the superstructure, the c(2 × 2)-A structure was found to be the most stable one for A = Ba, to be marginally stable for A = Sr, and to be unstable with respect to the (2 × 1)-A structure for A = Ca. Of course, DFT total energy calculations yield the groundstate for *T* = 0 K and as such are not able to predict or explain phase transitions. As judged from experiment, the equilibrium groundstate is more likely the (2 × 1) state for all three cases, but the trend in relative stability is apparently correctly represented in the DFT calculations (Cordin et al. 2012; Gao et al. 2010).

Our extended Landau model Eq. (4) yields an explanation for the phase transition as well as for the relative stability: since the first term in Eq. (4) representing the inter-adsorbate repulsion favours the c(2 × 2) structure and since the repulsive energy *J* is expected to be the largest for Ba and the smallest for Ca, the c(2 × 2) structure should indeed be the most stable for BaFe_{2}As_{2}, as predicted by Gao et al. (2010). The explanation of the different structures as a function of temperature in the present model is based on the assumption of a CDW instability in the substrates, viz. the AsFe_{2}As sandwiches.

_{2}As

_{2}compounds, at least not in many of the model band structures on which the analysis of antiferromagnetic correlations in the 122 compounds is usually based. Note that the ‘nesting’ vector underlying the CDW correlations postulated in the present model includes an angle of 45° with the nesting vector which is held responsible for the antiferromagnetic instability. Recent ARPES results, however, are at variance with the band topology at the surface Brillouin boundaries (SBZ) proposed in some simplified band structure models (Zabolotnyy et al. 2009; Kondo et al. 2010; de Jong et al. 2010). Points of high ARPES intensity and consequently high DOS are found at the Fermi level which resemble closely the ones seen in the Fermi surface map of Pt(110) shown in Fig. 6b (Cordin et al. 2012). Neither in Pt(110) nor in the 122 Fe arsenides do the connecting vectors precisely match half a reciprocal lattice vector as expected for a CDW of period 2. If this were the case, the result would presumably be a static CDW rather than CDW fluctuations. One should also be careful in applying nesting arguments too rigidly as pointed out by Mazin and coworkers (Johannes and Mazin 2008). Usually, the structures in the response function \( \chi \left( q \right) \) caused by Fermi surface nesting are not sharply peaked, since they result from integration over a finite energy interval around E

_{F}and in addition are weighted by the electron–phonon coupling.

The present model attributes the (2 × 1) phase to a CDW in the AsFe_{2}As sandwich layer which is stabilised in a bootstrap mechanism by the earth alkaline metal atoms. In the bulk compound, however, this mechanism cannot operate, since there is a full A layer separating the AsFe_{2}As sandwiches. Thus, instead of a static CDW, only charge density fluctuations are expected. The corresponding wave vector is oriented in the real space direction of the pairing interaction (Zhai et al. 2009). It is conceivable that a surface CDW stabilising the (2 × 1) structure could also originate from fluctuating orbital order (Kontani and Onari 2010) with \( {\mathbf{q}} = \left( {\pi ,0} \right) \) (Zhou et al. 2011).

## Conclusion

Tuning materials into a quasi-critical fluctuation regime offers the opportunity to make use of exotic properties associated with such a fluctuating state. Among these properties are a strong response to external perturbations and novel types of interactions, such as the Casimir force or Cooper pairing. The parameter range in which fluctuations persist is considerably enhanced in low-dimensional systems. Hence, precision tuning can be avoided, if low-dimensional systems are constructed. As an example, a metal–adsorbate system is analysed, in which charge-density fluctuations cause an unconventional phase transition. The phase transition is modelled in the spirit of Landau theory by expanding the free energy in terms of a CDW order parameter, but also adding terms representing the inter-adsorbate repulsion and the adsorbate–substrate coupling. Transferring the same model to the controversially discussed surface structures observed on 122 Fe-based superconductors a consistent explanation of both, the temperature dependence and the relative stability of the structures as a function of chemical composition is reached. As a consequence, it is suggested that CDW (or eventually orbital order) fluctuations are present in these compounds with a wave vector differing by an angular offset of 45° from that of the AFM–SDW fluctuations. On the one hand this underlines the pivotal role of fluctuations in unconventional superconductors, on the other hand it illustrates the rich phenomenology accessible by steering different order parameters into the fluctuation regime.

## Notes

### Acknowledgments

Financial support by the Austrian Science Fund is gratefully acknowledged.

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