continuous phase transitions can be characterised by an order parameter which changes from 
 zero above the critical temperature to a finite value below tc . 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:1\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ f\left ( { x , t } \right ) = f_{0 } \left ( { x , t } \right ) - \left [ 
 { hm\left ( { x , t } \right ) } \right ] + \alpha \left ( t \right)m^{2 } \left ( { x , t } \right ) + \lambda m^{4 } 
 \left ( { x , t } \right ) $ $ \end{document}f(x , t)=f0(x , t)[hm(x , t)]+(t)m2(x , t)+m4(x , t)here , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ f\left ( { x , t } \right ) $ $ \end{document}f(x , t ) is the free - energy density , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m\left ( { x , t } \right ) $ $ \end{document}m(x , t ) is the local order parameter , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \alpha \left ( t \right ) $ $ \end{document}(t ) is a coefficient , which depends linearly on \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t $ $ \end{document}t : \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \alpha \left ( t \right ) = \alpha_{0 } \left ( { t - t_{\text{c } } } 
 \right ) $ $ \end{document}(t)=0(ttc ) , and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \lambda > 0 $ $ \end{document}>0 is a constant , and h is an external 
 field . in field - free space , the expression contains only even powers of the order parameter , since 
 the sign of the order parameter is irrelevant for the free energy . consider for instance the order 
 parameter of a ferromagnetic material , i.e. the local magnetisation . obviously , the free energy 
 depends only on the magnitude , not on the direction of the magnetisation . in the presence of an 
 external field , however , the term given in square brackets in eq . it is linear in the order parameter , and hence 
 changes sign , as either the order parameter or the external field are reversed . now we concentrate on the phase transition at \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc in the absence of an external field . at \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc both the first and the second derivative with respect to \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m $ $ \end{document}m of the free energy are zero for \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m = 0 $ $ \end{document}m=0 . as a consequence , the order parameter is allowed to fluctuate around zero . thus , 
 the system is in a quite exceptional state , it exhibits critical fluctuations . this implies that the 
 local order parameter varies in time and space from point to point . however it does so in a very 
 peculiar manner , since the fluctuations are correlated . as \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t $ $ \end{document}t approaches \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc the correlation length diverges . if the phase with \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m = 0 $ $ \end{document}m=0 is denoted as phase 1 and the phase with \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m \ne 0 $ $ \end{document}m0 as phase 2 , then a diverging correlation length means that there are domains of 
 either phase in the system which momentarily span the whole system . moreover , at every moment , phase 
 2 is nucleating within phase 1 and vice versa . thus , at a given time , phase 2 domains of every size 
 are found in phase 1 and phase 1 domains of every size in phase 2 . consequently at \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc the correlation length \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \zeta $ $ \end{document} is the only characteristic length scale in the system . with \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \zeta $ $ \end{document} diverging at \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc , the system is said to be scale free . the proliferation of phase boundaries in 
 this state causes strong light scattering , the so - called critical opalescence . the fluctuation 
 dissipation theorem states that the material s response function \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \chi \left ( t \right ) $ $ \end{document}(t ) which characterises the response of a system to an external perturbation is 
 proportional to the fluctuations of the order parameter:2\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \chi \left ( { \omega = 0,t } \right ) = \frac{1}{{k_{b } t}}\left ( 
 { \left\langle { m^{2 } } \right\rangle - \left\langle m \right\rangle^{2 } } \right ) 
 $ $ \end{document}(=0,t)=1kbt(m2m2 ) 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 . 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 . this is a generalised casimir effect , the analogue to the interaction 
 resulting from a modification of the vacuum fluctuation spectrum between to dielectric 
 bodies . even more importantly , fluctuations are believed to be responsible for the cooper pairing 
 interactions in the unconventional superconductors , namely the cuprates and the iron - based 
 superconductors ( tru and kazuo 2003 ) . since in both 
 cases the undoped parent compounds are antiferromagnetic , it is not surprising that spin density 
 wave fluctuations are generally held responsible for the formation of the cooper pairs . figure  1 shows a cartoon of the interaction between charge 
 carriers with opposite spin in an antiferromagnetically fluctuating background ( monthoux et al . note that this type of interaction naturally 
 explains the d - wave symmetry of the pairing interaction experimentally found in the cuprates . one 
 should hasten to add that the subtle interactions governing the pairing and the formation of the 
 bose condensate require low temperatures . hence only critical fluctuations at a phase boundary which 
 in the phase diagram heads down to very low temperatures are relevant for superconductivity ( see 
 fig .  2 ) . if one extrapolates such a phase boundary through 
 the superconducting dome , it is seen to cross the zero - temperature axis . at this point the 
 fluctuations are obviously no longer thermally driven , but governed by quantum fluctuations . the 
 property of materials in the vicinity of such a quantum phase transition are not yet very well 
 understood ( sachdev 1999 ) . different scaling laws apply 
 and it is for instance not clear , at what temperature above the quantum critical point the signature 
 of quantum fluctuations disappears . it is often presumed that unconventional superconductivity is 
 not only a consequence of quasi - critical fluctuations of , let s say , the af order parameter , but 
 that it results from the system s reluctance to enter that bizarre quantum regime close to the 
 quantum critical point . instead the system tilts over into a new phase , the superconducting phase . 
 for the present purpose , it suffices to note that quasi - critical fluctuations along a phase boundary 
 which heads towards the zero temperature axis as a function of some experimental parameter ( doping , 
 pressure , external field , etc . ) 1schematic representation of spin - fluctuation mediated singlet pairing with d - wave 
 symmetry . orange ( blue ) sites 
 represent preference for spin up ( down ) in an antiferromagnetic 
 background . the interaction of a spin - up particle with a spin - down particle is attractive 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left ( { \left ( { \left ( { } \right ) } \right ) } \right ) 
 $ $ \end{document } ( ( ( ) ) ) , if the latter occupies a nearest - neighbour ( nn ) site , while it is repulsive 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left . { \left . { } \right ) } \right ) } \right)\left ( { \left ( 
 { \left ( { } \right . } $ $ \end{document } ) ) ) ( ( ( , if it occupies a nn site . 2schematic phase diagram of an unconventional superconductor schematic representation of spin - fluctuation mediated singlet pairing with d - wave 
 symmetry . orange ( blue ) sites 
 represent preference for spin up ( down ) in an antiferromagnetic 
 background . the interaction of a spin - up particle with a spin - down particle is attractive 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left ( { \left ( { \left ( { } \right ) } \right ) } \right ) 
 $ $ \end{document } ( ( ( ) ) ) , if the latter occupies a nearest - neighbour ( nn ) site , while it is repulsive 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left . { \left . { } \right ) } \right ) } \right)\left ( { \left ( 
 { \left ( { } \right . } $ $ \end{document } ) ) ) ( ( ( , if it occupies a nn site . as mentioned above , establishing quasi - critical fluctuations requires in general a precise 
 fine - tuning to near a continuous phase transition or a critical point . in the following discussion , 
 we concentrate on the temperature as the tuning parameter . in 3d , systems fluctuations are in 
 general rapidly suppressed as the temperature deviates only by a few tenths of a degree from 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc . this problem can be solved by constructing low - dimensional systems . as the 
 dimensionality is lowered , the temperature regime where appreciable fluctuations occur broadens 
 considerably . consider for instance a one - dimensional ( 1d ) ising chain of length 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:3\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \updelta f = \updelta u - t\updelta s = \left| j \right| - k_{b } 
 t\ln n $ $ \end{document}f=uts=|j|kbtlnnwith \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left| j \right| $ $ \end{document}|j| being the energy cost for the spin misalignment and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ k_{b } \ln n $ $ \end{document}kblnn the configurational entropy arising from the free choice on which of the n positions the spin is flipped . obviously , for t  >  0 the free energy is always lowered , provided that n is large enough . hence the correlation length in such a system with discrete symmetry 
 and solely nn interactions diverges only for \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t \to 0 $ $ \end{document}t0  k. similarly , in a 2d system with continuous symmetry , e.g. a heisenberg spin 
 array , the correlation length will also diverge only for \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t \to 0 $ $ \end{document}t0  k ( mermin and wagner 1966 ) . accordingly , fluctuations will prevail in such materials down to 0  k. the broad range where 
 fluctuations are expected to dominate the behaviour of quasi-1d systems is also illustrated by the 
 calculations of anderson and co - workers ( lee et al . 1973 ) who show that in their system , the correlation length diverges only at about 
 20  % of the critical temperature \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc . 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 . as a practical example we first consider a simple 2d system , i.e. an adsorbate on a transition 
 metal . the model system chosen here is half a monolayer ( ml ) of bromine on pt(110 ) . one ml is 
 defined here as the surface atom density of the unreconstructed pt(110 ) surface 
 ( 9.2    10 m ) . 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 . 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.fig . 3the c(2    2)-br / pt(110 ) surface : ball model and stm topographic image ( 3.8    3.8  nm ) 
 ( blum et al . ( color figure online ) the c(2    2)-br / pt(110 ) surface : ball model and stm topographic image ( 3.8    3.8  nm ) 
 ( blum et al . 2002 ) . ( color figure online ) the phase transition can be monitored by analysing the spot profile in low - energy electron 
 diffraction ( leed ) . for a long - range - ordered system , the spot profile is a gaussian and the height 
 of the gaussian serves as a measure of the order parameter . as shown in fig . 4a , the gaussian peak height drops precipitously as the disordering 
 temperature at ~370  k is reached indicating the loss of long - range order . as a result of defect 
 pinning at monatomic steps of the pt(110 ) close to the transition temperature the peak profile is not a pure gaussian . due to the local fluctuating order a lorentzian component appears and this is shown together with the 
 correlation length derived from the width of the lorentzian in fig . 4b and c. from the temperature dependence of the lorentzian peak height one can 
 conclude that fluctuations in the system are prominent within a range of about 100  k around 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc . 4spot profile analysis of the half - order ( red ) and an 
 integer - order ( green ) leed spot for the c(2    2)-br / pt(110 ) 
 structure . the profile has been recorded in the   direction perpendicular to the close - packed pt 
 atom rows . a gaussian component of the spot profile which is a 
 measure of the order parameter ( long - range order ) . insert 
 theoretical temperature dependence of order parameter near t 
 c. b lorentzian component which is a measure 
 of the short - range order in a fluctuating system . insert 
 monte - carlo calculation of the lorentzian amplitude at t 
 c in a finite 2d ising system . c correlation 
 length of the fluctuations derived from the width of the lorentzian component . ( color figure 
 online ) spot profile analysis of the half - order ( red ) and an 
 integer - order ( green ) leed spot for the c(2    2)-br / pt(110 ) 
 structure . the profile has been recorded in the   direction perpendicular to the close - packed pt 
 atom rows . a gaussian component of the spot profile which is a 
 measure of the order parameter ( long - range order ) . insert 
 theoretical temperature dependence of order parameter near t 
 c. b lorentzian component which is a measure 
 of the short - range order in a fluctuating system . insert 
 monte - carlo calculation of the lorentzian amplitude at t 
 c in a finite 2d ising system . c correlation 
 length of the fluctuations derived from the width of the lorentzian component . this is surprising , since cooling of an ordered structure usually 
 tends to improve the order . scanning tunnelling microscopy ( stm ) topographic images recorded at 50  k 
 ( fig . 5a , b ) reveal why order is lost : another phase 
 appears , with br atoms being arranged in a ( 2    1 ) unit cell . according to our dft calculations this 
 is the actual groundstate of the system in agreement with its appearance upon lowering the 
 temperature . it is somewhat unexpected that the ( 2    1 ) structure is most stable , since it should 
 exhibit a higher inter - adsorbate repulsive energy as compared to the c(2    2 ) . instead , heating the 
 sample is needed for rearranging the br atoms into the latter structure with extremely 
 well - developed long - range order at room temperature . the presence of a second long - range - ordered 
 structure for the same coverage at higher temperature is extremely rare in any adsorbate system . it 
 requires this structure to have a much higher entropy than the groundstate . since the long - range 
 order excludes a substantial contribution from configurational entropy , the entropy gain in the 
 adsorbate layer would have to arise from vibrational entropy . however the local bonding site is the 
 same in both structures , so the vibrational entropy difference can not explain this order the solution to the problem lies in the substrate contribution ( cordin et al . while the pt(110 ) surface is flat and essentially 
 bulk truncated , there is a pronounced buckling present in the ( 1    2)-br / pt(110 ) surface ( cordin et 
 al . this periodic lattice distortion ( pld ) is associated with a periodic charge 
 modulation ( cdw ) in the surface . the br rows decorate the charge density maxima of the cdw.fig .  5phase transition c(2    2 )    ( 2    1 ) occurring upon cooling . a the long - range ordered c(2    2 ) structure observed at room temperature decays into a 
 striped pattern of bright and dark domains . b a close - up reveals 
 the bright domains to be formed by c(2    2 ) and the dark domains by ( 2    1 ) order . c ball model of the ( 2    1 ) structure : grey 
 balls are substrate pt atoms , yellow balls br atoms . ( color figure online ) phase transition c(2    2 )    ( 2    1 ) occurring upon cooling . a the long - range ordered c(2    2 ) structure observed at room temperature decays into a 
 striped pattern of bright and dark domains . b a close - up reveals 
 the bright domains to be formed by c(2    2 ) and the dark domains by ( 2    1 ) order . c ball model of the ( 2    1 ) structure : grey 
 balls are substrate pt atoms , yellow balls br atoms . ( color figure online ) qualitatively , the observed phase transition can therefore be described as follows : to 
 establish the low - 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 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ i $ $ \end{document}i is assigned an occupation number \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ s_{i } = \pm { 1 \mathord{\left/ { \vphantom { 1 2 } } \right . \kern-0pt } 
 2 } $ $ \end{document}si=1/122 , depending on whether it is occupied ( + ) or not (  ) . since the br br distance 
 along the close - packed row direction ( the [ 1 1 0 ] direction ) is the same in both structures , the 
 difference in repulsive energies arises solely from the difference in the occupation of nn sites in 
 the   direction . accordingly , in a ( minimal ) 1d model the repulsive energy can be represented by 
 a term \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \sum\limits_{i } { js_{i } s_{i + 1 } } $ $ \end{document}ijsisi+1 , where \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ i $ $ \end{document}i counts the adsorption sites in   direction . for the c(2 2 ) structure , the 
 occupation numbers change sign from place to place , and therefore the contribution to the free 
 energy is negative . in the ( 2    1 ) structure in contrast , all \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ s_{i } $ $ \end{document}si 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 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \sum\limits_{i } { gm\left ( { s_{i } + s_{i + 1 } } \right ) } 
 $ $ \end{document}igm(si+si+1 ) . for \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ g < 0 $ $ \end{document}g<0 and \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m \ne 0 $ $ \end{document}m0 , this term lowers the total free energy , if the adsorbate forms a ( 2    1 ) phase , 
 but it is zero in the c(2    2 ) phase and also , if \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m = 0 $ $ \end{document}m=0 , i.e. if the substrate pld / cdw is suppressed . thus , one obtains in total ( ignoring 
 a constant contribution \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ f_{0 } $ $ \end{document}f0 to the free energy):4\documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ f = \sum\limits_{i } { js_{i } s_{i + 1 } + \sum\limits_{i } { gm\left ( 
 { s_{i } + s_{i + 1 } } \right ) } } + \alpha \left ( t \right)m^{2 } + \lambda m^{4 } $ $ \end{document}f=ijsisi+1+igm(si+si+1)+(t)m2+m4 the resulting free energy surface as a function of temperature and order parameter is 
 discussed in detail in ref ( cordin et al . on the flat substrate \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m = 0 $ $ \end{document}m=0 and owing to the first term the free energy is minimised , if \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ s_{i } = - s_{i + 1 } $ $ \end{document}si=si+1 . if the substrate is buckled 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \left ( { m \ne 0 } \right ) $ $ \end{document}(m0 ) , then the coupling term becomes effective . for strong enough coupling , it will 
 outweigh the first term and thus favour the ( 2    1 ) structure . note that the second term has the 
 form of an external field contribution to the free energy ( compare eq .  1 ) . thus , it stabilises a finite value of the order parameter even for 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t > t_{\text{c } } $ $ \end{document}t > tc . actually , \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t_{\text{c } } $ $ \end{document}tc could even be negative implying that the pld / cdw is unstable at all temperatures 
 on the clean surface . nevertheless , on the adsorbate covered surface an order parameter 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ m \ne 0 $ $ \end{document}m0 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 
 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 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ t = 50 $ $ \end{document}t=50  k , albeit on a time scale of several seconds . as pointed out before , this is not too 
 surprising , as a strong kohn anomaly has been observed even in the bulk ( tsunoda 2011 ) . very recently a discussion arose about the origin of surface structures observed on the 
 cleavage planes of 122 fe - based superconductors 
 afe2as2 , 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 
 bafe2as2 and 
 srfe2as2 , while the latter was found after 
 low - temperature cleavage . on cafe2as2 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 . 2010 ) , other groups proposed reconstructions of the arsenic top - layer ( niestemski et 
 al . this is motivated by ( i ) the 
 striking similarity of stm topographs obtained on br / pt(110 ) as compared to results from 
 afe2as2 cleavage planes [ see figs . 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 . ( 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 . 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 . ( 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 bafe2as2 , 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 asfe2as sandwiches . this leads to argument ( iii ) from above : in pt(110 ) , the cdw instability was attributed to a 
  nesting vector between two points of high density - of - states ( dos ) at the fermi level ( see 
 fig . a similar nesting condition is not anticipated for 
 the afe2as2 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 . 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 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ \chi \left ( q \right ) $ $ \end{document}(q ) caused by fermi surface nesting are not sharply peaked , since they result from 
 integration over a finite energy interval around ef and in addition are 
 weighted by the electron 6 
 a coexistence of c(2    2 ) and ( 2    1 ) domains showing a striking 
 similarity to surface structures observed on srfe2as2 
 ( niestemski et al . 2009 ) . b fermi surface map of pt(110 ) recorded in angle - resolved photoemission spectroscopy . the orange arrow indicates a wave vector 
 q of excitations between two points of high density of states at 
 the fermi level . ( color figure online ) 
 a coexistence of c(2    2 ) and ( 2    1 ) domains showing a striking 
 similarity to surface structures observed on srfe2as2 
 ( niestemski et al . b fermi surface map of pt(110 ) recorded in angle - resolved photoemission spectroscopy . the orange arrow indicates a wave vector 
 q of excitations between two points of high density of states at 
 the fermi level . ( color figure online ) the present model attributes the ( 2    1 ) phase to a cdw in the asfe2as 
 sandwich layer which is stabilised in a bootstrap mechanism by the earth alkaline metal atoms . in 
 the bulk compound , however , this mechanism can not operate , since there is a full a layer separating 
 the asfe2as 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 . it is 
 conceivable that a surface cdw stabilising the ( 2    1 ) structure could also originate from 
 fluctuating orbital order ( kontani and onari 2010 ) with 
 \documentclass[12pt]{minimal } 
				 \usepackage{amsmath } 
				 \usepackage{wasysym } 
				 \usepackage{amsfonts } 
				 \usepackage{amssymb } 
				 \usepackage{amsbsy } 
				 \usepackage{mathrsfs } 
				 \usepackage{upgreek } 
				 \setlength{\oddsidemargin}{-69pt } 
				 \begin{document}$$ { \mathbf{q } } = \left ( { \pi , 0 } \right ) $ $ \end{document}q=(,0 ) ( zhou et al . 2011 ) . 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 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 .
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 .