Critical exponents and scaling invariance in the absence of a critical point

Abstract

The paramagnetic-to-ferromagnetic phase transition is classified as a critical phenomenon due to the power-law behaviour shown by thermodynamic observables when the Curie point is approached. Here we report the observation of such a behaviour over extraordinarily many decades of suitable scaling variables in ultrathin Fe films, for certain ranges of temperature T and applied field B. This despite the fact that the underlying critical point is practically unreachable because protected by a phase with a modulated domain structure, induced by the dipole-dipole interaction. The modulated structure has a well-defined spatial period and is realized in a portion of the (T, B) plane that extends above the putative critical temperature, where thermodynamic quantities do not display any singularity. Our results imply that scaling behaviour of macroscopic observables is compatible with an avoided critical point.

ARTICLE

Received 22 Apr 2016 | Accepted 18 Oct 2016 | Published 5 Dec 2016

The paramagnetic-to-ferromagnetic phase transition is classied as a critical phenomenon due to the power-law behaviour shown by thermodynamic observables when the Curie point is approached. Here we report the observation of such a behaviour over extraordinarily many decades of suitable scaling variables in ultrathin Fe lms, for certain ranges of temperature T and applied eld B. This despite the fact that the underlying critical point is practically unreachable because protected by a phase with a modulated domain structure, induced by the dipoledipole interaction. The modulated structure has a well-dened spatial period and is realized in a portion of the (T, B) plane that extends above the putative critical temperature, where thermodynamic quantities do not display any singularity. Our results imply that scaling behaviour of macroscopic observables is compatible with an avoided critical point.

DOI: 10.1038/ncomms13611 OPEN

Critical exponents and scaling invariance in the absence of a critical point

N. Saratz1, D.A. Zanin1, U. Ramsperger1, S. Cannas2, D. Pescia3 & A. Vindigni1

1 Laboratorium fr Festkrperphysik, Eidgenssische Technische Hochschule Zrich, CH-8093 Zrich, Switzerland. 2 Facultad de Matemtica, Astronoma y Fsica y Computacion, Universidad Nacional de Crdoba, Instituto de Fsica Enrique Gaviola (IFEG-CONICET), Ciudad Universitaria, 5000 Crdoba, Argentina. 3 Laboratorium fr Festkrperphysik, Eidgenssische Technische Hochschule Zrich, and SIMDALEE2 Sources, Interaction with Matter, Detection and Analysis of Low Energy Electrons 2, Marie Sklodowska Curie FP7-PEOPLE-2013-ITN, CH-8093 Zrich, Switzerland. Correspondence and requests for materials should be addressed to A.V. (email: mailto:[email protected]

Web End [email protected] ).

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Magnetism, superconductivity and multiferroicity stand as examples of technologically relevant properties of matter resulting from a phase transition1,2. The

conventional understanding of ferromagnetism is based on models of collective order in which atomic magnetic moments interact via one single leadingtypically short-rangedexchange interaction36. Yet, realistic ferromagnets suffer from the unavoidable presence of dipoledipole interaction that, albeit weak, is long-ranged and frustrates the tendency to align magnetic moments parallel to each other promoted by the exchange interaction710. Probably, the most well-know consequence of this competition is the formation of magnetic domains, that is, a phase with modulated magnetization at low temperature. In the early 1980s, Wasilevsky studied the loss of modulated magnetic order with increasing temperature for an idealized sample geometry and within the mean-eld (MF) approximation. He showed that the uctuations of the local magnetization above the MF Curie temperature TMFc are also modulated and their spatial period matches exactly the width of domains entering the sample below TcMF. With his own words: ...

we can conclude that even in the paramagnetic phase [above TMFc]... the demagnetizing eld induces a domainization effect of the uctuations of magnetization11,12. It is important to point out that this domainization effect produced by the dipolar eld is not to be confused with the droplets that dene the spatial extent of statistical uctuations by means of the correlation length in the vicinity of an ordinary critical point3. In the prevailing understanding of ferromagnetismand of second-order phase transitions in generalbased on renormalization group, the divergence of the correlation length lies behind scaling behaviour and universality of critical exponents36. Within the theoretical scenario depicted by Wasilevsky the critical point of a ferro-magnet is replaced by a low-temperature phase with modulated magnetization that persists above TMFc. In particular, the spatial period of the modulated phase provides, alongside the correlation length, a second long length scale that transforms the critical point into a so-called avoided critical point. The more general theoretical framework of avoided criticality was developed in the 1990s (refs 1315) and foresees that even a small amount of frustration changes the critical point of an unfrustrated model of collective order into a completely different objectgenerally not a second-order transition pointdepending on details like the spatial range and the strength of the competing interactions involved, the dimensionality of the system and the number of components of the order parameter1622. A non-exhaustive list of pattern-forming systems for which the phenomenology of avoided criticality has been proposed comprises magnetic lms, ferrouids, diblock copolymers, amphiphilic solutions, systems undergoing Turing-like phase separating chemical reactions and charged stripes in cuprate high-Tc superconductors2226.

Here we report two novel aspects of the ferromagnetic phase transition, observed in perpendicularly magnetized ultrathin lms of Fe grown on Cu(001). First, partially in line Wasilevskys hypothesis11,12, we found that the width of stripe domains observed at low temperature remains nite at a presumed Tcnot simply dened as the MF critical temperature introduced by Wasilevskyand evolves continuously with temperature, up to at least 30 beyond Tc (that is, 10% of Tc itself), without displaying any singularity. (Note that ref. 22 predicts a divergence of the modulation length at some temperature.) This appears to represent a kind of conservation law for the spatial period of modulation in pattern-forming systems. Furthermore, it implies that the divergence of the susceptibility as a function of temperature expected for a standard ferromagnet at Tc is

replaced by an analytic behaviour. Second, we observe that the conventional critical point itself is indeed eliminated by the

presence of domains in line with the avoided-criticality scenario. However, the most distinctive feature of a second-order phase transition, namely the conventional scaling la Kadanoff36, is recovered in a temperature and magnetic-eld range sufciently away from the presumed critical point, over up to 80 orders of magnitude of the suitable scaling variable. In this respectbut only in this respectthe system behaves as if the critical point existed.

ResultsThe model Hamiltonian. Since the most relevant experimental outcomes of this work result from the analysis of the global magnetization as a function of the applied magnetic eld B and temperature T, it is convenient to dene from the outset the theoretical framework underlying this analysis. We consider the Hamiltonian27,28

H J X

hi;ji

SiSj g X

i;j

raij b X

i

SiSj

Si 1

where Si 1 are Ising spin variables disposed on a

two-dimensional (2D) lattice and associated with the two outof-plane directions along which magnetic moments preferentially point. The rst term on the right-hand side of equation (1) represents the exchange interaction (the sum runs over all pairs of nearest-neighbouring sites), while the second term represents a long-ranged frustrating interaction decaying with a power a of the distance between spin pairs. The third term represents the Zeeman energy with b m B, m being the atomic magnetic

moment. The equilibrium magnetization is dened as m

P

i

hSii=N, N being the total number of spins and hSii the

statistical average.

Let us rst recall the essential aspects of the unfrustrated model (g 0). The familiar description of the ferromagnetic phase

transition foresees that m(T, b) develops a discontinuity at b 0

below the Curie temperature, which denes a situation of spontaneously broken symmetry1 (black curves in Fig. 1a). Above the Curie temperature m(T, b) shows no discontinuity and behaves linearly for small enough elds (red curve in Fig. 1a). This behaviour is typically predicted by models in which ferromagnetic order is stabilized by a short-ranged exchange interaction, favouring parallel alignment of neighbouring spins36. In the absence of eld (b 0), congurations obtained

by ipping all the spins have the same energy, that is, the Hamiltonian possesses a Z2 symmetry (the symmetry elements being the identity and the simultaneous change of sign of all variables). The minimal energy is realized by the two congurations with all the spin pointing along the same direction (Fig. 1b). Passing from one of such congurations to the other requires the creation of a domain wall. For nite temperature and nite elds, the discontinuity obtained for ToTc specically reects the spontaneous breaking of the Z2 symmetry3,29. In the neighbourhood of the transition point (T, b) (Tc,0) observables behave critically. One implication of

criticality is that the equation of state m as a function of (T, b) has a scaling representation30,31:

m

jtj b

g

2

with g(x) being scaling functions, t (TTc)/Tc the reduced

temperature (g and g referring to positive and negative values,

respectively, of t)31. b and d are two critical exponents characteristic of a specic universality class, the 2D-Ising universality class for the model dened by Hamiltonian (1) with g 0. Another implication of criticality is that m(T, b) and

the magnetic susceptibility wT; b

@m@b j

b0as well as other

b

jtj bd

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Figure 1 | Properties of the model Hamiltonian. (a) Sketch of some equilibrium magnetization curves m(T, b) as a function of the applied eld b theoretically expected for an unfrustrated ferromagnet below Tc (black lines). The equilibrium curves are described coming from large elds, as indicated by arrows. Note the jump when b 0 is crossed for ToTc, highlighted by dots. A pictorial curve representing the paramagnetic behaviour, T4Tc, is also

displayed (red line). (b) Two ground-state congurations (linked by the Z2 symmetry) of the model dened by Hamiltonian (1) in the absence (g 0) and

(c) in the presence (ga0) of frustration. (d) Sketch of some equilibrium magnetization curves m as a function of the applied eld b for a frustrated system at different temperatures (blue lines). The discontinuity expected in the unfrustrated case at b 0 gives way to a linear behaviour, as a result of the

frustrating effect of dipolar interaction. The red line is the same as in the plot a.

observables not considered in this studybecome power-law functions of t and b while approaching the critical point.

In the presence of frustration (ga0), for ar3 the conguration with lowest energy is not the ferromagnetic phase but rather a modulated phase, typically striped32,33 (Fig. 1c). In this case, congurations obtained by a rigid translation of the whole striped pattern have the same energy. Concrete realizations of this model involve either Coulomb repulsion (a 1) or dipole

dipole antiferromagnetic interaction (a 3) that competes with a

ferromagnetic short-ranged interaction22. This competition leads to the formation of a striped ground state whose elementary excitations are described by an elastic-like Hamiltonian associated with the displacement of domain wallsas a result of the subtle interplay between the two interactions17,21. The experimental results presented in the next subsections refer to a ferromagnetic model system representative of the Hamiltonian (1) with a 3

(refs 18,19,27,3336). Some unconventional behaviours observed in these ferromagnetic lms, such as the systematic occurrence of inverse-symmetry-breaking transitions of magnetic-domain patterns, have been explained theoretically starting from the Hamiltonian (1)28,37. In the limit of Jcg, relevant for the experimental system, the spectrum of the elastic-like excitations mentioned above is gapless. In particular, a zero-energy mode (Goldstone mode) connects any pairs of stripe patterns related by a Z2 symmetry operation (Fig. 1c). Therefore, from the perspective of elementary excitations, the frustrated model is more akin to systems with continuous symmetry rather than to the unfrustrated Ising model with discrete Z2 symmetrywhose elementary excitations are domain walls with a nite energy. These differences suggest that the loss of the magnetic order realized in the ground state proceeds at nite temperature in a completely different way with respect to the unfrustrated case: a phase with nite spontaneous magnetization (m(T, b 0)a0) is

never realized at thermodynamic equilibrium and the system passes from a phase with domain patterns and global vanishing magnetization to the paramagnetic phase without showing any singularity related to the breaking of the discrete Z2 symmetry. As

a function of b, m does not develop any discontinuity, neither below nor above the transition temperature (Fig. 1d). The label avoided criticality is used to underline these major differences with respect to the unfrustrated model.

The low-temperature patterned phase has been investigated theoretically at different levels of approximation and focusing on aspects that range from self-generated glassiness14,15,20,22 to topological phase transitions1719,27,3336. All the literature we are aware of supports the absence of long-range positional order of domains at nite temperature17,18,21, thus suggesting that not even a kind of staggered magnetization associated with the striped pattern is expected to display the 2D-Ising critical behaviour (nor the MF critical behaviour foreseen by Wasilevsky11,12). In the prevailing understanding of critical phenomena the fullment of equation (2) and other scaling relations is associated directly with spontaneous breaking of a specic symmetry of the Hamiltonian in b 0. Since the arguments given above exclude

spontaneous breaking of the Z2 symmetry in the frustrated model, one should not expect to observe scaling relations with Ising critical exponents when ga0. With the present study we, instead, demonstrate that the critical scaling laws obeyed by the unfrustrated model are actually recovered in the frustrated case sufciently away from a putative critical point: numerical and experimental evidence supporting this statement is provided in the following.

Monte Carlo simulations. We performed a series of Monte Carlo simulations using the Hamiltonian (1) with a 3 (see the

subsection Technical details of Monte Carlo simulations in Methods). Several theoretical works investigated the modulated phases of this model27,3336. In this study we explored the transition between those modulated phases and the phase with uniform magnetization, specically searching for a region of the (T, b) plane, where the equations of state of ferromagnets were obeyed. The resulting magnetization curves are plotted in Fig. 2a and their scaling representation (2) in Fig. 2b. In Fig. 2b the upper

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Figure 2 | Monte Carlo simulations. (a) A family of m(T) isochamps obtained from Monte Carlo simulations for J/g 10 (kB 1 is assumed), Lx Ly 120

and different values of the Zeeman energy b m B. (b) Scaling plot m/|t|b versus b/|t|bd for the data shown in a using the 2D-Ising critical exponents:

collapsing is realized only when for b4bc (see the subsection Technical details of Monte Carlo simulations in Methods). Inset: the characteristic temperature Tc, used to optimize data collapsing in the Monte Carlo scaling plots, as a function of J/g. (c) Representative snapshots of Monte Carlo congurations obtained for Lx Ly 200, J/g 10, b/g 0.1 and T/g 10, 13, 16 (from left to right). (d) Sequence of selected SEMPA images recorded on

an Fe lm of 1.9 MLs deposited on Cu(001) while increasing the temperature from 315.3 to 332.2 K in a constant eld B 0.61 10 4 T. The length of the

bar is 10 mm.

compelling from the Monte Carlo simulations. For convenience, we draw it in line with the experimental results presented in the subsection Experimental phase diagram. For small enough elds magnetic lms are expected to be in the modulated phase, in which ferromagnetic scaling should not be obeyed anymore (dark grey in Fig. 3a). Instead, the relation between the magnetization and the applied eld B should be a linear one, that is, the graph of the experimental magnetization as a function of magnetic eld at nite temperatures should be qualitatively similar to the ones sketched in Fig. 1d, rather than the ones obtained in the absence of frustration (Fig. 1a). Further information about the expected phase diagram are sketched in Fig. 3a on the basis of previous experimental studies (see the subsection on Experimental scaling plots).

Experimental scaling plots. The ultrathin Fe lms investigated in this work are grown at room temperature by molecular-beam epitaxy onto the (001) surface of a Cu single crystal (see refs 3840 for details). The samples extend macroscopically along the directions dening the lm plane and have a thickness between 1.6 and 2.0 atomic monolayers (MLs). For this range of thickness these lms possess an out-of-plane magnetocrystalline anisotropystrong enough to overcome the shape anisotropy so that magnetic moments preferentially point perpendicularly to the plane41. In this sense, these lms represent an experimental counterpart of Hamiltonian (1) with a 3. Typically, modulated

phases in similar experimental systems consist of stripes and/or bubble domains of opposite magnetization4244 (some representative images are shown in Fig. 2d and later (Fig. 6af)). In previous studies magnetic imaging of these lms was performed with scanning electron microscopy with polarization analysis (SEMPA)39,40,45, see the subsection SEMPA imaging in Methods and Fig. 3b. SEMPA revealed, on one side, that above a temperature T* (sketched in Fig. 3a) domains become mobile4447. On the other side, below T* static domains were

branch contains the Monte Carlo data for negative values of t, the lower branch those for positive values of t (the two branches meet at large values of the variable on the horizontal axis). Note that the resulting Tc as a function of J/g can be expressed as the transition temperature of the 2D-Ising model without frustration, TIsingc 2J=ln 1 2

p

(assuming kB 1), subtracted by a value

which corresponds to about 11.2g (inset of Fig. 2b). This suggests that in the limit of vanishing frustration (g-0), the temperature Tc tends to TIsingc. Collapsing could be indeed realized for b4bc (coloured points in Fig. 2b), bc being a threshold eld above which the system is in a uniformly magnetized phase (see the subsection Technical details of Monte Carlo simulations in Methods). Signicant departure from collapsing was, instead, observed for bobc (open and full grey points corresponding to b/

g 0.1 and 0.2), that is, inside the region of the (T, b) plane where

a phase with modulated magnetization appears. Snapshots of representative Monte Carlo congurations computed for b/g 0.1 are given in Fig. 2c, along with selected magnetic

congurations observed in the experimental system (Fig. 2d). Note that the magnetization curves obtained for b4bc display qualitatively the same spreading with increasing elds expected for the unfrustrated model. The picture emerging from Fig. 2a,b underlines that scaling and power laws of the Ising type are also compatible with a scenario in which the Z2 symmetry is not spontaneously broken in the magnetization of any sublattice. Henceforth, we will use the wording ferromagnetic scaling region to indicate the region of the (T, b) parameter space in which equations of state of standard ferromagnets are obeyed even if a Curie transition point (Tc, b 0), in the conventional

understanding, is not realized.

We now summarize the scenario emerging from the Monte Carlo simulations. For large enough magnetic elds, the magnetization as a function of T and B should obey a scaling relation such as the one given in equation (2). This is the ferromagnetic scaling region, delimited schematically by a dashed violet line in Fig. 3a. The shape of the dashed line is not

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a c

b

M 2

n

g

r

e

g

i

o

n

B

Re-entrance 1

SEMPA MOKE

Laser beam

Secondary electrons

T *

Tc

T

Figure 3 | Illustration of experimental methods. (a) Phase diagram of a frustrated perpendicularly magnetized lm in the (T, B) parameter space. Dashed violet line and green line, the points 1 and 2 and the horizontal red and blue lines are explained in the main text. The portion of the (T, B) plane in which the scaling equations of state of ferromagnets are obeyed is indicated as ferromagnetic scaling region. (b) In SEMPA, the spatially focussed primary electron beam is scanned across the sample carrying magnetic domains (dark and bright stripes). The secondary electrons ejected from the surface (green) are spin polarized and are used for imaging the domains. (c) The MOKE experiment averages over all domains within the focus of the light beam (the violet circle, about 1 mm2). Measuring with MOKE the magnetization as a function of Talong the blue and red lines (a) produces the M(T) graphs in the inset given with the same colours.

observed that transformed into a uniform state on cooling (point 2 along the re-entrance red horizontal line in Fig. 3a). To complement the information about magnetic phases deduced from SEMPA imaging, the present work adds the detection of the spatially averaged magnetization as a function of B and T (Fig. 3c), measured using the magneto-optical Kerr effect (MOKE)48 (A full magnetization curve at constant T can be recorded with MOKE within seconds or less, while the time required to scan a SEMPA image is in the range of minutes.) We anticipate that analysing the MOKE data we were able to determine experimentally the line sketched as dashed violet in Fig. 3a and to locate the putative critical point at (T, B) (Tc, 0)

in spite of the fact that it fell in the region occupied by domains (Fig. 3a). Technically speaking, MOKE (Fig. 3c) measures a signal that is proportional to the spatially averaged magnetization within the size of the light beam (typically about 1 mm2)48 and is given in arbitrary units: we denote the measured quantity with the letter M. From now on, all quantities involving M must be considered in arbitrary units. On the basis of our Monte Carlo simulations, we expect two types of M(T) curves, depending on whether we vary the temperature in a situation of strong (dark blue horizontal line in Fig. 3a) or weak (red horizontal line in Fig. 3a) magnetic eld. Along the blue line, the lm is in a state of uniform magnetization and the M(T) curve follows the blue graph in the inset of Fig. 3c. Along the red line, at low temperatures, on the left-hand side of the green line in Fig. 3a a small magnetic eld is enough to establish the uniform magnetic state and M easily saturates. When the temperature is increased and the green line in the (T, B) parameter space is crossed (point 2 in Fig. 3a), the macroscopic magnetization abruptly drops to almost zero as the system enters the phase of static modulated order: the local magnetization within the domains is still large but M almost vanishes because of the cancellation of nite opposite values of the local magnetization within the domains. The corresponding M(T) is sketched in the inset of Fig. 3c with the same colour. At the point 1 (Fig. 3a,c) the M(T) is small and varies smoothly.

Figure 4a shows a measured M(T) family of isochamps for a lm with thickness of 1.75 MLs. The magnetic eld was swept with a frequency varying between 10 1 and 1 Hz. M(B) curves measured at xed temperature within this range of frequencies coincide, therefore we assume that we are observing properties related to thermodynamic equilibrium. At higher frequencies the shape of the magnetization curves depended of the sweeping rate

of B. Results of these studies will be reported in a separate paper. We distinguish two extreme sets of M(T) curves. On the right-hand side those corresponding to larger values of B, an example of which is highlighted in the data with a dashed orange line (the colour code used to label the values of B is indicated along the vertical bar in Fig. 4b). The horizontal line in the (T, B) parameter space, along which the orange M(T) curve was taken, is given in orange in the inset of Fig. 4a. It resides above the blue dotted curve, determined as explained in the subsection Experimental determination of Tc in Methods, and belongs to the set of curves illustrated in Fig. 3a by the horizontal blue line. This means that for these values of (T, B) the lm is in the uniformly magnetized phase and, accordingly, the familiar behaviour of M(T) curves separating out as B is increased, that is, moving away from the critical point, appears. On the left-hand side of Fig. 4a one nds the family of curves corresponding to low B, an example of which is highlighted in the data with a thick, red line also shown in the inset of Fig. 3a (and illustrated by the red graph in the inset to Fig. 3c). At low temperatures, on the left-hand side of the blue dotted line, a small magnetic eld is enough to establish the uniform magnetic phase and M easily saturates. When the temperature is increased and the blue line in the (T, B) parameter space is crossed, the macroscopic magnetization abruptly drops to almost zero as the system enters the phase of static modulated order: the local magnetization within the domains is still large but M almost vanishes because of the cancellation of nite opposite values of the local magnetization within the domains (this transition is marked with 2 in the sketch of Fig. 3a).

To produce the scaling plots of the data in Fig. 4a we determined Tc 3001 K (not to be confused with T*), and the

critical exponents b 0.150.03, d 132 usually dened in

the vicinity of an ordinary critical point36 (see the subsections Experimental determination of Tc and Experimental determination of b and d in Methods). The values of the critical exponents b and d found experimentally are close to those expected for the unfrustrated 2D-Ising universality class4951.

(We point out that some models of three-dimensional ferromagnetism4951 predict an ordinary critical point in the presence of the dipolar interaction, with critical exponents computable, for example, within the Renormalization Group approach.) Notice that in the simulations the 2D-Ising critical exponents were assumed. Here we obtain them from the experimental data. In Fig. 4b a scaling plot is attempted with

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Figure 4 | Scaling plots and experimental phase diagram. (a) Family of experimental M(T) isochamps (about 105 experimental data points). The lm thickness is about 1.75 MLs. The temperature T* is dened in the main text. Inset: paths on the (T,B) plane corresponding to the two M(T) curves highlighted in the main frame. Blue dots mark the boundary between the modulated and the uniform phase. (b) Scaling plot M/B1/d versus t/B1/(bd). The colour code used for the applied magnetic eld B in all the plots is given along the vertical bar. (c) Scaling plot M/|t|b versus B/|t|bd. The dashed line represents the scaling function of the 2D-Ising model taken from ref. 31. (d) The same experimental data points are transferred within the (T, B) parameter space. tc corresponds to the putative critical temperature. The vertical scale on the right-hand side gives the values of the magnetic eld B in units of the saturation magnetization of Fe, 2.16 T. The thick white vertical line and few thin white horizontal lines are due to the failure of data recording during data taking.

the variables M/B1/d (vertical axis) and t/B1/(bd) (horizontal axis)52. Except for a (substantial) set of data points, we observe that the spread of curves inherent to Fig. 4a has reduced considerably, that is, many M(T) isochamps have collapsed onto one thin graph in Fig. 4b. As anticipated, such a collapsing is considered to be a signature of scaling invariance realized while approaching the critical point (T, B) (Tc,0). But the emphasis

must be on the set of non-collapsed data points: they are better evidenced in the scaling representation given in equation (2)replacing m with M and b with Bplotted in Fig. 4c. The collapsed and non-collapsed data points were retrieved using an ad hoc software that distinguished the high-density pointswhich we consider as collapsedfrom the low-density oneswhich we consider as non-collapsed. The collapsed data points, which build the majority, are rendered with their original colours and are observed to be distributed onto the graphs of the 2D-Ising scaling functions g (x) (dashed line,

upper branch) and g (x) (dashed line, lower branch)31. The non-

collapsed data points are rendered in grey and are observed to fall outside the graphs of the 2D-Ising scaling functions.

Experimental phase diagram. The coloured (collapsed) and grey (non-collapsed data points, violating scaling) were subsequently transferred into their place in the (T, B) plane in Fig. 4d, where

they appear outside and, respectively, inside a bell-shaped region. As anticipated, this region, called the grey zone henceforth, surrounds and protects the putative critical point (Tc,0). The bell-shaped region starts on the left with the boundary line marking the transition from static modulated phase to the uniform phase (the blue line in the inset of Fig. 4a up to t 0).

The phase diagram obtained by SEMPA in a previous publication39 corresponds to this portion of the grey zone, residing below t 0. Slightly above t 0 stripes are observed to become

mobile45. The grey zone continues well beyond t 0: it is

conceivable that within the entire grey zone, where scaling is violated, some kind of modulated phase exists.

Further representations of scaling. In Fig. 5a the same data points shown in Fig. 4a are plotted in the GrifthsWidom representation

BMd f t=M1=b

3

with f(x) being a suitable scaling function30,31 with the same colour coding as in Fig. 4c. Particularly clear is how grey points deviate from scaling for the data plotted in the inset ( 2 1016rxr1017, with x t/M1/b). In the main frame the

non-collapsed, grey points decorate the coloured line of clearly collapsed points up to xB1023. For larger values of x, the line of

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104

M/B1/[afii9829]

M/|[afii9848]|[afii9826]

4

106

2

108

0 0 0.05 0.1 0.15 0.2

105 103 101

B[afii9840] / ([afii9826][afii9829]) /|[afii9848]|[afii9840] B[afii9840] / ([afii9826][afii9829]) /|[afii9848]|[afii9840]

Figure 5 | Further representations of scaling. (a) The GrifthsWidom representation of equation (3) uses the scaling variables B/Md versus x t/M1/b.

The scale is loglog in the main panel and loglinear in the inset. The dashed line represents the theoretical scaling function given in ref. 31 for the 2D-Ising model (shifted by non-universal, constant scaling factors). Note that collapsed data points overlap to this line throughout the range 6 1012uxu1023.

The slight deviations for x41040 are due to the experimental value of the exponent g being slightly different from the one expected for the 2D-Ising model (remember that g is the slope of the graph for very large values of x). (b) Representation of scaling given in equation (7) using the same data points shown in Fig. 4c, with the same colour coding. Coloured points are along the scaling function F1(x). (c) Representation of scaling given in equation (8) using the same data points shown in Fig. 4c, with the same colour coding. Coloured points are along the scaling function F2(x).

coloured dots broadens and deviates from the theoretical scaling function of the unfrustrated 2D-Ising model (dashed line) as well. In this GrifthsWidom representation data collapsing is realized over 40 orders of magnitude with respect to the x-variable dened above and 80 orders of magnitude with respect to the variable B/Md (note the remarkable agreement in the inset). Further representations of scaling are found in Fig. 5b,c (see the subsection Scaling and characteristic lengths in Methods for explanation).

Experimental stripe width. A typical evolution of magnetic-domain patterns as a function of temperature in B 0 imaged

with SEMPA is shown in Fig. 6af. The spatial resolution of the present SEMPA instrument is in the submicrometre range. Dark-and light-grey regions correspond to domains with opposite perpendicular magnetization. The in-plane components of the magnetization vanish within the experimental sensitivity. The images refer to a Fe lm with thickness of 2 MLs. The temperature is lowered while the image is acquired, starting from a to f. The limiting temperatures are always indicated at the edges

of each image and temperatures in-between are interpolated linearly. The left-hand side of the image in Fig. 6a appears almost contrastless, while on the right of the red linewhich denes the temperature T* 3331 Ka very weak, stripe-like contrast

develops. Further right, approaching the edge of the same image, the contrast becomes strong enough to allow the determination of the stripe width L(T). The transition from static to mobile stripes occurring at T* (red line in Fig. 6a) was already investigated in previous works4447. It is not clear yet how to call the phase with mobile stripes. Possibly the most appropriate denition is that of stripe liquid20 or oating solid53, which highlight the lack of positional order in this phase21. With certainty, the temperature corresponding to the red line cannot be identied with a Curie temperature. This in spite of the fact that when T* is approached from below the local magnetization inside the stripes decreases sizeably, mimicking the decrease of the spontaneous magnetization in a uniformly magnetized phase close to Tc (refs 39,45) (see the subsection Magnetization within domains in Methods).

The magenta squares in Fig. 6g correspond to the values of L determined by direct inspection of SEMPA images at the

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T * = 333 K

a

b

c

d

e

f

337.9 331.3 324.6 317.8 311.1 304.5 297.8

g h

4.0

101

7

tc

6

3.0

5

L(m)

101

4

L(m)

2.0

4

3

M

3

10 0 0.1 0.2

2

5 304.87 K 303.98 K 303.04 K 302.03 K 301.09 K 300.04 K 299.06 K 297.92 K 296.92 K 295.86 K 295.09 K

3

M

2

1.0

t = (T T *) /T *

1

1

0 0 5 10 15 20

0

0.0

B (104 T)

0 50 100 150

0.06 0.02 0.02 0.06 0.10

t = (T T *) /T *

B (104 T)

Figure 6 | Stripe width as a function of temperature. (af) SEMPA images of the spatially resolved magnetization taken while cooling: temperature is decreased from a to f. The vertical lines indicate the temperatures in K at which the values of the stripe width were extracted from each image and reported in the (g) plot. Images are taken in nominally zero applied eld and their linear size is 22 mm. (g) Stripe width L as a function of the reduced temperature t for a lm with thickness of 2 MLs (d 3.6 10 4 mm) obtained by visual inspection of SEMPA images (blue triangles and magenta squares) and from

equation (4) using the total magnetization measured with MOKE (green and red crosses). Inset: data produced applying equation (4) on the M(T,B) data

for B 0.5, 1.3, 2.5, 4.1 and 6.0 10 4 T (lm thickness 1.75 MLs, d 3.15 10 4 mm) are plotted in a loglinear scale together with the green and red

data points of the main frame, measured on a different lm (d 3.6 10 4 mm). Owing to the linear relation between M and B within the grey zone, the

graphs for different values of the eld practically coincide. (h) M versus B curves for temperatures close to Tc. The inset shows a zoom of the small-eld region for three selected temperatures, the black one corresponding to the putative critical isotherm. The small offset, of o0.510 4 T, visible in the inset is due to a small residual magnetic eld for the nominal value of B 0.

temperatures (ToT*) indicated by vertical lines in Fig. 6af. The horizontal scale gives the reduced temperature variable t (T T*)/T*. The blue triangles were obtained from a sequence

of images similar to the one shown in Fig. 6af but recorded while heating the sample. Note that at lower temperatures the L(T) values obtained on heating and cooling separate out clearly, indicating the development of an out-of-equilibrium situation towards low temperatures (discussed in ref. 40). This may result from quenched disorder, for example, due to local uctuations of the Fe thickness that hinders the motion of domain walls and thus prevents the system from adjusting L to its equilibrium value44. This out-of-equilibrium situation may also have an intrinsic origin as a manifestation of glassiness47, which has been predicted for the model Hamiltonian (1)14,15,20,22. (We recall that the M(T, B) data used to produce scaling plots and the phase diagram are, instead, reasonably equilibrated since they are independent of the sweeping rate of B.) Close to T* (t 0) the

values of L(T) merge within the experimental error.

The green and red crosses in the main frame of Fig. 6g and the coloured points in the inset were not obtained by imagingfor the simple reason that they extend above t 0, where mobility

makes stripe invisible to SEMPA imaging45. These additional, essential data on L(T) were deduced from an equation that relates L(T) to the spatially averaged magnetization MT; B in small

applied magnetic elds38,39 (see the subsection Experimental determination of L(T) using MT; B in Methods):

LT m0

spatially averaged magnetization measured by MOKE and is in arbitrary units). The response of M to B is well dened both below and above t 0, see, for example, Fig. 6h: accordingly,

M(B) can be used to determine L(T) via equation (4) in the range of temperatures where stripes are no longer observed in SEMPA. Figure 6g shows that below t 0, where both imaging (blue

triangles and pink squares) and M(T, B) (green and red crosses) can be used to deduce the stripe width, SEMPA and equation (4) give almost the same values for L(T). This correspondence strongly supports the validity of equation (4) itself. Remarkably, the data points represented by green and red crosses (and the coloured ones in the inset) continue into the temperature region where SEMPA imaging becomes contrastless, t40. This suggests that the stripe width, directly observed on static patterns below t 0, evolves smoothly up to the highest measured temperatures,

tE 0.16 corresponding to more than 30 above T* (inset of

Fig. 6g), and ranges from several micrometres at low T down to few tens of nanometres32. The estimates of L(T) represented with crosses, triangles and squares refer to a slightly thicker Fe lm(2 MLs) than the one (1.75 MLs) on which MOKE measurements and scaling analysis were performed. The Curie temperature determined in the thinner lm as described before is indicated in the inset as tc. Figure 6g thus establishes a characteristic long spatial scale L at every temperature, carrying neither a sign of the transition to mobile stripes (at T*) nor an anomaly at the putative critical temperature Tc. Equation (4) establishes a proportionality between L(T) and the magnetic susceptibility m0 @MT; B@B B0. As

L(T) is a non-singular quantity as a function of the temperature, we expect the susceptibility as well to be a non-singular quantity as a function of the temperature within the grey zone. Indeed, the nonlinear relationship between M and B, detected outside the grey zone, gives way to the linear, non-critical behaviour shown

p

4 d

@MT; B

@B

B0

4

d being the thickness of the lm. We point out that in equation (4) M is the spatially averaged magnetization expressed in physical units (to be distinguished from M, which is the

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in Fig. 6h (the susceptibility being the slope of the graphs in Fig. 6h at low magnetic elds). This remarkable difference has been anticipated in the sketches in Fig. 1a,d. From magnetic imaging we can associate the linear portion of the M(T, B) curve in Fig. 6h with the displacement of domain walls, which accompanies the increase of those magnetic domains whose magnetization is parallel to B at the expense of the size of domains with magnetization antiparallel to the magnetic eld39.

A legitimate question is whether the period of modulation actually acts as a natural cutoff that hinders the divergence of the correlation length x underlying the critical behaviour in the unfrustrated system. (We thank the reviewers for putting these issues under our attention (see Reports).) Should this be the case, then nite-size-scaling relations54,55, involving both x and L, should replace the equations of state of the 2D-Ising ferromagnet. In the subsection Scaling and characteristic lengths in Methods we show that nite-size-scaling relations are incompatible with the law M L B observed experimentally in the patterned

phase (inset of Fig. 6g) and provide an additional argument on this issue.

DiscussionWith the plots of Figs 2b, 4b,c and 5 we have discovered a strikingly simple but, surprisingly, yet unnoticed symmetry: even if an indenitely weak long-ranged and frustrating interaction the dipolar coupling in our caseis enough to eliminate the critical point of a standard second-order phase transition, sufciently away from it ordinary critical scaling la Kadanoff is fully recovered36. In our specic Fe lms, outside the region of the phase diagram corresponding to the patterned phase (grey zone) we observe 2D-Ising scaling, despite the fact that the true critical point at (T, B) (Tc, 0) isstrictly speakingavoided. It

remains to be determined what type of order is realized within the grey zone and how it relates to the physics outside of it, namely in the ferromagnetic scaling region. As already anticipated, strong theoretical arguments exclude the persistence of positional order of the stripe domain pattern at nite temperature21. For what concerns orientational order, the lowest-energy perturbations can be described by an orientational effective Hamiltonian that explicitly possesses O(2) symmetry21 (symmetric with respect to continuous rotations around a xed axis). Both these facts are compatible with the occurrence of a BerezinskiiKosterlitz Thouless phase transition from a nematic phase of magnetic domains to a disordered phase17,18,21 and our lms might, indeed, be good candidates to observe such a transition. The mixing of Z2 and O(2) symmetries is encountered in other models as well, like the 2D frustrated XY models5661. In the last ones, the O(2) symmetry is explicit in the Hamiltonian, while the Z2 symmetry emerges from the chiral degree of freedom of the ground state; the interplay between the two symmetries gives rise to a richand debatedscenario of phase transitions5761. However, the patterned phase of the model dened by equation (1) is expected to evolve differently with temperature with respect to frustrated XY models: in the model of our interest the Z2 symmetrybroken in one ground stateis restored at nite temperature by elastic-like excitations, while in genuine Ising-like transitions it is restored creating domain-wall excitations. These important open issues will certainly stimulate further work to better characterize the grey zone. From the present investigation we denitely conclude that whatever order is realized in the modulated phase of thin lms magnetized out of plane, it must be such that (i) standard scaling relations, like M/|t|b g

(B/|t|bd) or B/Md f(t/M1/b), are replaced by the

one-variable relation M/BBL(T) (In ref. 28 this scaling law is generalized to any dimension and to a generic power-law

exponent a in equation (1).) and (ii) a well-dened modulation length L(T) persists deep into the paramagnetic phase.

Methods

Technical details of Monte Carlo simulations. Data in Fig. 2ac were produced by Monte Carlo simulations on a square lattice with Lx Ly 120 using the

Hamiltonian (1). Some simulations were also run for Lx Ly 200 to exclude

possible nite-size effects and to produce the snapshots in Fig. 2c. Ewald sums technique was used to implement the long-range interaction with a 3 for periodic

boundary conditions27,62. This term represents the dipolar interaction in the Ising limit with the easy axis perpendicular to the lm plane, in which case dipoledipole coupling is antiferromagnetic for all spin pairs; the corresponding sum runs over all the (i,j) pairs of distinct sites on the lattice. The limiting eld bc above which the system is in a phase with mostly uniform spin prole was determined from the behaviour of the average magnetization m as a function of temperature for xed values of b, according to a zero-eld-cooledeld-cooled protocol. After having prepared the system in a uniformly saturated state (with m 1), we let the

temperature increase from very low values up to a predened Tf, lying above the transition from the modulated to the paramagnetic phase for b 0. Then, we

stopped the simulation and, starting from the nal conguration, the system was cooled down to the original temperature. When bobc a strong hysteresis in the m(T) curves was observed, below the temperature where modulated phases develop. When b4bc, instead, the curves were completely reversible in the whole temperature range. In this way we could estimate bc for rather large values ofJ/g (up to J/g 10), without the computational cost of complete phase-diagram

calculations27. To observe domain phases within the given Lx the ratio J/g cannot be as large as in the experiment, where Tc/gBO(102), because the width of domains increases exponentially with this ratio27,28, and realistic simulations can be performed up to about Lx Ly 200 (the necessity of summing over all pairs,

imposed by the dipolar interaction, limits the size of the simulation box signicantly with respect to the cans in which only short-ranged interactions are considered).

SEMPA imaging. Spatially resolved magnetic imaging shown in Fig. 6af is performed with SEMPA (Fig. 3b). A focused electron beam of primary electrons is directed towards the sample and the secondary electrons excited off the topmost surface layers by the primary electrons are sampled and analysed according to their spin polarization, which is proportional to the local magnetization vector within the beam focus. The spin polarization is rendered in the images by a grey scale, black and white corresponding to opposite spin polarization. Only the component perpendicular to the surface is displayed, as the components of the spin polarization parallel to the lm plane are vanishing. The images are built by horizontal line scans, consisting of about 200 pixels, from top to bottom, starting from the left-hand side of each image. For contrast to be detected, the magnetization distribution must be static over the times required to collect at least a dozen horizontal lines, that is, about 60 s.

Experimental determination of Tc. In conventional ferromagnets one nds Tc as the critical temperature at which, for example, the critical isochamp at zero applied magnetic eld M(T, B 0) vanishes. In the present system, the ordinary critical

point is obscured, at low magnetic elds, by the appearance of static and mobile domains, so that one has to nd a different way to determine Tc. One useful property of a conventional second-order phase transition is that the plot of the magnetic susceptibility wT; B m0 @MT; B=@B as a function of temperature

has a maximum at a temperature Tmax(B) that approaches Tc as B approaches zero63. Figure 7a shows the susceptibility w(T, b) of the unfrustrated (that is, without dipolar interactions) 2D-Ising model computed by means of Monte Carlo simulations. For any ba0 the maximum value of w(T, b) is nite butas expectedit is higher the weaker the elds. Notice that the true approach of Tmax(b) to Tc may not necessarily be linear in b, as pointed out in ref. 63, but our numerical simulations (Fig. 7b) show that a linear extrapolation of Tmax towards b 0 gives a fairly accurate estimate of Tc.

Experimentally, we obtained the w(T, B) by rst treating the raw M(T, B) data with a SavitzkyGolay nite-impulse-response smoothing lter implemented in MATLAB. After ltering, the derivative w(T, B) m0 qM(T, B)/qB could readily be

obtained. In Fig. 7c some of the resulting susceptibility curves are shown for selected values of B. The experimental B(Tmax) is plotted in Fig. 7d. This graph, in contrast to the one in Fig. 7b, shows two distinct regimes, depending on the range of B. We discuss rst the low eld regime. When cooling in weak enough elds, the system rst enters the grey zone, (see point 1 in Fig. 3a), without displaying any anomaly in the susceptibility in correspondence with this transition. On further cooling a re-entrant transition (point 2 in Fig. 3a) from the patterned to the uniform phase is encountered39: this second transition is accompanied by a sharp maximum in the susceptibility (abrupt increase of the magnetization). This type of maximahighlighted by a blue dashed line in Fig. 7dmark the transition from a phase with domains to the uniform phase. When extrapolated to B 0, they lead to

a temperature at which the sample consists of very large stripes carrying opposite but almost saturated values of the magnetization. Above this temperature, the sample keeps the modulated order up T*, where domains become mobile but the

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250

0.012

a

b

200

b/J = 0.0000 b/J = 0.0010 b/J = 0.0025 b/J = 0.0050 b/J = 0.0100

0.008

150

[afii9851]

[afii9851]

0

b/J

100

0.004

50

0

0.90 0.95 1.00 1.05 1.10 1.15 0.90 0.95 1.00 1.05 1.10 1.15

T/T cIsing T max/T cIsing

B (104 T)

0

c

100

d

40

100

80

80

30

B(104 T)

60

60

T

20

40

40

10

20

20

0

0 270 290 310 330 270 280 290 300 310 320 330

T (K) Tmax (K)

Figure 7 | Determination of the Curie temperature (a) Plot of the magnetic susceptibility w(T,B) qm/qb (b in units of J) a function of T/Tc for a lattice of

linear size Lx Ly 120, from Monte Carlo simulations for the unfrustrated 2D-Ising model obtained setting g 0. We recall that TcIsing 2.269J, assuming

kB 1; the selected values of b/J are given in the gure legend. (b) b/J versus Tmax(B)/TcIsing; for each eld, the value of Tmax(B) was extrapolated with FSS

using lattices of size Lx Ly 48, 64, 96, 128 and 200. (c) Plot of the magnetic susceptibility w(T,B) m0 qM(T,B)/qB of a 1.75-ML-thick Fe/Cu(001) lm

as a function of temperature for different magnetic elds. Different colours correspond to different applied elds (see colour map on the right). (d) Plot of B versus Tmax(B) obtained from the experimental susceptibility as described in the subsection Experimental determination of Tc in Methods.

magnetization within them is still substantial. Accordingly, Tc must be above T*. On the other side, for larger elds (the right-hand side portion of the graph) the grey zone of Fig. 3a is never entered and the system is in a uniformly magnetized phase: B(Tmax) for larger B was therefore taken to extrapolate towards a putative Tc.

Several linear ttings were performed by choosing different ranges of the eld B between 30 10 4 and 80 10 4 T. The different ttings produced the family of

red lines (Fig. 7d) from which the error on the Tc was estimated. The experimental Tc, for this particular sample, is 3001 K. Notice that the extrapolated Tc lies within the cross-over range (K) 298uTu302, in which the blue-line types of maxima transform into the red-line types of maxima.

Experimental determination of b and d. In conventional ferromagnets one nds the critical exponent b via the asymptotic behaviour of the critical isochamp near

Tc, that is, from M(T, B 0)B( t)b and the critical exponent d via the asymp

totic behaviour of the critical isotherm, that is, M(Tc,B)BB1/d. In the present

system, these ordinary power laws are obscured, at low magnetic elds, by the appearance of static and mobile domains. However, the conventional lore of scaling (ref. 1, p. 485) and our simulations for the specic system, indicate practical ways to nd b and d using experimental MOKE data of M(T, B) originating within the high-temperature, non-zero eld region of the (T, B) parameter space. The critical exponent b can be deduced from the experimentally determined values of the exponents g and d, using the relation bd b g. The exponent g determines the

magnetization in the region of weak elds (equation (148.8) in ref. 1) according to

M

B

jtj

7

and

g

5

The exponent d determines the magnetization in the region of strong elds according to the relation

M B1=d 6 (see equation (148.10) in ref. 1). The notion of weak and strong eld is, of course, dependent on which temperature interval is addressed. In Fig. 8a we plot log(M/B) versus log|t| for different values of B (for the colour code indicating the values of B see the horizontal scale in the inset). At sufciently high temperatures, a region of the graph emerges, where all the curves for different magnetic elds

almost collapse onto one single straight line, and thus full the scaling properties required by equation (5) for the quantity M/B. The negative of the slope of the resulting straight line is the sought-for exponent g. Several linear ttings were performed for xed elds ranging from 20 10 4 to 80 10 4 T. These

independent determinations of g are shown in the inset. After averaging, forTc 300 K we obtain g 1.780.09, with the error given by the s.d. of the mean.

As consistency check, the whole procedure was repeated varying the value of Tc.

The s.d. of the mean values of g goes through a minimum in the range (K) 298rTcr301, which is, accordingly, the range where the best collapsing of the

M/B curves is realized. When Tc is varied in this interval, g ranges from 1.7 to 1.9.

The log(M/B) versus logB plot of Fig. 8b in the temperature range (K) 299oTo301, reveals a low-eld region where the curves saturate to an almost constant value, indicating the linearity of M versus B for weak elds. In the strong-eld region the graphs are observed to almost collapse onto a single straight line, the slope of which amounts to (1 d)/d, consistently with equation (6). From these slopes

tted for different T in the appropriate regime of Fig. 8b we estimate d 132.

Scaling and characteristic lengths. As suggested by one of the reviewers, in a conventional ferromagnet scaling plots can also be written in such a way that the dependence on the correlation length x jtj n is made explicit (n being the

corresponding critical exponent). Let us consider the following equations of state of a ferromagnet:

MT; B

B1=d F1

B n=bd

jtj

n

8

with F1(x) and F2(x) some scaling functions. In Fig. 5b,c the same data points shown in Fig. 4c are plotted in the representations dened above. Points falling in the grey zone are rendered in grey. Indeed, collapsing is observed for the coloured points while the grey ones spread out in the plot planes. Figure 5c appears

MT; B jtj

b

F2

B n=bd

jtj

n

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a

b

104

2.4

105

M/B

299.06 K 299.28 K 299.44 K 299.72 K 300.04 K 300.33 K 300.48 K 300.83 K

2.1

M/B

[afii9828][notdef]1.8

104

106

1.5

20 40 60 80 100

B (104 T)

107103 102 101

100 101 102

|[afii9848]|

B (104 T)

Figure 8 | Determination of the exponents b and d. (a) Loglog plot of the ratio M/B versus t (T Tc)/Tc (with Tc 300 K). The dashed black line is a

guide to the eye with slope equal to the mean value of g. Different isochamps are plotted with different colours: specic values of the eld B can be identied from the horizontal axis of the inset. The inset plots the tted values of g as a function of the magnetic eld. The solid and dashed horizontal lines indicate the mean value of g and the s.d. from this average, respectively. The vertical dotted line marks the largest eld B 80 10 4 T used to determine

g. (b) Loglog plot of the ratio M/B versus B. Different isotherms are plotted with different colours (specic values of Tare given in the legend). The dashed black line is a guide to the eye with slope equal to mean value of (1 d)/d (tted exponents).

T/J

1.8

a

b

1.0

1.6

0.8

m L

0.6

m L /m

1.4

0.4

1.2

0.2

1.0

0.0

1.5 2 2.5 3 0 1 2 3

Lx |[afii9848]|[afii9840]

L (Tc)|[afii9848]|[afii9840]

1.0

c

d

0.8

1.0

0.8

0.6

M

M/ (A |[afii9848]|[afii9826] )

0.6

0.4

0.4

0.2

0.2

0.0

0.0

270 280 290 300 0 0.005 0.01

T (K)

Figure 9 | Test of FSS. (a) Monte Carlo simulations of mL(T) versus Tcomputed for the unfrustrated 2D-Ising model and b 0 for a square lattice of linear

size Lx 10, 20, 40 (colour code in the legend). The black curve is the exact Onsager solution mN(T). (b) Theoretical FSS function fM(x) deduced from the

data in a as described in the subsection Scaling and characteristic lengths in Methods. Inset: same data plotted in a loglinear scale. (c) Experimental M(T) data versus T measured for B 5 10 5 T. The green line corresponds to MN AN|t|b, with b 1/8 and AN 1.192 10 2. (d) Ratio ML(T)/|t|b versus

L(Tc) |t|n representing the function F4(x) dened in equation (14).

very similar to Fig. 4c and basically the same considerations apply. Equation (7) leads instead to the plot in Fig. 5b that has a different shape.

If the spreading of grey points observed in all the scaling plots was due tothe size of magnetic domains L acting as a cutoff for the correlation length, from equation (7) one would expect the following relation to be fullled in the patterned phase

MT; B

B1=d F3

B n=bd

L

~F3 Bn=bd L

9

with F3(x) and ~F3x appropriate scaling functions. This scaling relation is not

compatible with the one found in within the grey zone, MgreyT; B B L, which

can be rewritten as

MgreyT; BB1=d Bd 1=d L 10

Equations (9) and (10) can be simultaneously fullled only for

d 1

nb 11

This last relation is not obeyed by the 2D-Ising critical exponents b 1/8, d 15

and n 1.

A more proper framework to discuss whether L acts as a cutoff for the correlation length is provided by the nite-size scaling (FSS) ansatz54,55, according

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area occupied by up (fm), respectively, down (fk) perpendicular magnetization: M(T, B) M0(T, B) A(T, B). In the range to0 and for small B, M0(T, B) is a

smooth function of T and almost independent of B. A(T, B), instead, increases,

close to T*, linearly39 with B and describes a process where the width of stripes magnetized parallel to B increases at the expenses of the width of stripes magnetized antiparallel to B. Accordingly, the following scaling law has been demonstrated39: M(T, B)/BpL(T), that is, the response of M to B is linear(Fig. 6h) and the susceptibility is proportional to the sought-for equilibrium stripe width. The knowledge of the proportionality constant is thus crucial to determine L(T). This can be computed exactly for a square-like stripe prole (see equation (2.38) in ref. 38), which yields equation (4). A perfect square prole is not expected at nite T because the domain walls are certainly not atomically sharp. However, as shown in ref. 39, the assumption of an almost square prole explains precisely the response of A to B also in the very vicinity of t 0, so that we

are condent that the proportionality constant computed for a perfect square prole holds up to temperatures where L(T) behaves smoothly (inset, Fig. 6g). Note that M entering equation (4) is expressed in physical units, while in MOKE measurement we access M, which is in arbitrary units. For low temperatures and sufciently large applied elds the lm is in the uniform phase with all magnetic moments almost fully aligned, so that the value of the MOKE signal MS measured in this condition can be associated with the saturation magnetization of Fe, m0 MS 2.16 T. The other measured values were rescaled accordingly to obtain

the physical magnetization M(T, B).

Data availability. Data are available from the ETH Zurich Data Archive: http://doi.org/10.5905/ethz-1007-20

Web End =http:// http://doi.org/10.5905/ethz-1007-20

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0.3

0.2

P(T) (a.u.)

0.1

0.0

280 300 320 340

T (K)

Figure 10 | Local magnetization as a function of temperature. The local magnetization, given by the spin polarization P of the secondary electrons, is extracted from SEMPA images, adapted from ref. 39 (red) and ref. 45 (blue). Solid lines are guides to the eye obtained with a phenomenological power-law tting B(T*T)beff that yields an effective beff 0:25 in both

cases. Note that the two samples have different temperatures T* 326 and

347.2 K, resulting from the t and indicated by full dots.

to which the magnetization ML(T) of a nite system of linear size Lx should be related to the magnetization of the corresponding innite systems MN(T) by the law

MLT M1TfMLx=x 12 where B 0 is assumed and fM(x) is some FSS function of the single variable

x Lx/x such that

fMx ! 1 for Lx x

fMx ! const: for Lx x

13

Figure 9a displays the mL(T) as a function of T computed for the unfrustarted 2D-Ising model and B 0 for nite square lattices of size Lx 10, 20, 40.

Comparatively small lattices were chosen to enhance nite-size effects. The exact Onsager solution for the innite system, mN(T), is also shown as a solid black line.

From these data the theoretical FSS function fM(x) can be deduced simply by plotting mL/mN versus Lx|t|n, the latter being proportional to Lx/x (Fig. 9b). The same procedure was repeated for the experimental points assuming MN AN|t|b,

with b 1/8 and AN tting parameter. Figure 9c demonstrates that this law is

followed by the coloured points, while the grey points deviate from it. A speculative FSS function, in which the role of the lattice size Lx is played by the size of magnetic domains at the putative Curie temperature L(Tc), can be obtained directly from experimental points and reads

MT; B 0 jt

b

j

F4 LTc jtj n

14 The function F4(x) is plotted in Fig. 9d. The part of curve resulting from grey points deviates from fM(x) computed for the 2D-Ising model shown in Fig. 9b, which seems to conrm that L does not simply act as a cutoff for the correlation length x.

Magnetization within domains. Below T* domains are frozen and one can reliably extract from SEMPA images, like the ones shown in Figs 2d and 6af, the value of the local magnetization, that is, the magnetization within the domains. The data related to this quantity, published in two previous articles39,45, are reported in Fig. 10 for convenience. As anticipated in the main text, the local magnetization inside the stripe domains decreases sizeably while T* is approached. If one tries to capture this decrease using a power law, one obtains a curve, which vanishes in the vicinity of T*, with an effective critical exponent of about 0.25 (Fig. 10). However, from the present work we know that the putative Tc deduced from scaling analysis lies about 30 above T*. Therefore, the power-law behaviour represented by the continuous curves in Fig. 10 is not related to the scaling behaviour observed in the ferromagnetic scaling region. We point out that we are not aware of any prediction that assigns a power law to the local magnetization in the vicinity of T*. Moreover, we note that our experimental data in Fig. 10, strictly speaking, do not necessarily speak for a power-law vanishing of the local magnetization.

Experimental determination of L(T) using M(T, B). In the temperature region to0 stripe domains are static and can be imaged with SEMPA. By means of equation (4) the stripe width can also be determined from the behaviour of the spatially averaged magnetization M(T, B) at low magnetic elds. M is the product of |M0|the value of the local magnetization within the stripesand

AT; H

f" f#=f" f#, that is, the asymmetry between the lm

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Acknowledgements

We thank Thomas Bahler for technical assistance, G. M. Graf, A. Giuliani, O. V. Billoni and S. Ruffo for helpful discussions, as well as the Swiss National Science Foundation, ETH Zurich and CONICET (Argentina) for nancial support. D.P. dedicates the paper toV. Pokrovsky, on the occasion of his 85th birthday, for introducing to him the subject of low-dimensional physics.

Author contributions

N.S. developed the instruments, conceived the experiment, collected the experimental data and discussed the results; D.A.Z. developed the software for data analysis, analysed the data and discussed results; U.R. developed the instruments, collected the experimental data, discussed the results and supervised the experimental work; S.C. did the Monte Carlo simulations and discussed the paper; D.P. discussed the results and wrote the paper; A.V. conceived the experiment, analysed the data, discussed the results, wrote the paper and supervised research.

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How to cite this article: Saratz, N. et al. Critical exponents and scaling invariance in the absence of a critical point. Nat. Commun. 7, 13611 doi: 10.1038/ncomms13611 (2016).

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