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Sci Rep. 2017; 7: 45004.
Published online 2017 March 22. doi:  10.1038/srep45004
PMCID: PMC5361157

Continuously Varying Critical Exponents Beyond Weak Universality

Abstract

Renormalization group theory does not restrict the form of continuous variation of critical exponents which occurs in presence of a marginal operator. However, the continuous variation of critical exponents, observed in different contexts, usually follows a weak universality scenario where some of the exponents (e.g., β, γ, ν) vary keeping others (e.g., δ, η) fixed. Here we report ferromagnetic phase transition in (Sm1−yNdy)0.52Sr0.48MnO3 (0.5  y  1) single crystals where all three exponents β, γ, δ vary with Nd concentration y. Such a variation clearly violates both universality and weak universality hypothesis. We propose a new scaling theory that explains the present experimental results, reduces to the weak universality as a special case, and provides a generic route leading to continuous variation of critical exponents and multi-criticality.

Study of critical phenomena is based on two concepts: one is universality1,2 which states that the associated critical exponents and scaling functions are universal up to symmetries and space dimensionality, and another is scaling theory3 that describes the general properties of the scaling functions and relates different critical exponents. In the renormalization group approach4, the critical point is a fixed point governed by a unique set of relevant operators with scaling dimensions (critical exponents) which are fully independent of irrelevant operators. While a relevant perturbation may take the system to a new fixed point, the marginal one brings a possibility of continuous variation of exponents. Although the concept of universality has been verified experimentally time and again, starting from early 40s5 to present6, a continuous variation is rarely observed. A clear example is provided by Baxter7 who solved the eight vertex model8 (EVM) exactly, and Kadanoff and Wegner9 who provided a mapping of EVM to a two-layer Ising system with a marginal four-body interaction between the layers10 (similar to Ashkin Teller model11,12,13) that drives the continuous variation. In later years, Suzuki14 proposed a weak universality (WU) scenario where critical exponents (like β, γ, ν in EVM) change continuously but their ratios (An external file that holds a picture, illustration, etc.
Object name is srep45004-m1.jpg and consequently An external file that holds a picture, illustration, etc.
Object name is srep45004-m2.jpg) remain invariant. This WU scenario has been observed in frustrated spin systems15,16, interacting dimers17, magnetic hard squares18, Blume-Capel models19, reaction diffusion systems20, absorbing phase transitions21, percolation models22,23, fractal structures24, quantum critical points25, etc. The generic nature of the marginal interaction that leads to weak universality in all these different systems remains unclear.

To the best of our knowledge, most systems which show continuous variation of critical exponents obey weak universality - a few exceptions include criticality in Ising spin glass26, micellar solutions27,28, frustrated spin systems29, strong coupling QED30 etc. Experimentally, the continuous evolution of critical exponents with chemical substitution has been observed in URu2−xRexSi2 (0.2  x  0.6) single crystals31. With decreasing x, both γ and δ decrease linearly keeping β fixed [(β, γ, δ) = (0.8, 1.0, 2.25) for x = 0.6 and (β, γ, δ) = (0.8, 0.18, 1.23) for x = 0.2]. By extrapolation, it has been shown that γ  0 and δ  1 at x = 0.15 at which quantum phase transition occurs. Recently, Fuch et. al.32 have observed the linear variation of exponents in the polycrystalline samples of Sr1−zCazRuO3 from (β  0.5, γ  1, δ  3) for z = 0 to (β  1, γ  0.9, δ  1.6) for z = 0.6. They have suggested that the evolution of exponents may be originating from orthorhombic distortions or additional quantum fluctuations associated with quantum phase transition at z = 0.7 However, whether there is a quantum critical point in Sr1−zCazRuO3 at z = 0.7 is still under debate33,34,35,36,37.

Anomalous ferromagnetic (FM) transition has also been observed in mixed valance manganites, RE1−xAExMnO3 (RE: rare earth ions, AE: alkaline earth ions) either as a discontinuous transition or a continuous transition with a set of critical exponents that does not belong to any known universality or the weak universality38,39,40,41,42,43,44,45,46,47,48,49. In manganites, the nature of phases and transitions strongly depend on the bandwidth and disorder (namely quenched disorder) arising due to the size mismatch between A-site cations50,51. Such disorder reduces the carrier mobility and the formation energy for lattice polarons52, in effect TC reduces, rendering the FM transition towards first-order. A system with narrow bandwidth and large disorder such as Sm1−xSrxMnO3 shows a sharp first-order FM transition for x = 0.45  0.4842,43,44,45. The first-order transition is however extremely sensitive to external pressure, magnetic field, A-/B-site substitution, oxygen isotope exchange, etc. - with the application of external and internal pressure (chemical substitution) beyond a critical threshold, the transition becomes continuous43,44,45,53.

In this paper, we report two important results (i) a thermodynamic transition (i.e. FM phase transition in (Sm1−yNdy)0.52Sr0.48MnO3) where critical exponents β, γ, δ vary continuously, and (ii) a new scaling theory that explains the present experimental results and reduces to the weak universality as a special case. We propose that, to obey the scaling relations consistently, the variation of critical exponents are constrained to have specific forms. This scaling hypothesis naturally leads to two special cases which have been realized earlier, namely the weak universality14 (where δ is fixed) and the strong coupling QED30 (fixed γ). A more generic scenario is the one which allows simultaneous variation of all the critical exponents in an intrigue way, leading to a multi-critical point where the phase transition becomes discontinuous. This scenario is verified experimentally in a comprehensive and systematic study of FM phase transition in (Sm1−yNdy)0.52Sr0.48MnO3 single crystal. For higher doping concentration y > 0.4 the FM transition is found to be continuous, but to our surprise, the critical exponents exhibit continuous variation with Nd concentration y, starting from (β, γ, δ) = (0.16, 1.27, 9.30) at y = 0.5 to (0.36, 1.38, 4.72) at y = 1. Within error limits, y = 1 belongs to the universality class of Heisenberg model in three dimension (HM3d). The proposed scaling hypothesis successfully explains the continuous variation of exponents in the present system, and predicts that the transition is discontinuous for doping y [less, similar] 0.37 which has been observed experimentally43,44.

Results and Discussion

Critical temperature and exponents

Let us set notations by reminding that in absence of magnetic field (H) the spontaneous magnetization of the system vanishes as MS(0, ε) ~ (−ε)β and the initial susceptibility diverges as An external file that holds a picture, illustration, etc.
Object name is srep45004-m3.jpg as the critical point is approached, i.e., when (T/TC  1) [equivalent] ε  0. Again at T = TC, the magnetization varies as M(H, TC) ~ H1/δ54. To estimate the critical exponents β, γ and δ, we need to know TC accurately. To do so, we exploit the linearity in Arrott-Noakes equation of state55

An external file that holds a picture, illustration, etc.
Object name is srep45004-m4.jpg

where a, b are non-universal constants. The correct choice of β and γ can make the isotherms of M1 versus (H/M)1/γ a set of parallel straight lines with one unique critical isotherm that passes through the origin. This is explained in Fig. 1(a) for y = 0.5 and the self consistency is achieved for the values β = 0.16, γ = 1.30. The isotherm T = 192 K passes almost through the origin. From the intercepts of these parallel straight lines on M1 and (H/M)1/γ axes, we obtain MS and An external file that holds a picture, illustration, etc.
Object name is srep45004-m5.jpg for different temperatures which are shown in Fig. 1(b). The best power-law fit gives β = 0.16, γ = 1.26 and TC = 192.3 K. These estimates are in fact consistent with Kouvel-Fisher criteria56 which predict that in the scaling regime, both An external file that holds a picture, illustration, etc.
Object name is srep45004-m6.jpg and An external file that holds a picture, illustration, etc.
Object name is srep45004-m7.jpg are proportional to T with proportionality constants β−1 and γ−1 respectively; which is shown in Fig. 1(c). Another critical exponent δ is found from M-H isotherm at An external file that holds a picture, illustration, etc.
Object name is srep45004-m8.jpg (here T = 192 K). The log-scale plot of M vs. H, as shown in the inset of Fig. 1(c), is linear with slope δ = 9.3 Following the same method, we have determined the critical exponents and TC s’ for y = 0.6, 0.8 and 1.0 (see section I of Supplementary information for details) which are listed in Table 1 along with the critical exponents for HM3d. In the present system, the variation of TC with y can be explained in terms of bandwidth and A-site cation size disorder51 of the system. With increasing y, the bandwidth increases whereas disorder decreases; both of these effects enhance TC (see section II of Supplementary information for details). It is clear from the Table 1 that the exponents for y = 1 are very close to that of HM3d, whereas they deviate substantially and vary systematically when y is decreased. Clearly, β decreases by two-fold whereas δ increases almost by the same amount as one goes from y = 1 to y = 0.5 At the same time, the Widom scaling relation β + γ = βδ is satisfied for each y = 0.5, 0.6, 0.8, 1. This indicates that the change in γ has to be minimal, as observed here (see section IV of Supplementary information for details). The variation of critical exponents with respect to a system parameter contradicts universality, moreover it violates WU as δ varies along with β and γ. The focus of current work is to address this issue in details, but first let us ask whether the scaling hypothesis, crucial for the description of near critical systems, is valid here.

Figure 1
(a) Modified Arrott plot [M1/β vs An external file that holds a picture, illustration, etc.
Object name is srep45004-m28.jpg] isotherms (185 K  T  199 K in 1 K interval) of (Sm1−yNdy)0.52Sr0.48MnO3 (y = 0.5) single crystal. Solid lines ...
Table 1
Critical exponents of (Sm1−y Nd y )0.52Sr0.48MnO3.

For a thermodynamic system near FM transition, magnetization M depends on H and ε and follows a universal scaling form54

An external file that holds a picture, illustration, etc.
Object name is srep45004-m9.jpg

where F+ is for T > TC and F is for T < TC. The utility of universal scaling function lies in the fact that the M  H curves obtained for different T (near TC) can be collapsed onto a single curve when one plots An external file that holds a picture, illustration, etc.
Object name is srep45004-m10.jpg versus An external file that holds a picture, illustration, etc.
Object name is srep45004-m11.jpg This scaling collapse is shown in Fig. 1(d) for y = 0.5 (and in section I of Supplementary information, for other values of y); the two branches in this curve correspond to super- and sub-critical phases. For y = 0.6, 0.8 and 1.0, the data collapse is also shown in Fig. 2(a), where we use an alternative but equivalent form of Eq. (2),

Figure 2
(a) Scaling collapse of M  H curves for y = 0.6, 0.8 and 1 following Eq. (3). Although they appear different, the scaling functions for y = 0.5, 0.6, 0.8, 1 can be collapsed onto each other (shown ...
An external file that holds a picture, illustration, etc.
Object name is srep45004-m12.jpg

It is advantageous to use this form as the two branches of Eq. (2) are now merged to a single function F with its argument An external file that holds a picture, illustration, etc.
Object name is srep45004-m13.jpg extending from the sub-critical (x < 0) to the super-critical (x > 0) regimes. A very good quality data collapse confirms that the scaling hypothesis is in place and the estimated value of TC and critical exponents obtained through several prescriptions are unambiguous and self-consistent.

The scaling hypothesis

In the following, we propose a new scaling ansatz which explains the experimental findings presented here. Since diverging fluctuations are known to be the origin of power-laws (see ref. 57 for a proof for non-equilibrium systems, which also holds trivially for equilibrium), we start with a scaling relation that relates the exponents of energy and magnetic fluctuations (i.e., α and γ) with that of diverging correlation length associated with criticality:

An external file that holds a picture, illustration, etc.
Object name is srep45004-m14.jpg

where ν and α are exponents associated with correlation length An external file that holds a picture, illustration, etc.
Object name is srep45004-m15.jpg and specific heat An external file that holds a picture, illustration, etc.
Object name is srep45004-m16.jpg, respectively and d is the dimension. The first relation relates the diverging correlation length to the energy fluctuation whereas the second relation relates the same to the magnetic fluctuation (An external file that holds a picture, illustration, etc.
Object name is srep45004-m17.jpg and An external file that holds a picture, illustration, etc.
Object name is srep45004-m18.jpg). If a set of exponents originates from one underlying universality class then they must relate to the exponents of the parent universality (α0, ν0, γ0, δ0) as

An external file that holds a picture, illustration, etc.
Object name is srep45004-m19.jpg

so that the universal scaling relation [Eq. (4)] remains valid. Here, λ is the marginal parameter that drives the continuous variation and ω and κ are parameters yet to be determined. In Eq. (5), the variation of δ and γ are considered to be independent and equivalently one may vary η and ν independently and deduce the variation of other exponents from scaling relations (see section III of Supplementary information for details).

Clearly κ = 0 = ω is the parent universality class having unscaled (λ = 1) exponents. Other special cases are when κ or ω vanishes. When κ = 0, the continuous variation is governed by c = λω, In this case, we may set ω = 1 without loss of generality and identify c [equivalent] λ as the physical parameter that drives a continuous variation

An external file that holds a picture, illustration, etc.
Object name is srep45004-m20.jpg

This scenario is already known as weak universality where An external file that holds a picture, illustration, etc.
Object name is srep45004-m21.jpg and An external file that holds a picture, illustration, etc.
Object name is srep45004-m22.jpg are universal as in the old universality argument14. Another special case is ω = 0 and κ is set to be unity. Here γ remains invariant and

An external file that holds a picture, illustration, etc.
Object name is srep45004-m23.jpg

This variation, with λ being the gauge coupling constant, has been observed in strong coupling QED30. Now we turn our attention to the generic case where κ > 0 and ω is set to unity (generality is not compromised); the consequent variation of critical exponents is given by

An external file that holds a picture, illustration, etc.
Object name is srep45004-m24.jpg

This generic variation of exponents satisfies Widom scaling relation δ = 1 + γ/β for any λ, provided the parent universality class also obeys the same.

We further include a possibility that this generic variation may lead to β  0, a special limit where the phase transition becomes discontinuous. At this multi-critical limit, the usual scaling theory demands γ  1 and δ−1  058. The first requirement can be met if the multicritical point occurs at λ = γ0 and the second requirement determines

An external file that holds a picture, illustration, etc.
Object name is srep45004-m25.jpg

For the present experimental system, the parent class is HM3d (at y = 1) and accordingly κ = 1.29 (from Eq. (9) and Table 1). At doping y = 1, we may set λ = 1, but a general correspondence between y and the marginal parameter λ can not be obtained unless we know the correct interaction Hamiltonian. The best choice of λ that matches the exponents for y = 0.5, 0.6, 0.8 and 1 turns out to be 1.165, 1.087, 1.03 and 1.001 respectively. The best fit between λ and y with An external file that holds a picture, illustration, etc.
Object name is srep45004-m26.jpg yields A = 0.26, λ* = 0.97 and a = −0.4, which is shown in Fig. 3(b). This functional form An external file that holds a picture, illustration, etc.
Object name is srep45004-m27.jpg is used further in Eq. (8) to get continuous variation of β, γ and 1/δ with respect to y (shown as solid line in Fig. 3(c)). Moreover Eq. (8) suggests that the transition becomes discontinuous if λ > γ0 = 1.386 which corresponds to y ~ 0.37. In fact, a very sharp growth in the ordered moment just below TC along with thermal hysteresis (shown in Fig. 3(a)) has been observed in the present system for y < 0.444. In a related system (Sm1−yNdy)0.55Sr0.45MnO3, where Sr concentration differs slightly, the FM transition remains first order for 0  y  0.3343.

Figure 3
(a) Critical temperature TC for heating cycle (red circle) and the thermal hysteresis width ΔT (blue triangle) for different Nd concentration y. The values for y < 0.5 are taken from ref. 44. (b) The proposed scaling hypothesis ...

To emphasize that the new scaling hypothesis is indeed in work, we further investigate the scaling functions. If the critical behavior for different y is related to HM3d, one expects that the seemingly different universal scaling functions F in Fig. 2(a) must collapse onto each other if x and y axes are scaled by suitable constants. This is described in Fig. 2(b) - a good collapse of scaling functions for y = 0.5, 0.6, 0.8,1 to a unique universal form further supports that the observed criticality is only a rescaled form of Heisenberg fixed point.

Conclusions

In conclusion, we have made a comprehensive study of critical phenomenon in (Sm1−yNdy)0.52Sr0.48MnO3 single crystals with 0.5  y  1. The values of critical exponents (β, γ, δ) measured for y = 1 are consistent with Heisenberg universality class in three-dimension, whereas the same for y = 0.5, 0.6, 0.8 are far from any known universality class. All these exponents vary continuously with y, but they seem to obey the standard scaling laws following a single equation of state. The variation of exponents is not new to critical phenomena as it can be generated by a marginal interaction, but most examples (though there are a few exceptions) in both theoretical and experimental studies satisfy the weak universality14 where (β, γ, ν) vary but (η, δ) are fixed. We argue that, to be consistent with scaling, the continuous variation must occur in specific ways. Two special cases are, (a) δ remains unchanged, which leads to the weak universality, and (b) γ is unaltered, which results in a kind of variation observed in strong coupling QED30. This generic universality scenario, which leads to a multi-critical point, explains the continuous variation of critical exponents observed in (Sm1−yNdy)0.52Sr0.48MnO3 and correctly predicts the possibility of a discontinuous transition for doping y [less, similar] 0.37.

A marginal interaction that could provide variation of critical exponents beyond weak universality remains elusive. In particular, it is not clear, how or why a marginal operator is generated in all the above experimental conditions to drive continuous variations in specific ways. It is certainly challenging to devise a microscopic theory to accommodate this phenomenon.

Experimental Methods

The single crystals of (Sm1−yNdy)0.52Sr0.48MnO3 with y = 0.5, 0.6, 0.8 and 1.0 were prepared by floating zone technique under oxygen atmosphere59,60. Single crystallinity was confirmed by the Laue diffraction. The dc magnetization measurements were performed using a Quantum Design magnetic property measurement system (MPMS SQUID VSM) in fields up to 7 T. The data were collected after stabilizing the temperature for about 30 minutes. External magnetic field was applied along the longest sample direction and data were corrected for the demagnetization effect.

Additional Information

How to cite this article: Khan, N. et al. Continuously Varying Critical Exponents Beyond Weak Universality. Sci. Rep. 7, 45004; doi: 10.1038/srep45004 (2017).

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Material

Supplementary Information:

Acknowledgments

The authors would like to thank Mr. A. Pal for technical assistance.

Footnotes

The authors declare no competing financial interests.

Author Contributions P.M. conceptualized the problems, N.K., P.S. and A.M. did the experiment, N.K., P.S. and P.K.M. did the data analysis, P.S., P.M., and P.K.M. did the statistical analysis and wrote the manuscript.

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