Quantum critical fluctuations in layered YFe2Al10L. S. Wua, M. S. Kima, K. Parka, A. M. Tsvelikb, and M. C. Aronsona,b,1

aDepartment of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794; and bCondensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, NY 11973

Edited by Zachary Fisk, University of California, Irvine, CA, and approved August 8, 2014 (received for review July 14, 2014)

The absence of thermal fluctuations at T = 0 makes it possible toobserve the inherently quantum mechanical nature of systemswhere the competition among correlations leads to different typesof collective ground states. Our high precision measurements of themagnetic susceptibility, specific heat, and electrical resistivity in thelayered compound YFe2Al10 demonstrate robust field-temperaturescaling, evidence that this system is naturally poised without tuningon the verge of ferromagnetic order that occurs exactly at T = 0,where magnetic fields drive the system away from this quantumcritical point and restore normal metallic behavior.

quantum criticality | ferromagnet | dynamical scaling

The interplay of competing interactions is responsible forthe array of ground states that are possible in correlated

electron systems. It is of particular importance to understandhow one such ground state gives way to another when the systemis tuned at temperature T = 0, without the complications ofthermal fluctuations. Consequently, much interest has focused onquantum critical points (QCPs), where an ordered phase can becreated by an infinitesimal modifications of pressure, composi-tion, or field. The onset of ferromagnetic order is perhaps thesimplest T = 0 phase transition, and indeed much experimentaland theoretical effort has been directed toward understanding itsessential features (1–6). It is generally believed that ferro-magnetic order occurs at T = 0 via a discontinuous or firstorder transition, as is observed in the clean ferromagnetsZrZn2 (7) and MnSi (8) under pressure. Disorder is known torender the ferromagnetic transition continuous, leading to themean field behavior that is found when doping drives TC = 0 inNi1−xPdx (9), Zr1−xNbxZn2 (10), and Nb1−yFe2+y (11). Morecontroversial is the possibility that strong quantum fluctuations,such as those that destabilize order in low-dimensional systems,may be significant near the TC = 0 ferromagnetic transition andperhaps may even destroy its first-order character (5). A com-plete experimental investigation of the critical phenomena andtheir scaling behaviors in a carefully selected system wheredisorder is minimal is needed to establish that quantum criticalfluctuations are both present and relevant to the destabilizationof ferromagnetic order.Progress toward obtaining this information has been pains-

taking, although a number of systems have been identifiedwhere the Curie temperature TC has been driven to zero. Highpressure experiments evade the disorder that necessarilyaccompanies doping, but thus far only resistivity and suscepti-bility measurements have been reported. Complete experi-mental access is possible when doping is used to suppress TC →0, but even small amounts of compositional inhomogeneity canobscure any intrinsic critical fluctuations of TC = 0 ferromag-netic transitions, resulting in interesting complications such asthe Griffiths phase (12, 13, 14), as well as short ranged order,including spin glasses (15). Alternative ordered states that rangefrom unconventional superconductivity in UGe2 (16) and UCoGe(17), antiferromagnetic order in CeRu2Ge2 (18), hidden order inURu1−xRexSi2 (19), and spiral order in MnSi (20) may alsoemerge when ferromagnetism becomes sufficiently weak, poten-tially masking quantum critical fluctuations associated with theunderlying TC = 0 ferromagnetic transition. Finally, the electronic

delocalization via the Kondo effect in the f-electron based heavyfermions (21) or by proximity to a Mott transition in d-electron-based systems may also contribute to the destabilization of fer-romagnetic order. The discovery of a new ferromagnetic systemthat is naturally tuned to the onset of order at TC = 0 would bean enabling development, because a wide range of experimentaltools could then be used to assemble a holistic picture of thequantum critical fluctuations and their connection to the un-derlying criticality, providing important feedstock for futuretheoretical developments.We will present here experimental data and a detailed

scaling analysis of the magnetization, specific heat, andelectrical resistivity measured on YFe2Al10 single crystals thatestablish its properties are dominated at the lowest temper-atures and fields by the quantum critical fluctuations of aTC = 0 ferromagnetic transition. Although YFe2Al10 is not or-dered above 0.1 K (22), the correlations that drive this quasi-2D metal to the brink of ferromagnetic order are derived fromthe hybridization of Fe-based d-electrons with conductionelectrons. Disorder effects are expected to be minimal in thissystem, as single crystal X-ray diffraction measurements findno evidence of departures from stoichiometry or site disorder(22, 23).The layered nature of YFe2Al10 is evident from its crystal

structure (Fig. 1A), which features nearly square nets of Featoms that form the ac planes (23, 24). Magnetically, YFe2Al10can be considered quasi-2D. Measurements of the alternatingcurrent (AC) magnetic susceptibility χac(T) (Fig. 1B) finda strong temperature divergence with χac ∼ T−γ (γ = 1.4 ± 0.05),but only when the ac field Bac = 4.17 Oe lies in the ac plane.When Bac is parallel to the b axis, χac does not diverge, and itsvalue at 1.8 K is almost 30 times smaller than when the field isin the ac plane, where there is no measurable anisotropy. Thisdivergence in χac does not culminate in magnetic order, at leastfor temperatures above 0.1 K (22). Instead it indicates that the

Significance

Temperature-driven phase transitions, such as the melting ofice or the boiling of water, are a familiar part of daily life. Muchless is known about the most extreme phase transitions, whichhappen only at zero temperature, where quantum fluctuationslimit the stabilities of different collective ground states such asmagnetic order and superconductivity. We report an experi-mental investigation of YFe2Al10, where planes of Fe atoms arenaturally poised on the verge of ferromagnetic order, exactlyat T = 0. The thermal, magnetic, and electrical transportproperties in YFe2Al10 all diverge as T→ 0, a process that can bereversed by magnetic fields in a way that is dictated by theunderlying system energy.

Author contributions: L.S.W., K.P., and M.C.A. designed research; L.S.W., M.S.K., K.P., andM.C.A. performed research; L.S.W., A.M.T., and M.C.A. analyzed data; and L.S.W., A.M.T.,and M.C.A. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: maronson@bnl.gov.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1413112111/-/DCSupplemental.

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magnetic properties of YFe2Al10 are dominated below ∼20 K byproximity to a T = 0 phase transition.What type of T = 0 phase transition is responsible for the

quantum criticality in YFe2Al10? The magnitude of the uniformsusceptibility and the strength of its divergence χ(T) ∼ T−1.4, aswell as the divergence of the spin lattice relaxation time 1/T1Tfrom NMR measurements (25), together suggest that it involvesa uniform zero wave vector q = 0 or ferromagnetic instability.YFe2Al10 is a rare example of a system that forms naturally veryclose to a ferromagnetic QCP, without the need for tuning bypressure or composition.Our investigation of the field and temperature dependencies

of the magnetization M(T, B) reveals that the field B suppressesthe quantum critical fluctuations. Fig. 2A shows that the diver-gence of the direct current (DC) magnetic susceptibility χ = M/Bis suppressed in field and that χ(T) becomes increasingly tem-perature independent and smaller in magnitude with increasingfield, approaching the Fermi liquid (FL) behavior expected fora conventional metal, where χ is temperature independent. Thebroad maximum in χ(T), even more evident in the ac suscepti-bility χac (Fig. S1), defines a new scale T⋆(B) that marks thecross-over between the quantum critical (QC) (T � B) and FL(T � B) regimes, and the inset of Fig. 2A shows that T⋆(B) ∼B0.59. This duality between T and B suggests the possibility of

T/B0.59 scaling, and indeed a remarkable scaling collapse of themagnetic susceptibility M/B (Fig. 2B), is found for 1.8 K < T < 30K and B < 6 T, extending over more than three orders of mag-nitude in the scaling variable x = T/B0.59, where

−dχdT

B1:4 =ψ�

TB0:59

�: [1]

This scaling reveals an important property of the underlying freeenergy, which controls both the magnetic and thermodynamicproperties. Namely, the quantum critical T/B0.59 scaling can beunderstood on the basis of a generic free energy F that assumesthe validity of hyperscaling, where the spatial dimensionality d isaugmented by a dynamic exponent z, i.e., deff = d + z near thisT = 0 phase transition (2–4, 26):

FðB;TÞ=Td+zz ~f F

�B

T yb=z

�=B

d+zyb fF

�T

Bz=yb

�: [2]

Here yb is the scaling exponent that relates to the tuning param-eter B, which can alternatively be written in terms of the corre-lation length exponent ν, where z/yb = νz. Although it is believedthat there are different critical time scales for the order param-eter and for the fermionic degrees of freedom near a ferromag-netic QCP (5, 6), our analysis assumes that only one of thesetime scales dominates near the QCP in YFe2Al10, and conse-quently, only a single z is required. The field-temperature scalingof the magnetic susceptibility can be determined from that of thefree energy F, where x=T=Bz=yb and ~x=B=Tyb=z:

dχdT

=ddT

�d2FdB2

�=Bd=yb−2f ′χ

�T

Bz=yb

�: [3]

The critical exponents can be read directly from the sus-ceptibility scaling plot (Fig. 2B), giving d = z and νz = z/yb =0.59. The precision of the experimental data confers a corre-sponding precision to our determination of these criticalexponents, whose range of values is severely curtailed by thesharp minimum in the scaling residuals, shown in the inset ofFig. 2B and in more detail in Fig. S2. The values of d, z, and ybfound in YFe2Al10 imply a particularly simple scaling form forthe specific heat, which can be obtained from the same freeenergy F

Fig. 1. Quasi-2D quantum criticality. (A) Crystal structure of YFe2Al10,formed by stacking nearly square nets of Fe along the b axis. Gray solid lineindicates unit cell. Each Fe atom is surrounded by a distorted octahedron ofAl atoms, capped with Y atoms. (B) The ac susceptibility χac(T) diverges whenthe ac field Bac = 4.17Oe is in the ac plane, but not when Bac is parallel to theb axis. Red line is fit χac ∼ T−γ with γ = 1.4.

A B

Fig. 2. Field−temperature scaling and Fermi liquid breakdown: Magnetic susceptibility. (A) Temperature dependencies of the dc susceptibility χ = M/Bmeasured in different fixed fields, as indicated. Solid red line has slope γ = 1.4. (Inset) Contour plot of the same data with T⋆(B) taken from maxima in dM/dT(red rimmed circles; Fig. S1A) and dM/dB (blue rimmed circles; Fig. 4B). Red and blue solid lines have T⋆(B) ∼ B0.59, indicating the cross-over from QC to FLregimes. (B) Scaling collapse of the temperature derivative of χ as a function of the variable T/B0.59. Red line indicates fit from the scaling function fM(x), asdescribed in the text. An alternative plot of the data in B on linear scales is given in Fig. S2. (Inset) Net deviations from scaling determined for different valuesof the critical exponents 2 − d/yb and z/yb (Fig. S2E).

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CMðB;TÞT

=−∂2F∂T2 =T

d−zz ~f Cð~xÞ=B

d−zyb fCðxÞ: [4]

Using d = z, we find that

ΔCMðB;TÞT

=CMðB;TÞ

T−CMð0;TÞ

T= T0~f Cð~xÞ=B0fCðxÞ: [5]

Assuming that ~f Cð~xÞ can be separated into field-dependentand field-independent parts ~f Cð~xÞ=~f Cð0Þ+~f CðB=Tyb=zÞ, wesee from Eq. 5 that no power law divergence is possible forthe B = 0 specific heat CM/T, which is instead predicted to beconstant, implying that the observed temperature divergence ofCM/T is weaker than any power law, consistent with its observedlogarithmic temperature divergence (Fig. 3A and Fig. S3). Magneticfields suppress the divergence in ΔCM/T, much as they do for thesusceptibility χ(T), resulting in a FL-like temperature independenceof CM/T ∼ γ(B) (Fig. 3B). Eq. 5 predicts that T/B0.59 also controlsthe scaling of ΔCM/T. Fig. 3C confirms this expectation for T <10 K and fields as large as 7 T, over more than three orders ofmagnitude in the scaling variable x=T=Bz=yb . In addition to thefree energy (Eq. 2) that underlies the observed scaling, the in-ternal consistency of the magnetization and specific heat hasbeen verified in SI Text, where we demonstrate that their fieldand temperature dependencies obey a Maxwell relation, as isrequired by thermodynamics.An intriguingly simple expression for the scaling function

fF(x) can be found that not only describes the T/B0.59 scaling ofM/B, but which also reproduces the observed B = 0 and T =0 limits of both M/B and ΔCM/T. Inspired by similar expres-sions that were found for heavy fermions CeCu6−xAux (27) andβ-YbAlB4 (28), we propose

M =Bd+zyb

−1fM

�T

Bz=yb

�; [6]

with

fMðxÞ= c�x2 + a2

�−γ=2:

Here c and a are the fitting parameters. The parameter a is de-termined by fitting the maxima T⋆=Bz=yb = x⋆ =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2=ðγ + 1Þ

p≈ 3 in

dM/dT, where we have used the experimentally determined expo-nents d = z, z/yb = 0.59 and γ ∼ 1.4. The resulting parameter-freeexpression is Eq. 6 compared with the experimental data in Fig.2B, and the agreement is excellent over the full range of x. Thisscaling function fM(x) implies a certain scaling function for theunderlying free energy fF(x), which can in turn be related to thescaling function fC(x) for ΔCM/T. Although the details of thisprocedure are given in SI Text, its accuracy is demonstrated bythe excellent agreement between the deduced expression for thescaled specific heat ΔCM/T = fC(x) and the scaled data (Fig. 3C).The internal consistency of the scaling functions fF(x), fM(x), andfC(x) that is implied by this comparison suggests that specific ex-pressions for the temperature and field dependencies of ΔCM/Tand M/B can be deduced in their respective QC (T � B) and FL(T � B) limits.The experimental scaling relations that have been established

here provide a precise formulation of how the quantum criticalfluctuations lead to the breakdown of the high field FL phase.In the QC limit, the expected field-independent divergence inχac(T, B → 0) ∼ T−γ is recovered (Fig. 1B), as well as the B = 0logarithmic divergence CM/T ∼ –log(T) (Fig. 3A), albeit with acorrection B2T−γ−2 in small magnetic fields. Deep in the FL(T � B), the field dependencies of the Pauli susceptibility are

A B

DC

Fig. 3. Field−temperature scaling and Fermi liquid breakdown: Specific heat. (A) Temperature dependence of the B = 0 specific heat CM/T. An estimate of thephonon contribution has been subtracted to isolate the purely magnetic and electronic specific heat CM/T (22). (B) CM/T measured in different fixed fields, asindicated. (C) Scaling collapse of ΔCM/T = CM(B, T)/T − CM(0, T)/T, as a function of the variable T/B0.59. Red line indicates fit with scaling function fC(x), describedin the text. An alternative plot of these data on linear axes appears in Fig. S2. (D) Temperature dependence of the magnetic Grüneisen parameter Γ/B,determined in different fixed fields (SI Text).

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given by χðBÞ= dM=dB∼Bðd+zÞ=yb−2 (Fig. 4A) and the Som-merfeld coefficient is given by γ(B) = CM(B)/T ∼ −log(B) (Fig.4B, Inset). By comparing the QC and FL expressions, it is clearthat the field dependencies of CM(B)/T at fixed temperaturesare nonmonotonic, explaining the maxima that separate thehigh field FL and low field QC phases in Fig. 4B. The loga-rithmic field divergence of the T = 0 Sommerfeld coefficientγ(B, T) = CM(B, T)/T (Fig. 4B, Inset) implies a weaker di-vergence of the quasi-particle mass at the QCP than the powerlaw divergencies that are generally found in other QC systems(29, 30). However, the field divergence of the Pauli suscepti-bility (Fig. 4A) is too strong to be attributed entirely to that ofthe quasi-particle mass, reflecting instead a ferromagnetic en-hancement of the susceptibility that is echoed in the unusuallystrong divergence of χac(T) ∼ T−1.4 found for B = 0. By thesame token, the distinctly different temperature dependenciesof C/T and χ(T) also rules out the possibility of a Griffithsphase, where both would have the same critical exponents (14,31–33).It has been argued that the universal behaviors near different

types of QCPs can be established through the divergence of theGrüneisen ratio (34). For field-tuned magnetic systems, themagnetic Grüneisen ratio Γ is given by (SI Text)

ΓB=−

½dðM=BÞ=dT�=TC=T

∝T−γ−2

logT∼T−3:4

logT: [7]

Because the power law divergence of the numerator is muchstronger than the logarithmic divergence of the denominator,the magnetic Grüneisen parameter Γ/B is expected to be stronglydivergent at low temperatures, at least in low fields. This di-vergence is confirmed by the experimental data (Fig. 3D),where Γ is calculated indirectly from the measured ac mag-netic susceptibility χac with Bac = 4.17 Oe and the measuredzero field specific heat. The strong temperature divergencefor B = 0 establishes the quantum criticality of YFe2Al10, aswell as the role of magnetic field as a scaling variable thatsuppresses the QC fluctuations.The electrical resistivity ρ(T) is a measure of how strongly cou-

pled the quasi-particles are to the QC fluctuations that dominatethe magnetic susceptibility and specific heat at low temperaturesand fields. We carried out measurements of the anisotropy ofρ(T), using a modified Montgomery technique (35). Fig. 5A showsthat there is virtually no resistive anisotropy in the ac plane, wherethe resistivities ρIka ∼ ρIkc are approximately a factor of 2 largerthan the interplanar resistivity ρIkb, even at temperatures wherethe QC fluctuations no longer contribute to the magnetization orspecific heat. This anisotropy is much weaker than that found

in the magnetic susceptibility (Fig. 1B), suggesting that thequasi-particles are only weakly coupled to the critical fluctu-ations, reflecting instead the part of the quasi-2D character ofYFe2Al10 that can be traced to its layered crystal structure andits influence on the underlying electronic structure. The largemagnitudes of ρIka,c imply T = 0 sheet resistances for persquare Fe atoms in the ac plane Rsquare ≈ 2 kΩ, and consid-ering as well their relatively weak temperature dependencies,we infer that there is quite strong quasi-particle scattering inYFe2Al10, although little variation is found among crystalsfrom different batches (Fig. 5A, Inset), making the role of residualdisorder unclear. Both ρIkb and ρIka,c have metallic temperature

A B

Fig. 4. Field-driven Fermi liquid collapse. Field dependencies of χ = dM/dB (A) and CM/T (B) at different temperatures as indicated. Inset in B shows CM/T ∼−log(B) (red line), although the cross-over to quantum critical behavior at the lowest fields is an inevitable result of the nonzero measuring temperatureT = 0.55 K.

A B

C D

Fig. 5. Quantum critical electrical resistivity ρ(T). (A) The in-plane resistivi-ties ρIka and ρIkc are larger than the interplanar resistivity ρIkb. Data obtainedusing the modified Montgomery method (35). (Inset) ρ(T) measured for fourdifferent YFe2Al10 samples, in each case with Ika. (B) The effects of a 9-Tmagnetic field on ρ(T ) where Ika. (C ) ρ(T ) measured with Ika in differentfields Bka. (D) Scaling collapse of Δρ(B, T ) = ρ(B, T ) − ρ(0, T ) for tem-peratures 0.2–50 K and fields as large as 14 T. Different samples wereused for A–C. We estimate a systematic error of ∼10–20% arises fromuncertainties in sample dimensions for each voltage measurement,where the Montgomery method requires at each temperature six volt-age measurements with different current directions to arrive at ρIka, ρIka,and ρIka. The T → 0 resistivities for all measured samples are consistentwith the systematic errors.

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dependencies, indicating that the quasi-particles in YFe2Al10 aredefinitively not localized.The QC fluctuations provide an additional scattering mecha-

nism at low temperatures and fields that is strongest in the acplanes. The T → 0 resistive upturn in YFe2Al10 is suppressedby fields in the ac plane, but it is much less sensitive to fieldsapplied parallel to the b axis (Fig. 5B), the same anisotropyfound in the ac susceptibility χac (Fig. 1B). The part of theresistive upturn that is diminished with increasing field (Fig.5C) Δρ(B, T) = ρ(B, T) − ρ(0, T) undergoes a scaling collapsefor 0.2 K < T < 50 K and B < 14 T, with the same scalingvariable x = T/B0.59 that was found for M/B and CM/T (Fig.5D), where

ΔρðB;TÞT

= fR

�T

B0:59

�: [8]

This scaling confirms that the resistive upturn in YFe2Al10 resultsfrom increasingly strong scattering as the QCP is approached bydecreasing either T or B. Theories of quasi-particle scattering fromcritical fluctuations in ferromagnets find that the specific heatCM(B, T) is proportional to ρ(B, T) (∂ρ/∂T) when long (short)wavelength critical fluctuations dominate the quasi-particlescattering (36, 37). However, the scaling functions fC(x) andfR(x) in Eqs. 4 and 8 are not equal and cannot be simply related,ruling out these scenarios. Spin disorder scattering of quasi-particles from fluctuations of the moments enforces a proportion-ality between Δρ(B, T) and the magnetization M. This expla-nation is impossible in YFe2Al10, because the scaling form (Eq.8) cannot be reduced to a function of the magnetization (Eq. 6),due to the temperature prefactor of the latter. The traditionalpictures fail badly in YFe2Al10, suggesting that the fluctuatingmoments and the conduction electrons may not be entirelyseparable. Indeed, the low temperature resistive upturn is rem-iniscent of the Kondo effect in normal metals, and the B = 0 ρ(T)can be satisfactorily fitted (22) by the Kondo expression (38).However, the resistivity for a fixed number of Kondo compen-sated moments in a conventional metallic host scales with T/B(39, 40), and so a different explanation is needed for the T/B0.59

scaling found in YFe2Al10 (Fig. 5D). The observation of T/B0.59

scaling in the electrical resistivity is evidence that the order pa-rameter fluctuations are coupled to soft quasi-particle modes inYFe2Al10, although the influence of the QCP on the resistivity ismuch weaker than for the uniform susceptibility or specific heat,confirming that the quasi-particles are not coupled stronglyenough to the QC fluctuations to themselves become fully critical.Our primary experimental results in YFe2Al10 are the

divergencies in the uniform susceptibility χac ∼ T−γ (γ = 1.4) andin the specific heat CM/T ∝ −log(T), whereas field-temperaturescaling is found in the magnetization, specific heat, and elec-trical resistivity, in each case giving νz ≡ z/yb ∼ 0.59. Are thesecritical exponents consistent with theoretical predictions? Webegin with the standard Hertz-Moriya-Millis (HMM) theory ofmetals near QCPs (2–4), which is a mean-field theory thatignores the spatial and temporal fluctuations of the incipientordered phase. HMM theory for a d = z = 3 ferromagnet predictsboth a divergence of the magnetic susceptibility χac ∼ T−γ withγ = 4/3, close to the observed value γ = 1.4 ± 0.05 (Fig. 1B), andCM/T ∼ −log(T) (Fig. 3A). However, the observation of T/B0.59

scaling is itself inconsistent with HMM, where the truncatedequation of state B =M/χ0 + μBM3 with constant μ and χ0 ∼ T−4/3

implies a scaling variable x = T/B0.5. As is evident from the insetof Fig. 2B, this choice of z/yb = 0.5 leads to a scaling of themagnetic susceptibility data that is markedly inferior to that withz/yb = 0.59 (Fig. S2 E and F). This failure of HMM implies thatthe critical fluctuations of a T = 0 phase transition are responsible

in YFe2Al10 for the T = 0 divergencies of χ and CM/T and thatthey can be suppressed by magnetic fields to restore a featurelessand conventional metallic state.A field theory approach has been taken to derive values of

these QC exponents in clean and disordered ferromagnets indifferent dimensions d (5, 41). The theory reproduces the loga-rithmic divergence in C/T that we observe in YFe2Al10, as well asthe observed d = z, for disordered ferromagnets with 2 < d < 6.The situation for disordered ferromagnets is complicated be-cause the paramagnons and the diffusive quasi-particle modes havedifferent dynamical exponents zc = d and zq = 2, respectively (5),and thus two different critical time scales may in principle bepresent. The experimental observation d = z suggests that theorder parameter fluctuations may prevail in YFe2Al10, an ob-servation that is supported by the noncriticality of the overallresistivity. However, we stress that the experimental values ofthe susceptibility exponent γ = 1.4 and the scaling variable νz ≡z/yb ∼ 0.59 are not reproduced in this theory.The need to treat order parameter fluctuations and soft quasi-

particles on equal footing greatly increases the difficulty of theo-retical analysis in QC metallic ferromagnets, where a localGinzburg-Landau effective action for the order parameter fielddoes not exist (5). As this controversy continues, we resorted in-stead to a phenomenological description of the T/B0.59 scaling thatwas observed in YFe2Al10, based on a generic free energy withhyperscaling, where a single time scale was assumed to becomecritical. Augmented by a specific expression for the T/B0.59

scaling function, we reproduced the observed field and tem-perature dependencies of the quantum critical magnetizationand specific heat using a single set of critical exponents (d = z,γ = 1.4, νz = 0.59). The general success of this heuristic analysis,based on the hypothesis of hyperscaling, suggests that YFe2Al10is a system that is below its upper critical dimension, perhapsdue to its 2D character. Hyperscaling generally implies universality,where the form of the free energy does not depend on the details ofthe system, implying that critical points can be classified intouniversality classes based on the values of the critical exponentsthemselves. We propose that YFe2Al10 is, to our knowledge,the first confirmed member of such a universality class.Mean field theories predict that the collapse of ferromagnetic

order is via a first-order transition (5, 6) in the absence of dis-order or by a continuous mean field transition in the presence ofdisorder (7–10). According to ref. 5, it remains the case whenstrong quantum fluctuations are present, as may be realized inlow-dimensional systems. The data presented here establish that,although YFe2Al10 does not order, strong divergencies in theuniform susceptibility and specific heat indicate QC fluctuationsare indeed dominant as this system approaches a T = 0 ferro-magnetic transition. Of course, it would be of great interest touse compositional modifications to drive YFe2Al10 to the onset offerromagnetic order, provided this could be done without in-troducing significant disorder. Whether or not this putative TC =0 transition is ultimately found to be first order, it now seems likelythat ferromagnetic quantum criticality is protected in low-di-mensional systems like quasi-2D YFe2Al10, as it is in quasi-onedimensional YbNi4(P1−xAsx)2 (42).The evidence presented here shows that YFe2Al10 is a rare

example of a quasi-2D metallic system where critical fluctuationsassociated with a TC = 0 ferromagnetic transition dominate thespecific heat and uniform susceptibility at the lowest temper-atures and fields, without the need for compositional or pressuretuning. Our magnetization, specific heat, and electrical resistivitymeasurements have allowed us to propose and experimentallyestablish an expression for an underlying QC free energy thatunifies the scaling found in the susceptibility, specific heat, andthe QC part of the resistivity while assuring a strong divergenceof the magnetic Grüneisen parameter. These experimental re-sults extend a challenge to theory to develop new frameworks in

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which to understand the emergence of ferromagnetic order and,in particular, its impact on the underlying quasi-particles inmetallic systems.

ACKNOWLEDGMENTS. We thank M. Garst, M. Brando, and F. Steglich foruseful discussions. Work at Brookhaven National Laboratory was carried outunder the auspices of US Department of Energy, Office of Basic EnergySciences, Contract DE-AC02-98CH1886.

1. Stoner EC (1938) Collective electron ferromagnetism. Proc R Soc Math Phys Eng Sci165(922):372–414.

2. Hertz JA (1976) Quantum critical phenomena. Phys Rev B 14(3):1165–1184.3. Moriya T (1985) Spin Fluctuations in Itinerant Electron Magnetism, Springer Series in

Solid-State Sciences (Springer, Berlin), Vol 56.4. Millis AJ (1993) Effect of a nonzero temperature on quantum critical points in

itinerant fermion systems. Phys Rev B Condens Matter 48(10):7183–7196.5. Belitz D, Kirkpatrick TR, Vojta T (2005) How generic scale invariance influences

quantum and classical phase transitions. Rev Mod Phys 77(2):579–632.6. Lohneysen H, Rosch A, Vojta M, Wolfle P (2007) Fermi liquid instabilities at magnetic

quantum phase transitions. Rev Mod Phys 79(3):1015–1075.7. Uhlarz M, Pfleiderer C, Hayden SM (2004) Quantum phase transitions in the itinerant

ferromagnet ZrZn2. Phys Rev Lett 93(25):256404.8. Pfleiderer C, McMullen GJ, Julian SR, Lonzarich GG (1997) Magnetic quantum phase

transition in MnSi under hydrostatic pressure. Phys Rev B 55(13):8330–8338.9. Nicklas M, et al. (1999) Non-Fermi-Liquid behavior at a ferromagnetic quantum critical

point in NixPd1−x. Phys Rev Lett 82(21):4268–4271.10. Sokolov DA, Aronson MC, Gannon W, Fisk Z (2006) Critical Phenomena and the Quan-

tum Critical Point of Ferromagnetic Zr1−xNbxZn2. Phys Rev Lett 96(11):116404-1–116404-4.

11. Brando M, et al. (2008) Logarithmic Fermi-liquid breakdown in NbFe2. Phys Rev Lett101(2):026401.

12. Westerkamp T, et al. (2009) Kondo-cluster-glass state near a ferromagnetic quantumphase transition. Phys Rev Lett 102(20):206404.

13. Vojta T, et al. (2010) Quantum Griffiths effects and smeared phase transitions inmetals: Theory and experiment. J Low Temp Phys 161(1-2):299–323.

14. Ubaid-Kassis S, Vojta T, Schroeder A (2010) Quantum Griffiths phase in the weakitinerant ferromagnetic alloy Ni(1-x)V(x). Phys Rev Lett 104(6):066402.

15. Lausberg S, et al. (2012) Avoided ferromagnetic quantum critical point: Unusualshort-range ordered state in CeFePO. Phys Rev Lett 109(21):216402.

16. Saxena SS, et al. (2000) Superconductivity on the border of itinerant-electron ferro-magnetism in UGe2. Nature 406(6796):587–592.

17. Huy NT, et al. (2007) Superconductivity on the border of weak itinerant ferromag-netism in UCoGe. Phys Rev Lett 99(6):067006.

18. Süllow S, et al. (1999) Doniach phase diagram, revisited: From ferromagnet to Fermiliquid in pressurized CeRu2Ge2. Phys Rev Lett 82:2963–2966.

19. Butch NP, Maple MB (2009) Evolution of critical scaling behavior near a ferromagneticquantum phase transition. Phys Rev Lett 103(7):076404.

20. Pfleiderer C, et al. (2004) Partial order in the non-Fermi-liquid phase of MnSi. Nature427(6971):227–231.

21. Yamamoto SJ, Si Q (2010) Metallic ferromagnetism in the Kondo lattice. Proc NatlAcad Sci USA 107(36):15704–15707.

22. Park K, et al. (2011) Field-tuned Fermi liquid in quantum critical YFe2Al10. Phys Rev B84(9):094425-1–094425-7.

23. Kerkau A, et al. (2012) Crystal structure of yttrium iron aluminium (1/2/10), YFe2Al10.Z Kristallogr, New Cryst Struct 227:289–290.

24. Thiede VMT, Ebel T, Jeitschko W (1998) Ternary aluminides LnT2Al10 (Ln=Y, La-Nd,Sm, Gd-Lu and T=Fe, Ru, Os) with YbFe2Al10 type structure and magnetic propertiesof the iron-containing series. J Mater Chem 8(1):125–130.

25. Khuntia P, et al. (2012) Field-tuned critical fluctuations in YFe2Al10: Evidence frommagnetization, 27Al NMR, and NQR investigations. Phys Rev B 86(22):220401-1–220401-5.

26. Sachdev S (1999) Quantum Phase Transitions (Cambridge Univ Press, Cambridge, UK).27. Schroder A, et al.; v. Lohneysen H (2000) Onset of antiferromagnetism in heavy-fer-

mion metals. Nature 407(6802):351–355.28. Matsumoto Y, et al. (2011) Quantum criticality without tuning in the mixed valence

compound β-YbAlB4. Science 331(6015):316–319.29. Shishido H, Settai R, Harima H, Onuki Y (2005) A drastic change of the Fermi surface

at a critical pressure in CeRhIn5: dHvA study under pressure. J Phys Soc Jpn 74(4):1103–1106.

30. Gegenwart P, Custers J, Tokiwa Y, Geibel C, Steglich F (2005) Ferromagnetic quantumcritical fluctuations in YbRh2(Si0.95Ge0.05)2. Phys Rev Lett 94(7):076402–076404.

31. Castro Neto AH, Castilla G, Jones BA (1998) Non-Fermi liquid behavior and Griffithsphase in f-electron compounds. Phys Rev Lett 81(16):3531–3534.

32. Castro Neto AH, Jones BA (2000) Non-Fermi-liquid behavior in U and Ce alloys: Criticality,disorder, dissipation, and Griffiths-McCoy singularities. Phys Rev B 62(22):14975–15011.

33. Millis AJ, Morr DK, Schmalian J (2002) Quantum Griffiths effects in metallic systems.Phys Rev B 66(17):174433-1–174433-10.

34. Zhu L, Garst M, Rosch A, Si Q (2003) Universally diverging Grüneisen parameter andthe magnetocaloric effect close to quantum critical points. Phys Rev Lett 91(6):066404-1–066404-4.

35. Dos Santos CAM, et al. (2011) Procedure for measuring electrical resistivity of aniso-tropic materials: A revision of theMontgomery method. J Appl Phys 110(8):083703–083707.

36. de Gennes PG, Friedel J (1958) Anomalies de résistivité dans certains métaux mag-níques. J Phys Chem Solids 4(1):71–77.

37. Fisher ME, Langer JS (1968) Resistive anomalies at magnetic critical points. Phys RevLett 20(13):665–668.

38. Hamann DR (1967) New solution for exchange scattering in dilute alloys. Phys Rev158(3):570–580.

39. Schlottmann P (1983) Bethe-Ansatz solution of the ground-state of the SU (2j+1)Kondo (Coqblin-Schrieffer) model: Magnetization, magnetoresistance, and univer-sality. Z Phys B 51(3):223–235.

40. Batlogg B, et al. (1987) Superconductivity and heavy fermions. J Magn Magn Mater63-64:441–446.

41. Kirkpatrick TR, Belitz D (1996) Quantum critical behavior of disordered itinerantferromagnets. Phys Rev B Condens Matter 53(21):14364–14376.

42. Steppke A, et al. (2013) Ferromagnetic quantum critical point in the heavy-fermionmetal YbNi4(P(1-x)As(x))2. Science 339(6122):933–936.

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