Signatures of quantum criticalityin pure Cr at high pressureR. Jaramilloa, Yejun Fengb,c, J. Wangc, and T. F. Rosenbaumc,1

aSchool of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bThe Advanced Photon Source, Argonne National Laboratory,Argonne, IL 60439; and cThe James Franck Institute and Department of Physics, University of Chicago, Chicago, IL 60637

Edited by Laura H. Greene, University of Illinois at Urbana–Champaign, Urbana, IL, and approved June 28, 2010 (received for review April 13, 2010)

The elemental antiferromagnet Cr at high pressure presents a newtype of naked quantum critical point that is free of disorder andsymmetry-breaking fields. Here we measure magnetotransportin fine detail around the critical pressure, Pc ∼ 10 GPa, in a diamondanvil cell and reveal the role of quantum critical fluctuations at thephase transition. As the magnetism disappears and T → 0, themagntotransport scaling converges to a non-mean-field form thatillustrates the reconstruction of the magnetic Fermi surface, and isdistinct from the critical scaling measured in chemically disorderedCr∶V under pressure. The breakdown of itinerant antiferromagnet-ism only comes clearly into view in the clean limit, establishingdisorder as a relevant variable at a quantum phase transition.

antiferromagnetism ∣ spin density waves ∣ electric transport

Competition between magnetic and nonmagnetic states of mat-ter in the zero-temperature limit underlies the emergence of

exotic ground states such as non-Fermi liquid metals and uncon-ventional superconductors (1). This observation has motivatedseveral decades of work to understand the physics of magneticquantum phase transitions (QPT) (2–7). A substantial part ofthe effort has been directed at the materials science challengesthat are inherent to realizing nearly-magnetic states of matterand to the fine tuning of materials so that the phase transitionscan be probed systematically. The fundamental limitations thatremain are uncertainty over the role of disorder (2, 4, 8), as wellas a predilection for first-order transitions that shroud the quan-tum critical behavior (3, 5). Recent X-ray measurements identi-fied a continuous disappearance of magnetic order in theelemental antiferromagnet Cr near the critical pressure Pc∼10 GPa, and concurrent measurements of the crystal latticeacross the transition failed to detect any discontinuous changein symmetry or volume (9). These results identify Cr as a stoichio-metric itinerant magnet with a continuous QPT—where theeffects of the critical point should be manifest—and present arare opportunity to study quantum criticality in a theoreticallytractable system that is free from the effects of disorder. More-over, the use of hydrostatic pressure as a tuning parameter avoidsthe introduction of any confounding symmetry-breaking fields.

For the experimentalist, studying elemental Cr shifts the sig-nificant technical difficulties from solid state chemistry to highpressure experimentation. Here we report on high-resolutionmeasurements of the electrical resistivity and Hall coefficientof Cr as the system is tuned with pressure in a diamond anvil cellacross Pc. Magnetotransport is a sensitive probe of quantum cri-ticality and is widely used to identify and characterize quantummatter (4, 5, 8, 10). At ambient pressure Cr orders antiferromag-netically at the Néel temperature, TNðP ¼ 0Þ ¼ 311 K. BelowTN , electrons and holes form magnetic pairs and condense intoa spin density wave (SDW), in a process with strong analogies tothe Bardeen-Cooper-Schrieffer (BCS) formulation of electronpairing in superconductors (11). The quantum critical pointwhere TN → 0 can be reached either through applied pressure orchemical doping. Previous transport measurements of Cr1-xVx,x ¼ 3.2%, under pressure revealed a wide regime of quantum cri-tical scaling in this substitutionally disordered system (8). Doping

with electron-poor V to near-critical levels lowered the criticalpressure, making Pc accessible with a conventional clamp cell.Accessing the QPT in the pure system, on the other hand, re-quires high sensitivity measurements on submillimeter singlecrystals in a diamond anvil cell at low temperature (12, 13).

ResultsWe present here the results from experimental runs with sevendifferent samples, including two that were studied in fine detailin the critical regime. An overview of the resistivity for 0 < P <10 GPa is shown in Fig. 1A. The Néel transition is marked by asharp upturn in the resistivity, ρðTÞ, as the reduction in metalliccarrier density closely tracks the growth of the energy gap, gðTÞ,just below TN . This data is analyzed by first subtracting the para-magnetic background resistivity ρPMðTÞ, yielding the normalizedmagnetic excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ. This quantity isthen fit to a model function which accounts for the formationof a BCS-like energy gap below TN and the resulting freezing-out of carriers. This model function was successfully applied inan important early study of Cr under pressure by McWhanand Rice (13). By analyzing Δρ∕ρ (see Methods) we obtain thephase diagram of Fig. 1B. TNðPÞ evolves exponentially with pres-sure for P < 7 GPa with the form TNðPÞ ¼ TN;0 expð−cPÞ,TN;0 ¼ 310.9 0.9 K, c ¼ −0.163 0.001 GPa−1. Above 7 GPathis BCS-like exponential ground state breaks down as the systemapproaches the QPT.

The data analysis in the immediate vicinity of the QPT is pre-sented in logical progression in Fig. 2. We plot in Fig. 2A the elec-trical resistivity measured in fine detail in the quantum criticalregime. For T < 50 K the paramagnetic resistivity displays adominant T3 dependence. This is demonstrated in Fig. 2B wherewe plot ρðT3Þ, and for each pressure we limit the temperaturerange to T > TN in order to emphasize ρPMðTÞ. The T3 coeffi-cient b varies by less than 6% between samples and is welldescribed by metallic transport due to phonon scattering in thepresence of a weakly inelastic nonphonon scattering channel (14).Theory gives b∕d ¼ ð4.8∕Θ2Þ ¼ 1.74 × 10−5 K−2, where Θ is theDebye temperature (Θ ¼ 525 K (15)) and d is the linear tempera-ture coefficient of resistivity at high temperature. The coefficientd is determined from data for T > 315 K at P ¼ 0, and b is de-termined from data for T < 25 K in the paramagnetic phase athigh pressure. For the sample presented in Fig. 2 we findb∕d ¼ 1.95 0.15ð10−5 K−2Þ, in reasonable agreement with thetheoretical expectation. The T3 resistivity in this temperaturerange (vs. a T5 form) is typical for metallic samples with residualresistivities ρ0 ≥ 1 nΩ · cm (14, 16); our single-crystal Cr is99.996þ% pure and has ρ0 ≈ 0.1 μΩ · cm (compared to ρ0≈1.4 μΩ · cm in critically doped Cr∶V 3.2% (8)). The electronmean-free path in our samples is estimated to be λ > 400 Å atbase-T for all pressures P < Pc, where λ is calculated from the

Author contributions: R.J., Y.F., and T.F.R. designed research; R.J., Y.F., and J.W. performedresearch; R.J. and T.F.R. analyzed data; and R.J., Y.F., J.W., and T.F.R. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: tfr@uchicago.edu.

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measured Hall mobility. We note that the presence of finitequenched disorder in our samples is a necessary preconditionfor measuring a pressure-dependent residual resistivity. However,

the extremely low level of disorder suggests that pure Cr is abenchmark for how closely a QPT in a real solid state systemcan approach the clean limit.

We plot in Fig. 2C the excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρcalculated from the data in Fig. 2A and the ρPM background(see Fig. 2B and Methods). As P approaches Pc, it is preferableto analyze data for experimental cuts which are close to perpen-dicular to the increasingly steep phase diagram TNðPÞ. We extractsuch cuts from the data by considering isotherms of Δρ∕ρ. Theseisotherms Δρ∕ρjT are then fit to a power law Δρ∕ρjTðPÞ ¼aðPc;T − PÞβ, convolved with a Gaussian that accounts for thefinite pressure variation across the sample (see Methods: DataAnalysis in the Critical Regime). The 5 K isotherm and fit areplotted on the projected (P, Δρ∕ρ) plane in Fig. 2C, and a scalingplot of the data approaching the low-temperature limit is shownin Fig. 3A. The phase diagram is given by the fit parameters Pc;T ,and the exponent β relates to the breakdown of the SDW energygap and the reemergence of nested Fermi surface area; in theT → 0 limit β directly reflects the critical reconstruction of theFermi surface.

We present in Fig. 3 the resistivity scaling results for the quan-tum critical regime. The exponent β converges to 0.24 0.01 fortemperatures T ≤ 8 K (Fig. 3B). This exponent speaks to a rapidreconstitution of the Fermi surface that takes place in a narrowquantum critical regime, and stands in contrast to the value β ¼2∕3 which is seen at all temperatures in the pressure-driven quan-tum critical regime in Cr1-xVx, x ¼ 3.2% (8). β increases with tem-perature above 8 K, approaching the mean-field value, β ¼ 1∕2,or perhaps even β ¼ 2∕3, as the quantum critical point recedesfrom sight. However, due to limited data density and the difficultyof modeling ρPM at higher temperatures, we are not able to followβ to the point at which it settles at a high temperature limit. Thecrossover in temperature demonstrated in Fig. 3B is strongly re-miniscent of the crossover from quantum to classical critical scal-ing that is expected at finite temperatures in a system of itinerantfermions (6), although the applicability of the usual Landau-Ginzburg-Wilson (LGW) critical analysis to the case of nestedFermi surfaces remains in question (7). The critical phase dia-gram TNðPÞ ∝ ðPC − PÞγ is shown in Fig. 3C. The exponent γ de-termined from the two samples is 0.55 0.03 and 0.48 0.05,respectively, giving a best estimate γ ¼ 0.53 0.03, consistentwith the mean-field expectation, γ ¼ 1∕2, also observed forCr1-xVx, x ¼ 3.2%.

The critical reconstruction of the nested Fermi surface isfurther demonstrated by the Hall coefficient, RHðPÞ. The Halleffect is acutely sensitive to the quantum critical point, changing

P (GPa)

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3

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9

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50

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TF

P (GPa)

TN

(K

)

0 100 200 3000

5

10

15

T (K)

ρ (µ

Ω*c

m)

A

Fig. 1. Data overview and phase diagram for antiferromagnetic Cr underpressure. Data and results shown for all seven samples measured. (A)Resistivity ρðTÞ. (B) Antiferromagnetic phase diagram TNðPÞ. Black ¼determined directly from ρðTÞ curves; Blue ¼ determined indirectly fromX-ray measurements of the CDW diffraction intensity ICDW at low tempera-ture (9), from which the phase diagram can be calculated using the harmonicrelationship TN ∝ ICDW

1∕4. At low pressure the Néel transition is weakly firstorder, and is anticipated by thermal fluctuations (TF) for T > TN. At low tem-perature and for pressures P < 7 GPa the SDW is well described by the mean-field BCS-like theory and the phase diagram evolves exponentially with P (19,20). For pressures above 9 GPa this mean-field ground state is continuouslysuppressed by quantum critical (QC) fluctuations. Red shaded region indi-cates the quantum critical regime which is the focus of this paper.

0.1

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0.14

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P (GPa)9.2 9.4 9.6 9.8 10

Fig. 2. Data for 9 < P < 10 GPa for one of the two samples which were measured in detail in the quantum critical regime. The scaling results are the same forboth samples, but the different pressure conditions and residual resistivities make it difficult to clearly present raw data for both samples on the same plot. Thepressure colorbar applies to A–C. (A) Resistivity ρðTÞ. (B) ρðTÞ plotted against T3 for T < 50 K, with each curve truncated just above TN . Over this temperaturerange the paramagnetic background ρPMðTÞ is dominated by the shown T3 dependence. (C) The magnetic resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ, calculated from ρðTÞand the modeled ρPMðTÞ. Also shown (dashed red line) is the McWhan-Rice fit to the lowest pressure curve at 9.13 GPa, for which TN ¼ 37.9 0.03 K andg0∕kBTN ¼ 1.36 0.01 (error bars represent 1-σ variations from the nonlinear fit routine). (C, offset) Data and power law fit to the Δρ∕ρ isotherm at 5 K. Theexponent β ¼ 0.23 0.03 and the Gaussian pressure inhomogeneity is 0.24 GPa (FWHM).

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by 300% across the narrow critical regime at low temperature(Fig. 4). For P < Pc, the data can be described by the scaling formR−1H ∝ ð1 − Δρ∕ρÞ2, as predicted for a flat Fermi surface support-

ing a SDW (17). According to mean-field theory (17, 18), smalldeviations from ideal nesting will cause both RH and Δρ∕ρ toscale linearly with the SDWenergy gap g0 in the T → 0 limit. Un-der the conservative assumption that g0 ∝ TN the data indicatethat the gap scales with the mean-field exponent of 1∕2, whilethe Hall coefficient and the excess resistivity behave differently.The non-mean-field scaling which we observe for both RH andΔρ∕ρ implies that the observed critical behavior is driven notby the SDW energy gap, but by fluctuations that restore flat sec-tions of Fermi surface. Moreover, the lengthening of the SDWordering wavevector Q through the critical regime, in contrast tothe monotonically decreasing dependence of Q on P forP < 7 GPa, also has been interpreted as evidence for quantumcritical fluctuations (9).

DiscussionThe complete magnetic phase diagram of Cr can be considered inthree parts (illustrated in Fig. 1B). At low pressures the Néel tran-sition is weakly discontinuous, and the temperature regime T >TN is marked by incipient antiferromagnetic fluctuations that gobeyond the mean-field theory of the SDW (19). As a function ofpressure, however, this BCS-like theory accurately describes theobserved exponential dependence of both the phase boundaryTNðPÞ and the saturated order parameter at low temperature(19, 20). The exponential evolution results from a competitionbetween exchange energy and bandwidth that is tuned by appliedpressure while preserving the Fermi surface nesting condition(21). Above 7 GPa the exponential ground state breaks down andthe phase diagram approaches the QPT. Finally, at higher pres-sure the coherent SDW breaks down, quantum critical fluctua-tions dominate, and the nested Fermi surface reappears at acontinuous QPT.

We have shown that the critical scaling of this breakdown isdifferent in pure Cr than in Cr1-xVx, x ¼ 3.2%, which establishesthat substitutional disorder is a relevant variable at the pressure-driven QPT. Although the number and identity of relevant vari-ables are well known across the many categories of classical phasetransitions, the same is not true for their quantum counterparts.In both the clean and disordered limits we find that the exponentβ is distinct from the exponent γ ¼ 1∕2 that describes the scalingof the phase boundary, thus suggesting that the QPTs in both pureand V-doped Cr are driven by fluctuations that couple to theresistivity (18). However the value of β differs between the twocases, as does the scaling RHðPÞ ∝ ðPc − PÞα, which is mean-fieldin Cr∶V and distinctly non-mean-field with an exponent α close to1∕4 in pure Cr. V-doping is more efficient at driving the QPT, inthat the phase diagram departs from the exponential curve at alarger TN (or, equivalently, at a larger SDW coupling constant)for Cr1-xVx than for pure Cr under pressure (9). Furthermore,with V-doping the body centered cubic lattice expands and theSDW wavevector decreases monotonically, in contrast to thebehavior under pressure. This decrease in Q with V-doping re-sults from the fact that the band filling varies with electron-poordoping. However, barring the unrealistic scenario in which pureCr remains perfectly nested at all pressures, this change in bandfilling is not expected to alter the critical scaling at the QPT. Ourscaling results therefore demonstrate that the distinct micro-scopic effects of chemical doping (or “chemical pressure”) andhydrostatic pressure lead to distinct phase transitions, and

10−3

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10

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/2

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ρ)4 (

× 10

−3 )

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6 K

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/2

β = 1

/4

A

Fig. 3. Quantum critical scaling of the magnetic resistivity Δρ∕ρ and thephase boundary TN. (A) Scaling plot of Δρ∕ρ shows that the low-temperatureisotherms are well described by β ¼ 1∕4 and can be differentiated clearlyfrom the mean-field result, β ¼ 1∕2. (B) The Δρ∕ρ scaling exponent β con-verges to 0.24 0.01 for T ≤ 8 K. Results are shown for both samples (blackand gray, respectively) measured in detail in the critical regime. There is acrossover at higher T to a larger exponent, verging towards approachingeither mean-field behavior or the β ¼ 2∕3 found for Cr∶V. (C) The criticalscaling of TN , which is consistent with the mean-field exponent γ ¼ 1∕2for both samples (blue and red, respectively). A nearly constant offset of0.12 GPa was found between the phase diagrams measured for the two sam-ples. This offset characterizes the systematic uncertainty in our experiment,and although it does not affect error bars in the relative quantity (P-Pc), itdoes limit the accuracy to which we can determine Pc itself, which we reportas 9.71 0.08 GPa.

0 2 4 6 8 10 12

0.2

0.4

0.6

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1.2

P (GPa)

(RH

)−1 (

104 C

/cm

3 )

0.4 0.6 0.80.4

0.6

0.8

(RH

)−1 (104 C/cm3)

(ρP

M/ρ

AF)2

Fig. 4. Overview and critical scaling of the inverse Hall coefficient R−1H ðPÞ

measured at T ¼ 4.9 0.4 K. Inset shows R−1H ∝ ðρPM∕ρAFÞ2 scaling which is

predicted for a flat Fermi surface supporting a SDW; solid line is a linearfit to the data. Resistivity ρPM∕ρAF ¼ 1 − Δρ∕ρ (see Results) was recorded at5.0 0.1 K, and the plotted range is limited to P < Pc . The scaling relationshipis consistent with α ¼ β ≈ 1∕4, where the exponent α describes the T → 0

quantum critical scaling of the Hall coefficient.

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indicate that substitutional disorder must be considered a rele-vant variable for antiferromagnetic QPTs.

For superconducting copper oxides, the relevance of substitu-tional disorder at the postulated QPT remains an outstandingquestion. Recent transport measurements on La2-xSrxCuO4 athigh magnetic fields showed no clear signature of quantumcriticality near optimal hole doping, raising the prospect thatthe tuning parameter of the postulated QPT is substitutional dis-order (4). This situation bears similarities to the subject at hand:the scaling of Δρ∕ρ with pressure in disordered Cr1-xVx is broadand extends throughout the entire pressure-temperature plane,while pure Cr has a narrowly defined quantum critical regime.The role of substitutional disorder is somewhat better understoodin heavy fermion systems, and well characterized quantum criticalpoints have been found in a number of stoichiometric materials(7, 22). However, the critical spin density wave model, which un-doubtedly applies to Cr, does not capture the physics of heavyfermion quantum criticality. Crucially, the lack of local magneticmoments and the absence of effective mass divergences through-out the Brillouin zone separate the QPT in Cr from the heavyfermion examples (23).

Our results also stand in interesting contrast to a body of workon weak itinerant ferromagnets. For these systems, a line of con-tinuous thermal phase transitions terminates at a first-order QPT.The quantum critical regime is inaccessible, but both the mag-netic and nonmagnetic ground states are often characterizedby strong quantum fluctuations that destabilize the Fermi liquid(3, 5, 24). By contrast, the magnetic ground state of Cr is welldescribed by mean-field theory, with signatures of quantum fluc-tuations only developing within the narrow quantum critical re-gime. The outstanding feature common to both Cr and itinerantferromagnets appears to be a tricritical point in the pressure-tem-perature plane, where the quartic stiffness of the order parameterpasses through zero.

The nature of the quantum fluctuations at the QPTremains anopen question. Assuming the applicability of the traditional LGWformalism to nested fermions in three-dimensions, dimensionalarguments allow effects beyond mean-field in the quantum re-gime. The relation γ ¼ z∕ðdþ z − 2Þ implies a dynamical expo-nent z ¼ 1 and a scaling dimension dþ z ¼ 4, which is not overthe upper critical dimension (6, 22). Furthermore, the quasi-one-dimensional dispersion relation at the nested Fermi surface(which is the origin of the R−1

H ∝ ð1 − Δρ∕ρÞ2 scaling (17)) mayresult in a reduced effective dimension for the critical fluctua-tions, as has been observed for quantum critical heavy fermionsystems (25). The critical reconstruction of the nested Fermi sur-face in Cr is accompanied by the reemergence of nested fermionswith greatly enhanced exchange interactions and as the quantumcritical point is uncovered, it drives a weak coupling BCS-likestate towards strong-coupling physics (19, 26). The persistence ofstrongly interacting fermions above Pc also opens the possi-bility for the ground state that replaces the SDW to be charac-terized by short-coherence length pairing, akin to the BCS-BEC(Bose Einstein condensate) crossover observed in ultracoldgasses, or to a pseudogap-like state of dynamical pair fluctuations.

MethodsMagnetotransport in a Diamond Anvil Cell at Cryogenic Temperatures. Allmeasurements were performed in a low-temperature diamond anvil cellequipped with a He gas membrane for fine pressure control. The pressuremedium was a methanol∶ethanol 4∶1mixture. Pressure was measured in situusing the ruby fluorescence method. The pressure P is calculated from thewavelength λ of the ruby R1 fluorescence by P ¼ A · lnðλ∕λ0Þ, where λ0 isthe (temperature-dependent) R1 wavelength at ambient pressure. A has

been calibrated directly (12) at 5 K (A5 K ¼ 1;762 13 GPa) and room-T(A295 K ¼ 1;868 GPa). To interpolate between these two temperatures weassume that A is constant up to 100 K, above which it varies linearly withtemperature, in qualitative accordance with the temperature dependenceof the bulk modulus of Al2O3. Resistivity ρ and Hall coefficient RH were mea-sured in the four probe van der Pauw geometry on single-crystal Cr platesusing an ac resistance bridge. RH was derived from data taken in the range−3;500 < H < 3;500 Oe, which is in the low-field limit for all pressures. Themicroscopic samples, size ð200 × 200 × 40Þ μm3 with (111)-oriented faces,were derived from large Cr single-crystal discs (Alfa Aesar, 99.996þ%) bya procedure described elsewhere (27). The gold leads were spot-welded tothe sample and insulated from themetallic gasket using amixture of aluminapowder and epoxy.

McWhan-Rice Model. For pressures up to∼0.3 GPa below Pc the phase diagramwas determined by fitting Δρ∕ρ to the McWhan-Rice model (13). This modelhas three free parameters: theNéel temperature TN , the T → 0 energy gap g0,and the magnetic Fermi surface fraction q (note the typo in Eq. 6 of ref. 13,where E3∕2 is written instead of E3; for the correct expression see Eq. 6.16 ofref. 28). We implemented this model with an additional free parameter dTN

which describes the width of a Gaussian distribution in TN . This convolutionallows for pressure inhomogeneity and is valid as long as the variation in TN

withpressure is approximately linear over the rangedTN . The convolutionwasimplemented numerically, holding q constant and scaling g0 linearly with TN .Modeling ρPMðTÞ is easy at low temperatures (approximately TN < 50 K orP > 8.8 GPa), where it obeys the expected form ρPMðTÞ ¼ ρ0 þ bT3 þ cT5

and the T3 dependence dominates (14, 16). In this regime the McWhan-Ricefit parameters are robust. At higher T modeling ρPMðTÞ is difficult, and theMcWhan-Rice fit results for g0 and q are strongly correlated with the formassumed for ρPMðTÞ. The result for TN, however, remains robust. As a checkwe also estimated TN from ρðTÞ by simply finding the temperature at whichρðTÞ has the most negative slope. This point is assumed to correspond to thattemperature at which the energy gap gðTÞ grows the fastest, which is identi-fied with TN . For all P < 9 GPa the discrepancy between these results and theMcWhan-Rice approach is less than the size of the data points in Fig. 1B; forP > 9 GPa this simpler technique fails due to the increasing influence of thepressure inhomogeneity as the phase diagram steepens near Pc.

Data Analysis in the Critical Regime. For pressures within ∼0.3 GPa of PC theMcWhan-Rice fits fail for two reasons. First, our lowest measurement tem-perature of 4.5 K is too high for the Δρ∕ρ form to fully develop (Fig. 2C),and as a result the fit parameters are poorly determined. Second, the finitepressure inhomogeneity produces a smearing of TN that diverges at Pc . It ispreferable to consider the isotherms Δρ∕ρjT , which are fit to a power lawΔρ∕ρjT ðPÞ ¼ aðPc;T − PÞβ convolved with a Gaussian pressure distribution.The 5 K isotherm and best-fit curve are shown in Fig. 2C, and a scaling plotof the data approaching the low-temperature limit is shown in Fig. 3A. Thecritical exponents βðTÞ are plotted in Fig. 3B, and the fit parameters Pc;T

define the phase diagram which is plotted in Fig. 3C.For the two samples studied in fine detail in the critical regime, the best-fit

FWHM of the Gaussian pressure distribution was 0.24 and 0.33 GPa, respec-tively, for the T ¼ 5 K data. For a given sample this fit parameter is then heldconstant for fits to all T > 5 K isotherms to reduce systematic correlations be-tween fit parameters. The Gaussian pressure distributions correspond to a 2-σwidth of 0.43 and 0.58 GPa, respectively, somewhat smaller than the 0.72 GPabase width that was found for the pressure inhomogeneity over a ð200×200Þ μm2 area in a recent study of the pressure conditions in the samemethanol∶ethanol medium at 10 GPa and 5 K (12). The pressure conditionis characteristic of a given cell assembly and depends mainly on choice ofpressure medium and the sample-to-chamber volume ratio. A nearly con-stant offset of 0.12 GPa of the critical phase boundaries measured for twodifferent samples could result from several systematic issues, most notablythe position of the ruby chips (the ruby is positioned to the side of thesample, close to the gasket wall where the pressure gradients are highest).

ACKNOWLEDGMENTS.We acknowledge Arnab Banerjee and Peter Littlewoodfor enlightening discussions. The work at the University of Chicago wassupported by National Science Foundation (NSF) Grant DMR-0907025.

1. Sachdev S (2000) Quantum criticality: competing ground states in low dimensions.

Science 288:475–480.

2. Dai J, Si Q, Zhu J-X, Abrahams E (2009) Iron pnictides as a new setting for quantum

criticality. Proc Nat'l Acad Sci USA 106:4118–4121.

3. Pfleiderer C, et al. (2007) Non-Fermi liquid metal without quantum criticality. Science

316:1871–1874.

4. Cooper RA, et al. (2009) Anomalous criticality in the electrical resistivity of

La2-xSrxCuO4 . Science 323:603–607.

13634 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1005036107 Jaramillo et al.

Dow

nloa

ded

by g

uest

on

Janu

ary

6, 2

021

5. Smith RP, et al. (2008) Marginal breakdown of the Fermi-liquid state on the border ofmetallic ferromagnetism. Nature 455:1220–1223.

6. Millis AJ (1993) Effect of a nonzero temperature on quantum critical points in itinerantfermion systems. Phys Rev B 48:7183–7196.

7. Löhneysen HV, Rosch A, Vojta M, Wölfle P (2007) Fermi-liquid instabilities at magneticquantum phase transitions. Rev Mod Phys 79:1015–1075.

8. Lee M, Husmann A, Rosenbaum TF, Aeppli G (2004) High resolution study of magneticordering at absolute zero. Phys Rev Lett 92:187201.

9. Jaramillo R, et al. (2009) Breakdown of the Bardeen-Cooper-Schrieffer ground state ata quantum phase transition. Nature 459:405–409.

10. Yeh A, et al. (2002) Quantum phase transition in a common metal. Nature419:459–462.

11. Overhauser AW (1962) Spin density waves in an electron gas. Phys Rev 128:1437–1452.12. Feng Y, Jaramillo R, Wang J, Ren Y, Rosenbaum TF (2010) High pressure techniques for

condensed matter physics at low temperature. Rev Sci Instrum 81:041301.13. McWhan DB, Rice TM (1967) Pressure dependence of itinerant antiferromagnetism in

chromium. Phys Rev Lett 19:846–849.14. Campbell IA, Caplin AD, Rizzuto C (1971) Momentum nonconservation and the

low-temperature resistivity of alloys. Phys Rev Lett 26:239–242.15. Diana M, Mazzone G (1972) Absolute X-ray measurement of the atomic scattering

factor of chromium. Phys Rev B 5:3832–3836.16. Caplin AD, Rizzuto C (1970) Breakdown of Matthiessen’s rule in aluminum alloys.

Journal of Physics C: Solid State Physics 3:L117–L120.17. Norman MR, Qimiao S, Bazaliy YB, Ramazashvili R (2003) Hall effect in nested

antiferromagnets near the quantum critical point. Phys Rev Lett 90:116601.

18. Bazaliy YB, Ramazashvili R, Si Q, Norman MR (2004) Magnetotransport near a quan-tum critical point in a simple metal. Phys Rev B 69:144423.

19. Jaramillo R, et al. (2008) Chromium at high pressures: weak coupling and strongfluctuations in an itinerant antiferromagnet. Phys Rev B 77:184418.

20. Feng Y, et al. (2007) Pressure-tuned spin and charge ordering in an itinerant antiferro-magnet. Phys Rev Lett 99:137201.

21. Jaramillo R, Feng Y, Rosenbaum TF (2010) Diffraction line-shapes, Fermi surface nest-ing, and quantum criticality in antiferromagnetic chromium at high pressure. J AppPhys 107:09E116.

22. Varma CM, Nuzzinov Z, Saarloos WV (2002) Singular or non-Fermi liquids. Phys Rep361:267–417.

23. Hayden SM, Doubble R, Aeppli G, Perring TG, Fawcett E (2000) Strongly enhancedmagnetic excitations near the quantum critical point of Cr1-xVx and why strongexchange enhancement need not imply heavy fermion behavior. Phys Rev Lett84:999–1002.

24. Pfleiderer C (2007) On the identification of Fermi-liquid behavior in simple transitionmetal compounds. J Low Temp Phys 147:231–247.

25. Schroder A, Aeppli G, Bucher E, Ramazashvili R, Coleman P (1998) Scaling of magneticfluctuations near a quantum phase transition. Phys Rev Lett 80:5623–5626.

26. Aynajian P, et al. (2008) Energy gaps and Kohn anomalies in elemental superconduc-tors. Science 319:1509–1512.

27. Feng Y, et al. (2005) Energy dispersive X-ray diffraction of charge density waves via achemical filtering. Rev Sci Instrum 76:063913.

28. Jérome D, Rice TM, Kohn W (1967) Excitonic insulator. Phys Rev 158:462–475.

Jaramillo et al. PNAS ∣ August 3, 2010 ∣ vol. 107 ∣ no. 31 ∣ 13635

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