Signatures of quantum criticality in pure Cr at high pressure

R. 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 new with electron-poor V to near-critical levels lowered the critical type of naked quantum critical point that is free of disorder and pressure, making Pc accessible with a conventional clamp cell. symmetry-breaking fields. Here we measure magnetotransport Accessing the QPT in the pure system, on the other hand, re- in fine detail around the critical pressure, Pc ∼ 10 GPa, in a diamond quires high sensitivity measurements on submillimeter single anvil cell and reveal the role of quantum critical fluctuations at the crystals in a diamond anvil cell at low temperature (12, 13). . As the magnetism disappears and T → 0, the magntotransport scaling converges to a non-mean-field form that Results illustrates the reconstruction of the magnetic Fermi surface, and is We present here the results from experimental runs with seven distinct from the critical scaling measured in chemically disordered different samples, including two that were studied in fine detail 0 Cr∶V under pressure. The breakdown of itinerant antiferromagnet- in the critical regime. An overview of the resistivity for < P < 10 ism only comes clearly into view in the clean limit, establishing GPa is shown in Fig. 1A. The Néel transition is marked by a ρð Þ disorder as a relevant variable at a . sharp upturn in the resistivity, T , as the reduction in metallic carrier density closely tracks the growth of the energy gap, gðTÞ, ∣ spin density waves ∣ electric transport just below TN . This data is analyzed by first subtracting the para- ρ ð Þ magnetic background resistivity PM T , yielding the normalized

Δρ∕ρ ¼ðρ − ρ Þ∕ρ PHYSICS ompetition between magnetic and nonmagnetic states of mat- magnetic excess resistivity PM . This quantity is ter in the zero-temperature limit underlies the emergence of then fit to a model function which accounts for the formation C T exotic ground states such as non-Fermi liquid metals and uncon- of a BCS-like energy gap below N and the resulting freezing- out of carriers. This model function was successfully applied in ventional superconductors (1). This observation has motivated an important early study of Cr under pressure by McWhan several decades of work to understand the physics of magnetic Δρ∕ρ – and Rice (13). By analyzing (see Methods) we obtain the quantum phase transitions (QPT) (2 7). A substantial part of ð Þ phase diagram of Fig. 1B. TN P evolves exponentially with pres- the effort has been directed at the materials science challenges 7 ð Þ¼ ð− Þ sure for P < GPa with the form TN P TN;0 exp cP , that are inherent to realizing nearly-magnetic states of matter −1 T 0 ¼ 310.9 0.9 K, c ¼ −0.163 0.001 GPa . Above 7 GPa and to the fine tuning of materials so that the phase transitions N; this BCS-like exponential ground state breaks down as the system can be probed systematically. The fundamental limitations that approaches the QPT. remain are uncertainty over the role of disorder (2, 4, 8), as well The data analysis in the immediate vicinity of the QPT is pre- as a predilection for first-order transitions that shroud the quan- sented in logical progression in Fig. 2. We plot in Fig. 2A the elec- tum critical behavior (3, 5). Recent X-ray measurements identi- trical resistivity measured in fine detail in the quantum critical fied a continuous disappearance of magnetic order in the regime. For T < 50 K the paramagnetic resistivity displays a elemental antiferromagnet Cr near the critical pressure P ∼ 3 c dominant T dependence. This is demonstrated in Fig. 2B where 10 GPa, and concurrent measurements of the crystal lattice 3 we plot ρðT Þ, and for each pressure we limit the temperature across the transition failed to detect any discontinuous change 3 range to T>T in order to emphasize ρ ðTÞ. The T coeffi- in symmetry or volume (9). These results identify Cr as a stoichio- N PM cient b varies by less than 6% between samples and is well metric itinerant magnet with a continuous QPT—where the described by metallic transport due to scattering in the effects of the critical point should be manifest—and present a presence of a weakly inelastic nonphonon scattering channel (14). rare opportunity to study quantum criticality in a theoretically 2 −5 −2 Theory gives b∕d ¼ð4.8∕Θ Þ¼1.74 × 10 K , where Θ is the tractable system that is free from the effects of disorder. More- Debye temperature (Θ ¼ 525 K (15)) and d is the linear tempera- over, the use of hydrostatic pressure as a tuning parameter avoids ture coefficient of resistivity at high temperature. The coefficient the introduction of any confounding symmetry-breaking fields. d is determined from data for T>315 KatP ¼ 0, and b is de- For the experimentalist, studying elemental Cr shifts the sig- termined from data for T < 25 K in the paramagnetic phase at nificant technical difficulties from state chemistry to high high pressure. For the sample presented in Fig. 2 we find pressure experimentation. Here we report on high-resolution −5 −2 b∕d ¼ 1.95 0.15ð10 K Þ, in reasonable agreement with the measurements of the electrical resistivity and Hall coefficient theoretical expectation. The T3 resistivity in this temperature of Cr as the system is tuned with pressure in a diamond anvil cell range (vs. a T5 form) is typical for metallic samples with residual across P . Magnetotransport is a sensitive probe of quantum cri- c resistivities ρ0 ≥ 1 nΩ · cm (14, 16); our single-crystal Cr is ticality and is widely used to identify and characterize quantum 99.996 þ % pure and has ρ0 ≈ 0.1 μΩ · cm (compared to ρ0≈ matter (4, 5, 8, 10). At ambient pressure Cr orders antiferromag- 1 4 μΩ ∶ ð ¼ 0Þ¼311 . · cm in critically doped Cr V 3.2% (8)). The electron netically at the Néel temperature, TN P K. Below mean-free path in our samples is estimated to be λ > 400 Åat TN , electrons and holes form magnetic pairs and condense into base-T for all pressures P < P , where λ is calculated from the a spin density wave (SDW), in a process with strong analogies to c the Bardeen-Cooper-Schrieffer (BCS) formulation of electron pairing in superconductors (11). The quantum critical point Author contributions: R.J., Y.F., and T.F.R. designed research; R.J., Y.F., and J.W. performed → 0 research; R.J. and T.F.R. analyzed data; and R.J., Y.F., J.W., and T.F.R. wrote the paper. where TN can be reached either through applied pressure or The authors declare no conflict of interest. chemical doping. Previous transport measurements of Cr1-xVx, x ¼ 3.2%, under pressure revealed a wide regime of quantum cri- This article is a PNAS Direct Submission. tical scaling in this substitutionally disordered system (8). Doping 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.1005036107 PNAS Early Edition ∣ 1of5 Downloaded by guest on September 26, 2021 15 A P (GPa) the extremely low level of disorder suggests that pure Cr is a benchmark for how closely a QPT in a real solid state system 9 can approach the clean limit. Δρ∕ρ ¼ðρ − ρ Þ∕ρ We plot in Fig. 2C the excess resistivity PM 10 ρ calculated from the data in Fig. 2A and the PM background 6 (see Fig. 2B and Methods). As P approaches Pc, it is preferable

µΩ *cm) to analyze data for experimental cuts which are close to perpen- (

ρ ð Þ dicular to the increasingly steep phase diagram TN P . We extract 5 such cuts from the data by considering isotherms of Δρ∕ρ. These 3 Δρ∕ρj Δρ∕ρj ð Þ¼ isotherms T are then fit to a power law T P ð − Þβ a Pc;T P , convolved with a Gaussian that accounts for the finite pressure variation across the sample (see Methods: Data 0 0 0 100 200 300 Analysis in the Critical Regime). The 5 K isotherm and fit are Δρ∕ρ T (K) plotted on the projected (P, ) plane in Fig. 2C, and a scaling plot of the data approaching the low-temperature limit is shown in Fig. 3A. The phase diagram is given by the fit parameters P , B 300 ρ c;T TF (P,T) and the exponent β relates to the breakdown of the SDW energy 250 I (P, T < 8 K) gap and the reemergence of nested Fermi surface area; in the CDW T → 0 limit β directly reflects the critical reconstruction of the 200 Fermi surface. We present in Fig. 3 the resistivity scaling results for the quan- (K)

N 150 β 0 24 0 01

T tum critical regime. The exponent converges to . . for temperatures T ≤ 8 K (Fig. 3B). This exponent speaks to a rapid 100 reconstitution of the Fermi surface that takes place in a narrow quantum critical regime, and stands in contrast to the value β ¼ 50 BCS 2∕3 which is seen at all temperatures in the pressure-driven quan- QC ¼ 3 2% β 0 tum critical regime in Cr1-xVx, x . (8). increases with tem- 0 2 4 6 8 10 perature above 8 K, approaching the mean-field value, β ¼ 1∕2, P (GPa) or perhaps even β ¼ 2∕3, as the quantum critical point recedes from sight. However, due to limited data density and the difficulty Fig. 1. Data overview and phase diagram for antiferromagnetic Cr under ρ of modeling PM at higher temperatures, we are not able to follow pressure. Data and results shown for all seven samples measured. (A) β ρð Þ ð Þ ¼ to the point at which it settles at a high temperature limit. The Resistivity T .(B) Antiferromagnetic phase diagram T N P . Black determined directly from ρðTÞ curves; Blue ¼ determined indirectly from crossover in temperature demonstrated in Fig. 3B is strongly re-

X-ray measurements of the CDW diffraction intensity ICDW at low tempera- miniscent of the crossover from quantum to classical critical scal- ture (9), from which the phase diagram can be calculated using the harmonic ing that is expected at finite temperatures in a system of itinerant ∝ 1∕4 relationship T N ICDW . At low pressure the Néel transition is weakly first fermions (6), although the applicability of the usual Landau- order, and is anticipated by thermal fluctuations (TF) for T>TN. At low tem- Ginzburg-Wilson (LGW) critical analysis to the case of nested perature and for pressures P < 7 GPa the SDW is well described by the mean- Fermi surfaces remains in question (7). The critical phase dia- field BCS-like theory and the phase diagram evolves exponentially with P (19, ð Þ ∝ ð − Þγ γ gram TN P PC P is shown in Fig. 3C. The exponent de- 20). For pressures above 9 GPa this mean-field ground state is continuously termined from the two samples is 0.55 0.03 and 0.48 0.05, suppressed by quantum critical (QC) fluctuations. Red shaded region indi- γ ¼ 0 53 0 03 cates the quantum critical regime which is the focus of this paper. respectively, giving a best estimate . . , consistent with the mean-field expectation, γ ¼ 1∕2, also observed for ¼ 3 2% Cr1-xVx, x . . measured Hall mobility. We note that the presence of finite The critical reconstruction of the nested Fermi surface is ð Þ quenched disorder in our samples is a necessary precondition further demonstrated by the Hall coefficient, RH P . The Hall for measuring a pressure-dependent residual resistivity. However, effect is acutely sensitive to the quantum critical point, changing

A 0.18 B 0.18 C 0.3 0.16 0.14

µΩ *cm) 0.2 ( ∆ρ

0.14 ρ / ρ µΩ *cm) (

ρ 0.1 0.12 0.1 0 5 10 T = 5 K 3 4 3 T (10 K ) 9 0 P 9.5 20 40 0.1 (G 10 0 P a) T (K) 0 10 20 30 40 50 9.2 9.4 9.6 9.8 10 T (K) P (GPa)

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 for both samples, but the different pressure conditions and residual resistivities make it difficult to clearly present raw data for both samples on the same plot. The – ρð Þ ρð Þ 3 50 pressure colorbar applies to A C.(A) Resistivity T .(B) T plotted against T for T < K, with each curve truncated just above T N. Over this temperature ρ ð Þ 3 Δρ∕ρ ¼ðρ − ρ Þ∕ρ ρð Þ range the paramagnetic background PM T is dominated by the shown T dependence. (C) The magnetic resistivity PM , calculated from T ρ ð Þ ¼ 37 9 0 03 and the modeled PM T . Also shown (dashed red line) is the McWhan-Rice fit to the lowest pressure curve at 9.13 GPa, for which T N . . K and ∕ ¼ 1 36 0 01 σ Δρ∕ρ g0 kBT N . . (error bars represent 1- variations from the nonlinear fit routine). (C, offset) Data and power law fit to the isotherm at 5 K. The exponent β ¼ 0.23 0.03 and the Gaussian pressure inhomogeneity is 0.24 GPa (FWHM).

2of5 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1005036107 Jaramillo et al. Downloaded by guest on September 26, 2021 1.2 A 6 5 K 0.8 2 ) ) 6 K

1 AF −3 ρ /

4 ) 3 10 7 K 0.6 PM ×

= 1/4 ρ ( β 0.8 ( 4 8 K ) ρ C/cm /

2 4 0.4 ∆ρ

( = 1/2 0.6 0.4 0.6 0.8 β (10

−1 −1 4 3 0 ) (R ) (10 C/cm ) H H 0.4 −2 0 2 4 6 (R (P −P)/P (%) c c 0.2 1 B 024681012 β 0.75 (∆ρ/ρ) ∝ (P −P) P (GPa) T c,T −1ð Þ Fig. 4. Overview and critical scaling of the inverse Hall coefficient RH P β ¼ 4 9 0 4 −1 ∝ ðρ ∕ρ Þ2 0.5 measured at T . . K. Inset shows RH PM AF scaling which is predicted for a flat Fermi surface supporting a SDW; solid line is a linear ρ ∕ρ ¼ 1 − Δρ∕ρ fit to the data. Resistivity PM AF (see Results) was recorded at 0.25 5 0 0 1 . . K, and the plotted range is limited to P < Pc. The scaling relationship is consistent with α ¼ β ≈ 1∕4, where the exponent α describes the T → 0 0 quantum critical scaling of the Hall coefficient. 0 10 20 T (K) Discussion C 30 The complete magnetic phase diagram of Cr can be considered in three parts (illustrated in Fig. 1B). At low pressures the Néel tran-

sition is weakly discontinuous, and the temperature regime T> PHYSICS T is marked by incipient antiferromagnetic fluctuations that go γ = 1/2 N

(K) 10 beyond the mean-field theory of the SDW (19). As a function of N

T pressure, however, this BCS-like theory accurately describes the observed exponential dependence of both the phase boundary ð Þ TN P and the saturated order parameter at low temperature (19, 20). The exponential evolution results from a competition 3 −3 −2 10 10 between exchange energy and bandwidth that is tuned by applied pressure while preserving the Fermi surface nesting condition (P −P)/P C C (21). Above 7 GPa the exponential ground state breaks down and Fig. 3. Quantum critical scaling of the magnetic resistivity Δρ∕ρ and the the phase diagram approaches the QPT. Finally, at higher pres- Δρ∕ρ phase boundary T N.(A) Scaling plot of shows that the low-temperature sure the coherent SDW breaks down, quantum critical fluctua- isotherms are well described by β ¼ 1∕4 and can be differentiated clearly tions dominate, and the nested Fermi surface reappears at a from the mean-field result, β ¼ 1∕2.(B) The Δρ∕ρ scaling exponent β con- continuous QPT. 0 24 0 01 ≤ 8 verges to . . for T K. Results are shown for both samples (black We have shown that the critical scaling of this breakdown is and gray, respectively) measured in detail in the critical regime. There is a ¼ 3 2% crossover at higher T to a larger exponent, verging towards approaching different in pure Cr than in Cr1-xVx, x . , which establishes either mean-field behavior or the β ¼ 2∕3 found for Cr∶V. (C) The critical that substitutional disorder is a relevant variable at the pressure- γ ¼ 1∕2 scaling of T N, which is consistent with the mean-field exponent driven QPT. Although the number and identity of relevant vari- for both samples (blue and red, respectively). A nearly constant offset of ables are well known across the many categories of classical phase 0.12 GPa was found between the phase diagrams measured for the two sam- transitions, the same is not true for their quantum counterparts. ples. This offset characterizes the systematic uncertainty in our experiment, In both the clean and disordered limits we find that the exponent and although it does not affect error bars in the relative quantity (P-Pc), it β is distinct from the exponent γ ¼ 1∕2 that describes the scaling does limit the accuracy to which we can determine Pc itself, which we report as 9.71 0.08 GPa. of the phase boundary, thus suggesting that the QPTs in both pure and V-doped Cr are driven by fluctuations that couple to the resistivity (18). However the value of β differs between the two by 300% across the narrow critical regime at low temperature ð Þ ∝ ð − Þα cases, as does the scaling RH P Pc P , which is mean-field (Fig. 4). For P < Pc, the data can be described by the scaling form ∶ α −1 ∝ ð1 − Δρ∕ρÞ2 in Cr V and distinctly non-mean-field with an exponent close to RH , as predicted for a flat Fermi surface support- 1∕4 ing a SDW (17). According to mean-field theory (17, 18), small in pure Cr. V-doping is more efficient at driving the QPT, in deviations from ideal nesting will cause both R and Δρ∕ρ to that the phase diagram departs from the exponential curve at a H larger T (or, equivalently, at a larger SDW coupling constant) scale linearly with the SDW energy gap g0 in the T → 0 limit. Un- N ∝ for Cr1 xVx than for pure Cr under pressure (9). Furthermore, der the conservative assumption that g0 TN the data indicate - that the gap scales with the mean-field exponent of 1∕2, while with V-doping the body centered cubic lattice expands and the the Hall coefficient and the excess resistivity behave differently. SDW wavevector decreases monotonically, in contrast to the behavior under pressure. This decrease in Q with V-doping re- The non-mean-field scaling which we observe for both RH and Δρ∕ρ implies that the observed critical behavior is driven not sults from the fact that the band filling varies with electron-poor by the SDW energy gap, but by fluctuations that restore flat sec- doping. However, barring the unrealistic scenario in which pure tions of Fermi surface. Moreover, the lengthening of the SDW Cr remains perfectly nested at all pressures, this change in band ordering wavevector Q through the critical regime, in contrast to filling is not expected to alter the critical scaling at the QPT. Our the monotonically decreasing dependence of Q on P for scaling results therefore demonstrate that the distinct micro- P < 7 GPa, also has been interpreted as evidence for quantum scopic effects of chemical doping (or “chemical pressure”) and critical fluctuations (9). hydrostatic pressure lead to distinct phase transitions, and

Jaramillo et al. PNAS Early Edition ∣ 3of5 Downloaded by guest on September 26, 2021 ¼ 1 762 13 indicate that substitutional disorder must be considered a rele- been calibrated directly (12) at 5 K (A5 K ; GPa) and room-T ¼ 1 868 vant variable for antiferromagnetic QPTs. (A295 K ; GPa). To interpolate between these two temperatures we For superconducting copper oxides, the relevance of substitu- assume that A is constant up to 100 K, above which it varies linearly with tional disorder at the postulated QPT remains an outstanding temperature, in qualitative accordance with the temperature dependence ρ of the bulk modulus of Al2O3. Resistivity and Hall coefficient RH were mea- question. Recent transport measurements on La2-xSrxCuO4 at high magnetic fields showed no clear signature of quantum sured in the four probe van der Pauw geometry on single-crystal Cr plates using an ac resistance bridge. R was derived from data taken in the range criticality near optimal hole doping, raising the prospect that H −3;500 < H < 3;500 Oe, which is in the low-field limit for all pressures. The the tuning parameter of the postulated QPT is substitutional dis- microscopic samples, size ð200 × 200 × 40Þ μm3 with (111)-oriented faces, order (4). This situation bears similarities to the subject at hand: were derived from large Cr single-crystal discs (Alfa Aesar, 99.996 þ %)by Δρ∕ρ the scaling of with pressure in disordered Cr1-xVx is broad a procedure described elsewhere (27). The gold leads were spot-welded to and extends throughout the entire pressure-temperature plane, the sample and insulated from the metallic gasket using a mixture of alumina while pure Cr has a narrowly defined quantum critical regime. powder and epoxy. The role of substitutional disorder is somewhat better understood ∼0 3 in heavy fermion systems, and well characterized quantum critical McWhan-Rice Model. For pressures up to . GPa below Pc the phase diagram points have been found in a number of stoichiometric materials was determined by fitting Δρ∕ρ to the McWhan-Rice model (13). This model → 0 (7, 22). However, the critical spin density wave model, which un- has three free parameters: the Néel temperature T N,theT energy gap g0, and the magnetic Fermi surface fraction q (note the typo in Eq. 6 of ref. 13, doubtedly applies to Cr, does not capture the physics of heavy 3∕2 3 fermion quantum criticality. Crucially, the lack of local magnetic where E is written instead of E ; for the correct expression see Eq. 6.16 of moments and the absence of effective mass divergences through- ref. 28). We implemented this model with an additional free parameter dT N out the Brillouin zone separate the QPT in Cr from the heavy which describes the width of a Gaussian distribution in T N. This convolution allows for pressure inhomogeneity and is valid as long as the variation in T N fermion examples (23). with pressure is approximately linear over the range dT . The convolution was Our results also stand in interesting contrast to a body of work N implemented numerically, holding q constant and scaling g0 linearly with T N. on weak itinerant ferromagnets. For these systems, a line of con- ρ ð Þ 50 Modeling PM T is easy at low temperatures (approximately T N < Kor 8 8 ρ ð Þ¼ρ þ 3 þ 5 tinuous thermal phase transitions terminates at a first-order QPT. P> . GPa), where it obeys the expected form PM T 0 bT cT The quantum critical regime is inaccessible, but both the mag- and the T 3 dependence dominates (14, 16). In this regime the McWhan-Rice ρ ð Þ netic and nonmagnetic ground states are often characterized fit parameters are robust. At higher T modeling PM T is difficult, and the by strong quantum fluctuations that destabilize the Fermi liquid McWhan-Rice fit results for g0 and q are strongly correlated with the form ρ ð Þ (3, 5, 24). By contrast, the magnetic ground state of Cr is well assumed for PM T . The result for T N, however, remains robust. As a check ρð Þ described by mean-field theory, with signatures of quantum fluc- we also estimated T N from T by simply finding the temperature at which tuations only developing within the narrow quantum critical re- ρðTÞ has the most negative slope. This point is assumed to correspond to that gime. The outstanding feature common to both Cr and itinerant temperature at which the energy gap gðTÞ grows the fastest, which is identi- 9 ferromagnets appears to be a tricritical point in the pressure-tem- fied with T N. For all P < GPa the discrepancy between these results and the perature plane, where the quartic stiffness of the order parameter McWhan-Rice approach is less than the size of the data points in Fig. 1B;for P>9 GPa this simpler technique fails due to the increasing influence of the passes through zero. pressure inhomogeneity as the phase diagram steepens near P . The nature of the quantum fluctuations at the QPTremains an c open question. Assuming the applicability of the traditional LGW ∼0 3 Data Analysis in the Critical Regime. For pressures within . GPa of PC the formalism to nested fermions in three-dimensions, dimensional McWhan-Rice fits fail for two reasons. First, our lowest measurement tem- arguments allow effects beyond mean-field in the quantum re- perature of 4.5 K is too high for the Δρ∕ρ form to fully develop (Fig. 2C), gime. The relation γ ¼ z∕ðd þ z − 2Þ implies a dynamical expo- and as a result the fit parameters are poorly determined. Second, the finite

nent z ¼ 1 and a scaling dimension d þ z ¼ 4, which is not over pressure inhomogeneity produces a smearing of T N that diverges at Pc.Itis Δρ∕ρj the upper critical dimension (6, 22). Furthermore, the quasi-one- preferable to consider the isotherms T , which are fit to a power law Δρ∕ρj ð Þ¼ ð − Þβ dimensional dispersion relation at the nested Fermi surface T P a Pc;T P convolved with a Gaussian pressure distribution. −1 ∝ ð1 − Δρ∕ρÞ2 The 5 K isotherm and best-fit curve are shown in Fig. 2C, and a scaling plot (which is the origin of the RH scaling (17)) may result in a reduced effective dimension for the critical fluctua- of the data approaching the low-temperature limit is shown in Fig. 3A. The βð Þ tions, as has been observed for quantum critical heavy fermion critical exponents T are plotted in Fig. 3B, and the fit parameters Pc;T systems (25). The critical reconstruction of the nested Fermi sur- 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 face in Cr is accompanied by the reemergence of nested fermions FWHM of the Gaussian pressure distribution was 0.24 and 0.33 GPa, respec- with greatly enhanced exchange interactions and as the quantum tively, for the T ¼ 5 K data. For a given sample this fit parameter is then held critical point is uncovered, it drives a weak coupling BCS-like constant for fits to all T>5 K isotherms to reduce systematic correlations be- state towards strong-coupling physics (19, 26). The persistence of tween fit parameters. The Gaussian pressure distributions correspond to a 2-σ strongly interacting fermions above Pc also opens the possi- width of 0.43 and 0.58 GPa, respectively, somewhat smaller than the 0.72 GPa bility for the ground state that replaces the SDW to be charac- base width that was found for the pressure inhomogeneity over a ð200× terized by short-coherence length pairing, akin to the BCS-BEC 200Þ μm2 area in a recent study of the pressure conditions in the same (Bose Einstein condensate) crossover observed in ultracold methanol∶ethanol medium at 10 GPa and 5 K (12). The pressure condition gasses, or to a pseudogap-like state of dynamical pair fluctuations. is characteristic of a given cell assembly and depends mainly on choice of pressure medium and the sample-to-chamber volume ratio. A nearly con- Methods stant offset of 0.12 GPa of the critical phase boundaries measured for two Magnetotransport in a Diamond Anvil Cell at Cryogenic Temperatures. All different samples could result from several systematic issues, most notably measurements were performed in a low-temperature diamond anvil cell the position of the ruby chips (the ruby is positioned to the side of the equipped with a He gas membrane for fine pressure control. The pressure sample, close to the gasket wall where the pressure gradients are highest). medium was a methanol∶ethanol 4∶1 mixture. Pressure was measured in situ using the ruby fluorescence method. The pressure P is calculated from the ACKNOWLEDGMENTS. We acknowledge Arnab Banerjee and Peter Littlewood wavelength λ of the ruby R1 fluorescence by P ¼ A ·lnðλ∕λ0Þ, where λ0 is for enlightening discussions. The work at the University of Chicago was the (temperature-dependent) R1 wavelength at ambient pressure. A has supported by National Science Foundation (NSF) Grant DMR-0907025.

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