# Signatures of quantum criticality in pure Cr at high pressure.

**ABSTRACT** The elemental antiferromagnet Cr at high pressure presents a new type of naked quantum critical point that is free of disorder and symmetry-breaking fields. Here we measure magnetotransport in fine detail around the critical pressure, Pc approximately 10 GPa, in a diamond anvil cell and reveal the role of quantum critical fluctuations at the phase transition. As the magnetism disappears and T-->0, the magnetotransport scaling converges to a non-mean-field form that illustrates the reconstruction of the magnetic Fermi surface, and is distinct from the critical scaling measured in chemically disordered CrV under pressure. The breakdown of itinerant antiferromagnetism only comes clearly into view in the clean limit, establishing disorder as a relevant variable at a quantum phase transition.

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**ABSTRACT:**One of the common features of unconventional superconducting systems such as the heavy-fermion, high transition-temperature cuprate and iron-pnictide superconductors is that the superconductivity emerges in the vicinity of long-range antiferromagnetically ordered state. In addition to doping charge carriers, the application of external pressure is an effective and clean approach to induce unconventional superconductivity near a magnetic quantum critical point. Here we report on the discovery of superconductivity on the verge of antiferromagnetic order in CrAs via the application of external pressure. Bulk superconductivity with Tc≈2 K emerges at the critical pressure Pc≈8 kbar, where the first-order antiferromagnetic transition at TN≈265 K under ambient pressure is completely suppressed. The close proximity of superconductivity to an antiferromagnetic order suggests an unconventional pairing mechanism for CrAs. The present finding opens a new avenue for searching novel superconductors in the Cr and other transition metal-based systems.Nature Communications 06/2014; 5:5508. · 10.74 Impact Factor - Yejun Feng, Jiyang Wang, D M Silevitch, B Mihaila, J W Kim, J-Q Yan, R K Schulze, Nayoon Woo, A Palmer, Y Ren, Jasper van Wezel, P B Littlewood, T F Rosenbaum[Show abstract] [Hide abstract]

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**ABSTRACT:**We describe a technique for making electrical transport measurements in a diamond anvil cell using an alcohol pressure medium, permitting acute sensitivity while preserving sample fidelity. The sample is suspended in the liquid medium by four gold leads that are electrically isolated by a composite gasket made of stainless steel and an alumina-loaded epoxy. We demonstrate the technique with four-probe resistivity measurements of chromium single crystals at temperatures down to 4 K and pressures above 10 GPa. Our assembly is optimized for making high precision measurements of the magnetic phase diagram and quantum critical regime of chromium, which require repeated temperature sweeps and fine pressure steps while maintaining high sample quality. The high sample quality enabled by the quasi-hydrostatic pressure medium is evidenced by the residual resistivity below 0.1 μΩ cm and the relative resistivity ratio ρ(120 K)∕ρ(5 K) = 15.9 at 11.4 GPa. By studying the quality of Cr's antiferromagnetic transition over a range of pressures, we show that the pressure inhomogeneity experienced by the sample is always below 5%. Finally, we solve for the Debye temperature of Cr up to 11.4 GPa using the Bloch-Gruneisen formula and find it to be independent of pressure.The Review of scientific instruments 10/2012; 83(10):103902. · 1.52 Impact Factor

Page 1

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;

Argonne, IL 60439; and

bThe Advanced Photon Source, Argonne National Laboratory,

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

type of naked quantum critical point that is free of disorder and

symmetry-breaking fields. Here we measure magnetotransport

in fine detail around the critical pressure, Pc∼ 10 GPa, in a diamond

anvil cell and reveal the role of quantum critical fluctuations at the

phase transition. As the magnetism disappears and T → 0, the

magntotransport scaling converges to a non-mean-field form that

illustrates the reconstruction of the magnetic Fermi surface, and is

distinct from the critical scaling measured in chemically disordered

Cr∶V under pressure. The breakdown of itinerant antiferromagnet-

ism only comes clearly into view in the clean limit, establishing

disorder as a relevant variable at a quantum phase transition.

antiferromagnetism ∣ spin density waves ∣ electric transport

C

exotic ground states such as non-Fermi liquid metals and uncon-

ventional superconductors (1). This observation has motivated

several decades of work to understand the physics of magnetic

quantum phase transitions (QPT) (2–7). A substantial part of

the effort has been directed at the materials science challenges

that are inherent to realizing nearly-magnetic states of matter

and to the fine tuning of materials so that the phase transitions

can be probed systematically. The fundamental limitations that

remain are uncertainty over the role of disorder (2, 4, 8), as well

as 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 the

elemental antiferromagnet Cr near the critical pressure Pc∼

10 GPa, and concurrent measurements of the crystal lattice

across the transition failed to detect any discontinuous change

in symmetry or volume (9). These results identify Cr as a stoichio-

metric itinerant magnet with a continuous QPT—where the

effects of the critical point should be manifest—and present a

rare opportunity to study quantum criticality in a theoretically

tractable system that is free from the effects of disorder. More-

over, the use of hydrostatic pressure as a tuning parameter avoids

the introduction of any confounding symmetry-breaking fields.

For the experimentalist, studying elemental Cr shifts the sig-

nificant technical difficulties from solid state chemistry to high

pressure experimentation. Here we report on high-resolution

measurements of the electrical resistivity and Hall coefficient

of Cr as the system is tuned with pressure in a diamond anvil cell

across Pc. Magnetotransport is a sensitive probe of quantum cri-

ticality and is widely used to identify and characterize quantum

matter (4, 5, 8, 10). At ambient pressure Cr orders antiferromag-

netically at the Néel temperature, TNðP ¼ 0Þ ¼ 311 K. Below

TN, electrons and holes form magnetic pairs and condense into

a spin density wave (SDW), in a process with strong analogies to

the Bardeen-Cooper-Schrieffer (BCS) formulation of electron

pairing in superconductors (11). The quantum critical point

where TN→ 0 can be reached either through applied pressure or

chemical 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

ompetition between magnetic and nonmagnetic states of mat-

ter in the zero-temperature limit underlies the emergence of

with electron-poor V to near-critical levels lowered the critical

pressure, making Pcaccessible with a conventional clamp cell.

Accessing the QPT in the pure system, on the other hand, re-

quires high sensitivity measurements on submillimeter single

crystals in a diamond anvil cell at low temperature (12, 13).

Results

We present here the results from experimental runs with seven

different samples, including two that were studied in fine detail

in the critical regime. An overview of the resistivity for 0 < P <

10 GPa is shown in Fig. 1A. The Néel transition is marked by a

sharp upturn in the resistivity, ρðTÞ, as the reduction in metallic

carrier 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 normalized

magnetic excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ. This quantity is

then fit to a model function which accounts for the formation

of a BCS-like energy gap below TNand the resulting freezing-

out of carriers. This model function was successfully applied in

an important early study of Cr under pressure by McWhan

and Rice (13). By analyzing Δρ∕ρ (see Methods) we obtain the

phase diagram of Fig. 1B. TNðPÞ evolves exponentially with pres-

sure for P < 7 GPa with the form TNðPÞ ¼ TN;0expð−cPÞ,

TN;0¼ 310.9 ? 0.9 K, c ¼ −0.163 ? 0.001 GPa−1. Above 7 GPa

this BCS-like exponential ground state breaks down as the system

approaches 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. 2Athe elec-

trical resistivity measured in fine detail in the quantum critical

regime. For T < 50 K the paramagnetic resistivity displays a

dominant T3dependence. This is demonstrated in Fig. 2B where

we plot ρðT3Þ, and for each pressure we limit the temperature

range to T > TNin order to emphasize ρPMðTÞ. The T3coeffi-

cient b varies by less than 6% between samples and is well

described by metallic transport due to phonon scattering in the

presence of a weakly inelastic nonphonon scattering channel (14).

Theory gives b∕d ¼ ð4.8∕Θ2Þ ¼ 1.74 × 10−5K−2, where Θ is the

Debye temperature (Θ ¼ 525 K (15)) and d is the linear tempera-

ture coefficient of resistivity at high temperature. The coefficient

d 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 at

high pressure. For the sample presented in Fig. 2 we find

b∕d ¼ 1.95 ? 0.15ð10−5K−2Þ, in reasonable agreement with the

theoretical expectation. The T3resistivity in this temperature

range (vs. a T5form) is typical for metallic samples with residual

resistivities ρ0≥ 1 nΩ · cm (14, 16); our single-crystal Cr is

99.996 þ % pure and has ρ0≈ 0.1 μΩ · cm (compared to ρ0≈

1.4 μΩ · cm in critically doped Cr∶V 3.2% (8)). The electron

mean-free path in our samples is estimated to be λ > 400 Å at

base-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. performed

research; 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.

www.pnas.org/cgi/doi/10.1073/pnas.1005036107PNAS ∣ August 3, 2010 ∣ vol. 107 ∣ no. 31 ∣ 13631–13635

PHYSICS

Page 2

measured Hall mobility. We note that the presence of finite

quenched disorder in our samples is a necessary precondition

for measuring a pressure-dependent residual resistivity. However,

the extremely low level of disorder suggests that pure Cr is a

benchmark for how closely a QPT in a real solid state system

can approach the clean limit.

We plot in Fig. 2C the excess resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ

calculated from the data in Fig. 2A and the ρPMbackground

(see Fig. 2B and Methods). As P approaches Pc, it is preferable

to analyze data for experimental cuts which are close to perpen-

dicular to the increasingly steep phase diagram TNðPÞ. We extract

such cuts from the data by considering isotherms of Δρ∕ρ. These

isotherms Δρ∕ρjTare then fit to a power law Δρ∕ρjTðPÞ ¼

aðPc;T− PÞβ, convolved with a Gaussian that accounts for the

finite pressure variation across the sample (see Methods: Data

Analysis in the Critical Regime). The 5 K isotherm and fit are

plotted on the projected (P, Δρ∕ρ) 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 Pc;T,

and the exponent β relates to the breakdown of the SDW energy

gap and the reemergence of nested Fermi surface area; in the

T → 0 limit β directly reflects the critical reconstruction of the

Fermi 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 for

temperatures T ≤ 8 K (Fig. 3B). This exponent speaks to a rapid

reconstitution of the Fermi surface that takes place in a narrow

quantum critical regime, and stands in contrast to the value β ¼

2∕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 recedes

from sight. However, due to limited data density and the difficulty

of modeling ρPMat higher temperatures, we are not able to follow

β to the point at which it settles at a high temperature limit. The

crossover 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 itinerant

fermions (6), although the applicability of the usual Landau-

Ginzburg-Wilson (LGW) critical analysis to the case of nested

Fermi surfaces remains in question (7). The critical phase dia-

gram TNðPÞ ∝ ðPC− PÞγis 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, consistent

with the mean-field expectation, γ ¼ 1∕2, also observed for

Cr1-xVx, x ¼ 3.2%.

The critical reconstruction of the nested Fermi surface is

further demonstrated by the Hall coefficient, RHðPÞ. The Hall

effect is acutely sensitive to the quantum critical point, changing

P (GPa)

0

3

6

9

0246810

0

50

100

150

200

250

300

ICDW(P, T < 8 K)

ρ(P,T)

B

QC

BCS

TF

P (GPa)

TN (K)

0100200300

0

5

10

15

T (K)

ρ (µΩ*cm)

A

Fig. 1.

pressure. Data and results shown for all seven samples measured. (A)

Resistivity ρðTÞ. (B) Antiferromagnetic phase diagram TNðPÞ. Black ¼

determined directly fromρðTÞcurves;

X-ray measurements of the CDW diffraction intensity ICDWat low tempera-

ture (9), from which the phase diagram can be calculated using the harmonic

relationship TN∝ ICDW

order, 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 continuously

suppressed by quantum critical (QC) fluctuations. Red shaded region indi-

cates the quantum critical regime which is the focus of this paper.

Data overview and phase diagram for antiferromagnetic Cr under

Blue ¼ determined indirectly from

1∕4. At low pressure the Néel transition is weakly first

0.1

0.2

0.3

∆ρ/ρ

0

40

T (K)

20

0

C

P (GPa)

9.5

10

9

T = 5 K

0510

0.1

0.14

0.18

T3 (104 K3)

ρ (µΩ*cm)

B

01020304050

0.1

0.12

0.14

0.16

0.18

T (K)

ρ (µΩ*cm)

A

P (GPa)

9.29.49.69.810

Fig. 2.

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

pressure colorbar applies to A–C. (A) Resistivity ρðTÞ. (B) ρðTÞ plotted against T3for T < 50 K, with each curve truncated just above TN. Over this temperature

range the paramagnetic background ρPMðTÞ is dominated by the shown T3dependence. (C) The magnetic resistivity Δρ∕ρ ¼ ðρ − ρPMÞ∕ρ, calculated from ρðTÞ

and the modeled ρPMðTÞ. Also shown (dashed red line) is the McWhan-Rice fit to the lowest pressure curve at 9.13 GPa, for which TN¼ 37.9 ? 0.03 K and

g0∕kBTN¼ 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. The

exponent β ¼ 0.23 ? 0.03 and the Gaussian pressure inhomogeneity is 0.24 GPa (FWHM).

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

13632

∣

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

Page 3

by 300% across the narrow critical regime at low temperature

(Fig. 4). For P < Pc, the data can be described by the scaling form

R−1

ing a SDW (17). According to mean-field theory (17, 18), small

deviations from ideal nesting will cause both RHand Δρ∕ρ to

scale linearly with the SDWenergy gap g0in the T → 0 limit. Un-

der the conservative assumption that g0∝ TNthe data indicate

that the gap scales with the mean-field exponent of 1∕2, while

the Hall coefficient and the excess resistivity behave differently.

The non-mean-field scaling which we observe for both RHand

Δρ∕ρ implies that the observed critical behavior is driven not

by the SDW energy gap, but by fluctuations that restore flat sec-

tions of Fermi surface. Moreover, the lengthening of the SDW

ordering wavevector Q through the critical regime, in contrast to

the monotonically decreasing dependence of Q on P for

P < 7 GPa, also has been interpreted as evidence for quantum

critical fluctuations (9).

H∝ ð1 − Δρ∕ρÞ2, as predicted for a flat Fermi surface support-

Discussion

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 >

TNis marked by incipient antiferromagnetic fluctuations that go

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

pressure, however, this BCS-like theory accurately describes the

observed exponential dependence of both the phase boundary

TNðPÞ and the saturated order parameter at low temperature

(19, 20). The exponential evolution results from a competition

between exchange energy and bandwidth that is tuned by applied

pressure while preserving the Fermi surface nesting condition

(21). Above 7 GPa the exponential ground state breaks down and

the 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 a

continuous QPT.

We have shown that the critical scaling of this breakdown is

different in pure Cr than in Cr1-xVx, x ¼ 3.2%, which establishes

that 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 phase

transitions, the same is not true for their quantum counterparts.

In both the clean and disordered limits we find that the exponent

β is distinct from the exponent γ ¼ 1∕2 that describes the scaling

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

cases, as does the scaling RHðPÞ ∝ ðPc− PÞα, which is mean-field

in Cr∶V and distinctly non-mean-field with an exponent α close to

1∕4 in pure Cr. V-doping is more efficient at driving the QPT, in

that the phase diagram departs from the exponential curve at a

larger TN(or, equivalently, at a larger SDW coupling constant)

for Cr1-xVxthan for pure Cr under pressure (9). Furthermore,

with V-doping the body centered cubic lattice expands and the

SDW wavevector decreases monotonically, in contrast to the

behavior under pressure. This decrease in Q with V-doping re-

sults from the fact that the band filling varies with electron-poor

doping. However, barring the unrealistic scenario in which pure

Cr remains perfectly nested at all pressures, this change in band

filling is not expected to alter the critical scaling at the QPT. Our

scaling results therefore demonstrate that the distinct micro-

scopic effects of chemical doping (or “chemical pressure”) and

hydrostatic pressure lead to distinct phase transitions, and

10

−3

10

−2

10

(PC−P)/PC

TN (K)

γ = 1/2

3

30

C

01020

0

0.25

0.5

0.75

1

T (K)

β

(∆ρ/ρ)T∝ (Pc,T−P)β

B

−20246

0

2

4

6

(Pc−P)/Pc (%)

(∆ρ/ρ)4 (× 10−3)

5 K

6 K

7 K

8 K

β = 1/2

β = 1/4

A

Fig. 3.

phase boundary TN. (A) Scaling plot of Δρ∕ρ shows that the low-temperature

isotherms are well described by β ¼ 1∕4 and can be differentiated clearly

from 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 (black

and gray, respectively) measured in detail in the critical regime. There is a

crossover at higher T to a larger exponent, verging towards approaching

either mean-field behavior or the β ¼ 2∕3 found for Cr∶V. (C) The critical

scaling of TN, which is consistent with the mean-field exponent γ ¼ 1∕2

for both samples (blue and red, respectively). A nearly constant offset of

0.12 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), it

does limit the accuracy to which we can determine Pcitself, which we report

as 9.71 ? 0.08 GPa.

Quantum critical scaling of the magnetic resistivity Δρ∕ρ and the

0246810 12

0.2

0.4

0.6

0.8

1

1.2

P (GPa)

(RH)−1 (104 C/cm3)

0.4 0.60.8

0.4

0.6

0.8

(RH)−1 (104 C/cm3)

(ρPM/ρAF)2

Fig. 4.

measured at T ¼ 4.9 ? 0.4 K. Inset shows R−1

predicted for a flat Fermi surface supporting a SDW; solid line is a linear

fit to the data. Resistivity ρPM∕ρAF¼ 1 − Δρ∕ρ (see Results) was recorded at

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

quantum critical scaling of the Hall coefficient.

Overview and critical scaling of the inverse Hall coefficient R−1

HðPÞ

H∝ ðρPM∕ρAFÞ2scaling which is

Jaramillo et al.PNAS

∣

August 3, 2010

∣

vol. 107

∣

no. 31

∣

13633

PHYSICS

Page 4

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 outstanding

question. Recent transport measurements on La2-xSrxCuO4at

high magnetic fields showed no clear signature of quantum

criticality near optimal hole doping, raising the prospect that

the 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-xVxis broad

and 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 understood

in heavy fermion systems, and well characterized quantum critical

points 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 heavy

fermion quantum criticality. Crucially, the lack of local magnetic

moments and the absence of effective mass divergences through-

out the Brillouin zone separate the QPT in Cr from the heavy

fermion examples (23).

Our results also stand in interesting contrast to a body of work

on 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 characterized

by strong quantum fluctuations that destabilize the Fermi liquid

(3, 5, 24). By contrast, the magnetic ground state of Cr is well

described 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 itinerant

ferromagnets appears to be a tricritical point in the pressure-tem-

perature plane, where the quartic stiffness of the order parameter

passes through zero.

The nature of the quantum fluctuations at the QPTremains an

open question. Assuming the applicability of the traditional LGW

formalism to nested fermions in three-dimensions, dimensional

arguments allow effects beyond mean-field in the quantum re-

gime. The relation γ ¼ z∕ðd þ z − 2Þ implies a dynamical expo-

nent z ¼ 1 and a scaling dimension d þ z ¼ 4, which is not over

the 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

result in a reduced effective dimension for the critical fluctua-

tions, as has been observed for quantum critical heavy fermion

systems (25). The critical reconstruction of the nested Fermi sur-

face in Cr is accompanied by the reemergence of nested fermions

with greatly enhanced exchange interactions and as the quantum

critical point is uncovered, it drives a weak coupling BCS-like

state towards strong-coupling physics (19, 26). The persistence of

strongly interacting fermions above Pcalso 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 ultracold

gasses, or to a pseudogap-like state of dynamical pair fluctuations.

H∝ ð1 − Δρ∕ρÞ2scaling (17)) may

Methods

Magnetotransport in a Diamond Anvil Cell at Cryogenic Temperatures. All

measurements were performed in a low-temperature diamond anvil cell

equipped with a He gas membrane for fine pressure control. The pressure

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

wavelength λ of the ruby R1fluorescence by P ¼ A · lnðλ∕λ0Þ, where λ0is

the (temperature-dependent) R1wavelength at ambient pressure. A has

been calibrated directly (12) at 5 K (A5 K¼ 1;762 ? 13 GPa) and room-T

(A295 K¼ 1;868 GPa). To interpolate between these two temperatures we

assume that A is constant up to 100 K, above which it varies linearly with

temperature, in qualitative accordance with the temperature dependence

of the bulk modulus of Al2O3. Resistivity ρ and Hall coefficient RHwere mea-

sured in the four probe van der Pauw geometry on single-crystal Cr plates

using an ac resistance bridge. RHwas derived from data taken in the range

−3;500 < H < 3;500 Oe, which is in the low-field limit for all pressures. The

microscopic samples, size ð200 × 200 × 40Þ μm3with (111)-oriented faces,

were derived from large Cr single-crystal discs (Alfa Aesar, 99.996 þ %) by

a procedure described elsewhere (27). The gold leads were spot-welded to

the sample and insulated from the metallic gasket using a mixture of alumina

powder and epoxy.

McWhan-Rice Model. Forpressuresupto∼0.3 GPabelowPcthephasediagram

was determined by fitting Δρ∕ρ to the McWhan-Rice model (13). This model

hasthreefreeparameters:theNéeltemperatureTN,theT → 0energygapg0,

and the magnetic Fermi surface fraction q (note the typo in Eq. 6 of ref. 13,

where E3∕2is written instead of E3; for the correct expression see Eq. 6.16 of

ref. 28). We implemented this model with an additional free parameter dTN

which describes the width of a Gaussian distribution in TN. This convolution

allows for pressure inhomogeneity and is valid as long as the variation in TN

withpressureisapproximatelylinearovertherangedTN.Theconvolutionwas

implemented numerically, holding q constant and scaling g0linearly with TN.

Modeling ρPMðTÞ is easy at low temperatures (approximately TN< 50 K or

P > 8.8 GPa), where it obeys the expected form ρPMðTÞ ¼ ρ0þ bT3þ cT5

and the T3dependence dominates (14, 16). In this regime the McWhan-Rice

fit parameters are robust. At higher T modeling ρPMðTÞ is difficult, and the

McWhan-Rice fit results for g0and q are strongly correlated with the form

assumed for ρPMðTÞ. The result for TN, however, remains robust. As a check

we also estimated TNfrom ρðTÞ by simply finding the temperature at which

ρðTÞ has the most negative slope. This point is assumed to correspond to that

temperature 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 the

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

pressure inhomogeneity as the phase diagram steepens near Pc.

Data Analysis in the Critical Regime. For pressures within ∼0.3 GPa of PCthe

McWhan-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 finite

pressure inhomogeneity produces a smearing of TNthat diverges at Pc. It is

preferable to consider the isotherms Δρ∕ρjT, which are fit to a power law

Δρ∕ρjTðPÞ ¼ aðPc;T− PÞβconvolved with a Gaussian pressure distribution.

The 5 K isotherm and best-fit curve are shown in Fig. 2C, and a scaling plot

of the data approaching the low-temperature limit is shown in Fig. 3A. The

critical 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 held

constant for fits to all T > 5 K isotherms to reduce systematic correlations be-

tween fit parameters. The Gaussian pressure distributions correspond to a 2-σ

width of 0.43 and 0.58 GPa, respectively, somewhat smaller than the 0.72 GPa

base width that was found for the pressure inhomogeneity over a ð200×

200Þ μm2area in a recent study of the pressure conditions in the same

methanol∶ethanol medium at 10 GPa and 5 K (12). The pressure condition

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-

stant offset of 0.12 GPa of the critical phase boundaries measured for two

different samples could result from several systematic issues, most notably

the position of the ruby chips (the ruby is positioned to the side of the

sample, close to the gasket wall where the pressure gradients are highest).

ACKNOWLEDGMENTS. We acknowledge Arnab Banerjee and Peter Littlewood

for enlightening discussions. The work at the University of Chicago was

supported by National Science Foundation (NSF) Grant DMR-0907025.

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