ARTICLESPUBLISHED ONLINE: 4 MAY 2014 | DOI: 10.1038/NPHYS2961

Two-stage magnetic-field-tuned superconductor–insulator transition in underdoped La2−xSrxCuO4

Xiaoyan Shi1†, Ping V. Lin1, T. Sasagawa2, V. Dobrosavljević1 and Dragana Popović1*

In the underdoped pseudogap regime of cuprate superconductors, the normal state is commonly probed by applying amagneticfield (H). However, the nature of the H-induced resistive state has been the subject of a long-term debate, and clear evidencefor a zero-temperature H-tuned superconductor–insulator transition has proved elusive. Here we report magnetoresistancemeasurements on underdoped La2−xSrxCuO4, providing striking evidence for quantum-critical behaviour of the resistivity—the signature of a H-driven superconductor–insulator transition. The transition is not direct, being accompanied by theemergence of an intermediate state, which is a superconductor only at temperature T= 0. Our finding of a two-stageH-driven superconductor–insulator transition goes beyond the conventional scenario in which a single quantum critical pointseparates the superconductor and the insulator in the presence of a perpendicular magnetic field. Similar two-stage H-drivensuperconductor–insulator transitions, in which both disorder and quantum phase fluctuations play an important role, may alsobe expected in other copper-oxide high-temperature superconductors.

The superconductor–insulator transition (SIT) is an exampleof a quantum phase transition (QPT): a continuous phasetransition that occurs at T=0, controlled by some parameter

of the Hamiltonian of the system, such as doping or the externalmagnetic field1. A QPT can affect the behaviour of the system upto surprisingly high temperatures. In fact, many unusual propertiesof various strongly correlated materials have been attributed tothe proximity of quantum critical points (QCPs). An experimentalsignature of a QPT at nonzero T is the observation of scalingbehaviourwith relevant parameters in describing the data. Althoughthe SIT has been studied extensively2, even in conventionalsuperconductors many questions remain about the perpendicular-field-driven SIT in two-dimensional (2D) or quasi-2D systems3. Inhigh-Tc cuprates (here Tc is the transition temperature), which havea quasi-2D nature, early magnetoresistance experiments showedthe suppression of superconductivity with high H , revealing theinsulating behaviour4–6 and hinting at an underlying H -field-tunedSIT (ref. 7). However, even though understanding the effectsof H is believed to be essential to understanding high-Tc cupratesuperconductivity and continues to be a subject of intense research,the evidence for the H -field-driven SIT and the associated QPTscaling in cuprates remains scant and inconclusive.

In the conventional picture of type-II superconductors, Hpenetrates the sample in the form of a solid lattice of interactingvortex lines in the entire mixed state Hc1(T ) < H < Hc2(T ),where Hc1 is the Meissner field and Hc2 is the upper criticalfield. This picture, however, neglects fluctuations which, inhigh-Tc superconductors, are especially important. Indeed, thedelicate interplay of thermal fluctuations, quantum fluctuationsand disorder leads to a complex H–T phase diagram of vortexmatter8–10. Thermal fluctuations, for example, cause melting ofthe vortex solid into a vortex liquid for fields below what isnow a crossover line Hc2(T ). Quantum fluctuations could resultin a vortex liquid persisting down to T = 0. At very low T ,

the disorder becomes important and modifies the vortex phasediagram such that there are two distinct vortex solid phases9,10:a Bragg glass with Tc(H) > 0 at lower fields and a vortex glasswith Tc= 0 at higher fields. The SIT would then correspond to atransition from a vortex glass to an insulator at even higher H .However, the interplay of this vortex line physics and quantumcriticality, the key question in the H -field-tuned SIT, has remainedlargely unexplored.

In this study, we find strong evidence for the H -field-driven SITin underdoped La2−xSrxCuO4 (LSCO). The results are consistentwith the existence of three phases at T =0, although the behaviourof the in-plane resistivity ρ suggests the presence of the direct SITover a surprisingly wide range ofT andH . At the lowestT , however,the ρ(T ,H) data reveal an intermediate phase with Tc = 0 andthe true SIT at higher H . We focus on samples with a relativelylow Tc(H = 0) to ensure that experimentally attainable H arehigh enough to fully suppress superconductivity. Unlike most otherstudies, ours includes samples grown using different techniques(Methods), to separate out any effects that may depend on thesample preparation conditions from the more general behaviour.As well as providing evidence for the SIT, the magnetoresistancedata are used to calculate the contribution of superconductingfluctuations (SCFs) to conductivity, allowing a construction of theH–T phase diagram.

In-plane resistivity of La2−xSrxCuO4One sample was a film with the nominal compositionLa1.93Sr0.07CuO4 (ref. 11) and a measured Tc(H=0) of (3.8±0.1)K.The high-quality single crystal12 had Tc(H =0)= (5.2±0.1)K andthe nominal composition La1.94Sr0.06CuO4+y , with y not preciselyknown (see Methods for more details). Unless otherwise specified,Tc is defined throughout as the temperature at which the resistance(that is, ρ) becomes zero for a given H . (The method to determineTc(H) is illustrated in Supplementary Fig. 1.)

1National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32310, USA, 2Materials and StructuresLaboratory, Tokyo Institute of Technology, Kanagawa 226-8503, Japan. †Present address: Sandia National Laboratories, Albuquerque, New Mexico 87185,USA. *e-mail: dragana@magnet.fsu.edu

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2961

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Figure 1 | Temperature dependence of the in-plane resistivity ρ in di�erent magnetic fields H‖c for x=0.07 LSCO film. a, The ρ(T, H) data exhibit achange of the sign of dρ/dT as a function of H at high T & 5 K (R�/layer is resistance per square per CuO2 layer; see Methods). Except for H=0, the solidlines are guides for the eye. b, A selection of the ρ(T, H) data from a, shown on a log–log scale to focus on the low-T, intermediate-H regime. Short-dashedlines guide the eye. Solid lines represent power-law fits ρ(T, H)=ρ0(H)Tα(H). c, Fitting parameters ρ0(H) and α(H). Error bars indicate 1 standard deviation(s.d.) in the fits for ρ0(H) and α(H). Short-dashed lines guide the eye.

Figure 1a shows the ρ(T ) curves for different H that wereextracted from the magnetoresistance measurements at fixed T forthe x = 0.07 sample (see Supplementary Fig. 2 for similar dataon the x = 0.06 crystal). A small H clearly leads to a decrease ofTc(H), such that Tc→ 0 for H ≈ 4 T (Supplementary Informationand Supplementary Fig. 3). The survival of the superconductingphase with Tc>0 up to fields much higher than the Meissner fieldHc1 ∼ 100Oe is understood to be a consequence of the pinningof vortices by disorder8–10. As H increases further, a pronouncedmaximum appears in ρ(T ). The temperature at themaximum, Tmax,shifts to lower T with increasing H , similar to early observations5.At the highest H , ρ(T ) curves exhibit insulating behaviour.

The intermediate H regime, in which each ρ(T ) curve exhibitsa maximum, is especially intriguing. Here the system shows atendency towards insulating behaviour already at high T & 5K, butthen the sign of dρ/dT changes, suggesting that anothermechanismsets in at lower T and drives ρ(T→ 0) towards zero. The low-T , intermediate-H regime is more evident in Fig. 1b, where thesame data are presented on a log–log scale. For T < Tmax, thedata are described best with the phenomenological power-law fitsρ(T ,H)=ρ0(H)T α(H), which indicate that ρ(T = 0)= 0—that is,that the system is a superconductor only at T =0. The exponent αdepends on H (Fig. 1c) and goes to zero at H∼13.5 T. This impliesthe existence of aT -independent ρ at that field. The non-monotonicbehaviour of ρ(T ) in the intermediate H regime suggests thathigh-T (T > Tmax) and low-T (T < Tmax) regions should beexamined separately and more closely.

Scaling analysis of the high-temperature behaviourFigure 2a shows a more detailed set of magnetoresistancemeasurements on the x = 0.07 film, focused on the behaviourat 5 K.T < 10K. The magnetoresistance curves clearly exhibita well-defined, T -independent crossing point at µ0H ∗1 = 3.63 T.In other words, this is the field where dρ/dT changes sign frompositive or metallic at low H , to negative or insulating at H >H ∗1(see also Fig. 1a). We find that, near H ∗1 , ρ(T ) for different H canbe collapsed onto a single function by rescaling the temperature,as shown in Fig. 2b. Therefore, ρ(T , H) = ρ∗1 f1(T/T ∗1 ), wherethe scaling parameter T ∗1 is found to be the same function ofδ=(H−H ∗1 )/H ∗1 on both sides ofH ∗1 . In particular, T ∗1 ∝|δ|zν withzν≈ 0.73 (Fig. 2c) over a remarkably wide, more than two ordersof magnitude range of |δ|. Such a single-parameter scaling of theresistance is precisely what is expected13 near a T = 0 SIT in 2D,where z and ν are the dynamical and correlation length exponents,respectively2. Similar scaling on the x = 0.06 crystal sample(Supplementary Fig. 4) yields a comparable zν= 0.59± 0.08. It isinteresting that, although the critical fields H ∗1 in the two samplesdiffer by almost a factor of two (µ0H ∗1 = 3.63 T for x = 0.07 filmand µ0H ∗1 = 6.68 T for x = 0.06 crystal), the critical resistivitiesρ∗1 are almost the same (R�/layer ≈ (17.4 − 18.0) k�; see alsoSupplementary Information). The exponent product zν∼ 0.7 hasbeen observed for perpendicular H -field-tuned transitions also inconventional 2D superconductors (for example, in a-Bi (ref. 14)and a-NbSi (ref. 15)) and, more recently, in 2D superconductingLaTiO3/SrTiO3 interfaces16. The value zν∼0.7 is believed to be in

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NATURE PHYSICS DOI: 10.1038/NPHYS2961 ARTICLES2.0a

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Figure 2 | High-temperature (T&&& 5K) behaviour of the resistivity ρ in di�erent magnetic fields H‖c for x=0.07 LSCO film. a, Isothermal ρ(H) curves inthe high-T region show the existence of a T-independent crossing point at µ0H∗1 =3.63 T and ρ∗1 = 1.15 m� cm (or R�/layer≈ 17.4 k�). b, Scaling of the datain a with respect to a single variable T/T∗1 . The scaling region is shown in more detail in Fig. 4a. c, The scaling parameter T∗1 as a function of |H–H∗1 |/H∗1 onboth sides of H∗1 . The lines are fits with slopes of zν≈0.73, as shown.

the universality class of the 2D SIT in the clean limit, as describedby the (2+ 1)D XY model and assuming that z = 1 owing to thelong-range Coulomb interaction between charges13,17.

Scaling analysis of the low-temperature behaviourAs shown above, the behaviour of the system over a range of Tabove Tmax seems to be controlled by the QCP corresponding to thetransition from a superconductor to an insulator in the absence ofdisorder, and driven by quantumphase fluctuations. AsT is loweredbelow Tmax, however, this transition does not actually take place, assome of the insulating curves assume the power-law dependenceρ(T , H) = ρ0(H)T α(H) (Fig. 1), leading to a superconductingstate at T = 0. Figure 3a shows a set of magnetoresistancemeasurements carried out at very low T<Tmax, which exhibit aT -independent crossing point at µ0H ∗2 = 13.45 T, consistent withα(H=H ∗2 )≈0 (see also Fig. 1b,c). Near H ∗2 , an excellent scaling ofρ with temperature according to ρ(T ,H)=ρ∗2 f2(T/T ∗2 ) is obtained(Fig. 3b), where T ∗2 ∝ |δ|zν on both sides of H ∗2 (Fig. 3c). Hereδ=(H−H ∗2 )/H ∗2 and zν= 1.15± 0.05. Assuming z = 1, this typeof single-parameter scaling with ν>1 corresponds to the T=0 SITin a 2D disordered system13.

Superconducting fluctuationsThe extent of SCFs can be determined from the transverse(H ‖c) magnetoresistance by mapping out fields H ′c(T ) abovewhich the normal state is fully restored11,18–20. In the normal stateat low fields, the magnetoresistance increases as H 2 (ref. 21),so that the values of H ′c can be found from the downward

deviations from such quadratic dependence that arise fromsuperconductivity when H < H ′c . The H ′c(T ) line determinedusing this method (Supplementary Fig. 5) is shown in Fig. 4afor the x = 0.07 film11 (see Supplementary Information andSupplementary Fig. 6 for the x = 0.06 crystal sample). In bothcases, H ′c(T ) is well fitted by H ′c(T )=H ′c(0)[1−(T/T2)

2].

Figure 4a also shows the SCF contribution to the conductivity18,191σSCF(T ,H)=ρ−1(T ,H)−ρ−1n (T ,H), where the normal-stateresistivity ρn(T ,H) was obtained by extrapolating the region of H 2

magnetoresistance observed at high enough H and T , as illustratedin Supplementary Fig. 5.

Phase diagramIn addition toH ′c(T ) and1σSCF(T ,H), the phase diagram in Fig. 4aincludes Tc(H) and Tmax(H), as well as the critical fields H ∗1 andH ∗2 . For both samples,Tc(H=H0)→0 (Supplementary Informationand Supplementary Fig. 3) for fields H0≈H ∗1 (µ0H ∗1 =3.63 T forx=0.07 film and µ0H ∗1 = 6.68 T for x = 0.06 crystal). This isconsistent with the T = 0 transition from a pinned vortex solidto another phase at higher fields. For the x = 0.07 film sample,Tmax(H =H0)→ 0 for µ0H0 = (13.4± 0.7)T, meaning that Tmaxvanishes at the critical fieldµ0H ∗2 =13.45 Twithin themeasurementerror. In the x = 0.06 single crystal, Tmax(H) extrapolates to zeroat µ0H0= (14.0± 0.6)T— that is, at a field similar to that in thefilm sample, even though their H ′c(T = 0) are very different. Thevanishing of a characteristic energy scale, such as Tmax(H), in theT = 0 limit is consistent with the existence of a quantum phasetransition at H ∗2 .

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2961

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Figure 3 | Low-temperature (T...0.3 K) behaviour of the resistivity ρ in di�erent magnetic fields H‖c for x=0.07 LSCO film. a, Isothermal ρ(H) curves inthe low-T region show the existence of a T-independent crossing point at µ0H∗2= 13.45 T and ρ∗2=6.404 m� cm (or R�/layer≈97 k�). b, Scaling of thedata in a with respect to a single variable T/T∗2 . The scaling region, which is shown in more detail in Fig. 4a, includes the data at the lowest T≈0.09 K.c, The scaling parameter T∗2 as a function of |H–H∗2|/H∗2 on both sides of H∗2. The line is a fit with slope zν≈ 1.15.

The scaling regions associated with the critical fields H ∗1 andH ∗2 are also shown in Fig. 4a. It is striking that the ‘hidden’ criticalpoint at H ∗1 , corresponding to the SIT in the clean limit, dominatesa huge part of the phase diagram. (See also Supplementary Fig. 7for scaling of ρ(T ,H) in the x= 0.07 film over the entire scalingregion shown in Fig. 4a.) For H <H ∗1 , this scaling fails at low T ,precisely where SCFs increase markedly, leading to a large dropin ρ. For H >H ∗1 , the scaling works, of course, only for T >Tmax.The tendency towards insulating behaviour that is observed alreadyat low H &H ∗1 and T & 5K (Fig. 1a) clearly belongs to the samefamily of insulating curves that span the fields up to H≈H ′c at lowT . It is apparent from Fig. 2 (also Supplementary Fig. 7) that theinsulating ρ(T ) does not follow the lnT dependence4. The data areinstead consistent (not shown) with variable-range hopping (VRH),although the range of available fields is too small here to observe theorders-of-magnitude change in ρ that is characteristic of VRH. Alarger change was observed in some early studies5,22 of the H -field-induced localization in underdoped LSCOwith a similar Tc(H=0),and attributed to 2D or 3D Mott VRH.

The scaling region corresponding to the critical point at H ∗2 ,on the other hand, is remarkably small (Fig. 4a), but it dominatesthe behaviour at the lowest T . On the insulating side—that is, forH>H ∗2—there is evidence for the presence of SCFs, as the appliedH .H ′c . The lowest-T insulating behaviour (Fig. 3b) may also befitted to a VRH form, but the change of ρ is too small again todetermine the VRH exponent with high certainty.

A simplified version of the same phase diagram is shownin Fig. 4b, which includes crossover temperatures T ∗1 and T ∗2corresponding to the two QCPs H ∗1 and H ∗2 , respectively. Thelogarithmic T -scale also makes it more apparent that the vortex

solid phase with Tc(H)>0 extends precisely up toH ∗1—that is, thatthe QCP H ∗1 is associated with the quantum melting of the pinnedvortex solid. The other phases shown in Fig. 4b are discussed below.

DiscussionTo account for our observation of three distinct phases asT→0 andtwo QPTs, we discuss our results in the context of other relevantwork on the same material (see Fig. 5 for a sketch of the phasediagram, which is basedmainly on Fig. 4 and Supplementary Fig. 6).In particular, we note that two order parameters are needed for acomplete description of the behaviour of a type-II superconductor inthe presence ofH : one that describes superconductivity and anotherone that describes vortex matter (see, for example, ref. 23).

At low fields below the Tc(H) line, ρ(T )=0, which is attributedto the pinning of the vortex solid8–10. Tc(H) is known24,25 tocorrespond to the so-called irreversibility temperatureTirr(H) belowwhich the magnetization becomes hysteretic, indicating a transitionof the vortex system between a low-T , low-H pinned regime and anunpinned one10. This low-H vortex solid phase has been identifiedexperimentally as a Bragg glass in LSCO (ref. 26), as well asin other cuprates9,10 and some conventional superconductors (forexample, 2H -NbSe2 (ref. 27)). The Bragg glass forms when thedisorder is weak28,29: it retains the topological order of the Abrikosovvortex lattice (see sketch of a Bragg glass in Fig. 5) but yieldsbroadened diffraction peaks. As such a distorted Abrikosov latticehas many metastable states and barriers to motion, it is, strictlyspeaking, a glass.

At higherH , where the density of vortices is larger28,29, a topolog-ically disordered, amorphous vortex glass is expected (see sketch ofa vortex glass in Fig. 5). A transition from a Bragg glass to a vortex

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NATURE PHYSICS DOI: 10.1038/NPHYS2961 ARTICLES

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Figure 4 | Transport H−T phase diagram and scaling regions inunderdoped LSCO. The data are shown for x=0.07 film. a, The colour map(on a log scale) shows the contribution of superconducting fluctuations(SCF) to conductivity1σSCF versus T and H‖c. The dashed red line is a fitwith µ0H′c[T]= 15(1−(T[K]/29)2). The error bars indicate the uncertaintyin H′c that corresponds to 1 s.d. in the slopes of the linear fits inSupplementary Fig. 5. The horizontal dashed black lines mark the values ofthe T=0 critical fields H∗1 and H∗2 for scaling. The pink lines show the high-Tscaling region: the hashed symbols mark areas beyond which scaling fails,and the dots indicate areas beyond which the data are not available. Thegreen lines show the low-T scaling region. b, Simplified phase diagramshowing the crossover temperatures T∗1 and T∗2 corresponding to H∗1 and H∗2,respectively, and the three phases at T=0. The error bars for Tmax indicate3 s.d. from fitting ρ(T) in Fig. 1a.

glass with increasing H has been observed in LSCO (ref. 26) andother cuprates9,10, consistent with our conclusion about two distinctsuperconducting phases at T=0: a superconductor with Tc(H)>0forH<H ∗1 and a superconductor with Tc=0 forH ∗1 <H<H ∗2 . Thepower-law behaviour of ρ(T ) observed for T<Tmax and H >H ∗1is indeed reminiscent of the behaviour expected in a vortex liquidabove the glass transition30–32 occurring at Tg = 0, consistent withtheoretical considerations that have found the vortex glass phase tobe unstable at non-zero temperature33,34 even in 3D.

In general, the theoretical description of the T=0 melting of thevortex lattice (Bragg glass in Fig. 5), associated with the QCP at H ∗1(Fig. 4b), remains an open problem. However, it is known that, atfinite T , melting of the vortex lattice by proliferation of dislocationscorresponds to a phase transition in the 2D XY model9,10. It is thusplausible that the (2 + 1)D XY model in the clean limit coulddescribe the T = 0 melting of the pinned vortex solid, consistentwith our findings. TheQC scaling associatedwith this QPT does not

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Figure 5 | Sketch of the interplay of vortex physics and quantum criticalbehaviour in the H–T phase diagram. Two critical fields, H∗1 and H∗2, arereported, separating three distinct phases at T=0: a superconductor withρ=0 (dark blue) for all T<Tc(H) [Tc(H)>0] and H<H∗1 , a superconductorwith ρ=0 only at T=0 (that is, Tc=0) for H∗1 <H<H∗2, and an insulator(red), where ρ(T=0)→∞, for H∗2<H. The di�erence between the twosuperconducting ground states is in the ordering of the vortex matter: apinned vortex solid (Bragg glass) for H<H∗1 and a vortex glass for H>H∗1 ,as shown schematically. The quantum critical regions (QCRs)corresponding to H∗1 and H∗2 are also shown schematically (dashed lines).The QC scaling associated with H∗1 does not extend to the lowest T (seedotted lines); apparently, this quantum phase transition is ‘hidden’ at low Tby thermal fluctuations that cause the melting of the pinned vortex solid(Bragg glass) at Tm(H) for H<H∗1 , and by the e�ects of disorder for H>H∗1 .Tmax is the temperature at which dρ/dT changes sign.

extend all the way down to the lowest T . For H>H ∗1 , the data sug-gest that the effects of disorder become important at low T , causingthe freezing of the vortex liquid into a vortex glass phase with Tc=0.Tmax(H)may represent a crossover energy scale associated with theglassy freezing of vortices, although there is some evidence in othercuprates that a similar line is a continuous (second-order) glass tran-sition9,10. For H <H ∗1 , the scaling no longer works at low T , whereρ exhibits a large drop (Fig. 4a)—that is, for T lower than a field-dependent temperature scale resembling the Tm(H) line sketchedin Fig. 5. It is known that a sharp drop in ρ is associated with ajump in the reversible magnetization25, which is indicative of thethermal melting of the low-field vortex solid phase9,10,25. In general,the thermal melting takes place at Tm(H)>Tirr(H). Therefore, thefailure of scaling for H <H ∗1 on the low-T side (Fig. 4a) may beattributed to themelting of the pinned state.Indeed, our observationof QC scaling above the non-zero melting temperature is consistentwith general expectations1.

At even higherH , the transition from the superconducting vortexglass phase to an insulator with localized Cooper pairs occurs atH ∗2(whereH ∗1 <H ∗2 .H ′c), consistent with boson-dominated models ofthe SIT (ref. 13) and earlier arguments7. The QC behaviour associ-ated with this QPT is observed down to the lowest measured T .

The scaling behaviour observed near critical fields H ∗1 and H ∗2is consistent with a model where superconductivity is destroyedby quantum phase fluctuations in a 2D superconductor. We notethat such scaling is independent of the nature of the insulator1,13. Inparticular, the same scaling, except for the value of zν, is expectedwhen the insulating phase is due to disorder and when it is causedby inhomogeneous charge ordering, which is known to be present inlow-doped LSCO (refs 35–39), including charge-cluster glass11,40,41and charge density waves42.

We expect a similar two-stage H -field-tuned SIT to occur forother doping levels in the entire underdoped superconductingregime. This is supported by our observation of the same behaviourin samples that were prepared in two very different ways and inwhich the number of doped holes was probably not exactly the same.

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS2961

Also, the possibility of two critical points was suggested recently43for LSCO films with x = 0.08, 0.09 and 0.10 (SupplementaryInformation). Our results provide important insight into theinterplay of vortex line physics and quantum criticality. Apparently,the high-temperature, clean-limit SIT quantum critical fluid fallsinto the grip of disorder with decreasing T and splits into twotransitions: first, a T = 0 vortex lattice to vortex glass transition,followed by a genuine superconductor–insulator QPT at higher H .

MethodsSamples. The LSCO film with a nominal x=0.07 was described in detail inref. 11. The LSCO single crystal with a nominal x=0.06 was grown by thetravelling-solvent floating-zone technique12. Measurements were carried out on asample that was cut out along the main crystallographic axes and polished into abar with dimensions 3.02×0.42×0.34mm3 suitable for direct measurements ofthe in-plane resistance. Electrical contacts were made by evaporating gold onpolished crystal surfaces, followed by annealing in air at 700 ◦C. For currentcontacts, the whole area of two opposing side faces was covered with gold toensure a uniform current flow through the sample. In turn, the voltage contactswere made narrow (∼80 µm) to minimize the uncertainty in the absolute valuesof the resistance. The distance between the voltage contacts is 1.41mm. Goldleads were attached to the sample using Dupont 6838 silver paste. This wasfollowed by the heat treatment at 450 ◦C in an oxygen flow for 6min. Theresulting contact resistances were less than 1� at room temperature. As a resultof annealing, the sample composition is probably La1.94Sr0.06CuO4+y , with y notprecisely known.

Measurements. The in-plane sample resistance and magnetoresistance weremeasured with a standard four-probe a.c. method (∼11Hz) in the Ohmic regimeat current densities as low as (3×10−3−3×10−2)Acm−2 for the x=0.07 filmand 7×10−2 Acm−2 for the x=0.06 crystal. To cover a wide range of T and H ,several different cryostats were used: a 3He system at T down to 0.3 K, withmagnetic fields up to 9 T, using 0.02–0.05 Tmin−1 sweep rates; a dilutionrefrigerator with T down to 0.02K and a 3He system (0.3 K ≤T . 30K) insuperconducting magnets with fields up to 18 T at the National High MagneticField Laboratory (NHMFL), using 0.1−0.2 Tmin−1 sweep rates; and a 35 Tresistive magnet at the NHMFL with a variable-temperature insert (1.2 K≤T . 30K), using 1 Tmin−1 sweep rates. Therefore, the magnetoresistancemeasurements span more than two orders of magnitude in T , down to 0.09K,which is much lower than the temperatures commonly employed in studies ofunderdoped cuprates. For that reason, we use excitations (that is, currentdensities) that are orders of magnitude lower than in similar studies. It was notpossible to cool the samples below 0.09K. The fields, applied perpendicular tothe CuO2 planes, were swept at constant temperatures. The sweep rates werealways low enough to avoid any heating of the sample due to eddy currents.

The resistance per square R�=ρ/d=ρ/(nl) (d , sample thickness; n, numberof CuO2 layers; l=6.6 Å, thickness of each layer); the resistance per square perCuO2 layer R�/layer=nR�=ρ/l .

Received 21 August 2013; accepted 2 April 2014;published online 4 May 2014

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AcknowledgementsWe thank P. Baity for experimental assistance, and A. T. Bollinger and I. Bozovic for thefilm sample. The work by X.S., P.V.L. and D.P. was supported by NSF/DMR-0905843,NSF/DMR-1307075, and the NHMFL, which was supported by NSF/DMR-0654118,NSF/DMR-1157490 and the State of Florida. The work of T.S. was supported by an MSLCollaborative Research Project. V.D. was supported by NSF/DMR-1005751. V.D. and D.P.thank the Aspen Center for Physics, where part of this work was done, for hospitality andsupport under NSF/PHYS-1066293.

Author contributionsX.S. and D.P. conceived the project; the single crystal was grown by T.S.; X.S. and P.V.L.performed the measurements and analysed the data; V.D. and D.P. contributed to the dataanalysis and interpretation; X.S., P.V.L. and D.P. wrote the manuscript; D.P. planned andsupervised the investigation. All authors commented on the manuscript.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to D.P.

Competing financial interestsThe authors declare no competing financial interests.

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