Magnetic Field-Driven Quantum Criticality in Antiferromagnetic Ceptin4

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(35), al. crys- elevated et needle-shaped slightly Bławat to by contrast (Cmcm reported in structure tals Surprisingly, orthorhombic (34). 63) an group in crystallizing ogy phase a covered hypothetical CeT way including compounds the additional is pave for system observations look this unusual to These of 33). diagram (32, phase complex very field-induced a 25, Moreover, (24, criticality 31). quantum pressure-induced AFM- a an with with state ordered coexists superconductivity where example unique m nti otx ti oeotyt mention to noteworthy is pos- it context with 30). this fluctuations 29, In (18, spin character AFM Fulde–Ferrell–Larkin–Ovchinnikov by sible dominated behavior state ing is ulse etme 3 2019. 23, September published First (28). phase Q of formation and 27) Ce 26, (16, criticality quantum with (HFSC) (20). superconductor K heavy-fermion 2.3 distin- a is A which is 1) community. family this of research member guished condensed-matter the among d ulse ne the under Published Submission.y Direct PNAS a is article This interest.y wrote competing D.K. no performed and declare P.W., authors D.K. 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D.D., the and P.W., research; D.G., D.D., designed research; D.K. contributions: Author formula chemical owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To eeto rniinmtl 2–5 aeintda excitement an ignited have (20–25) metal) transition -electron xeddpafr o tdigqatmcriticality. quantum studying for platform extended end critical as quantum first-orderpoints 2 approaching into transitions splits transition metamagnetic the metamagnetic point, this antifer- intermediate Beyond separating sates. a and point triple at paramagnetic, a terminated also romagnetic, is is manner which field-like point tricritical mean of second-order suppression a continuous the in where diagram phase netic compound the in ordering CePtIn attempt antiferromagnetic our the In instability. suppress of to explore point precarious to this moti- community about has more research point condensed-matter critical the quantum understand the vated to at thrust physics unsettling enigmatic the the decades few past the For Significance norogigqett synthesize to quest ongoing our In ,adT and 1, = 2 T PdIn c . hc aiet o-em iudtp normal liquid-type non-Fermi a manifests which K 0.7 = y 4 8 ya ple ed eepoe acntn mag- fascinating a explored we field, applied an by (n CeCoIn PNAS T M and 2 = → NSlicense.y PNAS t d npriua,a ttrsott eqiea quite be to out turns it as particular, in Pd) Pt, = a,1 T | .Tu,orfidnso CePtIn on findings our Thus, 0. 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PHYSICS magnetic, electrical transport, and thermodynamic measure- the field dependence of differential susceptibility, dM /dH , for a ments on CePtIn4 accentuating a perplexing magnetic phase dia- few chosen temperatures (for all temperatures see SI Appendix, gram which exposes metamagnetic quantum critical end points. Fig. S1B). Here, the maxima in dM /dH vs. H correspond to This unique and remarkable finding expands the frontier of MMTs. The peaks in dM /dH become sharper and increase in this unconventional class of QCP, generating an opportunity height as the temperature is lowered, suggesting that quantum to exploit more experimental and theoretical understanding of fluctuations most likely play a key role in governing the breadth associated baffling physics. of field-driven transition as T → 0. One crucial observation that demands further attention is the temperature dependence of the Results and Discussion critical fields associated with these MMTs. While the position Fig. 1 A–D illustrates temperature dependence of magnetic of the first MMT (low-field transition) is hardly altered by the susceptibility (M /H ) of CePtIn4 with field applied along the temperature variation, the high-field MMT appears to be very crystallographic b axis. The AFM nature of the ordering is man- sensitive to the temperature. In Fig. 2B we have plotted the max- ifested by a prominent sharp peak in M /H at Néel temperature imum value of the differential susceptibility, dM /dH |max, as a (Fig. 1A). Notably, it is quite evident from Fig. 1A that in addi- function of temperature. At elevated temperatures, it tends to −n tion to the AFM transition at TN = 2.2 K, there is an upturn diverge with temperature as a power law; i.e., dM /dH |max ∼ T at ∼1 K which appears to be strongly influenced by the applied (with n = 3.1; red dashed line in Fig. 2B). This observation along field. In 1.4 T this upturn becomes distinctly visible and takes a with the fact that CePtIn4 undergoes a second-order–type AFM very pronounced shape (Fig. 1B), signaling a phase transition in transition at zero applied field indicates the possible involve- the system. In this context it must be emphasized that CePtIn4 ment of long-wavelength quantum critical fluctuations over the has only 1 Ce position in its crystal structure. Thus, this second H –T phase diagram in the vicinity of the critical field (36). At anomaly must be associated with the same Ce moments which lower temperature, dM /dH |max exhibits a saturation behavior order AFM at the first place. Surprisingly, with further increase which signals a broadening of the AFM transition. Such broad- in applied field (at 1.7 T, Fig. 1C) we observed a single transition. ening of transition can be caused either by a disorder effect At 5 T, M /H (Fig. 1D) reveals a saturation behavior evinc- or alternatively by thermal or quantum fluctuations. Given the ing the complete polarization of Ce moments. It is to be noted experimental evidence presented in the following section, quan- that zero-field–cooled (ZFC) and field-cooled (FC) data taken tum fluctuations are most likely to play a crucial role in governing at these fields do not show any characteristic difference originat- this behavior. ing from domain effect which in turn discards the possibility of a The temperature dependence of electrical resistivity ρ(T ) of ferromagnetic component within this second transition. It can be CePtIn4 measured in the temperature range 0.05 to 4 K in vari- engendered due to reorientation of magnetic moments forming ous fields applied along the b axis is presented in Fig. 3A. In zero another magnetic order. field, below TN = 2.3 K, ρ(T ) undergoes a drastic fall as a conse- In Fig. 1E, we present a few isothermal magnetization curves quence of reduction of spin disorder scattering due to AFM-type obtained in the AFM-ordered state with applied field parallel ordering of Ce moments. In this regard, it is worthwhile to men- to the crystallographic b axis (see SI Appendix, Fig. S1A for tion that in the earlier report on needle-like CePtIn4 single all isotherms). At very low temperatures (0.5 K ≤ T ≤ 0.8 K), crystals, no such drop in ρ(T ) was observed (35). With increasing M (H ) curves display 2 prominent step-like transitions. These field, TN first shifts toward lower temperature, featuring typical distinct features in M (H ) curves unambiguously demonstrate characteristics of AFM order. For µ0H ≥ 2 T the anomaly in metamagnetic transitions (MMTs) in the system. With increasing ρ(T ) starts to broaden. Simple analysis of low-temperature data temperature, these MMTs are no longer so prominently visible (below TN) considering only electron–electron interaction (i.e., in the M (H ) curves, even though a clear change in the slope is T 2 dependency) will be misleading in the present case due to visible for each isotherm. Furthermore, at sufficiently high fields a significant contribution from scattering of electrons on AFM all magnetization curves tend to saturate at a value ∼1.31 µB spin-wave excitations. As the actual magnon dispersion relation which is much smaller than that expected for a fully ordered mag- is not known for this material, we did not perform any analysis of netic moment of Ce3+ ions. This reduced value corresponds very ρ(T ) data. 2 well with a doublet ground state of F5/2 multiplet split in an To gain further insight into the anomalous behavior seen orthorhombic crystalline electric field potential. Fig. 2A depicts in ρ(T ) vs. T under different applied fields, we measured

AB E

Fig. 1. (A–D) Temperature dependence of magnetization divided by the applied field, M/H, measured at different fields as shown. Applied fields are parallel to the crystallographic b axis. (E) Field dependencies of magnetization at various fixed temperatures below TN as indicated.

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(Fig. visualized observation isotherms well magnetic Another in be witnessed 3 can those (Fig. with behavior plot agreement contour This color one. a single in transitions a distinct 2 T to temperature (1.39 converge increasing range With short field. very applied a over MMTs below sive well 3D that Fig. see feature in temperature extraordinary rigorous all presented A of data view MMT. A MR fied cycle. of the arrows) nature of by first-order inspection (indicated the field down decreasing evidencing and and increasing sweeps up between visible field is a hysteresis clear in K 0.1 at in MMT fest = (MR magnetoresistance with fitting the represents line dashed these Red as through eye. dependence lines the power-law to black guides The are maxima. symbols high-field represent triangles blue dM from susceptibility, differential derived Sharp of value field. maximum the applied of of dependence function dM a in as peaks text), (main susceptibility differential i.2. Fig. B A CePtIn il eiaieo antzto (dM magnetization of derivative Field (A) 4 /dH ut ovnigy h Rsnuaiisat singularities MR the convincingly, Quite . e qae orsodt h o-edmxm whereas maxima low-field the to correspond squares Red A. CePtIn vs. a H –c Temperature (B) system. the in MMT to correspond 4 ln.Fg 3B Fig. plane. nFg 3C Fig. In . ∼ T n lcrccretflwn ihnthe within flowing current electric and −n ρ(H . T .Teefidnsaei excellent in are findings These E). N )/ρ(0) ti vdn hti h AFM the in that evident is It . eel an reveals S2) Fig. Appendix, SI epeetM aameasured data MR present we rsnsM esrdat measured MR presents T − N of 1) Rehbt succes- 2 exhibits MR /dH ≤ ,as eerdt as to referred also ), CePtIn µ 0 H framagni- a (for ≤ .5T of T) 1.75 4 H ntrans- in c /dH mani- |max osblt fatirtclpit(C)at (TCP) point tricritical µ a of critical possibility the near zero phase under paramagnet nematic a field. is of applied compound land- latter formation in the entropy found the However, field. the also to was in due region field feature (12) critical similar the near A scape (40). conditions these obser- that fluctuation, This fact critical region. the of field existence suggesting critical the corroborates the strongly in region, vation field centered critical peak the to a increased exhibits is field the as that cates see a ese rmFg 4C, Fig. phase scans from field seen the magnetic be map out can capacity, to carried heat and we conjecture TCP, of this apparent around this diagram unveil states. to (IMM) metamagnetic So, intermediate and AFM, PM, rating T (1.4 region field S critical the the of showing vicinity landscape the entropy the resents in seen lowered, applied also high is For temperature effect. Zeeman from the the by as caused fields, likely strength. most field is position increasing whose It with temperature shape higher hump-like toward broadened moves a takes T, feature 2 the At this about respectively. excitations, information magnetic detailed and amass structure magnetic to essential in tempera- experiments are lower scattering fields neutron toward high inelastic shifts and and elastic to Both value starts ture. absolute then which its anomaly in single T, decrease a 1.55 into At very merge transition. anomalies becomes first-order–type both it possible a and signaling temperature sharp, lower- higher the toward field, systematically Inter- increasing (above). with fields applied estingly, various under resis- electrical measurements accordance considerably and tivity in magnetic becomes is from inferred This anomaly observations transitions. this the distinct with T, 2 observe 1.4 we up at and field sharp Notably, applied the T. by 1.3 influenced not to is which S4) Fig. Appendix, field, applied at at zero anomaly anomaly at sharp that the note from non–Fermi-liquid to apart important magnetic is in It systems. fluctuations (NFL) critical of state acteristic (PM) paramagnetic the investigated In C temperature correlations. electron lowest strong the of At field. C applied observed zero the in by see confirmed T is 7 ordering to up data the (for eprtr aito fteha aaiyoe temperature over capacity heat the (C ratio of variation single-crystalline of temperature capacity heat magnetotransport the CePtIn and measured electrical, we magnetic, properties, in seen transitions unclear. including and 33) systems, observed (32, such been Pd) few previously a has in to quite behavior worthwhile only is also similar which is that fields, It higher systems. emphasize AFM at heavy-fermion saturation in a unusual of tendency any MR(H out that Although note electrons. should conduction one can the field- of contribution the motion from positive cyclotron resulting induced the MR normal/ordinary (37), with field associated by field be critical magnetic explained the external usually an than in larger is fluctuations spin contribution of suppression negative the the While signs. 0 /T /T /T h etcpct aaa rsne nFg 4A Fig. in presented as data capacity heat The oeuiaeteitiscbl aueo aiu field-induced various of nature bulk intrinsic the elucidate To H .Tepeetdetoylnsaeindi- landscape entropy presented The ). S5 Fig. Appendix, SI .5Twihcicdswt antctil on sepa- point triple magnetic a with coincides which T 1.55 = sclrsae(o a nrp aaa oeseicfields specific some at data entropy raw (for scale color as −lnT ece infiatylrevle inln h presence the signaling value, large significantly a reaches olw a follows 4 /T ndfeetapidfilsHkb fields applied different in esrdfor measured ) eedne uhdvaina eyhg ed was fields high very at deviation Such dependence. frbte iulzto e e.3 and 34 ref. see visualization better (for K ∼1 Sr 3 −lnT Ru PNAS Ce C 2 2 (H O PdGa 7 | eednefralapidfilschar- fields applied all for dependence CePtIn tdfeetfie eprtrs As temperatures. fixed different at ), 3)and (39) coe ,2019 8, October CePtIn .Tebl AFM bulk The S3). Fig. Appendix, SI 12 C olw ierdpnec with- dependence linear a follows ) /T 3) e t culoii remains origin actual its yet (38), 4 (H T svr ls oteQPunder QCP the to close very is 4 CeAuSb N nmgei ed pt . T 2.5 to up fields magnetic in ) hr xssaohrbroad another exists there , ipasqieastonishing quite displays Ce | tp nml in anomaly λ-type i.4A Fig. . C o.116 vol. 3 T T /T 2 ≤ M T H nml rtshifts first anomaly 1) i.4B Fig. (15). In µ N –T trst deviate to starts 0 11 H | .8Kand K 1.28 = ersnsthe represents o 41 no. oo a in map color (T ≤ . )with T) 1.9 M Sr pr the spark 3 tor Pt = Ru | C S 20335 rep- 2 /T /T O SI 7

PHYSICS AB E

Fig. 3. (A) Temperature dependence of electrical resistivity, ρ(T), of single-crystalline CePtIn4 measured with electric current flowing in the a–c crystallographic plane, in different fields applied along the b axis. (B) Magnetic field dependencies of the transverse MR of single-crystalline CePtIn4 measured at several temperatures in the AFM state with electrical current flowing within the a–c plane and magnetic field applied along the crystallographic b axis. (C) Magnetic field scan of MR at 0.1 K. A clear hysteresis between 2 MMTs is indicative of the first-order nature of the transition. Vertical arrows indicate field up and down sweeps. (D) Magnified view of the MR data around the MMT for a few selective temperatures (see SI Appendix, Fig. S2 for all temperatures). (E) Transverse MR of CePtIn4 in the AFM state depicted in an H–T color map plot with values of MR represented as the color scale. As highlighted by arrows, the metamagnetic phase boundary is clearly visible in this plot.

behavior above and below this TCP (for C (H ) at all tempera- C /T (H ) is consistent with the metamagnetic phase formation tures investigated see SI Appendix, Fig. S6). At 1.3 K it shows a as probed by isothermal magnetization and MR measurements step-like feature at around 1.5 T but, as the temperature is low- (above). The second striking aspect of our C (H ) data is the ered, this step-like feature becomes very sharp and then splits very narrow spread of applied ﬁelds at which the peaks are into 2 sharp transitions whose magnitudes gradually decrease observed, indicating the fact that within the accuracy of our mea- with decreasing temperature. The appearance of 2 peaks in surements the AFM or MMT phase line becomes vertical in the

A B

Fig. 4. (A) Temperature variation of the heat capacity over temperature ratio (C/T) measured for single-crystalline CePtIn4 in various magnetic ﬁelds applied along the crystallographic b axis. (B) Entropy landscape presented as an H–T color map plot with entropy divided by temperature (S/T) represented by color scale. (C) Field scan of heat capacity (presented as C/T) at different ﬁxed temperatures.

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Rev. field. magnetic a with behavior (2010). 1166 (2001). 797–855 73, magnets. metallic metals. in criticality 1999). H PbCu(NO (H –T )/T ln as plane ) oo a ltwith plot map color antcpaedarmwt ple field applied with diagram phase Magnetic = 542(2002). 056402 89, srepresented is 4A) Fig. (from fields fixed different at T µ unu hs Transitions Phase Quantum N antcfil nue unu rtclpiti YbRh in point critical quantum induced Magnetic-field al., et 0 2 H (0)[1 ) T 6 T c hs e.Lett. Rev. Phys. N 1) and (19), hs e.Lett. Rev. Phys. N ≈ olw oe-a ma edlk)form field-like) (mean power-law a follows − → . .Hwvr nraiy uhaseai is scenario a such reality, in However, T. 2.4 (H ,laigt unu rtcledpoints end critical quantum to leading 0, /H CePtIn C hs e.B Rev. Phys. Yb /T 622(2017). 267202 119, T c N 124(2002). 217204 88, ) 2 3 ersne stecolor the as represented 4A) Fig. (from mi text). (main ] Pt akdb h lc ahdline dashed black the by marked CmrdeUiest rs,NwYr,NY, York, New Press, University (Cambridge 4 4 svr ls oafield-tuned a to close very is 49–40 (1998). R4198–R4201 57, (36). Nature 2–2 (2005). 226–229 433, C (H kb ) Sr xspeetdas presented axis sqierr in rare quite is T Science 3 Ru N e.Md Phys. Mod. Rev. as K 0 = 2 O 1161– 329, 2 Si 7 2 (12), . Phys. 4 .Hrio,M am,J .Mds,Retathde re tametamagnetic a at order hidden Reentrant Balicas Mydosh, L. A. 15. J. Jaime, M. Harrison, N. 14. land- Entropy Grigera, A. Lester S. C. Mackenzie, 13. P. A. Mercure, J.-F. Perry, S. R. Rost, W. A. 12. Grigera A. S. 11. Gegenwart P. 10. aziwc o sitn nsnhsso h rsasadtecytlstruc- crystal 2015/19/B/ST3/03158. the Grant Research National and under the crystals (Poland) by supported Center the was Science work of This respectively. synthesis characterization, in ture assisting for Daszkiewicz method ACKNOWLEDGMENTS. relaxation the using system. T refrigerator 7 to dilution up same fields very in PPMS the K per- 4 in QD were to 0.1 a measurements range capacity in the in Heat technique formed assembly. 4-probe to refrigerator current up dilution fields alternating T) magnetic standard (9 applied in a and using K T, measured 4 9 to was magnetometer 0.05 resistivity range SQUID electrical temperature MPMS the The over refrigerator. (QD) iHe3 Design an Quantum with measurements expanded a Magnetic cell in characterization. performed primary unit some were larger with in provided along a are 34 information crystals has structural ref. with its single and temperature it growth these ordering single-crystal (35), of Surprisingly, AFM details in al. one. enhancement reported et an the Bławat show to by along compared reported grown volume as crystals plate-like structure obtained tal We (34). crystallographic report the previous our CePtIn in of crystals Single Methods and Materials opens the by compound driven instabilities this field. quantum magnetic of in studies physics for prospects mystifying more of of perplexing content description a stantial such Our with diagram. points. antiferromagnets phase end become system critical electron lines quantum lated phase at where 2 terminate point point, Thus, and triple this like a Beyond first-order also metamag- coexist. is states intermediate which netic and TCP, paramagnetic, dia- the of phase antiferromagnetic, at suppression stunning terminated that a heat is yielded reveals specific fields, It with applied gram. together in observations, collected these data MR All fields. Remarkably, high isotherms. magnetoresistance below of the shape from of discerned easily proximity the be also the can transitions reflects Metamagnetic clearly QCP. by followed which divergence behavior, susceptibility power-law saturation transi- toward differential tendency metamagnetic temperature the exhibits subsequent decreasing 2 Upon tions. and anti- order of occurrence ferromagnetic the com- confirmed measurements antiferromagnetic Magnetization an of pound properties thermodynamic and for possibility. exciting essential are band electronic calculations with structure along Hall MMT van- Thus, the field. other across metamagnetic measurements the the effect at while transition Fermi field Lifshitz spin-polarized a increasing at the the ishes electronic of in enhanced one of is of evolution surfaces volume continuous the a where causes structure field magnetic the eainhpt h neligeetoi tutr.Smlrto Similar structure. possible for the electronic inferred is conclusions underlying diagram the the phase to observed the relationship of aspect tempting .P Gegenwart P. 9. ncnlso,w netgtdmgei,eetia transport, electrical magnetic, investigated we conclusion, In 642(2005). 064422 72, point. end critical quantum (2015). 373–378 Sr in criticality quantum with (2009). 1360–1363 associated 325, formation phase of scape Sr Matter Phys. J. New 3 Ru CePtIn 2 T CePtIn O 1418 (2008). 1184–1188 403, N 7 il-ual pndniywv hssi Sr in phases spin-density-wave Field-tunable al., et antcfil-ue unu rtclpiti CeAuSb in point critical quantum field-tuned Magnetic al., et . Science eel ut euirlna eddpnec at dependence field linear peculiar quite a reveals 7 (2006). 171 8, antcfil-ue unu rtclt ntemtli ruthenate metallic the in criticality quantum field-tuned Magnetic al., et novninlqatmciiaiyi YbRh in criticality quantum Unconventional al., et ihfil hs iga ftehayfrinmtlYbRh metal heavy-fermion the of diagram phase High-field al., et 4 4 un u ob nqecs fsrnl corre- strongly of case unique a be to out turns 2–3 (2001). 329–332 294, oepoeqatmciiaiyi hssystem. this in criticality quantum explore to PNAS b xs vntog tcytlie ntesm crys- same the in crystallizes it though Even axis. eaeidbe oAij akmradMarek and Hackemer Alicja to indebted are We 4 eegonuigteI-u ehdoutlined method In-flux the using grown were hs e.Lett. Rev. Phys. | coe ,2019 8, October CeRu 942(2003). 096402 90, 2 Si CePtIn 2 T | CePtIn 4) eatcpt that anticipate we (41), N o.116 vol. yteapidfield applied the by 4 3 niaigasub- a indicating Ru 4 2 | 2 Si oepoethis explore to O 2 o 41 no. T 7 . N . 3 hs Condens. B Phys. Ru a.Mater. Nat. . .The K. 2.3 = 2 . 2 hs e.B Rev. Phys. O 7 | . Science 20337 2 Si 14, 2 .

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