Field Driven Quantum Criticality in the Spinel Magnet ZnCr 2 Se 4
C. C. Gu, Z. Y. Zhao, X. L. Chen, M. Lee, E. S. Choi, Y. Y. Han, L. S. Ling, L. Pi, Y. H. Zhang, G. Chen, Z. R. Yang, H. D. Zhou, X. F. Sun
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Field Driven Quantum Criticality in the Spinel Magnet ZnCr Se C. C. Gu , Z. Y. Zhao , , X. L. Chen , M. Lee , , E. S. Choi , Y. Y. Han , L. S. Ling , L.Pi , , , Y. H. Zhang , , G. Chen , , ∗ Z. R. Yang , , , † H. D. Zhou , , ‡ and X. F. Sun , , § Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory, Chinese Academy of Sciences,Hefei, Anhui 230031, People’s Republic of China Department of Physics, Hefei National Laboratory for Physical Sciences at Microscale,and Key Laboratory of Strongly-Coupled Quantum Matter Physics (CAS),University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Fujian Institute of Research on the Structure of Matter,Chinese Academy of Sciences, Fuzhou, Fujian 350002, People’s Republic of China National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306-4005, USA Department of Physics, Florida State University, Tallahassee, FL 32306-3016, USA State Key Laboratory of Surface Physics, Center for Field Theory & ParticlePhysics Department of Physics, Fudan University, Shanghai, 200433, China Collaborative Innovation Center of Advanced Microstructures,Nanjing, Jiangsu 210093, People’s Republic of China Institute of Physical Science and Information Technology,Anhui University, Hefei, Anhui 230601, People’s Republic of China Key laboratory of Artificial Structures and Quantum Control (Ministry of Education),School of Physics and Astronomy, Shanghai JiaoTong University, Shanghai, 200240, China and Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA (Dated: September 12, 2018)We report detailed dc and ac magnetic susceptibilities, specific heat, and thermal conductivitymeasurements on the frustrated magnet ZnCr Se . At low temperatures, with increasing magneticfield, this spinel material goes through a series of spin state transitions from the helix spin state tothe spiral spin state and then to the fully polarized state. Our results indicate a direct quantumphase transition from the spiral spin state to the fully polarized state. As the system approachesthe quantum criticality, we find strong quantum fluctuations of the spins with the behaviors such asan unconventional T -dependent specific heat and temperature independent mean free path for thethermal transport. We complete the full phase diagram of ZnCr Se under the external magneticfield and propose the possibility of frustrated quantum criticality with extended densities of criticalmodes to account for the unusual low-energy excitations in the vicinity of the criticality. Our resultsreveal that ZnCr Se is a rare example of 3D magnet exhibiting a field-driven quantum criticalitywith unconventional properties. PACS numbers: 75.30.Kz, 75.40.-s, 75.47.Lx
Since the new centuary, quantum phase transition hasemerged as an important subject in modern condensedmatter physics [1]. Quantum phase transition and quan-tum criticality are associated with qualitative but con-tinuous changes in relevant physical properties of the un-derlying quantum many-body system at absolute zerotemperature [1, 2]. In the vicinity of quantum criti-cality, the low-energy and long-distance properties arecontrolled by the quantum fluctuation and the criticalmodes of the phase transition such that certain inter-esting and universal scaling laws could arise. It is well-known that quantum criticality often occurs in the sys-tem with competing interactions where different inter-actions favor distinct phases or orders. Many physicalsystems such as the high-temperature superconductingcuprates [2], heavy fermion and Kondo lattice materi-als [3], Fermi liquid metals with spin density wave insta-bility [4], and Mott insulators have been proposed to berealizations of quantum criticality [1]. For superconduc-tors and metals, the multiple low-energy degrees of free- dom and orders may complicate the critical phenomenaand the experimental interpretation. In contrast, Mottinsulators with large charge gaps are primarily describedby spin and/or orbital degrees of freedom and may havethe advantage of simplicity in revealing critical behaviors.The Ising magnets CoNb O and LiHoF in exter-nal magnetic fields realize the quantum Ising model andtransition [5–10]. External magnetic fields in dimer-ized magnets like han purple BaCuSi O [11, 12] inducea triplon Bose-Einstein condensation transition. In amore complicated example of the diamond lattice anti-ferromagnet FeSc S [13–17], it is the competition be-tween the superexchange interaction and the on-site spin-orbital coupling that drives a quantum phase transitionfrom the antiferromagnetic order to the spin-orbital sin-glet phase [18, 19]. These known examples of quan-tum phase transitions in strong Mott insulating mate-rials with spin degrees of freedom are described by sim-ple Ising or Gaussian criticality where there are discretenumber of critical modes governing the low-energy prop-erties. In this Letter, we explore the magnetic proper-ties of a three-dimensional frustrated magnetic materialZnCr Se . From the thermodynamic, dynamic suscepti-bility and thermal transport measurements, we demon-strate that there exists a field-driven quantum criticalitywith unusual properties such as a T -dependent specificheat and temperature independent mean free path for thethermal transport. Our quantum criticality has extendednumbers of critical modes and is beyond the simple Isingor Gaussian criticality among the existing materials thathave been reported before.In the spinel compound ZnCr Se , the Cr ion hoststhe localized electrons and give rise to the spin-3/2(Cr ) local moments that form a 3D pyrochlore lat-tice. The reported dielectric polarization [20], magne-tization and ultrasound [21], neutron and synchrotron x-ray [22, 23] studies have shown that, with increasing mag-netic field, this system goes from helix spin state to spiralspin state to an unidentified regime, and then fully polar-ized state at the measured temperatures. Two possibili-ties have been proposed for this unidentified regime, anumbrella state and a spin nematic state [21, 24]. Both theumbrella state and a spin nematic state break the spin ro-tational symmetry. We address this unidentified regimeby completing the magnetic phase diagram of ZnCr Se with dc and ac susceptibility, specific heat, and thermalconductivity measurements. We do not observe signa-tures of symmetry breaking in the previously unidenti-fied regime down to the lowest measured temperature.We attribute our experimental results to a quantum crit-ical point (QCP) between the spiral spin state and thepolarized state, and identify the previously unidentifiedregime as the quantum critical regime.The experimental details are listed in the online sup-plemental materials [25]. The dc magnetization mea-sured at 0.01 T in Fig. 1(a) shows a pronounced peakat T N = 21 K, corresponding to the antiferromagnetic(AFM) order accompanied by a cubic to tetragonal struc-tural transition as previsouly reported [22]. With increas-ing fields, the peak shifts to lower temperatures. The dcmagnetization measured at 0.5 K in Fig. 1(b) shows ananomaly near H C1 ∼ . M /d H curve. As previous studies reported,the magnetic domain reorientations occurs at this criti-cal field H C1 and above which, the helix spin structureis transformed into a tilted conical one [20–23]. Due tothe reorientation of magnetic domains, the magnetiza-tion displays hysteresis when the field is ramping downbelow H C1 . This reorientation is also revealed as an irre-versibility between the ZFC and FC curves below 8 K forsusceptibility measured at 0.01 T, while it is suppressedcompletely at H ≥ . χ ′ in Fig. 2(a) clearlyshows two peaks at H C1 and H C2 . Here, H C1 is consis-tent with the H C1 obtained from the magnetization dataabove. H C2 is consistent with the reported H C2 value, H C1 (b) T = 0.5 K H (T) M ( B /f. u . ) (a) ZFC: emptyFC: solid M ( B /f. u . ) T (K) d M / d H FIG. 1. (Color online.) (a) The temperature dependence ofzero field cooling (ZFC) and field cooling (FC) dc magneti-zations at different applied fields. (b) The dc magnetizationmeasured at 0.5 K and its d M /d H curve. above which the spiral spin structure is suppressed witha concomitant structure transition from tetragonal to cu-bic. Meanwhile, a small bump at H C1 , a sharp peak at H C2 , and a step-like anomaly near 9.5 T are clearly seenfor the imaginary part ( χ ′′ ) measured at 7.5 K. This step-like anomaly is in accordance with the plateau observedfrom the sound velocity measurements around 10 T at 2K, which has been correlated to the onset of fully polar-ized magnetic phase at H C3 [21]. Upon further cooling, H C3 moves to lower fields and is hardly discernible below1.5 K from the ac susceptibility measurement, while H C2 shifts to higher fields (see the inset of Fig. 2(b)).At zero magnetic field, the specific heat in Fig. 3(a)shows a sharp peak at T N = 21 K, which shifts to alower temperature with increasing magnetic field anddisappears completely at 6.5 T. Moreover, a small low-temperature hump around 1 ∼ H C2 H C1 ’ ( a r b . un i t s ) (a) (b) " ( a r b . un i t s ) H (T) H C3 " ( a r b . un i t s ) H (T) H C2 FIG. 2. (Color online.) The magnetic field dependence ofac susceptibility at several temperatures: (a) the real compo-nent; (b) the imaginary component. The inset of (b) showsthe zoom-in of the high-field data. The arrows indicate theevolution of high-field anomalies with increasing tempera-tures.
Below 1 K, we tend to fit the heat capacity data at 6.5T with a γT α behavior. The obtained result is T downto the lowest temperature of 0.06 K. Here we assume thelattice contribution of specific heat at so low tempera-tures is negligible, and then the T behavior for 6.5 Tdata is abnormal for a 3D magnet.To further manifest the dynamic properties of the sys-tem under the magnetic field, we carry out the thermalconductivity measurement. As we depict in Fig. 3(b),the thermal conductivity κ at 0 T shows a structural-transition-related anomaly at T N = 21 K and a strongweakness of the κ ( T ) slope around 1 K that should be re-lated to the spin fluctuations observed from specific heat.With increasing magnetic field, T N shifts to lower tem-peratures and disappears at H ≥ H ≥ . κ mainly shows agradual increase with increasing magnetic field at hightemperatures ( T >
T < κ ( H ) /κ (0) curve in Fig. 3(c) shows threeweak anomalies at ∼
1, 5.5 and 8 T, which correspondto H C1 , H C2 , and H C3 , respectively. At H C1 , a spinre-orientation appears, which is related to a minimizing -3 -2 -1 -3 -2 -1 (d)(a) ( W / K m ) T (K) (b) H C1 ( H ) / ( ) H (T) H C2 H C3 l ( mm ) T (K) (c) ~T C ( J / m o l * K ) T (K)
FIG. 3. (Color online.) (a) The temperature dependenceof specific heat at several magnetic fields from 0.06 K to 30K. The dashed line represents the T dependence. (b) Thetemperature dependence of thermal conductivity from 0.3 Kto 30 K at various magnetic fields. (c) The field dependenceof thermal conductivity at selected temperatures below 2 K.(d) The calculated mean free path. of the anisotropy gap and a sudden increase of the AFMmagnon excitations. This could cause an enhancement ofmagnon scattering on phonons and the low-field decreaseof κ . The second anomaly at H C2 , which becomes clearerat 0.97 K, is demonstrated as a dip-like suppression of κ and is likely due to the spin fluctuations at H C2 . Thethird anomaly at H C3 , identified as a quicker increase of κ , is apparently due to the strong suppression of spin fluc-tuations associated with the transition or crossover fromthat unidentified regime to the fully polarized spin state.The spin fluctuations are strongly suppressed in the fullypolarized spin state because the spin excitation is gappedat low energies. At lower temperatures that were not ac-cessed in the previous experiments [20–23], the anomaliesat H C2 and H C3 tend to merge, consistent with the oppo-site temperature dependencies of these two critical fieldsobserved from our ac susceptibility measurement. In par-ticular, at 0.5 K these two anomalies merge into a singleone at 6.5 T and the κ ( H ) /κ (0) curve shows a deep valleyat the background of field-induced enhancement. This isconsistent with the specific heat result showing that thespin fluctuation is the strongest around 6.5 T. As we willexplain in detail, both the specific heat and the thermaltransport results suggest the existence of the quantumcriticality at 6.5 T.Before getting onto our intepretation, we here calcu-late the phonon mean free path from κ using a standardmethod [30]. We choose the Debye temperature to be 308K [31] and assume κ is primarily phononic. The resultsare depicted in Fig. 3(d). At 0 T, the phonon mean free Quantum critical regime, C
Helix, T H C3 from ac M(H) H C2 from M(T) H C2 from ac M(H) H C1 from dc M(H) (T) Specific heat H C3 from (H) H C2 from (H) QCP
H (T) T ( K ) FM, C
Spiral, T
FIG. 4. (Color online.) The H - T phase diagram of ZnCr Se .“T” and “C” refer to the tetragonal and the cubic structure,respectively. “Helix”, “Spiral”, “FM” stand for the helix spinstate, spiral spin state, and spin-fully polarized state, respec-tively. A QCP is deduced between the spiral spin state andthe polarized phase. The solid (dashed) boundary refers to ac-tual phase transition (crossover). The pink region is markedas the quantum critical regime. See the main text for thedetailed discussion. path ( l ∼ − mm) is nearly two orders of magnitudesmaller than the sample size ( ∼ l , the magnetic excitations are not likely to make a siz-able contribution to the heat transport. With increasingmagnetic fields, l is generally enhanced, indicating a sup-pression of magnetic scatterings. Under the highest fieldof 14 T, the phonon mean free path approaches the sam-ple size, which indicates the complete suppression of spinfluctuations in the polarized state. This is consistent withthe gapped spin excitations for the fully polarized spinstate. In contract, at 6.5 T, l drops back to 10 − mmsize with no obvious temperature dependence below 1 K.A detailed H - T phase diagram of ZnCr Se was con-structed in Fig. 4 by using the phase transition tempera-tures and critical fields obtained from our above measure-ments. By comparing to the reported phase diagram [21],two important new features were observed in this fullphase diagram with lower temperatures and higher mag-netic fields. One is that the phase transition tempera-ture for the spiral spin structure is suppressed to zerotemperature with increasing fields before the system en-ters the fully polarized state. Therefore, there is a directquantum phase transition between the spiral spin state and the polarized phase, and this transition is marked asthe QCP in Fig. 4. The other one is that the previousunidentified regime between the spiral state and the fullypolarized state does not persist down to the lowest tem-perature. Note that our measurements were carried outat a much lower temperature than the previous reports.Thus, in Fig. 4 this previously unidentified regime is nat-urally identified as the quantum critical regime that is thefinite temperature extention of the quantum criticality.Why is the previously unidentified regime not an um-brella state or a spin nematic state? As we have pointedout earlier, both states break the spin rotational sym-metry, and the former may break the lattice translation.This is a 3D system, and this kind of symmetry break-ing should persist down to zero temperature and covera finite parameter regime. This finite-range phase is notobserved at the lowest temperature. For the same rea-son, the symmetry should be restored at high enoughtemperatures via a phase transition. Such a thermody-namic phase transition is clearly not observed in the heatcapacity and thermal transport measurements.The spin spiral state and the fully polarized state aredistinct phases with different symmetry properties. Thelatter is translational invariant and fully gapped, whilethe former breaks the lattice symmetry and spin rota-tional symmetry. There must be a phase transition sepa-rating them, and this quantum phase transition is man-ifested as the QCP at 6.5 T in Fig. 4. What is the prop-erty of this criticality? The heat capacity was found tobehave as T at low temperatures at the QCP, indicatingmuch larger density of states than the simple Gaussianfixed point. For a Gaussian fixed point, we would ex-pect the heat capacity as T up to a logarithmic correc-tion due to the critical fluctuations. The T heat capac-ity suggests that the low-energy density of states shouldscale as D ( ǫ ) ∼ ǫ with the energy ǫ . We know that thenodal line semimetal with symmetry and topologicallyprotected line degeneracies has this extended density ofstates when the Fermi energy is tuned to the degener-ate point [32]. However, our system is purely bosonicwith spin degrees of freedom, and there is no emergentfermionic statistics. To support D ( ǫ ) ∼ ǫ at the QCP,we would have the critical modes to be degenerate or al-most degenerate along the lines in the reciprocal spacesuch that the current thermodynamic measurement can-not resolve them. It has been known that the frustratedspin interactions could lead to such line degeneracies forthe critical modes and the resulting frustrated quantumcriticality [33, 34]. The possibility that infinite modeswith line degeneracies become critical at the same timeis an unconventional feature of this QCP. These criticalmodes scatter the phonon strongly and suppress the ther-mal transport near the criticality. It will be interesting todirectly probe these degenerate modes with inelastic neu-tron scattering and explore the fates of the critical modeson both sides of the QCP. Our thermal transport resultsalso call for further theoretical effects on the scatteringbetween the extended density of critical modes and thelow-energy phonons near the criticality.Finally the system displays different lattice structuresfor different magnetic phases in the phase diagram. Boththe helix and the spiral spin states have the tetragonalstructure, while the quantum critical regime and the fullypolarized state have the cubic structure. This is simplythe consequence of the spin-lattice coupling. The helixand the spiral spin states break the lattice cubic symme-try, and this symmetry is transmitted to the lattice viathe spin-lattice coupling. The quantum critical regimeand the fully polarized state are uniform states and re-store the lattice symmetry. The correlation between thesound velocity and the magnetic structure in the previousexperiments has a similar origin [21].In summary, by completing the H - T phase diagramof ZnCr Se , we demonstrate the existence of QCP andquantum critical regime induced by applied magneticphase in this 3D magnet. Our finding of the unconven-tional quantum criticality calls for future works and islikely to provide an unique example of frustrated quan-tum criticality for further studies. Acknowledgments. —This research was supported bythe National Key Research and Development Program ofChina (Grant No. 2016YFA0401804), the National Nat-ural Science Foundation of China (Grant Nos. 11574323,U1632275) and the Natural Science Foundation of AnhuiProvince (Grant No. 1708085QA19). X.F.S. acknowl-edges support from the National Natural Science Foun-dation of China (Grant Nos. 11374277 and U1532147),the National Basic Research Program of China (GrantNos. 2015CB921201 and 2016YFA0300103) and the In-novative Program of Development Foundation of HefeiCenter for Physical Science and Technology. G.C. thanksfor the support from the ministry of science and technol-ogy of China with the grant No. 2016YFA0301001, theinitiative research funds and the program of first-classUniversity construction of Fudan University, and thethousand-youth-talent program of China. H.D.Z. thanksfor the support from the Ministry of Science and Tech-nology of China with Grant No. 2016YFA0300500 andfrom NSF-DMR with Grant No. NSF-DMR-1350002.Z.Y.Z. acknowledges support from the National Natu-ral Science Foundation of China (Grant No. 51702320).M.L. and E.S.C. acknowledge the support from NSF-DMR-1309146. The work at NHMFL is supported byNSF-DMR-1157490 and the State of Florida. The x-raywork was performed at HPCAT (Sector 16), AdvancedPhoton Source, Argonne National Laboratory. HPCAToperations are supported by DOE-NNSA under AwardNo. de-na0001974 and DOE-BES under Award No. DE-FG02-99ER45775, with partial instrumentation fundingby NSF. The Advanced Photon Source is a US Depart-ment of Energy (DOE) Office of Science User Facility op-erated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-AC02-06CH11357.C.G. and Z.Y.Z. contributed equally to this work. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] S. Sachdev, Quantum Phase Transitions , 2nd ed. (Cam-bridge University Press, New York, USA, 2011).[2] S. Sachdev, Rev. Mod. Phys. , 913 (2003).[3] Q. Si and F. Steglich, Science , 1161 (2010).[4] H. v. L¨ohneysen, A. Rosch, M. Vojta, and P. W¨olfle, Rev.Mod. Phys. , 1015 (2007).[5] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzyn-ska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl,and K. Kiefer, Science , 177 (2010).[6] T. Liang, S. M. Koohpayeh, J. W. Krizan, T. M. Mc-Queen, R. J. Cava, and N. P. Ong, Nat. Commun. ,7611 (2015).[7] A. W. Kinross, M. Fu, T. J. Munsie, H. A. Dabkowska,G. M. Luke, Subir Sachdev, and T. Imai, Phys. Rev. X , 031008 (2014).[8] C. M. Morris, R. Vald´es Aguilar, A. Ghosh, S. M. Kooh-payeh, J. Krizan, R. J. Cava, O. Tchernyshyov, T. M.McQueen, and N. P. Armitage, Phys. Rev. Lett. ,137403 (2014).[9] H. M. Rønnow, J. Jensen, R. Parthasarathy, G. Aeppli,T. F. Rosenbaum, D. F. McMorrow, and C. Kraemer,Phys. Rev. B , 054426 (2007).[10] P. B. Chakraborty, P. Henelius, H. Kjønsberg, A. W.Sandvik, and S. M. Girvin, Phys. Rev. B , 144411(2004).[11] M. Jaime, V. F. Correa, N. Harrison, C. D. Batista, N.Kawashima, Y. Kazuma, G. A. Jorge, R. Stern, I. Hein-maa, S. A. Zvyagin, Y. Sasago, and K. Uchinokura, Phys.Rev. Lett. , 087203 (2004).[12] T. Giamarchi, C. R¨uegg, and O. Tchernyshyov, Nat.Phys. , 198 (2008).[13] V. Fritsch, J. Hemberger, N. B¨uttgen, E. W. Scheidt,H. A. Krug von Nidda, A. Loidl, and V. Tsurkan, Phys.Rev. Lett. , 116401 (2004).[14] A. Krimmel, M. M¨ucksch, V. Tsurkan, M. M. Koza, H.Mutka, and A. Loidl, Phys. Rev. Lett. , 237402 (2005).[15] N. J. Laurita, J. Deisenhofer, LiDong Pan, C. M. Morris,M. Schmidt, M. ohnsson, V. Tsurkan, A. Loidl, A. andN. P. Armitage, Phys. Rev. Lett. , 207201 (2015).[16] L. Mittelst¨adt, M. Schmidt, Zhe Wang, F. Mayr, V.Tsurkan, P. Lunkenheimer, D. Ish, L. Balents, J. Deisen-hofer, and A. Loidl, Phys. Rev. B , 125112 (2015).[17] A. Biffin, Ch. R¨uegg, J. Embs, T. Guidi, D. Cheptiakov,A. Loidl, V. Tsurkan, and R. Coldea, Phys. Rev. Lett. , 067205 (2017).[18] G. Chen, L. Balents, and A. P. Schnyder, Phys. Rev.Lett. , 096406 (2009).[19] G. Chen, A. P. Schnyder, and L. Balents, Phys. Rev. B , 224409 (2009).[20] H. Murakawa, Y. Onose, K. Ohgushi, S. Ishiwata, andY. Tokura, J. Phys. Soc. Jpn. , 043709 (2008).[21] V. Felea, S. Yasin, A. Gnther, J. Deisenhofer, H. A. Krugvon Nidda, S. Zherlitsyn, V. Tsurkan, P. Lemmens, J. Wosnitza, and A. Loidl, Phys. Rev. B , 104420 (2012).[22] F. Yokaichiya, A. Krimmel, V. Tsurkan, I. Margiolaki, P.Thompson, H. N. Bordallo, A. Buchsteiner, N. St¨u β er,D. N. Argyriou, and A. Loidl, Phys. Rev. B , 064423(2009).[23] J. Akimitsu, K. Siratori, G. Shirane, M. Iizumi, and T.Watanabe, J. Phys. Soc. Jpn. , 172 (1978).[24] A. Miyata, H. Ueda, Y. Ueda, Y. Motome, N. Shannon,K. Penc, and S. Takeyama, J. Phys. Soc. Jpn. , 114701(2012).[25] See Supplemental Material online for experimental de-tails, which includes Ref. [26-29].[26] C. C. Gu, Z. R. Yang, X. L. Chen, L. Pi, and Y. H.Zhang, J. Phys. Condens. Matt. , 18LT01 (2016).[27] P. Zajdel, W. Y. Li, W. van Beek, A. Lappas, A. Zi-olkowska, S. Jaskiewicz, C. Stock, and M. A. Green,Phys. Rev. B , 134401 (2017). [28] Z. L. Dun, M. Lee, E. S. Choi, A. M. Hallas, C. R. Wiebe,J. S. Gardner, E. Arrighi, R. S. Freitas, A. M. Arevalo-Lopez, J. P. Attfield, H. D. Zhou, and J. G. Cheng, Phys.Rev. B , 064401 (2014).[29] X. F. Sun, W. Tao, X. M. Wang, and C. Fan, Phys. Rev.Lett. , 167202 (2009).[30] Z. Y. Zhao, X. M. Wang, C. Fan, W. Tao, X. G. Liu, W.P. Ke, F. B. Zhang, X. Zhao, and X. F. Sun, Phys. Rev.B , 014414 (2011).[31] T. Rudolf, C. Kant, F. Mayr, J. Hemberger, V. Tsurkan,and A. Loidl, Phys. Rev. B , 052410 (2007).[32] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev.B , 235126 (2011).[33] A. Mulder, R. Ganesh, L. Capriotti, and A. Paramekanti,Phys. Rev. B , 214419 (2010).[34] G. Chen, M. Hermele, and L. Radzihovsky, Phys. Rev.Lett. , 016402 (2012). r X i v : . [ c ond - m a t . s t r- e l ] M a r Supplemental Online Material
Field Driven Quantum Criticality in the Spinel Magnet ZnCr Se C. C. Gu , Z. Y. Zhao , , X. L. Chen , M. Lee , , E. S. Choi , Y. Y. Han , L. S. Ling , L.Pi , , , Y. H. Zhang , , G. Chen , , ∗ Z. R. Yang , , , † H. D. Zhou , , ‡ and X. F. Sun , , § Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory, Chinese Academy of Sciences,Hefei, Anhui 230031, People’s Republic of China Department of Physics, Hefei National Laboratory for Physical Sciences at Microscale,and Key Laboratory of Strongly-Coupled Quantum Matter Physics (CAS),University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China Fujian Institute of Research on the Structure of Matter,Chinese Academy of Sciences, Fuzhou, Fujian 350002, People’s Republic of China National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306-4005, USA Department of Physics, Florida State University, Tallahassee, FL 32306-3016, USA State Key Laboratory of Surface Physics, Center for Field Theory & ParticlePhysics Department of Physics, Fudan University, Shanghai, 200433, China Collaborative Innovation Center of Advanced Microstructures,Nanjing, Jiangsu 210093, People’s Republic of China Institute of Physical Science and Information Technology,Anhui University, Hefei, Anhui 230601, People’s Republic of China Key laboratory of Artificial Structures and Quantum Control (Ministry of Education),School of Physics and Astronomy, Shanghai JiaoTong University, Shanghai, 200240, China and Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996-1200, USA (Dated: September 12, 2018)
Experimental Details
The single crystalline samples of ZnCr Se were grownby chemical vapor transport method using CrCl as thetransport agent in an evacuated sealed quartz tube [1].During the growth process, the temperatures at two endsof the quartz tube were kept at 850 ◦ C and 950 ◦ C, re-spectively. The crystal quality was investigated by high-resolution synchrotron x-ray diffraction (Fig. 1) per-formed at 16-BM-D, HPCAT of Advanced Photon Sourceof Argonne National Laboratory with λ = 0.4246 ˚A. TheRietveld refinement of this room temperature patternshows a lattice parameter a = 10.491(8) ˚A with sto-ichiometry as Zn . Cr . Se . . Within theerrors of the experiment, the as grown crystals were stoi-chiometric with negilable site disorder.This result is con-sistent with the reported synchrotron x-ray and neutronpowder diffraction studies on ZnCr Se [2]. The dc mag-netic properties were studied by a superconducting quan-tum interference device (SQUID) magnetometer. The acsusceptibility was measured at the National High Mag-netic Field Laboratory using the conventional mutual in-ductance technique [3]. The specific heat was measuredin a physical property measurement system (QuantumDesign PPMS). The thermal conductivity was measuredby using a “one heater, two thermometers” technique [4].In all these measurements, if the field is applied, the fieldis along the [111] axis. FIG. 1. (Color online) The room temperature powder syn-chrotron x-ray diffraction spectrum of ZnCr Se . ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] C. C. Gu, Z. R. Yang, X. L. Chen, L. Pi, and Y. H. Zhang,J. Phys. Condens. Matt. , 18LT01 (2016).[2] P. Zajdel, W. Y. Li, W. van Beek, A. Lappas, A. Zi-olkowska, S. Jaskiewicz, C. Stock, and M. A. Green, Phys.Rev. B , 134401 (2017). [3] Z. L. Dun, M. Lee, E. S. Choi, A. M. Hallas, C. R. Wiebe,J. S. Gardner, E. Arrighi, R. S. Freitas, A. M. Arevalo-Lopez, J. P. Attfield, H. D. Zhou, and J. G. Cheng, Phys.Rev. B , 064401 (2014). [4] X. F. Sun, W. Tao, X. M. Wang, and C. Fan, Phys. Rev.Lett.102