Spin Liquid State in the Disordered Triangular Lattice Sc{}_{2}Ga{}_{2}CuO{}_{7} Revealed by NMR

Quantum Spin Liquid State in the Disordered Triangular Lattice Sc2Ga2CuO7


We present microscopic magnetic properties of a two dimensional triangular lattice ScGaCuO, consisting of single and double triangular Cu planes. An antiferromagnetic (AFM) exchange interaction /k 35 K between Cu ( = 1/2) spins in the triangular bi-plane is obtained from the analysis of intrinsic magnetic susceptibility data. The intrinsic magnetic susceptibility, extracted from Ga NMR shift data, displays the presence of  AFM short range spin correlations and remains finite down to 50 mK suggesting a non-singlet ground state. The nuclear spin-lattice relaxation rate (1/) reveals a slowing down of Cu spin fluctuations with decreasing down to 100 mK. Magnetic specific heat (C) and 1/ exhibit a power law behavior at low temperatures implying gapless nature of the spin excitation spectrum. Absence of long range magnetic ordering down to /700, nonzero spin susceptibility at low , and power law behavior of C and 1/ suggest a gapless quantum spin liquid (QSL) state. Our results demonstrate that persistent spin dynamics induced by frustration maintain a quantum-disordered state at 0 in this triangular lattice antiferromagnet. This suggests that the low energy modes are dominated by spinon excitations in the QSL state due to randomness engendered by disorder and frustration.

NMR, frustration, spin liquid, quantum fluctuations.
75.40.Cx,75.10.Kt,76.60.-k, 76.60.Es, 74.40.Kb

year number number identifier Date text]: updated version on: LABEL:FirstPage1 LABEL:LastPage2

Collective excitations, frustration, and quantum fluctuations are key ingredients in driving novel ground state properties of correlated electron systems. Geometrically frustrated magnets harbor exotic physical phenomena such as spin glass, quantum spin liquid (QSL), spin ice, and superconductivity (1); (2); (3); (4); (5). The incompatibility of magnetic exchange interactions in achieving minimum energy yields degenerate ground states and the associated strong quantum fluctuations prevent the spin system from undergoing a symmetry breaking phase transition (1); (3); (4); (5); (6); (7); (8); (9). The experimental realization of novel states such as QSL in real materials is an exciting prospect in answering some of the key issues in condensed matter and set an enduring theme following Anderson’s resonance valence bond theory (10); (11). The most prominent QSL candidates reported so far are  = 1/2 kagomé lattices ZnCu(OH)Cl, CuZn(OH)Cl, [NH][CHN][VOF],  =1/2 hyperkagomé  PbCuTeO, NaIrO, and organic = 1/2 triangular lattice, EtMeSb[Pd(dmit)], -(BEDT-TTF)Cu(CN), and -(ET)Cu[N(CN)]Cl. The spin excitation spectra in the QSL state can be gapped or gapless with exicitng magnetic properties (4); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (32); (33); (34); (35); (36); (37). The frustrated quantum magnets are proposed to host emergent fractional excitations in the gapless QSL state, which is reflected as power law behavior in bulk and microscopic observables (1); (38); (39); (40); (41); (42); (43); (44). Recently, the observation of intriguing magnetic properties in BaTSbO (T=Cu, Co, Ni) and 5d iridates has rekindled enormous research activities in quantum materials in the context of emergent quantum states (1); (2); (23); (45); (46); (47); (48); (49); (50); (51). Among the frustrated magnets, the edge-shared triangular lattice AFM with = 1/2 offers the simplest archetype for QSL and to test theoretical models in other relatively complex lattices (13); (26); (28); (29). Furtheremore, the role of intersite distribution or disorder in stabilizing a QSL state in frustrated quantum magnets has recently been suggested (19); (49); (30); (31).

In view of the vastly evolving field of frustrated magnetism, significant attention has recently been paid to the growth and design of new quantum magnets which could epitomize as model materials for hosting exotic excitations pertinent to novel states and to test theoretical conjectures (1); (4); (5); (9). In the quest for novel states in frustrated magnets with low spin where inherent quantum effects lead to emergent phenomena, we synthesize and investigate an inorganic = 1/2 antiferromagnet ScGaCuO (henceforth SGCO). Recent detailed synchrotron x-ray and neutron diffraction measurements revealed that the magnetic lattice comprises of triangular bi-planes of correlated Cu spins dominated by 50 % Ga ions due to unavoidable intersite inversion and the single triangular plane is mainly occupied by non-magnetic Ga ions and 15 % Cu in the single triangular plane give rise to a paramagnetic behavior. The bulk magnetic susceptibility at low temperature is dominated by defect contributions and specific heat displays no signature of long range ordering down to 0.35 K, which invokes microscopic investigations (52). Absence of significant anisotropy and no appreciable spin-orbit coupling suggest that SGCO might be a promising quantum magnet to address low lying excitations intrinsic to the triangular lattice.

The microscopic details pertaining to the magnetic properties inherent to the magnetic lattice at very low temperature is a very crucial step forward for establishing the ground state convincingly and in exploring the nature of low lying excitations. Herein, we report the first nuclear magnetic resonance (NMR) studies on a new = 1/2 inorganic triangular lattice SGCO. NMR being a powerful local probe sheds light on the intrinsic spin susceptibility and the dynamic spin susceptibility via spectra and spin-lattice relaxation rate (1/) measurements, respectively, from a microscopic point of view. The intrinsic spin susceptibility suggests the presence of AFM spin correlations with /k 35 K  between Cu spins in the triangular bi-planes and non-singlet state without signature of long range magnetic ordering (LRO) down to 50 mK. The 1/ data suggest a slowing down of Cu spin fluctuations with decreasing temperature down to 100 mK and power law behavior of magnetic specific heat (C) and 1/T imply gapless spin excitations. Our comprehensive results establish a gapless quantum spin liquid state in SGCO.

Polycrystalline sample of ScGaCuO was prepared by a method described elsewhere (52). SGCO crystallizes in a hexagonal structure with a space group and lattice constants 3.30479(4) Å and = 28.1298(4)Å . The magnetic structure comprises of alternating single and double triangular planes. The interaction between the Cu spins is confined to the 2D triangular bi-plane only, with negligible interlayer interactions (52).

Shown in Fig. 1(a) is the temperature dependence of bulk magnetic susceptibility , which is found to be strongly enhanced at low temperatures without exhibiting any signature of long range magnetic ordering (LRO) down to 1.8 K. We did not observe ZFC and FC splitting in and no hysteresis was found in magnetization (52). The green dotted line in Fig. 1(a) shows the magnetic susceptibility after subtracting from  a contribution due to the presence of 15% Cu spins on the triangular plane assuming a simple Curie behavior of = 1/2 for the Cu spins. The Curie-Weiss (CW) fit of at high temperatures above 100 K yields a CW temperature 44 K, an effective magnetic moment () of 1.83 and  2. The negative value of indicates the presence of AFM interaction between Cu spins on the triangular bi-plane. The T-dependence of  magnetic specific heat (as shown in Fig. 1(b)) in different magnetic fields don’t display any sign of LRO. The magnetic specific heat (C) exhibits a power law (T) behavior indicating a non-singlet state (1); (23); (39); (40); (41); (42); (43); (44); (45); (52); (53).

Figure 2(a) shows the typical temperature evolution of field swept Ga NMR spectra of SGCO at a frequency = 69.5 MHz. With decreasing , although the Ga NMR spectra broaden, NMR shift for the main line shows a broad maximum around 70 K, which is a characteristic feature of low dimensional AFM spin systems due to short range spin correlations. Below the broad maximum, decreases and levels off at low and then remains nearly constant down to 50 mK. The frustration parameter ( ) is considered to be a measure of the depth of the spin liquid regime and is defined as  = /. In the present case we did not observe magnetic ordering down to 50 mK, so   /50 mK 900 (1); (73). This suggests the presence of strong magnetic frustration inspite of the large site inversion. The frustration between Cu spins residing in the 2D triangular bi-planes of SGCO might offer a route for the persistent spin dynamics of Cu spins down to 50 mK and these fluctuating spins preclude LRO. In addtion to the main Ga NMR line, we have observed a weak line (labeled as Ga(II) in Fig. 2(a)) whose NMR shift shows a CW behavior as shown in Fig. 2(b). Since the estimated signal internsity of the Ga(II) line is 19 % of the total Ga NMR intensity, which is in good agreement with an expected signal intensity of which 20 % Ga ions touching one Cu ion in the nearest neighbor of the single layer, the Ga(II) signal can be ascribed to Ga ions in single layers. The main Ga(I) line (Fig. 2(a)) is attributed to Ga ions in the triangular bi-plane. We were not able to detect Ga(II) line and hence at temperatures below 100 K due to inhomogeneous broadening of the spectra perhaps because of Cu spins in the single triangular planes.

The NMR shift consists of dependent spin shift and independent orbital (chemical) shift ; = + where is proportional to the spin part of magnetic susceptibility via hyperfine coupling constant , = . Here is Avogadro’s number. The hyperfine coupling constant is estimated to be = –3.8 0.2 kOe/ for  the main Ga(I) line from the slope of the so-called - plots using data at T 150 K.  values are estimated to be 0.049 for the main Ga(I) line. The T-dependence of the intrinsic magnetic susceptibility obtained from data for the main line is shown by solid spheres in Fig. 1(a). The shows a broad maximum around 70 K and decreases at low temperatures, but does not approach zero. The nonzero value of at low ( 40 of the maximum value) is strong evidence of the absence of spin gap in SGCO. Similar behavior of is reported in the well known spin liquid material ZnCu(OH)Cl(16); (17).

The dependence of above 30 K is reasonably reproduced by the high temperature series expansion (HTSE) of an = 1/2 triangular lattice Heisenberg model (56); (71) as shown in Fig. 1(a) by the red line where the (4,7) Padé approximant is adapted with an effective exchange coupling between Cu spins with = (35 3) K (see Supplemental Material (53)). The good fit indicates that, although more than 50% of Cu ions in the triangular bi-planes experienced site inversion, the intra-plane magnetic interaction is still maintained. It should be noted that does not coincide with at low . This indicates that the large enhancements of at low cannot be explained only by the 15 % Cu spins due to the antisite effects. The exact origin for the difference between and is not clear at present but might be associated with the site inversion between Cu and Ga sites in the system (52). As shown in Fig. 2(c), the full width at half maximum (FWHM = ) of the NMR spectrum for the main line increases with decreasing and saturates below 2 K. The  -independent  below 2 K is found to be independent of the applied magnetic field indicating both and -independent internal field at Ga sites. These results suggest that Cu spins fluctuate slowly i.e., at less than the NMR frequency ( 50 MHz) at low . From the saturated value at low T, we estimated the Cu magnetic moments of magnitude 0.19 , which is quite small compared to the total magnetic moment expected for = 1/2. The Sc NMR spectra, shift and also exhibit a similar -dependence with those of  Ga NMR results.

Figure 3 (a) depicts the dependence of spin-lattice relaxation rates of Ga, together with that of Sc. 1/ is almost independent of above 100 K and starts to decrease at low and then levels off below 10 K down to 2 K. With further decreasing , as shown in Fig. 3(a), independent of probing nuclei, 1/ decreases and displays a power law behavior i.e., 1/ down to 100 mK. 1/ is almost independent of magnetic field above 2 K, but is suppressed strongly with magnetic fields at low as shown in the Fig. 3(a).

A simple scenario for the decrease in 1/ due to suppression of magnetic fluctuations of isolated paramagnetic spins at high field and low cannot be attributed for the observed behavior. For the simple paramagnetic spin fluctuations of isolated spins, 1/ is known to be proportional to the first derivative of the Brillouin function, (=) which gives an exponential behavior of in following exp(-) function, in contrast to the power law behavior in the observed . As shown in Fig. 3(a), the exponent of the power law in 1/ is almost independent of magnetic fields implying the intrinsic and robust nature of the ground state properties. It is worth mentioning here that the power law dependence of spin-lattice relaxation rate 1/ has been discussed in the context of Dirac Fermion model in interpreting QSL (29); (58); (40). behavior in the = 1/2 triangular lattice -(BEDT-TTF)Cu(CN) has been reconciled in the framework of spin liquid (SL) with quantum critical spin excitations (26); (29); (39); (59); (60); (41); (61). Recently, another plausible theoretical conjecture in interpreting the role of randomness in driving a gapless SL state of -(BEDT-TTF)Cu(CN) and EtMeSb[Pd(dmit)] is proposed (64); (30). However, a general consensus in interpreting the dependence of in the SL materials is still lacking and little progress has been made in evolving a more generic and comprehensive framework. This is due to the unavailability of many model SL materials and experimental challenges in interpreting the implications of various subtle theoretical scenarios (1); (26); (30). Furthermore, one would expect a T- independent behavior of 1/ in the case of a spin liquid with a spinon Fermi surface and 1/ should drop exponentially in the case of gapped SL (1); (4); (39); (59); (60); (61); (65). Our results are not in accord with the above cited two scenarios but could be associated with the interpretation of not a fully gapless SL where at least some part of the q-space is gapped (28); (64). In view of the power law behavior of magnetic specific heat and 1/T, a detailed theoretical investigation call for in interepreting these results in the context of emergent excitations in the gapless QSL, which is beyond the scope of the present study, but renders a direction for further explorations (38); (39); (40); (41); (42); (43); (44).

Finally, it is important to point out that our data indicate a slowing down of Cu spin fluctuations at low temperature. 1/ is generally expressed by the Fourier transform of the time correlation function of the transverse component of the fluctuating local field at nuclear sites with respect to the nuclear Larmor frequency as (66); (67) , where is the gyromagnetic ratio of the nuclear spin. When the time correlation function is assumed to decay as e, one can write (eq.1) where is a parameter related to the hyperfine field at nuclear sites and is the magnetic susceptibility. In our case, would correspond to the inverse of the correlation time of the fluctuating hyperfine fields at the Ga or Sc sites, due to the Cu spins. When is much higher than , one finds that the is proportional to 1/. On the other hand, if , should depend on the magnetic field. When = , reaches a maximum value. Thus, the slowing down of the fluctuation frequency of Cu spins yields a peak in . Figure 3(b) represents the temperature dependence of , where the values are used for corresponding for each nucleus. When is independent of , 1/ should be constant, which is indeed observed above 50 K. This indicates 1/ above 50 K is explained by the paramagnetic fluctuations of the Cu spins, whereby the Cu spins fluctuate almost independently. Below 50 K, the 1/ starts to increase and shows dependent peaks at low T below 2 K. This can be explained by the slowing down in fluctuation frequency of spins at low T. These results indicate that the peak observed in 1/ originates from the slowing down (but not critical) of fluctuation frequency of Cu spins, whereby the fluctuation frequency below the peak temperature is less than the NMR frequency range (10 - 100 MHz). To derive the dependence of the fluctuation frequency of Cu spins in a wide temperature range, we extract the -dependence of from the -dependence of 1/, assuming eq. (1) is valid for all temperature regimes. The estimated dependences of for the different magnetic fields are shown in Fig. 4, together with the data estimated from Sc-. shows behavior at low T and is almost a constant with 310 Hz at high . At low T below 1 K, Cu spins fluctuate with low frequency. Such a slow spin dynamics is consistent with the observed broadening of the NMR spectra below 2 K. The absence of critical slowing down and no loss of NMR signal intensity rule out the possibility of spin glass phase down to 50 mK in SGCO. This is further substantiated by the absence of critical divergence of 1/T or cusp structure in 1/T generally observed in spin frozen states(30).

In summary, the intrinsic spin susceptibility ( obtained from NMR does not vanish and remains finite at T = 50 mK, reflecting a non-singlet ground state in ScGaCuO. The T-dependence of is well reproduced by the HTSE of the S = 1/2 Heisenberg model, indicating that the 2D magnetic interactions between Cu spins in the bi-plane are still maintained although more than 50% spins involved in the unavoidable intersite inversion. Quantum fluctuations enhanced by strong frustration between Cu spins in the 2D triangular bi-plane inhibit the LRO down to 50 mK despite an AFM exchange interaction 35 K. The spin-lattice relaxation rate exhibits a slowing down of  Cu spin fluctuations and short range spin correlations at low . The power law behavior of C and 1/ with decreasing temperature down to 100 mK infer gapless excitations consistent with  and suggest a quantum spin liquid state. The effect of site dilution, defect, and disorder in frustrated quantum magnets have been discussed in the context of novel magnetism such as spin liquids recently (19); (30); (31); (64).  The absence of spin freezing and no spin gap down to 50 mK in SGCO suggest that the low energy excitations might be mediated by deconfined spinons, which is generic to a gapless QSL state in frustrated quantum magnets. This point towards the predominant nature of deconfined spinons in QSL state in case of the randomness induced by disorder due to site inversion and frustration in realizing electron localization (1); (68). In this context SGCO offers a fertile ground for exploring the effect of dilution or disorder, and the role of control parameters in tuning emergent states in frustrated quantum magnets.

We acknowledge insightful discussions with P. Mendels, B. Koteswararao, and B. Roy. PK acknowledges support from the European Commission through Marie Curie International Incoming Fellowship (PIIF-GA-2013-627322). The research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering. Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.

Present address:  Laboratoire de Physique des Solides, Université Paris-Sud 11, UMR CNRS 8502, 91405 Orsay, France.


Figure Captions:

Fig. 1 (Color online) (a) Temperature dependence of the observed magnetic susceptibility (solid line) at 7 T and the subtracted magnetic susceptibility (dotted line) after subtraction of 15 % Cu spin contributions due to the site inversion as discussed in the text. The solid spheres depict the intrinsic magnetic susceptibility estimated from Ga-NMR shift. The red solid line is a fit as discussed in the text. (b) The inset shows the T-dependence of magnetic specific heat (C) in different magnetic fields and the solid line depicts the power law (T) behavior.

Fig. 2 (Color online) (a) Temperature evolution of field swept Ga NMR spectra at 69.5 MHz. The vertical broken line corresponds to zero-shift ( = 0) position. (b) dependence of both for main and Ga(II) lines. (c) dependence of NMR line width () at 69.5 MHz and 24.25 MHz.

Fig. 3 (a) (Color online) Temperature dependence of Ga and Sc 1/ at different frequencies. The solid line represents behavior (b) dependence of 1/ (1/ divided by temperature and respective spin susceptibilities and ).

Fig. 4 (Color online) Temperature dependence of  estimated from Ga and Sc 1/ as explained in the text. The solid line is the behavior.

.1 Methods

Polycrystalline ScCuGaO samples were synthesized by a method described elsewhere (69). Phase purity was confirmed by Rietveld refinement of x-ray diffraction (XRD), synchrotron and neutron diffraction data (69). The temperature dependence of dc magnetic susceptibility (=()/) was measured at 7 T in the temperature range 1.8 T 400 K using a Quantum Design, Physical Property Measurement System (PPMS). The absence of hysteresis and spin-freezing were confirmed from the magnetization measurements following the zero field and field cooled protocol in the sample studied in this work. The low temperature specific heat measurment at various applied magnetic fields was performed using the He option of QD, PPMS following thermal relaxation method. The magnetic specific heat was extracted from the measured specific heat by subtracting the lattice specific heat and contribution from nuclear Schttoky (69).The exchange interaction between the nearest neighbor  Cu spins is described by the Hamiltonian, =  . In order to estimate the exchange coupling () between the nearest neighbor Cu spins in the triangular biplane, we have fitted the intrinsic magnetic susceptibility data following the high temperature series expansion for = 1/2 system following the (4,7) Padé approximant appropriate for triangular lattice antiferromagnet

where are series coefficients and the values of which can be found in Ref.(70); (71). We obtained an exchange interaction of   = (35  3) K between Cu spins from the fit of intrinsic magnetic susceptibility of ScCuGaO.

Nuclear magnetic resonance (NMR) measurements down to 50 mK at various frequencies were carried out on Ga ( = 3/2, = 12.9847 MHz/T) and Sc ( = 7/2, = 10.343 MHz/T) by using a homemade phase-coherent spin-echo pulse spectrometer. The low temperature NMR measurements are performed with a Oxford Kelvinox dilution refrigerator installed at Ames Laboratory. NMR spectra were obtained by sweeping the magnetic field at a fixed frequency. The temperature dependence of NMR shift were obtained from the simulation of NMR spectra taken at different temperatures. The Ga NMR spectrum at high temperature were simulated with the superposition of two lines, one due to Ga(I) and other due to Ga(II) because of defect Cu spins. This appears to be due to antisite disorder between the Cu and the Ga atoms in the host lattice. The ratio of NMR intensities for these two lines is estimated to be 0.81:0.19 at different temperatures, which implies 19 % of Cu sits at the Ga site and is consistent with those deduced from magnetization, specific heat and neutron diffraction data. Site inversion between Cu and Zn is also observed in the well known kagomé spin liquid material herbersmithite(74); (75). The frustration parameter ( ), which is a measure of the depth of spin liquid regime and is defined as  =/, In the system presented here we didn’t observe magnetic ordering down to 50 mK, so   /50 mK 900(72); (73). The large value of  indicates the presence of strong magnetic frustration between Cu spins, which prevents LRO down to 50 mK and leads to exotic magnetic properties in ScCuGaO discussed here.

Furthermore, the role of perturbations such as lattice disorder due to site inversion, presence of defect spins and/or nonmagnetic substituents in frustrated antiferromagnets in modifying local environments by inducing magnetic moments in its immediate vicinity lead to interesting magnetic properties(74); (75); (76); (77); (78); (79); (80); (81).

In order to investigate dynamics of the Cu spins and the ground state properties, we have performed spin-lattice relaxation rate (1/) measurements in the wide temperature range 0.1 250 K for the central line. The recoveries of the longitudinal magnetization for both nuclei display stretched exponential behavior suggesting distributions of values. The at each is determined by fitting the nuclear magnetization () using the stretched double-exponential function 1-()/ = 0.1e+ 0.9e for the central line of the spectrum of the Ga ( = 3/2) nucleus(82). The recovery of  () at all temperatures could be fitted with 0.5 ( is found to be nearly independent of temperature). Here () and are the nuclear magnetization at time after saturation and the equilibrium nuclear magnetization at time , respectively. Similarly, for Sc ( = 7/2 ), spin-lattice relaxation rate was obtained by fitting () using stretched exponential function 1-()/ = 0.0119e+ 0.0682e+ 0.206e+ 0.7139e ( 0.5) valid for = 7/2(82).

We believe that the present work should open new avenues in frustrated magnetism and will stimulate further theoretical and experimental investigations exploring the nature of low lying excitations and the role of perturbations on the ground state of triangular lattice antiferromagnets.




  1. preprint: APS/123-QED


  1. L. Balents, Nature 464, 199 (2010) and references therein.
  2. P. A. Lee, Science 321,1306 (2008).
  3. S. Sachdev, Nature Physics 4, 173 (2008) and Nature Physics focus issue March 2008 Volume 4 No 3 pp167-204.
  4. C. Lacroix, P. Mendels, and F. Mila, Introduction to Frustrated Magnetism,(Springer, New York), Vol. 164.
  5. Frustrated Spin Systems, ed. H. T. Diep (World Scientific, Singapore, 2005).
  6. S. T. Bramwell and M. J. P. Gingras, Science 294, 1495 (2001).
  7. C. Nisoli, R. Moessner and P. Schiffer, Rev. Mod. Phys. 85, 1473 (2013) and references therein.
  8. J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev. Mod. Phys. 82, 53 (2010).
  9. C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature 451, 42 (2008).
  10. P.W. Anderson, Mater. Res. Bull. 8, 153 (1973).
  11. P.W. Anderson, Science 235, 1196 (1987).
  12. T.-H. Han, J. S. Helton, S. Chu, D. G. Nocera, J. A. Rodriguez-Rivera, C. Broholm, and Y. S. Lee, Nature 492, 406 (2012).
  13. M. Punk, D. Chowdhury, and S. Sachdev, Nat. Phys. 10, 289 (2014).
  14. J. S. Helton, H. Martinho, M. S. Sercheli, P. G. Pagliuso, D. D. Jackson, M. Torelli, J. W. Lynn, C. Rettori, Z. Fisk, and S. B. Oseroff, Phys. Rev. Lett. 98, 107204 (2007).
  15. M. Jeong, F. Bert, P. Mendels, F. Duc, J. C. Trombe, M. A. de Vries, and A. Harrison, Phys. Rev. Lett. 107, 237201 (2011).
  16. A. Olariu, P. Mendels, F. Bert, F. Duc, J. Trombe, M. de Vries, and A. Harrison, Phys. Rev. Lett. 100, 087202 (2008).
  17. T. Imai, E. A. Nytko, B. M. Bartlett, M. P. Shores, and D. G. Nocera, Phys. Rev. Lett. 100, 077203 (2008).
  18. M. Fu, T. Imai, T.-H. Han, and Y.S. Lee, Science 350, 655 (2015).
  19. B. Fåk, E. Kermarrec, L. Messio, B. Bernu, C. Lhuillier, F. Bert, P. Mendels, B. Koteswararao, F. Bouquet, J. Ollivier, A. D. Hillier, A. Amato, R. H. Colman, and A. S. Wills, Phys. Rev. Lett. 109, 037208 (2012).
  20. L. Clark et al., Phys. Rev. Lett. 110, 207208 (2013).
  21. S. E. Dutton, M. Kumar, M. Mourigal, Z. G. Soos, J.-J. Wen, C. L. Broholm, N. H. Andersen, Q. Huang, M. Zbiri, R. Toft-Petersen, and R. J. Cava, Phys. Rev. Lett. 108, 187206 (2012).
  22. P. Mendels and F. Bert, J. Phys. Soc. Jpn. 79, 011001 (2010).
  23. Y. Okamoto, M. Nohara, H. Aruga-Katori, and H. Takagi, Phys. Rev. Lett. 99, 137207 (2007).
  24. A. P. Ramirez, B. Hessen, and M. Winklemann, Phys. Rev. Lett. 84, 2957 (2000).
  25. Y. Shimizu, K. Miyagawa, K. Kanoda, M. Maesato, and G. Saito, Phys. Rev. Lett. 91, 107001 (2003).
  26. F. L. Pratt, P. J. Baker, S. J. Blundell, T. Lancaster, S. Ohira-Kawamura, C. Baines, Y. Shimizu, K. Kanoda, I. Watanabe and G. Saito, Nature 471,612 (2011).
  27. S. Yamashita, Y. Nakazawa, M. Oguni, Y. Oshima, H. Nojiri, Y. Shimizu, K. Miyagawa, and K. Kanoda, Nat. Phys. 4, 459 (2008).
  28. T. Itou, A. Oyamada, S. Maegawa, and R. Kato, Nature Physics 6, 673 (2010).
  29. Y. Qi and S. Sachdev, Phys. Rev. Lett. 102, 176401 (2009).
  30. T. Furukawa, K. Miyagawa, T. Itou, M. Ito, H. Taniguchi, M. Saito, S. Iguchi, T. Sasaki, and K. Kanoda, Phys. Rev. Lett. 115, 077001 (2015).
  31. A. Smerald and F. Mila, Phys. Rev. Lett. 115, 147202 (2015).
  32. S. Yan, D.A. Huse, and S.R. White, Science 332, 1173 (2011).
  33. A. E. Antipov, A. N. Rubtsov, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. B 83, 115126 (2011).
  34. Z. Hao and O. Tchernyshyov, Phys. Rev. Lett. 103, 187203 (2009).
  35. D. Watanabe, M. Yamashita, S. Tonegawa, Y. Oshima, H. M. Yamamoto, R. Kato. I. Sheikin, K. Behnia, T. Terashima, S. Uji, T. Shibauchi, and Y. Matsuda, Nature Commun. 3, 1090 (2012).
  36. X.-G. Wen, Phys. Rev. B 65, 165113 (2002).
  37. P. Khuntia et al., Phys. Rev. Lett. 116, 107203 (2016).
  38. Z. Y. Meng et al., Nature 464, 847 (2010).
  39. M. J. Lawler et al.,  Phys. Rev. Lett. 101, 197202 (2008).
  40. M. Hermele et al., Phys Rev. B 77, 224413(2008).
  41. Y. Ran et al., Phys. Rev. Lett. 98, 117205 (2007).
  42. M. Hermanns, S. Trebst, and A. Rosch, Phys. Rev. Lett. 115, 177205 (2015).
  43. M. Hermanns and S. Trebst, Phys. Rev. B 89, 235102 (2014).
  44. E. K.-H. Lee et al., Phys. Rev. B 89, 045117 (2014).
  45. H. D. Zhou, E. S. Choi, G. Li, L. Balicas, C. R. Wiebe, Y. Qiu, J. R. D. Copley, and J. S. Gardner, Phys. Rev. Lett. 106, 147204 (2011).
  46. S. Nakatsuji, K. Kuga, K. Kimura, R. Satake, N. Katayama, E. Nishibori, H. Sawa, R. Ishii, M. Hagiwara, F. Bridges, T. U. Ito, W. Higemoto, Y. Karaki, M. Halim, A. A. Nugroho, J. A. Rodriguez-Rivera, M. A. Green, and C. Broholm, Science 336, 559 (2012).
  47. J. A. Quilliam, F. Bert, E. Kermarrec, C. Payen, C. Guillot-Deudon, P. Bonville, C. Baines, H. Luetkens, and P. Mendels, Phys. Rev. Lett. 109, 117203 (2012).
  48. J. G. Cheng, G. Li, L. Balicas, J. S. Zhou, J. B. Goodenough, C. Xu, and H. D. Zhou, Phys. Rev. Lett. 107, 197204 (2011).
  49. T. Dey et al. , Phys. Rev. B 86, 140405(R) (2012).
  50. Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012).
  51. S. H. Chun et al., Nat. Phys. 11, 462 (2015).
  52. R. Kumar et al., Phys. Rev. B 92, 180411(R) (2015).
  53. See Supplemental Material at http://link.aps.org/supplemental/, for experimental details, results, and discussions on magnetization, specific heat, and NMR.
  54. Y. Shimizu, H. Takeda, M. Tanaka, M. Itoh, S. Niitaka, and H. Takagi, Nature Commun. 3, 981 (2012).
  55. A. P. Ramirez, Annu. Rev. Mater. Sci. 24, 453 (1994).
  56. H.E. Stanley, Phys. Rev.158, 546 (1967); N. Elstner, R. R. P. Singh, and A. P. Young, Phys. Rev. Lett. 71, 1629 (1993).
  57. K. Gregor and O. I. Motrunich, Phys. Rev. B 79, 024421 (2009).
  58. P. Calabrese, A. Pelissetto, and E. Vicari, Phys. Rev. B 67, 054505 (2003).
  59. M. J. Lawler, H.-Y. Kee, Y. B. Kim, and A. Vishwanath, Phys. Rev. Lett. 100, 227201 (2008).
  60. Y. Zhou et al., Phys. Rev. Lett. 101, 197201 (2008).
  61. S.-S. Lee, P. A. Lee and T. Senthil, Phys. Rev. Lett. 98, 067006 (2007).
  62. Z. Hao and O. Tchernyshyov, Phys. Rev. B 87, 214404 (2013).
  63. Y. Iqbal, F. Becca, S. Sorella, and D. Poilblanc, Phys. Rev. B 87, 060405(R) (2013).
  64. K. Watanabe, H. Kawamura, H. Nakano, and T.Sakai, J. Phys. Soc. Jpn. 83, 034714 (2014).
  65. T. Dey et al., Phys. Rev. B 88, 134425 (2013).
  66. N. Bloembergen, E. M. Purcell, and R. V. Pound, Nature 160, 475 (1947).
  67. A. Abragam, Principles of Nuclear Magnetism (Oxford University Press, New York, 1983).
  68. T. Shimokawa, K. Watanabe, and H. Kawamura,  Phys. Rev. B 92, 134407 (2015).
  69. R. Kumar et al., Phys. Rev. B 92, 180411(R) (2015).
  70. H.E. Stanley, Phys. Rev.158, 546 (1967); N. Elstner, R. R. P. Singh, and A. P. Young, Phys. Rev. Lett. 71, 1629 (1993).
  71. K. Gregor and O. I. Motrunich, Phys. Rev. B 79, 024421 (2009).
  72. L. Balents, Nature 464, 199 (2010).
  73. A. P. Ramirez, Annu. Rev. Mater. Sci. 24, 453 (1994).
  74. A. Olariu, P. Mendels, F. Bert, F. Duc, J. Trombe, M. de Vries, and A. Harrison, Phys. Rev. Lett. 100, 087202 (2008).
  75. T. Imai, E. A. Nytko, B. M. Bartlett, M. P. Shores, and D. G. Nocera, Phys. Rev. Lett. 100, 077203 (2008).
  76. S.-H. Lee, H. Kikuchi, Y. Qiu, B. Lake, Q. Huang, K. Habicht, and K. Kiefer, Nat. Mater. 6, 853 (2007).
  77. A.V. Mahajan, H. Alloul, G. Collin, and J. F. Marucco, Phys. Rev. Lett. 72, 3100 (1994).
  78. M. A. de Vries, K. V. Kamenev, W. A. Kockelmann, J. Sanchez-Benitez, and A. Harrison, Phys. Rev. Lett. 100, 157205 (2008).
  79. S. Dommange, M. Mambrini, B. Normand, and F. Mila , Phys. Rev. B 68, 224416 (2003).
  80. H. Alloul, J. Bobroff, M. Gabay, and P. J. Hirschfeld, Rev. Mod. Phys. 81, 45 (2009) and references therein.
  81. R. R. P Singh, Phys. Rev. Lett. 104,177203 (2010).
  82. E. R. Andrew and D. P. Tunstall, Proc. Phys. Soc. 78, 1 (1961); A. Narath, Phys. Rev. 162, 320 (1967); A. F. Mcdowell, J. Mag. Res. A 113, 242 (1995).
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description