Nematic Spin Liquid State in the S=\frac{5}{2} Heisenberg Kagomé Antiferromagnet Li_9Fe_3(P_2O_7)_3(PO_4)_2
E. Kermarrec, R. Kumar, R. Hénaff, B. Koteswararao, P. L. Paulose, P. Mendels, F. Bert
NNematic Spin Liquid State in the S = Heisenberg Kagom´e AntiferromagnetLi Fe (P O ) (PO ) E. Kermarrec, R. Kumar, R. H´enaff, B. Koteswararao, P. L. Paulose, P. Mendels, and F. Bert Universit´e Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405, Orsay, France Faculty in Physics, Indian Institute of Technology Tirupati Department of Condensed Matter Physics and Materials Science, TIFR, Mumbai
We investigate the low temperature magnetic properties of a S = Heisenberg kagom´e antifer-romagnet, the layered monodiphosphate Li Fe (P O ) (PO ) , using magnetization measurementsand P nuclear magnetic resonance. An antiferromagnetic-type order sets in at T N = 1 . T ∗ ∼ P NMR line broadening reveals the development of anisotropic short-range correlations within the plateau phase concomitantly with a gapless spin-lattice relaxationtime T ∼ k B T / (cid:126) S , which both point to the presence of a semiclassical nematic spin liquid statepredicted for the Heisenberg kagom´e antiferromagnetic model. PACS numbers: 75.10.Jm, 76.60.-k, 75.30.Cr, 75.10.Kt, 75.40.Gb,
Magnetic frustation sets as a playground for the dis-covery of new exotic phases of quantum matter. TheHeisenberg antiferromagnetic model on the kagom´e lat-tice (KHAF) constitutes the archetype of the frustrationin two dimensions, whose ground-state and exotic spindynamics still remain topical questions in quantum mag-netism even after 30 years of research [1–4].Early on, Chalker et al. [5] highlighted the distinctivemagnetic properties of the classical ( S = ∞ ) KHAF,for which thermal fluctuations select a spin nematic, orcoplanar, ground state via the order-by-disorder mecha-nism in zero field [6, 7], with spins on a single triangleoriented at 120 ◦ from each other (Fig.1e). Its spin dy-namics is governed by zero-energy soft modes, i.e. speciallines of weathervane defects [8]. The order-by-disordermechanism is also responsible for the remarkable exis-tence of semiclassical fractional plateaus of magnetiza-tion in frustrated magnets [1, 9–13]. In particular, the1/3 plateau –a hallmark of frustration[1, 14]–, arisingfrom collinear uud spin configurations (Fig.1d), has beenobserved in the triangular compounds Cs CuBr [15] andRbFe(MoO ) [16–18] but remains debated and/or dif-ficult to observe for quantum kagom´e antiferromagnetswith large interaction [21] or with deviations from thepure KHAF model [19, 20].More recently, the phase diagram of the KHAF in anexternal field was revisited with state-of-the art numer-ical methods, notably tensor network formalism[22, 23],DMRG[13] or exact diagonalization[12]. It encompassesfully “quantum” phases and semiclassical plateaus, in-cluding the 1/3 plateau that appears for all values of spin S . The specific aspect of the 1/3 plateau of the classi-cal kagom´e lattice under an applied field was pinpointedby finite-temperature Monte-Carlo studies which showedthat thermal fluctuations favor a a classical collinear spinliquid phase [24, 25] at M = M sat , made of a macro-scopic number of degenerate uud states.The comparison with classical experimental candi- dates has remained challenging for long as naturallyoccuring kagom´e lattice are rare. The jarosites fam-ily A Fe (OH) (SO ) (with A + typically an univalentcation) is held up as a paragon of the classical KHAF[1,26, 27]. They all (but A + =D O + [28]) display long rangeordering into a q = 0 structure, as a result of a defi-cient kagom´e network and/or of perturbative anisotropicterms. Furthermore, their large antiferromagnetic inter-action J ∼
600 K requires unattainable applied magneticfields in order to reach the 1/3 plateau.
FIG. 1. (Color online) (a) Schematic structure of LiFePOillustrating the corner-sharing triangular lattice made of Fe , S = (brown). Li and O atoms are omitted for clarity. (b),(c) Fe atoms are magnetically coupled through PO chemicalgroups with two different phosphorus sites: P (yellow) andP (blue). Illustration of the up-up-down uud (d) and of the120 ◦ ordered structure (e). In this Letter, we investigate the low temperature mag-netic properties of the layered iron monodiphosphateLi Fe (P O ) (PO ) (LiFePO), a S = Heisenbergkagom´e antiferromagnet material with a moderate ex-change J ∼ a r X i v : . [ c ond - m a t . s t r- e l ] F e b otic spin dynamics.LiFePO belongs to the family of monodiphosphates,which were primarily studied for their electronic trans-port properties in the context of battery materials[29]. LiFePO crystallizes in the centrosymmetric trigo-nal space group P3 c ( S = ) atoms formcorner sharing triangles, topologically equivalent to a reg-ular kagom´e lattice (Fig. 1), with a strong Heisenbergcharacter ( L (cid:39) C/ ( T − θ CW ), witha Curie-Weiss temperature θ CW = − J ∼ C = N A µ µ / k B yields aneffective moment µ eff = 5 . µ B , in very good agree-ment with the 5 . µ B value expected for a fully occupiedkagom´e lattice made of Fe ions. At T = T N = 1 . T N is reduced as compared to | θ CW | = 11 Kbecause of the frustration and the bidimensionality inher-ent to the kagom´e geometry. Li Fe (P O ) (PO ) thusshows a correlated paramagnetic regime between 1.3 and ∼
10 K.Magnetization versus field M( B ) curve is shown inFig. 2b. Above 10 K, S = 5 / B ) evolves almost linearly from0 to 16 T. As T is decreased, a pronounced curvatureis observed towards the saturated value M sat = 5 µ B .For T < T ∗ (cid:39) ∼ M sat /
3. Fig. 2c shows the derivative dM/dH which firmly confirms the existence of a narrow 1/3 mag-netization plateau developing below T ∗ . The evolutionof the 1/3 plateau is quantified by the size of the dip,∆( dM/dH ), and is shown as a function of T in Fig. 2.d.It displays a smooth crossover from a paramagnet for T > T ∗ to the plateau phase, i.e. a classical collinearspin liquid phase formed by the set of the uud states thatremain degenerate at M = M sat /
3, with no clear sign ofsymmetry-breaking like for a gas-liquid type transition.In the T → ∼ . ∼ . dM/dH singularity observed around 0.7 T at 0.5 K is close toa 1/9 fraction but is perhaps linked to a field-inducedmagnetic transition as found in RbFe(MoO ) with anincommensurate to commensurate transition along the c axis[18].The most recent theoretical phase diagram under ap-plied fields has been computed using numerical tensornetwork methods. It shows three different incompress- ible phases or magnetization plateaus besides the 1/3plateau[23]. The 1 / S = ). The others (one-magnonand two-magnon plateaus [23]) should appear for all val-ues of S and lie very close to the saturation, respectivelyat 43/45 and 41/45 for S = . These plateaus remainabsent in our 1.8 K data but a clear conclusion is dif-ficult and would require applying magnetic fields higherthan 16T. The absence of the C rotational symmetry ofour S = Fe lattice may also be detrimental to thesesemiclassical phases. FIG. 2. (Color online) Magnetization and susceptibilitymeasurements of LiFePO. (a) Magnetic susceptibility χ mea-sured under 100 Oe showing an antiferromagnetic transitionat T N = 1 . /χ (inset).(b) M vs B measured at T = 1 . M sat ∼ µ B /Fe and a 1/3-plateau for B ∼ dM/dH as a function of temperature (with a constant offsetfor clarity). The dip at ∼ T = T ∗ . (d) Evolution of the dip of the derivativeof the magnetization, ∆( dM/dH ), with temperature (orangesquares). Line is guide to the eye. The classical spin liquidphase emerges below T ∗ (cid:39) In order to gain more insights into the magneticproperties of LiFePO, we performed P NMR mea-surements. Li Fe (P O ) (PO ) naturally containstwo very sensitive NMR nuclei: P ( I = 1 / γ/ π = 17 . Li ( I = 3 / γ/ π =16 . Li was found to be mainly coupledto Fe through dipolar magnetic field resulting in broadlines which yield relatively poor accuracy on the determi-nation of the NMR shift. The NMR spectra were thus ob-tained with ν = γB / π = 109 .
732 MHz for the P lo-cal probe with I = 1 /
2, avoiding any quadrupolar effectson the NMR resonance line, using the standard π/ − τ − π pulse sequence, with τ = 20 µ s.Fig. 3 displays the evolution of the P NMR spec-trum with temperature. Two NMR lines, accounting forthe two P sites (labelled as 1 and 2) expected fromthe structure [29] (Fig.1a,b), are clearly visible below10 K. The NMR intensity relative to site (1) graduallydecreases as the temperature is increased, and finally es-capes our NMR time window above 50 K, due to a fastrelaxation, i.e. a shortening of the T transverse relax-ation time ( T < − µ s). The P NMR Knightshift K = A hf χA hf χA hf χ/ ( gµ B (cid:126) γ ) provides a direct measurementof the local magnetic susceptibility tensor χχχ . Follow-ing the common convention[30] in spherical coordinates,the eigenvalues of K relate to the shift of the NMR line∆ B = B − B in the applied field B through [31]:∆ BB = K iso + K ax (cid:0) θ − (cid:1) + K ani sin θ cos 2 φ (1)The spectra for sites (1) and (2) could be well fittedusing Eq.1 for an isotropic powder distribution (Fig. 3).The mismatch between observed (circles) and simulated(red line) intensities is due to an anisotropic T spin-spinrelaxation time, as confirmed by measurements using dif-ferent duration τ between the pulses (not shown). Whilesite (1) and site (2) probe the same local susceptibility,the anisotropic components of the Knight shift tensorare best revealed from site (2) and we thus focus on thatNMR line in the following. We then extract the temper-ature dependence of the components of the Knight shifttensor for site (2), as shown in Fig. 4.In the paramagnetic regime (300 −
10 K), the NMRlineshape of site (2) is anisotropic and shows a pro-nounced shoulder on its right-hand side. However, above10 K, K iso and K ani are found to scale with each other(Fig. 4) and thus the spectrum could be well fittedusing a unique T -dependent parameter –an isotropicmagnetic susceptibility χ ( T )–, with K iso = A isohf χ ( T ), K ani = A anihf χ ( T ) and K ax (cid:39)
0. The T -independentcomponents of the hyperfine tensor per Fe atom are A isohf = 2 . /µ B and A anihf = 0 . /µ B for site (2). An anisotropic hyperfine coupling tensor isa usual feature of P NMR.Upon cooling below T ∗ = 5 K the NMR line-shape evolves and reveals the development of a stronglytemperature-dependent anisotropy, which can thus onlybe ascribed to the magnetic susceptibility. The decreaseof K iso , signals the development of antiferromagneticcorrelations when T < θ CW = −
11 K, and clearly in-dicates the onset of a correlated magnetic phase. Bothsite (1) and site (2) reveal an anisotropic susceptibility as K ani and K ax undergo an abrupt change, a remarkablefeature for Heisenberg spins. The correlated magneticphase observed for B ∼ . < T <
10 K, forwhich M ∼ M sat , seems thus consistent with the theo-retically expected, strongly anisotropic, collinear phase.We now use P NMR to investigate the magnetic fluc-tuations in this 1/3-plateau phase. The spin-lattice relax-
FIG. 3. (Color online) Temperature dependence of the PNMR spectra (circles) measured as a function of applied mag-netic field B for LiFePO, from 80 K to 1.3 K. The phospho-rus located on site (1) is only observed at low T (magnifiedat 40 K ( × T relaxation effects. Red lineis a fit to Eq.1 (see text) from which the tensor components K are extracted. Vertical dashed line indicates the referencefield B = 6 . ation rate 1 /T probes the imaginary part of the dynamicsusceptibility χ (cid:48)(cid:48)⊥ ( ν, q ) at low energy ( ν ∼
100 MHz)through: 1 T = γ g µ B k B T (cid:88) q | A hf ( q ) | χ (cid:48)(cid:48)⊥ ( ν, q )2 πν (2)and was determined through the saturation recoverymethod, with a π/ − ∆ t − π/ − τ − π pulse sequence,from 1.3 to 300 K, at the maximal intensity of site (2)(Fig. 5). Four different regimes in temperature can beidentified. A peak is clearly observed for T ∼ . T N , which separates thequasi-static magnetic order from a classical spin-liquidstate for T N < T < T ∗ for which T ∼ T (red line inFig. 5). A crossover is observed at T = T ∗ ∼ T ∗ < T <
80 K, with an atyp-ical power law behavior α = 0 . | θ CW | , up to ∼
80 K. We specu-late that lattice vibrational modes[44] or dipole-dipoleinteractions are responsible for the unusual power lawbehavior of the spin dynamics observed here. Finally, T levels off in the high- T paramagnetic regime. TheMoriya paramagnetic limit gives the T -independent rate1 /T ∞ = γ g A S ( S + 1) / z √ πν e [42, 43] in mag-netic insulators, where ν e = J (cid:112) zS ( S + 1) / /h the ex-change frequency, with z = 2 the number of probed Feand z = 4 the number of magnetic nearest-neighboursof a Fe atom. Using A hf = A isohf and the experimentalvalue 1 /T ∞ = 6 ms − , the formula leads to J (cid:39) . J ∼ FIG. 4. (Color online) (a) Components of the NMR shift K as a function of temperature for site 2. (b) In the param-agnetic regime, K iso and K ani scale with each other showingthat the magnetic susceptibility is isotropic above T ∗ . Thered vertical dashed line indicates T ∗ . We now discuss more specifically the classical spin liq-uid phase observed in the T -range 1 . − − T dynamicsof the classical Heisenberg kagom´e antiferromagnet hasbeen investigated early on by Monte-Carlo simulations[5, 32, 33]. They gave evidence for the role of thermalfluctuations in selecting a coplanar short-range order viathe order-by-disorder mechanism for T →
0. Short-rangenematic spin correlations are predicted to survive at low T with non-dispersive soft magnon modes[34], a charac-teristic feature associated with the large degeneracy in-herent to the kagom´e lattice. Numerical simulations ofthe spin autocorrelation function yield (cid:104) S (0) S ( t ) (cid:105) ∼ e − νt for H = 0 [1, 33, 35, 36], with ν = ck B T / (cid:126) S ( c ∼ Ga O was the first experimental candidate to hint at suchspin dynamics, yet with an important contribution fromquantum fluctuations [33, 38–40] and spin vacancies [41].Later on, inelastic neutron scattering exposed a T -linearinverse relaxation rate in deuterium jarosite over the ex-tended temperature range 0 . < T /J cl < J cl = JS ( S + 1). Here, the T -dependence of 1 /T can beinferred from ν through the Redfield formula:1 T = 2 γ ∆ νν + γ B (cid:39) γ ∆ ν (3)where ∆ is the amplitude of the fluctuating field at thephosphorus position. Using the constant high- T value T ∞ set by the Moriya’s exchange frequency ν e [42], thisfinally leads to T = βT with β = ck B T ∞ /ν e (cid:126) S . InLiFePO, the dynamical behavior of T could be well fittedto such a law, with no adjustable parameter but c = 1 . T -range 0 . < T /J cl < . J cl = 8 .
25 K.In a more recent work[36], Taillefumier et al. charac-terized the dynamics for the long time scales appropri-ate for NMR using semiclassical numerical techniques,and predicted two different low temperature regimes inthe B → i ) a cooperative, classical, spin-liquidregime with T ∼ T in line with what is discussed aboveand ( ii ) a coplanar dynamical regime for T /J cl < . J cl = 8 .
25 K, our measure-ments down to 1.3 K in LiFePO give only access to theregion
T /J cl > .
2, i.e. the high- T paramagnetic and co-operative spin liquid phases, before residual interactionsapparently lift the degeneracy and produce the antiferro-magnetic transition. The observed thermal behavior ofour spin-lattice relaxation rate, T ∼ T , seems to showthat this dynamics unexpectedly persists under moder-ate applied fields ( B ∼ uud manifold of the 1/3 plateau. Furthermore, theshort-range anisotropic correlations that develop above T N according to our NMR local susceptibility measure-ments may indicate either that the nematic correlationsare still present for T N < T < T ∗ , or that we lie close toa quasi-static magnetic order. In the light of our results,a theoretical study of the spin dynamics under appliedfields, in the 1/3-phase, would be highly relevant to con-firm the origin of the dynamics observed in NMR.In conclusion we have identified a new classical S = Heisenberg kagom´e antiferromagnet that displays the 1/3magnetization plateau predicted a long time ago. Our P NMR measurements further identified a classicalspin-liquid regime for T N < T < T ∗ , for which the timescale of the spin dynamics is set by the temperature only,with T ∼ T , in very good agreement with numericalestimates. These fluctuations could naturally originatefrom the uud spin configurations manifold on each trian-gle that define the collinear spin liquid of the 1/3 plateau.The correlated low- T regime of LiFePO combines ther-mal fluctuations with pronounced anisotropic spin cor-relations and hence can be coined as a semiclassical ne-matic spin liquid . The discovery of archetypal magneticfrustration relevant to the kagom´e lattice in the layerediron monodisphophate LiFePO offers great perspectivesto improve our knowledge on classical and quantum fluc-tuations effects in zero and applied fields in the KHAF.In LiFePO, further elastic and inelastic neutron scatter-ing studies could help revisiting the central question ofthe selection of the q = 0 or q = √ × √ ( S = ), V ( S = 1)] willcertainly open new avenues to search for quantum phases FIG. 5. (Color online) 1 /T (up) and T (bottom) probedby P NMR as a function of temperature in LiFePO. Fourdifferent regimes can be identified: a quasi-static order (QSO,grey), a classical spin liquid phase (orange), an atypical high T regime (green) and high- T paramagnetic phase. The classicalspin liquid phase displays spin dynamics with a linear law T = βT (red line, see text). relevant to the KHAF. ACKNOWLEDGMENTS
It is a pleasure to acknowledge useful discussion withProf. S. Greenbaum on lithium diffusion, P. Berthet andC. Decorse for sharing their expertise on samples syn-thesis of lithium oxides. EK and RS acknowledge finan-cial support from the labex PALM for the QuantumPy-roMan project (ANR-10-LABX-0039-PALM). This workwas supported by the French Agence Nationale de laRecherche under grant ANR-18-CE30-0022-04 ‘LINK’. [1]
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