Weak Ferromagnetic Exchange and Anomalous Specific Heat in ZnCu3(OH)6Cl2
WWeak Ferromagnetic Exchange and Anomalous Specific Heat in ZnCu (OH) Cl Ookie Ma ∗ and J. B. Marston † Department of Physics, Brown University, Providence, RI USA 02912-1843 (Dated: December 1, 2018)Experimental evidence for a plethora of low energy spin excitations in the spin-1/2 kagome antifer-romagnet ZnCu (OH) Cl may be understandable in terms of an extended Fermi surface of spinonscoupled to a U(1) gauge field. We carry out variational calculations to examine the possibilitythat such a state may be energetically viable. A Gutzwiller-projected wavefunction reproduces thedimerization of a kagome strip found previously by DMRG. Application to the full kagome latticeshows that the inclusion of a small ferromagnetic next-nearest-neighbor interaction favors a groundstate with a spinon Fermi surface. PACS numbers: 75.10.-b, 75.10.Jm, 75.50.Ee
The spin-1/2 Heisenberg antiferromagnet on a kagomelattice is a proving ground for the existence of a two-dimensional spin liquid. Can the combination of lowspin, low coordination number, and geometric frustra-tion lead to spin disordered ground states and excita-tions with fractional statistics? If so, then it remainsto establish whether spin-spin correlations are short-ranged with a gap to triplet excitations or whether thecorrelations are quasi-long-ranged with gapless excita-tions. Experiment is the ultimate arbiter of these fun-damental questions. Thus, the recent experimental re-alization of a spin-1/2 kagome antiferromagnet (KAF),ZnCu (OH) Cl also known as Herbertsmithite[1], is ofgreat interest. Magnetic susceptibility, muon spin rota-tion, and NMR measurements show no evidence of mag-netic order[2, 3]. The bulk spin susceptibility exhibitsCurie behavior down to temperature T = 0 . J , where J ≈
170 K, but then increases sharply with decreasingtemperature, possibly saturating at T = 0. Likewise, thespecific heat shows significant enhancement at low tem-peratures. At very low temperatures, the temperaturedependence can be roughly fit to a power law with anexponent as small as α = 1 /
2. Over the range T = 5to 30 × − J , a best fit[4] to C v ∝ T α yields α = 2 / BaNiO and NENP[7, 8]. Powder NMR measurementsof the local spin susceptibility suggest a second possibil-ity. Although the averaged local susceptibility tracks the bulk, regions of the NMR spectrum actually show a de-crease in the susceptibility with decreasing temperature.The discrepancy between local and bulk measurementsmay indicate the presence of impurities or may instead beattributed to the sampling of the susceptibility along dif-ferent and distinct crystallographic directions[3, 9]. If thelatter case, the low temperature ferromagnetism may bedue to a Dyzaloshinskii-Moriya (DM) anisotropy[10, 11].To make a clear prediction for the specific heat atthe lowest experimental temperatures, we use Gutzwiller-projected mean-field theory. Our approach is motivatedin part by the appearance of a specific heat exponent α < C ( T ) ∝ T / (1)and thus, dominates conventional contributions such asthat due to phonons. This signature provides motivationto investigate candidate states that possess large spinonFermi surfaces.In the case of the nearest-neighbor (NN) Heisenbergmodel, our calculations reproduce the results of Ref. 5,namely, we find that the lowest energy state has no bro-ken symmetries and the corresponding mean-field stateexhibits Dirac fermions at nodal points. Under the hy-pothesis that the experimentally observed weak ferro-magnetism is intrinsic to the pure lattice of Cu spins,we investigate the effect of adding a small next-nearest-neighbor (NNN) interaction with negative (ferromag-netic) coupling H = J (cid:88) (cid:104) i,j (cid:105) S i · S j − J (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) S i · S j . (2)For convenience, we set J = 1 in the following. We a r X i v : . [ c ond - m a t . s t r- e l ] J a n note that establishing the sign of the NNN spin-spin in-teraction from first principles requires a detailed under-standing of the quantum chemistry of ZnCu (OH) Cl .An organic spin-1/2 KAF has been argued to have mixedferromagnetic NN and antiferromagnetic NNN bonds[15],but we leave open the possibility that other sub-leadingterms, like the DM anisotropy, may also be important ina realistic model of Herbertsmithite.A rational approach to the systematic constructionof variational spin liquid states can be realized by gen-eralizing the ordinary SU(2) Heisenberg antiferromag-net to SU(N) spins living in the self-conjugate repre-sentation, meaning that each site has N/2 fermions[16].The resulting model may then be solved exactly in thelarge-N limit by decoupling the spin-spin interactionwith the Hubbard-Stratonovich field χ ij = (cid:104) f † αi f jα (cid:105) .Appropriate variational states are then obtained bythe usual Gutzwiller-projection procedure[17]. To com-pare candidate variational wavefunctions in the physi-cal SU(2) limit, NN and NNN correlations are sampledby a standard determinantal wavefunction Monte Carloalgorithm[18, 19] that exactly enforces the single occu-pancy constraint f †↑ i f i ↑ + f †↓ i f i ↓ = 1. The χ ij field,interpreted as a bond amplitude order parameter, maythen be adjusted to optimize the ground state energy totest for various dimerization instabilities. This approachhas been shown to reproduce qualitatively the phase di-agrams of certain spin chains that are known exactly byother means[17].To test the reliability of this approach, we first studythe nearest-neighbor Heisenberg model on a 2-leg kagomestip. Because of its one-dimensional character, theground state has been accurately described by the es-sentially exact numerical density matrix renormalizationgroup (DMRG) method where it was shown[20] to bestrongly dimerized with five different values for the NNspin-spin correlations (cid:104) S i · S j (cid:105) . The strip employed inthe DMRG calculation has open boundaries that areterminated in such a way to remove triplet excitationsfrom the low energy spectrum; nevertheless the nearest-neighbor spin-spin correlations are well converged in theinterior. The NN spin-spin correlations (a) through (e),as labelled in Fig. 1, are − . − . − . − . − . E = − . π . NN correlations are sampled on a stripwith length 100 sites and 250 total spins. We find that . !! !! FIG. 1: The kagome strip. Top: The optimal variationalstate. Numerical values indicate the relative hopping magni-tudes along the links and the value of the flux through eachhexagon. There is no flux through the triangles. Bottom: Themagnitude of the five symmetry-unrelated nearest-neighborspin-spin correlation functions (cid:104) S i · S j (cid:105) , labeled (a) through(e), is indicated by the thickness of the lines. See text for theactual values. among states that at most double the size of the unitcell, the lowest energy state has a flux of π through ev-ery other hexagon and zero flux through the remaininghexagons and all triangles; thus, the variational state ex-hibits dimer order. The optimal arrangement of fluxesand bond magnitudes is depicted in the top panel of Fig.1. Along the two legs, alternating bond magnitudes ofapproximately 1.1 and 0.9 yield the lowest energy, withthe remaining third bond fixed at unit hopping ampli-tude. The doubled unit cell supports five symmetry-inequivalent values of the NN spin-spin correlation func-tion (cid:104) S i · S j (cid:105) in agreement with the DMRG results.Along bonds (a) through (e) the spin-spin correlationfunctions are − . − . − . − . − . E = − . .
5% higher thanthe DMRG value. Further relaxation of the energy to-wards the DMRG value may be expected if all five dis-tinct bonds were to be adjusted in magnitude.As the variational approach appears to give a good de-scription of the kagome strip, we now turn to the full2D kagome lattice. Several candidate spin liquid stateshave been examined[5]. One competitive state has flux π through the hexagons and no flux through the trian-gles. This π -hexagon state has Dirac nodes at the Fermilevel[5]. Another state of interest has no flux penetrat-ing through the lattice anywhere. As shown in Fig. 2,this uniform state possesses an extended Fermi surfaceat half-filling.The energies of the candidate states are calculated onan oblique lattice with 12 ×
12 3-site unit cells (432 totalsites). On finite lattices, the choice of boundary con-ditions is important. In the case of periodic boundaries,
FIG. 2: (Color online) Mean-field spinon dispersion of theuniform phase of zero flux. (a) Complete dispersion showingthat one of the three bands is completely flat. (b) Middleband cut by a plane at the Fermi energy corresponding tohalf-filling. Note that the Dirac nodes are not located atthe Fermi energy. (c) Rotation of (b) to highlight the nearlycircular Fermi surface. there is an ambiguity in the filling of independent-particlestates at the Fermi level, as only some of the degener-ate independent-particle states are filled. To resolve theambiguity, the degeneracy must be lifted. Mixed peri-odic and antiperiodic boundaries accomplishes this[23]at the cost of breaking lattice rotational symmetry. Con-sequently, the mixed boundary conditions induce modu-lation in observables such as (cid:104) S i · S j (cid:105) . The modulation,however, decreases with increasing system size, vanish-ing in the thermodynamic limit. For the relatively largelattice studied here, the modulation is smaller than theuncertainty in the calculated spin-spin correlation func-tion due to the statistics of finite samples.We check for instabilities of the variational states to-ward two different types of dimer patterns. The firstpattern is a dimer order of the type first proposed inRef. 24 based on 1 /N corrections to the large-N solution.Dimer coverings that maximize the number of hexagonswith exactly three dimers (“perfect hexagons”) are ener-getically preferred because the perfect hexagons resonatelike the alternating single and double bonds of a ben-zene ring. As the resonance is a second-order process,it lowers the energy. There can be at most one perfecthexagon for every 18 sites[24]. The top panel in Fig.3 shows an 18-site unit cell with one perfect hexagon.Larger unit cells support other possible dimer coveringsthat maximize the density of perfect hexagons includingthe honeycomb pattern shown in Fig. 1(b) of Ref. 24and the striped pattern shown in Fig. 5(b) of Ref. 25.For the NN Heisenberg model, a systematic dimer ex-pansion determines that the honeycomb and stripe pat- FIG. 3: Two possible dimer orderings with unit cells outlinedby dashed parallelograms. Thick lines indicate larger hoppingmagnitudes and hence, larger NN spin-spin correlations. Toppanel: Dimer order that maximizes the number of hexagonswith exactly three dimers[24]. Bottom panel: Plaquette orderthat outlines the shape of a star[22].TABLE I: The lattice-averaged spin-spin correlation function (cid:104) S i · S j (cid:105) for an oblique lattice of 12 ×
12 3-site unit cells (432sites total) with mixed periodic and antiperiodic boundaryconditions. The magnitudes of the bond amplitudes are mod-ulated by 5% for the ordered states. Numbers in parenthesesare the statistical errors due to the finite number of samples.( i, j ) ∈ NN ( i, j ) ∈ NNN π -hexagon state -0.42866(2) -0.02297(21)dimer order -0.42856(3) -0.02277(22)plaquette order -0.42841(4) -0.02271(17)uniform state -0.41215(2) 0.07724(13)dimer order -0.41205(2) 0.07873(10)plaquette order -0.41203(2) 0.07701(13) terns have ground state energies per site of -0.432088216and -0.4315321, respectively[26]. These energies comparequite well to the ground state energy of a 36 site clus-ter, -0.438377, found by exact diagonalization[27]. Thesecond dimer configuration has plaquette order of a typethat has been proposed as a potential instability of the π -hexagon state. As shown in the bottom panel of Fig.3, the plaquette takes the shape of a star formed by theouter edges of six triangles that share a common hexagon.This pattern of bonds would be induced by the dynam-ical formation of a non-chiral mass term that gaps outthe nodal fermions[22].Table I summarizes the variational results. Instabili-ties are tested by imposition of a small (5%) modulationin the magnitude of the χ ij bond amplitudes. In addi-tion to the tabulated results, we also find that imposi-tion of a 5% modulation of the honeycomb type uponthe π -hexagon state raises the energy only slightly, to -0.42860(3), while at the same time gapping out the Diracnodes. This result lends support to the idea[24, 25, 26]that the nearest-neighbor antiferromagnet is actuallydimer ordered. Furthermore, all of the dimer cover-ings we tested that maximize the number of perfecthexagons are energetically preferable to the star plaque-tte pattern[5, 22]. We speculate that further variationaltuning will yield an ordered state of perfect hexagonsthat is energetically competitive with the best estimatesfor the ground state energy of the NN spin-1 / π -hexagon phase but positive in the uniform phase,the uniform state is energetically favored over the π -hexagon state for sufficiently large ferromagnetic NNNcoupling, J > .
16 (the ground state remains antiferro-magnetic up to much larger values of J ). At J = 0.16,dimer modulation of the uniform state further lowers theenergy, with a minimum in the energy occurring at ap-proximately 4% bond modulation. This 4% bond modu-lation induces a 18% modulation in the value of (cid:104) S i · S j (cid:105) on symmetry-distinct bonds. Somewhat surprisingly, themean-field theory continues to support gapless excita-tions despite the broken translational and rotational sym-metries, with little change in the density of states at theFermi energy. This is because ordering wavevectors cor-responding to large (multiple of 18 sites) unit cells do notspan the spinon Fermi surface. Thus, a small ferromag-netic NNN interaction favors a spin-liquid phase with alarge spinon Fermi surface, suggesting that spinons canremain deconfined even in the presence of dimer order.We are not aware of any rigorous arguments (eg. Ref.28) that preclude this possibility.The large spinon Fermi surface exhibits at low temper-ature both a relatively large (but finite) magnetic suscep-tibility and an enhanced specific heat, Eq. 1, due to inter-actions between spinons and the U(1) gauge field. Theselow energy properties of the uniform state agree qualita-tively with those from available Herbertsmithite exper-iments, though to reproduce quantitatively the anoma-lously large susceptibility, it may be necessary to invokeadditional contributions from impurities[6] and/or theDM interaction[10, 11]. ACKNOWLEDGMENTS
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