Spiral ground state in the quasi-two-dimensional spin-1/2 system Cu2GeO4
Alexander A. Tsirlin, Ronald Zinke, Johannes Richter, Helge Rosner
aa r X i v : . [ c ond - m a t . s t r- e l ] M a r Spiral ground state in the quasi-two-dimensional spin- system Cu GeO Alexander A. Tsirlin, ∗ Ronald Zinke, Johannes Richter, and Helge Rosner † Max Planck Institute for Chemical Physics of Solids, N¨othnitzer Str. 40, 01187 Dresden, Germany Institute for Theoretical Physics, University of Magdeburg, P.O. Box 4120, 39016 Magdeburg, Germany
We apply density functional theory band structure calculations, the coupled-cluster method,and exact diagonalization to investigate the microscopic magnetic model of the spin- compoundCu GeO . The model is quasi-two-dimensional, with uniform spin chains along one direction andfrustrated spin chains along the other direction. The coupling along the uniform chains is antifer-romagnetic, J ≃
130 K. The couplings along the frustrated chains are J ≃ −
60 K and J ≃
80 Kbetween nearest neighbors and next-nearest neighbors, respectively. The ground state of the quan-tum model is a spiral, with the reduced sublattice magnetization of 0.62 µ B and the pitch angle of84 ◦ , both renormalized by quantum effects. The proposed spiral ground state of Cu GeO opens away to magnetoelectric effects in this compound. PACS numbers: 75.30.Et, 75.10.Jm, 71.20.Ps, 75.50.Ee
I. INTRODUCTION
Quantum magnetism is a field of fundamental researchfocused on exotic ground states and non-trivial low-temperature properties.
Nevertheless, certain effectsin quantum magnets are also relevant for applications.Spin-chain compounds show ballistic regime of heattransport, whereas frustrated magnets are capable of astrong magnetocaloric effect. Additionally, many of thefrustrated magnets undergo spiral or, in general, incom-mensurate ordering, and reveal ferroelectricity inducedby a magnetic field. A frustrated spin chain with com-peting ferromagnetic (FM) nearest-neighbor ( J ) and an-tiferromagnetic (AFM) next-nearest-neighbor ( J ) cou-plings is the simplest spin model giving rise to spiral mag-netic correlations at J /J < − (Ref. 7). This model iseasily realized experimentally and has a clear structuralfootprint, a chain of edge-sharing CuX plaquettes withX being oxygen, chlorine, or even nitrogen. Suchchains typically show FM J due to the nearly 90 ◦ Cu–X–Cu angle and AFM J due to the Cu–X–X–Cu superex-change. Indeed, many compounds of this type undergospiral magnetic ordering and sometimes exhibit magneticfield-induced ferroelectricity. However, the detailed mi-croscopic understanding of these effects remains challeng-ing, and even the electronic origin of ferroelectricity inspin-chain cuprates is vividly debated. Interchain couplings are an important feature of anyreal material. The couplings between spin chains canmodify the ground state qualitatively by inducing a long-range order with finite sublattice magnetization.
Inthe case of frustrated spin chains, such couplings influ-ence the behavior of doped systems, and play a decisiverole for the stability of exotic phases in high magneticfields. Regarding magnetoelectric effects, the inter-chain couplings naturally determine their temperaturescale by adjusting the magnetic ordering temperature.Theoretical studies of coupled frustrated spin chainsremain a challenge owing to the two-dimensional (2D)and frustrated nature of the problem. Therefore, experi- mental benchmarks are especially important. The avail-able frustrated-spin-chain compounds show relativelyweak interchain couplings, while the relevance of theopposite regime with strongly coupled frustrated spinchains remains unclear. A common and a somewhatnaive picture suggests that leading exchange couplingsshould run along the structural chains owing to shorterCu–Cu distances. In the following, we present a microscopic magneticmodel of Cu GeO . This compound is a unique exampleof a 2D system of strongly coupled frustrated spin chains.The coupling J between the frustrated chains is so strongthat the system can be equally viewed as uniform spinchains along J with the frustrated interchain couplings J and J (see Fig. 1). Both descriptions relate to certainfeatures of the magnetic behavior: while the uniform-chain model fits the magnetic susceptibility of Cu GeO down to T /J ≃ .
5, the ground state of the 2D model isa spiral, which is typical for the frustrated J − J spinchains.The crystal structure of Cu GeO belongs to thespinel type. Magnetic properties were studied in a rela-tion to the spin-Peierls compound CuGeO . The low-dimensional magnetic behavior of Cu GeO resemblesCuGeO indeed. However, no signatures of the struc-tural distortion or spin gap were found down to 10 K,and the long-range magnetic ordering at T N = 33 . Yamada et al . analyzed Cu GeO using theanisotropic pyrochlore lattice model with two inequiv-alent exchange couplings that are J and J c in our nota-tion (upper left panel of Fig. 1). This model arises froma straight-forward and naive geometrical consideration ofthe spinel structure, with inequivalent couplings drivenby the tetragonal distortion of the parent cubic system.At J c /J ≪
1, the anisotropic pyrochlore lattice splitsinto chains. According to Ref. 23, Cu GeO is close tothis limit, with J = 135 K and J c /J = 0 .
16. Starykh et al . studied the 2D analog of the model theoretically,and proposed a quantum-disordered valence-bond-solidground state. b g b ’ b ’ c a a ’ a ’ J JJ J J ab J J J c J FIG. 1. (Color online) Top panel: crystal structure ofCu GeO (left) and a single magnetic layer in the ab plane(right). Bottom panel: a sketch of the spin spiral with thepitch angle γ (left), and the magnetic model of J − J frus-trated spin chains coupled by J (right). Circles and dotsdenote the positions of the Cu atoms. Lines in the top leftpanel show the anisotropic pyrochlore lattice considered inRef. 23. II. BAND STRUCTURE
As a derivative of the spinel structure, Cu GeO mightbe thought of as a three-dimensional network of CuO octahedra. However, this description ignores essen-tial features of the electronic structure. In oxide com-pounds, Cu +2 tends to adopt a four-fold coordination(CuO plaquette) having dramatic influence on the or-bital ground state and magnetic properties. Such pla-quettes can be recognized in Cu GeO , and lead to apeculiar superexchange scenario. Four short bonds tooxygen (1.95 ˚A) form the CuO plaquettes in the ab plane, whereas the two remaining Cu–O bonds are muchlonger (2.50 ˚A). Edge-sharing CuO plaquettes comprisestructural chains that run along a or b , with parallelchains forming layers in the ab plane (upper right panelof Fig. 1). Equivalent layers with differently directedstructural chains alternate along the c axis. In the fol-lowing, we denote the direction of the structural chainsas b ′ and the perpendicular direction as a ′ , to distinguishthose from the crystallographic a and b axes. GeO tetra-hedra connect the chains into a three-dimensional (3D)framework (Fig. 1).To evaluate individual exchange couplings, we performscalar-relativistic density functional theory (DFT) bandstructure calculations using the FPLO9.00-33 code. Weapply the local density approximation (LDA) with theexchange-correlation potential by Perdew and Wang, and use a well-converged k mesh comprising 3350 pointsin the symmetry-irreducible part of the first Brillouinzone. With LDA calculations, we are able to identifyrelevant states, and to evaluate hopping parameters t i via a fit with an effective one-orbital tight-binding (TB)model. The hopping parameters are introduced into a TABLE I. Leading exchange couplings in Cu GeO : hoppingparameters t i of the TB model, AFM contributions to theexchange couplings J AFM i = 4 t i /U eff , and the total exchangeintegrals J i from LSDA+ U calculations with U d = 6 . t i J AFM i J i (˚A) (meV) (K) (K) J − J J J ab −
37 14 7 J c −
11 1 − Hubbard model with the effective on-site Coulomb repul-sion potential U eff = 4 . In the case of low-lyingexcitations, the Hubbard model is further reduced to aHeisenberg model under the conditions of half-filling andstrong correlations ( t i ≪ U eff ). Then, the AFM parts ofthe exchange integrals are evaluated as J AFM i = 4 t i /U eff .An alternative way to evaluate the exchange couplingsis to treat the strong correlations within DFT, via themean-field-like LSDA+ U approach. We calculate totalenergies for a set of collinear spin configurations, andmap these energies onto a classical Heisenberg model.Thus, total exchange integrals J i are estimated. In theLSDA+ U calculations, we use the Coulomb repulsionand exchange parameters U d = 6 . ± J d = 1 eV,respectively. The double-counting-correction (DCC)scheme was set to the around-mean-field (AMF) option.The application of the fully-localized-limit (FLL) DCChad little effect on the exchange couplings.The LDA energy spectrum of Cu GeO is typicalfor Cu +2 oxides. The mixed Cu 3 d – O 2 p valencebands extend down to − d x − y orbital (here, x and y align with the short Cu–O bonds). Germanium orbitals contribute to the bandsaround −
10 eV, and show negligible DOS at higher ener-gies. While LDA yields a metallic energy spectrum due tothe underestimation of electronic correlations in the Cu3 d shell, LSDA+ U restores the insulating scenario withthe band gap of E g = 2 . ± . U d = 6 . ± d x − y states are represented by four bandscrossing the Fermi level and arising from four Cu atoms inthe primitive cell of Cu GeO (Fig. 3). These bands areseparated from the rest of the valence bands by a pseudo-gap. To extract hopping parameters, we construct Wan-nier functions (WFs) based on the Cu d x − y character. This analysis evidences sizable nearest-neighbor ( t ) andnext-nearest-neighbor ( t ) hoppings along the structuralchains. However, the hopping t along a ′ is comparable to t and t . Additionally, a weak diagonal hopping in the ab plane is found (Table I). The nearest-neighbor hop-pings perpendicular to the ab plane ( t c ) are −
11 meV,yielding J AFM c as low as 1 K. The weak dispersion ofthe bands along Γ − Z also shows the pronounced two-
20 0Energy (eV) 4 - - D O S ( e V ) - FIG. 2. (Color online) LDA density of states for Cu GeO .The Fermi level is at zero energy. dimensionality of the system. Introducing the hoppingsinto an effective one-band Hubbard model, we evaluateAFM parts of the exchange integrals J AFM i (Table I).LSDA+ U calculations modify the LDA-based scenario.We find FM nearest-neighbor coupling within the struc-tural chains, J = − ∓
10 K for U d = 6 . ± J =80 ∓
20 K and the interchain coupling J = 130 ∓
30 K arebasically unchanged. Further couplings in the ab planeare below 10 K. The interplane coupling becomes FM andremains weak. Thus, we establish the quasi-2D J − J − J model with a weak interlayer coupling J c (Fig. 1).The quasi-2D model of Cu GeO results from thestrong tetragonal distortion of the spinel structure. Theplaquette description (Fig. 1), with the magnetic d x − y orbital coplanar to the CuO plaquette, clarifies the 2Dnature of the system. The couplings J c connect the pla-quettes lying in different planes, and therefore remainweak. By contrast, three sizable couplings in the ab planeestablish a frustrated spin lattice. Our model is dissimilarto the anisotropic pyrochlore lattice proposed by Yamada et al. The pyrochlore spin lattice omits the relevant ex-changes J and J , and should be discarded. Cu GeO isa frustrated magnet indeed, but the strong frustration isfound in the J − J chains rather than tetrahedral units.The FM nearest-neighbor coupling J should be re-ferred to the Cu–O–Cu angle of 91 . ◦ . The microscopicorigin of ferromagnetism is the Hund’s coupling on theoxygen site. The next-nearest-neighbor coupling J isthe AFM Cu–O–O–Cu superexchange. Similar valuesof 50 −
100 K for | J | and J have been established forthe archetype frustrated-spin-chain compounds, such asLiCu O and LiCuVO . Another remark on the structural implementation ofthe spin model regards the origin of the long-range cou-plings J and J . Since the Ge orbitals weakly contributeto the valence states, both couplings should be assignedto a Cu–O–O–Cu superexchange. Despite an identicalCu–Cu distance (Table I), a larger J value is caused by G G
X M Z - E n er gy ( e V ) FIG. 3. (Color online) LDA band structure of Cu GeO (thinlight lines) and the fit of the TB model (thick dark lines). TheFermi level is at zero energy. The k -path is defined as fol-lows: Γ(0 , , X (0 . , , M (0 . , . , Z (0 , , . π/a and 4 π/c . the co-planar arrangement of the plaquettes in the adja-cent chains. By contrast, the next-nearest-neighbor pla-quettes within the chain ( J ) lie in different planes due tothe buckled chain geometry (Fig. 1). It is worth to notethat the ab projections of the Cu GeO and LiCuVO structures are very similar. However, LiCuVO is aquasi-1D system with J ≪ | J | , J , while the spin sys-tem of Cu GeO is quasi-2D. III. MICROSCOPIC MODEL
In the following, we explore the ground state and finite-temperature properties of our model. We first considerthe purely 2D regime described by the Hamiltonian: H = X n n X i (cid:2) J s i,n · s i +1 ,n + J s i,n · s i +2 ,n (cid:3)o + X i X n J s i,n · s i,n +1 , (1)where the index n labels the structural chains (along b ′ ),and i denotes the lattice sites within a chain n . The effectof the interlayer coupling J c is discussed in Sec. III D.Our model can be viewed as frustrated J − J chains(along b ′ ) which are uniformly coupled by J (along a ′ ).Alternatively, one finds uniform spin chains along a ′ withfrustrated interchain couplings J and J along b ′ . Whileany of the parent 1D models is rather easy to handle,a rigorous treatment of their 2D combination is a chal-lenging problem. Below, we apply the Lanczos diagonal-ization and coupled cluster method to achieve an accu-rate description of the ground state. By contrast, finite-temperature properties of the quantum model can onlybe accessed at high temperatures by a series expansion(HTSE), whereas conventional techniques, such as quan-tum Monte-Carlo or exact diagonalization, fail becauseof the sign problem or finite-size effects. A. Magnetic susceptibility
Since the experimental information on Cu GeO isrestricted to the magnetic susceptibility and heat ca-pacity measurements in Ref. 23, we discuss thermody-namic properties first. The experimental specific heatcontains an unknown phonon contribution, hence themagnetic part can not be separated. Therefore, themagnetic susceptibility χ ( T ) (Fig. 4) remains the onlyquantity suitable for the comparison between theory andexperiment. The estimated Curie-Weiss temperature θ ≃ ( J + J + J ) = 75 K is in good agreement withthe experimental value of θ = 89 K. For a further comparison, we derive the HTSE for ourmodel: χ = N A g µ B k B T (cid:18) J + J + J T + J + J + J T (cid:19) − , (2)where N A is Avogadro’s number, µ B is Bohr magneton, g is the g -factor, and we used the expressions from Ref. 33up to the third order in temperature. The calculated ex-change couplings (Table I) are in reasonable agreementwith the experimental data down to 150 K (Fig. 4). Thedeviations at lower temperatures are likely related to thedivergence of the HTSE at T ≤ J . To improve the fitat higher temperatures, a slight adjustment of the ex-change couplings is required. However, the third-orderHTSE contains two J -dependent terms only, hence anunconstrained fit of three exchange parameters is impos-sible.To access temperatures below 150 K, a simplificationof the model is required. Since J exceeds | J | and J , thespin lattice is, to a first approximation, a set of uniformspin chains along a ′ . The respective 1D model fits theexperimental magnetic susceptibility down to 70 K with J = 140 K and g = 2 .
22. A similar fit with J = 135 Khas been given in Ref. 23. However, Yamada et al. er-roneously assign the spin chains to the structural chains(in their notation, J corresponds to J ). Our DFT calcu-lations show that the uniform spin chains run perpendic-ular to the structural chains, whereas J is FM. Such anintricate situation is not uncommon for low-dimensionalmagnets, see Refs. 34 and 35 for similar examples.Below 70 K, the 1D uniform-chain model overestimatesthe magnetic susceptibility of Cu GeO . This feature in-dicates an onset of 2D spin correlations. In contrast tothe uniform spin- chain having finite susceptibility atzero temperature, 2D and 3D systems usually developa long-range order with vanishing susceptibility at lowtemperatures. The onset temperature of 2D spin correla-tions is a rough measure of interchain couplings. Indeed,the temperature of 70 K conforms to our estimates of J = −
60 K and J = 80 K, the couplings between theuniform spin chains. c ( e m u / m o l C u ) - FIG. 4. (Color online) Fit of the experimental magnetic sus-ceptibility data with the uniform chain model (dashed line)and the comparison to the HTSE of Eq. (2) (solid line). Ex-perimental data are from Ref. 23.
B. Coupled cluster method
The coupled cluster method (CCM) and its applicationto frustrated spin systems have been previously reviewedin several articles, see, e.g., Refs. 21, 36–47. Therefore,we will give only a brief illustration of the main relevantfeatures of the method. For more general information onthe methodology of the CCM, see, e.g., Refs. 40 and 48,and references therein.The CCM is a universal quantum many-body method.The starting point for a CCM calculation is the choiceof a normalized reference or model state | Φ i , togetherwith a complete set of (mutually commuting) multi-configurational creation operators { C + L } and the corre-sponding set of their Hermitian adjoints { C L } . The CCMparametrizations of the ket- and bra- GSs are given by | Ψ i = e S | Φ i , S = X I =0 S I C + I ; h ˜Ψ | = h Φ | ˜ Se − S , ˜ S = 1 + X I =0 ˜ S I C I . (3)Using h Φ | C + I = 0 = C I | Φ i ∀ I = 0, C +0 ≡
1, the com-mutation rules [ C + L , C + K ] = 0 = [ C L , C K ], the orthonor-mality condition h Φ | C I C + J | Φ i = δ IJ , and completeness X I C + I | Φ ih Φ | C I = 1 = | Φ ih Φ | + X I =0 C + I | Φ ih Φ | C I , we geta set of non-linear and linear equations for the correlationcoefficients S I and ˜ S I , respectively. We choose a refer-ence state corresponding to the classical state of the spinmodel, i.e., a non-collinear reference state with up-downN´eel -type correlations along the a ′ -direction (uniform J chains) and with spiral correlations along the b ′ -direction(frustrated J − J chains). The spiral correlations arecharacterized by a pitch angle γ , i.e. | Φ i = | Φ( γ ) i . In thequantum model, the pitch angle is typically different fromthe corresponding classical value γ cl . Hence, we do notchoose the classical result for the pitch angle, and ratherconsider γ as a free parameter in the CCM calculation.The value of γ has to be determined by the minimizationof the GS energy (in a certain CCM approximation, seebelow) given by E ( γ ) = h Φ( γ ) | e − S He S | Φ( γ ) i , i.e., fromthe dE/dγ | γ = γ qu = 0 condition.In order to find an appropriate set of creation opera-tors, it is convenient to perform a rotation of the localaxes on each of the spins so that all spins in the referencestate align with the negative z -direction. This rotationby an appropriate local angle δ i,n = δ i,n ( γ ) of the spinon the lattice site ( i, n ) is equivalent to the spin-operatortransformation s xi,n = cos δ i,n ˆ s xi,n + sin δ i,n ˆ s zi,n ; s yi,n = ˆ s yi,n s zi,n = − sin δ i,n ˆ s xi,n + cos δ i,n ˆ s zi,n ) . (4)The reference state and the corresponding creation oper-ators C + L are given by | ˆΦ i = | ↓↓↓↓ · · · i ; C + L = ˆ s + i,n , ˆ s + i,n ˆ s + j,m , ˆ s + i,n ˆ s + j,m ˆ s + k,l , . . . , (5)where the indices ( i, n ) , ( j, m ) , ( k, l ) , . . . denote arbitrarylattice sites. This specified form of the creation opera-tors C + L and the corresponding reference state | ˆΦ i imme-diately make clear that the general relations listed belowEq. (4) are fulfilled. In the rotated coordinate frame, theHeisenberg Hamiltonian acquires a dependence on thepitch angle γ (see Ref. 21 for more details).The order parameter (sublattice magnetization) inthe rotated coordinate frame is given by m = − /N P Ni,n h ˜Ψ | s zi,n | Ψ i . The only approximation of theCCM is the truncation of the expansion of the corre-lation operators S and ˜ S . We use the well-establishedLSUB n scheme, where all multispin correlations on thelattice with n or fewer contiguous sites are taken intoaccount.In contrast to Ref. 21, which is focused on the J ≤| J | , J regime, we consider the case of J > | J | , J . Wealso evaluate LSUB n approximations of higher order, upto n = 8. In the highest order of approximation, LSUB8,we have 21124 configurations, i.e., 21124 coupled non-linear equations have to be solved numerically. More-over, the minimum of E ( γ ) has to be found numericallyto determine the quantum pitch angle γ qu . For the nu-merical calculations, we use the program package CCCM by D.J.J. Farnell and J. Schulenburg. Since the LSUB n becomes exact for n → ∞ , thenumerical result can be improved by extrapolating the“raw” LSUB n data to n → ∞ using the expression m n = m ∞ + a/n + b/n , cf. Refs. 38, 40, 45, and 48. C. Ground state
To find the classical ground state of the model givenby Eq. (1), we write the magnetic energy per lattice sitefor an arbitrary 2D propagation vector k = ( k x , k y ): E = 12 ( J cos k x + J cos k y + J cos(2 k y )) , (6) classical LSUB4Sublat. mag. ( ) in m m B Spin-spincorrelationS S < >
LSUB6LSUB8 m quantum Pitch angle ( / ) g p
J J y - = y )R (0, ) = y FIG. 5. (Color online) Left panel: the pitch angle ( γ ) andthe sublattice magnetization ( m ) calculated by the CCM for J = | J | . The shaded bar shows the coupling regime ofCu GeO . Note that the quantum pitch angles γ qu for theLSUB n approximations with n = 4 ,
6, and 8 almost coincide.Therefore, the shown LSUB8 curve represents effectively thelimit n → ∞ . Right panel: spin-spin correlation h S S R i , R = ( x, y ) along the J − J chains ( b ′ -axis) for a finite systemof N = 32 = 4 × J = | J | and J = | J | . where the unit cell of the spin lattice is used. The en-ergy minimum is found at k = ( π, arccos( − J J )) thatcorresponds to the AFM order along a ′ and spiral orderalong b ′ . The classical pitch angle is γ = arccos( − J J ) =79 . ◦ and does not depend on J , hence the a ′ and b ′ directions of the spin lattice are fully decoupled. The or-dering along a ′ is controlled by J , whereas the orderingalong b ′ is controlled by the competing couplings J and J . To check the validity of this result for the quantumcase, we use the CCM method and Lanczos diagonaliza-tion.In CCM, we reduce our exchange couplings (Table I) to J = − J = , and vary J (in Cu GeO , J = ). TheCCM results for γ and m as a function of J are shownin Fig. 5. In contrast to the classical pitch angle γ cl , thepitch angle of the quantum system ( γ qu ) slightly dependson J . In Cu GeO , we find γ qu = 83 . ◦ , which is about6% larger than the classical angle, but about 5% smallerthan the quantum pitch angle for the isolated chain, i.e.,at J = 0. The coupling J affects the dimensionality ofthe system and has a stronger effect on the sublatticemagnetization (see Fig. 5). The extrapolated value m ∞ has a maximum at J ≃ − . J . The calculated ex-change couplings in Cu GeO lead to m ∞ ∼ .
310 (i.e.,0.62 µ B ).The CCM results are confirmed by the Lanczos diag-onalization data for the spin-spin correlation functions h S S R i shown in the right panel of Fig. 5. We use a fi-nite lattice comprising four 8-spin J − J chains coupledby J . The correlations between nearest neighbors withinthe frustrated J − J chains [ R = (0 , R = (0 , ◦ (neighboring spins are nearly orthogo-nal). The correlations between the structural chains [at R = (1 , y )] follow the intra-chain correlations, yet show-ing the opposite sign. Thus, the ordering along a ′ isAFM. D. Long-range order
The 2D model given by Eq. (1) is ordered at zero tem-perature only. To account for the actual long-range orderin Cu GeO below T N = 33 K, the interlayer coupling J c should be considered. The FM coupling J c is compat-ible with J , yet competing with J and J . Assuming asimilar ground state with the 2D propagation vector, wefind that J c modifies the energy in Eq. (6) by∆ E = J c k x k y . (7)Using DFT estimates of individual exchange couplings(Table I), we arrive at the classical pitch angle modifiedby 0.15 %: 79 . ◦ vs. 79 . ◦ for the purely 2D model.The classical energy per lattice site is reduced by 0.5 %(about 0.6 K). This simplified analysis shows that theinterlayer coupling J c is capable of stabilizing the 3Dorder in Cu GeO . However, the classical model doesnot reflect all the features of the real quantum modelthat, unfortunately, remains unfeasible for an accuratenumerical study. In particular, the ordering temperature T N can not be determined with sufficient accuracy. IV. DISCUSSION AND SUMMARY
Although not obvious at first glance, the microscopicmagnetic model of Cu GeO can be deduced from sim-ple qualitative arguments. While the naive geometricalanalysis of the crystal structure suggests a 3D pyrochlore-lattice magnetism, a closer look at the crystal structureidentifies 2D features. In most of the Cu +2 oxides, elec-tronic structure and magnetism are controlled by the ar-rangement of CuO plaquettes, which are the basic struc-tural entities. Chains of edge-sharding plaquettes giverise to frustrated J − J spin chains, yet the coplanararrangement of the plaquettes in the neighboring struc-tural chains induces a strong AFM coupling along a ′ .Overall, we find magnetic layers in the ab plane, alongwith a weak FM interlayer coupling J c . By combiningthe frustration along b ′ with the strong unfrustrated ex-change along a ′ , Cu GeO expands the family of cupratesfeaturing frustrated spin chains.The dearth of the experimental data and the complex-ity of the 2D frustrated J − J − J lattice restrict theopportunities for an experimental verification of our mi-croscopic model. Nevertheless, the tangible success ofDFT in unraveling complex spin lattices for a range oftransition-metal compounds is a solid justifica-tion of our results. The 2D nature of the system and thefrustrated couplings along b ′ are confirmed by qualita-tive arguments and by a reference to similar Cu +2 com-pounds (see Sec. II). Further on, numerical estimates of individual exchanges conform to the experimental mag-netic susceptibility (Sec. III A). An ultimate test of theproposed model requires a study of the ground state byneutron or resonant x-ray scattering. Presently, we notethat our model does predict the long-range magnetic or-der, in contrast to the strongly anisotropic pyrochlorelattice that might have a quantum-disordered valence-bond-solid or a gapless spin-liquid ground state.The experimental data for Cu GeO and accurate the-oretical results for the ground state disclose the basicfeatures of our model. At high temperatures, thermody-namic properties are guided by the uniform spin chainsalong a ′ . The apparent spin-chain behavior is likely re-lated to the partial cancellation of FM J and AFM J inthe second-order term for the susceptibility (see Eq. (2)).A further evidence is the perfect fit of the experimentalmagnetic susceptibility down to 70 K ( T /J ≃ . b ′ come into play at lower tem-peratures, and essentially determine the ground state.The leading exchange J drives AFM ordering along a ′ .The ordering along b ′ has to satisfy the frustrated cou-plings J and J and is, therefore, a spiral, similar to asingle J − J frustrated spin chain. The interlayer cou-pling J c should stabilize the long-range order up to T N without changing the basic features of the ground state:the collinear AFM order along a ′ and the spiral orderalong b ′ .The highly accurate CCM approach provides reliableinformation on the ground state of the 2D system. Wefind the pitch angle of γ ≃ ◦ and the sublattice magne-tization close to 0.62 µ B . Both γ and m are renormalizedwith respect to the classical values, and suggest strongquantum fluctuations in the system. Enhanced quantumfluctuations should be ascribed to the reduced dimen-sionality and frustration. The magnetic ordering temper-ature T N /J ≃ .
25 is also suggestive of strong quantumfluctuations. For example, a quasi-2D system of squarelattices with a weak interlayer coupling J ⊥ /J = 0 . | J c | /J ≃ .
01) orders at a higher temper-ature of T N /J ≃ .
33 (Ref. 54). The quantum effects inthe system could be further probed by an experimentalstudy of the ground state.An interesting feature of the spiral magnetic order isthe possible emergence of electric polarization stronglycoupled to the magnetism. The direction of the electricpolarization depends on the twisting direction of the spi-ral. In Cu GeO , the AFM coupling J leads to oppositetwisting directions in the neighboring spirals, hence thepolarization is canceled. However, the proposed antifer-roelectricity of Cu GeO does not preclude the strongmagnetoelectric coupling, and should stimulate furtherexperimental investigation of the compound. We alsonote that the family of 2D frustrated materials repre-senting the J − J − J model can be further expandedby CuNCN lying in the limit of J ≫ | J | , J . In summary, we have shown that the electronic struc-ture of Cu GeO contradicts the previous, empirical-based spin model of the anisotropic pyrochlore lattice.The comprehensive computational study discloses thequasi-2D nature of this compound and suggests an orig-inal 2D spin model comprising frustrated and uniformspin chains along the two dimensions. Theoretical resultsfor this model show the robust nature of the spiral groundstate that is subject to strong quantum effects, evidencedby the reduced sublattice magnetization of 0.62 µ B andthe renormalized pitch angle of about 84 ◦ . ACKNOWLEDGMENTS
We are grateful to Oleg Janson and Deepa Kasi-nathan for fruitful discussions and careful reading of themanuscript. A. T. acknowledges financial support fromAlexander von Humboldt Foundation. J. R. appreciatesthe funding by DFG (project RI 615/16-1). ∗ [email protected] † [email protected] H. T. Diep, ed.,
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