Competing magnetic interactions in spin-1/2 square lattice: hidden order in Sr_2VO_4
Bongjae Kim, Sergii Khmelevskyi, Peter Mohn, Cesare Franchini
CCompeting magnetic interactions in spin-1/2 square lattice: hidden order in Sr VO Bongjae Kim , Sergii Khmelevskyi , , Peter Mohn , and Cesare Franchini University of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna A-1090, Austria and Center for Computational Materials Science, Institute for Applied Physics,Vienna University of Technology, Wiedner Hauptstrasse - , Vienna, Austria (Dated: August 24, 2017)With decreasing temperature Sr VO undergoes two structural phase transitions, tetragonal-to-orthorhombic-to-tetragonal, without long-range magnetic order. Recent experiments suggest,that only at very low temperature Sr VO might enter some, yet unknown, phase with long-rangemagnetic order, but without orthorhombic distortion. By combining relativistic density functionaltheory with an extended spin-1/2 compass-Heisenberg model we find an antiferromagnetic single-stripe ground state with highly competing exchange interactions, involving a non negligible inter-layer coupling, which places the system at the crossover between between the XY and Heisenbergpicture. Most strikingly, we find a strong two-site "spin-compass" exchange anisotropy which isrelieved by the orthorhombic distortion induced by the spin stripe order. Based on these resultswe discuss the origin of the hidden order phase and the possible formation of a spin-liquid at lowtemperatures. The Heisenberg model on a square lattice is one of themost widely studied models in statistical physics. Theapplicability of this model in modern solid-state physicshas become popular after the discovery of layered high- T c superconductors, since the parent magnetically orderedcompounds are often considered as quasi two-dimensional(q2D) systems [1], and, more recently, for the interest inthe interplay between magnetism and superconductivityin Fe-based superconductors [2]. In the 2D Heisenbergmodel the relative strength and competition of antiferro-magnetic nearest and next nearest neighbor interactions( J and J ) provides useful insights on the stability ofspecific types of magnetic order: J favors the Néel order(e.g., cuprates [3]), J favors the stripe (ST) order (e.g.few types of vanadates [4–7]), whereas a spin-liquid stateis expected to emerge near the classical phase-boundarybetween the Néel and the ST orderings [8–10]. The in-clusion of additional interactions ( J , J , . . . ) leads tomore exotic states [11], for example the enigmatic ne-matic phases recently found in Fe-based superconductorswhich arises from the strong competition between the STand Néel order [12–15].Tetragonal Sr VO , isostructural to the high- T c par-ent compound La CuO and similar to layered vanadateLi VOSiO [4], provides an opportunity to explore therole of the various types of magnetic interaction at playin a square lattice, owing to the presumably weak inter-plane interaction between spin-1/2 V ( d ) layers. Thecomplicated structural, magnetic, and electronic transi-tions observed in Sr VO , in fact, suggests a competitionand/or coexistence of different magnetic phases [16–24].Upon cooling, the crystal structure evolves from tetrago-nal to an intermediate phase (at T c ∼ K ), and againto a tetragonal phase ( T c ∼ K ) with a larger c/a ratio [16]. In the intermediate regime, an anomaly inthe susceptibility is observed at T M ∼ K , which wasinitially thought to originate from magnetic order [16]. No signals of spin-orbital order was detected down tolow temperature but, rather, a possible existence of a ne-matic phase, originating from the competition betweenmagnetic interaction and Jahn-Teller (JT) effect [23].Several proposals have been put forwarded for the elu-sive low-T ground state: orbitally ordered phase due toJT distortion [16], a competition between an orbitallyordered parquet-type and a double-stripe (DS) magneticordering [17, 18] (DS10 in Fig. 1(a)), an octupolar orderdriven by spin-orbit coupling (SOC) [19], and a Néel or-der with muted magnetic moment, where the spin andorbital moments cancel each other [21].The magnetic properties are highly dependent on thesample quality [20]. Recent experiments performed withhigh quality samples have shed some light on the physicsof Sr VO . X-ray diffraction (XRD) experiments has re-vealed that the intermediate structure ( T c < T < T c )is orthorhombic phase [22, 23]. Based on a muon spinrotation and relaxation ( µ + SR) study Sugiyama et. al found that the actual magnetic ordering temperature( T N ∼ K ) is much lower than the temperature ofthe susceptibility anomaly ( T M ∼ K ) and claimedthat the role of the SOC is marginal [24]. This is con-sistent with the reported persistence of inhomogeneousmagnetic states down to 30 mK, where the sizeable com-petition of ferromagnetic and antiferromagnetic correla-tions prevents the system to develop a long range orderedphase [22]. The current understanding of the magneticand structural properties of Sr VO is thus still debatedand a commonly agreed explanation is still lacking.In this Letter we investigate the peculiar magnetic be-havior of Sr VO by first-principles calculations includ-ing relativistic effects. We show that Sr VO can be con-sidered as a frustrated spin-1/2 Heisenberg system withhighly competing exchange interactions. Our data indi-cate that the onset of the magnetic order is controlled bytwo factors with similar order of magnitude: the relativis- a r X i v : . [ c ond - m a t . m t r l - s c i ] A ug (a) ST DS10 DS11 FMSD Neel Parquet−10 0102030 2 3 4 5 6 7 E ( m e V / V ) U (eV) (b) STDS10DS11FMSDNeelParquet J J J J R FIG. 1. (a) Sketch of the different magnetic orderings usedto fit the Heisenberg spin Hamiltonian. Single stripe (ST),double stripe along [100] and [110] (DS10 and DS11), fer-romagnetic (FM), staggered dimer (SD), Néel, and Parquetorder. (b) Total energy as a function of U within DFT+ U calculations. Inset: schematic description of the exchange in-teractions. tic two-site exchange anisotropy and a small, but non-vanishing, inter-plane coupling. In particular, we arguethat the absence of a static magnetic order down to low- T and the origin of the tetragonal-orthorhombic transition,interpreted earlier either as nematic transition or orbitalorder, can be understood as a competition of anisotropic"spin-compass" exchange interactions, exchange magne-tostriction and antiferromagnetic interlayer coupling.To study the magnetic interactions in Sr VO we haveperformed first principles calculations within the densityfunctional theory (DFT) plus an on-site Hubbard U usingthe Vienna Ab Initio Simulation Package [25, 26]. The experimental high- T tetragonal structure [27] was mod-eled with a supercell containing 16 unit cells, with whichwe have simulated different types of spin orderings: sin-gle stripe (ST), double stripe along [100] and [110] (DS10and DS11, respectively), ferromagnetic (FM), staggereddimer (SD), Néel order, and parquet order [see Fig. 1(a)].For the calculation of the exchange parameters we adoptthe value U =5 eV, which is consistent with the valueobtained fully ab initio within the constrained randomphase approximation (cRPA). Further methodologicaldetails are available in the Supplemental Materials [28].Fig 1(b) shows the DFT+U total energies for the con-sidered spin orderings as a function of U . The data for U < U DFTfinds a metallic ground state, in disagreement with ex-periment [29, 30]. We find that the ground state is theST configuration irrespective of the value of U . This issurprising because in earlier literature the ST phase hasnever been considered as a possible ground state mag-netic structure, even though other layered oxides suchas Li VOSiO exhibit the ST order [4]. Our data showsthat the relative energies does not depend strongly on U , and with increasing U the relative-energy window be-comes narrower indicating that the strength of the mag-netic interactions are progressively weakened. This canbe expected since the Js are inversely proportional to U , J ij ∼ t ij /U , where t ij is the hopping between two sites, i and j . The shrinking of the energy window clearly im-plies a strong competition among the various exchangeinteractions in action between neighboring spins (see in-set in Fig. 1).Independently on the magnetic state and on the U -value DFT deliver a spin moment on V of about 1 µ B /V,which suggests a spin-1/2 local moment state, in agree-ment with the experimental Curie-Weiss behavior of thehigh- T magnetic susceptibility which provides an effec-tive moment close to 1.36 µ B [24]. However, Sr VO doesnot show any magnetic order down to very low- T [20].The driving mechanism that stabilizes a magnetic or-der at finite temperatures is one of the most importantand subtle question related to q2D magnetic compounds,as stated in the famous Mermin-Wagner theorem [31].To analyze the physical reasons for the absence of mag-netic order in Sr VO we start by mapping the differ-ent spin-ordered DFT total energies onto the followingHeisenberg-like Hamiltonian, H H = J (cid:88) α, (cid:104) ij (cid:105) S iα · S jα + J (cid:88) α, (cid:104)(cid:104) ij (cid:105)(cid:105) S iα · S jα + J (cid:88) α, (cid:104)(cid:104)(cid:104) ij (cid:105)(cid:105)(cid:105) S iα · S jα + J (cid:88) α, (cid:104)(cid:104)(cid:104)(cid:104) ij (cid:105)(cid:105)(cid:105)(cid:105) S iα · S jα + J ⊥ (cid:88) α, (cid:104) ij (cid:105) S iα · S jα +1 + R (cid:88) α, plaquette [( S iα · S jα )( S kα · S lα ) + ( S iα · S l )( S kα · S jα ) + ( S iα · S kα )( S jα · S lα )] , (1)where (cid:104)(cid:105) , (cid:104)(cid:104)(cid:105)(cid:105) , (cid:104)(cid:104)(cid:104)(cid:105)(cid:105)(cid:105) , and (cid:104)(cid:104)(cid:104)(cid:104)(cid:105)(cid:105)(cid:105)(cid:105) represent the sum over first, second, third and fourth nearest neighbors (NNs), R is the multisite ring exchange summed within a squareplaquette as shown in the inset in Fig. 1(b), and theindex α denotes magnetic layers. A similar procedurehas been successfully applied for other 2D square latticessuch as the Fe-based superconductors [15], Sr RuO [32],and BaTi Sb O [33].The estimated exchange interactions are listed in Ta-ble I. The stability of the ST configuration can be ex-plained by the large AFM exchange coupling J =1.02meV, which is the dominating interaction [34], abouttwice larger than the first NN FM-interaction J (-0.65meV). However, the large values of J and J (-0.54 and0.46, respectively) suggest that the system deviates con-siderably from the J − J model [35]. By extrapolatingto large J the quantum Monte Carlo results obtainedfor the J − J − J model [36], one would expect thatour data ( J /J = − . , J /J = 0 . ) should fall inthe spin-liquid/spin-glass part of the quantum spin-1/2Heisenberg J − J − J phase diagram.Apart from the NNs Js, there are other two importantterms in our spin Hamiltonian, the multisite ring inter-action R and the interlayer coupling J ⊥ . Despite beingsmall, the ring coupling R is essential to explain the en-ergy difference between DS11 and the Parquet configura-tions (see Tab.I), because R produces the same contribu-tion to the energy for all orderings but DS11. It would betherefore interesting to try to interpret the absence of astatic classical magnetic order in Sr VO with a 2D quan-tum J − J − J − J − R Heisenberg model. Notably,the, ferromagnetic, interlayer coupling J ⊥ is not negli-gible, -0.12 meV, and brings the system on the 2D/3Dthreshold. J ⊥ favors the FM configuration and to a lesserextent DS10, DS11 and Parquet configurations leavingthe energy of ST and Néel order unchanged.It has been argued that even small value of J ⊥ canstabilize the magnetic order with a rather high transi-tion temperature [9]. However, due to the peculiarityof the body-centered tetragonal structure, where eachspin-site is connected to four spin-sites in the adjacentlayer, it might happen that for a given in-plane orderthe inter-plane interaction is fully frustrated. Examplesof such frustration are ordinary in-plane Néel order (e.g.cuprates) or single stripe order (e.g. Fe-based pnictides).In Fe-based superconductors it has been suggested thatthe critical role for the ordering might be played by atwo-site magnetic anisotropy [37] (note that for spin 1/2systems the single site anisotropy vanishes), which leadsto the crossover between 2D Heisenberg and either the2D Ising or the XY-Kosterlitz-Thouless model behavior(depending on sign of the anisotropy) [1]. For instance,recently it has been shown that the 3D magnetic orderin the novel q2D compound Sr TcO originates from atiny dipole-dipole coupling that leads to an out-of-planeanisotropy associated with a two-site exchange [38]. Con-sidering that our data on Sr VO shows that J ⊥ leavesthe ST order frustrated (i.e. unchanged) in the following part of the paper we will consider the possibility that thestabilization of a magnetic order at finite temperaturecould originates from anisotropic effects. Specifically, wewill analyze the anisotropy of the exchange interactionsdue to SOC.To this aim, we have computed the total energy ofthe ST phase with the moments oriented along the high-symmetry directions [001], [010] and [001] by means ofmagnetically constrained DFT+U+SOC calculations [39](see Fig.2. We found that E x is 2.75 meV and 2.63 meV(per V ion) lower than E y and E z , respectively, implyingstrong anisotropic effects. As these differences cannot beexplained by single-site anisotropy (since we have a goodspin-1/2 system) they must be attributed to symmetricanisotropy exchange, which can be taken into accountby including additional terms in the spin Hamiltonian(Eq. 1). Within a first NN approximation the resultingspin Hamiltonian reads: H rH = H H + (cid:88)
FIG. 2. (a) Schematic diagram showing the transition tem-peratures and structures. The system has a tetragonal struc-ture with active anisotropic exchange interactions for
T > T c ,while for the orthorhombic structure for T c > T > T c theanisotropy vanishes. Tetragonal symmetry is recovered below T c . Stripe order is shown with moments along (b) (100) and(c) (010), respectively. The squares denotes unit cell used. (ii) This short-range ST order causes a tetragonal to or-thorhombic structural transition and simultaneously killsthe spin-compass anisotropy; the system avoids a mag-netic transition to the ordered ST phase since the lat-ter remains fully frustrated. (iii)
When the tempera-ture is lowered further, the system re-enters the tetrago-nal phase, since other phases, like DS10, supported alsoby a non-frustrating inter-plane FM coupling, becomeenergetically equally favorable and partially destroy theshort-range ST order. (iv)
At this point the system en-ters an intermediate temperature range with different en-ergetically competing magnetic orders, in line with thecomplex behavior of the measured magnetic susceptibil-ity; (v)