Electronic structure and magnetic properties of the spin-1/2 Heisenberg system CuSe2O5

A microscopic magnetic model for the spin-1/2 Heisenberg chain compound CuSe2O5 is developed based on the results of a joint experimental and theoretical study. Magnetic susceptibility and specific heat data give evidence for quasi-1D magnetism with leading antiferromagnetic (AFM) couplings and an AFM ordering temperature of 17 K. For microscopic insight, full-potential DFT calculations within the local density approximation (LDA) were performed. Using the resulting band structure, a consistent set of transfer integrals for an effective one-band tight-binding model was obtained. Electronic correlations were treated on a mean-field level starting from LDA (LSDA+U method) and on a model level (Hubbard model). In excellent agreement of experiment and theory, we find that only two couplings in CuSe2O5 are relevant: the nearest-neighbour intra-chain interaction of 165 K and a non-frustrated inter-chain coupling of 20 K. From a comparison with structurally related systems (Sr2Cu(PO4)2, Bi2CuO4), general implications for a magnetic ordering in presence of inter-chain frustration are made.


Introduction
Low-dimensional spin-1/2 systems attract much interest due to a variety of ground states (GS) found in these systems which originates from an interplay between different exchange interactions and strong quantum fluctuations. There are, for instance, the spin-Peierls GS in CuGeO 3 [1], the helical GS in LiCuVO 4 [2], and the quantum critical behaviour in Li 2 ZrCuO 4 [3,4] etc. Besides, many of these materials (mostly cuprates, vanadates and titanates) appeared to be realizations of theoretically longstudied models in good approximation. One of the most prominent models is the spin-1/2 nearest-neighbour (NN) chain described by the Heisenberg Hamiltonian, for which the exact solution has been derived by Bethe [5]. The first compounds proposed to be good material realizations of this model were Sr 2 CuO 3 and Ca 2 CuO 3 [6,7]. Recently, Sr 2 Cu(PO 4 ) 2 and Ba 2 Cu(PO 4 ) 2 were suggested as even better realizations [8]- [12], followed by a study of K 2 CuP 2 O 7 [13] that qualified this compound to be the best realization of the spin-1/2 NN Heisenberg chain up to date. As a natural consequence of its simplicity, this model poorly describes one-dimensional and quasi-one-dimensional systems where additional interactions, like longer range couplings or anisotropies, are present. Thus, extensions of this model are required to allow an accurate description of real materials.
The simplest extension of the model is the inclusion of a next-nearest-neighbour (NNN) coupling J 2 leading to the so-called zig-zag chain model. In case of an antiferromagnetic (AFM) J 2 both NN and NNN couplings cannot be simultaneously satisfied, in other words, the system is magnetically frustrated. Here, the intra-chain frustration enriches the phase diagram with the spiral GS, the gapped AFM GS and a quantum critical point at J 2 /J 1 = −0.25 [14]- [17]. The evaluation of the two parameters in the zig-zag chain model allows to estimate quantities which can be directly measured or derived from experiments, namely spin-spin correlation functions, thermodynamic properties and the response in high magnetic fields. Nevertheless, this model fails to describe phenomena like long-range magnetic ordering, since one-or two-dimensional systems do not order at finite temperatures according to the Mermin-Wagner theorem. Thus, to account for magnetic ordering, the inter-chain (IC) coupling has to be included in the model. This problem has been addressed in a series of theoretical works [18]- [25], but the simplifications that had to be made to keep the models solvable (at least approximately) impede an accurate description of complex situations. The spin-1/2 Heisenberg chain system Sr 2 Cu(PO 4 ) 2 is the most prominent example for a huge discrepancy (two orders of magnitude) between the theoretical (adopting a simple NN inter-chain coupling) and the experimentally observed ordering temperature. The origin of this discrepancy is hidden in the effects of magnetic frustration and anisotropy, and the disentanglement of the effects is difficult. Thus, a reliable theoretical description of this phenomenon is still missing.
A natural way towards a deeper understanding is the search for real material realizations of "easy" models. For such systems, the experimental data can be supplemented by reliable microscopic models. This way, joint experimental and theoretical studies can challenge and improve the existing theoretical approaches.
Cu 2+ phosphates are up to date the best realizations of the NN spin-1/2 Heisenberg chain model [8,10,13] and could have a great potential for the discovery of further low-dimensional systems. Unfortunately, the experimental information about these materials is rather limited since they are up to now available as powders only, although several attempts have been made to grow single crystals required for advanced experimental studies.
In this paper, the structurally closely related compound, CuSe 2 O 5 , is investigated. Since selenites are often susceptible to chemical transport, the advantage of this material is the potential to grow large single crystals of high quality. Previous studies on a powder sample [26] may hint a 1D character of its magnetic properties, but the low-temperature data are strongly affected by impurities (figure 3 in [26]). Therefore, to probe the 1D nature of the system, a new detailed study on high quality samples with lower defect concentration is desirable.
Though the chain-like arrangement of CuO 4 squares in CuSe 2 O 5 is topologically similar to that in Cu 2+ phosphates, the geometry of magnetic coupling paths between the structural chains is essentially different. Thus, the role of magnetic frustration, which is ruled by the inter-chain coupling, can be evaluated in a comparative study.
The paper is organized as follows. In section 2 we describe the synthesis, sample characterization and experimental as well as theoretical methods used in this work. In section 3, we discuss the crystal structure of CuSe 2 O 5 in comparison to related systems. Section 4 reports the results of our measurements and theoretical calculations and proposes an appropriate microscopic model. A brief summary and an outlook are given in section 5.

Method and sample characterization
Single crystals of CuSe 2 O 5 were grown by chemical vapour transport using TeCl 4 as a transport agent. Using a micro-crystalline powder of CuSe 2 O 5 (obtained from a mixture of CuO and SeO 2 at 723 K) as a source, the transport experiments were carried out in an endothermic reaction of T 2 (source) 653 K to T 1 (sink) 553 K.
The obtained crystals have a green colour and form strongly elongated (along [001]) plates, which macroscopically look like needles. The typical length of a needle is 5-10 mm and the width does not exceed 1 mm and for most crystallites it is considerably smaller. The slight disorientation of plates forming a needle intricates a precise X-ray diffraction measurement on single crystals. Thus, the samples were characterized by Xray powder diffraction and energy-dispersive X-ray spectroscopy (EDXS) experiments. The lattice parameters of the synthesized crystals are similar to those reported for Magnetization was measured in a SQUID magnetometer (1.8 -350 K) in magnetic fields up to 1 T. Heat capacity (1.8 -100 K) was determined by a relaxation method up to µ 0 H = 9 T.
DFT calculations were carried out using the full potential local orbital code (FPLO) version 7.00-27 [27]. The standard basis set and the Perdew-Wang parameterization of the exchange-correlation potential were used [28]. Strong on-site Coulomb interaction in the Cu 3d orbitals, insufficiently described in the LDA, was taken into account independently (i) by mapping the LDA antibonding Cu-O dp σ bands onto a tightbinding model (Ĥ = i ǫ i + <ij>σ t ij (c † i,σ c j,σ + H.c.)) and subsequently via a Hubbard model (Ĥ = i ǫ i + <ij>σ t ij (ĉ i,σĉ † j,σ + H.c.) + U eff ini , ↑ni , ↓ ) onto a Heisenberg model (Ĥ = <ij> J ijˆ S iˆ S j ) (the procedure is well justified for spin excitations in the strongly correlated limit (U eff ≫ t ij ) at half-filling ( n i = 1)) and (ii) by using the LSDA+U method [29] (U d = 6.5 eV, J d = 1 eV). For the LDA calculations, we used a k-mesh of 1296 k-points (355 points in the irreducible wedge), for LSDA+U calculations of supercells irreducible k-meshes of 226, 242, 147 and 126 k-points were used. All kmeshes are well converged.
Quantum Monte-Carlo simulations have been preformed on N = 1200 sites clusters of S = 1/2 spins (30 coupled chains of 40 sites each) using the ALPS software package [30].

Crystal structure and empirical magnetic models
Crystal structures of cuprates are often subdivided into four large groups according to their dimensionality, which reflects how their elementary building blocks -CuO 4 plaquettes (planar or distorted) -are connected: they can be isolated (zerodimensional, 0D) or form chains (one-dimensional, 1D), layers (two-dimensional, 2D) or frameworks (three-dimensional, 3D). Although it is true for many systems that the magnetic dimensionality follows the dimensionality of the crystal structure, the real situations are often more complex, especially for 0D cases. There, the magnetic dimensionality is ruled by (i) the orientation of neighbouring plaquettes, and (ii) the position of anion groups formed by non-magnetic atoms that bridge the magnetic plaquettes. In most cases, the connection between structural peculiarities and the appropriate magnetic model cannot be accounted for by applying simple empirical rules (for instance, Goodenough-Kanamori-Anderson rules).
Therefore, an almost complete understanding of the macroscopic magnetic behaviour for a certain system of this class can be achieved only basing on a relevant microscopic model. The latter can be constructed either by using advanced experimental techniques (for instance, inelastic neutron scattering) or theoretical (DFT) calculations. Naturally, the most reliable approach is the combination of such a theory and experiment. Due to the complexity of such an analysis, it has been accomplished only for a rather limited number of real systems.
Two well studied systems of this class -Bi 2 CuO 4 [31] and Sr 2 Cu(PO 4 ) 2 [8] -are both structurally 0D cuprates, but antipodes with respect to their magnetic behaviour. A drastic change of the magnetic coupling regime originates from the arrangement of neighbouring plaquettes (figure 1, right panel: top and bottom): stacking (accompanied by additional twisting) of neighbouring plaquettes on top of each other makes Bi 2 CuO 4 a 3D magnet with T N ≈ 47 K [32] while in Sr 2 Cu(PO 4 ) 2 the plaquettes are arranged in a planar fashion (formally reminiscent of an edge-sharing chain with every second plaquette cut out), leading to a pronounced 1D behaviour and a very low Néel temperature T N = 0.085 K [9]. In this context we mention that the magnetic dimensionality of these systems is controlled by the dihedral angle φ between neighbouring plaquettes ( figure 1, right figure 1). This structural peculiarity is reflected in the morphology of the synthesized crystals (section 2): the needle-like shape with an elongation along [001] fits perfectly to structural chains along c (figure 1) formed by alternation of CuO 4 plaquettes and Se 2 O 5 polyanion groups. From the topological similarity of this structural chain to the one in the structure of Sr 2 Cu(PO 4 ) 2 (there, neighbouring plaquettes are bridged by two PO 4 tetrahedra) a 1D behaviour of CuSe 2 O 5 might be expected. A second argument supports this proposition: in CuSe 2 O 5 , the structural chains are not connected by covalent bonds, making a strong inter-chain coupling unlikely, similar to Sr 2 Cu(PO 4 ) 2 , where the neighbouring chains are well separated by Sr cations. These similarities of CuSe 2 O 5 and Sr 2 Cu(PO 4 ) 2 may lead to the conclusion that both systems imply essentially the same physics. However, a closer inspection of more subtle crystal chemical aspects immediately reveals an important difference related to the inter-chain coupling. As it follows from the microscopic model [8], in Sr 2 Cu(PO 4 ) 2 there are two relevant NN inter-chain couplings (2.7 K), which are equivalent by symmetry. Together with an intrachain NN coupling (187 K), they induce magnetic frustration which commonly leads to a considerable decrease of the ordering temperature (T N =0.085 K). In CuSe 2 O 5 , these two inter-chain couplings are not symmetry-equivalent. Therefore, by reducing the strength of one of them, the frustration can be lifted.
So far, crystal chemical considerations provided us with a qualitative insight. For a quantitative model, a microscopic analysis is required. Thus, in the next section a microscopic model basing on the results of DFT calculations is constructed. A prerequisite for an accurate modeling based on a band structure code is reliable structural information. For CuSe 2 O 5 , two refinements of the same structural model (space group C2/c with four formula units per cell) have been proposed so far [33,34]. Both structural data sets agree quite well with each other and with the lattice parameters of the synthesized samples (see table 1). ‡ The reliability of the structural data has been indirectly confirmed a posteriori by the good agreement of calculated and experimentally measured quantities.

Thermodynamical measurements
The first probe for the magnetic properties of a certain system is the measurement of magnetization (M) at various temperatures in a constant field (H) yielding ‡ It is well known, that X-ray diffraction analyses may result in considerable inaccuracies for internal coordinates of light elements (especially, hydrogen). These inaccuracies can have a large impact on the magnetic properties [35]. Since there are no light atoms in CuSe 2 O 5 , we rely on the diffraction analysis and therefore, no structural relaxation has been performed. the temperature dependence of magnetic susceptibility (χ(T ) = M(T )/H). This measurement already yields valuable information on the magnetic dimensionality, the sign and the energy scale of leading couplings, the presence of a spin gap, the spin anisotropy and the quality (defects, purity) of a sample. For CuSe 2 O 5 , the magnetic susceptibility curves for both field orientations (figure 2, left panel) have a broad maximum at T max ≈ 101 K and a finite value of χ at the lowest temperature measured (1.8 K), indicating the low-dimensional behaviour and the absence of a spin gap. The high-temperature parts of the curves obey the Curie-Weiss law χ(T ) = C/(T +θ) (figure 2, right panel; T > 220 K, H : θ = 165 K, C = 0.51, g = 2.32; H ⊥ : θ = 170 K, C = 0.43, g = 2.15). The positive Curie-Weiss temperature evidences that the dominating couplings in CuSe 2 O 5 are antiferromagnetic. The shape of the experimental curve reveals a close similarity to a spin-1/2 Heisenberg chain model. This model has an exact solution given by Bethe ansatz [5] and parametrized by Johnston et al. [36]. We have fitted the experimental curves using the parametrized solution ( figure 2, left panel).
To account for the deviation of the fitted curves from experimental ones we have fitted both curves independently § and varied the temperature window. As a result, the magnetic susceptibility measured perpendicular to needle-like crystallites can be perfectly fitted by a consistent set of parameters (J = 157.6 K, g = 2.00) in the whole temperature range down to the ordering temperature, while the fit to the susceptibility measured parallel to the chains (J = 157.1 K, g = 2.14) shows deviations at low temperatures (below T max ). This difference likely originates from a slight misalignment of microscopic plates in the needle-like crystallites.
A phase transition is observed at 17 K for both orientations of the magnetizing field. The nature of this magnetic transition can be understood by examination of the low-temperature part of the curve (below the transition). The interpretation is straightforward as soon as we account for (i) impurity effects and (ii) effects of misalignment of the sample (relevant especially for H , as shown above). Due to the high quality of samples, the temperature region between the kink at 17 K down to 10 K is practically unaffected by defects (no Curie tail). In this range, χ ⊥ decreases very slightly, while χ drops distinctly on cooling, following the theoretical result for ordered collinear antiferromagnets [37]. We assign the slight decrease of χ ⊥ (theory predicts it to be constant) to a small misalignment of crystallites, in agreement with the deviations of the Bethe ansatz fit. The small upturn in χ at about 6 K (a zero susceptibility at zero temperature follows from theory) is likely related to defects and paramagnetic impurities.
To get additional information about the magnetic properties, we measured the temperature dependence of the specific heat. The clear anomaly at 17 K (figure 3) and the linear behavior of C/T 2 (T ) below this temperature are typical for antiferromagnets [38]. Thus, we interpret this as a transition to an AFM ordered state (T N = 17 K). Remarkably, the anomaly does not shift nor decrease in amplitude in magnetic fields up to µ 0 H = 9 T. Prior to the analysis of the magnetic behaviour above T N , the specific heat should be decomposed into the magnetic contribution (which reflects the spectrum of magnetic excitations) and the phonon contribution (the spectrum of lattice vibrations). This decomposition is reliable only if the overlap of the two spectra (magnetic excitations and phonons) is relatively small (see [39] for an example). As the phonon contribution increases on temperature, the decomposition is possible for systems with weak magnetic couplings (J ij < 10 K). As we obtained from our susceptibility data, the energy scale of J in CuSe 2 O 5 is about 165 K. Thus, for a 1D Heisenberg chain we expect the maximum of the magnetic specific heat at 0.48J [36], i.e. § A simultaneous fit of both curves implying the same J value in the whole temperature range down to the phase transition yields considerable deviations from experiment. The origin of this deviation is discussed below in the text.
close to 80 K. At this temperature, the phonon contribution to the specific heat strongly dominates over the magnetic contribution. As a result, the experimental curve has no visible features in the vicinity of 80 K. For systems with large couplings (J ij > 10 K), the most accurate way to account for the phonon part is to measure an isostructural non-magnetic reference system (see [40] for an example). In case of CuSe 2 O 5 , it is not possible, as ZnSe 2 O 5 has a different crystal structure [41], and thus a different phonon spectrum. Therefore, the specific heat data provide a clear evidence of an AFM ordering at 17 K but do not allow an independent justification of the temperature scale for the leading magnetic interactions. The last remark concerns a pronounced kink at 7 K (figure 3, inset), i.e. the region of the ordered phase. The kink is stable at least up to µ 0 H = 9 T and thus not related to defects. Intriguingly, a similar feature has been observed for a related system Bi 2 CuO 4 (figure 3 in [42]) favouring the intrinsic nature of the kink rather than a sample dependent effect. For the magnetic contribution to the specific heat, such features have been proposed to mark a dying out of high frequency spin wave modes [43]. To elucidate this unusual feature, further experimental studies on CuSe 2 O 5 and similar systems as well as a careful theoretical analysis should be carried out.

Microscopic model
We start in our microscopic analysis with band structure calculations performed in the local density approximation (LDA). LDA yields a valence band of about 9 eV width formed mainly by Cu 3d, O 2p and Se 4p states ( figure 4, right panel). The well-separated double-peak at the Fermi level ε F contains two narrow, half-filled bands (figure 4, left panel). The width of this antibonding band complex (0.85 eV) is in between the widths of the same complex in Bi 2 CuO 4 (1.05 eV [31]) and Sr 2 Cu(PO 4 ) 2 (0.65 eV [8]). The LDA yields a metallic GS, contrary to the experimentally observed insulating behavior. This discrepancy is caused by the underestimate of strong on-site Coulomb interactions of the Cu 3d electrons. Nevertheless, LDA reliably yields the relevant orbitals and dispersions. Thus, we have a closer look to the band complex at ε F . The two bands, relevant for the low-lying magnetic excitations, are related to the antibonding dp σ orbital of a CuO 4 plaquette, i.e. the antibonding combination of Cu 3d x 2 −y 2 and O 2p σ states (orbitals are denoted with respect to the local coordinate system). The antibonding dp σ orbital is well separated (∆E ∼ 0.5 eV) from the lower lying Cu 3d and O 2p states. Thus, the most efficient way to describe the electronic structure is to construct an effective one-band tight-binding (TB) model (one band per plaquette), parametrized by a set of electron transfer integrals t ij . The correlation effects, insufficiently described by LDA and thus by the TB model, are accounted for by adopting a corresponding Hubbard model mapped subsequently onto a Heisenberg model (this mapping is valid for spin excitations in the strongly correlated limit at half-filling, both well justified for undoped cuprates with small magnetic exchange).
Prior to numerical calculations, we compare the dispersions of the two well separated bands at ε F (figure 5) to dispersions of the corresponding antibonding dp σ complexes of Sr 2 Cu(PO 4 ) 2 (figure 2 in [8]) and Bi 2 CuO 4 (figure 5 in [31]). Here, a close similarity of CuSe 2 O 5 and Sr 2 Cu(PO 4 ) 2 is revealed: both band structures have a dominating dispersion along the chain direction (for CuSe 2 O 5 , this is the c-axis in figure 1 and Γ-Z region in figure 4) and a weaker dispersion in other directions, unlike Bi 2 CuO 4 , where the dispersions along different directions in the k-space are comparable, indicating a 3D behaviour.
For a quantitative analysis, we constructed an effective one-band TB Hamiltonian and determined the set of transfer integrals t ij in order to get the best least-squares fit to the two LDA bands crossing ε F . As an alternative approach, we used a Wannier functions (WF) technique which implies the construction of WF for the Cu 3d x 2 −y 2 antibonding state, relevant for the magnetism, and the calculation of the overlap of the WF. The results of the latter method are affected by the overlap of the relevant 3d x 2 −y 2 antibonding state with other states. Therefore, for perfectly separated bands as in CuSe 2 O 5 , both methods should yield the same results within numerical accuracy. The numerical evaluation supports this statement: the difference between the transfer integrals obtained by the WF method and by the TB fit is tiny and does not exceed 2 meV for individual t ij values (the mean value for all t ij is 0.2 meV). This deviation can be considered as an error margin for the mapping procedure. Thus, for isolated bands the WF method should not be regarded more accurate than a direct TB fit, but rather as an independent alternative procedure [10,11,12]. The agreement of the results using the two independent mapping methods reflects the applicability of an effective one-band approach.
The resulting set of the transfer integrals (table 2, first column) yields perfect agreement with the LDA bands (figure 5). The hopping paths corresponding to the leading terms are shown in figure 5 (right panel).  We find that the leading couplings in CuSe 2 O 5 are the NN intra-chain coupling t c = 165 meV and one of the short inter-chain couplings t ab ≈ 45 meV. The corresponding To check the results for consistency, we have neglected all t ij smaller than 10 meV and repeated the fitting. The difference of leading terms in both approaches did not exceed 10%.  Wannier functions are pictured in figure 6. The value of the largest (NN intrachain) coupling in CuSe 2 O 5 is slightly larger than the corresponding coupling in Sr 2 Cu(PO 4 ) 2 (135 meV [8]). The difference in the largest inter-chain term is more pronounced: the size of the inter-chain coupling in CuSe 2 O 5 (45 meV) is considerably higher than in Sr 2 Cu(PO 4 ) 2 (16 meV [8]). Even more important is the difference in the specific coupling geometry -whether it is constructive towards the long-range ordering or not. As we stated while comparing the crystal structures of CuSe 2 O 5 and Sr 2 Cu(PO 4 ) 2 (see section 3), in both systems there are two short inter-chain coupling paths. The corresponding couplings are identical (symmetry-related) in Sr 2 Cu(PO 4 ) 2 , but independent (and in fact, considerably different) in CuSe 2 O 5 . The TB analysis reveals that only one (t ab ) of the two NN inter-chain couplings is relevant for CuSe 2 O 5 (table 2). Consequently, the essential difference between the two systems can be best understood in terms of the spin lattices that are formed by the strongest intra-chain and inter-chain couplings, as depicted in figure 7. In Sr 2 Cu(PO 4 ) 2 , three relevant couplings (the intra-chain NN coupling and two identical inter-chain couplings) are arranged on an anisotropic triangular lattice ( figure 7, left panel). By switching off one of the inter-chain couplings, the topology of the relevant couplings changes, and the system is described by two couplings forming an anisotropic square lattice ( figure 7, right panel). The main difference between the two topologies is that in Sr 2 Cu(PO 4 ) 2 the competition of relevant couplings, which can not be simultaneously satisfied, leads to strong magnetic frustration, while in CuSe 2 O 5 the inter-chain couplings are not frustrated. The lifting of frustration in CuSe 2 O 5 has a remarkable influence on the physical properties as will be discussed below.
The calculated transfer integrals provide valuable information on the coupling regime. To include the missing Coulomb interaction U eff , as described in the section 2,  (table 2, second column) and will be neglected in further discussion. Table 2. Leading transfer (first column) and exchange integrals (last column) of CuSe 2 O 5 . The AFM exchange (second column) is calculated via mapping the transfer integrals onto an extended Hubbard (U eff = 4.5 eV) and subsequently onto a Heisenberg model. The total exchange is taken from LSDA+U total energy calculations of supercells. The ferromagnetic exchange J FM ij is evaluated as the difference between J ij and J AFM The calculated leading magnetic exchange J AFM c = 285 K is considerably larger than our estimate from the Bethe ansatz fit (≈155 K) based on experimental χ(T ) data. Moreover, it is larger than the corresponding exchange integral in Sr 2 Cu(PO 4 ) 2 (187 K). This discrepancy originates from ferromagnetic (FM) contributions to the total magnetic exchange, which are neglected in the mapping procedure. For the NN exchange J c , we expect a considerable FM contribution originating from the overlap of O 2p wave functions of neighbouring plaquettes. Due to a dihedral angle φ = 64 • between the neighbouring plaquettes, this overlap has a sizable π contribution (this can be seen in the WF in figure 6, central panel) leading to a Hund's rule (FM) coupling. For the leading inter-chain coupling, the FM contribution is expected to be small due to a predominantly σ overlap of O 2p wave functions ( figure 6, right panel).
To get a numerical estimate for the FM contribution, we perform total energy calculations for various spin patterns of magnetic supercells using the LSDA+U method. The method is rather sensitive to the U d value. As U d = 6.5 eV yields agreement between the calculated and experimentally measured exchange integrals of the well studied La 2 CuO 4 and CuGeO 3 , we adopted this value in the calculations for CuSe 2 O 5 . ¶ The supercell method has limitations set by numerical accuracy for the small exchange integrals (J ij < 1 K) and the size of the required supercells. In our case, we constructed supercells and spin patterns that yield all exchange couplings which were found to be relevant from the TB analysis. The resulting total energies are mapped onto a Heisenberg model, which is parameterized by the total exchange integrals (table 2, last column) containing both AFM and FM contributions. Thus, by subtracting the AFM part J AFM ij from the total exchange J ij , the FM contribution J FM ij can be estimated (table 2, fourth column).
In general, LSDA+U calculations yield a reliable estimate for exchange integrals [8,13,31,39]. This reliability holds for CuSe 2 O 5 : we obtain J c = 165 K and J ab = 20 K, in almost perfect agreement with the estimates from magnetic susceptibility. In accordance with our expectations, J FM c = −120 K has a considerable contribution to the total exchange J c , while J FM ab = −7 K yields a smaller correction to the J ab value. We should note that the large ferromagnetic contribution J FM c = −120 K may originate, in addition to the mentioned π-overlap of O 2p wave functions, also from a destructive interference of coupling paths [44] or by a strong coupling to ligands [45]. Which of these mechanisms plays a leading role in CuSe 2 O 5 is an open question. This issue is, however, beyond the scope of the present paper and needs further theoretical investigation.
Though the calculated J c value (165 K) is very close to the estimate from the Bethe ansatz (157 K), we decided to check the exchange integrals for consistency by performing additional calculations for U d = 6.0 eV and U d = 7.0 eV. Besides the expected change of exchange integrals (0.5 eV increase of the U d results in about 20% decrease of J ij and vice versa), we found that the ratio α ≡ J ab /J c of the leading exchange integrals (α = 0.121 for U d = 6.0 eV, α = 0.129 for U d = 6.5 eV and α = 0.136 for U d = 7.0 eV) is rather stable with respect to the U d value.
Thus, the consideration of the FM contribution yielded a valuable improvement of the energy scale comparing to the AFM exchange values J AFM c and J AFM ab , but the ratio α of the two couplings, that is the most relevant for the magnetic ground state, stays almost unchanged. Moreover, this ratio is stable with respect to the model parameters U eff and U d , leading to a very reliable physical picture: CuSe 2 O 5 can be described as ¶ It is worth to note, that the U d parameter is not universal and depends on a calculational scheme and consequently on the basis set implemented in a code. Thus, different U d values adopted in this work for CuSe 2 O 5 (U d = 6.5 eV, the code fplo version 7.00-27) and for Sr 2 Cu(PO 4 ) 2 (U d = 8.0 eV [8], the code fplo version 5.00-18) originate from the different basis used in the codes. a quasi-1D system with AFM chains characterized by an NN intra-chain exchange of 165 K. Each chain is coupled to two neighbouring chains by the non-frustrated interchain exchange of 20 K (one order of magnitude smaller that the intra-chain coupling).

Simulations
Turning back to the discussion about the ordering temperature (section 1), it is reasonable to point out the advantages of CuSe 2 O 5 as a model system. First, we have evidence from both theory and experiment that the system is mainly 1D. Secondly, the microscopic analysis revealed that the NNN intra-chain coupling is practically absent leading to a valuable simplification for a theoretical analysis. Finally, there is only one relevant inter-chain coupling. The fact that this latter coupling is not frustrated allows to use the powerful quantum Monte Carlo (QMC) method for a simulation of thermodynamical data with a subsequent comparison to the experimentally measured curves. The results of the simulations are given in figure 8 in comparison with the Bethe ansatz fits (where the inter-chain coupling is neglected). Obviously, the inclusion of the inter-chain coupling yields only a tiny improvement with respect to the Bethe ansatz fits. This fact demonstrates a posteriori the importance of a microscopic model for systems like CuSe 2 O 5 : apart from the microscopic modelling, there is no reliable way to account for the small inter-chain coupling directly from measurements of the paramagnetic susceptibility.

Estimation of the Néel temperature
To benefit from the unique combination of the simple microscopic picture and the experimentally well-determined AFM ordering temperature T N , we make an attempt to estimate T N from an available simplified theory using the calculated exchange integrals. Still, there are three problems, intrinsic for quasi-1D systems, to be accounted for. The first is the spatial anisotropy of exchange couplings present in a system. This problem is resolved in CuSe 2 O 5 only partially. On one hand, there is only one relevant interchain coupling for every pair of neighbouring chains, but on the other hand, it couples a certain chain with only two of four neighbouring chains. Thus, the couplings to the other two chains are considerably smaller, resulting in the (spatial) exchange anisotropy. The second problem is the anisotropy in the spin space. In our microscopic approach, we used the isotropic Heisenberg model where this anisotropy is neglected. Although we observe a remarkable agreement between the microscopic model and the macroscopic behaviour, the spin anisotropy is present, as evidenced for instance by the strong dependence of the g-factor on the orientation of a magnetizing field (figure 2). In a common sense approach, the (spatial) exchange anisotropy is expected to lower T N , while the spin anisotropy raises it. In each system, these two effects are balanced. Attempts to find a suitable description for this balance were made in a number of advanced theoretical studies based on a mean-field formalism [18,19], [21]- [24]. Still, the problem seems not to be resolved, since a considerable disagreement remains between the numerical results yielded by different theories (none of which is generally accepted) and, even more important, due to the third problem -the problem of inter-chain magnetic frustration, which has not been addressed, so far.
Here, we make an empirical attempt to estimate how the frustration influences the magnetic ordering. For that purpose, we compare several quasi-1D magnetic compounds in a systematic way. The two well studied quasi-1D cuprates Sr 2 CuO 3 and Ca 2 CuO 3 are commonly referred as model systems in most theoretical studies regarding the T N problem. We should note that these two systems are essentially different from CuSe 2 O 5 due to the presence of corner-sharing chains of CuO 4 plaquettes, which results in one order of magnitude larger NN coupling. In addition, the NNN coupling is not negligible [6]. Nevertheless, they are referred here for the sake of completeness. In both, Sr 2 CuO 3 and Ca 2 CuO 3 , one of the relevant inter-chain couplings is frustrated. To calculate T N , we use formulas given by Schulz [19], Irkhin and Katanin [21] and Yasuda et al. [23]. The calculated T N are given in the three last columns of ( A theoretical approach is expected to work better for systems with more pronounced 1D nature -e.g. Sr 2 Cu(PO 4 ) 2 and K 2 CuP 2 O 7 . The structural peculiarities of these systems were discussed in section 3. The main issue here is the frustration caused by the leading inter-chain coupling. For Sr 2 Cu(PO 4 ) 2 , theory predicts an ordering temperature T N two orders of magnitude larger than the experimentally observed value. Table 3. Exchange integrals (columns 2 and 3) together with experimental (column 5) and theoretically calculated (columns 7-10) ordering temperatures T N for quasione-dimensional cuprates (3D magnet Bi 2 CuO 4 has a similar structural motive and was added for completeness). The ordering temperatures T N were calculated using formulas from references [19,21,23] and the exchange integrals from columns 2 (nearest neighbour intra-chain coupling J 1 ) and 3 (leading inter-chain coupling J ⊥ ). Note that the first four systems are frustrated due to inter-chain couplings, while in the last two the inter-chain couplings are not frustrated.
compound with coupling to four neighbouring chains). There is no unique way to take this feature into account. A simple approximation is to take the arithmetic average, which yields an effective inter-chain coupling value J ⊥ = J ab /2. * Using this value, theory yields a perfect agreement with experimental value (table 3, fifth row).
Obviously, the existing models describe the magnetic ordering in CuSe 2 O 5 much better than in Sr 2 CuO 3 and Ca 2 CuO 3 , and especially in Sr 2 Cu(PO 4 ) 2 . Despite our crude way of accounting for spatial exchange anisotropy and the neglect of spin anisotropy, the theoretical estimate of T N for CuSe 2 O 5 is in surprisingly good agreement with the experimental value. Though in general the magnetic ordering is affected by the spin anisotropy, CuSe 2 O 5 yields empirical evidence that for systems with a small spin anisotropy the isotropic model provides a rather accurate estimate of T N . Thus, it is unlikely that the disagreement between theoretical and experimentally observed T N values for Sr 2 Cu(PO 4 ) 2 originates from the neglect of spin anisotropy effects.
Finally, only the magnetic frustration is left to be a possible reason for a huge discrepancy between theory and experiment. Our analysis reveals that frustrated interchain couplings play a crucial role for the magnetic ordering. This fact explains why + For the theoretical estimation of T N , the formula 7 from [31] was used.
* Alternatively, the geometrical averaging can be used. This approach yields a correct limit with respect to the Mermin-Wagner theorem (zero ordering temperature for 1D and 2D systems). Then, calculational schemes from [19], [21] and [23] yield T N values of 14 K, 11 K and 10 K, respectively.
theoretical schemes fail to predict T N for frustrated systems.
To illustrate the influence of frustration, we use a simple formula from the spin wave theory in a random phase approximation, which connects Néel temperatures for two compounds A and B with the values of exchange integrals: Using the values of exchange integrals for Sr 2 Cu(PO 4 ) 2 and CuSe 2 O 5 and the experimental Néel temperature for Sr 2 Cu(PO 4 ) 2 (table 3), we obtain T N ≈ 0.146 K for CuSe 2 O 5 , almost 120 times smaller than the experimental value.
In the existing theoretical approaches, a parameter controlling the frustration caused by inter-chain couplings is missing. Therefore, new theories which would treat magnetic frustration as one of the key issues for the magnetic ordering, are needed. On the other hand, there is a lack of information from the experimental side, resulting in a very limited number of systems that challenge the theoretical predictions. Besides CuSe 2 O 5 , an almost perfect model system, synthesis and investigation of new systems with similar crystal chemistry are highly desirable.

Summary and outlook
The class of quasi-1D magnets attracts much attention as a field of search for prominent models and a playground for modern theories. Recently, by studying the magnetic properties of Cu 2+ phosphates, several systems of this class were found to exhibit the physics of a Heisenberg chain model. In these materials, the remarkable onedimensionality and the absence of long-range intra-chain interactions are ruled by a unique arrangement of magnetically active CuO 4 plaquettes: they form edge-sharing chains where every second plaquette is cut out. The chains are well separated by alkaline or alkaline earth cations (K, Sr). The magnetic susceptibility of these systems is perfectly described by the Bethe ansatz, which provides an exact solution for the nearest neighbour (NN) antiferromagnetic spin-1/2 Heisenberg chain. At the same time, the ordering temperature T N of the systems reveals a fundamental disagreement between theory and experiment. Unfortunately, the range of available experimental studies of these systems is rather limited, as the materials are presently available only as powders.
Therefore, we have synthesized CuSe 2 O 5 -a system implying a similar, isolated arrangement of neighbouring CuO 4 plaquettes (but tilted with respect to each other, unlike Cu 2+ phosphates) and allowing for a growth of high quality single crystals. Thermodynamic measurements reveal a quasi-1D behaviour with a leading antiferromagnetic coupling of about 160 K (obtained from the Bethe ansatz fit for the magnetic susceptibility). The system orders antiferromagnetically at 17 K, as evidenced by magnetic susceptibility and specific heat data. A microscopic analysis based on the results of DFT calculations reveals that CuSe 2 O 5 can be described in good approximation by only two relevant exchange integrals: NN intra-chain (J c = 165 K) and the leading inter-chain coupling (J ab = 20 K). The theoretical estimate of the ordering temperature T N is in perfect agreement with experimental value. This remarkable agreement is in sharp contrast with a huge overestimate of T N for Cu 2+ phosphates yielded by a formal application of the same theory. To reveal the origin of this difference on empirical grounds we analyzed systematically the factors affecting T N . Beyond the influence of the spatial exchange anisotropy and the spin anisotropy, we emphasize the role of the magnetic frustration due to equivalent inter-chain interactions in the latter compounds. Comparing theoretical and experimental data for related systems, we show that inter-chain frustrations have a crucial influence on T N and are likely the main cause for the failure of any theory which ignores them.
For an outlook, we propose further experimental studies (for instance, Raman spectroscopy and inelastic neutron scattering) in order to benefit from the availability of single crystals of CuSe 2 O 5 . Especially, additional experimental data are required in order to understand the nature of the specific heat anomaly at 7 K. Secondly, we hope that our work will inspire a directed search for new quasi-1D model systems. Last but not the least, we want to stimulate the development of more sophisticated theories for the estimation of T N . In particular, such theories should explicitly take into account the magnetic frustration arising from complex inter-chain interactions. The well understood system CuSe 2 O 5 could give valuable support for these theories.