Order from disorder phenomena in BaCoS₂

Abstract

At T_N ≃ 300K the layered insulator BaCoS₂ transitions to a columnar antiferromagnet that signals non-negligible magnetic frustration despite the relatively high T_N, all the more surprising given its quasi two-dimensional structure. Here, we show, by combining ab initio and model calculations, that the magnetic transition is an order-from-disorder phenomenon, which not only drives the columnar magnetic order, but also the inter-layer coherence responsible for the finite Néel transition temperature. This uncommon ordering mechanism, actively contributed by orbital degrees of freedom, hints at an abundance of low energy excitations above and across the Néel transition, in agreement with experimental evidence.

Introduction

Frustrated magnets often display a continuous accidental degeneracy of the classical ground state that leads to the appearance of pseudo-Goldstone modes within the harmonic spin-wave approximation¹. Since those modes are not protected by symmetry, they may acquire a mass once anharmonic terms are included in the spin-wave Hamiltonian. This mass, in turn, cuts off the singularities brought about by the pseudo-Goldstone modes, in that way stabilising ordered phases otherwise thwarted by fluctuations. To put it differently, let us imagine that the classical potential has a manifold of degenerate minima generally not invariant under the symmetry group of the Hamiltonian. It follows that the eigenvalues of the Hessian of the potential change from minimum to minimum. Allowing for quantum or thermal fluctuations is therefore expected to favour the minima with lowest Hessian determinant, although the two kinds of fluctuations not necessarily select the same ones². Moreover, it is reasonable to assume that the minima with lowest Hessian determinant are those that form subsets invariant under a symmetry transformation of the Hamiltonian so that choosing any of them corresponds to a spontaneous symmetry breaking. Such a phenomenon, also known as order from disorder³, emerges in many different contexts⁴, from particle physics^5,6 to condensed matter physics^7,8, even though frustrated magnets still provide the largest variety of physical realisations^{1,2,3,9,10,11,12,13,14,15}.

The layered insulator BaCoS₂ might be legitimately included in the class of frustrated magnets. Below a critical temperature T_N, BaCoS₂ becomes an antiferromagnet characterised by columnar spin-ordered planes, which we hereafter refer to as antiferromagnetic striped (AFS) order, a classic symptom of frustration. The planes are in turn stacked ferromagnetically along the c-axis, so called C-type stacking as opposed to the antiferromagnetic G-type one. Inelastic neutron scattering (INS) experiments show that magnetic excitations below T_N have pronounced two-dimensional (2D) character^16,17 implying strong quantum and thermal fluctuations that join with magnetic frustration to further hamper magnetic order. In spite of all that, the Néel temperature of BaCoS₂ is rather large, between 290K¹⁸ and 305K¹⁹, which is highly surprising. Indeed, a direct estimate of the spin exchange constants by neutron diffraction has been recently attempted in doped tetragonal BaCo_0.9Ni_0.1S_1.9 subject to an uniaxial strain¹⁷. This compound undergoes a Néel transition to the C-type AFS phase at 280 K¹⁷, not far from T_N of undoped BaCoS₂. The neutron data were fitted by a conventional J₁ − J₂ Heisenberg model¹³ on each plane plus an inter-plane ferromagnetic exchange J_c, yielding J₂ ~ 9.3 meV, J₁ ~ −2.3 meV and 0 < ∣J_c∣ < 0.04 meV, with the upper bound due to experimental resolution. The Néel temperature can be overestimated discarding J₁²⁰ and taking ∣J_c∣ equal to the upper bound. In that way, one obtains²¹ T_N ≃ 200K, which, despite supposedly being an overestimate, is 2/3 smaller than the observed value. This discrepancy is puzzling.

Another startling property is the anomalously broad peak of the magnetic susceptibility at T_N^18,19, which suggests a transition in the Ising universality class rather than the expected Heisenberg one¹⁹. A possible reason of this behaviour might be spin-orbit coupling¹⁹. Indeed, a Rashba effect due to the layered structure and the staggered sulfur pyramid orientation, see Fig. 1, has been found to yield sizeable band splittings at specific points within the Brillouin zone, at least in metallic tetragonal BaNiS₂²². The Rashba-like spin-orbit coupling strength may barely differ in BaCoS₂, or be weakened by strong correlations²³. In either case, its main effect is to introduce an easy plane anisotropy, as indeed observed experimentally¹⁹, which, at most, drives the transition towards the XY universality class. It is well possible that the weak orthorhombic distortion in BaCoS₂ may turn the easy plane into an easy axis, but the resulting magnetic anisotropy should be negligibly small and thus unable to convincingly explain the experimental observations.

Fig. 1: Crystal structure of BaCoS₂.

a Three-dimensional view of tetragonal BaCoS₂. Ba atoms are the large green spheres, while S atoms are shown in yellow. The cobalt atoms sit inside the blue square-based pyramids. b Top and lateral view of the structure, respectively. Note that, since the apexes of nearest neighbour pyramids point in opposite directions, there are two inequivalent Co atoms, shown as blue and grey spheres, with opposite vertical displacements from the a − b plane, which are connected by the non-symmorphic symmetry. c Magnetic order in the low-temperature orthorhombic phase. Dots represent Co atoms, arrows their spins and blue triangles indicate the orientation of the surrounding sulfur pyramids. Within each a − b plane the spins form a striped antiferromagnet (AFS) with ferromagnetic chains coupled antiferromagnetically. The ferromagnetic chains can be either along a (AFS-a) or along b (AFS-b). The planes are stacked ferromagnetically, C-type stacking, thus the two equivalent configurations C-AFS-a and C-AFS-b.

Lastly, BaCoS₂ shows a very strange semiconducting behaviour above T_N, with activated dc conductivity, no evidence of a Drude peak and, yet, an optical conductivity that grows linearly in frequency²⁴.

The relatively high T_N despite magnetic frustration and quasi-two dimensional character, as well as the abundance of low energy excitations above and across the Néel transition are pieces of evidence that some kind of order-from-disorder phenomenon takes place in BaCoS₂, a scenario that we here support by a thorough analysis combining ab initio and model calculations.

Results

Phase diagram of BaCoS₂

BaCoS₂ is a metastable layered compound that, quenched from high temperature, crystallises in an orthorhombic structure with space group Cmme, no. 67²⁵, characterised by in-plane primitive lattice vectors a ≠ b. However, we believe physically more significant to consider as reference structure the higher-symmetry non-symmorphic P4/nmm tetragonal one (a = b) of the opposite end member, BaNiS₂, and regard the orthorhombic distortion as an instability driven by the substitution of Ni with the more correlated Co. The hypothetical tetragonal phase of BaCoS₂ is shown in Fig. 1a. Each CoS a − b plane has two inequivalent cobalt atoms, Co(1) and Co(2), see Fig. 1b, which are related to each other by a non-symmorphic symmetry.

Below T_N, an AFS magnetically ordered phase sets in. In the a − b plane it consists of ferromagnetic chains, either along a (AFS-a) or b (AFS-b), coupled antiferromagnetically, see Fig. 1c. The stacking between the planes is C-type, i.e., ferromagnetic, thus the labels C-AFS-a and C-AFS-b that we shall use, as well as G-AFS-a and G-AFS-b whenever we discuss the G-type configurations with antiferromagnetic stacking. We mention that the orthorhombic distortion with b > a (a > b) is associated with C-AFS-a (C-AFS-b), i.e., ferromagnetic bonds along a (b)¹⁹, at odds with the expectation that ferromagnetic bonds are longer than antiferromagnetic ones. This counterintuitive behaviour represents a key test for the ab initio calculations that we later present.

Neutron scattering refinement and magnetic structure modelling in the low-temperature phase point to an ordered moment of μ_Co ~ 2.63 − 2.9μ_B^19,26, suggesting that each Co²⁺ is in a S = 3/2 spin configuration, in agreement with the high-temperature magnetic susceptibility¹⁹. Moreover, the form factor analysis of the neutron diffraction data²⁶ indicates that the three 1/2-spins lie one in the ${d}_{3{z}^{2}-{r}^{2}}$, the other in the ${d}_{{x}^{2}-{y}^{2}}$, and the third either in the d_xz or d_yz 3d-orbitals of Co. Since d_xz and d_yz, which we hereafter denote shortly as x and y orbitals, form in the P4/nmm tetragonal structure a degenerate E_g doublet occupied by a single hole, such degeneracy is going to be lifted at low-temperature. That hints at the existence of some kind of orbital order, besides the spin one, in the magnetic orthorhombic phase. Let us try to anticipate by symmetry arguments which kind of order can be stabilised. We observe that in the Cmme orthorhombic structure the cobalt atoms occupy the Wyckoff positions 4g, which, for convenience, we denote as Co(1) ≡ (0, 0, z), Co(2) ≡ (1/2, 0, −z), Co(3) ≡ (0, 1/2, −z), Co(4) ≡ (1/2, 1/2, z), and have symmetry mm2. As a consequence, the hole must occupy either the x orbital or the y one, but not a linear combination, and the chosen orbital must be the same for Co(1) and Co(4), as well as for Co(2) and Co(3). Therefore, we denote as d_n, d = x, y, the orbital occupied by the hole on Co(n), n = 1, …, 4, and as d₁d₂d₃d₄ a generic orbital configuration. Then, there are only four of them that are symmetry-allowed: xxxx, yyyy, xyyx and yxxy, see Fig. 2.

Fig. 2: Illustration of the orbital arrangements within BaCoS₂ allowed by the Cmme space group.

Co(1), Co(2), Co(3) and Co(4) correspond to the 4g Wyckoff positions occupied by the cobalt atoms. The label x(y) indicates that the hole occupies the d_xz(d_yz) orbital of the 3/4-filled d_xz/d_yz doublet of the corresponding cobalt atom.

We remark that xxxx is degenerate with yyyy in the tetragonal phase. The choice of either of them is associated with the same C₄ → C₂ symmetry breaking that characterises both the AFS-a or AFS-b spin order and the orthorhombic distortion, b > a or a > b. All these three choices can be associated with three Ising variables τ, σ and X such that τ = + 1 corresponds to xxxx, σ = +1 to AFS-a, X = +1 to b > a, and vice versa. Since they all have the same symmetry, odd under C₄, they would be coupled to each other should we describe the transition by a Landau-Ginzburg functional. We shall hereafter denote as Z₂(C₄) the Ising sector that describes the C₄ → C₂ symmetry breaking.

The other two allowed orbital configurations xyyx and yxxy (see Fig. 2) are instead degenerate both in the tetragonal and orthorhombic phases, but break the non-symmorphic symmetry (NS) that connects, e.g., Co(1) with Co(2) and Co(3). We can therefore associate to those configurations a new Ising sector Z₂(NS).

We emphasise that the above conclusions rely on the assumption of a Cmme space group. A mixing between x and y orbitals is instead allowed by the Pba2 space group proposed in ref. ²⁷ as an alternative scenario for BaCoS₂ at room temperature. As a matter of fact, the two symmetry-lowering routes, P4/nmm → Cmme and P4/nmm → Pba2, correspond to different Jahn-Teller-like distortions involving the d_xz-d_yz doublet and the E_g phonon mode of the P4/nmm structure at the M point, which is found to have imaginary frequency by ab initio calculations²⁷. However, latest high-accuracy X-ray diffraction data¹⁸ confirm the Cmme orthorhombic structure even at room temperature, thus supporting our assumption.

Ab initio analysis

Using density functional theory (DFT) and DFT + U calculations, in the first place, we checked if the tetragonal phase is unstable towards magnetism, considering both a conventional Néel order (AFM) compatible with the bipartite lattice and the observed AFS. We found, using a Hubbard interaction of U = 2.8 eV and a Hund’s coupling constant J = 0.95 eV for the Co-3d orbitals as motivated by constrained random phase approximation (cRPA)²⁴, that the lowest energy state is indeed the AFS, the AFM and non-magnetic phases lying above by about 0.5 eV and 2.3 eV, respectively. Let us therefore restrict our analysis to AFS and stick to U = 2.8 eV. Note, however, that the ordering of the four configurations with lowest energy within our DFT + U simulations does not change within 2 eV around that value. The main assumptions entering our ab initio modelling are thereby not sensitive to the precise choice of U around the value motivated by cRPA calculations. We use an 8-site unit cell that includes two planes, which allows us to compare C-AFS with G-AFS. In addition, we consider both the tetragonal structure with AFS-a, since AFS-b is degenerate, and the orthorhombic structure with b > a, in which case we analyse both AFS-a and AFS-b. For all cases, we investigate all four symmetry-allowed orbital configurations, xxxx, yyyy, xyyx and yxxy, assuming either a C-type or G-type orbital stacking between the two planes of the unit cell, so that, for instance, G(xxxx) means that one plane is in the xxxx configuration and the other in the yyyy one.

In Table 1, we report the energies per formula unit of several possible configurations in the tetragonal structure, including those that would be forbidden in the orthorhombic one. All energies are measured with respect to the lowest energy state and are expressed in Kelvin. In agreement with experiments, the lowest energy state T0 has spin order C-AFS, a or b being degenerate. In addition, it has C-type antiferro-orbital order, C(xyyx). We note that its G-type spin counterpart T1 is only 2 K above, supporting our observation that T_N = 290K is anomalously large if compared to these magnetic excitations. The abundance of nearly degenerate ground states is consistent with the seminal DFT + U study by Zainullina and Korotin²⁸, where the importance of different orbital configurations for a given stripe magnetic phase was studied for a larger value of U.

Table 1 Ab initio energies of the tetragonal phase.

The energy differences between C-type orbital stacked configurations and their G-type counterparts are too small to allow obtaining a reliable modelling of the inter-plane orbital coupling. On the contrary, the energy differences between in-plane orbital configurations can be accurately reproduced by a rather simple modelling. We assume on each Co-site an Ising variable τ₃ equal to the difference between the hole occupations of orbital x and of orbital y. The Ising variable on a given site is coupled only to those of the four nearest neighbour sites in the a − b plane, with exchange constants Γ_1a = Γ₁ + σ δΓ₁ and Γ_1b = Γ₁ − σ δΓ₁ along a and b, respectively. In addition, the Ising variables feel a uniform field B_τ σ. Here, σ is the Ising Z₂(C₄) order parameter that distinguishes AFS-a, σ = +1 from AFS-b, σ = −1. We find that the spectrum is well reproduced by the parameters in Table 2. It is worth noticing that the in-plane antiferro-orbital order is unexpected in light of the nematic columnar spin order that would rather suggest the ferro-orbital xxxx or yyyy configurations to have lowest energy. The explanation is that the antiferro-orbital order yields within DFT + U a larger insulating gap than the ferro-orbital order, see Fig. 3.

Table 2 Ising model for the tetragonal phase.

Fig. 3: Density of states for different orbital configurations.

The density of states (DOS) around the Fermi level is shown for the two orbital nematic configurations yyyy (full green line) and xxxx (dashed blue line) as well as for the orbital ordered configuration xyyx (dashed orange line), which is found to be lowest in energy within Hubbard-U corrected density functional theory (DFT + U) calculations.

We now move to the physical orthorhombic structure, assuming b > a with b/a = 1.008²⁵, and recalculate all above energies but considering only the orbital configurations allowed by the Cmme space group. In this case, we have to distinguish between AFS-a and AFS-b, which are no longer degenerate. The results are shown in Table 3.

Table 3 Ab initio energies of the orthorhombic phase.

The calculated magnetic moment per Co atom in the lowest energy state, O0 in Table 3, is μ_AFS ~ 2.65μ_B, in quite good agreement with experiments^19,26. We remark that the ab initio calculation correctly predicts that the lowest energy state O0 has ferromagnetic bonds along a despite b > a, which, as we mentioned, is an important test for the theory. The energy difference between AFS-a and AFS-b, i.e., O0 and O3, is about 20 K, and gives a measure of the spin-exchange spatial anisotropy in the a and b directions due to the orthorhombic distortion. This small value implies that the Néel transition temperature T_N ≃ 290K is largely insensitive to the orthorhombic distortion that exists also above T_N^18,19. In particular, the energy difference between C-type and G-type stacking, O0 and O1 in Table 3, remains the same tiny value found in the tetragonal phase. Table 3 thus suggests that the spin configurations C-AFS-a, C-AFS-b, G-AFS-a and G-AFS-b are almost equally probable at the Néel transition, and that despite the orthorhombic structure.

The orbital arrangement of the C-AFS-a configuration is also important to describe, e.g., the pressure-induced metal-insulator transition in BaCoS₂, see Supplementary Note 1.

Orthorhombic distortion

A further evidence of the marginal role played by the orthorhombic distortion at the Néel transition comes from the total energy as function of the parameter d = 2(a − b)/(a + b) that quantifies the distortion, shown in Fig. 4 for different orbital configurations assuming AFS-a magnetic order. We note that for all orbital configurations the energy gain due to a finite d is tiny with respect to d = 0. For instance, the lowest-energy orbital configuration xyyx reaches a minimum at about ${d}_{\min }^{xyyx} \sim -0.5 \%$, not far from the experimental value ${d}_{\exp } \sim -0.8 \%$²⁵, which reduces to ${d}_{\exp } \sim -0.4 \%$ under high pressure synthesis¹⁸. However, the energy gain with respect to d = 0 is less than 3 K. This result suggests that, despite the hypothetical P4/nmm structure of BaCoS₂ being inherently unstable to an orthorhombic Jahn-Teller distortion, the latter plays almost no role in stabilising the AFS magnetic order in contrast to naïve expectations.

Fig. 4: Phase stability upon orthorhombic distortions.

Energy ΔE per formula unit as function of the orthorhombic distortion d at fixed unit cell volume and measured with respect to the C(xyyx) phase at d = 0 (T0 in Table 1). We compare C(xyyx) with the two nematic configurations C(xxxx) and C(yyyy) assuming a magnetic order C-AFS-a. Experimental data are taken from refs. ^19,25,51. The phases T0, T7, and T17 refer to Table 1, phases O0, O12, and O17 to Table 3.

Wannierisation

To gain further insight into the mechanisms that drive the Néel transition, we generate two tight-binding Hamiltonians with maximally-localised Wannier functions for Co-d-like and Co-d_xz/yz-like orbitals, respectively. Both tight-binding models reproduce overall well the DFT band structure of the PM phase in the orthorhombic structure, see Fig. 5. Whereas the fit of the 5-orbital model is nearly perfect, the 2-orbital model shows small deviations along the M − Γ direction due to missing hybridisation with the other Co-d orbitals. Table 4 shows the leading hopping processes of the 5-band model restricted to the $({d}_{xz},{d}_{yz})$ subspace.

Fig. 5: Wannierisation of the density functional theory (DFT) band structure.

DFT band structure and eigenstates of our Wannier Hamiltonians along the high-symmetry path Γ − X − M − Γ − Y − M: (a) Co-d model and (b) d_xz-d_yz model with projection onto the d_xz-orbital character in red.

Table 4 Hopping amplitudes within the two-orbital submanifold.

We note that, because of the staggered shift of the Co atoms out of the sulfur basal plane, the largest intra-layer hopping is between next-nearest neighbour (NNN) cobalt atoms instead of nearest-neighbour (NN) ones. Moreover, the stacking of the sulfur pyramids and the position of the intercalated Ba atoms makes the inter-layer NN hopping negligible, contrary to the NNN one that is actually larger than the in-plane NN hopping, but still smaller than the in-plane NNN one (${t}_{NN}^{{{{{{{{\rm{inter}}}}}}}}}\ll {t}_{NN}^{{{{{{{{\rm{intra}}}}}}}}} \, < \, {t}_{NNN}^{{{{{{{{\rm{inter}}}}}}}}} \, < \, {t}_{NNN}^{{{{{{{{\rm{intra}}}}}}}}}$).

We finally remark that the orthorhombic distortion has a very weak effect on the inter-layer hopping, which is consistent with the tiny energy difference between C-AFS and G-AFS being insensitive to the distortion, compare, e.g., the energies of T1 and O1 in Tables 1 and 3, respectively.

Effective Heisenberg model

Armed with all the above ab initio results, we are now ready to address the main questions of this work, i.e., why T_N is so high and why the Néel transition looks Ising-like.

We already mentioned that the largest energy scales that emerge from the ab initio calculations are the magnetic ones separating the lowest-energy AFS configuration from the Néel and non-magnetic states. Therefore, even though BaCoS₂ seems not to lie deep inside a Mott insulating regime, we think it is worth discussing qualitatively the spin dynamics in terms of an effective S = 3/2 Heisenberg model. If we assume that the leading contribution to the exchange constants derives from the hopping processes within the d_xz − d_yz subspace, then Table 4 suggests the Heisenberg model shown in Fig. 6. According to this figure, the exchange constants J_1x/y, J₂, and J_3x/y are related to the hopping terms T_{(±1, 0, 0)/(0, ±1, 0)}, T_{±(1, −1, 0)/±(1, 1, 0)}, and T_{(±1, 0, 1)/(0, ±1, 1)}, reported in Table 4. This model consists of frustrated J₁ − J₂ planes^{13,29,30,31,32} coupled to each other by a still frustrating J₃ coupling, see Fig. 6b. In order to be consistent with the observed columnar magnetic order, the exchange constants have to satisfy the inequality $2{J}_{2} > \vert {J}_{3}\vert +\vert {J}_{1}\vert$. Moreover, J₃ forces to deal with a two sites unit cell, highlighted in yellow colour in Fig. 6a where the non-equivalent cobalt sites are referred to as Co(1), in blue, and Co(2), in red, respectively. The reason is that Co(1) on a plane is only coupled to Co(2) on the plane above but not below, and vice versa for Co(2).

Fig. 6: Effective Heisenberg spin model for BaCoS₂.

a shows the J₁ − J₂ model on the a − b plane, while (b) shows how nearest neighbour planes are coupled to each other by the exchange J₃. The latter forces to deal with a two-site unit cell, highlighted in yellow in (a). Blue and red balls indicate the two different Co sites of the unit cell.

To simplify the notation, we write, for a = 1, 3, ${J}_{ax}={J}_{a}\,\left(1-{\delta }_{a}\right)$ and ${J}_{ay}={J}_{a}\,\left(1+{\delta }_{a}\right)$, where, in analogy with the single-layer J₁ − J₂ model¹³, δ_a ≠ 0 are Ising order parameters associated with the C₄ → C₂ symmetry breaking. Moreover, we define the in-plane Fourier transforms of the spin operators:

$${{{{{{{{\boldsymbol{S}}}}}}}}}_{\ell ,n}({{{{{{{\bf{q}}}}}}}})=\mathop{\sum}\limits_{{{{{{{{\bf{R}}}}}}}}}\,{{{{{{{{\rm{e}}}}}}}}}^{-i{{{{{{{\bf{q}}}}}}}}\cdot {{{{{{{\bf{R}}}}}}}}}\ {{{{{{{{\boldsymbol{S}}}}}}}}}_{\ell ,n,{{{{{{{\bf{R}}}}}}}}}\,,$$

(1)

where R labels the N unit cells in the a − b plane, ℓ = 1, 2 the two sites (sublattices) within each unit cell, and n = 1, …, L the layer index. With those definitions, the Hamiltonian reads:

$$H = \frac{1}{N}\,\mathop{\sum}\limits_{n{{{{{{{\bf{q}}}}}}}}}\left\{{J}_{2}({{{{{{{\bf{q}}}}}}}})\,\left({{{{{{{{\bf{S}}}}}}}}}_{1,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{S}}}}}}}}}_{1,n}(-{{{{{{{\bf{q}}}}}}}})+{{{{{{{{\bf{S}}}}}}}}}_{2,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{S}}}}}}}}}_{2,n}(-{{{{{{{\bf{q}}}}}}}})\right)\right.\\ +{J}_{1}\,\left(\gamma ({{{{{{{\bf{q}}}}}}}},{\delta }_{1})\,{{{{{{{{\bf{S}}}}}}}}}_{1,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{S}}}}}}}}}_{2,n}(-{{{{{{{\bf{q}}}}}}}})+H.c.\right)\\ \left. +{J}_{3}\,\left(\gamma ({{{{{{{\bf{q}}}}}}}},{\delta }_{3})\,{{{{{{{{\bf{S}}}}}}}}}_{1,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{S}}}}}}}}}_{2,n+1}(-{{{{{{{\bf{q}}}}}}}})+H.c.\right)\right\}\\ \equiv \,{H}_{2}+{H}_{1}+{H}_{3},$$

(2)

where H_a is proportional to J_a, a = 1, 2, 3, and:

$${J}_{2}({{{{{{{\bf{q}}}}}}}}) = \,2{J}_{2}\,\cos {q}_{x}\,\cos {q}_{y}\,,\\ \gamma ({{{{{{{\bf{q}}}}}}}},\delta ) = \,{{{{{{{{\rm{e}}}}}}}}}^{i{q}_{x}}\,\left[\left(1-\delta \right)\,\cos {q}_{x}+\left(1+\delta \right)\,\cos {q}_{y}\right]\ .$$

(3)

The classical ground state corresponds to the three-dimensional modulation wave vector (π, 0, Q_z) ≡ (0, π, Q_z), which describes an antiferromagnetic order within each sublattice on each layer, and where the inter-plane Q_z is the value that minimises the classical energy per site, $E({Q}_{z})=-2{J}_{2}-2\sqrt{{J}_{1}^{2}{\delta }_{1}^{2}+{J}_{3}^{2}{\delta }_{3}^{2}+2{J}_{1}{J}_{3}{\delta }_{1}{\delta }_{3}\cos {Q}_{z}\,}$. The expression of E(Q_z) shows that inter-layer magnetic coherence sets in only when the two Ising-like order parameters, δ₁ and δ₃, lock together. Specifically, J₁J₃δ₁δ₃ > 0 stabilises C-AFS, Q_z = 0, otherwise G-AFS, Q_z = π. We already know that the former is lower in energy, though by only few Kelvins, see Table 3. Moreover, an orthorhombic distortion b > a favours AFS-a, which implies J₁ δ₁ + J₃ δ₃ > 0, even though AFS-b is higher by only 20K according to DFT + U, see O3 in Table 3.

Using our J₁ − J₂ − J₃ model to fit the INS data of ref. ¹⁷ at 200 K, we estimate J₂ ≃ 9.3 meV, J₁ + J₃ ≃ −2.34 meV, J₁δ₁ + J₃δ₃ ≃ 0.53 meV, and $0 \, < \, \sqrt{{J}_{1}{J}_{3}{\delta }_{1}{\delta }_{3}\,} < 0.14\,\,{{\mbox{meV}}}\,$, where, we recall, the upper bound is due to experimental resolution. Such small bound suggests that the two order parameters δ₁ and δ₃ are already formed at 200K, whereas their mutual locking is still suffering from fluctuations. We finally observe that the ferromagnetic sign of J₁ and J₃ is consistent with the diagonal hopping matrices in the corresponding directions (see Table 4) and the antiferro-orbital order. Estimations of the exchange constants based on the DFT + U energies can be found in Supplementary Note 2.

Spin-wave analysis of the ${{{{{{{{\mathcal{C}}}}}}}}}_{4}$ symmetric model

To better understand the interplay between the ${Z}_{2}\left({C}_{4}\right)$ Ising degrees of freedom and the magnetic order at T_N, we investigate in more detail the Hamiltonian (2) with δ₁ = δ₃ = 0, thus J_1x = J_1y = J₁ and J_3x = J_3y = J₃. Since J₂ > 0 is the dominant exchange process, the classical ground state corresponds to the spin configuration:

$${{{{{{{{\bf{S}}}}}}}}}_{i,n}({{{{{{{\bf{q}}}}}}}})=NS\,{{{{{{{{\boldsymbol{n}}}}}}}}}_{3,i,n}\,{\delta }_{{{{{{{{\bf{q}}}}}}}},{{{{{{{\bf{Q}}}}}}}}}\,,i=1,2\,,n=1,\ldots ,L\,,$$

(4)

where S = 3/2 is the spin magnitude, n_3,i,n is a unit vector, and ${{{{{{{\bf{Q}}}}}}}}=\left(\pi ,0\right)\equiv (0,\pi )$, the equivalence holding since G = (π, π) is a primitive in-plane lattice vector for the two-site unit cell. In other words, each sublattice on each plane is Néel ordered, and its staggered magnetisation n_3,i,n is arbitrary. We therefore expect that quantum and thermal fluctuations may yield a standard order-from-disorder phenomenon³.

Within spin-wave approximation, the spin operators can be written as:

$${{{{{{{{\bf{S}}}}}}}}}_{i,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,i,n}\simeq \,NS\,{\delta }_{{{{{{{{\bf{q}}}}}}}},{{{{{{{\bf{Q}}}}}}}}}-{{{\Pi }}}_{i,n}({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\,,\\ {{{{{{{{\bf{S}}}}}}}}}_{i,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{1,i,n}\simeq \sqrt{NS\,}\ {x}_{i,n}({{{{{{{\bf{q}}}}}}}})\,,\\ {{{{{{{{\bf{S}}}}}}}}}_{i,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{2,i,n}\simeq \sqrt{NS\,}\ {p}_{i,n}({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\,,$$

(5)

where n_1,i,n, n_2,i,n and n_3,i,n are orthogonal unit vectors, ${x}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{q}}}}}}}})={x}_{i,n}^{}(-{{{{{{{\bf{q}}}}}}}})$ and ${p}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{q}}}}}}}})={p}_{i,n}^{}(-{{{{{{{\bf{q}}}}}}}})$ are conjugate variables, i.e.:

$$\left[\,{x}_{i,n}^{}({{{{{{{\bf{q}}}}}}}})\,,\,{p}_{j,m}^{{{{\dagger}}} }({{{{{{{{\bf{q}}}}}}}}}^{{\prime} })\,\right]=i\,{\delta }_{i,j}\,{\delta }_{n,m}\,{\delta }_{{{{{{{{\bf{q}}}}}}}},{{{{{{{{\bf{q}}}}}}}}}^{{\prime} }}\,,$$

(6)

and

$${{{\Pi }}}_{i,n}({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}}) = \frac{1}{2}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}\left({x}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})\,{x}_{i,n}^{}({{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\right.\\ \left. +{p}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})\,{p}_{i,n}^{}({{{{{{{\bf{k}}}}}}}}+{{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})-{\delta }_{{{{{{{{\bf{q}}}}}}}},{{{{{{{\bf{Q}}}}}}}}}\right).$$

(7)

The three terms of the Hamiltonian (2) thus read, at leading order in quantum fluctuations, i.e., in the harmonic approximation:

$${H}_{2} \simeq \,{E}_{0}+S\mathop{\sum}\limits_{i,n\,{{{{{{{\bf{q}}}}}}}}}\left({J}_{2}({{{{{{{\bf{q}}}}}}}})-{J}_{2}({{{{{{{\bf{Q}}}}}}}})\right)\left({x}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{q}}}}}}}})\,{x}_{i,n}^{}({{{{{{{\bf{q}}}}}}}})\right.\\ \left. +{p}_{i,n}^{{{{\dagger}}} }({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\,{p}_{i,n}^{}({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\right)\,,\\ {H}_{1}\simeq \,S{J}_{1}\mathop{\sum}\limits_{n,{{{{{{{\bf{q}}}}}}}}}\left(\gamma ({{{{{{{\bf{q}}}}}}}})\,{{{{{{{{\bf{X}}}}}}}}}_{1,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{X}}}}}}}}}_{2,n}(-{{{{{{{\bf{q}}}}}}}})+H.c.\right)\,,\\ {H}_{3}\simeq \,S{J}_{3}\mathop{\sum}\limits_{n,{{{{{{{\bf{q}}}}}}}}}\left(\gamma ({{{{{{{\bf{q}}}}}}}})\,{{{{{{{{\bf{X}}}}}}}}}_{1,n}({{{{{{{\bf{q}}}}}}}})\cdot {{{{{{{{\bf{X}}}}}}}}}_{2,n+1}(-{{{{{{{\bf{q}}}}}}}})+H.c.\right),$$

(8)

where E₀ = 2NL S(S + 1) J₂(Q), γ(q) = γ(q, δ = 0), and:

$${{{{{{{{\bf{X}}}}}}}}}_{i,n}({{{{{{{\bf{q}}}}}}}})={{{{{{{{\boldsymbol{n}}}}}}}}}_{1,i,n}\,{x}_{i,n}({{{{{{{\bf{q}}}}}}}})+{{{{{{{{\boldsymbol{n}}}}}}}}}_{2,i,n}\,{p}_{i,n}({{{{{{{\bf{q}}}}}}}}-{{{{{{{\bf{Q}}}}}}}})\,.$$

(9)

We note that H₂ does not depend on the choice of n_3,i,n, reflecting the classical accidental degeneracy, unlike H₁ + H₃. We start treating H₁ and H₃ within perturbation theory. The unperturbed Hamiltonian H₂ can be diagonalised and yields the spin-wave dispersion:

$${\omega }_{2}({{{{{{{\bf{q}}}}}}}})=2S\,\sqrt{\,{J}_{2}{({{{{{{{\bf{0}}}}}}}})}^{2}-{J}_{2}{({{{{{{{\bf{q}}}}}}}})}^{2}\ }\ .$$

(10)

Free energy in perturbation theory and quadrupolar coupling

The free energy in perturbation theory can be written as F = ∑_ℓ F_ℓ, where F_ℓ is of ℓth order in H₁ + H₃, and F₀ is the unperturbed free energy of the Hamiltonian H₂. Notice that only even-order terms are non vanishing, thus ℓ = 0, 2, 4, … . Given the evolution operator in imaginary time:

$$S(\beta )={T}_{\tau }\left({{{{{{{{\rm{e}}}}}}}}}^{-\int\nolimits_{0}^{\beta }d\tau \left({H}_{1}(\tau )+{H}_{3}(\tau )\right)}\,\right)=\mathop{\sum}\limits_{\ell }\,{S}_{\ell }(\beta )\,,$$

(11)

where H_a(τ), a = 1, 3, evolves with the Hamiltonian H₂, the second order correction to the free energy is readily found to be:

$${F}_{2} = - T\,\langle \,{S}_{2}(\beta )\,\rangle \\ = - \frac{{{{\Xi }}}_{2}(T)}{{J}_{2}}\,\mathop{\sum}\limits_{n}\left[{J}_{1}^{2}\,{\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,2,n}\right)}^{2}+{J}_{3}^{2}\,{\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,2,n+1}\right)}^{2}\right],$$

(12)

where

$${{{\Xi }}}_{2}(T)={J}_{2}\,{S}^{2}\,\mathop{\sum}\limits_{{{{{{{{\bf{q}}}}}}}}}\,T\,\mathop{\sum}\limits_{\lambda }\,\left\vert \right.\gamma ({{{{{{{\bf{q}}}}}}}}){\left\vert \right.}^{2}\,\frac{\ {J}_{2}({{{{{{{\bf{0}}}}}}}})-{J}_{2}({{{{{{{\bf{q}}}}}}}})\ }{\ {J}_{2}({{{{{{{\bf{0}}}}}}}})+{J}_{2}({{{{{{{\bf{q}}}}}}}})\ }{\left(\frac{{\omega }_{2}({{{{{{{\bf{q}}}}}}}})}{{\omega }_{\lambda }^{2}+{\omega }_{2}{({{{{{{{\bf{q}}}}}}}})}^{2}}\right)}^{2} > 0\,,$$

(13)

with ω_λ = 2πλT, $\lambda \in {\mathbb{Z}}$, bosonic Matsubara frequencies. Even without explicitly evaluating Ξ₂, we can conclude that the free-energy gain at second order in H₁ + H₃ is maximised by n_3,1,n ⋅ n_3,2,m = ± 1, with m = n, n + 1, which reduces the classical degeneracy to 4^L configurations, where L is the total number of layers in the system. Such residual degeneracy is split by a fourth order correction to the free energy proportional to ${J}_{1}^{2}\,{J}_{3}^{2}$ that reads:

$${F}_{4} = -\frac{{J}_{1}^{2}\,{J}_{3}^{2}}{{J}_{2}^{3}}\ {{{\Xi }}}_{4}(T)\,\mathop{\sum}\limits_{n}\,\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,2,n}\right)\\ \left[\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,2,n+1}\right)+\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n-1}\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,2,n}\right)\right]\,,$$

(14)

where

$${{{\Xi }}}_{4}(T)=2{S}^{4}\,{J}_{2}^{3}\,T\,\mathop{\sum}\limits_{\lambda }\,\mathop{\sum}\limits_{{{{{{{{\bf{q}}}}}}}}}\,\left\vert \right.\gamma ({{{{{{{\bf{q}}}}}}}})\,\gamma ({{{{{{{\bf{q}}}}}}}}+{{{{{{{\bf{Q}}}}}}}}){\left\vert \right.}^{2}\ \frac{{\omega }_{\lambda }^{2}}{\ {\left({\omega }_{\lambda }^{2}+{\omega }_{2}{({{{{{{{\bf{q}}}}}}}})}^{2}\right)}^{3}\ } \, > \, 0\,.$$

(15)

We remark that, despite ω₂(q) vanishes linearly at q = 0, Q, both Ξ₂(T) and Ξ₄(T) are non-singular.

The fourth order correction F₄ in Eq. (14) has a twofold effect: it forces n_3,1,n ⋅ n_3,2,n to be the same on all layers and, in addition, stabilises a ferromagnetic inter-layer stacking. Therefore, the ground state manifold at fourth order in H₁ + H₃ is spanned by n_3,1,n = n₃ and n_3,2,n = σ n₃, where n₃ is an arbitrary unit vector reflecting the spin SU(2) symmetry, and σ = ± 1 is associated with the global C₄ → C₂ symmetry breaking.

Similarly to the single-plane J₁ − J₂ model¹³, the above results imply that an additional term must be added to the semiclassical spin action. Specifically, if we introduce the Ising-like fields σ_n(R) = n_3,1,n(R) ⋅ n_3,2,n(R) and σ_n+1/2(R) = n_3,1,n(R) ⋅ n_3,2,n+1(R), Eqs. (12) and (14) imply that, at the leading orders in J₁ and J₃, the effective action in the continuum limit includes the quadrupolar coupling term¹³:

$${A}_{{{{{{{{\rm{Q}}}}}}}}}\simeq - \mathop{\sum}\limits_{n}\,\int\,d{{{{{{{\bf{R}}}}}}}}\,\left\{\frac{{{{\Xi }}}_{2}(T)}{T{J}_{2}}\left({J}_{1}^{2}\,{\sigma }_{n}{({{{{{{{\bf{R}}}}}}}})}^{2}+{J}_{3}^{2}\,{\sigma }_{n+1/2}{({{{{{{{\bf{R}}}}}}}})}^{2}\right)\right.\\ \left. +\frac{{{{\Xi }}}_{4}(T){J}_{1}^{2}{J}_{3}^{2}}{T{J}_{2}^{3}}\ {\sigma }_{n}({{{{{{{\bf{R}}}}}}}})\left({\sigma }_{n+1/2}({{{{{{{\bf{R}}}}}}}})+{\sigma }_{n-1/2}({{{{{{{\bf{R}}}}}}}})\right)\right\}.$$

(16)

We expect a 3D Ising transition to occur at a critical temperature T_c, below which 〈σ_n(R)〉 = m₁, 〈σ_n+1/2(R)〉 = m₃, with m₁ m₃ > 0. In turn, the Ising order should bring along the 3D AFS one below a finite Néel temperature bounded from above by T_c²⁰. To get a rough estimate of the latter, based on Eq. (16) we assume that, upon integrating out the spin degrees of freedom, the classical action describes an anisotropic three-dimensional ferromagnetic Ising model with exchange constants I₁ on layers n, I₃ on layers n + 1/2, and I_⊥ < I₁, I₃ between layers. Hereafter, we take for simplicity J₁ = J₃, thus I₁ = I₃ ≡ I_∥.

We then note that, for the J₁ − J₂ − J₃ model with J₂ ≃ 9.3 meV and J₁ = J₃ ≃ −1.17 meV, the 2D Ising critical temperature with S = 3/2 of each layer n and n + 1/2 is about 0.4 (S + 1/2)²J₂ ≃ 173K³⁰. This critical temperature corresponds to I_∥ ≃ 6.6 meV in the 2D Ising model. The 3D Ising critical temperature T_c grows with I_⊥, reaching 280K and 345K at I_⊥ = 0.5I_∥ and I_⊥ = I_∥, respectively³³, which are reassuringly of the same order of magnitude as T_N.

However, BaCoS₂ remains orthorhombic above T_N, which implies that the structural C₄ → C₂ symmetry breaking occurs earlier than magnetic ordering upon cooling. Therefore, even though the effects of the orthorhombic distortion on the electronic structure are rather small, see Table 4, it is worth repeating the above discussion assuming from the start that σ_n(R) = n_3,1,n(R) ⋅ n_3,2,n(R) = 1 (AFS-a), so that σ_n+1/2(R) = n_3,1,n(R) ⋅ n_3,2,n+1(R) = n_3,1,n(R) ⋅ n_3,1,n+1(R). It follows that the quadrupolar term (16) becomes:

$${A}_{{{{{{{{\rm{Q}}}}}}}}}\simeq - \mathop{\sum}\limits_{n}\int\,d{{{{{{{\bf{R}}}}}}}}\,\left\{K(T)\,{\left({{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}({{{{{{{\bf{R}}}}}}}})\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n + 1}({{{{{{{\bf{R}}}}}}}})\right)}^{2}\right.\\ \left. + h(T)\,{{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n}({{{{{{{\bf{R}}}}}}}})\cdot {{{{{{{{\boldsymbol{n}}}}}}}}}_{3,1,n + 1}({{{{{{{\bf{R}}}}}}}})\right\},$$

(17)

with K(T) ≫ h(T) > 0. The term proportional to K(T) is bi-quadratic in the original spin operators and, alone, it would drive an Ising-like transition either towards C-AFS, n_3,1,n ⋅ n_3,1,n+1 = 1 or G-AFS, n_3,1,n ⋅ n_3,1,n+1 = − 1. On the contrary, the term proportional to h(T) is quadratic and yields an inter-layer ferromagnetic coupling that stabilises C-AFS, though by only two kelvins according to our DFT + U calculations. Therefore, (17) corresponds to an unusual model of coupled Heisenberg layers in which the dominant inter-layer coupling is bi-quadratic. We believe that this term is responsible of the higher T_N than the estimate obtained assuming just the small ferromagnetic exchange, as earlier discussed, as well as of the pronounced Ising-like character of the Néel transition. We also remark that which among C-AFS and G-AFS is lower in energy does depend on the orbital configurations, see Table 3. Therefore, we expect that the neglected coupling between spins and orbital fluctuations should further reduce the already small energy difference between C-AFS and G-AFS.

Conclusion

BaCoS₂ is a frustrated magnet with a pronounced two-dimensional character of the magnetic excitations that, nonetheless, orders magnetically at a Néel temperature of T_N ~ 300K^18,19 through a second order phase transition more similar to an Ising than a Heisenberg one. We have shown that these puzzling features can be pieced together within an order-from-disorder scenario that we have uncovered by a thorough ab initio analysis demonstrating the critical role of the specific Co d-orbitals involved in magnetism. Although specific to BaCoS₂, our results might be relevant to other spin-frustrated transition metal compounds that also crystallise in the non-symmorphic P4/nmm space group, like, e.g., the iron pnictides. In many (mainly electron-doped) iron-based superconductors the disordered phase at high temperature first spontaneously breaks C₄ symmetry when cooling below a critical temperature T_nem, thus entering a nematic phase³⁴. Only at a lower temperature T_N < T_nem, also spin SU(2) is broken and a stripe-ordered magnetic long-range order emerges^34,35. The Néel temperature can be quite large as in BaCoS₂ or even vanishing within experimental accuracy as in FeSe, where T_N ≠ 0 is observed only under pressure³⁶.

Also from a model point of view, our description of BaCoS₂ fits into the modelling of these materials. Indeed, the key role played by several order parameters in BaCoS₂ has parallels to the phenomenological Landau free energy description of FeSC³⁴. Besides itinerant multi-orbital Hubbard models^37,38,39, also spin-1 Heisenberg models conceptually similar to ours have been proposed for iron-based superconductors and FeSe^40,41. There, however, instead of achieving the C₄ symmetry breaking via an order-from-disorder phenomenon, it is often assumed from the start⁴² or by explicitly adding bi-quadratic spin exchanges⁴⁰ that mimic the quadrupolar terms (16).

Moreover, our ab initio simulations for BaCoS₂ predict that the lowest-energy phase has not ferro-orbital order, as often discussed in the context of FeSC, but rather an anti-ferro orbital ordering that breaks the non-symmorphic symmetry instead of C₄. Therefore, BaCoS₂ seems to realise a situation where collinear magnetism and orthorhombicity do not imply orbital nematicity, unless for specific crystal structures under pressure.

Methods

Ab initio calculations

We carried out ab initio DFT and DFT + U calculations using the Quantum ESPRESSO package^43,44. The density functional is of generalised gradient approximation type, namely the Perdew-Burke-Ernzerhof functional⁴⁵, on which local Hubbard interactions and Hund’s coupling terms were added to the Co atoms in case of the DFT + U within a fully rotational invariant framework^46,47. If not stated otherwise, the geometry of the unit cell and the internal coordinates of the atomic positions in the orthorhombic structure were those determined experimentally, taken from ref. ²⁵. For non-magnetic calculations, the relative atomic positions were kept fixed and the in-plane lattice constants a = b chosen such that the unit cell volume matched the one of the orthorhombic structure. Co and S atoms are described by norm-conserving pseudopotentials (PP) with non-linear core corrections, Ba atoms are described by ultrasoft pseudopotentials. The Co PP contains 13 valence electrons (3s²,3p⁶,3d⁷), the Ba PP 10 electrons (5s²,5p⁶,6s²), and S PPs are in a (3s²,3p³) configuration. The plane-waves cutoff has been set to 120Ry and we used a Gaussian smearing of 0.01Ry. The k-point sampling of the electron-momentum grid was at least 8 × 8 × 8 points in the 8 Co atom supercell.

Wannier interpolation

To determine the band structure and derive an effective low-energy model, we performed a Wannier interpolation with maximally localised Wannier functions^48,49 using the Wannier90 package⁵⁰. We constructed Wannier fits based on the non-magnetic DFT + U calculation using a 4 × 4 × 4 k-grid with a doubled in-plane unit cell comprising 4 Co atoms.