Magnetic inhomogeneity on a triangular lattice: the magnetic-exchange versus the elastic energy and the role of disorder

Inhomogeneity in the ground state is an intriguing, emergent phenomenon in magnetism. Recently, it has been observed in the magnetostructural channel of the geometrically frustrated α-NaMnO2, for the first time in the absence of active charge degrees of freedom. Here we report an in-depth numerical and local-probe experimental study of the isostructural sister compound CuMnO2 that emphasizes and provides an explanation for the crucial differences between the two systems. The experimentally verified, much more homogeneous, ground state of the stoichiometric CuMnO2 is attributed to the reduced magnetoelastic competition between the counteracting magnetic-exchange and elastic-energy contributions. The comparison of the two systems additionally highlights the role of disorder and allows the understanding of the puzzling phenomenon of phase separation in uniform antiferromagnets.

Inhomogeneity in the ground state is an intriguing, emergent phenomenon in magnetism. Recently, it has been observed in the magnetostructural channel of the geometrically frustrated a-NaMnO 2 , for the first time in the absence of active charge degrees of freedom. Here we report an in-depth numerical and local-probe experimental study of the isostructural sister compound CuMnO 2 that emphasizes and provides an explanation for the crucial differences between the two systems. The experimentally verified, much more homogeneous, ground state of the stoichiometric CuMnO 2 is attributed to the reduced magnetoelastic competition between the counteracting magnetic-exchange and elastic-energy contributions. The comparison of the two systems additionally highlights the role of disorder and allows the understanding of the puzzling phenomenon of phase separation in uniform antiferromagnets.
A lthough phase separation in a uniform system is a widespread phenomenon in diverse fields of matter 1-3 , ranging from biological systems [4][5][6] , to soft matter 7,8 , and strongly correlated electron systems [9][10][11][12][13][14][15] , in magnetism the microscopic pattering has been, until recently, almost exclusively limited to thin ferromagnetic (FM) films 16,17 . In this case, such a pattering is a trade-off between minimizing the exchange and the dipolar energies. It thus represents one possible manifestation of a general requirement of multiple competing phases that can lead to inhomogeneous states. Lately, it has become increasingly apparent that a similar competition between energetically nearly equivalent phases is also responsible for phase separation in geometrically frustrated spin systems [18][19][20][21] that are generically characterized by ground-state degeneracy 22 . However, the balance between the competing phases in these systems is generally much more delicate and, therefore, poorly understood.
Recently, the spatially anisotropic triangular antiferromagnet a-NaMnO 2 , with dominant intrachain (J 1 ) and geometrically frustrated interchain (J 2 ) antiferromagnetic (AFM) exchange interactions (inset in Figure 1a), has been highlighted as a paradigm of a phase-separated ground state in the absence of active charge degrees of freedom 18 . Its AFM order that sets in below the Néel temperature T N 5 45 K is accompanied by a simultaneous structural deformation 23 . This was initially suggested as being a phase transition from the high-temperature monoclinic (C2/m) to the low-temperature triclinic (P 1) crystal structure 23 . However, more detailed, recent experiments have shown that the magnetic order fails to drive this improper ferroelastic transition to completion 18 . Instead, an intricate magnetostructurally inhomogeneous state on the nano-scale has been discovered below the T N . Such a state was suggested to be an unforeseen consequence of the subtle interplay between the geometrical frustration and the competing structural phases 18 .
In order to fully understand this novel phenomenon, further theoretical studies and experimental investigations of related compounds are of paramount importance. In this respect, a comparison with the crystallographically 24 and magnetically 25 analogous sister compound CuMnO 2 , known as the mineral crednerite, is particularly relevant. Here, in contrast to a-NaMnO 2 , the emergent magnetic order below T N 5 65 K is believed to lift the macroscopic degeneracy in the spin space completely, by inducing the monoclinic-to-triclinic structural phase transition 26 . This spin-induced phase transition is witnessed by the splitting of several families of nuclear Bragg reflections 18,27 . It was suggested to reflect the strong magnetoelastic (ME) coupling that allows for the development of shear strain at a low energy cost 27 . Interestingly, the strain is significantly enhanced 28 in the offstoichiometric 29 Cu 11x Mn 12x O 2 , where T N is reduced and the structural transition temperature is further suppressed with increasing x even for small doping levels 28,30,31 . Moreover, in off-stoichiometric samples the interlayer ordering changes from AFM, observed in stoichiometric CuMnO 2 , to FM 28,30,31 , which was attributed to a partial substitution of Cu 21 for Mn 31 that should effectively change the interlayer exchange coupling from AFM to FM 32 . On the other hand, this implies that CuMnO 2 may be very close to an electronic instability, possibly of a similar kind to that found in stoichiometric a-NaMnO 2 .
Obtaining in-depth information about the magnetic and structural properties of CuMnO 2 on the local scale should clarify the differences with respect to the isostructural a-NaMnO 2 . Such knowledge would also help to address the pending issue of the microscopic origin of the phase-separation phenomenon in geometrically frustrated magnets. The most obvious ambiguities arise from questions like: why is the T N enhanced in CuMnO 2 compared to a-NaMnO 2 , despite the theoretically predicted sizably smaller exchange interactions in the former compound 33 ; what is the role of the ME coupling in establishing the structural distortion below the T N , what is the role of disorder; and ultimately, why does the structural phase transition appear to be fully developed in CuMnO 2 , while in a-NaMnO 2 it only manifests in a phase-separated state. Here, we answer these questions by combining numerical calculations with local-probe experimental investigations. First, we determine the dominant intralayer exchange interactions by modelling the magnetic susceptibility via exact-diagonalization calculations. We demonstrate that the difference in the T N for the two compounds can be understood via the mismatch of the two non-equivalent interchain exchange interactions, i.e., by the different extent of the frustration present in the two compounds, while the ME contribution is negligible in this respect. Moreover, we provide the first experimental microscopic insight into the magnetism of CuMnO 2 via 63,65 Cu nuclear magnetic resonance (NMR) and nuclear quadrupolar resonance (NQR) measurements, as well as complementary muon spin relaxation (mSR) measurements. These experiments clearly reveal that the ground state is more homogeneous in the Cu case than in the Na case and suggest that the stoichi-ometric CuMnO 2 is on the verge of a phase-separation instability. Finally, our ab-initio calculations suggest that the more homogeneous state of the stoichiometric CuMnO 2 originates from an enhanced energy difference (when compared to a-NaMnO 2 ) between the two competing phases, born out of the magneticexchange and the elastic-energy changes below the T N .

Results
Determination of the dominant exchange interactions and the T N . In order to understand the apparently significantly different properties of the two isostructural compounds, CuMnO 2 and a-NaMnO 2 , a proper determination of the dominant terms in the Hamiltonian is crucial. Therefore, we applied numerical finitetemperature Lanczos method (FTLM) simulations and densityfunctional theory (DFT) calculations. The former were aimed at quantifying the two main magnetic exchange interactions (J 1 and J 2 ) of the isotropic Heisenberg model on the spatially anisotropic two-dimensional (2D) triangular lattice (inset in Figure 1a), Here, the first sum runs over the (stronger) intrachain bonds in one direction, while the second sum runs over the (weaker) interchain bonds in the other two directions on the triangular lattice of spins S 5 2.
The temperature-dependent magnetic susceptibility x(T) of this model was calculated for various m 3 n spin clusters (see Methods). However, even for the largest reachable cluster sizes (7 3 2), some finite-size effects remain present at low T. With the use of results from many clusters (m 5 2-7) these effects can, however, be reduced and the value of x at the thermodynamic limit is thus approached. Both, the average susceptibility curve from the two largest-size clusters (that gives a better approximation than the individual clusters; see Methods) and the susceptibility curve obtained in an approach similar to finite-size scaling (see Methods) fit the experimental data above the T N very well (Figure 1a). A disagreement with the experimental data below the T N , on the other hand, is expected, because the 2D Heisenberg model cannot account for a finite T N . Both approaches yield very similar exchange-coupling constants, which we estimate to be J 1 5 53 K and J 2 /J 1 5 0.27 (2). These are in very good agreement with recent ab-initio predictions 33 , J 1 5 56 K and J 2 =J 1~0 :23, which, in principle, could be erroneous due to the unknown on-site repulsion 32 .
Our calculations thus confirm that the exchange interactions are indeed reduced for CuMnO 2 compared to a-NaMnO 2 , where 35 J 1 5 65 K and J 2 /J 1 5 0.44. Despite this fact, in CuMnO 2 the T N is increased with respect to that found in a-NaMnO 2 by more than 40%. So far, this has been attributed to a difference in the interlayer coupling 26 J9, which, however, is rather small 32,33 and can, therefore, only slightly affect the T N on 2D Heisenberg lattices 36 . Furthermore, the amount of frustration reflected in the J 2 /J 1 ratio should directly influence the T N , as the frustration is known to suppress the spin correlations 37 . In a-NaMnO 2 and CuMnO 2 the intrachain exchange coupling is dominant. Therefore, these compounds can be regarded as systems of coupled spin chains, which is manifested in the one-dimensional character of the magnetic excitations in a-NaMnO 2 38 . For such systems the T N can be determined with the use of a random-phase approximation 39,40 . In this approach the interchain coupling is treated at the mean-field level, whereas the intrachain interactions are treated exactly. For isotropic interchain coupling in a non-frustrated lattice the T N is determined by the condition zJ 2 x s (T N ) 5 1 (Ref. 36,[39][40][41], where x s (T) is the chain's staggered susceptibility. Within this approach, we generalize the above condition for T N to include the two interchain constants (J 2a . J 2b ) pertinent to the triclinic phase of CuMnO 2 and a-NaMnO 2 , as well as the interlayer coupling J9; Here, z 5 z9 5 2 corresponds to the number of neighbouring coupled chains and planes, respectively while J 2b adopts the minus sign because it frustrates the AFM order dictated by the larger J 2a . The constant k renormalizes the coordination numbers and is reduced from unity 41,42 because of quantum effects 36 . As (J 2a 2 J 2b )/J 1 5 0.083 and 0.035 in CuMnO 2 and a-NaMnO 2 , respectively 33 , while J9/J 1 is expected to be an order of magnitude smaller 32,33 , we estimate the T N by neglecting J9 in equation (2) and by taking k 5 0.7, which is appropriate for quasione dimensional cases (see Fig. 2 in Ref. 36). This gives T N 5 74 K in CuMnO 2 and T N 5 57 K in a-NaMnO 2 (see Figure 1b), which are in good agreement with the experimental values of 65 K and 45 K, respectively. We note that for anisotropic interchain couplings the constant k is expected to be further reduced and, ultimately, for J9 R 0 also k R 0 (Ref. 41,42), leading to T N R 0, which is consistent with the Mermin-Wagner theorem (no long-range order in the 2D Heisenberg model at finite T). However, it has been shown 43 that for quasi-2D systems the dependence of the T N and, in turn also of k, on J9 is sub-logarithmic. Therefore, for the exchange-coupling constants related to CuMnO 2 and a-NaMnO 2 , k will be somewhat, but not drastically, reduced from the value 0.7, which is perfectly in line with the small theoretical overestimates of the T N . This analysis reveals that the Néel transition is predominantly determined by the Heisenberg Hamiltonian of equation (1). Moreover, the ferrodistortive structural transition accompanying the magnetic ordering and leading to the splitting of the two interchain exchange constants (J 2a , J 2b ) in the triclinic phase is needed to ensure a finite T N . The extent to which frustration is relieved in the triclinic phase of the CuMnO 2 elevates its ordering temperature above the ordering temperature in a-NaMnO 2 . Other factors, such as the interlayer coupling, the magnetic anisotropy and the ME coupling, can, at best, only slightly shift the T N .
Total-energy change at the T N . Having established that the magnetic ordering at the T N is predominantly set by the 2D Heisenberg Hamiltonian and the tendency of both systems to remove magnetic degeneracy in the ground state by lattice deformation, the question that arises is what is the microscopic origin of such a complex transformation. In this respect, the ME coupling has been suggested as being the key factor 23,26,27 in both a-NaMnO 2 and CuMnO 2 . However, the ME coupling has been shown to be insubstantial in the former case 18 , and thus needs to be evaluated also in the CuMnO 2 . The total-energy change at the T N , associated solely with the magnetoelasticity, arises from the coupling terms 44 b ij E ij m i m j between the strain-tensor components E ij and the magnetization-direction vector m 5 (m x , m y , m z ). The strength of the ME coupling, and consequently the corresponding contribution to the total energy, is proportional to the ME-coupling coefficients b ij (i, j 5 x, y, z). The coefficients b xx 5 2.3 MJ/m 3 , b yy 5 1.6 MJ/m 3 and b xy 5 3.4 MJ/m 3 are determined as linear terms in the calculated dependence of the total-energy-density change Df ij on the strain (see Methods), which is shown in Figure 2a. The ME energy gain is the largest for the shear-strain component E xy , which is associated with the monoclinic-to-triclinic deformation. However, for the experimental strain E xy~0 :0028 (see Methods) the magnetoelastic energy change amounts to only 2.7 meV per triclinic unit cell, which is very similar to the value of 2.5 meV found in a-NaMnO 2 .
The contribution of the ME coupling to the total energy change at the T N is thus negligible in both compounds. Therefore, the complex phase transition at the T N has to reflect changes in the magneticexchange and elastic energies 18 . Our ab-initio calculations of both relevant contributions to the total energy in the CuMnO 2 for the non-spin-polarized case, relevant to non-magnetically ordered structures, reveal that the monoclinic structure is energetically lower than the triclinic one, although only by ,1 meV per 4 formula units (f.u.); see Figure 2b. This is in-line with the C2/m crystal symmetry found experimentally at room temperature. However, once the magnetic order sets in, the total energy of the triclinic structure is lowered below that of the monoclinic structure by about 4 meV per 4 f.u. This change of ,1.25 meV per Mn 31 ion is mainly a consequence of the exchange-energy decrease during the structural phase transition, associated with the removal of the degenerate magnetic states due to the interchain frustration. The resulting splitting of the interchain exchange constants by J 2a 2 J 2b 5 0.4 meV (Ref. 33) releases S 2 (J 2a 2 J 2b ) 5 1.6 meV of energy per Mn that is slightly larger than the totalenergy change at the T N , as it is partially spent to compensate for the elastic-energy increase in the triclinic phase.
The calculated total-energy difference below the T N of 1 meV per f.u. in CuMnO 2 between the two structures that is about 3-times above the calculation error bar (see Methods), is markedly larger than in the a-NaMnO 2 , where the calculated difference was below the calculation error bar; * v0:5 meV per f.u. 45 . However, in absolute terms this difference is small, even in the CuMnO 2 , so that a competition between the near-degenerate monoclinic and triclinic structures is expected for both compounds. Experimental local-probe magnetic techniques are then essential for highlighting possible differences between the two systems.
NMR/NQR insight to the magnetism. Information about the magnetic properties of CuMnO 2 on the local scale are revealed in the NQR/NMR experiments via the hyperfine (hf) coupling A hf of the electronic and the 63,65 Cu nuclear magnetic moments. Moreover, the quadrupolar splitting in the electric-field gradient (EFG) provides information about the material's structural properties. The NQR spectra measured in zero field correspond to a single line for each copper isotope 46 , while the powder NMR spectra are structured (see Methods for details). Our simultaneous fit of the NQR and NMR data at 80 K ( Figure 3a) yields a hf coupling constant 63 A hf 5 2.3(1) T/m B that is significantly larger than the  coupling constant 23 A hf 5 0.11(1) T/m B found 18 in a-NaMnO 2 . Since A hf scales with the orbital overlap, the charge transfer from the Mn 31 ions to the interlayer cations (Cu 1 or Na 1 ) is much larger in the CuMnO 2 than in the a-NaMnO 2 . This implies a stronger J9 in the former compound and thus is in line with the somewhat better agreement between the experimental and the predicted T N , as the renormalization factor k in equation (2) is closer to the used value of 0.7.
The width d of the NQR spectra at 80 K, amounting to 0.16 MHz/ 27.2 MHz 5 0.6% of the line-position value n c already in the paramagnetic phase (Figure 3b), is rather large. The line widths 63 d . 65 d reveal spectral broadening being in accordance with the quadrupole moments 63 Q . 65 Q and contradicting the gyromagnetic ratios 63 c , 65 c. Therefore, sizeable structural distortions of the local environments must be present. The temperature dependence of both n c and d shows a pronounced sudden increase below the T N (Figure 3b), clearly marking the phase transition. The anomaly in n c at the T N is attributed to the structural transformation of the CuMnO 2 sample, directly affecting the quadrupolar frequency n Q . Namely, static internal magnetic fields below the T N cause a symmetric broadening/splitting of the NQR line so that its center of gravity is unaffected 46 . On the other hand, the pronounced increase of d by a factor of ,2 at the T N , exceeding the change of n c by several orders of magnitude, can only be magnetic in origin.
We must emphasize that the existence of the NQR signal below the T N is unexpected. Namely, in the frame of the homogeneously ordered magnetic phase 27 with k~0,1=2,1=2 ð Þthe Cu nuclei would experience extremely large internal magnetic fields. Although the Cu site is a structural center of inversion, the magnetic order breaks this symmetry, as the spins at 6r from a given Cu site are FM ordered (see the inset in Figure 3b), in contrast to the a-NaMnO 2 , where the order of the two corresponding spins is AFM 18 . Such a spin configuration in CuMnO 2 yields a large local hf field B hf~6 3 A hf 3|m^2:2 T (m 5 3.05 m B is the size of the ordered 26 Mn 31 moment). This field leads to extremely broad NQR spectra, 63 d~6 3 c 2p|2B hf^5 0 MHz, being two orders of magnitude broader than the experimental ones. Indeed, the NQR signal below the T N corresponds to a minority fraction of all the 63 Cu nuclei, while a majority of the signal is lost at the T N due to the onset of large internal fields. Namely, the Boltzman-corrected intensity of the NQR signal at 4.6 K, when further corrected for nuclear relaxation effects, is smaller than the intensity at 80 K by a factor of ,17. This reveals that, unexpectedly, about 6% of all the Cu sites in our sample experience small or no internal magnetic fields and do not correspond to the reported homogeneous magnetic phase. We note that the AFM order of the moments positioned symmetrically with respect to the Cu site, or the absence of any order, result in a zero static local magnetic field at the Cu site, and would explain the NQR-observable sites below the T N . Since this minority signal exhibits clear anomalies at the T N (Figure 3b) it is obviously well coupled to the bulk that undergoes the magnetostructural transition. This is confirmed by the temperature dependence of the spin-lattice relaxation rate, 1/T 1 , that shows a maximum at the T N due to critical spin fluctuations 47 related to the magnetic instability of the bulk. However, in contrast to the monotonic decrease found in the Na-based compound below the T N , in the CuMnO 2 , another clear maximum in both the NMR and NQR 1/T 1 is observed at around 10 K. This reveals an, as yet, unobserved instability that could be either magnetic or structural in its nature.
Probing the magnetic disorder with mSR. In order to provide more insight into the magnetic state in the CuMnO 2 that is, according to the unexpected minority NQR signal, apparently not as homogeneous as inferred from previous bulk measurements, we resorted to the mSR local-probe technique. Moreover, this technique reveals details about the low-temperature anomaly in the NMR/NQR relaxation at 10 K. In contrast to NQR/NMR, which is limited because the intrinsic signal disappears below the T N , the mSR measurements can assess the magnetic properties of the entire CuMnO 2 sample also below the T N . This time a hf/ dipolar coupling between the electronic magnetic moments and the muon magnetic moment is utilized after a muon stops in the sample. The resulting local magnetic field B m at the muon site affects the mSR asymmetry A(t) that is proportional to the muon polarization precessing in B m . In CuMnO 2 , the weak-transverse-field (wTF) experiment that effectively keeps track of the temperature-dependent ordered part of the sample by measuring the amplitude of the oscillating mSR signal 48 , reveals that the fraction of the muons detecting large frozen internal fields starts growing already below 80 K ( Figure 4a); i.e., far above the T N , which can be attributed to developing short-range spin correlations 48 . Below the T N , the whole sample becomes magnetically ordered within only a few kelvins, leaving no room for a non-frozen fraction above the experimental error bar of a few percent. Similar information is obtained from the zero-field (ZF) mSR, where the initial asymmetry strongly decreases around the T N and at low temperatures reaches 1/3 of its high-temperature value (inset in Figure 4b). Such a reduction is characteristic of the establishment of strong static internal fields in powder samples. Statistically, in 1/3 of all cases the muon magnetic moment is aligned parallel to the B m and therefore exhibits no precession, while rapid oscillations of the asymmetry in other cases diminish the mSR signal on a coarse time scale.
A detailed look at the ZF relaxation curves below T N (Figure 4c) also allows for the detection of the quickly-oscillating component. Similar to the a-NaMnO 2 case 18 , the two-component model fits well with the experimental data. Here, f j denotes temperatureindependent probabilities that the muons stop at either of the two magnetically non-equivalent stopping sites j. The preferential site is occupied in 70(5)% cases. The internal field at this site is only slightly higher than at the second site (0.59 and 0.54 T at the first and the second site, respectively, at 5 K); however, a fit with only a single oscillating component (the dashed line in Figure 4c) results in a much poorer agreement with the data. The damping rate of the oscillations l T,j that is due to the finite width of the local-field distributions 18 in CuMnO 2 is reduced by a factor of ,3 when compared to the a-NaMnO 2 (Figure 4c), indicating more homogeneous magnetism.
In the ZF experiment, the magnetic phase transition at the T N is expressed as a maximum of the longitudinal muon-relaxation rate l L , like in the NQR/NMR relaxation experiments. Moreover, the second maximum observed in the NQR/NMR experiments at 10 K is also found in the mSR. Since the ZF mSR signal corresponds to the total volume of the sample and the muons are only sensitive to magnetism, this reveals that the low-temperature anomaly is of magnetic origin and is intrinsic to the CuMnO 2 system.

Discussion
The FTLM and DFT numerical calculations provide a solid basis for addressing the experimentally observed similarities and differences between the CuMnO 2 and the a-NaMnO 2 . Considering the latter calculations in the magnetically ordered state, the triclinic phase is energetically preferred in both compounds. With increasing temperature, the staggered susceptibility decreases and this leads to a finite T N . Above the T N the exchange-energy gain associated with the magnetically ordered state disappears, which in turn leads to a structural transformation to the monoclinic phase that is energetically preferred in the non-magnetic state. The isotropic Heisenberg Hamiltonian of the spatially anisotropic triangular lattice is dominantly responsible for elevating the T N in the CuMnO 2 with respect to the a-NaMnO 2 , while the magnetoelastic and the interlayer couplings play a less important role.
On the other hand, our local-probe experiments on the CuMnO 2 revealed some subtle, yet profound, features that should be carefully considered in the attempt to understand the presence/absence of nano-scale phase separation in the spatially anisotropic triangular lattice. Both, the NQR/NMR and the mSR investigations demonstrated that CuMnO 2 undergoes a magnetostructural phase transition at T N 5 65 K almost completely. The minority NQR component (,6%) that remains present below the T N can be explained by regions where the interlayer magnetic ordering is FM instead of being AFM, as the latter causes the disappearance of the NQR signal due to large local fields. The NQR signal exhibits a magnetic anomaly around 10 K, which is expressed by the increased relaxation rates of the NQR/NMR as well as the mSR. Since mSR, on the other hand, detects a bulk magnetic signal, the small NQR component is apparently coupled to the bulk magnetic phase. This is further confirmed by the line position and the width of the NQR spectra, changing considerably at the T N . The coupling with the bulk phase can then be regarded in the context of the nano-scale phase inhomogeneity. A comparison of the ZF mSR asymmetry curves of the CuMnO 2 and the a-NaMnO 2 is quite informative in this respect. The notably reduced damping of the oscillations in the ordered phase of the former compound provides evidence of much narrower field distributions, and hence less disorder. This conclusion is also in line with the number of the interlayer cation (Cu 1 and Na 1 ) sites experiencing internal fields that do not comply with the symmetry of the bulk magnetic order, which in the CuMnO 2 is decreased to 6%, from the 30% found 18 in the a-NaMnO 2 .
The magnetostructurally inhomogeneous ground state of the a-NaMnO 2 on the nano-scale has previously been attributed to the combined effects of geometrical frustration and near-degenerate monoclinic and triclinic structural phases 18 . We believe that the key factor controlling such an inhomogeneity is the difference in the total energy of the two competing phases in the magnetically ordered state. This difference is notably larger in the CuMnO 2 (1 meV per f.u.) than in the a-NaMnO 2 , where it is below the computational error bar (,0.5 meV per f.u. 45 ). In the latter compound, an infinitesimal quenched disorder, locally favouring one phase over the other, can then be held responsible for triggering the phase separation. Similar effects are suppressed in the stoichiometric CuMnO 2 , but would become enhanced for larger deviations from perfect system uniformity. Indeed, enhanced strain, acting as a precursor of the monoclinicto-triclinic structural phase transition, has been observed 18,23 in the high-temperature monoclinic phase in stoichiometric a-NaMnO 2 , while in the CuMnO 2 a Cu-Mn off-stoichiometry is required to produce such a strain 28 . Moreover, the diffuse magnetic scattering characteristic of 2D correlated regions that coexist with sharp magnetic Bragg peaks (one of the signatures of the inhomogeneity 18 found in the a-NaMnO 2 ) is also found 26,30,31 in the CuMnO 2 . However, in contrast to the a-NaMnO 2 , where it persists to low temperatures, in stoichiometric CuMnO 2 it gradually gives way to the 3D ordered phase below the T N . Interestingly though, in off-stoichiometric samples 30 the volume fraction of the 2D-correlated phase shows no decrease below the T N , implying that the 2D-ordered regions keep competing with the 3D order at low temperatures. The total-energy difference of the competing phases below the magnetostructural transition, reflect-ing the interplay of the magnetic-exchange and the elastic energies, then seems to determine the amount of disorder required to stabilize the inhomogeneous ground state on a geometrically frustrated triangular lattice. Systems with near-degenerate competing phases can be locally perturbed more easily. Such an inhomogeneity may, therefore, be a more general feature of geometrically frustrated magnets.

Methods
Finite-temperature Lanczos method simulations. Calculations of the spin susceptibility for the S 5 2 Heisenberg model on the anisotropic triangular lattice (equation (1)), were performed with the finite-temperature Lanczos method (FTLM) 49,50 and were used to determine the leading exchange couplings J 1 and J 2 in the CuMnO 2 . Within the FTLM finite-size clusters are diagonalized in a similar manner as for the standard exact diagonalization Lanczos method (at T 5 0) and additional random vector averaging over the R vectors is employed to determine the properties at T . 0. Typically, R , 10 suffices for the largest systems and the lowest T, while smaller systems require a larger R. The limitations of the method are mainly set by finite-size effects, which are the largest at low T and determine the lowest reachable T(,J 1 ). In order to reduce the finite-size effects we used periodic boundary conditions, adjusted cluster shapes, the largest reachable cluster sizes (up to N 5 14 sites), and additional approximations for the values in the thermodynamic limit.
The temperature-dependent magnetic susceptibility x(T) of the model given by equation (1) was calculated previously in Ref. 35 for a-NaMnO 2 . It was shown that in the regime of interest, elongated spin clusters are the most appropriate. In particular, if a cluster has m independent spins in the J 1 direction and n spins in the J 2 directions, it was realized that due to J 2 , J 1 and two competing J 2 bonds, x does not depend on n for n . 2 (see Fig. 2 in Ref. 35). This fact allows us to reduce the finite-size effects by using a larger m. We note that due to the alternating behaviour of x with m (see Figure 5), which originates in periodic boundary conditions and antiferromagnetic spin-spin correlations, the average susceptibility curve from the two largest-size clusters (m 5 6 and 7) is a better approximation than the m 5 7 curve.
Scaling-like approximation for the susceptibility. The results for several different sizes of finite clusters and their systematics, shown in Figure 5, allow a scaling-like analysis to obtain a better approximation of x in the thermodynamic limit. Typical scaling analyses use scaling functions of the form x(N) 5 a 1 b/N and additional higher terms when needed; e.g., c/N 2 . Since our calculations are limited to rather small maximum system sizes by S 5 2, we also use the results from small systems (starting with m 5 2). Consequently, such scaling functions are not appropriate. In particular, at higher T (see, for example, T . 5J 1 in Figure 5) x has already converged with N for systems with m $ 5, while for m , 5 notable finite-size effects are seen. Therefore, the scaling function should be close to a constant for 1/N smaller than some value, while at larger 1/N, the scaling function should allow for a stronger N dependence. For these reasons we use a generalized scaling function of the form x(N) 5 a 1 b [exp(c/N) 2 1], which corresponds to typically used functions in the limit of small 1/N. In order to also capture the alternating component of x with m ( Figure 5) we add, in a similar fashion, the term b 1 [exp(c 1 /N) 2 1] (21) N/2 . Such a scaling function also gives a correct (converged with N) result for high T, while typical scaling functions fail in this respect. We have performed such a scaling for each T separately. However, since we are limited to small systems with notable finite-size effects at low T * vJ 1 À Á and since the scaling function has many parameters, the result of such an analysis should not be taken as a strict thermodynamic limit. Rather, it should be regarded as a next approximation of it, compared to the result from simpler averaging of the two largest-cluster curves.
Density-functional theory calculations. The calculations of the total energies and the magneto-elastic (ME) coupling coefficients were performed within the framework of the density-functional theory (DFT) and the generalized-gradient approximation (GGA) 51 for the exchange-correlation contribution by applying the Quantum Espresso code 52 . The electron-ion interactions were described by the Vanderbilt ultrasoft potentials 53 including the spin-orbit coupling for the Mn atoms. The plane-wave cut-off parameters were set to 585 eV and 4678 eV for the expansion of the wave functions and the potential, respectively. In order to take into account the proper antiferromagnetic ordering, the 1 3 2 3 2 supercells of the monoclinic and the triclinic structures were used. The calculations of the total energies as a function of the unit-cell volume for the different types of magnetic ordering were carried out by using 4 3 8 3 2 reciprocal vectors in the full Brillouin zone (BZ) for the Methfessel-Paxton sampling 54 integration. The criterion for the self consistency was the total-energy difference between two subsequent iterations being less than 10 28 Ry. The monoclinic phase was further optimized by minimizing the total energy and the interatomic forces with respect to the lattice parameters and the atomic positions. The resulting structure served as the zero-strain reference for the calculations of the ME coefficients that are based on the evaluation of the total-energy differences of the order of ,10 24 Ry, which is also the accuracy for the determination of the totalenergy differences between the monoclinic and triclinic phases in Figure 2b, calculated per 4 f.u. The tests yielded 8 3 16 3 4 reciprocal vectors in the full BZ to be enough for well-converged results.
Determination of the magnetoelastic coupling. The magnetoelastic coupling constants b ij are calculated from the associated magnetoelastic energy density. This contains the products b ij E ij m i m j of the strain-tensor components E ij and the components m i of the normalized magnetization. The form of the magnetoelastic energy density is determined by the symmetry of a particular system 44 . The magnetism of the CuMnO 2 and the a-NaMnO 2 is essentially two-dimensional; therefore, only the terms with the lateral strain-tensor components are important. For the monoclinic symmetry, these include Individual magnetoelastic terms are then determined by specifically choosing strain components and magnetization directions and calculating the total-energy density f E xx ,E yy ,E xy ,m x ,m y ,m z À Á , from Df xx~f E 0 ,0,0,1,0,0 ð Þ {f E 0 ,0,0,0,0,1 ð Þ , The above total-energy differences are calculated ab-initio as a function of E 0 for relaxed crystal structures. In CuMnO 2 , Df xx and Df yy change linearly with increasing strain at least up to E 0~0 :03, while an additional quadratic term is observed in Df xy (Figure 2a). The experimental strain value E xy~0 :0028 that is obtained by calculating the relative shift of the Mn 21 ions in the triclinic structure, when compared to the monoclinic structure (based on high-resolution synchrotron XRD data analysis 18 ), is an order of magnitude lower. Therefore, the linear term is dominant for all three contributions and allows the extraction of the three magnetoelastic constants b xx 5 2.3 MJ/m 3 , b yy 5 1.6 MJ/m 3 and b xy 5 3.4 MJ/m 3 .
Nuclear magnetic/quadrupolar resonance. 63,65 Cu (I 5 3/2) NMR/NQR measurements were performed on a high-quality powder sample with the same phase purity and stoichiometry as in the study presented in Ref. 27. The NMR/NQR spectra and the spin-lattice relaxation were measured between 4.6 K and 120 K in a magnetic field of 8.9 T (NMR) and in zero magnetic field (NQR) on a custom-built spectrometer. Frequency sweeping and a solid-echo pulse sequence were used for recording the spectra, while a saturation recovery method was used for measuring the spin-lattice relaxation. Typical p/2-pulse lengths were 3.5 ms and 6 ms in the NMR and NQR experiments, respectively. The reference NMR Larmor frequencies of 63 n 0 5 100.728 MHz and 65 n 0 5 107.908 MHz were determined with a 0.1 M NaClsolution reference by taking into account the gyromagnetic ratios 23 c 5 2p 3 11.261 MHz/T, 63 c 5 2p 3 11.295 MHz/T and 65 c 5 2p 3 12.089 MHz/T. The NQR spectrum of each isotope is particularly simple, as it is given by a single line 46 at 63,65 n NQR~6 3,65 n Q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 1zg 2 =3 p , with the ratio of the quadrupolar frequencies 63 n Q / 65 n Q 5 1.08 fixed by the corresponding quadrupolar moments and the EFG tensor V ij asymmetry parameter being g 5 (V xx 2 V yy )/V zz . The NMR spectrum is more complicated, because the applied magnetic field B 0 breaks the symmetry in the spin space. The central-transition (21/2 « 1/2) powder NMR line adopts a characteristic structure because of the angular-dependent NMR shift K from the reference frequencies n 0 5 cB 0 /2p, 63,65 K~n { 63,65 n 0 63,65 n 0~K hf zK d z 63,65 K Q . In analogy 18 to the a-NaMnO 2 , we take the hf shift K hf~ hA hf m=n 0 ( h is the reduced Planck constant) to be isotropic, while the dipolar contribution K d and the quadrupolar 46 shift 63,65 K Q can be accurately calculated. The former has a uniaxial symmetry and is calculated 18 (K d 5 0.11% is the dominant eigenvalue) by taking into consideration all the Mn 31 paramagnetic spins around a given Cu site within a sphere large enough to ensure convergence.
A homogeneous life-time broadening of the NQR spectra is negligible. The spinspin relaxation time 63 T 2 5 46 ms at 80 K yields 63 d h 5 6.9 kHz, which is much smaller than the spectral width. The spin-lattice relaxation is of magnetic origin. We find the isotopic effect 65 T 1 / 63 T 1 5 0.86 that is in accordance with magnetic relaxation dictating 1=T 1 !c 2 A 2 hf .
Muon spin relaxation. The mSR investigation was carried out on the General Purpose Surface muon (GPS) instrument at the Paul Scherrer Institute, Villigen, using the same powder sample as in the NMR/NQR experiments. Zero-field (ZF) and weak-transverse-field (wTF) measurements in a 3 mT magnetic field were performed in the temperature range between 5 and 120 K. The veto mode was utilized to minimize the background signal. The ZF mSR measurements below the T N revealed that each muon stops at one of the two possible non-equivalent stopping sites, like was observed 18 in a-NaMnO 2 .