# Spin order and dynamics in the diamond-lattice Heisenberg antiferromagnets CuRhO and CoRhO

###### Abstract

Antiferromagnetic insulators on the diamond lattice are candidate materials to host exotic magnetic phenomena ranging from spin-orbital entanglement to degenerate spiral ground-states and topological paramagnetism. Compared to other three-dimensional networks of magnetic ions, such as the geometrically frustrated pyrochlore lattice, the investigation of diamond-lattice magnetism in real materials is less mature. In this work, we characterize the magnetic properties of model A-site spinels CoRhO (cobalt rhodite) and CuRhO (copper rhodite) by means of thermo-magnetic and neutron scattering measurements and perform group theory analysis, Rietveld refinement, mean-field theory, and spin wave theory calculations to analyze the experimental results. Our investigation reveals that cubic CoRhO is a canonical diamond-lattice Heisenberg antiferromagnet with a nearest neighbor exchange meV and a Néel ordered ground-state below a temperature of 25 K. In tetragonally distorted CuRhO, competiting exchange interactions between up to third nearest-neighbor spins lead to the development of an incommensurate spin helix at 24 K with a magnetic propagation vector . Strong reduction of the ordered moment is observed for the spins in CuRhO and captured by our corrections to the staggered magnetization. Our work identifies CoRhO and CuRhO as reference materials to guide future work searching for exotic quantum behavior in diamond-lattice antiferromagnets.

## I Introduction

Antiferromagnetic insulators often host novel forms of magnetic matter dominated by strong quantum fluctuations. Low dimensionality, Affleck (1989); Mikeska and Kolezhuk (2004); Lake et al. (2005); Coldea et al. (2010) geometrical frustration, Ramirez (1994); Lee (2008); Han et al. (2012); Savary and Balents (2016) spin-orbit coupling Jackeli and Khaliullin (2009); Banerjee et al. (2017) or topology Chisnell et al. (2015); Hirschberger et al. (2015); Chernyshev and Maksimov (2016) are known ingredients to suppress classical behavior in favor of more exotic spin order and dynamics. In three-dimensional (3D) magnets, the pyrochlore lattice has been a particularly fruitful platform to expose new physics, in particular in rare-earth compounds. Bramwell and Gingras (2001); Gardner et al. (2010); Fennell et al. (2009); Ross et al. (2011) Other three-dimensional lattice geometries, such as the diamond lattice, have been less extensively studied primarily because of the absence of obvious geometrical frustration.

Diamond-lattice Heisenberg antiferromagnets have attracted some recent attention, however, following the observation of a spin-liquid phase in the A-site spinel MnScS. Krimmel et al. (2006); Gao et al. (2016) This motivated detailed theoretical work that uncovered the existence of remarkable degenerate spin-spiral states when a dominant nearest-neighbor antiferromagnetic interaction competes with a small next-nearest neighbor exchange, Bergman et al. (2007); Bernier et al. (2008) i.e in presence of exchange frustration. It was also realized that spin-orbital degeneracy may play an important role in stabilizing exotic physics as for FeScS Fritsch et al. (2004); Krimmel et al. (2005); Laurita et al. (2015); Mittelstädt et al. (2015); Plumb et al. (2016); Biffin et al. (2017) in which spin-orbital entanglement Chen et al. (2009a, b) is an active ingredient. Furthermore, as demonstrated for CoAlO Suzuki et al. (2007); MacDougall et al. (2011); Zaharko et al. (2014); MacDougall et al. (2016), the combination of chemical disorder with the above effects can produce unique glassy magnetic behavior of great current interest. MacDougall et al. (2011)

The bipartite nature of the diamond-lattice may in fact be a favorable feature to create radically new forms of magnetism, such as the 3D topological paramagnetism recently proposed for frustrated diamond-lattice antiferromagnets. Wang et al. (2015) In that scenario, the ground-state is an exotic superposition of fluctuating Haldane () chains,Haldane (1983) and can be pictured as a 3D version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) construction Affleck et al. (1987) used in 1D. Remarkably, NiRhO Chamorro and McQueen (2017); Chen (2017) has already been identified as a promising candidate material to realize such topological paramagnetism, although the detailed role played by orbital degeneracy, spin-orbital entanglement, chemical disorder and exchange frustration in that material remains to be fully elucidated.

In this paper, we focus on the antiferromagnetic A-site spinels CoRhO (cobalt rhodite) and CuRhO (copper rhodite), the latter of which is isostructural with NiRhO. Our combined experimental and theoretical work relies primarily on a neutron scattering investigation of high-quality polycrystalline samples, and establishes the canonical magnetic behavior expected for diamond-lattice Heisenberg antiferromagnets in A-site spinels. In cubic CoRhO we show that the spins are unfrustrated and display static and dynamic properties in excellent agreement with mean-field and spin-wave theory predictions. In tetragonally-distorted CuRhO, however, we uncover an incommensurate magnetic order for the spins and the presence of sizable quantum effects. We provide detailed modeling of these observations using mean-field and spin-wave theory up to -order, and determine that the microscopic Hamiltonian for CuRhO involves sizable and competing magnetic exchange interactions up to the third nearest neighbor. Our results are an important reference point in the context of an accelerated search for exotic magnetic behavior on the diamond lattice.

This paper is organized as follows. Sec. II contains experimental details of our combined thermo-magnetic, X-ray and neutron characterization of polycrystalline samples of CoRhO and CuRhO. Sec. III presents and analyzes our results on CoRhO, demonstrating that this coumpound is a model realization of the diamond-lattice Heisenberg antiferromagnet with . Sec. IV, discusses CuRhO for which frustrated exchange interactions lead to the development of an helical ground-state with strong zero-point reduction of the moments. In Sec. V, we present mean-field and spin-wave theory results for the general Hamiltonian relevant for CuRhO and discuss quantum effects in distorted diamond-lattice Heisenberg antiferromagnets that might be relevant for other materials. Sec. VI concludes this work and additional details are provided in the Appendix.

## Ii Methods

### ii.1 Synthesis and determination of crystal structure

Black, polycrystalline samples were prepared by intimately mixing and grinding stoichiometric amounts of CoCO (Baker Adamson, 99.9%), CuO (Aldrich, 99.99%), and RhO in an agate mortar. The RhO was obtained by decomposing RhCl (Johnson Matthey, 99.9%) at 850\celsius for 12 hours under air flow. The samples were then pressed as pellets and sintered at 900-950\celsius for 36 hours (CuRhO) and 900-1000\celsius for 36 hours (CoRhO) with intermediate grinding.

Initial X-ray diffraction (XRD) characterization was performed using a Rigaku Miniflex II diffractometer using Cu K radiation and a graphite monochromator. Room temperature time-of-flight neutron diffraction data were collected on POWGEN at Oak Ridge National Laboratory’s (ORNL) Spallation Neutron Source (SNS) using 6-mm diameter vanadium sample cans. Rietveld analysis of the room-temperature X-ray and neutron diffraction data was carried out using the FULLPROF suite of programs. Rodriguez-Carvajal (1993)

### ii.2 Thermo-magnetic measurements

Magnetization measurements were performed using a SQUID magnetometer in an applied magnetic field of T. The temperature dependence of the magnetization was measured for 2 320 K on polycrystalline samples mounted in gelatin capsules. After removing the contribution from the gelatin, the magnetic susceptibility was obtained as where mol.emu and mol.emu are the calculated temperature independent ionic core contributions for CoRhO and CuRhO, respectively. Bain and Berry (2008)

Heat capacity measurements were performed using the relaxation method on a Quantum Design Physical Properties Measurement System (PPMS) equipped with a 14 T magnet. Polycrystalline samples were mixed with silver and pressed into pellets to increase their thermal conductivity. Contributions from the sample platform and grease, and from silver, were subtracted through separate measurements over the entire 1.6 100 K temperature range of our measurements.

### ii.3 Magnetic neutron diffraction

Low-temperature neutron powder diffraction measurements were performed on HB-2A at ORNL’s High Flux Isotope Reactor (HFIR). Garlea et al. (2010) Loose polycrystalline samples (4.0 g of each of CoRhO and CuRhO) were enclosed in narrow 6-mm diameter cylindrical aluminum cans to minimize the effects of neutron absorption in Rh, and sealed under one atmosphere of He at room temperature. The sample cans were mounted at the bottom of a close-cycled refrigerator reaching a base temperature K and measurements were conducted with two neutron wavelengths, Å from Ge(113) and Å from Ge(115).

### ii.4 Inelastic neutron scattering

Inelastic neutron scattering measurements were performed on the Fine-Resolution Fermi Chopper Spectrometer (SEQUOIA) at ORNL’s SNS. Granroth et al. (2010); Stone et al. (2014) The above samples and an empty aluminum can were mounted on a three-sample changer at the bottom of a close-cycle refrigerator reaching a base temperature of K. Incident neutron energies of meV and meV, used in combination with a Fermi chopper frequency of 360 Hz, provided full-width at half-maximum (FWHM) elastic energy resolutions of meV and meV, respectively. Measurements were taken from base temperature to K, and the contribution from the empty can has been subtracted from the inelastic neutron scattering measurements.

### ii.5 Spin dynamics simulations

Unless otherwise noted, we modeled the magnetic excitations of CoRhO and CuRhO using the numerical implementation of linear spin-wave theory Petit, S (2011) in the program SpinW. Toth and Lake (2015) In our simulations, we assume a diagonal form for Heisenberg exchange interactions, i.e. the Hamiltonian for -th nearest neighbors reads where the sum runs on all pairs of -th nearest neighbor spins twice. The reported neutron scattering intensity for neutron energy-transfer and momentum-transfer is proportional to the powder-averaged dynamical structure factor computed by SpinW, , where is the form-factor for Co or Cu and cm.

Our simulations are convoluted with a simple Gaussian lineshape to account for the and resolution of the spectrometer, which are assumed uncoupled. The -dependence of the -resolution is calculated from simple geometrical considerations and calibrated with the observed elastic -resolution. The -resolution is taken to be uniform across the whole -range and estimated from the width of the observed magnetic Bragg peaks.

## Iii Results on Cobalt Rhodite

### iii.1 Structural analysis

We start our experimental investigation by presenting the ideal diamond-lattice crystal structure of CoRhO. This material crystallizes in the cubic spinel structure [Fig. 1] with space group and room-temperature structural parameters reported in Tab. 1. With respect to the general spinel structure ABO, Co occupies the tetrahedrally coordinated A-site and Rh the octahedrally coordinated B-site. This results in a perfect diamond lattice for the Co ions with four nearest-neighbor Co atoms at a distance of Å. Nearest-neighbor magnetic exchange interactions are mediated by direct exchange or more likely by Co–O–Rh–O–Co superexchange paths Blasse (1963). Next-nearest-neighbor exchanges, if present, involve twelve equivalent superexchange pathways with Co–Co distances of Å.

CoRhO [ CoRhO ], K | ||||||
---|---|---|---|---|---|---|

Atom | Site | Occ. | (Å) | |||

Co | 0 | 0 | 0 | 1.0 | 0.0021(2) | |

Rh | 5/8 | 5/8 | 5/8 | 0.95(6) | 0.0002(1) | |

Co | 5/8 | 5/8 | 5/8 | 0.05(6) | 0.0002(1) | |

O | 0.2601(1) | 0.2601 | 0.2601 | 1.0 | 0.0023(1) |

The results of our refinement are consistent with previous reports Bertaut et al. (1959); Cascales and Rasines (1984) with two notable differences. First, the RhO octahedral are less distorted in our structure compared to previous reports; the shortened (respectively elongated) Co–O (respectively Rh–O) bonds lead to more chemically-reasonable bond-valence sums Brown (1981) of 1.79 for Co and 3.05 for Rh. Second, our refinements indicate a small degree of site mixing with 5.0(6)% of Co on the B-site and formally, a refined chemical formula of CoRhCoO. The Rh deficiency originates from the presence of a small RhO impurity phase. To maintain overall charge balance, either octahedral Co ions are , i.e. Rh(III)Co(III), or approximatively 5% of the Rh ions are , i.e. Rh(III)Rh(IV)Co(II). Since the ionic radii for either scenario are similar it is not possible to favor one scenario over the other based on structural refinements alone. Although formally CoRhO, we refer to our compound as CoRhO in the rest of this manuscript.

### iii.2 Thermo-magnetic properties

Magnetic and thermodynamic measurements for CoRhO are presented in Fig. 2. The inverse magnetic susceptibility [Fig. 2(a)] is linear over a broad range of temperatures K. A Curie-Weiss fit to the high-temperature paramagnetic regime ( K) yields a negative Weiss temperature K and an effective moment , consistent with previous reports. Blasse and Schipper (1963); Blasse (1963) In the undistorted tetrahedral crystal-field environment, Co adopts the electronic configuration with one unpaired electron in each , and orbitals. Bertaut et al. (1959) For such magnetic moments, the experimental value of yields a gyro-magnetic ratio after correcting for the presence of Co atoms per formula unit. At low temperatures, the magnetic susceptibility [Fig. 2(a)-inset] displays a sharp absolute maximum closely followed by an inflection point at K, attributed to long-range antiferromagnetic ordering. Blasse and Schipper (1963); Blasse (1963); Fiorani and Viticoli (1979)

These results are fully corroborated by heat-capacity measurements. The specific heat of CoRhO, plotted as [Fig. 2(b)], shows a sharp -shaped anomaly at K, indicative of a second-order phase transition. The precise correspondence between specific heat and magnetic susceptibility leaves no doubt as to its magnetic nature. Most of the specific heat above can be accounted for by a phonon model with two Debye temperatures, K and 742(9) K. Integrating the magnetic part of from 1.7 K to 50 K yields an entropy change J.K.mol, consistent with J.K.mol expected for degrees of freedom. Below , the magnetic contribution to the specific heat dominates and a broad feature is observed around K, which we attribute to magnon-magnon interactions. Below , the specific heat follows a behavior, as expected for gapless antiferromagnetic magnons. Given the relatively large energy scale set by K, a large applied magnetic field of T has almost no influence on the transition temperature. We observe a shift downward by a mere K [Fig. 2(b)-inset]. Overall, our measurements yield a frustration ratio Ramirez (1994) and suggest that CoRhO behaves as a canonical non-frustrated three-dimensional antiferromagnet with an average exchange interaction between nearest-neighbor magnetic moments () of meV.

### iii.3 Magnetic structure

Neutron powder diffraction allows one to determine the magnetic structure of CoRhO below the antiferromagnetic ordering transition at K [Fig. 3]. Upon cooling our sample from 40 K to 4 K, we observe a sizable change of intensity for some of the nuclear Bragg peaks, coinciding with the development of new Bragg peaks at nuclear positions forbidden by the space-group symmetry, for instance ( Å) and ( Å). The integrated intensity of the peak [Fig. 3-inset] follows an order-parameter behavior with a sharp onset at K, in close correspondence with the thermodynamic anomalies. We thus associate the change in Bragg scattering with the development of long-range magnetic ordering.

All the observed magnetic Bragg peaks can be indexed by the magnetic propagation vector with respect to the conventional unit cell. To determine the magnetic structure, we first investigate possible symmetry-allowed magnetic structures using the program Isodistort. Campbell et al. (2006) For CoRhO, there are two irreducible representations (irreps), labeled and in the notation of Miller and Love. Miller and Love (1967) These correspond to simple ferromagnetic and antiferromagnetic ordered pattern on the diamond lattice [Fig. 4(a)], respectively. As anticipated from the negative Curie-Weiss constant, only correctly accounts for the observed magnetic intensity. The resulting spin structure (magnetic space group ) is shown in Fig. 4(b). Our Rietveld refinement [Fig. 3] is in excellent agreement with the data () and yields an ordered magnetic moment , close to the value of expected for a ion with . Neutron powder diffraction thus demonstrates that CoRhO orders in a simple two-sublattice antiferromagnetic structure at K and places an upper bound of 5% on any reduction of the ordered moment due to quantum fluctuations at K.

### iii.4 Magnetic excitations

Inelastic neutron scattering measurements on CoRhO [Fig. 5(a)] reveal a simple magnetic excitation spectrum we associate with non-interacting magnons, i.e. spin fluctuations transverse to the ordered spin patterns of Fig. 4(b). The magnetic spectrum appears gapless within the resolution of our experiments, with characteristic acoustic spin-wave branches emerging from the strong magnetic Bragg peak positions. The bandwidth of the magnetic signal meV = 44 K matches well with the value of the Weiss constant K and corresponds to the energy of magnons at the Brillouin zone boundary. We obtain an excellent correspondence between the data and the calculated scattering intensity [Fig. 5(b)] with a single nearest-neighbor exchange parameter meV [Fig. 4(b)]. This matches very well with the average exchange value extracted from the magnetic susceptibility meV, indicating that further neighbor exchanges and magnon energy renormalization effects can be neglected in CoRhO.

The temperature dependence of the magnetic excitations [Fig. 6(a)] reveals a very rapid collapse of the magnetic excitations as is crossed. Unlike low-dimensional quasi-1D and quasi-2D magnets for which the overall bandwidth and shape of the magnetic excitations persists at and above , Lake et al. (2005); Rønnow et al. (1999) the excitations of CoRhO resemble that of a paramagnet already for . The top of the magnon band is considerably renormalized and broadened at , a temperature above which the excitations loose coherence and the inelastic signal becomes purely relaxational [Fig. 6(b)]. While the detailed analysis of the temperature dependence of these excitations is beyond the scope of this work, the simplicity of the spectrum and the presence of an unique energy scale meV makes CoRhO a model 3D antiferromagnetic material.

## Iv Results on Copper Rhodite

### iv.1 Structural analysis

CuRhO crystallizes in a lower-symmetry crystal structure than CoRhO due to a Jahn-Teller distortion around K Bertaut et al. (1959); Blasse (1963) lifting the degeneracy of the electronic configuration of Cu. The necessary destabilization of the magnetic orbital below leads to a compression of the oxygen tetrahedral with respect to the cubic cell. Bertaut et al. (1959) Indeed the structure of CuRhO has been described by both X-ray Khanolkar (1961) and neutron diffraction Ismunandar et al. (1999) as a tetragonally distorted spinel with space group Ismunandar et al. (1999) or .Khanolkar (1961)

CuRhO, K | ||||||
---|---|---|---|---|---|---|

Atom | Site | Occ. | (Å) | |||

Cu | 0 | 3/4 | 0.1368(3) | 1.0 | 0.0017(2) | |

Rh | 0 | 0 | 1/2 | 1.0 | 0.0005(1) | |

O | 0 | 0.0334(1) | 0.2430(1) | 1.0 | - | |

Anisotropic Atomic Displacement Parameters (Å) | ||||||

Atom | ||||||

O | 0.0021 | 0.0010 | 0.0022 | 0.0 | 0.0 |

Our room temperature neutron diffraction results for CuRhO are shown in Fig. 7. The results of our Rietveld refinement, reported in Table 2, yield as the appropriate room-temperature space group, consistent with the most recent studies. Ismunandar et al. (1999); Dollase and O’Neill (1997) Unlike CoRhO, we find no evidence for site mixing with bond valence sums of 3.05 for Rh, 1.97 for O and 1.79 for Cu. A close look at the crystal structure indicates that Rh octahedral are distorted with four distinct O—Rh—O bond angles of 98.23(4), 81.77(4), 92.83(5) and 87.17(5). In turn, the Cu tetrahedral are flattened with two distinct O—Cu—O bond angles of 128.8(2) and 102.6(1). For comparison, there are only two O-Rh-O angles of 85.10(3) and 94.90(3) and a single O-Cu-O angle of 109.47(3) in CoRhO. Our refined crystal structure also indicates Cu is displaced off the ideal site in a disordered manner. Instead, the copper position splits between two positions that are randomly occupied along the axis. Overall the tetragonal distortion leads to four nearest-neighbor Cu—Cu distances within of each other such that nearest-neighbor Cu ions in CuRhO effectively remain organized on a diamond lattice at an average distance of 3.61(5) Å. When compared to the cubic structure of CoRhO, however, next-nearest-neighbor Cu—Cu distances are strongly split into four short and eight long links. We will see below this has profound consequences for the magnetic properties of CuRhO.

### iv.2 Thermo-magnetic properties

Magnetic and thermodynamic measurements for CuRhO are presented in Fig. 8. Unlike CoRhO the inverse magnetic susceptibility [Fig. 8(a)] only becomes linear at high temperature after subtraction of a positive Van-Vleck contribution , associated with paramagnetic Rh. Endoh et al. (1999) Linearity of for K is obtained using emu.mol from which a Curie-Weiss fit yields K and . The obtained effective moment is somewhat too large for Cu. Using an empirical emu.mol, we obtain a good Curie-Weiss fit above K with values of K and , compatible with a previous report Endoh et al. (1999) and corresponding to a realistic gyro-magnetic ratio for the Cu ions. At low temperatures, the magnetic susceptibility [Fig. 8-inset] displays a local maximum with an inflection point at K, indicating antiferromagnetic ordering. Endoh et al. (1999)

The specific heat of CuRhO [Fig. 8(b)] displays a sharp -shaped peak at K in perfect correspondence with the susceptibility result. This peak shifts by less than K when a magnetic field of T is applied [Fig. 8(b)-inset]. A phonon model with two Debye temperatures, K and K, accounts for most of the specific heat for but overestimates the phonon contribution as the entropy change from 1.7 K to 50 K, J.K.mol, falls short of J.K.mol expected for degrees of freedom. Using the Debye model from CoRhO the magnetic entropy reaches J.K.mol at 50 K, the large value of which suggests possible magneto-elastic effects. Below K, the specific heat is well described by , where the small J.K.mol term may indicate weak glassiness in the low energy spectrum of otherwise gapless antiferromagnetic magnons. The large compared to suggests a moderate degree of frustration in CuRhO, with . In the following, we investigate the nature and consequences of competing (frustrated) exchange interactions in CuRhO.

### iv.3 Magnetic structure

More direct evidence for the presence of frustration in CuRhO comes from low-temperature neutron diffraction. After cooling our sample of CuRhO from 25 K to 4 K [Fig. 9], we observed new Bragg peaks at small wave-vectors ( Å). Given the known thermodynamic anomalies, we identify these peaks with the development of long-range magnetic order. As anticipated for a system, these magnetic Bragg peaks are very weak. In fact, we observed only a single magnetic peak above background (at Å) in our diffraction data taken with Å and optimized for high resolution. The temperature dependence of the integrated intensity of this peak [Fig. 9-inset] yields K. However, we were able to observe several magnetic Bragg peaks with good statistics by integrating our inelastic scattering data over the elastic energy resolution [Fig. 9], which we will henceforth refer to as “elastic scattering”.

The magnetic Bragg peaks are indexed by an incommensurate magnetic propagation vector with respect to the conventional unit cell, where . For the space-group of CuRhO and , there are three irreps, of which two are one-dimensional, and , and one is two-dimensional, . Campbell et al. (2006) However, the one-dimensional irreps can be discounted, because they correspond to amplitude-modulated spin-density waves with the ordered magnetic moment parallel to the axis of the tetragonal unit cell, which would lead to the Bragg peak ( Å) being absent, in conflict with experimental observations. The irrep corresponds to the ordered spin component lying in the plane and it contains two candidate magnetic structures for which all spins possess ordered magnetic moments of equal magnitude. Both structures are circular helices ( perpendicular to the spins’ plane of rotation), with the angle between adjacent spins along given by . Calculating the powder-diffraction patterns reveals that only the structure with shows good agreement with experimental data. We therefore identify the magnetic structure of CuRhO as a circular helix with . This structure (magnetic space group ), which probably originates from competing exchange interactions [Fig. 10(a)], is shown in Fig. 10(b).

We performed Rietveld refinements against our neutron data to obtain accurate values for and the ordered magnetic moment length . Because the elastic data have high statistics but relatively low resolution, while the opposite is true of the diffraction data, we fit to several datasets simultaneously; namely, the K elastic data (magnetic phase), the K elastic data (magnetic and nuclear phases), the K diffraction data (magnetic phase), and the K diffraction data (nuclear phase). The magnetic phase was excluded from the fit to the K diffraction data because of additional weak peaks from the sample environment, which may bias the magnetic refinement. The fit to the 4 K elastic data [Fig. 9] represents good agreement with the data (). The refined parameter values are and . The value of is significantly reduced from its maximum expected value of , which indicates strong quantum fluctuations, an effect we consider in detail below.

### iv.4 Magnetic excitations

CuRhO | meV | |
---|---|---|

To explain the origin of this incommensurate magnetic structure, we resort to inelastic neutron scattering to determine the values of possible magnetic exchange interactions for the distorted structure of CuRhO [Fig. 10(a)]. The magnetic spectrum of CuRhO appears gapless within the resolution of our experiments but unlike CoRhO we observe a complex landscape of high-energy excitations, with peaks in the density of magnetic scattering at meV and meV, several times greater than the excitation bandwidth of CoRhO. Given that the nearest-neighbor magnetic ion distances are very similar for the two compounds (3.68 Å and 3.61 Å), this suggests super-exchange interactions very sensitive to the details of the crystal structure. Furthermore, the presence of two apparent energy scales in CuRhO implies that several exchange interactions exist, and potentially compete, to stabilize the incommensurate magnetic structure.

To model the excitations of CuRhO, we consider a Heisenberg model with up to third-nearest neighbor interactions; see Fig. 10(a). The nearest-neighbor interaction defines a diamond lattice as in the cubic case. The next-nearest neighbor interaction, however, splits from a face-centered cubic connectivity into distinct and interactions that define body-centered and square networks, respectively. In turn, the third-neighbor interaction forms a diamond lattice. This model yields a large parameter space; we defer the study of its mean-field phase diagram and role of quantum fluctuations to Sec. V. With the propagation vector and the inelastic spectrum as a constraints, we obtain an excellent match between the data and the calculated scattering intensity for meV, , and [Fig. 11]. As we will see below, this set of parameters is uniquely constrained by the experimental data. We note that in the cubic case, the average value would yield a highly degenerate coplanar spiral state. Bergman et al. (2007) The Jahn-Teller distortion in CuRhO is thus crucial to stabilize a well-defined spin-helix with a unique propagation vector . In a trend already observed for CoRhO, the temperature dependence of the magnetic excitations of CuRhO [Fig. 12] is marked by a very rapid collapse of the magnetic bandwidth as is crossed.

## V Theoretical analysis

### v.1 Mean-field phase diagram

In this section we apply mean-field theory to relate the magnetic structure of CuRhO to a Heisenberg Hamiltonian with the exchange interactions of Fig. 10(a). Calculations are efficiently performed in a primitive unit cell, which is less symmetric than the conventional cell but contains the smallest possible number of atoms; see Appendix A. We proceed with the Heisenberg model,

(1) |

where denotes the -th spin of a primitive unit cell located at a lattice vector from the origin, and is the exchange interaction between spins and . We consider the four exchange interactions , , and shown in Fig. 10(a) and neglect possible exchange anisotropies.

Our mean-field theory follows the steps of Bertaut Bertaut (1962) and Chapon Chapon (2009) and proceeds by taking the Fourier transform of the exchange interactions,

(2) |

where labels the two Cu ions in the primitive unit cell. describes a Hermitian matrix for each momentum in the first Brillouin zone,

(3) |

where . The matrix elements are evaluated by identifying the lattice translation vectors that connect pairs of spins dressed by a given interaction. Using Eq. (A) to convert from primitive to conventional indices, we obtain

(4) | |||||

where are expressed in reciprocal lattice units of the conventional unit cell.

The interaction matrix has two eigenvalues at each wavevector , given by

(6) |

The wavevector for which reaches a global maximum in the first Brillouin zone is associated with the propagation vector of the ordered magnetic state. Only a small number of points related by symmetry usually fulfill this condition. Highly-frustrated systems are exceptions for which can be degenerate over large regions of the Brillouin zone. Reimers et al. (1991) Given the large parameter space, a systematic search for maximum eigenvalues as a function of , , and is very time consuming. Minimization of the classical ground-state energy can significantly reduce the computing burden by providing analytical solutions for the magnetic structure, see Appendix B.

Our mean-field phase diagram as a function of and for different values of and assuming all exchanges antiferromagnetic is shown in Fig. 13. As well as of the Néel phase, we identify three different incommensurate phases for which the magnetic propagation vector takes the form , (which is equivalent to ) or . The propagation vector observed for CuRhO, , is stabilized with an incommensurate value for a broad range of and values. A large , however, pins the spiral to the lattice and leads to . Critically, our results indicate that the value of is only affected by and but not by . Therefore, the experimentally-measured value of the propagation vector constraints the ratio of to , which eliminates one degree of freedom when simulating the excitations of CuRhO with linear spin wave theory.

### v.2 Linear spin-wave theory

With the knowledge of the possible magnetic structures of the model, we resort to linear spin-wave theory to simulate the dynamics of spins in both coumpounds and to refine further the exchanges parameters for CuRhO [Fig. 10(a)]. For CoRhO we only consider the nearest-neighbor coupling. While the simulated scattering intensities of Figs. 5 and 11 are obtained numerically using SpinW, (Toth and Lake, 2015) we proceed below with the explicit calculation of the magnon dispersion, a step necessary to calculate the effect of zero-point quantum fluctuations on the magnetic ordering.

We start with the general case of CuRhO for which we set the spins to lie in the - plane of the laboratory reference frame (the conventional unit cell). In order to align the quantization axis along the direction of each spin, we perform the following transformation,

(7) | |||

(8) | |||

(9) |

where is related to the propagation vector and is a function of and . We then introduce the Holstein-Primakoff -bosons, which to linear order relate to spin operators in the rotating frame as

(10) | |||||

(11) | |||||

(12) |

and Fourier transform as

(13) |

Keeping only quadratic terms in boson operators, we obtain the Hamiltonian

(14) |

where is a row vector of boson operators and the corresponding column vector. In this representation, is a Hermitian matrix which can be diagonalized provided the following bosonic commutation rules are preserved:

(15) |

For the magnetic structure observed in CuRhO, the matrix elements of read:

(16) | ||||

(17) | ||||

(18) | ||||

(19) | ||||

(20) | ||||

(21) | ||||

where ’s are the lattice harmonics associated with exchange ,

(22) | ||||

(23) | ||||

(24) | ||||

(25) | ||||

with in reciprocal lattice units of the conventional unit cell.

In general, it is not possible to give an analytical form for the above eigenvalue problem. We thus follow the numerical solution described by S. Petit. Petit, S (2011) First, we perform a Cholesky decomposition on to find that satisfies . The positive definiteness for is guaranteed provided the ground state minimizes the classical energy. Afterwards we numerically diagonalize . The eigenvalues of the resulting diagonal matrix provide the magnon energies (). To obtain the eigenvectors, we sort the positive eigenvalues in ascending order and sort the corresponding negative ones accordingly. The transformation matrix that leads to new boson operators from bosons is calculated in the following way

(26) |

where the unitary transformation matrix makes diagonal. Note that is not unitary and it is normalized through .

In the case of CoRhO, the Néel ground state allows to write an explicit analytical solution for the magnon energies. We can explicitly write down the quadratic Hamiltonian as

(27) |

such that the matrix reads

(28) |

with the identity matrix, and

(29) |

with .

From here, the calculation proceeds as for CuRhO, or alternatively a “two-step diagonalization” Chernyshev and Zhitomirsky (2015) can be applied due to the evident commutativity of and . We first apply the unitary transformation

(30) |

to rewrite the quadratic Hamiltonian as

(31) |

where are the eigenvalues of . This eliminates the cross terms between two types of boson operators and effectively leaves two independent single-boson Hamiltonians. From there, we perform the conventional Bogolyubov transformation for each individual species of -bosons,

(32) |

under the constraint . The solution for and is

(33) | ||||

(34) |

where

(35) |

is the two-fold degenerate dispersion relation for CoRhO.