PHYSICAL REVIEW B 102, 184431 (2020)

Magnetic field dependence of low-energy magnons, anisotropic heat conduction, and spontaneousrelaxation of magnetic domains in the cubic helimagnet ZnCr2Se4

D. S. Inosov ,1,2,* Y. O. Onykiienko,1 Y. V. Tymoshenko,1 A. Akopyan,3 D. Shukla,3 N. Prasai,3,† M. Doerr,1,2 D. Gorbunov,4

S. Zherlitsyn,4 D. J. Voneshen,5,6 M. Boehm,7 V. Tsurkan ,8,9 V. Felea,9 A. Loidl ,8 and J. L. Cohn 3,‡

1Institut für Festkörper und Materialphysik, Technische Universität Dresden, 01069 Dresden, Germany2Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter — ct.qmat,

Technische Universität Dresden, 01069 Dresden, Germany3Department of Physics, University of Miami, Coral Gables, Florida 33124, USA

4Hochfeld-Magnetlabor Dresden (HLD-EMFL), Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany5ISIS Facility, STFC, Rutherford Appleton Laboratory, Didcot, Oxfordshire OX11-0QX, United Kingdom

6Department of Physics, Royal Holloway University of London, Egham, TW20-0EX, United Kingdom7Institut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France

8Experimental Physics V, Center for Electronic Correlations and Magnetism, Institute of Physics,University of Augsburg, 86135 Augsburg, Germany

9Institute of Applied Physics, Chisinau MD-2028, Republic of Moldova

(Received 28 September 2020; revised 9 November 2020; accepted 12 November 2020; published 30 November 2020)

Anisotropic low-temperature properties of the cubic spinel helimagnet ZnCr2Se4 in the single-domain spin-spiral state are investigated by a combination of neutron scattering, thermal conductivity, ultrasound velocity,and dilatometry measurements. In an applied magnetic field, neutron spectroscopy shows a complex and non-monotonic evolution of the spin-wave spectrum across the quantum-critical point that separates the spin-spiralphase from the field-polarized ferromagnetic phase at high fields. A tiny spin gap of the pseudo-Goldstonemagnon mode, observed at wave vectors that are structurally equivalent but orthogonal to the propagationvector of the spin helix, vanishes at this quantum critical point, restoring the cubic symmetry in the magneticsubsystem. The anisotropy imposed by the spin helix has only a minor influence on the lattice structure andsound velocity but has a much stronger effect on the heat conductivities measured parallel and perpendicular tothe magnetic propagation vector. The thermal transport is anisotropic at T � 2 K, highly sensitive to an externalmagnetic field, and likely results directly from magnonic heat conduction. We also report long-time thermalrelaxation phenomena, revealed by capacitive dilatometry, which are due to magnetic domain motion relatedto the destruction of the single-domain magnetic state, initially stabilized in the sample by the application andremoval of magnetic field. Our results can be generalized to a broad class of helimagnetic materials in which adiscrete lattice symmetry is spontaneously broken by the magnetic order.

DOI: 10.1103/PhysRevB.102.184431

I. INTRODUCTION

The family of chromium spinels with the general formulaA2+Cr3+

2 X 2−4 (A = Mg, Zn, Cd, Hg — nonmagnetic metal

ion; X = O, S, Se — chalcogen ion) is a perfect playgroundto investigate different types of noncollinear magnetic orderresulting from frustrated magnetic interactions. The magneticCr3+ ions with S = 3

2 and L = 0 form an undistorted py-rochlore sublattice. Depending on the distance between Crnearest neighbors, the dominant nearest-neighbor exchangeinteraction J1 may vary from antiferro- (J1 > 0 in oxides) toferromagnetic (J1 < 0 in sulfides and selenides) [1].

*Corresponding author: dmytro.inosov@tu-dresden.de†Present address: Department of Physics, St. Mary’s College of

Maryland, St. Mary’s City, MD 20686, USA.‡Corresponding author: jcohn@miami.edu

On the one hand, a system of antiferromagnetically inter-acting classical Heisenberg spins on the pyrochlore lattice isgeometrically frustrated and serves as an example of a spin-liquid system [2,3]. However, ab initio calculations show thatCr spinels also have non-negligible further-neighbor exchangeinteractions [1], which should generally reduce the frustrationand stabilize long-range magnetic order [4]. On the otherhand, if the nearest-neighbor interaction is ferromagnetic(FM), the frustration is relieved and, unlike in the antiferro-magnetic (AFM) case, the introduction of further-neighborexchanges can promote bond frustration in the system andlead to noncollinear ordered ground states. In a recent study,the classical phase diagram of the pyrochlore lattice withFM nearest-neighbor interactions was constructed, taking intoaccount exchanges up to the fourth-nearest neighbor [5]. Itshows a rich variety of possible states depending on therelative strength of J2 and J3 exchange parameters, rangingfrom trivial collinear ferromagnetism to single- and multi-qnoncollinear spin structures.

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D. S. INOSOV et al. PHYSICAL REVIEW B 102, 184431 (2020)

Here we focus our attention on the ZnCr2Se4 com-pound — a typical member of the Cr-spinel family with FMnearest-neighbor interactions. It has a cubic crystal structuredescribed by the space group Fd3m with the lattice con-stant a = 10.497 Å [6,7]. Below TC = 21 K, the Cr magneticmoments order in an incommensurate proper-screw structurewith the propagation vector qh = (0 0 qh), with qh ≈ 0.44[8,9]. The high positive Curie-Weiss temperature �CW =90−110 K [10,11] indicates the dominant role of relativelystrong FM exchanges, although the presence of the competinginteractions postpones the formation of long-range order. Thedistinctive feature of a frustrated system is the presence ofshort-range spin correlations way above the ordering tem-perature. In ZnCr2Se4, they were observed by μSR [12] upto the Curie-Weiss temperature, which is around five timeshigher than the long-range ordering temperature. The negativethermal expansion that appears on the same temperature scale[11] can also be attributed to the correlated spin fluctuations,implying the presence of significant spin-lattice coupling.

Specific-heat and thermal-expansion data show λ-likeanomalies at TC, which are much sharper than for conventionalantiferromagnets, suggesting a weakly first-order transition[11], as it is usual for magnetic transitions coupled to a lat-tice instability [13]. It is accompanied by a minute structuraldistortion to either tetragonal [11,12,14–16] or orthorhombic[7,8] structure with (a − c)/a ≈ 6 × 10−4 [12], most clearlyevidenced by a strong splitting of certain phonon modes[17,18]. However, due to the underlying cubic symmetry,three equivalent magnetic domains are possible, with thepropagation vectors pointing along three different cubic axes.In addition, two possible chiralities are equally likely withineach domain. Therefore, if the sample is cooled down in theabsence of external symmetry-breaking fields, magnetic do-mains with all possible ordering-vector orientations and bothchiralities are equally populated and the structural distortionsare averaged out for a macroscopic sample.

The magnetic phase diagram of ZnCr2Se4 has been in-vestigated with magnetization, ultrasound, and small-angleneutron scattering [9,19]. Upon application of a smallmagnetic field, the helical structure acquires a weak FMcomponent along the field direction. After the field exceedsa certain critical value of approximately 1 T (at T = 2 K,decreasing toward higher temperatures), a domain-selectiontransition takes place, favoring the helix with the smallestangle between its propagation vector and the field axis. Ithas also been demonstrated that the rotation of magnetic-fielddirection at low temperatures induces ferroelectric polariza-tion, and a single-domain state with a unique chirality can beselected by the application of a sufficiently strong electric fieldin addition to the magnetic field [20,21].

With a further increase of magnetic field, the conical an-gle of the magnetic structure continues to decrease until itcollapses to zero at the critical field value of Bc ≈ 6.5 T.According to the magnetization and ultrasound data, a smallunsaturated spin component survives up to 10 T. Weak anoma-lies, observed at this field using both methods, lead to asuggestion about another high-field phase of so far unknownorigin [19]. While both domain-selection and the conical-to-collinear transition fields decrease upon warming, they areremarkably independent of the field direction. The ordering

wave vector also turns out to be nearly insensitive to themagnetic-field direction and strength, which means that thehelical pitch length does not depend on the conical angle [9].In the latest study by Gu et al. [22], the conical-to-collineartransition has been readdressed from the point of view offield-driven quantum criticality, which is evidenced by thelow-temperature ∝T 2 behavior of the specific heat and a sharpanomaly in thermal conductivity. In the updated phase dia-gram, the transition at Bc merges with the high-field transitionto the field-polarized FM state in the limit T → 0, leadingto a well-defined quantum critical point (QCP) and a broadquantum-critical region at elevated temperatures [22].

In this paper, we study the evolution of the low-energymagnon spectrum across the field-driven QCP and theanisotropic low-temperature heat conductivity in the single-domain helical phase. Using inelastic neutron scattering(INS), we uncover the mechanisms behind the formation ofthe QCP associated with the collapse of the conical anglein the spin-spiral structure of a bond-frustrated helimagnet,which can be associated with field-driven softening and con-densation of pseudo-Goldstone magnon modes and accidentalrestoration of cubic symmetry at the QCP. In line with ourearlier prediction [5], we also demonstrate that the highlyanisotropic and field-dependent low-energy magnon spectrumhas a profound influence on the low-temperature thermaltransport properties, resulting in a highly anisotropic heat con-ductance in an otherwise nearly cubic material. Further, wealso investigate the stability of the single-domain helical stateagainst thermal fluctuations. After such a state is stabilized inthe sample by field cooling or by application of a magneticfield above the domain-selection transition, as soon as theexternal magnetic field is removed, slow relaxation back tothe multidomain state (on the scale of hours) is observed evenat relatively low temperatures of about 0.5 TC.

II. SAMPLE PREPARATION ANDEXPERIMENTAL METHODS

All experiments presented in this paper have been carriedout on high-quality single crystals of ZnCr2Se4 grown bychemical vapor transport. The crystal growth and character-ization are described elsewhere [19]. For the purpose of INSmeasurements, we used a mosaic sample consisting of eightcoaligned single crystals with a total mass of approximately1 g (the same as in Ref. [5]). The crystals were oriented andcoaligned with an accuracy of about 1◦ using an x-ray Lauebackscattering camera.

The INS measurements were carried out using bothtime-of-flight (TOF) and triple-axis spectrometers (TASs),equipped with vertical-field cryomagnets. The sample wasaligned with its [001] axis vertical, so that the application ofmagnetic field promotes magnetic domains with the propaga-tion vector (0 0 qh) orthogonal to the scattering plane, therebyenabling INS measurements of the pseudo-Goldstone magnonmodes at the orthogonal wave vectors (qh 0 0) and (0 qh 0)in the single-domain state. The TOF experiment [23] wascarried out at LET (ISIS) [24] with the incident neutron energyEi = 2 meV, which yields the energy resolution of 0.04 meVat zero energy transfer. We collected the TOF data at 0, 3, 6,and 8 T. The advantage of the TOF method is that it allows us

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to cover a large volume of the energy-momentum space in asingle measurement. To fill the gaps in the field dependence,we complemented the measurements at the TAS instrumentThALES (ILL) [25,26], focusing our attention only at par-ticular points in reciprocal space [27]. The instrument wasoperated with a fixed final neutron wave vector, kf = 1.1 Å(2.5 meV). A pyrolytic-graphite monochromator and analyzerwere used in the double-focusing mode to increase the ef-fective neutron flux, and a cold beryllium filter was installedbetween the sample and analyzer to reduce the contamina-tion with higher-order neutrons from the monochromator. Forselected scans, we also installed a collimator to decrease thebackground and improve the resolution.

For ultrasound measurements, we polished two opposite{100} surfaces of a single crystal. Piezoelectric transducerswere glued to the polished surfaces to induce and detect ultra-sound vibrations with the generation frequency of 100 MHzand longitudinal polarization. In this configuration, both thewave vector of the ultrasound, ks, and its polarization wereparallel to the [100] cubic axis of the crystal. The same samplewas measured with two orientations of the magnetic field:B ‖ [100] ‖ ks and B ‖ [001] ⊥ ks, which are equivalent inthe cubic setting above TC.

The same polished single crystal was then used to measuredilatometric properties of the sample. We used a tilted-platecapacitive dilatometer with a sensitivity to relative lengthchanges of about 10−7 [28] to assess relative changes in thelattice constants as a function of magnetic field and tem-perature. The thermal expansion between 1.5 K and 300 Kand magnetostriction in steady fields were measured using asuperconducting 15 T magnet (Oxford Instruments) at the TUDresden. To determine the relaxation process from a single-domain to multidomain state, the sample was first cooleddown in a field of 7 T, then the field was ramped down withthe maximal rate of 1 T/min, and as soon as it reached zero,the dilatometer signal was measured over a long time.

Specimens for thermal-conductivity measurements werecut from as-grown single crystals, oriented by x-ray diffrac-tion, and polished into thin parallelepipeds with edges along〈100〉 directions. A two-thermometer – one-heater methodwas employed to measure the thermal conductivity using amatched pair of RuO sensors thermally linked to the spec-imens with ∅0.005′′ gold wires bonded with silver epoxy.The sensors were calibrated in each experiment against acalibrated reference sensor mounted in the Cu heat sink. Ex-periments with the applied field perpendicular and parallel tothe heat flow were performed in successive low-temperatureruns using the same thermometry attachments. The accuracyof the measurements is limited by uncertainty in the geometricfactor to approximately 10%.

III. ANISOTROPY IN THE LATTICE PARAMETERSAND SOUND VELOCITIES

Our subsequent discussion substantially relies on the un-derstanding of spin-lattice coupling and the origins of spinanisotropy in the magnetically ordered state. On the one hand,in the high-temperature cubic phase, the Cr t2g shell is halffilled, quenching the orbital moment to zero. Thus, no on-sitespin-orbit interaction is expected. On the other hand, partial

transfer of the magnetic moment to the neighboring Se siteswas suggested based on recent nuclear magnetic resonance(NMR) measurements [29]. This mechanism can potentiallyinduce a finite orbital component and, therefore, facilitate thespin-lattice interaction. The theory confirms this possibility,but the corresponding corrections are expected to be insignif-icant [1].

The response of the lattice to the onset of helimagneticorder is manifested in the tetragonal or orthorhombic latticedistortion below TC, which is mainly due to the magnetostric-tive displacements of Se2− ions resulting from the magneticinteraction of neighboring Cr3+ spins [7]. According to vari-ous estimates, this distortion does not exceed 0.06–0.08% atthe lowest measured temperature [11,12,14,15]. In view of thesmall spin-orbit coupling, any anisotropic terms in the spinHamiltonian that may result from this minor deviation fromthe cubic symmetry are usually considered negligible, leadingto the Heisenberg-type behavior of magnetic moments withno detectable spin-space anisotropy.

Structural distortions of the order of 10−4 are at the limitof sensitivity for conventional diffraction methods. Therefore,in our present work we used capacitive dilatometry with asensitivity of ∼10−7 to measure relative changes in the latticeconstants more accurately as a function of magnetic field andtemperature. The sample was mounted in the dilatometer tomeasure relative changes of its length, �l/l0, along one of its〈001〉 cubic axes. Depending on the direction of magnetic field(transverse or longitudinal with respect to the chosen axis),it corresponds to either the a or c axis, respectively, in thetetragonally distorted structure with a = b �= c. Therefore, wecan measure relative changes in both a and c lattice constantson the same single crystal without remounting it inside thedilatometer. All measurements are presented relative to the l0value, corresponding to the high-temperature cubic structureat B = 0 and T = 30 K.

Figure 1(a) shows the temperature dependence of �l/l0,measured in longitudinal and transverse magnetic fields (B =1, 3, 5, and 7 T) above the domain-selection transition. Inall measurements with B � 1 T, the sample always remainsin the single-domain state. The longitudinal and transversethermal-expansion curves essentially coincide at high temper-atures (T > TC) or in high magnetic fields above the saturation(B > Bc), suggesting that both paramagnetostriction (forcedmagnetostriction above the ordering temperature) and ther-mal expansion in the field-polarized FM state are isotropic.This behavior is typical for systems with spin-only magneticmoments [30–32], in contrast to strongly anisotropic mag-netostriction in some rare-earth local-moment systems thatfeature an aspherical 4 f electron shell with a nonzero orbitalangular momentum [33–36]. Spontaneous magnetostriction inthe helimagnetic state, on the contrary, is very anisotropic, re-sulting in the splitting of the thermal-expansion curves belowTC. The lattice contracts in the direction parallel to the spinspiral and expands in the orthogonal directions, which resultsin a tetragonal distortion with (a − c)/a ≈ 3.6 × 10−4 in thezero-field and zero-temperature limit, that is, even smallerthan in earlier estimates from x-ray diffraction [12,14–16].

The zero-field thermal expansion, measured after zero-fieldcooling (ZFC) and after field cooling in longitudinal andtransverse magnetic fields of 7 T, are shown separately in

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FIG. 1. (a) Thermal-expansion measurements in longitudinal andtransverse magnetic fields above the domain-selection transition. Allcurves are normalized to the same l0 value at T = 30 K and B = 0.(b) Zero-field thermal expansion, measured during zero-field cool-ing (ZFC) and after field-cooling in a longitudinal (FC 0 T ‖) ortransverse (FC 0 T ⊥) field of 7 T. The measurement steps (i)–(iii)are explained in the text. (c), (d) Magnetostriction measurements atseveral constant temperatures for longitudinal and transverse fielddirections, respectively.

Fig. 1(b) because they show qualitatively different behavior.The ZFC reference curve (black), which represents the ther-mal expansion of ZnCr2Se4 in the multidomain state, is inperfect agreement with earlier measurements by Gu et al. [37].(i) Because the field cooling takes place above the satura-tion field, where the magnetostriction is isotropic, the fielddoes not break the cubic symmetry of the lattice, and thecorresponding transverse and longitudinal thermal-expansioncurves (labeled FC 7 T) coincide. (ii) Then, at the base tem-perature of 2 K, the magnetic field is ramped down to zerowith a rate of 0.4 T/min, wherein the sample undergoes atransition to the helimagnetic tetragonal state with c ‖ qh ‖ B.In the longitudinal geometry (c axis), where the forced andspontaneous contributions to the magnetostriction have thesame sign, this results in a significant contraction of the sam-ple with a change in �l/l0 of approximately 4 × 10−4, whichappears in Fig. 1(b) as a vertical line. In the transverse geom-etry (a axis), the forced and spontaneous contributions nearlycancel each other, so the resulting change in �l/l0 is only0.5 × 10−4. (iii) After stabilizing the single-domain state ofthe sample by field cooling, thermal expansion is measured inzero field upon increasing the temperature. One could expectthat the single-domain state survives up to TC, which wouldresult in the order-parameter-like change in both lattice pa-rameters shown with dashed lines, similar to that observed in

finite fields [Fig. 1(a)]. However, measured thermal-expansioncurves (labeled FC 0 T ‖ and FC 0 T ⊥) deviate from thisbehavior at temperatures above 5 K, as they both converge tothe ZFC reference measurement. This suggests that the samplerelaxes back to the multidomain state at elevated temperatureson the timescale of our measurement, which was carried outwith a temperature sweep rate of 0.1 K/min. The result ismost pronounced in the transverse geometry, resulting in anonmonotonic sign-changing behavior of the magnetostrictivestrain.

The slow magnetic relaxation to the multidomain state isalso revealed in the measurements of magnetostriction at sev-eral constant temperatures, which are presented in Figs. 1(c)and 1(d) for the longitudinal and transverse field geometries,respectively. All measurements start in the single-domain stateachieved by cooling the sample in a 3 T field. The field isthen swept to the maximal field and back to zero, which inthe absence of relaxation processes should result in identicalmagnetostriction curves. One can see, however, that alreadyat 4 K the curves slightly deviate from each other, and aweak anomaly is observed at the domain-selection transitionaround 1 T, which is not expected if the sample remainedin the single-domain state after the field was switched off.This anomaly becomes much more pronounced at 8 K, wherethe domain relaxation is faster, although the curves measuredupon increasing and decreasing the field still show a hystere-sis, suggesting that the sample does not fully relax back to themultidomain state on the timescale of the measurement. Mostremarkably, a downturn (upturn) in the transverse (longitu-dinal) magnetostriction measured in decreasing field alreadystarts at 0.8 T, which implies that the domain relaxation takesplace not only in zero field but also in finite fields as long asthey remain below the domain-selection transition. At highertemperatures of 12 or 16 K, approaching TC, the relaxationbecomes so fast that the magnetostriction curves measured inupward and downward field sweeps nearly coincide. Finally,the measurements at T = 22 K > TC reveal nearly isotropicparamagnetostriction, which remains quadratic in field upto approximately 2 T with ε(B)/B2 ≈ 7 × 10−6 T−2, whereε(B) = [l (B, T ) − l (0, T )]/l (0, T ) is the parastrictive strain.

Before moving on to a more detailed consideration of theslow domain relaxation and discussing the possible mecha-nisms of such behavior, we would like to quantify the latticeanisotropy in the helimagnetic state of ZnCr2Se4 using acomplementary approach that consists of comparing soundvelocities in the directions parallel and orthogonal to themagnetic propagation vector. The ultrasound data in Fig. 2(a)show relative changes in the sound velocity, �v/v0, forthe longitudinal wave propagating along the [001] ‖ qh and[100] ⊥ qh directions, normalized to the respective values inthe paramagnetic state at T = 30 K. It is clear that the crystalis slightly stiffer in the direction orthogonal to the orderingvector of the spin helix, but the effect is relatively small, onthe order of 0.2%. This suggests that the difference in phononenergies and phononic densities of states along the two direc-tions must also be similarly small. For comparison, the inset toFig. 2(a) shows a similar temperature dependence, measuredin the field-polarized FM phase at B = 7 T in the directionsparallel and orthogonal to the field-induced magnetization.The two curves have been normalized to the respective v0

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5 10 15 20 25 30

Temperature (K)

−2

−1

0

1

2

Δv

v 0·1

0−3

(a)SD, ks ‖ qh

MDSD, ks ⊥ qh

MD

0 10 20 30

0

1

B

0 2 4 6 8 10 12

Magnetic field (T)

−20

−15

−10

−5

0

Δv

v 0·1

0−3

(b)

B‖ ks, T

B⊥ ks, T

B⊥ ks, T

FIG. 2. (a) The change in the sound velocity of the longitudinalacoustic mode, �v = v − v0, normalized to the value v0 at 30 K, inthe single-domain (SD, solid lines) and multidomain (MD, dashedlines) states. The top and bottom curves correspond to the ultra-sound wave vector ks perpendicular and parallel to the propagationvector of the spin helix, qh, respectively. The inset shows a similartemperature dependence, measured in the field-polarized FM phaseat B = 7 T for ks ⊥ B (top curve) and ks ‖ B (bottom curve), forcomparison. (b) Magnetic-field dependence of the relative changein the sound velocity, �v/v0, normalized to the respective v0 val-ues at 12 T. The measurements are done at a constant temperaturewith the ultrasound wave vector parallel and perpendicular to themagnetic-field direction. Note the tenfold difference in the rangesof the vertical scales between (a) and (b).

values at T = 30 K, where the sample remains cubic ac-cording to the paramagnetostriction measurements describedabove. The observed change in �v/v0 between 30 K and basetemperature differs by only 2 × 10−4 for the two directions ofthe ultrasound wave vector, which is within the reproducibilitymargin of our measurements.

The magnetic-field dependence of the relative change inthe sound velocity, �v/v0, measured for the same two di-rections at T = 2 K and normalized to the high-field value at12 T, is plotted in Fig. 2(b). One can see a very pronouncedsoftening of the longitudinal acoustic mode immediately be-low the critical field, which is much stronger for the acousticwaves propagating along the magnetic-field direction thanperpendicular to it. The maximal value of |�v/v0| reaches2%, which is ten times larger than the corresponding changesat base temperature and almost two orders of magnitude largerthan the spontaneous magnetostriction, suggesting that mag-netic interactions exert a much stronger relative effect onthe dynamic than static properties of the lattice. The secondanomaly that appears in both field-dependence curves around10 T represents the previously reported high-field transitionthat was previously associated either with a spin-nematicphase [19] or with a crossover from the quantum-critical re-gion to the fully field-polarized FM phase [22]. The new phasediagram suggested by Gu et al. [22] implies a very strong tem-perature dependence of this high-field anomaly so it mergeswith Bc in the zero-temperature limit. In contrast, we can seeessentially no difference between the 1.5 and 2 K curves inFig. 2(b), indicating that the crossover to the field-polarizedFM phase stays at approximately 10 T at both temperatures,which is hard to reconcile with the newly proposed phasediagram.

IV. SLOW RELAXATION OF HELIMAGNETIC DOMAINS

Having observed slow magnetic domain relaxation in mag-netostriction measurements, we proceed with estimating thefunctional dependence and characteristic times of such relax-ation by following the time-dependent changes in the samplelength at a constant temperature. After applying a 7 T field tothe sample in either longitudinal or transverse direction andstabilizing the temperature at T = 8 or 12 K, we swept thefield down to zero with the maximal rate of 1 T/min andfollowed the time-dependent change in the relative samplelength, �l/l0, both during and after the field sweep. Theresults are plotted in Fig. 3, where the bottom panel shows thetime dependence of the external field. During the downwardfield sweep, anomalies in �l/l0 are observed at the criticalfield (4.8 or 4.1 T, respectively) and at the domain-selectionfield (0.8 T). As soon as the magnetic field is reduced be-low the domain-selection field, slow relaxation toward themultidomain state is observed both in the transverse and longi-tudinal directions. The relaxation curves cannot be describedby a simple exponential function, which implies contributionsfrom processes with a broad spectrum of different relaxationtimes. Defining the time when magnetic field reaches zero ast0, we can describe the relaxation at times t > t0 using thestretched exponential function,

f (t ) = A + B exp

[−

(t − t0

τ

)β], (1)

with β = 0.5, where τ is the temperature-dependent decayconstant. This model is equivalent to a continuous weightedsum of simple exponential decays with a distribution of

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FIG. 3. Time dependence of the relative change in the samplelength �l/l0, measured during and after a rapid linear sweep ofmagnetic field from 7 T to zero in the transverse and longitudinaldirections. The time dependence of magnetic field is plotted at thebottom. At t < 0, the sample temperature is stabilized at T = 8 or12 K and a downward field sweep with 1 T/min starts. After thetime t0 = 420 s, when the field reaches zero, slow relaxation towardthe multidomain state can be described by a stretched exponential,Eq. (1), with β = 0.5 and the average decay constants τ8 K = 980 sand τ12 K = 190 s (thin white lines).

relaxation times described by the probability density:

ρ(θ ) = 1

2√

πθ−1/2 exp(−θ/4). (2)

Relaxation with a stretched-exponential behavior is typicalfor systems with randomly distributed magnetic domains andis commonly associated with magnetic domain-wall motion.Here it results in the destruction of the single-domain state onthe timescale of several minutes or hours (depending on thetemperature) with a tendency to restore the cubic symmetry ofthe sample on the macroscopic scale. At a lower temperatureof 8 K, the average relaxation rate is approximately five timesslower than at 12 K. Apart from that, the two data sets alsoshow a qualitative difference. At the higher temperature of12 K, both transverse and longitudinal curves relax to thesame asymptotic value of �l/l0 at t → ∞, suggesting thatan isotropic multidomain state is fully restored as a resultof such relaxation. In contrast, the 8 K data sets relax todifferent asymptotic values in the transverse and longitudi-nal directions, which implies that some partial anisotropy inthe distribution of magnetic domains survives arbitrarily longafter the external field is turned off. A crossover betweenthese two distinct behaviors must therefore take place at someintermediate temperature between 8 and 12 K.

The temperature dependence of the relaxation constant τ

is plotted in Fig. 4. Its rapid monotonic decrease by nearlyfive orders of magnitude between 4 and 16 K can be wellapproximated by the following empirical formula:

τ (T ) = A exp [(T0/T )p], (3)

with the best-fit parameters T0 ≈ 10.8 K and p ≈ 2.4. At tem-peratures below 4 K, the characteristic time of the domain

FIG. 4. The decay constant τ defined by Eq. (1) as a function oftemperature. Error bars are comparable to the size of data points. Thesolid line is a fit with Eq. (3).

relaxation exceeds one month, and therefore the single-domain state can be considered stable for the purposes ofmost experiments including thermal-transport and neutron-spectroscopy measurements.

Because every domain wall in a helimagnet costs ad-ditional energy, relaxation through nucleation of additionalmisaligned domains is only possible if some excess energyis already present in the initial state of the system. Indeed, byapplying magnetic field to the sample, we only select mag-netic domains with a single direction of the magnetic orderingvector, but such domains can still possess different chiralities.One therefore expects that domain walls separating right- andleft-handed helical domains are still present in the initial state,carrying excess energy that can be redistributed via nucleationof new magnetic domains with a misaligned orientation of themagnetic ordering vector as a result of relaxation processesfacilitated by thermal fluctuation. In other words, a domainwall separating helimagnetic domains with opposite chiralitycan decay into a larger number of domain walls betweendomains with orthogonal ordering vectors and either equal oropposite chiralities. It is natural to expect that such processesare governed by the corresponding domain-wall energies thatmust be different for all three types of domain walls and couldbe estimated from micromagnetic simulations.

V. INELASTIC NEUTRON SCATTERING

The main goal of our INS measurements is to investigatethe magnetic-field dependence of the previously discoveredpseudo-Goldstone magnon modes in ZnCr2Se4 [5] and revealtheir relationship to the field-driven QCP suggested by Guet al. [22]. In the paramagnetic phase above TC, the threepairs of wave vectors (±qh 0 0), (0 ±qh 0), and (0 0 ±qh) areequivalent because of the cubic symmetry. Helimagnetic orderbreaks this discrete symmetry by spontaneously selecting oneof the cubic directions as the propagation direction of thespin spiral. For the sake of unambiguity, let us assume that

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this direction is (0 0 ±qh). Consequently, all spins formingthe helix are constrained to the ab plane. Under the assump-tion of Heisenberg symmetry, which holds to a very goodapproximation for Cr3+ spins, the system remains invariantunder continuous U (1) spin rotations within the ab plane. As aconsequence, Goldstone modes emanating from the (0 0 ±qh)ordering vectors remain gapless down to the lowest tempera-tures (the upper limit for the spin gap at these points, set byprevious INS experiments, is 0.05 meV [12]).

Under the assumption that the lattice symmetry remainscubic, so that no spin-space anisotropy is imposed by thelattice distortions, linear spin-wave theory (LSWT) predictsgapless magnon modes also at the two orthogonal pairs ofwave vectors, (±qh 0 0) and (0 ±qh 0), which can be seenas an accidental degeneracy, because in contrast to the trueGoldstone modes, the zero spin gap at these points is notenforced by the symmetry of the Hamiltonian. As a conse-quence, the degeneracy can be weakly lifted by fluctuations,opening up a small magnon gap at the corresponding wavevectors [38] and resulting in pseudo-Goldstone modes thathave been experimentally observed in ZnCr2Se4 by neutronspectroscopy [5]. In the case of continuous degeneracies, thisphenomenon is known as order by disorder [39,40], whereasin ZnCr2Se4 the degeneracy is discrete, resulting in one pairof Goldstone and two pairs of pseudo-Goldstone modes, wellseparated in the Brillouin zone.

According to our spin-wave calculations in zero magneticfield, quantum-fluctuation corrections to the spin-wave spec-trum are qualitatively similar to those resulting from LSWTin the presence of weak easy-plane anisotropy (in the planeorthogonal to the spiral direction). Both effects break the cubicsymmetry of the magnon dispersions by opening a gap at thepseudo-Goldstone wave vectors while keeping the true Gold-stone modes gapless. In the first case, this symmetry breakingis spontaneous, whereas in the second case a preferred axisis explicitly introduced in the magnetic Hamiltonian to re-flect the tetragonal distortion resulting from the spin-latticecoupling. On the one hand, experimentally observed pseudo-Goldstone magnon gap �PG ≈ 0.17 meV shows reasonableagreement with the theoretical estimate that takes into accountquantum fluctuations (three- and four-magnon processes)even in the cubic setting [5]. This offers solid evidence thatthe anisotropy in the spin-wave spectrum results mainly fromquantum-fluctuation corrections beyond LSWT. On the otherhand, small easy-plane anisotropy cannot be fully excludedin the presence of a tetragonal lattice distortion, even if ourresults outlined in Sec. III indicate that the correspondingdistortions are minor, and due to the smallness of spin-orbitcoupling, the resulting spin anisotropy should be negligible.At the qualitative level, the contribution of magnetostrictivedistortions to the easy-plane anisotropy and, consequently,to the pseudo-Goldstone gap energy remains unknown. Es-timating it from first principles would require more elaboraterelativistic calculations, taking into account both lattice andspin degrees of freedom, which are currently beyond reach.

To shed light on this problem, we have measured thelow-energy magnon spectrum of ZnCr2Se4 in magnetic fieldsapplied along the [001] cubic axis. Figure 5 shows represen-tative cuts through the TOF data at 0, 3, 6, and 8 T. The toprow of panels [Figs. 5(a)–5(d)] presents constant-energy cuts

in the (H0L) plane at 0.17 meV. This energy correspondsto the pseudo-Goldstone magnon gap in zero field [5]. Thered dashed lines connect the center of the Brillouin zonewith the local minimum in the dispersion. In the panels be-low [Figs. 5(e)–5(h)], corresponding momentum-energy cutsalong these lines are shown.

The spectrum in Figs. 5(a) and 5(e) was measured in zerofield. In agreement with our previous report [5], we see clearintensity maxima at (±qh 0 0), near the bottom of the pseudo-Goldstone modes. Their positions are equivalent (in the cubicsetting) to the ordering vector, which is oriented along the caxis and cannot be reached in this experimental configuration.The dispersion has a sharp minimum, which is asymmetriceven in the direct neighborhood of the ordering vector. Onthe left side from the minimum, closer to the � point, thedispersion forms an arc with a maximum around 0.3 meV.On the right-hand side, the magnon branch rises linearly inenergy and has a much steeper dispersion, forming a beaklikeshape. The minimum of the dispersion corresponds to thepseudo-Goldstone gap energy. In the constant-energy plane inFig. 5(a), the magnetic signal appears in the form of sharpspots at (±qh 0 0), with no additional intensity maxima in thefield of view.

The application of a 3 T magnetic field, approximatelyhalf way to the QCP, qualitatively changes the shape of thedispersion. In the constant-energy plane in Fig. 5(b), the in-tensity now splits into four spots, which indicates that thedispersion minima are shifted up and down with respect to the(H K 0) plane and no longer coincide with the wave vectorsequivalent to the magnetic ordering vector in the cubic setting.The momentum-energy cut through one of these minima inthe dispersion, shown in Fig. 5(f), reveals a markedly differentspectrum to the beak-shaped one in zero field. Even though thespin gap remains nearly unchanged, the dispersion no longerhas a cusp and lacks the linear part on the right-hand side,approaching a parabolic shape. The comparison of 0 and 3 Tcuts in Fig. 6, taken through the corresponding minima in thedispersion along the transverse momentum direction [verticaldotted lines in Figs. 5(a) and 5(b)], indicates that the pseudo-Goldstone mode splits along the magnetic-field direction intotwo identical soft modes with a parabolic dispersion. Theappearance of a crossing point of the two dispersion branchesat the center of Fig. 6(b) suggests that there are two dif-ferent dispersion curves, possibly originating from magneticdomains with opposite chirality.

At a twice higher magnetic field of 6 T, the system isin immediate proximity to the field-driven QCP, where theincommensurate structure gets suppressed due to the collapseof the conical angle and continuously transforms into thefield-polarized FM phase. The corresponding INS spectrumis presented in Figs. 5(c) and 5(g). In contrast to the pre-vious cases, the gap in the spin-wave spectrum closes, anda ring of intensity can be seen in the constant-energy cut at0.17 meV. Despite the proximity of the system to a collinearfield-polarized state, spin fluctuations are still concentrated inthe neighborhood of a sphere with radius qh, centered at theorigin. Two hollow ellipses can be recognized around (±qh

0 0) and (0 0 ±qh), where the gapless Goldstone modes arenow located, and the spin-wave dispersion drops below thelevel of the constant-energy cut. Despite the magnetic field,

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0.0 0.2 0.40.0

0.2

0.4

0.6

(e) B

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inte

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y(a

rb.u

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)

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,qz)

(Å−1

)

qx in (qx, 0, 0) (Å−1)

Ene

rgy

(meV

)

Momentum (Å−1)

FIG. 5. The neutron TOF spectrum of ZnCr2Se4, measured in magnetic fields of 0, 3, 6, and 8 T, applied along the [001] direction. The toprow of panels (a)–(d) shows the corresponding constant-energy cuts in the (H0L) plane, integrated within ±0.03 meV around 0.17 meV energytransfer. The bottom row (e)–(h) shows energy-momentum cuts along the radial directions connecting the � point with the local minimum ofthe dispersion, as shown with dashed lines in (a)–(d). Integration ranges in momentum directions orthogonal to the plane of the figure areapproximately ±0.05 Å−1. The vertical dotted lines in (a) and (b) indicate the direction of energy-momentum cuts in Fig. 6.

which breaks the equivalency of the [100] and [001] axes, thespectrum restores its fourfold symmetry in the (H 0 L) plane atthe critical field. The momentum-energy cut in Fig. 5(g), par-allel to the qx direction, shows a nearly symmetric parabolicdispersion curve with a vanishing energy gap. There is asmall residual difference in the slopes on opposite sides of thedispersion curve, but neither shows signs of inflection withinthe available data range, unlike at smaller fields.

Finally, at even higher fields above Bc, i.e., in the field-polarized FM phase, the spin gap reopens again. This can

−0.2 0.0 0.20.0

0.2

0.4

0.6

(a)B

−0.2 0.0 0.2

(b)B

Ene

rgy

(meV

)

qz in (0.25, 0, qz) (Å−1) qz in (0.21, 0, qz) (Å−1)

FIG. 6. Energy-momentum cuts along the qz direction in theB = 0 and 3 T data sets, passing through the minima in the spin-wavedispersion, as indicated in Figs. 5(a) and 5(b), respectively, withvertical dotted lines. The integration range in qx and qy momentumdirections is approximately ±0.05 Å−1.

be well seen in the 8 T data shown in Figs. 5(d) and5(h). The value of the spin gap at this field is above the0.14–0.20 meV integration range of the constant-energy cut,hence no spin-wave intensity can be seen in Fig. 5(d). Themagnon dispersion in the momentum-energy cut in Fig. 5(h)is shaped as a symmetric parabola with an energy gapof 0.25 meV.

To follow the field dependence of the excitation spectrumin ZnCr2Se4 along the qx direction, we performed an ad-ditional TAS experiment, preserving the same experimentalgeometry. The measurements were taken at a single point (qh

0 0) in momentum space for a number of magnetic fieldsbetween 0 and 9.5 T. The results are shown in Fig. 7 in theform of a color map. Here, the horizontal axis corresponds tothe magnetic field strength, and the vertical one represents theneutron energy transfer E . The color indicates raw INS inten-sity, with the higher intensity shown in dark blue. The broadregion of oversaturated intensity near E = 0 comes from theincoherent elastic line. Every spectrum has been fitted with aGaussian profile, resulting in the peak positions shown withopen circles. In addition, several intermediate fields aroundthe 6 T transition were measured with tighter collimation toimprove the resolution. The corresponding fitted peak posi-tions are shown with red squares. Finally, orange star symbolsshow true values of the pseudo-Goldstone spin gap extractedfrom TOF data at the local minimum of the dispersion, whichdoes not fall within the (HK0) plane (see Fig. 6) and wastherefore inaccessible in our TAS measurements.

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0 2 4 6 8

Magnetic field (T)

0.0

0.1

0.2

0.3

0.4

0.5

Ene

rgy

tran

sfer

(meV

)

TAS

TAS +

TOF

FIG. 7. Magnetic-field dependence of the INS spectrum at (qh

0 0), measured as a function of magnetic field applied along the[001] direction. The color map shows INS intensity measured ona TAS instrument (dark-blue color corresponds to high intensity,white to the background level). Open circles show fitted positionsof the peak, resulting from Gaussian fits. Red squares show fits ofadditional TAS data, taken with tighter collimation to improve theresolution (the corresponding raw data are not shown). Orange starsymbols correspond to the spin-gap values, extracted from the TOFdata (see Fig. 6) at the local minimum of the dispersion, which wasnot accessible in our TAS experiment.

First, one notices the nonmonotonic behavior of the spin-wave energy at the (qh 0 0) wave vector. It starts from thepseudo-Goldstone magnon gap energy in zero field, then in-creases to approximately 0.3 meV at intermediate fields anddrops down to zero at the critical field, where the transitionto the field-polarized FM phase takes place. After that, thespin-wave spectrum is gapped again, and the magnon energyincreases linearly with a slope given by the Zeeman energy (gfactor). It should be emphasized that in the intermediate fieldrange, within the conical spin-spiral phase, the magnon energyat (qh 0 0) exceeds the actual spin-gap energy, because theminima in the magnon dispersion are shifted above and belowthe scattering plane and cannot be reached in the TAS config-uration. This splitting is the main reason for the maximum inthe spin-wave energy (open circles) around 3–4 T, which liesabove the actual spin-gap energy extracted from the TOF dataat the out-of-plane minima of the dispersion (star symbol at3 T).

The small discrepancy among the TOF and TAS data setsis explained by different momentum resolution. The TAS datawithout collimation were collected in the double-focusingmode of monochromator and analyzer, which increases theintensity at the expense of the resolution. As a result, theintensity is integrated in a broader vicinity of the dispersionminimum, shifting the center of mass of the peak to some-what higher energies. Vertical focusing mode with additionalcollimation reduces this effect, leading to a better agreementwith the TOF data.

To summarize, our INS results demonstrate a surprisinglycomplex nonmonotonic behavior of the low-energy magnonspectrum as a function of magnetic field. The previously re-ported pseudo-Goldstone mode [5], located at the (±qh 0 0)and (0 ±qh 0) wave vectors in zero field, splits into two sep-arate minima along the qz direction, and the correspondingspin gap is initially increased. Above 3–4 T, this tendencyreverses, and the two dispersion minima merge into a sin-gle gapless mode at the critical field, corresponding to thequantum-critical transition to the field-polarized FM state.The collapse of the conical angle of the spin spiral to zeroat Bc ≈ 6.25 T restores the cubic symmetry of the spin-wavespectrum, in spite of the external magnetic field that explicitlybreaks this symmetry. At even higher magnetic fields aboveBc, the spin-wave spectrum gets fully gapped, the spin gapincreasing linearly with the deviation from the QCP.

VI. ANISOTROPIC THERMAL CONDUCTIVITY

The nonmonotonic field dependence of pseudo-Goldstonemagnon modes are expected to have pronounced signaturesin the magnetic contributions to the specific heat and low-temperature thermal transport properties of ZnCr2Se4. Indeed,the spin-gap energy in zero magnetic field that results from fit-ting an exponent, A exp(−�PG/kBT ), to the specific-heat dataof Gu et al. [22] below 3 K is �PG = 0.186 meV, which per-fectly matches the pseudo-Goldstone magnon gap observed inour INS experiments. Note that the actual spin gap in the sys-tem (at the ordering wave vector) is expected to be below thisvalue [5,12], but because the true Goldstone mode carries only1/3 of the magnon spectral weight, thermodynamic propertiesare more sensitive to the pseudo-Goldstone magnon gap. Withincreasing magnetic field, the specific-heat anomaly graduallydisappears, resulting in a ∝T 2 behavior at the critical field[22]. It is natural to associate this behavior with the closingof the magnon gap observed in our INS measurements. Ata higher field of 10 T, the specific-heat anomaly reappearsat about 3.5 K, which is in reasonable agreement with themeasured spin-gap energy at this field.

Similar anomalies that resemble the nonmonotonic behav-ior of the spin gap were also observed in the low-temperaturethermal conductivity of ZnCr2Se4 as a function of magneticfield and temperature [22]. The zero-field thermal conduc-tivity, κ (T ), shows a change in behavior around 1 K, whichdisappears at the critical magnetic field. The correspondingfield dependence, κ (B), exhibits a nonmonotonic behaviorat temperatures below 1 K, with two clear anomalies atthe domain-selection field and at the transition to the field-polarized FM phase. In the thermal conductivity experimentfrom Ref. [22], the magnetic field was applied along one ofthe 〈111〉 crystallographic axes, while thermal current wasmeasured in an orthogonal direction [41]. In this geometry, theangle formed by the propagation vectors of all three magneticdomains with the magnetic field is approximately the same,hence the distribution of magnetic domains cannot be wellcontrolled (it would be determined only by small unavoidablemisalignments of the field direction). It is therefore unclearwhether the thermal conductivity was measured in a single- ormultidomain state, and how the thermal current was orientedwith respect to the magnetic propagation vector.

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100 101

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102

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(mW

m·K

2) ∼T 2

(a) ⊥ qh

‖ qh

Temperature (K)

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30

40

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(mW

m·K

)

100 101

Temperature (K)

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κ‖)

/κ‖

(b)

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mW

m·K

)

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15

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Wm·K

)

‖⊥‖⊥

‖

⊥(c)

0 2 4 6

Magnetic field (T)

0

2

4

6

κ⊥−

κ‖(

mW

m·K

)

(d)

FIG. 8. Anisotropy of the thermal conductivity with respect to the direction of the magnetic ordering vector. (a) Temperature dependenceof κ⊥/T (◦) and κ‖/T (•) at B = 0 and 5 T (main panel) and at two intermediate fields of 1.5 and 3 T (inset). The dotted line in the inset showsthe 3 T data set reproduced from Ref. [22], which we recalculated into κ (T )/T and normalized by a constant factor to match with our data inthe high-temperature region. (b) Temperature dependence of the relative thermal-conductivity anisotropy, (κ⊥ − κ‖)/κ‖, at different magneticfields. The inset shows the low-T part of the same data without normalization, i.e., κ⊥ − κ‖, on a linear temperature scale. (c) Magnetic-fielddependence of κ⊥ (◦) and κ‖ (•) for three different temperatures. (d) Magnetic-field dependence of the thermal-conductivity anisotropy, κ⊥ − κ‖,resulting from the pairwise subtraction of the data in panel (c).

The total thermal conductivity for a magnetic insulatorcan be represented as a sum of the phononic (lattice)and magnonic contributions, κ = κL + κm. The directionaldependence of κ , determined by the thermal gradient �T ,can yield additional information beyond that afforded byspecific heat. An elegant way to separate magnetic effectsfrom those arising entirely from the phonon dispersion is toexamine the difference in κ measured with �T parallel (κ‖)and perpendicular (κ⊥) to the magnetic ordering vector qh. Asseen in Sec. III, the deviations from cubic symmetry in bothstructural parameters and acoustic phonon frequencies in themagnetically ordered state of ZnCr2Se4 do not exceed 0.1%.Thus the difference, �κ = κ⊥ − κ‖, should be determinedexclusively by magnetic effects, which can include magnonheat conduction and anisotropic phonon scattering onmagnons and magnetic domain walls.

We have measured anisotropic thermal conductivity in thesingle-domain state of ZnCr2Se4 in the directions parallel

and orthogonal to the magnetic ordering vector in magneticfields up to 5 T and down to T ≈ 0.6 K. Figure 8(a) shows thetemperature dependence of κ⊥/T and κ‖/T at B = 0 and 5 T(main panel) and at two intermediate fields of 1.5 and 3 T (in-set). No anisotropy is detected in the zero-field measurement,but it can be clearly seen below 2 K in an applied magneticfield. It is more clearly represented by the relative difference,�κ/κ ≡ (κ⊥ − κ‖)/κ‖, which is plotted in Fig. 8(b). One cansee that �κ/κ reaches above 50% at the lowest measuredtemperature, which exceeds the tetragonal distortion by threeorders of magnitude and is 250 times higher compared to theanisotropy in the sound velocity at base temperature. Clearly,such a huge effect must have magnetic origin. Remarkably,the anisotropy disappears rapidly with increasing temperatureand can no longer be recognized above 2 K, which isstill ten times below the magnetic ordering temperature.To demonstrate that this rapid decrease is not an artefactof normalization to κ‖(T ), which increases rapidly with

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temperature, in the inset to Fig. 8(b) we show thelow-temperature part of �κ (T ) without normalization.Despite the increasing noise toward higher temperatures,the downward trend in the data can be still recognized,and above 2 K the �κ/κ becomes indistinguishable fromzero within the accuracy of our measurements. Neitherdilatometry nor ultrasound-velocity measurements presentedin Sec. III show anomalies around this temperature, whichpractically excludes any symmetry-lowering phase transitionas a possible origin of the anisotropy onset below 2 K.On the other hand, this temperature, converted to energyunits, coincides with the pseudo-Goldstone spin-gap energy,suggesting that a thermally activated process involvinglow-energy magnons could be responsible.

We have also followed the magnetic-field dependence ofκ‖ and κ⊥ at several temperatures, as shown in Fig. 8(c). Thefield dependence is nonmonotonic, showing a pronouncedmaximum around 2 or 2.5 T that exceeds the zero-field valueby a factor of 2.0 or 2.7 in the lowest-temperature data forκ‖(B) and κ⊥(B), respectively. This non-monotonic behavioris reminiscent of the field dependence of the spin gap dis-cussed in section V. In Fig. 8(d), we show the difference,�κ = κ⊥(B) − κ‖(B), for each temperature. It also exhibitsnonmonotonic behavior with a change in sign around 1.5 Tfor all three temperatures. Although our data are limited to 5 T,an extrapolation to higher fields suggests that the anisotropyshould vanish at the critical field of approximately 6 T, asexpected from the restored cubic symmetry of the magnonspectrum.

The field- and temperature-dependent data in Figs. 8(c)and 8(a) reveal that the pronounced low-temperature peak inκ (B) with a maximum at B ≈ 2−2.5 T and the associatedmaximum in the anisotropy at B ≈ 3 T [8(d)] are related to thelow-temperature upturn in κ (T )/T , which is most pronouncedin the 1.5 and 3 T data sets in the inset to Fig. 8(a). A compar-ison with the 3 and 5 T data sets published by Gu et al. [22],which extend to somewhat lower temperatures, reveals thatκ (T )/T has a tendency to decrease again below a certain tem-perature, and therefore the onset in our data must correspondto the right-hand side of a broad peak in κ (T )/T with a max-imum at approximately 0.4 K. This is well seen in the inset toFig. 8(a), where we reproduce the 3 T data set of Gu et al. forcomparison, after normalizing it so that it matches our data athigher temperatures. In spite of the different orientations ofthe field and thermal currents in the two experiments and alikely difference in the magnetic domain distribution, whichmay explain a mismatch in the absolute values of the thermalconductivity coefficients, the temperature dependence in bothdatasets is in qualitative agreement, revealing a pronouncedmaximum in κ (T )/T around 0.4 K. This maximum is mostpronounced in the transverse thermal conductivity, ∇T ⊥ B,and only in the intermediate field range around Bc/2, whichenhances κ⊥(0.5 K) at least sixfold (according to Ref. [22])compared to its zero-field value. In higher magnetic fieldsabove the field-induced critical point, the monotonic behaviorof κ (T )/T is restored [22].

VII. DISCUSSION AND CONCLUSIONS

Separating the magnonic (κm) and lattice (κL) contributionsto the thermal conductivity is a challenge given its complex

temperature and field dependencies. It is useful to compareκ for ZnCr2Se4 to that of the structurally similar helimagnetCu2OSeO3, which comprises an approximately fcc lattice ofCu4 tetrahedra [42]. Cu2OSeO3 has κ ≈ 300–400 W/mK at5 K [while κL ≈ (5–10)κm], three orders of magnitude largerthan that of ZnCr2Se4 [43]. This dramatic difference is clearlyrelated to the stronger coupling between the spin and latticesystems in ZnCr2Se4, as manifested, e.g., in the giant magne-tostriction [11,19] (see also Sec. III). Thus both κm and κL inZnCr2Se4 can be expected to be approximately three ordersof magnitude smaller than in Cu2OSeO3, though their relativemagnitudes may differ.

We first consider κL and make the reasonable assumptionthat it is limited at T � 5 K by one-phonon, two-magnonscattering. Such processes, in which a phonon decays intotwo magnons (and vice versa), are predominated by transversephonons in this temperature regime having typical velocitiesin the range 1–2 km/s. The scattering rate τ−1

ph−m is propor-tional to B2q, where B is an average magnetoelastic (ME)constant and q the phonon wave number [44]. Calculationsof κL for ZnCr2Se4 (see Figs. S1, S2 in the SupplementalMaterial [45]) provide an estimate, B � 300 meV, needed toproduce a κL equal to the measured κ at 5K. For comparison,this value is approximately 50 times larger than the averagevalue for yttrium-iron garnet [44], consistent with very strongspin-phonon coupling in ZnCr2Se4.

The temperature dependence of this scattering is dic-tated by the magnon occupation factors [44,45], τ−1

ph−m ∼(1 + nq−k + nk ), where nk is the Bose-Einstein distributionfunction for a magnon state k. Thus the scattering rate de-clines with decreasing temperature even for T � 2 K becausemagnon states satisfying momentum and energy conservationcontinue to be depopulated. This reduction in scattering canproduce an apparent increase of κL relative to the constantscattering rate behavior κL ∝ T 3 expected for an insulator inthe low-T limit. As T → 0, τ−1

ph−m becomes smaller than theboundary scattering rate, leading to κL ∝ T 3. Thus the overallbehavior appears as a plateau or shoulder in κL(T ) and amaximum in κ/T (Fig. S2).

Though the appearance of an upturn and maximum inκ/T [inset, Fig. 8(a)] may be at least partially associatedwith κL, the anisotropy in κ and its nonmonotonic behaviorwith field at T � 2 K are unlikely to be. The cubic symmetrydictates isotropy in the phonon dispersion. The isotropy of κ

at T � 2 K and absence, as noted above, of any identifiablesymmetry change in the lattice below 2 K from dilatometryand ultrasonic measurements imply isotropic ME constants aswell. To assess potential anisotropy in the phonon scattering,we computed [45] the phonon-magnon scattering rate assum-ing isotropic ME constants and using approximate analyticalforms for the three-dimensional magnon dispersion (Figs. S1,S2) [5] constrained by the neutron scattering results along themain symmetry directions—negligible anisotropy was found.The calculations also yield κL(B) (Fig. S2) incompatible withthe nonmonotonic behavior exhibited in Fig. 8(c). Gu et al.[22] proposed spin fluctuations as the primary scatterers ofphonons, but it is unclear how such scattering could producethe nonmonotonic behavior of κL in applied field.

On the other hand, the unusual magnon dispersion andits evolution with applied field described in Sec. V suggest

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that magnon heat conduction is a plausible source for boththe anisotropy and nonmonotonic field dependence of κ (B, T )(Figs. 8(c), 8(d), and Ref. [22]). The calculated κm(B, T ) [45],assuming a constant scattering time (Fig. S3), captures someof these key features, including κm,⊥ > κm,|| and nonmono-tonic κm(B) at low T . Not surprisingly, these features ariseprimarily from low-energy magnon states with momenta atsmall angles to qh. Notably, a maximum in κm,⊥(B) at B =3 T, consistent with the experiment, occurs due to a substan-tially larger modulation of the magnon energy at this field forrotations in the plane perpendicular to qh (Fig. S4)—a featurethat may be related to the splitting of the pseudo-Goldstonemode into two separate modes along the qz direction [Figs. 5,S1(b), and S1(d)].

While this correspondence between experiment and thecalculated κm is encouraging, two discrepancies remain: (i)the absence of detectable anisotropy in experiment at B = 0and (ii) the emergence of anisotropy in experiment only belowT ∼ 2 K. The former is true in spite of the single-domaincharacter of the specimens cooled in a 3 T field to the basetemperature (∼0.5 K) prior to measurements at B = 0 withincreasing T . Possible domain relaxation over the timescaleof the κ measurements up to 1 K (<1 h) appears insufficientto explain this discrepancy, as this relaxation only takes placeat much higher temperatures (see Sec. IV).

The constant magnon scattering time needed to producea calculated κm(1 K) equal to experimental values (κ � 10−2

W/mK) is τm ≈ 10−10 s [45]. This corresponds to an aver-age mean-free path m = 〈|vm|〉τm � 70 nm, where 〈|vm|〉 �700 m/s is the magnon velocity averaged over the thermal dis-tribution at T = 1 K. This value, m ≈ 30λh (λh ≡ 2π/qh ≈2.2 nm), is substantially shorter than the estimated scatter-ing length for magnon-phonon scattering, and thus anotherscattering mechanism must limit κm, e.g., magnetic disorder.Could this scattering explain both the absence of anisotropy atB = 0 and the onset of anisotropy in applied field only belowT � 2 K?

Intriguing candidates for the source of magnon scatteringon the appropriate length scale are chiral domain boundariesthat coexist within the magnetic single-domain specimen asdescribed in Sec. IV. A schematic of possible domain wallsseparating domains with the same qh direction but oppositechirality is depicted in Fig. 9 for the helical (B = 0) and coni-cal (1.5 T � B � 6.5 T) phases. The dominant FM interactionbetween nearest-neighbor spins makes domain-wall orienta-tions parallel to qh in the planar helical phase [Fig. 9(b)]energetically disfavored, given the periodic modulation of theneighboring spin orientations along the wall that inevitablyentail AFM arrangements (dashed area). The neighboring spinorientations in the conical phase along a wall like that ofFig. 9(d) are also modulated, with maximal deviation froma FM orientation equal to the cone angle, which decreaseswith increasing magnetic field. According to these consid-erations, domain walls are expected to have an anisotropicdistribution with the preferred orientation orthogonal to themagnetic ordering vector. For such a distribution, the absenceof anisotropy in κ at B = 0 might be explained if heat-carryingmagnons scatter from domain walls transverse to their mo-tion more strongly in the conical phase. Within an alternative

FIG. 9. Schematic of planar boundaries between helimagneticdomains of opposite chirality: (a), (b) at B = 0 and (c), (d) withinthe conical phase. The red dashed boxes in (b) and (d) highlight en-ergetically disfavored maximal deviations from FM nearest-neighborspin orientations.

scenario, in which magnon heat transport is increased alongthe walls, one might envision unknown dynamical proper-ties of the domain-wall spins on the timescale of τm thatcould be responsible for enhancing κm in the conical phaseat T � 2 K. This speculation provides motivation for furtherinvestigations where the chiral domain distribution might bemanipulated (for example, by electric fields). The proper-ties of domain walls in spiral magnets are not well knownand are a topic of broader interest, including for potentialapplications.

In summary, our studies reveal how the magnetoelasticproperties and novel spin-wave spectrum of ZnCr2Se4 evolvein applied magnetic field within its spin-spiral phase andidentify the anisotropic thermal conductivity to be a sensitiveprobe of the low-energy magnon spectrum. The nonmono-tonic field dependency of the latter at T � 2 K strongly favorsan interpretation in which transport by magnons is princi-pally responsible. The possible role of magnon scattering byor transport along two-dimensional domain walls associatedwith different spin-helix chiralities is proposed as an interest-ing issue for future investigation.

ACKNOWLEDGMENTS

We would like to thank C. Hess, S.-C. Lee, D. Efre-mov, A. Yaresko, and M. Vojta for stimulating discussionsand X.-F. Sun for sharing their recently published thermal-conductivity data [22]. This project was funded by theGerman Research Foundation (DFG) through the Collab-orative Research Center SFB 1143 in Dresden [projectsC03 (D.S.I.) and A06 (S.R.)], individual research GrantIN 209/4-1, and the Transregional Collaborative ResearchCenter TRR 80 (Augsburg, Munich, Stuttgart). T.M. andR.T. acknowledge support from DFG-SFB 1170 ToCoTronics(project B04) and the ERC Starting Grant ERC-StG-Thomale-336012 “Topolectrics”. Work at the University of Miami wassupported by the U.S. Department of Energy (DOE), Of-fice of Science, Basic Energy Sciences (BES), under AwardNo. DE-SC0008607.

184431-12

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184431-14

Supplemental Material for:

“Magnetic-field dependence of low-energy magnons, anisotropic heat conduction, andspontaneous relaxation of magnetic domains in the cubic helimagnet ZnCr2Se4”

D. S. Inosov, Y. O. Onykiienko, Y. V. Tymoshenko, A. Akopyan, D. P. Shukla, N. Prasai,M. Doerr, S. Zherlitsyn, D. Voneshen, M. Boehm, V. Tsurkan, V. Felea, A. Loidl, and J. L. Cohn

S1. Magnon dispersion

In prior work [5] an analytical form for the B = 0 magnondispersion was developed with parameters constrained byfitting to the inelastic neutron scattering results along themain symmetry directions. Figure S1 shows the results fromthis model in the kx − ky and kx − kz planes, extended toapplied fields B = 3, 6, 8 T based on the scattering datapresented in Sec. V. The axes are chosen with kz parallelto the spin spiral axis (designated qh in the main text); thepolar and azimuthal angles, θ and φ, are labeled in theB = 0 diagrams.

These approximations to the three-dimensional disper-sions E(φ,θ , k) were employed in calculations of thephonon-magnon scattering rate and magnon thermal con-ductivity as described further below.

𝜃

𝐸 𝑚𝑒𝑉

𝑘0.20.40.6

0.50.5𝑘𝜙

𝐸 𝑚𝑒𝑉0.20.40.6

0.50.5𝑘 𝑘∆

𝐸 𝑚𝑒𝑉

𝑘0.20.40.6

0.50.5𝑘

𝐸 𝑚𝑒𝑉0.20.40.6

0.50.5𝑘 𝑘

B=0

3 T

𝐸 𝑚𝑒𝑉

𝑘0.20.40.6

0.50.5𝑘

𝐸 𝑚𝑒𝑉0.20.40.6

0.50.5𝑘 𝑘

𝐸 𝑚𝑒𝑉

𝑘0.20.40.6

0.50.5𝑘

𝐸 𝑚𝑒𝑉0.20.40.6

0.50.5𝑘 𝑘

6 T

8 T

(c) (d)

(a) (b)

(g) (h)

(e) (f)

Fig. S1. Three-dimensional magnon dispersions, shown in thekx -ky (left) and kx -kz (right) planes for four values of magneticfield. Wave numbers are in units of 2π/a. The polar and azimuthalangles, θ ,φ, and the pseudo-Goldstone magnon gap (∆PG, seeFig. 6) are labeled in the plots for B = 0.

S2. Phonon-magnon scattering rate

Following the recent work of Streib et al. [39], we derivedthe scattering rate for 1-phonon, 2-magnon scattering as:

1τq,λ

=πħhmN

∑

k

|Γk,q−k,λ|2

εq,λ

�

1+nq−k+nk

�

δ�

Ek+Eq−k+εq,λ

�

,

where m and N are the mass per unit cell and the num-ber of unit cells, respectively, k and q are the magnon andphonon wave vectors (λ is a polarization index), nk is theBose-Einstein distribution function, Ek and εq,λ = ħhcλ|q|are magnon and phonon energies, respectively, and the δ-function enforces energy conservation. We used an approxi-mate analytic expression for Ek with parameters constrainedby neutron scattering measurements along the main sym-metry directions (see below). Γk,q−k,λ is the matrix elementfor phonon-magnon interaction that is linear in q with ascale set by the magnetoelastic constants. The latter wereassumed to be isotropic, and only transverse phonons wereconsidered (they should predominate in the scattering withmagnons given their lower energies), with both modes takento have the same velocity, cλ ≡ c = 2 km/s.

S3. Calculation of the phonon-magnon scattering rate

The sums over magnon momenta k were performed overa uniform k-space mesh (120 × 120 × 120) with spacing0.01 in the range from −0.595 to 0.595 [in reciprocal latticeunits, q/(2π/a)]. The scattering rates were found to be in-dependent of φ and very weakly dependent on θ . τ−1

q wascomputed for phonon coordinates q restricted to relativevalues in the range 0 to 0.422 with spacing 0.002 for theq range 0–0.104 and 0.008 for the remaining range. Thesubsequent integrations to determine κ employed interpola-tions on τ−1

q . The energy δ function was approximated [39]as δ(ε)≈ (1/

pπα)exp(−ε2/α2), with α= 0.005 meV.

The absence of anisotropy (θ dependence) in the scatter-ing rate is a consequence of the peculiar magnon dispersion.The dominant magnons in the scattering of phonons at lowT are those with momenta k ≈ 0.3 and energies in the rangeE ≈ 0.15–0.3 meV restricted to narrow conical regions ori-ented along the coordinate axes where the dispersion isflat and the magnon density of states correspondingly high(Fig. S1). Energy conservation dictates q� k and thus mo-mentum conservation, k+ k′ = q, is achieved with magnonstates having nearly opposing momenta, either along ortransverse to q.

S4. Calculation of lattice thermal conductivity

The total scattering rate employed in the κ calculationswas a sum of the phonon-magnon scattering rate and a

� S1 �

T (K)1 10

κ (W

/mK)

10-2

10-1

100

101

B=0

bndy. scatt.only

T (K)1 10

κ /T

(W/m

K2 )

10-2

10-1

calculated

B=6 T30

8

𝜅 (3 𝑇)

Fig. S2. Computed thermal lattice conductivities (open squares;κL,⊥ = κL,‖ to 1%) at various fields compared to the experimentalB = 0 data (filled circles). The magnetoelastic constant was setso that the T = 5 K conductivities matched the experimental data.Also shown is the computed thermal conductivity with only theboundary scattering term included (×). Inset: Computed κL/T forthe boundary scattering and B = 0, 3 T curves from the main paneland corresponding experimental 3 T data (filled circles) from theinset to Fig. 7 (c).

boundary scattering term, τ−1tot = τ

−1q + c/`0 (where `0 =

0.2 mm was taken as a typical dimension of the single-crystalspecimen transverse to the heat flow). We have ignored anyT dependence of the ME constant and possible scatteringfrom defects and phonons, but these should be reasonable as-sumptions at low T , particularly to assess changes with mag-netic field. The thermal conductivities were calculated as,

κL,⊥ =kB

(2π)3

�

kB Tħh

�2∫

(c cosθ sinφ)2τtotx4ex

(ex − 1)2d3q;

κL,‖ =kB

(2π)3

�

kB Tħh

�2∫

(c cosθ )2τtotx4ex

(ex − 1)2d3q,

where x = ħhcq/kBT . Figure S2 shows the calculated κL(open squares and curves) at the four field values correspond-ing to those for the dispersions. The computed anisotropywas very small (κL,⊥ − κL,‖ less than 1% of κL,⊥), so only asingle curve is shown at each field.

For the purpose of assessing changes in κL with field, weignored the shift in κ toward larger values throughout the Trange seen in experiment [Fig. 7 (a) and Ref. 22], and fixedthe magnetoelastic constant for the calculations so that theconductivities matched the experimental data at T = 5 K.Also shown in Fig. S2 is the thermal conductivity computedwith only the boundary scattering term included (×’s). Asthe phonon-magnon scattering declines with decreasing tem-perature, the computed κL’s interpolate toward the boundaryscattering curve, yielding a plateau or shoulder. This featureyields a peak in κL/T (inset, Fig. S2) qualitatively similar tothat of the data from Gu et al. [22] (though the maximumoccurs at somewhat higher T in the calculations). With theexception of the 3 T data which matches experiment well,the computed κL overestimates the experimental κ.

Most notably, the calculated κL at low-T decreases withincreasing field, failing to reproduce the nonmonotonic be-havior of experiment [Fig. 7 (a) and Ref. 22]. This trend inthe calculations is consistent with the dispersions (Fig. S1)which show increasing low-energy magnon spectral weight(from which phonons can scatter) as the field increases to 6 T.Though ∆PG increases significantly in going from B = 0 toB = 3 T [clearly seen in Fig. S1 (a, c)], the magnon spectralweight is enhanced at lower energy for B = 3 T, in part dueto the splitting of the pseudo-Goldstone mode into two sepa-rate modes along the kz direction [Fig. 5 and Fig. S1 (b, d)],and especially due to the magnon dispersion at small anglesθ (discussed further below in connection with calculationsof κm and shown in Fig. S4). At 8 T the entire spectrum isgapped and the phonon-magnon scattering is suppressed.

S5. Calculation of magnon thermal conductivity

The magnon velocities were computed from the gradientof the computed magnon dispersion function E(φ,θ , k),

ħh~vk =−→∇k E =

∂ E∂ k

k+1k∂ E∂ θ

θ +1

k sin(θ )∂ E∂ φ

φ.

With the applied field direction along z, the integral expres-sions for κ⊥ (

−→∇ T along x) and κ‖ (−→∇ T along z) include

scalar products (~vk · x)2 and (~vk · z)2, respectively:

ħh~vk · x = sinθ cosφ∂ E∂ k+

cosθ cosφk

∂ E∂ θ−

sinφk sinθ

∂ E∂ φ

;

ħh~vk · z = cosθ∂ E∂ k−

sinθk∂ E∂ θ

.

The magnon thermal conductivities are given as,

κm,⊥ =kB

4 (2π)3

∫

(~vk · x)2τmx2ex

(ex − 1)2d3k;

κm,‖ =kB

4 (2π)3

∫

(~vk · z)2τmx2ex

(ex − 1)2d3k,

where x = E/kBT , τm is the magnon relaxation time (takento be a constant, τm = 10−10 s), and the factor of 1/4 mul-tiplying the integrals arises from the smaller primitive cellof the fcc magnetic lattice compared to the crystallographiclattice.

The computed thermal conductivities are shown as func-tions of temperature and field in Fig. S3. We find κm,⊥ > κm,‖at all fields for T ® 5 K [Fig. S3 (a)]. The field dependence[Fig. S3 (c)] exhibits a maximum near B = 3 T for κm,⊥,quite similar to that observed experimentally, though the oc-currence of a maximum at higher field for the computed κm,‖is inconsistent with experiment where a somewhat lowerfield scale for the maximum (1.3 T) is observed for thisorientation.

The anisotropy and nonmonotonic behavior of κm at lowT arises primarily from features of the dispersion for mo-menta nearly parallel to the spin spiral axis (small θ), wheremagnon states appear at ever lower energies with decreasingθ as the dispersion evolves toward its gapless character atk = qh (Fig. S4). The nonmonotonic behavior of κm canbe traced to the nonmonotonic variation in ∂ E/∂φ with ap-plied field (the amplitude of the oscillations that determine

� S2 �

∂ E/∂ φ is largest for the B = 3 T dispersions, highlighted bythe vertical arrows in Fig. S4 — a substantial contribution

to κm,⊥ from this term arises from the (~vk · x)2 factor in theintegral expression above.

B (T)0 2 4 6 8

κ m(H

)/κm(0

)

0

1

2

3

||

⊥T=0.30 K

1.00

0.50

T (K)0.1 1 10

κ m (W

/mK)

10-6

10-5

10-4

10-3

10-2

10-1

100

τ=10-10s

T3

κ||

κ⊥

B=3 T

60

8

T (K)0.1 1 10

κ m/T

(W/m

K2 )10-5

10-4

10-3

10-2

10-1

κ⊥/T

κ||/T

B=0 T

36

8

(c)(a) (b)Fig. S3. Computed (a) κm and (b) κm/T perpendicular (κ⊥, blue curves) and parallel (κ‖, red curves) to applied magnetic field for B = 0,3, 6, and 8 T. (c) Normalized change in κ⊥ and κ‖ vs. magnetic field at T = 0.30, 0.50, and 1.00 K.

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.2

0.4

0.6

0.05

−0.05

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐵𝐵 = 3 𝑇𝑇

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐵𝐵 = 6 𝑇𝑇

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.2

0.4

0.6

0.05

−0.05

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.1

−0.1

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

0.2

0.4

0.6

𝐵𝐵 = 0

𝐸𝐸(𝑚𝑚𝑚𝑚𝑚𝑚)

0.2

0.4

0.6

0.05

−0.05

𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦

𝜃𝜃 =𝜋𝜋

16

𝜃𝜃 =𝜋𝜋

32

𝜃𝜃 =𝜋𝜋8

Fig. S4. Magnon dispersions at B = 0,3,6 T at angles (from top to bottom) θ = π/8,π/16,π/32, projected into the kx − ky plane (i.e.k =

q

k2x + k2

y/ sinθ) to show the φ dependencies. The vertical arrows for B = 3 T highlight the amplitude of the oscillations whichdetermine ∂ E/∂ φ.

� S3 �