arXiv:2012.14515v1 [cond-mat.str-el] 28 Dec 2020 and oigta h oi nl Ω angle solid respect, the this monopole. that In magnetic the Noting center. called texture sometimes is the it hedgehog in that the hedgehog point so sigular the space, con- a real dimensions, In has three-dimensional two the in structure. over skyrmion real-space extends the the distinction to in The is trast involved. two spins the the between 4 all by of subtended units gle in to, charge corresponds topological integer which the by characterized both tures in breathing frustration the the lattice. on by pyrochlore antiferromagnets induced Heisenberg far. be the so classical that can demonstrate reported lattice theoretically been will hedgehog we not paper, has this interaction other In DM mechanism the ordering counter hedgehog than the the however, Concerning lattice, i.e., [37–45]. interactions, frustration exchange re- magnetic between a as competitions appears of lattice sult skyrmion the without interaction, magnets Kawa- DM centrosymmetric the and in Chung, even realized that Okubo, [36] been by mura has work it pioneering Besides, the since [22–35]. 4–21] hedgehogs [1, the skyrmions the and of orders key crystalline a for is ingredient interaction accepted (DM) widely Dzyaloshinskii-Moriya nowadays of the is arrays that It periodic objects. devices are topological interest spin-electronic these particular to recent Of applications because [1–3]. possible attention much their mag- attracted of the have analogue, hedgehog, three-dimensional netic its and skyrmion 4] h eghgcnb nesoda topologically a as understood object. chirality be protected can hedgehog the [46], S i h antcsymo n eghgaesi tex- spin are hedgehog and skyrmion magnetic The magnetic the as such textures spin Topological h eghgltiei natraigproi array periodic alternating an is lattice hedgehog The · ( S S 1 eghgltiesi etr ncasclHiebr ant Heisenberg classical in texture spin Hedgehog-lattice k j eateto at n pc cec,Gaut colo Sc of School Graduate Science, Space and Earth of Department × srltdwt h clrsi chirality spin scalar the with related is S k ed h tutr fte( the of structure the field, elzdfralretidN niermgei interactio antiferromagnetic NN third large a for realized nanneontsi hrlt hc namtll ytm sh system, metalilc a in which chirality effect. spin Hall net nonzero a in yohoeltie ntemdl h rahn bond-alter int breathing ones, exchange the antiferromagnetic small Heisenb model, (NN) classical the nearest-neighbor in the In frustration but lattice. tha Dzya simulations interaction the pyrochlore Carlo of DM Monte presence the of the means not in by realized demonstrate we be Here, to known is monopoles, J eoetehdeo-atc tt ntebetigcs of case breathing the in state hedgehog-lattice the become i Ω via ) 1 ′ h eghgltie he-iesoa eidcaryo array periodic three-dimensional a lattice, hedgehog The /J .INTRODUCTION I. 1 ,i sacliersaefvrdb hra utain.I fluctuations. thermal by favored state collinear a is it 1, = ijk 2 ooaPyia n hmclRsac nttt,Aci 48 Aichi, Institute, Research Chemical and Physical Toyota J tan 2 = 1 ′ /J 1 quadruple- A . ijk − 1 o he spins three for  π 1+ auh Aoyama Kazushi h oa oi an- solid total the , 2 1 S , rahn yohoelattice pyrochlore breathing i · 2 1 S , j 2 1 + q eghgltiei hne rmcbct ergnl resu tetragonal, to cubic from changed is lattice hedgehog ) χ S ijk i tt ihteodrn etrof vector ordering the with state · Dtd aur ,2021) 1, January (Dated: S k + S χ S i ijk j , · S S 1 k = j  , n iauKawamura Hikaru and eigvectors dering httesisaeclieryainddet h ffc of above effect the the to Since due aligned fluctuations. collinearly thermal are spins the that ( r ree noaquadruple- a into ordered are iepaewsfudas nteD-recentrosymmet- DM-free lat- the SrFeO hedgehog in magnet also ric the found multiple- Recently, was of phase observation tice [22–28]. the topological reflections with the Bragg by together evidenced effect been Hall has lattice hedgehog directions tt,w r osprs h pncliert.Frthis For collinearity. spin the multiple- the suppress is to phase try ordered we the lat- that state, hedgehog sense the the in host tice potentially can antiferromagnet fsi tef h eghgltiei multiple- LRO a a is as lattice viewed long-range hedgehog When a the is itself, chirality. it spin the nonzero of words, of other negative (LRO) In or order chirality. positive spin having scalar (anti- each anti-hedgehogs and monopoles) (monopoles) hedgehogs the of fcre-hrn erhda ntepeec fntonly not teraction in- of exchange presence antiferromagnetic the nearest-neighbor(NN) the In tetrahedra. corner-sharing of frustrated lattice. pyrochlore of the Heisenberg example on classical antiferromagnets typical be the would A induces systems three-dimensional dimensions, [36]. two lattice effects in which skyrmion investigate hedgehog- frustration theoretically explo- the the we the hosting of towards texture, magnets work, of spin this classes lattice In have new compounds of far. candidate ration so other reported no and been unclear still is 4] .. h ocle oooia aleet nthe origin In chirality interac- effect. DM spin MnSi Hall the of tion topological having effect magnets so-called Hall non-centrosymmetric the the hedge- i.e., the is of [47], aspect lattice prominent ordering-wave-vectors A hog 30–35]. one 25–28, than 23, [22, more by described 2 1 h yohoeltiei he-iesoa network three-dimensional a is lattice pyrochlore The , 1 2 ln h oddirection bond the along n , ec,OaaUiest,Oaa5004,Japan 560-0043, Osaka University, Osaka ience, − rcinfrlrettaer ota for that to tetrahedra large for eraction J aini hrceie ytertoof ratio the by characterized is nation 2 1 r niermgeso h breathing the on antiferromagnets erg 1 ′ udyedacaatrsi topological characteristic a yield ould nuisof units in ) /J J 1 J ohnkiMry D)interaction. (DM) loshinskii-Moriya h eghgltiei nue by induced is lattice hedgehog the t − 1 1 3 antcmnplsadanti- and monopoles magnetic f x ttrsotta lsia esnegspins Heisenberg classical that out turns it , u lotetidN n ln h bond the along one NN third the also but < sas on hti magnetic a in that found also is t Ge q ,wiei h nfr aeof case uniform the in while 1, x q ( = 3 ih0 with 2 ( = 4,4] u h reigmechanism ordering the but 49], [48, -18 Japan 0-1118, 1 2 froant nthe on iferromagnets , 2 ± a π 1 2 1 2 , . , 25 ntecbcbss[05]and [50–52] basis cubic the in 1 2 ± ) , 2 1 x < J ( , q 3 − ± un u to out turns , 2 1 1 2 tt ihtefu or- four the with state h curneo the of occurrence the , ,wihis which ), , 2 1 , 2 1 ) lting J , 1 ( - 1 2 J , 3 − pyrochlore 2 1 , 2 1 q ,and ), state q ’s q q 2 purpose, we consider the so-called breathing pyrochlore lattice consisting of an alternating array of small and large tetrahedra (see Fig. 1) [51, 53]. In the breathing pyrochlore lattice, the bond alterna- tion should be reflected in different values of the NN in- teractions on small and large tetrahedra, J1 and J1′ , so that its strength could be measured by the ratio J1′ /J1. In the different context of the spin-lattice coupling, the effect of the breathing lattice structure has already been examined: within the collinear-spin manifold, the fully 1 1 1 ordered quadruple-q state with q = ( 2 , 2 , 2 ) is changed into a partially disordered one± for smaller± ± val- ues of J1′ /J1 [51]. This indicates that for the present 1 1 1 Heisenberg spins, the thermally-driven collinear ( 2 , 2 , 2 ) state would get unstable with decreasing J1′ /J1, possibly leading to the hedgehog lattice. As demonstrated in Fig. 1, this is actually the case. One can see from Fig. 1 1 1 1 that the quadruple-q ( 2 , 2 , 2 ) state on the breathing py- rochlore lattice is nothing but the hedgehog lattice (see 1 1 1 Secs. II and III for details). This ( 2 , 2 , 2 ) hedgehog lat- tice appears as a result of the frustration in the absence of the DM interaction. In this paper, we further clarify that the hedgehog lattice in a magnetic field possesses a nonzero net spin chirality which should yield emergent FIG. 1: Hedgehog-lattice spin texture at zero field on the phenomena such as the topological Hall effect. breathing pyrochlore lattice, where the spin configuration is obtained at T/J1 = 0.01 in the Monte Carlo (MC) simulation This paper is organized as follows: In Sec. II, we intro- ′ duce the spin Hamiltonian and discuss its ordering prop- for the Hamiltonian (1) with J1/J1 = 0.6, J2/J1 = 0 and erties based on analytical calculations. The results of our J3/J1 = 0.7. Red, green, blue, and yellow arrows represent spins on the corners 1, 2, 3, and 4 of the small tetrahedra MC simulations will be shown in Sec. III. In Sec. IV, we shown in the inset, respectively. The small tetrahedron at the will discuss the effect of a magnetic field and demon- center of the cubic unit cell is outlined by black dots. The strate the emergence of the net spin chirality which re- upper and lower central small tetrahedra correspond to the sults in an unconventional topological Hall effect quite hedgehog (monopole) and the anti-hedgehog (anti-monopole), different from the DM-induced one. The origin of this respectively (see the main text in Sec. III). field-induced topological Hall effect will be addressed in Sec. V. We end the paper with summary and discussions in Sec. VI.

bond direction J3 is crucial for the occurrence of the hedgehog lattice. Although the second NN interaction II. MODEL HAMILTONIAN AND ITS BASIC ORDERING PROPERTIES J2 is not essential, we have included J2 for complete- ness. When we discuss the in-field properties of the sys- tem in Sec. IV, we include the additional Zeeman term A. Model H i Si,z in the Hamiltonian (1), where H is the in- −tensity of a magnetic field. Throughout this paper, we The spin Hamiltonian we consider is given by takeP the cubic unit cell of edge length a = 1 shown in Fig. 1, and thus, the total number of spin N and the = J S S + J ′ S S 3 H 1 i · j 1 i · j linear system size L is related via N = 16L . i,j S i,j L hXi hXi +J S S + J S S , (1) 2 i · j 3 i · j i,j i,j hhXii hhhXiii B. Effect of the breathing lattice structure on the where Si is the classical Heisenberg spin at the site i, spin ordering S(L), i, j , and i, j denote the summations over sitehi pairshh onii the smallhhh (large)iii tetrahedra, the second NN pairs, and the third NN pairs along the bond direction, To get insight into roles of the breathing bond- respectively. As explained in Sec. I, the breathing bond- alternation, we first discuss ordering properties of this alternation is characterized by the ratio of the NN inter- model in the mean-field (MF) approximation [54, 55]. In- α action for large tetrahedra to that for small ones, J /J , troducing the Fourier transform Si = Sq exp(iq ri) 1′ 1 q · and the third NN antiferromagnetic interaction along the with the site index i = (α, r ), we can rewrite into the i P H 3 following form 9T 2 2 2 δf = A ΦQ ΦQ + ΦQ Φ Q 4 20 2 | µ | | ν | | µ · εs ν | 4  µ<ν εs= 1 N αβ α β X  X±  = J (q)Sq S q + const, (2) H 8 · − q i A3 (Φ Q1 ΦQµ )(ΦQν ΦQρ ) c.c. , X α,βX=1 − − · · − n µ=ν=ρ=1 o n o where the sublattice indices α =1, 2, 3, 4 correspond to 6 X6 6 2(1+3ε4) 16ε2(1 + ε2) 16ε3 the four corners of a tetrahedron [see the insets of Figs. A1 = 2 2 , A2 = 2 2 , A3 = 2 2 . 1 and 2 (d)]. By analyzing the lowest eigen value of (1+3ε ) (1+3ε ) (1+3ε ) the 4-by-4 matrix J αβ(q), λ , we find that for various q Only in the breathing case of J /J < 1, ε is nonzero sets of parameters we searched, possible ordering wave- 1′ 1 2π 1 1 1 2π and thus, δf4 is active. The most stable spin configura- vectors are a ( 2 , 2 , 2 ), a ( 1, 1, 1), and incom- ± ± ± ± ± ± tion is obtained by minimizing GL under the constraint mensurate ones (see Fig. 7 in Appendix A). For large J3, 2 4 F 2π 1 1 1 2π 1 1 1 2 Q Q S = 2 µ=1 ΦQµ . Noting that in the uniform case, λq is obtained at 1 = a ( 2 , 2 , 2 ), 2 = a ( 2 , 2 , 2 ), | | 2π 1 1 1 2π 1 1 1 − Q Q each ΦQ determines the spin orientation axis Pˆµ at each 3 = a ( 2 , 2 , 2 ), and 4 = a ( 2 , 2 , 2 ) as λQi = Pµ − 2 − 2 sublattice, we assume that even in the breathing case, J1 + J1′ 2J3 3(J1 J1′ ) + (J1 + J1′ +4J3) . The − − − S iθµ ˆ associated eigen vectors are given by ΦQµ takes the form of ΦQµ = e Pµ. Then, we have p √8

UQ1 = cN ( iε, 1, iε, iε), 2 − − − f2 = 3T + λQµ S , UQ2 = cN (iε, iε, 1, iε), 27T 4 U = c (iε, iε, iε, 1), f4 =  A1 S , Q3 N 160 UQ4 = cN (1, iε, iε, iε) (3) 9T 4 2 2 δf4 = S A2 3 + (Pˆ1 Pˆ2) + (Pˆ1 Pˆ3) with the normalization factor c = 1/√1+3ε2 and the 640 · · N  n dimensionless coefficient +(Pˆ Pˆ )2 + (Pˆ Pˆ )2 + (Pˆ Pˆ )2 + (Pˆ Pˆ )2 1 · 4 2 · 3 2 · 4 3 · 4 2 J1 + J1′ +4J3 J1 J1′ ˆ ˆ ˆ ˆ o ε = 1+3 − 1 . (4) 2A3 sin(θ1 θ2 θ3 θ4) (P1 P2)(P3 P4) − − − − · · 3(J1 J1′ ) s J1 + J1′ +4J3 − −     n +(Pˆ1 Pˆ3)(Pˆ2 Pˆ4) + (Pˆ1 Pˆ4)(Pˆ2 Pˆ3) . (7) ε is zero for the uniform pyrochlore lattice with J1′ /J1 = · · · · 1, and gradually increases with increasing the breathing o alternation (decreasing J1′ /J1). Note that J2 does not In the uniform case of J1′ /J1 = 1, δf4 = 0, so that U show up in λQµ and Qµ , and thus, irrelevant for the the relation among different θµ and Pˆµ cannot be deter- ordering properties. mined. In other words, spins at different sublattices are In the ( 1 , 1 , 1 ) ordered phase, the mean field Sα is 2 2 2 h qi uncorrelated at the MF level. As will be shown below, described as however, thermal fluctuations induce the collinear inter- 4 sublattice correlation. In the breathing case of J1′ /J1 < 1, α α α Sq = UQ ΦQµ δq,Qµ + [UQ ]∗ΦQ∗ δq, Qµ , (5) δf4 is active and the minimization condition for δf4 is h i µ µ µ − µ=1 θ1 θ2 θ3 θ4 = π/2 and Pˆµ Pˆν = 1/3 (µ = ν), sugges- X
− − − · − 6 α tive of a nonzero value of the chirality χijk = Si (Sj Sk) where U represents the αth component of UQ . In · × Qµ µ formed by any three spins on each tetrahedron. It will the uniform case of J1′ /J1 = 1 and thereby ε = 0, all be shown in the following numerical calculations that this the UQ ’s are orthogonal to one another, so that ΦQ1 , 1 1 1 µ quadruple-q ( 2 , 2 , 2 ) state with the nonzero chirality is ΦQ2 , ΦQ3 , and ΦQ4 correspond to the polarization vec- nothing but the hedgehog lattice. It should be empha- tors of the spins belonging to sublattice 2, 3, 4, and 1, sized that at least at the MF level, an infinitesimally respectively. In other words, to make spins exist at each small breathing alternation yields nonzero ε and resul- sublattice, the ordered phase should be the quadruple-q 1 1 1 tantly, can induce the ( 2 , 2 , 2 ) hedgehog lattice. state involving all the ΦQµ ’s, and thus, other multiple-q states such as single-q, double-q, and triple-q states are not realized in the present system. Then, the Ginzburg- III. RESULT OF THE MC SIMULATION AT Landau (GL) free energy in the form expanded with re- ZERO FIELD spect to ΦQ is obtained as /(N/4) = f + f + δf µ FGL 2 4 4 with Now that the role of the breathing bond-alternation 4 has been understood, we shall perform a rigorous analysis 2 f =2 3T + λQ ΦQ , (6)of the finite-temperature properties of the Hamiltonian 2 µ | µ | µ=1 (1) by means of Monte Carlo (MC) simulations. In this   X 4 section, we consider the zero-field case of H = 0. 9T 2 4 5 f = A ΦQ ΦQ +2 ΦQ , In our MC simulations, 2 10 sweeps are carried 4 20 1 | µ · µ | | µ | × µ=1 out under the periodic boundary condition and the X   4

3 1 2 1 2 N 2 i,j Si Sj 3 and the spin and chirality struc- · − 2 q 1 S iq ri q ture factorsP FS(
) = N i i e · and FC ( ) = 2 1 χ(R ) eiq Rl , where the scalar chirality of N/4 l l · P the lth tetrahedron with its center-of-mass position R P l is defined by χ(Rl) = i,j,k l th tetra χijk and the order of i, j, and k is defined in∈ the anticlockwise direction with respect to the normalP vector of the triangle formed by the three sites i, j, and k,n ˆijk, pointing outward from Rl. In the same manner, we define the solid an- gle subtended by the four spins on the tetrahedron as Ω(Rl)= i,j,k l th tetra Ωijk . ∈ The temperatureP dependence of the specific heat C and the spin collinearity P obtained in the MC simula- tions for J2/J1 = 0 and J3/J1 = 0.7 is shown in Fig. 2 (a). In the uniform pyrochlore lattice with J1′ /J1 = 1, P starts developing at the transition temperature indicated by the sharp peak in C, and approaches the maximum value of 1 at the lowest temperature. Since the Hamilto- nian (1) itself does not have an interaction to collinearize spins, thermal fluctuations should be relevant to the spin collinearity as is often the case with frustrated magnets [56, 57]. With increasing the breathing bond-alternation, i.e., decreasing J1′ /J1, such a temperature effect is grad- ually suppressed. At J1′ /J1 =0.6, P develops on cooling similarly to the case of J1′ /J1 = 1, but it finally goes to zero at T = 0 via an intermediate phase explained below, suggesting the occurrence of a noncollinear ground state. At the smaller value of J1′ /J1 =0.2, the ordered phase is completely masked by the noncollinear state with P = 0. To look into the details of the ordered phases, we ex- amine the spin and chirality structure factors F (q) and FIG. 2: MC results obtained for J2/J1 = 0 and J3/J1 = 0.7. S Note that a magnetic field H is not applied. (a) The tem- FC (q). perature dependence of the specific heat C (the first panel Figure 2 (b) shows FS(q) in the noncollinear state from the top), the spin collinearity P (the second one), and S with P = 0. This state is characterized by the mag- the average spin and chirality Bragg intensities O 1 1 1 and ( 2 2 2 ) netic Bragg peaks with the same intensity at the cubic- C O 1 1 1 (the third and fourth ones). Greenish, reddish, and 1 1 1 ( 2 2 2 ) symmetric families of ( 2 , 2 , 2 ), so that it is a cubic- ′ ± ± ± 1 1 1 bluish colored symbols denote the data for J1/J1 = 1, 0.6, symmetric quadruple-q state. Such a quadruple-( 2 , 2 , 2 ) and 0.2, respectively. (b)-(e) The results obtained in the or- spin structure can commonly be seen irrespective of the ′ dered phases for J /J1 = 0.6. (b) and (c) Spin and chirality 1 value of J1′ /J1. One can see from Fig. 2 (a) that structure factors FS (q) and FC (q) in the (h, h, l) plane at S the averaged intensity of the Bragg peaks O( 1 1 1 ) = T/J1 = 0.11. (d) Snapshots of spins mapped onto a unit 2 2 2 sphere in the hedgehog-lattice phase at T/J1 = 0.11 (top), 2 h,k,l= 1/2 FS (h,k,l) serves as the order parameter to ± the coplanar phase at T/J = 0.31 (middle), and the collinear characterize the transition from the paramagnetic phase 1 P phase at T/J1 = 0.41 (bottom), where red, green, blue, and in all the three cases of J1′ /J1 = 1, 0.6, and 0.2. Al- yellow dots represent spins on the corners 1, 2, 3, and 4 of the though whether the spins are collinear or noncollinear is small tetrahedra shown in the inset, respectively. indistinguishable in the spin sector FS (q), it is clearly visible in the chirality sector FC (q). In the uniform case of J1′ /J1 = 1, spins are collinearly ordered, so that the local scalar chirality χijk and the associated F (q) are vanishingly small. In contrast, in the breath- first half is discarded for thermalization, where our C ing case of J /J < 1, F (q) in the noncollinear state 1 MC sweep consists of 1 heatbath sweep and suc- 1′ 1 C with P = 0 exhibits Bragg peaks at ( 1 , 1 , 1 ) sug- cessive 10 overrelaxation sweeps. Observations are 2 2 2 gestive of a tetrahedron-based chirality± order± ± [see Fig. done at every MC sweep and the statistical average 2 (c)]. The averaged Bragg intensities in the chirality is taken over 4 independent runs starting from differ- C 1 sector O( 1 1 1 ) = 4 h,k,l= 1/2 FC (h,k,l) for J1′ /J1 = 1, ent random initial configurations. We identified the 2 2 2 ± low-temperature ordered phase by observing various 0.6, and 0.2 are shown in the bottom panel of Fig. 2 (a). PC One can see that O 1 1 1 serves as the order parameter physical quantities such as the spin collinearity P = ( 2 2 2 ) 5

hedgehog lattice, relative angles between the polarization axes for the four sublattices are changed with each up- down antiferromagnetic chain being almost unchanged. Figure 2 (d) shows the map of the spins in the three 1 1 1 different ( 2 , 2 , 2 ) phases obtained at J1′ /J1 = 0.6. One can see from the bottom (top) panel of Fig. 2 (d) that in the higher-temperature collinear (lower-temperature hedgehog-lattice) phase, the polarization axes for the four sublattices are the same (different). The spin ordering pattern in the collinear phase is up-up-down-down along all the bond directions [50, 51]. As shown in the middle of FIG. 3: The parameter dependence of the low-temperature Fig. 2 (d), the intermediate-temperature phase between ordered phases at H = 0. (a) J2/J1 = −0.2, (b) J2/J1 = 0, the two is a coplanar state in which four sublattices are and (c) J2/J1 = 0.2. The stability regions are determined divided into two pairs and spins belonging to each pair from the MC results at T/J = 0.01. The ( 1 , 1 , 1 ) hedgehog- 1 2 2 2 have the same polarization axis (see Fig. 8 in Appendix lattice and collinear phases are obtained at red and pink col- B for details of the real-space spin configurations). ored points, respectively, whereas LRO’s with ordering vec- 1 1 1 1 1 1 The stability region of the ( , , ) hedgehog lattice tors other than ( 2 , 2 , 2 ) are obtained at black colored points. 2 2 2 The stability regions are consistent with the result of the is summarized in Fig. 3. Among the three quadruple- αβ 1 1 1 J (q) analysis (see Fig. 7 in Appendix A) q ( 2 , 2 , 2 ) states, only the coplanar one cannot survive down to the lowest temperature, so that it does not ap- 1 1 1 pear in Fig. 3. The ( 2 , 2 , 2 ) hedgehog lattice is real- ized in the wide range of the parameter space even in 1 1 1 for the noncollinear ( 2 , 2 , 2 ) state with P = 0. the presence of J2. As expected from the MF analy- The real-space spin configuration of the noncollinear sis, a small breathing alternation, i.e., small deviation 1 J ′ /J , seems to be sufficient to stabilize the hedge- state is shown in Fig. 1. Spins belonging to each sublat- − 1 1 tice, which are represented by the same colored arrows hog lattice. in Fig. 1, constitute up-down antiferromagnetic chains along the bond directions, and four sublattices have dif- ferent polarization axes. In Fig. 1, since four outgoing IV. EFFECT OF A MAGNETIC FIELD (incoming) spins on a tetrahedron subtend the solid angle of Ω(Rl)=4π ( 4π), the upper (lower) small tetrahe- dron at the center− of the cubic unit cell has the topo- In this section, we will discuss the effect of a mag- 1 1 1 logical charge of +1 ( 1). As the spins are ordered in netic field H on the ( 2 , 2 , 2 ) hedgehog lattice. Figure 4 a way such that the two− cubic unit cells are alternately (a) shows the temperature and magnetic-field phase dia- arranged, the state is nothing but the hedgehog lattice gram for J1′ /J1 =0.2, J2/J1 = 0, and J3/J1 =0.7. The 1 1 1 in which the same number of the monopoles with +1 low-field and high-field phases are the ( 2 , 2 , 2 ) hedgehog- charge and the antimonopoles with 1 charge are equally lattice and coplanar states being subject to spin canting, − respectively. We note that for larger values of J /J , the populated and their densities n+ and n are given by 1′ 1 1 − canted collinear state also shows up (see Fig. 9 in Ap- n+ = n = 16 . For the monopole and antimonopole tetrahedra,− the total spin on each tetrahedron is zero as pendix C). Although a weak transition-like anomaly is found in the high-field phase [see open symbols in Fig. is expected from the NN antiferromagnetic interaction J1 4 (a)], the detailed analysis of the high-field phase is be- or J1′ [54], while not for other tetrahedra. This is because 1 1 1 yond the scope of this work, and hereafter, we will fo- the ( 2 , 2 , 2 ) spin ordering in the present system is driven by the relatively strong third NN antiferromagnetic in- cus on the low-field hedgehog-lattice phase. As shown in the left panels of Fig. 4 (b), the in-field hedge- teraction J3 rather than J1 or J1′ and thus, not all the tetrahedra satisfy the constraint of total spin zero stem- hog lattice is characterized by the enhanced (suppressed) ( 1 , 1 , 1 ) magnetic Bragg reflections for the spin ming from J1 and J1′ . In relation to this, as one can see 2 2 2 ± ± ± Sk from Fig. 1, the monopoles and antimonopoles, which component parallel (perpendicular) to the field O 1 1 1 ( 2 2 2 ) satisfy the constraint, are formed not on the large tetra- S⊥ O 1 1 1 as well as the nonzero chirality order parame- hedra but on the small tetrahedra, since the latter has the ( 2 2 2 ) C stronger NN antiferromagnetic interaction J > J and ter O 1 1 1 and the monopole or antimonopole density of 1 1′  ( 2 22 ) thus, tries to follow the constraint as much as possible. 1 n+ = n = 16 . Also, the magnetization m shows a weak We note that in the present system, the isotropic Heisen- downward− convex curve as a function of H in this low- berg spins spontaneously form this chirality order as a re- field phase. Besides, in a magnetic field, additional weak sult of the frustration, which is in sharp contrast to spin- Bragg peaks emerge at the wave vectors of the (0, 0, 1) ice systems with strong local magnetic anisotropy where and (1, 1, 0) families in both F (q) and F (q), which can 1 1 1 S C a similar ( 2 , 2 , 2 ) spin structure has been discussed [58]. clearly be seen in Fig. 4 (c). This in-field hedgehog lat- 1 1 1 In the quadruple-q ( 2 , 2 , 2 ) phases other than the tice is tetragonal symmetric in the sense that among the 6

are selected. As one can see from the right panels of Fig. 4 (b), the averaged intensities of these additional peaks in S⊥ S⊥ C the spin and chirality sectors O(001), O(110), O(001), and C O(110) are nonzero only in the in-field hedgehog lattice. When the system is metallic and localized spins Si are coupled to conduction electrons, the field-induced sym- metry reduction from cubic to tetragonal in the hedge- hog lattice is reflected in the Hall effect of spin chiral- ity origin, i.e., the topological Hall effect. In the weak- coupling theory [47], the topological Hall conductivity T σµν is proportional to the total spin chirality multi- plied by a geometrical factor χT = χ nˆ , i,j,k S,L ijk ijk h i namely, σT ǫ χT, where i, j, k denotes the sum- µν µνρ ρ S,LP mation over∝ all the three sitesh on eachi tetrahedron and T nˆijk represents the surface normal. χ corresponds to the emergent fictitious magnetic field. Although accord- ing to Ref. [47], χT should be calculated by taking fur- ther NN sites into account, we have restricted only to the NN triads because the dominant contribution comes from such short-distance triads. Indeed, we have checked that the original and present χT’s exhibit qualitatively the same field dependence. One can see from the bottom right panel of Fig. 4 (b) that with increasing the ex- ternal magnetic field, the fictitious field χT is gradually induced in the hedgehog-lattice phase and| it| vanishes out of this topological phase. It is also found that the direc- tion of χT is not arbitrary but rather takes one of the six degenerate directions, xˆ, yˆ, and zˆ. As inferred from ± ± ±S⊥ S⊥ C the similar field dependences of O(001), O(110), O(001), OC , and χT [see right panels of Fig. 4 (b)], the pre- (110) | | ferred direction of χT is determined from the tetragonal symmetry of the spin structure. For example, when the z-direction is picked up for the tetragonal spin structure like in the case of Fig. 4 (c), χT is along thez ˆ-direction, ′ so that σT = σT takes a finite value while other com- FIG. 4: MC results obtained for J1/J1 = 0.2, J2/J1 = 0, xy − yx and J3/J1 = 0.7. (a) The temperature and magnetic-field ponents are zero. In the next section, we will address the phase diagram. (b) Field dependence of various quantities at origin of this unique type of the topological Hall effect 1 1 1 T/J1 = 0.11. Top (center) panels: the average spin (chirality) characteristic of the ( , , ) hedgehog lattice. S 2 2 2 k S⊥ C S⊥ C Bragg intensities O 1 1 1 , O 1 1 1 [O 1 1 1 ], O [O(001) ], ( 2 2 2 ) ( 2 2 2 ) ( 2 2 2 ) (001) and OS⊥ [OC ], where S and S denote spin components (110) (110) k ⊥ V. ORIGIN OF THE FIELD-INDUCED parallel and perpendicular to the field, respectively. Bottom TOPOLOGICAL HALL EFFECT left panel: the monopole (or equivalently, antimonopole) den- sity n+ (= n−) and the magnetization m. Bottom right panel: the total spin chirality responsible for the topological Hall ef- To examine the association between the field-induced T fect |χ |. (c) Spin and chirality structure factors FS⊥ (q) tetragonal-symmetric hedgehog-lattice and the net chi- T and FC (q) in the (h, k, 0) (left), (h, 0,l) (center), and (0,k,l) rality χ , or equivalently, the emergent fictitious field for (right) planes obtained at T/J1 = 0.11 and H/J1 = 1. the topological Hall effect, we shall first discuss the mag- netic structure of the in-field hedgehog lattice. Figure 5 (a) shows the spin configuration associated with Fig. 4 1 1 1 (c). Since the ( 2 , 2 , 2 ) hedgehog lattice consists of the three cubic symmetric points of ( 1, 0, 0), (0, 1, 0), and alternating array of two cubic unit cells having oppositely (0, 0, 1), only one is selected, and± such a situation± is oriented spins [see Fig. 1], only one of the two is shown also the± case for the (1, 1, 0) family. In the case of Fig. in Fig. 5. One can see from the lower panels of Fig. 5 (a) 4 (c), z-direction is special. In the chirality sector [the that the sublattices 1 and 2, and 3 and 4 are paired up lower panels of Fig. 4 (c)], only (0, 0, 1) and (1, 1, 0) are and two spins in each pair are collinearly aligned in the picked up, whereas in the spin sector (the upper ones), SxSy spin plane. Due to this pairing, the symmetry of the rest four, i.e., (1, 0, 0), (0, 1, 0), (1, 0, 1) and (0, 1, 1), each tetrahedron is reduced to tetragonal with respect to 7

reduction to tetragonal occurs at the level of the cubic unit cell as well as each tetrahedron. Now that the spin structure of the in-field hedgehog lattice is clarified, we shall next discuss how it is re- lated to the total spin chirality χT = χ nˆ . i,j,k S,L ijk ijk As a first step, we consider the associatedh locali chirality T,tetra P χ = i,j,k tetra χijk nˆijk for the small tetrahedron shown in Fig. 6∈ (a), which is calculated as P

χ132 + χ234 χ124 χ143 T,tetra 1 − − χ = χ132 + χ234 + χ124 χ143 . (8) √3  −χ χ + χ −χ  132 − 234 124 − 143   When the tetrahedron is projected onto the two- dimensional planes perpendicular to the x-, y-, and z- axes [see Figs. 6 (b)-(d)], the chirality defined on each two-dimensional plane χ2D,tetra is given by a summation of four different χijk ’s with triad (i, j, k) ordered in the anticlockwise direction with respect to the surface nor- mal. Then, one finds that the so-obtained χ2D,tetra’s cor- respond to the x-, y-, and z-components of χT,tetra in FIG. 5: Tetragonal-symmetric spin structure in the in-field Eq. (8), which suggests that the ρ component of χT is hedgehog-lattice phase. (a) MC snapshots of spins mapped onto a unit sphere (upper panels) and their projection onto determined from the spin configurations on the projected the SxSy spin plane (lower panels) for the same parameters two-dimensional planes perpendicular to the ρ-direction. as those in Fig. 4 (c), where tetra’s I, II, III, and IV represent Thus, we next consider how the tetragonal symmetry in the four tetrahedra in the cubic unit cell shown in (b) and the each tetrahedron, i.e., the pairing pattern, looks on the color notations are the same as those in Figs. 1 and 2. The projected planes. spin configurations of tetra’s I and II (III and IV) are classi- One can see from wavy lines in Figs. 6 (a)-(d) that fied as AF-type (F-type) (see the main text). (c) Tetragonal- the paired spins are arranged diagonally on the plane symmetric pairing pattern in each tetrahedron, where wavy perpendicular to the tetragonal-symmetric direction (z- lines denote pair bonds. (d) Tetragonal-symmetric distribu- direction in Fig. 6), while adjacently on other planes. tion of the AF-type (green) and F-type (purple) tetrahedra Such a difference in the pairing arrangement yields an within the cubic unit cell. In (b) and (d), a monopole tetra- hedron is outlined by black dots. In (a), short-time average anisotropy in the local chirality. It turns out from a 2D,tetra over 10 MC steps has been made to reduce the thermal noise. simple calculation (see Appendix D) that χ on the projected plane becomes nonzero for the diagonal arrangement, while it is basically zero for the adjacent ones except a particular case of the F-type spin configu- ration, which corresponds to Fig. 10 (f) in Appendix D. the z-axis [see Fig, 5 (c)]. Possible three pairing patterns, Bearing the association between the tetragonal symme- i.e, stacking-directions of the pair bonds, correspond to try and the local chirality in our mind, we will turn to the possible three axes for the tetragonal spin structure. total chirality on the projected two-dimensional planes, Once a pairing symmetry is fixed, there exist two i.e. layers of one-tetrahedron width [see Figs. 6 (e) and types of the tetrahedral spin configurations, which we (f)]. call AF- and F-types. As shown in Fig. 5 (a), in the Figures 6 (g) and (h) show the layer-resolved distribu- T,tetra AF-type (F-type) tatrahedron, the SxSy components in tions of χ calculated from the spin configuration in each pair are antiferromagnetically (ferromagnetically) Fig. 5. One cansee from Fig. 6(g) that onthe layersper- aligned, whereas the Sz ones ferromagnetically (antifer- pendicular to the non-tetragonal-symmetric x-direction romagnetically). Also, the collinear axes for the two pairs where the paired spins are arranged adjacently, the local T,tetra are orthogonal to each other in the AF-type, while not in chirality χx for most of the tetrahedra takes a van- the F-type. The AF-type is further categorized into two: ishingly small value, which may be positive or negative one involves Sz’s of the same sign [tetra I in Fig. 5 (a)] depending on the thermal noise. Although an exceptional and the other involves Sz’s of opposite signs [tetra II in F-type tetrahedron, e.g., tetra III in the zoomed view in T,tetra Fig. 5 (a)]. The monopole and antimonopole correspond Fig. 6 (g), yields a nonzero large χx , it is completely to the latter. The distribution of these AF- and F-type canceled out by the contribution from the counter tetra- tetrahedra within the cubic unit cell is also tetragonal- hedron in the neighboring cubic unit cell, so that the net symmetric. One can see from Fig. 5 (d) that the AF-type chirality vanishes on any yz-plane layer. Such a situa- tetrahedron pair and the F-type one are stacking along tion is also the case for the layers perpendicular to the the tetragonal-symmetric z-axis. Thus, the symmetry- y-direction. 8

FIG. 6: Spin and chirality configurations on the projected two-dimensional planes. (a) A tetrahedron having a tetragonal- symmetric pairing with respect to the z-axis, and (b)-(d) its projections onto the two-dimensional planes perpendicular to x-, y-, and z-axes, respectively, where a red circular arrow denotes the direction in which the spin chirality χijk is defined and other notations are the same as those in Fig. 5. (e) and (f) Layers of one-small-tetrahedron width defined for the tetragonal-symmetric spin structure in Fig. 5 (d). (g) and (h) Layer-resolved distributions of the chirality calculated from the MC snapshot in Fig. T,tetra T,tetra 5. In (g) [(h)], χx (χz ) on the yz- (xy-)plane layers defined in (e) [(f)] is shown, where the left and right tetrahedra outlined by black dots correspond to the monopole and antimonopole, respectively. The xy-plane layer without the monopoles, which is enclosed by a red box in (h), yields a dominant contribution to the total chirality. In (g) and (h), short-time average over 200 MC steps has been made to reduce the thermal noise.

By contrast, on the layers perpendicular to the VI. SUMMARY AND DISCUSSION tetragonal-symmetric z-direction where the paired spins are arranged diagonally, not only the local chirality In this paper, we have demonstrated by means of MC χT,tetra but also the net chirality becomes nonzero. As z simulations that the hedgehog lattice characterized by shown in Fig. 6 (h), χT,tetra’s on the layer without z the ( 1 , 1 , 1 ) magnetic Bragg reflections is induced by monopoles [layer 1 in Fig. 6 (h)] take nonzero values 2 2 2 xy the frustration in classical Heisenberg antiferromagnets of the same sign, leading to a large total chirality, while on the breathing pyrochlore lattice, where the breath- χT,tetra’s on the layer with monopoles [layer 2 in Fig. 6 z xy ing bond-alternation quantified by the ratio of the NN (h)] take positive and negative signs. As one can see from antiferromagnetic interactions J /J and a large third the zoomed view focusing on the two cubic unit cells in 1′ 1 NN one along the bond directions J are essential. It is Fig. 6 (h), the monopole and antimonopole tetrahedra 3 T,tetra also found that in a magnetic field, the structure of the yield χz ’s of the same sign and their contributions 1 1 1 ( 2 , 2 , 2 ) hedgehog lattice is changed from cubic to tetrag- are canceled out by those from other tetrahedra. Thus, a T dominant contribution comes from the layers without the onal, resulting in the nonzero net spin chirality χ , or monopoles, which has actually been confirmed from the equivalently, nonzero emergent fictitious magnetic field, numerical data for Fig. 6 (h) together with the result of along the tetragonal-symmetric direction. analytical calculations shown in Appendix D. In the DM-induced hedgehog-lattice phase, the ficti- tious field is also caused by the magnetic field, but its 9 origin is the field-induced position shifts of monopoles and antimonopoles [33], which is in sharp contrast to the present frustrated system where their positions are un- changed. As a result, the total spin chirality χT appears in any of the possible three directions x, y, and z in the present system, while only along the applied field direc- tion in the DM system. In addition, the sign of χT can be positive or negative in the former, while it is fixed by the DM interaction in the latter. Such a degeneracy of the right-handed and left-handed chiralities is a unique as- pect of the frustration-induced topological spin textures distinct from the DM-induced ones [36]. FIG. 7: Ordering wave-vectors having the lowest eigen value αβ In experiments, although various classes of breathing of J (q) defined in Eq. (2). (a) J2/J1 = −0.2, (b) pyrochlore antiferromagnets such as the chromium ox- J2/J1 = 0, and (c) J2/J1 = 0.2. The commensurate wave- vectors ( 1 , 1 , 1 ) and (1, 1, 1) are represented in units of 2π , ides Li(Ga, In)Cr O [53, 59–66, 69, 70] and sulfides 2 2 2 a 4 8 where a is a side length of the cubic unit cell. Li(Ga, In)Cr4S8 [67, 68, 71] and the quantum magnet 1 1 1 Ba3Yb2Zn5O11 [72–76] have been studied, the ( 2 , 2 , 2 ) spin correlation has not been reported so far. On the αβ other hand, the uniform pyrochlore antiferromagnets lowest eigen value of J (q). For larger values of J3/J1, 1 1 1 GeB2O4 (B=Ni, Co, Fe, Cu) [77–87] do exhibit ( 2 , 2 , 2 ) the ground-state candidate is a state with the ordering magnetic LRO, although the experimentally proposed wave-vector of ( 1 , 1 , 1 ), whereas for smaller values ± 2 ± 2 ± 2 spin structure as well as the estimated exchange inter- of J3/J1, it is a state with an incommensurate wave- actions (e.g., ferromagnetic J ) seems to be different 1 vector. With increasing J2/J1 or decreasing J1′ /J1, the from the ones in the present theoretical model. If one 1 1 1 ( 2 , 2 , 2 ) state gets more stable. A commensurate (1, 1, 1) can introduce the breathing alternation in the GeB2O4 state is also possible for an antiferromagnetic J2 and a 1 1 1 family, modifying the exchange interactions, the ( 2 , 2 , 2 ) weak J3. hedgehog lattice might be realized. Although the above pyrochlore magnets are insulators, when the system is tuned to be metallic, the hedgehog lattice, if it is real- ized, should be evidenced distinctly by the topological Appendix B: Real-space spin configurations of the T Hall effect associated with the total spin chirality χ to- three different quadruple-q ( 1 , 1 , 1 ) states 1 1 1 2 2 2 gether with the ( 2 , 2 , 2 ) magnetic Bragg reflections. We believe that this work presenting the novel mech- In the MC simulations, we obtain the three different anism different from the conventional DM interaction to quadruple-q ( 1 , 1 , 1 ) states, the collinear, coplanar, and stabilize the hedgehog lattice will promote the explo- 2 2 2 hedgehog-lattice ones. Their real-space spin configura- ration of new classes of magnets hosting topological spin tions at H = 0 are shown in Fig. 8. In all the three textures. cases, spins belonging to each sublattice (the same col- ored arrows in Fig. 8) constitute up-down antiferromag- netic chains along the bond directions, although exactly Acknowledgments speaking, the spin directions in each chain are not per- fectly collinear when the lattice is not uniform, which will The authors thank K. Tomiyasu and S. Ishiwata for be explained below. useful discussions. We are thankful to ISSP, the Univer- The collinear phase can survive down to a low temper- sity of Tokyo and YITP, Kyoto University for providing ature basically for the uniform pyrochlore lattice with us with CPU time. This work is supported by JSPS J1′ /J1 = 1. Since the polarization axes of the differ- KAKENHI Grant Number JP17H06137. ent sublattices are collinearly aligned, the resultant spin configuration consists of the up-up-down-down antiferro- magnetic chains running along all the bond directions. In Appendix A: Ordering wave-vector with the lowest this case, the spin directions in each chain are perfectly αβ eigen value of J (q) collinear. The coplanar phase is realized only at moderate tem- In general, an ordering wave-vector minimizing the peratures. Indeed, when we evaluate the free energy of eigen value of J αβ(q) defined in Eq. (2) should be re- the coplanar state by using Eq. (7), it is higher than alized in the ground state, provided that the local spin- that of the hedgehog lattice. If the coefficient A3 de- length constraints Si = 1 are satisfied. Thus, search- fined in Eq. (6) were large enough, the coplanar state ing the ordering wave-vector| | with the lowest eigen value could be stable, but this is not the case in the present is a starting point to analyze ordering properties of the model. Thus, higher-order contributions in the GL free system. Figure 7 shows ordering wave-vectors with the energy, which have not been considered in Eq. (7), or 10

FIG. 9: The temperature and magnetic-field phase diagram ′ obtained in the MC simulations for J1/J1 = 0.6, J2/J1 = 0, and J3/J1 = 0.7.

FIG. 8: MC snapshots taken in the three different quadruple- 1 1 1 q ( 2 , 2 , 2 ) states for J2/J1 = 0, J3/J1 = 0.7, and H = 0. ′ the MF approximation. (a) the collinear state at J1/J1 = 1 and T/J1 = 0.01, (b) α ′ Since the Fourier component of spin Sq is obtained the coplanar state at J1/J1 = 0.6 and T/J1 = 0.31, and h i ′ S iθµ (c) the hedgehog-lattice state at J1/J1 = 0.2 and T/J1 = with the combined use of ΦQ = e Pˆ and Eqs. (3) µ √8 µ 0.001. In each figure, the upper panel shows the real-space and (5), spins belonging to sublattice α, Sα’s, are given spin configuration, and the lower one shows the associated all j by the spins mapped onto a unit sphere. The lower panel of (b) is exactly the same as the middle panel of Fig. 2 (d). The notations of symbols and colors are the same as those of Figs. 1 S S = Pˆ4 cos(Q4 rj + θ4)+ ε Pˆ1 cos(Q1 rj + θ1) 1 and 2. j √2 · − · h  +Pˆ cos(Q r + θ )+ Pˆ cos(Q r + θ ) , 2 2 · j 2 3 3 · j 3 i thermal fluctuations may be relevant for the stability of 2 S S = Pˆ1 cos(Q1 rj + θ1)+ ε Pˆ2 cos(Q2 rj + θ2) the coplanar phase. One can see from Fig. 8 (b) that in j √2 · · the coplanar phase, four sublattices represented by differ- h  +Pˆ cos(Q r + θ )+ Pˆ cos(Q r + θ ) , ent colors are divided into two pairs and spins belonging 3 3 · j 3 4 4 · j 4 to each pair have the same polarization axis, suggesting i 3 S ˆ ˆ that the spin state is coplanar although the coplanarity Sj = P2 cos(Q2 rj + θ2)+ ε P1 cos(Q1 rj + θ1) √2 · − · is disturbed by the thermal noise. We note that as we h  will discuss below, even if this state were realized at a low +Pˆ cos(Q r + θ )+ Pˆ cos(Q r + θ ) , 3 3 · j 3 4 4 · j 4 temperature escaping from the thermal noise, the spins i in each pair could not be perfectly collinear. 4 S ˆ ˆ Sj = P3 cos(Q3 rj + θ3)+ ε P1 cos(Q1 rj + θ1) The hedgehog-lattice phase is realized at low temper- √2 · − · h atures in the wide range of the parameter space. Al-  +Pˆ2 cos(Q2 rj + θ2)+ Pˆ4 cos(Q4 rj + θ4) . (B1) though the real-space spin configuration at J ′ /J1 = 0.6 · · 1 i and T/J1 =0.01 is presented in Fig. 1, here, we show the ˆ one for the stronger breathing alternation of J1′ /J1 =0.2 As long as Pµ’s are not collinear like in the cases of in Fig. 8 (c). The snapshot is obtained by cooling the the coplanar and hedgehog-lattice phases, spins on each system down to the very low temperature T/J1 =0.001 sublattice cannot be collinear because of the existence of to reduce the thermal noise. One can see from the lower ε. Particularly in the hedgehog-lattice phase, all the four panel of Fig. 8 (c) that the polarization vector of each spins on each sublattice within the cubic unit cell orient sublattice splits into four, although the real-space spin in the different directions, exhibiting the four spots when configuration in the upper panel does not differ so much the spins are mapped onto the sphere. Since ε is nonzero from Fig. 1. As we will address below, such a splitting only in the breathing case, such a splitting is inherent 1 1 1 1 1 1 is generic to the quadruple-q ( 2 , 2 , 2 ) noncollinear states to the ( 2 , 2 , 2 ) noncollinear states on the breathing on the breathing pyrochlore lattice. Actually, when the pyrochlore lattice. We note that ε is important for the spin configuration in Fig. 1 is further cooled down, the occurrence of the hedgehog-lattice phase as explained in splitting overcomes the thermal noise and becomes visi- Sec. II, but the resultant splitting itself is not. ble. The origin of the splitting can be understood within 11

FIG. 11: Real-space spin configuration associated with Figs. FIG. 10: Spin configurations of a tetragonal-symmetric tetra- 6 (g) and (h). The SxSy (Sz) component of a spin is repre- hedron projected onto a two-dimensional plane. (a)-(d) AF- sented by the arrow (color scale), and other color and symbol type tetrahedra, and (e)-(h) F-type tetrahedra. A red (blue) notations are the same as those in Figs. 6 (e) and (f). arrow represents a SxSy component of a spin vector pointing upward (downward), and a wavy line denotes a pair bond.

pair are given by

+ Appendix C: The temperature and magnetic-field ST1 = (0, Sxy, m + δSz) + (D1) phase diagram in the weaker breathing case of ST3 = (0, S , m + δSz) ′ xy J1/J1 = 0.6  − 2 with Sxy± = 1 (m δSz) . Then, the rest two spins Figure 9 shows the temperature and magnetic-field in the other pair− can be± expressed as p phase diagram for J1′ /J1 = 0.6, J2/J1 = 0, and + + J3/J1 = 0.7. Compared with the strongly breathing ST2 = (Sxy cos φ, Sxy sin φ, m + δSz) + + (D2) case of J1′ /J1 =0.2 [see Fig. 4 (a)], the canted collinear ST4 = ( Sxy cos φ, Sxy sin φ, m + δSz) state additionally shows up in the higher temperature  − − region, and the stability region of the hedgehog lattice or is suppressed. In the canted coplanar state, a weak ST2′ = (Sxy− cos φ, Sxy− sin φ, m δSz) transition-like anomaly is also found for J1′ /J1 = 0.6 as − (D3) ST4′ = ( Sxy− cos φ, Sxy− sin φ, m δSz), well as J1′ /J1 =0.2 (see open symbols in Fig. 9).  − − −

where φ determines the angle between the SxSy-plane collinear axes of the two pairs. The single-tetrahedron chirality on the two-dimensional plane is given by χ2D,tetra = χ , where i, j, k denotes the sum- Appendix D: Analytical calculation of the local and i,j,k ijk h i mation over allh thei four triads and χ = S (S S ) total chiralities in the in-field hedgehog-lattice phase P ijk i j k is defined in a way such that i, j, and k are· ordered× in the anticlockwise direction with respect to the surface As shown in Figs. 6 (a)-(d), when a tetragonal- normal. For the spin configurations shown in Figs. 10 symmetric tetrahedron is projected onto a two- (a)-(d), the single-tetrahedron chirality χ2D,tetra can be dimensional plane, two spins in each pair are arranged calculated as adjacently or diagonally on the plane. Here, we analyti- cally calculate the chirality on the two-dimensional plane 0 [Fig. 9(a)] for various spin arrangements of a single tetrahedron 2D,tetra 0 [Fig. 9(b)] χ =  + 2 shown in Fig. 10, and apply the result on the single 8 (Sxy) (m + δSz)cos φ [Fig. 9(c)]  − + tetrahedron to the in-field hedgehog lattice involving the  8 S S− m cos φ [Fig. 9(d)]. − xy xy 8 small tetrahedra. (D4)  Next, we consider the F-type tetrahedron in which We first consider the AF-type tetrahedron in which the SxSy components in each pair are ferromagnetically the SxSy components in each pair are antiferromagneti- aligned and the Sz ones antiferromagnetically. Figures cally aligned and the Sz ones ferromagnetically. Figures 10 (e)-(h) show the adjacent [(e) and (f)] and diagonal 10 (a)-(d) show the adjacent [(a) and (b)] and diagonal [(g) and (h)] arrangements for the F-type tetrahedron. [(c) and (d)] arrangements for the AF-type tetrahedron. In the F-type case, we assume that two spins in one pair Since the scalar chirality is invariant under any rotations are given by with respect to the applied field direction, i.e., Sz, we can fix a S S component of one spin to be, for example, in S = (0, S+ , m + δS ) x y T1 xy z . (D5) the Sy direction. Thus, we assume that two spins in one ST1′ = (0, S− , m δSz)  xy − 12

(two neighboring unit cells in Fig. 12) involve the con- tribution given by Eqs. (D4) and (D7) and the counter contribution obtained by carrying out the replacement

S± S∓ , δS δS (D8) xy →− xy z →− z in Eqs. (D4) and (D7). On the plane perpendicular to the non-tetragonal- symmetric direction [layers 1yz and 2yz in Fig. 12 (a)], the paired spins are arranged adjacently, so that χ2D,tetra becomes zero for almost all the tetrahedra except tetra III, which corresponds to Fig. 10 (f), and its counter tetrahedron in the neighboring unit cell. The tetra III yields the nonzero χ2D,tetra given by Eq. (D7) for Fig. 10 (f), but it is completely canceled out by the contribu- tion from the counter tetrahedron because of Eq. (D8). FIG. 12: Schematically-drawn layer-resolved spin configura- Hence, the total chirality on the two-dimensional plane tions of Fig. 11. (a) The simplified spin configuration of Fig. vanishes when the two paired spins are arranged adja- 11 on the yz-plane layers defined in Fig. 11, and (b) those on cently. the xy-plane layers, where a red (blue) arrow represents the In contrast, on the plane perpendicular to the SxSy component of a spin vector pointing upward (downward) and other notations are the same as those in Fig. 11. In each tetragonal-symmetric direction [layers 1xy and 2xy in Fig. 12 (b)] where the two paired spins are arranged diago- tetrahedron, the SxSy components of the two different pairs 2D,tetra are assumed to be orthogonal to each other for simplicity. nally, the local chirality χ becomes nonzero for all the tetrahedra. Since the four tetrahedra in the cubic unit cell [tetra’s I, II, III, and IV in Fig. 12 (b)] are not equivalent to one another, we will take the δSz com- Then, the rest two spins in the other pair can be ex- ponents for these tetrahedra, δSz,I, δSz,II, δSz,III, and pressed as δSz,IV, as independent variables. After averaging over all the tetrahedra within the two unit cells, we obtain S = (S+ cos φ, S+ sin φ, m + δS ) T2 xy xy z (D6) the total chirality per tetrahedron as ST2′ = (S− cos φ, S− sin φ, m δSz).  xy xy − + 2 2 m (S− S ) + 4(δS ) The single-tetrahedron chirality on the two-dimensional xy,III − xy,III z,III 2D,tetra plane χ can be calculated as h + 2 2 +(S− S ) + 4(δS ) (D9) xy,IV − xy,IV z,IV 0 [Fig. 9(e)] i 2 + 2 for the layer without the monopoles [layer 1xy in Fig. 12 2 (S− ) (S ) m  − xy − xy (b)] and  h + 2  +(Sxy− + Sxy) δSz cos φ [Fig. 9(f)]  2 + 2  + 2 i 2m (Sxy,− I) + (Sxy,I) χ2D,tetra =  2 (Sxy− Sxy) m  − −  2 + 2 h 2 +  +h (S− ) (S ) δSz cos φ [Fig. 9(g)] 4(δSz,I) 2S− S (D10) xy − xy − − xy,II xy,II + 2 i i  2 (Sxy− Sxy) m  − for the layer with the monopoles [layer 2xy in Fig. 12  h 2 + 2  + (Sxy− ) (Sxy) δSz cos φ [Fig. 9(h)]. (b)], where Sxy,λ± (λ =I, II, III, and IV) is defined by  − 2  (D7) S± = 1 (m δSz,λ) . One can see from Eq.   i xy,λ − ± As the chirality for a single tetrahedron χ2D,tetra is (D9) that the total chirality for the layer without the p obtained, we will turn to the total chirality in the hedge- monopoles definitely takes a nonzero value. In Eq. (D9), hog lattice phase. The spin configuration associated with the sign of the total chirality is positive since it is cal- Figs. 5 and 6 is shown in Fig. 11. Note that the spin culated in the specific case of Fig. 12 (b), but, in gen- structure is tetragonal-symmetric with respect to the z- eral, it can be positive or negative depending on the spin axis. This spin configuration is schematically drawn in configuration. Actually, when the spin-space inversion the layer-resolved form in Fig. 12, where for simplicity, φ Sx Sx is performed for all the spins in Fig. 12 (b), is assumed to be 0 or π such that the angle between the the→ state − is turned to the other chirality-degenerate state SxSy-plane collinear axes of the two pairs be orthogonal with its energy unchanged because the spin Hamiltonian to each other. Since the hedgehog lattice consists of the is invariant under the spin-space inversion, and then, the alternating array of the two cubic unit cells having op- total chirality changes its sign. On the other hand, on the positely oriented spins except the uniform magnetization layer with the monopoles [see Eq. (D10)], the cancella- m, the two neighboring unit cells on the projected plane tion occurs between the different kinds of tetrahedra, and 13 as a result, the net chirality on this layer becomes small. from Eqs. (D9) and (D10), the total chirality becomes Thus, the overall chirality is dominated by the contribu- non-vanishing only for nonzero m, the magnetic field as tion from the layers without the monopoles, which has well as the symmetry reduction to tetragonal is crucial actually been confirmed from the numerical data calcu- for the emergence of the topological Hall effect in the lated from the MC spin snapshots. Since as one can see present system.

[1] N. Nagaosa and Y. Tokura, Topological properties and Equilibrium Skyrmion Lattice Ground State in a Polar dynamics of magnetic skyrmions, Nat. Nanotech. 8, 899 Easy-plane Magnet, Sci. rep. 7, 7584 (2017). (2013). [15] T. Kurumaji, T. Nakajima, V. Ukleev, A. Feoktys- [2] W. Koshibae, Y. Kaneko, J. Iwasaki, M. Kawasaki, Y. tov, T. H. Arima, K. Kakurai, and Y. Tokura, Neel- Tokura, and N. Nagaosa, Memory functions of magnetic Type Skyrmion Lattice in the Tetragonal Polar Magnet skyrmions, Jpn. J. Appl. Phys. 54, 053001 (2015). VOSe2O5, Phys. Rev. Lett. 119, 237201 (2017). [3] Y. Fujishiro, N. Kanazawa, T. Shimojima, A. Nakamura, [16] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. K. Ishizuka, T. Koretsune, R. Arita, A. Miyake, H. Mi- Sahoo, F. Damay, U. K. Robler, C. Felser, and S. S. P. tamura, K. Akiba, M. Tokunaga, J. Shiogai, S. Kimura, Parkin, Magnetic antiskyrmions above room temperature S. Awaji, A. Tsukazaki, A. Kikkawa, Y. Taguchi, and in tetragonal Heusler materials, Nature 548, 561 (2017). Y. Tokura, Large magneto-thermopower in MnGe with [17] M. Kakihana, D. Aoki, A. Nakamura, F. Honda, M. topological spin texture, Nat. Commun. 9, 408 (2018). Nakashima, Y. Amako, S. Nakamura, T. Sakakibara, M. [4] K. Kadowaki, K. Okuda, and M. Date, Magnetization Hedo, T. Nakama, and Y. Onuki, Giant Hall Resistivity and Magnetoresistance of MnSi.I, J. Phys. Soc. Jpn. 51, and Magnetoresistance in Cubic Chiral Antiferromagnet 2433 (1982). EuPtSi, J. Phys. Soc. Jpn. 87, 023701 (2018). [5] S. Muhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. [18] K. Kaneko, M. D. Frontzek, M. Matsuda, A. Nakao, K. Rosch, A. Neubauer, R. Georgii, and P. Boni, Skyrmion Munakata, T. Ohhara, M. Kakihana, Y. Haga, M. Hedo, Lattice in a Chiral Magnet, Science 323, 915 (2009). T. Nakama, and Y. Onuki, Unique Helical Magnetic Or- [6] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. der and Field-Induced Phase in Trillium Lattice Antifer- G. Niklowitz, and P. Boni, Topological Hall Effect in the romagnet EuPtSi, J. Phys. Soc. Jpn. 88, 013702 (2019). A Phase of MnSi, Phys. Rev. Lett. 102, 186602 (2009). [19] U. K. Robler, A. N. Bogdanov, and C. Pfleiderer, Sponta- [7] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, neous skyrmion ground states in magnetic metals, Nature Y. Matsui, N. Nagaosa, and Y. Tokura, Real-space ob- 442, 797 (2006). servation of a two-dimensional skyrmion crystal, Nature [20] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, 465, 901 (2010). Skyrmions and anomalous Hall effect in a Dzyaloshinskii- [8] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Moriya spiral magnet, Phys. Rev. B 80, 054416 (2009). Zhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Near [21] S. Buhrandt and L. Fritz, Skyrmion lattice phase in three- room-temperature formation of a skyrmion crystal in dimensional chiral magnets from Monte Carlo simula- thin-films of the helimagnet FeGe, Nat. Mater. 10, 106 tions, Phys. Rev. B 88, 195137 (2013). (2011). [22] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K. [9] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Ku- Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and Y. betzka, R. Wiesendanger, G. Bihlmayer, and S. Blugel, Tokura, Large Topological Hall Effect in a Short-Period Spontaneous atomic-scale magnetic skyrmion lattice in Helimagnet MnGe, Phys. Rev. Lett. 106, 156603 (2011). two dimensions, Nat. Phys. 7, 713 (2011). [23] N. Kanazawa, J.-H. Kim, D. S. Inosov, J. S. White, N. [10] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Obser- Egetenmeyer, J. L. Gavilano, S. Ishiwata, Y. Onose, T. vation of Skyrmions in a Multiferroic Material, Science Arima, B. Keimer, and Y. Tokura, Possible skyrmion- 336, 198 (2012). lattice ground state in the B20 chiral-lattice magnet [11] Y. Tokunaga, X.Z. Yu, J.S. White, H.M. Ronnow, D. MnGe as seen via small-angle neutron scattering, Phys. Morikawa, Y. Taguchi, and Y. Tokura, A new class of Rev. B 86, 134425 (2012). chiral materials hosting magnetic skyrmions beyond room [24] Y. Shiomi, N. Kanazawa, K. Shibata, Y. Onose, and Y. temperature, Nat. Commun. 6, 7638 (2015). Tokura, Topological Nernst effect in a three-dimensional [12] I. Kezsmarki, S. Bordacs, P. Milde, E. Neuber, L. M. skyrmion-lattice phase, Phys. Rev. B 88, 064409 (2013). Eng, J. S. White, H. M. Ronnow, C. D. Dewhurst, [25] T. Tanigaki, K. Shibata, N. Kanazawa, X. Yu, Y. Onose, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. H. S. Park, D. Shindo, and Y. Tokura, Real-Space Ob- Tsurkan, and A. Loidl, Neel-type skyrmion lattice with servation of Short-Period Cubic Lattice of Skyrmions in confined orientation in the polar magnetic semiconduc- MnGe, Nano Lett. 15, 5438 (2015). tor GaV4S8, Nat. Mater. 14, 1116 (2015). [26] N. Kanazawa, Y. Nii, X.-X. Zhang, A. S. Mishchenko, [13] Y. Fujima, N. Abe, Y. Tokunaga, and T. Arima, Thermo- G. De Filippis, F. Kagawa, Y. Iwasa, N. Nagaosa, and dynamically stable skyrmion lattice at low temperatures Y. Tokura, Critical phenomena of emergent magnetic in a bulk crystal of lacunar spinel GaV4Se8, Phys. Rev. monopoles in a chiral magnet, Nat. Commun. 7, 11622 B 95, 180410(R) (2017). (2016). [14] S. Bordacs, A. Butykai, B. G. Szigeti, J. S. White, R. [27] N. Kanazawa, J. S. White, H. M. Ronnow, C. D. De- Cubitt, A. O. Leonov, S. Widmann, D. Ehlers, H.-A. whurst, D. Morikawa, K. Shibata, T. Arima, F. Kagawa, Krug von Nidda, V. Tsurkan, A. Loidl, and I. Kezsmarki, A. Tsukazaki, Y. Kozuka, M. Ichikawa, M. Kawasaki, and 14

Y. Tokura, Topological spin-hedgehog crystals of a chi- with a high topological number in itinerant magnets, ral magnet as engineered with magnetic anisotropy, Phys. Phys. Rev. B 99, 094420 (2019), and Z. Wang, Y. Su, Rev. B 96, 220414(R) (2017). S. Z. Lin, and C. D. Batista, Skyrmion Crystal from [28] Y. Fujishiro, N. Kanazawa, T. Nakajima, X. Z. Yu, K. RKKY Interaction Mediated by 2D Electron Gas, Phys. Ohishi, Y. Kawamura, K. Kakurai, T. Arima, H. Mita- Rev. Lett. 124, 207201 (2020). mura, A. Miyake, K. Akiba, M. Tokunaga, A. Matsuo, [44] T. Kurumaji, T. Nakajima, M. Hirschberger, A. K. Kindo, T. Koretsune, R. Arita, and Y. Tokura, Topo- Kikkawa, Y. Yamasaki, H. Sagayama, H. Nakao, Y. logical transitions among skyrmion- and hedgehog-lattice Taguchi, T. Arima, and Y. Tokura, Skyrmion lattice with states in cubic chiral magnets, Nat. Commun. 10, 1059 a giant topological Hall effect in a frustrated triangular- (2019). lattice magnet, Science 365, 914 (2019). [29] Y. Fujishiro, N. Kanazawa, and Y. Tokura, Engineering [45] N. D. Khanh, T. Nakajima, X. Z. Yu, S. Gao, K. Shi- skyrmions and emergent monopoles in topological spin bata, M. Hirschberger, Y. Yamasaki, H. Sagayama, H. crystals, Appl. Phys. Lett., 116, 090501 (2020). Nakao, L. C. Peng, K. Nakajima, R. Takagi, T. Arima, Y. [30] B. Binz, A. Vishwanath, and V. Aji, Theory of the Heli- Tokura, and S. Seki, Nanometric square skyrmion lattice cal Spin Crystal: A Candidate for the Partially Ordered in a centrosymmetric tetragonal magnet, Nat. Nanotech. State of MnSi, Phys. Rev. Lett. 96, 207202 (2006). 15, 444 (2020). [31] B. Binz and A. Vishwanath, Theory of helical spin crys- [46] A. van Oosterom and J. Strackee, The Solid Angle of tals: Phases, textures, and properties, Phys. Rev. B 74, a Plane Triangle, IEEE Trans. Biomed. Eng. BME-30, 214408 (2006). 125 (1983). [32] B. Binz and A. Vishwanath, Chirality induced [47] G. Tatara and H. Kawamura, Chirality-Driven Anoma- anomalous-Hall effect in helical spin crystals, Phys- lous Hall Effect in Weak Coupling Regime, J. Phys. Soc. ica B 403, 1336-1340 (2008). Jpn. 71, 2613 (2002). [33] J. H. Park and J. H. Han, Zero-temperature phases for [48] S. Ishiwata, M. Tokunaga, Y. Kaneko, D. Okuyama, chiral magnets in three dimensions, Phys. Rev. B 83, Y. Tokunaga, S. Wakimoto, K. Kakurai, T. Arima, Y. 184406 (2011). Taguchi, and Y. Tokura, Versatile helimagnetic phases [34] S-G. Yang, Y-H. Liu, and J. H. Han, Zero-temperature under magnetic fields in cubic perovskite SrFeO3, Phys. phases for chiral magnets in three dimensions, Phys. Rev. Rev. B 84, 054427 (2011). B 94, 054420 (2016). [49] S. Ishiwata, T. Nakajima, J. -H. Kim, D. S. Inosov, N. [35] S. Okumura, S. Hayami, Y. Kato, and Y. Motome, Kanazawa, J. S. White, J. L. Gavilano, R. Georgii, K. Magnetic hedgehog lattices in noncentrosymmetric met- M. Seemann, G. Brandl, P. Manuel, D. D. Khalyavin, als, Phys. Rev. B 101, 144416 (2020). S. Seki, Y. Tokunaga, M. Kinoshita, Y. W. Long, Y. [36] T. Okubo, S. Chung, and H. Kawamura, Multiple- Kaneko, Y. Taguchi, T. Arima, B. Keimer, and Y. q States and the Skyrmion Lattice of the Triangular- Tokura, Emergent topological spin structures in the cen- Lattice Heisenberg Antiferromagnet under Magnetic trosymmetric cubic perovskite SrFeO3, Phys. Rev. B 101, Fields, Phys. Rev. Lett. 108, 017206 (2012). 134406 (2020). [37] A. O. Leonov and M. Mostovoy, Multiply periodic states [50] K. Aoyama and H. Kawamura, Spin-Lattice-Coupled Or- and isolated skyrmions in an anisotropic frustrated mag- der in Heisenberg Antiferromagnets on the Pyrochlore net, Nat. Commun. 6, 8275 (2015). Lattice, Phys. Rev. Lett. 116, 257201 (2016). [38] S. Z. Lin and S. Hayami, Ginzburg-Landau theory for [51] K. Aoyama and H. Kawamura, Spin ordering induced by skyrmions in inversion-symmetric magnets with compet- lattice distortions in classical Heisenberg antiferromag- ing interactions, Phys. Rev. B 93, 064430 (2016). nets on the breathing pyrochlore lattice, Phys. Rev. B 99, [39] S. Hayami, S. Z. Lin, and C. D. Batista, Bubble and 144406 (2019). skyrmion crystals in frustrated magnets with easy-axis [52] T. Mizoguchi, L. D. C. Jaubert, R. Moessner, and M. anisotropy, Phys. Rev. B 93, 184413 (2016). Udagawa, Magnetic clustering, half-moons, and shadow [40] S. Hayami, S. Z. Lin, Y. Kamiya, and C. D. Batista, Vor- pinch points as signals of a proximate Coulomb phase in tices, skyrmions, and chirality waves in frustrated Mott frustrated Heisenberg magnets, Phys. Rev. B 98, 144446 insulators with a quenched periodic array of impurities, (2018). Phys. Rev. B 94, 174420 (2016). [53] Y. Okamoto, G. J. Nilsen, J. P. Attfield, and Z. Hiroi, [41] S. Z. Lin and C. D. Batista, Face Centered Cubic Breathing Pyrochlore Lattice Realized in A-Site Ordered and Hexagonal Close Packed Skyrmion Crystals in Cen- Spinel Oxides LiGaCr4O8 and LiInCr4O8, Phys. Rev. trosymmetric Magnets, Phys. Rev. Lett. 120, 077202 Lett. 110, 097203 (2013). (2018). [54] J. N. Reimers, A. J. Berlinsky, and A. C. Shi, Mean- [42] T. Shimokawa, T. Okubo, and H. Kawamura, Multiple-q field approach to magnetic ordering in highly frustrated states of the J1 − J2 classical honeycomb-lattice Heisen- pyrochlores, Phys. Rev. B 43, 865 (1991). berg antiferromagnet under a magnetic field, Phys. Rev. [55] T. Okubo, T. H. Nguyen, and H. Kawamura, Cubic and B 100, 224404 (2019). noncubic multiple-q states in the Heisenberg antiferro- [43] See also theoretical works applicable only to metallic sys- magnet on the pyrochlore lattice, Phys. Rev. B 84, 144432 tems:R. Ozawa, S. Hayami, and Y. Motome, Zero-Field (2011). Skyrmions with a High Topological Number in Itiner- [56] C. L. Henly, Ordering by disorder: Ground-state selection ant Magnets, Phys. Rev. Lett. 118, 147205 (2017), S. in fcc vector antiferromagnets, J. Appl. Phys. 61, 3962 Hayami, R. Ozawa, and Y. Motome, Effective bilinear- (1987). biquadratic model for noncoplanar ordering in itinerant [57] H. Kawamura and S. Miyashita, Phase Transition of the magnets, Phys. Rev. B 95, 224424 (2017), S. Hayami and Heisenberg Antiferromagnet on the Triangular Lattice in Y. Motome, Effect of magnetic anisotropy on skyrmions a Magnetic Field, J. Phys. Soc. Jpn. 85, 4530 (1985). 15

[58] H. Ishizuka, M. Udagawa, and Y. Motome, Magnetic Or- T, Phys. Rev. B 101, 054434 (2020). der and Charge Disproportionation in a Spin-Ice Type [72] K. Kimura, S. Nakatsuji, and T. Kimura, Experimental Kondo Lattice Model: Large Scale Monte Carlo Study, J. realization of a quantum breathing pyrochlore antiferro- Phys. Soc. Jpn. 81, 113706 (2011). magnet, Phys. Rev. B 90, 060414(R) (2014). [59] Y. Tanaka, M. Yoshida, M. Takigawa, Y. Okamoto, [73] T. Haku, K. Kimura, Y. Matsumoto, M. Soda, M. and Z. Hiroi, Novel Phase Transitions in the Breath- Sera, D. Yu, R. A. Mole, T. Takeuchi, S. Nakat- 7 ing Pyrochlore Lattice: Li-NMR on LiInCr4O8 and suji, Y. Kono, T. Sakakibara, L.-J. Chang, and T. LiGaCr4O8, Phys. Rev. Lett. 113, 227204 (2014). Masuda, Low-energy excitations and ground-state selec- [60] G. J. Nilsen, Y. Okamoto, T. Masuda, J. Rodriguez- tion in the quantum breathing pyrochlore antiferromagnet Carvajal, H. Mutka, T. Hansen, and Z. Hiroi, Com- Ba3Yb2Zn5O11, Phys. Rev. B 93, 220407(R) (2016). plex magnetostructural order in the frustrated spinel [74] T. Haku, M. Soda, M. Sera, K. Kimura, S. Itoh, T. LiInCr4O8, Phys. Rev. B 91, 174435 (2015). Yokoo, and T. Tasuda, Crystal Field Excitations in the [61] R. Saha, F. Fauth, M. Avdeev, P. Kayser, B. J. Kennedy, Breathing Pyrochlore Antiferromagnet Ba3Yb2Zn5O11, and A. Sundaresan, Magnetodielectric effects in A-site J. Phys. Soc. Jpn. 85, 034721 (2016). cation-ordered chromate spinels LiMCr4O8 (M=Ga and [75] J. G. Rau, L. S. Wu, A. F. May, L. Poudel, B. Winn, In), Phys. Rev. B 94, 064420 (2016). V. O. Garlea, A. Huq, P. Whitfield, A. E. Taylor, M. [62] S. Lee, S.-H. Do, W.-J. Lee, Y. S. Choi, M. Lee, E. S. D. Lumsden, M. J. P. Gingras, and A. D. Christian- Choi, A. P. Reyes, P. L. Kuhns, A. Ozarowski, and K.-Y. son, Anisotropic Exchange within Decoupled Tetrahedra Choi, Multistage symmetry breaking in the breathing py- in the Quantum Breathing Pyrochlore Ba3Yb2Zn5O11, rochlore lattice Li(Ga,In)Cr4O8, Phys. Rev. B 93, 174402 Phys. Rev. Lett. 116, 257204 (2016). (2016). [76] J. G. Rau, L. S. Wu, A. F. May, A. E. Taylor, I-Lin. Liu, [63] R. Saha, R. Dhanya, C. Bellin, K. Beneut, A. Bhat- J. Higgins, N. P. Butch, K. A. Ross, H. S. Nair, M. D. tacharyya, A. Shukla, C. Narayana, E. Suard, J. Lumsden, M. J. P. Gingras, and A. D. Christianson, Be- Rodriguez-Carvajal, and A. Sundaresan, Magnetostruc- havior of the breathing pyrochlore lattice Ba3Yb2Zn5O11 tural coupling and magnetodielectric effects in the A- in applied magnetic field, J. Phys.: Condens. Matter 30, site cation-ordered spinel LiFeCr4O8, Phys. Rev. B 96, 455801 (2018). 214439 (2017). [77] M. K. Crawford, R. L. Harlow, P. L. Lee, Y. Zhang, [64] Y. Okamoto, G. J. Nilsen, T. Nakazono, and Z. Hiroi, J. Hormadaly, R. Flippen, Q. Huang, J. W. Lynn, R. Magnetic Phase Diagram of the Breathing Pyrochlore An- Stevens, B. F. Woodfield, J. Boerio-Goates, and R. tiferromagnet LiGa1−xInxCr4O8, J. Phys. Soc. Jpn. 84, A. Fisher, Structure and properties of the integer-spin 043707 (2015). frustrated antiferromagnet GeNi2O4, Phys. Rev. B 68, [65] D. Wang, C. Tan, K. Huang, and L. Shu, The 220408(R) (2003). Complex Magnetism in the Breathing Pyrochlore [78] S. Diaz, S. de Brion, G. Chouteau, B. Canals, V. Simonet, LiIn(Cr1−xRhx)4O8, Chinese Phys. Lett. 33, 127501 and P. Strobel, Magnetic frustration in the spinel com- (2016). pounds GeCo2O4 and GeNi2O4, Phys. Rev. B 74, 092404 [66] R. Wawrzynczak, Y. Tanaka, M. Yoshida, Y. Okamoto, (2006). P. Manuel, N. Casati, Z. Hiroi, M. Takigawa, and [79] T. Lancaster, S. J. Blundell, D. Prabhakaran, P. J. Baker, G. J. Nilsen, Classical Spin Nematic Transition in W. Hayes, and F. L. Pratt, Magnetism in the S = 1 frus- LiGa0.95In0.05Cr4O8, Phys. Rev. Lett. 119, 087201 trated antiferromagnet GeNi2O4 studied using implanted (2017). muons, Phys. Rev. B 73, 184436 (2006). [67] Y. Okamoto, M. Mori, N. Katayama, A. Miyake, M. [80] M. Matsuda, J. -H. Chung, S. Park, T. J. Sato, K. Tokunaga, A. Matsuo, K. Kindo, and K. Takenaka, Matsuno, H. A. Katori, H. Takagi, K. Kakurai, K. Ka- Magnetic and Structural Properties of A-Site Ordered mazawa, Y. Tsunoda, I. Kagomiya, C. L. Henley, and Chromium Spinel Sulfides: Alternating Antiferromag- S. -H. Lee, Frustrated minority spins in GeNi2O4, Euro. netic and Ferromagnetic Interactions in the Breathing Phys. Lett. 82, 37006 (2008). Pyrochlore Lattice, J. Phys. Soc. Jpn. 87, 034709 (2018). [81] T. Watanabe, S. Hara, and S. I. Ikeda, Jahn-Teller inac- [68] G. Pokharel, A. F. May, D. S. Parker, S. Calder, G. tivity and magnetic frustration in GeCo2O4 probed by ul- Ehlers, A. Huq, S. A. J. Kimber, H. S. Arachchige, L. trasound velocity measurements, Phys. Rev. B 78, 094420 Poudel, M. A. McGuire, D. Mandrus, and A. D. Chris- (2008). tianson, Negative thermal expansion and magnetoelas- [82] T. Watanabe, S. Hara, S. I. Ikeda, and K. Tomiyasu, tic coupling in the breathing pyrochlore lattice material Elastic instabilities in an antiferromagnetically ordered LiGaCr4S8, Phys. Rev. B 97, 134117 (2018). phase of the orbitally frustrated spinel GeCo2O4, Phys. [69] Y. Okamoto, D. Nakamura, A. Miyake, S. Takeyama, M. Rev. B 84, 020409(R) (2011). Tokunaga, A. Matsuo, K. Kindo, and Z. Hiroi, Magnetic [83] K. Tomiyasu, M. K. Crawford, D. T. Adroja, P. Manuel, transitions under ultrahigh magnetic fields of up to 130 A. Tominaga, S. Hara, H. Sato, T. Watanabe, S. I. T in the breathing pyrochlore antiferromagnet LiInCr4O8, Ikeda, J. W. Lynn, K. Iwasa, and K. Yamada, Molecu- 1 Phys. Rev. B 95, 134438 (2017). lar spin-orbit excitations in the Jeff = 2 frustrated spinel [70] M. Gen, D. Nakamura, Y. Okamoto, and S. Takeyama, GeCo2O4, Phys. Rev. B 84, 054405 (2011). Ultra-high magnetic field magnetic phases up to 130 T [84] P. T. Barton, M. C. Kemei, M. W. Gaultois, S. L. Moffitt, in a breathing pyrochlore antiferromagnet LiInCr4O8, J. L. E. Darago, R. Seshadri, M. R. Suchomel, and B. C. Magn. Magn. Mater. 473, 387 (2019). Melot, Structural distortion below the Neel temperature [71] M. Gen, Y. Okamoto, M. Mori, K. Takenaka, and Y. Ko- in spinel GeCo2O4, Phys. Rev. B 90, 064105 (2014). hama, Magnetization process of the breathing pyrochlore [85] X. Fabreges, E. Ressouche, F. Duc, S. de Brion, M. magnet CuInCr4S8 in ultrahigh magnetic fields up to 150 Amara, C. Detlefs, L. Paolasini, E. Suard, L. -P. Reg- 16

nault, B. Canals, P. Strobel, and V. Simonet, Field- frustrated antiferromagnet GeCo2O4, Phys. Rev. B 99, driven magnetostructural transitions in GeCo2O4, Phys. 134422 (2019). Rev. B 95, 014428 (2017). [87] T. Zou, Y. -Q. Cai, C. R. delaCruz, V. O. Garlea, S. [86] P. Pramanik, S. Ghosh, P. Yanda, D. C. Joshi, S. Pit- D. Mahanti, J. -G. Cheng, and X. Ke, Up-up-down-down 1 tala, A. Sundaresan, P. K. Mishra, S. Thota, and M. magnetic chain structure of the spin- 2 tetragonally dis- S. Seehra, Magnetic ground state, field-induced transi- torted spinel GeCu2O4, Phys. Rev. B 94, 214406 (2016). tions, electronic structure, and optical band gap of the