Critical phenomena of emergent magnetic monopoles in a chiral magnet

Abstract

Second-order continuous phase transitions are characterized by symmetry breaking with order parameters. Topological orders of electrons, characterized by the topological index defined in momentum space, provide a distinct perspective for phase transitions, which are categorized as quantum phase transitions not being accompanied by symmetry breaking. However, there are still limited observations of counterparts in real space. Here we show a real-space topological phase transition in a chiral magnet MnGe, hosting a periodic array of hedgehog and antihedgehog topological spin singularities. This transition is driven by the pair annihilation of the hedgehogs and antihedgehogs acting as monopoles and antimonopoles of the emergent electromagnetic field. Observed anomalies in the magnetoresistivity and phonon softening are consistent with the theoretical prediction of critical phenomena associated with enhanced fluctuations of emergent field near the transition. This finding reveals a vital role of topology of the spins in strongly correlated systems.

ARTICLE

Received 25 Jan 2016 | Accepted 14 Apr 2016 | Published 16 May 2016

N. Kanazawa1, Y. Nii2,w, X.-X. Zhang1, A.S. Mishchenko2, G. De Filippis3, F. Kagawa2, Y. Iwasa1,2, N. Nagaosa1,2

& Y. Tokura1,2

Second-order continuous phase transitions are characterized by symmetry breaking with order parameters. Topological orders of electrons, characterized by the topological index dened in momentum space, provide a distinct perspective for phase transitions, which are categorized as quantum phase transitions not being accompanied by symmetry breaking. However, there are still limited observations of counterparts in real space. Here we show a real-space topological phase transition in a chiral magnet MnGe, hosting a periodic array of hedgehog and antihedgehog topological spin singularities. This transition is driven by the pair annihilation of the hedgehogs and antihedgehogs acting as monopoles and antimonopoles of the emergent electromagnetic eld. Observed anomalies in the magnetoresistivity and phonon softening are consistent with the theoretical prediction of critical phenomena associated with enhanced uctuations of emergent eld near the transition. This nding reveals a vital role of topology of the spins in strongly correlated systems.

DOI: 10.1038/ncomms11622 OPEN

Critical phenomena of emergent magnetic monopoles in a chiral magnet

1 Department of Applied Physics, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan. 2 RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan. 3 SPIN-CNR and Dipartimento di Fisica, Universit di Napoli Federico II, I-80126 Napoli, Italy. w Present address: Department of Basic Science, The University of Tokyo, Tokyo 153-8902, Japan (Y.N.). Correspondence and requests for materials should be addressed to N.K. (email: mailto:[email protected]

Web End [email protected] ) or to Y.T. (email: mailto:[email protected]

Web End [email protected] ).

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The search for phase transitions and critical phenomena beyond Landaus conventional scheme is one of the most fundamental issues in the physics of strongly correlated

systems and topological quantum matter1. Symmetry breaking is the key concept of Landau, which is missing in the topological phase transitions. The topological order is characterized by an index such as the Chern number or Z2 number dened in the rst

Brillouin zone, the edge or surface gapless states and also the ground-state degeneracies for the non-trivial topological geometry of the sample2,3. The topological phase transition, that is, the change of the topological index dened in momentum space, usually requires the gap closing at some momenta and gives quantum critical phenomena with diverging correlation length. However, the topological phase transition dened in real space and its associated critical phenomena have never been explored in the context of the electronic states.

Topological spin orders in real space can be transcribed into the electronic state via the gauge eld, that is, the emergent magnetic eld4. A real-space geometric arrangement of spins affects charge transport by exerting an additional quantum phase on the itinerant electrons, called Berry phase5, which acts as the effective magnetic eld. Among topological spin orders, hedgehogs and antihedgehogs play the roles of sources or sinks of emergent eld, that is, emergent monopoles or antimonopoles6,7. These hedgehog spin structures have been indeed realized as topological defects such as Bloch points811 and singular points where magnetic skyrmions coalesce with or split from one another12. However, topological transitions associated with creation and annihilation of these hedgehogs are random events in terms of time and space; observation and identication of the consequent critical phenomena in the presence of emergent monopoles remain unexplored because of difculty in controlling their behaviours.

Here we propose that the chiral magnet MnGe, where a periodic assembly of spin hedgehogs (monopoles) and anti-hedgehogs (antimonopoles) is realized as the magnetic ground state1315, is an ideal system showing the topological phase transition of the real-space spin conguration and critical uctuations acting on the conduction electrons and phonons. We have found that the Hall resistivity and magnetoresistivity (MR) in the magnetization processes show large deviations from conventional M- and M2-proportional proles, which are accounted for by static and dynamic proles of emergent elds, respectively. In particular, a prominent enhancement of positive MR and a large reduction in elastic constant are simultaneously observed on the pair annihilation of emergent monopoles, which highlights the role of the uctuations of emergent eld at the real-space topological phase transition.

ResultsHedgehog spin structures and emergent magnetic monopoles. The magnetic structure in MnGe (shown schematically in Fig. 1a) has been observed by neutron diffraction14 and Lorentz transmission electron microscopy15 below the magnetic transition temperature TNE170 K and is modelled by the superposition of three orthogonal helical structures induced by the DzyaloshinskiiMoriya interaction (DMI)16,

M0 r M0 sin qy cos qz; sin qz cos qx; sin qx cos qy

; where M0 and q are the amplitude and the wavenumber of the helical moment, respectively; the periodic modulation length l 2p/q is ranging from 3 nm (typically below 50 K) to 6 nm

(near TN)1315. In sharp contrast to the two-dimensional skyrmion crystal state17,18, there occur several points in the real space where M(r) 0 (ref. 19). However, this large modulation in

the magnitude of the spin moment is prohibited in the strong

correlation limit, where the spin moment is saturated at low temperatures. Those genuine topological magnetic defects with zero magnetization, which are surrounded by saturated moments pointing in all directions, can be realized by the topological protection even though it is energetically costly, as discussed theoretically8,9 and demonstrated experimentally10,11. This means that n r

M r

= M r

j j rather than M(r) is the more appropriate

eld that inuences the conduction electrons in the strongly correlated systems. We regard that MnGe belongs to this case; in fact, its saturated magnetization Ms 1.9 mB per f.u.13 is

approximately ve times enhanced as compared with the value of the related magnet MnSi20 with a larger one-electron bandwidth. Mathematically, M(r) belongs to B3 (inside the three-dimensional sphere), whereas n(r) belongs to S2 (surface of the three-dimensional sphere). The former is trivial, as all conguration of M(r) is smoothly connected to M(r) 0, whereas

S2 is non-trivial as characterized by the Berry curvature bk 12 Eijk n r

@in r

@jn r

, which corresponds to the

solid angle subtended by n(r) and the emergent magnetic eld acting on electrons in the same manner as the classical magnetic eld does4. It can be seen that the points with M(r) 0

correspond to the hedgehog or antihedgehog of n(r) with the effective magnetic charge Qm 14p R S dSkbk 1 (S is the

surface enclosing the singularity), namely the emergent magnetic monopole and antimonople (see Fig. 1c and also Supplementary Note 1)612,2123. Therefore, the strong correlation limit produces the non-trivial topological classication of the spin congurations and the topological phase transition as well. It is noteworthy that this topological phase transition and associated critical phenomena are dened for the classical spin conguration n(r) and its thermal uctuations at nite temperatures. It is also worth noting that the topology is dened for the continuous function n(r) and becomes less rigorous for ni dened on the atomic lattice. Therefore, the experimental signature should be detected as anomalies instead of the singularities of physical properties.

Figure 1 shows the exchange eld n(r) (Fig. 1c) and the corresponding emergent eld b(r) (Fig. 1cg). Zero points of M(r) dress hedgehog-type topological spin textures around them, resulting in the singular points of n(r), and behave as emergent monopoles (Fig. 1c), which are dotted in the magnetic structure exemplied in Fig. 1b. A stream plot of emergent magnetic eld of a monopoleantimonopole pair on the identical z-plane clearly demonstrates that the emergent monopole and antimonopole act as its source and sink, respectively (inset of Fig. 1b).

Now we turn to the magnetization process. The topological spin texture under the magnetic eld along the z direction is approximately expressed by the formula M(r) M0(r) (0,0,mz);

mz is introduced so as to concisely represent a uniform magnetization induced by an external magnetic eld along the z axis. An external magnetic eld does not spoil the topological singularity up to some critical eld, while shifting the positions of monopoles and antimonopoles along yellow and green trajectories, respectively, as shown in Fig. 1dg: their collision occurs at nz 0.457 and the pair annihilation

at nz 0.605, where we introduce the polarization factor

nz

1 Vunitcell

R unit cell Mz r

= M r

j jdV to compare the transport

data shown in Fig. 2. As it can be seen from contour mappings of the emergent eld in Fig. 1dg, the regions of strong emergent eld, which are constricted in small volumes and connect monopoles and antimonopoles, vary signicantly according to monopoles positions. At nite temperatures, the spin structure and the corresponding monopoles positions can thermally uctuate, thereby resulting in uctuating emergent magnetic elds. Such uctuations are expected to be greatly enhanced, in particular near the second-order phase transition associated

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a

b

c

b(r)

n(r)

Hedgehog

Monopole

Antihedgehog

Antimonopole

+1

1

b(r)

Qm = +1 Qm = 1

d nz = 0.245

e nz = 0.457

f nz = 0.605

nz = 0

g

Figure 1 | Spin conguration and corresponding emergent magnetic eld with monopole solution in MnGe. (a) Spin orientation in a 2 2 2 magnetic

unit cell. (b) Conguration of emergent monopoles (yellow dots) and antimonopoles (green dots) in a 1 1 1.5 magnetic unit cell corresponding to the

green box in a, displayed with constant z planes (blue and pink planes). A stream of emergent eld between a monopole and an antimonopole on the same z level is also exemplied (pink plane). (c) Hedgehog and antihedgehog spin arrangements n(r) realized in the spin structure of MnGe, which are regarded as quantized source (monopole; Qm 1) and sink (antimonopole; Qm 1) of emergent eld b(r), respectively, acting on conduction electrons.

(dg) Monopoles and antimonopoles positions and their trajectories (yellow and green dots and lines) and distributions of emergent magnetic eld under varying magnetic eld applied along the z direction. Monopoles and antimonopoles collide at white dots (f) and pair annihilate at black dots (g). Three contours of emergent-eld distribution, that is, regions with |bz| 10 and 2, and surfaces of green box of a with |bz|o0.1 are dyed according to strength and

sign of bzthicker colour for larger magnitude of bz, red and blue for positive and negative bz, respectively (dg).

with the monopoleantimonopole pair annihilation. As a hallmark of unconventional electromagnetic responses exemplifying such critical phenomena of the topological phase transition, below we discuss an unusually large, positive MR and elastic softening.

Critical phenomena in transport and elastic properties. We rst show anomalous magneto-transport and elastic properties observed at 30 K, which we attribute to consequences of static and dynamic properties of emergent elds. Figure 2ac shows magnetic-eld dependence of Hall resistivity ryx, MR r(H)/r(0) and relative elastic constant Dc(H)/c(0), respectively. In Fig. 2a, we reproduce the topological Hall effect in MnGe as reported in refs. 13 and 14. In addition to the normal and anomalous Hall effects, that is, conventional features appearing as H- and M-linear transverse voltages in metallic magnets24, the topological Hall effect due to the real-space emergent magnetic eld is observed as a hallmark of formation of a non-coplanar spin structure25,26. A clear deviation from the expected conventional Hall response ryx R0H SHr2M (red curve in

Fig. 2a) represents the averaged static emergent eld13. Here we note that transverse thermoelectric effect due to the emergent eld (topological Nernst effect) is also observed in MnGe, which further supports the existence of emergent magnetic elds27.

We show in Fig. 2b a large, positive MR in the course of magnetization alignment by magnetic eld; this is in stark contrast to cases of other isostructural helimagnets of MnSi28 and Mn1 xFexGe without emergent monopoles (see Supplementary

Note 2 and Supplementary Fig. 1), which show the conventional negative MR where spin-dependent scattering of conduction

electrons reduces in the magnetization process, resulting simply in decrease of the resistance, that is, negative MR29. Both longitudinal (H||I, red curve in Fig. 2b) and transverse (H?I, blue

curve in Fig. 2b) MRs largely depart from the estimated magnetic-eld dependence of the conventional negative MR (black curve in Fig. 2b, see Supplementary Fig. 2 and Supplementary Note 3 for the experimental procedure to estimate it) and reach the maximal value near the magnetic phase boundary to the induced ferromagnetic state.

Next, we present in Fig. 2c the ultrasonic responses with sound wave propagating parallel (H||k; k being the propagation vector) or orthogonal (H?k) to the gradually increasing

magnetic eld. A huge elastic softening around the magnetic transition, that is, the topological transition, as well as a broad tail below the critical eld are discerned in the parallel case, whereas the orthogonal case shows a moderate magnetic-eld dependence with a multi-peak-and-trench ne structure (see Supplementary Fig. 3 and Supplementary Note 4 for clarity of the ne structure). This is also in stark contrast to the known case of the magnetic skyrmions in MnSi30, where elastic responses occur in a stepwise manner in going from conical phase to skyrmion phase but show minimal magnetic-eld dependence within each phase. In the present case of MnGe, the elastic response is an order of magnitude larger than the conventional case (for example, in MnSi), showing up as the critical elastic softening around the topological phase transition (for H||k) and also as the ne structure in the magnetization process (for H?k). It is noteworthy that there is no distinct or

discontinuous anomaly in the corresponding magnetization curve (see Fig. 2a).

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a

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1.21.11.00.90.80.70.02

[afii9797][afii9845](H)/[afii9845](0) [p10] z / [p10] 0

H I

[afii9845](H)/[afii9845](0)

[afii9797][afii9845](H)/[afii9845](0)

0.30.20.10.0 0.1

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0.05

0.10

H || k

0.04

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[afii9839]0H (T) M/Ms nz

15

Figure 2 | Experimental and theoretical results on unusual magneto-transport and elastic properties originating from emergent monopoles.(ac) Experimental results on magnetic-eld dependences of magnetization (blue line), expected usual Hall resistivity (red line) due to normal and anomalous Hall effects and observed Hall resistivity (dots) (a); expected usual MR (black line) and observed longitudinal and transverse MRs (b) change in elastic constants with propagation directions parallel and perpendicular to the magnetic eld (c) at 30 K. Experimental estimates of topological Hall resistivity (d) positive contributions to MRs (e), which deviate from the usual MR prole (black curve in b), and relative elastic constants (f) at 30 K as functions of normalized magnetization M/Ms, Ms being the low-temperature and high-eld (for example, 2 K and 14 T) saturated value of M.

(gi) Theoretical calculations on static emergent magnetic ux fz in a unit of f0 h/e (g) longitudinal and transverse MRs originating from

emergent eld (h) elastic constants (i) as functions of the polarization factor nz.

Comparison with theoretical predictions. The above experimental results are consistent with theoretical predictions of unique features associated with the monopole dynamics in MnGe. To clarify the role of emergent magnetic eld, we extract the unusual contributions and compare them with theoretical calculations taking account of emergent-eld effect and its spatial and temporal variations via spin-wave excitations (see Methods for detail). Figure 2df respectively presents the deviations from conventional Hall resistivity and negative MR, and the elastic constants as functions of normalized magnetization M/Ms, where

Ms is a saturated magnetization dened as the value at low-temperature and high-magnetic eld, for example, at T 2 K and

m0H 14 T. Theoretical counterparts are shown in Fig. 2gi as

functions of nz, from which we nd the following characteristics. Variation of monopoles positions against magnetic-eld change (Fig. 1dg) results in continuous change in total emergent magnetic ux fz R unit cell bzdV. As shown in Fig. 2g, fz 0 at zero

magnetic eld (nz 0) due to cancellation between positive and

negative contributions, the maximum absolute value fz f0 at

the colliding point (nz 0.457) of monopoles and antimonopoles,

and fz 0 at the pair annihilation point (nz 0.605) (see also

Supplementary Fig. 4 and Supplementary Note 5), reproducing the experimental observation of the topological Hall resistivity as a function of M/Ms shown in Fig. 2d. (The disagreement between the corresponding experimental M/Ms and theoretical nz values arises from the crude assumption that M(r) M0(r) (0, 0, mz),

to simplify the magnetic-eld effect on the deformation of magnetic structure.) This continuity also ensures that the topological

phase transition due to the pair annihilation is of the second order; indeed, no hysteretic or discontinuous behaviour is experimentally observed13. The large, positive MR reaching the maximum around the magnetic phase boundary is ascribed to the uctuations of emergent magnetic eld that is critically enhanced around the monopoleantimonopole annihilation point, as represented in Fig. 2h. Although in general various mechanisms can simultaneously contribute to MR (see Supplementary Note 2 and Supplementary Fig. 1 for investigation of other possibilities), we found that the MR prole in MnGe can be well accounted for by considering uctuations of emergent magnetic eld, which are evaluated by calculating the correlation functions bk; bk h i

(k x, y, z). The calculated bk; bk

h i exhibits an anisotropy

bx; bxh i by; by

4 bz; bz

h i and gets larger towards the upsurge at

the phase transition (see Supplementary Fig. 5 and Supplementary Note 6), thus reproducing the experimental observations of MR (Fig. 2e). Spin-wave excitations in the monopoleantimonopole crystal, which are composed of three modes (linear, quadratic and gapped dispersions as functions of momentum), have much lower energies than the temperature and play a key role in a microscopic process of the inelastic scattering of conduction electrons by monopole uctuations (see Methods). In line with this model, modication of magnetic interactions due to the sound-wave-induced strain is responsible for the observed elastic anomalies (Fig. 2c). In reality, mixing up the three-mode magnon spectrum and the acoustic phonon spectrum results in a dramatic elastic softening around the phase transition for the parallel (H||k) case as well as a multi-peak-and-trench elastic

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a

response for the orthogonal (H?k) case, as shown Fig. 2i.

(See Methods, Supplementary Fig. 3 and Supplementary Note 4 as well.)

We thus nd good agreements between the experimentally observed anomalies and the theoretical predictions at these extremalfz and zerofz (pair annihilation) points. However, the calculated dip in MR around nz 0.457 (indicated with a vertical

dashed line in Fig. 2h), which is ascribed to the suppression of the uctuations of the emergent magnetic eld at the extremum of fz, is only barely discernible in the experimental results at a restricted temperature region (see Supplementary Fig. 2). Although the uctuation effect appears just around the phase transition (nz 0.605) in the calculations based on the random

phase approximation, the elastic softening is experimentally observed in a broader magnetic-eld range as the manifestation of extended uctuations of monopole and antimonopole. Such robust uctuations beyond the mean eld picture may obscure the MR dip structure expected to be seen at the intermediate magnetic eld with the extremum of fz and may also explain the broad tail of the elastic-constant anomaly below the critical eld Hc (Fig. 2f).

Figure 3 shows the data sets of the monopole-related transport properties at various temperatures. Although the characteristic properties listed above are observed in a broad range of temperature where the spin hedgehogantihedgehog crystal is formed1315, the following features are further identied: peak structures in topological Hall resistivity show up at the pair annihilation points above 50 K (open circles in Fig. 3a) and also additional kinks in MR and elastic constant appear around the extremum point of topological Hall effect, that is, the extremalfz point (open diamond and open up-triangle in Fig. 3b,c, respectively). However, those that remain unexplained may originate from emergent monopoles as well, given their close ties to the critical magnetic-eld points; the corresponding magnetic elds Hc2 and H0c2 for the additional anomalies in MR and elastic constant are later presented in

Fig. 4b,c.

1.2

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0

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35 %

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Hc

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HTHE:peak

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0 0 50 100 150

T (K)

b

Hc

Hc1

Hc2

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Hc

H'c1

H'c2

Figure 4 | Static and dynamic properties of emergent monopoles highlighted by contour mappings of unusual magneto-transport and elastic properties. Contour mappings of topological Hall resistivity (a), positive contribution to longitudinal MR (b) and elastic constant with propagation directions parallel to the magnetic eld (c) in TH plane. The critical elds (Hc) are dened as inection points in MH curves. The magnetic eld where the topological Hall resistivity shows its extremum (HTHE:peak) and other characteristic magnetic elds (Hc1, Hc2, H0c1 and H0c2)

are also indicated in the corresponding panels.

a b c

HTHE:peak

Hc1

Hc2 Hc2

Hc1

0.5

10 K

10 K

1.0

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10 K

30 K

0.8

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30 K

0.8

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(H)/[afii9845](0)

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0 0 0

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10 10 10

15

[afii9839]0H (T) [afii9839]0H (T) [afii9839]0H (T)

15 15

Figure 3 | Experimental results on temperature development of transport and elastic properties. Magnetic-eld dependences of topological Hall resistivity (a), longitudinal MR (b) and change in elastic constant with propagation directions parallel to the magnetic eld (c). Characteristic magnetic elds are marked as closed circles (negative peaks of topological Hall resistivity representing the maximal emergent magnetic eld |bz|; HTHE:peak), open

circles (positive peaks of topological Hall resistivity near the phase boundary), closed diamonds (peak in MR originating from uctuations around phase boundary; Hc1), open diamond (a peak in MR around the extremal-fz point; Hc2), closed triangles (negative peaks in elastic constant originating from uctuations around the phase boundary; H0c1) and open triangle (a kink in elastic constant around the extremal-fz point; H0c2).

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2 V from a double-exchange model of Hunds rule coupling

between electrons and localized spin moments. The interaction with spin-wave uctuations enters via the spatial and temporal dependence of the emergent vector potential a and a potential V on the spin congurations. We take these uctuations as rst-order deviations away from the ground state, adopt the memory function method32 to calculate the nite temperature correlation function Yii t Tt ji; Heff

t

ji; Heff

0

h i (i x, y, z) and extract the dc resistivity by a

numerical analytical continuation method33. This approach takes care of the gauge choice problem posed by the emergent monopolar eld b = a in a concise

manner. Moreover, equal-time nite temperature correlation functions of the emergent magnetic elds can be calculated as well, so as to compare the uctuations as stated in the main text.

Magnetoelastic interaction. Two types of magnetoelastic coupling originate from the expansion of the EXI/DMI strength J/D:

J =n

2 J0 aEXI @juj

With this reservation, we can separately assess these characteristic features from static and dynamic responses of emergent magnetic monopoles. We show in Fig. 4 contour mappings of the topological Hall resistivity (a), the positive MR (b) and the elastic constant (c) in the TH plane. In Fig. 4a, the strong signal of negative topological Hall effect rTyx, that is, emergent eld strength fz, shows up at intermediate external elds as characterized by the magnetic eld with the negative peak of rTyx (HTHE:peak) and continuously disappears at the phase boundary. In contrast, the strong signals of positive MR and elastic softening, which we assign to the outcomes of fz uctuations, appear around the critical eld Hc for the monopoleantimonopole pair annihilation as enhanced in an intermediate temperature region, for example, 2070 K below TN.

Coincidences of the critical eld Hc and characteristic elds with peak signals of MR and elastic softening (Hc1 and H0c1) in the wide temperature range further support that the observed anomalies are the critical phenomena associated with continuous phase transition on the monopoles pair annihilation.

The observed critical behaviours in MR and elastic property are well accounted for by the theoretical models, considering uctuations of emergent monopoles, and hence represent the topological phase transition associated with the pair annihilation of the real-space hedgehog spin singularities in MnGe. The topology of spin structure, guaranteed by the three-mode helical magnetic ordering, enables the topological phase transition to survive even at nite temperatures, which can realize versatile unique phenomena based on the dynamics of emergent magnetic monopoles.

Methods

Sample preparation. Polycrystalline MnGe samples were synthesized with a cubic-anvil-type high-pressure apparatus. Alloys with stoichiometric quantities of the constituents were prepared by arc melting under an argon atmosphere, followed by heat treatment at 1,073 K under a high pressure of 4.0 GPa for 1 h. The material was conrmed to be single phase of B20-type chiral cubic structure at room temperature by powder X-ray analyses.

Transport measurements. MR and Hall resistivity were measured by using AC-transport option in Physical Property Measurement System.

Elasticity measurements. A phase comparison method31 was used for measurements of relative change in elastic constant. The sample has a cuboid shape with dimensions of 3.8 1.8 1.8 mm3. Two 36 Y-cut LiNbO3 piezoelectric

transducers were attached on polished parallel surfaces for generation and detection of ultrasound. Longitudinal ultrasound having a frequency of 18 MHz was injected along the longest direction under the magnetic eld parallel or perpendicular to its propagation direction k.

Spin-wave theory. For a spin helix Mmi

^i m kiri Fi

Heff

12 p ea

@jni@jni

and

ni@jnk;

wherein we retain up to the rst-order coupling coefcient aEXI/DMI and incor

porate the longitudinal acoustic phonon degrees of freedom, the displacement eld uj. We then introduce spin-wave elds denoted collectively by a six-component eld jm (/, dm) and derive the interaction Lagrangian density of the form,

LME

Dn = n

Eijk D0 aDMI @juj

aEXI 2ad 20 CmjEXI@jjm@juj DnjEXIjn@juj

aDMI 2ad 10 CmjDMI@jjm@juj DnjDMIjn@juj

; m 1; . . . ; 6; u 4; 5; 6; j x; y; z

;wherein CEXI/DMI and DEXI/DMI are matrices of some form factors embedding

simultaneously the information of the ground-state spin conguration and the relevant form of magnetoelastic coupling derived from EXI/DMI. Together with

LSW and a standard longitudinal phonon theory, by integrating out the spin-wave elds, we can attain an effective phonon theory that well describes the new phonon excitations under the hybridization with magnon modes. A new renormalized k-linear phonon mode as identied by examining the spectral function takes direct charge of the elastic responses.

(Detailed calculation results of the MR and the ultrasonic responses for more general cases with the use of a broader range of parameter values can be found in the papers in preparation by Zhang et al., and by Zhang and Nagaosa, respectively.)

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propagating along the

i-direction (i x, y, z), we dene spin-wave elds fi and dmi as the uctuation part

of the phase Fi and the uniform magnetization mi, respectively. The imaginary-time spin-wave Lagrangian density for hedgehogantihedgehog lattice takes the form,

LSW X

i

iEijkAbifj _

fk iBbidmi

h i;

where A2eS 1a ; B a ; w DJ a ; rJ a , e is electric charge and S the spin

magnitude, a0 the lattice constant and d 3 the spatial dimensions. The emergent

magnetic eld bi are spatially averaged within a magnetic cell, as we are interested in the long-wavelength behaviour. The exchange interaction (EXI) and DMI stabilizing the hedgehogantihedgehog lattice herein enter via the their strengths J and D, respectively. The rst two terms originate from the spin Berry phase and the last two manifest the rigidity gained after symmetry breaking on the formation of hedgehogantihedgehog lattice. The key feature consists in a three-mode magnon spectrum (linear, quadratic in momentum (k) and gapped, respectively) of this theory (see also Supplementary Fig. 3 and Supplementary Note 4).

Resistivity calculation. Employing an adiabatic approximation for the real-space Berry phases produced by the spin moments of hedgehogs and meanwhile felt by itinerant electrons, one can derive an effective Hamiltonian for the electrons

_

fi wdm2i r rfi

2

6 NATURE COMMUNICATIONS | 7:11622 | DOI: 10.1038/ncomms11622 | http://www.nature.com/naturecommunications

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms11622 ARTICLE

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Acknowledgements

We thank T. Arima and T. Yokouchi for their fruitful discussions. This work was

partially supported by JSPS KAKENHI (grant numbers 24224009 and 15H05456).

X.-X.Z. was partially supported by the Panasonic Scholarship.

Author contributions

N.K. grew polycrystalline samples and performed transport measurements. Y.N.

performed elasticity measurements. X.-X.Z., A.S.M., G.D.F. and N.N. performed

theoretical calculations on magneto-transport and elastic properties. Y.T. conceived the

project. N.K., N.N. and Y.T. prepared the manuscript with assistance from Y.N., X.-X.Z.

and F.K. All authors discussed the results and commented on the manuscript.

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How to cite this article: Kanazawa, N. et al. Critical phenomena of emergent magnetic

monopoles in a chiral magnet. Nat. Commun. 7:11622 doi: 10.1038/ncomms11622 (2016).

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