Abstract
Secondorder 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 realspace 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.
Introduction
The search for phase transitions and critical phenomena beyond Landau’s conventional scheme is one of the most fundamental issues in the physics of strongly correlated systems and topological quantum matter^{1}. 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 Z_{2} number defined in the first Brillouin zone, the edge or surface gapless states and also the groundstate degeneracies for the nontrivial topological geometry of the sample^{2,3}. The topological phase transition, that is, the change of the topological index defined 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 defined 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 field, that is, the emergent magnetic field^{4}. A realspace geometric arrangement of spins affects charge transport by exerting an additional quantum phase on the itinerant electrons, called Berry phase^{5}, which acts as the effective magnetic field. Among topological spin orders, hedgehogs and antihedgehogs play the roles of sources or sinks of emergent field, that is, emergent monopoles or antimonopoles^{6,7}. These hedgehog spin structures have been indeed realized as topological defects such as Bloch points^{8,9,10,11} and singular points where magnetic skyrmions coalesce with or split from one another^{12}. However, topological transitions associated with creation and annihilation of these hedgehogs are random events in terms of time and space; observation and identification of the consequent critical phenomena in the presence of emergent monopoles remain unexplored because of difficulty in controlling their behaviours.
Here we propose that the chiral magnet MnGe, where a periodic assembly of spin hedgehogs (monopoles) and antihedgehogs (antimonopoles) is realized as the magnetic ground state^{13,14,15}, is an ideal system showing the topological phase transition of the realspace spin configuration and critical fluctuations 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 M^{2}proportional profiles, which are accounted for by static and dynamic profiles of emergent fields, 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 fluctuations of emergent field at the realspace topological phase transition.
Results
Hedgehog spin structures and emergent magnetic monopoles
The magnetic structure in MnGe (shown schematically in Fig. 1a) has been observed by neutron diffraction^{14} and Lorentz transmission electron microscopy^{15} below the magnetic transition temperature T_{N}≈170 K and is modelled by the superposition of three orthogonal helical structures induced by the Dzyaloshinskii–Moriya interaction (DMI)^{16},
where M_{0} and q are the amplitude and the wavenumber of the helical moment, respectively; the periodic modulation length λ=2π/q is ranging from 3 nm (typically below 50 K) to 6 nm (near T_{N})^{13,14,15}. In sharp contrast to the twodimensional skyrmion crystal state^{17,18}, 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 theoretically^{8,9} and demonstrated experimentally^{10,11}. This means that rather than M(r) is the more appropriate field that influences the conduction electrons in the strongly correlated systems. We regard that MnGe belongs to this case; in fact, its saturated magnetization M_{s}=1.9 μ_{B} per f.u.^{13} is approximately five times enhanced as compared with the value of the related magnet MnSi^{20} with a larger oneelectron bandwidth. Mathematically, M(r) belongs to B^{3} (inside the threedimensional sphere), whereas n(r) belongs to S^{2} (surface of the threedimensional sphere). The former is trivial, as all configuration of M(r) is smoothly connected to M(r)=0, whereas S^{2} is nontrivial as characterized by the Berry curvature , which corresponds to the solid angle subtended by n(r) and the emergent magnetic field acting on electrons in the same manner as the classical magnetic field does^{4}. 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 (S is the surface enclosing the singularity), namely the emergent magnetic monopole and antimonople (see Fig. 1c and also Supplementary Note 1)^{6,7,8,9,10,11,12,21,22,23}. Therefore, the strong correlation limit produces the nontrivial topological classification of the spin configurations and the topological phase transition as well. It is noteworthy that this topological phase transition and associated critical phenomena are defined for the classical spin configuration n(r) and its thermal fluctuations at finite temperatures. It is also worth noting that the topology is defined for the continuous function n(r) and becomes less rigorous for n_{i} defined 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 field n(r) (Fig. 1c) and the corresponding emergent field b(r) (Fig. 1c–g). Zero points of M(r) dress hedgehogtype 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 exemplified in Fig. 1b. A stream plot of emergent magnetic field of a monopole–antimonopole pair on the identical zplane 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 field along the z direction is approximately expressed by the formula M(r)=M_{0}(r)+(0,0,m_{z}); m_{z} is introduced so as to concisely represent a uniform magnetization induced by an external magnetic field along the z axis. An external magnetic field does not spoil the topological singularity up to some critical field, while shifting the positions of monopoles and antimonopoles along yellow and green trajectories, respectively, as shown in Fig. 1d–g: their collision occurs at n_{z}=0.457 and the pair annihilation at n_{z}=0.605, where we introduce the polarization factor to compare the transport data shown in Fig. 2. As it can be seen from contour mappings of the emergent field in Fig. 1d–g, the regions of strong emergent field, which are constricted in small volumes and connect monopoles and antimonopoles, vary significantly according to monopoles’ positions. At finite temperatures, the spin structure and the corresponding monopoles’ positions can thermally fluctuate, thereby resulting in fluctuating emergent magnetic fields. Such fluctuations are expected to be greatly enhanced, in particular near the secondorder phase transition associated with the monopole–antimonopole 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 first show anomalous magnetotransport and elastic properties observed at 30 K, which we attribute to consequences of static and dynamic properties of emergent fields. Figure 2a–c shows magneticfield dependence of Hall resistivity ρ_{yx}, MR ρ(H)/ρ(0) and relative elastic constant Δc(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 Mlinear transverse voltages in metallic magnets^{24}, the topological Hall effect due to the realspace emergent magnetic field is observed as a hallmark of formation of a noncoplanar spin structure^{25,26}. A clear deviation from the expected conventional Hall response ρ_{yx}=R_{0}H+S_{H}ρ^{2}M (red curve in Fig. 2a) represents the averaged static emergent field^{13}. Here we note that transverse thermoelectric effect due to the emergent field (topological Nernst effect) is also observed in MnGe, which further supports the existence of emergent magnetic fields^{27}.
We show in Fig. 2b a large, positive MR in the course of magnetization alignment by magnetic field; this is in stark contrast to cases of other isostructural helimagnets of MnSi^{28} and Mn_{1−x}Fe_{x}Ge without emergent monopoles (see Supplementary Note 2 and Supplementary Fig. 1), which show the conventional negative MR where spindependent scattering of conduction electrons reduces in the magnetization process, resulting simply in decrease of the resistance, that is, negative MR^{29}. Both longitudinal (HI, red curve in Fig. 2b) and transverse (H I, blue curve in Fig. 2b) MRs largely depart from the estimated magneticfield 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 (Hk; k being the propagation vector) or orthogonal (H k) to the gradually increasing magnetic field. A huge elastic softening around the magnetic transition, that is, the topological transition, as well as a broad tail below the critical field are discerned in the parallel case, whereas the orthogonal case shows a moderate magneticfield dependence with a multipeakandtrench fine structure (see Supplementary Fig. 3 and Supplementary Note 4 for clarity of the fine structure). This is also in stark contrast to the known case of the magnetic skyrmions in MnSi^{30}, where elastic responses occur in a stepwise manner in going from conical phase to skyrmion phase but show minimal magneticfield 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 Hk) and also as the fine 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).
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 field, we extract the unusual contributions and compare them with theoretical calculations taking account of emergentfield effect and its spatial and temporal variations via spinwave excitations (see Methods for detail). Figure 2d–f respectively presents the deviations from conventional Hall resistivity and negative MR, and the elastic constants as functions of normalized magnetization M/M_{s}, where M_{s} is a saturated magnetization defined as the value at lowtemperature and highmagnetic field, for example, at T=2 K and μ_{0}H=14 T. Theoretical counterparts are shown in Fig. 2g–i as functions of n_{z}, from which we find the following characteristics. Variation of monopoles’ positions against magneticfield change (Fig. 1d–g) results in continuous change in total emergent magnetic flux . As shown in Fig. 2g, φ_{z}=0 at zero magnetic field (n_{z}=0) due to cancellation between positive and negative contributions, the maximum absolute value φ_{z}=−φ_{0} at the colliding point (n_{z}=0.457) of monopoles and antimonopoles, and φ_{z}=0 at the pair annihilation point (n_{z}=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/M_{s} shown in Fig. 2d. (The disagreement between the corresponding experimental M/M_{s} and theoretical n_{z} values arises from the crude assumption that M(r)=M_{0}(r)+(0, 0, m_{z}), to simplify the magneticfield 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 observed^{13}. The large, positive MR reaching the maximum around the magnetic phase boundary is ascribed to the fluctuations of emergent magnetic field that is critically enhanced around the monopole–antimonopole 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 profile in MnGe can be well accounted for by considering fluctuations of emergent magnetic field, which are evaluated by calculating the correlation functions (k=x, y, z). The calculated exhibits an anisotropy 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). Spinwave excitations in the monopole–antimonopole 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 fluctuations (see Methods). In line with this model, modification of magnetic interactions due to the soundwaveinduced strain is responsible for the observed elastic anomalies (Fig. 2c). In reality, mixing up the threemode magnon spectrum and the acoustic phonon spectrum results in a dramatic elastic softening around the phase transition for the parallel (Hk) case as well as a multipeakandtrench elastic response for the orthogonal (H k) case, as shown Fig. 2i. (See Methods, Supplementary Fig. 3 and Supplementary Note 4 as well.)
We thus find good agreements between the experimentally observed anomalies and the theoretical predictions at these extremal–φ_{z} and zero–φ_{z} (pair annihilation) points. However, the calculated dip in MR around n_{z}=0.457 (indicated with a vertical dashed line in Fig. 2h), which is ascribed to the suppression of the fluctuations of the emergent magnetic field at the extremum of φ_{z}, is only barely discernible in the experimental results at a restricted temperature region (see Supplementary Fig. 2). Although the fluctuation effect appears just around the phase transition (n_{z}=0.605) in the calculations based on the random phase approximation, the elastic softening is experimentally observed in a broader magneticfield range as the manifestation of extended fluctuations of monopole and antimonopole. Such robust fluctuations beyond the mean field picture may obscure the MR dip structure expected to be seen at the intermediate magnetic field with the extremum of φ_{z} and may also explain the broad tail of the elasticconstant anomaly below the critical field H_{c} (Fig. 2f).
Figure 3 shows the data sets of the monopolerelated transport properties at various temperatures. Although the characteristic properties listed above are observed in a broad range of temperature where the spin hedgehog–antihedgehog crystal is formed^{13,14,15}, the following features are further identified: 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 extremal–φ_{z} point (open diamond and open uptriangle in Fig. 3b,c, respectively). However, those that remain unexplained may originate from emergent monopoles as well, given their close ties to the critical magneticfield points; the corresponding magnetic fields H_{c2} and for the additional anomalies in MR and elastic constant are later presented in Fig. 4b,c.
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 T–H plane. In Fig. 4a, the strong signal of negative topological Hall effect , that is, emergent field strength φ_{z}, shows up at intermediate external fields as characterized by the magnetic field with the negative peak of (H_{THE: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 φ_{z} fluctuations, appear around the critical field H_{c} for the monopole–antimonopole pair annihilation as enhanced in an intermediate temperature region, for example, 20–70 K below T_{N}. Coincidences of the critical field H_{c} and characteristic fields with peak signals of MR and elastic softening (H_{c1} and ) 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 fluctuations of emergent monopoles, and hence represent the topological phase transition associated with the pair annihilation of the realspace hedgehog spin singularities in MnGe. The topology of spin structure, guaranteed by the threemode helical magnetic ordering, enables the topological phase transition to survive even at finite 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 cubicanviltype highpressure 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 confirmed to be single phase of B20type chiral cubic structure at room temperature by powder Xray analyses.
Transport measurements
MR and Hall resistivity were measured by using ACtransport option in Physical Property Measurement System.
Elasticity measurements
A phase comparison method^{31} 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 mm^{3}. Two 36° Ycut LiNbO_{3} 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 field parallel or perpendicular to its propagation direction k.
Spinwave theory
For a spin helix propagating along the idirection (i=x, y, z), we define spinwave fields φ_{i} and δm_{i} as the fluctuation part of the phase Φ_{i} and the uniform magnetization m_{i}, respectively. The imaginarytime spinwave Lagrangian density for hedgehog–antihedgehog lattice takes the form,
where , e is electric charge and S the spin magnitude, a_{0} the lattice constant and d=3 the spatial dimensions. The emergent magnetic field b_{i} are spatially averaged within a magnetic cell, as we are interested in the longwavelength behaviour. The exchange interaction (EXI) and DMI stabilizing the hedgehog–antihedgehog lattice herein enter via the their strengths J and D, respectively. The first two terms originate from the spin Berry phase and the last two manifest the rigidity gained after symmetry breaking on the formation of hedgehog–antihedgehog lattice. The key feature consists in a threemode 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 realspace Berry phases produced by the spin moments of hedgehogs and meanwhile felt by itinerant electrons, one can derive an effective Hamiltonian for the electrons from a doubleexchange model of Hund’s rule coupling between electrons and localized spin moments. The interaction with spinwave fluctuations enters via the spatial and temporal dependence of the emergent vector potential a and a potential V on the spin configurations. We take these fluctuations as firstorder deviations away from the ground state, adopt the memory function method^{32} to calculate the finite temperature correlation function (i=x, y, z) and extract the dc resistivity by a numerical analytical continuation method^{33}. This approach takes care of the gauge choice problem posed by the emergent monopolar field b=∇ × a in a concise manner. Moreover, equaltime finite temperature correlation functions of the emergent magnetic fields can be calculated as well, so as to compare the fluctuations as stated in the main text.
Magnetoelastic interaction
Two types of magnetoelastic coupling originate from the expansion of the EXI/DMI strength J/D:
and
wherein we retain up to the firstorder coupling coefficient α_{EXI/DMI} and incorporate the longitudinal acoustic phonon degrees of freedom, the displacement field u_{j}. We then introduce spinwave fields denoted collectively by a sixcomponent field ϕ_{μ}=(φ, δm) and derive the interaction Lagrangian density of the form,
wherein C_{EXI/DMI} and D_{EXI/DMI} are matrices of some form factors embedding simultaneously the information of the groundstate spin configuration and the relevant form of magnetoelastic coupling derived from EXI/DMI. Together with and a standard longitudinal phonon theory, by integrating out the spinwave fields, we can attain an effective phonon theory that well describes the new phonon excitations under the hybridization with magnon modes. A new renormalized klinear phonon mode as identified 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.)
Additional information
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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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.
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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 magnetotransport 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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Supplementary Figures 15, Supplementary Notes 16 and Supplementary References (PDF 8352 kb)
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Kanazawa, N., Nii, Y., Zhang, XX. et al. Critical phenomena of emergent magnetic monopoles in a chiral magnet. Nat Commun 7, 11622 (2016). https://doi.org/10.1038/ncomms11622
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DOI: https://doi.org/10.1038/ncomms11622
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