Symmetry, Structure, and Dynamics of Monoaxial Chiral Magnets

Electrons have charge and thus are deflected by the Lorentz force when they pass through an area with an electric and/or magnetic field. Based on this simple mechanism, an electron beam probes electromagnetic structures in materials. Hence, the method is frequently called Lorentz microscopy.^67–71)

In the following subsections, we provide a brief introduction to the methods of observing and analyzing magnetic structures using an electron beam. In particular, the principle and capacities of real-space observation and reciprocal-space analysis in transmission electron microscopy (TEM) are given. As shown later, these methods are very useful for the detection of chiral magnetic orders in chiral magnets.²¹⁾

3.1.3 Fresnel mode

The Fresnel mode of Lorentz microscopy is used to analyze magnetic structures qualitatively. Figure 5 shows the principle of the Fresnel mode and how the twist of a chiral soliton in the CSL is visualized depending on its magnetic chirality.

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Figure 5. (Color online) Schematic drawings of electron beam deflection by magnetic fields in magnetic structures and the principle of the Fresnel and DPC modes of Lorentz microscopy. Ray diagrams of an electron beam illustrate the relation between the contrast formed in Lorentz micrographs obtained by the Fresnel and DPC modes and the twist of a chiral soliton in the CSL. The direction of the magnetization m transverse to the electron beam is determined from the bright and dark contrast. The intensity profile maps of the in-plane m and its spatial gradient \(\partial m/\partial x\) along the helical axis x correspond closely to micrographs of the soliton taken in the Fresnel and DPC modes, respectively, since the configuration is close to being divergence-free. The contrast is reversed when defocusing above and below the specimen in the Fresnel mode; \(\partial m/\partial x\) is observed under the overfocused condition while \(-\partial m/\partial x\) is observed under the underfocused condition. Magnetic chirality is determined by the contrast in each soliton because a soliton in the CSL deflects the electron beam in a manner depending on its magnetic chirality, reminiscent of the effect of cylindrical lenses on optical light. Figures adapted from Ref. 27. © 2015 American Physical Society.

An electron beam passing through the magnetic structure formed in a magnetic thin film is deflected by the Lorentz force in the region with magnetic moments transverse to electron beam propagation. Thus, in a defocused plane, a bright or dark contrast pattern appears in locations where the magnitude of the in-plane magnetization m is not uniform, such as at the boundary of magnetic domains.

The relation between the contrast pattern and the direction of magnetic moments is reversed depending on the defocusing condition as schematically drawn in Fig. 5. Therefore, by examining the bright or dark contrast pattern in Lorentz Fresnel micrographs taken under a fixed defocus condition, the magnetic structure is identified qualitatively. Roughly speaking, an intensity profile map of the spatial gradient of m is obtained in the Fresnel mode when the configuration of magnetic moments is close to being divergence-free.

A 2-D map of the in-plane m distribution can be reconstructed from a series of defocused Fresnel images. More precisely, the relative phase distribution is retrieved by using the transport-intensity equation (TIE) based on the propagation theory of an electron beam.^76–78) Then, the in-plane m map obtained is frequently presented in a color-wheel code to see the magnetic structure intuitively, although the information is essentially the same as the original phase map in a gray scale.

The TIE method can provide quantitative data of magnetic structures only when used under appropriate experimental conditions,^79–81) although there seem to have been many misleading statements on this method, particularly on its spatial resolution. The results obtained by TIE phase retrieval should be consistent with the original Fresnel images. In this sense, the proper consideration of Fresnel images is very important to understand the images reconstructed by the TIE method.

There have been many studies of the real-space imaging of magnetic structures in cubic helimagnetic crystals by Lorentz Fresnel microscopy including the TIE method since the first reports of Fresnel micrographs of Fe_0.5Co_0.5Si^82,83) and FeGe.^84,85) Again, it should be noted that Lorentz microscopy, in principle, only detects an in-plane component of magnetic moments in the specimen. Thus, additional experimental methods with careful consideration are required to analyze real-space images by Lorentz microscopy⁸⁶⁾ and specify a detailed magnetic structure even in the case of a CHM and CSL.

The Fresnel mode of Lorentz microscopy and SAES experiments have significantly contributed to confirming the existence of the CHM and CSL in CrNb₃S₆.²¹⁾

Figure 6 shows an example of Lorentz micrographs obtained at 110 K in the absence and presence of an external field perpendicular to the helical c-axis. A bright and dark contrast pattern repeatedly appears perpendicular to the c-axis and is very straight and regular in almost all the regions of the specimen observed. The contrast changes in a nearly sinusoidal manner with 48 nm period at 0 Oe. This sinusoidal pattern of in-plane magnetic moments is observed in a series of Lorentz Fresnel micrographs taken at various defocus values of less than 8 µm, and the contrast is reversed when defocusing in the opposite direction. As the field strength increases, the period \(L(H)\) monotonically increases. For instance, it expands to 92 nm at 2080 Oe in the TEM specimen presented in Fig. 6(b), suggesting that the CHM transforms into a CSL.

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Figure 6. Lorentz Fresnel micrographs of the CHM and CSL in CrNb₃S₆ at 110 K. (a) 0 Oe. (b) 2080 Oe. For TEM observations, thin specimens, typically of 5 µm width, 13 µm length, and 70 nm thickness, were fabricated using a focused ion beam (FIB) etching machine. A magnetic field was applied in the direction normal to the thin platelet specimen using the objective lens of a microscope. The scale was calibrated using SAES data. The defocus value is, respectively, 2 µm and 900 nm in the underfocused direction in (a) and (b). The wide curved dark fringes are bend contours caused by electron scattering through the slightly curved TEM specimen.

The sinusoidal magnetic pattern of the CHM was precisely examined in the reciprocal space. For the CHM, two characteristic features are expected to be observed in SAES data. First, magnetic diffraction spots should appear at \(\pm Q_{0}\) due to magnetic Bragg diffraction because the CHM consists of the harmonic sinusoidal modulation of magnetic moments with the single wave vector \(Q_{0}\) and acts as a sinusoidal phase grating for an electron beam. Second, the harmonic modulation of in-plane magnetic moments in the CHM should induce a change in the shape of the central round spot into an ellipse since electron waves are deflected by a Lorentz force in a sinusoidal manner in the CHM. The latter is reminiscent of SAES data of a 180° Bloch-type domain wall, in which the central spot is split into two symmetric spots with a diffusive streak in between.^67,93–96)

Figure 7 shows such expected behavior for the CHM. A pair of diffraction spots were found at (48 nm)⁻¹ along the c-axis close to the 000 diffraction spot. The spatial frequency detected was consistent with that obtained by SANS experiments⁴⁷⁾ as well as the period observed in real-space Lorentz images. More importantly, no higher harmonic spots were observed within the sensitivity of the SAES signal detection using films, a CCD camera, and imaging plates.

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Figure 7. (Color online) Series of SAES data at 0 Oe taken from CrNb₃S₆ with the camera length magnified by different orders of magnitude. (a) 30 cm, (b) 30 m, and (c) 300 m. The 000 and 001 spots of Bragg diffraction give a spatial frequency of (1.2 nm)⁻¹ in the reciprocal space. The CHM provides a pair of magnetic satellite spots with a spatial frequency of (48 nm)⁻¹ close to the 000 spot indicated by blue arrows. (c) Elliptical shape of the central spot due to magnetic deflection.

Magnetic deflection from the CHM is also detected in SAES experiments. Figure 7(c) presents the elliptical shape of the central spot along the c-axis in SAES taken at a very long camera length of 300 m. The size of the central spot is about \(7.4\times 10^{-6}\) rad along the c-axis, which gives 0.086 T for the saturation magnetization in TEM specimens with 70 nm thickness. The thickness is roughly estimated from an SEM cross-section image and the transparency of the specimen to electron beams. The value obtained from SAES data is in good agreement with 0.0862 T (1.5 \(\mu_{\text{B}}\)/Cr atom; \(\mu_{\text{B}}\) is Bohr's magneton) at 110 K obtained by magnetization measurements of bulk CrNb₃S₆ specimens.¹⁰³⁾

The CHM gradually transforms into the CSL in magnetic fields perpendicular to the c-axis. Figure 6(b) shows that the sinusoidal pattern of the CHM transforms into another periodic pattern segmented by dark contrast lines, which is ascribed to the CSL. Figure 8(a) presents a similar periodic pattern in a different TEM specimen taken at a larger magnification. It is partitioned by three contrast lines, that is, a central dark contrast line accompanied with two adjacent bright contrast lines under the underfocused condition, which is consistent with the contrast expected for a left-handed CSL as depicted in Fig. 5.

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Figure 8. Lorentz Fresnel micrograph and SAES data of the CSL at 2500 Oe in a CrNb₃S₆ specimen with \(H_{\text{c}}\) of 2550 Oe.

Namely, the analyses of the contrast pattern of the CSL in Lorentz microscopy are directly connected to the identification of the magnetic chirality of the CSL. As shown in Fig. 5, in the present TEM experimental configuration, forced FM domains in the CSL and the forced FM state have out-of-plane magnetic moments toward the applied vertical magnetic field. They are parallel to the propagation direction of the electron beam, thus do not cause any change in the specific contrast in Lorentz micrographs. Meanwhile, the \(2\pi\) twist solitons in the CSL give a striking contrast pattern: the solitons are identified by the presence of three lines with bright and dark contrast in the Fresnel mode, while they appear as paired lines in the DPC mode. Thus, the magnetic chirality of the CSL is uniquely determined from Lorentz micrographs.

It was clarified that the CSL has left-handed magnetic chirality in the specimens presented in Figs. 6 and 8. Moreover, the CHM should inherit the same magnetic chirality because the CSL develops from the CHM. It was found that most of the area of the crystal has left-handed magnetic chirality but very small regions have the opposite magnetic chirality (see Fig. 9) in a particular bulk CrNb₃S₆ single crystal when the magnetic chirality of the crystal is locally examined by Lorentz microscopy.

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Figure 9. Lorentz Fresnel micrograph of crystalline grains with different magnetic chirality taken under the underfocused condition at 1781 Oe at 100 K. The region with right-handed magnetic chirality is sandwiched by those with left-handed magnetic chirality.

Figure 10(a) shows an experimental plot of \(\Delta L(H)/L(0)\). Upon increasing the field strength, the CSL period \(L(H)\) increases monotonically and finally diverges at \(H_{\text{c}}\sim 0.230\) T for the specimen presented in Figs. 6 and 7. This behavior is consistent with the theoretical scenario based on the CSL formation. Indeed, the experimental data are well fitted by using a single function given by Eq. (39) later.

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Figure 10. (Color online) (a) Experimental plot of \(\Delta L(H)/L(0)=[L(H)-L(0)]/L(0)\) for the specimen presented in Figs. 6 and 7 with the fitting curve given by Eq. (39). \(H_{\text{c}}\) and \(L(0)\) are experimentally determined as ∼0.230 T and 48 nm, respectively. (b) Experimental plot of the soliton density \(L(0)/L(H)\), which plays the role of the order parameter of the IC–C phase transition. Figures adapted from Ref. 21. © 2012 American Physical Society.

No periodic pattern due to the CSL is observed above \(H_{\text{c}}\) in the Lorentz micrographs, indicating that the system undergoes the IC–C phase transition from the CSL to the forced FM state. The soliton density \(L(0)/L(H)\) is the order parameter of the IC–C phase transition that is described by the chiral sine-Gordon model. The plot of \(L(0)/L(H)\) in Fig. 10(b) shows that the data follows the behavior of the soliton density curve.

The CHM and CSL are very straight and perpendicular to the chiral helical axis and appear over most of the specimen as partially shown in Fig. 6. In particular, in most cases, the CHM has hardly any structural dislocations and is resistant to crystal defects, which potentially exist in specimens, and scratch defects extrinsically formed by the irradiation of an FIB along the [001] direction during TEM specimen fabrication (see Fig. 11). The specific features of high stability and robustness originate from the fact that both the CHM and CSL are manifestations of the macroscopic order of spin magnetic moments. Indeed, the chiral magnetic orders are macroscopically induced in CrNb₃S₆ by the DM interaction, which is allowed in the system because the hexagonal crystal of CrNb₃S₆ belongs to the noncentrosymmetric chiral space group.

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Figure 11. Lorentz Fresnel micrograph of CrNb₃S₆ at 0 Oe. The contrast modulation along the horizontal direction is due to a marked change in the thickness of the specimen caused by the irradiation of a focused gallium ion beam.

X-rays are known as one of the most powerful experimental probes for determining crystal structures of compounds. The chiral crystal structures are generally determined by an absolute structure analysis.

The intensities of a pair of reflections at \((h, k, l)\) and \((\bar{h},\bar{k},\bar{l})\), termed Bijvoet pairs,¹¹⁰⁾ are not equivalent because the atomic scattering amplitude in X-ray diffraction includes anomalous scattering, containing an imaginary part. By comparing the intensities of hundreds of Bragg reflections between \((h, k, l)\) and \((\bar{h},\bar{k},\bar{l})\), the deduced Flack parameter^111–113) reflects the ratio between the left- and right-handed crystalline domains.

We emphasize that the neutron diffraction technique cannot determine the crystal chirality. Most nuclear species (elements) have no anomalous scattering in terms of the nuclear scattering amplitude. Therefore, there is no difference in the nuclear scattering intensity between \((h, k, l)\) and \((\bar{h},\bar{k},\bar{l})\).

X-rays also play a very important role in magnetic structure studies in chiral science. It is well known that X-rays couple with the charge of electrons, which is independent of the magnetic structure in the compounds. However, X-rays are also scattered by magnetic electrons, and magnetic diffraction can be observed using X-rays.^114,115) The problem is that the signal of the magnetic scattering is extremely small compared with that of the charge scattering. Highly brilliant synchrotron radiation sources make it possible to detect X-ray magnetic diffraction signals. The magnetic Bragg peaks due to antiferromagnetic and helimagnetic ordering have been observed since the 1970s.^116,117) Using polarized X-ray beams, it is possible to detect the chirality of crystalline and helimagnetic structures. Here, we introduce some experimental results using circularly polarized X-rays.

3.4.1 Circularly polarized X-ray diffraction for detecting crystalline chirality

As we mentioned, the crystalline chirality can be determined by absolute structure analysis using X-rays. However, the problem is that the method can only be applied to samples of sub-mm order, because the samples must be smaller than the beam diameter. Moreover, in the case of racemic twinned crystals, the Flack parameter merely indicates the ratio between the left- and right-handed crystalline domains in the sample and does not give information on how the left- and right-handed domains are distributed in the sample.

However, the circularly polarized resonant X-ray diffraction technique can solve such problems. To detect the crystallographic chirality, Dmitrienko proposed a theoretical prediction by observing the anisotropy of the tensor of susceptibility (ATS) scattering.¹¹⁸⁾ Using the polarization of X-rays, the crystallographic chirality can be easily determined just by examining the intensity of one site of the reflection that is forbidden by the screw-axis in the crystal but recovered by the ATS scattering. This method follows the way that polarized neutron diffraction determines the chirality in helimagnetic structures.

The experimental results were first observed for α-quartz with the space group of \(P3_{1}21\) or \(P3_{2}21\) by means of soft X-ray diffraction experiments using ATS scattering on the Si K-edge and circular polarization.¹¹⁹⁾ In the case of chiral magnetic compounds, Kousaka et al. succeeded in detecting the crystalline chirality of CsCuCl₃ by observing the dependence of reflection intensities on the polarization of the beam at \((0,0,6n\pm 2)\), where the reflection is forbidden in principle. However, this extinction rule is violated if the ATS is taken into account.⁶¹⁾

Figure 16 shows longitudinal scan profiles of \((0, 0, l)\) reflections of CsCuCl₃ with the incident energy on the Cu K-edge, performed at BL19LXU in SPring-8. In Figs. 16(a) and 16(b), at \((0, 0, 10)\), where the reflection is forbidden by the screw-axis but recoved by the ATS scattering, the intensity for right-handed circularly polarized (RCP) beam was larger in the \(P6_{1}22\) sample and the intensity for the left-handed circularly polarized (LCP) beam was larger in the \(P6_{5}22\) sample. On the other hand, in Figs. 16(c) and 16(d), at \((0, 0, 14)\), which is also the screw-axis-forbidden reflection, the intensity for the LCP beam was larger in the \(P6_{1}22\) samples and the intensity for the RCP beam was larger in the \(P6_{5}22\) samples. Therefore, the crystallographic chirality is evaluated by comparing the difference of intensities for the RCP and LCP beams at the reflection sites forbidden by the screw-axis.

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Figure 16. (Color online) Reciprocal scan profiles of CsCuCl₃ at (a) \((0, 0, 10)\) and (c) \((0, 0, 14)\) reflections for \(P6_{1}22\) samples and of (b) \((0, 0, 10)\) and (d) \((0, 0, 14)\) reflections for \(P6_{5}22\) samples. Filled circles and open squares represent the intensities of a right-handed circularly polarized beam (RCP) and a left-handed circularly polarized beam (LCP), respectively.

Moreover, using a microbeam, the technique realizes scanning X-ray imaging of the crystalline chirality in a large area.⁵⁷⁾ It gives the chiral domain distribution in racemic twinned crystals. Therefore, it will be an essential method to evaluate new crystallization techniques for controlling chiral crystalline domains.

The existence of the IC magnetic phase and the transformation process into the forced FM state can be examined by magnetization measurements, particularly under an applied magnetic field normal to the helical c-axis. The nontrivial behavior of the magnetization is later described by Eq. (40) and shown in Fig. 39 in the theoretical part (Sect. 5) as well as in the literature.^124,125)

In 1965, the downward convex behavior of the magnetization was first calculated by Dzyaloshinskii.¹⁶⁾ For CrNb₃S₆, the magnetic structure was investigated experimentally by small-angle neutron diffraction at zero magnetic field.⁴⁷⁾ Below \(T_{\text{C}} = 127\) K, CrNb₃S₆ exhibits a magnetic Bragg peak of 0.13 nm⁻¹, which was considered as a manifestation of helical order with magnetic moments rotating in the ab-plane with a period of 48 nm along the c-axis.

Meanwhile, a steep change in the magnetization toward \(H_{\text{c}}\) under a magnetic field applied perpendicular to the helical axis was observed in CrNb₃S₆,^46,47) where the behavior was misinterpreted as a discontinuous phase transition between the CHM and forced FM states, and an intermediate phase in perpendicular fields below \(H_{\text{c}}\) was not identified as the CSL. The transition below \(H_{\text{c}}\) should be interpreted as the continuous IC–C phase transition theoretically proposed by Dzyaloshinskii.¹⁶⁾ This type of transition was actually reported to occur in Ba₂CuGe₂O₇.¹²⁶⁾ Magnetization profiles of CrNb₃S₆ were reexamined in the 2000's and it was found that the data indirectly suggest the formation of the CHM and CSL.^103,124)

4.1.1 Temperature and magnetic field dependence

Figure 17 shows the magnetization of a CrNb₃S₆ single crystal when a magnetic field is applied along the helical c-axis. When increasing the magnetic field at a fixed temperature, the magnetization exhibits a linear increase in its magnitude at a small magnetic field followed by a sudden rise toward a certain value of the magnetic field at temperatures below \(T_{\text{C}}\). The downward convex behavior of the magnetization is regarded as indirect evidence of the IC–C phase transition, that is, the transformation process of the CHM into the forced FM state via the CSL. There is little hysteresis of the magnetization during the increase (red solid circles) and decrease (blue open squares) in the magnetic field, as can be seen in Fig. 17(a). Meanwhile, the magnetization measured while changing the temperature at a fixed magnetic field shows a sharp peak structure in lower magnetic fields, which is also considered as an indication of the formation of magnetic orders such as the CHM and CSL.

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Figure 17. (Color online) Magnetization curves as a function of magnetic field and temperature. Black arrows in (a) and (b) indicate the positions of kinks and peaks in the M–H and M–T characteristics, respectively. (d) Magnetic phase diagram of CrNb₃S₆ single crystal based on the magnetization measurement.

The CSL couples with conduction electrons and gives rise to nontrivial physical properties. In terms of the magnetotransport, the CSL is a nonlinear array of magnetic soliton kinks, each of which acts as a strong scattering potential for conduction electrons with itinerant spins. Therefore, a nontrivial magnetoresistance (MR) is induced along the helical axis. Moreover, the number of magnetic soliton kinks correlates to the magnitude of the MR.

In CrNb₃S₆, Cr atoms are in the trivalent state and have localized electrons with spins of \(S = 3/2\), whereas conduction electrons are in an unfilled hybridized band of Nb and S. In this respect, CrNb₃S₆ is one of the ideal monoaxial chiral magnetic materials for examining magnetotransport properties.

In this subsection, the MR effects in bulk and micrometer-size single crystals of CrNb₃S₆ are introduced. It is shown that the nature of the MR strongly depends on the dimension of the crystal.

4.2.1 Magnetoresistance in a bulk crystal

Figure 18 shows the interlayer resistance as a function of temperature at zero magnetic field and the interlayer MR in a bulk CrNb₃S₆ single crystal. The crystal is the same as that used in the magnetization measurements presented in Fig. 17.

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Figure 18. (Color online) (a) Optical micrographs of CrNb₃S₆ single crystal. For interlayer MR measurements, two electrodes for voltage detection are attached to the \(ac\)-plane, while another two electrodes for current application are formed on the top and bottom surfaces of the ab-plane. (b) Interlayer resistance as a function of temperature at zero magnetic field. The insets show zoom-in view of the resistance close to \(T_{\text{C}}\) of 132 K and the temperature derivative of the resistance \(dR/dT\). (c) Interlayer MR with the magnetic field perpendicular to the helical c-axis for currents of 1 and 5 mA at 110 K. Figures adapted from Ref. 25. © 2013 American Physical Society.

The interlayer resistance clarifies that the phase transition occurs at a temperature of 132 K (\(T_{\text{C}}\)), which is defined as the location of the peak in the derivative of the resistance with respect to temperature \(dR/dT\). The value of \(T_{\text{C}}\) is slightly larger than typical values reported in the literature,²¹⁾ which might reflect the very high quality of the crystal investigated.

Figure 18(c) presents the interlayer MR examined with field steps of 2 Oe at 110 K. The magnetic field is applied by using a superconducting persistent current mode, so as to prevent the field from overshooting the target value, and kept as constant as possible during the MR measurements. The MR curves measured at 1 and 5 mA coincide with each other, indicating that no Joule heating occurs in this range of current.

The interlayer MR exhibits a rapid reduction toward 1570 Oe at 110 K. This value is consistent with \(H_{\text{c}}\) for the IC–C phase transition between the CSL and the forced FM state determined by the magnetization measurement. Thus, the onset of the steep kink in the interlayer MR can be regarded as an indication of \(H_{\text{c}}\).

Above \(H_{\text{c}}\), the interlayer MR decreases almost linearly with increasing magnetic field. This behavior is in accord with that observed in the forced FM state in conventional FM materials. Indeed, this tendency has been confirmed up to 20 kOe for the present CrNb₃S₆ crystal.

The contribution of the anisotropic magnetoresistance (AMR) to the interlayer MR is considered as a constant background in these experiments because the spin configuration of the CSL is predominantly oriented within the ab-plane and is thus always perpendicular to the current flow direction along the c-axis.

Figure 19 shows the interlayer MR measured at various temperatures at 5 mA. A negative change in the interlayer MR toward \(H_{\text{c}}\) is found in a wide range of temperatures up to 125 K. Significant hysteresis is not observed in the range of temperatures investigated. The change in MR becomes positive in the narrow range of temperatures between 125 K and \(T_{\text{C}}\). The position of \(H_{\text{c}}\), indicated by black arrows, shifts toward a lower magnetic field with increasing temperature.

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Figure 19. (Color online) Interlayer MR at 5 mA at various temperatures from 10 to 180 K. A negative MR is observed below \(H_{\text{c}}\), indicated by black arrows, in a wide range of temperature below \(T_{\text{C}}\). MR starts to increase around \(H_{\text{c}}\) above 125 K and decreases monotonically above \(T_{\text{C}}\). Each MR curve is normalized by \(R_{0}\) and given in an arbitrary unit in (a) and (b). Figures adapted from Ref. 25. © 2013 American Physical Society.

Figure 20 presents a data set of the normalized negative MR curves. All the MR curves almost lie on a single curve and are qualitatively fitted by a theoretical equation of the soliton density \(L(0)/L(H)\). This finding suggests that the MR is due to the magnetic scattering of conduction electrons by a nonlinear, periodic, and countable array of magnetic soliton kinks. In particular, the MR curves show good agreement with the change in the soliton density in a low magnetic field, where the total number of solitons in the CSL is as many as ∼4000 in the present crystal with 200 µm thickness alont the c-axis. At a high magnetic field, the MR curves deviate slightly close to \(H_{\text{c}}\) because the number of solitons in the CSL rapidly becomes very small and thus other factors will make a greater contribution to the MR.

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Figure 20. (Color online) Normalized MR curves in the temperature range where a negative MR is observed below \(H_{\text{c}}\). These curves are qualitatively fitted by a theoretical equation of the soliton density \(L(0)/L(H)\) as indicated by a red broken line in (b). Figures adapted from Ref. 25. © 2013 American Physical Society.

The empirical scaling law of the MR behavior indicates that the magnitude of MR is proportional to the number of magnetic soliton kinks, as naively expected. The change in MR due to CSL formation is noticeable since the magnitude of MR becomes extremely large with the accumulation of magnetic soliton kinks, for example, up to ∼4000 in the present crystal, although the resistance induced by a single magnetic soliton kink is much smaller than the total change in MR.

The CSL is a thermodynamic stable phase. Hence, CSL formation could be identified even when sweeping the temperature instead of the magnetic field, namely, regardless of the sweeping direction in the magnetic phase diagram.

Figure 21 shows the interlayer resistance as a function of temperature at various magnetic fields. A sharp peak in \(dR/dT\) is found at a low magnetic field, as shown in Fig. 21(a). This is also observed at zero magnetic field in Fig. 19(b). With increasing magnetic field, the sharp peak in \(dR/dT\) shifts slightly toward a lower temperature, although the shoulder structure remains around the location corresponding to \(T_{\text{C}}\), as shown in Fig. 21(b). As the magnetic field increases further, the sharp peak structure in \(dR/dT\) disappears. Instead, a stepwise change appears in \(d^{2}R/dT^{2}\), as can be seen in Fig. 21(c). The location coincides with \(H_{\text{c}}\) determined by the interlayer MR measurements shown in Fig. 19. At 2500 Oe, which is above \(H_{\text{c}}\) at low temperatures, the interlayer MR curve becomes featureless except that a shoulder structure appears after a maximum in \(dR/dT\). This structure remains even in higher magnetic fields above \(H_{\text{c}}\) and might be related to the spin fluctuation from the ordered forced FM state to a fluctuating paramagnetic state.

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Figure 21. (Color online) Interlayer resistance as a function of temperature with a current of 5 mA at various magnetic fields. (a) 300 Oe, (b) 900 Oe, (c) 1600 Oe, and (d) 2500 Oe. The sharp peak in \(dR/dT\) (magenta dots) is indicated by a black solid arrow in (a) and (b), while the stepwise anomaly in \(d^{2}R/dT^{2}\) (blue dots) is pointed out by a magenta solid arrow in (c). A shoulder structure (red arrowhead) appears after a maximum (blue dotted arrow) in \(dR/dT\) in (c) and (d). Figures adapted from Ref. 25. © 2013 American Physical Society.

The magnetic phase diagram constructed solely on the basis of the interlayer MR data is presented in Fig. 22(a). Solid squares indicate the onset of the change in the interlayer MR, while other symbols correspond to the locations of anomalies found in the R–T measurements, as described for Fig. 21.

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Figure 22. (Color online) Magnetic phase diagram of CrNb₃S₆. Various kinds of anomalies found in the interlayer MR measurements are given as a function of magnetic field and temperature in (a), while the critical fields identified by the magnetization measurements are shown in (b). Figures adapted from Ref. 25. © 2013 American Physical Society.

To compare the magnetic phase diagram with that obtained by the magnetization measurements, all the data obtained by the interlayer MR and magnetization measurements are plotted in the same magnetic phase diagram in Fig. 22(b). As mentioned before, it has already been reported in the literature that \(H_{\text{c}}\) for the IC–C phase transition between the CSL and the forced FM state can be identified by characteristic anomalies found in the magnetization data,^103,124) as indicated in Figs. 17(c) and 17(d). Importantly, the values of \(H_{\text{c}}\) determined in two different ways are consistent with each other, strongly indicating that the interlayer MR follows the continuous IC–C phase transition between the CSL and the forced FM state. That is, the interlayer conduction process directly couples with the magnetic superlattice associated with CSL formation and leads to the tunable interlayer MR.

4.2.2 Magnetoresistance in a micrometer-size crystal

The interlayer MR due to CSL formation is revealed to correlate to the population of solitons, namely, the soliton density in the bulk CrNb₃S₆ crystal.²⁵⁾ In this connection, it is interesting to see how the MR changes with the system size.²⁷⁾

Figure 23 shows the interlayer MR data for a micrometer-size crystal of CrNb₃S₆. Crystal pieces of typically \(10\times 10\times 1\) µm³ were cut from a bulk single crystal of CrNb₃S₆ and fixed to a Si substrate with patterned gold electrodes by using an FIB technique, as shown in Fig. 23(a). The crystal was placed in electrical contact with four gold electrodes by W deposition so that the current flowed along the helical c-axis. Two of the electrodes (of 1 µm width) were used for voltage detection and formed on the side and top surface (\(ac\)-plane) of the crystal with a typical separation of 1 µm, while the other two used for current injection were formed on the edges.

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Figure 23. (Color online) Interlayer MR data at various temperatures for a micrometer-size crystal of CrNb₃S₆. (a) Scanning ion micrograph of the crystal with electrodes. (b) MR curves obtained with increasing (red) and decreasing (blue) field at 10 K. The linear dependence above \(H_{\text{sat}}\) is subtracted as the MR background from the original MR data. Figures adapted from Ref. 27. © 2015 American Physical Society.

Hysteresis of the interlayer MR was clearly observed at low temperatures in the micrometer-size CrNb₃S₆ crystal, as shown in Fig. 23(b). The interlayer MR takes different paths with increasing (red) and decreasing (blue) field. With increasing field, the MR gradually decreases toward a minimum MR value at a saturation field \(H_{\text{sat}}\) of 2860 Oe, during which a few large steps appear at 1810, 1965, and 2165 Oe, followed by a cascade of small steps with a quasi-linear regime. Upon decreasing the field, the MR remains at the same value until just below 1965 Oe, where a distinct sudden jump occurs (\(H_{\text{jump}}\)). The MR eventually returns to the initial value at zero field as the field is reduced further.

The values of \(H_{\text{sat}}\) and \(H_{\text{jump}}\) are influenced by the demagnetization effect. Figure 24(a) presents interlayer MR curves for various angles between the applied field and platelet crystal with the field kept normal to the helical c-axis. Both \(H_{\text{sat}}\) and \(H_{\text{jump}}\) can be observed in all the MR curves. In the configuration where the applied field is perpendicular to both the plane and the c-axis, at which the angle is defined as 0°, \(H_{\text{sat}}\) and \(H_{\text{jump}}\) are large, for example, 2980 and 1900 Oe, respectively. With increasing angle, the MR curve shifts toward a lower field. When the magnetic field is applied along the plane and perpendicular to the c-axis (corresponding to the 90° configuration), \(H_{\text{sat}}\) and \(H_{\text{jump}}\) are reduced to about 1780 and 720 Oe, respectively. In the latter case, the value of \(H_{\text{sat}}\) is similar to that obtained in the configuration giving the smallest demagnetization field, for example, as reported for a bulk crystal.²⁵⁾ The demagnetization effect is maximum and minimum in the 0 and 90° configurations, respectively, and changes the range of fields where the MR associated with CSL formation appears, as summarized in Fig. 24(b).

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Figure 24. (Color online) Interlayer MR of the micrometer-size CrNb₃S₆ crystal for various angles between the applied field and the specimen plane. The field is kept perpendicular to the helical c-axis. The specimen dimensions are \(13\times 13\) µm² in the plane and 500 nm thick (13 µm width along the c-axis). The linear dependence above \(H_{\text{sat}}\) is subtracted as the MR background in each MR curve. \(H_{\text{sat}}\) and \(H_{\text{jump}}\) are indicated by solid and open arrows, respectively. Figures adapted from Ref. 27. © 2015 American Physical Society.

Many fine steps in the interlayer MR are found at low and high fields both with increasing and decreasing field as shown in Fig. 25, which shows a magnification of the data presented in Fig. 23(b). The steps have many different plateau widths and the magnitude of the jumps varies between 16 and 2060 µΩ. Seven steps, counted from zero field along branches representing the increasing and decreasing field, are found at almost the same MR value irrespective of the swept direction of the field. Steps also appear at higher fields close to \(H_{\text{sat}}\) along the branch representing the increasing field. The plateaus become wide near \(H_{\text{sat}}\) and the magnitudes of the jumps are integer multiples of \(26\pm 7\) µΩ for the present crystal shown in Fig. 23(a). At a low field, many steps have a jump of almost the same value or half the value.

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Figure 25. (Color online) Interlayer MR data at low and high fields with experimental error bars, obtained from the crystal shown in Fig. 23. The linear dependence above \(H_{\text{sat}}\) is subtracted as the MR background. Figures adapted from Ref. 27. © 2015 American Physical Society.

Figure 26 shows the interlayer MR data for a different CrNb₃S₆ crystal of 25 µm width along the c-axis. All the features observed are consistent with the results obtained for the crystal of 10 µm width along the c-axis presented in Fig. 23. Hysteresis and steps with many different plateau widths appear during CSL formation. The magnitudes of the jumps are integer multiples of \(19\pm 6\) µΩ at high fields close to \(H_{\text{sat}}\). The hysteresis is slightly smaller than that found in the crystal of 10 µm width. The resistance is small, which is partly due to the difference in the crystal dimensions and the short distance between the electrodes used for voltage detection. \(H_{\text{sat}}\) and \(H_{\text{jump}}\) are 3330 and 2058 Oe in the case of a decreasing field, respectively.

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Figure 26. (Color online) Interlayer MR for a different micrometer-size crystal of CrNb₃S₆. The dimensions of the crystal are \(25\times 12\) µm² in the plane and 900 nm thick. Figures adapted from Ref. 27. © 2015 American Physical Society.

A part of the MR curves obtained for the micrometer-size crystals lies on a single curve, as indicated by a dotted line in Fig. 26(b). This curve follows the behavior of the soliton density curve \(L(0)/L(H)\), which is qualitatively in agreement with that observed for the bulk crystal as described above.

Theoretical works¹²⁹⁾ predict that soliton confinement in a finite-size system induces discrete changes in the soliton period and density along the order parameter of the soliton density curve, which supports the experimental MR data of micrometer-size crystals. The deviation of the MR curve from the ideal soliton density curve is significant at low fields with increasing field and at high fields with decreasing field because of the hysteresis induced in micrometer-size crystals.

Minor hysteresis loops of the interlayer MR are examined for the micrometer-size CrNb₃S₆ crystal. Two kinds of minor MR loops were measured. The first type of minor loop was measured by the following procedure. The initial state was prepared by increasing the magnetic field from zero field to above \(H_{\text{sat}}\) and reducing the field to \(H_{\text{jump}}\). Then, the minor loop measurement was performed by cycling the field between \(H_{\text{jump}}\) and a field below \(H_{\text{sat}}\) with various field widths as shown in Figs. 27(a) and 27(b). The second type of minor loop was measured by a field procedure in which after every field cycle, the initialization was performed by reducing the field to zero, increasing the field to above \(H_{\text{sat}}\), and decreasing the field again to \(H_{\text{jump}}\) as shown in Figs. 27(c) and 27(d).

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Figure 27. (Color online) Minor hysteresis loop of the interlayer MR in the micrometer-size CrNb₃S₆ crystal. The measured crystal is the same as that in Fig. 23. The MR data above \(H_{\text{jump}}\) are shown in (a) and (c), while the entire MR curves are presented in (b) and (d). Figures adapted from Ref. 27. © 2015 American Physical Society.

The minor MR loops indicate that any point in the hysteresis region can be obtained by the procedure of changing the field, indicating that the number of solitons is tunable in the system. A part of the minor MR loops lies on a single curve as indicated by a dotted line in Figs. 27(b) and 27(d), which qualitatively follows the behavior of the soliton density curve \(L(0)/L(H)\). Similar behavior is observed in Fig. 26. Also, as discussed above, this tendency is in accord with the data for the bulk crystal²⁵⁾ and theoretical predictions.^129,130) The MR curves even for micrometer-size crystals are influenced by the order parameter of the soliton density curve, along which discrete stepwise changes in MR take place due to soliton confinement.

The interlayer MR curves for temperatures between 10 and 160 K are shown in Fig. 28. Hysteresis appear in the interlayer MR for a wide temperature range below \(T_{\text{C}}\) and becomes broader with decreasing temperature. The hysteresis disappears entirely for fields larger than \(H_{\text{sat}}\) as well as when the temperature increases very close to \(T_{\text{C}}\) (see the MR curve at 120 K). The hysteretic behavior observed in the 10-µm-size CrNb₃S₆ crystal contrasts with that found in the bulk crystal of 200 µm thickness along the c-axis, in which hysteresis is hardly found over the entire range of temperatures as shown above.²⁵⁾

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Figure 28. (Color online) Dependence of interlayer MR on temperature between 10 and 160 K. Figures adapted from Ref. 27. © 2015 American Physical Society.

The interlayer MR varies smoothly for the bulk CrNb₃S₆ crystal in accord with the calculated soliton density,²⁵⁾ while a part of the interlayer MR curves in the micrometer-size crystal exhibits a similar tendency with dozens of fine MR steps.²⁷⁾ The soliton density plays a role of the order parameter for the continuous IC–C phase transition^{14,18,21,124,131,132)} associated with CSL formation. The observations support the scenario that each soliton creates a spin dependent scattering potential for itinerant spins. At low temperatures, where the scattering path lengths are long, the interlayer MR involves scattering from all the solitons and therefore is used to measure the soliton density. To realize these phenomena, the perfect registry of solitons across the CSL is required. Indeed, at temperatures below 110 K a fine MR structure appears in the form of a sequence of steps along a portion of the branches obtained with increasing and decreasing field in Fig. 28, suggesting that discrete changes in the soliton density are responsible for the stepwise (discretized) changes in MR.

Lorentz TEM contributes to understanding the microscopic nature of the steps and plateaus observed in the interlayer MR curves. Evidence of multiple states of the CSL is observed directly within a grain having reversed crystalline chirality. This is a very interesting example of exchange pinning at a boundary where the DM interaction changes sign.

Analyses of high-resolution DPC micrographs^69,71) reveal the spin configuration in a region containing such a grain, as shown in Figs. 29(a)–29(c). A schematic illustration of the obtained spin configuration is given in Fig. 29(d). A magnetic field applied perpendicular to the plane causes magnetic moments to be oriented in the plane near the grain boundary, i.e., perpendicular to the applied field direction. Two \(\pi/2\) twists connect these spins with spins on either side of the boundary. The spin configuration of the \(\pi/2\) twist from the boundary has the appearance of the surface twist structure at free boundaries found in MnSi thin films.^130,133) The magnetic moments on both sides of the grain have opposite directions.

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Figure 29. (Color online) Lorentz DPC micrographs of the CSL in a narrow crystal grain of right-handed magnetic chirality sandwiched by wide regions of left-handed magnetic chirality in a magnetic field of 1450 Oe at 100 K. The grain boundaries are indicated by arrows in (a). The contrast and direction of the magnetic moment are schematically presented in (b) and (c). (d) Schematic of the spin configuration of the CSL around boundaries where the magnetic chirality switches. Solitons can be confined between the boundaries. Figures adapted from Ref. 27. © 2015 American Physical Society.

The Lorentz Fresnel micrographs in Fig. 30 clarify that the spin configuration at the boundaries remains the same even at fields above \(H_{\text{sat}}\), suggesting that the net direction of in-plane magnetic moments is fixed around the boundary throughout the field cycle. The existence of another such boundary some distance away creates a confining well for solitons trapped between the two interfaces. Indeed, stepwise changes in the soliton period in this confining well are observed upon changing the strength of the applied magnetic field, part of which are presented in Fig. 31.

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Figure 30. (Color online) Lorentz Fresnel micrograph, taken under the underfocused condition, at 100 K in a magnetic field of 2270 Oe, above \(H_{\text{sat}}\). The field of view is almost the same as that in Fig. 31. The positions of boundaries between grains with different chirality are indicated by red arrows. The contrast and direction of magnetization are schematically given in (a). (b) and (c) Line profiles of the contrast intensity integrated over the dashed squares in (a). Figures adapted from Ref. 27. © 2015 American Physical Society.

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Figure 31. (Color online) Lorentz Fresnel micrographs of the CSL, taken under the underfocused condition, around a 1-µm-width crystal grain of right-handed magnetic chirality in a magnetic field of 1560 Oe (a), 1771 Oe (b), and 1781 Oe (c) at 100 K. The chiral boundaries and confined right-handed solitons are indicated by red arrows and blue arrowheads, respectively. The number of solitons confined in the right-handed grain is given in (a) to (c). The initial number of confined solitons is 20. Figures adapted from Ref. 27. © 2015 American Physical Society.

Figure 32 provides solid evidence for confinement pinning. Namely, the direct observation of plateaus and stepwise jumps of the soliton period and density is presented for field-strength changes of 1 Oe. In the Lorentz Fresnel microscopy observations, the spacing between contrast lines within the grain remains constant within the experimental resolution independent of the magnetic field within a single plateau. Therefore, it is concluded that the solitons in the region with right-handed magnetic chirality are indeed confined.

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Figure 32. (Color online) Soliton period and corresponding soliton density as a function of field, which are measured in a series of Lorentz Fresnel images monitored with steps of 1 Oe during cycling of the magnetic field up to \(H_{\text{sat}}\). The initial number of confined solitons is 20. The number of confined solitons decreases one by one at particular field strengths with increasing field. Figures adapted from Ref. 27. © 2015 American Physical Society.

The original prediction of soliton confinement was made for a system with a very limited number of solitons,^129,130,133) in which the solitons were pinned at both edges of a finite-size specimen or thin film. The length of plateau in soliton numbers will increase as the confinement region and the number of solitons involved become smaller, which enables the detection of MR or magnetization anomalies in a MnSi thin film associated with the nucleation or annihilation processes of “two” soliton twists.¹³⁰⁾

Here, for crystals with macroscopic length scales, namely, in which tens or hundreds solitons are involved in the confinement, stepwise changes due to single-soliton events are distinguished during the IC–C transition to the forced FM state. Indeed, 20 solitons are confined in the case of the chirality grain shown in Fig. 31 and about 200 solitons are involved for the MR data in Figs. 23 and 25. These findings are consistent with microscopic observations via Lorentz micrographs and small-angle electron scattering data showing that solitons strongly correlate with each other over ten micrometers.²¹⁾

In an infinite soliton lattice, the period outside the confined region should change continuously. Outside a grain, the spacing indeed varies continuously with the field strength, such as for the case when 20 or 16 solitons are found in the right-handed grain. However, a sudden increase in the slope is also frequently observed outwith the grain, e.g., as seen at around 1300 and 1900 Oe in Fig. 33. This is consistent with the fact that the soliton lattice is not infinite outside the grain, and is confined over a distance on the order of the macroscopic size. The confinement pinning occurring outwith the grain, namely, between the chiral well and a free edge of the crystal, is over a length scale of 10 and 7 µm.

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Figure 33. (Color online) Soliton period as a function of applied magnetic field in three regions. The inset in (a) shows the sample dimensions. Data taken from inside the right-handed grain are given in (red) solid squares, while (magenta) open circles and (dark blue) solid circles are used to exhibit data taken from inside the left-handed grains on the left and right, respectively. Figures adapted from Ref. 27. © 2015 American Physical Society.

The length scales over which distinct plateaus are observed are striking. This requires a high degree of coherence of the CSL in order that the CSL at the separate edges works cooperatively to allow the escape or injection of a single soliton. The pinning mechanism in operation for the confinement in the 1 µm grain probably differs from the pinning that must occur at free edges, as observed in the 10 µm samples, but nevertheless confinement occurs in each situation. Single soliton changes are observed even in finite-size MR samples at high fields where the soliton density is smallest. This is consistent with single soliton decrease in the chirality grain throughout the increasing field branch, thereby reflecting the long-range coherence of the CSL. However, the spacing of the CSL becomes slightly irregular at high fields close to \(H_{\text{sat}}\), although solitons are very straight and perpendicular to the c-axis. The coherence of the CSL may begin to deteriorate at smaller soliton densities, partly because of random pinning and other factors that may be not negligible.

Soliton confinement and consequent discretization effects occur, depending on the system size. Figure 34 shows how the CSL changes the number of solitons in the chirality grains with widths of about 500 nm, 1 µm, and 1.5 µm, which initially contains 10, 20, and 32 confined solitons, respectively. With increasing the number of initial solitons in the chirality grains, the curve of the normalized soliton number shifts towards a lower field and approaches the ideal soliton density curve given by the dashed curve in Fig. 34(b). Similar behavior is found in micrometer-size crystals, as presented in Figs. 26 and 27, and bulk crystals, as shown in Fig. 18, indicating that TEM and MR data have the same origin of the soliton confinement and discretization effects. Figure 35 summarizes how discretization or quantization effects become prominent depending on the initial soliton number.

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Figure 34. (Color online) Normalized soliton number as a function of normalized field in TEM specimens with chirality grains of different widths with increasing field. Square (red), triangle (purple), and circle (green) dots represent data for 1.5-µm-wide, 1-µm-wide, and 500-nm-wide chirality grains, which initially contain 32, 20, and 10 confined solitons, respectively. Figures adapted from Ref. 27. © 2015 American Physical Society.

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Figure 35. (Color online) System size dependence of MR and soliton density. With reducing initial number of solitons in the system, discretization (quantization) effects become prominent.

From a technological viewpoint, the confinement discretizes the soliton density, thereby enabling the use of solitons as individual and countable objects. More practically, counting the number of solitons via electric measurements can be accomplished with relative ease, which allows us to exploit the discreteness of the soliton density for data storage applications.

The possible microscopic origins of the DM interaction are briefly summarized in this subsection. Chiral magnetic crystals are classified into three classes: magnetic insulators, metals where localized and itinerant spins coexist, and metals with only itinerant spins.

The first class corresponds to the case originally discussed by Moriya.¹⁷⁾ The Hamiltonian which describes two magnetic ions is \begin{equation} \mathcal{H}^{\prime} = \lambda \boldsymbol{{S}}_{1} \cdot \boldsymbol{{L}}_{1} + \lambda \boldsymbol{{S}}_{2} \cdot \boldsymbol{{L}}_{2} - J \boldsymbol{{S}}_{1} \cdot \boldsymbol{{S}}_{2}, \end{equation} (16) where λ and J are the strengths of spin–orbit and ferromagnetic couplings, respectively. The DM vector is obtained via second-order perturbation theory as \begin{equation} \boldsymbol{{D}}= - i \lambda J \left(\sum_{n_{1}} \frac{\langle g_{1} | \boldsymbol{{L}}_{1} | n_{1}\rangle}{E_{n_{1}}-E_{g_{1}}} - \sum_{n_{2}} \frac{\langle g_{2} | \boldsymbol{{L}}_{2}| n_{2}\rangle}{E_{n_{2}}-E_{g_{2}}}\right), \end{equation} (17) where g and n label the ground and excited states, respectively. CsCuCl₃ and Cu₂OSeO₃ are examples of compounds belonging to the first class.

In the second case, the particle–hole fluctuations of the itinerant electrons mediate the DM interaction between the localized spins. This corresponds to a generalized Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction. In this case, the crystal symmetry is embedded in the complex one-particle hopping, and the resultant DM interaction should appropriately reflect the crystal symmetry.¹³⁴⁾ CrNb₃S₆ and YbNi₃Al₉ belong to this category.

The third case is the most nontrivial.¹³⁵⁾ It is expected that after integrating out the one-particle degrees of freedom with spin–orbit coupling being treated as a perturbation, a coupling of the spin fluctuations eventually has the effective form of the DM interaction.¹³⁶⁾ B20 compounds such as MnSi, FeGe, and Fe\(_{1-x}\)Co_xSi are categorized into this class.

The CHM structure is an incommensurate magnetic structure with a single propagation vector. The magnetic structure is specified by the symmetry-adapted 2-D basis \((\hat{\boldsymbol{{e}}}_{1},\hat{\boldsymbol{{e}}}_{2})\) and the propagation vector \(\boldsymbol{{k}}=(0,0,k)\). The corresponding magnetization vector is represented by \begin{align} \boldsymbol{{M}}&= M\hat{\boldsymbol{{e}}}_{1}\cos (kz)\pm M\hat{\boldsymbol{{e}}}_{2}\sin (kz) \notag\\ &= M\text{Re} \{(\hat{\boldsymbol{{e}}}_{1}\mp i\hat{\boldsymbol{{e}}}_{2})e^{ikz}\}, \end{align} (18) where + and − signs respectively correspond to left- and right-handed helices.

The chiral space group \(\mathcal{G}_{\chi}\) consists of the elements \(\{g_{i}\}\), where all the elements \(g_{i}\) are represented by a proper orthogonal matrix, i.e., \(\det g=1\). Among the 230 space groups, there are 65 such chiral space groups. Among the \(g_{i}\), some elements leave the propagation vector \(\boldsymbol{{k}}=(0,0,k)\) invariant. These elements form the little group \(\mathcal{G}_{\boldsymbol{{k}}}\). The magnetic group \(\mathcal{G}_{\text{mag}}\) is written as \(\mathcal{G}_{\text{mag}}=\mathcal{G}_{\text{perm}}\otimes\mathcal{G}_{\text{axial}}\), where \(\mathcal{G}_{\text{perm}}\) and \(\mathcal{G}_{\text{axial}}\) represent the Wyckoff permutation group and the axial vector group, respectively.¹³⁷⁾ The corresponding magnetic representation \(\Gamma_{\text{mag}}\) is written as \(\Gamma_{\text{mag}}=\Gamma_{\text{perm}}\otimes\Gamma_{\text{axial}}\), where \(\Gamma_{\text{perm}}\) and \(\Gamma_{\text{axial}}\) are representations of \(\mathcal{G}_{\text{perm}}\) and \(\mathcal{G}_{\text{axial}}\), respectively. Then, \(\Gamma_{\text{mag}}\) is subduced into the irreducible representations of \(\mathcal{G}_{\boldsymbol{{k}}}\), \(\Gamma_{\text{mag}}=\sum_{i}n_{i}\Gamma_{i}^{\boldsymbol{{k}}}\), where \(\Gamma_{i}^{\boldsymbol{{k}}}\) is the irreducible representation of \(\mathcal{G}_{\boldsymbol{{k}}}\).

For the real 2-D (or complex 1-D) symmetry-adapted basis \(\hat{\boldsymbol{{e}}}_{1}\) and \(\hat{\boldsymbol{{e}}}_{2}\) to exist, it is required that the group elements of \(\mathcal{G}_{\boldsymbol{{k}}}\) include threefold (\(C_{3}\)), fourfold (\(C_{4}\)), or sixfold (\(C_{6}\)) rotations. Correspondingly, trigonal, tetragonal, hexagonal and cubic crystal classes contain such a rotation axis and are eligible to accommodate the CHM structure. This situation is depicted in Fig. 36. The cubic class is special because there are four \(C_{3}\) axes, although hexagonal, tetragonal, and trigonal crystals have only one principal axis. In the latter case, a monoaxial helimagnetic structure is expected to be favored.

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Figure 36. (Color online) For a CHM ordering to be realized, the crystal point group needs to have 2-D (or complex 1-D) irreducible representations. This means the point group elements must have a threefold (\(C_{3}\)), fourfold (\(C_{4}\)), or sixfold (\(C_{6}\)) axis. Correspondingly, (a) cubic, (b) hexagonal, (c) tetragonal, and (d) trigonal crystal classes are eligible to accommodate the CHM structure. Helices and arrows indicate how the helical axis or axes can reside in the crystal.

A necessary condition for the monoaxial DM vector to exist is that a magnetic crystal belongs to a chiral space group \(G_{\chi}\) whose symmetry elements only contain pure rotations, i.e., \(\forall g\in G_{\chi}\), \(\det g =1\). In the context of Landau's theory of phase transitions, an incommensurate helimagnetic structure is stabilized by an invariant term, the so-called Lifshitz invariant (LI), which is linear with respect to the gradient of slowly varying magnetization and has the general form¹³⁸⁾ \begin{equation} \mathcal{L}_{\alpha \beta}^{\gamma} = M_{\alpha}\frac{\partial M_{\beta}}{\partial x_{\gamma}}-M_{\beta}\frac{\partial M_{\alpha}}{\partial x_{\gamma}}, \end{equation} (19) where \(\alpha,\beta,\gamma =x,y,z\) represent spatial coordinates. The LI is a manifestation of the microscopic DM interaction in the Ginzburg–Landau free energy functional. The LI is allowed in the case of the trigonal \(C_{3}(3)\), \(C_{3v}(3m)\), \(D_{3}(32)\), tetragonal \(C_{4}(4)\), \(C_{4v}(4mm)\), \(D_{4}(422)\), \(D_{2d}(\bar{4}2m)\), \(S_{4}(\bar{4})\), hexagonal \(C_{6}(6)\), \(C_{6v}(6mm)\), \(D_{6}(622)\), and cubic \(T(23)\), \(O(432)\) point groups, where we give the Hermann–Mauguin notation in the parenthesis.

The most prominent feature of the LI is its ability to stabilize discommensuration in a background incommensurate helical structure. The LI-induced discommensuration has a topological texture protected by crystal chirality. In the pioneering work by Dzyaloshinskii,^14–16) it was shown that in the case of \(C_{n}\) and \(D_{n}\) (\(n=3\), 4, 6), \(\mathcal{L}_{xy}^{z}\) stabilizes the monoaxial CSL, which is also called a helicoid or kink crystal. On the other hand, as pointed out by Bogdanov and Yablonskii,¹³⁹⁾ a baby skyrmion is stabilized by a planar LI. Possible point groups and resultant LIs are listed in Table III (for a detailed discussion on LIs and consequent magnetic textures, see Refs. 140 and 141). From the symmetry viewpoint, the CSL and baby skyrmion are respectively axial and planar topological discommensurations stabilized by the LI.

»View table Table III. Point groups which admit the planar and axial Lifshitz invariants, where \(n=3,4,6\). \(c_{1}\)–\(c_{4}\) are arbitrary coefficients.

As an example, we consider the point group \(D_{6}(622)\), which CrNb₃S₆ belongs to. First we note that \(M_{x}=M_{x}^{\prime}+M_{y}^{\prime}/\sqrt{3}\) and \(M_{y}=\frac{2}{\sqrt{3}}M_{y}^{\prime}\), where \(M_{x}\) and \(M_{y}\) are the components in the hexagonal basis and \(M_{x}^{\prime}\) and \(M_{y}^{\prime}\) are the components in the orthogonal basis, as shown in Fig. 37. Then we see that \(\mathcal{L}_{x^{\prime}y^{\prime}}^{z}=(2/\sqrt{3})\mathcal{L}_{xy}^{z}\). Therefore, the symmetry operation in the hexagonal basis gives an additional factor of \(2/\sqrt{3}\) in the orthogonal basis. The symmetry elements of the \(D_{6}\) point group are listed as¹⁴²⁾ (1) 1; (2) \(3^{+}\) \(0,0,z\); (3) \(3^{-}\) \(0,0,z\); (4) 2 \(0,0,z\); (5) \(6^{-}\) \(0,0,z\); (6) \(6^{+}\) \(0,0,z\); (7) 2 \(x,x,0\); (8) 2 \(x,0,0\); (9) 2 \(0,y,0\); (10) 2 \(x,\bar{x},0\); (11) 2 \(x,2x,0\); (12) 2 \(2x,x,0\) in the hexagonal basis and each symmetry operation transforms \(x,y,z\) to (1) \(x,y,z\); (2) \(\bar{y},x-y,z\); (3) \(\bar{x}+y,\bar{x},z\); (4) \(\bar{x},\bar{y},z\); (5) \(y,\bar{x}+y,z\); (6) \(x-y,x,z\); (7) \(y,x,\bar{z}\); (8) \(x-y,\bar{y},\bar{z}\); (9) \(\bar{x},\bar{x}+y,\bar{z}\); (10) \(\bar{y},\bar{x},\bar{z}\); (11) \(\bar{x}+y,y,\bar{z}\); (12) \(x,x-y,\bar{z}\), respectively. For example, operation (2) transforms \(\mathcal{L}_{xy}^{z}\) as \(\mathcal{L}_{xy}^{z}\rightarrow (-M_{y})\partial (M_{x}-M_{y})/\partial z-(M_{x}-M_{y})\partial (-M_{y})/\partial z=\mathcal{L}_{xy}^{z}\), and (10) transforms \(\mathcal{L}_{xy}^{z}\) as \(\mathcal{L}_{xy}^{z}\rightarrow (-M_{y})\partial (-M_{x})/\partial (-z) -(-M_{x})\partial (-M_{y})/\partial (-z) =\mathcal{L}_{xy}^{z}\). We can easily verify that all the operations of the \(D_{6}\) point group leave \(\mathcal{L}_{xy}^{z}\) invariant.

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Figure 37. Stereographic projections to illustrate the structure of the point group \(D_{6}(622)\), where twofold rotation axes corresponding to the operations (7)–(12) are indicated. Hexagonal coordinate axes (x and y) and the orthogonal coordinate axes (\(x^{\prime}\) and \(y^{\prime}\)) are indicated.

From now on, as a canonical example of a monoaxial chiral helimagnet, we consider the case of CrNb₃S₆. The most prominent feature of this compound is its classical 1-D nature as a magnetic system, i.e., 2-D layered magnetic structures are weakly coupled via the interlayer exchange and DM interactions. Then, the system is well described as a classical 1-D chiral helimagnet.

Recently, Shinozaki et al.¹⁴³⁾ have applied a hybrid method called the 2dMC-1dMF method to take account of the spin correlation effect in the layer and they estimated the transition temperature. Consequently, the three interaction parameters in CrNb₃S₆ were estimated as the interlayer DM interaction strength \(D_{z}=2.9\) K, the interlayer FM exchange interaction strength \(J_{z}=18\) K, and intralayer exchange interaction strength \(J_{xy}=140\) K. The strong intralayer coupling should be responsible for causing the high transition temperature (\(T_{\text{c}}=127\) K) of CrNb₃S₆. It should be noted that the phase transitions and ordering structures of a model of a chiral helimagnet in three dimensions have intensively been studied by equilibrium Monte Carlo simulations for large systems of up to about \(10^{6}\) spins.¹⁴⁴⁾

In the following discussion, we consider only the ground state at zero temperature. Taking the limit as \(J_{xy}/J_{z}\rightarrow\infty\), we can omit the site dependence of the spin variables inside the layer and drop the \(J_{xy}\) term from the Hamiltonian. Then, a layered monoaxial chiral helimagnet is described by the effective 1-D Hamiltonian¹²⁵⁾ \begin{align} \mathcal{H} &= - J_{z}\sum_{i}\boldsymbol{{S}}_{i}\cdot \boldsymbol{{S}}_{i+1}-\boldsymbol{{D}}_{z}\cdot \sum_{i}\boldsymbol{{S}}_{i}\times \boldsymbol{{S}}_{i+1}\notag\\ &\quad +K_{\bot}\sum_{i}S_{z,i}^{2}+\tilde{\boldsymbol{{H}}}\cdot \sum_{i} \boldsymbol{{S}}_{i}, \end{align} (20) where \(\boldsymbol{{S}}_{i}\) is the local spin moment at site i, \(J>0\) is the nearest-neighbor FM exchange interaction, and \(\boldsymbol{{D}}_{z}=D_{z}\hat{\boldsymbol{{e}}}_{z}\) is the monoaxial DM interaction along a certain crystallographic helical axis (z-axis). We take the z-axis as the mono-axis and apply a static magnetic field \(\tilde{\boldsymbol{{H}}}=\tilde{H}_{0}^{x}\hat{\boldsymbol{{e}}}_{x}=g\mu_{\text{B}}H_{0}^{x}\hat{\boldsymbol{{e}}}_{x}\) to stabilize the CSL state, where g is the electron g-factor and \(\mu_{\text{B}}=| e|\hbar/2m\) is the Bohr magneton. We also include the easy-plane-type single-ion anisotropy, \(K_{\bot}>0\). Here we use the notation for the basis \(\{\hat{\boldsymbol{{e}}}_{x},\hat{\boldsymbol{{e}}}_{y},\hat{\boldsymbol{{e}}}_{z}\}\), which spans the orthogonal coordinate system (not the hexagonal coordinate system as discussed in the previous subsection).

Furthermore, the spatial modulation of the magnetic structure, \(L(0)\), is much larger than the atomic lattice constant, \(a_{0}\) (\(L_{0}/a_{0}\sim 40\) for CrNb₃S₆). This situation makes it legitimate to introduce the continuous classical spin variable along the helical axis, \(\boldsymbol{{S}}(z) =a_{0}\sum_{i}\boldsymbol{{S}}_{i}\delta (z-z_{i})=S\boldsymbol{{n}}(z)\). By taking the continuum limit \(\sum_{j}\rightarrow a_{0}^{-1}\int_{0}^{L}dz\), we obtain a continuum version of the Hamiltonian \(\mathcal{H} = a_{0}^{-1}\int_{0}^{L}\tilde{\mathcal{H}}\,dz\), where the Hamiltonian per unit length is \begin{align} \tilde{\mathcal{H}} &= \frac{J_{z}S^{2}a_{0}^{2}}{2}(\partial_{z} \boldsymbol{{n}})^{2}-a_{0}S^{2}\boldsymbol{{D}}_{z}\cdot \boldsymbol{{n}}\times \partial_{z}\boldsymbol{{n}}\notag\\ &\quad+K_{\bot}n_{z}^{2}+S\tilde{\boldsymbol{{H}}} \cdot \boldsymbol{{n}}\,dz, \end{align} (21) which is called the chiral sine-Gordon model. Using the spherical polar coordinates, \(\hat{\boldsymbol{{n}}}(z)=[\sin\theta(z)\cos\varphi (z),\sin\theta (z)\sin\varphi (z),\cos\theta(z)]\), the Hamiltonian is rewritten as \begin{align} \tilde{\mathcal{H}} &= \frac{J_{z}S^{2}a_{0}^{2}}{2}[(\partial_{z}\theta)^{2}+\sin^{2}\theta (\partial_{z}\varphi)^{2}]\notag\\ &\quad - a_{0}S^{2}D_{z}\sin^{2}\theta (\partial_{z}\varphi) +K_{\bot}S^{2}\cos^{2}\theta +S\tilde{\boldsymbol{{H}}}\cdot \boldsymbol{{n}}. \end{align} (22) The monoaxial DM interaction plays the role of easy-plane anisotropy and consequently all the spins are confined in the xy-plane. Therefore, the ground state property is described by the so-called chiral sine-Gordon Hamiltonian [Eq. (21)] with \(\theta (z)\) being fixed to \(\pi/2\), i.e., \begin{equation} \tilde{\mathcal{H}} = J_{z}S^{2}a_{0}^{2}\left[\frac{1}{2}(\partial_{z}\varphi)^{2}-q_{0}\partial_{z}\varphi+m^{2}\cos \varphi\right], \end{equation} (23) where the wave number of the zero-field helimagnetic structure is given by \begin{equation} q_{0} \equiv a_{0}^{-1}D_{z}/J_{z} \end{equation} (24) and \begin{equation} m^{2} \equiv \tilde{H}^{x}/(J_{z}Sa_{0}^{2}) \end{equation} (25) is a measure of nonlinear coupling and plays the role of the first breather mass in the context of sine-Gordon field theory.

The ground state of the model in Eq. (23) was first given by Dzyaloshinskii¹⁶⁾ in the magnetic system, followed by de Gennes¹³¹⁾ in the context of the magnetic-field-induced transition in a cholesteric liquid crystal. Variational analysis gives the stationary configuration of the φ field, which is given by \begin{equation} \sin (\varphi_{0}/2) = \mathop{\mathrm{sn}}\bar{z} \end{equation} (26) or, equivalently, \(\varphi_{0}(z)=2\mathop{\mathrm{am}}\bar{z}\), where \(\bar{z}=(m/\kappa)z\), sn is the Jacobi dn function, and am is the Jacobi's amplitude function with the elliptic modulus κ (\(0\leq\kappa <1\)).¹⁰⁸⁾ The CSL structure is explicitly given by \(\boldsymbol{{S}}_{0}(z) =S\boldsymbol{{n}}_{0}(z)\), where \begin{align} \boldsymbol{{n}}_{0}(z) &= (\cos \varphi_{0}(z), \sin \varphi_{0}(z), 0)\notag\\ &= \left(1-\frac{2}{\kappa^{2}}+\frac{2}{\kappa^{2}}\mathop{\mathrm{dn}}\nolimits^{2}\bar{z}, 2\mathop{\mathrm{sn}}\bar{z}\mathop{\mathrm{cn}}\bar{z}, 0\right). \end{align} (27)

In Fig. 38(a), we show the spatial distribution of \(\varphi_{0}(z)\). The solution (26) has the spatial period \begin{equation} L_{\text{CSL}} = 2\kappa K/m, \end{equation} (28) where \(K=K(\kappa)\) is the complete elliptic integral of the first kind. It is useful to see the asymptotic behavior of \(L_{\text{CSL}}\) near the IC–C transition (\(\kappa\rightarrow 1\)). Using the asymptotic form \(K(\kappa)\sim \log (4/\sqrt{1-\kappa^{2}})\), as κ approaches unity from below, we have \(L_{\text{CSL}}\rightarrow (2\kappa/m)\log (4/\sqrt{1-\kappa^{2}})\sim -m^{-1}\log (1-\kappa)\), i.e., \begin{equation} \kappa \sim 1-\exp (-mL_{\text{CSL}}). \end{equation} (29)

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Figure 38. (Color online) (a) Spatial distributions of the phase \(\varphi (z)\) in the CSL state and (b) corresponding topological charges. (c) Field dependences of the spatial period of the CSL and (d) soliton density.

Using \(L_{\text{CSL}}\), the dimensionless coordinate \(\bar{z}\) is related to z as \(\bar{z}=q_{\text{CSL}}z\), where \begin{equation} q_{\text{CSL}} = m/\kappa = 2K/L_{\text{CSL}} \end{equation} (30) has a physical meaning as the wave number of the CSL structure. As we will see shortly, the value of κ is determined by the field strength and therefore the period is a function of \(H^{x}\). The zero-field period is \(L_{0}=2\pi/q_{0}\). To visualize the lattice structure of the spin modulation, it is useful to introduce the topological charge \begin{equation} \mathcal{Q}(z) = \frac{1}{2\pi}\partial_{z}\varphi_{0}(z) = \frac{q_{\text{CSL}}}{\pi}\mathop{\mathrm{dn}}\bar{z}. \end{equation} (31) In Fig. 38(b), we show the partial distribution of \(Q(z)\). This figure is clearly reminiscent of an array of solitons. This is the reason why we call this state a chiral soliton lattice (CSL). In the CSL state, the chiral symmetry in the spin space is forced to be broken by the crystal symmetry. In an infinite system, without loss of generality, we can set \(\varphi_{0}(0)=0\) and \(\varphi_{0}(L)=2\pi n\), where L denotes the system length. The topological number n specifies the total number of solitons accommodated in the system and is determined by the winding number of the homotopy group \(\pi_{1}(S_{1})\).

In this state, the elliptic modulus κ is simply a constant of integral. Its value is determined by minimizing the energy with respect to κ. By noting that \(\partial_{z}\varphi_{0}=2q_{\text{CSL}}\mathop{\mathrm{dn}}\bar{z}\) and \(m^{2}\cos\varphi_{0}=2q_{\text{CSL}}^{2}\mathop{\mathrm{dn}}^{2}\bar{z}-2q_{\text{CSL}}^{2}+m^{2}\), each term in Eq. (23) is analyzed as \begin{align*} L^{-1}\int_{0}^{L}\frac{1}{2}(\partial_{z}\varphi_{0})^{2}\,dz &= 2q_{\text{CSL}}^{2}L^{-1}\int_{0}^{L}\mathop{\mathrm{dn}}\nolimits^{2}\bar{z}\,dz = 2q_{\text{CSL}}^{2}E/K,\\ L^{-1}\int_{0}^{L}q_{0}\partial_{z}\varphi_{0}\,dz &= 2\pi q_{0}/L_{\text{CSL}} = \pi q_{0}m/\kappa K,\\ L^{-1}\int_{0}^{L}m^{2}\cos \varphi_{0}\,dz &= 2q_{\text{CSL}}^{2}E/K-2q_{\text{CSL}}^{2}+m^{2} \end{align*} to give the energy per soliton. In the second line, we used the relation \(\varphi_{0}(L) -\varphi_{0}(0) =2\pi L/L_{\text{CSL}}\), where \(n=L/L_{\text{CSL}}\) is the total number of solitons. We also used the definition of the elliptic integral of the second kind, \(E=\int_{0}^{K}\mathop{\mathrm{dn}}^{2}x\,dx\). By collecting these terms and recalling that \(q_{\text{CSL}}=m/\kappa\), we obtain \begin{align} \mathcal{E}_{\text{soliton}}(\kappa) &= J_{z}S^{2}a_{0}^{2}\left(4q_{\text{CSL}}^{2}\frac{E}{K}-\frac{\pi q_{0}m}{\kappa K}-2q_{\text{CSL}}^{2}+m^{2}\right)\notag\\ &= 2J_{z}S^{2}m^{2}a_{0}^{2}\left(\frac{2E}{\kappa^{2}K}-\frac{\pi q_{0}}{2m\kappa K}-\frac{1}{\kappa^{2}}+\frac{1}{2}\right). \end{align} (32) Using the differential formulas \(dK/d\kappa =[E/(1-\kappa^{2})-K]/\kappa\) and \(dE/d\kappa = (E-K)/\kappa\), we obtain the condition for an external value, \(d\mathcal{E}_{\text{soliton}}(\kappa)/d\kappa =0\), which gives \begin{equation} \kappa = \frac{4E}{\pi q_{0}}m = \frac{4E}{\pi q_{0}}\sqrt{\frac{\tilde{H}^{x}}{J_{z}Sa_{0}^{2}}}. \end{equation} (33) This equation is also written as \begin{equation} \tilde{H}^{x}/\tilde{H}_{\text{c}}^{x} = (\kappa/E)^{2} \end{equation} (34) by introducing the critical field corresponding to \(\kappa =1\), \begin{equation} \tilde{H}_{\text{c}}^{x} = \left(\frac{\pi q_{0}a_{0}}{4}\right)^{2}J_{z}S\sim D_{z}^{2}/J_{z}, \end{equation} (35) at which an IC–C phase transition occurs. The spatial period of the CSL, given by Eq. (28), is now explicitly given by \begin{equation} L_{\text{CSL}} = 2\kappa K\sqrt{\frac{JS}{\tilde{H}^{x}}}a_{0} = \frac{8KE}{\pi q_{0}}, \end{equation} (36) and correspondingly, \begin{equation} q_{\text{CSL}} = \pi q_{0}/4E. \end{equation} (37) The reciprocal lattice constant of the soliton lattice is given by \begin{equation} G_{\text{CSL}} = \frac{2\pi}{L_{\text{CSL}}} = \frac{\pi}{K}q_{\text{CSL}} = \frac{\pi^{2}}{4KE}q_{0}. \end{equation} (38)

A set of Eqs. (34) and (36) gives the field dependence of \(L_{\text{CSL}}\), which continuously increases from \(L_{0}=2\pi/q_{0}\) [note that \(K(0)=E(0)=\pi/2\)] to infinity when the magnetic field increases from zero (\(\kappa =0\)) to \(H_{\text{c}}^{x}\) (\(\kappa\rightarrow 1\)). This behavior is indicated by the solid line in Fig. 38(c). In the IC–C phase transition, the soliton density \begin{equation} \frac{L_{0}}{L_{\text{CSL}}} = \frac{G_{\text{CSL}}}{q_{0}} = \frac{\pi^{2}}{4KE} \end{equation} (39) plays the role of the order parameter. To make clear this point, we show the plot of \(L_{0}/L_{\text{CSL}}\) in Fig. 38(d).

The magnetization averaged over the spatial period is given by \begin{equation} M = - \frac{g\mu_{\text{B}}}{L_{\text{CSL}}}\int_{0}^{L_{\text{CSL}}}S^{x}(z)\,dz = g\mu_{\text{B}}S\left(\frac{2}{\kappa^{2}}-\frac{2E}{\kappa^{2}K}-1\right), \end{equation} (40) where \(E=E(\kappa)\) is the complete elliptic integral of the second kind. Equation (40) gives the field dependence of the magnetization [(M–H) curve] shown in Fig. 39. This profile with the downward convex (M–H) curve is peculiar to the chiral helimagnet and is regarded as indirect evidence of chiral helimagnetism, which was first suggested by Dzyaloshinskii¹⁶⁾ and used to identify the transition to the CSL.¹²⁴⁾

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Figure 39. Magnetization curve of the CSL state.

Here we mention the stability of the CSL in the magnetic field plane formed by the components parallel and perpendicular to the helical axis. It was recently found¹⁴⁵⁾ that for field directions almost parallel or perpendicular to the helical axis, the transition from the incommensurate CSL to the commensurate forced-FM state is continuous, while for intermediate angles the transition is discontinuous and the IC and C states coexist on the transition line.

In the previous subsection, we considered the CSL in an infinite system. Now, it is natural to search for a confinement effect in a finite system, where spins on both ends are fixed. In Ref. 129, it was found by numerical analysis that in the case of \(\tilde{H}^{x}=0\), the system relaxes to a simple helical state with the maximal topological number \(n_{\text{max}}=[NQ_{0}/2\pi]\), where N is the total number of lattice sites. As \(\tilde{H}^{x}\) increases and a soliton lattice starts to form, the topological number for the stable ground state exhibits cascade transitions to \(n=n_{\text{max}}-1,n_{\text{max}}-2,\ldots, 1\) at the critical field strengths where the energy levels cross each other. The envelope of the ground-state energies provides a series of phase transitions from one topological sector to another with a smaller n. The period jumps at the energy-crossing points and exhibits steplike behavior. Each step gives the quantized period, \begin{equation} L_{n} = L/n = 2\kappa K/m. \end{equation} (41) This condition determines the discrete elliptic modulus \(\kappa_{n}\) as a function of \(\tilde{H}^{x}\).

Using Eq. (32), we obtain the total energy \(E[\varphi_{0}]\) over the whole chain with the phase winding given by \(\varphi_{0}(z)=2\mathop{\mathrm{am}}(\frac{m}{\kappa} z)\). By eliminating the length L from the quantization condition (41), we have \begin{align} &E_{\text{soliton}}(\kappa)/J_{z}S^{2}a_{0}\notag\\ &\quad = \int_{0}^{nL_{n}}\left[4\frac{m^{2}}{\kappa^{2}}\mathop{\mathrm{dn}}\nolimits^{2}\left(\frac{m}{\kappa}z\right) -q_{0}\frac{\partial \varphi_{0}}{\partial z}-2 \frac{m^{2}}{\kappa^{2}}+m^{2}\right]\,dz\notag\\ &\quad = 4n\frac{m}{\kappa}\left(2E-K+\frac{1}{2}\kappa^{2}K\right) -2\pi nq_{0}. \end{align} (42) In Fig. 40(b), we show the magnetization curves for \(n=6{\text{--}}10\). As \(m=\sqrt{\smash{\tilde{H}^{x}/(J_{z}Sa_{0}^{2})}\mathstrut}\) increases, the ground-state magnetization curve, indicated by the thick line, jumps at a critical field \(\tilde{H}_{n}^{x}\) from the \((n+1)\)th-sector to the nth-sector.

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Figure 40. (Color online) (a) Total energy of confined CSLs given by Eq. (42) as functions of h for \(n=10,9,8,7,6\). Energy-crossing points are indicated by circles. (b) Corresponding magnetization curves.

The CSL exhibits rich dynamical properties. First, the whole CSL system behaves as an elastic crystal lattice that consists of solitons, as shown in Fig. 41(a). The phonon-like excitations (CSL phonon) of the scalar sine-Gordon soliton lattice were first studied by Sutherland,¹⁴⁶⁾ and the case of a CSL was discussed in Refs. 23 and 147. He pointed out that the phonon mode obeys the Lamé equation of the Jacobi form¹⁰⁸⁾ as discussed below. Note that oscillations around individual solitons highly correlate with each other via the long-range interaction, which decays in an exponential manner at large distances. The CSL phonon is a direct manifestation of broken translational symmetry.

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Figure 41. (Color online) Dynamical motion associated with the CSL state. (a) Phonon-like excitations around the CSL, which correspond to the Goldstone mode associated with the translational symmetry breaking. (b) Collective translation of the whole CSL.

Second, because of broken Galilean symmetry, the CSL exhibits collective sliding motion, i.e., the whole CSL is translated in a coherent manner, as shown in Fig. 41(b). This motion corresponds to the Galilean boost of the CSL,¹⁴⁸⁾ and its physical consequences were discussed in Ref. 26. In this subsection, we discuss the CSL phonon, and the coherent sliding will be discussed in the next subsection. Note that there is an isolated soliton which surfs over the background CSL,¹⁴⁹⁾ which we do not discuss in this review.

We consider dynamical fluctuations around the stationary soliton lattice configuration, \begin{align} \varphi (z,t) &= \varphi_{0}(z) +\delta \varphi(z,t), \end{align} (43) \begin{align} \theta (z,t) &= \theta_{0}+\delta \theta (z,t), \end{align} (44) where \(\theta_{0}=\pi/2\). We depict these fluctuations in Fig. 42.

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Figure 42. (Color online) \(\delta\theta (z,t)\) (out-of-plane) and \(\delta\varphi (z,t)\) (in-plane) fluctuations of the local spins around the stationary soliton lattice configuration.

By expanding the Hamiltonian \(\mathcal{H}=\int_{0}^{L}dz\,\tilde{\mathcal{H}}\) up to the second order with respect to \(\delta\theta\) and \(\delta\varphi\), we have \(\tilde{\mathcal{H}}[\varphi,\theta]=\tilde{\mathcal{H}}[\varphi_{0},\theta_{0}]+\delta\tilde{\mathcal{H}}\), where \begin{equation} \delta \mathcal{H} = \frac{JS^{2}a_{0}^{2}}{2}\int_{0}^{L}\frac{dz}{a_{0}} (\delta \varphi\hat{\Lambda}_{\varphi}\delta \varphi +\delta \theta\hat{\Lambda}_{\theta}\delta \theta). \end{equation} (45) The linear differential operators are given by \begin{align} \hat{\Lambda}_{\varphi} &= - \partial_{z}^{2}-m^{2}\cos \varphi_{0} \notag\\ &= - \left(\frac{m}{\kappa}\right)^{2}(\partial_{\bar{z}}^{2}-2\kappa^{2} \mathop{\mathrm{sn}}\nolimits^{2}\bar{z}+\kappa^{2}) \end{align} (46) and \begin{equation} \hat{\Lambda}_{\theta} = \hat{\Lambda}_{\varphi}+\bar{K}_{\bot}+\Delta (z), \end{equation} (47) where \(\bar{K}_{\bot}=2K_{\bot}/(J_{z}S^{2}a_{0}^{2})\). Although the φ-mode is gapless, the θ-mode acquires a spatially varying energy gap, \begin{align} \Delta (z) &\equiv - (\partial_{z}\varphi_{0})^{2}+2q_{0}(\partial_{z}\varphi_{0}) \notag\\ &= - 4q_{\text{CSL}}^{2} \mathop{\mathrm{dn}}\nolimits^{2}\bar{z}+4q_{0}q_{\text{CSL}}\mathop{\mathrm{dn}}\bar{z}. \end{align} (48) This gap directly originates from the DM interaction. In Fig. 43, we show the spatial profile of \(\Delta (\bar{z})\) for some values of \(H^{z}/H_{\text{c}}^{z}\).

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Figure 43. Spatial profile of \(\Delta(z)\) for \(H^{z}/H_{\text{c}}^{z}\) = (a) 0, (b) 0.4, (c) 0.8, (d) 0.99, and (e) \(1-10^{-7}\).

We see that the soliton parts contribute to the gap formation, while the forced-FM (commensurate) domains do not. For an arbitrary value of \(H^{z}/H_{\text{c}}^{z}\), the average of \(\Delta (z)\) over its spatial period is computed as \begin{equation} \langle \Delta (H_{x})\rangle = \frac{1}{2K}\int_{0}^{2K}\Delta (\bar{z})\,d\bar{z} = q_{0}q_{\text{CSL}}. \end{equation} (49) Note that \(\langle\Delta (H_{x})\rangle\) is proportional to the soliton density \(L_{\text{CSL}}^{-1}=q_{\text{CSL}}/2\pi\). As \(H^{z}/H_{\text{c}}^{z}\rightarrow 1\), the gap becomes less relevant. This is because near the IC–C transition, the whole system is dominated by the FM domains and the effect of the DM interaction becomes irrelevant. On the other hand, for a weak-field regime, \(H^{z}/H_{\text{c}}^{z}\ll 1\), the gap becomes almost constant as \begin{equation} \Delta (z) \sim \Delta_{0} \equiv \langle \Delta (0)\rangle = q_{0}^{2}. \end{equation} (50) Noting Eq. (35), we have a scaling relation for the intrinsic gap due the DM interaction, \begin{equation} \delta_{0}^{(\theta)} = \frac{JS^{2}(q_{0}a_{0})^{2}}{2} = \frac{8S}{\pi^{2}}\tilde{H}_{\text{c}}^{x}. \end{equation} (51) For CrNb₃S₆ (\(S=3/2\)), using the experimentally observed \(\tilde{H}_{\text{c}}^{x}=2300\) Oe,²¹⁾ we estimate the gap as \(\delta_{0}^{(\theta)}\sim 0.19\) K.

In the region where the approximation (50) is valid, the eigenvalue equations \(\hat{\Lambda}_{\varphi}v(z)=\lambda^{(\varphi)}v(z)\) and \(\hat{\Lambda}_{\theta}u(z)=\lambda^{(\theta)}u(z)\) reduce to the Jacobi form of the Lamé equations,¹⁰⁸⁾ \begin{align} \partial_{\bar{z}}^{2}v(\bar{z}) &= \left[2\kappa^{2}\mathop{\mathrm{sn}}\nolimits^{2}\bar{z}-\kappa^{2}-\left(\frac{\kappa}{m}\right)^{2}\lambda^{(\varphi)}\right] v(z), \end{align} (52) \begin{align} \partial_{\bar{z}}^{2}u(\bar{z}) &= \left[2\kappa^{2}\mathop{\mathrm{sn}}\nolimits^{2}\bar{z}-\kappa^{2}-\left(\frac{\kappa}{m}\right)^{2}(\lambda^{(\theta)}-\bar{K}_{\bot}-\Delta_{0})\right] u(z). \end{align} (53) The eigenfunctions of Eq. (52) are given by Refs. 108, 146, and 107, \begin{equation} v(\bar{z}) = u(\bar{z}) = N\cfrac{\vartheta_{4}\biggl[\cfrac{\pi}{2K} (\bar{z}+\zeta)\biggr]}{\vartheta_{4}\biggl(\cfrac{\pi}{2K}\bar{z}\biggr)}\,e^{i\bar{q}\bar{z}}, \end{equation} (54) where \(\vartheta_{4}(x)\) is the theta function, N is a normalization constant, and ζ is a complex parameter. \(\bar{q}=(\kappa/m) q\) is the Floquet index, which has a physical meaning as a wave number. In Appendix, we present a pedagogical review of the propagating solution of the Lamé equation.

The φ-mode spectrum, \(\varepsilon_{q}^{(\varphi)}\), consists of two allowed bands, “valence” and “conduction” bands.¹⁴⁶⁾ The valence band is formed from correlated translations of individual solitons and is regarded as a Goldstone mode, while the conduction band corresponds to a renormalized Klein–Gordon mode.¹⁴⁶⁾

I. Valence band: The valence band is specified by \(\zeta =ia+K\) (\(-K^{\prime}\leq a\leq K^{\prime}\)), where \(K^{\prime}=K(\kappa^{\prime})\) with \(\kappa^{\prime}=\sqrt{1-\kappa^{2}}\). Using Eq. (A·17), the wave number is given by \begin{equation} q = \frac{m}{\kappa}\left[\bar{Z}(a)+\frac{\pi}{2KK^{\prime}}a\right], \end{equation} (55) where \(| q_{\alpha}|\) changes from 0 to \(G_{0}=\pi^{2}q_{0}/KE=\pi/L_{\text{CSL}}=\frac{m}{\kappa}\frac{\pi}{2K}\) as α changes from 0 to \(K^{\prime}\). The zeta function is defined as \(Z(z,\kappa)=E(z,\kappa)-(E/K) z\), where \(E(z,\kappa)\) is the fundamental elliptic integral of the second kind and \(\bar{Z}(a)=Z(a,\kappa^{\prime})=Z(a,\sqrt{1-\kappa^{2}})\). Accordingly, using Eq. (A·14), the energy spectrum is \begin{align} \varepsilon_{q}^{(\varphi)} &= \frac{JS^{2}a_{0}^{2}}{2} \lambda_{q}^{(\varphi)} = \frac{JS^{2}a_{0}^{2}}{2}\left(\frac{m}{\kappa}\right)^{2}\kappa^{\prime 2}\mathop{\mathrm{sn}}\nolimits^{2}(a,\kappa^{\prime})\notag\\ &= \frac{S}{2}\tilde{H}^{x}\left(\frac{\kappa^{\prime}}{\kappa}\right)^{2}\mathop{\mathrm{sn}}\nolimits^{2}(a,\kappa^{\prime}), \end{align} (56) where \(\overline{\mathop{\mathrm{sn}}}a=\mathop{\mathrm{sn}}(a,\kappa^{\prime})\). \(\varepsilon_{q}^{(\varphi)}\) changes from 0 to \begin{equation} W = \frac{S}{2}\tilde{H}^{x}\left(\frac{\kappa^{\prime}}{\kappa}\right)^{2}, \end{equation} (57) with W being the bandwidth of the valence band. Using the asymptotic form (29), we see that as \(\tilde{H}^{x}/\tilde{H}_{\text{c}}^{x}\) approaches unity from below, the band width exponentially decays as a function of \(L_{\text{CSL}}\), \begin{align} W &\sim S\tilde{H}^{x}(1-\kappa) \sim S\tilde{H}^{x}\exp (-mL_{\text{CSL}})\notag\\ &= S\tilde{H}^{x}\exp \left[-\frac{\pi^{2}}{2}\frac{L_{\text{CSL}}}{L(0)}\right], \end{align} (58) where \(L(0)\) is the spatial period of the CHM structure under zero magnetic field. Near the transition, the fluctuations are almost localized and consequently the valence band becomes almost flat.

We obtain the energy spectrum of the φ and θ fluctuations, which are depicted in Figs. 44(a) and 44(b), respectively. Here it should be noted that in the description of the φ-mode, we consider the fluctuations around the spatially modulating configuration \(\varphi_{0}\) with period \(L_{\text{CSL}}\). Therefore, \(q=0\) in the φ-mode corresponds to \(q=\pi/L_{\text{CSL}}\) in the laboratory frame. In Figs. 44(c) and 44(d), we depict the spatial distribution of φ corresponding to the band top and zero mode (which corresponds to the sliding mode discussed in the next subsection), respectively.

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Figure 44. (Color online) Schematic landscape of the energy spectra for (a) φ and (b) θ fluctuations. In (c) and (d), we depict snapshots of the spatial distribution of the φ fluctuation corresponding to the band top [white circle indicated in (a)] and the zero mode [black circle indicated in (a)].

II. Conduction band: The conduction band is specified by \(\zeta =ia\) (\(-K^{\prime}\leq a\leq K^{\prime}\)). Using Eq. (A·23), the wave number is given by \begin{equation} q_{a} = \frac{m}{\kappa}\left[\bar{Z}(a)+\frac{\pi}{2KK^{\prime}}a+\frac{\overline{\mathop{\mathrm{cn}}}a\overline{\mathop{\mathrm{dn}}}a}{\overline{\mathop{\mathrm{sn}}}a}\right], \end{equation} (59) where \(\overline{\mathop{\mathrm{cn}}}a=\mathop{\mathrm{cn}}(a,\kappa^{\prime})\) and \(\overline{\mathop{\mathrm{dn}}}a=\mathop{\mathrm{dn}}(a,\kappa^{\prime})\). \(q_{\alpha}\) changes from \(G_{0}\) to infinity as a changes from \(K^{\prime}\) to 0. Accordingly, using Eq. (A·20), the energy spectrum is \begin{align} \varepsilon_{q}^{(\varphi)} &= \frac{JS^{2}a_{0}^{2}}{2} \lambda_{q}^{(\varphi)} = \frac{JS^{2}a_{0}^{2}}{2}\left(\frac{m}{\kappa}\right)^{2}\overline{\mathop{\mathrm{sn}}}^{-2}a\notag\\ &= \frac{S}{2}\tilde{H}^{x}\frac{1}{\kappa^{2}\overline{\mathop{\mathrm{sn}}}^{2}a}, \end{align} (60) which changes from \(\frac{S}{2\kappa^{2}}\tilde{H}^{x}\) to infinity. The energy gap, \begin{equation} \delta = \frac{S}{2}\tilde{H}^{x}, \end{equation} (61) opens at the boundary of the Brillouin zone of the CSL.

The normalized wave function at the bottom (\(q=0\)) of the valence band is \begin{equation} v_{0}(z) = u_{0}(z) = L^{-1/2}\sqrt{\frac{K}{E}}\mathop{\mathrm{dn}}\bar{z} = \frac{1}{\sqrt{2\pi q_{0}\mathcal{N}}}\partial_{z}\varphi_{0}(z), \end{equation} (62) with dn being the Jacobi dn function and \(\mathcal{N}=L/L_{\text{CSL}}\) being the total number of solitons. \(v_{0}(z)\) and \(u_{0}(z)\) respectively correspond to the zero mode and quasi-zero mode, which are localized around each soliton.^23,147) In conventional terminology, the zero mode means a mode excited with no excess energy. In the present case, the in-plane \(v_{0}(z)\) mode exactly corresponds to this case, but the out-of-plane \(u_{0}(z)\) zero mode acquires the gap \begin{equation} \varepsilon_{0}^{(\theta)} = K_{\bot}+\delta_{0}^{(\theta)}, \end{equation} (63) where the intrinsic gap is given by Eq. (51). For this reason, the \(u_{0}(z)\) mode is called the quasi-zero mode.²²⁾ Note that the CSL phonon modes are orthogonal to the zero mode.

Using the orthonormal basis \(v_{q}(z)\) and \(u_{q}(z)\), the fluctuations (φ-mode and θ-mode) are spanned by the orthogonal eigenfunctions \(v_{q}(z)\) and \(u_{q}(z)\) as \begin{equation} \delta \varphi (z,t) = \sum_{q}\eta_{q}(t)v_{q}(z),\quad \delta \theta (z,t) = \sum_{q}\xi_{q}(t)u_{q}(z), \end{equation} (64) where \(\eta_{q}(t)\) and \(\xi_{q}(t)\) are the dynamical coordinates. Normalization gives \(\int_{0}^{L}dz\,v_{q}(z)v_{q^{\prime}}(z)=\int_{0}^{L}dz\,u_{q}(z)u_{q^{\prime}}(z)=\delta_{qq^{\prime}}\). Then, \(\delta H\) is diagonalized to give \begin{equation} \delta \mathcal{H} = \sum_{q}[\varepsilon_{q}^{(\varphi)}\eta_{q}^{2}(t)+\varepsilon_{q}^{(\theta)}\xi_{q}^{2}(t)]. \end{equation} (65)

We consider spin resonance in the CSL state, where, in addition to the static magnetic field to stabilize the CSL, a uniform rf field is applied parallel to the helical z-axis. In this case, the rf field couples with \(S^{z}=S\cos\theta \sim -S\delta\theta\) and the resonant absorption is caused by the CSL phonon modes with a series of special wave numbers, \begin{equation} q = q_{n} = nG_{\text{CSL}}. \end{equation} (66) The absorption strength of the nth-order resonance is written as \begin{equation} \mathcal{Q}_{n} \propto | u_{q_{n}}|^{2}\delta (\hbar \omega -\varepsilon_{q_{n}}^{(\theta)}), \end{equation} (67) where ω is the frequency of the rf field. The oscillator strength \(| u_{q_{n}}|^{2}\) is simply given by the Fourier transform of the wave function [Eq. (54)], \begin{align*} u_{n} &= \frac{1}{2K}\int_{-K}^{K}u(\bar{z})\exp\left(-in\frac{\pi}{K}\bar{z}\right)\,d\bar{z} \\ &= \frac{2p^{1/4}}{\vartheta_{1}^{\prime}}\cfrac{\vartheta_{4} \biggl[\cfrac{\pi}{2K}(iK^{\prime}+\zeta)\biggr]}{p^{n}-p^{-n}e^{-i\pi \zeta/K}}, \end{align*} where \(p\equiv e^{i\pi\tau}=e^{-\pi K^{\prime}/K}\), \(\vartheta_{1}^{\prime}=\vartheta_{2}\vartheta_{3}\vartheta_{4}=\sqrt{\kappa\kappa^{\prime}}(2K/\pi)^{3/2}\) with \(\vartheta_{2}=\vartheta_{2}(0)\) and so forth, \(\vartheta_{3}=\sqrt{2K/\pi}\), and \(\vartheta_{4}=\sqrt{2\kappa^{\prime}K/\pi}\).

For the valence band (\(n=0\)), \(a=0\) at the bottom of the band gives \begin{align} u_{n} &= \frac{1}{\vartheta_{1}^{\prime}}\cfrac{\vartheta_{2}\biggl(i \cfrac{\pi}{2K}a\biggr)}{\cosh \biggl[\cfrac{\pi}{2K}(a+2nK^{\prime})\biggr]} \notag\\ &= \frac{\vartheta_{2}}{\vartheta_{1}^{\prime}} = \frac{1}{\vartheta_{3}\vartheta_{4}} = \frac{\pi}{2\sqrt{\kappa^{\prime}}K}, \end{align} (68) where we used the relation \(\vartheta_{4}[\frac{\pi}{2K} (iK^{\prime}+ia+K)] = p^{-1/4}\exp(\frac{\pi a}{2K})\vartheta_{2}(i\frac{\pi}{2K})\). Note that \(u_{q_{0}}\rightarrow 1\) as \(\kappa\rightarrow 0\). For the conduction band (\(n\geq 0\)), \(\zeta=ia\) gives \begin{equation} u_{n} = - \frac{i}{\vartheta_{1}^{\prime}}\cfrac{\vartheta_{1}\biggl(i \cfrac{\pi}{2K}a_{n}\biggr)}{\sinh \biggl[\cfrac{\pi}{2K}(a_{n}+2nK^{\prime})\biggr]}, \end{equation} (69) where we used the relation \(\vartheta_{4}[\frac{\pi}{2K}(iK^{\prime}+ia)] =ip^{-1/4}\exp(\frac{\pi a}{2K})\vartheta_{1}(i\frac{\pi}{2K})\). Noting Eq. (A·23), \(a_{n}\) is determined by the condition \begin{equation} \bar{q}_{n} = \frac{\overline{\mathop{\mathrm{cn}}}a_{n}\overline{\mathop{\mathrm{dn}}}a_{n}}{\overline{\text{sn}}a_{n}}+\overline{Z}(a_{n}) +\frac{\pi}{2KK^{\prime}}a_{n} = \frac{2\pi}{K}n, \end{equation} (70) for \(n=1,2,\ldots\).

The first Brillouin zone of the CSL is \(| q|\leq G_{\text{CSL}}/2\) and the energy gap between the valence and conduction bands opens at the zone boundary. For \(n=0\), the bottom of the valence band (\(q=0\) and \(a=0\)) gives \(\hbar\omega_{0}=\varepsilon_{0}^{(\theta)}\). For \(n\geq 1\), the conduction band contributes to the resonance at \(\hbar\omega_{n}=\varepsilon_{q_{n}}^{(\theta)}\).

In Fig. 45(a), we schematically depict the distribution of the resonance energy levels, which become more dense upon increasing the magnetic field strength. In Fig. 45(b), we show the resonance frequencies \(\omega_{n}\) from \(n=0\) to 10 as functions of \(H^{x}/H_{\text{c}}^{x}\). Note that the strength \(| u_{n}|^{2}\) exponentially decreases as the order n increases. In Fig. 45(c), we show the ratios \(| u_{n}|^{2}/| u_{0}|^{2}\) on a logarithmic scale. At \(H^{x}=0\), the weight is completely dominated by the first (\(n=0\)) resonance, but as \(H^{x}/H_{\text{c}}^{x}\) approaches unity, the contribution from higher resonances becomes more conspicuous.

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Figure 45. Energy dispersion of the CSL phonon in the reduced zone scheme for (a-1) smaller and (a-2) larger magnetic field strengths. The vertical broken lines indicate the Brillouin zone boundaries \(q=\pm G_{\text{CSL}}/2\). (b) Expected resonance energy \(\omega_{n}\) for \(n=0\) to 10 as functions of \(H_{0}/H_{0}^{\ast}\). We took \(D/J=0.5\) and \(K_{\bot}/J=0\). (c) Ratios \(| u_{n}|^{2}/| u_{0}|^{2}\) for \(n=1\) to 6 as functions of \(H^{x}/H_{\text{c}}^{x}\) on a logarithmic scale.

Next we consider the sliding motion of the CSL [Fig. 41(b)]. In the continuum limit, the soliton lattice has continuous degeneracy related to the choice of the center of mass position, i.e., \(\boldsymbol{{n}}_{0}(z)\rightarrow\boldsymbol{{n}}_{0}(z-Z)\). To describe the collective dynamics, the parameter Z is promoted to a dynamical variable \(Z(t)\). However, the translational motion of the CSL always accompanies internal deformation, \(\xi_{0}(t) u_{0}[z-Z(t)]\), which is analogous to the Döring–Becker–Kittel mechanism of single domain-wall motion.¹⁵⁰⁾ As in the case of single domain-wall motion,¹⁵¹⁾ in the mode expansion given in Eq. (64), only the quasi-zero mode \(u_{0}\) contributes to the inertial mass of the soliton lattice,^23,147) because the spin-wave modes with \(q\neq 0\) are all orthogonal to the zero-mode wave functions and irrelevant to the sliding motion of the CSL. Taking account of these degrees of freedom, we write the fluctuation field variables as \begin{equation} \varphi(z,t) = \varphi_{0}[z-Z(t)] \end{equation} (71) and \begin{equation} \theta(z,t) = \pi/2+\xi_{0}(t)u_{0}[z-Z(t)]. \end{equation} (72) The two dynamical variables \(Z(t)\) and \(\xi_{0}(t)\) are relevant dynamical variables to describe the sliding motion, as shown in Fig. 46.

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Figure 46. \(Z(t)\) and \(\xi_{0}(t)\) as dynamical variables used to describe collective sliding of the CSL.

Because the φ-mode is gapless, there is no excess energy associated with \(\varphi(z,t)\). On the other hand, the θ-mode is protected by an energy gap (50) and the quasi-zero mode \(u_{0}\) has a finite energy \(\varepsilon_{0}^{(\theta)}\). This is the reason why only the quasi-zero mode \(u_{0}\) contributes to the sliding motion. The excitation energy given by Eq. (65) is now \begin{equation} \delta\mathcal{H}_{\text{sliding}} = \varepsilon_{0}^{(\theta)}\xi_{0}^{2}(t). \end{equation} (73)

Next we construct a Lagrangian for the collective dynamics. The kinetic (Berry phase) term is written as \begin{equation} \mathcal{L}_{\text{Berry}} = \frac{\hslash S}{a_{0}}\int_{0}^{L}dz(\cos \theta-1)\partial_{t}\varphi. \end{equation} (74) The total Lagrangian is then given by \begin{equation} \mathcal{L}_{\text{sliding}} = \mathcal{L}_{\text{Berry}}-\delta\mathcal{H}_{\text{sliding}}. \end{equation} (75)

Using the expansion (72) and the relation \(\partial_{t}\varphi [z-Z(t)] =-\dot{Z}\partial_{z}\varphi[z-Z(t)]\), \(\mathcal{L}_{\text{Berry}}\) is computed up to the linear order in \(\xi_{0}(t)\) and \(\dot{Z}(t)\) to give \begin{equation} \mathcal{L}_{\text{Berry}} \sim \frac{\hbar S}{a_{0}}\xi_{0}(t)\dot{Z} (t)\int_{0}^{L}u_{0}(z)\partial_{z}\varphi_{0}(z)+\mathcal{P}\dot{Z}(t), \end{equation} (76) where \(\mathcal{P}=2\pi\mathcal{N}a_{0}^{-1}\hbar S\dot{Z}(t)\) is the topological momentum with \(\mathcal{N} = L/L_{\text{CSL}}\) being the total number of solitons. Generally speaking, the Lagrangian is defined only to within an additive total time derivative of any function of space coordinates and time. Therefore, the second term of (76) does not affect the physical dynamics at all and we will drop it off from now on. Then, the zero-mode wave function (62) gives the overlap integral \begin{equation} \mathcal{K} \equiv \int_{0}^{L}u_{0}(z)\partial_{z}\varphi_{0}(z) = \sqrt{2\pi q_{0}\mathcal{N}}, \end{equation} (77) [where we used the relation (36)] and we obtain \begin{equation} \mathcal{L}_{\text{Berry}} = \frac{\hbar S}{a_{0}}\mathcal{K}\xi_{0}(t)\dot{Z} (t). \end{equation} (78) Now, the Lagrangian finally becomes \begin{equation} \mathcal{L}_{\text{sliding}} = a_{0}^{-1}\hbar S\mathcal{K}\xi_{0}(t)\dot{Z} (t)-\varepsilon_{0}^{(\theta)}\xi_{0}^{2}. \end{equation} (79) The form of the Berry phase term indicates that once the sliding motion starts, the velocity \(\dot{Z}(t)\) couples with the quasi-zero-mode coordinate \(\xi_{0}(t)\). This situation is simply the Döring–Becker–Kittel mechanism. The second term \(\varepsilon_{0}^{(\theta)}\xi_{0}^{2}\) has a physical meaning as the harmonic potential generated by confinement of the spins into the xy-plane by the DM integration.

Now that we have obtained the Lagrangian (79), we consider how to trigger the sliding motion. As pointed out in Ref. 26, the sliding is caused via the spin-torque transfer from the itinerant quantum spins to the localized magnetic moments which form the CSL. More directly, the sliding motion of the CSL is caused by applying a time-dependent magnetic field parallel to the helical axis.¹⁵²⁾ Here we briefly present the latter case.

The Zeeman coupling of the CSL with \(H^{z}(t)\) gives an additional term to the Lagrangian, \begin{equation} - \frac{S}{a_{0}}\tilde{H}^{z}(t)\int_{0}^{L}dz \cos\theta(z,t) \sim \frac{S \mathcal{K}}{q_{0}a_{0}}\tilde{H}^{z}(t)\xi_{0}(t), \end{equation} (80) where we have used Eqs. (62) and (72). Now, we have the Lagrangian \begin{equation} \mathcal{L} = a_{0}^{-1}\hbar S\mathcal{K}\xi_{0}(t)\dot{Z}(t)- \varepsilon_{0}^{(\theta)}\xi_{0}^{2}+\frac{S\mathcal{K}}{q_{0}a_{0}}\tilde{H}^{z}(t)\xi_{0}(t). \end{equation} (81) Furthermore, to incorporate the damping effect, the Rayleigh dissipation term \begin{equation} \mathcal{W} = \frac{\alpha\hbar S}{2a_{0}}\int_{0}^{L}dz(\partial_{t} \boldsymbol{{n}})^{2}\sim\frac{\alpha\hbar S}{2a_{0}}(\mathcal{M} \dot{Z}^{2}+\dot{\xi}_{0}^{2}) \end{equation} (82) is introduced, where α is a small coefficient specifying the Gilbert damping and the overlap integral is given by \begin{equation} \mathcal{M} \equiv \int_{0}^{L}dz(\partial_{z}\varphi_{0}(z))^{2} = 2\pi q_{0}\mathcal{N} = \mathcal{K}^{2}. \end{equation} (83)

A general form of the Euler–Lagrange–Rayleigh equation of motion for a generalized coordinate q is given by \(d(\partial\mathcal{L}/\partial\dot{q})/dt-\partial L/\partial\dot{q}=-\partial W/\partial\dot{q}\). In the present case, using Eqs. (81) and (82), the equations of motion for the collective coordinates are given by \begin{align} \hbar\mathcal{K}\dot{\xi}_{0} &= - \alpha\hbar\mathcal{M}\dot{Z}, \end{align} (84) \begin{align} \hbar\mathcal{K}\dot{Z} &= 2S^{-1}a_{0}\varepsilon_{0}^{(\theta)}\xi_{0}+\alpha\hbar\dot{\xi}_{0}-q_{0}^{-1}\mathcal{K}\tilde{H}^{z}. \end{align} (85) They are solved to give the solutions \begin{align} \dot{Z}(t) &= Ce^{-t/\tau_{\text{CSL}}}-\frac{e^{-t/\tau_{\text{CSL}}}}{(1+\alpha^{2})\hbar q_{0}}\int^{t}e^{t^{\prime}/\tau_{\text{CSL}}}\frac{d\tilde{H}_{z}(t^{\prime})}{dt^{\prime}}\,dt^{\prime}, \end{align} (86) \begin{align} \xi_{0}(t) &= De^{-t/\tau_{\text{CSL}}}+\frac{\alpha\mathcal{K}e^{-t/\tau_{\text{CSL}}}}{(1+\alpha^{2})\hbar q_{0}}\int^{t}e^{t^{\prime}/\tau_{\text{CSL}}}\tilde{H}_{z}(t^{\prime})\,dt^{\prime}, \end{align} (87) where we used the relation \(\mathcal{M}/\mathcal{K}^{2}=1\), which is valid for a weak field. The constants C and D are determined by the initial conditions \(\dot{Z}(0)=0\) and \(\xi_{0}(0)=0\). As can be seen from Eqs. (84) and (85), the two coordinates \(\xi_{0}\) and Z are coupled to each other via the Gilbert damping α. This means that the CSL never realizes dissipationless collective motion. In the case of a static \(\tilde{H}_{z}\), only trivial relaxational dynamics occurs when the sliding motion does not persist. Equation (87) means that the longitudinal field first directly couples to \(\xi_{0}\) and drives its growth via the Gilbert damping process. Then, the sliding motion follows the growth of \(\xi_{0}\).

The relaxation of the soliton lattice is caused by the Gilbert damping and its intrinsic relaxation time is expressed by \begin{equation} \tau_{\text{CSL}} = \frac{\hbar S(\alpha+\alpha^{-1})}{2a_{0} \varepsilon_{0}^{(\theta)}} \sim \frac{\hbar(\alpha+\alpha^{-1})}{S}\frac{J}{D^{2}}. \end{equation} (88) It is clear that the DM interaction gives rise to a finite relaxation time. Using the experimental data for CrNb₃S₆, the excitation gap is estimated as \(\varepsilon_{0}^{(\theta)}\sim 0.38\) K from Eq. (63). The intrinsic relaxation time of the soliton lattice is estimated as \(\tau_{\text{CSL}}\sim(\alpha+\alpha^{-1})\times 3.0\times 10^{-11}\) s. A small damping such as \(\alpha\sim 10^{-2}\) leads to \(\tau_{\text{CSL}}\sim 3.0\times 10^{-9}\) s.

Next, let us consider a typical case of an AC field, \(H^{z}(t)=H_{0}^{z}\sin(\Omega t)\). Equation (86) gives the velocity \begin{equation} \dot{Z}(t) = V_{0}[e^{-t/\tau_{\text{CSL}}}-\Omega \tau_{\text{CSL}}\sin(\Omega t)-\cos (\Omega t)], \end{equation} (89) with \(V_{0}=\tilde{H}_{0}^{z}\Omega\tau_{\text{CSL}}/[\hbar Q_{0}(1+\alpha^{2})(1+\tau_{\text{CSL}}^{2}\Omega^{2})]\). We see that the Gilbert damping causes out-of-phase oscillation. In Fig. 47, we show the response of the sliding velocity to the AC field. The transient state rapidly relaxes over the time scale of \(\tau_{\text{CSL}}\) to the stationary oscillation with a phase shift due to the damping. A salient feature of the sliding dynamics is that \(V_{0}\) changes its sign when \(Q_{0}\) is inverted to \(-Q_{0}\), i.e., the sign of \(V_{0}\) depends on the left- or right-handedness of the crystal. We may conclude that the crystal chirality is directly connected to the running direction of the CSL. The fact that the chiral geometry is converted to non-reciprocal dynamics is one of the most prominent features of chiral magnets.

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Figure 47. Time dependences of (a) longitudinal field \(H_{z}(t)=H_{z1}\sin(\Omega t)\) and (b) velocity \(\dot{Z}/V_{0}\) for \(\Omega^{-1}=0.5\,\tau_{\text{CSL}}\).

The CSL, being protected by the broken chiral symmetry, possesses a robust coherence. Owing to this circumstance, the CSL behaves as a coherent single domain, which can be referred to as a giant deformable body. Then it is natural to investigate its dynamics under a confinement potential caused by pinning. Within the small amplitude approximation, the confined dynamics reduces to a small oscillation under a nonlinear potential and resonant dynamics is expected to occur. The non-linearity stems from the nonlinear structure of the CSL.

To model the boundary pinning, we introduce the boundary pinning field \(H_{\text{p}}\) in the form \begin{equation} \delta V = \tilde{H}_{\text{p}}S_{-L/2}^{x}+\tilde{H}_{\text{p}}S_{L/2}^{x}, \end{equation} (90) where the system is confined over the region \(z\in [-L/2,L/2]\). It is also plausible to use \((S_{\pm L/2}^{x})^{2}\) instead of \(S_{\pm L/2}^{x}\). However, for the later numerical computation, we adopt the form of Eq. (90). This difference does not qualitatively affect the result (see Sect. 9.2 of Ref. 125).

Now, we consider the energy cost associated with the shift of Z. In the analytical approach presented in this section, we assume weak pinning and that the spatial profile of φ simply causes a parallel translation. The variable \(\xi_{0}\) is, however, dynamically generated upon sliding motion. We also assume that the whole system over \(z\in [-L/2,L/2]\) accommodates an integer number of solitons, i.e., \(\varphi (L/2,t)-\varphi (-L/2,t)=2\pi\mathcal{N}\), where \(\mathcal{N}\) is the number of solitons. Under these assumptions, the boundary spins are written as \begin{align} S_{\pm L/2}^{x} &= - S\cos \left[\xi_{0}(t)\sqrt{\frac{K}{LE}}\mathop{\mathrm{dn}} \left(\frac{\pi Q_{0}}{4E}Z(t)\right)\right]\notag\\ &\quad \times \cos \left[2\mathop{\mathrm{am}}\left(\frac{\pi Q_{0}}{4E}Z(t)\right)\right]. \end{align} (91)

Provided that the displacement \(Z(t)\) and the amplitude \(\xi_{0}\) are small, we expand Eq. (91) up to the second order with respect to \(\xi_{0}\) and Z. Noting that \(\mathop{\mathrm{am}}x\simeq x+\mathcal{O}(x^{3})\) and \(\mathop{\mathrm{dn}}x\simeq 1+\mathcal{O}(x^{3})\), we obtain \begin{equation} S_{\pm L/2}^{x} \simeq - S+\frac{S}{2}\left(\frac{\pi Q_{0}}{2E}\right)^{2}Z^{2}(t)+\frac{SK}{2LE}\xi_{0}^{2}(t). \end{equation} (92) Substituting this into Eq. (90), we obtain the confinement potential \begin{equation} \delta V = \frac{1}{2}A\xi_{0}^{2}(t)+\frac{1}{2}BZ^{2}(t), \end{equation} (93) where \(A=\tilde{H}_{\text{p}}S(K/LE)\) and \(B=\tilde{H}_{\text{p}}S(\pi Q_{0}/2E)^{2}\). By including this potential term in the Lagrangian (81), we can easily analyze the resonant oscillation under the AC field, \(H^{z}(t)=H_{0}^{z}\sin (\Omega t)\). The resonant frequency is given by \begin{equation} \Omega_{0}^{2} = \frac{Ba_{0}}{\hbar S^{2}\mathcal{K}}\frac{2\varepsilon_{0}^{(\theta)}+Aa_{0}}{\hbar \mathcal{K} +\alpha^{2}\hbar \mathcal{M}\mathcal{K}^{-1}}. \end{equation} (94) In the small-field limit, we have \begin{equation} \hbar \Omega_{0} \sim \sqrt{\smash{2\varepsilon_{0}^{(\theta)}\tilde{H}_{\text{p}}a_{0}/LS}\mathstrut}. \end{equation} (95)

To estimate the quantities, we choose \(Q_{0}=D/J=0.16\), as found for CrNb₃S₆, \(S=3/2\) (the spin of the Cr ions), \(J=100\) K, and \(\alpha=0.01\), which yield \(\varepsilon_{0}^{(\theta)} = 0.38\) K and \(\gamma=2.5\) GHz. For the boundary pinning field \(\tilde{H}_{\text{p}}\), the choice of \(\tilde{H}_{\text{p}}/JS=10^{-2}\) gives \(\tilde{H}_{\text{p}}=1.5\) K (or ∼10 kOe). According to the available experimental data, the number of periods in a domain with a definite chirality amounts to \(\mathcal{N}=20\), hence, \(L/a_{0}=2\pi\mathcal{N}/Q_{0}\simeq 785\). As a result, we deduce from Eq. (95) that \(\Omega_{0}\simeq 4\) GHz.

Because the sliding motion of the CSL accompanies the dynamical deformation of the spin texture, we naturally expect a spin motive force (SMF) to occur¹⁵²⁾ and to be strongly amplified in the configuration presented in Fig. 48. The external pinning may be realized by attaching FM bars on both sides of the monoaxial chiral helimagnet. The effective electric field that acts when the conduction electrons are adiabatically subjected to a spatially modulated spin structure along the z-axis has the form¹⁵³⁾ \begin{equation} E_{\sigma} = \frac{\hbar s}{2e}\sin \theta (\partial_{z}\theta \partial_{t}\varphi -\partial_{z}\varphi \partial_{t}\theta), \end{equation} (96) where \(\sigma =\pm\) denotes spin projections for majority (+) and minority (−) spins and e is the elementary charge. The general expression for the SMF is given by \begin{equation} \varepsilon_{\sigma}(t) = \int_{0}^{L}dz\,E_{s}(z,t). \end{equation} (97) Using Eqs. (71) and (72), the SMF acquires the form \begin{equation} \varepsilon_{\sigma}(t) = \frac{\hbar s}{2e}Q_{0}\dot{\xi}_{0}(t)\int_{0}^{L}dz\,u_{0}(z) = \alpha \frac{\pi \hbar}{e}Q_{0}\mathcal{N} \dot{Z}+\frac{B}{2eS}Z. \end{equation} (98) The second term is on the an order of \(\mathcal{O}(\mathcal{N}^{0})\) and can be neglected. Thus the eventual SMF for the majority spins is given as \begin{equation} \varepsilon (t) \simeq \alpha \Phi_{0}Q_{0}\mathcal{N}\dot{Z}, \end{equation} (99) where \(\Phi_{0}=h/2e\) is the magnetic flux quantum.

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Figure 48. (Color online) Schematic picture of amplification of the spin motive force by the resonant oscillation of the weakly pinned CSL.

Substituting the linear solution, (86), into Eq. (48), we obtain the SMF at the resonance frequency \(\Omega=\Omega_{0}\) as \(\varepsilon(t)=\varepsilon_{\text{max}}\cos(\Omega_{0}t)\) with the amplitude \begin{equation} \varepsilon_{\text{max}} \simeq \mathcal{N}\Phi_{0}\Omega_{0}\frac{\tilde{H}_{z0}S}{2\varepsilon_{0}^{(\theta)}}. \end{equation} (100) Taking the above values and the Josephson constant \(\Phi_{0}^{-1}=483.5979\) GHz/mV, we obtain the estimate \(\varepsilon_{\text{max}}=0.178\) mV.