Solid state physics, volume 66

Ovchinnikov† *Division of Natural and Environmental Sciences, The Open University of Japan, Chiba, Japan † Institute of Natural Sciences, Ural Federal University, Ekaterinburg, Russia 2.

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It is our great pleasure to present the 66th edition ofSolid State Physics Thevision statement for this series has not changed since its inception in 1955,andSolid State Physics continues to provide a “mechanism … whereby inves-tigators and students can readily obtain a balanced view of the whole field.”What has changed is the field and its extent As noted in 1955, the knowl-edge in areas associated with solid state physics has grown enormously, and it

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The topics reviewed in this volume illustrate the great breadth and sity of modern research into materials and complex systems, while providingthe reader with a context common to most physicists trained or working incondensed matter The editors and publishers hope that readers will find theintroductions and overviews useful and of benefit both as summaries forworkers in these fields, and as tutorials and explanations for those justentering

diver-ROBERTE CAMLEY ANDROBERTL STAMPS

ix

Theory of Monoaxial Chiral

Helimagnet

Jun-ichiro Kishine*,1, A.S Ovchinnikov†

*Division of Natural and Environmental Sciences, The Open University of Japan, Chiba, Japan

† Institute of Natural Sciences, Ural Federal University, Ekaterinburg, Russia

2.3 Microscopic Origins of the DM Interaction 11

3.2 Helimagnetic Structure for Zero Magnetic Field 14 3.3 Conical Structure Under a Magnetic Field Parallel to the Chiral Axis 15 3.4 Helimagnon Spectrum Around the Conical State 16

4.1 Chiral Soliton Lattice Under a Magnetic Field Perpendicular to the Chiral Axis 21 4.2 Commensuration, Incommensuration, and Discommensuration 26

4.4 Physical Origin of the Excitation Spectrum 33 4.5 Isolated Soliton Which Surfs Over the Background CSL 33

5 Experimental Probes of Structure and Dynamics of the CSL 35

6.2 Collective Sliding Caused by a Time-Dependent Magnetic Field 50 6.3 Mass Transport Associated with the Sliding CSL 55

8 Coupling of the CSL with Itinerant Electrons 73 8.1 Gauge Choice and One-Particle Spectrum 73

Solid State Physics, Volume 66 # 2015 Elsevier Inc.

8.2 Current-Driven CSL Sliding in the Hopping Gauge 77

Appendix A Brief Introduction to Jacobi Theta and Elliptic Functions 110

Appendix C Constrained Hamiltonian Dynamics 117 Appendix D Computation of the Spin Accumulation in Nonequilibrium State 120

1 INTRODUCTION

Symmetry-broken states with incommensurate modulation haveattracted considerable attention in condensed-matter physics Typical exam-ples are charge- and spin-density waves in metals, magnetic structures ininsulators, helicoidal structures in liquid crystals, and superconducting stateswith spatially nonuniform order parameters In spite of differences in micro-scopic origins, their physical properties are universally characterized by mac-roscopic phase coherence of the condensates and collective dynamicsassociated with them In particular, the condensates with multicomponentorder parameters are of special importance, because they have orientationaldegrees of freedom in physical space Consequently, not only amplitude butphase of the order parameter can exhibit long-range order Typical example

of such case is a helical magnetic structure (Fig 1), which is a main issue inthis article

The field of research on helimagnetic structure dates back to more than ahalf century ago Yoshimori [1], Kaplan [2], and Villain [3] interpreted anearlier report on magnetic structure of MnO2[4] as a helimagnetic structure.Since then, this field had been actively driven by neutron scattering mea-surements An early history of the field is well reviewed by Nagamiya [5].The microscopic origin of this class of helimagnets is the frustration amongdifferent superexchange interactions between localized spins or theRuderman–Kittel–Kasuya–Yosida (RKKY) interactions mediated by con-duction electrons Recently, the field of multiferroic materials has shed newlight on the frustration-driven noncollinear magnetic structures [6]

On the other hand, Dzyaloshinskii [7] found another class ofhelimagnetic structures which are stabilized by the antisymmetricDzyaloshinskii–Moriya (DM) interaction [8] The DM interaction origi-nates from the relativistic spin–orbit interaction [9] and imprint an asymmet-ric electronic structure to the antisymmetric spin–spin interactionDij
SiSj

between spins on sites i and j The constant vectorDijis called the DM tor The quantityχij¼SiSjis called the spin chirality which breaks chiralsymmetry The term “chiral symmetry breaking” means that space inversion(P) symmetry is broken, but time reversal (T ) symmetry combined with any properspatial rotation (R) is not broken, according to the definition of LaurenceBarron [10] Actuallyχijis odd under the parity transformationP, but evenunder time reversal operationT

vec-When the DMvectorDijhas a formDij¼ D^e with the D being constantand ^e being a unit vector along some crystallographic axis, competitionbetween DM interaction and the isotropic ferromagnetic (FM) coupling Jgives rise to a helical structure of spin magnetic moments Importantly,the direction of D determines whether spin magnetic moments rotate in

a left- or right-handed manner along the helical axis, thus providing chirality

to the given magnetic helix and creating a chiral helimagnetic (CHM) ture A necessary condition for this kind of DM vector to exist is that a mag-netic crystal belongs to a chiral space group Gχ whose symmetry elementscontain pure rotations only, i.e.,8g 2 Gχ, det g¼ 1 The concept of chirality,

struc-Mir ror

Figure 1 Left- and right-handed helimagnetic structure.

originally meaning left- or right-handedness, plays an essential role in metry properties of nature at all length scales from elementary particles tobiological systems.

sym-In a helimagnetic structure realized in a chiral crystal, the degeneracybetween the left- and right-handed helical structures, as shown in Fig 1,

is lifted at the level of Hamiltonian The macroscopic DM interaction comes

up in the Landau free energy as the Lifshitz invariant [7] Theoretical andexperimental achievements on this topic up to early 1980s are well reviewed

by Izyumov [11] Interestingly, Dzyaloshinskii’s work activated the researchfield of improper ferroelectricity where physical outcome of the Lifshitzinvariant had been intensively studied [12,13]

Despite the apparent similarity of spin structures, the helimagnetic tures of Yoshimori’s type and Dzyaloshinskii’s types have profound differ-ence in what level of chiral symmetry is broken In the Yoshimori type, chiralsymmetry is not broken at the level of Hamiltonian, but the helimagneticstructure spontaneously breaks chiral symmetry On the other hand, inthe Dzyaloshinskii’s (CHM) type, the Hamiltonian itself breaks chiral sym-metry because of the DM interaction and the magnetic structure is forced tobreak the chiral symmetry An essential feature of the chiral helimagneticstructure is that the structure is protected by crystal chirality The symmetrichelimagnet, however, does not have any macroscopic protectorate and iseasily fragmented into multidomains In Fig 2, we summarize basic prop-erties of symmetric and chiral helimagnets

This difference directly comes up in their magnetic structures undermagnetic fields and elementary excitations In particular, a significant differ-ence arises under a static magnetic field perpendicular to the helical axis Thesymmetric helimagnetic structure undergoes a discontinuous transition from

a helimagnet structure to a fan structure and then continuously approachesthe forced ferromagnetic configuration [5] On the other hand, in the chiralhelimagnet, the ground state continuously evolves into a periodic array ofthe commensurate (C) and incommensurate (IC) domains This state, a mainsubject of this article, has several names, i.e., chiral soliton lattice (CSL), heli-coid, or magnetic kink crystal (MKC) [7,11] Throughout this article, weuse the term chiral soliton lattice As the magnetic field strength increases,the spatial period of CSL increases and finally goes to infinity at the criticalfield strength This situation is depicted inFig 3

After almost a half century since the theoretical prediction [7], mental observation of the CSL was achieved by Togawa et al in the hexagonalhelimagnet CrNb3S6[14] which has magnetic phase transition temperature

perpendic-TC¼ 127 K and its helical pitch is 48 nm In this compound, ferromagneticlayers are coupled via interlayer weak exchange and DM interactions In thiscase, the formation of the CSL is observed by using Lorenz microscopy Thespatial period of the stripe corresponds to the period of the CSL The mag-netic field dependence of the period gives a clear evidence that a chiralhelimagnetic structure under zero filed continuously evolves into the CSLand finally undergoes a continuous phase transition to commensurateforced-ferromagnetic state at a critical field strength Hc 2300 Oe.

The CSL has some special features to be noted (1) In the CSL state, thetranslational symmetry along the helical axis is spontaneously broken There-fore, the corresponding Goldstone mode becomes phonon like [7,15] (2)The CSL state has infinite degeneracy associated with arbitrary choice of thecenter of mass position Consequently, the CSL can exhibit coherent slidingmotion [16] (3) The CSL exerts a magnetic super-lattice potential on theconduction electrons coupled to the CSL This coupling may cause amagnetoresistance effect [17,18] (4) Quantum spins carried by conductionelectrons cause spin-transfer torque on the CSL [19]

Here, we will review physical properties of the CSL from theoreticalviewpoints The remaining part of the review will be divided into nine sub-sections InSection 2, we will describe the symmetry-based views on chiralhelimagnetism InSection 3, we discuss helical and conical structures under

a magnetic field parallel to the chiral axis InSection 4we will describe theground state and elementary excitations associated with the CSL [feature (1)mentioned above] InSection 5, we will review some experimental probes

of structure and dynamics of the CSL In Section 6, we review physicalproperties of the sliding CSL [feature (2)] and discuss a possible spin motiveforce driven by the sliding motion (Section 7) InSection 8, we discuss thecoupling of the CSL with itinerant electrons [features (3) and (4)] In

Section 9, we consider the case where the CSL is confined in a finite system.Finally, we conclude and discuss the meaning of chirality in modern physicsfrom broader viewpoints (Section 10) We will leave some supplementary ortechnical materials to appendices

2 CHIRAL SYMMETRY BREAKING IN CRYSTAL ANDCHIRAL HELIMAGNETIC STRUCTURE

2.1 Magnetic Representation of Chiral HelimagneticStructure

Quantum spin state for spin S¼ 1/2 is described by a two-component spinor

in SU(2) space, parameterized by polar angles,

S ¼ χh j^S χj i ¼ S^n, ^n ¼ sinθ cosφ,sinθ sinφ,cosθð Þ: (2)This classical axial vector enters a macroscopic Maxwell equations as a mag-netic momentM¼gμBS It is to be noted that whenever we talk about M,permutation symmetry, which is purely quantum, is totally lost and insteadthe parametersφ and θ have meaning as polar angles φ(r) and θ(r) tied to aspatial positionr in O(3) space A purpose of magnetic representation theory

is to classify possible ordering of theM vector as an order parameter.The chiral helimagnetic structure is an incommensurate magnetic struc-ture with a single propagation vectork¼(0,0,k) The chiral space group Gχ

consists of the elements {gi} Among them, some elements leave the agation vectork¼(0,0,k) invariant, i.e., these elements form the little group

prop-Gk The magnetic representationΓmagis written asΓmag¼ Γperm Γaxial,whereΓperm and Γaxialrepresent the Wyckoff permutation representationand the axial vector representation, respectively [20] Then, Γmag isdecomposed into the nonzero irreducible representations of Gk Theincommensurate magnetic structure is determined by a “symmetry-adaptedbasis”of an axial vector space and the propagation vectork In a specific mag-netic ion, the decomposition becomesΓmag¼P

iniΓi, whereΓiis the ducible representations of Gk The chiral helimagnetic structure,

irre-M¼ Me1cosðkzÞ  Me2 sinðkzÞ ¼ MRe ðe 1 ie2Þeikz

(+ and signs correspond to left- and right-handed helix) requires real dimensional or complex one-dimensional symmetry-adapted basis,e1ande2.For these basis to exist, the group elements of Gk0include three- (C3), four-(C4), or sixfold (C6) rotations Therefore, among 65 chiral space groups whoseelements are all proper rotations (for 8g 2 G, det g ¼ 1), 52 space groupsbelonging to cubic, hexagonal, tetragonal, and trigonal crystal classes are eli-gible to accommodate the chiral helimagnetic structure This situation isdepicted inFig 4 The cubic class is special because there are four C3axes,although hexagonal, tetragonal, and trigonal crystals have only one principalaxis In the latter case, a monoaxial helimagnetic structure as shown inFig 1is

two-expected to be favored Difference in crystal symmetry comes up as a form ofthe Lifshitz invariant in the effective action (see next section).

2.2 Examples of Chiral Helimagnets

As stated above, chiral helimagnets are realized in a crystal with higher metry which contains atomic building blocks with low symmetry InTable 1,

sym-we give examples of chiral helimagnets which are known so far

In particular, we give detailed information about one of the most activelystudied magnetic compound CrNb3S6 This material has a hexagonal layeredstructure 2H-type NbS2, intercalated by Cr atoms, belonging to the spacegroup P6 22 as shown inFig 5 Studies of the material were started at the

Cubic

Figure 4 For a chiral helimagnetic ordering to be realized, the crystal point group needs

to have two-dimensional (or complex one-dimensional) irreducible representations This means it is required for the point group elements to have three- (C 3 ), four- (C 4 ),

or sixfold (C6) axis Correspondingly, (A) cubic, (B) hexagonal, (C) tetragonal, and (D) trigonal crystal classes are eligible to accommodate the chiral helimagnetic struc- ture Helices and arrows indicate how helical axis can reside in the crystal.

Table 1 Crystal class, space group, magnetic transition temperature to the helimagnetic state (Tc), helical pitch under the zero-field (L(0)) of known chiral helimagnets Classifications of the microscopic origins of the DM interaction are represented by A, B, C (see section 2.3 ) Note that CuB2O4does not belong to chiral space group but the possibility of an anti-helical structure is pointed out [see Ref 23 ]

Compound Crystal Class Space Group Tc[K] L(0)[nm] Type Refs.

beginning of 1970s, when the method of chemical gas transportation was used

to get single crystals CrNb3S6[26] In this work measurements of neutron andmagnetic properties were carried out, it has been found that it is a helimagnetwith a large period along the c-axis and spins rotating in the perpendicular(ab)-plane The Curie temperature is 127 K The saturation magnetization

is equal to 2.9μBper Cr atom Small-angle neutron scattering indicates anexistence of a helical structure with Q0¼ 0.013 A˚1, i.e., with the spatialperiod 480 A˚ The authors have concluded that such a long-periodic modu-lation is caused by the antisymmetric Dzyaloshinsky–Moriya exchange inter-action In the unit cell shown inFig 5, the Cr atoms have a trivalent state andthe localized electrons form a spin S¼ 3/2 Initially, data on the magnetizationprocess in a perpendicular magnetic field were misinterpreted as a manifesta-tion of first order phase transition between the helical ordering and the state offorced ferromagnetism [25] The interpretation of a sharp change of the mag-netization curve within the scenario of the magnetic soliton lattice was given

in the papers [31,32] Note that a similar situation arises with an explanation ofthe magnetic properties of the Ba2CuGe2O7, where an existence of the mag-netic soliton lattice was also confirmed [33]

InFig 6A, we sketch the original crystal structure of CrNb3S6 Nb and

Cr occupy special points with high symmetry (Wyckoff positions of two Nbsites and Cr sites are 2a/4f and 2d, respectively) On the other hand, the

S atom occupies a general point with the lowest symmetry (Wyckoff tion is 12i) and its atomic coordinate is (0.000350,0.667770,0.369130)

posi-We see a quite tiny chiral symmetry breaking arises To exaggerate the chiralsymmetry breaking, inFig 6B, we show the crystal structure with the S’s

Nb S

Cr A

B

b a c

Figure 5 The scheme of the crystal and magnetic structure in CrNb 3 S 6 (A) The tary cell of the crystal The spins of localized electrons rotate in the (ab) plane around the helical c-axis due to a presence of the Dzyaloshinsky –Moriya exchange (B).

elemen-atomic coordinate being modified to (0.100350,0.667770,0.369130)without changing its original space group symmetry In this case, we clearlyrecognize that there exists a helical arrangement of S atoms InFig 6C, weshow the same structure from another viewpoint It is seen that Cr ion issurrounded by S atoms in chiral manner.

Another important aspect of CrNb3S6 is its classical one-dimensionalnature as a magnetic network Quasi-one-dimensional systems are regarded

as a bunch of weakly coupled quantum 1D systems as shown inFig 7A Onthe other hand, as shown inFig 7B, when two-dimensional layered mag-netic structures are weakly coupled via the interlayer exchange and DMinteractions, the system is well described as a classical 1D system The lattercase is actually realized in CrNb3S6[34] This situation makes it legitimate totreat this system as a classical 1D chiral helimagnet

Another real system, that is in the focus of current investigations, is themanganese silicide (MnSi) The intermetallic compound belongs to the class

of band magnetic materials with a low Curie temperature (around 29 K) and

a low magnetic moment (approximately 0.4μB) per Mn atom This class ofmagnetic materials includes other popular compounds FeGe, Fe1xCoxSi,where the helimagnetic order was as well detected These systems possessthe space group P213, which does not contain a center of symmetry, andthe fact provides an appearance of chiral magnetic structures One of theremarkable features of MnSi is a deviation from the Fermi liquid behavior

in the paramagnetic phase [35,36] and a presence of a short-range spin order

in this phase [37]

c a

b c a

b c

Figure 6 (A) Original form of right-handed crystal of CrNb3S6 (B) Fictitious crystal out changing the original symmetry, where atomic coordinates of S atom is modified from its original (0.000350, 0.667770, 0.369130) to (0.100350, 0.667770, 0.369130)

with-to visualize chiral symmetry breaking (C) Side view of a unit cell of the fictitious crystal Helical arrangement of S atoms is clearly visible.

Helicoidal magnetic order in MnSi was discovered a long time ago [28].Recent studies of small angle neutron scattering in the so-called A-phasehave attracted an interest to the Skyrmion model [38] The magnetic struc-ture of MnSi in a zero magnetic field can be presented as a set of fer-romagnetically ordered planes arranged parallel to the crystallographicplane (111) As a result, in the magnetically ordered phase spins form aleft-handed helix with the incommensurate wave vector 0.036 A˚1 thatcorresponds to the spatial period 188 A˚ in the [111] direction A moredetailed survey of properties of the manganese silicide may be found inthe review [39].

2.3 Microscopic Origins of the DM Interaction

We here briefly summarize possible microscopic origins of the DM tion Chiral magnetic crystals are classified into three classes, i.e., Type A:insulator, Type B: metal with coexisting localized and itinerant spins, andType C: metal with only itinerant spins We depict these three cases in

interac-Fig 8 Type A corresponds to the case originally discussed by Moriya [9]

In this case, the Hamiltonian which describes two magnetic ions is

Figure 7 (A) A system of weakly coupled chains is treated as a quantum quasi dimensional system (B) The system of weakly coupled layers is treated as a (C) classical quasi one-dimensional system.

one-H0¼ λS1
L1+λS2
L2 JS1
S2, (4)where λ and J are strengths of spin–orbit and ferromagnetic couplings,respectively Then, the DM vector is obtained via the second-order pertur-bation theory as

where g and n label the ground and excited states, respectively

In the case of Type B, the particle-hole fluctuations of the itinerant trons, as shown inFig 8D, mediate the DM interaction between the local-ized spins This is a generalized RKKY interaction In this case, the crystalsymmetry is embedded in the complex one-particle hopping and the resul-tant DM interaction should appropriately reflect the crystal symmetry [40].The case of Type C is the most nontrivial [41] We expect that after inte-grating out the one-particle degrees of freedom with spin–orbit couplingbeing treated as a perturbation a coupling of the spin fluctuations, as shown

elec-inFig 8E, eventually has an effective form of the DM interaction [42] InTable1, we indicated which type real examples belong to

3 HELICAL AND CONICAL STRUCTURES

3.1 Model

From now on, as a canonical example of a monoaxial chiral helimagnet, weconsider the case of CrNb3S6 Then, we start with a Hamiltonian whichdescribes a weakly coupled layered system,

μB¼ ej jℏ=2m is the Bohr magneton.

Based on the picture of semiclassical 1D model as shown inFig 7C, weassume Jk J? Actually, a classical Monte-Carlo simulation [34] wasrecently done and it was shown that Jk’ 8.0K, J?’ 70K, and D ’ 1.3K

to describe experimentally found magnetization curve By taking a limit

J?! 1, the dynamical fluctuations inside the same layer are totally frozenand a rigid in-layer ferromagnetic arrangement is established Then, we canomit the site dependence of the spin variables inside the layer and drop the J?

term from the Hamiltonian (6) Now it is legitimate to make the reduction,

Taking Nxand Ny as the number of lattice sites along x and y directions,respectively, this simplification leads to the effective one-dimensionalHamiltonian,H ¼ H3D=NxNy, written as

represents an easy plane anisotropy energy which is

included only when we consider the spin wave spectrum

The representation expressed by (9) indicates that the DM interactionplays a role of a Peierls phase (or an external gauge field) connecting S+

j

and Sj + 1 The appearance of the Peierls phase in this representations is a ural consequence of the fact that the helimagnetic structure carries staticmomentum Q0 which characterizes the condensate This phase is gaugedaway by the global gauge transformation,

(note that when we

per-form this transper-formation, we should read Sjas a quantum operator ^Sj ) Thistransformation maps the helimagnetic chain to the ferromagnetic XXZ [43]

3.2 Helimagnetic Structure for Zero Magnetic Field

Because the effective spin ^Sjis regarded as a spin with large amplitude, tum fluctuations are strongly suppressed and therefore it is legitimate to treat

quan-it as the classical vector, i.e., the replacement ^Sj! Sj may be legitimate,whereSjis a semiclassical axial spin vector In this case, using the polar coor-dinatesθ and φ as shown inFig 9, we represent the spin vector asSj¼ S nj,where the unit vector field nj is nj¼ sinθjcosφj, sinθjsinφj, cosθj

: ForH¼0, the first term of the r.h.s of Eq (9) becomes minimum for

Sj¼ SeiQ 0 z jsinθj (zj¼ a0j) which gives the total energy

which becomes minimum for θj¼ π/2 (note J> J) This state corresponds

to a chiral helimagnetic structure, Sj¼ S cos Q0zj

48 nm in the case of CrNb3S6[14] Furthermore, in the case of CrNb3S6,

a0¼ 1.212 109

L ¼ 4.8 108m give D=J ¼ tanðQ a Þ ¼ 0:16

3.3 Conical Structure Under a Magnetic Field Parallel to theChiral Axis

To consider the magnetic structure under a magnetic field parallel to the ral axis,H¼(0,0,Hz), we add an easy plane anisotropy energy K?X

This state is depicted inFig 10 For Hz> Hz

c, the ground state is a forcedferromagnetic state, where all the spins are parallel toH In the conical state,chiral symmetry in the spin space is forced to be broken by the crystalsymmetry The magnetic field parallel to the chiral axis causes no additionalsymmetry breaking

z j z

3.4 Helimagnon Spectrum Around the Conical State

From now on, we consider the dynamical properties associated with theground states discussed in the previous section Reflecting the symmetrybreaking patterns, elementary excitations around the conical state andCSL state are totally different In the case of the conical state, the elementaryexcitations are described as a Goldstone mode to retrieve rotational symme-try breaking On the other hand, dynamical properties of the CSL state aremuch richer because of the translational symmetry breaking We will discussthem separately

In this section, we consider the spin wave spectrum around the conicalstate described by Sj ¼ SeiQ 0 z jsinθ0with a cone angleθ0being given by

Eq (14) We here stress that the broken symmetry associated with the ical state is the rotational symmetry around the helical axis In Eq (10), wecan allow the constant initial phase in the transverse spin distribution as

con-Sj¼ Sei Qð 0 z j + ϕ 0Þsinθ Then, the system has infinite numbers of ate ground states associated with arbitrarily choice ofϕ0 Then, we expectthat there appears gapless helimagnon mode as the Goldstone mode associ-ated with the broken continuous symmetry This point was discussed byElliott and Lange [44] for the case of symmetric (Yoshimori-type)helimagnet The same thing also happens in the chiral helimagnet but thedispersion spectrum quite differs from the symmetric case [41,45] At first,

degener-to capture the intuitive properties of the helimagnon excitations, we followthe equation-of-motion method adopted by Nagamiya [5] for the symmet-ric helimagnet

To compute the spectrum, we rotate the basis frame of the crystalcoordinate {e+

jg where the direction of ez

j points to the equilibrium spindirection at the j-th site The transformation to the local frame at the i-thsite is determined as

with a first rotation about z by an angle Q0j followed by a second rotationabout y by an angleθ0 The spin vector

Sj¼ S+

j e++ Sje+ Szez¼ Sj+e++ Sj e+ Szjez, (17)has the components

+ cos2θ 0 S z

j S z j

To find the propagating solutions, we use the ansatz

InFig 11A, we show the helimagnon dispersion for Hz¼ 0, 0:7Hz

Figure 11 (A) Spin wave (helimagnon) spectrum for different H0: The equilibrium state

is conical for H0< H0c , while forced ferromagnetic for H0 H0c (B) Amplitude ratio

continuously to the quadratic dispersionℏωk¼ 2 JS 1ð  coskaÞ at Hz¼ Hz

c.The Goldstone mode at k¼ 0 corresponds to the rigid rotation of the wholehelix For Hz Hz

c, the equilibrium state is the forced-ferromagnetic stateand the spin wave spectrum acquires the field-induced gap

To understand the nature of the spin wave excitations, it is useful to seethe ratio of the precession amplitudes,

qj ! 0 as q ! 0 for 0 Hz< Hz

c and the spins tend to be confined

to the helical plane However, as Hzapproaches the critical strength Hcz, thedip around q¼ 0 becomes narrower and eventually vanishes toward

Hz¼ Hz

c, where the precessional trajectory becomes a perfect circlecorresponding to the ferromagnetic spin wave InFig 11C, we show a typ-ical precession trajectory of the spin vector in the case of Hz¼ 0 As shown in

Fig 11D, it is clearly seen that as q departs from zero to π, the precessiontrajectory changes from a flat ellipsoidal shape to a more circular one Onthe other hand, the spin wave for q¼ 0 corresponds to rigid rotation ofthe whole helix around the chiral axis, which is a Goldstone mode toretrieve the broken rotational symmetry, just as in the case of symmetric hel-imagnet [5, 44] For Hz> Hz

c, the spin wave spectrum acquires the gap

ℏωq¼0¼ SHz The conical state has rotational degeneracy around the helicalaxis, but the ferromagnetic state is coaxial with the magnetic field and has nochance to feel rotational symmetry This is the reason why the gapless Gold-stone mode vanishes for Hz> Hz

c.The same result (27) is obtained by using the Holstein–Primakofftransformation

s

which reproduces Eq (27)

3.5 Spin Resonance in the Conical State

In the spin resonance experiments, the static magnetic fieldH0is applied tocause Larmor precession of magnetic spins Then supplying electromagneticenergy carried by microwave radiation, resonant absorption occurs at theprecession frequency The microwave is described as the uniform oscillatingmagnetic field, or the r.f field,h(t) polarized in the direction perpendicular

to H0 (Faraday configuration) The r.f field gives rise to the Zeemancoupling with spin,

The imaginary part of the dynamical susceptibility,

Now let us consider the spin resonance in the conical state In this case,the magnetic field is applied parallel to the helical axis (z-axis) and the r.f.field is polarized along the y-axis Then, the elementary excitations aredescribed by spin waves of the conical magnetic structure A quantized spinwave is called a helimagnon Then, the ESR spectrum is given by

uQ0¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

There is the single branch of resonance energy as shown inFig 12

4 CHIRAL SOLITON LATTICE

4.1 Chiral Soliton Lattice Under a Magnetic Field

Perpendicular to the Chiral Axis

Next we consider the case where the magnetic field is perpendicular to thechiral axis Because the spatial modulation of magnetic structure is quite slow

as compared with the atomic scales (L0/a0’ 40 for CrNb3S6), it is legitimate

to introduce the continuous field variable

0 dz, we obtain a continuumversion of the Hamiltonian (339) per unit area, H ¼RL

0 dzH, where Ldenotes the whole length of an effective 1D chain system (see Fig 7C).Here, we note that we are considering an effective 1D system and the inte-gration over x- and y-directions are implicitly taken into account TheHamiltonian density is then written as

is described by the Hamiltonian (44) withθ(z) being fixed to π/2, i.e.,

0 0.1 0.2 0.3 0.4 0.5

1

hwQ0 2JS∼

Figure 12 Field dependence of the helimagnon resonance energy.

where K¼ K κð Þ is the complete elliptic integral of the first kind Using

LCSL, the dimensionless coordinatez is related to z as z ¼ qCSLz, where

has a physical meaning as a wave number of the CSL structure As we will seeshortly, the value ofκ is determined by the field strength and therefore theperiod is a function of Hx The zero field period is

Using the relations, cn2z¼ 1  sn2

z and dn2z¼ 1 κ2

sn2z, we havealignments of the spins,S zð Þ ¼ S^n , where

^n0¼ cosφð 0ðzÞ,sinφ0ðzÞÞ ¼ 1  2snð 2z,2snzcnzÞ (53)which gives the CSL state For the later purpose, it is useful to note

0.4 0.6 0.8 1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

0.2 0.4 0.6 0.8 1.0

Figure 13 (A) Spatial distribution of the phase φ(z) in the CSL state and (B) the corresponding topological charges (C) Field dependence of the spatial period of the CSL and (D) the soliton density.

translational symmetry is spontaneously broken by forming an rate lattice of the topological charges.

incommensu-At this state, the elliptic modulusκ is nothing more than a constant ofintegral Its value is determined by minimizing the energy with respect

,

(58)

where we used the definition of the elliptic integral of the second kind,

Z K 0

LCSL¼ 2κK

ffiffiffiffiffiffiffiffiJSa20

c (κ ! 1) Thisbehavior is indicated by the solid line inFig 13C In the IC-C phase tran-sition, the soliton density L0/LCSLplays a role of the order parameter Tomake clear this point, we show the plot of L0/LCSLinFig 13D

The spin arrangements given by Eq (53), together with (63) and (65),are shown in Fig 3 The magnetization averaged over the spatial period

is quite peculiar to the chiral helimagnet and is regarded as an indirectevidence of chiral helimagnetism

4.2 Commensuration, Incommensuration, and

Discommensuration

It is here useful to have a view of the CSL in terms of monoaxial phase tures In the case of quasi-one-dimensional electron systems, it is wellknown that incommensurate spin/charge-density waves are formed bycoherent superposition of electron-hole pairs [48] This field of researchwas pioneered by an idea of sliding density-wave by Fr€olich [49] In such

tex-cases, the notions of C, IC and DC play essential roles [50] Naturally, thetopic of quasi-one-dimensional solids overlaps that of chiral helimagnetism.

InFig 15, we show typical phase textures in 1D system

4.3 Elementary Excitations Around the CSL

Because of translational symmetry breaking, the CSL exhibits rich dynamicalproperties There are mainly three types of excitations We show them sche-matically inFig 16 First, each soliton vibrates around its stable position andits correlated motion leads to phonon-like excitations of solitons (Fig 16A)

We call this “chiral soliton lattice phonon.” Second, there is an isolatedsoliton which surfs over the background CSL (Fig 16B) Third, the CSLexhibit collective sliding motion, i.e., the whole CSL moves in a coherentmanner (Fig 16C) We will discuss these three types of dynamical motions

To consider small dynamical fluctuations around the CSL, we introducetheδθ z,tð Þ (out-of-plane) and δφ z,tð Þ (in-plane) fluctuations of the localspins around the stationary soliton lattice configurationn0(z),

φ z,tð Þ ¼ φ0ð Þ + δφ z,tz ð Þ, θ z,tð Þ ¼ θ0+δθ z,tð Þ, (68)where θ0¼ π/2 The terms “out-of-plane” and “in-plane” are used withrespect to the helical xy-plane We depict these fluctuations in Fig 17

with the Gaussian fluctuations, 0,θ0

ation part is described by

δH ¼ JS2a0

2

Z L 0

dzδφ^Λφδφ + δθ^Λθδθ : (69)Here, the linear differential operators are given by

zφ0

+ 2q0ð@zφ0Þ ¼ 4q2

CSLdn2z + 4q0qCSLdnz, (72)where we used Eq (56) This gap has its direct origin in the DM interaction

InFig 18, we show a spatial profile ofΔðzÞ for some values of Hz=Hz

D(z)

Figure 18 Spatial profile of Δ(z) for H z =H z ¼ (a) 0, (b) 0.4, (c) 0.8, (d) 0.99, and (e) 1  10 7

We see that the soliton parts (kinks) contribute to the gap formation,while the forced-ferromagnetic (commensurate) domains do not For anarbitrary value of Hx=Hx

c, the average of Δ(z) over its spatial period iscomputed as

ΔðHxÞ

2K

Z 2K 0

vαðzÞ ¼ uαðzÞ ¼ Nϑ4 π

2Kðz  ζαÞ

ϑ4 π2Kz

where ϑ4 is the theta function and N is a normalization constant

qα¼ κ=mð Þqα is a Floquet index which has a physical meaning as a wavenumber ζαis a “shift parameter.” The solution is derived inAppendix B

(see Eq (B.7))

Using this orthonormal basis, the fluctuations (φ-mode and θ-mode) arespanned by the orthogonal eigenfunctions vq(z) and uq(z) as

Using Eq (B.9) inAppendix B, we obtain the energy spectrum ofφ and

θ fluctuations which are depicted in Fig 19A and B, respectively In theweak field approximation,εðθÞ

Theφ-mode spectrum, εðφÞ

q , consists of an “acoustic” and “optical” branch.The acoustic band is formed out of correlated translations of the individualkinks, while the optical band corresponds to renormalized Klein–Gordonbosons [51]

The acoustic branch has a shift parameterζα¼ iα + K (0 α < K0) and awave number

Acoustic : qα¼m

κhZðα,κ0Þ + π2KK0αi, (81)where qj j changes from 0 to Gα 0 2q0/8KE¼ π/LCSLas α changes from

0 to K0 The zeta-function is defined as Zðz,kÞ ¼ Eðz,kÞ  E=Kð Þz, where

E(z, k) is the fundamental elliptic integral of the second kind Accordingly,the energy spectrum is

 2

κ02sn2ðα,κ0Þ, (82)which changes from 0 toJS

 2

sn2ðα,κ0Þ, (84)which changes from JS

2a0

2

mκ

at the boundary of the Brillouin zone of CSL

The normalized wave function at the bottom (q¼ 0) of the acoustic band is

v0ðzÞ ¼ u0ðzÞ ¼ L1=2

ffiffiffiffiKE

In conventional terminology, the zero mode means a mode excited with

no excess energy In the present case, the in-plane v0(z) mode exactly sponds to this case, but the out-of-plane u0(z) zero mode acquires the gap

8S

where we used the relation (80) By this reason, the u0(z)-mode is called asquasi-zero mode [15] It is to be noted that the CSL phonon modes are

orthogonal to the zero mode For CrNb3S6, the excitation gap is estimated as

a0εðθÞ0 ’0.38 K by using Eq (87)

4.4 Physical Origin of the Excitation Spectrum

It may be instructive to mention a physical origin of the excitation spectrum

To see this, note the formula [47],

4.5 Isolated Soliton Which Surfs Over the Background CSL

This type of soliton solution (Fig 16B) is obtained by using the Ba¨cklundtransformation [53] The Ba¨cklund transformation is a powerful method

to construct nonlinear solutions from a given partial differential equation

In particular, multisoliton solutions are systematically obtained in thesine-Gordon model in 1 + 1ð Þ space-time dimensional Using light-cone

Reflectionless transmission

= optical band Reflectionless transmission

Wave function

Figure 20 Physical origin of the excitation spectrum of the CSL.

coordinates x+¼ x + tð Þ=2, x¼ x  tð Þ=2, the sine-Gordon model is ten simply as @+@φ ¼ sinφ: When there are two independent solutions

writ-φ0andφ1, they satisfy a relation,

ð Þ-dimensional sine-Gordon model, by choosing a constant solution

φ0¼const., Ba¨cklund transformation gives a soliton and antisoliton solution.Based on this algorithm, by choosing a CSL solution asφ0, we can construct

a soliton solution surfing over the CSL InFig 21, we show a profile of thissolution

The soliton we find is obtained as an output of the Ba¨cklund using theCSL solution as an input We emphasize that the soliton has definite chiral-ity, because of the presence of the DM term in the Hamiltonian density

Eq (203) The presence of this term lifts the degeneracy between theleft-handed soliton and the right-handed antisoliton solutions In the class

of the found solutions, we identify the soliton with the intrinsic boost formation An essential point is that the traveling soliton cannot exist without the

Figure 21 A spatial profile of an isolated soliton before (A) and after (B) the collision with a CSL.

soliton lattice as a topological background configuration It means that the nontrivialtopological object is excited over the topological vacuum The standing CSLenables the new soliton to emerge and transport the magnon density Ascompared with the motion of the whole kink crystal with a heavy mass[16, 54], our new soliton is a well localized object with a light mass Thisnew traveling soliton can be regarded as a promising candidate to transportmagnetic information by using chiral helimagnet.

5 EXPERIMENTAL PROBES OF STRUCTURE AND

DYNAMICS OF THE CSL

5.1 Transmission Electron Microscopy

To analyze the problem of the observation of the chiral helimagnet by mission electron microscopy, we calculate the optical phase shift by using theFourier method [55,56] A possibility to analyze a periodic array of magneticdomains, for example, stripe magnetic domains, is the main advantage of theapproach [57,58]

trans-The magnetic phase shift can be calculated from the standard Aharonov–Bohm expression [59,60]

ϕ ¼ eℏ

mag-We consider the specimen in form of a thin slab of constant thickness 2d

As customary in electron optics, the electron beam passing through the film isparallel to the y-axis, the object (x,z) plane is perpendicular to the incidentbeam The magnetization vector inside such a slab varies along the z-axis with

no magnetization variation in the (x,y)-plane Likewise we assume that thespecimen is infinite along the z-axis and has a finite width 2Lxalong the x-axis

In the geometry, the magnetic phase shift (91) depends on the y nent of the magnetic vector potential The latter is related with themagnetization

compo-AðrÞ ¼μ0

4π

ZMðr0Þ  r r0

r  r0

which takes a simple form in the Fourier space

AðkÞ ¼ iμk20M ðkÞ  k: (93)

In this way, the Fourier transform of the magnetic phase shift can beobtained and Fourier inverted to obtain a result in the real space

The magnetization can be described as

MðrÞ ¼ M0ðsinφðzÞ,cosφðzÞ,0Þ QðxÞUðyÞ, (94)where M0is the saturation magnetization, and the winding angle is given by(49) The magnetic field, that forms the soliton lattice, is directed along they-axis In Eq (94), the Heaviside’s step function

MxðkÞ ¼

Z

dr MxðrÞei k
r¼ M0HðkzÞ QðkxÞ ~UðkyÞ, (97)where QðkxÞ ¼ 2sin kð xLxÞ=kx, and ~UðkyÞ ¼ 2 sin k yd

=ky.The function

GCSL¼ 2π

KqCSL¼ π2

where K0 denotes the complete elliptic integral of the first kind with thecomplementary elliptic modulusκ0¼ ffiffiffiffiffiffiffiffiffiffiffiffi

1 κ2

p

.From Eqs (93) and (97), we obtain the Fourier transform for the vectorpotential

(100)Making the inverse Fourier transformation and plugging the result in

Eq (96) the magnetic phase shift can be written as follows in the limit

Lx! 1

ϕðx,zÞ ¼ πμ0M0d

ϕ0QCSL

2πκK

n¼1

cosðnGCSLzÞcoshðπnK0=KÞ

by the object For a generic input signal exp iφðx,zÞ½

image, square amplitude of the output signal,

c, the equilibrium state is the forced-ferromagnetic stateand the spin wave spectrum acquires the field-induced gap

To understand...

ℏωq¼0¼ SHz The conical state has rotational degeneracy around the helicalaxis, but the ferromagnetic state is coaxial with the magnetic field and has nochance... class="page_container" data-page="23">

s

which reproduces Eq (27)

3.5 Spin Resonance in the Conical State

In the spin resonance experiments, the static magnetic fieldH0is applied