Solid state physics, volume 66

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CONTRIBUTORS
Numbers in parenthesis indicate the pages on which the authors’ contributions begin
Ke Jiang (131)
Center for Biofrontiers Institute, University of Colorado at Colorado Springs, Colorado
Springs, Colorado, USA
Hamid Kachkachi (301)
PROMES, CNRS-UPR 8521, Universite´ de Perpignan Via Domitia, Rambla de la
Thermodynamique—Tecnosud, Perpignan, France
Jun-ichiro Kishine (1)
Division of Natural and Environmental Sciences, The Open University of Japan, Chiba,
Japan
A.S. Ovchinnikov (1)
Institute of Natural Sciences, Ural Federal University, Ekaterinburg, Russia
Anatoliy O. Pinchuk (131)
Center for Biofrontiers Institute, Department of Physics, University of Colorado at
Colorado Springs, Colorado Springs, Colorado, USA
Raymond C. Rumpf (213)
EM Lab, University of Texas at El Paso, El Paso, TX, USA
David S. Schmool (301)
PROMES, CNRS-UPR 8521, Universite´ de Perpignan Via Domitia, Rambla de la
Thermodynamique—Tecnosud, Perpignan, France

vii


PREFACE
It is our great pleasure to present the 66th edition of Solid State Physics. The
vision statement for this series has not changed since its inception in 1955,

and Solid State Physics continues to provide a “mechanism … whereby investigators 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 knowledge in areas associated with solid state physics has grown enormously, and it
is clear that boundaries have gone well beyond what was once, traditionally,
understood as solid state. Indeed, research on topics in materials physics,
applied and basic, now requires expertise across a remarkably wide range
of subjects and specialties. It is for this reason that there exists an important
need for up-to-date, compact reviews of topical areas. The intention of these
reviews is to provide a history and context for a topic that has matured sufficiently to warrant a guiding overview.
The topics reviewed in this volume illustrate the great breadth and diversity of modern research into materials and complex systems, while providing
the reader with a context common to most physicists trained or working in
condensed matter. The editors and publishers hope that readers will find the
introductions and overviews useful and of benefit both as summaries for
workers in these fields, and as tutorials and explanations for those just
entering.
ROBERT E. CAMLEY AND ROBERT L. STAMPS

ix


CHAPTER ONE

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
1
Corresponding author: e-mail address:

Contents
1. Introduction
2. Chiral Symmetry Breaking in Crystal and Chiral Helimagnetic Structure
2.1 Magnetic Representation of Chiral Helimagnetic Structure
2.2 Examples of Chiral Helimagnets
2.3 Microscopic Origins of the DM Interaction
3. Helical and Conical Structures
3.1 Model
3.2 Helimagnetic Structure for Zero Magnetic Field
3.3 Conical Structure Under a Magnetic Field Parallel to the Chiral Axis
3.4 Helimagnon Spectrum Around the Conical State
3.5 Spin Resonance in the Conical State
4. Chiral Soliton Lattice
4.1 Chiral Soliton Lattice Under a Magnetic Field Perpendicular to the Chiral Axis
4.2 Commensuration, Incommensuration, and Discommensuration
4.3 Elementary Excitations Around the CSL
4.4 Physical Origin of the Excitation Spectrum
4.5 Isolated Soliton Which Surfs Over the Background CSL
5. Experimental Probes of Structure and Dynamics of the CSL
5.1 Transmission Electron Microscopy
5.2 Magnetic Neutron Scattering
5.3 Muon Spin Relaxation
5.4 Spin Resonance in the CSL State
6. Sliding CSL Transport
6.1 Lagrangian for Sliding CSL
6.2 Collective Sliding Caused by a Time-Dependent Magnetic Field
6.3 Mass Transport Associated with the Sliding CSL
7. Spin Motive Force
7.1 General Formalism
7.2 Spin Motive Force by the CSL Sliding

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

Solid State Physics, Volume 66
ISSN 0081-1947
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Jun-ichiro Kishine and A.S. Ovchinnikov

8.2 Current-Driven CSL Sliding in the Hopping Gauge
8.3 Magnetoresistance in the sd Gauge
9. Confined CSL
9.1 Quantization of the CSL Period and Magnetization Jumps
9.2 Resonant Dynamics of Weakly Confined or Pinned CSL
10. Summary and Future Directions
Acknowledgments
Appendix A. Brief Introduction to Jacobi Theta and Elliptic Functions
Appendix B. LAME Equation
Appendix C. Constrained Hamiltonian Dynamics

Appendix D. Computation of the Spin Accumulation in Nonequilibrium State
References

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124

1. INTRODUCTION
Symmetry-broken states with incommensurate modulation have
attracted considerable attention in condensed-matter physics. Typical examples are charge- and spin-density waves in metals, magnetic structures in
insulators, helicoidal structures in liquid crystals, and superconducting states
with spatially nonuniform order parameters. In spite of differences in microscopic origins, their physical properties are universally characterized by macroscopic phase coherence of the condensates and collective dynamics
associated with them. In particular, the condensates with multicomponent
order parameters are of special importance, because they have orientational
degrees of freedom in physical space. Consequently, not only amplitude but
phase 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 in
this article.
The field of research on helimagnetic structure dates back to more than a
half century ago. Yoshimori [1], Kaplan [2], and Villain [3] interpreted an
earlier report on magnetic structure of MnO2 [4] as a helimagnetic structure.

Since then, this field had been actively driven by neutron scattering measurements. An early history of the field is well reviewed by Nagamiya [5].
The microscopic origin of this class of helimagnets is the frustration among
different superexchange interactions between localized spins or the
Ruderman–Kittel–Kasuya–Yosida (RKKY) interactions mediated by conduction electrons. Recently, the field of multiferroic materials has shed new
light on the frustration-driven noncollinear magnetic structures [6].


3

Theory of Monoaxial Chiral Helimagnet

r

o

r
ir

M

Figure 1 Left- and right-handed helimagnetic structure.

On the other hand, Dzyaloshinskii [7] found another class of
helimagnetic structures which are stabilized by the antisymmetric
Dzyaloshinskii–Moriya (DM) interaction [8]. The DM interaction originates from the relativistic spin–orbit interaction [9] and imprint an asymmetric electronic structure to the antisymmetric spin–spin interaction Dij ÁSi ÂSj
between spins on sites i and j. The constant vector Dij is called the DM vector. The quantity χ ij ¼Si ÂSj is called the spin chirality which breaks chiral
symmetry. The term “chiral symmetry breaking” means that space inversion
(P) symmetry is broken, but time reversal (T ) symmetry combined with any proper
spatial rotation (R) is not broken, according to the definition of Laurence
Barron [10]. Actually χ ij is odd under the parity transformation P, but even

under time reversal operation T .
When the DMvector Dij has a form Dij ¼ D^e with the D being constant
and ^e being a unit vector along some crystallographic axis, competition
between DM interaction and the isotropic ferromagnetic (FM) coupling J
gives 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) structure. A necessary condition for this kind of DM vector to exist is that a magnetic crystal belongs to a chiral space group Gχ whose symmetry elements
contain pure rotations only, i.e., 8g 2 Gχ , det g ¼ 1. The concept of chirality,


4

Jun-ichiro Kishine and A.S. Ovchinnikov

originally meaning left- or right-handedness, plays an essential role in symmetry properties of nature at all length scales from elementary particles to
biological systems.
In a helimagnetic structure realized in a chiral crystal, the degeneracy
between 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 and
experimental achievements on this topic up to early 1980s are well reviewed
by Izyumov [11]. Interestingly, Dzyaloshinskii’s work activated the research
field of improper ferroelectricity where physical outcome of the Lifshitz
invariant had been intensively studied [12, 13].
Despite the apparent similarity of spin structures, the helimagnetic structures of Yoshimori’s type and Dzyaloshinskii’s types have profound difference in what level of chiral symmetry is broken. In the Yoshimori type, chiral
symmetry is not broken at the level of Hamiltonian, but the helimagnetic
structure spontaneously breaks chiral symmetry. On the other hand, in
the Dzyaloshinskii’s (CHM) type, the Hamiltonian itself breaks chiral symmetry because of the DM interaction and the magnetic structure is forced to
break the chiral symmetry. An essential feature of the chiral helimagnetic

structure is that the structure is protected by crystal chirality. The symmetric
helimagnet, however, does not have any macroscopic protectorate and is
easily fragmented into multidomains. In Fig. 2, we summarize basic properties of symmetric and chiral helimagnets.
M–H curve

Mechanism
Symmetric (Yoshimori) Chiral (Dzyaloshinskii)

M

Continuum model

Ferro
Conical

M

Ferro

Helical

Fan

Pitch angle =
M

Ferro
Conical

M

Pitch angle =

to n
Soli

Ferro

la t

e
tic

Figure 2 Basic properties of symmetric and chiral helimagnets.

Spin wave


Theory of Monoaxial Chiral Helimagnet

5

This difference directly comes up in their magnetic structures under
magnetic fields and elementary excitations. In particular, a significant difference arises under a static magnetic field perpendicular to the helical axis. The
symmetric helimagnetic structure undergoes a discontinuous transition from
a helimagnet structure to a fan structure and then continuously approaches
the forced ferromagnetic configuration [5]. On the other hand, in the chiral
helimagnet, the ground state continuously evolves into a periodic array of
the commensurate (C) and incommensurate (IC) domains. This state, a main
subject of this article, has several names, i.e., chiral soliton lattice (CSL), helicoid, or magnetic kink crystal (MKC) [7, 11]. Throughout this article, we

use the term chiral soliton lattice. As the magnetic field strength increases,
the spatial period of CSL increases and finally goes to infinity at the critical
field strength. This situation is depicted in Fig. 3.
After almost a half century since the theoretical prediction [7], experimental observation of the CSL was achieved by Togawa et al. in the hexagonal
helimagnet CrNb3S6 [14] which has magnetic phase transition temperature
A

B

C

D

E

F

Figure 3 Formation of the chiral soliton lattice under a magnetic field applied perpendicular to the helical axis. As the magnetic field strength increases from (A) Hx ¼ 0 to (F) Hx ¼ Hxc ,
the spatial period of CSL increases and finally goes to infinity at the critical field strength.


6

Jun-ichiro Kishine and A.S. Ovchinnikov

TC ¼ 127 K and its helical pitch is 48 nm. In this compound, ferromagnetic
layers are coupled via interlayer weak exchange and DM interactions. In this
case, the formation of the CSL is observed by using Lorenz microscopy. The
spatial period of the stripe corresponds to the period of the CSL. The magnetic field dependence of the period gives a clear evidence that a chiral
helimagnetic structure under zero filed continuously evolves into the CSL

and finally undergoes a continuous phase transition to commensurate
forced-ferromagnetic state at a critical field strength Hc $ 2300 Oe.
The CSL has some special features to be noted. (1) In the CSL state, the
translational symmetry along the helical axis is spontaneously broken. Therefore, the corresponding Goldstone mode becomes phonon like [7, 15]. (2)
The CSL state has infinite degeneracy associated with arbitrary choice of the
center of mass position. Consequently, the CSL can exhibit coherent sliding
motion [16]. (3) The CSL exerts a magnetic super-lattice potential on the
conduction electrons coupled to the CSL. This coupling may cause a
magnetoresistance effect [17, 18]. (4) Quantum spins carried by conduction
electrons cause spin-transfer torque on the CSL [19].
Here, we will review physical properties of the CSL from theoretical
viewpoints. The remaining part of the review will be divided into nine subsections. In Section 2, we will describe the symmetry-based views on chiral
helimagnetism. In Section 3, we discuss helical and conical structures under
a magnetic field parallel to the chiral axis. In Section 4 we will describe the
ground state and elementary excitations associated with the CSL [feature (1)
mentioned above]. In Section 5, we will review some experimental probes
of structure and dynamics of the CSL. In Section 6, we review physical
properties of the sliding CSL [feature (2)] and discuss a possible spin motive
force driven by the sliding motion (Section 7). In Section 8, we discuss the
coupling 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 physics
from broader viewpoints (Section 10). We will leave some supplementary or
technical materials to appendices.

2. CHIRAL SYMMETRY BREAKING IN CRYSTAL AND
CHIRAL HELIMAGNETIC STRUCTURE
2.1 Magnetic Representation of Chiral Helimagnetic
Structure
Quantum spin state for spin S ¼ 1/2 is described by a two-component spinor

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


7

Theory of Monoaxial Chiral Helimagnet




eÀiφ=2 cos ðθ=2Þ
:
jχ i ¼ + iφ=2
e
sin ðθ=2Þ

(1)

Then, a spin operator as an observable is given by S^ ¼ S^
σ , where S ¼ ℏ=2
and σ^ denotes a Pauli spin operator and a corresponding spin polarization
vector is written as an axial vector in O(3) space,
S ¼ hχ jS^ jχ i ¼ S^
n, n
^ ¼ ðsinθ cosφ, sinθ sinφ,cos θÞ:

(2)

This classical axial vector enters a macroscopic Maxwell equations as a magnetic moment M ¼ ÀgμBS. It is to be noted that whenever we talk about M,
permutation symmetry, which is purely quantum, is totally lost and instead

the parameters φ and θ have meaning as polar angles φ(r) and θ(r) tied to a
spatial position r in O(3) space. A purpose of magnetic representation theory
is to classify possible ordering of the M vector as an order parameter.
The chiral helimagnetic structure is an incommensurate magnetic structure with a single propagation vector k ¼ (0,0,k). The chiral space group Gχ
consists of the elements {gi}. Among them, some elements leave the propagation vector k ¼ (0,0,k) invariant, i.e., these elements form the little group
Gk. The magnetic representation Γmag is written as Γmag ¼ Γperm  Γaxial,
where Γperm and Γaxial represent the Wyckoff permutation representation
and the axial vector representation, respectively [20]. Then, Γmag is
decomposed into the nonzero irreducible representations of Gk. The
incommensurate magnetic structure is determined by a “symmetry-adapted
basis”of an axial vector space and the propagation
Pvector k. In a specific magnetic ion, the decomposition becomes Γmag ¼ i ni Γi , where Γi is the irreducible representations of Gk. The chiral helimagnetic structure,
È
É
M Æ ¼ Me1 cos ðkzÞ Æ Me2 sin ðkzÞ ¼ MRe ðe1 Ç ie2 Þeikz ,

(3)

(+ and À signs correspond to left- and right-handed helix) requires real twodimensional or complex one-dimensional symmetry-adapted basis, e1 and e2.
For these basis to exist, the group elements of Gk0 include three- (C3), four(C4), or sixfold (C6) rotations. Therefore, among 65 chiral space groups whose
elements are all proper rotations (for 8g 2 G, det g ¼ 1), 52 space groups
belonging to cubic, hexagonal, tetragonal, and trigonal crystal classes are eligible to accommodate the chiral helimagnetic structure. This situation is
depicted in Fig. 4. The cubic class is special because there are four C3 axes,
although hexagonal, tetragonal, and trigonal crystals have only one principal
axis. In the latter case, a monoaxial helimagnetic structure as shown in Fig. 1 is


8

Jun-ichiro Kishine and A.S. Ovchinnikov

A

B

Cubic

Hexagonal

C

D

Trigonal

Tetragonal

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- (C3), four- (C4),
or sixfold (C6) axis. Correspondingly, (A) cubic, (B) hexagonal, (C) tetragonal, and
(D) trigonal crystal classes are eligible to accommodate the chiral helimagnetic structure. 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 CuB2O4 does 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.

CuB2O4

Tetragonal

I42d

9.35

77

A

[21–23]

CsCuCl3

Hexagonal

P6122 or P6522

10.5

22

A

[24]

CrNb3S6

Hexagonal

P6322

127

48

B

[25, 26]

YbNi3Al9

Trigonal

R32

3.4

34

B

[27]

MnSi

Cubic

P213

29.5

18

C

[28]

FeGe

Cubic

P213

279

70

C

[29]

Fe1ÀxCoxSi

Cubic

P213

59

25

C

[30]

expected to be favored. Difference in crystal symmetry comes up as a form of
the 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 symmetry which contains atomic building blocks with low symmetry. In Table 1,
we give examples of chiral helimagnets which are known so far.
In particular, we give detailed information about one of the most actively
studied magnetic compound CrNb3S6. This material has a hexagonal layered
structure 2H-type NbS2, intercalated by Cr atoms, belonging to the space
group P6322 as shown in Fig. 5. Studies of the material were started at the


9

Theory of Monoaxial Chiral Helimagnet

A

Nb
S

b

Cr

a
c

B

Figure 5 The scheme of the crystal and magnetic structure in CrNb3S6 (A). The elementary 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).

beginning of 1970s, when the method of chemical gas transportation was used
to get single crystals CrNb3S6 [26]. In this work measurements of neutron and
magnetic properties were carried out, it has been found that it is a helimagnet
with 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 2.9 μB per Cr atom. Small-angle neutron scattering indicates an
˚ À1, i.e., with the spatial
existence of a helical structure with Q0 ¼ 0.013 A
˚
period 480 A. The authors have concluded that such a long-periodic modulation is caused by the antisymmetric Dzyaloshinsky–Moriya exchange interaction. In the unit cell shown in Fig. 5, the Cr atoms have a trivalent state and
the localized electrons form a spin S ¼ 3/2. Initially, data on the magnetization
process in a perpendicular magnetic field were misinterpreted as a manifestation of first order phase transition between the helical ordering and the state of
forced ferromagnetism [25]. The interpretation of a sharp change of the magnetization 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 of
the magnetic properties of the Ba2CuGe2O7, where an existence of the magnetic soliton lattice was also confirmed [33].
In Fig. 6A, we sketch the original crystal structure of CrNb3S6. Nb and
Cr occupy special points with high symmetry (Wyckoff positions of two Nb
sites 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 position is 12i) and its atomic coordinate is (À0.000350,0.667770,0.369130).

We see a quite tiny chiral symmetry breaking arises. To exaggerate the chiral
symmetry breaking, in Fig. 6B, we show the crystal structure with the S’s


10

Jun-ichiro Kishine and A.S. Ovchinnikov

A

B

c
a

b

C

c
a

b

c
a

b

Figure 6 (A) Original form of right-handed crystal of CrNb3S6. (B) Fictitious crystal without changing the original symmetry, where atomic coordinates of S atom is modified

from its original (À0.000350, 0.667770, 0.369130) to (À0.100350, 0.667770, 0.369130)
to visualize chiral symmetry breaking. (C) Side view of a unit cell of the fictitious crystal.
Helical arrangement of S atoms is clearly visible.

atomic coordinate being modified to (À0.100350,0.667770,0.369130)
without changing its original space group symmetry. In this case, we clearly
recognize that there exists a helical arrangement of S atoms. In Fig. 6C, we
show the same structure from another viewpoint. It is seen that Cr ion is
surrounded by S atoms in chiral manner.
Another important aspect of CrNb3S6 is its classical one-dimensional
nature as a magnetic network. Quasi-one-dimensional systems are regarded
as a bunch of weakly coupled quantum 1D systems as shown in Fig. 7A. On
the other hand, as shown in Fig. 7B, when two-dimensional layered magnetic structures are weakly coupled via the interlayer exchange and DM
interactions, the system is well described as a classical 1D system. The latter
case is actually realized in CrNb3S6 [34]. This situation makes it legitimate to
treat this system as a classical 1D chiral helimagnet.
Another real system, that is in the focus of current investigations, is the
manganese 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 0.4 μB) per Mn atom. This class of
magnetic materials includes other popular compounds FeGe, Fe1ÀxCoxSi,
where the helimagnetic order was as well detected. These systems possess
the space group P213, which does not contain a center of symmetry, and
the fact provides an appearance of chiral magnetic structures. One of the
remarkable 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].


11

Theory of Monoaxial Chiral Helimagnet

A

B

C

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

Helicoidal magnetic order in MnSi was discovered a long time ago [28].
Recent studies of small angle neutron scattering in the so-called A-phase
have attracted an interest to the Skyrmion model [38]. The magnetic structure of MnSi in a zero magnetic field can be presented as a set of ferromagnetically ordered planes arranged parallel to the crystallographic
plane (111). As a result, in the magnetically ordered phase spins form a
left-handed helix with the incommensurate wave vector 0.036 A˚À1 that
˚ in the [111] direction. A more
corresponds to the spatial period 188 A
detailed survey of properties of the manganese silicide may be found in
the review [39].

2.3 Microscopic Origins of the DM Interaction
We here briefly summarize possible microscopic origins of the DM interaction. Chiral magnetic crystals are classified into three classes, i.e., Type A:
insulator, Type B: metal with coexisting localized and itinerant spins, and
Type C: metal with only itinerant spins. We depict these three cases in
Fig. 8. Type A corresponds to the case originally discussed by Moriya [9].
In this case, the Hamiltonian which describes two magnetic ions is

12

A

Jun-ichiro Kishine and A.S. Ovchinnikov

Type A

B

C

Type B

Type C

E
D
×

×

Figure 8 Schematic picture of origins of the DM interaction in (A) Type A: insulator,
(B) Type B: metal with coexisting localized and itinerant spins, and (C) Type C: metal with
only itinerant spins. In Type B and C, the DM interactions are caused by processes
represented by Feynman diagrams shown in (D) and (E), respectively.

H0 ¼ λS1 Á L1 + λS2 Á L2 À JS1 Á 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 perturbation theory as
!
X hg1 jL1 jn1 i X hg2 jL2 jn2 i
D ¼ ÀiλJ
,
À
(5)
En1 À Eg1
En2 À Eg2
n1
n2
where g and n label the ground and excited states, respectively.
In the case of Type B, the particle-hole fluctuations of the itinerant electrons, as shown in Fig. 8D, mediate the DM interaction between the localized spins. This is a generalized 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 [40].
The case of Type C is the most nontrivial [41]. We expect 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, as shown
in Fig. 8E, eventually has an effective form of the DM interaction [42]. In
Table 1, 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, we
consider the case of CrNb3S6. Then, we start with a Hamiltonian which
describes a weakly coupled layered system,


13

Theory of Monoaxial Chiral Helimagnet

H3D ¼ ÀJk

X

S^ m, n, j Á S^ m, n, j + 1 À D Á

m, n, j
X
$

+H Á

m, n, j

S^ m, n, j À J?

X
m, n, j

S^ m, n, j  S^ m, n, j + 1

X XX
hm, m0 i hn, n0 i

S^ m, n, j Á S^ m0 , n0 , j ,

(6)

j

where S^ m, n, j is a quantum spin at the site (m,n) on the j-th layer, Jk > 0 is the
ferromagnetic exchange interaction between the nearest layers, D ¼ D^e z is
the monoaxial Dzyaloshinskii–Moriya (DM) interaction along a certain
crystallographic chiral axis (taken as the z-axis) which is perpendicular to
the layers. J? > 0 is the ferromagnetic exchange interaction between the
nearest neighbor sites on the same layer. We take z-axis as the monoaxis
$

and apply magnetic field H ¼ gμB H, where g is the electron g-factor and
μB ¼ jejℏ=2m is the Bohr magneton.
Based on the picture of semiclassical 1D model as shown in Fig. 7C, we
assume Jk( J?. Actually, a classical Monte-Carlo simulation [34] was
recently 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 frozen
and a rigid in-layer ferromagnetic arrangement is established. Then, we can
omit 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,
S^ m, n, j ! S^ j :

(7)

Taking Nx and Ny as the number of lattice sites along x and y directions,
respectively, this simplification leads to the effective one-dimensional
Hamiltonian, H ¼ H3D =Nx Ny , written as
X
X
X

$
H ¼ ÀJ
S^ j Á S^ j + 1 À D Á
S^ j  S^ j + 1 + H Á
S^ j ,
(8)
j

j

j

where J reperensents the effective exchange interaction strength along
the axis.
In the case of the monoaxial DM interaction, D ¼ D^e z , the lattice
Hamiltonian (8) is rewritten as
$

À +
J X iQ0 a0 ^ + ^À
0 a0 ^ ^
Sj Sj + 1 Š
H ¼À2
½ej Sj Sj + 1 + eÀiQ
j

ÀJ

j
X

j

Æ
x
y
where S^j ¼ S^j Æ iS^j and

z z
S^j S^j + 1 + K?

X
j

$

ðS^j Þ2 + H Á
z

X
j

S^ j ,

(9)


14

Jun-ichiro Kishine and A.S. Ovchinnikov

(10)
Q0 ¼ aÀ1
0 arctan ðD=JÞ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
$
and J ¼ jJ + iDj ¼ J 2 + D2 : The lattice constant (interlayer spacing) is a0.
X z 2
The term K?
ðS^j Þ represents an easy plane anisotropy energy which is
j

included only when we consider the spin wave spectrum.
The representation expressed by (9) indicates that the DM interaction
plays a role of a Peierls phase (or an external gauge field) connecting Sj+
and SjÀ+ 1 . The appearance of the Peierls phase in this representations is a natural consequence of the fact that the helimagnetic structure carries static
momentum Q0 which characterizes the condensate. This phase is gauged
away by the global gauge transformation,
Æ
Æ À1
ÆiQ j Æ
(11)
S^j À!U^ S^j U^ ¼ ej 0 S^j ,
h P i
z
using the unitary operator, U^ ¼ exp iQ0 j jS^j (note that when we perÆ
form this transformation, we should read SjÆ as a quantum operator S^j ). This
transformation maps the helimagnetic chain to the ferromagnetic XXZ [43].

3.2 Helimagnetic Structure for Zero Magnetic Field
Because the effective spin S^ j is regarded as a spin with large amplitude, quantum fluctuations are strongly suppressed and therefore it is legitimate to treat

it as the classical vector, i.e., the replacement S^ j ! Sj may be legitimate,
where Sj is a semiclassical axial spin vector. In this case, using the polar coordinates θ and φ as shown in Fig. 9, we represent the spin vector as Sj ¼ S nj,


where the unit vector field nj is nj ¼ sinθj cosφj ,sin θj sin φj , cos θj : For
H ¼ 0, the first term of the r.h.s. of Eq. (9) becomes minimum for
SjÆ ¼ SeÆiQ0 zj sinθj (zj ¼ a0j) which gives the total energy
XÂ
Ã
e
J sinθj sin θj + 1 + J cos θj cos θj + 1
H ¼ ÀS2
(12)
j

$

which becomes minimum for θj ¼ π/2 (note J > J). This state corresponds
À À
Á
À
Á Á
to a chiral helimagnetic structure, Sj ¼ S cos Q0 zj ,sin Q0 zj ,0 .
The corresponding helical pitch is L0 ¼ 2π/Q0, which amounts to
48 nm in the case of CrNb3S6 [14]. Furthermore, in the case of CrNb3S6,
a0 ¼ 1.212 Â 10À9m and experimentally obtained helical pitch
L0 ¼ 4.8 Â 10À8m give D=J ¼ tanðQ0 a0 Þ ¼ 0:16.


15

Chiral axis

Theory of Monoaxial Chiral Helimagnet

z

y

zj
x

Figure 9 Spherical polar coordinate representation of the spin vector.

3.3 Conical Structure Under a Magnetic Field Parallel to the
Chiral Axis
To consider the magnetic structure under a magnetic field parallel toX
the chiz
ðSjz Þ2
ral axis, H ¼ (0, 0, H ), we add an easy plane anisotropy energy K?
j

to the Hamiltonian (8). Inclusion of this term is important to explain a global
profile of the spin wave spectrum. For Hz ¼ 0, the planar helical structure is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
stable under the condition, K? =J > 1 À 1 + ðD=J Þ2 , which is assumed to
be satisfied. For
$

$

$

z
z
0 < H < H c ¼ 2SðJ À J + K? Þ,

(13)

the ground state is described by SjÆ ¼ SeÆiQ0 zj sinθ0 , where the cone angle is
given by
!
$z
H
À1
(14)
θ0 ¼ cos
:
$
2Sð J ÀJ + K? Þ
$

$

This state is depicted in Fig. 10. For H z > H zc , the ground state is a forced
ferromagnetic state, where all the spins are parallel to H. In the conical state,
chiral symmetry in the spin space is forced to be broken by the crystal
symmetry. The magnetic field parallel to the chiral axis causes no additional
symmetry breaking.

16

Jun-ichiro Kishine and A.S. Ovchinnikov

Figure 10 Conical structure under the magnetic field parallel to the chiral axis.

3.4 Helimagnon Spectrum Around the Conical State
From now on, we consider the dynamical properties associated with the
ground states discussed in the previous section. Reflecting the symmetry
breaking patterns, elementary excitations around the conical state and
CSL state are totally different. In the case of the conical state, the elementary
excitations are described as a Goldstone mode to retrieve rotational symmetry breaking. On the other hand, dynamical properties of the CSL state are
much richer because of the translational symmetry breaking. We will discuss
them separately.
In this section, we consider the spin wave spectrum around the conical
state described by SjÆ ¼ SeÆiQ0 zj sinθ0 with a cone angle θ0 being given by
Eq. (14). We here stress that the broken symmetry associated with the conical state is the rotational symmetry around the helical axis. In Eq. (10), we
can allow the constant initial phase in the transverse spin distribution as
SÆ ¼ SeÆiðQ0 zj + ϕ0 Þ sin θ. Then, the system has infinite numbers of degenerj

ate ground states associated with arbitrarily choice of ϕ0. Then, we expect
that there appears gapless helimagnon mode as the Goldstone mode associated with the broken continuous symmetry. This point was discussed by
Elliott and Lange [44] for the case of symmetric (Yoshimori-type)
helimagnet. The same thing also happens in the chiral helimagnet but the
dispersion spectrum quite differs from the symmetric case [41, 45]. At first,
to capture the intuitive properties of the helimagnon excitations, we follow
the equation-of-motion method adopted by Nagamiya [5] for the symmetric helimagnet.
To compute the spectrum, we rotate the basis frame of the crystal
coordinate {e+, eÀ, ez} to the basis frame of the local coordinate

e zj g where the direction of e zj points to the equilibrium spin
fe j+ , e À
j ,
direction at the j-th site. The transformation to the local frame at the i-th
site is determined as
e zj ¼ ez cosθ0 À ðe + eiQ0 j + eÀ eÀiQ0 j Þ sin θ0 ,

e Æ
j ¼

Ã
1Â z
e sinθ0 + e + ðcosθ0 Æ 1ÞeiQ0 j + eÀ ðcos θ0 Ç 1ÞeÀiQ0 j ,
2

(15)
(16)


17

Theory of Monoaxial Chiral Helimagnet

with a first rotation about z by an angle Q0j followed by a second rotation
about y by an angle θ0. The spin vector
À

Sj ¼ Sj+ e + + SjÀ eÀ + SzÀ ez ¼ Sj e + + Sj e À + Sj e z ,
+

z

(17)

has the components


1
+
À
z
Sjz ¼ sinθ0 Sj + Sj + cosθ0 Sj ,
2
SjÆ

!
1
1
+
À
z ÆiQ0 j



¼ ðcos θ0 Æ 1ÞS j + ðcos θ0 Ç 1ÞS j À sin θ0 S j e
:
2
2

(18)

(19)

In the rotated basis frame, the Hamiltonian (8) acquires the form
!
 γ À 1

X γ + 1 + À
+
+ +
À À
z z




+
S
λ
S
+
+
Sj Sj + 1 + SÀ
S
S
S
S
S
j j+1
j

j+1
j j+1
j j+1
4
4
j
!


X 1
+ +
À À
+ À
À +
z z
sin 2 θ0 Sj Sj + Sj Sj + Sj Sj + Sj Sj + cos 2 θ0 Sj Sj
+ K?
(20)
4
j

H ¼ ÀJ

$

+ H z cosθ0

z
X
X z 2

z
ðSj Þ ,
Sj + K?
j

j

$

$

where λ ¼ sin 2 θ0 + ðJ= J Þcos 2 θ0 and γ ¼ ðJ= J Þsin 2 θ0 + cos 2 θ0 with the
cone-angle θ0 being given by Eq. (14).
To establish the equations of motion of the spin vectors, we recall the
relations,
h
i
h
i
+
À
z
z
Æ
Æ
Sj , Sj ¼ 2Sj ,
Sj , Sj ¼ ÆSj :
(21)
Æ

Using these, we obtain the Heisenberg equations of motion, dSj =
h
i
Æ
dt ¼ iℏÀ1 H, Sj , as
 1$z
y 2
y
1  y
2
y
y
¼ À JS SjÀ1 + Sj + 1 À 2λ Sj + H cos θ0 Sj À ℏ K? S cos θ0 Sj ,
dt
ℏ
ℏ
(22)

dSj

x


i 1$z
2
x
x
1 h  x
x
x

¼ JS γ SjÀ1 + Sj + 1 À 2λ Sj À H cosθ0 Sj + ℏ K? S cos2θ0 Sj :
dt
ℏ
ℏ
(23)

dSj

y


18

Jun-ichiro Kishine and A.S. Ovchinnikov

To find the propagating solutions, we use the ansatz
x
y
Sj ¼ uk eikxj Àiωt , Sj ¼ vk eikxj Àiωt

(24)

that yields
iℏωuk ¼ À2JSð1 À coskaÞvk ,


K? 2

sin θ À γcos ka uk :
iℏωvk ¼ 2J S 1 +

J

(25)
(26)

The system of equations results in
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


K
?
2
sin θ0 À γcos ka :
ℏωk ¼ 2JS ð1 À cos kaÞ 1 +
J

(27)

Note that when the static field is applied along the chiral axis, the spin wave
spectrum remains gapless. The mode with k ¼ 0 corresponds to a rigid rotation of the whole system. The condition γ < 1 leads to γ < 1. It means that
the conical structure is stable against spin wave excitations. This result
reduces to the one obtained by Kataoka [41] and Maleyev [45] using a
continuum approximation (k ! 0 limit).
In Fig. 11A, we show the helimagnon dispersion for Hz ¼ 0, 0:7Hcz , and
Hcz . Upon increasing the field, linear dispersions for 0 H z < Hcz cross over
Hz = 0
Hz= 0.7Hzc
Hz = Hzc
Hz > Hzc

C

Helical axis

wq

A

Hz = 0

x− i

Local frame
at the i th site

Spin
−π

π

0

Sqx Sqy

B

1

Helical plane

q

Hz = Hzc

D

y−i

z−i
Precession trajectory

Sqx

Hz = 0.9999Hzc
Hz = 0.7Hzc
Hz = 0
Sqy

−π

π

0

q=π
q = 0.5π
q = 0.05π
q=0

−0.5

1

q
$

Figure 11 (A) Spin wave (helimagnon) spectrum for different H 0 : The equilibrium state
$

$

$

$

is
forced ferromagnetic for H 0 ! H 0c . (B) Amplitude ratio
 for H 0 < H 0c$, while
 conical
$
 x  y 
S =S q  for different H 0 H 0c . (C) Typical precession trajectory of the spin vector in
q

$

the case of H 0 ¼ 0: (D) Wave-number dependence of the trajectories.


19

Theory of Monoaxial Chiral Helimagnet
$

continuously to the quadratic dispersion ℏωk ¼ 2 J Sð1 À coskaÞ at H z ¼ Hcz .
The Goldstone mode at k ¼ 0 corresponds to the rigid rotation of the whole
helix. For H z ! Hcz , the equilibrium state is the forced-ferromagnetic state
and the spin wave spectrum acquires the field-induced gap.
To understand the nature of the spin wave excitations, it is useful to see
the ratio of the precession amplitudes,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
1 À cosq
x y
x y


S j =S j ¼ S q =S q ¼ iu
:
t  $
(28)
1 + K= J sin 2 θ À γ cos q
x
y
Appearance of i means the Sq and Sq have phase difference π/2, just indix
y
cating precession. In Fig. 11B, we show jSq =Sq j as a function of q. We see
x
y
that jSq =Sq j ! 0 as q ! 0 for 0 H z < Hcz and the spins tend to be confined

to the helical plane. However, as Hz approaches the critical strength Hcz , the
dip around q ¼ 0 becomes narrower and eventually vanishes toward
H z ¼ Hcz , where the precessional trajectory becomes a perfect circle
corresponding to the ferromagnetic spin wave. In Fig. 11C, we show a typical precession trajectory of the spin vector in the case of Hz ¼ 0. As shown in
Fig. 11D, it is clearly seen that as q departs from zero to π, the precession
trajectory changes from a flat ellipsoidal shape to a more circular one. On
the other hand, the spin wave for q ¼ 0 corresponds to rigid rotation of
the whole helix around the chiral axis, which is a Goldstone mode to
retrieve the broken rotational symmetry, just as in the case of symmetric helimagnet [5, 44]. For H z > Hcz , the spin wave spectrum acquires the gap
ℏωq¼0 ¼ SH z . The conical state has rotational degeneracy around the helical
axis, but the ferromagnetic state is coaxial with the magnetic field and has no
chance to feel rotational symmetry. This is the reason why the gapless Goldstone mode vanishes for H z > Hcz .
The same result (27) is obtained by using the Holstein–Primakoff
transformation
pffiffiffiffiffi
pffiffiffiffiffi
z
+
À
(29)
Sj ¼ S À ^a{j ^aj , Sj ¼ 2S^aj , Sj ¼ 2S^a{j ,

where a{j and aj are Boson creation and annihilation operators, respectively.
P
After taking the Fourier transform aj ¼ k eikxj ak , we obtain


i
Xh  {
H ¼ JS

Ak ^ak ^ak + ^a{Àk ^aÀk + Bk ^ak ^aÀk + ^a{Àk ^a{k ,
(30)
k


20

Jun-ichiro Kishine and A.S. Ovchinnikov

where
Ak ¼ 1 À

γ +1
γ À1
cosk, Bk ¼ À
cosk:
2
2

(31)

The Bogoliubov transformation
^ k À sinh ϕk α
^ {Àk ,
^ak ¼ cosh ϕk α

(32)

^a{k

^ Àk ,
sinh ϕk α

(33)

diagonalizes the Hamiltonian to give
X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi {
^ k,
^kα
A2k À B2k α
H ¼ 2JS

(34)

¼

^ {k À
cosh ϕk α

k

where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
jAk j
1
jAk j
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + , sinhϕk ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi À ,

coshϕk ¼
2
2
2
2
2 Ak À B k 2
2 Ak À Bk 2

(35)

which reproduces Eq. (27).

3.5 Spin Resonance in the Conical State
In the spin resonance experiments, the static magnetic field H0 is applied to
cause Larmor precession of magnetic spins. Then supplying electromagnetic
energy carried by microwave radiation, resonant absorption occurs at the
precession frequency. The microwave is described as the uniform oscillating
magnetic 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 Zeeman
coupling with spin,
HZ ¼ ÀHðtÞ Á S0 ,

(36)

where H(t) ¼ geμBh(t) and S0 is the uniform (q ¼ 0) component of the spin
variable.
For
HðtÞ ¼ H μ eμ cos ðωt Þ

ðμ ¼ x,y,zÞ,

(37)

the ESR spectrum, namely, the absorbed energy per unit time, is given by
1
QðωÞ ¼ ωHμ2 χ 00μμ ðωÞ:
2

(38)


Theory of Monoaxial Chiral Helimagnet

21

The imaginary part of the dynamical susceptibility,
Á
1À
(39)
1 À eÀβω Cμν ðωÞ,
2


is related to the correlation function Cμν ðωÞ ¼ S0μ ðωÞS0ν through the
fluctuation–dissipation theorem.
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 are
described by spin waves of the conical magnetic structure. A quantized spin
wave is called a helimagnon. Then, the ESR spectrum is given by

χ 00μν ðωÞ ¼

1
Qhmag ðωÞ ¼ ωHy2 χ 00yy ðωÞ:
2

(40)

It is straightforward to obtain the helimagnon resonance spectrum,
Qhmag ðωÞ ¼

πS
2
2
+
À 2
ωHy2 δðω À ωQ0 Þ Â ½ðuQ+0 + uÀ
Q0 Þ + cos θ 0 ðuQ0 À uQ0 Þ Š,
8
(41)

where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi


1 P
Æ
Æ1
uQ0 ¼
2 ωQ0

and


(
)
 $

J


2
2
2
P ¼ S2 J + K? sin θ0 À J 1 + cos θ0 + $ sin θ0 :


J

(42)

(43)

There is the single branch of resonance energy as shown in Fig. 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 the
chiral 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


22

Jun-ichiro Kishine and A.S. Ovchinnikov

∼

hw Q 2JS
0.5
0

0.4
0.3
0.2
0.1

1

0

Figure 12 Field dependence of the helimagnon resonance energy.

SðzÞ ¼

X

À
Á

n ðzÞ
Sj δ z À zj ¼ S^

j

¼ SðsinθðzÞcos φðzÞ, sinθðzÞsinφðzÞ, cosθðzÞÞ:
P

! aÀ1
0

RL

dz, we obtain a continuum
RL
$
version of the Hamiltonian (339) per unit area, H ¼ 0 dz H , where L
denotes 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 integration over x- and y-directions are implicitly taken into account. The
Hamiltonian density is then written as
By taking a continuum limit

$

j

0

JS2 a0
S $

ð@z nÞ2 À S2 D Á n  @z n + H Á n
2
a0
JS2 a0
JS2 a0 2
2
¼
ð@z θÞ +
sin θð@z φÞ2
2
2
S $
S $
À S2 D sin 2 θ@z φ + H x sinθ cosφ + H z cos θ,
a0
a0

H¼

(44)

where we dropped the anisotropy term K?. For a nonzero transverse field
perpendicular to the chiral axis, H ¼ (Hx, 0, 0), there is no symmetrybreaking field parallel to the chiral axis. Furthermore, 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 Hamiltonian (44) with θ(z) being fixed to π/2, i.e.,