# Theory of spin current in chiral helimagnet

###### Abstract

We give detailed description of the transport spin current in the chiral helimagnet. Under the static magnetic field applied perpendicular to the helical axis, the magnetic kink crystal (chiral soliton lattice) is formed. Once the kink crystal begins to move under the Galilean boost, the spin-density accumulation occurs inside each kink and there emerges periodic arrays of the induced magnetic dipoles carrying the transport spin current. The coherent motion of the kink crystal dynamically generates the spontaneous demagnetization field. This mechanism is analogous to the Döring-Becker-Kittel mechanism of the domain wall motion in ferromagnets. To describe the kink crystal motion, we took account of not only the tangential -fluctuations but the longitudinal -fluctuations around the helimagnetic configuration. Based on the collective coordinate method and the Dirac’s canonical formulation for the singular Lagrangian system, we derived the closed formulae for the mass, spin current and induced magnetic dipole moment accompanied with the kink crystal motion. To materialize the theoretical model presented here, symmetry-adapted material synthesis would be required, where the interplay of crystallographic and magnetic chirality plays a key role there.

###### pacs:

Valid PACS appear here^{†}

^{†}preprint: APS/123-QED

## I Introduction

The core problem in the multidisciplinary field of spintronics is how to create, transport, and manipulate spin currents.Zutic04 () The key notions include the current-driven spin-transfer torqueSlonczewski96 (); Berger96 (); Stiles02 (); Stiles02b (); Tatara-Kohno04 () and resultant force acting on a domain wall (DW)Aharonov-Stern (); Bazaliy () in metallic ferromagnetic/nonmagnetic multilayers, the dissipationless spin currents in paramagnetic spin-orbit coupled systems,Rashba60 (); Murakami03 (); Sinova04 () and magnon transport in textured magnetic structures.Bruno05 () A fundamental query behind the issue is how to describe transport spin currents.Rashba05 () To make clear the meaning of the spin currents, we need to note the spin can appear in the macroscopic Maxwell equations only in the form of spin magnetization. In this viewpoint, the spin current is understood as the deviation of the spin projection from its equilibrium value. An emergence of the coherent collective transport in non-equilibrium state is then a manifestation of the dynamical off-diagonal long range order (DODLRO).Fomin91 (); Volovik07 ()

On the other hand, the physical currents are classified into two categories, i.e., the gauge current originating from the gauge invariance and the inertial current originating from the Galilean invariance. The electric current is the gauge current, where the electric charge is coupled to the electromagnetic U(1) gauge field. The electromagnetic field is a physical gauge field that has its own dynamics, i.e., we know the electromagnetic field energy. Then, the charge current and the charge density are related via the continuity equation . On the other hand, a typical example of the inertial current is the momentum current in a classical ideal fluid, where the momentum current satisfies the continuity equation, and given by with being equilibrium pressure.LLFluid () The non-equilibrium current is described by . In the spin current problem, at present, we have no known gauge field directly coupled to the spin current. Therefore, a promising candidate is the inertial current of the magnetization.

Historically, DöringDoring48 () pointed out that the longitudinal component of the slanted magnetic moment inside the Bloch DW emerges as a consequence of translational motion of the DW. An additional magnetic energy associated with the resultant demagnetization field is interpreted as the kinetic energy of the wall. This idea was simplified by BeckerBecker50 () and Kittel.Kittel50 () Recent progress of material synthesis sheds new light on this problem. In a series of magnets belonging to chiral space group without any rotoinversion symmetry elements, the crystallographic chirality gives rise to the asymmetric Dzyaloshinskii interaction that stabilizes either left-handed or right-handed chiral magnetic structures.Dzyaloshinskii58 () In these chiral helimagnets, magnetic field applied perpendicular to the helical axis stabilizes a periodic array of DWs with definite spin chirality forming kink crystal or chiral soliton lattice.Kishine_Inoue_Yoshida2005 ()

We recently proposed a new way to generate a spin current in the chiral helimagnets with magnetic field applied in the plain of rotation of magnetization.BKO08 () The mechanism is quite analogous to the Döring-Becker-Kittel mechanism. We showed that the periodic spin accumulation occurs as a dynamical effect caused by the moving magnetic kink crystal (chiral soliton lattice) formed in the chiral helimagnet under the static magnetic field applied perpendicular to the helical axis. The current is inertial flow triggered by the Galilean boost of the kink crystal. An emergence of the transport magnetic currents is then a consequence of the dynamical off-diagonal long range order along the helical axis.

In this paper, we give an extension of the results touched on in Ref. BKO08 (). In Sec. II, we give an overview of basic properties of the chiral magnets that materialize the theoretical model considered in this paper. In Sec. III, we present standard description of the kink crystal formation, and the vibrational modes around the kink crystal state. In Sec. IV, we apply the collective coordinate method to the moving kink crystal that makes clear the physical meaning of the mass and the magnon current carried by the moving system. In Sec. V, we perform quantitative estimates of the mass, magnetic current, and net magnetization induced by the movement. In Sec. VI, we discuss issues closely related to the present problem, including the background spin current problem, spin supercurrent in the superfluid He, and experimental aspects of our effects. Finally, we summarize the paper in Sec. VII.

## Ii Chiral helimagnet

In this section, we briefly review basic properties of chiral helimagnets that materialize our theoretical model. Recent progress of material synthesis promotes systematic researches on a series of magnets belonging to chiral space group without any rotoinversion symmetry elements.Kishine_Inoue_Yoshida2005 () In the chiral magnets, the crystallographic chirality possibly gives rise to the asymmetric Dzyaloshinskii interaction that stabilizes the chiral helimagnetic structure, where either left-handed or right-handed magnetic chiral helix is formed.Dzyaloshinskii58 () As we will see, in the chiral helimagnets, magnetic field applied perpendicular to the helical axis stabilizes a periodic array of DWs with definite spin chirality forming kink crystal or chiral soliton lattice.Kishine_Inoue_Yoshida2005 ()

The chiral helimagnetic structure is an incommensurate magnetic structure with a single propagation vector . The space group consists of the elements . Among them, some elements leave the propagation vector invariant, i.e., these elements form the little group .Izumov (); Kovalev () The magnetic representationKovalev () is written as , where and represent the Wyckoff permutation representation and the axial vector representation, respectively. Then, is decomposed into the non-zero irreducible representations of . The incommensurate magnetic structure is determined by a “magnetic basis frame”of an axial vector space and the propagation vector . In specific magnetic ion, the decomposition becomes , where is the irreducible representations of . Then, we have two cases leading to the chiral helimagnetic magnetic structure. Case I: The magnetic moments are described by two independent one-dimensional representations that form two-dimensional basis frames, or Case II: The magnetic moments are described by a single two-dimensional representations that form two-dimensional basis frames. In these cases, the symmetry condition allows the chiral helimagnetic structure to be realized. Then, the structure is stabilized by the generalized Dzyaloshinskii interaction. The generalized Dzyaloshinskii interaction means symmetry-adapted anti-symmetric exchange interaction, not restricted to conventional Dzyaloshinskii-Moriya (DM) interaction caused by the on-site spin-orbit coupling and the inter-site exchange interactions. The presence of this term is justified by the existence of the Lifshitz invariantDzyaloshinskii64 () for the little group .

Among the inorganic chiral helimagnets, the best known example is the metallic helimagnet MnSi (K) that belongs to the cubic space group 23(Å).Ishikawa76 () The metallic helimagnet CrNbS (K) belongs to the hexagonal space group 622 (Å Å).Moriya-Miyadai82 () The insulating copper metaborate, CuBO (K) has a larger unit cell and belongs to the tetragonal space group (Å, Å).Roessli01 (); Kousaka07 () As examples of molecular-based magnets, the structurally characterized green needle, [Cr(CN)][Mn( or )-pnH(HO)]HO (K), belongs to the orthorhombic space group (Å, Å, Å). The yellow needle, K[Cr(CN)][Mn()-pn]()-pnH: (()-pn = ()-1,2-diaminopropane) (K), belongs to the hexagonal space group (Å, Å).Kishine_Inoue_Yoshida2005 () From the symmetry-based viewpoints, these space groups are all eligible to realize the chiral helimagnetic order.

## Iii Kink crystal and vibrational modes around the kink-crystal state

As shown in Fig. 1, we consider a system of the chiral helimagnetic chains described by the model Hamiltonian

(1) |

where the first term represents the ferromagnetic coupling with the strength between the nearest neighbor spins and . The second term represents the parity-violating Dzyaloshinskii interaction between the nearest neighbors, characterized by the the mono-axial vector along a certain crystallographic chiral axis (taken as the -axis). The third term is the Zeeman coupling with the magnetic field applied perpendicular to the chiral axis. When we treat the model Hamiltonian (1), we implicitly assume that the magnetic atoms form a cubic lattice and the uniform ferromagnetic coupling exists between the adjacent chains to stabilize the long-range order. Then, the Hamiltonian (1) is interpreted as a quasi one-dimensional model based on the interchain mean field picture.SIP ()

When , the long-period incommensurate helimagnetic structure is stabilized with the definite chirality (left-handed or right-handed) fixed by the direction of the mono-axial -vector. The Hamiltonian (1) is the same as the model treated by LiuLiu73 () except that we ignore the single-ion anisotropy energy. Once we take into account the easy-axis type anisotropy term, , the mean field ground state configuration becomes either the chiral helimagnet for , or the Ising ferromagnet for . In this paper, we assume and left an effect of for a future study.

Taking the semiclassical parametrization of Heisenberg spins in the continuum limit by using the slowly varying polar angles and [see Fig. 2(a)], the Hamiltonian acquires the form

(2) |

where , and denotes the linear dimension of the system. From now on, all distances are measured in the lattice unit . The helical pitch in the zero field () is given by .

The magnetic kink crystal phase is described by the stationary soliton solution minimizing , and , where is the Jacobi elliptic function with the elliptic modulus ().Dzyaloshinskii64 (); Rubinstein70 () This solution corresponds to a periodic regular array of the magnetic kinks with the ”topological charge” density as shown in Figs. 2(b) and (c). The elliptic modulus is found from the minimization of energy per unit length that yields .Dzyaloshinskii64 () The period of the soliton lattice is given by

(3) |

where and denote the elliptic integrals of the first and second kind, respectively. The period increases from to infinity as increases from zero to unity. In the limit of , the function approaches and , i.e. as it should be in the case of zero field.

In the Hamiltonian (2), the exchange processes favor the incommensurate (IC) chiral helimagnetic order, while the Zeeman term favors the commensurate (C) phase. The C-IC transition occurs at provided , and the critical value of is given by . Bak (); Bulaevskii (); Pokrovskii () The critical field strength is determined from .

Next, we consider the fluctuations around the kink crystal state. The studies of collective excitations in the system have been focused on the phasons (-mode) presenting bending waves of domain walls of the soliton lattice.McMillan () In our analysis we are interested in the -modes also. We derive the spectrum of elementary exciations holding the scheme outlined in Ref.Izyumov-Laptev86 ()

In what follows, it is convenient to work with the dimensionless coordinate

(4) |

We introduce and and rewrite the Hamiltonian (2) as

(5) |

where the dimensionless Hamiltonian is defined by

(6) |

As the magnetic field increases from to , the parameter monotonously decreases from to . The fluctuations consist of the vibrational (phonon) modes and the translational mode, that are separately treated. In this section, we examine the phonon modes. We write

(7) |

and expand (6) up to and Then we have where corresponds to the stationary solution. The interaction part contains and terms that are neglected here. The vibrational term is given by , and where the differential operators, and are defined by

(8) |

The ”gap function” reads as

(9) |

where the relation was used. The minimum and maximum value of the gap are given by

(10) |

respectively, where is the complementary modulus. We see that the gap closes at the C-IC transition. In Fig.3 (a), we show the spatial variation of the gap function. The dependence of the minimum gap is shown in Fig.3 (b). For small ,we have , and . Therefore, and it is appropriate to approximate for the case of weak field. This approximation amounts to approximating .

If were we considered only the tangential -mode, our problem reduces to the case first investigated by Sutherland.Sutherland73 () Furthermore, the -mode is fully studied in the context of the chiral helimagnet.Izyumov-Laptev86 (); Aristov-Luther03 () However, to realize the longitudinal magnetic current, as we will see, it is essential to include into consideration the -mode. Even of zero-field, , the -mode acquires the energy gap .BKO08 () The -gap directly originates from the Dzyaloshinskii interaction that plays a role of easy plane anisotropy. On the other hand, the -mode is the massless Goldstone mode corresponding to rigid rotation of the whole helix around the helical axis.Elliot66 () Even after switching the perpendicular field, the -mode (-mode) remains to be massive (massless).

The mode expansion is

(11) |

where the orthonormal basis and is determined through the eigenvalue equations,

(12) |

with a mode index . The vibrational part is now given by

(13) |

In explicit form the eigensystem (12) present the Schrödinger-type equations,

(14) | ||||

In Eq. (LABEL:Lame_u) we consider the case of weak field corresponding to small that admit . In appendix A, we present the general scheme to treat the periodic potential having the spatial period and show that this approximation does not affect qualitative result presented below. Now, both equations (LABEL:Lame_u) and (14) reduce to the Jacobi form of the Lam equation,WW () and their solutions have been discussed by us previouslyBKO08 () (see also Appendix B). The analysis shows that both the and mode consist of two bands,Sutherland73 () i.e.,

(16) |

(17) |

where the real parameter runs over . Here, means the complete elliptic integral of the first kind with the complementary modulus .

By imposing the periodic boundary condition, the quasi-momentum (Floquet index) is introduced for the acoustic

(18) |

() and the optic

(19) |

() branches, respectively, where denotes the Jacobi’s Zeta-function.WW () The representation was given by Izyumov and Laptev,Izyumov-Laptev86 () and differs from a conventional representation.WW (); Sutherland73 ()

A dispersion relation is given by as a function of Floquet index . We show the excitation spectra and in Fig. 4. Because the energy gap of the mode, has a range The gap has a maximum value at zero field () and monotonously decreases as the field increases up to the critical field (). The normalized wave function at the bottom of the acoustic band is

(20) |

In the next section we demonstrate that exactly corresponds to the zero translational mode.

## Iv Galilean boost of the kink crystal

In the previous section, we determined the phonon modes. Next we consider the translational mode. The translational symmetry holds after the kink formation and gives rise to the Goldstone mode, i.e. zero mode . Although the Gaussian fluctuations around the kink crystal state are assumed to be small, this is not true for the zero mode which describes fluctuations without damping. Then, the center of mass coordinate is elevated to the status of the dynamical variable and the phonon modes are orthogonal to the zero mode. To describe this situation, we follow the collective coordinate method.Christ-Lee75 (); Rajaraman ()

At first, we construct the Lagrangian for the kink crystal system. We make use of the coherent states of spins,

(21) |

where

(22) |

with and are the generators of SU(2) in the spin- representation and satisfy The highest weight state satisfies and . The states form an overcomplete set and give Using this representation, the Berry phase contribution to the real-time Lagrangian per unit area is written as

(23) |

where we took the continuum limit in the second line. Now, we construct the Lagrangian,

(24) |

with the coefficients

(25) |

and expand and in the form,

(26) |

In the expansion of the -mode, it is not necessary to exclude , since the -mode does not contain zero mode. This description amounts to using the curvilinear basis, in functional space and taking the generalized coordinates , , . Since the zero mode is orthogonal to the phonon modes, we have

(27) |

for Noting that

and

and plugging these expressions into the Lagrangian (24), we obtain

(28) | ||||

(29) |

where higher order terms are dropped. The overlap coefficients are given by

(30) |

The Lagrangian (28) is singular because it does not contain any term of the form , and the rank of the Hessian matrix becomes zero. This means that there is no primary expressible velocities. Therefore we need to construct the Hamiltonian by using the Dirac’s algorithm for the constrained Hamiltonian systems.Dirac (); Gitman () The details of the treatment have been given in our previous treatment (see, also Appendix C). The final result is

(31) |

which means that only finite amplitude of the -mode,

(32) |

appears when the collective velocity is finite. This is exactly the manifestation of the ODLRO. In other words, the -field is interpreted as the demagnetization field that drives the inertial motion of the kink. Using Eq.(31), we reach the final form of the physical Hamiltonian,

(33) |

where the inertial mass of the kink crystal is introduced

(34) |

The physical Hamiltonian (33) describes the inertial motion of the kink crystal.

The linear momentum per unit area carried by the kink crystal may be presented in the formBKO08 () , where the topological charge

(35) |

is introduced. Apparently, the transverse magnetic field increases a period of the kink crystal lattice and diminishes the topological charge and therefore it affects only the background linear momentum (see discussion in Sec. VI). The physical momentum related with a mass transport due to the excitations around the kink crystal state is generated by the steady movement.

The “superfluid magnon current” transferred by the -fluctuations is determined through the definition of the accumulated magnon densityVolovik07 () in the total magnon density , where the superfluid part is conjugated with the magnon time-even current carried by the -fluctuations

(36) |

via a continuity equation.BKO08 () The important point is that the only massive -mode can carry the longitudinal magnon current as a manifestation of ordering in non-equilibrium state, i.e., dynamical off-diagonal long range order.Xiao ()

The net magnetization (magnetic dipole moment) induced by the movement is

(37) |

The minus sign means that the net magnetization produces a demagnetization field.

## V Quantitative Estimates

To compute the mass , the spin current and the magnetic dipole moment , we consider an array of parallel chains described by the model (1), where a number of chains per unit area is . In the case of the molecular-based chiral magnets, the crystal packing is usually loose ([m]) and the exchange interaction is rather weak ([K] [J]). On the other hand, in the case of the inorganic chiral magnets, the crystal packing is close ([m]) and the exchange interaction is rather strong ([K] [J]). We take these values as just typical parameter choices. The strength of the Dzyaloshinskii interaction is ambiguous and we simply take .

### v.1 Mass of the kink crystal

The mass of the kink crystal is given by Eq.(34). Evaluation of the overlap integral is performed in appendix D and yields , where

(38) |

Therefore we have

(39) |

The factor appears here after the MKS units [m] for distances are recovered in Eq.(33). The mass per unit area is given by

(40) |

that after simplification yields

The last relationship is reliable in the case of small fields, i.e. , and .

Noting that the period of kink measured in lattice units is given by Eq.(3), which turns into for small fields, the mass per one kink acquires the form

(41) |

As a typical example of the molecular-based chiral magnets, we have

For the chain length , we have the total mass [g/cm]. As a typical example of the inorganic chiral magnets, we have

For the chain length , we have the total mass [g/cm]. This heavy mass should be compared with the mass of conventional Bloch wall mass in ferromagnets. To make clear the difference, in appendix E, we gave a brief summary of this issue. In the present case, appearance of the heavy mass is easily understood, since the kink crystal consists of a macroscopic array of large numbers of local kinks.

### v.2 Spin current

As it follows from Eq.(36) the physical dimension of the spin current density is . Using the results of the appendix D the spin current density given by Eq.(36) transforms into

(42) |

The factor occurs after the MKS units for distances are recovered in the continuity equation and in the velocity . After simplifications with aid of Eqs.(3), (25), (20), and (38) we immediately have

(43) |

For the case of weak fields corresponding to small this yields

(44) |

We present a schematic view of an instant distribution of spins in the current-carrying state in Fig. 5. In Fig. 6, we present a snapshot of the position dependence of the current density in the weak field limit, given by Eq. (44). In Fig. 6, we depicted the cases of the magnetic field strengths , , and . Although the formula (44) is valid only for the case of weak field limit, but qualitative features are well demonstrated by just extrapolating the validity up to . As the field strength approaches the critical value, the current density is more and more localized.

For both the typical molecular-based and inorganic chiral magnets, we have

Taking the velocity of order we obtain finally

therefore the current through the unit area

### v.3 Magnetic dipole moment

The magnetic dipole moment [Eq.(37)] induced by the motion is given by

i.e. the relationship holds. By the same manner as it was made for the spin current we obtain in the case of the small fields

(45) |

Therefore, for both the molecular-based and inorganic chiral magnets, we have

(46) |

i.e. is of order . The total magnetic moment of the chain is

(47) |

We here used the relations and that leads to

(48) |

where is a topological charge introduced in Eq.(35).

Noting,

(49) |

we have the chain magnetization

The total moment per unit volume

As a typical example of the molecular-based chiral magnets, we have

As a typical example of the inorganic chiral magnets, we have

## Vi Discussions of related topics

### vi.1 Background spin current problem: SU(2) gauge invariant formulation

Heurich, König and MacDonaldHeurich03 () proposed that the external magnetic fields generate dissipationless spin currents in the ground state of systems with spiral magnetic order. Here, we comment on the relevance of the present work to this issue. In our model, the background spin current is given by

(50) |

i.e. there arises the misfit of the kink crystal to the helimagnetic modulation and consequently the current comes up. Below we prove that this current exists on a link between two sites but it causes no accumulation of magnon density (”magnetic charge”) at the site due to continuity equation, i.e. the current is not related to the magnon transport. This supports reasonings of arguments by Schütz, Kopietz, and M. KollarKollar () that appearance of finite spin currents is direct manifestation of quantum correlations in the system, and in the classical ground state the spin currents vanish.

The background spin current problem is best described by the SU(2) gauge invariant formulation developed by Chandra, Coleman and Larkin.CCL90 () By imposing the local SU(2) gauge invariance of the theory, we obtain the fictitious SU(2) gauge fields and that give the spin current , and the spin density