Spin current and chirality degrees of freedom inherent in localized electron orbitals Shintaro Hoshino1 Michi-To Suzuki23 and Hiroaki Ikeda4 1Department of Physics Saitama University Sakura Saitama 338-8570 Japan

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Spin current and chirality degrees of freedom inherent in localized electron orbitals

Shintaro Hoshino1, Michi-To Suzuki2,3, and Hiroaki Ikeda4

1Department of Physics, Saitama University, Sakura, Saitama 338-8570, Japan

2Center for Computational Materials Science, Institute for Materials Research,

Tohoku University, Sendai, Miyagi 980-8577, Japan

3Center for Spintronics Research Network, Graduate School of Engineering Science,

Osaka University, Toyonaka, Osaka 560-8531, Japan

4Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

(Dated: October 6, 2022)

In solid state physics, any phase transition is commonly observed as a change in the microscopic

distribution of charge, spin, or current. Here we report the nature of an exotic order parameter inher-

ent in the localized electron orbitals that cannot be primarily captured by these three fundamental

quantities. This order parameter is described as the electric toroidal multipoles connecting diﬀer-

ent total angular momenta under the spin-orbit coupling. The corresponding microscopic physical

quantity is the spin current tensor on an atomic scale, which induces spin-derived electric polariza-

tion and the chirality of the Dirac equation. We stress that the chirality intrinsic to the elementary

particle is the essence of electric toroidal multipoles. These ﬁndings link microscopic spin currents

and chirality in the Dirac theory to the concept of multipoles and provide a new perspective for

quantum states of matter.

Introduction.— In strongly correlated electron sys-

tems, the spin and orbital degrees of freedom of nearly lo-

calized electrons are activated by electronic correlations,

resulting in various intriguing phenomena, such as heavy

electrons, unconventional superconductivity, and exotic

magnetic orders. These quantum states are commonly

characterized by the spatial distribution of the funda-

mental microscopic physical quantities, charge, spin, and

current. They have been systematically studied using

the concept of multipole expansions [1–12]. These mul-

tipoles are classiﬁed into four categories (electric, mag-

netic, magnetic toroidal, and electric toroidal) according

to spatial parity and time-reversal parity [12]. Such a

classiﬁcation is useful for elucidating exotic orders and

predicting novel response to external ﬁelds.

In particular, electric toroidal multipole ordering has

recently attracted much attention as novel nonmagnetic

degree of freedom [13–17] which has not been strongly

recognized so far. In localized electron systems, such

degrees of freedom are inherent in the components con-

necting diﬀerent total angular momenta under the spin-

orbit coupling λ`·s. The simplest rank-1 electric toroidal

dipole is written as G(1) = `×s[15, 18–20]. When con-

sidering this electric toroidal dipole in analogy to a mag-

netic toroidal dipole [21], we would expect the electric

polarization to be circularly aligned. The ordinary elec-

tric polarization P(r) = rρ(r) (ρis a charge density),

however, cannot capture the toroidal structure, as P(r)

is a simple charge distribution. What physical quantity

characterizes the nature of electric toroidal multipoles

microscopically?

In the following, we clarify the underlying fundamental

physical quantity using the knowledge of the relativistic

quantum mechanics. It is shown that the electric polar-

ization PS(r) induced by the spin degrees of freedom is

responsible for the electric toroidal moment. The spin-

derived electric polarization PS(r) is also associated with

the microscopic spin current tensor, which is regarded as

one of the fundamental physical quantities analogous to

charge, spin, and electric current. The relation between

electric polarization and spin current has been discussed

in the context of spintronics based on weak-coupling itin-

erant Fermi liquid description [22]. We further ﬁnd the

relation between the electric toroidal multipole, spin cur-

rent, and chirality of the Dirac equation by employing

the localized electron picture. This chirality degree of

freedom is intimately related to the diagonal part of the

spin current, and is a more fundamental quantity cor-

responding to electric toroidal multipoles. We thus link

microscopic spin currents and chirality in Dirac theory

to the concept of multipoles.

Deﬁnition of multipoles.— Let us start with the deﬁni-

tion of multipoles in localized electron orbitals with the

angular momentum `. The multipole operators are de-

ﬁned as a complete matrix basis set to describe all oper-

ators of the type c†

mσcm0σ0, where cmσ is the annihilation

operator of the electron with the magnetic quantum num-

ber m∈[−`, `] and spin σ=↑,↓. Under strong spin-orbit

coupling, the multipole operators are usually considered

only for a ground-state j=`±1/2 multiplet. We here

consider the full space containing diﬀerent jmultiplets.

The classiﬁcation scheme recently formulated for `= 1

case [20] is applied to the general `[23]. In this case, the

generic rank-pmultipole is written as

Xγ(pη) = X

mm0σσ0

c†

mσOγ

mσ,m0σ0(pη)cm0σ0,(1)

where p= 0,··· ,2`+1, and Ois a matrix representation

of the multipole. γis a label to distinguish 2p+ 1 de-

generacies. We have separated the rank-pmultipole into

several pieces denoted by the η-index: η=a, b denote

intra-jmultiplet component, and η=c, d corresponds to

a component connecting diﬀerent jmultiplet [24]. For

η=a, b, c, the even (p= 2q) or odd (p= 2q+ 1) rank

arXiv:2210.02148v1 [cond-mat.str-el] 5 Oct 2022

ρMSj PSτZτX(∼ρ)τY(∼∇·MS)∇·PS

Multipole Type SI/TR +/+ +/− −/− −/+−/+ +/+−/−+/+

Electric Toroidal (2q+ 1)d+/+ 0 0 0 Nonzero Nonzero 0 0 0

Magnetic Toroidal (2q)d+/−0 Nonzero 0 0 0 0 0 0

Electric (2q)a,b,c +/+ Nonzero 0 0 Nonzero 0 Nonzero 0 Nonzero

Magnetic (2q+ 1)a,b,c +/−0 Nonzero Nonzero 0 0 0 Nonzero 0

TABLE I. List of the multipole distribution function f(r;pη, γ) (p= 2qor p= 2q+ 1) deﬁned in Eq. (3) where f=

ρ, MSµ, jµ, PSµ, τ Z,X,Y , and ∇·PS. The label ‘Nonzero’ (‘0’) for each cell indicates a non-zero (zero) multipole distribution

function in the leading-order contribution of the non-relativistic limit. The signs of the spatial inversion (SI) Pand time-reversal

(TR) Θ indicated in the third column and the second row are deﬁned as Pf(r)P−1=±f(−r) and Θf(r)Θ−1=±f(r).

coincides with the even/odd of the time reversal (TR),

but this relation is reversed for dcomponents. Note that

in the present intra-`case, the parity of spatial inversion

(SI) is always even. In addition, the type η=dmulti-

poles are categorized as the toroidal multipoles from its

symmetry [19]. These results are summarized in the left

three columns (‘Multipole’, ‘Type’, ‘SI/TR’) of Tab. I.

Charge and current distribution.— Since the multi-

poles obtained above were deﬁned based on localized elec-

tron orbitals, they can be described as microscopic charge

or current distributions in continuous space. With the

ﬁeld operator ψσ(r), the microscopic charge density is

given by ρ0=eψ†ψ, and the current density and magne-

tization operators by

j=e

2mψ†↔

pψ, MS=~e

2mcψ†σψ, (2)

where p=−i~∇, and A↔

∂B=A∂B −(∂A)B.

The spin summation is implicitly performed. We ex-

pand the operator by the atomic orbitals as ψσ(r) =

PmR(r)Y`m(ˆ

r)cmσ. Then, for example, the current op-

erator is written as

j(r) = X

pηγ

Xγ(pη)j(r;pη, γ).(3)

Namely, once the multipoles are given, the spatial distri-

bution of the current can be visualized through the prod-

uct with the distribution function, j(r;pη, γ) for each set

of (pη, γ). Similar expansions are possible for other mi-

croscopic physical quantities.

However, as will be shown later, there is no primary

change in the spacial distributions of ρ0(r), j(r), and

MS(r) in the ordered state of electric toroidal dipole

G(1). In order to obtain another microscopic quantity

having primary change, we start with the basic Hamil-

tonian Hwith relativistic corrections. Then the cur-

rent and charge are obtained by jtot =−δH

δA/c and

ρtot =δH

δΦ, where Aand Φ are vector and scalar po-

tentials [22, 23]. Based on the spin dependence, the total

current and charge can be uniquely separated into two

parts: jtot =j+c∇×MSand ρtot =ρ−∇·PS. The

current density and magnetization have already been de-

ﬁned above. The charge density and electric polarization

are given by

ρ=1 + ~2

8m2c2∇2eψ†ψ'eψ†ψ=ρ0,(4)

PS=~e

8m2c2ψ†↔

p×σψ. (5)

The second term in ρoriginates from the uncertainty

of the position in relativistic quantum mechanics. How-

ever, this second term has only a minor correction on

the charge distribution since it originates from the sec-

ond derivative of the large ﬁrst term (ρ0), and the spin

degrees of freedom are not directly involved. This cor-

rection term has the same origin as the Darwin term in

the Hamiltonian [25], which only aﬀects the selectron

(`= 0) and vanishes for ` > 0.

Here we make a few comments on the spin-derived elec-

tric polarization PS. The Gordon decomposition of the

charge and current also introduces the microscopic mag-

netization and electric polarization [25, 26]. We also note

that the presence of the microscopic electric polarization

has the same physical origin as the Aharonov-Casher ef-

fect [27], in which the particles with magnetization are

aﬀected by the electric ﬁeld.

The present spin-derived electric polarization PSis ex-

actly related to microscopic spin current. The connection

between the spin current and electric polarization has

been discussed for the non-colinear magnets [28] and the

electron gas model in the context of the spintronics [22].

The spin-derived electric polarization PScan be further

rewritten as

PSµ =~2e

8m2c2µνλjSνλ,(6)

where the spin current jSµν =−iψ†↔

∂µσνψand the an-

tisymmetric tensor µνλ are introduced (µ, ν, λ =x, y, z)

[22]. Furthermore, the spin current tensor jSµν may

be classiﬁed into the components with rank 0 (pseu-

doscalar), 1, and 2 [12]. We ﬁnd that the rank-0 pseu-

doscalar component is related to the chirality degrees of

freedom in the Dirac equation. The chirality density in

the second-quantized form is deﬁned by the annihilation

operators for right- and left-handed chiral fermions, ψR,L,

in the Weyl basis as follows,

τZ=ψ†

RψR−ψ†

LψL,(7)

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SpincurrentandchiralitydegreesoffreedominherentinlocalizedelectronorbitalsShintaroHoshino1,Michi-ToSuzuki2;3,andHiroakiIkeda41DepartmentofPhysics,SaitamaUniversity,Sakura,Saitama338-8570,Japan2CenterforComputationalMaterialsScience,InstituteforMaterialsResearch,TohokuUniversity,Sendai,Miyagi980-8577...

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Spin current and chirality degrees of freedom inherent in localized electron orbitals Shintaro Hoshino1 Michi-To Suzuki23 and Hiroaki Ikeda4 1Department of Physics Saitama University Sakura Saitama 338-8570 Japan

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