Spin currents, spin torques, and the concept of spin superfluidity

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by  Andreas Rückriegel

Spin currents, spin torques, and the concept of spin superfluidity
Andreas Ru¨ckriegel and Peter Kopietz
Institut fu¨r Theoretische Physik, Universita¨t Frankfurt,
Max-von-Laue Strasse 1, 60438 Frankfurt, Germany
(Dated: January 19, 2017)
In magnets with non-collinear spin configuration the expectation value of the conventionally
definedspincurrentoperatorcontainsacontributionwhichrenormalizesanexternalmagneticfield
and hence affects only the precessional motion of the spin polarization. This term, which has been
named angular spin current by Sun and Xie [Phys. Rev B 72, 245305 (2005)], does not describe
the translational motion of magnetic moments. We give a prescription how to separate these two
7
types of spin transport and show that the translational movement of the spin is always polarized
1
along the direction of the local magnetization. We also show that at vanishing temperature the
0
classicalmagneticorderparameterinmagneticinsulatorscannotcarryatranslationalspincurrent,
2
and elucidate how this affects the interpretation of spin supercurrents.

n
a
J
I. INTRODUCTION

9
1
The notion of spin currents describing the motion of

magnetic moments associated with the spins of the elec-
]
l trons in solids is of central importance in the field of
l
a spintronicswhereonetriestousethespindegreeoffree-
h
dom to store and process information. Unfortunately, in
-
s systems lacking spin-rotational invariance (which can be
e
broken by an external magnetic field or by relativistic
m
effects such as spin-orbit coupling or dipole-dipole inter-
. actions) the proper definition of the quantum mechani-
t
a cal operator representing the spin current is ambiguous, FIG. 1. (Color online) The two kinds of spin transport. (a)
m Translationalmotionofthemagneticmomentmwithaveloc-
because the magnetization does not satisfy a local con-
ity v, corresponding to the physical movement of a magnetic
- servation law. In the past decade several authors have
d moment with magnitude |m|. (b) Precessional motion of the
proposed resolutions of this ambiguity,1–16 but a gener-
n magnetic polarization mˆ =m/|m| with a frequency ω. The
o ally accepted agreement on the correct definition of the lattermotionispurelyangularandleavesthemagnitude|m|
c spin current operator in systems without spin conserva- of the magnetic moment invariant.
[ tion has not been found.

The purpose of this work is show that the distinction
1
v betweentranslationalandangularspincurrentsproposed
5 by Sun and Xie8 leads to a simple and unique definition that describes the precessional motion of the magnetiza-
0 oftheconceptofspintransportincondensedmattersys- tion. We will provide expressions for the translational
5 tems. Sun and Xie8 pointed out that spin currents de- spin current operator I(cid:107) and the precession frequency
5 i→j
scribe moving magnetic dipoles, and that generally the ω valid for itinerant as well as localized systems. Al-
0 i
transport of any vector can be decomposed into a trans- though the explicit construction of this decomposition is
.
1 lational part characterized by some velocity v(r) and an very simple, it entails profound physical consequences:
0 angular part described by some angular velocity ω(r), Since only the translational movement of magnetization
7
see Fig. 1. In the context of spin transport Sun and correspondstothephysicaldisplacementofmagneticmo-
1
: Xie8 called the latter contribution the angular spin cur- ments, in equilibrium only stationary translational spin
v rent, although this can be also viewed as the spin torque currents can generate an electrical field.17 Angular spin
i
X discussed earlier by Culcer et al. [6]. We will explicitly currents on the other hand only transport the polariza-
show below that the equation of motion of a magnetic tion,henceastationaryangularspincurrentissimplyan
r
a momentmi atlatticesiteRi withmagnitudemi =|mi| inhomogeneousconfiguration ofthelocalmagnetic order
and polarization mˆi =mi/|mi| can be decomposed into and does not create an electrical field. Furthermore, we
a translational part also show that in generic magnetic insulators the clas-
∂ m +(cid:88)(cid:104)I(cid:107) (cid:105)=0, (1) sical magnetic order cannot support translational spin
t i i→j transport at vanishing temperature; incoherent thermal
j or quantum fluctuations are mandatory for the physical
transport of magnetization in these systems. This also
that corresponds to the physical movement of magnetic
impliesthatspinsuperfluidityinmagneticinsulators18–26
dipole moments, and an angular part
must be angular spin transport that can be visualized as
∂ mˆ =ω ×mˆ , (2) transporting the spin polarization, but does not corre-
t i i i
2
spond to the physical movement of magnetic moments. where we have defined the operator
The remainder of this work is organized as follows: In
1 (cid:16) (cid:17)
Sec.IIwewillderivegeneralexpressionsfortheoperators I =− t c†σc −t∗c†σc
i→j 2i ij i j ij j i
correspondingtothetranslationalspincurrentandtothe
1(cid:16) (cid:17)
precession frequency of the magnetization, first for itin- − λ c†c +λ∗c†c
erant systems and then for localized magnetic moments. 2 ij i j ij j i
1 (cid:16) (cid:17)
We proceed to illustrate the usefulness of the decoupling − c†(σ×λ )c −c†(σ×λ∗)c . (7)
procedureinSec.III,wherewediscussspinsuperfluidity 2i i ij j j ij i
in easy-plane ferromagnets and persistent spin currents
Itistemptingtoassociatethisoperatorwiththespincur-
in Heisenberg rings. Finally, in Sec. IV we present our
rent describing the transport of spin from lattice site R
i
conclusions. The Appendix contains some additional de-
to lattice site R . It turns out, however, that a certain
j
tailsoftheself-consistentspin-waveexpansionweemploy
part of this operator simply renormalizes the external
to describe easy-plane ferromagnets.
magnetic field and therefore cannot be associated with
translational spin transport. To isolate this contribution
andidentifytheangularpartwhichrenormalizesthepre-
II. SEPARATING TRANSLATIONAL FROM cessional motion of the spins, we take the quantum me-
ANGULAR SPIN TRANSPORT chanical expectation value of both sides of the equation
of motion (6) and obtain a formally exact equation of
In this section, we explicitly show how translational motion for the magnetic moments m (t)=(cid:104)s (t)(cid:105),
i i
andangularspintransportcanbedefinedontheoperator
level. ∂tmi+Ti =−hi×mi, (8)
where the spin torque is defined by
A. Itinerant electrons (cid:88)
T = (cid:104)I (cid:105). (9)
i i→j
j
To construct the proper quantum mechanical defini-
tion of the translational spin transport operator let us To identify the contribution responsible for translational
consider a lattice model describing electrons with spin- spin transport, we further decompose the vector T into
dependent hopping tσijσ(cid:48) in an inhomogeneous magnetic a longitudinal and a transverse part, i
fieldh . ThesecondquantizedHamiltonianofourmodel
i
is T =T(cid:107)mˆ +T⊥, (10)
i i i i
H= (cid:88) tσijσ(cid:48)c†iσcjσ(cid:48) −(cid:88)hi·si+U, (3) where mˆi =mi/|mi| is the local spin polarization and
ijσσ(cid:48) i
T(cid:107) =(cid:88)mˆ ·(cid:104)I (cid:105), (11)
i i i→j
where U is some spin-rotationally invariant interaction,
j
c annihilatesafermionwithspin-projectionσatlattice
iσ T⊥ =T −(T ·mˆ )mˆ . (12)
site R , and the itinerant spin operators are defined by i i i i i
i
Writing
(cid:18) (cid:19)
1 c
s = c†σc , c = i↑ . (4)
i 2 i i i ci↓ Ti⊥ =(mˆi×Ti)×mˆi =δh⊥i ×mi, (13)
Here σ is the vector of Pauli matrices. The spin- where
dependent hopping energies tσσ(cid:48) are of the form
ij
δh⊥ = mˆi×Ti =(cid:88) mˆi ×(cid:104)I (cid:105) (14)
tσσ(cid:48) =t δ +i(λ ·σ) , (5) i |mi| |mi| i→j
ij ij σσ(cid:48) ij σσ(cid:48) j
where the vectors λ are proportional to the strength of is the induced magnetic field perpendicular to the direc-
ij
the spin-orbit coupling. The hermiticity of the Hamil- tionofmi,weseethatthetransversepartTi⊥ renormal-
tonian implies the symmetries t = t∗ and λ = izes the external magnetic field. The total angular fre-
ij ji ij
−λ∗. Using the canonical anticommutation relations quency relevant for the precessional motion of the mag-
ji
{c ,c† } = δ δ and the fact that the interaction netic moments is
iσ jσ(cid:48) ij σσ(cid:48)
is spin-rotationally invariant, [si,U] = 0, we obtain the ω =−h −δh⊥. (15)
Heisenberg equation of motion for the itinerant spins, i i i
Theterm−δh⊥×m canbecalledangularspincurrent8
∂si +(cid:88)I =−h ×s , (6) or spin torquei6 andi should be added to the external
∂t i→j i i
torque −h ×m acting on the magnetic moments. The
j i i
3
expectation value of the equation of motion (8) can now Here the isotropic exchange coupling J and the anti-
ij
be written as symmetric Dzyaloshinskii-Moriya vector D are given
ij
by
∂ m +mˆ T(cid:107) =ω ×m . (16)
t i i i i i
From this expression it is easy to show that the spin J = 4 (cid:0)|t |2−|λ |2(cid:1), (23)
ij U ij ij
torque does not contribute to the time-evolution of the
magnitude m = |m | of the magnetic moments, which D =−8Re[t λ ]= 8Re(cid:2)t λ∗(cid:3), (24)
satisfies the eqiuationiof motion5 ij U ij ji U ij ij
(cid:88) whileΓαβ isasymmetrictensorinspinspacewithmatrix
∂tmi+ mˆi·(cid:104)Ii→j(cid:105)=0. (17) ij
elements
j
4 (cid:16) (cid:17) 8 (cid:104) (cid:105)
In contrast, the precessional motion of the spin polariza- Γαβ =− λαλβ +λβλα = Re λα(λβ)∗ . (25)
tion is governed solely by the spin torque, ij U ij ji ij ji U ij ij
∂ mˆ =ω ×mˆ . (18) The Heisenberg equation of motion can be written as
t i i i
(cid:88)
Insummary,therenormalizedprecessionfrequencyas- ∂ S + Ispin =−h ×S , (26)
t i i→j i i
sociated with the angular spin current is
j
ω =−h −(cid:88) mˆi ×(cid:104)I (cid:105), (19) where the operator
i i |m | i→j
i
j
Ispin =S ×K S (27)
while the operator representing the translational spin i→j i ij j
current is
is the strong coupling limit of the operator I defined
i→j
Ii(cid:107)→j =mˆi·Ii→j. (20) in Eq. (7) in the reduced spin Hilbert space. Here Kij
is a tensor in spin space with matrix elements given by
Note that time-dependent changes in the length of the
Eq.(22). Alternatively,Eq.(27)canbeobtaineddirectly
magnetization are always accompanied by translational from Eq. (7) via a canonical transformation.27 With the
spin transport. On the other hand, stationary transla- substitution I → Ispin the expressions (19) and (20)
tional spin currents are also possible if the length of the i→j i→j
for the local precession frequency and the longitudinal
magnetization is constant.4
spin transport operator remain valid, so that we obtain
B. Localized spins ωi =−hi−(cid:88)|mmˆi| ×(cid:104)Si×KijSj(cid:105), (28)
i
j
The above expressions have been derived for a lattice (cid:104)I(cid:107) (cid:105)=mˆ ·(cid:104)S ×K S (cid:105). (29)
i→j i i ij j
model for itinerant electrons. It is instructive to work
out the explicit form of the rotation vector ωi and the In the classical limit and at vanishing temperature the
translational spin current operator I(cid:107) for a localized quantum mechanical expectation value (cid:104)S ×K S (cid:105) can
i→j i ij j
spin model containing only the spin degrees of freedom. be factorized,
For simplicity, let us specify the interaction to the on-
site Hubbard interaction U = U(cid:80) n n , where n = (cid:104)S ×K S (cid:105)→(cid:104)S (cid:105)×K (cid:104)S (cid:105)≡m ×K m . (30)
i i↑ i↓ iσ i ij j i ij j i ij j
c† c . Assuming U (cid:29) |tσσ(cid:48)| and a half-filled lattice,
iσ iσ ij
Consequently the expectation value of the longitudinal
we can use a canonical transformation27 to derive from
spin current operator vanishes identically in this limit.
Eq. (3) an effective Hamiltonian involving only spin 1/2
This implies that for a generic magnetic insulator with a
operatorsS actingonthereducedHilbertspaceofsingly
i spinHamiltonianoftheformofEq. (21)incoherentther-
occupiedlatticesites. TheeffectivespinHamiltoniancan
malorquantumfluctuationsareanecessaryprerequisite
be written as28
forthetransportofactualmagnetization. Thelocalpre-
Hspin =− 1 (cid:88)(cid:0)|t |2+|λ |2(cid:1) cession frequency reduces in the same limit to
2U ij ij
ij (cid:88)
ω =−h − mˆ ×(mˆ ×K m )
1(cid:88)(cid:88) (cid:88) i i i i ij j
+ KαβSαSβ − h ·S , (21)
2 ij i j i i j
ij αβ i =−h +(cid:88)[K m −mˆ (mˆ ·K m )]. (31)
i ij j i i ij j
where the spin-spin interaction tensor has three contri- j
butions,
ThelastterminEq.(31)isproportionaltomˆ andhence
i
Kαβ =δ J +(cid:15) Dγ +Γαβ. (22) does not contribute to ω × m so that we may write
ij αβ ij αβγ ij ij i i
4
ω ×m =(−h −δh )×m , where the renormalization the Goldstone modes associated with the spontaneous
i i i i i
of the magnetic field is given by breaking of the U(1)-symmetry in the superfluid state,
do not condense provided the condensate wave-function
(cid:88)
δh =− K m . (32) is self-consistently defined via the solution of the Gross-
i ij j
j Pitaevskii equation.
With the HP bosonization (35), we find that the lead-
Note that Eq. (32) can also be obtained by means of a ing order contributions to the precession frequency and
simple mean-field decoupling of the spin Hamiltonian in the spin current are of order S. Explicitly, the local pre-
Eq. (21). cession frequency (28) becomes
(cid:88)
ω =− SJ mˆ +SK(mˆ ·e )e . (36)
i ij j i z z
III. SPIN SUPERFLUIDITY AND PERSISTENT
j
SPIN CURRENTS
Tothisorder,thepolarizationequationofmotion(2)re-
To illustrate the differences between translational and casts the Landau-Lifshitz equation of classical spin dy-
angular spin transport and how this affects the interpre- namics. Assuming that the magnetic texture mˆ =
i
tation of spin supercurrents, it is instructive to consider mˆ(R ) varies only slowly in space, we can take the con-
i
simplemodelsystemsthatcansupporttranslationaland tinuumlimit. Theexchangecontributiontothepolariza-
angular spin currents in equilibrium or in metastable tion equation of motion can then be identified with the
states. divergence of the classical spin current,
(cid:88) (cid:88)
SJ mˆ ×mˆ → ∂ Jµ, (37)
ij j i µ
A. Easy-plane ferromagnet j µ
where the classical spin current is defined as
Let us first consider an easy-plane ferromagnet de-
scribed by the spin S Heisenberg Hamiltonian Jµ =−SJa2mˆ ×∂ mˆ. (38)
µ
1(cid:88) K (cid:88)
Hplane =− J S ·S + SzSz (33) Here ∂ = ∂/∂rµ, and a is the distance be-
2 ij i j 2 i i µ
tween nearest neighbors. An exact equilibrium solu-
ij i
tion for the polarization is then given by mˆ(r) =
on a simple cubic lattice, with exchange coupling Jij = [excosφ(r)+eysinφ(r)], with a local phase satisfying
J > 0 for nearest neighbors only, and an easy- ∇2φ(r) = 0. The U(1) freedom associated with the
plane anisotropy K > 0. This kind of systems has choice of φ(r) lies at the heart of the concept of spin
served in the literature as elementary example for spin superfluidity.20,26 Out of equilibrium we can make the
superfluidity.18,20,21,23,25,26 We will now calculate the lo- general ansatz
cal precession frequency (28) and the translational spin
(cid:112)
current(29)forthissystemtoleadingorderinan1/S ex- mˆ(r,t)= 1−ρ2(r,t)[e cosφ(r,t)+e sinφ(r,t)]
x y
pansion. To facilitate this we expand the spin operators
in a local basis defined by the instantaneous direction of +ezρ(r,t), (39)
the spin polarization mˆ (t)=(cid:104)S (t)(cid:105)/|(cid:104)S (t)(cid:105)|:
i i i
which is depicted graphically in Fig. 2. If the system is
S =S(cid:107)mˆ +S(1)e(1)+S(2)e(2). (34) only slightly driven out of equilibrium, the equation of
i i i i i i i motion (2) for the spin polarization becomes to lowest
Here,e(1)(t)ande(2)(t)areunitvectorschosensuchthat nonvanishing order in deviations from equilibrium
i i
{e(i1),e(i2),mˆi}formaright-handedbasisateverylattice ∂tφ=SKρ, (40a)
site. Inthisbasiswemaynowbosonizethespinoperators
∂ ρ=SJa2∇2φ. (40b)
by means of a Holstein-Primakoff (HP) transformation, t
The above Eqs. (40) are of the form of the Josephson
S(cid:107) =S−a†a , (35a)
i i i equations of superconductivity; hence they are conven-
√
S(1)+iS(2) = 2Sa +O(S−1/2), (35b) tionallyinterpretedasdescribingaspinsupercurrentcar-
i i i ried by the magnetization texture.20,23 However, since
where the a are canonical Bose operators. Especially this current does not correspond to translational trans-
i
note that since we self-consistently define the quan- port of the local magnetization |m |=|(cid:104)S (cid:105)|, this super-
i i
tization axis as the direction of the local magnetiza- current is not equivalent to the superfluid transport of
tion, mˆ (t) = (cid:104)S (t)(cid:105)/|(cid:104)S (t)(cid:105)|, by definition the HP magnetic moments. In particular, a stationary angular
i i i
bosons can never condense.29 This is completely anal- supercurrent with ∇φ = const is simply an inhomoge-
ogous to the fact that in the superfluid phase of inter- neous magnetic texture; while it can be viewed as a sta-
acting bosons the Bogoliubov quasi-particles, which are tionary transport of polarization, it does not correspond
5
i
ϑ
m
e i
z
ϑ
FIG. 3. Magnetization configuration of a ferromagnetic ring
in a radial magnetic field. Due to the exchange field the an-
FIG.2. Magneticpolarizationintheeasy-planeferromagnet. gle ϑ between the magnetization and the z-axis is slightly
m
ρdenotesthedeviationfromthegroundstatewiththemagne- smaller then the angle ϑ between the external field and the
tization lying in the x-y-plane. Superfluid angular transport z-axis, see Ref. [4].
is possible due to the U(1) freedom of the phase φ in this
plane.
spin transport like the one discussed in [25]. We empha-
size however that from the point of view we adopted,
to any physical movement of magnetic moments which
the superfluid does not arise due to the condensation
would generate an electrical field.17
of magnons. We rather consider the magnons as fluc-
ToevaluatetheO(S)contributiontothetranslational
tuations on top of the superfluid ground state; these
spin current (29), we will assume for simplicity a slowly
magnons cannot condense by definition.
varying magnetic texture mˆ in a metastable superfluid
i
state with ρ(cid:28)1. Applying the HP transformation (35)
to the spin current (29) then yields
B. Heisenberg ring
(cid:104) (cid:105)
(cid:104)I(cid:107) (cid:105)=Im −2SJ (cid:104)a†a (cid:105)+δ SK(cid:104)a a (cid:105) , (41)
i→j ij i j ij i i Next, let us consider a system where the translational
spin current is finite even in equilibrium, i.e., there is
which can to be evaluated to a persistent translational spin current corresponding to
(cid:20) (cid:18) (cid:19)(cid:21) the phyiscal movement of magnetic moments. Consider
(cid:104)I(cid:107) (cid:105)= 1 (cid:88)vµ n + 1 1− Ek , (42) a ferromagnetic spin S Heisenberg model in a radial in-
i→j Na k k 2 A
k k homogeneous magnetic field with quantum mechanical
Hamiltonian
whereR andR =R +ae arenearestneighborlattice
i j i µ
sites. Here n is the distribution functions of magnons 1(cid:88) (cid:88)
k Hring =− J S ·S − h ·S , (45)
with dispersion 2 ij i j i i
ij i
(cid:112)
Ek =S (K+Jk=0−Jk)(Jk=0−Jk) (43) wherethesumsareovertheN sitesofthelatticeoflocal-
izedspinsontheringcoupledbyferromagneticexchange
and velocity vµ = ∂E /∂kµ, and the remaining coeffi-
k k interactions Jij = J > 0 if i and j label nearest neigh-
cient of the quantum-mechanical zero-point fluctuations
bors. h isacrown-shapedinhomogeneousmagneticfield
i
is
of the form
A =S(J −J )+SK/2. (44)
k k=0 k h =h[sinϑ(e cosϕ +e sinϕ )+e cosϑ], (46)
i x i y i z
AdetailedderivationoftheHamiltonianinthelocalbasis
wheretheanglesϕ labelthepositionsofthespinsonthe
i
andofthemagnondispersion(43)isrelegatedtotheAp-
ring,asillustratedinFig.3. InRef.[4]ithasbeenshown
pendix. In equilibrium the translational current (42) of
that at finite temperature T > 0 spin-wave excitations
course vanishes by symmetry, i.e., (cid:104)I(cid:107) (cid:105)=0. Therefore carry a persistent equilibrium spin current
i→j
there is no translational movement of magnetic moment
in this superfluid spin state. (cid:104)I(cid:107) (cid:105)= 1 (cid:88) vn (47)
Lastly, let us note that the complete decoupling of the i→i+1 L e((cid:15)n+|h|)/T −1
n
magnetic texture and the incoherent magnons is an ar-
tifact of the lowest order approximation in the 1/S ex- circulating the ring. Here L is the length of the ring,
pansion, and of the assumption of a slowly varying tex- v = ∂(cid:15) /∂k is the magnon velocity, and the magnon
n n n
ture. If one relaxes either of these approximations, there dispersion is (cid:15) = SJa2k2, with lattice spacing a and
will be a coupling, resulting in a two-fluid description of quantized wavenvectors k n= 2π (cid:0)n− Ω(cid:1). Ω is the solid
n L 2π
6
angletracedoutbythelocalmagnetizationdirectionmˆ tization configuration is always accompanied by an elec-
i
on the unit sphere in order parameter space as it moves tric field, in contradiction with the elementary fact that
aroundthechain. ThisfinitesolidangleΩ,i.e.,thetopol- inclassicalelectromagnetismmagnetostaticsandelectro-
ogy of the spin configuration on the ring, is responsible statics are completely decoupled.
for the finiteness of the equilibrium current (47). The Finally,letusalsopointoutthatalthoughtheclassical
situation is completely analogous to persistent electri- groundstate(49)oftheHeisenbergringisverysimilarto
cal currents in mesoscopic metal rings pierced by a mag- the classical ground state of the easy-plane ferromagnet
neticflux.4 Theseelectricalcurrentsgenerateamagnetic discussed in the last section, it does not support spin su-
dipole field; one of us has shown in Ref. [4] that the perfluidity, i.e., angular spin supercurrents, because the
persistent spin current (47) similarly generates an elec- phase ϕ of the magnetization is pinned by the external
i
tric dipole field. This can be understood as follows:17 magnetic field h , Eq. (46).
i
A magnetic dipole moment m moving with velocity v
generates a magnetic field B in its rest frame. Lorentz-
transforming back to the laboratory frame, we find that IV. SUMMARY AND CONCLUSIONS
to lowest order in v/c (where c is the speed of light) this
magnetic field generates an electrical field E =−v ×B.
c Weconcludethatintheclassicallimittheangularspin
At zero temperature the equilibrium current (47) van-
currentintroducedbySunandXie8 canbeabsorbedinto
ishesbecausetherearenospinwavesintheferromagnetic
a renormalization of the external magnetic field which
ground state. However, Bruno and Dugaev7 pointed out
contributes to the torque acting on the magnetic mo-
that in this system the classical spin current Jµ defined
ments. In equilibrium, this term does not describe any
in Eq. (38) is finite and argued that therefore the sys-
current of magnetic moments because the equilibrium
tem exhibits an equilibrium spin supercurrent even at
configuration of the magnetization is such that the total
T = 0. While this interpretation is possible, we stress
torqueoneachmomentvanishes. Translationaltransport
that this classical current is angular and not transla-
of spins is described by the longitudinal spin current de-
tional; hence it cannot be associated with the motion
fined in Eq. (29), which is only finite due to thermal or
ofmagneticdipoles,butshouldratherbeconsideredasa
quantum fluctuations. An example for a system exhibit-
renormalizationoftheexternalmagneticfield.5Tounder-
ing a finite longitudinal spin current in equilibrium is a
stand this, let us explicitly calculate the inhomogeneous
mesoscopic Heisenberg ring in a crown-shaped magnetic
magnetizationconfigurationintheclassicalgroundstate
field, as discussed in Ref. [4].
of the Hamiltonian (45). Just as in the easy-plane ferro-
Our considerations imply that in equilibrium the clas-
magnet, the equation of motion for the magnetic texture
sicalspincurrentJµdefinedinEq.(38)doesnotdescribe
mˆ istoleadingorderin1/StheclassicalLandauLifshitz
i anymotionofmagnetization. Whileitcanbeinterpreted
equation
as a stationary current of magnetic polarization, such an
∂ mˆ =mˆ ×(h +δhex), (48) interpretation is by no means mandatory since there is
t i i i i no phyiscal movement. In nonequilibrium on the other
where the exchange field is δhex = S(cid:80) J mˆ . The hand,theclassicalspincurrentJµ contributestothean-
i ij ij j
gular spin current, i.e., to the precessional motion of the
equilibriumsolutionoftheLandau-Lifshitzequation(48)
spins, and does transport spin polarization. In particu-
is
lar,thisimpliesthatspinsuperfluidity,whichisbasedon
mˆ =sinϑ (e cosϕ +e sinϕ )+e cosϑ , (49) the formal similarity of Eq. (40) with a mass supercur-
i m x i y i z m
rentofsuperfluidbosons,20,26 doesnotcorrespondtothe
where the angle ϑ is slightly smaller than the angle ϑ physicalmovementofmagnetization, butofpolarization.
m
between the magnetic field and the z-axis, as discussed This means that in equilibrium, a superfluid spin state
in Ref. [4]. The deviation of ϑ from ϑ is determined willnotbeaccompaniedbyanelectricalfield,incontrast
m
by the exchange field δhex. As for the easy-plane ferro- to a persistent translational spin current.4 This physical
i
magnetdiscussedinthelastsection,thecontinuumlimit difference between the two types of spin transport per-
of the exchange torque is the divergence of the classical sists also out of equilibrium: As already shown by Sun
spin current, δhex ×mˆ → (cid:80) ∂ Jµ. For the classical and Xie,8 an angular spin current with finite ω×mˆ will
i i µ µ
ground state (49), the classical spin current has a finite generate an electrical field Eω ∝ 1/r2 for large distance
component ∝ e ∇ϕ. However, as in the easy-plane fer- r from the source, whereas the electrical field of transla-
z
romagnet, this finite current merely signals an inhomo- tional spin currents decays as E(cid:107) ∝1/r3.
geneous magnetization configuration and is not related We have also shown that spin superfluidity can be de-
to the physical transport of magnetization; hence it also scribed entirely without referring to off-diagonal long-
will not generate any electrical field. If one would on the range order30,31 and magnon condensation. This is
otherhandincorrectlyassociatetheclassicalspincurrent achieved by quantizing the spins in a self-consistently
Jµ with the stationary movement of physical dipoles, it defined frame of reference with the local z-axis pointing
wouldhavetobeaccompaniedbyanelectricalfield. This in the direction mˆ of the instantaneous magnetization.
would imply that a purely static inhomogeneous magne- Magnons defined with respect to this reference frame
7
can never condense or display off-diagonal long-range Expanding the Berry-phase magnetic field in this basis,
order, hence they are not superfluid. This is in agree- B =B(1)e(1)+B(2)e(3)+B(cid:107)mˆ , we explicitly find
i i i i i i i
ment with the general proof of Kohn and Sherrington31
that bosonic quasi-particles which are formed as bound
states of particle-hole pairs of the underlying fermionic B(1) =∂ θ =−e(2)·∂ mˆ , (A5a)
i t i i t i
system (such as excitons or magnons) do not exhibit off-
B(2) =sinθ ∂ φ =e(1)·∂ mˆ , (A5b)
diagonal long-range order in coordinate space. Hence, i i t i i t i
the Bose-Einstein condensation of this type of bosons is B(cid:107) =cosθ ∂ φ . (A5c)
i i t i
not accompanied by superfluidity. Although the change
in magnetic order in a magnetic insulator can be viewed
as Bose-Einstein condensation of magnons,25,26,32,33 the Applying the Holstein-Primakoff transformation (35) to
resulting state can always be characterized by magnons the rotated Hamiltonian then yields
thatexhibitneitheroff-diagonallong-rangeordernorsu-
perfluidity. H˜plane =E +H +H +O(S1/2). (A6)
0 1 2
ACKNOWLEDGMENTS Here the classical ground state energy is
This work was financially supported by the DFG via
SFB/TRR49. Our ideas on spin transport were sharp- E =−S2 (cid:88)J mˆ ·mˆ +S2K(cid:88)(mˆ ·e )2−S(cid:88)B(cid:107).
0 2 ij i j 2 i z i
ened during a workshop on Quantum Spintronics: Spin ij i i
Transport Through Quantum Magnetic Materials at the (A7)
SpinPhenomenaInterdisciplinaryCenter(SPICE)atthe The term linear in the Bose operators can be written as
University of Mainz, Germany.
(cid:114)
H = S (cid:88)a (cid:16)e(2)+ie(1)(cid:17)·(∂ mˆ −ω ×mˆ )+h.c.,
APPENDIX: MAGNONS IN THE ROTATING 1 2 i i i t i i i
i
REFERENCE FRAME
(A8)
whereω istheO(S)localprecessionfrequency,Eq. (36).
i
This Appendix is devoted to the derivation of the Because the magnetic polarization satisfies the equation
dispersion (43) of the magnons in the local reference of motion ∂ mˆ = ω ×mˆ up to this order in 1/S, we
t i i i
frame defined by the magnetic polarization mˆi(t) = conclude that H1 vanishes identically. Note that this
(cid:104)Si(t)(cid:105)/|(cid:104)Si(t)(cid:105)| . We have already derived the general implies that the Holstein-Primakoff bosons cannot con-
setup of a spin-wave expansion in this local and possibly dense, a statement which remains true to all orders in
time dependent reference frame in Ref. [29]. The first 1/S due to the self-consistency of the basis. Up to this
stepistorotatethez-axisofthelaboratoryframetothe point the discussion is completely general and applies to
direction mˆi(t) of the local magnetization by means of all magnetic insulators.
an unitary transformation U(t) acting on the spins. The
ToevaluatethequadraticpartoftherotatedHamilto-
explicit form of U(t) is given in Ref. [29]. It then turns
nianwewillassumeasinthemaintextthatthemagnetic
out that the rotated Hamiltonian H˜plane =Hplane+H
B texturevariesonlyslowlyinspaceandisonlyslightlyout
containsanadditionalBerry-phasetermactingasamag-
ofequilibrium. Thenwemayapproximatemˆ ·mˆ ≈1,
netic field due to the time dependence of mˆ (t), i i+1
i e(p)·e(p(cid:48)) ≈δpp(cid:48), e(p)·mˆ ≈0, with p,p(cid:48) ∈{1,2}, and
i i+1 i i+1
(cid:88) cosθ ≈0. With this simplifications we immediately ob-
H =−iU†∂ U =− B ·S . (A1) i
B t i i tain
i
Writing the spin polarization as (cid:34) (cid:18) (cid:19)
(cid:88) K
H = S J −J + a†a
mˆ =sinθ [e cosφ +e sinφ ]+e cosθ , (A2) 2 k=0 k 2 k k
i i x i y i z i k
(cid:35)
we can choose the transverse basis vectors as K K
−S (a a +h.c.)+S , (A9)
4 k −k 4
e(1) =−e sinφ +e cosφ , (A3)
i x i y i
e(2) =−cosθ [e cosφ +e sinφ ]+e sinθ . (A4)
i i x i y i z i which describes free magnons with the dispersion (43).
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