Omnidirectional spin Hall effect in a Weyl spin-orbit coupled atomic gas

0.58 MB·

by  J. Armaitis

Omnidirectional spin Hall effect in a Weyl spin-orbit coupled atomic gas
J. Armaitis,∗ J. Ruseckas,† and G. Juzeliu¯nas‡
Institute of Theoretical Physics and Astronomy, Vilnius University, Saule˙tekio Ave. 3, LT-10222 Vilnius, Lithuania
(Dated: April 4, 2017)
Weshowthatinthepresenceofathree-dimensional(Weyl)spin-orbitcoupling,atransversespin
current is generated in response to either a constant spin-independent force or a time-dependent
Zeeman field in an arbitrary direction. This effect is the non-Abelian counterpart of the univer-
sal intrinsic spin Hall effect characteristic to the two-dimensional Rashba spin-orbit coupling. We
quantify the strength of such an omnidirectional spin Hall effect by calculating the correspond-
7 ing conductivity for fermions and non-condensed bosons. The absence of any kind of disorder in
1 ultracold-atom systems makes the observation of this effect viable.
0
2 PACSnumbers: 03.65.Vf,03.75.Ss,05.30.Jp,85.75.-d

r
p
A I. INTRODUCTION

2 Gauge theories and related geometrical concepts play

a prominent role in the description of physics at a wide
]
s range of length scales covering all fundamental interac-
a tions [1]. On the other hand, when it comes to effective
g
models, many quantum-mechanical systems with adia-
-
t batically varying parameters are naturally described in
n
terms of Abelian gauge theories [2–4]. This geometric
a
u approach based on the Berry phase has paved the way
q to a multitude of both theoretical and experimental de-
. velopments covering molecular [3–5], solid-state [6–12],
t
a photonic[13–21],mechanical[22,23]andelectric[24,25] FIG. 1. In a Weyl spin-orbit coupled system, a spin-
m systems. Althoughthecorrespondingnon-Abeliangauge independent force pointing in an arbitrary direction drives
a spin current in the plane perpendicular to the force. Here
- structure in the presence of degenerate quantum states
d weshowthissituationinthemomentumspace. Particleswith
has been noticed promptly after the discovery of the
n spins (cid:104)σˆ(cid:105) respond to the force F by contributing to the spin
o Berry phase [26], a set of experimentally observed sig- current J which runs perpendicular to the force and car-
spin
c natures of the non-Abelian geometrical phases remains ries the spin orthogonal to both the current and the force,
[ limited [27, 28]. see Eq. (28). The independence of this phenomenon from
2 Ultracold-atom experiments have been recently gain- the driving direction unambiguously demonstrates the non-
v ing tools uniquely suited to address this elusive non- Abelian nature of the underlying dynamics.
3 Abelian gauge structure using the internal states of the
7 atom [29–44]. In particular, engineering various species
7 of spin-orbit coupling [45–63] in ultracold-atom systems two-dimensional (2D) geometry of the sample. There-
8
has seen rapid advances lately [64–74], allowing experi- fore, both the perturbation of the system (applied volt-
0
mentaldemonstrationof,e.g.,thephasediagramofspin- age) and the resulting spin current are confined to that
.
1 orbit coupled (SOC) bosons [75] and the spin Hall effect plane. Ontheotherhand,thespincurrentispolarizedin
0 [76]. However,despitetheexistenceofthisnoveltoolbox, the direction perpendicular to the SOC plane. This spin
7
there is a lack of concrete proposals to unambiguously Hall effect is induced by a spin-dependent Berry mag-
1
demonstrate the non-Abelian gauge structure. netic field (Berry curvature) perpendicular to the plane.
:
v The spin Hall effect (SHE), in which density currents Such a magnetic field is proportional to a single Pauli
Xi generate transverse spin currents has already played a matrixσz andhenceisAbelian, asitwillbediscussedin
prominentroleincondensed-matterphysicsandhaspro- more details in the next Section.
r
a vided an impetus to the field of spintronics [77]. The In the present article we put forward a three-
SHE has been detected experimentally in a wide variety dimensional (3D) version of the SHE based on the novel
of solid-state materials, which usualy possess a planar possibilitytoengineeranonplanarspin-orbitcouplingof
SOC of the Rashba-Dresselhaus type. In these solid- the Weyl type for ultracold atoms [55, 57–59, 78]. In the
state experiments, the SOC plane corresponds to the proposed setup, the atoms are affected by a 3D Berry
magnetic field which is non-Abelian and induces a spin-
dependent Lorentz-type force for all directions of atomic
∗ [email protected] motion. Perturbing the system along an arbitrary axis
† [email protected] produces a spin current perpendicular to the perturba-
‡ [email protected] tion(Fig.1). Sucharesponseisinastarkcontrasttothe
2
Abeliancase,wherethemagneticfieldunavoidablyhasa tial Aˆ. An extra term βe provides a spatially uniform
β
singlewell-defineddirection,andthe(spin)Halleffectoc- spin-independentdrivingforceβ˙e perturbingtheatoms
β
cursintheplaneperpendiculartoit. Wewillrefertothe alongaunitvectore ,thedotdenotingatimederivative.
β
presenteffectbasedonthe3DSOCasanomnidirectional HereU isaspin-independenttrappingpotential. Wesup-
spin Hall effect which is the non-Abelian counterpart of press the identity matrix in the spin space and set (cid:126)=1
the universal intrinsic spin Hall effect characteristic to at the outset. In Eq. (1) the bold font specifies a spatial
the two-dimensional Rashba SOC [79]. vector, whereas the hat indicates an operator acting on
In certain lattice systems [80–88], pairs of Weyl points the atomic internal (pseudo-) spin states. Moreover, p
of opposite topological charges arise governing the topo- is a momentum operator and m is an atomic mass. Al-
logical properties [80, 89] or interactions between parti- though for concreteness we consider (pseudo-) spin 1/2
cles [90] in the so-called Weyl semimetal regime. Since atoms,generalizationtoahigher-spinsystemisstraight-
the Weyl points have opposite topological charges, they forward and does not change the qualitative picture.
respond to driving in an opposite way, and the induced The Hamiltonian in Eq. (1) yields two dispersion
spin-currents cancel. Here we consider the SOC of the branches for an unperturbed atom (β = 0) affected by
Weyl type (also known as the Weyl-Rashba SOC) pro- the Weyl SOC:
duced by manipulating atomic internal states [55, 57–
59] rather than using a lattice, so only a single Weyl ε =(cid:0)p2±2χp+χ2(cid:1)/2m=(p±χ)2/2m, (2)
p±
point arises. This is an important feature for generating
anonzerospincurrentinresponsetoaspin-independent where the lower (upper) sign corresponds to the lower
force. (upper)dispersionbranchinwhichthespinpointsalong
In the present study we do not include the effects due (opposite to) the momentum p. In writing Eq. (2) we
to impurities. The impurities play a crucial role in the haveaddedaconstanttoplacetheminimumofthelower
spinHalleffectphysicsforelectronsinsolidstotheextent dispersion branch at the zero energy: εχ− =0.
of preventing the universal intrinsic spin Hall effect [77,
91,92]. However, ultracold-atomsarefreefromimpurity
scattering, both magnetic and nonmagnetic, so the spin- B. Equations of motion
Hall effect is not suppressed in these systems.
Furthermore,interactionsbetweentheultracold-atoms DefiningavelocityoperatorforanatomviatheHeisen-
are typically weak [93], and they can be further min- berg equation
imized by utilizing the Feshbach resonances [94]. We
therefore leave the detailed study of interaction effects vˆ =−i[r,Hˆ]=(p−Aˆ)/m, (3)
[95] for future work.
The paper is organized as follows. In Section II we one can rewrite the Hamiltonian in a concise manner:
define the atomic Hamiltonian with the Weyl SOC in- Hˆ = mvˆ2/2+U, where r is a position operator. The
cluded, and write down the equations of motion for the velocity operator vˆ contains a vector potential Aˆ which
spin and center-of-mass degrees of freedom. Section III is an operator acting in the spin space. Hence vˆ obeys
explores spin currents in this system, and presents the the following non-trivial Heisenberg equation of motion
possibility to generate a transverse spin current for any [62]
direction of the applied perturbation. The concluding
SectionIVsummarizesthefindingsandoutlinespossible mvˆ˙ =(vˆ×Bˆ−Bˆ×vˆ)/2−∇U +Eˆ, (4)
futuredirections. InAppendix,wediscussinmoredetail
the definition of the spin current used in the main text, where
andconsiderarelationshipbetweenspin(Stern-Gerlach)
projection measurement and and the spin Hall current. Eˆ=−∂tAˆ=−β˙eβ (5)
is the strength of the perturbing Berry electric field, and
II. THEORETICAL FRAMEWORK AND
Bˆ=−iAˆ×Aˆ=2χ2σˆ (6)
NON-ABELIAN DYNAMICS
is the strength of the Berry magnetic field.
A. Hamiltonian The latter magnetic field Bˆis proportional to the spin
operatorσˆ. Hence,ithasnon-commutingCartesiancom-
Let us consider an ensemble of atoms subjected to a ponents, showing a non-Abelian character of Bˆ. This is
Weyl (3D) SOC of a strength χ. Individual atoms are in contrast to the usual planar Rashba-type spin-orbit
described by the Hamiltonian coupling for which Aˆ ∝ σˆ e + σˆ e , so the Berry
x x y y
Hˆ =(p−Aˆ)2/2m+U, Aˆ=χσˆ +βe , (1) magnetic field strength Bˆ ∝ σˆzez contains commuting
β
Cartesian components, and is thus Abelian. Note that
where the Weyl SOC is due to the vector of Pauli matri- the components of the spatially uniform Berry magnetic
ces σˆ = σˆ e +σˆ e +σˆ e entering the vector poten- field (6) can be written in terms of the field strength
x x y y z z
3
Fˆ =−i[Aˆ ,Aˆ ][26],alsoknownastheYang-Millscur- ensures that (cid:104)σˆ(cid:105)2 = 1. The condition S (cid:39) 1 defines a
ab a b ±
vature [96]: Bˆ = (cid:80) (cid:15) Fˆ /2. Genuine non-Abelian range of validity of Eq. (10):
a b,c abc bc
dynamics occurs only in systems where [Fˆ ,Fˆ ] (cid:54)= 0,
ab cd β˙m/2χ|p(cid:48)|2 (cid:28)1. (13)
as discussed in detail in Ref. [62]. Indeed, our sys-
tem falls into the non-Abelian-dynamics class, as here
Fˆ =2χ2(cid:80) (cid:15) σˆ . Inthisadiabaticapproachthespinexpectationvalueis
ab c abc c determined by the momentum-dependent effective mag-
The spin-dependent part of the Hamiltonian (1) can
neticfieldM,aswellasbythecorrectiontermcontaining
be represented as −M ·σˆ, where we have introduced an the time-derivative M˙ due to the external force. In the
effective magnetic field
zero-order adiabatic approximation, the spin follows the
M =χp(cid:48)/m, p(cid:48) =p−βeβ, (7) effective magnetic field: (cid:104)σˆ(0)(cid:105)± = ±M/M = ±p(cid:48)/p(cid:48).
The first-order correction (cid:104)σˆ(1)(cid:105) = ±M ×M˙ /2M3 is
with p(cid:48) being a momentum shifted by the spin- ±
given by
independent driving term βe . The spin dynamics fol-
β
lows a Landau-Lifshitz type [97] equation (LLE) (cid:104)σˆ(1)(cid:105) =∓β˙mp(cid:48)×e /2χp(cid:48)3 ≈∓β˙mp×e /2χp3, (14)
± β β
σˆ˙ =−i[σˆ,Hˆ]=2σˆ ×M, (8)
where the last relations also assumes small momentum
changes: |p(cid:48)−p| = β (cid:28) p. The correction (cid:104)σˆ(1)(cid:105) tilts
where for our dissipationless cold-atom system we have ±
the spin in the direction orthogonal both to the momen-
notaddedtheGilbertdamping[98]termusuallypresent
tumoftheatomandalsoorthogonaltothedrivingforce
when describing the magnetization dynamics in solids.
whichcanpointinanarbitrarydirectione . Hence,this
One can now write down the following concise equation β
first-order correction term induces a transverse spin Hall
of motion for the velocity in terms of M:
current to be considered in detail in the next Section.
mvˆ˙ =−β˙e +2χM ×σˆ −∇U. (9) Theinducedspincurrentinturnprovidesadirectsigna-
β
ture of the omnidirectional spin Hall effect illustrated in
Equations (8) and (9) describe the full atomic dynam-
Fig. 1.
ics which involves both internal and center-of-mass de-
grees of freedom. From now on we consider a homoge-
neoussystemforwhichU =0. Thisconditionisviablein III. SPIN CURRENT
a harmonic trap in the local density approximation [93]
sense, or, alternatively, in a flat trap [99] away from the
Weuseananticommutator-baseddefinitionofthespin
boundaries of the trap. In such a homogeneous case, the
current tensor (see Appendix and Refs. [102–104] for a
momentum p is an integral of motion: p˙ =0. Therefore,
detailed discussion), namely,
dynamics in the spin sector completely determines the
evolution of the system. 1
(J )j = (cid:104){vˆ,σˆ }(cid:105)=(p(cid:48)(cid:104)σˆ (cid:105)−χn δ )/m, (15)
spin i 2 i j i j 3 ij
C. Adiabatic approach wheren isaparticledensityofour3Dsystem. Thesub-
3
script i labels the position-space components of the cur-
Since the momentum p is conserved, we will hence- rent defining the flow direction, whereas the superscript
forth work with momentum eigenstates and treat p and j indicates the spin components specifying the spin di-
M as ordinary vectors rather than operators. The equa- rection carried by the current. Angular brackets signify
tion of motion for the quantum expectation value (cid:104)σˆ(cid:105) a quantum average over the spin degrees of freedom for
then has the same form as Eq. (8) for the spin operator a fixed momentum p of an individual atom. An overline
σˆ/2. Hence one arrives at the following solution for the denotes a subsequent ensemble average, that is, a sta-
expectation value to the first order in time derivatives of tistical average over momentum eigenstates of the equi-
M (see Refs. [100, 101] for more details): librium atomic ensemble. Since we are working in the
Heisenbergrepresentation,thedynamicsiscontainedex-
(cid:18) (cid:19)
(cid:104)σˆ(cid:105) =±S M + 1 M ×M˙ , (10) clusively in the time dependence of the operators.
± M 2M3 The atoms in different dispersion branches contribute
differently to the spin current, so it is convenient to
where the upper (lower) sign pertains to upper (lower)
rewrite Eq. (15) as
dispersion branch in which spin points along (opposite
to) the effective magnetic field. Here (J )j =((cid:88)p(cid:48)(cid:104)σˆ (cid:105) −χn δ )/m, (16)
spin i i j η 3 ij
M ≡|M|=χ|p(cid:48)|/m, M˙ =−χβ˙e /m, (11) η=±
β
wheretheupper(lower)signcorrespondstoatomsinthe
and the normalization factor
upper (lower) dispersion branch labelled by the symbol
S =(cid:0)1+[β˙m/2χ|p(cid:48)|2]2(cid:1)−1/2 (cid:39)1 (12) η =±.
4
A. Equilibrium spin current force ∝ e and also to the spin that the current carries.
β
This holds for an arbitrarily directed driving force and
At equilibrium the external force is absent (β =0), so thus represents the omnidirectional spin Hall effect.
p(cid:48) = p and (cid:104)σˆ(cid:105) = (cid:104)σˆ(0)(cid:105) = ±p/p. Since the momen- It is noteworthy that the two dispersion branches pro-
± ±
vide opposite contribution to the spin conductivity in
tum distribution is spherically symmetric, the ensemble
Eq. (21). Therefore σ should be larger at low temper-
averagingyieldsforatomsinaselecteddispersionbranch: SH
atureswhentheatomspopulatepredominantlythelower
p p /p =δ p p /p =δ (p) /3. (17) dispersion branch. In the following Section we shall ex-
i j ± ij i i ± ij ±
plore this issue in more details.
Consequently the spin current (16) takes the form Atthispointitisusefultocontrasttheomnidirectional
spin Hall effect described by Eq. (21) to the usual spin
(cid:34) (cid:35)
HalleffectduetoRashbaSOCactinginthexyplane[79].
(cid:88)
(Js0pin)ji = η(p)η/3+χn3 δij/m. (18) lnthecaseoftheRashbaSOC,thespinHallresponseto
η=± an external force can be presented in a manner similar
to Eq. (21). Specifically, the spin current resulting from
Ascanbeseenfromthisexpression,theequilibriumspin
driving the system along e can be written as
current (J0 )j generally does not vanish in our system. β
spin i
This is usual for SOC systems in general [105] and has
(δJRashba)z ∼e ·(e ×e ), (23)
also been considered in the context of cold atoms in par- spin i z i β
ticular [106]. Note that at equilibrium, the spin current
sotheinducedspincurrentcarriesonlythez component
islongitudinal,i.e.,thespinispolarizedalongtheCarte-
of the spin, which is perpendicular to the SOC plane
sianvectore paralleltothedirectione ofthespincur-
j i (xy). The spin Hall current given by Eq. (23) is zero if
rent. This is reflected by the Kronecker delta function
thedrivingdirectione orthedirectione oftheinduced
β i
entering Eq. (18).
spin current are taken to be along e . In the case of the
z
Weyl SOC, the induced spin current given by Eq. (21)
carries spins polarized in any direction e . The induced
B. Spin Hall current j
spin current and the driving can then point in arbitrary
directions e and e as long as they are not parallel to
i β
In what follows, we concentrate on the spin currents each other.
brought about by driving. Specifically, we will consider
thedifferenceinthespincurrentsbetweenthedrivensys-
tem and the equilibrium situation, namely, the induced
C. Momentum averaging
spin current
Although we are dealing with a 3D system of atoms,
(δJ )j =p (cid:104)σˆ(1)(cid:105)/m. (19)
spin i i j it is convenient to define a generic D-dimensional parti-
cle density function n for a chemical potential µ at a
CallingonEq.(14)for(cid:104)σˆ(1)(cid:105)±,theinducedcurrenttakes temperature T: D
the form:
n =n +n , (24)
(δJ )j =− β˙ e ·(e ×e )(cid:88)η(cid:18)pipi(cid:19) . (20) D D+ D−
spin i 2χ j i β p3 where
η=± η
S (cid:90) ∞
Using the fact that the momentum distribution is spher- n = D pD−1dpf±(p) (25)
ically symmetric, one arrives at the following result: D± (2π)D 0
(δJ )j =β˙σ (e ×e )·e , (21) is a D-dimensional density of atoms in the upper or
spin i SH i j β
lower dispersion branch, S ≡ 2πD/2/Γ(D/2) is a D-
D
where dimensional unit-sphere area,
σ =− 1 (cid:88)η(1/p) (22) f±(p)=[e(εp±−µ)/kBT +α]−1 (26)
SH 6χ η
η=±
is a distribution function for fermions (α=1) or bosons
is the spin Hall conductivity. For instance, by choosing (α = −1) in the dispersion branch ε , and k is the
p± B
the driving to point along the z axis (e = e ) the spin Boltzmann constant.
β z
and its spin current will be in the xy plane, as in Fig. 1. Weconsiderasystem,withafixed3D particledensity
In this way, in contrast to the equilibrium spin cur- ν. The chemical potential µ at a certain temperature T
rent, the induced spin current given by Eqs. (21)–(22) is obtained from the condition
is transverse. Specifically, the spin current flows in the
direction e which is perpendicular both to the driving n (µ,T,χ)=ν. (27)
i 3
5
Usingthisnotation,thespinHallconductivity(22)takes 2.0 1.0
the form 0.8
1.5
0.6
n −n 1.0
σ = 2− 2+ , (28) 0.4
SH 6χπ 0.5 0.2
0.0 0.0
where n and n correspond to the 2D densities of 0 1 2 3 4 5 0 5 10 15 20
2+ 2−
atoms in the upper and lower dispersion branches, re-
spectively.
FIG. 2. Left: dimensionless spin Hall conductivity as a func-
In particular, an ensemble of fermions with a thermal
tion of the dimensionless SOC strength. Right: temperature
energyk T muchsmallerthattheFermienergyε pop-
B F dependenceofspinHallconductivityforafixedSOCstrength
ulatetheenergylevelsuptoε =µcorrespondingtothe
F χ¯ = 1 normalized to spin Hall conductivity at low temper-
F
zero temperature limit. If ε is below the band crossing,
F ature. Both plots show the results for fermions (blue solid)
only the lowest dispersion band is populated (n2+ = 0). and bosons (yellow dashed) in the absence of a condensate.
Ontheotherhand,ifε isabovethebandcrossing,both
F
dispersion band are populated. In both cases the differ-
ence in band densities is given by
in the dispersion [62, 63, 107]. Even small interactions
χ(cid:112) will have a large effect on the nature of the condensate
n2−−n2+ = π 2µm. (29) groundstate, and, in turn, on its transport properties.
Hence, the results presented here only hold for noncon-
Using Eqs. (28) and (29) one can see that for fermions densed gases with weak interactions.
√
the spin Hall conductivity σ ∝ µ depends on the
SH
SOC strength χ only through the chemical potential
µ = µ(χ,ν) in the zero temperature limit. This differs
D. Validity of approximation
from the previously considered 2D Rashba SOC where
thelowtemperaturespinHallconductivityσ(2D) takesa
SH In our analysis we have applied the adiabatic approxi-
universalvaluewhichisindependentoftheSOCstrength
mationwhichisapplicablewhenEq.(13)holds. Herewe
if both bands are populated [79].
explicitly check if this approximation holds for a typical
In general the spin-Hall conductivity σ depends on
SH experimental system in the range of interest of the pa-
the temperature, the statistical distribution, and the
rameters. Given a sufficiently low temperature and par-
SOC strength. We explore these dependencies in Fig. 2,
ticle density, when only the bottom of the lowest band
inwhichthespinHallconductivityisplottedforthefixed
is occupied, we can assume that |p| (cid:39) χ. Moreover, we
particle density ν as a function of the temperature and
assume for clarity that the driving is provided by some
the dimensionless SOC strength
harmonicpotentialwiththedisplacementofthecenterof
(cid:112) the system equal to the length of the trap. In that case
χ¯ =χ 2β /m, (30)
F F Eq. (13) immediately yields the condition ωβ (cid:28) χ¯2,
F F
where ω is the trap frequency. We consider a system
where β =1/k T and
F B F withaparticledensityof1018m−3,whichcorrespondsto
T ≡(3π2n)2/3/2mk (31) TF (cid:39) 20nK. For these parameter values, a driving force
F B provided by a ω = 10Hz trap leads to the adiabaticity
criterion χ¯ (cid:29)0.1.
isdefinedinthesamewaybothforbosonsandfermions. F
For fermions T corresponds to the Fermi temperature. Hence,ourapproximationiscertainlyvalidinasetting
F
In addition we define the de Broglie wavelength at the whendrivingisrelativelygentle,temperatureisverylow,
temperature TF and SOC strength is moderate. This regime, where σSH
is maximized and the approximation is robust, does not
Λ =(cid:112)2πβ /m=(16/πn2)1/6. (32) seem to put any extra challenges to the experimentalist,
F F
besides achieving the Weyl SOC. We note that temper-
The proposed effect is present both for bosons and atures as low as several nanokelvin have been demon-
fermions. Even though the induced conductivity is the strated[108],whileanoptically-generatedSOCroutinely
largest for χ¯F → 0, care must be taken in interpreting achieves χ¯F = 5 in the equal Rashba-Dresselhaus case
thisresult. Infact,inthisparameterrangetheadiabatic [76]. Thequestionofvalidityoftheadiabaticapproxima-
approximation becomes invalid, as it will be discussed tionis,however,separatefromthefeasibilityofdetecting
in detail in the next Subsection. Note that at a mean- this effect. The latter question is addressed in the next
field level, the conductivity would not be modified by subsection.
the presence of interactions, as they would merely shift In applying our adiabatic approximation we have im-
the chemical potential by a constant. Yet considering plicity assumed that the driving is switched on slowly.
a Bose-Einstein condensed state in this system is inher- However,ifthedrivingisswitchedonsuddenly,theadia-
ently nontrivial due to the absence of a single minimum baticapproximationisnotsufficient,andonehastosolve
6
the LLE at least to the second order in time derivatives. the same manner, these two ways of driving the system
We have checked, however, that the SHE is still present lead to the same effect for the spin dynamics. Therefore,
inthispost-adiabaticsolution. Theonlynewfeaturethat theaboveanalysisoftheinducedspincurrentduetothe
arises in this higher-order solution is a Zitterbewegung- spin-independent force can be transferred in a straight-
like beating between the two adiabatic solutions, which forward manner to the case of the Zeeman driving via
has been considered elsewhere [50, 69, 70, 109, 110]. the replacement of β by −mγ and e by e .
β γ
E. Detection of spin current IV. SUMMARY AND FUTURE WORK
As discussed above, the most direct signature of the In summary, we have put forward a proposal to ob-
omnidirectionalHalleffectisthespincurrentδJ . The serve inherently non-Abelian dynamics in the form of an
spin
experimental sequence needed to detect that current de- omnidirectionalspinHalleffectinadrivensysteminthe
pends on the precise details of the implementation of presence of a Weyl (three-dimensional) spin-orbit cou-
WeylSOC,asproposalstoachieveitutilizequalitatively pling. We have discussed two independent ways to drive
different physical means [55, 57, 59]. Nevertheless, sev- the system leading to the same effect for the spin dy-
eral general remarks can be made with no reference to namics: eitherthroughaconstantspin-independentforce
these experimental details. or a time-dependent Zeeman field. We have also evalu-
Inparticular,itispossibletoutilizethefactthatδJ ated the strength of this effect in terms of conductivity
spin
given by Eq. (21) is the transverse spin current. This is fornoninteractinguncondensedbosonicorfermionicgas.
beneficial, since the equilibrium spin current J0 is lon- Allofthecomponentsofthisproposalseemtobewithin
spin
gitudinal, and the spin of an atom in the upper (lower) the reach of cold-atom experiments in the near future,
band points along (opposite to) the momentum p. As and their combination has a potential to unambiguously
a result, one can filter out the δJ signal by choos- demonstratenon-Abeliandynamicsinacontinuum(non-
spin
ing a beneficial configuration of the driving direction e , lattice) cold-atom system for the first time.
β
the spin (Stern-Gerlach) projection axis e = e and Infuturework,weplantoinvestigatecollectivemodes
j SG
the direction of the momentum measurement e = e . of a trapped system and look for signatures of the non-
j det
Specifically,ifonetakesthesethreevectorstobeorthog- Abelian dynamics described here. Other promising av-
onal, the triple product enuesofresearchincludeamorecarefulaccountofinter-
actions, especially with the Bose-Einstein condensation
edet·(eSG×eβ) (33) in mind, and also considering the kinetic effects in this
system, e.g., therelaxationofspincurrentalsoknownas
featuredinEq.(21)forδJ ismaximizedandthesignal
spin spin drag [111], which was not considered here.
is the strongest. A relation between the spin current
and the spin (Stern-Gerlach) projection measurement is
considered in the last subsection of the Appendix.
ACKNOWLEDGMENTS
Moreover, one can estimate the size of the effect of
the omnidirectional spin Hall effect on the momentum
It is our pleasure to thank Egidijus Anisimovas, Alain
distribution. Since the SOC strength χ sets the charac-
Aspect, Denis Boiron, Rembert Duine, Gabriele Ferrari,
teristic momentum in the distribution of particles in the
Lars Fritz, Simonas Grubinskas, Krzysztof Jachymski,
system, the magnitude of the signal (the change of the
Wolfgang Ketterle, Oleksandr Marchukov, Bill Phillips,
momentum distribution due to the omnidirectional spin
Henk Stoof, Roland Winkler, and Ulrich Zuelicke for
Halleffect)isapproximatelyequaltotheratioβ˙σ /νχ.
SH stimulating discussions. J. A. has received funding from
theEuropeanUnion’sHorizon2020researchandinnova-
tionprogrammeundertheMarieSkl(cid:32)odowska-Curiegrant
F. Spin-current induced by a time-dependent
agreement No 706839 (SPINSOCS).
Zeeman term
The spin Hall effect can also be induced by a time-
Appendix: Spin current
dependentZeemanshiftratherthanbyatime-dependent
externalforce. Inthatcasethetermγe ·σˆ istobeadded
γ
InthisAppendixwemotivatethedefinitionofthespin
to the Hamiltonian, and the effective magnetic field M
current given in the text by deriving the spin continu-
determining the spin dynamics becomes
ity equation and considering the effect of a spin (Stern-
M =χp/m+γe . (34) Gerlach)projectiononthevelocityoperator. Incontrast
γ
to the main text, in this Appendix we use hats to label
Since the scalar driving β (due to a spin-independent not only the spin operators but all operators (including
force on an atom), and the Zeeman driving γ (due to a the coordinate and momentum operators rˆ and pˆ) in or-
magnetic pulse) enter the effective magnetic field M in der to make the Appendix as accessible as possible.
7
1. Continuity equation and spin current 2. Source term
The spin density is a vector field given by The vector potential Aˆ given by Eq. (1) describes
the 3D SOC and the driving. The space- and spin-
ρ(r)=Ψ†(r)σˆΨ(r)=(cid:104)Ψ|r(cid:105)σˆ(cid:104)r|Ψ(cid:105)=(cid:104)Ψ|ρˆ |Ψ(cid:105), (A.1)
r independent driving term β(t)e does not contribute to
β
where ρˆ = σˆδ(rˆ−r) is the corresponing spin density the commutators entering Eq. (A.4), giving
r
operator, and Ψ(r)=(cid:104)r|Ψ(cid:105) is a two component column-
spinor. Here the quantum average has been carried out [σˆj,Aˆl]vˆl =χ[σˆj,σˆl]vˆl =2iχεjlnσˆnvˆl =−2iχ(σˆ×vˆ)j.
over the full state-vector |Ψ(cid:105) accommodating both the (A.9)
motional and spin degrees of the atom. Furthermore, we In a similar way,
havecastedtheoperatorδ(rˆ−r)=|r(cid:105)(cid:104)r|intermsofthe
vˆ[σˆ ,Aˆ]=2iχ(vˆ×σˆ) . (A.10)
eigenstates |r(cid:105) of the coordinate operator: rˆ|r(cid:105)=r|r(cid:105). l j l j
The dynamics of the operator ρˆ is governed by the
r Consequently,
Hamiltonian given by Eq. (1) which contains the Weyl
SOC term, and thus Gˆ =χ(δ(rˆ−r)σˆ ×vˆ−vˆ×σˆδ(rˆ−r)) . (A.11)
r
d 1
ρˆ = [ρˆ ,Hˆ] (A.2)
dt r i r with mvˆ =pˆ−χσˆ −βeβ.
(cid:18) (cid:19) Inthecaseofthe3DSOC,theeigenstatesη describe
=12(cid:88) 1i[ρˆr,pˆl−Aˆl]vˆl+vˆl1i[ρˆr,pˆl−Aˆl] , the spin along the momentum: (cid:104)σˆ(cid:105) = ±p/pp±. There-
l fore, the source term vanishes after taking the quantum
expectation value (cid:104)Gˆ (cid:105) and averaging over an isotropic
wherethematrix-valuedvelocityoperatorvˆ isdefinedin r
momentum distribution.
Eq. (3). Since
[ρˆ ,pˆ −Aˆ]=−iσ∇ δ(rˆ−r)−[σ,Aˆ]δ(rˆ−r),
r l l l l
3. Spin projection measurement
one arrives at the following continuity equation:
dρˆ +∇ jˆ =Gˆ , (A.3) Hereweconsidertheeffectofthespin(Stern-Gerlach)
dt r l rl r projectionmeasurementonthevelocityvˆ alongtheunit
i
vector e . We will show that the spin current measured
where i
in this way is consistent with its previous definition. In
1 (cid:16) (cid:17)
Gˆ =− δ(rˆ−r)[σˆ,Aˆ]vˆ +vˆ[σˆ,Aˆ]δ(rˆ−r) particular, a Stern-Gerlach (SG) projection in the j di-
r 2i l l l l
rectionisgivenbytheprojector|s (cid:105)(cid:104)s |,wherethequan-
(A.4) j j
tum state |s (cid:105) describes the spin pointing along the unit
is the spin source operator and j
vectore . Calculatingtheexpectationvalueoftheveloc-
j
jˆ = 1(δ(rˆ−r)σˆvˆ +vˆσˆδ(rˆ−r)) (A.5) ityoperatorvˆi withrespecttosuchspin-projectedstates
rl 2 l l entails evaluating |s (cid:105)(cid:104)s |vˆ|s (cid:105)(cid:104)s |. Since the spin pro-
j j i j j
is the probability current operator. jection operator can be written down as
In what follows we shall consider the spin current for
1
momentum eigenstates of the Weyl SOC Hamiltonian, Iˆ =|s (cid:105)(cid:104)s |= (1+e ·σˆ), (A.12)
sj j j 2 j
Ψ(r)=Ψ (r)=V−1/2η eip·r, (A.6)
p± p±
we have
whereV isaquantizationvolumeandthespinorη de-
p±
scribesthequantumstatesforthespinalongoropposite Iˆ vˆIˆ =(cid:0)vˆ +(e ·σˆ)vˆ (e ·σˆ)(cid:1)/4
tothemomentum: p/p·σˆηp± =∓ηp±. Thecorrespond- sj i sj i +(cid:0)j(e ·σˆi)vˆj+vˆ (e ·σˆ)(cid:1)/4. (A.13)
ing expectation value of the spin current is j i i j
1 ComparingthisexpressionwithEq.(15),onecanseethat
j (r)=(cid:104)Ψ|jˆ |Ψ(cid:105)= (cid:104){σˆ,vˆ}(cid:105), (A.7)
l rl 2V l thesecondtermontheright-handsideisproportionalto
the spin current. Using the properties of Pauli matrices,
with {σˆ,vˆ} = σˆvˆ +vˆσˆ, where the brackets (cid:104)···(cid:105) la-
l l l the first term simplifies to
bel the quantum averaging over the spinor state η .
p±
Performing also a statistical averaging over the atomic
single-particledistributionf±(p),onearrivesatthespin vˆi+(ej ·σˆ)vˆi(ej ·σˆ)
current presented in Eq. (15) of the main text: =2[p −χ(e ·σˆ)(e ·e )]/m. (A.14)
i j i j
(J )j = (cid:88) fη(p)(cid:104){σˆ ,vˆ}(cid:105)= 1(cid:104){σˆ ,vˆ}(cid:105), (A.8) Consequently
spin i 2V j i 2 j i
p,η=± p −χ(e ·σˆ)(e ·e ) (Jˆ )j
wherethestatisticalaveragingisdenotedbyanoverline. IˆsjvˆiIˆsj = i j2m i j + sp2in i . (A.15)
8
As the projection direction is reversed, e → −e , the resulting from a spin (Stern-Gerlach) projection in the
j j
first term is unaffected, while the second term changes direction e , one measures the spin current exactly as
j
itssign. Thereforebyconsideringthedifferenceinveloc- defined in Eq. (15) in the main text.
ities between the spin up and the spin down components
[1] T. Frankel, The Geometry of Physics: An Introduction [25] V. V. Albert, L. I. Glazman, and L. Jiang, Phys. Rev.
(Cambridge University Press, 2004). Lett. 114, 173902 (2015).
[2] M. V. Berry, Proceedings of the Royal Society of Lon- [26] F.WilczekandA.Zee,Phys.Rev.Lett.52,2111(1984).
don. A. Mathematical and Physical Sciences 392, 45 [27] A. Zee, Phys. Rev. A 38, 1 (1988).
(1984). [28] J.W.Zwanziger,M.Koenig, andA.Pines,Phys.Rev.
[3] A. Shapere and F. Wilczek, eds., Geometric Phases in A 42, 3107 (1990).
Physics (World Scientific, Singapore, 1989). [29] Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips,
[4] A. Bohm, A. Mostafazadeh, H. Koizumi, Q. Niu, and J.V.Porto, andI.B.Spielman,Phys.Rev.Lett.102,
J. Zwanziger, Geometric Phases in Quantum Systems 130401 (2009).
(Springer, Berlin, Heidelberg, New York, 2003). [30] Y. J. Lin, R. L. Compton, K. Jimenez-Garcia, J. V.
[5] C. A. Mead, Rev. Mod. Phys. 64, 51 (1992). Porto, and I. B. Spielman, Nature 462, 628 (2009).
[6] D. Loss, P. Goldbart, and A. V. Balatsky, Phys. Rev. [31] M. Aidelsburger, M. Atala, S. Nascimb`ene, S. Trotzky,
Lett. 65, 1655 (1990). Y.-A.Chen, andI.Bloch,Phys.Rev.Lett.107,255301
[7] L. P. L´evy, G. Dolan, J. Dunsmuir, and H. Bouchiat, (2011).
Phys. Rev. Lett. 64, 2074 (1990). [32] J. Struck, C. O¨lschla¨ger, M. Weinberg, P. Hauke, J. Si-
[8] V. Chandrasekhar, R. A. Webb, M. J. Brady, M. B. monet, A. Eckardt, M. Lewenstein, K. Sengstock, and
Ketchen, W. J. Gallagher, and A. Kleinsasser, Phys. P.Windpassinger,Phys.Rev.Lett.108,225304(2012).
Rev. Lett. 67, 3578 (1991). [33] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,
[9] D. Mailly, C. Chapelier, and A. Benoit, Phys. Rev. B. Paredes, and I. Bloch, Phys. Rev. Lett. 111, 85301
Lett. 70, 2020 (1993). (2013).
[10] A.Neubauer,C.Pfleiderer,B.Binz,A.Rosch,R.Ritz, [34] H.Miyake,G.A.Siviloglou,C.J.Kennedy,W.C.Bur-
P. G. Niklowitz, and P. Bo¨ni, Phys. Rev. Lett. 102, ton, and W. Ketterle, Phys. Rev. Lett. 111, 185302
186602 (2009). (2013).
[11] Y. Li, N. Kanazawa, X. Z. Yu, A. Tsukazaki, [35] C.J.Kennedy,G.A.Siviloglou,H.Miyake,W.C.Bur-
M. Kawasaki, M. Ichikawa, X. F. Jin, F. Kagawa, and ton, and W. Ketterle, Phys. Rev. Lett. 111, 225301
Y. Tokura, Phys. Rev. Lett. 110, 117202 (2013). (2013).
[12] M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 [36] J. Armaitis, H. T. C. Stoof, and R. A. Duine, Phys.
(2010). Rev. Lett. 110, 260404 (2013).
[13] K.Y.Bliokh,A.Niv,V.Kleiner, andE.Hasman,Nat. [37] J.-y. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y.-i.
Photon. 2, 748 (2008). Shin, Phys. Rev. Lett. 111, 245301 (2013).
[14] M.Hafezi,E.A.Demler,M.D.Lukin, andJ.M.Taylor, [38] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,
Nature Phys. 7, 907 (2011). T. Uehlinger, D. Greif, and T. Esslinger, Nature 515,
[15] T.Kitagawa,M.A.Broome,A.Fedrizzi,M.S.Rudner, 237 (2014).
E. Berg, I. Kassal, A. Aspuru-Guzik, E. Demler, and [39] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala,
A. G. White, Nat. Commun. 3, 882 (2013). J. T. Barreiro, S. Nascimb`ene, N. R. Cooper, I. Bloch,
[16] M.C.Rechtsman,J.M.Zeuner,Y.Plotnik,Y.Lumer, and N. Goldman, Nature Phys. 11, 162 (2014).
D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and [40] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro,
A. Szameit, Nature 496, 196 (2013). B.Paredes, andI.Bloch,NaturePhys.10,588(2014).
[17] S. Mittal, J. Fan, S. Faez, A. Migdall, J. Taylor, and [41] N.Fla¨schner,B.S.Rem,M.Tarnowski,D.Vogel,D.-S.
M. Hafezi, Phys. Rev. Lett. 113, 087403 (2014). Lu¨hmann, K. Sengstock, and C. Weitenberg, Science
[18] L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and 352, 1091 (2016).
M. Lipson, Nat. Photon. 8, 701 (2014). [42] T. Li, L. Duca, M. Reitter, F. Grusdt, E. Demler,
[19] T. Dubˇcek, K. Lelas, D. Juki´c, R. Pezer, M. Soljaˇci´c, M.Endres,M.Schleier-Smith,I.Bloch, andU.Schnei-
and H. Buljan, New J. Phys. 17, 125002 (2015). der, Science 352, 1094 (2016).
[20] S. A. Skirlo, L. Lu, Y. Igarashi, Q. Yan, J. Joannopou- [43] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and
los, and M. Soljaˇci´c, Phys. Rev. Lett. 115, 253901 I. B. Spielman, Science 349, 1514 (2015).
(2015). [44] M. Mancini, G. Pagano, G. Cappellini, L. Livi,
[21] T.Ozawa,H.M.Price,N.Goldman,O.Zilberberg, and M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio,
I. Carusotto, Phys. Rev. A 93, 043827 (2016). M.Dalmonte, andL.Fallani,Science349,1510(2015).
[22] R. Su¨sstrunk and S. D. Huber, Science 349, 47 (2015). [45] A.M.Dudarev,R.B.Diener,I.Carusotto, andQ.Niu,
[23] G. Salerno, T. Ozawa, H. M. Price, and I. Carusotto, Phys. Rev. Lett. 92, 153005 (2004).
Phys. Rev. B 93, 085105 (2016). [46] J. Ruseckas, G. Juzeliu¯nas, P. O¨hberg, and M. Fleis-
[24] J. Ningyuan, C. Owens, A. Sommer, D. Schuster, and chhauer, Phys. Rev. Lett. 95, 010404 (2005).
J. Simon, Phys. Rev. X 5, 021031 (2015). [47] K. Osterloh, M. Baig, L. Santos, P. Zoller, and
M. Lewenstein, Phys. Rev. Lett. 95, 010403 (2005).
9
[48] G.Juzeliu¯nas,J.Ruseckas,M.Lindberg,L.Santos, and [76] M. C. Beeler, R. A. Williams, K. Jim´enez-Garc´ıa, L. J.
P. O¨hberg, Phys. Rev. A 77, 011802(R) (2008). LeBlanc,A.R.Perry, andI.B.Spielman,Nature498,
[49] T.D.Stanescu,C.Zhang, andV.Galitski,Phys.Rev. 201 (2013).
Lett. 99, 110403 (2007). [77] J.Sinova,S.O.Valenzuela,J.Wunderlich,C.H.Back,
[50] J.Y.VaishnavandC.W.Clark,Phys.Rev.Lett.100, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).
153002 (2008). [78] I. V. Tokatly and E. Y. Sherman, Phys. Rev. A 93,
[51] X.-J.Liu,M.F.Borunda,X.Liu, andJ.Sinova,Phys. 063635 (2016).
Rev. Lett. 102, 046402 (2009). [79] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jung-
[52] C. Zhang, Phys. Rev. A 82, 021607(R) (2010). wirth, and A. H. MacDonald, Phys. Rev. Lett. 92,
[53] D. L. Campbell, G. Juzeliu¯nas, and I. B. Spielman, 126603 (2004).
Phys. Rev. A 84, 025602 (2011). [80] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107,
[54] J.Dalibard,F.Gerbier,G.Juzeliu¯nas, andP.O¨hberg, 127205 (2011).
Rev. Mod. Phys. 83, 1523 (2011). [81] Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).
[55] B. M. Anderson, G. Juzeliu¯nas, V. M. Galitski, and [82] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolot-
I. B. Spielman, Phys. Rev. Lett. 108, 235301 (2012). nyy,B.Bu¨chner, andR.J.Cava,Phys.Rev.Lett.113,
[56] Z.-F. Xu, L. You, and M. Ueda, Phys. Rev. A 87, 027603 (2014).
063634 (2013). [83] Z.Liu,B.Zhou,Y.Zhang,Z.Wang,H.Weng,D.Prab-
[57] B. M. Anderson, I. B. Spielman, and G. Juzeliu¯nas, hakaran, S.-K. Mo, Z. Shen, Z. Fang, X. Dai, et al.,
Phys. Rev. Lett. 111, 125301 (2013). Science 343, 864 (2014).
[58] X. Zhou, Y. Li, Z. Cai, and C. Wu, J. Phys. B. 46, [84] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W.
134001 (2013). Krizan, I. Belopolski, M. Neupane, G. Bian, N. Ali-
[59] B.M.AndersonandC.W.Clark,J.Phys.B46,134003 doust, T.-R. Chang, et al., Science 347, 294 (2015).
(2013). [85] L.Lu,Z.Wang,D.Ye,L.Ran,L.Fu,J.D.Joannopou-
[60] Y.Li,G.I.Martone,L.P.Pitaevskii, andS.Stringari, los, and M. Soljaˇci´c, Science 349, 622 (2015).
Phys. Rev. Lett. 110, 235302 (2013). [86] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu,
[61] V. Galitski and I. B. Spielman, Nature 494, 49 (2013). M. Troyer, X. Dai, and B. A. Bernevig, Nature 527,
[62] N.Goldman,G.Juzeliu¯nas,P.O¨hberg, andI.B.Spiel- 495 (2015).
[87] B.Q.Lv,H.M.Weng,B.B.Fu,X.P.Wang,H.Miao,
man, Rep. Prog. Phys. 77, 126401 (2014).
J.Ma,P.Richard,X.C.Huang,L.X.Zhao,G.F.Chen,
[63] H. Zhai, Rep. Prog. Phys. 78, 026001 (2015).
Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X
[64] Y.-J. Lin, K. Jim´enez-Garc´ıa, and I. B. Spielman, Na-
5, 031013 (2015).
ture 471, 83 (2011).
[88] T. Dubˇcek, C. J. Kennedy, L. Lu, W. Ketterle,
[65] J.-Y. Zhang, S.-C. Ji, Z. Chen, L. Zhang, Z.-D. Du,
M. Soljaˇci´c, and H. Buljan, Phys. Rev. Lett. 114,
B.Yan,G.-S.Pan,B.Zhao,Y.-J.Deng,Z.H.,S.Chen,
225301 (2015).
and J.-W. Pan, Phys. Rev. Lett. 109, 115301 (2012).
[89] X. Wan, A. M. Turner, A. Vishwanath, and S. Y.
[66] P. Wang, Z.-Q. Yu, Z. Fu, L. Miao, L. Huang, S. Chai,
Savrasov, Phys. Rev. B 83, 205101 (2011).
H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301
[90] U. Gu¨rsoy, V. Jacobs, E. Plauschinn, H. Stoof, and
(2012).
S. Vandoren, J. High Energy Phys. 2013, 1 (2013).
[67] L.W.Cheuk,A.T.Sommer,Z.Hadzibabic,T.Yefsah,
[91] O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005).
W.S.Bakr, andJ.Zhang,Phys.Rev.Lett.109,095302
[92] A. Khaetskii, Phys. Rev. Lett. 96, 056602 (2006).
(2012).
[93] A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001).
[68] R.A.Williams,L.J.LeBlanc,K.Jim´enez-Garc´ıa,M.C.
[94] C.Chin,R.Grimm,P.Julienne, andE.Tiesinga,Rev.
Beeler,A.R.Perry,W.D.Phillips, andI.B.Spielman,
Mod. Phys. 82, 1225 (2010).
Science 335, 314 (2012).
[95] Q.GuanandD.Blume,Phys.Rev.A95,020702(2017).
[69] L. J. LeBlanc, M. C. Beeler, K. Jimenez-Garcia, A. R.
[96] T.Fujita,M.B.A.Jalil,S.G.Tan, andS.Murakami,
Perry, S. Sugawa, R. A. Williams, and I. B. Spielman,
J. Appl. Phys. 110, 121301 (2011).
New. J. Phys. 15, 073011 (2013).
[97] L. D. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8,
[70] C.Qu,C.Hamner,M.Gong,C.Zhang, andP.Engels,
101 (1935).
Phys. Rev. A 88, 021604(R) (2013).
[98] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).
[71] Z.Fu,L.Huang,Z.Meng,P.Wang,L.Zhang,S.Zhang,
[99] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P.
H.Zhai,P.Zhang, andJ.Zhang,NaturePhys.10,110
Smith, andZ.Hadzibabic,Phys.Rev.Lett.110,200406
(2014).
(2013).
[72] L. Huang, Z. Meng, P. Wang, P. Peng, S.-L. Zhang,
[100] E. van der Bijl and R. A. Duine, Phys. Rev. Lett. 107,
L. Chen, D. Li, Q. Zhou, and J. Zhang, Nature Phys.
195302 (2011).
12, 540 (2016).
[101] X.-G. Huang, Sci. Rep. 6, 20601 (2016).
[73] Z. Meng, L. Huang, P. Peng, D. Li, L. Chen, Y. Xu,
[102] E. I. Rashba, Phys. Rev. B 68, 241315(R) (2003).
C. Zhang, P. Wang, and J. Zhang, Phys. Rev. Lett.
[103] H.-J. Drouhin, G. Fishman, and J.-E. Wegrowe, Phys.
117, 235304 (2016).
Rev. B 83, 113307 (2011).
[74] Z.Wu,L.Zhang,W.Sun,X.-T.Xu,B.-Z.Wang,S.-C.
[104] E. Y. Sherman and D. Sokolovski, New J. Phys. 16,
Ji,Y.Deng,S.Chen,X.-J.Liu, andJ.-W.Pan,Science
015013 (2014).
354, 83 (2016).
[105] I. V. Tokatly, Phys. Rev. Lett. 101, 106601 (2008).
[75] S.-C. Ji, J.-Y. Zhang, L. Zhang, Z.-D. Du, W. Zheng,
[106] N. T. Phuc, G. Tatara, Y. Kawaguchi, and M. Ueda,
Y.-J. Deng, H. Zhai, S. Chen, and J.-W. Pan, Nature
Nat. Commun. 6 (2015).
Phys. 10, 314 (2014).
10
[107] T. D. Stanescu, B. Anderson, and V. Galitski, Phys. [110] U. Zu¨licke, J. Bolte, and R. Winkler, New J. Phys. 9,
Rev. A 78, 023616 (2008). 355 (2007).
[108] P. Medley, D. M. Weld, H. Miyake, D. E. Pritchard, [111] S.B.Koller,A.Groot,P.C.Bons,R.A.Duine,H.T.C.
and W. Ketterle, Phys. Rev. Lett. 106, 195301 (2011). Stoof, andP.vanderStraten,NewJ.Phys.17,113026
[109] M.Merkl,F.E.Zimmer,G.Juzeliu¯nas, andP.O¨hberg, (2015).
EPL 83, 54002 (2008).