Table Of ContentParity violating superfluidity in ultra-cold fermions
under the influence of artificial non-Abelian gauge fields
Kangjun Seo1,2, Li Han1 and C. A. R. Sa´ de Melo1
1. School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA and
2. Department of Physics, Clemson University, Clemson, South Carolina 29634, USA
(Dated: December 11, 2013)
WediscussthecreationofparityviolatingFermisuperfluidsinthepresenceofnon-Abeliangauge
3 fields involving spin-orbit coupling and crossed Zeeman fields. We focus on spin-orbit coupling
1 with equal Rashba and Dresselhaus (ERD) strengths which has been realized experimentally in
0 ultra-cold atoms, butwealso discuss thecase of arbitrary mixingof Rashbaand Dresselhaus (RD)
2 andofRashba-only(RO)spin-orbitcoupling. Toillustrate theemergence ofparityviolation in the
superfluid,weanalyzefirsttheexcitationspectruminthenormalstateandshowthatthegeneralized
n
helicitybandsdonothaveinversionsymmetryinmomentumspacewhencrossed Zeeman fieldsare
a
present. This is also reflected in the superfluid phase, where the order parameter tensor in the
J
generalized helicity basis violates parity. However, the pairing fields in singlet and triplet channels
7
of the generalized helicity basis are still parity even and odd, respectively. Parity violation is
further reflected on ground state properties such as the spin-resolved momentum distribution, and
]
s in excitation properties such as the spin-dependentspectral function and density of states.
a
g PACSnumbers: 03.75.Ss,67.85.Lm,67.85.-d
-
t
n
a Parity violating phenomena are very rare in physics, laboratory is
u
but a classical example is known from particle physics,
q where parity violating processes of the weak interaction HZSO(k)=−hzσz −[hy+hERD(k)]σy (1)
.
t wereproposed[1]andobservedinthe decayof60Cosev-
a
for an atom with center-of-mass momentum k and spin
m eral decades ago [2]. In this case, the weak interactions
basis |↑i, |↓i. The fields h = −Ω /2, h =−δ/2, and
allowforparityviolation,buttheparticlekineticenergies z R y
-
h (k) = vk can be controlled independently. Here,
d areparityeven,reflectingtheinversionsymmetryoftheir ERD x
n space. The Standard Model of particle physics, which is ΩR is the Raman coupling and δ is the detuning, which
o canbeadjustedtoexplorephasediagramsasachievedin
a non-Abelian gauge theory, incorporates parity viola-
c 87Rb experiments [7], or to study the high-temperature
[ tionsandpostulatesthatfornuclearbetadecayparityis
normal phases of Fermi atoms [20, 21].
maximally violated. Other examples of parity violation
1 In this letter, we show that ultra-cold Fermi superflu-
existforinstanceincondensedmatterphysics,wherepar-
v ids in the presence of non-Abelian gauge fields consist-
3 itybreakingisassociatedwithcrystalswithoutinversion
ing of artificially created spin-orbit and crossed Zeeman
5 symmetry [3] or with crystals which have inversionsym-
fields described in Eq.(1) canproduce a parity violating
3 metryinitially,butcandevelopspontaneouslypermanent
1 superfluid state when interactions are included. How-
electric polarization through lattice distortions leading
. ever,unlikethe caseofthe StandardModelwhereparity
1 to ferroelectric materials [4]. However, examples of par-
0 ity breaking in superfluids, such as those encountered in breaking is driven by the weak force, in our case, par-
3 ity breaking is driven by the effects of the non-Abelian
nuclear, atomic, condensed matter and astrophysics are
1 gaugefieldonthekineticenergy. Toillustratethelackof
hard to find, and to our knowledge there seems to be no
:
v confirmed example in nature. parity in physical observables, we analyze spectroscopic
i quantities such as the elementary excitation spectrum,
X Recently, it has been possible to create non-Abelian
momentumdistribution,spectralfunctionanddensityof
r gauge fields in ultra-cold atoms via artificial spin-orbit
a (SO) coupling of equal superposition of Rashba [5] states in the superfluid state.
Hamiltonian: To analyze parity violationin ultra-cold
h (k) = v (−k xˆ +k yˆ) and Dresselhaus [6] h (k) =
R R y x D
Fermi superfluids, we start from the Hamiltonian in mo-
v (k xˆ + k yˆ) terms, leading to the equal-Rashba-
D y x
mentum space as
Dresselhaus (ERD) form [7, 8] h (k) = vk yˆ, where
ERD x
v = v = v/2, for which parity preserving superfluid-
R D H = ψ†(k)H (k)ψ (k), (2)
ity is possible [9–11]. Other forms of SO fields, such as 0 s 0 s
Xks
the Rashba-only or Dresselhaus-only cases, require ad-
ditional lasers and create further experimental difficul- where H (k) = [K(k)1−h (k)·σ] with K(k) =
0 eff
ties [12], while several theory groups have investigated k2/2m−µ being the single particle kinetic energy rela-
the Rashba-only case [13–16] due to the connection to tive to the chemical potential µ; the vector-matrix σ de-
earlier condensed matter literature [17–19]. scribesthePaulimatrices(σ ,σ ,σ );h (k)istheeffec-
x y z eff
The current Zeeman-SO Hamiltonian created in the tivemagneticfieldwithcomponents[h (k),h (k),h (k)]
x y z
2
and ψ†(k) is the creation operator for fermions with (a) (b)
s " "
spin s and momentum k. In the ERD case, which is
readily available in ultra-cold atoms, the effective mag- #$% #$%
netic field is simply h (k) = [0,h +h (k),h ],
eff y ERD z # #
where h and h are Zeeman components correspond-
y z
ingtothe detuning δ andthe RamancouplingΩR,while F #$% #$%
!
h (k) = vk is the spin-orbit field. We define the !
ERD x ) " "
total number of fermions as N = N↑ +N↓, and the in- =0 ! " # " ! ! " # " !
z
ducedpopulationimbalanceasPind =(N↑−N↓)/N. We =k "(c) "(d)
choose our scales through the Fermi momentum kF de- ky
finedfromN/V =k3/(3π2),leadingtotheFermienergy k,x #$% #$%
F (
ǫF =kF2/2m and the Fermi velocity vF =kF/m. !! # #
Generalized Helicity Basis: The matrix H (k) can
0
be diagonalized in the generalized helicity (GH) basis #$% #$%
|k,αi≡Φ†(k)|0i via a momentum-dependent SU(2) ro-
α " "
tation generated by the unitary matrix ! " # " ! ! " # " !
k !k k !k
x F x F
u v
Uk =(cid:18)−vkk∗ ukk(cid:19), (3) (FbIlGue. 1li.ne)(caonldorǫ⇓on(kli)n/eǫ)FG(erenderlailnizee)dvehresluicsitmyobmaenndtsuǫm⇑(kkx)//kǫFF
with ky = kz = 0 and for ERD spin-orbit coupling v/vF =
where the normalization condition |uk|2 +|vk|2 = 1 is 0.4. The black dashed lines show the helicity bands for
imposed to satisfy the unitarity condition U†kUk = 1. v/vF = 0.4 with hz/ǫF = hy/ǫF = 0. The Zeeman fields
The corresponding eigenvectors are the spinors Φ(k) = are (a) hy/ǫF = 0 and hz/ǫF = 0.1, (b) hy/ǫF = 0 and
U†Ψ(k), where Φ(k) = [Φ (k),Φ (k)] is expressed in hz/ǫF = 0.7, (c) hy/ǫF =0.1 and hz/ǫF = 0.1, (d) hy/ǫF =
terkms of ψ(k)=[ψ (k),ψ (k⇑)] by t⇓he relations Φ (k)= 0.2 and hz/ǫF =0.7. Notice that ǫα(k)6=ǫα(−k) in (c) and
↑ ↓ ⇑ (d),indicating the absence of parity.
u c −v c andΦ (k)=v∗c +u c .Thecoherence
k k↑ k k↓ ⇓ k k↑ k k↓
factor u = 1 1+ hz is chosen to be real with-
k r2(cid:16) |heff(k)|(cid:17) Interactions and Order Parameter: In order to under-
stand the underlying physics of this system, it is im-
out loss of generality, and v = −eiϕk 1 1− hz
k r2(cid:16) |heff(k)|(cid:17) portant to rewrite the interaction Hamiltonian in the
is a complex function with phase ϕ defined by ϕ = generalized helicity basis. The starting interaction is
k k
Arg[h⊥(k)]. The complex field h⊥(k) = hx(k)−ihy(k) HI = −g qb†(q)b(q), where the pair creation op-
hascomponentshx(k)andhy(k)alongthexandydirec- erator withPcenter of mass momentum q is b†(q) =
tions,respectively. Themagnitudeoftheeffectivefieldis ψ†(k+q/2)ψ†(−k+q/2),canbewritteninthehelic-
k ↑ ↓
|heff(k)| = h2z +|h⊥(k)|2. In the ERD case hx(k)=0, iPty basis as H = −g B† (q)B (q), where the
and the raptio h⊥(k)/|h⊥(k)| = eiϕk = −isgn[hy(k)], indices α,β,γ,Iδ coverP⇑qαaβnγdδ⇓αsβtates.γδPairing is now
where h (k)=h +vk . e
y y x described by the operator
The generalized helicity spins α = (⇑,⇓) are aligned
orantialignedwithrespecttotheeffectivemagneticfield B (q)= Λ (k ,k )Φ (k )Φ (k ) (4)
αβ αβ + − α + β −
heff(k), and the corresponding eigenvalues of H0(k) are Xk
ξ (k) = ǫ (k)−µ and ξ (k) = ǫ (k)−µ. Here, the he-
⇑ ⇑ ⇓ ⇓ and its Hermitian conjugate, with momentum indices
licity energies are simply ǫ (k) = K(k)−|h (k)| and
⇑ eff k =±k+q/2. The matrix Λ (k ,k ) is directly re-
ǫ (k) = K(k) + |h (k)|. In the specific case of ERD ± αβ + −
⇓ eff latedtoproductsofcoherencefactorsu(k ), v(k )(and
coupling with non-zero detuning (h 6= 0) the effective ± ±
y their complex conjugates) of the momentum dependent
field is h (k) = h ˆz+[h +h (k)]yˆ, with magni-
eff z y ERD SU(2)rotationmatrixU(k ). Seeninthe GHbasis,the
±
tude |heff(k)| = h2z +(hy+vkx)2 and parity violation interactions reveal that the center of mass momentum
q
occurring along the x axis. This is illustrated in Fig. 1, k +k =q and the relative momentum k −k =2k
+ − + −
where for finite h (non-zerodetuning δ) the generalized are coupled and no longer independent, and thus do
y
helicity bands ǫ (k) and ǫ (k) do not have well defined not obey Galilean invariance. The interaction constant
⇑ ⇓
parityinmomentumspace. AsseeninFig.1(a)-(b),par- g is related to the scattering length via the Lippman-
ityispreservedforv 6=0ifh =0(zerodetuning). While Schwinger relation V/g =−Vm/(4πa )+ 1/(2ǫ ).
y s k k
as noted in Fig. 1(c)-(d), parity is violated for v 6= 0, if From Eq. (4) it is clear that pairing betwPeen fermions
h 6= 0 (finite detuning). Similar parity violation along ofmomenta k andk canoccur within the samehelic-
y + −
the xaxisoccursforothermixturesofRashbaandDres- ity band (intra-helicity pairing) or between two different
selhaus terms as long as h 6=0. helicitybands(inter-helicitypairing). Forpairingatzero
y
3
center-of-mass momentum q = 0, the order parameter have even parity, while a (k) has odd parity and is thus
1
for superfluidity is the tensor ∆ (k) = ∆ Λ (k,−k), responsiblefortheparityviolationthatoccursintheele-
αβ 0 αβ
where ∆ = −g hB (0)i, leading to components: mentaryexcitationspectrum. Furthermore,parityviola-
0 γδ γδ
∆⇑⇑(k) = ∆0(uPkv−k−vku−k) for total helicity pro- tion occurs only when both v and hy are non-zero,since
jection λ = +1; ∆ (k) = −∆ u u +v v∗ and wheneitherh =0orv =0thecoefficienta (k)vanishes
⇑⇓ 0 k −k k −k y 1
∆⇓⇑(k) = ∆0(uku−k+vk∗v−k) for(cid:0)total helicity p(cid:1)rojec- andparityinthe elementaryexcitationspectrumis fully
tion λ=0; and ∆ (k)=∆ u v∗ −v∗u for total restored. Fromthesecularequation,itfollowsthatwhen
⇓⇓ 0 k −k k −k
helicity projection λ=−1. Pa(cid:0)rity is violated i(cid:1)n ∆αβ(k) kx =0,thecoefficienta1(k)alsovanishesandtheexcita-
since they do not have well defined parity for non-zero tion energies E (0,k ,k ) have the same analytical form
i y z
spin-orbitcouplingandcrossedZeemanfields h andh . as in the case for h = 0, with the simple replacement
y z y
However, we may still define singlet and triplet sec- of h2 → h2 +h2. This property is just a consequence
z z y
tors in the generalizedhelicity basis, which are even and of the reflection symmetry of the Hamiltonian through
oddinmomentumspacerespectivelyforanyvalueofh . the k = 0 plane. However, parity is violated, be-
y x
The singlet sector is defined by the scalar order param- causeinversionsymmetrythroughtheoriginofmomenta
eter ∆ (k) = [∆ (k)−∆ (k)]/2 corresponding to does not exist, that is, E (−k) 6= E (k). In contrast,
S,0 ⇑⇓ ⇓⇑ i i
λ = 0. While the triplet sector is defined by the vector quasiparticle-quasihole symmetry is preserved since the
orderparameter∆ (k),byitsgeneralizedhelicitycom- corresponding quasiparticle-quasihole energies obey the
T,λ
ponents ∆ (k) = ∆ (k) corresponding to λ = +1; relations E (k)=−E (−k) and E (k)=−E (−k).
T,+1 ⇑⇑ 2 3 1 4
∆ (k)=[∆ (k)+∆ (k)]/2correspondingtoλ=0;
T,0 ⇑⇓ ⇓⇑ A simple inspection shows that gapless and fully
∆ (k)=∆ (k) corresponding to λ=−1.
T,−1 ⇑⇑ gapped phases emerge. A gapless phase with two rings
Superfluid Ground State and Elementary Excita- of nodes (US-2) appears when h2 + h2 − |∆ |2 > 0
y z 0
tions: The ground state for uniform superfluidity
and µ > h2+h2−|∆ |2. A gapless phase with one
can be expressed in terms of fermion pairs in the y z 0
q
GH basis as the many-body wavefunction |Gi = ring of nodes (US-1) occurs for h2 + h2 − |∆ |2 > 0
y z 0
k αβ Uαβ(k)+Vαβ(k)Φ†α(k)Φ†β(−k) |0i, where and |µ| < h2y+h2z −|∆0|2. A directly gapped phase
|Q0i insPthe vhacuum state with no particles. io (d-US-0) arqises for h2 + h2 − |∆ |2 > 0 and µ <
y z 0
The Hamiltonian matrix in the GH basis is
− h2 +h2−|∆ |2,whileanindirectlygappedphase(i-
y z 0
ξk⇑ 0 ∆⇑⇑(k) ∆⇑⇓(k) USq-0)emergesforh2+h2−|∆ |2 <0andµ>0. Lastly,
Heex(k)=∆∆∗⇑∗⇑⇓⇑0((kk)) ∆∆∗⇓∗⇓ξ⇓⇑k⇓((kk)) ∆−⇓ξ⇑0−(kk⇑) ∆−⇓ξ⇓0−(kk⇓), (5) iatnhtdieviceqauitnainsigcpeatrrhttaaiictnletmheexoycmuitneanitftoizuormnmergne0regoriugoynndsEws2t(haket)enbbhee2yccoom>mees|s∆nl0ee|gs2s-,
energetically favorable against the normal state [22].
which is traceless, showing that the sum of its eigenval-
ues is zero. We have obtained analytical solutions for Phase Diagram and Thermodynamic Poten-
the eigenvaluesofHex(k) forarbitraryRDspin-orbitor- tial: From the thermodynamic potential ΩUS =
bit and arbitrary Zeeman fields hy and hz, but we do −(T/2) k,jln[1+exp(−Ej(k)/T)] + kK(k) +
e
notlistthemhere,becausetheirexpressionsarecumber- |∆0|2/gPwe obtain self-consistently the zePro tempera-
some. However, for each momentum k, the determinant ture (T = 0) phase diagram as a function of crossed
Det ω1−H (k) , leads to the quartic equation Zeeman fields hy and hz for v/vF = 0.4 at unitarity
ex
h i 1/(kFas) = 0 in Fig. 2(a), and at the BEC regime
ω4+a (ek)ω3+a (k)ω2+a (k)ω+a (k)=0. (6) 1/(kFas) = 2.0 in Fig. 2(b), but a stability analysis
3 2 1 0
against non-uniform phases is necessary as in the
Inthe particularcaseofERDspin-orbitcouplingwith parity-preserving case [10, 11]. At unitarity the uniform
crossedZeemanfields,the coefficientsbecomea (k)=0, superfluid phases i-US-0, US-1, US-2 and the normal
3
the coefficient of the quadratic term takes the form (N) phase are present in the range shown, while in the
BEC regime only the d-US-0 occurs in the same range
a2(k)=−2 K2(k)+|∆0|2+|vkx|2+|hy|2+|hz|2 , of fields. The transitions between different US phases
(cid:0) (cid:1) is topological with no change in symmetry as in the
while the coefficient of the linear term is a (k) =
1 parity-preserving case [10, 11]. While the transitions
−8K(k)(vk )h , and lastly the coefficient of the zero-th
x y from US phases to the N phase involve a change in
order term is
symmetry, from broken to non-broken U(1), and are
discontinuous, as seen in the insets of Fig. 2.
a (k)=ξ (k)ξ (k)ξ (−k)ξ (−k)+|∆ |2α2(k),
0 ⇑ ⇓ ⇑ ⇓ 0 0
Detecting parity violation: A direct measurement
where α2(k) = 2K2(k)+|∆ |2+h2(k) with h2(k) = of parity violation in the superfluid state can be
0 0 0 0
2|vkx|2 − 2|hy|2(cid:0)− 2|hz|2. Notice that a(cid:1)2(k) and a0(k) made through the momentum distributions ns(k) =
4
(a) (b) festation of parity violation in the elementary excitation
11 11
0.8 2.0 spectrum for the US-1 superfluid phase, and the corre-
0.08.8 US-1 !!|!0F000...246 0.08.8 !!|!0F11..68 sponding implications for momentum integrated quanti-
!!F 0.06.6 US-2 | 00 0.25h0y!.5!F0.75 1 !!F 0.06.6 | 00 0.25h0y.!5!0F.75 1 tieksAsusc(hω,aks).thTehespmino-srtesiomlvpeodrtdaenntspitoyinotfissttahtaetsfρosr(ωfin)it=e
hz 0.04.4 hz 0.04.4 sPpin-orbit coupling v and when hy 6= 0, the excitation
d-US-0 energies E (k) 6= E (−k). This implies that degenerate
0.02.2 i-US-0 N 0.02.2 i i
peaks at h =0 (corresponding to minima or maxima of
y
00 00 theexcitationspectrum)areincreasinglysplitwithgrow-
00 00..22 00..44 00..66 00..88 11 00 00..22 00..44 00..66 00..88 11
hy!!F hy!!F inghy. This effectis illustratedinFig.4atthe locations
indicated by the small black arrows.
FIG. 2. The T = 0 phase diagram in the hy-hz parameter
space showing variousuniform superfluid phasesUS-2,US-1, (a) (b) (c) (d)
! ! ! !
d-US-0andi-US-0,andthenormalphaseforERDspin-orbit
" " " " h
coupling v/vF = 0.4 and at (a) unitarity 1/(kFas) = 0.0 # !F# !F# #
and in (b) the BEC regime 1/(kFas)=2.0. The insets show " /!" /!" " h
|∆0| as a function of hy for hz/ǫF = 0.2 (dotted line); for
! ! ! !
hz/ǫF = 0.4 (dot-dashed line); hz/ǫF = 0.6 (dashed line); ! " # " ! # " ! $ # " ! $ ! " # " !
and hz/ǫF = 0.8 (solid line). In the range shown, |∆0| is !! "kx#/kF" ! !# "!!("!) $ !# "!!("!) $ !! "ky#/kF" !
essentially independentof hy and hz in the BEC regime. !(e) !(f) !(g) !(h)
" !" !" " h
# !F# !F# #
hψ†(k)ψ (k)i. They are illustrated in Fig. 3 for US- " /!" /!" " h
s s
1 superfluid with spin-orbit v/v = 0.4 and interac- ! ! ! !
F ! " # " ! # " ! $ # " ! $ ! " # " !
tion 1/(kFas) = 0, in the parity-preserving case with kx/kF !!(") !!(") ky/kF
h /ǫ =0 and h /ǫ =0.7 in (a)-(d) and in the parity-
y F z F #
violating case with hy/ǫF = 0.2 and hz/ǫF = 0.7 in (e)- FIG.4. (coloronline)EigenvaluesEi(k)anddensityofstates
(h). Atfinitetemperatures,themomentumdistributions ρs(ω) (in units of ǫF and ǫ−F1, respectively) for 1/(kFas)=0
broaden, but parity violation is still self-evident. anUdv/vF =0.4intheUS-1phase,butclosetotheUS-1/US-2
boundary, with parameters hy/ǫF =0, hz/ǫF =0.7, µ/ǫF =
(a) (b) (c) (d) 0.h5803, |∆0|/ǫF = 0.3592, and Pind = 0.6592 in (a)-(d); and
" ! ! " hy/ǫF =0.2, hz/ǫF =0.7, µ/ǫF =0.5871, |∆0|/ǫF =0.3157,
###$$!&$%%% kk!yF#"" kk!yF#"" ###$$!&$%%% aaanrnhedds(Phhoin)w.dnI=ninp0(.aa6n)9e5al8sndfionr(e(ρ)e,s)(a-ω(nh)d).aalsComnugatls(l0ob,frkoEya,id0(ek)n)ainraeglosδnh/goǫwF(kn=xi,n00,(.0d01))
# ! ! # is used. The black arrows indicate examples of peaks that
! " # " ! ! " # " ! ! " # " ! ! " # " !
"! "ki!#kF" ! !! "kx#!kF" ! !! "kx#!kF" ! "! " k#i!kF" ! split when parity breakingoccurs for finite hy.
(e) (f) (g) (h)
" ! ! "
#$&% " " #$&% Conclusions: Weshowedthatnon-Abeliangaugefields
##$!$%% kk!yF#" kk!yF#" ##$!$%% cohnsisting of spin-orbit and crossed Zeeman fields lead
to parity violating superfluidity in ultra-cold Fermi sys-
# ! ! #
! " # " ! ! " # " ! ! " # " ! ! " # " ! tehms. We derived general relations that can be applied
ki!kF kx!kF kx!kF ki!kF
to spin-orbit couplings involving any linear combination
ofRashbaandDresselhausterms. We focusedmostlyon
FIG. 3. (color online) Momentum distributions (T = 0)
n↑(k) (two left-most columns) and n↓(k) (two right-most the case of equal Rashba-Dresselhaus (ERD) spin-orbit
columns) for 1/(kFas) = 0.0 and v/vF = 0.40 at the US-1 coupling. The presence of such fields produce a super-
phase. In(a)-(d)thefieldvaluesarehy/ǫF =0,hz/ǫF =0.7, fluid order parameter tensor whose components in the
withµ/ǫF =0.5803,|∆0|/ǫF =0.3592,andPind=0.6592. In generalized helicity basis are neither even nor odd un-
(e)-(h)thefieldvaluesarehy/ǫF =0.2andhz/ǫF =0.7,with der spatial inversion. Even though the elements of this
bµl/uǫeF-d=ash0e.5d8a7n1,d|r∆ed0|-/sǫoFlid=lin0e.3s1r5e7p,reasnedntPcinudts=of0n.6s9(5k8).alTonhge tensor written in generalized singlet or triplet helicity
channels have even or odd parity, respectively, the exci-
thedirections (0,ky,0) and (kx,0,0), respectively.
tation spectrum does not have well defined parity, but
preserves quasiparticle-quasihole symmetry. This parity
Parity violation is also manifested in other momen-
violation has important experimental signatures leading
tum resolved properties such as the spectral func-
to momentum distributions without inversion symmetry
tion A (ω,k) = −(1/π)ImG (iω = ω + iδ,k), where
s ss and to spin-resolved density of states that possess split
−1
Gss(iω,k) = iω1−Hex(k) , written in the s =↑,↓ peaks in frequency.
h i
basis. Instead, in Fig. 4, we choose to illustrate a mani- We thank ARO (W911NF-09-1-0220)for support.
e
5
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