Table Of ContentTopological Superfluidity of Spin-Orbit Coupled Bilayer
Fermi Gases
Liang-LiangWang, Wu-MingLiu⋆
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1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese
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J
AcademyofSciences,Beijing100190,China
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∗e-mail:[email protected]
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- Topologicalsuperfluid, new quantum matter that possesses gaplessexotic excitations known
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n
a
as Majorana fermions, has attracted extensive attention recently. These excitations, which
u
q
.
t can encode topological qubits, could be crucial ingredients for fault-tolerant quantum com-
a
m
- putation. However, creating and manipulating multiple Majorana fermions remain an on-
d
n
o going challenge. Loading a topologically protected system in multi-layer structures would
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be a natural and simple way to achieve this goal. Here we investigate the system of bilayer
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v
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Fermi gases with spin-orbit coupling and show that the topological condition is significantly
1
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3 influenced by theinter-layer tunneling, yieldingtwo noveltopologicalphases, whichsupport
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more Majorana Fermions. We demonstrate the existence of such novel topological phases
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and associated multiple Majorana fermions using bilayer Fermi gases trapped inside a har-
:
v
i
X
monic potential. This research pave a new way for generating multiple Majorana fermions
r
a
and wouldbea significantstep towards topologicalquantum computation.
Topological state of matter 1,2, as an activenew frontier in physics, has been attracting con-
siderable interest both experimentally and theoretically. The so-called Majorana fermions (MFs)
1
3–6, accompanied with topological states, have a good prospect for applications in topological
quantum computation, quantum memory and quantum random-number generation 7–9 since they
areintrinsicallyimmunetodecoherencecausedbylocalperturbations. Duetosuchpotentialappli-
cations, the realization of topological states in a well-controlled environment is highly desirable.
Recently, several theoretically predictions and experimentally observations of topological states
havebeen reported in a variety ofsystems,for example,InAs wires and banded carbon nanotubes
engineered by spin-orbit coupling of electrons 10–14, the surface of topological insulators 15–19 ,
ultracold atomic gases with strong non-Abelian synthetic gauge field 20–30 and so on. Spin-orbit
(SO) coupling plays a key role in all systems. In contrast to the other physical contexts, ultracold
atomic gases with more experimental controls over its parameters, such as high controllability in
interatomicinteraction,geometry,purity,etc, offeranexceptionalclean platformforexploringthe
topologicalsuperfluidity.
Recent experimental progresses in the synthetic SO coupling of ultracold Bose gases 31,32
and Fermi gases 33–35 have stimulated a lot of interest in exploring exotic topological quantum
properties of ultracold atoms. In particular, by deforming the Fermi surface and opening a topo-
logical band gap, SO coupled fermi gases become topological and support Majorana zero-energy
excitationsin thepresence of alarge perpendicular Zeeman field. However,based on theseexper-
iments, the existence of a topological band gap can only lead to a single topological phase and a
pairofMajoranazero modesassociatedwithtopologicalquantumphasetransition(TQPT). There
is a strong requirement in the realm of quantum information and quantum memory for proposing
a advanced system, which results in more types of topological phases, to create and manipulate
2
multipleMFs. Toachievethis,wegivearealisticscenariobaseduponengineeringatopologically
protected system in a bilayer or multi-layer structures. As we know, some intriguing quantum
effects, generated by layers’s extra degrees of freedom, have been widely studied in many multi-
layer condensed mattersystemsincluding fractional quantumHall states 40 and topological valley
transport in bilayergraphene 41. Interesting quantumeffects such as atomicJosephson effects and
macroscopic quantum self-trapping 42–47 have also been studied in ultracold Bose gases. How-
ever, for ultracold Fermi gases, mostof theinvestigationshavebeen developedmainlyin the field
ofsingle-layertopologicalsystem36–39, very littleisknownregardingthebilayerFermi gaseswith
SOcoupling. Itwouldbeinterestingtoexaminethepossibilityforrealizingandmanipulatingmul-
tiple MFs by loading SO coupled Fermi gases in a bilayer structure, providing a peculiar insight
intothetopologicalstates.
In this article, we concentrate on the potential topological properties of bilayer Fermi gases
with SO coupling and predict the existence of multiple Majorana zero states at the interfaces be-
tween distinct phases. The bilayer geometry can be readily realized by adding a two-dimensional
double-well optical lattice, and the atoms in each layer are affected by the same Raman SO cou-
pling,whilecoupledwitheachotherviatheinter-layertunneling. Notablydifferentfromprevious
meanfield studies,bytunningtheinter-layertunneling,traditionaltopologicalphaseconditionin-
duced by the collective effect of SO coupling and Zeeman field is shifted and causes two critical
transitionpoints[h ,h ]. Asaconsequence,thetopologicalphaseregionbecomesseparatedinto
c,1 c,2
twoparts: topo-Iandtopo-IIphases. Hereweprovideaphysicalpicturetoclassifythetwodiffer-
ent topological phases and discover that in suitable parameter regions, the system may undergo a
3
unique TQPT from a topo-I state to a topo-II state, which has not been previously identified. The
regionfortopologicalsuperfluiddependsonnotonlymagnitudeoftheZeemanfield,thechemical
potential,theSOcouplingstrength,butalsoontheinter-layertunneling,thusprovidesmoreknobs
in experiments. Furthermore, we give the self-consistent Bogoliubov-de Gennes (BdG) results of
thesystemtrappedinaharmonicpotentialanddiscusstheresultingmultipleMajoranazero-energy
modes in great detail. The number of exotic Majorana zero modes depends on the number of in-
terfaces between distinct phases that form in different areas of the trap, which are consistent with
analytical local-density approximation (LDA) expressions. Such SO coupled bilayer Fermi gases
offeran opportunitytocreate and manipulatemulti-MFsand aadvanced multi-layerexperimental
setupmay bethereforeimprovedtomeasuremoreMFs.
Results
Experimentalsetupofspin-orbitcoupledbilayerFermigases. SOcouplingforultracoldFermi
atoms has been successfully demonstrated in ultracold Fermi 40K and 6Li gases 34,35 at about the
same time, in which the Raman dressing scheme is based on coupling two magneticsub-levels of
thegroundstatemanifoldwithtwocounterpropagatingRamanlasers31,33. Thesystemconsidered
in this work is depicted in Fig. 1(a), where the Raman dressed 40K Fermi gases are loaded in a
bilayer geometry, which can be readily realized by adding a two-dimension double-well optical
lattice 48,49. Note that the optical lattice is spin-independent and can induce tunneling without
spin-flip. Sametwoatomicinternalspinstates 9/2,9/2 and 9/2,7/2 ofthegroundstateinj-th
| i | i
layer are selected to belabeled as spin-up( j, )and down( j, )states, where j = 1,2refers to
| ↑i | ↓i
4
anindividuallayer. Theatommovealongthexˆaxiswithinalayerandtwolayersareseparatedby
adistanced. ThetunnelingamplitudeJ = dzψ∗ (z)[V(z) δ]ψ (z)withV(z) = c(z2 d2)2
σ 1,σ ± 2,σ −
R
can be regulated by changing the intensity or relative phase of laser standing waves that engineer
thedouble-welllattice48. ThepositiveandnegativesignslabelthedetuningfromRamanresonance
to spin-up and spin-down respectively. Without loss of generality, we assume them to be real and
J = J = J. The four-level topology has been schematically shown in Fig. 1(b), where the
↑ ↓
bluesolidlinesrepresent theinter-layertunnelingandthegreen dashed circlesdenotetheRaman-
assistedintra-layerinteractionwithmomentumtransferk . Theinter-layerspin-fliptunnelingsare
0
negligible under current experimental conditions, therefore they are neglected in Fig. 1(b). Four
atomic states couple with each other in a cyclic manner with no momentum transferred during a
closedlooptransition.
The Hamiltonian for bilayer Fermi gases in the presence of SO coupling is given by H =
dx[H + H ], with the single-particle component H and the interacting component H =
S int S int
R
g Ψ† (x)Ψ† (x)Ψ (x)Ψ (x) describing the s-wave contact interaction between the two
j=1,2 j j↑ j↓ j↓ j↑
P
spinstatesin j-thlayer. Thesingle-particleHamiltonianis writtenas
ξ +V(x)+δ Ωeik0x J 0
k
Ωe−ik0x ξ +V(x) δ 0 J
k
H = Ψ†(x) − Ψ(x), (1)
S
J 0 ξ +V(x)+δ Ωeik0x
k
0 J Ωe−ik0x ξ +V(x) δ
k
−
with Ψ(x) = [Ψ (x),Ψ (x),Ψ (x),Ψ (x)]T being the atomic annihilation operators, ξ =
1,↑ 1,↓ 2,↑ 2,↓ k
ǫ µ, where ǫ = k2/2m denotes kinetic energy and µ is the chemical potential, m is mass
k − k x
5
of an atom, δ means the detuning of the Raman process, and V(x) is the trapping potential (for
convenience, we set ~ = k = 1). The constants Ω and k represent the coupling strength and
B 0
photon recoil momentum of the two-photon Raman coupling, respectively. After applying a local
gaugetransformation
1
Ψ (x) = [eik0x/2Φ (x) ieik0x/2Φ (x)],
j,↑ j,↑ j,↓
√2 −
(2)
1
Ψ (x) = [e−ik0x/2Φ (x)+ie−ik0x/2Φ (x)],
j,↓ j,↑ j,↓
√2
thesingle-particleHamiltonianbecomes
H = Φ†(x)H Φ (x)+J (Φ† (x)Φ (x)+H.c.), (3)
S j 0 j 1,σ 2,σ
Xj Xσ
whereΦ (x) = [Φ (x),Φ (x)]T isa singlelayerfield operator,and thesinglelayertermis
j j,↑ j,↓
H = ǫ µ+V(x)+(αk +h )σ h σ , (4)
0 k x y y z z
− −
where we define the chemical potential µ µ E /4 and σ , σ are the Pauli matrices acting
r y z
→ −
on the spins. The effective spin-orbit coupling constant α E /k and effective Zeeman field
r r
≡
(h δ, h Ω) are introduced. For convenience, the recoil momentum k k and recoil
y z r 0
≡ ≡ ≡
energy E k2/2m are taken as natural momentum and energy units. Notice that the interaction
r ≡ 0
HamiltonianH isinvariantunderthisgaugetransformation50.
int
WefirstconsiderhomogeneousFermigasestogivethesimplestqualitativepictureofthebi-
layer system. At thispoint, the physicsof thesingle-particleHamiltonianis simplein momentum
space using a Fourier decomposition. The four-band structure of the bilayer system is illustrated
in Fig. 1(c,d). Obviously, the in-plane Zeeman field h leads to a Fermi surface without inver-
y
sion symmetry and givesa so-called Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with a finite
6
total momentum pairing 27,28, but can not open a band gap at the Dirac point, as shown in Fig.
1(d). Forthe realizationof topologicalphase, aperpendicular Zeeman field is needed to break the
time-reversal (TR) symmetry and open a topological band gap. In Fig. 1(c), we report that the
topological band gaps occur at the Dirac point k = 0 induced by increasing h , which are similar
z
to thesingle-layermechanism. Thesuperfluid pairingis between atoms withoppositemomentum
k and k without in-plane Zeeman field h . A critical observation is that when the Zeeman field
y
−
is larger than the tunneling effect h > J, the increasing Zeeman field removes the level crossing
z
and open a gap between the two middle branches, as shown by the red curves in the Fig. 1(c,d).
Wewillgiveadetaileddiscussionin themean-field formalism.
Classification of topological superfluid phases. We consider the most simplified form of inter-
actions with superfluid pairing formed between atoms with opposite momentum ( k, k). Within
−
themean-field approximation,theinteractionterm can berewrittenas
∆ 2
H ∆ Φ† Φ† +∆∗Φ Φ | j| , (5)
int ≈ j j,k,↑ j,−k,↓ j j,−k,↓ j,k,↑− g
jX=1,2Xk j
with the order parameters ∆ = g Φ Φ . One can easily write the mean-field BdG
j j kh j,−k,↓ j,k,↑i
P
Hamiltonian,
H0(k) J 0 σ ∆ σ
S z 1 x
J H0(k) σ ∆ σ 0
S z 2 x
H (k) = , (6)
BdG
0 σ ∆ σ σ H0( k)σ J
− z 2 x − z S − z −
σ ∆ σ 0 J σ H0( k)σ
− z 1 x − − z S − z
wheretheNambuspinorbasisischosenas[c ,c ,c ,c ,c† ,c† ,c† ,c† ]T. The
k,1,↑ k,1,↓ k,2,↑ k,2,↓ k,2,↑ k,2,↓ k,1,↑ k,1,↓
elementaryexcitationscanbefoundbysolvingtheBdGequationH (k)W±(k) = E±(k)W±(k),
BdG η η η
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where E±(k) (η = 1,2,3,4) are the eigenvalues of the above 8 8 BdG Hamiltonian and corre-
η ×
spondingwavefunctionsareassumedasW±(k) = (u± ,u± ,u± ,u± ,v± ,v± ,v± ,v± )T
η 1,↑,η 1,↓,η 2,↑,η 2,↓,η 2,↑,η 2,↓,η 1,↑,η 1,↓,η
, where the label represent the particle as well as hole branches. Without loss of generality, we
±
set ∆ = ∆ = ∆ throughout the work. The spectrum of H (k) consists of eight bands given
1 2 BdG
by
E± (k) = b (k) 2 d (k),E± (k) = b (k) 2 d (k), (7)
η=1,2 ±q 1 ± 1 η=3,4 ±q 2 ± 2
p p
wheretheb (k) = (k2/2m µ J)2+∆2 +h2 +α2k2 andd (k) = (k2/2m µ J)2(α2k2+
1 − − z 1 − −
h2) + ∆2h2 with b (k) = (k2/2m µ + J)2 + ∆2 + h2 + α2k2 and d (k) = (k2/2m µ +
z z 2 − z 2 −
J)2(α2k2+h2)+∆2h2. BysolvingE+(0) = 0,weobtainallthegapcloseconditionsinanalytic
z z η
forms:
h4 C h2 +C = 0, (8)
z − 1 z 2
with C = 2(∆2 +J2 +µ2), C = ∆4 +J4 +µ4 +2∆2J2 +2∆2µ2 2J2µ2. The solutionsfor
1 2
−
theZeemanfield are
h = ∆2 +(J µ)2, (9)
z
±
p
here we just consider the parallel case h > 0. If there is no tunneling between two layers, the
z
critical Zeeman field reduces to the well-known h = µ2 +∆2, which has been discussed in
c
p
detail in earlier works 10,11. The system will be in a conventional superfluid at h < h and in
z c
a topological superfluid at h > h . Unlike the conventional phase transition in a single layer
z c
system, the tunneling shifts the chemical potential and gives two critical Zeeman fields: h =
c,1
∆2 +(J µ)2 and h = ∆2 +(J +µ)2 (or exchange to assure h < h ). The system
c,2 c,1 c,2
−
p p
undergoesaTQPTwhentheZeemanfieldcrossthesetwocriticalvalues,wherethesingle-particle
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excitation gap vanishes, representing a topological phase transition. To see the phase transitions
moreclearly, wegivethespinpopulationat k = 0, as showninFig. 2(b). Here wedefine thespin
vectorexpectationvaluealong thez-axis as
S (k) = Φ† Φ Φ† Φ
h z i h j,k,↑ j,k,↑− j,k,↓ j,k,↓i
Xj
(10)
= v+ 2 v+ 2 .
h| j,↑,η| −| j,↓,η| i
Xj,η
We find that S (0) changes discontinuously when the Zeeman field crosses the critical values
z
h i
[h ,h ], which implies thechange of thetopologyof thespin texture. S (0) = 0 if h < h ,
c,1 c,2 z z c,1
h i
which corresponds to thetrivialstate, while S (0) = 1 for h < h < h and S (0) = 2 for
z c,1 z c,2 z
h i h i
h > h denotediversetopologicalstates,asshowinFig. 2(b). Thereforewegiveaclassification
z c,2
oftopologicalsuperfluid phasesaccording tothestrengthofperpendicularZeeman field h :
z
trivial region :0 < h < h , ∆ = 0,
z c,1
6
topo I region :h < h < h , α∆ = 0, E > 0, (11)
c,1 z c,2 g
− 6
topo II region :h > h , α∆ = 0, E > 0,
z c,2 g
− 6
where the last condition E = min E+ (k) > 0 in the topological regimes ensures the bulk
g { j,σ }
quasi-particle excitations are gapped to protect the zero-energy MFs. The region for topological
superfluid depends not only on the chemical potential, pairing strength, but also on the tunneling
strength. It provides more control knobs for tuning the topological phase transition. Especially
when the tunneling can be comparable to the chemical potential, the perpendicular Zeeman field
threshold can be dramatically lowered. These new features may open a possibility for the experi-
mentalrealization oftopologicalsuperfluidundersmallZeemanfield.
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Phase diagram of spin-orbit coupled bilayer Fermi gases. To better understand the transition
fromonestatetoanotherdefinedbyequation(11),itisnecessarytoobservethecloseandreopenof
theexcitationgapE ,whichchangethetopologyofFermisurface. Thephasediagramintheplane
g
of the order parameter and the perpendicular Zeeman field is presented in Fig. 2(a). The graph is
colored according to the energy gap E , and white dotted curves mark contours of E = 0. There
g g
are three different topological regions determined by the behavior of the energy gap. The system
is topological trivial at first, as increasing the Zeeman field with a fixed ∆, the band gap may
first close and then reopen, signifying the transition from non-topological to topo I superfluid
−
(h [h ,h ]), and finally undergoes a topologicalphase transition from topo I to topo II
z c,1 c,2
∈ − −
phaseatthesecondgaplesspointh . InFig. 2(b-f),weplotthebandgapandenergyspectrumsto
c,2
illustratethe TQPT when ∆ = 0.5E . It should be emphasized that the range of region: topo I
r
−
ishighlydependentonthevalueoforderparameter∆, theincreasingoftheintensityof∆rapidly
shrinksthisregion.
As have discussed the evolution of the single-particle Hamiltonian as a function of h , here
z
we can further more rigorously demonstrate the topological phase transition with the change of
the chemical potential µ. We can sketch the phase diagram in the Fig. 3(a), which shows perfect
inversionsymmetry(µ µ)intheµ ∆plane. Forasmallorderparameter,theimportantfea-
→ − −
tureisthattherehavefourtopologicaltransitionpointsalongthechangeofthechemicalpotential.
Similar to the case in Fig. 2, by tuning through the transition point, the quasi-particle excitation
gap closes and open again, thus we can discriminatefive regions in accordance with the topologi-
cal condition (11) with central area being topo II. We can write down the scaling forms of the
−
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