Collective excitations across the BCS-BEC crossover induced by a synthetic Rashba spin-orbit coupling

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by  Jayantha P. Vyasanakere

Collective excitations across the BCS-BEC crossover induced by a synthetic Rashba
spin-orbit coupling
Jayantha P. Vyasanakere∗ and Vijay B. Shenoy†
Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560 012, India
(Dated: January 26, 2012)
Syntheticnon-AbeliangaugefieldsincoldatomsystemsproduceageneralizedRashbaspin-orbit
interaction described by a vector λ=(λ ,λ ,λ ) that influences the motion of spin-1 fermions. It
x y z 2
was recently shown [Phys. Rev. B 84, 014512 (2011)] that on increasing the strength of the spin-
orbit coupling λ=|λ|, a system of fermions at a finite density ρ≈k3 evolves to a BEC like state
2 F
1 even in the presence of a weak attractive interaction (described by a scattering length as). The
0 BEC obtained at large spin-orbit coupling (λ (cid:29) kF) is a condensate of rashbons – novel bosonic
2 bound pairs of fermions whose properties are determined solely by the gauge field. In this paper,
weinvestigatethecollectiveexcitationsofsuchsuperfluidsbyconstructingaGaussiantheoryusing
n
functional integral methods. We derive explicit expressions for superfluid phase stiffness, sound
a
speedandmassoftheAnderson-Higgsbosonthatarevalidforanyλandscatteringlength. Wefind
J
thatatfiniteλ,thephasestiffnessisalwayslowerthanthatsetbythedensityofparticles,consistent
5
with earlier work[arXiv:1110.3565] which attributed this to the lack of Galilean invariance of the
2
systematfiniteλ. Weshowthatthereisanemergent Galilean invarianceatlargeλ,andthephase

stiffness is determined by the rashbon density and mass, consistent with Leggett’s theorem. We
]
s further demonstrate that the rashbon BEC state is a superfluid of anisotropic rashbons interacting
a via a contact interaction characterized by a rashbon-rashbon scattering length a . We show that
R
g a goesasλ−1 andisessentiallyindependentofthescatteringlengthbetweenthefermionsaslong
R
-
as it is nonzero. Analytical results are presented for a rashbon BEC obtained in a spherical gauge
t
an field with λx =λy =λz = √λ3.
u
PACSnumbers: 03.75.Ss,05.30.Fk,67.85.-d,67.85.Lm,71.70.Ej
q
.
t
a
I. INTRODUCTION amined. Non-Abelian gauge fields in lower dimensions
m
and lattices have also been investigated.25–27 A review
-
d of these fast paced recent developments may be found
The simulation of quantum condensed matter
n in ref. [28]. Several aspects of the physics of spin-orbit
o systems1–3 with cold atoms has captivated the imagi- coupled fermions were reported earlier29,30 and were in-
c nation and efforts of many. Some of the most recent dependently discovered in the cold atoms context.16,17
[ new developments include the generation4–8 of synthetic
The motivating questions for this work pertain to the
gaugefieldsinbosons9–11 andrealizationoffermionicde-
1 properties of the RBEC that is obtained at large gauge
generacy in their presence.12
v coupling at a fixed scattering length a . In the usual
s
2 Uniform non-Abelian gauge fields produce spin-orbit BCS-BEC crossover31–35 in the absence of spin-orbit in-
3
interactions. The physics of bosons in spin-orbit cou- teraction, the BEC state for small positive scattering
3
pledsystemhasbeeninvestigatedbymanyauthors.13–15 length a is a condensate of bosons (fermionic dimer
5 s
The rich physics hidden in the fermion problem was re- molecules). This BEC state can be described by the Bo-
.
1 vealed by the solution of the two-body problem given in goliubovtheoryofinteractingbosons36,wheretheboson
0
ref. [16], where it was shown that for certain high sym- mass is twice the fermion mass and the effective boson-
2
metry gauge fields, a bound state appears even for an bosonscatteringlengthisproportionaltoa .35,37 Doesa
1 s
: infinitesimal attraction in the singlet channel. The key similardescriptionholdfortheRBECobtainedbytuning
v
outcome of this is that a BCS-BEC crossover is induced themagnitudeofthegaugecoupling? Howdoesrashbon-
i
X by increasing the strength of the gauge field even with a rashbonscatteringenterthedescription,i.e.,whatisthe
weak attractive interaction.17 The BEC that is realized effective rashbon-rashbon scattering length?
r
a was shown to be a condensate of a new type of boson – That collective excitations have interesting and un-
the rashbon – whose properties are determined solely by usual features was pointed out in ref. [38] which studied
the gauge field and not by the scattering length charac- phase stiffness Ks (superfluid density) for an extreme-
terizing the interaction between the fermions. This BEC oblate gauge field (see below for a definition). In the
realized at large gauge coupling is called the rashbon- regime λ (cid:46) k , the Ks decreases with increasing gauge
F
BEC (RBEC). Concurrently, anisotropic superfluidity of coupling. However, for λ (cid:38) k , Ks increases and satu-
F
rashbons18, zero-temperature BCS-BEC crossover in the rates as λ/k attains large values. For all λ, Ks is less
F
presence of Zeeman fields19,20 (imbalance) was studied, than ρ/4m, the value of phase stiffness for a superfluid
and transition temperatures were estimated21,22. Dres- without the spin-orbit interaction, where ρ the density
selhaus like spin-orbit interaction23,24 has also been ex- andmisthemassofthefermions. Thisisattributed38 to
2
the lack of Galilean invariance in systems with synthetic where, Cs and C†s are fermion operators,
non-Abeliangaugefields(seealso,ref.[39]). Whilethisis
true, we conjecture that Galilean will be approximately k2
ε (k)= αk , (3)
restored in the system for λ kF when an attractive α 2 − | λ|
(cid:29)
interaction, however weak, is present. The basis of this
α= 1 is the helicity, k =λ k e +λ k e +λ k e .
conjecture stems from the fact that at large λ the sys- λ x x x y y y z z z
±
tem with even a weak attraction can be thought of as The “vector” λ (λx,λy,λz) λλˆ describes the con-
≡ ≡
a collection of rashbons which disperse quadratically22, figuration of the gauge field that induces a generalized
εR(q)=−ER+(cid:80)i 2mqi2Ri , albeit with an anisotropic dis- Rmaasghnbitaudsepinof-otrhbeitgianutgereacctoiuopnl,inwghaenred λλˆ i=s a|λu|niitsvtehce-
persiondefinedbythedirectiondependentrashbonmass
tor. High symmetry gauge field configurations of in-
mR and ER is the rashbon binding energy, a result that
i terest include the extreme oblate (EO) gauge field with
is valid for q λ. This dispersion is Galilean invari- (cid:16) (cid:17)
ant, and the|re|fo(cid:28)re we expect to obtain a phase stiffness λ=λ √1 ,√1 ,0 andthespherical(S)gaugefieldwhich
2 2
tensor Kisj = mρRRδij (no sum on i), where ρR = ρ/2 is hasλ=λ(cid:16)√1 ,√1 ,√1 (cid:17). Weuseunitswherethefermion
therashbondensiity,consistentwithLeggett’sresult33,40. mass m and (cid:126)3 are3un3ity. We consider a finite density of
Testing this conjecture regarding emergent Galilean in-
fermionsρwhichdefinesamomentumscalek suchthat
F
variance and answering the questions raised in the pre- k3 k2
vious paragraph are the aims of this paper. ρ= 3πF2, and an energy scale EF = 2F.
Theinteractionpiece describesanattractioninthe
To this end, we investigate the collective excitations Hυ
singlet channel as
of superfluids induced by non-Abelian gauge fields using
aGaussianfluctuationstheorywithafunctionalintegral υ (cid:88)
framework. OurmainresultisthattherashbonBECcan Hυ = Ω C(†q2+k)↑C(†−q2+k(cid:48))↓Ck(cid:48)↓Ck↑ (4)
be described as a collection of weakly interacting rash- q,k,k(cid:48)
bons. We obtain an effective rashbon-rashbon scattering
length which we show is generically proportional to λ−1, where Ω is the volume of the system, υ is the bare in-
teraction parameter. The theory requires an ultraviolet
and is independent of the scattering length between the
cutoff Λ which can be eliminated by using 1 = 1 +Λ.
fermions to leading order. In addition, we show that 4πas υ
Using mean-field theory, it was shown in ref. [17] that
the phase stiffness has precisely the form as conjectured
increasing λ induces a BCS to BEC crossover even for
above. The RBEC state is a remarkable state where the
a weak attractive interaction (k a 1,a < 0). We
effective interaction between the emergent bosons (rash- F s s
| | (cid:28)
bons)isdeterminedbythekineticenergy(spin-orbitcou- aimtostudythecollectiveexcitationsofsuchsuperfluids
pling λ) of the constituent fermions, and not the attrac- across this crossover.
To this end we use a functional integral framework
tion between the fermions as long as it is non-vanishing.
which has been extensively used in the study of BCS-
Our theory also provides the phase stiffness, speed of
BEC crossover.35,41–45 Denoting inverse temperature as
soundandthemassoftheAnderson-Higgsbosonforany
β and chemical potential as µ, we write the action
gauge coupling.
Sec. II outlines the functional integral framework used
(cid:88) υ (cid:88)
in the analysis of the collective excitations and obtains S[Ψ]= Ψ(cid:63)(k)(−G−01(k,k(cid:48)))Ψ(k(cid:48))+βΩ S(cid:63)(q)S(q)
generalformulaeforthephasestiffness, soundspeedand k q
Anderson-HiggsmassforagenericRashbalikespin-orbit (5)
coupled system. Results for a spherical gauge field are where
discussed in sec. III, and sec. IV contains a discussion of
 c (k) 
the properties of rashbon BECs. The paper is summa- +
c(cid:63)( k)
rized in sec. V. Ψ(k)= + −  (6)
 c (k) 
−
c(cid:63)( k)
− −
II. FORMULATION
is a Nambu vector consisting of Grassmann variables
describing the fermions, k = (ik ,k) where ik is a
n n
We follow closely the notation and terminology intro- fermionic Matsubara frequency,
duced in16,17. The Hamiltonian of the system of interest
is made up of two pieces ikn−ξ+(k) 0 0 0 
G−01(k,k(cid:48))= 00 ikn+0ξ+(k) ikn−0ξ−(k) 00 δk,k(cid:48),
= R+ υ. (1) 0 0 0 ikn+ξ−(k)
H H H (7)
The kinetic energy of the spin-1 fermions is ξα(k)=εα(k) µ, and
2 −
HR =(cid:88)εα(k)Ck†αCkα (2) S(cid:63)(q)= (cid:88)Aαβ(q,k)c(cid:63)α(2q +k)c(cid:63)β(2q −k) (8)
k k,αβ
3
is the Fourier transform of the singlet density with q = With this ansatz for the saddle point, the Green’s func-
(iq ,q), iq is a bosonic Matsubara frequency. A (q,k) tion G(k,k(cid:48)) is
(cid:96) (cid:96) αβ
isthesingletamplitudeinatwoparticlestateofαandβ
helicities, with centre of mass momentum q and relative  Gp(k) Ga(k) 0 0 
+ +
mmoamnyensytummmekt.ryItprmopuesrttibeeswnohtiecdhathreatusAedαβe(xqt,ekn)sivsaeltyisifny G(k,k(cid:48))=−G0a+(k) Gh+0(k) Gp−0(k) Ga−0(k)δk,k(cid:48)
the work that follows. Moreover, care must be exercised 0 0 Ga(k) Gh(k)
− − −
in the definition of Aαβ(q,k) due to the non-zero Chern (16)
flux originating from the origin of the momentum space where
(see ref. [46]).
ik +ξ (k)
We now introduce a Hubbard-Stratanovich pair field Gp(k)= n α (17)
∆(q) to decouple the interaction term to obtain α (ikn)2−Eα2(k)
ik ξ (k)
S[Ψ,∆]=(cid:88)Ψ∗(k)(−G−1(k,k(cid:48)))Ψ(k(cid:48))−υ1 (cid:88)∆∗(q)∆(q) Ghα(k)= (iknn)2−−αEα2(k) (18)
k,k(cid:48) q Ga(k)= iα∆0 (19)
(9) α (ik )2 E2(k)
where G−1(k,k(cid:48)) is (cid:112) n − α
with E (k) = ξ (k)2+∆2. The saddle point condi-
α α 0
G−1(k,k(cid:48))=G (k,k(cid:48)) ∆(k,k(cid:48)), (10) tion, after appropriate frequency sums, is
0
−
1 1 (cid:88)tanhβEα(k)
∆(k,k(cid:48))= ∆˜++(00k,k(cid:48)) ∆∆−+++(0(kk,,kk(cid:48)(cid:48))) ∆˜+−00(k,k(cid:48)) ∆∆−+−−0((kk,,kk(cid:48)(cid:48))) − υ = 2Ω kα 2Eα(k2) (20)
∆˜−+(k,k(cid:48)) 0 ∆˜−−(k,k(cid:48)) 0 and agrees with the gap equation derived in ref. [17 and
(11)
22]. The saddle point number equation is
with
(cid:18) (cid:19)
∆αβ(k,k(cid:48))=(cid:88)√∆β(qΩ)Aαβ(q,k− q2)δq,k−k(cid:48) (12) ρ= 21Ω(cid:88) 1− Eξαα((kk)) (21)
q kα
∆˜αβ(k,k(cid:48))=(cid:88)∆∗(−q)Aβα( q,k q)δq,k−k(cid:48) (13) The values of ∆0 and µ are set by the simultaneous so-
√βΩ − − 2 lution of eqn. (20) and eqn. (21).
q
Collective excitations of the system are described by
We integrate out the fermions to obtain the action only fluctuations about the saddle point state. We treat
in terms of the pairing field them at Gaussian level by introducing “small oscilla-
tions” about the saddle point value of the pairing field,
1 (cid:88)
[∆]= ∆∗(q)∆(q) lndet[ G] (14) sp
S −υ − − ∆(q)=∆ (q)+η(q) (22)
q
Aftersomestraightforward,iflengthy,algebra,theaction
We now perform a saddle point analysis of the action
to quadratic order in η is
and look for static and homogeneous solutions via the
ansatz (cid:18) (cid:19)
[η]= sp+ 1(cid:88)(cid:0)η∗(q) η( q)(cid:1)Π(q) η(q)
∆sp(q)=(cid:112)βΩ√2∆0δq,0 (15) S S 2 q − η∗(−q)
(23)
where the factor of √2 is introduced for convenience. where
(cid:18) (cid:19)
Π (q) Π (q)
Π(q)= 11 12
Π (q) Π (q)
21 22
Π (q)=Π (−q)=−1 + 1 (cid:88)|A (q,k)|2Gp(iq +ik ,q +k)Gh(ik ,−q +k)
11 22 υ βΩ αβ α (cid:96) n 2 β n 2 (24)
k,αβ
Π (q)=Π (q)=− 1 (cid:88)αβ|A (q,k)|2Ga(iq +ik ,q +k)Ga(ik ,−q +k)=Π (−q)=Π (−q)
12 21 βΩ αβ α (cid:96) n 2 β n 2 12 21
k,αβ
Collective excitations of a superfluid can be conve- dent phase and amplitude oscillations. We, therefore,
niently described in terms of spatio-temporally depen- express η in terms of two other real fields ζ (amplitude
4
fluctuation) and φ (phase fluctuation) as WehavenotshowntheexpressionforV sinceitwillnot
ij
be used in the discussion below.
η(q)=∆ (ζ(q)+iφ(q)) (25)
0 The dispersion of the excitations can be obtained by
with ζ( q) = ζ∗(q) and φ( q) = φ∗(q). The action in first analytically continuing iq(cid:96) ω+ to real frequencies
terms o−f these two fields is − and solving detΓ(ω+,q) = 0. W→e obtain two modes for
a given q = qqˆ, one is a gapless sound mode and other
(cid:18) (cid:19)
[ζ,φ]= sp+1(cid:88)(cid:0)ζ∗(q) φ∗(q)(cid:1)Γ(q) ζ(q) (26) is the gapped Anderson-Higgs mode. The speed of sound
S S 2 φ(q) along direction qˆis given by
q
where, using eqn. (24), we find qˆKsqˆ
c2(qˆ)= i ij j (37)
(cid:18) Γ (q) Γ (q) (cid:19) s Z+ X2
Γ(q)= ζζ ζφ (27) U
Γ (q) Γ (q)
φζ φφ
Γ (q)=∆2(Π (q)+Π ( q)+2Π (q)) (28) and the mass of the Anderson-Higgs mode MAH is ob-
ζζ 0 11 11 − 12 tained as
Γ (q)=i∆2(Π (q) Π ( q))= Γ (q) (29)
ζφ 0 11 − 11 − − φζ
Γφφ(q)=∆20(Π11(q)+Π11(−q)−2Π12(q)) (30) MA2H = ZUZ+WX2 (38)
We now preform the necessary frequency sums to obtain
expressionsfortheΓs. Hereandhenceforthinthispaper, Itmustbenotedthattheamplitudeandphasemodesare
we focus at zero temperature (T = 0) and “small” q, coupled42; their coupling is determined by the quantity
and do not show the lengthy expressions valid for any X.
temperature and q. For small q at T =0, we have, Equations 34, 37 and 38 are the key results of this
Γ (iq ,q)=q Ksq Z(iq )2 (31) paper for the collective excitations of spin-orbit coupled
φφ (cid:96) i ij j − (cid:96) superfluidsthatareapplicabletoanyRashbagaugefield
Γζφ(iq(cid:96),q)= iq(cid:96)X (32) and scattering length at zero temperature. We have not
−
Γ (iq ,q)=U +q V q W(iq )2 (33) shownthefinitetemperatureresultsheretoavoidlengthy
ζζ (cid:96) i ij j (cid:96)
− expressions. In the remainder of the paper, we illustrate
where the quantities Ks,Z,X,U,V,W depend on the the physics of these formulae using the spherical gauge
saddlepointvaluesof∆0andµ. Kisj isthephasestiffness field (next section) and explore the consequences of our
given by results particularly for the rashbon-BEC (sec. IV).
Ks = ∆20 (cid:88)viα(k)vjα(k)
ij 2Ω 4E3(k)
kα α
III. COLLECTIVE EXCITATIONS FOR THE
+ 2∆20 (cid:88) (ε+(k)−ε−(k))2 S (k) SPHERICAL GAUGE FIELD
Ω 2E (k)E (k)(E (k)+E (k)) ij
+ − + −
k
(34) In this section we discuss collective excitations of su-
perfluids realized in a spherical gauge field with λ =
twhheesreinvgiαle(tka)m=pl∂iεt∂αuk(dike),Aand(Sqi,jk(k))foisrasmteanllsoqrathsatdefines λ(cid:16)√13,√13,√13(cid:17) as noted earlier. The two body prob-
+− lem for this gauge field was exhaustively investigated in
A (q,k)2 = A (q,k)2 q S (k)q . (35) ref. [16] where an analytical expression for the binding
+− −+ i ij j
| | | | ≈ energy valid for any scattering length is derived along
It must be noted that extensive use of the properties of with an analytical expression for the bound state wave
Aαβ(q,k) is made in arriving at this expression for the function. The binding energy of the rashbon16 is
phase stiffness tensor that is valid for any gauge field.
The other quantities in eqn. (31), λ2
ER = (39)
∆2 (cid:88) 1 3
Z = 0
2Ω 4E3(k)
kα α and the rashbon mass (in units of fermion mass) is22
X = ∆20 (cid:88) ξα(k)
2Ω kα 2Eα3(k) mR = 3(4+√2) (40)
(36) 7
∆4 (cid:88) 1
U = 0
2Ω E3(k) Aroutetoexperimentalrealizationofthisgaugefieldhas
kα α recently been suggested.47 A detailed study of two-body
∆4 (cid:88) 1
W =Z 0 . scatteringfromafiniterangeboxpotentialiscarriedout
− 2Ω kα 4Eα5(k) in ref. [48].
5
A. Analytical Results 1
Analytical results can be obtained in two regimes of 0.8
λ. These correspond to λ k , and the other to λ
F
(cid:28) (cid:29)
max(k ,1/a ).
F s
S0 0.6
K 1
/
S
1. λ(cid:28)k K 0.9
F 0.4
k a
F s
-1/4 1 0.8
Two regimes of a are tractable analytically for this
s
0.2 -1/2 1/2
regimeofλ,bothofwhicharewellknown;westatethem
-1 1/4 0.7
here for the sake of completion.
infinity 20 40 60 80
I. as <0, kFas 1: This regime is studied in detail in 0
ref. [17]. T| he ch|e(cid:28)mical potential in this regime is set by 0 2 4 6 8
λ/k
thevalueofthenoninteractingsystem(whichfallsbyan F
amount proportional to λ2). The gap ∆ is essentially
k2 0 FIG. 1. (color online) Phase stiffness - Evolution of phase
F
unaltered from the well known BCS value. Under these stiffnessKs withincreasingλforthesphericalgaugefieldfor
conditions, we obtain the phase stiffness to be ρ with a various scattering lengths. Ks = ρ/4. The inset shows that
4 0
falloforder λ2. Theleadingterminthespeedofsoundis Ks/K0tendsto2/mRforlargeλdemonstratingtheemergent
k2 Galilean invariance. The dashed vertical line corresponds to
F
k /√3asshownbyAnderson49 (withafallproportional λ=λ where there is a change in the topology of the Fermi
F T
toλ2/k2)andtheAnderson-Higgsmassisexponentially surface of the non-interacting system.17
F
small. ThislimitcorrespondsessentiallytotheBCSlimit
studied in ref. [42].
II. a > 0,k a 1: This corresponds to the usual and
s F s
(cid:28)
BEC regime (ref. [37 and 42]). Here the chemical po- 2
tential µ = −2a12s +2πasρ and the gap ∆20 = 4aπsρ. The MAH = 3λ2. (45)
phase stiffness Ks = ρ, speed of sound is c2 = 2πρa ,
4 s s Asexpected,theleadingtermsforallthequantitiesofin-
the M = 4 . In this regime, the amplitude and the
AH a2 terestareindependentofthescatteringlengthbetweenthe
s
phase modes are strongly mixed.
fermions; scattering length corrections (which we do not
show) appear as powers of (1/λa ), which in this regime
s
are small. We emphasize that in this RBEC regime the
2. λ(cid:29)k and λ(cid:29) 1
F as amplitudeandthephasemodearestronglycoupled,just
like in the usual BEC regime.
This is the regime of interest and corresponds to the
rashbon BEC. In this regime, we report new results for
the gap B. Numerical Results
2π λ
∆2 = (41) In this section we show the results of numerical calcu-
0 ρ √3 lations of evolution of Ks, c and M with increasing
s AH
λ for several scattering lengths.
and the chemical potential
ER √3
µ= +πρ . (42) 1. Superfluid Phase Stiffness
− 2 λ
By an analysis of the expression for the phase stiffness Fig. 1 shows a plot of the phase stiffness as a func-
(eqn. (34)) which is isotropic for this gauge field, we find tion of λ for various scattering lengths. We see that
that for small negative scattering lengths, the behaviour of
ρ Ks is non-monotonic; it decreases with increasing λ and
Ks = 2mR (43) attains a minimum near λ (cid:38) λT. This is fully con-
sistent with the finding of ref. [38] for the EO gauge
precisely as conjectured in the introductory section (see field. The new aspect uncovered in our work is that for
below for further discussion). Additional analysis pro- λ max(k ,1/a ),thephasestiffnesstendstothatofa
F s
vides co(cid:29)llection of interacting rashbons in exactly same way as
(cid:32) (cid:33) the motivating conjecture of this paper. In other words,
c2s = 2mπRρ √λ3 (44) bKesr(λde→nsi∞ty.)T=hmeρRRphwyshiecrsebρeRhin=dρt/h2isisisththeartatshhbeornasnhubmon-
6
0.6 kF as kF as
0.16 -1/4 -1/4
3kF -1/2 15 -1/2
0.5 / 1
c2λ S 0.155 infini-t1y 2/λ infini-t1y
c/k SF 0.4 0.15 20 40 λ/kF 60 80 11//124 2M/k FAH 10 M AH 0 .05 11//124
0.3 20 40 60 80
λ/k
F
5
0.2
0.1 0
0 2 4 λ/k 6 8 0 1 2 λ/k 3 4 5
F F
FIG.2. (coloronline)Soundspeed-Evolutionofthesound FIG.3. (coloronline)MassoftheAnderson-Higgsboson
speed c with increasing λ for the spherical gauge field for - Evolution of mass of the Anderson-Higgs boson M with
s AH
various scattering lengths. The inset shows that c2 has the increasingλforthesphericalgaugefieldforvariousscattering
s
behaviourobtainedineqn.(44),independentofthescattering lengths is shown in Fig. 3. The inset shows that M goes
AH
length. Thedashedverticallinecorrespondstoλ=λ where as λ2, independent of the scattering length (eqn. (45)). The
T
there is a change in the topology of the Fermi surface of the dashed vertical line corresponds to λ = λ where there is
T
non-interacting system. a change in the topology of the Fermi surface of the non-
interacting system.
dispersionε (q)= ER+ q2 isGalileaninvariant,and
hencethephRasestiff−nessas2fmouRndatλ isconsistent teracting rashbons. Since this regime is the raison d’etre
with Leggett’s result33,40. This is a re→ma∞rkable feature, ofthispaper,wedonotpausetoconsidertheinteresting
and corresponds to an emergent infrared symmetry, i. e., regime of λ λT which no doubt contains rich physics.
∼
in the presence of interactions however small, the system
organizes itself to posses a larger symmetry at low ener-
gies! A important point that can be inferred is that the IV. PROPERTIES OF RASHBON
BOSE-EINSTIEN CONDENSATES (RBEC)
nonzero phase stiffness implies that rashbons are inter-
actingbosons. Thenatureoftheinteractionisuncovered
in the next section. That the system evolves to a collection of interacting
rashbons with increasing λ is conclusively demonstrated
in the previous section. The rashbon dispersion derived
2. Sound Speed in ref. [22] provides the kinetic energy of the rashbons.
What about their interactions? Interestingly, the results
The variation of the sound speed with increasing λ is of the previous section allow us to answer this question.
showninfig.2. Weseethatthereisamonotonicdecrease Recall from the Bogoliubov theory36 that a collection
in the sound speed with increasing λ for all scattering of bosons of mass mB with number density ρB and a
lengths. At large λ, the sound speed is inversely pro- contact interaction described by a scattering length aB
portional to λ as obtained analytically (see eqn. (44)). has a superfluid ground state at zero temperature. The
Again, that there is sound propagation in the medium chemical potential of this system is
suggests the presence of interactions between the rash-
4πa
B
bons. µ = ρ (46)
B B
m
B
and the speed of sound is
3. Mass of the Anderson-Higgs boson
(cid:115)
(cid:114)
µ 4πa ρ
Forsmallgaugecoupling(λ kF )MAH corresponds cBs = mB = mB2 B. (47)
tothegapoftheamplitudemo(cid:28)deforsmallnegativescat- B B
tering lengths. This mass grows with increasing λ albeit
From eqn. (42), the rashbon chemical potential µR
with some features near λ λ for small negative scat-
tering lengths. At large λ∼weTfind the expected λ2 be- (measuredfromthebottomoftherashbonbandat−ER)
is
haviour.
The key result of this section is that at large λ, the √3
system behaves like a Galilean invariant collection of in- µR =2πρ (48)
λ
7
We see immediately that the speed of sound obtained R
in eqn. (44) is consistent with eqn. (47) from Bogoliubov λ
theory
µR
c2 = (49)
s mR
ThisclearlydemonstratesthattherashbonBECisacon-
densate of rashbons interacting with a contact interac- vR∗ vF∗
tion. Writing
v
(cid:115) vv == 11 v = 0
4πa ρ −−
c2 = R R (50)
s (mR)2 FIG. 4. (color online) Schematic two-body RG flow di-
agram - The rashbon state corresponds to the stable fixed
allows us to calculate the rashbon-rashbon scattering point R at λ = ∞ and v = −1. Flow from any point with
λ(cid:54)=0 and v(cid:54)=0 reaches R.
length as
3√3(4+√2)1
aR = (51) V. SUMMARY
7 λ
whichisapproximatelyequalto 4. Thisresultisremark- Inthispaper,weexplorethepropertiesofthesuperflu-
λ
able in the following sense that the effective interaction idsinducedbynon-Abeliangaugefieldsfocusingontheir
between rashbons is determined by a scale λ that en- collective excitations. We present results for superfluid
ters the kinetic energy of the constituent fermions, and phase stiffness, sound speed and Anderson-Higgs mass
not by the interaction between the constituent fermions valid for any Rashba gauge field and scattering length.
(scattering length as)! Our main results are
We emphasize that although our arguments used the
Superfluid phase stiffness has non-monotonic be-
sphericalgaugefields,theresultsobtainedareapplicable
•
haviour with increasing λ, the scale of the spin-
to other gauge field configurations described by a gen-
eral vector λ = λλˆ (except the extreme prolate gauge orbitinteraction. Thisisinagreementwithanear-
lierreport38 ofsuperfluiddensityfortheEOgauge
field which has only one nonvanishing component, see
field.
ref.[17]). Foragenericgaugefield,therashbonchemical
potential will be
A new result is that for large gauge coupling, i e.,
•
ρ in the rashbon BEC, the superfluid phase stiffness
µR =M(λˆ) (52) is determined by the rashbon mass22. This arises
λ
from an emergent Galilean invariance at infrared
where M(λˆ) is a dimensionless number that depends on energies for large gauge couplings, and the phase
λˆ, and the anisotropic speed of sound in the i-direction stiffness is consistent with Leggett’s result.
will be
The sound speed decreases monotonically with in-
•
µR creasinggaugecoupling. Atlargegaugecouplingit
c2s(i)= mR (53) goes as λ−1/2. The Anderson-Higgs mass increases
i with increasing λ and goes as λ2 in the rashbon-
where mR is the anisotropic rashbon mass22 that de- BEC.
i
pends, again, on λˆ. The rashbon-rashbon scattering
A key outcome of this work is that we show that
length will be •
therashbon-BECcanbedescribedasacollectionof
anisotropically dispersing rashbons interacting via
N(λˆ)
a = (54) a contact interaction. We obtain an analytical ex-
R
λ pressionfortherashbon-rashboninteractionforthe
spherical gauge field showing that it goes as λ−1.
where N(λˆ) is dimensionless number determined by λˆ
Wearguethatthisresultistrueforagenericgauge
The low energy properties of the rashbon BEC are sim-
field (spin-orbit interaction).
ilar to those of the usual Bogoliubov Bose fluid; in fact,
generically, RBEC is a superfluid of anisotropically dis- We conclude the paper by revisiting the RG flow di-
persingrashbonsinteractingwithacontactpotentialde- agram of the two body problem introduced in ref. [16].
scribedbyascatteringthatdependsinverselyonthespin Fig.4isaschematicRGflowdiagramintheλ-υplanefor
orbitcouplingstrengthofthefermions. Itmustbenoted thetwo-particleproblem. Thekeypointisthatflowfrom
thataccuratedeterminationofN(λˆ)mayrequirefurther any point with λ = 0 and υ = 0 reaches R which is the
self consistent treatment of the theory.45,50 stable rashbon fix(cid:54)ed point co(cid:54)rresponding to λ = and
∞
8
υ = 1. Indeed, the properties of the state attained by Acknowledgement
−
afinitedensityoffermionsatlargeλiscontrolledbythe
rashbon fixed point; it is therefore a weakly interacting JV acknowledges support from CSIR, India via a JRF
gas of rashbons – the rashbon BEC. grant. VBSisgratefultoDST,India(Ramanujangrant),
DAE, India (SRC grant) and IUSSTF for generous sup-
port.
∗ [email protected] 26 L. He and X.-G. Huang, ArXiv e-prints (2011),
† [email protected] arXiv:1109.5577 [cond-mat.quant-gas].
1 W. Ketterle and M. W. Zwierlein, (2008), 27 G. Chen, M. Gong, S. Jia, and C. Zhang, ArXiv e-prints
arXiv:0801.2500. (2012), arXiv:1201.2238 [cond-mat.quant-gas].
2 I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 28 H. Zhai, ArXiv e-prints (2011), arXiv:1110.6798 [cond-
80, 885 (2008). mat.quant-gas].
3 S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. 29 A. V. Chaplik and L. I. Magarill, Phys. Rev. Lett. 96,
Phys. 80, 1215 (2008). 126402 (2006).
4 D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003). 30 E. Cappelluti, C. Grimaldi, and F. Marsiglio, Phys. Rev.
5 K.Osterloh,M.Baig,L.Santos,P.Zoller, andM.Lewen- Lett. 98, 167002 (2007).
stein, Phys. Rev. Lett. 95, 010403 (2005). 31 D. M. Eagles, Phys. Rev. 186, 456 (1969).
6 J. Ruseckas, G. Juzeliu¯nas, P. O¨hberg, and M. Fleis- 32 A. J. Leggett, in Modern Trends in the Theory of Con-
chhauer, Phys. Rev. Lett. 95, 010404 (2005). densed Matter, edited by A. Pekalski and R. Przystawa
7 F. Gerbier and J. Dalibard, New J. Phys. 12, 033007 (Springer-Verlag, Berlin, 1980).
(2010). 33 A. J. Leggett, Quantum Liquids: Bose Condensation and
8 J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg, Cooper Pairing in Condensed-Matter Systems (Oxford
Rev. Mod. Phys. 83, 1523 (2011). University Press, 2006).
9 Y.-J. Lin, R. L. Compton, A. R. Perry, W. D. Phillips, 34 P. Nozi´eres and S. Schmitt-Rink, J. Low Temp. Phys. 59,
J. V. Porto, and I. B. Spielman, Phys. Rev. Lett. 102, 195 (1985).
130401 (2009). 35 M. Randeria, in Bose-Einstein Condensation, edited by
10 Y.-J.Lin,R.L.Compton,K.Jimenez-Garcia,J.V.Porto, A. Griffin, D. Snoke, and S. Stringari (Cambridge Uni-
and I. B. Spielman, Nature 462, 628 (2009). versity Press, 1995) Chap. 15.
11 Y.-J.Lin,K.Jimenez-Garcia, andI.B.Spielman,Nature 36 A. A. Abrikosov, L. P. Gor’kov, and I. Y. Dzyaloshinskii,
471, 83 (2011). Quantum field theoretical methods in statistical physics
12 H. Zhai, (2012), private communication. (Pergamon, 1965).
13 T.D.Stanescu,B.Anderson, andV.Galitski,Phys.Rev. 37 C. J. Pethick and H. Smith, Bose-Einstein Condensation
A 78, 023616 (2008). in Dilute Gases (Cambridge University Press, 2004).
14 C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. 38 K. Zhou and Z. Zhang, ArXiv e-prints (2011),
Lett. 105, 160403 (2010). arXiv:1110.3565 [cond-mat.quant-gas].
15 T.-L. Ho and S. Zhang, (2010), arXiv:1007.0650. 39 R. A. Williams, L. J. LeBlanc, K. Jimnez-Garca,
16 J. P. Vyasanakere and V. B. Shenoy, Phys. Rev. B 83, M. C. Beeler, A. R. Perry, W. D. Phillips,
094515 (2011). and I. B. Spielman, Science 335, 314 (2012),
17 J.P.Vyasanakere,S.Zhang, andV.B.Shenoy,Phys.Rev. http://www.sciencemag.org/content/335/6066/314.full.pdf.
B 84, 014512 (2011). 40 A.J.Leggett,JournalofStatisticalPhysics93,927(1998),
18 H. Hu, L. Jiang, X.-J. Liu, and H. Pu, (2011), 10.1023/B:JOSS.0000033170.38619.6c.
arXiv:1105.2488. 41 C.A.R.Sa´deMelo,M.Randeria, andJ.R.Engelbrecht,
19 M. Gong, S. Tewari, and C. Zhang, (2011), Phys. Rev. Lett. 71, 3202 (1993).
arXiv:1105.1796. 42 J.R.Engelbrecht,M.Randeria, andC.A.R.Sa´deMelo,
20 M. Iskin and A. L. Suba¸s ı, Phys. Rev. Lett. 107, 050402 Phys. Rev. B 55, 15153 (1997).
(2011). 43 S. De Palo, C. Castellani, C. Di Castro, and B. K.
21 Z.-Q. Yu and H. Zhai, (2011), arXiv:1105.2250. Chakraverty, Phys. Rev. B 60, 564 (1999).
22 J. P. Vyasanakere and V. B. Shenoy, (2011), 44 N. Dupuis, Phys. Rev. B 70, 134502 (2004).
arXiv:1108.4872. 45 R.B.Diener, R.Sensarma, andM.Randeria, Phys.Rev.
23 L. Han and C. A. R. S. de Melo, ArXiv e-prints (2011), A 77, 023626 (2008).
arXiv:1106.3613 [cond-mat.quant-gas]. 46 S. K. Ghosh, J. P. Vyasanakere, and V. B. Shenoy, Phys.
24 S. Takei, C.-H. Lin, B. M. Anderson, and V. Galitski, Rev. A 84, 053629 (2011).
ArXiv e-prints (2011), arXiv:1111.2340 [cond-mat.quant- 47 B.M.Anderson,G.Juzeliu¯nas,I.B.Spielman, andV.M.
gas]. Galitski, ArXiv e-prints (2011), arXiv:1112.6022 [cond-
25 N. Goldman, A. Kubasiak, A. Bermudez, P. Gaspard, mat.quant-gas].
M. Lewenstein, and M. A. Martin-Delgado, Phys. Rev. 48 X. Cui, ArXiv e-prints (2011), arXiv:1112.1122 [cond-
Lett. 103, 035301 (2009). mat.quant-gas].
49 P. W. Anderson, Phys. Rev. 112, 1900 (1958).
9
50 H. Hu, X.-J. Liu, and P. D. Drummond, Europhys. Lett.
74, 574 (2006).