Exact Solution for 1D Spin-Polarized Fermions with Resonant Interactions

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by  Adilet Imambekov

Exact Solution for 1D Spin-Polarized Fermions with Resonant Interactions
Adilet Imambekov,1 Alexander A. Lukyanov,2 Leonid I. Glazman,3 and Vladimir Gritsev4
1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA
2Abingdon Technology Centre, Schlumberger, Abingdon, OX14 1UJ, UK
3Department of Physics, Yale University, New Haven, Connecticut 06520, USA
4Physics Department, University of Fribourg, Chemin du Musee 3, 1700 Fribourg, Switzerland
(Dated: January 16, 2010)
0
Using the asymptotic Bethe Ansatz, we obtain an exact solution of the many-body problem for
1
1D spin-polarized fermions with resonant p-wave interactions, taking into account the effects of
0
both scattering volume and effective range. Under typical experimental conditions, accounting for
2
the effective range, the properties of the system are significantly modified due to the existence of

n “shape”resonances. Theexcitationspectrumoftheconsideredmodelhasunexpectedfeatures,such
a as the inverted position of the particle- and hole-like branches at small momenta, and roton-like
J minima. We find that the frequency of the “breathing” mode in the harmonic trap provides an

6 unambiguoussignature of theeffective range.
1
PACSnumbers: 03.75.-b,05.30.Fk

]
s
a
g Experimental progress in the cooling and trapping of betakenintoaccount. Weprovideanexactsolutionthat
- ultracold atomic gases makes it possible to investigate accounts for both scattering volume and effective range,
t
n their properties under strong transverse confinement, andobtain significantdeviations fromthe CS model [16]
a when the motion of atoms is effectively one-dimensional due to “shape” resonance in the p-wave scattering. We
u
(1D). In such experiments the 1D interaction parame- find severalnew effects, suchas the inversionofparticle-
q
ters are precisely known and can be tuned using Fesh- and hole-like spectra for low momenta, roton-like min-
.
at bach resonances [1] or by varying the harmonic trans- ima inthe excitationspectrum, andwe calculatedensity
m verse confinement strength. Recent experiments [2, 3] profiles and “breathing” modes in the harmonic trap.
have allowed for parameter-free comparison of 1D Bose We use the asymptotic Bethe Ansatz (BA) [18] which
-
d gaspropertieswithatheoreticaldescriptionbasedonthe is justified at sufficiently small densities, when only two-
n
exactly solvable Lieb-Liniger (LL) model [4]. The possi- particle collisions are important [19]. The underlying
o
bilitytocompareexperimentalresultswiththeoutcomes idea goes back to the earlier days of high energy physics
c
[ ofmany-bodycalculationsrevivedaninterestinthefield and was known as S-matrix theory [20] in the 1950s.

of exactly solvable 1D systems: spin-1/2 fermions [5], The BA method considers the scattering matrix be-
2
Bose-Fermi mixtures [6, 7], and spinor bosons [8] and tween asymptotic states as an alternative to a Hamil-
v
9 fermions [9]. tonian/Lagrangeandescription. The scattering of ultra-
6 cold atoms in 1D gases close to resonance can naturally
2 In this Letter, we obtain an exact solution for 1D be described by the scattering phase shift, whereas the
2 spin-polarized fermions under resonant scattering condi-
formulationofamicroscopicquantumHamiltonianisdif-
0. tions [10], which is relevant for 40K and 6Li atoms near ficult. It can be shown that the scattering matrix close
1 p-wave Feshbach resonances [11, 12]. Such a 1D experi- to resonance corresponds to a highly singular, although
9 mentalsystemhasbeenrealizedfor40K[13]. Therelated
local, two-body interaction in the spirit of Refs. [16, 21].
0
3D problem has also received significant theoretical at-
To avoid difficulties related to the determination of the
:
v tention recently [14]. For spin-polarized fermions only
operators and states in this case, we use an approach
i scatteringinoddpartialwavechannelsispresent,andat
X based entirely on the scattering phase shift.
low energies, p-wave scattering is the strongest. If only
r Let us start by briefly reviewing the 3D scattering
a the ”scattering volume” is taken into account and the properties in a p-wave channel. At low energies, the
”effective range” of p-wave scattering is neglected (see
phase shift δ (k) can be expanded as [10, 22]
p
Eq. (1) for definitions), then the projection to 1D [15]
results in a fermionic Cheon-Shigehara (CS) model [16], k3cotδp(k)= 1/w1 α1k2+O(k4), (1)
− −
whichisdualtothebosonicLLmodel. Forsuchamodel,
where w is the scattering volume, α is the effective
strongly interacting fermions with resonant interactions 1 1
range,andkistherelativemomentum. Forα w <0,the
are mapped to weakly interacting bosons, the so-called 1 1
scattering length obtained from δ (k) has a very sharp
fermionic Tonks-Girardeau(fTG) limit [17]. However, it p
shape resonance at the wave vector [22, 23]
wasshownbyL.Pricoupenko[10]thatunlikethe caseof
the strongly interacting bosonic TG limit [2, 3], the re-
k =1/√ α w . (2)
r 1 1
quirementsforthe observationofthe fTGlimitarequite −
stringent, and the effective range of scattering needs to Suchresonanceisabsentfors-wavescattering,butforthe
2
p-wave channel it exists due to the presence of the effec- thermodynamiclimit,wewillnotconsiderthecasewhere
tive range parameter. The higher order terms in Eq. (1) l > 0. For l < 0 we construct an exact wavefunction
p p
donotsignificantlyaffecttheshaperesonance,sincethey ψ (z ,...,z ) as a combination of plane waves, using
+ 1 M
aresuppressedbypowersofthesmallparameterkR 1, the BAmethod in a similar manner to the LL model. In
≪
where R is the characteristic radius of the 3D potential. ourcase,suchconstructionleadstothefollowingperiodic
For fermions, the typical momenta of scattering parti- boundary conditions on a circle of length L
cles are of the order of the Fermi momentum k = πn.
F M ξ (λ λ )2 1 +i(λ λ )
Therefore,theconditionkR≪1necessaryforneglecting eiλjL = p j − k − |lp| j − k , (7)
three-particle collisions [19] implies the low-density limit ξ (λ λ )2 1 i(λ λ )
nR 1. It is known [10, 22, 23] that α > 1/R > 0 kY=1 p j − k − |lp| − j − k
1
does≪notchangesignificantlyatthep-waveFe∼shbachres- and the total energy is given in terms of quasimomenta
onance, while w1 can be tuned to very large absolute λi as E = h¯2/(2m) λ2i. We prove that all solutions of
values compared to its characteristic values of the order Eq. (7) are real by wPriting the k-th term in the product
ofU|wn1d|e∼r Rtr3aanwsvaeyrsferomhatrhmeorneisconcaonncfien.ement with fre- as ((λλjj−−λλkk−−λλ∗++))((λλjj−−λλkk−−λλ∗−−)). Since Imλ± < 0 for ξp > 0
and l < 0, we then have (λ−λ+)(λ−λ−) 1( 1) for
quency ω⊥, only the lowest transverse mode is occupied p (cid:12)(λ−λ∗+)(λ−λ∗−)(cid:12) ≤ ≥
if the momenta of scattering fermions satisfies Imλ 0( 0). After that,(cid:12)the proof sim(cid:12)ply follows the
≤ ≥ (cid:12) (cid:12)
steps for the LL model described on p.11 of Ref. [26].
ka⊥ 1, (3) Toobtainathermodynamiclimit,wetakealogarithm
≪
of Eq. (7), which is written as Lλ + M θ(λ λ )=
where a⊥ = ¯h/(mω⊥), and m is the atomic mass. Un- j k=1 j− k
der such conpditions, the 1D scattering amplitude in an 2πnj, where nj are integer quantumPnumbers for odd
M. The phase shift θ(λ) is a monotonic antisymmetric
odd channel is given by [10]
function defined by
ik
fpodd = 1/lp+−ik+k2ξp, (4) θ(λ)=2Arg(iλ−ξpλ2+1/|lp|), (8)
and belongs to the interval ( 2π,2π), unlike the LL
where [24]
−
phase shift, which belongs to the interval ( π,π). We
a3 −1 α a2 thendirectlyfollowRef.[27]andshowthatre−alsolutions
l =3a ⊥ 3√2ζ( 1/2) , ξ = 1 ⊥ >0,(5)
p ⊥(cid:20)w − − (cid:21) p 3 oftheBAequationsexistforanychoiceofquantumnum-
1
bersn .Theirvaluesforthegroundstatecanbefixedby
j
and 3√2ζ( 1/2) 0.88. The notation adopted is that comparisonwith the LL model [4, 26, 27], and are given
− ≈−
ofRef.[19]. Usingestimates[10,25]ofα for6Liand40K by n =j (M +1)/2. Introducing a positive function
1 j
−
atoms at resonances with B 215G and 198.6G, and
0
3w0itkhHtzr[a1n3s]v,ewreseofbrteaqiuneξnpc/ieas⊥ω≈⊥≈5=02aπnd×≈20103k,Hrezspanecdti2vπel×y. K(λ,µ)=θ′(λ,µ)= [1−2|l|plp|ξ|p(cid:2)(1λ+−|lµp)|ξ2p]2(λ+−lp2µ(λ)2−(cid:3) µ)2,
Thus, under typical experimental conditions needed to
achievethe1Dregimeξ a andhenceallthreeterms we pass to the thermodynamic limit, and write an equa-
p ⊥
≫ tion for the ground state quasimomenta distribution in
are significant in the denominator of Eq. (4).
q
the usual way 2πρ(ν) K(ν,µ)ρ(µ)dµ = 1, where
The many-body fermionic wave function − −q
ψ(z ,...,z ) is anti-symmetric,ψ(...,z ,...,z ,...) = q is the highest(lowest)Rfilledquasimomentumandthe
1 M i j ± q
ψ(...,z ,...,z ,...),anddiscontinuouswhentwocoor- normalizationisgivenbyn=M/L= ρ(ν)dν. Apart
− j i −q
dinates coincide [16]. We define its symmetrized version from new definitions of θ(λ) and K(λR,µ), the structure
by ψ (z ,...,z ) = Sign(z z )ψ(z ,...,z ), of the theory is similar to the LL model, and we can
+ 1 M i<j i − j 1 M
which is continuous. TQhen Eq. (4) implies the following study the ground state energy, excitation spectra and fi-
boundary condition: nitetemperaturepropertiesusingstandardmethods[26].
We choose two dimensionless parameters that deter-
1
lim +∂ ξ ∂2 ψ (z ,...,z )=0. (6) mine the ground state properties in the stable region
z=zj−zi→0+(cid:18)lp z − p z(cid:19) + 1 M
1 1
γ = >0, γ = >0. (9)
Solving the two-body problemas ψ+(z1,z2)∝eiλ|z1−z2|, 1 −lpn 2 ξpn
we obtain two roots λ = i 1 4ξ /l /(2ξ ).
For l > 0, Imλ > ±0, wh(cid:16)i−ch±coprre−spo−ndsptop(cid:17)a boupnd In Fig. 1 we show the dimensionless groundstate energy
p + functional e(γ ,γ ) obtained by numerically solving the
1 2
state. The lowest energy state satisfying the boundary
equations for the ground state. The dimensionless form
condition(6)canthenbeconstructedasψ(z ,...,z )
1 M is given by the expression
∝
Sign(z z ) exp(iλ z z ). As in the at-
i<j i − j i<j +| i − j|
tQractive LL model,Qits energy does not have a proper E/L=e(γ1,γ2)(¯hn)2/(2m), (10)
3
weak interactions n/n0 ((cid:4)/(cid:4)z)2
6
CS model
e((cid:112)1,(cid:112)2)
5 fTG limit
2 4
fTG
3
1
2
0
1(cid:112)02 (cid:112)101 1 (cid:99)w1/a3(cid:34)
5 5 0.2 0.4 0.6 0.8 1.0 R/R0
super-fTG FIG. 2: (Color online) Density profiles under harmonic con-
00 finement, measured in units of Thomas-Fermi radius R0 and
FIG. 1: (Color online) Energy functional e(γ1,γ2) as a func- central densityn0 for thesame cloud in theabsenceof inter-
tionofdimensionlessparametersγ1 andγ2,seeEqs.(9)-(10). actions. Different curves correspond to {γ1(0),γ2(0)} in the
centerofthecloud 7.5,0.023 (blue,dotted), 97,0.058 (red,
Arrows indicate the regimes of weakly interacting fermions,
{ } { }
dashed), and free fermions (black, thick). For a cloud of
Cheon-Shigehara(CS)model[16],fermionicTonks-Girardeau
(fTG) gas [17], and super-fTG regime (see Eq. (11)) existing N = 100 6Li atoms with ω⊥ = 2π 200 kHz and ωz =
×
dueto thefinite effectiverange ξp. t2oπ×w210/0aH3⊥z=,w0e.0g5e,t0R.001,=0,4r1eµsmpecatnivdetlyh.esTehceurinvesestcoshrroewspsotnhde
−
and reduces to the LL functional e(γ ) for γ 1, since ratio of squares of the“breathing” and dipole mode frequen-
the CS model [16] obtained in this l1imit is2d≫ual to the cies (ω/ωz)2 for the same trap parameters. If a significant
part of the cloud is in the super-fTG regime (see text), this
LL model. The function e(γ ,γ ) equals 0 if γ = 0 or
1 2 1 ratio drops below the critical value 3 obtained in the fTG
γ2 =0. Expansion by methods of Ref. [28] at γ1,γ2 ≫1 limit. A critical value w1/a3⊥ 0.008 corresponds to a de-
yields e(γ1,γ2)≈e(γ1)−32π4/(15γ12γ2). tuningfrom the resona−nce ∆B≈≈50mG.
We use the function e(γ ,γ ) obtained numerically to
1 2
evaluate density profiles in a harmonic trap within the
k . In a similar manner to the super-TG gas of bosons,
r
local density approximation [29]. We also use the func-
the super-fTG regime can be experimentally identified
tion e(γ ,γ ) to find the “breathing” mode frequency ω
1 2 by measuring the ratio of the squares of the “breath-
by solving the hydrodynamic equations [6, 30]. The re-
ing” and dipole mode frequencies. In the CS model
sults are shown in Fig. 2 and depend on two dimension-
such a ratio is always larger than 3, similar to the LL
less parameters, γ (0) and γ (0), in the center of the
1 2 model [30], while the inset of Fig. 2 shows the regime
cloud. The presence of the effective range strongly af-
where it is smaller than 3. A sharpdecrease in this ratio
fects the shape of the profile compared to the CS model
for the super-fTG regime can be easily understood from
if γ (0) is small and γ (0) is not too large. This effect
2 1 the“sumrule”approachofRef.[30],sincethecloudden-
canbeunderstoodbyusingtheexpansionofe(γ ,γ )for
1 2 sityismuchmorecenteredinthesuper-fTGregimethan
γ ,γ 1. The leading term in the Taylor expansion
1 2 ≪ in the fTG regime. We can analytically estimate the
gives e(γ ,γ ) γ γ 1/n2 and hence via Eq.(10) the
energype1rp2art∝icle1fo2r∝thistermdoesnotdependonden- value of −w1/a3⊥ at which the center of the cloud enters
the super-fTG regime, and the drop in (ω/ω )2 occurs.
z
sity, i.e. the gas has a divergent compressibility. Higher
Forthat weuse Eq.(11)with the free fermiondensity in
order terms in the expansion lead to a finite but large
the center and obtain w1 = ω⊥/ωz , which
compressibility,whichdecreaseswithincreasingγ1 anda −a3⊥(cid:12)super−fTG 24Nξp/a⊥
constantratio γ /γ . Thus, the density profile exhibits a gives 0.008 for the paramet(cid:12)ers seen in Fig. 2.
2 1 (cid:12)
strong peak near the center where γ is smallest.
1 Theexcitationspectrumε(k)inauniformcloudisalso
Sincefermionsaremorestronglycorrelatedforγ 1
2 significantlymodifiedinthesuper-fTGregimecompared
≪
andnottoolargeγ comparedtothefTGregime,wesug-
1 to predictions of the CS model. In Fig. 3 we illustrate
gest calling such a regime a super-fTG gas, analogously
several qualitative features, which appear due to the fi-
to the super-TG gas of bosons [31]. It is realized if
nite effective range of interactions. Firstly, the system
γ γ hasa regimewhere the energyofthe particle-likeexcita-
Max(γ2, 41π22)<1. (11) tionissmallerthantheenergyofthehole-likeexcitation.
∼
Sincetheenergyoftheparticle-likeexcitationshouldap-
For w a3, the second condition corresponds to proach k2/(2m) at high momenta, there should also be
k <| 21π|n≪= 2⊥k . Thus, the super-fTG regime is real- an energy crossing. This crossing will manifest itself as
r F
ized∼if the largest relative momentum of non-interacting a kink in the k dependence of the lowest energy of the
−
fermions approaches the shape resonance wave vector densitywaveexcitations. Secondly,thespectrumofhole-
4
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