Possible quantum phase-manipulation of a two-leg ladder in mixed-dimensional fermionic cold atoms

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by  Wen-Min Huang

Possible quantum phase-manipulation of a two-leg ladder in mixed-dimensional
fermionic cold atoms
Wen-Min Huang, Kyle Irwin and Shan-Wen Tsai
Department of Physics and Astronomy, University of California, Riverside, CA 92521, USA.
(Dated: January 24, 2013)
The recent realization of mixed-dimensional systems of cold atoms has attracted much attention
frombothexperimentalistsandtheorists. Differenteffectiveinteractionsandnovelcorrelatedquan-
3 tummany-bodyphasesmaybeengineeredinthesesystems,withthedifferentphasesbeingtunable
1 viaexternalparameters. Inthisarticleweinvestigateatwo-speciesFermiatommixture: onespecies
0
of atom exists in two hyperfinestates and is confined to move in a two-leg ladder, interacting with
2
an on-site interaction, and theother movesfreely in a two dimensional square lattice that contains

n thetwo-legladder. Thetwospeciesofatomsinteractviaanon-siteinteractionontheladder. Inthe
a limit of weak inter-species interactions, the two-dimensional gas can be integrated out, leading to
J an effectivelong-range mediated interaction in theladder, generated by to the on-site inter-species

3 interaction. Weshow that theform of themediated interaction can becontrolled bythedensity of
2 thetwo-dimensionalgasandthatitenhancesthechargedensitywaveinstabilityinthetwo-leglad-
deraftertherenormalizationgrouptransformation. Parameterizingthephasediagramwithvarious

] experimentally controllable quantities, we discuss the possible tuning of the macroscopic quantum
s
many-bodyphases of the two-leg ladder in thismixed-dimensional fermionic cold atom system.
a
g
- PACSnumbers: 67.85.Lm,64.60.ae,71.10.Fd,71.10.Hf
t
n
a
Overthepastyears,Fermimixturesinultracoldatoms
u
q have been achieved experimentally[1–4], and have trig-
.
. gered intense studies by experimentalists and theorists ..
t .
a alike. A multitude of exotic phenomena, which nor- 3
m mallydonotoccurincondensedmattersystems,cannow
- be fabricated with exceptional tunable parameters[5– 2
d
11]. For instance, fermion mixtures can be manipu-
n =1
lated to carry different internal properties such as pop-
o
ulation densities, group symmetries, lattice geometries,
c
.
[ and dimensionalities. By cooling two fermionic iso- ..
.
topes of ytterbium with different nuclear spins, a de-
1
generateFermimixturewithanSU(2)×SU(6)symmetry
v
has recently been experimentally achieved[12]. Mixed
5 FIG. 1: (Color online)A two-leg ladder embedded in a 2D
6 two-species Fermi gases with unequal populations have squarelattice and the coordinate system are illustrated.
3 also been extensively studied[13, 14], as well as systems
5 of dipolar atoms and molecules[15–17]. Many-body ef-
1. fects in these systems can be further enhanced when
0 the system is loaded onto an optical lattice[18] or via
the higher dimensional species. For example, the phase
3 Feshbach resonance[19–21]. Fermions with imbalanced
diagram of a single chain is strongly modified by being
1 populations[22] and dipolar fermions[23, 24] on a square
: embedded in a two-dimensional lattice[32].
v lattice have been investigated and present a much richer
i phase diagram than their gas counterparts, with lattice Inspired by these experiments, we study a mixed-
X effects enhancing the transition temperature for various dimensional two-species fermionic system: one species
r phases. confined in a two-leg ladder with on-site repulsion, the
a
other moving freely in a two-dimensional (2D) square
Experimentally, it is also possible to confine com- lattice. An inter-species interaction is also introduced
ponents of a ultracold atom mixture in different as on-site due to the energy cost of double occupation.
dimensions[25,26], andthis hastriggedsome recentthe- Integrating out the 2D fermionic gas, a mediated long-
oretical studies[27–29]. As pointed out by Nishida[30], range interaction is generated in the ladder. We show
the mediated intra-species interaction generated by the that the mediated interaction moves the bare on-site in-
interaction between species moving in different dimen- teraction in the two-leg ladder off the symmetric point,
sions may reshape the phase diagram in a bilayer Fermi andenhancesthechargedensitywave(CDW)instability
gas. Furthermore, Efimov physics with fermions is inthe renormalizationgroup(RG) transformation. Here
alsoproposedto emerge in one/three-dimensionalmixed we use the term charge-density-wave to refer to density
systems[31]. The properties of the lower-dimensional modulations in analogy with nomenclature used in the
Fermi gas can be manipulated by tuning parameters in study of electronic systems, even though the atoms are
2
charge neutral. We find that, by controlling the filling effects,theGrassmannnumberϕisdecomposedintochi-
in the 2D gas, the phase of the ladder can be tuned. ral pairs[33–36, 38, 39], ϕy(x)≈ P,iTyiψPn(x)eiPkFix,
By mapping out the phase diagram for various parame- where P = R/L = +/− reprePsent right/left-moving
ters, we show the possible quantum phase-manipulation particles, and k is the Fermi wavelength of the band
Fi
of a two-leg ladder in mixed dimensional fermionic cold index i. The transformation matrix between leg-index
atoms. y and band-index i in the N-ladder is introduced as
A schematic of the system we consider here is illus- T = 2 sin π yi [35]. The interactions between
trated in Fig. 1. The action can be written as S = yi qN+1 hN+1 i
S + S0 + S , where S stands for the action for the these chiral fermions can be categorized as Cooper scat-
l c cl l terings cl , cs and forward scatterings fl , fs, where
two-leg ladder with on-site interaction Ul. The action of ij ij ij ij
we set f = 0 since f = c [36]. Thus, the mediated
the non-interacting 2D system is denoted as ii ii ii
interactions in a ladder system can be written as
Sc0 =Z dτhXr,r′iφ¯α(r,τ)[∂τδr,r′ +Hc(r,r′)]φα(r′,τ).(1) Smed ≃ UUc2l Z dτZ dx
l
where φ¯α,φα are Grassmann fields with (pseudo)spin in- fisjψ¯RiαψRiαψ¯LjβψLjβ +filjψ¯RiαψLjαψ¯LjβψRiβ
h
dexα,andtheHamiltonianisrepresentedasH (r,r′)=
[tc(r,r′)−µ2Dδr,r′]withchemicalpotentialµ2cD. More- +csijψ¯RiαψRjαψ¯LiβψLjβ +clijψ¯RiαψLjαψ¯LiβψRjβi,(4)
over,theuniformhoppingamplitudet (r,r′)=−1when
c
r and r′ represent nearest neighbor sites. The energy where the bare values of the couplings are expressed as
cost of for overlap of two atoms of different species can
fs =Umed(0), fl =Umed(k +k ),
be regarded as the on-site inter-species repulsion with ij iijj ij ijji Fi Fj
strength Ucl and represented as csij =Uimjiejd(kFi −kFj), clij =Uimjiejd(kFi +kFj), (5)
with the definition of dimensionless mediated interac-
S = dτ U n (r,τ)n (r,τ)δ , (2)
cl Z cl c l y,a tions, Umed(k) = U π dq Γ (q)χ (k,q). Here the
Xa Xr ijkl l −π 2π ijkl 0
particle-hole propagatRor is defined as
where n = φ¯ (r)φ (r) and n = ϕ¯ (x)ϕ (x)
c α α α l α yα yα
are the densPities for the 2D lattice anPd the ladder, re- dp nF ǫ(p) −nF ǫ(px+k,py+q)
spectively. The summation over a = 1,2,···,N stands χ0(k,q)=Z 4π2 ǫ(hp)−iǫ(p +hk,p +q)+i0+i, (6)
x y
forthepositionofN-legsalongthey-direction,asshown
as Fig. 1. Here we focus on the N =2 case. comingfromtheconvolutionofthetwomomentum-space
Considering the limit of weak inter-species interac- Green’s functions in Eq. (3), where ǫ(p) = −2(cosp +
x
tions, we expand the action in powers of Ucl and inte- cospy) − µ2D is the dispersion of the 2D gas and nF
grate out the non-interacting 2D gas, neglecting terms is the Fermi-Dirac distribution. Furthermore, the ex-
O(U3). The first term in the expansion gives a correc- tra kernel in the mediated interactions is defined as
cl
tiontothechemicalpotentialoftheladdersystem. Since Γ (q) = eiq(b−a)T∗T T∗T , summing over the
ijkl a,b ai aj bk bl
thisonlyslightlyshiftsthephaseboundaries,weignoreit leg indices Pa,b = 1,2,···,N. It is worthwhile to notice
here. The secondtermgeneratesa mediatedinteraction, thatthe mediatedinteractionsarethe Ruderman-Kittel-
modifying the profile of the total effective interaction in Kasuya-Yoshida (RKKY) type[37]. However, the exact
the two-legladder. The actionfor the laddercannowbe profile in real space can not be computed analytically
written as Seff =Sl+Smed, where since we consider a lattice model.
Intheabsenceofthemediatedinteraction,allbareval-
U2
S ≃ dτ dτ cl δ δ tr n (r ,τ ) ues of the couplings in RG equations would be the same
med Z 1Z 2 2 X y1,a y2,b h l 1 1 sinceonlyon-siteinteractionsareconsidered. Inthepres-
a,b
ence of the mediated interactions, this symmetry is bro-
G0(r1,τ1;r2,τ2)nl(r2,τ2)G0(r2,τ2;r1,τ1) , (3) kenandtheinitialcouplingsrenormalizeinverydifferent
i
ways, as shown as Eq. (5). However, since the mediated
with G0(r,τ;r′,τ′) = [∂τδr,r′ −Hc(r,r′)]−1 the interactions are determined only by the exchanged mo-
imaginary-time Green’s function. This shows that a menta in spin-conserving scattering processes, couplings
mediated interaction between particle n (r ,τ ) and sharing the same exchanged momenta remain the same
l 1 1
n (r ,τ )withretardationeffectsisgeneratedinthelad- bare values. For instance, zero momenta is exchangedin
l 2 2
der system. Retardation effects can be neglected when both csii and fisj and they therefore have the same bare
the Fermi velocity of the 2D fermions is large compared value, similarly,for cl and fl with exchangedmomenta
ij ij
with the one for the ladder. (k +k ).
Fi Fj
We now study the effects of the effective interac- The RG equations of a N-leg ladder can be found in
tion on the ladder system and determine its phase di- the literature[40], and are only solvednumerically. After
agram using a RG technique. By ignoring retardation taking the effective interactions as the initial condition
3
Umed a Ucl b Umed a Ucl b
Ul s-0S.55C Uls-SC
-0.1 2.0 -0.05 2.0 CDW
1.85 CDW 1.85
-0.15 1.5 -0.1 1.5
π d-SC d-SC
-0.2 1.0 2 d-SC -0.15 1.0 2
-π π
-0.250.5-π-π0.6 0.π7n0.8 0.9 1.0 0.50.5 0.6 0.7n 0.8 0.9 1.0 -0.20 0.5 1.0 1.5 2.0 0.50.0 0.5 1.0 1.5 2.0
μ μ
-0U.06med c UUcll CDW d Umed 2D c UUcl 2D d
2.0 l CDW
-0.05 2.0
-0.10 1.5 d-SC -0.1 1.5
d-SC
π 2 d-SC
1.0 -0.15
1.0
-0.14-π-π π 0.5 -0.2 -π π
0.5 0.6 0.7n0.8 0.9 1.0 0.5 0.6 0.7n 0.8 0.9 1.0 0 0.5 1.0 1.5 2.0 0.50.0 0.5 1.0 1.5 2.0
μ μ
2D 2D
FIG.2: (Coloronline)Interactionsmediatedbya2Dgaswith
FIG. 3: (Color online)The mediated interactions from a 2D
(a) µ2D =0 and (c) µ2D =1 versus density n of the ladder:
The blue solid line represents cs11, cs22 and f1s2. cl12 and f1l2 gpalosttweidthasdefunnsicttieiosnno=f th0.e55chaenmdicnal=po0t.e9n5tioanl µth2De loafddtherea2rDe
are denoted as the green dot-dash-dotted line. The red dot-
cds1a2shaendd, col2r2anrgesepdecottitveedly,.anTdhebrcoowrrnesdpaosnhdeidnglipnhesasdeednioatgeracml11s, s(tych)se,tersmeasmp(eeincatuisvneiilntys.FoTifgh.teh2es.t2roDCkoehrsorfeposprpinocnogudapinmlignpglpisthuiandsee(atdc)i)aaignnrda(m(ac)s)aaanrrdee
for (a) and (c) are illustrated in (b) and (d),respectively.
shown in (b) and (d).
and integrating the differential RG equations, the flows
of the couplings can be obtained. By analyzing these standard Hubbard model[35] is destroyed by the medi-
flowswiththe scalingAnsatz, g ∼1/(l −l)γi,wherel ated interactions. We find a three-phase “triple-point”
i d d
isthedivergentlengthscaleinone-loopRG,thehierarchy (CDW, d-SC and s-SC) at Ucl/Ul = 1.85 and n = 0.55.
of the relevant couplings can be directly read out from From there, increasing the Ucl/Ul ratio gives cl11(0) (or
theRGexponentγi[41,42]. CombiningwiththeAbelian cσ11(0)) negative, and an s-SC phase occurs, as expected
bosonizationmethod, the phase diagramof a laddersys- by BCS theory. Doping away from the “triple-point”,
tem is determined[35, 38, 39]. Furthermore, the rela- f1σ2(0) becomes negative and CDW dominates. Roughly
tive values of the chargeand spin gaps between different speaking, when Ucl/Ul > 1.85, the phase will be deter-
FermipointscanbedeterminedbytheRGexponents[41]. minedbythecompetitionbetweenthe negativevaluesof
This allowsus to distinguishthe d-wavesuperconductiv- f1σ2(0) and cσ11(0).
ity with an anisotopic spin-gap (referred as d-SC2 here) It is important to point out that the CDW phase
in a two-leg ladder with heavy doping, n<0.6. We note emerges due to enhanced mediated interactions fl near
12
that there is not a real phase transition between the d- half-filling. When n≃1,the momentum exchangeddur-
wave superconductivity with isotropic spin gaps (d-SC) ing fl processesapproachesπ. The particle-hole propa-
12
and d-SC2, because the number of spin and charge gaps gatorinthemediatedinteractionshowsamaximumcon-
andrelativesignofrelevantcouplingsarethesame. How- tribution at µ =0, resulting from maximum electron-
2D
ever,itisusefultoemphasizethisdifferenceherebecause hole pairing [from cosp and cos(p +π) = −cosp ] in
x x x
theanisotropicspingapscanmeasured,andthereforethe the 2D dispersion along the x-direction. Moving away
two phases can be distinguished experimentally. from n = 1, the mediated interaction of fl decreases,
12
To illustrate how the mediated interaction depends at which point the CDW phase ceases to occur. When
on the filling of the 2D fermions, we show results for we increase the 2D chemical potential, the same behav-
µ =0, when the 2D Fermi surface is nested (Fig. 2a), ior occurs. As shown in Fig. 2 (d), the regime of the
2D
and for µ = 1 (Fig. 2c). The corresponding phase di- CDW phase shrinks at µ =1 resulting fromweak me-
2D 2D
agrams are shown in Fig. 2b and Fig. 2d. In Fig. 2b, diated interactions. In this situation the mediated inter-
when the strength of the coupling U /U is larger than actions vary smoothly, and show little effect in the lad-
cl l
1.2, a CDW starts to emerge from d-SC near half-filling dersystem. Therefore,themediatedinteractiondepends
(n ≃ 1). Increasing U /U brings the phase boundary stronglyonthe chemicalpotentialinthe 2Dsystem,and
cl l
to higher hole-doping. It is interesting to note that the consequently, the phase diagram of a two-leg ladder can
Luttinger liquid phase that appears near n>∼0.5 for the bemodified/tunedviathefillinginthe2Dsquarelattice.
4
Tofurtherillustratetheinfluenceofthe2Ddensity,we neighbor repulsion exceeds the original on-site repulsive
showthe mediatedinteractionsandcorrespondingphase interaction. However, increasing the U /U ratio makes
cl l
diagramsparameterizedby the chemical potentialin the the magnitude of the effective on-site attraction larger
2Dsystemin Fig.3 (a)(b)and(c)(d) for fixeddensity in than the original on-site repulsion. When the average
the ladder, n = 0.55 and n = 0.95, respectively. When on-site attraction is weak, CDW and s-SC phases com-
U /U is larger than 1.2 at n = 0.95, the phase tran- pete. Nevertheless, the s-SC phase is dominant for large
cl l
sition between CDW and d-SC happens near µ = 0. U /U, andthe inter-species interactioncanbe regarded
2D cl l
However, the phase boundary moves to higher values of as a glue to pair the fermions, similar to the electron-
U /U when the chemical potential µ increases and phonon interaction in a conventionalsuperconductor.
cl l 2D
the mediated interaction decreases. In general, the ten- In conclusion, we studied a two-species Fermi mixture
dencyofthemediatedinteractionistodecreaseuponthe composedofaspeciesthatmovesinatwo-legladderwith
increasing of the filling in the 2D system (or µ ). This on-site interaction,and another,non-interacting,species
2D
is due to the non-interacting assumption used in the 2D moving on a 2D square lattice. By integrating the 2D
gas. Under this condition, the strength of the mediated Fermi gas in the limit of weak inter-species coupling,
interactions is roughly proportional to the density near a long-range mediated interaction is generated, and the
the Fermi surface, which is maximum at µ =0. CDWinstability inthe ladderis enhanced. We showthe
2D
Although the pattern of the mediated interactions in phase diagram parameterized by the filling in both the
real space is rather complicated, it can be approximated two-leg ladder and the 2D square lattice.
as an effective on-site attraction and a nearest-neighbor Acknowledgments - KI and SWT gratefully acknowl-
repulsion, since longer range terms are rapidly decaying. edge support from NSF under Grant DMR-0847801and
The phase diagram can then be understood from these from the UC-Lab FRP under Award number 09-LR-05-
two effective interactions. By tuning the chemical po- 118602. Meanwhile, WMH sincerely acknowledges sup-
tential in 2D, CDW emerges when the effective nearest- port from NSC Taiwan under Grant 101-2917-I-564-074.
[1] M. W. Zwierlein et al., Science 311, 492 (2006). Dipolar Fermi Gases”, arXiv:1209.2671.
[2] G. Partridge et al., Science 311, 503 (2006). [25] G. Lamporesi et. al., Phys. Rev. Lett. 104, 153202
[3] M.W.Zwierlein etal., Nature(London)442,54(2006). (2010).
[4] Y.Shin et al., Phys.Rev.Lett. 97, 030401 (2006). [26] E. Haller et. al., Phys. Rev.Lett. 104, 153203 (2010).
[5] D.Jaksch and P.Zoller, Ann.of Phys.315, 52 (2005). [27] Y. Nishida and S. Tan, Phys. Rev. Lett. 101, 170401
[6] I.Bloch, NaturePhys. 1, 23 (2005) (2008).
[7] M.Taglieberet.al.,Phys.Rev.Lett.100,010401(2008) [28] Y. Nishida and S. Tan, Phys.Rev.A 82, 062713 (2010)
[8] E. Wille et.al., Phys. Rev.Lett. 100, 053201 (2008) [29] M. Iskin and A. L. Subasi, Phys. Rev. A 82, 063628
[9] A.-C. Voigt et. al., Phys. Rev.Lett. 102, 020405 (2009) (2010).
[10] F.M.Spiegelhalderet.al.,Phys.Rev.Lett.103,223203 [30] Y. Nishida, Phys. Rev.A 82, 011605(R) (2010).
(2009). [31] Y. Nishida and S. Tan, Phys. Rev. A 79, 060701(R)
[11] T. G. Tiecke et.al., Phys. Rev.Lett. 104, 053202 (2010) (2009).
[12] S.Taie et. al., Phys.Rev.Lett. 105, 190401 (2010) [32] K. Irwin and S.-W. Tsai (unpublished).
[13] G. Partridge et al., Phys. Rev.Lett. 97, 190407 (2006). [33] L. Balents and M. P. A. Fisher, Phys. Rev. B 53, 12133
[14] Y.Shin et al., Nature451, 689 (2008). (1996).
[15] M. Lu, N. Q. Burdick, and B. L. Lev, Phys. Rev. Lett. [34] H.-H.Lin,L.Balents,andM.P.A.Fisher,Phys.Rev.B
108, 215301 (2012). 56, 6569 (1997).
[16] K.-K.Ni, et al.,Science 322, 231 (2008). [35] H.-H.Lin,L.Balents,andM.P.A.Fisher,Phys.Rev.B
[17] C.-H. Wu, et. al, “Ultracold Fermionic Feshbach 58, 1794 (1998).
Molecules of 23Na40K”, arXiv:1206.5023. [36] M.-H. Chang, W. Chen, and H.-H. Lin, Prog. Theor.
[18] I.Bloch, J. Dalibard, and W. Zwerger, Rev.Mod. Phys. Phys. Suppl.160, 79 (2005).
80, 885 (2008). [37] See e.g. N. W.Ashcroft and N. D.Mermin, “Solid State
[19] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Physics”, SaundersCollege Publishing (1976).
Mod. Phys. 82, 1225 (2010). [38] J. E. Bundler and H.-H. Lin, Phys. Rev. B 78, 035401
[20] E.Timmermans,P.Tommasini,M.Hussein,andA.Ker- (2008).
man, Phys.Rep.315, 199 (1999). [39] J. E. Bundler and H.-H. Lin, Phys. Rev. B 79, 045132
[21] A. Trenkwalder et. al. Phys. Rev. Lett. 106, 115304 (2009)
(2011). [40] A. Seidel, H.-H. Lin, and D.-H. Lee, Phys. Rev. B 71,
[22] Chen-YenLai, ChuntaiShi,andS.-W.Tsai, “Correlated 220501(R) (2005).
Phases of Population Imbalanced Fermi-Fermi Mixtures [41] H.-Y. Shih, W.-M. Huang, S.-B. Hsu, and H.-H. Lin,
on an Optical Lattice”, arXiv:1206.6797v1. Phys. Rev.B 81, 121107(R) (2010).
[23] S.G.Bhongale,L.Mathey,S.-W.Tsai,C.W.Clark,and [42] Y. Cai, W.-M. Huang, and H.-H. Lin, Phys. Rev. B 85,
E. Zhao, Phys. Rev.Lett. 108, 145301 (2012) 134502 (2012)
[24] S. G. Bhongale, L. Mathey, S.-W. Tsai, C. W. Clark,
and E. Zhao, “Unconventional Spin Density Waves in