Composite fermion basis for two-component Bose gases O. Liabøtrø and M. L. Meyer Department of Physics, University of Oslo, P.O. Box 1048 Blindern, 0316 Oslo, Norway (Dated: January 17, 2017) Despite its success, the composite fermion (CF) construction possesses some mathematical fea- turesthathave,untilrecently,notbeenfullyunderstood. Inparticular,itisknowntoproducewave functions that are not necessarily orthogonal, or even linearly independent, after projection to the lowest Landau level. While this is usually not a problem in practice in the quantum Hall regime, we have previously shown that it presents a technical challenge for rotating Bose gases with low 7 angular momentum. These are systems where the CF approach yields surprisingly good approx- 1 imations to the exact eigenstates of weak short-range interactions, and so solving the problem of 0 linearly dependent wave functions is of interest. It can also be useful for studying higher bands of 2 fermionicquantumHallstates. Herewepresentseveralwaysofconstructingabasisforthespaceof so-called“simple”CFstatesfortwo-componentrotatingBosegasesinthelowestLandaulevel,and n provethattheyallgivesetsoflinearlyindependentwavefunctionsthatspanthespace. Usingthis a J basis,westudythestructureofthelowest-lyingstateusingso-calledrestrictedwavefunctions. We also examine the scaling of the overlap between the exact and CF wave functions at the maximal 6 possible angular momentum for simple states. 1 ] s I. INTRODUCTION modelsandexperiments,calledpseudospin[18,19]. Typ- a ically this means multicomponent mixtures where the g different components are different internal states of the - Almost 20 years ago, the connection between the t atoms. Varying inter- and intraspecies interactions in- n physics of charged particles moving in two dimensions in dependently allows for novel behaviours [20]. This pseu- a a strong magnetic field, and dilute cold atoms rotating u dospin degree of freedom has been incorporated in the rapidly in a harmonic trap, was noticed [1]. Since then, q composite fermion scheme used for cold atom systems, a large body of theoretical and experimental work has . and has been used in the quantum Hall regime of high t accumulated that explore the various aspects of rapidly a angular momenta [21, 22]. On the other hand, near the m rotating atomic gases and the associated quantum phe- lowerendoftheangularmomentumscale, wheretheCF nomena; for reviews, see e.g. [2–4]. In particular, one - description is not a priori expected to work, it has been d expectsstronglycorrelatedphasessimilartothosefound shown that it actually works surprisingly well, both in n in the quantum Hall effect when the atoms experience o strongsyntheticmagneticfields. Theeffectofasynthetic scalar and two-component cases [23, 24]. c magnetic field can be generated by simply rotating the A property of the CF method of constructing wave [ cloud, or by more advanced techniques [5–9]. functions is that one typically needs to do a projection 1 For electron systems, one prominent way of theoreti- into the lowest Landau level (LLL) in order to either v callystudyingthequantumHalleffectinvolvesconstruct- compare different CF wave functions, or to compare a 8 ing (classes of) explicit trial wave functions that approx- CF to an eigenstate found by numerical diagonalization 7 imate the true low-energy eigenstates of the interacting oftheinteractingHamiltonian[25]. Thisprojectionleads 2 4 system. At least in the case of Coulomb interaction, the tonon-orthogonal,andoftenalsolinearlydependent,CF 0 true many-body eigenstates are extremely complicated. wave functions. This has been known in the context of . Still, many successful trial wave functions exist, most fa- electrons in the quantum Hall regime [26–28], but the 1 0 mously the Laughlin wave functions [10], the family of issueismuchmoreprominentforbosonswithlowangular 7 composite fermion (CF) states [11], and trial wave func- momentum: we have previously observed [29] one or two 1 tions addressing non-Abelian quantum Hall states [12– orders of magnitude difference between the number of : 14]. The success of these trial wave functions in explain- seeminglydistinctCFcandidatesandtheactualnumber v i ing various phenomena is linked to the way they capture of linearly independent wave functions. In some extreme X the important topological properties of the phases they cases the number of seemingly distinct CF candidates r describe. one can write down is even larger than the dimension of a therelevantsectorofHilbertspace,meaningtheycannot Manyofthemethodsmentionedabovehavebeenmod- possibly be independent. Until very recently, little was ified to be applicable to weakly interacting cold atom understood about the mechanisms responsible for these systems [15, 16]. The hope is to be able to experimen- lineardependenciesafterprojection. Inapreviouspaper tally study strongly correlated states in a cold atom set- [29],wediscussedthreetypesofrelationsbetweencertain ting, where parameters like density, disorder and scat- types of low-lying CF candidates, but were only able to tering lengths may be tuned much more finely than in giveexamplesdemonstratinghowtheserelationsseemto the semiconductor systems traditionally used in quan- explain all the linear dependencies. tum Hall experiments [17]. More recently, an additional degree of freedom is often taken into consideration in In this paper, we present sets of CF candidates for 2 two-component systems that we rigorously prove are ba- mayworkwithmany-bodywavefunctionsthatareeigen- sissetsforthesubspacethatminimizestheCFcyclotron states of L. These are homogeneous polynomials of de- energy for low angular momenta. We use these states greeL, symmetricinthecoordinatesofeachspeciessep- to study the real-space distribution of particles and vor- arately. As is common [2, 24] we will focus on transla- ticesofthelowest-lyingwavefunctionswhenwevarythe tionallyinvariantstates,i.e. polynomialsinvariantunder angular momentum, and also give additional attention a constant shift η0 of all coordinates, to certain special cases, comparing to known behaviour from scalar gases. Ψ({ηi+η0})=Ψ({ηi}). (4) As mentioned in the introduction, one may adapt the CF approach to produce wave functions for 2D bosons, II. TWO-COMPONENT ROTATING BOSE bothinthescalarandmulti-componentcases. ACFtrial GASES wavefunctionforthebosonictwo-speciessystemisofthe form [25] We first summarize the model for two-species Bose gases in the lowest Landau level, including their descrip- Ψ =P (Φ Φ J(z,w)q) (5) CF LLL Z W tion in terms of composite fermions. A more detailed introduction can be found in [24]. The two species of where q is an odd number, q = 1,3,5,.... ΦZ, ΦW are bosons experience a two-dimensional harmonic trap po- Slater determinants for each species of CFs, treated as tential of strength ω, and are rotating about the mini- non-interacting to lowest order. They consist of CF or- mum of the potential at frequency Ω. The Hamiltonian bitals is: (cid:18) (cid:19) ηη¯ ψ (η)=N ηmLm , m≥−n, (6) NX+M(cid:18)p2 1 (cid:19) n,m n,m n 2 H = i + mω2r2−Ωl 2m 2 i i where Lm is the associated Laguerre polynomial, and i=1 n (1) N is a normalization factor. The interpretation of N+M N+M n,m X X theseorbitalsisthattheyarecompositefermionsoccupy- + 2πg δ(r −r ). i,j i j ingLandau-likelevels,labeledbyn(oftencalledΛ-levels) i=1 j=i+1 inareducedeffectivemagneticfield. J isaJastrowfactor involving both species, Here N denotes the number of particles of the minority species, and M ≥ N denotes the number of particles of Y J(z,w)= (η −η ) the majority species, all with the same mass m. l is i j i theangularmomentumofparticlei. Thestrengthofthe i1,weobserveacorelessvortexof thefactthattheL stateisuniqueforgivenN,M,led winding N in the majority species at exactly L = MN. max ustocomputetheoverlapbetweentheexactlowest-lying From the plots in FIG. 8, however, we see that the exact state and the CF L state as function of M for given wavefunctionmusthavecontributionsfromCFstatesin max N =1,2,3. higher E bands. To answer whether or not this makes c In FIG. 8, the squared overlaps |hΨ |Ψ i|2 be- anyqualitativedifferencesfromtheresultspresentedhere CF exact tween the maximally rotating simple CF wave functions would require further investigation. at L=MN and the exact lowest-lying states are plottet As presented in [29], there are still linear dependence as functions of M for N =1,2,3 in (a) - (c) respectively. relationsbetweenCFcandidatesinhigherEc bandsthat In (a), we see exactly the same convergence to unity as are not understood. However, now that we have a good wasreportedin[23,36]foronecomponent. TheCFcan- understanding of the relations for simple states, it might didate in the one-component case is not simple (simple be possible to make further progress for these so-called statesdon’texistinthescalarcase),butitisuniqueand “compact states” [25]. They are the relevant candidates minimizes the CF cyclotron energy. The exact ground forL>MN inthesystemstudiedhere,buttheyarealso state of the single vortex, on the other hand, is known, relevant for electrons in strong magnetic fields, confined and its polynomial part is simply proportional to to quantum dots [38]. For other projection methods [39] and/or geometries [40], we expect some linear dependence relations similar Ψ =e (z˜) (25) s.v. M to the ones in this paper. The reason is that, as we have seen, it is in fact the Jastrow factor that is responsible P where z˜i = zi −1/M jzj are the particle coordinates for the equations that relate different Λ-level configura- relative to the center of mass, and ek is the elementary tion patterns. Certainly some rules will need to be mod- symmetricpolynomialofdegreek. Infact,forN =1,the ified, but the principle of translation invariance should exactlowest-lyingL=M stateisknownalsointhetwo- still hold. component case: it was given in [37] and its polynomial part is proportional to Acknowledgement Ψ =z˜e (w˜)−Me (w˜) (26) N=1,L=M M−1 M We would like to thank Susanne Viefers for helpful com- P Here η˜ =η −1/(N +M) η are the particle coordi- mentsonthemanuscript. Thisworkwasfinanciallysup- i i i i nates relative to the center of mass for all particles. ported by the Research Council of Norway. In(b)and(c)however,weclearlyseethatthesquared overlap decreases with system size, as is usually the case with most trial wave function approaches. It should Appendix A: Mathematical results be stressed that the dimension of the relevant sector of Hilbert space grows very rapidly with system size, so Lemma 1. There exists c∈Q|PN,M,Λ| such that given that Ψ is a unique state in this space, it L=MN is still surprising that the overlap with the true lowest- X Φ (a)= c ∆ (p)Φ (α), lying state is as large as it is. We should also remember Z p Z Z (A1) that we have restricted ourselves to simple states in this p∈PN,M,Λ analysis: the fact that the overlap decreases with system size therefore tells us that higher bands of CF cyclotron where α =i−1. Λ=PN a −α . i i=1 i i energy E contribute significantly for larger systems. c 9 1.00 1.000 0.95 rloveap 0.995 overlap 0.95 overlap rSquaed 0.990 Squared 0.90 Squared 0.90 0.85 0.85 0.985 0 5 10 15 20 0 2 4 6 8 10 12 14 3 4 5 6 7 8 Numberofmajorityparticles Numberofmajorityparticles Numberofmajorityparticles (a) (b) (c) FIG. 8: The squared overlap |hΨ |Ψ i|2 between the numerically exact lowest-lying states, and the CF states (16), at CF exact L=MN, for (a) N =1, (b) N =2, and (c) N =3, as function of M. Proof. Weusetheonetoonecorrespondencebetweenthe Eq. (A7) says that the Lemma holds for Φ (a) = Z elements of P and Φ (a) given by Φ (a(p ))andeq. (A8)saysthatitholdsforΦ (a)= N,M,Λ Z Z min Z Φ (a(p)) if it holds for all p0 K} (A10) greatest determinant with respect to the ordering occurs N,M,Λ|K N,M,Λ i when σ orders the partition non-decreasingly. This par- and note that trivially, for all p ∈ P ⊃ P , N,Λ,Λ|1 N,M,Λ|1 ticular determinant is ΦZ(a(p)) and we may therefore there exists dp ∈ Q|PN,Λ,Λ| such that eq. (A9) holds. write There is only one such p and ∆ (p) = ∆ . Now, we Z zΛ assume as induction hypothesis that (A9) holds for all X p∈PN,Λ,Λ|k when k ≤K. ∆ (p)Φ (α)= n Φ (a(p)) Z Z pp0 Z (A5) Let p∈P and consider the polynomial N,Λ,Λ|K+1 p0≤p N Y for some integers npp0. It is important that ∆zpi. (A11) i=1 n 6=0. (A6) pp It is a product of N factors. Each factor is a sum of Itisinfactpositive,sinceeverypermutationσthatgives N differentiation operators with respect to N different this determinant leaves the elements of α+PN p e variables. The resulting terms with the highest number k=1 σk k in increasing order. It follows that for the smallest par- ofnon-zeroexponentsarethosethatdonotmultiplydif- tition, p =min(P ), we have ferentiation operators for the same variable from several min N,M,Λ differentfactors∆ . ThisisexceptforthelastN−K−1 zpi 1 ∆ as ∂0 =∂0 and any of the NN−K−1 combinations ΦZ(a(pmin))= npminpmin∆Z(p)ΦZ(α) (A7) givzpeis thezsiame.zTj his restricted part of QNi=1∆zpi can be and in general described by permutations as Φ (a(p))= 1 ∆ (p)Φ (α)− X npp0Φ (a(p)). NN−K−1 X KY+1∂pi = NN−K−1 ∆ (p), Z n Z Z n Z ρi (N −K−1)! Z pp p0

0. this term tp, and it can be written as Ordinarytranslationinvarianceiscapturedinthecase N X Y n=1. t =K xM−pi, (A19) p p τi Proof. The operator (∆ + ∆ ) commutes with the τ∈SNi=1 zn wn Slater determinants, so it is enough to show that (∆ + zn where K is an integer that results from differentiation ∆ )J = 0. This result is independent of the splitting p wn andpossiblyapermutationsign. Thetermhastheprop- of particles into Z and W type. We can therefore use erty that the smallest exponent is as great as possible variables {ηi}Ni=+1M. We have that among terms in Ψ¯ . Given that, the second smallest ex- p ponent is as great as possible and so on. We use this to N N+M ∆ J =X∂n X (−1)|ρ| Y ηj−1 define an ordering on the tp terms, saying that ηn ηi ρj i=1 ρ∈SN+M j=1 t n j=1 least exponent of t and their k−1 least exponents are (A15) p0 pairwise equal. This is equivalent to p < p0, and if we Now, for each pair (i,ρ), there is a unique pair (i0,ρ0) pickanothernon-zerotermt0 ofΨ¯ bychoosingaσ that such that ρ −n = ρ and ρ0 −n = ρ0 and ρ = ρ0 for p p i i0 i0 i j j is not the identity, then we have t < t0. We index the all j 6= i,i0. Since ρ and ρ0 only differ by a permutation p p oftwoelements,(−1)|ρ|+|ρ0| =−1andthecorresponding partitions such that terms in eq. (A15) cancel out. Since this happens for all p

i. This means that if there exists a linear j dependence relation is a linearly independent set. Proof. If c Ψ¯ +...+c Ψ¯ =0, (A22) 1 p1 |PN,M,NM−L| p|PN,M,NM−L| Ψ =Φ (a(p))Φ (β)J ∈B , (A17) p Z W Z,L then the leftmost non-zero coeﬃcient of this relation mustbe0,andtherelationmustthereforebetrivial. then we denote the projection onto w = 0 ∀w by Ψ¯ , i i p and the set of projected states by B¯ . We will show Z,L [1] N.K. Wilkin, J.M.F. Gunn, and R.A. Smith, Phys. Rev. [3] N. Cooper, Advances in Physics 57, 539 (2008). Lett. 80, 2265 (1998). [4] A. L. Fetter, Rev. Mod. Phys. 81, 647 (2009) [2] S. Viefers, J. Phys.: Cond. Mat. 20, 123202 (2008). [5] V. Schweikhard, I. Coddington, P. Engels, V. P. Mogen-

Composite fermion basis for two-component Bose gases

Similar Composite fermion basis for two-component Bose gases