Model anisotropic quantum Hall states R.-Z. Qiu1, F. D. M. Haldane2, Xin Wan3, Kun Yang4, and Su Yi1 1State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics, Princeton University, Princeton NJ 08544-0708 3Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China and 4National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306, USA Model quantumHallstatesincludingLaughlin,Moore-Read andRead-Rezayistatesaregeneral- 2 ized into appropriate anisotropic form. The generalized states are exact zero-energy eigenstates of 1 corresponding anisotropic two- or multi-body Hamiltonians, and explicitly illustrate the existence 0 of geometric degrees of in the fractional quantum Hall effect. These generalized model quantum 2 HallstatescanprovideagooddescriptionofthequantumHallsystemwithanisotropicinteractions. Somenumeric results of theseanisotropic quantum Hall states are also presented. n a J PACSnumbers: 73.43.Nq,73.43.-f 0 1 I. INTRODUCTION system contain two parameters: the confining strength determined by the rotation frequency and the tilt angle ] l determinedbythe appliedfield. Thussuchsystemshave e Variational wave functions play a fundamental role in - our understanding of fractional quantum Hall (FQH) ef- highly tunable anisotropic interactions, and are ideal for tr fect, with the Laughlin wave function1 being the most studies of anisotropic FQH states. Results in Ref. 9 s clearly indicate the inadequacy of the known variational . prominent example. New classes of quantum Hall trial t wave functions in the description of such states. a wave functions have been proposed as correlation func- m tions in various conformal field theories,2,3 by general- As a matter of fact, consideration of possible - izing the clustering properties of the wave functions,4,5 anisotropicFQHstateshasalonghistory,11–13 withvari- d or as Jack polynomials with a negative parameter and a ational wave functions constructed that are straightfor- n matchingrootconfiguration.6Foralongtimeithasbeen ward modifications of the original Laughlin wave func- o understood that the Laughlin wave function, as well as tion.11,12 Unlike the originalLaughlinwavefunctionand c [ other FQH trial wave functions, contains no variational the states to be discussed here however, these earlier parameter. This understanding is the consequence of anisotropic FQH states are not exact eigenstates of spe- 1 our interest in searching for topologically distinct quan- cial2-bodyHamiltonians. Interestinsuchstateswaspar- v tum Hall wave functions; geometry, as oppose to topol- tiallymotivatedbytheobservationofcompressiblestates 3 ogy, is believed to be redundant. Recently one of us7 withstronganisotropictransportpropertiesinhighLan- 8 9 pointed out that such topological description of FQH dau levels (LLs).14,15 In all these states, the rotation 1 wave functions is not complete; in his geometrical de- symmetry is broken spontaneously. Very recently, an . scription the original Laughlin wave function is simply anisotropic FQH state has been observed at ν = 7/3.10 1 0 a member of a family of Laughlin states, parameterized Inthiscasetherotationsymmetryisbrokenexplicitlyby 2 by a hidden (continuous) geometricaldegree of freedom. an in-plane magnetic field, whose direction dictates the 1 The family of the Laughlin states, with the geometrical transport anisotropy. Generalizations of the FQH states : factor as a variational parameter, should provide a bet- to lattice models with zero net magnetic field and full v i ter description of the FQH effect in the presence of ei- latticetranslationsymmetryisalsocalledforto describe X theranisotropiceffectivemassoranisotropicinteraction, fractionalquantum anomalousHall states and fractional r which are present in real materials. topological insulators.16 a The anisotropic FQH states are of present interest, Ref. 7 pointed out the existence of a family of Laugh- both theoretically8,9 and experimentally.10 In Ref. 9, lin states that are zero energy ground states of a fam- some of us studied the quantum Hall effects in a fast ro- ily of Hamiltonians consisting of projection operators, tating quasi-two-dimensional gas of polarized fermionic which depend on a parameter called guiding center met- dipoles. The fast rotation is equivalent to the high mag- ric. However the states themselves were not constructed netic field according to the Larmor theorem. And since explicitly, and only some of their qualitative properties p-wave interaction for the polarized fermions is typically werementionedbriefly. Themainmotivationofthiswork very small unless in the resonance regime, the only sig- is to explore the construction of a family of wave func- nificant interaction is the dipole-dipole interaction that tionsin closed form withnumericalcomparisonsto facil- couldbetunedbyadjustingtheappliedelectricandmag- itate the study of the geometrical aspects of anisotropic neticfields. Bytuningthedirectionofthedipolemoment FQH states as a result of anisotropic interaction in the with respect to the z axis, the dipole-dipole interaction planer geometry. becomes anisotropic in the x-y plane. Specifically, the Therestofthepaperisorganizedasfollows. InSec.II 2 we introduce the anisotropic LL basis states using the where~ω =~2/mℓ2isthecyclotronenergyandtheLan- c unimodular transformation. In Sec. III we focus on the dau orbit ladder operators are given by recipe to generate the anisotropic many-particle states b=~−1ℓ(ω π), b† =~−1ℓ(ω∗ π), (7) duetotheunimodulartransformation. Section.IVcovers · · the numerical studies onvarious properties of the family with [b,b†] = 1. By introducing the “guiding center” of Laughlin states, including their density profile, pair coordinates R, correlationfunction, andvariationalenergy. We summa- rize the paper in Sec. V. Ra =ra+~−1ℓ2ǫabπ , (8) b which satisfy II. ANISOTROPIC LANDAU LEVEL BASIS [Ra,Rb]= iǫabℓ2, [Ra,π ]=0, (9) STATES − b we could define a second set of “guiding center” ladder The original Laughlin wave function was most easily operators, a and a† that commute with b and b† (and written down on a disc, using the symmetric gauge in hence with H0) whichthesingleparticlebasisstatesareangularmomen- a=ℓ−1(ω∗ R), a† =ℓ−1(ω R), [a,a†]=1.(10) tum eigenstates. The key to explicitly constructing the · · anisotropicLaughlinstatesproposedinRef.7(onadisc) To proceed further, let us choose the vector potential is the usage of a set of anisotropic LL basis states. We of the uniform magnetic field in the symmetric gauge, will, however, continue to use the symmetric gauge, in which the lowest LL (LLL) wave functions are holomor- A= 1B r = 1B( y,x), (11) 2 × 2 − phic. The same set of basis states appeared earlier in and define the complex coordinate, the consideration of deformation of shape of quantum Hall liquids in the context of Hall viscosity17, although ω r z = · . (12) the wave functions were not explicitly given. In the fol- ℓ lowing we generate them explicitly. In addition, we also Letusnotethatforthecaseofisotropicmassthecomplex usethemtoconstructthecorrespondingintegerquantum coordinate is explicitly given by Hall wave function, as a warm-upfor their later applica- tion to FQH states. x+iy z = , (13) Let us start from the eigenvalues problem of a two- √2ℓ dimensional charged particle subjected into a uniform which is not the most standard definition in the liter- magnetic field B = Bzˆ. The one-body Hamiltonian is ature. Then we could express the Landau orbit ladder given by operators as (neglecting the trivial i and i), − H0 = 12(m−1)abπaπb, π =p−eA, (1) b= 12z+∂z∗, b† = 21z∗−∂z, (14) where we used the Einstein convention, mab is the cy- where ∂zf(z,z∗) is the partial derivative ∂f/∂z z∗, etc, clotron effective mass tensor, e > 0 is the charge of the and the “guiding center” ladder operators as | particle, A is the vector potential of the uniform mag- netic field B, and the dynamical momentum π satisfies a= 21z∗+∂z, a† = 12z−∂z∗. (15) [π ,π ]=iǫ ~2ℓ−2 (2) The Hamiltonian H0 has a rotational symmetry gen- a b ab erated by with ǫ = ǫab the 2D antisymmetric Levi-Civita sym- ab L =a†a b†b, [L ,H ]=0, (16) bol and ℓ = ~/(eB) the magnetic length. In terms 0 − 0 0 of a complex vector ω, the effective mass tensor can be where p written as [L ,x]=iy, [L ,y]= ix, (17) 0 0 − mab = m (ωa∗ωb+ωb∗ωa), (3) [L0,px]=ipy, [L0,py]= ipx. (18) − (m−1)ab = m−1(ωa∗ωb+ωb∗ωa). (4) Simultaneous diagonalization of H and L allows a 0 0 we note that the assumption that it is isotropic in the complete orthonormal basis of one particle states ψnm | i (x,y) Cartesian coordinate system means which used to be constructed as (ω ,ω )=(ωx,ωy)= 1 (1,i). (5) H0|ψnmi=(n+ 21)~ωc|ψnmi, (19) x y √2 L0 ψnm =(m n)ψnm , (20) | i − | i (a†)m(b†)n The one-body Hamiltonian can now be expressed as ψ = ψ , (21) nm 00 | i √m!n! | i H = 1~ω b†b+bb† . (6) aψ =bψ =0. (22) 0 2 c | 00i | 00i (cid:0) (cid:1) 3 This is, of course, the commonly used basis states asso- Note that ψ (0) is the conventional basis ψ in nm nm | i | i ciated with the symmetric gauge. Eq. (21). However, it is possible to construct a set of closely Since the modelquantum Hallstates areoften defined relatedbut different setofbasisstates. Firstlet us write in the lowest Landau level (LLL) (n = 0), it is useful to examine the wave functions ψ (γ,r) = r ψ (γ) . L = L+L¯, (23) 0m h | 0m i 0 First of all, from the LLL property bψ (γ,r) = 0, we 0m L = 12(b†b+bb†), H0 =~ωcL, (24) have L¯ = 1(a†a+aa†). (25) 2 ψ0m(γ,r)=f(z)ψ00(0,r) (35) The conventional basis is then just the set of the simul- with taneous eigenstates of H and L¯, 0 L¯ ψ =(m+ 1)ψ . (26) ψ00(0,r)=(2π)−1/2e−|z|2/2. (36) | nmi 2 | nmi We may write L¯ = L¯(g0), where Second from aγψ00(γ,r)=0, we immediately obtain L¯(g)= 1 g RaRb, (27) ψ00(γ,r)=λ1/2e−21γz2ψ00(0,r) (37) 2ℓ2 ab with andg isapositive-definiteEuclideanmetrictensorwith ab detg = 1. Note that L¯(g0) uses the “Galilean metric” λ= 1 γ 2. (38) −| | derived from the effective mass tensor, Next let us note that ψp(γ,r), 0m m =m(ω∗ω +ω∗ω )=mg0 . (28) However, asahbas beenastrbessedbbya one ofaubs7,18, that be- 21z−∂z∗ +γ∗ 21z∗+∂z mψ00(γ,r), λm√m! forespecifyingthetwo-bodyinteractions,thereisnofun- (cid:2) (cid:0) (cid:1)(cid:3) damental reason to choose a guiding-center basis that is could be simplified by virtue of the property that adapted to the shape of the Landau orbits. There is a bψ (0,r)=0, or 00 moregeneralfamilyofbases,parameterizedbyacomplex number γ with γ <1, ∂z∗[ψ00(0,r)f(z)]=(−12z)ψ00(0,r)f(z) (39) | | ω¯ =(1 γ 2)−1/2(ω +γ∗ω∗), (29) and aψ00(0,r)=0, or a −| | a a that determines a unimodular metric gab(γ), ∂z[ψ00(0,r)f(z)]=ψ00(0,r)(∂z − 21z∗)f(z) (40) 1 (1+γ)(1+γ∗) i(γ γ∗) and 1−γ∗γ (cid:18) i(γ−γ∗) (1−γ)−(1−γ∗) (cid:19) ∂z e−21γz2f(z) =e−21γz2(∂z γz)f(z). (41) − according to h i Thus we obtain ω¯∗ω¯ = 1(g (γ)+iǫ ), (30) a b 2 ab ab λm andthengab(γ)=ω¯a∗ω¯b+ω¯aω¯b∗. Wecanthenconstructa ψ0m(γ,r) = ψ00(γ,r)√m!(z+z02∂z)m1 (42) basis of eigenstates ψ (γ) of H and L¯(γ) L¯(g(γ)), which is accomplish|ednbmy deifining0a new set o≡f “guiding = ψ (γ,r) λm W (z,z2) (43) 00 √m! m 0 center” ladder operators, also considered in Ref. 17, ω¯∗ R ω¯ R with a = · , a† = · , (31) γ √2ℓ γ √2ℓ z2 =γ∗/λ2 (44) 0 and explicitly as and W (z,z2) are the polynomials defined by W (z,z2) m 0 0 0 a 1 1 γ a = 1, and the recursion relations, γ = . (32) (cid:18)a†γ (cid:19) √1−γ∗γ (cid:18)γ∗ 1 (cid:19)(cid:18)a† (cid:19) Wm+1(z,z02) = (z+z02∂z)Wm(z,z02), (45) This is a Bogoliubov (or “squeezing”) transformation W (z,z2) = ∂ W (z,z2)/(m+1). (46) m 0 z m+1 0 that preserves the commutation relation [a ,a†] = 1. Then the new simultaneous eigenstates of H γanγd L¯ (γ) The first few of Wm(z,z02) are given by 0 0 are given by W (z,z2) = 1, 0 0 (b†)n(a†)m W (z,z2) = z, ψ (γ) = γ ψ (γ) , (33) 1 0 | nm i √n!m! | 00 i W2(z,z02) = z2+z02, a ψ (γ) =bψ (γ) =0. (34) W (z,z2) = z3+3z2z, γ| 00 i | 00 i 3 0 0 4 and the general solutions are given by 6 (a) (b) (c) (d) W (z,z2) = k!(2z2)kL(−12)( z2/2z2), (47) ℓ) 0 2k 0 0 k − 0 y( W (z,z2) = k!(2z2)kzL(12)( z2/2z2), (48) 2k+1 0 0 k − 0 −6 −6 0 6 where L(α)(x) are generalized Laguerre polynomials de- x(ℓ) k fined by FIG.1: (Color online)Thedensityprofilesof thegeneralized L(α)(x)= k Γ(α+k+1) (−x)i. (49) LLL wave functions ψ0m(γ =1/2,r) with m=0 (a), m=1 k Γ(α+i+1)Γ(k i+1) i! (b),m=2 (c) and m=3 (d). i=0 − X These expression do satisfy the reclusion relation, which 6 could be proved by using the following useful identities (a) (b) (c) (d) for L(α)(x), ℓ) 0 k ( y L(α)(x) = L(α+1)(x) L(α+1)(x), (50) −6 k k − k−1 −6 0 6 d L(α)(x) = L(α+1)(x), (51) x(ℓ) dx k − k−1 xL(α+1)(x) = (k+α)L(α) (x) kL(α)(x). (52) FIG. 2: (Color online) The density profiles of ψ (γ,r) with k−1 k−1 − k γ =eiφ/2forφ=0(a),φ=π/2(b),φ=π (c)a0n3dφ=3π/2 NoTtehethdaetnsfoitrie|sz|ψ2 ≫(|γz0,2r|,)W2mde(fizn,ze0d2)b→ytzhme.single-particle (d). Notethat thezeros are localized at z=0 and ±i√3z0. 0m | | orbitals have some noteworthy features. First, the den- sity profile is anisotropic for γ =0. Take a simple exam- where µ is a decreasing partition and det denotes the 6 ple like ψ00(γ ,r), matrix determinant. | | (1+ γ )x2+(1 γ )y2 As a special example, the Slater determinant slµ0(γ) ψ (γ ,r)2 exp | | −| | . with the partition µ = [N 1,N 2,...,1,0] is the | 00 | | | ∝ − 2ℓ2 0 − − (cid:20) (cid:21) generalized N-particle IQH state, whose (unnormalized) For this Gaussian wave packet, the width along y-axis is expression could be simplified as largerthanthewidthalongx-axis,sonon-zero γ causes | | stretching the wave function along some direction. Sec- Ψ (γ)= (z z ) ψ (γ,r ), (55) I i j 00 i ond, ψ (γ,r)2 has m zeros as can be seen from Fig. 1 − 0m i<j i in wh|ich we plo|t the density profiles ψ (1/2,r)2 for Y Y 0m m = 0,1,2,3. These zeros are roots o|f Wm(z,z02)|, and aslower-degreepolynomialsmustvanishduetothePauli they are aligned along the stretched direction. In the exclusion principle (this is generally not the case for limit of γ 0, they collapse to a multiple root at the other partitions, which is the key to understanding the → origin. Third, the stretching direction is determined by anisotropic FQH states). the phase of the complex number γ, arg(γ), which is The generalizedIQH state can also be viewed as a co- easy to understand by noting that W3(z,z02) = 0 gives herent superposition of the isotropic IQH state (or sim- rise to three roots, 0 and i√3z0. In Fig. 2, we plot ple Vandermonde determinant state) and its edge states ψ03(eiφ/2,r)2 with φ = 0±,π/2,π and 3π/2. Thus we with angular momentum differences being multiples of r|ealize that th|e density profile ψ0m(γ,r)2 is equivalent 2~. Thisis achievedbyexpandingthe exponentialfactor to rotating clockwisethe densit|y profile ψ|0m(γ ,r)2 by in Eq. (55). From the viewpoint of the superposition, | | | | arg(γ)/2. this is consistent with the numerical result in Ref. 9. To proceed further, let us define here the non- interacting basis states to prepare for the many-body problem. For bosonic system, the non-interacting basis III. ANISOTROPIC MANY-PARTICLE STATES state in the generalized LLL is given by (γ) perm[ψ (γ,r)], (53) µ 0µ Inthis section,we first considerthe two-particleprob- M ∝ lem and then simplify the interaction Hamiltonian pro- where µ = [µ ,µ ,...,µ ] is a non-decreasing parti- 1 2 N jected into the LLL in Subsec. IIIA. More importantly, tion and perm[ψ (γ,r)] is the permanent of the square 0µ we present the recipe to find the anisotropic counter- matrix whose matrix elements are ψ (γ,r ) (i,j = µi j part of a LLL state and elaborate it in Subsec. IIIB. 1,...,N). For the fermionic system, the non-interacting We specifically consider the anisotropic counterpart of basis state is defined by a Slater determinant, the prominent FQH states such as the Laughlin state in sl (γ)=det[ψ (γ,r)], (54) Subsec. IIIC. µ 0µ 5 A. Two-particle problem and the projected Thus the projected operator h (γ) for many-particle m two-body interaction Hamiltonian system could be rewritten as If the interaction is dominated by the energy gap ~ω c h (γ)= Mm Mm (i,j) (65) between the lowest and second Landau level, the inter- m | iγγ |! i<j M action Hamiltonian could be projected into the lowest X X Landau level. In terms of non-commuting “guiding cen- and in second quantized form as ter” coordinates R , the projected two-body interaction i Hamiltonian we consider here is given by7 1 m ,m mM mM m ,m γ 1 2 γγ 3 4 γ 2 h | i h | i H ( V ,γ)= V h (γ), (56) XM m1mX2m3m4 int m m m { } g† g† g g , (66) Xk m1 m2 m4 m3 where V is the anisotropic Haldane’s pseudopotential with m ,m = ψ (γ) ψ (γ) and g† being m | 1 2i | 0,m1 i⊗| 0,m2 i m and the projection operator h (γ) is given by the creation operator that creates a particle in state m ψ (γ) . 0m d2qℓ2 | For thie rotationally-invariantcase, γ =0, h (γ) = v (q,g(γ)) eiq·(Ri−Rj),(57) m m 2π Z Xi<j (z +z )M (z z )m 2 vm(q,g) = Lm(gabqaqbℓ2)exp(−12gabqaqbℓ2), (58) ΨMm(0,r1,r2)= √12MM2 ! √12−mm2! ψ00(0,ri) i=1 Y where L (x)=L(α=0)(x) are the Laguerrepolynomials. m m and the translation into the Heisenberg state is Inaddition,thetotalangularmomentumassociatedwith the guiding center is given by (a† +a†)M (a† a†)m Mm = 1 2 1− 2 00 . (67) γ γ 1 | i √2MM! √2mm! | i L¯(γ) = g RiaRib. (59) 2ℓ2 ab Thereforewefindtheanisotropicdeformationisnoweas- i X ily achieved by the simple replacement For two-particle, the LLL states that simultaneously diagonalize L¯(γ) and the interaction Hint({Vk},γ) are a†i →a†iγ, (68) completely fixed by symmetry. To clarify this, let us note that and the vacuum state Ψ (0) Ψ (γ) that satisfy 0 0 | i → | i a Ψ (γ) = 0. In the next subsection, we shall uti- iγ 0 L¯(γ) = a†1γa1γ +a†2γa2γ +1 lize|this idiea to consider the anisotropic deformation of = a† a +A† A +1 (60) a general LLL wave function. 12γ 12γ 12γ 12γ with a =(a a )/√2 and A =(a +a )/√2, 12γ 1γ− 2γ 12γ 1γ 2γ B. Anisotropic many-particle states and the only term containing operatorin the interaction Hamiltonian, R R , are given by 1 2 − Thegeneralformofarotationally-invariantN-particle ℓ LLL wave function in the symmetric gauge is Rx Rx = (1 γ∗)a +(1 γ)a† 1 − 2 λ − 12γ − 12γ ℓh i Ψ(r1,...,rN)=F(z1,...,zN) ψ00(ri), (69) Ry Ry = (1+γ)a† (1+γ∗)a 1 − 2 iλ 12γ − 12γ Yi h i whereF(z ,...,z )isahomogeneousmultivariatepoly- and contains only a and a† . Therefore, we could 1 N 12γ 12γ nomial with F(λz ,...λz ) = λMF(z ,...,z ) and expand L¯(γ) and Hint( Vk ,γ) in the orthonormalbasis L¯(0)Ψ = (M+1N1)Ψ . TNhe translation1to theNHeisen- { } | i 2 | i berg picture yields (A† )M(a† )m |Mmiγ = 12√γ M!m1!2γ |00iγ, (61) |Ψ(γ =0)i=F(a†1,...,a†N)|Ψ0(γ =0)i, a1γ|00iγ =a2γ|00iγ =0, (62) ai|Ψ0(γ =0)i=bi|Ψ0(γ =0)i=0. (70) and find that The deformation of this Heisenberg state becomes Hint({Vk},γ)|Mmiγ =Vm|Mmiγ, (63) |Ψ(γ)i=F(a†γ,1,...,a†γ,N)|Ψ0(γ)i, L¯(γ)Mm =(M +m+1)Mm . (64) a Ψ (γ) =b Ψ (γ) =0. (71) γ γ γ,i 0 i 0 | i | i | i | i 6 The Schrodinger wave function for this state Ψ(γ) will its anisotropic counterpart Ψ(γ,r ,...,r ) would be 1 N | i have the form, c(b) (γ), c sl (γ). (80) Ψ(γ,r ,...,r )=F (z ,...z ) ψ (r ). (72) µ Mµ µ µ 1 N γ 1 N 00 i µ µ X X i Y Here we find the coefficients c and c(b) are independent We now need the formal construction of F ( z ), µ µ γ i { } of γ. which is still holomorphic, but no longer a polynomial. The γ-independence of the coefficients results in γ- Using the homogeneity of the polynomial F, the wave independence of the occupation number for each orbital function is given by λ−MF({12zi−∂zi∗ +γ∗(21zi∗+∂zi)}) λ21e−12γzi2ψ00(ri). nm =hΨqL(γ)|gm† gm|ΨqL(γ)i. Yi This also suggests that the entanglement spectrum,19 Using the property a ψ (r ) = 0, or which encodes the topological properties of the state, is i 00 i also invariant, if the appropriate cut in the space of the ∂ ψ (r)f(z,z∗)=ψ (r)(∂ 1z∗)f(z,z∗), (73) total angular momentum associated with guiding cen- z 00 00 z − 2 ter is chosen. We see that topological properties of a and b ψ (r ) = 0, or i 00 i FQHstatearebuiltintothecoefficients(relativeweight), whicharemanifested,e.g.,intheentanglementspectrum ∂ ψ (r)f(z,z∗)=ψ (r)(∂ 1z)f(z,z∗), (74) z∗ 00 00 z∗− 2 calculation. On the other hand, geometrical properties the holomorphic part F ( z ) of the wave function be- of the anisotropic FQH states are encoded into the de- γ i comes { } formation of the non-interacting basis. Nevertheless, ge- ometrical information can be revealed by exploring the λ−MF({zi+γ∗∂zi}) λ12e−12γzi2. propertiesofthefamilyofvariationalwavefunctionsand i the corresponding variationalenergies; this point will be Y illustrated by examples in Sec. IV. Using the following equality, ∂z e−21γzi2f(z) =e−21γzi2(∂z −γz)f(z), (75) C. Anisotropic FQH states (cid:16) (cid:17) F turns into, γ Inthissubsection,letusconsidersomeprominentvari- ational functions such as the Laughlin wave function, i λ12e−12γzi2!λMF({zi+z02∂zi})1, MoTohree-νRe=ad1/sqtaLteauagnhdlinsowoanv.e function is given by Y which is easy to understand by noting Eq. (42). Using Ψq(γ =0)= (z z )q ψ (0,r ), (81) the homogeneity of F, we obtain the final result L i− j 00 i i<j i Y Y Fγ({λ−1zi})= iλ12e−21γzi2 F({zi+z02∂zi})1. (76) and its anisotropic counterpart is given by (cid:16) (cid:17) where λ and z2 arQe fixed by γ through Eqs. (38) and Ψq(γ)= ψ (γ,r ) z z +z2(∂ ∂ ) q1. 0 L 00 i i− j 0 zi − zj (44). This expression no longer requires homogeneity of i i<j F( z ), and is quite general for the deformation of the Y Y(cid:2) (cid:3) i (82) { } holomorphic part of a LLL wave function. Specially, the anisotropic counterpart of the TheanisotropicLaughlinstatesΨq(γ)arethe(minimum L rotationally-invariant non-interacting basis states total L¯(γ)) zero-energy ground state of the Haldane’s (53,54) pseudopotential Hamiltonian H ( V ,γ) with int m { } (γ =0), sl (γ =0), (77) µ µ V >0 and V =0. (83) M m<q m≥q are To prove this, we may consider the diagonalization of (γ), sl (γ). (78) the projected interaction Hamiltonian matrix and note µ µ M the γ-independence of the matrix elements Therefore, if we could expand the many-body state Ψ(γ = 0,r1,...,rN) in the non-interacting basis states hslµ(γ)|Hint({Vm},γ)|slµ′(γ)i, or µ(0) (53) or slµ(0) (54), µ(γ)Hint( Vm ,γ) µ′(γ) , M hM | { } |M i c(b) (0), c sl (0), (79) which results from the γ-independence of µ Mµ µ µ m ,m mM . µ µ γ 1 2 γ X X h | i 7 For two particles, the anisotropic counterpart of the 15 (a) 15 (b) 0.5 ν =1/q Laughlin state 0.4 Ψ0q(0,r1,r2)=(z1−z2)qΨ00(0,r1,r2) (84) y(ℓ) 0 0 00..23 is given by 0.1 −15 −15 0 Ψ (γ,r ,r ) −15 0 15 −15 0 15 0q 1 2 = ψ (γ,r )ψ (γ,r ) z z +z2(∂ ∂ ) q1, x(ℓ) 00 1 00 2 1− 2 0 z1 − z2 = ψ00(γ,r1)ψ00(γ,r2)((cid:2)z12+2z02∂z12)q1, (cid:3) FIG. 3: (Color online) The density profiles (in units of = ψ (γ,r )ψ (γ,r )W (z ,2z2), (85) 1/(2πℓ2)) of the ν = 1/3 anisotropic Laughlin state with 00 1 00 2 q 12 0 γ = 1/2 (a) and γ = 0 (b) for an N = 10 system. Note withz =z z and∂ = 1(∂ ∂ ). Amongthese theyellow bulkwhose valueapproximates 1/3. 12 1− 2 z12 2 z1 − z2 states,thestraightforward(anisotropic)generalizationof the ν = 1/2 bosonic Laughlin state is Ψ (γ,r ,r ), in 02 1 2 withthe3-bodybasisstate m ,m ,m = ψ (γ) whichthedoublezerooftheundeformedstateatz1 =z2 ψ (γ) ψ (γ) and| 1 2 3iγ | 0,m1 i⊗ is split into zeroes at z1 z2 = i√2z0. While for the | 0,m2 i⊗| 0,m3 i − ± ν =1/3fermionicLaughlinstateΨ03(γ,r1,r2),thetriple M zero at z1 =z2 is split into a single zero at z1 =z2, and a†γ1+a†γ2+a†γ3 displaced zeroes at z1−z2 =±i√6z0. This suggests the |Miγ = (cid:16) √3MM! (cid:17) |0,0,0iγ splitting in the pattern of zeros can be used as a tool to study the anisotropic deformation of a quantum Hall And the anisotropic ν = 1/2 fermionic Moore-Read wave function. states Ψq=2(γ) is the (minimum L¯(γ)) zero-energy MR The other FQH state, such as the Moore-Readstate2, ground state of the following anisotropic 3-body pseu- theRead-Rezayistate3,theHaldane-Rezayistate20,etc., dopotential Hamiltonian, could be generalized in this way, as long as the corre- H (γ)=V h (γ), V >0, (88) sponding wave functions could be expanded in the non- int 1 1 1 interactingLLLbasis. Inaddition,wealsoconcludethat where the projection operator h (γ) is given by 1 theanisotropicZ parafermionstates3 arealsothe(min- k ∞ imum total generalized angular momentum) zero-energy m ,m ,m 3,M 3 ground state of some special (k + 1)-body interaction. γh 1 2 3| − iγ This kind of interaction could be expressed in terms of MX=0m1m2mX3m4m5m6 thegeneralized(k+1)-bodypseudopotentials21 sincethe γh3,M −3|m4,m5,m6iγgm† 1gm† 2gm† 3gm6gm5gm4. Z parafermion states are the (minimum total angular k with the three-body state 3,M 3 being momentum) zero-energy ground state of (k + 1)-body | − iγ short-range interaction3,22. 3,M 3 = B (a† a† )(a† a† )(a† a† ) For example, the ν = 1/q anisotropic Moore-Read | − iγ M γ1− γ2 γ1− γ3 γ2− γ3 M−3 states are given by a† +a† +a† 0,0,0 γ1 γ2 γ3 | iγ 1 (cid:16) (cid:17) ΨqMR(γ) = ψ00(γ,ri)Pf z z +z2(∂ ∂ ) and BM being the appropriate normalization factor. i (cid:18) i− j 0 zi − zj (cid:19) Y zi−zj +z02(∂zi −∂zj) q1, (86) IV. NUMERICAL RESULTS FOR THE i<j ANISOTROPIC FQH STATES Y(cid:2) (cid:3) where Pf means the Pfaffian, the square root of the de- Inthissectionwenumericallystudysomepropertiesof terminant. Among these states, the anisotropic ν = 1 bosonicMoore-ReadstatesΨq=1(γ)isthe (minimumto- thetheanisotropicLaughlinstatesΨqL(γ)anditsapplica- tal L¯(γ)) zero-energy groundMstRate of the anisotropic 3- bilityinasystemwithdipole-dipoleinteraction. Wefirst demonstratethe deformationofthe FQHdropletinden- body pseudopotential Hamiltonian, sityprofile. Theanisotropyleadsalsotothedeformation H (γ)=V h (γ), V >0, (87) of the correlation hole, which can be understood by the int 0 0 0 split of the third-order zero to three adjacent first-order where the projection operator h (γ) is given by zeros. As a trivial example we show that the isotropic 0 Laughlin states, among the family of generalized states, ∞ is the ground state in the variational sense for isotropic m ,m ,m M γ 1 2 3 γ hard-coreinteraction. Ontheotherhand,theanisotropic h | i MX=0m1m2mX3m4m5m6 dipole-dipole interactionpicks avariationalgroundstate M m ,m ,m g† g† g† g g g with a finite γ parameter as expected. γh | 4 5 6iγ m1 m2 m3 m6 m5 m4 8 10 10 1.6 (a) (b) g(γ,0,y) 1 g(γ,x,0) ) g(0,x,0) ℓ 0 0 1.2 y( 0.5 −10 −10 0 0.8 −3 −10 0 10 −10 0 10 x 10 6 0.3 x(ℓ) (a) (b) 0.4 FIG. 4: (Color online) The pair correlation function for ν = 1/3 Laughlin state with γ = 1/2 (a) and γ = 0 (b) of an 0 0 N =10 system. 0 1 0 1 0 −15 −10 −5 0 5 10 15 x(ℓ) or y(ℓ) A. Density profiles FIG.5: (Coloronline)Thepaircorrelationfunctiong(γ,0,y), g(γ,x,0)withγ =1/2andg(0,x,0)ofanN =10anisotropic The density profile for the anisotropic Laughlin state Laughlin state. The inset (a) shows the asymptotic (green is most easily calculated using the Jack polynomials6,23. solid) line of g(0,x,0), which is proportional to x6. And the Explicitly, we can write inset (b) shows the asymptotic (green solid) line of g(γ,x,0) (g(γ,0,y)), which is proportional to x2 (y2). ̺(γ,r)= Ψq(γ)Ψˆ†(r)Ψˆ(r)Ψq(γ) (89) h L | | L i for a finite number of particles. In Fig. 3, we plot the very different from that of g(γ = 0,r 0). The differ- density profile ̺(γ,z) with γ = 1/2 for an N = 10 → encerootsinthetoymodelfortwoparticleswestudiedin fermionic system at ν = 1/3 and compare with that of Subsec.IIIC,fromwhichweexpectthatg(γ =0,r 0) the isotropicLaughlinwavefunction. Roughlyspeaking, 6 → vanishes as r 2, while g(γ =0,r 0) as r 6, as indi- intheanisotropiccasetheFQHdropletisstretchedalong | | → | | cated by Eqs. (84) and (85). The insets (a) and (b) in the y-direction, while it maintains its value around 1/3, Fig. 5 confirm the asymptotic behavior. in units of 1/(2πℓ2) in the bulk (indicated by the yellow Weemphasizethatintheanisotropiccasethepaircor- color in the color plot). relationfunctionisisotropicforsmallenough r ,reflect- | | ingthefirst-orderzerointhefermionicwavefunctionen- forcedby the Pauliexclusion principle. The two-particle B. Pair correlation function wave function can be regarded as the asymptotic wave function when two particles are significantly closer than Thedensity-densitycorrelationfunctionrepresentsthe their distance to any other particles. For r comparable conditionalprobabilityoffindoneparticleatr whenan- | | to γ , the pair correlationfunction becomes anisotropic. otherissimultaneouslyatr′. Forthe anisotropicLaugh- | | The observed anisotropy at short distances encodes the lin state Ψq(γ), the density-density correlation function L geometricaldeformationoftheLaughlinstate. Whentwo is defined by particlesareclosetoeachother,eachparticleseesqzeros (for the Laughlin state at ν =1/q);the spatial spreadof G(2)(γ,r,r′)= Ψq(γ)Ψˆ†(r)Ψˆ†(r′)Ψˆ(r′)Ψˆ(r)Ψq(γ). h L | | L them reflects the extent of the deformation. Topological properties, manifested in the isotropic case by the qth- Without loss of generality, we consider the pair correla- order zero,can be identified when one look not too close tion function with r′ fixed at the origin, (r γ ). | |≫| | G(2)(γ,r,0) g(γ,r)= . (90) ̺(γ,r)̺(γ,0) C. Variational energy In Fig. 4, we compare g(γ,r) of an N = 10 Laugh- lin state at ν = 1/3 with γ = 1/2 and 0. The two Given the family of Laughlin states Ψq(γ), we test L cases are clearly distinguishable, as the correlation hole the variational principle on a trivial Hamiltonian with for γ = 1/2 is stretched non-monotonically along the y- the isotropic hard-core interaction, which renders the direction, along which the density profile is stretched. isotropic Laughlin state as its exact zero-energy ground The centerof the hole is strictly zero but twoother min- state,andaHamiltonianwithanisotropicdipolar-dipolar ima developed in the y-direction are not. To explore interaction. In both cases the expectation value of the more details we analyze g(γ,x,y) as a function of x (or Hamiltonian develops a minimum, which may be iden- y) along the y-axis (x-axis) in Fig. 5. The comparison tified as the variational ground state energy. The mini- showsthatthe asymptoticbehaviorofg(γ =0,r 0)is mum occurs at γ = 0 for the isotropic interaction and a 6 → 9 0.01 NN ==87 3ℓ) 4x 10−4N =3 N =6 / N =4 )10.008 NN ==54 (cd 3 NN ==56 (V0.006 Nγ2=3 0) 2 NN ==78 , ) θ γ ( N)(c0.004 N()dd 1 (h E E − 0 0.002 ) γ θ,−1 00 0.02 0.04 0.06 0.08 0.1 N)(d γ (Ed−2 0 0.02 0.04 0.06 0.08 0.1 γ FIG. 6: (Color online) The hard-core interaction energy per particle (N)(γ) with respect to the distort parameter γ for N =3−E8hcsystem. Here we also plot γ2 (solid line) for com- FpeIGr .pa7r:ti(cCleolo(rNo)(nθlin=e)30T◦h,eγ)dwipiothle-rdeisppoelcetitnotetrhaectdioinstoernterpgay- parison. Edd rameter γ for N =3 8 system. Here axial oscillator length − d is set as 0.01ℓ. nonzero γ for the anisotropic interaction. Becausethe Laughlinwavefunction is the zero-energy ground state of the (isotropic) hard-core interaction, (θ =0). Thedipole-dipoleinteractionenergyperparticle 6 γ = 0 is naturally the minimum for the energy expec- in the anisotropic fermionic Laughlin state Ψq=3(γ) is L tation value. The variational energy per particle is ex- pectedtoincreaseasβ γ 2 forsmall γ ,ascanbequickly understood, e.g., from|th|e expansio|n|of ΨqL(γ) with re- (N)(θ,γ)= hΨqL(γ)| i<jV(θ,zi−zj)|ΨqL(γ)i. (93) spect to small γ , Edd N | | P Ψq(γ)= 1 1γ z2+ 1γ∗ ∂2 +... Ψq(0).(91) L − 2 i i 2 i zi L In Fig. 7, we plot (N)(θ = 30◦,γ) with respect to γ Here β, lik(cid:2)e stiffnePss, quantifiesPthe energy(cid:3)cost of the for finite-size systemEdwdith N = 3 8. We found that fluctuation of metric. In Fig. 6 we plot the hard-corein- thelowestgroundstateenergyofth−e dipole-dipoleinter- teractionenergyperparticlefortheanisotropicfermionic action occurs for the anisotropic Laughlin state with a Laughlin state ΨqL=3(γ) nonzeroγ. Thefurtherquantitativecomparisonbetween the model anisotropic quantum Hall state and the exact Ψq(γ)H ( V ,γ =0)Ψq(γ) (N)(γ) = h L | int { m} | L i, (92) ground state of the anisotropic dipole-dipole interaction Ehc N will be given elsewhere. withV =V δ . Forthesetoffinite-size systemswith m 1 m,1 N =3-8, we confirm that the minimum of the hard-core interaction energy occurs identically at γ = 0. There- fore,the (isotropic)Laughlinwavefunctionis indeedthe V. SUMMARY optimal state for the hard-core interaction. We now turn to the dipole-dipole interaction, which someofusstudiedinRef.9byexactdiagonalization. For In this work we have explicitly constructed families of thedipole-dipoleinteractionwiththedipolemomentsbe- anisotropicfractionalquantumHallstates,whichareex- ing polarized in the x-z plane, the potential in the x-y act zero energy ground states of appropriate anisotropic plane is given by short-range two- or multi-particle interactions. These e−ξ2/2d2 c families generalize the celebrated Laughlin, Moore-Read V(θ,x,y) = dξ d and Read-Rezayi states. Each family is parameterized √2πd2 (x2+y2+ξ2)5/2 by a single geometric parameter that describes the dis- Z x2+y2+ξ2 3(ξcosθ+xsinθ)2 , tortion of correlation hole in the density-density corre- − lation functions. These states thus explicitly illustrate wherecd isthein(cid:8)teractionstrengthandthemotiono(cid:9)fall theexistenceofgeometricdegreeoffreedominfractional theparticlesalongthez-axisisfrozentothegroundstate quantum Hall effect, as recently demonstrated in Ref. 7. of the axial harmonic oscillator π−1/4d−1/2e−z2/2d2 with Application of these states in studies of systems with dbeingtheaxialoscillatorlength. 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