Holomorphic quantum Hall states in higher Landau levels

Eigenstates of the planar magnetic Laplacian with homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems, in particular we represent effective Hamiltonians and particle densities in higher Landau levels by corresponding quantities in the lowest Landau level.


Introduction
The state space of a charged particle moving in a homogeneous magnetic field in a plane orthogonal to the field decomposes into Landau levels, differing in energy by integral multiples of the magnetic field strength. When the position coordinates are expressed as complex numbers in the symmetric gauge the states in the lowest Landau level form a Bargmann Date: December 2019. 1 space of holomorphic functions while wave functions in higher Landau levels involve also powers of the complex conjugate position variables in the standard representation.
As noted by many people since long, and emphasized in particular in [10,11,5], a holomorphic representation of states is not limited to the lowest Landau level, where it has proved to be important for deriving some basic properties, e.g. [18,19,22,23,27]. In fact, there is a natural unitary correspondence between states in different Landau levels, in particular between higher levels and the lowest one. The main application of this correspondence is to quantum Hall physics [8,12,16,28,29] In this expository paper we discuss several ways to arrive at the holomorphic representations and derive explicit formulas for particle densities and effective Hamiltonians in higher Landau levels, expressed in terms of corresponding quantities in the lowest Landau level. The methods have appeared in various disguises in the literature before but our aim is to present them in a coherent fashion that, we hope, will be found useful for students and researchers in quantum Hall physics.
A physically appealing starting point is the decomposition of the position variables into guiding center variables and variables associated with the cyclotron motion of the particle around the guiding centers. While the components of the position operator commute, the other two sets of variables are non-commutative and satisfy canonical commutation relations. They can be represented in terms of creation and annihilation operators for two distinct and mutually commuting harmonic oscillators. One way of arriving at a holomorphic representation of states is an expansion in terms of coherent states for the harmonic oscillator of the guiding center variables [11,5]. (These are the same as the "vortex eigenstates"in [2,4,3]). The transformation between position coordinates and the coherent state variables can also be expressed in terms of an integral operator with a kernel that is a modification of the reproducing kernel of a Bargmann space [2].
A formally simpler and more direct approach is to use the creation and annihilation operators of the cyclotron oscillator to define unitary mappings between different Landau levels 1 . This gives explicit formulas for particle densities of many-body states in one Landau level in terms of polynomials in the Laplacian applied to corresponding densities in the lowest Landau level. The same formulas can alternatively be obtained by a Fourier transformation, exploiting the factorization of the exponential factor in the guiding center and cyclotron variables respectively.
The main application of these considerations is in quantum Hall physics [8,12,16,28,29]. In this context, an electron gas is confined to two spatial dimensions and submitted to a magnetic field large enough to set the main energy scale. The quantization of the kinetic energy levels then becomes the salient feature. In the full plane, each level is infinitely degenerate, but for a finite area the degeneracy is proportional to the area times the field strength. For extremely large values of the latter, the lowest Landau level is degenerate enough to accommodate all electrons without violating the Pauli principle. For smaller values of the field several Landau levels can be completely filled with electrons and become inert in first approximation. The physics then boils down to the motion of the electrons in the last, partially filled, Landau level.
In both cases only one Landau level has to be taken into account, and an effective model of widespread use in the literature is given in terms of a Hamiltonian acting on holomorphic functions. We review this first, before describing in more details the unitary mappings between Landau levels. The remarkable fact is that the dependence of the effective Hamiltonian on the Landau level it corresponds to is quite simple and transparent. An intuitive explanation (albeit not the most direct one from a computational point of view) is that the good variables to use are not the position variables but rather those of the guiding centers. The Landau level index, which fixes the energy of the cyclotron motion, is encoded in a form factor in Fourier space that modifies external and interaction potentials via a differential operator. In particular, the unitary mappings between Landau levels map multiplication by potentials to operators of the same kind.
One salient feature of the effective operators acting on holomorphic functions is that they naturally suggest variational ansätze for their ground states, which become exact for certain truncated models. The Laughlin state [14,15] is the most emblematic of those, and much of our understanding of the fractional quantum Hall effect rests on its remarkable properties. In the last section we apply our formulas to Laughlin states in an arbitrary Landau level, computing their density profiles and extending rigidity results from [18,19,27,22].
Acknowledgements. We had helpful conversations regarding the material below with Thierry Champel, Søren Fournais and Alessandro Olgiati. We received funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement CORFRONMAT No 758620).

Projected Hamiltonians and densities in quantum Hall physics
Let us start from the many-body Hamiltonian (in symmetric gauge) for interacting 2D electrons in a constant perpendicular external magnetic field B and a one-body potential V Here w is the radial repulsive pair interaction potential, modeling 3D Coulomb interactions in QH physics, but more general choices are also of interest. The one-body potential V incorporates trapping in a finite size sample, plus the electrostatic potential generated by impurities. In the sequel vectors r = (x, y) ∈ R 2 will very often be identified with complex numbers z = x + iy ∈ C.
As appropriate for electrons we consider the action of H on the fermionic antisymmetric space In FQH physics, the energy scales are set, by order of importance: first by the magnetic field, second by the repulsive interactions, third by the one-body potential. Our discussion in the sequel will reflect this.
Landau levels. For large B it is relevant to restrict particles to live in an eigenspace of −i∇ r + B 2 r ⊥ 2 . Denote the n-th Landau level. The lowest level (n = 0) will be denoted LLL; it is made of analytic × gaussian functions: The corresponding fermionic spaces for N particles will be denoted by nLL N and LLL N : Hamiltonians in the LLL. Consider projecting (2.1) to the LLL. The first term is just a constant, the others can be expressed using the canonical basis Projecting (2.1) to the LLL leads to where (|ϕ m ϕ m |) ij projects 2 the relative coordinate r i − r j on the state ϕ m . Similarly |ϕ m ϕ ℓ | j is the operator mapping ϕ ℓ to ϕ m , acting on the j variable only. Usually in FQH physics one focuses attention on the interaction term. There are no off-diagonal terms in it because w is radially symmetric. The coefficients ϕ m |w|ϕ m are called "Haldane pseudo-potentials", and the main observation in the theory of the FQHE is that, if one truncates the sum (2.6) at m = ℓ − 1, then the Laughlin state is an exact ground state (L 2 -normalized by the constant in front). One can then argue, and prove to some extent [18,19,27,22], that such functions and natural variants are extremely robust, in particular to the addition of the external potential V .
Hamiltonians in higher Landau levels. Consider now a situation where n − 1 Landau levels are filled, so that additional electrons must sit in the higher ones, because of the Pauli principle. It is a common procedure in the FQH physics community [12,9,29] to model this situation using LLL functions again. The basis for this reduction is the following statement, contained in one form or another in a number of sources, in particular [11,5,29,12,21,6]. where P nLL orthogonally projects all particles into the nLL, i.e. it is the orthogonal projector from L 2 asym (R 2N ) to nLL N .
Then, for any n there exists an effective external potential V n and an effective (radial) interaction potential w n , depending only on V, w and n such that is unitarily equivalent to the LLL Hamiltonian H LLL Vn,wn , defined as in (2.6), and acting on LLL N . The effective n-th level potentials are as follows: where ∆ is the Laplacian and L n the Laguerre polynomial We shall give two proofs of the Theorem in Section 5. Note that the constant we subtract from H nLL is just the magnetic kinetic energy of N particles in the nLL.
What the Theorem says is that one can profit from the nice properties of the LLL to study phenomena in other Landau levels. This is particularly relevant because the main features are supposed not to depend very much on the potentials V n , w n entering (2.6). In particular the Laughlin states have equivalents in any Landau level.
Since potential energies are integrals of potentials against particle densities Theorem 2.1 can be seen as a corollary of a general result about particle densities of a many body states in different Landau levels. We recall that the k-particle density of an N -particle state with wave function Ψ(r 1 , . . . , r N ) is by definition Theorem 2.2 (Particle densities in the n-th Landau level).
There is a unitary mapping U N,n : nLL N → LLL N such that if Ψ 0 = U N,n Ψ n ∈ LLL N with Ψ n ∈ nLL N then for all k and for all test functions χ Theorem 2.1 follows as a corollary if one takes χ = V with k = 1, respectively χ = w with k = 2.
In the following we define in more detail (and in several related but distinct ways) the unitary mappings between Landau levels (see (4.12)), and discuss the proofs of Theorems 2.1 and 2.2. The physically most appealing way to interpret these unitaries is to see them as replacing the physical coordinates of electrons by the coordinates of the guiding centers of their cyclotron orbits, mathematically implemented through the use of coherent states.
Indeed, in the LLL the position coordinates and the guiding center coordinates are really two different names for the same thing as will be evident in Section 4.2. Moreover, the quantum mechanical spread of both coordinates is of the order of the magnetic length ∼ B −1/2 . The cyclotron radius in Landau level n has an extra factor √ n + 1. Thus it is plausible that for large B and small n the difference between position and guiding center coordinates, and the non-commutativity of the latter, is not of much significance in thermodynamically large systems, i.e., for large N , provided the magnetic length stays much smaller than the interparticle distance.
Although the coherent state approach offers a satisfactory physical picture it is not always the most convenient one from a computational point of view. This motivates our review of alternate routes to the mappings between levels.
We also take the example of Laughlin states to explain how to deduce properties of the actual wave-functions in nLL N minimizing effective energies from their representation in the LLL using the above unitary map. This amounts to saying that the density in guiding center coordinates can to a large extend indeed be identified with the true, physical, density in electron coordinates. We believe this is crucial for the understanding of the efficiency of the correspondence between Landau levels in FQH physics.

The Landau Hamiltonian and the two oscillators
3.1. The cyclotron oscillator. The magnetic Hamiltonian of a particle of charge q and effective mass m * , moving in a plane with position variables r = (x, y), is where π = (π x , π y ) = p − qA (3.2) is the gauge invariant kinetic momentum with A the magnetic vector potential and the canonical momentum. We assume a homogeneous magnetic field of strength B perpendicular to the plane and choose the symmetric gauge Moreover, we choose units and signs so that |q| = 1, qB ≡ B > 0, = 1 and m * = 1. Then and the kinetic momentum components satisfy the CCR In terms of the creation and annihilation operators with [a, a † ] = 1 the Hamiltonian is Powers of a † generate normalized eigenstates with aϕ 0 = 0 and the energy eigenvalues In position variables the corresponding wave functions are Choosing units so that B = 2, or equivalently, defining z = 1 Also, the gaussian factor e −(|x| 2 +|y| 2 )/4ℓ 2 B becomes e −|z| 2 /2 . For computations it is often convenient to use the corresponding operatorsâ † ,â, acting on the pre-factors to the gaussian and defined by (3.14) These areâ † =z − ∂ z ,â = ∂z.
(3.15) In the sequel we shall generally use the hatˆon operators and functions to indicate that the gaussian normalization factors are excluded.
Besides the standard definition z = x + iy, other complexifications of R 2 are possible and can be useful, as stressed in [11].
3.3. The guiding center oscillator. The classical 2D motion of a charged particle in a homogeneous magnetic field consists of a cyclotron rotation around "guiding centers". The quantization of the cyclotron motion is the physical basis for the energy spectrum (3.9), and the creation operators a † generate the corresponding harmonic oscillator eigenstates. Every energy eigenvalue is infinitely degenerate, due to the different possible positions of the guiding centers.
Quantum mechanically the dynamics of the guiding centers is described by another harmonic oscillator commuting with the first one. One arrives at this picture by splitting the (gauge invariant) position operator r into the guiding center part R and the cyclotron part with n the unit normal vector to the plane. Both R and R are gauge invariant and they commute with each other. On the other hand the two components of (R x , R y ) of R do not commute and likewise for the components of R. More precisely, we have and the commutation relations The creation and annihilation operators for R are the same as (3.6), Those for the guiding center, on the other hand, are Note the different signs compared to (3.21) due to the different signs in (3.20). We have [b, b † ] = 1 and in complex notation (3.25) The splitting (3.17) corresponds to While the operators a † , a increases or decrease the Landau level index, the operators b † , b leave each Landau level invariant. Pictorially speaking we can say that operators associated with the cyclotron oscillator move states "vertically"while those associated with the guiding center oscillator move them "horizontally". With ϕ 0,0 = ϕ 0 the common, normalized ground state for both oscillators, form a basis of common eigenstates of the oscillators with ϕ n,0 being the previously defined ϕ n . For fixed n the states ϕ n,m , m = 0, 1, . . . generate the Hilbert space of the n'th Landau level, which we shall denote by nLL. The lowest Landau level will be denoted LLL.
Using complex coordinates the wave functions with n = 0 respectively m = 0 are More generally, the wave functions can be written in terms of associated Laguerre polynomials. They are eigenfunctions of the angular momentum operator in the symmetric gauge (acting on the pre-factor to the gaussian)L = z∂ z −z∂z With coherent states. A coherent state associated with the guiding center oscillator in the nLL with parameter Z ∈ C is defined in a standard way [7,13] as The overlap of two coherent states is Moreover, is the projector on nLL, where d 2 Z := i 2 dZ ∧ dZ is the Lebesgue measure in the plane. Either (4.9) or (4.10) can be taken as the definition of U n . The unitary map is that used in Theorem 2.1.
The function Ψ(Z,Z) coincides with the LLL wave function of U n ψ if Z is identified with the complex position variable z = x + iy. Note, however, that Z is associated with the (non-commutative) components of the guiding center operator R rather than the (commutative) position operator r. By the definition (4.6) Ψ depends linearly on ψ; the alternative definition Ψ = ψ|Z, n , that is sometimes used, leads to an anti-unitary correspondence.

4.2.
With integral kernels. Consider the coherent state (4.1) without the gaussian normalization factors as as function of Z; z,z: The coherent state is an eigenstate of the annihilation operatorb = ∂ z with eigenvalue Z, so F (Z; z,z) = f (z,z)e Zz . (4.14) Furthermore, F is an eigenstate ofâ †â = (z − ∂ z )∂z to eigenvalue n which leads to   This formula was derived in a different way in [2] and appears there (in slightly different notation) as Equation (34). The inverse map is given by (4.20) Note that G can be written as g(z − Z) e −2i Im (zZ) (4.21) where g is essentially concentrated in a disc of radius ∼ √ n + 1 and the factor is a phase factor. Recall also that the length unit is √ 2ℓ B ∼ B −1/2 . A further remark is that for n = 0 G is the reproducing kernel in Bargmann space, confirming again that in the LLL Ψ and ψ are the same function on C just with different names for the variables. The phase factor in G is essential for this to hold.

With ladder operators.
A direct approach to the correspondence nLL ↔ LLL, bypassing the coherent states, starts from (4.9), noting that U n = (n!) −1/2 a n restricted to nLL (4.22) and hence U −1 n = (n!) −1/2 (a † ) n restricted to LLL. (4.23) Using the representations (3.15) for the creation and annihilation operators we conclude that the following holds:  Conversely, the wave function of ψ n = U −1 Note that Equation (4.25) implies in particular that the factor f n (z) to the highest power n ofz determines uniquely the factors to the lower powersz ν : The state is thus completely fixed by the holomorphic function f n and the Landau index n.
Incidentally, these considerations also lead to a method for projecting functions to the lowest Landau level: with arbitrary analytic functions g ν . It's orthogonal projection into LLL is where g (ν) = ∂ ν z g. This is well-known as the recipe "move allz factors to the left and replace them by derivatives in z", see e.g. [12]. For completeness we give the simple proof: Proof. The previous considerations lead to a method for splitting a state φ ∈ n k=0 kLL into its components in the different LL: Start with a wave function as in (4.27). It's component ψ n in the nLL is then given by (4.25) with f n := g n and f ν for 0 ≤ ν ≤ n − 1 defined by (4.26). The differenceφ = φ − ψ n is now in n−1 k=0 kLL and we can repeat the procedure with n replaced by n − 1, φ byφ etc. until we obtain the splitting ϕ = n k=0 ψ k with ψ k ∈ kLL. By induction over n, using that 3. Apply the differential operator ∂z n-times to the pre-factor of the Gaussian. Equivalently: Expand the pre-factor in powers ofz and keep only the highest power. The inverse mapping, LLL → nLL, is achieved by applying the differential operator (z − ∂ z ) n to the analytic function representing the state in the LLL. The last method is formally the simplest and in the next section we shall use it to discuss particle densities in higher Landau levels in term of their counterparts in the lowest Landau level.

k-particle densities and the nLL Hamiltonian, proofs of the Theorems
We now have all the necessary ingredients to prove Theorems 2.1 and 2.2. We provide two slightly different approaches. 5.1. Many body states and k-particle densities. All previous considerations carry straightforwardly over to many-body states in symmetric or anti-symmetric tensor powers nLL N ≡ nLL ⊗s,aN of single particle states by applying the single particle formulas to each tensor factor.
Let Ψ n be a state in nLL N with wave function Expanding in powers ofz i we can write The sum is here over N -tuples (ν 1 , . . . ν N ) such that ν k < n for at least one k. The functions f n and f ν 1 ,...,ν N are holomorphic and the latter are, in fact, derivatives of f n , cf. (4.26). The state Ψ 0 = U n Ψ n in LLL N has the wave function The wave function ψ n can now be written (5.2) Next we consider the k-particle density of Ψ n , defined by The density ρ (k) 0 of Ψ 0 = U n Ψ n is given by the same formula with n = 0. To prove Theorem 2.2 (which then implies Theorem 2.1) we have to compare ρ (k) n and ρ (k) 0 , integrated against trial functions χ(z 1 ,z 1 , . . . , z k ,z k ). We use the following lemma: Lemma 5.1 (Projecting potentials in the LLL). Let ψ(z,z) = f (z)e −|z| 2 /2 , ϕ(z,z)e −|z| 2 /2 ∈ L 2 with holomorphic f and g and χ : R 2 → R a smooth function. We have that Proof. This follows, using ladder operators, by partial integration and induction over n, recalling that ∂ zf (z) = ∂zf (z) = 0, and the same for g. Details are in Appendix A.
Applying this to each variable in a many-body wavefunction we obtain the connection between the densities ρ (k) n and ρ which proves Theorem 2.2. When χ is an external potential, V , we revert to the notation r for the position variables and write the potential energy for a state in the n-th Landau level as where V n (r) is the effective potential when the state is represented in the lowest Landau level. By (5.5) with k = 1 we have Likewise, for a two body interaction potential w(r 1 , r 2 ) we have ρ (2) n (r 1 , r 2 )w(r 1 , r 2 )d 2 r 1 d 2 r 2 = ρ

Lemma 5.2 (Plane waves projected in Landau levels).
For any q ∈ R 2 , identify e iq·r with the corresponding multiplication operator on L 2 (R 2 ), where r is the spatial variable. Let R be the guiding center operator defined in Section 3.3, Π n the orthogonal projector on nLL and U n : nLL → LLL the inter-LL unitary map.
We have that U n Π n e iq·r Π n U * n = L n |q| 2 4 e − |q| 2 8 Π 0 e iq·R Π 0 = L n |q| 2 4 Π 0 e iq·r Π 0 (5.10) with the Laguerre polynomial Let ψ n , φ n ∈ nLL and f, g be their holomorphic representations, ie Recalling Lemma 4.1, the matrix elements of U n Π n e iq·r Π n U * n are given by with V (r) = e iq·r . Thus, that the left-hand side of (5.10) equals the right-hand side follows from Lemma 5.1. We now provide another proof, using guiding center coordinates rather than ladder operators. This also connects with the middle expression in (5.10).
Proof. Some of the following computations can be found in a variety of sources, e.g. [12, Proof of Theorem 3.2] or [9]. First note that if A is a function of a † , a and B of b † , b, then This is a consequence of the fact that the two commuting harmonic oscillators (3.6) and (3.22) can be represented, in a unitarily equivalent way, in the tensor product of two spaces with basis vectors ϕ n,0 and ϕ 0,m respectively. In this representation ϕ n,m = ϕ n,0 ⊗ ϕ 0,m and the operators A and B act independently on each of the tensor factors. One can also pick directly A, B to be polynomials in creation and annihilation operators and use the CCR to prove the claim. We apply (5.12) to compute the matrix elements of exp(iq · r), q ∈ R 2 , in the nLL. With q = (q x , q y ), q = q x + iq y , r = (x, y), z = x + iy and further employing Since the a # 's and the b # 's commute it follows that exp (iq · r) = exp i 2 (qa † +qa) exp i 2 (qb † + qb) and thus by (5.12) ϕ n,m ′ | exp(iq · r)|ϕ n,m = ϕ n,0 | exp i 2 (qa † +qa) |ϕ n,0 ϕ 0,m ′ | exp i 2 (qb † + qb) |ϕ 0,m .
By the Baker-Campbell-Hausdorff formula e X+Y = e X e Y e − 1 for two operators commutating with their commutator (recall that [a, a † ] = 1) we can write and thus Expanding the exponential and using a k ϕ n,0 = (n − 1) · · · (n − k + 1) ϕ n−k,0 we obtaiñ Thus, recalling the definition of the guiding center coordinate R in Section 3.3, (5.15) implies the first equality in (5.10).
To obtain the second equality we subtract i 2 (qa † +qa) from (5.14) to get, employing the Campbell-Hausdorff formula again, On the LLL exp − i 2q a is the identity, so the second equality in (5.10) follows. To deduce Equations (5.6)-(5.9) and hence Theorem 2.1 it only remains to write the Fourier decompositions

Laughlin states in higher Landau levels
As already mentioned, a crucial approximation in FQH physics is to truncate the Haldane pseudo-potential series in the LLL Hamiltonian (2.6) to obtain the Laughlin state (2.7) as an exact ground state of the translation invariant problem V ≡ 0.
In view of Theorem 2.1, it is desirable to do the same in a higher Landau level, at the level of the effective Hamiltonian H LLL 0,wn and (2.7) then becomes an exact ground state after the unitary mapping to the LLL. In this section, we explain how the previous considerations allow to study the properties of the corresponding physical wave-function (that is, as expressed in the position coordinates, rather than in the guiding center coordinates).
6.1. Density estimates on mesoscopic scales. Consider a Laughlin state in the LLL N Here ℓ = 1, 3, . . . for fermions and ℓ = 2, 4, . . . for bosons. We denote by |Ψ 0,N (r, r 2 , . . . , r N )| 2 dr 2 . . . dr N (6.3) the corresponding 1-particle density. According to Laughlin's plasma analogy [14] the density profile is for large N well approximated by a droplet of radius (ℓN ) 1/2 and fixed density (πℓ) −1 , Indeed, by a rigorous mean-field analysis it was proved in [24] that this approximation holds in the sense of averages over discs of radius N α with 1/2 > α > 1/4. More generally, the k-particle densities are well approximated in this sense by the k-fold tensor power of the flat density if N → ∞. The more refined analysis of classical Coulomb systems in [17,1] leads to an extension of this result down to mesoscopic scales N α for all α > 0. We shall now use results from [17] to estimate the density of Laughlin states in higher Landau levels.
The Laughlin state corresponding to (6.1) in the n-th Landau level nLL N is with wave function (cf Lemma 4.1) This is, in electronic position variables, the exact ground state of a Hamiltonian obtained by • Projecting the physical starting point (2.1) in the nLL N .
• Unitarily mapping the result down to an effective Hamiltonian on LLL N using Theorem 2.1. • Neglecting the one-body potential V n and truncating the Haldane pseudo-potential series of the interaction potential w n .
The Hamiltonian obtained this way acts on LLL N , and its exact ground state is a LLL Laughlin state in guiding center variables. Lifting it back up to the nLL N results in (6.5): We now vindicate a natural expectation: the density of Ψ Lau n,N is very close, for large N , to that of Ψ Lau 0,N on length scales much larger than the magnetic length (1 in our units). This is because electron coordinates and guiding center coordinates differ only on the scale of a cyclotron orbit, which is much smaller than the thermodynamically large extent of the states themselves.
The potential in Equation (1.14) of [17] is here |z| 2 so Mean(ξ) = 0. Equation (1.17) in [17] and ̺ flat N χ r,ε,z 0 ∼ r 2 now lead directly to (6.8) above. The dependence of the error term on n and ε cannot be deduced from Equation (1.17) in Remark 1.2 alone, however.
For the fluctuations we need ∇ξ 2 , according to the "Mesoscopic case" of Theorem 1 in [17]. Since ξ = L n − 1 4 N −1 ∆ χ r,ε,r ′ 0 , Equation (6.9) is a consequence of this theorem and of a simple L 2 -estimate of the gradient of the test function χ r,ε,r ′ 0 .
6.2. Rigidity estimates. In [25,26,18,19,27,22] we have investigated rigidity/stability properties of the LLL Laughlin state. The question is now the response of the Laughlin function to a slight relaxation of the assumptions made in it's derivation, namely that one could in first approximation neglect the one-body potential and truncate the Haldane pseudo-potential series to a finite order. If one assumes the validity of a certain "spectral gap conjecture" (see [23,Appendix] and references therein), investigating this question basically means minimizing the one-body energy and the residual part of the interaction within the full ground eigenspace of the truncated interaction energy (cf degenerate perturbation theory). Our main conclusion was that this problem could be solved to leading order in the large N limit by generating quasi-holes on top of Laughlin's wave function. We now want to quickly explain how this can be generalized to Laughlin states in higher levels. We discuss only the adaptation of [19,27] for the response to one-body potential. One could consider as well the response to smooth long-range weak interactions as in [22], but for brevity we do not write this explicitly. We take v : R 2 → R + to be a smooth one-body potential, growing polynomially at infinity. We scale it so that it lives on the scale of the Laughlin wave-function: As discussed in the aforementioned references these assumptions can be relaxed to some extent. The main observation is that after the reduction of the nLL interacting Hamiltonian discussed in the previous subsection, any multiplication of the LLL Laughlin state by a symmetric analytic function F still yields an exact zero-energy eigenstate in guiding center variables. It is thus relevant to consider the action of the one-body potential V N on the ground-state space of the truncated interaction Hamiltonian. In electron variables the latter is Lau with F analytic and symmetric (6.10) where the LLL Laughlin state is as in (2.7). For any many-body wave-function Ψ N,n ∈ L ℓ N,n we define it's one-particle density as |Ψ N,n | 2 = 1 .
What is much less obvious is that this upper bound is optimal in the large N limit: Theorem 6.2 (Response of higher LL Laughlin states to external potentials). With the previous notation we have, for any fixed n, ℓ ∈ N E(N, n, ℓ) e(N, n, ℓ) → N →∞ 1 (6.13) The n = 0 version of the above was proved in [19,27]. The adaptation to higher n follows from the tools therein, together with the representation of U n V n U * n discussed at length in Section 5. We do not give details for brevity. We however point out that consequences for minimizing densities also follow, so that the density of a (quasi)-minimizer for (6.11) is approximately flat with value (πℓ) −1 on an open set to be optimized over, and quickly drops to 0 outside. This is in accordance with the physical picture of the system responding to external potentials by generating quasi-holes to accommodate their crests. Indeed, the interpretation of the states in (6.12) is that the zeroes of the analytic function f correspond to the location of quasi-holes in guiding center coordinates.

Appendix A. A calculation
We give here the details of the computation leading to (5.4). Define using induction over n.
using successively the discussion of Section 3.2, the relation aâ †(n+1) =â †(n+1)â + (n + 1)â †(n) , the fact thatâ cancels holomorphic functions and integration by parts in complex variables. The last equality in the computation above says that