Local two-body parent Hamiltonians for the entire Jain sequence

Using an algebra of second quantized operators, we develop local two-body parent Hamiltonians for all unprojected Jain states at filling factor $n/(2n{\sf p}+1)$, with integer $n$, and (half-)integer ${\sf p}$. We rigorously establish that these states are uniquely stabilized and that zero mode counting reproduces mode counting in the associated edge conformal field theory. We further establish an associated"entangled Pauli principle"describing these states and associated zero mode spaces, as well as an emergent SU($n$) symmetry.

Introduction. The fractional quantum Hall (FQH) effect enjoys a unique position in strongly correlated electron physics both as a fascinating physical effect [1] as well as a central juncture for the percolation of ideas between correlated electron physics and other areas of theoretical and mathematical physics. Originally, the success of the field owes much to construction principles for variational wave functions [2][3][4][5] and associated ideas to connect the latter to effective field theory [5,6]. In our opinion, the intimacy of the connection between microscopics and effective quantum field theory that is achievable in this field is, in some cases, essentially unparalleled. This is the case when the construction of a parent Hamiltonian [4,5,7,8] is possible that falls into what we term the "zero mode paradigm": The zero (energy) mode space of a positive Hamiltonian is comprised of an incompressible state as well as edge/quasi-hole excitations, where the counting of zero modes in each angular momentum sector (relative to the incompressible state) precisely matches [9,10] the mode counting in the conformal edge theory. This then unambiguously points to the edge conformal field theory associated to the state, and, thanks to bulk/edge correspondence, all universal physics are then essentially fixed through exact properties of the microscopic Hamiltonian.
While a considerable number of these very special Hamiltonians exist, they appear to be absent for a great wealth of phases that are of central importance to the theory of the Hall effect. The latter include almost all phases described by Jain composite fermion (CF) states [11], which have proven key to the understanding of the physics at Landau level (LL) filling factor ν < 1. While some Hamiltonians have been proposed for (non-Laughlin) Jain-type states [12][13][14], a zero mode paradigm has only been established at 2/5, Ref. [15]. There, some of us have argued that such a paradigm is possible in principle only for unprojected Jain states. In this case, traditional first-quantized construction principles for parent Hamiltonians face unusual challenges. The latter seek to enforce "analytic clustering properties" in the few body density matrices of zero modes [4,5,7,8,16,17]. In-deed, unprojected Jain states generally have a zero of order 2p + 1 when two particles meet at the same point. However, enforcing just this (2p + 1)-clustering property will generally lead to more exotic "parton" states [3,6,18] as the incompressible ground states when more than n = 2 LLs are present [19,20]. Actually, the (2p+1)clustering property comes from a purely holomorphic factor of the wave function while an anti-holomorphic dependence is also present. This is additional information that is not straightforwardly enforced through a local Hermitian few-body interaction.
In this work, we solve this problem for all Jain CF states at filling factors n/(2np + 1), with integer n, p. We utilize a recently developed operator formalism [21] that describes CFs as second quantized objects in Fock space. This leads to an algebraic construction of the parent Hamiltonian that represents a radical departure from the traditional constructions principles described above, and fully embraces the "guiding-center-only" approach to FQH physics that has recently been influential [22,23]. Our results have further important ramifications for the theory of frustration free lattice Hamiltonians, in that we establish a framework where these become tractable even if interactions are not strictly short ranged in generalized lattice coordinates. A close connection with the recently celebrated matrix-product structure of many FQH states [24][25][26] is anticipated, though we leave details for future work [27].
Composite fermions and zero modes. The unprojected Jain state at filling factor ν = n/(2np + 1) [28] can be defined in disk geometry as where Φ n (N ) denotes an integer quantum Hall (IQH) state of N particles in n LLs, and the z i = x i + iy i , z i = x i − iy i are the particles' complex coordinates. Φ n (N ) is by definition a state of "densest" possible electron configuration for given n and N , where ambiguities at the edge may arise for certain N that we will resolve in a manner to be made precise below. arXiv:1910.07725v1 [cond-mat.str-el] 17 Oct 2019 Equation (1) clearly has a "clustering property", where the wave function has an (2p + 1)th order zero when two particles converge to the same point. However, only for n=2 does Eq. (1) represent the densest (smallest filling factor) wave function(s) having this property. Related to that, for n=2, p=1 there is the aforementioned, welldocumented parent Hamiltonian satisfying the zero mode paradigm. To solve the general problem, we turn to an alternative characterization that has been given by some of us [21] in terms of an algebra of second quantized operators , which can be understood as "zero mode generators". We begin by summarizing the nuts and bolts of this formalism.
In first quantization, an orbital φ m, in the mth LL, m=0,1. . . , with angular momentum is a superposition of monomials of the form µ a, =z a z +a with 0 ≤ a ≤ m.
(We omit obligatory Gaussian factors.) Higher LL manybody wave functions such as Eq. (1) may be expanded in µ a, , adorned with additional particle indices. A significant advantage of the first quantized presentation is the fact that this expansion is essentially geometry independent, assuming that we limit ourselves to zero genus geometries (disk, cylinder, sphere) [23]. This is so since there is a one-to-one correspondence between the wave functions in these geometries, oncez, z (for the disk) are replaced with suitable functions of co-ordinates respecting the boundary conditions of the respective geometries. In other words, variational wave functions such as Eq. (1) are described by the same polynomials in the genus 0 geometries. To obtain a manifestly geometry independent language, and to the extent that the successful construction of a parent Hamiltonian is a direct consequence of the underlying polynomial structure, however complicated, it proves advantageous to make the monomials µ a, the essential degrees of freedom of the second quantized formalism also. For fixed a, we will think of these orbitals as constituting a "Λ-level". We thus introduce pseudo-fermion [29] operatorsc a, ,c * a, satisfying canonical anti-commutation relations where thec * a, create an electron in the orbital µ a, . These orbitals are not normalized or orthogonal (for fixed ), and hencec * a, andc a, are not Hermitian conjugates, but this will present no obstacle in the following. If desired, at the end we may always return to the canonical creation/annihilation operators c m, , c † m, of the orbitals φ m, via The (real) matrix A( ) is the only geometry-dependent aspect of this formalism. It is given in [30] for the disk/cylinder geometries.
The considerable advantage of the second quantized formalism [23], especially for multiple LLs, lies in the fact that it gives us control over an algebra of "zero mode generators" that we arguably do not have in first quantization. It is also much more conducive to recursive schemes in particle number which we will now heavily pursue. To this end we introduce the following operators, which we will think of as zero mode generators in a sense to be made precise: and which is a generalization of the operator O d introduced in Ref. [23] for multiple LLs. The operators in Eq. (4) generate an algebra (via taking sums and/or products) that we will denote by Z. The significance of this algebra is manifold [21]. It allows for a definition of composite fermion states recursimve in particle number, quite distinct from the recently fashionable matrix product presentation of fractional quantum Hall states [25,26], but it is in essence a generalization of Read's expression of the Laughlin state through an order parameter [31]. Indeed, the algebra allows for a microscopic definition of a complete set of order parameters for composite fermion states. In the present context, it will turn out that the algebra Z generates all possible zero modes when acting on the incompressible ground state. In that sense they are related to the first quantized formalism discussed by Stone [32] for the Laughlin state, possible there because n−1 a=0p a,a k (which, for n = 1 LL, is really all Eq. (4) boils down to) has a simple first quantized interpretation: It multiplies many-body wave functions with power-sum symmetric polynomials p z = z k i [23,33]. For multiple LLs, however, we need the full set p a,b k , which does not, in general, have a straightforward first quantized-interpretation [20].
Consider now Eq. (1). To resolve the "edge ambiguity" mentioned above, we will define the Slater determinant by successively filling the state µ a, with lowest available + a that has lowest not-yet-occupied a. We seek to establish a parent Hamiltonian such that Eq. (1), which we now also suitably write |Ψ n,p,N , is a zero mode of this Hamiltonian. Since general zero modes will describe edge excitations and, deeper in the bulk, quasi-hole excitations, one has the intuition [31] thatc a, |Ψ n,p,N , is also a zero mode of the Hamiltonian, namely, one describing a cluster of quasi-holes of total charge 1 inserted into |Ψ n,p,N . Anticipating that this is so, then, with the properties of thep a,b k as advertised, we must be able to interpret this as a zero mode generated by some combination ofp a,b k on top of the reference state |Ψ n,p,N −1 , or,c a, |Ψ n,p,N =Ẑ n,p,N,a, |Ψ n,p,N −1 , whereẐ n,p,N,a, is a suitable element of the algebra Z. Indeed, the relation betweenẐ n,p,N,a, and the generators (4) was made explicit in [21], but will not be needed in the following. Parent Hamiltonian for Composite Fermions. We are now ready to present the following Hamiltonian, where J runs over half-integer values with J ≥ −n, [34] are suitable generalizations of pseudopotentials, whose relation to Haldane pseudo-potentials for n = 1 was discussed in [23]. The E r a,b,J are positive constants and may be used to enforce desirable spatial symmetries. We show in [30] that positive E r a,b,J can always be chosen so as to render the resulting Hamiltonian local. The T r a,b,J may also be replaced with new linearly independent combinations without affecting the zero mode space.
Note that for fermions, T r a,b,J vanishes for even r and a = b, giving us pn 2 different pseudo-potentials at each pair-angular-momentum 2J. Assuming disk geometry, we use the conventionc a, ≡ 0 for a + < 0. A key observation is that the operators T r a,b,J andp a,b k satisfy the following commutation relation: This justifies the notion that thep a,b k are "zero mode generators": The condition for |ψ to be a zero mode of the semi-positive definite Hamiltonian (6) reads T r a,b,J |ψ = 0 for all r, J, a, b. The commutator (8) thus clearly vanishes within the zero mode subspace. Therefore, anyp a,b k acting on |ψ immediately generates another zero mode, with angular momentum increased by k. In the following, we first wish to (i) establish that the Jain state |Ψ n,p,N is a zero mode of Eq. (6), and (ii) find all zero modes of Eq. (6).
We will achieve these goals via a radical departure from established paradigms, i.e., not paying attention whatsoever to "analytic clustering properties". We will do so by utilizing the properties of the second-quantized operator algebras given above and in the following. For part (i), we give a simple induction proof in N which extends that of [35]. We give the induction step first, assuming that |Ψ n,p,N −1 is known to be a zero mode. One eas- We apply this to |Ψ n,p,N . Using Eq. (5) together with the fact that Z n,p,N,a, is a zero mode generator, i.e., T r a,b,J annihilates Eq. (5), and that a, c * a, c a, gives the total particle number N , yields T r a,b,J |Ψ n,p,N = N 2 T r a,b,J |Ψ n,p,N , or T r a,b,J |Ψ n,p,N = 0 for N > 2. So far, the only special property of the T r operators (0 ≤ r < 2p) that we have used is thatẐ n,p,N,a, is a zero mode generator as defined above. All that is left to do is to establish an induction beginning for N = 2. Indeed, the N = 2 state in the class of states |Ψ n,p,N has the wave function (z 1 − z 2 ) 2p (z 1 −z 2 ), or, in second quantization, This has angular momentum 2J = 2p−1, and the only Toperators that could possibly not annihilate the state are of the form T r 0,1,J . Acting with these operators produces for r < 2p, since indeed [36] 2p Entangled Pauli principle. Having now established that the Jain state |Ψ n,p,N is a ground state of the Hamiltonian H n,p , Eq. (6), we seek to understand the full zero mode space of these Hamiltonians. This will in particular establish the densest zero mode(s) of this Hamiltonian, whose existence is generally taken as the hallmark of incompressibility. The key to obtaining such results for Hamiltonians of the form (6) lies in the fact that there is a now well-established [15,19,23,35] general method to derive necessary conditions, in the form of "entangled Pauli principles" (EPPs) [19], on the "root states" for zero modes of such Hamiltonians. These root states encode the DNA of the incompressible fluids. Using these techniques we will now establish that a complete set of zero modes for H n,p is of the form (1), with the IQH-state Φ n replaced by S n , a generic Slater determinant with definite occupancies in n Landau-/Λ-levels (ΛLs). That indeed such states are zero modes follows easily from the fact that thep a,b k are zero mode generators, together with the convenient property that they commute [21] with the Laughlin-Jastrow flux-attachment operator. Acting on Eq. (1), thep a,b k may thus be thought of as acting directly on the IQH factor Φ n , thus, on ΛL degrees of freedom. It is easy to see that any S n can be generated out of Φ n by acting with appropriate products ofp a,b k 's. Consider now the expansion of any zero mode |ψ into Λ-level Slater-determinants: We define terms in the expansion (11) as "nonexpandable" [23] if the action with every possible "expansion" operator of the formc * a 1 , 1−xc * a 2 , 2+xc a1, 1ca2, 2 , 1 ≤ 2 , x > 0, leads to a term with zero coefficient. The root state of |ψ , |ψ root , is now defined as that part of the expansion (11) consisting only of non-expandable terms. |ψ root so defined is necessarily non-vanishing due to the finite dimensionality of the Hilbert space at a given angular momentum [19] [37]. As we show in [30], |ψ root is subject to the following EPP: i) The angular momenta of any two occupied single-particle states differ at least by 2p. ii) If they differ precisely by 2p, the root-level coefficients have the following anti-symmetry property in ΛL-indices: As in many known examples, the EPP immediately reveals the densest possible filling factor at which zero modes of the model (6) may exist. To this end, it is useful to translate the EPP into a language of SU(n)-spins, where each spin carries the fundamental representation. We may think of the ΛL index of a particle as an SU(n)index, and of its angular momentum as the position in a one-dimensional lattice. Then, permissible root states must be (linear combinations of) product states associated with certain clusters, each cluster containing up to n particles. Within each cluster, particles are 2p sites apart, and the "spin" wave function of each cluster is totally anti-symmetric. This renders the largest possible cluster an "SU(n)-singlet" of n spins (Fig. 1a), and clusters must be separated by at least 2p+1 sites. It is easy to see that the densest possible root state is just a product of such clusters at a filling factor of n/(2np+1). There are thus no zero modes whose filling factor can exceed this value in the thermodynamic limit, and the corresponding Jain state just satisfies this bound. One can, more generally, show [30] that the number of possible root-states sets an upper bound for the number of zero modes present in each angular momentum sector. A state counting argument shows, in turn, that the number of CF states of the form (Jastrow-factor)×S n precisely saturate this bound [30]. Therefore, such CF states form a complete set of zero modes of (6). It is further easy to see that the counting of such CF states in a given angular momentum sector (relative to a minimum angular momentum CF state) coincides with the number of modes in the expected edge theory of n branches of chiral fermions/bosons. This is pleasingly consistent with the fact that these zero modes are all generated by the application of the bosonic "density modes" (4) on the reference state (1), and that these modes have the simple action on the Slater-parts of CF-states stated above. The Hamiltonians constructed here are thus true representatives of the zero mode paradigm discussed initially.
Emergent SU(n) symmetry. In essence, the above establishes that root states, |Ψ root , come as products of representations of SU(n). Indeed, an underlying SU(n)symmetry is present not only at root level, but is an emergent property of the full zero mode space. To make this symmetry readily visible, we write the commutation relations of the zero-mode generators [21]: In a cylindrical geometry, where there is no constraint on the subscript k, the above commutator is just the loopalgebra of su(n). In particular, for k = k = 0, we recover the algebra of su(n) itself [38]. For the disk, we have the constraint a ≥ b − k, and the operatorsp a,b b−a still realize an su(n)-subalgebra. Therefore, the invariance of the the zero mode space under the infinite-dimensional algebra of zero-mode generators implies, in particular, its invariance under an su(n)-subalgebra. In view of the intimate connection between the zero mode generatorsp a,b k and the edge effective theory, it is not surprising that this SU(n) structure has long been associated to Jain CF states based on field theoretic grounds and/or variational constructions [39]. Through the present work, this structure becomes an exact feature of a solvable microscopic model for the Jain CF phases. For the special case n = 2, the similarity with the findings of Ref. [19] strongly suggests that much of the formalism presented here can be carried over to a rich class of "parton-like" states [3,6,18], which offer a large playground for the exploration of non-Abelian topological phases [20]. We leave this as an interesting challenge for future work.
Conclusions. The theory of the FQH effect traditionally rests on two pillars: (i) quantum-many body wave functions and (ii) effective field theories. Hamiltonians that are exactly solvable and fall into the zero mode paradigm provide a transparent connection between these pillars. Such a strong link between the microscopics and effective field theory has no counterpart in any other area of strongly correlated physics in more than one dimension. Even among the myriad phases of the FQH regime, the definitive parent Hamiltonians satisfying this paradigm cannot always be given. This used to be the situation for the most important class of phases in this regime, those described by Jain CF states. The present work exposes the underlying reasons for this and solves this problem by departing considerably from traditional Hamiltonian construction principles. The latter seek to describe a suitable few-body density-matrix via analytic clustering principles. This cannot be done adequately in the case at hand. Instead, we circumvent this problem by an algebraic characterization of few-body correlations in a suitable operator framework. Apart from giving a satisfying solution to the lack of parent Hamiltonians for Jain states [28], we expect the formalism presented here to be of profound value in the exploration of vast classes of more complicate mixed-LL wave functions realizing rich non-Abelian physics, as well as to complement traditional lowest-LL methods. We are hopeful that this angle will inspire exciting future developments.
SB Here, we establish that Hamiltonians in the general class discussed in the main text can be chosen to be local in real space coordinates. (As remarked in the introduction, it is not strictly local in the orbital "lattice" referred to by the c m , c † m operators.) This gives us opportunity to make contact with the first quantized picture of the Hamiltonian, and elaborate further why a definition in first quantization is prohibitive, very much unlike other familiar Hamiltonians in the field. We begin by slightly formalizing the setting of our main text. Let L n be the single particle Hilbert space of the nth LL. We work with single particle spaces which define the Fock spaces on which the Hamiltonians H n,M of the main text act. In the following, we will be particularly interested in the 2-particle spaces 2 H n . (We specialize again to fermions for brevity.) If we consider the limit n = ∞, i.e., including all LLs, we may give the familiar decomposition of 2 H ∞ into 2-particle subspaces of well defined total angular momentum J and relative angular momentum j r , H J,jr : The H J,jr have a basis with wave functions of the form On the other hand, it is important to appreciate that for finite n > 1, the spaces 2 H n cannot be given a basis of the form (S3): While for small enough a, b, |ψ a,b,J,jr will be contained in 2 H n , there are always those |ψ a,b,J,jr that are neither contained in 2 H n nor in its orthogonal complement (defined in 2 H ∞ ). Related to that, the 2 H n are not invariant subspaces of the relative angular momentum operator. Now, where, however, some terms on the righthand side may not be in 2 H n (though, of course, their components in the orthogonal complement will cancel.) These observations are intimately tied to the underlying reason why, unlike in the case n = 1 [S1], the Hamiltonians defined in the main text cannot be easily characterized in terms of relative angular momentum, or, more generally, clustering properties. We will still make use of the general expansion (S4) in the following.
We now first turn to the operators T r a,b,j of the main text and make their spatial dependence more explicit. We have whereψ † (z) creates an electron localized at z, and T (z 1 ,z 1 , z 2 ,z 2 ) is the wave function of the 2-particle state created by T r † a,b,j (whose dependence on r, a, b, j we leave understood for the moment). In a disk geometry, the expression of T r † a,b,j given in the main text (Eq. (6)) consists of a finite number of terms, so T (z 1 ,z 1 , z 2 ,z 2 ) is a polynomial (up to a Gaussian prefactor multiplying it), and its expansion of the form (S4) thus has a finite number of terms. Indeed, we can make that expansion explicit by Taylor expanding T in the new variables z 1 ± z 2 ,z 1 ±z 2 . For a given number of the n LLs, powers inz 1 ,z 2 are of order ≤ n − 1, and the aforementioned Taylor expansion then implies that a, b ≤ 2(n − 1) in (S4), while J = 2j is fixed. Since we are interested in the dependence on relative coordinates, we will now show that there is also a bound on j r that in particular is independent of J = 2j. Let W J ⊂ 2 H n be the (finite dimensional) 2-particle subspace at angular momentum J for n LLs. Let Y J ⊂ W J be the subspace of ii 2-particle zero modes of H n,p with angular momentum J, with orthogonal complement Y ⊥ J in W J . The results of the main text in particular imply that the 2-particle states T r † a,b,j |vac for fixed j = J/2 linearly generate this orthogonal complement Y ⊥ J . We will be interested in how Y ⊥ J fits into the total/relative angular momentum decomposition (S2). From Eq. (S3), it is clear that any element of H J,jr with j r ≥ 2p contains a factor (z 1 − z 2 ) 2p . Therefore, if the said element is also in 2 H n (i.e., in W J ) then, according to the results of the main text, it will be a zero mode of H n,p . This leads to the following inclusions: Once again, ⊥ denotes the orthogonal complement in W J , and we have used that the H J,jr are manifestly orthogonal to one another. Note that for n = 1, the last inclusion would be an identity, but not so for n > 1. This again explains why the zero mode space is not simply characterized by relative angular momentum quantum numbers. However, the inclusion does imply that j r < 2p in Eq. (S4) for all T r † a,b,j . In particular, for all of these operators, Eq. (S4) is a polynomial of bounded degree in the relative coordinates times exp( 1 8 |z 1 − z 2 | 2 ). Therefore, these are local operators. Next we consider the full Hamiltonian of Eq. (6). In what follows, we fix the values of the indices r, a, b and omit these to avoid a cumbersome notation. We will focus on a single term of the form We wish to demonstrate that positive coefficients E j can always be chosen such that the resulting operator is local. According to the above, the kernel has the following Gaussian/polynomial structure: with γ a,b,jr (j) being the polynomial coefficients describing T † j according to the above discussion. This then yields jr=−a |γ a,b,jr (j)||γ a ,b ,j r (j)| Next, we choose such that for the j-sum (over half-integer values) we have Inserting the above in Eq. (S9) gives where the degree of the polynomial in the various arguments is bounded by simple expressions in n and p as discussed above. Eq. (S7) then manifestly is a local interaction.

Geometry
Disk For fixed p and n, our results still apply to a large class of Hamiltonians. This is so since we may not only choose the constants E j within certain bounds (to preserve locality) but, as we point out in the main text, we may also replace the operators T r a,b,j with any suitable new (independent) linear combinations. In practice, one will often want to work with translationally invariant as well as local Hamiltonians. There is a canonical choice for a translationally invariant Hamiltonian: given any pair p, n, the class of Hamiltonians defined here contains exactly one member that is a 2particle projection operator. It is obtained by ortho-normalizing, at each j, the 2-particle states created by the T r † a,b,j , and forming the corresponding new linear combinations of these operators. We denote the resulting Hamiltonian by P p,n . When acting on 2-particle states, P p,n is the orthogonal projection onto the subspace of general 2-particle CF states. This subspace is translationally invariant; this implies that P p,n is also translationally invariant. This interaction can be represented by the form given on the righthand side of Eq. (S7) with a kernel K P (z 1 , z 2 , z 3 , z 4 ). We have numerically investigated the locality of this kernel for some choices of p, n, finding it to be exponentially decaying in a squared-distance-measure as suggested by Eq. (S12). The requisite ortho-normalization of T r † a,b,j -states has been carried out in terms of the matrix A( ) m,a defined in the main text. Table I provides explicit forms in various geometries. The results of calculations demonstrating the locality of K P are shown in Fig. S1.

ENTANGLED PAULI PRINCIPLE: DETAILED DERIVATION AND COMPLETENESS OF COMPOSITE FERMION ZERO MODES
Here, we will give rigorous derivations of the entangled Pauli principle and its main consequence, the completeness of the CF states as zero modes of their respective parent Hamiltonian. Note that the "root states" associated to the EPP necessarily agree with the state's thin cylinder limit, which is further identical (modulo boundary conditions) to the thin torus limit [S2-S12]. We will first derive a quintessential aspect of the EPP-that forbidding more than a single occupancy at the root level. We will establish this via a proof by contradiction. Towards this end, we let |ψ iv be a zero mode and, omitting normalization factors, write it as where a = b, |S is a N − 2-particle Slater determinant with definite occupancies in thec * -basis, and with the c * a,j , c * b,j states unoccupied. The sum over S is over N -particle Slater-determinants from the same basis, all having occupancies differing from the first term. All individual Slater determinants contain implicit phase and normalization factors. Let us now assume that the first term is non-expandable [S13] (as defined in the main text) thus contributing to the root state. Since |ψ is a zero mode, Each term in the above is, evidently, a Slater determinant, and all Slater determinants must cancel. In the first sum, every term is manifestly a Slater determinant different from |S . By 'different', we mean a different member, up to a phase, of the set of linearly independent Slater determinants generated by thec * operators. It thus follows that any term canceling the first term, |S , must originate from the second sum. Suppose then that c a,j+xcb,j−x |S ∝ |S , (S15) For x = 0, the LHS must vanish (otherwise, we will contradict the assumption that all |S have different occupancy configurations than the first term in Eq. (S13)). For x = 0, Eq. (S16) indicates that we can obtain the first term in Eq. (S13) by an "inward squeezing" process (moving two particles closer to each other while preserving angular momentum) applied to the Slater determinant |S appearing in the Slater-decomposition of |ψ , or conversely, that we can obtain |S from the first term in Eq. (S13) by an "expansion" process as defined in the main text. This contradicts the assumption that the first term in Eq. (S13) was unexpandable. Thus, no term in the root state can have double occupancies. We may proceed similarly to show that, assuming the first term is non-expandable, no decomposition of the form is possible, where 0 < ∆ < 2p is an even (odd) integer if j is an integer (half-odd integer). To this end, we consider the operators where p(x) is a polynomial. Obviously, so long as the degree of p is less than 2p, the operator T p a,b,j is a linear combination of the operators T r a,b,j of Eq. (6) (for a = b, the even part of p is irrelevant). For a given ∆, we may construct an even (odd) polynomial p e (p o ) of degree at most ∆ such that p e/o (∆/2) = 1 and p e/o (x) = 0 for with roles reversed for ∆ odd. We will now first consider the case in which a = b. By forming p + = (p e + p o )/2, we have p + (∆/2) = 1, p + (x) = 0 for x = −∆/2, . . . , ∆/2 − 1. There is then no obstruction for applying the reasoning of Eq. (S14) to 0 = T p+ a,b,j |ψ = . . . . Proceeding as before then shows that the first term in Eq. (S17) could not be part of the root state. When a = b, we can simply work with p o . It is noteworthy that this reasoning still applies when ∆ = 2p, since then p o is of degree 2p − 1 < 2p. The last case to be considered is thus that of a = b and ∆ = 2p. To this end, we expand where, again, all of the states |S have occupancies differing from those in the first two terms. Since p is integer, p o has degree 2p − 1 and p e has degree 2p. Thus only p o is available. Evaluating 0 = T po a,b,j |ψ = . . . , and using the v same arguments as in the above demonstrates that only the first two terms in Eq. (S19) can contribute to |S , yields A = −B (assuming that these first two terms are non-expandable, i.e., are root-level terms). Putting all of the pieces together thus proves the EPP of the main text. Valid root states are, therefore, linear combinations of products of clusters of the form (S20) where each cluster consists of at most n particles separated by 2p orbitals (in Eq. (S20), p = 1, n = 3 with ovals denoting an SU(n) singlet) and is totally antisymmetric in the ΛL-indices. In Eq. (S20), the ΛLs that are occupied in each cluster are indicated by subscripts 0 ≤ a < b . . . < n, etc.. (could be omitted for the singlet case [ovals] with n particles). Different clusters are separated by at least 2p + 1 orbitals. We now decompose the Hilbert space as follows where H r is spanned by all potential root states of the form (S20), and H is spanned by the following three types of states: i) single Slater determinants with two particles separated by less than 2p, ii) single Slater determinants with more than n consecutive spins separated by 2p orbitals, iii) product states similar to (S20) but where at least one of the clusters is of a different symmetry type (described by a different Young tableau), rather than totally anti-symmetric in ΛL-indices. Obviously, then, the sum in Eq. (S21) is direct. We may also withdraw to sub-spaces of given total angular momentum, so as to keep the Hilbert spaces appearing in Eq. (S21) finite dimensional, writing We can then prove the following, quite general Theorem: The number of linearly independent zero modes of the Hamiltonian (6) at given angular momentum L is at most equal to the dimension of the subspace of H r L . Proof: Let H Z L ⊂ H L be the sub space of zero modes of angular momentum L. Any |ψ ∈ H Z L can be uniquely decomposed as |ψ = |ψ r + |ψ with |ψ r ∈ H r L and |ψ ∈ H L . Assume now that n r L := dimH r L is less than n Z L := dimH Z L . The linear map P : H Z L → H r L , |ψ → |ψ r must then have non-zero kernel, so there is a nonzero |ψ ∈ H Z L with |ψ r = 0. Together with |ψ , the root state of this |ψ must then lie in H L , which violates our EPP, a contradiction Lastly, we will now show that, in the notation of the proof, n Z L = n r L for the models under consideration. As explained in the main text, this establishes a "zero mode paradigm" for these models. Recall that, in the main text, we demonstrated that CF states of the form 1≤i<j≤N (z i − z j ) 2p × "n − ΛL Slater determinant" (S23) are zero modes of H n,p . We may parametrize Slater determinants by an occupation number matrix n a,j of 1s and 0s, where 0 ≤ a < n and also n a,j = 0 for a < −j. Let N j := n−1 a=0 n a,j be the number of particles in the (angular momentum) j-column of the matrix. Assuming N j > 0, the particles in the j-column can be associated with a cluster of the form appearing in Eq. (S20) (see also Fig. (1) and caption), where the beginning orbital of the cluster has angular momentumj = 2p j <j N j + j, and the terminal orbital has angular momentumj + 2p(N j − 1), and the ΛL-indices occupied in the cluster are precisely the non-zero n a,j (j fixed!). It is easy to see that the product of the clusters associated to the Slater-determinant in this way gives a state of the from (S20), i.e., a possible root state, indeed one of the same angular momentum as the associated CF-state (S23). One could show that this product of clusters is indeed the root state of the associate CF-state. However, this is not necessary here, since it is easy to see that the mapping described here between all possible CF-states (S23) and the set of all possible root kets of the form (S20) is onto, i.e., for each such root ket, we can construct a Slater-determinant/CF state associated to it. Since all these CF states are linearly independent (as the underlying Slater determinants certainly are), and are zero modes, this proves n Z L ≥ n r L . Since we have already proven the opposite bound above, we must have n Z L = n r L .
This concludes the proof that there is a one-to-one correspondence between root "patterns" of the form (S20), and a set of linearly independent zero modes. In particular, the CF-states (S23) form a complete set of zero modes of the Hamiltonian H n,p , which is the chiefly desired property of the construction presented in this paper.