11Many-fermion wave functions: structure andexamples
D. K. Sunko
Department of Physics, Faculty of Science, University of Zagreb [email protected]
Abstract.
Many-fermion Hilbert space has the algebraic structure of a free modulegenerated by a finite number of antisymmetric functions called shapes . Physically,each shape is a many-body vacuum, whose excitations are described by symmetricfunctions (bosons). The infinity of bosonic excitations accounts for the infinity ofHilbert space, while all shapes can be generated algorithmically in closed form. Theshapes are geometric objects in wave-function space, such that any given many-bodyvacuum is their intersection. Correlation effects in laboratory space are geometricconstraints in wave-function space. Algebraic geometry is the natural mathematicalframework for the particle picture of quantum mechanics. Simple examples of thisscheme are given, and the current state of the art in generating shapes is describedfrom the viewpoint of treating very large function spaces.
The standard textbook picture of quantum mechanics is that one-body wavefunctions represent possible states of individual particles, while many-bodywave functions are constructed from the one-body functions by respectingindistinguishability for a given number of particles, leading, in the case offermions, to the well-known Slater determinants. This picture is called the particle picture of quantum mechanics. In advanced textbooks a field picture isintroduced, which corresponds, as Dirac put it, to a “deeper reality,” meaningit refers from the outset to an infinite number of degrees of freedom.The particle picture has a deeper reality of its own. Any wave function of N identical fermions in d dimensions may be written [1] Ψ = D (cid:88) i =1 Φ i Ψ i , (1.1)where the Ψ i are antisymmetric and Φ i are symmetric functions of N parti-cle coordinates. If the Φ i were c -numbers, the Ψ in Eq. (1.1) would form a D -dimensional vector space. As it stands, it presents a finitely generated free a r X i v : . [ phy s i c s . g e n - ph ] S e p D. K. Sunko module, where D = N ! d − is the dimension of the module (number of itsgenerators). The Ψ still belong to the full infinite-dimensional Hilbert spacespanned by all Slater determinants, because of the additional degrees of free-dom in the symmetric functions Φ i .The scheme (1.1) has an important geometric interpretation, which wasdiscovered by Menaechmos in his construction of the cube root √ a . He inter-preted the original observation by Hippocrates of Chios, namely y = x & y = ax = ⇒ x = ax, (1.2)to mean that the solution of x − a = 0 could be found by intersecting twoparabolas. This insight is fundamental to algebraic geometry: to representan unknown object as an intersection of known objects. Extending this idea,Omar Khayyam found 19 classes of cubics by constructing various intersec-tions of conics to solve them. From a modern viewpoint, due principally toHilbert, his classes may be presented as ideals generated by second-degreepolynomials in x and y , e.g. R = P · ( x − y ) + Q · ( ax − y ) , (1.3)where P and Q are arbitrary polynomials in x and y . This equation has thesame structure as Eq. (1.1). Its defining characteristic is that simultaneouszeros of the generators are necessarily zeros of all members R of the ideal.Shapes are to a fermion many-body wave function what conics are tocubics. They are generators of all solutions to the N -fermion wave equationwhich respect the Pauli principle. Like the conics, they are not arbitrary func-tions, and also like the conics, there is a finite number of them. The efficientgeneration of shapes is the subject of current research efforts, described in thesecond part of this chapter. In order to implement the scheme (1.1) most simply, a technical step is nec-essary. The Bargmann transform [2] reads B [ f ]( t ) = 1 π / (cid:90) R dx e − ( t + x ) + xt √ f ( x ) ≡ F ( t ) . (1.4)Here f ∈ L ( R ) and F ∈ F( C ), the Bargmann space of entire functions F : C → C such that (cid:90) C | F ( t ) | dλ ( t ) < ∞ , (1.5)where dλ ( t ) = 1 π e −| t | d Re t d Im t, (cid:90) C dλ ( t ) = 1 . (1.6)The inverse Bargmann transform is then Many-fermion wave functions 3 B − [ F ]( x ) = 1 π / (cid:90) C dλ ( t ) e − ( ¯ t + x ) + x ¯ t √ F ( t ) , (1.7)where the bar denotes complex conjugation.Specifically, the Bargmann transform of Hermite functions ψ n ( x ) is B [ ψ n ]( t ) = t n √ n ! . (1.8)It has the algebraically important property that quantum numbers (statelabels) n add when single-particle wave functions are multiplied, t n t m = t n + m .Because n ! m ! (cid:54) = ( n + m )!, one must use unnormalized single-particle wavefunctions t n , with scalar product( t n , t m ) = (cid:90) C ¯ t n t m dλ ( t ) = n ! δ nm . (1.9)In three dimensions, the Hermite functions in x , y , and z are mapped ontoBargmann-space variables t , u , and v , respectively. For N particles, the vari-ables acquire indices, e.g. t i , with i = 1 , . . . , N .The technical advantage of Bargmann space is that the factorizations Φ i Ψ i in Eq. (1.1) can be interpreted literally, as factorizations of polynomials. Thesame would not be so easy in laboratory space, where quantum numbers areindices of special functions, which have opaque properties under multiplica-tion. One should bear in mind, however, that the free-module structure (1.1)is an intrinsic feature of Hilbert space, irrespective of representation.Another important technicality is that the generating function (Hilbert se-ries [3]) which counts the shapes is known [1]. For N fermions in d dimensions,it is a polynomial P d ( N, q ), which satisfies
Svrtan’s recursion
N P d ( N, q ) = N (cid:88) k =1 ( − k +1 (cid:2) C Nk ( q ) (cid:3) d P d ( N − k, q ) , (1.10)where C Nk ( q ) = (1 − q N ) · · · (1 − q N − k +1 )(1 − q k ) (1.11)is a polynomial, and P d (0 , q ) = P d (1 , q ) = 1. For example, P (3 , q ) = q +4 q + q , meaning that, of the D = 3! − = 6 shapes of N = 3 fermions in d = 2 dimensions, one is a second-degree polynomial, four are third-degree,and one is fourth-degree. A particular example of the free-module structure (1.1) has been observed inthe context of the d = 2 fractional quantum Hall effect (FQHE), albeit without D. K. Sunko noticing its generality. Adopting the notation of Ref. [5] for this subsection,one of the six shapes counted by P (3 , q ) above is the (second-degree) ground-state Slater determinant, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x x y y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1.12)which is clearly not analytic in terms of the variables z j = x j − iy j . Onecombination of the four third-degree shapes found in Ref. [1] is Ψ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z z z z z z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ( z − z )( z − z )( z − z ) , (1.13)while the other three involve terms with ¯ z j on one or both rows of the deter-minant. The sixth (fourth-degree) shape goes into itself under the exchange x j ↔ y j , like the ground state (1.12), so it is not analytic in the z j either.On the other hand, Laughlin’s N = 3 wave function for the FQHE,Eq. (7.2.12) of Ref. [5], contains a factor( z a + iz b ) m − ( z a − iz b ) m = Φ m ( z − z )( z − z )( z − z ) = Φ m Ψ , (1.14)where z a = 1 √ z + z − z ) , z b = 1 √ z − z ) , (1.15)and Φ m is a symmetric polynomial in the z j . The factorization (1.14) is wellknown and easy to prove directly, which brings Laughlin’s wave function intothe scheme (1.1). This correspondence proves, by enumeration, Laughlin’sconjecture [5] that there is only one vacuum for N = 3 and d = 2 whichsatisfies the analyticity constraint. Two identical fermions in a three-dimensional harmonic potential are the sim-plest model of a finite system. It is easy to show that the Bargmann-spaceangular momentum operator has the same form as in laboratory space, L z = − i ( x∂ y − y∂ x ) B −→ − i ( t∂ u − u∂ t ) ≡ L v , (1.16)and cyclically. Therefore, solid harmonics in Bargmann space are the samepolynomials as in real space, with ( x, y, z ) simply replaced by ( t, u, v ).The generating function for this case is P (2 , q ) = 3 q + q . The ground-statetriplet is a vector in Bargmann space just as in laboratory space, Ψ = ( t − t , u − u , v − v ) = ( Ψ , Ψ , Ψ ) , (1.17) The localization terms exp( − x / − . . . ) are dropped for clarity. Many-fermion wave functions 5 while the fourth shape, appearing in the second-excited shell, is (cid:101) Ψ = Ψ Ψ Ψ , (1.18)geometrically a pseudoscalar (signed volume). Introducing a vector of sym-metric functions along the same lines, e = ( t + t , u + u , v + v ) = ( η , η , η ) (1.19)the first-excited shell is spanned by 9 vectors η i Ψ j , which is the scheme (1.1)again. Knowing the form of the solid harmonics, it is easy to recast thescheme in rotationally invariant combinations. E.g., e · Ψ is a scalar, while( − η + iη )( − Ψ + iΨ ) = e Ψ is a state with total angular momentum andprojection l = m = 2.The second-excited shell is more interesting. In addition to squares η i ,there appear a new type of symmetric-function excitations, the discriminants ∆ i = Ψ i , i = 1 , , , (1.20)which are excitations of relative motion. There is a total of 10 excitations in-volving relative motion alone, corresponding to the one-body oscillator stateswith three quanta. They are spanned by the fourth shape (cid:101) Ψ in addition to thenine states ∆ i Ψ j . Of the latter, one can easily construct a vector triplet,( ∆ + ∆ + ∆ ) Ψ , (1.21)noting that the sum of discriminants is a rotational scalar like r . The remain-ing 7 states constitute a rotational septiplet Ψ m with l = 3, where Ψ = Ψ and (cid:101) Ψ is embedded in the m = ± (cid:101) Ψ ∼ Ψ − Ψ , − . (1.22)Even in this simplest possible example of a finite system, there appear twobands in the spectrum, because there are two shapes constraining the motion:the vector Ψ , and the pseudoscalar (cid:101) Ψ . All states in the spectrum can beclassified according to whether they contain (cid:101) Ψ or not. The classification offinite-system spectra into bands is very like Omar Khayyam’s classification ofcubics by conics: bands are ideals generated by the shapes.This example is quite revealing of the kinematic (“off-shell”) nature of theconstraint (1.1). The classification into bands is traditionally presented in thecontext of dynamics, i.e. some concrete equations of motion. Here one sees thatbands are qualitative manifestations of geometric constraints in wave-functionspace, essentially many-body effects of the Pauli principle. Simulations of strongly correlated systems must contend with the well-knownfermion sign problem [6]: it is not known in general how to update a many-
D. K. Sunko fermion wave function consistently with an initial phase convention. Varia-tional approaches avoid this problem, but at the price of limiting the possiblerange of wave functions to the form of the initial ansatz .The shape approach has the potential to obviate both problems. Becausethe number of shapes is finite, and they can be generated algorithmically inclosed form, the expression (1.1) is effectively a variational expression whichis guaranteed to exhaust the whole wave-function space. It is natural to recastthis program in probabilistic language, because the spaces involved are verylarge, so it is generally impossible to have a complete expression like (1.1)stored in memory. The principal line of research into shapes at present is togenerate an arbitrary shape with equal a priori probability. In the remainderof this chapter, the current state of these efforts will be briefly presented.The principal results have not been published anywhere so far. In particular,the algorithm described in Ref. [7] is superseded here. From now on, thepresentation is limited to the case of three dimensions.
The very large number of shapes N ! ∼ N N must be put in perspective.The highest shape is unique and has degree G = 3 N ( N − / G -th oscillator shell is the sum of theshell degeneracies up to it: G (cid:88) n =0 (cid:18) n + 22 (cid:19) = (cid:18) G + 33 (cid:19) ∼ N , (1.23)so for N particles in the first G shells the total number of states is (cid:18) ∼ N N (cid:19) ∼ N N . (1.24)Even though the number of shapes is unimaginably large, it is vanishinglysmall in comparison with the total number of states spanning the same rangeof oscillator shells. Beyond the G -th shell, no new shapes appear, so the frac-tion of shapes in the total space can be made small at will.These considerations open the way to structured simulations of largespaces, in which one optimizes in the shape subspace before taking otherstates into account. It amounts to using Eq. (1.1) with Φ i ∈ C as a reducedvariational expression in a first step. Such an approach makes physical sensebecause the nodal surface of the ground-state many-body function is expectedto be an intersection of shapes alone: if the Φ i introduced new nodes, thesewould correspond to excited states. [In Omar Khayyam’s scheme (1.3), R cansimilarly have roots which are not solutions of the cubic, because of P and Q .] Thus one expects that corrections due to Φ i / ∈ C in the second step willchange the geometry but not the topology of the ground-state nodal surface. Many-fermion wave functions 7
The highest-degree shape S N of N identical fermions is a product of three 1Dground-state Slater determinants [7] ˜ ∆ N ( t ), cf. Eq. (1.18), S N = ˜ ∆ N ( t ) ˜ ∆ N ( u ) ˜ ∆ N ( v ) . (1.25)Slater determinants in Bargmann space are Vandermonde forms˜ ∆ N ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t N − / ( N − · · · t N − N / ( N − t / · · · t N / / · · · / (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:89) ≤ i Fig. 1.1. The number of shapes approaches the number of sentences asymptoticallywhen the number of particles N → ∞ , as shown here for N = 5 , , redundancy is illustrated in Fig. 1.1. The number of sentences is equal to thenumber of shapes only when the total number of letters in a sentence is equalor smaller than N , the number of particles. Higher and higher derivatives of S N eventually reach the ground state for any finite N , after which the numberof shapes is zero, as observed in the figure. The sentence notation is adapted to an arbitrary number of particles. Anysentence acting on S N will give a shape (or zero), irrespective of the value of N . A notation focused on a given finite N is introduced for the redundancyproblem now.Because derivatives act on columns of ˜ ∆ N , the factorials allow a compactexpression for the outcome of such operations:[ n . . . n N ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t n /n ! · · · t n N N /n N ! t n − / ( n − · · · t n N − N / ( n N − · · · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (1.30)For example, ˜ ∆ ( t ) = [2 2 2] and d ˜ ∆ ( t ) /dt = [1 2 2]. This determinant is aslight generalization of the so-called confluent Vandermonde form . Writing thethree terms in the product (1.25) vertically in the order t, u, v , the variablescan be left implicit, e.g. With all words having at least two distinct letters, because ( T a ) ∆ N ( t ) = 0 [7]. Many-fermion wave functions 9 [0 1 2][1 1 2][1 2 2] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t t / 20 1 t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u u u / 21 1 u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v v / v / v v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) =1 · ( u − u ) · ( v − v )( v − v / − v / . (1.31)A symmetrized derivative is obtained by summing all possible arrangementsof columns, e.g. = [1 1 2][1 1 2][2 2 2] + [1 2 1][1 2 1][2 2 2] + [2 1 1][2 1 1][2 2 2] . (1.32)The curly brackets denote a shape symbol , or symbol for short. For a given N ,sentences and symbols are related linearly, e.g.( T U ) S = + 2 , (1.33)where the first term corresponds to the i = j terms in the double sum ( T U ) ,i.e. the word ( T U ), while the second comes from the parts with i < j and i > j . The expansion of each sentence into symbols generates at least onedistinct symbol (the second one in the example above), so the symbols arejust as good a basis for the shapes as the sentences.Constraints can be deduced for the symbols from the underlying determi-nants. For example, a symbol will be zero if all entries on any row are lessthan N − 1, because the corresponding determinant then has a row of zeros.These constraints reduce the number of symbols with respect to the numberof sentences, which is practical enough for smaller problems, but there stillremain many more symbols than shapes of a given degree in general.If one could find a set of constraints which were both efficient in the sensethat they allowed every distinct shape to be generated exactly once, and operative in the sense that they could be implemented in polynomial time in N , the problem of generating all shapes with equal a priori probability wouldbe solved. Such progress is unlikely to happen by trial and error, becausethe redundancy problem has a structure and physical meaning of its own,described in the next section. Sentences can be classified according to the total powers of T , U , and V appearing in them: P ( T, U, V ) (cid:39) ( T a U b V c ) . (1.34)As an example, take N = 4 particles and generate all shapes P ( T, U, V ) S with ( a, b, c ) = (2 , , T U V ) can be spanned by five distinct symbols, such as , , , , . (1.35)Practically, they are found as follows. The highest shape S has [3 3 3 3] oneach row, so the row-sum is 12. A derivative such as ( T U V ) subtracts 2from the first and second rows, and 4 from the third, so all shapes in the sameclass can be generated by symbols whose row-sums are 10, 10, and 8 respec-tively. When all so-far known constraints and symmetries are implemented,21 symbols are allowed. One expands them one by one in the underlyingvariables t i , u i , v i until five distinct ones are found. The dimension of the sub-space (five) is known in advance from a refinement of Svrtan’s recursion: inthe formula (1.10), drop the ( − k +1 and replace (for d = 3) (cid:2) C Nk ( q ) (cid:3) → C Nk ( T ) · C Nk ( U ) · C Nk ( V ) . (1.36)With these modifications, the recursion gives a polynomial B ( N, T, U, V ), suchthat the coefficient of T a U b V c is the dimension of the shape subspace of class( T a U b V c ). In this particular example, B (4 , T, U, V ) = . . . + 5 · T U V + . . . , (1.37)so the algorithm can stop as soon as five distinct symbols are found. Amongall the 21 symbols, one finds relations such as + − = 0 . (1.38)In the theory of invariants, such relations are called syzygies [3]. They appearcharacteristically as polynomial expressions in some determinants, which van-ish when the determinants are expanded in the underlying variables. Whenthe determinants express geometrical constraints, a syzygy shows which con-straints imply one another. An example is Desargues’ theorem: adjusting somelines to intersect is the same as adjusting some points to be collinear. Thesyzygy (1.38) is a challenge to begin developing such geometric insights forconstraints induced by the Pauli principle in wave-function space. Notice, forexample, that this particular syzygy is one-dimensional, because the first tworows in all three symbols are the same.Syzygies are algebraic expressions of the fermion sign problem. It appearsin Eq. (1.38) as three different wave functions which interfere destructivelybecause there are actually only two distinct functions. Generating only distinctsymbols is solving the sign problem, because it fixes the expression (1.1) as avariational ansatz .Syzygies of shape symbols have physical meaning. Sentences represent de-excitations (loss of quanta) of the high-lying special state S N . In a fast deex-citation cascade starting with that state, expressions like (1.38) are physical Many-fermion wave functions 11 interference effects between different branches of the cascade: not to includethem would get the branching ratios wrong. Syzygies become a problem onlywhen one wants to model the equilibrium state, requiring that all distinctwave functions be included with equal a priori probabilities.Algebraically, syzygies generate their own ideal, the syzygy ideal of theinvariant ring. Calculations in the quotient ring modulo the syzygy ideal arereal in the sense that one is not inadvertently manipulating zeros. Methodsdeveloped for such computations are of two kinds. One requires “unpack-ing” the determinantal expressions in terms of the underlying variables, suchas the algorithm described above, which discovered the basis (1.35) and thesyzygy (1.38). The other is symbolic, in the sense that some rules at the levelof the symbols themselves determine which expressions are allowed. This sec-ond type of method is the goal of the present research, because expanding thesymbols in terms of the underlying variables rapidly becomes prohibitive whenthe number of fermions increases. It amounts to finding efficient constraintson the symbols, as discusssed in the previous section. It has been pointed out before [1] that the excitations Φ i in Eq. (1.1) arecounted in a manner strictly analogous to the quantization of the electro-magnetic field. Namely, the excitations in the three directions in space aremutually independent. Every Φ i can be expanded in monomials of the form Φ xj Φ yk Φ zl , where each Φ x,y,z is a symmetric one-dimensional function by itself .A 3D excitation is just an iteration of 1D excitations.The connection of these excitations to field theory is straightforward: justlet the number of particles N → ∞ . This limit is implicit in the theory ofsymmetric functions, where one assumes by default that N is fixed but maybe arbitrarily large, as if the “supply of variables” were inexhaustible [9].Physically, the heat capacity of the electromagnetic field increases withoutbound with temperature because the field has infinitely many degrees of free-dom in a finite volume, so the supply of degrees of freedom (wave-numbers)is inexhaustible, even if not all are excited.The important difference between the excitations Φ i and field degrees offreedom is that the former have no zero-point energy, which resides in theshapes [1]. In the limit N → ∞ , the mapping of sentences to shapes (Fig. 1.1)becomes one-to-one. Notably, all words in a sentence must have at least twodistinct letters, because one-dimensional words ( T a ) give zero when actingon S N [7]. While S N is a product of 1D forms, one needs to “tie together”the spatial directions in a word in order to obtain a shape in general. Thisproperty is in sharp contrast with the excitations, which factorize into one-dimensional functions Φ x,y,z only . A field-theoretic description of fermionicvacua should map to the space of (infinitely long) sentences (1.28), not to theexcitations Φ i . It remains to be seen whether such a connection is possible. The success of the Slater determinant basis rests on the physical fact that theground-state Slater determinant is a good starting point for the description ofthe ground state and low-lying excitations of many real systems. The simplealgebraic structure of this determinant then provides the formal advantage,that one can manipulate a very large expression — a Slater determinant of N = 10 particles — without ever having to “unpack” it in terms of the one-particle labels. This advantage is realized through the well-known formalismof second quantization.The advantage disappears in strongly correlated cases, which are formallycharacterized as those where the initial ansatz needs to be more complex thana single Slater determinant. These are known as configuration-interaction ap-proaches in quantum chemistry, going back to the Heitler-London wave func-tion. One can still use the second-quantized notation, but its cost-effectivenessis ultimately compromised by the physical necessity to calculate in terms ofthe individual orbitals. The end point of such a deconstruction are quantumMonte Carlo (QMC) simulations, which rapidly become prohibitive with alarger number of particles.The present work puts back some structure into the latter efforts, moti-vated by the desire to write a strongly correlated variational ansatz withoutprejudice. A QMC calculation deals with a very large Hilbert space. However,not all states in that space are born equal. A minority, the shapes, are distin-guished by the ability to act as vacuum states, or algebraically, as generatorsof the Hilbert space as a free module. These are natural generalizations ofthe ground-state Slater determinant. Learning to manipulate these states atthe formal level, using only the shape symbols as labels, is the correspondinggeneralization of the second-quantized formalism. It promises to mobilize theapparatus of algebraic geometry and classical invariant theory for many-bodyproblems with strong correlations.The interpretation of the fermion sign problem in terms of syzygies is acase in point. The syzygies are a nuisance in equilibrium calculations, which isthe sign problem, but they also have a real meaning. Physically, the sentencesand symbols represent chains of de-excitation of the initial (highest) shape S N .A syzygy like (1.38) means that the same wave function may be constructedby apparently different laboratory preparations, e.g. photon-loss sequences.In chemists’ language, it indicates different synthetic pathways to the sameentangled state.The daunting size of the shape space, N ! in three dimensions, has twomitigating factors. One is that current simulations work in a much larger spaceby default, so one who is not afraid of N N can hardly complain that N N is too much. However, that leaves open the possibility that such heavy workfor very large N may be unnecessary, which leads to the second mitigatingfactor. Experience with actual strongly correlated systems — the FQHE [5],large molecules [10], and modern functional materials [11] — indicates a hi- Many-fermion wave functions 13 erarchichal organization of the wave function, in which comparatively fewelectrons create a correlated template, which is then extended across the sys-tem. With the developments described above, the shape formalism is alreadyoperational for N ∼ 5, which is the number of identical fermions in a d shell,so it is more applicable to real-world problems than appears from the presentarticle.Investigating the geometry of wave-function space seems worthy both asan end in itself and as a practical tool. It is a new open frontier of fundamentalquantum mechanics. I thank J. Bonˇca and S. Kruchinin for the invitation to present these re-sults at the NATO Advanced Research Workshop in Odessa. Conversationswith M. Primc and D. Svrtan are gratefully acknowledged. This work wassupported by the Croatian Science Foundation under Project No. IP-2018-01-7828 and University of Zagreb Support Grant 20283207. References 1. D.K. Sunko, Phys. Rev. A , 062109 (2016). DOI10.1103/PhysRevA.93.0621092. V. Bargmann, Communications on Pure and Applied Mathematics (3), 187(1961). DOI 10.1002/cpa.31601403033. B. Sturmfels, Algorithms in Invariant Theory , 2nd edn. (Springer-Verlag,Wien, 2008)4. K. Roˇzman, D.K. Sunko, Eur. Phys. J. Plus , 30 (2020). DOI10.1140/epjp/s13360-019-00015-05. R.B. Laughlin, in The Quantum Hall Effect, Second Edition , ed. by R.E.Prange, S.E. Girvin (Springer, 1990), pp. 233–3016. J.E. Hirsch, Phys. Rev. B , 4403 (1985). DOI 10.1103/PhysRevB.31.44037. D.K. Sunko, Journal of Superconductivity and Novel Magnetism (2016). DOI10.1007/s10948-016-3799-18. F. Bergeron, L.F. Pr´eville-Ratelle, Journal of Combinatorics (3), 317 (2012).DOI 10.4310/JOC.2012.v3.n3.a49. R.P. 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