A Barycentric coordinates for many-body pseudopotentials

Geometric construction of Quantum Hall clustering Hamiltonians

Abstract

Many fractional quantum Hall wave functions are known to be unique highest-density zero modes of certain “pseudopotential” Hamiltonians. While a systematic method to construct such parent Hamiltonians has been available for the infinite plane and sphere geometries, the generalization to manifolds where relative angular momentum is not an exact quantum number, i.e. the cylinder or torus, has remained an open problem. This is particularly true for non-Abelian states, such as the Read-Rezayi series (in particular, the Moore-Read and Read-Rezayi states) and more exotic non-unitary (Haldane-Rezayi, Gaffnian) or irrational (Haffnian) states, whose parent Hamiltonians involve complicated many-body interactions. Here we develop a universal geometric approach for constructing pseudopotential Hamiltonians that is applicable to all geometries. Our method straightforwardly generalizes to the multicomponent SU() cases with a combination of spin or pseudospin (layer, subband, valley) degrees of freedom. We demonstrate the utility of our approach through several examples, some of which involve non-Abelian multicomponent states whose parent Hamiltonians were previously unknown, and verify the results by numerically computing their entanglement properties.

73.43.-f

I Introduction

Among the most striking emergent phenomena in condensed matter are the incompressible quantum fluids in the regime of the fractional quantum Hall effect Tsui et al. (1982). A key theoretical insight to understanding the many-body nature of such phases of matter was provided by Laughlin’s wave function Laughlin (1983). Shortly thereafter, Haldane Haldane (1983) realized that the Laughlin state also occurs as an exact zero-energy ground state of a certain positive semi-definite short-range parent Hamiltonian which annihilates any electron pair in a relative angular momentum state, but which assigns an energy penalty for each state with .

Haldane’s construction of the parent Hamiltonian for is just one aspect of a more general “pseudopotential” formalism that applies to quantum Hall systems in the infinite plane or on the surface of a sphere Haldane (1983). In those cases, the system is invariant under rotations around at least a single axis, and by virtue of the Wigner-Eckart theorem, any long range interaction (such as Coulomb interactions projected to Landau level) decomposes into a discrete sum of components . The different components are quantized according to the relative angular momentum , which is odd for spin-polarized fermions and even for spin-polarized bosons. The unique zero mode of the pseudopotential at is precisely the Laughlin state. Furthermore, it is the densest such mode because all the other states, at or any filling , are separated by a finite excitation gap as indicated by overwhelming numerical evidence. In numerical simulations, it is furthermore possible to selectively turn on the magnitude of longer range pseudopotentials (,, etc.), and verify that the Laughlin state adiabatically evolves to the exact ground state of the Coulomb interaction Haldane (1990). The corrections induced to the Laughlin state in this way are notably small (below 1% in finite systems containing about 10 particles), and the gap is maintained during the process Fano et al. (1986). These findings constitute an important support of Laughlin’s theory.

Remarkably, the existence of pseudopotential Hamiltonians is not limited to the Laughlin states. Generically, more complicated parent Hamiltonians arise in the case of quantum trial states with non-Abelian quasiparticles, which are one important route towards topological quantum computation Nayak et al. (2008); Pachos (2012). For example, the celebrated Moore-Read “Pfaffian” state Moore and Read (1991), believed to describe the quantum Hall plateau at filling fraction Greiter et al. (1991), was shown to possess a parent Hamiltonian which is the shortest-range repulsive potential acting on 3 particles at a time Greiter et al. (1991, 1992); Read and Rezayi (1996); Read (2001). This state is a member of a family of states – the parafermion Read-Rezayi (RR) sequence – where the parent Hamiltonian of the th member is the shortest-range -body pseudopotential Read and Rezayi (1999). Similar parent Hamiltonians have been formulated for non-Abelian chiral spin liquid states Greiter and Thomale (2009). More recent analysis has allowed to resolve the structure of quantum Hall trial states interpreted as null spaces of pseudopotential Hamiltonians Jackson et al. (2013); Ortiz et al. (2013); Chen and Seidel (2015); Mazaheri et al. (2015).

The knowledge of the parent Hamiltonian class is crucial for the complete characterization of a quantum Hall state. In certain cases, when the state can be represented as a correlator of a conformal field theory (CFT), the charged (quasihole) excitations can be constructed using the tools of the CFT Moore and Read (1991). (Note that such approaches become rather cumbersome for quasielectron excitations Hansson et al. (2009), and even more so in the torus geometry Greiter et al. ().) The tools of CFT relating wave functions to conformal blocks certainly become less useful when information about neutral excitations is needed. The knowledge of the parent Hamiltonian is thus indispensible, e.g. for estimating the neutral excitation gap of the system. In some cases, the neutral gap can be computed by the single-mode approximation Girvin et al. (1985, 1986), which is microscopically accurate for Abelian states Repellin et al. (2014), but requires non-trivial generalizations for the non-Abelian states Repellin et al. (2015). Similarly, entanglement spectra may help to characterize the elementary excitations solely deduced from the ground state wave function Li and Haldane (2008); Thomale et al. (2010); Sterdyniak et al. (2011), but only unfold their full strength as a complementary tool to the spectral analysis of the associated parent Hamiltonian. While a CFT underlies the structure of entanglement, conformal blocks, and clustering properties within most quantum Hall states of interest Jackson et al. (2013), it is desirable to have a general framework that complements it with a pseudopotential parent Hamiltonian.

An appealing “added value” of pseudopotentials as building blocks of parent Hamiltonians is the possibility of discovering new states by varying the considered pseudopotential terms and searching for new zero-mode ground states. This strategy is exemplified by the spin-singlet Haldane-Rezayi state Haldane and Rezayi (1988), initially discovered as a zero mode of the “hollow core” interaction between spinful fermions at half filling. Another example is the 3-body interaction of a slightly longer range than the one that gives rise to the Moore-Read state. It was found Greiter (1993) that such an interaction has the densest zero-energy ground state at filling factor , subsequently named the “Gaffnian” Simon et al. (2007a). These examples illustrate that a systematic description of an operator space of parent Hamiltonians holds the promise of the discovery of previously unknown quantum Hall trial states.

All appreciable aspects of parent Hamiltonians discussed so far are independent of the geometry of the manifold in which the quantum Hall state is embedded. For several purposes, however, knowing the parent Hamiltonian on geometries without rotation symmetry, such as the torus or cylinder, is particularly desirable. For example, quantum Hall states only exhibit topological ground state degeneracy on higher genus manifolds such as the torus Wen and Niu (1990). Accessing the set of topologically degenerate ground states further allows to extract the modular matrices which encode all topological information about the quasiparticles Wen (1990); Zhang et al. (2012). Furthermore, parent Hamiltonians on a cylinder or torus can be used to derive solvable models for quantum Hall states when one of the spatial dimensions becomes comparable to the magnetic length Lee and Leinaas (2004); Seidel et al. (2005); Jansen (2012); Wang and Nakamura (2013); Soulé and Jolicoeur (2012); Papić (2014). It has been shown that such models can be used to construct “matrix-product state” representations for quantum Hall states, and in some cases can be used to study the physics of “non-unitary” states and classify their gapless excitations Seidel and Yang (2011); Papić (2014); Weerasinghe and Seidel (2014).

A systematic construction of many-body parent Hamiltonians for quantum Hall states was first undertaken by Simon et al. for the infinite plane or sphere geometry Simon et al. (2007b); Davenport and Simon (2012). This approach relies on the relative angular momentum which in this case is an exact quantum number. As such, it cannot be applied to the cylinder, torus, or any quantum Hall lattice model such as fractional Chern insulators, for which the analogue of pseudopotentials has been developed recently Lee et al. (2013); Wu et al. (2013); Lee and Qi (2014); Claassen et al. (2015).

Ref. Lee et al., 2013 has introduced the closed-form expressions for all two-body Haldane pseudopotentials on the torus and cylinder. In this work, inspired by Refs. Simon et al., 2007b; Davenport and Simon, 2012; Lee et al., 2013 as the starting point of our analysis, we provide a complete framework for constructing general quantum Hall parent Hamiltonians involving -body pseudopotentials, for fermions as well as bosons, in cylindric and toroidal geometries. This advance proves particularly important for the non-Abelian states, most of which necessitate many-body pseudopotentials in their parent Hamiltonian class. From the construction scheme laid out in this work, all topological properties of the non-Abelian states such as their modular matrices and topological ground state degeneracy can now be conveniently studied from their associated toroidal parent Hamiltonian. Complementing previous results for the sphere and infinite plane, our formalism furthermore directly generalizes to multicomponent systems with an arbitrary number of “spin types” or “colors”. Therefore, our construction of many-body clustered Hamiltonians not only applies to arbitrary geometries, but also crucially simplifies previous approaches. We illustrate this by numerical examples, including non-Abelian multicomponent states whose parent Hamiltonians were previously unknown.

The article is organized as follows. In Sec. II, we start with a brief overview of Haldane’s pseudopotential formalism from the viewpoint of clustering conditions, and define our notation of generalized many-body multicomponent pseudopotentials. The clustering conditions, as well as the polynomial constraints following from them, form the core of our systematic geometric construction of many-body parent Hamiltonians described in Sec. III. The main result of that section is the appropriately chosen integral measure which allows us to generate all pseudopotentials through a direct Gram-Schmidt orthogonalization. The extension to many-body parent Hamiltonians for spinful states is described in Sec. IV. Technical details of the construction are delegated to the appendices. In Section V, explicit examples of parent Hamiltonian studies are worked out in detail. We discuss spin polarized states such as the Gaffnian, the Pfaffian and the Haffnian states, as well as spinful states such as the spin-singlet Gaffnian, the NASS and the Halperin-permanent states. Apart from deriving the second-quantized parent Hamiltonians, we also provide extensive numerical checks of our construction using exact diagonalization and analysis of entanglement spectra. In Sec. VI, we conclude that our geometric construction of parent Hamiltonians promises ubiquitous use in the analysis of fractional quantum Hall states and outline a few immediate future directions.

Ii Haldane pseudopotentials

Within a given Landau level, the kinetic energy term in the quantum Hall (QH) Hamiltonian is “quenched”, i.e. is effectively a constant. Hence the remaining effective Hamiltonian only depends on the interaction between particles (e.g., the Coulomb potential) projected to the given Landau level Haldane (1990). In the infinite plane, the lowest Landau level (LLL) projection amounts to evaluating the matrix elements of the Coulomb interaction between the single-particle states of the form

 ϕm(z)=zmexp(−|z|2/4ℓ2B), (1)

where is a complex parametrization of 2D electron coordinates and the magnetic length (Fig. 1a). For simplicity, we consider a QH system in the background of a fixed (isotropic) metric Haldane (2011), which allows us to write . The states in Eq. 1 are mutually orthonormal, and span the basis of the LLL. There are of these states, which is also the number of magnetic flux quanta through the system. The above will be assumed throughout this paper, which is appropriate in strong magnetic fields when the particle-hole excitations to other Landau levels are suppressed by the large cyclotron energy gap .

Restricting to a single LL, a large class of QH states can be classified by their clustering properties. (See Refs. Bernevig and Haldane, 2008; Wen and Wang, 2008 for classification schemes based on clustering.) These are a set of rules which describe how the wave function vanishes as particles are brought together in space. To define the clustering rules, it is essential to first consider the problem of two particles restricted to the LLL (Fig. 1b). As usual, the solution of the two-body problem proceeds by transforming from coordinates into the center of mass (COM) and the relative coordinate frame. In the new coordinates, the two-particle wave function decouples. As we are interested in translationally-invariant problems, only the relative wave function (which depends on ) will play a fundamental role in the following analysis. For any two particles, the relative wave function turns out to have an identical form to the single-particle wave function (1)

 ψm(z≡z1−z2)∝zmexp(−|z|2/8ℓ2B), (2)

up to the rescaling of the magnetic length . An important difference between Eqs. (2) and (1) is the new meaning of : since now represents the relative separation between two particles, in Eq. (2) is related to particle statistics, and therefore encodes the clustering properties. For spinless electrons, in Eq. (2) is only allowed to take odd integer values since the wave function must be antisymmetric with respect to , while for spinless bosons can be only be an even integer. Finally, is also the eigenvalue of the relative angular momentum for two particles (), as we can directly confirm from

 Lz=∑izi∂∂zi. (3)

After this two-particle analysis (summarized in Fig. 1), we are in position to introduce the notion of clustering properties for -particle states. Let us pick a pair of coordinates and of indistiguishable particles in a many-particle wave function. We say that these particles are in a state which obeys the clustering property with the power if vanishes as a polynomial of total power as the coordinates of the two particles approach each other:

 ψijm(zi→zj)∼(zi−zj)m. (4)

Similarly as before, we can relate the exponent to the angular momentum if the latter is a conserved quantity. Clustering conditions like this directly generalize to cases where more than particles approach each other, with the polynomial decay also specified by a power. For example, we say that an -tuple of particles is in a state with total relative angular momentum if vanishes as a polynomial of total degree as the coordinates of particles approach each other. Fixing an arbitrarily chosen reference particle of the -tuple, i.e. , we have:

 ψNm(z2,…,zN→zr)∼N∏j=2(zr−zj)mj, (5)

with as all remaining particles approach the reference particle . If the system is rotationally invariant about at least a single axis (such as for a disk or a sphere) it directly follows that the state in Eq. 5 is also an eigenstate of the corresponding relative angular momentum operator with eigenvalue .

The simplest illustration of the clustering condition is the fully filled Landau level. The wave function for such a state is the single Slater determinant of states in Eq. 1. Due to the Vandermonde identity, this wave function can be expressed as

 ψν=1=∏i

We see that when any pair of particles and is isolated, the relevant part of the wave function is . Therefore, the wave function of the filled Landau level vanishes with the exponent as particles are brought together. This is the minimal clustering constraint that any spinless fermionic wave function must satisfy. (As we will see below, interesting many-body physics results from stronger clustering conditions on the wave function.)

When properly orthogonalized, the states form an orthonormal basis in the space of magnetic translation invariant QH states. They allow to define the -body Haldane pseudopotentialsHaldane (1983) (PPs)

 UNm=∣∣ψNm⟩⟨ψNm∣∣, (7)

which obey the null space condition for . Since they are positive-definite, the PPs give energy penalties to -body states with total relative angular momentum . With a given many-body wave function, the Hamiltonian representation of will involve the sum over all -tuple subsets of particles. Concrete examples of this will be given in Secs. III and V.

For a given filling fraction, it can occur that a certain QH state is the unique and densest ground state lying in the null space of a certain linear combination of PPs, i.e. it is annihilated by a certain number of PPs. (The requirement of being the densest state is necessary to render the finding non-trivial, because it is simple in principle to construct additional zero modes of a given PP Hamiltonian by increasing the magnetic flux, i.e. by nucleating quasihole excitations.) The most elementary examples are the Laughlin states at filling, which lie in the null space of for all . As it also represents the densest configuration that is annihilated by the PP, the Laughlin state emerges as the unique ground state of a Hamiltonian at filling , , where the coefficients are arbitrary as long as . That is, the fermionic Laughlin state is the unique ground state of , while the fermionic state is the unique groudstate of any linear combination of and with positive weights. Note that the PPs of even are precluded by fermionic antisymmetry. For short-ranged two-body interactions on the disk where the degree of the clustering polynomial coincide with the exact relative angular momentum , the traditional notation by Haldane Haldane (1983) relates to ours via . As we elaborate below, many more exotic states can be realized as the highest density ground states of combinations of PPs involving bodies.

In view of other geometries than disk or sphere, one question immediately arises: Is there any hope of defining PPs in the absence of continous rotation symmetry and hence no exact relative angular momentum quantum number? This question is natural because one of the popular choices for the gauge of the magnetic field – the Landau gauge – is only compatible with periodic boundary conditions along one or both directions in the plane, which breaks continuous rotational symmetry.

The answer to the above question is affirmative, which is the central message of this paper. This is mainly because the clustering conditions, i.e. the polynomial exponent , are more fundamental than their interpretation as the relative angular momentum. The clustering conditions are short-distance properties: their support is the area associated with the fundamental droplet of particles . Assuming a one-to-one correspondence between a clustering power and a pseudopotential, the PPs should be independent of the specific geometry as long as they act in a manifold that is homogeneous and much larger than the fundamental -particle droplet. This viewpoint has been confirmed by the explicit constructions of quantum Hall trial wave functions. For instance, the successful generalization of the Laughlin wave function to the torus, given in the classic paper by Haldane and Rezayi Haldane and Rezayi (1985), has demonstrated that its short-distance properties are identical to its original version defined on the disk. The states in both geometries are uniquely characterized by their clustering properties and are locally indistinguishable; their main difference lies in the global properties, e.g., the fact that the torus Laughlin state is -fold degenerate due to its invariance under the center of mass (COM) translation. This degeneracy is of intrinsically topological origin Wen and Niu (1990). At filling , where are relatively co-prime integers, the quantum Hall state on a torus is invariant under the translation that moves every particle by orbitals (see Fig. 3c below). This symmetry guarantees an exact -fold degeneracy of the Laughlin state on the torus Haldane (1985). Additional topological degeneracies can arise for more complicated (non-Abelian) states Moore and Read (1991); Greiter et al. (1992).

In the following, we assume there exists, in general, a well-defined deformation of the null space specified by a planar QH trial wave function to the multi-dimensional null space specified by the associated set of topologically degenerate QH ground states on the torus. What we are then interested in is to find a suitable deformation of the planar Laplacian, whose bilinear form is the known planar parent Hamiltonian composed of the spherical PPs, to the toroidal Laplacian, whose bilinear form is the toroidal Hamiltonian. In the following sections, we solve this problem by what we refer to as the “geometric construction” of pseudopotentials.

Iii Geometric construction of pseudopotentials: spinless case

We describe the construction of generic QH pseudopotential Hamiltonians with a perpendicular magnetic field applied to the 2D electron gas. We assume the field to be sufficiently strong such that the spin is fully polarized, and that there are no further internal degrees of freedom for the particles. We first introduce a suitable single-particle basis for generic -body interactions that obeys the magnetic translation symmetry and conserves the center-of-mass (COM) momentum. Next, we show how the explicit functional form of Haldane PPs in Sec. II can be easily obtained from geometric principles and symmetry. We constrain ourselves to single-component PPs in this section, and generalize our construction to multicomponent (spinful) PPs in Sec. IV.

iii.1 Basis choice

A well-developed pseudopotential formalism is available in the literature Haldane (1983); Simon et al. (2007a) for the infinite plane or the sphere, where the -component of angular momentum is conserved. A different approach to pseudopotential Hamiltonian construction, however, is needed when the system is no longer invariant under continuous rotations around the -axis. This can occur when periodic boundary conditions are imposed [Fig.2], either along one direction (cylinder geometry) or both directions (torus).

Under periodic boundary conditions (PBCs) in one direction (say ), the single-particle Hilbert space is spanned by the Landau gauge basis wave function labelled by :

 ψn(x,y)≡⟨→r|c†n|0⟩∝eiκℓBnye−12(xℓB−κn)2, (8)

where is the magnetic length and is the second-quantized operator that creates a particle in the state . The parameter sets the effective separation between the one-body states in the -direction, as each one-body state is a Gaussian packet approximately localized around in -direction (Fig. 2).

Due to this simple one-parameter labeling of the one-body states, an -body interaction matrix element is labeled by indices. Additional constraints on its functional form arise when we consider interactions projected to a given Landau level (Fig. 3). Due to the magnetic translation invariance, the interactions projected to a Landau level only allow for scattering that conserves total momentum, or equivalently leaves the COM of the particles fixed. For example, the process in Fig. 3a is allowed, but the one in Fig. 3b is forbidden. This special structure in the interaction Hamiltonian gives rise to the symmetry under many-body translations that shifts every particle by orbitals at filling (Fig. 3c). This is the symmetry that underlies the topological ground state degeneracy.

The matrix elements of any interaction projected to a Landau level are given by

 V{nj},{n′j} = ∫dr1…drN (9) (∏jψ∗n′j(rj)ψnj(rj))V(r1,...,rN),

which corresponds to the second-quantized Hamiltonian

 H=∑α∑nbα†nbαn, bαn∝∑∑jnj=npα(κ¯n1,...,κ¯nN)e−κ22∑Nj=1¯n2jcn1…cnN.

Here denotes the orbital of particle with respect to the COM, , and is a polynomial in variables . The index , as will be clear from the explicit construction of below, specifies the degree of the polynomial and, for a multicomponent state discussed in Section IV, its spin sector. The polynomial can be chosen to be real, as will be evident from its geometric construction to follow. From now on, will refer to the COM, and not the index of a single particle previously appearing in Eq. 8. is the same operator as in Eq. 8, creating the th particle in state . The Hamiltonian thus consists of a product sum over all positions of the COM , as well as all polynomial degrees .

Eqs. 9 and LABEL:bn represent a general translationally invariant Hamiltonian projected to a Landau level. This Hamiltonian has a rather special form: it decomposes into a linear combination of positive-definite operators , such that the form factor in each depends only on the relative coordinates of particles. Any short-range Hamiltonian can be explictly expressed in this form Lee and Leinaas (2004); Seidel et al. (2005); Lee et al. (2013), which reveals its Laplacian structure.

Physically, Landau-level-projected Hamiltonians can be visualized as a long-range interacting 1D chain [Fig. 3]. The interaction terms can be interpreted as long-range (though Gaussian suppressed) hopping processes labeled by . For each , particles “hop” from sites to sites , according to a COM independent amplitude given by , such that the initial and final COM remains unchanged (Fig. 3c). Note that although we use the term “hopping”, there is no clear distinction between “hopping” and “interaction” in our case, as opposed to the Hubbard model. Rather, “hopping” is designated for any interaction term that is purely quantum (i.e., not of Hartree form). The goal of the remainder of this paper is to show how to systematically construct the polynomial amplitudes that need to be inserted into Eq. (LABEL:bn) to obtain the parent Hamiltonian for the desired quantum Hall state.

iii.2 Geometric derivation

We now specialize Eq. 9 to -body interactions that are Haldane PPs . One appealing feature of the second-quantized form of Eq. 9 is that we can construct the desired PPs from symmetry principles alone, without referring to the first-quantized form of the interaction .

The many-body PPs are, by definition, supposed to project onto orthogonal subspaces labeled by , where denotes the relative angular momentum:

 Missing or unrecognized delimiter for \left (11)

This requires that

 ⟨0|bm′†nbmn|0⟩ ∝ ∑¯n1+...+¯nN=0pm′(κ¯n1,...,κ¯nN) (12) × pm(κ¯n1,...,κ¯nN)exp(−κ2N∑j=1¯n2j) = δm′m.

If is to vanish with th total power as particles approach each other, the polynomial must be of degree . Hence the PPs will be completely determined once we find a set of polynomials such that: (1) is of total degree ; (2) has the correct symmetry property under exchange of particles, i.e. is totally (anti)symmetric for bosonic (fermionic) particles; (3) the ’s are orthonormal under the inner product measure

 ⟨pm′,pm⟩ = ∑¯n1+...+¯nN=0pm′(κ¯n1,...,κ¯nN) (13) × pm(κ¯n1,...,κ¯nN)exp(−κ2N∑j=1¯n2j).

Using the barycentric coordinates (Appendix A) to represent the tuple

 ⟨pm′,pm⟩ ≈ ∫RN−1dWdΩpm′(W,Ω)pm(W,Ω) (14) × exp(−N−1NW2)JN−1(W,Ω) = δm′m,

where and are the radial and angular coordinates of the vector representing the tuple , and is the Jacobian for the spherical coordinates in . We have exploited the magnetic translation symmetry of the problem in quotienting out the COM coordinate . (It is desirable to quotient out , since takes values on an infinite set when the particles lie on the 2D infinite plane, and that complicates the definition of the inner product measure.) Each quotient space is most elegantly represented as an -simplex in barycentric coordinates, where particle permutation symmetry (or subgroups of it) is manifest. Explicitly, the set of can be encoded in the vector

 →x=κN∑j=1¯nj→βj, (15)

where the basis vectors form a set that spans . A configuration is uniquely represented by a point that is independent of . Since should not favor any particular , any pair of vectors in the basis must form the same angle with each other. Specifically,

 →βj⋅→βk=NN−1δjk−1N−1, (16)

so that each vector points at the angle of from another.

With this parametrization, the Gaussian factor reduces to the simple form

 W2=|→x|2=NN−1κ2N∑j=1¯n2j. (17)

Further mathematical details can be found in the examples that follow, as well as in Appendix A.

The integral approximation in Eq. 14 becomes exact in the infinite plane limit, and is still very accurate for values of where the characteristic inter-particle separation is smaller than the smallest of the two linear dimensions of the QH system. In the following, we will assume this to be the case; otherwise, there can be significant effects from the interaction of a particle with its periodic images. This was systematically studied in the appendix of Ref. Lee et al., 2013 for . We note that the approximation in Eq. 14 does not affect the exact zero mode property of the trial Hamiltonians constructed below.

iii.3 Orthogonalization

We are now ready to evaluate the pseudopotentials. To find a second-quantized PPs with relative angular momentum , one needs to follow the rules listed in Table 1.

In the following, we shall execute this recipe explicitly for , and, to some extent, -body interactions.

2-body case

For two-body interactions, we have

 ⟨pm′,pm⟩=∫∞−∞pm′(W)pm(W)e−12W2dW, (18)

where . For this case, we have allowed to take negative values as the angular direction spans the 1D circle, which consists of just two points. The primitive polynomials for bosons are while those for fermions are . After performing the Gram-Schmidt orthogonalization procedure, the -body PPs are found to be , where

 bmn=∑¯n1+¯n2=0pm(κ¯n1,κ¯n2)e−12κ2(¯n21+¯n22)cn/2+¯n1cn/2+¯n2, (19)

are integers, and is a th degree Hermite polynomial given in Table 2. In particular, we recover the Laughlin bosonic or fermionic state for or respectively.

One can easily check that PPs become more delocalized in -space as increases. Indeed,

 κ2⟨∑j¯n2j⟩ = 12⟨W2⟩=12∫∞−∞pm(W)W2e−12W2dW (20) = m+12,

which is reminiscent of the interpretation of as the angular momentum of a pair of particles in rotationally-invariant geometries (Fig. 1). Obtaining the pseudopotentials in this second-quantized form is highly advantageous. In particular, note that this construction is free from ambiguities in the choice of , since several possible real-space interactions, e.g., those of the Trugman-Kivelson type Trugman and Kivelson (1985), can all be grouped into the same sector. This is discussed in more detail in Appendix B.

3-body case

For , the inner product measure takes the form

 ⟨pm′,pm⟩=∫∞0∫2π0pm′(W,θ)pm(W,θ)e−23W2WdθdW (21)

with

 ¯n1 =2W3κcosθ (22a) ¯n2 =W3κ(√3sinθ−cosθ) (22b) ¯n3 =W3κ(−√3sinθ−cosθ) (22c)

Each of the ’s are treated on equal footing, as one can easily check graphically. The above expressions are the simplest nontrivial cases of the general expressions for barycentric coordinates found in the Appendix (Eqs. 62-64).

The bosonic primitive polynomials are made up of elementary symmetric polynomials in the variables . Since , the only two symmetric primitive polynomials are

 −S2=−κ2∑i

and

 Y=S3 = κ3∏i¯ni (24) = κ327(2n1−n2−n3)(2n2−n3−n1) (2n3−n1−n2) = 227W3cos3θ.

The fermionic primitive polynomials are totally antisymmetric, and can always be writtenSimon et al. (2007b); Davenport and Simon (2012) as a symmetric polynomial multiplied by the Vandermonde determinant

 A = κ3(n1−n2)(n2−n3)(n3−n1) (25) = κ3(¯n1−¯n2)(¯n2−¯n3)(¯n3−¯n1) = −23√3W3sin3θ.

Note that is of degree 2 while and are of degree 3. All of them are independent of the COM coordinate , as they should be. PPs were derived in Ref. Lee et al., 2013 through explicit integration, and the approach discussed here considerably simplifies those computations by exploiting symmetry.

To generate the fermionic (bosonic) PPs up to , we need to orthogonalize the basis consisting of all possible (anti)symmetric primitive polynomials up to degree . For instance, the first seven (up to ) 3-body fermionic PPs are generated from the primitive basis . Note that the last two basis elements both contribute to the PP sector.

The 3-body PPs are found to be , where

 bmn=∑n1+n2+n3=npm(W,Y,A)e−13W2cn1cn2cn3, (26)

with the polynomials listed in Table 3. These results are fully compatible with those from Ref. Simon et al., 2007b. As mentioned, there can be more than one (anti)symmetric polynomial of the same degree for sufficiently large . This leads to the degenerate PP subspace, a specific example of which is presented in Sec. V.1.2.

N≥4-body case

For general PPs involving bodies, the inner product measure takes the form

 ⟨pm′,pm⟩ = ∫∞0e−N−1NW2WN−2dW ×∫2π0dϕN−2N−3∏k=1∫π0sinN−2−k(ϕk)dϕk ×pm′(W,ϕ1,...,ϕN−2)pm(W,ϕ1,...,ϕN−2),

where the Jacobian determinant from Eq. 14 has already been explicitly included. One transforms the tuple into -dim spherical coordinates via the barycentric coordinates detailed in Appendix A.

The bosonic primitive basis is spanned by the elementary symmetric polynomials and combinations thereof. For instance, with particles at degree , there are possible primitive polynomials: and . The fermionic primitive basis is spanned by all the symmetric polynomials as above, times the degree Vandermonde determinant shown in Fig. 4.

From the examples above, one easily deduces the degeneracy of the PPs to be for bosons and for fermions, where is the number of partitions of the integer into at most partsSimon et al. (2007b); Davenport and Simon (2012). In particular, the degeneracy is always nontrivial () whenever and .

Iv Geometric construction of pseudopotentials: spinful case

In the presence of “internal degrees of freedom” (DOFs) which we also refer to as “spins” or “components” for simplicity, there are considerably more possibilities for the diverse forms of the PPs. This is because the PPs consist of products of spatial and spin parts, and either part can have many possible symmetry types, as long as they conspire to produce an overall (anti)symmetric PP in the case of (fermions) bosons.

A generic multicomponent PP takes the form

 UNm=∑λ∑σUNm,λ∣∣σλ⟩⟨σλ∣∣. (28)

The notation here requires some explanation. defines a partition of all particles into several subgroups each of which imposing a symmetry constraint among particles of this subgroup. Associated with it is which is a spatial term (in -space) exhibiting this symmetry. refers to an (internal) spin basis that is consistent with the symmetry type . Each symmetry type corresponds to a partition of , with . For bosons, represents the situation where there is permutation symmetry among the first particles, among the next particles, etc. but no additional symmetry between the subsets. This is often represented by the Young Tableau with boxes in the row. For fermions, we use the conjugate representation , with the rows replaced by columns and symmetry conditions replaced by antisymmetry ones. For instance, the totally (anti)symmetric types are and , respectively.

Hence the space of PPs is specified by three parameters: – the number of particles interacting with each other, – the total relative angular momentum or the total polynomial degree in , and – the number of internal DOFs (spin). While the symmetry type and hence depends only on and , the set of possible also depends on . To further illustrate our notation, we specify parameters for the interactions relevant to some commonly known states: in the archetypical single-layer FQH states, we have components, and the PP interactions for the Laughlin state penalize pairs of particles with relative angular momentum , where is the filling fraction. For bilayer FQH states, we have and -body interactions. PPs as energy penalties in the sector with angular momentum and sectors with angular momentum give rise to the Halperin states. Here, the sector is also known as the triplet channel, as it is spanned by the following three basis vectors: . By contrast, the sector only contains as dictated by antisymmetry.

iv.1 Multicomponent pseudopotentials for a given symmetry

The construction of multicomponent PPs here parallels that of multicomponent wave functions described in Ref. Davenport and Simon, 2012. For completeness, we first review this construction, and proceed to show how an orthonormal multicomponent PP basis, adapted to the cylinder or torus, can be explicitly found through the geometric approach. We describe how to first find the spatial part of the PP , and second the spin basis .

Spatial part

For each symmetry type , we can construct the spatial part with elementary symmetric polynomials in subsets of the particle indices . They are, for instance, , , etc. Of course, we must have .

Like in the single-component case, the spatial part consists of a primitive polynomial which enforces the symmetry, and a totally symmetric factor that does not change the symmetry. Here, the main step in the multicomponent generalization is the replacement of primitive polynomials and by primitive polynomials consistent with the symmetry type . As the simplest example, the primitive polynomial in is (and cyclic permutations). It is the only possible degree expression symmetric in two (but not all three) of the indices.

In general, there can be more than one candidate monomial obeying a symmetry consistent with . For instance, for they are and (and cyclic permutations thereof). To find the primitive polynomials, we will have to construct one or more linear combinations of these terms which do not have any higher symmetry other than (i.e. in this case, this higher symmetry channel could be ). Elementary computation reveals that the only primitive polynomial should be , because it is the only linear combination that is manifestly symmetric in indices and disappears upon symmetrization over all three particles.

, , . As a more involved example, we demonstrate how to find the primitive polynomial corresponding to the symmetry type . Independent monomials that satisfy this symmetry include , and . The primitive polynomial is then given by the linear combination

 S21,12+μ1S2,12+μ2S2,34 (29)

with and to be determined by demanding that the linear combination disappears upon symmetrizing over permutations under and . The symmetrized sums are

 2((n2+n3)2+(n2+n4)2+(n3+n4)2+ μ1(n1n2+n1n3+n2n3)+μ2(n1n4+n2n4+n3n4))

and

 4(n22+n23+n24+n2n3+n3n4+n2n4)(4−μ1−μ2).

Setting them both to zero, we find , so the primitive polynomial is .

, , . The above procedure works for arbitrarily complicated cases, but quickly becomes cumbersome. This is when our geometric approach again becomes useful. We first write down the relevant monomials in barycentric coordinates given by Eq. 22 for bodies (or Eq. 71 for general ). The coefficients in the primitive polynomial can then be elegantly determined through graphical inspection. We demonstrate this explicitly by revisiting the example on . Recall that the primitive polynomial (call it ) is a linear combination of and , i.e.

 β2 = S21,12+μS2,12 = −W29κ(μ−2+(1+μ)(cos2θ−√3sin2θ))

where points towards the vertices favoring respectively. The correct value of will cause to disappear under symmetrization of the particles. Graphically, it means that the lobes of three copies of the plot of , each rotated an angle from each other, must cancel upon addition. This is illustrated in Fig. 5, where is readily identified as the correct value.

All in all, we have the primitive polynomial for