# Determination of many-electron basis functions for a Quantum Hall ground state using Schur polynomials

###### Abstract

A method for determining the ground state of a planar interacting many-electron system in a magnetic field perpendicular to the plane is described. The ground state wave-function is expressed as a linear combination of a set of basis functions. Given only the flux and the number of electrons describing an incompressible state, we use the combinatorics of partitioning the flux among the electrons to derive the basis wave-functions as linear combinations of Schur polynomials. The procedure ensures that the basis wave-functions form representations of the angular momentum algebra. We exemplify the method by deriving the basis functions for the quantum Hall state with a few particles.

Fractional quantum Hall effect at the filling factor is the first example in condensed matter systems wherein quasi-particles are predicted to obey a non-Abelian braiding statistics [1, 2]. At this filling, the second Landau level of spin-up electrons is half-filled. Moore and Read (MR) proposed a ground-state wave function [3] for this state with an effective filling factor which can be interpreted as a chiral -wave superfluid state of composite fermions [4] formed as a bound state of an electron and two flux quanta, each quantum being . This fractional quantum Hall state is also popularly known as the “Pfaffian state” as the real-space representation of the chiral -wave superfluid wave-function is a Pfaffian of a certain antisymmetric matrix depending on the relative positions of the particles. The edge of this state carries a neutral mode of Majorana fermions [5]. It occurs for a flux in units of in a system of electrons moving on the surface of a sphere[6] of radius . On a disc, the orbitals occupied by electrons have an angular momentum less than or equal to . The quasi-holes of the MR state obey a non-Abelian braiding statistics [1, 2] as the ground state of quasi-holes is -fold degenerate [7]. This wave-function is the exact ground state wave-function for a model three-body pseudo potential [1]. The MR wave-function has also been shown to be equivalent to the parafermion [8] wave-function. Jack polynomials characterized by occupation number configuration have been introduced [9] for the MR states. It naturally implements a squeezing rule in the configuration. However, no better alternative wave-function for the state has been derived from this type of configuration so far. A better wave-function is indeed necessary for understanding the ground state due for the Coulomb interaction.

In this article we propose a method to determine the basis wave-functions for a fractional quantum Hall state of electrons, with the maximum number of filled single-particle orbitals (same as the total number of flux quanta ). The ground state of the system is a suitable linear combination of these wave-functions. While the procedure is valid for arbitrary and , we consider, by way of example, the “Pfaffian flux-shift” , for which the MR wave-function [3] has been proposed as the ground state wave-function of the system. Numerical exact diagonalization studies [10], however, indicate that the MR wave-function is not the veritable ground state wave-function for the Coulomb interaction. The basis wave-functions obtained using our prescription should provide an accurate ground state.

Our method, being combinatorial in nature, has the advantage of being conceptually simple, if computationally demanding. The assumptions made are few and most natural, rendering the method very general. We obtain the basis wave-functions solely from the knowledge of the integral flux and the number of electrons.

Construction: Let us consider a collection of electrons on the complex plane , at positions in a magnetic field with a given total integral flux in units of the flux quantum . In other words, is a given, arbitrary, positive integer. The wave-function of this many-particle system is sought in the form , where is a polynomial in the coordinate ring . We assume that the physical quantities derived using this wave funciton remain unaltered as the electrons are shuffled. Assuming further that is distributed as flux lines between different pairs of electrons formed for a single electron, so that every unit of flux is divided between two electrons of the pair in two moieties, and every pair of electron feels at least one unit of flux, the total angular momentum of the collection is [6]

(1) |

Requiring the wave-function to furnish a representation of the angular momentum algebra implies that the polynomial is homogeneous of degree . Each electron feels this flux. The maximal index of any in is thus . Let us define . The number of such variables is

(2) |

the number of ways objects can be paired. Since two electrons are not allowed to be at the same position, the wave-function is supposed to vanish at , for every pair of and . This is ensured by assuming that the polynomial is antisymmetric under the exchange of and . Without any loss of generality we assume it to be of the form

(3) |

where is the Vandermonde polynomial. Since any physical quantity is evaluated in terms of the modulus of the wave-function, assuming the wave-function to be antisymmetric under the exchange of electrons is consistent with the invariance under shuffling. Each occurs in the Vandermonde polynomial as a factor only once. Hence is a symmetric homogeneous polynomial in containing each with maximal index

(4) |

and degree

(5) |

From now on we shall deal mostly with .

We propose a systematic way of deriving the symmetric polynomial.
It is useful to describe the system in terms of graphs consisting in
vertices and edges. We need to consider closed graphs, the ones in which
every edge connects a pair of vertices. The positions
of the electrons are associated to the vertices of a graph .
The edges are associated with , .
The edges are assigned weights, such that is given a weight
. The matrix is referred to as
the adjacency matrix of .
Each vertex is -valent, with the total weight of the edges attached to
it equal to . We therefore consider a directed -regular graph
without loops.
Let us note that since the requirement of every pair of electrons being
related by at least one unit of flux has been taken care of by the
Vandermonde polynomial, the graph , which is but associated to the
symmetric polynomial , need not be connected.
To the graph we associate a *graph monomial*

(6) |

We identify the symmetric polynomials in (3) with the symmetric graph polynomial obtained by symmetrizing the graph monomial,

(7) |

where denotes the permutation group of order . We determine the graph monomial by partitioning the total flux. The flux associated to each of the electrons is shared with every other electron, by assumption. We thus consider partitions of into parts. The number of such partitions is obtained as the coefficient of in the series expansion of the generating function

(8) |

For a given partition , of length , writing the wave-function (3) as the antisymmetrized form of

(9) |

ensures that the maximum index of each is , while the degree equals the total angular momentum in the polynomial. Taking out the Vandermonde polynomial, as in (3), is tantamount to reducing the index of each by unity. Thus, given a partition , the reduced entries determine the adjacency matrix of . Taking this into account, the graph monomial is written as

(10) |

The partitions can be obtained using `Mathematica`

, for example. We
present a Sudoku-like method to obtain the adjacency matrix from a
given partition. We demonstrate this for the case of .
Let us begin with a table of size . We consider a partition
. These numbers are to be arranged in the table
obeying the following rules. First, the diagonal boxes should be empty.
Secondly, no may be repeated along a row or a column.
Thirdly, the sum of every row and every column must equal .
To start with we fill the first row and the first column of the table
with – as shown, leaving the entry empty. The
successive rows are then filled with ’s keeping the first column intact,
which necessitates a protrusion of the table on the left.
At this stage we label the diagonal entries till the last label, viz., stops before the diagonal. This process fills the table from the top
left up to the skew-diagonal.
We then fix the entries in the white block,
except for the last row. Then the protruded blocks as well as the
diagonal entries are shifted inside the table as indicated by the colored
blocks. The last row is then filled up uniquely.

(11) |

Subtracting unity from each entry the upper triangular part of the table specifies the adjacency matrix

(12) |

Writing the wave-function in terms of the graph monomial serves purposes beyond semantic and book-keeping ones. The symmetric graph polynomial can be expressed as a polynomial in terms of the elementary symmetric polynomials [11]. For -regular graphs which we consider, the degree of the graph symmetric polynomial in terms of the elementary symmetric polynomials is . Homogenizing it (in terms of the symmetric polynomials) one obtains a homogeneous polynomial of degree which is -invariant if non-vanishing [12, 13]. At the level of algebra the may be identified with the angular momentum algebra [14] with total angular momentum , generated by

(13) |

satisfying

(14) |

Thus the construction of the wave-function guarantees that it is a representation of the angular momentum algebra.

This method yields as many symmetric polynomials , by (7),
as there are partitions , starting with a graph polynomial
written using (11) for each partition.
However, it turns out that these are not
linearly independent. The expressions of the
symmetric polynomials in terms of the elementary symmetric functions is not
particularly adept for finding the independent ones. It is more useful to
express the symmetric polynomials
as linear combinations of either the monomial symmetric polynomials as well
as Schur polynomials [11].
These can be achieved using combinations of
codes in `Mathematica`

and `Macaulay2`

[15].
At this point
let us recall that to a partition , with
,
is associated a Schur polynomial given by the Jacobi-Trudi formula
[16]

(15) |

However, not all the Schur polynomials associated to partitions may appear in . Let us recall that while is associated to the partitions , the symmetric polynomials are associated to , with different degrees. Indeed, as specified before, they are related to the generating function of the number of partitions of an integer into at most parts each of size at most , namely,

(16) |

The maximal number of Schur polynomials whose linear combination determines is thus obtained as the coefficient of in the series expansion of , using the expressions (4) and (5).

We shall now exemplify the procedure for the quantum hall state in a few examples. The computations become rather demanding in terms of computer memory as the number of electrons is increased.

Examples: Let us work out some examples using the procedure described above. We shall consider the cases . We restrict attention to the -state. Hence we have . Thus, only the knowledge of is necessary to derive the symmetric polynomials. Partitions for these cases are given in Table 1. The Young tableaux are all of height .

###### Example 1

Let us find the basis functions for electrons. The two partitions of are and , as shown in Table 1. For the first case the graph monomial and the graph are

(17) |

The graphs are directed with edges , for . We shall not indicate the directions explicitly in this article to avoid cluttering. For the second instance we have

(18) |

Let us emphasize that the graph monomial is directly obtained from the partitions using (10), which in turn can be used to draw the graphs. Moreover, the entries of the adjacency matrix is depicted as the number of lines between vertices. The resulting symmetric polynomials turn out to be related,

(19) |

in terms of the elementary symmetric polynomials [11],

(20) |

Hence for four electrons there is a single symmetric polynomial which determines in (3). Let us point out that the degree of the symmetric polynomial above in terms of the elementary symmetric polynomials is , hence the graphs are -regular. The symmetric polynomial can also be expressed in terms of the monomial symmetric polynomials, as well as the three Schur polynomials as

(21) |

where we indicated the partitions of as subscripts in both cases.
The maximal number of allowed
Schur polynomials obtained from (16) with
, and is indeed .
The symmetric polynomial is expressed in terms of
elementary and monomial symmetric polynomials using `Mathematica`

.
The former is then used to express
it in terms of Schur polynomials using `Macaulay2`

.

###### Example 2

Next, we consider particles. The partitions shown in Table 1 yield the graph monomials

(22) | |||

(23) | |||

(24) | |||

(25) | |||

(26) |

The corresponding graphs are, respectively,

(27) |

The symmetric polynomials can be written in terms of the elementary symmetric,
monomial symmetric, as well as the Schur polynomials, as before. We shall
abstain from writing all the
expressions since these are rather cumbersome. With , and
, it follows from (16) that there can be at most
Schur polynomials. We can thus express
the five symmetric polynomials as linear combinations of these Schur
polynomials. The rank of this matrix turns out to be . Using
the row reduction algorithm of `Mathematica`

we find three relations
among the five polynomials,

(28) | |||

(29) | |||

(30) |

We can thus choose and as the independent ones. We write the expressions for them below.

(31) |