# Exact closed-form analytic wave functions in two dimensions: Contact-
interacting fermionic spinful ultracold atoms in a rapidly rotating trap

Constantine Yannouleas<EMAIL_ADDRESS>Uzi Landman
<EMAIL_ADDRESS>School of Physics, Georgia Institute of
Technology, Atlanta, Georgia 30332-0430

(12 January 2021, Letter, Phys. Rev. Research, in press)

###### Abstract

Exact two-dimensional analytic wave functions for an arbitrary number $N$ of
contact-interacting lowest-Landau-level (LLL) spinful fermions are derived
with the use of combined numerical and symbolic computational approaches via
analysis of exact Hamiltonian numerical diagonalization data. Closed-form
analytic expressions are presented for two families of zero-interaction-energy
states at given total angular momentum and total spin $0\leq S\leq N/2$ in the
neighborhood of the $\nu=1$ filling, covering the range from the maximum
density droplet to the first quasihole. Our theoretical predictions for
higher-order spatial and momentum correlations reveal intrinsic polygonal,
multi-ring crystalline-type structures, which can be tested with ultracold-
atom experiments in rapidly rotating traps, simulating quantum Hall physics
(including quantum LLL skyrmions).

##  I Introduction

Exact analytic solutions for the quantum many-body problem, whether in a
closed-form algebraic expression or in the form of the Bethe ansatz, are
highly coveted and sought-after; however, they are available only for a few
cases. Among this select group (for early pioneering studies see Refs. gira60
; lieb63 ; mcgu64 ; calo71 ; suth71 ; mattisbook ; suthbook ), one-dimensional
(1D) assemblies of strongly contact-interacting ultracold atoms have attracted
much attention in the last few years gira01 ; gira07 ; gira10 ; guan09 ; cui14
; deur14 ; zinn14 ; bruu15 ; cui16 ; astr20 , motivated by rapid experimental
advances in the field of trapped ultracold atoms that allow direct
verification of theoretical results. In this context, in-situ and time-of-
flight single-atom measurements of real-space and momentum-space higher-order
correlations, respectively, hold a great promise grei02 ; gerb05 ; gerb05.2 ;
bloc15 ; ott16 ; hodg17 ; clem18 ; prei19 ; berg19 ; clem19 ; holt20 ;
yann07.2 ; yann19 ; yann20.2 .

Here we derive closed-form exact analytic wave functions (EAWFs) for two-
dimensional (2D) systems of spinful contact-interacting lowest-Landau-level
(LLL) fermions that simulate fractional quantum Hall (FQH) physics sanp07 ;
coop08 ; popp04 ; hazz08 ; geme10 ; caru18 ; palm20 ; yann20  with trapped
ultracold atoms. We first introduce a novel approach for the extraction of
EAWFs from the digital information provided via numerical exact-
diagonalization (i.e., the configuration interaction, CI shav98 ; yann20 ;
palm20 ; yann07.2 ) of the many-body LLL Hamiltonian. Subsequently, we present
illustrative examples, showing that such EAWFs exhibit intrinsic geometric
structures (ultracold Wigner molecules, UCWMs) in their higher-order
correlations, in line with earlier findings using numerical CI solutions (see,
e.g., Ref. yann20 ). The compact EAWFs enable consideration of larger
assemblies compared to the CI-computed UCWMs yann20 .

Starting with the Laughlin trial wave function laug83 , compact algebraic
forms have been extensively considered coop08 ; simochap ; yoso98 ; halp83 ;
jainbook ; brey96 ; roug14 ; lieb20  as approximations to the exact
diagonalization solutions, both for electrons in semiconductors simochap ;
yoso98 ; halp83 ; jainbook ; brey96  and for ultracold bosons in rotating
traps coop08 ; roug14 ; lieb20 . In several instances, like the Laughlin wave
functions, it was shown that the variational trial functions simochap ;
jainbook ; coop08  may be exact solutions, with zero-interaction energy (0IE
states), of specific short-range pseudopotential-type parent Hamiltonians
simochap ; coop08 .

Because of the fermionic statistics, this paper relates to electronic 2D
quantum LLL skyrmions brey96 ; jain96 ; note1 ; abol97 ; girv98 ,
multicomponent quantum Hall systems girv98 , 2D anyons caru18 , and rotating
electronic yann02 ; yann03 ; yann04 ; yann07  and ultracold-atom yann06 ;
yann07.2 ; yann20  Wigner molecules. Experimental realization of such 2D
systems (including bosonic analogs popp04 ; hazz08 ; geme10 ; caru18 ) with a
few ultracold fermionic atoms (e.g., 6Li) in rapidly rotating harmonic traps
is currently pursued palm20 . Importantly, unlike the skyrmion wave functions
used in the literature brey96 ; jain96 ; note1 ; abol97 ; yang06 , which are
not eigenstates of the total spin (see particularly Ref. abol97 , the
Appendix, and the Supplemental Material (SM) supp ), the EAWFs introduced here
provide total-spin preserving symmetric polynomials for the quantum LLL
skyrmions; for other spin-preserving polynomials (restricted to the spin-
singlet state), see Ref. note2 .

##  II Methodology

Extensions of Girardeau’s mapping between impenetrable bosons and non-
interacting spinless fermions gira60 , and similar mappings gira01 ; gira10
applied to spinful and spin-parallel fermions, led to the formulation of a
hard-core boundary condition for strongly-repelling 1D fermions guan09 ; cui14
. This entails vanishing of the many-body wave functions when two fermions
with antiparallel spins are at the same position (in addition to the vanishing
for parallel spins due to the Pauli exclusion principle). Concomitant of this
condition is the appearance of 0IE states.

Figure 1: CI-calculated, relative-ground-state LLL energy spectra for $N=9$
fermions associated with the contact-interaction term only; see third term of
$H_{\rm LLL}$ in Eq. (1). Spectra in a given spin sector $S=1/2,3/2,5/2,7/2$
and $S=9/2$ are expilcitly denoted. The spectra were calculated for
$S_{z}=1/2$; however, they are independent of the precise value of $S_{z}$.
The 0IE states of family $A$ are colored in red; those of family $B$ are
colored in white. The first quasi-hole state is also explicitly denoted. $L$
is the total angular momentum.

In CI calculations and for a given number $N$ of spinful LLL fermions, the 0IE
states emerge in each spin sector $(S,S_{z})$; see Fig. 1 for the case of
$N=9$ LLL fermions interacting with a $\sum_{i<j}\delta(z_{i}-z_{j})$ two-body
potential, where $z_{i}=x_{i}+iy_{i}$ (with $i=1,2,\ldots,N$). Interest in
such 0IE states arises from: (i) They can be prepared experimentally palm20 .
(For bosons, the experimental expectations for 0IE states include fittingly
the $\nu=1/2$ bosonic Laughlin state popp04 ; hazz08 ; dali11 .) (ii) They
represent microscopic states that describe quantum LLL skyrmions brey96 ;
note1 . (iii) The Laughlin states are 0IE eigenstates associated with short-
range pseudopotential-type Hamiltonians hald83 ; simochap . (iv) For fully
polarized fermions, 0IE states have been associated with the gapless edge
excitations of the Laughlin droplet wen92  in extended semiconductor samples.

The many-body Hamiltonian describing ultracold neutral atoms in a rapidly
rotating trap popp04 ; yann06 ; yann07.2 ; coop08 ; hazz08 ; palm20  is given
by

$\frac{H_{\rm LLL}}{\hbar\omega}=N+(1-\frac{\Omega}{\omega})L+2\pi
R_{\delta}\sum_{i<j}^{N}\delta(z_{i}-z_{j}),$ (1)

where $\omega$ and $\Omega$ are, respectively, the parabolic trapping and
rotational frequencies of the trap, and $L$ denotes the total angular
momentum, $L=\sum_{i=1}^{N}\ l_{i}$, normal to the rotating-trap plane; the
energies are in units of $\hbar\omega$ and the lengths in units of the
oscillator length $\Lambda=\sqrt{\hbar/(M\omega)}$, with $M$ being the fermion
mass. The first and second terms express the LLL kinetic energy, $H_{K}$, and
the third term represents the contact interaction, $H_{\rm int}$.

Our methodology integrating both numerical (e.g., fortran) and symbolic
(algebraic, e.g., MATHEMATICA math ) languages consists of two steps: (1)
numerical diagonalization of the Hamiltonian matrix problem employing the
ARPACK solver arpack ; arno51  of large-scale sparse eigenvalue problems,
followed by step (2) where the numerically exact CI wave functions

$\displaystyle\Phi_{\rm
CI}(z_{1}\sigma_{1},\ldots,z_{N}\sigma_{N})=\sum_{I}c_{\rm
CI}(I)\Psi_{I}(z_{1}\sigma_{1},\ldots,z_{N}\sigma_{N}),$ (2)

are analyzed and processed using symbolic scripts targeting extraction of the
corresponding exact analytical wave functions.

The basis Slater determinants that span the Hilbert space are

$\displaystyle\Psi_{I}={\rm
Det}[\varphi_{j_{r}}(z_{s})\sigma_{j_{r}}(s)]/\sqrt{N!},$ (3)

where $r,s=1,\ldots,N$, the LLL single-particle orbitals are

$\displaystyle\varphi_{j}(z)=z^{l_{j}}e^{-zz^{*}/2}/\sqrt{\pi l_{j}!},$ (4)

and $\sigma$ signifies an up ($\alpha$) or a down ($\beta$) spin. The master
index $I$ counts the number of ordered arrangements (lists)
$\\{j_{1},j_{2},\ldots,j_{N}\\}$ under the restriction that $1\leq
j_{1}<j_{2}<\ldots<j_{N}\leq K$; $K\in\mathbb{N}$ is chosen large enough to
provide numerical convergence. Below, explicit mention of the Gaussian factor
is omitted.

Step (2) starts with the rewriting of the CI wave function $\Phi_{\rm CI}$ in
Eq. (2) as

$\displaystyle\Phi_{\rm
alg}(z_{1}\sigma_{1},\ldots,z_{N}\sigma_{N})=\sum_{I}c_{\rm
alg}(I)\Psi_{I}(z_{1}\sigma_{1},\ldots,z_{N}\sigma_{N}),$ (5)

where the replacement of the subscript “CI” by “alg” corresponds to the fact
that, using the symbolic language code, one obtains an equivalent multivariate
homogeneous polynomial $\Phi_{\rm alg}$ with algebraic coefficients $c_{\rm
alg}$; see the transcription of coefficients for $N=4$ and $N=9$ in Tables STI
and STII in the SM supp .

Validation of our closed-form analytic wave functions (see below) is achieved
via direct comparison of the numerical CI coefficients, $c_{\rm CI}$, with
those in $\Phi_{\rm alg}$ [Eq. (5)], thus circumventing uncertainties,
associated with the common use of wave function overlap laug83 ; simochap ;
jainbook ; palm20 ; yoso98 , due to the van Vleck-Anderson orthogonality
catastrophe vlec36 ; ande67 ; kohn99 ; deml12 ; guzi14 ; ares18 ; gu19 .

Invariably, the symbolic code is able to simplify the derived multivariate
polynomial in Eq. (5) to the compact form of a product of a Vandermonde
determinant (VDdet), $\prod_{i<j}^{N}(z_{i}-z_{j})$, involving the space
coordinates only, with a symmetric polynomial (under two-particle exchange)
with mixed space and spin coordinates [see Eq. (6) below]. The factoring out
of the VDdet reflects the fact that $\Phi_{\rm alg}$ represents a 0IE LLL
state.

Using symbolic scripts, we verify further that the fully-algebraic $\Phi_{\rm
alg}$ [Eq. (6)] is indeed an eigenstate of the total spin, obeying the Fock
condition fock40 . The final closed form expressions [see Eq. (8) below] are
derived for $N\leq 9$, but they are valid for any $N$, thus circumventing the
CI numerical diagonalization of large matrices, which is not feasible for
$N\geq\sim 10$.

For the CI diagonalization, a small perturbing term $V_{P}$ (e.g., a small
trap deformation yann20 , or a small hard-wall boundary maca17 ) needs to be
added to the LLL Hamiltonian in Eq. (1). This has a negligible influence on
the numerical eigenvalues, but it is instrumental in lifting the degeneracies
among the 0IE states, and thus produce CI states whose total spin $S$ is a
good quantum number.

##  III Targeted total spins and angular momenta

For each size $N$, we provide analytic expressions for the maximum-spin
($S=S_{z}$) 0IE ground states with angular momenta $L=L_{0}+\Delta L$ [with
$L_{0}=N(N-1)/2$] from $\Delta L=0$ (maximum density droplet) to $\Delta L=N$
(first quasihole, 1QH); they form two families $A$ and $B$ (see Fig. 1 for an
illustration).

Using $k$ to denote the number of spin-up fermions and $p$ that of spin-down
fermions, and focusing on the case with $k\geq p$ (or equivalently $p\leq
N/2$), the states in both families are associated with the same set of total
spins specified as $S=S_{z}=(k-p)/2=N/2-p$. Furthermore, given a pair $(k,p)$:

(A) The states in family $A$ have $\Delta L=p$, with $\Delta L$ varying from 0
to $N/2$ for even $N$, and from 0 to $(N-1)/2$ for odd $N$.

(B) The states in family $B$ have $\Delta L=k$, with $\Delta L$ varying from
$N/2$ to $N$ for even $N$, and from $(N+1)/2$ to $N$ for odd $N$.

The states in family $A$ are unique ground states, whereas those in family $B$
are part of degenerate manifolds. (This degeneracy is lifted as described
above.)

##  IV The exact 0IE LLL wave functions

###  IV.0.1 Mathematical preliminary

The quantity $k$-subset($list$) is a subset containing exactly $k$ elements
out of the set of $n$ elements (named $list$). The number of $k$-subsets on
$n$ elements is given by $\left(\frac{n}{k}\right)=\frac{n!}{k!(n-k)!}$. The
set represented by $list$ is taken to be a list of cardinally ordered positive
integers. For example, there are 6 2-subsets when $list$={1,2,3,4}, namely
{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}.

###  IV.0.2 General form of the 0IE LLL wave functions

The compact algebraic expression has the general form

$\displaystyle\Phi_{\rm alg}\big{(}z_{1}\chi(1),$
$\displaystyle\ldots,z_{N}\chi(N)\big{)}\propto\Phi_{V}(l_{1},\ldots,l_{N};z_{1},\ldots,z_{N})$
$\displaystyle\times\Phi_{\rm
sym}\big{(}z_{1}\chi(1),\ldots,z_{N}\chi(N)\big{)},$ (6)

where $\chi(i)$ denotes an up spin, $\alpha$, or down spin, $\beta$, and
$i=1,\ldots,N$.

$\Phi_{V}$ is a Vandermonde determinant,

$\displaystyle\Phi_{V}([l];[z])={\rm
Det}[z_{i}^{l_{j}}]=\prod_{i<j}^{N}(z_{i}-z_{j}),$ (7)

where $l_{j}=(j-1)$ and $i,j=1,2,\ldots,N$. The product of Jastrow factors
above reflects the fact that the wave function in Eq. (6) is a 0IE eigenstate
of the contact-interaction term, $H_{\rm int}$, in Eq. (1).

Due to the fermionic symmetry of the $\Phi_{\rm alg}$, $\Phi_{\rm sym}$ has to
be symmetric under the exchange of any pair of indices $i$ and $j$.
Furthermore, $\Phi_{\rm sym}$ can be written as

$\displaystyle\Phi_{\rm
sym}\big{(}z_{1}\chi(1),\ldots,z_{N}\chi(N)\big{)}=\sum_{m=1}^{K}{\cal
P}_{m}^{o}[z]{\cal Z}_{m},$ (8)

where ${\cal P}_{m}^{o}$ (defined below) are homogeneous multivariate
polynomials of order $o=p$ (family $A$) or $o=k$ (family $B$), and

$\displaystyle{\cal
Z}_{m}=\alpha(i_{1})\alpha(i_{2})\ldots\alpha(i_{k})\beta(j_{k+1})\ldots\beta(j_{N}),$
(9)

is one of the $K=N!/(k!p!)$ distinct spin primitives having $k\leq N$ up and
$p=N-k\leq k$ down spins. The set of indices $\\{i_{1},\ldots,i_{k}\\}$ is the
$m$th element ($m=1,2,\ldots,K$) of the $k$-subsets of the cardinal list (top-
level $list$, see below) specified as $list$={1,2,…,$N$}. The set of indices
$\\{j_{k+1},\ldots,j_{N}\\}$ is complementary to the
$\\{i_{1},\ldots,i_{k}\\}$ set.

The $\Phi_{V}$ [Eq. (7)] corresponds to a filling factor $\nu=1$, whereas the
filling fraction corresponding to Eq. (6) [with $\Phi_{\rm sym}$ given in Eq.
(8) through polynomials of order $o$] is near $\nu=1$. These fractions are
indeed the ones most likely to be accessed first in upcoming experiments
palm20 .

###  IV.0.3 Algebraic expressions for the polynomials ${\cal P}_{m}^{o}([z])$

For each $S=S_{z}=(k-p)/2$, except when $k=p$ which has a single state, there
exists a pair of targeted LLL states, with one state of the pair belonging to
family $A$ and the other to family $B$ (see Fig. 1 for an example).

Figure 2:  $N$th-order spin-unresolved correlations for $N$ LLL fermions.
(a,b,c) $N=19$ with $L=180$, and $S=S_{z}=1/2$. (d) $N=25$ with $L=301$, and
$S=S_{z}=23/2$. (e) $N=27$ with $L=378$, and $S=S_{z}=27/2$ (1QH state). Fixed
fermions are marked by red dots for the outer ring, yellow dots for the middle
ring, and green dots for the inner ring. The white dot for the 1QH state
denotes the additional zero at the origin. Vertical axes: arbitrary units. See
text for details. In the LLL, momentum correlations coincide with the spatial
ones yann20 .

Family A: First, the following square matrices of rank $p$ (the number of
spin-down fermions) need to be considered:

$\displaystyle{\rm
M}_{q,m}=\left[\begin{array}[]{ccc}z_{i_{1}}-z_{j_{k+1}}&\dots&z_{i_{1}}-z_{j_{N}}\\\
\vdots&\ddots&\vdots\\\ z_{i_{p}}-z_{j_{k+1}}&\dots&z_{i_{p}}-z_{j_{N}}\\\
\end{array}\right],$ (13)

where the dummy indices $i_{1},\ldots,i_{p}$ here are associated with spin-up
fermions, and the set $\\{i_{1},\ldots,i_{p}\\}$ denotes the $q$th subset
among the $p$-subsets on a second-level $list$-2, with $list$-2 being the
$m$th element among the $k$-subsets on the $\\{1,2,\ldots,N\\}$ top-level
$list$. The number of $p$-subsets of any second-level $list$-2 is
$K_{2}=k!/(p!(k-p)!)$, and thus the $q$ subscript runs from 1 to $K_{2}$. The
set of indices $\\{j_{k+1},\ldots,j_{N}\\}$ is complementary to the
$\\{i_{1},\ldots,i_{k}\\}$ set, and thus it remains constant for a given $m$
index in the matrices defined in Eq. (13). (Recall that $k$ is the total
number of spin-up fermions, and that $\\{i_{1},\ldots,i_{k}\\}$ is also
referred to as a second-level list.)

The expression for the polynomial is given by

$\displaystyle{\cal P}_{m}^{p}([z])=\sum_{q=1}^{K_{2}}{\rm Perm}[{\rm
M}_{q,m}],$ (14)

where the symbol ”Perm” denotes a Permanent.

The analytic expressions of the states with $S_{z}<S$, in a given spin
multiplicity $2S+1$, are obtained by repeated application of the spin lowering
operator.

Example. We consider the state associated with $N=5$, $S=S_{z}=1/2$, and
$L=12$. Note that $L_{0}=N(N-1)/2=10$ in the corresponding fully polarized
case. There are $K=5!/(3!2!)=10$ spin primitives ${\cal Z}_{m}$, with
$m=1,2,\ldots,10$; they correspond to the ten 3-subsets on the top-level
$list$=$\\{1,2,3,4,5\\}$, i.e., $\\{1,2,3\\}$ ($m=1$), $\\{1,2,4\\}$ ($m=2$),
$\\{1,2,5\\}$ ($m=3$), $\\{1,3,4\\}$ ($m=4$), $\\{1,3,5\\}$ ($m=5$),
$\\{1,4,5\\}$ ($m=6$), $\\{2,3,4\\}$ ($m=7$), $\\{2,3,5\\}$ ($m=8$),
$\\{2,4,5\\}$ ($m=9$), $\\{3,4,5\\}$ ($m=10$).

Here $k=3$, $p=2$, and there are $K_{2}=3$ 2-subsets for each ($m$th) 3-subset
listed above. $K_{2}=3$ is also the number of permanents entering in
expression (14), i.e., $q=1,\dots,3$. Choosing $m=10$ as an example, the three
2-subsets are $\\{3,4\\}$, $\\{3,5\\}$, and $\\{4,5\\}$, and the three
associated matrices ${\rm M}_{q,10}$ are given by:

$\displaystyle{\rm
M}_{q,10}=\left[\begin{array}[]{ccc}\eta(q,1)-z_{1}&\eta(q,1)-z_{2}\\\
\eta(q,2)-z_{1}&\eta(q,2)-z_{2}\\\ \end{array}\right],$ (17)

with with $q=1,2,3$; $\eta(1,1)=z_{3}$, $\eta(1,2)=z_{4}$, $\eta(2,1)=z_{3}$,
$\eta(2,2)=z_{5}$, and $\eta(3,1)=z_{4}$, $\eta(3,2)=z_{5}$.

An additional example is presented in the SM supp .

Family B: Similarly, we found that the symmetric polynomials ${\cal
P}_{m}^{k}([z])$ related to the ground states of family $B$ consist always
[for any $m$ in the summation of Eq. (8)] of a single permanent associated
with a matrix of rank $k$ (the number of spin-up fermions). Namely

$\displaystyle{\cal P}_{m}^{k}([z])={\rm Perm}[{\rm M}_{m}^{B}],$ (18)

with

$\displaystyle{\rm M}_{m}^{B}=$
$\displaystyle\left[\begin{array}[]{cccccc}z_{i_{1}}-z_{j_{k+1}}&\dots&z_{i_{1}}-z_{j_{N}}&z_{i_{1}}-z_{j_{N+1}}&\ldots&z_{i_{1}}-z_{j_{2k}}\\\
\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\\
z_{i_{k}}-z_{j_{k+1}}&\dots&z_{i_{k}}-z_{j_{N}}&z_{i_{k}}-z_{j_{N+1}}&\ldots&z_{i_{k}}-z_{j_{2k}}\\\
\end{array}\right].$ (22)

Above, the set of indices $\\{i_{1},\ldots,i_{k}\\}$ is the $m$th element of
the $k$-subsets associated with the spin-up fermions [see Eq. (9)]. Because
$k>p$, the complimentary set of the $p$ spin-down indices
$\\{j_{k+1},\ldots,j_{N}\\}$ has been expanded to contain exactly $k$
elements, through the introduction of virtual fermion coordinates such that
$z_{j_{s}}=0$ for all $s>N$; see specific matrices ${\rm M}_{m}^{B}$, as well
as a comparison with the wave functions in Ref. brey96 , in the Appendix and
the SM supp .

Note that the first quasi-hole state (1QH) laug83 ; maca17  coincides with the
analytic expression associated with family $B$ above for $L=L_{0}+N$.

##  V Higher-order correlations

We used the analytic wave functions above to calculate spin-unresolved higher-
oder correlations for $N=19$, 25, and 27 fermions; see Fig. 2 (for
completeness, see Fig. SF3 for $N=9$ in the SM supp ). The $n$-body
correlations for spinful fermions were defined in detail in Sec. II C of Ref.
yann20 . For the $N$-body $\Phi_{\rm alg}$ [Eq. (6)], the spatial $n$-body
correlation is given in a compact form by

$\displaystyle{\cal G}^{n}$ $\displaystyle(N)=(1-\delta_{n,N})\int|\Phi_{\rm
alg}\big{(}z_{1}\chi(1),\ldots,z_{N}\chi(N)\big{)}|^{2}\times$ $\displaystyle
dz_{n+1}d\chi(n+1)\ldots dz_{N}d\chi(N)+\delta_{n,N}|\Phi_{\rm alg}|^{2},$
(23)

with $n=2,\dots,N$. ${\cal G}^{n}(N)$ gives the conditional probablility to
find particles $n,\ldots,N$ anywhere, for prespecified (fixed) locations of
particles $1,\ldots,n-1$ with predetermined (resolved) or unspecified
(unresolved) spins.

For $N=19$, Figs. 2(a,b,c) display structured $N$th-order correlations for the
spin state with $S=S_{z}=1/2$ and total angular momentum $L=180$. Extending
Ref. yann20 , we found similar crystalline structures also in the $N$th-order
correlations of the associated fully polarized, single VDdet state with
$S=19/2$, $S_{z}=19/2$, and $L_{0}=171$ (Pauli-exclusion-only case,
experimentally investigated holt20 ). Fig. 2(d) displays the structured $N$th-
order correlation for $N=25$ with $L=301$ and $S=S_{z}=23/2$, whereas Fig.
2(e) presents the structured $N$th-order correlations for the 1QH for $N=27$
(with $L=378$ and $S=S_{z}=27/2$). The implied intrinsic geometric structure
(UCWM) in Fig. 2 is a polygonal triple ring $(n_{1},n_{2},n_{3})$ of localized
fermions (with $n_{1}+n_{2}+n_{3}=N$); specifically (1,6,12), (3,9,13), and
(4,9,14) for Figs. 2(a,b,c), Fig. 2(d), and Fig. 2(e), respectively. We note
that in the LLL neighbohood of $\nu=1$ (expected in experiments with trapped
ultracold fermions palm20 ), the intrinsic ring geometry can be probed only
with higher-order correlations. Indeed in this case, the second-order
correlations are structureless; see the findings for $N=4$ [(0,4) single ring]
and $N=6$ [(1,5) double ring] in Ref. yann20 .

##  VI Conclusion

A novel approach for deriving exact closed-form analytic expressions for the
wave functions (beyond the Jastrow-factors paradigm) of an assembly of 2D
contact-interacting spinful LLL fermions (for any $N$) was introduced and
validated. Such expressions require as input only the three parameters $N$
(number of particles), $L$ (total angular momentum), and $S$ (total spin).
Examples were presented for two families of zero-interaction-energy states,
from the maximum density droplet to the first quasihole in the neighborhood of
$\nu=1$. Ensuing theoretical predictions for higher-order momentum
correlations for $N=19$, 25, and 27, revealing intrinsic polygonal, multi-ring
crystalline configurations, could be tested with ultracold-atom experiments in
rotating traps simulating spinful quantum Hall physics, including LLL
skyrmions. The present approach can be extended to the neighborhood of any
$\nu=1/m$ that starts with a Laughlin wave function.

##  VII Acknowledgments

This work has been supported by a grant from the Air Force Office of
Scientific Research (AFSOR) under Award No. FA9550-21-1-0198. Calculations
were carried out at the GATECH Center for Computational Materials Science.

*

##  Appendix A Comparison with the symmetric polynomials for quantum skyrmions
in Ref. brey96

We compare here with the symmetric polynomials for the seed skyrmions
specified in Eq. (6) of Ref. brey96  or Eq. (2) in Ref. jain96 .

Omitting the trivial Gaussian functions, these polynomials are given by the
single formula

$\displaystyle\Phi^{\rm
sk,MFB}_{p}=\sum_{m=1}^{K}z_{i_{1}}z_{i_{2}}...z_{i_{k}}{\cal Z}_{m},$ (24)

where ${\cal Z}_{m}$ are the spin primitives defined in Eq. (9), and the
superscript MFB stands for MacDonald-Fertig-Brey. The index $m$ runs also over
the $k$-subsets $\\{i_{1},\ldots,i_{k}\\}$ of the $list=\\{1,2,3,\ldots,N\\}$,
$k=N\\!\uparrow$ being the number of spin-up fermions, with
$p=N-k=N\\!\downarrow$ being that of the spin-down fermions. The number of
$k$-subsets is $K=N!/(k!p!)$. As is the case in Ref. brey96 , one can take the
index $m$ as running over the $p$-subsets associated with the spin-down
fermions, because there is a one-to-one correspondence to the $k$-subsets of
the spin-up fermions. Note that Ref. brey96  (Ref. jain96 ) uses the capital
letter $K$ ($R$) in place of our $p$.

We consider the case $N=5$, $k=4$, $p=1$, $S=S_{z}=3/2$, and $\Delta L=4$,
belonging to family $B$ in our exposition.

According to Eq. (24), the corresponding MFB symmetric polynomial is

$\displaystyle\begin{split}\Phi^{\rm sk,MFB}_{p=1}=&z_{1}z_{2}z_{3}z_{4}{\cal
Z}_{1}+z_{1}z_{2}z_{3}z_{5}{\cal Z}_{2}+z_{1}z_{2}z_{4}z_{5}{\cal Z}_{3}+\\\
&z_{1}z_{3}z_{4}z_{5}{\cal Z}_{4}+z_{2}z_{3}z_{4}z_{5}{\cal
Z}_{5}.\end{split}$ (25)

The corresponding exact symmetric polynomial derived in this paper is given by
Eqs. (8) and (18), namely

$\displaystyle\Phi^{\rm exact}_{\rm sym}(N=5,N\\!\uparrow=4,\Delta
L=4)=\sum_{m=1}^{5}{\cal P}^{4}_{m}[z]{\cal Z}_{m},$ (26)

Expanding the permanents, one obtains for the space-only polynomials ${\cal
P}^{4}_{m}[z]$ above (with $m=1,\ldots,5$, in front of the ${\cal Z}_{m}$ spin
primitives):

$\displaystyle\begin{split}{\cal
P}^{4}_{m}[z]=&c_{1}z_{1}z_{2}z_{3}z_{4}+c_{2}z_{1}z_{2}z_{3}z_{5}+c_{3}z_{1}z_{2}z_{4}z_{5}+\\\
&c_{4}z_{1}z_{3}z_{4}z_{5}+c_{5}z_{2}z_{3}z_{4}z_{5},\end{split}$ (27)

with $c_{i}=4$ when $i=m$ and $c_{i}=-1$ otherwise.

The polynomial in Eq. (26) is clearly different from the MFB one [Eq. (25)].
We verified that the wave functions derived here are eigenfunctions of the
square, $\hat{S}^{2}$, of the total-spin operator [with eigenvalue 15/4 and
$S=3/2$ for the case in this Appendix], whereas the MFB ones are not (see also
Ref. abol97 ); for details see Ref. supp .

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SUPPLEMENTAL MATERIAL

Exact closed-form analytic wave functions in two dimensions: Contact-
interacting fermionic spinful ultracold atoms in a rapidly rotating trap

Constantine Yannouleas† and Uzi Landman∗
School of Physics, Georgia Institute of Technology, Atlanta, Georgia
30332-0430

Contents:

1) Contrast with the symmetric polynomials for skyrmions in earlier
literature, p. 9

2) Additional specific examples of the compact analytic wave functions derived
in the main text, p. 10

3) Tables illustrating, for $N=4$ and $N=9$, the correspondence between
numerical CI and algebraic coefficients in the determinantal expansions of
$\Phi_{\rm CI}$ and $\Phi_{\rm alg}$, p. 11

4) Figure displaying additional higher-order correlations for $N=9$ LLL
fermions, p.12

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Comparison with symmetric polynomials for skyrmions in previous literature

We compare here with the symmetric polynomials for the seed skyrmions
specified in Eq. (6) of (R1) A. H. MacDonald, H. A. Fertig, and Luis Brey,
Skyrmions without Sigma Models in Quantum Hall Ferromagnets, Phys. Rev. Lett.
76, 2153 (1996) (https://doi.org/10.1103/PhysRevLett.76.2153); see also Eq.
(2) in (R2) R.K. Kamilla, X.G. Wu, and J.K. Jain, Skyrmions in the fractional
quantum Hall effect, Solid State Commun. 99, 289 (1996)
(https://doi.org/10.1016/0038-1098(96)00126-3).

Omitting the trivial Gaussian functions, these polynomials are given by the
single formula

$\displaystyle\Phi^{\rm
sk,MFB}_{p}=\sum_{m=1}^{K}z_{i_{1}}z_{i_{2}}...z_{i_{k}}{\cal Z}_{m},$ (S28)

where ${\cal Z}_{m}$ are the spin primitives defined in Eq. (9) of the main
text, i.e.,

$\displaystyle{\cal
Z}_{m}=\alpha(i_{1})\alpha(i_{2})\ldots\alpha(i_{k})\beta(j_{k+1})\ldots\beta(j_{N}).$
(S29)

The index $m$ runs also over the $k$-subsets $\\{i_{1},\ldots,i_{k}\\}$ of the
$list=\\{1,2,3,\ldots,N\\}$, $k=N\\!\uparrow$ being the number of spin-up
fermions, with $p=N-k=N\\!\downarrow$ being that of the spin-down fermions.
The number of $k$-subsets is $K=N!/(k!p!)$. As is the case in Ref. R1, one can
take the index $m$ as running over the $p$-subsets associated with the spin-
down fermions, because there is a one-to-one correspondence to the $k$-subsets
of the spin-up fermions. Note that Ref. R1 (Ref. R2) uses the capital letter
$K$ ($R$) in place of our $p$.

We present here a comparison for the case $N=5$, $k=4$, $p=1$, $S=S_{z}=3/2$,
and $\Delta L=4$, a state belonging to family B in our exposition.

According to Eq. (S28), the corresponding MacDonald-Fertig-Brey (MFB)
symmetric polynomial is

$\displaystyle\Phi^{\rm sk,MFB}_{p=1}=z_{1}z_{2}z_{3}z_{4}{\cal
Z}_{1}+z_{1}z_{2}z_{3}z_{5}{\cal Z}_{2}+z_{1}z_{2}z_{4}z_{5}{\cal
Z}_{3}+z_{1}z_{3}z_{4}z_{5}{\cal Z}_{4}+z_{2}z_{3}z_{4}z_{5}{\cal Z}_{5}.$
(S30)

The corresponding polynomial derived in this paper is given by Eqs. (8) and
(13) in the main text. Expanding the permanent, one obtains for the polynomial
with $k=4$ and $m=1$ (in front of the ${\cal Z}_{1}$ spin primitive):

$\displaystyle{\cal
P}^{4}_{1}[z]=4z_{1}z_{2}z_{3}z_{4}-z_{1}z_{2}z_{3}z_{5}-z_{1}z_{2}z_{4}z_{5}-z_{1}z_{3}z_{4}z_{5}-z_{2}z_{3}z_{4}z_{5}.$
(S31)

For the ${\cal P}^{4}_{2}$ polynomial in front of ${\cal Z}_{2}$, one obtains
similarly:

$\displaystyle{\cal
P}^{4}_{2}[z]=-z_{1}z_{2}z_{3}z_{4}+4z_{1}z_{2}z_{3}z_{5}-z_{1}z_{2}z_{4}z_{5}-z_{1}z_{3}z_{4}z_{5}-z_{2}z_{3}z_{4}z_{5}.$
(S32)

In general, one has

$\displaystyle\Phi^{\rm exact}_{\rm sym}(N=5,N\\!\uparrow=4,\Delta
L=4)=\sum_{m=1}^{5}{\cal P}^{4}_{m}[z]{\cal Z}_{m},$ (S33)

with

$\displaystyle{\cal
P}^{4}_{m}[z]=c_{1}z_{1}z_{2}z_{3}z_{4}+c_{2}z_{1}z_{2}z_{3}z_{5}+c_{3}z_{1}z_{2}z_{4}z_{5}+c_{4}z_{1}z_{3}z_{4}z_{5}+c_{5}z_{2}z_{3}z_{4}z_{5},$
(S34)

and $c_{i}=4$ when $i=m$ and $c_{i}=-1$ otherwise.

We note that the expressions associated with the ${\cal Z}_{i}$,
$i=1,\ldots,5$ in the MFB polynomial consist only of a single term with a
numerical factor +1 in front. This contrasts with our expressions in Eqs.
(S31), (S32), and (S34) which have five terms each with factors of +4 and -1
in front of them.

For $p$ (spin-down fermions) $>$ $k$ (spin-up fermions), the MFB expression in
Eq. (S28) is associated with a negative total-spin projection
$S_{z}=(k-p)/2<0$. In this case, the indices for the corresponding wave
function in this paper are found by reversing all $N$ spins, i.e., by
considering the case with $p\rightarrow k$, $k\rightarrow p$, and
$S_{z}=|(k-p)/2|$.

Using our algebraic scripts, we readily verified that the wave functions
derived in this work are indeed eigenfunctions of the total-spin square
operator [with eigenvalue 15/4 and $S=3/2$ for the case in this section],
whereas the MFB ones are not (as indeed has been discussed by M. Abolfath et
al., Phys. Rev. B 56, 6795 (1997) (https://doi.org/10.1103/PhysRevB.56.6795).

In particular, applying the spin-square, $\hat{S}^{2}$, operator, one gets

$\displaystyle\big{(}\hat{S}^{2}-\frac{15}{4}\big{)}\Phi^{\rm exact}_{\rm
sym}(N=5,N\\!\uparrow=4,\Delta L=4)=0.$ (S35)

On the contrary, for the MFB wave function, one gets

$\displaystyle\big{(}\hat{S}^{2}-\frac{15}{4}\big{)}\Phi^{\rm
sk,MFB}_{p=1}=(z_{1}z_{2}z_{3}z_{4}+z_{1}z_{2}z_{3}z_{5}+z_{1}z_{2}z_{4}z_{5}+z_{1}z_{3}z_{4}z_{5}+z_{2}z_{3}z_{4}z_{5})\sum_{m=1}^{5}{\cal
Z}_{m}.$ (S36)

=====================================================

Additional examples for the wave functions in families $A$ and $B$

Family $A$. As another example from family $A$, we consider the spin singlet
state associated with $N=4$, $S=0$, $S_{z}=0$, and $L=8$. Note that
$L_{0}=N(N-1)/2=6$ in the corresponding fully spin-polarized case. There are
$K=4!/(2!2!)=6$ spin primitives ${\cal Z}_{m}$, with $m=1,2,\ldots,6$; they
correspond to the six 2-subsets on the top-level $list=\\{1,2,3,4\\}$, i.e.,
$\\{1,2\\}$ ($m=1$), $\\{1,3\\}$ ($m=2$), $\\{1,4\\}$ ($m=3$), $\\{2,3\\}$
($m=4$), $\\{2,4\\}$ ($m=5$), $\\{3,4\\}$ ($m=6$).

Because $k=2$ and $p=2$, there is only one ($K_{2}=1$) 2-subset for each
($m$th) 2-subset listed above. $K_{2}=1$ is also the number of permanents
entering in expression (11) of the main text, i.e., the index $q$ takes only
the value of one. Choosing $m=2$ as an example, the single matrix ${\rm
M}_{1,2}$ is:

$\displaystyle{\rm
M}_{1,2}=\left[\begin{array}[]{ccc}z_{1}-z_{2}&z_{1}-z_{4}\\\
z_{3}-z_{2}&z_{3}-z_{4}\\\ \end{array}\right].$ (S39)

Family $B$. First example. As a first example from family $B$, we consider the
spin $S=S_{z}=3/2$ state associated with $N=9$ and $L=42$. Note that
$L_{0}=N(N-1)/2=36$. There are $k=6$ up spins, $p=3$ down spins, and
$K=9!/(6!3!)=84$ spin primitives ${\cal Z}_{m}$, with $m=1,2,\ldots,84$.

Focusing on the $m=1$ spin primitive, we have a subset of indices
$\\{i_{1},i_{2},\ldots,i_{6}\\}$ for the spin up fermions and a subset of
indices $\\{j_{7},j_{8},j_{9}\\}$ for the spin down fermions. The subset of
the spin down indices needs to be augmented by introducing three additional
indices $\\{j_{10},j_{11},j_{12}\\}$, which specify virtual fermions with
$z_{j_{10}}=z_{j_{11}}=z_{j_{12}}=0$. Then the corresponding matrix ${\rm
M}_{1}^{B}$ [see Eq. (14) in the main text] is given by

$\displaystyle{\rm
M}_{1}^{B}=\left[\begin{array}[]{cccccc}z_{i_{1}}-z_{j_{7}}&z_{i_{1}}-z_{j_{8}}&z_{i_{1}}-z_{j_{9}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}\\\
z_{i_{2}}-z_{j_{7}}&z_{i_{2}}-z_{j_{8}}&z_{i_{2}}-z_{j_{9}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}\\\
z_{i_{3}}-z_{j_{7}}&z_{i_{3}}-z_{j_{8}}&z_{i_{3}}-z_{j_{9}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}\\\
z_{i_{4}}-z_{j_{7}}&z_{i_{4}}-z_{j_{8}}&z_{i_{4}}-z_{j_{9}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}\\\
z_{i_{5}}-z_{j_{7}}&z_{i_{5}}-z_{j_{8}}&z_{i_{5}}-z_{j_{9}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}\\\
z_{i_{6}}-z_{j_{7}}&z_{i_{6}}-z_{j_{8}}&z_{i_{6}}-z_{j_{9}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}\\\
\end{array}\right].$ (S46)

Family $B$. Second example. As a second example from family $B$, we consider
the 1QH state for $N=9$. In this case, $S=S_{z}=9/2$ and $L=45$. Note that
$L_{0}=N(N-1)/2=36$. There are $k=9$ up spins, $p=0$ down spins, and
$K=9!/(9!0!)=1$ spin primitive ${\cal Z}_{1}$.

For this single spin primitive, we have a subset of indices
$\\{i_{1},i_{2},\ldots,i_{9}\\}$ for the spin up fermions and an empty subset
of indices for the spin down fermions. The subset of the spin down indices
needs to be augmented by introducing nine additional indices
$\\{j_{1},j_{2},\ldots,j_{9}\\}$, which specify virtual fermions with
$z_{j_{1}}=z_{j_{2}}=\ldots=z_{j_{9}}=0$. Then the corresponding matrix ${\rm
M}_{1}^{B}$ [see Eq. (14) in the main text] is given by

$\displaystyle{\rm
M}_{1}^{B}=\left[\begin{array}[]{ccccccccc}z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}&z_{i_{1}}\\\
z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}&z_{i_{2}}\\\
z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}&z_{i_{3}}\\\
z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}&z_{i_{4}}\\\
z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}&z_{i_{5}}\\\
z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}&z_{i_{6}}\\\
z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}&z_{i_{7}}\\\
z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}&z_{i_{8}}\\\
z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}&z_{i_{9}}\\\
\end{array}\right].$ (S56)

When expanded, the associated permanent yields one term only, namely,
$9!z_{i_{1}}z_{i_{2}}z_{i_{3}}z_{i_{4}}z_{i_{5}}z_{i_{6}}z_{i_{7}}z_{i_{8}}z_{i_{9}}$,
and thus the 1QH state here agrees with the analytic form introduced by
Laughlin; see https://doi.org/10.1103/PhysRevLett.50.1395.

==================================================

Table STI:  The 15 dominant numerical CI coefficients, $c_{\rm CI}(J)$, in the CI expansion of the relative LLL ground state, and the corresponding extracted algebraic ones, $c_{\rm alg}(J)$, for $N=4$ fermions with total angular momentum $L=8$ (first 0IE state) in the ($S=S_{z}=0$) spin sector of singlet states. The spinful-fermions Slater determinants are specified through the set of single-particle angular momenta and spins, $(l_{1}\uparrow,l_{2}\uparrow,l_{3}\downarrow,l_{4}\downarrow)$. The converged CI expansion included a much larger set of basis Slater determinants, but naturally the dominant ones are only relevant, namely those with coefficients $|c_{\rm CI}(I)|>0.001$. $J$ | $c_{\rm CI}(J)$ | $c_{\rm alg}(J)$ | $(l_{1}\uparrow,l_{2}\uparrow,l_{3}\downarrow,l_{4}\downarrow)$ | $\sum_{i=1}^{4}l_{i}$
---|---|---|---|---
1 | -0.28005598 | $\frac{2}{\sqrt{51}}$ | (0,1,3,4) | 8
2 | 0.34299719 | $-\sqrt{\frac{2}{17}}$ | (0,2,2,4) | 8
3 | -0.14002799 | $\frac{1}{\sqrt{51}}$ | (0,3,1,4) | 8
4 | -0.29704427 | $\sqrt{\frac{3}{34}}$ | (0,3,2,3) | 8
5 | 0.14002799 | $-\frac{1}{\sqrt{51}}$ | (0,4,1,3) | 8
6 | -0.24253563 | $\frac{1}{\sqrt{17}}$ | (1,2,1,4) | 8
7 | -0.17149861 | $\frac{1}{\sqrt{34}}$ | (1,2,2,3) | 8
8 | 0.14002799 | $-\frac{1}{\sqrt{51}}$ | (1,3,0,4) | 8
9 | 0.42008405 | $-\sqrt{\frac{3}{17}}$ | (1,3,1,3) | 8
10 | -0.14002799 | $\frac{1}{\sqrt{51}}$ | (1,4,0,3) | 8
11 | -0.24253562 | $\frac{1}{\sqrt{17}}$ | (1,4,1,2) | 8
12 | -0.29704427 | $\sqrt{\frac{3}{34}}$ | (2,3,0,3) | 8
13 | -0.17149861 | $\frac{1}{\sqrt{34}}$ | (2,3,1,2) | 8
14 | 0.34299718 | $-\sqrt{\frac{2}{17}}$ | (2,4,0,2) | 8
15 | -0.28005598 | $\frac{2}{\sqrt{51}}$ | (3,4,0,1) | 8
Table STII:  A sample of the 1551 dominant numerical CI coefficients, $c_{\rm CI}(J)$, in the CI expansion of the relative LLL ground state, and the corresponding extracted algebraic ones, $c_{\rm alg}(J)$, for $N=9$ fermions with total angular momentum $L=40$ (first 0IE state) in the ($S=S_{z}=1/2$) spin sector. The spinful-fermions Slater determinants are specified through the set of single-particle angular momenta and spins, $(l_{1}\uparrow,l_{2}\uparrow,l_{3}\uparrow,l_{4}\uparrow,l_{5}\uparrow,l_{6}\downarrow,l_{7}\downarrow,l_{8}\downarrow,l_{9}\downarrow)$. The converged CI expansion included a much larger set of basis Slater determinants, but naturally the dominant ones are only relevant, namely those with coefficients $|c_{\rm CI}(I)|>0.001$. $J$ | $c_{\rm CI}(J)$ | $c_{\rm alg}(J)$ | $(l_{1}\uparrow,l_{2}\uparrow,l_{3}\uparrow,l_{4}\uparrow,l_{5}\uparrow,l_{6}\downarrow,l_{7}\downarrow,l_{8}\downarrow,l_{9}\downarrow)$ | $\sum_{i=1}^{9}l_{i}$
---|---|---|---|---
1 | -0.10319005 | $4\sqrt{\frac{5}{7513}}$ | (0,1,2,3,4,6,7,8,9) | 40
2 | 0.11303906 | $-4\sqrt{\frac{6}{7513}}$ | (0,1,2,3,5,5,7,8,9) | 40
3 | -0.02063800 | $\frac{4}{\sqrt{37565}}$ | (0,1,2,3,6,4,7,8,9) | 40
4 | -0.10465386 | $\frac{24}{\sqrt{52591}}$ | (0,1,2,3,6,5,6,8,9) | 40
5 | 0.02063800 | $-\frac{4}{\sqrt{37565}}$ | (0,1,2,3,7,4,6,8,9) | 40
6 | 0.09789477 | $-6\sqrt{\frac{2}{7513}}$ | (0,1,2,3,7,5,6,7,9) | 40
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$
1546 | 0.00872117 | $-\frac{2}{\sqrt{52591}}$ | (4,5,6,7,9,0,1,3,5) | 40
1547 | 0.00827362 | $-3\sqrt{\frac{2}{262955}}$ | (4,5,6,7,9,0,2,3,4) | 40
1548 | -0.02136239 | $2\sqrt{\frac{6}{52591}}$ | (4,5,6,8,9,0,1,2,5) | 40
1549 | -0.01103148 | $4\sqrt{\frac{2}{262955}}$ | (4,5,6,8,9,0,1,3,4) | 40
1550 | 0.02527631 | $-2\sqrt{\frac{6}{37565}}$ | (4,5,7,8,9,0,1,2,4) | 40
1551 | -0.02063801 | $\frac{4}{\sqrt{37565}}$ | (4,6,7,8,9,0,1,2,3) | 40
Figure SF3: Higher-order correlations for $N=9$ ultracold LLL fermions. (a-d)
9th-order correlations for the 0IE lowest-spin state ($S=1/2$, $S_{z}=1/2$)
with total angular momentum $L=40$. Similar 9th-order correlations are found
for the fully polarized state ($S=S_{z}=9/2$) with $L_{0}=36$. (e,f) 8th-order
correlations for the fully polarized state ($S=9/2$, $S_{z}=9/2$) with
$L_{0}=36$, whose space part is a pure Vandermonde determinant. (g,h) 7th-
order correlations for the fully polarized state ($S=9/2$, $S_{z}=9/2$) with
$L_{0}=36$. The implied intrinsic geometric structure (ultracold Wigner
molecule) is a (2,7) double ring, with 2 fermions in the inner ring and 7
fermions in the outer ring. The fixed fermions are marked by solid dots (black
for those in the outer ring and red for those in the inner ring). The white
numbers denote the remaining, beyond the fixed ones, fermions. Vertical axes:
arbitrary units.