# Combined unitary and symmetric group approach applied to low-dimensional
spin systems

Werner Dobrautz<EMAIL_ADDRESS>Max Planck Institute for Solid State
Research, Heisenbergstr. 1, 70569 Stuttgart, Germany  Department of Chemistry
and Chemical Engineering, Chalmers University of Technology, 41296 Gothenburg,
Sweden  Vamshi M. Katukuri  Max Planck Institute for Solid State Research,
Heisenbergstr. 1, 70569 Stuttgart, Germany  Nikolay A. Bogdanov  Max Planck
Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany
Daniel Kats  Max Planck Institute for Solid State Research, Heisenbergstr. 1,
70569 Stuttgart, Germany  Giovanni Li Manni  Max Planck Institute for Solid
State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany  Ali Alavi  Max
Planck Institute for Solid State Research, Heisenbergstr. 1, 70569 Stuttgart,
Germany  Dept of Chemistry, University of Cambridge, Lensfield Road, Cambridge
CB2 1EW, United Kingdom

###### Abstract

A novel combined unitary and symmetric group approach is used to study the
spin-$\frac{1}{2}$ Heisenberg model and related Fermionic systems in a spin-
adapted representation, using a linearly-parameterised Ansatz for the many-
body wave function. We show that a more compact ground state wave function
representation is obtained when combining the symmetric group,
$\mathcal{S}_{n}$, in the form of permutations of the underlying lattice site
ordering, with the cumulative spin-coupling based on the unitary group,
$\mathrm{U}(n)$. In one-dimensional systems the observed compression of the
wave function is reminiscent of block-spin renormalization group approaches,
and allows us to study larger lattices (here taken up to 80 sites) with the
spin-adapted full configuration interaction quantum Monte Carlo method, which
benefits from the sparsity of the Hamiltonian matrix and the corresponding
sampled eigenstates that emerge from the reordering. We find that in an
optimal lattice ordering the configuration state function with highest weight
already captures with high accuracy the spin-spin correlation function of the
exact ground state wave function. This feature is found for more general
lattice models, such as the Hubbard model, and _ab initio_ quantum chemical
models, in this work exemplified by a one-dimensional hydrogen chain. We also
provide numerical evidence that the optimal lattice ordering for the unitary
group approach is not generally equivalent to the optimal ordering obtained
for methods based on matrix-product states, such as the density-matrix
renormalization group approach.

Heisenberg model, Spin systems, Quantum Monte Carlo, SU(2) symmetry, Unitary
Group, Symmetric Group

##  I Introduction

Symmetry is a concept of paramount importance in physics and chemistry.
Continuous symmetries are related to conservation laws by Noether’s theorem[1]
and are represented by Lie groups,[2] while discrete symmetries, given by an
operator $\hat{T}$ that commutes with the Hamiltonian $\hat{H}$ of a system,
are of special importance in electronic structure calculations. Since a set of
commuting operators can be simultaneously diagonalized, utilizing the
eigenfunctions $\ket{\Phi}$ of the operator $\hat{T}$ causes $\hat{H}$ to have
a block-diagonal structure in this basis. Common discrete symmetries used in
electronic structure calculations are the discrete translational symmetry on a
lattice (by the use of a momentum space basis / Bloch functions[3]), the
point-group symmetries of lattices and molecules (by the use of symmetry-
adapted molecular orbitals) or conservation of the number of electron
$n_{\mathrm{el}}$ and the projection of the total spin $\hat{S}_{z}$ (by the
use of a Slater-determinant (SD) basis with fixed $n_{\mathrm{el}}$ and
$m_{s}$). More elaborate symmetries, such as the global $\mathrm{SU}(2)$ spin-
rotation symmetry of spin-preserving nonrelativistic Hamiltonians, necessitate
a more elaborate consideration, with the unitary group approach (UGA)[4, 5, 6,
7] being a notable example.

The fundamental postulate of quantum mechanics which states that no observable
physical quantity must change after exchanging two indistinguishable particles
leads to the concept of exchange or permutation symmetry. The finite symmetric
group $\mathcal{S}_{n}$ consists of all $n!$ possible permutations of $n$
objects and, following the spin-statistics theorem,[8] fermionic wavefunctions
must transform as the antisymmetric irreducible representation of
$\mathcal{S}_{n}$. Additionally, Caley’s theorem states that every group and
thus symmetry, can be realized as a sub-group of a symmetric group.[9]

In addition to the exchange of particle labels, we can also consider the
effect of exchanging orbital or _lattice-site_ labels. An exchange of a pair
of orbitals can be seen as a 180° rotation between the two orbitals, a
particularly simple unitary transformation of the underlying basis.[10] Unlike
the exchange of particle labels, which in fermionic systems leads to the
aforementioned antisymmetric representation of $\mathcal{S}_{n}$ (and
trivially realised using Slater determinants), such exchange of orbitals leads
to transformations that span much larger irreducible representations of
$\mathcal{S}_{n}$ – in general the dimensions of these irreducible
representations scale combinatorially with the number of orbitals in the
problem. Therefore, different orderings of orbitals (or lattice sites) allow
to construct different, yet equivalent Hilbert spaces. It can be expected
that, for a particular system, an optimal permutational order can be found in
which the exact (e.g. ground-state) wavefunction can be expressed most
compactly.

The effect of permutations of orbital/site indices can be seen as a similarity
transformation of the Hamiltonian with an orthogonal permutation matrix,
$\hat{T}$, connecting the ordering schemes[11]

$\operatorname{e}^{-\hat{T}}\hat{H}\operatorname{e}^{\hat{T}}=\bar{H}.$ (1)

There is no change of the spectrum of $\bar{H}$, but the explicit form of
$\bar{H}$ in a spin-adapted basis does change, in contrast to a SD
formulation. The influence of orbital/site ordering into the structure of the
wave function was already observed in the application of the spin-adapted full
configuration interaction quantum Monte Carlo (GUGA-FCIQMC) method [12, 13] to
lattice systems. [14] However, the interplay between orbital type, ordering
and the unitary group, and their effects on the compactness of many-body wave
functions was discovered in our laboratory while solving _ab initio_
Hamiltonians for ground and excited states of poly-nuclear transition metal
clusters, exemplified by iron-sulfur clusters (dimers and cubanes) [15, 16,
17], and manganese-oxigen tri-nuclear molecular systems. [18] We found
physically and chemically motivated molecular orbital unitary transformations,
based on localization and reordering, that yield an increased compactness of
the ground- and excited-state wave functions (to the limit of single-reference
wave functions). Moreover, this approach leads to a unique (quasi)-block-
diagonal structure of the _ab initio_ Hamiltonian, that in practice allows
state-specific optimizations of electronic excited states. This approach is
extremely beneficial for spin-adapted methods that take advantage of the
sparseness of the wave function, including GUGA-FCIQMC, as the associated
computational costs are dramatically reduced.

In the present work we study, in spin-$\frac{1}{2}$ Heisenberg systems, the
combined effect of the symmetric group $\mathcal{S}_{n}$ in the form of the
permutations of orbital labels and the unitary group $\mathrm{U}(n)$ providing
a spin-adapted basis. Such systems exhibit large quantum fluctuations compared
to aforementioned large-$S$ problems, and therefore the benefit of such
transformations is not immediately obvious. Nevertheless we find that optimal
orderings do exist that both compactify the exact ground-state wavefunction
and in addition lead to single reference mean-field solutions whose physical
properties, such as spin-spin correlation functions, are very close to the
fully-correlated exact solutions. We will show a clear difference between the
optimal orderings found for density matrix renormalization group (DMRG) and
the one for compressing the many-body wave function within the UGA. We find
that, unlike in DMRG, it is not locality and entanglement that determine the
optimal ordering for the cumulatively spin-coupled UGA wave function, but a
mechanism reminiscent of renormalization. As shown in our earlier
investigations and in the present work, this finding is very general, and will
be shown here for to Hubbard model and a chemical _ab initio_ model,
exemplified by a chain of hydrogen atoms.

##  II The Heisenberg model

The Heisenberg model[19, 20, 21, 22, 23] describes the interaction of
localized quantum-mechanical spins on a lattice and is a long-studied model,
used to describe various aspects of magnetism in the solid state[24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35]. It is given by the Hamiltonian

$\hat{H}=\sum_{ij}^{n}J_{ij}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j},\quad\text{with}\quad\hat{\mathbf{S}}_{i}=\\{\hat{S}_{i}^{x},\hat{S}_{i}^{y},\hat{S}_{i}^{z}\\},$
(2)

where the indices $i$ and $j$ run over all $n$ lattice sites, $J_{ij}=J_{ji}$
are the symmetric exchange constants and $\hat{\mathbf{S}}_{i}$ are the
quantum mechanical spin operators with the corresponding quantum number
$s\in\\{\frac{1}{2},1,\frac{3}{2},\dots\\}$. In this work we focus on the
$s=\frac{1}{2}$ Heisenberg model with isotropic antiferromagnetic,
$J_{ij}=J>0$, nearest neighbor (NN) interactions only, indicated by the
summation subscript $\braket{i,j}$ in the rest of this work.

The one-dimensional (1D) Heisenberg model with NN interaction is exactly
solvable via the Bethe Ansatz[36, 37, 38], while an exact solution for higher
dimensions and/or long-range interactions is still an elusive problem.[39] In
1D systems matrix product state (MPS) based methods[40, 41, 42, 43], like the
DMRG approach[44, 45, 46] are very successful, even with periodic boundary
conditions[47, 48] and long range interactions, due to the area law
entanglement[49, 50, 51], while for higher dimensions tensor network state
approaches can be applied[52, 53, 54, 55, 56]. The model does not posses a
sign problem[57, 58, 59] for unfrustrated bipartite lattices[33, 60] and thus
quantum Monte Carlo[61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 80, 81, 82, 70, 83, 84, 85] and more recent neural
network-based approaches[86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96] are
highly effective in providing very accurate numerical solutions in higher
dimensions.

Due to impractical implementations, the total $\mathrm{SU}(2)$ spin symmetry
of the Heisenberg model is usually not used in numerical approaches[39, 97],
except the work from Flocke and Karwowski[98, 11, 99, 100, 101, 102] using the
symmetric group approach (SGA)[103, 104, 105, 106, 107], spin-symmetry adapted
MPS/DMRG studies[108, 109, 110, 111, 112, 113, 114, 115, 116] and occasional
ED[117, 97, 118] and real-space renormalization group studies[119].
Nevertheless the theoretical advantages of using a description conserving both
total spin projection, $m_{s}$, the total spin, $S$, are striking: (a) further
reduction of the Hilbert space size (by additional block diagonalization of
$\hat{H}$), (b) optimization of electronic states of desired spin, and (c)
separation of nearly degenerate states of different total spin.

There are multiple ways to create a spin-adapted basis or the so-called
configuration state functions (CSFs), see Refs. [106, 107] and references
therein. One of them is the above mentioned SGA, which relies on the
invariance of the Hamiltonian with respect to permutations of electrons, or
spins in the case of the Heisenberg model[101], and it’s connection to the
symmetric group $\mathcal{S}_{n}$, being the group of all permutations of $n$
elements. A different way to construct a spin-adapted basis is the UGA,
pioneered by Paldus [4, 5, 6, 7] and Shavitt [120, 121, 122, 123, 124], which
relies on the spin-free formulation of the electronic structure problem [125,
126]. Based on Shavitt’s graphical extension to UGA (GUGA), [120, 121, 122,
123, 124] we recently implemented a spin-adapted version of the FCIQMC method
[127, 128, 13, 12, 14, 17], which we will utilize in this work in the study of
large systems beyond the capabilities of exact diagonalisation.

To some extent the influence of the ordering of orbitals in the GUGA was
already noticed at its inception by Shavitt[122] and Brooks and Schaefer[129,
130]. However, this was mostly to circumvent technical limitations of the time
and did not concern any possible effect on the compactness of the ground state
wavefunction.

##  III The spin-free Heisenberg model

The Heisenberg Hamiltonian, see Eq. (2), can be expressed entirely in terms of
the spin-free excitation operators
$\hat{E}_{ij}=\sum_{\sigma=\uparrow,\downarrow}a_{i\sigma}^{\dagger}a_{j\sigma}$,
also called shift, replacement or singlet operators[131, 105, 132, 133], as
follows

$\hat{H}=-\frac{J}{2}\sum_{\braket{ij}}\hat{e}_{ij,ji}-\frac{JN_{b}}{4},$ (3)

where $N_{b}$ is the number of bonds in the lattice and
$\hat{e}_{ij,ji}=\hat{E}_{ij}\hat{E}_{ji}-\delta_{jj}\hat{E}_{ii}$. The
operator $\hat{E}_{ij}$ moves an electron or spin from lattice site $j$ to $i$
and fulfills the same commutation relations,
$\mathinner{\mathopen{[}\,\hat{E}_{ij},\hat{E}_{kl}\,\mathclose{]}}=\delta_{kj}\hat{E}_{il}-\delta_{il}\hat{E}_{kj}$,
as the generators of the unitary group[132]. Flocke and Karwowski[11],
employed the related SGA to study the Heisenberg model in a spin-adapted way.
The analogy between the SGA and UGA formulation of the Heisenberg Hamiltonian
is reviewed in Appendix B. Eq. (3) allows us to study the Heisenberg model in
a spin-adapted formalism via the UGA, as utilized in the GUGA-FCIQMC method.

##  IV Spin eigenfunctions and the action of permutation operators

Given a system of $n$ lattice sites, each associated with a spin-$\frac{1}{2}$
variable, $\sigma_{i}=\uparrow,\downarrow$, the Hilbert space of $2^{n}$
primitive spin functions $\ket{\sigma_{1}\sigma_{2}...\sigma_{n}}$ can be
subdivided into $\binom{n}{(n+m_{s})/2}$ states with a fixed spin polarization
$m_{s}$. From this set $g(n,S)=\binom{n}{n/2-S}-\binom{n}{n/2-S-1}$ spin
eigenfunctions[106] with spin $S$ and $m_{s}=S$ can be constructed. The set
forms a $g$-dimensional irreducible representation of the permutational
group.[134] In this section we outline, through some simple examples, the
consequences of this fact, which we will later exploit more generally.

Spin eigenfunctions can be constructed geneologically [135, 106] using the
addition theorem of angular momentum. Thus an $n$-electron CSF with spin $S$
can be constructed from an $(n-1)$ electron CSF with spin $S\pm{\frac{1}{2}}$,
by positively or negatively spin-coupling with a spin $s={\frac{1}{2}}$. A
positive spin coupling $\Delta S=+{\frac{1}{2}}$ is denoted with a symbol $u$
(up-spin) and a negative $\Delta S=-{\frac{1}{2}}$ with $d$ (down-spin). In
the GUGA method, this construction is carried out cumulatively, starting with
a single spin, and adding one spin at a time, until the $n$-electron CSF is
constructed. An $n$-electron CSF is therefore denoted as a string of $n$ $u$’s
and $d$’s, such as $\ket{uudd}$. At each intermediate step, say step $i$, a
pure-spin CSF is obtained, with cumulative spin $S_{i}=\sum_{j}^{i}\Delta
S_{j}\geq 0$. This means that first element of the CSF string must be a $u$,
and at each step of this cumulative construction, the number of $d$’s cannot
exceed that of $u$’s. Also, the final cumulative spin $S_{n}=S$. For example,
for an $n=3$ system with $S=\frac{1}{2}$, there are $g=2$ CSFs ($\ket{uud}$
and $\ket{udu}$) which can be constructed, from a 2-electron triplet and
singlet, respectively:

$\displaystyle\ket{uud}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\big{(}2\ket{\uparrow\uparrow\downarrow}-\ket{(\uparrow\downarrow+\downarrow\uparrow)\uparrow}\big{)}$
(4) $\displaystyle\ket{udu}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\ket{(\uparrow\downarrow-\downarrow\uparrow)\uparrow}$
(5)

Note that, in the above, we have not explicitly labelled the spins: we might
assume them to be in natural order $1,2,...,n$. However, this is not
necessary, and we are at liberty to construct the CSFs by coupling the spins
in any order we like. To be specific, let us insert the labels of the sites,
for example in natural order:

$\displaystyle\ket{u_{1}u_{2}d_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\big{(}2\ket{\uparrow_{1}\uparrow_{2}\downarrow_{3}}-\ket{(\uparrow_{1}\downarrow_{2}+\downarrow_{1}\uparrow_{2})\uparrow_{3}}\big{)}$
(6) $\displaystyle\ket{u_{1}d_{2}u_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\ket{(\uparrow_{1}\downarrow_{2}-\downarrow_{1}\uparrow_{2})\uparrow_{3}}$
(7)

Consider the action of the permutation operator $P_{23}$:

$\displaystyle P_{23}\ket{u_{1}u_{2}d_{3}}$ $\displaystyle=$
$\displaystyle\ket{u_{1}u_{3}d_{2}}$ (8) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\big{(}2\ket{\uparrow_{1}\uparrow_{3}\downarrow_{2}}-\ket{(\uparrow_{1}\downarrow_{3}+\downarrow_{1}\uparrow_{3})\uparrow_{2}}\big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\ket{u_{1}u_{2}d_{3}}+\frac{\sqrt{3}}{2}\ket{u_{1}d_{2}u_{3}}$
$\displaystyle P_{23}\ket{u_{1}d_{2}u_{3}}$ $\displaystyle=$
$\displaystyle\ket{u_{1}d_{3}u_{2}}$ (9) $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{2}\ket{u_{1}u_{2}d_{3}}+\frac{1}{2}\ket{u_{1}d_{2}u_{3}}$
(10)

In this way, we generate an equivalent set of spin eigenfunctions. However, if
we consider the Heisenberg Hamiltonian with $J=1$ for the 3-site chain with
open boundary conditions:

$\hat{H}=\hat{{\bf S}}_{1}\cdot\hat{{\bf S}}_{2}+\hat{{\bf
S}}_{2}\cdot\hat{{\bf S}}_{3}.$ (11)

In the natural order (1-2-3), the $S={\frac{1}{2}}$ sector of the Hamiltonian
is represented as

$H^{123}=-\begin{pmatrix}\frac{5}{4}&\frac{\sqrt{3}}{4}\\\
\frac{\sqrt{3}}{4}&\frac{3}{4}\\\ \end{pmatrix},$ (12)

while in the second order (1-3-2), it is already diagonal:

$H^{132}=\begin{pmatrix}-\frac{3}{2}&0\\\ 0&-\frac{1}{2}\\\ \end{pmatrix}.$
(13)

Therefore, through a process of mere site re-ordering in the CSFs, we have
diagonalized the Hamiltonian, meaning that a single CSF is able to fully
capture the exact eigenstates of the Hamiltonian in this spin sector of the
Hamiltonian. As the two representations of $\hat{H}$ in Eq. (12) and Eq. (13)
are related by a similarity transformation generated by the orthogonal
permutation matrix connecting the ordering schemes [11], see Eq. (1), the
eigenvalues do not change, while the eigenvectors undergo a very advantageous
compression.

Moving to the 4-site Heisenberg chain with periodic boundary conditions, we
find a similar behavior. Here the Hamiltonian is given by:

$\hat{H}=\hat{{\bf S}}_{1}\cdot\hat{{\bf S}}_{2}+\hat{{\bf
S}}_{2}\cdot\hat{{\bf S}}_{3}+\hat{{\bf S}}_{3}\cdot\hat{{\bf
S}}_{4}+\hat{{\bf S}}_{4}\cdot\hat{{\bf S}}_{1}$ (14)

The $S=0$ CSF Hilbert space is still only 2 dimensional ($g(4,0)=2$), with the
two states $\ket{uudd}$ and $\ket{udud}$. Employing the natural order 1-2-3-4,
we obtain the Hamiltonian

$H^{1234}=-\begin{pmatrix}\frac{1}{2}&\frac{\sqrt{3}}{2}\\\
\frac{\sqrt{3}}{2}&\frac{3}{2}\end{pmatrix}$ (15)

whilst the order 1-3-2-4 yields the diagonal Hamiltonian

$H^{1324}=-\begin{pmatrix}2&0\\\ 0&0\end{pmatrix},$ (16)

implying once again single-CSF exact eigenstates. This procedure is very
general and of broad applicability, as we have shown in Refs. [15, 16] for the
general non-relativistic _ab initio_ molecular Hamiltonian in the case of a
nitrogen dimer and an iron-sulfur cubane chemical model system.

It is worth noting that the 4-site chain with open boundary conditions is not
brought into diagonal form by any ordering. No low-spin ground state of a
Heisenberg chain/ring of size $n>4$ with NN interaction only is exactly
“single reference”. Nevertheless, as we shall show below, the underlying site
ordering largely influences the weight and nature of the most dominant CSF
(called the “reference CSF”). It is precisely this interplay between
permutations of site indices, represented by the symmetric group
$\mathcal{S}_{n}$, and the spin-adapted basis given by the unitary group
approach, $\mathrm{U}(n)$, which we wish to study in this paper.

We also note that Flocke and Karwowski[11] briefly mention the effect of the
numbering of the lattice sites on the SGA in their 1997 paper. They found that
the ordering affects the number of necessary matrix multiplications to
construct a transposition $(i,j)$. They illustrated the effect on a $4\times
2$ square lattice (ladder), and although we focus on 1D systems in this work,
we checked to find that they find a different ordering compared to our case,
since they were motivated to minimize the number of matrix multiplications to
construct the transposition $(i,j)$, see the Supplemental Material [136],
whereas our motivation is to find representations in which the exact
wavefunctions assume maximally compact forms.

###  IV.1 Extension to larger systems

Due to the factorial growth of the number of possible reorderings with the
system size, it is important to have a physically-motivated approach for the
relabeling for larger lattices. An obvious choice for a 6-site chain with open
boundary conditions (OBC), or ring with periodic boundary conditions (PBC),
would be a “bipartite” ordering, with one possible bipartite ordering given by
e.g. (1-3-5-2-4-6). Indeed, such a reordering, increases the weight of the
most dominant CSF, given by $\ket{u_{1}u_{3}u_{5}d_{2}d_{4}d_{6}}$ to 95.7%
compared to the natural order reference state
$\ket{u_{1}d_{2}u_{3}d_{4}u_{5}d_{6}}$ with a weight of 77.9% for PBC.
However, for OBC, this bipartite ordering actually decreases the weight of the
leading CSF to 89.9% compared with 92.2% for the natural ordering (see Table
1).

Table 1: 6-site orderings and leading CSF weights. |  |  | CI coefficient [%]
---|---|---|---
Order | Ref. CSF | PBC | OBC
Natural | 1-2-3-4-5-6 | $\ket{u_{1}d_{2}u_{3}d_{4}u_{5}d_{6}}$ | 77.9 | 92.2
Bipartite | 1-3-5-2-4-6 | $\ket{u_{1}u_{3}u_{5}d_{2}d_{4}d_{6}}$ | 95.7 | 89.9
Compact | 1-3-2-5-4-6 | $\ket{u_{1}u_{3}d_{2}u_{5}d_{4}d_{6}}$ | 97.1 | 94.7
SDs Néel state | $\ket{\uparrow_{1}\downarrow_{2}\uparrow_{3}\downarrow_{4}\uparrow_{5}\downarrow_{6}}$ | 47.9 | 44.9

Very interestingly, we find that there exists an even “more optimal” ordering,
which we term _compact_ ordering, shown in Fig. 1, to be explained in the
following: Let us take a closer look at the (1-3-2) and (1-3-2-4) orderings of
the 3- and 4-site lattice. The (1-3-2) ordering in the 3-site OBC case leads
to a single-CSF doublet ground state, and this fact is responsible for the
massively increased weight of the leading CSF also in larger 1D Heisenberg
systems. As we are dealing with a $S=\frac{1}{2}$ Heisenberg model, each
physical site is locally a doublet. Similar to renormalization group
approaches[137, 138, 139, 140], one can interpret that the first three sites
under the ordering (1-3-2) are coupled to a doublet state $S=\frac{1}{2}$ with
the CSF $\ket{uud}$, termed “three-site meta-spins $\frac{1}{2}$” by
Malrieu[138], and reminiscent of the block spin idea by Kadanoff[141] (see
Figure 2). To confirm this, we measured the _local spin_ expectation value of
the first three sites (described in the Appendix of Ref. [17])and obtained a
value very close the expected value of a doublet ($\frac{3}{4}$),
$\braket{(\hat{\mathbf{S}}_{1}+\hat{\mathbf{S}}_{3}+\hat{\mathbf{S}}_{2})^{2}}\approx
0.751$.

Figure 1: Weight of leading CSF for the compact ordered 6 site chain with OBC.
The labels on the lattice sites refer to the order in which the spins are
coupled in the GUGA CSF formalism. Thus, for example, the second site from
left with label “3” is coupled in the 3rd position in the CSF.

Thus, if we interpret sites (1-3-2) now as a renormalized site 1’, we can
again couple three “sites”, (1’-5-4) to a new doublet with index 2’, again
with $\ket{\mathbf{u\text{\bf'}}ud}$, where $\mathbf{u\text{\bf'}}$ indicates
the renormalized doublet. Finally, for the 6-site system the renormalized
doublet couples to singlet with the remaining 6th site, yielding the total
reference CSF as $\ket{u_{1}u_{3}d_{2}u_{5}d_{4}d_{6}}$. This process is
schematically displayed in Fig. 2 and the corresponding site ordering and
genealogical spin-coupling of the leading CSF in the compact ordering are
shown in Fig. 1.

Figure 2: Cumulative doublet coupling of “meta-spin-$\frac{1}{2}$” in the most
compact order.

The weight of this CSF in the compact ordering is 97.1% for PBC and 94.7% for
OBC in the 6-site lattice, a much larger weight as compared to both the
natural and bipartite orderings (see Table 1).

One important difference emerges from the comparison of the natural ordering
and the compact renormalized ordering. In the normal ordering the leading CSF
($\ket{ududud}$) is such that at every second site the cumulative spin
vanishes ($S_{i}=0$). The long-range spin correlation is therefore transferred
to the other CSFs of the ground state wave function. In the compact
renormalized ordering at no time $S=0$ in the leading CSF (except for the last
site for a total $S=0$ state with an even number of sites). We describe this
feature as a propagating doublet along the chain. Thus, already the leading
CSF carries information on the long range correlation. This aspect will be
discussed further in the following.

An expression for a total $n$-electron singlet CSF
$\ket{u(ud)^{\frac{n-2}{2}}d}$, arising from the coupling of the propagating
doublet with the last spin, reads in second quantized form

$\Psi_{n}^{S=0}=\frac{1}{\sqrt{2}}\left[\psi_{(n-1)\uparrow}a^{\dagger}_{n\downarrow}-\psi_{(n-1)\downarrow}a^{\dagger}_{n\uparrow}\right],$
(17)

where $a^{\dagger}_{i\sigma}$ is a creation operator at (ordered) position $i$
with spin $\sigma\\!\in\\!\left\\{\uparrow,\downarrow\right\\}$ and
$\psi_{i\sigma}$ is the propagating doublet defined by a recurrent formula

$\displaystyle\psi_{i\sigma}$
$\displaystyle=C^{uud}_{\sigma}\psi_{(i-2)\sigma}a^{\dagger}_{(i-1)\sigma}a^{\dagger}_{i\bar{\sigma}}$
(18)
$\displaystyle+C^{udu}_{\sigma}\left(\psi_{(i-2)\sigma}a^{\dagger}_{(i-1)\bar{\sigma}}+\psi_{(i-2)\bar{\sigma}}a^{\dagger}_{(i-1)\sigma}\right)a^{\dagger}_{i\sigma}$

with $i\\!\in\\!\left\\{1,3,5,...,n-1\right\\}$ and base
$\psi_{1\sigma}=a^{\dagger}_{1\sigma}$. The numerical coefficients
$C^{uud}_{\sigma}$ and $C^{udu}_{\sigma}$ are defined by the Clebsch-Gordan
coefficients
${\left\langle\vphantom{s_{1}m_{1};s_{2}m_{2}s_{\mathrm{tot}}m_{\mathrm{tot}}}s_{1}m_{1};s_{2}m_{2}\,\right|\left.\vphantom{s_{1}m_{1};s_{2}m_{2}s_{\mathrm{tot}}m_{\mathrm{tot}}}s_{\mathrm{tot}}m_{\mathrm{tot}}\right\rangle}$
arising from the angular momenta addition as

$\displaystyle
C^{uud}_{\uparrow}={\left\langle\vphantom{\frac{1}{2}\frac{1}{2};\frac{1}{2}\frac{1}{2}11}\frac{1}{2}\frac{1}{2};\frac{1}{2}\frac{1}{2}\,\right|\left.\vphantom{\frac{1}{2}\frac{1}{2};\frac{1}{2}\frac{1}{2}11}11\right\rangle}{\left\langle\vphantom{11;\frac{1}{2}\bar{\frac{1}{2}}\frac{1}{2}\frac{1}{2}}11;\frac{1}{2}\bar{\frac{1}{2}}\,\right|\left.\vphantom{11;\frac{1}{2}\bar{\frac{1}{2}}\frac{1}{2}\frac{1}{2}}\frac{1}{2}\frac{1}{2}\right\rangle}=\sqrt{\frac{2}{3}},$
(19) $\displaystyle
C^{uud}_{\downarrow}={\left\langle\vphantom{\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\bar{\frac{1}{2}}1\bar{1}}\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\bar{\frac{1}{2}}\,\right|\left.\vphantom{\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\bar{\frac{1}{2}}1\bar{1}}1\bar{1}\right\rangle}{\left\langle\vphantom{1\bar{1};\frac{1}{2}\frac{1}{2}\frac{1}{2}\bar{\frac{1}{2}}}1\bar{1};\frac{1}{2}\frac{1}{2}\,\right|\left.\vphantom{1\bar{1};\frac{1}{2}\frac{1}{2}\frac{1}{2}\bar{\frac{1}{2}}}\frac{1}{2}\bar{\frac{1}{2}}\right\rangle}=-\sqrt{\frac{2}{3}}$

and

$\displaystyle
C^{udu}_{\uparrow}={\left\langle\vphantom{\frac{1}{2}\frac{1}{2};\frac{1}{2}\bar{\frac{1}{2}}10}\frac{1}{2}\frac{1}{2};\frac{1}{2}\bar{\frac{1}{2}}\,\right|\left.\vphantom{\frac{1}{2}\frac{1}{2};\frac{1}{2}\bar{\frac{1}{2}}10}10\right\rangle}{\left\langle\vphantom{10;\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}}10;\frac{1}{2}\frac{1}{2}\,\right|\left.\vphantom{10;\frac{1}{2}\frac{1}{2}\frac{1}{2}\frac{1}{2}}\frac{1}{2}\frac{1}{2}\right\rangle}=-\sqrt{\frac{1}{6}},$
(20) $\displaystyle
C^{udu}_{\downarrow}={\left\langle\vphantom{\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\frac{1}{2}10}\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\frac{1}{2}\,\right|\left.\vphantom{\frac{1}{2}\bar{\frac{1}{2}};\frac{1}{2}\frac{1}{2}10}10\right\rangle}{\left\langle\vphantom{10;\frac{1}{2}\bar{\frac{1}{2}}\frac{1}{2}\bar{\frac{1}{2}}}10;\frac{1}{2}\bar{\frac{1}{2}}\,\right|\left.\vphantom{10;\frac{1}{2}\bar{\frac{1}{2}}\frac{1}{2}\bar{\frac{1}{2}}}\frac{1}{2}\bar{\frac{1}{2}}\right\rangle}=\sqrt{\frac{1}{6}}\,.$

One could also think about separate coupling of sites (1-3-2) and (6-5-4) to
two doublet states, and a consequent coupling of the two doublet states to an
overall singlet, as described in Ref. [138]. Interestingly, for the 6-site
ring, both these approaches are identical and lead exactly to the same
Hamiltonian representation. However, as our ultimate goal is to study these
systems with GUGA-FCIQMC method, we focus on the cumulative approach here.

With Eq. (17) and(18), the described ordering is easy to generalize to larger
lattice sites. Moreover, it is not restricted to a bipartite lattice with an
even number of sites, but is also applicable to inherently frustrated systems
with an odd number of sites, as shown for the weights of the leading CSFs for
a 7-site lattice with PBCs in Table 2.

Table 2: 7-site orderings and leading CSF weights. |  |  | CI coefficient [%]
---|---|---|---
Order | Ref. CSF | PBC | OBC
Natural | 1-2-3-4-5-6-7 | $\ket{u_{1}d_{2}u_{3}d_{4}u_{5}d_{6}u_{7}}$ | 79.5 | 67.0
Compact | 1-3-2-5-4-7-6 | $\ket{u_{1}u_{3}d_{2}u_{5}d_{4}u_{7}d_{6}}$ | 88.7 | 94.3
SDs “Néel state” | $\ket{\uparrow_{1}\downarrow_{2}\uparrow_{3}\downarrow_{4}\uparrow_{5}\uparrow_{6}\downarrow_{7}}$ | 40.5 | 54.4

###  IV.2 Exhaustive search study

We confirmed our renormalization-group motivated[138, 139] Ansatz, described
above, by considering all $n!$ possible permutations of site labels. For all
these permutations we exactly diagonalized the corresponding Heisenberg
Hamiltonian and investigated what the highest weighted CSF in the ground state
is. Due to the rather small Hilbert space size, we were able to exhaustively
search the full $n$-factorial permutational space up to 10 site systems and
confirm that this compact ordering holds for these system sizes. The optimal
ordering for a 10 site system is (1-3-2-5-4-7-6-9-8-10) and the dominant CSF
is $\ket{u_{1}u_{3}d_{2}u_{5}d_{4}u_{7}d_{6}u_{9}d_{8}d_{10}}$ with a 90.3%
weight, which again reflects the coupling of three consecutive sites to a
doublet in an iterative way, as depicted in Fig. 2.

For systems larger than 10 sites the combinatorial growth of permutations
($10!=3628800$ already) prevents an exhaustive search. At the same time
obtaining the highest weighted CSF requires a diagonalization of the
Hamiltonian, which also is not a feasible route to scale to bigger problems.
For this reason we investigated possible cheaper indicators of an optimal
ordering and the corresponding leading CSF for system sizes up to 10 sites.
For the spin-$\frac{1}{2}$ Heisenberg model with nearest neighbor interaction
the diagonal matrix elements are an indicator for the optimal ordering. Fig. 3
shows the diagonal matrix elements of the highest-weighted CSF as a function
of the possible non-cyclic permutations (only the first 100 are shown for
clarity) of a 6-site ring.

Figure 3: (a) Lowest single CSF energies and (b) corresponding weight $\lvert
c_{0}\rvert$ for the 6-site ring as a function of permutations.

The first 4 data points correspond to 4 equivalent compact orderings, followed
by 12 equivalent bipartite orderings of the site labels. The figure also shows
the single CSF energies of the natural ordering and the diagonal element of a
Néel state in a SD basis. One can see there is a stark decrease of the
diagonal matrix elements in the compact ordering, which is already quite close
to the ground state energy of this system. Thus, a cheaper option to find the
most compact representation, other than diagonalization of the full
Hamiltonian, is to minimize the diagonal matrix element over the space of
possible permutations

$\min_{\mathcal{S}_{n}}\min_{\mu}\,\langle{\mu}|{H}|{\mu}\rangle.$ (21)

In the GUGA-Heisenberg model the single CSF energies are given by the diagonal
exchange contributions[12, 121, 14]:

$\langle{\mu}|{\hat{H}}|{\mu}\rangle\sim\frac{1}{2}\sum_{j>i}J_{ij}X_{ij}(\mu),\quad
J_{ij}=\begin{cases}J&\text{for NN},\\\ 0&\text{else},\end{cases}$ (22)

where $X_{ij}(\mu)$ is a CSF dependent quantity and $J_{ij}$ is non-zero for
NN sites $i$ and $j$ depending on the chosen ordering. For NN interaction
only, the task to minimize Eq. (22) for a given CSF $\ket{\mu}$ is equivalent
to the traveling salesman problem (TSP)[142] (after scaling the possibly
negative $X_{ij}(\mu)$ to positive quantities). This can best be seen when
formulating the TSP as a integer linear program[143, 144] in the Miller-
Tucker-Zemlin[145] or Dantzig-Fulkerson-Johnson formulation[146]. For more
general Heisenberg (higher dimension, longer and anisotropic interactions), or
even _ab initio_ models, the minimization of the corresponding Eq. (22) can be
mapped to more general quadratic assignment problem[147].

Both quantities $X_{ij}(\mu)$ and $J_{ij}$ can be expressed in matrix form,
where the task is to construct an ordering of the 1D system, which changes
$J_{ij}$ to give the lowest possible diagonal matrix element, according to Eq.
(22). Fig. 4 shows $X_{ij}(\mu)$ for (a) $\ket{ududud}$ with the
original/natural ordering (1-2-3-4-5-6), (b) $\ket{uuuddd}$ with the bipartite
ordering (1-4-2-5-3-6) and (c) $\ket{uududd}$ with the _compact_ ordering
(1-3-2-5-4-6). The optimal $J_{ij}$ and thus orderings, which minimize the
diagonal matrix element, Eq. (22), are indicated by the red squares, which act
as a “mask” and determine which $X_{ij}$ elements contribute to the sum in Eq.
(22) (indicated by a red background color in Fig. 4).

Figure 4:  Exchange matrix elements, $X_{ij}$, for the 6 site chain reference
CSFs: (a) $\ket{ududud}$ (natural), (b) $\ket{uuuddd}$ (bipartite) and (c)
$\ket{uududd}$ (compact), with the respective order in parenthesis. The non-
zero $J_{ij}$ values, due to the corresponding orderings are indicated by the
black rectangles. They act as a “mask” in the product $X_{ij}J_{ij}$, see Eq.
(22), and for a given CSF, the “optimal” lattice ordering yields the lowest
possible diagonal matrix element, Eq. (22).

The direct relation between the lowest diagonal matrix element corresponding
to the highest weight in the ground state wavefunction can be used to
implement efficient approximate solvers to find the optimal permutation and
confirm our assumed renormalization structure. We implemented a simulated
annealing[148, 149, 150, 151, 152] minimizer with 2-[153, 154] and 3-opt
modifications[155] based on the Lin-Kernigham heuristic[156] to find the
optimal ordering for a given CSF. Additionally, the mapping to the TSP allowed
us to find the optimal ordering with a state-of-the-art solver by
Helsgaun[157, 158] (see the Supplemental Material for sample input
files[136]).

To deal with the exponentially growing Hilbert space with increasing system
size we combined our excitation generation routines of FCIQMC to
stochastically suggest new states, $\ket{\nu}$ for a given CSF, $\ket{\mu}$.
With this approach we were able to confirm our renormalized ordering Ansatz
for system sizes up to $n=20$ (where we were still able to enumerate the whole
Hilbert space), which gives us great confidence that this is not a result
restricted to small system sizes. More details on our simulated annealing
approach can be found in Appendix C.

###  IV.3 Spin-spin correlation functions

In addition to the renormalization behavior, another physical motivation can
be drawn for the dominant CSF in the optimal ordering: it shows a spin-spin
correlation function similar to the exact ground state wave function. This is
possible, since a CSF, $\ket{\mu}$, consists of a linear combination of
SDs[122, 159, 160, 161], $\\{\ket{I}\\}$, (see Supplemental Material [136] for
examples)

$\ket{\mu}=\sum_{I}c_{I}\ket{I},$ (23)

and thus can yield non-trivial spin-spin correlation functions, even for a
single CSF. Fig. 5 shows three spin-spin correlation functions
$\braket{\hat{S}_{j}^{z}\cdot\hat{S}_{x}^{z}}$, with $j=1,5$ and $j=10$ for a
10 site Heisenberg chain with OBC, for the different lattice orderings
considered in this work compared with the exact result. Both the natural and
bipartite single CSF spin-correlation function show a rather trivial behavior.
The former characterized by a quickly vanishing spin-spin correlation function
already after the first NN, and the latter exhibiting a alternating uniform
correlation function. In the case of the natural order leading CSF,
$\ket{u_{1}d_{2}u_{3}d_{4}u_{5}d_{6}u_{7}d_{8}u_{9}d_{10}}$, this behavior
stems from the above mentioned vanishing intermediate spin, $S_{i}=0$, at
every other lattice site. On the contrary, already the leading CSF in the
compact ordering carries information of the long-range spin-correlation and
exhibits a spin-correlation function that resembles the exact
$\braket{\hat{S}_{j}^{z}\cdot\hat{S}_{x}^{z}}$.

Figure 5:  (a) $\braket{S_{1}^{z}\cdot S_{x}^{z}}$, (b)
$\braket{S_{5}^{z}\cdot S_{x}^{z}}$ and (c) $\braket{S_{10}^{z}\cdot
S_{x}^{z}}$ exact and single-CSF spin-spin correlation functions for the 10
site chain with OBC.

We extended this study to larger lattices, as shown in Fig. 6a, where we find
that for odd- and even lattice spacings the spin-correlation function of the
compact reference CSF is exactly described by an exponential fit
$\braket{\hat{S}_{1}^{z}\cdot S_{x}^{z}}=a\cdot\operatorname{e}^{-bx}$ for $x$
even/odd (green and red dashed lines in Fig. 6a). The values of the fit are
given by $a_{even}=0.153$ and $b_{even}=0.203$ for even (excluding the first
$\braket{S_{1}^{z}\cdot S_{1}^{z}}$ data point) and $a_{odd}=-1/4$ and
$b_{odd}=0.203$ for odd lattice spacings.

Figure 6: (a) Single CSF spin-spin correlation function for the compact
ordering vs. chain size and an exponential fit to the even (green dashed line)
and odd spin-spin correlation (red dashed line) for the 36-site single CSF in
the compact order, with $a_{even}=0.153,b_{even}=0.203,a_{odd}=-0.25$ and
$b_{odd}=0.203$. (b) OBC Single CSF spin-spin correlation function for the
compact ordering compared with exact results obtained with the
$\mathcal{H}\Phi$ software package[162] for 30 sites. (c) Power fit to the
even and odd spin-spin correlation for the 30-site exact spin-spin correlation
function displayed on a double logarithmic scale, with
$a_{even}=0.133,b_{even}=-0.969,a_{odd}=-0.218$ and $b_{odd}=-1.049$.

The exact and compact-reference-CSF spin-spin correlation functions for a
30-site lattice with OBC are shown in Fig. 6b, which shows that the single CSF
results mimic the exact result even for large lattice sizes. However, the
short-range behavior of the exact spin-spin correlation function follows a
power law decay[39], $\braket{\hat{S}_{1}^{z}\cdot\hat{S}_{x}^{z}}\sim a\cdot
x^{b}$, with $a_{e}=0.133$ and $b_{e}=-0.969$ for even and $a_{o}=-0.218$ and
$b_{o}=-1.049$ for odd lattice spacings in the 30-site case, see Fig. 6c.

Using the method of _generating functions_ , Sato _et al._[163] found the
exact thermodynamic limit results for $\braket{S_{1}^{z}\cdot S_{r}^{z}}$ up
to $r=7$, which are shown in Table 3 along the single CSF and exact
diagonalization (ED) results of 30 sites with OBC.

Table 3: Exact thermodynamic limit spin-spin correlation functions, $\braket{S_{j}^{z}S_{j+k}^{z}}$[163], 30-site exact diagonalization (ED) and single CSF results. $k$ | Compact CSF | Exact OBC $n=30$ | Exact TDL[163]
---|---|---|---
1 | -0.166667 | -0.2174740 | -0.1477157
2 |  -0.083333 |  -0.0664877 |  -0.0606798
3 | -0.111111 | -0.0722632 | -0.0502486
4 |  -0.055556 |  -0.0370753 |  -0.0346528
5 | -0.074074 | -0.0417185 | -0.0308904
6 |  -0.037037 |  -0.0251398 |  -0.0244467
7 | -0.049383 | -0.0286345 | -0.0224982

Based on the entanglement perturbation theory (EPT) approach, Wang and
Chung[164], derived a relation of $A(r)=-0.1473r^{-0.9604}$ with an error of
$0.1\%$ for the odd separations, which also fits the even sites with an
opposite sign. Based on a bosonization approach, Hikihara and Furusaki[165]
find a critial exponent $b=-1$. We fitted a power law behavior to the
available TDL data[163], see Fig. 7, and gathered the resulting parameters $a$
and $b$ in Table 4, along with the data obtained based on the compact single
CSF and 30-site ED results with OBC. Even though the spin-spin correlation
function based on the leading CSF of the compact ordering is exponentially
decaying, as shown in Fig. 6a, the short-range behavior,
$\braket{\hat{S}_{j}^{z}\cdot\hat{S}_{j+1}^{z}}$ – especially for even lattice
spacings – is quite close to the exact TDL data, as shown in Fig. 7.

Figure 7: Analytic TDL spin-spin correlation function[163] with power law fits
(solid lines) for even and odd lattice sites, with
$a_{even}=0.107,\,b_{even}=-0.820,\,a_{odd}=-0.148$ and $b_{odd}=-0.975$,
compact single CSF spin-spin correlation function with exponential fit (dashed
lines) and $n=30$ exact results (ED) with OBC[162].

For this reason we also fitted the spin-spin correlation function obtained
from the compact-order leading CSF with a power law, $a\cdot x^{b}$, and
gathered the results in Table 4. The number of sites in Table 4 indicate how
many data points of $\braket{\hat{S}_{i}^{z}\cdot\hat{S}_{i+j}^{z}}$ were
taken into account for the fitting. Using only the first 18 lattices yields a
critical exponent $b_{even}=-0.979$ very close to the EPT result by Wang and
Chung. Increasing the considered number of sites to 96, the critical exponent
for even lattice separations overestimates the corresponding reference
results, but $b_{odd}$ comes closer. The fact that the spin-spin correlation
function obtained by a single CSF is so close to exact many-body results is
striking.

Table 4: Fit of $\braket{S_{0}^{z}\cdot S_{r}^{z}}$ to $a\cdot x^{b}$ for odd and even lattice sites. Method | Compact CSF | ED OBC |  EPT[164] |  TDL[163]
---|---|---|---|---
# sites | 18 | 96 | 30 | $\infty$ | 7
$a_{even}$ |  -0.174 |  -0.201 |  -0.133 | – |  -0.107
$b_{even}$ | -0.979 | -1.147 | -0.969 | -0.9604 | -0.820
Err[%] | 0.3 | 0.3 | 0.1 | 0.1 | 0.0
$a_{odd}$ | -0.178 | -0.183 | -0.218 | -0.1473 | -0.148
$b_{odd}$ | -0.726 | -0.880 | -1.049 | -0.9604 | -0.975
Err[%] | 0.8 | 0.7 | 0.1 | 0.1 | 0.0

##  V GUGA-FCIQMC calculations

In this section we study the scaling of the (increased) weight of the leading
CSF for larger lattice sizes and the effect it has on GUGA-FCIQMC
calculations, as the method usually benefits from a “more single reference and
sparse” character of the sampled wavefunction. Details on the GUGA-FCIQMC
method can be found in Appendix D and Refs. [12, 13] and computational details
and sample input files can be found in the Supplemental Material[136].

As we have identified the single CSF energy as a good indicator for a more
optimal ordering, we show it for the different orderings in Fig. 8a compared
to the exact energy of the Heisenberg model, obtained with the
$\mathcal{H}\Phi$ software package[162], with OBC as a function of the number
of sites. It can be seen that the single CSF energy in the compact ordering is
closest and in fact almost parallel to the exact energy.

Figure 8: (a) Open BC single CSF energies per site of different orderings vs.
the number of sites compared with exact Bethe Ansatz[166] and ED results,
obtained with $\mathcal{H}\Phi$[162]. (b) Open BC weights of the leading CSFs
of different orderings vs. the inverse lattice size.

In Fig. 8b we show the weights of the leading CSF, $\lvert c_{0}\rvert$,
obtained from GUGA-FCIQMC calculations for the different orderings as a
function of the inverse lattice size, $1/L$, with OBC. The effect that the
weight of the leading CSF in the _compact_ ordering is substantially larger
compared to the other orderings becomes even more pronounced for larger
lattices. This increased weight has a very beneficial influence on the
convergence of GUGA-FCIQMC calculation for finite lattices. To demonstrate
this, Fig. 9 shows the energy difference to numerically exact DMRG results,
obtained with BLOCK[167, 168, 108, 169], for the (a) 20- and (b) 30-site
Heisenberg model with OBC for the compact and natural ordering and a SD-based
calculations as a function of the number of walkers, $N_{w}$.

Using the spin-adapted GUGA-FCIQMC calculation with the natural ordering is an
order of magnitude more accurate for a given number of walkers $N_{w}$
compared to the standard SD-based implementation. In addition, using the
compact ordering yields an additional order of magnitude in accuracy. As we
show below, the increased weight of the leading CSF induced by site reordering
in the GUGA-scheme is not restricted to the Heisenberg model.

Figure 9: Difference of GUGA-FCIQMC energy per site results compared to ED
results[162] of the 20- (a) and 30-site (b) 1D Heisenberg model with open
boundary conditions for different orderings (natural and compact) and a SD-
based results (SDs) versus the number of walkers $N_{w}$.

###  V.1 Extension to Hubbard and _ab initio_ models.

Here we extend our study from a pure spin-model to Fermionic problems in form
of the Hubbard model and to _ab initio_ Hamiltonians in form of chains of
equally spaced hydrogen atoms. This entails a much larger Hilbert space size,
as the orbitals/sites can also be empty or doubly occupied. To stay as close
as possible to our Heisenberg study above, we choose the parameters of the
models in such a way (localized bases, large $U/t$ and hydrogen atom
separation) so that the ground states are dominated by states with entirely
singly-occupied/open-shell orbitals.

Fig. 10a-c shows the difference of GUGA-FCIQMC energy per site results
compared to numerically exact DMRG results[167, 168, 108, 169] for the 10-,
20- and 30-site Hubbard model with $U/t=16$ and OBC. Similar to the Heisenberg
results, using a spin-adapted formulation and the compact ordering yields an
order of magnitude more accurate results compared to a SD based calculation.

Figure 10:  Difference of GUGA-FCIQMC energy per site results compared to
$M=500$ DMRG reference results[167, 168, 108, 169] of the 10- (a), 20- (b) and
30-site (c) 1D Hubbard model with open boundary conditions for different
orderings (natural and compact) and SD-based results (SDs) versus the number
of walkers $N_{w}$.

Albeit not as drastic as for the Heisenberg model, also the weight of the
leading CSF, $\lvert c_{0}\rvert$, is substantially increased for the Hubbard
model calculations, as shown in Table 5.

Table 5: Weight of the leading CSF [%] for the Heisenberg, Hubbard and Hydrogen chains with OBC for the natural and compact ordering. Order |  | Heisenberg | Hubbard | Hydrogen
---|---|---|---|---
# sites |  | 20 | 30 | 10 | 20 | 30 | 10 | 20 | 30
Compact |  | 85.4 | 79.0 | 87.3 | 77.1 | 68.4 | 74.0 | 54.9 | 46.8
Natural |  | 60.2 | 43.0 | 78.9 | 54.7 | 43.3 | 67.2 | 41.3 | 36.1

As an example of an _ab initio_ model system, we study 1D hydrogen chains,
recently studied to benchmark various computational physics and chemistry
approaches[170, 171]. Fig. 11a-c shows the difference of GUGA-FCIQMC energy
per site results compared to numerically exact DMRG calculations[167, 168,
108, 169] for a 10-, 20- and 30-site hydrogen chain in a STO-3g basis set for
the different orderings and SD-based calculations as a function of the number
of walkers. The inter-hydrogen separation was 3.6 Å and we used localized
orbitals. The results are very similar to the Heisenberg and Hubbard model
discussed above, where again, performing spin-adapted calculations with a
compact ordering yields results an order of magnitude more accurate for a
given number of walkers compared to the standard SD-based FCIQMC method. In
addition the compact ordering increases the weight of the leading CSF compared
to the natural ordering, as shown in Table 5.

Figure 11: Energy difference per site to the $M=500$ DMRG reference results of
the 10- (a), 20- (b) and 30-site (c) hydrogen chain in a minimal basis for
different orderings (natural and compact) and SD-based results (SDs) versus
the number of walkers $N_{w}$.

###  V.2 Comparison with DMRG reordering

It is well known that orbital reordering is crucial in DMRG. However, as
discussed in our earlier works [15, 16] the reordering we seek in the context
of spin couplings differs from the one in DMRG, both in motivation and in aim.
In the context of DMRG, site reordering is very important for convergence with
respect to the bond dimension (M), and relies on concepts of entanglement and
quantum (mutual) information [172, 173, 174, 175, 168, 176, 177, 178, 167,
169]. Our reordering schemes are strictly motivated by the intrinsic
mechanisms of the cumulative spin couplings and aim at the compression of wave
functions expanded in CSFs. The former is bound to the concept of locality,
while our reordering is non-local.

In this section we provide a numerical proof that the best reordering in GUGA
is not necessarily the best in DMRG, by analyzing the DMRG convergence using
the natural and the optimal ordering for a chain of 30 hydrogen atoms. We used
the BLOCK DMRG code[167, 168, 108, 169], which is able to use $\mathrm{SU}(2)$
symmetry and allows user-defined orbital orderings. We used the standard
Fiedler algorithm [176, 179, 180, 181] to find the optimal order for the DMRG
calculation, which yielded the natural order as a result. (Computational
details and sample input files can be found in the Supplemental Material[136])
We then compared results obtained with the natural/Fiedler ordering and our
“optimal” compact ordering with and without $\mathrm{SU}(2)$ conservation. As
a reference we used a well-converged $M=400$ result using $\mathrm{SU}(2)$ and
natural ordering. Fig. 12 shows the results of this study.

Figure 12: 30-site hydrogen chain DMRG energy difference to $M=500$ Fiedler
(natural order) reference result for different orbital ordering with and
without spin-adaptation versus matrix dimension $M$.

A one-dimensional hydrogen chain, with a large atom separation of 3.6Å, is an
optimal case for a MPS based algorithm like DMRG. This is reflected in the
incredibly fast convergence of the natural ordered $\mathrm{SU}(2)$ results
with a matrix dimension of $M=100$. Similar to the GUGA-FCIQMC results, making
use of the inherent $\mathrm{SU}(2)$ symmetry is very beneficial for the DMRG
convergence. However, the 30-site hydrogen results indeed show a difference
between the natural and the compact ordering scheme. While the natural order
results with $\mathrm{SU}(2)$ symmetry (blue circles) are already converged
for $M=100$, the compact ordering scheme, which introduces some long range
interactions, requires a matrix dimension of $M=400$ to converge to similar
levels of accuracy ($<10^{-2}$ mH).

This demonstrates that the optimal ordering scheme for the GUGA framework
differs from the DMRG one. While the latter is based on locality and
entanglement arguments, the cumulative spin-coupling in the GUGA enables the
inclusion of renormalization-group concepts to render the description of
strongly-correlated many-body systems more compact (a non-local concept).

##  VI Conclusion

In this work we demonstrate a novel combined symmetric and unitary group
approach, applied to the one-dimensional spin-$\frac{1}{2}$ Heisenberg model,
which yields a more compact ground state wave function. We find that a
specific ordering of the underlying lattice sites, governed by the symmetric
group, $\mathcal{S}_{n}$, combined with the cumulative spin-coupling of the
unitary group approach, $\mathrm{U}(n)$, resembles a block-spin/real-space
renormalization group. This induces a more _compact_ description of the ground
state where the most important CSF has a much higher weight than in the
natural order. We derive an analytic formula for this compact CSF for the 1D
Heisenberg model, and find a general description of this _compact_ ordering
which is easily applicable to 1D lattices of arbitrary size. We find that this
state, up to leading order, already captures with high accuracy the spin-spin
correlation behavior of the exact ground state wave function. A more compact
ground state facilitates spin-adapted GUGA-FCIQMC calculations for larger
lattice sites. We compare this found compact ordering to optimal ordering for
DMRG calculations based on quantum mutual information, and find that they
differ. Finally, we show that this concept also applies for more general
lattice models, like the Hubbard model, and even to _ab initio_ quantum
chemical systems, in form of one-dimensional hydrogen chains. In future work
we will investigate the utility of this combined spin-adapted unitary and
symmetric group approach in more complex systems, including frustrated spin
systems.

## acknowledgments

The authors gratefully acknowledge the support of the Max Planck Society.

## Appendix

##  Appendix A The spin-free formulation of the Heisenberg model

Expressing the local spin operators as[6]

$\hat{S}_{i}^{k}=\frac{1}{2}\sum_{\mu,\nu=\uparrow,\downarrow}\sigma_{\mu,\nu}^{k}a_{i,\mu}^{\dagger}a_{i\nu},$
(24)

with the Pauli matrices[182]

$\sigma^{x}=\begin{pmatrix}0&\phantom{-}1\\\ 1&\phantom{-}0\\\
\end{pmatrix},\quad\sigma^{y}=\begin{pmatrix}0&-i\\\ i&\phantom{-}0\\\
\end{pmatrix},\quad\sigma^{z}=\begin{pmatrix}1&\phantom{-}0\\\ 0&-1\\\
\end{pmatrix}$ (25)

and the fermionic creation (annihilation) operators, $a_{i,\mu}^{(\dagger)}$
of electrons with spin $\mu$ in spatial orbital $i$. This results in the
explicit expressions

$\displaystyle\hat{S}_{i}^{x}$
$\displaystyle=\frac{1}{2}\left(a_{i\uparrow}^{\dagger}a_{i\downarrow}+a_{i\downarrow}^{\dagger}a_{i\uparrow}\right),$
(26) $\displaystyle\hat{S}_{i}^{y}$
$\displaystyle=\frac{i}{2}\left(a_{i\downarrow}^{\dagger}a_{i\uparrow}-a_{i\uparrow}^{\dagger}a_{i\downarrow}\right),$
(27) $\displaystyle\hat{S}_{i}^{z}$
$\displaystyle=\frac{1}{2}\left(n_{i\uparrow}-n_{i\downarrow}\right),$ (28)

where $n_{i\mu}=a_{i\mu}^{\dagger}a_{i\mu}$ is the fermionic number operator
of orbital $i$ and spin $\mu$. If we express
$\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}$ as

$\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}=\hat{{S}}_{i}^{z}\cdot\hat{{S}}_{j}^{z}+\hat{{S}}_{i}^{x}\cdot\hat{{S}}_{j}^{x}+\hat{{S}}_{i}^{y}\cdot\hat{{S}}_{j}^{y}$
(29)

and consequently the individual terms as

$\displaystyle\hat{{S}}_{i}^{z}\cdot\hat{{S}}_{j}^{z}=$
$\displaystyle\frac{1}{4}\left(n_{i\uparrow}-n_{i\downarrow}\right)\left(n_{j\uparrow}-n_{j\downarrow}\right),$
(30) $\displaystyle\hat{{S}}_{i}^{x}\cdot\hat{{S}}_{j}^{x}=$
$\displaystyle\frac{1}{4}\left(a_{i\uparrow}^{\dagger}a_{i\downarrow}+a_{i\downarrow}^{\dagger}a_{i\uparrow}\right)\left(a_{j\uparrow}^{\dagger}a_{j\downarrow}+a_{j\uparrow}^{\dagger}a_{j\downarrow}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}+a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}$
$\displaystyle+a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}+a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}\Big{)},$
(31) $\displaystyle\hat{{S}}_{i}^{y}\cdot\hat{{S}}_{j}^{y}=$
$\displaystyle-\frac{1}{4}\left(a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}-a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}\right)\left(a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}-a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{4}\Big{(}-a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}+a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}$
$\displaystyle+a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}-a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}\Big{)},$
(32)

we can combine the $x$ and $y$ terms as

$\displaystyle\hat{S}_{i}^{x}\cdot\hat{S}_{j}^{x}+\hat{S}_{i}^{y}\cdot\hat{S}_{j}^{y}=$
$\displaystyle\frac{1}{2}\left(a_{i\uparrow}^{\dagger}a_{i\downarrow}^{\phantom{\dagger}}a_{j\downarrow}^{\dagger}a_{j\uparrow}^{\phantom{\dagger}}+a_{i\downarrow}^{\dagger}a_{i\uparrow}^{\phantom{\dagger}}a_{j\uparrow}^{\dagger}a_{j\downarrow}^{\phantom{\dagger}}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\sigma}a_{i\sigma}^{\dagger}a_{i\bar{\sigma}}^{\phantom{\dagger}}a_{j\bar{\sigma}}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}.$
(33)

For $i\neq j$ we can transform Eq. (A) to

$\hat{S}_{i}^{x}\cdot\hat{S}_{j}^{x}+\hat{S}_{i}^{y}\cdot\hat{S}_{j}^{y}=-\frac{1}{2}\sum_{\sigma}a_{i\sigma}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}a_{j\bar{\sigma}}^{\dagger}a_{i\bar{\sigma}}^{\phantom{\dagger}}=A_{ij}.$
(34)

With the spin-free excitation operators,
$\hat{E}_{ij}=\sum_{\sigma=\uparrow,\downarrow}a_{i\sigma}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}$
and $i\neq j$ we can observe

$\displaystyle\hat{E}_{ij}\hat{E}_{ji}=$
$\displaystyle\left(\sum_{\sigma}a_{i\sigma}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}\right)\left(\sum_{\tau}a_{j\tau}^{\dagger}a_{i\tau}^{\phantom{\dagger}}\right)$
$\displaystyle=$
$\displaystyle\underbrace{\sum_{\sigma}a_{i\sigma}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}a_{j\bar{\sigma}}^{\dagger}a_{i\bar{\sigma}}^{\phantom{\dagger}}}_{-2A_{ij}}+\underbrace{\sum_{\sigma}a_{i\sigma}^{\dagger}a_{j\sigma}^{\phantom{\dagger}}a_{j\sigma}^{\dagger}a_{i\sigma}^{\phantom{\dagger}}}_{n_{i\sigma}(1-n_{j\sigma})}$
$\displaystyle=$
$\displaystyle-2A_{ij}+\sum_{\sigma}n_{i\sigma}-\sum_{\sigma}n_{i\sigma}n_{j\sigma}$
$\displaystyle=$
$\displaystyle-2A_{ij}+\hat{E}_{ii}-\sum_{\sigma}n_{i\sigma}n_{j\sigma},$ (35)

leading to the relation

$\displaystyle A_{ij}=$
$\displaystyle-\frac{1}{2}\left(\hat{E}_{ij}\hat{E}_{ji}-E_{ii}+\sum_{\sigma}n_{i\sigma}n_{j\sigma}\right)$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\left(\hat{e}_{ij,ji}+\sum_{\sigma}n_{i\sigma}n_{j\sigma}\right),$
(36)

where $\hat{e}_{ij,ji}=\hat{E}_{ij}\hat{E}_{ji}-\delta_{jj}\hat{E}_{ii}$. With
Eq. (A) we can express the spin-spin interaction, Eq. (29), as

$\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}=S_{i}^{z}\cdot
S_{j}^{z}-\frac{1}{2}\left(e_{ij,ji}+\sum_{\sigma}n_{i\sigma}n_{j\sigma}\right).$
(37)

To express Eq.(37) entirely in spin-free terms we can rewrite

$\displaystyle\hat{S}_{i}^{z}\cdot\hat{S}_{j}^{z}-$
$\displaystyle\frac{1}{2}\sum_{\sigma}n_{i\sigma}n_{j\sigma}=\frac{1}{4}(n_{i\uparrow}-n_{i\downarrow})(n_{j\uparrow}-n_{j\downarrow})$
$\displaystyle-\frac{1}{2}(n_{i\uparrow}n_{j\uparrow}+n_{i\downarrow}n_{j\downarrow})$
$\displaystyle=$
$\displaystyle\frac{1}{4}(n_{i\uparrow}n_{j\uparrow}-n_{i\uparrow}n_{j\downarrow}-n_{i\downarrow}n_{j\uparrow}+n_{i\downarrow}n_{j\downarrow})$
$\displaystyle-\frac{1}{2}(n_{i\uparrow}n_{j\uparrow}+n_{i\downarrow}n_{j\downarrow})$
(38) $\displaystyle=$
$\displaystyle-\frac{1}{4}\left(n_{i\uparrow}n_{j\uparrow}+n_{i\uparrow}n_{j\downarrow}+n_{i\downarrow}n_{j\uparrow}+n_{i\downarrow}n_{j\downarrow}\right)$
(39) $\displaystyle=$ $\displaystyle-\frac{\hat{e}_{ii,jj}}{4},$ (40)

which allows us to write the spin-spin correlation function entirely in spin-
free terms as

$\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}=-\frac{1}{2}\left(\hat{e}_{ij,ji}+\frac{\hat{e}_{ii,jj}}{2}\right).$
(41)

It is worth noting that the operator $\hat{e}_{ii,jj}$ is _diagonal_ , and for
a Heisenberg model, with explicitly singly occupied orbitals, it is identical
to one. This leads to the spin-free formulation of the Heisenberg model,
(erratum to Ref. [14])

$\displaystyle H=$ $\displaystyle
J\sum_{\braket{i,j}}\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}=-\frac{J}{2}\sum_{\braket{i,j}}\hat{e}_{ij,ji}-\frac{J}{4}\sum_{\braket{i,j}}1$
$\displaystyle=$
$\displaystyle-\frac{J}{2}\sum_{\braket{i,j}}\hat{e}_{ij,ji}-\frac{JN_{b}}{4},$
(42)

where $\braket{i,j}$ indicates the summation over nearest neighbors and
$N_{b}$ is the number of bonds. This enables a straightforward spin-free
implementation of the Heisenberg model in GUGA-FCIQMC, where only exchange-
type excitations $\hat{e}_{ij,ji}$ have to be considered.

##  Appendix B Dirac identity

There is an alternative way to derive the spin-free Heisenberg Hamiltonian in
terms unitary group generators, $\hat{E}_{ij}$, based on the Dirac
identity[101, 21]. The Dirac identity[21] is given by

$(i,j)=\frac{1}{2}+2\,\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}\quad\rightarrow\hat{\mathbf{S}}_{i}\cdot\hat{\mathbf{S}}_{j}=\frac{1}{2}\left[(i,j)-\frac{1}{2}\right],$
(43)

where $(i,j)$ indicates an exchange of electrons or spins $i$ and $j$. Eq. 43
allowed Flocke and Karwowski[11] to straightforwardly express the Heisenberg
Hamiltonian, Eq. 2, in terms of transpositions of spins $(i,j)$, as

$\hat{H}=\frac{1}{2}\sum_{ij}J_{ij}(i,j)-\frac{1}{4}\sum_{ij}J_{ij}.$ (44)

If we now use the observation by Flocke[101] that the action of the operator
$(\hat{E}_{ij}\hat{E}_{ji}-\mathds{1})$ on a “Heisenberg wavefunction”
(exclusively singly occupied orbitals /spins, no empty sites) $\ket{\phi}$ is
identical to a transposition

$(\hat{E}_{ij}\hat{E}_{ji}-\mathds{1})\ket{\phi}=(i,j)\ket{\phi},$ (45)

and similarly that the action of $\hat{E}_{ii}$ on $\ket{\phi}$ is just the
identity, $\hat{E}_{ii}\ket{\phi}=\ket{\phi}$, we can relate
$\hat{e}_{ij,ji}=(i,j)$ and obtain the same Hamiltonian in terms of unitary
group generators as in Eq. 3.

##  Appendix C Simulated annealing

Fig. 13 of our mixed stochastic simulated annealing (SA) approach. We start
with an arbitrary CSF $\mu$ and order $P$, usually taken as the natural order
and the corresponding highest weighted state with alternating $u$’s and $d$’s,
$\ket{\mu}=\ket{udududu\dots}$, and temperature $T_{\mu}$ and $T_{P}$. We then
use an _inner_ SA loop (blue box in Fig. 13), which finds the optimal order
for this CSF $\ket{\mu}$. This is done by suggesting a new order
$P^{\prime}\neq P$, with 2-[153, 154] and 3-opt modifications[155] based on
the Lin-Kernigham heuristic[156]. In the SA-spirit, this new order
$P^{\prime}$ is always accepted if the diagonal matrix element,
$E_{\mu}(P^{\prime})=\langle{\mu}|{\hat{H}(P^{\prime})}|{\mu}\rangle$ (Eq.
(22)), is lower than $E(P)$, or with probability
$p(P^{\prime}|P)=\operatorname{e}^{-\frac{\Delta
E_{\mu}(P,P^{\prime})}{T_{P}}}$, with $\Delta
E_{\mu}(P,P^{\prime})=E_{\mu}(P)-E_{\mu}(P^{\prime})$. We loop over this
process of suggesting new orderings $P^{\prime}$ and lower the temperature
$T_{P}$ every $n_{P}$ micro-iterations by a user defined ratio. This is done
until convergence or a set amount of micro-iterations is reached.

The final ordering $P$ is then fed-back to the main SA cycle (orange box in
13), where a new CSF $\ket{\mu^{\prime}}$ is suggested stochastically. For
this we use the _excitation generation_ routines in the GUGA-FCIQMC method[12,
13]. And for this new CSF $\ket{\mu^{\prime}}$ we again use the inner loop
(blue) to find the optimal ordering $P^{\prime}$ yielding the lowest
$E_{\mu^{\prime}}(P^{\prime})$. The new state $\ket{\mu^{\prime}}$ is then
accepted in the macro-cycle, if its energy,
$E_{\mu^{\prime}}(P^{\prime})=\langle{\mu^{\prime}}|{\hat{H}(P^{\prime})}|{\mu^{\prime}}\rangle$,
is lower than $E_{\mu}(P)$ or with probability
$p(\mu^{\prime},P^{\prime}|\mu,P)=\operatorname{e}^{-\frac{\Delta
E_{\mu}^{\mu^{\prime}}(P,P^{\prime})}{T_{\mu}}}$, with $\Delta
E_{\mu}^{\mu^{\prime}}(P,P^{\prime})=E_{\mu}(P)-E_{\mu^{\prime}}(P^{\prime})$.
We also loop over this macro-cycle, where the temperature $T_{\mu}$ is lowered
every $n_{\mu}$ macro-iterations by a user defined ratio, and do this until
convergence or a set amount of macro-iterations is reached.

For small enough systems ($<30$ sites), where the Hilbert space is not yet too
large, we can skip the stochastic part (orange box in 13) and perform the SA
to find the optimal order (blue box in Fig. 13) for every state in the Hilbert
space, as this is an embarrassingly parallel task.

Figure 13: Sketch of the mixed stochastic minimization procedure.

##  Appendix D The GUGA-FCIQMC method

Here, we concisely summarize the main concepts of the GUGA-FCIQMC method. An
in-depth description of the algorithm can be found in the literature [14, 12].
The FCIQMC algorithm [127, 128] is based on the imaginary-time
($\tau=\text{i}t$) version of the Schrödinger equation,

$\small\frac{\partial\mspace{2.0mu}\ket{\Psi(\tau)}}{\partial\mspace{2.0mu}\tau}=-\hat{H}\ket{\Psi(\tau)}\;\stackrel{{\scriptstyle\int\text{d}\tau}}{{\rightarrow}}\;\ket{\Psi(\tau)}=\operatorname{e}^{-\tau\hat{H}}\ket{\Phi(0)}.$
(46)

Integrating Eq. (46) and performing a first-order Taylor expansion yields an
iterable expression for the eigenstate, $\ket{\Psi(\tau)}$

$\Psi(\tau+\Delta\tau)\approx\left(1-\Delta\tau\hat{H}\right)\Psi(\tau),$ (47)

which yields the ground state, $\ket{\Psi_{0}}$, in the long time limit,
$\tau\rightarrow\infty$. In FCIQMC, the full wave function,
$\ket{\Psi(\tau)}$, is sampled stochastically by a set of so-called _walkers_
and yields estimates for the ground- and excited-state [183] energies and
properties [184] via the one- and two-body reduced density matrices. [185]
More details can be found in the literature, [127, 128] especially in the
recently published review article [13] and references therein.

In its original implementation FCIQMC is formulated in $m_{s}$-conserving SDs,
and thus, does not conserve the total spin quantum number, $S$. For
interpretability, control, and improved convergence properties, it is useful
to impose the $SU(2)$ spin-rotational symmetry. We recently developed a spin-
adapted implementation of FCIQMC in our lab[12, 14, 17], based on the UGA[4,
186, 5, 132, 187, 129, 130, 188, 126] and its graphical extension (GUGA).
[120, 121]. It was originally conceived for _ab initio_ quantum chemical
systems, but we recently applied it to study lattice models, like the two-
dimensional Hubbard model[189, 190, 191].

The UGA is based on the spin-free formulation of quantum chemistry [125] and
was pioneered by Paldus[4, 7], who found an efficient usage of the Gel’fand-
Tsetlin basis–a general basis for any unitary group $U(n)$[192, 193, 194]–to
the electronic structure problem[4, 5, 132]. Based on this work, Shavitt
developed the graphical unitary group approach (GUGA), [120, 121, 122] which
provides an elegant and highly effective way to calculate Hamiltonian matrix
elements between these spin-adapted basis states, also called configuration
state functions (CSFs).

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