#  An SU(3) Yang-Mills Structure for Electron-Phonon Interactions Resulting
from Strong Electron Correlations in 2D hexagonal lattices

J. M. Booth<EMAIL_ADDRESS>ARC Centre of Excellence in Exciton
Science, RMIT University, Melbourne, Australia

###### Abstract

Concise and powerful mathematical descriptions of the interplay of spin and
charge degrees of degrees of freedom with crystal lattice fluctuations are of
extreme importance in materials science. Such descriptions allow structured
approaches to optimizing material efficiencies resulting in considerable
resource savings and higher performance devices. In this work, by re-imagining
the the Gell-Mann matrices as 3$\times$3 linear transformations acting on a
column vector of position states, an SU(3) theory of the interplay between
lattice fluctuations and strong electron correlations in 2-dimensional
hexagonal materials such as graphene is formulated.

##  I Introduction

There is no doubt that two-dimensional hexagonal materials such as graphene
are of enormous significance in contemporary materials science. Graphene in
particular exhibits many useful electronic phenomena, and in addition has
attractive theoretical qualities. It exhibits extremely high electron mobility
at room temperature, and can sustain high densities of electric
currentsNovoselov  _et al._ (2012). It also has a high Young’s modulus and can
be readily chemically functionalizedNovoselov  _et al._ (2012).

Magic Angle twisted bilayer Graphene superlattices have been found to exhibit
both correlated insulating behaviorCao  _et al._ (2018a) and unconventional
superconductivityCao  _et al._ (2018b) with a Tc of up to $\sim$ 1.7 K. A
model for the metal-insulator transition in graphene superlattices was
proposed which took the form of a two-orbital Hubbard Model on an emergent
honeycomb latticeYuan and Fu (2018).

In a previous studyBooth and Russo (2020) it was found that an SU(2) Yang-
Mills description of electron-phonon interactions in linear systems such as
vanadium dioxide can be developed by assuming that the transverse phonons
couple to the electron spin via a Rashba-type mechanism, while charge ordering
which leaves the spins unaffected is carried out by the longitudinal modes
which constitute a “neutral” boson. The SU(2) interaction vertex described
there has many advantages over the standard U(1) approach to electron-phonon
coupling. It contains both charge and spin-ordering and manifests at
neighboring atomic sites and therefore can describe phase transitions in which
spin-ordering is also present.

In a subsequent studyBooth (2020) it was shown that the structure of the SU(2)
vertex arises from minimizing electron correlations. That is, the presence of
strong electron correlations, and thus electron physics obeying the Hubbard
Model, can generate phonons which are matrix operators acting on more than one
nucleus simultaneously. In that study a simple Hubbard Hamiltonian was used:

$H=-t\sum\limits_{\langle
ij\rangle}(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma})+\\\
U\sum\limits_{i}n_{i\uparrow}n_{i\downarrow}$ (1)

where $t$ is the electron hopping energy, and $U$ is the usual on-site
repulsion term which penalizes double occupancies. The phonons identified in
that study order the charge and spin in such a manner as to reduce the energy
of the electrons.

In quasi-linear systems such as vanadium dioxide it was shown that the matrix
valued electron phonon vertex was a 2$\times$2 linear transformation
parametrized by the Pauli matrices: an SU(2) gauge theory.

One of the biggest advantages of this approach was that by identifying the
phonons which correspond to charge and spin fluctuations which lower the
electron energy, the system near Tc can be described by an interacting liquid
of SU(2) bosons. This description can in turn be simply modelled using an
Ising-type Hamiltonian which leads to the formation of a phase coherent phonon
state which breaks the symmetry and opens a gapBooth (2020).

In this work we repeat the same process, but we examine the SU(3) gauge group
and find that it also describes charge and spin fluctuations in a low
dimensional structure, however unlike linear systems as per SU(2) it describes
2D hexagonal systems, such as graphene. Importantly, we assume the same
Rashba-type mechanism is active which couples the spin raising and lowering
operators to polarization vectors with components in the $x$ and $y$
directions, while polarizations along the $z$ axis are neutral.

While the Hubbard Model used here describes insulator to metal transitions,
the main point of this work is to show that the electron charge and spin
fluctuations of a hexagonal system described by the Hubbard Model can be
mapped to an SU(3) gauge theory. Therefore the correlated metallic state is an
interacting liquid of SU(3) bosons.

The properties of this liquid are then given by the scattering amplitudes of
the SU(3) bosons. Thus the strongly correlated electron system is mapped to a
Pure Yang-Mills theory: the interactions of the bosons contain a description
of both the electron and the lattice behavior.

This discussion presents the algebraic formalism of the SU(3) approach by
first presenting a review of the SU(2) formalismBooth and Russo (2020); Booth
(2020). This is then extended to SU(3) by identifying the eight generators of
the group with collective phonon modes which act on three site sub-units of
the hexagons. It is shown that like the SU(2) group of linear systems (e.g.
VO2), the SU(3) generators correspond to atomic motions which minimize the
energy of a system described by the Hubbard Model of Equation 1.

The calculation of boson-boson scattering terms is not included in this
discussion. It requires the formal machinery of Yang-Mills scattering
amplitudes to be adapted to a non-relativistic environment. Gluon scattering
amplitudes are highly non-trivial due to the number of generators, and the
redundancy built into the diagrammatic gauge theory formalism.Dixon (2013);
Arkani-Hamed  _et al._ (2017) This aspect of the approach is thus left to a
future study.

##  II SU(2) Gauge Theory Review

The key to viewing crystalline materials through the lens of non-Abelian gauge
theory is the coupling of the spinors to the atomic motions through the Pauli
matrices, and re-imagining the gauge transformations as linear transformations
acting on position states.

The tree-level form of a non-Abelian interaction vertex is given most
generally byZee (2003):

$\bar{\psi}\gamma^{\mu}\hat{G}_{\mu}^{a}\hat{T}^{a}\psi$ (2)

While unfamiliar for condensed matter physicists, its application to condensed
matter systems can be broken up into simple parts. The electron states $\psi$
are a column vector of position states, the number of which is given by the
dimension of the SU(N) group, i.e. for an SU(2) group there are two position
states. In the case of vanadium dioxide these are neighboring vanadium atoms
on the one-dimensional vanadium chains of the structure. Explicitly:

$\psi=\begin{pmatrix}\psi(x_{i})\\\ \psi(x_{j})\end{pmatrix}$ (3)

where $x_{i}$ and $x_{j}$ are neighbouring atomic sites on the chain.

Each position state electron wavefunction carries spinor indices, and the
gamma matrices $\gamma^{\mu}$ act on these spinor variables. Importantly each
$\psi(x_{i})$ is a four component spinor (examples of which are the Nambu
spinorsBooth and Russo (2020)), as it includes both electron and hole spinor
variables, which necessitates the use of gamma matrices. These are double-
stacked Pauli matrices, with a sign inversion ensuring the hole states satisfy
the Weyl equation. They are given by:

$\gamma^{0}=\begin{pmatrix}0&\mathbb{1}\\\
\mathbb{1}&0\end{pmatrix},\gamma^{i}=\begin{pmatrix}0&\sigma^{i}\\\
-\sigma^{i}&0\end{pmatrix}$ (4)

where $\mathbb{1}$ is the 2$\times$2 unit matrix, and the $\sigma^{i}$ are the
usual Pauli matrices.

The $\hat{G}^{a}_{\mu}$ are a set of $N^{2}-1$ vector fields, which are
quantized in the usual manner (note these may have non-linear dispersion,
unlike Lorentz invariant vector fields):

$\hat{G}^{a}_{\mu}(x)=\int\frac{d^{3}\mathbf{p}}{{2\pi}^{\frac{3}{2}}2E_{\mathbf{p}}^{\frac{1}{2}}}\sum_{\lambda}\big{[}\hat{a}_{\mathbf{p}}\epsilon^{\lambda}_{\mu}(p)e^{ipx}+\hat{a}_{\mathbf{p}}^{\dagger}\epsilon^{*\lambda}_{\mu}(p)e^{-ipx}\big{]}$
(5)

where $a$ (no hat) labels the vector field, and these multiply the
$\hat{T}^{a}$, which are the generators of the gauge group; $N\times N$
matrices. In the SU(2) theory developed for quasi-linear systems, the
generators of the SU(2) group are the Pauli matrices. However, as stated above
the key to the utility of the Yang-Mills formalism for crystal systems is in
re-imagining these generators as linear transformations which act on position
states.

What this means in practice is that the generators take the polarization
vector from the field $\hat{G}^{a}_{\mu}$ and apply it to $N$ sites
simultaneously, with coefficients given by the generator $\hat{T}^{a}$. The
simplest illustrative example of this would be $\hat{T}^{3}$ of the SU(2)
theory, contracted with a vector field (omitting the gamma matrices for
clarity):

$\hat{W}^{3}_{\mu}\sigma^{3}\psi=\begin{pmatrix}\hat{W}^{3}_{\mu}&0\\\
0&-\hat{W}^{3}_{\mu}\end{pmatrix}\begin{pmatrix}\psi(x_{i})\\\
\psi(x_{j})\end{pmatrix}$ (6)

This operation applies the polarization vector $\epsilon^{\lambda}_{\mu}$
(with the appropriate creation of annihilation operator) of the field
$\hat{W}^{3}_{\mu}$, to the positions of the nuclei to which the electron
wavefunctions are bound in the column vector. However with a crucial change of
sign for $\psi(x_{j})$, as the generator $\hat{T}^{3}$ is just the Pauli
matrix $\sigma^{3}$. Therefore, this interaction describes equal and opposite
motions of neighboring atoms, a Peierls pairing.

This describes the charge ordering, however the gamma matrices also include
spin ordering. For this to manifest in a convenient form we must choose a
coordinate system in which the diagonal elements of the generators act on the
$z$-components of the position states, while the off-diagonal terms act on the
$x$ and $y$ components. In particular, if we re-label the Pauli matrices
$\sigma^{1},\sigma^{2},\sigma^{3}\rightarrow\sigma^{x},\sigma^{y},\sigma^{z}$,
then the SU(N) generators which are derived from the Pauli matrices must
multiply the corresponding component of the boson polarization vector. When
this is contracted with the corresponding gamma matrix, the correct spin and
charge ordering occurs.

Again, taking SU(2) as an example, the first generator is
$\sigma^{1}=\sigma^{x}$. Choosing a coordinate system such that the
polarization vector $\epsilon^{1}_{\mu}$ of the corresponding boson field
$\hat{W}^{1}_{\mu}$ has only one non-zero component, the $x$ component, and
writing $\gamma^{1}=\gamma^{x}$ gives:

$\bar{\psi}\gamma^{\mu}\hat{W}_{\mu}^{a}\hat{T}^{a}\psi=\bar{\psi}\gamma^{x}\hat{W}^{1}_{x}\sigma^{x}\psi$
(7)

The operator sandwiched between the electron states is:

$\gamma^{x}\hat{W}^{1}_{x}\sigma^{x}=\begin{pmatrix}0&\epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}\\\ \epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}&0\end{pmatrix}$ (8)

The term in the first column applies a displacement to the nucleus position of
$\psi(x_{i})$, and operates on the electron spinor variables with $\sigma^{x}$
and on the hole spinor variables with $-\sigma^{x}$. The term in column two
applies the same operation to the nucleus position of $\psi(x_{j})$. The
generator has only positive components so the displacement vectors are in the
same direction, and this is depicted schematically in the first diagram of
Figure 1.

Figure 1: Atomic motions which correspond to the different SU(2) modes,
$\hat{W}^{1}$, $\hat{W}^{2}$ and $\hat{W}^{3}$.

The interaction vertex for the generator corresponding to $\sigma^{y}$, with
$\gamma^{2}$ relabeled as $\gamma^{y}$, is obtained in the same manner:

$\gamma^{y}\hat{W}^{2}_{y}\sigma^{y}=\begin{pmatrix}0&\epsilon_{y}\begin{pmatrix}0&-i\sigma^{y}\\\
i\sigma^{y}&0\end{pmatrix}\\\ \epsilon_{y}\begin{pmatrix}0&i\sigma^{y}\\\
-i\sigma^{y}&0\end{pmatrix}&0\end{pmatrix}$ (9)

where the imaginary units which of $\sigma^{y}$ have been grouped with the
Pauli matrices which make up $\gamma^{y}$. Comparing this with the result for
the $\sigma^{x}$ generator, we see that we can make linear combinations of
these two vertices which correspond to spin raising and lowering operators:

$\frac{1}{\sqrt{2}}\gamma^{\mu}\big{(}\hat{W}^{1}_{\mu}\sigma^{1}+\hat{W}^{2}_{\mu}\sigma^{2}\big{)}=\\\
\begin{pmatrix}0&\frac{1}{\sqrt{2}}\bigg{(}\epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}+\epsilon_{y}\begin{pmatrix}0&-i\sigma^{y}\\\
i\sigma^{y}&0\end{pmatrix}\bigg{)}\\\
\frac{1}{\sqrt{2}}\bigg{(}\epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}+\epsilon_{y}\begin{pmatrix}0&i\sigma^{y}\\\
-i\sigma^{y}&0\end{pmatrix}\bigg{)}&0\end{pmatrix}\\\
=\begin{pmatrix}0&\begin{pmatrix}0&\hat{S}^{-}\\\
-\hat{S}^{-}&0\end{pmatrix}\\\ \begin{pmatrix}0&\hat{S}^{+}\\\
-\hat{S}^{+}&0\end{pmatrix}&0\end{pmatrix}$

where we have set $\epsilon_{x}=\epsilon_{y}=1$ in the last line. When acting
on the column vector of spinor states this will flip the spins of neighboring
electrons into an antiferromagnetic order. This vertex, in which equal
polarization vectors in the $x$\- and $y$-directions when contracted with
gamma matrices gives antiferromagnetic ordering mirrors the ordering seen in
the M2 structure of vanadium dioxide.Pouget  _et al._ (1974) The M2 form
contains two alternating chain structures of vanadium atoms. One of which
pairs along the collinear axis, while the other orders antiferroelectrically.
The antiferroelectric ordering also coincides with anftiferromagnetic ordering
of the spins. This vertex describes a combination of antiferromagnetic and
antiferroelectric ordering, as required.

The third SU(2) generator describes Peierls pairing, however it leaves the
spins unchanged as it is contracted with the $\gamma^{3}$ matrix, which is
comprised on $\sigma^{z}$ matrices. Re-labeling $\gamma^{3}=\gamma^{z}$ and
$\sigma^{3}=\sigma^{z}$:

$\gamma^{z}\hat{W}^{3}_{z}\sigma^{z}=\begin{pmatrix}\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}&0\\\ 0&-\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}&0\end{pmatrix}$ (10)

Thus these three vertices which are derived from the three generators of the
SU(2) gauge group describe antiferroelectricity, Peierls pairing and
antiferromagnetic ordering when acting on a two component column vector of
neighboring spinor states.

In a previous work it was shown that the polarization vectors which correspond
to the SU(2) bosons are rather simply related to electron correlations arising
from the Hubbard ModelBooth (2020). This combination of antiferromagnetic
ordering and Peierls pairing will lower the energy of the electron liquid, and
therefore the Hubbard Model physics is the source of these SU(2) bosons.

As vanadium dioxide undergoes a metal-insulator transition which coincides
with a crystal structure transformation, the interest in that system is on
vacuum expectation values of the SU(2) bosons as the system passes through Tc.
However, above the critical Temperature, the tetragonal VO2 form will be a
liquid of strongly interacting electrons which are generating SU(2) lattice
fluctuations.

##  III 2d Hexagonal Lattices and SU(3)

The question then becomes, if pairing and antiferromagnetic spin ordering
lower the electron correlations, can this formalism which intrinsically
contains such phenomena, be used to describe systems which are not simple
linear chains? The answer, at least for strongly correlated 2D hexagonal
systems, is yes. However, there is a slight generalization needed, and the
gauge group must be expanded to SU(3).

The Gell-Mann matrices which parametrize the SU(3) gauge group of Quantum
Chromodynamics are given by:

$\lambda_{1}=\begin{pmatrix}0&1&0\\\ 1&0&0\\\
0&0&0\end{pmatrix},\quad\lambda_{2}=\begin{pmatrix}0&-i&0\\\ i&0&0\\\
0&0&0\end{pmatrix},\quad\lambda_{3}=\begin{pmatrix}1&0&0\\\ 0&-1&0\\\
0&0&0\end{pmatrix}\\\ \lambda_{4}=\begin{pmatrix}0&0&1\\\ 0&0&0\\\
1&0&0\end{pmatrix},\quad\lambda_{5}=\begin{pmatrix}0&0&-i\\\ 0&0&0\\\
i&0&0\end{pmatrix},\quad\lambda_{6}=\begin{pmatrix}0&0&0\\\ 0&0&1\\\
0&1&0\end{pmatrix}\\\ \lambda_{7}=\begin{pmatrix}0&0&0\\\ 0&0&-i\\\
0&i&0\end{pmatrix},\quad\lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\
0&1&0\\\ 0&0&-2\end{pmatrix}$ (11)

The key to understanding the physics of applying this approach to condensed
matter systems is identifying the three spin operators (i.e. the Pauli
matrices) which come from the gamma matrices, with the coefficients which
arise in the gauge group generators.

In the Gell-Mann matrices above, the interaction vertex multiplies each of the
entries in the matrix with a gamma matrix. The gamma matrices themselves are
comprised of the Pauli matrices which give the required spin ordering.

However, the Pauli matrices are tied to the polarization vectors of the bosons
as in the SU(2) theory above. So, to generate pairing which is a “neutral”
mode, i.e. doesn’t affect the spin, we need the Gell-Mann matrix to contain
$1$ and $-1$ for the paired polarization vectors, and for this to be
contracted with $\gamma^{3}$, which we re-iterate is:

$\gamma^{3}=\begin{pmatrix}0&\sigma^{z}\\\ -\sigma^{z}&0\end{pmatrix}$ (12)

Therefore the polarization vectors of this mode must be aligned along the
$z$-axis, in order for the spin operators to correspond to the coordinate axes
of the crystal.

To apply the spin raising and lowering operators to the structure we need
matrices which contain only terms equal to $1$ to contract with $\sigma^{x}$
which comes from $\gamma^{1}$, and separate generators which contain $\pm i$
to contract with $\gamma^{2}$ which contains $\sigma^{y}$ terms, to use in
linear combinations which give the spin raising and lowering operators.
However these must correspond to polarization vectors in the $x$, and $y$
directions.

Therefore, the coordinate system is heavily constrained. Thankfully a
coordinate system can be mapped to the SU(3) generators, and it describes a 2D
hexagonal structure.

Figure 2: Schematic of the lattice structure of a two-dimensional hexagonal
system such as graphene, and the coordinate system used in this work. The
system is Cartesian; the $z$-axis aligns with two of the six sides of each
hexagon, and the $x$\- and $y$-axis rotation with respect to this is a free
parameter determined by experiment.

Figure 2 gives a schematic illustration of a two-dimensional hexagonal
structure with the corresponding coordinate axes. The $z$-axis aligns with two
sides of each hexagon, while the $x$\- and $y$\- axes are orthogonal, with the
rotation with respect to the $z$-axis determined by experiment.

By orienting the coordinate axes in this manner, a remarkable coincidence
between the pairing motions and the diagonal generators of the SU(3) group
appears. As stated above the pairing modes must contain entries in their
respective generators which are opposite in sign. Obviously, only
$\lambda_{2},\lambda_{3},\lambda_{5},\lambda_{7}$, and $\lambda_{8}$ qualify.
However, $\lambda_{2},\lambda_{5}$ and $\lambda_{7}$ contain imaginary units
$\pm i$ needed for spin raising and lowering.

There is also the issue that as the structure is not linear, it is difficult
to write down generators that pair more than two of the six atoms in each ring
using polarization vectors which have only a non-zero $z$-component.

Figure 3: Schematic of the actions of the $\lambda_{3}$ and $\lambda_{8}$
generators when applied to the three-atom unit.

Figure 3 shows how this can be done. The first step is to break each 6 atom
ring into two chains of three atoms, an upper chain and a lower chain. This
gives the required three component column vector of nuclei position states for
the generator to act on. Then, looking at the action of $\lambda_{3}$ and
orienting the $z$-axis as shown, we see that like the $\sigma^{3}$ generator
of the SU(2) theory, it pairs the first two neighboring atoms on the chain.

Figure 4: Plot of the energy of a hexagonal Hubbard system in which the first
two atoms are paired, corresponding to the generator $\lambda_{3}$, with the
hopping energy between atoms 1 and 2 (denoted by -t1) decreasing from -1.0 eV
to -1.44 eV. The on-site energy $U$ is constant at 5 eV.

Figure 4 illustrates the energy of a single hexagonal system in which the
electron part of the Hamiltonian is given by Equation 1, and the first two
atoms of the hexagon are paired, thereby decreasing the hopping energy $t_{1}$
between them, corresponding to the $\lambda_{3}$ diagram of Figure 3, with the
on-site energy term constant at $U$ = 5 eV.

Despite this mode only pairing two of the six atoms of the hexagon, a
substantial decrease in the electronic energy of the system is observed
despite the corresponding increases in hopping energies from the neighbouring
atomic separations increasing. This figure suggests that even if a substantial
elastic potential penalty is incurred by this mode it is still likely to be a
significant component of the electron-phonon interactions of the system.

The problem of pairing the last two atoms using only z-components in the
polarization vectors is rather neatly solved by $\lambda_{8}$. This generator
applies vectors of equal magnitude and orientation to atoms 1 and 2,
maintaining their separation, however it applies a vector in the opposite
direction to atom 3. The opposite signs of the vectors acting on atoms 2 and 3
reduces the $z$-component of their separation. This is not as efficient as the
pairing applied to atoms 1 and 2 due to the angle the bond between atoms 2 and
3 makes with the $z$-axis, however the generator also applies a polarization
vector to atom 3 which is twice the length as those of the generator
$\lambda_{3}$, which compensates.

Figure 5: Plot of the energy of a hexagonal Hubbard system in which the second
two atoms are paired using polarization vectors with only $z$-components,
corresponding to the generator $\lambda_{8}$. The hopping energy between atoms
2 and 3 (denoted by -t2) decreases from -1.0 eV to -1.14 eV. The on-site
energy $U$ is constant at 5 eV.

Figure 5 illustrates the same type of calculation as for the generator
$\lambda_{3}$, instead applied to $\lambda_{8}$. Atoms 1, 2 and 3 of the
hexagonal system are shifted with eigenvectors corresponding to the
$\lambda_{8}$ diagram of Figure 3. Once again, despite only one separation of
the hexagonal system decreasing while slight increases in the separations
between atoms 3 and 4, and atoms 6 and 1 occur, the electronic energy of the
system decreases, although not to the same extent as for the mode
corresponding to $\lambda_{3}$.

As the eigenvectors of these modes are aligned along the z-axis, only the
$\gamma^{3}$ terms of Equation 2 survive, and as these contain $\sigma^{z}$
matrices acting on the spinor variables, these modes leave the spins
unchanged; they are neutral modes.

Thus the data of Figures 4 and 5 indicates that the Hubbard Hamiltonian,
Equation 1, is a source of pairing modes, which can be described by the
generators $\lambda_{3}$ and $\lambda_{8}$ of an SU(3) Yang-Mills theory.

Figure 6: Schematic of the actions of the $\lambda_{1}$, $\lambda_{4}$ and
$\lambda_{6}$ generators when applied to the three-atom unit.

However, such pairing modes will only be most effective at lowering the
electronic energy of the system if the spins of the hexagonal system can be
ordered to give maximum overlap between atoms. This can be achieved using the
remaining generators, which due to their orientations, when contracted with
the gamma matrices give spin raising and lowering operators.

Figure 6 illustrates the modes $\lambda_{1},\lambda_{4}$ and $\lambda_{6}$
corresponding to generators which are aligned with the $x$-axis. Each of these
contain the same entry, and this multiplies the $\gamma^{1}$ or equivalently
$\gamma^{x}$ matrix in the interaction vertex, giving the $\sigma^{x}$ terms
of the spin operators $\hat{S}^{\pm}=\frac{1}{\sqrt{2}}(\sigma^{x}\pm
i\sigma^{y})$.

Figure 7: Schematic of the actions of the $\lambda_{2}$, $\lambda_{5}$ and
$\lambda_{7}$ generators when applied to the three-atom unit.

Figure 7 illustrates the modes $\lambda_{2},\lambda_{5}$ and $\lambda_{7}$
which act on the $y$ components of the nuclei positions, which contain the
imaginary unit, and a pattern of sign changes which give antiferromagnetic
ordering. For example $\lambda_{2}$ applies a polarization vector of $+i$ to
atom 1, which when combined with $\lambda_{1}$ gives
$\sigma^{x}+i\sigma^{y}=\hat{S}^{+}$ acting on the spinor variables of the
electron states. On the hole states, the vertex gives
$-\sigma^{x}-i\sigma^{y}=-\hat{S}^{+}$.

On atom 2 the vertex applies a polarization vector of $-i$ which when combined
with $\lambda^{1}$ gives $\sigma^{x}-i\sigma^{y}=\hat{S}^{-}$ acting on the
electron states, and $-\sigma^{x}+i\sigma^{y}=-\hat{S}^{-}$ acting on the
spinor variables of the hole states.

This pattern is repeated for the generators $\lambda_{4}$,$\lambda_{5}$,
$\lambda_{6}$ and $\lambda_{7}$; combining $\lambda_{4}$ with $\lambda_{5}$
applies $\hat{S}^{+}$ and $-\hat{S}^{+}$ to the electron and hole states of
atom 1 respectively, and $\hat{S}^{-}$ and $-\hat{S}^{-}$ to the electron and
hole states of atom 3. The combination of $\lambda_{6}$ and $\lambda_{7}$
applies $\hat{S}^{+}$ and $-\hat{S}^{+}$ to atom 2 and $\hat{S}^{-}$ and
$-\hat{S}^{-}$ to atom 3.

Of course the oscillatory nature of the fields $\hat{G}^{a}_{\mu}$ will flip
the modes between $\hat{S}^{+}$ and $\hat{S}^{-}$ with a time dependence given
by the frequency of the mode, however the pattern of alternating spin
operations will remain.

In addition, the combination of $\lambda_{4}$ and $\lambda_{5}$ induces
antiferromagnetic ordering on atoms 1 and 3, this combined with the neutral
modes $\lambda_{3}$ and $\lambda_{8}$ will facilitate next-to-nearest-neighbor
hopping between sites 1 and 3 of the 3 atom sub-unit.

There is a slight subtlety required to give alternating spin raising and
lowering operators corresponding the antiferroelectricity. Since these are
composite bosons their actions are defined to be of the form:

$\hat{G}_{1,2}^{+}|\psi(x_{1})\rangle=(\hat{G}^{1}+i\hat{G}^{2})|\psi(x_{1})\rangle\sim\hat{a}^{\dagger}\epsilon_{x}\hat{a}^{\dagger}\epsilon_{y}|\psi(x_{1})\rangle$
(13)
$\hat{G}_{1,2}^{-}|\psi(x_{2})\rangle=(\hat{G}^{1}-i\hat{G}^{2})|\psi(x_{2})\rangle\sim-(\hat{a}^{\dagger}\epsilon_{x}\hat{a}^{\dagger}\epsilon_{y})|\psi(x_{2})\rangle$
(14)

where $\hat{G}_{1,2}$ is the linear combination of the field operators
$G^{1}_{\mu}\lambda^{1}$ and $\hat{G}^{2}\lambda^{2}$, and the imaginary unit
and details of the contraction with the gamma matrices are omitted for
clarity. Thus although only the $\lambda_{2}$ generator has a minus sign, the
operator products for $\hat{G}^{+}/\hat{G}^{-}$ produce polarization vectors
in opposite directions, which renders the linear combination
$\frac{1}{\sqrt{2}}(\hat{G}^{1}\pm i\hat{G}^{2})$ antiferroelectric. The
combinations $\frac{1}{\sqrt{2}}(\hat{G}^{4}\pm i\hat{G}^{5})$ and
$\frac{1}{\sqrt{2}}(\hat{G}^{6}\pm i\hat{G}^{7})$ are also antiferroelectric
modes for the same reason.

Thus, expanding the actions of the generators out to see their effects most
clearly, starting with $\lambda_{3}$, we have:

$\gamma^{\mu}\hat{G}^{3}_{\mu}\lambda_{3}\psi=\\\
\begin{pmatrix}\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}&0&0\\\
0&-\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\ -\sigma^{z}&0\end{pmatrix}&0\\\
0&0&0\end{pmatrix}\begin{pmatrix}\psi(x_{1})\\\ \psi(x_{2})\\\
\psi(x_{3})\end{pmatrix}$ (15)

In this vertex, the $\epsilon_{z}$ polarization vectors act on the positions
of the nuclei ($x_{1}$ and $x_{1}$), shifting them in the positive and
negative $z$ directions respectively, which gives a Peierls pairing of the two
atoms along the $z$-axis. The gamma matrix contains only $\sigma^{z}$ matrices
and therefore does not change the $z$ eigenstates of the spin. Identifying the
gauge charge with the electron spin, this mode is therefore neutral.

For $\lambda_{8}$ due to the non-zero entries lying on the diagonal, which are
contracted with $\gamma^{3}=\gamma^{z}$ we have another neutral mode:

$\gamma^{\mu}\hat{G}^{8}_{\mu}\lambda_{8}\psi=\\\
\frac{1}{\sqrt{3}}\begin{pmatrix}\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}&0&0\\\ 0&\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}&0\\\
0&0&-2\epsilon_{z}\begin{pmatrix}0&\sigma^{z}\\\
-\sigma^{z}&0\end{pmatrix}\end{pmatrix}\\\ \times\begin{pmatrix}\psi(x_{1})\\\
\psi(x_{2})\\\ \psi(x_{3})\end{pmatrix}$ (16)

The $\epsilon_{z}$ polarization vectors shift the nuclei of $\psi(x_{1})$ and
$\psi(x_{2})$ in the positive $z$-direction, while shifting the nucleus of
$\psi(x_{3})$ twice as far in the negative $z$-direction. This pairs the
nuclei at $\psi(x_{2})$ and $\psi(x_{3})$, however it will also facilitate
next-nearest-neighbor hopping between sites $x_{1}$ and $x_{3}$.

For the linear combination $\frac{1}{\sqrt{2}}(\lambda_{1}+\lambda_{2})$ we
have:

$\frac{1}{\sqrt{2}}\gamma^{\mu}\big{(}\hat{G}^{1}_{\mu}\lambda_{1}+\hat{G}^{2}_{\mu}\lambda_{2}\big{)}=\\\
\begin{pmatrix}0&\frac{1}{\sqrt{2}}\bigg{(}\epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}-\epsilon_{y}\begin{pmatrix}0&-i\sigma^{y}\\\
i\sigma^{y}&0\end{pmatrix}\bigg{)}&0\\\
\frac{1}{\sqrt{2}}\bigg{(}\epsilon_{x}\begin{pmatrix}0&\sigma^{x}\\\
-\sigma^{x}&0\end{pmatrix}+\epsilon_{y}\begin{pmatrix}0&i\sigma^{y}\\\
-i\sigma^{y}&0\end{pmatrix}\bigg{)}&0&0\\\
0&0&0\end{pmatrix}\begin{pmatrix}\psi(x_{1})\\\ \psi(x_{2})\\\
\psi(x_{3})\end{pmatrix}$

The $\epsilon_{x}$ and $\epsilon_{y}$ polarization vectors induce
antiferroelectricity of the nuclei at $x_{1}$ and $x_{2}$ as per the argument
above. However, as the generators are contracted as per:
$\gamma^{x}\lambda_{1}$ and $\gamma^{y}\lambda_{2}$. The combinations of the
$\sigma^{x}$ and $\sigma^{y}$ matrices from these contractions generate spin
raising and lowering operators.

The formation of spin raising and lowering operators for the combinations
$\frac{1}{\sqrt{2}}(\lambda_{4}+\lambda_{5})$ and
$\frac{1}{\sqrt{2}}(\lambda_{6}+\lambda_{7})$ arises in the same manner,
acting on sites 1 and 3, and 2 and 3 respectively.

##  IV Discussion

Thus the preceding argument demonstrates that when viewed as 3$\times$3 linear
transformations acting on 3 atom sub-units of 2-dimensional hexagonal
materials, the generators of the SU(3) group describe pairing, and spin
ordering which lower the electronic energy of the system (as given by the
Hubbard Hamiltonian) by facilitating hopping between sites, increasing the
admixture of the negative hopping energy in the total ground state energy.

Therefore, the SU(3) modes above can be considered as proxies for the
electron-electron interactions. An interacting liquid of SU(3) modes
represents a system of fluctuations which seek to minimize the electron
correlations of the system.

In a previous studyBooth (2020) which explored the application of the SU(2)
theory to vanadium dioxide, the SU(2) modes, which again were driven by
electron-electron interactions, exhibited vacuum expectation values. That is,
the quasi-linear system characterized by the SU(2) Yang-Mills theory
experiences a phase transition, in which the SU(2) modes describe the changes
in position of the metal atoms in the oxide. This allowed the system to be
described by an Ising-type Hamiltonian in which the SU(2) modes of adjacent
two-atom sites couple with increasing strength as the Temperature is lowered.

However, the hexagonal system is not expected to exhibit such a
transformation. The different modes cannot be oriented such that they are
orthogonal, for example, $\lambda_{1}$ and $\lambda_{4}$ cannot manifest
simultaneously, and perhaps more importantly, neither can the neutral pairing
modes $\lambda_{3}$ and $\lambda_{8}$.

Therefore, while the individual modes can combine to lower the electron
energy, there is no simple position space configuration which can be adopted
that, similarly to the quasi-linear vanadium dioxide, lowers the hopping
energy, which as a vacuum expectation value breaks the symmetry. Pairing the
atoms along linear chains is a relatively simple symmetry-breaking operation.
In hexagonal systems, this is not the case.

There may however be momentum-space configurations which can lower the energy,
such as Cooper pairing, however the emergence of such pairing requires a
detailed description of the scattering amplitudes of the SU(3) Yang-Mills
modes, which as mentioned above will be dealt with in a future study.

This is however a fascinating subject. In recent years remarkable mathematical
structures have been found to play unexpected roles in the calculation of
scattering amplitudes in planar $\mathcal{N}=4$ Super Yang-Mills theoryArkani-
Hamed  _et al._ (2012); Arkani-Hamed and Trnka (2013). A unification of the
formalism of scattering amplitudes in condensed matter and high energy physics
in this manner is an attractive theoretical prospect. An immediate
simplification in the condensed matter approach developed here is that the
generators are intimately related to the directions of the atomic motions in
space, and therefore the polarization vectors of the modes become largely
redundant.

This presents an exciting opportunity to both examine technologically
significant condensed matter systems through the lens of modern scattering
amplitude methods, and perhaps provide a more prosaic context for the SU(3)
Yang-Mills description of high energy physics which may be more amenable to
experiment.

##  V Methods

The data of Figures 4 and 5 were obtained by using the Python package
QuSpinWeinberg and Bukov (2019). The generators $\lambda_{3}$ and
$\lambda_{8}$ were applied to a single hexagonal unit with polarization
vectors giving the required hopping energy range. The energies of the system
were computed by computing the distances between the atomic sites after the
application of the polarization vectors, converting these into hopping
energies to give a set of $t_{i}$ where $i$ ranges from $1\rightarrow 6$, and
using exact diagonalization to compute the eigenvalues of the electron
Hamiltonian.

##  VI References

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