# Tight-Binding realization of non-abelian gauge fields: singular spectra and
wave confinement

Y. Hernández-Espinosa  Universidad Nacional Autónoma de México, Instituto de
Física, Apartado Postal, 04510 Ciudad de México  E. Sadurní  Benemérita
Universidad Autónoma de Puebla, Instituto de Física, Apartado Postal J-48,
72570 Puebla, México

###### Abstract

We present a geometric construction of a lattice that emulates the action of a
gauge field on a fermion. The construction consists of a square lattice made
of polymeric sites, where all clustered atoms are identical and represented by
potential wells or resonators supporting one bound state. The emulation covers
both abelian and non-abelian gauge fields. In the former case, Hofstadter’s
butterfly is reproduced by means of a chain made of rotating dimers, subject
to periodic boundary conditions parallel to the chain. A rigorous map between
this model and Harper’s Hamiltonian is derived. In the non-abelian case, band
mixing and wave confinement are obtained by interband coupling using SU(2) as
an internal group, i.e. the effects are due to non-commutability of field
components. A colored model with SU(3) made of trimers is also studied,
finding thereby the appearance of flat bands in special configurations. This
work constitutes the first all-geometric emulation of the Peierls
substitution, and is valid for many types of waves.

###### Contents

  1. I  Introduction
  2. II Presentation of results
  3. III Internal degrees of freedom: gauge group obtained from the lattice
    1. III.1 General theory on the lattice
    2. III.2 Decomposition in terms of external lattice couplings and internal polymeric couplings
  4. IV Emulation of Peierls substitution for U(1) gauge field
    1. IV.1 Linear chain
  5. V Emulation of SU(2) field
    1. V.1 A model with SU$(2)$
    2. V.2 Effective negative couplings and level inversion revisited
    3. V.3 Wave confinement in SU(2). Bound states
  6. VI Band mixing in SU(3) models with color
  7. VII Conclusion
  8. A Unitary map between translation operators in 1D; beyond Aubry’s duality.
  9. B Box-counting fractal dimension calculation

---
(a)
(b)
Figure 1:  (a) The spectrum of the decoupled symmetric band for a chain of 801
dimers. The Hofstadter butterfly emerges in the first quarter of the frequency
range $\omega(N-1)/2\pi$. The geometric values are $L=1.66$, $d=0.1$ and
$\lambda=1.66$ (b). The spectrum of the decoupled antisymmetric band where the
Harper’s Hamiltonian is emulated for $\Lambda=0.08\cos\omega$.The geometric
values are $L=2.8$, $d=0.35$ and $\lambda=1$. Energy is in arbitrary units.

##  I  Introduction

The realization of Yang-Mills theories in lattices is the ultimate goal of
spectral quantum emulations [1]. The standard model of particles and fields
rests on the assumption that internal degrees of freedom such as color are
dynamically responsible for the emergence of composite structures in the form
of baryons and mesons [2]. Lattice QCD is indeed a dynamical description of
such theories, where the numerical work is meant to show the emergence of
color confinement[3] and infrared slavery [4, 5, 6]. However, the
implementation of gauge symmetry on the lattice must be done with a proper
generalization of Peierls’ substitution that incorporates internal degrees of
freedom such as color or isospin. In this paper we take care of this aspect,
as the lattice emulations presented are sufficiently simple to illuminate
important aspects of gauge fields that can be implemented by purely
geometrical manipulations of sites on the crystal.

Previous steps towards this goal are the use of atomic levels in experiments
involving atomic fountains and the interaction of atoms with the radiation
field [7, 8]. We also find optical realizations of abelian gauge fields –e.g.
the typical electromagnetic field– where photons in coupled waveguides [9]
play the role of a Schrödinger wave under the action of an external force.
Complex hopping amplitudes have also been obtained by means of laser
techniques [10], and internal degrees of freedom in coupled waveguides have
been used with similar purposes [11]. However, as far as we can see, there has
been no attempt to produce these models on a homogeneous lattice without any
other assumption than a single quantum state per site. We also take this
opportunity to look at microwave realizations with dielectric resonators
supporting a single trapped mode. These have been experimentally successful in
the emulation of solid state systems such as graphene and boron nitride, which
constitute yet another mesoscopic testbed for relativistic spectral emulations
[12, 13, 14, 15, 16, 17, 18, 19] using the celebrated analogy between the
Dirac equation and structures with Dirac points [20, 21, 22, 23, 24].

Our task now is to emulate gauge fields using real scalar waves, starting with
the abelian case. We shall see, on the fly, that the procedure giving rise to
complex couplings can be employed to produce non-abelian theories as well.
With this, we take a step further in our quest for universal emulations using
a single-level atom and purely geometrical manipulations.

We focus on the fermionic part of a non-abelian Yang-Mills theory, where the
bosonic part is treated as a static field. The Yang-Mills Lagrangian for
vector fields interacting with fermions is:

$\displaystyle\mbox{$\cal
L\,$}=\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\bar{\psi}\psi+\bar{\psi}\gamma^{\mu}gA_{\mu}^{a}\tau_{a}\psi-\frac{1}{4}(F^{a}_{\mu\nu})^{2}$
(1) $\displaystyle
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-ig[A_{\mu},A_{\nu}],\quad
F_{\mu\nu}=F_{\mu\nu}^{a}\tau_{a}.$ (2)

As is well-known, the Dirac lagrangian
$\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi-m\bar{\psi}\psi$ has been emulated
in 2+1 dimensions using hexagonal lattices of one species (graphene, massless
case) and two species (boron nitride, massive case). For simplicity we pursue
an emulation of non-abelian fields acting on a fermion hopping on a square
lattice. Therefore, our attention is focused on the first and second terms of
(1), but this time formulated as a Peierls non-abelian gauge field. The
internal degrees of freedom due to the gauge group will be introduced by means
of polymeric sites, i.e. clusters of single-level atoms regarded as a single
cell in a crystal. In this respect, the term $(F^{q}_{\mu\nu})^{2}$ will be
assumed to be fixed and static.

Structure of the paper: section II presents the main results of our emulations
for fields U(1), SU(2) and SU(3) with different tight-binding systems made of
dimers and trimers. In section III we discuss the construction of lattices
with internal degrees of freedom using the energy levels of polymeric
structures. Section IV is dedicated to the emulation of the U(1) abelian field
using linear chains of rotating dimers. The emulation of the non-abelian field
SU(2) is presented in section V, where the phenomenon of wave confinement in a
square lattice of dimers is achieved. The non-abelian vector field ${\bf A}$
and the component $B_{z}$ of the field are plotted. Section VI contains some
numerical results of the spectrum obtained from linear chains of rotating
trimers.

Figure 2:  Hedgehog type lattice.  Figure 3:  Square lattice with localized
wave function at the center. The dimers are presented as a guide to the eye.
The highest energy state displays the localization effect. The dimers are made
of orange (dark) and yellow (light) dots, presented as a guide to the eye. The
parameters are $L=2.9$, $d=1.3$ and $\lambda=1$.

##  II Presentation of results

In quantum field theory the magnetic field is considered the starting point in
the study of gauge invariance. Our first step is the emulation of the abelian
U(1) theory. Historically, the effects of an external magnetic field acting on
a two-dimensional lattice were studied by Peierls [25] and later by
Harper[26]. The Harper model is a particular case of the Almost Mathieu
Operator (AMO) [27], where the irrationality of the magnetic flux per unit
cell plays an important role in the shape of the spectral set. In 1964 [28]
Azbel found that in a lattice with an external magnetic field, each Landau
level in the spectrum of the particle splits successively into sublevels. In
1976 Hofstadter found strong evidence of the fractal nature of the spectrum
and subsequently it was proved that the spectral set must be a Cantor dust
with zero Lebesgue measure [29]. Emulations of the so-called Hofstadter
butterfly have been proposed in acoustic systems [30, 31] and microwave
scatterers [32].

U(1) emulation with dimer chains. We proceed with a construction of linear
chains made of rotating dimers within a nearest-neighbor tight-binding
description. In this artificial realization, the number of complete turns $p$
of one dimer along the chain is the analog of the $\omega$ frequency –or
magnetic flux– in Harper’s Hamiltonian. These chains exhibit singular spectra
for specific lattice parameters, such as the distance between centers of
neighboring dimers and the length of each dimer. Due to the internal degrees
of freedom of each array site, the spectrum shows coupled bands. Two cases can
be distinguished here: i) if the bands are strongly coupled, the non-abelian
nature of the field emerges and ii) when these two bands are completely
decoupled, we recover independent U(1) theories. The latter case is
illustrated in Fig. 1. The electronic structure can be studied properly if we
write the Hamiltonian $H$ in the basis of symmetric and antisymmetric states
of each dimer, and reorganize it into a block-diagonal form of couplings
between symmetric levels (S–S) and antisymmetric levels (A–A). The spectrum
generated by the S–S block is presented in panel (a) of Fig. 1, where the
Hofstadter butterfly appears. This case emulates a magnetic field piercing
through a square lattice and corresponds to $\Lambda=1$ in the AMO. On the
other hand, panel (b) shows the spectrum of a magnetic field in a rectangular
lattice, in this case $\Lambda$ is approximately 0.08$\cos\omega$. Both
spectra show invariance when periodic boundary conditions are imposed, which
is in agreement with the independence of the spectrum on Bloch’s quasi-
momentum $k$ along the periodic direction of a square lattice under the effect
of an external magnetic field in the Landau gauge. It has been proved that the
spectrum of the AMO is a Cantor set [33], so in order to complete the analogy
of our system with the Harper’s model, we report the fractal dimensions of the
antisymmetric and symmetric bands for fixed values of $\omega(N-1)/2\pi$ in
Fig. 14 and Table 1 respectively.

Figure 4:  Spectrum of the hedgehog type lattice as a function of lattice
parameter $L$. Blue points represent localized states at the center of the
array as shown in Fig. 3. This state merges with one of the bands as the
interdimer distance increases.

SU(2) emulation and wave confinement. The hypothesis of color confinement in
quantum chromodynamics (QCD) establishes that only color-neutral multiplets of
SU(3) can be observed. In the same spirit of the previous section, we
construct a square lattice made of dimers in order to emulate the phenomenon
of localization in SU(2). Although wave localization has been studied also in
disordered arrays –Anderson localization– we are interested here in stronger
effects, e.g. bound states localized around a central point. The lattice is
illustrated in Fig. 2 and the details of its construction will be discussed in
section V. Figure 3 shows the localized wave function of the highest energy
state on a lattice made of 400 dimers, whose sites are included as spatial
reference. We can observe that the confinement manifests around the region
with the strongest couplings. This localized state coalesces with the
propagation band and disappears in the limit when the lattice constant
–distance between dimer centers– is much greater than the length of the dimer
i.e. the fully periodic case without any internal structure.

Linear chain with SU(3). Analogously to the linear chain of dimers, the SU(3)
case can be studied by adding more degrees of freedom to each site. We propose
a polymer of nine sites illustrated in Fig. 13. Some interesting results arise
from this model. The spectrum of the system consists of three bands and two of
them are overlapped as shown in Fig. 5 panels a) and b). The upper band shows
fractal structure consistent with Hofstadter’s butterfly, while the lower band
shows a dense structure with no recognizable pattern. The unitary cell of the
chain consists of a trimer made of trimers as shown in Fig. 13. The separation
of the bands can be performed by symmetry breaking of i) the global $C_{3}$
symmetry by rotation of only one internal trimer and ii) internal $C_{3}$
symmetry by deformation of the internal trimers. In panel c) of Fig. 5 the
lower part of the spectrum shows four decoupled bands with very small
bandwidths. The study of superconductivity associated with flat bands might be
of interest, as shown elsewhere[34]. In our case, the spectrum shows localized
states around specific energy values when the magnetic flux varies.

---
(a)
(b)
(c)
Figure 5:  The spectra of a linear chain made of trimers. The unit cell of the
chain is illustrated in Fig. 13. a) The upper part of the spectrum is made of
two overlapped bands. A fractal structure can be distinguished, resembling a
Hofstadter’s butterfly. b) Lower part of the spectrum. The band is dense and
shows no recognizable structure. c) Spectrum with internal and global $C_{3}$
broken symmetry. The lower bands are flat.  Figure 6: Neighboring polymer
sites. Two types of couplings can be distinguished here: i) intradimeric,
which connects two sites of the same polymer (solid yellow line) and ii)
interdimeric, which couples two sites from different polymers (blue dashed
line). The distance between polymer centers (green dashed line) is assumed
constant.  Figure 7: Chains of rotating dimers. The arrows go from site 1 to
site 2 of each dimer. The index $p$ stands for the number of complete
rotations of one dimer along the chain. This parameter is analogous to the
frequency $\omega$ in Harper’s Hamiltonian as shown in eq. (58) and eq. (59).

##  III Internal degrees of freedom: gauge group obtained from the lattice

The results presented in section II provide evidence of a U(1) field emulation
on two-dimensional lattices and the localization phenomenon in the SU(2) case.
In order to properly formulate gauge fields in tight-binding arrays, the
analog of minimal coupling in discrete systems must be introduced. This
section is devoted to such formalism in discrete space with their correct laws
of transformation, including non-abelian fields.

###  III.1 General theory on the lattice

The usual treatment in continuous variables includes non-abelian fields
$A_{\mu}=A_{\mu}^{\sigma}\tau_{\sigma}$, with $\mu$ a space-time index,
$\sigma$ a group index and $\tau$ the generators of SU$(N)$. The minimal
coupling prescription for space and time, ensures the gauge invariance of a
theory involving charged particles and dynamical bosonic fields

$\displaystyle p_{\mu}\mapsto p_{\mu}+gA_{\mu},\qquad{\bf p}\mapsto{\bf
p}+g{\bf A}.$ (3)

This can be specialized to static fields where the 0-th component is
irrelevant (purely magnetic); this is expressed in the second relation above
and it is useful when considering static realizations in crystals. Upon the
action of a unitary group U$(N)$, one has the transformations (here $\hbar=1$)

$\displaystyle p_{\mu}\mapsto\tilde{p}_{\mu}=U^{\dagger}p_{\mu}U,$ (4)
$\displaystyle
A_{\mu}\mapsto\tilde{A}_{\mu}=UA_{\mu}U^{\dagger}-\frac{i}{g}U\partial_{\mu}U^{\dagger},$
(5) $\displaystyle|\psi\rangle\mapsto U|\psi\rangle$ (6)

and they are such that $p_{\mu}+gA_{\mu}\mapsto
U^{\dagger}(p_{\mu}+g\tilde{A}_{\mu})U$ remains an invariant, i.e. the
transformation of $A$ compensates for the unitary transformation of the
operators and physical states. In our discussions, this standard formulation
has to be deduced in the spirit of (3): A stationary Hamiltonian formulation
in first quantization necessitates a Schrödinger-like equation coming from
$p_{0}$, so a gauge invariant spectral problem involves now the transformation

$\displaystyle\tilde{H}=U^{\dagger}HU,\qquad\partial_{0}U=0$ (7)

and a static field such that

$\displaystyle A_{0}={\rm constant},\quad\tilde{{\bf A}}=U{\bf
A}U^{\dagger}-\frac{i}{g}U{\bf\nabla}U^{\dagger}.$ (8)

When it comes to derivatives on the lattice, we have to consider finite-
difference operators. The aim is to identify the functional dependence of $H$
on the field $A$ using the correct form of minimal coupling in discrete
variables. For Hamiltonians that depend of translation operators $T_{i}$ and
lattice vectors ${\bf R}$, we use the notation

$\displaystyle{\bf R}=\sum_{i=1}^{m}n_{i}{\bf a}_{i}$ (9) $\displaystyle
N_{i}|n_{1},...,n_{m}\rangle=n_{i}|n_{1},...,n_{m}\rangle,$ (10)
$\displaystyle
T_{i}|n_{1},...,n_{i},...,n_{m}\rangle=|n_{1},...,n_{i}+1,...,n_{m}\rangle.$
(11)

where ${\bf a}_{i}$ stand for primitive vectors and $m$ for the number of such
vectors in the array. In our realizations, the lattice states
$|n_{1},...,n_{m}\rangle$ are also represented by localized wave functions
around sites –e.g. a trapped mode inside a cylindrical dielectric cavity. In
this language, the operators $T$ have a differential form

$\displaystyle\langle{\bf r}|T_{i}|n_{1},...,n_{2}\rangle$ $\displaystyle=$
$\displaystyle\langle{\bf r}|n_{1},...,n_{i}+1,...,n_{2}\rangle$ (12)
$\displaystyle=$ $\displaystyle\langle{\bf r}-{\bf
a}_{i}|n_{1},...,n_{m}\rangle$ $\displaystyle=$ $\displaystyle\exp(-i{\bf
a}_{i}\cdot{\bf p})\langle{\bf r}|n_{1},...,n_{m}\rangle.$

It is not difficult to guess that minimal couplings should be given by complex
exponentials in front of $T$: Indeed, under a static gauge transformation

$\displaystyle\tilde{T}_{i}=\exp(-i{\bf a}_{i}\cdot U^{\dagger}{\bf
p}U)=U^{\dagger}T_{i}U.$ (13)

Moreover, a local dependence of the transformation $U(N_{1},...,N_{m})\equiv
U({\bf N})$ allows to write

$\displaystyle\tilde{T}_{i}=U^{\dagger}(N_{1},...,N_{2})U(N_{1},...,N_{i}-1,...,N_{m})T_{i},$
(14)

or $\tilde{T}_{i}=U^{\dagger}({\bf N})U({\bf N}-{\bf e}_{i})T_{i}$, which are
the transformations we want to emulate. In order to compensate for this
transformation, one writes space-dependent couplings in front of translation
operator as

$\displaystyle H=\sum_{k,i}\Delta_{k,i}({\bf N})(T_{i})^{k}+{\rm h.c.}+V({\bf
N}),$ (15)

where $V$ is a local potential, $k$ is the range of the interaction, and
$\Delta_{k,i}$ are the coupling constants as functions of site operators; in
our case, they are real and positive. The aforementioned invariance imposes a
transformation of $\Delta$ of the type

$\displaystyle\tilde{\Delta}_{k,i}({\bf N})=\Delta_{k,i}({\bf
N})U^{\dagger}({\bf N}-{\bf e}_{i})U({\bf N}),$ (16)

thus compensating the extra factors appearing in (14). If we want to know the
specific dependence of $H$ on $A$, the abelian case yields simply

$\displaystyle\Delta_{k,i}({\bf N})=|\Delta_{k,i}({\bf
N})|\times\exp\\{-ig\,{\bf a}_{i}\cdot{\bf A}({\bf N})\\},$ (17)
$\displaystyle|\Delta_{k,i}({\bf N})|\equiv|\Delta_{k,i}({\bf N})|^{\dagger}.$
(18)

Here, the hermitian part of $\Delta$ has been written explicitly to show that
the field appears inside a unitary operator, while the modulus  is still
allowed to depend on ${\bf N}$ due to lattice deformations, such as those
considered in [35, 19, 36]. A gauge transformation can be performed now at the
level of $A$; by defining a set of orthogonal vectors –not necessarily on the
physical lattice– $\hat{b}_{1}\perp{\bf a}_{1},\hat{b}_{2}\perp{\bf a}_{2}$,
the displacement by a discretized derivative

$\displaystyle{\bf A}({\bf N})\mapsto{\bf A}({\bf
N})+\frac{\hat{b}_{1}}{\hat{b}_{1}\cdot{\bf a}_{2}}\left[\Phi({\bf N}-{\bf
e}_{1})-\Phi({\bf N})\right]+\frac{\hat{b}_{2}}{\hat{b}_{2}\cdot{\bf
a}_{1}}\left[\Phi({\bf N}-{\bf e}_{2})-\Phi({\bf N})\right]$ (19)

produces exactly the transformation (14) when substituted in (17). The unitary
operator in this case is $U({\bf N})=e^{ig\Phi({\bf N})}$.

For non-abelian fields, one does not express $\Delta$ directly as an
exponential of $A$. Instead, one has to solve a relation of the type

$\displaystyle U^{\dagger}({\bf N})\Delta_{k,i}\left[\tilde{{\bf
A}}\right](T_{i})^{k}U({\bf N})=\Delta_{k,i}\left[{\bf A}\right](T_{i})^{k}$
(20)

which must compensate once more (14), so now

$\displaystyle U^{\dagger}({\bf N})\Delta_{k,i}\left[\tilde{{\bf
A}}\right]U({\bf N})=\Delta_{k,i}\left[{\bf A}\right]U^{\dagger}({\bf N}-{\bf
e}_{i})U({\bf N})$ (21)

and since

$\displaystyle\tilde{A_{j}}=U({\bf
N})\left[A_{j}-\frac{i}{g}(1-U^{\dagger}({\bf N}-{\bf e}_{j})U({\bf
N}))\right]U^{\dagger}({\bf N})$ (22)

one arrives to the functional relation

$\displaystyle\Delta_{k,i}\left[A_{j}-\frac{i}{g}(1-\exp\left[-ig\,\Phi({\bf
N}-{\bf e}_{j})\right]\exp\left[-ig\,\Phi({\bf N})\right])\right]$
$\displaystyle=\Delta_{k,i}\left[A_{j}\right]\times\exp\left[-ig\,\Phi({\bf
N}-{\bf e}_{i})\right]$ (23)

where $\Phi=\Phi^{\sigma}\tau_{\sigma}$ is now an arbitrary non-abelian
operator. This functional relation determines $\Delta$ in terms of $A$
recursively, but its explicit solution is difficult to obtain in general.
Instead we say that there is a non-trivial gauge field acting on a particle
described by Hamiltonian $H$ if there is no transformation $U$ for which

$\displaystyle\Delta_{k,i}\left[0\right]U({\bf N}-{\bf e}_{i})=U({\bf
N})\Delta_{k,i}\left[{\bf A}\right].$ (24)

Even in the abelian case, this demands the use of non-trivial structures such
as complex hopping amplitudes and, as a consequence, singular spectra via
Harper’s equation.

We have achieved the introduction of minimal coupling and gauge theory in the
context of tight-binding systems with single-level sites, e.g. arrays of
resonators. The U(1) field will be implemented in section IV.1 with the
construction of a linear chain made of dimers. This system possesses two
internal degrees of freedom and the couplings $\Delta$ are modulated by the
rotation of each dimer along the chain. We will show that in the appropriate
basis of symmetric and antisymmetric dimeric states, two decoupled blocks
emerge, both emulating singular spectra similar to those obtained from
Harper’s Hamiltonian. The phenomenon of wave confinement with the SU(2) field
is presented in section V.

Figure 8: (a) Geometric parameters for two adjacent dimers. (b) All pairwise
inter dimer couplings. Each dimer is rotated by an angle $\theta_{n}$ which is
a function of the number $N$ of dimers in the chain and the number $p$ of
complete rotations of one dimer along the chain. (b) All possible pairwise
couplings between sites in the same array of dimers as in (a). The couplings
depend on parameters $d$, $L$ and $\theta_{n}$.

###  III.2 Decomposition in terms of external lattice couplings and internal
polymeric couplings

The general idea to include an internal SU$(N)$ group, consists in the
introduction of internal levels corresponding to each lattice site. Since one
site corresponds to a single level resonator, we consider a set or cluster of
such resonators put tightly together, resembling a polymer. Then, each of
these sets, when regarded as a single site with a complex structure, shall
have as many levels as sites in the polymer. See Fig. 6. In connection with
couplings, it is also reasonable to decompose them into two parts using a
diagonalization trick acting exclusively on each polymer. This helps to keep
track of their contribution to the global lattice defined by the location of
each polymer, or its geometric center. The distance between centers shall be
associated with a coupling $\Delta$ referred to as external coupling. The
distance between a geometric center and all other sites of the same polymer
shall be regarded as an internal distance, therefore we refer to the
corresponding $\Delta$ as internal coupling. Some triangular inequalities can
be established for such couplings, as long as they are given by monotonically
decreasing functions of the separation distances, e.g. exponential functions
as in [37, 38, 39] or even more careful approximations involving modified
Bessel functions. Based on geometric relations for the distances, we can say
that the internal couplings in each polymer are more important in magnitude
than the external ones. However, the external couplings shall reflect the
desired crystalline structure, while internal bonds shall give rise to a local
non-abelian field acting as a perturbation. Such a perturbation can be either
weak or strong, depending on the geometry. Once this decomposition is
achieved, we will be ready to use an appropriate basis to describe the
polymeric states that interact with other neighboring polymeric states. In
fact, this has a close resemblance with standard tight-binding models applied
to solid state physics, where individual sites have more than one conduction
orbital [40]. The diagonalization of isolated polymers does the job,
eliminating thus the internal couplings between polymeric eigenstates and
revealing new site-to-site interactions.

As it is evident, the number of sites in a polymer is equivalent to the
dimensionality of the internal group. We note that the resulting theory is
abelian, if the polymeric states are coupled only to their neighboring
counterparts with the same index (i.e. first level with first level, second
level with second level and so forth). This gives rise to a number of $N$
decoupled abelian theories on the same lattice. On the other hand, if the
polymeric states are connected with different internal eigenstate indices of a
neighboring site, the corresponding Hamiltonian shall contain non-diagonal
operators defined on the generating algebra su$(N)$. This shall give rise to a
non-abelian theory, where the external field appears in the coupling constants
instead of a local potential; this is precisely what we showed in (24).

We use the following notation: $\sigma_{\pm}$ is the operator that rises
(lowers) site number inside a polymer, $T_{i}$ is the translation operator
between neighbors in the direction of a primitive vector ${\bf a}_{i}$, $d$ is
the distance between the center of a polymer and one of its sites (for
simplicity, we take all these as equal), $L$ is the distance between geometric
centers of neighboring polymers, and
$\Delta(d_{A,B}^{m,n})=\Delta_{A,B}^{m,n}$ is the coupling as a function of
the distance as Fig. 6 shows.

---
(a)
(b)
Figure 9:  Deformed fractal spectrum due to off-diagonal couplings produced by
an SU(2) field. (a) Strong corrections to the symmetric mode band. (b) Mild
corrections to the antisymmetric mode band.  Figure 10: Linear chains made of
trimers. The unit cell rotates a number $p$ of complete turns along the chain.
The global $C_{3}$ symmetry is broken if one internal trimer is rotated with
respect to the other two. When the symmetry $C_{3}$ is restored a flat band
appears in the inferior part of the spectrum. This symmetry breaking splits
the resulting energy bands.

##  IV Emulation of Peierls substitution for U(1) gauge field

|
---|---
|
Figure 11:  Portraits of ${\bf A}$ on the square lattice. The components
Im$\\{A_{0}\\}$ and Im$\\{A_{2}\\}$ are presented, as
Re$\\{A_{0}\\}=$Re$\\{A_{2}\\}=0$. Similarly the components Re$\\{A_{1}\\}$
and Re$\\{A_{3}\\}$ are presented, while Im$\\{A_{1}\\}=$Im$\\{A_{3}\\}=0$.

###  IV.1 Linear chain

As announced in section II, we introduce the construction of dimeric chains,
as can be visualized in Fig. 7. Each chain is characterized by the parameter
$p$ that corresponds to the number of complete turns of one dimer along the
array. The localized states around each site are denoted by $|n,i\rangle$
where $n$ stands for the dimer position and $i$ is the intradimeric site. The
couplings are given by:

$\displaystyle\Delta_{n,m}^{ij}$ $\displaystyle=$ $\displaystyle\langle
n,i|H|m,j\rangle$ (25) $\displaystyle=$
$\displaystyle\int\psi_{i}^{*}(r)H(-\nabla^{2},{\bf r})\psi_{j}(r)dr$
$\displaystyle\sim$ $\displaystyle\Delta_{0}\exp(-d_{n,m}^{i,j}/\lambda).$

where $d_{n,m}^{i,j}$ is the distance between sites (n,i) and (m,j) and
$\lambda$ is the evanescence length. In our description only nearest neighbors
are considered. The notation for the couplings is simplified as follows:
$\Delta_{n,n+1}^{ij}\mapsto\Delta_{n}^{i,j}$. We write the Hamiltonian of one
chain as:

$H_{p}=\left(\begin{array}[]{cccccc}H_{0}&h_{1}&0&\cdots&&0\\\
h_{1}^{\dagger}&H_{0}&h_{2}&\ddots&&0\\\ 0&h_{2}^{\dagger}&H_{0}&&&0\\\
\vdots&\ddots&&\ddots&&0\\\ &&&&H_{0}&h_{n-1}\\\
0&&&0&h_{n-1}^{\dagger}&H_{0}\end{array}\right),$ (26)

where

$H_{0}=\left(\begin{array}[]{cc}0&\Delta_{0}\\\
\Delta_{0}&0\end{array}\right),\quad
h_{n}=\left(\begin{array}[]{cc}\Delta_{n}^{11}&\Delta_{n}^{12}\\\
\Delta_{n}^{21}&\Delta_{n}^{22}\end{array}\right).$ (27)

In this notation, $H_{0}$ represents the Hamiltonian of one dimer with its on-
site energies shifted to zero for simplicity. Now we can diagonalize each
dimeric block in the main diagonal using a Hadamard matrix:

$\small H_{p}^{(1)}=\left(\mathbf{1}_{N}\otimes
U\right)^{\dagger}H_{p}\left(\mathbf{1}_{N}\otimes U\right),\quad
U=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ll}+1&+1\\\ +1&-1\\\
\end{array}\right).$ (28)

The transformed Hamiltonian in the symmetric and antisymmetric basis is:

$H_{p}^{(1)}=\left(\begin{array}[]{cccccc}H_{D}&\tilde{h}_{1}&0&\cdots&&0\\\
\tilde{h}_{1}^{\dagger}&H_{D}&\tilde{h}_{2}&\ddots&&0\\\
0&\tilde{h}_{2}^{\dagger}&H_{D}&&&0\\\ \vdots&\ddots&&\ddots&&0\\\
&&&&H_{D}&\tilde{h}_{n-1}\\\
0&&&0&\tilde{h}_{n-1}^{\dagger}&H_{D}\end{array}\right),$ (29)

with $H_{D}$ and $\tilde{h}_{i}$ given by:

$H_{D}=\left(\begin{array}[]{cc}\Delta_{0}&0\\\
0&-\Delta_{0}\end{array}\right),\quad\tilde{h}_{n}=\left(\begin{array}[]{cc}\Delta_{n}^{AA}&\Delta_{n}^{AS}\\\
\Delta_{n}^{SA}&\Delta_{n}^{SS}\end{array}\right).$ (30)

The superscript of $\Delta_{n}$ in $\tilde{h}_{n}$ now represents the
couplings between antisymmetric (A) and symmetric (S) states. We express
$\Delta_{n}^{AA},\Delta_{n}^{SA},\Delta_{n}^{AS}$ and $\Delta_{n}^{SS}$ in
terms of site-to-site couplings:

$\displaystyle\Delta_{n}^{AA}=\frac{1}{2}\left(\Delta_{n}^{11}+\Delta_{n}^{12}+\Delta_{n}^{21}+\Delta_{n}^{22}\right)$
$\displaystyle\Delta_{n}^{AS}=\frac{1}{2}\left(\Delta_{n}^{11}-\Delta_{n}^{12}+\Delta_{n}^{21}-\Delta_{n}^{22}\right)$
$\displaystyle\Delta_{n}^{SA}=\frac{1}{2}\left(\Delta_{n}^{11}+\Delta_{n}^{12}-\Delta_{n}^{21}-\Delta_{n}^{22}\right)$
$\displaystyle\Delta_{n}^{SS}=\frac{1}{2}\left(\Delta_{n}^{11}-\Delta_{n}^{12}-\Delta_{n}^{21}+\Delta_{n}^{22}\right).$
(31)

Reorganization of the transformed Hamiltonian $H_{p}^{(1)}$ into A and S
blocks obtains

$H_{p}^{(2)}=\left(\begin{array}[]{cc}H_{A}&\delta_{AS}\\\
\delta_{AS}^{\dagger}&H_{S}\end{array}\right),$ (32)

where $H_{A}$ and $H_{S}$ contain only A–A and S–S couplings respectively:

$\displaystyle
H_{A}=\left(\begin{array}[]{cccccc}\Delta_{0}&\Delta_{1}^{AA}&0&\cdots&&0\\\
\Delta_{1}^{AA}&\Delta_{0}&\Delta_{2}^{AA}&\ddots&&\vdots\\\
0&\Delta_{2}^{AA}&\Delta_{0}&&&\\\ \vdots&\ddots&&\ddots&\\\
&&&&\Delta_{0}&\Delta_{n-1}^{AA}\\\
0&&&&\Delta_{n-1}^{AA}&\Delta_{0}\end{array}\right)$ (39) $\displaystyle
H_{S}=\left(\begin{array}[]{cccccc}\Delta_{0}&\Delta_{1}^{SS}&0&\cdots&&0\\\
\Delta_{1}^{SS}&\Delta_{0}&\Delta_{2}^{SS}&\ddots&&\vdots\\\
0&\Delta_{2}^{SS}&\Delta_{0}&&&\\\ \vdots&\ddots&&\ddots&\\\
&&&&\Delta_{0}&\Delta_{n-1}^{SS}\\\
0&&&&\Delta_{n-1}^{SS}&\Delta_{0}\end{array}\right)$ (46)
$\displaystyle\delta_{AS}=\left(\begin{array}[]{cccccc}0&\Delta_{1}^{AS}&0&\cdots&&0\\\
\Delta_{1}^{SA}&0&\Delta_{2}^{AS}&\ddots&&\vdots\\\
0&\Delta_{2}^{SA}&{}_{0}&&&\\\ \vdots&\ddots&&\ddots&\\\
&&&&0&\Delta_{n-1}^{AS}\\\ 0&&&&\Delta_{n-1}^{SA}&0\end{array}\right)$ (53)

We observe from (IV.1) that the elements of $\delta_{AS}$ depend on the
couplings between neighboring sites, and this dependence is purely geometrical
due to the definition of $\Delta_{n,m}^{ij}$ in (25). The ideal case was
announced in section II where $\delta_{AS}$ is identically zero; the
corresponding spectrum is illustrated in Fig. 1.

In general, the Hamiltonian $H_{p}^{(2)}$ possesses two coupled blocks,
although some approximations can be made. If the elements of $\delta_{AS}$ are
smaller than the ones of $H_{A}$ and $H_{S}$, the spectrum consists of
decoupled bands with a Cantor spectrum; in the more general case, the
Hofstadter butterfly appears slightly deformed, as the reader may visualize in
Fig. 9.

In order to obtain the couplings as functions of site number, we must
calculate the four distances between the dimer $n$ and the dimer $n+1$. Let us
denote by $\mathbf{r}_{n}^{(i)},\,i=1,2$ the position vector of site $i$ in
dimer $n$, so all the pairwise distances are given by the following
expression:

$d_{n}^{ij}=|{\bf r}_{n}^{(i)}-{\bf
r}_{n+1}^{(j)}|=L\left\\{1+2\alpha^{2}A_{n}^{ij}-2\alpha
B_{n}^{ij}\right\\}^{1/2},$ (54)

where $A_{n}^{ij}$ and $B_{n}^{ij}$ contain information on the rotations of
each dimer $\theta_{n}=2\pi P(n-1)/(N-1)$ as

$\displaystyle A_{n}^{ij}$ $\displaystyle=$ $\displaystyle
1-(-1)^{i+j}\cos(\theta_{n}-\theta_{n+1})$ $\displaystyle B_{n}^{ij}$
$\displaystyle=$
$\displaystyle(-1)^{i}\sin\theta_{n}-(-1)^{j}\sin\theta_{n+1},$ (55)

and $\alpha=d/L$. The corresponding couplings are given as follows:

$\Delta_{n}^{ij}\simeq\left\\{1+B_{n}^{ij}\frac{d\alpha}{\lambda}+\left[\frac{B_{n}^{ij^{2}}}{2}\left(1+\frac{L}{\lambda}\right)-A_{n}^{ij}\right]\frac{d\alpha^{2}}{\lambda}\right\\}\Delta_{L}.$
(56)

where a second order approximation for $\alpha\ll 1$ has been used and
$\Delta_{L}=\Delta_{0}\exp(-L/\lambda)$. From (IV.1) the second order
approximation in $\alpha$ leads to

$\Delta_{n}^{SS}=\frac{2d^{2}}{\lambda
L}\left\\{\cos\theta_{n}\cos\theta_{n+1}-\frac{L}{\lambda}\sin\theta_{n}\theta_{n+1}\right\\}\Delta_{L}.$
(57)

In the special case when $L/\lambda=1$, we obtain

$\Delta_{n}^{SS}=\frac{2d^{2}}{L^{2}}\cos\left[\omega(2n-1)\right]\Delta_{L},$
(58)

with the angular frequency $\omega$ defined as $2\pi p/(N-1)$. It can be
demonstrated that there is a unitary map between the Harper’s Hamiltonian and
a tight-binding model without potential and with site-dependent couplings, as
obtained in (58). This rigorous map is of utmost importance, and it is
included in Appendix A. The effective parameter is $\Lambda=1$ for the
symmetric mode band. On the other hand, for $\Delta_{n}^{AA}$ couplings we
have:

$\Delta_{n}^{AA}=2\gamma\left\\{1-\frac{\beta\alpha^{2}(\beta+1)}{\gamma}\cos\omega\cos[\omega(2n-1)]\right\\}\Delta_{L}.$
(59)

where

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
2+\beta\alpha^{2}(\beta-1),$ $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{d}{\lambda},\quad\alpha=\frac{L}{d}.$ (60)

In this case, $\Lambda$ in the Harper’s model must be assumed small as shown
in (124). The estimated effective is $\Lambda=0.08\cos\omega$ for the
antisymmetric mode band. It is worth noting the non-removable frequency
dependence of $\Lambda$. The Lebesgue measure for the AMO is $|4-4\Lambda|$
[41]. The symmetric and antisymmetric bands have zero and 3.68$|\cos\omega|$
Lebesgue measure respectively.

A more general treatment to find the couplings as functions of site numbers,
consists in a decomposition of $\tilde{h}_{n}$ in polar form:

$\displaystyle\tilde{h}_{n}$ $\displaystyle=$ $\displaystyle
UP=\exp(i\sigma_{2}\phi)P$ (65) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\cos\phi&\sin\phi\\\
-\sin\phi&\cos\phi\end{array}\right)\left(\begin{array}[]{cc}e^{\theta}\cosh\chi&\sinh\chi\\\
\sinh\chi&e^{-\theta}\cosh\chi,\end{array}\right)$

obtaining thus the following relations

$\displaystyle\Delta_{n}^{AA}$ $\displaystyle=$ $\displaystyle
e^{\theta}\cos\phi\cosh\chi+\sin\phi\sinh\chi$ $\displaystyle\Delta_{n}^{AS}$
$\displaystyle=$ $\displaystyle
e^{-\theta}\sin\phi\cosh\chi+\cos\phi\sinh\chi$ $\displaystyle\Delta_{n}^{SA}$
$\displaystyle=$ $\displaystyle-e^{\theta}\sin\phi\cosh\chi+\cos\phi\sinh\chi$
$\displaystyle\Delta_{n}^{SS}$ $\displaystyle=$ $\displaystyle
e^{-\theta}\cos\phi\cosh\chi-\sin\phi\sinh\chi.$ (66)

where the parameters $\chi$, $\phi$ and $\theta$ are given in terms of the
couplings as:

$\displaystyle e^{\theta}\cosh\chi$ $\displaystyle=$
$\displaystyle\frac{2(\Delta_{n}^{11}\Delta_{n}^{12}+\Delta_{n}^{21}\Delta_{n}^{22})+\gamma}{2\gamma^{1/2}},$
(67) $\displaystyle e^{-\theta}\cosh\chi$ $\displaystyle=$
$\displaystyle\frac{-2(\Delta_{n}^{11}\Delta_{n}^{12}+\Delta_{n}^{21}\Delta_{n}^{22})+\gamma}{2\gamma^{1/2}},$
(68) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle(\Delta_{n}^{12}-\Delta_{n}^{21})^{2}+(\Delta_{n}^{11}+\Delta_{n}^{22})^{2}.$
(69)

We have seen how to obtain a Hamiltonian equivalent to Harper’s model in an
abelian sector of the theory. As presented in previous sections, this produces
indeed a butterfly coming from an AMO.

On the other hand, non-negligible elements in $\delta_{SA}$ reveal the non-
abelian nature of the field as we shall explore in the next section.

##  V Emulation of SU(2) field

The present section shows that the geometric dependence of couplings can be
decomposed into a function of the distance between dimer centers and a
function of each dimer orientation. The latter dependence is crucial in the
construction of arrays with localized wave functions as the one presented in
section V.3.

Figure 12: Component $\sigma_{1}$ of the field $B_{z}$. This field is the result of the commutator $[A_{x},A_{y}]$ in eq. (83) and constitutes the purely non-abelian contribution of the field. The other components of $B_{z}$ are small in comparison.  Table 1:  Fractal dimensions of the symmetric mode band. $\omega 2\pi/(N-1)$ | Fractal dimension
---|---
0 | 0.9606
1 | 0.9601
2 | 0.4980
3 | 0.4884
5 | 0.4244
7 | 0.6525
11 | 0.6848
13 | 0.6995
17 | 0.6941
19 | 0.7010
23 | 0.7233
29 | 0.6897
31 | 0.7308
37 | 0.7516
41 | 0.7389

###  V.1 A model with SU$(2)$

Following the general formalism exposed in sections III.1 and III.2, we study
the construction of a square lattice made of dimers. We define the projectors
$P_{\pm}=(1\pm\sigma_{3})/2$ that select the state 1 or 2 inside one dimer.
The coupling between state 1 of a lattice site with state 2 belonging to its
neighbor at the right would correspond to the operator $T_{1}\sigma_{+}$ and
its hermitian conjugate, by reciprocity. A square lattice with nearest lattice
couplings, internal dimer couplings and arbitrary on-site energies is
described by the following Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle(\Delta_{1}^{+}\sigma_{+}+\Delta_{1}^{-}\sigma_{-}+\tilde{\Delta}_{1}^{+}P_{+}+\tilde{\Delta}_{1}^{-}P_{-})T_{1}+h.c.$
(70) $\displaystyle+$
$\displaystyle(\Delta_{2}^{+}\sigma_{+}+\Delta_{2}^{-}\sigma_{-}+\tilde{\Delta}_{2}^{+}P_{+}+\tilde{\Delta}_{2}^{-}P_{-})T_{2}+h.c.$
$\displaystyle+$
$\displaystyle\Delta_{0}^{+}\sigma_{+}+\Delta_{0}^{-}\sigma_{-}+\tilde{\Delta}_{0}^{+}P_{+}+\tilde{\Delta}_{0}^{-}P_{-}+V$

whereas a triangular lattice (which has coordination number 6) made of dimeric
sites has a Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle((\Delta_{1}^{+}\sigma_{+}+\Delta_{1}^{-}\sigma_{-}+\tilde{\Delta}_{1}^{+}P_{+}+\tilde{\Delta}_{1}^{-}P_{-})T_{1}+h.c.$
(71) $\displaystyle+$
$\displaystyle(\Delta_{2}^{+}\sigma_{+}+\Delta_{2}^{-}\sigma_{-}+\tilde{\Delta}_{2}^{+}P_{+}+\tilde{\Delta}_{2}^{-}P_{-})T_{2}+h.c.$
$\displaystyle+$
$\displaystyle(\Delta_{3}^{+}\sigma_{+}+\Delta_{3}^{-}\sigma_{-}+\tilde{\Delta}_{3}^{+}P_{+}+\tilde{\Delta}_{3}^{-}P_{-})T^{\dagger}_{2}T_{1}+h.c.$
$\displaystyle+$
$\displaystyle\Delta_{0}^{+}\sigma_{+}+\Delta_{0}^{-}\sigma_{-}+\tilde{\Delta}_{0}^{+}P_{+}+\tilde{\Delta}_{0}^{-}P_{-}+V$

where $T_{3}$ is actually $T^{\dagger}_{2}T_{1}$. The six neighbors are
obtained here if we recognize that the hermitian conjugate of the third row
reverses the sign of the vector ${\bf a}_{1}-{\bf a}_{2}$. The Hamiltonian of
a particle in a nearest neighbor lattice of coordination number $c$ can be
written in the same fashion. Moreover, we choose to express $\sigma_{\pm}$ in
terms of $\sigma_{1,2}$, $P_{\pm}$ in terms of $\sigma_{3}$ and
$\sigma_{0}={\bf 1}$. The resulting expression is refreshingly simple if we
adopt the Einstein convention for greek indices in $\sigma_{\mu}$, i.e.

$\displaystyle
H=\sum_{k=1}^{c}\Delta_{k}^{\mu}\sigma_{\mu}T_{k}+\Delta_{0}^{\mu}\sigma_{\mu}+h.c.,$
(72)

albeit no Lorentzian metric has been required for this theory.

If the operators $\Delta_{k}^{\mu}$ representing local-dependent couplings are
all parallel, then $\Delta_{k}^{\mu}=\Delta^{\mu}$ for all $j$, and the
Hamiltonian (72) will represent a lattice with site-dependent deformations but
without a gauge field. We have the following factorization property:

$\displaystyle
H=\Delta^{\mu}\sigma_{\mu}\sum_{k=1}^{c}T_{k}+h.c.+\Delta_{0}^{\mu}\sigma_{\mu}$
(73)

Another case of reduction is the decoupling of identical lattices, i.e.
$\Delta_{k}^{1}=\Delta_{k}^{2}=\Delta_{k}^{3}=0$ for all $k$, leading to

$\displaystyle H={\bf
1}\otimes\left\\{\sum_{k}\Delta^{0}_{k}T_{k}+h.c.+V\right\\},$ (74)

i.e. a theory without internal structure. It is important to note that in
(72), the coupling constants are complex in general; however, in specific
realizations, the use of real and positive $\Delta_{k}^{\pm}$ for each site
shall impose some restrictions on the values of $\Delta_{k}^{\mu}$. Evidently,
the result is aperiodic in general, as described in Fig. 7.

Before diagonalizing each dimer, we decompose each site-to-site coupling in
center-to-center couplings plus fluctuations:

$\displaystyle\Delta_{k}^{\mu}(d_{n,m}^{i,j})=\Delta(L)+\delta_{k}^{\mu}(d_{n,m}^{i,j})$
(75)

The distances $d_{n}^{i,j}$ explained in Fig. 8, can be expressed in terms of
reference lengths $L,d$ and the angles $\theta_{n},\theta_{n+1}$, such that
the dependence of $\delta_{k}^{\mu}$ on such parameters can be exhibited.
Assuming only orientational dependence but no deformation, and the fact that
$\Delta$ must be real and positive, one has $\Delta^{3}_{k}=\Delta^{2}_{k}=0$,
uniform on-site energies (i.e. $V=0$ can be imposed) and the Hamiltonian
becomes

$\displaystyle H=$ $\displaystyle({\bf
1}+\sigma_{1})\Delta(L)\sum_{k}T_{k}+h.c.$ (76)
$\displaystyle+\sum_{k}\delta^{\mu}_{k}\sigma_{\mu}T_{k}+h.c.+\Delta(d)\sigma_{1},$

where the first term represents the periodic lattice, the second term contains
all orientational fluctuations (disorder could be introduced in this part) and
the last term is the intra-dimer coupling contribution. Finally, we introduce
a change of basis, in order to transform each dimer into an effective two-
level potential well. It suffices to diagonalize $\sigma_{1}$, leading to the
following Hamiltonian:

$\displaystyle\tilde{H}$ $\displaystyle=$ $\displaystyle
P_{+}\left\\{\sum_{k}(\Delta(L)+\delta_{k}^{1}+\delta_{k}^{0})T_{k}+h.c.+\Delta(d)\right\\}$
(77) $\displaystyle+$ $\displaystyle
P_{+}\left\\{\sum_{k}(\delta_{k}^{0}-\delta_{k}^{1})T_{k}+h.c.-\Delta(d)\right\\}$
$\displaystyle+$
$\displaystyle\sigma_{1}\sum_{k}\delta_{k}^{3}(T_{k}+T_{k}^{\dagger})$

or in block form

$\displaystyle\tilde{H}=\left(\begin{array}[]{cc}\sum_{k}(\Delta(L)+\delta_{k}^{1}+\delta_{k}^{0})T_{k}+h.c.+\Delta(d)&\sum_{j}\delta_{k}^{3}(T_{k}+T_{k}^{\dagger})\\\
\sum_{k}\delta_{k}^{3}(T_{k}+T_{k}^{\dagger})&\sum_{k}(\delta_{k}^{0}-\delta_{k}^{1})T_{k}+h.c.-\Delta(d)\end{array}\right)$
(80)

where each block is a Hamiltonian for the global lattice. The off-diagonal
blocks represent the non-abelian nature of the resulting theory, in the basis
of dimer eigenfunctions. The fact that this dependence cannot be removed,
indicates that the emulation of interactions with an external non-abelian
field depend crucially on the orientational fluctuations
$\delta_{k}^{1},\delta_{k}^{3}$, which in turn can be geometrically designed
as indicated in Fig. 7.

By comparing the Hamiltonians (80) and (32) we can conclude that the non-
abelian behavior of the field is originated by the dimer orientations of the
chains.

Figure 13:  Unit cell for the chain of trimers. Each internal triangle can
rotate independently with angles $\theta_{1}$, $\theta_{2}$ and $\theta_{3}$.
The dashed blue lines represent the distances between centers.

###  V.2 Effective negative couplings and level inversion revisited

When the fluctuations are small, we may try to find an approximate
representation that eliminates the off-diagonal blocks. Here, we must recover
an abelian gauge field, but by the same token, this field must be described by
the fluctuations in the orientations. If we simply eliminate $\delta_{j}^{3}$,
the second block has a peculiarity already noted in [42, 43]: level inversion
can be achieved whenever $\delta_{j}^{0}<\delta_{j}^{1}$, and even a
degeneracy point occurs when $\delta_{j}^{0}=\delta_{j}^{1}$, i.e. a hidden
symmetry that does not pertain to the underlying lattice symmetries. The level
inversion corresponds to the realization of negative couplings using
exclusively positive on-site waves. Another way to look at this, according to
[42], is to recognize that the basis of a dimer contains both symmetric and
antisymmetric functions, which means that couplings between dimeric excited
states can change sign if the antisymmetric waves enter with opposite signs in
the overlap integrals.

In the case where the fluctuations are indeed small, but their effect is not
completely neglected to second order, a perturbative approach can help to
reabsorb them in the diagonal part by a unitary transformation.

###  V.3 Wave confinement in SU(2). Bound states

In section II we showed examples of lattices that exhibit the confinement
phenomenon of its wave function in the center of the array. In this section we
present the construction of the corresponding square lattice.

Our method of construction is based on the intensities of the couplings
between two adjacent dimers in the direction of the primitive vectors. The
strongest couplings must be located at the center of the array while the
orientation of the dimers out of this region must be disposed in a
configuration that favors small coupling values. We achieve these conditions
with four central dimers illustrated in Fig. 3. In general, the position of
each site is given by the following expressions:

$\displaystyle\mathbf{a_{1}^{({\bf n})}}$ $\displaystyle=$
$\displaystyle\left((n_{x}-1)l-d\sin\left[\frac{\pi}{4}+\frac{\pi}{2(n-2)}\right](n_{y}-n_{x}),-(n_{y}-1)l-d\cos\left[\frac{\pi}{4}+\frac{\pi}{2(n-2)}\right](n_{y}-n_{x})\right)$
(81) $\displaystyle\mathbf{a_{2}^{({\bf n})}}$ $\displaystyle=$
$\displaystyle\left((n_{x}-1)l-d\sin\left[\frac{5\pi}{4}+\frac{\pi}{2(n-2)}\right](n_{y}-n_{x}),-(n_{y}-1)l-d\cos\left[\frac{5\pi}{4}+\frac{\pi}{2(n-2)}\right](n_{y}-n_{x})\right),$
(82)

where ${\bf a}_{1}^{({\bf n})}$ and ${\bf a}_{2}^{({\bf n})}$ stand for the
positions 1 and 2 of the dimer at ${\bf n}=(n_{x},n_{y})$. Now we vary the
lattice parameter and find a localized stated at the top of the energy band.
This localized state moves downwards, as the lattice constant increases with
fixed dimeric distance. In fact, it merges with the propagation band in the
limit of negligible intradimeric distance –small compared to the lattice
constant– and reveals the non abelian nature of the effect. An analogy with
solitons can be made here, as the localized state dwells in a region produced
by a Hedgehog-type external field. This configuration of dimers has some
characteristics in common with solutions of the classical non-linear field
equations possessing a finite energy and localized non-dispersive energy
density i.e. this wave travels with its shape unchanged[44, 45]. Moreover,
such solution falls to zero at infinity. In the non-linear theory we pretend
to emulate, the solutions perturb our discrete Schrödinger equation.

The portrait of the wave function for the higher energy state was shown in
section 3 in Fig. 4. We observe that the electronic state is confined at the
center of the array, but we are also interested in the shape of the emulated
non-abelian field. In the direction perpendicular to the lattice plane,
$B_{z}$ is obtained as follows:

$B_{z}=\partial_{y}A_{x}-\partial_{x}A_{y}-i[A_{x},A_{y}],$ (83)

where the commutator plays here an important role. In order to calculate the
components of $A$, first we must express each $2\times 2$ coupling block as an
exponential:

$\Delta({\bf N})=\exp(\mbox{\boldmath$\Phi$\unboldmath}^{(0)}{\bf
1}+\mbox{\boldmath$\Phi$\unboldmath}\cdot\mbox{\boldmath$\sigma$\unboldmath}).$
(84)

The component $A_{i}$ is the difference between
$\mbox{\boldmath$\Phi$\unboldmath}({\bf N})$ and
$\mbox{\boldmath$\Phi$\unboldmath}({\bf N}-\hat{{\bf e}}_{i})$. The
projections on the Pauli basis $\sigma_{0},\sigma_{1},\sigma_{2}$ and
$\sigma_{3}$ of ${\bf A}$ are readily found by computing Tr$(A\sigma_{i})$ and
are illustrated in Fig 11. The real part of $A^{(0)}$, $A^{(2)}$ and the
imaginary part of $A^{(1)}$, $A^{(3)}$ are zero.

|
---|---
(a) | (b)
Figure 14:  Panels (a) and (b) present the fractal dimensions of the
antisymmetric mode band and upper band respectively. Both spectra show
similarities in the fractal dimension values.

##  VI Band mixing in SU(3) models with color

The emulation of SU(3) fields can be performed in a similar way as presented
for the previous SU(2) case. This time we construct a liner chain made of
trimers, where the unit cell is shown in Fig. 13. The energy levels of each
internal trimer represent three internal degrees of freedom in the cell. As
the triangles rotate independently with angles $\theta_{1},\theta_{2}$ and
$\theta_{3}$, we can break or restore the local polygonal symmetry of each
cell, and thus manipulate the emergent band structure. When these angles
vanish, the system has $C_{3}\times C_{3}$ symmetry. The Hamiltonian of the
array is, in matrix form:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}H_{0}&\Delta_{AB}&\Delta_{AC}\\\
\Delta^{\dagger}_{AB}&H_{0}&\Delta_{BC}\\\
\Delta^{\dagger}_{AC}&\Delta^{\dagger}_{BC}&H_{0}\end{array}\right),$ (88)
$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}E&\Delta&\Delta\\\ \Delta&E&\Delta\\\
\Delta&\Delta&E\end{array}\right),$ (92) $\displaystyle\Delta_{mn}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\Delta_{m1,n1}&\Delta_{m1,n2}&\Delta_{m1,n3}\\\
\Delta_{m2.n1}&\Delta_{m2,n2}&\Delta_{m2,n3}\\\
\Delta_{m3,n1}&\Delta_{m3,n2}&\Delta_{m3,n3}\end{array}\right),$ (96)

where $m,n=A,B,C$ and $\Delta_{ml,nk}$ is the coupling between the $l$-th site
of the trimer $m$ with the $k$-th site of trimer $n$. A transformed
Hamiltonian is obtained if we use the $3\times 3$ matrix that diagonalizes
$H_{0}$, i.e. $\tilde{H}=U^{\dagger}H_{0}U$ where $U=\mathbf{1}\otimes
U_{3\times 3}$. The transformed matrix elements are calculated analytically
below:

$\displaystyle\left(U_{3\times 3}\right)_{p,q}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\exp\left[\frac{i2\pi}{3}(2+p)q\right],$ (97)
$\displaystyle\left(\tilde{\Delta}_{mn}\right)_{pq}$ $\displaystyle=$
$\displaystyle\frac{e^{4\pi i(q-p)/3}}{3}\sum_{k,k^{\prime}=1}^{3}e^{2\pi
i(k^{\prime}q-kp)/3}\Delta_{mk,nk^{\prime}}.$

Now, the Hamiltonian is expressed in the eigenbasis of $C_{3}$: Reorganizing
matrix elements leads to a Hamiltonian dived into three blocks:

$\displaystyle\small
H=\left(\begin{array}[]{ccccccccc}S_{1}&(\Delta^{\prime}_{AB})_{33}&(\Delta^{\prime}_{AC})_{33}&0&(\Delta^{\prime}_{AB})_{31}&(\Delta^{\prime}_{AC})_{31}&0&(\Delta^{\prime}_{AB})_{32}&(\Delta^{\prime}_{AC})_{32}\\\
(\Delta^{\prime}_{AB})^{*}_{33}&S_{2}&(\Delta^{\prime}_{BC})_{33}&(\Delta^{\prime}_{AC})^{*}_{13}&0&(\Delta^{\prime}_{BC})_{31}&(\Delta^{\prime}_{AC})^{*}_{23}&0&(\Delta^{\prime}_{BC})_{32}\\\
(\Delta^{\prime}_{AC})^{*}_{33}&(\Delta^{\prime}_{BC})^{*}_{33}&S_{3}&(\Delta^{\prime}_{AC})^{*}_{13}&(\Delta^{\prime}_{BC})^{*}_{13}&0&(\Delta^{\prime}_{AC})^{*}_{23}&(\Delta^{\prime}_{BC})^{*}_{23}&0\\\
0&(\Delta^{\prime}_{AB})_{13}&(\Delta^{\prime}_{AC})_{13}&D_{11}&(\Delta^{\prime}_{AB})_{11}&(\Delta^{\prime}_{AC})_{11}&0&(\Delta^{\prime}_{AB})_{12}&(\Delta^{\prime}_{AC})_{12}\\\
(\Delta^{\prime}_{AB})^{*}_{31}&0&(\Delta^{\prime}_{BC})_{13}&(\Delta^{\prime}_{AB})^{*}_{11}&D_{21}&(\Delta^{\prime}_{BC})^{*}_{11}&(\Delta^{\prime}_{AB})^{*}_{21}&0&(\Delta^{\prime}_{BC})_{12}\\\
(\Delta^{\prime}_{AC})^{*}_{31}&(\Delta^{\prime}_{BC})^{*}_{31}&0&(\Delta^{\prime}_{AC})^{*}_{11}&(\Delta^{\prime}_{BC})^{*}_{11}&D_{31}&(\Delta^{\prime}_{AC})^{*}_{21}&(\Delta^{\prime}_{BC})^{*}_{21}&0\\\
0&(\Delta^{\prime}_{AB})_{23}&(\Delta^{\prime}_{AC})_{23}&0&(\Delta^{\prime}_{AB})_{21}&(\Delta^{\prime}_{AC})_{21}&D_{12}&(\Delta^{\prime}_{AB})_{22}&(\Delta^{\prime}_{AC})_{22}\\\
(\Delta^{\prime}_{AB})^{*}_{32}&0&(\Delta^{\prime}_{BC})_{23}&(\Delta^{\prime}_{AB})^{*}_{12}&0&(\Delta^{\prime}_{BC})_{21}&(\Delta^{\prime}_{AB})^{*}_{22}&D_{22}&(\Delta^{\prime}_{BC})_{22}\\\
(\Delta^{\prime}_{AC})^{*}_{32}&(\Delta^{\prime}_{BC})^{*}_{32}&0&(\Delta^{\prime}_{AC})^{*}_{12}&(\Delta^{\prime}_{BC})^{*}_{12}&0&(\Delta^{\prime}_{AC})^{*}_{22}&(\Delta^{\prime}_{BC})^{*}_{22}&D_{32}\end{array}\right).$
(108)

The symmetry breaking of a global $C_{3}$ of each trimer is introduced by a
rotation of only one of the internal dimers; as a consequence, triple
degeneracy points emerge as announced previously. Note though that doublets
are never split. The internal $C_{3}$ breaking is achieved by deformation of
the internal trimers, and in this case lifted doublets appear. Following the
same path as in the case of SU(2), we construct a linear chain made of rotated
unit cells. This system produces the spectrum of the Fig. 5 where we find
overlapped bands. The superior band contains nine copies of the Hofstadter
butterfly. The model shows emergent flat bands, of potential interest in
recent studies on superconductors.

Finally, the model can be generalized also to a 2D lattice, in a similar way
to the SU(2) case, with the aim of emulating confinement due to SU(3) color
fields.

##  VII Conclusion

In this work, the emulation of non abelian fields was studied in periodic,
quasi-periodic and defective lattices with internal degrees of freedom given
by the energy levels of a polymer. The full lattice is homogeneous, i.e. all
sites were identical. For a given set of geometric parameters in a linear
chain of rotating dimers, the projections of the energy levels in the
uncoupled symmetric and antisymmetric states reproduced well Harper’s
Hamiltonian spectra with $\Lambda=1.7$ and $\Lambda=1$ for the two emergent
bands. It was also possible to show that the spectrum was independent of
Bloch’ quasimomentum, as in the standard theory of ergodic operators. This
gave us a clear indication of an emergent magnetic field produced by mere
deformations.

In the general description of coupled bands, the orientations of the couplings
between neighboring sites were responsible for non zero elements in the
diagonal block of the Hamiltonian; this showed that our construction was a
genuine realization of a non-abelian theory in a regime of strong couplings
between bands. The phenomenon of confinement due to SU(2), was emulated in a
square array with dimers where the hopping amplitudes were tuned in order to
produce an effective (non-diagonal) defect; the wave function as shown to be
located at the center of the crystal and its energy was found on the superior
part of the propagation band. This localized state merged with the latter, as
the lattice constant increased, which resulted in a fully periodic array in
the limit of negligible dimer size. It is important to note that the dimeric
structure could be shown to be indispensable to determine whether a non-
abelian group SU(2) entered or not in the description. The bound state located
at the lattice defect would simply disappear in the absence of a non-abelian
theory.

A few words are in order in connection with technological applications. The
magnetic field emulated with deformations allows to study similar effects in
mesoscopic systems and wave transport. Its applications to wave propagation in
elastic tight-binding emulations [46] and photonic crystals offers itself as a
type of metallic-to-insulator transition in systems where external fields that
manipulate waves are not available. We have also shown that the introduction
of extra orbitals in solid state physics leads to deformations of Cantor sets.
As an outlook, we plan to analyze the implications of such spectral
deformations on the Quantum Hall effect [47, 48], whose mysteries are yet to
be fully explained.

###### Acknowledgements.

Beca 291197 CONACYT and project 100518931-VIEP BUAP-CA-289.

##  Appendix A Unitary map between translation operators in 1D; beyond Aubry’s
duality.

In this section we show explicitly the unitary map of the Harper Hamiltonian
into a tight-binding operator without potential, but with variable couplings.
For ease in the calculations, we shall work in a representation where the
transformation an site number operators act on smooth functions, i.e.
$T\rightarrow e^{iap}$, $p=-i\hbar\partial_{x}$, $N\rightarrow x/a$. If we act
on a complete basis of Wannier functions $\phi_{b}(x-a)$ this represents no
loss of generality. However, we must make sure that our unitary map takes us
from periodic functions (period $a$, index band $b$) to periodic functions.
Let us take the following unitary transformation on the Hamiltonian

$\tilde{H}=UHU^{\dagger},\quad UU^{\dagger}=\mathbf{1}.$ (109)

The unitary operator $U$ can be expressed as:

$U=V(N)\exp\\{i(\alpha p^{2}+\beta p)\\},$ (110)

such that $V(N)V^{\dagger}(N)=\mathbf{1},\alpha,\beta\in\mbox{$\cal R$}$. We
shall see that $U$ does not change the nature of the lattice if $\alpha$ and
$\beta$ are properly chosen. We have

$T=e^{iap},\quad E\equiv e^{i\omega N}=e^{i\omega x/a}$ (111)

and defining $v\equiv\omega/a$, $E$ can be written as $E\equiv e^{ivx}$, where
$x=i\hbar\partial_{p}$. With these definitions, the Harper’s Hamiltonian
(Almost Mathieu Operator) has the form:

$H=T+T^{\dagger}+\Lambda\left(E+E^{\dagger}\right),$ (112)

for $\nu=0$, which is isospectral to any $\nu\neq 0$. Direct substitution
leads to

$\displaystyle UTU^{\dagger}$ $\displaystyle=$ $\displaystyle V(N)e^{i(\alpha
p^{2}+\beta p)}Te^{-i(\alpha p^{2}+\beta p)}V^{\dagger}(N)$ (113)
$\displaystyle=$ $\displaystyle V(N)TV^{\dagger}(N)=V(x/a)TV^{\dagger}(x/a)$
$\displaystyle=$ $\displaystyle V(N)V^{\dagger}(N+1)T.$

In addition

$UEU^{\dagger}=V(N)e^{i(\alpha p^{2}+\beta p)}Ee^{-i(\alpha p^{2}+\beta
p)}V^{\dagger}(N)$ (114)

and

$Ef(p)=f(p-v)E,$ (115)

so that

$\displaystyle UEU^{\dagger}$ $\displaystyle=$ $\displaystyle V(N)e^{i(\alpha
p^{2}+\beta p)}e^{-i[\alpha(p-v)^{2}+\beta(p-v)]}EV^{\dagger}(N)$ (116)
$\displaystyle=$ $\displaystyle V(N)e^{i(2\alpha vp+\beta v-\alpha
v^{2})}EV^{\dagger}(N)$ $\displaystyle=$ $\displaystyle e^{i(\beta v-\alpha
v^{2})}V(x/a)e^{i2\alpha vp}EV^{\dagger}(x/a)$ $\displaystyle=$ $\displaystyle
e^{i(\beta v-\alpha v^{2})}V(x/a)e^{i2\alpha vp}EV^{\dagger}(x/a)$
$\displaystyle=$ $\displaystyle e^{i\beta v-\alpha
v^{2}}V(x/a)V^{\dagger}\left(\frac{x+2\alpha v}{a}i\right)e^{iqap}E.$

This is a closed result within the space of lattice translations if $2\alpha
v=qa,\ q\in\mbox{$\cal Z$}$. The transformed $E$ is:

$\displaystyle UEU^{\dagger}$ $\displaystyle=$ $\displaystyle e^{i\beta
v-\alpha v^{2}}V(N)V^{\dagger}(N+q)T^{q}E$ (117) $\displaystyle=$
$\displaystyle e^{i\beta v-\alpha v^{2}}V(N)V^{\dagger}(N+q)ET^{q}e^{ivaq}$
$\displaystyle=$ $\displaystyle e^{i(\beta v-\alpha
v^{2}+vaq)}V(N)V(N+q)ET^{q}$ $\displaystyle=$ $\displaystyle e^{i(\beta
v-qav/2+vaq)}V(N)V^{\dagger}(N+q)ET^{q}$ $\displaystyle=$ $\displaystyle
e^{i\chi_{q}}V(N)V(N+q)ET^{q}.$

Therefore, the new Hamiltonian becomes:

$\tilde{H}=V(N)V^{\dagger}(N+1)T+\Lambda
e^{i\chi_{q}}V(N)V^{\dagger}(N+q)ET^{q}+h.c.$ (118)

which is a multiple neighbor tight-binding operator. Since the parameters in
$U$ can chosen at will, we have that for $q=1$ or $q=-1$, $H$ reduces to a
nearest-neighbor Hamiltonian. Finally, we recognize that every nearest-
neighbor tight-binding model in 1D can be cast in the form of a Hamiltonian
with real and positive couplings by means of gauge transformation. In general,
we have

$\Delta=V(N)V^{\dagger}(N+1)\left(1+\Lambda e^{\pm i\chi_{q}}e^{\pm i\omega
N}\right).$ (119)

We recognize the unitary nature of $V(N)$ so $VV^{\dagger}=\mathbf{1}$ implies
$V=\exp(i\Phi(N))$ and
$V(N)V^{\dagger}(N+1)=\exp\\{i(\Phi(N)-\Phi(N+1)\\}\equiv\exp(-i\delta\Phi(N))\\}$
which is only a phase factor, and then

$\displaystyle\Delta=e^{i\Omega(N)}|\Delta(N)|,\
\tilde{U}=e^{i\sum_{j=-\infty}^{n}\Omega(N+j)}$ (120) $\displaystyle
T\tilde{U}^{\dagger}=e^{-i\sum_{j=\infty}^{n+1}\Omega(N+j)i}T$ (121)
$\displaystyle\tilde{U}T\tilde{U}^{\dagger}=e^{-i\Omega(N+n+1)}T.$ (122)

If we choose $n=-1$ we obtain the following expression for $|\Delta|$

$|\Delta|=\sqrt{1+\Lambda^{2}+2\Lambda\cos(\omega N+\chi_{\pm})},$ (123)

which is the sought expression. Geometric deformations are therefore
equivalent to on-site quasiperiodic energies.

In the case of a very faint intensity of the potential, we can analyze the
limit $\Lambda\ll 1$ and provide an expression for the coupling:

$|\Delta|\simeq 1+\Lambda\cos(\omega N+\chi_{\pm}),\quad\Lambda\ll 1,$ (124)

which is in full agreement with the first corrections to the effective
coupling in the problem of rotating dimers.

##  Appendix B Box-counting fractal dimension calculation

The fractal dimensions of the symmetric and antisymmetric spectra of the chain
made of dimers were calculated using the method of box counting:

$N(s)=\left(\frac{1}{s}\right)^{D},$ (125)

where $N$ is the number of boxes and $s$ is the size of one box. Thus the
fractal dimension can be calculated as follows:

$D=\frac{\log N(s)}{\log(1/s)}.$ (126)

The fractal dimensions for specific values of $\omega(N-1)/2\pi$ of the
spectrum showed Fig. 1 (b) are presented in Fig. 14. The range of values of
the fractal dimensions for the symmetric mode band are reported in Table 1.

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