# Localization via Quasi-Periodic Bulk-Bulk Correspondence

Dan S. Borgnia<EMAIL_ADDRESS>Department of Physics, Harvard
University, Cambridge, MA 02138  Robert-Jan Slager<EMAIL_ADDRESS>TCM
Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue,
Cambridge CB3 0HE, United Kingdom  Department of Physics, Harvard University,
Cambridge, MA 02138

###### Abstract

We report on a direct connection between quasi-periodic topology and the
Almost Mathieu (Andre-Aubry) metal insulator transition (MIT). By constructing
quasi-periodic transfer matrix equations from the limit of rational
approximate projected Green’s functions, we relate results from
$\text{SL}(2,\mathbb{R})$ co-cycle theory (transfer matrix eigenvalue scaling)
to consequences of rational band theory. This reduction links the
eigenfunction localization of the MIT to the chiral edge modes of the
Hofstadter Hamiltonian, implying the localized phase roots in a topological
“bulk-bulk” correspondence, a bulk-boundary correspondence between the 1D AAH
system (boundary) and its 2D parent Hamiltonian (bulk). This differentiates
quasi-periodic localization from Anderson localization in disordered systems.
Our results are widely applicable to systems beyond this paradigmatic model.

Introduction.— Quasi-periodic systems have played and continue to play a
prominent role in condensed matter physics [1, 2, 3, 4, 5, 6, 7, 8, 9]. The
Andre-Aubry-Harper (AAH) model, or the almost Mathieu Operator in the
mathematics community, is the prototypical example of quasi-periodic systems.
In contrast to periodic or randomly disordered systems, the AAH model hosts
many exotic properties, including a metal insulator transition (MIT), a well
established duality between the two phases [10, 11, 12], and a connection to a
2D quantum Hall system on a lattice, nearest neighbor (n.n.) Hofstadter model.

Here, we report a new relation between this 2D quasi-periodic topology and the
famous metal insulator transition. Following the ideas in Ref. [13] and
constructing a “rational approximate” transfer matrix equation (TME) sequence
from projected Green’s functions (pGF) of the rational approximates, we
construct an explicit map between the $\text{SL}(2,\mathbb{R})$ co-cycle
theory [14, 15, 16, 17, 18, 19] describing the eigenfunction localization and
these higher dimensional topological invariants. In particular, we show that
the quasi-periodic localization is an example of 1D bulk localized states
reflecting a virtual 2D bulk topological invariant, see Table 1. This exotic
behavior does not occur in fully disordered systems as non-trivial Chern
markers are incompatible with localization [20], thereby both adding to the
rapporteur of exotic quasi-periodic phenomena, such as the 1D MIT, higher
dimensional topology, self-duality, as well as lending a deeper explanation to
the differences between random and quasi-periodic disorder.

We proceed by reviewing the AAH model, including the known phase diagram [14,
21, 18], the connection between the 1D AAH model and the 2D Hofstadter model,
and accordingly a discussion of its topological classification [22, 23, 2, 1,
11, 24]. We then outline the projected Green’s function method for
constructing 2D rational approximate TMEs from Ref. [13] and detail a
procedure for projecting 2D TMEs back into 1D quasi-periodic TMEs. This links
2D chiral topological edge modes to a lack of 1D TME solutions and reproduces
the full AAH phase diagram directly from the quasi-periodic bulk-boundary
correspondence, a “bulk-bulk” correspondence.

| Horizontal | Vertical | Phase | Chiral Projection
---|---|---|---|---
$V<t$ | Yes | No | Metal | One
$V>t$ | No | Yes | Insulator | All
$V=t$ | Yes/No | Yes/No | Transition | N/A
Table 1: For Diophantine $\alpha$, table lists AAH phases, the convergent 2D
parent Hamiltonian unit cell, and number of chiral edge modes from 2D parent
Hamiltonian surviving the 1D projection. Chiral edge modes isolate spectrum
for $V>t$. Figure 1:  Top panel shows Magnetic unit cells chosen in the 2D
parent Hamiltonian (center). For Diophantine $\alpha$ in Eq. (2), the regimes
$V<t$ and $V>t$ dictate a horizontal or vertical cell respectively, defining
allowed projections back to 1D (left/right), i.e. horizontal unit cells
naturally project back to 1D, while vertical unit cells do not. Bottom panel
shows the transfer matrix eigenvalues corresponding to 1D projections and the
according numerically computed AAH spectrum (green), corresponding spectral
gaps (grey), and 2D pGF zeros (red x) for parameters
$t=1,N=2048,\alpha=\frac{1}{2}(\sqrt{5}-1)$ and $V=2$ (left) or $V=0.5$
(right). The spectrum (green) has transfer matrix eigenvalues on the unit
circle, $|\lambda_{T}|=1$. Zero eigenvalues of the pGF (red x) imply the
transfer matrix is rank deficient, $\lambda_{T}=0$. Note the many small gaps
(grey) and zeros in almost (up to numerical precision) every gap for the
horizontal projection, and the continuum of rank deficient points for the
vertical projection.

Almost Mathieu Review.— The AAH model presents a rich playground for new
techniques in analysis and single-particle physics. Setting
$\Theta=2\pi\alpha$ and $\alpha\in\mathbb{R}-\mathbb{Q}$,

$\displaystyle\hat{H}=\sum_{x}t(\hat{c}^{\dagger}_{x+1}\hat{c}_{x}+\hat{c}_{x+1}\hat{c}^{\dagger}_{x})+2V\cos(\Theta
x+\delta)\hat{c}^{\dagger}_{x}\hat{c}_{x}.\quad$ (1)

The model is a particular example of a 1D quasi-periodic system that has a
non-trivial bulk topology, see S.I. VI.4 [13, 1]. We illustrate this with a 2D
parent Hamiltonian formulation for the AAH model arising from phase shift
degrees of freedom in the quasi-periodic pattern, i.e.
$\delta\rightarrow\delta+\Theta$ is equivalent to $x\rightarrow x+1$ [1, 22,
23]. Stacking de-coupled AAH chains and parameterizing each layer by the phase
choice $\delta_{y}$ [21, 1, 22, 2] we can take an inverse Fourier transform
along $\delta_{y}$.

$\displaystyle\mathcal{H}_{2D}$ $\displaystyle=$
$\displaystyle\sum_{x,\delta_{y}}t\hat{c}_{x+1,\delta_{y}}^{\dagger}\hat{c}_{x,\delta_{y}}+t^{*}\hat{c}_{x,\delta_{y}}^{\dagger}\hat{c}_{x+1,\delta_{y}}$
(2) $\displaystyle+$ $\displaystyle 2V\cos(\Theta
x+\delta_{y})\hat{c}_{x,\delta_{y}}^{\dagger}\hat{c}_{x,\delta_{y}},$
$\displaystyle\tilde{\mathcal{H}}_{2D}$ $\displaystyle=$
$\displaystyle\sum_{x,y}t(\hat{c}_{x+1,y}^{\dagger}\hat{c}_{x,y}+\hat{c}_{x,y}^{\dagger}\hat{c}_{x+1,y})$
(3) $\displaystyle+$ $\displaystyle V(e^{i\Theta
x}\hat{c}_{x,y+1}^{\dagger}\hat{c}_{x,y}+e^{-i\Theta
x}\hat{c}_{x,y-1}^{\dagger}\hat{c}_{x,y}).$

The result corresponds to a 2D tight-binding model with irrational magnetic
flux per plaquette, $\Theta$, see Refs. [13, 1]. Varying $\Theta$ produces the
Hofstadter butterfly, Fig. S2, with gaps labeled by integers,
$\\{m+n\Theta|m,n\in\mathbb{Z}\\}$ [25].

The AAH model famously hosts a 1D metal-insulator transition (MIT), first
shown through the duality of Eq. (1) under a Fourier transform-like,
$\hat{c}_{k}=\sum_{x}\exp(i\Theta kx)\hat{c}_{x}$,

$\displaystyle\tilde{H}=\sum_{k}V(\hat{c}^{\dagger}_{k+1}\hat{c}_{k}+\hat{c}_{k+1}\hat{c}^{\dagger}_{k})+2t\cos(\Theta
k+\delta_{k})\hat{c}^{\dagger}_{k}\hat{c}_{k}.\quad$ (4)

The model is self-dual for $V=t$, fixing a transition from momentum-like
(metallic) to position-like (insulating) eigenfunctions, S.I. II [13, 10].

A rigorous formulation of the MIT relies on the notion of spectral measure,
formally defined in S.I. I [14]. Intuitively, a spectrum is absolutely
continuous if the inner-product measure of an energy window
$(\omega-\epsilon,\omega+\epsilon)$ continuously with $\epsilon$. This implies
the presence of arbitrarily close eigenvalues which requires participation of
arbitrarily many sites, guaranteeing de-localized eigenfunctions. By contrast,
a spectrum is purely point-like if there is a discontinuity in the inner-
product measure as a function of $\epsilon$. This implies localized
eigenfunctions because discrete eigenvalues require fewer participating sites.
For Almost-Mathieu operators, the spectrum is

  1. 1.

Absolutely continuous (metallic) for all $\Theta$ and all $\delta$ when $V<t$

  2. 2.

Singularly continuous (critical) for all $\Theta$ and all $\delta$ when $V=t$

  3. 3.

Pure point-like (insulator) for almost all $\Theta$ and almost all $\delta$
when $V>t$

with the first “almost” in (3) corresponding to Liouville irrationals (see
below) and the second “almost” corresponding to $\delta/2=\Theta
N+2\pi\mathbb{Z}$ for any $N\in\mathbb{N}$.

These results were originally obtained through the construction of
$\text{SL}(2,\mathbb{R})$ co-cycles – transfer matrices indexed by scalars
($\delta$) – associated with the AAH Hamiltonian, and developed results from
co-cycle theory to classify the scaling behavior of transfer matrix solutions,
i.e. Lyapunov exponents [21, 14, 19, 3, 15, 16, 17, 26, 18]. Instead, we
construct these same transfer matrices as the limit of a sequence of rational
approximate transfer matrices following the methods recently developed in Ref.
[13]

Rational Approximate Transfer Matrix Equations.— We define a sequence of
transfer matrix equations (TMEs) for quasi-periodic Hamiltonian operators
whose quasi-periodic parameter, $\alpha\notin\mathbb{Q}$ is replaced by a
rational $\frac{p_{n}}{q_{n}}\in\mathbb{Q}$ such that
$\lim_{n\rightarrow\infty}\frac{p_{n}}{q_{n}}=\alpha$. The fastest converging
sequence is the continued fraction approximation,

$\displaystyle\frac{p_{n}}{q_{n}}=a_{0}+1/\left(a_{1}+1/\left(a_{2}+1/\left(\ldots+1/a_{n}\right)\right)\right).$
(5)

The golden mean is an example of a Diophantine number meaning its continued
fraction approximation converges with at worst
$|\alpha-\frac{p_{n}}{q_{n}}|<\frac{1}{\sqrt{5}q_{n}^{2}}$. By contrast, for
Liouville numbers there exists a $\beta(\alpha)$ such that
$|\alpha-\frac{p_{n}}{q_{n}}|<e^{-\beta(\alpha)q_{n}}$.

We will take sequences of rational approximate operators by substituting
$\Theta=2\pi\alpha$ in Eq. (2) with its continued fraction approximation
$\Theta_{N}=2\pi p_{N}/q_{N}$. For any rational flux,
$\Theta/2\pi=p/q\in\mathbb{Q}$, one can define a magnetic unit cell specifying
bands that have a Chern number, which sum to zero, and the irrational bulk
topology is indeed the continued fraction limit.

For each rational approximate, $H_{2D,N}$, we construct a TME, see S.I. III
and- Refs. [27, 13] for more details. The unit cell for the N-th rational
approximate is $q_{N}$ sites long, and the eigenfunction on the n-th unit cell
is

$\displaystyle\Psi_{n,N}=\begin{pmatrix}\psi_{n+1}&\ldots&\psi_{n+q_{N}}\end{pmatrix}^{T}.$
(6)

This translates the eigenvalue equation into

$\displaystyle
J_{N}\Psi_{n+1}+M_{N}\Psi_{n}+J_{N}^{\dagger}\Psi_{n-1}=E\Psi_{n}.$ (7)

Here $J_{N}$ is the hopping matrix connecting the $q_{N}$-th site of $n$-th
unit cell to the $1$-st site of the $n+1$-th unit cell, $J_{N}^{\dagger}$ does
the opposite, and $M_{N}$ is the intra-unit cell term which acts internally on
the $q_{N}$ internal sites of $\Psi_{n}$.

For larger unit cells, the nearest neighbor (rank 1) hopping matrix, $J_{N}$,
is not invertible. Taking advantage of the detailed work in [27], we consider
the reduced singular value decomposition of $J_{N}=V_{N}D_{N}W_{N}^{\dagger}$,
and reduce Eq. 7 from a $2q_{N}\times 2q_{N}$ to a $2\times 2$ matrix
equation.

Defining, $G_{N}=(\omega-M_{N})^{-1}$ (projected Green’s function) and
$V_{N},W_{N}$ by the reduced singular value decomposition, we construct our
TME (setting $t=1$) [27, 13],

$\displaystyle\overbrace{(W_{N}^{\dagger}G_{N}V_{N})^{-1}\begin{pmatrix}1&-(W_{N}^{\dagger}G_{N}W_{N})\\\
V_{N}^{\dagger}G_{N}V_{N}&V_{N}^{\dagger}G_{N}W_{N}(W_{N}^{\dagger}G_{N}V_{N})-V_{N}^{\dagger}G_{N}V_{N}W_{N}^{\dagger}G_{N}W_{N}\end{pmatrix}}^{\hat{T}_{q_{N},n}}\begin{pmatrix}V_{N}^{\dagger}\Psi_{n}\\\
W_{N}^{\dagger}\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}V_{N}^{\dagger}\Psi_{n+1}\\\
W_{N}^{\dagger}\Psi_{n}\end{pmatrix}.$ (8)

When $W_{N}^{\dagger}G_{N}W_{N}\neq 0$ and $V_{N}^{\dagger}G_{N}V_{N}\neq 0$,
$\hat{T}_{q_{N},n}$ is unitary and has reciprocal eigenvalues,
$\lambda_{T,1}\lambda_{T,2}=1$. The spectrum, $E\in\Sigma$, is formed by
energies for which $\left|\lambda_{T}\right|=1$. By contrast, energies for
which $\left|\lambda_{T}\right|\in(0,1)\cup(1,\infty)$ form the spectral gaps,
$E\in\mathbb{R}-\Sigma$, see Fig. 1 (bottom).

The TMEs are determined by the translation invariant intra-cell projected
Green’s function (pGF), $G_{N}(\omega)$ in Eq. (8). We construct it by inverse
Fourier transforming the momentum-space bulk Green’s function back into real
space along one momentum dimension, $G(\omega,k_{\parallel},x_{i},x_{f})$,
acting on a single unit cell $x_{i}=x_{f}$,

$\displaystyle\hat{G}_{\perp}(\omega,k_{\parallel},n)=\int
dk_{\perp}G(\omega,k_{\parallel},k_{\perp})e^{ik_{\perp}\cdot(x_{f}-x_{i})},$
(9)

In general, $k_{\parallel},k_{\perp}$ represent momenta in parallel and
transverse directions to the projection. The projection direction is fixed in
1D, but depends on the magnetic unit cell for the 2D rational approximates.
E.g. a horizontal unit cell is the choice
$G_{N}(\omega,\delta_{y})=\hat{G}_{\perp,N}(\omega,k_{\parallel})$ with
$\delta_{y}$ defined in Eq. (2).

The key point is that zeros of the pGF arise from topological winding of the
bulk Green’s function in Eq. 9 [28, 29, 30, 31, 32, 33] and correspond to edge
modes, see S.I. V and Refs. [30, 33, 29, 13]. In Eq. 8, zeros of the pGF
correspond to $W_{N}^{\dagger}G_{N}W_{N}=0$ (or
$V_{N}^{\dagger}G_{N}V_{N}=0$), such that $\hat{T}_{q_{N},n}$ is no longer
unitary and $\lambda_{T}=0$ (no normalizeable solutions). The pGF,
$G_{N}(\omega)$, thus determines the existence of solutions to the rational
approximate TMEs. However, we must check that as $N\rightarrow\infty$, these
rational approximates converge to the irrational TME.

When rational pGFs converge to the irrational limit as discussed below, the
band topology of the rational approximates can generate rank deficient points
in the irrational TME, and the 2D parent Hamiltonian topology generates
obstructions for the 1D quasi-periodic spectrum, see Fig. 1(bottom). In
practice, convergence/divergence depends on the parameter ratio $V/t$ and the
choice of magnetic unit cell. For Diophantine $\alpha$, we find a horizontal
(vertical) magnetic unit cell converges for $V<t$ ($V>t$) and diverges for
$V>t$ ($V<t$), see Table 1 and Figs. 1, S1. We note that while the unit cell
choice appears symmetric, the original 1D system aligns with a horizontal unit
cell arrangement.

Convergence Criteria.— Following the argument in [13], we check convergence by
evaluating the (operator norm) difference between the rational pGF,
$G_{N}(\omega)$, and the full irrational, $G_{\alpha}(\omega)$, defined by
substituting $2\pi\alpha$ for $2\pi\frac{p_{N}}{q_{N}}$ in $G_{N}(\omega)$. We
evaluate this difference by bounding (from above and below) the difference
between the rational and irrational bulk Green’s functions on a single unit
cell, which reduces to bounding $||(\omega-\tilde{H}_{2D,N})||$ (S.I. IV).

The operator norm obeys the triangle inequality,

$\displaystyle\left||\omega|-||\tilde{H}_{2D,N}||\right|\leq||(\omega-\tilde{H}_{2D,N})||\leq|\omega|+||\tilde{H}_{2D,N}||\quad\quad$
(10)

We can then use the lower bound to prove divergence for $V>t$ and the upper
bound to prove convergence for $V<t$ (for horizontal unit cell). Convergence
and divergence are rigorously shown in S.I. IV, but the main idea rests on the
fact that if all Gershgorin circles,

$\displaystyle\mathcal{R}_{i}=\left\\{r\in\mathbb{R}:|r-\tilde{H}_{2D,N,ii}|<t\right\\}$
(11)

where $t=\sum_{j}|\tilde{H}_{2D,N,ij}|=2t$ and $\tilde{H}_{2D,N,ij}$ are
matrix elements of $\tilde{H}_{2D,N}$ – of an infinite matrix overlap, the
eigenvalues are contained in their union [34]. Then for $V<t$, $|2V\cos{\Theta
x}|<2t$ implies all circles overlap.

However, when $V>t$, we prove the determinant of $H_{N}$ diverges with
$q_{N}\rightarrow\infty$, S.I. IV. This coupled with the fact that all but one
eigenvalue can diverge [3] implies the lower bound in Eq. (10) diverges.
Similarly, choosing a vertical unit cell inverts these cases and $V>t$ ($V<t$)
leads to a bound (divergence), S.I IV. This generates the phase diagram for
Diophantine irrationals.

We obtain a more general convergence criteria by considering Liouville
$\alpha$ with $\delta_{N}=e^{-\beta q_{N}}$. Even as the circles diverge,
$||H_{2D,N}\delta_{N}q_{N}||$ converges for $V\lesssim te^{\beta}$. Thus, for
$t<V<te^{\beta}$, both the horizontal and vertical unit cells are convergent,
reproducing results from Ref. [18] as shown in Fig. S1. Note, vertical unit
cells also converge for $te^{-\beta}<V<t$, but projection back to 1D
(discussed below) keeps the phase diagram unaltered, see S.I. IV.1.

Another subtle exception occurs when $\delta_{y}/2=\Theta+2\pi\mathbb{Z}$, for
which the 2D rational approximates, $H_{2D,N}$ have a chiral symmetry (class
AIII) and are topologically trivial in 2D (no edge modes). In fact, any
resonance $\delta_{y}/2-k\Theta=2\pi\mathbb{Z}$ suppresses the 2D edge modes
and causes the MIT to vanish, again corroborating results in Ref. [18].

Aside from these subtle exceptions that in fact are invisible to numerical
precision, bulk-boundary correspondence [35, 1] guarantees the 2D rational
approximates will host edge modes corresponding to the non-trivial magnetic
flux per plaquette (the IQHE), see Fig. 2. Thus, any magnetic unit cell choice
results in a rational pGF with zeros for each energy in the spectral gap, Fig
2.

Figure 2: (top) Concatenated Transfer matrices for vertical unit cells. For
each $\omega$ there exists a $\delta_{y}$ such that
$\det{(\hat{T}_{q_{N},\delta_{y}})}=0$ (bottom) Rational approximate pGF zeros
plotted as a function of $\delta_{y}$ ($k_{y}$) for $q_{N}=8$ in golden mean
approximation. Notice the presence of a zero for each energy (up to numerical
precision) in the spectral gaps.

Return to 1D.— Having generated a sequence of TME rational approximates in 2D,
we construct the 1D physics by projecting back, fixing $\delta_{y}$. The 1D
projection step notably is not symmetric between the two unit cell choices.

Projecting a horizontal unit cell into 1D is well defined for the rational
approximates and in the irrational limit $N\rightarrow\infty$ (we are free to
fix $\delta_{y}$ in the Eq. (2)). Solutions to the horizontal unit cell TMEs
thus form an absolutely continuous spectrum, the limit of the 2D spectrum.

A vertical unit cell as constructed lacks a natural projection into 1D as the
$\delta_{y}$ quantum number is unwrapped by the transfer matrix construction.
To preserve $\delta_{y}$ in Eq. (2), we choose vertical unit cells, but
construct the rational TMEs by unwrapping the degenerate $k_{x}$ quantum
number as done for horizontal unit cells, see Fig. 2(top). The corresponding
rational pGF approximates still have a good $\delta_{y}$ quantum number,
$G_{\perp,N}(\omega,\delta_{y},x_{n})$. However, the rational approximates do
not grow in real-space and $\hat{T}_{q_{N},n}$ spans a single real-space site,
$x_{n}$, see Fig. 2. To avoid a dependence on $x_{n}$, we absorb the site
information into the phase choice, $\delta_{y}(n)=\delta_{y}+\Theta n$ – the
same intuition as $\text{SL}(2,\mathbb{R})$ co-cycles [17, 21, 14, 15]. We
remark that doing otherwise will cause the resulting spectrum to deviate from
the horizontal unit cell spectrum as guaranteed by the duality. Assuming
$\delta_{y}/2-k\Theta\neq 2\pi\mathbb{Z}$, all $\delta_{y}$ are included by
the associated translations. We then concatenate the resulting transfer
matrices into,

$\displaystyle\hat{T}_{q_{N}}=\prod_{n=-\infty}^{\infty}\hat{T}_{q_{N},n}(\delta_{y}(n)).$
(12)

Notice, that $\det{(\hat{T}_{q_{N}})}=0$ if for any $\delta_{y}$,
$\det{(\hat{T}_{q_{N},n})}=0$. Thus if there exists an energy in the spectrum,
$\omega\in\Sigma$, such that $\det{G_{\perp,N}(\omega,\delta_{y})}=0$ and
$\det{(\hat{T}_{q_{N}})}=0$, bulk normalizable solutions do not exist.

Topological Localization and emergent ”bulk-bulk- correspondence.— The 2D
rational approximates are Hofstadter Hamiltonians with rational fluxes per
plaquette hosting chiral edge modes. Correspondingly, the vertical rational
approximate pGFs have zero eigenvalues for each $\omega$ in the rational
spectral gaps, Fig. 2. This implies the rational transfer matrices have
deficient points for every $\omega$ with at least one corresponding
$\delta_{y}$. Since the rational pGF zeros converge in an operator norm sense
to irrational pGF zeros, see S.I. IV, the rational TME rank deficient energies
correspond to irrational TME rank deficient energies. Thus, for a convergent
vertical unit cell, the irrational 1D TME only has solutions at isolated
energies with a discontinuity in transfer matrix rank across any spectral gap.
This forms a pure-point like spectrum.

As discussed in S.I. I and Refs. [13, 14, 21], a pure point-like spectrum
corresponds to localized eigenfunctions, while an absolute continuous spectrum
corresponds to delocalized solutions. In this way, the transition from a
horizontal to a vertical unit cell corresponds to a localization transition,
and bulk-boundary correspondence – the quasi-periodic bulk topology and the
corresponding chiral edge modes – forces bulk localization, a “bulk-bulk”
correspondence. In fact, the Fibonacci quasi-crystal lacks bulk-boundary
correspondence [1, 25] and, consistent with the discussion above, is power-law
(not exponentially) localized for any parameter regime [36, 37].

Conclusions.— In summary, we report that the quasi-periodic metal insulator
transition is indeed the consequence of bulk-boundary correspondence and
quasi-periodic topology, generated by the chiral edge modes of the rational
approximates. Beyond the novelty of topological bulk localization, the
connection between quasi-periodic topology and eigenfunction localization
makes clear the need to separate random disorder physics from quasi-periodic
“disorder” physics. While similar from the perspective of numerical
simulation, random disorder does not impose these topological constraints.

Moreover, metallic eigenfunctions are invariant under phase shifts of the AAH
on-site potential, while the localized eigenfunctions are not – a phase shift
is a translation and localized solutions are not translation invariant. This
resembles a $U(1)$ symmetry breaking transition, as pointed out by Andre and
Aubry in [10, 38].

In practice, this opens the door to many future questions regarding the role
of topology in other applications of quasi-periodicity. The methods in this
letter have already been shown to generalize beyond the AAH model [13], and we
leave to future work the connection between localization and higher order
topological invariants in higher dimensional quasi-periodic systems.

###### Acknowledgements.

We cordially thank Ashvin Vishwanath for many helpful discussions and advice.
We also thank Vir B. Bulchandani, Ruben Verresen, Matthew Gilbert, Nick G.
Jones, Joaquin Rodriguez-Nieva, Dominic Else, Ioannis Petrides, Daniel E.
Parker, Eitan Borgnia, Madeline McCann, William E. Conway, Saul K. Wilson,
Will Vega-Brown, and especially Matthew Brennan for insightful discussions.
R.-J.S. acknowledges funding from the Winton Programme for the Physics of
Sustainability and from the Marie Skłodowska-Curie programme under EC Grant
No. 842901 as well as from Trinity College at the University of Cambridge.

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Supplemental Information - Localization via Quasi-Periodic Bulk-Bulk
Correspondence

##  I S.I. AAH Background

The main results of this work generalize to multiple classes of quasiperiodic
models, but the Andre-Aubry model (Almost-Mathieu operator) is the most well
studied. We introduce it and some background on current methods in the study
of single-particle quasiperiodic models. The Hamiltonian is simple, but
presents a rich playground for new techniques in analysis and single-particle
physics:

$\displaystyle\hat{H}=\sum_{x}t(\hat{c}^{\dagger}_{x+1}\hat{c}_{x}+\hat{c}_{x+1}\hat{c}^{\dagger}_{x})+2V\cos(\Theta
x+\delta_{y})\hat{c}^{\dagger}_{x}\hat{c}_{x}.\quad$ (S1)

Here $\Theta=2\pi a$, and we’ll take $a\in\mathbb{R}-\mathbb{Q}$.

To understand why the Almost-Mathieu operator is mathematically interesting,
beyond the physically interesting metal-insulator transition, we introduce the
notion of a spectral measure:

For any self-adjoint linear operator, $T$, one can decompose its measure on
the target Hilbert space, $\mathcal{H}$ as an absolutely continuous,
singularly continuous, and pure point-like components. The spectral measure of
$T$ is defined with respect to a vector $h\in\mathcal{H}$ and a positive
linear functional
$f:T\rightarrow\mathinner{\langle{h}|}f(T)\mathinner{|{h}\rangle}=\int_{\sigma(T)}fd\mu_{h}$,
where $\sigma(T)$ is the spectrum of the operator T and $\mu_{h}$ is the
unique measure associated with $h$ and $T$.

The portion of the Hilbert space, i.e. the subspace of vectors, for which
$\mu_{h}$ is dominated by the Lebesgue measure on the same subspace – for
every measureable set $A$, if the Lebesgue measure $L(A)=0$, $\mu_{h}(A)=0$ –
is absolutely continuous. By contrast, the pure point like component is the
discrete portion of the spectrum where points can have finite measure in terms
of $\mu_{h}$, but points have zero Lebesgue measure. The singularly continuous
part of the spectrum is defined as the singular part of the spectrum – the
subspace which can be formed by a disjoint union of sets $A$ and $B$ for which
$\mu_{h}(A)=0$ when $L(B)=0$ – which is not pure point like.

The original metal-insulator transition was shown non-rigorously through the
duality of the Andre-Aubry under a Fourier transform-like operation,
$\hat{c}_{k}=\sum_{x}\exp(i\Theta kx)\hat{c}_{x}$,

$\displaystyle{\tilde{\hat{H}}}=\sum_{k}V(\hat{c}^{\dagger}_{k+1}\hat{c}_{k}+\hat{c}_{k+1}\hat{c}^{\dagger}_{k})+2t\cos(\Theta
k+\delta_{k})\hat{c}^{\dagger}_{k}\hat{c}_{k}\quad,$ (S2)

see S.I. II. The model has a self-dual point for $V=t$, fixing a transition
from momentum-like to position-like eigenfunctions. A more complete
formulation of the problem, however, was constructed and proven for Almost-
Mathieu operators. It was proven that the spectrum of the Almost-Mathieu
operator is (setting $t=1$)

  1. 1.

Absolutely continuous for all $\Theta$ and $\delta_{x}$ when $V<1$

  2. 2.

Singularly continuous for all $\Theta$ and $\delta_{x}$ when $V=1$

  3. 3.

Pure point-like for almost all $\Theta$ and $\delta_{x}$ when $V>1$

A pure point-like spectrum guarantees Anderson localization as it corresponds
to eigenfunctions having finite measure at the eigenvalues and zero measure
elsewhere. Intuitively, only a finite number can effectively participate
(exponentially decaying weight) in generating discretely separated
eigenvalues, and the eigenfunction is exponentially decaying on the lattice.
More formally, the pure-point like spectrum forces eigenfunctions to be Semi-
Uniformly-Localized-Eigenstates [39]. By contrast, an absolutely continuous
spectrum guarantees delocalization if the spectrum has finite measure, which
has been shown to be the case for the Almost-Mathieu operator. Much less is
known about the singularly continuous case, and it has been the topic of
multiple famous problems proposed by Barry Simon [40, 41, 42]. One of the few
results on the singularly continuous spectrum is its existence deep in the
pure-point like regime for Liouville $a=2\pi/\Theta$ – sequence of rational
approximates $\\{p_{n}/q_{n}\\}$ exists such that $|a-p_{n}/q_{n}|<n^{-q_{n}}$
[43]. In fact, for Liouville numbers, the pure-point like transition occurs
for $\lambda=e^{\beta}$ with
$\beta=\lim_{n\rightarrow\infty}\ln(q_{n})/q_{n+1}$ [18].

In this language, the almost-Mathieu operator becomes a clear bridge between
the well understood Mathieu operator (periodic operators) and random disorder.
Understanding localization for the almost-Mathieu operator directly links to
our understanding of chaos and localization in disordered systems. And yet, we
still do not understand the full parameter space of a 1D nearest neighbor
hopping lattice model with a cosine potential. The almost-everywhere part of
this problem is important as it determines the physical stability of the
model. Modern techniques in the field rely on cocycle theory [15, 17, 14, 44],
and the absolutely continuous part of the spectrum is conjectured to be
equivalent to the almost-reduciblity of the corresponding cocycle [15]. The
connection with cocycle theory further highlights the importance of this
problem, as the reducibility classes of $SL(2,\mathbb{R})$ cocycles are known
to describe the onset of quantum chaos and directly link to the Lyapunov
exponent [15, 16, 44].

##  II S.I. The Andre Aubry’s Argument

Early studies of quasi-periodic system dynamics focused on the construction of
eigenstates from sequences of rational approximates, inductively [45, 11].
While the original work by Andre and Aubry [10] relied on the continuity of
the Thouless parameter and self-dual models to explain the transition. The RG
like induction methods proved rigorously the existence of a localized phase.
For the AAH model[45, 11] and similar quasiperiodic potentials[11], these
methods demonstrated the emergence of a pure point-like spectrum for strong
enough onsite potentials (relative to hopping terms). A pure point-like
spectrum enforces localized eigenstates as eigenstates lack support across any
continuous energy windows [45, 11, 21, 14, 3]. Below we introduce the simple
AAH model and note key insights about the breakdown of Eigenstate ergodicity.

The original paper by Andre and Aubry [10] rests on two fundamental
requirements for a quasi-periodic Hamiltonian, its self-duality and its
fidelity to a sequence of rational approximates. It proposed a Hamiltonian,
the AAH model, which satisfies a self-duality constraint under a real-space to
dual-space (momentum space in the continuum limit) transformation,
$\hat{c}_{k}=\sum_{x}\exp(i\Theta kx)\hat{c}_{x}$:

$\displaystyle\hat{H}=\sum_{x}t(\hat{c}^{\dagger}_{x+1}\hat{c}_{x}+\hat{c}_{x+1}\hat{c}^{\dagger}_{y})+2V\cos(\Theta
x+\delta_{x})\hat{c}^{\dagger}_{x}\hat{c}_{x}\quad$ (S3)
$\displaystyle{\tilde{\hat{H}}}=\sum_{k}V(\hat{c}^{\dagger}_{k+1}\hat{c}_{k}+\hat{c}_{k+1}\hat{c}^{\dagger}_{k})+2t\cos(\Theta
k+\delta_{k})\hat{c}^{\dagger}_{k}\hat{c}_{k}\quad$ (S4)

Here $\Theta$ is some irrational parameter relative to $\pi$ and clearly for
$t=V$ the Hamiltonian is self-dual, indicating the existence of a transition.
One can introduce a sequence of rational approximates,
$\\{a_{n}/b_{n}\\}_{n\in\mathbb{N}}$ with $a_{n},b_{n}\in\mathbb{Z}$ and
$\lim_{n\rightarrow\infty}a_{n}/b_{n}=\Theta$. The sequence of Hamiltonians
with periodic potentials links the density of states on either side of the
duality transformation because there are well-defined bands. One can then
write down the corresponding Thouless exponent for each side of the
transition:

$\displaystyle\gamma(E)=\int_{-\infty}^{\infty}\log|E-E^{\prime}|dN(E^{\prime})$
(S5)

For rational $a_{n}/b_{n}$, with $t=1$ and $V=\lambda$, the transformation
from Eq. (S3) to Eq. (S4) takes $\tilde{V}\rightarrow 1/\lambda$ and
$E\rightarrow\tilde{E}/\lambda$, which implies
$N_{\lambda,k}(E)=\tilde{N}_{1/\lambda,k}(E/\lambda)$ [10]. So,

$\displaystyle\gamma(E)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}\log|\frac{E-E^{\prime}}{\lambda}|d\tilde{N}\left(\frac{E^{\prime}}{\lambda}\right)+\log|\lambda|$
$\displaystyle\gamma(E)$ $\displaystyle=$
$\displaystyle\tilde{\gamma}(\frac{E}{\lambda})+\log|\lambda|.$ (S6)

Since quasiperiodic systems do not have bands, but rather protected band gaps
(discussed below [21, 14, 26, 46, 47, 3]), the Thouless exponent must be non-
negative by construction in 1D [10]. Thus, for $\lambda>1$, $\gamma(E)>0$ and
states are exponentially localized. This all relies on the continuity of the
Thouless exponent, only proven in 2002 [44]. Here, the rational approximates
differ drastically from the irrational limit, but the density of state is well
described by the approximation. In fact, these spectral properties are
topologically protected by the quasiperiodic pattern’s robustness [1], further
expanded below.

Via the above arguments, Andre and Aubry demonstrated the existence of non-
zero Thouless parameter for $V>1$. And, by the duality of the model,
$\gamma(E)$ must be zero for $V<1$. This transition is unusually sharp,
exhibiting exponential localization on either side due to the relation between
the Thouless parameters of the self-dual models. Further, the methodology is
quite general in 1D and can be extended to other self-dual models, even if the
duality is energy dependent, [13]. The argument breaks down in higher
dimensions as the Thouless exponent is no longer guaranteed to be non-negative
[10].

Returning to the Hamiltonian in Eq. S3, note the phase $\delta_{x}$ sets the
”origin” of the pattern. The eigenvalues cannot depend on the phase
$\delta_{x}$ in the thermodynamic limit. However, for $\lambda>1$, if a state
of energy $E$ is localized to site $x$ when $\delta_{x}=0$, then the state
localized to site $x-\delta/\Theta$ has energy $E$ for the shifted Hamiltonian
with phase $\delta_{x}=\delta$. Thus, the eigenstates of each eigenvalue do
depend directly on the phase. In [10, 38], this is described as a gauge-group
symmetry breaking transition.

##  III S.I. Transfer Matrix

The explicit construction of a $2\times 2$ transfer matrix starts with a
reduced SVD of the hopping matrix $J_{N}$,

$\displaystyle J_{N}$ $\displaystyle=$ $\displaystyle
V_{N}D_{N}W^{\dagger}_{N}$ (S7) $\displaystyle J_{N}^{\dagger}$
$\displaystyle=$ $\displaystyle W_{N}D_{N}^{\dagger}V^{\dagger}_{N}$ (S8)

with $V^{\dagger}V=W^{\dagger}W=\mathbb{1}$ and $W^{\dagger}V=0$, and in the
case of $J_{N}$,

$\displaystyle D_{N}=D_{N}^{\dagger}=\begin{pmatrix}t&0&\ldots&0\\\
0&0&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&0\end{pmatrix},$
(S9)

Since our hopping matrix is rank 1, we truncate $D_{N}=t$ and and
correspondingly, the $q_{N}\times 1$ dimensional operators

$\displaystyle W_{N}=\begin{pmatrix}0\\\ \vdots\\\ 0\\\
(-1)^{q_{N}}\end{pmatrix},V_{N}=\begin{pmatrix}(-1)^{q_{N}}\\\ 0\\\ \vdots\\\
0\end{pmatrix}$ (S10)

Rewriting our unit cell Green’s function as

$\displaystyle G_{N}(\omega)=(\omega-M_{N})^{-1}$ (S11)

The transfer matrix equation reduces to

$\displaystyle\Psi_{n}$ $\displaystyle=$ $\displaystyle
G_{N}J_{N}\Psi_{n+1}+G_{N}J_{N}^{\dagger}\Psi_{n-1}$ (S12)
$\displaystyle\Psi_{n}$ $\displaystyle=$ $\displaystyle
G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}$
(S13)

Projecting into the $V_{N},W_{N}$ subspaces of $\Psi_{n}$,

$\displaystyle V^{\dagger}_{N}\Psi_{n}$ $\displaystyle=$ $\displaystyle
V^{\dagger}G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+V^{\dagger}G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}$
(S14) $\displaystyle W^{\dagger}_{N}\Psi_{n}$ $\displaystyle=$ $\displaystyle
W^{\dagger}G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+W^{\dagger}G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}.$
(S15)

This reduces the Transfer matrix equation to [27], setting $t=1$,

$\displaystyle\begin{pmatrix}(W_{N}^{\dagger}G_{N}V_{N})^{-1}&-(W_{N}^{\dagger}G_{N}V_{N})^{-1}(W_{N}^{\dagger}G_{N}W_{N})\\\
(V_{N}^{\dagger}G_{N}V_{N})(W_{N}^{\dagger}G_{N}V_{N})^{-1}&V_{N}^{\dagger}G_{N}W_{N}-V_{N}^{\dagger}G_{N}V_{N}(W_{N}^{\dagger}G_{N}V_{N})^{-1}W_{N}^{\dagger}G_{N}W_{N}\end{pmatrix}\begin{pmatrix}V_{N}^{\dagger}\Psi_{n}\\\
W_{N}^{\dagger}\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}V_{N}^{\dagger}\Psi_{n+1}\\\
W_{N}^{\dagger}\Psi_{n}\end{pmatrix}$ (S16)

Notice all of the elements in the $2\times 2$ transfer matrix are effectively
scalars and thus commute with each other and we can just factor out the common
factor $(W_{N}^{\dagger}G_{N}V_{N})^{-1}$,

$\displaystyle(W_{N}^{\dagger}G_{N}V_{N})^{-1}\begin{pmatrix}1&-(W_{N}^{\dagger}G_{N}W_{N})\\\
V_{N}^{\dagger}G_{N}V_{N}&V_{N}^{\dagger}G_{N}W_{N}(W_{N}^{\dagger}G_{N}V_{N})-V_{N}^{\dagger}G_{N}V_{N}W_{N}^{\dagger}G_{N}W_{N}\end{pmatrix}\begin{pmatrix}V_{N}^{\dagger}\Psi_{n}\\\
W_{N}^{\dagger}\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}V_{N}^{\dagger}\Psi_{n+1}\\\
W_{N}^{\dagger}\Psi_{n}\end{pmatrix}$ (S17)

Now, if ever $(V_{N}^{\dagger}G_{N}V_{N})=0$ and
$(V_{N}^{\dagger}G^{2}_{N}V_{N})=0$ or $(W_{N}^{\dagger}G_{N}W_{N})=0$ and
$(W_{N}^{\dagger}G_{N}^{2}W_{N})=0$ the transfer matrix has rank 1 and the
only non-vanishing solution is localized at the edge. The corresponding energy
is not in the bulk spectrum, $\omega\notin\Sigma$. The above condition happens
when $G_{N}(\omega)=(\omega-M_{N})^{-1}=0$. These zeros correspond to
protected obstructions of the spectrum and are important to quasi-periodic
localization.

##  IV S.I. Bulk to Projected Green’s Function

Our rational approximation to the irrational Hamiltonian follows the continued
fraction approximation of irrational parameter $\alpha$ described above. We
define,

$\displaystyle\hat{H}_{N}=\sum_{x}t(\hat{c}^{\dagger}_{x+1}\hat{c}_{x}+\hat{c}^{\dagger}_{x}\hat{c}_{x+1})+2V\cos{(2\pi\frac{p_{N}}{q_{N}}x+\phi)}\hat{c}_{x}^{\dagger}\hat{c}_{x}$
(S18)

for which we can define a $q_{N}$ site unit cell and Fourier transform into
(setting $t=1$)

$\displaystyle\tilde{H}_{N}=\sum_{k}e^{ikx}c_{k}^{\dagger}c_{k}\begin{pmatrix}2V\cos{(2\pi\frac{p_{N}}{q_{N}}1+\phi)}&1&0&\ldots&e^{ik}\\\
1&2V\cos{(2\pi\frac{p_{N}}{q_{N}}2+\phi)}&1&\ldots&0\\\
0&1&2V\cos{(2\pi\frac{p_{N}}{q_{N}}3+\phi)}&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&1\\\
e^{-ik}&\ldots&0&1&V\cos{(2\pi\frac{p_{N}}{q_{N}}q_{N}+\phi)}\end{pmatrix}.$
(S19)

The bulk Green’s function is just the natural
$G(\omega,k)=(\omega-\tilde{H}_{N}(k))^{-1}$ and the corresponding “projected”
Green’s function is simply

$\displaystyle G_{\perp,N}(\omega)=\int\frac{dk}{2\pi}G(\omega,k)$ (S20)

We examine difference between the almost tri-diagonal matrix in E.Q. (S19) and
the irrational Hamiltonian on a given unit cell. First, we set $\phi=0$
w.l.o.g. and notice that

$\displaystyle V\cos(2\pi\alpha
x)=V\cos(2\pi(\frac{p_{N}}{q_{N}}+\delta_{N})x)=2V\cos(2\pi\frac{p_{N}}{q_{N}}x)-2V\sin{(\pi\delta_{N}x)}\sin{(2\pi\frac{p_{N}}{q_{N}}x+\pi\delta_{N}x)}$
(S21)

So,

$\displaystyle|2V\cos(2\pi\alpha
x)-2V\cos(2\pi\frac{p_{N}}{q_{N}}x)|<|4V\sin{(\pi\delta_{N}x)}|=|4V\sum_{n=1}^{\infty}\frac{(-1)^{n}}{(2n-1)!}(\pi\delta_{N}x)^{2n-1}|<|4V\pi\delta_{N}q_{N}|$
(S22)

where in the last inequality we used that an alternating and uniformly
convergent series is bounded at any step and that $x\leq q_{N}$. Notice, that
for any irrational number $\delta_{N}<\frac{1}{\sqrt{5}q_{N}^{2}}$ and thus

$|4V\pi\delta_{N}q_{N}|<\frac{4V\pi}{\sqrt{5}q_{N}}\rightarrow 0$

in the limit of large $q_{N}$. However, we keep $\delta_{N}$ going forward, to
account for special cases of stronger bounds on $\delta_{N}$, i.e. Liouville
numbers. Expanding the full irrational parameter on-site Green’s function,

$\displaystyle
G_{\perp,\alpha}(\omega)=\int\frac{dk}{2\pi}G_{N}(\omega)\left(\mathbb{1}+G_{N}(\omega)\left(\sum_{n=1}^{q_{N}}\left[2V\cos(2\pi\alpha
n)-2V\cos(2\pi\frac{p_{N}}{q_{N}}n)\right]\mathinner{|{n}\rangle}\mathinner{\langle{n}|}\right)\right)^{-1}$
(S23)

we see that

$\displaystyle G_{\perp,\alpha}(\omega)-G_{\perp,N}(\omega)$ $\displaystyle=$
$\displaystyle-\int\frac{dk}{2\pi}G_{N}^{2}(\omega)\frac{\left(\sum_{n=1}^{q_{N}}\left[2V\cos(2\pi\alpha
n)-2V\cos(2\pi\frac{p_{N}}{q_{N}}n)\right]\mathinner{|{n}\rangle}\mathinner{\langle{n}|}\right)}{\left(\mathbb{1}+G_{N}(\omega)\left(\sum_{n=1}^{q_{N}}\left[2V\cos(2\pi\alpha
n)-2V\cos(2\pi\frac{p_{N}}{q_{N}}n)\right]\mathinner{|{n}\rangle}\mathinner{\langle{n}|}\right)\right)}.$
(S24)

We can bound the above expression in terms of the operator norm by using our
above bound on

$|2V\cos(2\pi\alpha
x)-2V\cos(2\pi\frac{p_{N}}{q_{N}}x)|<|4V\pi\delta_{N}q_{N}|.$

Convergence then depends on the uniform convergence of

$\displaystyle\left\|G_{N}(\omega)\left(\sum_{n=1}^{q_{N}}\left[2V\cos(2\pi\alpha
n)-2V\cos(2\pi\frac{p_{N}}{q_{N}}n)\right]\mathinner{|{n}\rangle}\mathinner{\langle{n}|}\right)\right\|^{k}<\|\left(G_{N}(\omega)4V\pi\delta_{N}q_{N}\right)\|^{k}$
(S25)

as $k\rightarrow\infty$, where we have used the operator norm
$||A||=\sup_{\psi\in\mathcal{H}}(||A\psi||/||\psi||)$. We need

$\displaystyle\|\left(G_{N}(\omega)4V\pi\delta_{N}q_{N}\right)\|<1$ (S26)

We have taken $\omega$ not in the spectrum, $\Sigma$ of the operator and
$G_{N}(\omega)<\infty$, but this doesn’t necessarily bound
$G_{N}(\omega)<1/(\delta_{N}q_{N})$. The operator norm of $G_{N}(\omega)$ is
at most the inverse of the gap half-width. The minimum gap width for the
rational Hofstadter approximates is known, when $V<1$, $\sim V^{q_{N}/2}$
[48], which means $G_{N}(\omega)\sim V^{-q_{N}/2}$. Taking this approach,
there will always be gaps for which the pGF doesn’t converge.

Instead, consider a small shift of $\omega$ towards the upper and lower half
of complex plane $\omega\rightarrow\omega\pm i\epsilon$. This fixes the
maximum of $G_{N}(\omega\pm i\epsilon)<\epsilon^{-1}$. We can always take a
large enough $q_{N}$ such that $\delta_{N}q_{N}<\epsilon$ for any
$\epsilon>0$. However, we now need both the imaginary and real parts of the
offset pGF to converge.

$\displaystyle\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=-\int\frac{dk}{2\pi}\frac{\left[(\omega-\tilde{H}_{N})^{2}+(\omega-\tilde{H}_{N})(\tilde{H}_{N}-\tilde{H}_{\alpha})-\epsilon^{2}\right](\tilde{H}_{\alpha}-\tilde{H}_{N})}{((\omega-\tilde{H}_{N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{N}-H_{\alpha})+\tilde{H}_{N}^{2}-\tilde{H}_{\alpha}^{2}}{((\omega-\tilde{H}_{N})^{2}+\epsilon^{2})}\right)}$
(S27) $\displaystyle\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=\int\frac{dk}{2\pi}\frac{\left[\pm
2i\epsilon(\omega-\tilde{H}_{N})+i\epsilon(H_{N}-\tilde{H}_{\alpha})\right](\tilde{H}_{\alpha}-\tilde{H}_{N})}{((\omega-\tilde{H}_{N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{N}-H_{\alpha})+\tilde{H}_{N}^{2}-\tilde{H}_{\alpha}^{2}}{((\omega-\tilde{H}_{N})^{2}+\epsilon^{2})}\right)}$
(S28)

Now we use that $||(\omega-\tilde{H}_{N})^{2}+\epsilon^{2}||>\epsilon^{2}$, to
cancel the bottom factors $\sim\frac{1}{\epsilon^{2}}$. And, we can use the
bound from above, $||\tilde{H}_{N}-\tilde{H}_{\alpha}||<4V\pi\delta_{N}q_{N}$
to get

$\displaystyle||\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\int\frac{dk}{2\pi}||\epsilon^{-4}\left[-\epsilon^{2}4V\pi\delta_{N}q_{N}+(\omega-\tilde{H}_{N})\left(4V\pi\delta_{N}q_{N}\right)^{2}\right]||$
(S29) $\displaystyle||\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\int\frac{dk}{2\pi}\epsilon^{-3}\left[8V\pi\delta_{N}q_{N}(\omega-\tilde{H}_{N})+\left(4V\pi\delta_{N}q_{N}\right)^{2}\right].$
(S30)

Clearly the imaginary part will converge to zero for large $q_{N}$. We bound
the real part by showing

$\displaystyle||(\omega-\tilde{H}_{N})||<Cq_{N},$ (S31)

so that we can always take $q_{N}$ big enough to make
$\left(4V\pi\delta_{N}q_{N}\right)Cq_{N}<\epsilon^{3}$ for any $\epsilon>0$ to
get the uniform convergence above, for all Diophantine $\alpha$.

The operator norm obeys the triangle inequality,

$\displaystyle\left||\omega|-||\tilde{H}_{N}||\right|\leq||(\omega-\tilde{H}_{N})||\leq|\omega|+||\tilde{H}_{N}||$
(S32)

We can then use the lower bound to prove divergence for $V>1$ and the upper
bound to prove convergence for $V<1$.

While any finite matrix has finite eigenvalues contained in Gershgorin
circles,
$\mathcal{R}_{i}=\left\\{r\in\mathbb{R}:|r-\tilde{H}_{N,ii}|<t\right\\}$ where
$t=\sum_{j}|\tilde{H}_{N,ij}|=2t$, the limit of infinite matrices are not as
simple. Indeed, in [3] the semi-infinite spectrum of $\tilde{H}_{N}$ is noted
to contain up to 2 eigenvalues inside each gap and up to 1 in each of the
infinite intervals above and below the bulk. We need to bound the eigenvalues
above and below the gap. We will bound them as constant when $V<t$ and as
exponentially divergent for $V>t$

Fortunately, for $V<t$ all Gershgorin circles intersect for all $q_{N}$ as
$|2\cos{\Theta x}|<2t$. Thus, we are guaranteed that the infinite spectrum
lies in the union of the circles. By contrast, for $V>t$ there exist
Gershgorin circles which do not intersect, as $\max_{x}|2\cos{\Theta x}|>2t$.
Here we can even show a divergence of the largest eigenvalue, by considering
the determinant of $\tilde{H}_{N}$. In particular, if $V>t$ and $t=1$

$\displaystyle\det\tilde{H}_{N}=\sum_{1\geq n\leq
q_{N}}\binom{q_{N}}{n}(2V)^{n}\left(\int_{0}^{2\pi}\prod_{j=1}^{n}(dx_{j}\cos{x_{j}})\right)>\sum_{n\leq\lfloor{q_{N}/2}\rfloor}\binom{q_{N}}{2n}V^{2n}>2^{\lfloor{q_{N}/2}\rfloor}$
(S33)

with odd terms canceling in the integral. As the determinant diverges, the
determinant of $\tilde{H}_{N}^{-1}$ approaches 0 as $q_{N}\rightarrow\infty$,
implying the existence of zero eigenvalues, implying $||\tilde{H}_{N}||$ is
unbounded in the limit of $q_{N}\rightarrow\infty$.

Thus, for $V<1$ and any $\epsilon>0$ both the imaginary and real parts of the
rational pGFs converge for $q_{N}$ such that $\delta_{N}q_{N}<\epsilon^{4}$.
This implies the uniform convergence of the rational approximates and the
Green’s function of the irrational limit is arbitrarily close to that of the
translation invariant Green’s function. Using the transfer matrix construction
above, the quasi-periodic transfer matrix is arbitrarily close to the
translation invariant transfer matrix. This is similar to almost-reducibility
as defined by Artur Avilla in [15].

Now, consider the case of $\delta_{N}=e^{-\beta q_{N}}$, when $\alpha$ is
Liouville. Here,

$\displaystyle\|\left(G_{N}(\omega)4V\pi\delta_{N}q_{N}\right)\|<\|\left(G_{N}(\omega)4V\pi
e^{-\beta q_{N}}q_{N}\right)\|$ (S34)

Now, even if $V>1$ we can bound the determinant of
$\det\tilde{H}_{N}\delta_{N}q_{N}$

$\displaystyle\det\tilde{H}_{N}\delta_{N}q_{N}=\sum_{1\geq n\leq
q_{N}}\binom{q_{N}}{n}(2V)^{n}\left(\int_{0}^{2\pi}\prod_{j=1}^{n}(dx_{j}\cos{x_{j}})\right)<\sum_{n\leq\lfloor{q_{N}/2}\rfloor}\binom{q_{N}}{2n}V^{2n}<(2V)^{q_{N}}e^{-\beta
q_{N}}q_{N},$ (S35)

and the rational approximate pGFs converge for $V\lesssim e^{\beta}$. Thus,
for $1<V<e^{\beta}$, both the horizontal and vertical unit cells are
convergent – this reproduces results in [18].

Figure S1: (a) Plot of max (operator norm) difference over all $\omega$
between the rational and irrational pGFs for $\alpha=(\sqrt{5}-1)/2$.
Convergence is defined by the ability to take
$||G_{\alpha}(\omega+i\epsilon)-G_{N}(\omega+i\epsilon)||<\epsilon^{2}.$ For
$V/t<1$ ($V/t>1$) horizontal (vertical) unit cell converges. Taking larger
system sizes N, sharpens convergence/divergence [shrinking divergence for
large $V/t$ caused by finite system size]. (inset) AAH phase diagram for
Liouville $\alpha$, with $\beta$ the Liouville exponent and transition at
$\beta=\ln{(V)}$ (blue: metal, red: insulator, gray: transition).
(b) Transfer matrix trace generated by pGF (top row) and direct calculation
(bottom row) plotted for corresponding regimes, demonstrating convergence of
Eq. (8). Note the divergence of the trace for the insulating regime, implying
only localized solutions.

###  IV.1 S.I. Gauge Transformation

If $V>1$, the above convergence proof fails and our rational Green’s function
approximates are no longer guaranteed converge to the irrational Green’s
function. Consequentially, the rational approximate transfer matrices no
longer converge to the quasi-periodic transfer matrix and the quasi-periodic
eigenfunctions are no longer the limit of the de-localized eigenfunctions of
the rational approximate transfer matrices, defined by unit cells along the
$x$-direction of the 2D parent Hamiltonian lattice.

Instead we must apply a 2D gauge transformation to the 2D Hamiltonian, such
that the “magnetic” flux is contained in the $x$-direction and the unit cell
is constructed in the $y$-direction. The resulting transfer matrix will be
constructed from on-site projected Green’s functions of the form

$\displaystyle
G_{\perp,N}=\int\frac{d\phi}{2\pi}\left[\omega\mathbb{1}-\begin{pmatrix}2\cos{(2\pi\frac{p_{N}}{q_{N}}1+k_{x})}&V&0&\ldots&Ve^{i\phi}\\\
V&2\cos{(2\pi\frac{p_{N}}{q_{N}}2+k_{x})}&V&\ldots&0\\\
0&V&2\cos{(2\pi\frac{p_{N}}{q_{N}}3+k_{x})}&\ddots&\vdots\\\
\vdots&\vdots&\ddots&\ddots&V\\\
Ve^{-i\phi}&\ldots&0&V&2\cos{(2\pi\frac{p_{N}}{q_{N}}q_{N}+k_{x})}\end{pmatrix}\right]^{-1}.$

We can factor out a V from the entire equation above and generate the same
Green’s function as above with $V\rightarrow 1/V$ and
$\omega\rightarrow\omega/V$. As a consequence, we have a clear convergence for
$1/V<1$ or $V>1$ for the projected Green’s function along a vertical unit cell
in the limit of $q_{N}\rightarrow\infty$.

As a subtle point, while this gauge transformation is well defined in 2D for
Liouville numbers, it fails to manifest in 1D, as the “magnetic flux” can only
shift to the hopping elements in $\hat{H}$ under a quasi-Fourier
transformation. When the rational approximates in the horizontal unit cell
converge, the quasi-Fourier transform is close (exponentially for $\alpha$
Liouville) to a rational Fourier transform, and we simply obtain the
horizontal unit cell rational approximates. This point only manifests itself
for Liouville choices of $\alpha$, such that the convergence regime extends
into the convergence regime of the gauge transformed Green’s function, $V>1$.

For Liouville $\alpha$, multiple gauge choices and unit cell configurations
define convergent approximations to the full quasi-periodic projected Green’s
functions, but the in the projection to 1D is biased towards the horizontal
unit cells as the eigenfunctions survive this projection and the “gauge group
symmetry” – phase shift invariance – is preserved.

More precisely, the Greshgorin circles force convergence for $V<1$, meaning
any TME solutions must preserve the phase shift invariance. However, for
Liouville irrationals, not only does a single unit cell converge, but any
polynomial number of unit cells converge, i.e. $\delta_{N}q_{N}^{M}=e^{-\beta
q_{N}}q_{N}^{M}\rightarrow 0$ in Eq. S22. Thus, even for the vertical unit
cell choice, we preserve the $k_{y}$ quantum number in the 1D projection and
we retain phase-shift invariance. Correspondingly, the phase diagram is biased
towards the metallic phase in Fig. S1

##  V S.I. Projected Green’s Function Formalism

In translation-invariant systems, the Brillouin zone allows for flexibility in
writing locally computable formulas for topological invariants. In this
language, Green’s function zeros are singular and carry topological
significance [49, 30, 29, 33, 28, 50, 51, 32]. More recently, it was noticed
that bound state formation criteria along an edge are also defined by Green’s
function zeros [30, 29, 33, 3, 32, 28], thereby tracking both topological
invariants and their corresponding edge modes.

Extending this methodology beyond translation invariant systems consists of
two steps. One must show both that Green’s function zeros are still of
topological significance and that edge formation criteria are still described
by the presence of in-gap zeros. We first show the latter.

The poles of the Green’s function restricted to a particular site in position
space correspond to an energy state at that particular site. Here restricted
refers to the projection of the system Green’s function, $G$, to a single
site,

$\displaystyle
G(\omega,\mathbf{r}_{\perp},\alpha_{\parallel})=\sum_{\alpha}|\mathinner{\langle{\alpha|\mathbf{r}_{\perp}}\rangle}|^{2}G(\omega,\alpha),$
(S36)

where $\alpha$ generically labels the Eigenvalues and $\alpha_{\parallel}$ is
the remaining index post the contraction with $\mathbf{r}_{\perp}$.
Generically, there will be many poles corresponding to the spectrum at
$\mathbf{r}_{\perp}$, but they are not universal.

By adding on-site impurities and considering
$G(\omega,\mathbf{r}_{\perp},\alpha_{\parallel})$ only in the band gap of the
bare Green’s function, any poles will be a result of the impurity potential,
$\mathcal{V}(r)=\mathcal{V}\delta(r-\mathbf{r}_{\perp})$, binding a state in
the gap. Then, by constructing an appropriate impurity geometry, and taking
$|\mathcal{V}|\rightarrow\infty$, an edge is formed. Therefore, the condition
for impurity localized states as $|\mathcal{V}|\rightarrow\infty$ is
equivalent to the criteria for the formation of edge localized modes. And,
impurity bound states correspond to zeros of the restricted in-gap Green’s
function. This is most readily seen by factoring the full Green’s function,
$G$ of some system with Hamiltonian $H_{0}$ and an impurity potential
$\mathcal{V}$. That is, the full Green’s function $G$ can be written in terms
of the Green’s function $G_{0}=(\omega-H_{0})^{\text{-}1}$ of the original
system without the impurity,

$\displaystyle G(\omega,\alpha)$ $\displaystyle=$
$\displaystyle(\omega-(H_{0}+\mathcal{V}))^{\text{-}1}=(1+\mathcal{V}G_{0})^{\text{-}1}G_{0}.$
(S37)

Correspondingly, impurity bound states (poles of $G$) in the gap (not a pole
of $G_{0}$) must be a pole of $(1-\mathcal{V}G_{0})^{\text{-}1}$,

$\det\left[G_{0}(\omega,\alpha)\mathcal{V}-\mathbf{1}\right]=0.$ (S38)

For $|\mathcal{V}|\rightarrow\infty$, solutions require $G_{0}\rightarrow 0$,
and the zeros of $G_{0}$ correspond to poles of $G$. Hence, the zeros of the
restricted in-gap Green’s function,
$G(\omega,\mathbf{r}_{\perp},\alpha_{\parallel})$, correspond to edge modes,
just as in the translation-invariant case [30].

The above requires in-gap bound states of an aperiodic system to appear as
zeros of the projected Green’s function. We now derive the conditions under
which such states are fixed by the pattern topology. In these cases, in-gap
states survive small disorder [1], and impose constraints on system dynamics.
The fundamental difference between the translation invariant and aperiodic
cases comes down to the existence of a good momentum quantum number. For
translation invariant systems, the $x$-basis is dual to the Brillouin zone
momentum basis. Thus, each momentum eigenstate is equally weighted in the
projection, Eq. (S36) with position $0$, and singular points of the projected
Green’s function directly relate to an obstruction of consistently writing the
Green’s function over $k$-space. Heuristically, for two bands separated by a
topologically non-trivial gap, the eigenstates switch eigenvalues [30, 33,
29].

There is no such guarantee for generic aperiodic systems, but we can reduce
the the constraints of translation invariance to a single condition for which
the sum in Eq. (S36) is in fact reduce-able to a sum over the topologically-
fixed IDoS. We now focus on 1D systems (higher dimensions generalize by
choosing codimension-1 surface [33, 30, 29]), where the projected Green’s
function on the site $x_{0}$ reduces to

$\displaystyle
G_{\perp}(\omega,x_{0})=\mathinner{\langle{x_{0}}|}\left[\sum_{\alpha}G(\omega,\alpha)\mathinner{|{\alpha}\rangle}\mathinner{\langle{\alpha}|}\right]\mathinner{|{x_{0}}\rangle},$
(S39)

with $\alpha$ indexing the eigenstates of the system. In 1D, it is clear that
any shift of the projection site $x_{0}$ can be absorbed as a phase shift in
the pattern. We illustrate this with the AAH model, whose Hamiltonian reads

$\displaystyle\hat{H}=\sum_{x}t(\hat{c}^{\dagger}_{x+1}\hat{c}_{x}+\hat{c}_{x+1}\hat{c}^{\dagger}_{x})+2V\cos(\Theta
x+\delta_{x})\hat{c}^{\dagger}_{x}\hat{c}_{x}.$ (S40)

Here $\Theta$ is some irrational multiple of $2\pi$ generating the
quasiperiodic pattern. The dynamics are generated by translations $\tau_{x}$,
taking $V\cos(\Theta x+\delta_{x})\rightarrow V\cos(\Theta(x+1)+\delta_{x})$.
And, we can shift $x_{0}\rightarrow x$ by taking $\delta_{x}=\Theta(x-x_{0})$.
We can clearly index the Hamiltonian (and corresponding Green’s function) by
the real-space phase $\delta_{x}$. This is a general property of quasiperiodic
patterns, and the spectrum is invariant under this shift [25]. This is
guaranteed by choosing the hull of the pattern to define our unital algebra,
as the system dynamics guarantee any initial point can be translated into any
other point on the hull. We can therefore rewrite Eq. (S39) as

$\displaystyle
G_{\perp}(\omega,x_{0})=\frac{1}{N}\sum_{x}\mathinner{\langle{x}|}\left[\sum_{\alpha}G(\omega,\alpha)\mathinner{|{\alpha(\delta_{x})}\rangle}\mathinner{\langle{\alpha(\delta_{x})}|}\right]\mathinner{|{x}\rangle},$

with $\delta_{x}=-\Theta(x-x_{0})$ depending on $x$, and the eigenstates
depend on the choice of phase, i.e. an eigenstate localized at $x_{0}$ becomes
localized at site $x$, but with the same energy and corresponding Green’s
function component, $G(\omega,\alpha)$. If this were a translation-invariant
setting, it would be clear that the choice of $x_{0}$ cannot matter, and, thus
the sum in Eq. (V) must reduce to

$\displaystyle
G_{\perp}(\omega,x_{0})=\frac{1}{N}\sum_{\alpha}G(\omega,\alpha).$ (S42)

If this holds for the projected Green’s function of an aperiodic system with
non-trivial pattern topology, then the quantization of the IDoS fixes the sum
in Eq. (S42). All states are summed over. As a consequence, for an $\omega$ in
a spectral gap, states above and below the gap contribute to the sum, such
that for some $\omega_{*}$, $G(\omega)|_{\omega=\omega_{*}}=0$. In translation
invariant systems, the location of $\omega_{*}$ is usually protected by
symmetries such as chirality, fixing states to be at equal energies above and
below the gap. In quasiperiodic systems, the IDoS fixes the number of states
above and below the gap[25]. For some gap labeled, $F$, the integrated density
of states (IDoS) below each gap is fixed, i.e. for the AAH model
$\text{IDoS}(F)=(m+n\Theta)\cup[0,N]$ [25]. Thus, for each gap, $F$, the
relevant $\omega_{F}$ for which $G(\omega\in F)|_{\omega=\omega_{F}}=0$ would
also be fixed. This observation motivates our generalization to aperiodic
systems. In particular, the ability to induce real-space translations as
pattern phase shifts suggests that full translation invariance is not
necessary.

##  VI S.I. Projected Green’s Function Zeros in 1D

Having proven the pGF zeros are preserved in the irrational limit, we can
project back down from 2D into 1D and construct the full solutions for the 1D
problem. Notice this projection is natural with the horizontal unit cell
choice and problematic in the gauge transformed vertical unit cells.

###  VI.1 Horizontal Unit Cell

For $V<t$ ($V<te^{\beta(\alpha)}$ when $\alpha$ is Liouville), we proved the
convergence of rational approximates to the incommensurate projected Green’s
function along a growing set of horizontal unit cells of size $q_{N}$, chosen
via the continued fraction approximation for an irrational parameter,
$\alpha$. In this context, a horizontal unit cell corresponds a periodic
structure along the real-space coordinate direction, $x$ from the original 1D
problem. As such the projection back down to 1D simply corresponds to the
projection onto a single phase choice $\phi$ or $k_{y}$ in the rational
approximates. Since phase is irrelevant in the limit of $N\rightarrow\infty$,
the projected Green’s function and by extension the transfer matrix of the 2D
rational approximates is directly projected into 1D and the 2D irrational
limit is equivalent to the 1D irrational transfer matrix.

Note, a direct consequence of this projection is the passing of bulk boundary
correspondence via the zeros of the projected Green’s function defining
equivalence classes of transfer matrices and the existence of edge-localized
modes as per Ref. [13].

###  VI.2 Vertical Unit Cell

For $V>t$ ($V>te^{\beta(\alpha)}$), the convergent vertical unit cells are not
as naturally projected into 1D. Choosing a particular phase, $\phi$, along the
vertical unit cell corresponds to a momentum cut along $\delta_{y}$ in the 2D
rational approximates. The cut is unnatural, however, because the $\delta_{y}$
quantum number doesn’t survive the limit $N\rightarrow\infty$. Instead, we
focus on the spectral gaps and the presence of pGF zeros.

As discussed above, pGF zeros survive the limit $N\rightarrow\infty$ and
correspond to edge localized modes (rank deficient points in the transfer
matrix). The rational approximates are known to host edge localized modes on
open boundary conditions. The edge mode spectra span the bulk spectral gaps,
filling the incommensurate spectral gaps in the limit $N\rightarrow\infty$.
Thus, for each $\omega\notin\Sigma$ there exists a $\phi=\delta_{y}(\omega)$
such that

$\displaystyle
G_{\perp,\alpha}(\omega,x_{0},\phi)=\lim_{N\rightarrow\infty}G_{\perp,N}(\omega,x_{0},\phi)=0.$
(S43)

Projection to 1D now requires choosing a different $\delta_{y}$ for each site
corresponding to the phase induced by lattice translation in the original
quasi-periodic pattern. As all $\delta_{y}$ are included by the associated
translations, if for any energy in the spectrum, $\omega\in\Sigma$,
$\det{G_{\perp,N}(\omega,\delta_{y})}=0$, $\det{(\hat{T}_{q_{N}})}=0$ and
normalizable solutions do not exist (without an edge).

###  VI.3 Spectral Measure

Bringing these ideas together, for any $\omega\in\Sigma$, there exists a
$\gamma$ such that $\omega\pm\epsilon\notin\Sigma$ for any
$0<\epsilon<\gamma$. In other words, the 1D spectrum for $V>t$
($V>te^{\beta(\alpha)}$ is pure point-like as only the $\lambda$ such that
$\det(\lambda\mathbb{1}-H_{\alpha})=0$ produce normalizable solutions, $\psi$,
such that $||\hat{T}\psi||<\infty$, with $T$ the incommensurate transfer
matrix.

By contrast, for $V<t$ ($V<te^{\beta(\alpha)}$, $\alpha$ Liouville) only one
$\delta_{y}$ is fixed and not all $\omega$ are disallowed, one per gap in the
limit of $N\rightarrow\infty$. More precisely, the 2D incommensurate transfer
matrix is the same as the $T_{2D}=T_{1D}$. Thus, all 2D horizontal unit cell
eigenfunctions are 1D eigenfunctions. And, since the translation invariant
approximates converge in the limit of $N\rightarrow\infty$ the innerproduct
measure of the 2D eigenstates is absolutely continuous by Floquet Theory [15,
16, 17, 45]. Thus, the 1D spectrum is abs. continuous for $V<t$
($V<te^{\beta(\alpha)}$).

Finally, the transition point ($V=t$ when $\alpha$ is Diophantine) is more
complicated, and we save discussions for a future work. We do note that
neither the horizontal nor vertical unit cells will converge in this case. We
instead need a chiral gauge choice [3].

###  VI.4 S.I. AAH Algebra

We follow the work of Prodan [1] in deriving explicit topological invariants
in the AAH model context. We construct the unital algebra and use it to label
the resulting spectral gaps of the AAH Hamiltonian. Recall that it reads

$\displaystyle\mathcal{H}_{\delta_{x}}=\sum_{n}t\hat{c}_{n+1}^{\dagger}\hat{c}_{n}+\textnormal{h.c.}+2V\cos(\Theta
n+\delta_{x})\hat{c}_{n}^{\dagger}\hat{c}_{n},$ (S44)

in terms of the creation operators $c^{\dagger}_{i}$, lattice constant $a$,
potential $V$ that depends on the position $n$ and is indexed by phase
$\delta_{x}$. The model exhibits a duality under the pseudo-Fourier
transformation $c_{k}=\sum_{n}\exp(-ikn)c_{k}$ [10]. Considering,
$\delta_{x}=0$, one obtains

$\displaystyle\tilde{\mathcal{H}}(k)$ $\displaystyle=$
$\displaystyle\sum_{k,k^{\prime},n}te^{ik(n+1)-ik^{\prime}n}\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}}+t^{*}e^{ikn-
ik^{\prime}(n+1)}\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}}+V(e^{2\pi
ian+i(k-k^{\prime})n}+e^{-2\pi
ian+i(k-k^{\prime})n})\hat{c}_{k}^{\dagger}\hat{c}_{k^{\prime}}$
$\displaystyle\tilde{\mathcal{H}}_{\phi}$ $\displaystyle=$
$\displaystyle\sum_{k}2t\cos(\Theta
k)\hat{c}_{k}^{\dagger}\hat{c}_{k}+V(\hat{c}_{k+1}^{\dagger}\hat{c}_{k}+\textnormal{h.c.}).$
(S45)

where in the last line we have set $k=\Theta m$ and defined
$\sum_{n}\exp(i\Theta n(m-m^{\prime}))=\delta(m-m^{\prime})$ in the limit
$n\rightarrow\infty$. A natural equivalence emerges between $\mathcal{H}$ and
$\tilde{\mathcal{H}}$ under $V\rightarrow t$, implying the model undergoes a
transition for $V=t$, being the well known 1D metal-insulator transition.
Considering the limits $V=0$ and $t=0$, the duality relates extended
(momentum-localized) eigenstates to position-localized states.

The duality in the AAH model has been focus of many localization studies, past
and present [10, 3]. The model took on new light, however, when [2] noticed it
could be parameterized by the phase choice $\delta_{x}$ [2, 3, 16, 22].
Naively, this phase choice is irrelevant as it corresponds to a shift in
initial position of an infinite chain, but the 2D parent Hamiltonian, as
function of $x$ and $\delta_{x}$, has a topological notion. In particular, it
corresponds to a 2D tight-binding model with an irrational magnetic flux per
plaquette. Explicitly,

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{n,\delta_{x}}t\hat{c}_{n+1,\delta_{x}}^{\dagger}\hat{c}_{n,\delta_{x}}+t^{*}\hat{c}_{n,\delta_{x}}^{\dagger}\hat{c}_{n+1,\delta_{x}}+2V\cos(\Theta
x+\delta_{x})\hat{c}_{n,\delta_{x}}^{\dagger}\hat{c}_{n,\delta_{x}},$
$\displaystyle\tilde{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\sum_{n,m,m^{\prime}}t\delta_{m,m^{\prime}}(\hat{c}_{n+1,m}^{\dagger}\hat{c}_{n,m^{\prime}}+t^{*}\hat{c}_{n,m}^{\dagger}\hat{c}_{n+1,m^{\prime}})+V\left(e^{i\Theta
x}\delta_{m+1,m^{\prime}}+e^{-i\Theta
x}\delta_{m-1,m^{\prime}}\right)\hat{c}_{n,m}^{\dagger}\hat{c}_{n,m^{\prime}},$
$\displaystyle\tilde{\mathcal{H}}$ $\displaystyle=$
$\displaystyle\sum_{n,m}t(\hat{c}_{n+1,m}^{\dagger}\hat{c}_{n,m}+\hat{c}_{n,m}^{\dagger}\hat{c}_{n+1,m})+V(e^{i\Theta
n}\hat{c}_{n,m+1}^{\dagger}\hat{c}_{n,m}+e^{-i\Theta
n}\hat{c}_{n,m-1}^{\dagger}\hat{c}_{n,m}).$ (S46)

The 2D spectrum amounts to a Hofstadter butterfly when varying the flux per
plaquette, $\Theta$. For any rational flux, $\Theta/2\pi=p/q\in\mathbb{Q}$,
one can define a magnetic unit cell specifying bands that have a Chern number,
which sum to zero. This is however not possible for an irrational flux. In
this case strategies outlining sequences of rational approximates, with
similar band gaps, to find topological invariants were employed [52].

Hamiltonian (VI.4) is manifestly topological for all rational fluxes, $a$
[52]. We can therefore create a sequence of rational approximates to an
irrational flux $\\{a_{n}\\}$, such that
$\lim_{n\rightarrow\infty}a_{n}=a_{*}$. The problem, however, arises when
projecting back down into one dimension. This is made most clear by
considering the spectrum for different phase choices $\delta_{x}$, contrasting
the irrational and rational case. For example, for $a=1/2$, the choice
$\delta_{x}$ changes the maximum amplitude of the on-site potential. In the
irrational case, however, it has no effect on the spectrum and acts as a
translation. Hence, the projection of the sequence of rational approximates
does not create a 1D sequence of rational approximates to the AAH model [1].
Instead more recent works leverage powerful tools from non-commutative
geometry to tackle the problem conclusively, finding deep connections between
non-commutative topological invariants and the inherited topology for the 1D
projection, i.e the AAH model. In fact, methods described below describe both
the inheritance of a 2D topological invariant and a bulk-boundary
correspondence in the model [1].

Figure S2: (a) Quasi-periodic spectrum as function of plaquette flux. Note the
gap labeling, where gaps are generated by lines of different slopes (lines
cured in spectrum, but straight in IDoS) (b) Illustration of quasi-periodic
pattern generating a minimal surface (hull). This forms the underlying unital
algebra, taking the place of a Brillouin zone.

####  VI.4.1 Non-commutative topological characterization

We formalize the ideas behind a parent Hamiltonian for the AAH model and show
the AAH model obeys the same algebra as the Hofstader Hamiltonian (defined
below), forming a non-commutative torus. We show these two Hamiltonians are
equivalent up to representation and all topological properties carry over [1].

Consider the translation operator, acting on $\mathcal{H}_{\delta_{x}}$ as

$\displaystyle T^{n}\mathcal{H}_{\delta_{x}}T^{\dagger
n}=\mathcal{H}_{\delta_{x}+n\Theta}.$ (S47)

For $\Theta/2\pi\notin\mathbb{Q}$, the repeated action of the translation
operator parameterizes the motion of the phase, $\phi\in\mathbf{S}$, along the
unit circle. On the set of continuous functions over $\mathbf{S}$,
$C(\mathbf{S}):\mathbf{S}\rightarrow\mathbb{C}$, the translation operator acts
similarly. That is, for some $f\in C(\mathbf{S})$,

$\displaystyle\alpha_{n}:f(\phi)\rightarrow f(\phi+\Theta n).$ (S48)

More formally, one paramterizes the action of $\mathbb{Z}$ on $\mathbf{S}$ as
a dynamical system and then constructs its dual, i.e. the pattern hull
described in the main text. The key step, however, is to define a unitary
which acts in place of the translation operator on functions $f\in
C(\mathbf{S})$.

$\displaystyle u^{n}f(\phi)u^{-n}=f(\phi+\Theta n)$ (S49)

In this language, we can define elements of the space
$C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z}$, the semi-direct product between
complex continuous functions of $\mathbf{S}$ and the translations $\mathbb{Z}$
generated by $\alpha:f(\phi)\xrightarrow{\alpha_{n}}f(\phi+\Theta n)$ as

$\displaystyle\mathbf{a}=\sum_{n\in\mathbb{Z}}a_{n}u^{n}.$ (S50)

In the above $a_{n}\in C(\mathbf{S})$, and $u^{n}$ corresponds to a
translation along $\mathbb{Z}$. The main benefit of defining this operator
algebra corresponding to the 1D translations is that we can pick a
representation of the Hilbert space,

$\displaystyle\pi_{x}(\mathbf{a})=\sum_{n,x\in\mathbb{Z}}a_{n}(\delta_{x}+\Theta
x)\mathinner{|{x}\rangle}\mathinner{\langle{x}|}T^{n},$ (S51)

with $\phi\in[0,2\pi]$. In this representation, the elements,
$\mathcal{H}_{\delta_{x}}$, are simply represented as
$\pi_{\phi}(\mathbf{h})$. Here,

$\displaystyle\mathbf{h}=u+u^{-1}+2Vcos(\phi)$ (S52)

is an element of $C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z}$, and the potential
shifts when $T$ acts on $\phi$ at each site, i.e. $(\phi+\Theta
n)\textnormal{mod}_{2\pi}$.

As done for Eq. (VI.4), we now show that this element $h\in
C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z}$ also generates the Hofstader
Hamiltonian, see also [1]. We first rewrite the Hofstader Hamiltonian as

$\displaystyle H_{\Theta}=\sum_{x,y}T_{x}+T_{x}^{-1}+V(T_{y}+T_{y}^{-1}),$
(S53)

where

$\displaystyle T_{x}\mathinner{|{x,y}\rangle}=\mathinner{|{x+1,y}\rangle}\ \
\textnormal{and}\ \ T_{y}\mathinner{|{x,y}\rangle}=e^{-i\Theta
x}\mathinner{|{x,y+1}\rangle}$

are magnetic translations with commutation relations
$T_{x}T_{y}=e^{i\Theta}T_{y}T_{x}$. In this form, we define unitary operators
$u$, as before, and $z=\exp{i\phi}$ acting on $C(\mathbf{S})$ corresponding to
the translations along $x$ and $y$, respectively. These have the same
commutation relations $uz=e^{i\Theta}zu$ and allow for a representation of the
Hilbert space of $l^{2}$-normed functions on $\mathbb{Z}^{2}:$

$\displaystyle\pi^{\prime}(\textbf{a})=\sum_{n,m}f_{n}T_{x}^{n}T_{y}^{m}.$
(S54)

Then, $H_{\Theta}=\pi^{\prime}(\textbf{h})$, for

$\displaystyle\mathbf{h}=u+u^{-1}+V(z+z^{-1})=u+u^{-1}+V(e^{i\phi}+e^{-i\phi})$
(S55)

as in Eq. (S52). Framing the problem in terms of the operator algebra
$C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z}$ allows one to use techniques from
non-commutative geometry to solve the problem immediately. In particular, this
is a unital *-algebra for which a non-commutative calculus can be defined. The
elements of the algebra are

$\displaystyle\mathbf{a}=\sum_{m,n\in\mathbb{Z}}f_{m,n}z^{m}u^{n},$ (S56)

We can further revert to tools from non-commutative geometry to compute
topological invariants. Details can be found in [1]. We simply state the
results here. One can define differentiation intuitively along each direction:

$\displaystyle\partial_{1}\mathbf{a}$ $\displaystyle=$ $\displaystyle
i\sum_{m,n\in\mathbb{Z}}mf_{m,n}z^{m}u^{n},$
$\displaystyle\partial_{2}\mathbf{a}$ $\displaystyle=$ $\displaystyle
i\sum_{m,n\in\mathbb{Z}}nf_{m,n}z^{m}u^{n}.$ (S57)

And, then integration follows as the inverse operation,
$\mathcal{I}(\mathbf{a})=f_{00}$, i.e. the constant term. These operations
along with the algebra define the non-commutative Brillouin torus [23, 53,
54], $(C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z},\partial,\mathcal{I})$, and
form a special case of a spectral triple.

Thus far, this section has only been a formalization of the concepts explained
above. However, expressing the system as a spectral triple allows us to bring
down the hammer of non-commutative geometry. In particular, there has been a
careful formulation of K-theory in the case of spectral triples [53, 25]. For
the particularly simple case of a non-commutative torus, one can write down a
locally computable index formula [25, 22, 23, 54, 53], and compute a Chern
number. We introduce a projection operator,
$\mathbf{p}=1/2(1+\mathrm{sgn}(\epsilon_{F}-\mathbf{h}))$, which defines a
filling of the spectrum below some Fermi level, $\epsilon_{F}$. The first non-
commutative Chern number is then given by[1, 23, 53, 54, 55]

$\displaystyle\textnormal{Ch}_{1}(\mathbf{p})=2\pi\mathcal{I}(\mathbf{p}[\partial_{1}\mathbf{p},\partial_{2}\mathbf{p}]).$
(S58)

Like the normal Chern invariant, this is well defined as long as there is a
finite spectral gap. Here we replicate the key result of [1], by computing
this integral in the representation given by Eq. (S51). where
$\mathcal{I}(\mathbf{a})=1/(2\pi)\int_{\mathbf{S}}d\phi f_{0}(\phi)$. In this
simple case, the integral reduces to

$\displaystyle\mathcal{I}(\mathbf{a})$ $\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\frac{1}{2N}\sum_{-N\leq x\leq
N}f_{0}(\phi+\Theta x)=\mathrm{Tr}_{L}(\pi_{\phi}(\mathbf{a}))$ (S59)

where we have used that $f_{0}(\phi+\Theta
x)=\mathinner{\langle{x}|}\pi_{\phi}(\mathbf{a})\mathinner{|{x}\rangle}$, and
$\mathrm{Tr}_{L}$ is the normalized trace. In this representation

$\displaystyle\textnormal{Ch}_{1}=2\pi
i\mathrm{Tr}_{L}\left(\pi_{\phi}(\mathbf{p}[\partial_{1}\mathbf{p},\partial_{2}\mathbf{p})\right)$
(S60)

Using that,

$\displaystyle\pi_{\phi}(\partial_{1}\mathbf{p})=\partial_{\phi}\pi_{\phi}(\mathbf{p})\
\ \textnormal{and}\ \
\pi_{\phi}(\partial_{2}\mathbf{p})=i\left[X,\pi_{\phi}(\mathbf{p})\right],$

Defining $P_{\phi}=\pi_{\phi}(\mathbf{p})$, the projection operator in the AAH
representation, the Chern number takes on a simple form,

$\displaystyle\textnormal{Ch}_{1}=-2\pi\mathrm{Tr}_{L}\left(P_{\phi}[\partial_{\phi}P_{\phi},[X,P_{\phi}]]\right).$
(S61)

Therefore, the Hofstader and AAH Hamiltonians are generated by the same
element of $C(\mathbf{S})\rtimes_{\alpha}\mathbb{Z}$. This proves that the
topological invariant of the 2D Hofstadter Hamiltonian is inherited by the 1D
AAH model and explains the natural appearance of bulk-boundary correspondence
– the existence of boundary localized states reflecting the bulk topological
invariant. The topological invariant is robust to disorder, and the edge
spectrum is gapless when cycling through $\phi$ [1].

An interesting consequence of this pGF transfer matrix formalism is the clear
connection via the transfer matrix poles between the non-commutative geometry
of the system and the spectral measure. Interestingly, this topological
criteria was also noticed by [3] using a different set of techniques to
analyze the semi-infinite AAH model.

###  VI.5 S.I. Gap-Labeling Theorems

A well studied question arises from this topological criterion. Non-
commutative geometry predicts the existence of gaps in the quasi-periodic
integrated density of states (IDoS) depending on the irrational parameter
$\alpha$ in the AAH Hamiltonian. Given the clear role topology plays in the
dynamics, it was predicted and shown that these gaps in the IDoS form open
sets in the complement of the quasi-periodic spectrum [23, 22, 53].

Quasi-periodic systems can be indexed by patterns generating a ”deterministic”
disorder [25, 54, 55]. In Fig. S2b peaks can be labeled by a coordinate,
$P=\\{p_{i}\\}_{i\in\mathbb{Z}}$, forming a pattern. In the absence of a
Brillouin zone, we consider the pattern as a dynamical system and find its
convex hull - the minimal surface into which it can be embedded, $\Omega$. For
example, for a simple generator such as $G=\cos(\Theta x)$ with
$\Theta/2\pi\notin\mathbb{Q}$ it forms a ellipse. However, we can similarly
find the hull for more complicated patterns upon defining the map,
$f=\\{p_{i}\in P|f(p_{i})=(p_{i+1}-p_{i},p_{i+2}-p_{i+1},\ldots)\in X\\}$
where $X$ is a hyper-cube of edge length defined by the pattern $P$. Although
each element of the pattern is assigned a coordinate in an arbitrarily high
dimensional space, there are only as many linearly independent coordinates as
there are generators of the pattern. For the aforementioned sinusoidal
generator of fixed amplitude, the element $p_{i+1}-p_{i}$ sets the period and
further elements - $p_{i+2}-p_{i+1},\ldots$ \- are linearly dependent on the
first two by a translation. Consequently, it forms the anticipated ellipse in
any dimensional hypercube rather than a higher dimensional surface, see Fig.
S2b. Ergodicity on this minimal embedding implies there exists a trajectory
between any initial approaching (arbitrarily close) any other point on the
surface. On the pattern hull, the notion of gauge invariance for quasiperiodic
eigenstates mentioned above is similar to gauge invariance in a Brillouin
zone, taking
$\mathinner{|{k}\rangle}\rightarrow\mathinner{|{k+\delta}\rangle}$.

Thus, we consider the space of continuous functions on the hull of the pattern
$\mathcal{C}(\Omega)$, the direct analog of a Brillouin zone, and introduce
dynamics by defining the action of pattern translations, $\tau$, on these
functions, defining a so-called $C^{*}$-Algebra,
$\mathcal{C}(\Omega)\rtimes_{\tau}\mathbb{G}$, where $\mathbb{G}$ is the group
generated by translations. Elements of this unital algebra are the non-
commutative analogs for Hamiltonians on the momentum space torus. Adding
information about the on-site Hilbert space - bands in translation invariant
case - and a differential operator extracts sufficient information from the
quasiperiodic system to define topology in the same way as done in
conventional translation invariant band topology. More specifically, one
relies on a generalization of the Atiyah-Singer Index theorem to spectral
triples by Connes and Teleman [24].

As discussed in S.I.VI.4, the unital algebra generated by a quasiperiodic
pattern with real-space translations is a non-commutative n-torus – dimension
corresponding to the number of generators for the pattern and system dynamics.
As a direct consequence, the spectral gaps of quasiperiodic systems as a
function of the incommensurate parameter, $\Theta\in[0,1]$, can be labeled by
integers, see Fig. S2a, i.e. $\\{m+n\Theta|m,n\in\mathbb{Z}\\}\cap[0,N]$ with
N is system size, label the gaps in the IDoS of the AAH Model. By
construction, this labeling is invariant to small disorder as long as the
pattern is well defined and gaps remain open [25, 1, 55].