data points (blue transparent dots, 25 for each value of $\delta$). The
transparent circles partly lay on top of each other because the state of the
simulation was logged at a frequency of 0.001.

Figure S6:  Supplement to maintext fig. 6, where measured data collapse onto a
linear dependence upon rescaling to effective neighbour coupling strength, cf.
maintext eq. 24. The small remaining scatter of the data points around the
black line in maintext fig. 6B is largely given by a $\delta$-dependent offset
as the ordered colours of the data points indicate. This $\delta$-dependent
offset is analytically known for $\epsilon_{n}=0$ by maintext eq. 21.
Subtracting the offset at $\epsilon_{n}=0$, where
$T_{mo}\left(\epsilon_{n}=0\right)=\hat{T}_{mo}\left(\epsilon_{n}=0\right)=\frac{\log
2}{\delta+\sum_{k=1}^{8}\exp\left(\epsilon_{s}\left\langle\mathbf{s},\mathbf{e}_{k}\right\rangle\right)}$
is taken from the analytical expression in maintext eq. 21, reduces the
scatter further. A. All data of the mean-field model, $\beta=1$. Colour code
of cell death rate $\delta$ applies to both panels. B. All data of the IPS
simulations, $\beta=1$. The black line obeys $\qquad\qquad
T_{mo}=T_{mo}\left(\epsilon_{n}^{eff}\right)=(0.0516\pm
0.0001)\underbrace{\frac{\beta}{\beta+\delta}}_{=p_{eq}}\epsilon_{n}+T_{mo}\left(\epsilon_{n}=0\right)$
where values in brackets denote $(\textnormal{mean}\pm\textnormal{std})$ of an
orthogonal distance regression.

Figure S7:  The time of minimal order $T_{mo}$ depends smoothly on the
coupling strength $\epsilon_{s}$ to the global signal, and IPS and mean-field
model agree closely. A. IPS simulation results for mean $T_{mo}$ of 25
repetitions shown as heatmap with contourlines (isotemporales at marked
levels). B. Comparison of time of minimal order as contourlines for IPS model
($T_{mo}$, solid) and mean-field model ($\hat{T}_{mo}$, dashed). Colour code
of the time of minimal order applies to both panels. For all values
$\epsilon_{s}\in\left\\{0.1,\,0.2,\,0.5,\,1.0,\,2.0,\,3.0,\,4.0,\,5.0\right\\}$
the tissue polarity pattern reorganises, both in the IPS simulations and in
the mean-field model. The time of minimal order decreases for higher coupling
strength $\epsilon_{s}$ to the global signal, as predicted. For
$\epsilon_{s}=0.0$, there is no influence of a global signal and no dominant
polarisation direction develops (grey shaded areas in A,B). Note that by the
choice of our initial condition, cf. maintext eq. 10, no dominant polarisation
direction exists for $\epsilon_{s}=0.0$ in the initial state either. Other
parameters $\epsilon_{n}=1.0$, $\beta=1.0$.


# Qubit from the classical collision entropy

Kelvin Onggadinata<EMAIL_ADDRESS>Centre for Quantum
Technologies, National University of Singapore, 3 Science Drive 2, 117543
Singapore, Singapore  Department of Physics, National University of Singapore,
3 Science Drive 2, 117543 Singapore, Singapore  Paweł Kurzyński  Institute of
Spintronics and Quantum Information, Faculty of Physics, Adam Mickiewicz
University, Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland  Centre for
Quantum Technologies, National University of Singapore, 3 Science Drive 2,
117543 Singapore, Singapore  Dagomir Kaszlikowski<EMAIL_ADDRESS>Centre for
Quantum Technologies, National University of Singapore, 3 Science Drive 2,
117543 Singapore, Singapore  Department of Physics, National University of
Singapore, 3 Science Drive 2, 117543 Singapore, Singapore

###### Abstract

An orthodox formulation of quantum mechanics relies on a set of postulates in
Hilbert space supplemented with rules to connect it with classical mechanics
such as quantisation techniques, correspondence principle, etc. Here we deduce
a qubit and its dynamics straightforwardly from a discrete deterministic
dynamics and conservation of the classical collision entropy. No Hilbert space
is required although it can be inferred from this approach if necessary.

##  I Introduction

Soon after its inception, physicists encoded quantum theory in complex Hilbert
space together with a set of postulates to link it with experiment. Departure
from classical physics was bridged with a set of techniques and prescriptions
of how to ‘quantise’ familiar classical objects such as Hamiltonian,
Lagrangian, momentum, position, etc. Although this approach works ‘good
enough’, it lacks clear physical meaning unlike, for instance, general
relativity theory, entirely derived from two experimentally falsifiable
assumptions: relativity and equivalence principle. One’s desire for more
intuitive and simpler postulates to construct quantum theory is understandable
in this context.

In the last few decades, some researchers found alternative axioms for quantum
theory. Hardy [1] proposed five axioms which later expanded and framed as
generalized probability theory (GPT) by Barrett [2]. All these alternative
formulations of quanta are forced to use quasi-probability theory (also known
as signed measures [3] or, colloquially, as negative probabilities) to account
for observed randomness of microscopic phenomena [4]. Note that GPT is
constructed such that one never has to assign negative probabilities to
measurement outcomes. This eliminates any ontological discussions about the
meaning of negative probabilities, a practice we follow in this paper.

Quasi-probabilities are almost as old as quanta itself thanks to Wigner who
introduced them in 1932 [5]. Since then the idea has been discussed by many
authors in diverse contexts [3, 6, 7, 8, 9, 10]. The most recent developments
in the field suggest that quasi-probabilities could be viewed as a fundamental
resource in quantum non-locality [11, 12, 13, 14, 15] and quantum computation
[16, 17, 18, 19, 20, 21], and used extensively in quantum optics [22, 23, 24].

Here we propose a simple information-theoretic postulate from which we can
derive a discrete four dimensional quasi-stochastic system equivalent to a
qubit and its dynamics. Curiously, with no further restrictions, our postulate
allows universal-NOT operation but we can get rid of it if we assume
continuity of reversible quasi-stochastic processes. One of the interesting
aspects of this approach is that we can reconstruct discrete quantum system
without invoking Hilbert space at all. The information-theoretic flavour of
our proposal falls closely within the proximity of information causality and
its subsequent generalisations [25, 26, 27].

##  II Deterministic dynamics and Renyi entropies

Consider a particle on a one-dimensional lattice with $d$ vertices (enumerated
by $i=1,2,\dots,d$) and periodic boundary conditions. Or, if you prefer, you
can see it as an abstract dynamical system whose states are single-vertex
occupations, i.e., bit strings like this $(000\dots 01000\dots 0)$, where $1$
denotes the $i$-th vertex occupation.

The simplest possible discrete deterministic dynamics is when the particle
starts at some vertex $i$ and in every step hops to the next vertex:
$i\rightarrow i+1\rightarrow i+2\rightarrow\dots$. We represent the particle’s
state as a $d$-dimensional basis vector $\mathbf{e}$ (only one entry equals to
$1$ with others equal to $0$) and its dynamics as a permutation matrix $\Pi$.
After $k$ steps, the initial state $\mathbf{e}$ evolves to
$\mathbf{e}^{(k)}=\Pi^{k}\mathbf{e}$.

If the initial conditions are uncertain, $\mathbf{e}$ becomes a probability
vector $\mathbf{p}=[p_{1},p_{2},\dots,p_{d}]^{T}$ where $p_{i}$ ($p_{i}\geq 0$
and $\sum_{i=1}^{d}p_{i}=1$) is the probability of finding the particle in the
vertex $i$. The dynamics stays the same, i.e., after $k$ steps the state
changes to $\mathbf{p}^{(k)}=\Pi^{k}\mathbf{p}$. It is rather obvious to
observe a simple information-theoretic property of such dynamics:

$\sum_{i=1}^{d}p_{i}^{\alpha}=\sum_{i=1}^{d}\left([\Pi^{k}\mathbf{p}]_{i}\right)^{\alpha}$

for every $k$ and $\alpha\geq 0$, i.e., $\sum_{i=1}^{d}p_{i}^{\alpha}$ is
constant for any initial probability distribution $\mathbf{p}$. More
generality we can express this with Renyi-$\alpha$ entropies,
$H_{\alpha}(\mathbf{p})$, as

$H_{\alpha}(\Pi^{k}\mathbf{p})=H_{\alpha}(\mathbf{p})\,,$ (1)

where $H_{\alpha}$ is

$H_{\alpha}(\mathbf{p})=\frac{1}{1-\alpha}\log{\left(\sum_{i=1}^{d}p_{i}^{\alpha}\right)}\,.$
(2)

In fact, any permutation conserves all the Renyi entropies. Introducing Renyi
entropies may look unnecessary but it does give us an option to study Shannon
entropy for $\alpha=1$. It is physically obvious that Shannon entropy is
constant for any initial state $\mathbf{p}$ and deterministic dynamics.

We now ask the fundamental question in this paper: Can we extend deterministic
dynamics in our model to some other dynamics, not necessarily deterministic,
such that some Renyi entropies remain constant for any given initial state
$\mathbf{p}$?

A ‘yes’ answer would mean there is a family of dynamics $S$ and states such
that (i) $S\mathbf{p}$ remains a proper positive probability distribution and,
(ii) there is a range of $\alpha$ where

$H_{\alpha}(S\mathbf{p})=H_{\alpha}(\mathbf{p})$ (3)

for any $\mathbf{p}$ for which $S\mathbf{p}$ is a proper positive probability
distribution.

We can already expect certain features of this extension. First, any extended
dynamics $S$ must be a $d\times d$ matrix whose rows sum up to one or else the
probabilities $S\mathbf{p}$ would not be normalized. However, we cannot
guarantee that $S$’s elements are all positive, making $S$ at least a quasi-
stochastic matrix if not a quasi-bistochastic one.

To sum up, at this moment, we have a well defined, physically motivated
mathematical problem. In the next section, we provide a solution of deep
physical significance: we recover qubit and its dynamics without Hilbert
space.

##  III Generalised dynamics and Renyi entropies

The problem formulated in the previous section is difficult to solve. However,
we found an important solution for the Renyi entropy with $\alpha=2$. This
entropy is called collision entropy in the literature and it reads:

$H_{2}(\mathbf{p})=-\log{|\mathbf{p}|^{2}}\,,$ (4)

where $|\mathbf{p}|=\sqrt{\sum_{i=1}^{d}p_{i}^{2}}$, i.e., it is the geometric
length of a $d$-dimensional vector $\mathbf{p}$.

If you want to satisfy Eq. (3), i.e., keep the collision entropy invariant
under a generalised dynamics $S$, you must have

$|S\mathbf{p}|^{2}=|\mathbf{p}|^{2}\,.$

This is possible only if transposition of the matrix $S$ is its own inverse
because of a trivial observation

$|S\mathbf{p}|^{2}=(S\mathbf{p})\cdot(S\mathbf{p})=\mathbf{p}\cdot(S^{T}S\mathbf{p})=\mathbf{p}\cdot\mathbf{p}=|\mathbf{p}|^{2}$
$\Rightarrow\quad S^{T}=S^{-1}$

In other words, orthogonal matrices produce the new generalized dynamics.

Next, we need to find the minimal dimension $d$ where this is possible. It
helps to note that if $S$ is a valid dynamics so is $S^{T}$ and thus $S$ must
have columns and rows summing up to one if we want to keep $S\mathbf{p}$ a
proper probability vector. This is only possible if some of the elements in
rows and columns are negative because the off-diagonal terms in $SS^{T}$ must
be equal to zero. Thus $S$ must be a quasi-bistochastic matrix.

For $d=2$ the most general quasi-bistochastic matrix reads

$S=\begin{bmatrix}q&1-q\\\ 1-q&q\end{bmatrix}\,,$ (5)

where $q$ is an arbitrary real number. It is easy to see that orthogonality is
only true for $q=0$ or $q=1$, which makes it a permutation. Hence, no new
dynamics is observed for $d=2$.

The situation changes for $d=3$. We start with the generalised Birkhoff-von
Neumann decomposition [15] of quasi-bistochastic matrices

$S=\sum_{k=0}^{2}q_{k}\Pi^{k}+\sum_{k=0}^{2}r_{k}\Pi^{k}R\,,$ (6)

where $\Pi$ is a permutation matrix such that $123\rightarrow 312$, $R$
permutes $123\to 132$ and $\sum_{k=0}^{2}(q_{k}+r_{k})=1$ ($q_{k},r_{k}$ can
be negative). It is easy to see that $\Pi^{k}$ and $\Pi^{k}R$ cover all $6$
permutations of the string $123$. For convenience we put
$Q=\sum_{k=0}^{2}q_{k}\Pi^{k}$ and $Q^{\prime}=\sum_{k=0}^{2}r_{k}\Pi^{k}$ so
that

$S=Q+Q^{\prime}R\,.$ (7)

Orthogonality mean $SS^{T}=\mathbb{1}$ and thus

$\displaystyle\mathbb{1}$
$\displaystyle=QQ^{T}+Q^{\prime}(Q^{\prime})^{T}+QR(Q^{\prime})^{T}+Q^{\prime}RQ^{T}$
(8) $\displaystyle=QQ^{T}+Q^{\prime}(Q^{\prime})^{T}+2QQ^{\prime}R\,,$

where the last equality is obtained from using the fact that
$R\Pi^{-k}=\Pi^{k}R$. This equation can be true if and only if (i)
$QQ^{\prime}=0$ or (ii) $QQ^{\prime}=R$. However, the latter is not possible
because determinants of $Q,Q^{\prime}$ are +1 whereas the $R$’s determinant is
$-1$. We are left with two distinct possibilities: either $Q$ or $Q^{\prime}$
is zero, giving us $S_{+}=Q$ or $S_{-}=Q^{\prime}R$. The subscript $\pm$ is to
indicate the respective solution has determinant $\pm 1$. Note that $S_{-}$
corresponds to a discontinuous dynamics that is not physical.

In Appendix A we show that $S_{+}$ has a unique form that reads

$S_{+}(\phi)=q_{0}I+q_{1}\Pi+q_{2}\Pi^{2}\,,$ (9)

where
$q_{k}(\phi)=\frac{1}{3}\left(1+2\text{Re}{\left(\omega^{k}e^{i\phi}\right)}\right)$,
$\omega=e^{i2\pi/3}$ is the cubic root of unity. Note that for any $\phi$ only
one of the quasi-probabilities is negative and because
$\sum_{k=0}^{2}\omega^{k}=0$ we have $\sum_{k=0}^{2}q_{k}=1$. As such, this
non-deterministic dynamics is reversible as the dynamics it generalizes.

What we need to fix now is the domain of the state space to which
$S_{+}(\phi)$ is a valid transformation. As $\mathbf{p}$ describes a
probability distribution on the lattice it must stay positive for any
$S_{+}(\phi)$. Again, the proof is in the Appendix A and we give here the
solution:

$p_{k}(\theta)=\frac{1}{3}(1+t\hat{\mathbf{a}}_{k}\cdot[\sin\theta,\cos\theta])\,,\quad
k=0,1,2\,,$ (10)

where $0\leq t\leq 1$, $0\leq\theta\leq 2\pi$, and $\hat{\mathbf{a}}_{k}$’s
are 2-dimensional real unit vectors such that
$\sum_{k=0}^{2}\hat{\mathbf{a}}_{k}=0$.

We arrived at a generalised dynamics for $d=3$, conserving the collision
entropy of any initial positive probability distribution $\mathbf{p}$. This is
a quasi-bistochastic dynamics that preserves positivity of $\mathbf{p}$ and
thus resembles quasi-probability representations of quantum theory discussed
in [28]. Indeed, this dynamics and the set of admissible probability
distributions is equivalent to rotations around the axis $\hat{z}$ of qubit
states with the Bloch vector $\mathbf{s}=t[\sin\theta,\cos\theta,0]$ confined
to the $xy$-plane.

Dropping the continuity of $S$, we end up with a bigger dynamical system that
still describes a qubit but with a larger dynamics that contains
experimentally impossible operations. They correspond to reflections of
qubit’s Bloch vector (including forbidden universal-NOT gate) if we map them
to two-dimensional Hilbert space.

Of course, laboratory measurements on a qubit give only two outcomes (qubit in
the state $|0\rangle$ or $|1\rangle$ along the measurement direction), so we
need to show how to interpret the lattice probability distribution
$\mathbf{p}$. We easily recover the measurement probabilities along an
arbitrary direction on the Bloch sphere’s equator $\hat{\mathbf{m}}$ if we use
the over-completeness of the vectors $\hat{\mathbf{a}}_{k}$,
$\sum_{k=0}^{2}\hat{\mathbf{a}}_{k}\hat{\mathbf{a}}_{k}=\frac{3}{2}I$ (here
$\mathbf{a}\mathbf{b}$ denotes a dyadic product of two vectors). We have

$\displaystyle p(\pm|\hat{\mathbf{m}})$
$\displaystyle=\frac{1}{2}\left(1\pm\hat{\mathbf{m}}\cdot\mathbf{s}\right)$
(11a)
$\displaystyle=\frac{1}{2}\left(\mathbf{1}\pm\mathbf{v}\right)\cdot\mathbf{p}$
(11b)

where $\mathbf{1}$ is vectors of all ones and
$\mathbf{v}=[\hat{\mathbf{m}}\cdot\hat{\mathbf{a}}_{0},\hat{\mathbf{m}}\cdot\hat{\mathbf{a}}_{1},\hat{\mathbf{m}}\cdot\hat{\mathbf{a}}_{2}]$.
We define $\mathbf{e}(\pm|\hat{\mathbf{m}})=\frac{1}{2}(1\pm\mathbf{v})$ as
the effect corresponding to the measurement along $\hat{\mathbf{m}}$ with
outcome $\pm$. Effects formalism is not the primary concern of this paper but
there is extensive literature on this topic [29] the reader can consult.

In Appendix B we show how to extend the continuous dynamics found for $d=3$ to
$d=4$. The significance of this extension is a full reconstruction of qubit
states and their physical transformations.

A natural question at this point is if we can derive a two-qubit dynamics and
thus, using a set of two-qubit universal gates, dynamics of any
$D$-dimensional quantum system. We already know such collision entropy
preserving quasi-bistochastic dynamics equivalent to two qubits — it can be
constructed using a SIC-POVM frame [30], a mapping from a four-dimensional
Hilbert space to quasi-probabilistic space. This means we can at worst get a
larger class of systems, some of which may not correspond to two qubits. What
would such systems be? These are for now open questions we will address in the
future work.

##  IV Conclusions

The gist of this paper is that one can deduce the existence of a qubit
together with its full dynamics from a deterministic (reversible) dynamics of
a particle hopping on a one-dimensional lattice with $4$ vertices if one
postulates the collision entropy conservation. Hilbert space is not needed but
you can recover it if you need to.

Using this information-theoretic postulate we get qubit’s dynamics as an
orthogonal quasi-bistochastic continuous processes of a particle hopping on a
one-dimensional 4-vertex lattice. The particle’s states are restricted to non-
negative probability distributions that can be uniquely mapped to qubit’s
measurement probabilities.

Our results can be positioned in the ongoing research to derive quantum
mechanics from some basic, information-theoretic principles without invoking
orthodox Hilbert space axioms.

Some open questions:

  1. 1.

It is not clear at the moment what the physical significance of the collision
entropy is. Technically, it enforces orthogonality of the quasi-bistochastic
dynamics $S$ and thus its reversibility: $S^{-1}=S^{T}$. However, you can
imagine a more general reversibility where $S^{-1}\neq S^{T}$. You can also
notice that the $S$’s orthogonality is equivalent to the conservation of
qubit’s purity by unitary dynamics.

  2. 2.

How to recover a dissipative qubit dynamics?

  3. 3.

Can we get some other, perhaps post-quantum dynamics (for instance, PR-boxes
[31]) conserving Renyi entropies for other $\alpha$?

  4. 4.

Can we extend this approach to continuous variables systems?

After finishing this work, we learnt about a paper by Brandenburger et al.
[32] where the authors also use Renyi entropies to connect qubit with quasi-
probability distributions via quantum uncertainty principle. How Brandenburger
et al.’s results are related to ours requires in-depth study.

## Acknowledgements

P.K. is supported by the Polish National Science Centre (NCN) under the
Maestro Grant no. DEC-2019/34/A/ST2/00081. D.K. is supported by the National
Research Foundation, Singapore, and A*Star under the CQT Bridging Grant.

##  Appendix A Derivation of dynamics and states for $d=3$

Here, we show the parameterization of the orthogonal quasi-bistochastic matrix
for $d=3$. There are two forms of the solution. The first with $+1$
determinant has the form

$S_{+}=\begin{bmatrix}q_{0}&q_{2}&q_{1}\\\ q_{1}&q_{0}&q_{2}\\\
q_{2}&q_{1}&q_{0}\end{bmatrix}$ (12)

with the constraints

$\displaystyle q_{0}+q_{1}+q_{2}=1\,,$ (13a) $\displaystyle
q_{0}^{2}+q_{1}^{2}+q_{2}^{2}=1\,,$ (13b) $\displaystyle
q_{0}q_{1}+q_{0}q_{2}+q_{1}q_{2}=0\,.$ (13c)

Parameterizing the solution space based on the constraints above means that we
are solving the problem of intersection between 3-dimensional hypersphere with
a hyperplane of the same dimension. The general procedure for solving this
problem is presented in Appendix C. The solution for the parameter space is

$\displaystyle q_{0}(\phi)$ $\displaystyle=\frac{1}{3}+\frac{2}{3}\cos\phi\,,$
(14a) $\displaystyle q_{1}(\phi)$
$\displaystyle=\frac{1}{3}-\frac{2}{3}\sin(\frac{\pi}{6}+\phi)\,,$ (14b)
$\displaystyle q_{2}(\phi)$
$\displaystyle=\frac{1}{3}-\frac{2}{3}\sin(\frac{\pi}{6}-\phi)\,.$ (14c)

The above can be compactly expressed as
$q_{k}=q_{k}(\phi)=\frac{1}{3}\left[1+2\text{Re}(\omega^{k}e^{i\phi})\right]$,
$k=0,1,2$, where $\omega$ is the cube root of unity. These quantities can take
values from $-\frac{1}{3}\leq q_{k}\leq 1$ and clearly satisfies unit sum for
all $\phi$. $S_{+}=S_{+}(\phi)$ here forms the group of quasi-bistochastic
SO(3) matrices.

The other solution is orthogonal quasi-bistochastic matrix with $-1$
determinant of the form

$S_{-}=\begin{bmatrix}q_{0}&q_{1}&q_{2}\\\ q_{1}&q_{2}&q_{0}\\\
q_{2}&q_{0}&q_{1}\end{bmatrix}$ (15)

with similar constraint as in Eq. (13). Consequently, the solution space can
be parameterize similarly as in Eq. (14). Contrary to the previous case,
$S_{-}$ here is not continuous and do not form a group.

The next thing that we will find is the domain of the state space, where we
have specifically chose to be nonnegative and will behave consistently under
the $S(\phi)$ above. The reason we can do this is due to the self-duality
relation between the state and effect [33]. This choice of construction will
not change the behavior of the system. With this in mind, let us now construct
the state space $\mathcal{S}\subset\mathbb{R}^{3}_{+}$ based on the quantum
dynamic that we just obtained.

The problem of finding the state space can be stated as follows. Suppose that
we have a quasi-bistochastic matrix $S_{+}$ from Eq. (12) with $q_{k}$ given
by Eq. (14). The goal is then to find the domain
$\mathcal{S}\subset\mathbb{R}^{3}_{+}$ where $\mathbf{p}\in\mathcal{S}$
satisfies

$S_{+}\mathbf{p}=\mathbf{p}^{\prime}\in\mathcal{S}\quad\forall\,\mathbf{p},\,\forall\,\phi\,.$
(16)

Since $\mathbf{p}=[p_{0},p_{1},p_{2}]^{T}$ is a probability distribution, it
is then constraint to have unit sum, $\sum_{k}p_{k}=1$. Since the matrix can
take negative values, the state space $\mathcal{S}$ is then a subset of the
probability simplex.

To solve this, we bring up the fact that $S_{+}$ is an orthogonal matrix so it
leaves the squared-norm of the state vector invariant after the
transformation, i.e., $|S_{+}\mathbf{p}|^{2}=|\mathbf{p}|^{2}$. A typical
probability vector has squared-norm that is less or equal to 1 but we know
that $S_{+}$ can potentially bring probability vector with squared-norm of 1
into negative probability distribution. Therefore, it implies that there exist
an upper bound $K<1$ for the squared-norm of $\mathbf{p}\in\mathcal{S}$. To
find this bound, we consider the parameterized state:

$\displaystyle\mathbf{r}$ $\displaystyle=$
$\displaystyle\lambda\begin{bmatrix}1\\\ 0\\\
0\end{bmatrix}+(1-\lambda)\begin{bmatrix}\frac{1}{3}\\\ \frac{1}{3}\\\
\frac{1}{3}\end{bmatrix}$ (17) $\displaystyle=$
$\displaystyle\frac{1}{3}\begin{bmatrix}1+2\lambda\\\ 1-\lambda\\\
1-\lambda\end{bmatrix}\,,\quad 0<\lambda<1\,.$

The squared-norm of $\mathbf{r}$ can be easily calculated to be
$(1+2\lambda^{2})/3$. The goal here is to find the largest $\lambda$ such that

$S_{+}\mathbf{r}\geq 0\quad\forall\,\phi\,.$ (18)

The above positivity criteria then implies that

$\displaystyle\lambda$ $\displaystyle\geq\frac{1}{1-3q_{0}}\,,$ (19a)
$\displaystyle\lambda$ $\displaystyle\geq\frac{1}{1-3q_{1}}\,,$ (19b)
$\displaystyle\lambda$ $\displaystyle\geq\frac{1}{1-3q_{2}}\,.$ (19c)

Since it needs to be in the range $0<\lambda<1$ and works for all $\phi$, one
can easily deduce that in the end we have $\lambda\geq\frac{1}{2}$. Therefore,
we can infer that the squared-norm of $\mathbf{p}$ takes the range
$\frac{1}{3}\leq\sum_{k}p_{k}^{2}\leq K=\frac{1}{2}$. The extremal states
(pure states) of $\mathcal{S}$ are $\mathbf{p}\geq 0$, $\sum_{k}p_{k}=1$, with
$|\mathbf{p}|^{2}=\frac{1}{2}$.

From here we can parameterize $\mathbf{p}$ as

$p_{k}=\frac{1}{3}(1+t\hat{\mathbf{a}}_{k}\cdot[\sin\theta,\cos\theta])\,,\quad
k=0,1,2\,,$ (20)

where $0\leq t\leq 1$, $0\leq\theta\leq 2\pi$, and $\hat{\mathbf{a}}_{k}$’s
are 2-dimensional real unit vector and $\sum_{k}\hat{\mathbf{a}}_{k}=0$. If we
choose $\hat{\mathbf{a}}_{0}=[0,1]$, then it is natural to have
$\hat{\mathbf{a}}_{1}=[\frac{\sqrt{3}}{2},-\frac{1}{2}]$ and
$\hat{\mathbf{a}}_{2}=[-\frac{\sqrt{3}}{2},-\frac{1}{2}]$. We can re-
parameterise $[t\sin\theta,t\cos\theta]\to[x,y]$ with the condition
$x^{2}+y^{2}\leq 1$. It can also be shown easily that $\mathbf{p}$ above is
closed under transformation of $S_{-}$, i.e., $S_{-}\mathbf{p}\in\mathcal{S}$.

With the hindsight of Hilbert space quantum mechanics, we already know the
degrees of freedom $[x,y]$ corresponds to the components of the Bloch sphere
in unit circle. In fact, with the choice of $\hat{\mathbf{a}}_{k}$ above, we
recovered the trine quasi-probability representation of a qubit in the
$xy$-plane [30].

As for the measurement space, one only needs to find the real vector
$\mathbf{m}$ that satisfies

$0\leq\mathbf{m}\cdot\mathbf{p}\leq
1\quad\forall\,\mathbf{p}\in\mathcal{S}\,.$ (21)

The shrinking of the state space from the classical simplex and the deformity
of the geometry allows the effect space to be larger than the classical effect
space, and hence take on negative values. This trade-off is known as (strong)
self-duality [33].

##  Appendix B Derivation of dynamics and states for $d=4$

The construction of the extended theory in $d=4$ can be done in a similar
manner as how it is done in $d=3$. However, finding the general form for the
quasi-bistochastic SO(4) group using the method above can be quite tedious and
complicated. Instead, we will construct it through some basic assumptions
about its properties. First we note that the quasi-bistochastic SO(3) matrix
is a subgroup of the quasi-bistochastic SO(4) matrices. Hence, there exists 4
elementary rotation matrices

$\displaystyle R_{1}$ $\displaystyle=\begin{bmatrix}1&0&0&0\\\
0&q_{0}&q_{2}&q_{1}\\\ 0&q_{1}&q_{0}&q_{2}\\\
0&q_{2}&q_{1}&q_{0}\end{bmatrix}\,,$ (22a) $\displaystyle R_{2}$
$\displaystyle=\begin{bmatrix}q_{0}&0&q_{2}&q_{1}\\\ 0&1&0&0\\\
q_{1}&0&q_{0}&q_{2}\\\ q_{2}&0&q_{1}&q_{0}\end{bmatrix}\,,$ (22b)
$\displaystyle R_{3}$ $\displaystyle=\begin{bmatrix}q_{0}&q_{2}&0&q_{1}\\\
q_{1}&q_{0}&0&q_{2}\\\ 0&0&1&0\\\ q_{2}&q_{1}&0&q_{0}\end{bmatrix}\,,$ (22c)
$\displaystyle R_{4}$ $\displaystyle=\begin{bmatrix}q_{0}&q_{2}&q_{1}&0\\\
q_{1}&q_{0}&q_{2}&0\\\ q_{2}&q_{1}&q_{0}&0\\\ 0&0&0&1\end{bmatrix}\,,$ (22d)

with $R_{k}=R_{k}(\phi)$ and $q_{k}=q_{k}(\phi)$ exactly having the same
expression in Eq. (14). Then, the general quasi-bistochastic SO(4) matrices
$S$ can be written in terms of

$S=R_{1}(\phi_{1})R_{2}(\phi_{2})R_{3}(\phi_{3})R_{4}(\phi_{4})\,,$ (23)

where $\phi_{k}$’s are real parameters that can be found for any quasi-
bistochastic SO(4) matrices.

We employ the same method to find the state space and find that extremal
states have squared-norm $|\mathbf{p}|^{2}=\frac{1}{3}$. Hence, the
parameterization of the extremal state follows a similar form as the $d=3$
case:

$p_{k}=\frac{1}{4}\left(1+\hat{\mathbf{b}}_{k}\cdot[x,y,z]\right)\,,\quad
k=0,1,2,3\,,$ (24)

where we have the constraint $x^{2}+y^{2}+z^{2}\leq 1$ and
$\hat{\mathbf{b}}_{k}$’s are 3-dimensional real unit vector that satisfy
$\sum_{k}\hat{\mathbf{b}}_{k}=0$. As a generalization from the $d=3$ case that
has vectors of equilateral triangle, we then can have $\hat{\mathbf{b}}_{k}$’s
to be vectors of tetrahedron:

$\displaystyle\hat{\mathbf{b}}_{0}$ $\displaystyle=[0,0,1]\,,$ (25a)
$\displaystyle\hat{\mathbf{b}}_{1}$
$\displaystyle=\left[\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right]\,,$ (25b)
$\displaystyle\hat{\mathbf{b}}_{2}$
$\displaystyle=\left[-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right]\,,$
(25c) $\displaystyle\hat{\mathbf{b}}_{3}$
$\displaystyle=\left[-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right]\,.$
(25d)

We have recovered the full Bloch vectors $[x,y,z]$ and hence the most
elementary system in discrete quantum system — the qubit. In fact, this
quasiprobability representation corresponds to the frame representation with
SIC-POVM frames [30].

Lastly, the effect space is also constructed in the similar manner as the
previous section.

##  Appendix C Solution to the intersection between $n$-dimensional
hypersphere and hyperplane

Here, we will show how to obtain the solution to the intersection of
$n$-dimensional hyperplane and $n$-dimensional hypersphere with unit radius.
This problem can be formulated as finding the solution space of
$\mathbf{a}=[a_{1},a_{2},\dots,a_{n}]\in\mathbb{R}^{n}$ given that it
satisfies two equations:

$\displaystyle\sum_{i}a_{i}$ $\displaystyle=a_{1}+a_{2}+\dots+a_{n}=1\,,$
(26a) $\displaystyle\sum_{i}a_{i}^{2}$
$\displaystyle=a_{1}^{2}+a_{2}^{2}+\dots+a_{n}^{2}=1\,.$ (26b)

Before going on how to obtain the solution space, let us first learn how to
parameterize the solution for $n$-dimensional hypersphere with radius $r$:

$b_{1}^{2}+b_{2}^{2}+\dots+b_{n}^{2}=r^{2}\,.$ (27)

The trick for this is to iteratively reduce the problem into a 2-dimensional
sphere equation, which we already know how to parameterize. Let
$x_{1}^{2}=b_{1}^{2}$, $y_{1}^{2}=b_{2}^{2}+b_{3}^{2}+\dots+b_{n}^{2}$,
$r_{1}=r$ and we have reduced the Eq. (27) into equation of a circle:

$x_{1}^{2}+y_{1}^{2}=r_{1}^{2}\,.$ (28)

Therefore, the solution to this can be parameterized as

$x_{1}=r_{1}\cos t_{1}\,,\quad y_{1}=r_{1}\sin t_{1}\,.$ (29)

Then, we can repeat the same thing again for

$y_{1}^{2}=b_{2}^{2}+b_{3}^{2}+\dots+b_{n}^{2}=r_{1}^{2}\sin^{2}t_{1}\,.$ (30)

Letting $x^{2}_{2}=b_{2}^{2}$, $y_{2}^{2}=b_{3}^{2}+\dots+b_{n}^{2}$,
$r_{2}^{2}=r_{1}^{2}\sin^{2}t_{1}$, we obtain another level of parameter

$x_{2}=r_{2}\cos t_{2}\,,\quad y_{2}=r_{2}\sin t_{2}\,.$ (31)

Doing this $n-1$ times will resolve the parameterization problem.

The following describes the method to solve the intersection problem. Suppose
that $U=[\mathbf{u}_{1},\mathbf{u}_{2},\dots,\mathbf{u}_{n}]$ is an orthogonal
matrix with $\\{\mathbf{u}_{1},\mathbf{u}_{2},\dots,\mathbf{u}_{n}\\}$ forming
an orthonormal basis with $\mathbf{u}_{n}=(1,1,\dots,1)/\sqrt{n}$. We then can
write

$\mathbf{a}=U\mathbf{b}=\begin{bmatrix}\uparrow&\uparrow&&\uparrow\\\
\mathbf{u}_{1}&\mathbf{u}_{2}&\dots&\mathbf{u}_{n}\\\
\downarrow&\downarrow&&\downarrow\end{bmatrix}\begin{bmatrix}b_{1}\\\ b_{2}\\\
\vdots\\\ b_{n}\end{bmatrix}\,.$ (32)

From the above expression, we have the following two equations

$\begin{cases}\frac{1}{\sqrt{n}}=\mathbf{u}_{n}^{T}\mathbf{a}=b_{n}\,,\\\
b_{1}^{2}+b_{2}^{2}+\dots+b_{n-1}^{2}=\sqrt{1-\frac{1}{n}}\,.\end{cases}$ (33)

The second equation becomes a problem of $(n-1)$-dimensional hypercube with
radius $r=1-\frac{1}{n}$. The solution can be parameterized using the
technique discussed above. Upon parameterizing $\mathbf{b}$, the form of
$\mathbf{a}$ in Eq. (32) immediately satisfies Eq. (26)

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# The Dry Ten Martini Problem at Criticality

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 apply recently developed methods for the construction of quasi-periodic
transfer matrices to the Dry Ten Martini problem for the critical almost-
Mathieu Operator, also known as the Aubry-Andre-Harper (AAH) model.

##  I Introduction

The almost-Mathieu operator,

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

has presented a rich playground for deep mathematical questions about linear
operators. While many question regarding this deceptively simple linear
operator have been answered [1, 2, 3, 4, 5, 6, 7, 8], the critical point,
$V=t$, is difficult to access via existing analytic methods. Of particular
interest is the stability of the model’s topological properties, namely
whether or not the spectral gaps labeled by the gap-labeling theorem [9] form
open sets in the almost-Mathieu spectrum. This was formally posed as the “dry
ten martini problem”:

Dry Ten Martini Problem. Consider an energy in the spectrum of the critical
almost-Mathieu operator $E\in\Sigma$, satisfying
$\hat{H}\mathinner{|{\psi}\rangle}=E\mathinner{|{\psi}\rangle}$ with $\hat{H}$
as in Eq. (1) and $V=1$. If the integrated density of states, $N(E)=m\alpha+n$
with $m,n\in\mathbb{Z}$ and $\Theta=2\pi\alpha\in$, then E belongs to the
boundary of a component of $\mathbb{R}-\Sigma$.

Remark: This is equivalent to saying the compliment of the spectrum,
$\mathbb{R}-\Sigma$, is composed of open sets for gaps labeled by the gap-
labeling theorem.

This problem has been tackled for quasi-periodic models without self-duality
[10] and proven for the absolutely continuous and pure-point like regimes of
the spectrum [11, 12]. The self-dual version was an open problem [13, 11, 10]
until recently (during the writing of this manuscript). Ref. [2] proved the
purely the critical almost-Mathieu operator is purely singularly continuous
and thus has no eigenvalues. This directly implies all spectral gaps inherited
from rational approximates remain open in the irrational limit.

We present here an alternative proof for the above statement for the critical
almost-Mathieu operator in Section IV. Unlike, Ref. [2] and others, which
approach the problem in the irrational limit, we take advantage of the
recently formulated rational transfer matrix approximates [14, 15] to
construct the irrational almost-Mathieu transfer matrix from a continued
fraction sequence of rational approximates. These approximates are taken in
the higher dimensional parent Hamiltonian representation for the almost-
Mathieu operator:

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

This preserves the topological invariants associated with the quasi-periodic
patter [5]. We choose a chiral gauge, inspired by [13], and substitute the
continued fraction sequence for $\Theta$,

$\displaystyle\mathcal{H}_{ch}=\sum_{x,\delta_{y}}2\sin{(\frac{\Theta}{2}x+\delta_{y})}\hat{c}^{\dagger}_{x+1,\delta_{y}}\hat{c}_{x,\delta_{y}}+2\sin{(\frac{\Theta}{2}x+\delta_{y})}\hat{c}_{x+1,\delta_{y}}\hat{c}^{\dagger}_{x,\delta_{y}}.$
(3)

Then, leveraging the bulk-boundary correspondence of the topologically non-
trivial rational approximates and Lemma 1 (defined below), we obtain the
desired result.

To this end, section II defines our rational approximate transfer matrices
(TMs) in terms of projected Green’s functions (pGFs). Section II.1 [14, 15].
Section II.2 proves Lemma 1 linking open sets to pGF zeros. Section III.1
applies the chiral gauge transformation, Eq. (3), and proves the convergence
of the rational transfer matrix sequence via pGF convergence. Section IV puts
everything together in the context of the dry ten martini problem.

Figure 1: Illustration of chiral gauge transformation. Note the increased unit
cell increases flux per plaquette to $2\Theta$. Hamiltonian in Eq. (3)
obtained by Fourier Transforming back.

##  II Rational Transfer Matrix Sequence

Quasi-periodic systems can be readily analyzed with sequences of rational
approximates [14]. These are sequences of linear operators whose quasi-
periodic parameter, $\alpha\notin\mathbb{Q}$ is replaced by a rational
parameter $\frac{p_{n}}{q_{n}}\in\mathbb{Q}$ such that
$\lim_{n\rightarrow\infty}\frac{p_{n}}{q_{n}}=\alpha$. The fastest converging
sequence is the continued fraction approximation with the n-th approximate
defined by

$\frac{p_{n}}{q_{n}}=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{\ddots+\cfrac{1}{a_{n}}}}.$

The golden mean and $\sqrt{2}$ are Diophantine numbers and their continued
fraction approximation converges with at worst
$|\alpha-\frac{p_{n}}{q_{n}}|<\frac{1}{\sqrt{5}q_{n}^{2}}$ – saturated by the
golden mean. By contrast Liouville numbers, are very well approximated by
their continued fractions. This means that there exists a $\beta(\alpha)$ such
that $|\alpha-\frac{p_{n}}{q_{n}}|<e^{-\beta(\alpha)q_{n}}$. In this work, we
focus on the Diophantine case, but these methods can be generalized.

We will take sequences of rational approximate operators by taking the
$\Theta=2\pi\alpha$ in Eq. (2) to be its continued fraction approximation
$\Theta_{N}=2\pi p_{N}/q_{N}$. For each rational approximate, $H_{N}$, we
construct a transfer matrix equation (TME), see [16, 14]. The unit cell for
the N-th rational approximate is $q_{N}$ sites long, and a wavefunction on the
n-th unit cell is defined by

$\displaystyle\Psi_{n,N}=\begin{pmatrix}\psi_{n+1}&\ldots&\psi_{n+q_{N}}\end{pmatrix}^{T}.$
(4)

This translates the eigenvalue equation into a simple form,

$\displaystyle
J_{N}\Psi_{n+1}+M_{N}\Psi_{n}+J_{N}^{\dagger}\Psi_{n-1}=E\Psi_{n}.$ (5)

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 $\Psi_{n}$.

From this we construct a transfer matrix equation (TME),

$\displaystyle\begin{pmatrix}J_{N}^{-1}(E-M_{N})&-J_{N}^{-1}J_{N}^{\dagger}\\\
1&0\end{pmatrix}\begin{pmatrix}\Psi_{n}\\\
\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}\Psi_{n+1}\\\ \Psi_{n}\end{pmatrix}.$
(6)

$J_{N}$ will in general not be invertible, however, we can reduce the transfer
matrix in Eq (6), from a $2q_{N}\times 2q_{N}$ matrix to a $2\times 2$ matrix
for each rational approximate. The explicit construction of a $2\times 2$
transfer matrix begins with a reduced SVD of the hopping matrix $J_{N}$,

$\displaystyle J_{N}$ $\displaystyle=$ $\displaystyle
V_{N}D_{N}W^{\dagger}_{N}$ (7) $\displaystyle J_{N}^{\dagger}$
$\displaystyle=$ $\displaystyle W_{N}D_{N}^{\dagger}V^{\dagger}_{N}$ (8)

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

$\displaystyle D_{N}=D_{N}^{\dagger}=\begin{pmatrix}t&0&\ldots&0\\\
0&0&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&0\end{pmatrix},$ (9)

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

$\displaystyle
W_{N}^{\dagger}=\begin{pmatrix}0&\ldots&0&(-1)^{q_{N}}\end{pmatrix},V_{N}^{\dagger}=\begin{pmatrix}(-1)^{q_{N}}&0&\ldots&0\end{pmatrix}$
(10)

Rewriting our intra-unit cell term as a projected Green’s function (pGF)

$\displaystyle G_{N}(\omega)=(\omega-M_{N})^{-1}$ (11)

The transfer matrix equation reduces to

$\displaystyle\Psi_{n}$ $\displaystyle=$ $\displaystyle
G_{N}J_{N}\Psi_{n+1}+G_{N}J_{N}^{\dagger}\Psi_{n-1}$ (12)
$\displaystyle\Psi_{n}$ $\displaystyle=$ $\displaystyle
G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}$
(13)

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

$\displaystyle V^{\dagger}_{N}\Psi_{n}$ $\displaystyle=$ $\displaystyle
V^{\dagger}G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+V^{\dagger}G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}$
(14) $\displaystyle W^{\dagger}_{N}\Psi_{n}$ $\displaystyle=$ $\displaystyle
W^{\dagger}G_{N}V_{N}D_{N}W^{\dagger}_{N}\Psi_{n+1}+W^{\dagger}G_{N}W_{N}D_{N}V^{\dagger}_{N}\Psi_{n-1}.$
(15)

This reduces the Transfer matrix equation to [16, 14], setting $t=1$,

$\displaystyle\begin{pmatrix}(W_{N}^{\dagger}G_{N}V_{N})^{-1}&-(W_{N}^{\dagger}G_{N}V_{N})^{-1}(W_{N}^{\dagger}G_{N}W_{N})\\\
(V_{N}^{\dagger}G_{N}V_{N})(W_{N}^{\dagger}G_{N}V_{N})^{-1}&V_{N}^{\dagger}G_{N}W_{N}-V_{N}^{\dagger}G_{N}V_{N}(W_{N}^{\dagger}G_{N}V_{N})^{-1}W_{N}^{\dagger}G_{N}W_{N}\end{pmatrix}\begin{pmatrix}V_{N}^{\dagger}\Psi_{n}\\\
W_{N}^{\dagger}\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}V_{N}^{\dagger}\Psi_{n+1}\\\
W_{N}^{\dagger}\Psi_{n}\end{pmatrix}$ (16)

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 $(W_{N}^{\dagger}G_{N}V_{N})^{-1}$,

$\displaystyle\overbrace{(W_{N}^{\dagger}G_{N}V_{N})^{-1}\begin{pmatrix}1&-(W_{N}^{\dagger}G_{N}W_{N})\\\
V_{N}^{\dagger}G_{N}V_{N}&V_{N}^{\dagger}G_{N}W_{N}(W_{N}^{\dagger}G_{N}V_{N})-V_{N}^{\dagger}G_{N}V_{N}W_{N}^{\dagger}G_{N}W_{N}\end{pmatrix}}^{\hat{T}_{q_{N},n}}\begin{pmatrix}V_{N}^{\dagger}\Psi_{n}\\\
W_{N}^{\dagger}\Psi_{n-1}\end{pmatrix}=\begin{pmatrix}V_{N}^{\dagger}\Psi_{n+1}\\\
W_{N}^{\dagger}\Psi_{n}\end{pmatrix}.$ (17)

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,
$E\in\mathbb{R}-\Sigma$.

In Eq. 17, 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. This reduces to checking
that $G_{N}(\omega)\rightarrow(\omega-\hat{H}_{\alpha})^{-1}$ as
$N\rightarrow\infty$ [14, 15].

###  II.1 Projected Green’s Function

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 [17, 18, 19, 20, 21, 22, 23, 24, 25]. More recently, it was
noticed that bound state formation criteria along an edge are also defined by
Green’s function zeros [18, 19, 20, 13, 25, 21], 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,

$\displaystyle
G(\omega,\mathbf{r}_{\perp},\alpha_{\parallel})=\sum_{\alpha}|\mathinner{\langle{\alpha|\mathbf{r}_{\perp}}\rangle}|^{2}G(\omega,\alpha),$
(18)

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. We make the impurity bound states into edge modes
by tuning the impurity into an edge. This is done by constructing an
appropriate impurity geometry and taking $|\mathcal{V}|\rightarrow\infty$.
Therefore, the condition for impurity localized states as
$|\mathcal{V}|\rightarrow\infty$ is equivalent to the criteria for the
formation of edge localized modes.

As $|\mathcal{V}|\rightarrow\infty$, 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,

$\displaystyle G(\omega,\alpha)$ $\displaystyle=$
$\displaystyle(\omega-(H_{0}+\mathcal{V}))^{\text{-}1}=(1+\mathcal{V}G_{0})^{\text{-}1}G_{0}.$
(19)

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

$\det\left[G_{0}(\omega,\alpha)\mathcal{V}-\mathbf{1}\right]=0.$ (20)

For $|\mathcal{V}|\rightarrow\infty$, solutions require $G_{0}\rightarrow 0$.
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 [18].

We show the pGF zeros are still of topological significance in the almost-
Mathieu operator by proving the convergence of the rational pGF zeros to the
irrational pGF zeros (see below). The rational pGF zeros correspond to
deficient points in the transfer matrix equations, thereby defining
reducibility classes for the transfer matrix – the deficient point cannot be
removed. As such, if the rational pGFs converge to the irrational pGF in an
operator norm sense, then the irrational transfer matrix has the same
reducibility classes as the limit of the rational sequence generated by the
irrational pGF zeros. Proving this convergence for the critical almost-Mathieu
operator is non-trivial and requires a special gauge choice (see below).

Thus we define the translation invariant intra-cell projected Green’s
function, $G_{N}(\omega)$ in Eq. (17) as

$\displaystyle\hat{G}_{N}(\omega,\mathbf{k}_{\parallel})=\int
d\mathbf{k}_{\perp}G(\omega,\mathbf{k}_{\parallel},\mathbf{k}_{\perp})$ (21)

where $\mathbf{k}_{\parallel},\mathbf{k}_{\perp}$ represent momenta in
parallel and transverse directions to the projection. We then take the limit
of $N\rightarrow\infty$, to construct the almost-Mathieu transfer matrix.

###  II.2 Irrational PGF Zeros and Open Sets

Having constructed irrational pGFs, consider the case where the irrational TME
is indeed the convergent limit of the rational TME sequence. In such a case,
allowed eigenstates correspond to the limit of rational approximate
eigenstates and are effectively invariant under the choice of phase, $\delta$,
in Eq. (1) [14, 15].

Lemma 1. If all almost-Mathieu eigenstates are translation invariant, pGF
zeros occur if and only if there exists an open complement of the spectrum.

Proof. Consider the pGF for translation invariant eigenstates, and average
over all initial sites $x_{0}$ by shifting the phase $\delta$ accordingly

$\displaystyle G_{\perp,\alpha}(\omega)$ $\displaystyle=$
$\displaystyle\mathinner{\langle{x_{0}}|}\sum_{\xi}G(\omega,\xi)\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}\mathinner{|{x_{0}}\rangle}=\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{x_{0}}\mathinner{\langle{x_{0}}|}\sum_{\xi}G(\omega,\xi)\mathinner{|{\xi}\rangle}\mathinner{\langle{\xi}|}\mathinner{|{x_{0}}\rangle}$
(22) $\displaystyle=$
$\displaystyle\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{\xi}G(\omega,\xi)\mathinner{\langle{\xi}|}(\sum_{x_{0}}\mathinner{|{x_{0}}\rangle}\mathinner{\langle{x_{0}}|})\mathinner{|{\xi}\rangle}=\lim_{L\rightarrow\infty}\frac{1}{L}\sum_{\xi}\frac{1}{\omega-
E_{\xi}}$

Given $\omega$ in a spectral gap between two energies $E_{1}<E_{2}$,
$\frac{1}{\omega-E_{1}}>0$ and $\frac{1}{\omega-E_{2}}<0$. Since $\omega$
interopolates between a positive and negative pole, by the Intermediate Value
Theorem, it must cross zero. Thus, open sets imply pGF zeros. Note: bulk-
boundary correspondence found in [5] follows from this direction.

The reverse direction follows a similar reasoning, assuming the existence of a
pGF zero at some $\omega$, this $\omega$ cannot be a pole of $G(\omega,\xi)$,
or else we could take an $\omega+i\epsilon$ and then discontinuously remove
the zero (see convergence argument below). In the context of the almost-
Mathieu operator, this implies that if for some $\omega$,
$G_{\perp,\alpha}(\omega)=0$, there must be a ball of radius $\epsilon>0$ for
which there are no poles. Otherwise, $\omega$ is the limit point of a sequence
of poles and not a pole itself, and the spectrum of $\hat{H}_{\alpha}$ would
not contain all of its limit points. However, the almost-Mathieu spectrum is
the limit of its rational approximates (see below), and $\omega$ would be the
limit of the limit points of a convergent rational sequence. Thus, the almost-
Mathieu spectrum must contain its limit points (its spectrum is a Cantor set
[13, 7]). This would be a contradiction. Therefore, pGF zeros occur only on
open complements of the spectrum. $\hfill\square$

##  III Almost-Mathieu Convergence

With Lemma 1 in hand, the convergence of the rational pGF sequence to the
irrational pGF becomes a connection between the gap topology and open sets in
the quasi-periodic spectrum.

Away from the metal-insulator transition, we can choose a magnetic unit cell
of the 2D parent Hamiltonian in Eq. (2) such that the rational approximate pGF
sequence converges in operator norm to the irrational pGF [14, 15]: for any
$\epsilon>0$ there exists an $M\in\mathbb{N}$ such that for $N>M$,

$\displaystyle||G_{\perp,N}(\omega)-G_{\perp,\alpha}(\omega)||<\epsilon.$ (23)

However, at criticality, $V=1$ in Eq. (1), the convergence is marginal, [14,
15]. Thus, this parameter regime requires a different gauge choice. To this
end, we take advantage of the chiral gauge introduced by Ref. [13] and used by
Ref. [2].

###  III.1 The Chiral Gauge

Defining the operators

$\displaystyle\hat{V}:\hat{c}^{\dagger}_{x,\delta_{y}}\hat{c}_{x,\delta_{y}}\rightarrow\hat{c}^{\dagger}_{2x,\delta_{y}}\hat{c}_{2x,\delta_{y}}$
(24)
$\displaystyle\hat{U}_{a}:\hat{c}^{\dagger}_{x,\delta_{y}}\hat{c}_{x,\delta_{y}}\rightarrow
e^{2\pi
ixa(\delta_{y}+\Theta/2x)}\hat{c}^{\dagger}_{x,\delta_{y}}\hat{c}_{x,\delta_{y}}$
(25)
$\displaystyle\hat{R}:\hat{c}^{\dagger}_{x,\delta_{y}}\hat{c}_{x,\delta_{y}}\rightarrow\sum_{k\in\mathbb{Z}}e^{-2\pi
ik(\Theta x+\delta_{y})}\int_{\mathbb{T}}d\beta e^{-2\pi i\beta
x}\hat{c}^{\dagger}_{k,\beta}\hat{c}_{k,\beta}$ (26)

and the composite operator $\hat{Q}=\hat{U}_{1}\hat{R}\hat{U}_{1/2}$, it was
shown in [13] that

$\displaystyle\mathcal{H}_{2D,\Theta}=\hat{Q}\hat{V}^{-1}\mathcal{H}_{ch,\Theta/2}\hat{V}\hat{Q}^{-1}$
(27)

where $\mathcal{H}_{2D}$ and $\mathcal{H}_{ch}$ are defined in Eq. (2) and Eq.
(3) respectively. This highly non-local transformation ($\hat{V}$ maps
$x\rightarrow 2x$) is exact and we can construct a rational approximating
sequence for $\mathcal{H}_{ch}$,

$\displaystyle\mathcal{H}_{ch,N}=\sum_{x,\delta_{y}}2\sin{(\pi\frac{p_{N}}{q_{N}}x+\delta_{y})}\hat{c}^{\dagger}_{x+1,\delta_{y}}\hat{c}_{x,\delta_{y}}+2\sin{(\pi\frac{p_{N}}{q_{N}}x+\delta_{y})}\hat{c}_{x+1,\delta_{y}}\hat{c}_{x,\delta_{y}}^{\dagger}$
(28)

with $\frac{p_{N}}{q_{N}}$ being the $N$-th continued fraction approximation
to $\alpha$. This sequence of rational approximate Hamiltonians,
$\\{\mathcal{H}_{ch,N}\\}_{N\in\mathbb{N}}$, generates a sequence of rational
approximate pGFs and we can use similar methods as in [14, 15] to prove
convergence.

####  III.1.1 Persistence of pGF Zeros in Chiral Gauge

The chiral gauge choice is arbitrary in 2D, and although it makes the rational
approximates appear to be in class AIII, which is general topologically
trivial in 2D, the chiral edge modes persist. More precisely, the
transformation is only exact for an infinite system, and any rational
approximate is translation invariant on the open plane. This translation
invariance protects the topological invariant as a stack of chiral 1D models.
And, since we are choosing a single $k_{y}$ in our “projection” to the 1D
Hamiltonian in Eq. (1), the chiral symmetry actually protects the 1D edge mode
even while projecting each of the rational approximates to 1D. Thus, each
rational approximate will still have the topologically protected deficient
points in its transfer matrix, Eq. 17, for each spectral gap.

###  III.2 Chiral PGF Convergence

We define the corresponding pGF for each $\mathcal{H}_{ch,N}$,

$\displaystyle
G_{\perp,N}=\int\frac{dk}{2\pi}(\omega-\mathcal{H}_{ch,N})^{-1}.$ (29)

As done for $\tilde{\mathcal{H}}_{2D}$ in [14, 15], we show
$||G_{\perp,N}(\omega)-G_{\perp,\alpha}(\omega)||$ converges for all $\omega$,

$\displaystyle\left(G_{\perp,N}(\omega)-G_{\perp,\alpha}(\omega)\right)$
$\displaystyle=$
$\displaystyle\int\frac{dk}{2\pi}\left[(\omega-\mathcal{H}_{ch,N})^{-1}-(\omega-\mathcal{H}_{ch,\alpha})^{-1}\right]$
(30) $\displaystyle=$
$\displaystyle\int\frac{dk}{2\pi}\left[\frac{\mathcal{H}_{ch,N}-\mathcal{H}_{ch,\alpha}}{(\omega-\mathcal{H}_{ch,N})^{2}+(\omega-\mathcal{H}_{ch,N})(\mathcal{H}_{ch,N}-\mathcal{H}_{ch,\alpha})}\right].$

We first bound $\Delta_{N}=||(\mathcal{H}_{ch,N}-\mathcal{H}_{ch,\alpha})||$
by considering

$\displaystyle|2\sin{(\pi\frac{p_{N}}{q_{N}}x-\delta_{y})}-2\sin{(\pi\alpha
x-\delta_{y})}|<2\pi|\frac{p_{N}}{q_{N}}-\alpha|x$ (31)

At worst for diophantine $\alpha$,
$|\frac{p_{N}}{q_{N}}-\alpha|<\frac{1}{\sqrt{5}q_{N}^{2}}$, so

$\displaystyle|2\sin{(\pi\frac{p_{N}}{q_{N}}x-\delta_{y})}-2\sin{(\pi\alpha
x-\delta_{y})}|<\frac{2\pi x}{\sqrt{5}q_{N}^{2}}.$ (32)

Plugging back into $\Delta_{N}$, we have by Samuelson’s inequality

$\displaystyle\Delta_{N}<2\sqrt{2\sum_{x=1}^{q_{N}}|2\sin{(\pi\frac{p_{N}}{q_{N}}x-\delta_{y})}-2\sin{(\pi\alpha
x-\delta_{y})}|^{2}}<2\sqrt{2\sum_{x=1}^{q_{N}}\left(\frac{\pi
x}{\sqrt{5}q_{N}^{2}}\right)^{2}}<2\sqrt{\frac{2\pi^{2}}{30q_{N}}}=2\pi\sqrt{\frac{1}{15q_{N}}}$
(33)

This bounds the numerator in Eq. (30), but we need to bound the denominator
from below as well. This is difficult as the rational spectral gaps become
exponentially small in the limit of $N\rightarrow\infty$ [26, 13]. We avoid
this problem by considering a small deviation into the complex plane to bound
$(\omega\pm i\epsilon-\mathcal{H}_{ch,N})^{-1}<|\epsilon|$, by the Hermiticity
of $\mathcal{H}_{ch,N}$.

####  III.2.1 Complex pGF Convergence

We consider $\omega\rightarrow\omega\pm i\epsilon$ and prove that for any
$\epsilon>0$, we can take $q_{N}$ sufficiently large such that for any
$\delta>0$

$\displaystyle||\mathrm{Re}\,(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm i\epsilon))||$ $\displaystyle<$
$\displaystyle\delta,\quad||\mathrm{Im}\,(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm i\epsilon))||<\delta.$ (34)

Or,

$\displaystyle\lim_{\epsilon\rightarrow
0}\lim_{q_{N}\rightarrow\infty}||G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm i\epsilon)||=0.$ (35)

As the Green’s function is continuous on the upper and lower half of the
complex plane, with the same limit,

$\displaystyle\lim_{q_{N}\rightarrow\infty}||G_{\perp,\alpha}(\omega)-G_{\perp,N}(\omega)||=0.$
(36)

We need the real and imaginary parts to converge separately,

$\displaystyle\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=-\int\frac{dk}{2\pi}\frac{\left[(\omega-\tilde{H}_{ch,N})^{2}+(\omega-\tilde{H}_{ch,N})(\tilde{H}_{ch,N}-\tilde{H}_{ch,\alpha})-\epsilon^{2}\right](\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}$
(37) $\displaystyle\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=-\int\frac{dk}{2\pi}\frac{(\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}$
$\displaystyle-\int\frac{dk}{2\pi}\frac{\left[(\omega-\tilde{H}_{ch,N})(\tilde{H}_{ch,N}-\tilde{H}_{ch,\alpha})-2\epsilon^{2}\right](\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}$
(38) $\displaystyle\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=\int\frac{dk}{2\pi}\frac{\left[\pm
2i\epsilon(\omega-\tilde{H}_{ch,N})+i\epsilon(H_{ch,N}-\tilde{H}_{ch,\alpha})\right](\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}$
$\displaystyle\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}=\int\frac{dk}{2\pi}\frac{2i\epsilon(\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})((\omega-\tilde{H}_{ch,N})\pm
i\epsilon)\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}$
$\displaystyle+\int\frac{dk}{2\pi}\frac{\left[2\epsilon^{2}+i\epsilon(H_{ch,N}-\tilde{H}_{ch,\alpha})\right](\tilde{H}_{ch,\alpha}-\tilde{H}_{ch,N})}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})^{2}\left(\mathbb{1}+\frac{2\omega(H_{ch,N}-H_{\alpha})+\tilde{H}_{ch,N}^{2}-\tilde{H}_{ch,\alpha}^{2}}{((\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2})}\right)}.$
(39)

Now we use that $||(\omega-\tilde{H}_{ch,N})^{2}+\epsilon^{2}||>\epsilon^{2}$,
to cancel the bottom factors $\sim\frac{1}{\epsilon^{2}}$. And, we can use the
bound from above,
$||\tilde{H}_{ch,N}-\tilde{H}_{ch,\alpha}||<2\pi\sqrt{\frac{1}{15q_{N}}}$ to
get

$\displaystyle||\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\int\frac{dk}{2\pi}||\epsilon^{-4}\left[-\epsilon^{2}2\pi\sqrt{\frac{1}{15q_{N}}}+(\omega-\tilde{H}_{ch,N})\left(2\pi\sqrt{\frac{1}{15q_{N}}}\right)^{2}\right]||$
(40) $\displaystyle||\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\int\frac{dk}{2\pi}\epsilon^{-3}\left[\epsilon
8\pi\sqrt{\frac{1}{15q_{N}}}+\left(2\pi\sqrt{\frac{1}{15q_{N}}}\right)^{2}\right].$
(41)

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

$\displaystyle||(\omega-\tilde{H}_{ch,N})||<C\sqrt{q_{N}},$ (42)

so that we can always take $q_{N}$ big enough to make
$\left(2\pi\sqrt{\frac{1}{15q_{N}}}\right)^{2}C\sqrt{q_{N}}<\epsilon^{4}$ for
any $\epsilon>0$ to get the uniform convergence above, for all Diophantine
$\alpha$.

Thus, the inequalities in Eq. (34) reduce to bounding $||(\omega-H_{ch,N})||$,
and the operator norm obeys the triangle inequality. So,

$\displaystyle||(\omega-\tilde{H}_{ch,N})||\leq|\omega|+||\tilde{H}_{ch,N}||$
(43)

We the use Samuelson’s inequality to bound the chiral gauge eigenvalues.
Notice that the characteristic polynomial of $\tilde{H}_{ch,N}$ has the
coefficients

$\displaystyle
P_{N}(\lambda)=\lambda^{q_{N}}+0\lambda^{q_{N}-1}-\sum_{i=1}^{q_{N}}4\sin^{2}{(\pi\frac{p_{N}}{q_{N}}i-\delta_{y})}\lambda^{q_{N}-2}+\ldots+\det{\tilde{H}_{ch,N}}\lambda^{0}$
(44)

Therefore, by Samuelson’s inequality, the largest eigenvalue and
consequentially the operator norm is at most,

$\displaystyle||\tilde{H}_{ch,N}||<2\sqrt{2\sum_{i=1}^{q_{N}}\sin^{2}{(\pi\frac{p_{N}}{q_{N}}i-\delta_{y})}}=2\sqrt{q_{N}}$
(45)

and thus

$\displaystyle||\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\left[\epsilon^{-2}2\pi\sqrt{\frac{1}{15q_{N}}}+\epsilon^{-4}(|\omega|+2\sqrt{q_{N}})\left(2\pi\sqrt{\frac{1}{15q_{N}}}\right)^{2}\right].$
(46)

Then, choosing $q_{N}>\epsilon^{-4}\delta^{-2}$,

$\displaystyle||\mathrm{Re}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm
i\epsilon))}||<\delta,\quad||\mathrm{Im}\,{(G_{\perp,\alpha}(\omega\pm
i\epsilon)-G_{\perp,N}(\omega\pm i\epsilon))}||<\delta,$ (47)

as desired. $\hfill\square$

###  III.3 Effective Translation Invariance

The pGF convergence proven in Section III.2 directly implies the rational
transfer matrix equation converges to the irrational transfer matrix equation
(TME). Thus, if they exist, the eigenfunctions for the critical almost-Mathieu
operator, defined as the limit points of the rational sequence, are in fact
the irrational limit of the rational approximate eigenfunctions.

The initial choice of phase $\delta$ in Eq. 1 can be absorbed by choosing a
different starting point from which to approximate the irrational TME, i.e.
$\cos{[\Theta x+\delta]}=\cos{[\Theta(x-1)+\delta+\theta]}$. Since the the
irrational TME is the limit of the rational TME, these phase shifts are
absorbed by the rational sequence. The rational eigenfunctions are thus
effectively invariant under these phase shifts, and consequentially, their
limit is invariant as well. However, in 1D these phase shifts amount to
translations of the lattice. Therefore, the critical almost-Mathieu operator
has translation invariant eigenfunctions and Lemma 1 applies.

##  IV Dry Ten Martini Problem

We restate the problem statement for convenience:

Dry Ten Martini Problem. Consider an energy in the spectrum of the critical
almost-Mathieu operator $E\in\Sigma$, satisfying
$\hat{H}\mathinner{|{\psi}\rangle}=E\mathinner{|{\psi}\rangle}$ with $\hat{H}$
as in Eq. (1) and $V=1$. If $N(E)=m\alpha+n$ with $m,n\in\mathbb{Z}$ and
$\Theta=2\pi\alpha\in$, then E belongs to the boundary of a component of
$\mathbb{R}-\Sigma$.

Proof. As discussed in Section III.3, the convergence of the rational transfer
matrix sequence to the irrational transfer matrix, implies the almost-Mathieu
eigenfunctions are invariant under phase shifts
$\delta\rightarrow\delta+\Theta$ and Lemma 1 holds. So, we combine Lemma 1
with the transfer matrix results above.

First, by the convergence proof above, all eigenvalues of the irrational pGF
are generated by the limit of rational 2D approximates, including zeros. Thus
all irrational pGF zeros are generated by zeros occurring in spectral gaps of
rational approximates. Then, by Lemma 1, the sequence of rational pGF zeros
form open sets in the compliment of the almost-Mathieu operator spectrum
$\mathbb{R}-\Sigma$.

Second, for any rational approximate, each gap has a non-trivial Chern number
thanks to the magnetic flux per plaquette, even in the chiral gauge, Section
III.1.1. By the results in [5, 27, 9], these Chern numbers (slopes) form the
irrational gap-labels in the continued fraction limit. Bulk-boundary
correspondence [28, 29, 5] implies the existence of edge modes (pGF zeros)
corresponding to the non-trivial Chern numbers in each gap. So, for any gap
labeled by the gap-labeling theorem, we can take a large enough rational
approximate, $q_{N}$, such that for all future approximates the rational band
gap with the same label has a pGF zero. Thus, for large enough $q_{N}$, and
any gaps labeled by the gap-labeling theorem, there exists an $\omega_{g}$ in
the gap, such that $G_{\perp,\alpha}(\omega_{g})=0$.

Together, these statements imply that every gap-labeling theorem labeled ”gap”
will form an open set in the compliment of the almost-Mathieu spectrum,
$\mathbb{R}-\Sigma$. Or, as phrased above, if the density of states
$N(E)=m\Theta+n$, then $E$ must form the boundary of an open set.
$\hfill\square$

##  V Conclusion

In summary, the central idea in this work was the construction of the almost-
Mathieu transfer matrix from a sequence of rational approximate transfer
matrices using the reduced SVD approach in [16, 14, 15]. The projected Green’s
functions appear as a consequence of the reduced SVD method and allow us to
import powerful ideas from band topology to show the spectral gaps form open
sets. A clear benefit of this approach is the geometric intuition provided by
the rational approximates, which can be readily extended to other quasi-
periodic patterns, see [14].

While only a single point in parameter space, the critical almost-Mathieu
operator is precisely the isotropic Hofstadter Hamiltonian. And, since there
is an irrational flux arbitrarily close to any rational flux (irrationals are
dense in the reals), the dry ten martini problem asks whether or not
topological invariants are stable for IQHE on a lattice. We push the answer to
the affirmative.

###### Acknowledgements.

While writing this paper, [2] was published. The authors have proven the
spectrum to be purely singularly continuous. Their proof also relies on the
chiral gauge transformation and implies all gap-labeled gaps are open sets.
Our approach is, however, substantially different and, we believe, offers an
intuition/generalizablity beyond the arguments in [2], albeit we fully
recognize the formality of the proof presented in that paper. 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, 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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