# A Topological Classification of Finite Chiral Structures using Complete
Matchings

Maxine M. McCarthy  D. M. Whittaker  University of Sheffield

###### Abstract

We present the theory and experimental demonstration of a topological
classification of finite tight binding Hamiltonians with chiral symmetry.
Using the graph-theoretic notion of complete matchings, we show that many
chiral tight binding structures can be divided into a number of sections, each
of which has independent topological phases. Hence the overall classification
is $N\mathbb{Z}_{2}$, corresponding to $2^{N}$ distinct phases, where $N$ is
the number of sections with a non-trivial $\mathbb{Z}_{2}$ classification. In
our classification, distinct topological phases are separated by exact
closures in the energy spectrum of the Hamiltonian, with degenerate pairs of
zero energy states. We show that that these zero energy states have an unusual
localisation across distinct regions of the structure, determined by the
manner in which the sections are connected together. We use this localisation
to provide an experimental demonstration of the validity of the
classification, through radio frequency measurements on a coaxial cable
network which maps onto a tight binding system. The structure we investigate
is a cable analogue of an ideal graphene ribbon, which divides into four
sections and has a $4\mathbb{Z}_{2}$ topological classification.

##  I Introduction

Topology has become ubiquitous in modern physics, with many lattice structures
and materials shown to have non-trivial topological attributes. A topological
material is characterised by properties which remain unchanged during
adiabatic evolution, making them robust in the presence of disorder. For
instance, boundary states in one-dimensional topological lattices may have
energies which are resistant to disorder [1], while in two-dimensions they may
provide directional transport which is protected against backscattering [2, 3,
4, 5]. In order to investigate these potentially useful properties, it is
necessary to be able to classify the different topological phases of a given
Hamiltonian.

Distinct topological phases are generally understood from two perspectives.
The first is that two phases are distinct if, and only if, they cannot be
related by (symmetry respecting) adiabatic evolution. The second is that a
non-trivial topological phase has interesting boundary properties. The first
perspective allows us to assign topological indices to distinct phases, and
the second gives a physical significance to a classification. Both are
connected by the famous bulk-boundary correspondence [6] which, since the
discovery that the quantum Hall effect is topological [2, 7] has proven a
powerful tool for the prediction of topological boundary phenomena. Most
approaches to classification make use of the bulk-boundary correspondence in
some way.

A ubiquitous topological classification is that of the 10-fold way [8], in
which K-theoretic methods are used to classify each of the Altland-Zirnbauer
(AZ) symmetry classes of momentum-space Hamiltonians [9]. Another approach is
topological quantum chemistry [10, 11], where the classification is obtained
by looking for topologically non-trivial states. In particular, the presence
of states which cannot be adiabatically deformed to the atomic limit indicates
a topologically non-trivial band in the bulk [12]. These methods provide a
classification of stable and fragile topological phases in a huge number of
systems [13].

A useful approach for strong disorder relies on defining a non-commutative
Brillouin zone [14, 15, 16]. Using non-commutative geometry, the Brillouin
zone is modified to allow for variations in each unit cell of the structure,
by defining a configuration space for the set of distinct unit cells.
Translational invariance is no longer required, giving a disordered bulk-
boundary correspondence.

For finite structures, a spectral localiser [17, 18] can predict the presence
of approximately zero energy boundary states (also by demonstrating states
cannot be localised to an atomic limit), therefore classifying a structure via
the converse of the bulk-boundary correspondence. Alternatively local
realspace markers may be used [19, 20] whose average gives a topological index
for the entire structure. While this averages to a quantised value in the
thermodynamic limit, such local markers are not exactly quantised in finite
structures.

It has been shown that finite size effects may cause a rich sequence of
topological phase transitions, corresponding to gap closures separating
topological bubbles [21] in inversion symmetric structures. The number of
approximately zero energy boundary states in such a bubble takes an integer
value corresponding to a $\mathbb{Z}$ classification in finite, inversion-
symmetric Hamiltonians. These states are resistant to small amounts of
disorder. Such an approach has also been extended to time-reversal symmetric
systems [22]. Alternatively the finite structure may be repeated as a periodic
supercell [23, 24] allowing the use of momentum-space methods to classify the
topological phases.

These approaches leave open a problem: how may a structure be classified when
we completely lose the bulk-boundary correspondence? That is, we no longer
have any way to define a boundary or a bulk. This situation may occur in a
small finite structure, particularly with strong disorder and/or no underlying
lattice structure (for instance, a random finite network). Strongly disordered
finite systems may have a non-trivial topological classification [25, 26, 17]
although the number of distinct topological phases is not always clear,
motivating a general approach to topologically classify finite structures
which have lost the bulk-boundary correspondence. In this paper, we propose a
partial solution to this problem. Using graph-theoretic methods, we give a
topological classification for finite chiral symmetric structures. We achieve
this by considering equivalence classes of finite real space Hamiltonians with
arbitrary values for the hopping terms.

In a finite structure at sufficient disorder, a bulk may not be possible to
define, however non-adiabatic evolution may still be defined. For a finite
system, a closing in the energy spectrum is analogous to a band gap closing in
momentum space. In structures with chiral symmetry, and thus a symmetric
energy spectrum, this corresponds to the appearance of pairs of zero energy
states. Determining the conditions on the hopping terms which lead to the
presence of these states allows us to define equivalence classes of
Hamiltonians. A similar approach to defining topological phase boundaries has
been used in [25, 26, 17].

Using this definition of equivalence classes we find that many structures have
a rich classification resulting from the fact that they can be divided into a
number of sections, each of which can be assigned an independent topological
phase. These sections are identified as corresponding to factors of the
determinant. Such factors may be determined by the hopping terms that appear
in the expansion of the determinant of the Hamiltonian, and therefore play a
role in defining the conditions for zero energy states. Any other hopping
terms in the Hamiltonian can be removed without affecting the classification.
This removal process results in the structure separating into disconnected
pieces, each constituting a section. For each section we can determine an
independent 0 or $\mathbb{Z}_{2}$ classification, using a sub-Hamiltonian
defined on the section.

The existence of sections depends on the connectivity of the network
representing the hopping terms in the Hamiltonian. They can occur in
structures with some regularity, as in the graphene related examples which are
considered in this work. However, they can also appear in more randomly
connected structures. Checking 154131 non-isomorphic graphs which represent
some of the connected chiral structures with 18 sites or fewer, and using
randomised searches on larger finite chiral structures, we find that the
classification of a structure is the most rich when each site has an average
of three hopping terms. For an average of two the underlying connectivity is
generally too sparse to provide a rich classification, and at four or greater,
the structure is often too constrained by its connectivity to allow the
division into many sections.

Of the 154131 graphs classified, for all positive and real hopping terms, 14%
had no distinct topological phases, with the remaining being topologically
non-trivial. 82% had two topological phases, and 4% had 4 to 16 distinct
topological phases, with only 18 of the classified graphs having 16 phases.
Including trivial and non-trivial sections 8% of structures has no sections,
68% had one section, and 24% had 2 to 9 sections (trivial sections can still
affect physical properties, such as localisation).

The infinite and finite classification of the sub-Hamiltonian corresponding to
a section may be different. Considering a section as a one-dimensional
supercell of an infinite lattice a transfer matrix treatment leads to one
trivial and one non-trivial topological phase, separated by a gap closure
somewhere in the Brillouin zone. Hence, with this definition, every section
has a $\mathbb{Z}_{2}$ classification. Furthermore, the classes AIII and BDI
have a $\mathbb{Z}$ classification using the 10-fold way. In this infinite
interpretation, a section having a non-trivial topological phase results in
boundary localised topologically robust zero modes to one end of that section.
This gives a connection to higher order topology arising through stacked
chiral structures.

We seek to classify the actual finite structure, so we adopt the convention
that only unavoidable gap closures which are observable (corresponding to
closure at zero momentum) separate topological phases. This means that some
sections will have a $\mathbb{Z}_{2}$ classification, while others will be
topologically trivial. The complete structure behaves as a stack of sections,
connected together in such a way that the topological phase of each section is
independent. This gives a classification as the direct sum
$\bigoplus_{1}^{N}\mathbb{Z}_{2}\cong N\mathbb{Z}_{2}$ where $N$ is the number
of sections with a non-trivial classification, and depends on the underlying
connectivity of the structure. For periodic materials, the stacking is not
unlike that seen in [27]. Others have also shown that a rich classification
can follow from the underlying connectivity of a structure [28, 29].

We also show that the localisation of the zero energy states that accompany a
topological phase transition is determined by the connections between the
sections. We define a partial ordering of the sections based on this
connectivity, such that each of the zero energy states created by making one
section topologically marginal spreads in only one direction through the
remaining sections. This provides an experimentally accessible verification of
the existence of the sections. We perform such an experiment using a coaxial
cable network, which has been shown to map onto a tight-binding Hamiltonian
[26, 30, 31, 32]. The structure we consider is a zigzag ‘graphene’ nanoribbon
consisting of four rows of sites, each of which forms a separate section,
leading to a $4\mathbb{Z}_{2}$ classification.

We begin by discussing the theory behind the classification in section II
before discussing experimental results in a small graphene structure in
section III, and concluding in IV. Supplementary material is provided with
more mathematical and experimental details.

##  II Classifying finite chiral structures

Figure 1: A two row ribbon zigzag graphene, looped to form a cylinder. The
structure has chiral symmetry, with the two sublattices indicated by the black
and white sites. The topological classification of structures like this can be
found using the methods we describe: this one is $2\mathbb{Z}_{2}$.

Chiral symmetry is one of the fundamental symmetries of the Altland-Zirnbeaur
symmetry classes [9]. A chiral Hamiltonian $H$ anticommutes with a unitary
$C$, ensuring non-zero eigenvalues come in $\pm\varepsilon$ pairs. When
allowing algebraically independent hopping terms (thereby letting the
structure have arbitrary hopping disorder) a Hamiltonian with chiral symmetry
necessarily has two sublattices of sites, which we label ‘black’ and ‘white’
(a consequence of the Harary-Sachs theorem [33, 34]). Each black site is only
connected (through non-zero hopping terms) to white sites, and each white site
is only connected to black sites. An example is shown in Fig. 1.

For a structure with $n_{B}$ black sites and $n_{W}$ white sites, there are
are always at least $|n_{B}-n_{W}|$ zero energy states. Since these exist
independently of the values of the hopping terms in $H$, they are described as
topologically protected states. Such states can also occur in structures with
$n_{B}=n_{W}$. If a structure has such protected states, the determinant of
the Hamiltonian is necessarily zero for all values of the hopping terms, so a
classification based on the conditions for $|H|=0$ is not possible. In this
work, we consider only structures with equal numbers of black and sites and no
topologically protected states.

Two Hamiltonians $H_{1}$ and $H_{2}$, are considered to have a different
topological phase if they cannot be related under adiabatic evolution. That
is, there exists no continuous way to evolve between $H_{1}$ and $H_{2}$
without a gap closing in the spectrum. For a chiral Hamiltonian with an even
number of sites, zero energy states necessarily occur in degenerate pairs.
This degeneracy corresponds to a closed gap in the spectrum, indicating non-
adiabatic evolution. Therefore finding the conditions for a singular
Hamiltonian determines the topological phase boundaries in a finite structure.

To explore distinct topological phases on a finite chiral structure we
consider a set of sites connected by non-zero hopping terms. Formally this
defines a graph $G$. On $G$ we define a tight binding (TB) Hamiltonian, $H$,
where each hopping term may be continuously evolved, algebraically
independently. To ensure changes to $G$ only include continuous evolution we
do not allow new connections to be introduced, or existing connections to be
broken. In terms of $H$, this means non-zero hopping terms must remain non-
zero, and hopping terms which are zero cannot be modified. Otherwise we allow
complete freedom to continuously evolve hopping terms of $H$, giving access to
strongly disordered Hamiltonians defined on $G$. We do, however, restrict the
hopping terms to be real: any topological phase boundary in a finite structure
can be avoided, by evolving the Hamiltonian through a path involving complex
hopping terms. Although we can allow negative values, the requirement for non
zero real hopping terms means that they cannot change sign as they evolve.

The tuple of algebraically independent hopping terms on $G$, defines an affine
space, $\xi$. Each Hamiltonian $H$ on $G$ defines a point in $\xi$.
Continuously evolving hopping terms in $H$ corresponds to following a path in
$\xi$. We refer to $\xi$ as the parameter space of $G$. Topological phase
boundaries in $\xi$ correspond to boundary free surfaces one dimension lower
than $\xi$. An example of a slice of a parameter space is given in Fig. 3.

The classification problem can be completely reduced to finding solutions to
$|H|=0$, which necessarily corresponds to degenerate zero energy states.
Although degeneracies can occur at non-zero energies, in a system with
algebraically independent hopping terms, such a gap closure requires at least
two constraints on the hopping terms, a consequence of the Harary-Sachs
theorem [33, 34]. Such constraints are described by surfaces which are at
least two dimensions lower than $\xi$, and therefore cannot divide it: they
are always avoidable gap closures. This ensures topological phase boundaries
are only given by a collection of surfaces in $\xi$ corresponding to $|H|=0$.

The Hamiltonian for a chiral structure has an antidiagonal block basis
corresponding to ordering the sites by sublattice,

$\displaystyle H=\begin{pmatrix}0&Q\\\ Q^{\dagger}&0\end{pmatrix}\;,$ (1)

so the determinant $|H|=-|Q||Q^{\dagger}|$. For equal numbers of black and
white sites, $Q$ is a square matrix, so $|Q|=|Q^{\dagger}|$, and we can use
either to determine the classification.

Every term in the expansion of $Q$ is algebraically independent, so $|Q|$ is a
multi linear form of the hopping terms. That is $|Q|$ varies linearly with
respect to each hopping term in its expansion. For each hopping term in $|Q|$,
solutions to $|Q|=0$ follow from solving a linear equation, constraining
exactly one hopping term. This ensures that every solution to $|Q|=0$
corresponds to a surface in $\xi$ which is both unbounded, and one dimension
lower than that of the parameter space, splitting $\xi$ in to distinct
regions. This linear behaviour also ensures that $|Q|$ changes sign as we go
over a phase boundary, meaning $\text{Sign}\left[|Q|\right]$ is a topological
invariant [17]. This is in contrast to $\text{Sign}\left[|H|\right]$, which
does not change at a phase boundary.

To classify a structure, we must understand how to set $|Q|$ to zero by
continuously evolving hopping terms, which may be done by finding the
irreducible factors of $|Q|$. If a factorisation of $|Q|=\prod|q_{i}|$ exists,
and it is possible to solve $|q_{i}|=0$, then the factor $|q_{i}|$ defines a
pair of distinct phases, where $\text{Sign}\left[|q_{i}|\right]$ is a
topological invariant. We say such a factor is non-trivial. The classification
of the structure is then given by $\bigoplus^{N}\mathbb{Z}_{2}\cong
N\mathbb{Z}_{2}$ where $N$ is the number of non-trivial factors.

The determinant $|Q|$ defines a polynomial in a unique factorisation domain
(details are given in the supplementary material section A) ensuring the
maximum number of non-trivial factors, $N$, of $|Q|$ is also a topological
invariant, so the classification is well defined. It is only possible to
change $N$ by removing or introducing new hopping terms or sites, that is,
under discontinuous evolution of $H$.

$\circ$$\bullet$$\circ$$\bullet$$a_{1}$$a_{2}$$a_{3}$$a_{4}$$\bullet$$\circ$$\bullet$$\circ$$b_{1}$$b_{2}$$b_{3}$$b_{4}$$c_{1}$$c_{2}$
Figure 2:  A simple two row graphene zigzag structure with the hopping terms
labelled as in Eq.(2). This structure has two sections and four distinct
topological phases, corresponding to a $2\mathbb{Z}_{2}$ classification.

To show a simple example, we consider a 2 row zigzag graphene sructure, with 4
sites on each row, see Fig. 2. This is described by the matrix

$Q=\begin{pmatrix}a_{1}&a_{4}&c_{1}&0\\\ a_{2}&a_{3}&0&c_{2}\\\
0&0&b_{1}&b_{2}\\\ 0&0&b_{4}&b_{3}\end{pmatrix}$ (2)

giving the determinant,

$|Q|=(a_{1}a_{3}-a_{2}a_{4})(b_{1}b_{3}-b_{2}b_{4})=|q_{1}||q_{2}|.$ (3)

The condition $|Q|=0$ can be satisfied by making either factor, $|q_{1}|$ or
$|q_{2}|$, equal to zero. This gives four distinct topological phases
corresponding to
$(\text{Sign}[|q_{1}|],\text{Sign}[|q_{1}|])=(+1,+1),(+1,-1),(-1,+1),(-1,-1)$.
Therefore this structure has the classification $2\mathbb{Z}_{2}$.

It is also apparent from this simple calculation that some of the hopping
terms – $c_{1}$ and $c_{2}$ – do not appear in the expansion of $|Q|$. This
means that the topological classification will be the same as for a structure
in which these hopping terms have been set to zero. Physically, removing these
links creates two completely separate lattices, the top and bottom loops in
Fig.2. We call these the two sections of the original structure.

The Hamiltonian, and thus $Q$, for a separated structure is block diagonal, so
the determinant, $|Q|$, is simply a product of the two block determinants, as
the expansion shows. The absence of the $c$ terms from the expansion of $|Q|$
is a consequence of the block triangular form of Eq.(2). However, this pattern
is only explicit if the sites are ordered correctly, which depends on finding
the sections defining the blocks in the matrix. As we prove in the
supplementary material section A.1, the determinant is factored if and only if
there exists a block triangular form of $Q$, so finding an appropriate
ordering of sites allows us to find the number of sections in a structure.

A graph-theoretic approach to this problem involves enumerating the complete
matchings of a structure. A complete matching consists of a set of pairings of
sites, or matchings, which are connected (by a non-zero hopping term), such
that every site in the structure is included in exactly one pair. Examples of
such complete matchings are shown in Fig. 4, for the four row graphene ribbon
which we investigate experimentally in section III. A complete matching has an
algebraic interpretation: if we index separately the black and white sites
such that each each matched pair has the same index, the corresponding hopping
terms appear along the diagonal of the matrix $Q$. The terms in the expansion
of the determinant of a matrix correspond to the product of the diagonal
elements for every possible permutation of the columns (or rows). Hence
finding the set of complete matchings for a structure gives all the terms in
the expansion (to get the signs, it is also necessary to keep track of the
number of swaps required to go between matchings). This is the Harary-Sachs
theorem [33, 34] for weighted bipartite graphs.

A consequence of the relationship between complete matchings and the expansion
of $|Q|$ is that any connection which is not included in any matching does not
appear in the expansion, so can be removed to reveal the sections. In the case
of the structure in Fig. 4, all the connections between the rows can be
removed, so the structure has four sections. For each section, there are two
complete matchings, so each factor $|q_{i}|$ of $|Q|$ contains two terms.
Since these have opposite signs, every factor can be made to pass through
zero, so the structure has classification $4\mathbb{Z}_{2}$. Note that even
small changes in connectivity can change this completely. If we add just one
connection between the a white site of top row and a black site of the bottom
row of the structure, we find that there are complete matchings which include
every connection, so there are no sections and the classification is
$\mathbb{Z}_{2}$.

Figure 3: A slice of the parameter space for 2 row ribbon zigzag graphene,
which has the classification $2\mathbb{Z}_{2}$. This slice is defined by
setting all the hopping terms equal to 1, apart from one in each row, $a$ and
$b$, which are allowed to evolve independently. $[H_{x}]$ denotes a set of
topologically equivalent Hamiltonians, and the dashed lines denote phase
boundaries. The path shown undergoes 3 phase transitions between the phases
$[H_{1}]$, $[H_{2}]$, and $[H_{3}]$, where each phase transition is identified
by the appearance of a pair of unavoidable degenerate zero energy states.

(a) (b)

Figure 4: (a) The four row ribbon graphene structure looped to form a
cylinder. (b) The complete matchings of this structure, flattened for clarity.
A complete matching consists of a solid or dashed set of matchings from each
row. The choice for each row is independent so there are $2^{4}$ complete
matchings. The connections between the rows do not appear in any of the
matchings, so each row forms a separate section. the topological
classification for this structure is $4\mathbb{Z}_{2}$.

The relationship between the number of sections of a structure and a block
triangular form of $Q$ in Eq.(2) can be made rigorous. In the supplementary
material section A.1, we prove that the factorisation of $|Q|$ according to
sections, $|Q|=\prod_{i}|q_{i}|$, can be made if, and only if $Q$ can be
written in the block triangular form

$Q=\begin{pmatrix}q_{j}&\cdots&c_{i+1,j}&c_{i,j}\\\ &\ddots&\vdots&\vdots\\\
&&q_{i+1}&c_{i,i+1}\\\ \text{\huge 0}&&&q_{i}\\\ \end{pmatrix}\;,$ (4)

where the $q_{i}$ and $c_{ij}$ are matrix blocks. Therefore by finding an
appropriate order of the sites the number of sections will be revealed in a
structure.

The diagonal blocks, $q_{i}$, in Eq.(4) define the sections of the structure.
They include the hopping terms which connect the black and white sites within
the section. The off diagonal blocks, $c_{i,j}$, contain the connections
between the sections. The hopping terms in the off diagonal blocks do not
appear in the expansion of $|Q|$. Note that the connecting terms only appear
within superdiagonal blocks – the corresponding subdiagonal is necessarily
zero. Thus the connections between sections are always between the
sublattices. For instance if $Q$ maps from the black to the white sublattice,
then only the white sites of $g_{j}$ may connect to the black sites of
$g_{i}$. We will make use of this shortly to define a partial ordering of the
sections.

The relationship between the block structure of a matrix and the topological
classification of a chiral structure has allowed us to find an algorithm to
classify a random chiral network. Details of this algorithm are given in the
supplementary material section B. Of our classification algorithm, the most
numerically expensive part is finding a basis with the largest number of
triangular blocks. An alternative algorithm for this part of the
classification is detailed in [35] although we have not compared the
complexity of our algorithm to this alternative approach.

The factorisation and triangular form of $Q$ tells us more than just the
topological classification of the Hamiltonian. Physically the nature of
localisation at criticality (that is, at a topological phase boundary) is
affected, with nullstates exactly restricted to a particular subset of
sections. We consider a structure represented by the graph $G$, with a section
$g_{i}$ corresponding to the factor $q_{i}$. For a topologically marginal
chiral structure each nullstate can be localised to the black or white sub
lattice. If only one section is singular, the black state may be non-zero only
on a subset $G_{b}$ of non-critical sections, and the white state may be non-
zero on a subset $G_{w}$ of non-critical sections. The sets $G_{w},\,\,G_{b}$
and the critical section $g_{c}$ are disjoint, so that the only section with
support of both the black and white states is $g_{c}$. This non-trivial
localisation yields an experimental method to find the classification of a
structure, discussed in section III.

To understand this localisation, we define a partial ordering of sections,
$(g\in G,\leq)$, which provides the upper triangular form in Eq.(4). We say
that $g_{i}<g_{j}$, or $g_{i}$ is lower than $g_{j}$, if white sites of
$g_{i}$ connect to black sites of $g_{j}$, and $g_{i}>g_{j}$, or $g_{i}$ is
higher than $g_{j}$, if black sites of $g_{i}$ connect to white sites of
$g_{j}$. If the blocks $q_{i}$ and $q_{j}$ can be permuted in $Q$ to swap
places on the diagonal in such a way that maintains an upper triangular block
matrix $Q$, then $g_{i}=g_{j}$. Otherwise if $g_{i}$ and $g_{j}$ are not
directly connected, we look to neighbouring sections to define the partial
ordering. As the structure is connected, we may always iterate to neighbours
of neighbours until we have the partial ordering relationship between any two
sections.

In order to demonstrate how the localisation of nullstates in a critical
section is altered by the partial ordering of sections, consider a 4 section
structure where $Q$ has the form

$Q=\begin{pmatrix}q_{1}&c_{12}&c_{13}&0\\\ 0&q_{2}&0&c_{24}\\\
0&0&q_{3}&c_{34}\\\ 0&0&0&q_{4}\end{pmatrix}$ (5)

where $q_{i}$ and $c_{i,j}$ are block matrices. The sections have the partial
ordering $g_{1}<g_{2}=g_{3}<g_{4}$. Suppose that section $g_{3}$ is marginal,
so $|q_{3}|=0$ and $|q_{1}|,|q_{2}|,|q_{4}|\neq 0$. Then there exists a null
eigenvector $\ket{\phi}$ such that $q_{3}\ket{\phi}=0$. The solution over all
of $Q$ is then given by

$\begin{pmatrix}q_{1}&c_{12}&c_{13}&0\\\ 0&q_{2}&0&c_{24}\\\
0&0&q_{3}&c_{34}\\\
0&0&0&q_{4}\end{pmatrix}\begin{pmatrix}-q_{1}^{-1}c_{13}\ket{\phi}\\\ 0\\\
\ket{\phi}\\\ 0\end{pmatrix}=0$ (6)

demonstrating the nullstate can have non-zero support on the black sites of
$g_{1}$ and $g_{3}$ only. Similarly $|q_{3}^{\dagger}|=0$ so we have a similar
solution for $Q^{\dagger}$ where

$\begin{pmatrix}q_{1}^{\dagger}&0&0&0\\\
c_{12}^{\dagger}&q_{2}^{\dagger}&0&0\\\
c_{13}^{\dagger}&0&q_{3}^{\dagger}&0\\\
0&c_{24}^{\dagger}&c_{34}^{\dagger}&q_{4}^{\dagger}\end{pmatrix}\begin{pmatrix}0\\\
0\\\ \ket{\psi}\\\
-(q_{4}^{\dagger})^{-1}c_{34}^{\dagger}\ket{\psi}\end{pmatrix}=0$ (7)

and $q^{\dagger}_{3}\ket{\psi}=0$. Hence this nullstate has non-zero support
on the white sites of $g_{3}$ and $g_{4}$ only. The localisation of the
support by sublattices is a direct consequence of the partial ordering, in
this example $g_{1}<g_{2}=g_{3}<g_{4}$. These rules generalise in a
straightforward way when there are more sections: the black zero energy state
is localised on the marginal section and those lower in the partial ordering,
while the white state is confined to the marginal section and those which are
higher.

It should be noted that within the critical section itself, for a randomly
selected distribution of hopping terms that satisfy the condition for a
section to be critical, the zero energy states have, with probability one,
support on every site of the section. Proofs of this, and of the general
relationship between the factorisation and localisation, are given in the
supplementary material section A.1.

It is natural to ask what happens when a structure has more than one critical
section. In some instances it is possible to have more than two zero energy
states. However, for most structures, for almost all sets of hopping terms,
there remain only two zero energy states when there are multiple critical
sections. This is because the null states originating from each critical
section must be orthogonal to each other, which imposes additional constraints
on the hopping terms, including those which connect the sections. The
conditions for such higher nullity will be explored in future work.

##  III Experimental demonstration of classification

To demonstrate the topological classification defined in Section III, we have
performed experiments on a coaxial cable network which represents the 4 row
ribbon graphene structure of Fig. 4. A network where all coaxial cables have
the same transmission time, $\tau$, maps on to a tight binding model [26, 30]
where the sites are the junctions in the network. The ‘energy’, $\varepsilon$,
is given by $\varepsilon=\cos\omega\tau$ where $\omega$ is the driving
frequency. This yields a Schrödinger type eigenvalue equation
$H\psi=\varepsilon\psi$. Entries in $\psi$ correspond to scaled voltages at
the junctions. Up to this scaling factor, individual hopping terms are given
by the reciprocal of the impedance of a cable connecting the sites. Swapping
between cables of different impedances thus allows us to traverse a
structure’s parameter space $\xi$.

Experimental measurements are made with a vector network analyser (VNA) and
takes two forms. From on-site reflectance measurements, we determine the
structure’s impedance, the real part of which is proportional to the local
density of states (LDOS) at that site [26]. We use this to demonstrate the
localisation of the null states. Transmittance measurement give the
relationship between the magnitude of a state on two sites. This gives a
direct experimental determination of the block triangular form of $Q$, Eq.(2),
and thus the classification of the structure.

####  III.0.1 Measuring the Local Density of States

Measuring the local density of states on every site allows us to characterise
the localisation properties of the zero energy states in a marginal structure
with multiple sections, of which only one is critical. Fig. 5 displays the
LDOS measurements of the disordered looped 4 row ribbon zigzag graphene in
Fig. 4. According to the classification of Sec. II, this structure has four
non-trivials sections, corresponding to the horizontal rows, so its
topological classification is $4\mathbb{Z}_{2}$. Individual cables are chosen
randomly from a binary distribution of $50\Omega$ and $93\Omega$ cables (see
more details in supplementary material section C). One section is critical,
the second row in the figure, while the other three are not. The localisation
of the nullstates is clear: the white state only has significant strength on
the top two rows, while the black state appears only on the lower three. This
is in agreement with the predicted localisation, given the ordering of the
rows $g_{1}>g_{2}>g_{3}>g_{4}$.

Figure 5: Experimental measurement of the impedance spectra in a topologically
marginal 4 row ribbon graphene structure. The strucure has four sections,
correponding to the rows of the structure. Spectra are shown for two sites on
the black sublattice, labelled $a$ and $b$ in the structure diagram in the
inset. In the plot of the structure the diameter of the circle representing
each site is proportional the amplitude of the zero energy state at that
point, which is derived from the strength of the corresponding impedance peak.
The widths of the lines showing the connections indicates the impedance of the
corresponding cable, with wide lines being 93$\Omega$ and narrow lines
$50\Omega$ cables. The red lines pick out the second row which is the critical
section. The localisation of the zero energy eigenstates onto distinct
sublattices on the upper and lower sections relative to the critical row
agrees with our theoretical prediction. Note that the structure has
cylindrical boundary conditions as displayed in Fig. 4 (b), with the hanging
connections at either side linked together.

####  III.0.2 Topological Classification through Transmittance

Measurements of the transmittance can be used to verify the triangular form of
$Q$, Eq.(2), and thus the division of the structure into sections. We describe
here an experiment where we make a cut in each section of the structure, by
disconnecting one end of a cable, and measure transmittance between various
cuts. We show that transmittance at zero energy ($\cos\omega\tau=0$) only
occurs when the measurement is within one section: the transmittance between
sections is zero.

In order to describe the transmittance of the coaxial cable network, we use a
transfer matrix formalism to relate the voltages and currents at the output
sites to those at the input sites. If we cut a site on every section (creating
an input and output site on each section), the transfer matrix will have
dimensions $2\tilde{N}\times 2\tilde{N}$, corresponding to the voltage and
current variables for each of the $\tilde{N}$ sections. Here we use
$\tilde{N}$ for the total number of sections, to distinguish from $N$, the
number of topologically non-trivial sections from earlier.

For a chiral structure at zero energy, all the variables can be coloured black
or white, according to the two sublattices, in such a way that the matrix
consists of two $\tilde{N}\times\tilde{N}$ diagonal blocks. At zero energy the
voltage at a given site is determined entirely by the currents flowing out of
the neighbouring sites. The neighbours in a chiral structure are on the other
sublattice, so, if we assign the voltage the same colour as its site, and the
currents flowing into and out of a site the opposite colour to the site, the
variables of the two colours are completely independent, giving the two
blocks.

To see how the form of the transfer matrix for the cut structure is related to
the sections, consider making a cut within a section, forming an input and
output site. The cut adds a site to the structure, unbalancing the black and
white sublattices in the section, and thus creating a new zero energy state.
For instance cutting on a white site creates one extra white site, resulting
in a zero energy state that only has support on the white sites of that
section and of any higher sections in the partial ordering. This means that
the block triangular form of the matrix $Q$ in Eq.(2) translates to a
triangular form of each block of the transfer matrix. This triangular form
persists regardless of the site at which the cut is made in each section; it
can shown to exits if, and only if, the uncut structure has at least
$\tilde{N}$ sections.

$g_{2}$$\bullet$$\bullet$$\bullet$$\circ$$\circ$$\circ$$\circ$$\circ$$g_{1}$$\bullet$$\bullet$$\bullet$$\circ$$\circ$$\circ$$\circ$$\circ$InputsOutputs
Figure 6: A structure with two sections, $g_{1}$ and $g_{2}$, with the partial
ordering of $g_{1}<g_{2}$ since white sites in $g_{1}$ connect to black sites
in $g_{2}$. The sections correspond to the ellipses, and contain arbitrary
numbers of sites, which are not shown individually. The hopping terms
connecting the two sections are denoted schematically by bunches of three
lines, but they may join any white sites of $g_{1}$ to any black sites of
$g_{2}$. Each section has been cut to create input and output sites for
transmittance measurements.

As a simple example, consider a structure with just two sections, $g_{1}$ and
$g_{2}$, as shown in Fig. 6. The white sites in $g_{1}$ connect to the black
sites in $g_{2}$, so in the partial ordering, $g_{1}<g_{2}$. Each section is
cut at a white site, so the white block of the transfer matrix relates the
input and output voltages, while the black block connects the currents. The
transfer matrix is thus

$\begin{pmatrix}I^{g_{1}}_{\text{out}}\\\ I^{g_{2}}_{\text{out}}\\\
V^{g_{1}}_{\text{out}}\\\ V^{g_{2}}_{\text{out}}\\\
\end{pmatrix}=\begin{pmatrix}\gamma&\delta&0&0\\\ 0&\beta&0&0\\\
0&0&\alpha&0\\\
0&0&\nu&\mu\end{pmatrix}\begin{pmatrix}I^{g_{1}}_{\text{in}}\\\
I^{g_{2}}_{\text{in}}\\\ V^{g_{1}}_{\text{in}}\\\ V^{g_{2}}_{\text{in}}\\\
\end{pmatrix},$ (8)

where the non-zero matrix entries represented by Greek letters are functions
of the hopping amplitudes in the Hamiltonian, and depend on the details of the
actual structure.

It is not possible to measure directly the form of such a transfer matrix
using a two-port VNA, which can only give the transmittance between one input
and one output site. The voltages and currents at the other input and output
sites cannot simply be made zero: we can either leave them open circuit, in
which case there can be a voltage on the site, but no current flowing in or
out, or they can be shorted, giving a current but no voltage. In our
experiment, the output site is always open circuit, and the choice of whether
to short or leave open the input site is made according to the partial
ordering.

To see how this works, consider the case where the input and output are
connected to section $g_{1}$ and the input to $g_{2}$ is open circuit. Then
$I_{\text{in}}^{g_{2}}=0$ and $I_{\text{out}}^{g_{2}}=0$. From Eq.(8), we get
$I_{\text{out}}^{g_{1}}=\gamma I_{\text{in}}^{g_{1}}$ and
$V_{\text{out}}^{g_{1}}=\alpha V_{\text{in}}^{g_{1}}$, so non-zero
transmittance occurs. However, if the input to $g_{2}$ is shorted, the
boundary conditions instead become $V_{\text{in}}^{g_{2}}=0$ and
$I_{\text{out}}^{g_{2}}=0$. This gives $I_{\text{out}}^{g_{1}}=0$ and
$V_{\text{out}}^{g_{1}}=\alpha V_{\text{in}}^{g_{1}}$, so no transmittance can
be seen, because this requires both the output voltage and current to be non-
zero. Hence, for our classification experiment to work, with intra-section
transmittance non-zero, we need the $g_{2}$ input site to be open circuit.

Proceeding in the same way, we find that if the input port is connected to the
input site of $g_{1}$ and the output port to the output site of $g_{2}$, we
get non-zero transmittance if the input to $g_{2}$ is shorted, but not if it
is open circuit. Thus, to see no transmittance between sections, we again need
$g_{2}$ to be open circuit.

If we do these experiments with input port connected to the input site of
$g_{2}$, the requirement is reversed: the $g_{1}$ input site has to be shorted
to get non-zero transmittance within the section $g_{2}$, and no transmittance
between sections $g_{1}$ and $g_{2}$. The reason for this difference is the
partial ordering of the sections $g_{1}<g_{2}$. However, these rules only work
because we have cut the sections on the white sites. Going through the
different cases, we find that the requirement for shorting or leaving open the
unused input sites depends on the sublattice of the site and the position of
the section in the partial ordering relative to the input site connected to
the VNA. These requirements are summarised in Table 1. When these rules are
satisfied, there is non-zero transmittance between the input and output sites
when they are on the same section, but not when they are on different
sections. This pattern is a direct consequence of the triangular blocks in the
transfer matrix, and does not occur otherwise. For example, in a structure
without sections but with two cuts on white sites, shorting one of the input
sites results in no transmittance for either output site.

Input site | Higher | Lower | Equal
---|---|---|---
Black | Short | Open | Open/Short
White | Open | Short | Open/Short
Table 1: Conditions for shorting or leaving open circuit the inputs to the
sections where the input site is not connected to the VNA. The requirements
depend on the sublattice which the input belongs to, black or white, and its
position in the partial ordering, higher or lower than the connected section.
With these choices, non-zero transmittance occurs only when the output site is
in the same section as the input. Figure 7:  (a) The looped four row zigzag
ribbon structure cut so it forms a flat sheet. The looped structure is
predicted to consist of four sections, corresponding to the four rows of
sites. Input sites for the transmittance measurements are labelled on the
right hand side, and outputs on the left hand side. For the disordered
structure, dashed lines indicate 93$\Omega$ RG62 cables, while solid lines
indicate 50$\Omega$ RG58 cables. The clean structure had only 50$\Omega$ RG58
cables. (b) Experimental transmittance data. Each subplot corresponds to a
different output site according to the labels in (a), with spectra shown for
all four inputs in each case. The blue spectrum is for the case where the
input and output site are on the same section, with the dashed red spectra
corresponding to transmittance between sections. The transmittance at zero
energy is expected to be zero if the input and output sites are in different
sections, but non-zero when they are in the same section. The experiment thus
confirms that each row is a separate section.

Having established these rules, we use transmittance measurements to verify
the presence of the sections in the looped four row zigzag graphene ribbon
shown in Fig. 4 (which has been used as an example elsewhere in this paper).
Recall that this structure has four sections, corresponding to the four rows
of sites. Where the subscript of each row is sequentially labelled, the
partial ordering of the sections is $g_{1}>g_{2}>g_{3}>g_{4}$. A cut is made
in each row, so the looped structure is transformed into a sheet, shown in
Fig. 7 (a), creating an extra site on each row. Note, however, that the
sections we find correspond to the uncut loop rather than the sheet. In two of
the sections, the cut is on a white site, while in the other two the site is
black. We perform transmittance measurements going through the four output
sites, and measuring a spectrum for every input sites in each case. The rules
in Table 1 determine whether the unconnected input sites are left open circuit
or shorted. For example, when the VNA is connected to the input site on the
second row, the first row, which is higher in the ordering and has a white
input site, is left open, while the third and fourth rows are, respectively,
shorted and open circuit. Using the transfer matrix for the clean structure,
where all the cables have impedance $Z_{0}=50\Omega$, this input gives

$\begin{split}\begin{pmatrix}V^{g_{1}}_{\text{out}}\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}I^{g_{2}}_{\text{out}}}}\\\
V^{g_{3}}_{\text{out}}\\\ I^{g_{4}}_{\text{out}}\\\ I^{g_{1}}_{\text{out}}\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}V^{g_{2}}_{\text{out}}}}\\\
I^{g_{3}}_{\text{out}}\\\
V^{g_{4}}_{\text{out}}\end{pmatrix}=\begin{pmatrix}1&Z_{0}i&-1&-Z_{0}i&&&&\\\
0&1&\frac{1}{Z_{0}}i&-1&&&&\\\ 0&0&1&Z_{0}i&&&&\\\ 0&0&0&1&&&&\\\
&&&&1&0&0&0\\\ &&&&Z_{0}i&1&0&0\\\ &&&&-1&\frac{1}{Z_{0}}i&1&0\\\
&&&&-Z_{0}i&-1&Z_{0}i&1\end{pmatrix}\begin{pmatrix}V^{g_{1}}\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}I^{g_{2}}_{\text{in}}}}\\\
0\\\ 0\\\ 0\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}V^{g_{2}}_{\text{in}}}}\\\
I^{g_{3}}\\\
V^{g_{4}}\end{pmatrix}=\begin{pmatrix}V^{g_{1}}+Z_{0}iI^{g_{2}}_{\text{in}}\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}I^{g_{2}}_{\text{in}}}}\\\
0\\\ 0\\\ 0\\\
\bm{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}V^{g_{2}}_{\text{in}}}}\\\
I^{g_{3}}+\frac{1}{Z_{0}}iV^{g_{2}}_{\text{in}}\\\
V^{g_{4}}+Z_{0}iI^{g_{3}}-V^{g_{2}}_{\text{in}}\end{pmatrix}\;.\end{split}$
(9)

For the disordered structure, the blocks of the matrix are still triangular,
but the expression for the elements are more complicated. The numerical value
for this matrix if given in the supplementary material section C.

If the other port of the VNA is not attached to the output site on the second
row, it is left open circuit, that is $I_{\text{out}}^{g_{2}}=0$. However,
this forces $I_{\text{in}}^{g_{2}}$ also to be zero, so no current enters the
structure, and no transmittance occurs. However, for the output on the same
row, $I_{\text{out}}^{g_{2}}=I_{\text{in}}^{g_{2}}$ and
$V_{\text{out}}^{g_{2}}=V_{\text{in}}^{g_{2}}$, so, in the absence of
absorption, there is a perfect transmittance of 1.

The transmittance experiment was carried out twice, first using a ‘clean’
structure with all 50$\Omega$ (RG58) cables, and then for a disordered
structure where hopping terms were randomly selected from a binary
distribution of 50$\Omega$ (RG58) and 93$\Omega$ (RG62) cables, identical to
the one in Sec. III.0.1. The clean structure has reflection symmetry about a
line between the second and third rows, so we need only need to use two
outputs, $\alpha$ and $\beta$ (labelled in Fig.7 (a)), but in each case we
measure transmittance for all four input sites. In the disordered structure
this symmetry is broken, requiring measurements for every input and output
site in order to complete the experiment.

The measured transmittance spectra are shown in Fig. 7 (b). Each panel shows
data for a particular output site, with the four spectra corresponding to the
different input sites. As predicted, in both the clean and disordered
structures, the transmittance at $\varepsilon=\cos\omega\tau=0$ is non-zero
only when the output site is on the same section as the input (blue curves).
This confirms that the structure has $4$ sections, experimentally verifying
the $4\mathbb{Z}_{2}$ classification of this structure. The actual value of
the zero-energy transmittance between the sites within the section is
determined by how close the section is to being topologically marginal. For a
marginal section, the ideal transmittance, in the absence of losses, would be
expected to have a value of unity [26]. Although the differences are not
great, it can be seen that the transmittance is highest in the clean
structure, where all the sections are marginal, and for the second row section
of the disordered structure (output (b)), which is also marginal.

As the structure has chiral symmetry, we expect a symmetry in the
transmittance spectra around zero energy. The slight asymmetry in the data is
mainly a result of losses in the cables, which have more effect at higher
frequencies, but there is also some chiral symmetry breaking due to
imperfections in the structure. These necessarily occur because the mapping of
the coaxial cable structure to the tight binding model requires the sections
of cable between the sites to be of uniform impedance. However, the SMA
connectors which form the junctions between the cables are $50\Omega$
components, so for the connections with $93\Omega$ cables this uniformity is
necessarily unachievable. As a result, the symmetry breaking is generally
greater for the disordered structure than the clean one, where minor
variations in the cable lengths are the likely cause.

##  IV Conclusion

We have described an approach to determining exact topological phases in
finite chiral structures, identifying the phase boundaries by the appearance
of degenerate pairs of zero energy states. This has been shown, in many cases,
to lead to a rich topological classification, obtained by finding a division
into sections which correspond to irreducible factors of the determinant of
the Hamiltonian. The topological classification is then $N\mathbb{Z}_{2}$,
where $N$ is the number of topologically non-trivial sections or factors. The
sections can be identified using the complete matchings of the underlying
structure, relating the topological classification to the structures
underlying connectivity. Each complete matching is related to a term in the
expansion of the determinant, so a hopping term which does not appear in any
matching can be omitted without changing the determinant. The sections
correspond to parts of the structure which become separated when these
connections are removed. We also give, in section B of the supplementary
material, a simple computational algorithm for finding the sections of a
structure.

We have defined a partial ordering of the sections in a structure which gives
rise to an unusual localisation of the zero energy states which are present
when one of the sections is topologically marginal. The zero energy states can
be separated so they each have support on a single sublattice. With our
definition, the white state is confined to the marginal section and those
which are higher in the ordering, while the black state appears on the
marginal section and lower sections. This localisation can be seen as a
property of finite structures which has some equivalence to the bulk boundary
correspondence in infinite structures.

This localisation provides a way in which the sections, and corresponding
topological classification can be demonstrated experimentally. We have
performed such experiments on simple coaxial cable networks which map directly
onto chiral tight binding models. When we make one section marginal, we can
use impedance measurements to map out the local density of states on each
site. This provides a direct demonstration of the predicted localisation
related to the partial ordering. Even without a marginal section, we can use
transmittance measurements to show that the structure separates into the
expected sections. We have used this method to confirm the $4\mathbb{Z}_{2}$
classification which our theory predicts for a four row ribbon graphene
structure.

## Acknowledgements

We are enormously grateful for help in the experimental work from Qingqing
Duan, Ben Kinvig, and Elena Callus, and also to Belle Darling and Phillip
Graham for huge help in finding space to run the experiments. Many thanks also
to Patrick Fowler, Barry Pickup, Qingqing Duan, Ben Kinvig, and Elena Callus
for many illuminating and insightful discussions while completing this work.

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Supplementary Materials: A Topological Classification of Finite Chiral
Structures using Complete Matchings

##  A Factorisation theorem

In this section we prove a theorem relating the triangular block form of $Q$
to the factorisation of $|Q|$. Consider a real or complex matrix $Q$, where
$Q$ is a chiral block of a Hamiltonian

$H=\begin{pmatrix}0&Q\\\ Q^{\dagger}&0\end{pmatrix}.$ (S1)

Below we show that, for algebraically independent hopping terms, then $|Q|$ is
reducible if and only if there exists a way of ordering the sites such that
$Q$ is upper block triangular. We refer to the resultant form of $Q$ as the
maximal triangular block basis, or triangular block basis of $Q$, but it
should be emphasised that this basis specifically corresponds to a permutation
of $H$. We will use this in section B to define an algorithm that classifies a
particular structure.
Note that often in the following arguments we will refer to a property that
occurs almost always. By this we mean that if hopping terms were selected
randomly from a continuous distribution (possibly one that satisfies a certain
constraint) then this property occurs with probability 1.

We proceed by showing that the factorisation is well defined, before proving
that a square weighted matrix has a factored determinant (for all matrix
entries) if and only if it is block triangular. In proving this relationship
we will show that when a section is singular, for almost all hopping terms a
section has non-zero support of an eigenstate on all sites.
In order to interpret determinants of a matrix as a polynomial, we consider a
polynomial ring that contains them. Formally this ring is larger then the set
of polynomials corresponding to determinants, but this is unimportant for our
discussion. In particular we are interested in determinants of matrices, so we
consider an $N\times N$ matrix,

$Q_{i,j}=X_{i,j}$ (S2)

where $X_{i,j}$ is an indeterminate over some field, or else fixed at 0, and
all non-zero $X_{i,j}$ are algebraically independent. Generically we take this
to be $\mathbb{R}$ or $\mathbb{C}$, but more arbitrary fields are perfectly
reasonable to consider. We then take the polynomial ring $P[X_{i,j}]$ over the
indeterminates $X_{i,j}$. The determinants $|Q|$ are polynomials in this ring
which are linear for each hopping term in $|Q|$. To consider the matrices that
may only represent tight binding models undergoing continuous evolution, then
we need to restrict the domain for each indeterminate to be $\mathbb{R}^{\pm}$
or $\mathbb{C}\setminus\\{0\\}$ or else fixed at 0. Non-zero indeterminates
are then in a semiring.
Formally when we classify a structure we are interested in the subspace
$E_{0}$ of $\xi$ which has no (exactly) zero energy states. We then wish to
calculate the number of ways we can map a Hamiltonian to this subspace, which
corresponds to calculating the zeroth homotopy group of $E_{0}$ under the
usual topology. The zeroth homotopy group of a space counts the number of
disconnected components of that space. The irreducible factorisation of $|Q|$
tells us the number of path connected components in $E_{0}$.

######  Definition A.1.

Let $X$ be the subspace of $\xi$ corresponding to a solution to $|Q|=0$. Then
$E_{0}:=\xi\setminus X$.

In order for our classification to be well defined, we need there to exist a
factorisation of $|Q|$ that is irreducible and unique, that is we need the
polynomial ring to be prime. This follows directly from the fact that the
indeterminates are taken over a field. In other words $P$ is a unique
factorisation domain. For a less abstract argument, consider the following:

######  Proposition A.2.

If for some hopping terms $|Q|\neq 0$ then $|Q|$ has a unique irreducible
factorisation in $P$.

###### Proof.

Suppose $|Q|=\prod a_{i}$ and $|Q|=\prod b_{i}$ such that $a_{i},b_{i}$ are
irreducible. Assume that for the irreducible factor $a_{j}$ there is no
irreducible factor $b_{k}$ such that $a_{j}=mb_{k}$ for some non zero constant
$m$, then

$\prod_{i\neq j}a_{i}\neq\frac{\prod_{i\neq k}b_{i}}{m}.$ (S3)

multiplying by $a_{j}=mb_{k}$ we see

$a_{j}\prod_{i\neq j}a_{i}\neq b_{k}\prod_{i\neq k}b_{i}$ (S4)

which is a contradiction. Hence for every $j$ there is a $k$ such that
$a_{j}=mb_{k}$ for some constant $m$, showing a factorisation of $|Q|$ is
unique and irreducible. ∎

Given that the irreducible factorisation of $|Q|$ is well defined, we now give
a proof that $|Q|=\prod|q_{i}|$ if and only if there exists a block triangular
structure of $Q$ with $q_{i}$ as diagonal blocks.

### A.1

Here we aim to prove that a determinant of $Q$ is factored if and only if
there exists a block triangular basis of $Q$. A corollary of one of the
propositions is that a critical section almost always has non-zero support of
the nullstates on every site. To prove this we first show that two sections,
pairwise, may only connect on one sublattice. Then we show that for a
structure with $N$ sections there must always be at least one section that may
only connect to any other section on one sublattice. As this is true for any
number of sections $N$ this implies that if $|Q|$ is factored, then there
exists a triangular block structure of $Q$.
More specifically, given a section $g_{i}$, if every first minor of $q_{i}$ is
almost always non-zero then deleting a black and a white site from $g_{i}$
will result in a structure with a complete matching, because the determinant
of the remaining structure is almost always non-zero. Therefore if we have two
sections, $g_{i}$ and $g_{j}$ and they connect to one another on both sub
lattices, then we can delete a black and a white site from both sections
(each) that are connected to one another. The remaining structure has a
complete matching because the associated minor is factored by a first minor of
$q_{i}$ and a first minor of $q_{j}$. Consequently $g_{i}$ and $g_{j}$ can
connect to one another on only one sublattice.
Our proof relies on knowing when hopping terms are in a complete matching of
some graph or not. For this we need to define a type of matching.

######  Definition A.3.

A valid matching is a matching between two sites on some graph $g$ such that
the matching is part of a complete matching of $g$.

We then show that there cannot exist a particular type of cycle of sections
(see Definition A.8). This ensures at least one section can only connect to
other sections from one sublattice. This is true for an arbitrary number of
sections, giving the triangular structure to $Q$.
For consistency if two sections $g_{i}$ and $g_{j}$ connect from a black site
in $g_{i}$ to a white site in $g_{j}$ we say $g_{i}$ connects to $g_{j}$ on
the black sub lattice.

######  Proposition A.4.

Given a section $g_{i}$ with tight binding model
$h_{i}=\begin{bmatrix}0&q_{i}\\\ q_{i}^{\dagger}&0\end{bmatrix}$ such that
$|q_{i}|$ is irreducible, then every first minor of $q_{i}$ is almost always
nonzero.

###### Proof.

Given that $h_{i}$ is chiral, a non-zero term in $|q_{i}|$ corresponds to a
complete matching of $g_{i}$. Any first order minor of $q_{i}$ can be accessed
by deleting a white and a black site from $g_{i}$, with the first order minors
corresponding to determinants of the $(N-1)\times(N-1)$ sub matrices of
$q_{i}$. So if there exists a complete matching of every structure where we
delete a black and a white site from $g_{i}$ then for almost all hopping terms
the minor is non-zero.
Suppose that we have a section with the set of complete matchings
$\mathscr{C}=\\{C_{i}\\}$ and delete a black site $b_{1}$. Match all the
remaining sites possible from the complete matching $C_{i}$, leaving a single
unmatched white site $w_{1}$. If an individual site is in only one valid
matching, as every complete matching must include a matching on every site,
this would factorise the determinant. So every site is in at least two
distinct valid matchings contained in the complete matchings $\mathscr{C}$. So
$w_{1}$ may be matched to a black site $b_{2}$ with a matching in $C_{j}$.
Removing the matching containing $b_{2}$ which is in the complete matching
$C_{i}$ leaves a second white site $w_{2}$ unmatched. This process has changed
which site is the unmatched white site, see Fig. S1 for an example. Iterating
this step defines a walk over the structure. If we delete the unmatched white
site, we automatically get a complete matching of the remaining structure,
showing the associated minor is almost always non-zero.
We now demonstrate any white site in a section may be unmatched by such a
walk. Assume there are a set $\mathscr{W}_{m}$ of white sites that cannot be
unmatched by iterating this walk, and a set $\bar{\mathscr{W}}_{u}$ of white
sites that can be unmatched by iterating this walk. In $C_{i}$ all
$\mathscr{W}_{m}$ white sites are matched to a black site. If there exists a
valid matching from a black site $b_{1}$ to a white site in $\mathscr{W}_{m}$
and a valid matching from $b_{1}$ to a white site in $\bar{\mathscr{W}}_{u}$,
then the site in $\mathscr{W}_{m}$ is unmatchable. So the black neighbours of
$\mathscr{W}_{m}$ sites do not have a valid matching to any white site in
$\bar{\mathscr{W}}_{u}$. If a white site has a valid matching to $b_{1}$ it is
possible to unmatch, so no white site $w_{m}\in\mathscr{W}_{m}$ has a valid
matching to $b_{1}$. This partitions the sites in to two sets: one with white
sites that can be unmatched $U$, and one with white sites that cannot be
unmatched $M$, with valid matchings. The valid matchings over $M$ are
therefore independent of the valid matchings over $U$, giving a factorisation
of $|q_{i}|$. By contradiction all white sites in a section are possible to
unmatch via such a walk. As the choice of $b_{1}$ was arbitrary, this proves
the existence of a complete matching of a section when deleting one white and
one black site, that is, every first minor of $|q_{i}|$ is non-zero. ∎

$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$(a)$\circ$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\circ$$\bullet$$\bullet$$\circ$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$(b)
Figure S1: (a) 3 rung ladder graphene, which has the classification
$\mathbb{Z}_{2}$, and all 3 complete matchings of the structure denoted with a
solid line, dashed line, and dash dot line. (b) Displays the structure having
deleted one of the black vertices, and matching all but one white vertex. By
swapping a single matching that neighbours the unmatched vertex at a time, it
is possible to leave any of the white sites unmatched. Deleting the unmatched
white vertex shows that for the deleted black vertex, deleting any of the
white vertices leaves a structure with a complete matching.

######  Corollary A.5.

Suppose a section is singular, then $q_{i}\ket{\psi}=0$ for some non-zero
$\ket{\psi}$, and for almost all hopping terms $\ket{\psi_{j}}$ indexed over
the $j$ sites of $g_{i}$ is non-zero for every $j$.

###### Proof.

By Prop. A.4 every first minor of $q_{i}$ is almost always non-zero. Therefore
any submatrix $\bar{q}_{i}$ resulting from deleting a white and black site of
$g_{i}$ is almost always non-singular. Projecting the eigenstate
$q_{i}\ket{\psi}=0$ on to the submatrix gives

$\bar{q}_{i}\ket{\bar{\psi}}\neq 0$ (S5)

for all $\bar{q}_{i}$. Therefore given any submatrix of
$\begin{bmatrix}\bar{q}_{i}&\alpha\end{bmatrix}$ of $q_{i}$ such that

$\begin{split}q_{i}&=\begin{bmatrix}\bar{q}_{i}&\alpha\\\ \hskip
10.00002ptA\end{bmatrix}\\\
\text{and}\,\,\,\ket{\psi}&=\begin{bmatrix}\ket{\bar{\psi}}\\\
\ket{\psi_{\alpha}}\end{bmatrix}\,\,\,\text{for}\,\,\,q_{i}\ket{\psi}=0\end{split}$
(S6)

we see

$\bar{q}_{i}\ket{\bar{\psi}}=-\alpha\ket{\psi_{\alpha}}.$ (S7)

As every first minor is non-zero then for almost all hopping term values
$\ket{\psi}$ has non-zero support on every site of $g_{i}$. ∎

We now wish to show that, given a pair of sections $g_{i},g_{j}$ then they may
only connect on one sublattice.

######  Proposition A.6.

Given two section $g_{i},g_{j}$ then non-zero hopping terms may only be
between black sites on $g_{i}$ to white sites on $g_{j}$.

###### Proof.

Given two sections $g_{i},g_{j}$ assume they connect to one another on both
sublattices. Delete four sites, a black and a white site in $g_{i}$ and a
white and a black site in $g_{j}$ that connect to the original white and black
site $g_{i}$. By Prop. A.4 the resulting structure has a complete matching.
Therefore if we put back in the four deleted sites, and match them via their
associated hopping terms, then there exists a complete matching of $g_{i}$ and
$g_{j}$ that is not factored by $|q_{i}|$ and $|q_{j}|$. So by contradiction
$g_{i}$ and $g_{j}$ connect on only one sublattice. ∎

To show that given $N$ sections there exists a section which can only connect
to any other section from one sub lattice, we first define an edge section.

######  Definition A.7.

An edge section is a section that may only connect to all other sections from
one sub lattice.

To prove there always exists an edge section we consider a cycle of sections.

######  Definition A.8.

A section cycle is a path through a subset of sections in $G$ such that it
starts and ends on the same section following hopping terms in $g_{i}$ and
leaves each section on a different sub lattice to the one it entered on.

We now show there cannot exist a section cycle.

######  Proposition A.9.

A section cycle cannot exist in any structure.

###### Proof.

Suppose every section is in a cycle, then every section connects to a distinct
section on both sub lattices. For each distinct section cycle we can delete a
pair of sites, one black and one white, from each section in the cycle such
that the black and white sites connect to different sections in the cycle. By
Prop. A.4 the resulting structure has a complete matching. Putting back in the
deleted sites and hopping terms, we can now construct a complete matching that
is not factored by any of the sections in that cycle. Therefore by
contradiction there exist no section cycles. ∎

######  Corollary A.10.

In a structure with $N$ sections there exists at least one edge section.

###### Proof.

Suppose every section $g_{i}$ connects to at least one other section $g_{j}$
on the white sub lattice and one different section $g_{k}$ on the black sub
lattice. Given such a requirement, then if no edge section exists there must
be at least $N+1$ sections. Therefore by the pigeonhole principle, if every
section connects on both sub lattices a section cycle exists, which is not
possible by Prop. A.8 giving a contradiction. ∎

We now wish to show that the existence of an edge section gives the triangular
basis to $Q$.

######  Theorem A.11.

If $|Q|=\prod|q_{i}|$ for $|Q|,|q_{i}|\in P[X_{i,j}]$ then there exists a
permutation of $H$ that gives a triangular block basis for $Q$.

###### Proof.

Given a structure with $N$ sections then by corollary A.10 there is at least
one section, $g_{1}$, which only connects to all others from one sub lattice.
Taking $G$ and deleting $g_{1}$ yields a new structure, $\bar{G}$, with $N-1$
sections. By corollary A.10 this structure also has an edge section $g_{2}$.
This may be iterated until only one section remains. This gives the partial
ordering of the structure and therefore defines a triangular block basis of
$Q$. ∎

##  B A classification algorithm

We present an algorithm to find the classification of a structure based on
Theorem A.11. That is, we take some input structure, with a chiral Hamiltonian
and find an ordering of the sites where $Q$ is block triangular, and every
diagonal block corresponds to a section of the structure. We refer to the
corresponding basis as the maximal triangular block basis of $Q$. Then, given
the domain of each hopping term and numerical bounds on the determinant, we
check if a section is trivial or not. We anticipate that this algorithm can be
significantly optimised, but currently we are able to analyse up to random
networks with around 50 sites. Of course, given a more consistent underlying
lattice structure it often makes sense to classify a large (say many thousand
sites or more) system by proving some simple results about the complete
matchings of that lattice, coupled with some boundary conditions.
There are two main parts of the algorithm: to find a triangular block basis of
$Q$ with all sections on the diagonal, and then to classify each section. The
former is more technically challenging. The first part uses the fact that to
get a triangular block structure to and $N\times N$ size $Q$ there needs to be
a $j\times(N-j)$ block of zeros, so by taking the complement of $Q$ (with all
unit hopping terms) then we can find overlaps in zeros by taking dot products
between column vectors, or a generalised product over several column vectors.
The generalised product is defined as follows.

######  Definition B.1.

The generalised product of $N$ column vectors $\\{c^{i}\\}$ is given by

$p=\sum_{j}\prod_{i}c^{i}_{j}$ (S8)

where $j$ denotes the $j$th entry of the column vector $c^{i}$.

Given $N-j$ columns, if $p\geq j$ then there is a $j\times(N-j)$ block of
zeros in $Q$. The algorithm then follows the following general outline

  1. 1.

Compute $Q^{\circ}$ where $Q^{\circ}_{i,j}=1$ if $Q_{i,j}=0$, otherwise
$Q^{\circ}_{i,j}=0$

  2. 2.

Check for $1\times(N-1)$ blocks of zeros by checking $\sum_{i}Q^{\circ}_{i,j}$
for every $j$. Take indices of all blocks of this size.

  3. 3.

Find pairwise overlap matrix, A, of two column vectors of $Q^{\circ}_{i,j}=1$,

$A_{i,j}=\sum_{k}Q^{\circ}_{k,i}Q^{\circ}_{k,j}$ (S9)

  4. 4.

If any $A_{i,j}\geq N-2$ there is a $2\times(N-2)$ block of zeros. Take
indices of all blocks of this size.

  5. 5.

For $j\geq 2$ take all columns that have overlap greater than $N-j$. If number
of columns with such an overlap $\geq N-j$ set $\text{Check}=1$.

  6. If Check==1:

  7. For all combinations of $N-j$ columns with pairwise overlap $\geq N-j$ compute p.

  8. If $p\geq N-j$ return indices of $j\times(N-j)$ block of zeros.

A simple way to optimise this algorithm slightly is to project on to a sub
matrix of $Q$ that excludes any individual section, whenever a relevant
$j\times(N-j)$ block of zeros is found. Then the combinations of column
vectors that need to be checked to find more blocks of zeros is significantly
smaller. Furthermore the complexity of the search scales with the number of
potential column vectors to check. So if the search for zero blocks can be
done up until $j\leq\frac{N}{2}$ for the column vectors, and then switched to
do a search over the remaining row vectors the algorithm may be faster.
The second part of the algorithm is much simpler, and only applies when
hopping terms are restricted to being real. By the fundamental theorem of
algebra if all hopping terms are complex then a solution to $|q_{i}|=0$ always
exists for at least a $2\times 2$ size block $q_{i}$ corresponding to a
section. For hopping terms restricted to $\mathbb{R}^{\pm}$ we require some
more computation. The first part of the algorithm computes the number of
sections of $Q$, and by indexing blocks of zeros finds a maximal triangular
block basis for $Q$. So we now need to see if each factor can be set to 0.
This works by considering bounds for the largest value of $|q_{i}|$ for all
$|u|\in(0.5,1]$ hopping terms, where the sign of $u$ set by the domain of that
individual hopping term (i.e. if $u\in\mathbb{R}^{+}$ then $u>0.5$ and if
$u\in\mathbb{R}^{-}$ then $u<-0.5$). All hopping terms are then selected as a
random float in $\pm(0.5,1]$, and it is checked if this gives a singular
section, if not we proceed, otherwise we reselect randomly reselect hopping
terms. Suppose all hopping terms in $q_{i}$ are non-zero, then by Prop. A.4
every hopping term appears in the expansion of $|q_{i}|$. The maximum number
of non-zero terms in the determinant of an $N\times N$ matrix is $N!$ and so
$|q_{i}|\leq N!$. Therefore if a particular hopping term, $a$, is left free to
vary then $|q_{i}|=aA+B$. If solutions to $|q_{i}|=0$ exist for some $a$ then
for $|a|\times(0.5)^{N-1}>(N-1)!$ and $|q_{i}|$ has one sign, and for $a=0$
then $|q_{i}|=B$ and has a different sign. This is then repeated for every
hopping term until a solution is shown to exist, or it is shown to not exist
for all sections. That is

  1. 1.

Randomly select all hopping terms $u$ so that $|u|\in(0.5,1]$ and the sign
corresponds to the domain of that hopping term. If $|q_{i}|\neq 0$ proceed,
otherwise randomly reselect all hopping terms until $|q_{i}|\neq 0$.

  2. 2.

Keeping the hopping terms as in the first step set $a=\pm 2^{N}(N-1)!$ where
the sign is specific to the domain of $a$. Calculate $\text{Sign}(|q_{i}|)$.

  3. 3.

Keeping the hopping terms as in the first step set $a=0$ where the sign is
specific to the domain of $a$. Calculate $\text{Sign}(|q_{i}|)$.

  4. 4.

If a change of sign is found:

  5. Return section non-trivial

  6. 5.

Else:

  7. If all hopping terms checked:

  8. Return section trivial

  9. Else:

  10. Set $a$ to its value from step 1 and repeat from step 2 for a different hopping term $b$

This part of the algorithm can be optimised by finding a set of sites
connected on a loop within the section that is itself non-trivial, then only
one hopping term would need to be varied (one from this ring) for $q_{i}$ to
be non-trivial.

##  C Experimental details

Experimental data were collected with a NanoVNA V2 Plus 4 and NanoVNA V2 Plus
4 pro. Two measurements were considered: two port measurements for
transmittance and a single port measurement for reflectance. To operate the
VNA, the software NanoVNA-Saver was used. In making a structure, cables were
taken from a binary distribution of 50$\Omega$ RG58 SMA cables, and 93$\Omega$
RG62 SMA cables. SMA connecters are generically available at only 50$\Omega$
impedances and so for the clean structure all cables were chosen to be
50$\Omega$. The exact cables used in the disordered structures are displayed
in Fig. S2.
Data were collected between $1-240$MHz with $\approx 114MHz$ being the
frequency at which $\varepsilon=\cos\omega\tau=0$.
For transmittance data, the four input states described in Table 1 were
measured in both the clean and disordered structure, and transmittance data
taken in separate experiments on each of the four output sites.

Input site | Input 1 | Input 2 | Input 3 | Input 4
---|---|---|---|---
$g_{1}$ | $(I,V)$ | Open | Open | Open
$g_{2}$ | Open | $(I,V)$ | Short | Short
$g_{3}$ | Short | Short | $(I,V)$ | Open
$g_{4}$ | Open | Open | Open | $(I,V)$
Table 1: The input states for the classification experiments in the graphene
CCN. An entry of $(I,V)$ denotes that this is the site where the input port of
the VNA is attached to the CCN.

At zero energy, the transfer matrix for the disordered 4 row ribbon graphene
is given by the following

$\begin{pmatrix}\frac{50}{93}&100i&-\frac{143}{93}&-\frac{2500}{804357}i&&&&\\\
0&1&\frac{143}{2500}i&-\frac{8649}{2500}&&&&\\\
0&0&\frac{93}{50}&\frac{557450}{8649}i&&&&\\\ 0&0&0&\frac{50}{93}&&&&\\\
&&&&\frac{93}{50}&0&0&0\\\ &&&&\frac{557450}{8649}i&\frac{50}{93}&0&0\\\
&&&&-\frac{1161857}{804357}&\frac{143}{8649}i&1&0\\\
&&&&-\frac{125000}{8649}i&-\frac{50}{93}&186i&\frac{93}{50}\end{pmatrix}$
(S10)

where numerical values are a consequence of the $50$ $\Omega$ and $93$
$\Omega$ coaxial cables. By applying the four input states of Table 1 to
equation (S10) shows the transmittance is only non-zero for sites in the same
section.

$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\circ$$\bullet$$\bullet$$\circ$$\circ$$\bullet$$\bullet$$\circ$$\bullet$$\bullet$$\circ$$\circ$$\bullet$$\circ$InputsOutputs
Figure S2: The disordered structure the transmittance experiment was performed
on, as discussed in section III 2 of the paper, with sites in each section
already cut for measurements. The dashed lines denote the 93$\Omega$ RG62
cables, and the solid lines denote 50$\Omega$ RG58 cables. The same structure
with all 50$\Omega$ RG58 cables is the clean structure used for the
transmittance experiments. The dash-doted ellipses denote the input and output
sites. The same structure with uncut sites was used for the localisation
experiment in section III 1 of the paper. This structure has 4 sections and a
4$\mathbb{Z}_{2}$ classification.