# Deep learning for disordered topological insulators through entanglement
spectrum

Alejandro José Uría-Álvarez<EMAIL_ADDRESS>Departamento de Física de
la Materia Condensada, Condensed Matter Physics Center (IFIMAC), and Instituto
Nicolás Cabrera (INC), Universidad Autónoma de Madrid, Cantoblanco 28049,
Spain  Daniel Molpeceres-Mingo  Departamento de Física de la Materia
Condensada, Condensed Matter Physics Center (IFIMAC), and Instituto Nicolás
Cabrera (INC), Universidad Autónoma de Madrid, Cantoblanco 28049, Spain  Juan
José Palacios<EMAIL_ADDRESS>Departamento de Física de la Materia
Condensada, Condensed Matter Physics Center (IFIMAC), and Instituto Nicolás
Cabrera (INC), Universidad Autónoma de Madrid, Cantoblanco 28049, Spain

###### Abstract

Calculation of topological invariants for crystalline systems is well
understood in reciprocal space, allowing for the topological classification of
a wide spectrum of materials. In this work, we present a technique based on
the entanglement spectrum, which can be used to identify the hidden topology
of systems without translational invariance. By training a neural network to
distinguish between trivial and topological phases using the entanglement
spectrum obtained from crystalline or weakly disordered phases, we can predict
the topological phase diagram for generic disordered systems. This approach
becomes particularly useful for gapless systems, while providing a
computational speed-up compared to the commonly used Wilson loop technique for
gapful situations. Our methodology is illustrated in two-dimensional models
based on the Wilson-Dirac lattice Hamiltonian.

††preprint: APS/123-QED

Introduction— Identification of topological materials requires the computation
of quantities such as the Berry phase, the Chern number [1] or the
$\mathbb{Z}_{2}$ index [2], the so-called topological invariants. The
evaluation of Wilson loops [3], the most common methodology used to unveil the
presence of invariants, works well for crystalline systems since they are
performed in reciprocal space, to the point of allowing for high-throughput
screening of materials [4, 5]. Other approaches based on reciprocal space to
condensed matter topology exist such as topological quantum chemistry [6, 7],
symmetry indicators [8, 9, 10], or the scattering invariant approach [11].
Alternatively, one can resort to the bulk-boundary correspondence: if the
system is topological, we expect the presence of conducting surface states. By
calculating different observables, such as the conductance or the density of
states, one strives to find evidence of the topology, without directly
computing any invariant [12].

The use of the Wilson loop technique requires the existence of a direct band
gap everywhere in the Brillouin zone (e.g., no band crossings) near the Fermi
level. An overall gap does not need to exist and the system may still be
topologically non-trivial. For disordered or non-translationally invariant
systems where the bands are not well-defined (e.g. in an open system), or are
defined in a very small Brillouin zone (a large supercell where the spectrum
becomes essentially discrete) the direct gap concept is lost. In the latter
case, if an overall gap is clearly visible, the Wilson loop can still be
calculated [13]. However, the absence of an spectral gap does not preclude a
non-trivial topology [14].

The recent introduction of topological markers such as the Chern marker [15]
and the Bott index [16, 17], has changed our view that working in the
reciprocal space is essential since they manage to provide information on the
topology of the system based exclusively on real-space computations. For time-
reversal topological insulators, on the other hand, much less has been
reported in this regard, with the exception of a variant of the Bott index
which enables a real-space study of the $\mathbb{Z}_{2}$ number [16] or a spin
Bott index [18] in analogy with the spin Chern number [19].

Some quantities that have been shown to be related to the topology of the
system are the entanglement spectrum and entropy [20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31]. These magnitudes measure the degree of entanglement of
our ground state between two halves of the system. In the presence of
translational symmetry, the $\mathbb{Z}_{2}$ index can be defined from the
entanglement spectrum [22] through its flow, much in analogy with the hybrid
Wannier charge center (HWCC) flow [32]. Resorting to the insulating picture,
it is known that a topologically non-trivial ground state presents in-gap
single-particle states that circulate around the material, wrapping it. The
flow of the entanglement spectrum thus reflects the appearance of surface
states upon the separation of the two halves. Again, however, for disordered
systems where translational symmetry is lost, we cannot resolve the spectral
flow, unless one resorts to inconveniently large supercells.

Here we will show that one can still use the entanglement spectrum for
disordered systems where no momentum component is conserved with the aid of
artificial neural networks (ANNs). Machine learning (ML) algorithms, and ANNs
in particular, have been shown to accurately predict topological phases on a
wide range on inputs, such as wavefunctions [33, 34, 35], density matrices
[36, 37], Berry curvature [38], Hamiltonians [39, 40]. It was also
demonstrated that the entanglement spectrum can be used to train ML algorithms
to identify topology in translationally invariant systems [41, 42], as a
function of disorder in one-dimensional AIII models [43], or localization
phases in interacting systems [44]. In our case, we consider disordered two-
dimensional (2D) time-reversal topological insulators. By training an ANN to
differentiate between topological and trivial entanglement spectra in models
whose invariant is known or can be computed through the Wilson loop technique,
we show that we can predict the topology of the system without resorting to
the calculation of momentum-space flows. More importantly, our ANN is blind to
the absence or existence of a gap in the system.

Model— Our choice is the Wilson-Dirac lattice fermion model, which is a
discretized version of the Dirac Hamiltonian on a cubic lattice, with an
additional mass term to remove unphysical states when comparing with the
continuum Hamiltonian [45]. In real-space, this tight-binding model reads in
the following way [46]:

$\displaystyle H=\sum_{i,\mu}$
$\displaystyle\left[i\frac{t}{2}c^{\dagger}_{i+\mu}\alpha_{\mu}c_{i}+\frac{r}{2}c^{\dagger}_{i+\mu}\beta
c_{i}+\text{h.c.}\right]$ $\displaystyle+(M-3r)\sum_{i}c^{\dagger}_{i}\beta
c_{i}$ (1)

where the index $i$ sums over lattice positions, and $\mu$ sums over spatial
coordinates ($\mu=x,y,z$). In other words, hopping is only considered between
first neighbours along the cartesian axis (a cubic lattice is assumed).
$\\{\alpha_{\mu}\\}_{\mu},\beta$ denote gamma matrices, given by
$\alpha_{\mu}=\sigma_{x}\otimes\sigma_{\mu}$, $\beta=\sigma_{z}\otimes I$.

In the following, we will fix $r=1$ and $t=1$ and we will restrict ourselves
to a 2D square lattice. This model is known to be able to describe topological
insulators [47], with the parameter $M$ tuning the different topological
phases. For $1>M>3$ and $3>M>5$, it describes a topological insulator, while
for any other value it corresponds to a trivial insulator. This can be readily
checked by computing the $\mathbb{Z}_{2}$ invariant with the help of the HWCC
or eigenvalues of the Wilson loop
$W(k_{y})=\prod_{k_{i}\in\text{path}}M_{k_{i},k_{i+1}}$, where
$M_{k_{i},k_{i+1}}=U^{\dagger}(k_{i},k_{y})U(k_{i+1},k_{y})$ and
$U(k_{i},k_{y})$ is the unitary matrix that diagonalizes the Bloch Hamiltonian
in the atomic gauge [48].

It has also been shown that this model can also realize non-trivial topology
in disordered systems [33, 49, 50, 51]. Following [49], we introduce a
generalized version of the Wilson-Dirac model (1) to describe crystalline
disorder or amorphous solids:

$\displaystyle H=\sum_{i,j}$
$\displaystyle\frac{i}{2}t(R)c^{\dagger}_{i}(\sin\phi\sin\theta\alpha_{x}+\sin\phi\cos\theta\alpha_{y}+$
$\displaystyle\cos\theta\alpha_{z}-i\beta)c_{j}+\sum_{i}\beta(M-3)c^{\dagger}_{i}c_{i}.$
(2)

Here, the variables $(R,\phi,\theta)$ denote the spherical coordinates of the
vector determined by the relative position between lattice sites $i,j$,
randomly placed near their original crystal positions. The degree of disorder
is characterized by the parameter $\Delta r$, which measures the
characteristic distance of the site from its crystal position. As for the
hopping amplitude $t\equiv t(R)$, we introduce a dependence with the bond
length $R$ through an exponential law:

$t(R)=\exp\left({\frac{a-R}{a}}\right)\theta_{H}(R-R_{c}).$

where $a$ is some reference lattice spacing (which here will be set $a=1$).
Thus, when $R=a$, we recover $t=1$ as in the original model. Note that there
is also a Heaviside step function $\theta_{H}$, which serves as a cut-off for
bonds between atoms that are too far apart, $R_{c}$ being the cut-off
distance. It can be seen that when the lattice sites are restricted to the
cubic lattice, we recover the same Hamiltonian in (1).

Entanglement— Entanglement can provide information about correlations in
complex systems, depending on the chosen vector space, such as orbital, spin
or spatial [28, 29]. In the context of topological materials, it was shown
that an entanglement cut of the system into two halves reveals a non-trivial
flow of the entanglement spectrum, which can be understood as edge states
appearing due to the virtual edge formed by the entanglement cut [24, 25, 26,
22, 27]. Alternatively, it was shown that the entanglement spectrum can be
interpreted as coarse-grained hybrid Wannier centers [52, 53, 54], therefore
measuring charge pumping across the cut [32, 55].

The entanglement cut means partitioning our $N$-particle Hilbert space
$\mathcal{H}$ into two subspaces $\mathcal{H_{A,B}}$ such that
$\mathcal{H}=\mathcal{H_{A}}\otimes\mathcal{H_{B}}$. According to the Schmidt
decomposition theorem, any state $\ket{\psi}\in\mathcal{H}$ can be written
uniquely as
$\ket{\psi}=\sum_{i}e^{-E_{i}}\ket{\alpha_{i}}\otimes\ket{\beta_{i}}$, where
$\\{\ket{\alpha_{i}}\\}_{i}$, $\\{\ket{\beta_{i}}\\}_{i}$ denote some basis
states from the corresponding Hilbert spaces $\mathcal{H}_{A/B}$, and
$\\{E_{i}\\}_{i}$ form the many-body entanglement spectrum. From the reduced
density matrix, defined as $\rho_{A}=\mathrm{Tr}_{B}\rho=e^{-H_{e}}$, $H_{e}$
being the entanglement Hamiltonian, we extract the entanglement spectrum as
its many-body eigenvalues.

While in principle we could use this theorem to decompose the ground state, in
practice it is better to use an alternative approach. It was proven that for
non-interacting systems, the entanglement spectrum can be obtained from the
one-particle correlation function [56]:

$C_{ij}=\braket{\Psi}{c^{\dagger}_{i}c_{j}}{\Psi},\ i,j\in A$ (3)

where $\ket{\Psi}=\prod_{n,k}c^{\dagger}_{nk}\ket{0}$ is the Fermi sea. The
eigenvalues of $C$ are called single-particle entanglement spectrum, and they
are related to the spectrum of $\rho_{A}$ via a monotonic function [56],
meaning that we can use either indistinctly. For our purposes, it suffices to
use the single-particle entanglement spectrum, although the many-body one can
be obtained by filling states in the partitions in all possible combinations
[22].

If we define $H^{\prime}=\frac{1}{2}-C$, $C$ evaluated over the whole system
can be regarded as a flattened Hamiltonian [27], with an energy band
$\varepsilon=-1/2$ for occupied states, and $\varepsilon=1/2$ for unoccupied
states. Then, by virtue of the adiabatic theorem [57], since both $H$ and
$H^{\prime}$ have a gap, they share the same topology as there is a family of
Hamiltonians that interpolate between them [25]. Thus, the restriction of $C$
to a half-space results in edge states appearing in its spectrum, connecting
the valence and conduction bands.

Following the model we have introduced, the rest of the discussion will be in
2D. In case that our system has translational symmetry, the entanglement cut
will preserve it in one direction, meaning that, e.g., $k_{y}$ is a good
quantum number. The one-particle correlation matrix can be calculated as a
function of $k_{y}$:

$C_{ij}(k_{y})=\braket{\Psi}{c^{\dagger}_{ik_{y}}c_{jk_{y}}}{\Psi}=\sum_{k_{x}}e^{k_{x}(i-j)}P_{ij}(\textbf{k})$
(4)

allowing to obtain the entanglement spectrum as a function of $k_{y}$, thus
resolving the spectral flow characteristic of the edge states [22]. Here,
$P_{ij}(\textbf{k})$ denotes the matrix elements of the projector onto the
ground state.

From a computational standpoint, the ${k}$-resolved entanglement spectrum
approach does not provide an advantage relative to the computation of HWCC,
since both scale as $\mathcal{O}((N\times N)^{3}\times nk\times npaths)$,
where $N$ defines the bidimensional system size. However, for amorphous or
disordered systems, where a supercell is needed, the use of the real-space
entanglement spectrum proves to be advantageous as it drastically reduces the
number of diagonalizations needed, while still allowing us to determine the
$\mathbb{Z}_{2}$ invariant. Furthermore, its use is mandatory when the systems
become gapless, as it usually happens for strong disorder.

Neural Network— Our aim now is to extract qualitative features from the
entanglement spectrum where the topology is expected to be encoded. To be able
to tell apart the spectrum of a trivial case from that of a topological one,
on top of the automated invariant computation, we introduce an ANN. For
topological materials one would naively expect states within the gap of the
entanglement spectrum, although flattened Hamiltonians might present midgap
energies as well without being necessarily topological [26]. Otherwise, the
spectrum does not present intricate features, meaning that a simple network
suffices to classify correctly the phases. As for the network architecture,
our election was a one-dimensional convolutional neural network, with a
sigmoid output for binary classification. To tackle the three-dimensional
case, the output would have to be a softmax with three neurons, to distinguish
between weak and strong topological phases, and trivial. Training with more
complex networks (e.g. more convolutional layers and/or more dense layers)
does not seem to improve the performance. One should be aware that inversion-
symmetric materials can present non-trivial spectral flows [24], which could
result in false positives. Since we are dealing with disordered topological
insulators, this will not be problematic.

The training of the ANN is done by explicitly computing the $\mathbb{Z}_{2}$
invariant (when possible) and associating it with its corresponding
entanglement spectrum. We first use data from the crystalline regime (zero
disorder), whose invariant is easy to compute, and then associate it with the
entanglement spectrum corresponding to a supercell big enough so that it is
representative of the actual samples for which we want to determine their
topology. For completeness, we also use spectra corresponding to the low
disorder region ($\Delta r<0.05$) to train the ANN. The only difficulty here
is that the computation of the topological invariant is expensive since a
supercell is needed and cannot be done for strong disorder where the gap is
lost. Nevertheless, as we show below, non-trivial topology can be predicted
beyond the training set.

Two different systems will be studied with this methodology: first, a square
lattice which is increasingly deformed by random displacements of the atoms,
to the point of the crystal becoming amorphous. Second, a fractal lattice,
specifically the Bethe lattice. We will show that in both cases we are able to
predict the presence of topological phases by means of the entanglement
spectrum, as confirmed by the direct observation of edge states.

Figure 1: (Top) Gap diagrams for the Wilson-Fermion model on a square lattice
as a function of the mass parameter $M$ and the displacement $\Delta r$ of the
atoms, for two different cutoff distances. (Bottom) Topological phase diagrams
predicted by the ANN, in terms of the outputted probability. Black lines
correspond to contour lines from the gap diagram for $0.1$ eV. (Left)
$R_{c}=1.1$, (Right) $R_{c}=1.4$. The cell size is $N=30$ (30 unit cells in
each direction).

Amorphous solid— The specific disorder model we use is set in the following
way. Given a maximum displacement value $\Delta r$, which we take as the
disorder parameter, we define the following random variables:

$r\sim U(0,\Delta r),\ \theta\sim U(0,\pi/2),\ \phi\sim U(0,2\pi)$

where $U(a,b)$ denotes a uniform distribution between $a,b$, $a<b$. Note that
out-of-plane displacements are allowed. Out of one sample of these variables,
we generate a displacement vector given by
$\Delta\textbf{r}=r(\sin\theta\sin\phi,\sin\theta\cos\phi,cos\theta)$, so the
final position of each atom in the supercell is
$\textbf{r}_{i}=\textbf{r}^{0}_{i}+\Delta\textbf{r}$, where
$\textbf{r}^{0}_{i}$ denotes the position if the lattice were crystalline. As
we increase the value of $\Delta r$ , the lattice becomes more disordered
until long-range order is lost. The topological phase diagram will be obtained
as a function of the maximum displacement $\Delta r$ and the mass parameter
$M$. In the following we will work with one supercell only, imposing periodic
boundary conditions.

To obtain the phase diagrams, first we have to generate data, both for
training and for prediction. We compute the entanglement spectrum for all
combinations of $M$ and $\Delta r$ [see Figs. 2(c) and 4(c) and Suppl.
Material for examples]. The training set will be given by the points
corresponding to zero or very low disorder for $R_{c}=1.1$. As long as the
functional form of the Hamiltonian remains the same, we expect the neural
network to be valid even if it has not seen data from the system with
$R_{c}=1.4$.

Figure 2: (a) Edge state, (b) Energy spectrum for both open and periodic
boundary conditions and (c) Entanglement spectrum for $M=3$, $R_{c}=1.4$,
$\Delta r=0.5$.

In Fig. 1 we show the topological phase diagrams obtained with the trained
model for the largest systems studied (lower panels). We also show the gap
(upper panels), which shows a weak dependence with the system size [see Fig.
3(a)]. Here, we choose to plot the output of the ANN as an estimator of the
probability of being in the topological phase. As commonly accepted, for
probabilities higher than $0.5$ the system is considered to be topological.
The model predicts the existence of topological states even in regions where
the gap has vanished.

Figure 3: (a) Gap for $\Delta r=0.25$ and different system sizes. (b)
Coordination number of the solid for different cutoff distances as the
disorder increases. (c) Average edge occupation of the lowest 21 eigenstates
in energy for $R_{c}=1.1$ (d) Same as (c) for $R_{c}=1.4$.

To verify the predictions of the ANN, we look for edge states near the Fermi
energy where the ANN predicts non-trivial topology for disorder values outside
the training set. In Fig. 2(a) we show an edge state obtained from one
instance of the model near zero energy for $R_{c}=1.4$ and disorder $\Delta
r=0.5$. The gapful spectrum for this particular case is shown in Fig. 2(b) for
periodic and open boundary conditions. We see a standard edge state in the
sense that it is delocalized around the edge of the solid, as it would be
expected. If we take a look at the bonds between atoms, the crystal has a high
percolation due to more bonds appearing as we increase disorder. However, for
the smaller cutoff distance $R_{c}=1.1$, the solid starts to break, as shown
by the diminishing coordination number in Fig. 3(b). This means that there are
less paths available for a state to spread along, or equivalently that it has
a lower percolation. If we now take a look at some edge state in the regions
predicted to be non-trivial by the ANN, as in Fig. 4(a), we see that the
occupation is not that we would have expected for an edge state, that is,
around the borders of the solid. Still, looking closely we see that the
electronic density appears mainly at the end of chains, which is the behaviour
we would expect for one-dimensional topological systems. This clearly
indicates that the system has undergone a transition from 2D to quasi-1D as it
becomes increasingly disordered (due to the imposed cutoff between
neighbours), while keeping a non-trivial topological nature. In this case, as
Fig. 4(b) shows, there is no gap.

Finally, we can quantify the edge character in the transition from trivial to
topological by looking at the average edge localization of eigenstates near
zero energy as a function of $M$, as shown in Fig. 3(c) and (d). As we
approach the boundary between phases predicted by the ANN, there is a drastic
change in edge localization, which is indicative of the phase transition.

Figure 4: (a) Edge state, (b) energy spectrum for both OBC and PBC and (c)
entanglement spectrum for $M=3$, $R_{c}=1.1$, $\Delta r=0.5$. Figure 5: (Left)
Gap diagram for the Bethe lattice with $z=3$ and depth$=8$. (Right)
Topological phase diagram for the same model as predicted by the ANN trained
with the Wilson-fermion model. Black lines correspond to contour lines from
the gap diagram for $0.1$ eV.

Bethe lattice— We consider next a different type of system without
translational symmetry, namely a Bethe lattice, which is a type of fractal
lattice where the $\mathbb{Z}_{2}$ invariant cannot be computed by standard
means. Since the underlying model Hamiltonian is the same, we expect the
previously trained neural network to predict the topological phase diagram of
this system as well. The Bette lattice is defined by a coordination number
$z$, which specifies the number of neighbours each atom has. Then, starting
from a central node, the number of nodes in each consecutive layer is given
by:

$N_{k}=z(z-1)^{k-1},\ k>0$ (5)

where $k$ denotes the $k$-th layer, e.g. in layer 1 there are three nodes. It
is important to mention that, mathematically speaking, the Bethe lattice is
realized if the above equation is fulfilled. For us, however, the specific
arrangement of the atoms is relevant since the generalized Wilson model
depends explicitly on the angles between the atoms. Thus, we arrange the atoms
of each layer such that the angular spacing between them is uniform. Also, the
distance between every connected pair of atoms is fixed. These two
constraints, with the coordination number, reproduce the lattice shown in
Fig.6.

With the model established, we proceed in a analogous fashion as with the
amorphous lattice. To obtain an equivalent topological phase diagram, first we
must choose some parameters that define the parameter space. As opposed to the
amorphous model, for the Bethe lattice there is not a disorder parameter since
the lattice is fixed. Instead, we choose the bond length. The bond length
affects the hopping amplitude, effectively changing the electronic structure.
The corresponding gap diagram is shown in Fig. 5(a).

The entanglement spectrum for different combinations of the mass and bond
length parameters is fed into the neural network, which predicts the phase
diagram shown in Fig. 5(b). In this case, the whole topological region has a
negligible gap. To verify that the neural network is predicting correctly the
different phases, we may represent eigenstates near the Fermi energy. In Fig.
6(a) we see how an edge state appears, similar to the ones in the amorphous
system for $R_{c}=1.1$ (note that the probability density is located mainly at
the end of the different branches, which in this case happens to be also the
outermost atoms). For comparison, a trivial state is also shown in Fig. 6(b),
as well as the average edge occupation for several eigenstates close to the
Fermi level. In all cases the results are consistent with the diagram
predicted by the ANN.

Figure 6: (a) Edge state for $M=2.5$, $l=0.7$. (b) Lowest absolute energy
eigenstate for $M=1$, $l=1.4$. Inset: Edge occupation as a function of $M$ for
different bond lengths $l$

Conclusions— We have shown that it is possible to use the entanglement
spectrum of a non-translationally invariant system to train a neural network
to predict topological and trivial phases. We have applied it, in particular,
to the case of a disordered, even amorphous lattice and to a Bethe lattice
with an underlying Wilson-Dirac fermion model. This method reduces the
computational time for the calculation of the invariant once we have an ANN
already trained, as opposed to using, e.g., the Wilson loop technique. More
importantly, it can be used with gapless systems where no other method is
currently available. We expect that this method will allow to explore
realistic models of disordered topological insulators, such as alloys as a
function of the composition, topological metals and disorder-induced phase
transitions in general.

###### Acknowledgements.

The authors acknowledge financial support from Spanish MICIN through Grant No.
PID2019- 109539GB-C43, the María de Maeztu Program for Units of Excellence in
R&D (Grant No. CEX2018-000805-M), the Comunidad Autónoma de Madrid through the
Nanomag COST-CM Program (Grant No. S2018/NMT-4321), the Generalitat Valenciana
through Programa Prometeo/2021/01, the Centro de Computación Científica of the
Universidad Autónoma de Madrid, and the computer resources of the Red Española
de Supercomputación.

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