Machine learning guided discovery of stable, spin-resolved topological insulators

Identification of a non-trivial $\mathbb{Z}_{2}$ index in a spinful two dimensional insulator indicates the presence of an odd, quantized (pseudo)spin-resolved Chern number, $C_{s}=(C_{\uparrow}-C_{\downarrow})/2$. However, the statement is not biconditional. An odd spin-Chern number can survive when the familiar $\mathbb{Z}_{2}$ index vanishes. Identification of solid-state systems hosting an odd, quantized $C_{s}$ and trivial $\mathbb{Z}_{2}$ index is a pressing issue due to the potential for such insulators to admit band gaps optimal for experiments and quantum devices. Nevertheless, they have proven elusive due to the computational expense associated with their discovery. In this work, a neural network capable of identifying the spin-Chern number is developed and used to identify the first solid-state systems hosting a trivial $\mathbb{Z}_{2}$ index and odd $C_{s}$. We demonstrate the potential of one such system, Ti$_{2}$CO$_{2}$, to support Majorana corner modes via the superconducting proximity effect.


I. INTRODUCTION
The comprehensive cataloguing of materials supporting non-trivial band-topology, as diagnosed via elementary band representations (EBRs) and other efficient protocols relying on analysis of symmetry eigenvalues represents a remarkable milestone [1][2][3][4][5].However, it is now clear that these works can not represent a comprehensive analysis of bulk topology.This is due to the existence of symmetry non-indicative phases (SNIPs) [6][7][8][9][10][11][12][13][14][15].Importantly, it has been demonstrated that there exist a number of two-dimensional higher-order insulators which fall in this category for which the ground-state bulk invariant is a non-zero spin-Chern number, as defined by Prodan [6].
The spin-Chern number, C s = (C ↑ − C ↓ )/2, has been shown to be robust both in the presence of impurity effects and the absence of spin-rotation symmetry.It is protected by both the energetic bulk gap as well as the spin-gap.The spin-gap is identified by constructing the projected spin operator, P SO = PŝP, where P is the projector over occupied states and ŝ is the preferred spin direction.In the presence of spin-rotation symmetry, the eigenvalues of the PSO are fixed to be ±1.When spin-rotation symmetry is broken, the eigenvalues adiabatically deviate, but as long as the spectra of the PSO remains gapped the spin-Chern number is robust.It is further shown by Lin et.al [16], that for an odd, ground-state spin-Chern number, only the bulkgap need be maintained to protect the band topology.Under this formulation, the Z 2 index and mirror Chern number both represent special cases of the spin-Chern number, however this statement is not biconditional.A finite spin-Chern number need not imply a finite Z 2 index or mirror Chern number.
Identification of a non-trivial spin-Chern number in systems lacking enhanced symmetry has proven computationally demanding, particularly in the context of ab initio simulations of many-band systems [15][16][17].While density-functional theory software exists for efficient symmetry analysis of the wavefunctions and computation of Wannier center spectra [18][19][20][21], these systems often fall under the category of higher-order topological insulators (HOTIs) and can thus be "invisible" to the Z 2 index.In correspondence with a trivialized Z 2 index, the surface spectra does not support gapless modes which can be used to diagnose topology.In certain cases, corner localized states can be used to diagnose topology [22][23][24][25][26], however corner modes are notoriously difficult to identify as their presence depends on the geometry of the sample and in the absence of chiral symmetry they are not pinned to zero energy, often becoming hidden among bulk states.
Alternatively, in a recent work by Tyner and Goswami [17], the proposal of Qi and Zhang [27] and Ran et.al [28] to utilize magnetic flux tubes (π-flux vortices) as real-space probes of both Chern and spin-Chern number was expanded to ab initio simulations.It was shown to be robust in both symmetry indicative and non-indicative phases.An automated workflow was developed to scan 141 two-dimensional, spinful (T 2 = −1), insulators.The result was identification of 21, novel quantum spin-Hall insulators, falling under the category of SNIPs, but supporting a non-trivial, even spin-Chern number.While automated and basis agnostic, this work remained computationally demanding.Furthermore, a two-dimensional insulator supporting odd spin-Chern number and trivial Z 2 index remained elusive.
This situation motivates a novel search strategy relying on development of a neural network capable of diagnosing the presence of bulk topological order in T preserving systems with T 2 = −1.While a machine learning based approach does not eliminate the need for density functional theory computations as they are necessary to validate the results; this approach allows for creation of an optimal list of candidate materials that can be validated within a reasonable timescale.Furthermore, prior intuition or knowledge of the electronic properties of a given material is not necessary to determine whether it is an optimal candidate for non-trivial topology, allowing for a more diverse material landscape to be considered.This neural network should posses the capability to identify In this work we construct two convolutional neural networks (CNNs), both relying on voxel encoding of the crystal structure [29,30].The first produces a binary classification, dictating whether the spin-Chern number is zero or finite.The second produces a multi-class classification, dictating whether the spin-Chern number is zero, odd, or non-zero and even.These CNNs are then applied to a set of experimentally synthesized, two-dimensional insulators in the computational twodimensional materials database (C2DB) [31,32].Materials of interest predicted to support non-trivial, odd ground-state spin-Chern number are subsequently selected and analyzed in depth, leading to identification of the first two-dimensional insulator supporting |C s,G | = 1 and a trivial Z 2 index.This work serves as a proof that the apparent computational expense present in identification of spin-resolved band topology can be overcome through the use of machine learning techniques.
There is a particular need for such efficient tools to diagnose (pseudo)spin topology given the ongoing experimental interest in two-dimensional hetero-structures and twisted materials.In these systems additional degrees of freedom such as valley can give rise to pseudospin-resolved topology [33,34].

II. CONSTRUCTION OF TRAINING DATA-SET
The training dataset consists of spinful (T 2 = −1), two-dimensional insulators.The corresponding groundstate spin-Chern number is labeled as zero, odd, or finite and even.This dataset is made distinct from existing databases by enhanced criteria for labelling a system as topologically trivial.Following Ref. [17], a compound is labelled topologically trivial only if it is demonstrated that an inserted magnetic flux tube (vortex) admits no mid-gap bound modes.
It has been shown that for a spin-Hall insulator supporting |C s,G | = N , when flux is tuned to ϕ = hc/(2e) (π-flux) there exists 2N mid-gap vortex bound modes (VBMs) [15,16,27,28].The spin-Chern number can then determined by computing quantized induced charge on the vortex.If the VBMs are half-filled, the vortex acquires induced spin but no induced charge.If we dope by N e ∈ [−N, +N ] electrons away from half-filling, occupying all VBMs, the vortex acquires induced charge δQ = N e × e.If this condition is satisfied, the spin Chern number can be directly calculated by fixing N e = N such that δQ = |C s,G | × e.While the introduction of spin-orbit coupling causes the spin bound to the vortex to become finite but non-quantized, the quantization of bound charge remains robust.For this reason, quantization of bound charge was used in Ref. [17] as the criteria for topological classification.We choose to follow this procedure for construction of the training dataset in this work, screening all materials in the database of Mounet FIG.2: Construction of crystal graphs: Schematic detailing the construction of two-dimensional crystal graphs for each compound in the training set.This process ensures a continuous representation of the structure that can be presented to the neural network.et al. [35] with less than 10 atoms in the unit cell which have not previously been labeled as topological via existing symmetry based methods.
Spin-orbit coupling is considered in all calculations.
The Wannier90 [19], Z2pack [20], and BerryEasy [42] software packages were utilized in calculation of all topological invariants.In order to facilitate automated analysis of the bulk topology, Wannier tightbinding (WTB) models are constructed through use of the SCDM method introduced by Vitale et.al [43].Manipulation of Wannier tight-binding models for vortex insertion is done with a custom python program which will be made publicly available upon being developed into a stand-alone package.The result is a database of 246 symmetry non-indicative compounds with the full results available at Ref. [44].Despite this expanded search, in each system analyzed the bulk invariant is found to be either zero or an even spin-Chern number.A twodimensional insulator supporting odd spin-Chern number and trivial Z 2 index remained elusive.
To expand the dataset, all insulators supporting a nontrivial mirror Chern number or Z 2 index identified in Refs.[45][46][47] are incorporated into the training set.This is possible due to the fact that a non-trivial Z 2 index or mirror Chern number can be considered as a special case of the (pseudo)spin-Chern number.We remark that we cannot include any materials labelled trivial from Refs.[45][46][47] in the training data as this may cause mislabelling of SNIPs.The resulting aggregate dataset consists of 443 two-dimensional materials.Further information regarding the composition of the dataset is visible in Fig. (2).

III. DATA AUGMENTATION AND PROCESSING
A number of strategies for presentation of lattice structure to CNNs have been developed and tested in recent years [48][49][50][51][52][53].In this work, we utilize the strategy of forming a continuous representation by autoencoding voxel images of the crystal structure to create a 2D crystal graph.We account for the possibility of 79 different elements in the crystal structure, specifically atomic numbers 1-84 removing the noble gases.Details of the training set are shown in Fig. (2).As a result, regardless of the number of elements in a single crystal structure, 80 voxel images will be produced.An autoencoder then translates each image into a vector.A similar process is done to form a voxel image of the lattice which again is translated into a one-dimensional array through use of an autoencoder.These one-dimensional arrays are then reshaped into 2D crytal images which can be presented to the CNN.In this way the crystal structure obtains a continuous and reversible representation.
Importantly, this process allows for implementation of  a data-augmentation strategy, expanding the dataset to include ≈ 10 3 data-points such that the convolutional neural network may achieve a level of accuracy sufficient to benefit our materials search [48].Namely, we permute the primitive lattice vectors, altering the atomic positions accordingly and generating new 2D crystal images.This process allows for training data to be augmented by a factor of two if only the a and b primitive vectors are permuted, and up to a factor of six if all are permuted.For further details of voxel image production and autoencoding please consult the supplementary material [54].Details of the CNNs architecture for binary and multiclass classification are visible in Fig. (3).

IV. NEURAL NETWORK PERFORMANCE
We begin with the CNN for binary classification, given the limited training data available it is expected that this model will achieve higher accuracy.The data set is randomized and an 80%/10%/10% train/validate/test split is utilized.Early stopping based on validation loss is implemented and the batch size is set to 128.The CNN reaches a train/validation/test accuracy of 96%/88.7%/88.5%.While extraordinary accuracy for neural networks has become commonplace, we note that these values are quite high given the limited training dataset.For context, we compare these values with those produced utilizing a convolutional crystal graph neural network (CGCNN) in the supplementary material [48,54,55].The CGCNN architecture is commonly regarded as the state-of-the-art for machine-learning based property prediction in computational materials science.It is important to note that CGCNN has not been selected as the primary approach in this work as it disallows the use of the data-augmentation technique of exchanging lattice parameters.This limits the size of the total training data set.For a larger dataset, the CGCNN method could be advantageous.Other common machine learning architectures for performing topological classification of crystal structures rely primarily on generation of an input vector constructed from features of the constituent atoms as well properties of the nearest neighbors rather than a direct continuous image of the crystal structure [56][57][58].In models of this type, details of the crystal structure have also been incorporated by dividing the training dataset such that all constituent compounds correspond to a selected prototypical crystal structure [59].This approach is again problematic for the purposes of this work as it would result in severely limiting the size of the training dataset.We place emphasis on the use of a continuous and reversible image of the crystal structure as input to lay a foundation for the future incorporation of the model in a generative artificial intelligence architecture [30].
For the multi-class classification model, we adjust the CNN architecture to the form seen in Fig. (2).We again employ the same split for training, test and validation data and find a train/validation/test categorical accuracy of 95%/89.5%/88%.This represents a significant step towards isolating optimal candidate SNIPs.
At this point it is important to discuss potential sources of bias in the neural network.The most prominent source of bias is likely to be the limited quantity of training data available.This bias is expected as not all elements in the training data set will appear with the same frequency.Those which appear sparsely have the potential to have a significant impact on the model, particularly if all compounds in which the element is found have the same topological classification.Such biases and trends are explored in detail in the supplementary material [54].

V. APPLICATION OF CNN
We begin by selecting candidate compounds from the computational two-dimensional materials database (C2DB) [31,32] with the criteria that a compound must not be included in the training database or previously identified as topological via symmetry-based methods.Each compound must also admit a bulk band gap greater than 0.1 eV and be dynamically stable.Two-dimensional crystal graphs are then constructed and analyzed using the binary classification CNN.Compounds labelled as non-trivial are isolated and subsequently fed to the multiclass CNN.We filter this final list by number of atoms in the unit cell, selecting two compounds with less than 10 atoms in the unit cell predicted to support |C s | = 1, Os 2 Te 4 and Ti 2 CO 2 .
Plots of the crystal structure and band structure for both compounds are visible in Fig. (4).In order to perform topological analysis of the structures, a Wannier tight-binding model is produced [19], exactly replicating only the Kramers-degenerate bands nearest to the Fermi energy using carefully selected orbitals.As these bands are significantly energetically separated from all other bands, the TB model reproduces the DFT data precisely.For more computational details please see the supplementary material [54].The spin Chern number is then computed directly via the method established by Prodan [6,42].This procedure requires defining the projected spin operator (PSO), P (k)ŝP (k), where P (k) is the projector onto occupied bands and ŝ is the preferred spin axis.We identify the preferred spin axis supporting a spin-gap through a computationally expensive trial and error procedure.However, once identified we are able to produce the results shown in Fig. ( 4), displaying calculation of the spin-Chern number via spin-resolved Wilson loop as detailed in Lin et.al [16].Remarkably both compounds demonstrate proper labeling by the neural network and support of |C s | = 1.
These results are of significance due to the robust nature of the bulk invariant.Unlike other SNIPs which require the preservation of the bulk spin-gap, a challenging task experimentally, the bulk topological invariant of these systems requires only preservation of the bulk energetic gap to remain intact.As a result, the bulk-invariant is robust to the influence of disorder [6] and other perturbations, much like traditional Z 2 topological insulators.Unlike existing topological insulators, both systems support sizeable bulk energetic gaps, 0.325eV and 0.713eV for Ti 2 CO 2 and Os 2 Te 4 respectively.These large energetic gaps make both materials primary candidates for experiments and quantum devices as we explore in the following section.Furthermore, synthesis of Ti 2 CO 2 has been reported in the literature [60] as well as single crystal Os 2 Te 4 [61].In the case of Ti 2 CO 2 , synthesis is accelerated by the commercial availability of TiC.

VI. ACCESSING MAJORANA CORNER MODES VIA SUPERCONDUCTING PROXIMITY EFFECT
Prior works have explored the superconducting proximity effect in topological insulators [62][63][64][65][66][67][68][69].In particular such proposals have focused on use of the superconducting proximity effect to realize Majorana bound states (MBSs) by layering a two-dimensional topological insulator on the surface of a superconductor.The experimental realization of MBSs is important due to their proposed utilization as a platform for topological quantum computing [70].A common issue in the proposed platforms for realizing MBSs via the superconducting prox- imity effect in topological insulators is an extremely small topological gap.Experimental realizations generally rely on known two-dimensional topological insulators [68,69] such as 1T'-WTe 2 [71] and PbTe [72,73] which support normal state topological band gaps on the order of tens of meV .
Similarly, proposals of platforms for the realization of Majorana corner states generally rely on an induced dwave pairing.Experimental estimates for the induced pairing gap are on the order of several meV [74][75][76].Control of this small topological gap poses a significant experimental challenge, particularly as it requires extremely small levels of disorder to be present in a given sample.
In this section we briefly investigate the superconducting proximity effect in one of the proposed spin-resolved topological insulators listed above, Ti 2 CO 2 .In order to investigate the superconducting proximity effect we consider a device constructed from a two-dimensional slab of Ti 2 CO 2 placed on top of a superconductor as shown schematically in Fig. (5(a)).To model the system we construct an effective Bouligobov-de-Gennes (BdG) Hamiltonian, H ef f , of the form, where H W T B is the Wannier tight-binding model constructed in the previous section for analysis of Ti 2 CO 2 .Following Ref. [66], we include the superconducting proximity effect by introducing a simplistic nondissipative pairing term, ∆, which we fix as a constant.The resulting band structure of H ef f is shown in Fig. (

5(b)).
We investigate the presence of MBSs by performing exact diagonalization of H ef f for a system of 49 × 37 unit cells, and ∆ = 180µeV .Fascinatingly, examining the states nearest to zero energy, the results in Fig. (5(c)) demonstrate the presence of two zero energy MBSs.Investigating the localization of the MBSs we find the results in Fig. (5(d)) demonstrating that these states are corner bound and exist on opposite sides of the sample.Importantly, as seen in Fig. (5(b)), the bulk energetic gap, ∆E gap , is on the order of ≈ 0.45eV .This is significantly larger than the typical value for known alternatives and underscores the implications of identifying large band-gap SNIPs.Beyond the intriguing properties which they support on a theoretical level, SNIPs provide a viable route to overcome current practical experimental issues in the identification of MBSs.
Furthermore, we highlight that in this setup, Majorana corner modes survive when the pairing term is suppressed.Such pairing is crucial for the existence of corner modes in alternative proposals based on the proximity effect in two-dimensional FOTIs [68,73,77].This result underscores the potential utility of spin-resolved topological insulators in experimental platforms.

VII. SUMMARY
In this work, machine learning guided discovery of symmetry non-indicative topological phases in twodimensions has been shown to be a promising route to the identification of large band-gap quantum spin-Hall insulators.Furthermore, the resulting network can leverage a limited amount of training data through augmentation techniques.In the past, machine learning techniques for identification of non-trivial topology were dismissed due to the computational efficiency of directly calculating the bulk invariant using symmetry indicator techniques.The immense computational expense associated with direct calculation of band topology in SNIPs warrants the use of a novel machine learning approach.By limiting the associated computational expense, it is now possible to go beyond analysis of known compounds that have been successfully synthesized and explore large datasets such as the virtual two-dimensional material database (V2DB) [78].The neural network produced in this work can also serve as a building block in the future development of generative artificial intelligence models for inverse design of topological materials.Finally, we expect that such a network can find extensive use in analyzing the growing number of two-dimensional hetero-structures and twisted architectures which have attracted experimental interest but where the normal state band topol-ogy is unknown due to the complexity of a many-atom unit cell which will be explored in a future work.
the projector onto occupied bands and ŝ = σ 3 ⊗ τ 0 is the preferred spin axis.As the PSO supports a spectral gap, the spin-Chern number can be computed via spin-resolved Wilson loop as detailed in Lin et.al [16].The results shown in Fig. (6(f)), demonstrate the WCC spectra when performing the spin-resolved Wilson loop along the x axis as a function of transverse momenta k y for the band corresponding to negative eigenvalues of the PSO, θ − x .The conclusion is in alignment with the earlier determination that the ground states supports |C s | = 1.
The spin-Chern number could further have been diagnosed using the method of magnetic flux tube insertion.In order to accomplish flux insertion, we consider a slab of 60 × 60 unit cells and introduce the Peierls factor, H ij e iϕij where ϕ ij = 2π(ϕ/ϕ 0 ) × Θ(x j − x i )δ(y i ).Examining the local density of states on the flux tube as a function of ϕ/ϕ 0 , we find the results shown in Fig. (6(g)) detailing a pair of vortex-bound modes (VBMs) which traverse the bulk gap.The induced charge on the vortex is then measured at ϕ = ϕ 0 /2 as a function of filling with the results shown in Fig. (6(h)), detailing that when the VBMs are fully occupied or unoccupied, the vortex acquires a quantized induced charge in accordance with Ref. [17].The resulting one-dimensional arrays are combined to form a 80 × 512 image that is reshaped into size 160 × 256 before being presented to the convolutional neural network.We implement early stopping based on validation loss in both cases to avoid over-fitting.This early-stopping begins after 60 epochs and is implemented with a patience of 120 epochs.The early-stopping location for which the model weights are restored after training is denoted by a vertical dashed line.We note that for epochs beyond this location the validation accuracy is generally stagnant while the training accuracy improves, a sign of the over-fitting we wish to avoid.
It is also important to discuss the choice of hyper- parameters in the model.The initial hyperparameter choices were guided by the published data in Ref. [30].These parameters were then adjusted adiabatically to optimize for the model at hand based on computational considerations such as GPU ram as well as the empirical judgement of model performance.It was found that, for the given architecture, the performance of the model was reasonable equivalent for a range of hyperparameters.
The same was true when utilizing the CGCNN [48,55] architecture as will be discussed in the section VI.

Appendix E: Trends in network classification
Given the limited training dataset, it is natural to inquire as to whether obvious trends appear in the behavior of the classifiers.Two such trends we wish to investigate are associated with (a) the band gap at the Fermi energy and (b) the presence of certain elements in the unit cell.
We begin by analyzing trends associated with the band-gap at the Fermi energy.As stated in the main body, large band-gap topological insulators are rare due to the fact that the band gap is generally induced via spin-orbit coupling.We expect this correlation between small band-gaps and non-trivial topology to manifest in the behavior of the binary and multiclass predictor.To observe whether this trend is present in the classifier we supply all two-dimensional materials in the C2DB database to the binary and multiclass classifier.The results are then plotted as a function of the binary classifier classification as well as band gap at zero energy.The points are further colored by the multiclass classification.The resulting plot, Fig. (9(a)), demonstrates a clear bias towards small band-gap insulators supporting non-trivial topology.Interestingly, this figure also demonstrates that while the multiclass and binary networks are overall consistent, there are instances in which the binary classifier labels a system as trivial and the multiclass classifier does not, and vice versa.In this way these two networks can be used as a form of cross validation.
Next, we wish to search for clear trends in topological classification as a function of constituent elements.This is important given the limited training dataset quantity as it could be possible that if, for example, element X appears in only three entries in the training data which are all labeled trivial, that any system containing element X will be labeled trivial.To search for these trends we supply all two-dimensional materials in the C2DB database to the multiclass classifier and arrange the results based on the presence of the elements in each crystal that is classified.The results are shown in Fig. (9(b)).From these results it is most informative to observe those elements which display large bias.For example, all systems containing boron and rubidium have been marked trivial with |C s | = 0.In the case of these two elements, this aligns with the example scenario.All materials with boron or rubidium in the training set are labeled topologically trivial.This is a clear bias in the dataset and as such predictions made by the model on materials which include these elements should be treated cautiously.Similarly, we note all systems containing potassium and niobium are labeled as supporting |C s | = 0 and |C s | = 1 respectively.These represent more subtle cases in which the training dataset incorporates both topological and trivial systems which include these elements, however the C2DB dataset which we analyze has only four and two systems containing these elements respectively, creating large bias in the analysis.

Appendix F: Comparison with CGCNN
In this work a strategy of creating two-dimensional crystal graphs to present to a convolutional neural net-work has been utilized.This choice was primarily made as it allowed for ease in adopting the data augmentation technique presented in the main body.This is important given the limited quantity of training data available.A clear alternate choice of neural network is a crystal graph convolutional neural network (CGCNN).CGCNNs have emerged as one of the most effective methods for application of neural networks to crystalline data.As such, it is natural to inquire as to the effectiveness of CGCNNs in our work.We have trained a CGCNN using the code made publicly available by Xie [48].
We note that a drawback of the CGCNN in our context is that the data augmentation procedure presented in the main-body is no longer effective.We thus have less data points then is recommended for a CGCNN.Nevertheless, training the model with the same train/test/validation split used in the main body and using the default setting for a classification model, we find that the CGCNN reaches a train/validation/test accuracy of 98%/85%/72%.While not as high as the model presented in the main-body, it remains an impressive result and demonstrates the potential effectiveness of this powerful method if more data is made available.

Appendix G: Surface states of Os2Te4 and Ti2CO2
Surface states: As the Wannier center spectra is gapped for both Os 2 Te 4 and Ti 2 CO 2 , the bulk-boundary correspondence principle dictates that the surface spectra will be gapped.The surface spectra for both compounds is generated using the WannierTools software package [21] which utilizes the recursive Greens function algorithm of Ref. [79].The spectral density on the (01) surface for Os 2 Te 4 and Ti 2 CO 2 is shown in Fig. (10(a)) and (10(b)) respectively.We note the presence of surface-bound midgap states and the absence of gapless states in both compounds.Due to the absence of gapless modes, determination of the bulk topology can not be made solely through examination of the surface spectra.

FIG. 1 :
FIG. 1: Statistics of training dataset: Statistics of compounds contained within the training dataset.The periodic table is color coded according to the number of times each element occurs in the dataset on a log scale.Elements in white do not appear in the dataset.The circular plot details the breakdown of how many compounds support |C s | = 2, 1, 0, and how many of the non-trivial materials are further identified by a Z 2 index or mirror Chern number.The breakdown of spacegroups present and their occurrences is further provided.

FIG. 3 :
FIG. 3: Convolutional neural network architecture: Architecture of convolutional neural network for binary and multiclass classification.The binary and multiclass classification utilize a sigmoid and softmax activation function respectively for the final dense layer.The leakyReLu function is fixed to α = 0.2 in each case.

FIG. 4 :
FIG.4: Analysis of spin-resolved topological insulator candidates: Structure of (a) Os 2 Te 4 and (e) Ti 2 CO 2 as given via the C2DB database[31,32].The computed band structure along a high-symmetry path in the Brillouin zone detailing the bands nearest to the Fermi energy can be seen in (b) and (f) for Os 2 Te 4 and Ti 2 CO 2 respectively.The gapped wannier center spectra of both compounds, seen in (c) and (g) indicates these systems are trivial under the Z 2 index.The ground state spin-Chern number is computed via spin-resolved Wilson loop, detailing a single winding for both compounds, confirming |C s | = 1.

FIG. 5 :
FIG. 5: Majorana corner modes via superconducting proximity effect: (a) Schematic of setup for generating superconducting proximity effect in Ti 2 CO 2 .(b) Band structure of BdG Hamiltonian, H ef f .(c) States nearest zero energy for slab of H ef f shown in (d) implementing open boundary conditions along both principal axes.(d) Localization of zero energy states shown in (c), demonstrating that they are corner bound.Local density of states on a given site is displayed as a function of color and site size for clarity.A larger site size indicates increased local density of states.

FIG. 6 :FIG. 7 :
FIG. 6: (a) Band structure of the tight-binding model in eq.(A1).The band structure considering a 20 × 1 slab with open(periodic)-boundary conditions along the x(y) direction, detailing the gapped edge spectra.(c) Gapped Wannier center charge spectra disallowing non-trivial topological classification.(d) Spectra under two-dimensional open-boundary conditions for a slab of size 50 × 50 unit cells.Four degenerate zero modes are colored in red.Localization of the zero modes is shown in (e), demonstrating that they are corner bound.(f) Spin-resolved Wannier center charge spectra demonstrating |C s | = 1.(g) Local density of states on inserted flux tube as a function of flux strength.(h) Induced charge on inserted flux tube as a function of occupied vortex-bound modes.

FIG. 8 :HCoFIG. 9 :
FIG. 8: (a)[c] Training and validation loss as a function of epoch for binary[multiclass] classifier.Dashed lines marks location of minimum validation loss from which weights are restored for final model.(b)[d] Training and validation accuracy for binary[multiclass] classifier as a function of epoch.Dashed lines marks location of minimum validation loss from which weights are restored for final model.

FIG. 10 :
FIG. 10: Spectral density on (a) the (10) surface of Os 2 Te 4 the (b) zigzag surface of Ti 2 CO 2 .In both cases the surface spectra is gapped, but surface bound mid-gap states can be seen.