Ulrich Vickos
email: vickos.ulrich@gmx.com

Marianna Camasta
email: marianna.camasta@unica.it

Nicole Grandi
email: nicole.grandi@unica.it

Sante Scognamiglio
email: sante.scognamiglio@unica.it

Tobias Schindler
email: tobias.schindler@swisstph.ch

Marie Roseline

Darnycka Belizaire
email: belizairem@who.int

Ernest Lango-Yaya
email: elangoyaya@gmail.com

Wilfried Giscard

Oscar Koyaweda
email: gkoyaweda@gmail.com

Simon Senzongo

Pounguinza

Marianna Camas

Koyaweda

Oscar Senzongo
email: oscarsenzongo@yahoo.fr

Simon Pounguinza

Kaleb Kandou Jephté

Francis Estimé

Stephanie N'yetobouko
email: nyetoboukostephanie@yahoo.com

Christelle Luce
email: bchristelleluce@yahoo.fr

Bobossi Gadia

Domin-Alfred Feiganazoui
email: fedominosalfred@gmail.com

Alain Le Faou
email: alainlefaou@gmail.com

Massimiliano Orsini
email: morsini@izsvenezie.it

Carlo Federico Perno

Luca Zinzula
email: zinzula@biochem.mpg.de

Clotaire Donatien
email: clotairerafai@yahoo.fr

COVID-19 Genomic Surveillance in Bangui (Central African Republic) Reveals a Landscape of Circulating Variants Linked to Validated Antiviral Targets of SARS-CoV-2 Proteome

Keywords: Central African Republic, severe acute respiratory syndrome coronavirus 2, SARS-CoV-2, COVID-19, Delta variant, screening

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) emerged as a human pathogen in December 2019 in Wuhan, the capital of the Chinese Hubei Province, with an outbreak of severe pneumonia, the clinical manifestations of which took the name coronavirus disease 2019 (COVID-19) [START_REF] Wu | A new coronavirus associated with human respiratory disease in China[END_REF][START_REF] Zhou | A pneumonia outbreak associated with a new coronavirus of probable bat origin[END_REF]. Thereafter, the virus spread worldwide as a pandemic and after more than three years has been responsible for 676 million cases and more than 6.8 million deaths [START_REF]COVID-19 Dashboard by the Center for Systems Science and Engineering[END_REF]. Moreover, despite the availability of vaccines with more than thirteen billion doses administered so far, COVID-19 still poses a threat to global health. This is at least due in part to the lack of approved antiviral therapeutics administrable to nonhospitalized patients, as well as to the non-equitable access to COVID-19 vaccines among countries [START_REF] Edwards | Stopping pandemics before they start: Lessons learned from SARS-CoV-2[END_REF]. Furthermore, disparities in the application of, or adherence to, restriction policies, containment measures, diagnosis, and surveillance further exacerbate the risk of emergence of more virulent, transmissible, and vaccine-resistant SARS-CoV-2 variants [START_REF] Robinson | COVID-19 therapeutics: Challenges and directions for the future[END_REF]. Particularly, Africa was considered at high risk for COVID-19 spread due to the high number and diffusion of infectious diseases, inefficient health systems, and poverty, having only limited available resources to contain the virus [START_REF] Lone | COVID-19 pandemic-An African perspective[END_REF]. In light of such a scenario, grim predictions had been made for the spread of COVID-19 across the African continent, although they were contradicted by relatively fewer cases in the continent as compared to the rest of the world, and with a lower case fatality ratio (CFR) during the first wave of the pandemic [START_REF] Musa | Addressing Africa's pandemic puzzle: Perspectives on COVID-19 transmission and mortality in sub-Saharan Africa[END_REF]. Indicators such as a younger population, comorbidity rates, urbanisation, population density, and poor virus detection capabilities have been proposed as drivers for this phenomenon, albeit with discrepancies in the impact of variables among countries [START_REF] James | An in-depth statistical analysis of the COVID-19 pandemic's initial spread in the WHO African region[END_REF]. During the second wave of infections, an overall increase in CFR was reported for all African Union member states, although disease severity and progression varied among them [START_REF] Salyer | The first and second waves of the COVID-19 pandemic in Africa: A cross-sectional study[END_REF], thereby emphasizing the need to conduct studies that focus on regional levels for a better understanding of the impact of COVID-19 across Africa.

The Central African Republic (CAR) is located in the centre of the African continent, bordering Chad to the north, the Republic of Congo (ROC) and the Democratic Republic of Congo (DRC) to the south, Sudan and South Sudan to the east, and Cameroon to the west. Bangui, with an estimated population of 1,145,280 inhabitants in 2021, is the administrative capital and the main city of the CAR, with a multicultural mix resulting from being the main point of entry for expatriates and displaced people from central African territories [START_REF] Green | The Central African Republic's silent health crisis[END_REF]. Starting from the first case of COVID-19 reported on 14 March 2020, and referred to an immigrant patient, CAR has recorded, as of 2 July 2021, a total of 11,064 cumulative confirmed cases, of which 7142 were symptomatic (including 581 imported ones) and 3922 were asymptomatic, which were confirmed as resulting from local community transmission. Among 7103 patients treated, 98 died: 50 inside and 48 outside a hospital setting. In the main districts of the area selected for this study, i.e., Bangui, 78,685 people were vaccinated against COVID-19 [START_REF]Rapport de Situation Journalière de COVID-19 en République Centrafricaine[END_REF]. Despite the spread of the virus in the country, the incidence of recorded cases was not as high as in countries like South Africa, Brazil, Italy, and the United States [START_REF] Nguimkeu | Why is the number of COVID-19 cases lower than expected in Sub-Saharan Africa? A cross-sectional analysis of the role of demographic and geographic factors[END_REF]. Most cases in the country have been reported in towns such as Bangui, Bouar, Berberati, and Mbaïki, but apparently with a slower spread since the lack of tests and the cost of analyses had led the CAR Ministry of Public Health to prioritize the testing of symptomatic subjects. These considerations suggest that official statistics are underestimated, as confirmed by a serological study that assessed the prevalence of antibodies (IgG Abs) against SARS-CoV-2 in 74.1% of the 799 asymptomatic participants [START_REF] Manirakiza | Very high relative seroprevalence of anti-SARS-CoV-2 antibodies among communities in Bangui, Central African Republic[END_REF].

As the virus spread and evolved, its genome accumulated many mutations identified in many of the 16,065,326 genome sequences submitted to GISAID (accessed on Viruses 2023, 15, 2309 3 of 14 2 October 2023) [START_REF] Shu | Global initiative on sharing all influenza data-From vision to reality[END_REF]15]. Interestingly, B.1.620 SARS-CoV-2 lineage was identified as originating specifically in the CAR, which brings mutations (like D614G on spike protein) and deletions already encountered in Variants of Concern (VoCs), increasing infectivity and resistance to the immune response [START_REF] Dudas | Emergence and spread of SARS-CoV-2 lineage B.1.620 with variant of concern-like mutations and deletions[END_REF]. Considered as a Variant Under Monitoring (VUM), B.1.620 lineage was also encountered in the Republic of Congo as one of the most common lineages circulating in the country [17]. However, after June 2021, B.1.620 was replaced by the Delta variant in the CAR [START_REF] Tegally | The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance[END_REF].

Because of the lack of information, we performed a regional epidemiological investigation along with genomic surveillance in the CAR. We sampled 2687 individuals at the main entry points in the city of Bangui and the positive samples were selected for viral genome sequencing and determination of the mutations at the amino acid level.

Materials and Methods

Sampling Procedures

An epidemiological mass survey was conducted between 2 and 22 July 2021 to assess the spread of SARS-CoV-2 in the general population of Bangui at the main entry points of the city, which are Port-Beach, PK9 Bimbo, PK26, and LNBCSP. A total of 2687 participants were randomly included. Swabs were used to obtain the nasopharyngeal samples.

LumiraDx Antigen Detection Rapid Diagnostic Tests (Ag-RDT)

Samples were tested for SARS-CoV-2 antigens within hours using a microfluidic immunofluorescence assay on the LumiraDx™ platform (LumiraDx GmbH, Cologne, Germany). The tests were carried out according to the manufacturer's protocol. Since the samples were taken on a viral transport medium (VTM), a mixture of equal volume of the VTM containing the swab and the diluent of the test on LumiraDx was prepared. This dilution rate was made according to the manufacturer's recommendations and immediately tested on the instrument.

RT-PCR Test

All samples positive for the LumiraDx Test were retested using RT-PCR. Viral RNA was extracted from nasopharyngeal samples using the QIAamp Viral RNA Mini Kit (Qiagen, Hilden, Germany) according to the manufacturer's instructions. A real-time PCR targeting SARS-CoV-2 ORF_1ab and N genes was performed using a specific PCR kit (SANSURE Biotech Kit, Changsha, People's Republic of China) according to the manufacturer's protocol using the 7500 Fast Dx Real-Time PCR Instrument (Applied Biosystems, Waltham, MA, USA).

Sample Selection and Sequencing

Positive samples for both antigen detection and RT-PCR test (CT value < 30) were retained for sequencing. These samples were sent to the National Research Institute of Kinshasa (DRC) for whole genome sequencing by Next Generation Sequencing (NGS).

NGS Sequencing

Libraries were prepared according to the COVID-19 ARTIC v3 Illumina Library Construction and Sequencing Protocol V.5 [START_REF] Dnapipelines | COVID-19 ARTIC V3 Illumina Library Construction and Sequencing Protocol V2[END_REF], using the NEBNext Ultra DNA Library Preparation Kit II (New England Biolabs Inc., Ipswich, MA, USA). Libraries were quantified (Qubit DNA BR, Thermo Scientific, Waltham, MA, USA), normalized, and pooled, and sequencing was performed using an Illumina MiSeq 100 (Illumina, San Diego, CA, USA).

Assembly

Raw reads were quality checked using the FastQC tool with default parameters [START_REF] Andrews | FastQC: A Quality Control Tool for High Throughput Sequence Data[END_REF]. Read quality was improved by read trimming using fastp software, version v0.23.4 [START_REF] Chen | fastp: An ultra-fast all-in-one FASTQ preprocessor[END_REF]. Cleaned reads were assembled by reference-guided mapping using the Wuhan-1 reference sequence (accession ID: MN908947) and bowtie2 with default parameters [START_REF] Langmead | Fast gapped-read alignment with Bowtie 2[END_REF].

Viruses 2023, 15, 2309 4 of 14 A consensus sequence for the SARS-CoV-2 genome of each sample was obtained using the samtools/bcftool pipeline [START_REF] Danecek | BCFtools/csq: Haplotype-aware variant consequences[END_REF], and then polished by Pilon software, version 1.19 [START_REF] Walker | An integrated tool for comprehensive microbial variant detection and genome assembly improvement[END_REF]. Sequences were annotated by searching for Open Reading Frames (ORFs) potentially encoding SARS-CoV-2 viral proteins as annotated in the Uniprot database [START_REF]The UniProt Consortium UniProt: The Universal Protein Knowledgebase in 2023[END_REF]. Due to the low quality of assembled regions, ORFs showing > 50% of coverage were retained.

Phylogenetic Analyses and Variant/Lineage Identification

Alignments including low-coverage CAR sequences generated for this study (n = 29) as well as CAR and Africa sequences retrieved from GISAID and a set of reference sequences representing all main Variants of Concern (VoCs) from the NCBI database (Supplementary Table S1) were constructed using MAFFT v7 [START_REF] Katoh | MAFFT online service: Multiple sequence alignment, interactive sequence choice and visualization[END_REF]. A Neighbour Joining (NJ) phylogenetic tree of CAR genomes was built using MEGA X software [START_REF] Kumar | Molecular Evolutionary Genetics Analysis across Computing Platforms[END_REF], using a p-distance model, and applying a pairwise deletion option. A maximum likelihood (ML) phylogenetic tree of African genomes was inferred using FastTree software, version 2.1.11 [START_REF] Price | FastTree 2-Approximately maximum-likelihood trees for large alignments[END_REF], which allows the computation of approximate ML trees for very large alignments, applying a generalized time-reversible (GTR) model of nucleotide evolution and CAT approximation for the varying rates of evolution across sites. Local support values were computed with the Shimodaira-Hasegawa test. Trees were visualized and analysed using FigTree v.1.4.4 [29]. For variant identification and mutation annotation, sequences were analysed using Nextclade [START_REF] Aksamentov | Nextclade: Clade assignment, mutation calling and quality control for viral genomes[END_REF].

Structural Analysis of SARS-CoV-2 Protein Mutants

Aminoacidic sequences of SARS-CoV-2 proteins were derived by analysing the Open Reading Frames (ORFs) from genome assembly via the online server tool ExPASy Translate (https://web.expasy.org/translate/, accessed on 14 November 2023) [START_REF] Gasteiger | ExPASy: The Proteomics Server for In-Depth Protein Knowledge and Analysis[END_REF]. Reference sequences of the SARS-CoV-2 genome (NC_045512.2) and of the proteins NSP3 (YP_009725299.1) NSP5 (YP_009725301.1), NSP7 (YP_009725303.1), NSP8 (YP_009725304.1), NSP12 (YP_009725307.1), S (YP_009724390.1), and N (YP_009724397.2) were retrieved as FASTA format files from the NCBI Nucleotide and Protein databases, respectively. For each protein, the structural organization and amino acid positions corresponding to the boundaries of functional domains were defined according to the published literature and rendered as cartoon schemes onto which point mutations from the dataset were mapped on scale. Files for the 3D molecular viewing of experimentally solved protein structures available in Protein Data Bank (PDB) (NSP3: 7KAG (Ubl-1), 7JRN (Ubl-2-PLpro), 6Z5T (ADRP), 7LGO (NAB), 7RQG (Y3); NSP5: 6Y2E; NSP7/8/12 RdRp complex: 6M71; N: 7CDZ (NTD), 6YUN (CTD); S: 6VXX) were selected according to the highest resolution and earliest deposition criteria. Protein sequences were edited using PyMOL v.2.5.2 software [START_REF]PyMOL The PyMOL Molecular Graphics System, Version 2.0 Schrödinger[END_REF] for the annotation of point mutations using the software mutagenesis tool, and protein structures were rendered as superimposed ribbon-backbone and isosurface representations shown at different angle orientations. Phylogenetic trees based on protein sequences were rendered using FigTree v.1.4.4 [29], and mutations identified within the dataset were mapped on the corresponding branches of each tree.

Results

Sampling, Processing, and Demographic Distribution

From April 2021, at the time of the second wave of the pandemic (n = 8294 confirmed cases and 79 deaths) to the beginning of July 2021, when we initiated this study, the number of cases increased rapidly, registering an amount of 11,075 confirmed cases and 98 deaths [START_REF][END_REF]. We carried out the SARS-CoV-2 screening study from 2 to 22 July 2021, with the enrolment of 2687 participants and the collection of their nasopharyngeal samples at the four main checkpoints of Bangui (Figure 1A). In total, 56.5% of the samples were collected at the Laboratoire National de Biologie Clinique et de Santé Publique (LNBCSP) from air-travelling passengers; 18.2% at Port Beach from travellers from North DRC; 15.7% at the North-PK26 entry point from road travellers and carriers from Cameroon, Nigeria, or south-west Chad, and the entry point; while 9.6% at the South-PK9 entry from participants coming from the border with the ROC and the CAR forest zone (Figure 1B). Of the 2687 samples tested, 109 (4.0%) resulted positive following the LumiraDx antigen test, and then only 53 (2.0%) were confirmed positive for SARS-CoV-2 by RT-PCR with a low Cq value. Among the latter, 13 were females and 40 males, all asymptomatic. The 31-50 years old age group was the most affected in the population, with an average age of 38 years (Figure 1C). A total of 29 of the 53 samples had a SARS-CoV-2 sequence available.

were collected at the Laboratoire National de Biologie Clinique et de Santé (LNBCSP) from air-travelling passengers; 18.2% at Port Beach from travellers f DRC; 15.7% at the North-PK26 entry point from road travellers and car Cameroon, Nigeria, or south-west Chad, and the entry point; while 9.6% at the entry from participants coming from the border with the ROC and the CAR f (Figure 1B). Of the 2687 samples tested, 109 (4.0%) resulted positive foll LumiraDx antigen test, and then only 53 (2.0%) were confirmed positive for SA by RT-PCR with a low Cq value. Among the latter, 13 were females and 40 asymptomatic. The 31-50 years old age group was the most affected in the p with an average age of 38 years (Figure 1C). A total of 29 of the 53 samples ha CoV-2 sequence available.

Phylogenetic Analysis

Phylogenetic Analysis

SARS-CoV-2 whole genome sequences from CAR samples collected between 2 and 22 July 2021 (n = 29) were compared in a Neighbour Joining (NJ) phylogenetic tree with the other sequences from CAR from the November 2020-January 2022 period, as retrieved from the GISAID database [15] (n = 138, low coverage excluded, listed in Supplementary Table S1) (Figure 2). Reference sequences for Wuhan wild type SARS-CoV-2 and the different VoCs (Supplementary Table S1) have also been included in the analysis. Both the SARS-CoV-2 genome sequences originated in this study and the ones retrieved from GISAID clustered together based on the lineage and clade, according to Nextclade analysis (which uses Pangolin lineage assignment) (Figure 2). 2). The sequences from the present study (n = 29) clustered within the Delta clade, being split into the two subclusters of lineages AY.46.4 (n = 4) and AY.122 (n = 24, one clustered together with the Delta reference), except for one genome (SARS-CoV-2_CAR_73) that was associated with the 19A clade (Figure 2). Two additional major phylogenetic clusters comprised viral genomes from GISAID and were classified into lineages B.1.620 (n = 33, 100% bootstrap) and B.1.640.1 (n = 42, 99.9% bootstrap), both belonging to the 20A clade: the former circulated in the CAR between March and April 2021 while the latter spanned the whole year (Figure 2). Finally, a minority of sequences reported in the CAR were grouped into the Eta clade cluster (21D, lineage B.1.525, 100% bootstrap) and the B.1.619 lineage cluster (clade 20A, 100% bootstrap) (Figure 2). An additional maximum likelihood (ML) phylogenetic tree was built including our 29 sequences plus all the SARS-CoV-2 genomes, as reported in GISAID for Africa within the same period (2-22 July 2021, only high coverage, n = 2983, Supplementary Table S1) (Supplementary Figure S1). In line with CAR epidemiology, the analysis confirmed the extensive diffusion of the Delta VoC in African countries, representing 77% of all the reported sequences (n = 2311) and including different lineages from clades 21A, 21I, and 21J (Supplementary Figure S1). In addition, the African panorama appeared to be more heterogeneous, showing a wider diffusion of the Beta VoC that accounted for 19% of the genomes circulating in July 2021 (Supplementary Figure S1). The remaining 4% of sequences fell into clusters belonging to clades 19A and 19B (n = 6), 20A (n = 21), 20B (n = 17), 20C (n = 1), 20D (n = 42), 20I (Alpha VoC, n = 40), 21B (Kappa, n = 1), 21D (Eta, n = 2), and 21H (Mu, n = 2) (Supplementary Figure S1).

Structural Analysis

The SARS-CoV-2 30kb genome encodes sixteen non-structural proteins (nsps) and four structural proteins (N, nucleocapsid, E, envelope, M, membrane, and S, spike). We decided to focus our analysis on the proteins validated as therapeutic targets, as NSP3 (multifunctional protein, including papain-like protease function), NSP5 (main protease), NSP8 (polymerase cofactor), NSP12 (RNA dependent RNA polymerase), and N and S protein. As an NSP12 cofactor, NSP7 was also included in the study, but mutations were absent in the respective aminoacidic sequence. We found nine mutations in the NSP3 sequence [START_REF] Von Soosten | The Swiss army knife of SARS-CoV-2: The structures and functions of NSP3[END_REF] (Figure 3A,B), specifically T73I and T86I in the ubiquitin-like domain 1 (Ubl-1), E134G in the Glu-rich acidic region (AC domain), A488S in the SARS-unique domain (SUD), Y1185N in the nucleic-acid-binding domain (NAB domain), P1228L in the Betacoronavirus-specific marker domain (G2M domain), and, finally, Y1463H, P1469S, and A1502V in the NSP3 ectodomain (3Ecto), also called zinc-finger domain (ZF domain). Among them, only A488S, P1228L, and P1469S mutations were reported as non-defining mutations for the Delta variant [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF]. We located the single mutation S139L in the II 3C protease-like domain on NSP5 [START_REF] Zhang | Crystal structure of SARS-CoV-2 main protease provides a basis for design of improved α-ketoamide inhibitors[END_REF][START_REF] Costanzi | Structural and Biochemical Analysis of the Dual Inhibition of MG-132 against SARS-CoV-2 Main Protease (Mpro/3CLpro) and Human Cathepsin-L[END_REF] (Figure 4A,B), present in only one of our samples (Supplementary Figure S2).

Structural Analysis

The SARS-CoV-2 30kb genome encodes sixteen non-structural proteins (nsps) and four structural proteins (N, nucleocapsid, E, envelope, M, membrane, and S, spike). We decided to focus our analysis on the proteins validated as therapeutic targets, as NSP3 (multifunctional protein, including papain-like protease function), NSP5 (main protease), NSP8 (polymerase cofactor), NSP12 (RNA dependent RNA polymerase), and N and S protein. As an NSP12 cofactor, NSP7 was also included in the study, but mutations were absent in the respective aminoacidic sequence. We found nine mutations in the NSP3 sequence [START_REF] Von Soosten | The Swiss army knife of SARS-CoV-2: The structures and functions of NSP3[END_REF] (Figure 3A,B), specifically T73I and T86I in the ubiquitin-like domain 1 (Ubl-1), E134G in the Glu-rich acidic region (AC domain), A488S in the SARS-unique domain (SUD), Y1185N in the nucleic-acid-binding domain (NAB domain), P1228L in the Betacoronavirus-specific marker domain (G2M domain), and, finally, Y1463H, P1469S, and We identified three mutations on NSP12 [START_REF] Gao | Structure of the RNA-dependent RNA polymerase from COVID-19 virus[END_REF] (Figure 5A,B): V42A in the β-hairpin region, P323L on the interface domain, and G671S on the fingers' subdomain. In addition, two mutations were found at the C-terminal domain of its cofactor NSP8: R111A and T145I. Particularly, P323L and G671S mutations presented a high frequency among the genomes we studied, as a confirmation of the presence of these mutations in the Delta variant [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF], while V42A and the NSP8 mutations presented a very low frequency among our dataset (Supplementary Figure S3).

A1502V in the NSP3 ectodomain (3Ecto), also called zinc-finger domain (ZF domain). Among them, only A488S, P1228L, and P1469S mutations were reported as non-defining mutations for the Delta variant [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF]. We located the single mutation S139L in the II 3C protease-like domain on NSP5 [START_REF] Zhang | Crystal structure of SARS-CoV-2 main protease provides a basis for design of improved α-ketoamide inhibitors[END_REF][START_REF] Costanzi | Structural and Biochemical Analysis of the Dual Inhibition of MG-132 against SARS-CoV-2 Main Protease (Mpro/3CLpro) and Human Cathepsin-L[END_REF] (Figure 4A,B), present in only one of our samples (Supplementary Figure S2). We identified three mutations on NSP12 [START_REF] Gao | Structure of the RNA-dependent RNA polymerase from COVID-19 virus[END_REF] (Figure 5A,B): V42A in the β-hairpin region, P323L on the interface domain, and G671S on the fingers' subdomain. In addition, two mutations were found at the C-terminal domain of its cofactor NSP8: R111A and T145I. Particularly, P323L and G671S mutations presented a high frequency among the The N protein [START_REF] Peng | Structures of the SARS-CoV-2 nucleocapsid and their perspectives for drug design[END_REF][START_REF] Zinzula | High-resolution structure and biophysical characterization of the nucleocapsid phosphoprotein dimerization domain from the Covid-19 severe acute respiratory syndrome coronavirus 2[END_REF][START_REF] Boniardi | Suramin inhibits SARS-CoV-2 nucleocapsid phosphoprotein genome packaging function[END_REF] showed seven mutations (Figure 6A,B): D63G in the Nterminal Domain (NTD); S193N, R203M, and G215C in the intrinsically disordered region (IDR) 1; T334I in the C-terminal domain (CTD); and D377Y and T393Y in the IDR2. We observed the same pattern, as for the other targets, with four of the seven mutations encountered in the Delta VoC, specifically D63G, R203M, and D377Y as defining mutations, and G215C as a non-defining mutation [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF] (Supplementary Figure S4). The N protein [START_REF] Peng | Structures of the SARS-CoV-2 nucleocapsid and their perspectives for drug design[END_REF][START_REF] Zinzula | High-resolution structure and biophysical characterization of the nucleocapsid phosphoprotein dimerization domain from the Covid-19 severe acute respiratory syndrome coronavirus 2[END_REF][START_REF] Boniardi | Suramin inhibits SARS-CoV-2 nucleocapsid phosphoprotein genome packaging function[END_REF] showed seven mutations (Figure 6A,B): D63G in the N-terminal Domain (NTD); S193N, R203M, and G215C in the intrinsically disordered region (IDR) 1; T334I in the C-terminal domain (CTD); and D377Y and T393Y in the IDR2. We observed the same pattern, as for the other targets, with four of the seven mutations encountered in the Delta VoC, specifically D63G, R203M, and D377Y as defining mutations, and G215C as a non-defining mutation [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF] (Supplementary Figure S4). Among our targets, we identified the highest number of mutations on the spike protein (S) [START_REF] Walls | Structure, Function, and Antigenicity of the SARS-CoV-2 Spike Glycoprotein[END_REF] (Figure 7A,B). On the one hand, ten mutations were located on the subunit S1, particularly T19R, G142D, and R158G in the N-terminal Domain (NTD); T315Y, D614G, E661D, and P681R in disordered regions; and L452R, T478K, and G482D in the Receptor Binding Domain (RBD). On the other hand, only the mutation D950N was located on the subunit S2 heptad repeat 1 (HR1). Except for T315Y, E661D, and G482D mutations that were present in a low frequency among our samples, all the others were Among our targets, we identified the highest number of mutations on the spike protein (S) [START_REF] Walls | Structure, Function, and Antigenicity of the SARS-CoV-2 Spike Glycoprotein[END_REF] (Figure 7A,B). On the one hand, ten mutations were located on the subunit S1, particularly T19R, G142D, and R158G in the N-terminal Domain (NTD); T315Y, D614G, E661D, and P681R in disordered regions; and L452R, T478K, and G482D in the Receptor Binding Domain (RBD). On the other hand, only the mutation D950N was located on the subunit S2 heptad repeat 1 (HR1). Except for T315Y, E661D, and G482D mutations that were present in a low frequency among our samples, all the others were classified as defining mutations for the Delta variant [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF].

Discussion

To evaluate the presence of SARS-CoV-2 infection in the participants, we analysed the samples with two different techniques, the LumiraDx antigen test and the real-time PCR (RT-PCR). As expected, based on the different sensibility and specificity of the two techniques applied in attesting positive results, PCR was confirmed as the reference pointof-care (Figure 1).

The age of participants spanned from 1 to 50 years old, and the most affected age groups were 31-40 and 41-50 (Figure 1). Indeed, these age groups represent the most active part of the population because of their involvement in informal income-generating activities, and therefore their need to travel regularly.

In general, because of the very low number of positive samples in this study (circa 2%), we can state that their geographic distribution highlights a low circulation and spread of SARS-CoV-2 across the considered Bangui border checkpoints.

Overall, phylogenetic analyses showed that the Delta VoC had a widespread diffusion in the CAR as well as in other African countries, representing the great majority of sequences in our survey during July 2021 (Figure 2 and Supplementary Figure S1). While in the CAR these genomes were related mostly to the 21J clade, including lineages AY.46.4 and AY.122, the African scenario included 21A and 21I clades too. In addition, the Africa tree revealed a major presence of the Beta VoC, which was instead absent in the CAR in the same period. In line with its former presence in the CAR, the respective tree showed the abundant presence of B.1.620 (clade 20A): such a lineage was not represented in July 2021 since it was replaced by Delta VoC in the CAR after June 2021 [START_REF] Tegally | The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance[END_REF]. Similarly, even if lineage B.1.640 (clade 20A) was first identified in France in April 2021 and was reported in the ROC since September 2021 [START_REF] Colson | Emergence in southern France of a new SARS-CoV-2 variant harbouring both N501Y and E484K substitutions in the spike protein[END_REF], it was present in the CAR from January 2021 (Figure 2), likely suggesting earlier diffusion in the African continent. Finally, both analyses suggested the continuous residual circulation of the original Wuhan clades 19A and B (Figure 2 and Supplementary Figure S1).

Coherently with phylogenetic analyses, the mutations found in a high number of genomes from the dataset were mostly related to Delta VoC, both defining and non-defining the variant according to CoVariant [START_REF] Hodcroft | SARS-CoV-2 Mutations and Variants of Interest[END_REF]. Hence, their presence in the majority of sequences was expected. Contrarily, other mutations are not related to the variant widely circulating at the time of study and are present in only a few genomes, likely reflecting independent clusters of infection. An intermediate situation was found for NSP3, which harbours mutations that are not specifically linked to the Delta VoC, albeit showing a frequency similar to the ones indicated to be associated (as defining or non-defining) to this variant (Supplementary Figure S2). Among the Delta-associated mutations found in the different viral proteins, a number were reported to have an impact on viral fitness. For example, NSP12 mutations P323L and G671S were found to be associated with a higher activity and stability of the NSP7/8/12 complex [START_REF] Kim | SARS-CoV-2 variants with NSP12 P323L/G671S mutations display enhanced virus replication in ferret upper airways and higher transmissibility[END_REF]. R203M mutation in N protein plays a role in a higher spreading of the virus [START_REF] Syed | Rapid assessment of SARS-CoV-2-evolved variants using virus-like particles[END_REF]. Likewise, many of the mutations found in the spike protein play key roles in helping the virus become more efficient in escaping the immune system. Particularly, T19R mutation in the NTD has been suggested to affect the binding of neutralizing antibodies [START_REF] Planas | Reduced sensitivity of SARS-CoV-2 variant Delta to antibody neutralization[END_REF], as well as L452R and T478K mutations in the RBD. More in detail, together with G142D and R158G mutations, T19R can avoid antibody binding helping variant transmission [START_REF] Kannan | Evolutionary analysis of the Delta and Delta Plus variants of the SARS-CoV-2 viruses[END_REF], while T478K can enhance ACE2 binding, as previously shown for D614G [START_REF] Singh | Genomic characterization unravelling the causative role of SARS-CoV-2 Delta variant of lineage B.1.617.2 in 2nd wave of COVID-19 pandemic in Chhattisgarh, India[END_REF]. P681R mutation, at the boundary between S1 and S2 subunits, and D950N, on S2 subunit, can both increase the S1/S2 cleavage, enhancing viral replication [START_REF] Singh | Genomic characterization unravelling the causative role of SARS-CoV-2 Delta variant of lineage B.1.617.2 in 2nd wave of COVID-19 pandemic in Chhattisgarh, India[END_REF][START_REF] Liu | Delta spike P681R mutation enhances SARS-CoV-2 fitness over Alpha variant[END_REF]. Among the mutations not related to the Delta variant, only the E661D mutation has already been described elsewhere, as belonging to a Gamma-like-II lineage in the State of Paraná (Brazil) [START_REF] Oliveira | Re-emergence of Gamma-like-II and emergence of Gamma-S:E661D SARS-CoV-2 lineages in the south of Brazil after the 2021 outbreak[END_REF].

Conclusions

To date, only a few studies took into account SARS-CoV-2 diversity in Africa [START_REF] Tegally | The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance[END_REF][START_REF] Wilkinson | A year of genomic surveillance reveals how the SARS-CoV-2 pandemic unfolded in Africa[END_REF], and none of them were specifically focused on CAR. Therefore, despite the limited number of strains sequenced, regional studies like the present one could be highly informative. Indeed, both phylogenetic and structural analyses of the mutations found in the dataset showed that the major variant circulating in the CAR at the time of the study was the

Figure 1 .

 1 Figure 1. Geographic and demographic distribution of the samples. (A). Geographic dis the four different sampling checkpoints in Bangui, Central African Republic (CAR). distribution at the checkpoints (left); diverse percentage of subjects positive to Se Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) depending on the test performed (m depending on the collection point (right). (C). Demographic characteristics of the posi compared to the totality of the participants: general gender distribution (left) and gend and age range (right) distribution according to the tests performed.

Figure 1 .

 1 Figure 1. Geographic and demographic distribution of the samples. (A). Geographic distribution of the four different sampling checkpoints in Bangui, Central African Republic (CAR). (B). Sample distribution at the checkpoints (left); diverse percentage of subjects positive to Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) depending on the test performed (middle) and depending on the collection point (right). (C). Demographic characteristics of the positive subjects compared to the totality of the participants: general gender distribution (left) and gender (middle) and age range (right) distribution according to the tests performed.

Figure 2 .

 2 Figure 2. Phylogenetic tree of SARS-CoV-2 genomes circulating in the CAR. Neighbour Joining (NJ) phylogenetic tree of whole genome sequences of SARS-CoV-2 detected in the CAR, spanning from November 2020 to January 2022. Tree was built with MEGA X software. Bootstrap values ≥ 70 are shown on the branches. Pangolin clades and lineages are indicated with colours referred on the right side of the tree, with the Variants of Concern (VoCs) highlighted in bold. Reference sequences representing SARS-CoV-2 VoCs and CAR sequences from this study are labelled with an empty and a filled dot coloured according to the phylogenetic cluster they belong to, respectively. For VoCs, the date and country of the first report according to the World Health Organisation (WHO) are indicated between round brackets.

Figure 2 .

 2 Figure 2. Phylogenetic tree of SARS-CoV-2 genomes circulating in the CAR. Neighbour Joining (NJ) phylogenetic tree of whole genome sequences of SARS-CoV-2 detected in the CAR, spanning from November 2020 to January 2022. Tree was built with MEGA X software. Bootstrap values ≥ 70 are shown on the branches. Pangolin clades and lineages are indicated with colours referred on the right side of the tree, with the Variants of Concern (VoCs) highlighted in bold. Reference sequences representing SARS-CoV-2 VoCs and CAR sequences from this study are labelled with an empty and a filled dot coloured according to the phylogenetic cluster they belong to, respectively. For VoCs, the date and country of the first report according to the World Health Organisation (WHO) are indicated between round brackets.

Figure 3 .

 3 Figure 3. Non-structural proteins 3 (NSP3) schematic and tridimensional structure with reported mutations. (A). NSP3 structure scheme showing domain boundaries and non-synonymous mutations (coloured triangles). (B). NSP3 3D structure with the solved domains connected by dashed lines representing unsolved regions of the protein where most of the mutations found are indicated.

Figure 3 .

 3 Figure 3. Non-structural proteins 3 (NSP3) schematic and tridimensional structure with reported mutations. (A). NSP3 structure scheme showing domain boundaries and non-synonymous mutations (coloured triangles). (B). NSP3 3D structure with the solved domains connected by dashed lines representing unsolved regions of the protein where most of the mutations found are indicated. Viruses 2023, 15, 2309 9 of 16

Figure 4 .

 4 Figure 4. NSP5 schematic and tridimensional structure with reported mutations. (A). NSP5 structure scheme showing domain boundaries and the only one non-synonymous mutation found (in red). (B). NSP5 3D structure in the biological assembly as a dimer shown from different perspectives with the coloured mutation on each monomer.

Figure 4 .

 4 Figure 4. NSP5 schematic and tridimensional structure with reported mutations. (A) NSP5 structure scheme showing domain boundaries and the only one non-synonymous mutation found (in red). (B) NSP5 3D structure in the biological assembly as a dimer shown from different perspectives with the coloured mutation on each monomer.

Figure 5 .

 5 Figure 5. NSP12-7-8 schematic and tridimensional structure with reported mutations. (A). NSP12 and its cofactors NSP7 and NSP8 structure schemes showing domain boundaries and nonsynonymous mutations as coloured triangles. (B). RNA dependent RNA polymerase (RdRp) complex (NSP12-NSP7-NSP8-NSP8) 3D structure in different orientations to show the mutations found.

Figure 5 .

 5 Figure 5. NSP12-7-8 schematic and tridimensional structure with reported mutations. (A) NSP12 and its cofactors NSP7 and NSP8 structure schemes showing domain boundaries and non-synonymous mutations as coloured triangles. (B) RNA dependent RNA polymerase (RdRp) complex (NSP12-NSP7-NSP8-NSP8) 3D structure in different orientations to show the mutations found.

Figure 6 .

 6 Figure 6. Nucleocapsid (N) schematic and tridimensional structure with reported mutations. (A). N structure scheme showing domain boundaries and non-synonymous mutations found as coloured triangles. (B). N 3D structure in the biological assembly as a dimer showing N-terminal Domain (NTD) and C-terminal Domain (CTD) solved portions with the coloured mutations on each monomer.

Figure 6 .

 6 Figure 6. Nucleocapsid (N) schematic and tridimensional structure with reported mutations. (A) N structure scheme showing domain boundaries and non-synonymous mutations found as coloured triangles. (B) N 3D structure in the biological assembly as a dimer showing N-terminal Domain (NTD) and C-terminal Domain solved portions with the coloured mutations on each monomer.

Figure 7 .

 7 Figure 7. Spike protein (S) schematic and tridimensional structure with reported mutations. (A) S structure scheme showing domain boundaries and non-synonymous mutations as coloured triangles. (B) S 3D structure in its biological assembly as a trimer shown from two different perspectives with the coloured mutations on each monomer.

  The phylogenetic clusters identified in the CAR included one of the haplotypes originally circulating in Wuhan in December 2019 (lineage B, clade 19A, 88.8% bootstrap) and the VoCs Alpha (lineage B.1.1.7, clade 20I, 100% bootstrap), Beta (B.1.351, 20H, 90.2% bootstrap), Delta (lineage B.1.617.2 from clade 21A and lineages AY.46.4 and AY.122 from clade 21J), and Omicron from December 2021 (lineages BA.1, BA.1.1, BA.1.17.2, and BA.1.18 from clade 21K, 100% bootstrap) (Figure

Acknowledgments: M.C. contributed to this research under the award of the scholarship "Max Planck Summer Internship Program", edition 2021.

AY.122 20.0 hCoV-19/Central_African_Republic/RCA-01/2021|EPI_ISL_1762052|2021 MW598419.1_beta hCoV-19/Central_African_Republic/24377/2021|EPI_ISL_2281252|2021-03-04 hCoV-19/Central_African_Republic/24398/2021|EPI_ISL_2281258|2021-03-09 hCoV-19/Central_African_Republic/24256/2021|EPI_ISL_2281245|2021-02-12 hCoV-19/Central_African_Republic/235/2021|EPI_ISL_2365410|2021-03-31 hCoV-19/Central_African_Republic/RCA-60/2021|EPI_ISL_3398832|2021 MZ344997.1_alpha hCoV-19/Central_African_Republic/24507/2021|EPI_ISL_2281277|2021-03-18 hCoV-19/Central_African_Republic/24343/2021|EPI_ISL_2281248|2021-02-26 hCoV-19/Central_African_Republic/LNBCSP017/2021|EPI_ISL_17423772|2021-12-02 SARS-CoV-2_CAR_98 hCoV-19/Central_African_Republic/LNBCSP067/2021|EPI_ISL_17418849|2021-12-14 hCoV-19/Central_African_Republic/RCA-45/2021|EPI_ISL_1762087|2021 hCoV-19/Central_African_Republic/LNBCSP060/2021|EPI_ISL_17418842|2021-12-30 hCoV-19/Central_African_Republic/24172/2021|EPI_ISL_2281237|2021-01-20 hCoV-19/Central_African_Republic/LNBCSP086/2021|EPI_ISL_17418863|2021-12-30 hCoV-19/Central_African_Republic/24435/2021|EPI_ISL_2281264|2021-03-13 hCoV-19/Central_African_Republic/23800/2020|EPI_ISL_2281230|2020-12-02 hCoV-19/Central_African_Republic/24476/2021|EPI_ISL_2281280|2021-03-18 hCoV-19/Central_African_Republic/23749/2020|EPI_ISL_2281229|2020-11-24 hCoV-19/Central_African_Republic/LNBCSP062/2021|EPI_ISL_17418844|2021-12-30 hCoV-19/Central_African_Republic/23743/2020|EPI_ISL_2281228|2020-11-24 SARS-COV-2_CAR_57 hCoV-19/Central_African_Republic/LNBCSP054/2021|EPI_ISL_17418836|2021-02-18 hCoV-19/Central_African_Republic/234/2021|EPI_ISL_2365409|2021-03-31 SARS-CoV-2_CAR_82 hCoV-19/Central_African_Republic/LNBCSP129/2021|EPI_ISL_17475278|2021-12-29 hCoV-19/Central_African_Republic/LNBCSP048/2021|EPI_ISL_17418830|2021-02-16 SARS-CoV-2_CAR_66 hCoV-19/Central_African_Republic/LNBCSP044/2021|EPI_ISL_17418826|2021-02-12 hCoV-19/Central_African_Republic/LNBCSP064/2021|EPI_ISL_17418846|2021-12-30 hCoV-19/Central_African_Republic/LNBCSP057/2021|EPI_ISL_17418839|2021-02-18 hCoV-19/Central_African_Republic/24418/2021|EPI_ISL_2281261|2021-03-12 SARS-CoV-2_CAR_86 hCoV-19/Central_African_Republic/24264/2021|EPI_ISL_2281246|2021-02-16 hCoV-19/Central_African_Republic/LNBCSP076/2021|EPI_ISL_17418857|2021-12-15 hCoV-19/Central_African_Republic/LNBCSP097/2021|EPI_ISL_17475267|2021-03-16 hCoV-19/Central_African_Republic/LNBCSP071/2021|EPI_ISL_17418853|2021-12-15 hCoV-19/Central_African_Republic/RCA-34/2021|EPI_ISL_1762076|2021 hCoV-19/Central_African_Republic/LNBCSP070/2021|EPI_ISL_17418852|2021-12-15 hCoV-19/Central_African_Republic/RCA-51/2021|EPI_ISL_3398824|2021 SARS-CoV-2_CAR_59 hCoV-19/Central_African_Republic/LNBCSP015/2021|EPI_ISL_17423770|2021-12-02 hCoV-19/Central_African_Republic/LNBCSP007/2021|EPI_ISL_17423767|2021-12-20 hCoV-19/Central_African_Republic/LNBCSP069/2021|EPI_ISL_17418851|2021-12-15 hCoV-19/Central_African_Republic/RCA-026/2022|EPI_ISL_10854834|2022-01-03 hCoV-19/Central_African_Republic/RCA-46/2021|EPI_ISL_1762088|2021 hCoV-19/Central_African_Republic/LNBCSP091/2021|EPI_ISL_17475265|2021-01-25 hCoV-19/Central_African_Republic/LNBCSP072/2021|EPI_ISL_17418854|2021-12-15 hCoV-19/Central_African_Republic/LNBCSP075/2021|EPI_ISL_17418856|2021-12-15 hCoV-19/Central_African_Republic/24492/2021|EPI_ISL_2281273|2021-03-17 NC_045512_SARS2_Wuhan hCoV-19/Central_African_Republic/LNBCSP080/2021|EPI_ISL_17418859|2021-12-30 hCoV-19/Central_African_Republic/LNBCSP083/2021|EPI_ISL_17418861|2021-12-30 hCoV-19/Central_African_Republic/24397/2021|EPI_ISL_2281257|2021-03-09 hCoV-19/Central_African_Republic/LNBCSP065/2021|EPI_ISL_17418847|2021-12-14 hCoV-19/Central_African_Republic/326/2021|EPI_ISL_2365417|2021-04-26 SARS-CoV-2_CAR_73 hCoV-19/Central_African_Republic/24014/2020|EPI_ISL_2281233|2020-12-15 hCoV-19/Central_African_Republic/RCA-71/2021|EPI_ISL_3398839|2021 OM958567.1_omicron_BA.2.12.1 hCoV-19/Central_African_Republic/RCA-27/2021|EPI_ISL_2967034|2021 hCoV-19/Central_African_Republic/RCA-063/2022|EPI_ISL_10854856|2022-01-03 hCoV-19/Central_African_Republic/24216/2021|EPI_ISL_2281239|2021-02-01 hCoV-19/Central_African_Republic/LNBCSP084/2021|EPI_ISL_17423705|2021-12-30 hCoV-19/Central_African_Republic/RCA-55/2021|EPI_ISL_3398827|2021 hCoV-19/Central_African_Republic/24396/2021|EPI_ISL_2281256|2021-03-09 ON249995.1_omicron_BA.5 hCoV-19/Central_African_Republic/24452/2021|EPI_ISL_2281269|2021-03-15 hCoV-19/Central_African_Republic/266/2021|EPI_ISL_2365415|2021-04-12 hCoV-19/Central_African_Republic/JXCDC-85/2021|EPI_ISL_8917340|2021-12-12 SARS-CoV-2_CAR_94 hCoV-19/Central_African_Republic/24449/2021|EPI_ISL_2281267|2021-03-15 hCoV-19/Central_African_Republic/LNBCSP078/2021|EPI_ISL_17418858|2021-12-30 SARS_COV-2_CAR_51 hCoV-19/Central_African_Republic/LNBCSP018/2021|EPI_ISL_17423773|2021-12-02 hCoV-19/Central_African_Republic/RCA-023/2022|EPI_ISL_10854831|2022-01-03 hCoV-19/Central_African_Republic/237/2021|EPI_ISL_2365411|2021-03-31 hCoV-19/Central_African_Republic/LNBCSP068/2021|EPI_ISL_17418850|2021-06-09 hCoV-19/Central_African_Republic/LNBCSP110/2021|EPI_ISL_17475272|2021-12-08 hCoV-19/Central_African_Republic/LNBCSP043/2021|EPI_ISL_17418825|2021-12-14 SARS-CoV-2_CAR_62 hCoV-19/Central_African_Republic/RCA-68/2021|EPI_ISL_3398838|2021 hCoV-19/Central_African_Republic/LNBCSP081/2021|EPI_ISL_17418860|2021-12-30 SARS-CoV-2_CAR_97 SARS_COV-2_CAR_54 SARS-CoV-2_CAR_84 hCoV-19/Central_African_Republic/24387/2021|EPI_ISL_2281254|2021-03-06 hCoV-19/Central_African_Republic/245/2021|EPI_ISL_2365413|2021-03-02 ON373214.1_omicron_BA.4 SARS_COV-2_CAR_56 hCoV-19/Central_African_Republic/LNBCSP055/2021|EPI_ISL_17418837|2021-02-18 hCoV-19/Central_African_Republic/RCA-54/2021|EPI_ISL_3398826|2021 hCoV-19/Central_African_Republic/24196/2021|EPI_ISL_2281238|2021-01-27 hCoV-19/Central_African_Republic/RCA-16/2021|EPI_ISL_1762064|2021 hCoV-19/Central_African_Republic/LNBCSP073/2021|EPI_ISL_17418855|2021-12-15 hCoV-19/Central_African_Republic/LNBCSP124/2021|EPI_ISL_17475276|2021-12-29 hCoV-19/Central_African_Republic/RCA-62/2021|EPI_ISL_3398833|2021 hCoV-19/Central_African_Republic/LNBCSP117/2021|EPI_ISL_17475274|2021-12-20 hCoV-19/Central_African_Republic/LNBCSP066/2021|EPI_ISL_17418848|2021-12-14 hCoV-19/Central_African_Republic/LNBCSP042/2021|EPI_ISL_17418824|2021-12-14 SARS_COV-2_CAR_52 SARS-CoV-2_CAR_85 hCoV-19/Central_African_Republic/24439/2021|EPI_ISL_2281265|2021-03-15 hCoV-19/Central_African_Republic/RCA-29/2021|EPI_ISL_1762073|2021 hCoV-19/Central_African_Republic/LNBCSP056/2021|EPI_ISL_17418838|2021-02-18 hCoV-19/Central_African_Republic/24458/2021|EPI_ISL_2281270|2021-03-17 hCoV-19/Central_African_Republic/LNBCSP051/2021|EPI_ISL_17418833|2021-02-18 hCoV-19/Central_African_Republic/24494/2021|EPI_ISL_2281274|2021-03-17 hCoV-19/Central_African_Republic/24371/2021|EPI_ISL_2281250|2021-03-03 hCoV-19/Central_African_Republic/24451/2021|EPI_ISL_2281268|2021-03-15 hCoV-19/Central_African_Republic/RCA-48/2021|EPI_ISL_1764145|2021 hCoV-19/Central_African_Republic/RCA-02/2021|EPI_ISL_1762053|2021 SARS-CoV-2_CAR_64 hCoV-19/Central_African_Republic/LNBCSP050/2021|EPI_ISL_17418832|2021-02-17 hCoV-19/Central_African_Republic/LNBCSP085/2021|EPI_ISL_17418862|2021-12-30 hCoV-19/Central_African_Republic/RCA-010/2022|EPI_ISL_10854820|2022-01-03 hCoV-19/Central_African_Republic/RCA-23/2021|EPI_ISL_1762069|2021 SARS-CoV-2_CAR_58 hCoV-19/Central_African_Republic/24407/2021|EPI_ISL_2281260|2021-03-10 hCoV-19/Central_African_Republic/RCA-53/2021|EPI_ISL_3398825|2021 hCoV-19/Central_African_Republic/RCA-32/2021|EPI_ISL_1762075|2021 MZ009823.1_delta hCoV-19/Central_African_Republic/LNBCSP001/2021|EPI_ISL_17423763|2021-11-25 SARS-CoV-2_CAR_67 hCoV-19/Central_African_Republic/RCA-025/2022|EPI_ISL_10854833|2022-01-03 OL672836.1_omicron_BA.1 SARS-CoV-2_CAR_71 hCoV-19/Central_African_Republic/RCA-043/2022|EPI_ISL_10854845|2022-01-03 hCoV-19/Central_African_Republic/24474/2021|EPI_ISL_2281272|2021-03-16 hCoV-19/Central_African_Republic/24253/2021|EPI_ISL_2281243|2021-02-11 hCoV-19/Central_African_Republic/LNBCSP046/2021|EPI_ISL_17418828|2021-02-15 hCoV-19/Central_African_Republic/24505/2021|EPI_ISL_2281276|2021-03-18 hCoV-19/Central_African_Republic/RCA-30/2021|EPI_ISL_1762074|2021 hCoV-19/Central_African_Republic/24332/2021|EPI_ISL_2281247|2021-02-25 hCoV-19/Central_African_Republic/RCA-47/2021|EPI_ISL_1764144|2021 hCoV-19/Central_African_Republic/24386/2021|EPI_ISL_2281253|2021-03-06 hCoV-19/Central_African_Republic/24432/2021|EPI_ISL_2281263|2021-03-13 hCoV-19/Central_African_Republic/301/2021|EPI_ISL_2365416|2021-04-22 MN908947.3_SARS2_Wuhan hCoV-19/Central_African_Republic/LNBCSP049/2021|EPI_ISL_17418831|2021-02-17 hCoV-19/Central_African_Republic/LNBCSP061/2021|EPI_ISL_17418843|2021-12-30 OM371884.1_omicron_BA.2 hCoV-19/Central_African_Republic/23648/2020|EPI_ISL_2281227|2020-11-11 hCoV-19/Central_African_Republic/RCA-024/2022|EPI_ISL_10854832|2022-01-03 hCoV-19/Central_African_Republic/RCA-022/2022|EPI_ISL_10854830|2022-01-03 hCoV-19/Central_African_Republic/LNBCSP053/2021|EPI_ISL_17418835|2021-02-18 SARS_COV-2_CAR_12 hCoV-19/Central_African_Republic/LNBCSP010/2021|EPI_ISL_17423768|2021-11-30 MW642250.1_gamma SARS_COV-2_CAR_53 hCoV-19/Central_African_Republic/LNBCSP059/2021|EPI_ISL_17418841|2021-03-11 hCoV-19/Central_African_Republic/RCA-22/2021|EPI_ISL_1762068|2021 SARS-CoV-2_CAR_95 hCoV-19/Central_African_Republic/LNBCSP063/2021|EPI_ISL_17418845|2021-12-30 hCoV-19/Central_African_Republic/229/2021|EPI_ISL_2365408|2021-03-29 SARS-CoV-2_CAR_76 hCoV-19/Central_African_Republic/24368/2021|EPI_ISL_2281249|2021-03-02 SARS-CoV-2_CAR_61 hCoV-19/Central_African_Republic/RCA-28/2021|EPI_ISL_1762072|2021 hCoV-19/Central_African_Republic/LNBCSP052/2021|EPI_ISL_17418834|2021-02-18 hCoV-19/Central_African_Republic/RCA-021/2022|EPI_ISL_10854829|2022-01-03 hCoV-19/Central_African_Republic/RCA-57/2021|EPI_ISL_3398829|2021 hCoV-19/Central_African_Republic/LNBCSP045/2021|EPI_ISL_17418827|2021-02-15 hCoV-19/Central_African_Republic/24402/2021|EPI_ISL_2281278|2021 hCoV-19/Central_African_Republic/249/2021|EPI_ISL_2365414|2021-04-01 hCoV-19/Central_African_Republic/24245/2021|EPI_ISL_2281242|2021-02-08 hCoV-19/Central_African_Republic/242/2021|EPI_ISL_2365412|2021-03-31 SARS-CoV-2-CAR_80 hCoV-19/Central_African_Republic/24404/2021|EPI_ISL_2281279|2021 hCoV-19/Central_African_Republic/RCA-66/2021|EPI_ISL_3398837|2021 hCoV-19/Central_African_Republic/24473/2021|EPI_ISL_2281271|2021-03-16 SARS-CoV-2-CAR_72 SARS-CoV-2_CAR_65 hCoV-19/Central_African_Republic/RCA-006/2022|EPI_ISL_10854816|2022-01-03 Data Availability Statement: SARS-CoV-2 genome sequences generated in this study are available upon request.

Delta one. Hence, enhanced epidemiological investigations and genomic characterization help to monitor viral evolution, detect new variants, elucidate transmission patterns, and identify clusters in outbreaks. Such actionable insights can be used to implement strategies to fight against the spread of SARS-CoV-2 variants, especially at borders and entry points. Admittedly, the first strain reported in the CAR was in an immigrant patient, identified thanks to the national strategy put in place for the detection of the various VoCs by the Ministry of Health (MoH) and its partners, mainly the World Health Organization (WHO). In this regard, this study may serve as a model for the better understanding and control of endemic disease circulation in African countries and advocate for the development of such study elsewhere in the continent. For this purpose, the accessibility of genomic tools (e.g., SARS-CoV-2 Africa dashboard for real-time COVID-19 information) [START_REF] Xavier | SARS-CoV-2 Africa dashboard for real-time COVID-19 information[END_REF] is essential to provide epidemiologic data about ongoing spreading events and to prevent future outbreaks of pathogens with epidemic potential, especially in areas with health management issues such as low-income countries.

Supplementary Materials:

The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/v15122309/s1, Figure S1 Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.

Conflicts of Interest:

The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.


Vardan Kaladzhyan
email: vardan.kaladzhyan@cea.fr

Julien Despres

Ipsita Mandal

Cristina Bena

Majorana fermions in finite-size strips with in-plane magnetic fields

published or not. The documents may come

I. INTRODUCTION

There has been a lot of progress recently in realizing Majorana bound states (MBS) in various low-dimensional systems both theoretically and experimentally. [28][29][30][31][32] The key ingredients are a strong Rashba spin-orbit coupling and Zeeman magnetic fields. However, the inevitable orbital effects of the magnetic field drastically modify the topological phase diagram, eventually destroying the MBS. [33][34][35][36] One possible way to avoid orbital effects is to use in-plane magnetic fields. [37][38][39][40] In Refs. [41-44] various low-dimensional systems, such as infinite ribbons and finite-size strips, have been studied. These systems respect particle-hole symmetry (PHS), but violate time-reversal symmetry (TRS) due to the presence of a magnetic field. Therefore, the corresponding topological phase diagrams have been obtained by computing the Z 2 topological invariant (in accordance with the well-known tenfold topological classification 45 ), as well as numerical methods. However, only the case of magnetic fields perpendicular to the plane of the system has been considered.

In this paper we consider the formation of MBS in infinite ribbons and finite-size strips, while focusing on inplane magnetic fields. We thus show that infinite ribbons can host one or two pairs of chiral Majorana modes [46][47][48][49][50] in the presence of an in-plane magnetic field perpendicular to the edges, and we calculate the corresponding topological phase diagram. For finite-size strips we first use a numerical diagonalization of the tight-binding Hamiltonian 51 which allows us to evaluate the Majorana polarization (MP). 41,43,44 We show that one or multiple MBS pairs can form along the short edges of the system for an in-plane magnetic field parallel to the longer dimension of the finite-size strips, and we calculate the topological phase diagrams of these systems. Second we use the singular points (SP) technique introduced in Refs. [1-4], based on the momentum values where the determinant of the Hamiltonian vanishes, and we show that it yields results consistent with the numerical ones.

We study the stability of the resulting topological states with respect to disorder, 42 and we confirm that the states with even numbers of Majorana fermions are not protected, whereas those with odd numbers are. We also perform a calculation of the Z 2 invariant which should give one access to the parity of the number of Majorana modes. Indeed we find that this calculation predicts correctly a topologically non-trivial character for the phasespace regions shown numerically to have an odd number of MBS pairs and to survive the effects of disorder.

The paper is organized as follows: in Sec. II we introduce the general model and give a concise description of the tight-binding and singular points techniques, as well as of the Majorana polarization definition. In Sec. III we present the results for 1D wires and infinite ribbons. In Sec. IV we present the phase diagrams for finite-size strips of different widths, using the numerical diagonalization and singular points methods. In Sec V we present the effects of disorder and we compare the disordered results with those obtained using a topological-invariant calculation. In Section VI we consider finite-size square systems. Finally, we conclude in Sec. VII leaving the technical details of topological-invariant calculations for the Appendix.

II. MODEL AND METHODS

We introduce a model that can describe different 1D and 2D systems with an intrinsic (or proximityinduced) s-wave superconducting pairing ∆, longitudinal and transversal Rashba spin-orbit couplings λ x,y , and a Zeeman magnetic field B = (B x , B y , B z ). We write the Hamiltonian in the Nambu basis Ψ r = arXiv:1611.09367v2 [cond-mat.mes-hall] 9 Dec 2017 c † r↑ , c † r↓ , c r↓ , -c r↑ , where c rσ (c † rσ ) annihilates (creates) a particle of spin σ at site r = (i, j) in a square lattice:

H = r Ψ † r (-µτ z + ∆τ x + B • σ) Ψ r + (1) + Ψ † r (-t x -iλ x σ y ) τ z Ψ r+x + H.c. + + Ψ † r (-t y + iλ y σ x ) τ z Ψ r+y + H.c
. , where t is the hopping amplitude, µ denotes the chemical potential, x,y are unit vectors for the x and y directions correspondingly, and the lattice spacing is set to unity.

A. Numerical tight-binding techniques and the Majorana polarization

The eigenstates of the tight-binding Hamiltonian described above can be obtained using a numerical diagonalization (here performed using the MatQ code 51 ). In the Nambu basis, an eigenstate j of the tight binding Hamiltonian can be written as ψ j T r = u j r↑ , u j r↓ , v j r↓ , -v j r↑ , where u and v denote the electron and hole components respectively. The vector of the local Majorana polarization 41,43,44 on each site r = (x, y), for the eigenstate j is given by:

P j (r) ≡ P j x (r) P j y (r) ≡   -2 Re u j r↑ v j r↑ + u j r↓ v j r↓ -2 Im u j r↑ v j r↑ + u j r↓ v j r↓   (2)
This quantity allows to discriminate locally pure electron (hole) states from Majorana-like states. It is easy to see that for pure electron (v r↑ = v r↓ = 0) and pure hole states (u r↑ = u r↓ = 0) the local Majorana polarization equals to zero. For our purposes it is more practical to use the integral of the MP vector over a spatial region R defined as:

C j = r∈R P j x (r) + iP j y (r) 2 (3)
Note that in Eqs. (2,3) we assume that the wave function is normalized.

To obtain the topological phase diagram we first find the lowest energy states of the given system. If these states have energies close to zero they may be MBS. We divide our system into two halves (along the shorter length), and we compute the integral of the MP vector in each of these halves defined by r ∈ R for each 'zero'energy state. The states that have C = 1 are MBS, and those with C = 0 are regular electron or hole states. Note that we may have only a pair of state with C = 1, or multiple degenerate zero-energy MBS states with C = 1. In the results that we present here we sum the MP over all the lowest energy states. Note however that the states with even C (even number of MBS pairs) are not topologically protected, and thus any small disorder introduced into the system destroys such Majorana states.

B. Singular points of the Hamiltonian

To find a MBS in a system described by a PHS Hamiltonian H(k), we seek for localized zero-energy solutions of the Schrödinger equation: H(k)Φ = 0. In the most general case these solutions are of the form e ikr that can be rewritten as e ik r •e -zr ⊥ , where k denotes the "good" quantum number and z is defined below. We analytically continue k to the complex plane and we consider the solutions of the following equation det

H(k) = 0, (4)
defining the so-called singular points k i ≡ k + iz in the complex plane at which the determinant of the Hamiltonian vanishes. By definition z is given by the imaginary part of the singular point k i . The practical use of these complex momentum values is the following: by continuously changing the parameters of our Hamiltonian, we continuously change the corresponding k i 's. If we are in a topological phase, z must be positive (in other words the solution is localized). As soon as z crosses zero and becomes negative, the solution becomes delocalized, and therefore we enter a non-topological phase. Further details can be found in Refs. [1-3].

We propose the following way of constructing a phase diagram. The parameter space is given by the chemical potential µ and the magnetic field B = |B|. Firstly, we find all the k i 's as a function of the parameters in the Hamiltonian, such that k i = k i (µ, B), i ∈ 1, 2N , where 2N is the total number of k i solutions. We then sort them at each point in the parameter space with respect to their imaginary parts as follows: Im k i < Im k i+1 , i ∈ 1, 2N -1. Sorting is one way to construct continuous functions Im k i (µ, B) in the parameter space. Although it is possible to deal with discontinuities analytically, 3 in numerical simulations the continuity of Im k i 's becomes crucial since it is hard to discriminate the zeros of Im k i (µ, B) from its discontinuities. Subsequently we look for the functions k i (µ, B) whose imaginary part crosses zero. This can be done by plotting their imaginary part as a function of the system parameters. Since Eq. ( 4) yields pairs of solutions with opposite imaginary parts, and since these pairs have been sorted according to their imaginary parts, it is therefore sufficient to plot the imaginary part of either the smallest positive root (i = N + 1) or of the largest negative root (i = N ). The set of points (µ 0 , B 0 ) where Im k N (µ 0 , B 0 ) = 0 (or equivalently Im k N +1 (µ 0 , B 0 ) = 0) yield thus the phase transition lines between the topological and non-topological regions in the phase diagram.

Note that this technique can in principle used also to count the number of MBS present in a given phase. The corresponding counting formula is extremely simplified when "Exceptional points" are present 1,2 for a system with unbroken chiral symmetry. In such a case, the Hamiltonian can be brought to a block off-diagonal form, however, when the chiral symmetry is broken, this blockoff diagonal form cannot be achieved in any basis and it becomes cumbersome to isolate two sets of N continuous solutions k i (with opposite imaginary parts) and plug them into the counting formula to get the total number of MBS. Thus we will use the singular points method only to find the phase-transition lines. In order to find the number of MBS pairs in a given phase we will rely on the numerical tight-binding calculations.

III. 1D WIRES AND INFINITE RIBBONS A. 1D wire

We start by describing the well-known phase diagram of a 1D SC wire which we take to be lying along the xaxis (N y = 1 and N x 1). In the presence of a magnetic field the time-reversal symmetry (TRS) is broken, and only the particle-hole symmetry (PHS) holds, therefore the system is in the topological class D described by a Z 2 invariant. 45 If the applied magnetic field is perpendicular to the spin-orbit direction, i.e. either B = (0, 0, B z ) or B = (B x , 0, 0), the SC wire enters a gapful topological phase as soon as B x or B z become larger than (µ -2t x ) 2 + ∆ 2 . The corresponding phase diagram is shown in Fig. 1. Further details of the Z 2 invariant calculation can be found in the first subsection of Appendix A.

B. Infinite ribbon

In this subsection we study an infinite ribbon with a finite but large number of sites in the y-direction (N y 1), and infinite in the x-direction N x → ∞ (see Fig. 2). We set λ x = λ y = λ and t x = t y = 1. We are interested in the formation of zero-energy edge states parallel to the x axis (see the black circles in Fig. 2). We consider open boundary conditions (also referred to as "zero boundary conditions") at the edges of the ribbon (in the y direction), and that the ribbon is infinite (or equivalently periodic boundary conditions) in the x direction. In Fig. 3 we plot the band structure of this system for an in-plane magnetic field B y parallel to the y-axis (perpendicular to the ribbon edges), as well as the topological phase diagram of such a ribbon obtained using the tight-binding numerical diagonalization and the evaluation of the MP as described in section II. First of all, we note that the spectrum is PHS even though the band structure is not. Second, as we can see from the band structure, the system may become gapless, i.e. there are region in the momentum space in which the gap in the spectrum is closing. However, despite the fact that there is no overall gap, chiral MBS 50 do form, and they correspond to values of momenta for which the bulk spectrum remains gapped (e.g. k x a = 0 and k x a = π). Such states are dispersive and propagate along the edges of the ribbon. We should note that similar situations in which the closing the gap can occur for certain regions in the parameter space have been previously studied, and it has been shown that, despite the absence of an overall gap, the system can still be topological, and support MBS. [52][53][54][55][56] The number of MBS pairs varies from 0 to 2, depending on the parameters of the system (see Fig. 3). However, the case of two Majorana fermions propagating at the same boundary is not stable, and in the absence of protection by TRS, for example in the presence of small disorder, such states would combine to form a conventional fermionic state. Thus the system is topologically non-trivial only when the number of MBS pairs is equal to 1. It is also worth mentioning that the number of sites in the y direction must be large enough so that the overlap of the wave functions of the two Majorana states localized on the two opposite edges of the ribbon is exponentially small, and that these states cannot hybridize and acquire a finite energy.

Note that if the magnetic field is applied along the xaxis the system is also gapless, however, in this case no Majorana modes form, for any region in the parameter space, and the system is fully trivial.

IV. FINITE-SIZE STRIPS

In what follows we focus on finite-size strips, i.e. systems made-up of N y > 1 coupled wires each with a finite but large number of sites N x 1 and N x N y . We consider an in-plane magnetic field B x parallel to the long edge of the system (see Fig. 4). Note that for a magnetic field parallel to the y-axis, no Majorana states can form, since the magnetic field would in this case be parallel to the direction of the spin-orbit coupling in the wires. A similar system was considered in Ref. [44], but only for magnetic fields perpendicular to the plane of the system. We apply open boundary conditions in both x and y directions.

As described in Section II, we will use two main tools to obtain the phase diagram for these systems. The first is to numerically diagonalize the tight-binding Hamiltonian and employ the integrated MP by plotting its value as a function of the chemical potential and the magnetic field. The second is to assume that the momentum k x along x is a "good" quantum number, and exploit the SP technique. In Fig. 5 we show numerically that the results of the first two methods are fully consistent. Each phase transition boundary defined as a change in the number of MBS pairs obtained via the MP technique corresponds to a line of zeroes in the SP plot. The only apparent exception is the special case of the white lines in the N y = 4 close to µ = t and µ = 2.2t and B = 0.5t. They seem to correspond to the special case of a zero-size non-topological line-like region between two topological regions with one pair of MBS. Such situations, i.e. topological regions divided by a line of non-topological points, can arise, and are well captured here by the SP method. The numerical method is a bit less precise in this case, and the non-topological lines acquire a finite width, mostly because of the finite length of the considered systems (in the infinite-length limit the width of these regions should go to zero).

Note that the integrated MP allows not only to show the phase transition boundaries, but also to give access to the number of emerging MBS. Since the chiral symmetry (a combination of the PHS and the TRS) is absent, we cannot easily use the counting formula introduced in Ref. [3] to obtain the number of Majorana modes using this method. The counting formula is in principle also applicable in the absence of the chiral symmetry, but the broken TRS case is very cumbersome and much harder to implement numerically. Therefore, we use here the SP technique only to obtain the phase transition lines; the actual number of the MBS pairs and the topological char- acter of a given phase space region are obtained numerically via the calculation of the total MP. We should point out that some segments of the phase transition lines in the right column of Fig. 5 are almost non-visible. This is not due to the failure of the method, but to the numerical grid: the regions in which the zeroes of the determinant occur are very thin and the number of points required in the grid would be too large to capture them entirely. We did check though that the phase transition lines are present everywhere as expected, even if not fully shown in Fig. 5.

Note also that by increasing the number of wires we can In the left column we plot the total MP summed over all low-energy states with a MP larger than a given cutoff, here taken to be 0.8. The color scheme indicates the number of MBS pairs. In the right column we plot the results of the SP calculation; the phase transition lines are given by the zeroes of the plot. We consider finite-size strips with 2, 3 and 4 coupled wires. The results of these two methods are consistent, and the phase transition lines coincide. We set ∆ = 0.2t, λx = λy = 0.5t.

increase the number of MBS. However, as we will show in the next section only the states with an odd number of MBS pairs are topologically protected.

V. EFFECTS OF DISORDER AND TOPOLOGICAL INVARIANT CALCULATIONS

In what follows we show that a small amount of disorder makes the Majorana modes re-combine and form regular electronic states in all the regions in the parameter space with an even number of Majorana modes; thus these regions are not topologically protected. However, in all the regions with odd numbers of Majorana modes one MBS pair survives the effects of disorder. In the left column of Fig. 6 we show the phase diagrams for finite-size strips in the presence of disorder. The disorder considered here is a random variation of the value of the Zeeman magnetic field with an intensity of 5% around its average value. 42 Indeed, we see that in the even-parity regions of the phase space the Majorana modes are destroyed, confirming their non-topological character.

We also present the corresponding phase diagrams computed using the topological invariant (TI) (for the details of the derivations see Appendix A, as well as Refs. [44] and [57]). In Fig. 6 we compare the phase diagrams showing the topological regions surviving the effects of disorder (left column) and those obtained using the topological invariant (right column). Indeed, up to some sets of lines of special points, the topological regions, as predicted by the topological invariant, coincide with the regions in the phase diagram shown numerically to exhibit an odd-parity of MBS pairs and to survive the presence of disorder.

It is also worth discussing how the value of the topological gap protecting the zero-energy states changes in the presence of disorder. In Fig. 7 we plot the energy spectra of a 4-wire finite-size strip for a fixed value of the chemical potential as a function of an in-plane magnetic field B x , both in the absence and in the presence of disorder. This corresponds to taking vertical cuts of the lower left panels of Figs. 5 and6. Without disorder, in full accordance with Fig. 5, we have MBS for magnetic fields B x from ∼ 0.45t to ∼ 0.73t and ∼ 0.92t to ∼ 1.25t. It is worth mentioning that certain regions contain more than one pair of MBS. The number of pairs is shown above the corresponding Majorana zero energy lines, highlighted in red. First, we note that, consistent with the phase diagrams presented in Figs. 5 and6, all the regions with odd numbers of MBS are protected against disorder, exhibiting one stable zero-energy mode (cf. regions from ∼ 0.55t to ∼ 0.64t and ∼ 0.92t to ∼ 1.25t respectively), whereas in the regions with even numbers these states acquire a finite energy in the presence of disorder, confirming that these regions in the phase diagram are not topologically protected. Moreover we see that the gap protecting these zero-energy states is affected slightly by disorder, more significantly for the states with even numbers of Majoranas, consistent with the lack of topological protection for these states. In the left column we depict the MP of the lowest-energy mode for a disordered system. We impose a MP cutoff of 0.95 (the states with MP smaller than 1 cannot be considered actual Majoranas and usually correspond to non-zero energies, even if they remain Majorana-like). In the right column we depict the phase diagram as obtained using the topological invariant calculation (without disorder) (the topological regions are shown in violet). Note that, up to some special-points lines, the TI results are fully consistent with the those for the MP in the disordered system, which is expected since the TI gives access to the parity of the number of the MBS. In all the panels ∆ = 0.2t, λx = λy = 0.5t.

VI. FINITE-SIZE SQUARES

If both N x,y 1 and are comparable in size then we are dealing with a finite-size square. In Ref. [44] it has been shown that for perpendicular Zeeman magnetic fields finite-energy quasi-Majorana-like states (C = √ 2/2) may form, localized mostly in the corners of these square flakes, for a set of parameters inside the 2D bulk topological phase. However, for in-plane magnetic fields (e.g. along x-axis) the situation is very different since the rotation symmetry is broken and we always have an in-plane special direction, and we can no longer expect quasi-Majorana states with rotationally symmetric MP.

By analyzing the MP of these systems we note that the generic situation that emerges is that depicted in Fig. 8: quais-disordered edge states localized on the edges of the system perpendicular to the magnetic field. Such states have also a quasi-disordered MP, and the integral of the MP over one of these edges states is finite (for the case in Fig. 8 this is of the order of 0.9). This tendency to form a Majorana state is larger for values of the magnetic field close to the transition, and for systems with a very large N x we can actually recover actual Majorana states in these systems on the edges perpendicular to the magnetic field. The systems required to recover a full Majorana are too large for our numerical abilities, but even for smaller systems we have managed to tune up the parameters to get a MP up to 0.9, and increasing the size will improve this value. This is important from an experimental perspective, since it indicates that for in-plane fields actual Majorana states can form even in wide square systems, while for perpendicular fields this can never be the case unless one dimension is much larger than the other one.

VII. CONCLUSIONS

We have studied the formation of Majorana bound states in infinite ribbons, finite-size strips and squares with Rashba spin-orbit coupling and an in-plane magnetic field. We have shown that in infinite ribbons chiral Majorana fermions may form when the magnetic field is perpendicular to the edges of the ribbon. Furthermore, we have studied finite-size strips exploiting a numerical diagonalization technique and the Majorana polarization, as well as the singular points technique, and we have proven the qualitative equivalence of these two methods in constructing the phase diagrams of these sys-tems. We have also evaluated the topological invariant for the finite-size strips and we have shown that its usage allows one to obtain the correct phase diagrams for the parity of the number of MBS pairs. Moreover, we have confirmed numerically that the phases with even number of MBS pairs are not stable in the presence of disorder, and are thus topologically trivial, while the phases with an odd number of MBS preserve their topological character.

collinear with the spin-orbit coupling (B z or B x , but not B y ), the system stays gapful and thus topological invariants are well-defined. Time-reversal symmetry is broken in the presence of a magnetic field, therefore we expect to have a Z 2 topological invariant in accordance with the classification in Ref. [45]. On the contrary, if we consider the case of a non-zero B y along the y-axis, we end up having a gapless trivial phase. Thus, below we set B y = 0.

To compute the topological invariant we seek the Gamma-points of the Hamiltonian (A1), i.e. the points for which L(k) = 0. It is easy to see that within the first Brillouin zone we have two such points: Γ 1 = 0, Γ 2 = π. At each of these points one can define a skew-symmetric matrix W (k = Γ i ) = H(k = Γ i )Λ with an associated Pfaffian. The topological invariant is given by ) This expression can be simplified using the identity (Pf A) 2 = det A, thus yielding:

δ = k=Γi det [W (k)] Pf [W (k)] . ( A3
δ = k=Γi sgn Pf [W (k)] = sgn Pf [W (0)] sgn Pf [W (π)] . (A4)
We compute the corresponding Pfaffians

Pf [W (0)] = B 2 -∆ 2 -(µ + 2t) 2 , Pf [W (π)] = B 2 -∆ 2 -(µ -2t) 2 ,
where

B 2 ≡ |B| 2 = B 2 x + B 2 z .
Therefore, the topological invariant is given by:

δ = sgn (B 2 -∆ 2 -(µ + 2t) 2 )(B 2 -∆ 2 -(µ -2t) 2 ) . (A5)
The corresponding phase diagram is given in Fig. 1 as well as on the left panel of Fig. 9.

2D systems

For this system f (k) ≡ -2t(cos k x + cos k y ) -µ and L(k) ≡ iλ(sin k x -i sin k y ) is the Rashba spin-orbit coupling term. Despite the fact that in parallel magnetic fields the system can become gapless for certain regions in the parameter space, below we compute formally a Z 2 topological invariant. This invariant should indicate the parity of the number of Majorana modes arising at the edges introduced into the system. To compute the topological invariant we find the Gamma-points of the Hamiltonian (A1), i.e. the points where L(k) = 0. Within the first Brillouin zone there are four such points: Γ 1 = (0, 0), Γ 2 = (π, 0), Γ 3 = (0, π), Γ 4 = (π, π). At each of these points one can define a skew-symmetric matrix W (k = Γ i ) = H(k = Γ i )Λ with an associated Pfaffian. Exactly as in the previous subsection the topological invariant is given by δ

= k=Γi sgn Pf [W (k)] . (A6)
Computing the corresponding Pfaffians we get:

Pf [W (Γ 1 )] = B 2 -∆ 2 s -(µ + 4t) 2 , Pf [W (Γ 2 )] = Pf [W (Γ 3 )] = B 2 -∆ 2 -µ 2 , Pf [W (Γ 4 )] = B 2 -∆ 2 -(µ -4t) 2 ,
where B 2 ≡ |B| 2 = B 2 x + B 2 y + B 2 z . Therefore, the topological invariant is given by:

δ = (B 2 -∆ 2 -µ 2 ) 2 sgn (B 2 -∆ 2 -(µ + 4t) 2 )(B 2 -∆ 2 -(µ -4t) 2 ) . (A7)
The corresponding phase diagram is given in the right panel of the Fig. 9. It is clear that there is no difference between the cases of a magnetic field perpendicular to the plane and an in-plane magnetic field perpendicular to the edges of the ribbon (note however that an in-plane magnetic field parallel to the edges of the ribbon does not give rise to a topological phase for any region in the parameter space). Despite the closing of the gap, this calculation yields the correct result for the parity of the number of Majorana modes (compare with the results of the numerical simulations presented in Fig. 3).

Finite-size strips

We follow the section about finite-size strips in Ref. [44], but instead of considering a magnetic field perpendicular to the plane of the system, we consider also non-zero in-plane components of the magnetic field, B x and B y . The Fourier-transformed Hamiltonian of this system in the case of N y coupled wires can be written in the Nambu basis ψ ↑ , ψ ↓ , ψ † ↓ , -ψ † ↑ in the following manner: where the N y × N y square matrices I Ny , M 1Ny , M 2Ny act in the sublattice space and are defined as follows:

H(k) =      f (
I Ny ij = δ ij , M 1Ny
ij = δ i,j-1 + δ i-1,j , M 2Ny ij = -δ i,j-1 + δ i-1,j , ∀i, j ∈ 1, N y , with δ denoting the Kronecker delta. We define also the spectrum of the free electrons f (k) = -2t cos k -µ, and the spin-orbit coupling term L(k) = -2iλ sin k. In this system the PHS operator can be written as follows: where K is the complex conjugation operator. We take the same path as in the previous subsections: firstly, we find the Gamma-points of the Hamiltonian (A9), i.e. the points where L(k) = 0. We find only two such points in the first Brillouin zone: Γ 1 = 0, Γ 2 = π. A skew-symmetric matrix W (k = Γ i ) ≡ H(k = Γ i )Λ is defined at each of those points along with an associated Pfaffian. Exactly as before, the topological invariant is given by δ = sgn Pf [W (0)] sgn Pf [W (π)] (A11)

  FIG. 1. (color online) The phase diagram of a 1D superconducting nanowire obtained with topological invariant calculation as a function of the chemical potential µ and the magnetic field along the wire B = Bx (the phase diagram remains the same in the case of a magnetic field perpendicular to the wire B = Bz). We set ∆ = 0.2t, λx = 0.5t.

FIG. 2 .

 2 FIG. 2. (color online) A sketch of an infinite ribbon along xaxis with a magnetic field B = By perpendicular to its edges. The black sites denote the edges of the ribbon where the chiral Majorana modes are localized.

FIG. 3 .

 3 FIG. 3. (color online) Band structure of an infinite ribbon with a magnetic field B = By perpendicular to the edges for µ = 4t, By = 0.3t (upper panel) and µ = 0t, By = 0.3t (middle panel). Note that the system may host either one or two pairs of chiral Majorana modes. The corresponding topological phase diagram (lower panel) depicts the number of Majorana modes (as evaluated from the total MP) as a function of µ and B. In all the examples we set ∆ = 0.2t, λx = λy = 0.5t.

FIG. 4 .FIG. 5 .

 45 FIG. 4. (color online)A sketch of a finite-size strip with a magnetic field B = Bx along the x-axis. The black sites denote the short edges of the system where the Majorana modes would be localized. This system can be thought of as a set of 1D wires coupled in the y-direction.

1 FIG. 6 .

 16 FIG.6. (color online) Topological phase diagrams in the (µ, Bx) plane for 2 and 4 coupled wires. In the left column we depict the MP of the lowest-energy mode for a disordered system. We impose a MP cutoff of 0.95 (the states with MP smaller than 1 cannot be considered actual Majoranas and usually correspond to non-zero energies, even if they remain Majorana-like). In the right column we depict the phase diagram as obtained using the topological invariant calculation (without disorder) (the topological regions are shown in violet). Note that, up to some special-points lines, the TI results are fully consistent with the those for the MP in the disordered system, which is expected since the TI gives access to the parity of the number of the MBS. In all the panels ∆ = 0.2t, λx = λy = 0.5t.

FIG. 7 .

 7 FIG. 7. (color online)Energy spectra for 4-wires finite-size strips with and without disorder (lower and upper panel respectively) as a function of an in-plane magnetic field Bx, varying from 0.25t to 1.25t. We restrict ourselves to plotting only the lowest 60 energy levels and we set ∆ = 0.2t, λx = λy = 0.5t, and µ = 0.69t. These panels correspond to vertical cuts of lower left panels in Fig.5 and 6respectively. The number of pairs is shown above the corresponding Majorana zero energy lines, highlighted in red.

FIG. 8 .

 8 FIG. 8. (color online)The MP vector for a finite-size square consisting of 100 × 100 sites in a magnetic field along the xaxis. We choose a set of parameters µ = 4t, Bx = 0.28t, ∆ = 0.2t, λx = λy = 0.5t.

FIG. 9 .

 9 FIG. 9. (color online) The phase diagram of a 1D nanowire (left panel) and of an infinite ribbon (right panel) as a function of the chemical potential µ and the magnetic field B, obtained using a topological invariant calculation. In the case of a wire the magnetic field is either along the wire B = Bx or perpendicular to the wire B = Bz, whereas in the case of an infinite ribbon it is perpendicular to the edge of the ribbon, i.e. B = By. The black lines correspond to the values for which the topological invariant δ = 0. The topological regions are colored in violet. We set ∆ = 0.2t, λx = 0.5t, λy = 0t for the left panel and ∆ = 0.2t, λx = λy = 0.5t for the right one.

  k)I Ny -tM 1Ny L(k)I Ny + iλM 2Ny ∆I Ny 0 -L(k)I Ny -iλM 2Ny T f (k)I Ny -tM 1Ny 0 ∆I Ny ∆I Ny 0 -f (k)I Ny + tM 1Ny L(-k)I Ny + iλM 2Ny T 0 ∆I Ny -L(-k)I Ny -iλM 2Ny -f (k)I Ny + tM 1Ny Ny (B x -iB y )I Ny 0 0 (B x + iB y )I Ny -B z I Ny 0 0 0 0 B z I Ny (B x -iB y )I Ny 0 0 (B x + iB y )I Ny -B z I Ny

Ξ

VIII. ACKNOWLEDGEMENTS

We would like to thank Nicholas Sedlmayr for fruitful discussions and useful comments on our manuscript. VK is grateful for the hospitality provided by the Perimeter Institute where a part of this work was performed. The research at the Perimeter Institute was supported in part by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Information. The research at IPhT was supported in part by the ERC Starting Independent Researcher Grant NANOGRAPHENE 256965.

IX. AUTHOR CONTRIBUTION STATEMENT

Cristina Bena suggested the issue, Ipsita Mandal helped with implementing the singular points technique, and Vardan Kaladzhyan, Julien Despres and Cristina Bena carried out the calculations. All the authors wrote and revised this article.

Appendix A: Topological invariant calculation

We write the Bogoliubov-de Gennes Hamiltonian on a square lattice in the Nambu basis ψ ↑ , ψ ↓ , ψ † ↓ , -ψ † ↑ as:

where B = (B x , B y , B z ) is the magnetic field, f (k) is the dimension-dependent dispersion for electrons on the lattice with a chemical potential µ and a hopping parameter t, while L(k) is the Rashba spin-orbit coupling term, which also depends on the dimensionality of the lattice. We disregard the orbital effects of the magnetic field. The Hamiltonian given by Eq. (A1) is particle-hole symmetric, i.e.

ΞH(k)Ξ

where K is the complex conjugate operator, and we set Λ ≡ σ y ⊗ τ y .

1D wires

We consider a 1D superconducting nanowire with Rashba spin-orbit coupling and with an arbitrary direction of the magnetic field. We thus have f (k) ≡ -2t cos k -µ and L(k) ≡ iλ sin k. As long as the magnetic field is not