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We present an approach for the calculation of the $\mathbb{Z}_2$ topological invariant in non-crystalline two-dimensional quantum spin Hall insulators. While topological invariants were originally mathematically introduced for crystalline periodic systems, and crucially hinge on tracking the evolution of occupied states through the Brillouin zone, the introduction of disorder or dynamical effects can break the translational symmetry and imply the use of larger simulation cells, where the $\bf{k}$-point sampling is typically reduced to the single $\Gamma$-point. Here, we introduce a single-point formula for the spin Chern number that enables to adopt the supercell framework, where a single Hamiltonian diagonalisation is performed. Inspired by the work of E. Prodan [Phys. Rev. B, $\textbf{80}$, 12 (2009)], our single-point approach allows to calculate the spin Chern number even when the spin operator $\hat{s}_z$ does not commute with the Hamiltonian, as in the presence of Rashba spin-orbit coupling. We validate our method on the Kane-Mele model, both pristine and in the presence of Anderson disorder. Finally, we investigate the disorder-driven transition from the trivial phase to the topological state known as topological Anderson insulator. Beyond disordered systems, our approach is particularly useful to investigate the role of defects, to study topological alloys and in the context of ab-initio molecular dynamics simulations at finite temperature., Comment: 10 pages, 7 figures, code freely available at https://github.com/roberta-favata/spinv

Monolayer 1T'-WTe$_2$ has been the first two-dimensional crystal where a quantum spin Hall phase was experimentally observed. In addition, recent experiments and theoretical modeling reported the presence of a robust excitonic insulating phase. While first-principles calculations with hybrid functionals and several measurements at low temperatures suggest the presence of a band gap of the order of 50 meV, experiments could confirm the presence of the helical edge states only up to 100 K. Here, we study with first-principle simulations the temperature effects on the electronic structure of monolayer 1T'-WTe$_2$ and consider the contributions of both thermal expansion and electron-phonon coupling. First, we show that thermal expansion is weak but tends to increase the indirect band gap. Then, we calculate the effect of electron-phonon coupling on the band structure with non-perturbative methods and observe a small reduction of the band inversion with increasing temperature. Notably, the topological phase and the presence of a finite gap are found to be particularly robust to thermal effects up to and above room temperature., 6 pages, 4 figures + supplementary material (1 page, 1 figure)

Taking the clue from the modern theory of polarization [Rev. Mod. Phys. 66, 899 (1994)], we identify an operator to distinguish between Z2-even (trivial) and Z2-odd (topological) insulators in two spatial dimen- sions. Its definition extends the position operator [Phys. Rev. Lett. 82, 370 (1999)], which was introduced in one-dimensional systems. We first show a few examples of noninteracting models where single-particle wave functions are defined and allow for a direct comparison with standard techniques on large system sizes. Then, we illustrate its applicability for an interacting model on a small cluster where exact diagonalizations are available. Its formulation in the Fock space allows a direct computation of expectation values over the ground-state wave function (or any approximation of it), thus, allowing us to investigate generic interacting systems, such as strongly correlated topological insulators.

Quantum spin Hall insulators (QSHI) have been proposed to power a number of applications, many of which rely on the possibility to switch on and off the non-trivial topology. Typically this control is achieved through strain or external electric fields, which require energy consumption to be maintained. On the contrary, a non-volatile mechanism would be highly beneficial and could be realized through ferroelectricity if opposite polarization states are associated with different topological phases. While this is not possible in a single ferroelectric material where the two polarization states are related by inversion, the necessary asymmetry could be introduced by combining a ferroelectric layer with another two-dimensional (2D) trivial insulator. Here, by means of first-principles simulations, not only we propose that this is a promising strategy to engineer non-volatile ferroelectric control of topological order in 2D heterostructures, but also that the effect is robust and can survive up to room temperature, irrespective of the weak van der Waals coupling between the layers. We illustrate the general idea by considering a heterostructure made of a well-known ferroelectric material, In$_2$Se$_3$, and a suitably chosen, easily exfoliable trivial insulator, CuI. In one polarization state the system is trivial, while it becomes a QSHI with a robust band gap upon polarization reversal. Remarkably, the topological band gap is mediated by the interlayer hybridization and allows to maximise the effect of intralayer spin-orbit coupling, promoting a robust ferroelectric topological phase that could not exist in monolayer materials and is resilient against relative orientation and lattice matching between the layers., Comment: 13 pages, 8 figures + supplementary (6 pages, 3 figures)

We report high-resolution angle-resolved photoemission measurements on single crystals of Pt_{2}HgSe_{3} grown by high-pressure synthesis. Our data reveal a gapped Dirac nodal line whose (001) projection separates the surface Brillouin zone in topological and trivial areas. In the nontrivial k-space range, we find surface states with multiple saddle points in the dispersion, resulting in two van Hove singularities in the surface density of states. Based on density-functional theory calculations, we identify these surface states as signatures of a topological crystalline state, which coexists with a weak topological phase.

Funder: European Union's Horizon 2020 research and innovation program (project E-CAM). Grant agreement no. 676531, Funder: NCCR MARVEL of the Swiss National Science Foundation and the European Union’s Centre of Excellence MaX “Materials design at the Exascale”. Grant no. 824143, Maximally-localised Wannier functions (MLWFs) are routinely used to compute from first-principles advanced materials properties that require very dense Brillouin zone integration and to build accurate tight-binding models for scale-bridging simulations. At the same time, high-throughput (HT) computational materials design is an emergent field that promises to accelerate reliable and cost-effective design and optimisation of new materials with target properties. The use of MLWFs in HT workflows has been hampered by the fact that generating MLWFs automatically and robustly without any user intervention and for arbitrary materials is, in general, very challenging. We address this problem directly by proposing a procedure for automatically generating MLWFs for HT frameworks. Our approach is based on the selected columns of the density matrix method and we present the details of its implementation in an AiiDA workflow. We apply our approach to a dataset of 200 bulk crystalline materials that span a wide structural and chemical space. We assess the quality of our MLWFs in terms of the accuracy of the band-structure interpolation that they provide as compared to the band-structure obtained via full first-principles calculations. Finally, we provide a downloadable virtual machine that can be used to reproduce the results of this paper, including all first-principles and atomistic simulations as well as the computational workflows.

We search for novel two-dimensional materials that can be easily exfoliated from their parent compounds. Starting from 108423 unique, experimentally known three-dimensional compounds we identify a subset of 5619 that appear layered according to robust geometric and bonding criteria. High-throughput calculations using van-der-Waals density-functional theory, validated against experimental structural data and calculated random-phase-approximation binding energies, allow to identify 1825 compounds that are either easily or potentially exfoliable, including all that are commonly exfoliated experimentally. In particular, the subset of 1036 easily exfoliable cases---layered materials held together mostly by dispersion interactions and with binding energies up to $30-35$ meV$\cdot\text{\AA}^{-2}$---provides a wealth of novel structural prototypes and simple ternary compounds, and a large portfolio to search materials for optimal properties. For the 258 compounds with up to 6 atoms per primitive cell we comprehensively explore vibrational, electronic, magnetic, and topological properties, identifying in particular 56 ferromagnetic and antiferromagnetic systems, including half-metals and half-semiconductors., Comment: 15 pages, 5 figures, supplementary material available at https://www.nature.com/articles/s41565-017-0035-5#Sec15. Data available on the Materials Cloud https://www.materialscloud.org/discover/2dstructures/dashboard/ptable

Quantum spin Hall insulators make up a class of two-dimensional materials with a finite electronic band gap in the bulk and gapless helical edge states. In the presence of time-reversal symmetry, [Formula: see text] topological order distinguishes the topological phase from the ordinary insulating one. Some of the phenomena that can be hosted in these materials, from one-dimensional low-dissipation electronic transport to spin filtering, could be promising for many technological applications in the fields of electronics, spintronics, and topological quantum computing. Nevertheless, the rarity of two-dimensional materials that can exhibit nontrivial [Formula: see text] topological order at room temperature hinders development. Here, we screen a comprehensive database we recently identified of 1825 monolayers that can be exfoliated from experimentally known compounds to search for novel quantum spin Hall insulators. Using density-functional and many-body perturbation theory simulations, we identify 13 monolayers that are candidates for quantum spin Hall insulators including high-performing materials such as AsCuLi

Recently, the very first large-gap Kane-Mele quantum spin Hall insulator was predicted to be monolayer jacutingaite (Pt$_2$HgSe$_3$), a naturally-occurring exfoliable mineral discovered in Brazil in 2008. The stacking of quantum spin Hall monolayers into a van-der-Waals layered crystal typically leads to a (0;001) weak topological phase, which does not protect the existence of surface states on the (001) surface. Unexpectedly, recent angle-resolved photoemission spectroscopy experiments revealed the presence of surface states dispersing over large areas of the 001-surface Brillouin zone of jacutingaite single crystals. The 001-surface states have been shown to be topologically protected by a mirror Chern number $C_M=-2$, associated with a nodal line gapped by spin-orbit interactions. Here, we extend the two-dimensional Kane-Mele model to bulk jacutingaite and unveil the microscopic origin of the gapped nodal line and the emerging crystalline topological order. By using maximally-localized Wannier functions, we identify a large non-trivial second nearest-layer hopping term that breaks the standard paradigm of weak topological insulators. Complemented by this term, the predictions of the Kane-Mele model are in remarkable agreement with recent experiments and first-principles simulations, providing an appealing conceptual framework also relevant for other layered materials made of stacked honeycomb lattices., Comment: main text: 7 pages, 3 figures; supplementary: 3 pages, 2 figures

Quantum spin Hall insulators make up a class of two-dimensional materials with a finite electronic band gap in the bulk and gapless helical edge states. In the presence of time-reversal symmetry, Z(2) topological order distinguishes the topological phase from the ordinary insulating one. Some of the phenomena that can be hosted in these materials, from one-dimensional low-dissipation electronic transport to spin filtering, could be promising for many technological applications in the fields of electronics, spintronics, and topological quantum computing. Nevertheless, the rarity of two-dimensional materials that can exhibit nontrivial Z(2) topological order at room temperature hinders development. Here, we screen a comprehensive database we recently identified of 1825 monolayers that can be exfoliated from experimentally known compounds to search for novel quantum spin Hall insulators. Using density-functional and many-body perturbation theory simulations, we identify 13 monolayers that are candidates for quantum spin Hall insulators including high-performing materials such as AsCuL2 and (platinum) jacutingaite (P2HgS3). We also identify monolayer Pd2HgSe3 (palladium jacutingaite) as a novel Kane-Mele quantum spin Hall insulator and compare it with platinum jacutingaite. A handful of promising materials are mechanically stable and exhibit Z(2) topological order, either unperturbed or driven by small amounts of strain. Such screening highlights a relative abundance of Z(2) topological order of around 1% and provides an optimal set of candidates for experimental efforts.

Despite the enormous success and popularity of density-functional theory, systematic verification and validation studies are still limited in number and scope. Here, we propose a protocol to test publicly available pseudopotential libraries, based on several independent criteria including verification against all-electron equations of state and plane-wave convergence tests for phonon frequencies, band structure, cohesive energy and pressure. Adopting these criteria we obtain curated pseudopotential libraries (named SSSP or standard solid-state pseudopotential libraries), that we target for high-throughput materials screening (“SSSP efficiency”) and high-precision materials modelling (“SSSP precision”). This latter scores highest among open-source pseudopotential libraries available in the Δ-factor test of equations of states of elemental solids. Curated pseudopotential libraries obtained by systematic testing of available pseudopotentials are obtained using a newly proposed testing protocol. Density functional theory is very popular, but little attention has been devoted so far to the verification of the underlying pseudopotentials and projector augmented-wave approximations. The issue of performance is also of importance, as smoother pseudopotentials would enable faster calculations. Now, Nicola Marzari and colleagues from the Ecole Polytechnique Federale de Lausanne in Switzerland introduce a testing protocol for pseudopotentials in publicly available libraries, and select the optimal pseudopotential for 85 elements. The protocol includes both a verification step and performance evaluation step. Finding the right balance between precision and performance is particularly important for high-throughput materials searches, which are currently the focus of big efforts worldwide

The geometrical intrinsic contribution to the anomalous Hall conductivity (AHC) of a metal is commonly expressed as a reciprocal-space integral: as such, it only addresses unbounded and macroscopically homogeneous samples. Here we show that the geometrical AHC has an equivalent expression as a local property. We define a "geometrical marker" which actually probes the AHC in inhomogeneous systems (e.g. heterojunctions), as well as in bounded samples. The marker may even include extrinsic contributions of geometrical nature., Comment: 4 pages, 4 figures

Fundamental research and technological applications of topological insulators are hindered by the rarity of materials exhibiting a robust topologically nontrivial phase, especially in two dimensions. Here, by means of extensive first-principles calculations, we propose a novel quantum spin Hall insulator with a sizable band gap of $\ensuremath{\sim}0.5\text{ }\text{ }\mathrm{eV}$ that is a monolayer of jacutingaite, a naturally occurring layered mineral first discovered in 2008 in Brazil and recently synthesized. This system realizes the paradigmatic Kane-Mele model for quantum spin Hall insulators in a potentially exfoliable two-dimensional monolayer, with helical edge states that are robust and that can be manipulated exploiting a unique strong interplay between spin-orbit coupling, crystal-symmetry breaking, and dielectric response.

The automation of ab initio simulations is essential in view of performing high-throughput (HT) computational screenings oriented to the discovery of novel materials with desired physical properties. In this work, we propose algorithms and implementations that are relevant to extend this approach beyond density functional theory (DFT), in order to automate many-body perturbation theory (MBPT) calculations. Notably, a novel algorithm pursuing the goal of an efficient and robust convergence procedure for GW and BSE simulations is provided, together with its implementation in a fully automated framework. This is accompanied by an automatic GW band interpolation scheme based on maximally-localized Wannier functions, aiming at a reduction of the computational burden of quasiparticle band structures while preserving high accuracy. The proposed developments are validated on a set of representative semiconductor and metallic systems., Comment: 14 pages, 6 figures

Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronic-structure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials., Journal of Physics: Condensed Matter, 32 (16), ISSN:0953-8984, ISSN:1361-648X