Showing 1–21 of 21 results for author: Eek, L

Emergent $\mathbb{Z}$-type topology in a quasi-one-dimensional extended QWZ model

Authors: Zebedeus F. Osseweijer, Lumen Eek, Cristiane Morais Smith

Abstract: We investigate the emergence of zero-dimensional topological end states in nanoribbons described by the Qi-Wu-Zhang (QWZ) model and its extensions with longer-range couplings. While dimensional reduction from two to one dimension is often assumed to preserve the symmetry classification of the parent system, here an additional symmetry can emerge originating from the real-space geometry of the ribb… ▽ More We investigate the emergence of zero-dimensional topological end states in nanoribbons described by the Qi-Wu-Zhang (QWZ) model and its extensions with longer-range couplings. While dimensional reduction from two to one dimension is often assumed to preserve the symmetry classification of the parent system, here an additional symmetry can emerge originating from the real-space geometry of the ribbon. This symmetry acts as a chiral symmetry, combining orbital and spatial transformations, and promotes the effective one-dimensional system from symmetry class D to class BDI. We demonstrate that the existence of such a symmetry depends both on the long and end termination of the ribbon and exhibits an even-odd effect with respect to ribbon width, revealing that the commonly studied rectangular ribbons constitute a special high-symmetry case. For the conventional QWZ model, we derive analytic expressions for the topological phase boundaries of finite-width nanoribbons and characterize the resulting hybridization-gap phases through ($\mathbb{Z}_2 $) and winding-number invariants. We further show that extended QWZ models with longer-range couplings support phases with multiple topological end states and higher winding numbers. These phases arise through distinct mechanisms, including the hybridization of multiple edge modes inherited from higher-Chern-number bulk phases. Our results demonstrate that both long and end termination can fundamentally alter the topological classification of confined Chern insulators, highlighting the interplay between crystalline geometry, emergent symmetries, and dimensional reduction. △ Less

Submitted 3 July, 2026; originally announced July 2026.

Comments: 10 pages, 6 figures

Universality of dimensional crossovers in topological insulators

Authors: Lumen Eek, Zeb Osseweijer, Cristiane Morais Smith

Abstract: We investigate dimensional crossovers in minimal tight-binding models of three-dimensional (3D) topological insulators subject to geometric confinement. While thin films are commonly understood to host a crossover from a 3D strong topological insulator to a two-dimensional (2D) quantum spin Hall phase via hybridization of surface states, we demonstrate that this picture is incomplete once bulk con… ▽ More We investigate dimensional crossovers in minimal tight-binding models of three-dimensional (3D) topological insulators subject to geometric confinement. While thin films are commonly understood to host a crossover from a 3D strong topological insulator to a two-dimensional (2D) quantum spin Hall phase via hybridization of surface states, we demonstrate that this picture is incomplete once bulk confinement effects and boundary termination are fully taken into account. Using lattice models, we show that reducing the system size induces a strongly non-monotonic dependence of the topology on thickness and microscopic parameters, leading to a sequence of topological phase transitions that is highly sensitive to surface termination. In particular, we find a cascade of dimensional reduction from a 3D topological insulator to a 2D quantum spin Hall phase and ultimately to a one-dimensional phase consisting of end states of Kramers pairs protected by inversion symmetry. Remarkably, we show that both the 2D and 1D topological phases can emerge even when the corresponding 3D bulk phase is topologically trivial. Our results reveal an unexpected universality in the phase diagrams of 3D-to-2D and 2D-to-1D crossovers, pointing toward a unified framework for topology under dimensional reduction. △ Less

Submitted 22 June, 2026; originally announced June 2026.

Comments: 8 pages, 5 figures

Topology of honeycomb nanoribbons revisited

Authors: Zebedeus F. Osseweijer, Lumen Eek, Harold J. W. Zandvliet, Pantelis Bampoulis, Cristiane Morais Smith

Abstract: We present an in-depth study of end states in honeycomb nanoribbons, focusing on the interplay between nanoribbon termination, chiral symmetry, and complex next-nearest-neighbor hopping in the framework of the Haldane model. Although previous work has identified zero-dimensional end states in such systems, this analysis is incomplete. Here, we systematically investigate zigzag and armchair nanorib… ▽ More We present an in-depth study of end states in honeycomb nanoribbons, focusing on the interplay between nanoribbon termination, chiral symmetry, and complex next-nearest-neighbor hopping in the framework of the Haldane model. Although previous work has identified zero-dimensional end states in such systems, this analysis is incomplete. Here, we systematically investigate zigzag and armchair nanoribbons of various widths, using the multiband Zak phase to characterize the topological properties of the occupied bands. We show that the Zak phase is quantized only for certain ribbon terminations, and we elucidate how this termination dependence governs the existence and robustness of end states. Furthermore, we explore the effect of varying the complex next-nearest-neighbor hopping phase, demonstrating the breakdown of chiral symmetry, the evolution of the bulk gap, and the resulting depinning of end-state energies. Finally, we place our findings in the context of previous studies and discuss connections to the Kane-Mele model, including the role of Rashba spin-orbit coupling. Our work provides a more detailed analysis of topological end states in nanoribbons described by the Haldane and Kane-Mele models and offers a framework for their characterization in related systems. △ Less

Submitted 26 March, 2026; originally announced March 2026.

Comments: 14 pages, 9 figures

Fragile topology for six-fold rotation symmetry indicated by the concentric Wilson loop spectrum

Authors: Xinyang Li, Lumen Eek, Jasper van Wezel, Cristiane Morais Smith

Abstract: We investigate topological phase transitions for the Haldane and Kane-Mele model in a lattice with $p6$ symmetry, which consists of triangles and hexagons arranged in a two-dimensional geometry. For the Haldane model, which breaks time-reversal symmetry, we calculate the Chern number using a multi-band non-Abelian Wilson loop formalism. By varying the hopping parameters in the triangles and hexago… ▽ More We investigate topological phase transitions for the Haldane and Kane-Mele model in a lattice with $p6$ symmetry, which consists of triangles and hexagons arranged in a two-dimensional geometry. For the Haldane model, which breaks time-reversal symmetry, we calculate the Chern number using a multi-band non-Abelian Wilson loop formalism. By varying the hopping parameters in the triangles and hexagons independently, a large variety of topological phases emerge. In the presence of a next-next-nearest neighbor hopping, the phase diagram becomes even richer, with regions exhibiting high Chern numbers. Then, we consider the Kane-Mele model, for which time-reversal symmetry is preserved, and calculate the number of $π$-crossings in the Concentric Wilson Loop Spectrum (CWLS). This method is appropriate to determine the topological invariant for systems hosting time-reversal and rotational symmetry, but lacking all other symmetries. According to a classification based on $K$-theory, the CWLS invariant reveals topological properties even when more conventional invariants fail to detect them. The formalism was previously successfully applied to systems with 3- and 4-fold symmetry. Here, we surprisingly find that for the 6-fold-symmetry model investigated, the topology identified by this invariant is fragile, therefore questioning the claim that this should be the strong invariant missing in a complete classification of topological insulators. △ Less

Submitted 25 March, 2026; originally announced March 2026.

Comments: 8 pages, 8 figures

Enhanced spin-current generation in Dirac altermagnets through Klein tunneling

Authors: Tomas T. Osterholt, Lumen Eek, Cristiane Morais Smith, Rembert A. Duine

Abstract: Altermagnets have recently emerged as a new platform for spintronics applications, offering spin-split electronic bands despite vanishing net magnetization. Here, we investigate spin-current generation in Dirac altermagnets and identify Klein tunneling as an efficient mechanism for enhancing spin transport. Using a low-energy Dirac model combined with scattering theory, we demonstrate that Klein t… ▽ More Altermagnets have recently emerged as a new platform for spintronics applications, offering spin-split electronic bands despite vanishing net magnetization. Here, we investigate spin-current generation in Dirac altermagnets and identify Klein tunneling as an efficient mechanism for enhancing spin transport. Using a low-energy Dirac model combined with scattering theory, we demonstrate that Klein tunneling in altermagnets is strongly spin-dependent and can be used to effectively control the electronic spin-current polarization by, for instance, adjusting the height, width and orientation of the potential barrier. Finally, we explore how the l-wave symmetry of the Dirac altermagnet shapes the spin-current polarization and transmission, focusing especially on the d- and g-wave cases. Particularly promising results are obtained for the g-wave Dirac altermagnet, as it is found that the presence of a potential barrier can significantly boost the spin-current polarization, even when the intrinsic polarization due to the spin-split band structure is vanishingly small. For a barrier implemented via electrostatic gating, such a mechanism would in turn allow the spin-current polarization to be switched on and off via a gate voltage. △ Less

Submitted 24 March, 2026; originally announced March 2026.

Comments: 16 pages, 6 figures

Tensor-network methodology for real-space super-moiré excitons

Authors: Anouar Moustaj, Yitao Sun, Tiago V. C. Antão, Lumen Eek, Jose L. Lado

Abstract: Computing excitonic spectra in quasicrystal and super-moiré systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic $10^{18}$-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectra… ▽ More Computing excitonic spectra in quasicrystal and super-moiré systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic $10^{18}$-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moiré systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moiré-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moiré quantum matter. △ Less

Submitted 20 August, 2026; v1 submitted 2 March, 2026; originally announced March 2026.

Comments: 16 pages (8 main text, 8 supplementary material), 7 figures (4 main text and 3 supplementary material). Article submitted to SciPost Phys

Real-Space Imaging of Moiré-Confined Excitons in Twisted Bilayer MoS$_2$

Authors: Laurens J. M. Westenberg, Lumen Eek, Jort D. Verbakel, Kevin Vonk, Stijn J. H. Borggreve, Kenji Watanabe, Takashi Taniguchi, Paul de Boeij, Rodrigo Arouca, Cristiane Morais Smith, Pantelis Bampoulis

Abstract: Twisted two-dimensional semiconductors generate a moiré landscape that confines excitons (bound electron-hole pairs) into programmable lattices, offering routes to efficient light sources, sensing, and room-temperature information processing. However, direct real-space imaging of confined excitonic species within a moiré unit cell remains challenging; existing claims are inferred from spatially av… ▽ More Twisted two-dimensional semiconductors generate a moiré landscape that confines excitons (bound electron-hole pairs) into programmable lattices, offering routes to efficient light sources, sensing, and room-temperature information processing. However, direct real-space imaging of confined excitonic species within a moiré unit cell remains challenging; existing claims are inferred from spatially averaged far-field signals that are intrinsically insufficient to resolve nanometre-scale variations. Here, we imaged excitons across the moiré of a 2$^{\circ}$ twisted bilayer MoS$_2$ with nanometre resolution using room-temperature photocurrent atomic force microscopy. We directly resolved site-selective confinement: direct and indirect excitons localize at different stacking registries of the moiré, with contrast governed by alignment between site-selective generation and confinement minima. A Wannier-based moiré-exciton model reproduces the measured energies and the moiré-induced localization of the exciton wavefunction. These species-specific, unit-cell-resolved measurements constrain microscopic models of moiré excitons, provide benchmarks for excitonic order, and establish a device-compatible route to engineering excitonic lattices in van der Waals heterostructures. △ Less

Submitted 25 November, 2025; originally announced November 2025.

Electric field-induced spin-valley locking in twisted bilayer buckled honeycomb materials

Authors: Harold J. W. Zandvliet, Pantelis Bampoulis, Cristiane Morais Smith, Lumen Eek

Abstract: A twisted honeycomb bilayer exhibits a moiré superstructure that is composed of a hexagonal arrangement of AB and BA stacked domains separated by domain boundaries. In the case of twisted bilayer graphene, the application of an electric field normal to the bilayer leads to the opening of inverted band gaps in the AB and BA stacked domains. The inverted band gaps result in the formation of a two-di… ▽ More A twisted honeycomb bilayer exhibits a moiré superstructure that is composed of a hexagonal arrangement of AB and BA stacked domains separated by domain boundaries. In the case of twisted bilayer graphene, the application of an electric field normal to the bilayer leads to the opening of inverted band gaps in the AB and BA stacked domains. The inverted band gaps result in the formation of a two-dimensional triangular network of counterpropagating valley protected helical domain boundary states, also referred to as the quantum valley Hall effect. Owing to spin-orbit coupling and buckling, the quantum valley Hall effect in twisted bilayer silicene and germanene is more complex than in twisted bilayer graphene. We found that there is a range of electric fields for which the spin degree of freedom is locked to the valley degree of freedom of the electrons in the quantum valley Hall states, resulting in a stronger topological protection. For electric fields smaller than the aforementioned range the twisted bilayer does not exhibit the quantum valley Hall effect, whereas for larger electric fields the spin-valley locking is lifted and the emergent quantum valley Hall states are only valley-protected. △ Less

Submitted 16 October, 2025; originally announced October 2025.

Comments: 18 pages, 6 figures

Twist-modulated magnetic interactions in bilayer van der Waals materials

Authors: Tomas T. Osterholt, D. O. Oriekhov, Lumen Eek, Cristiane Morais Smith, Rembert A. Duine

Abstract: The ability to control magnetic interactions at the nanoscale is crucial for the development of next-generation spintronic devices and functional magnetic materials. In this work, we investigate theoretically, by means of many-body perturbation theory, how interlayer twisting modulates magnetic interactions in bilayer van der Waals systems composed of two ferromagnetic layers. We demonstrate that… ▽ More The ability to control magnetic interactions at the nanoscale is crucial for the development of next-generation spintronic devices and functional magnetic materials. In this work, we investigate theoretically, by means of many-body perturbation theory, how interlayer twisting modulates magnetic interactions in bilayer van der Waals systems composed of two ferromagnetic layers. We demonstrate that the relative strengths of the interlayer Heisenberg exchange interaction, the Dzyaloshinskii-Moriya interaction, and the anisotropic exchange interaction can be significantly altered by varying the twist angle between the layers, thus leading to tunable magnetic textures. We further show that these interactions are strongly dependent on the chemical potential, enabling additional control via electrostatic gating or doping. Importantly, our approach is applicable to arbitrary twist angles and does not rely on the construction of a Moiré supercell, making it particularly efficient even at small twist angles. △ Less

Submitted 17 September, 2025; originally announced September 2025.

Electric-field control of zero-dimensional topological states in ultranarrow germanene nanoribbons

Authors: Lumen Eek, Esra D. van 't Westende, Dennis J. Klaassen, Harold J. W. Zandvliet, Pantelis Bampoulis, Cristiane Morais Smith

Abstract: Reversible, all-electric control of symmetry-protected zero-dimensional modes has been a long-standing goal. In buckled honeycomb lattices, a perpendicular field couples to the staggered sublattice potential providing the required handle. We combine scanning tunneling microscopy and tight-binding theory to switch zero-dimensional topological end states reversibly on and off in ultranarrow germanen… ▽ More Reversible, all-electric control of symmetry-protected zero-dimensional modes has been a long-standing goal. In buckled honeycomb lattices, a perpendicular field couples to the staggered sublattice potential providing the required handle. We combine scanning tunneling microscopy and tight-binding theory to switch zero-dimensional topological end states reversibly on and off in ultranarrow germanene nanoribbons by tuning the electric field in the tunnel junction. Increasing the field switches off the end modes of topological two-hexagon wide ribbons, while the same field switches on zero-dimensional states in initially trivial three- and four-hexagon wide ribbons. This atomic scale platform realizes a proof-of-principle for a zero-dimensional topological field effect device, opening a path for ultrasmall memory, controllable qubits, and neuromorphic architectures. △ Less

Submitted 20 October, 2025; v1 submitted 19 June, 2025; originally announced June 2025.

Emergence of Imaginary Time Crystals in the non-Hermitian Su-Schrieffer-Heeger model

Authors: E. Slootman, L. Eek, C. Morais Smith, R. Arouca

Abstract: Parity-time symmetry constrains the spectrum of non-Hermitian systems to be either real or come in complex conjugate pairs. The transition between a symmetry-preserving phase with real energies and a symmetry-broken phase with complex energies is marked by exceptional points, one of the hallmarks of non-Hermitian systems. Because of these properties, these systems are widely studied, both theoreti… ▽ More Parity-time symmetry constrains the spectrum of non-Hermitian systems to be either real or come in complex conjugate pairs. The transition between a symmetry-preserving phase with real energies and a symmetry-broken phase with complex energies is marked by exceptional points, one of the hallmarks of non-Hermitian systems. Because of these properties, these systems are widely studied, both theoretically and experimentally. In this work, we investigate the thermodynamic properties of the gain and loss Su-Schrieffer-Heeger model for both bosons and fermions, and establish the existence of an imaginary time crystal phase, an imaginary time analogue of a time crystal. This phase occurs when there is a resonance condition between the Matsubara frequencies and the spectrum of the system, making the Green's function of the system oscillate in imaginary time with the Matsubara frequency. We show that this phase appears in the symmetry-broken region. In particular, the topological edge states of this system exhibit oscillations that are present for bosons. Finally, we discuss the applicability of our results for experiments. We examine signatures of these phases in terms of correlation functions in real time and oscillations in thermodynamic potential in inverse temperature $β$, and explore possible experimental platforms to realize this system. △ Less

Submitted 27 April, 2025; originally announced April 2025.

Comments: 9 pages, 6 figures

Journal ref: Phys. Rev. B 112, 085149 (2025)

Realization of a one-dimensional topological insulator in ultrathin germanene nanoribbons

Authors: Dennis J. Klaassen, Lumen A. G. Eek, Alexander N. Rudenko, Esra D. van`t Westende, Carolien Castenmiller, Zhiguo Zhang, Paul de Boeij, Arie van Houselt, Motohiko Ezawa, Harold J. W. Zandvliet, Cristiane Morais Smith, Pantelis Bampoulis

Abstract: Realizing a one-dimensional (1D) topological insulator and identifying the lower dimensional limit of two-dimensional (2D) behavior are crucial steps toward developing high-density quantum state networks, advancing topological quantum computing, and exploring dimensionality effects in topological materials. Although 2D topological insulators have been experimentally realized, their lower dimension… ▽ More Realizing a one-dimensional (1D) topological insulator and identifying the lower dimensional limit of two-dimensional (2D) behavior are crucial steps toward developing high-density quantum state networks, advancing topological quantum computing, and exploring dimensionality effects in topological materials. Although 2D topological insulators have been experimentally realized, their lower dimensional limit and 1D counterparts remain elusive. Here, we fabricated and characterized arrays of zigzag-terminated germanene nanoribbons, a 2D topological insulator with a large topological bulk gap. The electronic properties of these nanoribbons strongly depend on their width, with topological edge states persisting down to a critical width (approx. 2 nm), defining the limit of 2D topology. Below this threshold, contrary to the tenfold way classification, we observe zero-dimensional (0D) states localized at the ends of the ultrathin nanoribbons. These end states, topologically protected by time-reversal and mirror symmetries, mark the first realization of a 1D topological insulator with strong spin-orbit coupling. Our findings establish germanene nanoribbons as a platform for investigating 1D topology and dimensionality effects in topological materials. △ Less

Submitted 27 November, 2024; originally announced November 2024.

Journal ref: Nat Commun 16, 2059 (2025)

Fractality-induced Topology

Authors: L. Eek, Z. F. Osseweijer, C. Morais Smith

Abstract: Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. In this work, we introduce a theoretical framework that leverages isospectral reduction to effectively simplify complex fractal structures, revealing the presence of topologically protected boundary and corner states. Our approach demonst… ▽ More Fractal geometries, characterized by self-similar patterns and non-integer dimensions, provide an intriguing platform for exploring topological phases of matter. In this work, we introduce a theoretical framework that leverages isospectral reduction to effectively simplify complex fractal structures, revealing the presence of topologically protected boundary and corner states. Our approach demonstrates that fractals can support topological phases, even in the absence of traditional driving mechanisms such as magnetic fields or spin-orbit coupling. The isospectral reduction not only elucidates the underlying topological features but also makes this framework broadly applicable to a variety of fractal systems. Furthermore, our findings suggest that these topological phases may naturally occur in materials with fractal structures found in nature. This work opens new avenues for designing fractal-based topological materials, advancing both theoretical understanding and experimental exploration of topology in complex, self-similar geometries. △ Less

Submitted 19 November, 2024; originally announced November 2024.

Latent Haldane Models

Authors: Anouar Moustaj, Lumen Eek, Malte Rontgen, Cristiane Morais Smith

Abstract: Latent symmetries, which materialize after performing isospectral reductions, have recently been shown to be instrumental in revealing novel topological phases in one-dimensional systems, among many other applications. In this work, we explore how to construct a family of seemingly complicated two-dimensional models that result in energy-dependent Haldane models upon performing an isospectral redu… ▽ More Latent symmetries, which materialize after performing isospectral reductions, have recently been shown to be instrumental in revealing novel topological phases in one-dimensional systems, among many other applications. In this work, we explore how to construct a family of seemingly complicated two-dimensional models that result in energy-dependent Haldane models upon performing an isospectral reduction. In these models, we find energy-dependent latent Semenoff masses without introducing a staggered on-site potential. In addition, energy-dependent latent Haldane masses also emerge in decorated lattices with nearest-neighbor complex hoppings. Using the Haldane model's properties, we then predict the location of the topological gaps in the aforementioned family of models and construct phase diagrams to determine where the topological phases lie in parameter space. This idea yielded, for instance, useful insights in the case of a modified version of $α$-graphyne and hexagonal plaquettes with additional decorations, where the gap-closing energies can be calculated using the ISR to predict topological phase transitions. △ Less

Submitted 19 February, 2025; v1 submitted 12 November, 2024; originally announced November 2024.

Comments: 13 pages, 14 figures, regular article to be submitted to Physical Review B

Haldane model on the Sierpiński gasket

Authors: Zebedeus Osseweijer, Lumen Eek, Anouar Moustaj, Mikael Fremling, Cristiane Morais Smith

Abstract: We investigate the topological phases of the Haldane model on the Sierpiński gasket. As a consequence of the fractal geometry, multiple fractal gaps arise. Additionally, a flat band appears, and due to a complex next-nearest neighbour hopping, this band splits and multiple topological flux-induced gaps emerge. Owing to the fractal nature of the model, conventional momentum-space topological invari… ▽ More We investigate the topological phases of the Haldane model on the Sierpiński gasket. As a consequence of the fractal geometry, multiple fractal gaps arise. Additionally, a flat band appears, and due to a complex next-nearest neighbour hopping, this band splits and multiple topological flux-induced gaps emerge. Owing to the fractal nature of the model, conventional momentum-space topological invariants cannot be used. Therefore, we characterise the system's topology in terms of a real-space Chern number. In addition, we verify the robustness of the topological states to disorder. Finally, we present phase diagrams for both a fractal gap and a flux-induced gap. Previous work on a similar system claims that fractality "squeezes" the well-known Haldane phase diagram. However, this result arises because a doubled system was considered with two Sierpiński gaskets glued together. We consider only a single copy of the Sierpiński gasket, keeping global self-similarity. In contrast with these previous results, we find intricate and complex patterns in the phase diagram of this single fractal. Our work shows that the fractality of the model greatly influences the phase space of these structures, and can drive topological phases in the multitude of fractal and flux-induced gaps, providing a richer platform than a conventional integer dimensional geometry. △ Less

Submitted 2 February, 2025; v1 submitted 29 July, 2024; originally announced July 2024.

Comments: 18 pages, 17 figures

Higher-order topology protected by latent crystalline symmetries

Authors: L. Eek, M. Röntgen, A. Moustaj, C. Morais Smith

Abstract: We demonstrate that rotation symmetry is not a necessary requirement for the existence of fractional corner charges in Cn-symmetric higher-order topological crystalline insulators. Instead, it is sufficient to have a latent rotation symmetry, which may be revealed upon performing an isospectral reduction on the system. We introduce the concept of a filling anomaly for latent crystalline symmetric… ▽ More We demonstrate that rotation symmetry is not a necessary requirement for the existence of fractional corner charges in Cn-symmetric higher-order topological crystalline insulators. Instead, it is sufficient to have a latent rotation symmetry, which may be revealed upon performing an isospectral reduction on the system. We introduce the concept of a filling anomaly for latent crystalline symmetric systems, and propose modified topological invariants. The notion of higher-order topology in two dimensions protected by Cn symmetry is thus generalized to a protection by latent symmetry. Our claims are corroborated by concrete examples of models that show non-trivial corner charge in the absence of Cn-symmetry. This work extends the classification of topological crystalline insulators to include latent symmetries. △ Less

Submitted 23 December, 2024; v1 submitted 4 May, 2024; originally announced May 2024.

Comments: 25 pages, 13 figures. 22 pages main text and 3 pages appendix. Regular article to be submitted to SciPost

Journal ref: SciPost Phys. 18, 061 (2025)

Emergent non-Hermitian models

Authors: Lumen Eek, Anouar Moustaj, Malte Röntgen, Vincent Pagneux, Vassos Achilleos, Cristiane Morais Smith

Abstract: The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger model are paradigmatic examples of non-Hermitian systems that host non-trivial boundary phenomena. In this work, we use recently developed graph-theoretical tools to design systems whose isospectral reduction -- akin to an effective Hamiltonian -- has the form of either of these two models. In the reduced version, the couplings and on-si… ▽ More The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger model are paradigmatic examples of non-Hermitian systems that host non-trivial boundary phenomena. In this work, we use recently developed graph-theoretical tools to design systems whose isospectral reduction -- akin to an effective Hamiltonian -- has the form of either of these two models. In the reduced version, the couplings and on-site potentials become energy-dependent. We show that this leads to interesting phenomena such as an energy-dependent non-Hermitian skin effect, where eigenstates can simultaneously localize on either ends of the systems, with different localization lengths. Moreover, we predict the existence of various topological edge states, pinned at non-zero energies, with different exponential envelopes, depending on their energy. Overall, our work sheds new light on the nature of topological phases and the non-Hermitian skin effect in one-dimensional systems. △ Less

Submitted 18 October, 2023; originally announced October 2023.

Comments: two-column article, 15 pages, 9 figures, comments are welcome

Journal ref: Phys. Rev. B 109, 045122 (2024)

Topological edge and corner states in Bi fractals on InSb

Authors: R. Canyellas, Chen Liu, R. Arouca, L. Eek, Guanyong Wang, Yin Yin, Dandan Guan, Yaoyi Li, Shiyong Wang, Hao Zheng, Canhua Liu, Jinfeng Jia, C. Morais Smith

Abstract: Topological materials hosting metallic edges characterized by integer quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. However, only integer-dimensional models have been classified, and the interplay of topology and fractals, which may have a non-inte… ▽ More Topological materials hosting metallic edges characterized by integer quantized conductivity in an insulating bulk have revolutionized our understanding of transport in matter. The topological protection of these edge states is based on symmetries and dimensionality. However, only integer-dimensional models have been classified, and the interplay of topology and fractals, which may have a non-integer dimension, remained largely unexplored. Quantum fractals have recently been engineered in metamaterials, but up to present no topological states were unveiled in fractals realized in real materials. Here, we show theoretically and experimentally that topological edge and corner modes arise in fractals formed upon depositing thin layers of bismuth on an indium antimonide substrate. Scanning tunneling microscopy reveals the appearance of (nearly) zero-energy modes at the corners of Sierpiński triangles, as well as the formation of outer and inner edge modes at higher energies. Unexpectedly, a robust and sharp depleted mode appears at the outer and inner edges of the samples at negative bias voltages. The experimental findings are corroborated by theoretical calculations in the framework of a continuum muffin-tin and a lattice tight-binding model. The stability of the topological features to the introduction of a Rashba spin-orbit coupling and disorder is discussed. This work opens the perspective to novel electronics in real materials at non-integer dimensions with robust and protected topological states. △ Less

Submitted 26 September, 2023; v1 submitted 18 September, 2023; originally announced September 2023.

Comments: Main manuscript 14 pages, supplementary material 34 pages

Journal ref: Nat. Phys. 20, 1421-1428 (2024)

Breaking and resurgence of symmetry in the non-Hermitian Su-Schrieffer-Heeger model in photonic waveguides

Authors: E. Slootman, W. Cherifi, L. Eek, R. Arouca, E. J. Bergholtz, M. Bourennane, C. Morais Smith

Abstract: Symmetry is one of the cornerstones of modern physics and has profound implications in different areas. In symmetry-protected topological systems, symmetries are responsible for protecting surface states, which are at the heart of the fascinating properties exhibited by these materials. When the symmetry protecting the edge mode is broken, the topological phase becomes trivial. By engineering loss… ▽ More Symmetry is one of the cornerstones of modern physics and has profound implications in different areas. In symmetry-protected topological systems, symmetries are responsible for protecting surface states, which are at the heart of the fascinating properties exhibited by these materials. When the symmetry protecting the edge mode is broken, the topological phase becomes trivial. By engineering losses that break the symmetry protecting a topological Hermitian phase, we show that a new genuinely non-Hermitian symmetry emerges, which protects and selects one of the boundary modes: the topological monomode. Moreover, the topology of the non-Hermitian system can be characterized by an effective Hermitian Hamiltonian in a higher dimension. To corroborate the theory, we experimentally investigated the non-Hermitian 1D and 2D SSH models using photonic lattices and observed dynamically generated monomodes in both cases. We classify the systems in terms of the (non-Hermitian) symmetries that are present and calculate the corresponding topological invariants. △ Less

Submitted 9 May, 2024; v1 submitted 12 April, 2023; originally announced April 2023.

Comments: 24 pages, 17 figures

Journal ref: Phys. Rev. Research 6, 023140 (2024)

Compact localized boundary states in a quasi-1D electronic diamond-necklace chain

Authors: S. N. Kempkes, P. Capiod, S. Ismaili, J. Mulkens, L. Eek, I. Swart, C. Morais Smith

Abstract: Zero-energy modes localized at the ends of one-dimensional (1D) wires hold great potential as qubits for fault-tolerant quantum computing. However, all the candidates known to date exhibit a wave function that decays exponentially into the bulk and hybridizes with other nearby zero-modes, thus hampering their use for braiding operations. Here, we show that a quasi-1D diamond-necklace chain exhibit… ▽ More Zero-energy modes localized at the ends of one-dimensional (1D) wires hold great potential as qubits for fault-tolerant quantum computing. However, all the candidates known to date exhibit a wave function that decays exponentially into the bulk and hybridizes with other nearby zero-modes, thus hampering their use for braiding operations. Here, we show that a quasi-1D diamond-necklace chain exhibits a completely unforeseen type of robust boundary state, namely compact localized zero-energy modes that do not decay into the bulk. We theoretically engineer a lattice geometry to access this mode, and experimentally realize it in an electronic quantum simulator setup. Our work provides a general route for the realization of robust and compact localized zero-energy modes that could potentially be braided without the drawbacks of hybridization. △ Less

Submitted 11 October, 2023; v1 submitted 6 January, 2022; originally announced January 2022.

Comments: 15 pages, 5 figures

Journal ref: Quantum Front 2, 1 (2023)

Field Theoretical Study of Disorder in Non-Hermitian Topological Models

Authors: Anouar Moustaj, Lumen Eek, Cristiane Morais Smith

Abstract: Non-Hermitian systems have provided a rich platform to study unconventional topological phases.These phases are usually robust against external perturbations that respect certain symmetries of thesystem. In this work, we provide a new method to analytically study the effect of disorder, usingtools from quantum field theory applied to discrete models around phase-transition points. Weinvestigate tw… ▽ More Non-Hermitian systems have provided a rich platform to study unconventional topological phases.These phases are usually robust against external perturbations that respect certain symmetries of thesystem. In this work, we provide a new method to analytically study the effect of disorder, usingtools from quantum field theory applied to discrete models around phase-transition points. Weinvestigate two different one-dimensional models, the paradigmatic non-Hermitian SSH model andas-wave superconductor with imbalanced pairing. These analytic results are compared to numericalsimulations in the discrete models. An universal behavior is found for the two investigated models,namely that the systems are driven from a topological to a trivial phase for disorder strengths equalto about four times the energy scale of the model. △ Less

Submitted 22 June, 2022; v1 submitted 29 July, 2021; originally announced July 2021.

Comments: 6 pages, 2 figures, article. Supplemental material: 5 pages, 1 figure

Journal ref: Phys. Rev. B 105, L180503 (2022)