Topological Materials: From Weak to Strong Correlations

The topological phases of periodically driven systems have been classified across all dimensions in the periodic table of Floquet topological insulators.The Floquet multiplicity of bands implies the emergence of anomalous edge states which cross bulk gaps that do not occur in static systems. Here, we present our studies on the non-interacting topological Qi-Wu-Zhang model under the influence of a periodic drive, and analyze its drive-induced edge modes in two regimes of the driving frequency $Qmega$; higher or lower than the static bandwidth, For the experimental detection of edge states, we, calculate the dI/dV spectra at non-zero DC bias voltage $V$, using the Keldysh-Floquet formalism. We predict quantized conductance plateaus when the transport voltage is within a normal gap ($V$ centered around $V=0$, normal edge mode) or within an anomalous gap (around $V=pm Omega/2$, anomalous edge mode). We also perfom a spatially resolved computation of the chiral tranmission channels of the finite-size system with finite bias applied, showing that the transport is along an edge and that it is spatially modulated corresponding to the wave number $pi$ of the (anomalous) edge mode.

The recently formulated hyperbolic band theory allows to describe single-particle energy states of tight-binding models in negatively curved spaces. The most salient feature of this theory is the unusually large dimension of the momentum space: the spectrum of particles on a two-dimensional hyperbolic lattice necessitates a characterization with an at least four-dimensional Brillouin zone. Such higher-dimensional momentum spaces imply the existence of a larger set of topological band invariants than for the Euclidean lattices, suggesting potentially new types of topological hyperbolic matter. In this theoretical work, we formulate hyperbolic versions of two paradigm topological tight-binding models, namely of (1) the Haldane model of Chern insulator, and of (2) the Kane-Mele model of time-reversal-symmetric topological insulator. These modifications are achieved by replacing the hexagonal cells of the honeycomb lattice by octagons. For both models, we analyze the correspondence between the bulk invariants in the 4D Brillouin zone, the real-space invariants in the 2D position space, and the appearance of metallic in-gap states at the boundaries. Our results provide pivotal steps towards unravelling topological aspects of models on hyperbolic lattices. Authors of the work: David M. Urwyler, Patrick M. Lenggenhager, Titus Neupert, Tomáš Bzdušek

The current accepted theory of hyperfine coupling in Weyl systems predicts universal power-law scaling of key NMR parameters. This theory takes into account the universal contribution from Weyl fermions, valid in the low-energy long-wavelength limit. One important limitation of this theory is that it ignores the additional short-range terms to the hyperfine coupling that can also arise from non-universal, system-specific processes within the real-space unit cell. In this study, we show that the NMR parameters in CuTlSe$_2$ are different for different nuclei ($^{63}$Cu,$^{65}$Cu $^{205}$Tl, $^{77}$Se). Additional evidence from bulk measurement techniques such as $\mu$SR and transport measurements show no unusual features in the bulk and hence confirm that the observed differences are indeed due to these short-range processes. This is an important result because it highlights the limits of applicability of the currently accepted theory and shows that CuTlSe$_{2}$ a model system to study short-range terms in hyperfine coupling in weyl systems.

The kagome lattice — a lattice consisting of corner-sharing triangles — has been studied in the context of quantum physics for seventy years. The particular geometry of this lattice means that the electronic structure features a flat band, inflection points called ‘van Hove singularities’, and Dirac cones. Flat bands and inflection points in the band structure are natural drivers for strong interactions, while Dirac cones promote non-trivial topological effects. However, a material realizing the kagome interplay between frustrated geometry, correlations, and topology has long been in waiting. The discovery of the family of kagome metals KV3Sb5, CsV3Sb5 and RbV3Sb5 — commonly abbreviated to AV3Sb5 — recently brought this search to a successful end: Realizing the kagome band structure, these materials have been shown to exhibit superconductivity at low temperature and an unusual charge order at high temperature. I will highlight these discoveries, place them in the context of wider research efforts in topological physics and superconductivity, and add our own approaches. Specifically, I will show that the sublattice interference mechanism is central to understanding the formation of both charge order and superconductivity in these kagome metals. From there, I will outline how an unconventional charge order — exhibiting orbital currents and nematicity — can evolve, and what implications this imposes on the preferred superconducting pairing symmetry. If the hypothesis of unconventional order is substantiated, AV3Sb5 could constitute a valuable resource for building quantum matter by design and open several avenues for future research.

The simultaneous co-existence and gate-tunability of the superconducting, magnetic and topological orders in magic angle twisted bilayer graphene (MATBG) open up entirely new possibilities for the creation of complex hybrid Josephson junctions (JJ). Here we report on the creation of gate-defined, magnetic Josephson junctions in MATBG, where the weak link is gate-tuned close to the correlated state at a moiré filling factor of ν = −2. A highly unconventional Fraunhofer pattern emerges, which is phase-shifted and asymmetric with respect to the current and magnetic field directions, and shows a pronounced magnetic hysteresis. We demonstrate how the combination of magnetization and its current induced magnetization switching in the MATBG JJ allows us to realize a programmable zero field superconducting diode, which represents a major building block for a new generation of superconducting quantum electronics.

In this work, we describe the phenomenon of Weyl-point teleportation. Weyl points usually move continuously in the configuration parameter space of a quantum system when the control parameters are varied continuously. However, there are special transition points in the control space where the continuous motion of the Weyl points is disrupted. In such transition points, an extended nodal structure (nodal line or nodal surface) emerges, serving as a wormhole for the Weyl points, allowing their teleportation in the configuration space. A characteristic side effect of the teleportation is that the motional susceptibility of the Weyl point diverges in the vicinity of the transition point, and this divergence is characterized by a universal scaling law. We exemplify these effects via a two-spin model and a Weyl Josephson circuit model. We expect that these effects generalize to many other settings including electronic band structures of topological semimetals. Reference: https://arxiv.org/abs/2112.14556 [2112.14556] Weyl-point teleportation In this work, we describe the phenomenon of Weyl-point teleportation. Weyl points usually move continuously in the configuration parameter space of a quantum system when the control parameters are varied continuously. However, there are special transition points in the control space where the continuous motion of the Weyl points is disrupted. In such transition points, an extended nodal ... arxiv.org

In this poster I present a toy model, developed together with my collaborators, of the surface electronic band structure of the material NbGeSb. This band structure shows an unusual '3+1' motif along the edge of the surface Brillouin zone, where a Rashba-split pair of holelike bands crosses a Rashba-split pair of electron-like ones but only one of the four crossing points appears to develop a gap. The toy model explains this feature in terms of an interesting interplay between the spin and orbital content of the bands. A result of the model is the existence of Weyl-like points where it is the orbital angular momentum rather than the spin angular momentum that exhibits a non-trivial winding as the crossing point is encircled.

Half-quantum vortices (HQVs) have a Majorana bound state associated with them that mean they obey non-Abelian statistics. When moved around each other this generates a quantum entanglement that could be exploited to make a robust quantum computer. Stable half quantum vortices (HQVs) have been observed in geometrically confined superfluid helium-3 and in polariton condensates. The observation of mobile HQVs in superconductors however remains elusive. Localised half-flux quanta have been detected in unconventional superconductors at tri-crystal grain boundaries in the cuprates and arguably in mesoscopic rings of Sr$_2$RuO$_4$and polycrystalline $\beta$-Bi$_2$Pd. However, the non-integer flux in these cases is a consequence of the samples engineered geometry. In this poster/talk we will explore which materials could be favourable candidates to host unconfined half quantum vortices.

A universal phenomenon in phase transitions is critical slowing down (CSD) -- systems, after an initial perturbation, take an exceptionally long time to return to equilibrium. It is universally observed in the dynamics of bosonic excitations, like order-parameter collective modes, but has never been seen directly in the elementary fermionic excitations or quasiparticles. In heavy-fermion (HF) materials, these quasiparticles are compound objects with a strongly enhanced effective mass, composed of itinerant and localized electronic states. Their CSD would be a hallmark of quasiparticle breakdown near a quantum phase transition. Here, we observe fermionic CSD in the HF compound YbRh$_2$Si$_2$ by measuring the quasiparticle weight and excitation rate across the antiferromagnetic quantum phase transition (QPT) by terahertz time-domain spectroscopy. Apart from the expected build-up of spectral weight with decreasing temperature reaching a maximum near the Kondo temperature of $T_K*\approx 25$~K, we observe on the heavy Fermi liquid (HFL) side of the QPT a logarithmic rise of the quasiparticle excitation rate below $10$~K. A critical two-band HFL theory shows that this is indicative of fermionic quasiparticle CSD. Our results shed light on a new classification of HF QPTs in terms of the critical behaviors of fermionic quasiparticles.

We show that fully localized, three-dimensional, time-reversal-symmetry-broken Anderson insulators support topologically distinct phases that can be labelled by integers. Any two such topologically localized phases are separated by a metallic phase. We find that these novel topological phases are fundamentally distinct from insulators without disorder: they are guaranteed to host delocalized states along their insulating boundaries, giving rise to the quantized boundary Hall conductance whose value is determined by the integer invariant assigned to the bulk.

Triple nodal points are degeneracies of energy bands in momentum space at which three Hamiltonian eigenstates coalesce at a single eigenenergy. We present a classification of triple nodal points for spinless particles in all space groups (including magnetic and non-symmorphic ones) based on symmetry properties. This provides criteria for the stability of such band nodes and predicts the intricate nodal structures observed in first-principle data obtained for real materials. Based on this classification, we derive a universal higher-order bulk-boundary correspondence, where pairs of triple nodal points are characterized by fractional jumps of the hinge charge. In the presence of space-time-inversion symmetry, this result is enriched by a further correspondence to the Stiefel-Whitney monopole invariant and by additional non-Abelian multiband topology. We demonstrate the bulk-hinge correspondence for the various species of triple nodal points carrying higher-order topology on tight-binding models and using first-principle predictions for real materials.

The two-dimensional Su-Schrieffer-Heeger model is endowed with rich topological physics. First we show that the random flux can induce a metal-insulator transition in the two-dimensional Su-Schrieffer-Heeger model, thus reporting the first example of such a transition. Remarkably, we find that the resulting insulating phase can even be a higher-order topological insulator with zero-energy corner modes and fractional corner charges. Employing both level statistics and finite-size scaling analysis, we characterize the metal-insulator transition and numerically extract its critical exponent. By proposing another inclined two-dimensional Su-Schrieffer-Heeger, a deformed one, we show that a pair of Dirac points protected by space-time inversion symmetry appear in the semimetallic phase. Remarkably, the locations of these Dirac points are not pinned to any high-symmetry points of the Brillouin zone but highly tunable through parameter modulations. Moreover, the merging of two Dirac points undergoes a topological phase transition, which leads to either an anisotropic topological insulating phase or a nodal-line metallic phase. We provide a systematic analysis of these topological phases from both bulk and boundary perspectives combined with symmetry arguments.

Non-trivial topological phases are much pursued by the researchers in condensed matter physics due to its novel transport properties, which may enable the development of future technologies in the field of spintronics. Furthermore, the addition of Ce ions in the compounds leads to other complex properties, such as crystalline electrical field effects, magnetism, and the Kondo effect. The interaction of these effects with topological phases are not fully understood yet and might enhance the observation of unusual transport properties, for instance, the Kondo effect may pin band crossings close to the Fermi energy favoring non-trivial topological properties [1]. In particular, the family of compounds LnAlX (Ln = lanthanides, X = Ge, Si) is promising to host non-trivial topological phases. Its members crystallize in the noncentrosymmetric structure (I41md), which breaks the space-inversion symmetry, and many present magnetic orders, that break time-reversal symmetry. The breaking of these symmetries are the main ingredients to the formation of magnetic Weyl semimetals. Here we focus on CeAlSi, in which a noncollinear ferromagnetic order takes place below 8.2 K [2] and chiral domain walls were recently observed [3,4]. We investigated the non-trivial topological properties of CeAlSi by combining electrical resistivity, Hall effect, and magnetization measurements with pressure tuning and DFT calculations [5]. A large contribution of domain wall scattering to the anomalous Hall effect and an atypical temperature response of the quantum oscillations amplitude were observed. Applying external pressure suppresses both behaviors; whereas it enhances TC to 9.4 K at 2.7 GPa. Magnetization measurements show no evidence of changes in the magnetic structure as a function of pressure. Finally, DFT calculations revealed a negligible effect of external pressure on the position of the Weyl nodes. Therefore, we suppose that the suppression of the anomalous responses of the Hall effect and the quantum oscillations amplitude are related to changes in the domain wall landscape of CeAlSi. [1] S. E. Grefe, H.-H. Lai, S. Paschen, and Q. Si, Phys. Rev. B, 101, 075138 (2020). [2] H.-Y. Yang et al., Phys. Rev. B, 103, 115143 (2021). [3] B. Xu, J. Franklin, A. Jayakody, H.-Y. Yang, F. Tafti, I. Sochnikov, Adv. Quantum Technol., 4, 2000101 (2021). [4] Y. Sun, C. Lee, H.-Y. Yang, D. H. Torchinsky, F. Tafti, J. Orenstein, Phys. Rev. B, 104, 235119 (2021). [5] M. M. Piva, J. C. Souza, V. Brousseau-Couture, K. R. Pakuszewski, Janas K. John, C. Adriano, M. Côté, P. G. Pagliuso, M. Nicklas, arXiv:2111.05742v1 (2021).

Identifying new materials that have a large thermoelectric response would be beneficial for global energy production by converting waste heat into useful electric power. While conventional thermoelectric have regularly faced challenges with regards to vanishing electronic entropy at lower temperatures, topological materials offer an alternative pathway to surpass them through the topological protection of electronic states. Notably, theories have proposed that topological semimetals can large to a large, non-saturating thermopower and quantized thermoelectric Hall conductivity at the quantum limit. We experimentally demonstrate these unique features in topological Weyl semimetal tantalum phosphide (TaP), showcasing an ultrahigh longitudinal thermopower and a giant power factor which are predominantly attributable to the quantized thermoelectric Hall effect. Our findings demonstrate a confluence in the pursuit of effective thermoelectrics and novel topological materials in addition to the realization of the quantum thermoelectric Hall effect towards low-temperature energy harvesting applications. [1] Han, F., Andrejevic, N., Nguyen, T., Kozii, V., et al. Quantized thermoelectric Hall effect induces giant power factor in a topological semimetal. Nat Commun 11, 6167 (2020).

Flat-band superconductivity demonstrates the importance of band topology and geometry to correlated phases. The geometry of fragile topological and obstructed atomic bands enhances the superfluid weight and hence the superconducting critical temperature. Here, we derive general lower bounds for the superfluid weight in terms of momentum space irreps in all 2D space groups, extending the reach of topological quantum chemistry to superconducting states. Using exact Monte Carlo simulations we prove the validity of our bounds beyond mean-field.

The chiral topological semimetals RhSi and CoSi exhibit band degeneracies near the Fermi level enforced by the crystal symmetries. The low-energy quasiparticles emerging near these band degeneracies, referred to as multifold fermions, have no counterpart as elementary fermionic particles. We calculate the linear optical conductivity of all chiral multifold fermions [1] and show that it provides an experimental fingerprint for each type of multifold fermion. We use a tight-binding model for space group 198, where RhSi and CoSi crystallize, revealing that the location of the chemical potential is crucial to understand the optical response seen in experiments [2,3], determined at low energies by the threefold fermion at the $\Gamma$ point in both materials, and providing signatures of the existence of a spin-3/2 fourfold fermion in CoSi. Finally, we study the second-harmonic generation of RhSi. We analyze the experimental results using a second-order $k\cdot p$ Hamiltonian and compare our results with density functional theory calculations to provide a comprehensive description of the origin of the different features in the second-harmonic response and their relation to the topological character of the bands in RhSi [1] M.-Á. Sánchez-Martínez, F. de Juan, and A.G. Grushin, "Linear and nonlinear optical conductivity of chiral multifold fermions", Phys. Rev. B 99, 155145 (2019) [2] B. Xu, Z. Fang, M. Á. Sánchez-Martínez, J. W. F. Venderbos, Z. Ni, T. Qiu, K. Manna, K. Wang, J. Paglione, C. Bernhard, C. Felser, E. J. Mele, A. G. Grushin, A. M. Rappe, L. Wu, "Optical signatures of multifold fermions in the chiral topological semimetal CoSi'', Proceedings of the National Academy of Sciences 202010752 (2020). [3] Z. Ni, B. Xu, M. Á. Sánchez-Martínez, Y. Zhang, K. Manna, C. Bernhard, J. W. F. Venderbos, F. de Juan, C. Felser, A. G. Grushin, L. Wu, "Linear and nonlinear optical responses in the chiral multifold semimetal RhSi'', npj Quantum Mater. 5, 96 (2020). [4] B. Lu, S. Sayyad, M. Á. Sánchez-Martínez, K. Manna, C. Felser, A. G. Grushin, D. Torchinski, "Suppression of second-harmonic generation from linear bands in the topological multifold semimetal RhSi'', arXiv:2102.07416 (2021).

Magnetic adatoms deposited onto the surface of a superconducting crystal can give rise to the Yu-Shiba-Rusinov bound states, whose energies lie within the superconducting gap of the bulk. Here, we theoretically study the properties of experimentally measured terminations where adatoms are placed to form two-dimensional lattices. These give rise to band structures with non-trivial topology protected by crystalline symmetries. In the topological regime, we predict the presence of zero energy edge-modes in the lattice, matching the experimental measurements of the local density of states of the termination.

The structure of edge modes at the boundary of quantum Hall (QH) phases forms the basis for understanding low energy transport properties. In particular, the presence of ``upstream'' modes, moving against the direction of charge current flow, is critical for the emergence of renormalized modes with exotic quantum statistics. Detection of excess noise at the edge is a smoking gun for the presence of upstream modes. Here we report on noise measurements at the edges of fractional QH (FQH) phases realized in dual graphite-gated bilayer graphene devices. A noiseless dc current is injected at one of the edge contacts, and the noise generated at contacts at L=4 μm or 10 μm away along the upstream direction is studied. For integer and particle-like FQH states, no detectable noise is measured. By contrast, for ``hole-conjugate'' FQH states, we detect a strong noise proportional to the injected current, unambiguously proving the existence of upstream modes. The noise magnitude remaining independent of length together with a remarkable agreement with our theoretical analysis demonstrates the ballistic nature of upstream energy transport, quite distinct from the diffusive propagation reported earlier in GaAs-based systems. Our investigation opens the door to the study of upstream transport in more complex geometries and in edges of non-Abelian phases in graphene.

The symmetry-indicators provide valuable information about the topological properties of band structures in real materials. For inversion-symmetric, non-magnetic materials, the pattern of parity eigenvalues of various Kramers-degenerate bands at the time-reversal-invariant momentum points are generally analyzed with the combination of strong and weak indices. Can the symmetry indicators identify the tunneling configurations of SU(2) Berry connections or the three-dimensional, winding numbers of topologically non-trivial bands? We perform detailed analytical and numerical calculations on various effective tight-binding models to answer this question. If the parity eigenvalues are regarded as fictitious Ising spins, located at the vertices of Miller hypercube, the strong index describes the net ferro-magnetic moment, which is shown to be inadequate for identifying non-trivial bands, supporting even integer winding numbers. We demonstrate that an anti-ferromagnetic index, measuring the staggered magnetization can distinguish between bands possessing zero, odd, and even integer winding numbers. The coarse-grained analysis of symmetry-indicators is substantiated by computing the change in rotational-symmetry-protected, quantized Berry flux and Wilson loops along various high-symmetry axes. By simultaneously computing ferromagnetic and anti-ferromagnetic indices, we categorize various bands of bismuth, antimony, rhombohedral phosphorus, and Bi2Se3.

We investigate the full doping and strain-dependent phase diagram (absent superconductivity) of magic-angle twisted bilayer graphene (TBG). Using comprehensive Hartree-Fock calculations, we show that at temperatures where superconductivity is absent the global phase structure can be understood based on the competition and coexistence between three types of intertwined orders: a fully symmetric phase, spatially uniform flavor-symmetry-breaking states, and an incommensurate Kekule spiral (IKS) order. For small strain, the IKS phase, recently proposed as a candidate order at all non-zero integer fillings of the moir\'e unit cell, is found to be ubiquitous for non-integer doping as well. We demonstrate that the corresponding electronic compressibility and Fermi surface structure are consistent with the 'cascade' physics and Landau fans observed experimentally.

Zirconium pentatelluride (ZrTe5) is a 3D Dirac semimetal, which has recently gained attention due to various intriguing experimental observations including the chiral magnetic effect [1] and the quasi-quantized 3D quantum Hall effect [2]. However, its topological categorization remains challenging because experiments cannot easily probe the characteristic features of gapless surface and edge states. Addressing this challenge, we devised an optoelectronic detection scheme based on scanning photocurrent imaging. The technique relies on a global electrical read-out of locally-induced currents, and contrasts previous all-optical studies of photogalvanic effects. We applied this approach to characterize ZrTe5 single crystals, and visualized rich spatial patterns of photoinduced currents as function of temperature and magnetic field. Most remarkably, current maps obtained in the quasi-quantized Hall regime display long-range photoresponse over mmdistances and oscillations in magnetic field, hinting towards charge movement of ambient carriers in the system [3] instead of solely local photogalvanic effects. We anticipate that our spatially resolved photoinduced imaging scheme provides a generalizable platform for studying the non-trivial physics of gapless edge and surface states in topological materials. References [1] Q. Li et. al.; Nature Physics 12, 550–554 (2016) [2] S. Galeski et. al.; Nat Commun 12, 3197 (2021) [3] J. C. W. Song and L. S. Levitov; Phys. Rev. B 90, 075415 (2014)

Topological materials have emerged as a new frontier in physics and materials science. The non-trivial band topology gives rise to robust, spin-polarized electronic states at the edge or surface of the materials, that are, topologically protected surface states (TSSs). They are attractive for energy-related catalysis. Our interest is to design the high-efficiency hydrogen evolution reaction (HER) catalyst under the guidance of topology, reveal the role of TSSs on surface catalytic reactions, and develop a simple bulk band-based descriptor to predict the catalytic performance for various materials via ab-initio calculations in collaboration with electrochemical and photoemission experiments. 1) Topological engineering of Pt-group-metal-based chiral crystals toward high-efficiency hydrogen evolution catalysts. 2) Activating inert basal plane activity via joint utilization of trivial and non-trivial surface states in \ce{Nb2S2C}. 3) Projected Berry phase as an activity descriptor for hydrogen evolution catalysts.