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Phase-change materials are ubiquitous in technology because of their ability to transition between amorphous and crystalline phases fast and reversibly, upon shining light or passing a current. Here we argue that to fully understand their electronic properties it is necessary to define a novel electronic phase: the amorphous obstructed insulator. It differs from an obstructed insulator crystal in that it presents localized edge or surface states irrespective of the sample termination. Consequently, we show that obstructed amorphous insulators in three-dimensions host a surface two-dimensional flat band, detectable using ARPES. Our work establishes basic models for materials where topological and obstructed properties can be switched on and off externally, including two-dimensional surface flatbands., 7 pages + 6 pages of supplementary material and 6 figures

Crystalline symmetries have played a central role in the identification of topological materials. The use of symmetry indicators and band representations have enabled a classification scheme for crystalline topological materials, leading to large scale topological materials discovery. In this work we address whether amorphous topological materials, which lie beyond this classification due to the lack of long-range structural order, exist in the solid state. We study amorphous Bi$_2$Se$_3$ thin films, which show a metallic behavior and an increased bulk resistance. The observed low field magnetoresistance due to weak antilocalization demonstrates a significant number of two dimensional surface conduction channels. Our angle-resolved photoemission spectroscopy data is consistent with a dispersive two-dimensional surface state that crosses the bulk gap. Spin resolved photoemission spectroscopy shows this state has an anti-symmetric spin texture resembling that of the surface state of crystalline Bi$_2$Se$_3$. These experimental results are consistent with theoretical photoemission spectra obtained with an amorphous tight-binding model that utilizes a realistic amorphous structure. This discovery of amorphous materials with topological properties uncovers an overlooked subset of topological matter outside the current classification scheme, enabling a new route to discover materials that can enhance the development of scalable topological devices., Comment: 30 pages, 4 figures

Electronic edge states in topological insulators have become a major paradigm in physics. The oldest and primary example is that of quantum Hall (QH) edge channels that propagate along the periphery of two-dimensional electron gases (2DEGs) under perpendicular magnetic field. Yet, despite 40 years of intensive studies using a variety of transport and scanning probe techniques, imaging the real-space structure of QH edge channels has proven difficult, mainly due to the buried nature of most 2DEGs in semiconductors. Here, we show that QH edge states in graphene are confined to a few magnetic lengths at the crystal edges by performing scanning tunneling spectroscopy up to the edge of a graphene flake on hexagonal boron nitride. These findings indicate that QH edge states are defined by boundary conditions of vanishing electronic wavefunctions at the crystal edges, resulting in ideal one-dimensional chiral channels, free of electrostatic reconstruction. We further evidence a uniform charge carrier density at the edges, contrasting with conjectures on the existence of non-topological upstream modes. The absence of electrostatic reconstruction of quantum Hall edge states has profound implications for the universality of electron and heat transport experiments in graphene-based systems and other 2D crystalline materials., New theoretical simulations of the tunneling DOS added

The relation between thermal transport and gravity was highlighted in the seminal work by Luttinger in 1964, and has been extensively developed to understand thermal transport, most notably the thermal Hall effect. Here we review the novel concepts that relate thermal transport, the geometry of space-time and quantum field theory anomalies. We give emphasis to the cross-pollination between emergent ideas in condensed matter, notably Weyl and Dirac semimetals, and the understanding of gravitational and scale anomalies stemming from high-energy physics. We finish by relating to recent experimental advances and presenting a perspective of several open problems., 68 pages, 13 figures; comments welcome!

The chiral anomaly underlies a broad number of phenomena, from enhanced electronic transport in topological metals to anomalous currents in the quark-gluon plasma. The discovery of topological states of matter in non-Hermitian systems -- effective descriptions of dissipative systems -- raises the question of whether there are anomalous conservation laws that remain unaccounted for. To answer this question, we consider both $1+1$ and $3+1$ dimensions, presenting a unified formulation to calculate anomalous responses in Hermitianized, anti-Hermitianized and non-Hermitian systems of massless electrons with complex Fermi velocities coupled to non-Hermitian gauge fields. Our results indicate that the quantum conservation laws of chiral currents of non-Hermitian systems are not related to those in Hermitianized and anti-Hermitianized systems, as would be expected classically, due to novel anomalous terms that we derive. These may have implications for a broad class of emerging experimental systems that realize non-Hermitian Hamiltonians., Comment: 7+16 pages, 1+5 figure

In Weyl semimetals the location of linear band crossings, the Weyl cones, is not bound to any high symmetry point of the Brillouin zone, unlike the Dirac nodes in graphene. This flexibility is advantageous for valleytronics, where information is encoded in the valleys of the band structure when intervalley scattering is weak. However, if numerous Weyl cones coexist the encoded information can decohere rapidly because of band mixing. Here, we investigate how the helical iso-spin texture of Weyl cones affects valleytronics in heterojunctions of Weyl materials, and show how the chirality of this iso-spin texture can serve to encode information., Comment: 13 pages, 7 figures ; supplementary material included

When electrons populate a flat band their kinetic energy becomes negligible, forcing them to organize in exotic many-body states to minimize their Coulomb energy. The zeroth Landau level of graphene under magnetic field is a particularly interesting strongly interacting flat band because inter-electron interactions are predicted to induce a rich variety of broken-symmetry states with distinct topological and lattice-scale orders. Evidence for these stems mostly from indirect transport experiments that suggest that broken-symmetry states are tunable by boosting the Zeeman energy or by dielectric screening of the Coulomb interaction. However, confirming the existence of these ground states requires a direct visualization of their lattice-scale orders. Here, we image three distinct broken-symmetry phases in graphene using scanning tunneling spectroscopy. We explore the phase diagram by tuning the screening of the Coulomb interaction by a low or high dielectric constant environment, and with a magnetic field. In the unscreened case, we unveil a Kekul\'e bond order, consistent with observations of an insulating state undergoing a magnetic-field driven Kosterlitz-Thouless transition. Under dielectric screening, a sublattice-unpolarized ground state emerges at low magnetic fields, and transits to a charge-density-wave order with partial sublattice polarization at higher magnetic fields. The Kekul\'e and charge-density-wave orders furthermore coexist with additional, secondary lattice-scale orders that enrich the phase diagram beyond current theory predictions. This screening-induced tunability of broken-symmetry orders may prove valuable to uncover correlated phases of matter in other quantum materials., Comment: Main text + SI

For many materials, a precise knowledge of their dispersion spectra is insufficient to predict their ordered phases and physical responses. Instead, these materials are classified by the geometrical and topological properties of their wavefunctions. A key challenge is to identify and implement experiments that probe or control these quantum properties. In this review, we describe recent progress in this direction, focusing on nonlinear electromagnetic responses that arise directly from quantum geometry and topology. We give an overview of the field by discussing new theoretical ideas, groundbreaking experiments, and the novel materials that drive them. We conclude by discussing how these techniques can be combined with new device architectures to uncover, probe, and ultimately control novel quantum phases with emergent topological and correlated properties., Comment: Nature Materials (In Press)

Collective guidance of out-of-equilibrium systems without using external fields is a challenge of paramount importance in active matter, ranging from bacterial colonies to swarms of self-propelled particles. Designing strategies to guide active matter and exploiting enhanced diffusion associated to its motion will provide insights for application from sensing, drug delivery to water remediation. However, achieving directed motion without breaking detailed balance, for example by asymmetric topographical patterning, is challenging. Here we engineer a two-dimensional periodic topographical design with detailed balance in its unit cell where we observe spontaneous particle edge guidance and corner accumulation of self-propelled particles. This emergent behaviour is guaranteed by a second-order non-Hermitian skin effect, a topologically robust non-equilibrium phenomenon, that we use to dynamically break detailed balance. Our stochastic circuit model predicts, without fitting parameters, how guidance and accumulation can be controlled and enhanced by design: a device guides particles more efficiently if the topological invariant characterizing it is non-zero. Our work establishes a fruitful bridge between active and topological matter, and our design principles offer a blueprint to design devices that display spontaneous, robust and predictable guided motion and accumulation, guaranteed by out-of-equilibrium topology., Sustainable strategies for shepherding active particles are at the heart of many prospective applications. Here, Palacios et al. use the emerging topological properties of a microfluidic maze array to passively guide self-propelled colloids from the interior to the edges of the device.

The absence of mirror symmetry, or chirality, is behind striking natural phenomena found in systems as diverse as DNA and crystalline solids. A remarkable example occurs when chiral semimetals with topologically protected band degeneracies are illuminated with circularly polarized light. Under the right conditions, the part of the generated photocurrent that switches sign upon reversal of the light’s polarization, known as the circular photo-galvanic effect, is predicted to depend only on fundamental constants. The conditions to observe quantization are non-universal, and depend on material parameters and the incident frequency. In this work, we perform terahertz emission spectroscopy with tunable photon energy from 0.2 –1.1 eV in the chiral topological semimetal CoSi. We identify a large longitudinal photocurrent peaked at 0.4 eV reaching ~550 μ A/V2, which is much larger than the photocurrent in any chiral crystal reported in the literature. Using first-principles calculations we establish that the peak originates only from topological band crossings, reaching 3.3 ± 0.3 in units of the quantization constant. Our calculations indicate that the quantized circular photo-galvanic effect is within reach in CoSi upon doping and increase of the hot-carrier lifetime. The large photo-conductivity suggests that topological semimetals could potentially be used as novel mid-infrared detectors., Quantized circular photogalvanic effect (CPGE) is predicted in chiral topological semimetals, but the experimental observation remains challenging. Here, Ni et al. observe a large topological longitudinal photocurrent in CoSi, which is much larger than the photocurrent in any other chiral crystals, indicating quantized CPGE within reach upon doping and increase of the hot-carrier lifetime.

From Feynman diagrams to tensor networks, diagrammatic representations of computations in quantum mechanics have catalysed progress in physics. These diagrams represent the underlying mathematical operations and aid physical interpretation, but cannot generally be computed with directly. In this paper we introduce the ZXH-calculus, a graphical language based on the ZX-calculus, that we use to represent and reason about many-body states entirely graphically. As a demonstration, we express the 1D AKLT state, a symmetry protected topological state, in the ZXH-calculus by developing a representation of spins higher than 1/2 within the calculus. By exploiting the simplifying power of the ZXH-calculus rules we show how this representation straightforwardly recovers the AKLT matrix-product state representation, the existence of topologically protected edge states, and the non-vanishing of a string order parameter. Extending beyond these known properties, our diagrammatic approach also allows us to analytically derive that the Berry phase of any finite-length 1D AKLT chain is $\pi$. In addition, we provide an alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum computing resource. Lastly, we build 2D higher-order topological phases diagrammatically, which we use to illustrate a symmetry-breaking phase transition. Our results show that the ZXH-calculus is a powerful language for representing and computing with physical states entirely graphically, paving the way to develop more efficient many-body algorithms and giving a novel diagrammatic perspective on quantum phase transitions., Comment: Updated version accepted in Phys. Rev. X Quantum

Amorphous solids remain outside of the classification and systematic discovery of new topological materials, partially due to the lack of realistic models that are analytically tractable. Here we introduce the topological Weaire-Thorpe class of models, which are defined on amorphous lattices with fixed coordination number, a realistic feature of covalently bonded amorphous solids. Their short-range properties allow us to analytically predict spectral gaps. Their symmetry under permutation of orbitals allows us to analytically compute topological phase diagrams, which determine quantized observables like circular dichroism, by introducing symmetry indicators for the first time in amorphous systems. These models and our procedures to define invariants are generalizable to higher coordination number and dimensions, opening a route towards a complete classification of amorphous topological states in real space using quasilocal properties., Comment: 6+11 pages, 9 figures. Upgraded discussion, including coordination $z=4$ phase diagram and symmetry indicators

We report the optical conductivity in high-quality crystals of the chiral topological semimetal CoSi, which hosts exotic quasiparticles known as multifold fermions. We find that the optical response is separated into several distinct regions as a function of frequency, each dominated by different types of quasiparticles. The low-frequency intraband response is captured by a narrow Drude peak from a high-mobility electron pocket of double Weyl quasiparticles, and the temperature dependence of the spectral weight is consistent with its Fermi velocity. By subtracting the low-frequency sharp Drude and phonon peaks at low temperatures, we reveal two intermediate quasilinear interband contributions separated by a kink at 0.2 eV. Using Wannier tight-binding models based on first-principle calculations, we link the optical conductivity above and below 0.2 eV to interband transitions near the double Weyl fermion and a threefold fermion, respectively. We analyze and determine the chemical potential relative to the energy of the threefold fermion, revealing the importance of transitions between a linearly dispersing band and a flat band. More strikingly, below 0.1 eV our data are best explained if spin-orbit coupling is included, suggesting that at these energies, the optical response is governed by transitions between a previously unobserved fourfold spin-3/2 node and a Weyl node. Our comprehensive combined experimental and theoretical study provides a way to resolve different types of multifold fermions in CoSi at different energy. More broadly, our results provide the necessary basis to interpret the burgeoning set of optical and transport experiments in chiral topological semimetals.

We present an experimental set-up for the controlled application of strain gradients by mechanical piezoactuation on 3D crystalline microcantilevers that were fabricated by focused ion beam machining. A simple sample design tailored for transport characterization under strain at cryogenic temperatures is proposed. The topological semi-metal Cd3As2 serves as a test bed for the method, and we report extreme strain gradients of up to 1.3 % μ m−1 at a surface strain value of ≈ 0.65 % at 4 K. Interestingly, the unchanged quantum transport of the cantilever suggests that the bending cycle does not induce defects via plastic deformation. This approach is a first step towards realizing transport phenomena based on structural gradients, such as artificial gauge fields in topological materials.

We develop a stochastic description of the topological properties in an interacting Chern insulator. We confirm the Mott transition's first-order nature in the interacting Haldane model on the honeycomb geometry, from a mean-field variational approach, supported by density matrix renormalization group results and Ginzburg-Landau arguments. From the Bloch sphere, we make predictions for circular dichroism of light related to the quantum Hall conductivity on the lattice and in the presence of interactions. This analysis shows that the topological number can be measured from the light response at the Dirac points. Electron-electron interactions can also produce a substantial number of particle-hole pairs above the band gap, which leads us to propose a "stochastic Chern number" as an interacting measure of the topology. The stochastic Chern number can describe disordered situations with a fluctuating staggered potential and we build an analogy between interaction-induced particle-hole pairs and temperature effects. Our stochastic approach is physically intuitive, easy to implement and leads the way to further studies of interaction effects., 20 pages, 13 Figures, Table I, version to be published in Physical Review B

When two lasers are applied to a non-centrosymmetric material, light can be generated at the difference of the incoming frequencies $\Delta\omega$, a phenomenon known as difference frequency generation (DFG), well characterized in semiconductors. In this work, we derive a general expression for DFG in metals, which we use to show that the DFG in chiral topological semimetals under circular polarized light is quantized in units of $e^3/h^2$ and independent of material parameters, including the scattering time $\tau$, when $\Delta\omega \gg \tau^{-1}$. In this regime, DFG provides a simpler alternative to measure a quantized response in metals compared to previous proposals based on single frequency experiments. Our general derivation unmasks, in addition, a free-carrier contribution to the circular DFG beyond the semiclassical one. This contribution can be written as a Fermi surface integral, features strong frequency dependence, and oscillates with a $\pi/2$ shift with respect to the quantized contribution. We make predictions for the circular DFG of chiral and non-chiral materials using generic effective models, and ab-initio calculations for TaAs and RhSi. Our work provides a complete picture of the DFG in the length gauge approach, in the clean, non-interacting limit, and highlights a plausible experiment to measure topologically quantizated photocurrents in metals., Comment: 4+epsilon+10 pages, 3 figures

The dualities that map hard-to-solve, interacting theories to free, non-interacting ones often trigger a deeper understanding of the systems to which they apply. However, simplifying assumptions such as Lorentz invariance, low dimensionality, or the absence of axial gauge fields, limit their application to a broad class of systems, including topological semimetals. Here we derive several axial field theory dualities in 2+1 and 3+1 dimensions by developing an axial slave-rotor approach capable of accounting for the axial anomaly. Our 2+1-dimensional duality suggests the existence of a dual, critical surface theory for strained three-dimensional non-symmorphic topological insulators. Our 3+1-dimensional duality maps free Dirac fermions to Dirac fermions coupled to emergent U(1) and Kalb-Ramond vector and axial gauge fields. Upon fixing an axial field configuration that breaks Lorentz invariance, this duality maps free to interacting Weyl semimetals, thereby suggesting that the quantization of the non-linear circular photogalvanic effect can be robust to certain interactions. Our work emphasizes how axial and Lorentz-breaking dualities improve our understanding of topological matter., 11+3 pages, 3 figures, minor changes, accepted in Phys. Rev. B

The half-Heusler rare-earth intermetallic GdPtBi has recently gained attention due to peculiar magnetotransport phenomena that have been associated with the possible existence of Weyl fermions, thought to arise from the crossings of spin-split conduction and valence bands. On the other hand, similar magnetotransport phenomena observed in other rare-earth intermetallics have often been attributed to the interaction of itinerant carriers with localized magnetic moments stemming from the $4f$-shell of the rare-earth element. In order to address the origin of the magnetotransport phenomena in GdPtBi, we performed a comprehensive study of the magnetization, electrical and thermal magnetoresistivity on two single-crystalline GdPtBi samples. In addition, we performed an analysis of the Fermi surface via Shubnikov-de Haas oscillations in one of the samples and compared the results to \emph{ab initio} band structure calculations. Our findings indicate that the electrical and thermal magnetotransport in GdPtBi cannot be solely explained by Weyl physics and is strongly influenced by the interaction of both itinerant charge carriers and phonons with localized magnetic Gd-ions and possibly also paramagnetic impurities., Comment: 11 figures

Chiral topological semimetals are materials that break both inversion and mirror symmetries. They host interesting phenomena such as the quantized circular photogalvanic effect (CPGE) and the chiral magnetic effect. In this work, we report a comprehensive theoretical and experimental analysis of the linear and non-linear optical responses of the chiral topological semimetal RhSi, which is known to host multifold fermions. We show that the characteristic features of the optical conductivity, which display two distinct quasi-linear regimes above and below 0.4 eV, can be linked to excitations of different kinds of multifold fermions. The characteristic features of the CPGE, which displays a sign change at 0.4 eV and a large non-quantized response peak of around 160 $\mu \textrm{A V}^{-2}$ at 0.7 eV, are explained by assuming that the chemical potential crosses a flat hole band at the Brillouin zone center. Our theory predicts that, in order to observe a quantized CPGE in RhSi, it is necessary to increase the chemical potential as well as the quasiparticle lifetime. More broadly our methodology, especially the development of the broadband terahertz emission spectroscopy, could be widely applied to study photo-galvanic effects in noncentrosymmetric materials and in topological insulators in a contact-less way and accelerate the technological development of efficient infrared detectors based on topological semimetals., Comment: Accepted in npj Quantum Materials; Abstract updated

In this work we study the electronic structure of Ag3AuSe2 and Ag3AuTe2, two chiral insulators whose gap can be tuned through small changes in the lattice parameter by applying hydrostatic pressure or choosing different growth protocols. Based on first principles calculations we compute their band structure for different values of the lattice parameters and show that while Ag3AuSe2 retains its direct narrow gap at the Γ point, Ag3AuTe2 can turn into a metal. Focusing on Ag3AuSe2 we derive a low energy model around Γ using group theory, which we use to calculate the optical conductivity for different values of the lattice constant. We discuss our results in the context of detection of light dark matter particles, which have masses of the order of a keV, and conclude that Ag3AuSe2 satisfies three important requirements for a suitable detector: small Fermi velocities, meV band gap, and low photon screening. Our work motivates the growth of high-quality and large samples of Ag3AuSe2 to be used as target materials in dark matter detectors., We acknowledge support from the European Union’s Horizon 2020 research and innovation programme under the Marie–Sklodowska–Curie grant agreement No. 754303 and the GreQuE Cofund programme (MASM). AGG is also supported by the ANR under the grant ANR-18-CE30-0001-01 and the European FET-OPEN SCHINES project No. 829044. MGV acknowledges the IS2016-75862-P national project of the Spanish MINECO. AB acknowledges financial support from the Spanish Ministry of Economy and Competitiveness (FIS2016-76617-P) and the Department of Education, Universities and Research of the Basque Government and the University of the Basque Country (IT756-13)

Dirac and Weyl semimetals react to position-dependent and time-dependent perturbations, such as strain, as if subject to emergent electromagnetic fields, known as pseudo-fields. Pseudo-fields differ from external electromagnetic fields in their symmetries and phenomenology and enable a simple and unified description of a variety of inhomogeneous systems. We review the different physical means of generating pseudo-fields, the observable consequences of pseudo-fields and their similarities to and differences from electromagnetic fields. Among these differences is their effect on quantum anomalies — absences of classical symmetries in the quantum theory — which we revisit from a quantum field theory and a semi-classical viewpoint. We conclude with predicted observable signatures of the pseudo-fields and the status of the nascent experimental research. Pseudo-electromagnetic fields emerge in inhomogeneous materials. This Review discusses the properties of such fields in the context of 3D topological semimetals, the origin and consequences of pseudo-fields in real materials and their field theory description.

Condensed matter systems realizing Weyl fermions exhibit striking phenomenology derived from their topologically protected surface states as well as chiral anomalies induced by electromagnetic fields. More recently, inhomogeneous strain or magnetization were predicted to result in chiral electric ${\mathbf{E}}_{5}$ and magnetic ${\mathbf{B}}_{5}$ fields, which modify and enrich the chiral anomaly with additional terms. In this Rapid Communication, we develop a lattice-based approach to describe the chiral anomaly, which involves Landau and pseudo-Landau levels and treats all anomalous terms on equal footing, while naturally incorporating Fermi arcs. We exemplify its potential by physically interpreting the largely overlooked role of Fermi arcs in the covariant (Fermi level) contribution to the anomaly and revisiting the factor of $1/3$ difference between the covariant and consistent (complete band) contributions to the ${\mathbf{E}}_{5}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathbf{B}}_{5}$ term in the anomaly. Our framework provides a versatile tool for the analysis of anomalies in realistic lattice models as well as a source of simple physical intuition for understanding strained and magnetized inhomogeneous Weyl semimetals.

Multifold fermions are generalizations of two-fold degenerate Weyl fermions with three-, four-, six- or eight-fold degeneracies protected by crystal symmetries, of which only the last type is necessarily non-chiral. Their low energy degrees of freedom can be described as emergent particles not present in the Standard Model of particle physics. We propose a range of experimental probes for multifold fermions in chiral symmetry groups based on the gyrotropic magnetic effect (GME) and the circular photo-galvanic effect (CPGE). We find that, in contrast to Weyl fermions, multifold fermions can have zero Berry curvature yet a finite GME, leading to an enhanced response. The CPGE is quantized and independent of frequency provided that the frequency region at which it is probed defines closed optically-activated momentum surfaces. We confirm the above properties by calculations in symmetry-restricted tight binding models with realistic density functional theory parameters. We identify a range of previously-unidentified ternary compounds able to exhibit chiral multifold fermions of all types (including a range of materials in the families AsBaPt and Gd$_3$Cl$_3$C), and provide specific predictions for the known multifold material RhSi., 19+12 pages, 8+1 figures. This version: minor typos corrected

The experimental realization of the Harper-Hofstadter model in ultra-cold atomic gases has placed fractional states of matter in these systems within reach---a fractional Chern insulator state (FCI) is expected to emerge for sufficiently strong interactions when half-filling the lowest band. The experimental setups naturally allow to probe the dynamics of this topological state, yet little is known about its out-of-equilibrium properties. We explore, using density matrix renormalization group (DMRG) simulations, the response of the FCI state to spatially localized perturbations. After confirming the static properties of the phase we show that the characteristic, gapless features are clearly visible in the edge dynamics. We find that a local edge perturbation in this model propagates chirally independent of the perturbation strength. This contrasts the behavior of single particle models with counter-propagating edge states, such as the non-interacting Harper-Hofstadter model, where the chirality is manifest only for weak perturbations. Additionally, our simulations show that there is inevitable density leakage from the first row of sites into the bulk, preventing a naive chiral Luttinger theory interpretation of the dynamics., 4+epsilon pages, 4 pages of supplementary material and a total of 8 figures. Published version with updated title, discussion, references, and supplementary information

In this work, we describe a toolbox to realize and probe synthetic axial gauge fields in engineered Weyl semimetals. These synthetic electromagnetic fields, which are sensitive to the chirality associated with Weyl nodes, emerge due to spatially and temporally dependent shifts of the corresponding Weyl momenta. First, we introduce two realistic models, inspired by recent cold-atom developments, which are particularly suitable for the exploration of these synthetic axial gauge fields. Second, we describe how to realize and measure the effects of such axial fields through center-ofmass observables, based on semiclassical equations of motion and exact numerical simulations. In particular, we suggest realistic protocols to reveal an axial Hall response due to the axial electric field E5, as well as axial cyclotron orbits and chiral pseudo-magnetic effect due to the axial magnetic field B5., SCOPUS: ar.j, info:eu-repo/semantics/published

This chapter is a collection of techniques, warnings, facts and ideas that are sometimes regarded as theoretical curiosities in high-energy physics but have important consequences in condensed matter physics. In particular, we describe theories that have the property of having finite but undetermined radiative corrections that also happen to describe topological semi-metallic phases in condensed matter. In the process, we describe typical methods in high-energy physics that illustrate the working principles to describe a given phase of matter and its response to external fields., Comment: Chapter in "Topological Matter". Springer Series in Solid-State Sciences, vol 190. Springer. Based on three lectures given at the 2017 Topological Matter School, available at https://www.youtube.com/watch?v=ZmmOe61Ea4M , https://www.youtube.com/watch?v=Gw6dmVPSI_g , and https://www.youtube.com/watch?v=UG6rwZdQabA

We propose the use of three-dimensional Dirac materials as targets for direct detection of sub-MeV dark matter. Dirac materials are characterized by a linear dispersion for low-energy electronic excitations, with a small band gap of O(meV) if lattice symmetries are broken. Dark matter at the keV scale carrying kinetic energy as small as a few meV can scatter and excite an electron across the gap. Alternatively, bosonic dark matter as light as a few meV can be absorbed by the electrons in the target. We develop the formalism for dark matter scattering and absorption in Dirac materials and calculate the experimental reach of these target materials. We find that Dirac materials can play a crucial role in detecting dark matter in the keV to MeV mass range that scatters with electrons via a kinetically mixed dark photon, as the dark photon does not develop an in-medium effective mass. The same target materials provide excellent sensitivity to absorption of light bosonic dark matter in the meV to hundreds of meV mass range, superior to all other existing proposals when the dark matter is a kinetically mixed dark photon., Comment: 24+25 pages, 4 appendices, 12 figures, 1 table. v2: references added

We reveal an intriguing manifestation of topology, which appears in the depletion rate of topological states of matter in response to an external drive. This phenomenon is presented by analyzing the response of a generic two-dimensional (2D) Chern insulator subjected to a circular time-periodic perturbation. Because of the system's chiral nature, the depletion rate is shown to depend on the orientation of the circular shake; taking the difference between the rates obtained from two opposite orientations of the drive, and integrating over a proper drive-frequency range, provides a direct measure of the topological Chern number (n) of the populated band: This "differential integrated rate" is directly related to the strength of the driving field through the quantized coefficient η0 = ν/ℏ2, where h = 2π ℏ is Planck's constant. Contrary to the integer quantum Hall effect, this quantized response is found to be nonlinear with respect to the strength of the driving field, and it explicitly involves interband transitions. We investigate the possibility of probing this phenomenon in ultracold gases and highlight the crucial role played by edge states in this effect. We extend our results to 3D lattices, establishing a link between depletion rates and the nonlinear photogalvanic effect predicted for Weyl semimetals. The quantized circular dichroism revealed in this work designates depletion rate measurements as a universal probe for topological order in quantum matter., SCOPUS: ar.j, info:eu-repo/semantics/published

The circular photogalvanic effect (CPGE) is the part of a photocurrent that switches depending on the sense of circular polarization of the incident light. It has been consistently observed in systems without inversion symmetry and depends on non-universal material details. Here we find that in a class of Weyl semimetals (e.g. SrSi$_2$) and three-dimensional Rashba materials (e.g. doped Te) without inversion and mirror symmetries, the injection contribution to the CPGE trace is effectively quantized in terms of the fundamental constants $e, h, c$ and $\epsilon_0$ with no material-dependent parameters. This is so because the CPGE directly measures the topological charge of Weyl points, and non-quantized corrections from disorder and additional bands can be small over a significant range of incident frequencies. Moreover, the magnitude of the CPGE induced by a Weyl node is relatively large, which enables the direct detection of the monopole charge with current techniques., Comment: 4+5 pages, 3 figures, published version

Weyl semimetals could be used to build a range of electronic devices, from superlenses for scanning tunneling microscopes to transistors.

The existence and topological classification of lower-dimensional Fermi surfaces is often tied to the crystal symmetries of the underlying lattice systems. Artificially engineered lattices, such as heterostructures and other superlattices, provide promising avenues to realize desired crystal symmetries that protect lower-dimensional Fermi surface, such as nodal lines. In this work, we investigate a Weyl semimetal subjected to spatially periodic onsite potential, giving rise to several phases, including a nodal-line semimetal phase. In contrast to proposals that purely focus on lattice symmetries, the emergence of the nodal line in this setup does not require small spin-orbit coupling, but rather relies on its presence. We show that the stability of the nodal line is understood from reflection symmetry and a combination of a fractional lattice translation and charge-conjugation symmetry. Depending on the choice of parameters, this model exhibits drumhead surface states that are exponentially localized at the surface, or weakly localized surface states that decay into the bulk at all energies., Comment: 11 pages, 8 figures, Editors' Suggestion

The Hofstadter butterfly of lattice electrons in a strong magnetic field is a cornerstone of condensed matter physics, exploring the competition between periodicities imposed by the lattice and the field. In this work we introduce and characterize the Weyl butterfly, which emerges when a large magnetic field is applied to a three-dimensional Weyl semimetal. Using an experimentally motivated lattice model for cold atomic systems, we solve this problem numerically. We find that Weyl nodes reemerge at commensurate fluxes and propose using wavepackets dynamics to reveal their chirality and location. Moreover, we show that the chiral anomaly -- a hallmark of the topological Weyl semimetal -- does not remain proportional to magnetic field at large fields, but rather inherits a fractal structure of linear regimes as a function of external field. The slope of each linear regime is determined by the difference of two Chern numbers in gaps of the Weyl butterfly and can be measured experimentally in time of flight., Comment: 4 pages, 4 figures + supplementary material, version accepted in Phys. Rev. B as a Rapid Communication

Weyl semimetals (WSMs) are topological quantum states wherein the electronic bands linearly disperse around pairs of nodes, the Weyl points, of fixed (left or right) chirality. The recent discovery of WSM materials triggered an experimental search for the exotic quantum phenomenon known as the chiral anomaly. Via the chiral anomaly nonorthogonal electric and magnetic fields induce a chiral density imbalance that results in an unconventional negative longitudinal magnetoresistance, the chiral magnetic effect. Recent theoretical work suggests that this effect does not require well-defined Weyl nodes. Experimentally however, it remains an open question to what extent it survives when chirality is not well-defined, for example when the Fermi energy is far away from the Weyl points. Here, we establish the detailed Fermi surface topology of the recently identified WSM TaP via a combination of angle-resolved quantum oscillation spectra and band structure calculations. The Fermi surface forms spin-polarized banana-shaped electron and hole pockets attached to pairs of Weyl points. Although the chiral anomaly is therefore ill-defined, we observe a large negative magnetoresistance (NMR) appearing for collinear magnetic and electric fields as observed in other WSMs. In addition, we show experimental signatures indicating that such longitudinal magnetoresistance measurements can be affected by an inhomogeneous current distribution inside the sample in a magnetic field. Our results provide a clear framework how to detect the chiral magnetic effect., Comment: 57 pages including the supplementary information. 4 figures in the main manuscript and 21 figures in the supplementary information

The analysis of the electronic properties of strained or lattice deformed graphene combines ideas from classical condensed matter physics, soft matter, and geometrical aspects of quantum field theory (QFT) in curved spaces. Recent theoretical and experimental work shows the influence of strains in many properties of graphene not considered before, such as electronic transport, spin-orbit coupling, the formation of Moiré patterns and optics. There is also significant evidence of anharmonic effects, which can modify the structural properties of graphene. These phenomena are not restricted to graphene, and they are being intensively studied in other two dimensional materials, such as the transition metal dichalcogenides. We review here recent developments related to the role of strains in the structural and electronic properties of graphene and other two dimensional compounds., We acknowledge the financial support of the following institutions: Fundação para a Ciência e a Tecnologia, Portugal, through Grant No. SFRH/BD/78987/2011 (BA); European Union through ESF funded-JAE doc program (AC); JAE-Pre (CSIC, Spain) (AGR); Juan de la Cierva Program (MINECO, Spain) and FCT-Portugal through Grant No. EXPL/FIS-NAN/1728/2013 (RR); ERC Advanced Grant 290846 (FG, VP); Spanish Ministry of Economy (MINECO) through Grant Nos. FIS2011-23713 (PSJ) and ` (RR); CONICET (PIP 0747) and ANPCyT (PICT 2012-1724) (MS); Spanish MECD grant PIB2010BZ-00512, and the European Union Seventh Framework Programme under Grant Agreement No. 604391 Graphene Flagship (MV, JS).

Formulating consistent theories describing strongly correlated metallic topological phases is an outstanding problem in condensed matter physics. In this work we derive a theory defining a fractionalized analogue of the Weyl semimetal state: the fractional chiral metal. Our approach is to construct a 4+1D quantum Hall insulator by stacking 3+1D Weyl semimetals in a magnetic field. In a strong enough field the low-energy physics is determined by the lowest Landau level of each Weyl semimetal, which is highly degenerate and chiral, motivating us to use a coupled-wire approach. The one-dimensional dispersion of the lowest Landau level allows us to model the system as a set of degenerate 1+1D quantum wires that can be bosonized in the presence of electron-electron interactions and coupled such that a gapped phase is obtained, whose response to an electromagnetic field is given in terms of a Chern-Simons field theory. At the boundary of this phase we obtain the field theory of a 3+1D gapless fractional chiral state, which we show is consistent with a previous theory for the surface of a 4+1D Chern-Simons theory. The boundary's response to an external electromagnetic field is determined by a chiral anomaly with a fractionalized coefficient. We suggest that such anomalous response can be taken as a working definition of a fractionalized strongly correlated analogue of the Weyl semimetal state., Comment: 15 pages, 6 figures, extended discussion about potential realizations of 3+1D fractional chiral metals, final version