Multifold Fermions Research Articles

Floquet engineering of multifold fermions

Using Floquet theory, we investigate the effect of light on triplefold fermions. We study a low-energy model as well as a simplified tight-binding model under illumination. We find that the threefold degeneracy remains symmetry protected even after applying light if the original band structure is rotationally symmetric around the degeneracy. Otherwise, light can lift the degeneracy and open up a gap. We further investigate the effect of light on the topological Fermi arcs by means of numerical computations. The changes caused by illumination are reflected in experimentally detectable signatures, such as the anomalous Hall conductivity, which we calculate.

Linear and nonlinear optical responses in the chiral multifold semimetal RhSi

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 nonlinear 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 μA/V2 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 photogalvanic 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.

Symmetry protection and giant Fermi arcs from multifold fermions in binary, ternary, and quaternary compounds

Higher-fold chiral fermions that go beyond two-fold Weyl fermions have recently been reported in crystalline systems. Here, we focus on such excitations in several binary, ternary and quaternary alloys/compounds with CoGe, BiSbPt and KMgBO3 as the representative examples that belong to the crystal space group (SG) 198. We found distinct three-fold, four-fold and six-fold chiral fermions in the bulk via Density Functional computations. We provide general symmetry arguments for the protection of these degeneracies with special emphasis on the four-fold fermions. Our surface spectra simulations show that the size of Fermi arcs resulting from these chiral fermions are large, robust and untouched from the bulk states due to the near absence of trivial bulk Fermi pockets. All these features make these systems -- especially CoGe and KMgBO3 -- promising topological semimetal candidates to realize higher-fold fermions in future photo-emission and transport experiments.

Optical signatures of multifold fermions in the chiral topological semimetal CoSi

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.

Ab initio study of quantized circular photogalvanic effect in chiral multifold semimetals

So far, the circular photogalvanic effect (CPGE) is the only possible quantized signal in Weyl semimetals. With inversion and mirror symmetries broken, Weyl and multifold fermions in band structures with opposite chiralities can stay at different energies and generate a net topological charge. Such a kind of net topological charge can present as a quantized signal in the circular polarized light-induced injection current. According to current theoretical understanding, RhSi and its counterparts are believed to be the most promising candidates for the experimental observation of the quantized CPGE. However, a real quantized signal has yet to be experimentally observed. Since previous theoretical studies for the quantized CPGE were based on an effective model but not realistic band structures, it should lose some crucial details that influence the quantized signal. The current status motivates us to perform a realistic ab initio study for the CPGE. Our result shows that RhSi and PtAl in chiral multifold semimetals are alternative materials for obtaining the quantized CPGE which is very easy to be interfered with by trivial band-related optic transitions, and a fine tuning of the chemical potential by doping is essential for the observation of the quantized CPGE. We perform an ab initio analysis for the quantized CPGE based on a realistic electronic band structure and provide an effective way to solve the current problem for the given materials.

Optical method to detect the relationship between chirality of reciprocal space chiral multifold fermions and real space chiral crystals

The chirality of chiral multifold fermions in reciprocal space is related to the chirality of crystal lattice structures in real space. In this study, we propose a strategy to detect and identify multifold fermions of opposite chirality in nonmagnetic systems using second-order optical transports. Chiral crystals related with inversion operations cannot be made to overlap with each other via any experimental operation. Further, chiral multifold fermions within such crystals host opposite chiralities corresponding to a given $k$ point. A change in chirality is indicated by a corresponding change in the sign of the second-order charge current dominated by chiral fermions. This property can be exploited to study the relationship between chiralities in reciprocal and real spaces by utilizing bulk transport.

Observation of giant spin-split Fermi-arc with maximal Chern number in the chiral topological semimetal PtGa

Non-symmorphic chiral topological crystals host exotic multifold fermions, and their associated Fermi arcs helically wrap around and expand throughout the Brillouin zone between the high-symmetry center and surface-corner momenta. However, Fermi-arc splitting and realization of the theoretically proposed maximal Chern number rely heavily on the spin-orbit coupling (SOC) strength. In the present work, we investigate the topological states of a new chiral crystal, PtGa, which has the strongest SOC among all chiral crystals reported to date. With a comprehensive investigation using high-resolution angle-resolved photoemission spectroscopy, quantum-oscillation measurements, and state-of-the-art ab initio calculations, we report a giant SOC-induced splitting of both Fermi arcs and bulk states. Consequently, this study experimentally confirms the realization of a maximal Chern number equal to ±4 in multifold fermionic systems, thereby providing a platform to observe large-quantized photogalvanic currents in optical experiments.

Optical conductivity of multifold fermions: The case of RhSi

We measured the reflectivity of the multifold semimetal RhSi in a frequency range from 80 to 20000 cm$^{-1}$ (10 meV - 2.5 eV) at temperatures down to 10 K. The optical conductivity, calculated from the reflectivity, is dominated by the free-carrier (Drude) contribution below 1000 cm$^{-1}$ (120 meV) and by interband transitions at higher frequencies. The temperature-induced changes in the spectra are generally weak: only the Drude bands narrow upon cooling, with an unscreened plasma frequency that is constant with temperature at approximately 1.4 eV, in agreement with a weak temperature dependence of the free-carrier concentration determined by Hall measurements. The interband portion of conductivity exhibits two linear-in-frequency regions below 5000 cm$^{-1}$ ($\sim$ 600 meV), a broad flat maximum at around 6000 cm$^{-1}$ (750 meV), and a further increase starting around 10000 cm$^{-1}$ ($\sim$ 1.2 eV). We assign the linear behavior of the interband conductivity to transitions between the linear bands near the band crossing points. Our findings are in accord with the predictions for the low-energy conductivity behavior in multifold semimetals and with earlier computations based on band structure calculations for RhSi.

Non-Abelian anomalies in multi-Weyl semimetals

We construct the effective field theory for time-reversal symmetry breaking multi-Weyl semimetals (mWSMs), composed of a single pair of Weyl nodes of (anti-)monopole charge $n$, with $n=1,2,3$ in crystalline environment. From both the continuum and lattice models, we show that a mWSM with $n>1$ can be constructed by placing $n$ flavors of linearly dispersing simple Weyl fermions (with $n=1$) in a bath of an $SU(2)$ non-Abelian static background gauge field. Such an $SU(2)$ field preserves certain crystalline symmetry (four-fold rotational or $C_4$ in our construction), but breaks the Lorentz symmetry, resulting in nonlinear band spectra (namely, $E \sim (p^2_x + p^2_y)^{n/2}$, but $E \sim |p_z|$, for example, where momenta ${\bf p}$ is measured from the Weyl nodes). Consequently, the effective field theory displays $U(1) \times SU(2)$ non-Abelian anomaly, yielding anomalous Hall effect, its non-Abelian generalization, and various chiral conductivities. The anomalous violation of conservation laws is determined by the monopole charge $n$ and a specific algebraic property of the $SU(2)$ Lie group, which we further substantiate by numerically computing the regular and "isospin" densities from the lattice models of mWSMs. These predictions are also supported from a strongly coupled (holographic) description of mWSMs. Altogether our findings unify the field theoretic descriptions of mWSMs of arbitrary monopole charge $n$ (featuring $n$ copies of the Fermi arc surface states), predict signatures of non-Abelian anomaly in table-top experiments, and pave the route to explore anomaly structures for multi-fold fermions, transforming under arbitrary half-integer or integer spin representations.

Strong bulk photovoltaic effect in chiral crystals in the visible spectrum

Structurally chiral materials hosting multifold fermions with large topological number have attracted considerable attention because of their naturally long surface Fermi arcs and bulk quantized circular photogalvanic effect (CPGE). Multifold fermions only appear in metallic states, and therefore, most studies so far have only focused on the semimetals in compounds with chiral crystal structures. In this work, we show that the structurally chiral topological trivial insulators are also exotic states, which is interesting from the application point of view, owing to their natural advantage to host a large bulk photovoltaic effect in the visible wavelength region. In the last decades, the shift current in the visible wavelength region was limited to be 10 uA/V2 . By scanning the insulators with chiral structure, we found a class of compounds with photoconductivity ranging from 20 to 80 uA/V2 , which is approximately one order of magnitude larger than that reported in other real materials. This work illustrates that the compounds with chiral structure can host both quantum CPGE and a strong shift current in the second order optical response. Moreover, this work offers a good platform for the study of the shift current and its future application by putting the focus on insulator with chiral lattices, so far overlooked in photovoltaic technologies.

Multifold nodal points in magnetic materials

We describe the symmetry protected nodal points that can exist in magnetic space groups and show that only three-, six-, and eightfold degeneracies are possible (in addition to the two- and fourfold degeneracies that have already been studied). The three- and sixfold degeneracies are derived from “spin-1” Weyl fermions. The eightfold degeneracies come in different flavors. In particular, we distinguish between eightfold fermions that realize nonchiral “Rarita-Schwinger fermions” and those that can be described as four degenerate Weyl fermions. We list the (magnetic and nonmagnetic) space groups where these exotic fermions can be found. We further show that in several cases, a magnetic translation symmetry pins the Hamiltonian of the multifold fermion to an idealized exactly solvable point that is not achievable in nonmagnetic crystals without fine-tuning. Finally, we present known compounds that may host these fermions and methods for systematically finding more candidate materials.

Topological Semimetals from First Principles

We review recent theoretical progress in the understanding and prediction of novel topological semimetals. Topological semimetals define a class of gapless electronic phases exhibiting topologically stable crossings of energy bands. Different types of topological semimetals can be distinguished on the basis of the degeneracy of the band crossings, their codimension (e.g., point or line nodes), and the crystal space group symmetries on which the protection of stable band crossings relies. The dispersion near the band crossing is a further discriminating characteristic. These properties give rise to a wide range of distinct semimetal phases such as Dirac or Weyl semimetals, point or line node semimetals, and type I or type II semimetals. In this review we give a general description of various families of topological semimetals, with an emphasis on proposed material realizations from first-principles calculations. The conceptual framework for studying topological gapless electronic phases is reviewed, with a particular focus on the symmetry requirements of energy band crossings, and the relation between the different families of topological semimetals is elucidated. In addition to the paradigmatic Dirac and Weyl semimetals, we pay particular attention to more recent examples of topological semimetals, which include nodal line semimetals, multifold fermion semimetals, and triple-point semimetals. Less emphasis is placed on their surface state properties, their responses to external probes, and recent experimental developments.

Chiral topological semimetal with multifold band crossings and long Fermi arcs

Topological semimetals in crystals with a chiral structure (which possess a handedness due to a lack of mirror and inversion symmetries) are expected to display numerous exotic physical phenomena, including fermionic excitations with large topological charge [1], long Fermi arc surface states [2,3], unusual magnetotransport [4] and lattice dynamics [5], as well as a quantized response to circularly polarized light [6]. To date, all experimentally confirmed topological semimetals exist in crystals that contain mirror operations, meaning that these properties do not appear. Here, we show that AlPt is a structurally chiral topological semimetal that hosts new fourfold and sixfold fermions, which can be viewed as a higher spin eneralization of Weyl fermions without equivalence in elementary particle physics. These multifold fermions are located at high symmetry points and have Chern numbers larger than those in Weyl semimetals, thus resulting in multiple Fermi arcs that span the full diagonal of the surface Brillouin zone. By imaging these long Fermi arcs, we experimentally determine the magnitude and sign of their Chern number, allowing us to relate their dispersion to the handedness of their host crystal.

Linear optical conductivity of chiral multifold fermions

Chiral multifold fermions are quasiparticles described by higher spin generalizations of the Weyl equation, and are realized as low energy excitations near symmetry protected band crossings in certain chiral crystals. In this work we calculate the linear optical conductivity of all chiral multifold fermions. We show that it is enhanced with respect to that of Weyl fermions with the same Fermi velocity, and features characteristic activation frequencies for each multifold fermion class, providing an experimental fingerprint to detect them. We calculate the conductivity for realistic chiral multifold semimetals by using lattice tight-binding Hamiltonians that match the effective models of multifold fermions at low energies, for space groups 199 and 198. The latter includes RhSi, for which we give quantitative predictions, and also CoSi and AlPt. Our predictions can be tested in absorption or penetration depth measurements, and are necessary to extract the recently proposed quantized photocurrents from experiments.

Chiral optical response of multifold fermions

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.