Linear and nonlinear optical responses in the chiral multifold semimetal RhSi

Chiral topological metals are materials that can quantize the circular photogalvanic effect (CPGE), a universal photocurrent that is generated by circularly polarized light. However, to determine quantization, a precise knowledge of the linear and non-linear optical responses are necessary. In this work we report a broad theoretical and experimental analysis of the linear and non-linear optical conductivity of RhSi that establishes the road map to measure quantization in this material. Combining \textit{ab-initio} and tight-binding calculations, with broadband optical conductivity and terahertz emission measurements, we develop a consistent picture of the optical response of RhSi. The features of the linear optical conductivity, which displays two distinct quasi-linear regimes above and below 0.4 eV, and the features of the CPGE, which displays a sign change at 0.4 eV and a large, non-quantized response peak of 163 ($\pm$19) $\mu A/V^2$ at 0.7 eV, are explained by assuming that the chemical potential crosses a flat hole band at the $\Gamma$ point, and a short hot-carrier scattering time of $\tau \sim 4$-$7$ fs. We link these features to interband excitations around topological band crossings of multiple bands, known as multifold fermions. Our results indicate that to observe a quantized CPGE in RhSi it is necessary to increase significantly the quasiparticle lifetime as well as the carrier concentration. Our methodology can be applied to other chiral topological multifold semimetals, and establishes the conditions to observe quantization in chiral topological metals.


Introduction
The robust and intrinsic electronic properties of topological metals-a new class of quantum materials-can potentially protect or enhance useful electromagnetic responses [1][2][3][4] . However, direct and unambiguous detection of these properties is often challenging. For example, in Dirac semimetals such as Cd 3 As 2 and Na 3 Bi 5,6 two doubly degenerate bands cross linearly at a single point, the Dirac point, and this crossing is protected by rotational symmetry [7][8][9] . A Dirac point can be understood as two coincident topological crossings with equal but opposite topological charge 4 , and as a result, the topological contributions to the response to external probes cancel in this class of materials, rendering external probes insensitive to the topological charge.
Weyl semimetals may offer an alternative, as this class of topological metals is defined by the presence of isolated twofold topological band crossings, separated in momentum space from a partner crossing with opposite topological charge. This requires the breaking of either timereversal or inversion symmetry. The Weyl semimetal phases discovered in materials of the transition monopnictide family such as TaAs 10-17 lack inversion symmetry, which allows for nonzero second-order nonlinear optical responses and has motivated the search for topological responses using techniques of nonlinear optics. This search has resulted in the observation of giant second harmonic generation 18,19 , as well as interesting photogalvanic effects [20][21][22][23][24] . However, neither response can be directly attributed to the topological charge of a single band crossing, since mirror symmetry-present in most known Weyl semimetals-imposes that charges with opposite sign lie at the same energy and thus contribute equally 19,25 . This is similar for other types of Weyl semimetal materials, such as type-II Weyl semimetals 26 . Type-II Weyl semimetals display open Fermi surfaces to lowest order in momentum [26][27][28][29][30][31][32] , giving rise to remarkable photogalvanic effects [33][34][35][36][37] , but not directly linked to their topological charge.
Materials with even lower symmetry can hold the key to measuring the topological charge directly. Chiral topological metals do not possess any inversion or mirror symmetries [38][39][40][41] , and as a result, the topological band crossings do not only occur at different momenta but also at different energies, making them accessible to external probes. Notably, the circular photogalvanic effect (CPGE), i.e., the part of the photocurrent that reverses sign with the sense of polarization, was predicted to be quantized in chiral Weyl semimetals 42 . However, chiral Weyl semimetals with sizable Weyl node separations, such as SrSi 2 43 , have not been synthesized as single crystals.
Recently, a new class of chiral single crystals has emerged as a promising venue for studying topologi-cal semimetallic behavior deriving from topological band crossings. Following a theoretical prediction [38][39][40][41]44,45 , experimental evidence provided proof that a family of silicides hosts topological band crossings with nonzero topological charge at which more than two bands meet. Such band crossings, known as multifold nodes, may be viewed as generalizations of Weyl points and are enforced by crystal symmetries.
The prediction of a quantized CPGE was extended to materials in this class, specifically to RhSi in space group 198 44,46,47 . These materials display a protected three-band crossing of topological charge 2, known as a threefold fermion, at the Γ point, and a protected double Weyl node of opposite topological charge at the R point. In RhSi, theory predicts that below 0.7 eV, only the Γ point is excited, resulting in a CPGE plateau when the chemical potential is above the threefold node 44,46,47 . Above 0.7 eV, the R point contribution of opposite charge compensates it, resulting in a vanishing CPGE at large frequencies 44,46,47 . The predicted energy dependence above 0.5 eV is qualitatively consistent with a recent experiment in RhSi performed within photon energies ranging between 0.5 eV to 1.1 eV, but with a non-quantized plateau 48 .
Despite this progress, the challenge is to find if and how quantization can be observed in practice in these materials, since multiple effects can conspire to destroy it. Firstly, a finer analysis and density functional theory calculations 46,47 indicates that, unlike in the case of Weyl nodes, quadratic corrections can spoil quantization beyond the linear dispersion regime in multifold fermion materials. Secondly, a short hot-carrier scattering time will rapidly relax the quantized CPGE into a nonuniversal value dependent on this scattering time 42,49 . Without precise knowledge of this scattering time, extracting the universal quantized value is a challenging task. To this end, a precise knowledge of the linear optical conductivity can be used to estimate the relative size of the hot-carrier scattering time, and reveal the energy range where the multifold fermions dominate optical transitions.
In this work we report the measurement of the linear and non-linear response of RhSi, analyzed by different theoretical models of increasing complexity to provide a consistent picture of how quantization can be observed. We performed optical conductivity measurements from 4 meV to 6 eV and 10 K to 300 K, as well as terahertz (THz) emission spectroscopy with incident photon energy from 0.2 eV to 1.1 eV at 300 K. Our optical conductivity measurements, combined with tight-binding and ab-initio calculations, show that interband transitions 0.4 eV are mainly dominated by the vertical transitions at the Brillouin zone center, the Γ point. We found that the transport lifetime is relatively short in RhSi, ≈ 13 fs for the hole pocket at the Γ point and ≈ 80-160 fs for the electron pocket at the R point. The measured CPGE response shows a sign change and no clear plateau. Our optical conductivity and CPGE experiments are qualitatively reproduced by tight-binding and first-principle calculations when the chemical potential lies below the threefold node at the Γ point, crossing a relatively flat band, and when the hot-electron lifetime is chosen to be ≈ 4-7 fs. These observations are behind the absence of quantization. Our broad analysis indicates that a quantized CPGE could be observed by increasing the electronic doping by 100 meV with respect to the chemical potential in the current generation of samples 48,50,51 , if it is accompanied by an improvement in the sample quality that can significantly increase the hot carrier lifetime. Our methodology can be applied to other multifold materials in the same spacegroup, and thus serves as a benchmark for future studies.

Results and Discussion
Optical conductivity measurement The measured frequency-dependent reflectivity R(ω) by a FTIR spectrometer (see methods) is shown in Fig. 1(a) in the frequency range from 0 to 8 000 cm −1 for several selected temperatures. R(ω) at room temperature is shown over a much larger range up to 50 000 cm −1 in the inset. In the low-frequency range, R(ω) is rather high and has a R = 1 − A √ ω response characteristic of a metal in the Hagen-Rubens regime. Around 2 000 cm −1 a temperature-independent plasma frequency is observed in the reflectivity. For ω > 8000 cm −1 the reflectivity is approximately temperature independent.
The results of the Kramers-Kronig analysis of R(ω) are shown in Figs. 1(b) and 1(c) in terms of the real part of the dielectric function ε 1 (ω) and the real part of optical conductivity σ 1 (ω). At low frequencies, ε 1 (ω) is negative, a defining property of a metal. With increasing photon energy ω, ε 1 (ω) crosses zero around 1 600 cm −1 and reaches values up to 33 around 4 000 cm −1 . The crossing point, where ε 1 (ω) = 0, is related to the screened plasma frequency ω scr p of free carriers. As shown by the inset of Fig. 1(b), ω scr p is almost temperature independent. Similar temperature dependence and values of ω scr p have been recently reported in another work for RhSi 51 , indicating similar large carrier densities and small transport lifetime in naturally grown samples. Fig. 1(c) shows the temperature dependence of σ 1 (ω) for RhSi. Overall, σ 1 (ω) is dominated by a narrow Dudelike peak in the far-infrared region, followed by a relativity flat tail in the frequency region between 1 000 and 3 500 cm −1 . As the temperature decreases, the Drudelike peak narrows with a concomitant increase of the low-frequency optical conductivity. In addition, the inset shows the σ 1 (ω) spectrum at room temperature over the entire measurement range, in which the high-frequency σ 1 (ω) is dominated by two interband transition peaks around 8 000 cm −1 and 20 000 cm −1 .
To perform a quantitative analysis of the optical data at low frequencies, we fit the σ 1 (ω) spectra with a Drude-  (color online) (a) Optical conductivity spectrum of RhSi up to 12 000 cm −1 (1.5 eV) at 10 K. The thin red line through the data is the Drude-Lorentz fitting result, which consists of the contributions from a narrow Drude peak (green line), a broad Drude peak (orange line) and several Lorentz terms that accounts for the phonons (gray) and the interband transitions (light blue, magenta, and wine lines). Temperature dependence of (b) the plasma frequency Ωp,D and (c) the transport scattering rate 1/τD of the Drude terms. (d) Optical conductivity spectrum of RhSi at 10 K, and the corresponding spectrum after the Drude response and the sharp phonon modes have been subtracted. Black dashed lines are eye guidance for different quasi-linear regimes.

Lorentz model
where Z 0 is the vacuum impedance. The first sum of Drude-terms describes the response of the itinerant carriers in the different bands that are crossing the Fermilevel, each characterized by a plasma frequency Ω pD,j and a transport scattering rate 1/τ D,j . The second term contains a sum of Lorentz oscillators, each with a different resonance frequency ω 0,k , a line width γ k , and an oscillator strength S k . The corresponding fit to the conductivity at 10 K (thick blue line) using the function of Eq. (1) (red line) is shown in Fig. 2(a) up to 12 000 cm −1 . As shown by the thin colored lines, the fitting curve is composed of two Drude terms with small and large transport scattering rates, respectively, and several Lorentz terms that account for the phonons at low energy and the interband transitions at higher energy. Fits of the σ 1 (ω) curves at all the measured temperatures return the temperature dependence of the fitting parameters. Figure 2(b) shows the temperature dependence of the plasma frequencies Ω p,D of the two Drude terms, which remain constant within the error bar of the measurement, indicating that the band structure hardly changes with temperature. Figure 2(c) displays the temperature dependence of the corresponding transport scattering rates 1/τ D of the two Drude terms. The transport scattering rate of the broad Drude term remains temperature independent, while that of the narrow Drude decreases at low temperature. Note that the temperature dependence of the Drude responses appears to be much stronger than in Ref. 51 probably due to the difference of natural facets in our study versus polished surface in Ref. 51 as also found in previous studies on TaAs 52,53 .
The need for two Drude terms indicates that RhSi has two types of charge carriers with very different transport scattering rates. Such a two-Drude fit is often used to describe the optical response of multiband systems. A prominent example of such multiband materials are the iron-based superconductors for which the narrow Drude-peak is assigned to the electron-like bands around the M point of the Brillouin zone, while the broad peak is assigned to the hole-like bands at the Γ point [54][55][56] . As we discuss below, in the case of RhSi two main pockets are expected to cross the Fermi level, centered around the Γ (heavy hole pocket) and the R point (electron pocket) 44,45 . Note that there might be a small hole pocket at the M point as well. Accordingly, the two-Drude fit can be assigned to the intraband response around the Γ (broad Drude term) and R (narrow Drude term) points of the Brillouin zone. A third Drude peak for the pocked at M could be included but its contribution must be very small.
Having examined the evolution of the two-Drude response with temperature, we next investigate the σ 1 (ω) spectrum associated with interband transitions. To single out this contribution, we show in Fig. 2(c) the σ 1 (ω) spectra, after subtracting the two-Drude response and the sharp phonon modes. With the subtraction of two Drude peaks with transport scattering rates of 200 cm −1 and 2 400 cm −1 (Drude fit 1 in Fig. 2(c)), we reveal a linear behavior of σ 1 (ω) in the low-frequency regime (up to about 3 500 cm −1 ). Such behavior is a strong indication for the presence of three-dimensional linearly dispersing bands near the Fermi level 57 . Indeed, from band structure calculations (see Fig. 3), we see that this lowenergy ω-linear interband conductivity (ω < 3 500 cm −1 ) could be attributed to the interband transitions around the Γ point. At higher energy, the interband contributions around the R point become allowed and can be responsible for the second ω-linear interband conductivity region (3 500 cm −1 < ω < 6 500 cm −1 ). At ω > 6 500 cm −1 , the optical conductivity flattens and forms a broad maximum around 8 000 cm −1 . From Fig. 2(a) we see that this maximum is a consequence of the Lorentian peak around 0.85 eV (light blue) and around 1.1 eV (magenta). As analyzed by density-functional theory below, the peak around 0.85 eV is most likely attributed to interband transitions centered at the M point, which was previously systematically studied in CoSi 58 .
Before analyzing these further, it is important to note that the fit to the broader Drude peak suffers from more uncertainty than that of the narrow Drude peak. Small changes in its width result in appreciable changes when subtracting it from the full data set to obtain the interband response. By subtracting the broad Drude peak this time with a smaller transport scatter rate of 1 350 cm −1 (Drude fit 2 in Fig. 2(c)), the onset frequency at which the interband conductivity emerges decreases and the magnitude of σ 1 below 4 000 cm −1 increases, with respect to the Drude fit 1. However, the resulting linear slope below 3 500 cm −1 is not significantly modified as the wide Drude response contributes as a flat background in this regime. As a consequence of this analysis, we expect the true optical conductivity to lie between that resulting from our Drude Fit 1 and 2, which we will take into account as lower and upper bounds. Nevertheless, the slope of the low-frequency conductivity is the most important feature that we will be analyzed next in our theoretical modeling.
Linear optical conductivity calculation To gather insight on the low-frequency interband optical conductivity of RhSi, we now put to test the predictions based on a low-energy linearized model around the Γ point, a fourband tight-binding model, and ab-initio calculations (see Materials and Methods).
The band structure of RhSi calculated using Density Functional Theory (DFT) is shown in Fig. 3. Without spin-orbit coupling, three low-energy bands cross at Γ forming a protected threefold crossing (see Fig. 3(b)) that splits, upon the addition of spin-orbit coupling into a fourfold node and and a Weyl node at this high symmetry point ( Fig. 3(a)). The band structure calculations of Fig. 3 suggest that at low frequencies, ω 0.4 eV, the optical conductivity is dominated by interband transitions close to the Γ point. To linear order in momentum, the middle band at Γ is flat (see Fig. 3(b)), and the only parameter is the Fermi velocity of the upper and lower dispersing bands v F . To match previous conventions 44,57 , we define v F = v p /2, that we set by fitting the band structure shown in Fig. 3(b), resulting in v p = 0.77 eV. Given the Fermi velocity, the optical conductivity of a threefold fermion is σ 3f 1 = 4σ W , written in terms of the conductivity of a single Weyl node σ W = g s e 2 ω/(24π v F ) which is linear in ω and accounts for spin degeneracy through g s = 2 57 .
The position of the chemical potential in Fig. 3 indicates that the deviations from linearity of the central band at Γ can play a significant role in optical transitions. To include them we use a four-band tight-binding lattice model that incorporates the lattice symmetries of space-group 198 44,46 (see Materials and Methods for details). Without spin-orbit coupling, an assumption that we justify later on, this model is completely determined by four-material dependent parameters that are set by a fit to the band-structure, and specifying the real space positions of the atoms in RhSi 46 (see Materials and Meth-  ods for details). The resulting tight-binding band structure is shown in Fig. 4(a). The horizontal dashed lines signal different trial chemical potentials discussed next.
In Fig. 4(b) we compare the experimental optical conductivity in the interval ω ∈ [0, 0.7] eV, obtained by subtracting the Drude fit 2, with the linear threefold node conductivity σ 3f 1 (solid gray line), and the tight-binding model with different chemical potentials. The low energy σ 3f 1 shows a smaller average slope compared to the low-energy part of our experimental data. In addition, a purely linear conductivity is insufficient to describe a small shoulder at around 200 meV (see a more zoom-in data in Fig. 5(a) compared to the linear guide to the eye).
These deviations suggest that the chemical potential is below the threefold node, as in Fig. 3(a) and (b). Indeed, by sweeping the chemical potential across the node (see Fig. 4(a) and (b)), and without yet including a hotcarrier scattering time τ , we observe that if the chemical potential is below the threefold node (see µ = 0, −100 meV curves in Fig. 4 (b)), a peak appears (around 200 meV for µ = −100 meV), followed by a decrease in the optical conductivity at larger frequencies, before the activation of the transitions centered at the R point.
The decrease and increase of the optical conductivity is a pattern that can be traced back to features in the joint density of states (JDOS). This quantity exhibits an increase and decrease of the number of allowed optical transitions to and from the central threefold band. This happens when the allowed transitions switch from connecting the lower and middle bands to connecting the middle and upper bands at the Γ point 58 . These transitions are allowed by quadratic corrections and are absent in the linear low energy model. They are also absent if the doping level is above the threefold node since the middle and lower threefold bands would be completely filled. In such case the tight-binding model recovers the linear optical conductivity of a threefold fermion (see µ = 100 meV curve in Fig. 4(b)). Lastly, we observe that as the Energy (eV) chemical potential is lowered the R point activates at lower frequencies.
Although placing the chemical potential below the Γ node results in a marked peak around 0.2 eV, it is clearly sharper, and overshoots compared to the data, for which the sudden drop at frequencies above the peak is also absent. It is likely that this drop is masked by the finite and relatively larger disorder related broadening η = /τ . This scale is expected to be large for RhSi given the broad nature of the low-energy Drude peaks. Note that τ is the hot-electron lifetime, which is different from, but is usually bounded by, the transport lifetime τ D estimated from the Drude peaks. In Fig. 4(c) we compare different hot-carrier scattering times for µ = −100 meV. Upon increasing η the sharp features in Fig. 4(b) are broadened, turning the sharp peak into a shoulder, which was observed in the experimental data. When η = 100 − 150 meV the resulting optical conductivity falls close to our experimental data, including the upturn at 0.4 eV, associated to the broadened transitions around the R point.
The solid line in Fig. 4(c) shows the best agreement to the experimental optical conductivity data, indicating that the chemical potential is situated below the node, that the R point transitions are activated around 0.4 eV, and that the broadening scale η ≈ 100 − 150 meV is relatively large. The latter scale is of the order, if not larger, than the spin-orbit coupling energy scale, that we can estimate from the splitting between the bands at the Γ point in Fig. 3(a), of ≈ 100 meV. The large disorder scale washes out any feature narrower than 100 meV and justifies our discussion based on a tight-binding model without spin-orbit coupling.
We note that despite the general agreement, the tight-binding calculations deviate from the data at frequencies above 0.5 eV. This is likely due to the tight-binding model's known limitations, which fails to accurately capture the band structure curvature and orbital character at other high-symmetry points such as the M point. These limitations will also play a role in our discussion of the CPGE.
To improve on these aspects and examine further the role of spin-orbit coupling, we have used the DFT method to calculate the optical conductivity, including spin-orbit coupling on a wider frequency range up to ω = 4 eV. The optical conductivity we obtained is compared to our data in Fig. 5(a) for a wide range of frequencies, and in Fig.5(b) within a low energy frequency window. The smaller broadening factors, η = 10, 50 meV, reveal fine features due to spin-orbit coupling such as the peaks at low energies, due to the spin-orbit splitting of the Γ and R points. These are smoothened as the broadening is increased (see Fig. 5(b)), consistent with our discussion above. As shown in Fig. 5(a), the low-energy conductivity below 0.2 eV is better explained if the chemical potential is at -30 meV. The dip at around 0.5 eV, seen in Fig. 5(b) for low broadening, is filled with spectral weight as broadening is increased, consistent with our tight-binding calculations. In the low frequency window both the tight-binding (see Fig. 4(c)) and DFT results (see Fig. 5(a)) agree qualitatively. The quantitative differences are expected, due to the different curvatures of the bands of the DFT compared to the tight-binding calculations, exposing the limitations of the latter calculation.
At higher energies the DFT calculation recovers a peak at ω ≈ 1 eV, seen also in our data (see Fig. 5(b)). If the broadening is small (η=10 meV), the peak is sharper compared to the measurement and exhibits a double feature with a shoulder around 0.85 eV and a sharp peak around 1.08 eV . Although smoothened by disordered, this double feature is consistent with the need of two Lorentzian functions (Lorentz 1 and Lorentz 2 in Fig. 2(a)) to model this peak phenomenologically with Eq. (1). Fig. 3(a) shows that the excitation energy at the M point is around 0.7 eV. In a recent work on CoSi, in the same space group 198 as RhSi, a momentum-resolved study shows that the dominant contribution of this Lorentzian function (Lorentz 1) arises from the saddle point at M 58 . The peak around 1.1 eV could arise from another van Hove singularity with a gap size larger than that at the M point. Lastly, the measured peak at ω = 2.5 eV that comes from van Hove singularities with even higher energy is also recovered by the DFT calculation, but the magnitude differs by 15%, due to inaccuracy of the DFT to capture higher conduction bands.
Overall, the curves with broadening factor η = 100 meV in both the tight-binding and DFT calculations show a good qualitative agreement with the experimentally measured curve in a wide frequency range, especially the conductivity below 0.2 eV by the DFT and below 0.6 eV by the tight-binding analysis. This observation determines approximately the hot-electron lifetime in RhSi to be τ = /η ≈ 6.6 fs.

CPGE measurement
As a first step of CPGE experiment, we determined the high symmetry axes, [0,0,1] and [1,-1,0] directions of the RhSi (110) sample respectively by second harmonic generation (SHG). To stimulate SHG, pulses of 800 nm wavelength were focused at near-normal incidence to a spot with a 10µm diameter on the natural (110) facet. Fig. 6(a) shows the polar patterns of SHG as a a function of the polarization of the linear light in the co-rotating parallel-polarizer (red) and crossed-polarizer (blue) configurations 18,19 . The solid lines are the fit constrained by the point group symmetry with only one nonzero term χ (2) xyz and the angle dependence are: where θ is the angle between the polarization of the incident light and the [1,-1,0] axis. Next, we perform THz emission spectroscopy to measure the longitudinal CPGE in RhSi. An ultrafast circularly-polarized laser pulse is incident on the sample at 45 degrees to generate a transient photocurrent. Due to the longitudinal direction of the CPGE, the transient current flows along the light propogation direction inside RhSi, and therefore, in the incident plane 42,44,46 . The THz electric fields radiated by the time-dependent photocurrent is collected and measured by a standard electro-optical sampling method with a ZnTe detector 59 . The component in the incident plane is measured by placing a THz wire-grid polarizer before the detector. Fig. 6(b) shows a typical response of the component of emitted THz pulses in the incident plane under lefthanded and right-handed circularly polarized light at an incident energy of 0.425 eV. The nearly opposite curves demonstrate the dominating CPGE contribution to the photocurrent with an almost vanishing linear photogalvanic effect (LPGE) at this incident energy. The CPGE contribution can be extracted by taking the difference between the two emitted THz pulses under circularly polarized light with opposite helicity. During our measurement, the [0 0 1] axis is kept horizontally, even though the CPGE signal does not depend on the crystal orientation due to the cubic crystalline structure.
In order to measure the amplitude of the CPGE photocurrent, we use a motorized delay stage to move a standard candle ZnTe at the same position to perform the THz emission experiment right after measuring RhSi for every photon energy between 0.2 eV to 1.1 eV 60 . ZnTe is a good benchmark due to its relatively flat frequency dependence on the electric-optical sampling coefficient for photon energy below the gap 61 . Its use circumvents assumptions regarding the incident pulse length, the wavelength dependent focus spot size on the sample, and the calculation of collection efficiency of the off-axis parabolic mirrors. A spectrum of CPGE photo-conductivity as a function of incident photon energy is shown in Fig. 6(c) in units of µA/V 2 (squares). Upon decreasing the incident photon energy from 1.1 eV to 0.7 eV, we observe a rapid increase of CPGE response with a peak value of 163 (±19) µA/V 2 at 0.7 eV. The features of this line resemble those observed in Ref. 48. Further decreasing the photon energy from 0.7 eV to 0.2 eV, the CPGE conductivity displays a sharp drop with a striking sign change at 0.4 eV, which was not seen in a previous study as the lowest photon energy measured was around 0.5 eV 48 . Interestingly, the peak photoconductivity at 0.7 eV is much larger than the photo-galvanic effect in BaTiO 3 62 , single-layer monochalcogenides 62,63 other chiral crystals 64 and it is comparable to the colossal bulk-photovoltaic response in TaAs 22 . It is also one order of magnitude larger than the previous study on RhSi 48 probably due to the fact that the carrier lifetime is around 10 times larger as the Drude transport lifetime in our sample is around 10 times larger than the one in Ref. 48. Interestingly, the sign change at 0.4 eV was not predicted in previous theory studies either 44,46,47 . The quantized CPGE below 0.7 eV predicted in RhSi in previous theory studies is completely absent 44,46,47 .

CPGE calculation
In order to understand the absense of quantized CPGE and the origin of the sign change in our CPGE data, we have calculated the CPGE response, β ij , using the tight-binding model discussed earlier, and using a first-principle calculation via FPLO (full-potential local-orbital minimum-basis) 65,66 (See Materials and Methods for details). Due to cubic symmetry, the only finite CPGE component is β xx 46 . In our conventions, the tensor β ij determines the photocur-  at T = 0 K, and a broadening parameter η = 10 meV. While µ = 0 and -30 meV sit below the Γ threefold node (see Fig. 3(a) and (b)), the two remaining chemical potentials lie above the Γ node. rent rate. When the hot-electron lifetime τ is short compared to the pulse width, and τ is a constant as a function of energy, the total photocurrent is given β xx τ 47 .
In Fig. 6(c) we plot the only independent non-zero component for the CPGE in this space group, β xx , multiplied by the hot-carrier scattering time corresponding to a broadening η = /τ = 100 meV (τ ≈ 6.6 fs) calculated using the four-band tight-binding model and the DFT ( µ = −30 meV, T = 300 K). The DFT calculation captures quantitatively the features seen in the CPGE data: the existence of a peak at around 0.7 eV, its broadening provided we choose η = 100 meV, and the sign change of the response. Together they support the conclusion that the chemical potential lies below the Γ node (see Fig. 3(a)), consistent with the features of the optical conductivity.
It is illustrative to compare these results especially the striking sign change with tight-binding calculations for the CPGE based on the four-band model introduced earlier as the latter is the simplest model to capture both the multifold fermions at the Γ and R points. Following Ref. 46 we computed the CPGE with η = 100 meV and µ = −100 meV (Fig. 6(c) orange line). For a chemical potential and broadening that match the optical conductivity (see Fig. 4(c), solid line) it underestimates the position of the peak and the overall magnitude of the CPGE. This is mainly attributed to the failure of the tight-binding to capture the curvature of the bands, especially the central flat band at Γ, but also the dispersion around the M point. However, it shows similar features to both the data and the DFT, most notably the overall peak-dip structure of the response and its sign change around 0.4 eV. By lowering the chemical potential, the tight-binding result can be made to match the DFT and the data, paying the price that the optical conductivity will no longer be reproduced. The combined analysis of both linear and nonlinear responses illustrates the crucial role played by the curvature of the flat-band at the Γ point and the saddle point at M . These are accurately captured by DFT, which matches both responses over a broad frequency range, but not by the tight-binding model. Nevertheless, the analysis of the tight-binding CPGE and optical conductivity spectrum together shows that the CPGE sign change occurs when the allowed excitations switch between those around the threefold node at Γ, to those around the double Weyl node at R .
Despite the reasonable agreement, the remaining differences between the data and the calculations suggests that a constant, energy independent hot-carrier scattering time τ is an oversimplified phenomenological model for the disorder. In general, the hot-carrier scattering time is energy and momentum dependent, a functional dependence that can enter the CPGE and affect its magnitude 49 . Including these effects might help bridge the gap between the magnitude of the calculated and experimentally observed CPGE around the peak and at low energy.
We end by discussing the possibility of observing a quantized CPGE in this sample. In Fig. 6(d) we show the effect of changing the chemical potential on the DFT calculated CPGE tensor trace β = Tr[β ij ] in units of the quantization constant β 0 = πe 3 /h 2 . For an ideal multifold fermion, and taking into account spin-degeneracy, the CPGE is expected to be quantized to a Chern number of four, as β = 4iβ 0 , corresponding to the charge of the node 46 , C = 2, times the spindegeneracy. Upon decreasing the broadening to 10 meV and changing the Fermi level by 50 -100 meV compared to the chemical potential found in our DFT calculations a close-to-quantized CPGE frequency window emerges for ω ∈ (0.3, 0.65) eV 47 , shown as light-blue and purple lines in Fig. 6(d). We note that when the chemical potential is above the nodes, there is no sign change below 0.7 eV.

Conclusions
Using a broad set of theoretical models applied to interpret the linear and nonlinear optical responses we have established a consistent picture of the optical transitions in RhSi. Our data is explained if the chemical potential crosses a large hole-like band at Γ, and with a relatively short hot-electron lifetime ≈ 4.4 − 6.6 fs.
We arrived to these conclusions by first fitting the freecarrier contribution in the linear conductivity with two Drude peaks: a narrow Drude peak of 200 cm −1 width at low temperatures, and a broad Drude peak of width in the interval 1300-2 400 cm −1 . Subtracting these from the optical conductivity revealed the interband conduc-tivity, which shows two quasi-linear regions where the conductivity increases smoothly with frequency and a slope change around 0.4 eV. The slope in the first region is determined by a disorder broadened Γ point contribution associated with a threefold fermion, with the chemical potential lying below the node, crossing a large hole-like band. The slope in the second region is determined by the onset of a broadened R point conductivity. We captured the features of the optical conductivity by a disorder broadening, η ∼ 100 − 150 meV, equivalent to a hot-electron lifetime of τ ∼ 4.4 − 6.6 fs.
Assuming the same chemical potential and broadening our calculations also reproduces the observed circular photogalvanic effect, including its sign change close to ω ∼ 0.4 eV, and its a non-quantized peak at ≈ 0.7 eV. The magnitude of the CPGE response is approximately captured by our ab-initio calculations for a wide range of frequencies, but not by our tight-binding calculations, which fail to capture adequately the curvature of the abinitio band structure. The observed differences between the measured and calculated CPGE suggest the need for a more elaborate, frequency-dependent model of disorder scattering. Lastly, our calculations suggest that by electron-doping RhSi by ≈ 100 meV a close to quantized value could be observed, if the hot-carrier scattering time is significantly increased.
In conclusion, our analysis provides a consistent picture of the optical response of RhSi, with a short hot-electron lifetime, similar to recent observations in the 0.5-1 eV energy range 48 , and a chemical potential that lies below the threefold node at the Γ point, different from previous studies 48,51 .. These two observations hinder the measurement of an exactly quantized CPGE in RhSi. Nonetheless, our theoretical results indicate that electron doping of cleaner samples can result in the observation of an extended, close-to-quantized, CPGE region. Our systematic methodology can be applied to other chiral multifold semimetals in space gorup 198, like CoSi, AlPt or PdGa which share a similar band structure similar to RhSi, as well as other materials in the related space-groups 212 and 213 46 . We expect our work to aid future studies that investigate the CPGE quantization in chiral topological metals.

Materials and Methods
Crystal growth The high-quality single crystal of RhSi was grown by the Bridgeman method 50 . 2 mm×5 mm large RhSi with a (110) natural facet is used in this study.
Optical conductivity measurement The in-plane reflectivity R(ω) was measured at a near-normal angle of incidence using a Bruker VERTEX 70v FTIR spectrometer with an in situ gold overfilling technique 67 . Data from 30 to 12 000 cm −1 ( 4 meV to 1.5 eV) were collected at different temperatures from 10 to 300 K with a ARS-Helitran cryostat. The optical response function in the near-infrared to the ultraviolet range (4 000 -50 000 cm −1 ) was extended by a commercial ellipsome-ter (Woollam VASE) in order to obtain more accurate results for the Kramers-Kronig analysis of R(ω) 68 .
THz emission experiment The THz emission experiment is performed at dry air environment with relative humidity less than 3% at room temperature. An ultrafast laser pulse is incident on the sample at 45 degrees to the surface normal and is focused down to 1 mm 2 to induce THz emission. The THz pulse is focused by a pair of 3-inch off-axis parabolic mirrors on the ZnTe (110) detector. The temporal THz electric field can be directly measured with a gated probe pulse of 1.55 eV and 35 fs duration 59 . The polarization of incident light is controlled by either a near-infrared achromatic or a midinfrared quarter-wave plate. A wire-grid THz polarizer is utilized to pick out the THz electric field component in the incident plane. The photon energy of the incident light is tunable from 0.2 eV to 1.1 eV by an optical parametric amplifier and difference frequency generation. Pulse energy of 12 µJ is used for 0.4-1.1 eV and 6 µJ is used for 0.2-0.4 eV.
Four-band tight-binding model In the main text we use a four-band tight-binding model introduced in Ref. 44 and further expanded in Ref. 46. Since this model has been extensively studied before, we briefly review its main features relevant to this work. Our notation and the model is detailed in appendix F of Ref. 57.
Without spin-orbit coupling, an assumption that is justified in the main text, the four-band tight-binding model is determined by three-material dependent parameters, v 1 , v p , and v 2 . By fitting the band structure in Fig 3(b) we set v 1 = 1.95, v p = 0.77 and v 2 = 0.4. Our DFT fits deliver parameters that differ from those obtained in Ref. 44, subsequently used in Ref. 57. Our new parameters result in a better agreement to the observed optical conductivity and circular photogalvanic effect data. Additionally, we rigidly shift the zero of energies of the tight-binding model by 0.78 meV with respect to Ref. 57 to facilitate comparison with the DFT calculation. As described in Ref. 46 we additionally incorporate the orbital embedding in this model, which takes into account the position of the atoms in the unit cell. It amounts to a unitary transformation of the Hamiltonian which depends on the atomic coordinates through a material dependent parameter x, where x RhSi = 0.3959 for RhSi 46,57 .
Details of the linear optical conductivity calculations Since RhSi crystallizes in a cubic lattice it is sufficient to calculate one component of the real part of the longitudinal optical conductivity, σ 1 (ω), given by the expression 69 defined for a system of volume V described by a Hamiltonian H with eigenvalues and eigenvectors n and |n , respectively, and a velocity matrix element v x nm = 1 n| ∂ ki H |m . The chemical potential µ and temperature T enter through the difference in Fermi functions f nm = f n − f m , and we define nm = n − m . We have replaced the sharp Dirac delta function that governs the allowed transitions by a Lorentzian distribution L τ (ω) = 1 π η ω 2 +η 2 to phenomenologically incorporate disorder with a constant hot-carrier scattering time τ = /η.
The DFT calculations of the optical conductivity are also performed via Eq. (4), with the ab-initio tight binding Hamiltonian constructed from DFT calculations. We use a dense 300 × 300 × 300 momentum grid for the small broadening factor η = 10 meV, and a 200 × 200 × 200 momentum grid for broadenings η ≥ 50 meV.
Details of the CPGE calculations For our abinitio CPGE calculations we projected the ab-initio DFT Bloch wave function into atomic-orbital-like Wannier functions 70 . To ensure the accuracy of the Wannier projection, we have included the outermost d, s, and p orbitals for transition metals (4d, 5s, and 5p orbitals for Rh) and the outermost s and p orbitals for main-group elements. Based on the highly symmetric Wannier functions, we constructed an effective tight-binding model Hamiltonian and calculated the CPGE evaluating 47,69 using a dense 480 × 480 × 480 momentum grid. Here ∆ a mn ≡ ∂ ka mn / , and r a mn ≡ i m|∂ ka n = v nm /i nm is the interband transition matrix element or off-diagonal Berry connection. As for the linear optical conductivity, the chemical potential and temperature are considered via the Fermi-Dirac distribution f n , and the Lorentzian function L τ (ω) accounts for a finite hot-carrier scattering time.
Forschungsgemeinschaft (Project-ID 258499086 and FE 63330-1). This research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The DFT calculations were carried out on the Draco cluster of MPCDF, Max Planck society.