Introduction

The past decade has vividly witnessed fundamental discoveries of relativistic phenomena in a large variety of novel solid-state topological materials1,2,3. Astonishing physical properties are anticipated in the Weyl semimetals, for which their topology derives from the split of the doubly degenerate, massless Dirac fermions because of broken time-reversal (TR) or space-inversion (SI) symmetry or even a combination of both4,5,6.

Lately, the research focus moves to the crucial link between magnetism and topology7. Early evidence for magnetic Weyl semimetals encounters for instance the non-collinear antiferromagnet Mn3Sn8 and the Kagomé ferromagnetic (FM) Co3Sn2S29, both being centrosymmetric. The noncentrosymmetric RAlGe compounds (R = rare earth) recently advance as a new family of magnetic materials paving the way to intriguing quantum states and hosting emergent topological properties of Weyl fermions10. Of particular interest is PrAlGe (with Pr 4f2 orbitals), which exhibits a FM order below TC ~ 15 K arising from the coupling between the f-electrons local moments10,11,12,13,14. The latter serve as an effective Zeeman field, making the conduction bands (s, p and d orbitals) spin-polarised10. The resulting broken TR symmetry displaces the Weyl nodes in the reciprocal space, therefore ensuring, together with the energy displacement due to the broken SI symmetry, the lift of the degeneracy at the original Dirac states. Angle-resolved-photoemission-spectroscopy (ARPES) experiments reveal the topological semimetallic nature of PrAlGe and highlight the linear energy-dispersion at its Weyl nodes11. The alliance between topology and magnetism renders PrAlGe even a suitable candidate for fundamental studies in view of innovative applications, as when designing spintronic devices15,16.

The prediction from first-principles calculations10 that the Weyl fermions in PrAlGe are in proximity of the Fermi level (EF) and its established large anomalous Hall conductivity (AHC) below TC11,12 have a significant implication: the enhanced Berry curvature fields associated with those Weyl nodes and concomitant with the broken TR symmetry may deem necessary to generate a large, intrinsic (topological) anomalous Hall effect17. PrAlGe is thus an ideal playground in order to chase the elusive ingredients framing the relationship between topology and peculiar transport properties.

By exploring the temperature (T) dependence of the real part (σ1(ω)) of the optical conductivity18, achieved over an extremely broad spectral range and combined with a dedicated first-principles calculation, we ultimately capture the spectroscopic hallmark of the intrinsic anomalous Hall effect in PrAlGe. Henceforth, this is of significance even from a methodological point of view, since our mode of operation may represent a straightforward, broadband yet alternative approach to magneto-optical methods19,20,21,22,23, when keeping track of the impact of the electronic band structure on the anomalous transport properties across a magnetic transition.

Results

Optical conductivity

We commence our data survey with Fig. 1, which displays the T dependence of σ1(ω) below 3000 cm−1. Its inset compares the spectra at 5 and 300 K over the whole measured spectral range. We refer to Methods and to the Supplementary Note 3 for more details. Upon increasing the frequency from zero, we first observe the metallic-like (Drude) zero-energy mode, followed by a rather broad absorption at far-infrared (FIR) energy scales. These features undergo the most prominent T dependence: both narrow with decreasing T and upon crossing the FM transition the high frequency shoulder of the FIR absorption gets suppressed, mainly leading to a cusp-like feature (peaked at about 500 cm−1) and to an accumulation of spectral weight (SW) (i.e., SW ~ ∫σ1(ω)dω, see Supplementary Note 4) at FIR energies. We will later argue that this adumbrated SW redistribution across TC is remarkably univocal, when compared to the situation in the non-magnetic LaAlGe. A smooth SW removal with decreasing T is then observed in the mid-infrared energy interval (i.e., 1500–3000 cm−1), with accompanying SW redistribution at higher energies, up to the near-infrared spectral range (i.e., 4000–10,000 cm−1, inset of Fig. 1). The σ1(ω) spectra at all T finally merge at ultra-violet energies (inset of Fig. 1) into the T-independent absorptions, which are almost identical in all RAlGe compounds24.

Fig. 1: Optical conductivity of PrAlGe.
figure 1

T dependence of σ1(ω) of PrAlGe below 3000 cm−1 (1 eV = 8.06548 × 103 cm−1). The thick blue dashed line accentuates the intraband \({\sigma }_{1}^{{\rm{Drude}}}(\omega )\) component at 5 K, obtained within the two-Drude approach (i.e., sum of the narrow (D1) and broad (D2) Drude terms, Supplementary Note 4, Supplementary Equation 1 and Supplementary Fig. 3). The same procedure was applied at all T. The inset shows σ1(ω) in the total measured energy interval at 300 and 5 K (please note the logarithmic energy scale).

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In analogy to the analysis of the charge dynamics recently reported for LaAlGe and CeAlGe24, the zero-energy mode in σ1(ω) (i.e., the intraband component, \({\sigma }_{1}^{{\rm{Drude}}}(\omega )\), thick blue dashed line in Fig. 1) of PrAlGe is well accounted for by two Drude terms (Supplementary Note 4, Supplementary Equation 1 and Supplementary Fig. 3). By subtracting \({\sigma }_{1}^{{\rm{Drude}}}(\omega )\) from σ1(ω) at each T, we can better emphasise its interband contribution (i.e., \({{\Delta }}{\sigma }_{1}(\omega )={\sigma }_{1}(\omega )-{\sigma }_{1}^{{\rm{Drude}}}(\omega )\)), shown in the inset of Fig. 2 at 5 K for the FIR energy scales. The resulting Δσ1(ω) comprises a characteristic (single) linear frequency dependence at its low energy tail, which is distinct from the kink feature observed in Δσ1(ω) of LaAlGe and CeAlGe24 and is definitely reminiscent of the typical optical response for type-I Weyl semimetals25. The linear frequency dependence of σ1(ω) is the direct consequence of the linear dispersion relation at the Dirac cones, from which Weyl states originate26. That the Weyl cones located near EF are not tilted (as it is the case for type-II) is also consistent with both theoretical predictions and experimental observations; after ref. 10, one may expect about 32 type-I Weyl nodes and about 8 type-II Weyl ones at energies below 50 meV, even though the latter are not observed in ARPES data11. Moreover, we remark the intercept at finite frequency of the (extrapolated) linear frequency dependence in Δσ1(ω) (thick orange dashed line in the inset of Fig. 2), which indicates a tiny gap (2Δ ~ 2–6 meV) at the Dirac/Weyl nodes, similar to graphene26,27, ZrTe528 and Co3Sn2S229,30. It could be here generated for instance by the reinforced virtue of the spin-orbit coupling (SOC) particularly below TC31 or as a consequence of electronic correlations32.

Fig. 2: Fermi velocity in PrAlGe.
figure 2

T dependence of vF, extracted from Δσ1(ω) ~ ω/vF (see text). The vertical dashed line marks TC ~ 15 K12. The error bars estimation is provided at Supplementary Note 5. The inset displays the interband contribution \({{\Delta }}{\sigma }_{1}(\omega )={\sigma }_{1}(\omega )-{\sigma }_{1}^{{\rm{Drude}}}(\omega )\) at 5 K (\({\sigma }_{1}^{{\rm{Drude}}}(\omega )\) is shown in Fig. 1). The red line is the fit of Δσ1(ω) after Supplementary Equation 2, which merges into a linear frequency dependence (thick orange dashed line) at ω > 2Δ.

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Fermi velocity

We can exploit the steep linear frequency dependence of Δσ1(ω) in order to extract the Fermi velocity (vF). Since the non-trivial bands mostly shape the electronic band structure close to EF10,11, we are confident that a superposition of trivial interband transitions33 is not relevant in PrAlGe and therefore will not invalidate our following analysis. In fact, after Supplementary Equation 2 we can state that the slope of Δσ1(ω) at ω > 2Δ is given by \(\frac{N{G}_{0}}{24{v}_{{\rm{F}}}}\)25,26,27,28,34 (Supplementary Note 5), with G0 being the quantum conductance and N = 32 and 16 the number of type-I (W3 and W4) Weyl points seating close to EF below and above TC, respectively10. The extracted T-dependent vF is then shown in Fig. 2.

With decreasing T > TC, vF tends to be suppressed, similar to the Weyl semimetal YbMnBi235, yet in a less pronounced fashion than in CeAlGe below 100 K24. The low T values of vF in PrAlGe are overall smaller than those in typical weakly interacting Weyl semimetals36,37. Moreover, the T dependence of vF is here opposite to the findings in LaAlGe (without f-electrons), showing a continuous increase, compatible with the T dependence of a two and three dimensional Dirac material within the Hartree–Fock approximation (i.e., vF ~ ln(1/T))24,38. Such a logarithmic divergence of vF(T) is supposed to vanish if correlation effects are taken into account39,40,41,42. Therefore, we advance that the rather flat vF(T) in quite the entire T range above TC and its still suppressed values at low T with respect to the non-correlated LaAlGe material (i.e., with vF = 3.2 × 104 m/s at 5 K24) could hint at the major impact of correlation effects, similar to earlier observations in YbPtBi42 and CeAlGe24 as well as in nodal-line Dirac semimetals43. The renormalisation of vF in PrAlGe, compared to the bare band value supplied by LaAlGe24, also pairs with the strongly reduced kinetic energy estimated from the Drude SW at low T (see details in Supplementary Note 4), another well-established signature of electronic correlations29,44,45. Conversely, the suppressed scattering from magnetic fluctuations in the FM state, as suggested by the drop of the resistivity (Supplementary Fig. 1), causes a renewed, moderate increase of vF across TC (Fig. 2).

Spectral weight and AHC

Before going any further with the data interpretation, we highlight that Δσ1(ω) can be phenomenologically described by a combination of Lorentz (L) harmonic oscillators (HOs in Supplementary Equation 1 and Supplementary Fig. 3), which also help on elucidating the SW redistribution beyond the itinerant components of the charge dynamics. The SW of each HO corresponds to the square of its strength (Ωj) (Supplementary Note 4 and Supplementary Equation 1). We follow the same fit procedure successfully applied for the La and Ce compositions24, which then guarantees a consistent data elaboration and a robust analysis of the T dependence of σ1(ω) across TC, as evinced in Fig. 1. Herewith, we bring out HOs L1 and L2 (Supplementary Fig. 3), which devise the minimal approach of Δσ1(ω) below 0.1 eV and experience a SW reshuffling. In particular at this point, we underscore that HO L1, which forges the low frequency edge of the FIR and Weyl states related absorption in Δσ1(ω), is prone to considerably gain \({\rm{SW}} \sim {{{\Omega }}}_{1}^{2}\) at low T (inset of Fig. 3 and for more insights see discussion of Supplementary Fig. 6 in Supplementary Note 4). Interestingly enough, the sudden increase of \({{{\Omega }}}_{1}^{2}\) starts at the Curie–Weiss temperature TΘ ~ 30 K12, below which the intrinsic AHC (i.e., \({\sigma }_{xy}^{A}\)) is already different from zero (Fig. 3). We speculate that this might signal the relevance of magnetic fluctuations46 for both optical response and transport properties at TC < T < TΘ. The enhancement of SW for HO L1 going hand in hand with the onset of the large \({\sigma }_{xy}^{A}\) in PrAlGe at T ≲ 30 K is not detected in the paramagnetic (PM) LaAlGe (Supplementary Note 4) and is our salient finding. It may allude to the association of the topological resonance with states of intense Berry curvature22,23, upon which we are going to argue for the rest of this paper.

Fig. 3: Spectral weight versus AHC in PrAlGe.
figure 3

T dependence of the squared strength (\({{{\Omega }}}_{1}^{2}\)) of HO L1 (Supplementary Equation 1 and Supplementary Fig. 3) and of the Hall conductivity σxy, reproduced from ref. 12. σxy is read at the magnetic field H = 6 T (H∥c-axis), where it saturates. It is known that \({\sigma }_{xy} \sim {\sigma }_{xy}^{A}\)12. The linear fits (dashed lines) are guide to the eyes. The vertical dashed and dashed-dotted lines mark TC and TΘ12, respectively. The error bars of \({{{\Omega }}}_{1}^{2}\) are estimated numerically within the non-linear least-squares fit technique. Inset: Δσ1(ω) at 5 and 300 K with HO L1, showing its enhanced SW (colour shaded area) at T < TC (see also Supplementary Note 4 and Supplementary Fig. 6).

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Discussion

The first-principles calculation of the electronic band structure is very instrumental, in order to address the origin and consequences of the absorption features in σ1(ω), particularly at the energy scales involving the Weyl states. As detailed in Supplementary Fig. 7, the electronic band structure, achieved in the PM and FM state upon including SOC, agrees with similar investigations10,47 and is broadly confirmed by ARPES11. In the energy interval of about 100 meV around EF, the non-trivial bands have been postulated to accommodate large Berry curvatures10,11, so that changes of their occupation across TC is supposed to determine the magnitude of AHC. Therefore, we primarily look at regions in the Brillouin zone (BZ), where Weyl nodes group together (Fig. 4a, b) and Fermi surface pockets mainly exist47, and consider those cuts in the reciprocal space (i.e., off the high-symmetry directions) with k-paths running nearby representative Weyl nodes (black stars PM1/2 and FM1/2/3 in Fig. 4a, b and Supplementary Table I)10,11. Figure 4c–g then emphasise the evolution of the resulting band structure between the PM and FM state (for details see Supplementary Note 6). The overall band structure calculation sets the stage in order to achieve a reliable estimation of the interband part of σ1(ω) (i.e., Δσ1(ω)) within the Kubo–Greenwood formula (Supplementary Equation 3), which is displayed in Fig. 4h and can be mapped onto its experimental counterpart (Fig. 1 and inset of Figs. 2 and 3). Two features at ~250 and 500 cm−1 (green and red arrows in Fig. 4h) catch our attention in the calculated Δσ1(ω) of the PM state, since they occur at energies very close to the dominant FIR absorption in the experimental Δσ1(ω). By inspecting the electronic band structure, we actually learn that these absorptions may correspond to direct interband transitions (green and red bars in Fig. 4c, d and Supplementary Fig. 8) at k-points in BZ meshing the Weyl nodes10. Into the FM state, the electronic band reconstruction, because of the effects due to the Zeeman splitting, leads to a differentiation of the states occupation near EF and to a new distribution of direct interband transitions (blue bars in Fig. 4e–g). This merges into a broad convolution of absorptions in Δσ1(ω), which strengthen and induce a SW redistribution within the energy interval 130–400 cm−1 (Fig. 4h) below TC. The theoretical Δσ1(ω) (Fig. 4h) fairly copies with the main trend of the experimental quantity at least below 1000 cm−1 (Fig. 1 and inset of Fig. 3). In Supplementary Note 7, we deal with the deviations encountered above 0.1 eV between the experimental and calculated σ1(ω).

Fig. 4: First-principles calculations in PrAlGe.
figure 4

a, b Schemes of BZ indicating the locations of the Weyl nodes in the PM and FM state10. c–g First-principles calculation of the electronic band structure with SOC in the PM (c, d) as well as FM (e–g) state along k-paths in BZ, near some specific Weyl nodes (i.e., black stars PM1 and PM2 as well as FM1, FM2 and FM3 in a, b and i for the PM and FM state, respectively). The coloured bars in c–g point out a selection of representative interband transitions, then mapped onto Δσ1(ω). The green and red bars refer to the transitions in the PM state, while the blue ones to those in the FM state. h Calculation of the interband part of σ1(ω) (i.e., Δσ1(ω)) in both the PM and FM state, derived from the electronic band structure (Supplementary Equation 3). The colour code for the arrows matches the one of the bars in c, d, thus establishing their relationship. i Effective Berry curvature (i.e., its difference between the PM and FM state, Supplementary Fig. 9) integrated along kz as a function of kx and ky from our ab initio calculations, which emphasises its strong enhancement in the FM state at those k-points of BZ almost coincident with the (projected) Weyl nodes (green symbols). Black stars foreground the selected Weyl nodes in the FM state, as in b. We refer to Supplementary Notes 6, 7 and 8 for details about all calculations.

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The reconstruction of the electronic band structure across TC then triggers the evolution of the (integrated) Berry curvature (see Supplementary Note 8), so that, as exemplified by Fig. 4i, there is an imbalance between its negative and positive values47. This effectively happens at k-points in BZ situated in proximity to the Weyl nodes, like those (black stars in Fig. 4a, b, i) at which the pronounced FIR interband transitions are taking place, in accordance with Fig. 4h. The just mentioned imbalance is a direct corollary of the broken TR symmetry in the FM state, as a consequence of which relevant parts of the electronic band structure get asymmetric (i.e., between −k and +k directions) below TC (Fig. 4e–g) than above TC (Fig. 4c, d). In conjunction with these arguments, we offer a complementary discussion at Supplementary Notes 4 and 8 and with Supplementary Fig. 9. In the end, this conspires in promoting the enhancement of \({\sigma }_{xy}^{A}\)11,12,47. Since \({\sigma }_{xy}^{A}\) is proportional to the BZ integral of the convolution of the Berry curvature with the T dependent occupation function48,49, Fig. 3 indeed conveys the connection between optics and \({\sigma }_{xy}^{A}\): the observed increasing strength (Ω1) in HO L1 at T < TΘ (Fig. 3) generally mimics the empowered matrix-element of the dipole-active interband transition (see Supplementary Note 4) at topologically driven band points upon entering the FM state47 and equally indicates the increasing occupation of the significant states participating to the intrinsic AHC22,23 (further details at Supplementary Note 6). Finally, the intimate relationship between \({\sigma }_{xy}^{A}\) and the SW reshuffling (Fig. 3), pertinent to the excitations around the Weyl nodes and imprinting the Berry curvature, evidences a strong similarity with our previous findings in the van der Waals Fe3GeTe2 ferromagnet50. We conjecture that this seems to be a generic, common feature in materials displaying large AHC, even benefiting from electronic correlations at low T42,51.

In conclusion, measurements of the charge dynamics of the noncentrosymmetric, topological correlated PrAlGe prove themselves to be a superior approach in order to satisfy the quest for the spectroscopic link between the reconstructed electronic band structure across the FM transition and the onset of the large AHC. The presence of the non-trivial Weyl states near EF with large Berry curvature turns out to be decisive, when making an impact in the anomalous transport properties of the title material. At last, we may also envisage our robust experiment as future best practice with additional tunable variables, like pressure and magnetic field.

Methods

Sample preparation

The single crystals of PrAlGe were grown by the high temperature self-flux method52. Further details about the sample preparation can be found at Supplementary Note 1.

Optical experiment

The optical reflectivity R(ω) as a function of temperature was first measured from the far-infrared up to the ultra-violet (i.e., ~30–5 × 104 cm−1). The real part σ1(ω) of the optical conductivity was then obtained via the Kramers–Kronig transformation of R(ω) by applying suitable extrapolations at low and high frequencies18. Further details about this experiment can be found at Supplementary Note 3.

First-principles calculations

Calculations of the electronic band structure were performed using density functional theory as implemented in the Vienna ab initio simulation package code53,54,55. Further details about the band structure calculations can be found at Supplementary Note 6.