Introduction

Topological semimetals1,2,3,4, a class of quantum states with symmetry-protected band crossings, have attracted tremendous interest recently because of their nontrivial topology, the presence of the peculiar surface states, and the resultant exotic electromagnetic responses5,6. Among many types of topological semimetals, nodal-line semimetals (NLSMs) arguably offer the most fascinating quantum system with rich topological structures7,8,9 and electronic correlations10,11. In NLSMs, the crossings of conduction and valence bands extend along one-dimensional lines in the momentum space, which can have various topologically distinct forms, e.g., an extended line across the entire Brillouin zone (BZ), a single closed loop inside the BZ, or a chain of multiple loops knotted or linked together12,13,14. In real systems with a finite carrier density, these nodal lines are enclosed by a thin tubular Fermi surface (FS), on which the associated π Berry flux imprints the characteristic smoke-ring-shaped pseudospin texture15,16,17. These unique topological characteristics of nodal-line fermions are expected to induce exotic charge and spin transport phenomena such as electric-field-induced anomalous transverse current17,18, large weak antilocalization19, spin-polarized filtering20, and anomalous Andreev reflection21,22, most of which are yet to be realized in experiments.

One major obstacle to investigating the unique transport phenomena of nodal-line fermions is the lack of suitable materials. Thus far, experimental studies on NLSMs have focused on the verification of nodal-line electronic structures, using angle-resolved photoemission spectroscopy (ARPES)23,24,25,26 or de Haas-van Alphen (dHvA) oscillations27,28,29,30, not on their unique transport properties. This is because most NLSM candidates possess complex multiple nodal loops linked together23,24,25,28; only a handful of candidates31,32 are expected to have a single loop and the corresponding torus-shaped FS in the BZ. Furthermore, in many cases, there exist topologically trivial states at the Fermi level (EF) that provide additional conduction channels, which hampers the identification of the characteristic transport properties of nodal-line fermions alone. Here, we present an NLSM candidate SrAs3 as a model system in which the quantum transport responses are entirely dictated by nodal-line fermions from a single torus-shaped FS without other trivial states. Shubnikov-de Hass (SdH) oscillations confirm dominant charge conduction by nodal-line fermions in slightly hole-doped SrAs3 and identify its tubular FS, thinnest among those of known NLSMs, and the characteristic smoke-ring-type pseudospin texture. These unique characters of nodal-line fermions are further corroborated by the quantum interference effect with disorder-induced scattering, resulting in unusual two-dimensional behaviors of weak antilocalization (WAL) and its strong variation to the FS characters.

Results

Nodal-line electronic structure of SrAs3

SrAs3, a member of the material class AEPn3 (AE = Ca, Sr, Ba, and Pn = P, As)33,34,35, has been suggested as a promising candidate NLSM32. It consists of buckled As planes, sandwiched by Sr atoms and staked along the c-axis in a monoclinic structure (space group C2/m) (Fig. 1a). Band crossings accidentally occur between the conduction and valence bands, derived from the p orbital states of two inequivalent As sites32 (Supplementary Fig. 1). The resultant single nodal-loop is expected to be located around the Y point in the BZ on the a–c plane, or (kx,ky) plane in the momentum space, where kx, ky and kz denote the orthogonal basis vectors parallel to the crystal axis a, the reciprocal lattice vector kc, and the crystal axis b, respectively (Fig. 1d). This nodal-loop structure at low-energy states has been suggested by band calculations32 and ARPES36 and partly by quantum oscillations37,38. Our ARPES intensity plots in a wide energy region along the kx and ky directions clearly show that the low-energy states are only located near the Y point (Fig. 1e, h), in good agreement with the band structure calculations using the modified Becke–Johnson exchange potential (Supplementary Fig. 3). Since the nodal-loop is expected to be located in the (kx,ky) plane centered at the Y point, a series of ARPES data taken along the kx axis was collected at different ky’s across the Y point (Fig. 1h). The evolution of ARPES spectra along the ky direction clearly reveals that band crossings only occur near the Y point of the BZ. The radius of the nodal-loop K0 is estimated to be ~0.057 Å−1 (Fig. 1i), consistent with the results of quantum oscillations, discussed below.

Fig. 1: Crystal and electronic structures of a nodal-line semimetal SrAs3.
figure 1

a The crystal structure of SrAs3. b The schematic band crossing for asymmetric nodal-line states with a tilted energy dispersion (Etilt), a finite spin–orbit-coupling gap (ΔSOC) and a band overlap energy (Δ). The corresponding Fermi surfaces at different Fermi levels (EF) are shown in the right, a crescent-type for EF,1 and a torus-type for EF,2. c The smoke-ring-type pseudospin texture imprinted on the Fermi surface. d The Brillouin zone of SrAs3 with a single nodal ring (red circle) centered at the Y point. e The ARPES spectra of SrAs3 taken at the Y point along kx with the photon energy of 99 eV. The overlaid red and blue lines indicate the conduction and valence bands, respectively. f The temperature dependence of the in-plane resistivity (ρ). The inset shows the carrier densities (n) for electron (e) and hole (h). g The magnetic field-dependent Hall resistivity (ρxy) of SrAs3 at different temperatures. h A series of ARPES spectra taken along kx at different photon energies (85-104 eV) corresponding to ky marked on top of each panel. i The nodal-ring of the crossing points between the conduction and valence bands in ARPES data, with dashed red circle as a guide to the eye.

Full size image

Having established the nodal-line electronic structure in SrAs3, we now focus on the details of the FS, which cannot be directly resolved by APRES due to its small size. In SrAs3, unlike the ideal nodal-loop, the conduction and valence bands have asymmetric dispersion, which introduces a tilting of the nodal-loop with a characteristic energy scale Etilt, smaller than the band overlap energy Δ (Fig. 1b). Furthermore, the finite spin–orbit-coupling (SOC) lifts the band degeneracy at the nodal-loop and induces a small momentum-dependent SOC gap ΔSOC. Thus, the torus-shaped FS is only established when εF, the energy difference between EF and the band crossing point in the momentum-energy space, is larger than ΔSOC/2 and Etilt but smaller than Δ/2. This characteristic torus-shaped FS possesses the smoke-ring-type pseudospin texture (Fig. 1c) and the associated π Berry flux, which disappears, e.g., in a drum-shaped FS for εF > Δ/2. Therefore, proper adjustment of εF is needed to access the unique transport properties of nodal-line fermions in SrAs3. We carefully selected the crystals that showed a dominant carrier type at low temperatures, using the in-plane resistivity, ρ, (Fig. 1f) and Hall resistivity, ρxy (Fig. 1g). At high temperatures, all crystals exhibit the nonlinear field dependence of ρxy(H), which originates from the two-band conduction of thermally excited electrons and holes, as commonly observed in many topological semimetals with low carrier densities39,40. Using the two-band conduction model, we estimated the temperature dependence of electron (ne) and hole (nh) carrier densities. Some samples show that the electron density ne is drastically reduced at low temperatures, smaller by one or two orders of magnitude than the hole density nh = 3–7 × 1017 cm−3 (Supplementary Fig. 4 and Table 1). These samples are used to measure both SdH oscillations and quantum interference effect for H∥ky, while two representative samples with a relatively large hole carrier density (S1 and S2) were used for investigating full angle-dependent SdH oscillations.

Torus-shaped Fermi surface

The magnetoresistance (MR), Δρ(H)/ρ(0), taken at high-magnetic fields up to 31.6 T for various field directions, is presented for the selected crystals (S1 and S2) in Fig. 2a, b. As compared to dHvA oscillations, SdH oscillations in the MR directly access the FSs responsible for charge conduction. For a torus-shaped FS (the inset of Fig. 2c), a small cyclotron orbit (α) on the poloidal plane is expected under magnetic fields in the nodal-loop plane, here H∥ (kx,ky) plane. For the magnetic field normal to the nodal-loop plane, H∥kz, two extremal inner (β) and outer toroidal (δ) orbits, significantly different in size, are expected41. These characteristic behaviors of SdH oscillations are indeed observed in experiments (Fig. 2a, b). As the magnetic field orientation changes in the (ky,kz) plane, SdH oscillations with a small frequency F vary systematically with the polar angle (θ) as the cyclotron orbit changes from α to β. Near H∥kz, additional oscillations with a high F ~ 129 T are detected, which is more clearly visible in the second derivative curve of ρ(H), -\({d}^{2}\rho /d{({H}^{-1})}^{2}\) (Fig. 2a). This additional cyclotron orbit with a large size corresponds to the outer toroidal orbit (δ). For H∥ (kx,ky) plane (Fig. 2b), SdH oscillations are well described by a single SdH frequency, corresponding to the poloidal orbit (α). The SdH oscillations with a single frequency F are described by the Lifshitz–Kosevich (LK) formula42,43,44,

$${{\Delta }}{\sigma }_{xx}\propto {R}_{T}{R}_{D}\left[\cos 2\pi \left(\frac{F}{H}+{\phi }_{0}+\frac{{\phi }_{s}}{2}\right)+\cos 2\pi \left(\frac{F}{H}+{\phi }_{0}-\frac{{\phi }_{s}}{2}\right)\right],$$
(1)

where RT and RD are damping factors due to a finite temperature and scattering, respectively. The characteristic phase ϕ0 and spin-splitting phase ϕs are two major components determining the phase offset of SdH oscillations ϕSdH, as discussed below. From the temperature-dependent SdH oscillations, we estimate the cyclotron effective mass of each orbit, yielding m*/me = 0.076(5), 0.23(1), and 0.079(3) for the α, β, and δ orbits, respectively (Fig. 2d, e, and f). The estimated quantum scattering times from the field-dependent SdH oscillations are τq = 0.085(8), 0.075(7), and 0.010(1) ps for the α, β, and δ orbits, respectively, which are relatively long, as typically observed in topological semimetals.

Fig. 2: Shubnikov-de Hass oscillations of SrAs3.
figure 2

a, b Magnetoresistance (MR) Δρ(H)/ρ(0) of SrAs3 with different magnetic field orientations in the (ky,kz) plane (a, S1) and in the (kx,ky) plane (b, S2). The second derivative of ρ(H) with respect to 1/H at H∥kz is also shown in a. The overlaid dashed gray curve corresponds to the coexisting SdH oscillations with a lower frequency. The black arrows indicate peaks and deeps of the higher-frequency oscillations. The polar (θ) and azimuthal (ϕ) angles are defined with respect to the torus-shaped Fermi surface as shown in the insets. c SdH oscillations (Δρosc./ρ(0)) at various magnetic field orientations in the planes of (kx,ky), (ky,kz) and (kx,kz) for S1. The inset shows torus-shaped Fermi surface of SrAs3 with the poloidal orbit (α) and the inner (β) and outer (δ) toroidal orbits. d, e, f Fast Fourier transform (FFT) amplitudes for α orbit (d), and β orbit (d) and δ orbit (f), taken at various temperatures for H∥ky (d) and H∥kz (e, f). The insets show the temperature-dependent FFT amplitudes, together with the fits (red lines) to the Lifshitz–Kosevich equation. In e, the FFT amplitude of the δ orbit, F > 100 T, is magnified for comparison.

Full size image

A small deviation from the ideal torus-shaped FS is well resolved in the detailed angle dependence of the SdH frequency (Fig. 3a). To this end, we constructed a general two-band model Hamiltonian near the Y point \(H({{{{{{{\boldsymbol{k}}}}}}}})=\mathop{\sum }\nolimits_{i=0}^{3}{g}_{i}({{{{{{{\boldsymbol{k}}}}}}}}){\sigma }_{i}\), where σ0 is the identity matrix, σ1,2,3 are Pauli matrices, and gi(k) is the real function of k.32 Considering three symmetries at Y point, time-reversal symmetry \(\hat{T}=K\) with spinless complex conjugate operator K, inversion symmetry \(\hat{P}={\sigma }_{z}\), and mirror symmetry \(\hat{M}:{k}_{x}\leftrightarrow {k}_{x};\,{k}_{y}\leftrightarrow {k}_{y};{k}_{z}\leftrightarrow -{k}_{z}\), the coefficients gi(k) for the lowest orders of k are described by \({g}_{0}({{{{{{{\boldsymbol{k}}}}}}}})={a}_{0}+{a}_{1}{k}_{x}^{2}+{a}_{2}{k}_{y}^{2}+{a}_{3}{k}_{z}^{2}\), g2(k) = b3kz, and \({g}_{3}({{{{{{{\boldsymbol{k}}}}}}}})={m}_{0}+{m}_{1}{k}_{x}^{2}+{m}_{2}{k}_{y}^{2}+{m}_{3}{k}_{z}^{2}\). The parameters ai, b3, mi are obtained to match the calculated cross-sectional size of FS with the measured SdH frequency as a function of the polar (θ) and azimuthal (ϕ) angles (Fig. 3a and Supplementary Table 2). Unlike the ideal torus-shaped FS, the resultant FS of SrAs3 has a crescent-shaped poloidal cross-section, rather than the circular one and exhibits a small momentum-dependent asymmetry within the nodal plane. Along the toroidal direction, a finite tilting energy Δtilt ~ 5 meV, far smaller than the band overlap energy Δ ~ 120 meV and the Fermi level EF ~ 50 meV, introduces ϕ-dependent distortion, leading to a weak variation of the SdH frequency. A detailed comparison between model calculations and experiments is provided in the Supplementary Note 4. The volume of FS and the corresponding carrier density nh ~ 1.7 × 1018 cm−3 are in reasonable agreement with nh ~ 7 × 1017 cm−3 from the Hall effect. The radius of the nodal-loop K0 ~ 0.065 Å−1, estimated from the constructed FS, agrees well with the APRES results (Fig. 1i). In addition, the calculated cyclotron masses using the constructed Hamiltonian are consistent with the experimental values for H∥kx, ky, and kz (Supplementary Table 2). Moreover, the band overlap energy Δ ~ 120 meV from our model calculations is consistent with that obtained by the optical conductivity measurements on our crystal. These agreements reveal that the magnetotransport response in SrAs3 is determined by the single torus-shaped FS, consistent with APRES results (Fig. 1).

Fig. 3: Toroidal Fermi surface and Berry phase evolution of SrAs3.
figure 3

a Angle-dependent SdH frequency (F) and the phase offset of SdH oscillation (ϕSdH) for two samples S1 (black) and S2 (red). The spin-splitting phase (ϕs) and the characteristic phase (ϕ0) are also shown in the lower panels. The calculated F using the model Hamiltonian is overlaid with red lines. The corresponding extremal orbits on the torus-shaped Fermi surface are also presented for selected field orientations in the inset. b Torus-shaped Fermi surface of SrAs3 with the poloidal orbit (α) and the inner (β) and outer (δ) toroidal orbits. c Poloidal cross-section of the Fermi surface (α) with pseudospin textures indicated by the arrows. d–g Landau fan diagram for various field orientations with different polar (θ) angles (d, f) and azimuthal (ϕ) angles (e, g) for S1. The maxima (solid circles) and minima (open circles) of Δρ(H)/ρ(0) are assigned with integer and half-integer of the Landau index. h, i The second derivative of ρ(H), − d2ρ/dH2, as a function of the nomalized F/H for various magnetic field orientations with different polar (θ) (h) and azimuthal (ϕ) angles (i) for S2. The spin-splitting peaks of SdH oscillations are indicated by triangle symbols. The shaded dashed lines correspond to the spin-split Landau levels, indicated by the color-coded integer index and the + and – symbols.

Full size image

Smoke-ring-type pseudospin texture

The smoke-ring-type pseudospin texture, imprinted on the torus-shaped FSs, is confirmed by SdH oscillations. By assigning the maxima and minima of Δρ(H) as integers and half-integers of the Landau index (Supplementary Note 3), respectively, we plot the Landau fan diagrams for different field orientations and extract the phase offset of SdH oscillations ϕSdH from the interception of the linear fit (Fig. 3d–g). For both crystals (S1 and S2), ϕSdH as a function of the polar angle θ on the (ky,kz) plane exhibits a clear change from − (0.3–0.4) to 0 near θ ~ 10∘, when the poloidal orbit (α) is converted to the inner toroidal (β) orbit (Fig. 3a). As the phase offset ϕSdH is partly determined by the Berry phase ϕB, the observed change in ϕSdH may indicate the additional Berry phase for the poloidal orbit (α), due to the associated the π Berry flux and smoke-ring-type pseudospin texture. In contrast, both the inner (β) and outer (δ) toroidal orbits are expected to have a zero Berry phase41. Consistently, for the outer toroidal orbit (δ) near H∥kz, we also observed the same ϕSdH as the inner orbit (β), as shown in Fig. 3a.

In order to clarify that the observed change in ϕSdH is due to the Berry phase change expected for the smoke-ring-type pseudospin texture, we consider other contributions to ϕSdH, including the correction term for three-dimensional (3D) FS (ϕ3D) and the spin-splitting effect (ϕs). The phase ϕ0 in Eq. (1) is determined by ϕB and ϕ3D with a relation of ϕ0 = − 1/2 + ϕB/2π + ϕ3D. For hole carriers, ϕ3D is ± 1/8 for the maximum and minimum cross-sections42. In addition, the spin-splitting of the Landau levels (LLs) by the Zeeman effect introduces the phase shift of SdH oscillations by ± ϕs = ± gm*/2me, where g is g-factor, m* is the effective mass, and me is the free electron mass. Usually, at relatively small magnetic fields, this Zeeman spin splitting introduces the so-called spin-splitting factor \({R}_{s}=\cos (\pi g{m}^{*}/2{m}_{e})\), and its sign change is equivalent to the phase shift of π, which often hampers precise estimation of ϕ0. For high-magnetic fields near the quantum limit, however, the spin splitting of LLs can be directly resolved by the additional peak splitting in SdH oscillations, which have been indeed observed in our SrAs3 crystals (Fig. 3h, i). We found systematic dependence of the spin splitting of LLs on polar (θ) and azimuthal angles (ϕ), presumably due to changes in the g-factor and effective mass (Supplementary Note 5), as observed in other topological semimetals29,44. Then the extracted ϕs enables to determine the remaining phase ϕ0, shown in the lower panels of Fig. 3a.

In hole-doped SrAs3, the poloidal orbit (α) is expected to have an additional Berry phase (ϕB = π) and minimum cross-section (ϕ3D = − 1/8), resulting in ϕ0 = − 1/8. On the other hand, the inner (β) and outer (δ) toroidal orbits have zero Berry phase (ϕB = 0) and the maximum cross-section (ϕ3D = + 1/8) due to the crescent-shaped cross-section in the poloidal planes, which leads to the same ϕ0 = − 3/8. Thus, near θ ~ 10∘, when the poloidal orbit (α) is converted to the inner toroidal orbit (β), a phase shift by Δϕ0 = − 1/4 is expected, which is in good agreement with the observed shift Δϕ0 = –0.26(6) (lower panels of Fig. 3a). We note that without considering the Berry phase change, Δϕ0 = + 1/4 is expected when the α orbit changes to the β orbit near θ ~ 10∘, opposite to experiments.

For the azimuthal angle (ϕ) dependence, we found that ϕ0 is nearly constant for different poloidal orbits around the torus-shaped FS, which is consistent with the smoke-ring-type pseudospin texture. A slight variation of ϕ0 with field orientation in the (kx, ky) plane can be attributed to asymmetries in the Fermi velocity and the spin–orbit coupling (SOC), expected in SrAs3 due to the low crystalline symmetry. For the Dirac node with a finite ΔSOC, the Berry phase is not quantized but varies from π to zero, as described by \({\phi }_{B}=\pi \left(1-\frac{{{{\Delta }}}_{{{{{{{{\rm{SOC}}}}}}}}}}{2|{\varepsilon }_{{{{{{{{\rm{F}}}}}}}}}|}\right)\)41. Thus, the ϕ-dependence in both the SOC gap (ΔSOC) and the εF introduces modulation of ϕB for each poloidal cyclotron orbit. Upon rotating magnetic field from H∥ky to H∥kx, the SdH frequency decreases gradually, implying that the energy position of the Dirac node corresponding to the extremal poloidal orbit becomes closer to EF, reducing ∣εF∣. This induces a slight decrease of ϕB and thus ϕ0, as the magnetic field approaches to H∥kx. Together with the torus-shaped FS, this Berry phase evolution with magnetic field orientation provides compelling evidence for nodal-line fermions in SrAs3.

Quantum interference effect of nodal-line fermions

Now we discuss a unique quantum interference effect for the well-isolated nodal-line fermions in SrAs3. Figure 4b presents the low-field magnetoconductivity, Δσ(H)/σ(0), in transverse configuration under magnetic fields H∥kc for SrAs3 crystals with different hole carrier densities (nh). The sharp peak in Δσ(H) is attributed to weak antilocalization (WAL) due to quantum interference of electrons with impurity scattering, as found in topological semimetals45,46,47,48. From the magnetoconductivity data of SrAs3 (Fig. 4b) and other topological semimetals (Supplementary Fig. 11), we estimate the excess conductivity ΔσWAL and the semi-classical conductivity σ0, with and without quantum interference effect, respectively, by fitting the high field data to the H2 dependent conductivity from the orbital MR or the chiral anomaly effects49,50 (Supplementary Note 6). In topological semimetals, e.g., Weyl semimetals, dominant small-angle (intravalley) scattering leads to WAL due to π Berry phase of the back-scattering trajectories encircling a Weyl point. However, a finite large-angle (intervalley) scattering without the associated Berry phase induces the competing weak localization (WL) and reduces the resulting ΔσWAL49. Usually the large-angle scattering is more effective to reduce the semi-classical conductivity σ0 than the small-angle scattering, the measured ΔσWAL is likely to decrease with lowering σ0. Such a trend of ΔσWAL with variation of σ0 is observed for various topological semimetals, as shown in Fig. 4e.

Fig. 4: Weak antilocalization of nodal-line fermions in SrAs3.
figure 4

a Back-scattering processes of nodal-line fermions on the poloidal plane of the torus-shaped Fermi surface in the momentum space (upper panel). The π Berry flux (yellow line) along the nodal-loop leads to weak antilocalization (WAL). The corresponding diffusion of nodal-fermions in the real space is two-dimensional (lower panel), which significantly enhances the quantum interference effect. b The low-field transverse magnetoconductivity Δσ(H)/σ(0), taken at T = 2 K and H⊥J, from eleven SrAs3 crystals with different hole carrier densities (nh) and the ratio (K0/κ) between the radii of the nodal-loop (K0) and the poloidal orbit (κ). c The transverse magnetoconductivity Δσ(H) for S1 together with the fits to the 2D WAL (red line) and 3D WAL (blue line) models. d Temperature-dependent phase coherence length lϕ for S1, following T−1 dependence (blue dashed line) at high temperatures. The fit to the 2D WAL model is also shown (green solid line). e The excess conductivity ΔσWAL as a function of σ0 for various topological semimetals. The inset shows the ΔσWAL of SrAs3 crystals taken at 2 K with variation of the ratio K0/κ.

Full size image

What is unique for SrAs3 is the unusual magnetic field and temperature dependences of the magnetoconductivity Δσ(H, T), that can be attributed to two main characters of the nodal-line fermions. First, when the poloidal orbit of radius κ is smaller than the nodal-loop of radius K0, i.e., κ < K0,19 the tubular FS has the local two-dimensionality in the momentum space (Fig. 4a). In SrAs3, the loop radius K0 ~ 0.065(8) Å−1 is fixed, as estimated by ARPES (Fig. 1) and SdH oscillations (Fig. 3), while reducing nh makes the tubular part of the torus-shaped FS thinner with a smaller radius κ. For SrAs3 crystals with different nh, we found that the SdH frequency of the poloidal orbit for H∥c systematically decreases as nh reduces (Supplementary Fig. 4) and reaches the smallest size AF found among NLSM candidates so far (Supplementary Table 3). Using F = ℏAF/2eπ = ℏκ2/2e, we estimate the averaged κ and K0/κ ~ 3.3 − 4.5. This is much larger than, e.g., K0/κ ~ 1.56 of CaAgAs, a recent NLSM candidate30, indicating the strong two-dimensional (2D) nature of the tubular FS in SrAs3. Second, due to the unusual screening effect of NLSMs16, the impurity potential is a long-range type, and the impurity scattering mainly involves with a small momentum change (small-angle scattering) at low temperatures. Therefore, the back-scattering trajectories of the electron’s diffusive motion mostly lie on the 2D poloidal plane, encircling the π Berry flux (Fig. 4a), rather than along the toroidal direction without involving the Berry flux. In this case, the dominant 2D WAL is expected to determine the magnetoconductivity Δσ(H, T) of SrAs3 at low-magnetic fields.

For the 2D WAL, Δσ(H) is described by the Hikami-Larkin-Nagaoka model51, roughly following the—\(\ln H\) dependence, and the temperature-dependent phase coherence length lϕ follows lϕ ∝ T−p/2 with the exponents p = 1 or p = 2 due to electron–electron or electron–phonon interactions, respectively. These behaviors are clearly distinguished from the 3D behaviors with \({{\Delta }}\sigma (H) \sim -\sqrt{H}\) and lϕ ∝ T−p/2 with exponents p = 3/2 or p = 3 for electron–electron or electron–phonon interactions, respectively (Supplementary Note 7)49. The stiff drop of Δσ(H) of SrAs3 at low-magnetic fields is well reproduced by the fit to the 2D WAL model rather than the 3D WAL model (Fig. 4c). Consistently, the temperature-dependent lϕ follows the 2D model with the exponent of p = 2 at high temperatures, corresponding to the 2D electron–phonon interactions. The temperature dependence of lϕ is well reproduced by the fit to the equation, \(1/{l}_{\phi }^{2}=1/{l}_{\phi 0}^{2}+{A}_{ep}{T}^{2}\), where lϕ0 = 83(1) nm is the zero-temperature dephasing length and Aep = 7.0(6) × 10−8 nm−2 K−2 is the coefficient for electron–phonon scattering (Fig. 4d)52. Furthermore, since the key parameter for describing the local 2D nature of the torus-shaped FS is the ratio between the radii of the poloidal orbit (κ) and the nodal-loop (K0), systematic variation of ΔσWAL is expected with variation of the ratio K0/κ. As κ becomes close to K0, the contribution of weak localization by scattering along the toroidal direction without associated with Berry flux becomes sizable. Then the competition between WAL and WL determines the size of ΔσWAL, leading to increase of ΔσWAL with the ratio K0/κ, in good agreement with experiments (the inset of Fig. 4e). These results strongly indicate the 2D nature of the WAL induced by nodal-line fermions in SrAs3.

Discussion

The quantum transport signatures of nodal-line fermions, quantum oscillations and quantum interference presented in this work, consistently evince the dominant transport of nodal-line fermions in slightly hole-doped SrAs3 crystals without any sizable contribution from other topologically trivial states at the Fermi level. There are several questions remained to be investigated, including quantitative understanding on the competing WAL and WL processes upon varying K0/κ and observation of the possible crossover between them19. Nevertheless, our findings highlight SrAs3 as a unique platform of nodal-line fermions with the thinnest tubular FS and the largest K0/κ among the NLSMs candidates and thus establish SrAs3 as a desirable system for studying various unique transport phenomena of nodal-line fermions, theoretically proposed but not yet realized in experiments17,18,19,20,21,22. Our study also emphasizes that precise control of the size difference between the radii of the nodal-loop and the poloidal cross-section is crucial for unveiling the otherwise hidden transport signature of nodal-line fermions. These findings provide a guideline for designing NLSMs suitable for novel topological electronic applications, by tuning the chemical doping or external perturbations such as strain or pressure.

Methods

Single-crystal growth and characterization

Single crystals of SrAs3 were grown by the Bridgman method (Supplementary Note 1). The resistivity of single crystals was measured using the standard six-probe method with a Physical Property Measurement System (PPMS-14T, Quantum Design) to measure the in-plane and Hall resistivities.

Angle-resolved photoemission spectroscopy

ARPES experiments were carried out with the Beamline 4.0.3, Advanced Light Source (Supplementary Note 2). The ARPES end-station (MERLIN) is equipped with a hemispherical electron analyzer. The energy and momentum resolutions were better than 20 meV and 0.01 Å−1, respectively. We used the photon energy of 30–125 eV with linear-horizontal polarization. Samples were cryogenically cooled to 30–40 K and cleaved in the ultrahigh vacuum chamber with the base pressure of 1.5 × 10−11 torr.

Magnetotransport property measurements at high-magnetic fields

Shubnikov de Haas oscillations of SrAs3 were measured using the magnetoresistivity measurements in high-magnetic fields up to 31.6 T in National High Magnetic Field Laboratory (NHMFL), Tallahassee and up to 56.7 T in International MegaGauss Science Laboratory at the Institute for Solid State Physics (ISSP), University of Tokyo (Supplementary Note 3).

Electronic structure calculations

Electronic structures were calculated using WIEN2K code53, which uses a full-potential augmented plane base method. The Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) was used for the exchange-correlation functional54 and spin–orbit coupling (SOC) was included in the calculations. The modified Becke–Johnson potential (mbJ) was also employed to overcome the shortcoming of the PBE-GGA method in the underestimation of the band gap55 (Supplementary Note 2). Two-thousand k-points were used for self-consistent calculations.