Coexistence of nontrivial topological properties and strong ferromagnetic fluctuations in quasi-one-dimensional A₂Cr₃As₃

Abstract

Superconductivity in crystals without inversion symmetry has received extensive attention due to its unconventional pairing and possible nontrivial topological properties. Using first-principles calculations, we systemically study the electronic structure of noncentrosymmetric superconductors A₂Cr₃As₃ (A = Na, K, Rb, and Cs). Topologically protected triply degenerate points connected by one-dimensional arcs appear along the C₃ axis, coexisting with strong ferromagnetic (FM) fluctuations in the non-superconducting state. Within random phase approximation, our calculations show that strong enhancements of spin fluctuations are present in K₂Cr₃As₃ and Rb₂Cr₃As₃ and are substantially reduced in Na₂Cr₃As₃ and Cs₂Cr₃As₃. Symmetry analysis of pairing gap Δ(k) and spin–orbit coupling g_k suggest that the arc surface states may also exist in the superconducting state, giving rise to possible nontrivial topological properties.

Introduction

Materials with nontrivial topological properties have been extensively studied over the past two decades. Initially, comprehensive attentions were paid to topological insulators (TIs)^1,2,3, which have an insulating gap in the bulk and metallic surface states at the boundary. Fully gapped superconductors with the topological protected gapless surface mode, a close analogy with TI, are regarded as promising candidates for hosting Majorana fermions. Since the discovery of Weyl, Dirac, nodal line semimetals, and triply degenerate point (TP) topological metals^{4,5,6,7,8,9,10,11,12,13,14,15}, it was shown that gapless systems can possess novel topology as well. Simultaneously, it is also possible for superconductors with nodes (e.g., CePt₃Si, UPt₃) to have topologically protected edge states, which are guaranteed by momentum-dependent topological numbers^16,17,18. Among these exotic superconductors, nodal noncentrosymmetric superconductors (NCSs) with topological stable nodes have fascinating properties, i.e., the zero-energy boundary modes^19,20. These zero-energy boundary modes are believed to be closely associated with the nodal gap structure via the so-called bulk–boundary correspondence²¹.

The Cr-based arsenides A₂Cr₃As₃ (A = Na, K, Rb, and Cs) are of great interest in terms of low dimensionality, strong ferromagnetic (FM) fluctuations, and noncentrosymmetric superconductivity. Their nodal, unconventional superconductivity was suggested by London penetration depth, nuclear magnetic resonance (NMR), muon spin spectroscopy, and specific-heat measurements^{22,23,24,25,26,27,28,29}. Besides, spin–orbit coupling (SOC) has a remarkable effect on β and γ bands in K₂Cr₃As₃ with a band spin-splitting much larger than the superconducting gap³⁰. The coalescence of considerable SOC effect and strong FM fluctuations in A₂Cr₃As₃ NCSs is crucial to the superconducting pairing symmetry, leading to a predominant spin–triplet component, which is distinguished from the pairing in the isotropic channel of an usual s-wave superconductor. More recently, NMR experiments on A₂Cr₃As₃ suggest that the compounds in this family are possibly a solid-state analog of superfluid ³He, implying that the unconventional superconductor A₂Cr₃As₃ may host nontrivial topological properties³¹. In addition, the NMR measurements of Cs₂Cr₃As₃ were distinct from that of K₂Cr₃As₃ (Rb₂Cr₃As₃), which displayed suppression of FM fluctuations in the former³². Therefore, a systematic study of the SOC effect, the topological properties as well as the FM fluctuations, and a thorough comparison among the family is in need.

In this article, we report our latest first-principles results on the A₂Cr₃As₃ family. Our results show: (1) The variations in band structures and Fermi surfaces (FSs) due to alkaline element substitution do not show apparent systematic behavior; (2) The anti-symmetric spin–orbit coupling (ASOC) splitting is the largest in K₂Cr₃As₃, but its effect is most significant in Cs₂Cr₃As₃ and enhances its one dimensionality; (3) All compounds of this family host TPs along Γ–A and the surface states emerging from the TPs form one-dimensional (1D) Fermi arcs; (4) The magnetic susceptibility spectrum exhibits a strong peak of the spin susceptibilities at the Γ point in K₂Cr₃As₃, followed by Rb₂Cr₃As₃, while in Na₂Cr₃As₃ and Cs₂Cr₃As₃ the enhancement of spin susceptibilities at the Γ point is not obvious. Inclusion of dynamic self-energy due to spin fluctuation does not alter the topology of electronic spectrum, and thus the existence of TPs is robust against the dynamic spin fluctuation at random phase approximation (RPA) level. Finally, we discuss the possibility of the existence of topologically stable arc states in the superconducting phase.

Results

Electronic structure and topological properties

The crystal structure and Brillouin zone of K₂Cr₃As₃ are illustrated in Fig. 1a, b. For the compounds of this family, each primitive unit cell consists of [(Cr₆As₆)]_∞ sub-nanotube along the c axis forming a quasi-one-dimensional (Q1D) structure^{29,33,34,35,36}. In contrast to ACr₃As₃ (A = K, Rb)^37,38,39,40, the A⁺ ions around the [(Cr₆As₆)]_∞ sub-nanotube break the inversion symmetry, rendering A₂Cr₃As₃ noncentrosymmetric (with symmetry D_3h, space group 187). From Na₂Cr₃As₃, K₂Cr₃As₃, Rb₂Cr₃As₃ to Cs₂Cr₃As₃, the lattice expands in-plane while the average Cr-Cr bond length barely changes, implying increased distances between the [(Cr₆As₆)]_∞ tubes.

Fig. 1: Geometry and Brillouin zone.

a Crystal structure of A₂Cr₃As₃. The red, blue, and purple are alkali, chromium, and arsenic atoms, respectively. b Brillouin zone (BZ) and high symmetry points. The blue lines represent the [010] surface BZ.

Despite this systematic structural variation, the band structures (Fig. 2a–d) of A₂Cr₃As₃ around the Fermi level resemble each other and do not exhibit apparent change except for Cs₂Cr₃As₃. In the absence of SOC, the α and β bands cross the Fermi level along Γ–A, forming two Q1D FSs (see Supplementary Fig. 1 for details). In contrast, the γ band forms one three-dimensional (3D) FS around the Γ point (Fig. 3a–c) for Na₂Cr₃As₃ and Rb₂Cr₃As₃, similar to K₂Cr₃As₃^30,41. For Cs₂Cr₃As₃, it is worth noting that the γ band do not cross the Fermi level in the k_z = 0 plane. As a result, this band forms one deformed Q1D FS. In addition, a fourth band ($\gamma ^{\prime}$) around the Γ point emerges, creating a new 3D FS (Fig. 3d). Once the SOC effect is included, for all members in this family, the β and γ bands further split due to the ASOC effect.

Fig. 2: Electronic band structure of A₂Cr₃As₃.

a, e A = Na; b, f A = K; c, g A = Rb; d, h A = Cs. The upper/lower panels are results without/with spin–orbit coupling.

Fig. 3: Fermi-surface sheets of A₂Cr₃As₃.

a Na₂Cr₃As₃, b K₂Cr₃As₃, c Rb₂Cr₃As₃, and d Cs₂Cr₃As₃. In Na₂Cr₃As₃, K₂Cr₃As₃, and Rb₂Cr₃As₃, there are two Q1D FSs and one 3D FS, whereas in Cs₂Cr₃As₃ one more FS emerges.

Remarkably, the compounds of A₂Cr₃As₃ host TPs along Γ–A. In particular, the TPs in Na₂Cr₃As₃ (8 meV above the Fermi level ϵ_F) and K₂Cr₃As₃ (0.6 meV below ϵ_F) around the Γ point are very close to the Fermi level. Considering the presence of time reversal symmetry T and mirror symmetry σ_h (orthogonal to the C₃ axis) in A₂Cr₃As₃, the TP fermions in current compounds belong to type A¹⁴, which is accompanied by one Weyl nodal line instead of four in type B^15,42. Including SOC, along the C₃ rotation axis, the singly degenerate band (Λ₅ or Λ₆ state) belongs to the 1D representation of C_3v symmetry, while the doubly degenerate bands (Λ₄ state) form the two-dimensional representation, and these TPs along k_z are due to band inversion between the Λ₅/Λ₆ state and the Λ₄ state. Around the Fermi level, band inversion occurs for α, β, and γ bands in Na₂Cr₃As₃ (Fig. 4a) as well as α and β in K₂Cr₃As₃ (see Supplementary Fig. 2), and these TPs are protected by the C_3v symmetry. We compare the band structures of A₂Cr₃As₃ along k_z (Supplementary Fig. 2) as well as TPs and (010) surface states (see Supplementary Fig. 3a–c). Owing to the overwhelming bulk states, only the surface states in Na₂Cr₃As₃ can be clearly distinguished from the bulk band continuum, resulting in clear Fermi arc structures on the (010) surface (Fig. 4b, c). We also performed calculations using the modified Becke–Johnson (mBJ) potentials⁴³ and obtained results similar to that of Perdew, Burke, and Ernzerhoff (PBE) for Na₂Cr₃As₃ (for details, see “Methods” below). In K₂Cr₃As₃, however, the Λ₄ state near ϵ_F, which is lower than the Λ₅ and Λ₆ states in PBE calculations (see Supplementary Fig. 2b), is now elevated higher around Γ points, leading to extra band inversions between β and γ band and hence two new TPs (located at ϵ_F+90 and ϵ_F-22 meV), as shown in Fig. 4d. Figure 4b, c, e, f show the surface states in Na₂Cr₃As₃ (PBE results) and K₂Cr₃As₃ (mBJ results), respectively. For both of these two compounds, two surface states, SS₁ and SS₂, emerge from two TPs (TP₁ and TP₂) near the Γ point, respectively, while the surface states from the TPs (TP₃ and TP₄ in Na₂Cr₃As₃, TP₃ in K₂Cr₃As₃) away from Γ point are mixed with surrounding bulk states and cannot be easily distinguished. The iso-energy surface states at TP₁ in Na₂Cr₃As₃ (ϵ_F+8 meV) and K₂Cr₃As₃ (ϵ_F+90 meV) are shown in Fig. 4c, f. Akin to ZrTe family of compounds, the TPs (TP₁ in Na₂Cr₃As₃ and K₂Cr₃As₃) marked as blue (Fig. 4c) and red (Fig. 4f) dots are connected by double 1D Fermi arcs on the (010) surface. We note that, for K₂Cr₃As₃, the number of TPs are different in PBE and mBJ calculations. Nevertheless, as long as the band inversions between the Λ₄ and Λ₅ (Λ₆) exist, these intrinsic TPs will be protected by the extra global symmetry (C_3v). In the next section, we will further discuss the dynamic correlation effect due to the spin fluctuation on these TPs.

Fig. 4: Electronic band structure and the (010) surface state along k_z.

a–c Na₂Cr₃As₃ within PBE calculation. d, e K₂Cr₃As₃ within mBJ calculation. The triply degenerate points are marked as solid dots. The iso-energy surface state is located at ϵ_F+8 meV for Na₂Cr₃As₃ and at ϵ_F+90 meV for K₂Cr₃As₃. The inset of f shows surface states only, emerging from TP₁ and TP₂ separately without the bulk states.

Spin fluctuations and multiorbital susceptibilities

The imaginary part of the bare electron susceptibility χ₀ of K₂Cr₃As₃ exhibited a strong peak at the Γ point³⁰. In order to investigate the variation of electron susceptibility, we also calculate susceptibility χ₀ using the Lindhard response function for all compounds of this family. As shown in Fig. 5a, b, the bare susceptibilities of Na₂Cr₃As₃, Rb₂Cr₃As₃, and Cs₂Cr₃As₃ resemble that of K₂Cr₃As₃. Specifically, the real part $\chi ^{\prime}_0$ is relatively featureless, while the imaginary part $\chi ^{\prime \prime}_0$ is dominated by a resonance peak at the Γ point. As reported in previous NMR studies^24,25,32, there were clear differences in the Knight shift and NMR relaxation rate (1/T₁T) of K₂Cr₃As₃ (Rb₂Cr₃As₃) and Cs₂Cr₃As₃, implying that the spin susceptibility of K₂Cr₃As₃ (Rb₂Cr₃As₃) strikingly differs from that of Cs₂Cr₃As₃. Therefore, electron–electron interactions must be taken into consideration to explain the aforementioned NMR experiments. We calculate both spin and charge susceptibilities within RPA with intra-orbital Coulomb (U), inter-orbital Coulomb ($U^{\prime}$), Hund’s coupling (J), and pair-hopping ($J^{\prime}$) interactions involved (see Supplementary Method for details). In Fig. 5c–f, the real part of charge (spin) susceptibility $\chi ^{\prime}_c$ ($\chi ^{\prime}_s$) is presented for U = 2 eV and J = 0.3 eV in comparison to that of bare ones. For all members in this family, the spin susceptibilities are significantly enhanced, while the charge susceptibilities are suppressed. The sharp peak present at the Γ point of K₂Cr₃As₃ indicates that it is very close to the critical point with certain values of U and J. Naively, using the simplest approximation, the enhancement of the imaginary part χ″ can be written as $\chi ^{\prime\prime} ({\bf{q}},\omega )\approx {\chi }_{0}({\bf{q}},\omega )/{\left[1-\bar{U}{\chi ^{\prime} }_{0}({\bf{q}},\omega )\right]}^{2}$ ^44,45, where $\overline{U}$ is the Stoner factor. Applying this approximation, the imaginary part χ″ of K₂Cr₃As₃ is also expected to be strongly enhanced at the Γ point when $1-U{\chi }_{0}^{\prime}({\bf{q}},\omega )$ approaches zero. The scattering peak of $\chi ^{\prime}$ in Rb₂Cr₃As₃ still locates at the Γ point but is lower and broader compared to K₂Cr₃As₃. In the case of Cs₂Cr₃As₃ (Na₂Cr₃As₃), in contrast, only a broadened and plateau-like structure can be found around the M(π, 0, 0) and $K\left(\frac{2}{3}\pi ,\frac{2}{3}\pi ,0\right)$ points. Interestingly, there exists another broad hump around Q^* = (0, 0, 0.6π) in Cs₂Cr₃As₃, which is possibly due to intraband scattering within the γ band. Similar to previous report in iron-based superconductor LaOFeAs⁴⁶, such a broad hump might be related to spin–density–wave (SDW) fluctuations. In K₂Cr₃As₃ (Rb₂Cr₃As₃), the SDW hump at Q^* also exists but not apparent, overwhelmed by the large peak located at q(0, 0, 0). It is worth noting that the strong peak of $\chi ^{\prime}$ at the Γ point of K₂Cr₃As₃ (Rb₂Cr₃As₃) is robust against the Hund’s coupling (J). It will also be present even if the Hund’s coupling is completely turned off (J = 0). In addition, the spin susceptibility appears to be insensitive to either the inter-orbital Coulomb repulsion ($U^{\prime}$) or the pair-hopping interaction ($J^{\prime}$).

Fig. 5: Electron susceptibility of A₂Cr₃As₃.

a Real and b imaginary part of the bare electron susceptibility of the A₂Cr₃As₃ family. c–f The real part of magnetic susceptibility calculated at the RPA level for c Na₂Cr₃As₃, d K₂Cr₃As₃, e Rb₂Cr₃As₃, and f Cs₂Cr₃As₃, respectively. The red, blue, and cyan solid line are spin, charge, and bare (in comparison) susceptibilities, respectively, along high symmetry lines for the compounds of this family with U = 2.0 eV and J = 0.3 eV.

Considering the striking difference in $\chi ^{\prime}_s$ between K₂Cr₃As₃ and Cs₂Cr₃As₃, we analyze the dynamic spin susceptibility of these two compounds at the Γ, M, and Q^* points as shown in Fig. 6a, b. The spin susceptibility exhibits a larger enhancement at the Γ point in K₂Cr₃As₃, while in Cs₂Cr₃As₃ the spin susceptibility seems to be more enhanced at a finite q. For all members in this family, we further compare the enhancement of spin susceptibility $\chi ^{\prime}_s$ and $\chi ^{\prime \prime}_s$ at q = (0,0,0) in Fig. 6c, d. The results show that the enhancement is most significant in K₂Cr₃As₃, followed by Rb₂Cr₃As₃ and Cs₂Cr₃As₃. Such a systematic trend can also be found in the temperature-dependent spin susceptibility enhancement (Fig. 6e, f). Note that the enhancement of χ″ is always larger than that of $\chi ^{\prime}$ for both the T- and U-dependent spin susceptibilities, indicating that the system is away from an FM ordered state.

Fig. 6: Comparison of spin susceptibility.

a, b Imaginary part of the spin susceptibility (χ_s/χ₀) at Γ, M, and Q^* (marked in Fig. 5f) for a K₂Cr₃As₃ and b Cs₂Cr₃As₃, respectively. c, d show c the real and d imaginary part of the spin susceptibility (χ_s/χ₀) at Γ as a function of intra-orbital Coulomb interaction U for A₂Cr₃As₃ (A = Na, K, Rb, and Cs). e, f Real and imaginary part of the spin susceptibility (χ_s/χ₀) as a function of temperature at Γ of A₂Cr₃As₃. The interaction parameters have been chosen to be U = 1.2 eV and J = 0.2 U, and the results for other interaction region (U < U_c) are similar.

We now consider the NMR relaxation rate 1/T₁T. It is proportional to $\frac{1}{N}{\sum }_{{\bf{q}}}\frac{A({\bf{q}}){\rm{Im}}[\chi ({\bf{q}},\,\omega,\,T)]}{\omega }$, where A(q) is a geometrical structure factor. We chose A(q) = 1 since the geometrical structure factor has little effect on 1/T₁T⁴⁷. The NMR frequency ω is close to zero, which is chosen to be 1 × 10⁻⁵ eV in our calculation. Intriguingly, if we assume that the NMR relaxation rate 1/T₁T is dominated by the imaginary part of the spin susceptibility at q = 0 (or the so-called long wavelength approximation^48,49), the calculated 1/T₁T for Na₂Cr₃As₃, K₂Cr₃As₃, and Rb₂Cr₃As₃ (Fig. 6e, f) are qualitatively similar to the NMR experiments of^24,25,31, although such a Curie–Weiss-like temperature dependence is more accurate within a self-consistent renormalization theory⁵⁰. Nevertheless, in contrast to K₂Cr₃As₃ and Rb₂Cr₃As₃, the enhancement of the spin susceptibility in Cs₂Cr₃As₃ (Fig. 6e, f) exhibits a weak temperature dependence. This implies that the large enhancement of the spin susceptibility at the Γ point is substantially suppressed in Cs₂Cr₃As₃, consistent with ref. ³². Similar T dependence of the spin susceptibility were also obtained by Graser et al. in LaFeAsO⁴⁷ and 26% Co-doped BaFe₂As₂⁵¹.

It is also informative to consider the impact of FM fluctuation on the electronic band structure. As the spin fluctuation is most significant for K₂Cr₃As₃ in our calculations, we calculated its dynamic RPA self-energy Σ(ω) (see Supplementary Fig. 4) and obtained its dynamic correlated electronic spectrum. In Fig. 7, we show the renormalized quasi-particle energy spectrum. Away from the Fermi level, we see significantly increased scattering rate, most prominent close to A around E_F − 0.4 eV. However, the renormalization to the quasi-particle states close to the Fermi level is negligible within RPA. Thus the TPs remain and are only slightly shifted in K₂Cr₃As₃. Our results are in agreement with previous theoretical studies that correlation effect may be greatly reduced owing to the formation of molecular orbitals^52,53,54,55 and are in line with the experimental observation that no appreciable magnetic local moment is found in these systems. In addition, the comparison between the RPA and PBE results also illustrates that the dynamic RPA self-energy has little effect on the existence of TPs in these systems. Despite of its moderate effect at normal state, the FM fluctuation would influence the topological properties in superconducting state by allowing for the nontrivial spin–triplet pairing (p- or f-wave), as has been conjectured theoretically^{52,53,54,55,56} and investigated experimentally^{22,23,24,25,57}.

Fig. 7: The RPA renormalized quasi-particle spectrum of K₂As₃Cr₃ based on PBE results.

Around the Fermi level, the dynamic correlation has relative small effect on the quasi-particle spectrum. The inset shows the same spectrum with smaller energy range. The triply degenerate points are marked as red solid dots.

Discussion

In the normal state, the TP topological metal has been regarded as the intermediate phase between Dirac and Weyl semimetals¹⁴. The TPs can split into Weyl points by breaking σ_v containing the C₃ axis or merge into Dirac point after imposing inversion symmetry. Although the 233-type A₂Cr₃As₃ is absent of an inversion center, the 133-type ACr₃As₃ is centrosymmetric. It is interesting to notice that a Dirac-like crossing point appears between the α and β bands along k_z in the recently reported KHCr₃As₃^58,59. In Na₂Cr₃As₃, the surface state lies only 8 meV above the Fermi level, and it is possible for this surface state to become superconducting through proximity effect. In experiment, such a surface superconducting state might have influence on the in-plane H_c2⁶⁰, although such a possibility has been ruled out in K₂Cr₃As₃^57,61. One possible reason is that the surface state is far away from the Fermi level in stoichiometric K₂Cr₃As₃.

In the superconducting state, lacking inversion symmetry gives rise to Rashba-type ASOC interactions so that the single-particle Hamiltonian takes the form h(k) = ξ(k)σ₀ + g_k⋅σ. With an extra parity-breaking term g_k⋅σ, the mixture of singlet and triplet pairing is allowed and the order parameter takes the general form Δ(k) = Ψ_kσ₀ + d_k⋅σ, where Ψ_k = Ψ_−k and d_k = −d_−k. The triplet pairing state can be stable as long as d_k is parallel to g_k⁶². In the following, we discuss the possible nontrivial topological property in superconducting state by the symmetry analysis of superconducting gap Δ(k) and the spin–orbit coupling g_k. First, considering the gap symmetry, the large FM fluctuations favor the spin–triplet pairing component than the spin-singlet one. For the gap, symmetry belongs to the A″ representation of D_3h group (e.g., p_z-wave⁵³), Δ(k) satisfies ${U}_{{\bf{k}}}({S}_{6}){\rm{\Delta }}({\bf{k}}){U}_{{\bf{-k}}}{({S}_{6})}^{T}={\chi }_{{S}_{6}}{{\rm{\Delta }}}_{{S}_{6}}({\bf{k}})$, where ${\chi }_{{S}_{6}}=-1$ and the sixfold rotoinversion symmetry S₆ = iC₆. We calculate the ${{\mathbb{Z}}}_{6}$ indexes ($z^{\prime}_6$ and $z^{\prime\prime}_6$) associated with the S₆ according to the symmetry indicators for topological superconductors defined in refs ^63,64. The resulting $z^{\prime}_6$ and $z^{\prime\prime}_6$ are both nonzeros ($z^{\prime}_6 =1$ and $z^{\prime \prime}_6$ = −1) for mBJ results of Na₂Cr₃As₃ and K₂Cr₃As₃ (see Supplementary Table 1 for details). Second, considering the large 3D FS around the Γ point in K₂Cr₃As₃ (Rb₂Cr₃As₃), the corresponding representation of D_3h is Γ₇ (Γ₈), yielding ${{\bf{g}}}_{{\bf{k}}}={\beta }_{1}{k}_{z}\left[\left({k}_{x}^{2}-{k}_{y}^{2}\right){\sigma }_{x}-2{k}_{x}{k}_{y}{\sigma }_{y}\right]+{\beta }_{2}{k}_{x}\left({k}_{x}^{2}-3{k}_{y}^{2}\right){\sigma }_{z}$⁶⁵, where β₁ and β₂ are linear combination coefficients. Moreover, the ASOC splitting around $K\left(\frac{2}{3}\pi ,\frac{2}{3}\pi ,0\right)$ is roughly 60 meV³⁰, while along A(0, 0, π)–H(π, 0, π) it is nearly negligible. Therefore the ASOC splitting should be dominated by the β₂ term, i.e., ${{\bf{g}}}_{{\bf{k}}}={k}_{x}\left({k}_{x}^{2}-3{k}_{y}^{2}\right){\sigma }_{z}$, and thus it is invariant under ${{\bf{g}}}_{{{\bf{k}}}_{i},+{k}_{0}}={{\bf{g}}}_{{{\bf{k}}}_{i},-{k}_{0}}$. It has been demonstrated by Schnyder et al. ¹⁶ that the presence of such type of symmetry gives rise to topological nontrivial features of line nodes in the gap of NCSs. More importantly, the Majorana surface states can exist at time-reversal-invariant momenta of the surface Brillouin zone. Since in the non-superconducting state the 1D arc surface states connecting the TPs already exist, it will be more interesting to further investigate the topological properties in the superconducting state of A₂Cr₃As₃.

In summary, we have performed first-principles calculations on A₂Cr₃As₃ (A = Na, K, Rb and Cs) and analyzed the systematic variations of electronic structures, FSs, topological properties, and magnetic spin susceptibilities of the compounds in this family. Using the surface Green’s function method, we calculate the (010) surface state and find that the TPs in Na₂Cr₃As₃ and K₂Cr₃As₃ are connected by double 1D Fermi arcs. To explore the behavior of magnetic spin response, charge and spin susceptibility are obtained by employing RPA calculations. We demonstrate that the strong enhancement of spin fluctuations is present at the Γ point in K₂Cr₃As₃ and Rb₂Cr₃As₃, while in Cs₂Cr₃As₃ and Na₂Cr₃As₃, the FM spin fluctuations are not apparently enhanced. The existence of TPs is robust against the FM spin fluctuation under RPA. Based on symmetry analysis, the A₂Cr₃As₃ compounds are possible candidates for topological superconductor.

Note added

During the preparation of our manuscript, we became aware of a recent paper⁶⁶. In addition to the scenario we discussed here, they have proposed that triplet p_z-pairing can also be topological superconductors with weak SOC.

Methods

Calculation parameters

The calculations were carried out using density functional theory as implemented in the Vienna Abinitio Simulation Package^67,68. The PBE parameterization of generalized gradient approximation to the exchange correlation functional was employed⁶⁹. The energy cutoff of the plane-wave basis was up to 450 eV, and 6 × 6 × 12 Γ-centered Monkhorst–Pack⁷⁰ k-point mesh was chosen to ensure that the total energy converges to 1 meV/cell. We also performed a comparison with the mBJ exchange potentials⁴³ for Na₂Cr₃As₃ and K₂Cr₃As₃.

Structural parameters

The calculations of K₂Cr₃As₃, Rb₂Cr₃As₃, and Cs₂Cr₃As₃ were performed with experimental crystal structure. For Na₂Cr₃As₃, only the lattice constants are available from experiment, therefore we employed its lattice constants from experiment, while the atomic positions are fully relaxed with dynamical mean-field theory at 300 K^71,72.

Tight binding (TB) Hamiltonian

The band structures obtained with the PBE and the mBJ methods were fitted to a TB model Hamiltonian with maximally projected Wannier function method^73,74 using 48 atomic orbitals including Cr 3d and As 4p. The resulting Hamiltonian was then used to calculate the FS, charge (spin) susceptibility, self-energy, and surface state with surface Green’s function⁷⁵.