Odd-parity topological superconductivity in kagome metal RbV₃Sb₅

Introduction.—The quasi-2D kagome superconductors AV₃Sb₅ (A=K, Rb, Cs) are widely studied due to the presence of many intriguing phases in it [2, 3, 4, 5, 6], such as time-reversal-symmetry-breaking (TRSB) charge density wave [7, 8, 9, 10, 11, 12, 13, 14, 15], charge bond orders [16, 17, 18], nematic phase [19, 20, 21, 22, 23], superconductivity [24, 25, 26, 27, 28, 29, 30, 31, 32] and pair density wave [33, 34, 35, 36]. Interestingly, the in-plane magnetic response of superconductivity in RbV₃Sb₅ has recently been found to be highly non-trivial, as reported in Ref. [1]. In this experiment, the in-plane upper critical field exhibits anisotropy and demonstrates a two-fold symmetry. Additionally, hysteresis in resistance and the reentrance of superconductivity are observed under an in-plane magnetic field. These findings impose strong constraints on the pairing symmetry and rule out the conventional superconductivity in RbV₃Sb₅, necessitating a new theory to elucidate the superconductivity in this kagome superconductor.

The hysteresis of resistance observed in RbV₃Sb₅ indicates the breaking of time-reversal symmetry in the superconducting state [1]. This type of superconductivity is characterized by odd-parity and spin-polarized Cooper pairs (i.e. nonunitary). By taking into account the spin polarization of the superconducting states, the hysteresis can be explained using a domain picture. As illustrated in Fig. 1, the superconducting domains with spin-polarization resemble ferromagnetic domains. In this context, the superconducting domains with spin-polarization along the positive and negative directions are referred to as positive and negative domains, respectively. Switching the direction of the magnetic field induces a metastable state that contains two types of domains with opposite spin polarization, as shown at step \small{3}⃝ in Fig. 1 (b). Compared with the superconducting state at step \small{4}⃝ which only contains negative domains, the metastable state at step \small{3}⃝ exhibits higher energy and a lower transition temperature when the magnetic field is along the negative direction. Therefore, the hysteresis is observed during the field-sweeping process  (\small{2}⃝ $\rightarrow$ \small{3}⃝ $\rightarrow$ \small{4}⃝ $\rightarrow$ \small{5}⃝). More importantly, the existence of the metastable state is strongly supported by the current quenching experiment. Initially, a large current can be applied to fully suppress the metastable state with finite resistance at step \small{3}⃝. After removing the current, due to the presence of a magnetic field along the negative direction, the system transitions into the superconducting state with only negative domains, which exhibit zero resistance, as shown at step \small{4}⃝.

Moreover, the reentrance of the superconductivity observed in the experiment at around 400 mK can also be understood by the domain picture. When the magnetic field changes sign from negative to positive, the negative domains become less energetically favorable. Due to thermal effects near the superconducting transition temperature, these negative domains shrink and become disconnected, resulting in a finite-resistance state. As the magnetic field continues to increase in the positive direction, the positive domains become energetically favorable, grow larger, and connect with each other, leading to the reappearance of a zero-resistance state.

The minimal model for the normal state of kagome superconductor.—First of all, we construct a four-orbital (eight-band) minimal model to describe the normal state of the kagome superconductor RbV₃Sb₅. As discussed in the previous work [37], the $d$-orbitals of V atoms and the $p_{z}$-orbitals of the in-plane Sb atoms have different parities, leading to the spin-orbital-parity coupling (SOPC) terms between them [38, 39]. Fig. 2 (a) shows the real space structure of the kagome plane in RbV₃Sb₅, and the Brillouin zone of the kagome structure is shown in Fig. 2 (b).

The corresponding Hamiltonian can be expressed as:

where the specific form of $H^{4-orbital}_{\mathbf{k},ss^{\prime}}$ can be found in Supplemental Material [40]. The basis of the Hamiltonian is $\Psi_{\mathbf{k},s}=(c_{\mathbf{k},V_{A},s},c_{\mathbf{k},V_{B},s},c_{\mathbf{k},V_{C},s},c_{\mathbf{k},Sb,s})^{T}$. In the normal state, the band structures without and with SOPC along the high-symmetry line are shown on the left and right sides of Fig. 2 (c), respectively.

The construction of pseudospin basis.—In the presence of SOPC, spin is no longer a good quantum number. However, due to the presence of inversion and time-reversal symmetry, the bands of the 4-orbital minimal model $H^{4-orbital}_{\mathbf{k},ss^{\prime}}$ remain doubly degenerate at every $\mathbf{k}$ point. Therefore, we can define a pseudospin basis in which the pseudospin transforms identically to the real spin (or magnetic moment) under symmetry operations. This pseudospin basis is referred to as manifestly covariant Bloch basis (MCBB) [41, 42, 43, 39, 38, 44]. To construct the MCBB, we project the four-orbital (eight-band) model onto the band basis and select two degenerate bands near the Fermi energy as target bands. Then the pseudospin operators $\tilde{\sigma}_{i}(\mathbf{k}),~{}(i=x,y,z)$ are derived by applying the projector onto the spin operators $\sigma_{i}\otimes I_{4\times 4}~{}(i=x,y,z)$ in the real spin basis. Subsequently, we can expand these pseudospin operators using linear combinations of the Pauli matrices $\rho_{i}\,(i=x,y,z)$ (Further details can be found in the Supplemental Material [40]).

When applying a magnetic field $\mathbf{B}=(B_{x},B_{y},B_{z})$, the effective two-band Hamiltonian can be written as:

where $a_{ij}(\mathbf{k})$ arises from the expansion of pseudospin operators using Pauli matrices. In the Eq. (2), we define an effective magnetic field: $g_{j}(\mathbf{k})=-\mu_{B}\sum_{i}B_{i}a_{ij}(\mathbf{k})$, which acts on the pseudospin space. $\xi_{\mathbf{k}}$ represents the dispersion of the band near the Fermi surface.

The pairing symmetry and superconducting order parameter—Considering the nematic normal state [19], the point group describing the symmetry of normal states in the kagome superconductor becomes $D_{2h}$. Under MCBB, since we focus on the pseudospin triplet pairing states, only the irreducible representations (irreps) with odd parity are considered. The corresponding order parameters in these channels are listed in Table  5.

It can be concluded that if the superconducting state belongs to a single irreducible representation (irrep), this superconducting state cannot be nonunitary and, therefore, is not consistent with the aforementioned mechanism for the observed hysteresis and reentrance in superconducting states. Thus, a natural consideration to construct the nonunitary paring is to combine different irreps, by assuming they have similar critical temperatures. This approach is an analog to the theoretical framework proposed for the heavy fermion superconductor [45, 46, 47, 48]. By considering the nodal properties of the superconducting states [49], we choose, without loss of generality, the order parameters with point nodes located along the $k_{1}=0$ axis. The nonunitary order parameter with an in-plane magnetic moment can generally be written as:

where the parameter $\theta$ controls the direction of the spin polarization of the Cooper pairs in MCBB. For example, the order parameter $\mathbf{d}=\mathbf{d}_{A_{u}+iB_{3u}}=\sin(k_{1})(\cos(\phi),0,i\sin(\phi))$ represents a nodal superconducting state with spin polarization along the $y$-direction, corresponding to $\theta=0$. The parameter $\phi$ controls the amplitude of the Cooper pair’s spin polarization, labeled by $\eta\propto sin(2\phi)$, with $\phi$ ranging from $0$ to $\pi/2$. For $\phi=0$ or $\phi=\pi/2$, the amplitude of the Cooper pair’s spin-polarization is zero, resulting in a unitary superconducting state. In contrast, for $\phi=\pi/4$, the superconducting state is non-unitary and fully polarized, with half of the electrons remaining unpaired.

In our consideration, both of the two Fermi surfaces composed of the $p$ and $d$ orbitals (annular Fermi surfaces as shown in Fig.2 (d)) take part in the superconducting state. The order parameter given in Eq. (3) naturally gives rise to different gap amplitudes for the inner and outer Fermi surfaces, then the superconducting states are expected to exhibit a two-gap behavior. This is consistent with experimental observations of multi-gap signatures in kagome superconductors [49, 50].

The upper critical field of the superconductivity in monolayer limit.—In this section, within the MCBB, we calculate the upper critical field of the superconductivity in monolayer limit by using the gap equation. From the linearized Eliashberg equation, the gap equation of superconductivity can be derived as [51, 52]:

where the $G^{0}_{s,s^{\prime}}(\mathbf{k},i\omega_{n})$ is the Matsubara Greens function constructed by the effective two-band Hamiltonian in Eq. (2).

We simplify the gap equation by using the decoupling of the interaction [52]:

The decoupling constant is represented by $v$, which is proportional to the strength of the interaction.

Furthermore, in the experiment [1], the hysteresis behavior remains unchanged even when the initial superconducting state is established through the zero-field cooling process. This fact confirms the intrinsic unbalance between the amplitudes of two equal spin pairing states with opposite spin-polarization. Therefore, we can assume the intrinsic polarization $\eta_{0}=sin(2\phi_{0})~{}(\phi_{0}=0.3\pi$ or $0.2\pi)$ is selected by the form of the decoupling constant $v=2v_{0}(1-(\eta-\eta_{0})^{2})$. With this assumption, we can numerically solve the gap equation, accounting for the presence of the nematic normal state. In Fig. 3 (a), a two-fold symmetric upper critical field is shown. During this calculation, the direction of the magnetic moment for the superconducting order parameter is chosen to align with the applied magnetic field, minimizing the total energy of this superconductor. However, during the field-sweeping process, when the magnetic field changes sign, the superconductor enters into a metastable state which is formed by two superconducting domains with opposite magnetic moments. Thus, we also calculate the upper critical field for superconducting states with magnetic moments opposite to the applied magnetic field. Compared to the superconducting states where the magnetic moments align with the applied magnetic field, those with opposite magnetic moments have higher energy and a lower upper critical field, as shown in Fig. 3 (b).

Although the calculated upper critical fields in the monolayer limit exceed the Pauli limit, experiments show that the upper critical field of kagome RbV₃Sb₅ does not exceed this limit. This discrepancy is attributed to the suppression of the upper critical field induced by interlayer coupling [53, 54, 55]. The violation of the Pauli limit observed in another kagome superconductor, Cs$V_{3}$Sb₅ [50], which exhibits much weaker interlayer coupling [27], further supports this mechanism.

Topological properties.—In the previous section, we introduced the superconducting order parameter and discussed the superconducting properties of the kagome material within the MCBB. The results indicate that the superconductivity in this material exhibits point nodes and breaks time-reversal symmetry. Building on these nodal features and the TRSB behavior, in this section, we will explore the topological properties of the superconducting state described by the order parameter in Eq. (3). For convenience, we rewrite the order parameter as $\mathbf{d}=\sin(k_{1})(\delta_{1},\delta_{2},i\delta_{3})$, the relation between $\delta_{i}$ and $\theta$, $\phi$ can be obtained by comparing it with Eq. (3). The Bogoliubov-de Gennes (BdG) Hamiltonian of the superconductivity at zero magnetic field case can be expressed as:

where $\tau_{i}$ and $\rho_{i}$ represent the Pauli matrices in Nambu and pseudospin space, respectively. In $H_{0}(\mathbf{k})$, the magnetic field $B_{i}$ is taken to be $0$. This Hamiltonian can be diagonalized into two decoupled blocks by a unitary matrix $U_{TH}$ (whose specific form can be found in Supplemental Material [40]):

where $H_{1,2}(\mathbf{k})=$

In the normal state, there are two annular Fermi surfaces around the $\Gamma$ point of the Brillouin zone. Additionally, the zeros of the order parameter $\mathbf{d}$ are located along the $k_{1}=0$ axis. As a result, the superconductivity described by $\mathbf{d}$ exhibits four point nodes. In Fig. 4(a), the open boundary calculation at zero filed case shows the presence of Majorana zero modes (so-called edge Majorana flat band, EMFB) [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] between two point nodes belonging to different Fermi surfaces. The $k_{2}$-coordinates of the four point nodes are $-k_{2,1}$, $-k_{2,2}$, $k_{2,2}$ and $k_{2,1}$, respectively, where the latter subscript $i~{}(=1,2)$ in momentum $k_{2,i}$ represents different Fermi surfaces ($i=1$ represents outer Fermi surface, and $i=2$ represents inner Fermi surface). The winding numbers for the blocks of the Hamiltonian $H_{1}(\mathbf{k})$ and $H_{2}(\mathbf{k})$ are labeled as $W_{1}$ and $W_{2}$, respectively. We calculate these winding numbers at some representative $k_{2}$ under two different conditions: $\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}>\delta_{3}~{}(\phi=0.2\pi)$ and $\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}<\delta_{3}~{}(\phi=0.4\pi)$. The results are presented in Table. 2.

Because two blocks of the Hamiltonian do not couple to each other, the EMFBs are robust in zero-field cases. Both two blocks of the Hamiltonian preserve the ”effective time-reversal-symmetry” $T^{{}^{\prime}}$ ($T^{{}^{\prime}}=i\sigma_{z}K$, $T^{{}^{\prime}2}=1$) and the particle-hole symmetry $P$ ($P=\sigma_{x}K$), indicating that both blocks belong to $BDI$ class [67, 68, 57, 62, 69, 70, 71].

The two blocks of the Hamiltonian couple to each other when a magnetic field is applied in an in-plane direction. Without considering the momentum dependence of the effective magnetic field $\mathbf{g}(\mathbf{k})$, the Hamiltonian for a magnetic field applied along the $\bm{a}_{1}$-axis can be expressed as: $H_{BdG}(\mathbf{k})+H_{B}$, where $H_{B}=g_{x,0}\tau_{z}\otimes\rho_{x}$. We then perform the open boundary calculation under two different conditions: $\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}>\delta_{3}$ and $\sqrt{\delta_{1}^{2}+\delta_{2}^{2}}<\delta_{3}$. The results are shown in Fig. 4 (b) and (c), respectively. The evolution of the EMFB under a magnetic field differs between these two conditions. However, in both cases, the open boundary spectrum reveals the existence of a single EMFB. The winding numbers of both cases at some representative $k_{2}$ can also be calculated, and the results are shown in Table. 3.

When considering the momentum dependence of the effective magnetic field, although obtaining the open boundary spectrum is difficult, we can still calculate the winding number of the Hamiltonian. The results are the same as those presented in Table 3, indicating that the properties of the EMFB remain unchanged compared to the case where the momentum dependence of the effective magnetic field is not considered. Details of the open boundary spectrum and the winding number calculation in this section can be found in the Supplemental Material [40].

Tunneling properties.—In this section, we propose a tunnel-junction experiment to detect the nodal topological superconductivity in kagome superconductor RbV₃Sb₅. At zero-field case, we calculate the tunneling spectra of the model in Eq. (6) using the recursive Green’s function method [72, 73, 74, 75] (The details for the calculation can be found in Supplemental Material [40]). As shown in Fig. 4(d), due to the existence of the EMFB, a zero-bias conductance peak emerges as the consequence of the resonant Andreev reflection processes [76, 77], which can be evidence for the nodal topological superconductivity. Additionally, apart from the zero-bias peak, the tunneling conductance also exhibits additional peaks around $E=\pm 0.2\Delta_{0}$ and $E=\pm 0.13\Delta_{0}$, as shown in Fig. 4 (d). These additional peaks are attributed to the multigap behavior of the quasiparticle bands in the superconducting state, arising from the different gap amplitudes on the two annular Fermi surfaces (details can be found in the Supplemental Material [40]).

Conclusion and discussion.—In conclusion, RbV₃Sb₅ is an odd parity superconductor with non-trivial topological properties. The observed hysteresis in resistance and reentrance of superconductivity are attributed to the spin-triplet pairing components with finite spin-polarization. This type of spin-triplet pairing breaks time-reversal symmetry, leading to the formation of two types of superconducting domains with opposite spin-polarization. The formation and evolution of these two types of superconducting domains under an in-plane magnetic field can account for the hysteresis in magnetoresistance and the reentrant superconductivity observed in the experiment.

Within a pseudo-spin basis, we investigate the topological properties of this kagome superconductor. The open boundary spectrum reveals the presence of Majorana flat bands localized at the boundary of the superconductor. This suggests that a large local density of states can be observed at the boundary through tunneling experiments. Additionally, by combining the superconducting order parameter with the two annular Fermi surfaces in the normal state, we anticipate observing a two-gap behavior in this kagome superconductor, consistent with experimental observations [49]. This kind of two-gap behavior can also lead to additional peaks in the tunneling conductance, which could help identify the superconducting order parameters when the Fermi surfaces of the normal state are known. Interestingly, when considering the repulsive interaction, the two annular Fermi surfaces naturally favor $p$-wave superconductivity within the framework of the Kohn-Luttinger mechanism [78, 79, 80, 81, 82, 83], potentially explaining the origin of the odd-parity superconductivity observed in kagome RbV₃Sb₅. This will be further explored in our future work.

Acknowledgements—K. T. L. acknowledges the support of the Ministry of Science and Technology, China, and Hong Kong Research Grant Council through Grants No. 2020YFA0309600, No. RFS2021-6S03, No. C6025-19G, No. AoE/P-701/20, No. 16310520, No. 16307622, and No. 16309223.