Intertwining orbital current order and superconductivity in Kagome metal

Author’s response 1

We appreciate referee for reviewing our manuscript with a great summary of our work. In this study, we consider a simplified model to argue that the orbital current order is responsible for both the non-trivial band topology and unconventional superconducting (SC) orders.
Most preceding works on CDW (for example, we cited PRB 104, 045122 (2021), PRB 104, 035142 (2021)) in the spinless system introduces various types of commensurate modulation on kagome metal. Here, we particularly choose 3Q-states and attach the spin degrees of freedom to classify them and associate them with SC orders. Then, we argue that the iCDW order is closely related to unconventional SC orders through the phase relationship in u_4-term.


"I find that the most important theoretical step, the structure of iCDW state, Eq.(5), is purely explained in the manuscript. It remains unclear whether four states (5) are distinct iCDW states with different broken symmetry whose relative stability is determined by higher-order GL terms beyond quartic order or they are just multiple domains of the same state?"

Author’s response 2

We thank the referee for pointing out this issue. As we arrange in Table. I, the classification of four iCDW patterns is based on different lattice symmetries combined with time-reversal (TR) which they preserve. For 4 different iCDW patterns, the relative bond currents between up- and down-spin electrons exhibit distinctive features which are manifested in the Chern numbers in each spin sector. The degeneracy of different types of iCDW patterns in the GL theory (beyond the quartic order) is based on,
(1) the symmetry properties in Table. I in the bulk and
(2) the spin polarization preserved by the CDW order parameter.

In our work, the spin degrees of freedom is included to lift the degeneracy. In the presence of SC order parameter, we explicitly derive the u_4-term which couples the up- and down-spin CDW order parameters. Then, the degeneracy of different iCDW patterns is lifted and a particular pattern is chosen for a SC order parameter by minimizing f_(CDW&SC).
Based on the referee’s comment, we move the Figs. 5 and 6 from the appendix to the Section 3 in the main text for clarification of our explanation. Here, Fig. 5 is the graphical representation of four iCDW patterns in the main text. Fig. 6 emphasizes the difference in four edge spectrum although their GL free energy in bulk are degenerate.


"I also find that the chosen structure of the manuscript with all technical details are moved to Appendices is not very optimal. The authors may rethink the way they present technical and qualitative aspects of their study. Improved manuscript can be, in my view, accepted for publication."

Author’s response 3

We thank referee’s suggestion for improvement of our manuscript. Based on the referee’s comment, we add more detailed explanation in the main text by moving the Figs. 5 and 6, (the flux pattern and edge spectrum of 4 iCDW patterns) from the appendix to Section 3. We hope this change leads to clearer and more understandable explanation.

Author’s response 1

We are grateful for the referee’s review with interesting comments. As the referee clarified, we study the intertwined relation between iCDW and superconducting (SC) order parameters on kagome metal at the van Hove singularity (vHs). Below, we respond to the referee’s comment and present the list of changes.

"Compared to the present manuscript, at least three recent studies -- [PRB 104, 045122 (2021)], [PRB 104, 035142 (2021)], and [arXiv:2207.12820] -- reported a far more extensive GL analysis of possible CDW states, including conventional (rCDW) and iCDW, as well as their coexistence and competition, as relevant to real-world materials. The only novelty of the current work lies in its Section 4.2 that introduces superconducting order parameters and relates them to the type of iCDW, while completely neglecting rCDW and its role in the physics. In my opinion, this tiny original part restricted to one page of the manuscript out of 20, is simply too meager to justify publication. Taking into account acceptance criteria of SciPost Phys. ("detail a groundbreaking discovery, present a breakthrough, open a new pathway with clear potential for multipronged follow-up work..."), I would expect a more extensive analysis resulting in more significant insights and implications, as well as a more realistic consideration of the experimental situation in AV3Sb5."

Author’s response 2

We thank the referee for considerate comment and pointing out important references. We acknowledge that extensive studies of CDW considering both the rCDW and iCDW with their mixtures have been done. Various types of CDW modulation in the spinless system have been considered, which might be relevant to experimental results. Indeed, to understand the experimental situation of AV3Sb5, several degrees of freedom such as the multi-orbital nature and rCDW orders should be considered. In our work, however, we rather focus on how the iCDW order is closely related to both the non-trivial band topology and unconventional SC order parameters in the simplified model, which can be widely applicable to the Kagome metal system with the van-Hove singularities near the Fermi level.
Instead of considering the full sets of CDW orders, our study focuses on 3Q-iCDW, and take into account the spin degrees of freedom to associate them with the SC. It is important to note that the iCDW orders can be simply classified according to the relative bond current between up- and down-spin sectors. We show that although the different types of iCDW are degenerate in energy, they can be characterized by different edge states and Chern numbers. To remind and clarify our results on orbital currents orders, we move the figs. 5 and 6 from the Appdendix to the main text, Section 3. Then we relate the types of iCDW to the particular SC orders in Section. 4.2 as the referee pointed out. Since the Ginzburg-Landau theory (GL) free energy (especially, u_4-term) is formulated based on the symmetry properties of order parameters, our analysis is not restricted to Eq. (12) and might be extended to include other exotic CDW orders beyond 3Q-states. For example, with the ansatz of nematic charge order with orbital current such as 〖ReΦ〗(↑1)≠0,〖ReΦ〗(↑2)≠0,〖ImΦ〗(↓1)≠0,〖ImΦ〗(↓2)≠0, and zero otherwise, the SC order parameter obtained by minimizing f_(CDW&SC) with respect to constant 〖|Δ〗i | would be Δ_patch=i/3 Δ_s+(1-i)/√6 Δ(d_(x^2-y^2 ) )-1/3 √(4+3i) Δ_(d_xy ).

"Considering AV3Sb5 as the main motivation for this and many other recent theory studies of kagome metals, the following points may be of importance: 1) In real materials, iCDW does not exist by itself and always coincides with rCDW. This rCDW is typically accessed experimentally via its charge order that strongly modifies the bands and Fermi surface, especially around the M-point. It is hard to imagine that completely neglecting this rCDW and its effect on charge carriers could lead to realistic predictions for the superconductivity."

Author’s response 3

We acknowledge the referee’s comment emphasizing the importance of rCDW. Of course, to understand the precise modulation in real space, the inclusion of rCDW is required. If the modified band by rCDW keeps the vHs (so that the diverging density of states close to the M-point exists), one can speculate the effect of rCDW based on our GL theory as the following. First, because of the modified Fermi surface, the dispersion close to M-points would be deviated from the perfect nesting, which results in a correction to the free energy coefficients u_i. However, we emphasize that the GL formalism in Eqs. (18)-(20) is still valid regardless of microscopic details and the real part of CDW. Second, the rCDW barely brings about unconventional SC order parameter in f_(CDW&SC). If both CDW order parameters, Φ_↑ and Φ_↓ are purely real, then f_(CDW&SC) always favours the s-wave SC order parameter (real Δ_(i=1,2,3)) and f_(CDW&SC) simply controls the relative amplitude of order parameters. In other words, the anisotropic pairing in SC order parameter is mostly contributed from the imaginary part of CDW.

"2) Recent experimental work shows that different types of rCDW may exist and possibly coexist in CsV3Sb5, see [PRB 105, 195136 (2022)], [arXiv:2203.15057], [arXiv:2203.12317], [arXiv:2201.05211]. Therefore, it is an experimentally relevant question what happens with iCDW and superconductivity in this case. Could different types of iCDW be combined with each other? What will be implications of this coexistence for superconductivity?"

Author’s response 4

We appreciate referee’s comment with a reference. The cubic coupling of rCDW and iCDW ~ (ReΦ_1σImΦ_2σImΦ_3σ+...) with spin σ might relate different types of iCDW due to the coexistence of rCDW orders. The coexistence of CDW might introduce new kinds of modulations and also diversify the SC order parameters other than Eq. (12) in our simple estimate. The coexistence of cases (i) ~ (iv) on the same ab-plane is possible for a specific choice of ReΦ_iσ, e.g. Φ_↑ = (i,i,i),Φ_↓ = (0,-i,i) Then a similar analysis in the main text leads to Δ_patch=√(2/3) Δ_s-1/2 Δ_(d_(x^2-y^2 ) )+1/(2√3) Δ_(d_xy ). assuming the ReΦ_iσ is small compared to ImΦ_iσ. Thus, we expect that the coexistence of rCDW might diversify both the iCDW type and SC orders beyond our ansatz. The detailed analysis including rCDW is in progress.

"3) In AV3Sb5, superconductivity exists over a much broader doping and pressure range than rCDW. Therefore, it is not granted that the type of the CDW prescribes the superconducting order parameter at all. In fact, itinerant charge carriers are only weakly affected by the CDW formation, as shown by the nearly unchanged plasma frequency measured in optics, see [PRB 104, 045130 (2021)] and [PRB 105, 245123 (2022)]. It would be natural to ask what experimental signatures could signal the relevance of the authors' scenario of superconducting order parameter due to iCDW. For example, what range of Tc's should be expected? Could multiple superconducting gaps occur?"

Author’s response 5

We thank referee’s comment. For multigap SC, it requires another set of vHs in addition to our model. The multi-orbital model as an extension of our study covers this possibility, which we leave as a future research. Our simplified model considers the regime of small doping and pressure where the prescribed CDW is dominant. Of course, it is also possible that a well-developed SC order parameter prescribes the CDW order if the SC order is primary. For a wider range of doping and pressure, it is required to estimate the free energy coefficients to determine which order is predominant than others. (and similarly for the range of Tc) It needs comprehensive understanding of microscopic details beyond the perfect nesting in the main text. Even the predominance among collective orders is not settled and the relation between the CDW and SC order parameter still needs to be investigated. In the main text, we have suggested angle-resolved Josephson junction experiment to identify the SC pairings and the local density of states which manefest the symmetry breaking patterns by CDW order. In addition, we plot the spectral function to display the edge states contributed from both CDW and SC order parameters.

"4) Several experimental studies suggest s-wave superconductivity with two gaps in CsV3Sb5, see [Science China Phys. Mech. Astron. 64, 107462 (2021)], [npj Quant. Mater. 7, 49 (2022)], [arXiv:2203.05770]. How does this fit to the theory presented here? Could one determine the exact type of iCDW, make concrete predictions for the anomalous Hall effect (or some other measurable property), and compare them to the experiment?"

Author’s response 6

We thank referee for pointing out important points. Since our simplified model considers a single band at the van Hove filling (one-gap model), the multigap picture is not considered in our study. To consider multigap SC, multiple bands close to vHs are needed to associate them with SC pairings. We expect the multi-orbital system is more appropriate to fit it to the references. For measurable quantity related to different types of iCDW, we have shown distinct LDOS in Fig.5 in the main text.

Author’s response 1
We thank the referee for the careful review and helpful feedback. As clarified by the referee, we study the charge density wave order (CDW) and superconducting (SC) orders on the Kagome metal with a p-type van Hove singularity (vHs).


"There is a slight mismatch between the microscopic model employed and the experimental connections drawn: there is no clear unanimous sign for orbital currents in kagome metals. "

Author’s response 2

We appreciate the referee for pointing out an important issue. Our motivations for the orbital current order (iCDW) are the following. One is the experimental observation of anomalous Hall effect (AHE) along with the CDW transition without any static magnetic order (for example, we cited Phys. Rev. B 104, L041103 (2021), Journal of Physics: Condensed Matter 33(23), 235801 (2021) and Science Advances 6(31), eabb6003 (2020)). It motivates us to focus on the time-reversal (TR) symmetry breaking contributed from CDW order itself. The orbital currents characterized by iCDW orders naturally give rise to the nonzero Chern numbers which might be associated with the AHE. Additionally, we are motivated by the following experiments (for example, Nature Materials 20.10 (2021): 1353-1357 and Nature 602(7896), 245 (2022)), which report the chirality of CDW orders or equivalently, the TR breaking CDW.

The second reason is the non-trivial phase relationship between the iCDW and the SC order parameters in the free energy f_(CDW&SC). We speculate that the unconventional SC orders are closely related to the complex CDW orders. Our Ginzburg-Landau analysis verifies the stability of 3Q-CDW states |Φ_1σ |=|Φ_2σ |=|Φ_3σ | rather than 1Q or 2Q states (Eqs. (6) and (7) in the main text). If the 3Q-CDW is purely real, then the interaction term Eq. (10) between the CDW and SC order parameters (especially, the u_4-term) always favors the s-wave SC order parameter and the interaction term simply shifts their critical temperatures. (Here, the formulation in Eq. (10) is valid regardless of the presence of the real or imaginary part of CDW). In other words, the non-trivial phase relationship in u_4-term implies that the imaginary part of CDW is essential to promote the non-trivial phases of SC order parameters Δ_α or equivalently, unconventional SC orders.

In summary, our viewpoints are
(1)The TR symmetry breaking occurs without static magnetic order so that it can be contributed from CDW only,
and
(2)Instead of the rigorous explanation of experiments, we focus on the imaginary part of CDW since it is closely related to both the band topology and the unconventional SC orders.

For clarity, we supplement our viewpoints in the Introduction part (specified in List of changes).


"Even for the Kagome Hubbard model at van Hove filling, the situation is still largely undetermined theoretically - it is a question of microscopic parameters whether the current-carrying or the T-invariant CDW is energetically favoured. "

Author’s response 3

We thank the referee for a valuable comment. Following the Ginzburg Landau free energy, Eq. (18) in the main text, the instabilities of T-invariant and the current-carrying CDWs are signaled by
1/2|I_rCDW | +χ_CDW=0 and 1/2|I_iCDW | +χ_CDW=0,
where I_rCDW and I_iCDW are the interaction strengths in the patch model and χ_CDW is the particle-hole susceptibility so that the critical temperature becomes
T_rCDW ~ t exp[-A/√(|I_rCDW | )] and T_iCDW ~ t exp[-A'/√(|I_iCDW | )].

As the referee pointed out, the explicit dependence of I_rCDW and I_iCDW on microscopic parameters such as on-site interaction, Hund coupling, or evenly the pressure are still elusive. Despite the complexity of microscopic details, we can still infer meaningful outcomes in the regime where the experimental studies predict the presence of current-carrying CDW (Author’s response 2). Since our free energy analysis is based on the symmetry properties of the CDW and SC order parameters only, we can still draw meaningful conclusions from f_(CDW&SC). For example, the opposing behaviors between the two critical temperature f_(CDW&SC)>0 and the relationship with SC orders (Eq. (12) in the main text) work well even if we do not know the precise microscopic parameters.

"Similarly, there is no clear cut motivation why to exclude f-wave pairing as a candidate for superconductivity in the kagome Hubbard model. In fact, it is rather f-wave superconductivity which is advocated by a significant part of the literature."

Author’s response 4

We thank the referee for bringing out this subject. Following the viewpoint in Author’s response 2, we assume that only iCDW is responsible for the TR symmetry breaking so that the SC order parameter is spin-singlet (s- or d-wave SC orders). However, as the referee commented, several significant papers have presented the importance of f-wave SC order while our patch model considers only the spin-singlet SC, s- and d-waves. Our motivation for choosing this ansatz in the patch model is as follows.

We have compared the particle-particle susceptibilities for intra-patch pairing, χ_intra, and inter-patch pairings, χ_inter. The former is responsible for the spin-singlet s- and d-wave while the latter is responsible for the f-wave SC order. In the limit of perfect nesting, it turns out that (Eq. (25) in the appendix)
χ_intra ~- t〖(log Λ/T)〗^2 and χ_inter ~-t(log Λ/T),
so that the perfect nesting vHs which we evaluated in the main text is highly susceptible to the spin-singlet SC orders. In future, we expect that our simple model analysis offers a new insights to study more thorough microscopic details.