Undamped Higgs Modes in Strongly Interacting Superconductors

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this work the authors studied analytically Higgs modes 

in strongly interacting superconductors. This work is a 

continuation of a previous numerical study of the same 

problem.

 

I liked the main idea and I enjoyed reading the paper. 

However, before I can recommend publication, I'd like the

authors to address an important issue that is obviously 

missing form the paper. What are experimental implications 

of this work? There have been several reports of Higgs mode 

in superconductors, such as Nature Physics 11, 188 (2015). 

The authors need to discuss those previous works in their

paper. How do their results compare with experiments? What 

other experiments can one perform to prove or disprove 

their calculations?

 

After this issue is addressed, I'd be happy to recommend 

publication. 

Comments on the Quality of English Language

Minor English corrections are needed. 

Author Response

Reviewer 1: In this work the authors studied analytically Higgs modes  in strongly interacting superconductors. This work is a continuation of a previous numerical study of the same problem. I liked the main idea and I enjoyed reading the paper. However, before I can recommend publication, I'd like the authors to address an important issue that is obviously missing form the paper. What are experimental implications of this work? There have been several reports of Higgs mode in superconductors, such as Nature Physics 11, 188 (2015). The authors need to discuss those previous works in their paper. How do their results compare with experiments? What other experiments can one perform to prove or disprove their calculations? 

After this issue is addressed, I'd be happy to recommend publication. 

 

Our reply: 

We would like to thank the referee for her/his positive assessment of our work and her/his suggestion to comment in more detail on experimental implications. 

The reviewer mentions the paper by D. Sherman et al. (Nature Physics 11, 188 (2015)) in which the measured subgap absorption of strongly disordered NbN and InO films is attributed to Higgs modes. This interpretation is motivated by Lorentz-invariant models [S. Doniach and M. Inui, Phys. Rev. B 41, 6668 (1990). K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). D. Podolsky, A. Auerbach, and D. P. Arovas, Phys. Rev. B 84, 174522 (2011)] which suggest that the amplitude fluctuations of the SC order parameter can give rise to subgap absorption even in the clean case when the quantum critical point for SC destruction is approached. However, these processes involve excitations of multiple amplitude bosonic modes, so that they are expected to be subleading with respect to direct one-mode amplitude excitations, which is the issue discussed in the present work. In addition, the behavior of amplitude fluctuations in the presence of disorder in a system far from criticality, where the dynamics of amplitude fluctuations is known not to be Lorentz invariant is another question still open. Instead, the subgap absorption observed by Sherman et al. [and also by B. Cheng et al., Phys. Rev. B 93, 180511 (2016)] is most probably due to phase modes which in strongly disordered superconductors can become relevant in the subgap region due to the low superfluid density which shifts down the plasma mode in these systems, see e.g. T. Cea et al., Phys. Rev. B 89, 17406 (2014). 

 In any case, the question of the referee about the experimental implications of our work is certainly important. Our main finding is that in strongly coupled SC’s the Higgs mode appears as a single pole inside the SC gap in contrast to the weak coupling case, where it occurs at the energy of the spectral gap and therefore is strongly damped by quasiparticle excitations. Our work is only relevant for superconductors where the interaction becomes a significant fraction of the bandwidth. This is clearly not the case for condensed matter systems but ultracold fermionic quantum gases can provide a platform to investigate in a controlled way the coherent modes of superfluid systems where the interaction strength can be tuned via Feshbach resonances. In fact, in a recent experiment which we discuss in the conclusions (Ref. 14) a well-defined collective mode has been observed in an ultracold Fermi gas in the strongly correlated regime. It should be noted that in this experiment also the confinement has been modulated which may contribute to the appearance of an undamped Higgs mode but Ref. 14 reports definitely the type of experiment which is relevant for our theory. In future work it is therefore important to disentangle the role of confinement modulations vs. strong correlations in this experiment. 

Our theory does not only apply to superconductors but a similar analysis within the time-dependent Gutzwiller approximation can also be applied to magnetic systems. In Ref. 13 we have shown that our theory predicts undamped spin amplitude modes in undoped cuprate superconductors which is expected to induce a frequency-dependent Faraday rotated optical signal and therefore can be detected by magneto-optical spectroscopy. Since this experimental implication has already proposed in Ref. 13 we only briefly refer to it in the Conclusions. 

In our revised manuscript we comment in the introduction on the paper mentioned by the Reviewer and state again that our theory has implications for experiments in ultracold Fermi gases. 

Reviewer 2 Report

Comments and Suggestions for Authors

This is a very clear and well written manuscript. The work appears correct and the conclusions are well supported. I appreciate that the approximations made (flat DOS, weak-coupling limit) are in order to be able to obtain analytic formulae. It would be interesting if the authors were willing to speculate a bit about what might happen when some of these approximations are relaxed. For instance, if one uses a more realistic tight binding on square lattice DOS, which has a van Hove singularity for the half filled system (in the absence of interactions), maybe this could push the Higgs mode more strongly inside the gap? Or if not, is there some way to infer what other conditions might lead to that?

I appreciate that these questions may not be at all trivial to answer and I leave it entirely to the authors to decide how to respond. I believe that this is a good manuscript and can be published as is.

 

Author Response

Comments 1: This is a very clear and well written manuscript. The work appears correct and the conclusions are well supported. 

Reply 1:  We would like to thank the reviewer for her/his positive assessment of our work. 

Comments 2:  I appreciate that the approximations made (flat DOS, weak-coupling limit) are in order to be able to obtain analytic formulae. It would be interesting if the authors were willing to speculate a bit about what might happen when some of these approximations are relaxed. For instance, if one uses a more realistic tight binding on square lattice DOS, which has a van Hove singularity for the half filled system (in the absence of interactions), maybe this could push the Higgs mode more strongly inside the gap? Or if not, is there some way to infer what other conditions might lead to that? 

I appreciate that these questions may not be at all trivial to answer and I leave it entirely to the authors to decide how to respond. I believe that this is a good manuscript and can be published as is. 

Reply 2:  In our previous numerical analysis (PRL 132, 026501 (2024)) we have in fact investigated the dependence of the Higgs mode energy on DOS, charge carrier concentration and interaction strength. For example, it turned out that for the half-filled system (?=1) and interaction |?|?=2 (? is the bandwidth parameter) the relative shift (2Δ−Ω?????)/2Δ is by approximately a factor ‘two’ larger for a semielliptic DOS than in case of a square lattice DOS with van Hove singularity. We agree that it would be interesting to study in detail the mechanisms how the various parameters influence on the position of the Higgs mode inside the gap, however, this would only be possible within a comprehensive numerical plus analytical analysis. The purpose of the present manuscript instead is to answer the question, how the Higgs pole develops in the weak coupling limit, i.e. if there exists a critical interaction strength for the appearance of the mode or if it continuously reduces to the BCS limit. We have shown (at least for a constant DOS) that the latter scenario is realized. In our revised manuscript we briefly comment on the issue of the DOS in the Conclusions. 

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

In the first round of refereeing I liked the paper, but asked

the authors to address a certain issue that found to be missing

from the paper. The authors have resubmitted the paper, and 

in the new version they address this issue. I believe that the 

paper now reads much better and I am happy to recommend publication. 

Comments on the Quality of English Language

Minor corrections are required.