FL* Interpretation of a Dichotomy in the Spin Susceptibility of the Cuprates

Round 1

Reviewer 1 Report

The author proposes a way to reconcile different, and seemingly
opposing results in the spin susceptibility in the cuprates. The
idea is based on the so-called FL* fractionalized Fermi liquid
concept, proposed earlier in Ref.[7] and [8]. I liked the idea
of the paper, however before I recommend publication, I'd like
the author to address the following issues:

1) In Fig.1 the author compares the results of his calculations
with experimental results on spin susceptibility from Ref.[5].
The agreement is only qualitative. What can be done to improve
the agreement? Does FL* theory has any parameters that can be
fine tuned in order to match experimental data better?

2) Does FL* theory can be used to calculate some other experimentally
accessible quantities? Surely other physical quantities could
also be described as either Fermi or non-Fermi liquids, and
the FL* theory might be able to help distinguish between them.

After these issues are addressed, I'd be happy to recommended
publication.

Author Response

I thank you for your Referee report and below I reply point to point to your remarks, quoting some modifications in the text motivated by them.

            In Fig.1 the author compares the results of his calculations
            with experimental results on spin susceptibility from Ref.[5].
            The agreement is only qualitative. What can be done to improve
             the agreement? Does FL* theory has any parameters that can be
             fine tuned in order to match experimental data better?

I added in the text the following clarification:

“Let us remark that within such approximations the doping and temperature dependences are completely  determined by the theory only up to the scales  involved which have been optimized for one doping and never then changed.”

One could certainly improve the description of the Fermi surface and the related Fermi momenta, which in the paper are just considered as an average, but in my opinion too many aspects of the cuprates are not taken into account by the t-t’-J model that ask just in terms of it  for an accurate description is to demand too much.

The idea here is that nevertheless the key low-energy features are inddeed captured by this FL* approach to the model.

2) Does FL* theory can be used to calculate some other experimentally accessible  quantities? Surely other physical quantities could also be described as either Fermi or non-Fermi liquids, and the FL* theory might be able to help distinguish between them

I added a related comment in the section Discussion:

“ Following the strategy here advocated for the spin susceptibility the FL* approach presented allows to solve FL vs NFL dichotomies appearing for  several physical quantities and in different regions of the phase diagram of the cuprates, such as the dichotomy between in-plane resistivity and ARPES in PG quoted in the introduction. Computations in this direction are in progress, with preliminary positive results \cite{Marchetti6}.”

Reviewer 2 Report

The manuscript "FL* interpretation of a dichotomy in the spin susceptibility of the cuprates", by Pieralberto Marchetti, seems to contain results that are interesting and sound enough to warrant publication.

The properties of cuprates are a matter of intense debate, many issues are still controversial and many competing scenarios have been put forward by various groups. Thus, very likely, some of the assumptions made by the author of this piece of work will not be unanimously accepted within the scientific community, but I find that they are clearly stated and therefore can be the object of forthcoming physical debate.

Overall, the manuscript is clearly written and the list of references provides a reasonably complete overview on the subject at issue in this piece of work, namely the Fermi-liquid vs. non-Fermi-liquid controversy about cuprates.

Maybe, my only suggestion to improve the quality of the presentation of this piece of work concerns the concluding Sec. 4, which appears way too succinct to be of use to the reader. I wonder whether the author can make an extra effort to expand the final discussion, putting his work in a wider perspective, maybe in connection with other research lines, or with presumable developments of the author's research line.

I would also suggest the author to provide a bibliographic reference for the Reizer momentum.

With the above changes made, I think that the manuscript may be accepted for publication.

Author Response

I thank you for the Referee report and I reply below to your remarks.

 Maybe, my only suggestion to improve the quality of the presentation of this   piece of work concerns the concluding Sec. 4, which appears way too succinct to be of use to the reader. I wonder whether the author can make an extra effort to expand the final discussion, putting his work in a wider perspective, maybe in connection with other research lines, or with presumable developments of the author's research line.

I extended the final section with some comments and an announcement of some developments of the strategy outlined in the paper adding two related sentences

"We find a qualitative agreement with data for the doping and temperature behaviour of the the uniform susceptibility, dominated by the hole-resonance and exhibiting at high temperature  behaviour essentially of FL  type, modified at low temperature by high pseudogap effects which we attribute to charge pairing. Actually one can recover an improvement of the comparison at small dopings and temperature, where it is worse, by considering a crossover to the low pseudogap "phase" PG. \cite{Marchetti6}."

(I do not introduce this addition in the paper because it would need a discussion of the PG “phase” that was not considered here.)

“ Following the strategy here advocated for the spin susceptibility the FL* approach presented allows to solve FL vs NFL dichotomies appearing for  several physical quantities and in different regions of the phase diagram of the cuprates, such as the dichotomy between in-plane resistivity and ARPES in PG quoted in the introduction. Computations in this direction are in progress, with preliminary positive results \cite{Marchetti6}.”

I would also suggest the author to provide a bibliographic reference for the Reizer momentum.

I provided the reference to Reizer's paper.

Reviewer 3 Report

I attached a PDF file with my review.

Comments for author File: Comments.pdf

Author Response

I thank you for the Referee report and for your useful comments. I try below to reply point to point to your remarks, quoting some modifications in the text motivated by them.

Although I understand that most of the theoretical formalism developed is published elsewhere, I believe that the manuscript would benefit from the introduction of more details regarding the mathematical calculations performed during the spin-charge decomposition and in the introduction of the charge and spin fluxes. Moreover, the physical consequences derived from the calculations shown in the text are not clear for the general reader. Instead of making the manuscript focused on those who are familiar with the formalism, I would like to see a more robust line of reasoning that led the author to claim the results, mainly in section 3, without pointing only to other references. For example, a complementary discussion regarding the Reizer momentum and, if possible, other references discussing its dependence on temperature and how this affects the resonant states of holons and magnons.

I added a reference and expanded the related text as

“As a consequence of the $T$-dependence of the Reizer momentum, the hole and the magnon resonances  have a strongly $T$-dependent life-time leading to a behaviour of these excitations less coherent than in a standard Fermi-liquid. In particular in the "strange metal phase" it behaves as $T^{-4/3}$. To take into account the effect of gauge fluctuations beyond perturbation theory  as a very rough approximation we apply a kind of eikonal resummation. This resummation is obtained by treating first $a_\mu$ as an external field, expanding the correlation function in terms of quantum mechanical paths of spinons and, for the hole, also of holons, then integrating out the leading transverse component of $a_\mu$ to obtain an interaction between paths, controlled by the Reizer momentum. The interaction is then treated in the eikonal approximation. Finally a Fourier transform is performed to get the retarded correlation function, treating the short scales via a multiplicative scale-renormalization, assuming as UV cutoff the Reizer momentum, see \cite{Marchetti2} and references therein.”

Concerning the spin-lattice relaxation rate I enlarge the text including an additional equation (eq. 7) on which the calculation is based.

Furthermore, I believe the manuscript would benefit from a more detailed discussion and conclusion sections. Although qualitatively the results are in agreement with experimental measurements, some deviations are pointed by the author and are clear in the comparison shown for the spin susceptibility, for example for the starting of the downward behavior for doping values above 0.14.

I added a comment about an improvement for low dopings taking into account the crossover to PG:

“Actually one can recover an improvement of the comparison at small dopings and temperature, where it is worse, by considering a crossover to the low pseudogap "phase" PG. \cite{Marchetti6}.”

I do not introduce this addition in the paper because it would need a discussion of the PG “phase” that was not considered here.

Now, some specific questions regarding the content:

1) There is a proposition to explain the dichotomy of FL vs NFL behavior in cuprates in terms of the FL* fractionalized Fermi liquid theory, that is, in terms of holons and spinons. The application of such theory is said to be given in terms of the t-t’-J model, that the author explicitly points to be insufficient to explain other features of the phase diagram. My question regarding this topic is divided into two parts: first, is it possible to apply the same formalism, namely spin-charge decomposition of hole operators, to other models that are claimed to explain some features of cuprates superconductors, as for example the extended Hubbard model or other approaches? Second: the cuprates are known to be dirty systems under the doping with oxygen atoms for example. Is the formalism capable of dealing with heterogeneous dopant distributions? I mean, if explicitly introduced in the Hamiltonian a disorder term or if the square lattice is not perfect due to impurity interstitial states, is the formalism still applicable?

 4) I would like to have a clearer explanation on how the spin-charge decomposition in the 2D t-t’-J model is able to recover an exact rewriting of the model in terms of the change of statistics. For the 1D case, one can compare the results from the decomposition with the exact solution of the Bethe ansatz. How is this made for the 2D case? Furthermore, I would like to have a more detailed description of how the statistics changes of U(1) and SU(2) compensate each other

I added a related comment specifying under which condition the rewriting with U(1) and SU(2) gauge fields can be implemented  and I expanded the text as:

“These fluxes don’t modify the dynamics and the statistics changes of $U(1)$ and $SU(2)$ compensate each other thus providing an exact rewriting of the model. This is rigorously proved in the $t$-$J$ model in the previous references using the euclidean path-integral approach, where the introduction of the charge and spin fluxes is implemented by minimal coupling the fermions of the $t$-$J$ model to Chern-Simons gauge fields. The proof is based on the representation of partition and correlation functions in terms of quantum mechanical paths of the fermions, where the Chern-Simons gauge fields appear in phase-factors associated to the fermion worldlines. Since the Chern-Simons gauge theory is topological, the only effect of the $U(1)$ and $SU(2)$ gauge fields is the introduction of phase factors, respectively $i$ for $U(1)$ and $-i$ for $SU(2)$, for each undercrossing and the opposite factors for each overcrossing of the fermion worldlines. Therefore the two contributions cancel in every crossing among each other, thus providing an exact rewriting of the model. The crucial ingredients for such rewriting is the existence of global $U(1)$ charge and $SU(2)$ spin symmetries of the model, allowing the gauging by Chern-Simons gauge fields and, for lattice models, the no-double occupation constraint, forbidding finite intersections in the worldlines of fermions , so that the crossings are well-defined. Therefore this procedure can be applied to any model with the above features and in particular the extension to the $t$-$t'$-$J$ model considered here is straightforward.”

According to the above description one can guess that a rigorous implementation of spin-charge decomposition in the Hubbard model can only be performed when the model is transformed in such a way that in the low-energy sector one can apply the no-double occupation constraint. The idea is that, however, the relevant low-energy excitations can still be described in terms of a t-J model and its generalization. Furthermore if one believes that the Zhang-Rice singlets provide a good description of the low-energy physics,  then one might argue that for such description one can be directly brought to such models, without going through the Hubbard one.

Again accordig to the above description, there is instead no problem for the spin-charge decomposition with the introduction of disorder.

2) How does orbital degrees of freedom could enter in the description of the model introduced in the manuscript? Could one make the decomposition of the hole field interms of not only spinons and holons, but also orbitons, describing a possible multi-orbital scenario?

I must admit that I never thought about this issue, which , however, I do not believe essential for the topics discussed in this paper.

3) I would like to point out a recent article that the author cites in Ref. 9, (G. Seibold et al, Communications Physics 4, 7 2021), where the authors claim that the strange metal behavior of cuprates is explained by charge-density fluctuations, related to the stablished charge-density waves of the cuprates phase diagram. How does the author relate the formalism developed in the manuscript with the recent experimental findings of X-ray scattering experiments, since the t-t’-J model does not reproduce the charge-density waves scenario?

To present my point of view on this issue I added in the text:
“Of course it is known that this model is insufficient to explain some phenomena appearing in the cuprates, such as charge density waves and fluctuations (see e.g. \cite{Seibold}),  and a more complete description is needed including  doubly occupied sites, oxigen, phonons and disorder. However, apparently these phenomena are qualitatively irrelevant for the issue of the present paper, where we briefly discuss the above mentioned dichotomy for the spin susceptibility.”

 5) I would like to understand if this approach used to the spin susceptibility can also be used to derive and understand the behavior of other quantities in the SM phase. For example, it is known that the resistivity has a linear dependence with temperature in the SM phase. Is this formalism capable of reproducing this result?

I added a comment on this issue:

“As a side remark we notice that with the same formalism a linear in $T$ contribution to the in-plane resistivity is reproduced in SM, but with the slope decreasing with doping, as in the experiments \cite{Marchetti2}.”

6) How does the author explain the missing upward shift with temperature of the spin- lattice relaxation time in the calculations, when compared to the experimental results?

I added a comment about this issue:

“In the above calculation of ${}^{63}T_1T$ the upward shift with doping of the experimental curves is not reproduced, but one can conjecture that it might arise from a contribution of weak antiferromagnetism of the hole resonance.”

Also, some notes regarding the writing:

• Below Eq. (2), when averaging the gradient of spin flux, appears the quantity ??. It is not introduced elsewhere in the text. Please specify what this quantity represents.

Thank you, I specify it in the text.

• In Eq. (5), the square brackets are not rightly closed. I believe that all the square root is inside the hyperbolic tangent

Thank you, I correct the misprint.

Round 2

Reviewer 2 Report

The author has taken into account the main suggestions and comments of the reviewers in a reasonable manner, the manuscript has been modified accordingly, the revised version has been definitely improved with respect to the previous version, therefore I do now recommend publication in the present form.

Reviewer 3 Report

I thank the author for the modifications and further explanations. I am satisfied with the new version of the manuscript and I recommend publication on Condensed Matter.