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> ----------------------------------------------------------------------
> Report of the First Referee -- BN13654/Kent-Dobias
> ----------------------------------------------------------------------
>
> The work deals with a purely phenomenological model for the “hidden”
> order parameter if URu2Si2, with particular emphasis on the expected
> elastic properties. The work might eventually be suitable for Phys.
> Rev. B, but some aspects are not clear to me.
>
> The main result is Fig. 2, where the behavior around TN is difficult
> to see. I suggest to add zooms on that crucial T-range, where it seems
> to me that there is a qualitative difference between model and
> experiments. The justification given by the Authors (“mean field
> theory—which is based on a small-eta expansion—will not work
> quantitatively far below the transition where eta has a large nonzero
> value and higher powers in the free energy become important”) does not
> look plausible as the disagreement does not appear to develop slowly
> as T decreases, but appears immediately below TN, where eta is small.
The disagreement between the theory at low temperature is largely resolved by
the addition of an additional interaction in the mean-field free energy of the
form ε²η². In a new appendix, we have worked through the mean field modulus
implied with this new interaction and a fit is now shown in Fig. 2 as a thin
black line.
An inset with a zoom on the critical region has been added to the figure.
> Is it not clear how discriminatory is the agreement above TN in 2a, 2b
> and 2c. Are calculation results robust over a wide range of fitting
> parameters, or does the agreement result from a fine-tuning? (e.g.,
> the presence of a maximum at 120 K in 2b).
In order to check the agreement in our fit, we preformed the fit with a moving
temperature window that cuts off at T_max. Our fit's parameters are x_i for i =
1, …, 5 for C₀ = x₁ - x₂ (T / K), x₃ = a/b², x₄ = b²/Dq⁴, and x₅ = b √(-g/u).
The variation of these parameters as a function of T_max are shown in
referee_response_cutoff_parameters.pdf. The parameter x₁ is fairly stable at
all temperature cutoffs, while the rest vary by at most 20–60% of their 275K
value down to cutoffs of ~90K.
More insight into the consistency of the fit comes from examining the linear
combinations of parameters that form eigenvectors of the fit covariance matrix,
since these have uncorrelated uncertainties. For the fit including all
temperatures (up to 275K), these are (in order of fit uncertainty):
y₁ = -0.00198126 x₁ + 2.16869 10⁻⁶ x₂ - 0.99998 x₃ - 0.00227756 x₄ - 0.00560291 x₅
y₂ = -0.0151198 x₁ + 0.0000415145 x₂ - 0.00552438 x₃ - 0.0205384 x₄ + 0.999659 x₅
y₃ = 0.635138 x₁ - 0.00196902 x₂ - 0.00315925 x₃ + 0.77197 x₄ + 0.0254495 x₅
y₄ = 0.772222 x₁ - 0.00663886 x₂ - 0.0000753204 x₃ - 0.635317 x₄ - 0.00137316 x₅
y₅ = 0.00637806 x₁ + 0.999976 x₂ - 4.32279 10⁻⁶ x₃ - 0.00269696 x₄ - 4.93718 10⁻⁷ x₅
The variation of these parameter combinations as a function of T_max are shown
in referee_response_cutoff_eigenvectors.pdf. The parameter y₁, which is
principally x₃ = a/b², varies the most with the cutoff, at most around 60% of
its value until ~90K. The parameter y₂, which is principally x₅ = b √(-g/u),
varies at most around 15% of its value until ~90K. The other three parameters
are stable at any cutoff, and are likewise mixed combinations of x₁, x₂, and
x₄.
Plots of the fits performed between 90 and 275 K are shown in
referee_response_cutoff_curves.pdf.
> Is it possible to say something about the c/a ratio, which displays a
> non-trivial T-dependence?
The c/a ratio is governed by the behavior of the A1g moduli, which exhibit no
novel behavior in our theory. We therefore have nothing to say about it.
[Add something to cite about why we shouldn't need to say something about this?]
> At last, I understand that the model is meant to be purely
> phenomenological, but given the plethora of publications on URu2Si2
> over 30 years, where any conceivable order parameter has been proposed
> as candidate, the Authors should make a connection between their
> abstract OP and possible physical realizations. For instance, in the
> simplest framework of localized f-electrons, what ionic moments would
> fit the present proposal?
In a picture of localized U-5f electrons this
This is a consistent with hexadecapolar order
[Mike, can you help with this?] [Yes I think he can! And it's a good point]
> ----------------------------------------------------------------------
> Report of the Second Referee -- BN13654/Kent-Dobias
> ----------------------------------------------------------------------
>
> In this paper, possible elastic properties of URu2Si2 are studied with
> focusing on the long-standing hidden order (HO) problem. The authors
> introduce a generic form of the free energy density for the elastic
> energy, a modulated order parameter, and their mutual coupling, and
> analyze the temperature dependences of the elastic constants by
> minimizing the free energy. It is shown that the B1g component
> exhibits a remarkable softening with decreasing temperature and a cusp
> singularity at the HO transition point, and these results are compared
> with recent ultrasound experiments. From the comparison, the authors
> conclude that the HO phase of URu2Si2 originates from the modulated
> B1g order parameter.
>
> In the course of evaluation, the referee does not recommend the paper
> to be published in PRB, mainly based on the following reason.
>
> 1) The scheme for the coupled strains in this paper is quite standard
> within the mean-field treatment and does not provide a novel
> theoretical advance.
[hmmmm what can you say to this. i guess we can emphasize the part you did that IS new, at least we think it's new. Also, what's wrong with using existing, working tools for sovling problem?!?!!!!?!?!?! Does everyone have to come up with some black-hole-based-nonsense every time they solve a problem?]
>
> 2) One can generically expect several sources for softening elastic
> constants. For example, the authors in ref.25 also succeeded in the
> quantitative fits in the framework of a 4f crystal field model for T >
> T_HO. Thus, the fitting is not regarded as the decisive evidence on
> the validity of the model.
[Not sure how to respond to this; Brad?] [yeah Curie-weiss is generic, that's true. ref 25 is also purely phenomenological (I'm guessing, haven't looked yet), we have testable predictions and connect to other experiments. ]
> 3) The agreement of C[B1g] in the region T<T_HO is poor, though only
> the cusp at T_HO seems qualitatively consistent with the experiment.
> Moreover, the referee expects that even a cusp structure in the
> elastic constants is not unique to this model; it can be obtained from
> more general models beyond the linear coupling (4), within the
> mean-field level. Therefore, the referee thinks that this analysis
> does not lead uniquely to the authors' arguments on the realization of
> the B1g order parameter.
The disagreement between the theory at low temperature is resolved by the
addition of an additional interaction in the mean-field free energy of the form
ε²η², now shown in Fig. 2 as a thin black line.
While terms like this provide cusp-like features in the modulus for each strain
symmetry, they cannot explain the 1/ΔT softening seen in the high-temperature
phase, since their contribution to the response function is zero above T_c.
> 4) The most important point in the HO problem is the microscopic
> identification of symmetry breaking and the order parameter. In spite
> of the long history in research over almost 40 years, there is no
> experimental evidence of the formation of any superlattice structure
> at least at ambient pressure. So, the proposed modulated order is not
> consistent with the absence or identification of symmetry breaking.
> The authors do not provide any resolution on that point which is the
> most relevant in this problem.
The articles below, also cited in our work, all provide experimental evidence
for the formation of superlattice structure along the c-axis at ambient
pressure.
[I pulled these from our citation on the estimate for q_*. Do they actually
provide the evidence we need? Can someone who knows more about these techniques
elaborate?]
https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.111.127002
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.43.12809
https://journals.jps.jp/doi/10.1143/JPSJ.79.064719
https://www.nature.com/articles/nphys522
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