On the dynamic distinguishibality of nodal quasi-particles in overdoped cuprates

This is a thought provoking paper on an interesting topic. It deserves to be read. At several points, however, I found the logic hard to follow. I would recommend a partial re-writing to try to bring out the ideas as cleanly as possible.

First let me summarize my understanding of what I think paper is saying:

1. At dopings immediately above the onset of the superconducting dome, the resistivity of LSCO shows a T^2 temperature dependence at the lowest temperatures. This is a well-known fact.

2. The author observes that a dimensionless measure of the size of the T^2 resistivity shows that it is larger than in other metals with a similar behavior. Instead, the magnitude of the T^2 term in LSCO at this doping is comparable to He-3 on the verge of solidification.

3. The author then argues that — based on the concept of dynamic distinguishability from refs 1,2,3,4 — at the onset of solidification in He-3, Fermi-Dirac statistics doesn’t apply to the nearly-solidified fermions because they are localized in space and unable to exchange places with other fermions.

4. The author then suggests that a similar phenomenon may apply to the nodal electrons in LSCO, so that some of the nodal electrons are “excluded from the Fermi sea”.

5. The author then argues that if this phenomenon does occur it could explain a variety of observed facts at slightly lower doping including (i) planckian T-linear dissipation, by scattering off the classicalized electrons, (ii) isotropy of T-linear scattering, (iii) the evolution of carrier density with doping and (iv) the onset of superconductivity.

There are two things that I find broadly attractive about this line of argumentation.

The first is that it is interesting to compare the dimensionless magnitude of T^2 resistivity between materials, with the idea that there might be a maximal scattering rate analogous to ideas of a Planckian bound on T-linear scattering. At that this maximum might have some interesting physics in it.

The second is that electrons in cuprates are indeed, broadly speaking, on the verge of localization and/or solidification of various kinds. I didn’t see this connection made explicitly in the paper, where solidification is discussed explicitly in He-3 but not in cuprates. I think that the paper is arguing that the vibrations of localized, classical electrons could provide the T-linear scattering that is observed. I’m not sure if this is very plausible (see below), but the general idea is thought-provoking.

It is not completely clear to me what the language of distinguishability brings to this discussion (especially in the absence of a clear computational framework). Once the electrons start to localize it’s obvious that they should not be described by Bloch vectors in a Fermi surface. In the discussion of distinguishability, just before the start of section IV, the author seems to be conflating two things: the onset of solidification, and increase in scattering rate. I understand the idea that a particle in a solid is localized and “sees” its neighbors less and may perhaps be more classical. However, the author then states that (i) it is collisions that lead to indistinguishability in a liquid and (ii) that at the onset of solidification an increase in collisions “confines” the particle to a unit cell, making it distinguishable. These statements are in tension: do the collisions help or hinder distinguishability?

Related to this point, the nodal electrons don’t seem to be close to localization at a doping of 0.3. The mean free path of these electrons (the discussion is at very low temperatures) are long (compared to the lattice spacing and to the de broglie wavelength). They are still good metals and a Bloch representation of the electronic states should be good. Now, if, nonetheless, there were some kind of localization or solidification going on, there should be some evidence for that at these dopings.

I’m actually a bit confused why the author wishes to make the nodal electrons the ones that are potentially becoming classical. The author states “the nodal quasiparticles which have the fastest Fermi velocity suffer the largest electron-electron collision rate”. It’s true of course that if all the electrons have the same mean free path, then the faster ones have the shorter lifetime. However, the mean free paths are not the same here. In general, the anti-nodal electrons in cuprates are more incoherent and therefore have shorter lifetimes. The modeling around equation (3) is probably not correct — there should be a sum of channels that make up the conductivity, allowing for different lifetimes/mean free paths/velocities for the nodal and anti-nodal contributions.

If there are excitations that are classical at these low temperatures, these would have a big signature in the specific heat. From equipartition c/T ~ 1/T is large. I do not believe there is evidence for this temperature dependence (or even more weakly, some kind of increase in the density of states) in cuprates.

Given that the paper is structured around an analogy with He-3 it would be more compelling if some of the phenomenology the author wishes to explain (T-linear scattering, superconductivity, etc) were also visible there.

I understand that the author is presenting a speculation rather than a theory. However, all of the statements in the final section that are supposed to follow from the classical electrons are quite vague: the scattering mechanism “will find a straightforward explanation”, the isotropy of scattering “would not be surprising”, the evolution of carried density “becomes less surprising”, pairing “would naturally vanish” along the nodal directions. None of these statement are especially obvious at the level of detail given.

Finally, the most solid aspect of the paper is the evaluation of a dimensionless strength of T^2 scattering. However, the presentation here is a little confusing at points. In particular, the legnthscale plotted in figure 2 is not especially natural to compare between different materials. As is noted below equation (4), this length scale needs to be compared to different quantities in 2d and 3d materials, and this quantity of comparison also depends (e.g. in 3d) on the density of electrons. We cannot say whether it is “large” or not without this comparison. The more natural quantity to plot between materials would be the dimensionless zeta quantity which can be defined without any reference to a legnthscale — it is just defined in terms of the scattering rate and E_F. But this quantity is only compared to one other material in table I. I think that for the author to really make the case that LSCO is special in terms of the stregnth of T^2 scattering — and again, this is the more solid part of the paper — zeta should be computed for as many of the materials in figure 2b (and, for that matter, fig 2a) as possible. Personally I see no need to introduce the legnthscale l_quad.

In short, I think that it’s okay to have a paper that is a speculation based on an observation. Nonetheless, the observation needs to be fleshed out more in terms of the dimensionless zeta across materials, otherwise it’s a little thin, and the consequences need to be fleshed out at least a little bit more so that it’s clear how they actually connect to the observation. And, finally, potential difficulties should be addressed to some extent, notably the lack of a signature of classicalized and solidified electrons in either thermodynamics or via some signature of localization.

See attached report

See attached report

This manuscript presents a new angle on the enigma of high-temperature superconductivity in the cuprates and in particular on the coincident emergence of superconductivity and a T-linear component in the low-temperature electrical resistivity upon reducing doping from the far overdoped side of the cuprate phase diagram. The novel approach taken by the author is to compare the absolute magnitude of the scattering time in overdoped LSCO with that deduced from thermal transport measurements close to the liquid/solid transition in He-3. To my knowledge, no-one has previously considered the magnitude of the (T^2) scattering rate as a driver of the emergence of superconductivity and T-linear resistivity and for this reason (combined with the absence of a viable candidate for the origin of low-T T-linear scattering over such a broad doping range), I am minded to recommend publication of a suitably modified version of this manuscript in SciPost Physics.

The first reviewer has made some pertinent comments for the author to address about the comparison between metallic LSCO and He-3 and I do not have anything to add in this respect. Instead, I would like to ask the author to consider and address these additional points in any subsequent revision:

(i) The same phenomenology, viz a viz the advent of a low-T T-linear resistivity at the SC/non-SC boundary is also seen in the electron-doped cuprates. Comparison of the magnitude of the T^2 scattering rate in the electron-doped counterparts would arguably constitute a robust test of the author's claim and therefore should be considered for inclusion. I believe there is sufficient data on these systems to enable such a comparison.

(ii) The author has argued that the scattering time becomes pathologically short only for the nodal quasiparticles. As far as I can see, the reasoning behind this postulate is that the Fermi velocity is largest along the nodal directions. Two earlier experimental studies, however, are not consistent with this assumption of an isotropic mean-free-path: a) The temperature dependence of the Hall coefficient of overdoped LSCO at the same or similar doping level was successfully modeled assuming an isotropic T^2 scattering rate combined with a large scattering rate at the vertices of the diamond-shaped Fermi surface closest to the van Hove singularity that indicated a strong violation of the isotropic-l approximation [Narduzzo et al., PRB vol. 77, 220502 (08)]. b) ARPES measurements on LSCO (x = 0.23) with a very similar Fermi surface geometry indicated that the nodal single-particle lifetime was in fact longest and Fermi-liquid-like (varying as 1/(E-E_F)^2), while away from the nodes, the lifetime became shorter and exhibited a 1/(E-E_F) dependence [Chang et al., Nat. Commun. vol. 4, 2559 (13)].

The author's arguments are essentially equivalent whether the distinguishable particles are located preferentially near the nodes or at the 'anti-nodes', so for the sake of consistency with existing experimental data, the author might want to consider also this alternative scenario. If the author has good grounds to claim that the nodal quasiparticles really are the ones where this distinguishability is manifest, then at least he should present such arguments in light of these earlier experimental studies.

I have two other minor changes to request.

(iii) In Eq. (3) and in the calculation of n - the carrier density - for LSCO, the correct c-axis lattice parameter is 0.65 nm, not 1.3 nm. Alternatively, there should be an extra factor of two in both expressions to take into account the body-centred-tetragonal nature of the lattice.

(iv) It's (in)distinguishability, not (in)distinguishibality!

See attached report

See enclosed report

See enclosed report

This is an interesting paper from an interesting thinker who always provides refreshing new angles from which to picture important problems. I have a number of questions for the author, both fundamental and more trivial. At the fundamental level, I guess I don’t understand the link that the author is making between distinguishability and whether or not quantum statistics is obeyed. For me, indistinguishability can (and does!) occur in the infinite-time limit of thermodynamics even when the system is ‘classical’ in all other ways.

1. I am not sure I fully buy the argument for distinguishability in solid 3He. In a standard solid it comes from the probability of atomic exchange being vanishingly small. I don’t see how one can postulate strong atomic exchange and mobility and still postulate strict distinguishability. It seems to me like trying to ‘have your cake and eat it’.

2. The extension of the argument to a system where no spatial localization is implied by strong inter-electron scattering is even more puzzling to me. Is information not lost in each such a quantum mechanical scattering event, and is that not the fundamental cause of indistinguishability? For me, the Gibbs paradox shows that there is a difference between a classical gas of distinguishable particles and one of indistinguishable particles, and the fact that entropy is properly extensive in the classical limit shows that even classical particles are fundamentally indistinguishable. This is an absolutely key point of the paper, so I think that it needs to be more clearly addressed, with less reliance on simply citing ‘dynamical distinguishability’ papers by Trachenko and collaborator.

3. Even if we ignore comments 1 and 2, the existence of a classical sub-system of mobile particles in the T -> 0 limit would lead to a non-integrable T^-1 divergence of the heat capacity. Is there any evidence of this in strongly overdoped LSCO? I doubt it.

4. When reading the article, I was confused at the part describing l_dd in the construction of the dimensionless parameter in Eq. 4. I guess the author means the ‘dd’ to refer to dimension, i.e. 2d and 3d, but it was not immediately clear to me how to understand that the 2d version goes as 4pi.c . I am being lazy in not figuring this out for myself, but it would be convenient if the author just expanded on the derivation himself in the manuscript.

I would like to see the author’s responses to the above points before I would feel myself able to judge on whether a revised manuscript would be suitable for publication.

See enclosed report