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

by Kamran Behnia

This is not the latest submitted version.

This Submission thread is now published as SciPost Phys. 12, 200 (2022)

Submission summary

As Contributors: Kamran Behnia
Arxiv Link: (pdf)
Date submitted: 2022-02-22 11:30
Submitted by: Behnia, Kamran
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Experiment
  • Condensed Matter Physics - Theory
Approaches: Theoretical, Experimental


La$_{1.67}$Sr$_{0.33}$CuO$_4$ is not a superconductor and its resistivity follows a purely T$^2$ temperature dependence at very low temperatures. La$_{1.71}$Sr$_{0.29}$CuO$_4$, on the other hand, has a superconducting ground state together with a T-Linear term in its resistivity. The concomitant emergence of these two features below a critical doping is mystifying. Here, I begin by noticing that the electron-electron collision rate in the Fermi liquid above the doping threshold is unusually large. Therefore, the scattering time of nodal quasi-particles is close to the threshold for dynamic indistinguishibality, which is documented in liquid $^3$He at its zero-temperature melting pressure. Failing this requirement of Fermi-Dirac statistics will exclude nodal electrons from the Fermi sea. Becoming classical, they will scatter other carriers within a phase space growing linearly with temperature.

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Reports on this Submission

Anonymous Report 3 on 2022-5-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2202.10128v1, delivered 2022-05-02, doi: 10.21468/SciPost.Report.5015


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.

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Author:  Kamran Behnia  on 2022-05-15  [id 2469]

(in reply to Report 3 on 2022-05-02)

I am grateful to the referee for the careful reading and very constructive criticisms and for finding the line of argumentation “broadly attractive.”
Let me begin by commenting on this sentence: “I think that the paper is arguing that the vibrations of localized, classical electrons could provide the T-linear scattering that is observed. “
What I propose is partially inspired by what was suggested in the case of Sr3Ru2O7 by Mousatov et al (PNAS 117, 2852 (2020)). There, T-linear resistivity is explained by invoking the scattering of degenerate electrons by ‘hot’ electrons unbound by Fermi-Dirac statistics. The multiplicity of Fermi pockets in Sr3Ru2O7 makes this conceivable. However, cuprates have a single Fermi surface. Therefore, if there are ‘hot’ (i.e. non-degenerate) electrons, one needs a mechanism to generate them inside the Fermi sea.

The referee wonders “what the language of distinguishability brings to this discussion”. I have revised Fig.4 and the discussion around the issue of dynamic distinguishability. The main point is that Landau parameters of a Fermi liquid cannot increase indefinitely, because a particle have a finite number of first neighbors. What differentiates a liquid from a gas is the finite value of the two-particle correlation function in the liquid. The latter keeps increasing with increasing pressure, but saturates to a finite value when the liquid becomes a solid. I agree with the referee that only a rigorous computation can settle what this maximum value for the two-particle correlation can be for a quantum liquid. Presumably, it depends on the details of the interaction. However, assuming that two-particle correlator is bound in any realistic system is not a particularly outrageous conjecture.

The referee states: “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?”
I did not mean or write that “it is collisions that lead to indistinguishability in a liquid.” During a collision, energy and momentum are exchanged between those particles which happen to be close to each other in the real space. Therefore, the higher the collision rate in the liquid, the harder for the atoms to stay indistinguishable (Please also see item 2 in my response to referee 1).

I agree with the referee that electrons in cuprates are not close to Anderson localization. Their mean-free-path is much longer than both the lattice spacing and their wave-vector. But this is also the case of 3He. At the onset of solidification, the mean-free-path of 3He atoms is much longer than the wavelength of the Fermions. Solidification of 3He is not driven by this kind of localization. Both LSCO (x=0.33) and 3He (P=3.4Mpa) are Fermi liquids with exceedingly large Landau parameters, where fermions travel a long distance before being scattered. Following the referee’s comments, the new version includes a detailed discussion in the end of section 3 arguing that solidification is driven by the impossibility of two-particle correlators to become arbitrarily large.

As mentioned in my response to referee 2 above, I don’t assume the mean-free-path to be isotropic. The mean-free-path foe electron-electron scattering is LONGER along the nodal orientation, but the corresponding scattering time is SHORTER along the nodal orientation. This is a consequence of a peculiarity of the Fermi surface. As seen in Fig. 5, the anisotropy of the Fermi velocity is much larger than the anisotropy of the Fermi radius. What is assumed to be isotropic is the in-plane resistivity in presence of tetragonal symmetry.

For a discussion of specific heat, please see point 3 in my response to referee 1.

I have followed the referee’s recommendation and added more details to the speculations formulated in the last section. The length scale l_quad was introduced in ref. 21 (lin eta l. 2015). I have begun by invoking it for this historical reason. I agree with the referee that the dimensionless zeta is a more adequate concept for our purpose. However, as discussed in the new version its quantification in multiband fermi liquids is problematic. You can quantify the Kadowaki-Woods ratio using two quantities directly accessible to the experimentalist (A and gamma). Quantifying l_quad when you have multiple bands is harder, but you can still use the average Fermi energy. In the case of zeta, you need to go further in oversimplification by taking average values for the Fermi energy and the Fermi wave-vector. This is why quantifying zeta in metals with complicated Fermi surfaces is a nightmare. Nevertheless, I have followed the referee’s recommendation, and introduced a new section (section V) estimating the rough magnitude of zeta for a few other metals. Table 1 includes now a comparison between the cuprate and four other Fermi liquids.

Anonymous Report 2 on 2022-4-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2202.10128v1, delivered 2022-04-01, doi: 10.21468/SciPost.Report.4841


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!

Requested changes

See attached report

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Author:  Kamran Behnia  on 2022-05-18  [id 2490]

(in reply to Report 2 on 2022-04-01)

I would like to clarify an issue regarding point i, namely, the T-linear/T-square boundary in electron-doped cuprates.

I was recently informed about a paper (Jin et al., Nature 476, 73 (2011), which I had totally overlooked.

I apologize for this shortcoming. The referee is right.

The final version of the paper will include a reference to electron-doped LCCO, which confirms the observation made in the case of hole-doped LSCO.

Author:  Kamran Behnia  on 2022-05-15  [id 2468]

(in reply to Report 2 on 2022-04-01)

I am grateful to the referee for the time devoted to reading the manuscript, the positive assessment, and the useful comments. Here are my comments.
(i) The referee is entirely right that electron-doped cuprates display T-square resistivity and T-linear resistivity in different regimes of temperature and doping and deserve attention. The reason they are absent in the present manuscript is that I could not find any study pinpointing the emergence of T-linear resistivity at a critical doping, let alone its concomitance with a superconducting ground state, in the context of a well-established single Fermi surface (comparable to what was reported by Hussey’s group). This may be due to my ignorance.
(ii) Yes, I argue that “the scattering time becomes pathologically short only for the nodal quasiparticles.” However, I do NOT assume that the mean-free-path is isotropic. The mean-free-path is anisotropic and LONGER for nodal quasi-particles. I do not contest the experimental findings invoked by the referee. The issue here is the electron-electron scattering time. This scattering time would have been isotropic if the Fermi velocity and the Fermi wave-vector had the same anisotropy with opposite signs. This is not the case here because of the proximity of a van Hove singularity. The new version discusses in more detail. The in-plane resistivity is isotropic, constrained by the symmetry of the underlying lattice and found by experiment. The mean-free-path, the scattering time, the velocity, and the wavevector and can be and are all anisotropic.
(iii) I use 1.3nm for the lattice parameter of LSCO because this is what is most often used in literature. For a neutron diffraction study devoted to measuring this lattice parameter, see V. Voronin et al., Physica C: Superconductivity, 218, 407 (1993). I understand that the distance between two adjacent CuO sheets is 0.65nm.
(iv) Many thanks! Corrected.

Anonymous Report 1 on 2022-3-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2202.10128v1, delivered 2022-03-15, doi: 10.21468/SciPost.Report.4684


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.

Requested changes

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Author:  Kamran Behnia  on 2022-05-15  [id 2467]

(in reply to Report 1 on 2022-03-15)

I would like to thank the referee for these kind remarks and very helpful comments. Here are my comments regarding the four points.

1- I agree that distinguishability and strong atomic exchange are opposite to each other. Here is my picture: 3He atoms are indistinguishable, no matter the phase (gas, liquid or solid) they belong with. In contrast to the liquid, however, each atom in the solid belongs to a site with a definite position in the real space. These sites are distinguishable and as far as this correspondence (between an atom and its site) holds, the solid state survives. At pressures much higher than the melting pressure (about 30 atmospheres), 3He is like any “standard solid” with “the probability of atomic exchange being vanishingly small”. As the pressure decreases towards the melting pressure, atoms start tunneling from one site to another. This weakens the one-to-one correspondence between an atom and its designated site. As a result, the solid eventually melts. Indeed, one cannot ‘have a cake and eat it.’

2- In the case of 3He, starting from the solid and moving towards the liquid, the solid becomes unstable beyond a threshold. This is because atoms cannot stick to their position in real space. Now, starting from the liquid and moving towards the solid, the liquid becomes unstable, because collisions in real space become so frequent that the wave-vector ceases to be a good quantum number. Now, let us consider the role of fermion-fermion scattering in this first order phase transition. With increasing pressure, the collisions between two fermions become more frequent and end up by solidifying, i.e. «confining [the] atom to the neighbourhood of a particular lattice site.” (Thouless, Proc. Phys. Soc. 86, 893 (1965)).

What drives these collisions is the interaction between fermions. The intensity of interaction between two colliding fermions depends on their position in the real space and therefore collisions favor spatial localization. If the time between two collisions in real space becomes too short compared to the finite required for a permutation between two fermions, then indistinguishability, (which requires avoiding confinement to a real-space neighbourhood) will become impossible. As far as I know, Thouless, who argued that the quantum solid near the melting transition is a “system of N! identical cavities joined by a system of ducts” (Proc. Phys. Soc. 86, 893 (1965), was the first to point his finger on the role of distinguishability in this context. His interest was focused on the limited relevance of the Pauli exclusion principle in the solid and not collisions in the liquid state.

The new version has an extended discussion of the issue of dynamic indistinguishability. The idea is that the two-particle correlators of the Fermi liquid, its Landau parameters, cannot become arbitrarily large. Spatial confinement of the mobile becomes unavoidable if collision rate becomes exceedingly large. Now, let us consider the relevance of this idea to electrons in metals: let us postulate that the Landau parameters of a Fermi liquid cannot become arbitrarily large. Then, in the case of overdoped cuprates, nodal quasi-particles are the first to touch this threshold. In the case of 3He, fermions meet this threshold simultaneously and this leads to solidification. In contrast, what can happen if this ceiling is attained by a subset of fermions can only be guessed. If metallicity is preserved with a subset of carriers excluded from the Fermi liquid, the latter will scatter the remaining carriers without being bound by the T-square size of the phase space. Please see the modified Figure 4.

3- Regarding the specific heat, a similar point is raised by referee 3. Indeed, equipartition implies that if mobile particles survive down to zero temperature, then there will be a finite specific heat ~k_B and a diverging C/T. Of course, as far as I know, no classical liquid (electronic or otherwise) is known to survive at zero -temperature, and I do not dare to suggest that cuprates are an exception. The fate of quasi-particles kicked out of the Fermi Sea can only be guessed. They may localize (at least partially) or remain mobile. In the latter case, they will be particles of a liquid with a low degeneracy temperature and only classical above this temperature scale. Their contribution to the total specific heat is small given their relative population. On the other hand, given their weight in generating T-square resistivity, they will continue to scatter other electrons significantly.

4- This section has been entirely re-written in the new version. I hope it becomes clear how zeta is related to the Fermi energy and the Fermi wave-vector both in 3D and in 2D, albeit differently.

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