# Hall anomaly and moving vortex charge in layered superconductors

### Submission summary

 As Contributors: Assa Auerbach Arxiv Link: https://arxiv.org/abs/1912.10062v3 (pdf) Date accepted: 2020-03-25 Date submitted: 2020-03-06 01:00 Submitted by: Auerbach, Assa Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Approaches: Theoretical, Phenomenological

### Abstract

Magnetotransport theory of layered superconductors in the flux flow steady state is revisited. Longstanding controversies concerning observed Hall sign reversals are resolved. The conductivity separates into a Bardeen-Stephen vortex core contribution, and a Hall conductivity due to moving vortex charge. This charge, which is responsible for Hall anomaly, diverges logarithmically at weak magnetic field. Its values can be extracted from magetoresistivity data by extrapolation of vortex core Hall angle from the normal phase. Hall anomalies in YBCO, BSCCO, and NCCO data are consistent with theoretical estimates based on doping dependence of London penetration depths. In the appendices, we derive the Streda formula for the hydrodynamical Hall conductivity, and refute previously assumed relevance of Galilean symmetry to Hall anomalies.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 8, 061 (2020)

Dear Editor,

We hereby resubmit our revised Manuscript entitled "Hall anomaly and moving vortex charge in layered superconductors", by Assa Auerbach and Daniel P. Arovas. In response to reports of Referees 1 and 2, this version incorporates the answers to their questions. The discussion section has been significantly expanded to explain the advantages of our approach, where the current is derived by imposing an external electric field into the Ginzburg-Landau Hamiltonian. We show how this approach resolves longstanding controversies related to determination of vortex forces and dynamics. A Galilean symmetry argument, which was invoked by previous papers to determine the flux-flow Hall conductivity, is refuted in Appendix B.

### List of changes

1. We add a comment in the abstract about the content of the two appendices.

2. Section 2: After Eq. (4) we add a note about fixing the gauge by setting A_0=0.

3. Section 8: Last paragraph adds a discussion of the reason for experimentally observed Hall sign reversal due to vortices existing even slightly above the nominal T_c.

4. Section 9: The expanded discussion section now includes: a) The inherent difficulty of setting up a microscopic basis for vortex dynamics equations. b) The invalidity of Galilean symmetry arguments for the Hall effect, as applied to the flux flow regime of real superconductors. c) An explanation of two Hall sign reversals as effects of inelastic scattering effecting the vortex core conductivity.

5. Appendix B: A new appendix which addresses a log time misconception in the superconducting literature. We prove by three complementary arguments, that even for Galilean invariant Hamiltonians, short range superconducting stiffness prevents the vortices from "going with the flow" as expected by Kelvin's theorem for classical liquids.

### Submission & Refereeing History

#### Published as SciPost Phys. 8, 061 (2020)

Resubmission 1912.10062v3 on 6 March 2020
Submission 1912.10062v2 on 16 January 2020

## Reports on this Submission

### Report

Accept in present form

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -

### Anonymous Report 2 on 2020-3-9 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1912.10062v3, delivered 2020-03-09, doi: 10.21468/SciPost.Report.1562

### Strengths

A potentially mportant contribution to a confused field.

### Weaknesses

It is not at all clear in this paper what is being $assumed$ in the physics, and what is somewhere (but not in the text) $derived$ from assumptions. This
makes the paper hard to understand.

### Report

I do not want to delay publication, but the paper really would be be improved
if some further explanation of the core ideas were given.

The key equation ${\mathcal E}=- {\hbar/2e} \nabla\dot \phi$ is neither derived, explained, or cited from
elsewhere. Is it a definition of ${\mathcal E}$? If so, how does ${\mathcal E}$ act as a true EMF? If it is a genuine electric field which Maxwell equation does it come from? Or is it
an electrochemical potential of some sort? Does ${\mathcal E}$ drive the time evolution of the phase or does the time evolution of the phase somehow create ${\mathcal E}$?

I can see ${\mathcal E}$ coming from ${\mathcal E}= -\dot {\bf A}/c$ and
$${\bf j}\propto \left(\nabla \phi - \frac {2e}{\hbar c} {\bf A}\right)$$
if we assert that
$${\bf j}_v\equiv {\rm const.} \left(\nabla \phi_v - \frac {2e}{\hbar c} {\bf A}\right)$$
is pointwise constant, but why should this be the case?
Further this relation cannot universally true because dissipative processes can allow vortices to move against the flow even in neutral superfluids. In that
case the time evolution of $\phi$ is driven by the contribution to the Josephson
phase evolution equation from the interplay between appplied current and the vortex current in the gradient-squared kinematic part of the
energy. Are we ignoring such effects?

All this is particularly interesting because of equation 9. This is a mathematical identity given the definition of ${\mathcal E}$ and $\phi_v$. It also looks like Josephson's
relation between the electric field and the vortex velocity, but Josephson explicitly assumes that the moving flux lines lead to a time dependent magnetic
field and so his EMF is an genuine electric field from Faraday's flux rule. In their response to my initial review the authors say that the ${\bf B}$ field is not being advected with the vortex flow. So what generates the EMF?

Eq 9 has a trivial typo: The first equality has $\bar {\mathcal E}$ parallel to ${\bf V}$ while the last
one has it perpendicular. A $\hat z \times$ is missing, I think.

• validity: -
• significance: high
• originality: high
• clarity: low
• formatting: good
• grammar: excellent

### Author:  Assa Auerbach  on 2020-03-12  [id 763]

(in reply to Report 2 on 2020-03-09)

# Response to Second Referee Report 2

We thank the referee for the followup report. We clarify the remaining issues below.

## 1. Referee comment:

It is not at all clear in this paper what is being assumed in the physics and what is somewhere (but not in the text) derived from assumptions. This makes the paper hard to understand.

Our assumptions (stated in the text) are: (i) The existence of a vortex liquid regime between the vortex melting field and normal metal regime, as written in Eq. (3). (ii) The Ginzburg-Landau free energy (Eq. (21)) governs this regime. (iii) The set up for transport calculations is in Fig. 3. The electric field is externally determined, and the current can flow freely in both longitudinal and transverse directions. This is realized in a Corbino geometry - the geometry of conductivity measurements. (iv) For the moving vortex charge effect (and Hall anomaly) we require a layered structure of low mobility dopant layers between superconducting planes, shown in Fig. 5.

## 2. Referee question:

Is $\varepsilon = -\hbar/2e \nabla\dot{\phi}$ a definition of $\varepsilon$? Is it a genuine electric field ... or is it an electrochemical potential of some sort? Does the time evolution of the phase somehow create $\varepsilon$? why should that be the case that ${\bf j}_v={\rm const}$?

Yes, $\varepsilon = -\hbar/2e \nabla\dot{\phi}$ is a definition. It is not the electric field, but the hydrodynamical (chemical potential ) stress field due to moving vortices. It is ab manifestation of Jospehson's relation. Only in the flux flow steady state $\langle \varepsilon\rangle = {\bf E}$, where ${\bf E}$ is the external electric field. This equality allows ${\bf j} ={\rm const}$. In the pinned vortices phase, for example, $0=\varepsilon\ne {\bf E}$, and an instability occurs because the energy and current increase with time.

## 3. Referee question:

The time evolution of $\phi$ is driven by... the interplay between applied current and the vortex current .... Are we ignoring such effects?

In our flux flow transport theory we do not consider a bias current, but an imposed electric field. The values of $\dot{\phi},{\bf V}, \varepsilon$ are constrained (in the steady state) by ${\bf E}$. This eliminates uncertainties about vortex forces and interplay between them. We are not neglecting any effects, its just that our approach does not require us to define them.

## 4. Referee question:

Equation 9 ... looks like Josephson's relation between the electric field and the vortex velocity, but Josephson explicitly assumes that the moving flux lines lead to a time dependent magnetic field and so his EMF is a genuine electric field from Faraday's flux rule. In their response to my initial review the authors say that ${\bf B}$ field is not being advected with the vortex flow. So what generates the EMF?

Indeed Josephson Phys. Lett. 1965 paper explains ${\bf E} =-{1\over c} {\bf V} \times {\bf B}$ as if produced by motion of flux tubes, as the Referee correctly points out. However, its been long realized that this equation holds even when $\lambda \gg l_B$, and the magnetic field is nearly uniform in space. This is especially true in thin films, where $\lambda$ can exceed the dimensions of the film. The general justification of Josephson's equation is due to the hydrodynamical stress field induced by moving vorticity. Vorticity and flux tubes only move together in the strong screening $\lambda << l_B$ regime. Motion of vorticity however always induces a transverse pressure gradient, even in neutral superfluids. In normal fluids it is related to "Crocco's theorem".

### Anonymous Report 1 on 2020-3-7 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1912.10062v3, delivered 2020-03-07, doi: 10.21468/SciPost.Report.1560

### Report

The authors adequately replied to my comments. And although some further counterarguments can be raised, I find that this further scientific discussion should be exercised in the format of the whole interested community and that the reviewer must not delay the publication on the ground of arguable comments. As I stated before, the paper is well written and will definitely stimulate further progress in the field. I thus recommend publication of the manuscript.

• validity: -
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• originality: -
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### Anonymous on 2020-03-09  [id 755]

Category:
question
objection

What is the justification for treating the vortex core as a uniform & continuous metal when the coherence length is of the order of the lattice spacing?

### Anonymous on 2020-03-09  [id 756]

(in reply to Anonymous Comment on 2020-03-09 [id 755])
Category:
You are certainly right that the large metallic core model is unjustified for cuprates at low temperatures, when $k_F\xi \sim O(1)$. That is why we restricted our experimental analysis of BSCCO in Section 7 to $T> 65 K$, where the coherence length satisfies $k_F \xi(T) >k_F l > 1$. Therefore near $T_c$, our extrapolation of the Hall angle (Fig. 6) from above $T_c$ is justified. The moving vortex charge Hall conductivity is only weakly dependent on the vortex core via the parameter $B_{c_2}(T)$.