Response to Referee Report 2

We thank the referee for a thorough report, and for the clarifying questions.

1. Referee comment:

The ideas in the paper build on the original Bardeen Stephen (B-S) model whose theoretical basis is phenomenological, and not fully convincing. In particular, when the superfluid is uncharged and the model Galilean invariant, a correct model should reproduce Kelvin's theorem that the vortices move with the flow. This property is preserved by Nozieres and Vinen and by the time dependent Landau Ginzburg model considered by Dorsey, but not by B-S.

Our answer:

We have not followed Bardeen and Stephen's (BS) approach, although we have assumed their (microscopically justifiable) model of a metallic vortex core of radius $k_F\xi \gg 1$. Their vortex dynamics equation (as does Nozieres and Vinen's (NV)) suffers from several lingering problems on which we elaborate in our answer to Question 3 of Referee 1, and in our expanded discussion in the new Section 9.

Indeed, it is quite surprising that with our orthogonal (current response) approach to flux flow transport theory, we have recovered BS conductivities $\sigma_{xx}(B)$ and $\sigma_{xy}(B)$ as the first term in Eq. (1). Our paper contradicts NV's result for $\sigma_{xy}$ and for the Hall angle, which we believe is due to invalid assumptions in their vortex forces. In our response to Question 4 of Referee 1, we explain why the Galilean invariance argument put forth by NV and adopted by many subsequent authors, is not relevant to flux flow regime of real superconductors. The aforementioned "go with the flow'' rule (Kelvin's circulation theorem for classical fluids), and therefore also the equation $\sigma_{xy} = nec/B$, are invalidated by the periodic lattice potential, impurities, and most importantly by the emergence of short range superfluid stiffness below $B_{\rm c2}$. Superconducting stiffness distinguishes the flux flow from the normal metal regimes, and has a different effect on the Hall conductivity than derived from Galilean symmetry arguments. This point is now well explained in Appendix B.

2. Referee comment:

The origin of the key equation $\varepsilon= - \hbar/(2e) \nabla \dot{\phi}$ is unclear. Usually one thinks of the superfluid fluid being accelerated by the EMF rather than it creating and EMF. Indeed as it stands the equation cannot be quite correct as the EMF should be a gauge invariant quantity and $\nabla\dot{\phi}$ is missing the ${\bf A}$ field required for gauge covariance.

Our answer:

The current density is gauge invariant ${\bf j} = \rho_s (\nabla \phi - {2e\over \hbar c} {\bf A})$. Throughout the paper we choose the gauge $A_0({\bf x},t) =0$ to define the external electric field as ${\bf E} = - {1\over c} \dot{\bf A}$ and vorticity produced EMF $\varepsilon= - \hbar/(2e) \nabla \dot{\phi}$. We apologise for previously neglecting to state our gauge choice, which is now stated after Eq. [4], and thank the Referee for the comment.

3. Referee comment:

I always thought that as the $B$ field is being advected with the vortex we could take it to be ${\bf A}({\bf r}- {\bf v}_L t)$, and it is the ${\bf A}= - \dot{\bf E}$ which ensures that the ${\bf A}$ field carries the circulation with it.

Our answer:

We beg to differ. The external magnetic field ${\bf B}$ is not advected, because it is approximately uniform in the (unscreened) thin film where $\lambda \gg l_B$. Therefore, the vector potential ${\bf A}_B = -{1\over 2} {\bf r}\times {\bf B}$ is time independent. The motion of vortices in the steady state is dictated by a uniform external electric field ${\bf E}$ which we introduce with ${\bf A}_E = -c {\bf E}t$. The vortices in the uniform (DC current) steady state must move at the Josephson velocity ${\bf V} = c {\bf E} \times {\bf B}/B^2$, as derived in Section 2.

Response to Referee Report 1

We thank the referee for a thorough report, and for urging us to clarify several important points concerning experiments and other theoretical work.

1. Referee question:

1. The Hall sign reversal phenomenon includes also a double sign change ... Can the proposed phenomenological consideration reveal the reason for that and explain this double change?

We answer:

Whether there are one or two sign reversals depends on the individual temperature dependence of each of the two terms in Eqn. (1). The first term (core conductivity) depends on the scattering time of the normal state quasiparticles in the vortex core. The second term, due to moving vortex charge (MVC) generally has the opposite sign, (see our answer to Q.4 below) and depends on the superfluid stiffness, which saturates at low temperatures. The (lower temperature) sign reversal reflects a rapid increase of core conductivity at low temperatures if it is dominated by inelastic (electron-electron and electron phonon) scattering. This point is clarified in Section 9.

2. Referee question:

According to the beautiful sketch of Figure 1 of the manuscript, the sign change is expected to occur below Bc2. However, there are many works (including, again, Ref. [7] of the manuscript,...reporting that the sign change starts already above Bc2. How should the sketch of Figure 1 reconcile with these observations?

We answer:

The curves in Fig. 1 depict Bardeen-Stephen mean field theory, which expects $B_{\rm c2}(T)$ to be sharp phase transition. Experimentally, $B_{\rm c2}$ is merely the midpoint in the magnetoresistivity rise between the flux flow and normal metal regimes. Evidently, short range phase correlations persist within a limited range above $T_c(B)$, as adduced from the broadened magnetoresistivity curves in Fig. (5) of the supplementary material of Zhao et.al. (Ref. [7]). A few layers of BSCCO films are expected to undergo a zero field Kosterlitz-Thouless (KT) transition. Halperin and Nelson (J. Low. Temp 36, 599 (1975)) theory describes mobile vortices in a limited range above $T_{\rm c}$. The MVC carried by these vortices can reverse the Hall sign even slightly above the nominal $T_{\rm c}(B)$. We discuss this point in Section 8.

3. Referee comment:

Reference [7] of the manuscript presents a quantitative description of the experimental data in the framework of vortex dynamics (Reference [15] of the manuscript). Based on the fair agreement between the theory and experiment, Reference [7] claims that the theory developed in [15] completely explains the phenomenon of the Hall sign change. One could then conclude that Ref. [15] of the manuscript together with the subsequent extension by Geshkenbein, Ioffe, and Larkin, Superconductivity in a system with preformed pairs, PRB 55, 3173 (1997), have already resolved all the controversies existing before. It would be useful for a reader to learn which deficiencies of these (Ref [15] and PRB of 1997) works were resolved in the present manuscript.

We answer:

The BSCCO data of Zhao et al., Ref. [7], was indeed fit by the theories of Feigel'man et al. in Ref. [15] and PRB 1997. However, our (different) theory also fits the data of Zhao et al., as well as that of Hagen et al. (Ref. [20]), which includes Hall anomalies in both hole and electron doped cuprates. That said, even good fits cannot prove any theory.

As mentioned in the introduction, and discussed now more thoroughly in Section 9, our paper addresses unresolved issues concerning the widely used vortex dynamics approach, which includes the theory of Ref. [15]. The old controversy between Bardeen and Stephen versus Nozieres and Vinen concerning the flux flow Hall effect has not yet been settled. We now understand that this controversy originates from an inherent shortcoming of the vortex dynamics equation. The vortex forces are chosen by phenomenological considerations, such as power dissipation and symmetries. A basic problem is that the equation is set up to calculate the generated EMF in response to a bias current. However, conductivities in the presence of potentials and interactions are feasibly calculated only by inserting an electric field into the Hamiltonian and evaluating the generated current. This is the approach we adopt in the present paper. The main advantage of this approach is that we can avoid the determination of vortex forces where momentum is not conserved. We do not need to calculate the vortex velocity from the vortex dynamics equation, since it is strictly constrained by Josephson equation to be ${\bf V} = c{\bf E} \times {\bf B}/B^2$, with no free parameters, as derived in Section 2. In addition, (as explained below) we derive the opposite sign of the vortex core charge than obtained by the Galilean symmetry consideration in Ref. [15].

4. Referee question:

... the authors apply the theory of the vortex screening by Khomskii and Freimuth (Ref. [10] of the manuscript). However, this theory gives the extra charge of the opposite sign than that obtained by other authors. How does this discrepancy may influence the conclusions of the manuscript?

We answer:

Feigel'man et al. (Ref. [15]) obtained a Hall conductivity contribution $\Delta\sigma_{xy} = -\delta n ec /B$. We refer to this specific result below as Eqn A. In Eqn. A, $\delta n = n_0 - n_{\infty}$ is the charge accumulated at the vortex core. Khomskii and Freimuth (KF), along with our work, arrived at the opposite conclusion, i.e. that Hall sign reversal is related to vortex core charge depletion rather than accumulation. Ref. [15], and references therein invoked Galilean symmetry which is supposed to be recovered for zero core charge. In that limit Eqn A yields $\sigma_{xy} = n_\infty ec /B$.

However, we dispute the validity of Eqn A. In Section 9 and Appendix B we explain why Galilean symmetry is not applicable to the flux flow regime of general superconductors, for two main reasons:

(1) A real superconductor is composed of band electrons in a periodic potential, and thus is not Galilean invariant. For example, Galilean symmetry dramatically fails near a half-filled band (e.g. at $n \approx 1$ for the square lattice tight binding model), where the Hall conductivity must vanish by particle-hole symmetry. de Gennes and Nozieres, in Phys. Lett. 15, 216 (1965), noted that Galilean fluid effects (e.g. vortex mutual precession) may be observable (perhaps) only for $\omega_c\tau \gg 1$. In BSCCO, $\omega_c \tau \sim 10^{-3}$ as shown by the Hall angle in our Fig. (6). These lie outside the Galilean effects regime.

(2) Even for a quadratic band, vortices will not "go with the flow", as expected by Kelvin's circulation theorem for classical fluids. The emergence of short range superconducting stiffness below $T_{\rm c}$, differentiates between the flux flow and normal metal transport. Even for parabolic bands, Galilean symmetry is broken in a superconductor due to rigidity of the vortex lattice pinned by arbitrary weak potentials. In the flux flow regime, where short range stiffness survives, we show in Appendix B that the vortices do not drag the background condensate. Hence the Hall conductivity is not given by $\sigma_{xy} = n_\infty ec /B$. Interestingly, it is given instead by the charge-flux attachment of the vortex liquid, i.e. the Streda formula which is derived from the Kubo formula in Appendix A: $\Delta\sigma_{xy} = ec {\partial n\over \partial B}$. This expression generally yields a sign opposite to that of Eqn. A.

A compelling phenomenological argument against Eqn. A, is that for BCS theory (as shown by KF), the vortex charge is lower than the uniform charge, i.e. $\delta n <0$. Hence Hall sign reversals which were observed in conventional superconductors, e.g. Nb and V by Noto et al. in Ref. [2], are not consistent with the positive sign of Eqn. A. As far as BSCCO is concerned, we hope that the sign of the vortex core charge may be directly measured by upcoming experiments in the near future.