Kondo spectral functions at low-temperatures: A dynamical-exchange-correlation-field perspective.

This paper applies a novel method, the so called "dynamical exchange-correlation field" (Ref. 37), to the Anderson impurity model in order to compute spectral functions in the Kondo regime.

I find the paper quite interesting. But I think the paper lacks clarity in a couple of places. The paper would also benefit from better explanations in some parts, since the dynamical exchange-correlation field is a novel approach. I have a couple of comments and questions that the author should address before I can agree to publication:

(1) I was at first confused by Eq. (11): How could the dynamic exchange correlation hole for $r^{\prime\prime}=r$ become time-independent and just equal to the negative density? But this follows from Eq.(8) and the fact that the second order Green's function $G^{(2)}(r,r^\prime,r^{\prime\prime};t)$ (using the notation of Ref. 37) vanishes for $r^{\prime\prime}=r$, and thus also the correlation function $g(r,r^\prime,r;t)=0$. I think the author should give this explanation after Eq. (11) to help the reader.

(2) Eqs. (13) and (14) are the Lehmann representations of the Green's function (which should be mentioned), and the denominators are just the partition functions $Z$. I think the equations would become clearer if $Z$ was introduced and used. In the following equation (20), the denominators cancel anyway.

(3) To help the reader, it should be explicitly stated that Eq. (20) follows from applying the equation of motion (15) to the Lehmann representation and solving for Vxc.

(4) Sec. 3.1, after Eq. (22): I am not sure whether it is appropriate to speak of "Kondo regime" in the context of the Anderson dimer. The Kondo effect is usually associated with an impurity coupled to a continuous band of conduction electrons.

(5) The last two sentences of Sec. 3.1, p. 8: I think this explanation for the temperature induced broadening follows simply from the Lehmann representation of the GF (13,14) which the author used to obtain the approximation for the dynamic Vxc.

(6) Is the Vxc given by Eq. (30) valid only for $t>0$? If so, what is the corresponding equation for $t<0$? I think it would also be interesting to see Vxc in the frequency domain, i.e. the Fourier transform of Eq. (30), which could then be compared to the self-energy for the SIAM. I suspect they must be very similar in the case of the SIAM.

(7) How did the author arrive at the hyperbolic-tangent form for $R(L)$ fitted to the data in Fig. 3b? Is that based on some theoretical background? Otherwise I think the actual functional form cannot be extrapolated from the calculated data, since the data is still largely in the linear regime. Very different functional forms leading to very different limits $R(L=\infty)$could be compatible with the data.

(8) It would be nice if in Figs. 4 and 5 the calculated spectra would be directly compared to the NRG spectra of Refs. [28] and [47].

Ask for major revision

The present manuscript applies the formalism of the so-called
"dynamical exchange-correlation (xc) field" of Ref.[37] to the single-impurity Anderson model. The authors proposes a rather simple ansatz (Eq.(30)) for this dynamical xc field to obtain the spectral function of the Anderson model in the Kondo regime where the parameters are fixed by using the known peak positions and widths of the spectral peaks. This ansatz seems to work surprisingly well given its simplicity.

While I find the paper interesting in general, I still have a number of
points which I would like the author to address:

1. The coefficient a_{n+,m} is defined just after Eq.(14). Shouldn't this also be sigma-dependent?

2. On Eq.(20): first, I suppose it is only meant to be valid for t>0, no?
Second, I am a bit confused about its form: why is there no explicit
dependence on the interaction U? Shouldn't it (loosely speaking) be
something like U G^(2)(t)/G(t) where G^(2) is the two-particle Green
function? Also, I don't understand the factor \aN{n+,m} \omega_{n^+,m} in the denominator. I would have expected this to be
<m| \hat{n}_{-\sigma} f_{\sigma} |n+><n+|f^{\dagger}_{\sigma} |m>.
Please clarify!

3.Please give more details on what is actually done in Sec. 3.2 and how, such that interested readers could repeat the calculations. The time-dependent variational principle is used to obtain which quantity, the one-particle Green function of the cluster?

4.In Fig.4: could the author plot the NRG results on top of the present results for better comparison? The same applies for Fig.5.

5.In Eq.(37): I assume that the parameter \Omega_T is temperature dependent? How is this parameter determined in practice? Is it used as a fit parameter to reproduce known spectral functions? Please show its evolution as function of temperature!

6.Finally, I noticed a typo in line 141: it should be "emphasize" instead of
"emphasis"

To summarize, before I can recommend this manuscript for publication in SciPost Physics I would like to see the issues raised above being addressed.

Ask for major revision