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Kondo spectral functions at low-temperatures: A dynamical-exchange-correlation-field perspective.
by Zhen Zhao
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Zhen Zhao |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202407_00039v2 (pdf) |
| Date submitted: | 2024-10-25 20:45 |
| Submitted by: | Zhao, Zhen |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We calculate the low-temperature spectral function of the symmetric single impurity Anderson model using a recently proposed dynamical exchange-correlation (xc) field formalism. The xc field, coupled to the one-particle Green's function, is obtained through analytic analysis and numerical extrapolation based on finite clusters. In the Kondo regime, the xc field is modeled by an ansatz that takes into account the different asymptotic behaviors in the small- and large-time regimes. The small-time xc field contributes to the Hubbard side-band, whereas the large-time to the Kondo resonance. We illustrate these features in terms of analytical and numerical calculations for small- and medium-size finite clusters, and in the thermodynamic limit. The results indicate that the xc field formalism provides a good trade-off between accuracy and complexity in solving impurity problems. Consequently, it can significantly reduce the complexity of the many-body problem faced by first-principles approaches to strongly correlated materials.
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- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
I herewith resubmit the manuscript "Kondo spectral functions at low-temperatures: A dynamical-exchange-correlation-field perspective" for publication in SciPost Physics.
I would like to thank the referees for their insightful reports. The referees provided general observations as well as detailed questions, some of which helped us improve our original approach and inspired us to deepen our understanding of the results. In accordance with our replies sent to the referee reports, text changes have been made in the revised manuscript, addressing the referees' observations and suggestions to make the manuscript more suitable for publication in SciPost Physics.
Finally, I would like to thank you for your consideration and handling of this manuscript.
With best regards,
Z. Zhao
List of changes
1. Changes in Sec. 2
A relation between the dynamical xc field and the self-energy in frequency domain has been added.
An explanation of the exact constraint of the dynamical xc hole has been included.
2. Changes in Sec. 2.1
The use of the Lehmann representation of the Green's function is now explicitly pointed out.
Notations related to spin indices have been revised for consistency through the manuscript.
3. Changes in Sec. 2.2
It is now explicitly stated that the xc field (Eq. 27 in the revised manuscript) is derived by applying the Lehmann representation of the Green's function to its equation of motion and solving for the xc field.
4. Changes in Sec. 3.1
For the dimer, the term "Kondo regime" has been replaced with "large interaction regime" for precision.
The statement in the end of this subsection regarding the temperature-induced broadening of the spectral peaks has been revised.
5. Changes in Sec. 3.2
In the end of this subsection, additional discussions have been added regarding the fact that finite cluster results provide insights about the xc field in the small-time regime. For the SIAM, the Kondo resonance is related to the large-time behavior of the xc field.
6. Changes in Sec. 3.3
An ansatz capturing both small- and large-time behavior of the xc field has been proposed. The Green's function can now be calculated by directly integrating the ansatz over time, which improves the treatment in the original manuscript, where a low-order expansion was applied.
The way of calculating the xc field with negative time using the particle-hole symmetry is now explicitly described.
Interpretations of the ansatz parameters (Eqs. 43-45 in the revised manuscript) are explained in more details.
Additional comments on the hyperbolic-tangent functional fitting are provided.
The treatment of the ansatz parameters at low-temperatures (Eqs. 47-51 in the revised manuscript) is now improved. A Fermi-liquid-theory-based expression for the Kondo peak half-width in the literature is now used as reference.
NRG results adapted from some literature have been added in Fig. 4 and Fig.5 for better comparison. With the improved treatment at low-temperatures, we now plot the spectral functions in Fig. 5 for additional temperature values. We also expand discussions about the detailed difference between our results and those from NRG.
7. Changes in Sec. 4
Discussions on the asymptotic behaviors of the xc field in the small- and large-time regimes have been added.
8. Changes in Appendix C
A detailed explanation of how the Green's function on a finite cluster at zero-temperature can be calculated using the TDVP method is provided.
9. Changes in Appendix D
We derive the Green's function using the improved ansatz of the xc field, showing that no low-order expansion is involved.
10. Changes in References
Additional citations have been added regarding the reference expression for the Kondo peak half-width in Sec. 3.3.
Current status:
Reports on this Submission
Report
The author has answered to the comments and questions of both referees satisfactory, and revised the manuscript accordingly. All my doubts have been answered, and I think the manuscript is much clearer now. I am glad that our comments helped the author to improve his Ansatz for the dynamical xc potential of the Anderson model. The results are very nice and compare very well to NRG data.
However, I have noticed two mistakes introduced in the newly added part in Sec. 3.3 related to the width of the Kondo peak and its relation to the Kondo temperature (see below for details). These mistakes are not essential, but should be corrected before the paper can be published, in order to avoid confusion. I therefore recommend publication after these mistakes have been corrected.
Requested changes
1) The first mistake concerns the relation between the Kondo peak width at T=0, and the Kondo temperature (lines 329-330). The proportionality factor ~3.7 was found numerically from NRG calculations in Ref. 49. But an exact analytic relationship in terms of the Wilson number was given in Ref. 52 which results in a proportionality factor of 3.92, close to the numerical one. In this context it is also confusing to use w=3.7, since "w" usually would denote the Wilson number (w=0.4128).
2) This is probably related to the first one: Eq. (50) reproduces Eq. (10) of Ref. 52, but with $\tilde{T}_K$ instead of $\Delta_K$ (the Frota width parameter). It is not clear to me what is $\tilde{T}_K$ and why can it be set to ~0.491?! Is it a typo? The value of $\Delta_K$ is linked to the Kondo temperature by $\Delta_K=1.542\,T_K$ (as correctly used in the prefactor to the square-root in Eq. (50)), and therefore depends on the interaction U and the bare width $\Gamma$. Also as stated above $\Gamma_K(T=0)\approx3.92\,T_K$, so the rescaling should be done with 3.92 instead of 3.7, as stated after Eq. (50). Maybe this rescaling is also somewhat confusing, and the author should think about using $T_K$ as the Kondo temperature (as defined by Wislon), and use $\Gamma_K^0$ for the Kondo peak width.
Recommendation
Ask for minor revision
Report
I am happy with the author's reply to my previous report and also
to the report of the other referee and now recommend publication of
this manuscript in SciPost Physics.
Just one tiny detail which needs to be corrected: in line 355, for
the quantity \tilde{T}_K the energy unit is missing which should be T_K, i.e., \tilde{T}_K \approx 0.491 T_K.
Requested changes
1) in line 355, for the quantity \tilde{T}_K the energy unit is missing which should be T_K, i.e., \tilde{T}_K \approx 0.491 T_K.
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
I thank the referee for giving positive evaluation for the manuscript and for pointing out the typo.
Eq. (50) is from Eq. (10) of Ref. 52, and as the referee noticed, in line 355, it should be \tilde{T}_K\approx 0.491 T_K.
This typo will be corrected in the revised manuscript.
Author: Zhen Zhao on 2024-11-20 [id 4968]
(in reply to Report 2 on 2024-11-19)I thank the referee for the positive evaluation for the manuscript and for pointing out the typo and the statement that may lead to confusion.
Regarding the first point, we followed Eq. (10) of Ref. 52 to determine the relation between the Kondo temperature $T_K$ and the Kondo peak half-width $\Gamma_K$, which means only the factor 3.92 is considered. We agree that mentioning the numerical factor 3.7 and using the symbol "w" may cause confusion. These issues will be corrected in the revised manuscript.
Concerning the second point, following Eq. (10) of Ref. 52, it should be $\tilde{T}_K\approx1.542T_K/\pi=0.491T_K$. We apologize for the typo. As mentioned in the first point, we have used the factor 3.92 for the rescaling. By comparing with NRG results, we observe that $\Gamma_K^0=3.92 T_K$ is a good description for the SIAM in the strong Kondo regime, but not when $\Gamma$ is large. Therefore, we used $\Gamma_K^0\sim T_K$ instead of $\Gamma_K^0=3.92 T_K$ for our zero-$T$ treatment. This clarification will be included in the revised manuscript to explain our rescaling approach.
Finally, I wish to thank the referee again for his/her remarks and suggestions. I hope to have properly addressed them and clarified the manuscript.