Resolving the nonequilibrium Kondo singlet in energy- and position-space using quantum measurements

(1) Detailed analysis of singlet correlations in the Anderson impurity model out-of-equilibrium

(1) The authors focus on an Anderson impurity model in a one-dimensional configuration for which more accurate methods are available.

The authors analyze singlet correlations in an Anderson impurity model out-of-equilibrium. These nonlocal singlets are what is usually called the Kondo screening cloud.
Using the NCA, the authors calculate the time-evolution of the energy-dependent and position-dependent singlet weights for two different initial states. Furthermore, they analyze the time-evolution of the singlet correlations for a driven system, where both leads have different chemical potential. They observe that the singlet weight is enhanced in this situation due to the bias and give a physical picture.

I think the paper is interesting and well written, presenting some novel results about the Kondo effect out-of-equilibrium.
I will recommend publication after the authors have addressed the following questions and comments. However, after having read the acceptance criteria and expectations for SciPost, I have the feeling that this paper is more suited for SciPost Physics Core than SciPost Physics, particularly when comparing with references 29 and 30 in the current manuscript.

Furthermore, I have the following comments and questions:

Concerning the method:
In equation 12, why is the operator on the right-hand side time-dependent, although there is the time evolution operator. If this equation is in the interaction picture, the authors should define this.

Below equation 8, the projector P^sigma,sigma^\prime selects sigma_D sigma^\prime_chi seems to be the opposite. Should it say select sigma^\prime_ D sigma_chi ?

I think it would be helpful if the authors can give a short derivation of equation 24.

The authors use the NCA to calculate nonlocal singlet correlations. As in this case, the operator describing the singlet includes a hybridization, I think that the NCA is a severe approximation. Is there a way to justify or confirm their results?
Can the authors compare their results to existing studies about the Kondo effect out-of-equilibrium?

Concerning the results:
What sets the time scale for the relaxation to the equilibrium?
The Kondo temperature is set to 0.8 Gamma, nearly Gamma. So naively, I would have expected that this energy (or better its inverse) also sets the relaxation time.
But it seems to be much longer.

How do the correlations look for an uncorrelated quantum dot (U=0)? Does it look like the results at high temperatures?

1) This paper contains a new twist in the investigation of the Kondo Problem, since perviously typically the spin correlation function in real space (Refs[8-10]) or the fixed point properties of the total system (Ref [5]) were addressed.
2) The authors extend their analysis to a two-lead finite bias out of equilibrium steady state situation.
3)Overall, I am impressed by the evolution of the singlet projector expectation value as function of energy. Starting at short times, correlations live in the vicinity of the single occupied excitation energy $e_D$ and evolve towards the chemical potential.
4)The authors excellently present their rather complicated calculation scheme for non-equilibrium situation, but also
use these real-time integral equation to approach the thermal steady state.
5)In non-equilibrium it is nicely demonstrated that the singlet weight is redistributed by temperature as well as
by finite bias voltage but in distinctly different ways. In the real space representation the authors demonstrate that the singlet expectation value oscillated between even and odd sites typical for a particle-hole symmetric lead,
and the major weight is located around the quantum dot.

1)The NCA has a long history, developed independently by Grewe and Kuramoto in their seminal papers from 1983. Unfortunately, the authors do not really address the shortcomings of SNCA for finite U in reproducing the proper Kondo temperature. Vertex correction are required for including the Schrieffer-Wulff limit of the Anderson model, see Z Phys. B Condensed Matter 74, 439 (1989)
2) I do not fully understand how the Kondo temperature was extracted. Either I did not understand the parameter (using the definition found in Ref [3] I found $T_K\approx 0.05\Gamma$) or there is a problem with the estimated TK. The differentical conductance plotted in Fig 5, however, indicates that T>Tk in the calculations.
3) The y-axis in Fig 5 does not have any units.
4)The plots in Fig 7, however are so small that one cannot see the tails of s(x) at large values of x

Much of the report has been included already in the summary above.

In this paper the author address the question of how a singlet in the Kondo problem
is distributed in real-space and in energy space at finite temperature. This is a new twist in the investigation of the Kondo Problem, since perviously typically the spin correlation function in real space (Refs[8-10]) or the fixed point properties of the total system (Ref [5]) were addressed. The authors extend their analysis to a two-lead finite bias out of equilibrium steady state situation.
For that purpose the authors introduce two composite operators products, $P$ and $E$, to construct a many-body generalization of a two-particle spin singlet projector. This operator is correlating the quantum dot spin state with a spin configuration of a lead orbital in real or energy space.

In order to calculate the expectation values of these projection operator the authors resort to
the simple-NCA (SNCA) in a real-time out-of equilibrium formulation.
The authors excellently present their rather complicated calculation scheme for non-equilibrium situation, but also
use these real-time integral equation to approach the thermal steady state.

The NCA has a very long history which is only presented from a perspective of recent papers. I do not want to criticise the history part of this methodology section to much since different people have different personal perspective onto the history of this almost 40 year approach. I just would like to recommend that the authors briefly addressing the relevant question of the influence of vertex correction onto equilibrium low energy scale. Maybe I overlooked something in their complicated equations such that vertex corrections are somewhere hidden in their formulation. Overall, the section 3 is well written and documents nicely the integral equation used for obtaining the final results.

I mentioned already, that I had a problem understanding the claim of $T_K\approx 0.8\Gamma$ in the paper . Using the definition found in Ref [3] one finds about $T_K\approx 0.05\Gamma$ for the parameters used by the authors which would indicate that that all plots are performed in the Kondo limit of the Anderson model hence above $T_K$ and not in the strong coupling regime of the model.
This observation is backed by the differential conductance depicted in Figure 5: the zero-bias conductance is very low compared to the charge fluctuation peak visible at about $V=U$, which is typical in the Kondo regime ($T\approx T_K$) of the model. In the strong coupling limit, however, $dI/dV$ should
approach the unitary limit of $2e^2/h$ which will be overshot by any (S)NCA treatment due to its violation of the Fermi liquid properties. Unfortunately, Fig 5 lacks the units on the y-axis so we cannot judge how far off the calculations are from the unitary conductance limit.

I have to conclude that the selected parameter operates the finite U NCA in the crossover regime about $T_K$ where we see already some onset of Kondo correlations but have not reached the strong coupling regime.
The authors can also check their numerics: The propagator spectral functions in equilibrium (not plotted here)
should exhibit a clear x-ray edge threshold behavior which can be fitted to a power law at $T\ll T_K$. At finite $T$ this power-law is washed out (see Mueller-Hartmann 1984 or the spectra plotted in Z Phys. B Condensed Matter 74, 439 (1989).

Overall, I am impressed by the evolution of the singlet projector expectation value as function of energy. Starting at short times, correlations live in the vicinity of the single occupied excitation energy $e_D$ and evolve towards the chemical potential. The interesting question arises whether there exist a sum rule by integrating over the spectrum $s_{L/R}(e)$ or $s_{L/R}(x)$ in a similar fashion as for the spatially resolved spin correlation function as pointed out by Affleck and others, for example Ref [9] for a singlet ground state. Deviations from such a sum rule could serve as an indicator for the deviations from the ground state by finite temperature excitation.

In non-equilibrium it is nicely demonstrated that the singlet weight is redistributed by temperature as well as
by finite bias voltage but in distinctly different ways. In the real space representation the authors demonstrate that the singlet expectation value oscillated between even and odd sites typical for a particle-hole symmetric lead,
and the major weight is located around the quantum dot. The plots in Fig 7, however are so small that one cannot see the large x tails of s: it would be useful to provide a double logarithmic plot of $s(x)$
to check for potential power law decays, change of the exponent once x exceeds the estimated Kondo cloud size
as well as whether the even values change from negative to positive value.

Overall the paper is a very solid and excellent piece of work which adds a new twisted to the investigation of correlations in the Kondo problem. I can highly recommend the paper, after some minor points listed in the requested change section are addressed.

1) Check Tk
2) Quote the original finite U ENCA paper Z Phys. B Condensed Matter 74, 439 (1989) which states the value of$ T_K\approx \Gamma/10$ for $U/\Gamma=6$. Hence $T_K$ should be smaller for the parameters used in this paper. My guess: $T_K\approx 0.05\Gamma$.
3) Discuss the difference between a simple NCA and the ENCA with respect to the low energy scale
4) Discuss the unitary limit of the differential conduction in the model and provide proper units in Fig 5.
5) Optional: Optional: provide a log-log plot of the data in Fig 7 top for $V=0$ for revealing a potential power-low tails
which would be characteristic for the strong coupling regime with $T\ll T_K$,
and an a change of exponents inside and outside of the Kondo cloud. At high temperature I expect an exponential decay driven by temperature.

6) Optional: I am wondering about a spatial or energy sum rule of these projector weights analog to the sum rule for spin-density correlation function (see Affleck et al or Borda 2007)