Heat and charge transport in interacting nanoconductors driven by time-modulated temperatures

1. The subject is very timely.
2. The technique presented in this paper is promising for the study of thermal and thermoelectric transport in quantum systems with many-body interactions.

1. This approach is valid only in the framework of linear response.

The authors investigate quantum transport of charge and heat in a mesoscopic structure with a quantum dot
Coupled to fermonic reservoirs with time-dependent temperatures. They formulate the problem in the framework of
Luttinger’s Hamiltonian representation of the temperature bias. They solve the corresponding non-equilibrium problem
by means of Keldysh non-equilibrium Green’s functions. They focus on the strong-coupling limit between the quantum dot and
The reservoirs and they discuss the importance of properly taking into account the energy temporarily stored at the tunneling
contacts.

The problem is timely, the article is well written and the implementation of the Luttinger representation is a useful tool in the
calculation of energy transport in systems with many-body interactions. I think that this manuscript deserves publication in
Scipost Physics after authors better explain the following points:

1. In Ref. 52, a gauge invariant formulation Luttinger’s representation was presented. I think that the authors follow this route, as they
Introduce the ‘’gravitational field’’ in the hopping term. I also guess that Eq. (21) results from expanding an exponential after a gauge
Transformation. Is that correct? In any case, it would be useful to have more details on the steps from Eq. (8) to Eq. (21).

2. In addition and somehow related to the previous item, it is not completely clear how the ‘’gravitational field’’ is related to the temperature
bias. In Luttinger’s approach, there is only one field associated to the difference of temperature between the two reservoirs. In Ref. 52, this was substituted
by the time-derivative of a vector potential (following the analogy with electromagnetism). In the present paper, I’m not able to find the explicit relation
between \psi_\ell and the temperatures. Are two fields necessary instead of a single one? Why? More discussion on these points is most welcome.

3. It would be perhaps interesting and useful to analyze the dc limit, corresponding to zero frequency.

4. Minor details I detected:

* There is a missing operator d in Eq. (44)
* There is a missing ‘prime’ in the argument of the second t_{lk..} in Eq. (24)

See items 1-4 of the report.

1. Fully quantum coherent treatment of charge and heat transfer through an interacting quantum dot

1. Only a small temperature bias is considered

The authors address the transport of charge and heat through a single-level quantum dot under a time-dependent modulation of temperatures of its contacts. To preserve the quantum coherence of the problem, the authors use the Luttinger approach, where a thermodynamic quantity, temperature, is represented through a dynamical quantity, the gravitational field, which is coupled to the energy density of the system. This is an interesting approach, which is useful for studying various problems in the intensively developing field of Quantum Thermodynamics and its emerging sub-fields. The authors present a number of interesting results, which can be verified using modern experiments. I can recommend this manuscript for publication after the authors take into account the following comments:

1. If the authors find it useful, can they comment on the possible relationship (or lack thereof) between their approach to time-dependent temperature bias and the Tien-Gordon approach to time-dependent voltage bias?

2. Why in Eq.(24) are the two tunneling amplitudes tlkσ,m and (tlkσ,m')^* calculated at the same times t (not at t and t')?

3. After Eq.(26), the authors write: "While this value can diverge in the wide-band limit, it is irrelevant in our study..". To me, this saying is a bit confusing: If this term would not be subtracted on the left hand side of Eq.(26), then the right hand side of this equation would diverge. Therefore, this term is relevant in order to obtain physically meaningful finite result. May be the authors mean something different.

4. What is E_{T\ell 0} in the additional unphysical term of Eq.(37)in the linear response? If this is the energy stored in the tunneling barrier, then the additional unphysical term is zero. Since in a linear response, E_{T\ell 0} is calculated at \Psi_{\ell}=0, that is, in the static case. But the energy stored in the tunneling barrier is zero in the static case.

5. The quantities R,C, and Z have indices that do not match in Figure 2 and in its caption.

6. Which equations are used to produce plots shown in Figure 3 ?

7. On page 16, the authors write: "It should be noted that the fluctuation-dissipation theorem applied to the heat transport through two-contact systems is no longer valid because scattering events that connect two different terminals induce a nonvanishing term for the equilibrium heat-heat correlation function at the low temperature limit, which is incompatible with the expected behavior of Klr(Ω) [63,64]"

Could the authors be more specific by showing an example of what FDT predicts, what they predict, and what the difference is between the two predictions?

Possible misprints:

i. After Eq.(8) : (In my view, "a" implies any, but "same" is definitely not any)
with a same frequency -> with the same frequency
by a same Fermi distribution -> by the same Fermi distribution

ii. After Eq.(30):
NEQF -> NEGF

see Report