Thermoelectric coefficients and the figure of merit for large open quantum dots

1. The paper quantitatively addresses the problem of weak thermoelectric response of large open quantum dots which was left open by reviewed prior works.

2. The averages of thermoelectric quantities, their powers and even their full statistics are explicitly derived. The used diagrammatic technique is very helpful in understanding the derivation, in particular the nontrivial factorization into pair contributions (Fig. 6).

3. The results indicate that a formula indicated in previous work without derivation (Ref. [10]) although correct is of quite limited relevance. Instead, the relevant temperature dependence is almost completely governed by higher order terms in $k T/ E_{Th}$ computed in this paper.

4 The paper is clearly written and good to follow, the figures are very good and the appendix is helpful.

None

This is an nicely written and interesting work and I recommend its publication. Apart from the unproblematic changes requested below I have two optional questions /suggestions:

1.
The cancellations that causes only diagram I and I-TR to contribute make me wonder if there is a simple deeper principle causing this. Do the authors have an idea / suggestion?

2.
The authors are careful to mention that all discussed effects are small and emphasize thermoelectrics as a probe for the physics of dots. Can they comment briefly in the outlook on electron-electron interactions perhaps with some pertinent reference? I guess these break the symmetry that causes $\langle S \rangle =0$? Is their any prospect of including these with their / a similar approach?

1.
Abstract. The "asymmetry between the left and right leads" is easily misunderstood in the abstract for the standard multipicative asymmetry. I suggest some modification to distinguish this:
"...does not depend on the _additive_ asymmetry between the left and right leads."

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2.
Eq. (19): the paper stresses that the previously reported result (19) is essentially irrelevant since it holds only for very small numerical ratios of $T/E_{Th}$. It is not so clear what the simple behavior at low T now is, in particular the N dependence.
Looking at Fig. 2, it seems to me that a rough linear interpolation between the origin and the maximum is the relevant result, i.e.,

$\langle S^2\rangle \propto \langle S^2\rangle_{max} (kT/E_Th)$

which at scales as $1/N^3$ rather than $1/N^4$ or $1/N^2$. It is only when fixing an $N$-dependent temperature $kT \sim E_{Th} \sim N \Delta$ that one gets $\langle S^2 \rangle_{max} \propto 1/N^2$ as in Eq. (20). Please comment on this.
I would even suggest to point out more clearly in the introduction and conclusions hat the result of Ref. [10] in some way puts one on the wrong track. This does not become clear until p. 6. and actually is key insight of this work in my view.

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3. After Eq. (20):
It may be worth to comment on the $T$-position of the maximum of (17):
it seems to be the same for low and high fields and but in between makes a (small?) excursion for finite fields $B \sim B_c$?

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4. After Eq. (21):
I was surprised the authors did not comment on the fact that their result (21) is up to constant factors just Eq. (17):

$\langle ZT \rangle = (3/\pi^2) (e/k_B)^2 \langle S^2 \rangle = \langle S^2 \rangle / L $.

where L is the Lorenz number. The constant factor derives from $\langle I_2\rangle /\langle I_0\rangle e^2 T^2$ by Eq. (7-8). I was not sure if Eq. (21) is now an unexpected result given (17) or should one expect it (does Wiedemann-Franz apply)? Please clarify.
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5.
On p. 7: "because the leading term in the expansion of .... is of order $x^2$"
I guess the leading correction (next-to-leading term) is meant here, the leading term is 1/b.

-----6.
On p. 8, Eq. (10): the $lim_{T\to 0}$ is confusing here (the limit is just zero), write $k_B T \ll E_{Th}$ instead.

-----7.
On p. 11: "it is only take one line of algebra" needs a fix.

-----8.
On p. 12, Fig. 3: the rules mention "an encounter inside the dot". I cannot make out from the diagrams why lines I-IV do not run through the dot ($\kappa=0$) and V-VI do run through it ($\kappa \neq 0$). Where is the dot to be seen ?
It is not stated in the caption where the dot is to be seen and that the boxes are the "encounters".

-----9.
On p. 13: Fig. 5 is referenced here but it appears only on p. 16 after unrelated text. Move Fig. 5 to p. 14.

-----10.
On p. 17: "those with E_J" should read "with E_j" ?

-----11.
On p. 17: Summary: "We have provided" ... "We showed ..." perhaps? Fix: "have long been knowN to...", "thermoeleCTRIC figure of merit"