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

by Rosa López, Pascal Simon, Minchul Lee

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Submission summary

Authors (as registered SciPost users): Minchul Lee
Submission information
Preprint Link:  (pdf)
Date accepted: 2024-03-04
Date submitted: 2024-02-22 05:11
Submitted by: Lee, Minchul
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


We investigate the quantum transport of the heat and the charge through a quantum dot coupled to fermionic contacts under the influence of time modulation of temperatures. We derive, within the nonequilibrium Keldysh Green's function formalism, generic formulas for the charge and heat currents by extending the concept of gravitational field introduced by Luttinger to the dynamically driven system and by identifying the correct form of dynamical contact energy. In linear response regime our formalism is validated from satisfying the Onsager reciprocity relations and demonstrates its utility to reveal nontrivial dynamical effects of the Coulomb interaction on charge and energy relaxations.

Author comments upon resubmission

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Dear Editor,

We are grateful for sending us the review reports of our manuscript entitled
“Heat and charge transport in interacting nanoconductors driven by
time-modulated temperatures” by R. Lopez, P. Simon and M. Lee. The Referees
consider that our work presents a number of interesting results that can be
verified using modern experiments. We are very happy with their constructive
and positive reports that will allow to our work to improve.

We have answered to the Referees' questions/comments. After this exhaustive
review, we want to transmit to the Editor that we are very satisfied with the
new version that we believe that is now ready for publication as an open access
article in SciPost.

# Reply to Referee 1

We thank the referee for reviewing our manuscript and for his/her criticism
that will help to improve our manuscript. We appreciate very much the comments
about our results that are considered interesting and susceptible to being
tested in experiments. Please, find below our responses.

> 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?

In the Tien-Gordon approach the effect of a microwave field consists of adding
a (spatially constant and) time-dependent electric potential $V_{AC}
\cos\omega_0 t$ which is coupled to the charge number operator in leads. It
transforms the wave function $\Psi(x)$ to the time-dependent one $\Psi(x,t) =
e^{iV_{AC} \cos\omega_0 t} \Psi(x)$, which yields the appearance of
quasi-energies $\epsilon \rightarrow \epsilon + n\omega_0$. This approach was
initially applied to the tunneling between thin superconducting films through a
barrier and revealed that tunneling occurs not only at energy $\epsilon$ but
also at the quasi-energy values $\epsilon + n\omega_0$ due to absorption and
emission processes of quanta of energy (photons). The current through the
barrier is then composed of the sum of all possible absorption/emission
processes, each of which is weighted with some probability given by the Bessel
functions. This behavior is maintained in the case of quantum dots driven by an
ac potential but only in the non-interacting case and in the wide-band limit
(when the tunneling broadening is considered constant). When interactions are
considered the quantum dot charge needs to be computed self-consistently and
the Tien-Gordon description is no longer valid [see Rosa López et al.,
Low-temperature transport in ac-driven quantum dots in the Kondo regime,
Physical Review B 64, 075319 (2001)]. In such a case the current is no longer
simply the sum of individual current events evaluated at the quasi-energies
weighted by special functions, but much more involved.

The Tien-Gordon approach cannot be directly applied to the time-dependent
temperature-driven case which is the main focus of our work. The key difference
is that in the Luttinger’s scheme, the time-dependent field is now coupled to
the excitation energies as well as the change numbers. Then, the simple
analysis based on the quasi-energies is no longer valid: the photo-assisted
processes with different n become intermingled with each other. So the current
cannot be interpreted in terms of the sum of the processes. Therefore, the
Tien-Gordon approach seems not appropriate in our case.

Also, there is one more issue. In the original Tien-Gordon approach, the field
is spatially constant. But our study shows that the time-dependent field should
not be spatially constant, but instead, a part of the field should be coupled to
the barrier as well, in order to predict the correct heat current.

As the referee has recommended we have added a couple of sentences to make
clear about the main difference between our formalism and the Tien-Gordon
picture [see page 9 in the revised manuscript].

> 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’)?

Surely the time for (tlkσ,m’)^* should be t’.

We thank the referee for alerting us about this typo. We have corrected it in
the revised version.

> 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. Maybe the authors mean something different.

The relevant physical quantity which is to be measured in experiments is not
the energy stored in the leads, but its time derivative, that is, heat current
(the change rate of the energy). Hence, the constant, whether it diverges or
not, is eliminated once the time derivative is taken. Also, the wide-band limit
is a theoretical artifact used to make the results simpler (by eliminating
non-essential parts). So, the constant should be finite.

We have stressed better that the term is relevant only for the heat flow in the
revised version [see below Eq. (26)].

> 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.

The constant $E_{T\ell0}$, which is not zero, is the energy stored in the
barrier for the static case. Under the same reasoning as in the answer for
Comment 3, this static value does not contribute to the current. However, this
constant $E_{T\ell0}$ also appears in the first-order term of the current. We
attribute it to the artifact of our Luttinger’s scheme. We have devoted one
paragraph (below Eq. (37)) to the explanation of its appearance and the reason
why it can be safely ignored.

In our setup, we apply the field $\Psi(t)$ to the lead, and the additional field
$\Psi(t)/2$ to the barrier. So, an effective dynamic energy capacitor is
formed, which should not be present in the original system. This effective
energy capacitor induces an additional ac heat current between two energy
reservoirs, which is the last term in Eq. (37). So, this term should be
ignored. We have confirmed this fact by applying our formalism to the
non-interacting case. Moreover, the presence of the interaction in the quantum
dot does not affect this unphysical term, so the heat current expression,
Eq. (37) without the unphysical term is generally correct as long as the linear
response regime is considered.

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

The referee is right. We have corrected the indices in Figure 2 to match with
those in the main text.

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

The Referee's remark is very useful. Indeed it was not clear from the caption
which equations are used to make the lines in Fig. 3. The cross thermal
resistance (Fig. 3a) and capacitance (Fig. 3b) are evaluated via Eq. (50) with
Y = K, where the cross thermal admittance K is given by Eq. (48b). The cross
thermoelectric resistance (Fig. 3c) and capacitance (Fig. 3d) are evaluated via
Eq. (50) with Y = L, where the cross thermoelectric admittance L is given by
Eq. (48a). In both the cases, the temperatures are finite but the value of the
frequency is chosen numerically as small as possible so that the low-frequency
limit is taken. For the case of low temperature (solid black lines) there is a
comparison with the analytical expressions [Eq. 54(a,b) and 55(a,b)] obtained
from the Sommerfeld approach as stated in the caption. We have included this
information in the caption of the revised version.

> 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?

Usually, it is assumed that FDT holds for heat transport in which the spectral
density of the energy current and the ac heat conductance are related
accordingly to

= \hbar \omega T \Re G_{th} \coth \left(\frac{\hbar \omega }{2T}\right)
S(0) = 2T^2 G_{th}(0)

And this assumption is based on the zero frequency limit for the static
case. However, Ref [63] (see Reference List of our work) demonstrated that this
is not the case. In the case of heat transport, the heat conductance is not
given solely by the energy fluctuations and it has an extra term. Physically,
the origin of this extra term can be traced back to finite coupling between the
reservoirs, which creates quantum fluctuations of their energy even at $T=0$,
when the thermal conductance in the FDT relation vanishes since there are no
real excitations that could irreversibly transfer energy between the

To show explicitly the departure of the FDT we need to extend our theory to the
calculation of fluctuations which is a very interesting extension for our work
in the short future. At this stage, we can solely affirm that our results seem
to indicate that FDT cannot hold for ac transport in the heat flow.

> 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):


We thank the referee for alerting us about these misprints. We have corrected
all of them in the new version.

# Reply to Referee 2

We thank the Referee for reviewing our manuscript and for his/her
questions/comments that will help to improve our manuscript. Please, find below
our responses.

> 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).

The formulation in Ref. [52], called as the thermal vector potential theory,
rewrites the coupling term between the gravitational field $\Psi(x,t)$ to the
energy in terms of the (thermal) vector potential, by using the energy
conservation law. One can then find the relation

\partial_t A(x,t) = - \frac{\nabla T}{T}.

That is, the spatial variation of the temperature is replaced by the temporal
(and spatial) variation of the (thermal) vector potential. The formulation is
usually applied to the bulk case, and it is known that this kind of
transformation has some pros: the elimination of $T=0$ divergence and the
ability to incorporate the magnetic currents.

In our scheme, our gravitational field follows its original definition in the
Luttinger's idea

\nabla \Psi(x,t) = - \frac{\nabla T}{T}.

That is, our field has nothing to do with the vector potential.

We didn't follow the formulation based on the thermal vector potential theory
because we focus on the thermal transport through nanostructures and the
dynamic thermal scattering by the nanoconductors such as quantum dots. Then,
the original formulation of Luttinger is enough.

> 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.

From the relation $\nabla \Psi(x,t) = - (\nabla T)/T$, one may regard the
gravitational field as the inverse of the temperature (as long as the
temperature variation is small enough). Since the temperature is (in a rather
macroscopic scale) a function of both the time and the position, the
gravitational field can vary with the time and the position. In our scheme, the
spatial dependence is denoted by the lead index $\ell$, assuming that the
temperature is homogenous in each lead, and we focus on the temporally
sinusoidal variation of the temperature.

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

The strictly dc case does not require the Luttinger's scheme because the
Landauer-Buttiker formalism works successfully in this case.

In the demonstrations of the non-interacting and the interacting cases (with
the Hatree approximation), we indeed studied the low-frequency limit. The
resistances, capacitances and RC times shown in Figs. 3, 4, 5 and 6 are the
values in the low-frequency limit. It should be noted that the low-frequency
limit does not necessarily saturate to the dc case. For example, as shown in
Fig. 2, the self-resistance $R_{Y,\ell}$ and self-capacitance $C_{Y,\ell}$ are
relevant only in the ac case. It is because the dc current cannot flow through
the capacitance. The same reasoning applies to the cross capacitance
$C_{Y,lr}$. So, only the low-frequency quantity which is relevant in the dc
case is the cross resistance $R_{Y,lr}$. We already have a short discussion on
$R_{Y,lr}$ in the first paragraph of Sec. 4.2, which relates the resistances to
the density of states in the quantum dot.

> 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)

We thank the Referee for alerting us about these misprints. We have corrected
all of them in the new version.

List of changes

1. [At pages 8--9, below Eq. (26)] The sentences are changed to reflect the
point (comment 3) raised by the Referee 1.

2. [At page 9, below Eq. (30)] A new paragraph is added to explain the point
(comment 1) raised by the Referee 1.

3. [Figure 2] The figure is redrawn to have correct mathematical symbols as
pointed by the Referee 1 (comment 5).

4. [At the caption of Fig. 3] A new sentence is added to explain how the plots
are produced, as requested by the Referee 1 (comment 6)

5. Correction of typoes

[Below Eq. (9)]
with a same frequency -> with the same frequency
by a same Fermi distribution -> by the same Fermi distribution

[At Eq. (24)]
a missing ‘prime’ in the argument of the second is added: (t) -> (t')

[Below Eq. (30)]

[At Eq. (44)]
a missing operator d is added

Published as SciPost Phys. 16, 094 (2024)

Reports on this Submission

Anonymous Report 2 on 2024-2-27 (Invited Report)


Same as in previous report


Same as in previous report


I read the manuscript and the Author's responses.
I recommend the present version for publication in Scipost Physics.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2024-2-27 (Invited 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 responded satisfactorily to all the comments I made. Therefore, I can now recommend this manuscript for publication.

Requested changes

no changes

  • validity: high
  • significance: high
  • originality: top
  • clarity: high
  • formatting: perfect
  • grammar: excellent

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