Classical and quantum harmonic mean-field models coupled intensively and extensively with external baths

We changed the paper following the suggestions and comments made by the referees: see the list of changes for the most relevant modifications. We also corrected several typos and grammatical errors which are not listed below.

The following paragraph was added to the introduction on page 2:
" It is now understood that one and two-dimensional, non-linear, non-
integrable, short-range models conserving energy, momentum and stretch, superdiffusive
transport occurs generically. Admittedly, there exist also instances where diffusive regimes
are unexpectedly found numerically [14]. If one conservation law is broken, as in the case
of coupled rotors [15, 16], standard diffusion is restored."

The following sentence in the introduction on page 3 was changed:
"In the present work, we study one of simplest cases of long-range interacting non-equilibrium, quantum many-body system"
->
"In the present work, we study one of simplest cases of long-range interacting non-
equilibrium systems"

The following paragraph was added at the end of the introduction on page 3:
"Models with flat interaction have a paradigmatic importance in the study of
long-range systems due to the fact that they may allow for an analytic solution. Moreover,
they may be seen as providing the limit σ → −d of systems with spatial couplings of the
form ∼ 1/rd+σ, with r being the inter-constituents distance, and a generally interesting
question is whether this limit is well defined or singular. Furthermore, spin models with
flat interactions have been experimentally realized in trapped-ions systems [22] and for
cold atoms in double wells described by the Lipkin-Meshkov-Glick model [41], with the
stationary state of the Hamiltonian Mean Field model simulated with cold atoms in
a cavity [42]. Recent progress in such systems allows for the study of dynamical and
transport properties, despite the effective inherent absence of spatial distance [22]"

On page 10 the following line was added under eq. (22):
"where o(x) indicates a quantity that goes to zero faster than x."

In section 3.3 on page 10 the following paragraph was changed:
"We remark that the crucial point is the cancellation of the pole in the dispersion relation
of the system, which stems from two properties of the model. The first one is the conser-
vation of the total magnetization M in absence of external baths, which stems from the
mean-field nature of the system: from a mathematical point of view, this is related to the
(N − 1)-fold degeneracy of the spectrum of the matrix Φ. The second ingredient is the
linearity of the system, that allows the equations of motion to be solved exactly in terms
of the Green’s function, which in this analysis is given by Q(ω)/ω. By lifting either of
these properties, the temperature profile flattens on the average of the temperatures of
the baths."
->
"We remark that the crucial point is the cancellation of the pole in the dispersion relation
of the system, which stems from two properties of the model. The first one is the linearity
of the system, that allows the equations of motion to be solved exactly, in terms of the
Green’s function, which in this analysis is given by Q(ω)/ω.
The second one is the conservation of the total magnetization M in absence of external
baths, which stems from the mean-field nature of the system: from a mathematical
point of view, this is related to the (N − 1)-fold degeneracy of the spectrum of the
matrix Φ. "

The following sentence was changed on page 11:
"We performed some simulation adding a term −x3 to the right-hand side of Eqs.31 [40] and found that kinetic temperatures settle to the the average."
->
"We performed simulations with the
same method used to obtain the data plotted in fig.2, i.e. via the numerical solution
of the Langevin equations of motion, now adding a term −x3 to the right-hand side of
Eqs.31 [45]. We found that kinetic temperatures settle to the average of the temperatures
of the baths."

The following paragraph was added on page 11:
"Indeed, for the temperature profile to vanish, it suffices that the degeneracy of the
matrix Φ is of order N . For example, if we add a pinning potential to the first and last
site, the degeneracy of Φ is N − 3, that is still of order N . One can then examine the
equations of motion of the system as we did above: the result is that once again we have
a number of degrees of freedom of order N completely decoupled from the baths. Thus
we guess that in order to have thermalization one needs to break the degeneracy of Φ to
a quantity of order 1."

We added the value of k1 to the caption of figure 8 on page 19.