Spin-chain based quantum thermal machines

Report This paper presents a formalism to study systems that thermalise with a bath via local couplings. The formalism is applied to a spin chain engine, and the different regimes of operation are obtained. This result is potentially interesting within the emerging field of quantum heat engines, particularly in the context of many-body working substances. However, I do not think the significance is clear enough to warrant publication in SciPost Physics. Unless this can be improved upon, a more specialised journal such as SciPost Physics Core would be more appropriate. The significance could be made clearer by, for example, including a discussion on the performance of this engine compared to one with the entire chain coupled to a reservoir. The analysis of power output would be interesting, and could include an optimisation of the function in Eq. (49). This optimisation is presumably easier than cases where Eq. (49) does not hold. REPLY: We thank the referee for acknowledging the potential interest of our work. We respectfully disagree that comparing our set-up with one where the entire chain is coupled to a reservoir would clarify the significance of our results. What is most significant in our work is the finding that the presence of symmetries (in our case total magnetization is conserved) may allow for tremendous simplifications in the calculations, and so allow to study very big chains (up to thousands of spins), which is far beyond what can normally be achieved (including the case where the whole chain is coupled to a reservoir).

Requested changes 1. Could the authors describe in a little more detail the quantum channel formalism so that the presentation is self-contained? Is the essence of this formalism in the identification of a LCPTC that acts on a subsystem, or in the discretising of the evolution into n cyclic steps, or both? REPLY: the evolution of the global system density matrix is given by the concatenation of unitary and thermalisation steps. Such concatenations generally result in a LCPTC map. For example, the operator that advances of 1 cycle is an LCPTC map. So yes, discretization is an important aspect of our analysis and the LCPTC property of the map is in fact of crucial relevance. We believe that a reader can understand the dynamics of advancement of one cycle with the notions provided in the text, even without knowing much about channel-formalism. We slightly changed the presentation in order not to give the impression channel formalism is crucial to understand how to advance of one cycle. We now observe, a posteriori, that the transformations that advance the state of one cycle are LCTPC maps, and refer the reader to according references on the topic of channel formalism.

1. Could the authors provide more interpretation of the ansatz Eq. (49)? The function g presumably arises due to the state prior to each work step being thermal. How then does the function f relate to the transition probabilities |⟨i|U(τ)|j⟩| for states i and j of the spins coupled to the reservoirs? How are these probabilities constrained by the symmetries of the setup? REPLY: our exactly solved examples with N=2,3, provide explicit expression of f in terms of the transition probabilities |⟨i|U(τ)|j⟩|. This is particularly clear in Eq. (61). Our low-T analysis also provides the explicit expression of f in terms of transition probabilities, see Eq. (80). As mentioned in the text the symmetry imposes selection rules by which the total magnetization is preserved.

2. Related to the point above, the discussion of Eq. (49) at the end of page 8 suggests the single excitation expression Eq. (83) may hold even for high temperatures. This is surprising, as higher excitation sectors of the Hilbert space will in general be occupied. Can the authors provide additional arguments as to why these higher excitation regions are unimportant when determining heat flows and work outputs? Or if the higher excitation sectors are important, how can this be reconciled with Eq. (83) and Eq. (49)? REPLY: From Eq. (49) we see that high excitation regions are indeed very important in the evaluation of heats and work: only their effect is lumped in the “g” function rather than in the “f” function. The “f” function alone on the other hand is determined only by the low T sector of the Hilbert space, Eq. (83) and this per se does not mean that high T sectors are unimportant for calculating Q and W. The physical reason why the low T sector suffices for the calculation of “f”, is that the chain is such that it allows for at most one excitation at a time to be transferred, i.e., the transfer mechanism is mediated by single excitations only. We added a discussion of this point for the benefit of the readers.

3. The forcing of the system via turning on and off local couplings is a little subtle and should be emphasised. In the results, could the authors include more of a discussion on the effects of this on the energy of the system? When is the work an input or an output, or does this vary with the duration of the intermediate, energy conserving evolution? I think the statement, “the working fluid is not driven by external forces, except for turning on and off local couplings” in the introduction is misleading, as it relegates rather than emphasises the forcing done by tuning the couplings. REPLY: the turning on and off of the coupling has a great impact on the thermodynamics of the system, as it is the only source of work at play here. The sign of work (and heat as well) depends indeed on the duration of the unitary evolutions, whether the magnetization is conserved, and in fact on all parameters entering the Hamiltonian, and the temperatures. It is an extremely rich interplay. Fig 2 and 6, are devoted exactly to illustrate those changes of sign: each of the operation modes is determined by certain combinations of signs for W,Q_H and Q_C as extensively discussed on page 4. We think that those figures are already sufficient to illustrate this point. In order to emphasise the role of the on/off switching we have changed the sentence “the working fluid is not driven by external forces, except for turning on and off local couplings” into “the working fluid is driven by an external force that is responsible for tuning on and off the local couplings. Besides that, no other external force is applied onto the system”.

4. Figure 3 is very hard to decipher. Separate frames for the three surfaces would be clearer. We have replotted the figure as a grid of three distinct plots. It should be easier now to decipher it.

5. Why is the refrigerator region not present in Fig. 6(a) despite the text arguing that all four regions are possible? REPLY: when E_1 and E_2 have opposite signs, as in Fig.6(a) all operation modes are possible. The fact that they are not all represented in Fig 6(a) is only a mere accident and has no special reason. It only means that for the chosen parameters, and ranges of \tau_1, \tau_2, the [R] operation, was not realized. Note that we can see such [R] regions in Fig. 5 (c), relative to E_1 and E_2 with opposite signs, which confirms the validity of our statement that all four regions are possible.

Optional changes:

1. Figure sizing is inconsistent with the amount of detail. Fig. 4, for example, is very large relative to the detail, whereas Fig. 5 and 6 are very small relative to the detail.

We have decreased the display size of fig. 4. We have amended Fig.5,6 so as to make the axes labels better readable.

1. Section III B 2. The derivation of f at low temperatures is quite technical. I wonder if it can be presented more simply by the commonly used approximation of replacing spin raising operators by bosonic raising operators, following a Holstein-Primakoff transformation? We believe that maybe the low T derivation could be done using mappings onto possibly bosonic or fermionic operators, by means of appropriate transformations as suggested. We prefer to stick with the current derivation because it is very much in line with the LCPTP formalism which as the basis of our approach.

2. The coloured regions in Fig. 2 do not quite align with their boundaries (very minor, however I was initially confused by this). We edited the figure so that the boundaries now are better aligned.

• validity: good • significance: ok • originality: good • clarity: ok • formatting: good • grammar: good

The paper by Centamori et al. is an original analysis of many-body quantum thermal machines with interesting results. It is clearly written, except for a few too many typos/omissions and improvable figures. It also shows very interesting results which seem solid to me. I like the paper, however before recommending I would like the authors to consider the following questions/comments:

1) The authors consider the turning on and off of the System Hamiltonians H_AC and H_BC, and how they affect work. However, what about the coupling and decoupling to the baths which then thermalize one of the spins at the extremities? That also typically contributes to some work exchange, which may be small for very small system-bath coupling (although this would result in a very slow relaxation). I would like to understand more about this from the authors. REPLY: we agree that turning on/off the coupling of the end-spins to the baths contributes to some work. Our results are obtained, as the referee observes, in the regime of weak coupling. In this regime the associated work is negligible as compared to all other relevant energy scales. To illustrate this point, imagine H= H_0 +\epsilon H_I where H_I is coupling Hamiltonian and \epsilon is a dimensionless parameter. If you abruptly turn on the coupling energy, the according work is W= \Tr \rho (H-H_0) = \epsilon \Tr \rho H_1. This can be made arbitrarily small by decreasing \epsilon. Our assumption is that \epsilon is so small that W is negligible compared to the work done to attach/detach the end spins to the rest of the chain. This may result indeed in a slow relaxation, hence long thermalization time, meaning very low power. These are all aspects that need to be taken into account in practical realizations of the machine. We added a discussion of this point for the benefit of the readers, see added footnote [42].

2) in the results by the authors, it seems that the integrable or not character of the spin Hamiltonian does not play a role. Only the presence of a symmetry. Can the authors stress this more emphatically, and maybe also suggest a reason of why that is the case? I think it is because this is a many-body open system and what is more important is that the maps are mixing, which does not require non-integrability of the Hamiltonian. REPLY: we agree with the referee: what counts the most here is the fact that the map is mixing, and that may happen regardless of whether the Hamiltonian is or is not integrable. We agree that this needs to be further emphasized. We added a discussion of this point in the conclusions of the revised ms.

3) Another point I would like to have a comment from the authors is about the insight that a low temperature analysis can give. The larger the systems are, the smaller are the energy gaps, and hence the regime of small temperature is significantly reduced. However this is not said explicitely. REPLY. We agree with the referee that the regime of small T would be significantly reduced. This bears no consequences in regard to our estimation of the function “f”, which does not depend on the temperatures. As long as a low T region exists, no matter how small that is, our argument would be applicable and valid. We remark that “f” would generally depend on the size of the chain, but that is a different issue. We added a discussion of this point the new ms for the benefit of the readership

4) Last but not least, the result of Eq.(49) is very enticing, and I wonder whether it is valid only in the limits analyzed by the authors, mainly small systems or low temperature (still in presence of a symmetry). Especially, typical behavior in many-body system occur for relatively high energies (that one would reach at high temperature), while near the ground state, behavior is not typical. So I would like to understand a bit better why one could confidently extend results at low temperatures to be generic. Related to this, what properties of the spectrum one would need/like to have for the conjecture to be verified? Does it work for both gapped and gapless systems? Is it really only necessary that the maps are mixing and the system has a conserved quantity? REPLY: Regarding Eq. (49) we have analytical evidence for small systems, and for chains of arbitrary size at low temperatures. We also have numerical evidence of intermediate sizes at finite temperature. All this suggests that the ansatz may be generally valid. In our understanding the structure of Eq. (49) is a consequence of the symmetry and mixing, combined with the very structure of the thermodynamic cycle. Thus symmetry and mixing are sufficient but not necessary conditions. We stress that the calculation of heat and work markedly depend on the bath temperatures: the interesting point is that such dependence is all umped into the “g” function, so it would not be correct to state that low T evaluation of heat and work extend to high T: only those contributions to such quantities that depend on single excitation physics, which are all lumped into the factor “f” do extend to high T. As mentioned above, as long as there is a low T region, no matter how small that is, our ansatz would be obeyed. Thus as long as the spectrum is gapped, that is, as long as the system size is finite (no matter how large) Eq. (49) would be valid.
We added a remark paragraph, after Eq. (83), in order to clarify this crucial point for the benefit of the readers.

Small comments: there are a bit too many typos.

For instance a few [?] CORRECTED Eq.s CORRECTED of the of the CORRECTED of of CORRECTED extra white spaces CORRECTED "the explicit evaluation the heat exchanges ..." CORRECTED the quality of the figures can be improved: lines are indistinguishable in black and white, some labels are too small, some are too large and pixelated, some figures are huge CORRECTED as mentioned above on experimental realization of heat engines with ion trap I do not see the work by Van Horne from M. Mukherjee's group https://www.nature.com/articles/s41534-020-0264-6 ADDED between the many-body heat engine papers, I am suprised not to see the paper by one the authors on phase transition and heat engines https://www.nature.com/articles/ncomms11895 ADDED • validity: high • significance: good • originality: good • clarity: high • formatting: good • grammar: excellent