Ground State Analysis of the Spin-1/2 XX Chain Model with Anisotropic Three-Spin Interaction

1. Interesting variation on previous models with three spin interactions
2. Very interesting to discuss 'entanglement measures' such as concurrence and quantum discord.
3. Very interesting result that one of the phases doesn't seem to obey the usual CFT entanglement scaling.

1. Extension of model doesn't seem to provide any new phases not seen in previous literature
2. I find some of the characterisation of the phases misleading
3. The most interesting result about not obeying CFT entanglement scaling needs further explanation.

This paper studies a model with a three spin interaction that maps to free fermions by a Jordan-Wigner transformation. They study the phase diagram, along with a number of ‘entanglement based’ probes including concurrence and quantum discord.

While the model is somewhat artificial, it is an obvious deformation of previously studied models, and studying phase diagrams via entanglement probes is a relatively new and upcoming field that I believe is offering new insights into correlated phases of matter. I find the paper interesting, however have a number of queries about what is done.

1. In the plot of the spectrum in Fig. 1 (and equation 14), there is not a consistency about whether the spectrum is positive or negative (or in other words, whether to take the positive or negative square root). I believe one valid approach is to use the positive square root for |k|<pi/2 and the negative one for |k|>pi/2. Another might be to define epsilon(k) as only between -pi/2 and pi/2, and also take the of the spectrum (analogous to the reduced zone scheme). But I don’t completely understand the plot shown the way it is. I also think it might be very useful to see what happens exactly at the transition boundary — i.e. how the new fermi points appear.
2. The two phases are referred to as spin-nematic-I and spin-nematic-II; however I do not see where this terminology comes from. My understanding of spin-nematic states is that they break spin rotation symmetry but not time-reversal — and I see no indication (or discussion in the manuscript) that these phases do that (potentially they break spin rotation because the Hamiltonian does, but I don’t think this is a good use of the term spin-nematic). The spin-nematic-I phase is also labelled chiral so potentially breaks some reflection symmetry (although this could be discussed more in the manuscript if this is correct), but I don’t see what symmetries the spin-nematic-II phase breaks.
3. On a similar note, the notation of ‘cluster order parameter’ I think is a bit misleading. A bit more background could be given as to what they mean, and are they really order parameters? For a generic beta, these seem like terms in the Hamiltonian — so certainly are not indicative of any symmetry breaking (as one expects from a conventional order parameter). Indeed, following on from point 1, if the Hamiltonian is chiral, the surprise may not be that one of the phases is chiral, but that the other one isn’t. I feel this could be discussed a lot more in the manuscript.
4. Another point about the phase diagram (Fig. 2c) — at beta = -1, the model reduces to the one studied in Ref. 11 — and already along this line in the phase diagram, one sees both phases. If this is correct, what can one see in the extension to the model presented in this work that one can’t see in the earlier paper? This is not clear.
5. One final point about the phase diagram — the spin-nematic-II phase includes the line alpha=0, which is just the XX model. If there are no further phase transitions, then surely this entire phase is the same as the ground state of the XX model, which one usually calls quantum critical or quantum disordered (or something to that effect). Is there anything in this phase that one can’t see in the XX model?
6. On a technical note, I’m not convinced how appropriate it is to call the transition between two gapless phases a ‘second-order quantum critical line’. In the fermionic language, this would better be called a Lifshitz transition. Maybe there are other names for it, but it definitely shouldn’t be characterised as a traditional second order quantum critical line. It’s worthy of note that the boundary between the gapless XXZ and gapped XXZ at Delta=1 is of Berezinkii-Kosterlitz-Thouless universality class, so also not second order.
7. Could more be said about where concurrence goes to zero? This is kind of curious as it is not associated with any of the phase transition lines. Could the authors expound a physical reason for this to enhance understanding?
8. About the central charge: naively, I would expect the entire phase labelled spin-nematic-II to have central charge 1 (as this phase encompasses the XX point). Is this correct? I would then expect the spin-nematic-I phase with an extra fermi point to have central charge 2 — however the result seems to be that it doesn’t obey the standard logarithmic scaling of a gapless theory. This is very surprising (and interesting) if correct so needs a lot more explanation. Can this entanglement be explained? And how does one get around the standard Calabrese-Cardy result from CFT? Does it mean this phase is not described by a CFT — maybe there is something coming from the Jordan-Wigner transformation, but I’ve not seen anything like this before.
9. Finally, almost all of the plots in the paper are at alpha=1. I think this is not entirely representative, as at alpha=1, the critical point is at beta=-1 — and beta=-1 is a special place as it is where the superconducting terms in the free fermion model go away. For example, the phase transition at alpha=1 beta=-1 involves the fermi velocity becoming zero at one of the fermi points. I don’t think this is representative for other values of alpha. It also makes me a bit suspicious of the central charge calculation at this point — according to the fermionic spectrum, the model shouldn’t be described by a CFT at all at this point. At other values of alpha though, I don’t think the fermi velocity goes to zero (although I haven’t carefully checked this) so this may be a slightly special point in the phase diagram.

In a few other minor points, it seems strange in Fig 1a when plotting the spectrum to say ‘for a chain system with N=1000’. Isn’t this just the analytic formula 14 being plotted? Similarly, there are many other places where the chain length is mentioned where it seems it is an analytical formula being plotted. I also didn’t understand the step in Eq.19 going from the definition of fidelity susceptibility to an expression in terms of the Bogolyubov angles — perhaps this can be expanded on further. There is also a minor error in the Hamiltonian, Eq.11 where alpha is written when it should be J’ (or the overall factor of J has to be in common for all of the terms which is probably what was intended). Similarly in Eq.15.

In summary, there appears to be some interesting results in this work, but many of them need to be explained more fully, and the work should be placed better in the context of previous work — in particular Ref. 11 which has exactly the same two phases.

1. More comparison with the phases in Ref. 11 -- and more clarity on what the phases are
2. Results for some value other than alpha=1 as this isn't necessarily representative.
3. Suggest some changing of terminology, particularly about order parameters and second order critical lines.
4. More physical explanation of what results mean.
5. I would suggest some rewriting of the introduction - I find it quite long winded and not very focussed on the present study.
5. Other minor corrections mentioned in report above.

Ask for major revision

1. Valid solution of the proposed model is given.
2. A detailed discussion of various physical properties of the model is given.
3. The presentation of results is clear and comprehensible.

1. From the theoretical point of view the proposed model is only a marginal generalization of the models discussed previously in the literature.
2. The method of solution (standard Jordan-Wigner) is quite routine and requires no new ideas or methods as compared to the previous studies.
3. There is no discussion of possible applications of the model to real physical systems and materials.

This work introduces and solves one more 1-D spin 1/2 chain model, exactly solvable by means of the Jordan-Wigner transformation. Besides the standard two-spin interaction this model involves also three-spin terms of a special form. From the point of view of the theory of exactly solvable systems this work is a quite routine one, it does not propose any new ideas or methods. At the same time a detailed study of the models physical properties reveals quite reach zero-temperature behaviour including a number of different phases and phase transitions. The derivation of corresponding results from the exact solution of the model is, however, straightforward, it does not constitute a serious challenge for any qualified theorist. It should also be mentioned that the authors do not compare their results to any existing experimental data and do not discuss possible physical candidates for implementation of the model under study.

I do not recommend publishing this paper in SciPost Physics.

On the other hand the results of the present study are solid and, though being rather abstract, may be of certain interest for specialists. In my opinion it may be fit for SciPost Physics Core

Accept in alternative Journal (see Report)