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Front dynamics in the XY chain after local excitations
by Viktor Eisler, Florian Maislinger
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We study the time evolution of magnetization and entanglement for initial states with local excitations, created upon the ferromagnetic ground state of the XY chain. For excitations corresponding to a single or two well separated domain walls, the magnetization profile has a simple hydrodynamic limit, which has a standard interpretation in terms of quasiparticles. In contrast, for a spin-flip we obtain an interference term, which has to do with the nonlocality of the excitation in the fermionic basis. Surprisingly, for the single domain wall the hydrodynamic limit of the entropy and magnetization profiles are found to be directly related. Furthermore, the entropy profile is additive for the double domain wall, whereas in case of the spin-flip excitation one has a nontrivial behaviour.
Submission & Refereeing History
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Reports on this Submission
- Cite as: Anonymous, Report on arXiv:1909.02841v1, delivered 2020-02-13, doi: 10.21468/SciPost.Report.1505
1- the problem addressed is general
2- the analytical results are supported by numerical simulations
3- the paper is very readable
1- the model is not interacting
This paper reports analytical and numerical investigations into the dynamics of a particular class of "excitations" in the XY chain. The authors consider three cases (single domain wall, double domain wall, and single spin flip) in which the ground state is perturbed by simple operators that have a local representation in terms of the Jordan-Wigner fermions in terms of which the Hamiltonian is quadratic. The hydrodynamic limit of the magnetisation profile is computed exactly. The equal-time two-point function of the x component of the spin is obtained at the lowest order of a form-factor expansion. The bipartite entanglement profile is conjectured and the prediction is numerically checked using tensor network algorithms. In the final example, the authors investigate the time evolution of the aforementioned quantities after joining together the two ground states of the ferromagnetic Hamiltonian.
I think that the paper is well written and meets the criteria of acceptance of SciPost. My only criticism is that the examples considered lack a bit of courage, investigating only cases that could have been addressed by rather standard means.
1- In my opinion the authors should comment a bit more on whether some of their results might be strong enough to hold true in the presence of interactions.
- Cite as: Anonymous, Report on arXiv:1909.02841v1, delivered 2019-10-20, doi: 10.21468/SciPost.Report.1246
1- Potentially interesting generalizations to interacting systems
2- Clearly stated results
1- Some of the results already appeared in Ref.  by the same authors.
In this paper, the authors investigate the out of equilibrium dynamics of the XY spin chain, in a regime for which a simple hydrodynamic picture is expected. The initial states are however not standard, with domain wall created on to of the (symmetry-broken) ground states. This means that a hydrodynamic description is, a priori, not obvious.
Nevertheless, the authors use the exact solvability of the model and known form factors to compute simple observable such as magnetization. This is done by saddle point analysis. The final result sometimes admits a simple hydrodynamic description, but there can also be more complicated interference effects. Results are also obtained for the edge of the obtained hydrodynamic profiles, which are described by the Airy kernel.
Perhaps the most interesting results are for the entanglement. It is described by a simple scaling ansatz for certain protocols (such as 'double domain wall'), and checked numerically with good agreement. Other setups are left open to future analytical investigations.
Overall the paper is clear and reasonably well written. Given the timely nature of the results I recommend publication in Scipost, provided the comments below are be addressed:
1) Page 5, is the use of MPS techniques mere convenience, or are there difficulties associated to using Pfaffian techniques for more complicated correlations than magnetization?
2) The authors should be more careful about the use of the word local in several places. For instance, after (18), the operator is not local in terms of fermions in the limit considered right after (19), by any reasonable definition of local. Similar comment at the top of page 9.
3) When doing form factors expansions, only the lowest order contributions are kept. Then, the authors perform a saddle point analysis of the resulting integrals, to get hydrodynamic behavior. However, it should be possible to show that higher order form factor contributions become subleading in the long time limit. Can the authors comment on that?
4) In section 5.1, it is not really clear why massive QFT results are relevant to the problem discussed here, since massive field theories describe only the vicinity of a critical point. Perhaps there is a better justification of the binary entropy ansatz of (46).
5) Page 16, second paragraph. I do not understand the meaning of 'critical point' or 'critical slowing down' for '$h_c=0.75$'. Can the authors also comment on the use that can be made of (34)?
Here is also a list of misprints:
a) Before (15), 'In turn' should be replaced by something else.
a) End of section 3.2. 'is due the fact' should read 'is due to the fact'
b) Beginning ofsection 5. 'are interested about' should read 'are interested in'
c) After (40), what are 'the ranges' of $\rho_i$ ?
d) After (41), 'thus' should be removed