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Dynamical signatures of topological order in the drivendissipative Kitaev chain
by Moos van Caspel, Sergio Enrique Tapias Arze, Isaac Pérez Castillo
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Submission summary
As Contributors:  Sergio Enrique Tapias Arze · Moos van Caspel 
Arxiv Link:  https://arxiv.org/abs/1812.02126v1 (pdf) 
Date submitted:  20181206 01:00 
Submitted by:  van Caspel, Moos 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate the effects of dissipation and driving on topological order in superconducting nanowires. Rather than studying the nonequilibrium steady state, we propose a method to classify and detect dynamical signatures of topological order in open quantum systems. Bulk winding numbers for the Lindblad generator $\hat{\mathcal{L}}$ of the dissipative Kitaev chain are found to be linked to the presence of Majorana edge master modes  localized eigenmodes of $\hat{\mathcal{L}}$. Despite decaying in time, these modes provide dynamical fingerprints of the topological phases of the closed system, which are now separated by intermediate regions where winding numbers are illdefined and the bulkboundary correspondence breaks down. Combining these techniques with the Floquet formalism reveals higher winding numbers and different types of edge modes under periodic driving. Finally, we link the presence of edge modes to a steady state current. This work aims to demonstrate a powerful and widely applicable approach to topology in drivendissipative systems.
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Reports on this Submission
Anonymous Report 2 on 201914 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1812.02126v1, delivered 20190104, doi: 10.21468/SciPost.Report.777
Strengths
Completeness and clarity
Weaknesses
Free Lindbladian models
Report
The manuscript studies a new concept in condensed matter and quantum physics: open topological quantum matter. This is a timely article due to the fact that in the past it has been studied the concept of topological order or the dynamics of driven and dissipative quantum systems. Nonetheless, in recent years, and motivated by new experimental results, new concepts, fields or setups have appeared. (see for instance: Nature Physics volume 4, pages 878–883 (2008) and Nature Physics volume 7, pages 971–977 (2011)).
The article is clearly written and complete in the discussion. It discusses the different dynamical elements that appear in a free Lindbladian evolution. Nonetheless, this part has already appeared in the literature (see for instance: Phys. Rev. Lett. 109, 130402 (2012) and New J. Phys. 15, 085001 (2013)).
One of the new results is to define the winding number in a "biorthogonal basis". Here, a question appears, how different or how similar is this measure of topological order compare with the one given in Phys. Rev. Lett. 112, 130401 (2014)?
Requested changes
Could the authors clarify their definition of the winding number comparing with previous ones?
Anonymous Report 1 on 20181219 (Invited Report)
 Cite as: Anonymous, Report on arXiv:1812.02126v1, delivered 20181219, doi: 10.21468/SciPost.Report.755
Strengths
1) Originality
Weaknesses
1) Not so general result
Report
The paper is interesting. The authors take the Kitaev chain, they add dissipation so that the system is still quadratic in fermionic operators (and then integrable) and construct a generalization of the winding number to study the topological properties. They still find a topological phase with edge modes which decay in time, and a normal phase. There is also an intermediate phase with no topology and gapless imaginary part of the rapidities. The topological phase can be experimentally seen in the properties of the steadystate current. The authirs also extend this picture to drivendissipative systems finding evidence of nontrivial winding numbers in the Floquet eigenmode bands (imaginary parts of the Floquet quasirapidities). The paper is worth of publication in SciPost. I have just some comments
1) Is there some dependence of the winding number Eq.26 on the choice of the rapidity band where it is evaluated?
2) Just a curiosity. How does the width of the intermediate region with gapless imaginary part of the rapidity depend on the dissipation strength? Is there a value of the dissipation strength where the gapped windows in Figs. 2right and 5bottom close up? At that point, I imagine that the systems shows no topological properties at all. How does this reflects in the behaviour of the current?
3) There are small typos: p.8 ``pointss'', p.15 ``more that two'', p.16 ``Brillioun''
Requested changes
1) In the abstract the authors write about ``powerful and widely applicable approach to topology in drivendissipative systems''. If I understand correctly the approach they describe is strictly restricted to quadratic fermionic Hamiltonians. The authors should clarify why their approach is ``widely applicable'' or withdraw this sentence.
Author: Moos van Caspel on 20190128 [id 417]
(in reply to Report 1 on 20181219)
Thank you for your kind comments. Below, we try to answer the questions your raised.
1) There is indeed a dependence on the choice of rapidity band. However, we note that $\sum_i \nu_i \mod 2 = 0$. For the purpose of defining topological phases and the bulkboundary correspondence, it is therefore sufficient to focus on one band. We have added a line below Eq.26 to clarify this.
2) Yes, the width of the intermediate region scales linearly with the dissipation strength $g$, as described in section 3.1 for the undriven system. With driving, this is not simple to see analytically. The referee correctly remarks that the topologically nontrivial phase vanishes for sufficiently strong dissipation. The paper assumes a relatively weak dissipation (see footnote on page 7), which is consistent with the BornMarkov approximation that underlies the Lindblad master equation. The steady state current's dependence on $\mu$ (figure 4 bottom panels, and figure 7) will be further smoothened out if the dissipation increases, but shows no novel behavior when the gapped windows close.
3) The typos have been corrected. The last one appeared in more than one place but we have made sure that it has been amended in all places.
Author: Moos van Caspel on 20190128 [id 416]
(in reply to Report 2 on 20190104)We thank the referee for pointing out these two references. We have cited the very relevant second paper at the start of section 3 in our resubmission. While we understand that the first paper suggested is a pioneering work in the field, we believe it does not relate directly to our submission.
The two references given (Phys. Rev. Lett. 109, 130402 (2012) and New J. Phys. 15, 085001 (2013)) take an approach that is very different from ours. To the best of our understanding, they aim to construct fully dissipative systems (with no intrinsic Hamiltonian) in which the steady state(s) exhibits some form of topological order. Our paper, on the hand, seeks to understand the dynamical behavior of a known Hamiltonian system with topological order once dissipation is added. See also our reply to the next question. It is conceivable that the decaying edge modes we find are somehow related to the zeropurity modes of New J. Phys. 15, 085001 (2013), which would be an interesting direction for future work. We have added these references at the start of section 3.
This is an interesting point. The Uhlmann phase from the cited paper is used to characterize topological order for mixed states which result when the system is coupled to a thermal bath in the "traditional" way: by the thermal ensemble. Firstly, this approach has been shown in \url{https://doi.org/10.1098/rsta.2015.0231} to lead to inconsistencies in the Kitaev chain at finite temperature. Secondly, while techniques to study the topology of mixed states can be applied to the steady state of an open quantum system, and can possibly lead to interesting new types of topological order, there is no relation to the topological order of the original, closed system (tied to the ground state) if the steady state is unique. Because we are interested in the effect of dissipation on topological order of the Kitaev chain, we use an entirely different approach in which a winding number is associated with the generator of the nonunitary time evolution, rather than with any one mixed quantum state. This leads to the dynamical signatures we described, including edge modes with exponential decay.
In our case, the steady state is unique and does not contain (zerodamping) Majorana modes, as we argued in the paper. Thus, we expect that the application of the Uhlmann phase would always render a trivial topological phase. It would be indeed interesting to confirm our intuition, but this falls beyond the scope of the present paper. We have expanded upon this in the conclusions and outlook section, also including the corresponding reference.