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On the stability of the infinite Projected Entangled Pair Operator ansatz for driven-dissipative 2D lattices
by Dainius Kilda, Alberto Biella, Marco Schiro, Rosario Fazio, Jonathan Keeling
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|Alberto Biella · Dainius Kilda · Marco Schirò
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We present calculations of the time-evolution of the driven-dissipative XYZ model using the infinite Projected Entangled Pair Operator (iPEPO) method, introduced by [A. Kshetrimayum, H. Weimer and R. Orús, Nat. Commun. 8, 1291 (2017)]. We explore the conditions under which this approach reaches a steady state. In particular, we study the conditions where apparently converged calculations may become unstable with increasing bond dimension of the tensor-network ansatz. We discuss how more reliable results could be obtained.
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- Cite as: Anonymous, Report on arXiv:scipost_202012_00006v1, delivered 2021-01-21, doi: 10.21468/SciPost.Report.2446
The manuscript ‘On the stability of the infinite Projected Entangled Pair Operator ansatz for driven-dissipative 2D lattices’ by D. Kilda et al reproduces the earlier work of Kshetrimayum et al (Nat. Comm. 8, 1291 (2017)) and discusses the stability of the iPEPO algorithm at certain parameter regimes by looking at the dissipative XYZ model. They look at this model for different bond dimensions of the iPEPO and different initial states.
The study of open quantum system in 2D is a very challenging and important problem in physics and such works suggest that much remains to be done towards the numerical investigation of dissipative systems specially in the critical regimes. The paper is short but clear. While the findings of the paper are not surprising, I recommend its publication in SciPost Physics. However, I have some comments for which I would like to seek clarification from the authors.
1) Looking at Figure 1, the red regions correspond to the parameter regime where the iPEPO does not reach a steady state. They also correspond to the critical regimes which are difficult for Tensor networks in general even in closed systems. Similar result has been found in the original work of Nat. Comm. 8, 1291 (2017) where the authors compute the expectation value of the vectorized Liouvillian operator. Can the authors comment more elaborately on this?
2) In Fig. 1 again, the authors present results obtained using different Trotter steps. Can the authors explain why the results are so different for these particular regions? The trotter error coming from large step is of the order of \delta t^2 and this can be minimized by taking higher order trotterization. I recommend the authors to implement this and see if taking larger trotter steps help the calculation and if so, why.
3) In Fig. 2, the singular values for different parameter regimes are shown. However, it is not clear to me why the authors have shown only one singular value for each parameter. Is this the largest singular value? What happened to the rest of the singular values?
4) In Fig. 6, the authors take different bond dimensions and comment that some of the simulations could get unstable with increasing bond dimension. For simulations with D=10, 12, 14, 15, what is the bond dimension of the environment used here? Are these results correctly converged with the bond dimension of the environment?
In general, I feel that the authors present the result but does not offer much explanation or interpretation of the results. For example, why do different trotter step give different results, how their figure of merit is related to the results obtained by Kshetrimayum et al, how will the full update provide better stability, etc. I suggest the authors to take care of these things before publishing the manuscript.