Demonstration of a two-dimensional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetric crystal

With the discovery of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetric quantum mechanics, it was shown that even non-Hermitian systems may exhibit entirely real eigenvalue spectra. This finding did not only change the perception of quantum mechanics itself, it also significantly influenced the field of photonics. By appropriately designing one-dimensional distributions of gain and loss, it was possible to experimentally verify some of the hallmark features of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetry using electromagnetic waves. Nevertheless, an experimental platform to study the impact of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT -symmetry in two spatial dimensions has so far remained elusive. We break new grounds by devising a two-dimensional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal P}{\cal T}$$\end{document}PT-symmetric system based on photonic waveguide lattices with judiciously designed refractive index landscape and alternating loss. With this system at hand, we demonstrate a non-Hermitian two-dimensional topological phase transition that is closely linked to the emergence of topological mid-gap edge states.

concepts and phenomena that have been predicted to emerge in this type of photonic platforms. For example, the authors developed a new technology to efficiently and controllably introduce artificial losses into the system, and exploited it to demonstrate non-Hermitian two-dimensional topological phase transitions. Overall, the paper is convincing, well written, and accessible to a broad audience. There are, however, several important issues that should be addressed, as discussed in the following.
In summary, for the reasons outlined above, I support the publication of the submitted manuscript provided that the following points are properly addressed: (1) In the abstract, the authors mention "non-reciprocal light evolution [7]" as one of the "hallmark features of PT symmetry". However, the system studied in [7] is perfectly reciprocal! Nonreciprocity is generally unrelated to PT symmetry, as the presence of loss and gain does not break Lorentz reciprocity theorem for electromagnetic waves. Indeed, the scattering matrix of any linear PT-symmetric system is symmetric (S12=S21). The authors should remove or fix this statement.
(2) The labels of Fig. 1b,c,d,Fig. 2b,and Fig. 4b,c,d are too small. Also, the authors should explicitly define the symbols in these figures.
(3) Under "Additional information", the authors write "Supplementary information is available in the online version of the paper." However, no supplementary document was provided.
(4) It is not clear how the "strain" is implemented experimentally. The author should clarify this point, perhaps also including images of their fabricated samples. (6) The authors write: "In contrast, in the unbroken PT-phase all eigenvalues exhibit the same imaginary part and, for an infinte system, the power decay in the lattice were independent of the excited waveguide. Therefore, as the strain is increased above the critical value, the standard deviation of the transmitted power will eventually vanish [12]." Since this is key to the demonstration of a phase transition, the authors should elaborate more on this point. Also, they should explain how the theoretical results in Fig. 3 have been calculated.
(7) I was expecting to see a more drastic difference between the intensity plots in Fig. 4c and Fig.  4d, right panels, since a phase transition occurs between them. C an the authors clarify why these intensity plots look so similar? Also, these panels need a colobar.
(8) A typo: "the full potential of PT-symmetric in higher dimensions" -> "PT symmetry" or "PTsymmetric systems" Reviewer #3: Remarks to the Author: In this manuscript entitled "Demonstration of a two-dimensional PT-symmetric crystal: Bulk dynamics, topology, and edge states", the authors reported their experimental demonstration of PT-symmetry phase transition and topological phase transition in a photonic waveguide lattice. The waveguide lattice is fabricated in fused silica glass using laser direct writing technique, where parameters of the system, e.g. additional loss, coupling strength, strain strength, lattice structure, can be adjusted. By judiciously choosing parameters, a two-dimensional PT-symmetric waveguides lattice with honeycomb structure is achieved. This PT-symmetric system not only has broken/unbroken PT-symmetry phase transition, but also has topological phase transition due to its special band structure. The merit of this work lies in the 2D lattices which could demonstrate some features beyond what's observed in 1D PT-symmetric structures. Overall the manuscript is well organized. It could be improved if the following issues are addressed or corrected: 1. In Eq. (2), it seems that the unit of the term on the left hand side does not match with that of the right hand side.
2. Eq. (1) and Eq. (2) set up the framework to study PT symmetry in two-dimensional photonic crystals, based on which the constraints on the complex refractive index can be derived. Yet in the following discussion, the condition of PT symmetry is given with respect to a one-dimensional optical potential, as shown in Eq. (3). It will be helpful if the authors could provide more discussion on the condition of PT symmetry for a two-dimensional system.
3. There is a lack of argument or demonstration explaining why the refractive index potential in this two-dimensional photonic graphene lattice satisfies the condition of PT symmetry.
4. In Fig. 3, the number of data points provided is not enough to show a complete trend in which the standard deviation of the output intensity changes. In particular, more data points are needed to show the eventually vanishing behaviour when the system is in the unbroken PT regime 5. In Fig. 3, the experimental data seem to have a slight deviation from the theory. In theory, the standard deviation does not exhibit a sharp decrease to zero after the phase transition point (τ=2.316 as derived from the parameters provided). But in the experiment, the change appears to be abrupt around the phase transition point. The reason for this difference should be provided.
6. In this experiment, the PT-symmetric optical potential is realized with an alternating loss, and thus the system exhibits an overall lossy feature. How is the topological phase transition point, as mentioned on page 6, influenced by the global loss of the system? 7. On page 7, the authors state that "Moreover, we highlighted the close connection of a PTsymmetry phase transition to a topological phase transition in our graphene lattice." Except for the conventional argument that "the topological mid-gap state spontaneously breaks PT-symmetry" (page 6), it is not apparent how these two kinds of phase transitions are related. The link between them should be clarified.
8. It will be helpful if the authors could provide more information on the advantageous features and applications of the two-dimensional PT-symmetric crystal compared to the previous onedimensional platforms with PT symmetry. 9 When this system undergoes a PT-symmetry phase transition, it is theoretically predicted that a topological phase transition will be accompanied. More interesting, when the loss difference γ > 0, these two transitions can be observed at different values of strain τ, which is shown in Fig. 4. But the observed results in the right panel of Fig. 4(b)-(d) are not very clear. The paper says in Fig.  4(c) & 4(d) the light injected is spreading into the bulk due to the absence of edge states, and in Fig. 4(b) the edge states exist thus light stays close to the edge. However, the lattice structure presented in this manuscript only has three sublattices, thus the definitions of "edge part" and "bulk region" of the system is not very clear. In Fig. 4(b), it does not seem like the light is just "remaining at the edge", one can also see some light penetrates into the bulk. In order to convincingly state that the topological edge states are present, is there a better way to visualize and quantify this effect? It is important to clarify this issue in Fig. 4, because "the demonstration of a non-Hermitian two-dimensional topological phase transition" is a key result of this work, as said in the abstract. It seems a larger system with more sublattices will make it clear that the light in the waveguides is really confined to the edge.
10 There are some typos in the manuscript. For example, on the left hand side of Eq. (1), ∂/∂z should be corrected as ∂/∂t, i.e., the partial z should be partial t. In Figure caption 4, "(b)" is repeated twice where "(c)" and "(d)" should be used.

Comment:
The authors experimentally create a two-dimensional PT symmetric lattice and use it to demonstration the existence of a non-Hermitian two-dimensional topological phase transition that coincides with the emergence of mid-gap edge states. PT symmetric physics is an important and active research area; in this regard, the present manuscript contributes positively.

Response:
We thank the reviewer for this evaluation and are happy to see that he/she recognises our manuscript as part of an active and important field of research.

Comment:
I find the results reported in the manuscript rather "thin". The authors did not explain why their key result (the demonstration of a non-Hermitian two-dimensional topological phase transition that coincides with the emergence of mid-gap edge states) is very important that warrant publication in Nature Communications? The authors did not explain the physical ramification of such a phase transition. Unless the authors make a clear case explaining why the experimental results reported in this manuscript are very important, I cannot recommend publication of this manuscript in Nature Communication in its current form.

Response:
The reviewer himself/herself pointed out earlier that PT symmetric physics is an important and active research area. In this vein, we aim to contribute to this exciting field by introducing and characterising an experimental platform that allows verification of numerous recent theoretical proposals. To this date, there is no experimental platform available to study and verify PT-symmetric physics in two spatial dimensions. In order to show that we offer a suitable and versatile platform we study bulk and edge effects and the interplay of PT-symmetry and topology. With this palette of opportunities at hand we pave the way for a plethora of future research. Let us mention a few contexts in which the study of 2D PT structures will provide access to exciting physical effects: