Observation of non-local impedance response in a passive electrical circuit

Dear Editor and Referees,
We express our sincere gratitude for overseeing the review process of our manuscript. Our thanks are extended to the referees for their meticulous review and insightful comments. We are particularly thankful for the referees' acknowledgement of the originality and importance of our work. In response to their feedback, we have thoroughly revised and improved our manuscript accordingly.

Specifically, we concur with Referee 1's observation that the critical skin effect (cited as Ref 37) inherently relates to finite-size phenomena. Consequently, we have adjusted the subtitle of the relevant subsection to "Size-dependent appearance of topological modes from parasitic resistances" and have expanded our discussion on this topic within both the introduction and the conclusion sections.

Moreover, we acknowledge and appreciate Referee 2's constructive remarks regarding the clarity and presentation of our manuscript, particularly the elucidation of our results and the specific electrical component values employed in our study. We have incorporated all the modifications recommended by Referee 2 into the manuscript.

Warm regards,
Xiao Zhang
On behalf of the authors

Response to referee 1:

The manuscript presents results on a very interesting and physically highly relevant topic of simulating topological quantum materials and transitions in electrical circuits. The text in general is well written and structured. Other contributions in this very active and competitive field are properly cited and acknowledged. The manuscript in my opinion meets at least one acceptance criterion which must be satisfied, according to the journal guidelines: "(3) - open a pathway in an existing or a new research direction...."

I have however one concern regarding the "topological phase crossover" (or "critical non-Hermitian skin effect"). The nature of this phenomenon needs to be clarified and explained in a self-contained form in the manuscript. As discussed in Ref. 55, it can be related to the behavior of energy zeros on the complex plane of wave numbers z=eik. Is it a disorder line of the first or second kind? (According to the definitions of Stephenson, PRB 1, 4405 (1970)). If it is a finite-size effect, where are the parametric boundaries for it to disappear and how it is related to the positions of zeros z on the complex plane?

I think this issue needs to be addressed before the manuscript can be recommended for publication.

Response: We greatly appreciate the reviewer’s recognition on the importance of the work, and our research approaching combining theory and experiments, as well as our organization of citations in the main text. Indeed, it is important to find that the modified spectral bulk-boundary correspondence reported in other literature only holds for the impedance between certain pairs of points, as measured in our experiment. From a physical perspective, it is also interesting to know that broken bulk-boundary correspondence can already be achieved just with resistors, without the need for asymmetric couplings.

The appearance of topological modes only at certain system sizes is a manifestation of the critical skin effect, which we have now explained in more depth and rigor in the text. We agree that the critical skin effect is by definition a finite-size effect, and we have also changed the title of that subsection to “Size-dependent appearance of topological modes from parasitic resistances”, and made similar changes to the discussion.

However, it is not related to the mechanism of complex zeroes as in Stephenson, PRB 1, 4405 (1970)). In PRB 1, 4405 (1970), the physical system consists of interacting spins, and the size dependence is indeed related to critical power-law behavior governed by conformal field theory. In contrast, in the critical skin effect, the notion of “criticality” has a different meaning, unrelated to disorder lines. Here, it is known as “critical” because the system is in the proximity of two solutions of the generalized Brillouin zones. Physically, that means that the system is in the proximity of two different manners by which the NHSE can occur. For instance, if we have two weakly coupled chains with NHSE in different directions, then if the coupling is weaker than a certain threshold, the two chains would be in the weak coupling regime and the NHSE in each chain occurs independently. However, above that threshold and the NHSE between the two chains almost cancel, leading to a different OBC spectrum. Now, because the NHSE is a strongly non-local effect, it is sensitive to the system size. Hence whether we are in the weak or strong coupled regime depends on the system size. But as the referee correctly pointed out, this is more like a continuous evolution between the two regimes, and not a sharp transition that persists in the thermodynamic limit. Hence we no longer call it a “phase transition” or a “crossover”.

Response to referee 2:

This work investigates the impedance response of an L-R-C circuit that realizes the geometry of a two-leg ladder with both open (OBC) and periodic boundary conditions (PBC) along the legs. A non-local response is found in the cross-leg impedance response ratio ZPBC/ZOBC.

Response: We greatly appreciate the reviewer's recognition of the novelty and significance of our manuscript, and we fully agree with the reviewer’s comments on the clarity and formatting, and have made detailed improvement in accordance with them. We would like to thank the reviewer’s careful reading and detailed comments.

I understand that the authors have fabricated an N=8 device and performed complementary simulations for different, in particular larger values of N. However, it is not always clear from the text which results are experimental findings and which ones obtained by simulation. I believe that this point needs clarification.

Response: We sincerely thank the reviewer pointing out this ambiguity. Indeed, our results include three different kinds of results: experiments, circuit simulation and calculation from formulas. We have updated the manuscript and detailed whether the plots are from measured data or are simulated or computed.

On another note, the manuscript seems to have been prepared for a different target journal, and I think that it would be useful if the authors could reformat it according to the SciPost Physics guidelines, see https://scipost.org/SciPostPhys/authoring. In particular, the "Methods" sections should be either integrated into the main text (no length limitation) or moved to Appendices.

Response: We sincerely thank the reviewer’s advice and have reformatted the draft with a SciPost template.

I have one question: the values C=1nF, L=1mH, R=5kΩ, r=50kΩ are nominal values (?). Have the authors verified if their components agree with the specifications? In view of parasitic resistances on the order of 0.2 to 1.7%, one might imagine deviations from nominal values on the same order of magnitude.

Response: We greatly appreciate the reviewer pointing this out, indeed this is an important fact that we should have explained in the main text. Yes, those are nominal values, and those components with maximum error are 1% for the 1nF capacitors, 2% for the 1mH inductors, and 0.5% for both the 5k and 50k resistors, as measured. Through circuit simulation, we discovered that the deviations minimally affect the impedance and fail to account for the discrepancies observed between the simulated and measured results. Consequently, we chose to overlook these discrepancies, incorporating parasitic resistances into the simulation, which led to a nice alignment between our simulation and experimental data.

Requested changes by referee 1:
1. Subsection "Size-dependent topological phase crossover..." needs to be revised as discussed above.

Response: We thank the referee for the comment, and we have now provided a more in-depth discussion of the size dependence. We agree that the critical skin effect (now Ref 37) is by definition a finite-size effect, and we have also changed the title of that subsection to “Size-dependent appearance of topological modes from parasitic resistances”, and made similar changes to the discussion.

2. Consequently, some related comments on the physical nature of the phenomenon in question would improve the Introduction and Conclusion.

Response: We have added some comments on this phenomenon in the introduction and conclusion, including:

On page 2, “When larger system sizes are accessible, it would even be possible for topological modes to appear due to the interplay between parasitic resistances, the sublattice structure of the circuit lattice and the competition between the emergent non-local influences in different sublattices.”

On page 9 to 11, “The critical non-Hermitian skin effect (NHSE) in this circuit is best understood in the rotated basis as shown in Eq. 6, where it assumes the form of two coupled effective chains. For sufficiently long chains of length N, NHSE modes in each effective chain (Fig. 1a) grow exponentially large in N, such the hopping probability across the chain may be non-negligible even if the inter-chain couplings are very small. In this case, the NHSE modes can dynamically hop across to the other chain whenever it hits the end of each chain, thereby continuing the amplification cycle. The existence of such an amplification cycle at large N, and their absence at smaller N, therefore leads to qualitatively different long-time behavior that is reflected in the value in the imaginary part of the spectrum. Exactly when this threshold occurs has been evaluated for the simplest critical NHSE model in Refs. [56,57].”

On page 11, “Physically, the parasitic resistances introduce another manner by which non-Hermiticity enters the system and, together with the sublattice asymmetry intrinsic in the designed circuit, controls the extent to which the topological modes are allowed to exist by virtue of the critical non-Hermitian skin effect.”

Requested changes by referee 2:
1- State clearly which results are experimental ones and which one come from simulations. A comparison between the two may also be appropriate.

Response: We sincerely thank the reviewer pointing this out, and have detailed the source of results as “calculation”, “simulation”, and “experiment” from main text, we also have moved the subsection “Parasitic resistances and accurate modeling of our circuit Laplacian” from Appendices to main text on page 6 to make a comparison.

Such updates include on page 7 caption of Fig2 “impedances obtained by circuit simulation with the Cadence virtuoso software……”, on page 8 caption of Fig3 “The 2-point impedance experimentally measured……PBC and OBC impedance ratios extrapolated to larger system sizes across various types of intervals, using the formula in Eq.7……”, on page 9 ”Calculated in Fig.4a……”; on page 10 caption of Fig 4 ” Calculated Laplacian spectrum……”, on page 14 caption of Fig6 “Calculation results of ZPBC/ZOBC vs N by Eq.7……”, on page 14 “In general for inter-ladder intervals calculated by Eq.7……”, on page 14 “from 20 to 150 through calculation……”, on page 15 caption of Fig 7 “Two-point impedance ratios ZPBC/ZOBC calculated from Eq.7……”, on page 16 caption of Fig 8 “Shown are the calculated……”.

2- Reformat the manuscript according to SciPost Physics guidelines, https://scipost.org/SciPostPhys/authoring.

Response: We have reformatted the draft according to the SciPost Physics template.

3- Clarify if the values for C, L, R, and r are nominal ones, or if the components actually have these values.

Response: Indeed they are nominal values, and we have explained it the text.

We have added on page 5 “In our setup, we utilized eight unit cells, each comprising capacitors (C) with a nominal value of 1nF, inductors (L) of 1mH, and resistors (R) of 5k and (r) of 50k. These components led to the dimensionless parameters t (defined as RC), with a value of 5, and v=R/(2r)), with a value of 0.05. We observed the maximum deviation in component values to be 1% for the 1nF capacitors, 2% for the 1mH inductors, and 0.5% for both the 5k and 50k resistors, as per our measurements.” and on page 6 “Through circuit simulation by the Cadence virtuoso software, we discovered that the deviations minimally affect the impedance and fail to account for the discrepancies observed between the simulated and measured results. Consequently, we chose to overlook these discrepancies, incorporating parasitic resistances into the simulation, which led to a nice alignment between our simulation and experimental data.”

We also added in the Fig4 caption of page 10: “Nominal parameters are C=1nF, L=1mH, R=5k and r=50k with parasitic resistances Rpc=2 and Rpl=17 determined through fitting simulation with experiments.”

4- First paragraph of the Introduction: The list of 33 references [6-38] is not very helpful to the reader. The authors should either break this down into smaller units with appropriate comments, or reduce the list to really relevant references.

Response: We sincerely thank the reviewer pointing out the redundancy of the references. We have cut the list into 14, and reduced the list into [6-20].

5- At the bottom left of page 2, the authors say that the Laplacian J relates V to I, but then they write the opposite equation I=JV. As long as J is invertible, the two are of course equivalent, but presentation should nevertheless be coherent.

Response: We thank the reviewer’s careful reading, and have changed the description on page 3 as “a Laplacian J describes the steady-state relationship between the electrical potentials V and input currents I across the nodes. Explicitly, we write I = JV, which can be thought of as the matrix form of Kirchhoff's law, with the matrix element Jij describing the linear relationship between the potential Vj at node j and the input current Ii at node i.”

6- Format Eq. (4) properly, i.e., it should start with a new paragraph, not in line with the text.

Response: We sincerely thank the reviewer’s careful reading on the paper and have rewrite the equation not in line with the text.

7- Some figures use fonts that are too small. This applies in particular to Figs. 2, 5, and 7, and their legends.

Response: We thank the reviewer’s comments on the format of figures and have updated them. Now they should be much more legible than before.

8- On page 4, there is a discussion of "parasitic" resistance, but this is not properly explained before page 6 (one point where the manuscript would benefit from restructuring).

Response: We sincerely thank the reviewer’s great advice on restructuring the main text. We have moved the subsection “Parasitic resistances and accurate modeling of our circuit Laplacian” from Appendices to main text on page 6 to explain the effects parasitic resistance brings and make a comparison between simulation and experiments.
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9- Units are missing after the 106 below Eq. (9) (Hz?).
Response: We greatly thank the reviewer’s careful reading and have added the units (Hz).

10- Caption of Fig. 6: the definition of t is hidden somewhere in the text. For clarity, I recommend to recall it here.

Response: We thank the reviewer’s reminder on this, and have added “whereas dimensionless parameter t is controlling the effective coupling asymmetry for the hidden (canceled) directed amplification” in the caption.

11- Update references, specifically:
[29] Science Bulletin 67, 1865-1873 (2022)
[30] Phys. Rev. B 106, 075158 (2022)
[34] Phys. Rev. A 107, L010202 (2023)
[35] Phys. Rev. B 107, L220301 (2023)
[61] Phys. Rev. Research 5, L012041 (2023)
[70] Nature Photonics, 120–125 (2023)
... and add DOIs.

Response: We appreciate the reviewer’s reminder on the references, we have updated all the references and added DOIs.