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Reciprocal skin effect and its realization in a topolectrical circuit

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Pith's one-line read A reciprocal, non-Hermitian two-dimensional lattice can push eigenmodes of opposite momenta to opposite edges, and a passive resistor–inductor–capacitor circuit realizes it.

desk verdict A genuinely new reciprocal 2D skin effect with a solid theoretical core, but the circuit experiment is only as good as the LTSPICE calibration and needs more transparency. read the letter →

arxiv 1908.02759 v2 pith:AQUQ4BOF submitted 2019-08-07 cond-mat.mes-hall

Tobias Hofmann , Tobias Helbig , Frank Schindler , Nora Salgo , Marta Brzezińska , Martin Greiter , Tobias Kiessling , David Wolf
show 6 more authors
Achim Vollhardt Anton Kabaši Ching Hua Lee Ante Bilušić Ronny Thomale Titus Neupert
This is my paper · ORCID
classification cond-mat.mes-hall
keywords reciprocalskineffectnon-Hermitiantopologytopolectricalcircuitsexceptionalpointspi-fluxmodelbulk-boundarycorrespondencepassiveRLCcircuit
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces and experimentally demonstrates the reciprocal skin effect: in a non-Hermitian but reciprocal two-dimensional system, eigenmodes with opposite momentum along the boundary become exponentially localized on opposite transverse edges. A system is non-Hermitian when it exchanges energy with its environment, and reciprocal when it responds the same way upon interchange of input and response; earlier skin effects were thought to require non-reciprocity, typically implemented by direction-dependent hoppings or active amplifiers. The paper shows that a reciprocal lattice with a lossy, complex-valued coupling can instead be sliced by momentum, so that each fixed-momentum slice behaves like a non-reciprocal one-dimensional chain, with opposite momenta pushed to opposite edges. The authors build a passive circuit of capacitors, inductors, and resistors whose circuit Laplacian reproduces the model, and measure eigenstates that localize right at $k_y\approx\pi/2$ and left at $k_y\approx 3\pi/2$. If correct, the finding widens the platforms for skin-mode localization and points toward passive directional and polarization detectors for electromagnetic waves.

What carries the argument

The carrying object is the momentum-dependent non-Hermitian coupling term $-ir\begin{pmatrix}0&e^{ik_y}\\e^{-ik_y}&0\end{pmatrix}$, a reciprocal off-diagonal hopping in which the forward and reverse amplitudes are equal rather than conjugate. Fixing $k_y$ as a parameter converts the two-dimensional model into an effectively one-dimensional chain with an induced non-reciprocal hopping proportional to $r\sin(k_y)$; because $\sin(k_y)$ is odd, opposite momenta see opposite effective non-reciprocity and therefore accumulate on opposite edges, while the translational symmetry of the full two-dimensional system prevents the reciprocal partners from hybridizing. In the circuit, the same term is produced by a resistor across the diagonal of each plaquette, whose admittance contribution enters the circuit Laplacian as the same matrix; the circuit Laplacian $J(\omega_0)$ is the measured object whose eigenvalues and eigenstates represent the model Hamiltonian.

What would settle it

Drive the circuit from two bulk sites with a $+\pi/2$ phase difference and separately with a $-\pi/2$ phase difference, following the protocol of Appendix F: if the $+\pi/2$ drive does not build up a voltage concentrated at the right edge and the $-\pi/2$ drive at the left edge, the reciprocal skin effect is not present in the realized circuit. A second check is to sweep the drive frequency away from the calibrated value and confirm that the localization direction and edge accumulation follow the sign of $\sin(k_y)$ rather than reflecting a parasitic resonance.

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Extended reading notes

Core claim

The central claim is that non-Hermiticity alone, without non-reciprocity, can produce extensive skin-mode localization in two dimensions. Starting from the Hermitian $\pi$-flux model on a square lattice, the paper adds a single reciprocal, non-Hermitian diagonal hopping term, giving $H(k_x,k_y) = H_\pi(k_x,k_y) - i r\begin{pmatrix}0&e^{ik_y}\\e^{-ik_y}&0\end{pmatrix}$. The same complex amplitude $ir$ multiplies a hop and its reverse, so the model satisfies $H(k_x,k_y)^T = H(-k_x,-k_y)$, yet the eigenvalues become complex and each Dirac point splits into a pair of exceptional points, degeneracies where two eigenstates coalesce. With open boundaries in $x$ and periodic boundaries in $y$, treating $k_y$ as a parameter turns the model into an effective one-dimensional non-reciprocal chain: for $k_y\in(0,\pi)$ all bulk modes localize on the right edge, for $k_y\in(\pi,2\pi)$ they localize on the left edge, and at $k_y=0,\pi$ they are delocalized; the localization is strongest near the former Dirac points at $k_y=\pi/2$ and $3\pi/2$, and the localization length vanishes at $r=1$. The experiment realizes this in a $10\times20$ unit-cell circuit: a resistor connecting sublattice nodes supplies the lossy reciprocal coupling, and measured Green's functions give eigenstates whose inverse participation ratio, a measure of localization, shows the predicted momentum-dependent edge accumulation.

Load-bearing premise

The experimental demonstration assumes that the assembled RLC network at the chosen drive frequency realizes the intended tight-binding Hamiltonian, including cancellation of the sublattice diagonal terms by grounding inductors and the exact mapping of the resistor term to $-ir\begin{pmatrix}0&e^{ik_y}\\e^{-ik_y}&0\end{pmatrix}$; parasitic effects and component tolerances are fixed by circuit simulation rather than by independently measuring the realized Hamiltonian, so a calibration error would change which Hamiltonian's eigenstates are observed.

Editorial extensions

If this is right

  • The reciprocal skin effect should appear in any non-Hermitian reciprocal two-dimensional lattice whose couplings connect different internal degrees of freedom, because the momentum-slice argument depends only on the sign of $\sin(k_y)$, not on the electrical implementation.
  • The breakdown of bulk-boundary correspondence is now observable without active or non-reciprocal elements: a passive RLC network suffices, so the effect transfers to optical, acoustic, and mechanical metamaterials built from lossy reciprocal components.
  • Because localization direction is fixed by the sign of $\sin(k_y)$, an incident wave with a definite propagation direction and polarization drives a voltage buildup at a selectable edge, making the same circuit a passive direction or polarization detector for electromagnetic waves.
  • Tuning the resistance so that $r=1$ makes the localization length vanish at $k_y=\pi/2$ and $3\pi/2$, giving infinitely localized skin modes there while Su-Schrieffer-Heeger-type one-dimensional edge modes coexist in the gapped bulk spectrum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The momentum-slice picture suggests a three-dimensional extension: with open boundaries in two directions, different patches of the surface Brillouin zone should localize modes on different edges or hinges, forming higher-dimensional reciprocal skin accumulations that the paper does not explicitly formulate.
  • The Appendix F response calculation implies a concrete device test: a two-port input with a $\pm\pi/2$ phase difference should act as a passive switch that routes a signal to one edge or the other; measuring output impedance versus input phase would quantify bandwidth and efficiency.
  • Because only reciprocal loss is required, a photonic lattice with absorption or an acoustic lattice with damping should reproduce the effect as long as the internal degrees of freedom carry the complex phase, making the phenomenon platform-independent.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces a 'reciprocal skin effect' in non-Hermitian but reciprocal two-dimensional systems. Starting from the π-flux tight-binding model on a square lattice, the authors add a complex diagonal hopping term that breaks Hermiticity while preserving reciprocity, and show that for open boundary conditions along x, each fixed transverse momentum ky behaves as an effectively non-reciprocal one-dimensional model exhibiting skin localization. The localization length is derived analytically in Eq. (C8), and opposite ky values localize to opposite edges, with ky=0,π delocalized. The paper reports an experimental realization in a passive RLC circuit, using Green's-function reconstruction to extract the circuit Laplacian spectrum and eigenstates, and claims observation of exceptional points and of the reciprocal skin effect through inverse participation ratio (IPR) coloring of eigenstates. The authors also present a non-unitary transformation to an SSH-type model and propose a polarization-detection application. The theoretical derivation is coherent, while the experimental demonstration and the stated direction of localization contain issues that need to be resolved.

Significance. If the experimental mapping is sound, this is a conceptually important advance: it shows that extensive skin localization does not require non-reciprocal couplings, but can arise in reciprocal systems through momentum-resolved effective non-reciprocity in two dimensions. The analytic localization length in Eq. (C8) is a clean, checkable prediction, and the passive-circuit platform is a practical and accessible implementation. The measured IPR-based observation of opposite localization at opposite ky is suggestive and, if reproduced with error bars and independent component calibration, would be a valuable demonstration. The paper also provides a useful derivation of the circuit-Laplacian-to-Hamiltonian mapping and an SSH-type interpretation of the non-Hermitian model.

major comments (3)
  1. [II (Theory), Fig. 1c, Fig. 2d, and Appendix C 3] The direction of the predicted skin localization is stated inconsistently. Section II says 'For ky∈(0,π), all OBC bulk modes localize at the right edge, while the localization switches to the left edge for ky∈(π,2π)', and Fig. 2d colors states near ky=π/2 as right-localized and near ky=3π/2 as left-localized. However, the caption of Fig. 1c assigns ky=π/2 to the left and ky=3π/2 to the right, and Appendix C, in the text following Eq. (C8), concludes 'This leads to left edge localized modes, if ky∈(0,π) and to right edge localization, if ky∈(π,2π)'. With ψx∼e^{-x/ξ}, a positive ξ (ky∈(0,π)) corresponds to decay toward larger x and hence left-edge localization, so the Appendix C convention is the one consistent with the derivation. The authors must fix a single sign convention and state explicitly how the colors in Fig. 2d are assigned; this is load-bearing for the experimental claim and for the directional-detector application in Fig. 1c and Appendix F.
  2. [Appendix D and Fig. 2d] The experimental identification of the realized Hamiltonian is under-determined. Appendix D states that the resonance frequency f0=87.25 kHz was found by matching features of a frequency sweep to an LTSPICE simulation that includes parasitic resistances, and that 'capacitors and resistors ... were therefore not characterized', with only inductors pre-characterized. The mapping H=[J(ω0)-ir1]/(iω0C) requires the diagonal terms to cancel at f0 (Eq. C3) and the resistor term to realize exactly the -ir e^{±iky} off-diagonal contribution (Eqs. C4 and C6). Any error in the parasitic model or in component values introduces unknown diagonal (σz) and hopping renormalizations. Because no error bars, raw Green's-function data, or independent component measurements are presented, it is not possible to verify that the measured eigenstates at 87.25 kHz are eigenstates of Eq. (2) rather than of a nearby off-resonance Hamiltonian. This is the least secured link of the central experimental claim.
  3. [Eq. (C8), Fig. 2d] The quantitative prediction of the reciprocal skin effect, the localization length ξ(ky) in Eq. (C8), is not compared with the experimental eigenstates. Fig. 2d reports only a binary coloring based on IPR; no spatial profiles, fitted localization lengths, or comparison to the theoretical ξ(ky) curve are shown. Since the paper emphasizes 'exponentially localized with localization length ξ', a profile fit or at least a scatter of IPR versus the predicted ξ would substantiate the claim and would help discriminate the skin effect from generic edge weighting or disorder. The IPR threshold used for the color assignment is also not defined. This comparison should be added, or the experimental claim should be softened to a qualitative statement.
minor comments (5)
  1. [III] In §III, 'contstraints' should be 'constraints' in the phrase 'local connectivity contstraints'.
  2. [Appendix D] In Appendix D, 'LCR Brige' should be 'LCR Bridge'.
  3. [Appendix E] The sentence 'We observe, for PBC, a clear from slices at constant ky with a single circle and two circles' is missing a word; it should read 'a clear change from slices' or similar.
  4. [Fig. 2d] The precise definition of the IPR used and the threshold separating red and blue points in Fig. 2d are not given; please specify them.
  5. [Appendix D] The manuscript does not state whether raw measurement data and the LTSPICE netlist will be made available; a data availability statement would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reciprocal skin effect is derived from an explicit model Hamiltonian and the circuit experiment is an independent implementation, not a fit to the prediction.

full rationale

The paper's central theoretical claim is derived from a stated non-Hermitian but reciprocal tight-binding Hamiltonian, Eq. (2), with the localization property obtained via the explicitly written non-Bloch analysis leading to Eq. (C8), ξ^{-1} = (1/4) ln[(1+r²+2r sin ky)/(1+r²−2r sin ky)]. This is a genuine prediction from the model, not an input fitted to data. The circuit mapping is likewise explicit in Appendix C: the Hamiltonian is identified with the circuit Laplacian as H = [J(ω0) − ir1]/(iω0C) with r = 1/(ω0RC), and r is simply chosen as r = 1. The resonance frequency in Appendix D is calibrated from LTSPICE features, but it is not tuned to reproduce the observed left/right localization pattern; the eigenmode localization is then extracted independently from the measured Green's function. Self-citations to prior topolectrical-circuit and skin-effect work (Refs. 16, 19, 32, 39) support standard techniques and previously established effects; they are not used as the load-bearing derivation of the reciprocal skin effect. The experimental calibration dependence noted in Appendix D is a legitimate uncertainty about component parasitics, but it is an implementation concern, not circularity: the predicted and observed phenomena could in principle disagree, and the paper reports their agreement. No self-definitional reduction, fitted-input-called-prediction, or author-imported uniqueness argument is present. The theoretical derivation is self-contained, and the experimental observation is an independent test of the model's prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends on one hand-chosen model parameter (r=1) and one calibrated experimental frequency. The theoretical framework adds no new free parameters beyond the standard tight-binding and circuit assumptions. No new physical entities are introduced.

free parameters (2)
  • r = 1
    Dimensionless non-Hermitian coupling strength in Eq. (C6), r = 1/(omega0 R C). The resistor R is chosen so that r = 1; this is a hand-picked value to make the skin effect strong and is not fitted to reproduce the predicted localization.
  • resonance frequency f0 = 87.25 kHz
    The measurement frequency is identified by comparing a frequency sweep to LTSPICE simulation with parasitic resistances (Appendix D). This calibration is necessary for the circuit to realize the intended model, but it is not a theoretical free parameter.
assumptions (4)
  • domain assumption The non-Bloch band theory / generalized Brillouin zone framework of Ref. 16 applies to the effective 1D model at fixed ky.
    The OBC spectrum and localization length are derived via a non-unitary transformation following Ref. 16 (Section II and Appendix C.3). If this framework were invalid for the model, the predicted skin effect could fail.
  • domain assumption The circuit at the chosen resonance frequency realizes the tight-binding model H = [J(omega0) - ir 1]/(i omega0 C).
    This mapping requires the inductor to act as a negative capacitor and the grounding inductors Lg to cancel sublattice diagonal terms (Appendix C, Eqs. C3 and C6). Parasitic effects are calibrated against LTSPICE, not independently measured.
  • domain assumption Translation invariance in y direction, with periodic boundary conditions in y, so ky is a good quantum number.
    The reciprocal skin effect is formulated for a strip with OBC in x and PBC in y. Without a conserved ky, the momentum-resolved opposite localization is not defined (Sections II and III).
  • domain assumption Right eigenvectors of the circuit Laplacian, extracted from the measured Green's function, determine the localization of physical modes.
    The IPR is computed from right eigenstates of J(omega0) (Fig. 2d). For non-Hermitian systems, left and right eigenvectors differ; the measured response involves both, so the physical interpretation of the right-eigenvector IPR is an assumption.

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Cite this review

Pith. "Pith review of Reciprocal skin effect and its realization in a topolectrical circuit." pith.science (2026). https://pith.science/paper/AQUQ4BOF

@misc{pith2026190802759,
  author       = {Pith},
  title        = {Pith review of: Reciprocal skin effect and its realization in a topolectrical circuit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AQUQ4BOF}},
  note         = {Machine review of arXiv:1908.02759}
}
read the original abstract

A system is non-Hermitian when it exchanges energy with its environment and non-reciprocal when it behaves differently upon the interchange of input and response. Within the field of metamaterial research on synthetic topological matter, the skin effect describes the conspiracy of non-Hermiticity and non-reciprocity to yield extensive anomalous localization of all eigenmodes in a (quasi) one-dimensional geometry. Here, we introduce the reciprocal skin effect, which occurs in non-Hermitian but reciprocal systems in two or more dimensions: Eigenmodes with opposite longitudinal momentum exhibit opposite transverse anomalous localization. We experimentally demonstrate the reciprocal skin effect in a passive RLC circuit, suggesting convenient alternative implementations in optical, acoustic, mechanical, and related platforms. Skin mode localization brings forth potential applications in directional and polarization detectors for electromagnetic waves.

Figures

Figures reproduced from arXiv: 1908.02759 by the authors.

Figure 1
Figure 1. FIG. 1. Theory of the reciprocal skin effect. a) Left: unit cell and spectrum of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Experimental topolectrical circuit realization of the reciprocal skin effect and exceptional points. a) Bulk unit cell and boundary [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Numerical calculation of the complex eigenspectrum of the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Representation of the circuit Laplacian spectra for periodic boundary conditions (PBC) in the complex plane as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Reciprocal skin effect voltage response due to a localized bulk driving current with phase shift. The sites where the current is applied [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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