Observation of continuum Landau modes in non-Hermitian electric circuits

Continuum Landau modes — predicted recently in a non-Hermitian Dirac Hamiltonian under a uniform magnetic field — are continuous bound states with no counterparts in Hermitian systems. However, they have still not been confirmed in experiments. Here, we report an experimental observation of continuum Landau modes in non-Hermitian electric circuits, in which the non-Hermitian Dirac Hamiltonian is simulated by non-reciprocal hoppings and the pseudomagnetic field is introduced by inhomogeneous complex on-site potentials. Through measuring the admittance spectrum and the eigenstates, we successfully verify key features of continuum Landau modes. Particularly, we observe the exotic voltage response acting as a rainbow trap or wave funnel through full-field excitation. This response originates from the linear relationship between the modes’ center position and complex eigenvalues. Our work builds a bridge between non-Hermiticity and magnetic fields, and thus opens an avenue to explore exotic non-Hermitian physics.

The main achievement of the paper is in demonstrating a clear signature of a CLM analogue in an electrical circuit.This is no mean feat due to experimental issues like device tolerances and losses of the circuit elements.It also opens a new avenue of research into the effect of gauge terms in non-Hermitian systems, and its realization in actual physical/electrical circuits.As such, I am of the opinion that it is worthy of publication in Nature Comms.
However, could the authors comment on how other magnetic field induced/Landau level features in Hermitian circuits (e.g.skipping orbits) are transformed in non-Hermitian systems, and whether these can also give rise to electrical signatures in an electrical circuit.
Reviewer #3 (Remarks to the Author): In the manuscript "Observation of continuum Landau modes in non-Hermitian electric circuits", the authors construct non-Hermitian electric circuit networks to simulate a non-Hermitian Dirac Hamiltonian under uniform magnetic field.They successfully capture the key characteristics of a recently identified novel state, referred to as the "Continuum Landau Modes (CLMs)", by measuring the admittance spectrum and eigenstates of these electric circuit networks.Furthermore, they observe phenomena akin to rainbow trapping or wave funneling in the voltage responses of 1D cases.To the best of my knowledge, this manuscript represents the first report of the experimental determination of the CLMs, an endeavor that is commendable.However, there are a few points that I believe the authors need to address before a final decision can be made.
i.There is a slight discrepancy between the parameters used in the calculations and simulations and those used in the experiment.This difference hinders an intuitive comparison between the results presented in Fig. 1 and Fig. 2. If it is not too time-consuming, I would recommend using the same parameters as those in the experimental setup for Fig. 1.These parameters also appear to be inconsistent with the content of the Methods section.The author should revise the manuscript to rectify errors that undermine its credibility.
ii.Moreover, the calculated spectra depicted in Fig. 1 (c) or Fig. 2 (b) and (c) of Ref. (37) exhibit highly symmetrical pattern.However, this characteristic is not observed in the measured admittance data.It would be beneficial if the author could make a simulation by LTspice or other similar software with/without consideration of components errors to investigate this discrepancy and provide a succinct discussion on the matter.We also notice the high-precision electronic components were utilized in experiment.
iii.In the section titled "Experiments in 1D Circuit Lattices", the reference to Fig. 4 should be amended to Fig. 3. iv.As shown in Fig. 3(b) and (e) of Ref. (37), 'the eigenenergies fill the complex plane' for the 1D case.However, the calculated results in the supplementary material, along with the experimental data, primarily show eigenenergies localized around a distinct line within the complex plane.What differentiates the model proposed by the authors from those described in Reference (37)?"

Reviewer #1 (Remarks to the Author):
- This paper reports on an experimental realization of the recently-discovered phenomenon of Continuum Landau Modes (CLMs), using an electrical circuit lattice.This is a notable and significant work for the non-Hermitian physics community, since it demonstrates that non-Hermitian systems can violate the usual expectation that bound states (spatially localized eigenstates) must form a discrete spectrum.
The experiment sticks rather closely to the earlier theory paper [37], including in the design of the lattice and the set of experimental signatures.Much of the theoretical discussion from pages 3 to 5 recapitulate the discussions of the theory paper, though the text is succinct enough that this is acceptable.The new element is the implementation of the non-Hermitian Hamiltonian by means of a circuit Laplacian, which is based on previous developments in the field of circuit lattices [38,39].

Comment:
The paper has a few shortcomings that ought to be corrected before it can be published.These are grouped into two categories.

Reply:
We thank the referee for the positive comments of our work.In the revised manuscript, we have improved it significantly in terms of the referee's suggestions and comments.We hope that this improved version can be suitable for publication in Nature Communications.
Comment 1: First, it is advisable for the authors to do a slightly better job of articulating how their circuit implementation deviates from the theory paper [37].There are a few places where there are interesting deviations, but they are not discussed prominently enough: Reply 1: We thank the referee for the helpful comments.In the following reply and 2 / 17 revised manuscript, we have clarified the role of the frequency in our circuit system and demonstrated its effects on the CLMs.The continuum of the complex admittance and eigenfrequency spectra have been also analyzed.
Comment 2: In the circuit formulation, the frequency is a tuning parameter rather than an eigenvalue of the Hamiltonian.Since this was not anticipated in [37], the meaning of the frequency parameter (and particularly its effects on the CLMs) should be explained more clearly.
Reply 2: We thank the referee for the invaluable comment.Indeed, the frequency is a tuning parameter incorporated in the circuit Laplacian [Eq.( 1) in the main text] with the coefficient  0 () and the pseudomagnetic filed   ().In the following, we illustrate the effects of these two frequency-dependent parameters on the admittance spectrum and eigenstates, respectively.
(i) The frequency modifies the admittance spectrum.We first note that, from the circuit Laplacian ()/(), the diagonal components  0 () and   () induce the shift of the admittance spectrum along the directions of Re[/()] and Im[/()] , respectively.Due to the existence of the site index  before   (), the bandwidth of Im[/()] may vary with the frequency.On the other hand, by requiring the center position  0 of CLMs [Eq.(2) in the main text] to lie in the lattice, we can obtain the boundaries of the admittance spectrum as where   and   denote the size of the circuit lattice in the  and  directions, respectively.It can be seen that  0 () causes the shift of Re[/()], while   () not only gives rise to the shift of Im[/()], but also influences its bandwidth.(ii) The frequency affects the localization of the admittance eigenstates as well as the slope of the linearity between the CLM's center position and complex eigenvalues.On one hand, from the wavefunction  0 of the CLM, the localization of the Gaussian envelope is characterized by   = −  ()/(2  ) and   =   /(2  ).That is to say, the frequency can only affect the localization of the CLMs in the  direction.Figures R2a and R2b show the calculated and simulated amplitude distributions along lines passing through the center of one eigenstate (marked by star in Fig. R1) for the different frequencies, respectively.These calculated and simulated results agree well with the theoretical predictions.However, the effect (~1/) is very weak and is thus hard to be observed in experiment.

Figures R1a and R1b
On the other hand, according to Eq. ( 2) in the main text, the linear relationships between complex admittance eigenvalues and the CLM's center position are given by

/ 17
Comment 3: Conversely, since the complex eigenvalues of the admittance matrix are not frequencies, the meaning of having a "continuum" of these quantities ought to be analyzed.
Reply 3: We thank the referee for the helpful suggestion.As an external parameter, the frequency in the circuit Laplacian can be tuned at will.For each frequency, the eigenvalues of the admittance matrix fill the complex energy plane if the lattice is infinite, i.e., they form a continuum.The roots of the admittance spectrum () = 0 corresponds to the complex eigenfrequency spectrum of the system.As shown in Fig. R3, this complex eigenfrequency spectrum has the same number of the eigenstates as the complex admittance spectrum, and can thus form a continuum filling the complex frequency space.When the complex admittance or eigenfrequency spectra form a continuum, the voltage response is continuous, i.e., any frequency can excite the corresponding eigenmode of the circuit.Comment 4: The fact that the complex admittance eigenvalues can be directly 7 / 17 measured in an experiment (Fig. 2b) is surely an important advantage of the circuit approach.Complex eigenfrequency spectra normally cannot be retrieved experimentally like this.Yet this was hardly emphasized in the text.
Reply 4: We thank the referee for the comment.For the complex eigenfrequency, its positive (negative) imaginary part indicates the dissipation (amplification) of the voltage.Since the oscilloscope could not capture the fast-changing dynamics of the exponentially oscillating amplitudes, it is hard to measure the complex eigenfrequency spectrum in experiments.However, the admittance matrix can be experimentally reconstructed by measuring the voltage response at each node to a local a.c.input.
Moreover, the admittance eigenvalues and eigenstates can be extracted through numerical diagonalization.
In the revised manuscript, we have emphasized the above point in Paragraph 2 on Page 6. Reply 5: We thank the referee for the valuable suggestion.In the revised manuscript, we have extended key physical signatures of the 2D lattice, including replotting Figs.
1f and 2f of the original manuscript (Figs.3c and 3e of the revised manuscript), illustrating the significance of the frequency dependence of the voltage response, and redesigning experimental method to observe the CLMs' spatial characteristics.
Comment 6: First of all, these subplots are too cramped, given their importance.The "3D slice" plots are hard to read, given that a lot of the data is obscured behind the slices.I would recommend splitting these subplots into a separate figure, or finding some other way of improving the presentation.
the present studies about the interplay of non-Hermiticity and magnetic field, we present three comments as follows.
(i) For the non-Hermitian Dirac Hamiltonians under a magnetic field considered by our work, the eigenstates of the system have Gaussian spatial envelopes and form a continuum filling the complex energy plane.These modes can map to the zeroth Landau level modes of the Hermitian Dirac Hamiltonian.Therefore, the Landau quantization and the edge states (i.e., skipping orbits in semi-classical view) are missing.
(ii) For the non-reciprocal model under a magnetic field (see, for example, References [46][47][49][50] of the revised manuscript), the semiclassical trajectories of the wavepacket may turn out to be closed/skipping orbits in the 4D complex space [47].
The Landau levels exhibit the usual quantized spectra and the Hall-like edge states are still found.
(iii) Benefiting from a wide range of passive and active circuit elements in electric circuit networks, it is interesting to experimentally investigate these unique non-Hermitian effects controlled by magnetic field.For example, the pseudomagnetic filed can be mimicked through inhomogeneous strain of the graphene-like circuit lattice and the non-Hermitian terms can be achieved by choosing appropriate circuit elements.The Landau levels can be measured through admittance and impedance spectra.The localized Landau modes and the Hall-like edge states can be observed by steady-state voltage response or dynamics of the excitation.
In Paragraph 4 on Page 10 of the revised manuscript, we have added some discussions and outlook about other magnetic field induced effects controlled by non-Hermiticity.

Reply:
We thank the referee for the positive comments of our work.In the revised manuscript, we have improved it significantly in terms of the referee's suggestions and comments.We hope the referee can recommend our revised manuscript to be published in Nature Communications.
Comment 1: There is a slight discrepancy between the parameters used in the calculations and simulations and those used in the experiment.This difference hinders an intuitive comparison between the results presented in Fig. 1 and Fig. 2. If it is not too time-consuming, I would recommend using the same parameters as those in the experimental setup for Fig. 1.These parameters also appear to be inconsistent with the content of the Methods section.The author should revise the manuscript to rectify errors that undermine its credibility.
Reply 1: We thank the referee for the helpful suggestion.In the revised manuscript, we have replotted Fig. 1 according to the experimental parameters (i.e., those in Fig. 2).In addition, we have also corrected the parameters in the Method section.

/ 17
Comment 2: Moreover, the calculated spectra depicted in Fig. 1(c 37) exhibit highly symmetrical pattern.However, this characteristic is not observed in the measured admittance data.It would be beneficial if the author could make a simulation by LTspice or other similar software with/without consideration of components errors to investigate this discrepancy and provide a succinct discussion on the matter.We also notice the high-precision electronic components were utilized in experiment.
Reply 2: We thank the referee for the nice suggestion.In Fig. 1c or Figs.2b and 2c of Reference (37), the clean circuit Laplacian/Hamiltonian is considered, and the admittance/energy spectra are thus highly symmetric.In the real experiments, the errors of the electronic components usually exist.For example, in our experiment with the high-precision electronic components, these errors are about ±1% .In order to investigate the effects induced by the errors, we use the LTspice software to simulate the admittance spectra and the eigenstate's energy against the position expected values by introducing ±1% (a, b) and ±5% (c, d) disorders to all the circuit components.
As shown in Fig. R6a. the admittance spectrum turns out to be cluttered, even if the high-precision electronic components with the errors of ±1% are introduced.
However, the key features of the CLMs, the localization of the eigenstates and the linear relationship between the eigenstates' center position and eigenvalues, still exist (Fig. R6b).When the errors increase (±5%), the similar properties of the admittance spectra and the linear relationship between the eigenstates' center position and eigenvalues are found (Fig. R6c and R6d), which means that the features of the CLMs are robust to the errors of the electronic components.
We have pointed out the discrepancy of the experimental results of the admittance spectrum in Paragraph 2 on Page 7 of the revised manuscript.The related discussions and simulation results have been added in Supplementary S-VI.
These discussions have been added in Paragraph 2 on Page 6 of the revised manuscript.

Fig. R3 a
Fig. R3 a Calculated admittance spectrum of the circuit Laplacian in Eq. (1) fed by the frequency  = 162 kHz.The color of each point indicates the participation ratio of the corresponding eigenstate.b Complex eigenfrequency spectrum ( ′ ) close to the frequency  = 162 kHz.The other parameters are the same as those in Fig. R1.

Comment 5 :
The second area where the paper falls short involves the key physical signatures of the 2D lattice, shown in Fig. 1f (numerical data) and Fig. 2f (experimental data).

Fig. R6 a
Fig. R6 a Simulated admittance spectrum of the circuit Laplacian by considering the error of ±1% for all circuit components.The color of each point indicates the participation ratio of the corresponding eigenstate.b Simulated results of Re[/()] (left panel) and Im[/()] (right panel) versus the expectation values of the eigenstate's position, 〈〉 and 〈〉 , respectively.c, d The simulated results corresponding to a and b with the error of ±5% for all circuit components.The other parameters are the same as those in Fig. 2 of the revised manuscript.