Evolution of the quantum Hall bulk spectrum into chiral edge states

One of the most intriguing and fundamental properties of topological systems is the correspondence between the conducting edge states and the gapped bulk spectrum. Here, we use a GaAs cleaved edge quantum wire to perform momentum-resolved spectroscopy of the quantum Hall edge states in a tunnel-coupled 2D electron gas. This reveals the momentum and position of the edge states with unprecedented precision and shows the evolution from very low magnetic fields all the way to high fields where depopulation occurs. We present consistent analytical and numerical models, inferring the edge states from the well-known bulk spectrum, finding excellent agreement with the experiment—thus providing direct evidence for the bulk to edge correspondence. In addition, we observe various features beyond the single-particle picture, such as Fermi level pinning, exchange-enhanced spin splitting and signatures of edge-state reconstruction.

This work is extremely relevant to the present manuscript since it discusses wavefunctions and dispersions that involve a hybridization of a sharp quantum Hall edge and a deeply bound accumulation wire at the sharp edge. Steinke et al. shows that when a Landau-level edge state coexists with a deeply bound wire ground state, the exact solution to the quantum mechanical wavefunction is almost identical to taking the Landau-level edge state and projecting out the deeply bound wire state. This concept is exactly what the authors propose in this discussion, but Steinke et al have already conducted such an analysis for a very similar system and have proven that the wavefunctions so derived are extremely accurate. However, it is worth noting that Steinke et al also demonstrated that as the magnetic field strength increases, the anticrossing energy scale becomes quite large and the naive perturbative coupling implied in Fig. 1e is no longer valid, as anticrossing gaps become of order hbar omega_c. Such a limit is reached in the extreme quantum limit, as soon as the magnetic length starts to become as small as the triangular wire confinement length scale.
(3) On p. 3 within the first full paragraph, after the sentence, "only little momentum transfer and correspondingly small |BY| is required to bring the modes into resonance", perhaps the clarifying comment or something similar would be in order: "And because the UW and LW states both share the same real space position in Y, the resonant tunneling condition is independent of Bz, thus generating a horizontal stripe in Fig. 2." (4) Fig. 3: The colors in panel 3e appear to have inverted: red => black, black => red. This causes some confusion as all the other panels seem to be direct scaled zooms of the existing main panel 3a without any color adjustments.
(5) P. 4, 2nd to last full paragraph: "First, all LL resonances terminate..." note that the resonances terminate both at high field AND at low field. It is helpful to explicitly mention that the termination of interest here is the high-field limit so that people are looking at the right end of each resonance curve.
(6) Also, the sentence "Note that the LL index i denotes the orbital states counting from i = 0, while the filling factor includes the spin degeneracy" seems misplaced. The fundamental description of how the Landau levels and filling factors are indexed should be stated earlier in the definition of the system, not mentioned parenthetically as an aside in the middle of a data description. At the point where these terms are defined, there needs to be an explicit description: nu = 2*i + g where nu = 1, 2, 3 ... is the filling factor, i = 0, 1, 2,... is the Landau level index, and g = 1, 2 is the spin occupancy -1 for spin polarized Landau level, 2 for spin up + spin down both occupied. Some sort of mathematical definition like this needs to be explicitly stated so that the reader knows the relation between nu and i.
(7) Then in the paragraph on p. 4, 2nd to last full paragraph, here is where an explicit reference to the top axis, where the filling factor nu is plotted, needs to be made. Then one can mention explicitly how the filling factor is related to the Landau level index i, which is the subscript of the LL label. An explicit mention that the spin-unresolved case (g = 2 in the above formula), then can be described to pair up i = 1 to nu = 4, and i = 2 to nu = 6 and i = 3 to nu = 8, as seen for the termination of the curves LL1, LL2 and LL3. The description as it stands is very sparse and confusing and it is very easy to lose the reader early if explicit care is not taken.
(8) P. 4, last full paragraph: "Second, a set of vertical lines appears in the upper half of Fig. 3a (dashed lines in Fig. 3b)," Very confusing. Are the authors referring to the dashed vertical lines that are labeled by the filling factor nu? If so, why do the authors not simply refer to them as being the dashed lines that are labeled by the filling factor nu? It is awkward to have this description of an experimental feature, and nowhere in the paragraph do you explicitly mention that these are indexed by the filling factor. The closest that the authors come is to say that the vertical lines "reflect the bulk filling factor". Very obtuse choice of words.
(9) P. 5: "the real space resolution of this spectroscopy technique improves with perpendicular magnetic field (white bars in Fig. 3d)" -how is it an improvement? All three white scale bars are labeled 2 nm, so the resolution seems to be the same.
(10) Fig. 3d: The panel 3d is labeled with "LL depletion" and "subband depletion". To my mind, "LL depletion" and "spin resolved depletion" would be much better adjectives to describe the difference between the two conditions here.
(11) P. 6: On the top of page 6, the lead sentence "In the last part of this article, we develop an analytical model inspired by the work of Halperin [9],..." This analytical model was already developed before by Huber,et al. [11,12,13] for quantum Hall edges and by Steinke, et al [Refs above] for quantum Hall edges coincident with a deeply bound edge wire. These works need to be referenced as well since the fundamental ideas of the model described by the authors were already laid out in these prior works.
(12) Fig. 4a: The vertical axis labeled Bz of Fig. 4a is confusing. The first interval is 0.34 -0.16 = 0.18. The next interval is 1.00 -0.34 = 0.66. The next interval is 2.78 -1.00 = 1.78, and the final interval is 2.91 -2.78 = 0.13. There is no logic to why these Bz fields are chosen and why they are plotted as being equally spaced, when the actual intervals in Bz that are being covered differ by over an order of magnitude.
(13) P. 6: Equation (1) is not rigorous, it is just a hand-waving approximation to the actual dispersion that is correct in the limit form kx >> 0 and kx << 0. There is nothing wrong with this approximation. But the word "approximation" is never stated in association with this Ansatz, making the argument misleading. The authors need to explain that Eq. (1) is just an useful analytical estimate of the dispersion, not a rigorous derivation.
Reviewer #3 (Remarks to the Author): The paper is the first to show rather precise direct momentum resolved tunneling of edge and bulk states and, as this is an important first, I think the paper should be published in Nature Communications.
I think it would be helpful if the authors would make a couple of changes. First, there is a large prior body of work focusing on 1d-1d tunneling from Amir Yacoby's group (refs. 17-24). There should be a few sentences describing the relation of this work to the prior work and the difference between this work and the prior work from Yacoby's group. In this regard, it would be useful for me and the reader to understand what technical hurdles, if any, were overcome to do this work -why didn't Yacoby's group do this experiment 15 years ago? That said, the 1D-2D nature of the work here is clearly different from the prior work.
I don't like the use of the word "topological" in this paper. It will confuse readers in thinking that there is a quantum spin Hall effect in these samples or some such other physics rather than the physics of 2D edge states that was first described well before the development of "topological" theories of Kane and Male, etc. The paper is about old-fashioned integer quantum Hall edge states. While the authors may be technically right in calling them "topological", I think it is a new use of the word that will just confuse some readers.
The paper overall is well written and clearly explains what is going on. It would be nice to include in Fig. 1 a picture of the wavefunction of LL0 trapped in the magnetic potential and with a hard wall to show the reader the definitions of guiding center and center of mass. There is an attempt to describe this in Fig. 3s of the supplement, but I think a simple intuitive picture in Fig. 1 would help many readers.
Part of the power of this spectroscopy is that it is all done at zero bias. There are no heating or lifetime effects. It might be useful to point this out. That said, I do wonder what happens as a function of energy. Have the authors attempted to look at what happens with applied DC voltage? Is there a magnetic field induced tunneling gap (or Luttinger behavior similar to what is seen with only one edge state occupied) for edge states similar to that in the bulk?
While the paper represents and important step forward, there aren't big surprises in it. Aside from DC voltage, do the authors see anything interesting with varying temperature? The exchange gaps will close at high temperature -that would be expected. But does anything happen between say, 10 mK and 100 mK? It seems that the authors worked very hard to get the samples very cold in this experiment, but we don't know if it matters at all. Identification of physics only appearing at very low temperatures would be an interesting addition to this paper.
The authors describe future experiments looking for fractional quantum Hall states, but one wonders why they haven't looked at higher Bz in this paper? The features at high Bz appear to be disappearing, and I'm wondering if this is the result of the development of the magnetic field induced tunneling gap suppressing the signal. Again, it would be very interesting to understand the nature of any gaps for tunneling into edge states.
In short, I recommend publication of this paper in Nature Communications. The paper provides a new window into wire and edge states. One can see a strong agreement with theory and there impressive clarity in the data, showing things like the results of momentum boosts from both + and -k-states in the wire tunneling into the chiral 2D edge states. There is a lot of new detail here and a strong understanding of most of it.

RESPONSES TO REFEREE 1
Reviewer 1: I found this paper a joy to read. It describes in detail the plethora of experimental data, and presents a clear physical interpretation. The data are really striking -very strong signal of the edge states of the various Landau levels, with several copies due to tunneling from different wire states. The analysis and the interpretation explain most of the data, and overall the paper is an important contribution, which should be definitely published in Nature Physics. I have two minor questions, which the authors should clarify before publication: 1.The distance between the wire state and the edge states in the 2DEG is increasing with the Landau level index, as the edge states associated with these Landau levels are deeper in the 2DEG. I would expect that the signal would decrease exponentially with the LL index, but I see only weak dependence in the data. Can the authors deduce from the dependence of the signal on the LL index the distance of the different edge states from the edge of the system? Reply: We would like to thank the Referee for the positive comments on our work. The signal strength depends on the wave function overlap between the edge state and the wire mode. Because of the hard wall confinement, the lowest LL edge states are pulled towards the edge of the sample (see Fig. 4(a) of the main text or Figs. 3S(f), 4S of the supplement), in contrast to the case of soft confinement, which leads to the formation of compressible and incompressible strips. In the latter case, the edge states of different Landau levels are sitting in different places in space, whereas in the case of a hard wall, all the edge state wave functions start at the hard wall and interpenetrate each other. The wave function of the lower wire is very narrow compared to the wave function of the Landau level edge states at small magnetic fields and is localized close to the sample edge. This leads to the strong overlap between the wire mode and the last bump of different Landau level wave functions. As a result, only an algebraic decrease of the tunneling signal is observed. To deduce the position of the edge states from the tunneling signal strength, self-consistent calculations are required. Currently, such calculations are being worked out.
Reply: We agree the density of electrons in the wires does create an additional potential. This potential, however, compensates the approx. triangular potential close to the sample edge that results from the side dopants present in the CEO samples (introduced in the overgrowth process after sample cleaving in order to attract charge to the sample edge and form quantum wires). Thus, the remaining potential "after filling up the quantum wire states" can be well approximated with an approximately flat bottom potential, as also previously shown by Ref. 31 (revised manuscript).
An exact treatment of the full electrostatic problem (top + side dopants) results in the formation of hybrid states rather than separate Landau levels and quantum wire modes, shown in Fig.1(e) of the manuscript. The hybridization leads to opening of gaps that could be potentially interesting and would be visible at high magnetic fields using this type of spectroscopy. While we are currently working on self-consistent solutions for the problem (which goes beyond the scope of the current work), we note that already the simple single particle picture gives a good quantitative account of the experimental data.
Rev1: Can one change the density in the upper wire so as to affect the actual boundary potential the electrons in the 2DEG see? Maybe this can be used to study actual edge reconstruction in the 2DEG.
Reply: It would be very interesting to see the evolution of the edge reconstruction as a function of confinement potential. Potentially this could be done if we had a range of samples with different doping levels and correspondingly different wire densities or a sample with a side gate where the density can be changed in-situ. Unfortunately, such samples are currently not available.

RESPONSES TO REVIEWER 2
Reviewer 2: This paper reports evidence of momentum-resolved resonant tunneling between quantum Hall edge states and a deeply bound wire at the sharp edge of a cleaved-edge overgrown quantum Hall system. The experiment is challenging and the fact that the data is so convincing and clear is a testament to the quality of the work presented here. The results deserve publication in a prominent venue such as Nature Communications.
However, the manuscript has some drawbacks. The description tends to be a bit rambling, with some important aspects like defining the relevant quantum numbers being put into incidental remarks in the middle of a data description instead of being placed front and center in the problem description. Also key aspects of some of the conceptual plots are quantitatively inaccurate (Fig. 1) or highly confusing (Fig. 4a). At some points, the manuscript reads more like a thesis chapter than a well-honed journal article. The description is occasionally incomplete or underdefined, leaving confusing questions in the mind of the reader. A critical editing of the article for conciseness and clarity would improve the readability and therefore the likelihood that this work will be cited.
Reply: We thank the referee for these points and acknowledge these problems in the manuscript. We have revised the manuscript and have tried to implement all suggested changes. In particular, we have moved forward and merged two paragraphs which are now paragraphs 2 and three (left column) on the first page, marked in blue in the revised manuscript. The paragraphs now begins: "Previously, tunneling spectroscopy of cleaved edge overgrowth wires has established the system as one of the best realization of a 1D ballistic conductor..." We have also added the definition of the filling factor early in the paper (first paragraph on page 2) as requested.
"Throughout the paper the filling factor is defined as $\nu=2 n + g$, where $n=0, 1, 2, ...$ is the orbital Landau level index, and $g$ is the spin occupancy." Rev2: Several explicit points that require clarity are described below: (1) Fig. 1: The Fig. 1a and 1b are helpful conceptually, but are quantitatively incorrect and must be corrected. These quantitative inaccuracies confuse the issue by creating physically impossible dispersion scenarios. (Figs. 1d, 1e and the insets of Fig2, etc. are all quantitatively accurate, on the other hand.) In Fig. 1a, b, for a simple hard-wall potential, the guiding center dispersion (dark blue line) must cross the energy E = 1.5 hbar omega_c at guiding center coordinate Y = 0. This is because the ground state energy of a state that is bisected by a hard wall has a node at its guiding center position Y = 0, yielding a wavefunction at the wall that is identical to the first excited state in the bulk which is antisymmetric around Y = 0 and therefore also has a node in its center. The width of the ground-state bulk gaussian wavefunction in Fig. 1b is furthermore not accurate in the following manner. The width of the wavefunction (blue) for the ground state Landau level is far too narrow. The wavefunction's energy in the guiding center dispersion (bold blue line) will only increase once the finite tail of the wavefunction (light blue gaussian) overlaps with the hard-wall confinement potential at Y = 0. As it is currently drawn, the guiding center dispersion starts to curve upward when the wavefunction is much too far away from the edge. As a rule of thumb, the dark blue and light blue dispersions in panel 1b MUST BE IDENTICAL TO THE CORRESPONDING DISPERSIONS IN PANEL 1a, but simply scaled in Y by the smaller magnetic length and scaled in E by the increased cyclotron energy. Thus the dispersion in 1b must ALSO cross Y = 0 at the energy E = 1.5 hbar omega_c. In all cases, it would be better if these curves and the corresponding wave functions were simply calculated rather than misrepresented by inaccurate hand-drawn approximations, but at the very least, respect for the correct magnetic length scale and energy scale must be preserved.
Reply: Thank you for pointing this out. We were just making a qualitative figure and we have now changed it to be also quantitatively accurate. We have made all suggested changes in Fig.1a and   Letters 193, 193117 (2008). This work is extremely relevant to the present manuscript since it discusses wavefunctions and dispersions that involve a hybridization of a sharp quantum Hall edge and a deeply bound accumulation wire at the sharp edge. Steinke et al. shows that when a Landau-level edge state coexists with a deeply bound wire ground state, the exact solution to the quantum mechanical wavefunction is almost identical to taking the Landau-level edge state and projecting out the deeply bound wire state. This concept is exactly what the authors propose in this discussion, but Steinke et al have already conducted such an analysis for a very similar system and have proven that the wavefunctions so derived are extremely accurate. However, it is worth noting that Steinke et al also demonstrated that as the magnetic field strength increases, the anticrossing energy scale becomes quite large and the naive perturbative coupling implied in Fig. 1e is no longer valid, as anticrossing gaps become of order hbar omega_c. Such a limit is reached in the extreme quantum limit, as soon as the magnetic length starts to become as small as the triangular wire confinement length scale.
Reply: Yes, thank you very much for this comment. We are now citing the first suggested paper: (reference [30]) addressing similar issues several years earlier, so we are also citing this as well. The second paper suggested by the referee is predominantly about the characterization of 2DEG corner structures and seems not that relevant for our work and we have therefore decided not to cite it. We have also found another important paper which we haven't cited so far and we have added it to the citations (  B 29, 1616 (1984).
"In the last part of this article, we develop an analytical model [17,18,22,30] ..." Rev2: (3) On p. 3 within the first full paragraph, after the sentence, "only little momentum transfer and correspondingly small |BY| is required to bring the modes into resonance", perhaps the clarifying comment or something similar would be in order: "And because the UW and LW states both share the same real space position in Y, the resonant tunneling condition is independent of Bz, thus generating a horizontal stripe in Fig. 2

."
Reply: Thank you for this comment; we have added a very similar sentence to the manuscript.
On page 3, in the first column, last paragraph, we have inserted: "These resonances are independent of $B_Z$ because the $Y$ coordinates of both UW$_1$ and LW$_1$ modes are very similar" Rev2: (4) Fig. 3: The colors in panel 3e appear to have inverted: red => black, black => red. This causes some confusion as all the other panels seem to be direct scaled zooms of the existing main panel 3a without any color adjustments.
Reply: Thank you for pointing this out. Panel 3e shows undifferentiated raw data while panel 3a shows the second derivative of the raw data with respect to magnetic field By. So these panels show two different physical quantities and have color bars which can't be compared directly.
We changed the caption to highlight this fact: "LL spin splitting clearly visible even in undifferentiated raw data (tunneling conductance $g_T$)." Rev2: (5) P. 4, 2nd to last full paragraph: "First, all LL resonances terminate..." note that the resonances terminate both at high field AND at low field. It is helpful to explicitly mention that the termination of interest here is the high-field limit so that people are looking at the right end of each resonance curve.
Reply: Thank you for this comment. We changed the following sentence to include your suggestion: "First, all LL resonances terminate on the right end at a specific bulk filling factor …" Rev2: (6) Also, the sentence "Note that the LL index i denotes the orbital states counting from i = 0, while the filling factor includes the spin degeneracy" seems misplaced. The fundamental description of how the Landau levels and filling factors are indexed should be stated earlier in the definition of the system, not mentioned parenthetically as an aside in the middle of a data description. At the point where these terms are defined, there needs to be an explicit description: where nu = 1, 2, 3 ... is the filling factor, i = 0, 1, 2,... is the Landau level index, and g = 1, 2 is the spin occupancy -1 for spin polarized Landau level, 2 for spin up + spin down both occupied. Some sort of mathematical definition like this needs to be explicitly stated so that the reader knows the relation between nu and i.
Reply: Thank you for your suggestion. We have added the following sentence to define the filling factor (page 2, left column, first paragraph): "Throughout the paper, the filling factor is defined as $\nu=2 n + g$, where $n=0, 1, 2, ...$ is the orbital Landau level index, and $g$ is the spin occupancy." Rev2: (7) Then in the paragraph on p. 4, 2nd to last full paragraph, here is where an explicit reference to the top axis, where the filling factor nu is plotted, needs to be made. Then one can mention explicitly how the filling factor is related to the Landau level index i, which is the subscript of the LL label. An explicit mention that the spin-unresolved case (g = 2 in the above formula), then can be described to pair up i = 1 to nu = 4, and i = 2 to nu = 6 and i = 3 to nu = 8, as seen for the termination of the curves LL1, LL2 and LL3. The description as it stands is very sparse and confusing and it is very easy to lose the reader early if explicit care is not taken.
Reply: Thank you for the comment. We have taken it into account and modified the corresponding sentence: "In particular, tunneling involving LL$_2$ with $n=2$ is observed up to $B_Z\approx 1.1\,$T, terminating at the corresponding bulk filling factor $\nu=6$, labeled on the top axes in Fig.\,\ref{fig:3}b. Here, spin occupancy $g=2$ because both spins are populated." Rev2: (8) P. 4, last full paragraph: "Second, a set of vertical lines appears in the upper half of Fig. 3a (dashed lines in Fig. 3b)," Very confusing. Are the authors referring to the dashed vertical lines that are labeled by the filling factor nu? If so, why do the authors not simply refer to them as being the dashed lines that are labeled by the filling factor nu? It is awkward to have this description of an experimental feature, and nowhere in the paragraph do you explicitly mention that these are indexed by the filling factor. The closest that the authors come is to say that the vertical lines "reflect the bulk filling factor". Very obtuse choice of words.
Reply: In this paragraph, we refer to the bright vertical features present only in the region of By>0 and Bz>0 in Fig.3a. To make our point clearer, we have changed this paragraph and included the reference to the supplementary figure (undifferentiated raw data) where the corresponding features are even more visible: Rev2: (9) P. 5: "the real space resolution of this spectroscopy technique improves with perpendicular magnetic field (white bars in Fig. 3d)" -how is it an improvement? All three white scale bars are labeled 2 nm, so the resolution seems to be the same.
Reply: All three scale bars correspond to a real space displacement of 2nm, but the length of these scale bars are different and increases for larger magnetic field Bz, thus the sensitivity is increasing. We modified Fig.3d and the caption to emphasize this. "The three vertical bars of growing height indicate a distance of $2\,$nm in real space. The height $\Delta B_Y$ of the bar is given by $\Delta B_Y = \Delta Y B_Z / d$, where $\Delta Y$ is the distance in real space. Thus, the real space resolution is improving with increasing magnetic field $B_Z$." Rev2: (10) Fig. 3d: The panel 3d is labeled with "LL depletion" and "subband depletion". To my mind, "LL depletion" and "spin resolved depletion" would be much better adjectives to describe the difference between the two conditions here.
Reply: We agree and have changed "subband depletion" to "spin resolved depletion" in panel 3d.
Rev2: (12) Fig. 4a: The vertical axis labeled Bz of Fig. 4a is confusing. The first interval is 0.34 -0.16 = 0.18. The next interval is 1.00 -0.34 = 0.66. The next interval is 2.78 -1.00 = 1.78, and the final interval is 2.91 -2.78 = 0.13. There is no logic to why these Bz fields are chosen and why they are plotted as being equally spaced, when the actual intervals in Bz that are being covered differ by over an order of magnitude.
Reply: Thank you for this comment. Please note that the evolution of the wave function is highly nonlinear when the corresponding Landau Level is close to the Fermi level. We have added a small section to the last paragraph on page 6 to explain this behavior: "Throughout the process of increasing magnetic field Bz, the electron wave function is progressively compressed (from green to red curves). There are two stages of the edge state motion as magnetic field Bz increases: first, motion of the center of mass towards the hard wall (empty circles for $B_Z<2.78\,$T) and motion away from the hard wall at larger fields. During the latter stage, the center of mass merges with the guiding center position (black and blue curves approach and then coincide for larger $B_Z$ in Fig.\ref{fig:4}a), followed by depopulation of the corresponding LL." We have also modified the caption of Fig.4a to emphasize this.
"Landau Level wave functions for particular values of $B_Z$ chosen to visualize the important stages of magnetic field evolution. Note that the resulting vertical scale is highly nonlinear." Rev2: (13) P. 6: Equation (1) is not rigorous, it is just a hand-waving approximation to the actual dispersion that is correct in the limit form kx >> 0 and kx << 0. There is nothing wrong with this approximation. But the word "approximation" is never stated in association with this Ansatz, making the argument misleading. The authors need to explain that Eq. (1) is just an useful analytical estimate of the dispersion, not a rigorous derivation.
Reply: Thank you for pointing this out. We have modified the sentence with the model description (page 6, first column, last paragraph): "We assume that upon approaching the hard wall, LLs remain at their bulk energy …" We also added the word "approximations" before the formula (1): "Using these approximations the LL dispersion …"

RESPONSES TO REVIEWER 3
Reviewer 3: The paper is the first to show rather precise direct momentum resolved tunneling of edge and bulk states and, as this is an important first, I think the paper should be published in Nature Communications.
I think it would be helpful if the authors would make a couple of changes. First, there is a large prior body of work focusing on 1d-1d tunneling from Amir Yacoby's group (refs. 17-24). There should be a few sentences describing the relation of this work to the prior work and the difference between this work and the prior work from Yacoby's group. In this regard, it would be useful for me and the reader to understand what technical hurdles, if any, were overcome to do this work -why didn't Yacoby's group do this experiment 15 years ago? That said, the 1D-2D nature of the work here is clearly different from the prior work.
Reply: Thank you for the comment. This work was only possible because of the ability to independently and smoothly controlling two orthogonal magnetic fields using a vector magnet. This allows us to form the quantum Hall edge states applying some value of a magnetic field B_Z perpendicular to the 2D electron gas and scan an in plane field B_Y to perform momentum resolved tunneling from states with different momentum in the upper system.
To make this point also clear in the paper we have changed the section of the first paragraph on page two "Such wires are among ..." and converted it into a new paragraph about all previous works done on similar samples (first page, first column, second paragraph). We also have added a sentence about the importance of having independent control over two orthogonal magnetic fields: "Here we use a vector magnet to independently control two orthogonal magnetic fields: one to form quantum Hall edge states and another to perform tunneling spectroscopy." Rev3: I don't like the use of the word "topological" in this paper. It will confuse readers in thinking that there is a quantum spin Hall effect in these samples or some such other physics rather than the physics of 2D edge states that was first described well before the development of "topological" theories of Kane and Male, etc. The paper is about oldfashioned integer quantum Hall edge states. While the authors may be technically right in calling them "topological", I think it is a new use of the word that will just confuse some readers.
Reply: Thank you for your comment. We have added a sentence (first page, second column, first paragraph) to clarify this: "Note that in this work we are studying integer quantum Hall edge states and not the spin Hall effect or any other topological state. However, this technique is also applicable to the latter states." Rev3: The paper overall is well written and clearly explains what is going on. It would be nice to include in Fig. 1 a picture of the wavefunction of LL0 trapped in the magnetic potential and with a hard wall to show the reader the definitions of guiding center and center of mass. There is an attempt to describe this in Fig. 3s of the supplement, but I think a simple intuitive picture in Fig. 1 would help many readers.
Reply: Thank you for your suggestion. We have added a magnetic confinement parabola to Fig.1b (gray dashed parabolas) for two guiding center positions and show the corresponding wave functions.
Rev3: Part of the power of this spectroscopy is that it is all done at zero bias. There are no heating or lifetime effects. It might be useful to point this out. That said, I do wonder what happens as a function of energy. Have the authors attempted to look at what happens with applied DC voltage? Is there a magnetic field induced tunneling gap (or Luttinger behavior similar to what is seen with only one edge state occupied) for edge states similar to that in the bulk?
Reply: Thank you for this important comment. The DC bias is an additional control knob which we haven't explored yet in great detail. We plan to do bias dependent measurements in future experiments. We have added a sentence (first page second column first paragraph): "We emphasize that this spectroscopy is done at zero bias, thus eliminating heating or lifetime effects." Rev3: While the paper represents an important step forward, there aren't big surprises in it. Aside from DC voltage, do the authors see anything interesting with varying temperature? The exchange gaps will close at high temperature -that would be expected. But does anything happen between say, 10 mK and 100 mK? It seems that the authors worked very hard to get the samples very cold in this experiment, but we don't know if it matters at all. Identification of physics only appearing at very low temperatures would be an interesting addition to this paper.
Reply: This new technique opens new avenues for the direct and precise study of edge state reconstruction, spin exchange and Fermi level pinning. For a substantial part of the experiment presented here, it turns out that temperature plays a minor role. For example, the data in Fig. 2 looks essentially the same when measured at 0.5 K. There is a little bit more smearing, but resonances for all the Landau level edge states are still visible (apart possibly from the very faint ones). Having low temperatures is essential to study more fragile states, such as fractional quantum Hall edge states, on which we would like to focus in the future. In addition, the present sample was used in an earlier study to investigate helical nuclear order in the quantum wires, which is fully developed only at temperatures below roughly 100 mK [15]. The present spectroscopy technique may also be used to investigate the associated helical gaps in the electronic spectrum of the quantum wires, either through intra wire tunneling or using the integer quantum Hall edge states as a sensor.