Unusual Transport Phenomena in Spatially Modulated Correlated Electron Waveguides

A very interesting theoretical work on a novel model that definitely merits publication in SciPost Physics.

None.

Report on Shavit and Oreg's manuscript "Unusual Transport Phenomena in Spatially Modulated Correlated Electron Waveguides"

This is a very interesting theoretical work which I recommend for publication in SciPost Physics. It is on the conductance of a 1D system, modelled as a Luttinger liquid. The specificity of this work is that the inter-particle interaction U(x-y) is has a specific spatial-modulated, such that the Luttinger liquid has two bosonic modes with one bosonic mode's velocity being an INTEGER MULTIPLE of the other bosonic mode's velocity. The manuscript presents two main results. The first (see fig 2) is the observation of a conductance plateau at 9/5 e^2/h, which the authors associate with the formation of a Luttinger liquid of electronic "trions". The second is a conductance dip below 1 e^2/h.

TYPOS:
I did not search carefully for typos, but I saw one
in the first sentence of Section "B. Fractional conductance". The word "faith" should probably be "fate" in the sentence "The physics we are interested in concerns the faith of ..."

1. Novel ideas/mechanism of attraction-enhanced backscattering
2. Clear experimental motivation and relevance
3. Elegant theoretical analysis and presentation

1. Approximations/simplifications are not always clear to me.
2. Some technicalities need to be clarified.

This article theoretically discusses the effects of spatially modulated attractive interactions in quantum wires.

For vertical modulation it is pointed out that (i) Strong backscattering leads to a fractional conductance plateau with varying chemical potential which may evolve into a peak-dip structure as a gap opens up (ii) A perturbative, spatially modulated interaction enhances this backscattering (iii) For strong attractive interactions, arguments are provided for formation of a trion phase in the two-mode wire. The results suggest that in the discussed interesting experiments the attractive interaction is modulated at twice the wave vector of one of the modes of the wire.
For lateral modulation and high magnetic field (Zeeman term) it is shown that (i) Spatially modulated Rashba-spin-orbit coupling leads to a gap with a very similar peak-dip structure --but not at a fractional conductance value-- (ii) Interactions are argued to enhance the periodic modulation leading to a non-interacting backscattering impurity by RG arguments

The paper presents a comprehensive set of interesting, apparently new theoretical ideas in a clear way. These are well motivated by experiments on state of the art attractive quantum wires which are discussed at several places in the paper.

The paper satisfies all general acceptance criteria for SciPost Physics. Concerning the additional requirements ("expectations") for SciPost Physics, I would say it identifies an important new explanation of puzzling transport effects identified experimentally (expectation 2). The topic of attractive interactions in nanostructures is very timely, currently showing both experimental and theoretical progress. Also, this work provides a clear starting point for follow up work by complementary theoretical methods to further investigate the proposed backscattering effects in multimode attractive quantum wires, e.g., taking into account more experimental details (expectation 3).

I am therefore inclined to recommend the paper for publication in SciPost Physics, provided the authors make some improvements and answer some detailed questions that came up. Since the paper is written in a constructive style, nicely piecing together together a plausible model based on experimental clues, some assumptions / approximations that were not clear to me need to be clarified. Also, simple improvements can be made to make the paper more accessible to readers who are not experts in the theory of interacting quantum wires (like myself).
See the requested list of changes for all of these points as well as some suggestions that the authors may optionally consider.

I particularly suggest the authors to consider making the title of the paper more specific to its contents, for example,
"Backscattering and Spatially Modulated attractive interaction in transport through Electron Waveguides"
or something better. In particular, I find putting "unusual" or "interesting" in the title not very informative. In fact, I find the transport not so unusual or unexpected (some spectral feature just affects transport...). The more surprising insight to me seems to be the "fate" of the backscattering feature in the spectrum in the presence of modulation and interactions.
I also suggest the abstract and summarizing part of the summary are improved. For example, in the abstract, the line
"The second peculiar observation is a sharp conductance dip below a conductance of $1 \times e^2 /h$"
aims to summarize the lateral case (Section IV) as showing a "dip". I found this quite confusing since Fig. 4 and Fig. 2 are qualitatively very similar. I'd say the vertical case also shows a dip (but for very different reasons it seems).

p. 1
"In the laterally modulated waveguides we detect a different anomaly in the transport data,"
The phrasing suggest the authors "detected" something the experimentalist did not notice. If so, it should be made more clear, if not then please rephrase.

"We show that the interplay of strong interactions in the waveguide and short length of the conductor results in the “travelling” nature of this dip."
It is unclear what "traveling" means at this point: traveling as as function of what?

p. 2
"The spin label of these two modes and their spatial distribution in the cross-section of the waveguide is immaterial for the purposes of this work."
Can the authors motivate this? It is not apriori obvious when spin is unimportant in interacting systems.

The definition of the (total) Hamiltonian density is missing,
$\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_\text{int} + \mathcal{H}_\lambda$ where in $\mathcal{H}_\text{int}$ the $g_\text{bs}$ term dropped. These and other definitions are important to follow the paper since terms are being added and omitted as the analysis progresses.

p. 3
It would be helpful if the section title read something like
$\textit{B. Fractional conductance for strong backscattering}$
since backscattering is treated only perturbatively in the section that follows.

p. 4
The details of the RG treatement are omitted (which is OK). It may be useful for the general reader to mention that here short-distance spatial degrees of freedom are integrated out and mention the role of alpha. A pertinent reference to a review of this particular RG technique would be welcome.

"We make a plausible simplification, by assuming all other intra-mode interaction matrix elements are of comparable strength,..."
If this is indeed plausible, then please provide an argument. If not, then write "For simplicity, we assume" as is done elsewhere in the manuscript (which is OK).

"In the vicinity of $K_g^∗ = 1/5$ , which suggests the presence rather strong interactions, we may supplement ..."
The value $K_g^∗ = 1/5$ simply corresponds to a point of attractive interactions which happens to be solvable, right? I don't understand what "which suggests the presence of ..". Do you mean "which corresponds to.."?

"The point .. represents a generalization of the exactly solvable Luther-Emery point [28] of attractive spin-degenerate electrons in a quantum wire."
I'm confused here that $K_g* < 1/5<1$ is said to correspond to attraction, whereas after Eq. (34), $K > 1$ corresponds to attraction. Please clarify at both positions.

The transmission function Eq. (19) is written down without comment, but later on, the transmission in Eq. (33) is introduced as "the approximated transmission function". It is not clear which approximation was made at Eq. (33) and it makes me wonder whether in Eq. (19) also some approximations was made? Please clarify at both positions.

I found the title of section II D "Experimental consequences" not very appropriate since it just continues the theoretical analysis. It is also unnecessary since such consequences are discussed throughout the paper (which is good).

"In order to more reliably relate our calculations to the experimental observation,"
Please state explicitly what was not "reliable" so far. (This confused me since it cannot be treatment of interactions since $K_g^*=1/5$ was just discussed a few lines back).

Eq. (19): the quantities b_1, b_2 are not defined in terms of model parameters or explained.

Fig. 2: lower value of the vertical scale is missing.

p. 5
"with $W$ a typical bandwidth parameter."
The bandwidth was not introduced when discussing the model. Does $W$ relate to short distance scales integrated out by the RG? This seems to be the case as later on after Eq. (36) it says "once again W=\tilde{v}/alpha" where the "again" confuses me since this was not mentioned earlier.

"when the bandwidth of the two modes is sufficiently larger than the strength of the interactions"
Bandwidth meant is $W$? Which interactions, the $U^ij$ or $g$'s (I noted that the $g_i$ are differences of local interactions). Please specify.

Eq. (21): A comment on how one arrives at the alpha-offset in Eq. (21) would be helpful. I can imagine, but I'd like to understand how.

p. 6
Although the regime of $m V_z \gg m \mu + k_\text{SO}^2/2$ is considered, the expression for $k_F$ is not expanded to consistently to reflect this approximation:
$k_F \approx \sqrt(2mV_Z) + (m\mu + k_\text{SO}^2/2)/\sqrt(2mV_Z) +...$.
Why not? The later analysis should not depend on (not) doing this.

I'm confused about the need to include the chemical potential $\mu$ in the (grand-)canonical Hamiltonian density (27). Why is $\mu$ not included in $\mathcal{H}_0$ in Section II ?

"...thus we can use Landauer’s two-terminal conductance formula in Eq. (18) to calculate the conductance"
This seems to be correct only when changing variable to $\epsilon'=\epsilon-\mu$ since now $\mu$ is included in H? It is unclear which $\epsilon$ is meant in Eq. (33).
Related to this: How is $\epsilon$ in Eq. (B2) defined in Appendix B?

As mentioned earlier, why is Eq. (33) "approximate" ? Please clarify.

In this discussion of Section III it seems to me that $\mu$ is included in $H$ and then apparently removed again after Eq. (31). In my naive understanding, the transmission in Landauer's formula does not depend on $\mu$. $\mu$ comes from the linear-response fermi-function difference [cosh(..)]^{-2} of the adiabatically connected leads. Including $\mu$ in the Hamiltonian and removing it implicitly seems to confuse the matter in relation to Section II. Please clarify.

p. 7
"we neglect the effect of different Fermi velocities in different regions of the system."
Can one motivate this? Is it known (in other contexts) what effects such differences can cause when they are included? If there is no such motivation, write "for simplicity we neglect..." to make clear there is a potentially relevant effect to consider in further work.

Before Eq. (36): $\Delta^0_Q$ is not defined / mentioned.

p. 10
Eq. (B2): What is the definition of epsilon here?

"from which we recover Eq. (32) for $\epsilon=\tilde{\mu}=0$."
As stated this makes no sense (negative argument square root). In Eq. (B3), why not give the solution in terms of cosh and sinh, i.e., for parameter values in the gap ($|\epsilon + \tilde \mu| \leq \Delta_Q$) which are of interest?


----- TYPOS SPOTTED-----------------------
replace faith -> fate throughout the text.

p. 3
Typo:
current ... in both modes may be expressed as $J = (1, 1) · (I_R − O_l ).$
replice $O_l$ -> $O_L$

p. 4
pre-plateua” conductance -> pre-platEAU” conductance

p. 5:
for readability insert:
"in the experiment THE VALUE OF q ∗ around which the interaction is peaked, remains constant.}
The formula for $K_f$ needs to be "displayed" format, not as inline equation.
interaction tend to male K g small
interaction tend to MAKE K g small

p. 6:
insert missing word:
We examine the Hamiltonian DENSITY $\mathcal{H}_0 + \mathcal{H}_int$ [Eqs. (2) and (3)]

p. 7
develops to a pronounce dip -> develops to a PRONOUNCED dip
the plateuas observed in Ref. [21] -> the plateAUS observed in Ref. [21]

1. Relevant -explains intriguing features seen in a recent experiment
2. Interesting - suggests that fractional conductance quantization can arise in 1D systems with attractive interactions
3. Carefully presented - it is easy to read, with the authors providing physical insight for technical results

I miss a discussion of some specific theory's predictions that could confirm or dismiss the proposed scenario.

The article is a theoretical investigation of a one-dimensional two-subband conductor within a Luttinget liquid formalism. It is motivated by conductance plateaus and dips at fractional values observed in experiments. It explains these as due to the electron-electron interaction strength, and spin-orbit interaction strength, respectively, being highly oscillatory (in space).

I find the article interesting, of high scientific level, and well presented, and I recommend the publication. I suggest that the authors discuss two additional questions: a possible verification of the proposed explanation and magnetic field effects. Finally, I have some minor suggestions concerning the presentation.

Suggestions for further discussion:
1) The analysis of this type (the RG) relies on examining the system parameters tendencies and has to remain rather qualitative. Here, the interaction constants Kg of Kf are functions of several matrix elements of U, which are unknown. Authors find that their scenario requires that U_2kF dominates other U's in Eq. 17 but is only "relatively dominant" (meaning negligible if divided by 5) in the equation for Kf on page 5. While this is well believable, I am wondering whether the authors' explanation can be made more verifiable.
Therefore: Is there a prediction that could be checked experimentally to confirm or dismiss this specific explanation?

2) In the first part, the effects of the magnetic field are ignored, both the Zeeman energy and the orbital effects. In the second part, it is the opposite, and the wire is supposed to be fully spin-polarized. Not easy to reconcile these two views. Especially the first part, where plateaus seen in the experiment at 3-7 Tesla and even 9 Tesla are discussed, but the Zeeman energy is not considered.
Therefore: What is the Zeeman energy at these fields, e.g., compared to the gap of 2-20 ueV used in Figure 2? Why can it be neglected in the first part?

To the presentation:
I had problems with the backscattering term in Eq. 3. First, there seems to be a typo in Eq. 3 as this term transfers electron across subbands 1 and 2, unlike Eq. 1. One could figure out which term was meant if the "k_1/2,F" was defined, but I can't find its definition. Finally, the last sentence on page 2 says that the term does not conserve the momentum. But I understood that U is modulated in space and therefore, it should have an additional Fourier space index, which could compensate any momentum mismatch. Please clarify the "modulation" of the interaction U(x,y) better.

"the faith of ... some variable" should read "the fate of ... some variable" (appears twice)
"to male" should read "to make"
"Rasbha" should read "Rashba"

See report.