SciPost logo

SciPost Submission Page

A proposal to extract and enhance four-Majorana interactions in hybrid nanowires

by Tasnum Reza, Sergey M. Frolov, David Pekker

Submission summary

As Contributors: Sergey Frolov
Arxiv Link: (pdf)
Code repository:
Date submitted: 2021-11-01 13:31
Submitted by: Frolov, Sergey
Submitted to: SciPost Physics
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


We simulate the smallest building block of the Sachdev-Ye-Kitaev (SYK) model, a system of four interacting Majorana modes. We propose a 1D Kitaev chain that has been split into three segments, i.e., two topological segments separated by a non-topological segment in the middle, hosting four Majorana Zero Modes at the ends of the topological segments. We add a non-local interaction term to this Hamiltonian which produces both bilinear (two-body) interactions and a quartic (four-body) interaction between the Majorana modes. We further tune the parameters in the Hamiltonian to reach the regime with a finite quartic interaction strength and close to zero bilinear interaction strength, as required by the SYK model. To achieve this, we map the Hamiltonian from Majorana basis to a complex fermion basis, and extract the interaction strengths using a method of characterization of low-lying energy levels and then finding the differences in energies between odd and even parity levels. We show that the interaction strengths can be tuned using two methods - (i) an approximate method of tuning overlapping Majorana wave functions (without non-local interactions) to a zero energy point followed by addition of a non-local interaction, and (ii) a direct parameter space optimization method using a genetic algorithm. We propose that this model could be further extended to more Majorana modes, and show a 6-Majorana model as an example. Since eigenspectral characterization of one-dimensional nanowire devices can be done via tunneling spectroscopy in quantum transport measurements, this study could be performed in experiment.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 3 on 2021-12-21 (Invited Report)


I find that this manuscript is not appropriate for the publication in SciPost. I also doubt that in its present form it may be resubmitted to any other scientific journal.

The idea of authors was to propose a mesoscopic realization of the SYK model. The latter requires a large number N>>1 of Majorana fermions with all-to-all random interaction. (In fact the requirement N>>1 is needed to show the equivalence of SYK low energy physics to the the so-called Schwarzian quantum mechanics, which in turn implies the correspondence to the JT gravity, black-hole physics, many-body chaos and etc.).

The cases discussed by authors are restricted to N=4 and N=6. And from the very idea (splitting of a 1D spin-orbit quantum wire into trivial and topological segments), I do not see a room to scalability of this proposal for larger N. So, in fact, the manuscript discusses (almost) regular spectrum of just few subgap states in the superconducting mesoscopic wire. Increasing the wire length (to get more Majoranas) would also require the usage of very long-ranged Coulomb interaction which unevitibly will be screened by nearby gates which should control the electron concentration in different segments.

  • validity: poor
  • significance: poor
  • originality: low
  • clarity: high
  • formatting: good
  • grammar: good

Anonymous Report 2 on 2021-12-8 (Invited Report)


The Authors characterize the four-Majorana interaction in a system hosting a few Majorana zero modes by its energy spectrum.
The central result of the paper is that for a one-dimensional Kitaev chain hosting four Majoranas in the presence of all-to-all density-density interaction, one can suppress the bilinear terms and maximize the four-Majorana interaction in the low-energy theory. To do so, the Authors systematically tune the original Hamiltonian using the relation between the coupling constants in the low-energy theory to the four lowest energy levels.

The goal is to provide a minimal building block for the experimental realization of the Sachdev-Ye-Kitaev model - a system with a four-Majorana interaction with suppressed bilinear terms. The Authors show a way to eliminate the bilinear contribution in a few-Majorana system.

The paper is very well structured and written clearly.
However, I think some issues would need to be addressed before suggesting the paper for publication:

1) What is the origin of the non-local interaction term (12)? Does it appear as a low-energy projection of density-density interaction with the Coulomb potential?
Are there any conditions for the nanowire parameters to provide this type of long-range interaction?

2) The SYK model describes the collective behavior of large-$N_\gamma$ randomly interacting zero modes. Providing that the interaction (12) can be made non-local for $N_\gamma>4$, is there a way to make the resulting four-Majorana interaction random in the suggested setup?

3) According to Fig. 8a, the amplitudes of the two-body interaction for $N_\gamma = 6$ are equal $u_1=u_2=u_3$. Does it imply equal $J_{ijkl}$ in the Hamiltonian (13)? If yes, can it be violated by adding disorder?

Apart from that, I would like to ask for two brief clarifications:

- The energy characteristic curves Figs. 4-6, 8,9 are Energy vs $U$, where $U$ is the strength of the original two-body interaction (12). However, in Section VII the Authors mention that $U$ is the hardest to tune among the parameters of the problem and, hence, the hardest to variate in the experiment. I am wondering if similar energy characteristic curves can be potentially done for a different parameter along the x-axis?

- In the second paragraph of Section V.B. the Authors write: "Our optimization problem now consists of a six parameter space $\lbrace\mu_{t1}, \mu_{t2}, t, \Delta, U\rbrace$ ..." Though, there are only five parameters listed. Is there an extra parameter?

In summary, the paper demonstrates that it is possible to engineer a few-MZM system with four-Majorana interaction and suppressed bilinear terms in one dimension. It definitely meets SciPost's general acceptance criteria.
However, it is not clear to me yet if the results can be extended to a low $N_\gamma$ realization of the SYK model.
As I mentioned above in (1-3), It seems to me that the paper in the current version requires some additions. Namely, adding the discussion on the applicability of the long-range interaction Hamiltonian (12) and a possible mechanism to make the resulting four-Majorana interaction random for $N_\gamma>4$. If these issues can be resolved, then the paper would check off "3. Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work" among Scipost expectations as a novel proposal for realizing the SYK model in one dimension.

Requested changes

1. Adding the discussion on the applicability of the all-to-all density-density Hamiltonian (12).

2. Adding a condition that would allow for random four-Majorana terms at the SYK point.

  • validity: high
  • significance: ok
  • originality: ok
  • clarity: top
  • formatting: perfect
  • grammar: perfect

Anonymous Report 1 on 2021-11-26 (Invited Report)


1- Clearly written
2-Physically sound


1-Limited applicability


In this paper, the authors study a system of four Majorana modes, which can be written as two complex fermions. Their goal is to find a way to build the SYK model using these four-Majorana-systems as building blocks. By creating a model of an interacting Kitaev chain and including interactions, they show that by tuning several parameters they are able to minimize the bilinear terms in the low-energy Hamiltonian, a requirement for the SYK model. They demonstrate their procedure with numerics for a 4-Majorana and 6-Majorana model.

This paper meets all of the general acceptance criteria: it is clearly written and appropriately references and cites its sources. However, it must also meet one of SciPost's expectations, which I feel at the moment it currently does not. The paper hopes its calculations may be applied to simulate the SYK model, which would be an appropriate criterion for publication, but there are several important considerations that need to be taken into account for this to happen:

First, the paper only considers 4 and 6 Majoranas, or 2 and 3 complex fermions. As the number of fermion grows, the Hilbert space will grow exponentially (and number of bilinears quadratically), while the number of parameters available to tune will grow linearly. I suspect that this minimization procedure will not work as N grows large, which is the regime of SYK; there are far too many bilinears to minimize.

Second, the SYK model not only requires the absence of bilinear terms, but also a random four-body interaction between all Majorana modes. It is insufficient to prove that the four-body term is nonzero. As interactions typically decay with distance, it is difficult to see how such a term will arise in the Kitaev chain suggested in the paper.

Third, the parameters tuned in the model include the pairing strength and the interaction. How are these tuned in actual physical systems? In addition, the authors mention that tunneling spectroscopy will be able to determine the energy parameters in Eq. 5, but how does one determine which energy corresponds to which state E_{a,b,c,...}, especially in the presence of superconductivity, where total fermion number is not a good quantum number? The authors show this is possible for 2 complex fermions, with only knowledge of the parity of the states to determine the parameters \lambda and u. But is this true for higher numbers of fermions?

In summary, the paper is well-written and clear. However, in order to have the necessary relevance, it needs to show the 4-Majorana analysis can be extended to the SYK model (with large numbers of Majoranas). If the appropriate points above are addressed, I would be happy to recommend publication.

Requested changes

I address specific points by section.
1-From an experimental spectrum of the energy eigenstates, how does one determine which occupations a,b,c,... in E_{a,b,c...} a certain energy level corresponds to, for more than 2 complex fermions?
2-How physical is the interaction term in Eq. 12? It does not decay with distance, as most interactions should. By not decaying with distance, it potentially can generate the all-to-all interactions required for SYK, but such interactions should not occur in nature.
3-How does one tune \mu, \Delta, and U in experiment? In general, are there enough parameters to tune in an experimental model to reach the SYK point?
4-Is the analysis for the 4-Majorana model extensible to N-Majorana models, where N is very large? Can all bilinears be suppressed by tuning appropriate terms, and can the four-body interaction generated be all-to-all and random?

  • validity: high
  • significance: low
  • originality: ok
  • clarity: top
  • formatting: good
  • grammar: perfect

Login to report or comment