# Disorder effects on the so-called Andreev band in Majorana nanowires

Sankar Das Sarma  Condensed Matter Theory Center and Joint Quantum Institute,
Department of Physics, University of Maryland, College Park, Maryland 20742,
USA  Haining Pan  Department of Physics, Cornell University, Ithaca, NY 14850,
USA

###### Abstract

We comment on a recent publication Phys. Rev. Lett. 130, 207001 (2023),
pointing out that the periodic model for the superconducting gap and/or the
spin splitting used by the authors is artificial and does not apply to any
real systems. In addition, we show that the resulting Andreev band introduced
by this artificial and unrealistic periodicity is suppressed by the potential
disorder invariably present in all experimental systems. The results of this
model are therefore contrived and do not apply to any experimental system.

We point out that the effects of so-called Andreev band physics in Majorana
nanowires proposed in [1] are suppressed by realistic disorder. Reference 1
introduces an Andreev band into the bulk nanowire by putting in an artificial
periodic (Fig. 1 in [1]) array of superconducting gap and/or Zeeman splitting
throughout the nanowire and by assuming vanishing spin-orbit coupling, causing
a seeming closing/opening of a gap-like structure in the nonlocal tunneling
conductance measured across the wire ends. Additionally, Ref. 1 also manually
introduces another feature in the local tunneling, leading to a zero-bias
conductance peak (ZBCP) in the tunneling spectroscopy. These two features
appear incompatible with the methodology of using local and nonlocal tunneling
spectroscopies together [2, 3] to ascertain topological Majorana zero modes in
nanowires. It is well-established that Majorana nanowires are dominated by
random unintentional disorder [4, 5, 6], and the artificial finetuned Andreev
band results presented in [1] are suppressed by disorder. We show our
calculated representative results in Fig. 1, depicting Andreev band results
for a periodic array of varying $g$-factor in the bulk nanowire (following
[1]) with potential disorder (and quantum dots at ends in order to produce
ZBCPs [7]). We note that the artificial features of the periodic pristine
system in [1] disappear under realistic potential disorder. The periodic model
of Ref. 1 with zero spin-orbit coupling and without realistic disorder is
unjustified as a realistic approximation, and therefore, it cannot be used to
interpret experimental results. We conclude by pointing out that the realistic
disorder situation and its effects on Majorana nanowire experiments have
already been extensively discussed in the literature [3, 4, 5, 6, 7], and Ref.
1 does not add any substantive physics to the elucidation of disorder effects
in Majorana nanowires through its artificial periodic Andreev band model.

Figure 1:  Left (right) two panels show calculated local (nonlocal)
conductance from both ends in wires with bulk trivial Andreev bands in the
pristine system (as in [1]) by a finetuned periodic array of varying
$g$-factor. (a-d) pristine results showing physics in Ref. 1 with ZBCPs in
local tunneling from both ends and gap closing/opening-like features in
nonlocal conductance. (e-h)/(i-l): results with a small (larger) chemical
potential disorder of 0.5 (1) meV. Features in [1] are suppressed in (k-l).
All features are nontopological, and ZBCPs in local tunneling stem from
manually inserted quantum dots at wire ends (Ref. 1). The 3-$\mu$m wire has a
bulk chemical potential of 1 meV and 0.5 meV at both ends (dot length scale
$0.3~{}\mu$m) with random chemical potential disorder. The $g$-factor profile
comprises 5 periodic peaks of $g^{*}=1$, dividing the wire into disjointed
parts spaced by $0.4~{}\mu$m, and zero elsewhere except ends set to 1. The
spin-orbit coupling is 0, and the induced superconductivity is 0.2 meV.

## References

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