# Realizing Majorana Kramers pairs in two-channel InAs-Al nanowires with
highly misaligned electric fields

Benjamin D. Woods  Mark Friesen  Department of Physics, University of
Wisconsin-Madison, Madison, WI 53706, USA

###### Abstract

Common proposals for realizing topological superconductivity and Majorana zero
modes in semiconductor-superconductor hybrids require large magnetic fields,
which paradoxically suppress the superconducting gap of the parent
superconductor. Although two-channel schemes have been proposed as a way to
eliminate magnetic fields, geometric constraints make their implementation
challenging, since the channels should be immersed in nearly antiparallel
electric fields. Here, we propose an experimentally favorable scheme for
realizing field-free topological superconductivity, in two-channel InAs-Al
nanowires, that overcomes such growth constraints. Crucially, we show that
antiparallel fields are not required, if the channels are energetically
detuned. We compute topological phase diagrams for realistically modeled
nanowires, finding a broad range of parameters that could potentially harbor
Majorana zero modes. This work, therefore, solves a major technical challenge
and opens the door to near-term experiments.

##  I Introduction

Majorana zero modes (MZMs) are zero-energy modes localized at the ends of
topological superconductors and a potential building block for topological
qubits [1, 2, 3, 4]. Over the past decade, there has been considerable
progress in realizing MZMs in semiconductor-superconductor hybrids [5, 6, 7,
8, 9, 10, 11, 12], primarily in the context of the Lutchyn-Oreg model [13, 14]
which requires applying a large magnetic field to the hybrid system. The
application of magnetic fields is worrisome, however, since it suppresses the
superconductivity [15, 16]. This has motivated research into topological
superconductivity in the absence of magnetic fields, so-called time-reversal
invariant topological superconductivity (TRITSC) [17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], where the MZMs on either end
of a wire occur in pairs called Majorana Kramers pairs (MKP), due to the
preservation of time-reversal symmetry. Of particular interest are proposals
that rely on two channels with approximately opposite spin-orbit coupling
vectors, such as Rashba bilayers [18, 19, 20] and two-channel nanowires [24,
25, 26, 27, 28], where the system is essentially composed of two copies of the
Lutchyn-Oreg model with the spin degree of freedom replaced by the channel
degree of freedom.

The planar geometry of Rashba bilayers naturally provides antiparallel spin-
orbit vectors in the two channels, as illustrated in Fig. 1(a), because the
spin-orbit vectors align with the local electric field. Creating such
structures in semiconductor-superconductor hybrids is challenging, however,
because it requires growing a high-quality semiconducting crystal layer
between two superconductors, which has yet to be demonstrated. In contrast,
epitaxial growth of superconductors on semiconductor nanowires has already
been accomplished [36]. In this case, however, it is difficult to engineer
antiparallel spin-orbit vectors in the two channels. This is a serious problem
because small misalignments of the spin-orbit vectors are expected to collapse
the topological gap [24], by effectively coupling the two copies of the
Lutchyn-Oreg model present in the wire.

Figure 1: Structures for realizing TRITSC. (a) Schematic cross section of a
proposed Rashba bilayer system, where two superconductors are coupled to two
tunnel-coupled channels, with opposite spin-orbit vectors:
$\bar{\boldsymbol{\alpha}}_{2}=-\bar{\boldsymbol{\alpha}}_{1}$. Here, the
spin-orbit vectors align to the local electric field produced by charge inside
of the channels. (b) Schematic cross section of the InAs-Al hybrid nanowire
system proposed here, where two channels (blue ovals) with highly misaligned
spin-orbit vectors, $\bar{\boldsymbol{\alpha}}_{L}$ and
$\bar{\boldsymbol{\alpha}}_{R}$, form near the InAs-Al interfaces. Various
parameters appearing in the low-energy effective Hamiltonian, Eq. (4), are
labeled, including the tunnel coupling $t$, energy detuning $g$, spin-orbit
misalignment angle $\theta$, and superconductive pairing $\pm\Delta_{0}$. (c)
Zoomed out view (not drawn to scale) of the system in (b), showing the gates
(gold) used to control the electrostatic environment. The potential $-e\phi$
calculated for the gate voltages
$(V_{T},V_{B},V_{L},V_{R})=(-5.5,-3,0,0)~{}\text{V}$ and the InAs-Al interface
boundary condition $\phi=0.3~{}\text{V}$ is plotted in the cross section of
the nanowire.

In this work, we turn the problem on its head by showing that TRITSC can be
realized in experimentally feasible [37, 38] two-channel hybrid nanowires
where the spin-orbit vectors of the two channels are highly misaligned, as
depicted in Fig. 1(b). The key to our proposal is to introduce an energy
detuning between the two channels, on order of the superconducting gap.
Essentially, the detuning separates the two copies of the Lutchyn-Oreg model
in momentum space, which significantly reduces the coupling induced by the
spin-orbit vector misalignment, allowing the topological phase to persist.
Since the success of our proposal hinges on the degree of spin-orbit vector
misalignment and other device parameters, we consider a more rigorous device
model than the Hamiltonian-level models typically used in the literature,
incorporating both geometric and electrostatic details. This ensures that our
conclusions do not result from inaccurate estimates of the model parameters.

##  II Model

We consider the system shown in Fig. 1(c) in which a semiconducting InAs
nanowire with a triangular cross section is coupled to two Al superconductors
that cover half of the upper facets. The system is translationally invariant
along the $\hat{z}$ axis. Similar to previous TRITSC proposals [24, 25, 26,
27], an external flux loop [39] imposes an order parameter phase difference of
$\pi$ between the two superconductors. In addition, gates surround the
nanowire to control the electrostatics. The system is modeled by a Bogoliubov-
de-Gennes (BdG) Hamiltonian of the canonical form [40, 41],

$H_{\text{BdG}}(k_{z})=\begin{pmatrix}H_{N}(k_{z})&\widetilde{\Delta}\\\
-\widetilde{\Delta}^{*}&-H_{N}^{*}(-k_{z})\end{pmatrix},$ (1)

where $H_{N}$ and $\widetilde{\Delta}$ are the normal and superconductive
pairing components, respectively, $k_{z}$ is the momentum along the $\hat{z}$
axis, and the first and second columns act upon particle and hole degrees of
freedom, respectively. The normal Hamiltonian takes the form
$H_{N}=H_{0}+H_{\text{SO}}$, where $H_{0}$ is an effective mass Hamiltonian
given by

$H_{0}(k_{z})=\frac{\hbar^{2}}{2m^{*}}\left(-\partial_{x}^{2}-\partial_{y}^{2}+k_{z}^{2}\right)-e\phi(x,y).$
(2)

Here, $m^{*}$ is the effective mass and $\phi$ is the electrostatic potential
that satisfies Poisson’s equation,
$\nabla\cdot\left[\epsilon(\mathbf{r})\nabla\phi(\mathbf{r})\right]=-\rho(\mathbf{r})$,
where $\epsilon$ is a material-dependent dielectric constant and $\rho$ is the
free-charge density that is calculated self-consistently [42]. The potential
$\phi$ satisfies Dirichlet boundary conditions on the gates shown in Fig.
1(c). The surfaces of the Al superconductors are also set to $\phi=0$, except
for the InAs-Al interfaces, which are set to $\phi=0.3$ V to account for the
band offset between the two materials [43, 44, 45, 46]. The spin-orbit term
$H_{\text{SO}}$ is given by

$H_{\text{SO}}(k_{z})=\bigl{[}\widetilde{\alpha}_{x}(x,y)\sigma_{y}-\widetilde{\alpha}_{y}(x,y)\sigma_{x}\bigr{]}k_{z},$
(3)

where $\widetilde{\alpha}_{j}$ are Rashba fields and $\sigma_{j}$ are Pauli
spin matrices. We neglect Dresselhaus and transverse-Rashba spin-orbit
coupling contributions because they are weak, as explained in Appendix D
Importantly, the Rashba fields are position-dependent and related to the
potential by $\boldsymbol{\widetilde{\alpha}}(x,y)=eC\nabla\phi(x,y)$ [47,
48], where $C$ is a material-dependent constant. Finally, the superconductive
pairing is treated as an induced pairing within the InAs, with
$\widetilde{\Delta}=i\Delta(x,y)\sigma_{y}$. (See details below.)

###  II.1 Two-Channel Effective Hamiltonian

We derive the low-energy effective Hamiltonian of the system by considering
the electrostatic potential shown in Fig. 1(c), for typical gate voltages.
Here, electrons are attracted to the InAs-Al interfaces, primarily because of
the band offset between the materials, and we observe channels forming near
each InAs-Al interface, as illustrated in Fig. 1(b). We then project Eq. (1)
onto the two lowest-energy orbital subbands, $\varphi_{1}$ and $\varphi_{2}$,
defined by $H_{0}(0)\varphi_{n}(x,y)=\varepsilon_{n}\varphi_{n}(x,y)$, where
$\varphi_{1}$ and $\varphi_{2}$ can be expressed as superpositions of the left
and right channel wave functions, $\chi_{L}$ and $\chi_{R}$, depicted in Fig.
1(b): $\varphi_{n}=a_{n}\chi_{L}+b_{n}\chi_{R}$ for $n=1,2$. In this two-
channel basis, as shown in Appendix A, the effective BdG Hamiltonian becomes

$\begin{split}H_{\text{eff}}(k_{z})=&\left[\varepsilon(k_{z})\lambda_{0}+t\lambda_{x}+g\lambda_{z}\right]\sigma_{0}\tau_{z}-\Delta_{0}\sigma_{y}\lambda_{z}\tau_{y}\\\
&+\alpha
k_{z}\left(\cos\theta\sigma_{y}\lambda_{z}\tau_{z}+\sin\theta\sigma_{x}\lambda_{0}\tau_{0}\right),\end{split}$
(4)

where $\sigma_{j}$, $\lambda_{j}$, and $\tau_{j}$ are Pauli matrices acting on
spin, channel, and particle-hole space, respectively, and
$\varepsilon(k_{z})=\hbar^{2}k_{z}^{2}/(2m^{*})-\mu$ is the bare dispersion.
By defining $h_{ij}=\matrixelement{\chi_{i}}{H_{0}(0)}{\chi_{j}}$, we can
express the chemical potential as $\mu=(h_{LL}\\!+\\!h_{RR})/2$, the
interchannel tunnel coupling as $t=h_{RL}$, and the channel energy detuning as
$g=(h_{LL}\\!-\\!h_{RR})/2$. We note that these parameters all depend on the
local electrostatics. For example, $t$ characterizes the localization of the
channels near the InAs-Al interfaces, while $g$ is largely determined by the
voltage bias $V_{L}-V_{R}$ between the side gates, shown in Fig. 1(c). To
understand the parameters $\alpha$ and $\theta$ in Eq. (4), it is helpful to
define the spin-orbit vector of channel $i$ as
$\bar{\boldsymbol{\alpha}}_{i}=\int|\chi_{i}(x,y)|^{2}\boldsymbol{\widetilde{\alpha}}(x,y)\,dxdy$,
which averages the electric field over the channel wave function. The
parameters $\alpha$ and $\theta$ appearing in Eq. (4) represent the magnitude
and direction of the spin-orbit vectors, as illustrated in Fig. 1(b).
Crucially, the sample geometry ensures that $\bar{\boldsymbol{\alpha}}_{L}$
and $\bar{\boldsymbol{\alpha}}_{R}$ are generically misaligned. Additionally,
the symmetry of the nanowire cross section imparts approximate mirror symmetry
to the two spin-orbit vectors, such that
$\bar{\boldsymbol{\alpha}}_{L}\cdot\hat{x}\approx-\bar{\boldsymbol{\alpha}}_{R}\cdot\hat{x}$
and
$\bar{\boldsymbol{\alpha}}_{L}\cdot\hat{y}\approx\bar{\boldsymbol{\alpha}}_{R}\cdot\hat{y}$.
This is reflected in the $\lambda_{z}$ and $\lambda_{0}$ factors in the second
line of Eq. (4). The angle $\theta$ characterizes the misalignment between the
spin-orbit vectors, and how far the system is from the ideal limit
($\theta=0$), typically assumed in the literature [18, 19, 20, 25, 26, 27,
28]. In a real device, the spin-orbit vectors will not be perfectly mirror
symmetric, due to the voltage bias $V_{L}-V_{R}$ and device imperfections. We
neglect such symmetry-breaking terms here for simplicity, and because they do
not significantly affect the physics, as shown in Appendix A. Finally, we note
that the superconductive pairing for channel $i$ could also be computed as
$\Delta_{i}=\int|\chi_{i}(x,y)|^{2}\Delta(x,y)\,dxdy$, in principle. However,
a rigorous calculation of $\Delta(x,y)$ involves complicating factors, such as
details about disorder and the InAs-Al interface [49], which are beyond the
scope of this work. Here, we simply define
($\Delta_{L},\Delta_{R})=(\Delta_{0},-\Delta_{0}$), which captures the $\pi$
phase difference between the superconducting order parameters, and we adopt a
typical value of $\Delta_{0}=0.2~{}\text{meV}$ for such InAs-Al hybrids [9].

##  III Topological Phase Diagrams

Before focusing on the device shown in Fig. 1(c), we first explore how the
channel degree of freedom enables the realization of TRITSC. Additionally, we
show how misalignment of the spin-orbit vector ($\theta\neq 0$) is detrimental
to the topological phase and how introducing a channel detuning ($g\neq 0$)
can alleviate this issue. For now, the parameters $\mu$, $t$, $g$, and
$\theta$ in Eq. (4) are treated as free parameters, to elucidate the physics
of the effective model. However, they are calculated more realistically, later
in the paper. Other parameters used here are $m^{*}=0.026m_{0}$ and
$\alpha=20~{}\text{meV}\cdot\text{nm}$ [9].

Figure 2: Topological phase diagrams in the absence of channel detuning. (a)
Topological phase diagram for antiparallel spin-orbit vectors ($\theta=0$) and
no channel detuning ($g=0$). The colormap shows the computed topological gap
$\Delta_{\text{top}}$ for the topological phase. (b) Bulk energy spectrum for
parameters corresponding to the green star in (a). All bands are two-fold
degenerate because Eq. (5) decomposes into two identical spin blocks when
$\theta\\!=g\\!=0$. The shaded energy gap corresponds to
$2\Delta_{\text{top}}$. (c) Phase diagram for the same system as (a), but with
spin-orbit misalignment $\theta=30^{\circ}$, yielding a large gapless region
(gray). (d) A typical gapless bulk spectrum, for parameters corresponding to
the red triangle in (c).

To begin, it is helpful to change the basis of the effective Hamiltonian, Eq.
(4), as explained in Appendix B, yielding

$\begin{split}H_{\text{eff}}^{\prime}(k_{z})=&[\varepsilon(k_{z})\lambda_{0}\tau_{z}+t\lambda_{x}\tau_{z}-\Delta_{0}\lambda_{y}\tau_{y}\\\
&+\alpha
k_{z}\cos\theta\lambda_{z}\tau_{0}]\sigma_{0}+g\sigma_{z}\lambda_{z}\tau_{0}\\\
&+\alpha k_{z}\sin\theta\sigma_{x}\lambda_{x}\tau_{0}.\end{split}$ (5)

We first consider the limit of antiparallel spin-orbit vectors ($\theta=0$),
as typically assumed in the literature [25, 26, 27, 28], with no channel
detuning ($g=0$). Equation (5) then decomposes into two identical spin blocks.
[This is the reason for the basis change in Eq. (5).] As noted previously
[24], each of the spin blocks corresponds to an independent Lutchyn-Oreg model
[13, 14] of a superconducting nanowire with spin-orbit coupling in an external
magnetic field, except that the spin degrees of freedom are now replaced by
channels $\sigma_{j}\rightarrow\lambda_{j}$, and the Zeeman energy is replaced
by the interchannel tunnel coupling $t$. In analogy with the Lutchyn-Oreg
model, the system transitions from a trivial phase without MZMs to a
topological phase with MZMs when $t^{2}>\mu^{2}+\Delta_{0}^{2}$. In contrast
to the Lutchyn-Oreg model, however, two MZMs (corresponding to one MKP) appear
at each end of the nanowire, due to the presence of dual spin blocks in the
TRITSC case. (Note that the degeneracy of the two pairs of MZMs at zero energy
is protected by time-reversal symmetry [17].) An example phase diagram is
shown in Fig. 2(a) for the case of $\theta\\!=g\\!=0$. (See Appendix C for a
discussion of our method for calculating phase diagrams, which involves
determining whether an MKP exists at the end of a semi-infinite wire.) Like
other MZM schemes, several device parameters (including the chemical
potential) must be tuned simultaneously, to enter the topological phase.
Indeed, the range of the chemical potentials yielding a topological phase in
Fig. 2(a) is the same range found for Majorana nanowires with broken time-
reversal symmetry [13, 14]. In Fig. 2(a), the colormap indicates the size of
the topological gap $\Delta_{\text{top}}$, defined by the red region in Fig.
2(b), where we plot a typical bulk energy spectrum. We note that this spectrum
is everywhere two-fold degenerate, due to the identical spin blocks.

Figure 3: Effect of channel detuning $g$ on topological phase diagrams. (a),
(b) Topological phase diagrams, as function of the chemical potential $\mu$
and spin-orbit misalignment $\theta$, for $t=0.5~{}\text{meV}$, and (a) $g=0$,
or (b) $g=\Delta_{0}$. Increasing $\theta$ in (a) suppresses the topological
gap $\Delta_{\text{top}}$, yielding a gapless phase (gray) when $\theta\gtrsim
22^{\circ}$. Introducing the channel detuning $g$ in (b) removes the gapless
phase and extends the topological region out to large $\theta$. (c), (d) Bulk
spectra for parameters corresponding to the green circle ($\theta=0$) and
green cross ($\theta=30^{\circ}$) in (b). In (c), the channel detuning $g$
lifts the spin degeneracy when $k_{z}\neq 0$, as shown with orange (purple)
for the spin $\uparrow$ ($\downarrow$) bands. Lifting the degeneracy reduces
the level repulsion between spins, caused by misaligned spin-orbit vectors,
and allows the topological phase to survive up to large $\theta$.

Next, we consider the case of misaligned spin-orbit vectors ($\theta\neq 0$),
although we still assume $g=0$. The term $\alpha
k_{z}\\!\sin\theta\sigma_{x}\lambda_{x}\tau_{0}$ in Eq. (5) now plays an
important role, coupling the two spin blocks. This has a dramatic effect on
the phase diagram, as shown in Fig. 2(c) for $\theta=30^{\circ}$. Here, the
topological phase boundary is unchanged at small $t$; however, the topological
gap is significantly reduced, disappearing all together for large enough $t$.
Importantly, the closing of the gap destroys the topological phase and
accompanying MZMs because a topological phase is only well-defined in the
presence of a bulk gap [50]. This explains why misaligned spin-orbit vectors
are considered dangerous for TRITSC [24]. A typical gapless bulk spectrum is
shown in Fig. 2(d). Here, the coupling between the spin blocks, due to spin-
orbit vector misalignment, causes the degenerate bands to split, when
$k_{z}\neq 0$.

The key result of this paper is that the topological gap can remain robust,
even for highly misaligned spin-orbit vectors, if we introduce an energy
detuning $g$ between the two channels. This is demonstrated in Figs. 3(a) and
3(b), where we report topological phase diagrams for different channel
detunings, as a function of the spin-orbit misalignment angle $\theta$ and
chemical potential $\mu$. For the case $g=0$ [Fig. 3(a)], we see that
increasing $\theta$ suppresses the topological gap, until the system becomes
gapless for $\theta\gtrsim 22^{\circ}$, as consistent with Fig. 2(c). In stark
contrast for $g=\Delta_{0}$ [Fig. 3(b)], there is no gapless phase, and the
topological phase remains intact for a range of chemical potentials that is
nearly constant for $\theta$ between 0 and $40^{\circ}$. In fact, the
topological phase extends nearly to the worst-case limit of
$\theta=90^{\circ}$, where the spin-orbit vectors of the two channels are
parallel. Detuning is therefore found to be crucial for maintaining a
topological phase in TRITSC when $\theta\neq 0$.

To understand the mechanism providing robustness to the topological gap, we
first consider the case of $\theta=0$ and $g\neq 0$. Solving for the gap-
closing condition, we obtain the topological criterion, $t^{2}>(\mu-
E_{g})^{2}+\Delta_{0}^{2}$, where $E_{g}=\hbar^{2}g^{2}/(2m^{*}\alpha^{2})$.
Interestingly, we find that the bulk gap does not close at $k_{z}=0$, as in
the Lutchyn-Oreg model, but rather at $k_{z}=\mp g/\alpha$. More importantly,
the topological criterion is the same as the $g=0$ case, except for an overall
shift to larger chemical potentials. Therefore, introducing the channel
detuning $g$ by itself does not alter the size of the topological region. We
then consider the spectrum for $\theta=0$ and $g=\Delta_{0}$, in the
topological regime [Fig. 3(c)]. Comparing this to the $g=0$ spectrum in Fig.
2(b), we see that the channel detuning $g$ breaks the two-fold spin degeneracy
when $k_{z}\neq 0$. This occurs because the $g\sigma_{z}\lambda_{z}\tau_{0}$
term in Eq. (5) renders the two spin blocks inequivalent. If we also misalign
the spin orbit vectors ($\theta\neq 0$), as shown in Fig. 3(d) for the case
$\theta=30^{\circ}$, we find that the topological gap is reduced, as expected
from Fig. 2. However, the topological gap is more robust, because the band
minima are separated in $k_{z}$ space [see Fig. 3(c)], effectively reducing
the level repulsion between the two spin blocks when $\theta\neq 0$. Finally,
we note that, to be effective, $g$ must be comparable to other energies in the
problem, which are of order of the induced gap $\Delta_{0}$.

##  IV Full Device Calculations

We now compute topological phase diagrams for the device shown in Fig. 1(c),
as well as the parameters $t$, $\theta$, and $\alpha$ appearing in the
effective Hamiltonian, Eq. (5). The results suggest that prospects for
realizing TRITSC in such a device are very promising, and validate the
conclusions of Figs. 2 and 3 in a realistic setting. For these calculations,
we adopt the geometric parameters $(L,d,h,w)=(65,50,90,100)~{}\text{nm}$ and
assume a 45∘ angle at the base of the nanowire, as consistent with recent
experiments [38]. The triangular cross section is found to be beneficial for
TRITSC by providing single-subband occupancy in the conduction band, while
maintaining a large interchannel tunnel coupling $t$, due to the large
surface-to-volume ratio. As before, the band offset at the InAs-Al interfaces
is set to $0.3~{}\text{V}$ [51], and other parameters are given by
$C=1.17~{}\text{nm\textsuperscript{2}}$ and
$(\epsilon_{\text{InAs}},\epsilon_{\text{Si}\text{O}_{2}})=(14.6,3.9)$ [52].
Our numerical procedure begins by self-consistently [42] solving the
Schrödinger-Poisson equations, treating the charge density in the Hartree
approximation. Here, the orbital subbands $\varphi_{n}$ of $H_{0}$ in Eq. (2)
are solved using finite-element methods [53] and the electrostatic potential
is solved with help from the FEniCS software library [54]. Note that we focus
on the very-low-density regime, where self-consistency has only a modest
impact on the results. We then use the solutions obtained for $\varphi_{n}$ to
calculate the matrix elements of $H_{\text{SO}}$ and $\widetilde{\Delta}$.
Projecting onto just a few ($n\lesssim 10$) low-energy subbands, we finally
calculate the local density of states near zero energy, for a semi-infinite
wire, as described in Appendices A and C. This procedure is repeated for a
range of system parameters to obtain a topological phase diagram.

Figure 4: Topological phase diagrams for full device. Phase diagrams are shown
for the device in Fig. 1(c), as functions of topgate voltage $V_{T}$ and
chemical potential $\mu$, for channel detuning values of (a) $g=0$, or (b)
$g=\Delta_{0}$. Similar to Figs. 3(a) and 3(b), we find a large gapless region
(gray) in (a) for $g=0$, which is nearly removed in (b) by introducing a
channel detuning $g=\Delta_{0}$. Inset: Normalized tunnel coupling
$t/\Delta_{0}$ (blue) and spin-orbit misalignment angle $\theta$ (red), which
characterize the effective Hamiltonian in Eq. (4), and are numerically
extracted as a function of topgate voltage $V_{T}$, for $\mu=g=0$. Note that
the width of the topological region, as a function of $\mu$, is largely
determined by $t$. Also note that $\theta\approx 35^{\circ}$ for all $V_{T}$,
indicating highly misaligned spin-orbit vectors. Parameters like $t$ and
$\theta$ depend strongly on the device geometry and electrostatics.

Figures 4(a) and 4(b) compare the topological phase diagrams obtained for the
different channel detuning values, (a) $g=0$ and (b) $g=\Delta_{0}$. Here, the
backgate and topgate provide full control over the chemical potential $\mu$,
where $\mu=0$ is defined as the Fermi energy lying halfway between the first
two subband energies, at $k_{z}=0$. The phase diagrams can be understood in
analogy to Figs. 2 and 3. In short, Fig. 4(a) shows a large gapless region,
due to the absence of channel detuning ($g=0$). However, this region almost
disappears in 4(b), when we include a channel detuning $g=\Delta_{0}$ by
applying a voltage bias $V_{L}-V_{R}$. The striking difference between Figs.
4(a) and 4(b) can be attributed to the channel detuning in the presence of
highly misaligned spin-orbit vectors. Here, the computed misalignment angle
$\theta$ is shown in the inset (red curve), and is found to range from
$32^{\circ}$ to $39^{\circ}$, remaining close to $45^{\circ}$ due to the
geometry. The topological gap in Fig. 4(b) is also large, indicating a robust
topological phase. Moreover, the phase diagram is quite broad with respect to
$\mu$, particularly for larger $V_{T}$ values, due to the exponential
dependence of the interchannel tunnel coupling $t$ on $V_{T}$, as shown in the
inset (blue curve). We note that a broad phase diagram is particularly
conducive for forming MZMs, since larger fluctuations in $\mu$ can be
tolerated along the length of the wire, without admitting localized Andreev
bound states [55, 56, 57, 58, 59, 60, 61, 62, 63]. Finally, we obtain a spin-
orbit magnitude of $\alpha\approx 25~{}\text{meV}\cdot\text{nm}$ for the range
of voltages shown in Fig. 4, a spin-orbit energy of
$E_{\text{SO}}=m^{*}\alpha^{2}/(2\hbar^{2})\approx 0.11~{}\text{meV}$, and a
spin-orbit length of $l_{\text{SO}}=\hbar^{2}/(m^{*}\alpha)\approx
115~{}\text{nm}$. This large spin-orbit coupling arises mainly from the
electric field induced by the band offset at the InAs-Al interfaces.

##  V Conclusion

Contrary to previous expectations [24], we have shown how TRITSC may be
realized in InAs-Al two-channel nanowires with highly misaligned spin-orbit
vectors. The key ingredient in our scheme is an energy detuning $g$ between
the two channels. By incorporating geometrical and electrostatic details into
a realistic device model, we have shown that MZMs can be realized, without
magnetic fields, using currently existing InAs-Al technology [37, 38].

We caution however, that the proposed scheme still requires a high degree of
uniformity along the length of the wire to avoid forming low-energy Andreev
bound states, similar to other MZM schemes. Moreover, fluctuations of the
interchannel tunnel coupling $t$ and the detuning parameter $g$ could also
affect the formation of Andreev bound states, in addition to more well-studied
fluctuations of the chemical potential $\mu$ coming from electrostatic
nonuniformities [55, 56, 57, 58, 59, 60], including smooth confinement
potentials at the edges of the wire [61, 62, 63]. Finally, we note that the
decoupling of the MKP in TRITSC relies upon time-reversal symmetry. Magnetic
impurities, such as Overhauser fields from nuclear spins, familiar to the
quantum dot community [64], could therefore be detrimental and will be
addressed in future work.

###### Acknowledgements.

Acknowledgments – We are grateful to M. A. Eriksson and T. D. Stanescu for
helpful discussions. This research was sponsored in part by the Army Research
Office (ARO) under Award No. W911NF-22-1-0090, and by the National Science
Foundation (NSF) through QLCI-HQAN (Award No. 2016136). The views,
conclusions, and recommendations contained in this document are those of the
authors and are not necessarily endorsed nor should they be interpreted as
representing the official policies, either expressed or implied, of the Army
Research Office (ARO) or the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for Government purposes
notwithstanding any copyright notation herein.

##  Appendix A Derivation of the two-channel effective Hamiltonian

In the main text, we stated that the effective Hamiltonian $H_{\text{eff}}$ in
Eq. (4) can be derived by projecting the BdG Hamiltonian $H_{BdG}$ in Eq. (1)
onto the two channel orbital wave functions $\chi_{L}$ and $\chi_{R}$, which
are superpositions of the two lowest-energy orbital subbands $\varphi_{1}$ and
$\varphi_{2}$ of the effective mass Hamiltonian $H_{0}$. In this section, we
provide details regarding this derivation. For clarity, some statements from
the main text will be repeated here.

To begin, we define an orbital subband $\varphi_{n}$ as the eigenstate of the
effective mass Hamiltonian $H_{0}$ for $k_{z}=0$, i.e.

$H_{0}(0)\varphi_{n}(x,y)=\varepsilon_{n}\varphi_{n}(x,y),$ (6)

where $\varepsilon_{n}$ is the subband energy and $n\in\mathbb{N}^{+}$. Note
that $\varphi_{n}(x,y)$ can be chosen to be real-valued, which we adopt. Also
note that the orbital subband $\varphi_{n}$ is only an orbital wave function,
which does not contain spin. This is well-defined because the effective mass
Hamiltonian $H_{0}$ is spin independent, i.e. $H_{0}$ is proportional to the
identity operator $\sigma_{0}$ in spin space. We can construct a subband wave
function with spin using a simple tensor product,
$\varphi_{n,\sigma}=\varphi_{n}(x,y)\ket{\sigma}$, where
$\sigma=\uparrow,\downarrow$. Furthermore, we can define a BdG subband wave
function that specifies both the spin and particle-hole degrees of freedom as
the tensor product,
$\varphi_{n,\sigma,\tau}=\varphi_{n}(x,y)\ket{\sigma\tau}$, where $\tau=p$ for
particle and $\tau=h$ for hole.

We now rewrite our BdG Hamiltonian using the basis consisting of the BdG
subband wave functions $\left\\{\varphi_{n,\sigma,\tau}\right\\}$. This leads
to the BdG Hamiltonian,

$H^{\prime}_{BdG}(k_{z})=\begin{pmatrix}H^{\prime}_{N}(k_{z})&\widetilde{\Delta}^{\prime}\\\
-\widetilde{\Delta}^{\prime*}&-H_{N}^{\prime*}(-k_{z})\end{pmatrix},$ (7)

where $H_{N}^{\prime}(k_{z})$ and $\widetilde{\Delta}^{\prime}$ are matrices
whose elements are found by evaluating the expressions,
$\matrixelement{\varphi_{m,\sigma}}{H_{N}(k_{z})}{\varphi_{n,\sigma^{\prime}}}$
and
$\matrixelement{\varphi_{m,\sigma}}{\widetilde{\Delta}}{\varphi_{n,\sigma^{\prime}}}$,
respectively. Performing these calculations, we find $H_{N}^{\prime}(k_{z})$
and $\widetilde{\Delta}^{\prime}$ can be expressed as

$\displaystyle H_{N}^{\prime}(k_{z})$
$\displaystyle=\bar{\Lambda}(k_{z})\sigma_{0}+\bar{\alpha}_{x}k_{z}\sigma_{y}-\bar{\alpha}_{y}k_{z}\sigma_{x},$
(8) $\displaystyle\widetilde{\Delta}^{\prime}$
$\displaystyle=i\bar{\Delta}\sigma_{y},$ (9)

where $\sigma_{j}$ are Pauli spin matrices. Here,
$\bar{\Lambda},\bar{\alpha}_{x}$, $\bar{\alpha}_{y}$, and $\bar{\Delta}$ are
real-symmetric matrices with elements,

$\displaystyle\bar{\Lambda}_{mn}(k_{z})$
$\displaystyle=\delta_{mn}\left(\varepsilon_{n}+\frac{\hbar^{2}k_{z}^{2}}{2m^{*}}\right),$
(10) $\displaystyle\left(\bar{\alpha}_{j}\right)_{mn}$
$\displaystyle=eC\int\varphi_{m}(x,y)(\partial_{j}\phi)\varphi_{n}(x,y)\,dxdy,$
(11) $\displaystyle\bar{\Delta}_{mn}$
$\displaystyle=\int\varphi_{m}(x,y)\Delta(x,y)\varphi_{n}(x,y)\,dxdy.$ (12)

Notice that $\bar{\Lambda}$ is diagonal in subband space since the kinetic
energy term $\hbar^{2}k_{z}^{2}/(2m^{*})$ in the effective mass Hamiltonian
$H_{0}(k_{z})$ has no spatial dependence. The spin-orbit $\bar{\alpha}_{j}$
and superconductive pairing $\bar{\Delta}$ matrices, in contrast, couple
different subbands because of the spatial inhomogeneity of $\phi(x,y)$ and
$\Delta(x,y)$. The full BdG Hamiltonian $H_{\text{BdG}}^{\prime}(k_{z})$ can
then be expressed as

$\begin{split}H_{\text{BdG}}^{\prime}(k_{z})=&\bar{\Lambda}(k_{z})\sigma_{0}\tau_{z}+\bar{\alpha}_{x}k_{z}\sigma_{y}\tau_{z}-\bar{\alpha}_{y}k_{z}\sigma_{x}\tau_{0}\\\
&-\bar{\Delta}\sigma_{y}\tau_{y},\end{split}$ (13)

where $\tau_{j}$ are Pauli matrices acting in particle-hole space.

The largest energy scale present in Eq. (13) is the energy separation between
most subbands, i.e. $\varepsilon_{m}-\varepsilon_{n}$ for $m\neq n$. In
comparison, the $\bar{\alpha}_{j}$ and $\bar{\Delta}$ terms that provide
inter-subband coupling are small perturbations. We can then arrive at a low-
energy effective Hamiltonian that captures all of the relevant low-energy
physics by projecting $H_{\text{BdG}}^{\prime}(k_{z})$ onto a few low-energy
subbands. As stated in the main text, the two lowest-energy subbands
$\varphi_{1}$ and $\varphi_{2}$ are superpositions of the left and right
channel wave functions $\chi_{L}$ and $\chi_{R}$, depicted in Fig. 1(b) of the
main text: $\varphi_{n}=a_{n}\chi_{L}+b_{n}\chi_{R}$ for $n=1,2$. The channel
wave functions, being strongly localized near the InAs-Al interfaces as shown
in Fig. 1(b) of the main text, have relatively weak tunnel coupling
$t=\matrixelement{\chi_{R}}{H_{0}(0)}{\chi_{L}}$, which leads to a small
subband energy difference $\varepsilon_{2}-\varepsilon_{1}\approx 2t$ between
the two lowest-energy subbands. Indeed, $t$ is typically on the same energy
scale as $\bar{\alpha}_{j}k_{z}$ and $\bar{\Delta}$. We, therefore, need to
keep both $\varphi_{1}$ and $\varphi_{2}$ in the low-energy basis to capture
the relevant low-energy physics. In contrast,
$\varepsilon_{3}-\varepsilon_{2}\gg\varepsilon_{2}-\varepsilon_{1}$ since
$\varepsilon_{3}-\varepsilon_{2}\sim\hbar^{2}/(m^{*}\ell^{2})$, where $\ell$
is the small length scale associated with the localization of the channel wave
functions near each InAs-Al interface. Therefore, the relevant low-energy
physics can be fully captured while ignoring subbands $\varphi_{n}$ for $n\geq
3$. In other words, we only keep the first two rows and columns of the
$\bar{\Lambda},\bar{\alpha}_{x}$, $\bar{\alpha}_{y}$, and $\bar{\Delta}$
matrices in Eq. (13) for our low-energy effective Hamiltonian.

Figure A.1: Wave function profiles for the two lowest-energy orbital subbands
$\varphi_{1}$ and $\varphi_{2}$ are shown in (a) and (b), respectively. These
subbands can be decomposed into left and right channels whose wave functions
$\chi_{1}$ and $\chi_{2}$ are shown in (c) and (d), respectively. Note that
these channel wave functions are localized near the InAs-Al interfaces because
of the strong band bending near these interfaces, which is primarily due to
the band offset between the materials. The gate voltages for this calculation
are $V_{T}=-5.5~{}\text{V}$, $V_{B}=-3~{}\text{V}$, $V_{L}=V_{R}=0$, and the
boundary condition at the InAs-Al interfaces is $\phi=0.3~{}\text{V}$. Note
that the electrostatic potential $\phi(x,y)$ within the InAs nanowire for
these gate settings is shown in Fig. 1(c) of the main text.

Finally, we transform the low-energy effective Hamiltonian from the basis
containing the first two orbital subbands $\varphi_{1}$ and $\varphi_{2}$ to
the basis containing the two channels $\chi_{L}$ and $\chi_{R}$. To do so, we
explicitly define the channel wave functions as

$\begin{pmatrix}\chi_{L}(x,y)\\\
\chi_{R}(x,y)\end{pmatrix}=O\begin{pmatrix}\varphi_{1}(x,y)\\\
\varphi_{2}(x,y)\end{pmatrix},$ (14)

where $O$ is an orthogonal matrix. Specifically, we choose $O$ such that the
channel wave functions are maximally separated from one another in the $x$
direction, i.e. $\Delta
x=\matrixelement{\chi_{R}}{x}{\chi_{R}}-\matrixelement{\chi_{L}}{x}{\chi_{L}}$
is maximized. To find such an $O$, we first define $X$ as

$X=\begin{pmatrix}\matrixelement{\varphi_{1}}{x}{\varphi_{1}}&\matrixelement{\varphi_{1}}{x}{\varphi_{2}}\\\
\matrixelement{\varphi_{2}}{x}{\varphi_{1}}&\matrixelement{\varphi_{2}}{x}{\varphi_{2}}\end{pmatrix},$
(15)

i.e. the matrix representation of the $x$ operator within the subspace
containing the two lowest-energy orbital subbands. $\Delta x$ is then
maximized by choosing $O$ to be the matrix which diagonalizes $X$,
$O^{T}XO=\text{diag}(\widetilde{x}_{1},\widetilde{x}_{2})$ where
$\widetilde{x}_{1}<\widetilde{x}_{2}$. An example is shown in Fig. A.1 of the
wave functions of the two lowest-energy orbital subbands, $\varphi_{1}$ and
$\varphi_{2}$, and how they decompose into left and right channel wave
functions, $\chi_{L}$ and $\chi_{R}$. Note that in the case where we have
mirror symmetry about the $\hat{y}$ axis ($g=0$), the channel wave functions,
$\chi_{L}$ and $\chi_{R}$, are simply the even and odd equal superpositions of
the orbital subband wave functions, $\varphi_{1}$ and $\varphi_{2}$, as is the
case in Fig. A.1. Evaluating the effective BdG Hamiltonian in this new basis,
where the channel wave functions $\chi_{L}$ and $\chi_{R}$ replace the orbital
subbands $\varphi_{1}$ and $\varphi_{2}$, then yields the two-channel
effective Hamiltonian,

$\begin{split}H_{BdG}^{\text{eff}}(k_{z})=&\left[\left(\frac{\hbar^{2}k_{z}^{2}}{2m^{*}}-\mu\right)\lambda_{0}+t\lambda_{x}+g\lambda_{z}\right]\sigma_{0}\tau_{z}-\Delta_{0}\sigma_{y}\lambda_{z}\tau_{y}+\alpha
k_{z}\left(\cos\theta\sigma_{y}\lambda_{z}\tau_{z}+\sin\theta\sigma_{x}\lambda_{0}\tau_{0}\right)\\\
&+\left(\eta_{1}\lambda_{0}+\xi_{1}\lambda_{x}\right)k_{z}\sigma_{y}\tau_{z}+\left(\eta_{2}\lambda_{z}+\xi_{2}\lambda_{x}\right)k_{z}\sigma_{x}\tau_{0}+(\Delta_{1}\lambda_{0}+\Delta_{2}\lambda_{x})\sigma_{y}\tau_{y}\end{split},$
(16)

where $\sigma_{j}$, $\lambda_{j}$, and $\tau_{j}$ with $j=0,x,y,z$ are Pauli
matrices acting in spin, channel, and particle-hole space, respectively. In
other words, the first and second columns of a $\lambda_{j}$ matrix correspond
to the left and right channel, respectively. The parameters in Eq. (16) are
given by the expressions,

$\displaystyle\mu$ $\displaystyle=(h_{LL}+h_{RR})/2,$ (17) $\displaystyle t$
$\displaystyle=h_{RL},$ (18) $\displaystyle g$
$\displaystyle=(h_{LL}-h_{RR})/2,$ (19) $\displaystyle\Delta_{0}$
$\displaystyle=(d_{LL}-d_{RR})/2,$ (20) $\displaystyle\alpha\cos\theta$
$\displaystyle=(s^{x}_{LL}-s^{x}_{RR})/2,$ (21)
$\displaystyle\alpha\sin\theta$ $\displaystyle=-(s^{y}_{LL}+s^{y}_{RR})/2,$
(22) $\displaystyle\eta_{1}$ $\displaystyle=(s^{x}_{LL}+s^{x}_{RR})/2,$ (23)
$\displaystyle\eta_{2}$ $\displaystyle=-(s^{y}_{LL}-s^{y}_{RR})/2,$ (24)
$\displaystyle\xi_{1}$ $\displaystyle=s^{x}_{RL},$ (25) $\displaystyle\xi_{2}$
$\displaystyle=-s^{y}_{RL},$ (26) $\displaystyle\Delta_{1}$
$\displaystyle=(d_{LL}+d_{RR})/2,$ (27) $\displaystyle\Delta_{2}$
$\displaystyle=d_{RL},$ (28)

where $h_{ij}=\matrixelement{\chi_{i}}{H_{0}(k_{z})}{\chi_{j}}$,
$s^{x}_{ij}=\matrixelement{\chi_{i}}{\widetilde{\alpha}_{x}(x,y)}{\chi_{j}}$,
$s^{y}_{ij}=\matrixelement{\chi_{i}}{\widetilde{\alpha}_{y}(x,y)}{\chi_{j}}$,
and $d_{ij}=\matrixelement{\chi_{i}}{\Delta(x,y)}{\chi_{j}}$ with $i,j=L,R$.
Note that Eq. (16) contains additional terms involving $\eta_{1}$, $\eta_{2}$,
$\xi_{1}$, $\xi_{2}$, $\Delta_{1}$, and $\Delta_{2}$ that are not show in Eq.
(4) of the main text. Here, $\eta_{1}$ and $\eta_{2}$ account for the fact
that the $x$ and $y$ components of the electric field at the positions of the
two channels are only approximately opposite and equal, respectively. The
terms $\xi_{1}$ and $\xi_{2}$ account for the interchannel spin-orbit
coupling, as indicated by the $\lambda_{x}$ matrix. The $\Delta_{1}$ and
$\Delta_{2}$ terms allow for differences between the two channels in pairing
strength and interchannel pairing, respectively. These six terms in our
calculations are all found to be small and do not qualitatively impact the
physics of the effective model. For simplicity, we have ignored these terms
when showing the effective Hamiltonian in Eq. (4) of the main text. However,
these terms have been included in the calculation of the topological phase
diagrams shown in Figs. 4(a) and (b) of the main text. Furthermore, we find
that ignoring these terms only slightly changes the phase diagrams with no
qualitative differences.

##  Appendix B Transformation of the two-channel effective Hamiltonian

In the main text, we stated that the two-channel effective Hamiltonian
$H_{\text{eff}}(k_{z})$ shown in Eq. (4) transforms into
$H^{\prime}_{\text{eff}}(k_{z})$ shown in Eq. (5). The Hamiltonians are
connected by the relation
$H^{\prime}_{\text{eff}}(k_{z})=U^{\dagger}H_{\text{eff}}(k_{z})U$, where $U$
is a unitary matrix. Here, we provide the form of $U$ and discuss its
structure. The goal of the transformation is to bring the Hamiltonian into a
form where the spins are decoupled for the ideal limit of antiparallel spin-
orbit vectors ($\theta=0$).

The unitary matrix $U$ is best understood as the product of four simple
transformations,
$U=R^{\sigma}_{x}S^{\sigma}_{h}S^{\lambda}_{h}S^{\lambda}_{\downarrow}$, where
$R$ and $S$ stand for rotation and swap operations that we now explain. The
first operation $R^{\sigma}_{x}$ is a rotation of the spins by $90^{\circ}$
about the $x$ axis,

$R^{\sigma}_{x}=\frac{1}{\sqrt{2}}\left(1+i\sigma_{x}\tau_{z}\right).$ (29)

This results in the transformed Hamiltonian
$H_{1}(k_{z})={R^{\sigma}_{x}}^{\dagger}H_{\text{eff}}(k_{z})R^{\sigma}_{x}$
given by

$\begin{split}H_{1}(k_{z})=&\left[\varepsilon(k_{z})\lambda_{0}+t\lambda_{x}+g\lambda_{z}\right]\sigma_{0}\tau_{z}-\Delta_{0}\sigma_{y}\lambda_{z}\tau_{y}\\\
&+\alpha
k_{z}\left(\cos\theta\sigma_{z}\lambda_{z}\tau_{o}+\sin\theta\sigma_{x}\lambda_{0}\tau_{0}\right),\end{split}$
(30)

where we see that the spin-orbit coupling aligns with the $z$ axis for the
ideal limit of antiparallel spin-orbit vectors ($\theta=0$). Next, we notice
that the superconductive pairing exists between particles and holes of the
opposite spin species, as evident in the
$\Delta_{0}\sigma_{y}\lambda_{z}\tau_{y}$ term. We want, however, the pairing
to exist between the same spin species since we are seeking a final
Hamiltonian that is block diagonal in spin space for the case of $\theta=0$.
To accomplish this, we swap the spin degrees of freedom in the hole sector
using the operator,

$S^{\sigma}_{h}=\frac{1}{2}\left[\sigma_{0}\left(\tau_{0}+\tau_{z}\right)+\sigma_{x}\left(\tau_{0}-\tau_{z}\right)\right].$
(31)

This results in the transformed Hamiltonian
$H_{2}(k_{z})={S^{\sigma}_{h}}^{\dagger}H_{1}(k_{z})S^{\sigma}_{h}$ given by

$\begin{split}H_{2}(k_{z})=&\left[\varepsilon(k_{z})\lambda_{0}+t\lambda_{x}+g\lambda_{z}\right]\sigma_{0}\tau_{z}+\Delta_{0}\sigma_{z}\lambda_{z}\tau_{x}\\\
&+\alpha
k_{z}\left(\cos\theta\sigma_{z}\lambda_{z}\tau_{z}+\sin\theta\sigma_{x}\lambda_{0}\tau_{0}\right),\end{split}$
(32)

where we see that the Hamiltonian is indeed block diagonal in spin space for
the case of $\theta=0$. Next, we notice that the superconductive pairing
within each spin block is diagonal in channel space. We want, however, the
channels to play the role of ordinary spins in an s-wave superconductor, which
have superconductive pairing between particles and holes of opposite spin. In
order to make the channel degree of freedom better resemble the ordinary spin
degree of freedom, we swap the channel degrees of freedom in the hole sector
using the operator,

$S^{\lambda}_{h}=\frac{1}{2}\left[\lambda_{0}\left(\tau_{0}+\tau_{z}\right)+\lambda_{x}\left(\tau_{0}-\tau_{z}\right)\right].$
(33)

This results in the transformed Hamiltonian
$H_{3}(k_{z})={S^{\lambda}_{h}}^{\dagger}H_{2}(k_{z})S^{\lambda}_{h}$ given by

$\begin{split}H_{3}(k_{z})=&\left[\varepsilon(k_{z})\lambda_{0}+t\lambda_{x}\right]\sigma_{0}\tau_{z}+g\sigma_{0}\lambda_{z}\tau_{0}-\Delta_{0}\sigma_{z}\lambda_{y}\tau_{y}\\\
&+\alpha
k_{z}\left(\cos\theta\sigma_{z}\lambda_{z}\tau_{o}+\sin\theta\sigma_{x}\lambda_{0}\tau_{0}\right).\end{split}$
(34)

Interestingly for $g=0$, each diagonal spin block of Eq. (34) is exactly the
Lutchyn-Oreg Hamiltonian of Refs. [13, 14] for a superconducting nanowire with
spin-orbit coupling in an external magnetic field, except that the spin
degrees of freedom are replaced by channels
$\sigma_{j}\rightarrow\lambda_{j}$, and the Zeeman energy is replaced by the
interchannel tunnel coupling $t$. The diagonal spin blocks are not identical,
however, since they have opposite superconductive pairing and spin-orbit
coupling coefficients, as evident by the $\sigma_{z}$ in the
$\Delta_{0}\sigma_{z}\lambda_{y}\tau_{y}$ and $\alpha
k_{z}\cos\theta\sigma_{z}\lambda_{z}\tau_{o}$ terms. The spin blocks become
identical if we swap the channel degrees of freedom within the spin
$\downarrow$ sector using the operator,

$S^{\lambda}_{\downarrow}=\frac{1}{2}\left[\left(\sigma_{0}+\sigma_{z}\right)\lambda_{0}+\left(\sigma_{0}-\sigma_{z}\right)\lambda_{x}\right].$
(35)

Finally, this results in the effective Hamiltonian
$H^{\prime}_{\text{eff}}(k_{z})={S^{\lambda}_{h}}^{\dagger}H_{3}(k_{z})S^{\lambda}_{h}=U^{\dagger}H_{\text{eff}}(k_{z})U$
shown in Eq. (5) of the main text. Notice that this last operation transforms
the $g\sigma_{0}\lambda_{z}\tau_{o}$ term in $H_{3}(k_{z})$ into
$g\sigma_{z}\lambda_{z}\tau_{o}$, where the channel detuning $g$ is
effectively opposite for the two spin blocks. As stressed in the main text,
this difference between the spin blocks arising from the channel detuning $g$
is responsible for the increased robustness of the topological phase to spin-
orbit vector misalignment ($\theta=0$). Lastly, we note that the product of
these four simple operations can be algebraically manipulated into the more
compact expression,

$U=\frac{1}{4}(1+i[\sigma_{x}+\sigma_{y}+\sigma_{z}])(1-i\sigma_{z}\lambda_{x})(1+i\lambda_{x}\tau_{z}).$
(36)

##  Appendix C Phase diagram calculation method

Figure A.2: Surface (red) and bulk (blue) local density of states for a system
in the topological phase. The presence of Majorana zero modes is evident from
the peak in the surface local density of states at zero energy $\omega=0$. The
absence or presence of such a peak can be used to determine if the system is
in the trivial or topological phase for any given choice of parameters. System
parameters are $m^{*}=0.026m_{o}$, $\Delta_{0}=0.2~{}\text{meV}$,
$t=0.5~{}\text{meV}$, $\alpha=20~{}\text{meV}\cdot\text{nm}$, and $\theta=0$.
All other parameters in Eq. (16) are zero.

In the main text, we provide several phase diagrams showing the boundary
between the trivial and topological phases, as well as the topological gap
$\Delta_{\text{top}}$ within the topological regions. Often times, the
assignment of trivial and topological phases is done by calculating a
topological invariant [41]. In this work, we use an alternative technique
where we look for the presence of zero-energy Majorana zero modes at the end
of a semi-infinite system. To do so, we first discretize the Hamiltonian in
Eq. (16) on a 1D lattice with a small lattice spacing of $a=0.5~{}\text{nm}$.
Note that each site of the 1D lattice has eight orbitals to account for the
spin, channel, and particle-hole degrees of freedom. Information about the
local density of states is then contained in the Green’s function matrix
$G(\omega)=(\omega I-H_{dis}+i\eta)^{-1}$, where $\omega\in\mathbb{R}$ is the
input energy, $H_{dis}$ is the discretized Hamiltonian matrix, $I$ is the
identity matrix, and $\eta=10^{-5}~{}\text{meV}$ is a small energy to bring
the Green’s function into the upper half of the complex plane. While generic
matrix inversion is numerically expensive, finding the diagonal elements of
$G(\omega)$ can be done in a numerically efficient manner for systems with
translation invariance by using the decimation technique of Ref. [65]. Using
this technique, the surface Green’s function block $G_{s}$ (which is just the
block of the Green’s function matrix for the first lattice site) and the bulk
Green’s function block $G_{b}$ can be found for a system with $2^{N}$ lattice
sites using only $N$ decimation iterations. These, in turn, give us the
surface and bulk local density of states,

$\displaystyle LDOS_{s}(\omega)$
$\displaystyle=-\frac{1}{\pi}\imaginary(\Tr[G_{s}(\omega)]),$ (37)
$\displaystyle LDOS_{b}(\omega)$
$\displaystyle=-\frac{1}{\pi}\imaginary(\Tr[G_{b}(\omega)]).$ (38)

Using $N=50$ iterations, we can then obtain the surface and bulk density of
states for system of length $L\approx 10^{6}~{}\text{m}$, which is semi-
infinite for all practical purposes. An example of the surface and bulk local
density of states for a system in the topological phase is shown in Fig. A.2.
We see a peak in the surface local density of states at zero energy $\omega=0$
coming from the presence of Majorana zero modes at the ends of the system. The
absence or presence of such a peak can then be used to determine if the system
is in the trivial or topological phase for any given choice of parameters. In
addition, we can read off the topological gap $\Delta_{\text{top}}$ of the
topological phase from the location of the coherence peaks in the bulk local
density of states (blue curve in Fig. A.2).

##  Appendix D Quantifying Dresselhaus and transverse-Rashba spin-orbit
coupling

In the main text, we only included Rashba spin-orbit coupling and limited it
to the component involving the longitudinal momentum $k_{z}$ (see Eq. (3) of
main text). However, there also exists Rashba spin-orbit coupling involving
the transverse momenta $k_{x}$ and $k_{y}$, along with Dresselhaus spin-orbit
coupling. Within in the main text, we neglected these terms due to their
contributions yielding only small quantitative effects while not affecting the
qualitative aspects of the physics. In this section, we quantify these other
forms of spin-orbit coupling to show that neglecting them is well justified.

Let us begin with the Dresselhaus spin-orbit coupling, which arises from the
bulk inversion asymmetry of the InAs zinc blende lattice. Its Hamiltonian
component, which is to be added to $H_{\text{SO}}$ in Eq. (3) of the main
text, takes the form [52],

$\begin{split}H_{\text{BIA}}=\gamma_{D}\big{[}&\left(k_{y}^{2}-k_{z}^{2}\right)k_{x}\sigma_{x}+\left(k_{z}^{2}-k_{x}^{2}\right)k_{y}\sigma_{y}\\\
&+\left(k_{x}^{2}-k_{y}^{2}\right)k_{z}\sigma_{z}big],\end{split}$ (39)

where $\gamma_{D}$ is a material-dependent constant given by
$\gamma_{D}=27.18~{}\text{meV}\cdot\text{nm}^{3}$ for InAs [52]. (Ref. [66]
gives a similar value of $\gamma_{D}=21.7~{}\text{meV}\cdot\text{nm}^{3}$.)
Note that the Dresselhaus spin-orbit Hamiltonian takes the particular form
shown in Eq. (39) because the triangular InAs nanowires studied in this work
are grown along the $\left[001\right]$ crystallographic direction with the
upper facets of the nanowire being within the $[110]$ and $[\bar{1}10]$ planes
[47]. Otherwise, the appropriate coordinate rotation would have to be applied
to Eq. (39). Projecting the Dresselhaus Hamiltonian onto the two-channel basis
defined in Sec. A leads to the term,

$H_{\text{BdG}}^{\text{eff,BIA}}=\nu_{x}\sigma_{x}\lambda_{y}\tau_{0}+\nu_{y}\sigma_{y}\lambda_{y}\tau_{z}+\left[\left(\frac{\beta_{LL}+\beta_{RR}}{2}\right)\lambda_{0}+\left(\frac{\beta_{LL}-\beta_{RR}}{2}\right)\lambda_{z}+\beta_{LR}\lambda_{x}\right]k_{z}\sigma_{z}\tau_{0}+\mathcal{O}(k_{z}^{2}),$
(40)

which should be added to $H_{\text{BdG}}^{\text{eff}}$ in Eq. (16). Here,
$\nu_{x}$ and $\nu_{y}$ are interchannel spin-orbit coupling coefficients that
come from the two terms in Eq. (39) not involving the longitudinal momentum
$k_{z}$ and are given by

$\displaystyle\nu_{x}$
$\displaystyle=-i\gamma_{D}\matrixelement{\chi_{R}}{k_{x}k_{y}^{2}}{\chi_{L}},$
(41) $\displaystyle\nu_{y}$
$\displaystyle=i\gamma_{D}\matrixelement{\chi_{R}}{k_{x}^{2}k_{y}}{\chi_{L}},$
(42)

where $\chi_{L}$ and $\chi_{R}$ are the channel wave functions. Note that
these these terms do not produce analogous intrachannel spin-orbit coupling
coefficients. This is because the expectation values of the operators
$k_{x}k_{y}^{2}$ and $k_{x}^{2}k_{y}$ automatically vanish since they both
have an imaginary prefactor when expressed as differential operators and the
channel wave functions $\chi_{i}$ are real-valued. The $\beta_{ij}$
coefficients come from the term involving $\sigma_{z}$ in Eq. (39) and are
given by

$\beta_{ij}=\gamma_{D}\matrixelement{\chi_{i}}{k_{x}^{2}-k_{y}^{2}}{\chi_{j}}.$
(43)

Note that both interchannel $\beta_{LR}$ and intrachannel $\beta_{ii}$
coefficients exist since the expectation value of the operator
$k_{x}^{2}-k_{y}^{2}$ for each channel wave function generically does not
vanish.

Figure A.3: Spin-orbit coefficients from Dresselhaus and transverse-Rashba
spin-orbit coupling as a function of topgate voltage $V_{T}$ for $\mu=g=0$.
(a) Intrachannel $\beta_{ii}$ (solid gold) and interchannel $\beta_{LR}$
Dresselhaus coefficients (dashed blue). The intrachannel Dresselhaus
coefficient $\beta_{ii}$ dominates for the interchannel Dresselhaus
coefficient $\beta_{LR}$. Furthermore, the intrachannel Dresselhaus
coefficient $\beta_{ii}$ is smaller than the numerically extracted Rashba
spin-orbit coefficient of $\alpha\approx 25~{}\text{meV}\cdot\text{nm}$ (given
in the main text) by over a factor of 40. (b) Interchannel spin-orbit
coefficients $\nu_{x}$ and $\nu_{z}$ that come from Dresselhaus and
transverse-Rashba spin-orbit coupling, respectively. (c) Ratio of interchannel
tunnel coupling $t$ and interchannel spin-orbit coefficient $\nu_{z}$, showing
that the spin-independent coupling $t$ dominates over the spin-dependent
coupling $\nu_{z}$.

The values of the $\beta_{ij}$ coefficients are shown in Fig. A.3(a) as a
function of topgate voltage $V_{T}$ for $\mu=g=0$. The interchannel
coefficient $\beta_{LR}$ and intrachannel coefficient $\beta_{ii}$ are shown
as blue (dashed) and gold (solid) lines, respectively. Note that
$\beta_{LL}=\beta_{RR}$ because of mirror symmetry about the $\hat{y}$ axis
when $g=0$. Notice that the interchannel coefficient $\beta_{LR}$ is smaller
than the intrachannel coefficient $\beta_{ii}$ by at least an order of
magnitude over the entire voltage range. Furthermore, comparing the
intrachannel Dresselhaus coefficient $\beta_{ii}$ to the extracted Rashba
spin-orbit coefficient of $\alpha\approx 25~{}\text{meV}\cdot\text{nm}$ from
the main text, we see that $\beta_{ii}$ is smaller by over a factor of $40$.
Therefore, we conclude that the Dresselhaus terms involving $\beta_{ij}$ have
a negligible effect on the physics. In addition, the interchannel spin-orbit
coefficient $\nu_{x}$ is shown in Fig. A.3(b) as a function of $V_{T}$, as a
red (dashed) line. We numerically find that $\nu_{x}$ is always small
($\nu_{x}\lesssim 1~{}\mu\text{eV}$), and is therefore safely neglected.
Finally, we also numerically found that $\nu_{y}$ from Dresselhaus spin-orbit
coupling vanishes, and, therefore, plays no role in our system.

It is interesting to note that the physical reason why the intrachannel
coefficients $\beta_{ii}$ are so small in this system is because the upper
facets of the InAs nanowire are within the $[110]$ and $[\bar{1}10]$ planes.
To understand this, notice first that the left channel wave function
$\chi_{L}$ in Fig. A.1(c) closely resembles a particle-in-a-box state with
quantization axes that are $\approx 45^{\circ}$ with respect to the $x$ and
$y$ axes, respectively, that is, along the $[110]$ and $[\bar{1}10]$
crystallographic directions. Next, we can rewrite the $k_{x}^{2}-k_{y}^{2}$
operator appearing in Eq. (43) as

$k_{x}^{2}-k_{y}^{2}=2k_{x^{\prime}}k_{y^{\prime}},$ (44)

where the $x^{\prime}$ and $y^{\prime}$ axes align with the $[110]$ and
$[\bar{1}10]$ crystallographic directions. Importantly, a particle-in-a-box
state with quantization axes along the $x^{\prime}$ and $y^{\prime}$ axes has
vanishing expectation value for the operator $k_{x^{\prime}}k_{y^{\prime}}$.
This is because the the operators $k_{x^{\prime}}$ and $k_{y^{\prime}}$ have
imaginary prefactors when expressed as differential operators, and the
particle-in-a-box state decomposes into a product of real wave functions in
the $x^{\prime}$ and $y^{\prime}$ direction. Now, $\chi_{L}$ is not perfectly
captured by a product state separable in the $x^{\prime}$ and $y^{\prime}$
direction, so $\beta_{LL}$ does not exactly vanish. Its close resemblance,
however, keeps $\beta_{LL}$ very small.

Next, let us consider the transverse-Rashba spin-orbit coupling. The Rashba
spin-orbit coupling included within $H_{\text{SO}}$ in Eq. (3) of the main
text only involves longitudinal momentum $k_{z}$, whereas transverse-Rashba
spin-orbit coupling involves the transverse momenta $k_{x}$ and $k_{y}$. Its
Hamiltonian component, which is to be added to $H_{\text{SO}}$ in Eq. (3) of
the main text, is given by [48]

$\begin{split}H_{R_{\perp}}=\frac{1}{2}\Big{(}&\left[\alpha_{y}(x,y)k_{x}+k_{x}\alpha_{y}(x,y)\right]\\\
&-\left[\alpha_{x}(x,y)k_{y}+k_{y}\alpha_{x}(x,y)\right]\Big{)}\sigma_{z},\end{split}$
(45)

where care has been taken to properly symmetrize the operators such that the
Hamiltonian is Hermitean. Projecting the transverse-Rashba Hamiltonian onto
the two-channel basis defined in Sec. A leads to the term,

$H_{\text{BdG}}^{\text{eff},R_{\perp}}=\nu_{z}\sigma_{z}\lambda_{y}\tau_{0},$
(46)

which should be added to $H_{\text{BdG}}^{\text{eff}}$ in Eq. (16). Here,
$\nu_{z}$ is an interchannel spin-orbit coupling coefficient given by

$\begin{split}\nu_{z}=&\frac{i}{2}\matrixelement{\chi_{R}}{\left(\alpha_{y}(x,y)k_{x}+k_{x}\alpha_{y}(x,y)\right)}{\chi_{L}}\\\
&-\frac{i}{2}\matrixelement{\chi_{R}}{\left(\alpha_{x}(x,y)k_{y}+k_{y}\alpha_{x}(x,y)\right)}{\chi_{L}}.\end{split}$
(47)

Note that the transverse-Rashba Hamiltonian does not produce an analogous
intrachannel spin-orbit coupling coefficient. This is because the expectation
values of the operators $k_{x}$ and $k_{y}$ automatically vanish for the same
reason given above for the $k_{x}k_{y}^{2}$ and $k_{x}^{2}k_{y}$ operators.

Figure A.4: Topological phase diagram of the same system as Fig. 4 of the main
text except that the $\nu_{z}\sigma_{z}\lambda_{y}\tau_{0}$ term from
transverse-Rashba spin-orbit coupling is included in the model. Comparing the
results to Figs. 4(a) and (b) of the main text, we find that inclusion of the
transverse-Rashba spin-orbit coupling slightly reduces the area of the
topological phase and the topological gap $\Delta_{\text{top}}$ within the
topological region. The effect is quite modest although.

The interchannel spin-orbit coefficient $\nu_{z}$ is shown in Fig. A.3(b) as a
function of $V_{T}$, as a black (solid) line. We find that the interchannel
spin-orbit coefficient $\nu_{z}$ coming from transverse-Rashba spin-orbit
coupling is comparable to the superconducting gap
$\Delta_{0}=0.2~{}\text{meV}$ for the larger top gate voltages in Fig. A.3(b).
It may then at first seem we cannot ignore the transverse-Rashba spin-orbit
coupling. However, if we compare this coupling to the interchannel coupling
$t$, as shown in Fig. A.3(c), we see that $\nu_{z}$ is smaller by roughly an
order of magnitude within the entire voltage range. We therefore do not expect
$\nu_{z}$ to play a significant role in the physics of the system. To test
this expectation, we redo the calculation of the topological phase diagrams
shown in Figs. 4(a) and (b) of the main text with the inclusion of the
$\nu_{z}\sigma_{z}\lambda_{y}\tau_{0}$ term into the effective two-channel
Hamiltonian given in Eq. (16). The results are shown in Fig. A.4, again for
two values of channel detuning, (a) $g=0$ and (b) $g=\Delta_{0}$. Comparing
the results to Figs. 4(a) and (b) of the main text, we see that the addition
of transverse-Rashba spin-orbit coupling has a modest effect on the
topological phase diagrams. The results are qualitatively the same as the
results in Figs. 4(a) and (b) of the main text, with the only quantitative
differences being a small reduction of the topological region and the
reduction of the topological gap $\Delta_{\text{top}}$ within the topological
region. Therefore, we conclude that while transverse-Rashba spin-orbit
coupling is slightly detrimental to topological superconductivity in our
system, the effect is quite small and does not qualitatively impact the
physics.

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  *[RL]: Reinforcement Learning
  *[GRL]: Graph Reinforcement Learning
  *[GNNs]: Graph Neural Network
  *[GNN]: Graph Neural Network
  *[EV]: Electric Vehicle
  *[CPES]: cyber-physical energy system
  *[OPF]: Optimal Power Flow
  *[OPFs]: Optimal Power Flow
  *[DRL]: Deep Reinforcement Learning
  *[CNN]: Convolutional Neural Networks
  *[CNNs]: Convolutional Neural Networks
  *[GAT]: Graph Attention Network
  *[MDPs]: Markov Decision Process
  *[MDP]: Markov Decision Process
  *[MCTS]: Monte Carlo Tree Search
  *[DDQN]: Double Deep Q-Network
  *[DQN]: Deep Q-Network
  *[A3C]: Asynchronous Advantage Actor-Critic
  *[DDPG]: Deep Deterministic Policy Gradient
  *[PPO]: Proximal Policy Optimization
  *[SAC]: Soft Actor-Critic
  *[BBF]: Bigger, Better, Faster
  *[GCN]: Graph Convolutional Network
  *[RES]: Renewable Energy Sources
  *[GATs]: Graph Attention Network
  *[STGCN]: Spatio-Temporal Graph Convolutional Network
  *[BESS]: Battery Energy Storage Systems
  *[SMAAC]: Semi-Markov Afterstate Actor-Critic
  *[SACD]: Soft Actor-Critic Discrete
  *[ESS]: Energy Storage Systems
  *[GCNs]: Graph Convolutional Network
  *[DNR]: Distribution Network Reconfiguration
  *[GCAPCN]: Capsule-based Graph Convolutional Network
  *[MISOCP]: Mixed-integer Second-order Conic Programming
  *[BPSO]: binary particle swarm optimization
  *[OSPI]: Optimal Solution with Perfect Information
  *[MPC]: Model Predictive Control
  *[GSOs]: Graph Structure Operators
  *[EVs]: Electric Vehicle
  *[GRLs]: Graph Reinforcement Learning
  *[RLs]: Reinforcement Learning