# Majorana nanowires, Kitaev chains, and spin models

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

###### Abstract

Motivated by the fact that the idealized Kitaev chain toy model and the
experimental semiconductor-superconductor Majorana nanowire can both host the
Majorana zero modes, we theoretically investigate the question to what extent
the two models are equivalent or similar, using the perspective of the
corresponding dual spin models for both. We start with the duality between the
Kitaev chain and the transverse-field XY spin model through the Jordan-Wigner
transformation with the goal of establishing the connection between the Kitaev
chain and the nanowire. By applying the Jordan-Wigner transformation to the
nanowire, we find that the corresponding bosonic spin model is a generalized
spin cluster model, containing 3- and 4-spin terms, with staggered couplings.
By projecting out the upper band of the bare semiconductor with higher energy,
we obtain an effective low-energy spinless system from the spinful nanowire
system deep in the topological regime. Finally, we establish the connection
between the Kitaev chain and Majorana nanowire by showing that the spinless
Kitaev chain can be viewed as the first-order approximation of the spinful
Majorana nanowire deep in the topological regime.

##  I Introduction

The Majorana zero modes (MZMs) are non-Abelian Ising anyons with potential
application to fault-tolerant topological quantum computing because of the
robustness of the quantum degenerate ground state against local perturbations
protected by a topological gap [1, 2]. One prototypical model that was shown
to host boundary MZMs is the Kitaev chain [3], which contains the nearest-
neighbor hopping and $p$-wave superconductivity in a spinless system [4].
Given appropriate parameters and conditions, a pair of Majorana zero modes can
emerge at both ends of the Kitaev chain by construction. Although, the Kitaev
chain is an idealized toy model to show a proof of principle, because
spinlessness and $p$-wave superconductivity are hard to find in nature, the
model has been influential in focusing attention on non-Abelian MZMs,
topological superconductivity, topological quantum phase transition (TQPT),
and topological quantum computing using Majorana zero modes. Therefore,
realistic experimentally feasible models have been proposed to search for
MZMs. The most prevailing scheme used in experiments—see, e.g., Ref. 5 and
references therein for a recent effort to experimentally realize Majorana zero
modes in semiconductor-superconductor hybrid structures— is the one-
dimensional (1D) semiconductor-superconductor (SM-SC) nanowire [6, 7, 8, 9],
where the semiconductor nanowire with a large Rashba-type spin-orbit coupling
(SOC) acquires $p$-wave superconductivity from an $s$-wave superconductor due
to the proximity effect, and a pair of MZMs should appear at the ends of the
nanowire given a sufficiently large magnetic field. This realistic SM-SC
system is often referred to as a Majorana nanowire since the MZMs are
predicted to arise at the two ends of the nanowire under suitable conditions.
However, although both Kitaev chain and Majorana nanowire systems can host
MZMs, they are two completely different models which, at first sight, appear
to have little to do with each other except for their one-dimensionality and
the possible existence of MZMs. For example, the Kitaev chain is a lattice
model made of spinless fermions whereas the nanowire is a continuum system
with real spinful electrons. Also, in the Kitaev chain, $p$-wave
superconductivity is assumed a priori in the model whereas in the Majorana
nanowire the effective topological $p$-wave superconductivity is emergent
under the subtle combined effects of induced $s$-wave SC, Rashba SO-coupling,
and Zeeman spin splitting. Historically, the Majorana nanowire proposal arose
not motivated by the Kitaev chain idea, but from various 2D proposals
involving SO-coupling, $s$-wave SC, and spin splitting [10, 11, 8]. After the
fact, however, it seems obvious that the two models, Kitaev chain and Majorana
nanowire, should be related since both depend on having an effectively
spinless $p$-wave SC leading to MZMs localized at the ends of a 1D system.
However, the extent to which they are generically equivalent to each other
remains unclear despite some isolated efforts [12, 13, 14, 15] to connect them
through specific properties. In particular, References 16, 17 were among the
first to have preliminary discussions on possible connections between the
Kitaev chain and the Majorana nanowire models.

This lack of clarity sometimes leaves a misconception that these two models,
Kitaev chain and Majorana nanowire, are always equivalent, which is incorrect,
as we will show later under which circumstances these two models are
equivalent and when they cannot be mapped into each other. This lack of
complete equivalence between the two models is rather obvious if one considers
that the Kitaev model is a spinless one band model whereas the nanowire model
is a spinful multiband model. Thus, a claim that the nanowire model can be
obtained by taking the continuum limit of the Kitaev chain and projecting onto
the low-energy degrees of freedom is technically incorrect since such a low-
energy projection is meaningless for the original Kitaev chain model. Thus, we
are motivated to study the underlying general connection between the Kitaev
chain and the SM-SC nanowire try to see whether we can find an exact Majorana
mapping between them. Mapping both onto effective spin models by eliminating
the fermionic degrees of freedom enables a direct one-to-one comparison
between the models to figure out how they are connected and how they are not.
This is what we accomplish in the this paper.

We first recall the duality [3, 18, 19] between the Kitaev chain and
transverse-field XY model (which can be further tuned to the transverse-field
Ising model given appropriate parameters) as this is the origin of the term
Ising anyon for MZMs (which is often used [1]). This mapping of the Kitaev
chain to the corresponding transverse-field XY/Ising model is achieved by a
nonlocal Jordan-Wigner transformation converting the fermions to bosonic spin
operators. For the same reason, to see the connection between the Kitaev chain
and SM-SC Majorana nanowire, we should ask first how to map an SM-SC Majorana
nanowire to a corresponding spin model through the Jordan-Wigner
transformation. However, we find that the resulting spin model is, unlike the
relatively simple form of the transverse-field XY model (as for the Kitaev
chain), rather complicated — it is a generalized cluster model containing up
to 4-spin interactions with staggered couplings, which do not exist in the
mapping of the Kitaev chain to the transverse-field XY model. Thus the
connection between the two models appears much more obscure in the spin
language than in the fermionic language, where at least, both models at their
cores are free fermion “band” problems.

We, therefore, approach the SM-SC Majorana nanowire from another angle: The
first step is to convert the spinful nanowire into an effective low-energy
model by projecting out the upper band of the bare semiconductor with higher
energy in momentum space in the limit of the large magnetic field. We can then
manually construct a spinless model by ignoring the band index and
incorporating the superconducting term. We find that the form of the effective
low-energy Hamiltonian in momentum space is already a spinless $p$-wave
superconductor that contains a normal SM part plus an antisymmetric SC part,
which is superficially similar to the Kitaev chain. To make a more direct
comparison, we expand the Hamiltonian in momentum space in the limit of a
large magnetic field, and recover the real space Hamiltonian up to the third
order of approximation. We find that the first order of this high-field
expansion can be exactly mapped into the Kitaev chain, and the higher order
approximation will result in longer-range hoppings and pairings. Therefore,
the Kitaev chain can be viewed at least as a first-order approximation of the
SM-SC nanowire model at a high magnetic field deep in the topological regime.
This also makes physical sense because, in the high-field limit, the low-
energy Majorana nanowire projection essentially produces a spinless $p$-wave
SC as assumed by construction in the Kitaev model. The two models obviously
cannot be equivalent at a low magnetic field deep in the trivial phase since
the nanowire-induced proximity SC is essentially $s$-wave in the trivial phase
whereas by construction the SC is $p$-wave in the Kitaev chain.

This paper is organized as follows. In Sec. II, we state the conventions for
the Jordan-Wigner transformation adopted throughout this paper. In Sec. III,
we recapitulate and expand on the duality between the Kitaev chain and the
transverse-field XY model through the Jordan-Wigner transformation. Then we
generalize the Jordan-Wigner transformation for the spinful model, and
directly apply it to the SM-SC Majorana nanowire Hamiltonian to obtain a
generalized spin cluster model with staggered couplings in Sec. IV.1. Finally,
we present the direct mapping between the Kitaev chain and the SM-SC nanowire
by projecting out the higher energy band in Sec. IV.2. Our conclusion is in
Sec. V.

##  II Jordan-Wigner transformation

The Jordan-Wigner transformation [20] is a nonlocal transformation that
connects a fermionic operator and a series of nonlocal bosonic operators (in
$\frac{1}{2}$-spin systems) while keeping the canonical commutation relation
the same. We define the Jordan-Wigner transformation following the conventions
as

$\begin{split}f_{i}^{\dagger}&=\prod_{j=1}^{i-1}\left(-\sigma_{j}^{z}\right)\sigma_{i}^{+},\\\
f_{i}&=\prod_{j=1}^{i-1}\left(-\sigma_{j}^{z}\right)\sigma_{i}^{-},\end{split}$
(1)

where $f_{i}$ ($f_{i}^{\dagger}$) is the fermionic annihilation (creation)
operator acting on site $i$ in real space, and $\sigma_{i}^{+}$
($\sigma_{i}^{-}$) is the bosonic ladder operator that raises (lowers) the
spin by one unit at site $i$ in $\frac{1}{2}$-spin systems, which can be
expressed in terms of the Pauli matrix:

$\sigma_{i}^{\pm}=\frac{1}{2}\left(\sigma_{i}^{x}\pm i\sigma_{i}^{y}\right).$
(2)

It is straightforward to verify that the anti commutating relation of the
fermionic operators defined in Eq. (1) is also restored by the commuting
bosonic spin operators.

For a general noninteracting fermionic Hamiltonian, the bilinear form of a
local product of two fermionic operators can be transformed into the product
of a series of local spin operators through the Jordan-Wigner transformation,
e.g.,

$\begin{split}f_{i}^{\dagger}f_{i}&=\frac{1}{2}\left(1+\sigma_{i}^{z}\right),\\\
f_{i+1}^{\dagger}f_{i}&=\sigma_{i+1}^{+}\sigma_{i}^{-},\\\
f_{i+1}^{\dagger}f_{i}^{\dagger}&=-\sigma_{i+1}^{+}\sigma_{i}^{+},\end{split}$
(3)

which preserves the locality.

##  III Kitaev chain

We first briefly review, for the sake of clarity and completeness, the duality
between the Kitaev chain and the $s=\frac{1}{2}$ 1D transverse-field XY model
[21], which can be mapped into each other through the Jordan-Wigner
transformation of Eq. (1). The Kitaev chain [3] is a spinless fermionic
lattice tight-binding model with the nearest-neighbor hopping $t$, and
$p$-wave superconducting pairing energy $\Delta$, where its Hamiltonian in
real space is

$\hat{H}_{\text{KC}}^{(f)}=\sum_{i=1}^{N}-\mu
f_{i}^{\dagger}f_{i}+\left(-tf_{i+1}^{\dagger}f_{i}+\Delta
f_{i+1}^{\dagger}f_{i}^{\dagger}+\text{h.c.}\right)$ (4)

Here, $\mu$ is the onsite chemical potential, $N$ is the total number of sites
in the Kitaev chain, and we take the periodic boundary condition for
simplicity.

By defining the Fourier transformation
$\tilde{f}_{k}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{j}f_{j}^{\dagger}e^{ikj}$, we
can rewrite the Hamiltonian in momentum space as

$\hat{H}_{\text{KC}}^{(k)}=\sum_{k}\left(-2t\cos
k-\mu\right)\tilde{f}_{k}^{\dagger}\tilde{f}_{k}-\left(i\Delta\sin
k\tilde{f}_{k}^{\dagger}f_{-k}^{\dagger}+\text{h.c.}\right).$ (5)

To obtain the band structure, we construct the Bogoliubov-de Gennes
Hamiltonian in momentum space,
$\hat{H}_{\text{KC}}^{(k)}=\frac{1}{2}\sum_{k}\tilde{\bm{f}}_{k}^{\dagger}H_{\text{KC}}^{\text{BdG}}(k)\tilde{\bm{f}}_{k}-2t\cos
k-\mu$, where
$\tilde{\bm{f}}_{k}=\left(\tilde{f}_{k},\tilde{f}_{-k}^{\dagger}\right)^{\intercal}$
and

$H_{\text{KC}}^{\text{BdG}}(k)=(-2t\cos k-\mu)\tau_{z}+2\Delta\sin k\tau_{y}.$
(6)

Therefore, by squaring the BdG Hamiltonian, the band structure for the Kitaev
chain in Eq. (4) is $E_{\text{KC}}^{2}(k)=\left(2t\cos
k+\mu\right)^{2}+4\Delta^{2}\sin^{2}k$ .

By applying the Jordan-Wigner transformation Eq. (1), we obtain the
corresponding bosonic spin model by substituting Eq. (3) into Eq. (4),

$\begin{split}\hat{H}_{\text{KC}}^{(\sigma)}&=\sum_{i=1}^{N}-\frac{\mu}{2}\left(1+\sigma_{i}^{z}\right)-\left(t\sigma_{i+1}^{+}\sigma_{i}^{-}+\Delta\sigma_{i+1}^{+}\sigma_{i}^{+}+\text{h.c.}\right)\\\
&=-\frac{\mu
N}{2}-\sum_{i=1}^{N}\frac{t+\Delta}{2}\sigma_{i+1}^{x}\sigma_{i}^{x}+\frac{t-\Delta}{2}\sigma_{i+1}^{y}\sigma_{i}^{y}+\frac{\mu}{2}\sigma_{i}^{z},\end{split}$
(7)

which is the $s=\frac{1}{2}$ 1D transverse-field XY model. Note that Eq. (7)
may be easily converted to the simpler transverse-field Ising model by making
special choices for $t$, $\Delta$, etc., but the generic mapping of the Kitaev
chain is always to the transverse-field XY model, a distinction (i.e., Ising
or XY) often overlooked in discussions of the Kitaev chain.

The direct mapping from the Kitaev chain in Eq. (4) to the transverse-field XY
model in Eq. (7) indicates that they have the same band structures and wave
functions. Furthermore, the two phases in the Kitaev chain also correspond to
the two different phases in the transverse-field XY model, where the phase
transition happens at the same point. In both cases, there are two gapped
phases, separated by a critical point. For the Kitaev chain, the band gap
closes at $2\absolutevalue{t}=\absolutevalue{\mu}$ where two different phases,
trivial and topological, exist for $2\absolutevalue{t}<\absolutevalue{\mu}$
and $2\absolutevalue{t}>\absolutevalue{\mu}$, respectively. For the
transverse-field XY model, if we can reduce it to a transverse-field Ising
model [22] by setting $t=\pm\Delta$ (as we can tell that the change of
$\Delta$ cannot close the band gap from the fermionic Hamiltonian), the two
phases, the paramagnetic and ferromagnetic state, exist for
$2\absolutevalue{t}<\absolutevalue{\mu}$ and
$2\absolutevalue{t}>\absolutevalue{\mu}$, respectively. This paramagnetic to
ferromagnetic transition is, however, not a topological quantum phase
transition (TQPT)– it is a standard breaking of the SU(2) spin symmetry at a
phase transition. Ising model does not possess Ising anyons! Despite their
dual nature revealed by the Jordan- Wigner transformation converting one to
the other, the physics of the two models is qualitatively different. The
Kitaev chain manifests a topological phase transition protected by the
particle-hole symmetry throughout the entire change of parameters, while the
transverse-field XY model is a conventional spontaneous symmetry breaking
phase transition, where the spin $\mathbb{Z}_{2}$ symmetry breaks as
$\absolutevalue{t}$ increases. The spin symmetry is not protected as any
random local magnetic field fluctuation can break it as was already clearly
stated in a footnote in Ref. 3.

##  IV Semiconductor-superconductor nanowire

All of the above are standard procedures to show the duality between the
Kitaev chain and transverse-field XY (or Ising) model via the Jordan-Wigner
transformation. However, the Kitaev chain is a mere toy model to establish the
proof of principle that $p$-wave spinless superconductor can support Majorana
zero modes, which was in fact already pointed out in Ref. 4 in the context of
2D systems and fractional quantum Hall effects. In experiments, the SM-SC
nanowire is a real spinful electronic 1D Majorana nanowire utilizing a
semiconductor with a large Rashba-type SOC in proximity to a conventional
$s$-wave superconductor in the presence of a magnetic Zeeman spin splitting
field to implement an effective spinless $p$-wave superconductor. The minimal
model of a one-dimensional single channel SM-SC nanowire can be described in
continuum real space as

$\hat{H}_{\text{NW}}=\int_{0}^{L}\left[\sum_{s,s^{\prime}}f_{s}^{\dagger}(x)\left(-\frac{\hbar^{2}\partial_{x}^{2}}{2m^{*}}-\mu-i\alpha\sigma_{y}\partial_{x}+V_{\text{Z}}\sigma_{x}\right)_{s,s^{\prime}}f_{s^{\prime}}(x)+\left(\Delta
f_{\uparrow}^{\dagger}(x)f_{\downarrow}^{\dagger}(x)+\text{h.c.}\right)\right]\differential{x},$
(8)

where $L$ is the total length of the nanowire, $f_{s}(x)^{\dagger}$
[$f_{s}(x)$] creates (annihilates) an electron at the position $x$ with spin s
($s=\uparrow$/$\downarrow$), $m^{*}$ is the effective mass of electrons in the
conduction band, $\mu$ is the chemical potential, $\alpha$ is the magnitude of
the Rashba-type SOC which is aligned perpendicular to the direction of the 1D
nanowire, $V_{\text{Z}}$ is the Zeeman splitting energy arising from the
magnetic field applied along the direction of the 1D nanowire, and $\Delta$ is
the proximitized $s$-wave SC pairing term. In Hamiltonian (8), the term in the
first parentheses accounts for the semiconductor denoted as
$\hat{H}_{\text{SM}}$, and the term in the second parenthesis is for the
superconductor denoted as $\hat{H}_{\text{SC}}$. Note that although Eq. (8) is
a realistic Hamiltonian for experimental SM-SC hybrid nanowires, it is still a
free fermion Hamiltonian within the mean-field BdG description of the induced
SC.

Although the SM-SC nanowire serves as a realistic model of the Kitaev chain to
host a pair of Majorana zero modes at the wire ends, the connection between
the Kitaev chain and the SM-SC nanowire remains nonobvious. For example, the
SM-SC nanowire is established in a spinful system with more terms in the
normal (SM) part than the Kitaev chain. Therefore, we wonder how these two
models are connected, e.g., whether there exists any duality between the two
models that allows us to map one model into the other. To answer this
question, we consider two approaches: First, we follow the same procedure as
in the Kitaev chain by converting the fermionic Hamiltonian of the SM-SC
nanowire into the bosonic spin Hamiltonian through the Jordan-Wigner
transformation, and comparing the converted spin model with the transverse-
field XY model (which is what the Kitaev chain (4) is formally equivalent to)
in the hope of finding some hints for how they might be connected.
Alternatively, we can first project the complete spinful Hamiltonian for the
SM-SC nanowire onto an effective low-energy Hamiltonian to obtain an effective
spinless system, and then establish the connection since both systems are
spinless now. We will explain the details of the first approach in Sec. IV.1,
and the second approach in Sec. IV.2.

Since we establish the Kitaev chain in a lattice model Eq. (4), we need to
study the lattice version of the continuous Hamiltonian for the SM-SC nanowire
in Eq. (8) for a direct comparison. The discretization procedure is standard
and used extensively. By replacing the differential operator with the finite
difference (i.e.,
$\partial_{x}f_{s}^{\dagger}(x)\mapsto\left[f_{s}^{\dagger}\left(x+a\right)-f_{s}^{\dagger}(x-a)\right]/{2a}$,
where $a$ is the fictitious lattice constant in the discretization), we obtain
a lattice model for the SM-SC nanowire as

$\hat{H}_{\text{NW}}^{(f)}=\sum_{i=1}^{N}\sum_{s,s^{\prime}}\left[f_{i,s}^{\dagger}\left(2t-\mu+V_{\text{Z}}\sigma_{x}\right)_{s,s^{\prime}}f_{i,s^{\prime}}+f_{i,s}^{\dagger}\left(-t-i\alpha_{R}\sigma_{y}\right)_{s,s^{\prime}}f_{i+1,s^{\prime}}+\text{h.c.}\right]+\left(\Delta
f_{i,\uparrow}^{\dagger}f_{i,\downarrow}^{\dagger}+\text{h.c.}\right)$ (9)

where $f_{i,s}^{\dagger}=f_{s}^{\dagger}(ia)$ and $f_{i,s}=f_{s}(ia)$ are the
creation and annihilation operators defined at the lattice site $i$, effective
nearest neighbor hopping $t=\frac{\hbar^{2}}{2m^{*}a^{2}}$, and reduced SOC
$\alpha_{R}=\frac{\alpha}{2a}$. Here, for a direct comparison between the
lattice model for the SM-SC nanowire and the Kitaev chain in Eq. (4), we
impose the periodic boundary condition and drop the factor of $\frac{L}{N}$
before Eq. (9) which arises from the discretization.

Like the Kitaev chain, the SM-SC nanowire also carries two topologically
distinct phases that cannot be adiabatically connected, which requires us to
first solve the band structure of the SM-SC nanowire to find the vanishing of
the band gap. Therefore, we rewrite the Hamiltonian (9) in momentum space by
imposing the periodic boundary condition as

$\begin{split}&\hat{H}_{\text{NW}}^{(k)}=\sum_{k}\left[2t\left(1-\cos
k\right)-\mu\right]\left(\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{k,\uparrow}+\tilde{f}_{k,\downarrow}^{\dagger}\tilde{f}_{k,\downarrow}\right)\\\
&+\left[\left(V_{\text{Z}}-2i\alpha_{R}\sin
k\right)\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{k,\downarrow}+\Delta\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{-k,\downarrow}^{\dagger}+\text{h.c.}\right]\end{split}$
(10)

where
$\tilde{f}_{k,s}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{j}f_{j,s}^{\dagger}e^{ikj}$,
and we set the fictitious lattice constant $a=1$ for notational simplicity. To
diagonalize Hamiltonian (10), we construct an effective Bogoliubov-de Gennes
(BdG) Hamiltonian,
$\hat{H}_{\text{NW}}^{(k)}=\sum_{k}\frac{1}{2}\Psi(k)^{\dagger}H_{\text{NW}}^{\text{BdG}}(k)\Psi(k)+2t\left(1-\cos
k\right)-\mu$ , where

$\begin{split}H_{\text{NW}}^{\text{BdG}}(k)&=\left[2t\left(1-\cos
k\right)-\mu+V_{\text{Z}}\sigma_{x}+2\alpha_{R}\sin
k\sigma_{y}\right]\tau_{z}\\\ &-\Delta\sigma_{y}\tau_{y},\end{split}$ (11)

and
$\Psi(k)=\left(\tilde{f}_{k,\uparrow},\tilde{f}_{k,\downarrow},\tilde{f}_{-k,\uparrow}^{\dagger},\tilde{f}_{-k,\downarrow}^{\dagger}\right)^{\intercal}$.
Here, $\bm{\sigma}$ and $\bm{\tau}$ are vectors of Pauli matrices acting on
the spin and particle-hole space respectively. Therefore, the band structure
of the SM-SC is

$\begin{split}E_{\text{NW},\pm}^{2}(k)&=\xi^{2}(k)+V_{\text{Z}}^{2}+\Delta^{2}+\alpha^{2}(k)\\\
&\pm
2\sqrt{V_{\text{Z}}^{2}\Delta^{2}+V_{\text{Z}}^{2}\xi^{2}(k)+\alpha^{2}(k)\xi^{2}(k)},\end{split}$
(12)

where $\xi(k)=2t\left(1-\cos k\right)-\mu$, and $\alpha(k)=2\alpha_{R}\sin k$.
By tracking the vanishing of the band gap at $k=0$, we find that the TQPT
happens at $V_{\text{Z}}^{2}=\mu^{2}+\Delta^{2}$ with the trivial
(topological) phase being $V_{\text{Z}}^{2}<\mu^{2}+\Delta^{2}$
($V_{\text{Z}}^{2}>\mu^{2}+\Delta^{2}$) to be contrasted with
$2\absolutevalue{t}<\absolutevalue{\mu}$ (trivial) and
$2\absolutevalue{t}>\absolutevalue{\mu}$ (topological) in the Kitaev chain.

###  IV.1 Complete spinful model and generalized cluster model

We start by directly applying the Jordan-Wigner transformation to the SM-SC
nanowire in Eq. (9). First, we need to generalize the previous definition of
the Jordan-Wigner transformation for the spinless system to a spinful system
with $f_{i,\uparrow}$ and $f_{i,\downarrow}$. We define the nonlocal spin
operators using the same relation for the Jordan-Wigner transformation Eq. (1)
following a designated order as per

$f_{1,\uparrow}~{}f_{1,\downarrow}~{}\cdots~{}f_{i,\uparrow}~{}f_{i,\downarrow}~{}f_{i+1,\uparrow}~{}f_{i+1,\downarrow}~{}\cdots~{}f_{N,\uparrow}~{}f_{N,\downarrow}.$
(13)

Therefore, the Jordan-Wigner transformation for a spinful fermionic system
takes different forms depending on its spin, i.e.,

$\begin{split}f_{i,\uparrow}^{\dagger}&=\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{+},\\\
f_{i,\downarrow}^{\dagger}&=-\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{z}\sigma_{i,\downarrow}^{+},\end{split}$
(14)

where $\sigma_{i,\uparrow}^{+}$ and $\sigma_{i,\downarrow}^{+}$ can be viewed
as the raising operators at site $i$ with spin up and down, respectively.

Since the Hamiltonian for the SM-SC nanowire not only couples the nearest
neighbors but also the further sites [“further” in the sense of the designated
order in Eq. (13)], e.g., the SOC couples $f_{i,\uparrow}^{\dagger}$ and
$f_{i+1,\downarrow}$, which are separated by a distance of 3, it is worth
considering the general form of the spin model arising from the coupling
between any two fermionic sites separated by a distance of $\alpha$ in the
basis of Majorana operators (due to the existence of the SC term),
$\bar{\gamma}_{i,s}\gamma_{i+\alpha,s^{\prime}}$, where
$\gamma_{i,s}=f_{i,s}^{\dagger}+f_{i,s},\bar{\gamma}_{i,s}=i\left(f_{i,s}-f_{i,s}^{\dagger}\right)$.
Following the Jordan-Wigner transformation for spinful fermionic operators in
Eq. (14), we can define the transformation for Majorana operators as,

$\begin{split}\gamma_{i,\uparrow}&=\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{x},\\\
\bar{\gamma}_{i,\uparrow}&=\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{y},\\\
\gamma_{i,\downarrow}&=-\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{z}\sigma_{i,\downarrow}^{x},\\\
\bar{\gamma}_{i,\downarrow}&=-\prod_{j=1}^{i-1}\left(\sigma_{j,\uparrow}^{z}\sigma_{j,\downarrow}^{z}\right)\sigma_{i,\uparrow}^{z}\sigma_{i,\downarrow}^{y}.\end{split}$
(15)

We summarize in Table 1 that the product of two Majorana operators (in the
form of $i\bar{\gamma}\gamma$) that are separated by a distance of $\alpha$
(the index of $\gamma$ minus that of $\bar{\gamma}$) will result in an
$(\alpha+1)$-spin interaction in the form of
$\sigma^{x}\underbrace{\sigma^{z}\dots\sigma^{z}}_{\absolutevalue{\alpha}-1}\sigma^{x}$
or
$\sigma^{y}\underbrace{\sigma^{z}\dots\sigma^{z}}_{\absolutevalue{\alpha}-1}\sigma^{y}$
depending on the sign of $\alpha$, except for $\alpha=0$, which simply
corresponds to a polarizing field along $z$ direction.

Table 1: Spin models for the coupling between two Majorana operators separated by a distance of $\alpha$. Fermionic models | $\alpha$ | Spin models
---|---|---
$i\bar{\gamma}_{i+1,\downarrow}\gamma_{i,\uparrow}$ | -3 | $\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{y}$
$i\bar{\gamma}_{i+1,\uparrow}\gamma_{i,\uparrow}/i\gamma_{i+1,\downarrow}\gamma_{i,\downarrow}$ | -2 | $-\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{y}/-\sigma_{i,\downarrow}^{y}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{y}$
$i\bar{\gamma}_{i,\downarrow}\gamma_{i,\uparrow}/i\bar{\gamma}_{i+1,\uparrow}\gamma_{i,\downarrow}$ | -1 | $\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{y}/\sigma_{i,\downarrow}^{y}\sigma_{i+1,\uparrow}^{y}$
$i\bar{\gamma}_{i,\uparrow}\gamma_{i,\uparrow}/i\bar{\gamma}_{i,\downarrow}\gamma_{i,\downarrow}$ | $0$ | $\sigma_{i,\uparrow}^{z}/\sigma_{i,\downarrow}^{z}$
$i\bar{\gamma}_{i,\uparrow}\gamma_{i,\downarrow}/i\bar{\gamma}_{i,\downarrow}\gamma_{i+1,\uparrow}$ | $1$ | $\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{x}/\sigma_{i,\downarrow}^{x}\sigma_{i+1,\uparrow}^{x}$
$i\bar{\gamma}_{i,\uparrow}\gamma_{i+1,\uparrow}/i\bar{\gamma}_{i,\downarrow}\gamma_{i+1,\downarrow}$ | $2$ | $-\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{x}/-\sigma_{i,\downarrow}^{x}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{x}$
$i\bar{\gamma}_{i,\uparrow}\gamma_{i+1,\downarrow}$ | 3 | $\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{x}$

Therefore, owing to the second term in Eq. (9) describing the hopping between
$f_{i,s}^{\dagger}$ and $f_{i+1,s^{\prime}}^{\dagger}$, we expect many spin-
interactions, i.e., $n$-spin interactions with $n\geq 2$, which looks
superficially similar to the cluster model [21, 23]. Therefore, to obtain the
spin model corresponding to Eq. (9), we first rewrite Eq. (9) in terms of
Majorana operators by replacing $f_{i,s}^{\dagger}f_{i,s}$ with
$\left(1+i\bar{\gamma}_{i,s}\gamma_{i,s}\right)/2$,
$f_{i,s}^{\dagger}f_{j,s^{\prime}}+\text{h.c.}$ with
$i\left(\bar{\gamma}_{i,s}\gamma_{j,s^{\prime}}+\bar{\gamma}_{j,s^{\prime}}\gamma_{i,s}\right)/2$
[$(i,s)\neq(j,s^{\prime})$], and
$f_{i,s}^{\dagger}f_{j,s^{\prime}}^{\dagger}+\text{h.c.}$ with
$i\left(\bar{\gamma}_{i,s}\gamma_{j,s^{\prime}}-\bar{\gamma}_{j,s^{\prime}}\gamma_{i,s}\right)$
[$(i,s)\neq(j,s^{\prime})$]. The Hamiltonian in the Majorana basis is still
simply quadratic (because Hamiltonian (9) is indeed a free fermion problem in
the mean-field level) as per

$\begin{split}&\hat{H}_{\text{NW}}^{(\gamma)}=\left(2t-\mu\right)N+\frac{i}{2}\sum_{i=1}^{N}\\\
&{\left(2t-\mu\right)}\left(\bar{\gamma}_{i,\uparrow}\gamma_{i,\uparrow}+\bar{\gamma}_{i,\downarrow}\gamma_{i,\downarrow}\right)+V_{\text{Z}}\left(\bar{\gamma}_{i,\uparrow}\gamma_{i,\downarrow}+\bar{\gamma}_{i,\downarrow}\gamma_{i,\uparrow}\right)\\\
&-t\left(\bar{\gamma}_{i,\uparrow}\gamma_{i+1,\uparrow}+\bar{\gamma}_{i,\downarrow}\gamma_{i+1,\downarrow}+\bar{\gamma}_{i+1,\uparrow}\gamma_{i,\uparrow}+\bar{\gamma}_{i+1,\downarrow}\gamma_{i,\downarrow}\right)\\\
&-\alpha_{R}\left(\bar{\gamma}_{i,\uparrow}\gamma_{i+1,\downarrow}+\bar{\gamma}_{i+1,\downarrow}\gamma_{i,\uparrow}-\bar{\gamma}_{i,\downarrow}\gamma_{i+1,\uparrow}-\bar{\gamma}_{i+1,\uparrow}\gamma_{i,\downarrow}\right)\\\
&+\Delta\left(\bar{\gamma}_{i,\uparrow}\gamma_{i,\downarrow}-\bar{\gamma}_{i,\downarrow}\gamma_{i,\uparrow}\right).\end{split}$
(16)

Next, we refer to Table 1, and substitute all products of Majorana operators
in Eq. (16) to obtain the Hamiltonian for the corresponding bosonic spin model
for the SM-SC nanowire as per

$\begin{split}&\hat{H}_{\text{NW}}^{(\sigma)}=\left(2t-\mu\right)N+\frac{1}{2}\sum_{i=1}^{N}\\\
&\left(2t-\mu\right)\left(\sigma_{i,\uparrow}^{z}+\sigma_{i,\downarrow}^{z}\right)+\left(V_{\text{Z}}+\Delta\right)\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{x}+\left(V_{\text{Z}}-\Delta\right)\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{y}\\\
&+\alpha_{R}\left(\sigma_{i,\downarrow}^{x}\sigma_{i+1,\uparrow}^{x}+\sigma_{i,\downarrow}^{y}\sigma_{i+1,\uparrow}^{y}\right)+t\left(\sigma_{i,\downarrow}^{x}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{x}\right.\\\
&\left.+\sigma_{i,\downarrow}^{y}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{y}+\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{x}+\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{y}\right)\\\
&-\alpha_{R}\left(\sigma_{i,\uparrow}^{x}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{x}+\sigma_{i,\uparrow}^{y}\sigma_{i,\downarrow}^{z}\sigma_{i+1,\uparrow}^{z}\sigma_{i+1,\downarrow}^{y}\right).\end{split}$
(17)

Equation (17) is essentially a generalized cluster model with staggered
couplings if we relabel the indices of spin operators according to the order
in Eq. (13). By redefining $\sigma_{i,\uparrow}^{x,y,z}=\sigma_{2i-1}^{x,y,z}$
and $\sigma_{i,\downarrow}^{x,y,z}=\sigma_{2i}^{x,y,z}$, the generalized
cluster model becomes obvious as

$\hat{H}_{\text{C}}^{(\sigma)}=\left(2t-\mu\right)N+\sum_{i=1}^{2N}\left(J_{1,i}\sigma_{i}^{z}+\sum_{s=\\{x,y\\}}J_{2,i}^{s}\sigma_{i}^{s}\sigma_{i+1}^{s}+J_{3,i}^{s}\sigma_{i}^{s}\sigma_{i+1}^{z}\sigma_{i+2}^{s}+J_{4,i}^{s}\sigma_{i}^{s}\sigma_{i+1}^{z}\sigma_{i+2}^{z}\sigma_{i+3}^{s}\right),$
(18)

with constant couplings for $J_{1,i}^{x,y}$ and $J_{3,i}^{x,y}$, and the
staggered couplings for $J_{2,i}^{x,y}$ and $J_{4,i}^{x,y}$

$J_{1,i}=\frac{2t-\mu}{2},~{}J_{2,i}^{x}=\begin{cases}\frac{\alpha_{R}}{2}&i\in
2\mathbb{Z}^{+}\\\ \frac{V_{\text{Z}}+\Delta}{2}&i\in
2\mathbb{Z}^{+}-1\end{cases},~{}J_{2,i}^{y}=\begin{cases}\frac{\alpha_{R}}{2}&i\in
2\mathbb{Z}^{+}\\\ \frac{V_{\text{Z}}-\Delta}{2}&i\in
2\mathbb{Z}^{+}-1\end{cases},~{}J_{3,i}^{x}=J_{3,i}^{y}=\frac{t}{2},~{}J_{4,i}^{x}=J_{4,i}^{y}=\begin{cases}0&i\in
2\mathbb{Z}^{+}\\\ -\frac{\alpha_{R}}{2}&i\in 2\mathbb{Z}^{+}-1\end{cases}.$
(19)

Due to the 3-spin and 4-spin interactions, we see that the spin model for the
SM-SC through a direct Jordan-Wigner transformation is much more complicated
than that of the Kitaev chain, i.e., a transverse-field XY model in Eq. (7)
which only contains the polarizing term and 2-spin interactions. Thus, one key
formal difference between the Kitaev chain and Majorana nanowire is that while
the former becomes more transparent theoretically through the Jordan-Wigner
transformation by mapping into a spin problem well-understood for more than 50
years [22], the latter is more transparent, in fact, in the original fermionic
picture, as its spin dual model is essentially an unknown and complex spin
cluster model.

For a generalized cluster model in $1$D, it has been shown [24] that a chain
composed of even-number spin interactions (in the form of
$\sigma^{x}\sigma^{z}\dots\sigma^{z}\sigma^{x}$ ) has $\mathbb{Z}_{2}$
symmetry breaking (SB) while a chain composed of odd-number spin interactions
can carry symmetry-protected topological (SPT) phases. However, in our spin
model in Eqs. (18) and (19), i.e., with a collection of both even number spin
interactions and odd number spin interactions with staggered spin couplings,
the connection to the SPT phases or SB phase, and how the transition happens
are not obvious in the spin language.

###  IV.2 Effective low-energy spinless model and Kitaev chain

Although the direct application of the Jordan-Wigner transformation to the SM-
SC nanowire does not provide a mapping to the Kitaev chain due to the 3-spin
and 4-spin interactions, we can see their connection by projecting the spinful
SM-SC nanowire onto an effective spinless low-energy model. The logic is that:
(1) we first solve the band structure of the SM part in Hamiltonian (9) by
rewriting it in momentum space; (2) then we obtain an effective low-energy
model by projecting out the upper band with higher energy; (3) finally, we
rewrite the projected effective low-energy Hamiltonian back into the real
space and compare that with the Kitaev chain. The logic is based on the idea
that eliminating the spinfulness of the Majorana nanowire in a systematic
manner converting it into an effective low-energy spinless system should
provide a direct clue to its connection with the spinless Kitaev chain.

To solve the SM (normal) term $\hat{H}_{\text{SM}}^{(k)}$ (obtained by setting
$\Delta=0$ in Eq. (10)) as the first step, we can directly diagonalize it to
obtain the band structure of the bare SM, $E_{\text{SM},\pm}(k)=2t\left(1-\cos
k\right)-\mu\pm\sqrt{4\alpha_{R}^{2}\sin^{2}k+V_{\text{Z}}^{2}}$. Therefore,
the Hamiltonian for the SM part can be reconstructed as

$\hat{H}_{\text{SM}}^{(k)}=\sum_{k}E_{+}(k)\tilde{f}_{k,+}^{\dagger}\tilde{f}_{k,+}+E_{-}(k)\tilde{f}_{k,-}^{\dagger}\tilde{f}_{k,-},$
(20)

where $\tilde{f}_{k,+}^{\dagger}$ and $\tilde{f}_{k,-}^{\dagger}$ create an
electron in the upper and lower bands, respectively, and are connected to the
original (physical) $\tilde{f}_{k,\uparrow}^{\dagger}$ and
$\tilde{f}_{k,\downarrow}^{\dagger}$ through the unitary transformation

$\begin{pmatrix}\tilde{f}_{k,+}^{\dagger}\\\
\tilde{f}_{k,-}^{\dagger}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&e^{i\phi(k)}\\\
e^{-i\phi(k)}&-1\end{pmatrix}\begin{pmatrix}\tilde{f}_{k,\uparrow}^{\dagger}\\\
\tilde{f}_{k,\downarrow}^{\dagger}\end{pmatrix},$ (21)

where $\phi(k)=\arctan(\dfrac{2\alpha_{R}\sin k}{V_{\text{Z}}})$.

Since we are only interested in low-energy physics, we can project out the
upper band $E_{+}(k)$ by simply setting $\tilde{f}_{k,+}^{\dagger}$ to 0.
Thus, the effective low-energy model can be obtained by substituting Eq. (21)
into the SC part of Hamiltonian (10) as

$\begin{split}\hat{H}_{\text{eff}}^{(k)}&=\sum_{k}\left[2t\left(1-\cos
k\right)-\mu-\sqrt{4\alpha_{R}^{2}\sin^{2}k+V_{\text{Z}}^{2}}\right]\tilde{f}_{k}^{\dagger}\tilde{f}_{k}\\\
&+\frac{2i\Delta\alpha_{R}\sin k}{\sqrt{(2\alpha_{R}\sin
k)^{2}+V_{\text{Z}}^{2}}}\left(\tilde{f}_{k}^{\dagger}\tilde{f}_{-k}^{\dagger}+\tilde{f}_{k}\tilde{f}_{-k}\right),\end{split}$
(22)

where we drop the band index $\pm$ in fermionic operators from now on due to
the elimination of the upper band. Thus, the effective low-energy model Eq.
(22) can be also viewed as a spinless system, which takes a more similar form
to the Kitaev chain.

However, such a projection has implicitly assumed the large $V_{\text{Z}}$
limit in order to minimize the occupancy on the upper band
$E_{\text{SM},+}(k)$, which further indicates the deep topological regime. For
smaller $V_{\text{Z}}$, the effective low-energy model fails to faithfully
characterize the band bottom of $E_{\text{NW},-}(k)$ near zero momentum, which
is where the band gap vanishes. Therefore, this low-energy model is not able
to reproduce the real TQPT at $V_{\text{Z}}^{2}=\mu^{2}+\Delta^{2}$. Despite
this low-energy model being unsuitable to describe the physics of the SM-SC
nanowire in all regimes, we may still treat the effective low-energy model as
an independent model from a purely theoretical viewpoint by ignoring the fact
that this approximation relies on large $V_{\text{Z}}$.

Therefore, we obtain the band structure and track the vanishing of the band
gap by writing down the Bogoliubov-de Gennes Hamiltonian for Eq. (22),
$\hat{H}_{\text{eff}}^{(k)}=\frac{1}{2}\sum_{k}\tilde{\bm{f}}_{k}^{\dagger}H_{\text{eff}}^{\text{BdG}}(k)\tilde{\bm{f}}_{k}+2t\left(1-\cos
k\right)-\mu-\sqrt{4\alpha_{R}^{2}\sin^{2}k+V_{\text{Z}}^{2}}$, where
$\tilde{\bm{f}}_{k}=\left(\tilde{f}_{k},\tilde{f}_{-k}^{\dagger}\right)^{\intercal}$,
and $H_{\text{eff}}^{\text{BdG}}(k)=\bm{h}(k)\cdot\bm{\tau}$. Here,
$\bm{h}(k)=\left(h_{x}(k),h_{y}(k),h_{z}(k)\right)$ where

$\begin{split}h_{x}(k)&=0,\\\ h_{y}(k)&=-\frac{4\Delta\alpha_{R}\sin
k}{\sqrt{(2\alpha_{R}\sin k)^{2}+V_{\text{Z}}^{2}}},\\\
h_{z}(k)&=2t\left(1-\cos
k\right)-\mu-\sqrt{4\alpha_{R}^{2}\sin^{2}k+V_{\text{Z}}^{2}}.\end{split}$
(23)

Therefore, the band structure is simply
$E_{\text{eff},\pm}(k)=\pm\sqrt{h_{y}(k)^{2}+h_{z}^{2}(k)}$, and band gap
vanishes when $V_{\text{Z}}=-\mu$ at $k=0$ or $V_{\text{Z}}=4t-\mu$ at
$k=\pi$, which divides the phase diagram into three topologically distinctive
regimes ($V_{\text{Z}}<-\mu$, $-\mu<V_{\text{Z}}<4t-\mu$, and
$V_{\text{Z}}>4t-\mu$).

To study the topological property in each regime, we define the winding number
$\mathcal{W}=\frac{1}{2\pi}\oint d\theta(k)$ in the $h_{y}$-$h_{z}$ plane as
the topological invariant, where $\theta(k)=\arctan(h_{z}(k)/h_{y}(k))$, which
is further equivalent to a simpler form

$\mathcal{W}=\text{sign}\left(h_{z}(0)h_{z}(\pi)\right).$ (24)

Therefore, by applying Eq. (24) to the three aforementioned regimes, we find
two phases: the topological phase with $\absolutevalue{\mathcal{W}}=1$ for
$-\mu<V_{\text{Z}}<4t-\mu$, and the trivial phase with
$\absolutevalue{\mathcal{W}}=0$ for $V_{\text{Z}}>4t-\mu$ or
$V_{\text{Z}}<-\mu$. We note that the trivial regime for $V_{\text{Z}}>4t-\mu$
can be eliminated in practice by setting a sufficiently small fictitious
lattice constant $a$. This second trivial phase appears to be an artifact of
discretization. So our statement of large $V_{\text{Z}}$ indicating deep
topological regime still holds in the ideal case, where $t$ approaches
infinity. Therefore, the “physical” phase boundary is just
$V_{\text{Z}}=-\mu$, which is different from the original SM-SC nanowire
($V_{\text{Z}}^{2}=\mu^{2}+\Delta^{2}$) because of the inaccurate projection
from the complete spinful Hamiltonian (10) onto the effective low-energy
spinless Hamiltonian (22) near the real TQPT. Thus, we will focus on the model
deep inside the topological regime with a reasonably good approximation of the
projection, and restore the real space Hamiltonian to uncover the connection
between the SM-SC nanowire and Kitaev chain by establishing a direct mapping.

So far, from the effective low-energy model in momentum space in Eq. (22), we
can already tell that it is a spinless $p$-wave superconductor with a normal
part in the first line, and an antisymmetric SC part in the second line.
However, there are still some slight differences between Eq. (22) and the
Kitaev chain in Eq. (5), e.g., Equation (22) contains a much more complicated
form of the $p$-wave superconducting pairing than Eq. (5). To see the
underlying connection between the two models, we need to obtain the effective
low-energy spinless model in real space by expanding the Hamiltonian (22)
(i.e., $\sqrt{4\alpha_{R}^{2}\sin^{2}k+V_{\text{Z}}^{2}}$ in the normal part
and $\frac{2i\Delta\alpha_{R}\sin k}{\sqrt{(2\alpha_{R}\sin
k)^{2}+V_{\text{Z}}^{2}}}$ in the SC part) near
$\frac{\alpha_{R}}{V_{\text{Z}}}\sim 0$ (assuming the large $V_{\text{Z}}$
limit) to the $n$-th order as shown the second column in Table 2. From the
form of the expansion, we notice the long-range hoppings and pairings arising
from high-frequency modes (e.g., $\cos 2k,\sin 3k$) for the expansion with an
order $>$ 2\. By performing the inverse Fourier transformation, we restore the
effective low-energy spinless model in real space as shown in the third column
in Table 2.

In Table 2, the zeroth order approximation is simply a trivial SM without the
SC pairing term, which is not of any interest. However, more interesting
mapping emerges once we go to the first order, where the Hamiltonian
$\hat{H}_{\text{eff},1}^{(r)}$ becomes an exact Kitaev chain if we map
$(\mu_{\text{KC}},t_{\text{KC}},\Delta_{\text{KC}})\mapsto(\mu-2t+V_{\text{Z}},t,-\frac{2\Delta\alpha_{R}}{V_{\text{Z}}})$.
The TQPT of this mapped Kitaev chain from the first order approximation is
$\absolutevalue{\mu-2t+V_{\text{Z}}}=2t$ (as
$t=\frac{\hbar^{2}}{2m^{*}a^{2}}>0$ ), which also coincides with the TQPT of
the original effective low-energy spinless model (22). Therefore, the Kitaev
chain can be at least viewed as a first-order approximation of the SM-SC
nanowire deep inside the topological regime. For the higher orders of
approximation, more detailed corrections to the normal (SC) part are added at
an even (odd) order of approximation, which corresponds to the Kitaev chain
with a longer range of hoppings and pairings. In principle, the original
spinless effective low-energy model Eq. (22) should correspond to a real space
model with infinite couplings. However, as opposed to the general Kitaev chain
with long-range couplings, which can carry multiple pairs of MZMs [25, 26],
all orders of Kitaev chains in Table 2 are topologically equivalent because
their resulting band structures are all adiabatically connected due to the
exact vanishing of the high-frequency mode at $k=0$ or $k=\pi$ (which is also
where the band inversion happens). In principle, one can have multiple pairs
of MZMs in Majorana nanowires if one allows the multi-subband occupancy [27,
28] (which is actually a more experimentally realistic model), because
Hamiltonian (8) can be constructed completely real in 1D (as long as Zeeman
field is parallel to the nanowire as it is now), leading to an effective
chiral symmetry in class BDI characterized by $\mathbb{Z}$ invariant [29, 30,
31] in 1D system by redefining the time-reversal operator to a mere complex
conjugate $\mathcal{K}$ [32]. Therefore, the multi-subband Hamiltonian Eq. (8)
with $N$ pairs of MZMs is equivalent to $N$ occupied Kitaev chains Eq. (4)
along with $N$ unoccupied Kitaev chains being energetically pushed upward by
the sufficiently large Zeeman splitting field (these $N$ unoccupied bands are
also the bands that we projected out as shown in Eqs. (20) and (21)), which
can be further considered as a single Kitaev chain with long-range hoppings
and pairings in class BDI in 1D due to the real SC pairings in the Hamiltonian
(4). However, in this paper, because we restrict the Majorana nanowire Eq. (8)
in the strict one subband limit, we can only have one pair of MZMs on both
ends in the Majorana nanowire, topologically equivalent to the case of a
single Kitaev chain where next-nearest-neighbor couplings along with all
further couplings in the Kitaev chain vanish [33], which can only carry at
most one pair of MZMs. Furthermore, the fact that the leading terms in the
long-range couplings in Table 2 decay exponentially as
$\left(\frac{\alpha_{R}}{V_{\text{Z}}}\right)^{d}$ with $d$ being the distance
between two coupled sites indicates that this model should essentially be
thought of as a short-range model equivalent to the Kitaev chain with only the
nearest-neighbor hoppings and pairings. Therefore, all effective low-energy
spinless models in Table 2 have the same TQPT, and thus the same phase
diagram. Their essential physics of a spinless $p$-wave topological
superconductor is unchanged despite the formal differences in terms of the
range of hoppings and pairings in the Hamiltonian at different orders of
approximation.

Table 2: Effective low-energy spinless models up to $n$-th order approximation. $n$ |  $k$-space $\hat{H}_{\text{eff},n}^{(k)}=\sum_{k}\dots$ |  $r$-space $\hat{H}_{\text{eff},n}^{(r)}=\sum_{i=1}^{L}\dots$
---|---|---
0 | $\left[2t\left(1-\cos k\right)-\mu-V_{\text{Z}}\right]\tilde{f}_{k}^{\dagger}\tilde{f}_{k}$ | $\left(2t-\mu-V_{\text{Z}}\right)f_{i}^{\dagger}f_{i}-t\left(f_{i}^{\dagger}f_{i+1}+\text{h.c.}\right)$
1 | $\left[2t\left(1-\cos k\right)-\mu-V_{\text{Z}}\right]\tilde{f}_{k}^{\dagger}\tilde{f}_{k}+\left(\frac{2i\Delta\alpha_{R}\sin k}{V_{\text{Z}}}\tilde{f}_{k}^{\dagger}\tilde{f}_{-k}^{\dagger}+\text{h.c.}\right)$ | $\left(2t-\mu-V_{\text{Z}}\right)f_{i}^{\dagger}f_{i}+\left(-tf_{i}^{\dagger}f_{i+1}+\frac{2\Delta\alpha_{R}}{V_{\text{Z}}}f_{i}^{\dagger}f_{i+1}^{\dagger}+\text{h.c.}\right)$
2 | $\matrixquantity{\left[2t\left(1-\cos k\right)-\mu-V_{\text{Z}}-\frac{2\alpha_{R}^{2}\sin^{2}k}{V_{\text{Z}}}\right]\tilde{f}_{k}^{\dagger}\tilde{f}_{k}\\\ +\left(\frac{2i\alpha_{R}\Delta\sin k}{V_{\text{Z}}}\tilde{f}_{k}^{\dagger}\tilde{f}_{-k}^{\dagger}+\text{h.c.}\right)}$ | $\matrixquantity{\left(2t-\mu-V_{\text{Z}}-\frac{\alpha_{R}^{2}}{V_{\text{Z}}}\right)f_{i}^{\dagger}f_{i}+\left(-tf_{i}^{\dagger}f_{i+1}+\frac{\alpha_{R}^{2}}{2V_{\text{Z}}}f_{i}^{\dagger}f_{i+2}+\text{h.c.}\right)\\\ +\left(\frac{2\Delta\alpha_{R}}{V_{\text{Z}}}f_{i}^{\dagger}f_{i+1}^{\dagger}+\text{h.c.}\right)}$
3 | $\matrixquantity{\left[2t\left(1-\cos k\right)-\mu-V_{\text{Z}}-\frac{2\alpha_{R}^{2}\sin^{2}k}{V_{\text{Z}}}\right]\tilde{f}_{k}^{\dagger}\tilde{f}_{k}\\\ +\left[\left(\frac{2i\alpha_{R}\Delta\sin k}{V_{\text{Z}}}-\frac{4i\alpha_{R}^{3}\Delta\sin^{3}k}{V_{\text{Z}}^{3}}\right)\tilde{f}_{k}^{\dagger}\tilde{f}_{-k}^{\dagger}+\text{h.c.}\right]}$ | $\matrixquantity{\left(2t-\mu-V_{\text{Z}}-\frac{\alpha_{R}^{2}}{V_{\text{Z}}}\right)f_{i}^{\dagger}f_{i}+\left(-tf_{i}^{\dagger}f_{i+1}+\frac{\alpha_{R}^{2}}{2V_{\text{Z}}}f_{i}^{\dagger}f_{i+2}+\text{h.c.}\right)\\\ +\left[\left(\frac{2\alpha_{R}\Delta}{V_{\text{Z}}}-\frac{3\alpha_{R}^{3}\Delta}{V_{\text{Z}}^{3}}\right)f_{i}^{\dagger}f_{i+1}^{\dagger}+\frac{\Delta\alpha_{R}^{3}}{V_{\text{Z}}^{3}}f_{i}^{\dagger}f_{i+3}^{\dagger}+\text{h.c.}\right]}$

##  V Conclusion

In this paper, we study the underlying connection between the Kitaev chain toy
model and the SM-SC realistic Majorana nanowire model, motivated by the fact
that both of these 1D systems can host Majorana zero modes at the boundaries
in the open geometry. We first establish the known duality between the Kitaev
chain and the transverse-field XY model (which can be further fine-tuned to a
transverse-field Ising model under a specific set of parameters) through the
Jordan-Wigner transformation, and then ask the question what the corresponding
spin dual model for the SM-SC nanowire looks like if we apply the Jordan-
Wigner transformation directly. Following this logic, we eventually find that,
unlike the Kitaev chain, which can be transformed into a well-known spin model
with only 2-spin interactions, the SM-SC nanowire corresponds to a complicated
dual generalized spin cluster model with staggered couplings, which indicates
that the connection between the Kitaev chain and SM-SC nanowire is not a
priori obvious.

Therefore, we resort to a different approach, which is to project out the
upper band of the bare SM with higher energy to obtain an effective low-energy
spinless nanowire model in the large $V_{\text{Z}}$ limit. This spinless model
in momentum space is already a spinless $p$-wave SC, which can be further
Fourier transformed into the real space to show a direct mapping to the
original Kitaev chain. Finally, we find that the Kitaev chain can be at least
viewed as a first-order approximation of the Majorana nanowire deep in the
topological regime. The two models are identical deep in the MZM-carrying
topological regime. However, the real TQPT cannot be faithfully reproduced in
this effective low-energy model as the projection from a complete spinful
model to the effective low-energy spinless model is only accurate in the large
$V_{\text{Z}}$ limit.

We emphasize that the statement that the Kitaev chain is isomorphic to the
nanowire model [6], as is often claimed in the literature, is incorrect or at
least incomplete since the Kitaev chain is spinless and the nanowire is
spinful, and one cannot thus obtain the nanowire model ever from the Kitaev
chain. However, the opposite statement that the Kitaev chain may be a low-
energy projection of the nanowire model into a single spin subband could be
correct in some situations. In the this paper, we explicitly point out when
this is true, and when it is not, and the necessary conditions for this
restricted isomorphism. In general, the two models are very different except
that they belong to the same Ising anyon universality class for the
topological quantum critical transition between the trivial and the
topological phases.

In principle, we can find more quantitatively accurate projections considering
the SC term as well by constructing a pseudo “time-reversal” symmetry as a
function of momentum to block-diagonalize the complete spinful SM-SC BdG
Hamiltonian such that spin-up and spin-down channels do not couple. However,
an extra degree of freedom then emerges as one can separate the normal part
and SC part arbitrarily, leading to non-unique effective models. But once we
select one effective model among them, we again have a spinless model which
can be used to compare with the Kitaev chain throughout the trivial and
topological regime (not just deep in the topological regime). However, the
choice of this gauge as a function of $k$ is a nontrivial question, which is
left for future work. In fact, it is unclear to us on physical grounds (i.e.,
the assumption of spinless $p$-wave SC in the Kitaev chain) that it is
possible to find a complete theoretical equivalence between these two models
except deep in the topological regime.

## Acknowledgment

We thank Jay Deep Sau, Fengcheng Wu, Jiabin Yu, Yu-An Chen, Fnu Setiawan,
Chun-Xiao Liu, and Dinhduy Vu for helpful discussions. This paper is supported
by the Laboratory for Physical Sciences.

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