# Precursors to Topological Phase Transition in Topological Ladders

Seungju Han  Department of Physics, Korea University, Seoul 02841, South Korea
Mahn-Soo Choi<EMAIL_ADDRESS>Department of Physics, Korea University,
Seoul 02841, South Korea

###### Abstract

We study the coupling of two topologcal subsystems in distinct topological
states, and show that it leads to a precursor behavior of the topological
phase transition in the overall system. This behavior is solely determined by
the symmetry classes of the subsystem Hamiltonians and coupling terms, and is
marked by the persistent existence of subgap states within the bulk energy
gap. By investigating the critical current of Josephson junctions involving
topological superconductors, we also illustrate that such subgap states play
crucial roles in physical properties of nanoscale devices or materials.

Keywords: topological insulator, topological supercondcutors, zero-energy edge
states, suggap states

##  I Introduction

In the more traditional frameworks of the first-order or second-order phase
transitions, the states of matter have been classified by symmetry [1]. For
topological materials, the states are determined by topology, more
specifically, the topological properties of the momentum space that are
independent of deformation [2]. By now, there are a plethora of studies on
topological states of matter and physical phenomena resulting from the
particular topological states [3, 4, 5]. The signature property of gapped
topological materials, i.e., topological insulators, is the so-called zero-
energy states, i.e, the states pinned at the zero energy in the middle of the
insulating energy gap and localized at the boundary of the material [4].
Ultimately related to the bulk-boundary correspondence of the topological
states, the zero-energy states are almost always presumed to be the only
subgap states (if any) in the energy spectra of topological insulators
(including topological superconductors). That is, it is generally accepted
that a topological insulator is characterized by continua of bulk energies
and, in the topologically non-trivial phase, the discrete zero-energy states.

In this work, we demonstrate a generic situation where a topological insulator
retains subgap states of topological origin other than the zero-energy states.
In particular, we show that the interplay between topological states of matter
in two coupled topological materials leads to a genuine _precursor behavior_
of the topological phase transition in the overall system. The precursor
behavior is governed completely by the symmetry classes of the subsystem
Hamiltonians and the coupling term, and are characterized by the persistent
presence of subgap states inside the bulk gap. It is crucial to take into
account such subgap states in many cases such as the analysis of the critical
current of Josephson junctions involving topological superconductors [6, 7,
8]. The effects of these additional subgap states get more pronounced in
nanostructures or low-dimensional materials such as nanowires [9, 10, 11, 12].

##  II Model

We consider a coupled system of two bulk-gapped topological subsystems
governed by Hamiltonian of the form

$\displaystyle\hat{H}_{\text{total}}=\hat{H}_{A}+\hat{H}_{B}+\lambda\hat{V},$
(1)

where $\hat{H}_{A}$ and $\hat{H}_{B}$ represent the Hamiltonians of the
individual subsystems $A$ and $B$ and $\hat{V}$ describes the coupling between
the two subsystems. Each subsystem is assumed to be in a topologically non-
trivial phase. Here, we have introduced an explicit parameter $\lambda$ to
indicate the coupling strength. We will investigate the regime where $\lambda$
is turned on from zero to a small but finite value.

Later, we will show that the coupling $\hat{V}$ leads to a generic interplay
between topologically distinct states in the two topological materials $A$ and
$B$, which gives rise to persistent subgap states inside the bulk gap. As a
brief overview, let us consider a specific case where two subsystems belonging
to the BDI symmetry class. When the subsystems are decoupled ($\lambda=0$),
the system belongs to the BDI $\oplus$ BDI symmetry class. Suppose that both
subsystems are in the topological phase. Then, there are edge states
associated with each of the subsystems localized at the boundaries. We find
that with for weak coupling $\lambda$, there must be genuine subgap states
(including the zero-energy states) inside the bulk gap in the total system.
This is regardless of whether the total system is in the topological or
trivial phase. However, it is instructive to distinguish the two cases. If the
total system is in topologically trivial state after turning on $\lambda$,
then the subgap states must have finite energy (still inside the bulk gap). In
this case, the persistence of the subgap states (well separated from the bulk
spectrum) is guaranteed energetically because the weak coupling cannot lift
the degenerate states from the subsystems into the bulk continuum with a
finite gap. On the other hand, if the total system is in the topological
phase, then the subgap states are at the zero energy. Obviously, in this case,
they are protected for the topological reasons. In both cases, unless
$\lambda$ is too large compared with the bulk-energy gap, the subgap states
are localized at the boundaries of the system.

---
Figure 1: (color online) (a) A schematic of a nanowire system with an external
magnetic field on a superconductor. The dotted double-headed arrow indicates
the direction of the magnetic field induced by the Rashba-type spin-orbit
coupling on the nanowire. (b) A dispersion relation without superconductivity,
$E^{\pm}_{so}(k)$. Blue arrows indicate the directions of the effective
magnetic field at given momentum. At $k=0,\pm\pi$, the effective field is
aligned to the direction of the Zeeman field (positive $x$-axis) due to the
absence of the Rashba field. At $k=\pm\pi/2$, the effective field is most
affected by the Rashba field, which has an opposite direction for opposite
momentum.

###  II.1 Nanowire as an independent two-band model

In many physical systems, the interplay mentioned above is not immediately
apparent. The most prominent example is the nanowire with spin-orbit coupling
in proximity to a superconductor and in an external magnetic field as shown in
Fig. 1 (a). This system is widely recognized as a physical realization of the
theoretical model of Kitaev chain, and is known to demonstrate the topological
phase transition and associated Majorana edge states predicted in the Kitaev
chain [13, 14, 15]. Throughout this work, we will argue that in fact, the
nanowire system involves two topological subsystems (more precisely, two bands
in topologically non-trivial phase as elaborated below) and the interplay
between them gives rise to a precursor behavior, which has been widely ignored
so far, before the topological phase transition.

Here, we briefly review a common picture where the two _uncoupled_ bands in
the nanowire system leads to the topological phase transition predicted in the
Kitaev chain. Later, we will point out that the residual coupling between the
two subsystems plays an important role. The tight-binding Hamiltonian for the
system in the momentum space representation reads

$\displaystyle\hat{H}_{\mathrm{bulk}}$
$\displaystyle=\frac{1}{2}\sum_{k}\hat{\Psi}^{\dagger}_{k}H_{\mathrm{bulk}}(k)\hat{\Psi}_{k}$
(2)

with

$H_{\mathrm{bulk}}(k)=\begin{bmatrix}H_{so}(k)&\Delta_{sc}\sigma_{0}\\\
\Delta_{sc}^{*}\sigma_{0}&-H_{so}^{*}(-k)\end{bmatrix},$ (3)

where

$H_{so}(k)=\left\\{-\mu+t(1-\cos
k)\right\\}\sigma_{0}+\Delta_{Z}\sigma_{x}+\Delta_{so}\sin k\sigma_{y}$ (4)

is the single-particle Rashba Hamiltonian and

$\hat{\Psi}^{\dagger}_{k}=\begin{bmatrix}\hat{c}^{\dagger}_{k\uparrow}&\hat{c}^{\dagger}_{k\downarrow}&\hat{c}_{-k\uparrow}&\hat{c}_{-k\downarrow}\end{bmatrix}$
(5)

is the spinor field in the Nambu notation, $\mu$ is the chemical potential of
the system, $\Delta_{so}$ is the Rashba type spin-orbit coupling along
$y$-direction, $\Delta_{Z}$ is the Zeeman field along $x$-direction, and
$\Delta_{sc}$ is the proximity-induced superconducting pairing potential.
Without loss of generality, all the parameters are assumed to be positive
(with proper choice of gauge if necessary). Band width $2t$ and chemical
potential $\mu$ are typically the largest energy scales in this system, and
accordingly, we will assume that both are much larger than both the spin-orbit
coupling $\alpha_{R}$ and superconducting pairing potential $\Delta_{sc}$
($t,\mu>\Delta_{so},\Delta_{sc}$).

As well known [14], the above system exhibits a topological phase transition
(TPT) at the critical Zeeman field
$\Delta_{Z}^{*}:=\sqrt{\mu^{2}+\Delta_{sc}^{2}}.$ For Zeeman field
$\Delta_{Z}$ greater than $\Delta_{Z}^{*}$, the system is in a topological
regime hosting a pair of Majoranas with one Majorana at each end. We call this
the ‘main’ topological phase transition to distinguish it from similar
transitions in individual subsystems.

The typical picture describing the main topological phase transition starts
with identifying two bands. The single-particle Rashba Hamiltonian in Eq. (4)
gives rise to the two momentum-locked spin bands due to the spin-orbit
coupling, i.e.,

$\hat{H}_{so}(k)=\sum_{\pm}E_{so}^{\pm}(k)\,\hat{c}_{k,\pm}^{\dagger}\hat{c}_{k,\pm}.$
(6)

where the dispersions $E_{so}^{\pm}$ are given by

$E^{\pm}_{so}(k)=-\mu+t(1-\cos
k)\pm\sqrt{\Delta_{Z}^{2}+\Delta_{so}^{2}\sin^{2}k}$ (7)

The two electron operators $\hat{c}_{k,\pm}$ on the two bands $\pm$ are
related to the bare electron operators $\hat{c}_{k,\uparrow/\downarrow}$ via
the following unitary transformation

$\begin{bmatrix}\hat{c}_{k,+}\\\
\hat{c}_{k,-}\end{bmatrix}=U_{k}\begin{bmatrix}\hat{c}_{k,\uparrow}\\\
\hat{c}_{k,\downarrow}\end{bmatrix}\,,\quad
U_{k}:=\frac{1}{\sqrt{2}}\begin{bmatrix}e^{i\phi_{k}}&1\\\
-e^{i\phi_{k}}&1\end{bmatrix},$ (8)

where $\phi_{k}$ is the angle between the directions of the Zeeman field and
the effective magnetic field that electron with momentum $k$ feels, i.e.,
$\sin\phi_{k}=\Delta_{so}\sin k/\Delta_{Z};$ see the blue arrows in Fig. 1
(b). Their typical dispersion relations are as shown in Fig. 1 (b). In these
bands, the spin quantization axes are oriented parallel and anti-parallel,
respectively, to the effective magnetic field consisting of the external
Zeeman field $\Delta_{Z}$ in the $x$-direction and the momentum-dependent
spin-orbit field $\Delta_{so}\sin{k}$ in the $y$-direction.

|
---|---
Figure 2:  (color online) Relation between the nanowire and coupled Kitaev chains. (a) Dispersion relation without superconductivity. Superconducting pairing is represented as arrows. Curved red arrows indicate the pairing within the same band. The black arrow is for the pairing between the different bands. (b) Two coupled Kitaev chains as a model for nanowire. The blue(orange) bar with an illustration of a tight-binding model represents a Kitaev chain that corresponds to the upper(lower) band of the nanowire and the intra-band pairing of the band. Black vertical arrows are for the parametric coupling between two Kitaev chains, which corresponds to the inter-band pairing of the nanowire. |
---|---
Figure 3:  (color online) (a) Schematic illustration of reflections at the
Fermi energy. Spin directions The arrows in black circles represent the spin
directions, and the blue and orange curves display the energy dispersions for
both bands. The dotted curved arrow is for the intra-band reflection, and the
solid curved arrow is for the inter-band reflection. (b) The inter-band (solid
arrow) and intra-band (dotted arrow) reflections in coupled Kitaev chains. In
particular, the inter-band reflection is modeled as hopping with amplitude
$t_{\text{int}}$ between two chains at a single site near the boundary.

Due to the locking of spin to momentum, mathematically, each of the above two
bands $E_{so}^{\pm}$ can be regarded as describing a _spinless_ fermion.
Therefore, turning on the proximity-induced superconducting pairing potential
$\Delta_{sc}$ leads effectively to a $p$-wave pairing within each band.
Overall, the bulk Hamiltonian for the nanowire in Eq. (2) can be effectively
described by

$\hat{H}_{\mathrm{bulk}}\approx\hat{H}_{+}+\hat{H}_{-}$ (9)

with

$\hat{H}_{\pm}:=\frac{1}{2}\sum_{k}\hat{\Psi}_{\pm}^{\dagger}(k)\begin{bmatrix}E_{so}^{\pm}(k)&\pm
i\Delta_{k}^{P}\\\ \mp
i\left(\Delta_{k}^{P}\right)^{*}&-E_{so}^{\pm}(k)\end{bmatrix}\hat{\Psi}_{\pm}(k)\,,$
(10)

where $\Delta_{k}^{P}:=\Delta_{sc}\sin\phi_{k}$ is the strength of the
$p$-wave pairing potentials within individual bands $\pm$, and we have
introduced new spinor fields

$\hat{\Psi}_{\pm}(k):=\begin{bmatrix}\hat{c}^{\dagger}_{k,\pm}&\hat{c}_{-k,\pm}\end{bmatrix}^{T}.$
(11)

In this simplified picture, the two bands may be in the topological or trivial
phase separately, depending on the values of the chemical potential $\mu$ and
the external Zeeman splitting $\Delta_{Z}$. For example, if
$\Delta_{Z}<\Delta_{Z}^{*}$ (with $\mu<2t$), then both bands cross the Fermi
level and would be in the topologically non-trivial phase, leading to two
Majorana edge states (from respective bands) at each end of the nanowire.
Since two Majoranas occupying the same location merge into one Dirac fermion,
the topological state of the nanowire in this case is trivial. On the other
hand, if $\Delta_{Z}^{*}<\Delta_{Z}<\sqrt{(2t-\mu)^{2}+\Delta_{sc}^{2}},$ then
the upper band $E_{so}^{+}$ is off the Fermi level while the lower band
$E_{so}^{-}$ remains crossing the Fermi level, leaving one Majorana edge state
at each end, and hence the nanowire is in the topologically non-trivial phase.
This explains the main topological phase transition at
$\Delta_{Z}=\Delta_{Z}^{*}$.

In reality, however, the two bands are coupled with each other and cannot be
treated separately as in the above argument.

###  II.2 Nanowire as a coupled two-band model

The coupling between the two momentum-locked spin bands $H_{so}^{\pm}$ arises
for two different reasons. One is the bulk effect. Note that due to the
external Zeeman field, the spin quantization axes for opposite momenta are not
exactly opposite. Considering that superconducting pairing occurs between
electrons with opposite spins and momenta, this leads to off-diagonal
components of the pairing in the momentum-locked spin basis. More
specifically, the paring potential can be divided into two parts, the intra-
band pairing, $\Delta^{P}$, already seen in Eq. (10) and inter-band pairing,
$\Delta_{k}^{S}:=\Delta_{sc}\cos\phi_{k},$ depending on whether it is diagonal
or off-diagonal; see Fig. 2 (a). This leads to a more complete description of
the nanowire Hamiltonian in Eq. (2) in the form

$\hat{H}_{\mathrm{bulk}}=\hat{H}_{+}+\hat{H}_{-}+\hat{V},$ (12)

where $\hat{H}_{\pm}$ describe the two bands $\pm$ as defined in Eq. (10) and

$\hat{V}:=\frac{1}{2}\sum_{\pm}\hat{\Psi}_{\pm}^{\dagger}(k)\begin{bmatrix}0&\pm\Delta_{k}^{S}\\\
\mp\left(\Delta_{k}^{S}\right)^{*}&0\end{bmatrix}\hat{\Psi}_{\mp}(k)$ (13)

is responsible for the inter-band coupling.

The other source of the coupling between two bands is the reflection at the
boundary of the nanowire. This coupling originates in the spin mismatch
between left-going and right-going electrons in the same band. When a left-
going electron in the lower band, for example, reflects at the boundary, it
cannot be reflected exclusively into the lower band due to the spin
conservation (see Fig. 3 (a)). There should be a reflected wave in the upper
band as well. This can be clearly seen when we express the annihilation
operators, $\hat{c}_{x,\uparrow}$ and $\hat{c}_{x,\downarrow}$, for electrons
with spin up and down, respectively, in terms of the operators
$\hat{c}_{y,\pm}$ for electrons with the momentum-locked spins $\pm$,

$\begin{bmatrix}\hat{c}_{x,\uparrow}\\\
\hat{c}_{x,\downarrow}\end{bmatrix}=\sum_{y}F(x-y)\begin{bmatrix}\hat{c}_{y,+}\\\
\hat{c}_{y,-}\end{bmatrix},$ (14)

where

$F(x-y):=\frac{1}{N}\sum_{k}e^{ik(x-y)}U_{k}^{\dagger}$ (15)

with unitary matrix $U_{k}$ defined in Eq. (8). Since the spin rotation matrix
$U_{k}$ depends on $k$, $F(x-y)$ is not diagonal in general. This means that
even a simple boundary condition in position space, such as an open or hard
boundary, induces reflections from one band to the other.

###  II.3 Coupled Kitaev Chains

Before we discuss the effects of the inter-band coupling on the topological
properties of the nanowire system, to avoid the unnecessary complexity and to
focus on the main physics, we examine the same effects in a toy model that has
the key features of nanowire. Our toy model is composed of two coupled Kitaev
chains as schematically depicted in Fig. 2 (b) and Fig. 3 (b), its Hamiltonian
reads as

$\hat{H}_{\text{CKC}}=\hat{H}_{\text{KC,1}}+\hat{H}_{\text{KC,2}}+\hat{V}_{\text{D}}+\hat{V}_{\text{R}}.$
(16)

Here, the isolated upper ($\alpha=1$) and lower ($\alpha=2$) Kitaev chains are
described by

$\displaystyle\hat{H}_{\text{KC},\alpha}$
$\displaystyle=-\frac{t_{\alpha}}{2}\sum_{x=1}^{N-1}\hat{c}^{\dagger}_{x+1,\alpha}\hat{c}_{x,\alpha}+\frac{\Delta_{\alpha}}{2}\sum_{x=1}^{N-1}\hat{c}^{\dagger}_{x+1,\alpha}\hat{c}^{\dagger}_{x,\alpha}+h.c.$
$\displaystyle-\mu_{\alpha}\sum_{x=1}^{N}\left(\hat{c}^{\dagger}_{x,\alpha}\hat{c}_{x,\alpha}-\frac{1}{2}\right),$
(17)

where $\hat{c}^{\dagger}_{x,\alpha}$ $(\hat{c}_{x,\alpha})$ is a fermion
creation (annihilation) operator at site $x$, $t_{\alpha}>0$ the hopping
amplitude, $\Delta_{\alpha}$ the pairing potential, and $\mu_{\alpha}$ the
chemical potential on chain $\alpha$. We set $t_{\alpha}>|\mu_{\alpha}|$, such
that without any coupling between the chains, both chains are in the
topological phase. The term

$\hat{V}_{\text{D}}=\Delta_{\text{int}}\sum_{x=1}^{N}\hat{c}^{\dagger}_{x,1}\hat{c}^{\dagger}_{x,2}+h.c.$
(18)

is responsible for the coupling due to the inter-chain pairing potential
$\Delta_{\mathrm{inter}}$ [see Figure 2 (b)] whereas the term

$\hat{V}_{\text{R}}=\left(-t_{\text{int}}\hat{c}^{\dagger}_{1,1}\hat{c}_{1,2}+t_{\text{int}}\hat{c}^{\dagger}_{N,1}\hat{c}_{N,2}\right)+h.c.$
(19)

describes the reflection from one chain to the other at the boundaries
($x=1,N$) with reflection amplitude $\mp t_{\text{int}}$ [see Fig. (3) (b)].

We take the toy model in Eq. (16) as a simplified model of realistic
nanowires, and the connection between the two models is already clear from
corresponding and resembling terms. However, some of the simplifications is
worth mentioning. In the nanowire model, the inter-band pairing,
$\Delta_{k}^{S}=\Delta_{\text{sc}}\cos\phi_{k}$ in Eq. (13), depends on
momentum $k$. However, the complex phase (including signs) does not change as
the momentum $k$ varies because $-\pi/2<\phi_{k}<\pi/2$. Therefore, we take
the inter-chain coupling $\Delta_{\mathrm{inter}}$ constant in $k$ as shown in
Fig. 2 (b). The relative phases of all three pairing potentials, $\Delta_{1}$,
$\Delta_{2}$ and $\Delta_{\text{int}}$ cannot be eliminated by a U(1) gauge
transformation, but the relative phases of any two of them can be fixed by
choosing a proper gauge. Throughout the work, we fix the complex phases of
intra-chain pairing $\Delta_{1}$ and $\Delta_{2}$ to $\pi$ and $0$,
respectively, i.e., $\Delta_{1}<0$ and $\Delta_{2}>0$.

##  III Persistent subgap states

###  III.1 Two Coupled Kitaev Chains

As the first examples of physical implications of the interplay of two
subsystems in topologically distinct states, we first investigate the toy
model of coupled Kitaev chains described in Eq. (16). As we assumed in Section
II.3, both Kitaev chains are in the topological phase when they are decoupled,
and each chain hosts one Majorana bound state at its each end. It will be
instructive to regard the inter-chain pairing $\Delta_{\text{int}}$ and
hopping $t_{\text{int}}$ as independent parameters. When we eventually compare
this toy model to the nanowire model, the inter-chain pairing
$\Delta_{\text{int}}$ must be real-valued. We lift this constraint and allow
it to take _arbitrary_ complex phase to investigate a broader impacts of
coupling on the symmetry classes of the total system.

First, let us turn on only the inter-chain pairing; $\Delta_{\mathrm{in}}\neq
0$ and $t_{\text{int}}=0$. The ratio between bulk gap energy and subgap energy
is shown in Fig. 4 (a). When the coupling is zero, $\Delta_{\text{int}}=0$,
the two Majoranas at each end exist at zero energy(by ignoring the small
coupling proportional to $e^{-L}$), such that the ratio between the bulk gap
and subgap energy, $E_{\text{Subgap}}/E_{\text{Bulk gap}}$ is zero. As we turn
on the coupling, the energy starts to change. Note that if the inter-chain
pairing can have an arbitrary complex phase, the symmetry class of the coupled
system is D with its allowed topological invariant $\mathcal{Z}_{2}$. It means
that at least one of the two Majorana bound states at one end should
disappear. However, as it is well known, a single Majorana bound state cannot
disappear through the interaction with bulk states because of the finite bulk
gap energy. Therefore, the system should be in a topologically trivial phase,
such that the two Majorana bound states are to interact with each other and be
merged into a Dirac state with its energy within the bulk gap, which is shown
in Fig. 4 (a). The nanowire analogy result is shown in Fig. 4 (b). As the
parameter, $\Delta_{\text{int}}$, changes along the positive real axis, we
have nonzero subgap state energy within the bulk gap.

Note that an accidental degeneracy could arise, i.e., an accidental symmetry
may prevent the two Majoranas from interacting with each other. Under the
exclusion of the inter-band hopping, it happens when the inter-chain pairing
is pure imaginary, $\Delta_{\text{int}}=\pm i|\Delta_{\text{int}}|$. In this
case, the subgap states are fixed at zero energy until the coupling becomes
sufficiently large to close the band gap, and the subgap states disappear
through the band touching. The detailed discussion of this regime will be
presented in Appendix Sec. A with relevant figures in Fig. A.1.

|  |
---|---|---
Figure 4:  (color online) Energy profiles of the two coupled Kitaev chains
model as a function of inter-chain parametric pairing, $\Delta_{\text{int}}$.
Inter-chain hopping is turned off. (a) Ratio between the subgap-state energy
and bulk gap energy. The subgap-state energy is measured from the Fermi level,
such that the ratio $E_{\text{Subgap}}/E_{\text{Bulk gap}}$ converges to $0.5$
when the subgap energy is merged into bulk. Inset arrows (Appx.1) is for the
appendix. (b,c) Energy levels along the arrows in panel (a), i.e. complex
phase of the pairing is fixed to $0,\pi/4$, and $\pi/2$ in (b) and (c),
respectively. The pure real $\Delta_{\text{int}}$ corresponds to the nanowire
model. Only the positive energy levels are shown as the system has particle-
hole symmetry. Parameters: The number of sites for a chains $N=60$, on-site
energy $\mu_{1}=0$, $\mu_{2}=1$ (unit energy scale), intra-chain pairings
$\Delta_{1}=-\Delta_{2}=-\mu_{2}$, hoppings $t_{1}=t_{2}=2\mu_{2}$.

Second, we set the parametric pairing equal to zero and consider the inter-
chain reflection as the only coupling between the chains. The results are
presented in Fig. 5. Unlike the varying $\Delta_{\text{int}}$ case, the inter-
chain hopping exists only at the boundaries, such that the bulk band does not
show any significant changes. Besides this difference, the overall behavior of
the energy of the subgap state can be seen within the similar context. The set
of Hamiltonians with the arbitrary complex number $t_{\text{int}}$ is in the
symmetry class D, so the system is in a topologically trivial phase for the
same reason as mentioned above. Starting with zero energy states at
$t_{\text{int}}=0$, we have finite subgap energy well separated from the bulk
band.

|  |
---|---|---
Figure 5:  (color online) Energy profiles of the two coupled Kitaev chains
model as a function of inter-chain hopping, $t_{\text{int}}$. The inter-chain
pairing is set to zero, $\Delta_{\text{int}}=0$. (a) Ratio between the subgap
state energy and bulk gap energy. The subgap energy is measured from the Fermi
level. (b,c) Energy levels along the arrows in panel (a), i.e. complex phase
of the hopping is fixed to $0,\pi/4$, and $\pi/2$ in (b) and (c),
respectively. The pure real $t_{\text{int}}$ corresponds to the nanowire
model. Only the positive energy levels are shown. Parameters: The number of
sites for a chains $N=60$, on-site energy $\mu_{1}=0$, $\mu_{2}=1$ (unit
energy scale), intra-chain pairings $\Delta_{1}=-\Delta_{2}=-\mu_{2}$,
hoppings $t_{1}=t_{2}=2\mu_{2}$.

Finally, in Fig. 6, we take both inter-chain pairing and hopping into account
simultaneously. We set the pure real inter-chain hopping
$t_{\text{int}}=|\Delta_{\text{int}}|$ and vary the pairing with an arbitrary
complex phase as we did previously. Unlike the previous results, the interplay
between two couplings, $t_{\text{int}}$ and $\Delta_{\text{int}}$, makes
subgap energy now depends on the complex phase of $\Delta_{\text{int}}$ more
asymmetrically.

|  |
---|---|---
Figure 6:  (color online) Energy profiles of the two coupled Kitaev chains
model as a function of inter-chain parametric pairing, $\Delta_{\text{int}}$,
with inter-chain hopping $t_{\text{int}}=|\Delta_{\text{int}}|$ at the
boundaries. (a) Ratio between the subgap state energy and bulk gap energy.
Inset arrows (Appx.1) is for the appendix. (b,c) Energy levels as function of
$\Delta_{\text{int}}$ with its complex phase are indicated as an arrow in
panel (a). Parameters: The number of sites for a chains $N=60$, on-site energy
$\mu_{1}=0$, $\mu_{2}=1$ (unit energy scale), intra-chain pairings
$\Delta_{1}=-\Delta_{2}=-\mu_{2}$, hoppings $t_{1}=t_{2}=2\mu_{2}$.

###  III.2 Nanowire

In this section, we will consider a more realistic model, that is, the
nanowire. Subgap energy and the bulk band of the nanowire system as a function
of the Zeeman field are given in Fig. 7. As mentioned earlier in Sec. II.1,
when the Zeeman field is larger than the critical field $\Delta_{Z}^{*}$, the
system is in the topological phase. The Majorana bound states exist at the
zero energy (ignoring exponentially small coupling $e^{-N}$ between Majorana
states at opposite edges). With the Zeeman field smaller than the critical
field, we find the subgap state(black curve) originates in the interplay of
two Majoranas from upper and lower bands, respectively.

Before we discuss the result, let us consider inter-band pairing and
reflection separately, as we did in the previous section.

First, the case with truncated inter-band reflection is following. To
eliminate the inter-band reflection, we express the Hamiltonian in momentum
space using the momentum-locked spin basis depicted in Eq. (11). Then, we take
the inverse Fourier transformation, such that the position space operator,
$\hat{c}_{x_{i},\pm}$, will be written as Eq. (14). Because of the non-local
nature of this basis, the open boundary can no longer be written in a simple
form. Instead of using the usual open boundary condition, as we are interested
in ignoring the inter-band reflection, we impose a new open boundary condition
directly on this basis. In this way, we obtain the red curve in Fig. 7, the
subgap state energy without the inter-band reflection. When the Zeeman field
is zero, there will be no inter-band pairing due to the exact spin matching
between opposite spin electrons with the opposite momentum. So, the subgap
energy converges to zero. As the Zeeman field increases, the inter-band
pairing also increases, such that we have the finite subgap state energy.
After the peak near $\Delta_{Z}\approx 0.1\mu$, it is hard to make a
qualitative description.

To verify that the inter-band reflection is properly truncated and show the
finite subgap energy originates in the inter-band pairing, we calculate energy
levels as a function $\lambda_{S}$, where $\lambda_{S}$ for a modifier in
front of the inter-band pairing,
$\Delta^{S}_{k}\rightarrow\lambda_{S}\Delta^{S}_{k}$ for all $k$. See the
inset of Fig. 7. As the parameter $\lambda_{S}$ goes to zero, there will be no
coupling at all between the two bands, such that the subgap state energy goes
to zero as we have seen in the two coupled Kitaev chains model.

Second, let us describe the case that the inter-band reflection is the only
coupling between two bands. The result is obtained by imposing the usual open
boundary condition to the bulk model with $\lambda_{S}=0$. See the blue curve
in Fig. 7 where the subgap energy solely comes from the inter-band reflection.
When the Zeeman field vanishes, the direction of the effective magnetic field
is simply determined by the Rashba-field. As the direction of the Rashba field
is opposite for electrons with opposite momentum, there will be total
reflection between two bands so that we have large inter-band coupling with
large subgap energy merging into the bulk.

In summary, we find the subgap energy of the nanowire in the parameter regime
before the topological phase transition occurs. Together with the finite
subgap energy, we find the origins of the subgap energy, inter-band pairing
and reflection. The effect of the inter-band pairing is studied with modified
pairing $\lambda_{S}\Delta_{\text{int}}$, while the effect of inter-band
reflection is discussed qualitatively.

Figure 7:  (color online) Subgap energy with different conditions and bulk
energy levels of the nanowire as a function of the Zeeman field is given. Gray
area indicate bulk state energy, and black line is for the subgap state
energy. The red line is the subgap energy when inter-band reflection is
forbidden. The blue line is for the subgap energy without the inter-band
pairing, i.e. the inter-band reflection is the only coupling between bands.
The vertical dashed line is for the inset figure. We used tight binding model
with $N=500,\Delta_{\text{SC}}=0.3\mu$, and $\Delta_{\text{SO}}=0.3\mu$. The
critical Zeeman field is approximately $1.044\mu.$ (Inset) Energy level versus
modified inter-band pairing strength
$\Delta^{S}_{k}\rightarrow\lambda^{S}\Delta^{S}_{k}$ for all $k$. The inter-
band reflection is turned off. The vertical dashed line indicate
$\lambda_{S}=1$. Parameters are $N=500,\Delta_{sc}=0.3\mu,\Delta_{so}=0.3\mu$,
and $\Delta_{Z}=0.25$.

##  IV Critical current

In this section, we now present the physical implication of the subgap states
discussed above on the critical current of the nanowire junction.

We calculate the zero-bias critical current by using linear response theory
for the nanowire connected to a normal superconductor or a topological
superconductor as a function of the Zeeman field and temperature. Both the
normal and topological superconductors, as well as the nanowire, are expressed
by tight-binding models Eq. (2) with appropriate parameters. We assumed a
tunnel junction, expressed by the tunneling Hamiltonian,

$H_{\text{tunnel}}=t\sum_{\sigma}\hat{c}^{\dagger}_{L,\sigma}\hat{c}_{R,\sigma}+h.c.,$
(20)

where $\hat{c}^{\dagger}_{L(R),\sigma}$ is an electron creation operator on
the junction side $L(R)$ with its and spin $\sigma$ and $t$ is a tunneling
amplitude. Without loss of generality, we set $R$ to be the nanowire and $L$
to be a normal superconductor or a topological superconductor.

To find the contribution of the subgap state to the critical current, we
divide the critical current into two parts, localized states contribution and
the rest. The meaning of these division will be presented below.

For the normal superconductor-nanowire junction(NS-NW), the normal
superconductor does not have any localized state. Therefore, as shown in Fig.
8, the critical current can be divided into two terms, bulk-bulk and
localized-bulk. We find a finite contribution to critical current from the
localized state. Note that when the Zeeman field exceeds the critical value,
i.e., when the nanowire is in the topological phase, there is no supercurrent
through the localized state (Majorana bound state) [16]. The finite
contribution of the localized state can be found only in the topologically
trivial regime.

For the topological superconductor-nanowire junction(TS-NW), we find that most
supercurrent flow through the localized states (see Fig. 9). Interestingly,
even when the nanowire is in the trivial phase one cannot ignore the subgap
state; it has a finite contribution to the critical current. Unlike the NS-NW
junction, the temperature severely affect the critical current since the
majority of critical current flows through the lowest energy states, namely,
the subgap state in the nanowire and Majorana bound state in the topological
superconductor.

Figure 8:  (color online) Zero bias critical current $I_{c}/I_{c}^{0}$, where
$I_{c}^{0}:=2e\Delta_{sc}/\hbar$ is the critical current in the short-junction
limit, between a normal superconductor and the nanowire obtained by the tight-
binding model is presented. Parameters are $N=500,\Delta_{\text{SC}}=0.3\mu$
for the normal superconductor and
$N=500,\Delta_{sc}=0.3\mu,\Delta_{so}=0.3\mu$ for the nanowire. Tunneling
amplitude is $t=0.02\mu$. Temperatures are $k_{B}T/\mu=0.04,0.005$. (Inset)
Critical current through the bulk states of the normal superconductor and the
nanowire(Solid line) and through the subgap state at the nanowire and bulk
states of the normal superconductor(Dashed line) with the low temperature is
given. Axis units are the same as used in the main plot.  Figure 9:  (color
online) Zero bias critical current between a topological superconductor and
the nanowire obtained by the tight-binding model is given. Solid lines
indicate critical current. Dotted lines are for current through the localized
states. We use topological superconductor parameters,
$N=500,\Delta_{sc}=0.3\mu,\Delta_{so}=0.3\mu,\Delta_{Z}=1.5$, and the nanowire
parameters = $N=500,\Delta_{sc}=0.3\mu,\Delta_{so}=0.3\mu$. Tunneling
amplitude is $t=0.02\mu$. Temperatures are $k_{B}T/\mu=0.04,0.005$. A small
kink near the phase transition, $\Delta_{Z}/\mu\approx 1.04$, is due to a
systematic error, i.e. the localized state and bulk states are not well
separated near the point. (Inset) The critical current that bulk states
contribute, i.e., bulk states to bulk states and localized states to bulk
states.

##  V Conclusion

We have examined coupled topological subsystems, each in different topological
states of matter, and reveal that this interaction results in a precursor
effect during the topological phase transition within the entire system. This
effect is entirely governed by the symmetry classes of the subsystem
Hamiltonians and the coupling term, and it is characterized by the continuous
presence of subgap states within the bulk energy gap.

By analyzing the critical current in Josephson junctions incorporating
topological superconductors, we have demonstrated the essential roles these
subgap states play in the physical properties of nanostructures and low-
dimensional materials.

###### Acknowledgements.

This work was supported by the National Research Function (NRF) of Korea
(Grant Nos. 2022M3H3A106307411 and 2023R1A2C1005588) and by the Ministry of
Education through the BK21 Four program.

##  Appendix A Accidental symmetry of the two coupled Kitaev chains model

In Section III.1, we investigatd the effects of inter-chain coupling with
arbitrary phase. Here, we will discuss the special case that the inter-chain
pairing is pure imaginary. The point is that this constraint on the inter-
chain coupling can lead to additional anti-unitary symmetries which affect the
topological class of the model. However, such a constraint and result
topological class is illusive in the sense that it easily breaks down in
realistic situation such as local disorders and next nearest-neighbor (NNN)
couplings. On this ground, we emphasize that the classification and
corresponding discussions in Section III.1 are physically valid in general.
Nevertheless, we present the analysis of the topological class of this special
case to avoid potential confusion when one encounters the similar situation
with illusive anti-unitary symmetries.

Recall that the phase of intra-chain pairings are fixed to
$\Delta_{1}=-|\Delta_{1}|$ and $\Delta_{2}=|\Delta_{2}|$ without loss of
generality; in other words, the phase of the inter-chain pairing is a gauge-
fixed quantity.

With the given constraint (and assumption $t_{\text{int}}=0$), the model
acquires a time-reversal-like symmetry operation $\hat{\mathcal{T}}$,

$\displaystyle\hat{\mathcal{T}}\hat{c}_{j,\alpha}\hat{\mathcal{T}}^{-1}$
$\displaystyle=(-1)^{\alpha}\hat{c}_{j,\alpha},\quad\alpha=1,2$
$\displaystyle\hat{\mathcal{T}}i\hat{\mathcal{T}}^{-1}$ $\displaystyle=-i,$
(21)

such that the Hamiltonian satisfies

$\displaystyle\hat{\mathcal{T}}\hat{H}_{\text{CKC}}\hat{\mathcal{T}}^{-1}$
$\displaystyle=\hat{H}_{\text{CKC}}.$ (22)

It is stressed that this symmetry operation is _not_ related to the _physical_
time-reversal symmetry. Since we consider the model of coupled Kitaev chains
as an analogy to the nanowire model which is composed of electrons, the
_physical_ time reversal symmetry operator should satisfy the relation
$\hat{\mathcal{T}}_{\text{physical}}^{2}=(-1)^{\hat{N}}$, where $\hat{N}$ for
electron number operator. To the contrary, the symmetry operator
$\hat{\mathcal{T}}$ of our interest here satisfies $\hat{\mathcal{T}}^{2}=1$.

Though the symmetry described by $\hat{\mathcal{T}}$ is not a _physical_ time-
reversal symmetry, it determines the symmetry class of the Hamiltonian.
Together with the particle-hole constraint of the BdG Hamiltonian, the
relevant symmetry class of the system is BDI and the relevant topological
invariant is the winding number. This winding number captures the number of
the _chiral Majorana bound states_ localized at one end; that is, it
represents the number of Majorana bound states with different chiralities
(i.e., eigenvalues of the chiral operator) at one end. In our case, as briefly
mentioned in Sec. III.1, if we change the strength of the inter-chain pairing
$\Delta_{\text{int}}$, the two chiral Majoranas at each end remain at zero
energy until the gap closes. The winding number is $2$ before the gap closes,
so the two Majorana bound states on the left end of the chains have chirality
$-1$ while the other two states on the right end have chirality $+1$. After
the gap closes, the winding number is changed to 0 and there are no Majorana
bound states anymore; see Fig. A.1 (a).

However, unlike the usual unpaired Majorana bound states, the energy of the
above states can be easily lifted from zero as the time-reversal-like symmetry
can be easily broken if we introduce additional interactions or local
disorder. For example, inter-chain hopping terms,
$t_{\text{inter}}\hat{c}_{j,1}^{\dagger}\hat{c}_{j,2}$, or NNN hopping terms
with nonzero complex phase, i.e.
$t^{(2)}\hat{c}_{j+2,\alpha}^{\dagger}\hat{c}_{j,\alpha}$, would break the
time-reversal-like symmetry, such that the energy of boundary states are
lifted from zero and have finite values.

In Fig. A.1 (b), we consider the model with boundary hopping
$t_{\text{int}}=|\Delta_{\text{int}}|$ turned on, which is already considered
in Sec. II.3 and Fig. 6. As the inter-chain pairing $\Delta_{\text{int}}$
increases, the zero energy is lifted from zero before the band touching. It is
due to direct coupling between two chiral Majorana bound states via the
boundary hopping term. In Fig. A.1 (c), we introduce NNN hopping terms for
each chain,
$\sum_{j,\alpha}t^{(2)}\hat{c}_{j+2,\alpha}^{\dagger}\hat{c}_{j,\alpha}+h.c.$
where $t^{(2)}=0.1e^{i\pi/4}\mu_{2}$. Though the NNN hopping term does not
couple the two chiral Majorana bound states directly as it is intra-chain
coupling, it changes the character of Majorana bound states itself for each
chain, such that the inter-chain pairing couples the two Majorana bound states
and gives them finite energy. In the both cases, the additional terms break
the time-reversal-like symmetry. The relevant symmetry classes of the systems
are D and relevant topological invariants $Z_{2}$(Chern-Simon integral) are 0,
i.e. the systems are in a topologically trivial phase.

By considering other constraints rather than pure imaginary
$\Delta_{\text{int}}$, one may find similar time-reversal-like symmetry. For
example, without boundary hopping terms, the pure real inter-chain pairing
implants a time-reversal like symmetry. The detailed properties of the
symmetry operation and corresponding winding number would change with the pure
imaginary $\Delta_{\text{int}}$ case. Another example is pure imaginary
boundary hopping without any inter-chain pairings.

|  |
---|---|---
Figure A.1:  Energy levels as a function $|\Delta_{\text{int}}|$, where
$|\Delta_{\text{int}}|$ is pure imaginary. (a) Energy levels along the
positive vertical axis of Fig. 4 (a). (b) Energy levels along the positive
vertical axis Fig. 6 (a). (c) Energy levels with next nearest neighbor hopping
term,
$\sum_{j,\alpha}t^{(2)}\hat{c}_{j+2,\alpha}^{\dagger}\hat{c}_{j,\alpha}+h.c.$
where $t^{(2)}=0.1e^{i\pi/4}\mu_{2}$. Parameters: The number of sites for a
chains $N=60$, on-site energy $\mu_{1}=0$, $\mu_{2}=1$ (unit energy scale),
intra-chain pairings $\Delta_{1}=-\Delta_{2}=-\mu_{2}$, hoppings
$t_{1}=t_{2}=2\mu_{2}$.

## References

  * Goldenfeld [1992] N. Goldenfeld, _Phase Transitions and the Renormalization Group_ (Addison-Wesley, New York, 1992).
  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010), URL http://link.aps.org/doi/10.1103/RevModPhys.82.3045.
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Chiu et al. [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016), URL https://link.aps.org/doi/10.1103/RevModPhys.88.035005.
  * Armitage et al. [2018] N. Armitage, E. Mele, and A. Vishwanath, Reviews of Modern Physics 90, 015001 (2018).
  * Tiira et al. [2017] J. Tiira, E. Strambini, M. Amado, S. Roddaro, P. San-Jose, R. Aguado, F. S. Bergeret, D. Ercolani, L. Sorba, and F. Giazotto, Nature Communications 8, 14984 (2017), URL https://doi.org/10.1038/ncomms14984.
  * Gül et al. [2018] Ö. Gül, H. Zhang, J. D. S. Bommer, M. W. A. de Moor, D. Car, S. R. Plissard, E. P. A. M. Bakkers, A. Geresdi, K. Watanabe, T. Taniguchi, et al., Nature Nanotechnology 13, 192 (2018), URL https://doi.org/10.1038/s41565-017-0032-8.
  * Ren et al. [2021] J.-T. Ren, S.-S. Ke, Y. Guo, H.-W. Zhang, and H.-F. Lü, Phys. Rev. B 103, 045428 (2021), URL https://link-aps-org-ssl.oca.korea.ac.kr/doi/10.1103/PhysRevB.103.045428.
  * Kells et al. [2012] G. Kells, D. Meidan, and P. W. Brouwer, Phys. Rev. B 86, 100503 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.86.100503.
  * Bagrets and Altland [2012] D. Bagrets and A. Altland, Phys. Rev. Lett. 109, 227005 (2012), URL https://link.aps.org/doi/10.1103/PhysRevLett.109.227005.
  * Aguado [2017] R. Aguado, La Rivisita del Nuovo Cimento 40, 523 (2017).
  * Peñaranda et al. [2018] F. Peñaranda, R. Aguado, P. San-Jose, and E. Prada, Phys. Rev. B 98, 235406 (2018), URL https://link.aps.org/doi/10.1103/PhysRevB.98.235406.
  * Kitaev [2001] A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001), URL https://dx.doi.org/10.1070/1063-7869/44/10S/S29.
  * Oreg et al. [2010] Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.105.177002.
  * Lutchyn et al. [2010] R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010), URL https://link.aps.org/doi/10.1103/PhysRevLett.105.077001.
  * Zazunov and Egger [2012] A. Zazunov and R. Egger, Phys. Rev. B 85, 104514 (2012), URL https://link.aps.org/doi/10.1103/PhysRevB.85.104514.