# Can non-local conductance spectra conclusively signal Majorana zero modes?
Insights from von Neumann entropy

Abhishek Kejriwal  Department of Physics, Indian Institute of Technology
Bombay, Powai, Mumbai-400076, India  Bhaskaran Muralidharan  Department of
Electrical Engineering, Indian Institute of Technology Bombay, Powai,
Mumbai-400076, India<EMAIL_ADDRESS>
###### Abstract

The topological origin of the zero bias conductance signatures obtained via
conductance spectroscopy in topological superconductor hybrid systems is a
much contended issue. Recently, non-local conductance signatures that exploit
the non-locality of the zero modes in three terminal hybrid setups have been
proposed as means to ascertain the definitive presence of Majorana modes. The
topological entanglement entropy, which is based on the von Neumann entropy,
is yet another way to gauge the non-locality in connection with the bulk-
boundary correspondence of a topological phase. We show that while both the
entanglement entropy and the non-local conductance exhibit a clear topological
phase transition signature for long enough pristine nanowires, non-local
conductance fails to signal a topological phase transition for shorter
disordered wires. While recent experiments have indeed shown premature gap-
closure signatures in the non-local conductance spectra, we believe that the
entanglement entropy can indeed signal a genuine transition, regardless of the
constituent non-idealities in an experimental situation. Our results thereby
point toward furthering the development of experimental techniques beyond
conductance measurements to achieve a conclusive detection of Majorana zero
modes.

Rashba nanowire-superconductor hybrid systems [1, 2, 3, 4, 5, 6, 7] are
preeminent candidates for detecting and manipulating Majorana zero modes
(MZMs) [8], with the ultimate goal of actualizing fault-tolerant topological
quantum computation [9, 10, 11, 12]. Most experimental efforts aimed at the
unambiguous detection of MZMs [13, 14, 15, 16, 17, 18, 17] are based on
observing quantized zero-bias conductance peaks (ZBCPs) in two terminal
normal-topological superconductor (N-TS) links. However, there has been
intense scrutiny and debate due to the non-topological origins of the observed
ZBCPs [19, 20, 21, 22]. Near-zero energy Andreev bound states (ABSs), often
termed as quasi-MZMs, mimic most of the local MZM signatures [23, 24, 25, 26,
27], which calls for other methodologies to distinguish between trivial and
topological zero-energy modes. Recent proposals have focused on exploiting the
non-locality of MZMs via non-local transport measurements on three-terminal
normal-topological superconductor-normal (N-TS-N) setups [28, 29, 30, 31, 32,
33]. These non-local signatures could augment the certainty of MZM detection
by gauging the non-trivial correlations in the system. Theoretical claims have
presented the ability of non-local conductance spectra to detect topological
phase transition based on the closing of the bulk gap at the phase transition
point [30]. However, experiments present contradicting results [33] with the
gap closing well before the topological phase transition point. The object of
this Letter is to gauge the effectiveness of non-local conductance spectra
itself by considering a three terminal setup based on the archetypical TS, the
Kitaev chain, in conjunction with the topological entanglement entropy.

Figure 1: The Experimental Setup. (a) The N-TS-N system considered in this
work. The topological superconductor is the archetypical Kitaev chain
connected to the ground. The normal contacts are connected to potential
$V_{L}$ and $V_{R}$ respectively. Three electrostatic gates dictate the
potential profile of the setup. Two gate voltages represented by $V_{B1}$ and
$V_{B2}$ control the tunneling into the Kitaev chain and the third gate
voltage $V_{G}$ controls the on-site potential $\mu$ in the Kitaev chain. (b)
The pristine setup with no disorder in the on-site potential. The disorder
potential is zero throughout the chain leading to a homogenous on-site
potential. (c) The disordered system with local inhomogeneity in on-site
potential. The disorder potential is gaussian and appears at the contacts.
Such an inhomogeneity occurs due to Fermi energy mismatch as well as charge
inhomogeneities in the system

Figure 2: Eigenspectrum as a function of $\mu/\Delta$ for pristine and
disordered chains with different system parameters. (a) Eigenspectrum for a
pristine setup with N=100 and $t=\Delta$. (b) Eigenspectrum for a disordered
setup with N=100 and $t=\Delta$. (c) Eigenspectrum for a pristine setup with
N=100 and $t=4.1\Delta$. (d) Eigenspectrum for a disordered setup with N=100
and $t=4.1\Delta$. (e) Eigenspectrum for a pristine setup with N=21 and
$t=\Delta$. (f) Eigenspectrum for a disordered setup with N=21 and $t=\Delta$.
(g) Eigenspectrum for a pristine setup with N=21 and $t=4.1\Delta$. (h)
Eigenspectrum for a disordered setup with N=21 and $t=4.1\Delta$ For the
pristine chains, zero energy modes are observed in the topological regime,
i.e, $\mu/\Delta<2t/\Delta$. However, for the disordered chain, zero energy
modes are observed both in the topological regime as well as the trivial
regime.

Non-local correlations related to topological order and the resulting bulk-
boundary correspondence can also be suitably quantified using the degree of
entanglement [34, 35, 36, 37], quantified by the von Neumann entropy [38]. The
idea of using this construct is that entanglement in a higher dimensional
Hilbert space is reflected via the mixedness of the density matrix in a
reduced dimensional Hilbert space. By dividing the composite system into two
subsystems, A and B, each of which having similar internal degrees of freedom,
we calculate the mixedness of either subsystem using the von Neumann entropy
using the partially traced density matrix :
$S_{ent}=-\operatorname{Tr}\left(\rho_{A/B}\ln\rho_{A/B}\right)$. As pointed
out in [34], away from the phase transition point, the entanglement entropy is
directly related to the Berry phase of the ground state. Systems with a non-
trivial Berry phase of $\pi\times$ (odd integer), host topologically protected
boundary modes and have a non-trivial value of entanglement entropy $>\ln 2$
per boundary mode. However, for systems with a trivial Berry phase, the
entanglement entropy vanishes. At the point of phase transition, the value of
entanglement entropy is dictated by the correlation length, which leads to a
spike in the magnitude of the entanglement entropy [39], making it an ideal
metric for gauging the onset of a topological phase transition.
We now present the numerical results performed on a three-terminal N-TS-N
system, as shown in Fig. 1. The three terminal setup [33, 32] consists of a
grounded superconductor [40] connected to the chain, and two contacts from
which independent voltages $(V_{L},V_{R})$ can be applied. We consider two
situations: pristine and disordered, as discussed in previous works [41, 42].
As shown in Fig. 1, the pristine setup has a homogenous on-site potential,
whereas the disordered setup, as shown in Fig. 1, has a local inhomogeneity in
the on-site potential at the ends of the TS chain [42]. The presence of local
inhomogeneity in on-site potential leads to the manifestation of trivial ABSs
in the disordered setup. Our calculations reveal that both non-local
conductance and the entanglement entropy provide a clear phase transition
signature for a long-enough chain. However, the non-local conductance fails to
signal a phase transition for smaller TS chain sizes, whereas the entanglement
entropy stays robust to chain size.
Energy Spectra: We begin by presenting the equilibrium energy spectra for the
pristine and disordered topological superconductor setup. We chose the Kitaev
chain, a archetypical topological superconductor as our choice for the TS
chain. The energy spectrum for the Kitaev chain is well known in literature
and is obtained by numerically diagonalizing the tight-binding Hamiltonian:

$\hat{H}=-\sum_{i=1}^{N}(\mu+V(i))c_{i}^{\dagger}c_{i}+\sum_{i=1}^{N-1}\left(\Delta
c_{i}^{\dagger}c_{i+1}^{\dagger}-tc_{i+1}^{\dagger}c_{i}+\text{h.c.}\right),$
(1)

where, $c_{i}^{(\dagger)}$ represents the annihilation (creation) operator on
a site i, $\mu$ is the electrochemical potential for the setup which remains
constant throughout the chain, $V(i)$ is the additional electrostatic
potential at the $i$th site, $t$ is the nearest neighbor inter-site hopping
term, h.c., stands for the hermitian conjugate, and $\Delta$ is the
superconducting pairing term coupling two neighboring sites as dictated by the
Kitaev chain model.
In the topological phase space, the region $\mu/\Delta<2t/\Delta$ represents
the topological phase and the region $\mu/\Delta>2t/\Delta$ represents the
trivial phase. Figure 2 shows the eigenspectrum for different setup
configurations as a function of $\mu$ with constant $t$ and $\Delta$. Figure 2
shows the eigenspectrum for a pristine chain of length $N=100$ with
$t=\Delta$, which corresponds to the Kitaev point [43, 44]. As expected, the
topological phase ($\mu/\Delta<2$) hosts two MZMs and in the trivial phase
($\mu/\Delta>2$) the MZMs vanish. However, for a disordered chain of length
$N=100$ with $t=\Delta$, as shown in Fig. 2, zero-energy modes are observed in
the trivial regime as well. The figure clearly shows two zero-energy modes for
$\mu/t>2$, which is within the trivial phase. These zero-energy modes are
disorder-induced ABSs [45].
Figure 2 depicts the eigenspectrum for a pristine chain of length $N=100$ with
$t=4.1\Delta$, where the system hosts two MZMs in the topological regime
($\mu/\Delta<8.2$). For a disordered setup, as shown in Fig. 2, the system
hosts near-zero energy modes in the trivial regime as well. We observe similar
features for the systems with smaller chain size of $N=21$ as shown in Figs.
2-2. The disordered setups host near-zero energy ABSs in the trivial regime
and true MZMs in the topological regime. From these results, we can conclude
that the presence of disorder induces zero energy modes in the trivial regime.
Conductance Matrix : We now present the relation for the conductance matrix
for the setup. As shown in Fig. 1, we apply separate voltages $V_{L(R)}$ and
measure terminal currents $I_{L(R)}$ at the left and the right contacts,
respectively. Before defining the conductance matrix ($[G]$), we need the
relation for the terminal currents $I_{L}$ and $I_{R}$.
We use the Keldysh non-equilibrium Green’s function formalism to evaluate the
terminal currents [44, 46]. The Keldysh Green’s functions are defined over the
Keldysh contour, which typically involves the retarded (advanced) Green’s
function $G^{r(a)}$ and the lesser (greater) Green’s function $G^{<(>)}$. The
Fourier transform of the Green’s function into the energy domain gives the
standard steady-state prescription for the transport calculations [44, 46]
involving the density of states, the correlators, and the currents across the
setup. Details of this calculation are presented in the supplementary
material. In the wide band approximation limit, $G^{<}$ is calculated using
the contact broadening matrices and the contact Fermi energies. Once we have
the Green’s functions, we can derive the terminal currents using the Keldysh
current operator [47]. For instance, the terminal electron current at the left
contact can be derived (see supplementary material) in the Landauer Büttiker
form as:

$\displaystyle I_{L}^{(e)}=$ $\displaystyle-\frac{e}{h}\left\\{\int
dET_{A}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E+eV_{L}\right)\right]\right.$
(2) $\displaystyle+\int
dET_{CAR}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E+eV_{R}\right)\right]$
$\displaystyle\left.+\int
dET_{D}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E-eV_{R}\right)\right]\right\\},$

where, $T^{(e)}_{D}(E)$, $T^{(e)}_{A}(E)$, and $T^{(e)}_{CAR}(E)$ represent
the energy resolved transmission probabilities for the direct, Andreev and
crossed-Andreev processes involving the left and right contacts for the
electronic sector of the Nambu space.

Figure 3: Results for the setup with N=100 and $t=\Delta$. The black vertical
lines represent the topological phase boundary. Local conductance signature
for (a) the pristine setup, and (b) for the disordered setup. The ZBCP is
observed in the topological regime in (a) and is observed for both the trivial
and the topological regimes in (b). Non-local conductance signature for (c)
the pristine setup, and (d) the disordered setup. The gap closing is observed
at the topological phase boundary in both (c) and (d). Entanglement entropy
signatures for (e) the pristine setup, and (f) the disordered setup. Observed
features are similar in both setups. The non local conductance shows a gap
closing signature at topological phase transition point for both
configurations. Thus, both entanglement entropy and non-local conductance can
faithfully detect a topological phase transition for the pristine and
disordered setups.

Using the terminal currents, the conductance matrix $[G]$ can be defined as:

$\mathrm{[G]}=\left(\begin{array}[]{cc}G_{LL}&G_{LR}\\\
G_{RL}&G_{RR}\end{array}\right)=\left(\begin{array}[]{ll}\left.\frac{\partial
I_{L}}{\partial V_{L}}\right|_{V_{R}=0}&\left.\frac{\partial I_{L}}{\partial
V_{R}}\right|_{V_{L}=0}\\\ \left.\frac{\partial I_{R}}{\partial
V_{L}}\right|_{V_{R}=0}&\left.\frac{\partial I_{R}}{\partial
V_{R}}\right|_{V_{L}=0}\end{array}\right),$ (3)

where the diagonal matrix elements represent the local conductance at the left
and right contacts, whereas the off-diagonal components represent the non-
local conductance.

Figure 4: Results for the setup with N=100 and $t=4.1\Delta$. The black
vertical lines represent the topological phase boundary. Non-local conductance
signature for (a) the pristine setup, and (b) the disordered setup. The gap
closing is observed at the topological phase boundary in both (a) and (b) .
Entanglement entropy signatures for (c) the pristine setup, and (d) the
disordered setup. The non local conductance shows a gap closing signature at
topological phase transition point for both configurations. Observed features
are similar in both setups. Thus, both entanglement entropy and non-local
conductance can faithfully detect a topological phase transition for the
pristine and disordered setups.

The local conductance at the left contact can be derived by taking a partial
derivative of the left terminal current ($I_{L}$), as given in (2), with the
left contact voltage ($V_{L}$), and the right contact voltage ($V_{R}$) set to
zero, and is given by : $G_{LL}=\left.\frac{\partial I_{L}}{\partial
V_{L}}\right|_{V_{R}=0}$. This translates to the following, in the Landauer-
Büttiker form:

$\displaystyle\left.G_{LL}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{A}(E=eV)+T_{A}(E=-eV)\right.+$ (4)
$\displaystyle\left.T_{CAR}(E=eV)+T_{D}(E=eV)+G_{LL}^{Gnd}(V)\right],$

where the last term $G_{LL}^{Gnd}(V)$ comes as an additional factor due to
currents flowing into the grounded terminal (see supplementary material).

Figure 5: Conductance matrix and entanglement entropy for the setup with N=21
and $t=\Delta$. The black vertical lines represent the topological phase
boundary. Non-local conductance signatures for (a) the pristine setup, and (b)
for the disordered setup. The gap closing is observed at the topological phase
boundary in (a) and is observed before the topological phase transition in
(b). Entanglement entropy signatures for (c) the pristine setup, and (d) for
the disordered setup. The observed entanglement entropy features are similar
in both setups. The non-local conductance gap closing occurs before the
topological phase transition for the disordered setup. Thus, entanglement
entropy can faithfully detect a topological phase transition for both pristine
and disordered setups whereas non-local conductance fails to do so.

The non-local conductance formula can be derived by taking a partial
derivative of the left terminal current ($I_{L}$), as given in (2), over the
right terminal voltage ($V_{R}$), with the left terminal voltage ($V_{L}$) set
to zero, such that, $G_{LR}=\left.\frac{\partial I_{L}}{\partial
V_{R}}\right|_{V_{L}=0}$. This gives us the non-local conductance formula in
terms of the transmission probabilities as:

$\displaystyle\left.G_{LR}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{D}(E=eV)-T_{CAR}(E=-eV)\right],$ (5)

from which, we note that the non-local conductance can show negative values
[30] that can be used to signal a topological phase transition.
Entanglement Entropy : We finally define the entanglement entropy metric for
the TS chain. It is important to note that the entanglement entropy is
calculated for the isolated TS chain. To calculate the entanglement entropy,
we first evaluate the correlation matrix for the chain. The correlation matrix
($C$) is defined as:

$C_{nm}=\left(\begin{array}[]{cc}\left\langle
c^{\dagger}c\right\rangle&\left\langle c^{\dagger}c^{\dagger}\right\rangle\\\
\langle cc\rangle&\left\langle
cc^{\dagger}\right\rangle\end{array}\right)_{nm},$ (6)

where the matrix elements of the correlation matrix can be calculated by
evaluating the expectation values of the operators quadratic in $c$,
$c^{\dagger}$, in the BCS ground state. Once we evaluate the correlation
matrix, we partition the chain into two subsystems, A and B, and assess the
truncated correlation matrix for either of the subsystems. The matrix elements
for the truncated correlation matrix ($\tilde{C}$) for subsystem $A$ can be
defined as follows:

$\displaystyle\tilde{C}_{nm}$ $\displaystyle=C_{(n,m)\in A}.$ (7)

Following the calculation detailed in the supplementary material [48], we can
prove that the entanglement entropy and the eigenvalues of the truncated
correlation matrix are related through the formula:

$\displaystyle
S_{ent}=-\sum_{k=1}^{M}\left[\xi_{k}\log\xi_{k}+\left(1-\xi_{k}\right)\log\left(1-\xi_{k}\right)\right],$
(8)

where $\xi_{k}$ denote the eigenvalues of the truncated correlation matrix.

Figure 6: Conductance matrix and entanglement entropy for the setup with N=21
and $t=4.1\Delta$. The black vertical lines represent the topological phase
boundary. Non-local conductance signature for (a) the pristine setup, and (b)
the disordered setup. The gap closing is observed at the topological phase
boundary in (a) and is observed before the topological phase transition in (b)
. Entanglement entropy signatures for (c) the pristine setup, and (d) the
disordered setup. The observed entanglement entropy features are similar in
both setups. The non-local conductance gap closing occurs before the
topological phase transition for the disordered setup. Thus, entanglement
entropy can faithfully detect a topological phase transition for both pristine
and disordered setups whereas non-local conductance fails to do so.

Now that we have the conductance matrix and entanglement entropy equations, we
illustrate the numerical results for our setup for different system
parameters. Figure 3 depicts the numerical simulations performed for the
Kitaev chain with system size $N=100$ and $t/\Delta=1$ . Using the local
conductance formula (4), we present the local conductance signatures for the
pristine and disordered setups in Figs. 3 and 3. We first re-emphasize the
well-known fact that true MBSs show up as ZBCPs with a quantized conductance
of $2e^{2}/h$ in the local conductance spectrum in the tunneling regime. This
phenomenon can be clearly observed in Fig. 3, where the topological regime
($\mu/\Delta<2$) exhibits a sharp ZBCP. However, the ZBCP signature is also
observed in the trivial regime($\mu/\Delta>2$) for a disordered setup, as
shown in Fig. 3. This indicates that disorder-induced ABSs can also give rise
to ZBCPs in the local conductance spectrum.
Next, we present the non-local conductance signatures for the pristine and
disordered setups using (5). As pointed out in [30], the bulk gap in the non-
local conductance spectrum closes at the topological phase transition point
irrespective of the setup configuration. For the above choice of system
parameters, we clearly observe the gap closing at the topological phase
transition point for pristine and disordered setups, as shown in Figs. 3 and
3, and thus we can say that the non-local conductance can faithfully detect a
topological phase transition. Figures 3 and 3 depict the entanglement entropy
spectrum for the pristine and disordered setups, calculated numerically using
(8). We observe that for both the setup configurations, the entanglement
entropy has a non-trivial value in the topological regime and drops to a near-
zero value at the topological phase transition point. The non-trivial value of
entanglement entropy is associated with the fact that the topological regime
hosts topologically protected MZMs that are non-locally correlated. The
correlation length governs the height of the peak at the phase transition
point for the setup [34]. We re-emphasize the fact that the disordered chain
hosts near-zero energy quasi-MZMs in the trivial regime. Nevertheless, these
quasi-MZMs give a near-zero entanglement entropy value due to their local
nature and thus can be clearly distinguished from true MZMs using entanglement
entropy.
Figure 4 depicts the numerical simulation results for the chain with system
size $N=100$ and $t/\Delta=4.1$. An important point to note is that the local
conductance signatures give similar features as presented in Figs. 3 and 3
irrespective of the system parameters. Thus, for the above system parameters,
we directly start with illustrating the non-local conductance signatures. As
pointed out in Figs. 4 and 4, we again observe a gap-closing signature at the
topological phase transition point irrespective of the presence of disorder.
Thus, for the above choice of parameters, the non-local conductance faithfully
detects a topological phase transition. Figures 4 and 4 present the
entanglement entropy signatures for the pristine and disordered
configurations. Here, we again observe a non-trivial entanglement entropy
value in the topological regime and a near-zero value in the trivial regime
for both pristine and disordered setups with the non-trivial value attributed
to the non-local correlations for true MZMs.
We now present the calculations performed for a system with a smaller size. In
Fig. 5 we illustrate the numerical simulation results for a system with system
size $N=21$ and $t/\Delta=1$. Based on the original proposal [30], we again
expect the gap in the non-local conductance spectrum to close at the
topological phase point. We observe this feature for the pristine chain, as
shown in Fig. 5. However, contrary to what was shown in [30] and what we
observed for a larger system size of $N=100$ as depicted in Figs. 3 and 4, we
observe a gap-closing well before the topological phase transition for the
disordered setup, as shown in Fig. 5. This observation shows that the non-
local conductance is prone to the finite size effect and fails to detect a
topological phase transition for smaller system sizes. The entanglement
entropy spectrum, as shown in Figs. 5 and 5, shows a non-trivial value for the
topological regime and a near-zero value for the trivial regime for both
pristine and disordered configurations. This observation establishes the
robustness of the entanglement entropy metric to system parameters.
Finally, Fig. 6 illustrates the numerical simulation results for the chain
with system size $N=21$ and $t/\Delta=4.1$. The observed results have similar
features as the system with $N=21$ and $t/\Delta=1$, as shown in Fig. 5 . The
pristine setup shows a gap-closing signature at the topological phase
transition point, as shown in Fig. 6. However, for a disordered setup, as
shown in Fig. 6, we observe the gap closing well before the topological phase
transition. This result reiterates that non-local conductance as a metric to
detect topological phase transition is prone to the finite-size effect. The
entanglement entropy again has a non-trivial value in the topological regime
and a near-zero value in the topological regime for both pristine and
disordered configurations, as shown in Figs. 6 and 6. This observation further
emphasizes the robustness of entanglement entropy as a metric to distinguish
between the trivial and topological phases.
Experimental non-local conductance measurements for disordered physical
systems have signaled a gap-closure signature before the appearance of ZBCPs
in the local conductance spectrum [33]. This rules out the connection between
gap-closure and ABSs. One of the possible reasons for the gap-closure can be
related to the characteristic length scales of the disorder-induced subgap
states being longer than the typical device lengths. For our device setup with
local inhomogeneity, we observe a gap closure in the trivial regime for device
lengths smaller than the characteristic localization length of the ABSs.
However, the entanglement entropy signature remains robust to the device size
and faithfully detects a topological phase transition.
The ideas presented here pose the challenge of looking beyond conductance
spectra as means to ascertaining the presence of topological MZMs and
establishing experimental measurement techniques for the entanglement entropy
spectra. One of the possible ways is to figure out the exact ground state
density matrix through quantum state tomography. However, quantum state
tomography is resource-intensive, with the cost scaling exponentially with the
subsystem size. Estimating the statistical correlations in the subsystem
through the technique of random measurements provides an alternate route for
assessing the entanglement entropy [49, 50, 51, 52]. The key idea behind this
protocol is that the second-order Rényi entropies can be quantified using
statistical correlations between the outcomes of measurements performed in
random bases. Other works have presented the connection between entanglement
entropy and the electrical shot noise generated by opening and closing a
quantum point contact periodically in time [53]. It may also be possible to
device setups based on quantum thermodynamics [54, 55] to detect such entropic
signatures.
Conclusion: To summarize, we theoretically proposed the use of topological
entanglement entropy as a metric to distinguish between the topological MZM
and the trivial ABS. We clearly demonstrated that under various experimental
conditions related to the setup, the non-local conductance measurements could
yield false positives and premature gap closures. We showed that while both
the entanglement entropy and the non-local conductance exhibit a clear
topological phase transition signature for long enough pristine nanowires,
non-local conductance fails to signal a topological phase transition for
shorter disordered wires. While recent experiments have indeed shown premature
gap-closure signatures in the non-local conductance spectra, we believe that
the entanglement entropy can indeed signal a genuine transition, regardless of
the constituent non-idealities in an experimental situation. Our results
thereby point toward furthering the development of experimental techniques
beyond conductance measurements to achieve a conclusive detection of Majorana
zero modes.
Acknowedgements: The authors acknowledge Adhip Agarwala for insightful
discussions. The research and development work undertaken in the project under
the Visvesvaraya Ph.D Scheme of the Ministry of Electronics and Information
Technology (MEITY), Government of India, is implemented by Digital India
Corporation (formerly Media Lab Asia). This work is also supported by the
Science and Engineering Research Board (SERB), Government of India, Grant No.
Grant No. STR/2019/000030, the Ministry of Human Resource Development (MHRD),
Government of India, Grant No. STARS/APR2019/NS/226/FS under the STARS scheme.

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## Supplementary Material

###  .1 Entanglement Entropy

####  .1.1 Method 1

Let’s start with considering the system defined by the Hilbert space
$\mathcal{H}$. To calculate the entanglement entropy we divide the system into
two subsystems $A$ and $B$, such that
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. The entanglement entropy
for the entire system is given by the von Neumann entropy of subsystem $A$ or
$B$. The von Neumann entropy for a subsystem can be calculated using the
reduced density matrix for the subsystem which is evaluated as:

$S=-\operatorname{Tr}\left[\rho_{A}\ln{\rho_{A}}\right]=-\sum_{k}\alpha_{k}\ln{\alpha_{k}},$
(S1)

where $\alpha_{k}$ represents the $k^{th}$ eigenvalue of the reduced density
matrix $\rho_{A}$. Now, we need to know what is the reduced density matrix
$\rho_{A}$ for our problem and then we need a way to find its eigenvalues.

To find $\rho_{A}$, we first need to find the ground state wavefunction
$|\Phi_{0}\rangle$ for the entire chain. Starting with the general form of a
Hamiltonian quadratic in the fermionic operators we can write:

$\hat{H}=\sum_{i,j}^{N}\left[c_{i}^{\dagger}A_{ij}c_{j}+\frac{1}{2}\left(c_{i}^{\dagger}B_{ij}c_{j}^{\dagger}-c_{i}B_{ij}c_{j}\right)\right].$
(S2)

For the Kitaev chain setup, the matrix $A$ and $B$ are given by:

$A=\left[\begin{array}[]{ccccccc}-\mu&-t&&&&&\\\ -t&-\mu&-t&&&&\\\
&-t&-\mu&-t&&&\\\ &&\ddots&\ddots&\ddots&&\\\ &&&-t&-\mu&-t&\\\
&&&&-t&-\mu&-t\\\ &&&&&-t&-\mu\end{array}\right]\mspace{20.0mu}\quad
B=\left[\begin{array}[]{ccccccc}0&\Delta&&&&&\\\ -\Delta&0&\Delta&&&&\\\
&-\Delta&0&\Delta&&&\\\ &&\ddots&\ddots&\ddots&&\\\ &&&-\Delta&0&\Delta&\\\
&&&&-\Delta&0&\Delta\\\ &&&&&-\Delta&0\end{array}\right].$ (S3)

Although this Hamiltonian does not conserve particle number, it does conserve
parity. We use the Bugoliubov transformation to get the eigenstates of the
Hamiltonian. This technique uses matrices $g$ and $h$ such that the
Hamiltonian is diagonalized:

$\displaystyle\eta_{k}=\sum_{j}g_{kj}c_{j}+h_{kj}c_{j}^{\dagger},\mspace{30.0mu}\eta_{k}^{\dagger}=\sum_{j}g_{kj}c_{j}^{\dagger}+h_{kj}c_{j},$
(S4)

such that,

$\hat{H}=\sum_{k}\Lambda_{k}\eta_{k}^{\dagger}\eta_{k}+\mathrm{constant}.$
(S5)

The matrices $g$ and $h$ can be chosen using the condition that $\eta_{k}$ and
$\eta_{k}^{\dagger}$ anti-commute. The anti-commutation relation imposes the
condition:

$\left[\eta_{k},\hat{H}\right]=\Lambda_{k}\eta_{k}.$ (S6)

Substituting (S4) and (S5) to (S6) imposes constraints on $g$ and $h$ that can
be written as:

$\begin{array}[]{l}\sum_{j}\left(g_{kj}A_{jn}-h_{kj}B_{jn}\right)=\Lambda_{k}g_{kn}\\\
\sum_{j}\left(g_{kj}B_{jn}-h_{kj}A_{jn}\right)=\Lambda_{k}h_{kn}.\end{array}$
(S7)

To solve for $g$ and $h$, two new matrices are defined,

$\displaystyle\Phi=g+h,\mspace{50.0mu}\Psi=g-h.$ (S8)

This reduces (S7) to an eigenvalue problem in $\Phi$ and $\Psi$.

$\begin{array}[]{l}\Phi(A-B)(A+B)=\Lambda^{2}\Phi\\\
\Psi(A+B)(A-B)=\Lambda^{2}\Psi.\end{array}$ (S9)

The eigenvectors for the above eigenvalue problem give us $\Phi$ and $\Psi$.
We then use (S8) to get back the matrices $g$ and $h$ and thus we have
successfully found a way to diagonalize $\hat{H}$. Since $\hat{H}$ conserves
parity, its ground state can either have odd or even number of fermions.
Without loss of generality, we can assume the ground state to have even number
of fermions. The general form of ground state for a Hamiltonian quadratic in
fermionic operators and with definite parity is given by the BCS wavefunction
which represents a superposition of either odd or even number of particles
depending upon the parity of the system,

$\left|\Phi_{0}\right\rangle=C\exp\left(\frac{1}{2}\sum_{ij}c_{i}^{\dagger}G_{ij}c_{j}^{\dagger}\right)|0\rangle,$
(S10)

where $|0\rangle$ is the vacuum state of the operators $c_{j}$. Since the
ground state $|\Phi_{0}\rangle$ is the vacuum state of opertors $\eta_{j}$,
$\eta_{j}|\Phi_{0}\rangle=0$. This relation can be used to find the matrix $G$
in terms of $g$ and $h$ and some tedious calculation leads us to the relation:
$gG=h$.

Now that we have the ground state wavefunction for the system
$|\Phi_{0}\rangle$, the density matrix is given by
$\rho=|\Phi_{0}\rangle\langle\Phi_{0}|$ which when substituted with (S10)
yields:

$\rho=|C|^{2}\exp\left(\frac{1}{2}\sum_{ij}c_{i}^{\dagger}G_{ij}c_{j}^{\dagger}\right)|0\rangle\langle
0|\exp\left(-\frac{1}{2}\sum_{ij}c_{i}G_{ij}c_{j}\right).$ (S11)

The next step is to get the reduced density matrix for the sub-system A, using
$\rho_{A}=\operatorname{Tr}_{B}(\rho)$. The final step is to directly
diagonalize $\rho_{A}$ to get its eigenvalues. These eigenvalues can be
subsituted to (S1) to get the entanglement entropy. But this method requires
calculating all the intermediate matrices defined above like $g$, $h$, $G$ and
can be quite tedious.

####  .1.2 Method 2

Another way to get the entanglement entropy is to use the correlation matrix
for the system. A rigorous calculation using Grassman algebra [48] simplifies
the reduced density matrix derived in (S11) to:

$\rho_{A}=Kexp(-\mathcal{H}_{A}),$ (S12)

where $K=|C|^{2}$ and $\mathcal{H}_{A}$ is the effective Hamiltonian for the
sub-system $A$, which has the same general form as the original Hamiltonian
$\hat{H}$, the only difference being the matrices $A$ and $B$. If we could
figure out a way to get the eigenvalues of $\mathcal{H}_{A}$, then we can
easily get the eigenvalues of $\rho_{A}$ and thus the entanglement entropy.

Let’s assume that the subsystem $A$ has $M$ sites and the net system has $N$
sites. The effective Hamiltonian $\mathcal{H}_{A}$ can be written as:

$\mathcal{H}_{A}=\sum_{i,j=1}^{2M}\tilde{\mathcal{C}}_{i}^{\dagger}H_{ij}\tilde{\mathcal{C}}_{j},$
(S13)

where
$\tilde{\mathcal{C^{\dagger}}}=(c_{1}^{\dagger},...,c_{M}^{\dagger},c_{1},...,c_{M})$.
$\mathcal{H}_{A}$ can now be diagonalized by defining
$\Psi^{\dagger}=\tilde{\mathcal{C^{\dagger}}}S=(a_{M},...,a_{1},a_{1}^{\dagger},...,a_{M}^{\dagger})$
with $S$ such that
$S^{\dagger}HS=D=diag(-\varepsilon_{M},...,-\varepsilon_{1},\varepsilon_{1},...,\varepsilon_{M})$.
This simple substitution yeilds:

$\mathcal{H}_{A}=\sum_{ij=1}^{2M}\tilde{\mathcal{C}}_{i}^{\dagger}H_{ij}\tilde{\mathcal{C}}_{j}=\sum_{j=1}^{2M}\Psi_{j}^{\dagger}D_{jj}\Psi_{j}=\sum_{j=1}^{M}a_{j}^{\dagger}\varepsilon_{j}a_{j}-a_{j}\varepsilon_{j}a_{j}^{\dagger}=2\sum_{j=1}^{M}\varepsilon_{j}a_{j}^{\dagger}a_{j}-E_{0},$
(S14)

where $E_{0}=\sum_{j=1}^{M}\varepsilon_{j}$. Since the matrix $H$ is not
known, an alternate method is required to find the eigenvalues $\varepsilon$.
This is where the concept of correlation matrix comes in. We’ll see later in
this section how the eigenvalues of the truncated correlation matrix are
related to the eigenvalues of $\mathcal{H}_{A}$. But first, we start by
defining the correlation matrix and the truncated correlation matrix and
describe how it’s calculated.

The correlation matrix for the entire system with size $N$ is defined as:

$C_{nm}=\left(\begin{array}[]{cc}\left\langle
c^{\dagger}c\right\rangle&\left\langle c^{\dagger}c^{\dagger}\right\rangle\\\
\langle cc\rangle&\left\langle
cc^{\dagger}\right\rangle\end{array}\right)_{nm}=\left\langle\mathcal{C}_{n}^{\dagger}\mathcal{C}_{m}\right\rangle=\operatorname{Tr}\left[\rho\mathcal{C}_{n}^{\dagger}\mathcal{C}_{m}\right],$
(S15)

where
$\mathcal{C^{\dagger}}=(c_{1}^{\dagger},....,c_{N}^{\dagger},c_{1},....,c_{N})$.
The elements of the correlation matrix have the form $\left\langle
c_{n}^{\dagger}c_{m}\right\rangle=\left\langle\Phi_{0}\left|c_{n}^{\dagger}c_{m}\right|\Phi_{0}\right\rangle$.
To evaluate these expectation values, we follow a procedure exactly similar to
what we did in (S14). The Hamiltonian for the net system $\hat{H}$ given by
(S2) can be written as $\hat{H}=\mathcal{C}^{\dagger}H_{BdG}\mathcal{C}$ where

$H_{BdG}=\frac{1}{2}\left[\begin{array}[]{cc}A&B\\\ -B&-A\end{array}\right].$
(S16)

Just as (S13), we can define
$\Psi^{\dagger}=C^{\dagger}S=(a_{N},...,a_{1},a_{1}^{\dagger},...,a_{N}^{\dagger})$
to diagonalize $H_{BdG}$, and we finally get:

$\displaystyle\left\langle\Phi_{0}\left|c_{n}^{\dagger}c_{m}\right|\Phi_{0}\right\rangle$
$\displaystyle=\sum_{kl}\left\langle\Phi_{0}\left|\Psi_{k}^{\dagger}S_{kn}^{T}S_{ml}\Psi_{l}\right|\Phi_{0}\right\rangle$
(S17) $\displaystyle=\sum_{kl\leq
N}\left\langle\Phi_{0}\left|a_{N+1-k}S_{kn}^{T}S_{ml}a_{N+1-l}^{+}\right|\Phi_{0}\right\rangle$
$\displaystyle=\sum_{k=1}^{N}S_{k,n}^{T}S_{m,k}.$

Similarly we can get the other terms:

$\displaystyle\left\langle\Phi_{0}\left|c_{n}c_{m}\right|\Phi_{0}\right\rangle=\sum_{k>N,l\leq
N}S_{n,k}S_{m,l}\left\langle\Phi_{0}\left|a_{k-N}a_{N+1-l}^{\dagger}\right|\Phi_{0}\right\rangle=\sum_{k=1}^{N}S_{2N+1-k,n}^{T}S_{m,k},$
(S18)

and

$\displaystyle\left\langle\Phi_{0}\left|c_{n}^{\dagger}c_{m}^{\dagger}\right|\Phi_{0}\right\rangle=\sum_{k=1}^{N}S_{k,n}^{T}S_{m,2N+1-k}\mspace{50.0mu}\left\langle\Phi_{0}\left|c_{n}c_{m}^{\dagger}\right|\Phi_{0}\right\rangle$
$\displaystyle=\sum_{k=1}^{N}S_{k+N,n}^{T}S_{m,k+N}.$ (S19)

The above relations give us the entire correlation matrix $C$. The truncated
correlation matrix $\tilde{C}$ is a sub-matrix of the net correlation matrix
$C$ which is defined as follows:

$\tilde{C}_{nm}=\left(\begin{array}[]{cc}\left\langle
c^{\dagger}c\right\rangle&\left\langle c^{\dagger}c^{\dagger}\right\rangle\\\
\langle cc\rangle&\left\langle
cc^{\dagger}\right\rangle\end{array}\right)_{n,m\in
A}=\operatorname{Tr}\left[\rho_{A}\tilde{\mathcal{C}}_{n}^{\dagger}\tilde{\mathcal{C}}_{m}\right].$
(S20)

Now that we have the truncated correlation matrix, our final task is to derive
a relation between the eigenvalues of the reduced density matrix and the
eigenvalues of the truncated correlation matrix. We start by substituting
(S12) in (S20) to get:

$\tilde{C}_{nm}=\operatorname{Tr}\left[K\exp\left(-\mathcal{H}_{A}\right)\tilde{\mathcal{C}}_{n}^{\dagger}\tilde{\mathcal{C}}_{m}\right].$
(S21)

Substituting (S14) in (S21) gives:

$\tilde{C}=K\operatorname{Tr}\left[\exp\left(-2\sum_{j}\varepsilon_{j}a_{j}^{\dagger}a_{j}+E_{0}\right)\sum_{kl}\Psi_{k}^{+}S_{kn}^{T}S_{ml}\Psi_{l}\right].$
(S22)

This can be simplified as:

$\displaystyle\tilde{C}$
$\displaystyle=Ke^{E_{0}}\sum_{k=1}^{2M}S_{kn}^{T}S_{mk}\operatorname{Tr}\left[\exp\left(-2\sum_{j}\varepsilon_{j}n_{j}\right)\Psi_{k}^{\dagger}\Psi_{k}\right]$
(S23) $\displaystyle=Ke^{E_{0}}\sum_{k\leq
M}S_{kn}^{T}S_{mk}\operatorname{Tr}\left[\exp\left(-2\sum_{j}\varepsilon_{j}n_{j}\right)\left(1-n_{k}\right)\right]+Ke^{E_{0}}\sum_{k>M}S_{kn}^{T}S_{mk}\operatorname{Tr}\left[\exp\left(-2\sum_{j}\varepsilon_{j}n_{j}\right)n_{k}\right],$

where $n_{k}=a^{\dagger}_{k}a_{k}$. The value of K can be evaluated by
imposing the condition $Tr[\rho_{A}]=1$, which is a fundamental property of
density matrices. We can now uncover the following:

$\displaystyle\operatorname{Tr}\left[\rho_{A}\right]$
$\displaystyle=Ke^{E_{0}}\operatorname{Tr}\left[\exp\left(-2\sum_{j}\varepsilon_{j}n_{j}\right)\right]=1$
(S24)
$\displaystyle=Ke^{E_{0}}\left(\prod_{l=1}^{M}\sum_{n_{l}=0,1}\right)\left\langle\Phi_{0}\left|\prod_{l=1}^{M}a_{l}^{n_{l}}e^{-2\varepsilon_{l}n_{l}}a_{l}^{\dagger
n_{l}}\right|\Psi_{0}\right\rangle=Ke^{E_{0}}\prod_{l=1}^{M}\sum_{n_{l}=0,1}e^{-2\varepsilon_{l}n_{l}}=Ke^{E_{0}}\prod_{l=1}^{M}\left(1+e^{-2\varepsilon_{l}}\right)$
$\displaystyle\Longrightarrow
K=e^{-E_{0}}\prod_{l=1}^{M}\frac{1}{1+e^{-2\varepsilon_{l}}}.$

Using the above two relations we get:

$\displaystyle\tilde{C}_{nm}$ $\displaystyle=\sum_{k\leq
M}S_{kn}^{T}S_{mk}\left(1-\frac{e^{-2\varepsilon_{M+1-k}}}{1+e^{-2\varepsilon_{M+1-k}}}\right)+\sum_{k>M}S_{kn}^{T}S_{mk}\frac{e^{-2\varepsilon_{k-M}}}{1+e^{-2\varepsilon_{k-M}}}$
(S25) $\displaystyle=\sum_{k\leq
M}S_{kn}^{T}S_{mk}\frac{1}{1+e^{-2\varepsilon_{M+1-k}}}+\sum_{k>M}S_{kn}^{T}S_{mk}\frac{1}{1+e^{2\varepsilon_{k-M}}}.$

Comparing this with $H_{nm}$ in (S13), which was,

$\displaystyle H_{nm}$
$\displaystyle=\sum_{k}S_{nk}\lambda_{k}S_{km}^{T}=\sum_{k\leq
M}S_{kn}^{T}S_{mk}\left(-\varepsilon_{M+1-k}\right)+\sum_{k>M}S_{kn}^{T}S_{mk}\varepsilon_{k-M},$
(S26)

where $\lambda_{k}=D_{kk}$. Thus in terms of $\lambda_{k}$,

$\displaystyle H_{nm}$ $\displaystyle=\sum_{k}\lambda_{k}S_{kn}^{T}S_{mk}$
(S27) $\displaystyle\tilde{C}_{nm}$
$\displaystyle=\sum_{k}\frac{1}{1+e^{2\lambda_{k}}}S_{kn}^{T}S_{mk}=\sum_{k}\xi_{k}S_{kn}^{T}S_{mk}.$

We can clearly see that $H$ and $\tilde{C}$ have the same set of eigenvectors
and their eigenvalues $\lambda_{k}$ and $\varepsilon_{k}$ are related as:

$\begin{array}[]{l}\xi_{k}=\frac{1}{1+e^{2\lambda_{k}}},\mspace{50.0mu}\lambda_{k}=\frac{1}{2}\log\left(\frac{1-\xi_{k}}{\xi_{k}}\right).\end{array}$
(S28)

These relations finally lead us to the equation for entanglement entropy in
terms of the eigenvalues of the truncated correlation matrix
$\varepsilon_{k}$:

$\displaystyle S_{ent}$
$\displaystyle=-\operatorname{Tr}\left[\rho_{A}\log\rho_{A}\right]=\sum_{k=1}^{M}\left[\frac{2\varepsilon_{k}}{1+e^{2\varepsilon_{k}}}+\log\left(1+e^{-2\varepsilon_{k}}\right)\right]$
(S29)
$\displaystyle=-\sum_{k=1}^{M}\left[\xi_{k}\log\xi_{k}+\left(1-\xi_{k}\right)\log\left(1-\xi_{k}\right)\right].$

###  .2  The Keldysh NEGF approach

####  .2.1 Current calculations

First, we start with the assumption that the leads are large enough so that
they can be considered semi-infinite, and thus be effectively modelled by
infinite dimensional matrices. This assumption allows using the NEGF approach
by partitioning the channel and the leads and working with the channel Green’s
function and incorporating the leads via self energies. The retarded channel
Green’s function can now be written as:

$G^{r}(E)=\left[(E+i\eta)I-H_{BdG}-\Sigma_{L}^{r}-\Sigma_{R}^{r}\right]^{-1},$
(B.1)

where E is the free variable energy, and I is the identity matrix of the
dimension of the Hamiltonian, $\eta$ is a small positive damping parameter,
and $\Sigma^{r}_{L}$ and $\Sigma^{r}_{R}$ represent the retarted self energies
for the semi-infinite contacts.

Figure S1: NEGF

The self energies can be represented in two possible choice of basis: (i) In
the eigenbasis of the real-space Hamiltonian, (ii) the eigen basis with wide-
band approximation, where the self energy is encapsulated via a parameter
$\Gamma_{L/R}$. In the basis of the real-space hamiltonian, the self-energy
matrices can be evaluated by a straightforward calculation of the matrix
elements $\Sigma^{r}_{\alpha}\left(i,i\right)=\sigma_{\alpha\in L/R}$ :

$\sigma_{\alpha\in
L/R}(E)=\sum_{k}\frac{\left|\tau_{\alpha,k}\right|^{2}}{\left(E-\epsilon_{k\alpha}+i\eta\right)}=P\int
d\epsilon_{k\alpha}\frac{\left|\tau_{\alpha,k}\right|^{2}}{\left(E-\epsilon_{k\alpha}\right)}-i\pi\int
d\epsilon_{k\alpha}\left|\tau_{\alpha,k}\right|^{2}\delta\left(E-\epsilon_{k\alpha}\right),$
(B.2)

where P stands for the Cauchy principal value. Recasting this integral into a
real and an imaginary part
($\sigma_{\alpha}=\epsilon_{\alpha}\left(E\right)-i\gamma_{\alpha}/2$)
provides a better physical picture , with the imaginary part
$\gamma_{\alpha}=i\left(\sigma^{r}_{\alpha}-\sigma^{a}_{\alpha}\right)$
representing the level broadening, and the real part $\epsilon_{\alpha}$
calculated from the principal value integral.

In this work, we stick to the wide-band approximation basis. Under this
approximation, $\tau_{k\alpha,m}$ is independent of energy and thus the
principal value integral representing the real-part of the self energy
vanishes and we are just left with the broadening matrix
$\Gamma_{\alpha}\left(E\right)$ such that the self energies can be represented
as $\Sigma^{r}_{\alpha}=-i\Gamma_{\alpha}/2$, which can be treated as an input
parameter. Once we are done calculating the self energies, we need to evaluate
the “lesser” Green’s function:

$G^{<}(E)=G^{r}(E)\left(\Sigma_{L}^{<}(E)+\Sigma_{R}^{<}(E)\right)G^{a}(E),$
(B.3)

where $G^{a}$ is the advanced Green’s function (hermitian conjugate of the
retarded Green’s function calculated from (B.1)). Calculating the lesser
Green’s function is equivalent to calculating the electron (hole) correlation
function. Here, $\Sigma^{<}_{L,R}(E)$ represent the in-scattering functions
from leads L and R respectively which can be evaluated using:

$\displaystyle\Sigma_{\alpha}^{<}(E)=-\left[\Sigma_{\alpha}^{r}(E)-\Sigma_{\alpha}^{a}(E)\right]f_{\alpha}(E)=i\Gamma_{\alpha}(E)f_{\alpha}(E),$
(B.4)

where $f_{\alpha}=f(E-\mu_{\alpha})$ represents the Fermi-Dirac distribution
in either leads $\alpha=L(R)$ with a chemical potential $\mu_{\alpha}$. In
this formulation, all different components of the currents can then be deduced
from the current operator formula. The elements of the current operator matrix
can be deduced using the correlation function and the hopping elements of the
Hamiltonian. A rigorous but straightforward calculation based on fundamental
consideration gives the current operator matrix:

$\displaystyle
I_{L}^{op}(E)=\frac{e}{h}\left[G^{r}(E)\Sigma_{L}^{<}-\Sigma_{L}^{<}G^{a}(E)\right.\left.+G^{<}(E)\Sigma_{L}^{a}(E)-\Sigma_{L}^{r}(E)G^{<}(E)\right].$
(B.5)

Taking the trace of the above equation gives the net charge current, which is
given by the current formula:

$I_{L}(E)=\frac{ie}{h}\text{ Trace
}\left[\Gamma_{L}(E)f_{L}(E)\left(G^{r}(E)-G^{a}(E)\right)+\Gamma_{L}(E)G^{<}(E)\right].$
(B.6)

The above formula can be further simplified by the notation introduced in with
the spectral function $A(E)=i(G^{r}-G^{a})$ and the electron correlation
matrix $G^{n}(E)=-iG^{<}(E)$ as

$I_{L}(E)=\frac{e}{h}\text{ Trace
}\left[\Gamma_{L}(E)f_{L}(E)A(E)-\Gamma_{L}(E)G^{n}(E)\right].$ (B.7)

Using simple manipulations as given below we get:

$\displaystyle A(E)$
$\displaystyle=i\left(G^{r}-G^{a}\right)=iG^{r}\left[\left(G^{a}\right)^{-1}-\left(G^{r}\right)^{-1}\right]G^{a}=iG^{r}\left[\Sigma^{r}-\Sigma^{a}\right]G^{a}=G^{r}\Gamma
G^{a},$ (B.8)

where $\Gamma(E)=\Gamma_{L}(E)+\Gamma_{R}(E)$. Using the above in conjunction
with (B.3) and (B.4), along with the properties of the trace operation, yields
the Landauer transmission formula for the current as

$I_{L}=\int
dETrace\left[\Gamma_{L}G^{r}\Gamma_{R}G^{a}\right]\left(f_{L}(E)-f_{R}(E)\right),$
(B.9)

where $f_{L(R)}(E)$ represents the Fermi-Dirac distribution in either lead.
However, in the case involving superconductors, either functioning as a
contact or as the central system or both, the net current through the contact
$\alpha$ is the difference between electron and hole currents in Nambu space.
We can now use the full fledged current operator in (B.5) to evaluate the
current from first principles as

$I_{L}^{op}(E)=\frac{e}{h}\tau_{z}\left[G^{r}(E)\Sigma_{L}^{<}(E)-\Sigma_{L}^{<}(E)G^{a}(E)+G^{<}(E)\Sigma_{L}^{a}(E)-\Sigma_{L}^{r}(E)G^{<}(E)\right],$
(B.10)

where $\tau_{z}=\sigma_{z}\otimes I_{N\times N}$ , where is the Pauli-z matrix
and is the identity matrix of dimension $N\times N$. The next step is to take
a trace of the above equation, which in general need not take the form of
(B.6), typically when the contacts are superconducting due to the non-diagonal
structure of $\Sigma_{L(R)}$, and may lead to erroneous results, specifically
when evaluating the Josephson currents. However, in our case, as the contacts
are normal, and hence diagonal, (B.10) indeed takes the form of (B.6) with the
net current becoming

$I^{\alpha}=\int dE\frac{(I_{\alpha}^{(e)}-I_{\alpha}^{(h)})}{2},$ (B.11)

where each current $I^{e(h)}_{\alpha}$ can be evaluated separately using the
form in (B.6) or (B.7). The factor of a half comes since the original
Hamiltonian been doubled while writing the BdG Hamiltonian in order to write
it consistently in the Nambu space. It is then instructive in our context to
note that various matrices defined within the approach will have a matrix
structure due to the electron-hole Nambu space. In particular, the contact
broadening matrices etc., can be written with a general diagonal structure
(due to the diagonal structure of normal contacts in Nambu space) as
$\Gamma_{\alpha}=\Gamma^{ee}_{\alpha}+\Gamma^{hh}_{\alpha}$, where the
superscripts $ee(hh)$ represent the electron (hole) diagonal part of the self
energy or broadening matrix. Following this and using the above observations
on the current operator formula in (B.7), we obtain the electron (hole)
current across the TS as a sum of three components:

$\displaystyle I_{L}^{e(h)}(E)$ $\displaystyle=\frac{e}{h}\left(\text{ Trace
}\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{ee(hh)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{ee(hh)}(E)\right]\right)\longrightarrow(i)$
(B.12) $\displaystyle+\frac{e}{h}\left(\text{ Trace
}\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{L}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{L}^{hh(ee)}(E)\right]\right)\longrightarrow(ii)$
$\displaystyle+\frac{e}{h}\left(\text{ Trace
}\left(\Gamma_{L}^{ee(hh)}G^{r}\Gamma_{R}^{hh(ee)}G^{a}\right)\left[f_{L}^{ee(hh)}(E)-f_{R}^{hh(ee)}(E)\right]\right),\longrightarrow(iii)$

where the term (i) represents the direct transmission process of either the
electron or the hole, (ii) represents the direct Andreev transmission and
(iii) represents the crossed Andreev transmission. At this point it is worth
noting that $f^{ee}_{\alpha}=f(E-\mu_{\alpha})$,
$f^{hh}_{\alpha}=f(E+\mu_{\alpha})$. The matrix structure of the broadening
matrix $\Gamma$ itself is diagonal such that
$\Gamma^{ee}_{L}=\Gamma_{L}(1,1)=\gamma$,
$\Gamma^{hh}_{L}=\Gamma_{L}(2,2)=\gamma$,
$\Gamma^{ee}_{R}=\Gamma_{R}(2N-1,2N-1)=\gamma$,
$\Gamma^{hh}_{R}=\Gamma_{R}(2N,2N)=\gamma$ and zeros otherwise. The Andreev
transmission can then be written as
$T_{A}(E)=\gamma^{2}|G^{r,eh}_{11}(E)|^{2}$ and the crossed Andreeev
transmission as $T_{CA}(E)=\gamma^{2}|G^{r,eh}_{1N}(E)|^{2}$ the direct
transmission can be written as $T_{D}(E)=\gamma^{2}|G^{r,ee}_{1N}(E)|^{2}$.
Here $G^{r,ee(hh)}_{ij}$ represents the i, j element of the electron (hole)
diagonal block of the retarded Green’s function in Nambu space and
$G^{r,eh(he)}_{ij}$ represents that of the off-diagonal block of the retarded
Green’s function in Nambu space.

###  .3 Local and non-local conductances

####  .3.1 Floating superconductor configuration

We start with the current formula for a generic bias situation,
($V_{L}$,$V_{R}$) for the left and right contacts with $\mu_{L}=eV_{L}$ and
$\mu_{R}=eV_{R}$

The current through one of the contacts has contributions from both electron
and hole flow. The net current is given by:

$I^{L}=\frac{I_{L}^{(e)}-I_{L}^{(h)}}{2}.$ (B.13)

The individual electron and hole components can then be derived using:

$\displaystyle I_{L}^{(e)}=-\frac{e}{h}$ $\displaystyle\left\\{\int
dET_{A}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E+eV_{L}\right)\right]\right.$
(B.14) $\displaystyle+\int
dET_{CAR}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E+eV_{R}\right)\right]$
$\displaystyle\left.+\int
dwET_{D}^{(e)}(E)\left[f\left(E-eV_{L}\right)-f\left(E-eV_{R}\right)\right]\right\\},$

$\displaystyle I_{L}^{(h)}=-\frac{e}{h}$ $\displaystyle\left\\{\int
dET_{A}^{(h)}(E)\left[f\left(E+eV_{L}\right)-f\left(E-eV_{L}\right)\right]\right.$
(B.15) $\displaystyle+\int
dET_{CAR}^{(h)}(E)\left[f\left(E+eV_{L}\right)-f\left(E-eV_{R}\right)\right]$
$\displaystyle\left.+\int
dET_{D}^{(h)}(E)\left[f\left(E+eV_{L}\right)-f\left(E+eV_{R}\right)\right]\right\\}.$

Here, $I^{(e)}_{L}$ and $I^{(h)}_{L}$ can be shown to be equal and opposite
using symmetry conditions and especially:

$\displaystyle T_{CAR}^{(e)}(E)\equiv
T^{(h)}_{CAR}(E),\mspace{50.0mu}T_{D}^{(e)}(E)\equiv T_{D}^{(h)}(-E).$ (B.16)

It is thus sufficient to consider either one of $I^{(e)}_{L}$ and
$I^{(h)}_{L}$ while calculating the net current through a contact. Let us now
consider the conductance matrix:

$\mathrm{G}=\left(\begin{array}[]{cc}G_{LL}&G_{LR}\\\
G_{RL}&G_{RR}\end{array}\right)=\left(\begin{array}[]{cc}\left.\frac{\partial
I_{L}}{\partial V_{L}}\right|_{V_{R}=0}&\left.\frac{\partial I_{L}}{\partial
V_{R}}\right|_{V_{L}=0}\\\ \left.\frac{\partial I_{R}}{\partial
V_{L}}\right|_{V_{R}=0}&\left.\frac{\partial I_{R}}{\partial
V_{R}}\right|_{V_{L}=0}.\end{array}\right)$ (B.17)

Now let us consider only the electron current for now since
$I_{L}=I^{(e)}_{L}$. Using this fact we can evaluate the Local conductance as:

$\displaystyle G_{LL}$ $\displaystyle\equiv\left.\frac{\partial
I_{L}}{\partial V_{L}}\right|_{V_{R}\equiv 0}$ (B.18)
$\displaystyle\equiv\frac{e}{h}\left\\{\frac{\partial}{\partial
V_{L}}\left[\int dET_{A}(E)[f(E-eV_{L})-f(E+eV_{L})]\right.\right.+\int
dET_{CAR}(E)[f(E-eV_{L})-f(E)]$
$\displaystyle\mspace{450.0mu}+\left.\left.\int
dET_{D}(E)[f(E-eV_{L})-f(E)]\right]\right\\}$
$\displaystyle\equiv\frac{e}{h}\left[\int
dE\right.T_{A}(E)\left[\frac{\partial f\left(E-eV_{L}\right)}{\partial
V_{L}}-\frac{\partial f(E+eV_{L})}{\partial V_{L}}\right]+\int
dET_{CAR}(E)\left[\frac{\partial f(E-eV_{L})}{\partial V_{L}}\right]$
$\displaystyle\mspace{450.0mu}+\left.\int dET_{D}(E)\left[\frac{\partial
f(E-eV_{L})}{\partial V_{L}}\right]\right].$

At zero temperature, the $f\left(x\right)=\Theta\left(-x\right)$ where
$\Theta$ is the Heaviside step function. This implies
$f\left(E-eV\right)=\Theta\left(eV-E\right)$ and similarly we can write the
other terms. Since the derivative of the heaviside function is the Dirac Delta
function, we get:

$\displaystyle\left.G_{LL}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{A}(E=eV)+T_{A}(E=-eV)\right.+\left.T_{CAR}(E=eV)+T_{D}(E=eV)\right].$
(B.19)

Now we can similarly get the non local conductance.

$\displaystyle G_{LR}=$ $\displaystyle\left.\frac{\partial I_{L}}{\partial
V_{R}}\right|_{V_{L}\equiv 0}$ (B.20) $\displaystyle=$
$\displaystyle\frac{e}{h}\left\\{\frac{\partial}{\partial V_{R}}\left[\int
dET_{CAR}(E)[f(E)-f(E+eV_{R})]\right.\right.\left.\left.+\int
dET_{D}(E)\left[f\left(E\right)-f\left(E-eV_{R}\right)\right]\right]\right\\}$
$\displaystyle=$ $\displaystyle\frac{e}{h}\left[\int
dET_{CAR}(E)\left[\frac{-\partial f(E+eV_{R})}{\partial
V_{R}}\right]\right.+\left.\int dET_{D}(E)\left[\frac{-\partial
f(E-eV_{R})}{\partial V_{R}}\right].\right]$

Once again, using the property of fermi dirac functions as $T\rightarrow 0$ as
$f(x)=\Theta(-x)$ we get:

$\displaystyle\left.G_{LR}(V)\right|_{T\rightarrow
0}\equiv\frac{e^{2}}{h}\left[T_{D}(E=eV)-T_{CAR}(E=-eV)\right].$ (B.21)

####  .3.2 Grounded superconductor configuration

When the proximitizing superconductor is connected to the ground, current can
flow into the superconducting terminal as well. This leads to an additional
self-energy term ($\Sigma_{S}$) in the Green’s function. The Green’s function
for the grounded setup is thus given by :

$G^{r}(E)=\left[(E+i\eta)I-H_{BdG}-\Sigma_{L}^{r}-\Sigma_{R}^{r}-\Sigma_{S}\right]^{-1},$
(B.22)

where $\Sigma_{S}$ can be derived using a method similar (B.2) and is given
by:

$\Sigma_{S}(E)=-\gamma\frac{E+\Delta\tau_{x}}{\sqrt{\Delta^{2}-E^{2}}},$
(B.23)

where $\gamma$ represents the coupling strength between the Kitaev chain and
the superconducting contact, and $\tau_{x}$ represents the Pauli-sigma $x$
matrix.

Due to the presence of the superconducting contact, as we’ll see below, an
additional term gets added to the previously derived formula for local
conductance (B.19), and no additional term gets added to the non-local
conductance formula (B.21).

Starting with the general current operator for the left contact (B.10), we get
the electron current for the left contact as:

$\displaystyle I_{L}^{e}(E)$
$\displaystyle=\frac{e}{h}\operatorname{Tr}\left[\Sigma_{L}^{<}(E)\left(G^{r}(E)-G^{a}(E)\right)\right.\left.-G^{<}(E)\left(\Sigma_{L}^{a}-\Sigma_{L}^{r}\right)\right]$
(B.24)
$\displaystyle=\frac{e}{h}\operatorname{Tr}\left[\Gamma_{L}f_{L}\left(G^{r}(E)\Gamma
G^{a}(E)\right)\right.\left.+iG^{r}(E)\left(\Sigma_{L}^{<}+\Sigma_{R}^{<}+\Sigma_{S}^{<}\right)G^{a}(E)\Gamma_{L}\right],$

where $\Sigma^{<}_{S}$ is defined by the (B.4) with the Fermi energy of the
superconducting contact given by $f_{S}$. The broadening matrix $\Gamma$ is
the sum of the broadening matrices for left,right and superconducting
contacts: $\Gamma=\Gamma_{L}+\Gamma_{R}+\Gamma_{S}$. On further expanding the
terms in (B.24) we get,

$I_{L}^{e}(E)=\frac{e}{h}\operatorname{Tr}\left[\Gamma_{L}f_{L}G^{r}(E)\left(\Gamma_{L}+\Gamma_{R}+\Gamma_{S}\right)G^{a}(E)+iG^{r}(E)\left(i\Gamma_{L}f_{L}(E)+i\Gamma_{R}f_{R}(E)+i\Gamma_{S}f_{S}(E)\right)G^{a}(E)\Gamma_{L}\right].$
(B.25)

The additional terms due to the presence of the superconducting contact are:

$\frac{e}{h}\operatorname{Tr}\left[\underbrace{\Gamma_{L}f_{L}G^{r}(E)\Gamma_{S}G^{a}(E)}_{(\mathrm{i})}-\underbrace{G^{r}(E)\Gamma_{S}f_{S}(E)G^{a}(E)\Gamma_{L}}_{(\mathrm{ii})}\right].$
(B.26)

The term (i) here has a $V_{L}$ dependence since $f_{L}=f(E-eV_{L})$, whereas
the term (ii) has neither $V_{L}$ nor $V_{R}$ dependence. Since the first term
has a $V_{L}$ dependence it adds an additional term to the local conductance
formula (B.19):

$G^{Gnd}_{LL}(V_{L})=\frac{e}{h}\operatorname{Tr}\left[\Gamma_{L}\frac{df_{L}}{dV_{L}}G^{r}(E)\Gamma_{S}G^{a}(E)\right].$
(B.27)

Using the fact that the derivative of Fermi function at zero temperature is
the Dirac-delta function, we get the additional term to local conductance as:

$G^{Gnd}_{LL}(V)=\frac{e^{2}}{h}\operatorname{Tr}\left[\Gamma_{L}G^{r}(V)\Gamma_{S}G^{a}(V)\right].$
(B.28)

Since neither of the terms in (B.26) have $V_{R}$ dependence, no additional
term gets added to the non-local conductance equation (B.21) on introducing
the grounded superconductor.