# Conductance matrix symmetries of multiterminal semiconductor-superconductor
devices

Andrea Maiani  Center for Quantum Devices, Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  Max Geier  Center for Quantum
Devices, Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen,
Denmark  Karsten Flensberg  Center for Quantum Devices, Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark

###### Abstract

Nonlocal tunneling spectroscopy of multiterminal semiconductor-superconductor
hybrid devices is a powerful tool to investigate the Andreev bound states
below the parent superconducting gap. We examine how to exploit both
microscopic and geometrical symmetries of the system to extract information on
the normal and Andreev transmission probabilities from the multiterminal
electric or thermoelectric differential conductance matrix under the
assumption of an electrostatic potential landscape independent of the bias
voltages, as well as the absence of leakage currents. These assumptions lead
to several symmetry relations on the conductance matrix. Next, by considering
a numerical model of a proximitized semiconductor wire with spin-orbit
coupling and two normal contacts at its ends, we show how such symmetries can
be used to identify the direction and relative strength of Rashba versus
Dresselhaus spin-orbit coupling. Finally, we study how a voltage-bias-
dependent electrostatic potential as well as quasiparticle leakage breaks the
derived symmetry relations and investigate characteristic signatures of these
two effects.

##  I Introduction

Tunneling spectroscopy is a powerful tool for studying superconductor-
semiconductor hybrid devices as it provides a clear signature for Andreev
bound states (ABS). Nonlocal conductance spectroscopy is the natural extension
of local two-probe spectroscopy and overcomes some of its limitations.
Initially used in the context of the search for signatures of Cooper-pair
splitting [1, 2, 3, 4, 5], this type of measurement has been recently
considered in the context of topological superconductivity (TS) [6, 7, 8]
leading to its use in experiments [9, 10], its inclusion in identification
protocols for Majorana bound states (MBSs) in nanowires [11] and
unconventional superconductors vortex cores [12], as well as for the
characterization of chiral Majorana edge states in two-dimensional (2D) TS
[13] and the helical gap in two-dimensional electron gases (2DEGs) [14].
Moreover, the same concept appears in experiments involving quantum dots to
probe the non-equilibrium dynamics of quasiparticles [4, 15].

When an electron current flows across the device, aside from the electric
charge current, energy and heat currents flow too. For this reason, the
spectral features of the device, including peaks connected to the onset of the
topological phase, can also be identified when analyzing the thermal
conductance [16, 17]. Nevertheless, measurements of thermal transport require
a very complex and delicate experimental setup. Easier experiments are the
ones that study thermoelectric transport, where the system is driven out of
equilibrium by using leads thermalized at different temperatures while the
measured output is still a charge current. Thermoelectric measurements have
been proposed as an additional tool to investigate subgap features and
identify MBSs [18].

Motivated by recent experimental success in measuring multiterminal electric
differential conductance [2, 10, 9, 19, 20, 5, 21, 22, 23] and the need to
characterize hybrid superconductor-semiconductor devices, we here extend the
theory of multiterminal tunneling spectroscopy to extract additional
information on the electronic and Andreev transmission processes from linear
combinations of local and nonlocal differential conductance measurements at
different bias voltage or magnetic fields. These linear combinations are
derived using conditions that follow from quasiparticle-number conservation,
microreversibility, and particle-hole conjugation in the presence of
superconductivity. Further relations can be derived in the presence of
geometrical symmetries, such as mirror symmetry, or less general Hamiltonian
symmetries like additional antiunitary symmetries.

Figure 1: Examples of multiterminal superconductor-semiconductor devices used
as toy models in this work: (a) double-dot Josephson junction, and (b) a spin-
orbit coupled semiconductor nanowire (blue) proximitized by an $s$-wave
superconductor (gray). The metallic contacts are depicted in yellow. All the
superconductive leads are grounded.

For the specific case of a semiconductor nanowire proximity coupled to an
$s$-wave superconductor, we show that the resulting symmetry relations of the
conductance matrix can be used to identify the relative strength of Rashba
versus Dresselhaus spin-orbit coupling (SOC). We furthermore show that these
symmetry relations can be employed to identify signatures of deviations from
the assumed symmetries, in particular, voltage bias-dependent electric
potential landscapes and quasiparticle dissipation into environmental baths.

To achieve this objective, we discuss an extended version of the Landauer-
Büttiker theory that accounts for bias-voltage-dependent electric potentials.
These results are compared to the linear Landauer-Büttiker theory, where it is
assumed that the potential landscape, in which the scattering occurs, does not
depend on the bias voltages. We refer to this assumption as the constant
landscape approximation (CLA).

Extensions of the CLA theory for both nonlinear electric conductance [24, 25]
and thermoelectric conductance [26, 27, 28] have been considered before.
Several previous works focused on the quadratic correction (in voltage bias)
to differential conductance obtained by the method of characteristic
potential, e.g. [25, 29, 30, 27, 31, 32], that in practical applications
relies on Thomas-Fermi approximation [33, 32]. This method is more suited for
mesoscopic metallic devices with a high density of states, in which the
finite-size effects can be neglected.

In the case of superconductor-semiconductor nanoelectronic devices, instead,
the electrostatic potential can be a complicated function of the gate voltage
that in general requires the solution of the complete Schrödinger-Poisson
problem [34, 35, 36, 37, 38, 39, 40]. An initial characterization of finite-
bias effects in the fully nonlinear regime was obtained by a combination of
approximate analytic and numerical methods in Ref. [41].

This paper is organized as follows. In Sec. II, we describe the general theory
of nonlinear charge transport in multiterminal hybrid devices and the
difference with CLA results. In Sec. III, we show how fundamental symmetries
such as microreversibility and particle-hole conjugation in the $S$ matrix
generate conductance symmetries valid in CLA and how to use these to extract
additional information on the transmission processes from the differential
conductance matrix. As an example application, in Sec. IV we demonstrate how
these symmetries can be exploited to identify the spin-orbit direction in a
semiconductor nanowire proximitized by a superconductor. Finally, in Sec. V,
with simple numerical simulations we show the effect of finite-bias
deformation of the electrostatic potential and dissipation and discuss how
violation of conductance symmetry can be used to distinguish between the two.
We also illustrate an example of spin-orbit coupling characterization, and
discuss how thermoelectric differential conductance can be used as a probe to
avoid the finite-bias effect.

##  II Scattering transport theory

The Landauer-Büttiker formalism is a simple yet powerful technique to model
transport phenomena. While usually employed in CLA, the nonlinear version can
be constructed easily while paying attention to preserving the gauge
invariance of the theory [25].

When the motion of quasiparticles is a coherent process and the interactions
between quasiparticles beyond mean-field theory can be neglected,
quasiparticle transport phenomena in a device are completely described by the
single-particle scattering matrix $S$ that relates the probability amplitude
of the incoming and outgoing quasiparticles in the leads. More specifically,
the relation between the average currents and biases of a multiterminal device
depends only on the transmission probability defined as

$T_{\alpha\beta}^{\gamma\delta}(\varepsilon;P)\equiv\tr\absolutevalue{S_{\alpha\beta}^{\gamma\delta}(\varepsilon;P)}^{2}\,,$
(1)

where $S_{\alpha\beta}^{\gamma\delta}$ is the subblock of the scattering
matrix that connects the channel of the incoming particles of type $\delta$
from lead $\beta$ to the channel of outgoing particles of type $\gamma$ in
lead $\alpha$ for scattering events at energy $\varepsilon$. Since the device
we are describing is a superconductor-semiconductor hybrid, the transmission
matrix features a Nambu structure with particle and hole sectors,
$\delta,\gamma\in\\{e,h\\}$. The transmission probabilities depend on the set
of electric potentials applied to all the electrodes in the system
$\\{V_{\eta}\\}$ and other external parameters like the applied magnetic field
$\mathbf{B}$ and the pairing amplitude of superconductive leads
$\\{\Delta_{\nu}\\}$. We denote the set of external parameters by
$P=\quantity{V_{\eta}}\cup\quantity{\Delta_{\nu}}\cup\quantity{\mathbf{B}}$.
In principle also the temperature of the leads can enter as a parameter of the
system. For example, temperature could affect the size of the superconductive
gap or induce charge accumulation in the semiconductor. We will neglect these
effects as we are assuming that the temperature differences involved are much
smaller than the critical temperature of the superconductor and are too small
to induce a relevant change in the electrostatic potential landscape. Indeed,
while voltage bias enters directly into the Poisson equation as boundary
conditions, the temperature can enter the electrostatic problem only through
charge accumulation.

A generic multiterminal system comprises a number of normal and
superconductive terminals, suggesting a division of the $S$ matrix into
subblocks as follows:

$S=\begin{pmatrix}S_{NN}&S_{NS}\\\ S_{SN}&S_{SS}\end{pmatrix}\,.$ (2)

In this paper, we will consider the simplest case in which all the
superconductive terminals are grounded together and we denote the voltage of
the common superconductive lead as $V_{S}$.

If the system conserves the quasiparticle number, the $S$ matrix is unitary
and it follows that

$\displaystyle
R_{\alpha}^{ee}+R_{\alpha}^{he}+\sum_{\beta\neq\alpha}\quantity(T_{\beta\alpha}^{ee}+T_{\beta\alpha}^{he})$
$\displaystyle=N^{e}_{\alpha}(\varepsilon)\,,$ (3) $\displaystyle
R_{\alpha}^{ee}+R_{\alpha}^{eh}+\sum_{\beta\neq\alpha}\quantity(T_{\alpha\beta}^{ee}+T_{\alpha\beta}^{eh})$
$\displaystyle=N^{e}_{\alpha}(\varepsilon)\,,$ (4)

where we defined for clarity the reflection matrix
$R^{\gamma\delta}_{\alpha}\equiv T^{\gamma\delta}_{\alpha\alpha}$ while
$N^{e}_{\alpha}(\varepsilon)$ is the number of eigenmodes for electrons in
lead $\alpha$. Note that, unless the density in the lead is so low to break
particle-hole symmetry, $N_{\alpha}(\varepsilon)\equiv
N^{e}_{\alpha}(\varepsilon)=N^{h}_{\alpha}(\varepsilon)$. Equation (3)
represents the conservation of incoming quasiparticles from lead $\alpha$
while Eq. (4) is the same for outgoing quasiparticles. The conservation law
breaks down when dissipation effects are included as they do not preserve the
particle number. When restricted to energies below the smallest parent
superconductor gap $\varepsilon<\min\\{\absolutevalue{\Delta_{\nu}}\\}$, the
matrices $S_{SS}(\varepsilon)$, $S_{SN}(\varepsilon)$, $S_{NS}(\varepsilon)$
are null and therefore a stronger relation holds where the sum over $\beta$ is
restricted to the non-superconductive leads.

As mentioned above, in the superconductive version of the Landauer-Büttiker
theory quasiparticle conservation takes the place of electron conservation for
conventional devices. This implies that the electric charge is not explicitly
conserved. Indeed, we can distinguish between the charge-conserving normal
processes $T^{ee}_{\alpha\beta}$, and $T^{hh}_{\alpha\beta}$, and the non-
charge-conserving Andreev processes $T^{eh}_{\alpha\beta}$, and
$T^{he}_{\alpha\beta}$. The last two imply, respectively, the creation and the
destruction of one Cooper pair in the superconductive leads.

A correct theory of nonlinear conductance of a multiterminal device needs to
be gauge invariant. This means that the transmission probabilities are not
changed by the addition of a constant offset to all the voltages and particle
energy:

$T_{\alpha\beta}^{\gamma\delta}(\varepsilon+\Delta E;\\{V_{\eta}+\Delta
E\\})=T_{\alpha\beta}^{\gamma\delta}(\varepsilon;\\{V_{\eta}\\})$ (5)

The easiest way to guarantee gauge invariance is to define a reference
voltage. We define the superconductor lead to be our reference voltage and set
$V_{S}=0$. The first argument of the transmission function is then the energy
of the scattering particle with respect to the superconductive lead chemical
potential $\varepsilon=E_{p}+eV_{S}$. The parameters $V_{\alpha S}$ and
$V_{\beta S}$ are the bias of the leads with respect to $V_{S}$. These biases,
together with the voltage biases applied to the capacitively coupled gates,
determine the electrostatic potential landscape in which the scattering events
take place. Therefore, they enter here as parameters of the $S$ matrix as well
as the transmission probability matrix.

###  II.1 Currents

The nonlinear Landauer-Büttiker approach consists in first solving the
electrostatic problem for the potential landscape given the applied biases.
With the calculated potential landscape, the $S$ matrix for the scattering
processes can be evaluated. Finally, once the static scattering problem has
been solved, from the $S$ matrix all the transport properties of the system
can be derived. In particular, we can use the transmission functions to
determine the average currents in the nonequilibrium steady state. In this
last step, the terminal biases $V_{\alpha S}$ appear again also as parameters
of the distribution functions of the leads. The application of this approach
for superconductive systems has been derived on several occasions [42, 43, 44,
45, 46], and in the most general formulation the average electric current
through a lead $\alpha$ is then

$\begin{split}I^{q}_{\alpha}(P)=+&\frac{e}{h}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\alpha S},\theta_{\alpha})-f(\varepsilon,\theta_{S})]\\\
&\times\quantity[N_{\alpha}-R_{\alpha}^{ee}(\varepsilon;P)+R_{\alpha}^{he}(\varepsilon;P)]\\\
-&\sum_{\beta\neq\alpha}\frac{e}{h}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\beta S},\theta_{\beta})-f(\varepsilon,\theta_{S})]\\\
&\times\quantity[T_{\alpha\beta}^{ee}(\varepsilon;P)-T_{\alpha\beta}^{he}(\varepsilon;P)]\,,\end{split}$
(6)

where $f(\varepsilon,\theta)=\quantity(1+e^{\varepsilon/k_{B}\theta})^{-1}$ is
the Fermi-Dirac distribution, and the parameters $\\{\theta_{\eta}\\}$ are the
temperatures of the leads. We assumed for simplicity that the temperature of
all the superconductive leads is equal to $\theta_{S}$. A similar expression
can be written for the energy current $I^{\varepsilon}_{\alpha}$ while the
heat current $I_{\alpha}^{h}=I_{\alpha}^{\varepsilon}+eV_{\alpha
S}I_{\alpha}^{q}$ follows easily from the first law of thermodynamics [28].

We can write, for the $I^{q}$ and $I^{h}$ vectors of currents, the following
differential relation with the voltages $V$ and temperatures $\theta$ vectors
[47, 27, 30]:

$\begin{pmatrix}\differential I^{q}_{\alpha}\\\ \differential
I^{h}_{\alpha}\end{pmatrix}=\begin{pmatrix}G_{\alpha\beta}&L_{\alpha\beta}\\\
M_{\alpha\beta}&N_{\alpha\beta}\end{pmatrix}\begin{pmatrix}\differential
V_{\beta}\\\ \differential\theta_{\beta}\end{pmatrix}$ (7)

where $G$ is the differential electric conductance, $N$ is the differential
thermal conductance, while $L$ is the thermoelectric and $M$ is the
electrothermal differential conductance. We do not consider thermal transport
in this work.

An important observation about the charge current is that it does not satisfy
Kirchhoff’s current law. This is because the expression derived describes only
the quasiparticle current while the supercurrent is not captured in this
formalism. By imposing charge conservation, the net supercurrent flowing into
the device is $I^{s}=-\sum_{\alpha}I^{q}_{\alpha}$. The net supercurrent can
be evaluated as

$\begin{split}I^{s}=-\frac{2e}{h}\sum_{\alpha\beta}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\beta S},\theta_{\beta})-f(\varepsilon,\theta_{S})]\\\
\times\quantity[R_{\beta}^{he}(\varepsilon;P)+T_{\alpha\beta}^{he}(\varepsilon;P)]\,.\end{split}$
(8)

In the case of a single superconductive lead, this is equal to the
supercurrent flowing into the device. In the case of multiple superconductive
leads, instead, the supercurrent divides between the different superconductive
leads.

###  II.2 Constant landscape approximation

In the CLA, the change in the potential landscape when a voltage bias is
applied is neglected. We denote this by writing that $P=P_{0}$ where $P_{0}$
is the set of parameters at equilibrium. In this case, the CLA result for
electrical conductance is

$\begin{split}G_{\alpha\alpha}&=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[-\partial_{\varepsilon}f(\varepsilon-
eV_{\alpha S},\theta_{\alpha})]\\\
&\times\quantity[N_{\alpha}-R_{\alpha}^{ee}(\varepsilon;P=P_{0})+R_{\alpha}^{he}(\varepsilon;P=P_{0})]\,,\end{split}$
(9)

for local differential conductance, while for the nonlocal differential
conductance we have

$\begin{split}G_{\alpha\beta}&=-G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[-\partial_{\varepsilon}f(\varepsilon-
eV_{\beta S},\theta_{\beta})]\\\
&\times\quantity[T_{\alpha\beta}^{ee}(\varepsilon;P=P_{0})-T_{\alpha\beta}^{he}(\varepsilon;P=P_{0})]\,,\end{split}$
(10)

where $G_{0}=\frac{e^{2}}{h}$ is the conductance quantum and
$\partial_{\varepsilon}f(\varepsilon,\theta)=-\frac{1}{2k_{B}\theta}\frac{1}{1+\cosh(\varepsilon/k_{B}\theta)}$.

A similar expression can be obtained for thermoelectric conductance. The local
and nonlocal thermoelectric conductance reads as

$\begin{split}L_{\alpha\alpha}&=+L_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,k_{B}^{-1}\partial_{\theta}f(\varepsilon-
eV_{\alpha S},\theta_{\alpha})\\\
&\times\quantity[N_{\alpha}-R_{\alpha}^{ee}(\varepsilon;P=P_{0})+R_{\alpha}^{he}(\varepsilon;P=P_{0})]\end{split}$
(11)

and

$\begin{split}L_{\alpha\beta}&=-L_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,k_{B}^{-1}\partial_{\theta}f(\varepsilon-
eV_{\beta S},\theta_{\beta})\\\
&\times\quantity[T_{\alpha\beta}^{ee}(\varepsilon;P=P_{0})-T_{\alpha\beta}^{he}(\varepsilon;P=P_{0})]\,,\end{split}$
(12)

where $L_{0}=\frac{ek_{B}}{h}$ is the thermoelectric conductance quantum and
we used the derivative of the distribution function with respect to
temperature, that is
$k_{B}^{-1}\partial_{\theta}f(\varepsilon,\theta)=-\frac{\varepsilon}{k_{B}\theta}\partial_{\varepsilon}f(\varepsilon,\theta)$.
Note that since $\partial_{\varepsilon}f$ is an odd function of the energy,
thermoelectric conductance is sensitive only to the antisymmetric component of
the transmission spectrum of the device.

###  II.3 Differential conductance in the nonlinear theory

In principle, the average current given by Eq. (6) is exact for a non-
interacting system if the potential landscape is calculated self-consistently
from the set of parameters $P$. We do not attempt such a calculation here,
since the devices treated often have a complicated three-dimensional geometry
that cannot easily be reduced to a simple one-dimensional model when taking
the electrostatic environment into account. Instead, in this section, we focus
on general considerations, while in Sec. V we parametrize a potential
landscape in a physically motivated way and look at differences with CLA
results.

Evaluating the full derivatives of the charge current $I^{q}$ with respect to
a terminal voltage bias we find that the electric differential conductance can
be split into two parts

$\begin{split}G_{\alpha\beta}(\quantity{V_{\eta}})=\derivative{I^{q}_{\alpha}}{V_{\beta
S}}=G^{\mathrm{(m)}}_{\alpha\beta}+G^{\mathrm{(def)}}_{\alpha\beta}\end{split}$
(13)

where the first term is the marginal contribution that reads as, i.e. for the
local conductance,

$\begin{split}G^{\mathrm{(m)}}_{\alpha\alpha}=+G_{0}&\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[-\partial_{\varepsilon}f(\varepsilon-
eV_{\alpha S},\theta_{\alpha})]\\\
&\times\quantity[N_{\alpha}-R_{\alpha}^{ee}(\varepsilon;P)+R_{\alpha}^{he}(\varepsilon;P)]\,.\end{split}$
(14)

This term can be interpreted as the fact that when evaluating the additional
current carried by higher-energy states, the $S$ matrix has to be calculated
using the potential landscape that takes into account the modified voltage
bias. The second term accounts for the deformation of the $S$ matrix for the
already filled channels due to the effect of the biasing itself. The
deformation contribution reads as

$\begin{split}G^{\mathrm{(def)}}_{\alpha\beta}=&G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\alpha
S},\theta_{\alpha})-f(\varepsilon,\theta_{S})]\quantity[-\partialderivative{R_{\alpha}^{ee}(\varepsilon;P)}{V_{\beta
S}}+\partialderivative{R_{\alpha}^{he}(\varepsilon;P)}{V_{\beta S}}]\\\
+&G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\beta
S},\theta_{\beta})-f(\varepsilon,\theta_{S})]\quantity[-\partialderivative{T_{\alpha\beta}^{ee}(\varepsilon;P)}{V_{\beta
S}}+\partialderivative{T_{\alpha\beta}^{he}(\varepsilon;P)}{V_{\beta
S}}]\,.\end{split}$ (15)

This correction is often neglected in previous works, e.g. Ref. [25]. The
reason is that in the case of symmetric biasing $V_{\alpha}=V_{\beta}=V$ and
fixed temperature $\theta_{\alpha}=\theta_{\beta}=\theta_{S}$, we can rewrite
$G^{\mathrm{(def)}}_{\alpha\beta}$ as

$\begin{split}G^{\mathrm{(def)}}_{\alpha\beta}(V)=-G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,&\quantity[f(\varepsilon-
eV,\theta_{S})-f(\varepsilon,\theta_{S})]\partialderivative{V_{\beta}}\\\
&\times\quantity[N_{\alpha}(\varepsilon)-R_{\alpha}^{eh}(\varepsilon;P)-R_{\alpha}^{he}(\varepsilon;P)-\sum_{\beta^{\prime}}(T_{\alpha\beta^{\prime}}^{eh}(\varepsilon;P)+T_{\alpha\beta^{\prime}}^{he}(\varepsilon;P))]\,.\end{split}$
(16)

Since $N_{\alpha}$ does not depend on the bias (it is a property of the leads)
and all the other terms in the brackets are probabilities of Andreev’s
processes, it is clear that this quantity vanishes for non-superconductive
devices.

However, this contribution shows an interesting interplay between
electrostatic behavior and superconductivity.

For example, let us consider a simple N-S junction. The differential
conductance can be written as

$G_{\alpha\alpha}=+2G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[-\partial_{\varepsilon}f(\varepsilon-
eV_{\alpha
S},\theta_{\alpha})]R_{\alpha}^{he}(\varepsilon;P)+2G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\,\quantity[f(\varepsilon-
eV_{\alpha
S},\theta_{\alpha})-f(\varepsilon,\theta_{S})]\partialderivative{R_{\alpha}^{he}(\varepsilon;P)}{V_{\alpha
S}}\,.$ (17)

It is evident that in case
$\partialderivative{R_{\alpha}^{he}(\varepsilon;P)}{V_{\alpha S}}>0$ the
second term in the sum can overcome the first one, which is always positive,
resulting in negative local differential conductance. A simple case of this
can be when the coupling of the scattering region with the superconductor
decreases with the bias [48, 41].

##  III Conductance symmetries

In this section, we consider how the symmetries of the system manifest
themselves first as symmetries of the $S$ matrix and, consequently, in the
differential conductance matrix. We consider the ideal CLA case and, for this
reason, we drop the biases as arguments in the $S$ matrix.

We choose the Nambu basis of time-reversed holes
$\Psi^{T}=(\psi,\mathcal{T}\psi)^{T}=\begin{pmatrix}\psi_{\uparrow}&\psi_{\downarrow}&-\psi^{\dagger}_{\downarrow}&\psi^{\dagger}_{\uparrow}\end{pmatrix}^{T}$
where we have chosen for the time-reversal symmetry
$\mathcal{T}=-i\sigma_{y}\mathcal{K}$, where $\mathcal{K}$ is the complex-
conjugation operator, and therefore for the particle-hole symmetry
$\mathcal{P}=i\tau_{y}\mathcal{T}$.

In the following analysis, we consider symmetric and antisymmetric linear
combinations of the conductance matrix elements at opposite voltage bias and
magnetic field. We have investigated all linear combinations. However, in the
following, we only present the interesting cases in which the linear
combination leads to a reduction in the number of terms.

###  III.1 Particle-hole symmetry

If the system features particle-hole symmetry (PHS), the energy-resolved
scattering matrix satisfies the following relation

$S(\varepsilon)=\mathcal{P}S(-\varepsilon)\mathcal{P}^{\dagger}=\sigma_{y}\tau_{y}S^{*}(-\varepsilon)\sigma_{y}\tau_{y}\,.$
(18)

This poses an additional constraint on Andreev transmission probabilities that
read as

$T^{\gamma\delta}_{\alpha\beta}(+\varepsilon)=T^{\bar{\gamma}\bar{\delta}}_{\alpha\beta}(-\varepsilon)\,,$
(19)

where the overbar indicates that the index should be flipped $e\leftrightarrow
h$.

This property of the transmission probabilities has a number of consequences
on differential conductance. For example, in a system with a single normal
terminal (e.g., an NS junction), or for a terminal that is completely isolated
from others such that there are no propagating channels (normal or Andreev)
connecting it to other leads, the reflection coefficients have to be energy
symmetric below the gap. As a consequence, the conductance has to be a
symmetric function of the voltage for ideal devices. For the same reason, in
these cases the thermoelectric conductance is always exponentially suppressed
for temperatures much smaller than the superconducting gap.

Dissipation, inelastic scattering, and coupling to other leads are known
effects that break this symmetry of the transmission matrix [49, 50, 51] while
finite-bias effects can lead to the break-down of the symmetry at conductance-
matrix level [52, 41].

A generalization of this conductance-matrix symmetry for the multiterminal
case can be obtained by considering the quantity

$G^{\mathrm{sum}}_{\alpha}(V)\equiv
G_{\alpha\alpha}(V)+\sum_{\beta\neq\alpha}G_{\alpha\beta}(V)\,,$ (20)

that is the sum of the local conductance at terminal $\alpha$ and the nonlocal
conductances obtained measuring the current at $\alpha$ while applying a
voltage bias to all the other normal leads. It follows that, as a consequence
of Eq. (19),

$G_{\alpha}^{\mathrm{sum}}(V)=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon-
eV)]H_{\alpha}(\varepsilon)\,,$ (21)

where we defined the quantity

$\begin{split}H_{\alpha}(\varepsilon)=&R_{\alpha}^{he}(+\varepsilon)+R_{\alpha}^{he}(-\varepsilon)\\\
+&\sum_{\beta\neq\alpha}\quantity[T_{\alpha\beta}^{he}(+\varepsilon)+T_{\alpha\beta}^{he}(-\varepsilon)]\\\
+&\sum_{\nu}\quantity[T_{\alpha\nu}^{ee}(+\varepsilon)+T_{\alpha\nu}^{eh}(+\varepsilon)]\,.\end{split}$
(22)

The first two terms in $H_{\alpha}(\varepsilon)$ are explicitly symmetric in
$\varepsilon$, while the last sum is null for
$\varepsilon<\min{\absolutevalue{\Delta_{\nu}}}$. Therefore, as a consequence,

$G^{\mathrm{sa}}_{\alpha}(V)\equiv
G^{\mathrm{sum}}_{\alpha}(V)-G^{\mathrm{sum}}_{\alpha}(-V)=0\,.$ (23)

This relation is a generalization of the three-terminal case derived in Ref.
[8]. This result has been derived for non-interacting systems in CLA,
therefore, any deviation from zero in $G^{\mathrm{sa}}$ can be used as a tool
to inspect deviations from the CLA and the contributions of quasiparticle
dissipation or Coulomb repulsion between quasiparticles. We will discuss these
effects in Sec. V.

We can split the local and nonlocal differential conductance into symmetric
and antisymmetric components

$\displaystyle G^{\mathrm{sym}}_{\alpha\beta}(V)$
$\displaystyle\equiv\frac{G_{\alpha\beta}(V)+G_{\alpha\beta}(-V)}{2}\,,$ (24)
$\displaystyle G^{\mathrm{anti}}_{\alpha\beta}(V)$
$\displaystyle\equiv\frac{G_{\alpha\beta}(V)-G_{\alpha\beta}(-V)}{2}\,.$ (25)

It has been shown that one can extract the BCS charge, i.e.
$\expectationvalue{\tau_{z}}$, of each ABS from the antisymmetric combination
$G_{\alpha\beta}^{\mathrm{anti}}(V)$, given the ABSs are sufficiently
separated in the spectrum [8].

A similar relation can be derived for thermoelectric differential conductance.
Indeed, under the same conditions, we can define

$\begin{split}L^{\mathrm{sum}}_{\alpha}(\theta)\equiv
L_{\alpha\alpha}+\sum_{\beta\neq\alpha}L_{\alpha\beta}\\\
=L_{0}\int_{-\infty}^{+\infty}\differential\varepsilon
k_{B}^{-1}\partial_{\theta}f(\varepsilon,\theta)H_{\alpha}(\varepsilon)\,.\end{split}$
(26)

Since $H_{\alpha}(\varepsilon)$ is an even function of $\varepsilon$ while
$\partial_{\theta}f(\varepsilon)$ is an odd function, we have that
$L^{\mathrm{sum}}\simeq 0$ for $k_{B}\theta\ll\min{\Delta_{\nu}}$. This holds,
again, for non-interacting systems but only for temperatures low enough to
exclude excitations of states above the parent gap. Since the thermoelectric
conductance is connected to the antisymmetric part of the transmission
spectrum, its sign at low temperature can be linked to the BCS charge
$\expectationvalue{\tau_{z}}$ of ABSs, following a similar argument as for
$G^{\mathrm{anti}}$ as presented in Ref. [8].

In the presence of dissipation, the previous transport symmetries do not hold.
Here by dissipation, we mean the presence of a reservoir at the Fermi level
that induces quasiparticle leakage. This can be due to various reasons, such
as the presence of subgap states in the superconductor causing a softening of
the gap or some other leakage mechanism that connects the scattering region to
the common ground. A simple way to model quasiparticle leakage is by
considering a fictitious lead $\beta^{\prime}$ that is excluded when taking
the calculation of $G^{\mathrm{sum}}_{\alpha}$. Focusing on energies below the
gap, one finds that the antisymmetric part does not vanish but equals to

$\begin{split}&G^{\mathrm{sa}}_{\alpha}(V)=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon)]\big{[}T_{\alpha\beta^{\prime}}^{eh}(\varepsilon-
eV)\\\ +&T_{\alpha\beta^{\prime}}^{ee}(\varepsilon-
eV)-T_{\alpha\beta^{\prime}}^{eh}(\varepsilon+eV)-T_{\alpha\beta^{\prime}}^{ee}(\varepsilon+eV)\big{]}\,.\end{split}$
(27)

A similar relation can be obtained for thermoelectric conductance. To assess
in a more quantitative way the effect of dissipation later we will switch to
numerical simulations (see Sec. V).

###  III.2 Microreversibility

The microreversibility of the scattering process is a consequence of global
time-reversal symmetry and implies that, upon inversion of the time-reversal
breaking fields and spin direction, the motion can be reversed111 Note that we
are considering systems that do not break time-reversal symmetry internally..
As a consequence, the scattering matrix is equal to its transpose

$S(\mathbf{B},\Delta_{\nu})=\mathcal{T}S(-\mathbf{B},\Delta_{\nu}^{*})\mathcal{T}^{\dagger}=\sigma_{y}S^{T}(-\mathbf{B},\Delta_{\nu}^{*})\sigma_{y}\,,$
(28)

which, expanding in particle and lead labels, becomes

$S^{\gamma\delta}_{\alpha\beta}(\mathbf{B},\Delta_{\nu}^{*})=\sigma_{y}\quantity[S^{\delta\gamma}_{\beta\alpha}(-\mathbf{B},\Delta_{\nu}^{*})]^{T}\sigma_{y}\,.$
(29)

If we consider only non-spin-polarized leads, we can take the trace over the
internal spin indices and get the following symmetry relation:

$T^{\gamma\delta}_{\alpha\beta}(\varepsilon,\mathbf{B})=T^{\delta\gamma}_{\beta\alpha}(\varepsilon,-\mathbf{B},\Delta_{\nu}^{*})\,.$
(30)

Combining this relation with the one derived from particle-hole symmetry [Eq.
(19)], we find that, for Andreev reflections and transmission coefficients,

$T^{eh}_{\alpha\beta}(\varepsilon,\mathbf{B})=T^{eh}_{\beta\alpha}(-\varepsilon,-\mathbf{B},\Delta_{\nu}^{*})\,.$
(31)

This has implications both in the electric and thermoelectric nonlocal
conductance.

The connection between microreversibility and thermoelectric quantities has
been explored on general grounds both in theory [53, 54] and experiments [55].
In particular, microreversibility is the microscopic explanation of the
Onsager-Casimir relations that lead to other transport symmetries in the
charge, heat, and spin channels valid in linear response [56, 57].

Microreversibilty can be exploited to study separately normal and Andreev
processes. To do so, we introduce two new quantities, $G^{\mathrm{ra}}$ and
$G^{\mathrm{re}}$, both in the local and nonlocal versions, that we call
reciprocal conductances and that can be extracted from the electric
differential conductance matrix:

$\displaystyle G^{\mathrm{ra}}_{\alpha\beta}(V,\mathbf{B})$
$\displaystyle\equiv
G_{\alpha\beta}(V,\mathbf{B})-G_{\beta\alpha}(+V,-\mathbf{B})$ (32)
$\displaystyle G^{\mathrm{re}}_{\alpha\beta}(V,\mathbf{B})$
$\displaystyle\equiv
G_{\alpha\beta}(V,\mathbf{B})-G_{\beta\alpha}(-V,-\mathbf{B})$ (33)

By using microreversibility and particle-hole symmetry it is possible to show
that

$\displaystyle
G^{\mathrm{ra}}_{\alpha\beta}(V,\mathbf{B})=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon)]\quantity[T^{he}_{\alpha\beta}(\varepsilon-
eV,\mathbf{B})-T^{he}_{\alpha\beta}(\varepsilon+eV,\mathbf{B})]\,,$ (34)
$\displaystyle
G^{\mathrm{re}}_{\alpha\beta}(V,\mathbf{B})=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon)]\big{[}T^{ee}_{\alpha\beta}(\varepsilon-
eV,\mathbf{B})-T^{ee}_{\alpha\beta}(\varepsilon+eV,\mathbf{B})\big{]}\,,$ (35)

where $G^{\mathrm{ra}}(V)$ is proportional to the antisymmetric part of the
Andreev transmission probability while $G^{\mathrm{re}}(V)$ is proportional to
the antisymmetric part of the normal electron transmission probability. For
this reason, these two quantities can be used to analyze separately the two
types of transport processes. Moreover, it can be verified from their
definitions that these two quantities are the decomposition of the
antisymmetric part of the local differential conductance:

$G^{\mathrm{anti}}_{\alpha\beta}=G^{\mathrm{ra}}_{\beta\alpha}+G^{\mathrm{re}}_{\beta\alpha}.$
(36)

The local versions, $G^{\mathrm{re}}_{\alpha\alpha}$ and
$G^{\mathrm{ra}}_{\alpha\alpha}$ are proportional only to the antisymmetric
part of the reflection probabilities. As mentioned before, if a lead is
sufficiently isolated from the others such that there are no propagating
channels connecting it to other leads, the reflection probabilities are bound
to be energy-symmetric making the defined quantities null in absence of
inelastic scattering.

The quantities (32), and (33) are the only symmetric or antisymmetric
combinations of conductance matrix elements $G_{\alpha\beta}(V,\mathbf{B})$
that simplify to a difference of two transmission or reflection probabilities
under the constraints imposed by unitarity [Eqs. (3) and (4)], particle-hole
symmetry [Eq. (19)], and time-reversal symmetry [Eq. (30)]. Contrarily to PHS-
derived conductance symmetries, the results in Eqs. (32) and (33) are not
affected by dissipation since the derivation does not make use of the
unitarity of the $S$ matrix.

Note that $\lim_{V\to 0}G^{\mathrm{ra}}(V)=0$ in agreement with Onsager-
Casimir relation. The vanishing of $G^{\mathrm{ra}}$ for normal (non-
superconductive) devices can be explained as an extension of Onsager-Casimir
relations beyond the linear-response regime.

###  III.3 Additional antiunitary symmetry

Several widely used models in the context of proximitized devices, e.g., the
Lutchyn-Oreg Hamiltonian describing a topological phase transition in a
proximitized semiconductor nanowire [58, 59] satisfy an additional antiunitary
symmetry $\mathcal{A}=U_{\mathcal{A}}\mathcal{K}$ aside from
microreversibility that persists even in the presence of a Zeeman field. This
symmetry implies additional constraints on the conductance matrix. In case the
antiunitary symmetry is inherited from the normal state (i.e., it holds
separately for electron and hole parts of the wavefunction), then the matrix
$U_{\mathcal{A}}$ does not mix the particle-hole and lead indices. In this
case, the symmetry condition for the scattering matrix can be written as

$S(\mathbf{B},\Delta_{\nu})=U_{\mathcal{A}}^{T}S(\mathbf{B},\Delta_{\nu})^{T}U_{\mathcal{A}}^{*}\,.$
(37)

As a consequence, the transmission probabilities satisfy the symmetry
relations

$T^{\gamma\delta}_{\alpha\beta}(\mathbf{B},\Delta_{\nu})=T^{\delta\gamma}_{\beta\alpha}(\mathbf{B},\Delta_{\nu})\,,$
(38)

The validity of this symmetry on the transmission probabilities is due to the
block-diagonal structure of the unitary $U_{\mathcal{A}}$ combined with the
definition of $T^{\gamma\delta}_{\alpha\beta}(\mathbf{B},\Delta_{\nu})$ in Eq.
(1) that contains a trace over all single-lead indices that are present in the
normal state.

In combination with PHS [Eq. (19)], we find

$T^{\gamma\delta}_{\alpha\beta}(+\varepsilon,+\mathbf{B},\Delta_{\nu})=T^{\bar{\delta}\bar{\gamma}}_{\beta\alpha}(-\varepsilon,+\mathbf{B},\Delta_{\nu})\,.$
(39)

and, in particular for the Andreev transmission,

$T^{eh}_{\alpha\beta}(+\varepsilon,+\mathbf{B},\Delta_{\nu})=T^{eh}_{\beta\alpha}(-\varepsilon,+\mathbf{B},\Delta_{\nu})\,.$
(40)

The combination with microreversibility in Eq. (30) instead gives

$T^{\gamma\delta}_{\alpha\beta}(+\varepsilon,+\mathbf{B},\Delta_{\nu})=T^{\gamma\delta}_{\alpha\beta}(+\varepsilon,-\mathbf{B},\Delta_{\nu}^{*})\,.$
(41)

As a result, the conductance magnetic asymmetry, that we define as

$G^{\mathrm{m}}_{\alpha\beta}(V,\mathbf{B},\Delta_{\nu})\equiv
G_{\alpha\beta}(V,\mathbf{B},\Delta_{\nu})-G_{\alpha\beta}(V,-\mathbf{B},\Delta_{\nu}^{*})\,.$
(42)

vanishes.222Note that for local quantities
$G^{\mathrm{m}}_{\alpha\alpha}=G^{\mathrm{ra}}_{\alpha\alpha}$. Violations of
this symmetry relation can be attributed to perturbations that break the
antiunitary symmetry $\mathcal{A}$. This can be the result o, e.g., orbital
effects or phase inhomogeneities.

Similar considerations can be drawn for thermoelectric conductance. In the
same fashion, we can define the thermoelectric conductance magnetic asymmetry

$L^{\mathrm{m}}_{\alpha\beta}(V,\mathbf{B},\Delta_{\nu})\equiv
L_{\alpha\beta}(V,\mathbf{B},\Delta_{\nu})-L_{\alpha\beta}(V,-\mathbf{B},\Delta_{\nu}^{*})\,,$
(43)

this quantity vanishes under the same assumptions.

Note that, in simpler systems like two-terminal metallic wires, the
conductance is expected to be a symmetric function of the magnetic field.
Therefore, the presence of a conductance magnetic asymmetry can be used as a
probe of electron-electron interactions in the system [32]. In multiterminal
superconductive devices instead, the symmetry is not expected even in the non-
interacting system where the presence of the additional antiunitary symmetry
is necessary to have a vanishing $G^{\mathrm{m}}$.

###  III.4 Geometrical symmetries

Geometrical symmetries of the device can also be exploited to build quantities
that select only specific components of the transmission matrix. These can be
useful in case the geometry of the system can be controlled to some degree of
accuracy such that it may feature approximate geometrical symmetries.

For introducing a concrete example, let us consider a 2DEG wire like the one
in Figs. 1(a) and 2(a). The wire is aligned along the $x$ direction, with two
symmetric leads $L$ and $R$. Suppose the system is symmetric upon mirroring
along the $x$ and $y$ directions, $\mathcal{M}_{x}$, and $\mathcal{M}_{y}$,
and it features a rotation symmetry $\mathcal{R}_{z}(\pi)$. By using only
$\mathcal{T}$, $\mathcal{P}$ and $\mathcal{M}_{x}$ symmetries we have

$T^{eh}_{RL}(\varepsilon,B_{x},B_{y},B_{z})\xrightarrow{\mathcal{T}\mathcal{P}\mathcal{M}_{x}}T^{eh}_{RL}(-\varepsilon,-B_{x},B_{y},B_{z})\\\
$ (44)
$T^{ee}_{RL}(\varepsilon,B_{x},B_{y},B_{z})\xrightarrow{\mathcal{T}\mathcal{M}_{x}}T^{ee}_{RL}(\varepsilon,-B_{x},B_{y},B_{z})$
(45)

and we can use this result to define new quantities similar to the reciprocal
differential conductances, but with the advantage that is evaluated at only
one lead and reversing only one component of the magnetic field

$\displaystyle G^{\mathrm{xa}}_{LR}(V,\mathbf{B})$ $\displaystyle\equiv
G_{LR}(V,B_{x})-G_{LR}(V,-B_{x})\,,$ (46) $\displaystyle
G^{\mathrm{xe}}_{LR}(V,\mathbf{B})$ $\displaystyle\equiv
G_{LR}(V,B_{x})-G_{LR}(-V,-B_{x})\,.$ (47)

By using Eqs. (44) and (45) it is possible to show that

$\displaystyle
G^{\mathrm{xa}}_{LR}(V,\mathbf{B})=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon)]\quantity[T^{he}_{LR}(\varepsilon-
eV,\mathbf{B})-T^{he}_{LR}(\varepsilon+eV,\mathbf{B})]$ (48) $\displaystyle
G^{\mathrm{xe}}_{LR}(V,\mathbf{B})=G_{0}\int_{-\infty}^{+\infty}\differential{\varepsilon}\quantity[-\partial_{\varepsilon}f(\varepsilon)]\big{[}T^{ee}_{LR}(\varepsilon+eV,\mathbf{B})-T^{ee}_{LR}(\varepsilon-
eV,\mathbf{B})\big{]},$ (49)

Similarly, for the thermoelectric conductance, we have

$L^{\mathrm{x}}_{LR}(\theta,B_{x})\equiv
L_{LR}(\theta,B_{x})-L_{LR}(\theta,-B_{x})\simeq 0\,.$ (50)

Again, this quantity is exactly zero if we neglect the energies above the
parent gap and thus deviation at low temperatures can be directly linked to
dissipation effects.

If a system featuring mirror symmetry satisfies an additional antiunitary
symmetry as discussed in Sec. III.3, then it follows that
$G^{\mathrm{ra}}_{\alpha\beta}=0$. In absence of the additional antiunitary
symmetry, the conductance symmetry $G^{\mathrm{ra}}_{\alpha\beta}=0$ is
present in case the magnetic field lies in the plane orthogonal to the mirror
symmetry axis. These properties can be used as an indication of whether the
system satisfies mirror symmetry.

##  IV Additional antiunitary symmetry in a proximitized semiconductor
nanowire

Figure 2: Sketch of a proximitized semiconductor nanowire with spin-orbit
coupling. In (a), using as a reference a simple 2DEG geometry, we introduce a
Rashba field ${\bf\it\alpha}$ transverse to the nanowire direction
$\mathbf{e}_{\mathrm{x}}$ due to the interface electric field. The Rashba SOC
manifest in a momentum-dependent spin splitting in the
$\mathbf{e}_{\mathrm{x}}\times{\bf\it\alpha}$ direction. A native Dresselhaus
spin-orbit coupling can also be present; this acts as momentum-dependent
$\mathbf{e}_{\mathrm{x}}$ spin splitting. The combination of the two effects
can be described by a generalized spin-orbit direction ${\bf\it\kappa}$ that
defines an orthogonal plane shown in green in (b). When the magnetic field
$\mathbf{B}$ lays in the orthogonal plane, the system features an additional
antiunitary symmetry. In (c) we introduce a coordinate system to discuss the
dependence of transport on the direction of the external magnetic field
$\mathbf{B}$.

To introduce a concrete example application of the additional antiunitary
symmetry discussed in Sec. III.3, we now consider a quasi-1D semiconductor
nanowire proximitized by a superconductor. An important question for these
devices is the characterization of spin-orbit coupling. This can be achieved
by leveraging the symmetry relations previously introduced.

We consider a system represented by the following low-energy effective
Hamiltonian

$\begin{split}\mathcal{H}=&\quantity[\frac{\hbar^{2}\mathbf{k}^{2}}{2m^{*}}+V(\mathbf{r})+\mathcal{H}_{\mathrm{SOC}}]\tau_{z}\\\
&+\mathbf{b}\cdot{\bf\it\sigma}\tau_{0}+\tilde{\Delta}\tau_{x}\,,\end{split}$
(51)

where $\mathbf{k}=-i\gradient$ is the wavevector, $m^{*}$ is the effective
mass, $V(x)=-\mu(x)$ is the potential landscape (which can include disorder),
$\tilde{\Delta}$ is the proximity-induced pairing potential in the weak
coupling limit, and $\mathbf{b}$ is the Zeeman spin splitting in the
semiconductor. Finally, the spin-orbit coupling
$\mathcal{H}_{\mathrm{SOC}}=\mathcal{H}_{R}+\mathcal{H}_{D}$ is the sum of the
Rashba and the Dresselhaus term.

The Rashba interaction reads
$\mathcal{H}_{R}=\mathbf{k}\crossproduct{\bf\it\alpha}\cdot{\bf\it\sigma}$
where the Rashba field ${\bf\it\alpha}$ is proportional to the electric field
in the device. The Dresselhaus spin-orbit coupling arises from the lack of
inversion symmetry of the material and can be written as
$\mathcal{H}_{D}=\gamma_{D}\mathbf{l}\cdot{\bf\it\sigma}$. In zinc-blende
crystals, the $\mathbf{l}$ vector components are
$l_{a}=k_{a}(k_{b}^{2}-k_{c}^{2})$ where $(a,b,c)$ are cyclic permutations of
the coordinates $(x,y,z)$ [60].

Here we consider a quasi-1D system in the $\mathbf{e}_{x}$ direction as shown
in Fig. 2. For sufficiently thin wires with sufficiently energy-separated
eigenmodes with different radial momenta, we can replace the radial momentum
operators by their expectation values evaluated on the transverse eigenmode
wave function
$\mathbf{k}\simeq(k_{x},\expectationvalue{k_{y}},\expectationvalue{k_{z}})$.
Under this assumption, we can rewrite the Dresselhaus Hamiltonian as
$\mathcal{H}_{D}=\gamma_{D}\sigma_{x}k_{x}\quantity[-\langle
k_{z}^{2}\rangle+\langle k_{y}^{2}\rangle]=\beta k_{x}\sigma_{x}$, while for
the Rashba SOC,
$\mathcal{H}_{R}=(\alpha_{y}\sigma_{z}-\alpha_{z}\sigma_{y})k_{x}$, where a
term
$\langle\mathbf{k}_{\perp}\rangle\crossproduct{\bf\it\alpha}_{\parallel}\cdot{\bf\it\sigma}$
vanishes due to $\langle\mathbf{k}_{\perp}\rangle=0$ for confined eigenmodes.

First, consider the case of pure Rashba spin-orbit coupling, i.e.,
$\gamma_{D}=0$. The Hamiltonian satisfies an antiunitary symmetry when the
magnetic field points within the plane spanned by the Rashba field
${\bf\it\alpha}$ and the direction of the wire. For
${\bf\it\alpha}=\alpha\mathbf{e}_{z}$, the antiunitary symmetry is complex
conjugation and the real-space Hamiltonian in Eq. (51) is real.

If the system features both Rashba and Dresselhaus spin-orbit coupling, the
plane spanned by the magnetic fields that preserve an antiunitary symmetry is
tilted. Without loss of generality we choose a Rashba field perpendicular to
the wire pointing along $z$, i.e.
${\bf\it\alpha}_{\perp}=\alpha_{\perp}\hat{e}_{z}$. We introduce a coordinate
system for the magnetic field defined as
$\mathbf{b}=b\quantity(\cos\theta\cos\phi,\cos\theta\sin\phi,\sin\theta)$
where $\theta$ is the elevation and $\phi$ the azimuth with respect to the
wire direction. In this case, the spin-orbit coupling term reads
$k_{x}(\alpha_{\perp}\sigma_{y}+\beta\sigma_{x})$. A rotation
$e^{i\phi_{\kappa}\sigma_{z}/2}$ in spin space by the angle
$\tan\phi_{\kappa}=\frac{\beta}{\alpha_{\perp}}$ transforms
$e^{-i\phi_{\kappa}\sigma_{z}/2}k_{x}(\alpha_{\perp}\sigma_{y}+\beta\sigma_{x})e^{i\phi_{\kappa}\sigma_{z}/2}=k_{x}\sqrt{\alpha_{\perp}^{2}+\beta^{2}}\sigma_{y}$.
In this basis, spin-orbit coupling is real and the Hamiltonian satisfies
$\mathcal{A}=\mathcal{K}$.

This antiunitary symmetry is preserved by a Zeeman field
$\mathbf{b}=b_{\perp}\sigma_{z}+b_{\parallel}(\cos\phi_{\kappa}\sigma_{x}-\sin\phi_{\kappa}\sigma_{y})$,
such that
$e^{-i\phi_{\kappa}\sigma_{z}/2}\left[b_{\perp}\sigma_{z}+b_{\parallel}(\cos\phi_{\kappa}\sigma_{x}-\sin\phi_{\kappa}\sigma_{y})\right]e^{i\phi_{\kappa}\sigma_{z}/2}=b_{\perp}\sigma_{z}+b_{\parallel}\sigma_{x}$
is real. Here, $\mathbf{b}_{\perp}$ is the component of the magnetic field
parallel to ${\bf\it\alpha}_{\perp}$ and $\mathbf{b}_{\parallel}$ is the
component pointing in the direction orthogonal to ${\bf\it\alpha}_{\perp}$ and
$\beta\hat{x}+\alpha_{\perp}\hat{y}$. In other words, the combined Rashba and
Dresselhaus spin-orbit coupling terms can be written as
${\bf\it\kappa}\cdot{\bf\it\sigma}k_{x}$ where
${\bf\it\kappa}=\mathbf{e}_{x}\times{\bf\it\alpha}_{\perp}+\beta\mathbf{e}_{x}$
is the generalized spin-orbit direction. The antiunitary symmetry is preserved
by a Zeeman field $\mathbf{b}\cdot{\bf\it\kappa}=0$. The plane spanned by the
$\mathbf{b}$ vectors that satisfy the orthogonality condition can be
identified by an angle $\phi_{0}=\phi_{\kappa}+\pi/2$.

##  V Numerical models

In this section, we introduce two numerical models to show examples of how
microscopic symmetries of the systems manifest themselves in the transport
properties. In both cases, we model the grounded leads in the system with the
method of self-energies. This can be useful when the lead is a metal with high
density compared to the scattering region as the self-energy takes the simple
form of a local complex-valued potential. This is added to the Hamiltonian to
generate an effective non-Hermitian energy-dependent Hamiltonian that can be
studied with the scattering approach.

We consider the most general case of a grounded soft-gap superconductor that
can be described by the Dynes superconductor model [61, 62, 63]. In the case
of a superconductive lead with a high density of states, the intermediate
coupling regime can be adequately described by the following local self-energy

$\Sigma_{\nu}(\varepsilon,\mathbf{r})=\gamma_{\nu}(\mathbf{r})\frac{-\quantity(\varepsilon+i\Gamma_{\nu})\tau_{0}\sigma_{0}+\quantity(\Delta_{\nu}\tau_{+}+\Delta_{\nu}^{\dagger}\tau_{-})\sigma_{0}}{\sqrt{\absolutevalue{\Delta_{\nu}}^{2}\tau_{0}\sigma_{0}-\quantity[(\varepsilon+i\Gamma_{\nu})\tau_{0}\sigma_{0}]^{2}}}\,,$
(52)

where $\Delta_{\nu}=|\Delta_{\nu}|e^{i\phi_{\nu}}$ is the pairing amplitude
with phase $\phi_{\nu}$ (that we assume constant in space), $\Gamma_{\nu}$ is
the Dynes parameter that models pair-breaking scattering processes, the local
coupling strength is $\gamma_{\nu}=\pi\mathcal{D}_{\nu}t_{\nu}^{2}$ with
$\mathcal{D}_{\nu}$ being the density of states in the lead $\nu$, and
$t_{\nu}$ is the interface hopping amplitude between the scattering region and
the superconductive lead $\nu$. For a normal lead, this reduces to

$\Sigma_{\mathrm{n}}(\varepsilon,\mathbf{r})=-i\gamma_{\mathrm{n}}(\mathbf{r})\sigma_{0}\tau_{0}\,,$
(53)

that is an imaginary potential that causes the decay of the quasiparticle wave
function.

###  V.1 Double-dot Josephson junction

To study the effect of the additional antiunitary symmetry discussed in Sec.
III.3, we first consider a double-dot Josephson junction illustrated in Fig.
1. The effective Hamiltonian of the system is

$\begin{split}\mathcal{H}=&\begin{pmatrix}-\mu_{1}&-t\\\
-t&-\mu_{2}\end{pmatrix}\tau_{z}\\\ &+\begin{pmatrix}1&0\\\
0&0\end{pmatrix}\Sigma_{1}(\varepsilon)+\begin{pmatrix}0&0\\\
0&1\end{pmatrix}\Sigma_{2}(\varepsilon)\end{split}$ (54)

where $\mu_{i}$ are the local chemical potentials in the dots, $t$ is the
hopping amplitude between the dots, and $\Sigma_{i}$ are the local self-
energies induced by the superconductive leads $i=1,2$. The scattering matrix
can be obtained by using the Weidenmüller formula (see, e.g., [64])

$S(\varepsilon)=\mathbb{I}-2\pi
iW^{\dagger}\frac{1}{\varepsilon-\mathcal{H}+\mathrm{i}\pi WW^{\dagger}}W\,,$
(55)

where $W_{i}(E)\equiv\sqrt{\rho_{i}(\varepsilon)}t_{i}\Pi_{i}(\varepsilon)$
with $t_{i}$ the tunneling amplitude from the device to lead $i$,
$\rho_{i}(\varepsilon)$ the density of states in lead $i$, and
$\Pi_{i}(\varepsilon)$ the projector onto the eigenstates of lead $i$ at
energy $\varepsilon$. In our model, we approximate the tunnel coupling between
lead $L$ and dot $1$ (lead $R$ and dot $2$) by two energy-independent
parameters, such that $W=\begin{pmatrix}w_{L}&w_{R}\end{pmatrix}\tau_{z}$.

When the phase difference between the two superconductive terminals
$\phi_{12}=\phi_{1}-\phi_{2}$ is zero or $\pi$, the system features the
antiunitary symmetry $\mathcal{A}=\mathcal{K}$. As a consequence, the Andreev
process probabilities are symmetric in the energy axis, i.e.,
$T^{eh}(\varepsilon)=T^{eh}(-\varepsilon)$, and
$R^{eh}(\varepsilon)=R^{eh}(-\varepsilon)$. This can be verified by the zero
in the conductance magnetic asymmetry $G^{\mathrm{m}}$, as shown in the second
column of Fig. 3.

The reciprocal conductance can also be used to verify the presence of a mirror
symmetry of the device. A mirror symmetry $\mathcal{M}_{x}$ exchanges the two
dots and reverses the sign of the phase difference $\phi_{i}\to-\phi_{i}$. The
latter can be seen by noticing that the phase difference can be created by a
magnetic field $B_{z}$ piercing a superconducting loop within the
$x$-$y$-plane connecting to the two dots. This mirror symmetry implies a zero
in the non-local reciprocal conductances $G^{\mathrm{ra}}_{LR}$,
$G^{\mathrm{ra}}_{RL}$. Indeed, by calculating $G^{\mathrm{ra}}_{LR}$ as a
function of the dots’ levels asymmetry
$\delta\mu=(\mu_{1}-\mu_{2})/(\mu_{1}+\mu_{2})$ (see the rightmost column of
Fig. 3), we verify the presence of a zero in $G^{\mathrm{ra}}$ for
$\delta\mu=0$, that is when the system feature a mirror symmetry.

As a function of phase difference $\phi_{12}$, the symmetric configuration
$\mu_{1}=\mu_{2}$ exhibits a zero in the reciprocal conductances
$G^{\mathrm{ra}}_{LL}$, $G^{\mathrm{ra}}_{LR}$ at $\phi_{12}=0$ due to the
mirror symmetry (c.f. the fourth column of Fig. 3). The zero in
$G^{\mathrm{ra}}_{LL}$, $G^{\mathrm{ra}}_{LR}$ signals energy-symmetric
Andreev reflection and transmission amplitudes [c.f. Eq. 32]. This mirror
symmetry is broken at finite $|\phi_{12}|>0$, including $\phi_{12}=\pi$, due
to the different superconducting phase at the two dots.

Figure 3:  Local conductance $G_{LL}$ and combinations $G_{LL}^{\mathrm{m}}$,
$G_{LL}^{\mathrm{re}}$, $G_{LL}^{\mathrm{ra}}$ (top row, from left to right)
and non-local conductance $G_{LR}$ and combinations $G_{LR}^{\mathrm{m}}$,
$G_{LR}^{\mathrm{re}}$, $G_{LR}^{\mathrm{ra}}$ (bottom row, from left to
right) in a double-dot Josephson junction as a function of the phase
difference $\phi_{12}=\phi_{1}-\phi_{2}$ for a symmetric configuration
$\mu_{1}=\mu_{2}$ (columns one to four). The rightmost column displays the
reciprocal conductance $G_{LL}^{ra}$ for $\phi_{LR}=\pi/4$ as a function of
the chemical potential asymmetry
$\delta\mu=(\mu_{1}-\mu_{2})/(\mu_{1}+\mu_{2})$. The parameters used are
$t=$0.2\text{\,}\mathrm{meV}$$,
$w_{L}^{2}=w_{R}^{2}=$0.01\text{\,}\mathrm{meV}$$, $\mu_{1}=\mu(1+\delta\mu)$,
$\mu_{2}=\mu(1-\delta\mu)$, $\mu=$0.10\text{\,}\mathrm{meV}$$,
$\gamma_{1}=\gamma_{2}=$0.3\text{\,}\mathrm{meV}$$,
$|\Delta_{1}|=|\Delta_{2}|=$1\text{\,}\mathrm{meV}$$.

###  V.2 Proximitized semiconductor nanowire

As an example of a three-terminal device, we consider the case of a
semiconductor nanowire proximitized by an $s$-wave superconductor as shown in
Fig. 4. We demonstrate how an antiunitary symmetry persisting at a finite
magnetic field for specific directions can be employed to extract the ratio
between Dresselhaus and Rashba spin-orbit coupling, as introduced in Sec. IV.
We further study the effects of dissipation and voltage-bias-dependent
potentials on the symmetry relations derived under CLA.

The Hamiltonian is similar to the one in Eq. (51), but here we treat the
superconductive lead using the self-energy model

$\begin{split}\mathcal{H}(\varepsilon)=&\quantity[\frac{\hbar^{2}k_{x}^{2}}{2m^{*}}+V(x)]\tau_{z}\\\
&+\quantity[(\alpha_{y}\sigma_{z}-\alpha_{z}\sigma_{y})k_{x}+\beta
k_{x}\sigma_{x}]\tau_{z}\\\
&+\mathbf{b}\cdot{\bf\it\sigma}\tau_{0}+\Sigma(\varepsilon)\,,\end{split}$
(56)

where we take $m^{*}=0.026m_{e}$, consistent with an InAs nanowire, and
$\Sigma(\varepsilon)$ is the superconductive lead self-energy as given in Eq.
(52).

Figure 4: Sketch of one contact of a three-terminals device. The monomodal
wire (blue, green) is coated with a superconductive shell (gray), which
induces a superconducting pairing potential in the central region and fixes
the chemical potential due to the high density. Two barriers (quantum point
contacts) are introduced on the side (green), while barrier gates can
partially control the shape of the potential drop here. Two bias voltages are
applied on the left and right leads (yellow) with respect to the grounded
superconductor. In the lower panel, we sketch the potential landscape of the
system $V(x)$ in the unbiased regime (dashed blue line) and biased one (solid
blue line) showing all the quantities used to parametrize the potential
landscape.

To include the effect of a finite bias, we model the deformation of the
potential landscape in a simplified effective manner. Following an approach
proposed by Ref. [41], we assume that the density of states in the
superconductor shell is high enough to guarantee perfect screening of the
electric field. This means that the potential drop falls entirely in the
(depleted) barrier region. The case of imperfect screening is analyzed in Ref.
[11].

To provide a gauge-invariant description of the potential landscape we select
the voltage applied to the superconductor as the reference voltage
$V_{\mathrm{S}}=0$ such that $\varepsilon=E_{p}-\mu_{S}=E_{p}+eV_{\mathrm{S}}$
is the energy of the scattering particle. We define the left and right biases
as $V_{LS}$ and $V_{RS}$. The other parameters that enter in the effective
potential landscape are the chemical potential in the lead and the wire, that
we define as $\mu_{\mathrm{l}}=-e(V_{\mathrm{l}}-V_{\mathrm{S}})$ and
$\mu_{\mathrm{w}}=-e(V_{\mathrm{w}}-V_{\mathrm{S}})$, and the zero-bias
barrier height $\Delta V_{\mathrm{b}}=e(V_{\mathrm{b}}-V_{\mathrm{S}})$. We
assume the absence of built-in biases in the junction by considering a flat
potential barrier at zero bias. The effect of the zero-bias barrier $\Delta
V_{\mathrm{b}}$ can be connected to a reduction of the coupling with the leads
that causes a reduction of the height of the peaks and increased sharpness in
the differential conductance. All these parameters are shown in the sketch of
the landscape shown in Fig. 4. We modeled the effect of the finite bias as a
linear voltage drop (i.e., a constant electric field) within the barrier, and
we smoothed the potential using a sigmoid function instead of Heaviside steps
to avoid sharp transitions between the different parts of the system.

We used a nanowire length of $L_{\mathrm{w}}=$500\text{\,}\mathrm{nm}$$ with
barriers of length $L_{\mathrm{b}}=$50\text{\,}\mathrm{nm}$$. The local
chemical potential in the nanowire is set to
$\mu_{\mathrm{w}}=$0.5\text{\,}\mathrm{meV}$$ while the zero-bias barrier
height is set to $\Delta V_{\mathrm{b}}=$0.3\text{\,}\mathrm{meV}$$. The lead
have $\mu_{\mathrm{l}}=$25\text{\,}\mathrm{meV}$$. For the superconductive
lead, we set $\Delta=$0.35\text{\,}\mathrm{meV}$$ and $\gamma_{\rm
Sc}=$0.2\text{\,}\mathrm{meV}$$. To simplify the evaluation of reciprocal
conductance, we restrict the elevation angle $\theta\in[-\pi/2,\pi/2]$ while
allowing the magnitude $b$ to take negative values. We discretized the
Hamiltonian using the finite-differences method with step lengths
$a_{x}=$1\text{\,}\mathrm{nm}$$, then evaluated the scattering matrix
$S[\varepsilon,\mathcal{H}(\varepsilon;P)]$ using the Kwant package for
quantum transport [65]. After evaluation of the $S$ matrix, the conductance is
calculated for the CLA case following Eqs. (9)-(12), while in the nonlinear
case, the electric charge current is calculated by numerical integration of
Eq. (6).

####  V.2.1 Identification of the spin-orbit coupling direction

We first focus on the newly introduced quantities, reciprocal differential
conductances and conductance magnetic asymmetry, and their use for the
determination of the spin-orbit coupling direction. To emphasize the effect
and maximize $G^{\mathrm{ra}}$, we consider a strongly asymmetric case in
which the left barrier is set to $\Delta
V_{\mathrm{b},L}=$0.3\text{\,}\mathrm{meV}$$ while the right barrier is in the
open regime $\Delta V_{\mathrm{b},R}=0$. We also choose to align the Rashba
field in the out-of-plane direction
$\alpha_{R}=(0,0,-10)\,$\mathrm{meV}\text{\,}\mathrm{nm}$$ while we set the
Dresselhaus energy to $\beta=$5\text{\,}\mathrm{meV}\text{\,}\mathrm{nm}$$.

A sweep in Zeeman energy $b$ is shown in Fig. 5. We find that
$G^{\mathrm{re}}$ is much larger than $G^{\mathrm{ra}}$. By Eqs. (32) and
(33), this indicates that the antisymmetric part of the electron-electron
transmission probability dominates over the antisymmetric part of the crossed
Andreev reflection probability. Note that in the presence of both mirror
symmetry $\mathcal{M}_{x}$ (inverting the wire direction) and an antiunitary
symmetry, the Andreev transmission probabilities are symmetric in energy such
that $G^{\mathrm{ra}}_{LR}$ vanishes. For our device, in the presence of only
Rashba SOC and a magnetic field oriented in the wire direction, a mirror-
symmetric device satisfies an antiunitary symmetry $\mathcal{A}=\mathcal{K}$
and a mirror symmetry $\mathcal{M}_{x}=\sigma_{x}\tau_{0}$. Therefore, a
signal in $G^{\mathrm{ra}}_{LR}$ is correlated to the mirror symmetry breaking
terms in the device geometry, such as the asymmetric barrier configuration
used here, and Dresselhaus spin-orbit coupling $\propto k_{x}\sigma_{x}$,
breaking both the antiunitary symmetry and the mirror symmetry.

Figure 5: Reciprocal conductance and conductance magnetic asymmetry in a
super-semi nanowire. We show the typical signature of the reciprocal
conductance and conductance magnetic asymmetry [Eqs. (32), (33), and (42)] for
a proximitized semiconductor nanowire with Rashba spin-orbit coupling as a
function of the Zeeman energy $b$. We set the direction of the magnetic field
as $\theta=\frac{\pi}{4}$ and $\phi=0$. In this simulation,
$V_{\mathrm{b},L}=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$,
$V_{\mathrm{b},R}=$0\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, and
$\beta=$5\text{\,}\mathrm{meV}\text{\,}\mathrm{nm}$$.

The reciprocal conductances $G^{\mathrm{ra}}$ and $G^{\mathrm{re}}$ can be
used to characterize the spin-orbit coupling of the nanowire. Indeed both are
symmetric under reversal of magnetic field only if the system satisfies an
antiunitary symmetry that persists at a finite magnetic field. An alternative
and easier measurement is the conductance magnetic asymmetry $G^{\mathrm{m}}$,
Eq. (42), since it requires the combination of only two differential
conductances at the same terminal. This quantity vanishes if there is an
antiunitary symmetry persisting at a finite magnetic field. Measuring this
quantity while rotating the azimuth of the magnetic field allows for the
identification of the direction of the generalized spin-orbit coupling vector
$\mathbf{e}_{\kappa}$, as shown in Fig. 6.

The zero of the quantity $G^{\mathrm{m}}(\phi=\phi_{0})=0$ is achieved only
when the orthogonality condition $\mathbf{b}\cdot{\bf\it\kappa}=0$ is
satisfied. Therefore, with the measured set of directions for which
$G^{\mathrm{m}}=0$ it is possible to determine $\mathbf{e}_{\kappa}$ and its
relative angle with the wire direction $\phi_{\kappa}=\phi_{0}-\pi/2$. With
this information, it is possible to determine both the direction of the Rashba
field and the ratio of the orthogonal Rashba and Dresselhaus SOC. The
orthogonal Rashba field ${\bf\it\alpha}_{\perp}$ is oriented in the direction
$\mathbf{e}_{\kappa}\crossproduct\mathbf{e}_{x}$ while the ratio of the two
fields is connected to the angle by
$\beta/\alpha_{\perp}=\tan(\phi_{\kappa})$.

Note that $G^{\mathrm{m}}(\phi)$ shows a linear behavior in $\phi$ near
$\phi_{0}$ (marked by a change of sign in the neighborhood). In the
simulations we noticed an additional zero in the direction
$(\phi_{\kappa},\theta_{\kappa})$ that is
${\bf\it\kappa}\crossproduct\mathbf{b}=0$. In this case,
$G^{\mathrm{m}}(\phi,\theta)$ has a quadratic behavior in both $\phi$ and
$\theta$ in the neighborhood of $(\phi_{\kappa},\theta_{\kappa})$. Note that
$\theta_{\kappa}=0$ in the chosen coordinate system.

Figure 6:  Identification of the spin-orbit orientation. We show how local
($G^{\mathrm{m}}_{LL}$) and nonlocal ($G^{\mathrm{m}}_{LR}$) conductance
magnetic asymmetry, Eq. (42), can be used to identify the spin-orbit coupling
orientation by measuring these quantities while rotating the magnetic field,
i.e. the angles $\quantity(\theta,\phi)$ in the reference frame introduced in
Fig. 2. Both these quantities vanish linearly when the condition
$\mathbf{b}_{\bot}\crossproduct{\bf\it\kappa}=0$ is met. In the case shown,
the Rashba spin-orbit coupling $\alpha_{R}$ is oriented in the $z$-direction
and the relative strength of Dresselhaus over Rashba spin-orbit coupling is
$\beta/\alpha_{\perp}=1/2=\tan(\phi_{\kappa})$. Therefore $G^{\mathrm{m}}$
vanishes only for the plane identified by the angle
$\phi_{0}=\phi_{\kappa}+\pi/2$ and is thus observed both in the case of
$\theta=0$ shown in the left panels and the case $\theta=\pi/4$ shown in the
right ones. In this simulation, $V_{\mathrm{b},L}=$0.3\text{\,}\mathrm{meV}$$,
$V_{\mathrm{b},R}=$0\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$,
$\beta=$5\text{\,}\mathrm{meV}\text{\,}\mathrm{nm}$$, while the Zeeman field
strength is $b=$0.42\text{\,}\mathrm{meV}$$.

####  V.2.2 Finite-bias effect and dissipation

Transport symmetries can be also exploited to assess the presence of non-ideal
effects and possibly distinguish between them. To illustrate the idea in this
example system, we consider the dissipation and finite-bias effect. Indeed, in
an ideal system the antisymmetric components of the local and nonlocal
electrical differential conductance as a function of bias voltage are opposite
to each other, such that $G_{\alpha}^{\mathrm{sa}}(V)\equiv
G^{\mathrm{anti}}_{LL}(V)+G^{\mathrm{anti}}_{LR}(V)=0$ [c.f. Eq. (23)]. This
is illustrated in Fig. 7.

Figure 7: Splitting into symmetric and antisymmetric parts of electric
differential conductance. The antisymmetric parts of local and nonlocal
electric conductance are opposite. This corresponds to the symmetry relation
$G^{\mathrm{sa}}(V)\equiv
G^{\mathrm{anti}}_{LL}(V)+G^{\mathrm{anti}}_{LR}(V)=0$ in ideal systems.
Deviations from this symmetry relation can be used to identify non-ideal
effects like dissipation or finite-bias effect.

This symmetry relation is broken by finite-bias effects, dissipation, and
Coulomb scattering between quasiparticles. We verify the possibility of
distinguishing between finite-bias and dissipation effects by calculating the
quantity $G_{L}^{\mathrm{sa}}$ in presence of these effects. We consider the
same system as before in Sec. V.2.1 with the only difference of considering
symmetric barrier of $\Delta
V_{\mathrm{b}}=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ and we set the
Dresselhaus SOC $\beta=0$.

First, we introduce the finite-bias effect by manually introducing the voltage
drop in the barrier regions as shown in Fig. 4. The nonlinear differential
conductance is then obtained by numerical differentiation of the total current
calculated with Eq. (6). The comparison of the full-nonlinear theory and the
CLA can be seen in Fig. 8. It is possible to distinguish two corrections, one
general background correction in the local conductance that can be attributed
to an increase in the average barrier height as the potential is raised. On
top of this, we can identify a shift in the position of the peaks. The effect
of a finite bias gets stronger and more evident as the barrier length is
increased since the effect of the voltage drop is distributed in a greater
area of the device.

Figure 8: Electric differential conductance in the symmetric setup calculated
with nonlinear theory (left panels) and correction to differential conductance
in CLA. It is possible to distinguish two contributions. One general
background correction in the local conductance that can be attributed to an
increase in the average barrier height as the potential is raised. On top of
this, we can identify a correction that moves and changes the position of the
peaks. The finite-bias effect gets stronger and more evident as the barrier
length is increased. In this simulation,
$V_{\mathrm{b},L}=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$,
$V_{\mathrm{b},R}=$30\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$, $\beta=0$,
while we set the Zeeman field to
$\mathbf{b}=(1,0,0)$40\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$.

To introduce dissipation, we compare two cases: in the first case we consider
an additional normal lead acting as a quasiparticle reservoir described by a
self-energy $\Sigma_{\mathrm{Sm}}$ of the form Eq. (53) with parameter
$\gamma_{n}$ describing the coupling strength between system and reservoir,
while in the second case we introduce a soft gap in the superconductor through
the parameter $\Gamma_{\nu}=\Gamma_{\mathrm{Sc}}$ in the Dynes model for the
self-energy $\Sigma_{\mathrm{Sc}}$ of proximity induced superconductivity, Eq.
(52). Note that the first case has been already assessed in Ref. [51] for
local conductance.

We consider two leads such that
$\Sigma=\Sigma_{\mathrm{Sc}}+\Sigma_{\mathrm{Sm}}$. In the quasiparticle
reservoir case we assume $\Gamma_{\mathrm{Sc}}=0$ and
$\gamma_{\mathrm{Sm}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ while in the
soft gap case $\Gamma_{\mathrm{Sc}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro
eV}$$ while $\gamma_{\mathrm{Sm}}=0$.

The left and right plots in the top row of Fig. 9 shows, within CLA, the
effect of a dissipation term in the wire Hamiltonian (setting
$\gamma_{\mathrm{Sm}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$,
$\Gamma_{\mathrm{Sc}}=0$) and of inelastic scattering processes in the
superconductor modeled with the Dynes model (setting $\gamma_{\mathrm{Sm}}=0$,
$\Gamma_{\mathrm{Sc}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$),
respectively. The effects of the two dissipation terms are very similar and
consistent with the result in Eq. (27). Therefore, it is not possible to
distinguish between the two effects with this measurement. In the lower plots,
nonlinear theory within perfect metallic screening approximation is
considered. The effect of finite bias on the symmetry relation appears
qualitatively different from dissipation also in this case. It can be
described by a background contribution that depends on the sign of the applied
voltage together with a localized correction in the position of the peaks.
More strikingly, after the topological transition, there is no evident
oscillation in the sign connected to the local BCS charge like in the
dissipation case. Therefore measurements of electrical conductance offer the
possibility of distinguishing between the effect of finite bias and
dissipation. These results are consistent with previous analyses [51, 41].

Figure 9: The plots show the value of $G^{\mathrm{sa}}_{L}$ when different
symmetry-breaking mechanisms are introduced. In the upper plots, two distinct
dissipation terms are considered within CLA, in the left
$\gamma_{\mathrm{Sm}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$ while in the
right one $\Gamma_{\mathrm{Sc}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$.
The lower plots shows the same quantity calculated using the nonlinear theory
on the left $\gamma_{\mathrm{Sm}}=0$ while on the right
$\Gamma_{\mathrm{Sc}}=$5\text{\,}\mathrm{\SIUnitSymbolMicro eV}$$.

####  V.2.3 Thermoelectric conductance

The clear advantage of thermoelectric conductance is that temperature-induced
charge accumulation, which leads to potential change modification, can be
safely ignored in the regime of interest. Therefore it represents an
alternative measurement free of problems related to the finite-bias effect. We
stress here that by thermoelectric measurements we mean the measurement of the
current as a change in the temperature of the leads. We note that for
negligible inelastic scattering we expect no local thermalization, such that
the device parameters should remain unchanged by the temperatures in the
leads.

As in the case of electric differential conductance, we can define local and
nonlocal thermoelectric conductance. These satisfy the symmetry relation

$L^{\mathrm{sum}}_{L}(\theta)=L_{LL}(\theta_{L}=\theta)+L_{LR}(\theta_{R}=\theta)\simeq
0$ (57)

if we restrict the integral over the energy to values below the parent gap
region. The latter just introduces a non-exactly balanced background
contribution. As can be seen in Fig. 10, the interesting features are the
lobes with an oscillating sign at low temperatures. These features can be
linked to the BCS charge $\expectationvalue{\tau_{z}}$ of the Andreev bound
states at the end of the wire by a straightforward extension of the derivation
using non-local electric conductance presented in Ref. [8].

Figure 10: Local and nonlocal thermoelectric conductance in a proximitized
semiconductor nanowire. The low-temperature lobes with alternating signs can
be associated with the BCS charge $\expectationvalue{\tau_{z}}$ similarly to
the interpretation of $G^{\mathrm{anti}}$.

Finite-bias effects can also affect the procedure for the determination of the
spin-orbit coupling outlined in Section V.2.1. The same information can be
obtained by thermoelectric measurements by evaluating the thermoelectric
conductance magnetic asymmetry $L^{\mathrm{m}}$ while rotating the magnetic
field as shown in Fig. 11. As expected, when the magnetic field lies in the
plane orthogonal to the generalized spin-orbit coupling vector
${\bf\it\kappa}$, identified by the angle $\phi_{0}$, we observe a zero in
$L^{\mathrm{m}}$. In contrast to the electric conductance combination,
$G^{m}_{\alpha\beta}$, the thermoelectric conductance combination $L_{LR}^{m}$
displays a quadratic behavior in $\phi$ around $\phi_{0}$ at the magnetic
field angle $\theta=\pi/4$. Also for $L^{m}$ we observe an additional
quadratic zero at $(\phi_{\kappa},\theta_{\kappa})$, i.e. when
${\bf\it\kappa}\crossproduct\mathbf{b}=0$.

Figure 11:  Identification of the spin-orbit orientation by thermoelectric
measurements. We show how local ($L^{\mathrm{m}}_{LL}$) and nonlocal
($L^{\mathrm{m}}_{LR}$) thermoelectric conductance magnetic asymmetry can be
used to identify the spin-orbit coupling orientation by measuring these
quantities while rotating the magnetic field, i.e. the angles
$\quantity(\theta,\phi)$ in the reference frame introduced in Fig. 2. Both
these quantities vanish when the condition
$\mathbf{b}_{\bot}\crossproduct{\bf\it\kappa}=0$ is met. In the case shown,
the Rashba spin-orbit coupling $\alpha_{R}$ is oriented in the $z$-direction
while $\beta/\alpha_{\perp}=\tan(\phi_{\kappa})=1/2$. Therefore
$L^{\mathrm{m}}$ vanishes only for the plane identified by the angle
$\phi_{0}=\phi_{\kappa}+\pi/2$ and is thus observed both in the case of
$\theta=0$ shown in the left panels and the case $\theta=\pi/4$ shown in the
right ones. The parameters are the same as Fig. 6.

##  VI Conclusions

In this work, we have explored the limits of local and nonlocal tunneling
spectroscopy of hybrid devices within the extended Landauer-Büttiker
formalism. We have derived symmetry constraints on the multiterminal
conductance matrix that follow from the fundamental microreversibility and
particle-hole conjugation in the presence of superconductivity. Our first
result shows that the reciprocal conductances $G^{\mathrm{ra}}$ and
$G^{\mathrm{re}}$, defined in Eqs. (32) and (33), can be employed to extract
the antisymmetric-in-voltage parts of the individual electron and Andreev
transmission and reflection probabilities.

In the presence of an additional antiunitary symmetry that persists at a
finite Zeeman field, a further relation can be derived for the conductance
magnetic asymmetry $G^{\mathrm{m}}$ in Eq. (42). This relation is particularly
useful in the study of spin-orbit coupled semiconductor nanowires proximitized
by an $s$-wave superconductor since it allows extracting the ratio between the
Rashba and Dresselhaus spin-orbit coupling strength. We have demonstrated this
result in an explicit numerical model. This result may be useful for material
and device characterization because the characterization of the spin-orbit
coupling in proximitized semiconductor devices is an open research question.
Future work can study these quantities in a more realistic scenario modeling
the cross-section of the superconductor-semiconductor heterostructure to
include multiple transverse modes and the orbital coupling of the magnetic
field.

Furthermore, we have studied the effects of dissipation and the dependence of
the electric potential on the bias voltage on the symmetry relations at an
explicit model of a proximitized semiconductor nanowire. Generally, these
symmetry relations are broken by these non-idealities. However, the two
effects yield distinct signatures in the conductance matrix elements and their
linear combinations.

In conclusion, nonlocal tunneling spectroscopy is a powerful tool employed in
the study of Andreev bound states [9, 10, 8, 12, 21, 22, 23]. We hope that our
work contributes to the interpretation of the experimental measurements and
expands the scope of the method by allowing access to more detailed
information on the system under study.

##  VII Acknowledgments

The authors want to thank Alisa Danilenko, Andreas Pöschl, and Charles Marcus
for useful discussions. This work was supported by the Danish National
Research Foundation, the Danish Council for Independent Research |Natural
Sciences. The authors acknowledge Microsoft research for support and
computational resources.

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