# Steady-state dc transport through an Anderson impurity coupled to leads with
spin-orbit coupling

Anirudha Mirmira  Jawaharlal Nehru Centre for Advanced Scientific Research,
Bangalore 560 064, India.  N. S. Vidhyadhiraja<EMAIL_ADDRESS>Jawaharlal
Nehru Centre for Advanced Scientific Research, Bangalore 560 064, India.

###### Abstract

We study the steady-state dc transport characteristics of a system comprised
of an interacting quantum dot, modelled as an Anderson impurity, coupled to
two metallic non-interacting leads with Rashba SOC, using an interpolative
perturbative approach (IPA). The single-particle spectra, current and
differential conductance are obtained in weak and strong coupling regimes over
a wide range of SOC and bias values. Extensive benchmarking of the IPA
validates the method in the linear as well as non-linear response regime. The
universal, zero bias ($V_{sd}=0$) peak with a width proportional to the Kondo
scale ($T_{K}$) and two non-universal finite bias peaks around $V_{sd}=\pm U$
in the zero temperature differential conductance show a clear separation with
increasing $U$ or increasing SOC. In the strong coupling regime, increasing
temperature induces melting of the zero bias peak leading to a crossover from
a three-peak conductance to a two-peak conductance. Recent experiments find
the emergence of a two-peak structure by increasing SOC at a fixed
temperature. Our results appear to provide a qualitative explanation of these
observations as a SOC tuned crossover from weak/intermediate to strong
coupling, and a simultaneous crossover from low $T/T_{K}$ to high $T/T_{K}$
ratio. We also reproduce the experimentally observed temperature dependence of
the zero bias conductance.

##  I Introduction

The Kondo effect and its interplay with bias in the leads is a rich and well-
studied problem. This non-equilibrium Kondo effect has been observed in a wide
variety of systems like quantum dotsGoldhaber-Gordon  _et al._ (1998);
Cronenwett  _et al._ (1998); Chorley  _et al._ (2012), molecular systemsLiang
_et al._ (2002); Yu and Natelson (2004), carbon nano tubesGrigorian  _et al._
(1999); Nygård  _et al._ (2000); Eichler  _et al._ (2009); Laird  _et al._
(2015), and quantum point contactsvan Wees  _et al._ (1988); Thomas  _et al._
(1996); Brun  _et al._ (2014); Heedt  _et al._ (2016); Smith  _et al._ (2022).
Correspondingly, experiments also observe a wide range of phenomena in these
systems like quantum interferenceAradhya and Venkataraman (2013), spin-
selective trasportKobayashi  _et al._ (2010), etc. The single impurity
Anderson model (SIAM) has been the standard paradigm for modelling such
systems both in and out of equilibrium. The SIAM in equilibrium has been
studied extensively, with the Bethe ansatzTsvelick and Wiegmann (1983) and
Numerical Renormalization groupKrishna-murthy  _et al._ (1980) results being
considered authoritative.

In recent years, a whole range of theoretical methods have been developed and
used to study Kondo systems out of equilibrium. However, unlike in the case of
equilibrium, where a reasonable consensus has been achieved, the physics of
nonequilibrium interacting systems is still under debate. In case of the SIAM,
the splitting of the Kondo peaks, effects of magnetic field and spin-orbit
coupling (SOC) on the Kondo effect, the effects of bias and temperature have
been studied using various methods like numerical renormalization groupAnders
(2008); Anders and Schiller (2005); Fritsch and Kehrein (2010), non-crossing
approximationWingreen and Meir (1994), perturbative methodsFujii and Ueda
(2003); Lopes and Anda (2019); Joura  _et al._ (2015), quantum Monte Carlo
methodsWerner  _et al._ (2010), exact diagonalisation methodsNuss  _et al._
(2013), Bethe ansatzMehta and Andrei (2006), hierarchical equation of
motionCheng  _et al._ (2015) and master equation approachesDorda  _et al._
(2014) among many others.

Spin-orbit coupling is an important parameter, and is well known to be an
integral aspect of exotic materials like topological insulators, topological
magnets and is useful in many spintronics applications. Spin-orbit coupling in
quantum dot systems has been studied both with SOC on the dot and with SOC in
the leads. Experiments with the SOC on the dot include directly measuring the
SOC in two-atom quantum dotsFasth  _et al._ (2007), the observation of an
anomalous Josephson currentZazunov  _et al._ (2009), and controlling the SOC
using magnetic field direction Tanttu  _et al._ (2019) among many others. The
case with spin-orbit coupling in the leads has also been studied with
phenomena like the gate voltage tunability of SOCGuo  _et al._ (2021); Smith
_et al._ (2022), magnetotransportSchäpers  _et al._ (2004), etc. having been
observed.

In equilibrium, the effect of the spin-orbit coupling on the Kondo resonance
has been theoretically studied in detailZarea  _et al._ (2012); Wong  _et al._
(2016); Žitko and Bonča (2011).A driven quantum dot, coupled to normal
metallic leads, with SOC on the dot has been investigated using the finite-U
slave boson methodLü and Guo (2007) and the findings show that the RSOC
introduces new conductance peaks next to the Kondo peak while suppressing the
Kondo peak. Quantum wire systems have also been studied, including the
destruction of spin accumulation due to an impurityZhang  _et al._ (2011)and
multichannel effectsGelabert  _et al._ (2010). The quantum dot connected to a
nanoribbon with SOC has also been studied using the Hubbard III
approximationLopes and Anda (2019).

A powerful tool in experimentally studying the effect of bias on quantum
systems is the paradigm of quantum point contacts (QPCs). QPCs are realised by
constricting a 2D electron gas between contacts which have a source-drain and
gate voltage applied. Multiple experiments on QPCs have observed quantized
conductance along with some well-known anomalies Topinka  _et al._ (2000);
Meir  _et al._ (2002); Iqbal  _et al._ (2013). Recent experiments in systems
with quantum point contacts have also seen the interplay of SOC and Kondo
physics. In particular, Smith et al. Smith  _et al._ (2022), have realised a
QPC setup where they can tune the SOC and have observed a two-peak conductance
in the large SOC regime along with an increase in the zero bias conductance
with temperature.

In this paper, we study the interplay between Rashba spin-orbit coupling on
the leads, a constant bias and interactions on the steady-state dc transport
through a quantum dot system. As mentioned before, the effect of Rashba SOC in
the leads on the Kondo effect in equilibrium has received significant
attentionZarea  _et al._ (2012); Wong  _et al._ (2016); Žitko and Bonča
(2011). When subjected to a DC bias, the system studied will be out of
equilibrium and the interplay of bias and SOC in the leads becomes an
important consideration. We employ the Interpolative ApproximationAligia
(2006) (IPA), which reduces in the particle-hole (p-h) symmetric limit to the
Keldysh Second Order Perturbative theory (KPT2), to investigate transport
through the dot with Rashba SOC on normal metallic two-dimensional leads.
Since the method is approximate, we begin with a benchmarking of the method
against exact methods like the time-evolving block decimationNuss  _et al._
(2013) (TEBD) and identify the regimes within which the results from KPT2 are
reliable. We have investigated the effect of SOC on (i) universality and
scaling in the linear response regime, and subsequently, on (ii) differential
conductance in the non-linear bias regime. We find that the equilibrium
universal scale, i.e. the quasiparticle weight controls the extent of the
linear response regime and is also crucial in determining the effective
temperature of the system when comparing the theoretical and experimental
results. The features in differential conductance are investigated in a wide
parameter space, and characteristic signatures of the interplay of bias,
interactions, Rashba SOC and temperature are identified. Finally, we compare
some of our results corresponding to the strong coupling regime to recent
experiments, and offer a qualitative explanation for some of the observations.

We begin with a brief outline of the model and formalism in the next section.
Subsequently, we present our results in three parts in section III. A short
discussion and the conclusions are presented in the final section. A few
details regarding the derivation of some results are presented in an appendix.

##  II Formalism

The Hamiltonian for a quantum dot system connected to two leads with Rashba
Spin-orbit Coupling (RSOC) can be written as

$H=H_{0}+H_{\rm d}+H_{\rm RSOC}+H_{\rm hyb}\,,$ (1)

where the two-dimensional conduction band reservoirs (L/R) are represented by
$H_{0}=\sum_{\alpha{\mathbf{k}}\sigma}\epsilon_{\alpha{\mathbf{k}}}^{\phantom{\dagger}}c^{\dagger}_{\alpha{\mathbf{k}}\sigma}c_{\alpha{\mathbf{k}}\sigma}^{\phantom{\dagger}}$
and the Hamiltonian for the quantum dot is given by $H_{\rm
d}=\sum_{\sigma}\epsilon_{d}d^{\dagger}_{\sigma}d_{\sigma}^{\phantom{\dagger}}+Un_{d\uparrow}n_{d\downarrow}$,
where $\alpha=L/R$ and $\sigma=\uparrow/\downarrow$ are the lead and spin
indices respectively. The RSOC term is represented byZarea  _et al._ (2012)

$H_{\rm
RSOC}=\sum_{\alpha{\mathbf{k}}}\lambda\psi^{\dagger}_{\alpha{\mathbf{k}}}({\mathbf{k}}\times\vec{\sigma})_{z}\psi_{\alpha{\mathbf{k}}}\,,$
(2)

where ${\mathbf{k}}=(k_{x},k_{y})$, and
$\psi^{\dagger}_{\alpha{\mathbf{k}}}=(c^{\dagger}_{\alpha{\mathbf{k}},\uparrow}\;c^{\dagger}_{\alpha{\mathbf{k}},\downarrow})$.
Finally, the hybridization between the quantum dot and the leads is given by

$H_{\rm
hyb}=\sum_{\alpha{\mathbf{k}}\sigma}\left(V_{\mathbf{k}}c^{\dagger}_{\alpha{\mathbf{k}}\sigma}d_{\sigma}^{\phantom{\dagger}}+{\rm
h.c}\right)\,.$ (3)

The conduction band terms, namely $H_{0}$ and $H_{\rm RSOC}$ may be
combinedZarea  _et al._ (2012), which leads to the emergence of chiral
conduction bands. This is accomplished using an angular momentum expansion for
the conduction band operators as:

$c_{{\mathbf{k}}\sigma}=c_{k_{x}k_{y}\sigma}=\frac{1}{\sqrt{2\pi
k}}\sum_{m=-\infty}^{\infty}c_{km\sigma}\exp(im\theta_{\mathbf{k}})\,,$ (4)

where $k=|{\mathbf{k}}|$, with inverse transform being defined as
$c_{km\sigma}=\sqrt{k/2\pi}\int^{2\pi}_{0}d\theta_{{\mathbf{k}}}c_{{\mathbf{k}}\sigma}e^{-im\theta_{{\mathbf{k}}}}$.

Substituting the above expansion (equation 4) into the Hamiltonian, and
assuming an isotropic dispersion, given by
$\epsilon_{\mathbf{k}}=\hbar^{2}k^{2}/2m$, the following form of the
Hamiltonian is obtainedZarea  _et al._ (2012):

$\begin{split}H&=\sum_{khm}\tilde{\epsilon}_{kh}(c_{kh}^{m+\frac{1}{2}})^{\dagger}c_{kh}^{m+\frac{1}{2}}\\\
&+\sum_{khm}\delta_{m,0}\tilde{V_{k}}\left((c_{kh}^{m+\frac{1}{2}})^{\dagger}d_{\uparrow}+h(c_{kh}^{m-\frac{1}{2}})^{\dagger}d_{\downarrow}+{\rm
h.c}\right)\\\ &+H_{d}\,,\end{split}$ (5)

where $h=\pm 1$ is an emergent chiral quantum number. This also allows us to
define an emergent angular momentum quantum number, given by
$j_{m}=m+\frac{h}{2}$, with only the bands corresponding to
$j_{m}=\pm\frac{1}{2}$ coupling to the dot, while the rest of the bands are
decoupled. The renormalized dispersion $\tilde{\epsilon}_{kh}$, now depends on
$h$ and the spin-orbit interaction $\lambda$, as given by the expression
$\tilde{\epsilon}_{kh}=(\epsilon_{k}+h\lambda
k)/k=\tilde{\epsilon}_{k}+h\lambda$. For a free-electron like dispersion,
$\tilde{\epsilon}_{k}$ will be linear in $k$. Hence RSOC introduces a Zeeman-
type splitting of the conduction band, without breaking the time-reversal
symmetry. Further, the hybridization matrix elements are assumed to be
isotropic, i.e $V_{\mathbf{k}}=V_{k}$, and $\tilde{V}_{k}=V_{k}\sqrt{2\pi/k}$.

In order to study the interplay between the RSOC and a constant voltage bias,
we consider the left and right leads to have a lead-dependent chemical
potential, $\mu_{\alpha}$ applied to all the emergent $h,j_{m}$ channels in
each lead, such that the voltage bias is given by $V_{sd}=\mu_{L}-\mu_{R}$.
This leads to the Hamiltonian

$H_{eff}=\sum_{\alpha=L,R}H_{\alpha}+H_{hyb}+H_{dot}\,,$ (6)

where the individual terms are given by

$\displaystyle H_{\alpha}$
$\displaystyle=\sum_{khj_{m}}\tilde{\epsilon}^{\phantom{\dagger}}_{k\alpha
h}c^{\dagger}_{\alpha khj_{m}}c^{\phantom{\dagger}}_{\alpha khj_{m}}\,,$ (7)
$\displaystyle H_{hyb}$ $\displaystyle=\sum_{\alpha
kh}\tilde{V}^{\phantom{\dagger}}_{k\alpha}\big{[}c^{\dagger}_{\alpha
kh\frac{+1}{2}}d^{\phantom{\dagger}}_{\uparrow}+hc^{\dagger}_{\alpha
kh\frac{-1}{2}}d^{\phantom{\dagger}}_{\downarrow}+h.c.\big{]}\,,$ (8)
$\displaystyle H_{dot}$
$\displaystyle=\sum_{\sigma}\epsilon_{d}n_{\sigma}+Un_{\uparrow}n_{\downarrow}\,,$
(9)

with $\tilde{\epsilon}_{k\alpha h}=\tilde{\epsilon}_{kh}+\mu_{\alpha}$. The
model described above can be visualised by the schematic in figure 1. We note
that this model was investigated using a quantum master equation approach in a
recent work Smith  _et al._ (2022), and the results were used to explain
certain aspects of differential conductance.

Figure 1: Schematic of the correlated quantum dot connected to two leads,
split further by spin-orbit coupling into chiral bands, subject to a voltage
bias, $V_{sd}=\mu_{L}-\mu_{R}$.

In the presence of a finite bias ($V_{sd}\neq 0$), the steady state current
($J$) can be computed using the symmetrized Meir-Wingreen formula, given by
Meir and Wingreen (1992)

$\displaystyle J=$ $\displaystyle\frac{2ie}{h}\sum_{\sigma}\int
d\omega\,\Big{[}\left(\Gamma_{L}(\omega)-\Gamma_{R}(\omega)\right)G^{<}_{d\sigma}(\omega){}$
$\displaystyle+$ $\displaystyle
2i\left(f_{L}(\omega)\Gamma_{L}(\omega)-f_{R}(\omega)\Gamma_{R}(\omega)\right){\rm
Im}G^{r}_{d\sigma}(\omega)\Big{]}\,.$ (10)

In the above, the $G^{<}_{d\sigma}$ and $G^{r}_{d\sigma}$ represent the lesser
and retarded dot Green’s functions computed within Keldysh formalism, and
$f_{\alpha}$ is the Fermi-Dirac distribution function of the $\alpha^{\rm th}$
lead. The $\Gamma_{\alpha}(\omega)$ represents the dot-lead coupling and is
given by

$\Gamma_{\alpha}(\omega)=-{\rm Im}\Delta_{\alpha}(\omega)\,,$ (11)

where the hybridization function for the $\alpha^{\rm th}$ lead,
$\Delta_{\alpha}(\omega)$, is given by

$\begin{split}\Delta_{\alpha}(\omega)&=\sum_{kh}\frac{{\tilde{V}_{k}}^{2}}{\omega^{+}-\epsilon_{k\alpha
h}}\\\
&=\sum_{k}{\tilde{V}_{k}}^{2}\left[\frac{1}{\omega^{+}-\tilde{\epsilon}_{k\alpha}+\lambda}+\frac{1}{\omega^{+}-\tilde{\epsilon}_{k\alpha}-\lambda}\right]\\\
&=\sum_{h}\Delta_{h\alpha}(\omega)\,=E_{\alpha}(\omega)-i\Gamma_{\alpha}(\omega).\end{split}$
(12)

With a suitable choice of the $\tilde{V}_{k}$-dependence on $k$ as
$\tilde{V}_{k}^{2}=V_{0}^{2}f(k)$, we can transform the summation over $k$
into an integral that has the form of a Hilbert transform, and hence the
hybridization function can be obtained as:

$\Delta_{h\alpha}(\omega)=V_{0}^{2}{\rm
H}\left[\omega^{+}-\mu_{\alpha}-h\lambda\right]$ (13)

where ${\rm H}[z]$ is the Hilbert transform with respect to a ‘density of
states’, $\rho_{0}(\epsilon)$, given by

${\rm H}[z]=\int\,d\epsilon\,\frac{\rho_{0}(\epsilon)}{z-\epsilon}\,.$ (14)

Note that the ‘density of states’, $\rho_{0}(\epsilon)$ in the above equation
stems from the $k$-dependence of the hybridization matrix element. The
conduction band density of states is already incorporated through the electron
dispersion of a free-electron form (see the discussion below equation (4)). We
have explored three types of $\rho_{0}(\epsilon)$: (i)a Gaussian (G), which is
not bounded, but can be interpreted as having an effective finite bandwidth,
(ii) a semi-elliptic (S) form which has a compact support, and (iii) a wide
flat band (F). The expressions for the three forms are

$\displaystyle\rho^{\rm G}_{0}(\epsilon)$
$\displaystyle=\frac{1}{\sqrt{\pi}t_{*}}\exp\\!\left(-\frac{\epsilon^{2}}{t_{*}^{2}}\right)\,$
(15) $\displaystyle\rho_{0}^{\rm S}(\epsilon)$ $\displaystyle=\frac{1}{\pi
t_{*}}\left(1-\frac{\epsilon^{2}}{4t_{*}^{2}}\right)^{1/2}\,$ (16)
$\displaystyle\rho_{0}^{\rm F}(\epsilon)$
$\displaystyle=\frac{1}{2t_{*}}\theta(t_{*}-|\epsilon|)\,.$ (17)

We define an energy scale, $\Delta_{0}=\pi V_{0}^{2}\rho_{0}(0)$ which is
known to determine the scaling of dynamics and transport properties in
equilibriumHewson (1993), and hence can be expected to play an important role
in the steady-state as well. In terms of this scale, the dot-lead coupling is
given by,
$\Gamma_{h\alpha}(\omega)=\Delta_{0}\rho_{0}(\omega-\mu_{\alpha}-h\lambda)/\rho_{0}(0)$.
For all our calculations, we have used the semi-elliptic or the Gaussian forms
with $t_{*}=1$, except for benchmarking where we have used the flat-band form
with a large $t_{*}$, but a finite $\Delta_{0}$. In all the calculations
described in this work, we have chosen $\Delta_{0}=0.1$, implying that the
choice of $V_{0}$ is not the same for the three forms. The infinitely wide
flat band hybridization is obtained by using the limit
$t_{*}\rightarrow\infty$, and a concomitant scaling of
$V_{0}\sim\sqrt{t_{*}}$, such that $\Delta_{0}=0.1$. Thus, in such a limit,
the hybridization does not have a bias or spin-orbit coupling dependence while
the semi-elliptic and the Gaussian forms do, and this difference leads to
several observable consequences as we will discuss in the results section.

The dot Green’s functions used in the current expression, equation II may be
computed using the Dyson’s equations given by Aligia (2006)

$\displaystyle[G^{r}_{d}(\omega)]^{-1}=[g^{r}_{d}(\omega)]^{-1}-\Sigma^{r}(\omega)\,,$
(18) $\displaystyle
G^{<}_{d}(\omega)=|G^{r}_{d}(\omega)|^{2}\Bigg{(}\frac{g^{<}_{d}(\omega)}{|g^{r}_{d}(\omega)|^{2}}-\Sigma^{<}(\omega)\Bigg{)}\,,$
(19)

where $g^{r}_{d},g^{<}_{d}$ represent the non-interacting ($U=0$) Green’s
functions of the dot, and $\Sigma^{r},\Sigma^{<}$ are the retarded and lesser
self-energies respectively. Obtaining the self-energies represents the
greatest challenge in computing the current. We have employed the
Interpolative Perturbative Approximation (IPA), introduced by AligiaAligia
(2006), which is equivalent to the second-order Keldysh perturbation theory
(KPT2) in the steady-state p-h symmetric limit, and to the IPT (Kajueter-
KotliarKajueter and Kotliar (1996)) in the equilibrium limit, to get the self-
energies. The second-order expressions for the retarded and lesser self-
energies are as follows:

$\Sigma^{r}(\omega)=U^{2}\int\left(\prod_{i=1}^{3}d\epsilon_{i}D(\epsilon_{i})\right)\,(\omega^{+}+\epsilon_{3}-\epsilon_{2}-\epsilon_{1})^{-1}\\\
\times\left[\tilde{f}(-\epsilon_{1})\tilde{f}(-\epsilon_{2})\tilde{f}(\epsilon_{3})+\tilde{f}(\epsilon_{1})\tilde{f}(\epsilon_{2})\tilde{f}(-\epsilon_{3})\right]$
(20)

and

$\Sigma^{<}(\omega)=-2i\pi U^{2}\int
d\epsilon_{1}d\epsilon_{2}D(\epsilon_{1})D(\epsilon_{2})D(\epsilon_{1}+\epsilon_{2}-\omega)\\\
\times\left[\tilde{f}(\epsilon_{1})\tilde{f}(\epsilon_{2})\tilde{f}(\omega-\epsilon_{1}-\epsilon_{2})\right]$
(21)

where $D(\omega)=-(1/\pi)\Im{\tilde{g}^{r}_{d}(\omega)}$ is the spectral
function calculated from the Hartree-corrected, retarded dot Green’s function
given by

$[\tilde{g}^{r}_{d\sigma}(\omega)]^{-1}=\omega^{+}-\sum_{\alpha}\Delta_{\alpha}(\omega)\,,$
(22)

and
$\tilde{f}(\omega)=\sum_{\alpha}\Gamma_{\alpha}(\omega)f_{\alpha}(\omega)/\sum_{\alpha}\Gamma_{\alpha}(\omega)$
is the weighted Fermi function, with $\Gamma_{\alpha}(\omega)=-{\rm
Im}\Delta_{\alpha}(\omega)$ (equation 12), and
$f_{\alpha}(\omega)=f(\omega-\mu_{\alpha})$ is the Fermi function for the
$\alpha^{\rm{th}}$ lead. In writing the above, we have assumed the p-h
symmetric limit, where $\epsilon_{d}=-U/2$ cancels the first order Hartree
contribution. To obtain the lesser Green’s function ($G^{<}_{d}(\omega)$), we
need the Hartree-corrected lesser Green’s function of the dot, which is given
by

$\tilde{g}^{<}_{d\sigma}(\omega)=2i|\tilde{g}^{r}_{d\sigma}(\omega)|^{2}\sum_{\alpha}\Gamma_{\alpha}(\omega)f_{\alpha}(\omega)\,.$
(23)

Solutions of equations II to 23 yield the physical picture of the interplay of
bias, interactions and SOC on the spectra and current-voltage characteristics.
The self-energies in equation (20) and (21) are evaluated as convolutions
using the Fourier transform. The details of the numerical implementation of
the convolution can be found in earlier worksBarman and Vidhyadhiraja (2011).
The differential conductance is calculated as $G=dJ/dV_{sd}$, and numerically
implemented through a derivative of the splined current. We present our
results for spectra and transport quantities in the next section.

##  III Results

In equilibrium studies of the p-h symmetric Anderson model, the IPTKajueter
and Kotliar (1996), based on the second-order perturbation theory, is a good
approximation in the weak coupling limit, and by coincidence also reproduces
the atomic limit. Hence, the IPT has been extensively used as an interpolating
approximation for lattice models like the Hubbard model and the periodic
Anderson model within dynamical mean-field theory to investigate Mott
transition and heavy fermion physicsGeorges  _et al._ (1996). Various Keldysh
perturbation theory based approximations including IPA have been used quite
widely in the out-of-equilibrium case as well Joura  _et al._ (2015); Fujii
and Ueda (2003); Aligia (2006). In this work, we carry out a simple
benchmarking exercise to ascertain the regime of validity of the IPA and
subsequently use it to investigate the effect of spin-orbit coupling.

Figure 2: The current-voltage curves computed within IPA ($\lambda=0$) for
various values of $U$ (mentioned as legends) compared to TEBD results of Nuss
et al. Nuss  _et al._ (2013). The solid lines correspond to the semi-elliptic
hybridization, while the dashed lines correspond to the flat band
hybridization. The symbols are data extracted from Nuss et al. Nuss  _et al._
(2013) and the current is scaled by a factor of 3.33. Figure 3: The density of
states of the dot computed within IPA for $U=6\Delta_{0}$ and the flat band
hybridization in the absence of SOC. A splitting of the zero frequency peak
with increasing bias (indicated as legend) is seen, which matches with the
results of a fourth-order calculation of Fujii et al. Fujii and Ueda (2003).

###  III.1 Benchmarking

As a first benchmark, we compare the current-voltage (IV) curves for the flat
band and the semi-elliptic DoS case with the steady state results from Nuss et
al. Nuss  _et al._ (2013) who have used the time-evolving block decimation
(TEBD) method on a model with 150 sites comprising two leads and a quantum dot
subjected to a DC bias. As shown in figure 2 we find that the $J-V_{sd}$
curves match the TEBD results very well111For a fixed energy scale
$\Delta_{0}$, the hybridisation strength $V_{0}$, needs to have a
multiplicative factor of $\sqrt{2}$ for a quantitative comparison with the
TEBD results. The presence of a peak in the current at a specific bias roughly
around half the bandwidth is seen in the case of the finite bandwidth semi-
elliptic DoS case. The peak position moves to higher bias values for
increasing interaction strength and concomitantly the peak current magnitude
decreases. In the case of the infinitely wide, flat conduction band, we see a
saturation of the current at high bias values and the saturation current
decreases with increasing $U$. Both these results show that the IPA captures
the current characteristics very well over a wide range of interaction
strengths.

As a second benchmark, we compare our results with fourth-order perturbation
theory by Fujii et al.  Fujii and Ueda (2003) (for a flat band hybridization,
with $U=6\Delta_{0}$ and SOC strength $\lambda=0$) in figure 3 and see that
the interacting dot DoS matches very well with their results. We see that
there is a reduction in the zero frequency weight with increasing bias and
also splitting and broadening of the central peak. Next, we examine the effect
of equilibrium scales on the transport characteristics in the linear response
regime.

###  III.2 Linear response regime: scaling and universality

The excellent agreement with TEBD and fourth-order perturbation theory yield
strong confidence in the validity of IPA in a wide bias and interaction range.
We now explore the extent of the linear response regime and examine if
equilibrium quantities describe the response of the system at finite bias, but
close to equilibrium.

As we can see from figure 2, the linear dependence of the current at low
values of bias agrees well with the TEBD calculations and the extent of the
linear response regime depends on the value of interaction and SOC strengths.
In equilibrium, the quasiparticle weight, which can be calculated as
$Z_{0}=[1-{\rm Re}(d\Sigma(\omega)/d\omega|_{\omega=0})]^{-1}$, where
$\Sigma(\omega)$ is the self energy, happens to be proportional to the Kondo
scale, and hence determines the extent of the universal regimeHewson (1993).
In order to understand the dependence of the self-energy on SOC and bias, we
have carried out a simple analysis (refer to the Appendix) of the second-order
expression assuming that the Hartree-corrected density of states is not
varying rapidly in the neighbourhood of $\omega=0$ for low values of bias, and
SOC, i.e $D(\omega)\simeq D_{0}=-{\rm Im}(\tilde{g}^{r}_{d\sigma}(\omega=0)$
for $|\omega|,|V_{sd}|,\lambda<<t_{*}$. $D_{0}$ may be found by considering
the expressions for the retarded Green’s functions in section II, as shown
below. From equation 22, we see that

$\left[\tilde{g}^{r}_{d\sigma}(\omega=0)\right]^{-1}=V_{0}^{2}\sum_{\alpha
h}\int\,d\epsilon\,\frac{\rho_{0}(\epsilon)}{\mu_{\alpha}+h\lambda+\epsilon-i\eta}\,$
(24)

where $\eta\rightarrow 0^{+}$. Using the above expression, we note that at
$\omega=0$, the Hartree-corrected non-interacting Green’s function,
$\tilde{g}^{r}_{d\sigma}$ is purely imaginary, since

${\rm
Re}\left(\tilde{g}^{r}_{d\sigma}(\omega=0)\right)^{-1}=V_{0}^{2}\sum_{\alpha
h}{\cal{P}}\int\,d\epsilon\,\frac{\rho_{0}(\epsilon)}{\mu_{\alpha}+h\lambda+\epsilon}=0\,$
(25)

for a p-h symmetric $\rho_{0}(\epsilon)$. In the above equations,
$\mu_{L}=V_{sd}/2,\mu_{R}=-V_{sd}/2,h=\pm 1$, and ${\cal{P}}$ denotes the
principal value. Thus,

$\tilde{g}^{r}_{d\sigma}(\omega=0)=-\frac{i}{\pi
V_{0}^{2}\sum_{h\alpha}\rho_{0}(\mu_{\alpha}+h\lambda)}\,$ (26)

using which we can write, for zero bias,
$D_{0}=(4\pi^{2}V_{0}^{2}\rho_{0}(\lambda))^{-1}$. We find that the imaginary
part of the retarded self-energy, in terms of $D_{0}$ is given by (for
$\omega\rightarrow 0$ and $T=0$) :

$\\-\frac{1}{\pi}\imaginary\Sigma^{r}(\omega)=\frac{U^{2}D^{3}_{0}}{2}\left[\omega^{2}+\frac{3V_{sd}^{2}}{4}\right]$
(27)

Using the Kramers-Krönig transformation, the real part of the retarded self-
energy is given by

$\real\Sigma^{r}(\omega)=-\frac{U^{2}D^{3}_{0}}{\Lambda}\left[\Lambda^{2}-\frac{3V_{sd}^{2}}{4}\right]\omega\,,$
(28)

where $\Lambda$ is a high-energy cutoff that represents the extent of the
quadratic dependence of the imaginary part of the self-energy.

Thus, using the definition of the quasiparticle weight as
$\real\Sigma^{r}(\omega)=\omega(1-\frac{1}{Z})$ we get the analytical
expression (please refer to the Appendix for details of the calculation),

$Z(U,\lambda,V_{sd})=\left(1+\frac{U^{2}D^{3}_{0}}{\Lambda}\left[\Lambda^{2}-\frac{3V_{sd}^{2}}{4}\right]\right)^{-1}\,.$
(29)

Using this, we note that, since the IPA is based on second-order perturbation
theory, the equilibrium quasiparticle weight, $Z_{0}=Z(U,\lambda,V_{sd}=0)$
decays algebraically with increasing interaction strength (at zero bias) as
$U^{-2}$. The dependence on SOC enters through $D_{0}$. For a flat-band
density of states, since $D_{0}$ does not depend on $\lambda$, the
quasiparticle weight, $Z_{0}$ will be independent of SOC. The Kondo scale is
proportional to the product of bandwidth and $Z_{0}$, and since the chiral
bands move outward with increasing $\lambda$, the effective bandwidth will
increase and hence will give rise to a linearly increasing Kondo scale.
However, for a strongly frequency-dependent density of states like the semi-
elliptic or Gaussian density of states, $D_{0}$ increases as
$\rho_{0}^{-3}(\lambda)$. For example, for the Gaussian and semi-elliptic
forms of the hybridization, the $D_{0}$ is proportional to
$\exp(3\lambda^{2}/t_{*}^{2})$ and $(1-\lambda^{2}/4t_{*}^{2})^{-3/2}$
respectively. Hence the quasiparticle weight will decrease sharply with
increasing $\lambda$ as $Z_{0}\propto D_{0}^{-3}$ (from equation 29). So, even
though the effective bandwidth increases linearly with increasing SOC in
parallel to the flat band case, the strong decrease due to the factor of
$D_{0}^{-3}$ dominates. The equilibrium quasiparticle weight, $Z_{0}$ shown in
the bottom left panel of figure 4, for the semi-elliptic and the Gaussian
hybridization functions (denoted as SE and G respectively), is seen to
decrease sharply with increasing $\lambda/\Delta_{0}$ in agreement with the
arguments above.

Figure 4: Current ($J/\Delta_{0}$) as a function of bias ($V_{sd}/\Delta_{0}$)
is shown in the top two panels. The left panel shows the result for various
values of interaction strength $U/\Delta_{0}$, and SOC strength $\lambda=0$,
while the right panel shows the result for $U=20\Delta_{0}$ and various values
of $\lambda/\Delta_{0}$. The bottom left panel shows the dependence of
equilibrium ($V_{sd}=0$) quasiparticle weight $Z_{0}$ on the spin-orbit
coupling for the semi-elliptic (SE) and Gaussian (G) forms of hybridization.
The bottom right panel shows the collapse of all the curves in the top two
panels up to a bias given by $V_{sd}\simeq Z_{0}\Delta_{0}$, when the current
and bias are scaled by $Z_{0}$, thus determining the linear response regime.

Consistent with the decrease of $Z_{0}$ with $\lambda$, the linear response
regime in the current-voltage ($J-V_{sd}$) relationship shrinks considerably
with increasing interaction strength (for $\lambda=0$) as seen in the top left
panel of figure 4 and with increasing $\lambda$ for $U=20\Delta_{0}$ as seen
in the top right panel. However, when the current and bias are scaled by
$Z_{0}$, all the curves of the top two panels collapse up to a bias of
$V_{sd}\simeq Z_{0}\Delta_{0}$ (shown in the bottom right panel), which
confirms that the equilibrium quasiparticle weight, or equivalently, $T_{K}$,
determines the extent of the linear response regime Grobis  _et al._ (2008).

Figure 5: Variation of the differential conductance with interaction ($U$),
spin-orbit coupling (SOC) and bias. The colours correspond to the values of
the conductance $G=dJ/dV_{sd}$ as mentioned in the colour bar with certain
values marked by the black contours. The dashed lines represent the negative
differential resistance regime The top left and right panels correspond to
varying $U$ and fixed SOC equal to zero and $15\Delta_{0}$ respectively. The
bottom left and right panels correspond to varying SOC and fixed $U$ equal to
zero and $15\Delta_{0}$ respectively. Note that the top and bottom right
panels show the x-axis on the log scale for clarity, since the linear response
regime is highly compressed for large $U$ and/or SOC.

###  III.3 Conductance

Subsequent to the benchmarking and a study of the scales and the linear
response regime, we now present the main results of our work, which focus on
the interplay of spin-orbit coupling and electron-electron interactions on the
finite bias differential conductance. Since the SOC does not alter the
hybridization functions in the case of the infinite flat band, the results do
not show any dependence on $\lambda$. Therefore, we will consider the semi-
elliptic and Gaussian hybridization forms where the effective bandwidth is
modified by $\lambda$ as seen from equations 13\- 17.

Figure 5 shows contour plots of the conductance for various values of $U$ and
$\lambda$ at $T=0$, computed with the semi-elliptic hybridization function.
The conductance is computed by fitting a cubic spline to the current vs
$V_{sd}$ data and taking the first derivative. At the outset, we see that both
interactions and SOC have a strong effect on the conductance. The top left and
right panels represent varying interaction strength on the y-axis, but fixed
SOC of zero and $15\Delta_{0}$ respectively, while the bottom left and right
panels represent varying SOC on the y-axis, but fixed $U$ of zero and
$15\Delta_{0}$ respectively. The zero bias conductance peak is seen to be a
universal feature in all panels, and it is interesting to note that the width
of this peak decreases with increasing $U$ with or without $\lambda$, while
the effect of $\lambda$ on the peak width at $U=0$ is very weak (bottom left
panel), but in the presence of $U$, $\lambda$ leads to a sharp narrowing of
the zero bias peak (bottom right panel), which is consistent with the bottom
left panel of figure 4.

With increasing bias, a negative differential conductance (NDC) regime is seen
in all panels. In order to understand the origin of the NDC, we first focus on
the $U=0$ results, since the calculation of conductance for $U=0$ does not
depend on the second order approximation for the self-energy, and is hence
exact. The non-interacting regime also exhibits NDC at high bias values as
seen in the top panels and the bottom left panel. We confirm that the
finiteness of the bandwidth of the hybridization is responsible for the
NDCEvlashin  _et al._ (2021), since the wide flat band hybridization does not
yield NDC. Additionally, the top right panel shows an intermediate bias regime
where NDC is obtained at large $U$ values. In the strong coupling regime,
reached by increasing $U$ and/or $\lambda$, the width of the central Kondo
peak in the dot DoS, being proportional to $T_{K}$, shrinks exponentially.
However, adiabatic continuity to the non-interacting limit ensures that the
Kondo peak has precisely the same form as the non-interacting dot DoS if the
frequency is scaled as $\omega/Z_{0}\Delta_{0}$Logan  _et al._ (1998). This
fact implies that the system has an effective bandwidth of $Z_{0}\Delta_{0}$
in strong coupling, and hence as the bias value crosses the linear response
regime, an NDC regime may be obtained in parallel to the non-interacting
regime. Since IPA is perturbative, and not accurate in strong coupling the NDC
seen in strong coupling at intermediate bias values could be due to both the
finite support of the hybridization and the approximation used for computing
the self-energy.

The top left panel of figure 5 shows that in the absence of SOC, the
conductance decreases monotonically with increasing bias, and the linear
response regime (over which the conductance is close to 1) shrinks with
increasing interaction strength. The top right panel shows that in the
presence of strong SOC ($15\Delta_{0}$), the conductance becomes highly non-
monotonic, and a light band (for $U\gtrsim 10\Delta_{0}$) appears signifying a
peak at a non-zero bias, that shifts to higher bias with increasing
interaction strength. In fact, we observe that the bias value at which the
peak occurs is almost the same as the interaction strength. Another
interesting observation can be made from the bottom right panel of figure 5,
which shows the existence of a finite bias conductance peak for
$\lambda\gtrsim 10\Delta_{0}$, whose position (on the bias-axis) is almost
independent of the value of SOC. Thus, we see that the position of the finite
bias peak correlates very well with the interaction strength, but is
independent of $\lambda$, which suggests that the Hubbard bands and their
increasing prominence with increasing $\lambda$ are giving rise to this peak.
Below, we provide a physical basis for the emergence of this finite bias
conductance peak in terms of the spectral functions and the hybridization
functions.

In equilibrium (zero bias), and in the absence of SOC, the dot spectral
function acquires Hubbard bands at $\omega\gtrsim\pm U/2$ as the top panel of
figure 6 shows. For a flat band hybridization , the Hubbard bands are known to
lie close to $\pm U/2$Syaina and Majidi (2018), but for a dispersive, semi-
elliptic or Gaussian density of states, these incoherent peaks lie beyond $\pm
U/2$. If we now turn on SOC, then as the middle panel shows, the left and
right lead hybridization functions broaden significantly. This is because the
chiral bands that are split by the SOC, when superimposed give rise to a
hybridization, that has a width equal to $D+2\lambda$, and for the middle
panel, since $D=40\Delta_{0}$, and $\lambda=15\Delta_{0}$, the band-edges are
at $\pm(D/2+\lambda)=\pm 35\Delta_{0}$. Since, for this larger bandwidth, the
hybridization approaches more closely the flat band approximation, the Hubbard
bands become more prominent, and their location is almost at $\pm U/2$. As
bias is increased, the band centres (chemical potentials) of the two leads
move apart ($\mu_{L}=V_{sd}/2$ and $\mu_{R}=-V_{sd}/2$). As seen in the bottom
panel of figure 6, the red dashed and blue dot-dashed lines, which are the
left and right lead hybridizations, have small bumps at precisely these
energies for $V_{sd}=20\Delta_{0}$. Since the Hubbard bands are also at $\pm
U/2$, which are $\pm 10\Delta_{0}$ in figure 6, a maximum in conductance is
seen when the bias becomes equal to the peak position difference of the
Hubbard bands Fujii and Ueda (2003).

So an argument for the conductance peak at a bias value equal to interaction,
is that the chemical potential of the left lead matches with one Hubbard band,
while that of the right lead matches with the other, and hence the conductance
is peaked due to a resonant situation when the SOC is large ($\gtrsim
10\Delta_{0}$), as the bottom right panel of figure 5 shows. Indeed, the left
bottom panel, for which $U=0$, shows that the conductance decreases
monotonically with increasing bias, while the bottom right panel shows a peak
at bias around $15\Delta_{0}$ which is present only for large SOC ($\gtrsim
10\Delta_{0}$), but independent of the value of SOC. Thus, we find that, at
$T=0$, in addition to the zero bias peak, a finite bias peak arises in the
presence of strong spin-orbit coupling, which is positioned at a bias roughly
equal to the value of the interaction strength.

Figure 6: The dot spectral function (solid black line), and the left and right
lead hybridization functions (red dashed and blue dot-dashed respectively) as
a function of $\omega/\Delta_{0}$ showing the variation of spectral features
with increasing SOC and bias for a fixed interaction strength,
$U=20\Delta_{0}$. The top panel is for $\lambda=V_{sd}=0$, while
$\lambda=15\Delta_{0}$ is turned on in the middle panel keeping bias zero, and
the bottom panel has $\lambda=15\Delta_{0}$ and $V_{sd}=20\Delta_{0}$.  Figure
7: The main panel shows the differential conductance in units of $G_{0}$ as a
function of scaled bias ($V_{sd}/\Delta_{0}$) for three values of interaction
strength, namely $U=0$ (dot-dashed), $10\Delta_{0}$ (dashed), $20\Delta_{0}$
(solid) and $\lambda=15\Delta_{0}$ for a semi-elliptic hybridization. The left
inset shows $G$ vs $V_{sd}/\Delta_{0}$ for $U=20\Delta_{0}$ at four different
temperatures, mentioned as legends. The central peak is seen to melt with
increasing temperature. The right inset shows the zero bias conductance as a
function of $T/T_{K}$; The black symbols represent IPA data, and the red
dashed line represents a fit, $G(T)=G_{0}(1+(2^{1/s}-1)(T/T_{K})^{2})^{-s}$
with $s=0.22$ and $T_{K}=0.58\Delta_{0}$ as appropriate for universal Kondo
behaviour

We now focus on this finite bias conductance peak, which appears only for
relatively large values of interaction strength and SOC. We extend the bias to
negative values, and show the conductance as a function of bias, for various
$U$ values and $\lambda=15\Delta_{0}$ in figure 7. The conductance at $T=0$ is
seen to have a three-peak form consistent with the p-h symmetric regime, and
the results of figure 5. Again, the locations of the side peaks correspond
closely to the value of $U/\Delta_{0}$, while the zero bias peak is
characteristic of the linear response regime. We can also see that the full
width at half maximum (FWHM) of the central peak decreases sharply with
increasing interaction, while the conductance at zero bias is pinned at unity.
The width of the central peak is roughly proportional to the zero bias
quasiparticle weightKretinin  _et al._ (2011), which decreases exponentially
(algebraically within IPA) with increasing $U/\Delta_{0}$. The left inset of
figure 7 shows the temperature dependence of the conductance for the largest
$U/\Delta_{0}$ considered in the main panel. Interestingly, the central peak
melts and thus, at modest temperatures, we see a two-peak structure, that
resembles the results of a recent experimental study Smith  _et al._ (2022).
However, the right inset shows that the conductance at zero bias decreases
monotonically with increasing temperature in accordance with the universal
Kondo behaviour Goldhaber-Gordon  _et al._ (1998), while the ‘double zero bias
peak’ (DZBA) feature seen in experiments Smith  _et al._ (2022) exhibits the
opposite behaviour, namely an increase of $G$ with temperature. Hence, the
finite bias conductance peak in the intermediate-$U$ and $\lambda$ regime
fails to explain the experimental results. As we will show below, the latter
are best understood within the IPA framework from a strong correlation
perspective.

Figure 8: Schematic of the hybridization and the Dot density of states, with
the IPA computed results. $\Gamma_{\alpha}$ is the lead hybridization with
$\alpha=L/R$ for left (black)/right (blue), and the central (red) curve shows
the Dot density of states. Similarly, $\mu_{\alpha}$ represents the chemical
potential of the $\alpha^{\rm th}$ lead. The interaction strength is fixed at
$U=20\Delta_{0}$, while SOC, temperature and bias are turned on sequentially
from panels (a)-(d) as follows: (a) $\lambda=T=V_{sd}=0$; (b)
$\lambda=18\Delta_{0}$, $T=V_{sd}=0$; (c)
$\lambda=18\Delta_{0},T=0.05\Delta_{0}$, $V_{sd}=0$;
(d)$\lambda=18\Delta_{0},T=0.05\Delta_{0}$, $V_{sd}=25\Delta_{0}$. The shaded
regions represent the $T=0$ occupied states. The dotted and dashed lines are
guides to the eye.

Equilibrium investigations of the SIAM have shown that the Kondo scale
decreases exponentially with increasing $U$ for $\lambda=0$ Tsvelick and
Wiegmann (1983). In the presence of $\lambda$ on the leads and $U$ on the dot,
the Kondo scale has been shown to increase or decrease depending on whether
the leads have a wide flat band or a finite bandwidthZarea  _et al._ (2012);
Žitko and Bonča (2011) respectively. In our work, we have shown (see equation
29) that the quasiparticle weight, and hence the Kondo scale depends on $U$
algebraically (decreasing as $U^{-2}$) and on $\lambda$ through the density of
states, namely as $(\rho_{0}(\lambda))^{3}$. The algebraic decrease of the
scale with $U$ is known to be an artefact of the perturbative approximation
employed to obtain the self-energy, while exact methods like Bethe
AnsatzTsvelick and Wiegmann (1983) find an exponential decrease. Since
$\rho_{0}(\epsilon)$ is usually chosen to have a maximum at the chemical
potential, and decreases monotonically with increasing $|\epsilon|$, the Kondo
scale will also decrease monotonically with increasing $\lambda$ as discussed
above. The precise form of this decrease will depend on the form of the
density of states. We see from figure 4, bottom-left panel, that the
quasiparticle weight scale decreases sharply with increasing $\lambda$.
Although the qualitative trend is the same for the semi-elliptic as well as
the Gaussian hybridization, the scale decreases much more rapidly for the
latter. For the same value of the parameters, the scale corresponding to the
Gaussian hybridization is much smaller than that of the semi-elliptic case.
Hence for the strong coupling regime, we choose to work with the Gaussian
hybridization. The zero bias conductance peak has a width of the order of
$T_{K}$, while the finite bias peaks are positioned at $\sim U$ (as discussed
in the previous sub-section). Hence in strong coupling ($T_{K}\rightarrow 0$),
these two features get clearly separated. This clear separation of scales,
which is the best distinguishing feature of the strong coupling regime, has
important consequences on the evolution of spectra and conductance with
increasing temperature, as discussed later. We will begin with a discussion of
the changes in the spectra and hybridization as we sequentially turn on SOC,
temperature, and bias for $U=20\Delta_{0}$.

The schematic shown in figure 8 shows the left lead and the right lead
hybridization functions as $\Gamma_{L}$ and $\Gamma_{R}$ in black and blue
colours respectively. The dot spectral function is shown in the centre as a
red solid line. The vertical axis is energy/frequency. The schematic has been
made using data obtained from IPA, and the curves in the four panels are not
mere sketches. Panel(a) represents a situation where $U=20\Delta_{0}$, but
$\lambda=T=V_{sd}=0$. The dot spectral function has a three-peak structure,
with the central peak being the Kondo peak, and the other two being the
Hubbard bands at $\pm U/2$. As we turn on $\lambda=18\Delta_{0}$, keeping
$T=V_{sd}=0$, panel(b) shows that the central peak in the dot spectral
function becomes extremely narrow, while $\Gamma_{L}$ and $\Gamma_{R}$ show
the development of chiral bands due to the SOC. The clear separation of the
Kondo peak and the Hubbard bands is evident. When we turn on a small, but
finite temperature, panel (c) shows that the central peak melts leaving the
Hubbard bands as the only distinct features in the spectrum. Finally, when a
bias of $V_{sd}=25\Delta_{0}$ is turned on, there are minor changes in the
positions of the spectral peaks, while the $\Gamma_{L}$ and $\Gamma_{R}$ move
up and down respectively. The occupied region of the left lead is seen to
overlap with the region between the Hubbard bands, and the unoccupied region
of the right lead. This overlap is likely the reason for the finite bias
peaks.

Figure 9: Scaled conductance, $G(V_{sd},T)$ as a function of bias,
$V_{sd}/\Delta_{0}$ for various temperatures, $T/\Delta_{0}$ indicated as
legends in the main panel. The parameters (except for bias) are the same as
those considered in figure 8(c). The left inset shows the zero temperature
conductance as a function of scaled bias, while the right inset shows the zero
bias conductance, $G(V_{sd}=0,T)$ as a function of scaled temperature.

The temperature interval over which the resistivity shows universal behaviour
when viewed on an absolute scale, i.e as $T/\Delta_{0}$ (and not as $T/T_{K}$)
shrinks drastically in the strong interaction and strong spin-orbit coupling
regime, because $T/\Delta_{0}=(T/T_{K})(T_{K}/\Delta_{0})$, and the latter
term is exponentially small in the strong coupling regime. In such a regime, a
clear separation of scales happens in the conductivity as well as in the
spectra. We show this in the top left panel of figure 9, where the conductance
as a function of bias, computed within IPA at $T=0$ for
$U=20\Delta_{0},\lambda=18\Delta_{0}$, using the Gaussian form of
hybridization. The quasiparticle weight that determines the linear response
regime, and the $U$ scale that determines the finite bias conductance peak
show up as distinct features in the form of an extremely narrow zero bias
peak, and broad, finite bias peaks. The top right panel shows that the zero
bias conductance decreases rapidly with increasing temperature, and reaches a
value three orders of magnitude smaller than the $T=0$ unitary limit at $T\sim
10^{-2}\Delta_{0}$. The zero bias conductance follows the evolution of the dot
spectral function and as the schematic figure 8 shows, the melting of the
Kondo peak manifests in the rapid decrease of the ZBC. However, as the
temperature reaches non-universal values beyond $0.5\Delta_{0}$, the Hubbard
bands become accessible, and the ZBC starts to rise again. The bottom panel
shows the conductance as a function of bias for various temperatures in this
rising ZBC regime. The two finite bias peaks are the only distinct features of
the conductance in a wide temperature interval of
$T\in(10^{-3}\Delta_{0},0.5\Delta_{0})$. For $T\gtrsim 0.5\Delta_{0}$,
consistent with the top right panel, the ZBC starts rising, filling in the gap
between the finite bias peaks, and concomitantly the latter diminishes in
magnitude slightly. An isosbestic point, commonly seen in many correlated
systems, is visible around $V_{sd}\simeq 10\Delta_{0}$.

In a recent experiment by Smith et al. Smith  _et al._ (2022), the conductance
through a point contact in an InGaAs/InAlAs heterostructure and a split-gate
geometry, was measured as a function of source-drain bias with varying in-
plane and transverse magnetic fields. The highlight of this study was the
tunability of the Kondo effect through the tuning of the Rashba spin-orbit
coupling of the leads. The split-gate voltage, $V_{sg}$, was used to control
the electron density, which in turn, determines the strength of the RSOC
($\lambda$). At a fixed temperature of about $25{\rm mK}$, a single zero bias
peak was observed at low $\lambda$, while a ’double zero bias peak’ was
observed at high $\lambda$ (see fig1(i) of Smith et al. Smith  _et al._
(2022)).

The zero bias peak is known to be related to the equilibrium Kondo scale, and
in the absence of a magnetic field, the zero bias peak must be present in the
conductance at $T=0$, no matter how strong the correlation is. But since the
experiment is performed at a fixed and finite absolute temperature, the
scenario becomes drastically different. As $\lambda$ is increased, the $T_{K}$
value decreases, so the ratio of $T/T_{K}$ increases exponentially. This
implies that the system with low $\lambda$ experiences a low $T/T_{K}$ ratio,
while the system with high $\lambda$ is at a very high $T/T_{K}$ ratio. We
have shown in figure 9 that the central zero bias peak melts rapidly on a
scale of $T_{K}$ at high $\lambda$, while the finite bias peaks retain their
form. Thus, we conjecture that the experimental finding of a crossover from a
single peak to a two-peak structure by increasing $\lambda$ is arising through
a combined crossover of a weak to strong coupling (exponentially decreasing
$T_{K}$) and from a low $T/T_{K}$ ratio to a high value of $T/T_{K}$ (due to a
fixed $T$).

Figure 10: The conductance plotted as a function of $V_{sd}/\Delta_{0}$ for
various values of SOC strengths as marked on the curves. The interaction and
temperature are kept fixed at $U=20\Delta_{0},T=0.05\Delta_{0}$ respectively.
The crosses track the location of the $V_{sd}=\pm U$ peak, while the stars
track a satellite feature of the zero bias peak. The left and right insets
correspond to the low ($\lambda=2\Delta_{0}$) and high
($\lambda=18\Delta_{0}$) SOC strengths respectively. The insets show the
different single and double peak conductance in these regimes.

We consolidate this speculation in figure 10, where the top two panels show
the conductance as a function of bias at low $\lambda=2\Delta_{0}$ (left
panel) and high $\lambda=18\Delta_{0}$ (right panel) computed within IPA at a
fixed temperature of $T=0.05\Delta_{0}$, and $U=20\Delta_{0}$ for a Gaussian
hybridization. We see that a single zero bias peak has transformed into a
double peak structure. We emphasise that the single zero bias peak is a
universal feature implying that its form and temperature dependence are
determined by the Kondo scale, but the finite bias peaks are non-universal
features whose position, and form are determined by non-universal parameters
like $V_{sd},U$ and $\lambda$. The bottom panel shows the conductance for the
same parameters as the top two panels, but with varying $\lambda$ marked as
numbers. The conductance curves are shifted vertically by a constant number to
provide clarity. The graph shows the gradual evolution of the single peak
structure into a two-peak structure. The dashed line marks the evolution of
these finite bias peaks (marked by crosses). Interestingly, the central peak
is flanked by two satellite structures, which we identify visually (marked by
stars). These peaks drift to higher $V_{sd}$ with increasing $\lambda$, and
merge with the ‘Hubbard peaks’, and continue the blue shift for higher
$\lambda$ values. Similar features are observed in the experimental results
(see figure 4(c) of Smith et al.  Smith  _et al._ (2022)). Naturally,
experimental results are far richer than what our simple model and the
perturbative calculation reveal. However, the theoretical model used for
explaining the experimental results Smith  _et al._ (2022) is exactly the same
as the one used in the present work, and our results based on IPA may be
viewed as complementary to those found by the quantum master equation
approach.

##  IV Discussion and conclusions

We have investigated the interplay of electron-electron interactions, bias,
and spin-orbit coupling on the conductance through an Anderson impurity, using
a second-order Keldysh approach. We validated the method and our
implementation through extensive benchmarks. The linear response regime and
its relation with the equilibrium quasiparticle weight ($Z_{0}$) is explored,
and the decrease of $Z_{0}$ with increasing spin-orbit coupling ($\lambda$),
for finite bandwidth hybridization functions, is highlighted and discussed in
detail. This decrease turns out to be crucial for explaining the experimental
observation of a crossover of a single zero bias peak into a double peak
structure with increasing $\lambda$. The quantum master equation approach
Smith  _et al._ (2022), which was used for explaining the experimental results
Smith  _et al._ (2022), while recovering the split peak structure, addresses
the finite field effects. However, the dependence of the Kondo scale on
$\lambda$ was not incorporated, and finite temperature effects have not been
addressed, and these two aspects have been incorporated and investigated in
detail in our work.

One of the inferences made from the experimental results Smith  _et al._
(2022) is that the Kondo scale increases with increasing $\lambda$. This
inference directly contradicts our results, and hence deserves a discussion.
We note that the $T_{K}$ is determined experimentally for the system with the
two peak structure (at high $\lambda$) by increasing magnetic field until the
two peaks merge into a single zero bias peak structure, and then fitting the
temperature dependence of the zero bias conductance. It is well known from
equilibrium studies that the quasiparticle weight increases monotonically with
increasing magnetic fieldLogan and Dickens (2001), so the determination of
$T_{K}$ for the high magnetic field system should not be taken to represent
the $T_{K}$ for the zero-field system with the double peak conductance. In
fact, for this system, if our results are any indication, then the $T_{K}$ is
probably around the lower limit of the temperatures considered in the
experiments. Increasing magnetic field is akin to a crossover from strong to
weak coupling regime, and hence is equivalent to decreasing spin-orbit
coupling. And hence the merging of the double peak conductance to a single
peak with increasing magnetic field may be viewed as decreasing $\lambda$ in
figure 9.

While the IPA has been shown to work well in the parameter regime under
consideration, it fails to capture many subtler aspects like the exponential
decrease of $Z_{0}$ with increasing interaction strength, which requires a
method capable of capturing strong coupling physics. The IPA has been
generalized for finite magnetic fields Aligia (2006), incorporating which will
yield fresh insights into magnetotransport measurements. We plan to pursue
both of these directions.

###### Acknowledgements.

The authors would like to thank Vinayak M. Kulkarni and Arijit Dutta for
discussions. We also acknowledge funding from JNCASR and the national
supercomputing mission (DST/NSM/R&D_HPC_Applications/2021/26).

*

##  Appendix A Derivation of an analytical expression for the quasiparticle
weight

In order to get more clarity, we calculate an analytical expression for the
quasiparticle weight using the second-order self-energy expression (20).

$\begin{split}\Sigma^{r}(\omega)&=U^{2}D_{0}^{3}\int
d\epsilon_{1}d\epsilon_{2}d\epsilon_{3}\left(\underbrace{\frac{\tilde{f}(-\epsilon_{1})\tilde{f}(-\epsilon_{2})\tilde{f}(\epsilon_{3})}{\omega^{+}+\epsilon_{3}-\epsilon_{2}-\epsilon_{1}}}_{\Sigma^{r}_{1}}\right.\\\
&\left.+\underbrace{\frac{\tilde{f}(\epsilon_{1})\tilde{f}(\epsilon_{2})\tilde{f}(-\epsilon_{3})}{\omega^{+}+\epsilon_{3}-\epsilon_{2}-\epsilon_{1}}}_{\Sigma^{r}_{2}}\right)\,,\end{split}$
(30)

where $D_{0}$ is the Hartree-corrected density of states at $\omega=0$. The
two terms $\Sigma^{r}_{1,2}$ are not independent and we can show that
$\Sigma^{r}_{2}(\omega)=(\Sigma^{r}_{1}(-\omega))^{*}$. When we consider the
zero temperature limit and define
$\rho_{\Sigma}=-1/\pi\Im{\Sigma^{r}_{1}(\omega)}$, the expression then reduces
to a product of Heaviside step functions given by

$\begin{split}\rho^{r}_{\Sigma 1}(\omega)&=\frac{U^{2}D^{3}_{0}}{8}\int
d\epsilon_{1}d\epsilon_{2}\left(\left[\underbrace{\Theta\left(V_{sd}/2-\epsilon_{1}\right)}_{a_{1}}+\underbrace{\Theta\left(-V_{sd}/2-\epsilon_{1}\right)}_{a_{2}}\right]\right.\\\
&\left.\left[\underbrace{\Theta\left(V_{sd}/2-\epsilon_{2}\right)}_{b_{1}}+\underbrace{\Theta\left(-V_{sd}/2-\epsilon_{2}\right)}_{b_{2}}\right]\right.\\\
&\left.\left[\underbrace{\Theta\left(V_{sd}/2+\omega-\epsilon_{1}-\epsilon_{2}\right)}_{c_{1}}+\underbrace{\Theta\left(-V_{sd}/2+\omega-\epsilon_{1}-\epsilon_{2}\right)}_{c_{2}}\right]\right)\end{split}$
(31)

The various products of the terms $a_{i},b_{i},c_{i}$ are tabulated below

$\begin{array}[]{|c|c|}\hline\cr{a_{i}b_{i}c_{i}}&8\rho^{r}_{\Sigma
1}|_{a_{i}b_{i}c_{i}}/(U^{2}D_{0}^{3})\\\
\hline\cr\hline\cr{a_{1}b_{1}c_{1}}&\omega^{2}/2+3\omega
V_{sd}/2+V_{sd}^{2}+V_{sd}^{2}/8\\\ {a_{1}b_{1}c_{2}}&\omega^{2}/2+\omega
V_{sd}/2+V_{sd}^{2}/8\\\ {a_{1}b_{2}c_{1}}&\omega^{2}/2+\omega
V_{sd}/2+V_{sd}^{2}/8\\\ {a_{1}b_{2}c_{2}}&(\omega^{2}/2-\omega
V_{sd}/2+V_{sd}^{2}/8)\Theta(\omega-V_{sd}/2)\\\
{a_{2}b_{1}c_{1}}&\omega^{2}/2+\omega V_{sd}/2+V_{sd}^{2}/8\\\
{a_{2}b_{1}c_{2}}&(\omega^{2}/2-\omega V_{sd}/2+V_{sd}^{2}/8)\Theta(\omega-
V_{sd}/2)\\\ {a_{2}b_{2}c_{1}}&(\omega^{2}/2-\omega
V_{sd}/2+V_{sd}^{2}/8)\Theta(\omega-V_{sd}/2)\\\
{a_{2}b_{2}c_{2}}&(\omega^{2}/2-3\omega
V_{sd}/2+V_{sd}^{2}+V_{sd}^{2}/8)\Theta(\omega-3V_{sd}/2)\\\
\hline\cr\end{array}$ (32)

Adding all the terms we get

$\rho^{r}_{\Sigma}=\frac{U^{2}D^{3}_{0}}{2}\left[\omega^{2}+\frac{3V_{sd}^{2}}{4}\right]\,.$
(33)

We can use the Kramer-Krönig transform to calculate the real part of self-
energy as

$\real\Sigma^{r}(\omega)=\frac{1}{\pi}\mathcal{P}\int
d\omega^{\prime}\frac{\imaginary\Sigma^{r}(\omega^{\prime})}{\omega^{\prime}-\omega}\,$
(34)

which we can write by considering a high energy cutoff $\Lambda$ as

$\real\Sigma^{r}(\omega)=\frac{U^{2}D_{0}^{3}}{2}\int_{-\Lambda}^{\Lambda}d\omega^{\prime}\frac{\omega^{\prime
2}+3V_{sd}^{2}/4}{\omega^{\prime}-\omega}\,.$ (35)

Considering particle-hole symmetry and truncating terms of
$\mathcal{O}(\omega^{2})$, we get the expression

$\real\Sigma^{r}(\omega)=-\frac{U^{2}D^{3}_{0}}{\Lambda}\left[\Lambda^{2}-\frac{3V_{sd}^{2}}{4}\right]\omega\,.$
(36)

Using the definition of the quasiparticle weight as
$\real\Sigma^{r}(\omega)=\omega(1-\frac{1}{Z})$ we get the analytical
expression (equation (29) in the main text) as

$Z=\left(1+\frac{U^{2}D^{3}_{0}}{\Lambda}\left[\Lambda^{2}-\frac{3V_{sd}^{2}}{4}\right]\right)^{-1}\,.$
(37)

## References

  * Goldhaber-Gordon  _et al._ (1998) D. Goldhaber-Gordon, J. Göres, M. A. Kastner, H. Shtrikman, D. Mahalu, and U. Meirav, Phys. Rev. Lett. 81, 5225 (1998).
  * Cronenwett  _et al._ (1998) S. M. Cronenwett, T. H. Oosterkamp, and L. P. Kouwenhoven, Science 281, 540 (1998).
  * Chorley  _et al._ (2012) S. J. Chorley, M. R. Galpin, F. W. Jayatilaka, C. G. Smith, D. E. Logan, and M. R. Buitelaar, Phys. Rev. Lett. 109, 156804 (2012).
  * Liang  _et al._ (2002) W. Liang, M. P. Shores, M. Bockrath, J. R. Long, and H. Park, Nature 417, 725 (2002).
  * Yu and Natelson (2004) L. H. Yu and D. Natelson, Nano Letters 4, 79 (2004).
  * Grigorian  _et al._ (1999) L. Grigorian, G. U. Sumanasekera, A. L. Loper, S. L. Fang, J. L. Allen, and P. C. Eklund, Phys. Rev. B 60, R11309 (1999).
  * Nygård  _et al._ (2000) J. Nygård, D. H. Cobden, and P. E. Lindelof, Nature 408, 342 (2000).
  * Eichler  _et al._ (2009) A. Eichler, R. Deblock, M. Weiss, C. Karrasch, V. Meden, C. Schönenberger, and H. Bouchiat, Phys. Rev. B 79, 161407 (2009).
  * Laird  _et al._ (2015) E. A. Laird, F. Kuemmeth, G. A. Steele, K. Grove-Rasmussen, J. Nygård, K. Flensberg, and L. P. Kouwenhoven, Rev. Mod. Phys. 87, 703 (2015).
  * van Wees  _et al._ (1988) B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L. P. Kouwenhoven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett. 60, 848 (1988).
  * Thomas  _et al._ (1996) K. J. Thomas, J. T. Nicholls, M. Y. Simmons, M. Pepper, D. R. Mace, and D. A. Ritchie, Phys. Rev. Lett. 77, 135 (1996).
  * Brun  _et al._ (2014) B. Brun, F. Martins, S. Faniel, B. Hackens, G. Bachelier, A. Cavanna, C. Ulysse, A. Ouerghi, U. Gennser, D. Mailly, S. Huant, V. Bayot, M. Sanquer, and H. Sellier, Nature Communications 5, 4290 (2014).
  * Heedt  _et al._ (2016) S. Heedt, W. Prost, J. Schubert, D. Grützmacher, and T. Schäpers, Nano Letters 16, 3116 (2016), pMID: 27104768.
  * Smith  _et al._ (2022) L. W. Smith, H.-B. Chen, C.-W. Chang, C.-W. Wu, S.-T. Lo, S.-H. Chao, I. Farrer, H. E. Beere, J. P. Griffiths, G. A. C. Jones, D. A. Ritchie, Y.-N. Chen, and T.-M. Chen, Phys. Rev. Lett. 128, 027701 (2022).
  * Aradhya and Venkataraman (2013) S. V. Aradhya and L. Venkataraman, Nature Nanotechnology 8, 399 (2013).
  * Kobayashi  _et al._ (2010) T. Kobayashi, S. Tsuruta, S. Sasaki, T. Fujisawa, Y. Tokura, and T. Akazaki, Phys. Rev. Lett. 104, 036804 (2010).
  * Tsvelick and Wiegmann (1983) A. Tsvelick and P. Wiegmann, Advances in Physics 32, 453 (1983).
  * Krishna-murthy  _et al._ (1980) H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980).
  * Anders (2008) F. B. Anders, Phys. Rev. Lett. 101, 066804 (2008).
  * Anders and Schiller (2005) F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005).
  * Fritsch and Kehrein (2010) P. Fritsch and S. Kehrein, Phys. Rev. B 81, 035113 (2010).
  * Wingreen and Meir (1994) N. S. Wingreen and Y. Meir, Phys. Rev. B 49, 11040 (1994).
  * Fujii and Ueda (2003) T. Fujii and K. Ueda, Phys. Rev. B 68, 155310 (2003).
  * Lopes and Anda (2019) V. Lopes and E. V. Anda, Journal of Physics and Chemistry of Solids 128, 188 (2019), spin-Orbit Coupled Materials.
  * Joura  _et al._ (2015) A. V. Joura, J. K. Freericks, and A. I. Lichtenstein, Phys. Rev. B 91, 245153 (2015).
  * Werner  _et al._ (2010) P. Werner, T. Oka, M. Eckstein, and A. J. Millis, Phys. Rev. B 81, 035108 (2010).
  * Nuss  _et al._ (2013) M. Nuss, M. Ganahl, H. G. Evertz, E. Arrigoni, and W. von der Linden, Phys. Rev. B 88, 045132 (2013).
  * Mehta and Andrei (2006) P. Mehta and N. Andrei, Phys. Rev. Lett. 96, 216802 (2006).
  * Cheng  _et al._ (2015) Y. Cheng, W. Hou, Y. Wang, Z. Li, J. Wei, and Y. Yan, New Journal of Physics 17, 033009 (2015).
  * Dorda  _et al._ (2014) A. Dorda, M. Nuss, W. von der Linden, and E. Arrigoni, Phys. Rev. B 89, 165105 (2014).
  * Fasth  _et al._ (2007) C. Fasth, A. Fuhrer, L. Samuelson, V. N. Golovach, and D. Loss, Phys. Rev. Lett. 98, 266801 (2007).
  * Zazunov  _et al._ (2009) A. Zazunov, R. Egger, T. Jonckheere, and T. Martin, Phys. Rev. Lett. 103, 147004 (2009).
  * Tanttu  _et al._ (2019) T. Tanttu, B. Hensen, K. W. Chan, C. H. Yang, W. W. Huang, M. Fogarty, F. Hudson, K. Itoh, D. Culcer, A. Laucht, A. Morello, and A. Dzurak, Phys. Rev. X 9, 021028 (2019).
  * Guo  _et al._ (2021) L. Guo, Y. Yan, R. Xu, J. Li, and C. Zeng, Phys. Rev. Lett. 126, 057701 (2021).
  * Schäpers  _et al._ (2004) T. Schäpers, J. Knobbe, and V. A. Guzenko, Phys. Rev. B 69, 235323 (2004).
  * Zarea  _et al._ (2012) M. Zarea, S. E. Ulloa, and N. Sandler, Phys. Rev. Lett. 108, 046601 (2012).
  * Wong  _et al._ (2016) A. Wong, S. E. Ulloa, N. Sandler, and K. Ingersent, Phys. Rev. B 93, 075148 (2016).
  * Žitko and Bonča (2011) R. Žitko and J. Bonča, Phys. Rev. B 84, 193411 (2011).
  * Lü and Guo (2007) H.-F. Lü and Y. Guo, Phys. Rev. B 76, 045120 (2007).
  * Zhang  _et al._ (2011) S. Zhang, W. Gong, G. Wei, and A. Du, Journal of Applied Physics 109, 023704 (2011).
  * Gelabert  _et al._ (2010) M. M. Gelabert, L. m. c. Serra, D. Sánchez, and R. López, Phys. Rev. B 81, 165317 (2010).
  * Topinka  _et al._ (2000) M. A. Topinka, B. J. LeRoy, S. E. J. Shaw, E. J. Heller, R. M. Westervelt, K. D. Maranowski, and A. C. Gossard, Science 289, 2323 (2000).
  * Meir  _et al._ (2002) Y. Meir, K. Hirose, and N. S. Wingreen, Phys. Rev. Lett. 89, 196802 (2002).
  * Iqbal  _et al._ (2013) M. J. Iqbal, R. Levy, E. J. Koop, J. B. Dekker, J. P. de Jong, J. H. M. van der Velde, D. Reuter, A. D. Wieck, R. Aguado, Y. Meir, and C. H. van der Wal, Nature 501, 79 (2013).
  * Aligia (2006) A. A. Aligia, Phys. Rev. B 74, 155125 (2006).
  * Meir and Wingreen (1992) Y. Meir and N. S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992).
  * Hewson (1993) A. C. Hewson,  _The Kondo Problem to Heavy Fermions_, Cambridge Studies in Magnetism (Cambridge University Press, 1993).
  * Kajueter and Kotliar (1996) H. Kajueter and G. Kotliar, Phys. Rev. Lett. 77, 131 (1996).
  * Barman and Vidhyadhiraja (2011) H. Barman and N. S. Vidhyadhiraja, International Journal of Modern Physics B 25, 2461 (2011).
  * Georges  _et al._ (1996) A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * Note (1) For a fixed energy scale $\Delta_{0}$, the hybridisation strength $V_{0}$, needs to have a multiplicative factor of $\sqrt{2}$ for a quantitative comparison with the TEBD results.
  * Grobis  _et al._ (2008) M. Grobis, I. G. Rau, R. M. Potok, H. Shtrikman, and D. Goldhaber-Gordon, Phys. Rev. Lett. 100, 246601 (2008).
  * Evlashin  _et al._ (2021) S. Evlashin, M. Tarkhov, D. Chernodubov, A. Inyushkin, A. Pilevsky, P. Dyakonov, A. Pavlov, N. Suetin, I. Akhatov, and V. Perebeinos, Phys. Rev. Applied 15, 054057 (2021).
  * Logan  _et al._ (1998) D. E. Logan, M. P. Eastwood, and M. A. Tusch, Journal of Physics: Condensed Matter 10, 2673 (1998).
  * Syaina and Majidi (2018) L. P. Syaina and M. A. Majidi, Journal of Physics: Conference Series 1011, 012072 (2018).
  * Kretinin  _et al._ (2011) A. V. Kretinin, H. Shtrikman, D. Goldhaber-Gordon, M. Hanl, A. Weichselbaum, J. von Delft, T. Costi, and D. Mahalu, Phys. Rev. B 84, 245316 (2011).
  * Logan and Dickens (2001) D. E. Logan and N. L. Dickens, Europhysics Letters (EPL) 54, 227 (2001).