# Protection of a non-Fermi liquid by spin-orbit interaction

T. K. T. Nguyen  Institute of Physics, Vietnam Academy of Science and
Technology, 10 Dao Tan, Hanoi, Vietnam  M. N. Kiselev  The Abdus Salam
International Centre for Theoretical Physics, Strada Costiera 11, I-34151,
Trieste, Italy

###### Abstract

We show that a thermoelectric transport through a Quantum Dot (QD) - single-
mode Quantum Point Contact (QPC) nano-device demonstrating pronounced
fingerprints of Non-Fermi Liquid (NFL) behaviour in the absence of external
magnetic field is protected from magnetic field NFL destruction by strong
spin-orbit interaction (SOI). The mechanism of protection is associated with
appearance of additional scattering processes due to lack of spin conservation
in the presence of both SOI and small Zeeman field. The interplay between in-
plane magnetic field $\vec{B}$ and SOI is controlled by the angle between
$\vec{B}$ and $\vec{B}_{SOI}$. We predict strong dependence of the
thermoelectric coefficients on the orientation of the magnetic field and
discuss a window of parameters for experimental observation of NFL effects.

###### pacs:

73.23.Hk, 73.50.Lw, 72.15.Qm, 73.21.La

##  I Introduction

The paradigm of Landau Fermi Liquid (FL) LandauFL  is one of the cornerstones
of modern condensed matter theory. Based on the concepts of quasiparticles -
well defined excitations whose energy in the long-wave limit is greater than
their decay rate, the FL theory successfully explains the behaviour of normal
and superconducting metals giving universal predictions for thermodynamic and
transport properties Anderson . The FL phenomenology is justified in many
microscopic models describing interacting fermions in- and out- of
equilibrium. However, there are several cases where a violation of the FL
picture is observed experimentally (e.g. in strongly correlated electron
systems such as heavy fermion compounds HF , unconventional superconductors
Anderson  and quantum transport through nano-structures Ralph ; Goldhaber ).
The pronounced Non-Fermi Liquid (NFL) behaviour of these systems is attributed
to a breakdown the quasiparticle concept: the decay rate of low-energy
excitations becomes greater than the energy of the excitations itself.

Figure 1: (Color online) Typical setup for thermoelectric measurements: ”cold”
contact (light orange area) at reference temperature $T$ \- Quantum Dot -
Quantum Point Contact electrostatically defined by gates (blue boxes),
separated by a tunnel barrier from a ”hot” (deep orange) contact at
temperature $T+\Delta T$. The voltage $V$ is applied across the device for
zero current measurements (see text for the details).

While the fingerprints of NFL physics in thermodynamics of strongly correlated
systems and quantum transport had been seen experimentally HF ; Goldhaber , it
is also generally accepted that the NFL picture is extremely sensitive to
variation of external parameters being unstable against the FL ground state.
Thus, the stability of the NFL domain and the possibility to observe strong
deviations from the Landau FL paradigm posses major challenges including the
development of theoretical models and predictions for the stabilization of
NFL-states. Important questions are: is it possible at all to protect unstable
NFLs? What are the physical observables which demonstrate the most pronounced
manifestation of the NFL physics?

In this paper we present an example based on a window of parameters within
which the observation of strong deviation from the FL picture can be protected
and extended by effects of strong spin-orbit interaction (SOI). The physical
observables we consider in this work are thermoelectric coefficients of a
nano-device (Fig. 1). Our theoretical model justifying NFL behaviour is a two-
channel Kondo (2CK) model NozBla80 ; Zawad ; Affleck91 . While the scattering
of single orbital channel electrons on a resonance quantum impurity itself
leads to strong modification of the thermoelectric transport properties within
the Landau FL paradigm through strong renormalization of the FL energy scale
Fazio ; Costi2010 ; Zitko , the detour from the FL picture is predicted to
change completely both electric Matveev95 ; Flensberg ; FM_theory ; LeHur  and
thermoelectric transports MA_theory ; NKK10 . For example, one of the
manifestations of the NFL behaviour in quantum transport is associated with
the logarithmic enhancement of the thermoelectric power MA_theory  in the
situation when the 2CK model originates from the charge Kondo effect in a
single mode quantum point contact (QPC) - quantum dot (QD) setup tuned by gate
voltages to the Coulomb blockade (CB) peak regime Matveev95 ; Flensberg ;
FM_theory ; LeHur ; MA_theory . In that case two channels are the electron
spin degrees of freedom while the almost transparent QPC (weak back-scatterer)
works as a quantum impurity. The 2CK physics is known to be unstable with
respect to any effects which can potentially break (statically or dynamically)
the symmetry between the channels Goldhaber ; NKK10 . In particular, it has
been shown that the effects associated with time-reversal symmetry breaking
(TRS) due to an external magnetic field restore the FL properties at
temperatures below $T_{\rm eff}$ tunable by the field NKK10 .  The
universality class of unstable 2CK model than changes to single channel Kondo
problem (1CK). Fully screened 1CK is characterized by stable local  FL
properties. Therefore, while being very attractive from theoretical point of
view, the 2CK physics suffers from serious experimental obstacles Ralph ;
Goldhaber ; GG11  impeding a direct observation of NFL behaviour.

The paper is organized as follows: we describe possible experimental setup for
observing the NFL transport in Section II. A theoretical model accounting for
the interplay between SOI, external magnetic field and effects of Coulomb
blockade in Quantum Dot is presented in Section III. We discuss the solution
of one-dimensional quantum-mechanical scattering problem in the presence of
strong SOI in Section IV. An effective model describing a low-energy physics
of the problem and its exact solution is presented in Section V. The transport
coefficients computed with the help of the exact solutions are discussed in
Section VI. Section VII is devoted to discussion of the key results of the
paper including estimation for the parameters and definition of the conditions
necessary for experimental observations of the NFL physics. Summary and
Conclusions are given in the Section VIII. Details of derivation of the
effective model are presented in the Appendix A.

##  II Proposed experimental setup

We consider a two-terminal nano-device (see Fig. 1) designed to be used for
thermoelectric measurements Mol1 ; Mol2 . The QD - QPC contains 2-d electron
gas (2DEG) confined in $z$-direction (light orange area on the Fig. 1). The
open QPC connects it to the drain at the reference temperature $T$. We assume
that the Rashba SOI SOIbook ; Spintronics  (caused by the gradient of the
confining potential in $z$-direction) leads to appreciable effects which we
will discuss in this paper. The source is separated from the QD by a tunnel
barrier with low transparency $|t|$$\ll$$1$. The temperature of the source
(deep orange) is adjusted by the Joule heat controlled by the current $I_{J}$
flowing along the lead (black arrow). The temperature difference $\Delta T$
across the tunnel barrier is assumed to be small compared to the reference
temperature $T$ to guarantee the linear response operation regime for the
device. The QD is electrostatically controlled by two plunger gates (blue
rectangles) to adjust the size of the electron island. The device is operated
in the steady state of zero source-drain current
$I_{sd}$$=$$0$$=$$G\Delta$$V_{th}$$+$$G_{T}$$\Delta T$, controlled by applying
a thermo-voltage $\Delta V_{th}$ between the source and the drain. The QPC
(denoted by the cross in the light orange area) is tuned to the single mode
regime characterized by a controllable small reflectivity $|r|$$\ll$$1$.
Under this assumption and neglecting the resistance of the ”metallic” QD we
assume that the voltage difference $\Delta V_{th}$ arises across the tunnel
barrier between the source and QD. The transport coefficients: electric
conductance $G$ and thermoelectric coefficient $G_{T}$ (measured
independently) define the thermoelectric power (TP) $S$:

$G_{T}=\frac{\partial I_{sd}}{\partial\Delta T},\;\;\;G=\frac{\partial
I_{sd}}{\partial V},\;\;S=-\left.\frac{\Delta V_{th}}{\Delta
T}\right|_{I_{sd}=0}=\frac{G_{T}}{G}.$

We assume that the magnetic field (blue arrow) is applied parallel to the
plane of 2DEG to avoid orbital effects.

##  III Theoretical model

The theoretical description of setup (see Fig 1) is formulated in terms of the
Hamiltonian:

$\displaystyle H=H_{s}+H_{d}+H_{tun}+H_{z}.$ (1)

Here $H_{s}$ and $H_{d}$ are the Hamiltonians of the source  (”hot” contact)
and the drain  (”cold” contact), respectively. $H_{tun}$ describes tunneling
between the source and the drain and $H_{z}$ accounts for the Zeeman effect in
both contacts. We assume that the source can be described by a standard FL
approach

$H_{s}=\sum_{k,\sigma}\epsilon_{k\sigma}c^{\dagger}_{k\sigma}c_{k\sigma}$ (2)

here $c^{\dagger}$ and $c$ are creation/annihilation operators of quasi-
particles  (we adopt a system of units $\hbar=k_{B}=1$). The drain
$H_{d}=H_{c}+H_{QPC}$ includes the Coulomb blockaded QD described by charging
Hamiltonian $H_{c}$ and QPC represented by

$H_{QPC}=H_{0}+H_{SOI}+H_{BS}$ (3)

We assume that the charge  $\hat{Q}=e(\hat{n}_{s}+\hat{n}_{d})$ in the QD is
weakly quantized (mesoscopic CB regime AG ) and controlled by the gate voltage
$V_{g}$:

$\displaystyle H_{c}=E_{c}\left[\hat{n}_{s}+\hat{n}_{d}-N(V_{g})\right]^{2},$
(4)

here  $\hat{n}_{s}$ and $\hat{n}_{d}$ are the operators of the number of
electrons that entered the dot through the source and the drain respectively,
$E_{c}\sim e^{2}/L_{QD}$ is the charging energy of QD with geometric size
$L_{QD}$. Below we ignore effects associated with finite mean-level spacing in
the dot. While charge is only weakly quantized in the mesoscopic CB regime,
the spin remains a good quantum number in the absence of SOI. However, when
the SOI is present, two spin sub-bands are split horizontally in k-space and
while spin is no more conserved, the sub-band index characterizes quantized
states instead. The single mode QPC being a short quantum wire can be viewed
as 1-d electron system in the presence of Rashba SOI GJPB05 ; Sun07 ; Miranda
; Cheng
$H_{SOI}=\alpha_{R}\left[\vec{k}\times\vec{n}_{z}\right]\cdot\vec{\sigma}$:

$\displaystyle
H_{0}=-iv_{F}\sum_{\lambda\sigma}\lambda\int_{-\infty}^{\infty}dy\Psi_{\lambda,\sigma}^{\dagger}\left(y\right)\partial_{y}\Psi_{\lambda,\sigma}\left(y\right),$
(5) $\displaystyle
H_{SOI}=\alpha_{R}k_{F}\sum_{\lambda\sigma}\lambda\int_{-\infty}^{\infty}dy\left[\Psi_{\lambda,\uparrow}^{\dagger}\Psi_{\lambda,\downarrow}+\Psi_{\lambda,\downarrow}^{\dagger}\Psi_{\lambda,\uparrow}\right].$
(6)

We denote here by $\Psi_{\lambda,\sigma}$ the left ($\lambda$$=$$-$) and right
($\lambda$$=$$+$) movers with spin $\sigma=\uparrow,\downarrow$. The constant
$\alpha_{R}$ characterizes Rashba SOI strength. The $k_{F}$ and
$v_{F}$$=$$k_{F}$$/$$m^{*}$ correspond to the Fermi momentum and Fermi
velocity (here $m^{*}$ is a fermion’s mass). The 1-d electron transport
through the QPC is along the $y$-axis (see Fig.1). The Rashba SOI
$H_{SOI}$$=$$\alpha_{R}$$k_{y}$$\sigma_{x}$ is associated with the electric
field gradient along the $z$-axis and can be characterized by the effective
SOI field $g$$\mu_{B}$$\vec{B}_{SOI}$$/$$2$$=$$\alpha_{R}$$k_{F}$$\vec{e}_{x}$
perpendicular to the direction of electron transport ($g$ is the Lande factor,
$\mu_{B}$ is the Bohr magneton). Notice, that the SOI field alone does not
lead to the TRS breaking.

The backscattering (BS) Hamiltonian describes a scattering of electrons with
momentum transfer $2k_{F}$ on a non-magnetic quantum impurity located at the
origin and characterized by a short-range potential $V(y)$:

$\displaystyle H_{BS}=\sum_{\lambda,\sigma}\int
dy\Psi_{\lambda,\sigma}^{\dagger}\left(y\right)V(y)\Psi_{\bar{\lambda},\sigma}\left(y\right)e^{-2i\lambda
k_{F}y}.$ (7)

The Hamiltonian $H_{tun}$ represents the weak tunneling
$|t_{k}|$$=$$|t|$$\ll$$1$ of the electrons from the left contact to QD:

$\displaystyle
H_{tun}=\sum_{k\lambda\sigma}\left[t_{k}c^{\dagger}_{k\sigma}\Psi_{\lambda\sigma}(-\infty)+h.c.\right].$
(8)

The Zeeman Hamiltonian $H_{z}$ describes the effects of the external magnetic
field $H_{z}=-g\mu_{B}\vec{B}(\vec{s}_{s}+\vec{s}_{d})$, where $\vec{s}_{s}$
and $\vec{s}_{d}$ are the spin densities of electrons in the source and drain
respectively. We consider a situation when both sizes of the QD  ($L_{QD}$)
and QPC  ($L_{QPC}$) are small compared to the SOI length scale $L_{QD}\sim
L_{QPC}$$\ll$$l_{SOI}$$=$$1$$/$$(m^{*}$$\alpha_{R})$.  Since the effective
energy scale determining the behaviour of the transport coefficients of the
model (2-8) which will be referred below as the Kondo temperature $T_{K}$ is
FM_theory  $\sim E_{c}$ (see Appendix A), the condition $l_{SOI}$$\gg$$L_{QD}$
is equivalent to $g$$\mu_{B}$$B_{SOI}$$\ll$$T_{K}$ (see a discussion about
interplay between Kondo effect and SOI in Ref. Kikoin12, ). We also assume
that the SOI effects in the QD are already taken into account by using the
approach developed in Ref. Aleiner01, .

Figure 2: (Color online) Top panels: two sub-band spectra (left) - Zeeman
splitting in the absence of SOI, (center) - $\vec{B}\perp\vec{B}_{SOI}$, angle
between magnetic field and $y$-direction $\varphi$=$0$; (right) - arbitrary
angle $-\pi/2$$<$$\varphi$$<$$\pi/2$. Bottom panels: magnetic field dependence
of reflection amplitudes $|r_{\mu\nu}|$ for the spectra shown on top panels.
For illustration we performed all calculations with model barrier
$V(y)=V_{0}\exp(-|y|/L_{QPC})$, $k_{0F}L_{QPC}=3.6$ and hight of the barrier
$V_{0}$ is tuned to get $r_{0}^{2}=0.1$ (see details in the text). Insert
shows relative orientation of $\vec{B}$ and $\vec{B}_{SOI}$.

##  IV Scattering in the presence of SOI and magnetic field

We consider 1d scattering problem in the presence of SOI SOIbook  and Zeeman
field applied parallel to the plane of 2DEG. The Hamiltonian is given by

${\cal H}={\cal
H}_{0}+V(y)=\frac{k^{2}}{2m^{*}}+\alpha_{R}\sigma_{x}k-\gamma\vec{\sigma}\cdot\vec{B}+V\left(y\right).$
(9)

The short-range potential $V(y)$ describes a non-magnetic impurity located at
the origin Miranda ; Sun07 . The electron’s transport is along $y$-direction,
$k$$=$$k_{y}$. Angle $\varphi$ characterizes the orientation of magnetic field
$\vec{B}$ with respect to $y$\- axis (see Fig. 2 left bottom panel insert).

The kinetic energy term of (9) is given by

$\displaystyle{\cal
H}_{0}=\left(\begin{array}[]{cc}\frac{k^{2}}{2m^{*}}&\alpha_{R}k+i\gamma
Be^{i\varphi}\\\ \alpha_{R}k-i\gamma
Be^{-i\varphi}&\frac{k^{2}}{2m^{*}}\end{array}\right).$ (12)

The Hamiltonian ${\cal H}_{0}$ is promptly diagonalized in $k$-space. The
eigenvalues (spectra) describe two sub-bands ($\nu=+$ and $\nu=-$) split both
horizontally due to the SOI and vertically due to Zeeman effect (Fig. 2):

$E_{\pm}(k)=\frac{k^{2}}{2m^{*}}\pm\sqrt{\left(\alpha_{R}k\right)^{2}+\left(\gamma
B\right)^{2}-2\alpha_{R}\gamma kB\sin\varphi}.$ (13)

(we use the short-hand notation $\gamma=g\mu_{B}/2$). In the presence of both
fields there are two sub-bands while the spin polarization changes
continuously as one moves from one Fermi point to the other along each sub-
band. We assume that magnetic field is applied parallel to the plane of the
2DEG to decouple it from the orbital degrees of freedom and concentrate on the
Zeeman effect only. The angle $\varphi$ characterizes the orientation of
$\vec{B}$ with respect to the axis of 1-d transport (y). The spectra for
$\varphi$$=$$0$ \- perpendicular orientation of $\vec{B}$ and $\vec{B}_{SOI}$
describe the situation when magnetic field $\vec{B}$ is oriented along
direction of the transport. If $B$ is larger than $B_{SOI}$, the most
important effects on thermoelectric transport are due to the Zeeman splitting
of two sub-bands (Fig 2 upper left panel) NKK10 . In that limit the effects of
$B$ are associated with breaking of the channel symmetry,
$|r_{\uparrow}|$$\neq$$|r_{\downarrow}|$ (Fig 2, lower left panel) which is
crucial for the fate of NFL NKK10 . For the case $B<B_{SOI}$ we distinguish
two cases: i) $\varphi=0$ (Fig. 2, central panel) and ii)
$-\pi/2$$<$$\varphi$$<$$\pi/2$, $\varphi\neq 0$ (Fig. 2, right panel). Since
the orbital effects are negligible for small magnetic fields if $B<B_{SOI}$,
the case i) $\varphi=0$ (Fig. 2, central panel) can also be realized when
magnetic field is perpendicular to the plane of 2DEG. Besides, the theory
discussed in the paper is also applicable when both Rashba and Dresselhaus SOI
SOIbook ; Spintronics  are present. The transport coefficients for generic
situation of in-plane $B$-field are fully determined by the angle
$\Phi$$=$$\varphi_{0}$$-$$\varphi$ between $\vec{B}$ and $\vec{B}_{SOI}$,
where $\varphi_{0}$ ($\varphi$) are the angles between $\vec{B}_{SOI}$
($\vec{B}$) and axis of 1d motion respectively.

The eigenfunctions of $H_{1d}$ are momentum-dependent spinors
$\Psi_{\nu}\left(y\right)=e^{ik\cdot y}\chi_{\nu}\left(k\right)$

$\displaystyle\chi_{\pm}\left(k\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\pm
ie^{-i\vartheta\left(k\right)}\\\ 1\end{array}\right),$ (16)

where

$\displaystyle\vartheta\left(k\right)=\arctan\left(\frac{\alpha_{R}k-\gamma
B\sin\varphi}{\gamma B\cos\varphi}\right).$ (17)

The four reflection amplitudes in the first order of the backscattering
potential are determined by $2k_{F}$ momentum transfer and given by the matrix
elements of $V(y)$ in spinor basis $\Psi_{\nu}$. The diagonal matrix elements

$|r_{\mu\mu}|=\left|\frac{V\left(k_{F+}^{\mu}-k_{F-}^{\mu}\right)}{v_{0F}}\cos\left(\frac{\vartheta\left(k_{F+}^{\mu}\right)-\vartheta\left(k_{F-}^{\mu}\right)}{2}\right)\right|$
(18)

characterize the intra-band scattering (we shall use the short-hand notations
$|r_{\mu\mu}|$$\equiv$$|r_{\mu}|$ below). Here $k_{F+}^{\mu}$$>$$0$ and
$k_{F-}^{\mu}$$<$$0$ stand for the right and left Fermi points of a sub-band
$\mu$ respectively, $v_{0F}$$\sim$$(m^{*}a)^{-1}$ originates from the high
energy cutoff, $a$ is a lattice constant. The off-diagonal matrix elements
($|r_{+-}|$$\equiv$$|r_{\pm}|$ and $|r_{-+}|$$\equiv$$|r_{\mp}|$):

$|r_{\mu\nu}|=\left|\frac{V\left(k_{F+}^{\mu}-k_{F-}^{\nu}\right)}{v_{0F}}\sin\left(\frac{\vartheta\left(k_{F+}^{\mu}\right)-\vartheta\left(k_{F-}^{\nu}\right)}{2}\right)\right|$
(19)

describe the inter-band scattering.

The backscattering Hamiltonian (7) in the basis of eigenfunctions (16) casts
the following form:

$\displaystyle
H_{BS}=v_{F}\sum_{\lambda\mu\nu}|r_{\mu\nu}|\left[\Psi_{\lambda,\mu}^{\dagger}(0)\Psi_{\bar{\lambda},\nu}(0)+h.c.\right].$
(20)

Let us analyse various limits of the back-scattering corresponding to
different orientation of the in-plane magnetic field $\vec{B}$ in the regime
of strong interplay with the effects of SOI.

For the most generic case of interplay between SOI and Zeeman magnetic field
there exist four independent scattering processes resulting in four different
reflection amplitudes (Fig. 2, right lower panel). The reflection amplitudes
for the intra-band scattering $|r_{+}|$ and $|r_{-}|$ (black and red dashed
arrow on Fig. 2, right upper panel) in the first order of the backscattering
potential are proportional to the amplitude of $B$ (Ref. Miranda, )

$\displaystyle|r_{+/-}|=r_{0}\left[\frac{k_{0F}}{k_{0F}\mp\delta}\right]\left(\frac{B}{B_{SOI}}\right)\cos\varphi,$
(21)

where $k_{0F}$ is the Fermi momentum at zero splitting ($\delta$$=$$0$),
$\delta$$=$$m^{*}\alpha_{R}$, $r_{0}\propto|V(2k_{0F})m^{*}a|\ll 1$ is a
coefficient characterizing the transparency of the barrier. The intra-band
scattering is completely suppresses for $\varphi$$=$$\pi$$/$$2$ since the
angle $\vartheta(k_{F\pm})$$=$$\pm\pi/2$ and the eigenfunctions do not depend
on $B$. The intra-band reflection amplitudes $|r_{+}|$$=$$|r_{-}|$$=$$0$ while
inter-band $(|r_{\pm}|$,$|r_{\mp}|)$$\neq$$0$. Thus, for
$\varphi$$=$$\pi$$/$$2$ we have only two non-zero reflection amplitudes
$|r_{\pm}|$ and $|r_{\mp}|$ and therefore the thermoelectric transport can be
described by equations of Ref. NKK10,  if replacing
$|r_{\uparrow}|$$\to$$|r_{\pm}|$ and $|r_{\downarrow}|$$\to$$|r_{\mp}|$.

The inter-band scattering amplitudes (blue dashed arrows on central upper
panel of Fig. 2) for the case $-\pi/2$$<$$\varphi$$<$$\pi/2$ are given by

$\displaystyle|r_{\pm/\mp}|=r_{0}\left[1\mp
b\left(\frac{B\sin\varphi}{B_{SOI}}\right)-c(\varphi)\left(\frac{B}{B_{SOI}}\right)^{2}\right].$
(22)

Here coefficients $(b,c(\varphi))\sim 1$ depend on the geometry of the QPC.
One can see that for $\varphi=0$, the scattering term linear in $B$ (linear
Zeeman effect) disappears and additional symmetry $|r_{\pm}|=|r_{\mp}|$
emerges (Fig. 2 central panel). The reflection amplitudes depend on magnetic
field quadratically (quadratic Zeeman effect). Thus, the scattering
Hamiltonian in that case contains three independent scattering parameters.

##  V Effective model

We recapitulate briefly the main steps of the derivation of transport
coefficients (for details see Appendix A): i) we bosonize the 1-d Hamiltonian
(5-7) using a standard approach Thierry ; GNT . The effective bosonic
Hamiltonian gives us a boundary sine-Gordon (BSG) model GNT  with four
different backscattering amplitudes. The high-$T$ results are obtained by
perturbative expansion (in reflection amplitudes) around the strong-coupling
fixed point of the model. ii) The non-perturbative results in the low-$T$
regime are obtained by re-fermionization procedure through the mapping the BSG
model onto the effective Anderson model Matveev95 ; MA_theory ; LeHur ; NKK10
. iii) The effects of the Zeeman field at the QPC resulting in TRS breaking
caused by the asymmetry of reflection amplitudes NKK10  are accounted by a
magnetic field dependent  resonance width $\Gamma$ at the CB peaks NKK10 .
The resonance width $\Gamma$ in the presence of Zeeman field remains finite
for whole range of the gate voltages, cuts the temperature-dependent logarithm
and therefore restores FL properties. The width $\Gamma$ (Refs. Matveev95, ;
FM_theory, ; GNT, ) is attributed to a single local Majorana mode interacting
with a single mode of chiral fermions Matveev95 ; MA_theory  in the theory
containing only two (intra-band) scattering processes. The  scattering on
quantum impurity in the presence of SOI and Zeeman fields  involves four
”$2k_{F}$” processes which can be accounted for by two local Majorana modes
interacting with four modes of two species of the chiral fermions. As a
result, two different resonance widths enter the transport coefficients. The
interplay between two widths associated with inter- and intra-band processes
leads to remarkable effects in thermoelectric transport.

The effective Anderson model which describes a hybridization of two local
Majorana fermions $\eta_{1}$ and $\eta_{2}$ with two species of  chiral
fermions (see Appendix A) is a direct generalization of Matveev95 ; FM_theory
; MA_theory ; NKK10  for a case of interplay between Zeeman and SOI fields:

$\displaystyle
H_{\tau}(t)=\int_{-\infty}^{\infty}dk\left[\sum_{\alpha=1,2}(k\cdot
v_{F})c_{\alpha,k}^{\dagger}c_{\alpha,k}\right.$ (23)
$\displaystyle-\sqrt{2}\left.\left(\omega_{s\tau}(t)\eta_{1}(c_{1,k}-c_{1,k}^{\dagger})-i\omega_{a\tau}(t)\eta_{1}(c_{1,k}+c_{1,k}^{\dagger})\right.\right.$
$\displaystyle\left.\left.+\omega_{ms\tau}(t)\eta_{2}(c_{2,k}-c_{2,k}^{\dagger})-i\omega_{ma\tau}(t)\eta_{2}(c_{2,k}+c_{2,k}^{\dagger})\right)\right].$

where following Ref. MA_theory,  we define

$\displaystyle\omega_{s/a\;\tau}(t)$ $\displaystyle=$
$\displaystyle\Omega_{s/a}\cdot f_{s/a\;\tau}(t),$
$\displaystyle\omega_{ms/ma\;\tau}(t)$ $\displaystyle=$
$\displaystyle\Omega_{ms/ma}\cdot f_{s/a\;\tau}(t),$ (24)

with

$\displaystyle\Omega_{s/a}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{v_{F}E_{c}e^{C}}{2\pi^{3}}}||r_{+}|\pm|r_{-}||,$
$\displaystyle\Omega_{ms/ma}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{v_{F}E_{c}e^{C}}{2\pi^{3}}}||r_{\pm}|\pm|r_{\mp}||,$
(25)

and the time-dependent functions

$f_{s/a\;\tau}=(-1)^{n_{\tau}(t)}{\rm Re}/{\rm
Im}\left[\exp\left\\{i\left(\delta\chi_{\tau}(t)-\pi
N\right)\right\\}\right].$ (26)

Function $\delta\chi_{\tau}(t)$ describes the deviation of the phase of the
charge mode mean value from $\pi n_{\tau}(t)$ (Ref. MA_theory, ):

$\displaystyle\delta\chi_{\tau}(t)\approx\frac{\pi^{2}T}{2E_{c}}\left(\cot[\pi
T(t-\tau)]-\cot[\pi Tt]\right),$ (27)

here $N$ is a function of a gate voltage $V_{g}$: $N$ is integer in the
Coulomb blockade valleys and half-integer in the Coulomb blockade peaks,
$n_{\tau}(t)=\theta(t)\theta(\tau-t)$ ($\theta(t)$ is a step function) and
$C\approx 0.577$ is the Euler’s constant.

The original Matveev model Matveev95  corresponds to the case
$\Omega_{a}$$=$$\Omega_{ma}$$=$$\Omega_{ms}$$=$$0$ and describes a single
Majorana fermion $\eta_{1}$ coupled to the odd combination of
creation/annihilation operators of chiral fermions. Contrast to conventional
Anderson model which preserves $U(1)$ symmetry, the model Matveev95 ;
FM_theory  is characterized by $Z_{2}$ symmetry instead. As a result, the NFL
properties associated with the two-channel Kondo physics emerge. The NFL
behaviour of the two-channel Kondo model are attributed to overscreened regime
realized when the number of orbital channels ${\cal N}$ exceeds twice a spin
of a quantum impurity. As it was shown in Ref. NKK10, , Zeeman in-plane
magnetic field restores the $U(1)$ symmetry through appearance of non-zero
$\Omega_{a}$ and therefore leads to the restoration of the FL behaviour
characteristic for the single-channel fully screened Kondo model in both
thermodynamic LeHur  and transport NKK10  coefficients. The Kondo temperature
$T_{K}$ is of the order of the charging energy $E_{c}$ (see Appendix A). The
effective Hamiltonian (23) describing scattering in the presence of both
Zeeman and SOI fields has a structure of two copies of the two-channel Kondo
model where coupling constants $\omega_{i\tau}$ depend on the magnetic field.
Thus, when all reflection amplitudes are different, the model (23) is
characterized by generic FL properties. However, if accidental degeneracy
fine-tuned by the orientation of in-plane magnetic field appears, one of the
non-identical copies of the two-channel Kondo model preserves the NFL
properties.

The effects of interplay between the backscattering at the QPC and Coulomb
interaction in the QD can be accounted by the correlator $K(\tau)=\langle
T_{\tau}F(\tau)F^{\dagger}(0)\rangle$ (see details in Refs. MA_theory, ;
NKK10, ).  The operator $F(\tau)$ accounts for the weak charge quantization in
the mesoscopically Coulomb blockaded QD. Following Ref. MA_theory,  we define
the charge of QD $\hat{Q}=e(\hat{n}_{\tau}+\hat{n}_{d})$, where
$\hat{n}_{\tau}$ is an integer valued operator which commutes with the
annihilation operator of the electron in the dot at the position of the source
Ref. FM_theory, . Since by definition $[F(\tau),\hat{n}_{\tau}]=F(\tau)$, the
role of operator $F(\tau)$ is to account for the effects of interaction in QD:
$\Psi_{\lambda}(\tau)$$=$$F(\tau)\Psi_{0\lambda}(\tau)$ where $\Psi_{\lambda}$
and $\Psi_{0\lambda}$ correspond to interacting and non-interacting left/right
fermions, respectively. Thus, the dressed Green’s Function (GF) ${\cal
G}(\tau)=-\langle
T_{\tau}\Psi_{\lambda}(\tau)\Psi^{\dagger}_{\lambda}(0)\rangle$ and free
fermionic Green’s function $G_{0}(\tau)$$=$$-\langle
T_{\tau}\Psi_{0\lambda}(\tau)\Psi^{\dagger}_{0\lambda}(0)\rangle$$=$$-\pi\nu_{0}T/\sin(\pi
T\tau)$ are connected MA_theory  by simple relation ${\cal
G}(\tau)=K(\tau)G_{0}(\tau)$ (here $\nu_{0}$ is a density of states in QD).
The transport coefficients of the model are determined by the Green’s function
${\cal G}$ (see next Section).

In order to compute the Green’s function ${\cal G}$ (or, equivalently, compute
the correlator $K$) we define the operator
$U_{\tau}$$=$$\left(-1\right)^{d^{\dagger}d}$ where
$d$$=$$($$\eta_{1}$$+$$i$$\eta_{2}$$)$$/\sqrt{2}$ and
$d^{\dagger}$$=$$($$\eta_{1}$$-$$i$$\eta_{2}$$)$$/\sqrt{2}$. We keep notations
of the Matveev and Andreev work Ref. MA_theory,  for
$U_{\tau}$$=$$2i$$\eta_{2}$$\eta_{1}$. Notice, that the ”spin” and charge are
completely disentangled in the correlator
$K(\tau)$$=$$K_{c}(\tau)K_{s}(\tau)$.

While
$K_{c}$$($$\tau$$)$$=$$\pi^{2}$$T$$e^{-C}$$/$$($$2$$E_{c}$$|$$\sin$$($$\pi$$T$$\tau$$)$$|)$
(see Ref. FM_theory, ; MA_theory,  for details of calculations), the
$K_{s}(\tau)$ in zero-th order in $H_{\tau}(t)$$-$$H_{\tau=0}(t)$ is defined
by the correlator

$\displaystyle K^{(0)}_{s}(\tau)=\langle T_{t}U(\tau)U(0)\rangle_{0}.$ (28)

Here $\langle...\rangle_{0}$ denotes an averaging with (23) taken at
$\tau$$=$$0$. (Notice obvious correspondence
$\eta_{1}$$\to$$\sigma_{x}$$/$$\sqrt{2}$,
$\eta_{2}$$\to$$\sigma_{y}$$/$$\sqrt{2}$ and
$2$$i$$\eta_{2}$$\eta_{1}$$\to$$\sigma_{z}$, where $\sigma_{i}$ are spin
$s$$=$$1$$/2$ operators ($i$$=$$x$$,y$$,z$). Therefore,
$K^{0}_{s}(\tau)$$=$$\langle$$T_{t}\sigma_{z}$$(\tau)$$\sigma_{z}$$(0)$$\rangle$.)

The first non-vanishing order in $H_{\tau}(t)$$-$$H_{\tau=0}(t)$ correction to
the correlator $K_{s}(\tau)$ is given by:

$\displaystyle K^{(1)}_{s}(\tau)=-\int_{0}^{1/T}dt\langle
T_{t}H^{\prime}_{\tau}(t)U(\tau)U(0)\rangle_{0},$ (29)

where the Hamiltonian $H^{\prime}_{\tau}(t)$ has the form:

$\displaystyle
H^{\prime}_{\tau}(t)=-2i\delta\chi_{\tau}(t)\left(\Omega_{s}\sin(\pi
N)\eta_{1}\varsigma_{1}-\Omega_{a}\cos(\pi N)\eta_{1}\zeta_{1}\right.$
$\displaystyle\left.+\Omega_{ms}\sin(\pi
N)\eta_{2}\varsigma_{2}-\Omega_{ma}\cos(\pi N)\eta_{2}\zeta_{2}\right).$ (30)

Here we define four additional Majorana fermions ($\alpha$$=$$1,2$) through a
$k$-Fourier transform of the even/odd combinations of creation/annihilation
operators of two species of chiral fermions $c_{\alpha,k}$ taken at the
position of the quantum impurity $y$$=$$0$:

$\displaystyle\zeta_{\alpha}$
$\displaystyle=\displaystyle\int_{-\infty}^{\infty}dk\zeta_{\alpha
k}=\frac{1}{\sqrt{2}}\int_{-\infty}^{\infty}dk\left(c_{\alpha,k}+c_{\alpha,k}^{\dagger}\right),$
$\displaystyle\varsigma_{\alpha}$
$\displaystyle=\displaystyle\int_{-\infty}^{\infty}dk\varsigma_{\alpha
k}=\frac{1}{i\sqrt{2}}\int_{-\infty}^{\infty}dk\left(c_{\alpha,k}-c_{\alpha,k}^{\dagger}\right).$
(31)

In order to compute the correlators (28, 29) we apply the Wick’s theorem to
the product of even number of fermions and express the result in terms of the
products of the single particle GF’s. The imaginary time (Matsubara) GF’s form
a $6\times 6$ matrix with 21 independent components (6 diagonal and 15 off-
diagonal):

$\displaystyle G_{\mu\nu}^{\eta\eta}(\tau)$ $\displaystyle=$
$\displaystyle-\langle
T_{\tau}\eta_{\mu}(\tau)\eta_{\nu}(0)\rangle,\;G_{\mu\nu}^{\zeta\zeta}(\tau)=-\langle
T_{\tau}\zeta_{\mu}(\tau)\zeta_{\nu}(0)\rangle,$ $\displaystyle
G_{\mu\nu}^{\varsigma\varsigma}(\tau)$ $\displaystyle=$ $\displaystyle-\langle
T_{\tau}\varsigma_{\mu}(\tau)\varsigma_{\nu}(0)\rangle,\;G_{\mu\nu}^{\zeta\varsigma}(\tau)=-\langle
T_{\tau}\zeta_{\mu}(\tau)\varsigma_{\nu}(0)\rangle,$ $\displaystyle
G_{\mu\nu}^{\zeta\eta}(\tau)$ $\displaystyle=$ $\displaystyle-\langle
T_{\tau}\zeta_{\mu}(\tau)\eta_{\nu}(0)\rangle,\;G_{\mu\nu}^{\varsigma\eta}(\tau)=-\langle
T_{\tau}\varsigma_{\mu}(\tau)\eta_{\nu}(0)\rangle.$

The GF ’s of quadratic Anderson-type Hamiltonian (23) can be found exactly
(e.g. by solving equations of motion for the Majorana fermions). For computing
the correlators (28, 29) we need only 6 GF’, namely, two diagonal local
Majorana’s GF (here $R$ denotes the retarded GF’s):

$\displaystyle
G_{11,R}^{\eta\eta}(\epsilon)=\frac{1}{\epsilon+i\Gamma_{B}},\;G_{22,R}^{\eta\eta}(\epsilon)=\frac{1}{\epsilon+i\Gamma_{A}}$
(32)

and four off-diagonal hybridized GF’s:

$\displaystyle G_{11,R}^{\zeta\eta}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\Omega_{a}\sin(\pi N)2\pi/v_{F}}{\epsilon+i\Gamma_{B}},$
$\displaystyle G_{11,R}^{\varsigma\eta}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\Omega_{s}\cos(\pi N)2\pi/v_{F}}{\epsilon+i\Gamma_{B}},$
$\displaystyle G_{22,R}^{\zeta\eta}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\Omega_{ma}\sin(\pi N)2\pi/v_{F}}{\epsilon+i\Gamma_{A}},$
$\displaystyle G_{22,R}^{\varsigma\eta}(\epsilon)$ $\displaystyle=$
$\displaystyle\frac{\Omega_{ms}\cos(\pi N)2\pi/v_{F}}{\epsilon+i\Gamma_{A}}.$
(33)

Here we denote the resonance widths associated with symmetric and
antisymmetric combination of reflection amplitudes as:

$\displaystyle\Gamma_{s/ms}$ $\displaystyle=$
$\displaystyle\Omega_{s/ms}^{2}\cos^{2}(\pi N)4\pi/v_{F},$
$\displaystyle\Gamma_{a/ma}$ $\displaystyle=$
$\displaystyle\Omega_{a/ma}^{2}\sin^{2}(\pi N)4\pi/v_{F}.$ (34)

Two resonance Kondo widths entering the transport coefficients are given by:

$\displaystyle\Gamma_{A}=\Gamma_{ms}+\Gamma_{ma},\;\;\;\;\;\Gamma_{B}=\Gamma_{s}+\Gamma_{a}.$
(35)

Notice that ten GF’s, namely $G_{\mu\nu}^{\zeta\zeta}$,
$G_{\mu\nu}^{\varsigma\varsigma}$ and $G_{\mu\nu}^{\zeta\varsigma}$ do not
depend on the local Majorana fermions describing the quantum impurity. These
GF renormalise the correlations between the conduction electrons, but do not
enter the Eqs. (28, 29). Another five GF’s allowed by the symmetry of the
Hamiltonian (23) do not appear in the equations (28, 29) due to specific form
of fermionic correlations in the Hamiltonian $H^{\prime}_{\tau}(t)$.

##  VI Transport coefficients

The thermoelectric coefficient $G_{T}$ and electric conductance $G$

$\displaystyle G_{T}$ $\displaystyle=$
$\displaystyle-\frac{i\pi^{2}G_{L}T}{2e}\int_{-\infty}^{\infty}\frac{\sinh(\pi
Tt)}{\cosh^{3}(\pi Tt)}K\left(\frac{1}{2T}+it\right)dt,\;\;\;\;\;\;\;$ (36)
$\displaystyle G$ $\displaystyle=$ $\displaystyle\frac{\pi
G_{L}T}{2}\int_{-\infty}^{\infty}\frac{1}{\cosh^{2}(\pi
Tt)}K\left(\frac{1}{2T}+it\right)dt$ (37)

are here calculated by accounting for interaction effects in the QD through
the correlator $K(\tau)$ defined in the previous Section. The conductance of
the barrier between the source and QD $G_{L}=2\pi e^{2}\nu_{0}\nu_{L}|t|^{2}$
is expressed through Fermi’s golden rule as function of the density of states
(DoS) of the source $\nu_{L}$, the DoS of the QD $\nu_{0}$ and the weak
tunnelling amplitude $|t|$ (Ref. comSOI, ).

The correlator $K^{(0)}_{s}(1/(2T)+it)$ defined by (28) is an even function of
time. This correlator determines the behaviour of the electric conductance
$G$, but does not contribute to $G_{T}$:

$\displaystyle G^{(0)}=\frac{G_{L}\Gamma_{A}\Gamma_{B}e^{-C}}{32\pi
TE_{c}}F_{G}\left(\frac{\Gamma_{A}}{T},\frac{\Gamma_{B}}{T}\right).$ (38)

The equation for the thermoelectric coefficient $G_{T}$ is given by an odd
function $K^{(1)}_{s}(1/(2T)+it)$ defined by (29):

$G_{T}^{\left(1\right)}=-\frac{G_{L}\Gamma_{A}\Gamma_{B}\sin\left(2\pi
N\right)}{6e\pi^{2}E_{c}}\left[\frac{{\color[rgb]{0,0,0}|r_{+}r_{-}|}}{\Gamma_{B}}\ln\left(\frac{E_{c}}{T+\Gamma_{B}}\right)F\left(\frac{\Gamma_{B}}{T},\frac{\Gamma_{A}}{T}\right)+\frac{{\color[rgb]{0,0,0}|r_{\pm}r_{\mp}|}}{\Gamma_{A}}\ln\left(\frac{E_{c}}{T+\Gamma_{A}}\right)F\left(\frac{\Gamma_{A}}{T},\frac{\Gamma_{B}}{T}\right)\right].$
(39)

The ratio of $G_{T}$ and $G$ defines the thermoelectric power:

$S=-\frac{16e^{C}\sin\left(2\pi
N\right)T}{3e\pi}\left[\frac{{\color[rgb]{0,0,0}|r_{+}r_{-}|}}{\Gamma_{B}}\ln\left(\frac{E_{c}}{T+\Gamma_{B}}\right)\frac{F\left(\frac{\Gamma_{B}}{T},\frac{\Gamma_{A}}{T}\right)}{F_{G}\left(\frac{\Gamma_{A}}{T},\frac{\Gamma_{B}}{T}\right)}+\frac{{\color[rgb]{0,0,0}|r_{\pm}r_{\mp}|}}{\Gamma_{A}}\ln\left(\frac{E_{c}}{T+\Gamma_{A}}\right)\frac{F\left(\frac{\Gamma_{A}}{T},\frac{\Gamma_{B}}{T}\right)}{F_{G}\left(\frac{\Gamma_{A}}{T},\frac{\Gamma_{B}}{T}\right)}\right].$
(40)

The functions $F_{G}$ and $F$ universally depend on the ratio of the resonance
Kondo widths $\Gamma_{A}$, $\Gamma_{B}$ and the temperature:

$\displaystyle
F_{G}\left(x,y\right)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dzdz^{\prime}\frac{\left[\left(z+z^{\prime}\right)^{2}+\pi^{2}\right]}{\left[\left(z^{\prime}\right)^{2}+x^{2}\right]\left[z^{2}+y^{2}\right]}\frac{1}{\cosh\left(\frac{z}{2}\right)\cosh\left(\frac{z^{\prime}}{2}\right)\cosh\left(\frac{z+z^{\prime}}{2}\right)},$
(41) $\displaystyle
F\left(x,y\right)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dzdz^{\prime}\frac{z\left(z+z^{\prime}\right)\left[\left(z+z^{\prime}\right)^{2}+\pi^{2}\right]}{\left[z^{2}+x^{2}\right]\left[\left(z^{\prime}\right)^{2}+y^{2}\right]}\frac{1}{\cosh\left(\frac{z}{2}\right)\cosh\left(\frac{z^{\prime}}{2}\right)\cosh\left(\frac{z+z^{\prime}}{2}\right)}.$
(42)

Figure 3: (Color online) Left panel: Gate voltage and magnetic field
dependence of $\Gamma_{\alpha}/E_{c}$: red - $\Gamma_{A}$, blue -
$\Gamma_{B}$. Right panel: a ”phase diagram” - four main regimes of the
thermoelectric transport are inside the green, red, blue and magenta domains;
A - perturbative NFL, B - weak partial NFL, C- strong partial NFL, D - non-
perturbative FL (see details in Section VII). Domains boundaries are defined
by the crossover conditions $\Gamma_{A}(B,N)$$=$$T$ and
$\Gamma_{B}(B,N)$$=$$T$. For all plots $\alpha_{R}$$=$$0.15$$v_{F}$,
$\varphi$$=$$\pi/4$, $r_{0}^{2}$$=$$0.1$, $k_{0F}L_{QPC}$$=$$3.6$,
$T$$=$$0.05E_{c}$.

##  VII Results and discussion

###  VII.1 Four main regimes of thermoelectric transport

The non-perturbative equation for the resonance width $\Gamma$ related to the
inter-band scattering (34-35) demonstrates a weak dependence of $\Gamma$ on
the magnetic field away from the CB peaks (see Fig.3 left panel):

$\displaystyle\Gamma_{A}\propto\Gamma_{0}\left[\left(1-\Lambda_{A}^{2}\right)\cos^{2}(\pi
N)+\Delta_{A}^{2}\right],$ (43)

where $\Gamma_{0}$$=$$r_{0}^{2}$$E_{c}$,
$\Delta_{A}(B,\varphi)$$=$$b$$\left(B/B_{SOI}\right)\sin\varphi$ and
$\Lambda_{A}^{2}=\Delta_{A}^{2}+2c(\varphi)(B/B_{SOI})^{2}$.

In contrast, the resonance width $\Gamma$ associated with the intra-band
scattering (34-35) strongly depends on $B$ at all gate voltages:

$\displaystyle\Gamma_{B}\propto\Gamma_{0}\left[(1-\Delta^{2}_{B})\cos^{2}(\pi
N)+\Delta^{2}_{B}\right]\left(\frac{B\cos\varphi}{B_{SOI}}\right)^{2}.$ (44)

The
$\Gamma^{min}_{B}$$\equiv$$\Gamma_{B}$$($$N$$=$$\frac{1}{2}$$)$$\propto$$\Gamma_{0}$$\cdot$$(B\cos\varphi/B_{c})^{2}$
is a minimal resonance width, $\Delta_{B}$$=$$\delta/k_{0F}$ and
$B_{c}$$\sim$$D$, where $D$$\sim$$(m^{*}a^{2})^{-1}$ is the bandwidth. Thus,
$B_{c}$ corresponds to the field strength that is necessary to reach full spin
polarization of the conduction channel.

Varying the temperature, gate voltage, amplitude and direction of the magnetic
field one can achieve four different regimes of thermoelectric transport (Fig.
3 right panel):
A) $(\Gamma_{A},\Gamma_{B})$$\ll$$T$ \- fully perturbative NFL regime. While
$\Gamma_{B}$ is gapped and the gap is $\Gamma^{min}_{B}$$\sim$$B^{2}$,
$\Gamma_{A}$ could be gapless if the gate voltage is fine-tuned to the
positions of CB peaks $N$$\to$$1$$/$$2$ and $\varphi$$\to$$0$. The TP (38-40)
demonstrates fingerprints of weak NFL logarithmic behaviour:

$\displaystyle S\propto r_{0}^{2}\ln\left(\frac{E_{c}}{T}\right)\sin(2\pi N).$
(45)

B) $\Gamma_{B}$$\ll$$T$$\ll$$\Gamma_{A}$ \- perturbative in $\Gamma_{B}$$/T$
and non-perturbative in $\Gamma_{A}$$/$$T$ (see (38-40)). This regime can be
reached either by fine-tuning the gate voltage away from the CB peaks or by
tuning the direction of Zeeman field to be parallel to SOI in order to
suppress the intra-band scattering:

$S\propto\left[|r_{+}r_{-}|\ln\left(\frac{E_{c}}{T}\right)+|r_{\pm}r_{\mp}|\frac{T}{\Gamma_{A}}\ln\left(\frac{E_{c}}{\Gamma_{A}}\right)\right]\sin(2\pi
N).$ (46)

The weak NFL effects are manifested in the TP log- behaviour originated from
the intra-band scattering. The inter-band processes result in appreciable FL
corrections to the TP (38-40). The NFL effects are weak since the amplitude of
intra-band scattering is small at $B<B_{SOI}$.
C) $\Gamma_{A}$$\ll$$T$$\ll$$\Gamma_{B}$ \- perturbative in $\Gamma_{A}$$/$$T$
and non-perturbative in $\Gamma_{B}$$/$$T$ (see 38-40). This regime is
achieved in the vicinity of CB peaks and characterized by strong NFL effects
due to a weak magnetic field dependence of $|r_{\pm}|$ and $|r_{\mp}|$
protected by SOI. Thus, by fine-tuning the orientation of magnetic field
perpendicular to SOI $\varphi$$=$$0$ one can controllably protect the NFL
behaviour of TP in the regime $B$$<$$B_{SOI}$. The magnetic field controlled
gap associated with the intra-band scattering weakly depends on the gate
voltage and results in small FL corrections to TP (compared to NFL effects):

$S\propto\left[|r_{+}r_{-}|\frac{T}{\Gamma_{B}}\ln\left(\frac{E_{c}}{\Gamma_{B}}\right)+|r_{\pm}r_{\mp}|\ln\left(\frac{E_{c}}{T}\right)\right]\sin(2\pi
N).$ (47)

D) $T$$\ll$$(\Gamma_{A},\Gamma_{B})$ \- FL non-perturbative regime. The NFL
logs associated with the intra- and inter- band scattering processes are cut
by the corresponding resonance widths (38-40):

$S\propto
T\left[\frac{|r_{+}r_{-}|}{\Gamma_{B}}\ln\left(\frac{E_{C}}{\Gamma_{B}}\right)+\frac{|r_{\pm}r_{\mp}|}{\Gamma_{A}}\ln\left(\frac{E_{C}}{\Gamma_{A}}\right)\right]\sin\left(2\pi
N\right)$ (48)

The TP is linear function of the temperature. However, the coefficient in
front of $T$ strongly depends on both gate voltage and magnetic field.

Figure 4: (Color online) Top left: $eS_{max}$ as function of $B/B_{SOI}$ for
different angles $\varphi$ at $T$$=$$0.001E_{c}$ and
$\alpha_{R}$$=$$0.15v_{F}$, from top to bottom
$\varphi$$=$$\pi/12$,$\pi/6$,$\pi/3$,$5\pi/12$. Insert: $eS(N)$ for
$\varphi$$=$$5\pi/12$ for $B$$=$$0.5B_{SOI}$ (black), $B$$=$$B_{SOI}$ (red),
$B$$=$$1.5B_{SOI}$ (green). Top right: $eS_{max}$ as function of the angle
$\varphi$ for different amplitudes of $B/B_{SOI}$ at
$\alpha_{R}$$=$$0.15$$v_{F}$ and $T=0.001E_{c}$ from top to bottom:
$B/B_{SOI}=$$0.6,0.8,1.0,1.2.$. Insert: $eS(N)$ for $\varphi$$=$$5\pi/12$,
$B$$=$$0.5B_{SOI}$, $T$$=$$0.001$ (black), $0.01$ (red), $0.1$ (blue). Bottom
left: $eS_{max}$ as function of $B/B_{SOI}$ for different $T/E_{c}$ at
$\alpha_{R}$$=$$0.15v_{F}$ and $\varphi$$=$$\pi/6$; $T/E_{c}$$=$$10^{-1}$\-
blue (regime $A$), $10^{-2}$\- green (crossover $A$$\to$$C$), $10^{-3}$\- red
($B$$\to$$D$$\to$$C$), $10^{-4}$ \- black ($B$$\to$$D$) . Bottom right:
$e$$S_{max}$ as a function of $T/$$\Gamma_{B}^{min}$
($\Gamma_{B}^{min}$$=$$\Gamma_{B}$$($$1/2)$) with $a_{R}$$=$$0.15$$v_{F}$,
$B$$=$$0.5$$B_{SOI}$, for $\varphi$$=$$5\pi/12$ (black), $\varphi$$=$$\pi/3$
(red), $\varphi$$=$$\pi/6$ (green), $\varphi$$=$$\pi$$/12$ (blue),
($r^{2}_{0}$$=$$0.1$ and $k_{0F}$$L_{QPC}$$=$$3.6$).

###  VII.2 Possible experimental realization and ”smoking gun” predictions

Choice of a material. We suggest to use narrow-gap semiconductors, e.g. $InSb$
or $InAs$ for the observation of the NFL fingerprints in the quantum
transport. Both materials are characterized by large bulk $g$-factors, e.g.
$|g|\sim 10$ in InAs (see Ref. Petta11, ) and $|g|\sim 50$ in InSb (see Ref.
Mourik12, ). The domain of parameters favourable for the observation of the
NFL regime is defined as: $\delta_{0}<T<\gamma B_{SOI}<E_{c}<\epsilon_{F}$,
where $\delta_{0}\sim 1/(\nu_{0}V_{QD})$ is a single-particle mean-level
spacing in the QD of the size $L_{QD}$ and ”volume” $V_{QD}$:
$\delta_{0}^{2D}\sim\hbar^{2}/(m^{*}L_{QD}^{2})$ and
$\delta_{0}^{3D}\sim(k_{F}L_{QD})^{-1}\cdot\hbar^{2}/(m^{*}L_{QD}^{2})$,
$m^{*}$ is an effective mass of the carrier in a semiconductor, other
parameters are defined in the previous Sections. The condition
$L_{QD}<l_{SOI}$ allows to disregard the effects of the SOI in the QD (Ref.
Aleiner01, ), while the condition $L_{QPC}<l_{mfp}$ defines a ballistic regime
of quantum transport through the QPC (here $l_{mfp}$ is an elastic mean-free
path).

According to Ref. Mourik12, , the parameters for $InSb$ QD - QPC are as
follows: $\hbar\alpha_{R}\sim 0.1eV\cdot\AA\div 0.2eV\cdot\AA$,
$l_{SOI}=\hbar/(m^{*}\alpha_{R})\sim 200nm\div 400nm$, $|m^{*}|\sim
0.015m_{e}$ ($m_{e}$ is electron’s mass), $E_{SOI}=m^{*}\alpha_{R}^{2}/2\sim
50\mu eV$, the typical charging energy $E_{c}\sim 1meV$ and typical Fermi
velocities are $\hbar v_{F}\sim 0.5eV\cdot\AA\div 1eV\cdot\AA$, while the mean
level spacing $\delta_{0}<10\mu eV$ for $L_{QD}\sim l_{SOI}$. The mean-free
path of the QPC of a width $d_{QPC}\sim 10nm$ is $l_{mfp}\sim 300nm\div 1\mu
m$. Typical parameters for the $InAs$ QD - QPC are not much different Petta11
: $|m^{*}|\sim 0.03m_{e}$, $\hbar\alpha_{R}\sim 0.05eV\cdot\AA\div
0.3eV\cdot\AA$, $l_{SOI}\sim 200nm\div 1\mu m$ and $l_{mfp}\sim 300nm\div 1\mu
m$. Therefore, if we assume that $L_{QD}\approx L_{QPC}\sim 300nm\div 500nm$,
our predictions could be verified at magnetic fields $B<500mT$ and
temperatures $T\sim 100mK\div 300mK$ for typical densities of 2DEG
$n_{2DEG}\sim 10^{11}cm^{-2}\div 10^{12}cm^{-2}$. This estimation for
parameters is taken from available literature (to our best knowledge), but may
vary due to anisotropic character of the $g$-factor which in turn depends on
external magnetic field Petta11  and may also be strongly reduced in confined
geometries of the nano-structures.

Testing a Mott-Cutler law. The first important test of the interplay between
effects of the SOI and Zeeman field is to verify the Mott-Cutler (MC) law
Cutler69  at external in-plane magnetic field. The MC-law is a standard
benchmark for the FL properties MA_theory . The MC-law says that the TP is
proportional to a log-derivative of the electric conductance with respect to a
position of the chemical potential (gate voltage $V_{g}$):

$S\propto\frac{T}{E_{c}}\frac{\partial\ln G}{\partial N(V_{g})}$ (49)

In the limit $T\ll(\Gamma_{A},\Gamma_{B})\ll E_{c}$ corresponding to the FL
regime we get:

$\frac{\partial\ln G}{\partial N(V_{g})}\propto
E_{c}\left[\frac{|r_{+}r_{-}|}{\Gamma_{B}}+\frac{|r_{\pm}r_{\mp}|}{\Gamma_{A}}\right]\sin\left(2\pi
N\right)$ (50)

while the TP is given by (48). Thus, a strong deviation from the MC-law in the
FL regime at finite in-plane magnetic fields is a pre-cursor for the NFL
behaviour discussed in the paper. Notice that break down of the MC-law
indicates that there exists no equivalent classic electric circuit consisting
of the resistances connected in parallel or in series and therefore the
effects of both intra- and inter-band scattering play an important role in the
quantum transport. The violation of MC-law in the thermo-electric transport
through a single-electron transistor has been reported in Ref. Mol1, . We are
not aware of existence of theoretical explanation of this effect in the
framework of the FL theory.

Thermopower in the presence of the external B-field. The next step is to
measure the thermopower of a prototype nano-device (Fig.1). The magnitude and
orientation of the in-plane magnetic field can be controlled in a standard way
by four magnetic coils (not shown in the picture). The TP maximum
$eS_{max}(B)$ demonstrates a non-monotonic magnetic field dependence (strong
NFL) which is most pronounced when $\vec{B}$ is orthogonal to $\vec{B}_{SOI}$
(black curve on Fig. 4 top left and right panels.) This has to be contrasted
to almost monotonic TP maximum behaviour (blue curves) characteristic for weak
NFL - FL regimes. The non-monotonic behaviour of TP maximum as a function of
magnetic field is a central prediction of our paper. The non-monotonicity
indicates that the NFL regime of TP is protected by SOI contrast to FL-like
behaviour demonstrating rapid decrease of TP when magnetic field increases
NKK10 . Another indication of the NFL behaviour is attributed to the gate
voltage dependence (Fig. 4 inserts). According to MA_theory ; NKK10  it is
characterized by strongly non-sinusoidal form (Fig. 4 inserts). The TP maximum
at zero field, NFL regime, scales according to MA_theory  as
$eS_{max}$$\sim$$r_{0}$$\sqrt{T/E_{c}}$$\ln(E_{c}/T)$. The TP maximum in the
FL regime scales as $eS$$\sim$$T/T_{\rm eff}$ with $T_{\rm
eff}$$/E_{c}$$=$$B$$/$$B_{c}$$\ln^{-1}(B_{c}/(B|r_{0}|)$ (Ref. NKK10, ). The
$B$-field dependence of TP maximum measured at different temperatures (Fig. 4
left bottom panel) allows to distinguish between four main regimes A-D
discussed in the previous subsection. This measurement can be used for
identification of crossovers between different domains. The TP maximum depends
linearly on $T$ in the FL regime for
$T$$<$$\Gamma^{min}_{B}$$=$$\Gamma_{B}$$($$N$$=$$1$$/2$$)$ (Ref. NKK10, ).
This regime holds for $\varphi$$\to$$\pi$$/$$2$. In contrast to FL regime, the
temperature dependence of TP maximum pronouncedly departs from the linear
behaviour (see Fig. 4 right bottom panel) when $\varphi$ is detuned from
$\pi/2$. We suggest to test experimentally this effect as a benchmark for the
NFL physics.

##  VIII Summary and conclusions

We have demonstrated that the theory describing scattering of electrons
characterized by two orbital degrees of freedom on a spin $s$$=$$1/2$ quantum
impurity (two channel Kondo model) is strongly modified in the presence of
both appreciable spin-orbit interaction and Zeeman splitting. It is shown
that, on the one hand, the lack of spin conservation due to SOI leads to the
appearance of new (extra) scattering channels which potentially enhance the
thermoelectric transport. On the other hand, the Zeeman splitting produces
non-zero resonance widths of Majorana modes describing the quantum impurity
and thus suppresses the NFL effects. The interplay between these two
tendencies can be controlled by fine-tuning the angle between Zeeman and SOI
fields. Our calculations predict a strong dependence of the thermoelectric
power on the angle between $\vec{B}$ and $\vec{B}_{SOI}$ and thus open a
possibility to control the scattering mechanism by changing between four,
three or two independent scattering processes. While the cases of four- and
two- weak back-scattering do favour the FL behaviour, the additional
degeneracy in scattering amplitudes appearing for three- scattering models due
to SOI effects protects the NFL behaviour for the range of magnetic fields
$B$$<$$B_{SOI}$. We conclude therefore, that SOI can indeed protect the NFL
against the destructive effects associated with breaking of channel symmetry.

##  IX Acknowledgements

We are grateful to J.C. Egues, V. Fal’ko, L. Glazman, K. Kikoin, A. Komnik, S.
Ludwig, C. Marcus, K. Matveev, L.W. Molenkamp and O. Starykh for illuminating
discussions. T.K.T.N acknowledges support through short-term visiting program
of ICTP and Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under grant number 103.01-2014.24.

##  Appendix A Effective Hamiltonian

### Backscattering: from fermions to bosons

The backscattering Hamiltonian mixes the left- and right- moving fermions:

$\displaystyle
H_{BS}=v_{F}\sum_{\lambda\mu\nu}|r_{\mu\nu}|\left[\Psi_{\lambda,\mu}^{\dagger}(0)\Psi_{\bar{\lambda},\nu}(0)+h.c.\right].$
(51)

The Hamiltonians Eqs. (3-4) of the main text and the Hamiltonian (51) can be
bosonized Thierry ; GNT  in terms of dual fields $\phi_{\nu}(y)$ and
$\theta_{\nu}(y)$ satisfying commutation relations
$\left[\phi_{\nu}(y),\theta_{\mu}(y^{\prime})\right]=-i\pi\delta_{\nu\mu}\text{{sgn}}\left(y-y^{\prime}\right)/2$
(Refs. Thierry, ; GNT, ):

$\displaystyle\Psi_{\lambda,\nu}(y)=\frac{u_{\lambda,\nu}}{\sqrt{2\pi
a}}e^{i\lambda
k_{F}^{\nu}y}\exp\\{i[-\lambda\phi_{\nu}(y)+\theta_{\nu}(y)]\\},$ (52)

where $u_{\lambda,\nu}$ are Klein factors Thierry ; GNT  introduced to ensure
proper anticommutation relations between the right- and left- moving fermions.

Using a standard procedure Thierry ; GNT  we introduce the symmetric (charge)
and antisymmetric (”spin”) dual variables
$\phi_{c,s}(y)$$=$$[\phi_{+}(y)$$\pm$$\phi_{-}(y)]$$/$$\sqrt{2}$ and
$\theta_{c,s}(y)$$=$$[\theta_{+}(y)$$\pm$$\theta_{-}(y)]$$/$$\sqrt{2}$
satisfying the commutation relations
$\left[\phi_{c/s}(y),\theta_{c/s}(y^{\prime})\right]$$=$$-i$$\pi$sgn$\left(y-y^{\prime}\right)/2$
(notice that we still refer to the antisymmetric in the band index bosonic
field as ”spin”). We rewrite the backscattering Hamiltonian (51) in terms of
the charge and spin bosonic fields as follows:

$\displaystyle H_{BS}$ $\displaystyle=$
$\displaystyle-\frac{2D}{\pi}\left(r_{s}\cos[\sqrt{2}\phi_{c}(0)]\cos[\sqrt{2}\phi_{s}(0)]\right.$
(53)
$\displaystyle\left.+r_{a}\sin[\sqrt{2}\phi_{c}(0)]\sin[\sqrt{2}\phi_{s}(0)]\right.$
$\displaystyle\left.+r_{ms}\cos[\sqrt{2}\phi_{c}(0)]\cos[\sqrt{2}\theta_{s}(0)]\right.$
$\displaystyle\left.+r_{ma}\sin[\sqrt{2}\phi_{c}(0)]\sin[\sqrt{2}\theta_{s}(0)]\right),$

where $r_{s}=||r_{+}|+|r_{-}||/2$, $r_{a}=||r_{+}|-|r_{-}||/2$,
$r_{ms}=||r_{\pm}|+|r_{\mp}||/2$, $r_{ma}=||r_{\pm}|-|r_{\mp}||/2$.

### Backscattering: Majorana fermions

As a first step we replace the charge mode by its mean value averaged over
fast charge degrees of freedom using the functional integral technique
developed in Ref. MA_theory,  and obtain the Hamiltonian:

$\displaystyle
H_{\tau}(t)=\frac{v_{F}}{2\pi}\int_{-\infty}^{\infty}\\{[\partial_{y}\theta_{s}(y)]^{2}+[\partial_{y}\phi_{s}(y)]^{2}\\}dy$
$\displaystyle-\sqrt{\frac{4D}{v_{F}}}\left(\omega_{s\tau}(t)\cos[\sqrt{2}\phi_{s}(0)]+\omega_{a\tau}(t)\sin[\sqrt{2}\phi_{s}(0)]\right.$
$\displaystyle\left.+\omega_{ms\tau}(t)\cos[\sqrt{2}\theta_{s}(0)]+\omega_{ma\tau}(t)\sin[\sqrt{2}\theta_{s}(0)]\right),$
(54)

where we use the notations (24,25) of the Section V.

As a next step we introduce the even and odd combinations of the ”spin” (aka
sub-band) bosonic fields
$\phi_{e/o}$$($$y$$)$$=$$[$$\phi_{s}(y)$$\pm$$\phi_{s}(-y)$$]$$/$$\sqrt{2}$,
$\theta_{e/o}$$(y$$)$$=$$[$$\theta_{s}(y)$$\pm$$\theta_{s}(-y)$$]$$/$$\sqrt{2}$.
As a result, we obtain new chiral fields
$\Phi_{1/2}$$($$y$$)$$=$$\theta_{o/e}(y)$$-$$\phi_{e/o}$$($$y$$)$ satisfying
the commutation relations:
$[$$\Phi_{\alpha}(y)$,$\Phi_{\alpha^{\prime}}(y^{\prime})$$]$$=$$i$$\pi$$\delta_{\alpha\alpha^{\prime}}$sgn$\left(y-y^{\prime}\right)$
where $\alpha$,$\alpha^{\prime}$$=$$1,2$. We define new fermionic fields
$\Psi_{\alpha}\left(y\right)$$=$$(\eta_{\alpha}/\sqrt{2\pi
a})\exp{(-i\Phi_{\alpha}\left(y\right))}$ with a help of two local Majorana
fermions $\eta_{1}$$=$$(d$$+$$d^{\dagger})$$/$$\sqrt{2}$ and
$\eta_{2}$$=$$(d$$-$$d^{\dagger})$$/$$(i\sqrt{2})$ representing the quantum
impurity MA_theory .

Finally, we integrate out the fluctuations of the spin degree of freedom with
the frequencies exceeding $E_{c}$ (Ref. MA_theory, ). This procedure is
equivalent to the poor man’s scaling approach originally used for the Kondo
problem Hewson  and leads to replacement of the bandwidth $D$ by the new
bandwidth $T_{K}\sim E_{c}$. As a result, we derive the effective Anderson
model which describes a hybridization of two local Majorana fermions
$\eta_{1}$ and $\eta_{2}$ with two species of conduction electrons. The
effective Hamiltonian (23) has a structure of two copies of the two-channel
Kondo model where coupling constants $\omega_{i\tau}$ depend on both Zeeman
and SOI fields:

$\displaystyle
H_{\tau}(t)=\int_{-\infty}^{\infty}dk\left[\sum_{\alpha=1,2}(k\cdot
v_{F})c_{\alpha,k}^{\dagger}c_{\alpha,k}\right.$ (55)
$\displaystyle-\sqrt{2}\left.\left(\omega_{s\tau}(t)\eta_{1}(c_{1,k}-c_{1,k}^{\dagger})-i\omega_{a\tau}(t)\eta_{1}(c_{1,k}+c_{1,k}^{\dagger})\right.\right.$
$\displaystyle\left.\left.+\omega_{ms\tau}(t)\eta_{2}(c_{2,k}-c_{2,k}^{\dagger})-i\omega_{ma\tau}(t)\eta_{2}(c_{2,k}+c_{2,k}^{\dagger})\right)\right].$

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