#  Nonequilibrium Green’s function approach to multi-band Cooper-pair
transport:
linear magnetoresistance effect due to nonunitary superconductivity

G. Tkachov  Institute of Physics, Augsburg University, 86135 Augsburg, Germany

###### Abstract

Many-body transport has emerged as an efficient tool for understanding
interaction effects in quantum materials with a multi-band electronic
structure. This paper proposes a formula for the two-particle transmission
coefficient for Cooper-pair transport between multi-band normal and
superconducting materials. The approach employs a tight-binding nonequilibrium
Green’s function technique, allowing a direct calculation of the two-particle
current, without invoking the paradigm of Andreev reflection. As an
application of the theory, we demonstrate a low-field linear magnetoresistance
effect for superconductors with an induced nonunitary order parameter. These
results uncover an unexplored route for detecting unconventional nonunitary
superconductivity in quantum materials of current theoretical and experimental
interest.

##  I Introduction

Quantum materials are a burgeoning frontier of condensed matter physics and
materials science Keimer17 ; Giustino20 . In such materials, quantum effects
emerge at macroscopic scales as a result of reduced dimensionality,
interactions, band structure specifics or a combination of these factors. The
recent flurry of results on topological insulators Liu16 , Weyl semimetals
Armitage18 , transition metal dichalcogenides (TMDs) Choi17 , topological and
unconventional superconductors Tanaka12 ; Sato17 ; Ghosh20 ; Culcer20 ;
Ramires22  is a case in point. A common thread that links these groups of
materials is their multi-band electronic structure associated with internal
degrees of freedom (DOF). These include the electron spin and, depending on
the context, atomic orbitals, crystal sublattices or/and the particle-hole
degree of freedom.

The purpose of this paper is twofold. First, we revisit some many-body aspects
of electron transport theory for multi-band materials. On the one-particle
level or within one-particle approaches, the progress in understanding multi-
band transport properties is significant (see e.g. review Culcer20  and
references therein). Many-body transport, on the other hand, has been explored
to a much lesser extent even at the two-particle level. A typical and
important example is the transport of Cooper pairs across an interface between
normal and superconducting materials. The Cooper-pair transport offers
insights into the superconducting order parameter and is traditionally treated
in the language of Andreev reflection Blonder82 . For multi-band materials,
however, the number of channels available for Andreev reflection can be very
large. First of all, because of the spin degree of freedom, two electrons in a
Cooper pair can form one singlet and three triplet states, which yields a
total of four possible scattering channels. If, additionally, the pairing
involves multiple orbitals or sublattices, the number of the scattering
channels increases greatly, e.g. for two atomic orbitals (acting as
pseudospin) it jumps to sixteen since the pseudospin singlet and triplets come
now into play as well. A large number of transport channels makes the problem
intriguing, but keeping track of them in the Andreev reflection language
becomes increasingly difficult.

Instead, in this paper we directly compute the two-particle transmission
coefficient for Cooper pairs, using the tight-binding approach in combination
with the nonequilibrium Green’s function (NEGF) technique. The calculation
builds upon the original proposal of Ref. Cuevas96  and more recent work GT17
, taking into account the lattice structure, multiple internal DOF and pairing
states as well as the position dependence of real-space NEGFs. We obtain the
pair transmission in a compact analytic form applicable to diverse multi-band
materials interfaces.

Another goal of this paper is to examine the pair transport for unconventional
triplet superconductors with broken time-reversal symmetry (TRS). Such
superconductors can often be identified as nonunitary superconductors and are
currently the subjects of intense experimental and theoretical effort in a
broader context of quantum materials (see e.g. Hillier09 ; Hillier12 ; Lado19
; Shang22 ; Ghosh22 ; Wolf22  and reviews Ghosh20 ; Culcer20 ; Ramires22 ).
From the perspective of a transport study, it is convenient to define unitary
and nonunitary superconducting states, using the matrix condensate Green’s
function $\hat{F}$.111This definition (see e.g. Culcer20 ) is complementary to
that using the gap function, cf. Sigrist91  and recent reviews Ghosh20 ;
Ramires22 . The state is then called unitary if $\hat{F}\hat{F}^{\dagger}$ is
proportional to a unit matrix, otherwise it is called nonunitary. For example,
in the singlet-triplet basis
$\hat{F}=\hat{f}_{0}i\sigma_{y}+\hat{\bm{f}}\cdot{\bm{\sigma}}i\sigma_{y},$
where ${\bm{\sigma}}$ is the vector of Pauli matrices $\sigma_{x},\sigma_{y}$
and $\sigma_{z}$, while $\hat{f}_{0}$ and
$\hat{\bm{f}}=[\hat{f}_{x},\hat{f}_{y},\hat{f}_{z}]$ are the amplitudes of the
singlet and triplet pairing, being matrices in all DOF other than
spin.222Hereafter, direct products of the $f$\- and $\sigma$-matrices are
assumed. The “dot” and “cross” mean the usual scalar and vector products. We
have then

$\hat{F}\hat{F}^{\dagger}=(\hat{f}_{0}\hat{f}^{\dagger}_{0}+\hat{\bm{f}}\cdot\hat{\bm{f}}^{\dagger})\sigma_{0}+(\hat{f}_{0}\hat{\bm{f}}^{\dagger}+\hat{\bm{f}}\hat{f}^{\dagger}_{0}+i\hat{\bm{f}}\times\hat{\bm{f}}^{\dagger})\cdot{\bm{\sigma}},$
(1)

where the first and second terms correspond to unitary and nonunitary pairing,
respectively, in spin space ($\sigma_{0}$ is the unit spin matrix). In
particular, the axial vector $i\hat{\bm{f}}\times\hat{\bm{f}}^{\dagger}$
indicates the TRS breaking due to emergent magnetization of the triplet pairs.

Beyond the single-orbital picture, one encounters a bigger family of the
candidate nonunitary order parameters (see e.g. Ghosh20 ; Ramires22 ; Wolf22
), so the ability to access and characterize them becomes instrumental.
Experimentally, the states with broken TRS can be detected by using the muon
spin relaxation technique Ghosh20 . Theoretical efforts to identify nonunitary
states in observables have been focused on the electric conductance of normal-
superconductor interfaces Honerkamp98 ; Linder07 ,GT17 ,GT19  and on finite-
energy gap structures in the spectrum of superconductors Lado19 ,Ramires22 .

Figure 1:  Interface magnetoresistance (2) versus Zeeman splitting in the
superconductor. The linear magnetoresistance for $|h|\ll\Delta$ indicates a
TRS-breaking nonunitary triplet state. The chemical potential increases from
$\mu=\Delta$ to $\mu=10\Delta$ as shown by the arrow (see also Eqs. 42 and
43).

Generally, signatures of nonunitary states remain elusive if there is no means
of controlling the TRS breaking in the superconducting state. In that regard,
it would be helpful to identify a magnetotransport effect associated with and
controllable through the axial vector
$i\hat{\bm{f}}\times\hat{\bm{f}}^{\dagger}$. We propose a prototype of such an
effect in a bilayer system comprising a spin-orbit-coupled superconductor and
a quantum anomalous Hall insulator (QAHI). In this case, the nonunitarity
emerges as the combined effect of spin-orbit coupling (SOC) and Zeeman field,
${\bm{h}}$, caused by the magnetization of the QAHI layer. Furthermore, the
QAHI acts as a spin filter, enabling the observation of the triplet pair
magnetoresistance which we find to be

$\frac{R(h)-R(0)}{R(h)}=-\alpha\frac{|h|}{\Delta}\left(\beta+\frac{|h|}{\Delta}\right).$
(2)

Above, $h$ is the projection of ${\bm{h}}$ on the interface normal, $\Delta$
is the energy gap of the superconductor, and $\alpha$ and $\beta$ are material
parameters.

A distinct feature of Eq. (2) is a linear low-field dependence on $|h|$ (see
also Fig. 1). It indicates the broken TRS associated with the axial vector
$i\hat{\bm{f}}\times\hat{\bm{f}}^{\dagger}\propto{\bm{h}}$ and, therefore,
provides direct evidence for nonuntary triplet pairing. Further details of the
model and calculation as well as a comprehensive discussion of the results are
given in the sections below.

##  II NEGF technique for multi-band interface transport

We begin by briefly outlining the NEGF method for interface transport. The
goal is to introduce matrix notations that enable efficient computation of the
interface current for multi-band materials.

###  II.1 Interface Hamiltonian and current operator

Figure 2:  Interface between two layered TMDs as an example of a multi-band
materials system for NEGF transport calculation (see Sec. II.1). The tight-
binding model accounts for internal DOF consisting of the spin states and
atomic orbitals (for $d$ \- electrons in the case of TMDs).

We consider two generic crystalline materials (here $a$ and $b$) forming a
planar interface (see also Fig. 2). The structure is described by the
Hamiltonian $H=H_{a}+H_{b}+H_{ab},$ where $H_{a}$ and $H_{b}$ are the tight-
binding Hamiltonians of the constituents, while $H_{ab}$ accounts for tight-
binding hopping across the interface:

$H_{ab}=\sum\limits_{\bm{r}\bm{r}^{\prime}\\\ \sigma\sigma^{\prime}\\\
jj^{\prime}}\left[b^{\dagger}_{\bm{r}\sigma
j}\left(t_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{jj^{\prime}}a_{\bm{r}^{\prime}\sigma^{\prime}j^{\prime}}+h.c.\right].$
(3)

Operator $a_{\bm{r}\sigma j}$ ($a^{\dagger}_{\bm{r}\sigma j}$) annihilates
(creates) an electron in material $a$ on lattice site $\bm{r}$ with spin
projection $\sigma=\uparrow,\downarrow$ in atomic orbital $j=1,2,...$;
$b_{\bm{r}\sigma j}$ and $b^{\dagger}_{\bm{r}\sigma j}$ are the analogous
operators for material $b$, and
$\left(t_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{jj^{\prime}}$
is the hopping energy between the states with $\bm{r}\sigma j$ and
$\bm{r}^{\prime}\sigma^{\prime}j^{\prime}$. Correspondingly, the operator of
the electric current between $a$ and $b$ is given by

$I=\frac{ie}{\hbar}\sum\limits_{\bm{r}\bm{r}^{\prime}\sigma\sigma^{\prime}jj^{\prime}}\left[b^{\dagger}_{\bm{r}\sigma
j}\left(t_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{jj^{\prime}}a_{\bm{r}^{\prime}\sigma^{\prime}j^{\prime}}-h.c.\right].$
(4)

We proceed by recasting the interface Hamiltonian into a compact quadratic
form,

$H_{ab}=\frac{1}{2}\sum\limits_{\bm{r}\bm{r}^{\prime}}C^{\dagger}_{\bm{r}}\hat{V}_{\bm{r}\bm{r}^{\prime}}C_{\bm{r}^{\prime}},$
(5)

using matrix notations for electron operators $C_{\bm{r}}$,
$C^{\dagger}_{\bm{r}}$, and hopping energy $\hat{V}_{\bm{r}\bm{r}^{\prime}}$
which absorb all discrete DOF, i.e. the material constituents ($a,b$),
particle and hole states (p,h), spin species ($\uparrow,\downarrow$) and
atomic orbitals ($1,2,...$). The hierarchy of the DOF is chosen as follows.
First, $C_{\bm{r}}$ is two-component in $ab$ space:

$C_{\bm{r}}=\left[\begin{matrix}A_{\bm{r}}\\\ B_{\bm{r}}\end{matrix}\right],$
(6)

where operators $A_{\bm{r}}$ and $B_{\bm{r}}$ act in the subsystems. Each of
these operators is, in turn, two-component in ph space:

$A_{\bm{r}}=\left[\begin{matrix}a_{\bm{r}}\\\
a^{\dagger}_{\bm{r}}\end{matrix}\right],\quad
B_{\bm{r}}=\left[\begin{matrix}b_{\bm{r}}\\\
b^{\dagger}_{\bm{r}}\end{matrix}\right].$ (7)

This (Nambu) representation applies equally to normal and superconducting
materials and brings along the particle-hole symmetry [which explains the
prefactor of $1/2$ in Eq. (5)]. Further, both operators $a_{\bm{r}}$ and
$b_{\bm{r}}$ are naturally two-component spinors:

$a_{\bm{r}}=\left[\begin{matrix}a_{\uparrow\bm{r}}\\\
a_{\downarrow\bm{r}}\end{matrix}\right],\quad
b_{\bm{r}}=\left[\begin{matrix}b_{\uparrow\bm{r}}\\\
b_{\downarrow\bm{r}}\end{matrix}\right],$ (8)

and, lastly, $a_{\bm{r}\sigma}$ and $b_{\bm{r}\sigma}$ have the orbital
components:

$\displaystyle a_{\bm{r}\sigma}=\left[\begin{matrix}a_{\bm{r}\sigma 1}\\\
a_{\bm{r}\sigma 2}\\\ ...\end{matrix}\right],\quad
b_{\bm{r}\sigma}=\left[\begin{matrix}b_{\bm{r}\sigma 1}\\\ b_{\bm{r}\sigma
2}\\\ ...\end{matrix}\right].$ (9)

The hopping matrix $\hat{V}_{\bm{r}\bm{r}^{\prime}}$ has the same hierarchy of
the discrete DOF and unfolds as follows

$\displaystyle\hat{V}_{\bm{r}\bm{r}^{\prime}}=\left[\begin{matrix}0&\hat{T}^{\dagger}_{\bm{r}\bm{r}^{\prime}}\\\
\hat{T}_{\bm{r}\bm{r}^{\prime}}&0\end{matrix}\right],$ (10)
$\displaystyle\hat{T}_{\bm{r}\bm{r}^{\prime}}=\left[\begin{matrix}\hat{t}_{\bm{r}\bm{r}^{\prime}}&0\\\
0&-\hat{t}^{*}_{\bm{r}\bm{r}^{\prime}}\end{matrix}\right],$ (11)
$\displaystyle\hat{t}_{\bm{r}\bm{r}^{\prime}}=\left[\begin{matrix}(\hat{t}_{\bm{r}\bm{r}^{\prime}})^{\uparrow\uparrow}&(\hat{t}_{\bm{r}\bm{r}^{\prime}})^{\uparrow\downarrow}\\\
(\hat{t}_{\bm{r}\bm{r}^{\prime}})^{\downarrow\uparrow}&(\hat{t}_{\bm{r}\bm{r}^{\prime}})^{\downarrow\downarrow}\end{matrix}\right],$
(12)
$\displaystyle\quad(\hat{t}_{\bm{r}\bm{r}^{\prime}})^{\sigma\sigma^{\prime}}=\left[\begin{matrix}\left(\hat{t}_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{11}&\left(\hat{t}_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{12}&...\\\
\left(\hat{t}_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{21}&\left(\hat{t}_{\bm{r}\bm{r}^{\prime}}\right)^{\sigma\sigma^{\prime}}_{22}&...\\\
...&...&...\end{matrix}\right].$ (13)

Here, matrices (10) – (13) act in $ab$, ph, spin and orbital spaces,
respectively. Obviously, the Hamiltonian of the constituents, $H_{a}+H_{b}$,
can be written as in terms of the same operators $C_{\bm{r}}$ and
$C^{\dagger}_{\bm{r}}$, so the Hamiltonian of the entire system is

$H=\frac{1}{2}\sum\limits_{\bm{r}\bm{r}^{\prime}}C^{\dagger}_{\bm{r}}\left(\hat{H}^{0}_{\bm{r}\bm{r}^{\prime}}+\hat{V}_{\bm{r}\bm{r}^{\prime}}\right)C_{\bm{r}^{\prime}}.$
(14)

As the hopping between the constituents is included in
$\hat{V}_{\bm{r}\bm{r}^{\prime}}$, matrix
$\hat{H}^{0}_{\bm{r}\bm{r}^{\prime}}$ is diagonal in $ab$ space,

$\hat{H}^{0}_{\bm{r}\bm{r}^{\prime}}=\left[\begin{matrix}\hat{H}^{a}_{\bm{r}\bm{r}^{\prime}}&0\\\
0&\hat{H}^{b}_{\bm{r}\bm{r}^{\prime}}\end{matrix}\right],$ (15)

where $\hat{H}^{a,b}_{\bm{r}\bm{r}^{\prime}}$ are matrices in the internal DOF
of materials $a$ and $b$. Finally, in the same matrix notations the current
operator in Eq. (4) takes the form

$I=-\frac{ie}{2\hbar}\sum\limits_{\bm{r}\bm{r}^{\prime}}C^{\dagger}_{\bm{r}}\nu_{3}\tau_{3}i_{0}\hat{V}_{\bm{r}\bm{r}^{\prime}}C_{\bm{r}^{\prime}},$
(16)

where $\nu_{3}$ and $\tau_{3}$ are Pauli matrices in $ab$ and ph spaces;
$i_{0}$ is the unit matrix in the remaning discrete DOF, and the product of
these three matrices is direct.

###  II.2 Nonequilibrium Green’s functions

The observable current, ${\cal I}$, is defined as the expectation value of
operator $I$ (16) over the nonequilibrium state of the system and can be
expressed as an integral over the energy ($E$) axis:

${\cal I}=-\frac{e}{4\pi\hbar}\int\sum\limits_{\bm{r}\bm{r}^{\prime}}{\rm
Tr}[\nu_{3}\tau_{3}i_{0}\hat{V}_{\bm{r}^{\prime}\bm{r}}\hat{G}^{>}_{\bm{r}\bm{r}^{\prime}}(E)]dE.$
(17)

Here, $\hat{G}^{>}_{\bm{r}\bm{r}^{\prime}}(E)$ is the greater function, being
a matrix element of the time-ordered NEGF (see e.g. Cuevas96 ):

$\displaystyle\breve{G}_{\bm{r}\bm{r}^{\prime}}(E)=\left[\begin{matrix}\hat{G}_{\bm{r}\bm{r}^{\prime}}(E)&\hat{G}^{<}_{\bm{r}\bm{r}^{\prime}}(E)\\\
\hat{G}^{>}_{\bm{r}\bm{r}^{\prime}}(E)&\hat{\overline{G}}_{\bm{r}\bm{r}^{\prime}}(E)\end{matrix}\right].$
(18)

Its elements can be expressed in terms of the advanced (A), retarded (R), and
Keldysh (K) Green’s functions:

$\displaystyle\hat{G}=\frac{1}{2}\left[\hat{G}^{K}+\hat{G}^{A}+\hat{G}^{R}\right],\hat{\overline{G}}=\frac{1}{2}\left[\hat{G}^{K}-\hat{G}^{A}-\hat{G}^{R}\right],$
$\displaystyle\hat{G}^{<}=\frac{1}{2}\left[\hat{G}^{K}+\hat{G}^{A}-\hat{G}^{R}\right],\hat{G}^{>}=\frac{1}{2}\left[\hat{G}^{K}-\hat{G}^{A}+\hat{G}^{R}\right].$

Their arguments have been omitted for brevity. The summations over ${\bm{r}}$
and ${\bm{r}}^{\prime}$ can be absorbed into the trace operation, ${\rm Tr}$,
so the expression for the current is in fact independent of the initially
chosen representation:

${\cal I}=-\frac{e}{4\pi\hbar}\int{\rm
Tr}[\nu_{3}\tau_{3}i_{0}\hat{V}\hat{G}^{>}(E)]dE.$ (19)

The next stage is to determine the greater function $\hat{G}^{>}(E)$ and trace
out the $ab$ and ph DOF. These calculations (see Appendix A) yield the
interface current for the magnetotransport problem considered next.

##  III Pair magnetotransport across a QAHI/superconductor interface

Summarizing the results of Appendix A, we write the interface current as the
sum of one-particle (${\cal I}_{1}$), two-particle (${\cal I}_{2}$) and
higher-order many-particle terms: ${\cal I}={\cal I}_{1}+{\cal I}_{2}+...$. Of
particular interest is the low-energy transport inside the excitation gap, a
regime in which ${\cal I}_{1}$ is negligible compared to the many-particle
contributions. We also assume a low-transparency interface333This case is
relevant for order-parameter spectroscopy (e.g. Kashiwaya00 ). and focus on
the main contribution – the two-particle current:

$\displaystyle{\cal I}_{2}=\frac{e}{2\pi\hbar}\int{\cal
T}_{2}(E)[f_{{}_{N}}(E)+f_{{}_{N}}(-E)-1]dE.$ (20)

Here,

$\displaystyle{\cal T}_{2}(E)=(2\pi)^{2}{\rm
Tr}\Bigl{\\{}\hat{t}^{\dagger}\hat{A}(E)\hat{t}\hat{F}(E)\bigl{[}\hat{t}^{\dagger}\hat{A}(-E)\hat{t}\bigr{]}^{*}\hat{F}^{\dagger}(E)\Bigr{\\}}\,\,\,$
(21)

is the two-particle transmission coefficient. It is given in terms of the
condensate Green’s function of the superconductor, $\hat{F}(E)$, the spectral
function of the normal system, $\hat{A}(E)$, and the interface hopping matrix,
$\hat{t}$. As expected, a Cooper pair is transmitted via TRS-partner states
with energies $\pm E$ and occupation numbers $f_{{}_{N}}(\pm E)$ in the normal
system.444We assume a small bias voltage producing
$f_{{}_{N}}(E)\not=1-f_{{}_{N}}(-E)$.

In the following, we evaluate ${\cal T}_{2}(E)$ for a planar interface between
two-dimensional (2D) systems: a QAHI and a superconductor (S) with Rashba and
Zeeman interactions (see also Fig. 3).

###  III.1 QAHI: The case of a layered magnetic TMD

The advantage of a QAHI is that its spectral function $\hat{A}(E)$ is robust
against disorder because it is determined by a chiral edge state Liu16 . As a
suitable candidate material we choose a layered TMD because it is effectively
2D and its band-structure near the Fermi level can be modelled by a four-band
Hamiltonian. We start with the Hamiltonian for a nonmagnetic TMD (see e.g.
Shi19 ) and add the Zeeman interaction to it as follows

$\displaystyle\hat{H}=(\epsilon_{0}-\epsilon_{x}\partial^{2}_{x}+\epsilon_{y}k^{2})s_{0}\sigma_{0}-iv_{x}\partial_{x}s_{x}\sigma_{z}-v_{y}ks_{y}\sigma_{0}+$
$\displaystyle(\delta_{0}-\delta_{x}\partial^{2}_{x}+\delta_{y}k^{2})s_{z}\sigma_{0}-Js_{0}\sigma_{z}-ms_{z}\sigma_{z}.$
(22)

This Hamiltonian acts in the space of two orbital and two spin states, where
Pauli matrices $s_{i}$ and $\sigma_{i}$ (with $i=x,y,$ or $z$) represent the
orbital and spin DOF, respectively; $s_{0}$ and $\sigma_{0}$ are the
corresponding unit matrices, $\epsilon_{0}$, $\delta_{0}$, $\epsilon_{x,y}$,
$\delta_{x,y}$ and $v_{x,y}$ are the band structure parameters of the model,
while energies $J$ and $m$ characterize the Zeeman splitting, such that
$2(J+m)$ and $2(J-m)$ are the splittings of the conduction and valence bands,
respectively. Also, Eq. (22) implies that the edge is in the $y$ \- direction,
and $k$ is the conserved wave number.

Figure 3:  Top: Schematic of a planar interface between a QAHI (a layered
magnetic TMD) and a superconducting layer (S). Bottom: Band structure of
spin-$\uparrow$ electronic states in a magnetic TMD from Eqs. (22) and (23).
The two bulk valleys are connected by a chiral spin-polarized edge mode
crossing the Fermi level at energy $E=0$. The spin-$\downarrow$ states (not
shown) exhibit a band gap around $E=0$ and do not contribute to transport at
the Fermi level. The plot shows the band dispersion along the $y$ direction
for representative parameters $\epsilon_{0}=0.2$ eV, $\epsilon_{y}=1$
eV$\mathring{\rm A}^{2}$, $v_{y}=0.5$ eV$\mathring{\rm A}$, $\delta_{0}=-0.2$
eV, $\delta_{y}=12$ eV$\mathring{\rm A}^{2}$, and $J=m=0.25$ eV.

Solving the edge problem (see e.g. Liu16 ; GT22 ), we readily find an edge
mode as

$E_{k}=\epsilon_{0}-|J|+\epsilon_{y}k^{2}+{\rm sgn}(m)v_{y}k.$ (23)

For $J,m>0$ its spin is in the $z$ \- direction ($\uparrow$), so is the
magnetization. In the band structure, the edge mode crosses the gap between
the conduction and valence bands of the TMD, connecting its two valleys, as
depicted in Fig. 3. At the same time, all spin-$\downarrow$ states remain
gapped near the Fermi level ($E=0$), so the in-gap spectral function
$\hat{A}(E)$ is determined solely by the edge mode. For $J,m<0$ the
magnetization is antiparallel to the $z$ \- axis. The edge mode has then spin
$\downarrow$ and opposite chirality ${\rm sgn}(m)=-1$. In both cases,
$|m|>|\delta_{0}|$.

The knowledge of the spectrum and wave function (see Liu16 ; GT22 ) makes the
calculation of $\hat{A}(E)$ straightforward. The result can be written as a
direct product of the matrices in coordinate, spin and orbital spaces:

$\displaystyle\hat{A}(E)=\hat{\rho}(E)\frac{\sigma_{0}+{\bm{\sigma}}\cdot{\bm{m}}}{2}\frac{s_{0}+s_{y}}{2},\quad{\bm{m}}={\rm
sgn}(m){\bm{z}},$ (24)

where the coordinate matrix, $\hat{\rho}(E)$, is defined by its matrix
elements

$\displaystyle\rho_{\bm{r}\bm{r}^{\prime}}(E)=\sum_{k}c_{k}(\bm{r})c^{*}_{k}(\bm{r}^{\prime})\delta(E-E_{k}),$
(25) $\displaystyle
c_{k}(\bm{r})=\sqrt{N_{k}}[e^{-\varkappa_{1}(k)(x-x_{0})}-e^{-\varkappa_{2}(k)(x-x_{0})}]e^{iky},\qquad$
(26)

with
$\varkappa_{1,2}(k)=\left|\frac{v_{x}}{2\delta_{x}}\right|\pm\sqrt{\left(\frac{v_{x}}{2\delta_{x}}\right)^{2}+\frac{\delta_{0}-|m|+\delta_{y}k^{2}}{\delta_{x}}},$
and $N_{k}$ being the normalizing factor. The matrix structure of Eq. (24)
reflects the key properties of the edge mode: (i) the coordinate matrix
$\hat{\rho}(E)$ accounts for its exponential decay from the edge (located at
$x=x_{0}$), (ii) the spin matrix
$\frac{1}{2}(\sigma_{0}+{\bm{\sigma}}\cdot{\bm{m}})$ is the projector on the
magnetization direction specified by unit vector ${\bm{m}}$, and (iii) the
orbital matrix $\frac{1}{2}(s_{0}+s_{y})$ indicates the interband character of
the edge mode as a mixture of the atomic orbitals of the conduction and
valence bands.

###  III.2 Superconductor: Nonunitarity from Rashba and Zeeman effects

In the chosen geometry, the QAHI contacts mainly the interior of the S layer,
which minimizes the role of boundary effects in S (see Fig. 3). We can
therefore use a Bogoliubov-de Gennes (BdG) Hamiltonian in a 2D ${\bm{k}}$
space:

$H_{BdG}=\left[\begin{matrix}{\cal H}_{\bm{k}}&\Delta i\sigma_{y}\\\ -\Delta
i\sigma_{y}&-{\cal
H}^{*}_{-{\bm{k}}}\end{matrix}\right],\qquad\bm{k}=[k_{x},k_{y},0],$ (27)

where $\Delta$ is an $s$-wave pair potential (assumed real valued), while

${\cal
H}_{\bm{k}}=(\varepsilon_{0}+\varepsilon_{1}{\bm{k}}^{2})\sigma_{0}s_{0}+(\lambda_{0}+\lambda_{1}{\bm{k}}^{2})\sigma_{0}s_{z}+(\bm{\gamma}_{\bm{k}}-\bm{h})\cdot\bm{\sigma}s_{0}$
(28)

is the four-band normal-state Hamiltonian of the layer, in which orbital
matrices $s_{z}$ and $s_{0}$ act in the conduction-valence band space, and
$\bm{\gamma}_{\bm{k}}$ and $\bm{h}$ are the Rashba and Zeeman fields,
respectively, given by

$\bm{\gamma}_{\bm{k}}=\alpha_{\rm so}[k_{y},-k_{x},0],\qquad\bm{h}=[0,0,h].$
(29)

Above, $\alpha_{\rm so}$ is the SOC constant, while parameters
$\varepsilon_{0},\varepsilon_{1},\lambda_{0}$ and $\lambda_{1}$ (all positive
with $\lambda_{1}>\varepsilon_{1}$) yield the positions and curvatures of the
conduction and valence bands. Typically, SOC and Zeeman terms are both smaller
than $\Delta$, and in the first approximation the pair potential in Eq. (27)
remains unaffected by the spin interactions. Also, since the QAHI layer is
magnetized out of plane, the choice of the perpendicular $\bm{h}$ is
justified, albeit in-plane magnetic fields offer an interesting alternative
that has been explored recently in the context of topological
superconductivity Loder13 ; Loder15 .

Here, we need only the condensate Green’s function, $\hat{F}(E)$. It can be
obtained by solving the BdG equation for the Green’s function in ph space, a
standard calculation (see e.g. GT17 ; GT19 ) which yields

$\hat{F}(E)=[\hat{f}_{0}(E)i\sigma_{y}+\hat{{\bm{f}}}(E)\cdot{\bm{\sigma}}i\sigma_{y}]\frac{s_{0}+s_{z}}{2}.$
(30)

We leave the technical details aside, focusing rather on the physical picture
behind this result.

It is assumed that the Fermi level lies in the conduction band and the pairing
affects only the conduction-band states. The latter assumption is justified
since the pairing energy $\Delta$ is typically much smaller than the band gap
$2\lambda_{0}$. This explains the presence of projector
$\frac{1}{2}(s_{0}+s_{z})$ on the conduction band in the orbital space.
Further, the brackets in Eq. (30) enclose the singlet and triplet pair
functions, which are the direct products of the spin matrices and matrices
$\hat{f}_{0}(E)$ and $\hat{{\bm{f}}}(E)$ in ${\bm{k}}$ space, e.g.

$\displaystyle{\bm{f}}_{{\bm{k}}{\bm{k}}^{\prime}}(E)={\bm{f}}(E,{\bm{k}})\delta_{{\bm{k}}{\bm{k}}^{\prime}},$
(31)
$\displaystyle{\bm{f}}(E,{\bm{k}})=\frac{2\Delta}{\Pi(E,{\bm{k}})}\,(\xi_{\bm{k}}{\bm{\gamma}}_{\bm{k}}-E\bm{h}-i{\bm{\gamma}}_{\bm{k}}\times\bm{h}),$
(32)

where vector ${\bm{f}}(E,{\bm{k}})$ represents the triplet order parameter.
Note that any weak SOC and Zeeman fields can induce a finite ${\bm{f}}$ \-
vector. For this reason and in order to handle the upcoming calculations, we
neglect ${\bm{\gamma}}_{\bm{k}}$ and $\bm{h}$ in the denominator,
$\Pi(E,{\bm{k}})$, which takes then the form

$\displaystyle\Pi(E,{\bm{k}})$ $\displaystyle\approx$
$\displaystyle(E^{2}-\xi_{\bm{k}}^{2}-\Delta^{2})^{2}.$ (33)

Above, $\xi_{\bm{k}}$ denotes the conduction-band dispersion given by
$\xi_{\bm{k}}=(\varepsilon_{1}+\lambda_{1}){\bm{k}}^{2}-\mu$, where
$\mu=-\varepsilon_{0}-\lambda_{0}$ is the chemical potential measured from the
bottom of the band.

As for the singlet amplitude $\hat{f}_{0}(E)$, it plays no role in the
interface transport as long as the QAHI filters out pairs with zero spin. For
that, we choose the Fermi level at energy $E=0$ which lies in the gaps of both
S and QAHI. This situation is most suited for probing nonunitary states
possessing a nonvanishing axial vector
$i{\bm{f}}(0,{\bm{k}})\times{\bm{f}}^{*}(0,{\bm{k}})$. In the case at hand,

$i{\bm{f}}(0,{\bm{k}})\times{\bm{f}}^{*}(0,{\bm{k}})=\frac{8\Delta^{2}{\bm{\gamma}}_{\bm{k}}^{2}\xi_{\bm{k}}}{\Pi(0,{\bm{k}})^{2}}\bm{h}.$
(34)

We will argue that this axial vector acts as pair magnetization, causing a
distinct magnetoresistance effect.

###  III.3 Triplet magnetoresistance effect

The magnetoresistance effect is defined as the dependence of the interface
resistance on an applied magnetic field. In our case, it is applied
perpendicular to the interface, serving the twofold purpose of stabilizing the
magnetization in the QAHI and tuning Zeeman splitting $h$ in the S (29). The
magnetoresistance effect is caused by the dependence of transmission ${\cal
T}_{2}$ (21) on $h$ which comes primarily from the triplet ${\bm{f}}$ \-
vector. To compute ${\cal T}_{2}(h)$, we need, apart from $\hat{A}$ (24) and
$\hat{F}$ (30), the hopping matrix $\hat{t}$. A widely used model is the
hopping that depends neither on the spin nor on the orbital DOF. In this case,
the hopping matrix can be written as the direct product of the coordinate-
space matrix, $\hat{\zeta}$, and the unit matrices in spin and orbital spaces:

$\hat{t}=\hat{\zeta}\sigma_{0}s_{0}.$ (35)

Since all matrices in Eq. (21) are now separable, we can trace out the spin
and orbital DOF one after the other, using the usual algebra of Pauli
matrices. As the calculation repeats that in GT17 , we just quote the result:

$\displaystyle{\cal T}_{2}(E)$ $\displaystyle=$ $\displaystyle\frac{1}{4}{\rm
Tr}\Bigl{\\{}\hat{\Gamma}(E)\bigl{[}\hat{\bm{f}}(E)\times\bm{m}\bigr{]}\cdot\hat{\Gamma}^{*}(-E)\bigl{[}\hat{\bm{f}}^{\dagger}(E)\times\bm{m}\bigr{]}$
(36) $\displaystyle+$ $\displaystyle
i\bm{m}\cdot\bigl{[}\hat{\Gamma}(E)\hat{\bm{f}}(E)\times\hat{\Gamma}^{*}(-E)\hat{\bm{f}}^{\dagger}(E)\bigr{]}\Bigr{\\}},$

where matrix
$\hat{\Gamma}(E)=2\pi\hat{\zeta}^{\dagger}\hat{\rho}(E)\hat{\zeta}$ yields (up
to Planck’s constant) single-particle hopping rates from or into the normal
system, and the remaining trace operation pertains to a configuration space
(i.e. position or ${\bm{k}}$ space).

Equation (36) is one of the main results and deserves a separate discussion.
It is valid for any triplet amplitude $\hat{\bm{f}}$ and unit magnetization
$\bm{m}$. The vanishing of transmission coefficient ${\cal T}_{2}$ for
$\hat{\bm{f}}=0$ indicates transport of pure triplet pairs. In the S, the spin
of the pair is perpendicular to $\hat{\bm{f}}$, while in the normal system it
is parallel to magnetization direction $\bm{m}$. Therefore, the triplet
transport is characterized by a noncollinear orientation of vectors
$\hat{\bm{f}}$ and $\bm{m}$, which we indeed see in Eq. (36). Also, the
noncollinearity of $\hat{\bm{f}}$ and $\bm{m}$ generally implies that the
triplet transport is accompanied by torques on the magnetization.

Furthermore, Eq. (36) accounts for nonunitary superconductivity with broken
TRS. Wee see that the vector product of $\hat{\bm{f}}$ and
$\hat{\bm{f}}^{\dagger}$ can indeed be ascribed to the pair magnetization, as
it couples directly to the magnetization of the normal system. The magnetic
coupling is a nonequilibrium transport effect which involves the pair hopping
rates given by matrices $\hat{\Gamma}(E)$ and $\hat{\Gamma}^{*}(-E)$.

Besides, Eq. (36) provides a generalization of the two-particle transmission
coefficient obtained in GT17 . The necessity of this generalization is
dictated by several reasons. For instance, Ref. GT17  employs a single-orbital
model (plus the spin), while the majority of quantum materials are multi-
orbital systems. In Eq. (36), the orbital DOF result in the prefactor of
$1/4$. Such a substantial suppression of the transmission can be traced back
to the fact that the edge mode is an equal-weight mixture of the conduction-
and valence-band orbitals [see Eq. (24)]. Therefore, only half of the orbital
state per electron can be transmitted from the QAHI into the S conduction
band. Alternatively, this can be explained by the mismatch of the orbital
pseudospin, ${\bm{s}}=\frac{1}{2}[s_{x},s_{y},s_{z}]$, which points in the $y$
and $z$ directions in the QAHI and S, respectively. Another limitation of GT17
is the use of the translationally invariant Green’s functions in ${\bm{k}}$
space. In contrast, Eq. (36) does not invoke any translational invariance,
allowing both position and ${\bm{k}}$ representations on an equal footing.

To proceed, we take advantage of the 2D character of the QAHI and S layers,
allowing the evaluation of the trace in the basis of the in-plane ${\bm{k}}$
states as

$\displaystyle{\cal
T}_{2}(E)=\frac{1}{4}\sum_{{\bm{k}}_{1}{\bm{k}}_{2}{\bm{k}}_{3}{\bm{k}}_{4}}\Gamma_{{\bm{k}}_{1}{\bm{k}}_{2}}(E)\Gamma^{*}_{{\bm{k}}_{3}{\bm{k}}_{4}}(-E)\Bigl{\\{}\bigl{[}\bm{f}_{{\bm{k}}_{2}{\bm{k}}_{3}}(E)\times\bm{m}\bigr{]}\cdot\bigl{[}\bm{f}^{*}_{{\bm{k}}_{1}{\bm{k}}_{4}}(E)\times\bm{m}\bigr{]}+i\bm{m}\cdot\bigl{[}\bm{f}_{{\bm{k}}_{2}{\bm{k}}_{3}}(E)\times\bm{f}^{*}_{{\bm{k}}_{1}{\bm{k}}_{4}}(E)\bigr{]}\Bigr{\\}}.$
(37)

Here, ${\bm{f}}_{{\bm{k}}{\bm{k}}^{\prime}}(E)$ is a diagonal matrix given by
Eq. (31), and
$\Gamma_{{\bm{k}}{\bm{k}}^{\prime}}(E)=2\pi\sum_{{\bm{k}}_{1}{\bm{k}}_{2}}\zeta^{*}_{{\bm{k}}_{1}{\bm{k}}}\rho_{{\bm{k}}_{1}{\bm{k}}_{2}}(E)\zeta_{{\bm{k}}_{2}{\bm{k}}^{\prime}}$
is the rate matrix, where $\zeta_{{\bm{k}}{\bm{k}}^{\prime}}$ is the
interlayer hopping matrix, and $\rho_{{\bm{k}}_{1}{\bm{k}}_{2}}(E)$ is the
Fourier transform of Eq. (25) in the 2D ${\bm{k}}$ space. Generally, the
hopping matrix is not diagonal, neither is the spectral matrix
$\rho_{{\bm{k}}_{1}{\bm{k}}_{2}}(E)$ because the QAHI has no translational
invariance due to the boundary. We can however adopt the widely used “broad-
band” approximation, $\zeta_{{\bm{k}}{\bm{k}}^{\prime}}\approx\zeta=const$,
making the calculation easier.555The underlying physics is independent of this
approximation and is associated with the terms $\propto
i{\bm{f}}\times{\bm{f}}^{*}$ present for any hopping. In this approximation,
the elements of the rate matrix are ${\bm{k}}$ \- independent:
$\Gamma_{{\bm{k}}{\bm{k}}^{\prime}}(E)\approx\Gamma(E)=2\pi|\zeta|^{2}\sum_{{\bm{k}}_{1}{\bm{k}}_{2}}\rho_{{\bm{k}}_{1}{\bm{k}}_{2}}(E)=2\pi|\zeta|^{2}{\cal
N}\rho_{{\bm{r}}={\bm{r}}^{\prime}=0}(E)$, where we used the Fourier
transformation back to position space (${\cal N}$ is the number of unit cells
in the normal layer). After some algebra involving Eqs. 32 and 34, we find

$\displaystyle{\cal T}_{2}(E,h)$ $\displaystyle=$
$\displaystyle\Delta^{2}\Gamma(E)\Gamma(-E)\sum_{\bm{k}}\frac{{\bm{\gamma}}^{2}_{\bm{k}}(\xi_{\bm{k}}+{\bm{m}}\cdot\bm{h})^{2}}{\Pi(E,{\bm{k}})^{2}}$
(38) $\displaystyle=$
$\displaystyle\Delta^{2}\Gamma(E)\Gamma(-E)\sum_{\bm{k}}\frac{{\bm{\gamma}}^{2}_{\bm{k}}(\xi_{\bm{k}}+{\rm
sgn}(m)h)^{2}}{\Pi(E,{\bm{k}})^{2}},\quad\,\,\,$ (39)

with ${\bm{m}}={\rm sgn}(m){\bm{z}}$ from Eq. (24). Finally, we define the
zero-bias and -temperature electric conductance,
$G(h)=\frac{e^{2}}{\pi\hbar}{\cal T}_{2}(0,h)$, resistance, $R(h)=1/G(h)$, and
the relative change in the resistance

$\displaystyle\frac{R(h)-R(0)}{R(h)}=\frac{G(0)-G(h)}{G(0)}=\frac{{\cal
T}_{2}(0,0)-{\cal T}_{2}(0,h)}{{\cal T}_{2}(0,0)}.$ (40)

This ratio quantifies the triplet magnetoresistance effect discussed in detail
below.

##  IV Discussion and conclusions

A striking feature of the triplet magnetoresistance effect is its dependence
on the relative orientation of the magnetizations in the QAHI and S, which is
captured by the signs of $m$ and $h$ in Eq. (39). For parallel magnetizations,
Zeeman field $h$ has the same sign as $m$, i.e.

$h=|h|{\rm sgn}(m),$ (41)

and we recover magnetoresistance formula (2) announced in Introduction, where
parameters $\alpha$ and $\beta$ depend on the Fermi energy and excitation gap
in the S:

$\displaystyle\alpha\left(\frac{\mu}{\Delta}\right)$ $\displaystyle=$
$\displaystyle\int^{\infty}_{-\frac{\mu}{\Delta}}\frac{\left(\frac{\mu}{\Delta}+x\right)dx}{(1+x^{2})^{4}}/\int^{\infty}_{-\frac{\mu}{\Delta}}\frac{\left(\frac{\mu}{\Delta}+x\right)x^{2}dx}{(1+x^{2})^{4}}\quad$
(42)

and

$\displaystyle\beta\left(\frac{\mu}{\Delta}\right)$ $\displaystyle=$
$\displaystyle
2\int^{\infty}_{-\frac{\mu}{\Delta}}\frac{\left(\frac{\mu}{\Delta}+x\right)xdx}{(1+x^{2})^{4}}/\int^{\infty}_{-\frac{\mu}{\Delta}}\frac{\left(\frac{\mu}{\Delta}+x\right)dx}{(1+x^{2})^{4}}.\quad$
(43)

Asymptotically, $\alpha\approx 5$ and
$\beta\approx\frac{2}{5}\frac{\Delta}{\mu}$ for $\mu\gg\Delta$. The
magnetoresistance behaviour plotted in Fig. 1 is specific to the spin-triplet
transport. For small Zeeman splitting, $|h|\ll\Delta$, the relative
magnetoresistance varies linearly with $|h|$, which is a direct indicator of
the nonunitary order parameter in the S. With increasing $\frac{\mu}{\Delta}$,
the quadratic $h$ \- dependence becomes visibly pronounced, although for small
enough $h$ the magnetoresistance is always linear.

Figure 4:  Interface magnetoresistance (45) versus Zeeman splitting in the
superconductor. The arrow indicates an increasing chemical potential from
$\mu=\Delta$ to $\mu=10\Delta$ (see also Eqs. 42 and 43).

The parallel orientation of the magnetizations is favoured because the
magnetic field emerging from the QAHI naturally tends to magnetize the S in
the same direction. This, however, does not exclude the case of the
antiparallel magnetizations, e.g. if one of the materials has a negative
$g$-factor. In this case, the sign of Zeeman field $h$ is opposite to ${\rm
sgn}(m)$:

$h=-|h|{\rm sgn}(m),$ (44)

and the resistance ratio in Eq. (40) takes the form

$\frac{R(h)-R(0)}{R(h)}=\alpha\frac{|h|}{\Delta}\left(\beta-\frac{|h|}{\Delta}\right),$
(45)

with same $\alpha$ and $\beta$. Unlike the parallel case, the low-field
magnetoresistance is now positive, turning negative only at large enough $h$,
as plotted in Fig. 4. Still, the linear magnetoresistance for $|h|\ll\Delta$
unmistakably signals the nonunitary superconductivity.

It is worth emphasizing that the above results pertain to an intraband
nonunitary pairing in the spin space. Albeit not involved in the pairing, the
band DOF affect the transport due to the mismatch of the “band pseudospins” in
the QAHI and S, thereby reducing the pair transmission by the factor of 4.
That is, the pair transport studied here is a multi-band problem in which both
spin and band DOF need to be taken into account. Another specific of the model
is the absence of the electron interaction in the triplet channel. In this
case, the nonunitary superconductivity is an emergent phenomenon due to the
combined effect of Rashba and Zeeman interactions, allowing one to build a
prototype theory of the linear triplet magnetoresistance effect. The proposed
formula for the pair transmission, Eq. (36), is however valid beyond that
particular model. Also for intrinsic nonunitary states, the Onsager symmetry
requires that an external magnetization couples to the odd powers of a
magnetic field in the superconductor via an axial vector
$i\hat{\bm{f}}\times\hat{\bm{f}}^{\dagger}$. The presented theory can
therefore be implemented for transport identification of TRS - breaking
nonunitary states in intrinsic multi-band superconductors as well.

###### Acknowledgements.

The author thanks Thilo Kopp and Arno Kampf for their valuable comments. This
work was partially supported by the German Research Foundation (DFG) through
TRR 80.

##  Appendix A One- and two-particle currents from NEGFs

Since the interface hopping in Hamiltonian (14) is treated as a potential, we
obtain a Lippmann-Schwinger-like equation for NEGF $\breve{G}(E)$ and can
therefore write the solution in the form of an iterative series:

$\displaystyle\breve{G}(E)$ $\displaystyle=$
$\displaystyle\breve{g}(E)+\breve{g}(E)\breve{V}\breve{g}(E)+\breve{g}(E)\breve{V}\breve{g}(E)\breve{V}\breve{g}(E)$
(46) $\displaystyle+$
$\displaystyle\breve{g}(E)\breve{V}\breve{g}(E)\breve{V}\breve{g}(E)\breve{V}\breve{g}(E)+...$

Here, $\breve{g}(E)$ is the Green’s function in the absence of the interface
hopping, and $\breve{V}$ is the hopping matrix in the NEGF representation:

$\breve{V}=\left[\begin{matrix}+\hat{V}&0\\\ 0&-\hat{V}\end{matrix}\right],$
(47)

where signs $\pm$ reflect the opposite directions of the time integration on
the upper and lower branches of the Keldysh contour. The matrix structure of
$\breve{g}(E)$ is analogous to that in Eq. (18), so, by doing the matrix
multiplication in Eq. (46), one can obtain the required (greater) function
$\hat{G}^{>}(E)$ to any order in hopping:

$\hat{G}^{>}(E)=...+\hat{G}^{>}_{1}(E)+...+\hat{G}^{>}_{2}(E)+...,$ (48)

where labels $1,2,...$ refer to one-particle, two-particle and higher-order
many-particle hoppings. In the following, we focus on the one- and two-
particle currents.

###  A.1 One-particle current

It arises from the second term in the series in Eq. (46). The expression for
$\hat{G}^{>}_{1}(E)$ is

$\displaystyle\hat{G}^{>}_{1}(E)=\frac{1}{2}$
$\displaystyle\left(\left[\hat{g}^{K}(E)-\hat{g}^{A}(E)\right]\hat{V}\hat{g}^{A}(E)+\right.$
(49)
$\displaystyle\left.\hat{g}^{R}(E)\hat{V}\left[\hat{g}^{K}(E)+\hat{g}^{R}(E)\right]\right),$

which yields the single-particle current [see Eq. (19)]

$\displaystyle{\cal I}_{1}=-\frac{e}{8\pi\hbar}\int$ $\displaystyle{\rm
Tr}\Bigl{[}\nu_{3}\tau_{3}i_{0}\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)+$
(50)
$\displaystyle\nu_{3}\tau_{3}i_{0}\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\Bigr{]}dE.$

The integrals with $\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)$ and
$\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)$ vanish due to the analytical properties
of the retarded and advanced functions.

Next, we trace over SN space, using Eq. (10) for $\hat{V}$ and the diagonal
Green’s functions

$\hat{g}^{K,A,R}=\left[\begin{matrix}\hat{g}^{K,A,R}_{{}_{S}}&0\\\
0&\hat{g}^{K,A,R}_{{}_{N}}\end{matrix}\right],$ (51)

with the following result

$\displaystyle{\cal I}_{1}=\frac{e}{8\pi\hbar}$ $\displaystyle\int{\rm
Tr}\Bigl{(}\tau_{3}i_{0}\hat{T}\hat{g}^{K}_{{}_{S}}(E)\hat{T}^{\dagger}[\hat{g}^{A}_{{}_{N}}(E)-\hat{g}^{R}_{{}_{N}}(E)]-$
(52)
$\displaystyle\tau_{3}i_{0}\hat{T}[\hat{g}^{A}_{{}_{S}}(E)-\hat{g}^{R}_{{}_{S}}(E)]\hat{T}^{\dagger}\hat{g}^{K}_{{}_{N}}(E)\Bigr{)}dE.$

Further, we insert $\hat{T}$ from Eq. (11) and $\hat{g}^{K,A,R}_{{}_{N}}(E)$
given (in ph space) by

$\hat{g}^{A,R}_{{}_{N}}(E)=\left[\begin{matrix}\hat{G}^{A,R}_{{}_{N}}(E)&0\\\
0&-\hat{G}^{A,R}_{{}_{N}}(-E)^{*}\end{matrix}\right]$ (53)

and

$\hat{g}^{K}_{{}_{N}}(E)=\left[\begin{matrix}\hat{G}^{K}_{{}_{N}}(E)&0\\\
0&\hat{G}^{K}_{{}_{N}}(-E)^{*}\end{matrix}\right],$ (54)

where

$\displaystyle\hat{G}^{K}_{{}_{N}}(E)=-2\pi
i\hat{A}_{{}_{N}}(E)[1-2f_{{}_{N}}(E)],$ (55)

with the spectral function

$\displaystyle\hat{A}_{{}_{N}}(E)=\frac{\hat{G}^{A}_{{}_{N}}(E)-\hat{G}^{R}_{{}_{N}}(E)}{2\pi
i}$ (56)

and the fermion distribution function $f_{{}_{N}}(E)$. Since $\hat{T}$ and
$\hat{g}^{K,A,R}_{{}_{N}}(E)$ are diagonal, only the diagonal parts of
matrices $\hat{g}^{K,A,R}_{{}_{S}}(E)$ contribute to the trace over ph space.
These have the same structure as in Eqs. (53) – (56), where label N is
replaced with S. The one-particle current can then be cast in the form

$\displaystyle{\cal I}_{1}$ $\displaystyle=$
$\displaystyle\frac{e}{4\pi\hbar}\int{\cal
T}_{1}(E)[f_{{}_{N}}(E)-f_{{}_{S}}(E)]dE$ (57) $\displaystyle+$
$\displaystyle\frac{e}{4\pi\hbar}\int{\cal
T}^{*}_{1}(-E)[f_{{}_{N}}(-E)-f_{{}_{S}}(-E)]dE\quad$ $\displaystyle=$
$\displaystyle\frac{e}{2\pi\hbar}\int{\cal
T}_{1}(E)[f_{{}_{N}}(E)-f_{{}_{S}}(E)],$

with the one-particle transmission coefficient

${\cal T}_{1}(E)=(2\pi)^{2}{\rm
Tr}\Bigl{[}\hat{t}^{\dagger}\hat{A}_{{}_{N}}(E)\hat{t}\hat{A}_{{}_{S}}(E)\Bigr{]}.$
(59)

Since the latter is real-valued, the particle and hole contributions to ${\cal
I}_{1}$ are identical [cf. Eqs. (57) and (A.1)].

###  A.2 Two-particle current

The two-particle current is related to the correction $\hat{G}^{>}_{2}(E)$
from the fourth term in the series in Eq. (46). The direct matrix
multiplication yields

$\displaystyle\hat{G}^{>}_{2}(E)=\frac{1}{2}$
$\displaystyle\Bigl{[}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)$
(60)
$\displaystyle+\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)$
$\displaystyle+\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)$
$\displaystyle+\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)$
$\displaystyle+\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)$
$\displaystyle-\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)\Bigr{]}.$

The last two terms do not contribute to current (17) for the same reason as in
the one-particle case. Hence, the two-particle current

$\displaystyle{\cal I}_{2}$ $\displaystyle=$
$\displaystyle-\frac{e}{8\pi\hbar}\int{\rm
Tr}\Bigl{\\{}\nu_{3}\tau_{3}i_{0}\Bigl{[}$ (61)
$\displaystyle\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)+$
$\displaystyle\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)\hat{V}\hat{g}^{A}(E)+$
$\displaystyle\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\hat{V}\hat{g}^{A}(E)+$
$\displaystyle\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{R}(E)\hat{V}\hat{g}^{K}(E)\Bigr{]}\Bigr{\\}}dE.\quad$

To calculate the trace in SN space, we again use Eq. (10) for $\hat{V}$ and
Eq. (51) for $\hat{g}^{K,A,R}$ and find

$\displaystyle{\cal I}_{2}$ $\displaystyle=$
$\displaystyle-\frac{e}{8\pi\hbar}\int{\rm Tr}\Bigl{\\{}\tau_{3}i_{0}\Bigl{[}$
(62)
$\displaystyle\hat{T}\hat{g}^{K}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{A}_{{}_{N}}(E)\hat{T}\hat{g}^{A}_{{}_{S}}(E)\hat{T}^{\dagger}[\hat{g}^{A}_{{}_{N}}(E)-\hat{g}^{R}_{{}_{N}}(E)]+$
$\displaystyle\hat{T}\hat{g}^{R}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{K}_{{}_{N}}(E)\hat{T}\hat{g}^{A}_{{}_{S}}(E)\hat{T}^{\dagger}[\hat{g}^{A}_{{}_{N}}(E)-\hat{g}^{R}_{{}_{N}}(E)]+$
$\displaystyle\hat{T}\hat{g}^{R}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{R}_{{}_{N}}(E)\hat{T}\hat{g}^{K}_{{}_{S}}(E)\hat{T}^{\dagger}[\hat{g}^{A}_{{}_{N}}(E)-\hat{g}^{R}_{{}_{N}}(E)]+$
$\displaystyle\hat{T}\hat{g}^{R}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{R}_{{}_{N}}(E)\hat{T}\hat{g}^{R}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{K}_{{}_{N}}(E)-$
$\displaystyle\hat{T}\hat{g}^{A}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{A}_{{}_{N}}(E)\hat{T}\hat{g}^{A}_{{}_{S}}(E)\hat{T}^{\dagger}\hat{g}^{K}_{{}_{N}}(E)\Bigr{]}\Bigr{\\}}dE.\quad$

Generally, the two-particle current is a subleading correction. However, under
conditions

$\hat{g}^{A}_{{}_{S}}(E)=\hat{g}^{R}_{{}_{S}}(E)\equiv\hat{g}_{{}_{S}}(E),\qquad\hat{g}^{K}_{{}_{S}}(E)=0,$
(63)

the one-particle current in Eq. (52) is suppressed, and ${\cal I}_{2}$ becomes
the leading order. This is the case if S has a single-particle gap $\Delta$
and the energies of interest are much smaller than $\Delta$. In the following,
we specifically focus on this regime. It turns out that under conditions (63)
only the off-diagonal part of $\hat{g}_{{}_{S}}(E)$ contributes to the trace
in ph space. We therefore use

$\hat{g}_{{}_{S}}(E)=\left[\begin{matrix}0&\hat{F}_{{}_{S}}(E)\\\
\hat{F}^{\dagger}_{{}_{S}}(E)&0\end{matrix}\right]$ (64)

along with Eq. (11) for $\hat{T}$ and Eqs. (53) and (54) for
$\hat{g}^{K,A,R}_{{}_{N}}(E)$ to find

$\displaystyle{\cal I}_{2}=\frac{e}{4\pi\hbar}\int{\rm Tr}\Bigl{[}$ (65)
$\displaystyle\hat{t}^{\dagger*}\hat{G}^{K}_{{}_{N}}(-E)^{*}\hat{t}^{*}\hat{F}^{\dagger}_{{}_{S}}(E)\hat{t}^{\dagger}[\hat{G}^{A}_{{}_{N}}(E)-\hat{G}^{R}_{{}_{N}}(E)]\hat{t}\hat{F}_{{}_{S}}(E)+$
$\displaystyle\hat{t}^{\dagger}\hat{G}^{K}_{{}_{N}}(E)\hat{t}\hat{F}_{{}_{S}}(E)\hat{t}^{\dagger*}[\hat{G}^{A}_{{}_{N}}(-E)-\hat{G}^{R}_{{}_{N}}(-E)]^{*}\hat{t}^{*}\hat{F}^{\dagger}_{{}_{S}}(E)\Bigr{]}dE,$

where $\hat{F}_{{}_{S}}(E)$ is the anomalous (condensate) Green’s function.
Finally, in view of Eqs. (55) and (56), the two-particle current can be cast
into the form presented in the main text [see Eqs. (20) and (21)]. To ease the
notation, we omit there subscripts $S$ and $N$ at the Green’s functions.

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#  Averaging operators over nondegenerate quadratic surfaces in finite fields

Doowon Koh  Department of Mathematics
Chungbuk National University
Cheongju city, Chungbuk-Do 361-763 Korea<EMAIL_ADDRESS>
###### Abstract.

We study mapping properties of the averaging operator related to the variety
$V=\\{x\in\mathbb{F}_{q}^{d}:Q(x)=0\\},$ where $Q(x)$ is a nondegenerate
quadratic polynomial over a finite field $\mathbb{F}_{q}$ with $q$ elements.
This paper is devoted to eliminating the logarithmic bound appearing in the
paper [5]. As a consequence, we settle down the averaging problems over the
quadratic surfaces $V$ in the case when the dimensions $d\geq 4$ are even and
$V$ contains a $d/2$-dimensional subspace.

###### 2010 Mathematics Subject Classification:

43A32, 43A15, 11T99

Key words and phrases: averaging operator, finite fields.
This research was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology(2012010487)

##  1\. Introduction

Let $V\subset\mathbb{R}^{d}$ be a smooth hypersurface and $d\sigma$ a smooth,
compactly supported surface measure on $V.$ An averaging operator $A$ over $V$
is given by

$Af(x)=f\ast d\sigma(x)=\int_{V}f(x-y)d\sigma(y)$

where $f$ is a complex valued function on $\mathbb{R}^{d}.$ In this Euclidean
setting, the averaging problem is to determine the optimal range of exponents
$1\leq p,r\leq\infty$ such that

(1.1) $\|f\ast d\sigma\|_{L^{r}(\mathbb{R}^{d})}\leq
C_{p,r,d}\|f\|_{L^{p}(\mathbb{R}^{d})},~{}~{}f\in\mathcal{S}(\mathbb{R}^{d})$

where $\mathcal{S}(\mathbb{R}^{d})$ denotes the space of Schwartz functions.
When $V$ is the unit sphere $\mathbb{S}^{d-1},$ this problem is closely
related to regularity estimates of the solutions to the wave equation at time
$t=1$, and it was studied by R.S. Strichartz [12]. It is well known that
$L^{p}-L^{r}$ averaging results can be obtained by the decay estimates of the
Fourier transform of the surface measure $d\sigma$ on $V$. For instance, if
$|\widehat{d\sigma}(\xi)|=\left|\int_{V}e^{-2\pi
ix\cdot\xi}d\sigma(x)\right|\lesssim(1+|\xi|)^{-\alpha}$ for some $\alpha>0$,
then the averaging inequality (1.1) holds whenever

$1\leq p\leq
2,\quad\frac{1}{p}-\frac{1}{2}\leq\frac{1}{2}\left(\frac{\alpha}{\alpha+1}\right),~{}~{}\mbox{and}~{}~{}r=p^{\prime},$

where $p^{\prime}$ denotes the exponent conjugate to $p$ (see [6, 11]). Thus,
if $|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-(d-1)/2}$ and
$(1/p,1/r)=(d/(d+1),1/(d+1)),$ then the averaging estimate (1.1) holds. Since
$L^{1}-L^{1}$ and $L^{\infty}-L^{\infty}$ estimates are clearly possible, we
see from the interpolation theorem that if
$|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-(d-1)/2},$ then $L^{p}-L^{r}$
estimates hold whenever $(1/p,1/r)$ lies in the triangle $\Delta_{d}$ with
vertices $(0,0),(1,1),$ and $(d/(d+1),1/(d+1)).$ Moreover, it is well known
that $L^{p}-L^{r}$ estimates are impossible if $(1/p,1/r)$ lies outside of the
triangle $\Delta_{d}.$ Such analogous phenomena were also observed in the
finite field setting (see, for example, [1, 4, 5]).
On the other hand, if the optimal Fourier decay estimate of the surface
measure $d\sigma$ is given by

$|\widehat{d\sigma}(\xi)|\lesssim(1+|\xi|)^{-\alpha}\quad\mbox{for
some}~{}~{}\alpha<(d-1)/2,$

then it is in general hard to prove sharp averaging results and some technical
arguments are required to deal with the case (see [3]). In the finite field
case, this was also pointed out by the authors in [1].

As an analogue of Euclidean averaging problems, Carbery, Stones, and Wright
[1] initially introduced and studied the averaging problem in finite fields
and the author with Shen has recently investigated the averaging problem over
homogeneous varieties. In this introduction, we shortly review notation and
definitions for averaging problems in the finite field setting and readers are
referred to [5] for further information and motivation on the averaging
problem. Let $\mathbb{F}_{q}^{d}$ be a $d$-dimensional vector space over a
finite field $\mathbb{F}_{q}$ with $q$ elements. We endow the space
$\mathbb{F}_{q}^{d}$ with a normalized counting measure $``dx"$. Let
$V\subset\mathbb{F}_{q}^{d}$ be an algebraic variety. Then a normalized
surface measure $\sigma$ supported on $V$ can be defined by the relation

$\int f(x)~{}d\sigma(x)=\frac{1}{|V|}\sum_{x\in V}f(x)$

where $|V|$ denotes the cardinality of $V$ (see [9]). An averaging operator
$A$ can be defined by

$Af(x)=f\ast d\sigma(x)=\int f(x-y)~{}d\sigma(y)=\frac{1}{|V|}\sum_{y\in
V}f(x-y)$

where both $f$ and $Af$ are functions on $(\mathbb{F}_{q}^{d},dx).$ In this
setting the averaging problem over $V$ is to determine $1\leq p,r\leq\infty$
such that

(1.2) $\|Af\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq
C\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)},$

where the constant $C>0$ is independent of functions $f$ and $q$, the
cardinality of the underlying finite field $\mathbb{F}_{q}.$

######  Definition 1.1.

Let $1\leq p,r\leq\infty.$ We denote by $A(p\to r)\lesssim 1$ to indicate that
the averaging inequality (1.2) holds.

The main purpose of this paper is to obtain the complete $L^{p}-L^{r}$
estimates of the averaging operators over varieties determined by
nondegenerate quadratic form over $\mathbb{F}_{q}$. Let
$Q(x)\in\mathbb{F}_{q}[x_{1},\dots,x_{d}]$ be a nondegenerate quadratic form.
Define a variety

$S=\\{x\in\mathbb{F}_{q}^{d}:Q(x)=0\\}.$

We shall name this kind of varieties as a nondegenerate quadratic surface in
$\mathbb{F}_{q}^{d}.$ Since $Q(x)$ is a nondegenerate quadratic form, it can
be transformed into a diagonal form $a_{1}x_{1}^{2}+\dots+a_{d}x_{d}^{2}$ with
$a_{j}\neq 0$ by means of a linear substitution (see [8]). Therefore, we may
assume that any nondegenerate quadratic surface can be written by the form

(1.3) $S=\\{x\in\mathbb{F}_{q}^{d}:a_{1}x_{1}^{2}+\cdots+a_{d}x_{d}^{2}=0\\}$

where $a_{j}\in\mathbb{F}_{q}\setminus\\{0\\},j=1,\dots,d.$ The necessary
conditions for the averaging estimates over $S$ were given in [1, 5]. In fact,
$A(p\to r)\lesssim 1$ only if $(1/p,1/r)$ lies in the convex hull of
$(0,0),(0,1),(1,1),$ and $(d/(d+1),1/(d+1)).$ It is known from [5] that this
necessary conditions for $A(p\to r)\lesssim 1$ are sufficient conditions if
the dimension $d\geq 3$ is odd. On the other hand, it was observed in [5] that
if $d\geq 4$ is even and $S$ contains a subspace $H$ with $|H|=q^{d/2},$ then
$A(p\to r)\lesssim 1$ only if $(1/p,1/r)$ lies in the convex hull of

(1.4)
$(0,0),(0,1),(1,1),\left(\frac{d^{2}-2d+2}{d(d-1)},~{}\frac{1}{d-1}\right),~{}~{}\mbox{and}~{}~{}\left(\frac{d-2}{d-1},~{}\frac{d-2}{d(d-1)}\right).$

In this paper we show that (1.4) is also the sufficient conditions for $A(p\to
r)\lesssim 1$ in the specific case when the variety $S$ contains
$d/2$-dimensional subspace with $d\geq 4$ even. See Figure 1.

Figure 1. Necessary and Sufficient Conditions for $A(p\to r)\lesssim 1$ in the
case that $d\geq 4$ is even and $S$ contains a $d/2$-dimensional subspace

###  1.1. Statement of main result

######  Theorem 1.2.

Let $d\sigma$ be the normalized surface measure on the nondegenerate quadratic
surface $S\subset\mathbb{F}_{q}^{d},$ as defined in (1.3). Suppose that $d\geq
4$ is an even integer and $S$ contains a $d/2$-dimensional subspace. Then
$A(p\to r)\lesssim 1$ if and only if $(1/p,1/r)$ lies in the convex hull of

$(0,0),(0,1),(1,1),\left(\frac{d^{2}-2d+2}{d(d-1)},~{}\frac{1}{d-1}\right),~{}~{}\mbox{and}~{}~{}\left(\frac{d-2}{d-1},~{}\frac{d-2}{d(d-1)}\right).$

######  Remark 1.3.

If the dimension $d\geq 4$ is even, then the diagonal entries $a_{j}$ can be
properly chosen so that $S$ contains a $d/2$-dimensional subspace $H.$ Such an
example is essentially the following one (see Theorem 4.5.1 of [10]):
$S=\\{x\in\mathbb{F}_{q}^{d}:\sum_{k=1}^{d}(-1)^{k+1}x_{k}^{2}=0\\}$ and
$H=\\{(t_{1},t_{1},t_{2},t_{2},\dots,t_{d/2},t_{d/2})\in\mathbb{F}_{q}^{d}:t_{1},t_{2},\dots,t_{d/2}\in\mathbb{F}_{q}\\}.$

######  Remark 1.4.

From the observation (1.4), we only need to prove the “ if ” part of Theorem
1.2. Since $dx$ is the normalized counting measure on $\mathbb{F}_{q}^{d}$, it
follows from Young’s inequality that $A(p\to r)\lesssim 1$ for $1\leq r\leq
p\leq\infty.$ Thus, by duality and the interpolation theorem, it will be
enough to prove that

(1.5) $\|f\ast
d\sigma\|_{L^{d-1}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{d(d-1)/(d^{2}-2d+2)}(\mathbb{F}_{q}^{d},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d}.$

The authors in [5] showed that this inequality holds for all characteristic
functions on subsets of $\mathbb{F}_{q}^{d}.$ Here, we improve upon their work
by obtaining the strong-type estimate.

###  1.2. Outline of this paper

In the remaining parts of this paper, we focus on providing the detail proof
of Theorem 1.2. In Section 2, we review the Fourier analysis in finite fields
and prove key lemmas which are essential in proving our main theorem. The
proof of Theorem 1.2 for even dimensions $d\geq 6$ will be completed in
Section 3. Namely, when $d\geq 6$ is any even integer, the inequality (1.5)
will be proved in Section 3. In the final section, we finish the proof of
Theorem 1.2 by proving the inequality (1.5) for $d=4.$

##  2\. key lemmas

In this section we drive key lemmas which play a crucial role in proving
Theorem 1.2. We begin by reviewing the Discrete Fourier analysis and readers
can be reffered to [5] for more information on it. Let $\mathbb{F}_{q}$ be a
finite field with $q$ elements. Throughout this paper, we assume that $q$ is a
power of odd prime so that the characteristic of $\mathbb{F}_{q}$ is greater
than two. We denote by $\chi$ a nontrivial additive character of
$\mathbb{F}_{q}.$ Recall that the orthogonality relation of the canonical
additive character $\chi$ says that

$\sum_{x\in\mathbb{F}_{q}^{d}}\chi(m\cdot
x)=\left\\{\begin{array}[]{ll}0&\mbox{if}~{}~{}m\neq(0,\dots,0)\\\
q^{d}&\mbox{if}~{}~{}m=(0,\dots,0),\end{array}\right.$

where $\mathbb{F}_{q}^{d}$ denotes the $d$-dimensional vector space over
$\mathbb{F}_{q}$ and $m\cdot x$ is the usual dot-product notation. Denote by
$(\mathbb{F}_{q}^{d},dx)$ the vector space over $\mathbb{F}_{q}$, endowed with
the normalized counting measure $``dx".$ Its dual space will be denoted by
$(\mathbb{F}_{q}^{d},dm)$ and we endow it with a counting measure $``dm".$ If
$f:(\mathbb{F}_{q}^{d},dx)\rightarrow\mathbb{C},$ then the Fourier transform
of the function $f$ is defined on $(\mathbb{F}_{q}^{d},dm)$:

$\widehat{f}(m)=\int_{\mathbb{F}_{q}^{d}}f(x)\chi(-x\cdot
m)~{}dx=\frac{1}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}f(x)\chi(-x\cdot
m)\quad\mbox{for}~{}~{}m\in\mathbb{F}_{q}^{d}.$

We also recall the Plancherel theorem:

$\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{f}(m)|^{2}=\frac{1}{q^{d}}\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{2}.$

Throughout this paper, we identify the set $E\subset\mathbb{F}_{q}^{d}$ with
the characteristic function on the set $E.$ We denote by $(d\sigma)^{\vee}$
the inverse Fourier transform of the normalized surface measure $d\sigma$ on
$S$ in (1.3) . Recall that

$(d\sigma)^{\vee}(m)=\int_{S}\chi(m\cdot
x)~{}d\sigma(x)=\frac{1}{|S|}\sum_{x\in S}\chi(m\cdot x).$

###  2.1. Gauss sums and estimates of $(d\sigma)^{\vee}$

Let $\eta$ denote the quadratic character of ${\mathbb{F}_{q}}.$ For each
$t\in{\mathbb{F}_{q}},$ the Gauss sum $G_{t}(\eta,\chi)$ is defined by

$G_{t}(\eta,\chi)=\sum_{s\in{\mathbb{F}_{q}}\setminus\\{0\\}}\eta(s)\chi(ts).$

The absolute value of the Gauss sum is given as follows (see [8, 2]):

$|G_{t}(\eta,\chi)|=\left\\{\begin{array}[]{ll}q^{\frac{1}{2}}&\mbox{if}~{}~{}t\neq
0\\\ 0&\mbox{if}~{}~{}t=0,\end{array}\right..$

It turns out that the inverse Fourier transform of $d\sigma$ can be written in
terms of the Gauss sum. The following was given in Lemma 4.1 of [5].

######  Lemma 2.1.

Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3), and let
$d\sigma$ be the normalized surface measure on $S.$ If $d\geq 2$ is even, then
we have

$(d\sigma)^{\vee}(m)=\left\\{\begin{array}[]{ll}q^{d-1}|S|^{-1}+\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m=(0,\dots,0)\\\
\frac{G_{1}^{d}}{|S|}(1-q^{-1})\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
-\frac{G_{1}^{d}}{q|S|}\eta(a_{1}\cdots
a_{d})&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0.\end{array}\right.$

Here and throughout this paper we denote by $\eta$ the quadratic character of
${\mathbb{F}_{q}}$ and we write $G_{1}$ for the Gauss sum $G_{1}(\eta,\chi).$

Lemma 2.1 yields the following corollary.

######  Corollary 2.2.

Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3) and let
$d\sigma$ be the normalized surface measure on $S.$ If $d\geq 4$ is even, then
we have

(2.1) $|S|=q^{d-1}+G_{1}^{d}(1-q^{-1})\eta(a_{1}\cdots a_{d})\sim q^{d-1},$

and

(2.2)
$|(d\sigma)^{\vee}(m)|\sim\left\\{\begin{array}[]{ll}q^{-\frac{(d-2)}{2}}&\mbox{if}~{}~{}m\neq(0,\dots,0),~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\\
q^{-\frac{d}{2}}&\mbox{if}~{}~{}\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0.\end{array}\right.$

###### Proof.

By the definition of $(d\sigma)^{\vee}(0,\ldots,0)$, it is clear that
$(d\sigma)^{\vee}(0,\ldots,0)=1.$ Comparison with Lemma 2.1 shows that

$|S|=q^{d-1}+G_{1}^{d}(1-q^{-1})\eta(a_{1}\cdots a_{d}).$

Since $|G_{1}|=q^{1/2}$, it follows that $|S|\sim q^{d-1}$ for $d\geq 4$ even.
This proves (2.1). The inequality (2.2) follows immediately from Lemma 2.1
because $|G_{1}|=q^{1/2}$ and $|S|\sim q^{d-1}$ for $d\geq 4$ even. ∎

######  Remark 2.3.

It is clear from (2.2) that if $d\geq 4$ is even and $S$ is any nondegenerate
quadratic surface in $\mathbb{F}_{q}^{d},$ then

(2.3) $|(d\sigma)^{\vee}(m)|=\left|\frac{1}{|S|}\sum_{x\in S}\chi(m\cdot
x)\right|\sim\frac{1}{q^{d-1}}\left|\sum_{x\in S}\chi(m\cdot x)\right|\lesssim
q^{-\frac{(d-2)}{2}}~{}~{}\mbox{for}~{}~{}m\in\mathbb{F}_{q}^{d}\setminus\\{(0,\dots,0)\\}.$

###  2.2.  Bochner-Riesz kernel

Recall that $d\sigma$ is the normalized surface measure on the nondegenerate
quadratic surface $S$ . In the finite field setting, the Bochner-Riesz kernel
$K$ is a function on $(\mathbb{F}_{q}^{d},dm)$ and it satisfies that
$K=(d\sigma)^{\vee}-\delta_{0}.$ Recall that $dm$ denotes the counting measure
on $\mathbb{F}_{q}^{d}.$ Notice that $K(m)=0$ if $m=(0,\ldots,0)$, and
$K(m)=(d\sigma)^{\vee}(m)$ otherwise. Also observe that

$d\sigma=\widehat{K}+\widehat{\delta_{0}}=\widehat{K}+1.$

Here, the last equality follows because $\delta_{0}$ is defined on the vector
space with the counting measure $dm,$ and its Fourier transform
$\widehat{\delta_{0}}$ is defined on the dual space with the normalized
counting measure $dx.$ More precisely, if $x\in(\mathbb{F}_{q}^{d},dx),$ then

$\widehat{\delta_{0}}(x)=\int_{m\in\mathbb{F}_{q}^{d}}\chi(-m\cdot
x)\delta_{0}(m)~{}dm=\sum_{m\in\mathbb{F}_{q}^{d}}\chi(-m\cdot
x)\delta_{0}(m)=1.$

Our main lemma is as follows.

######  Lemma 2.4.

Suppose that $d\geq 6$ is even. Then, for every $E\subset\mathbb{F}_{q}^{d},$
we have

(2.4)
$\|E\ast\widehat{K}\|_{L^{\frac{d-1}{2}}(\mathbb{F}_{q}^{d},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{\frac{-d^{2}+2d-3}{d-1}}|E|^{\frac{d-3}{d-1}}&\mbox{if}~{}~{}1\leq|E|\leq
q^{\frac{d-2}{2}}\\\
q^{\frac{-d^{2}+d-1}{d-1}}|E|&\mbox{if}~{}~{}q^{\frac{d-2}{2}}\leq|E|\leq
q^{\frac{d}{2}}\\\
q^{-d+1}|E|^{\frac{d-3}{d-1}}&\mbox{if}~{}~{}q^{\frac{d}{2}}\leq|E|\leq
q^{d}.\end{array}\right.$

where $K$ is the Bochner-Riesz kernel. On the other hand, for every
$E\subset\mathbb{F}_{q}^{4},$ it follows that

(2.5)
$\|E\ast\widehat{K}\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{19}{6}}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-\frac{10}{3}}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4}.\end{array}\right.$

###### Proof.

Using the interpolation theorem, it suffices to prove that the following two
inequalities hold for all d$\geq 4$ even:

(2.6) $\|E\ast\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\lesssim
q^{-d+1}|E|$

and

(2.7)
$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{\frac{-2d+1}{2}}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}1\leq|E|\leq
q^{\frac{d-2}{2}}\\\
q^{\frac{-5d+4}{4}}|E|&\mbox{if}~{}~{}q^{\frac{d-2}{2}}\leq|E|\leq
q^{\frac{d}{2}}\\\
q^{-d+1}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}q^{\frac{d}{2}}\leq|E|\leq
q^{d}.\end{array}\right.$

The estimate (2.6) can be obtained by applying Young’s inequality. In fact, we
see that

$\|E\ast\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\leq\|\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\|E\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}.$

Since $\|\widehat{K}\|_{L^{\infty}(\mathbb{F}_{q}^{d},dx)}\lesssim q$ and
$\|E\|_{L^{1}(\mathbb{F}_{q}^{d},dx)}=q^{-d}|E|,$ the inequality (2.6) is
established. To prove the inequality (2.7), first use the Plancherel theorem.
It follows that

$\|E\ast\widehat{K}\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dx)}=\|\widehat{E}K\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dm)}.$

Now, we recall that $dx$ is the normalized counting measure but $dm$ is the
counting measure. Thus, the expression above is given by

$\displaystyle\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}|K(m)|^{2}=$
$\displaystyle\sum_{m\neq(0,\dots,0)}|\widehat{E}(m)|^{2}|(d\sigma)^{\vee}(m)|^{2}$
$\displaystyle\sim\frac{1}{q^{d-2}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\end{subarray}}|\widehat{E}(m)|^{2}+\frac{1}{q^{d}}\sum_{\begin{subarray}{c}m\neq(0,\dots,0):\\\
\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}\neq
0\end{subarray}}|\widehat{E}(m)|^{2}=\mbox{I}+\mbox{II},$

where the first line and the second line follow from the definition of $K$ and
the inequality (2.2) in Corollary 2.2, respectively. Applying the Plancherel
theorem, it is clear that

(2.8)
$\mbox{II}\leq\frac{1}{q^{d}}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}=q^{-2d}|E|.$

In order to obtain a good upper bound of I, we shall conduct two different
estimates on $\mbox{I}.$ First, the Plancherel theorem yields

(2.9)
$\mbox{I}\leq\frac{1}{q^{d-2}}\sum_{m\in\mathbb{F}_{q}^{d}}|\widehat{E}(m)|^{2}=\frac{|E|}{q^{2d-2}}.$

On the other hand, it follows that

$\mbox{I}\leq\frac{1}{q^{d-2}}\sum_{\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0}|\widehat{E}(m)|^{2}=\frac{1}{q^{3d-2}}\sum_{\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0}\sum_{x,y\in
E}\chi(-m\cdot(x-y)).$

Now, let
$S_{a}=\\{m\in\mathbb{F}_{q}^{d}:\frac{m_{1}^{2}}{a_{1}}+\cdots+\frac{m_{d}^{2}}{a_{d}}=0\\}$
which is also a nondegenerate quadratic surface with $|S_{a}|\sim q^{d-1}.$
Then the expression above can be written by

$\frac{1}{q^{3d-2}}\sum_{x,y\in E:x=y}|S_{a}|+\frac{1}{q^{3d-2}}\sum_{x,y\in
E:x\neq y}\left(\sum_{m\in S_{a}}\chi(-m\cdot(x-y))\right).$

Now, we see from (2.3) that if $x\neq y$, then $\left|\sum_{m\in
S_{a}}\chi(-m\cdot(x-y))\right|\lesssim q^{\frac{d}{2}}.$ Thus, we obtain that

$\mbox{I}\lesssim q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}.$

Combining this with the inequality (2.9) gives

$\mbox{I}\lesssim\min\left(\frac{|E|}{q^{2d-2}},~{}q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}\right).$

In conjunction with the inequality (2.8), this shows that

$\|E\ast\widehat{K}\|^{2}_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\min\left(\frac{|E|}{q^{2d-2}},~{}q^{-2d+1}|E|+q^{\frac{-5d+4}{2}}|E|^{2}\right)~{}+q^{-2d}|E|.$

Since $(\alpha+\beta)^{1/2}\sim\alpha^{1/2}+\beta^{1/2}$ for $\alpha,\beta\geq
0$, it also follows that

$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{d},dx)}\lesssim\min\left(q^{-d+1}|E|^{\frac{1}{2}},~{}q^{\frac{-2d+1}{2}}|E|^{\frac{1}{2}}+q^{\frac{-5d+4}{4}}|E|\right)~{}+q^{-d}|E|^{\frac{1}{2}}.$

A direct computation shows that this implies the inequality (2.7). We complete
the proof of Lemma 2.4.

∎

##  3\. Proof of Theorem 1.2 for $d\geq 6$

In this section we provide the complete proof of Theorem 1.2 in the case that
$d\geq 6$ is even. The proof for $d=4$ shall be independently given in the
following section. The main reason is as follows. Lemma 2.4 shall be used to
prove Theorem 1.2. If $d\geq 6$ is even, then we have seen that the inequality
(2.4) of Lemma 2.4 follows by interpolating (2.6) and (2.7). However, if $d$
is four, then such an interpolation is too meaningless to assert that (2.4)
holds for $d=4.$ As an alternative approach, the inequality (2.5) of Lemma 2.4
shall be applied to complete the proof for $d=4.$ In this case we need more
delicate estimates.

Now we start proving Theorem 1.2 for $d\geq 6$ even. As mentioned in Remark
1.4, it is enough to prove the following statement.

######  Theorem 3.1.

Let $S$ be the variety in $\mathbb{F}_{q}^{d}$ as defined in (1.3). If $d\geq
6$ is even, then we have

$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{d},$

where $(p,r)=\left(\frac{d^{2}-d}{d^{2}-2d+2},d-1\right).$

###### Proof.

Let $p=\frac{d^{2}-d}{d^{2}-2d+2}$ and $r=d-1.$ We aim to prove that for every
complex-valued function $f$ on $\mathbb{F}_{q}^{d},$

$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}=\left(q^{-d}\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}\right)^{\frac{1}{p}}.$

As in [7] we proceed with the proof by decomposing the function $f$ to which
the operator is applied into level sets. Without loss of generality, we may
assume that $f$ is a nonnegative real-valued function and

$\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}=1.$

Therefore, we may also assume that

(3.1) $f=\sum_{k=0}^{\infty}2^{-k}E_{k},$

where $E_{0},E_{1},\dots$ are disjoint subsets of $\mathbb{F}_{q}^{d}.$ It
follows from these assumptions that

(3.2) $\sum_{j=0}^{\infty}2^{-pj}|E_{j}|=1,$

and hence for every $j=0,1,\dots,$

(3.3) $|E_{j}|\leq 2^{pj}.$

Recall that $d\sigma=\widehat{K}+1$ where $K$ is the Bochner-Riesz kernel. It
follows that

$\|f\ast
d\sigma\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq\|f\ast\widehat{K}\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}+\|f\ast
1\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}.$

Since $r>p$ and $dx$ is the normalized counting measure on
$\mathbb{F}_{q}^{d},$ it is clear from Young’s inequality that

$\|f\ast
1\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\leq\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}.$

Therefore, it suffices to prove the following inequality

$\|f\ast\widehat{K}\|_{L^{r}(\mathbb{F}_{q}^{d},dx)}\lesssim\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}.$

Since we have assumed that $\sum_{x\in\mathbb{F}_{q}^{d}}|f(x)|^{p}=1$, we see
that $\|f\|_{L^{p}(\mathbb{F}_{q}^{d},dx)}=q^{-d/p}.$ Also observe that

$\|f\ast\widehat{K}\|^{2}_{L^{r}(\mathbb{F}_{q}^{d},dx)}=\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}.$

From these observations, our task is to show that

(3.4)
$q^{\frac{2d}{p}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}\lesssim
1.$

Using (3.1), we see that

$\displaystyle
q^{\frac{2d}{p}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}$
$\displaystyle\leq
q^{\frac{2d}{p}}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})(E_{j}\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)}$
$\displaystyle\lesssim
q^{\frac{2d}{p}}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-k-j}|E_{k}|\|(E_{j}\ast\widehat{K})\|_{L^{\frac{r}{2}}(\mathbb{F}_{q}^{d},dx)},$

where the last line follows by the symmetry of $k$ and $j,$ and the inequality
(2.6). Now, for each $j=0,1,2,\dots,$ we consider the following three sets:

$J_{1}=\\{j:1\leq|E_{j}|\leq q^{\frac{d-2}{2}}\\},$
$J_{2}=\\{j:q^{\frac{d-2}{2}}<|E_{j}|\leq q^{\frac{d}{2}}\\},$

and

$J_{3}=\\{j:q^{\frac{d}{2}}<|E_{j}|\leq q^{d}\\}.$

Since $r/2=(d-1)/2,$ it is clear from (2.4) in Lemma 2.4 that our goal is to
prove the following three inequalities:

(3.5)
$A_{1}:=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+2d-3}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{1}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\frac{d-3}{d-1}}\lesssim 1,$ (3.6)
$A_{2}:=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}||E_{j}|\lesssim 1,$ (3.7)
$A_{3}:=q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\frac{d-3}{d-1}}\lesssim 1.$

First, we prove that the inequality (3.5) holds. Since
$p=\frac{d^{2}-d}{d^{2}-2d+2},$ a direct computation shows that
$q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+2d-3}{d-1}}=1.$ Now recall from (3.3)
that $|E_{j}|\leq 2^{pj}$ for all $j=0,1,\dots.$ Therefore, it follows that

$A_{1}\leq\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{1}}^{\infty}2^{-k-j}|E_{k}|2^{\frac{jp(d-3)}{d-1}}\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{j\left(-1+\frac{p(d-3)}{d-1}\right)}.$

Since $-1+\frac{p(d-3)}{d-1}=\frac{-d-2}{d^{2}-2d+2}<0$ and the sum over $j$
is a geometric series, we see that
$\sum_{j=k}^{\infty}2^{j\left(-1+\frac{p(d-3)}{d-1}\right)}\sim
2^{k\left(-1+\frac{p(d-3)}{d-1}\right)}.$ Thus, the inequality (3.5) is
established as follows:

$A_{1}\lesssim\sum_{k=0}^{\infty}|E_{k}|2^{k\left(-2+\frac{p(d-3)}{d-1}\right)}\leq\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1,$

where we used the simple observation that $-2+\frac{p(d-3)}{d-1}<-p,$ and then
the assumption $(\ref{as1}).$
Second, we prove that the inequality (3.6) holds. Let
$\varepsilon=\frac{2d-4}{d^{2}-d}.$ Since $d\geq 6,$ we see that
$0<\varepsilon<1.$ Write $A_{2}$ as follows:

$A_{2}=q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{1-\varepsilon}|E_{j}|^{\varepsilon}.$

Since $0<\varepsilon<1,$ we notice from (3.3) that
$|E_{j}|^{1-\varepsilon}\leq 2^{p(1-\varepsilon)j}.$ By the definition of the
set $J_{2}$, we also see that $|E_{j}|^{\varepsilon}\leq
q^{\frac{d\varepsilon}{2}}$ for all $j\in J_{2}.$ Then, we have

$\displaystyle A_{2}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{2}}^{\infty}2^{-k-j}|E_{k}|2^{p(1-\varepsilon)j}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{j\left(-1+p(1-\varepsilon)\right)}.$

Notice that
$q^{\frac{2d}{p}}q^{-d+1}q^{\frac{-d^{2}+d-1}{d-1}}q^{\frac{d\varepsilon}{2}}=1,$
and the geometric series over $j$ converges to $\sim
2^{k\left(-1+p(1-\varepsilon)\right)}$ because
$-1+p(1-\varepsilon)=\frac{-d+2}{d^{2}-2d+2}<0$ for $d\geq 6.$ From this
observation and (3.2), the inequality (3.6) follows because we have

$A_{2}\lesssim\sum_{k=0}^{\infty}|E_{k}|2^{k\left(-2+p(1-\varepsilon)\right)}=\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1.$

Finally, we show that the inequality (3.7) holds. As in the proof of the
inequality (3.6), we let $0<\delta=\frac{4}{d^{2}-d}<1$ for $d\geq 6.$ The
value $A_{3}$ is written by

$A_{3}=q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}||E_{j}|^{\delta+\frac{d-3}{d-1}}|E_{j}|^{-\delta}.$

Notice from (3.3) that $|E_{j}|^{\delta+\frac{d-3}{d-1}}\leq
2^{p\left(\delta+\frac{d-3}{d-1}\right)j}$ for all $j=0,1,2,\dots.$ By the
definition of $J_{3}$, it is easy to notice that $|E_{j}|^{-\delta}\leq
q^{\frac{-d\delta}{2}}$ for $j\in J_{3}.$ It therefore follows that

$\displaystyle A_{3}$ $\displaystyle\leq
q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}q^{\frac{-d\delta}{2}}\sum_{k=0}^{\infty}\sum_{j=k:j\in
J_{3}}^{\infty}2^{-k-j}|E_{k}|2^{p\left(\delta+\frac{d-3}{d-1}\right)j}$
$\displaystyle\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{\left(-1+p\left(\delta+\frac{d-3}{d-1}\right)\right)j}$
$\displaystyle\sim\sum_{k=0}^{\infty}|E_{k}|2^{\left(-2+p\left(\delta+\frac{d-3}{d-1}\right)\right)k}=\sum_{k=0}^{\infty}|E_{k}|2^{-pk}=1,$

where we used the facts that
$q^{\frac{2d}{p}}q^{-d+1}q^{-d+1}q^{\frac{-d\delta}{2}}=1,$
$\left(-1+p\left(\delta+\frac{d-3}{d-1}\right)\right)=\frac{-d+2}{d^{2}-2d+2}<0$
for $d\geq 6,$ and $\left(-2+p\left(\delta+\frac{d-3}{d-1}\right)\right)=-p,$
and then the assumption (3.2) for the last equality. Thus, the inequality
(3.7) holds and the proof of Theorem 3.1 is complete.∎

##  4\. Proof of Theorem 1.2 for $d=4$

As observed in Remark 1.4, it amounts to showing the following statement.

######  Theorem 4.1.

Let $S$ be the variety in $\mathbb{F}_{q}^{4}$ as defined in (1.3). Then, we
have

$\|f\ast
d\sigma\|_{L^{3}(\mathbb{F}_{q}^{4},dx)}\lesssim\|f\|_{L^{\frac{6}{5}}(\mathbb{F}_{q}^{4},dx)}~{}~{}\mbox{for
all functions}~{}~{}f~{}~{}\mbox{on}~{}~{}\mathbb{F}_{q}^{4}.$

###### Proof.

We will proceed by the similar ways as in the previous section. However, the
proof of the theorem will be based on (2.5), rather than (2.4) in Lemma 2.4.
We begin by recalling from (2.5) and (2.7) that

(4.1)
$\|E\ast\widehat{K}\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{19}{6}}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-\frac{10}{3}}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{5}{6}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4},\end{array}\right.$

and

(4.2)
$\|E\ast\widehat{K}\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}\lesssim\left\\{\begin{array}[]{ll}q^{-\frac{7}{2}}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}1\leq|E|\leq
q\\\ q^{-4}|E|&\mbox{if}~{}~{}q\leq|E|\leq q^{2}\\\
q^{-3}|E|^{\frac{1}{2}}&\mbox{if}~{}~{}q^{2}\leq|E|\leq
q^{4}.\end{array}\right.$

We must show that for all complex-valued functions $f$ on
$\mathbb{F}_{q}^{4},$

$\|f\ast
d\sigma\|_{L^{3}(\mathbb{F}_{q}^{4},dx)}\lesssim\|f\|_{L^{\frac{6}{5}}(\mathbb{F}_{q}^{4},dx)}.$

As noticed in the previous section, it suffices to prove this inequality under
the following assumptions:

$\sum_{x\in\mathbb{F}_{q}^{4}}|f(x)|^{\frac{6}{5}}=1\quad\mbox{and}\quad
f=\sum_{k=0}^{\infty}2^{-k}E_{k},$

where $E_{0},E_{1},\dots$ are disjoint subsets of $\mathbb{F}_{q}^{4}.$ From
these assumptions, it is clear that

(4.3) $\sum_{j=0}^{\infty}2^{-\frac{6j}{5}}|E_{j}|=1\quad\mbox{for
all}~{}j=0,1,\dots.$

This clearly implies that

(4.4) $|E_{j}|\leq 2^{\frac{6j}{5}}\quad\mbox{for all}~{}j=0,1,\dots.$

According to (3.4), it is enough to prove that

$q^{\frac{20}{3}}\|(f\ast\widehat{K})(f\ast\widehat{K})\|_{L^{\frac{3}{2}}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$

Since $f=\sum_{k=0}^{\infty}2^{-k}E_{k},$ it is enough to show that

$q^{\frac{20}{3}}\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})(E_{j}\ast\widehat{K})\|_{L^{\frac{3}{2}}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$

By the symmetry of $k$ and $j,$ and the Hölder inequality, our task is to
prove

$q^{\frac{20}{3}}\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}\lesssim
1.$

Main steps to prove this inequality are summarized as follows. By considering
the sizes of $|E_{k}|$ and $|E_{j}|$, we first decompose
$\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}$ as nine parts. Next, using the
estimates (4.1),(4.2), (4.4), (4.3), and a convergence property of a geometric
series, we show that each part of them is $\lesssim 1,$ which completes the
proof of Theorem 4.1. For the sake of completeness, we shall give full
details. To do this, let us define the following $9$ sets: for
$N=\\{0,1,\dots\\},$

$I_{1}=\\{(k,j)\in N\times N:k\leq j,~{}1\leq|E_{k}|\leq q,~{}1\leq|E_{j}|\leq
q\\},$ $I_{2}=\\{(k,j)\in N\times N:k\leq j,~{}1\leq|E_{k}|\leq
q,~{}q<|E_{j}|\leq q^{2}\\},$ $I_{3}=\\{(k,j)\in N\times N:k\leq
j,~{}1\leq|E_{k}|\leq q,~{}q^{2}<|E_{j}|\leq q^{4}\\},$ $I_{4}=\\{(k,j)\in
N\times N:k\leq j,~{}q<|E_{k}|\leq q^{2},~{}1\leq|E_{j}|\leq q\\},$
$I_{5}=\\{(k,j)\in N\times N:k\leq j,~{}q<|E_{k}|\leq q^{2},~{}q<|E_{j}|\leq
q^{2}\\},$ $I_{6}=\\{(k,j)\in N\times N:k\leq j,~{}q<|E_{k}|\leq
q^{2},~{}q^{2}<|E_{j}|\leq q^{4}\\},$ $I_{7}=\\{(k,j)\in N\times N:k\leq
j,~{}q^{2}<|E_{k}|\leq q^{4},~{}1\leq|E_{j}|\leq q\\},$ $I_{8}=\\{(k,j)\in
N\times N:k\leq j,~{}q^{2}<|E_{k}|\leq q^{4},~{}q<|E_{j}|\leq q^{2}\\},$
$I_{9}=\\{(k,j)\in N\times N:k\leq j,~{}q^{2}<|E_{k}|\leq
q^{4},~{}q^{2}<|E_{j}|\leq q^{4}\\}.$

###  4.1.  Estimate of the sum over $I_{1}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{1}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle\sum_{(k,j)\in
I_{1}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}\leq\sum_{(k,j)\in
I_{1}}2^{-k-j}|E_{k}|2^{\frac{3j}{5}}\quad\mbox{since}\quad|E_{j}|^{\frac{1}{2}}\leq
2^{\frac{3j}{5}}\quad\mbox{by (\ref{size})}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}|E_{k}|2^{-k}\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim\sum_{k=0}^{\infty}|E_{k}|2^{-\frac{7k}{5}}\leq\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1\quad\mbox{by
(\ref{one})}.$

###  4.2. Estimate of the sum over $I_{2}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{2}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{2}}\sum_{(k,j)\in
I_{2}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|\leq q^{-\frac{1}{2}}\sum_{(k,j)\in
I_{2}}2^{-j}|E_{j}|\quad\mbox{since}\quad 2^{-k}|E_{k}|^{\frac{5}{6}}\leq
1\quad\mbox{by (\ref{size})}$ $\displaystyle\leq$ $\displaystyle\sum_{(k,j)\in
I_{2}}2^{-j}|E_{j}|^{\frac{3}{4}}\quad\mbox{since}\quad|E_{j}|^{\frac{1}{4}}\leq
q^{\frac{1}{2}}\quad\mbox{for}~{}~{}(k,j)\in I_{2}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-\frac{j}{10}}\quad\mbox{by
(\ref{size})}$ $\displaystyle\sim$
$\displaystyle\sum_{k=0}^{\infty}2^{-\frac{k}{10}}\sim 1.$

###  4.3. Estimate of the sum over $I_{3}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{3}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{2}}\sum_{(k,j)\in
I_{3}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}\leq
q^{\frac{1}{2}}\sum_{(k,j)\in I_{3}}2^{-j}|E_{j}|^{\frac{1}{2}}\quad\mbox{by
(\ref{size})}$ $\displaystyle=$ $\displaystyle\sum_{(k,j)\in
I_{3}}2^{-j}|E_{j}|^{\frac{3}{4}}q^{\frac{1}{2}}|E_{j}|^{-\frac{1}{4}}<\sum_{(k,j)\in
I_{3}}2^{-j}|E_{j}|^{\frac{3}{4}}\quad\mbox{since}\quad
q^{2}<|E_{j}|~{}~{}\mbox{for}~{}~{}(k,j)\in I_{3}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}\sum_{j=k}^{\infty}2^{-\frac{j}{10}}\sim\sum_{k=0}^{\infty}2^{-\frac{k}{10}}\sim
1.$

where (4.4) was also used to obtain the last inequality .

###  4.4.  Estimate of the sum over $I_{4}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{4}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{6}}\sum_{(k,j)\in
I_{4}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}}\leq q^{-\frac{1}{6}}\sum_{(k,j)\in
I_{4}}2^{-k}|E_{k}|2^{-j}2^{\frac{3j}{5}}\quad\mbox{by (\ref{size})}$
$\displaystyle\leq$ $\displaystyle
q^{-\frac{1}{6}}\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim
q^{-\frac{1}{6}}\sum_{k=0}^{\infty}2^{-\frac{7k}{5}}|E_{k}|$
$\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1\quad\mbox{by
(\ref{one})}.$

###  4.5.  Estimate of the sum over $I_{5}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{5}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{2}{3}}\sum_{(k,j)\in
I_{5}}2^{-k-j}|E_{k}||E_{j}|\leq\sum_{(k,j)\in
I_{5}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}\quad\mbox{since}~{}~{}|E_{j}|^{\frac{1}{3}}\leq
q^{\frac{2}{3}}\quad\mbox{for}~{}~{}(k,j)\in I_{5}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$

where we used (4.4), the convergence of a geometric series, and (4.3) in the
last line.

###  4.6.  Estimate of the sum over $I_{6}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{6}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{3}}\sum_{(k,j)\in
I_{6}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}}<\sum_{(k,j)\in
I_{6}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}\quad\mbox{since}~{}~{}|E_{j}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}\quad\mbox{for}~{}~{}(k,j)\in
I_{6}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$

where (4.4), the convergence of a geometric series, and (4.3) were also
applied in the last line.

###  4.7.  Estimate of the sum over $I_{7}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{7}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{1}{6}}\sum_{(k,j)\in
I_{7}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}=q^{\frac{1}{6}}\sum_{(k,j)\in
I_{7}}2^{-k-j}|E_{k}||E_{k}|^{-\frac{1}{6}}|E_{j}|^{\frac{1}{2}}$
$\displaystyle<$ $\displaystyle q^{\frac{1}{6}-\frac{1}{3}}\sum_{(k,j)\in
I_{7}}2^{-k}|E_{k}|2^{-\frac{2j}{5}}\quad\mbox{by
observing}\quad|E_{k}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}\quad{for}~{}~{}(k,j)\in
I_{7}~{}~{}\mbox{and by (\ref{size})}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{2j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{7k}{5}}|E_{k}|\leq\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1.$

###  4.8.  Estimate of the sum over $I_{8}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{-\frac{1}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|=q^{-\frac{1}{3}}\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}|E_{k}|^{-\frac{1}{6}}|E_{j}|^{\frac{1}{3}}$
$\displaystyle<$ $\displaystyle\sum_{(k,j)\in
I_{8}}2^{-k-j}|E_{k}||E_{j}|^{\frac{2}{3}}\quad\mbox{since}~{}~{}|E_{k}|^{-\frac{1}{6}}<q^{-\frac{1}{3}},\quad|E_{j}|^{\frac{1}{3}}\leq
q^{\frac{2}{3}}~{}~{}\mbox{for}~{}~{}(k,j)\in I_{8}$ $\displaystyle\leq$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1.$

###  4.9.  Estimate of the sum over $I_{9}$

It follows from (4.1) and (4.2) that

$\displaystyle q^{\frac{20}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}\|(E_{k}\ast\widehat{K})\|_{L^{6}(\mathbb{F}_{q}^{4},dx)}\|(E_{j}\ast\widehat{K})\|_{L^{2}(\mathbb{F}_{q}^{4},dx)}$
$\displaystyle\lesssim$ $\displaystyle q^{\frac{2}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}|^{\frac{5}{6}}|E_{j}|^{\frac{1}{2}}=q^{\frac{2}{3}}\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}|E_{k}|^{-\frac{1}{6}}|E_{j}|^{-\frac{1}{6}}$
$\displaystyle<$ $\displaystyle\sum_{(k,j)\in
I_{9}}2^{-k-j}|E_{k}||E_{j}|^{\frac{1}{2}+\frac{1}{6}}\quad\mbox{since}~{}~{}|E_{k}|^{-\frac{1}{6}},|E_{j}|^{-\frac{1}{6}}<q^{-\frac{1}{3}}~{}~{}\mbox{for}~{}~{}(k,j)\in
I_{9}$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-j}|E_{j}|^{\frac{2}{3}}\leq\sum_{k=0}^{\infty}2^{-k}|E_{k}|\sum_{j=k}^{\infty}2^{-\frac{j}{5}}\sim\sum_{k=0}^{\infty}2^{-\frac{6k}{5}}|E_{k}|=1,$

where we also used (4.4), the convergence of a geometric series, and (4.3) in
the last line. ∎

Acknowledgment : The author would like to thank the referee for his/her
valuable comments for developing the final version of this paper.

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