text:
\section{Introduction}

Electronic topological transition of the Fermi surface with no broken symmetries, known as Lifshitz
transition plays a significant role in numerous branches of condensed mater physics. For example,
the collapse of the normal state pseudogap at a Lifshitz transition in Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$
cuprate superconductor \cite{Benhabib}, Lifshitz transition in underdoped cuprates \cite{Norman},
Dirac semi metal \cite{Su-Yang Xu}, relativistic insulator NaOsO$_3$ \cite{Bongjae}, discontinuous
Lifshitz transition in  Na$_x$CoO$_2$ \cite{Okamoto}, Zeeman driven Lifshitz transition in
YbRh$_2$Si$_2$ \cite{Hackl}, in two-dimensional Hubbard model \cite{Chen}, bilayer graphene \cite{Lemonik},
Quantum Hall Liquids \cite{Varlet}, to mention a few.
Lifshitz transition is recently considered as a quantum phase transition in strongly
correlated electron systems also plays a remarkable role in Fe-based high T$_c$ superconductors
owing to its special `Fermiology' and multi-band nature \cite{LiuA,Khan}.
The phase diagrams of Fe-based superconductors (SCs) consist of a number of exhilarating features
apart from its novel superconducting phase \cite{Nandi,Avci,Ni,Thaler,Kasahara}.
Various phases are very much sensitive to external parameters like impurity, doping,
temperature and pressure.
On the other hand, Lifshitz transitions are experimentally observed with respect to some of the
above mentioned external parameters \cite{Liu,Xu,LiuB}.
Therefore, topological transitions, magnetism and superconductivity which occurs
in a variety of materials in condensed matter physics is a
subject of general interest.

 The Fe-based high T$_c$ SCs being a multiband system has multiple Fermi surfaces (FS) (two electron like and
three hole like), any of the FSs may collapse (become smaller ones) either because, some of the bands
that crosses the Fermi level (FL) may move away (above or below) from the FL or due to topological modifications
({\it e.g}, electron like band transforming to hole like or vice versa) under external perturbation.
Lifshitz transition (LT) or electronic topological transition (ETT), where FSs/electronic bands alter
topology, have been usually investigated as zero temperature phenomena arising due to impurity, doping
or pressure etc. However, temperature dependent LT/ETT is observed recently in WTe$_2$ \cite{Kaminski}.
The consequences of these LT/ETT are innumerable. It can lead to reduced interband scattering (affecting mechanism of SC), nesting of FS (affecting magnetism), anomalies or singularities in the density of states at FL and in general anomalies in the kinematics,
dynamics and thermodynamics of electrons, which would affect various physical
properties \cite{Lifshitz}. LT/ETT are also observed in Fe-based SCs induced by pressure,
impurity and doping  \cite{Quader,Sn,Khan,Liu,Wang}.
In particular, for BaFe$_2$As$_2$ (Ba122) system, LT is predicted theoretically due to small but
unintentional Sn impurity in contrast to other  compounds like SrFe$_2$As$_2$ (Sr122) and CaFe$_2$As$_2$ (Ca122) \cite{Sn}.
In hole doped Ba122, LT is  found in the heavily doped regime as
evident from the theoretical work of Khan {\it et al}., \cite{Khan}. On the other hand,
LT in electron (Co in place of Fe) doped Ba122 is observed experimentally \cite{Kaminski}.
But the temperature induced LT in general is rare;
we have predicted temperature dependent LT in Ru doped Ba122 system.
In this letter, we show with explicit demonstrations in a series of Fe-based SCs
that the LT / ETT occur at a doping concentration where SC is maximum and magnetism/structural transition vanishes.

Determining LT/ETT experimentally is challenging specially in systems that constitute multi-orbital-derived FS like Fe-based SCs; we show that
the FS area (FSA) carries sensitivity of topological modifications more acutely than the band structures (BS) and this can be used
as a better experimental tool to identify ETT/LT.
As a result of LT or ETT the FSA of a particular FS gets largely affected and hence a systematic study
of FSA as a function of doping is most desirable.  A detailed study on the variations of FSA, {\it e.g.}, variations of (i) areas of each individual FSs, (ii) sum total areas of all the electron FSs,  (iii) sum total areas of all the hole
FSs, (iv) sum total areas of all the five FSs, (v) difference of all hole and all electron FS areas as a
function of doping is a rare wealth of information that can be verified by the de Haas-van Alphen and allied effects (i.e, Shubnikov-de Haas effect) are presented in this letter. All the above FS areas with doping show distinctly different behaviors below and above the ETT.
Clear deviation in the variations of FSA with doping marks the occurrence of
topological transition (TT). This can therefore lead to a step forward progress in the experimental detection of
LT/ETT in general. It is worthwhile mentioning here that in the overdoped regime of high-T$_c$ cuprate SCs,
 ARPES and quantum oscillation studies revealed large FS but not only its shape and size change with doping
 but also breaks up into Fermi arcs \cite{Norman1} which turned out to be closed hole
 Fermi pocket \cite{hbyang,hbyang1,LeBoeuf,Sebastian}.
 All these points to a question, are high T$_c$ SCs topological or whether the high  T$_c$ is
 restricted by TT ?!

  \begin{figure*}[t]

  \flushleft \includegraphics [width=\textwidth,height=5cm]{Fig1.eps}
    \caption{(Colour online) Calculated FSs in four pair of columns for (i) Ba$_{1-x}$K$_x$Fe$_2$As$_2$  (a \----c); (ii) Ba$_{1-x}$Na$_x$Fe$_2$As$_2$ (d \---- f) in the low temperature orthorhombic
        phase; (iii) BaFe$_{2-x}$Co$_x$As$_2$ (g\----i) in the high temperature
          tetragonal phase and (iv) BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (j\----l)  in the low temperature orthorhombic phase for various dopings as indicated in the figure. In the left columns of each pair, all the five FSs around $\Gamma$ (hole like),
     X and Y (electron like) points are shown using different colours for each compounds whereas in the right columns of the same pair, topologically modified FSs are shown separately.}
   \label{KFS}
  \end{figure*}
   \begin{figure}[ht]
     \centering
 \includegraphics [height=3.0cm,width=0.45\textwidth]{Fig2.eps}
     \includegraphics [height=3.0cm,width=0.45\textwidth]{fig2a.eps}
     \caption{(Colour online) Zoomed orbital resolved electronic BSs around the high symmetry points where TTs are found in Figs.\ref{KFS} for (i) Ba$_{1-x}$K$_x$Fe$_2$As$_2$  (a \--b); (ii) Ba$_{1-x}$Na$_x$Fe$_2$As$_2$ (c \-- d) ; (iii) BaFe$_{2-x}$Co$_x$As$_2$ (e\--f) and (iv) BaFe$_2$(As$_{1-x}$P$_x$)$_2$ (g\----l). Topological modifications in the d$_{xz}$, d$_{xy}$ bands from electron to hole like dispersion is observed as doping is increased (cf. (a,b) and (c,d)). LT in the d$_{xy}$ bands of BaFe$_{2-x}$Co$_x$As$_2$ [Figs.(e\--f)] and BaFe$_2$(As$_{1-x}$P$_x$)$_2$ [Figs.(g\----l)] are worth observing.
         }
     \label{BS-e-h-iso}
     \end{figure}
First principles density functional theories (DFT) can produce correct solutions of the many electron Schr\"{o}dinger equation if exact electronic density is  used as input. Various modern X-ray diffraction techniques {\it e.g}., using Synchrotrons
radiation source etc. that determines crystallographic information at different external perturbations are essentially
result of diffraction from various atomic charge densities (Bragg's diffraction). Considering experimentally determined
structural parameters at different temperatures (doping)  as input thus in turn provides temperature (doping) dependent
exact densities in our first principles calculation. These input structural parameters are kept fixed through out the calculation
for a fixed temperature (doping). The main effect on the electronic structure from finite temperature is the underlying crystal structure,
and the average crystal structure at finite temperature can usually be reliably determined from diffraction experiment at a given temperature. Our first principles calculations are performed by implementing ultrasoft pseudopotential with plane wave basis set based on DFT \cite{CASTEP}. Electronic
exchange correlation is treated within the generalised gradient approximation (GGA) using
Perdew-Burke-Ernzerhof (PBE) functional \cite{PBE}.
Electronic structures (BS, FS {\it etc.}),
calculated using optimized lattice parameters ($a$, $b$, $c$ and $z_{As}$) do not resemble with that of the
experimentally measured one \cite{pla,zAs}. This insist us to employ experimental
lattice parameters {\it i.e.,} $a$, $b$, $c$ and $z_{As}$
\cite{Avci,thesis,Acta,Sefat,Ni} as a function of doping and temperature both in the low temperature
orthorhombic as well as in the high temperature tetragonal phases
as inputs of our first principles calculations.
In order to dope, we use Virtual crystal approximation (VCA) \cite{Bellaiche}.
Non-spin-polarized and spin polarised single point energy calculations
are performed for tetragonal phase with space group symmetry I4/mmm (No. 139) and
orthorhombic phase with space group symmetry Fmmm (No. 69) respectively
with energy cut off 500 eV and higher as well as
self-consistent field (SCF) tolerance as $10^{-7}$ eV/atom.
Brillouin zone is sampled in the k-space within Monkhorst–Pack scheme
and grid size for SCF calculation is chosen as per requirement of the calculation
for different systems. For simulating FS, grid size of SCF calculation is chosen as
$26\times 26\times 31$.

\begin{figure}[ht]
  \centering
  \includegraphics [height=5.0cm,width=0.45\textwidth]{Fig3.eps}

  \caption{(Colour online) Calculated FS of BaFe$_{2-x}$Ru$_x$As$_2$ below structural transition
  (pair of left columns) and
  that of undoped BaFe$_2$As$_2$ for different z$_{As}$ (pair of right columns). From each pair of columns, in the left column all the five FSs around $\Gamma$ (hole like), X and Y (electron like) points are displayed using different colours while in the right column topologically modified inner hole FSs are indicated by arrows.}
  \label{RuFS}
 \end{figure}

We investigate the role of various kinds of dopings ({\it e.g}, electron, hole or iso-electronic)  in
ETT / LT of various Fe-based superconductors [Ba$_{1-x}$M$_x$Fe$_2$As$_2$ (M= K, Na) (see Figs. \ref{KFS} (a\---f)), BaFe$_{2-x}$Co$_x$As$_2$ (see Figs. \ref{KFS} (g\---i)),  BaFe$_2$As$_{2-x}$P$_x$ (see Figs. \ref{KFS} (j\---l))].
In Fig.\ref{KFS}(a\---c) leftmost pair of columns, we depict the FSs of K doped Ba122 systems.
It is very clear from Fig.\ref{KFS} that there are five Fermi pockets,
two around X and Y points (electron like) and three around $\Gamma$ point
(hole like).
We find that one of the electron like FSs around X/Y point
transform into a different topology ({\it cf}. right column Fig. \ref{KFS}(c)).
We display orbital projected BS
of Ba$_{1-x}$K$_x$Fe$_2$As$_2$ around X point separately in Fig.\ref{BS-e-h-iso}(a,b) for $x=0.2$ and $x=0.5$ respectively. Same colour codes are used for all orbital resolved BSs throughout.
From Fig.\ref{BS-e-h-iso} (a,b) it is quite clear that the  electron like nature of d$_{xz}$ band around X (Y) points transform into hole like band; one of the hole like d$_{xy}$ bands also gets modified to an electron like band as a result of K-doping. These topological transformations of FSs/BSs are identified as ETT or LT.
 \begin{figure}[ht]
  \centering
  \includegraphics [height=1.75cm,width=0.42\textwidth]{fig4.eps}
  \includegraphics [height=1.75cm,width=0.43\textwidth]{fig4a.eps}
  \caption{(Colour online) Calculated orbitally resolved zoomed BSs of  BaFe$_{2-x}$Ru$_x$As$_2$  in the low temperature orthorhombic phase for three different temperatures (top row). Calculated BSs of BaFe$_2$As$_2$ for three different values of z$_{As}$ around $\Gamma$ point.
  }
  \label{RuBS}
 \end{figure}

 \begin{figure}[h]
\centering
\includegraphics[width=0.5\textwidth,height=5.75cm]{fs.eps}
\caption{(Colour online) FSAs of various doped Fe-based SCs
namely Co doped (1st column), K doped (2nd column), Na doped (3rd column) and P doped (4th column),
as a function of doping concentration. First row, FSA of each individual FSs
with doping; sudden change in slopes at the Lifshitz point is worth noting. Second row, total electron and
hole FSAs as a function of doping; jumps/change in slopes at the Lifshitz doping is noticeable.
Third row, total FSAs and the difference in the FSAs of electron and hole FSs.
Remarkable modification of the later at the Lifshitz doping is seen.}
\label{FSA}
\end{figure}

In Fig.\ref{KFS}(d \--- f) we
present the FSs of Ba$_{1-x}$Na$_x$Fe$_2$As$_2$ for
various  doping concentrations ($x$). Similar to the ETT found in Ba$_{1-x}$K$_x$Fe$_2$As$_2$ materials, Ba$_{1-x}$Na$_x$Fe$_2$As$_2$ also exhibits
the same at around the same $x$.
BSs of Na doped Ba122 systems presented in Fig.\ref{BS-e-h-iso} (c,d) are
qualitatively very similar to that of the K doped Ba122 system.
So we conclude from these two hole doped superconductors that the ETT are found in the electron like bands around the X and Y points at nearly 50\% hole doping. Remarkably, the established phase diagrams of the hole doped Ba122 SCs indicate that the highest  achievable T$_c$ is also around 50\% K (Na) doping where we found LT.

In Fig.\ref{KFS} (g\---i), we display FSs of BaFe$_{2-x}$Co$_x$As$_2$ (electron doped Fe based SC) in the tetragonal phase. It is quite evident from the right column figures \ref{KFS} (g\---i) that one of the inner hole like FSs around $\Gamma$ point is modified remarkably due to Co doping. For $x=0.228$ ({\it i.e}, around 11 $\%$ of Co doping) in BaFe$_{2-x}$Co$_x$As$_2$ system, the FS around $\Gamma$ point vanishes which corresponds to a topological change in the FS, bears the signature of LT. These findings are quite consistent with experimental observation \cite{LiuA}. In Fig.\ref{BS-e-h-iso} (e\---f), we show electronic BS of BaFe$_{2-x}$Co$_x$As$_2$. The hole like (d$_{xy}$) band that lies well above FL for $x=0.026$ (Fig. \ref{BS-e-h-iso}e) moves below the FL with higher Co doping $x=0.228$ (Fig. \ref{BS-e-h-iso}f) causing doping induced topological change in the FS. Note BaFe$_{2-x}$Co$_x$As$_2$ SCs have maximum T$_c$ at around 11$ \%$ doping where LT is found from our calculation. Therefore, both the hole doped as well as electron doped systems exhibit LT where SC attains maximum T$_c$ and magnetism vanishes in Fe-based SCs. It  should also be noted here that on the contrary to hole
doped Ba122 systems, LT in the electron doped Fe-based SCs occurs in the hole FS around $\Gamma$ point.
Our findings are very much consistent with the previous experimental findings \cite{Liu}.

We study the FSs and BSs of BaFe$_2$(As$_{1-x}$P$_x$)$_2$ for various isovalent P dopings in two different
structural phases at different temperatures (low temperature orthorhombic phase and high temperature tetragonal phase).
It is quite evident from the right column of Figs.\ref{KFS} (j\---l) that the topology of one of the hole like FSs is getting modified with increasing P doping. Our calculated orbital resolved BSs shown in Figs.\ref{BS-e-h-iso} (g\---l) distinctly reflect signatures of LT as a result of larger P doping. One of the bands, namely the d$_{xy}$ band around $\Gamma$ point, well above the FL moves below the FL.
We compare the signatures of LT/ETT between two structural phases through orbitally resolved BSs of BaFe$_2$(As$_{1-x}$P$_x$)$_2$ in the low temperature orthorhombic phase (Fig.\ref{BS-e-h-iso}(g,h,i)) and the high temperature tetragonal phase (Fig.\ref{BS-e-h-iso} (j, k, l)) for various $x$. This part of the work thus demonstrates combined doping and temperature induced LT.
In both the phases (10K and 300K), d$_{xy}$ band around $\Gamma$ point goes below FL at $x=0.6$. But a closer look at the BSs of BaFe$_2$(As$_{1-x}$P$_x$)$_2$
with $x=0.37$ at 10K and at 300K, reveal
that there are important differences in the BSs. In case of BS of orthorhombic phase (10K) (which is in SC phase), the d$_{xy}$
band is about to cross the FL (Lifshitz point) but on the other hand in case of non-SC
tetragonal phase (300K) d$_{xy}$ lies above
the FL (FSAs in the two cases are different, see below). This provides an indication of the influence of temperature and structural phase, on ETT/LT. {\em Note} in this case also LT occurs at a doping where SC is maximum and no magnetism.
From Figs.\ref{BS-e-h-iso}(g\---l) one can see that the d$_{xz}$ and d$_{yz}$ bands are hardly affected by P doping at As sites, but d$_{xy}$ bands are very sensitive to doping concentration. One of the possible reasons of this fact may arise from the structural modification caused by P doping on As site (Fe-As hybridization modifies anion height characterized by z$_{As}$). This fact is adequately demonstrated in Fig.\ref{RuBS}.

In case of BaFe$_{2-x}$Ru$_x$As$_2$ also, we find LT (not shown here for brevity) at relatively higher Ru doping
concentration (\textgreater 50\%) where
SC T$_c$ reaches its maximum.
This is consistent with the findings of earlier theoretical work on Sr122 system \cite{Wang}. Below we present the temperature  induced LT in  5\% Ru doped BaFe$_2$As$_2$.
It is well established that the electronic structure of Fe-based SCs is highly sensitive
to temperature \cite{Sen,Dhaka,Acta,sust}. It should also be noted that the structural parameters which essentially control the electronic structures of these systems, have significant temperature dependencies \cite{Acta,Avci,Avci2,Allred}. We found temperature dependent LT in 5\% Ru doped Ba122
system (BaFe$_{2-x}$Ru$_x$As$_2$ with $x=0.1$)  ({\it cf} Fig.\ref{RuFS}) where inner
hole FS at 90 K nearly vanishes.
In Fig.\ref{RuBS}, we depict the BSs of BaFe$_{2-x}$Ru$_x$As$_2$  ($x=0.1$) around $\Gamma$ point
which clearly show that the d$_{xy}$ band near FL goes below FL as temperature is lowered from 120K to 90K. Temperature induced LT is not very common in Fe-based superconductors.
To find the root cause of temperature dependent LT, we carry out further calculations on undoped
Ba122 \cite{Rotter} with various z$_{As}$ around the experimental value.

Essentially we find from Figs.\ref{RuFS} (d\---f),\ref{RuBS} (d\---f) that
electronic structures very near to the FL are heavily influenced by z$_{As}$ and it reproduces the overall essential features of temperature induced LT in Ru as well as P doped Ba122 SCs.

 FSAs as a function of various kinds of doping shown in Fig.\ref{FSA} can be experimentally measured and turns out to be
 a unique way of identifying ETT/LT. Normally, hole doping would enlarge (reduce) hole (electron) FSA in contrast to electron doping
  that would enlarge (reduce) electron (hole) FSA. But in case of any particular FS undergoing TT
  it would follow against the above and such signatures are remarkably visible in  Figs.\ref{FSA}(d),(e),(g),(h),(k)).

  The red curves in Fig.\ref{FSA} (d,g) suggest electron-FSA initially gets reduced with hole doping but increases after certain hole doping because that FS undergoes topological modification from electron like to hole like.
  This has caused sudden increase in the FSAs of hole-FSs see {\it e.g}, green, blue, pink curves of Fig.\ref{FSA} (d,e) around the LT. Thus FSA are found to carry sensitivity of topological modifications more acutely than the BSs and can be used as a better experimental tool to identify ETT/LT. The difference in FSAs of hole and electron FSs indicate net electronic charge density which increases with doping ({\it cf} red curves in Fig.\ref{FSA} (c,f,i,l)) as expected, but the slope of the curves are different for different materials. This is consistent with experiments \cite{Ideta}. The doping induced net charge density vary differently below and above the LT, much slowly above the LT indicating less effective doping above LT ({\it cf} Fig.\ref{FSA} (i,f,l))); absence of a particular FS due to LT further reduces density of states at FL and thereby high T$_c$ would be limited \cite{setti}.

 We have systematically shown in a variety of high T$_c$ Fe-based SCs that SC occurs
 at the verge of LT/ETT where magnetism disappears. This is achieved through detailed evaluation of FSs through first principles simulations with experimental structural parameters as a function of doping (temperature) as inputs and detailed demonstrations of LT/ETT. The doping induced net charge density is found to be suppressed at the LT/ETT. This definitely indicates the intriguing heed to
  the inter-relationship between SC and LT in Fe-based SCs. We have also provided a new way of detecting LT/ETT, through evaluation of FSA as a function of doping or temperature as the case may be which can be experimentally performed and is applicable in general to any system. This may modify the experimental phase diagram of Fe-SCs as far as the location of ETT/LT are concerned. We believe our work will open up many theoretical as well as experimental research activities in this direction.

  {\bf Acknowledgements} We thank Dr. P. A. Naik and Dr. P. D. Gupta for their encouragement in this work. One of us (SS) acknowledges the HBNI, RRCAT for financial support and encouragements.


text:
\section{Model system}\label{sec:JW}

In this work, we map a half-filled chain of spinless fermions interacting through nearest neighbor Coulomb repulsion onto a corresponsing spin Hamiltonian. The initial model Hamiltonian is given by:
\begin{equation}
\label{eq:Hamiltonian_fermionic}
    \hat{\mathcal{H}}(t)=-\frac{J}{2}\sum_j
	(\hat{c}_{j}^\dagger \hat{c}_{j+1}+\mathrm{H.c.})+V(t)\sum_j\tilde{n}_{j}\tilde{n}_{j+1},
\end{equation}
where $\hat{c}_{j}^\dagger$ ($\hat{c}_{j}$) is a fermionic creation (annihilation) operator at site $j$, $\tilde{n}_{j}=\hat{c}_{j}^{\dagger}\hat{c}_{j}-1/2$ is the number operator relative to half filling, $J$ is a constant hopping amplitude, and $V(t)$ is a time-dependent, nearest-neighbor Coulomb interaction.
Through the Jordan-Wigner transformation \cite{Giamarchi2004quantum}, this charge Hamiltonian maps onto an equivalent spin-1/2 anisotropic Heisenberg (XXZ) chain (Fig.~1a in the main text):
\begin{equation}\label{eq:Hamiltonian_Spin}
	\hat{\mathcal{H}}_{\mathrm{sp}}(t)
	=\sum_j\left[-\frac{J}{2}\left(\hat{S}_{j}^{+}  \hat{S}_{j+1}^{-}+\mathrm{H.c.}\right) + V(t)\hat{S}_{j}^{z}\hat{S}_{j+1}^{z}\right],
\end{equation}
where $\hat{S}_{j}^{\pm}= \frac{1}{2} \left(\hat{S}_{j}^x\pm \hat{S}_{j}^y\right)$ and $\hat{S}_{j}^\alpha$ ($\alpha={x,y,z})$ are the usual spin operators defined in terms of the Pauli matrices $\hat{S}^\alpha = \tfrac{1}{2}\hat{\sigma}_\alpha$. In this picture, the hopping amplitude $J$ becomes the exchange coupling while $\Delta(t) =  V(t) / J$ quantifies the anisotropy of the spin interactions.
\par At equilibrium, this dual quantum chain exhibits well-known quantum phase transitions. Upon increasing $\Delta=V/J$, the fermionic chain evolves from a gapless Luttinger liquid (LL) phase with short-range correlations to a charge density wave (CDW) phase with long-range correlations. The XXZ chain instead undergoes two separate transitions into an Ising ferromagnet ($\Delta<-1$) and antiferromagnet ($\Delta>1$), while for ($|\Delta|<1$) it exhibits an XY phase \cite{Giamarchi2004quantum}.
Since the dual charge and spin formulations are one-to-one equivalent, we choose to study the time-dependent dynamics of the quantum spin chain using exact diagonalization (ED). The ED calculations for finite-size chains extending up to $L=24$ sites are performed using the Qu\-Spin \cite{weinberg2017quspin,weinberg2019quspin} as well as the HPhi~\cite{kawamura2017} packages.

The antiferromagnetic XXZ ground state in a finite-size system contains a mixture of nearly-degenerate states (notably $\ket{\uparrow\downarrow\ldots\uparrow\downarrow}\pm\ket{\downarrow\uparrow\ldots\downarrow\uparrow}$). In order to break this near degeneracy, which turns into complete degeneracy in the thermodynamic limit, we also introduce a small staggered magnetic field $\hat{\mathcal{H}}_{\mathrm{ext}}=\sum_j(-1)^jh_z\hat{S}_{j}^{z}$ to select a specific spin configuration (cf. next section). This ensures that the disconnected part of the spin correlation function in Eq.~(2) in the main text vanishes in the limit $\Delta\rightarrow\infty$. Our spin sector quench dynamics is then benchmarked for selected conditions against real-time density matrix renormalization group (DMRG) calculations \cite{Kennes_2018} of the time evolution of an infinite chain.

\section{Finite-size effects on the Quantum Fisher Information calculations}\label{sec:Finite-size}
In the antiferromagnetic (AFM) Ising limit $\Delta\rightarrow\infty$, the ground state of the XXZ chain in the spin sector is degenerate in the thermodynamic limit ($L\rightarrow\infty$) and is given by the two possible N\'{e}el states $\ket{\uparrow\downarrow\uparrow\downarrow\ldots}$ and $\ket{\downarrow\uparrow\downarrow\uparrow\ldots}$, related to each other via a site translation. At finite $L$, the two lowest-energy spin configurations $\ket{\uparrow\downarrow\uparrow\downarrow\ldots}\pm\ket{\downarrow\uparrow\downarrow\uparrow\ldots}$ are split by a finite energy gap that decreases as $\propto e^{-\alpha L}$ with increasing system size (with $\alpha>0$). For the AFM ground state at finite $L$, the disconnected part of the spin correlation function in Eq.~(2) of the main text, i.e.\ $\sum_l(-1)^l\left\langle\hat{S}^z_l\right\rangle\left\langle\hat{S}^z_{l+1}\right\rangle$, vanishes, leading to an asymptotically finite value of the QFI $F_Q$. In order to reinstate the proper behaviour of the QFI certified by the DMRG calculations in the charge sector, we introduce a staggered degeneracy-breaking magnetic field term $\hat{\mathcal{H}}_{\mathrm{ext}}=\sum_j(-1)^jh_z\hat{S}_{j}^{z}$ to select a specific N\'{e}el component of the ground state and restore the proper limit of $F_Q$. Consequently, the position of the maximum of the equilibrium QFI density shown in Fig.~1b of the main text thus becomes dependent on the magnitude $h_z$ of this additional term. In addition, both the magnitude of the QFI maximum as well as its position are dependent on the chain length $L$. In Fig.~\ref{fig:Finite_Size}, we plot the equilibrium QFI density in the spin sector as a function of the staggered magnetic field for a fixed chain length (panel~a) and as a function of the system size for a fixed $h_z$.

\begin{figure}[h]
	\centering
	\includegraphics[scale = 1]{fig_Finite_Size_V.pdf}
	\caption{(a) Equilibrium QFI density $f_Q$ for a spin chain with $L=10$ for  different values of the staggered magnetic field $h_z$. (b) QFI density as a function of the chain length for a fixed $h_z =1.414$.}\label{fig:Finite_Size}
\end{figure}

\section{Time-frequency analysis of the non-equilibrium QFI}\label{sec:Time_Frequency}
In Fig.~2 of the main text and the accompanying discussion, we outline the main features of the nonequilibrium QFI dynamics. Upon crossing the critical point $\Delta=1$, the driven system passes through a more  entangled state compared to the initial condtion. Depending on the ramp velocity, the QFI density $f_Q(t) = F_Q(t)/L$ either maps onto the equilibrium phase diagram, thereby decaying asymptotically to zero as $t\rightarrow\infty$, or exhibits a rich oscillatory behaviour when the driving protocol reaches the impulsive limit at sufficiently high velocities. In this section, we focus on origin of the oscillatory dynamics, while the transition between adiabatic and impulsive regimes is discussed in Sec.~\ref{sec:Limits}.\\

\par  We start by focusing our attention on the 1D linecuts of $f_Q(t)$ plotted in Fig.~2c of the main text. In panel~a of Fig.~\ref{fig:Time_Frequency}, we present three of these curves as a function of physical time instead of the time-dependent nearest-neighbour Coulomb repulsion $V(t)$. Both representations are related through the transformation $t=V(t)/v$. The time-domain representation allows us to better appreciate characteristic dynamical features of the non-adiabatic regime (\textcolor{black}{$v>3.5$}). First, all curves in Fig.~\ref{fig:Time_Frequency}a reveal a slow oscillation at $\nu_\mathrm{osc}\sim\textcolor{black}{0.36}$, independent of the driving speed. Furthermore, all curves also contain very fast oscillations, evident at early times and featuring a pronounced chirp. In order to isolate the different frequency contributions and characterize the change of the non-stationary contributions as a function of time, we perform a short-time Fourier transform (STFT) analysis. In the resulting spectrogram, a stationary frequency manifests itself as a horizontal line, whereas a linear change of the frequency over time corresponds to a linear chirp. \\

\par In panels b-d of Fig.~\ref{fig:Time_Frequency}, we plot the spectrograms corresponding to the three  velocities $v=2.0$, $v=10.0$, and $v=40.0$. The $v$-independent oscillations correspond to the low-frequency mode at $\nu_\mathrm{osc} \sim \textcolor{black}{2\pi\Delta_{01} \approx 0.36}$. The fast oscillations show a linear chirp, more clearly pronounced at faster ramp velocities, with a slope given by $\textcolor{black}{\overline{b}\sim \tfrac{v}{2\pi}}$.\\

\begin{figure}[h]
	\centering
	\includegraphics[scale = 1]{fig_Spectrograms_V.pdf}
	\caption{ (a) Linecuts from \textcolor{black}{Fig.~2c} in the main text, plotted as a function of time. (b) - (d) spectrograms of the time-dependent QFI density for three selected velocities: $v=2$ (b), $v=10$ (c), and $v=40$ (d). Note that the color scale is logarithmic.}\label{fig:Time_Frequency}
\end{figure}

\par We now discuss the microscopic origin of these features with the aid of Fig.~\ref{fig:Projections}. By analyzing the energy level diagram of the $L=10$ chain (with $h_z$ fixed at $0.005$), we can attribute the $\nu_\mathrm{osc}\sim \textcolor{black}{0.36}$ oscillation to a transition between the ground and the first excited level. The corresponding energy separation $\Delta_{01}$ is shown as a function of the nearest-neighbour Coulomb repulsion for different chain sized in Fig.~\ref{fig:Projections}a. For $L=10$ and sufficiently large anisotropies ($\Delta>3$), the gap size assumes an asymtoptic value of $\textcolor{black}{\tfrac{0.314}{2\pi}}$, hence, the oscillation frequency in the spectrograms in Fig.~\ref{fig:Time_Frequency} is largely time-independent. \\
\begin{figure}[h]
	\centering
	\includegraphics[scale = 0.75]{fig_Projections_scheme_V.pdf}
	\caption{(a)-(b) Energy separations between the ground state and the first (a) resp.\ the fifth (b) excited states of the chain, plotted as a function of the nearest-neighbour Coulomb repulsion for different chain lengths. A staggered magnetic field value of $h_z=0.005$ has been used. Grey vertical lines in (a) indicate the positions of the critical points. (c) Projection of the time-evolved state ${\psi}_0(t)$ onto selected eigenstates of the \textit{time-dependent} Hamiltonian at the end of the time-evolution $\hat{\mathcal{H}}(t=t_\mathrm{max})$. The dark-purple curve shows the projection of ${\psi}_0$ onto the initial state at $t_0$. Panel (d) Time-dependence of selected matrix elements $b_{ij}$, given by products of overlaps of the time-propagated ${\psi}_0(t)$ with the eigenstates of the \textit{instantaneous} Hamiltonian  $\hat{\mathcal{H}}(t)$ at time $t$. Panel (e) energy diagram showing the relevant energy levels dominating the dynamics. Some of the energy levels are degenerate.}\label{fig:Projections}
\end{figure}

The second dominant feature, the linearly-chirped rapid oscillations,  has its origin in a transition between the ground state and one of the higher-lying excited levels ($j=5$). The corresponding gap $\Delta_{05}$ increases linearly with $V$ (see Fig.~\ref{fig:Projections}b), its slope for $L=10$ matches well the slope of the corresponding feature in the short-time Fourier transform. The relationship between the two is $\textcolor{black}{\tfrac{\overline{b}}{v}\sim \tfrac{1}{2\pi}\tfrac{\mathrm{d}\Delta_{05}}{\mathrm{d} V}}$. The excitation to this (and others) specific state(s) occurs while driving the system through the critical point $\Delta=1$. In panel c of Fig.~\ref{fig:Projections}, we plot the projection of the time-evolved initial pure state ${\psi}_0(t)$ (coinciding with the GS of the LL-model at $t_0=0$)  onto the $j^\mathrm{th}$-eigenstate of the time-dependent Hamiltonian $\hat{\mathcal{H}}(t)$ (denoted by $\psi_t^{(j)}$) at the \textit{end of the time evolution} $t_\mathrm{max}$. Apart from the nearly-degenerate low-lying states with $j=1,2$, only the state with $j=5$ has a significant contribution, and beatings with the ${\psi}_0(t)$ dominate the time-frequency structure of the non-equilibrium QFI. The state $\psi_t^{(j=5)}$ is dominated by spin configurations featuring two domain walls (e.g. $\ket{\uparrow\su\downarrow\uparrow\su\downarrow\uparrow\downarrow\uparrow\downarrow}$) for all values of $t$, whereas the time-evolved ${\psi}_0(t)$, especially after crossing $\Delta=1$, becomes progressively dominated by the defect-free two N\`{e}el configurations. The projection of  ${\psi}_0(t)$ onto the initial state at $t_0$, plotted as a dark purple line in Fig.~\ref{fig:Projections}c, exhibits an interesting dynamics consisting of a decay followed by multiple revivals. For completeness, in panel d of the same figure we show the temporal evolution of several matrix elements $b_{ij}=\left\langle \psi_t^{(i)} \right. \left| {\psi}_0(t)\right\rangle \left\langle  {\psi}_0(t) \right. \left|\psi_t^{(j)}\right\rangle$, i.e. the projection of ${\psi}_0(t)$ over the instantaneous eigenstate at each $t$. These results imply that the non-equilibrium QFI in the impulsive regime is determined by the non-adiabatic excitation of multiple states at the critical point and the associated increased domain wall density.

\section{Lindblad master equation}\label{sec:Lindblad}

In Fig.~\ref{fig:Decoherence_Rates}, we present calculation results showcasing the influence of the decoherence rate on the  QFI during non-unitary time evolution ($L=10$, $h_z = 0.005$). The main features are preserved for coupling strengths up to $\gamma\sim 0.05$, and become progressively obliterated as $\gamma\geq 0.1$. The dissipative Lindblad jump operator acting on each site $l$ is given by $\hat{\mathcal{L}}^{(l)}\equiv \hat{\mathcal{L}}_{z} = \hat{\sigma}^z_l$. \\
\begin{figure}[h]
	\centering
	\includegraphics[scale = 0.85]{fig_DecoherenceRate_V.pdf}
	\caption{(a)-(c) Density matrix purity as a function of effective nearest-neighbour Coulomb repulsion and ramp speed. The decoherence rate is set to $\gamma=0.005$ in panels (a)/(d), to $\gamma=0.05$ in (b)/(e), and to $\gamma =0.1$ in (c)/(f).
	(d)-(f) Non-equilibrium QFI in the presence of decoherence for three different decoherence rates $\gamma$, displayed as a function of the effective nearest-neighbour Coulomb repulsion and ramp speed. The colormap is normalized with respect to the unitary QFI density, shown in Fig.~\textcolor{black}{2a} of the main text.   }\label{fig:Decoherence_Rates}
\end{figure}
\par The QFI in the presence of decoherence is calculated by first propagating the density matrix $\hat{\rho}(t)$ obeying the initial condition:
\begin{equation}
    \hat{\rho}(t_0) = \ket{{\psi}_0(t_0)}\bra{{\psi}_0(t_0)}. \label{eq:rhoDM_t0}
\end{equation}
The QFI is obtained as~\cite{Hauke2016measuring}:
\begin{equation}
    \mathcal{F}_Q(t) = 2 \sum_{i,j}\frac{\left(\epsilon^i_t-\epsilon^j_t\right)^2}{\epsilon^i_t+\epsilon^j_t}
    \left|\bra{\varphi^i_t}\hat{\mathcal{O}}_\pi\ket{\varphi^j_t}\right|^2\label{eq:QFI_DM},
\end{equation}
where $\epsilon^i_t$ and $\ket{\varphi^i_t}$ are the eigenvalues and the eigenvectors of $\hat{\rho}$ at time $t$.

\section{Kibble-Zurek analysis of the nonequilibrium QFI}\label{sec:Limits}
\begin{figure}[h]
	\centering
	\includegraphics[scale = 0.85]{fig_Limits_labeled_V.pdf}
	\caption{(a)-(c) Positions of the maxima of the time-dependent QFI density $f_Q$ as a function of the ramp velocity, displayed as a function of the effective nearest-neighbour Coulomb repulsion [panels (a) and (b)] or time (c). Points falling into the adiabatic and (partially) to the ``intermediate'' regime are shown in panel (a). Panels (b) and (c) show the ``intermediate'' and the non-adiabatic regions on a double-logarithmic scale. The onset of the non-adiabatic regime is manifested by a marked change in the slope of the curves (see text). The symbols indicate the positions of the three different ramp velocities corresponding to the data in the bottom row. (d)-(f) Time-dependent matrix elements $b_{ij}$, given by products of overlaps of the time-propagated ${\psi}_0(t)$ with the eigenstates of the \textit{instantaneous} Hamiltonian  $\hat{\mathcal{H}}(t)$ at time $t$, shown for three different velocities (see legends). }\label{fig:Limits}
\end{figure}

The nonequilibrium QFI dynamics (see. Fig.~\textcolor{black}{2a} of the main text) of the spinless fermion chain subject to a linear ramp can be broadly divided into three regions depending on the ramp velocity. These regimes can be delineated by tracking the the time ($t^\ast$) or the effective  nearest-neighbour Coulomb interaction ($V(t^\ast)$) which maximize the  time-dependent $f_Q$ at a given ramp speed $\mathit{v}$. The resulting curve (s. white line in Fig.~\textcolor{black}{2a} or Fig.~\ref{fig:Limits}a-c) effectively tracks the position of the main QFI ``crest'' as it propagates towards larger $V(t^\ast)$ (smaller $t^\ast$) on increasing the ramp speed. The positions of the maxima are extracted by fitting a skewed Voigt distribution to the time profile of $f_Q$ at each velocity in a region centered around the main peak.   \\

\par The two limiting cases are defined by the adiabatic ($v<0.025$, cf.\ Fig.~\ref{fig:Limits}a) and the nonadiabatic ($v>3.5$, cf.\ Fig.~\ref{fig:Limits}b-c) limits, separated by an intermediate region. Whereas the onset of the non-adiabatic region is rather sudden and well-marked by a ``kink'' in the $\mathit{v}-V(t^\ast)$ resp. $\mathit{v}-t^\ast$ plots in Fig.~\ref{fig:Limits}b-c, the separation between adiabatic and intermediate regions is rather fluid.  In the deep adiabatic regime, the position of the $f_Q$-maximum is weakly dependent on the driving speed, and the driven system essentially traces out the equilibrium phase diagram. This regime (illustrated for $v=0.001$ in Fig.~\ref{fig:Limits}d) holds as long as excitations to the first excited state on crossing the critical point $\Delta=1$ can be neglected, i.e. $b_{ij}\approx \delta_{0,i}\delta_{0,j}$ for the entire time evolution. Examining the time-resolved data underlying Fig.~\textcolor{black}{2a}, we estimate a value of $v^{\ast,1}\sim0.025$ for the maximum ramp speed that warrants purely adiabatic evolution before oscillations due to excitations to the first excited state develop.
Invoking simple Landau-Zener (LZ) arguments, we can estimate the corresponding excitation probability to the first excited state as $P_{01}=1-e^{-\frac{\pi\tilde{\delta}\Delta_{01}^2}{2v}}$, where $\tilde{\delta}$ is a constant related to the rate of change of the gap at $\Delta=1$ as a function of the critical parameter $V(t)$. The threshold for the breakdown of adiabaticity in the LZ picture ($\Delta_{01}^2\approx v|\mathrm{d}\Delta_{01}/\mathrm{d}V|$), however,  largely overestimates $v^{\ast,1}$, indicating a more complex behaviour.  \\

\par For ramp velocities above $v^{\ast,1}$ but not exceeding $v^{\ast,2}$, the dynamics falls into the intermediate regime, characterized by an extensive scaling of the QFI density $f_Q\sim L$ in the unitary evolution case. Due to the rapidly oscillating and saturating behaviour of $f_Q$, the corresponding maxima could not be extracted for velocities ranging from $0.175$ up to shortly before the onset of the non-adiabatic region $v^{\ast,2}\sim 3.5$ (cf. square/diamond symbols in Fig.~\ref{fig:Limits}b-c). Here, the dynamics are mainly governed by the ground and the first excited states of the system, as one can deduce from the dominant $b_{ij}$-coefficients in Fig.~\ref{fig:Limits}e. Recalling that in the presence of the staggered magnetic field of amplitude $h_z$ the ground state at $t_0$ is dominated by one of the two N\'e{e}l configurations, creating excitations across the gap mixes the  wavefunction with the state where the other  N\'e{e}l is dominant. The resulting superposition contains these two components with a similar amplitude, hence the growth of the entanglement. \\

\par The non-adiabatic regime has a sharply delineated onset, marked by a rapid change of the slopes of $\Delta v / \Delta V(t^\ast)$ resp.\ $\Delta v / \Delta t^\ast$. The nonadiabatic limit commences when the probability of an excitation to the higher-lying states (in particular $j=5$ for the $L=10$ chain) exceeds the corresponding excitation fraction to the first excited state $j=1$ (cf. Fig.~\ref{fig:Limits}f), leading to proliferation of defects. In this regime, the maximum of the QFI density follows a regular behaviour, and its functional dependence can be extracted by fitting the corresponding slopes in Fig.~\ref{fig:Limits}b-c: $V(t^\ast)\propto \sqrt{v}$ resp.\ $t^\ast\propto v^{-1/2}$. The exact values of the slopes are \textcolor{black}{$0.5295\pm0.0009$} and \textcolor{black}{$-0.5937\pm0.0019$}, respectively. With the aid of the critical exponents of the system at $\Delta=1$, we can relate the scaling of the entanglement jet to the corresponding scaling of the density of defects in a Kibble-Zurek type of analysis:
    \begin{equation}
        d_\mathrm{KZ}\propto \frac{a_0}{\xi_0}\left(\tau_0 v\right)^{1/2}\label{eq:dKZ},
    \end{equation}
where $a_0$, $\xi_0$, $\tau_0$ are the characteristic length scale, correlation length, and relaxation rate of the system. Note that a KZ-analysis has a limited validity at the phase transition at the $\Delta=1$ critical point is of BKT type instead of a second-order one.

\section{Nonequilibrium QFI in presence of next-nearest-neighbor interactions}\label{sec:Next_nearest_neighbor}
Finally, we corroborate the stability (or universality) of the nonequilibrium QFI features by also considering the role of additional interactions, that break integrability in the infinite system~\cite{patrick2019}. We specifically study the dynamical QFI for the spinless fermion chain at half-filling (Eq.~\eqref{eq:Hamiltonian_fermionic}) with the inclusion of next-nearest-neighbor interactions of the form $\hat{\mathcal{H}}_{(2)} = V^{(2)}\sum_l \tilde{n}_l\tilde{n}_{l+2}$. The results presented in Fig.~\ref{fig:Second_Neighbours}c for a second-order coupling of $V^{(2)}=2$ illustrate that the main traits of the nonequilibrium QFI dynamics are preserved. Importantly, this observation underscores the generic character and robustness of our main findings and suggests that they should be observable across many different experimental platforms, irrespective of microscopic details.

\begin{figure}[h]
	\centering
	\includegraphics[scale = 0.85]{fig_integ_2D_map_1D_cuts_2pack_broken_V.pdf}
	\caption{(a) Non-equilibrium QFI for a Heisenberg chain featuring additional next-nearest neighbour interaction term $\hat{\mathcal{H}}_{(2)}$ as a function of the nearest-neighbour Coulomb repulsion and the ramp speed. (b) Selected 1D line cuts for several ramp velocities.}\label{fig:Second_Neighbours}
\end{figure}

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}
Let $\mathbb{F}_q$ be a finite field of order $q$, where $q$ is an odd prime power. For  $x=(x_1, \ldots, x_d)$ and $(y_1, \ldots, y_d)$ in $E$, define a distance between $x$ and $y$  by
\[||x-y||:=(x_1-y_1)^2+\cdots+(x_d-y_d)^2,\]
which is the square of the Euclidean distance. We denote by $\Delta(E)$ the set of distances determined by pairs of points in $E$, namely,
\[\Delta(E):=\{||x-y||\colon x, y\in E\}.\]
The Erd\H{o}s-Falconer distance problem in $\mathbb{F}_q^d$ asks for the smallest exponent $N$ such that for any $E\subset \mathbb{F}_q^d$ with at least $q^N$ elements, the number of distinct distances determined by pairs of points in $E$ is at least $cq$, for some constant $0<c<1$.

Iosevich and Rudnev \cite{io} showed that if $|E|\ge 4q^{(d+1)/2}$, then $|\Delta(E)|=q$, which means that for any $\lambda\in \mathbb{F}_q$, there exist two points $x, y\in E$ such that $||x-y||=\lambda$.  Hart et al. \cite{hart} proved that the exponent $\frac{d+1}{2}$ is essentially sharp in odd dimensions, even though we wish to cover a positive proportion of all distances. However, in even dimensions, it is conjectured that the right exponent should be $\frac{d}{2}$. We refer the interested reader to \cite{mot, mu} and references therein for most recent progress on this conjecture.

Let $E$ and $F$ be sets in $\mathbb{F}_q^d$, in this paper, we focus on the following analogue: How large do $E$ and $F$ need to be
to guarantee the inequality
\[|\Delta(E)+\Delta(F)|\gg q?\]

Here, and throughout this paper, for simplicity, we will use  $C$ to denote  a sufficiently large constant independent of the field size $ q.$ We also use the following notations: $X \ll Y$ means that there exists  some absolute constant $C_1>0$ such that $X \leq C_1Y$, the notation $X \gtrsim Y$ means $X\gg (\log Y)^{-C_2} Y$ for some absolute constant $C_2>0$, and  $X\sim Y$ means $Y\ll X\ll Y$.

One can easily check that $\Delta(E)+\Delta(F)=\Delta(E\times F)$, where $\Delta(E\times F)$ is obviously defined as the distance set determined by the product set $E\times F$ in $\mathbb F_q^{2d}.$ Thus, it follows trivially from Iosevich-Rudnev's result that if $|E||F|\ge4 q^{\frac{2d+1}{2}}$, then $\Delta(E)+\Delta(F)=\mathbb{F}_q$. In a recent paper, by using results on additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh \cite{kohmz} showed that the exponent $d+\frac{1}{2}$ can be improved to $d+\frac{1}{3}$ as follows.

\begin{theorem}[\cite{kohmz}]\label{thm1mz}Let $E$ and  $F$ be sets in $\mathbb{F}_q^d$. Suppose either $d=4k-2$ and $q\equiv 3\mod 4$ or $d\ge 3$ is odd. If $|E||F|\ge Cq^{d+\frac{1}{3}}$ for some big positive constant $C$, then
$|\Delta(E)+\Delta(F)|> \frac{q}{2}.$
\end{theorem}

They also constructed examples to show that the exponent $d+\frac{1}{3}$ is optimal for the case $|E|=|F|$ in odd dimensions. Using the same approach, they indicated that the Erd\H{o}s-Falconer distance conjecture holds for sets of the form $A^4\subset \mathbb{F}_q^4$, where $q$ is a prime number and $A$ is a multiplicative subgroup in $\mathbb{F}_q^*.$

We note that in the form of $\Delta(E, F)+\Delta(E, F)$, where $\Delta(E, F):=\{||x-y||\colon x\in E, y\in F\}$, it was first proved by  Shparlinski \cite[Corollary 2]{shparlinski} that
\begin{equation} \label{Shparlinski} |\Delta(E, F)+\Delta(E, F)|\ge \frac{1}{3}\min\left\lbrace q, \frac{|E||F|}{q^{d-1}}, \frac{|E||F|^2}{q^{\frac{3d}{2}}}\right\rbrace,\end{equation}
which is non-trivial when the sizes of sets $E$ and $F$ differ significantly. A  simple  graph theoretic proof of this result and applications can be found in the work of Hegyv\'{a}ri and  P\'{a}lfy in \cite{HP}.

From the Iosevich-Rudnev result \cite{io}, we observe   that if the size of $E$ (resp. $F$) is at least $4q^{\frac{d+1}{2}}$, then $|\Delta(E)|= q$ (resp. $|\Delta(F)|= q$),
 and so we have $|\Delta(E)+\Delta(F)|= q$. Hence, in the rest of this paper, without loss of generality, we assume that $|E|, |F|< 4q^{\frac{d+1}{2}}$.

The main purpose of this paper is to give improvements of Theorem \ref{thm1mz} in the unbalanced case, namely, $|E|\ne |F|$. The most important tool in our proofs  is  an optimal $L^2$ restriction theorem for the sphere of zero radius. More precisely, our first result is as follows.

\begin{theorem}\label{thm0}
Let $E$ and $F$ be sets in $\mathbb{F}_q^d$.
Suppose either $d=4k-2$ and $q\equiv 3\mod 4$ or $d\ge 3$ is odd. Then there exists a large positive constant $C$ such that  if either $|E||F|^2\ge Cq^{\frac{3d+1}{2}}$  or $|E|^2|F|\ge Cq^{\frac{3d+1}{2}}$, then we have
\[|\Delta(E)+\Delta(F)|> \frac{q}{2}.\]
\end{theorem}

To see how much Theorem \ref{thm0} is better than Theorem \ref{thm1mz}, let us make a brief comparison.

If $|E||F|\ge  Cq^{d+\frac{1}{3}}$, then either $|E||F|^2\ge Cq^{\frac{3d+1}{2}}$ or $|E|^2|F|\ge Cq^{\frac{3d+1}{2}}$. Indeed, otherwise, one has $|E||F|< C^{\frac{2}{3}}q^{d+\frac{1}{3}}$, a contradiction. When $|E|=|F|$, both Theorems \ref{thm1mz} and \ref{thm0} give the same exponent $\frac{d}{2}+\frac{1}{6}$. For $0<\epsilon<\frac{1}{3}$, one can check that for $E, F\subset \mathbb{F}_q^d$ with $|E|=q^{\frac{d-1}{2}+2\epsilon}$ and $|F|=q^{\frac{d+1}{2}-\epsilon}$, we have $|E||F|=q^{d+\epsilon}$. For these sets, Theorem \ref{thm1mz} does not tell us about the size of $\Delta(E)+\Delta(F)$, but Theorem \ref{thm0} gives the expected lower bound $cq$.

Theorem \ref{thm0} is essentially sharp in odd dimensions. To show this, let us recall the following result from \cite{hart}. If either $d\ge 2$ is even and $q\equiv 1\mod 4$ or $d=4k$, $k\in \mathbb{N}$, then there exist $\frac{d}{2}$ independent vectors $\{v_1, \ldots, v_{\frac{d}{2}}\}$ in $\mathbb{F}_q^d$ such that $v_i\cdot v_j=0$ for all $1\le i, j\le \frac{d}{2}$. Thus, if either $d=4k+3$ and $q\equiv 1\mod 4$ or $d=4k+1$, $k\in \mathbb{N}$, then we always can choose a subspace $V$ of $\frac{d-1}{2}$ vectors in $\mathbb{F}_q^{d-1}\times \{0\}$ such that $u\cdot v=0$ for all $u, v\in V$. Set $E=V$. It is not hard to check that $\Delta(E)=\{0\}$.
It has also been indicated in \cite{hart} that for any $\epsilon>0$, there exists a set $F\subset \mathbb{F}_q^d$ such that $|F|\sim q^{\frac{d+1}{2}-\epsilon}$ and $|\Delta(F)|\sim q^{1-\epsilon}$. In other words, we have $|E||F|^2\sim q^{\frac{3d+1}{2}-2\epsilon}$ and $|\Delta(E)+\Delta(F)|\sim q^{1-\epsilon}$ for any $\epsilon>0$.

It is worth noting that one can not hope to prove Theorem \ref{thm0} in all even dimensions with $q\equiv 1\mod 4$, namely, in Section \ref{prof}, we will construct examples which tell us that there are sets $E,F\subset \mathbb{F}_q^d$ with $d$ even and $q\equiv 1\mod 4$ such that $|E||F|^2\sim q^{\frac{3d}{2}+\frac{2}{3}}$ and $|\Delta(E)+\Delta(F)|\le q/2$.

While Theorem \ref{thm0} is sharp in odd dimensions, we believe that in the corresponding even dimensions the right condition should be $|E||F|^2\ge C q^\frac{3d}{2}$ or $|E|^2|F|\ge C q^\frac{3d}{2}$, which is in line with the Erd\H{o}s-Falconer distance conjecture. The difference between these cases will be seen clearly in our coming proofs.  Let us discuss shortly the main obstacle here. In our method, we will reduce the problem to making a spherical $L^2$ restriction estimate, more precisely, to finding a good upper bound of   $\max_{r\in \mathbb F_q} \sum_{m\in S_r^{d-1}} |\widehat{F}(m)|^2$ for $F\subset \mathbb{F}_q^d$, which might be attained at $r=0$ since the sphere $S_0^{d-1}$ of zero radius has the structure which is similar to that of cones whose Fourier decay is generally not very good (for example, see \cite{kohcone}). More precisely, if we rewrite
\[\sum_{m\in S_0^{d-1}} |\widehat{F}(m)|^2=q^{-d}\sum_{x, y\in F}\widehat{S_0^{d-1}}(x-y)\le q^{-d}|F|^2\cdot\max_{m\ne \mathbf{0}}|\widehat{S_0^{d-1}}(m)|\]
then, it is known that $\max_{m\ne \mathbf{0}}|\widehat{S_0^{d-1}}(m)|\sim q^{-\frac{d+1}{2}}$ for $d\ge 3$ odd, but the maximal value is close to $q^{-\frac{d}{2}}$ for $d\ge 4$ even. Therefore, to overcome this situation, one can think of two ways: either removing the contribution from $S_0^{d-1}$  or finding conditions on $q$ and $d$ in which the corresponding Fourier decay is not worse. In the former case,
by following the proof of Theorem 3.5 in \cite{kohsun} with some slight modifications, we have $|\Delta(E)+\Delta(F)|\ge q/144$ under two conditions $|E|^2|F|\ge Cq^{\frac{3d+1}{2}}$ and $|F|^2|E|\ge Cq^{\frac{3d+1}{2}}$, for some large positive constant $C$. This result holds without any conditions on $q$ and $d$. However, it is clear that this result is weaker than both Theorems \ref{thm1mz} and \ref{thm0}. Hence, it seems that the latter will be our last chance to get some improvements. It has been recently indicated in \cite{IKSPS} that when $d=4k-2$ and $q\equiv 3\mod 4$, the zero radius sphere gives us an optimal $L^2$ restriction theorem, which is much better than that of spheres of non-zero radii. This interesting aspect comes from the fact that in the Fourier decay of $S_0^{d-1}$, the Gauss sum, whose explicit form is known, plays the crucial role. Gluing these observations together gives us Theorem \ref{thm0}.

Restricting our attention to prime fields,  we obtain an improvement of Theorem \ref{thm0} in two dimensions. The key ingredient in our proof comes from the very recent paper on the Erd\H{o}s-Falconer distance conjecture in the plane, which says that for any set $E\subset \mathbb{F}_q^2$ with $|E|\ge q^{5/4}$, one has $|\Delta(E)|\gg q$. We refer the interested reader to \cite{mu} for more details.
\begin{theorem}\label{primecase}
Let $q\equiv 3\mod 4$ be a prime number, $E, F$ be sets in $\mathbb{F}_q^2$, and $C$ be a large positive constant. If either $|E|^4|F|^6\ge Cq^{11}$ or $|E|^6|F|^4\ge Cq^{11},$ then we have \[|\Delta(E)+\Delta(F)|\gg q.\]
\end{theorem}
The following corollary is an immediate consequence of the above theorem.
\begin{corollary}
Let $q\equiv 3\mod 4$ be a prime number, $E$ be a set in $\mathbb{F}_q^2$, and $C$ be a large positive constant. If $|E|\ge Cq^{11/10}$, then we have \[|\Delta(E)+\Delta(E)|\gg q.\]
\end{corollary}

\section{ preliminary lemmas}

In this section, we review some basics on the discrete Fourier analysis, and then use these to  derive  some lemmas  that will be used in proving  Theorem 2.1.

\subsection{Discrete Fourier analysis and Gauss sums}
The Fourier transform is defined by
$$ \widehat{f}(\alpha)=q^{-n} \sum_{\beta\in \mathbb F_q^n} \chi(-\alpha\cdot \beta) f(\beta),$$
where $f$ is a complex valued function on $\mathbb F_q^n.$
Here, and throughout this paper,  $\chi$ denotes the principle additive character of $\mathbb F_q.$  Recall that
the orthogonality of the additive character $\chi$ says that
$$ \sum_{\alpha\in \mathbb F_q^n} \chi(\beta\cdot \alpha)
=\left\{\begin{array}{ll} 0\quad &\mbox{if}\quad \beta\ne (0,\ldots, 0),\\
q^n\quad &\mbox{if}\quad \beta=(0,\ldots,0). \end{array}\right.$$
As a direct application of the orthogonality of $\chi$, the following Plancherel theorem can be proved:
 for any set $\Omega$ in $\mathbb F_q^n,$
$$ \sum_{\alpha\in \mathbb F_q^n} |\widehat{\Omega}(\alpha)|^2= q^{-n}|\Omega|.$$
In this paper,  we identify a set $\Omega$ with the indicator function $1_\Omega$ on $\Omega.$
It is not hard to prove the following formula which is referred as the Fourier inversion theorem
$$f(\beta)=\sum_{\alpha\in \mathbb F_q^n} \chi(\alpha\cdot \beta) \widehat{f}(\alpha).$$

Throughout this paper,  we denote by $\eta$  the quadratic character of $\mathbb F_q.$   For $a\in \mathbb F_q^*,$ the Gauss sum $G_a$ is defined by
$$G_a=\sum_{s\in \mathbb F_q^*}\eta(s) \chi(as).$$
The Gauss sum $G_a$ is also written as
$$ G_a=\sum_{s\in \mathbb F_q} \chi(as^2)=\eta(a) G_1.$$
The absolute value of the Gauss sum $G_a$ is exactly $q.$ Moreover, the explicit
value of the Gauss sum $G_1$ is well--known.
\begin{lemma}[\cite{LN97}, Theorem 5.15]\label{ExplicitGauss}
Let $\mathbb F_q$ be a finite field with $ q= p^{\ell},$ where $p$ is an odd prime and $\ell \in {\mathbb N}.$
Then we have
$$G_1= \left\{\begin{array}{ll}  {(-1)}^{\ell-1} q^{\frac{1}{2}} \quad &\mbox{if} \quad p \equiv 1 \mod 4 \\
{(-1)}^{\ell-1} i^\ell q^{\frac{1}{2}} \quad &\mbox{if} \quad p\equiv 3 \mod 4.\end{array}\right. $$
\end{lemma}
The following corollary  follows from the explicit value of the  Gauss sum $G_1$. For the sake of completeness, we include a proof here.
\begin{corollary} \label{Corm}Let $\eta$ be the quadratic character of $\mathbb F_q^*.$ Then, for any positive integer $n\equiv 2 \mod {4}$ and $q\equiv 3 \mod{4},$ we have
$$ G_1^n= -q^{\frac{n}{2}}.$$
\end{corollary}
\begin{proof} Since $q\equiv 3 \mod{4}$, Lemma \ref{ExplicitGauss} implies that
$ G_1= {(-1)}^{\ell-1} i^\ell q^{\frac{1}{2}}$ for some odd integer $\ell \ge 1.$
Since $n=4k-2$  with $k\in \mathbb N,$ we have
$$ G_1^n=  (-1)^{(\ell-1) (4k-2)}   i^{\ell (4k-2)}  q^{n/2}= (-1)^\ell q^{n/2}= -q^{n/2}.$$ \end{proof}

Completing the square and using a change of variables, it is not hard to show that
\begin{equation}\label{ComSqu}
 \sum_{s\in \mathbb F_q} \chi(as^2+bs)= \eta(a)G_1 \chi\left(\frac{b^2}{-4a}\right),\end{equation}
for any $a\in \mathbb{F}_q^*$ and $b\in \mathbb{F}_q$.

\subsection{ $L^2$ Fourier restriction estimates for spheres}
We recall that  for each $ r\in \mathbb F_q,$  the sphere $S^{d-1}_r$ in $\mathbb F_q^d$ with radius $r$ is defined by
$$ S^{d-1}_r=\{x\in \mathbb F_q^d: \sum_{i=1}^d x_i^2=r\}.$$
Notice that  compared to the Euclidean setting, we have no condition on $r$.

It is well--known that the Fourier transform $\widehat{S^{d-1}_j}(m)$   is closely related to  the Kloosterman sum
$$K(a, b):= \sum_{s\in \mathbb F_q^*}  \chi(as+ b/s),$$
or  the twisted Kloosterman sum
$$TK(a,b):=  \sum_{s\in \mathbb F_q^*} \eta(s) \chi(as+ b/s),$$
where $a, b\in \mathbb F_q,$ and $\eta$ denotes the quadratic character of $\mathbb F_q.$ In particular, the next lemma  was given in \cite[Lemma 4]{IK10}.
\begin{lemma} \label{FTFS}  For $m\in \mathbb F_q^d,$  let $\delta_0(m)=1$  if $m=(0,\ldots, 0)$ and  $0$ otherwise.
\begin{enumerate}
\item  If $d\ge 3$ is an odd integer, then  for $m\in \mathbb F_q^d,$
$$ \widehat{S^{d-1}_j}(m)=  q^{-1} \delta_0(m) + q^{-d-1} \eta(-1)G_1^d TK\left(j, \frac{||m||}{4}\right).$$
\item  If $d\ge 2$ is an even integer, then  for $m\in \mathbb F_q^d,$
$$ \widehat{S^{d-1}_j}(m)=  q^{-1} \delta_0(m) + q^{-d-1} G_1^d  K\left(j, \frac{||m||}{4}\right).$$
\end{enumerate}
\end{lemma}
Recall that   $|K(a,b)|\le 2 q^{1/2}$ if $ab\ne 0,$ and  $|TK(a,b)|\le 2 q^{1/2}$ if $a, b\in \mathbb F_q$ (for example, see \cite{LN97}).

For $F\subset \mathbb F_q^d$  and $j\in \mathbb F_q, $ consider the  $L^2$ Fourier restriction  for the sphere $S_j^{d-1}$
$$ \mathcal{M}_j(F):=\sum_{m\in S_j^{d-1}} |\widehat{F}(m)|^2.$$
It follows from the definition of the Fourier transform that
\begin{equation}\label{MjF}\mathcal{M}_j(F) =  q^{-d} \sum_{x,y\in F} \widehat{S_j^{d-1}}(x-y).\end{equation}
In the next result, we give an upper bound for this quantity.
\begin{proposition}\label{pro2.4} Let $F$ be a subset of $\mathbb F_q^d.$  Suppose that  either $d=4k-2$ and $q\equiv 3\mod 4$ or $d\ge 3$ is odd. Then we have
$$ \max_{j\in \mathbb F_q} \mathcal{M}_j(F) \le q^{-d-1} |F| + 2 q^{\frac{-3d-1}{2}} |F|^2.$$
\end{proposition}
\begin{proof}
{\bf Case $1$:} Suppose that  $d$ is odd.  Combining \eqref{MjF} with  the first part of  Lemma \ref{FTFS},  we get
$$ \mathcal{M}_j(F) =q^{-d-1} |F| + q^{-2d-1}G_1^d \eta(-1) \sum_{x,y\in F}  TK(j, ||x-y||/4).$$
Since the absolute value of the twisted Kloosterman sum is bounded by $2 q^{1/2}$ and $ |G_1|=q^{1/2},$    we have
$$ \mathcal{M}_j(F)  \le  q^{-d-1} |F| + 2 q^{\frac{-3d-1}{2}} |F|^2.$$
Since this bound is independent of $r\in \mathbb F_q,$   we complete the proof.\\
{\bf Case $2$:}  Assume that  $d$ is even  with $ d=4k-2,$  and   $q\equiv 3 \mod{4}.$    By combining \eqref{MjF} with  the second part of  Lemma \ref{FTFS},   we have
$$\mathcal{M}_j(F) =q^{-d-1} |F| + q^{-2d-1}G_1^d  \sum_{x,y\in F}  K(j, ||x-y||/4).$$
If $j\ne 0,$  then  the Kloosterman sum $ | K(j, ||x-y||/4) | $ is bounded by $2 q^{1/2}.$
Hence,   as in Case 1,  we obtain that
$$ \max_{t\ne 0} \mathcal{M}_j(F) \le q^{-d-1} |F| + 2 q^{\frac{-3d-1}{2}} |F|^2.$$
To complete the proof,  it therefore remains to show that
\begin{equation}\label{RT1}  \mathcal{M}_0(F) \le q^{-d-1} |F| + 2 q^{\frac{-3d-1}{2}} |F|^2.\end{equation}
In fact, we can prove that
\begin{equation}\label{RT}\mathcal{M}_0(F) \le q^{-d-1} |F| +  q^{\frac{-3d-2}{2}} |F|^2,\end{equation}
which is much stronger. Indeed,
$$\mathcal{M}_0(F)=q^{-d-1} |F| + q^{-2d-1}G_1^d  \sum_{x,y\in F}  K(0, ||x-y||/4)$$
$$=q^{-d-1} |F| + q^{-2d-1}G_1^d  \sum_{x,y\in F: ||x-y||=0}  K(0, 0) +   q^{-2d-1}G_1^d  \sum_{x,y\in F: ||x-y|| \ne0}K(0, ||x-y||/4).$$
Using the facts that $K(0,0)=q-1,$ and  $K(0, s)=-1$ for $s\ne 0,$
 \begin{align*} \mathcal{M}_0(F) &= q^{-d-1} |F|+ q^{-2d-1}G_1^d  \sum_{x,y\in F: ||x-y||=0} (q-1) +   q^{-2d-1}G_1^d  \sum_{x,y\in F: ||x-y|| \ne0} (-1)\\
&=q^{-d-1} |F| + q^{-2d} G_1^d \sum_{x, y\in F: ||x-y||=0} 1   - q^{-2d-1} G_1^d  \sum_{x,y\in F} 1. \end{align*}
Since $G_1^d=-q^{d/2}$ by Corollary \ref{Corm},  the second term above  is negative and the third term equals  $q^{\frac{-3d-2}{2}} |F|^2.$
Thus, we obtain that
$$ \mathcal{M}_0(F)\le q^{-d-1} |F|+q^{\frac{-3d-2}{2}} |F|^2,$$
as required.

We remark here that one can apply directly Theorem 1.3 in \cite{IKSPS} for characteristic functions to give a bound which is better than (\ref{RT1}), but weaker than (\ref{RT}).
\end{proof}

\section{Proof of Theorem \ref{thm0}}\label{prof}
In this section, we devote ourselves to giving a proof of Theorem \ref{thm0}.
Our idea is to involve a suitable algebraic variety in the  Fourier analysis. An advantage in using an algebraic variety argument is that it offers a new form for the  upper bound of the $L^2$-norm of a certain counting function, which is more manageable for our purpose afterwards.

\subsection{Algebraic variety and related Fourier transform}
  Let $X=(\mathbf{x},\mathbf{y})$ be the coordinates of  $ \mathbb F_q^{2d} \times \mathbb F_q^{2d}=\mathbb F_q^{4d},$ and let $ ||X||_*$ be the homogeneous polynomial defined by
$$ ||X||_*:= ||\mathbf{x}||-||\mathbf{z}||=x_1^2+\cdots + x_{2d}^2-z_1^2-\cdots -z_{2d}^2.$$

\begin{definition} \label{defV0} Let $V_0$ be the subvariety of $\mathbb F_q^{4d}$ cut out by the equation $ ||X||_*=0$, i.e.,
$$ V_0:=\{\ X\in \mathbb F_q^{4d}: ||X||_*=0\ \}.$$
\end{definition}
We need the following Fourier transform of the  variety $V_0$ in $\mathbb F_q^{4d}.$
\begin{lemma} \label{defVFT} If $M\in \mathbb F_q^{4d},$  then we have
$$ \widehat{V_0}(M):=q^{-4d} \sum_{X\in V_0} \chi(-M\cdot X)=\left\{\begin{array}{ll} q^{-1}\delta_0(M) +  q^{-2d-1} (q-1) \quad &\mbox{if}\quad||M||_*=0,\\
   -  q^{-2d-1}\quad &\mbox{if} \quad||M||_*\ne 0. \end{array} \right.$$                -
\end{lemma}
\begin{proof}
It follows from the orthogonality of $\chi$ that
$$ \widehat{V_0}(M)=q^{-4d} \sum_{X\in V_0} \chi(-M\cdot X)=q^{-1} \delta_0(M) + q^{-4d-1} \sum_{X\in \mathbb F_q^{4d}} \sum_{s\ne 0} \chi(s||X||_* - M\cdot X).$$
From the formula \eqref{ComSqu},  it follows
$$\widehat{V_0}(M)=q^{-1}\delta_0(M) + q^{-4d-1} {G}_1^{4d} \sum_{s\ne 0} \eta^{2d}(s) \eta^{2d}(-s) \chi\left(\frac{||M||_*}{-4s}\right).$$
Since $\eta^{2d}=1,$ by a change of variables, one has
$$\widehat{V_0}(M)=q^{-1}\delta_0(M) +  q^{-4d-1} {G}_1^{4d} \sum_{r\ne 0}\chi(r||M||_*).$$
By the orthogonality of $\chi$,
$$ \widehat{V_0}(M)=\left\{\begin{array}{ll} q^{-1}\delta_0(M) +{G}_1^{4d}  q^{-4d-1} (q-1) \quad &\mbox{if}\quad||M||_*=0,\\
   -{G}_1^{4d}  q^{-4d-1}\quad &\mbox{if} \quad||M||_*\ne 0. \end{array} \right.$$
Since $G_1^{4d}= q^{2d},$  the proof is complete.
\end{proof}

By invoking Lemma \ref{defVFT},  we are able to deduce the following lemma.
\begin{lemma}  \label{goodLem}
Let $\mathcal{D} \subset \mathbb F_q^{2d}.$  For each $t\in \mathbb{F}_q$, let $\nu(t)$ be the number of pairs $(\mathbf{x}, \mathbf{y})\in \mathcal{D}\times \mathcal{D}$ such that $||\mathbf{x}-\mathbf{y}||=t$. Then we have
$$  \sum_{t\in \mathbb F_q} \nu(t)^2 \le \frac{|\mathcal{D}|^4}{q}  +  q^{6d}\sum_{||M||_*=0} |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2.$$
\end{lemma}
\begin{proof}
Since $\nu(t)=\sum_{\mathbf{x}, \mathbf{y}\in \mathcal{D}: ||\mathbf{x}-\mathbf{y}||=t} 1,$ we have
$$ \sum_{t\in \mathbb F_q} \nu(t)^2=\sum_{t\in \mathbb F_q} \left(\sum_{\mathbf{x}, \mathbf{y}\in \mathcal{D}: ||\mathbf{x}-\mathbf{y}||=t} 1   \right)^2 = \sum_{\mathbf{x,y,z,w} \in \mathcal{D}: ||\mathbf{x}-\mathbf{y}||=||\mathbf{z}-\mathbf{w}||} 1.$$
We will relate  the value $\sum_{t} \nu(t)^2$  to  the Fourier transform  on the variety $V_0$  in $\mathbb F_q^{4d}.$
To do this,  we let $X=(\mathbf{x}, \mathbf{z}),  Y=(\mathbf{y}, \mathbf{w}) \in \mathcal{D}\times \mathcal{D}. $  Using these notations with $|| \cdot ||_*$,  we can write
$$ \sum_{t\in \mathbb F_q} \nu(t)^2=  \sum_{X, Y\in \mathcal{D} \times \mathcal{D} : ||X-Y||_*=0}   1= \sum_{X, Y\in \mathcal{D} \times \mathcal{D} }  V_0(X-Y),$$
where  we recall from Definition \ref{defV0} that  the  variety $V_0$  is given by
$$ V_0= \{ X\in \mathbb F_q^{4d}:  ||X||_*=0\}.$$
Applying the Fourier inversion theorem to the characteristic function $V_0(X-Y),$ we get
$$ \sum_{t\in \mathbb F_q} \nu(t)^2=\sum_{X,Y\in \mathcal{D}\times \mathcal{D}} V_0(X-Y)= q^{8d} \sum_{M\in \mathbb F_q^{4d}} \widehat{V_0}(M) |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2.$$
Replacing $\widehat{V_0}(M)$ by the explicit value given  in  Lemma \ref{defVFT},   we get
$$ \sum_{t\in \mathbb F_q} \nu(t)^2= q^{8d-1} \sum_{||M||_*=0} \delta_0(M) |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2 + q^{6d}\sum_{||M||_*=0} |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2 - q^{6d-1} \sum_{M\in \mathbb F_q^{4d}} |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2. $$
Since the last term of the RHS  is negative and  the first term   is  $|\mathcal{D}|^4/q,$     we  have
$$ \sum_{t\in \mathbb F_q} \nu(t)^2 \le \frac{|\mathcal{D}|^4}{q}  +  q^{6d}\sum_{||M||_*=0} |\widehat{\mathcal{D}\times \mathcal{D}}(M)|^2,$$
as desired.
\end{proof}

We are ready to prove Theorem \ref{thm0}.
\bigskip

\paragraph{\textbf{Proof of Theorem \ref{thm0}}:}
Define $\mathcal{D}=E\times F\subset \mathbb F_q^d \times \mathbb F_q^d$. For $t\in \mathbb{F}_q$, let $\nu(t)$ be the number of pairs $(\mathbf{x}, \mathbf{y})\in \mathcal{D}\times \mathcal{D}$ such that $||\mathbf{x}-\mathbf{y}||=t$.
By the Cauchy-Schwarz inequality, we have
\begin{equation}\label{formudis}  |\Delta(E)+\Delta(F)|=|\Delta(E\times F)|\ge \frac{|E|^4|F|^4}{\sum\limits_{t\in  \mathbb F_q}\nu(t)^2}.\end{equation}
Since $|\mathcal{D}|=|E||F|,$   Lemma \ref{goodLem} implies that
$$\sum_{t\in \mathbb F_q} \nu(t)^2 \le  \frac{|E|^4|F|^4}{q} + q^{6d} \sum_{M'\in \mathbb F_q^{3d}}  |\widehat{E\times F \times E} (M')|^2  \max_{r\in \mathbb F_q} \sum_{m\in S_r^{d-1}} |\widehat{F}(m)|^2. $$
Using Proposition \ref{pro2.4} and the Plancherel theorem, we get
$$\sum_{t\in \mathbb F_q} \nu(t)^2\le \frac{|E|^4|F|^4}{q} +  q^{6d} \left(q^{-3d} |E|^2|F|\right) \left( q^{-d-1} |F| + 2 q^{\frac{-3d-1}{2}} |F|^2\right).$$
Simplifying  the RHS gives us
$$ \sum_{t\in \mathbb F_q} \nu(t)^2 \le   \frac{|E|^4|F|^4}{q} + q^{2d-1}|E|^2|F|^2 + 2 q^{\frac{3d-1}{2}} |E|^2|F|^3.$$
This estimate can be combined with \eqref{formudis} to deduce that
$$ |\Delta(E)+\Delta(F)| >q/2$$
under the conditions that  $|E||F|\ge 2 q^d$ and $|E|^2|F| \ge C q^{(3d+1)/2}$ for a sufficiently large constant $C.$
Now we show that  the condition $|E||F|\ge 2q^d$ is not necessary.
Recall that  we  can assume that  $|E|< 4 q^{(d+1)/2}$, otherwise   $|\Delta(E)|=q,$  which is  the consequence due to Iosevich and Rudnev \cite{io}.
We claim  that  if $|E|^2|F| \ge C q^{(3d+1)/2}$, then $|E||F|\ge 2 q^d.$
If not, then  $|E||F|< 2 q^d$  and  $|E|^2|F| \ge C q^{(3d+1)/2}.$ These two conditions clearly imply that $|E|> \frac{C}{2} q^{(d+1)/2},$ which contradicts  our assumption that $|E| < 4 q^{(d+1)/2}.$

A symmetric argument by switching the roles of $E$ and $F$  also yields that
if $|E||F|^2 \ge C q^{(3d+1)/2},$ then $|\Delta(E) + \Delta(F)|> q/2.$  This completes the proof. $\square$

In the following construction, we show that Theorem \ref{thm0} can not hold when $d\ge 2$ is even and $q\equiv 1\mod 4$.
\begin{construction}\label{sharp-odd} Let $d\ge 2$ is even.
\begin{enumerate}
\item Suppose $q=p^l$ with $p\equiv 1\mod 4$ and $l=3k$. There exist sets $E, F\subset \mathbb{F}_q^d$ such that $|E||F|^2\sim q^{d+\frac{2}{3}}$, and
$|\Delta(E)+\Delta(F)|\le q/2.$
\item Suppose that $q=p^l$ with $p\equiv 3\mod 4$ and $l=6k$. There exist sets $E, F\subset \mathbb{F}_q^d$ such that $|E||F|^2\sim q^{d+\frac{2}{3}}$, and
$|\Delta(E)+\Delta(F)|\le q/2.$
\end{enumerate}

\end{construction}
\begin{proof}
We first recall the following result from \cite{mur} due to Murphy and Petridis.
\begin{itemize}
\item If $q=p^l$ with $p\equiv 1\mod 4$ and $l=3k$, then there exists a set $A\subset \mathbb{F}_q^2$ such that $|A|\sim q^{4/3}$ and $|\Delta(A)|\le q/2$.
\item If $q=p^l$ with $p\equiv 3\mod 4$ and $l=6k$, then there exists a set $A\subset \mathbb{F}_q^2$ such that $|A|\sim q^{4/3}$ and
$|\Delta(A)|\le q/2$.
\end{itemize}
We note that $q=p^l\equiv 3\mod 4$ if and only if $p\equiv 3\mod 4$ and $l$ is odd.

Since $d\ge 2$ is even and $q\equiv 1\mod 4$, it is known in \cite{hart} that one can find $\frac{d}{2}$ independent vectors in $\mathbb{F}_q^d$, say $\{v_1, \ldots, v_{\frac{d}{2}}\}$, such that $v_i\cdot v_j=0$ for all $1\le i, j\le \frac{d}{2}$, and $\frac{d-2}{2}$ independent vectors in $\mathbb{F}_q^{d-2}$, say $\{v_1', \ldots, v_{\frac{d-2}{2}}'\}$, such that $v_i'\cdot v_j'=0$ for all $1\le i, j\le \frac{d-2}{2}$. Define $E=\mathtt{Span}(v_1, \ldots, v_{\frac{d}{2}})$ and $F=\mathtt{Span}(v_1', \ldots, v_{\frac{d-2}{2}}')\times A$. It is clear that $|E|=q^{d/2}$ and $|F|\sim q^{\frac{d-2}{2}+\frac{4}{3}}$. So, $|E||F|^2\sim q^{\frac{3d}{2}+\frac{2}{3}}$. It follows from definitions of $E$ and $F$ that $\Delta(E)=\{0\}$ and $\Delta(F)=\Delta(A)$. Thus, $|\Delta(E)+\Delta(F)|=|\Delta(A)|\le q/2$.
\end{proof}

\section{Proof of Theorem \ref{primecase}}

In this section,  we give a proof of Theorem \ref{primecase}. To do this, we first recall the following proposition from \cite{pham}, which essentially says that the $L^2$-norm of the distance measure on the Cartesian product set $E\times F$ can be reduced to the $L^2$-norm of the distance measure on each component.

\begin{proposition}\label{proLove} Let $E, F \subset \mathbb F_q^d.$ For any $r\in \mathbb{F}_q$, let $\nu(r)$ be the number of pairs $((e_1, f_1), (e_2, f_2))\in (E\times F)^2$ such that $||e_1-e_2||+||f_1-f_2||=r$, and let $\mu(r)$ be the number of pairs $(x, y)\in E\times E$ such that $||x-y||=r$. Then we have
 \[\sum_{r\in \mathbb F_q}\nu(r)^2\le \frac{|E|^4|F|^4}{q}+q^d|F|^2\sum_{r\in \mathbb F_q}\mu(r)^2.\]
\end{proposition}
\paragraph{\textbf{Proof of Theorem \ref{primecase}}:}
If $|E|\ge q^{5/4}$, then it has been proved in \cite[Theorem 1]{mu} that $|\Delta(E)|\gg q$. Therefore, without loss of generality, we can assume that $|E|\le q^{5/4}$.

Let $T(E)$ be the number of triples $(x, y, z)\in E\times E\times E$ such that $||x-y||=||x-z||$ with $||y-z||\ne 0$. Since $q\equiv 3\mod 4$, there are no two distinct points $y, z\in E$ such that $||y-z||=0$.
It has been proved in \cite[Theorem 4]{mu} that there exists a large enough constant $C_2$ such that
\[T(E)\le C_2\left(\frac{|E|^3}{q}+q^{2/3}|E|^{5/3}+q^{1/4}|E|^2\right)\ll q^{2/3}|E|^{5/3},\]
for  $ |E|\le q^{5/4}$.
By the Cauchy-Schwarz inequality, we have
\[\sum_{r\in \mathbb F_q}\mu(r)^2\le |E|\cdot (T(E)+|E|^2) \le C_2\left( \frac{|E|^4}{q}+q^{2/3}|E|^{8/3}+q^{1/4}|E|^3\right)+|E|^3\ll q^{2/3}|E|^{8/3},\]
for $|E|\le q^{5/4}$.
By Proposition \ref{proLove},  it follows that
\[\sum_{r\in \mathbb F_q}\nu(r)^2\le \frac{|E|^4|F|^4}{q}+q^2|F|^2\sum_{r\in \mathbb F_q}\mu(r)^2.\]
Therefore,
\[\sum_{r\in \mathbb F_q}\nu(r)^2\ll \frac{|E|^4|F|^4}{q}+q^\frac{8}{3}|F|^2|E|^{\frac{8}{3}}\ll \frac{|E|^4|F|^4}{q},\]
whenever $|E|^4|F|^6\gg q^{11}$.

As in the proof of Theorem \ref{thm0}, we have
\[|\Delta(E)+\Delta(F)|\ge \frac{|E|^4|F|^4}{\sum_{r}\nu(r)^2}\gg q,\]
under the condition $|E|^4|F|^6\gg q^{11}$.

We also change the roles of $E$ and $F$  in the above proof. Hence, we also see that
if $|E|^6|F|^4 \gg q^{11},$ then $|\Delta(E) + \Delta(F)|\gg q/2.$  This completes the proof. $\square$

\section*{Acknowledgments}
Daewoong Cheong and Doowon Koh were supported by Basic Science Research Programs through National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B07045594 and NRF-2018R1D1A1B07044469, respectively). Thang Pham was supported by Swiss National Science Foundation grant P400P2-183916. The authors would like to thank Vietnam Institute for Advanced Study in Mathematics for hospitality during their visit.