# Large anomalous Hall, Nernst effect and topological phases in the 3d-4d/5d
based
oxide double perovskites

Kartik Samanta∗ Max Planck Institute for Chemical Physics of Solids, 01187
Dresden, Germany<EMAIL_ADDRESS>Jonathan Noky  Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany  Iñigo
Robredo  Max Planck Institute for Chemical Physics of Solids, 01187 Dresden,
Germany  Donostia International Physics Center, 20018 Donostia - San
Sebastián, Spain  Juergen Kuebler  Technische Universitaet Darmstadt, 64289
Darmstadt, Germany  Maia G. Vergniory∗ Max Planck Institute for Chemical
Physics of Solids, 01187 Dresden, Germany  Donostia International Physics
Center, 20018 Donostia - San Sebastián, Spain<EMAIL_ADDRESS>Claudia Felser∗ Max Planck Institute for Chemical Physics of Solids, 01187
Dresden, Germany<EMAIL_ADDRESS>
###### Abstract

Magnetic topological quantum materials are attracting considerable attention
owing to their potential technological applications. However, only a small
number of these materials have been experimentally realized, thereby giving
rise to the need for new stable magnetic topological quantum materials.
Magnetism and spin–orbit coupling, two essential ingredients of the oxide
materials, lead to various topological transport phenomena such as the
anomalous Hall and anomalous Nernst effects, which can be significantly
enhanced by designing an electronic structure with a large Berry curvature. In
that respect, double perovskites with the general formula A2BB′O6 with an
alternating ordered arrangement of two transition metal sites, B(3d) and
B′(4d/5d), present attractive possibilities as they are robustly stable
against oxidation under ambient conditions and versatile. These double
perovskites also offer a high energy scale for magnetism as well as strong
spin-orbit coupling with a high magnetic ordering temperature. Here, using
first-principles density functional theory calculations, we present a
comprehensive study of the intrinsic anomalous transport for 3d-4d/5d based
cubic and tetragonal stable double perovskite (DP) compounds. A few of the DPs
exhibit a very large anomalous Hall effect with a distinct topological band
crossing in the vicinity of the Fermi energy. Our results show the importance
of symmetries, particularly the mirror planes, as well as the clean
topological band crossing near the Fermi energy, which is primarily
contributed by the 5d-t2g for large anomalous Hall and Nernst effects.

##  I Introduction

Since the theoretical prediction of topological insulatorsTop-insulator-2005 ;
Bernevig-2006 ; Hassan-Kane-2010 ; Weider_review  and topological semi-metals
PhysRevX.5.011029 ; Rev-Yan-2017 ; RevModPhys.90.015001 , researchers in the
field of condensed matter and materials science have extensively endeavored to
design, characterize, and synthesize novel topological materials Rev-Yan-2017
; Hassan-Kane-2010 ; Science-2015 ; Wieder-2022  considering their potential
applicability in the field of quantum informationKitaev-2003 , coherent spin
transportNature-Mat-2012 , and high-efficiency catalysisAdv-Mat-2017 . Recent
advances in elementary band representation theory have accelerated the
prediction of novel insulating and (semi-)metallic non-magnetic and magnetic
topological phases of matter in real materialsBradly-2017 ; Cano-2018 ; Cano-
PRB-2017 ; Maia-2019 ; Maia-2022 ; Nature-Mag-topo-2020 ; Nature-review-Mag-
topo-2022 . However, the prediction of new stable magnetic topological
materials with large transport characteristics such as like large anomalous
Hall conductivity (AHC) and Nernst conductivity (ANC) in strongly correlated
magnetic oxides is more challenging and systematic calculations have been
conducted only for a few selected strongly correlated oxides SrRuO3-Science ;
CrO2-2018 ; Oxides-2019 ; Jap-Cry-2020 ; Jap-strain-2021 ; Wang-2018 ; DP-
topo1 ; DP-topo2 ; Rev-topo-oxide ; DP-SFMO .

In correlated magnetic systems, the interplay between magnetism, crystal
symmetry, and spin-orbit coupling (SOC) underlies the various topological
transport phenomena and are highly intriguing from an experimental
perspectiveBurkov-2014 ; Silvia-2020 ; Ye-2018 . In three-dimensional (3D)
ferromagnets, spin-polarized bands often display topologically non-trivial
nodal points and nodal lines (NLs)Kim-2018 ; Hasan-2016 ; Noky-2019 ;
Noky-2020 ; Shekhar-2021 ; nodal2 ; CoS2-exp . The lifting of band degeneracy
because of the SOC around a nodal structure gives rise to increased Berry
curvature around it, leading to a variety of topological transport phenomena,
including the anomalous Hall effect (AHE) as well as the anomalous Nernst
effect (ANE) Nagaosa-2010 ; Noky-2020 ; Noky-linear-2019 ; SrRuO3-2d .

In the case of magnetic topological materials with large anomalous Hall (AHC)
and Nernst (ANC) conductivities, it is important to have a magnetic site with
a localized magnetic moment, which can provide a high energy scale for
magnetism, as well as a magnetic site occupied by a heavy element with strong
SOC. From the perspective of designing new magnetic topological materials, it
is also important to have materials with physically separated ions that host
magnetism and ions that host strong SOC, to avoid the problem resulting from
the interplay between the correlation effect and SOC at the same site. A
promising class of materials in this context is the rocksalt-ordered Rock-salt
double perovskite (DP) compounds with the general formula A2BB′O6. These
compounds present the possibility of hosting a 3d transition metal (TM) ion at
the B site and 4d/5d TM ions at the B′ site, offering a suitable playground
for exploring exotic properties that depend on the competing energy scales
associated with the bandwidth, Coulomb correlation, and SOC for the 3d and
4d/5d orbitals at these two sites. From an experimental point of view, these
compounds are easy to produce, are robustly stable against oxidation in
ambient air, and possess a high magnetic ordering temperature. However,
because most conventional topological materials have low transition
temperatures, their application to potential technologies is complicated.
Several prominent 3d-4d/5d DP compounds with remarkable properties, such as
half-metallicity at room temperature Half-metal  and colossal
magnetoresistance SFRO-mag  are known. These compounds are excellent
candidates for real-life applications, such as spintronics and low dissipation
devicesCoey2001 . In addition, in many of these 3d-4d/5d DP materials, the
low-energy physics is primarily governed by the t2g-orbital bands of the 4d/5d
TM ions with strong SOC, with the possibility of a topological band crossing
near the Fermi energy with highly efficient transport characteristics, such as
AHC and ANC. In spite of the extensive studies of the electronic and magnetic
properties of these materials DP-review ; DP-review-TSD ; samanta-CaFeCoNiO ;
samanta-CaCoNiOsO , the topological characteristics of these materials have
not yet been adequately investigated. This motivated our study, in which we
considered selected cubic and tetragonal 3d-4d/5d-based experimentally
reported double perovskites with the aim to explore and understand new stable
topological materials with large AHC and ANC. Our results show the possible
existence of a clean topological band crossing near the Fermi energy,
primarily contributed by the 5d-t2g orbitals with large anomalous Hall and
Nernst conductivities. In certain DP compounds, particularly in Ba2NiReO6 a
mirror symmetry protected topological nodal line and their drumhead surface
state were observed very close to the Fermi energy. In addition, the AHC and
ANC of these compounds are almost three times larger than those of the
extensively studied prototype metallic ferromagnetic oxide material SrRuO3
SrRuO3-Science ; SrRuO3-2d .

##  II Computational details

For the calculations, we considered stable cubic and tetragonal 3d-4d/5d-based
DP compounds whose structural data was taken from the ICSD databaseICSD  with
a 3d magnetic element at the B site and 4d/5d magnetic element at the B′ site
to enable us to investigate topological transport characteristics such as the
anomalous Hall and Nernst effects. We considered the experimentally measured
lattice constant, atomic position, and space group as inputs for the density
functional theory (DFT) calculations.

The DFT-based plane-wave projected augmented wave (PAW) method, as implemented
in the Vienna ab initio simulation package (VASP), was employed vasp ; paw1 ;
paw2 . We used a $12\times 12\times 12$ ($12\times 12\times 10$) k-points mesh
and a plane wave cut-off of 520 eV, for the self-consistent calculations of
cubic (tegragonal) compounds. These selections of the k-mesh and the plane-
wave cut-off were found to result in good convergence of the total energy. We
used the Perdew-Burke-Ernzerhof (PBE) pbe  exchange correlation functional
within the generalized gradient approximation (GGA). The electron-electron
correlation effects beyond GGA at the strongly correlated 3d and 4d/5d
magnetic sites were taken into account using the supplemented on-site Hubbard
U correction via the GGA+U method lda+u . Considering that the B site is
occupied by a 3d transition metal and the B′ site by a 4d/5d transition metal
with wider bands compared to the B site, the correlation effect would be
expected to be stronger at the B site compared to the B′ site. Thus, we chose
U-values of 4.0 eV and 1.6 eV for the B and B′ sites of the considered
3d-4d/5d-based DPs, respectively. The selection of these values was motivated
by the U-values used for 3d-4d/5d-based DP compounds in previous studies
Kartik-CaCoNiOsO6 ; Kartik-FeCoNiOSO6 ; Samanta-Nadoped ; DP-review-TSD , in
which the electronic structure and magnetic ground state were found to be in
good agreement with the experimental results. We initially checked our
calculations by varying the U-values of a few compounds, but overall, the
electronic structure and magnetic ground state remained unaltered. The effect
of the SOC is included in the fully relativistic schemes and for all
calculations the magnetic moments are chosen to be parallel to the (001)
direction.

We calculated the contribution of the intrinsic Berry curvature to the
anomalous Hall conductivity by employing the Wannier interpolation
techniquewan1 . To compute the Berry curvature, we first constructed a
maximally localized Wannier functions (MLWFs) Hamiltonian projected from the
GGA+$U$+$SOC$ Bloch wave functions using the Wannier90 tool wan2 ; wan3 ; wan4
. Atomic orbital-like MLWFs of the A$-d$, B$-3d$, B${}^{{}^{\prime}}-4d/5d$
and O$-p$ states were considered to construct the tight-binding Hamiltonian,
which reproduced the spectrum of the system accurately in the energy window of
2.0 eV around the Fermi energy. The wannierization was repeated with different
inner and outer energy windows until it reproduced the spectrum of the system
quite accurately in the energy window of $\pm$ 2.0 eV around the Fermi energy.

With the tight-binding model Hamiltonian, we calculated the intrinsic AHC
using the linear response Kubo formula approach as follows Kubo-2004 :

$\displaystyle\Omega^{z}_{n}(\mathbf{k})=-\hslash^{2}\sum_{n\neq
m}\frac{\operatorname{2Im}\langle
u_{n\mathbf{k}}|\hat{v}_{x}|u_{m\mathbf{k}}\rangle\langle
u_{m\mathbf{k}}|\hat{v}_{y}|u_{n\mathbf{k}}\rangle}{(\epsilon_{m\mathbf{k}}-\epsilon_{n\mathbf{k}})^{2}},$
(1)

where $\Omega^{z}_{n}(\mathbf{k})$ is the Berry curvature of band $n$,
$\hslash\hat{v}_{i}={\partial\hat{H}(\mathbf{k})}/{\partial k_{i}}$ is the
$i$’th velocity operator, $u_{n\mathbf{k}}$ and $\epsilon_{n\mathbf{k}}$ are
the eigenstates and eigenvalues of the Hamiltonian $\hat{H}(\mathbf{k})$,
respectively.

Subsequently, we calculated the AHC, given by:

$\displaystyle\sigma^{A}_{H}=$
$\displaystyle-\frac{e^{2}}{\hbar}\sum_{n}\int_{BZ}\frac{d{\bf
k}}{\left(2\pi\right)^{3}}f_{n}\Omega^{z}_{n}(\mathbf{k}),$ (2)

and the ANC, $\alpha^{A}_{H}$, as proposed by Xiao et al. ANE-prl ; ANE-2010

$\displaystyle\alpha^{A}_{H}=$
$\displaystyle-\frac{1}{T}\frac{e^{2}}{\hbar}\sum_{n}\int_{BZ}\frac{d{\bf
k}}{\left(2\pi\right)^{3}}\Omega^{z}_{n}(\mathbf{k})$ (3)
$\displaystyle[(E_{n}-E_{F})f_{n}+k_{B}Tln(1+exp(\frac{E_{n}-E_{F}}{-k_{B}T}))],$

where T is the actual temperature, $f_{n}$ is the Fermi distribution, and EF
is the Fermi level. We used a k-point mesh of $300\times 300\times 300$ for
the calculation of the AHC and ANC using the Eq. 2 and Eq. 3, respectively,
and T=300 K for the calculation of the ANC. The surface states were calculated
using tight-binding methods by the combination of the Wannier90wan2 ; wan3 ;
wan4  and WannierTools software packagesWanTool1 ; WanTool2 .

##  III Result and discussion

Figure 1: Crystal structure of (a) cubic space group Fm-3m (No. 225) and (b) tetragonal space group I4/m (No. 87) double perovskites. The arrows indicate the direction of magnetic moment at the B and B′ magnetic sites. (c) Schematic energy level diagram of the d-levels in the cubic octahedron crystal field at the 3d-B and 4d/5d B′ sites. Table 1: Summary of the results of selected materials considered in this study. Listed are the space group (SG), charge state, and electron occupancy at the B-3d and B′-4d/5d sites, magnetic moment at the B-3d and B′-4d/5d sites, theoretical magnetization per formula unit (f.u.), AHC, maximum AHC, ANC, and maximum ANC. The maximum values are obtained in an energy window of 200 meV around the Fermi level. $\Delta$E indicates the energy distance in eV of the maximum values with regard to the Fermi level. Material | SG | B-3d | B′-4d/5d |  | B |  | B′ |  | M |  |  | AHC |  | AHCmax($\Delta$E) |  | ANC |  | ANCmax($\Delta$E)
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  | site | site |  | ($\mu_{B}$) |  | ($\mu_{B}$) |  | ($\mu_{B}$/f.u.) |  |  | (S cm-1) |  | (S cm-1) |  | (A m-1 K-1) |  | (A m-1 K-1)

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
Ba2FeMoO6 | 225 | Fe3+ \- 3d5 | Mo5+ \- 4d1 |  | 3.971 |  | -0.399 |  | 3.99 |  |  | 6.94 |  | 37(0.2) |  | 0.043 |  | -0.18(0.2)
Ba2CoMoO6 | 225 | Co2+ \- 3d7 | Mo6+ \- 4d0 |  | 2.697 |  | 0.029 |  | 2.99 |  |  | 0.0 |  | 0.00 |  | 0.00 |  | 0.00
Ba2MnMoO6 | 225 | Mn2+ \- 3d5 | Mo6+ \- 4d0 |  | 4.537 |  | 0.109 |  | 4.99 |  |  | 0.0 |  | 0.00 |  | 0.00 |  | 0.00
Ba2FeReO6 | 225 | Fe3+ \- 3d5 | Re5+ \- 5d2 |  | 3.897 |  | -0.920 |  | 3.06 |  |  | 33.43 |  | 329(0.2) |  | 0.04 |  | -1.2(0.14)
Ba2MnReO6 | 225 | Mn2+ \- 3d5 | Re6+ \- 5d1 |  | 4.516 |  | -0.6611 |  | 4.032 |  |  | 278 |  | 536(0.03) |  | 1.52 |  | -2.1(0.10)
Ba2NiReO6 | 225 | Ni2+ \- 3d8 | Re6+ \- 5d1 |  | 1.702 |  | -0.706 |  | 1.027 |  |  | 495 |  | 571 (0.01) |  | -1.120 |  | -2.83(0.05)
Ba2NiWO6 | 225 | Ni2+ \- 3d8 | W6+ \- 5d0 |  | 1.710 |  | 0.061 |  | 2.013 |  |  | 0.0 |  | 0.0 |  | 0.00 |  | 0.0
Sr2NiMoO6 | 87 | Ni2+ \- 3d8 | Mo6+ \- 4d0 |  | 1.673 |  | 0.073 |  | 1.996 |  |  | 0.00 |  | 0.0 |  | 0.00 |  | 0.00
Sr2FeMoO6 | 87 | Fe3+ \- 3d5 | Mo5+ \- 4d1 |  | 4.058 |  | -0.554 |  | 3.99 |  |  | 73.68 |  | 99(0.17) |  | 0.03 |  | 0.94(0.16)
Sr2CoOsO6 | 87 | Co2+ \- 3d7 | Os6+ \- 5d2 |  | 2.678 |  | -1.183 |  | 1.155 |  |  | 300 |  | 517(0.12) |  | -1.610 |  | 3.13(0.2)
Sr2CoReO6 | 87 | Co2+ \- 3d7 | Re6+ \- 5d1 |  | 2.444 |  | -0.438 |  | 2.027 |  |  | -22 |  | 200(0.19) |  | -0.52 |  | -1.97(0.2)
Sr2CrOsO6 | 87 | Cr3+ \- 3d3 | Os5+ \- 4d3 |  | 2.741 |  | -1.795 |  | 0.184 |  |  | 0.00 |  | 0.00 |  | 0.00 |  | 0.00
Sr2NiOsO6 | 87 | Ni2+ \- 3d8 | Os6+ \- 5d2 |  | 1.677 |  | -0.957 |  | 0.369 |  |  | -3.10 |  | 48 (0.08) |  | 0.24 |  | -1.12(0.19)
Sr2NiReO6 | 87 | Ni2+ \- 3d8 | Re6+ \- 5d1 |  | 1.671 |  | -0.649 |  | 1.031 |  |  | 447 |  | 447 |  | -0.3 |  | 2.54(0.01)

In the present study, we systematically investigated 3d-4d/5d-based cubic and
tetragonal stable DP compounds in an attempt to identify and further our
understanding of new stable magnetic topological quantum materials with large
AHE. The primary structural building blocks of the considered compounds are
the corner-sharing B(3d)-O6 and B′(4d/5d)-O6 octahedra (cf. Fig. 1(a,b)). As a
result of the tetragonal distortion, the octahedra in a tetragonal DP are
slightly distorted, whereas they are perfect octahedra in a cubic DP. In the
cubic crystal field of regular octahedra, the $d$-orbitals split into a higher
energy level of two-fold (four-fold including spin) degenerate eg orbitals and
a lower energy level of three fold (six-fold including spin) degenerate t2g
orbitals, as schematically shown in the Fig. 1c. The electron-electron
correlation is expected to decrease from 3d to 4d to 5d transition metal (TM)
elements, whereas the bandwidth and spin-orbit coupling strength increases
from 3d to 4d to 5d TM elements owing to the enhanced delocalization of
electronic wave functions and the higher atomic number (Z), respectively.
Consequently, exchange splitting at the 3d-B site is likely to be quite large.
However, crystal field splitting at the 4d/5d-B’ site is expected to be very
large relative to the exchange splitting owing to the strong delocalization of
the electronic wave function.

The Large crystal field splitting at the 4d/5d-B′ site results in the
stabilization of the low spin state, and when the 4d/5d-B′ orbital is less
than or equal to half filled, the empty eg orbitals would lie far above the
Fermi level, with a negligible impact on the electronic structure and
transport properties. The main contributing orbitals around the Fermi energy
would be the three 4d/5d-t2g orbitals with strong SOC. In general, the
energetically filled 3d orbitals of the B sites, lie below the Fermi energy,
which has a finite degree of hybridization with the occupied 4d/5d-t2g
orbitals through the O-p orbitals.

From both an experimental and theoretical perspective, it is important to have
a topological material with a clean band structure with a topological crossing
near the Fermi energy. Depending on the electron occupation of the 3d orbitals
of the B site (thereby effectively controlling the exchange splitting), which
lie below the Fermi energy, we can effectively control the mixing of the
occupied 3d orbitals with the partially occupied 4d/5d-t2g states. In the
present study, we considered 3d-4d/5d-based cubic and tetragonal DPs with 0,
1, 2, or 3 electrons in the 4d/5d-t2g state near the Fermi energy, which can
provide strong SOC and many electrons in the 3d B sites. Table I presents the
calculated results of the compounds with a large AHC and/or ANC.

Our calculations indicated that whenever B-3d magnetic sites have fully half-
filled d-shells or half-filled eg states, because of the enhanced exchange
splitting, the filled 3d-orbitals are pushed below the Fermi energy,
significantly reducing mixing with the occupied 4d/5d-t2g states. As a result,
a clean band structure with predominant 4d/5d-t2g orbital character is
observed around the Fermi energy. For example, in Fig. 2(a,b), we present the
band structure of Sr2CoReO6 and Ba2NiReO6 with electron occupancy at the B and
B′-sites of Co-d7, Ni-d8 and Re-d1, respectively. Owing to the completely
half-filled Ni-eg states in the compound, we observed a clean band structure
with predominantly partially filled Re-t2g states near the Fermi energy
compared to the Sr2CoReO6 compound.

Figure 2: Schematic energy level diagram, orbital filling of the d levels in
the octahedron crystal field at the Co/Ni-3d and Re-5d sites and band
structure with SOC for (a) tetragonal double perovskites, Sr2CoReO6, and (b)
cubic double perovskites, Ba2NiReO6.

In the following section, we briefly describe the charge state and electron
occupancies of the d-orbitals for the considered compounds. In the Fe based
cubic DP, we find a magnetic moment of around 4.0 $\mu_{B}$, indicating a
nominal valency of Fe3+ with a high-spin state d5 occupancy [t2g(3$\uparrow$),
eg(2$\uparrow$)] consistent with the experimental result reported for the Fe
based compounds Fe-based-DP-exp1 ; Fe-based-DP-exp2 ; Fe-based-DP-exp3 . On
the other hand, the calculated density of states (DOS) and magnetic moment
indicated a nominal valency of ”2+” at the Mn, Co, and Ni with the following
occupancies for d5 [t2g(3$\uparrow$), eg(2$\uparrow$)], d7
[t2g(3$\uparrow$,2$\downarrow$), eg(2$\uparrow$)] and d8 [t2g(3$\uparrow$,
3$\downarrow$), eg(2$\uparrow$)], respectively. The d-level occupancy at the
4d/5d-B′ site of Mo, W, and Re varies from 0 to 2 (cf. Table I). The low
energy state is primarily governed by the 4d/5d-t2g states. So it is expected
to have a negligible contribution from the 4d/5d-t2g orbital near the Fermi
energy for the completely unoccupied 4d/5d-t2g state. Therefore, compounds
with half-filled 3d-B orbitals and completely unoccupied 4d/5d-t2g orbitals
usually lead to an insulating solution as found in Ba2MnMoO6 and Ba2NiWO6.

In the tetragonal group of DP compounds, from the calculated DOS and magnetic
moment, the sites occupied by Co, Ni and Fe, Cr are found to be in the ”2+”
and ”3+” charge state, respectively, with occupancies of d7
[t2g(3$\uparrow$,2$\downarrow$), eg(2$\uparrow$)], d8 [t2g(3$\uparrow$,
3$\downarrow$), eg(2$\uparrow$)], d5 [t2g(3$\uparrow$), eg(2$\uparrow$)], and
d3 [t2g(3$\uparrow$)], respectively. In the group of tetragonal DP compounds,
Sr2NiMoO6 and Sr2CrOsO6 were found to be insulating. In the Sr2CrOsO6
compound, both Cr and Os are in the d3 configuration and aligned anti-parallel
to each other. Both completely filled, the Cr-t2g up spin channel and Os-t2g
down spin channel lie below the Fermi energy and the empty eg states of Cr and
Os lie above the Fermi energy. This leads to an insulating solution where as
in the Sr2NiMoO6 compound Mo is the d0 state, thus a negligible contribution
around the Fermi energy. The half filled Ni-eg states lie below the Fermi
energy and give rise to an insulating solution.

Based on the knowledge we gained of the electronic structure for all the
considered DP compounds, we subsequently investigated the topological
transport characteristics, AHC and ANC. Table I summarizes the calculated AHC
and ANC for all the compounds at the Fermi energy and supplementary Figure S1
shows a graphic representation. It is clearly evident that the value of AHC
and ANC around the Fermi energy is more or less of the same order of magnitude
for the cubic and tetragonal group of DP compounds, for example the cubic DP
Ba2NiReO6 and tetragonal DP Sr2NiReO6. To correlate the similar trend that was
observed for the AHC and ANC in the cubic and tetragonal group of DP
compounds, we investigated their magnetic space group in detail using the
FINDSYM findsym1 ; findsym2  software package. Considering the ferrimagnetic
alignment of the B-3d and B′-4d/5d sites, the magnetic space groups (MSG) of
the cubic and tetragonal DP compounds were determined to be I4/m m′m′ and I4/m
, respectively. Both MSG have a mirror mz along the c axis, and lack mx and
my, which is why both cubic and tetragonal DPs present a gapless nodal line at
kz = 0 but gapped nodal lines at kx = 0 and ky = 0. However, these gapped
nodal lines have a different origin, in the case of cubic DPs mx and my are
broken because of the coupling between SOC and magnetism, whereas in the case
of tetragonal DPs they were never present. Hence, for the cubic DP compounds,
the main source of the Berry curvature around the Fermi energy is found to be
the gapped topological nodal line in the presence of SOC and is discussed in
more detail in section A.

The calculated value of the AHC of the Ba2NiReO6 is almost three times larger
than that of the ferromagnetic SrRuO3,which is a prototype of ferromagnetic
oxide materials and has been studied extensively SrRuO3-Science ; SrRuO3-2d .
In addition, the determination of the energy dependence of the AHC and ANC is
also important to account for the energy shifts away from the Fermi energy.
Therefore, we additionally show the largest possible value in a range of 200
meV around the Fermi energy in Table 1. This occurs because minor oxygen
nonstoichiometry is commonly found in perovskite oxideschen-ang-oxygen-
nonstoichiometry ; lima-magnetic ; wei-tong-double-exchange  and has the
effect of a small degree of doping and, consequently, a shift in the Fermi
level. The largest values of AHC and ANC we found are 571 S/cm and -2.83 A m-1
K-1 in Ba2NiReO6. These results show that changes in the Fermi energy can have
a large influence on the anomalous transport coefficients, AHC, and ANC. To
understand the large anomalous transport coefficients of Ba2NiReO6, we
investigated its electronic and magnetic structure and topological features in
more detail, and present the results in the following section.

Figure 3: Spin-polarized GGA+$U$ density of states for the DP Ba2NiReO6,
projected onto Ni $-$d (blue), Re$-$d, (cyan), and O$-$p (orange) states. Zero
energy is specified to be equal to the GGA+$U4$ Fermi energy. Figure 4: (a)
Band structure in the presence of spin–orbit coupling (SOC) and Hubbard U. (b)
anomalous Hall conductivity (AHC), $\sigma$, and (c) anomalous Nernst
conductivity (ANC), $\alpha$, for the DP Ba2NiReO6. Figure 5: Band structure
calculated along the $K-\Gamma$ direction in the planes kz = 0 (a), ky = 0
(b), and kx = 0 (c) for the DP Ba2NiReO6. Green lines: band structure with
GGA+$U$, blue lines: band structure with GGA+U+SOC. In the kz = 0 plane, the
crossing indicated by the red circle is intact due to the protected mirror
plane while in the kx = 0 or ky = 0 planes, nodal lines gapped out in the
presence of SOC (magnetization, M $\parallel$ (001). Figure 6: (a) Band gap in
the kx = 0 plane with magnetization, M$\parallel$ (001). The Gaped nodal line
is visible. (b) Berry curvature in the kx = 0 plane with M$\parallel$ (001).
Strong Berry curvature contribution is visible along the gapped nodal line.

###  III.1 Electronic structure and magnetic ground state of Ba2NiReO6

First, using the GGA+$U$ method, we compared the total energies of the
different magnetic structures $-$ non-magnetic, ferromagnetic, and
ferrimagnetic $-$ to determine the magnetic ground state of DP compounds. The
calculations revealed the ferrimagnetic ground state to be energetically
favorable. The calculated spin polarized DOS, considering the ferrimagnetic
ground state with the GGA+$U$ calculation, is shown in Fig. 3. Within the
crystal field of a regular octahedron, the d-levels split into t2g and eg
states and the d-states of Ni and the Re ions are exchanged and the crystal
field splits. The exchange splitting at the 3d-Ni site is expected to be high
compared to the 5d-Re site, whereas the large crystal field splitting at the
5d-Re site stabilizes in the low spin state. The calculations with the GGA+$U$
showed that the localized magnetic moment at the Ni$-3d$ and Re$-5d$ sites is
found to be 1.70 and -0.7 $\mu_{B}$, respectively, which indicates that
Ni2+(3d8) is in the high spin ([t2g(3$\uparrow$, 3$\downarrow$)
eg(2$\uparrow$)]) and Re6+(5d1) in the low spin [t2g(1$\downarrow$)] states.
In the DP Ba2NiReO6 compound, as Ni is in the high spin d8 configuration, both
the majority and minority spin channels of t2g and the majority spin channel
of eg are completely filled, whereas the minority spin channel of eg remains
completely empty and lies above the Fermi energy (cf. Fig. 3). On the other
hand, low spin Re6+(5d1) has only one electron and, from Fig. 3, it is evident
that, the states close to the Fermi level (EF) are mainly contributed by the
partially filled Re-[t2g(1$\downarrow$)] states in the down spin channel. The
partially filled Re-t2g states cross the Fermi level in the minority spin
channel, which causes this spin channel to become conducting. In contrast, a
clear gap arises in the majority spin channel, leading to a half-metallic
ferrimagnetic ground state with full spin polarization. From the orbital-
resolved partial DOS (Fig. 3), it is clearly evident that a trivial band gap
of 1.2 eV exists in the majority spin channel of Ba2NiReO6, whereas that in
the minority spin channel contributes significant states to form peak-like DOS
near the Fermi level, which consist mainly of Re-t2g states hybridized with
O-2p states.

As the states close to the Fermi energy are dominated by the partially filled
Re-t2g states, the effect of the SOC in the electronic structure is expected
to be high. The electronic band structure, with the GGA+$U$+SOC calculations
considering the ferrimagnetic ground state, is shown in the Fig. 4a. With the
constructed tight binding Hamiltonian considering the atomic orbital-like
MLWFs of the Ni$-d$ , Re$-d$, Sr$-d$ and O$-p$ states, the calculated
intrinsic contribution of the AHC and the ANC as a function of the Fermi
energy is shown in the Fig. 4(b,c), respectively.

From the computed value of the AHC as a function of Fermi energy, we observed
a large peaks of the AHC around the Fermi energy (cf. Fig. 4b). We also
crosschecked our electronic structure, AHC and ANC calculations for a
different choice of $U$ values at the Ni$-3d$ and Re$-5d$ site, but no
significant changes were observed in the basic electronic structure and in the
peak of AHC and ANC near the Fermi energy (cf. Supplementary Figure S2). This
demonstrates the robustness of our calculation. In the next section, we
investigated the peak in the AHC in detail, taking into account the values of
U $=$ 4.0 and 1.6 eV at Ni$-3d$ and Re$-5d$ sites, respectively.

Figure 7: (a) Band dispersion of Ba2NiReO6 on the projected (001) surface. (b)
Brillouin zones (BZs) of bulk and (001) surfaces with high-symmetry points of
the fcc structure.

###  III.2 Topological analysis of Ba2NiReO6

The band structure of the DP Ba2NiReO6 undergoes a very distinct change
because of the strong SOC of the Re$-$5d states. Near the Fermi level, most of
the band touching points are gapped. Along the $K-\Gamma$ directions, the
valence and conduction bands of Re-$t_{2g}$ intersect with each other. To
understand the topological nature of the band crossing along $K-\Gamma$, we
examined the electronic structure in more detail. In the absence of magnetism
and the SOC, the cubic DP Ba2NiReO6 possesses mx, my, mz, C4x, C4y, and C4z
symmetries. Without SOC, the internal spin-space, and the lattice space are
decoupled from each other and spins can be separated into spin-up and spin-
down sectors. Thus, in the absence of the SOC, cubic Ba2NiReO6 possesses three
mirror planes at kz = 0, ky = 0, kx = 0 and symmetry-protected nodal lines can
appear in any of the three planes nodal1 ; nodal2 ; nodal3 . Considering the
magnetic ground state with GGA+$U$, we observe the symmetry-protected nodal
line in each of the three planes (kx = 0, ky = 0, kz = 0) along the $K-\Gamma$
direction, as shown with green lines in Fig. 5(a,b,c).

The situation is different when we take into account the SOC, in the presence
of which the spin-space (coupled up- and down-spin channels) and the lattice
space are coupled with each other, thus breaking the spin-rotational symmetry.
Particularly, when considering the moments aligned along the z$-$direction,
i.e., the SOC along the 001-direction, the mx and my symmetries are broken,
thus breaking the symmetry protection of nodal lines in the kx = 0 and ky = 0
planes, leading to a gapped nodal line along the $K-\Gamma$ high symmetry line
(cf. Fig. 5(c,b)). Instead, for the moment aligned along the z-direction, the
mz and C4z symmetries are still preserved, which leads to the gapless nodal
line in the kz = 0 plane (cf. Fig. 5a). As a result of the gapped nodal lines,
a large Berry curvature could be induced in the Brillouin zone (BZ),
consequently enhancing the AHCnodal1 ; nodal2 ; nodal3 .

We calculated the Berry curvature contribution in these planes to further
investigate the contribution of these gapped nodal lines to the AHC. As
expected, due to the presence of the mirror symmetry at the $k_{z}$ = 0 plane
with SOC (M $\parallel$ 001), a closed nodal line would be visible and,
consequently, the closed nodal line results in an almost zero Berry curvature
contribution in the $k_{z}$ = 0 plane. In the Fig. 6a, we show the band gap in
the $k_{x}$ = 0 plane by considering the two bands that form the nodal line
for the DP Ba2NiReO6. In the $k_{x}$ = 0 plane, in which the mirror symmetry
is broken due to the magnetization along the z-direction (001), the nodal
lines are separated by a gap, as shown in Fig. 6a. Consequently, the gapped
nodal lines make a large contribution to the Berry curvature in the system, as
shown in Fig. 6b. To quantify the contributions that the gapped nodal lines
make to the total AHC, we restricted the integration of Eq. 2 to slices around
the $k_{x}$ = 0 and $k_{y}$ = 0 planes, as these are the planes in which the
gapped lines reside. Considering the thickness of the slices at 20$\%$ of the
full BZ, we observe that 80 $\%$ of the AHC value can be attributed to the
gapped nodal line structure of the DP Ba2NiReO6, which suggests that the
gapped nodal lines are the primary source of AHC.

###  III.3 Topological surface state

One of the hallmark features of the nodal line compounds are the protected
drumhead surface states, consisting of surface states covering the projection
of the nodal line on the surface. Therefore, to find the topologically
protected surface state of DP Ba2NiReO6, the surface states were calculated
using the WannierTools software package WanTool1 ; WanTool2 , which employs
tight-binding Hamiltonians constructed by maximally localized Wannier
functions wan1 ; wan4 . In the topological nodal line compounds, the
topological invariants Nodal-invariant  and, thus, the surface spectrum are
protected by crystalline symmetries. Therefore only surfaces which preserve
them will show topologically protected surface statesNodal-sur1 ; Nodal-sur2 ;
Nodal-sur3 . In the case of the DP Ba2NiReO6, the mirror symmetry protected
nodal line is observed in the plane kz = 0, thus one could expect a surface
state to prevail on the (001) surface.

In Fig. 7a we show the projection of the band dispersion onto the (001)
surface, following the high-symmetry k paths shown in the Fig. 7b. On the
projected (001) surface, topological nontrivial surface states are clearly
visible, connecting the two gapless points within $\Gamma_{1}-X_{1}$ and
$M_{1}-\Gamma_{1}$ to form a two-dimensional drumhead surface. Moreover, the
dispersion of the drumhead surface states are observed to be small and located
very close to the Fermi energy, contributing a large surface DOS at the Fermi
energy, which is greatly beneficial for experimental realization as well as in
spintronics applicationsNodal-sur-application .

##  IV Summary

Using the first-principles density functional, we investigated the topological
transport characteristics, AHC and ANC for stable cubic and tetragonal rock-
salt ordered DPs (A2BB′O6) considering the 3d TM magnetic ions at the B site
and 4d/5d TM magnetic ions at the B′ site. Using first-principles DFT
calculations and Wannier interpolation, we have investigated the topological
properties, AHC and ANC of a list of representative materials. Our
calculations of the electronic structure showed that DP compounds with a half-
filled B-site 3d-t2g or eg and partially filled 4d/5d-t2g states, lead to the
possibility of a clean topological band crossing. This is primarily attributed
to strong SOC in the 5d-t2g orbital near the Fermi energy with large AHC.

The magnetic space group of the cubic and tetragonal DP compounds considering
the ferri-magnetic aligned of B-3d and B′-4d/5d site, is found to be in the
I4/m m′m′ and I4/m , respectively, with a mirror symmetry mz along the c axis
in the $k_{z}$ = 0 plane. Prior to the inclusion of SOC and magnetism, the
cubic structure also had $m_{x}$ and $m_{y}$ mirror symmetries, this is why
the primary source of the Berry curvature for the cubic DP compounds around
the Fermi energy was to be the gapped topological nodal line in the $k_{x}$=0
and $k_{y}$=0 planes. The presence of mirror symmetries and a clean
topological band crossing primarily resulting from the 5d-t2g state, were
found to be crucial for large AHC and ANC, which, in combination with the
magnetism, produces a large BC in the band structure. In certain DP compounds,
particularly in Ba2NiReO6, we found a symmetry-protected nodal line 10 meV
below the Fermi energy with very large AHC and/or ANC values that are three
times larger than the extensively studied prototype metallic ferromagnetic
oxide material SrRuO3 SrRuO3-Science ; SrRuO3-2d . In the DP Ba2NiReO6,
slightly dispersing mirror symmetry protected drumhead surface states is
observed around the Fermi energy making it an excellent candidates for using
the topological properties in the spintronics applications. We additionally
found that both AHC and ANC are strongly dependent on the position of the
Fermi level and therefore on the doping level of the investigated material.
This work demonstrates the versatility and usability of double perovskites
based on 3d and 4d/5d TM ions to realize a clean topological band crossing
primarily resulting from the 4d/5d-t2g state with a large AHC. Our findings
pave the way for new high-performance stable magnetic topological materials
based on oxide double perovskites.

##  V Acknowledgements

This work is financially supported by the European Research Council (ERC
Advanced Grant No. 742068 ‘TOPMAT’). We also acknowledge funding by the DFG
through SFB 1143 (project ID. 247310070) and the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC2147,
project ID. 39085490). M.G.V. and I.R. thanks support from the Spanish
Ministerio de Ciencia e Innovacion (grant PID2019-109905GB-C21) and European
Research Council (ERC) grant agreement no. 101020833. M.G.V. and C.F.
acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – FOR 5249 (QUAST).

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# Collisionless and Decentralized Formation Control for Strings 111This paper
was not presented at any IFAC meeting. Corresponding author D. Kalise Tel. +44
738 039 1139.

Young-Pil Choi<EMAIL_ADDRESS>Dante Kalise
<EMAIL_ADDRESS>Andrés A. Peters<EMAIL_ADDRESS>Department of Mathematics, Yonsei University, 50 Yonsei-Ro, Seodaemun-Gu,
Seoul 03722, Republic of Korea  School of Mathematical Sciences, University of
Nottingham, University Park, Nottingham NG7 2QL, United Kingdom  Faculty of
Engineering and Sciences, Universidad Adolfo Ibáñez, Santiago 7941169, Chile

###### Abstract

A decentralized feedback controller for multi-agent systems, inspired by
vehicle platooning, is proposed. The closed-loop resulting from the
decentralized control action has three distinctive features: the generation of
collision-free trajectories, flocking of the system towards a consensus state
in velocity, and asymptotic convergence to a prescribed pattern of distances
between agents. For each feature, a rigorous dynamical analysis is provided,
yielding a characterization of the set of parameters and initial
configurations where collision avoidance, flocking, and pattern formation is
guaranteed. Numerical tests assess the theoretical results presented.

###### keywords:

Multi-agent systems, decentralized control, nonlinear control, consensus and
pattern formation control, autonomous vehicles.

##  1 Introduction

Multi-agent systems (MAS) have proven to be a versatile framework for studying
diverse scalability problems in Science and Engineering, such as dynamic
networks [olfati], autonomous vehicles [balch], collective behaviour of humans
or animals [sumpter, vicsek], and many others. Mathematically, MAS are often
modelled as large-scale dynamical systems where each agent can be considered
as a subset of states, updated via interaction forces such as attraction,
repulsion, alignment, etc., [helbing, cs07] or through the optimization of a
pay-off function in a control/game framework [lasry, huang].

In this work, we approach the study of MAS from a control viewpoint. We study
a class of sparsely interconnected agents in one dimension, interacting
through nonlinear couplings and a decentralized control law. The elementary
building block of our approach is the celebrated Cucker-Smale model for
consensus dynamics [cs07], which corresponds to a MAS where each agent is
endowed with second-order nonlinear dynamics for velocity alignment, and where
the influence of neighbouring agents decays with distance. The Cucker-Smale
model and variants can represent the physical motion of agents on the real
line, inspired by autonomous vehicle formations in platooning with a nearest-
neighbour interaction scheme [sepahe04, wang2019survey]. The couplings to be
studied are motivated by the more general setting of the Cucker-Smale dynamics
in arbitrary dimension. The original Cucker-Smale dynamics considers full
network connectivity in the agent interactions, generating flocking dynamics
capable of exhibiting emergent consensus behavior, that is, agents that may
reach a common velocity in steady state, without the action of external
forces. This framework has been extended in several directions, being most
notable the inclusion of forcing terms and control [cuckerhuepe, HaHaKim,
cfpt, hk18], optimal control [bbck, bfk, akkjmlr] formation control [pkh10,
perea, ckpp19], leadership [dalmao] and collision-avoidance capabilities [CD,
achl, bf, Choi2017], the latter being an increasingly sought after property of
formation control schemes for autonomous fleets of vehicles.

The main contribution of this paper is to propose a nearest neighbour
interaction of agents on the real line which exhibits emergent consensus and
collision avoidance under the action of a simple decentralised control law.
The proposed feedback enforces a desired inter-agent distance in steady state
and is inspired by formation control in vehicular platoons [sepahe04, Peters].
Such a model has the potential to achieve the goals of platooning
applications, that is, the coordinated, scalable, secure and efficient travel
of automated vehicles [feng2019], while at the same time offering flocking,
pattern formation, and collision avoidance features from the non-linear
dynamics. Moreover, we provide collision-avoidance guarantees that, from a
safety viewpoint, do not rely on traditional concepts in platooning such as
string stability. For the derivation of collision-avoidance results we
consider the framework developed in [cch14, CCMP, mp18], which uses singular
interaction kernels that blow-up whenever two agents are located at the same
position. We modify this setup to also consider agents with a _volume_ by
including a threshold inter-agent distance where the kernel becomes singular.

Our main result is a rigorous characterization of flocking, collision-
avoidance, and platooning behaviour for the proposed nonlinear model, in terms
of the initial configuration of the system, interaction and control law
parameters. Cucker-Smale models usually consider full state measurements of
every agent, available to every agent, which reflects a free interaction
between them giving rise to self-organization properties (see for example the
recent work [ZHANG2020201] where the full interaction Cucker-Smale model is
characterised on the real line). We show that these properties are still
present when a highly sparse nearest neighbour interaction is considered in
the 1D case.

The remainder of the paper is structured as follows. In Section 2 we present
the proposed model to be studied, a Cucker-Smale model with nearest neighbor
singular interactions and a decentralized feedback control. We also define
here a total energy functional $E(x,v)$ for the model and show that it is not
increasing in time. In Section 3 we give the results ensuring the collision-
avoidance behaviour of the controlled system, and Section LABEL:flocking
includes a flocking estimate showing that the velocity alignment between
individuals and the inter-agent distances are uniformly bounded in time. In
Section LABEL:formation we present the main formation control result. We
provide different numerical experiments that illustrate our theoretical
results in Section LABEL:numerics along with some concluding remarks.

##  2 Problem description and preliminary results

We consider a string of $N$ agents each characterized by a pair
$(x_{i}(t),v_{i}(t))$ in $\mathbb{R}^{2}$ evolving in time $t$ through second-
order dynamics of the form

$\displaystyle\left\\{\begin{aligned} \frac{dx_{i}(t)}{dt}&=v_{i}(t),\quad
i=1,\dots,N,\quad
t>0,\cr\frac{dv_{i}(t)}{dt}&=I_{i}(t)+u_{i}(t),\end{aligned}\right.$ (1)

subject to initial data

$\displaystyle(x_{i}(0),v_{i}(0))=:(x_{i}^{0},v_{i}^{0})\quad\mbox{for}\quad
i=1,\dots,N.$ (2)

Figure 1: Diagram at a particular instant of three consecutive agents for the
considered MAS. The singular interactions, providing _barriers_ to the agents,
are indicated with the radii $\delta_{i}$ of the semicircles.

Here, the term $I_{i}$ describes nonlocal velocity interactions between
individuals which are weighted by a singular communication function
$\psi(r):\mathbb{R}^{+}\longrightarrow\mathbb{R}$,

$\displaystyle\left\\{\begin{aligned}
&I_{1}=\psi(|x_{2}-x_{1}|-\delta_{1})(v_{2}-v_{1}),&\cr&I_{k}=\psi(|x_{k}-x_{k-1}|-\delta_{k-1})(v_{k-1}-v_{k})&\cr&\hphantom{I_{N}=}+\psi(|x_{k+1}-x_{k}|-\delta_{k})(v_{k+1}-v_{k}),&\cr&I_{N}=\psi(|x_{N}-x_{N-1}|-\delta_{N-1})(v_{N-1}-v_{N}),&\end{aligned}\right.\qquad
k=2,\dots,N-1,$

where the parameters $\delta_{i}>0$ are fixed. This interaction term induces
consensus in the velocities of the agents while preventing collisions. The
second term $u_{i}$ serves as a decentralized feedback control depending on
weight function $\phi(r):\mathbb{R}^{+}\longrightarrow\mathbb{R}$ and is given
by

$\displaystyle\left\\{\begin{aligned}
&u_{1}=-\phi(|x_{1}-x_{2}-z_{1}|^{2})(x_{1}-x_{2}-z_{1}),\cr&u_{k}=\phi(|x_{k-1}-x_{k}-z_{k-1}|^{2})(x_{k-1}-x_{k}-z_{k-1})\\\
&\hphantom{u_{k}=}-\phi(|x_{k}-x_{k+1}-z_{k}|^{2})(x_{k}-x_{k+1}-z_{k}),\cr&u_{N}=\phi(|x_{N-1}-x_{N}-z_{N-1}|^{2})(x_{N-1}-x_{N}-z_{N-1}),\end{aligned}\right.\qquad
k=2,\dots,N-1\,.$

This feedback also depends on a vector of relative distances
$z:=(z_{1},\dots,z_{N-1})\in\mathbb{R}^{N-1}$. The objective of this control
law is to induce the formation of a string pattern characterized by $z$. The
complete setting is depicted in Figure 1. For the sake of clarity, the weight
functions $\psi$ and $\phi$ are chosen as

$\displaystyle\psi(r)=\frac{1}{r^{\alpha}},\quad\text{and}\quad\phi(r)=\frac{1}{(1+r)^{\beta}},\quad\alpha,\beta>0\,.$
(3)

However, the results we will state in the forthcoming sections can extended to
the case in which $\phi$ is bounded and Lipschitz continuous. We will
establish conditions under which the string (1) converge to a consensus state
with a prescribed formation while avoiding collisions between agents. For
this, we begin by stating an a priori energy estimate which will be
significantly used for estimating consensus emergence. We first define the
total energy functional

$\displaystyle E(x,v)$
$\displaystyle:=E_{1}(v)+E_{2}(x)=\frac{1}{4N}\sum_{i,j=1}^{N}|v_{i}-v_{j}|^{2}+\frac{1}{2}\sum_{i=2}^{N}\int_{0}^{|x_{i-1}-x_{i}-z_{i-1}|^{2}}\phi(r)\,dr,$

and its dissipation rate

$D(x,v):=\sum_{i=2}^{N}\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i}-v_{i-1})^{2}$

###### Lemma 1

Let $\\{(x_{i},v_{i})\\}_{i=1}^{N}$ be a smooth solution to the system (1) on
the time interval $[0,T]$. Then:

  * (i)

the mean velocity is conserved in time:

$v_{c}(t):=\frac{1}{N}\sum_{i=1}^{N}v_{i}(t)=v_{c}(0)\,.$

  * (ii)

the total energy is not increasing in time:

$\frac{d}{dt}E(x(t),v(t))+D(x(t),v(t))=0\,.$

###### Proof 1

(i) A straightforward computation yields

$\sum_{i=1}^{N}I_{i}=\sum_{i=1}^{N}u_{i}=0.$

This implies

$\frac{d}{dt}v_{c}(t)=0,\quad\mbox{i.e.}\quad v_{c}(t)=v_{c}(0)$ (4)

for $t\in[0,T]$.

(ii) Let us first begin with the estimate for the kinetic energy:

$\displaystyle\begin{aligned}
\frac{1}{4N}\frac{d}{dt}\sum_{i,j=1}^{N}(v_{i}-v_{j})^{2}&=\frac{1}{2N}\sum_{i,j=1}^{N}(v_{i}-v_{j})\left(\frac{dv_{i}}{dt}-\frac{dv_{j}}{dt}\right)=\frac{1}{2N}\sum_{i,j=1}^{N}\left(v_{i}\frac{dv_{i}}{dt}+v_{j}\frac{dv_{j}}{dt}\right)\cr&=\sum_{i=1}^{N}v_{i}\frac{dv_{i}}{dt}=\sum_{i=1}^{N}v_{i}(I_{i}+u_{i}),\end{aligned}$
(5)

where we used (4). Here, we use the same idea of [ckpp19, Lemma 3.1] to obtain

$\displaystyle\begin{aligned}
\sum_{i=1}^{N}v_{i}u_{i}=-\frac{1}{2}\frac{d}{dt}\sum_{i=2}^{N}\int_{0}^{|x_{i-1}-x_{i}-z_{i-1}|^{2}}\phi(r)\,dr.\end{aligned}$
(6)

On the other hand, we estimate the term with $I_{i}$ as

$\displaystyle\sum_{i=2}^{N-1}v_{i}I_{i}=$
$\displaystyle\sum_{i=2}^{N-1}v_{i}\left(\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})\right.\left.+\psi(|x_{i}-x_{i+1}|-\delta_{i})(v_{i+1}-v_{i})\right)$
$\displaystyle=$
$\displaystyle\sum_{i=2}^{N-1}v_{i}\left(\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})\right)+\sum_{i=3}^{N}v_{i-1}\left(\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i}-v_{i-1})\right)$
$\displaystyle=$ $\displaystyle
v_{2}\psi(|x_{2}-x_{1}|-\delta_{1})(v_{1}-v_{2})-\sum_{i=3}^{N-1}\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})^{2}+v_{N-1}\psi(|x_{N}-x_{N-1}|-\delta_{N-1})(v_{N}-v_{N-1})$
$\displaystyle=$ $\displaystyle-
v_{2}I_{1}-v_{N-1}I_{N}-\sum_{i=3}^{N-1}\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})^{2}.$

This asserts

$\displaystyle\begin{aligned}
\sum_{i=1}^{N}v_{i}I_{i}&=v_{1}I_{1}+\sum_{i=2}^{N-1}v_{i}I_{i}+v_{N}I_{N}=-(v_{2}-v_{1})I_{1}-\sum_{i=3}^{N-1}\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})^{2}-(v_{N-1}-v_{N})I_{N}\cr&=-\sum_{i=2}^{N}\psi(|x_{i}-x_{i-1}|-\delta_{i-1})(v_{i-1}-v_{i})^{2}.\end{aligned}$
(7)

Combining the estimates (5), (7), and (6), we conclude the desired result.

##  3 Global and local existence of solutions

In this section, we show that system (1), under certain parametric and initial
conditions, exhibits a non-collisional behaviour, which together with Cauchy-
Lipschitz theory, subsequently provides global-in-time existence and
uniqueness of smooth solutions to the system (1)-(2). We present two results
regarding existence of non-collisional trajectories. The first theorem
requires a prescribed ordering for the initial datum $x_{i}^{0}$ and the power
$\alpha$ in $\eqref{kers}$ to be $\alpha\geq 1$. The second result requires
$\alpha\geq 2$, but the initial ordering assumption is removed from the
string. A third result characterizes a pathological case, where a 2-agent
string blows up in finite time.

###### Theorem 1

Suppose that $\alpha\geq 1$ and the initial configuration $x_{0}$ satisfies
$x_{i+1}^{0}>x_{i}^{0}+\delta_{i}$ for all $i=1,\dots,N-1$. Then, there exists
the global unique smooth solution to the system (1)-(2) satisfying
$x_{i+1}(t)>x_{i}(t)+\delta_{i}$ for all $i=1,\dots,N-1$ and all $t>0$.


that privacy cost and then compute the maximum variance to target variance
ratio for each method. The results for the uniform targets are shown in Table
5, with SM-II achieving $\approx 25\%$ improvement over the best competitor.
The results for the random targets are shown in Table 6, with SM-II improving
by at least 40% over competitors. In the uniform case, the _sum squared error_
of SM-II ranges from 1.04 to 1.09 times that of CA, depending on the value of
$d$.

Table 5. Maximum ratio of achieved variance to target variance on Range Queries with uniform accuracy targets. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
d = 64 | 10.63 | 2.84 | 1.27 | 1.27 | 1.27 | 1.00
d = 128 | 17.45 | 3.48 | 1.25 | 1.25 | 1.25 | 1.00
d = 256 | 28.43 | 3.90 | 1.29 | 1.29 | 1.29 | 1.00
d = 512 | 47.36 | 4.68 | 1.23 | 1.23 | 1.23 | 1.00
d = 1024 | 77.26 | 5.42 | 1.24 | 1.24 | 1.24 | 1.00
Table 6. Maximum ratio of achieved variance to target variance on Range Queries with random accuracy targets. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
d = 64 | 16.04 | 4.63 | 2.53 | 1.62 | 1.98 | 1.00
d = 128 | 21.28 | 5.87 | 2.14 | 1.49 | 1.75 | 1.00
d = 256 | 41.76 | 6.05 | 1.91 | 1.45 | 1.69 | 1.00
d = 512 | 71.51 | 7.35 | 1.80 | 1.46 | 1.52 | 1.00
d = 1024 | 109.34 | 7.68 | 1.69 | 1.43 | 1.49 | 1.00
Table 7. Maximum achieved variance to target variance ratio on Random Queries with a Uniform target variance. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
d = 64 | 1.14 | 2.29 | 1.14 | 1.14 | 1.14 | 1.00
d = 128 | 1.14 | 2.29 | 1.07 | 1.07 | 1.07 | 1.00
d = 256 | 1.14 | 3.39 | 1.06 | 1.06 | 1.06 | 1.00
d = 512 | 1.15 | 3.40 | 1.05 | 1.05 | 1.05 | 1.00
d = 1024 | 1.15 | 3.41 | 1.04 | 1.04 | 1.04 | 1.00
Table 8. Maximum achieved variance to target variance ratio on Random Queries with a Random target variance. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
d = 64 | 2.58 | 7.44 | 2.43 | 1.15 | 1.61 | 1.00
d = 128 | 2.82 | 8.28 | 2.58 | 1.18 | 1.72 | 1.00
d = 256 | 2.82 | 8.36 | 2.58 | 1.17 | 1.70 | 1.00
d = 512 | 2.79 | 8.27 | 2.51 | 1.16 | 1.68 | 1.00
d = 1024 | 2.76 | 8.17 | 2.49 | 1.16 | 1.67 | 1.00

###  7.5. Random Queries

We next consider a random query workload similar to the setting in (Yuan et
al., 2012). Here for each entry in each $d$-dimensional query vector, we flip
a coin with $P(\text{heads})=0.2$. If it lands heads, the entry is set to 1,
otherwise it is set to $-1$. Again we consider two scenarios. (1) the uniform
cases where the target variance bound is equal to $1$ for each query, and (2)
the random case where the target variance bound for each query is randomly
sampled from a uniform$(1,10)$ distribution. We vary
$d\in\\{64,128,256,512,1024\\}$, and the number of queries is $m=2*d$. The
basis matrix is $\mathbf{B}=\mathbf{I}$.

Table 7 lists the maximum variance ratio among all queries under the same
privacy cost for different methods for the uniform scenario. Table 8 shows the
corresponding results for random targets. In both cases, the best competitor
is not much worse than SM-II. We speculate that this is because the queries do
not have much structure that can be exploited to reduce privacy cost. In terms
of _sum squared error_ , for the case of uniform target variances, the total
error of SM-II was at most $1.005$ times that of CA.

Table 9. Maximum achieved variance to target variance ratio for Age pyramids with uniform accuracy targets. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
AGE | 32.49 | 6.2 | 1.58 | 1.58 | 1.58 | 1.00

###  7.6. Age Pyramids

Demographers often study the distribution of ages in a population by gender.
We consider a dataset schema similar to the Census, where gender is a binary
attribute and there are 116 ages (0-115). The range queries we consider are
prefix queries (i.e., age $\in[0,x]$ for all $x$) and age $\in[18,115]$ (the
voting age population). The workload consists of (1) range queries for males,
(2) range queries for females, (3) range queries  for all. We set uniform
target variance bounds of 1. We use the basis matrix $\mathbf{B}=\mathbf{U}$
(upper triangular). Table 9 shows the maximum ratio of achieved variance to
target variance for each algorithm, with SM-II clearly outperforming the
competitors by at least 58%. Meanwhile the sum squared error of SM-II is 1.07
times that of CA.

###  7.7. Marginals

We next consider histograms $H$ on $r$ variables, where each variable can take
on one of $t$ values, resulting in a domain size of $d=t^{r}$. We consider the
workload consisting of all one-way and two-way marginals with uniform target
variance of 1. For this set of experiments, we set $r=3$ and varied the values
of $t$. We used the basis matrix is $\mathbf{B}=\mathbf{I}$ for SM-II. Table
10 shows the maximum target variance ratio of different algorithms. SM-II
outperforms the competitors, especially as $t$ increases.

Table 10. Maximum achieved variance to target variance ratio on Marginal Queries when $r=3$ Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
t = 2 | 1.82 | 3.02 | 1.14 | 1.14 | 1.14 | 1.00
t = 4 | 4.55 | 10.28 | 1.42 | 1.42 | 1.42 | 1.00
t = 6 | 8.6 | 21.38 | 1.66 | 1.66 | 1.66 | 1.00
t = 8 | 14.03 | 36.72 | 1.88 | 1.88 | 1.88 | 1.00
t = 10 | 20.84 | 56.23 | 2.08 | 2.08 | 2.08 | 1.00
t = 12 | 28.76 | 78.96 | 2.25 | 2.25 | 2.25 | 1.00
t = 14 | 38.17 | 106.22 | 2.41 | 2.41 | 2.41 | 1.00
t = 16 | 48.85 | 137.41 | 2.57 | 2.57 | 2.57 | 1.00

The ratio of sum squared error of SM-II to CA ranged from 1.1 ($t=2$) to 1.34
($t=16$).

###  7.8. WRelated Queries

We next consider the one-dimensional WRelated workload of (Yuan et al., 2012).
The workload matrix is $W=CA$. Here matrix $C$ has size $m\times s$ and matrix
$A$ has size $s\times d$, each follows the Gaussian(0, 1) distribution. In the
experiment we set $m=d/2,s=d/2$. Due to space constraints, we only show
results for the uniform target variance bound of 1. We vary the domain size
$d\in\\{64,128,256,512,1024\\}$, and use the basis matrix
$\mathbf{B}=\mathbf{I}$. The results in Table 11 shows SM-II outperforming
competitors by around 31% on the smaller domain sizes. The ratio of total
squared error for SM-II to that of CA was at most 1.08

Table 11. Maximum achieved variance to target variance ratio on WRelated Queries with a Uniform target variance. Mechanism | IP | HM | CA | wCA-I | wCA-II | SM-II
---|---|---|---|---|---|---
d = 64 | 3.51 | 9.59 | 1.31 | 1.31 | 1.31 | 1.00
d = 128 | 4.16 | 11.16 | 1.31 | 1.31 | 1.31 | 1.00
d = 256 | 4.28 | 12.60 | 1.19 | 1.19 | 1.19 | 1.00
d = 512 | 3.62 | 10.46 | 1.12 | 1.12 | 1.12 | 1.00
d = 1024 | 3.50 | 10.05 | 1.07 | 1.07 | 1.07 | 1.00

##  8\. Conclusions and Future Work

In this work we introduce the fitness-for-use problem, where the goal is to
calculate minimal privacy cost under accuracy constraints for each query.
After theoretical analysis of the problem, we proposed an algorithm named SM-
II to solve it. While our algorithm used variance as the accuracy measure for
each query, other applications may require their own specific accuracy
measures. This consideration leads to two important directions for future
work. The first is to create optimized differentially private algorithms that
meet fitness-for-use goals under measures other than squared error. The second
direction is to achieve the same kind of extreme scalability as the high-
dimensional matrix mechanism (McKenna et al., 2018) that provided a tradeoff
between optimality and scalability.

## Acknowledgments

This work was supported by NSF Awards CNS-1702760 and CNS-1931686. We are
grateful to Aleksandar Nikolov for discussions about factorization mechanisms.

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##  Appendix A Proofs from Section 3

[section3]

##  Appendix B Proofs from Section 5

[section5]

###  B.1. Proofs of Uniqueness for Solution Results

[uniqueness]

###  B.2. Proofs from Section 6

[approx]