# More on Jensen functional and convexity

Shoshana Abramovich  Department of Mathematics, University of Haifa, Haifa,
Israel<EMAIL_ADDRESS>
(Date: May 22, 2024)

###### Abstract.

In this paper we prove results on the difference between a normalized Jensen
functional and the sum of other normalized Jensen functionals for convex
function.

###### Key words and phrases:

Jensen Functionals, Convexity, Uniform Convexity

###### 1991 Mathematics Subject Classification:

26D15, 26A51, 39B62, 47A63, 47A64

##  1\.
Introduction

In this paper we prove results on the difference between a normalized Jensen
functional and the sum of other normalized Jensen functionals for convex
function.

Jensen functional is:

$J_{n}\left(f,\mathbf{x},\mathbf{p}\right)=\sum_{i=1}^{n}p_{i}f\left(x_{i}\right)-f\left(\sum_{i=1}^{n}p_{i}x_{i}\right).$

We start with the theorem of S. S. Dragomir:

######  Theorem 1.

[2] Consider the normalized Jensen functional where
$f:C\longrightarrow\mathbb{R}$ is a convex function on the convex set $C$ in a
real linear space, $\mathbf{x}=\left(x_{1},...,x_{n}\right)\in C^{n},$ and
$\mathbf{p}=\left(p_{1},...,p_{n}\right),$
$\mathbf{q}=\left(q_{1},...,q_{n}\right)$ are non-negative n-tuples satisfying
$\sum_{i=1}^{n}p_{i}=1,$ $\sum_{i=1}^{n}q_{i}=1,$ $p_{i},$ $q_{i}>0,$
$i=1,...,n$. Then

$MJ_{n}\left(f,\mathbf{x},\mathbf{q}\right)\geq
J_{n}\left(f,\mathbf{x},\mathbf{p}\right)\geq
mJ_{n}\left(f,\mathbf{x},\mathbf{q}\right),\qquad$

provided that

$m=\min_{1\leq i\leq n}\left(\frac{p_{i}}{q_{i}}\right),\quad M=\max_{1\leq
i\leq n}\left(\frac{p_{i}}{q_{i}}\right).$

This important theorem was extended in [1], [3], [4] and [5] using a similar
technique as we use here.

In Theorem 2 the right handside of the inequality in Theorem 1 is extended.

In Theorem 2 the following conditions and notations are used:

$0\leq p_{i,1}\leq 1$, $0<q_{i}\leq 1,\
\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i}=1$.

Denote $m_{1}=\underset{i=1,...,n}{\min}\left(\frac{p_{i,1}}{q_{i}}\right)$,
and $s_{1}$ the number of $i$-th for which $m_{1}$ occur.

Define recoursively

$\displaystyle p_{i,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}p_{i,k-1}-m_{k-1}q_{i},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i,}}\\\
\frac{1}{s_{k-1}}m_{k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i}}\end{array}\right.,$
$\displaystyle m_{k-1}$ $\displaystyle=$
$\displaystyle\begin{array}[]{cc}\underset{1\leq i\leq
n}{\min}\left(\frac{p_{i,k-1}}{q_{i}}\right),&k=2,...,N\end{array}\qquad\qquad$

and denote $s_{k-1}$ as the number of cases for which $m_{k-1}$ occurs.

Let also
$\mathbf{x}_{1}=\left(x_{1,1},x_{2,1},...,x_{n,1}\right)\in\left(a,b\right)^{n}$,
and define recoursively

$x_{i,k}=\left\\{\begin{array}[]{cc}x_{i,k-1},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i}}\\\
\sum_{i=1}^{n}q_{i}x_{i,k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i}}\end{array}\right.,$

$i=1,...n$, $k=2,...,N$.

######  Theorem 2.

[1, Theorem 5] Suppose that $f:\left[a,b\right)\rightarrow\mathbb{R},$
$a<b\leq\infty$ is a convex function. Then, for every integer $N,$

(1.3)
$J_{n}\left(f,\mathbf{x}_{1},\mathbf{p}_{1}\right)-\sum_{k=1}^{N}m_{k}J_{n}\left(f,\mathbf{x}_{k},\mathbf{q}\right)\geq
0,$

where $m_{k}$ and $\mathbf{x}_{k}=$ $\left(x_{1,k},...,x_{n,k}\right),$
$k=1,...,N$ are as defined above, and
$\mathbf{p}_{1}=\left(p_{1,1},...,p_{n,1}\right),$
$\mathbf{q=}\left(q_{1},...,q_{n}\right)\mathbf{,}$ and
$\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i}=1,$ and $p_{i,1}\geq 0,$ $q_{i}>0,$
$\ \ i=1,...,n$, $m_{1}=\underset{1\leq i\leq
n}{\min}\left(\frac{p_{i,1}}{q_{i}}\right).$

In Theorem 3 the left hand-side inequality in Theorem 1 is extended.

Denote

$\displaystyle\mathbf{p}_{1}$ $\displaystyle=$
$\displaystyle\left(p_{1,1},...,p_{1,n}\right),\quad\mathbf{q=}\left(q_{1},...,q_{n}\right),$
$\displaystyle\quad\mathbf{x}_{k}$ $\displaystyle=$
$\displaystyle\left(x_{1,k},...,x_{n,k}\right),\quad k=1,...,N$ $\displaystyle
p_{i,1}$ $\displaystyle\geq$ $\displaystyle 0,\quad q_{i}>0,\quad
i=1,...,n,\quad\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i}=1,$ $\displaystyle
M_{1}$ $\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n}{Max}\left(\frac{p_{i,1}}{q_{i}}\right)=\frac{p_{j_{1},1}}{q_{j_{1}}},$

where $j_{1}$ is a fixed specific integer for which $M_{1}$ holds.

Define recoursively

$\displaystyle p_{i,k}$ $\displaystyle=$ $\displaystyle
p_{i,k-1}-M_{k-1}q_{i},\ x_{i,k}=x_{i,k-1},\ \text{when}\
M_{k-1}\neq\frac{p_{i,k-1}}{q_{i}},\ k=2,...,N$ $\displaystyle p_{i,k}$
$\displaystyle=$ $\displaystyle p_{i,k-1}-M_{k-1}q_{i},\ x_{i,k}=x_{i,k-1},\
\text{when}\ M_{k-1}=\frac{p_{i,k-1}}{q_{i}},\ i\neq j_{k},\ k=2,...,N$
$\displaystyle p_{j_{k},k}$ $\displaystyle=$ $\displaystyle M_{k-1},\
x_{j_{k},k}=\sum_{i=1}^{n}q_{i}x_{i,k-1},\ \text{when}\
M_{k-1}=\frac{p_{j_{k},k-1}}{q_{j_{k-1}}},$ $\displaystyle M_{k}$
$\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n}{Max}\left(\frac{p_{i,k}}{q_{i}}\right)=\frac{p_{j_{k},k}}{q_{j_{k}}},\qquad
k=1,...,N,$

where $j_{k}$ is a specific index for which $M_{k}$ holds.

With the notations and conditions above the following is obtained:

######  Theorem 3.

Let $f:\left[a,b\right)\rightarrow\mathbb{R},$ $a\leq b\leq\infty,$ be a
convex function, and let the notations and conditions above hold. Then, for
every integer $N$

$J_{n}\left(f,\mathbf{x}_{1},\mathbf{p}_{1}\right)-\sum_{k=1}^{N}M_{k}J_{n}\left(f,\mathbf{x}_{k},\mathbf{q}\right)\leq
0,$

and

$M_{k}=\frac{p_{j_{1},1}}{q_{j_{1}}^{k}},\qquad k=1,...,N$

hold, where $j_{1}$ is a fixed specific integer for which
$M_{1}=\underset{1\leq i\leq
n}{\max}\left(\frac{p_{i,1}}{q_{i}}\right)=\frac{p_{j_{1},1}}{q_{j_{1}}}$ is
satisfied.

In Section 2, theorems 6 and 7 we extend Theorem 2 and Theorem 3, hence also
[3, Theorem 1] by replacing $\mathbf{q=}\left(q_{1},...,q_{n}\right)$ in the
$N$ Jensen functionals with $\mathbf{q}_{k},$ $k=1,...,N$.

The extensions in theorems 8 and 9 are obtained by replacing in Theorem 2 and
Theorem 3 $\mathbf{q=}\left(q_{1},...,q_{n}\right)$ in the $N$ Jensen
functional with $\mathbf{q}_{k}=\left(q_{1,k},...,q_{n_{k},k}\right)$, when
$n_{k+1}\leq n_{k}\leq n_{1},$ $k=1,...,N$. These theorems extend the
following theorem in [4] and [5].

######  Theorem 4.

[4]. Let $\alpha=\left(\alpha_{1},...\alpha_{n}\right),$
$\beta=\left(\beta,...,\beta_{n}\right),$
$\gamma=\left(\gamma_{1},...,\gamma_{n}\right)\in p_{n}$ satisfy that
$\beta_{i}+\gamma_{i}>0$ for each $i=1,...,n$. If $f$ is a convex function on
an interval $I:=\left[a,b\right]$ and
$\mathbf{x}=\left(x_{1},...,x_{n}\right)\in I^{n}$, then

$\displaystyle\underset{1\leq i\leq
n}{\min}\left\\{\frac{\alpha_{i}}{\beta_{i}+\gamma_{i}}\right\\}\left[J_{n}\left(f,\mathbf{x},\beta\right)+J_{n}\left(f,\mathbf{x},\gamma\right)\right]$
$\displaystyle\leq$ $\displaystyle J_{n}\left(f,\mathbf{x},\alpha\right)\leq
2\underset{1\leq i\leq
n}{\max}\left\\{\frac{\alpha_{i}}{\beta_{i}+\gamma_{i}}\right\\}J_{n}\left(f,\mathbf{x},\left(\frac{\beta+\gamma}{2}\right)\right).$

######  Theorem 5.

[5, Theorem 2.1] Under the hypotheses and notations as in Theorem 1, we have
the Jensen-Dragomir type inequalties

$mJ_{n}\left(f,\mathbf{x},\beta\right)+\mathbf{m}^{\ast}\left(\left|J\right|+1\right)H_{j}\leq
J_{n}\left(f,\mathbf{x},\alpha\right)\leq
mJ_{n}\left(f,\mathbf{x},\beta\right)+\mathbf{M}^{\ast}\left(\left|J\right|+1\right)H_{j},$

where $J=\left\\{i:\alpha_{i}-\beta_{i}\neq 0\right\\},$ $\left|J\right|$ is
the cardinal of $J$, $\mathbf{m}^{\ast}=$ $\underset{i\in
J}{min}\left\\{m,\alpha_{i}-m\beta_{i}\right\\},$
$\mathbf{M}^{\ast}=\underset{i\in
J}{\max}\left\\{m,\alpha_{i}-m\beta_{i}\right\\}$, and

$H_{J}:=\frac{1}{\left|J\right|+1}\left[\sum_{i\in
J}f\left(x_{i}\right)+f\left(\sum_{i=1}^{n}\beta_{i}x_{i}\right)\right]-f\left(\frac{1}{\left|J\right|+1}\left(\sum_{i\in
J}x_{i}+\sum_{i=1}^{n}\beta_{i}x_{i}\right)\right).$

##  2\.
Convexity and extended normalized Jensen functional

We start this section with theorems 6 and 7 which extend theorems 2 and 3. All
we need to do in order to prove theorems 6 and 7 is to replace in in theoems 2
and 3 $\mathbf{q=}\left(q_{1},...,q_{n}\right)$ with
$\mathbf{q}_{k}\mathbf{=}\left(q_{1,k},...,q_{n,k}\right),$ $k=1,...,N$.
Therefore we do not find it necessary to prove these two theorems but only to
emphasize where the notations, definitions and conditions differ.

In Theorem 6 we use the following conditions and notations:

$0\leq p_{i,1}\leq 1$, $0<q_{i,k}\leq 1,\ i=1,...,n$, $\
\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i,k}=1,$ $k=1,...,N$.

Denote $m_{1}=\underset{1\leq i\leq
n}{\min}\left(\frac{p_{i,1}}{q_{i,1}}\right)$ and $s_{1}$ the number of $i$-th
for which $m_{1}$ occur.

Define recoursively

(2.3) $\displaystyle p_{i,k}$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}p_{i,k-1}-m_{k-1}q_{i,k-1},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}}\\\
\frac{1}{s_{k-1}}m_{k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i,k-1}}\end{array}\right.,\qquad
k=2,...$ (2.5) $\displaystyle m_{k-1}$ $\displaystyle=$
$\displaystyle\begin{array}[]{cc}\underset{1\leq i\leq
n}{\min}\left(\frac{p_{i,k-1}}{q_{i,k-1}}\right),&k=2,...,\end{array}\qquad\qquad$

and denote $s_{k-1}$ as the number of cases for which $m_{k-1}$ occurs.

Let also $x_{i,1}\in\left(a,b\right),$ $i=1,...,n$ and define recoursively

(2.6)
$x_{i,k}={\Large\\{}\begin{array}[]{cc}x_{i,k-1},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}}\\\
\sum_{i=1}^{n}q_{i,k-1}x_{i,k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i,k-1}}\end{array},$

$i=1,...n$, $k=2,...,N$.

We emphasize that $x_{i,k},$ $m_{k}$ in (2.6) means that $\mathbf{x}_{k},\
$and $m_{k}$, $k=2,...,N$ are recoursively constructed, but $\mathbf{q}_{k},$
$k=1,...,N$ are apriori given and not constructed recoursively.

######  Theorem 6.

Suppose that $f:\left[a,b\right)\rightarrow\mathbb{R},$ $a<b\leq\infty$ is a
convex function. Then, for every integer $N,$

(2.7)
$J_{n}\left(f,\mathbf{x}_{1},\mathbf{p}_{1}\right)-\sum_{k=1}^{N}m_{k}J_{n}\left(f,\mathbf{x}_{k},\mathbf{q}_{k}\right)\geq
0,$

where $\mathbf{p}_{1}=\left(p_{1,1},...,p_{n,1}\right),$
$\mathbf{q}_{k}\mathbf{=}\left(q_{1,k},...,q_{n,k}\right)\mathbf{,}$
$\mathbf{x}_{k}=$ $\left(x_{1,k},...,x_{n,k}\right),$ $k=1,...,N$, ,
$\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i,k}=1,$ $k=1,...,N$, $p_{i,1}\geq 0,$
$q_{i,k}>0,$ $i=1,...,n$, and $m_{k}=\underset{1\leq i\leq
n}{\min}\left(\frac{p_{i,k}}{q_{i,k}}\right)$ $k=1,...,N$, where $p_{i,k}$,
$m_{k},$ $x_{i,k},$ are as denoted in (2.3) and (2.6).

Replacing in Theorem 6 $\mathbf{q}_{i}=\mathbf{q}$, $i=1,...,N$, we get
Theorem 2. Replacing $\mathbf{q}_{i}=\mathbf{q}$, $i=1,...,N$ and replacing
$q_{i},$ $i=1,...,n$ by $\frac{1}{n}$ in Theorem 6 we get [3, Theorem 1].

Similarly to the extenssions in Theorem 6 of Theorem 2 ([1, Theorem 5]) we
extend now in Theorem 7 ([1, Theorem 6]), Theorem 3 and the left hand-side
inequality in Theorem 1.

We denote

$\displaystyle\mathbf{p}_{1}$ $\displaystyle=$
$\displaystyle\left(p_{1,1},...,p_{n,1}\right),\quad\mathbf{q}_{k}\mathbf{=}\left(q_{1,k},...,q_{n,k}\right),\quad
k=1,...,N$ $\displaystyle\text{where \ }p_{i,1}$ $\displaystyle\geq$
$\displaystyle 0,\ q_{i,k}>0,\ i=1,...,n,\
\sum_{i=1}^{n}p_{i,1}=\sum_{i=1}^{n}q_{i,k}=1,\ k=1,...,N$ $\displaystyle
M_{1}$ $\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n}{Max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}},\quad$

where $j_{1}$ is a fixed specific integer for which $M_{1}$ holds.

We define recoursively $p_{k}=\left(p_{1,k},...,p_{n,k}\right),$
$\mathbf{x}_{k}=\left(x_{1,k},...,x_{n,k}\right),$ and $M_{k}$, $\ k=1,...,N$
as follows:

$\displaystyle p_{i,k}$ $\displaystyle=$ $\displaystyle
p_{i,k-1}-M_{k-1}q_{i,k},\ x_{i,k}=x_{i,k-1},\ \text{when}\
M_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}},\ k=2,...,N$ $\displaystyle p_{i,k}$
$\displaystyle=$ $\displaystyle p_{i,k-1}-M_{k-1}q_{i,k-1},\
x_{i,k}=x_{i,k-1},\ \text{when}\ M_{k-1}=\frac{p_{i,k-1}}{q_{i,k-1}},\ i\neq
j_{k},\ k=2,...,N$ $\displaystyle p_{j_{k},k}$ $\displaystyle=$ $\displaystyle
M_{k-1},\ x_{j_{k},k}=\sum_{i=1}^{n}q_{i,k-1}x_{i,k-1},\ \text{when}\
M_{k-1}=\frac{p_{j_{k-1},k-1}}{q_{j_{k-1},k-1}},$ $\displaystyle M_{k}$
$\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n}{Max}\left(\frac{p_{i,k}}{q_{i,k}}\right)=\frac{p_{j_{k},k}}{q_{j_{k},k}},\qquad
k=1,...,N,$

where $j_{k}$ are specific indices for which $M_{k},$ $k=1,...,N$ hold.

With the notations and conditions in (LABEL:2.4) and (LABEL:2.5) we get:

######  Theorem 7.

Let $f:\left[a,b\right]\rightarrow\mathbb{R},$ be a convex function, and let
(LABEL:2.4) and (LABEL:2.5) hold. Then, for every integer $N$

(2.10)
$J_{n}\left(f,\mathbf{x}_{1},\mathbf{p}_{1}\right)-\sum_{k=1}^{N}M_{k}J_{n}\left(f,\mathbf{x}_{k},\mathbf{q}_{k}\right)\leq
0,$

and

(2.11) $M_{k}=\frac{p_{j_{1},1}}{\Pi_{m=1}^{k}\left(q_{j_{1},m}\right)},\qquad
k=1,...,N$

hold, where $j_{1}$ is a fixed specific integer for which
$M_{1}=\underset{1\leq i\leq
n}{Max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}}$
is satisfied.

Theorems 8 and 9 extend theorems1, 4 and 5.

The difference between Theorem 6 and Theorem 8 is that inTheorem 8 we
eliminate all the terms that satisfy
$\left(p_{j_{i},k}-m_{k}q_{j_{i},k}\right)f\left(x_{i}\right)$ where
$p_{j_{i},k}-m_{k}q_{j_{i},k}=0$ and add instead only one term equal to
$m_{k}f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)$ therefore
$\mathbf{q}_{k}=\left(q_{1},...,q_{n_{k}}\right),$ $k=1,...,N$ satisfy
$n_{1}\geq n_{2}\geq...\geq n_{N}$.

In Theorem 8 we use the following conditions and notations:

$0\leq p_{i,1}\leq 1$, $0<q_{i,1}\leq 1$, $i=1,...,n_{1},\
\sum_{i=1}^{n_{1}}p_{i,1}=\sum_{i=1}^{n_{1}}q_{i,1}=1,$
$m_{1}=\underset{i=1,...,n_{1}}{\min}\left(\frac{p_{i,1}}{q_{i,1}}\right)$.

We define recoursively,

(2.14)
$\displaystyle\begin{array}[]{cccc}p_{i,k}=&p_{j_{i},k-1}-m_{k-1}q_{j_{i},k-1},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}}&i=1,...,n_{k-1}-1\\\
p_{n_{k},k}=&m_{k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i,k-1}}&\end{array}$ (2.16)
$\displaystyle\begin{array}[]{ccc}q_{i,k-1}=q_{j_{i},k-1},&i=1,...,n_{k-1},&k=2,...,N\end{array}\qquad$
$m_{k-1}=\begin{array}[]{cc}\underset{1\leq i\leq
n_{k-1}}{\min}\left(\frac{p_{i,k-1}}{q_{i,k-1}}\right),&k=2,...N,\end{array}\qquad\qquad\qquad\qquad\qquad\qquad$

and

(2.17)
$\begin{array}[]{cccc}x_{i,k}=&x_{j_{i},k-1},&m_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}}&i=1,...,n_{k-1}-1\\\
x_{n_{k},k}=&\sum_{i=1}^{n}q_{i,k-1}x_{i,k-1},&m_{k-1}=\frac{p_{i,k-1}}{q_{i,k-1}}&\end{array}$

$k=2,...,N$.

Because of the similar techniques to those in the proof of Theorem 2 (see
[1]), we show here only an outline of the proof.

######  Theorem 8.

Suppose that $f:\left[a,b\right]\rightarrow\mathbb{R},$ is a convex function.
Then, for every integer $N,$

$J_{n_{1}}\left(f,\mathbf{x}_{1},\mathbf{p}_{1}\right)-\sum_{k=1}^{N}m_{k}J_{n_{k}}\left(f,\mathbf{x}_{k},\mathbf{q}_{k}\right)\geq
0,$

holds where $\mathbf{p}_{1}=\left(p_{1,1},...,p_{n_{1},1}\right),$
$\mathbf{q}_{k}\mathbf{=}\left(q_{1,k},...,q_{n_{k},k}\right)\mathbf{,}$
$\mathbf{x}_{k}=$ $\left(x_{1,k},...,x_{n_{k},k}\right),$ $n_{1}\geq
n_{2}\geq,...,\geq n_{k}\geq n_{k+1}\geq n_{N},\ k=1,...,N$, and $p_{i,k}$,
$m_{k},$ $x_{i,k},$ are as denoted in (2.14) and (2.17),
$\sum_{i=1}^{n_{1}}p_{i,1}=\sum_{i=1}^{n_{k}}q_{i,k}=1,$ $k=1,...,N$ and
$p_{i,1}\geq 0,\ \ i=1,...,n_{1}$, $q_{i,k}>0,\ i=1,...,n_{k}$,
$m_{k}=\underset{1\leq i\leq
n_{k}}{\min}\left(\frac{p_{i,k}}{q_{i,k}}\right),$ $k=1,...,N.$

###### Proof.

According to (2.14) and (2.17)

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-m_{1}\left(\sum_{i=1}^{n_{1}}q_{i,1}f\left(x_{i,1}\right)-f\left(\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{1}}\left(p_{i,1}-m_{1}q_{i,1}\right)f\left(x_{i,1}\right)+m_{1}f\left(\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}-1}\left(p_{i,1}-m_{1}q_{i,1}\right)f\left(x_{i,1}\right)+m_{1}f\left(\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1}\right),$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}}p_{i,2}f\left(x_{i,2}\right),$

The expression
$\sum_{i=1}^{n_{2}-1}\left(p_{i,1}-m_{1}q_{i,1}\right)f\left(x_{i,1}\right)$
includes only the terms $\left(p_{i,1}-m_{1}q_{i,1}\right)>0$.

Therefore, from (2) we get that

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-m_{1}\left(\sum_{i=1}^{n_{1}}q_{i,1}f\left(x_{i,1}\right)-f\left(\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}}p_{i,2}f\left(x_{i,2}\right),$

and it is easy to verify that

(2.19) $\sum_{i=1}^{n_{2}}p_{i,2}=1,\quad p_{i,2}\geq
0,\quad\sum_{i=1}^{n_{2}}p_{i,2}x_{i,2}=\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}$

Hence,

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-\sum_{k=1}^{N}m_{k}\left(\sum_{i=1}^{n_{k}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}}p_{i,2}f\left(x_{i,2}\right)-\sum_{k=2}^{N}m_{k}\left(\sum_{i=1}^{n_{k}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\qquad\qquad\cdot\qquad\qquad\cdot\qquad\qquad\cdot\qquad\qquad\qquad\qquad\cdot$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N}}p_{i,N}f\left(x_{i,N}\right)-m_{N}\left(\sum_{i=1}^{n_{N}}q_{i,N}f\left(x_{i,N}\right)-f\left(\sum_{i=1}^{n_{N}}q_{i,N}x_{i,N}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N}}\left(p_{i,N}-m_{N}q_{i,N}\right)f\left(x_{i,N}\right)+m_{N}f\left(\sum_{i=1}^{n_{N}}q_{i,N}x_{i,N}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N+1}-1}\left(p_{i,N}-m_{N}q_{i,N}\right)f\left(x_{i,N}\right)+m_{N}f\left(\sum_{i=1}^{n_{N}}q_{i,N}x_{i,N}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N+1}}p_{i,N+1}f\left(x_{i,N+1}\right).$

Also, it is clear, the same way as in (2.19) that

(2.20) $\displaystyle\sum_{i=1}^{n_{k}}p_{i,k}$ $\displaystyle=$
$\displaystyle 1,\quad p_{i,k}\geq 0,\quad i=1,...,n_{k},\quad k=1,...,N+1,$
$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{k}}p_{i,k}x_{i,k},\quad k=1,...,N+1.$

Therefore, as a result we have to show in order to prove the theorem that

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-\sum_{k=1}^{N}m_{k}\left(\sum_{i=1}^{n_{k}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N+1}}p_{i,N+1}f\left(x_{i,N+1}\right)\geq
f\left(\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}\right)=f\left(\sum_{i=1}^{n_{N+1}}p_{i,N+1}x_{i,N+1}\right),$

holds, and inequality (2) is satisfied because of the convexity of $f$, (2.20)
and
$\sum_{i=1}^{n_{N+1}}p_{i,N+1}\left(x_{i,N+1}\right)=\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}$.

The proof of the theorem is complete. ∎

In Theorem 9, similarly to Theorem 8, where we extend Theorem 2 and the right
handside of the inequality in Theorem 1, we extend now Theorem 3 and the left
hand-side inequality in Theorem 1.

The difference between Theorem 7 and Theorem 9 is that in Theorem 9 we
eliminate all the terms that satisfy $p_{j_{i},k}-M_{k}q_{j_{i},k}=0$ and add
instead only one term equal to
$m_{k}f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)$. Therefore,
$\mathbf{q}_{k}=\left(q_{1,k},...,q_{n_{k},k}\right),$
$\mathbf{x}_{k}=\left(x_{1,k},...,x_{n_{k},k}\right),$ $k=1,...,N$ satisfy
$n_{1}\geq n_{2}\geq,...,\geq n_{N}$. Moreover, we prove that $n_{k}=n_{2},\
k=2,...,N.$

We denote

(2.22) $\displaystyle\mathbf{p}_{1}$ $\displaystyle=$
$\displaystyle\left(p_{1,1},...,p_{n_{1},1}\right),\quad\mathbf{q}_{1}\mathbf{=}\left(q_{1,1},...,q_{n_{1},1}\right),$
$\displaystyle\text{where\quad}p_{i,1}$ $\displaystyle\geq$ $\displaystyle
0,\quad q_{i,1}>0,\quad
i=1,...,n_{1},\quad\sum_{i=1}^{n_{1}}p_{i,1}=\sum_{i=1}^{n_{1}}q_{i,1}=1,$
$\displaystyle M_{1}$ $\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n_{1}}{Max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}},$

for a fixed specific integer $j_{1}$ which satisfies $M_{1}$.

We also define recoursively

(2.23) $\displaystyle\mathbf{x}_{k}$ $\displaystyle=$
$\displaystyle\left(x_{1,k},...,x_{n_{k},k}\right),\quad k=1,...,N$
$\displaystyle p_{i,k}$ $\displaystyle=$ $\displaystyle
p_{i,k-1}-M_{k-1}q_{i,k-1},\ x_{i,k}=x_{i,k-1},\ \text{when}\
M_{k-1}\neq\frac{p_{i,k-1}}{q_{i,k-1}},\ k=2,...,N$ $\displaystyle
p_{j_{k},k}$ $\displaystyle=$ $\displaystyle M_{k-1},\
x_{j_{k},k}=\sum_{i=1}^{n_{k-1}}q_{i,k-1}x_{i,k-1},\ \text{when}\
M_{k-1}=\frac{p_{j_{k-1},k-1}}{q_{j_{k-1},k-1}},$ $\displaystyle M_{k}$
$\displaystyle=$ $\displaystyle\underset{1\leq i\leq
n_{k}}{Max}\left(\frac{p_{i,k}}{q_{i,k}}\right)=\frac{p_{j_{k},k}}{q_{j_{k},k}},\qquad
k=1,...,N,$

where each $j_{k},$is a specific index for which $M_{k}$ holds, $k=1,...,N$.

As in Theorem 8, because of the similar techniques to those in [1] we show in
Theorem 9 only an outline of the proof.

With the notations, definitions and conditions in (2.22) and (2.23) we get:

######  Theorem 9.

Let $f:\left[a,b\right)\rightarrow\mathbb{R},$ $a\leq b<\infty,$ be a convex
function, and let (2.22) and (2.23) hold. Then, for every integer $N$

(2.24)
$J_{n_{1}}\left(f,\mathbf{x}_{1,}\mathbf{p}_{1}\right)-\sum_{k=1}^{N}M_{k}J_{k}\left(f,\mathbf{x}_{k},\mathbf{q}_{k}\right)\leq
0,$

(2.25)
$M_{k}=\frac{p_{j_{1},1}}{\Pi_{m=1}^{k}\left(q_{j_{1},m}\right)},Ck=1,...,N$

and

(2.26) $n_{1}\geq n_{2}=n_{k},=n_{N},\quad k=2,...,N$

hold, where $j_{1}$ is a fixed specific integer for which
$M_{1}=\underset{1\leq i\leq
n_{1}}{\max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}}$
is satisfied.

###### Proof.

As $j_{1}$ is a specific integer for which $M_{1}=\underset{1\leq i\leq
n_{1}}{Max}\left(\frac{p_{i,1}}{q_{i},_{1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}},$
it is easy to see that

$M_{k-1}=\underset{1\leq i\leq
n_{k-1}}{Max}\frac{p_{i,k-1}}{q_{i,k-1}}=\frac{p_{j_{1},1}}{\Pi_{m=1}^{k-1}q_{j_{1}},m},\quad
k=2,...,N+1$

because the only positive $p_{i,k},$ $k=2,...$ is $p_{j_{1},k}$, as

$\text{when }\frac{p_{i,1}}{q_{i,1}}<\frac{p_{j_{1},1}}{q_{j_{1},1}},\quad
i\neq j_{1},\quad\text{then }p_{i,2}<0,\quad x_{i,2}=x_{i,1},$

$\text{when }i=j_{1},\quad\text{then }p_{i,2}\ >0,\quad$

and

$\sum_{i=1}^{n_{2}}p_{i,2}x_{i,2}=1,\quad
x_{j_{1},2}=\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1.}.$

Hence, also for $k=2,...,N$ the only positive $p_{i,k},$ $i=2,...,n_{k}$ is
$p_{j_{1},k}$ where $j_{1}$ is the fixed integer that satisfies
$M_{1}=\underset{1\leq i\leq
n_{1}}{\max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}},$
and therefore we can replace in (2.23) $j_{k}$ with $j_{1}$, which means that
(2.25) holds. Also, as the only positive $p_{i,k}$, $k=2,...,N$ is
$p_{j_{k},k}$, it is obvious that $n_{k}=n_{2}$, $k=2,...,N$.

In order to complete the proof of the theorem, we proceed now with proving
(2.24):

In a similar way as in Theorem 7 together with (2.26) we get

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-\sum_{k=1}^{N}M_{k}\left(\sum_{i=1}^{n_{k}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{2}}p_{i,2}f\left(x_{i,2}\right)-\sum_{k=2}^{N}M_{k}\left(\sum_{i=1}^{n_{k}=n_{2}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}=n_{2}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\qquad\cdot\qquad\qquad\qquad\cdot\qquad\qquad\cdot\qquad\qquad\qquad\qquad\cdot$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N}=n_{2}}p_{i,N}f\left(x_{i,N}\right)-M_{N}\left(\sum_{i=1}^{n_{N}=n_{2}}q_{i,N}f\left(x_{i,N}\right)-f\left(\sum_{i=1}^{n_{N}=n_{2}}q_{i,N}x_{i,N}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N}=n_{2}}\left(p_{i,N}-M_{N}q_{i}\right)f\left(x_{i,N}\right)+M_{N}f\left(\sum_{i=1}^{n_{N}=n_{2}}q_{i,N}x_{i,N}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N+1}=n_{2}}p_{i,N+1}f\left(x_{i,N+1}\right)$

Also it is clear that

$\displaystyle p_{i,k}$ $\displaystyle\leq$ $\displaystyle 0,\quad
i=1,...,n_{k},\quad i\neq j_{1},\quad p_{j_{1},k}>0,\quad k=2,...,N+1,$
$\displaystyle\sum_{i=1}^{n_{k+1}=n_{2}}p_{i,k+1}$ $\displaystyle=$
$\displaystyle
1,\quad\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}=\sum_{i=1}^{n_{k}=n_{2}}p_{i,k}x_{i,k},\quad
k=2,...,N+1.$

From (LABEL:2.19) it follows that in order to prove (2.24) we have to show
that

$\displaystyle\sum_{i=1}^{n_{1}}p_{i,1}f\left(x_{i,1}\right)-M_{1}\left(\sum_{i=1}^{n_{1}}q_{i,1}f\left(x_{i,1}\right)-f\left(\sum_{i=1}^{n_{1}}q_{i,1}x_{i,1}\right)\right)$
$\displaystyle-\sum_{k=2}^{N}M_{k}\left(\sum_{i=1}^{n_{k}=n_{2}}q_{i,k}f\left(x_{i,k}\right)-f\left(\sum_{i=1}^{n_{k}=n_{2}}q_{i,k}x_{i,k}\right)\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{N+1}=n_{2}}p_{i,N+1}f\left(x_{i,N+1}\right)\leq
f\left(\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}\right),$

that is, we have to show that

$\sum_{i=1}^{n_{N}=n_{2}}\left(p_{i,N}-M_{N}q_{i,N}\right)f\left(x_{i,N}\right)+M_{N}f\left(\sum_{i=1}^{n_{N}=n_{2}}q_{i,N}x_{i,N}\right)\leq
f\left(\sum_{i=1}^{n_{1}}p_{i,1}x_{i,1}\right).$

In other words, we have to show that

$\displaystyle\frac{1}{M_{N}}f\left(\sum_{i=1}^{n_{N}=n_{2}}p_{i,N}x_{i,N}\right)+\sum_{i=1,\
i\neq
j_{1}}^{n_{N}=n_{2}}\left(q_{i,N}-\frac{p_{i,N}}{M_{N}}\right)f\left(x_{i,N}\right)$
$\displaystyle\geq$ $\displaystyle
f\left(\sum_{i=1}^{n_{N+1}=n_{2}}q_{i,N}x_{i,N.}\right)$

holds.

The last inequality follows from the convexity of $f$ because
$q_{i,N}-\frac{p_{i,N}}{M_{N}}\geq 0\ ,$ $i=1,...,n_{N},$ $i\neq j_{1}$ and
$\frac{1}{M_{N}}>0$, therefore, the inequality (2.24) holds.

The proof of the theorem is complete. ∎

######  Remark 1.

If in Theorem 9 $M_{1}=\underset{1\leq i\leq
n_{1}}{Max}\left(\frac{p_{i,1}}{q_{i,1}}\right)=\frac{p_{j_{1},1}}{q_{j_{1},1}}>\frac{p_{i,1}}{q_{i,1}},$
$i=1,...,n,$ $i\neq j_{1}$, then Theorem 9 is the same as Theorem 7.

Concluding Remarks. In Theorem 6 we eliminate all $s_{k-1}$ terms for which
$\left(p_{i,k-1}-m_{k-1}q_{i,k-1}\right)f\left(x_{i,k-1}\right)=0$ and replace
all of these $s_{k-1}$ with terms
$\frac{m_{k-1}}{s_{k-1}}f\left(\sum_{i=1}^{n}q_{i,k-1}x_{i,k-1}\right)$. On
the other hand, in Theorem 8 we eliminate all the $s_{k-1}$ terms for which
$\left(p_{i,k-1}-m_{k-1}q_{i,k-1}\right)=0$ and replace only one of them with
$m_{k-1}f\left(\sum_{i=1}^{n}q_{i,k-1}x_{i,k-1}\right)$.

From theorems 6 and 8 it is easy to realize that we can prove similarly to
these two theorems a theorem in which we eliminate all $s_{k-1}$ terms for
which $\left(p_{i,k-1}-m_{k-1}q_{i,k-1}\right)=0$ and replace $r_{k-1}$ terms
$1\leq r_{k-1}\leq s_{k-1}$ and get similar results. The same can be done as a
consequence of what is done in theorems 7 and 9.

## References

  * [1] S. Abramovich, Extended Normalized Jensen Functional, related to convexity, 1 -quasiconvexity and superquadracity, JMI Vol. 12 no.3, (2018), 753-764.
  * [2] S. S. Dragomir, ”Bounds of the normalised Jensen functional”, Bull. Austral. Math. Soc., 74 (2006), 471-478.
  * [3] M. Sababheh, ”Improved Jensen’s inequality”, Math. Inequal. Appl., 20 (2017), no2, 389-403.
  * [4] Y. Sayyarri, M. Dehghanian, H. Barsam, ”New bounds for Jensen-Dragomir functionalwith applications in analysis”, J. Indian Math. Soc., (N.S.) 90 (2023, no. 1-2, 175-185.
  * [5] Pham Thanh The, Doan Thi Thuy Van, Duong Quoc Huy, ” Improved Jensen-Dragomir type inequalities and applications”, Preprint.


Dated: ]

# Interplay between multipolar spin interactions, Jahn-Teller effect and
electronic correlation in a $J_{eff}=\frac{3}{2}$ insulator

Dario Fiore Mosca  University of Vienna, Faculty of Physics and Center for
Computational Materials Science, Vienna, Austria  Leonid V. Pourovskii  Centre
de Physique Théorique, Ecole Polytechnique, CNRS, Institut Polytechnique de
Paris, 91128 Palaiseau Cedex, France  Collège de France, 11 place Marcelin
Berthelot, 75005 Paris, France  Beom Hyun Kim  Korea Institute for Advanced
Study, Seoul 02455, South Korea  Peitao Liu  University of Vienna, Faculty of
Physics and Center for Computational Materials Science, Vienna, Austria
Samuele Sanna  Department of Physics and Astronomy, Alma Mater Studiorum -
Università di Bologna, Bologna, 40127 Italy  Federico Boscherini  Department
of Physics and Astronomy, Alma Mater Studiorum - Università di Bologna,
Bologna, 40127 Italy  Sergii Khmelevskyi  Center for Computational Materials
Science, Institute for Applied Physics, Vienna University of Technology,
Wiedner Hauptstrasse $8$ \- $10$, $1040$ Vienna, Austria  Cesare Franchini
<EMAIL_ADDRESS>University of Vienna, Faculty of Physics and
Center for Computational Materials Science, Vienna, Austria  Department of
Physics and Astronomy, Alma Mater Studiorum - Università di Bologna, Bologna,
40127 Italy

([)

###### Abstract

In this work we study the complex entanglement between spin interactions,
electron correlation and Janh-Teller structural instabilities in the 5d1
$J_{eff}=\frac{3}{2}$ spin-orbit coupled double perovskite $\rm
Ba_{2}NaOsO_{6}$ using first principles approaches. By combining non-collinear
magnetic calculations with multipolar pseudospin Hamiltonian analysis and
many-body techniques we elucidate the origin of the observed quadrupolar
canted antifferomagnetic. We show that the non-collinear magnetic order
originates from Jahn-Teller distortions due to the cooperation of Heisenberg
exchange, quadrupolar spin-spin terms and both dipolar and multipolar
Dzyaloshinskii-Moriya interactions. We find a strong competition between
ferromagnetic and antiferromagnetic canted and collinear quadrupolar magnetic
phases: the transition from one magnetic order to another can be controlled by
the strength of the electronic correlation ($U$) and by the degree of Jahn-
Teller distortions.

Double perovskites with strong spin-orbit coupling (SOC) represent a unique
playground for the emergence of exotic spin-orbital-lattice entangled phases
Chen  _et al._ (2010); Ishizuka and Balents (2014); Iwahara  _et al._ (2018);
Romhányi  _et al._ (2017); Martins  _et al._ (2017). Structural and orbital
frustrations Khomskii and Mostovoy (2003), SOC-induced formation of
_effective_ J manifolds Goodenough (1968) and relatively strong electronic
correlation give rise to a wide range of unusual electronic and magnetic
phases including so-called Dirac-Mott insulating states Lee and Pickett
(2007); Xiang and Whangbo (2007), multipolar spin interactions Chen  _et al._
(2010); Ishizuka and Balents (2014); Lee and Pickett (2007) and entangled SOC-
Jahn-Teller effects Iwahara  _et al._ (2018); Xu  _et al._ (2016).

A particularly interesting case is that of 5$d^{1}$ double perovskites
characterized by an effective total angular momentum $J_{eff}=3/2$, an example
of which is $\rm Ba_{2}NaOsO_{6}$ (BNOO). BNOO is Dirac-Mott insulator Feng
_et al._ (2016); Lee and Pickett (2007) with strong SOC and electronic
correlation effects Gangopadhyay and Pickett (2015, 2016); Wang  _et al._
(2019). Magnetization measurements indicate that below 6.8 K BNOO exhibits
weak ferromagnetic interactions and a small ordered spin moment of 0.2
$\mu_{B}$ Stitzer  _et al._ (2002); Erickson  _et al._ (2007); Steele  _et
al._ (2011); Ahn  _et al._ (2017). Refined NMR analysis confirmed the
theoretically predicted canted spin state with two magnetic sublattices
Ishizuka and Balents (2014); Lu  _et al._ (2017). In this spin model, depicted
in Fig.1(a), the spins align ferromagnetically within the $xy$ plane with
alternating canting angle $\phi$ in adjacent planes along the [001] axis [see
Fig.1(c)]. As the optimal canting angle extracted from the NMR data,
$\phi_{expt}$=67∘, is closer to the AFM limit (90∘) than to the FM one (0∘) it
is more appropriate to use the jargon canted-antiferromagnetic (c-AFM), rather
than c-FM as initially proposed Lu  _et al._ (2017); Ishizuka and Balents
(2014); Willa  _et al._ (2019). The most stunning aspect of BNOO is that this
c-AFM phase is associated with a local point symmetry breaking manifested by
local deformations of the BO6 octahedra Lu  _et al._ (2017); Liu  _et al._
(2018). However, x-ray diffraction shows that the cubic symmetry is maintained
even at low temperature Erickson  _et al._ (2007) and no direct evidence of
Jahn-Teller (JT) instabilities have been detected by NMR Lu  _et al._ (2017);
Liu  _et al._ (2018), even though JT instabilities would be expected in a
$d^{1}$ electronic configuration Khomskii and Mostovoy (2003); Khomskii (2014)
(one of the models proposed in Ref. Lu  _et al._ (2017) involves a contraction
of the OsO6 octahedra, assimilable to a local Q1-JT distortion, see Fig. 1).
This apparent violation of the JT theorem Jahn and Bragg (1938); Gehring and
Gehring (1975) has been explained by the entanglement of vibronic dynamics
with SOC, suggesting the presence of a dynamical Jahn-Teller effect with small
static component Iwahara  _et al._ (2018); Xu  _et al._ (2016) and orbital
selective quadrupolar charge ordering Cong  _et al._ (2019). This explains the
breaking of local point symmetry and suggests that the canted magnetic ground
state of BNOO should be described in terms of the spin-orbital-lattice
entangled states involving high-rank spin interactions.

Figure 1: (Color online) Graphical representation of (a) the canted-AFM
magnetic ground state with layer-dependent (1 & 2) spin ordering and (b)
associated JT vibrational mode. Following Van Vleck notation Van Vleck (1939);
sup  we obtain $Q_{1}=0.031~{}\AA$ and
$\theta=\tan^{-1}(Q_{2}/Q_{3})=79.35^{\circ}$ (plane 1) and -79.35∘ (plane 2)
Kanamori (1960). This structural motif is compatible with model A proposed by
Liu _et al._ Liu  _et al._ (2018). (c) Visualization of the staggered-type JT-
distortion in the two layers. (d) Schematic definition of the canting angle
$\phi$: $\phi$=0 corresponds to the FM limit; by convention $0<\phi<\pi/4$
refers to c-FM state whereas $\pi/4<\phi<\pi/2$ identifies a c-AFM spin
arrangement. Optimized structural data and JT distortions are given in the
Supplementary Materials (SM) sup .

With this study we aim to clarify the link between JT instabilities and spin
couplings by computing the variation of the multipolar interactions due to
local structural distortions and explain how this JT-multipolar coupling
affects the spin deflection and dictates the formation of the c-AFM ground
state. We do so by employing first principles non-collinear relativistic
density functional theory (DFT) Hobbs  _et al._ (2000); Liu  _et al._ (2015);
Dudarev  _et al._ (2019) within the Perdew-Burke-Ernzerhof approximation
Perdew  _et al._ (1996) using the VASP code Kresse and Hafner (1993); Kresse
and Furthmüller (1996); Blöchl (1994) and many-body techniques. The inter-site
interactions have been extracted from magnetically constrained DFT+U+SOC
calculations, from DFT+dynamical mean-field theory (DMFT) Georges  _et al._
(1996); Anisimov  _et al._ (1997) in the Hubbard-I approximation (HI) Hubbard
(1963); Lichtenstein and Katsnelson (1998) in conjunction with the magnetic
force theorem (FT) to effective intersite interactions (abbreviated, FT-HI)
Pourovskii (2016), as well as from exact diagonalization (ED) of a microscopic
effective Hamiltonians Kim  _et al._ (2012). The charge self-consistent DFT+HI
approachAichhorn  _et al._ (2009, 2011) is implemented using the Wien-2k
package Blaha  _et al._ (2018) and TRIQS library Parcollet  _et al._ (2015);
Aichhorn  _et al._ (2016) using the projective technique of Refs. Amadon  _et
al._ (2008); Aichhorn  _et al._ (2009). A more detailed description of the
computational protocol is given in the method section and in the SM sup .

We find that in the fully undistorted (JT-quenched) limit the spins prefer to
align ferromagnetically but the onset of staggered JT distortions activates
both first-order (dipole-dipole) and second order (quadrupole-quadrupole +
dipole-octupole) Dzyaloshinskii-Moriya (DM) interactions Hosoi  _et al._ which
drive the development of the observed c-AFM phase at the optimum JT
distortion. Moreover, our data show that the multipolar c-AFM ground state is
highly sensitive to the degree of JT distortions and to the strength of the
Coulomb interaction $U$: slightly lower $U$ or larger JT distortions shift the
system to a competing canted-ferromagnetic (c-FM) state.

We start by comparing the dependence of the energy on the canting angle for
the JT-distorted and JT-quenched (undistorted) phases in Fig.2, obtained by
changing stepwise the canting angle $\phi$ from the FM ($\phi=0^{\circ}$) to
the AFM ($\phi=90^{\circ}$) limits. The JT structure exhibits two minima, a
lower c-AFM ground state found at $\phi=66^{\circ}$, in excellent agreement
with experiment (67∘Lu  _et al._ (2017)) and a c-FM phase just 0.05 meV/u.c.
higher in energy. Conversely, the undistorted structure shows a clear minimum
for a parallel spin alignment followed by an AFM state. The size of the spin
($m_{s}$) and orbital ($m_{o}$) moments are essentially the same in both
structures ($m_{s}\approx 0.8~{}\mu_{B}$ and $m_{o}\approx 0.6\mu_{B}$). In
the ground state JT-distorted structure the resulting net ordered moment is
$0.23\mu_{B}$, in very good agreement with measurements (0.2$\mu_{B}$)
Erickson  _et al._ (2007); Lu  _et al._ (2017). These results clearly indicate
that the JT effect is essential to stabilize the observed canted ground state.

To decipher the role of JT distortions on the spin deflection we have
extracted the intersite spin interactions by fitting the canting energies
curves with a pseudospin Hamiltonian in the general multipolar form Arima  _et
al._ (1969); Santini  _et al._ (2009):

$H_{ij}=\sum_{K,K^{\prime}}\sum_{Q,Q^{\prime}}I_{KK^{\prime}}^{QQ^{\prime}}O_{Q}^{K}(J_{i})O_{Q^{\prime}}^{K^{\prime}}(J_{j}),$
(1)

where $i$,$j$ are the site indexes, $I_{KK^{\prime}}^{QQ^{\prime}}$ the
coupling constants and $O_{Q}^{K}(J_{i})$ and
$O_{Q^{\prime}}^{K^{\prime}}(J_{j})$ are the multipolar tensor operators of
rank $K$ and $K^{\prime}$, with $K\leq 2J_{i}$, $Q=-K,...,K$, $K^{\prime}\leq
2J_{j}$, $Q^{\prime}=-K^{\prime},...,K^{\prime}$. The multipolar operators $O$
are estimated within the mean field approximation, i.e.
$O(J_{i})O(J_{j})\approx\langle{O(J_{i})}\rangle{O(J_{j})}+O(J_{i})\langle{O(J_{j})}\rangle-\langle{O(J_{i})}\rangle\langle{O(J_{j})}\rangle$.
After applying symmetry properties of pseudovectors and multipolar operators
for the $J_{eff}=\frac{3}{2}$ (or $\Gamma_{8}$) state Shiina  _et al._ (1998);
Chen  _et al._ (2010) as well as crystal symmetry we obtained the following
fitting formula which expresses the magnetic energy as a function of the
canting angle $\phi$ (the derivation is given in the SM sup ):

$\displaystyle E(\theta)$ $\displaystyle=$ $\displaystyle
A_{1}\cos(2\phi)+A_{2}\cos(4\phi)+A_{3}\cos(6\phi)$ (2)
$\displaystyle+B_{1}\sin(2\phi)+B_{2}\sin(4\phi)$

Figure 2: (Color online) Dependence of the energy on the canting angle and
spin-spin interaction matrix. (a) Calculated DFT+U+SOC $\phi$ dependent
energies for the optimized JT distorted phase (circles) and corresponding
fitting curve obtained using Eq. 2. The fitting curves obtained by removing
the A3 (o-o), the A2 (q-q) and A2-A3-B2 (q-q- and o-o) coefficients are also
displayed; (b) Comparison between DFT, FT-HI and ED canting energies for the
undistorted (Q1=0) phase. (c) FT-HI interaction matrix showing the non-zero
and non-constant mean values of the tensor operators $\langle{O}\rangle$ in
the pseudo spin eigenbasis (see Eq. 1) summed over the out-of plane
interacting bonds (see SM for details sup ). To guarantee a quantitatively
consistent comparison between DFT and FT-HI the DFT FM-AFM energy difference
is aligned to that obtained by FT-HI (see SM for details sup ).

To validate the DFT-mapping and decode the specific type of spin couplings at
play we have performed a direct calculation of the intersite interactions
using the FT-HI method proposed by Pourovskii Pourovskii (2016) and ED Kim
_et al._ (2012), taking the undistorted phase as a reference (FT-HI and ED
cannot be applied to the JT-distorted phase). All possible on-site
fluctuations within the paramagnetic ground state of a shell, encoded by the
corresponding multipole moments $O_{Q}^{K}(J_{i})$, are included; all relevant
inter-site couplings can thus be extracted. In the present case of the
$J_{eff}=\frac{3}{2}$ shell of Os, those are all inter-site interactions
between dipole, quadrupole, and octupole moments that are permitted by the
symmetry. With the effective Hamiltonian thus obtained we evaluated the
dependence of zero-temperature total energy versus $\phi$; the spin-spin
coupling parameters $A_{i}$ are subsequently extracted by fitting it to the
form (2).

Spin-spin Interactions
---
|  | Undistorted | JT-distorted
|  | cFM | cAFM
Coefficients | FT-HI | ED | DFT | DFT
A1 |  | -0.72 | -1.06 | -0.66 | 0.13
| d-d | 0.84 | 0.39 |  |
| q-q | -0.03 | 0.00 |  |
| d-o | -1.32 | -1.25 |  |
| o-o | -0.21 | -0.20 |  |
A2 | q-q | -0.40 | -0.48 | -0.69 | -0.38
A3 | o-o | 0.05 | 0.08 | -0.03 | -0.08
B1 | DM d-d | 0 | 0 | 0 | -0.99
B2 | DM q-q, d-o | 0 | 0 | 0 | -0.08
Table 1: Collection of spin-spin interactions (in meV) as derived from DFT,
FT-HI and ED approaches for the undistorted Q1=0 FM phase and corresponding
DFT values for the fully distorted c-AFM phase. The coefficients are
decomposed over their distinct d-d, d-o, o-o, and q-q character.

The fitting curves follow very well the first principles data for both the
distorted and undistorted phases as shown in Fig. 2(a,b). The optimal fitting
parameters and the corresponding FT-HI and ED results are collected in Tab. 1
where each coefficient is associated with the specific type of dipolar or
multipolar couplings. Both phases exhibit relatively large quadrupole-
quadurpole (q-q) couplings and non negligible octupole-octupole (o-o)
interactions. The inclusion of o-o terms in the JT-phase is essential to shift
the minimum from 70∘ to 66∘, whereas the formation of the two competing canted
minima is mostly governed by the q-q term $A_{2}$. We also note that DM
interactions, embedded in coefficients $B_{1}$ and $B_{2}$, are inoperative in
the undistorted phase (not allowed by symmetry), but the breaking of the local
symmetry caused by the JT effect activates the DM channel which is the main
responsible for stabilizing of the canted phases. Our data indicate that,
besides the standard first order d-d DM interaction, an additional second
order DM coupling emerges involving both q-q and d-o terms. This type of
interaction can be classified as multipolar DM interaction, namely an
antisymmetric DM exchange among multipolar tensors.

Figure 3: (Color online) Magnetic canting energy as a function of (a) $U$
(relative global minimum set to zero energy) and (b) $\widetilde{Q_{1}}$
($\widetilde{Q_{1}}=1$ refers to the fully optimized JT phase). (c) Global
magnetic phase diagram showing the dependence of the ground state magnetic
ordering (canting angle) for different values of $U$ and $\widetilde{Q_{1}}$.
Decreasing $\widetilde{Q_{1}}$ drives the system to a collinear FM state
whereas a reduction of $U$ washes-out the c-AFM system and stabilizes a c-FM
configuration.

After clarifying the essential features of the c-AFM ground state we move now
to the analysis of the competition between the c-AFM and c-FM phases and how
this competition is controlled by the strength of $U$ and the degree of JT
mode $Q_{1}$. Being almost degenerate in energy, it is expected that small
perturbations could change the relative stability of these two phases,
providing insights on the mechanism favoring one or the other spin ordering.
To explore this possibility we have computed the canting energies for
different values of the relative JT distortion $\widetilde{Q_{1}}$ (from the
fully distorted structure, here identified as $\widetilde{Q_{1}}=1$ to the
undistorted $\widetilde{Q_{1}}$=0 limit) and $U$ (from 3.4 to 2.4 eV). The
results, displayed in Fig. 3, indicate that both degrees of freedom act on the
spin deflection and can trigger a magnetic transition from the c-AFM to c-FM.

By decreasing $U$ the c-AFM phase is progressively destabilized in favor of
the c-FM phase which becomes the global minimum. A further decrease of $U$
washes-out completely the AFM canting. Similarly, reducing $\widetilde{Q_{1}}$
acts on the canting angle by inverting the stability of the two canted minima
and drives the systems towards the collinear limits as $\widetilde{Q_{1}}$
approaches 0. In order to explore the combined action of JT effect and
electronic correlation we have repeated similar calculations and constructed a
complete magnetic phase diagram as a function of $\widetilde{Q_{1}}$ and $U$,
displayed in Fig. 3(c). The phase diagram shows three different phases
characterized by specific value of the canting angle: (i) a FM phase which is
favored in the limit of vanishing JT-distortion and small $U$; (ii) a c-FM
phase emerges beyond a small critical JT distortion in the whole considered
$U$-regime and finally (iii) the c-AFM phase, corresponding to the
experimentally observed ground state, which covers the top portion of the
phase diagram and is favored by increasing the JT mode and $U$. For
$\widetilde{Q_{1}}$ =1 the c-AFM phase represents the ground state for $U$
larger than 3.1 eV.

The interplay between $\widetilde{Q_{1}}$ and $U$ affects the exchange
couplings. The transition from one phase to the other can be rationalized by
inspecting the variation of the spin-spin interactions upon $U$ and
$\widetilde{Q_{1}}$ shown in Fig. 4. First we note that the tendency to move
toward a FM-type canting is mostly triggered by the steep decrease of the d-d
interaction embedded in the exchange coefficient $A_{1}$, which changes sign
rather quickly when $U$ or $\widetilde{Q_{1}}$ are reduced, thereby driving
the AFM-to-FM transition. We note that coefficients $A_{3}$ and $B_{2}$
besides being relatively small remain essentially unaffected by tuning $U$ at
the optimized JT distorted structure, implying that the o-o character is
mostly linked to the structural instabilities. This is confirmed by the
observation that $A_{3}$ is rapidly diminished with decreasing
$\widetilde{Q_{1}}$ thus revealing a direct link between o-o interaction and
structural distortion. The JT distortion is therefore a determinant factor for
the onset of the dipolar and multipolar DM interactions expressed by the
coefficient $B_{1}$ and $B_{2}$, respectively. Similarly to the multipolar DM
component also the dipolar one ($B_{1}$), which is zero by symmetry in the
undistorted limit, rapidly increases vs. $\widetilde{Q_{1}}$ and reaches a
large value in the fully JT-distorted phase ($B_{1}=-0.99$ meV). Finally, from
the trend of $A_{2}$ we infer that q-q interactions become stronger with
decreasing $\widetilde{Q_{1}}$ and are responsible for the preservation of the
two minima in the undistorted phase. Conversely, reducing $U$ leads to a
linear reduction of $A_{2}$, which contributes to the disruption of the
large-$\phi$ minimum. The above analysis can give insights on the role of
dynamical JT effects in BNOO, that should be manifested by two types of
magnetic fluctuations: FM fluctuations, controlled by the changes in $A_{1}$,
and canted-FM to canted-AFM fluctuations governed by the variation of $B_{1}$.

Figure 4: (Color online) Evolution of the DFT+U spin-spin exchange
coefficients as a function of (left) $U$ and (right) $\widetilde{Q_{1}}$.

In conclusion, by combining many-body effective Hamiltonians with material
specific DFT-based descriptions we have decoded the complex entanglement
between structural frustration and electronic correlation in the spin-orbit
coupled and spin-canted double perovskite BNOO. We show that this exotic
magnetic order originates from the anomalous Dzyaloshinskii-Morya interactions
generated by JT -distortions, and elucidate the direct effect of U and JT on
the spin-spin interactions. This work contributes to the raising interest on
atypical multipolar orderings in relativistic oxides Maharaj  _et al._ (2020);
Paramekanti  _et al._ (2020) with a transparent analysis that helps to light
up the intricate physics at play in this class of materials.

##  I Methods

To study the cross coupling between spin exchange interaction, Jahn-Teller
distortions and electronic correlation we have integrated three different
schemes: magnetically constrained non-collinear DFT+U, DMFT in the Hubbard-I
approximation (FT-HI) combined with the magnetic force theorem and exact
diagonalization. Since the direct computation of spin exchanges at FT-HI and
ED for JT-distorted structures is at present not possible, we have first
verified the validity of the DFT+U data for the undistorted phase by comparing
the spin exchanges extracted from DFT+U total energies with those calculated
explicitly at FT-HI and ED level (Fig.2(b)). In DFT+U the spin interactions
are estimated by fitting the total energies curves obtained at different
canting angle (via magnetically constrained calculations) with the spin
Hamiltonian given in Eq. 2 (derived in the SM sup . After normalizing the
DFT+U data to the FT-HI/ED values (see SM sup  for further details) we have
then addressed the JT-induced modifications at DFT+U level, by performing a
series of calculation stepwise from the ideal undistorted phase to the fully
distorted real structure (Fig.3(a-b)). In order to account for the role of
electron-electron correlation and construct the phase diagram displayed in
Fig. 3(c), these calculations have been conducted for different values of $U$,
from 2.4 eV to 3.4 eV.

In studying the JT distortion we have followed the Van Vleck notation for the
definition of the main JT distortions $Q_{1}$, $Q_{2}$ and $Q_{3}$. Since the
value of $Q_{1}$ is significantly larger than the corresponding $Q_{2}$ and
$Q_{3}$ value (see SM sup ) we have considered the variation of the the spin
interactions as a function of the JT parameter $\tilde{Q}_{1}$, defined as the
ratio between Q1 for the interpolated structure and the corresponding value
for the optimal one. Additional details can be found in the SM sup , which
includes Refs. Winter  _et al._ (2016); Kim  _et al._ (2019); T. Inui and
Onodera (1990); Horvat  _et al._ (2017); Franchini  _et al._ (2011); Archer
_et al._ (2011); leo ; Horvat  _et al._ (2017); Pourovskii and Khmelevskyi
(2019).

##  II Acknowledgements

We gratefully acknowledge the helpful discussions with V. Mitrovic. C.F.
thanks S. Streltsov for useful insights on the Jahn-Teller effect. DFM
acknowledges support from the Vienna Doctoral School in Physics and from the
University of Bologna (Thesis Abroad Scholarship). LVP acknowledges the
support by the European Research Council grants ERC-319286-”QMAC” and is
grateful to the computer team at CPHT for support. B.H.K was supported by a
KIAS Individual Grant (CG068701) at Korea Institute for Advanced Study. The
computational results were achieved by using the Vienna Scientific Cluster
(VSC).

## References

  * Chen  _et al._ (2010) G. Chen, R. Pereira, and L. Balents, Phys. Rev. B 82, 174440 (2010).
  * Ishizuka and Balents (2014) H. Ishizuka and L. Balents, Phys. Rev. B 90, 184422 (2014).
  * Iwahara  _et al._ (2018) N. Iwahara, V. Vieru, and L. F. Chibotaru, Phys. Rev. B 98, 075138 (2018).
  * Romhányi  _et al._ (2017) J. Romhányi, L. Balents, and G. Jackeli, Phys. Rev. Lett. 118, 217202 (2017).
  * Martins  _et al._ (2017) C. Martins, M. Aichhorn, and S. Biermann, Journal of Physics: Condensed Matter 29, 263001 (2017).
  * Khomskii and Mostovoy (2003) D. I. Khomskii and M. V. Mostovoy, Journal of Physics A: Mathematical and General 36, 9197 (2003).
  * Goodenough (1968) J. B. Goodenough, Phys. Rev. 171, 466 (1968).
  * Lee and Pickett (2007) K.-W. Lee and W. E. Pickett, Europhysics Letters (EPL) 80, 37008 (2007).
  * Xiang and Whangbo (2007) H. J. Xiang and M.-H. Whangbo, Phys. Rev. B 75, 052407 (2007).
  * Xu  _et al._ (2016) L. Xu, N. A. Bogdanov, A. Princep, P. Fulde, J. van den Brink, and L. Hozoi, npj Quantum Materials 1, 16029 (2016).
  * Feng  _et al._ (2016) H. L. Feng, S. Calder, M. P. Ghimire, Y.-H. Yuan, Y. Shirako, Y. Tsujimoto, Y. Matsushita, Z. Hu, C.-Y. Kuo, L. H. Tjeng, T.-W. Pi, Y.-L. Soo, J. He, M. Tanaka, Y. Katsuya, M. Richter, and K. Yamaura, Phys. Rev. B 94, 235158 (2016).
  * Gangopadhyay and Pickett (2015) S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 91, 045133 (2015).
  * Gangopadhyay and Pickett (2016) S. Gangopadhyay and W. E. Pickett, Phys. Rev. B 93, 155126 (2016).
  * Wang  _et al._ (2019) C. Wang, Y. Xu, W. Hao, R. Wang, X. Zhang, K. Sun, and X. Hao, Phys. Rev. B 99, 035126 (2019).
  * Stitzer  _et al._ (2002) K. E. Stitzer, M. D. Smith, and H.-C. zur Loye, Solid State Sciences 4, 311 (2002).
  * Erickson  _et al._ (2007) A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta, Z. Schlesinger, W. A. Harrison, J. M. Kim, and I. R. Fisher, Phys. Rev. Lett. 99, 016404 (2007).
  * Steele  _et al._ (2011) A. J. Steele, P. J. Baker, T. Lancaster, F. L. Pratt, I. Franke, S. Ghannadzadeh, P. A. Goddard, W. Hayes, D. Prabhakaran, and S. J. Blundell, Phys. Rev. B 84, 144416 (2011).
  * Ahn  _et al._ (2017) K.-H. Ahn, K. Pajskr, K.-W. Lee, and J. Kuneš, Phys. Rev. B 95, 064416 (2017).
  * Lu  _et al._ (2017) L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee, I. R. Fisher, and V. F. Mitrovic, Nature Communications 8, 14407 (2017).
  * Willa  _et al._ (2019) K. Willa, R. Willa, U. Welp, I. R. Fisher, A. Rydh, W.-K. Kwok, and Z. Islam, Phys. Rev. B 100, 041108 (2019).
  * Liu  _et al._ (2018) W. Liu, R. Cong, A. P. Reyes, I. R. Fisher, and V. F. Mitrović, Phys. Rev. B 97, 224103 (2018).
  * Khomskii (2014) D. I. Khomskii,  _Transition Metal Compounds_ (Cambridge University Press, 2014).
  * Jahn and Bragg (1938) H. A. Jahn and W. H. Bragg, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences 164, 117 (1938).
  * Gehring and Gehring (1975) G. A. Gehring and K. A. Gehring, Reports on Progress in Physics 38, 1 (1975).
  * Cong  _et al._ (2019) R. Cong, R. Nanguneri, B. Rubenstein, and V. F. Mitrović, Phys. Rev. B 100, 245141 (2019).
  * Van Vleck (1939) J. H. Van Vleck, The Journal of Chemical Physics 7, 72 (1939), https://doi.org/10.1063/1.1750327 .
  * (27) Supplementary Material.
  * Kanamori (1960) J. Kanamori, Journal of Applied Physics 31, S14 (1960), https://doi.org/10.1063/1.1984590 .
  * Hobbs  _et al._ (2000) D. Hobbs, G. Kresse, and J. Hafner, Phys. Rev. B 62, 11556 (2000).
  * Liu  _et al._ (2015) P. Liu, S. Khmelevskyi, B. Kim, M. Marsman, D. Li, X.-Q. Chen, D. D. Sarma, G. Kresse, and C. Franchini, Phys. Rev. B 92, 054428 (2015).
  * Dudarev  _et al._ (2019) S. L. Dudarev, P. Liu, D. A. Andersson, C. R. Stanek, T. Ozaki, and C. Franchini, Phys. Rev. Materials 3, 083802 (2019).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).
  * Georges  _et al._ (1996) A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996).
  * Anisimov  _et al._ (1997) V. I. Anisimov, A. I. Poteryaev, M. A. Korotin, A. O. Anokhin, and G. Kotliar, Journal of Physics: Condensed Matter 9, 7359 (1997).
  * Hubbard (1963) J. Hubbard, Proc. Roy. Soc. (London) A 276, 238 (1963).
  * Lichtenstein and Katsnelson (1998) A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57, 6884 (1998).
  * Pourovskii (2016) L. V. Pourovskii, Phys. Rev. B 94, 115117 (2016).
  * Kim  _et al._ (2012) B. H. Kim, G. Khaliullin, and B. I. Min, Phys. Rev. Lett. 109, 167205 (2012).
  * Aichhorn  _et al._ (2009) M. Aichhorn, L. Pourovskii, V. Vildosola, M. Ferrero, O. Parcollet, T. Miyake, A. Georges, and S. Biermann, Phys. Rev. B 80, 085101 (2009).
  * Aichhorn  _et al._ (2011) M. Aichhorn, L. Pourovskii, and A. Georges, Phys. Rev. B 84, 054529 (2011).
  * Blaha  _et al._ (2018) P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, R. Laskowski, F. Tran, and L. D. Marks,  _WIEN2k, An augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties_ (Karlheinz Schwarz, Techn. Universität Wien, Austria,ISBN 3-9501031-1-2, 2018).
  * Parcollet  _et al._ (2015) O. Parcollet, M. Ferrero, T. Ayral, H. Hafermann, I. Krivenko, L. Messio, and P. Seth, Computer Physics Communications 196, 398 (2015).
  * Aichhorn  _et al._ (2016) M. Aichhorn, L. Pourovskii, P. Seth, V. Vildosola, M. Zingl, O. E. Peil, X. Deng, J. Mravlje, G. J. Kraberger, C. Martins, _et al._ , Computer Physics Communications 204, 200 (2016).
  * Amadon  _et al._ (2008) B. Amadon, F. Lechermann, A. Georges, F. Jollet, T. O. Wehling, and A. I. Lichtenstein, Phys. Rev. B 77, 205112 (2008).
  * (48) M. Hosoi, T. Mizoguchi, T. Hinokihara, H. Hiroyasu Matsuura, and M. Ogata, arXiv:1804.04874 [cond-mat.str-el] .
  * Arima  _et al._ (1969) A. Arima, M. Harvey, and K. Shimizu, Physics Letters B 30, 517 (1969).
  * Santini  _et al._ (2009) P. Santini, S. Carretta, G. Amoretti, R. Caciuffo, N. Magnani, and G. H. Lander, Rev. Mod. Phys. 81, 807 (2009).
  * Shiina  _et al._ (1998) R. Shiina, O. Sakai, H. Shiba, and P. Thalmeier, Journal of the Physical Society of Japan 67, 941 (1998), https://doi.org/10.1143/JPSJ.67.941 .
  * Maharaj  _et al._ (2020) D. D. Maharaj, G. Sala, M. B. Stone, E. Kermarrec, C. Ritter, F. Fauth, C. A. Marjerrison, J. E. Greedan, A. Paramekanti, and B. D. Gaulin, Phys. Rev. Lett. 124, 087206 (2020).
  * Paramekanti  _et al._ (2020) A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, Phys. Rev. B 101, 054439 (2020).
  * Winter  _et al._ (2016) S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Phys. Rev. B 93, 214431 (2016).
  * Kim  _et al._ (2019) B. H. Kim, D. V. Efremov, and J. van den Brink, Phys. Rev. Materials 3, 014414 (2019).
  * T. Inui and Onodera (1990) Y. T. T. Inui and Y. Onodera,  _Group theory and its applications in physics_ (Springer-Verlag Berlin Heidelberg, 1990).
  * Horvat  _et al._ (2017) A. Horvat, L. Pourovskii, M. Aichhorn, and J. Mravlje, Phys. Rev. B 95, 205115 (2017).
  * Franchini  _et al._ (2011) C. Franchini, T. Archer, J. He, X.-Q. Chen, A. Filippetti, and S. Sanvito, Phys. Rev. B 83, 220402 (2011).
  * Archer  _et al._ (2011) T. Archer, C. D. Pemmaraju, S. Sanvito, C. Franchini, J. He, A. Filippetti, P. Delugas, D. Puggioni, V. Fiorentini, R. Tiwari, and P. Majumdar, Phys. Rev. B 84, 115114 (2011).
  * (60) L. V. Pourovskii, private communication.
  * Pourovskii and Khmelevskyi (2019) L. V. Pourovskii and S. Khmelevskyi, Phys. Rev. B 99, 094439 (2019).

##  III Supplementary Material

##  IV Methods

In this section the computational approaches used to investigate the
multipolar interactions are described.

###  IV.1 Computational details

Density Functional Theory (DFT) calculations were done within the projector
augmented wave method Blöchl (1994) as implemented in the Vienna ab initio
Simulation Package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller
(1996). The Perdew-Burke-Ernzerhof approximation Perdew  _et al._ (1996) for
the exchange-correlation functional was used, and, in order to take into
account the relativistic effects, we considered the fully relativistic scheme
that includes Spin Orbit Coupling (SOC), with a SOC energy of 482 meV
(computed by mapping the DFT energy difference calculated with and without SOC
onto a relativistic atomic Hamiltonian for $d$ orbitals Dudarev  _et al._
(2019)). Furthermore, we added an on-site Hubbard U correction as implemented
in the Dudarev’s approach for noncollinear spin configurations Dudarev  _et
al._ (2019):

$E_{DFT+U}=E_{DFT}+\frac{U}{2}\sum_{\sigma}\Big{[}\Big{(}\sum_{m_{1}}n^{\sigma}_{m_{1},m_{1}}\Big{)}-\Big{(}\sum_{m_{1},m_{2}}\hat{n}^{\sigma}_{m_{1},m_{2}}\hat{n}^{\sigma}_{m_{2},m_{1}}\Big{)}\Big{]}\
,$ (3)

where $\hat{n}^{\sigma}_{m_{1},m_{2}}$ is density matrix with $m_{1}$ and
$m_{2}$ running over the orbital angular momentum and $\sigma$ over the spin
angular momentum and U is the effective Hubbard U interaction. We used a U of
3.4 eV, in line with previous experimental estimations Erickson  _et al._
(2007). As a reference we show in Fig. 8 the optical spectra obtained with
this value of $U$ at DFT+U+SOC and ED level.

The supercell used is shown in Figure 5, where the lattice parameters are
$a/\sqrt{2}\times a/\sqrt{2}\times a$, with $a$ = 8.287 Å being the
experimental room-temperature value Stitzer  _et al._ (2002). It contains two
formula units, or, equivalently, two osmium atoms on parallel planes along the
z direction. The reciprocal space was sampled with a $k$-point mesh of
6$\times$6$\times$6 and a high accuracy was achieved by setting the energy
cutoff to the plane waves of 600 eV, together with a electronic convergence
criterion of $\sim$ 10-8 eV.

Figure 5: Magnetic unit cell of BNOO.

The local structure was studied by means of selective dynamics routine, fixing
the positions of the Na, Os and Ba sites and allowing the oxygens to be
relaxed to their ground state through the direct inversion in the iterative
subspace (RMM-DIIS) algorithm. The magnetic configuration in the canted phase
was set with noncollinear magnetic moments constrained as in Figure 1 (b)
(main text), with $\phi=67^{o}$. Both DFT, DFT + U and DFT + U + SOC were used
in order to probe the role of the different degrees of freedom. We verified
that the most stable local structure is the one defined as Model A in Ref. Liu
_et al._ (2018).

The canted procedure used for studying the exchange interactions – keeping the
structure fixed and changing the direction of the local magnetic moments –
allows to make use of the magnetic force theorem. To do so, we tilted the
angles between magnetic moments in parallel planes in steps of 2.5 degree
starting from a AFM [$\bar{1}$10] and ending in the FM [110] configuration
(see Figure 6)

Figure 6: Canting angles for BNOO

This can be achieved in VASP by means of the constrained magnetic moment
approach which constrains the direction of the local spin magnetic moment by
adding an extra penalty energy, which has the following expression Liu  _et
al._ (2015)

$E(\lambda)=\sum_{I}\lambda\Big{[}\mathbf{\vec{M}}_{I}-\mathbf{\hat{M}}_{I}^{0}\big{(}\mathbf{\hat{M}}_{I}^{0}\cdot\mathbf{M}_{I}\big{)}\Big{]}^{2}\
.$ (4)

Here the sum runs over the ionic sites $I$, $\mathbf{\hat{M}}_{I}^{0}$ is the
unit vector that gives the constrained direction of the magnetic moment at
site $I$, and $\vec{M}_{I}$ is the integrated magnetic moment inside the
Muffin-tin sphere $\Omega_{I}$. The constant $\lambda$ gives the strength of
the penalty energy and was chosen manually in order to get a value of
$E(\lambda)\ll E(DFT)$, a necessary requirement for having meaningful DFT
total energies. Our values of $E(\lambda)$ were always lower than 10-6 eV.

The result is a energy variation as a function of the canting angle, that, as
we will show in the next section, can be mapped onto a Pseudo Spin Hamiltonian
(PSH) with multipolar interactions.

###  IV.2 Calculations of exchange interactions within DFT+HI

Our charge self-consistent DFT+DMFT calculations using the Hubbard-I (HI)
approximation for Os 5$d$, abbriviated as DFT+HI, were carried out for the
paramagnetic phase and the undistored cubic structure of BNOO with $a=8.287$ Å
and the Os-O distance of 1.869 Å. We employed the Wien-2k electronic structure
package in conjunction with ”TRIQS” library implementations for the DMFT cycle
and Hubbard-I method. The spin-orbit coupling was included in Wien2k within
the standard second-variation treatment. The Brillouin zone (BZ) integration
was carried out using 400 k-points in the full BZ.

The Wannier orbitals representing $t_{2g}$ states of Os 5$d$ were constructed
by the projective technique of Refs. Amadon  _et al._ (2008); Aichhorn  _et
al._ (2009) using the bands enclosed by the energy window $[-1.09:2.04]$ eV;
this window thus encloses all $t_{2g}$-like bands. The Coulomb interaction was
specified by the Slater parameter $F_{0}=$3.2 eV and the Hund’s rule coupling
$J_{H}$=0.5 eV. We employed the rotationally invariant Kanamori Coulomb
repulsion between Os $t_{2g}$, which was correspondingly defined by the intra-
orbital repulsion $U_{K}=F_{0}+4J_{H}/5$ and the Kanamori Hund’s rule
$J_{K}=3J_{H}/5$. The double counting was evaluated using the fully-localized-
limit expression and the atomic occupancy $N=1$ of Os 5$d$ shell. The DFT+HI
calculations were converged to 0.01 mRy in the total energy.

The DFT+HI calculations for the paramagnetic BNOO phase predict the ground
state quadruplet $J_{eff}=3/2$ for the Os 5$d^{1}$ shell, as expected; the
excited doublet $J_{eff}=1/2$ ws predicted to be 0.45 eV higher in energy,
corresponding to the spin-orbit coupling strength $\lambda$ for the $t_{2g}$
subshell equal to 0.3 eV. We subsequently employed the FT-HI method of Ref.
Pourovskii (2016) to evaluate all nearest-neighbour exchange interactions
between the $J_{eff}=3/2$ ground state multiplets. Namely, matrix elements of
inter-site coupling $V^{\bf RR^{\prime}}$ between sites ${\bf R}$ and ${\bf
R}^{\prime}$ read:

$\langle M_{1}M_{3}|V^{\bf
RR^{\prime}}|M_{2}M_{4}\rangle=\mathrm{Tr}\left[G_{\bf
RR^{\prime}}\frac{\delta\Sigma^{at}_{\bf
R^{\prime}}}{\delta\rho^{M_{3}M_{4}}_{{\bf R}^{\prime}}}G_{\bf
R^{\prime}R}\frac{\delta\Sigma^{at}_{\bf R}}{\delta\rho^{M_{1}M_{2}}_{{\bf
R}}}\right],$ (5)

where $M$ is the projection quantum number, $M=-3/2...3/2$, for the multiplet
$J_{eff}=3/2$, $\rho^{M_{i}M_{j}}_{{\bf R}}$ is the corresponding element of
the $J_{eff}=3/2$ density matrix on site $R$, $\frac{\delta\Sigma^{at}_{\bf
R}}{\delta\rho^{M_{i}M_{j}}_{{\bf R}}}$ is the derivative of atomic
(Hubbard-I) self-energy $\Sigma^{at}_{\bf R}$ over a fluctuation of the
$\rho^{M_{i}M_{j}}_{{\bf R}}$ element, $G_{\bf RR^{\prime}}$ is the inter-site
Green’s function evaluated within the DFT+HI. Once all matrix elements (5) are
calculated, they are transformed to the couplings
$I_{KK^{\prime}}^{QQ^{\prime}}$ between on-site moments (eq. 1 of the main
text) as follows:

$I^{QQ^{\prime}}_{KK^{\prime}}({\bf
RR^{\prime}})=\sum_{\begin{subarray}{c}M_{1}M_{2}\\\
M_{3}M_{4}\end{subarray}}\langle M_{1}M_{3}|V^{\bf
RR^{\prime}}|M_{2}M_{4}\rangle\left[O_{Q}^{K}(J)\right]_{M_{2}M_{1}}\left[O_{Q^{\prime}}^{K^{\prime}}(J)\right]_{M_{4}M_{3}},$

where $\left[O_{Q}^{K}(J)\right]_{M_{1}M_{2}}$ is the $M_{1}M_{2}$ matrix
element of the corresponding multipolar tensor.

The resulting 15$\times$15 interaction matrix $I^{QQ^{\prime}}_{KK^{\prime}}$
for the [1/2,1/2,0] Os-Os bond (omitting the irrelevant monopole term) is
shown in Table 3. Interaction matrices for other NN bonds can be obtained from
it using the corresponding symmetry operations; the interaction matrix summed
over all interlayer bonds and omitting the terms irrelevant for the planar
magnetic order of BNOO is graphically depicted in Fig. 2c of the main text.

Table 2:  The interaction matrix $I^{QQ^{\prime}}_{KK^{\prime}}$ calculated by DFT+HI for the nearest-neighbour Os-Os bond [1/2,1/2,0], in meV. The order of $K$ and $Q$ is indicated in the first two columns. | -1 | -0.96 | -0.03 | -0.74 | 0 | 0 | 0 | 0 | 0 | 0.14 | -0.01 | -1.05 | 0 | 0.19 | 0 | -0.54
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 0 | -0.03 | 1.36 | -0.02 | 0 | 0 | 0 | 0 | 0 | 0 | -0.83 | -0.01 | 0.44 | 0 | 0.01 | 0
| 1 | -0.74 | -0.02 | -0.85 | 0 | 0 | 0 | 0 | 0 | 0.52 | 0 | 0.17 | 0 | -1.05 | 0 | -0.09
| -2 | 0 | 0 | 0 | -1.04 | 0 | -1.02 | -0.02 | -0.01 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| -1 | 0 | 0 | 0 | 0 | -1.65 | 0.01 | 0.10 | -0.01 | 0 | 0 | 0 | 0 | 0 | 0 | 0
2 | 0 | 0 | 0 | 0 | -1.02 | 0.01 | 1.43 | 0.01 | 0.07 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| 1 | 0 | 0 | 0 | -0.02 | 0.10 | 0.01 | -1.76 | 0.02 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| 2 | 0 | 0 | 0 | -0.01 | -0.01 | 0.07 | 0.02 | -1.42 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| -3 | 0.14 | 0 | 0.52 | 0 | 0 | 0 | 0 | 0 | 1.69 | -0.01 | -0.20 | 0.01 | 0.49 | -0.01 | 0
| -2 | -0.01 | -0.83 | 0 | 0 | 0 | 0 | 0 | 0 | -0.01 | -1.14 | 0 | 0.32 | 0.02 | 0 | -0.02
| -1 | -1.05 | -0.01 | 0.17 | 0 | 0 | 0 | 0 | 0 | -0.20 | 0 | 0.19 | -0.01 | 0.58 | 0.01 | -0.49
3 | 0 | 0 | 0.44 | 0 | 0 | 0 | 0 | 0 | 0 | 0.01 | 0.32 | -0.01 | 1.34 | 0 | -0.09 | 0
| 1 | 0.19 | 0 | -1.05 | 0 | 0 | 0 | 0 | 0 | 0.49 | 0.02 | 0.58 | 0 | 0.32 | 0 | 0.08
| 2 | 0 | 0.01 | 0 | 0 | 0 | 0 | 0 | 0 | -0.01 | 0 | 0.01 | -0.09 | 0 | -1.49 | 0.01
| 3 | -0.54 | 0 | -0.09 | 0 | 0 | 0 | 0 | 0 | 0 | -0.02 | -0.49 | 0 | 0.08 | 0.01 | 1.68

###  IV.3 Exact Diagonalization

To calculate the multipolar interactions between neighboring Os7+ ions, we
considered two-site $t_{2g}$-orbital systems with following Hamiltonian

$\displaystyle H$
$\displaystyle=\sum_{i\mu\nu\sigma}\epsilon_{\mu\nu}^{i}c^{\dagger}_{i\mu\sigma}c_{i\nu\sigma}+\lambda\sum_{i\mu\nu\sigma\sigma^{\prime}}(\mathbf{l}\cdot\mathbf{s})_{\mu\sigma,\nu\sigma^{\prime}}c_{i\mu\sigma}^{\dagger}c_{i\nu\sigma^{\prime}}+\frac{1}{2}\sum_{i\sigma\sigma^{\prime}\mu\nu}U_{\mu\nu}c_{i\mu\sigma}^{\dagger}c_{i\nu\sigma^{\prime}}^{\dagger}c_{i\nu\sigma^{\prime}}c_{i\mu\sigma}+\frac{1}{2}\sum_{\begin{subarray}{c}i\sigma\sigma^{\prime}\\\
\mu\neq\nu\end{subarray}}J_{\mu\nu}c_{i\mu\sigma}^{\dagger}c_{i\nu\sigma^{\prime}}^{\dagger}c_{i\mu\sigma^{\prime}}c_{i\nu\sigma}$
$\displaystyle+\frac{1}{2}\sum_{\begin{subarray}{c}\sigma\\\
\mu\neq\nu\end{subarray}}J_{\mu\nu}^{\prime}c_{i\mu\sigma}^{\dagger}c_{i\mu,-\sigma}^{\dagger}c_{i\nu,-\sigma}c_{i\nu\sigma}+\sum_{\mu\nu\sigma}\left(t_{\mu\nu}^{12}c^{\dagger}_{1\mu\sigma}c_{2\nu\sigma}+h.c.\right),$
(6)

where $c^{\dagger}_{i\mu\sigma}$ is the creation operator of $t_{2g}$
electrons with $\mu$ ($\mu=xy,yz,zx$) orbital and $\sigma$ ($\sigma=+,-$) spin
states at the $i$-the site ($i=1,2$). First and second terms describe local
level splitting of $t_{2g}$ orbitals due to the crystal field and spin-orbit
coupling, respectively. The spin-orbit coupling strength $\lambda$ is set to
be $0.3$ eV. Third, fourth, and fifth terms refer to the Kanamori-type Coulomb
interactions with $U_{\mu\mu}=U$, $U_{\mu\neq\nu}=U-2J_{H}$, and
$J_{\mu\nu}=J_{\mu\nu}^{\prime}=J_{H}$. We set $U=3.4$ and $J_{H}=0.3$ eV.
Last term is the hopping Hamiltonian between $t_{2g}$ orbitals at 1 and 2
sites. The hopping integral $t_{\mu\nu}^{12}$ and local crystal splitting
$\epsilon_{\mu\nu}^{i}$ are estimated with the Wannier projection of the DFT
calculation. We took into account all possible states of two electrons in two-
site cluster. Employing the exact diagonalization method, we solved Eq. IV.3
and got eigenvalues and eigenstates.

Because $U$ is much larger than strength of hopping integrals, lowet $16$
eigenstates are dominantly determined by unperturbative eigenstates
$|M_{1}M_{2}\rangle$’s ($-3/2\leq M_{1},M_{2}\leq 3/2$). Let $E_{n}$ and
$\left|\Psi_{n}\right>$ be $n$-th eigenvalue and eigenstate, respectively. The
spin-orbital dynamics of $J_{eff}=3/2$ multiplets can be depicted with
following effective Hamiltonian

$H_{\rm
eff}=\sum_{M_{1}M_{2}M_{1}^{\prime}M_{2}^{\prime}}h^{M_{1}M_{2}}_{M_{1}^{\prime}M_{2}^{\prime}}|M_{1}M_{2}\rangle\langle
M_{1}^{\prime}M_{2}^{\prime}|.$ (7)

The overlap matrix $\mathbf{S}$ is defined as
$S_{nm}=\sum_{M_{1}M_{2}}\left<\Psi_{n}\right|M_{1}M_{2}\left>\right<M_{1}M_{2}\left|\Psi_{m}\right>$.
Coefficient $h^{M_{1}M_{2}}_{M_{1}^{\prime}M_{2}^{\prime}}$ is approximately
estimated as

$h^{M_{1}M_{2}}_{M_{1}^{\prime}M_{2}^{\prime}}=\sum_{n=1}^{16}\sum_{m=1}^{16}\sum_{m^{\prime}=1}^{16}E_{n}[S^{-1/2}]_{mn}[S^{-1/2}]_{nm^{\prime}}\langle
M_{1}M_{2}|\Psi_{m}\rangle\langle\Psi_{m^{\prime}}|M_{1}^{\prime}M_{2}^{\prime}\rangle,$
(8)

where $\mathbf{S}^{-1/2}$ is the inverse of square-root matrix of $\mathbf{S}$
Winter  _et al._ (2016); Kim  _et al._ (2019).

Let $T^{K}_{Q}(J)$ be the irreducible tensor operator in a $J$ manifold
($K=0,\cdots,2J$ and $Q=-K,\cdots,K$). According to the Wigner-Eckart theorem,
$T^{K}_{Q}(J)$ is given as

$T^{K}_{Q}(J)=\sum_{MM^{\prime}}\langle
J||T^{K}(J)||J\rangle\frac{C_{J^{\prime}M^{\prime}KQ}^{JM}}{\sqrt{2J+1}}\left|J,M\right>\left<J,M^{\prime}\right|,$
(9)

where,$\langle J||T^{K}(J)||J\rangle$ is the reduced matrix element of rank
$K$ tensors and $C_{J^{\prime}M^{\prime}KQ}^{JM}$ is the Clebsch-Gordan (CG)
coefficients defined as $C_{J^{\prime}M^{\prime}KQ}^{JM}=\langle
JM|J^{\prime}M^{\prime}KQ\rangle$ T. Inui and Onodera (1990). Note that CG
coefficients are satisfied with following orthogonality relation

$\sum_{KQ}\frac{2K+1}{2J+1}C^{JM_{1}}_{JM_{2}KQ}C^{JM_{1}^{\prime}}_{JM_{2}^{\prime}KQ}=\delta_{M_{1}M_{2}}\delta_{M^{\prime}_{1}M^{\prime}_{2}}.$
(10)

We can easily show the following relation

$\left|J,M\right>\left<J,M^{\prime}\right|=\sum_{KQ}\frac{2K+1}{\sqrt{2J+1}}\frac{C^{JM}_{JM^{\prime}KQ}}{\langle
J||T^{K}(J)||J\rangle}T^{K}_{Q}(J)=\sum_{KQ}A^{JMM^{\prime}}_{KQ}T^{K}_{Q}(J),$
(11)

where
$A^{JMM^{\prime}}_{KQ}=\frac{2K+1}{\sqrt{2J+1}}\frac{C^{JM}_{JM^{\prime}KQ}}{\langle
J||T^{K}||J\rangle}$. The multipolar tensor operator $O^{K}_{Q}(J)$ of real-
valued (tesseral) harmonics are built as following

$\displaystyle O^{K}_{0}(J)$ $\displaystyle=T^{K}_{0}(J)$ (12a) $\displaystyle
O^{K}_{Q}(J)$
$\displaystyle=\frac{1}{\sqrt{2}}\Big{(}T^{K}_{-Q}(J)+(-1)^{Q}T^{K}_{Q}(J)\Big{)}~{}(Q>0)$
(12b) $\displaystyle O^{K}_{-Q}(J)$
$\displaystyle=\frac{i}{\sqrt{2}}\Big{(}T^{K}_{-Q}(J)-(-1)^{Q}T^{K}_{Q}(J)\Big{)}~{}(Q>0).$
(12c)

The projection operator $\left|J,M\right>\left<J,M^{\prime}\right|$ is given
as

$\left|J,M\right>\left<J,M^{\prime}\right|=\sum_{KQ}B^{JMM^{\prime}}_{KQ}O^{K}_{Q}(J),$
(13)

where $B^{JMM^{\prime}}_{K0}=A^{JMM^{\prime}}_{K0}$,
$B^{JMM^{\prime}}_{KQ}=\frac{A^{JMM^{\prime}}_{K,-Q}+(-1)^{Q}A^{JMM^{\prime}}_{KQ}}{\sqrt{2}}$,
and
$B^{JMM^{\prime}}_{K,-Q}=\frac{A^{JMM^{\prime}}_{K,-Q}-(-1)^{Q}A^{JMM^{\prime}}_{KQ}}{i\sqrt{2}}$
($Q>0$). Accordingly, the effective Hamiltonian in Eq. 7 is expressed in terms
of $O^{K}_{Q}$ operators as following

$H_{\rm
eff}=I^{00}_{00}\mathbf{I}+\sideset{}{{}^{\prime}}{\sum}_{KQ}I^{K0}_{Q0}O^{K}_{Q}(J_{1})+\sideset{}{{}^{\prime}}{\sum}_{KQ}I^{0K}_{0Q}O^{K}_{Q}(J_{2})+\sideset{}{{}^{\prime}}{\sum}_{K_{1}Q_{1}}\sideset{}{{}^{\prime}}{\sum}_{K_{2}Q_{2}}I^{K_{1}K_{2}}_{Q_{1}Q_{2}}O^{K_{1}}_{Q_{1}}(J_{1})O^{K_{2}}_{Q_{2}}(J_{2}),$
(14)

where
$I^{K_{1}K_{2}}_{Q_{1}Q_{2}}=\sum_{M_{1}M_{2}M^{\prime}_{1}M^{\prime}_{2}}h^{M_{1}M_{2}}_{M^{\prime}_{1}M^{\prime}_{2}}B^{JM_{1}M^{\prime}_{1}}_{K_{1}Q_{1}}B^{JM_{2}M^{\prime}_{2}}_{K_{2}Q_{2}}$
and $\sideset{}{{}^{\prime}}{\sum}_{KQ}$ refers to the summation over all $K$
and $Q$ values except for $K=Q=0$.

Table 3:  The interaction matrix $I^{K_{1}K_{2}}_{Q_{1}Q_{2}}$ calculated by ED for the nearest-neighbour Os-Os bond [1/2,1/2,0], in meV. The order of $K$ and $Q$ is indicated in the first two columns. | -1 | -0.68 | 0 | -0.61 | 0 | 0 | 0 | 0 | 0 | 0.06 | 0 | -1.07 | 0 | 0.11 | 0 | -0.28
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 0 | 0 | 0.90 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.53 | 0 | 0.15 | 0 | 0 | 0
| 1 | -0.61 | 0 | -0.68 | 0 | 0 | 0 | 0 | 0 | 0.28 | 0 | 0.11 | 0 | -1.07 | 0 | -0.06
| -2 | 0 | 0 | 0 | -1.17 | 0 | 0.80 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| -1 | 0 | 0 | 0 | 0 | -1.63 | 0 | 0.08 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
2 | 0 | 0 | 0 | 0 | 0.80 | 0 | 1.51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| 1 | 0 | 0 | 0 | 0 | 0.08 | 0 | -1.63 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1.42 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| -3 | 0.06 | 0 | 0.28 | 0 | 0 | 0 | 0 | 0 | 1.09 | 0 | -0.07 | 0 | 0.28 | 0 | 0
| -2 | 0 | -0.53 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1.08 | 0 | 0.31 | 0 | 0 | 0
| -1 | -1.07 | 0 | 0.11 | 0 | 0 | 0 | 0 | 0 | -0.07 | 0 | 0.33 | 0 | 0.47 | 0 | -0.28
3 | 0 | 0 | 0.15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.31 | 0 | 1.30 | 0 | 0 | 0
| 1 | 0.11 | 0 | -1.07 | 0 | 0 | 0 | 0 | 0 | 0.28 | 0 | 0.47 | 0 | 0.33 | 0 | 0.07
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1.26 | 0
| 3 | -0.28 | 0 | -0.06 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.28 | 0 | 0.07 | 0 | 1.09

##  V Optimal Structural Data

The energy minimum is found for an expansion of the Os-O octahedra along the
x, y, and z directions with different magnitude along the different
crystallographic axis. Between parallel planes along z, the distortions are
staggered in the xy plane, meaning that the Os(1)-O bond length along x (y) is
equal to the Os(2)-O bond length along y (x). Furthermore, the angles between
bonds in the xy plane are slightly changed from the perfect 90 degrees,
forming alternated angles of $\sim$ 90.5o and $\sim$ 89.5o.

The corresponding Point Group is D2h-12 or Pmnn and allows for anti-symmetric
exchange interactions. The structural CIF file for this compound is in
Appendix.

Atoms | x | y | z
---|---|---|---
Na | 0 | 0 | 0
| 0.5 | 0.5 | 0.5
Os | 0.5 | 0.5 | 0
| 0 | 0 | 0.5
Ba | 0.5 | 0 | 0.75
| 0 | 0.5 | 0.75
| 0.5 | 0 | 0.25
| 0 | 0.5 | 0.25
O | 0.27126 | 0.26896 | 0.0
| 0.72874 | 0.73104 | 0.0
| 0.72667 | 0.27112 | 0.0
| 0.27333 | 0.72888 | 0.0
| 0.22667 | 0.22888 | 0.5
| 0.77333 | 0.77112 | 0.5
| 0.77126 | 0.23104 | 0.5
| 0.22874 | 0.76896 | 0.5
| 0 | 0 | 0.72849
| 0 | 0 | 0.27151
| 0.5 | 0.5 | 0.77151
| 0.5 | 0.5 | 0.22849
Table 4:  Optimized structural positions in direct coordinates.

###  V.1 Jahn-Teller distortions

The JT vibrational mode Q1, Q2 and Q3, are defined following the standard Van
Vleck notation Van Vleck (1939):

$\displaystyle Q_{1}$ $\displaystyle=\frac{dx+dy+dz}{\sqrt{6}}$ (15)
$\displaystyle Q_{2}$ $\displaystyle=\frac{dx-dy}{2}$ (16) $\displaystyle
Q_{3}$ $\displaystyle=\frac{dx+dy-2dz}{2\sqrt{3}}$ (17) $\displaystyle\theta$
$\displaystyle=\tan^{-1}\Big{(}\frac{Q_{2}}{Q_{3}}\Big{)}$ (18) $\displaystyle
Q$ $\displaystyle=\sqrt{Q_{2}^{2}+Q_{3}^{2}}\ ,$ (19)

where dx (dy, dz) is xi \- x0 (yi \- y0 , zi \- z0 ) i.e. the difference
between the distorted Os-O bond length and the undistorted one.

To construct the phase diagram we have introduced the relative JT parameter
$\tilde{Q}_{1}$, defined as the ratio between Q1 for the interpolated
structure and the corresponding value for the optimal one. The values of Q1,
Q2 and Q3, as well as of $\theta$ and Q as a function of $\tilde{Q}_{1}$ are
in Table 5.

Table 5: Values of Q1, Q2 and Q3 as a function of $\tilde{Q}_{1}$ in plane 1 & 2. $\tilde{Q}_{1}$ | Q1 (Å) | Q2 (Å) | Q3 (Å) |  $\theta$ (deg) | Q (Å) | Q1 (Å) | Q2 (Å) | Q3 (Å) |  $\theta$ (deg) | Q (Å)
---|---|---|---|---|---|---|---|---|---|---
| Plane 1 (0 0 0) | Plane 2 (0 0 0.5)
0 | 0 | 0 | 0 |  | 0 | 0 | 0 | 0 |  | 0
0.25 | 0.00789 | 0.00219 | 0.00041 | 79.35 | 0.00222 | 0.00789 | -0.00219 | 0.00041 | -79.35 | 0.00222
0.5 | 0.01579 | 0.00437 | 0.00082 | 79.35 | 0.00445 | 0.01579 | -0.00437 | 0.00082 | -79.35 | 0.00445
0.75 | 0.02369 | 0.00656 | 0.00123 | 79.35 | 0.00667 | 0.02369 | -0.00656 | 0.00123 | -79.35 | 0.00667
1 | 0.03159 | 0.00875 | 0.00164 | 79.35 | 0.00890 | 0.03159 | -0.00875 | 0.00164 | -79.35 | 0.00890

###  V.2 Phase Diagram

The phase diagram was derived by combining ab initio calculations and linear
interpolation of the PSH parameters.

The first-principles calculations were done by relaxing the structures with
the cAFM configuration for U=2.8, 3.0, 3.2 and 3.4 eV. Then, all the local
structural parameters were linearly interpolated from the distorted to the
undistorted (experimental-high-temperature) case. The parameters that are
interpolated include the xy angles and the variation of the bond length which
define the $\tilde{Q}_{1}$ parameter in Figure 3(c) of the main text.

We have performed a linear interpolation of the fitting parameters, by
linearly interpolating the fitting coefficients from the distorted to the
undistorted structure. Being a linear model suitable for the transition (See
Figure 3(b) of the main text), we extrapolated the full phase diagram at
higher values of $\tilde{Q}_{1}$.

##  VI Band structure and optical spectra

The Dirac-Mott nature of this compound is confirmed by our ab-initio
calculations with DFT, DFT+SOC, DFT+U and DFT+U+SOC methods (see Figure 7).
Even with a high U of 5 eV we are not able to open the gap within the DFT+U
scheme and only by including the relativistic effects and a U larger than 1.5
eV, is possible to open the gap.

The band structure within DFT+U+SOC does not show significant changes upon
different magnetic configurations, as well as by changing the local structure.
The corresponding optical spectra at DFT+U*SOC level is shown in Fig. 8.

Figure 7: Band structure of BNOO with DFT (a), DFT+SOC (b), DFT+U (c) and
DFT+U+ SOC (d). U is 3.4 eV. Figure 8: DFT+U+SOC calculated optical spectra
for Ba2NAaOsO6 compared with the ED prediction of the main Hubbard peak
(dominated by $t_{2g}-t_{2g}$ transitions.).

##  VII Pseudo Spin Hamiltonian

The exchange interactions were studied by mapping the DFT total energies onto
Pseudo Spin Hamiltonian (PSH). The PSH used is given as a multipolar expansion
of tensor products of dipolar, quadrupolar and octupolar operators that can be
written, in its general form, as

$H_{ij}=\sum_{K,Q}I_{Q_{i},Q_{j}}^{K_{i},K_{j}}(i,j)O_{Q_{i}}^{K_{i}}(J)O_{Q_{j}}^{K_{j}}(J)\
,$ (20)

where i,j label ionic sites, $K\leq 2J+1$, $Q=-K,...,K$ and $O_{Q}^{K}$ are
multipolar tensor operators written in the spherical harmonic superbasis. The
rank of such operators is given by K; where for K = 1 we have a _dipole_ , K=2
identifies a _quadrupole_ and K=3 the _octupole_. The corresponding operators
are written in Table 6.

Table 6: Multipolar operators for the BNOO Hamiltonian. The Multipole moments within a cubic $\Gamma_{8}$ quartet. Bars over symbols correspond to permutations of dipole operators, e.g. $\overline{j_{x}(j_{y})^{2}}=j_{x}j_{y}j_{y}+j_{y}j_{x}j_{y}+j_{y}j_{y}j_{x}$. Moment | Spherical Superbasis | Moment | Spherical Superbasis
---|---|---|---
Dipole | $O_{-1}^{1}=j_{y}$ | Octupole | $O_{-3}^{3}=\sqrt{5/8}\ (\overline{j^{2}_{x}j_{y}}-j_{y}^{3})$
| $O_{0}^{1}=j_{z}$ |  | $O_{-2}^{3}=\sqrt{5/12}\ \overline{j_{x}j_{y}j_{z}}$
| $O_{1}^{1}=j_{x}$ |  | $O_{-1}^{3}=\sqrt{3/8}\ (4/3\ \overline{j^{2}_{z}j_{y}}-1/3\ \overline{j^{2}_{x}j_{y}}-j_{y}^{3})$
Quadrupole | $O_{-2}^{2}=\sqrt{3}/2\ \overline{j_{x}j_{y}}$ |  | $O_{0}^{3}=j_{z}^{3}-1/2\ (\overline{j^{2}_{x}j_{z}}+\overline{j^{2}_{y}j_{z}})$
| $O_{-1}^{2}=\sqrt{3}/2\ \overline{j_{y}j_{z}}$ |  | $O_{1}^{3}=\sqrt{3/8}\ (4/3\ \overline{j^{2}_{z}j_{x}}-1/3\ \overline{j^{2}_{y}j_{x}}-j_{x}^{3})$
| $O_{0}^{2}=[3(j_{z})^{2}-j(j+1)]/2$ |  | $O_{2}^{3}=\sqrt{5/12}\ (\overline{j^{2}_{x}j_{z}}-\overline{j^{2}_{y}j_{z}})$
| $O_{1}^{2}=\sqrt{3}/2\ \overline{j_{z}j_{x}}$ |  | $O_{3}^{3}=\sqrt{5/8}\ (-\overline{j^{2}_{y}j_{x}}+j_{x}^{3})$
| $O_{2}^{2}=\sqrt{3}/2\ (j_{x}^{2}-j_{y}^{2})$ |  |

The PSH calculation proceeded in two steps. The first was to write the Pseudo
Spin Matrices for a J = 3/2 Pauli pseudo spin and to project them onto the
direction of the osmium magnetic moment. Due to the canting procedure
described above, the coordinate system used is the global one of Figure 9, and
therefore, the unit vectors that describe the two osmium pseudo spins are
given by

$\hat{n}_{1}=(sin(\alpha),cos(\alpha),0)\ \ \ \ \ \ \
\hat{n}_{2}=(cos(\alpha),sin(\alpha),0)\ .$ (21)

where $\alpha$ is defined in Figure 9 and it is related to $\phi$ via
$\alpha=45^{o}-\phi$. Once this projection has been done, the new pseudo spin
matrix was diagonalized and the eigenstate with highest eigenvalue was chosen.

Figure 9: Global reference frame for the canting angles.

Then we applied the mean field analysis, summarized in equation
$O(1)O(2)\approx\langle O(1)\rangle O(2)+O(1)\langle O(2)\rangle-\langle
O(1)\rangle\langle O(2)\rangle$, which allows to rewrite the total energy as a
function of the mean values over pseudo eigenstates of the tensors operators,
leading to

$E=\frac{1}{2}\sum_{\langle
i,j\rangle}\sum_{K,Q}I_{Q_{i},Q_{j}}^{K_{i},K_{j}}(i,j)\langle
O_{Q_{i}}^{K_{i}}(J)\rangle\langle O_{Q_{j}}^{K_{j}}(J)\rangle\ .$ (22)

From Table 6 it is clear that every tensor contains powers of $j_{x}$, $j_{y}$
and $j_{z}$, and therefore, powers of the canting angle. Due to time reversal
symmetry, only the even powers of pseudo spins are allowed, and, through
algebraic simplifications, Eq. (22) can be rewritten as

$E(\phi)=A_{0}+\sum_{n=1}^{N}A_{n}cos(2n\phi)+\sum_{m=1}^{M}B_{m}sin(2m\phi)\
.$ (23)

Other restrictions come from the symmetry of the magnetic cell, which allows
symmetric exchange-interaction matrices only in-plane (i.e. when Os(1) and
Os(2) belong to the same plane), while anti-symmetric exchanges are allowed
between out-of-plane interactions.

The energy can be further separated into

$E=A_{0}+E_{d-d}+E_{q-q}+E_{d-o}+E_{o-o},$ (24)

where $E_{d-d}$ is the dipole-dipole contribution, $E_{q-q}$ is the
quadrupole-quadrupole contribution,$E_{d-o}$ is the dipole-octupole
contribution, $E_{o-o}$ is the octupole-octupole contribution. Converting Eq.
(24) into angular dependencies gives

$E(\phi)=A_{0}+\sum_{n=1}^{3}A_{n}cos(2n\phi)+\sum_{m=1}^{2}B_{m}sin(2m\phi)\
,$ (25)

where the coefficients $A_{n}$ are symmetric, while the coefficients $B_{m}$
come from broken inversion symmetry.

###  VII.1 Comparison DFT vs HI vs ED

The comparison between DFT and HI/ED was achieved by reproducing the DFT+U+SOC
canting curve starting from the calculated exchange coefficients obtained by
HI and ED. This was done by reducing the full 15 $\times$ 15 HI and ED
interaction matrices to the relevant coefficients acting within the xy plane
canting, based on a pseudo spin analysis. As previously specified, the mean
values $\langle O_{Q_{i}}^{K_{i}}(J)\rangle\langle
O_{Q_{j}}^{K_{j}}(J)\rangle$ are powers of $cos(\phi)$ and $sin(\phi)$. These
mean values were multiplied by the corresponding coefficients, collected and
added, so to obtain the final expression with coefficients $A_{n}$ and
$B_{m}$.

As mentioned in the main text, in order to achieve a quantitatively consistent
comparison between DFT+U and DFT+HI, the DFT+U+SOC AFM ($\phi$=90) - FM
($\phi=0$) energy difference has been aligned to the corresponding DFT+HI
value. Specifically, the DFT+U+SOC curves (and therefore the corresponding
magnetic interactions) have been divided by a factor of 10. The ED results
have not been rescaled, which guarantees a full consistency between DFT+HI and
ED data, manifested by the very similar interaction matrices (see Tab. 2 and
Tab. 3) and corresponding curves and exchange coefficients discussed in the
main text. In the following we discuss the source of difference between DFT+U
and effective Hamiltonian approaches and justify our choice to take DFT-HI as
quantitative reference.

First we notice that (i) the two types of approaches employ different
correlated basis set: PAW/FLAPW ’local’ basis in VASP/Wien2k and $t_{2g}$
Wannier functions in DFT+HubI and ED; moreover, (ii) while the DFT+U method
calculates the total energies of competing magnetic phases, the ED and HubI-
based approaches extract effective Hamiltonians by a perturbation-theory
treatment of the local-moment paramagnetic phase.

Despite this fundamental difference between these two approaches, for
localized $d$ system (in particular 3$d$ and to a lesser extent 4$d$) there is
generally a good correspondence between these two basis and the results
obtained are typically consistent, see for instance the results for SrTcO3
Horvat  _et al._ (2017); Franchini  _et al._ (2011) and NiO Archer  _et al._
(2011); leo . However, for more extended $d$ orbitals, where hybridization
effects are generally large the degree of comparability between the two models
is less robust. In effective Hamiltonian approaches the occupation of the
orbital subspace is well defined and corresponds to the ’formal charge’ (in
our case one electron in the $t_{2g}$ orbitals), whereas in DFT the ’full
occupation’ is difficult to identify objectively as it depends on the choice
of the integrated spheres and hybridization effects. This ambiguity between
’formal charge’ and (DFT) ’full charge’ is particularly evident in Os-double
perovskites as discussed by Pickett for BaCaOs2O6 (BCOO) Gangopadhyay and
Pickett (2016). In BCOO the formal charge is 2, but DFT leads to a full
occupation of about 5 electrons, due to the highly dispersive character of the
Os 5$d$ orbital and the large degree of hybridization with the O-$p$ states.
In our calculations we fall precisely in this situation as the full charge
computed by VASP is 6 electrons, significantly larger than the corresponding
formal charge considered in the model calculations (1 electron).

Provided these significant differences in the correlated subspace occupancies
between VASP and DFT+HubI/ED, the resulting large difference in the ordered
state energetics is not surprising. We would like to underline that formal and
full $d$-shell occupation are virtually identical in NiO and (to a lesser
extent) in SrTcO3; in the latter DFT+U occupancy is 3.43 as compared to the
formal 4$d$ occupancy of 3 Horvat  _et al._ (2017). The similarity of formal
and full $d$-shell occupancies in those cases results in the exchange energies
being very similar in the both approaches. Conversely, for the 5$d$ oxide UO2
quantitative difference between DFT+U+SOC and HI were already reported in
literature Dudarev  _et al._ (2019); Pourovskii and Khmelevskyi (2019).

Support on the quantitative validity of the effective Hamiltonian approaches
can be inferred from the comparison of calculated and experimental Néel
temperature $T_{N}$. The mean-field value of $T_{N}$=17.5 K obtained from the
calculated FT-HI interactions for the undistorted phase is in a reasonable
agreement, taking into account the usual mean-field overestimation, with its
experimental value of 6.7 K. For lattice distortions to occur at the AFM
transition, the ordering energy for the distorted structure must be larger
than that for the undistorted one. Hence, one would expect the ordering energy
(and, similarly, $T_{N}$) for the distorted phase to be at least not smaller
than that of the undistorted one. The ordering energy and (to a lesser extent)
transition temperature of the undistorted case thus provide a lower limit for
those quantitites in the distorted case. Since undistorted FT-HI $T_{N}$
already has a correct (and even somewhat overestimated) magnitude, one would
not expect the magnetic energy to be significantly larger than that given by
FT-HI (and ED).

From these arguments the overall magnitude of FT-HI/ED interactions seems to
be in a better correspondence to experiment as compared to DFT+U, hence, we
normalized the magnitude of DFT+U interactions accordingly. However, notice
that all three approaches predict virtually the same non-trivial shape of
magnetic energy vs. angle $\phi$ (Fig. 2b). Correspondingly, the relative
magnitudes of coefficients $A_{1}$, $A_{2}$ and $A_{3}$ in the undistorted
phase also agree well. Basing on these observations one may be confident that
the qualitative shape of magnetic energy vs. $\phi$ and the relative magnitude
of various multipolar contributions into it are correctly resolved by the
DFT+U method.

##  VIII Appendix

CIF file created by FINDSYM, version 7.0
data_findsym-output
_audit_creation_method FINDSYM

_cell_length_a 5.8597900000
_cell_length_b 5.8597900000
_cell_length_c 8.2870000000
_cell_angle_alpha 90.0000000000
_cell_angle_beta 90.0000000000
_cell_angle_gamma 90.0000000000
_cell_volume 284.5518696011

_symmetry_space_group_name_H-M ”P 21/n 21/n 2/m”
_symmetry_Int_Tables_number 58
_space_group.reference_setting ’058:-P 2 2n’
_space_group.transform_Pp_abc a,b,c;0,0,0

loop_
_space_group_symop_id
_space_group_symop_operation_xyz
1 x,y,z
2 x+1/2,-y+1/2,-z+1/2
3 -x+1/2,y+1/2,-z+1/2
4 -x,-y,z
5 -x,-y,-z
6 -x+1/2,y+1/2,z+1/2
7 x+1/2,-y+1/2,z+1/2
8 x,y,-z

loop_
_atom_site_label

_atom_site_type_symbol
_atom_site_symmetry_multiplicity
_atom_site_Wyckoff_label
_atom_site_fract_x
_atom_site_fract_y
_atom_site_fract_z
_atom_site_occupancy
_atom_site_fract_symmform
Na1 Na 2 b 0.00000 0.00000 0.50000 1.00000 0,0,0
Os1 Os 2 a 0.00000 0.00000 0.00000 1.00000 0,0,0
Ba1 Ba 4 f 0.00000 0.50000 0.25000 1.00000 0,0,Dz
O1 O 4 g 0.77126 0.23104 0.00000 1.00000 Dx,Dy,0
O2 O 4 g 0.22667 0.22888 0.00000 1.00000 Dx,Dy,0
O3 O 4 e 0.00000 0.00000 0.22849 1.00000 0,0,Dz