# Hyperarithmetic numerals

Caleb M.H. Camrud  CEIP Maria Zambrano
Seville, Spain<EMAIL_ADDRESS>Department of Mathematics
Iowa State University
Ames, Iowa 50011<EMAIL_ADDRESS>and  Timothy H. McNicholl  Department of
Mathematics
Iowa State University
Ames, Iowa 50011<EMAIL_ADDRESS>
###### Abstract.

Within the framework of computable infinitary continuous logic, we develop a
system of hyperarithmetic numerals. These numerals are infinitary sentences in
a metric language $L$ that have the same truth value in every interpretation
of $L$. We prove that every hyperarithmetic real can be represented by a
hyperarithmetic numeral at the same level of complexity.

##  1\. Introduction

A _numeral_ is a symbolic representation of a number. The common notions which
may come to mind are the Arabic digits “0” through “9”, though also commonly
used are the Roman “I”, “V”, “X”, _etc._ A numeral for our purposes, however,
is defined as the following.

######  Definition 1.1.

A possibly infinitary $L$-sentence $\varphi$ is a _numeral_ for $r$ if for
every $L$-structure $\mathcal{M}$, $\varphi^{\mathcal{M}}=r$.

In classical logic, the space of truth values includes only two numbers, $0$
and $1$, with $0$ normally interpreted as falsity and $1$ as truth. As such,
in classical logic, we can consider sentences yielding _contradictions_ as
numerals for $0$ and wffs giving _tautologies_ as numerals for $1$. To give a
pair of examples, for every first order structure $\mathcal{M}$,

$(\exists x\ x\neq x)^{\mathcal{M}}=0\ \ \ \ \text{ and }\ \ \ \ (\forall x\
x=x)^{\mathcal{M}}=1.$

But in the continuous logic of [2], the space of truth values is $[0,1]$. A
natural question arises: for which $r\in[0,1]$ is there a wff $\varphi$ such
that for every $L$-structure $\mathcal{M}$, $\varphi^{\mathcal{M}}=r$? That
is, which numbers have $L$-numerals?

In the original version of this continuous logic, there is a connective $u$
for every continuous mapping on $[0,1]$. Notably, such versions of continuous
logic have trivial numerals: for every $r\in[0,1]$, there is a connective
$\underline{u}_{r}$ corresponding to the constant map $u_{r}:[0,1]\to\\{r\\}$.
Hence, for every structure $\mathcal{M}$ and sentence $\varphi$,
$(\underline{u}_{r}\varphi)^{\mathcal{M}}=u_{r}(\varphi^{\mathcal{M}})=r$.

In subsequent formulations of this logic, however, a restricted set of
connectives consisting only of $\neg$,  . $-$ , and $\frac{1}{2}$ was used [3,
4, 5]. This was done for two primary reasons. First, $\neg$ plays precisely
the role of classical negation ($\neg$) and  . $-$ of reverse implication
($\leftarrow$), while the interpretation of the $\frac{1}{2}$ operator is
similarly intuitive, halving the truth-value of the wff it is attached to.
Second, in [5], it was shown that after interpretation, $\neg$, $\frac{1}{2}$,
and  . $-$ are dense in the set of all continuous maps on $[0,1]$. Thus
finitary well-formed formulas in these connectives can approximate those in
the wider set of connectives arbitrarily well. Such an approximation is,
moreover, sufficient for completeness (as was shown in [4]). To avoid
trivialities, this is the logic we will be investigating in this paper.

Unfortunately, with this reduced set of connectives, the set of reals with
$L$-numerals may be very small. For example, if $L$ is the language of metric
spaces, then only the dyadic rational numbers in $[0,1]$ have $L$-numerals.
So, in order to expand the space of numerals, we consider infinitary sentences
of $L$ in the continuous infinitary logic developed by Eagle [7]. With this
expansion, it is not hard to see that every number in $[0,1]$ has an
$L$-numeral; in fact every such number has an infinitary $\Sigma_{1}$ numeral.
So, to sharpen the question, we assume $L$ is computably numbered and we focus
on computable infinitary formulas. In fact, we confine our attention to the
language of metric spaces $L_{M}$ which consists only of a symbol
$\underline{d}$ for the metric. Since $L_{M}$ is a sub-language of all other
metric languages (_i.e._ , signatures of metric structures as in [2]), our
results apply to any computably numbered metric language. We then investigate
which reals have computable infinitary $L$-numerals. We also investigate the
relationship between the complexity of a real number and the complexity of its
$L$-numerals. We measure the complexity of a real by the complexity of its
right and left Dedekind cuts in the hyperarithmetic hierarchy.

Our main result is the following.

######  Theorem 1.2.

Suppose $0<\alpha<\omega_{1}^{\operatorname{CK}}$, and let $r\in[0,1]$.

  1. (1)

If the right Dedekind cut of $r$ is $\Sigma^{0}_{\alpha}$, then $r$ has a
computable $\Sigma_{\alpha}$ $L_{M}$-numeral $\Phi$. Furthermore, a code of
$\Phi$ can be computed from a $\Sigma^{0}_{\alpha}$ index of the right
Dedekind cut of $r$.

  2. (2)

If the left Dedekind cut of $r$ is $\Sigma^{0}_{\alpha}$, then $r$ has a
computable $\Pi_{\alpha}$ $L_{M}$-numeral $\Phi$. Furthermore, a code of
$\Phi$ can be computed from a $\Sigma^{0}_{\alpha}$ index of the left Dedekind
cut of $r$.

By means of a straightforward transfinite induction, it follows that the
converses of the statements in Theorem 1.2 hold.

Before moving to the proof of Theorem 1.2, we present some background material
and a few preliminary developments.

##  2\. Background and preliminaries

Let $\mathbb{D}$ denote the set of all dyadic rational numbers.

When $s\in\mathbb{R}$, let $D^{<}(s)$ and $D^{>}(s)$ denote the left and right
Dedekind cuts of $s$ respectively. Let $D^{\leq}(s)$ and $D^{\geq}(s)$ denote
the complements of the right and left Dedekind cuts of $s$ respectively.

###  2.1. Background from computability theory

We presume familiarity with the hyperarithmetic hierarchy as constructed in
Chapter 5 of [1]. See also Chapter II of [8]. In [6], continuous computable
infinitary logic is developed based on the classical computable infinitary
logic (see Chapter 7 of [1]) and the continuous infinitary logic developed by
Eagle in [7].

Let $\langle,\rangle$ be a computable bijection of
$\mathbb{N}\times\mathbb{N}$ onto $\mathbb{N}$, and let $(\ )_{0}$ and $(\
)_{1}$ denote its left and right inverses respectively so that
$\langle(n)_{0},(n)_{1}\rangle=n$ for all $n\in\mathbb{N}$.

Fix a computable enumeration $(q_{n})_{n\in\mathbb{N}}$ of $\mathbb{Q}$.

Let $m,n$ be positive integers. Suppose $R\subseteq\mathbb{N}^{m}$, and let
$X\subseteq\mathbb{N}$. We say that $e\in\mathbb{N}$ is a _$\Sigma_{n}^{0}(X)$
index_ of $R$ if $e$ is a $\Delta^{0}_{1}$ index of an
$S\subseteq\mathcal{P}(\mathbb{N})\times\mathbb{N}^{n+m}$ so that for all
$y_{1},\ldots,y_{m}\in\mathbb{N}$,

$R(y_{1},\ldots,y_{m})\ \Leftrightarrow\exists x_{1}\forall x_{2}\ldots
Qx_{n}S(X;y_{1},\ldots,y_{m},x_{1},\ldots,x_{n})$

where $Q\in\\{\forall,\exists\\}$ is $\forall$ just in case $n$ is even. By a
_$\Sigma^{0}_{n}$ index_ of $R$ we mean a $\Sigma^{0}_{n}(\emptyset)$ index of
$R$.

Suppose $\alpha<\omega_{1}^{\operatorname{CK}}$ and $A\subseteq\mathbb{N}$. We
say that $e\in\mathbb{N}$ is a _$\Sigma^{0}_{\alpha}$ index of $A$_ if
$(n)_{1}\in\mathcal{O}$ and $A=W_{(n)_{0}}^{\mathcal{H}((n)_{1})}$.

Suppose $h:\mathbb{N}\rightarrow\omega_{1}^{\operatorname{CK}}$. We say $h$ is
_computable_ if there is an $\hat{h}:\mathbb{N}\rightarrow\mathcal{O}$ so that
$h(n)=|\hat{h}(n)|_{\mathcal{O}}$ for all $n\in\ N$ and so that
$\operatorname{ran}(\hat{h})$ is linearly ordered by $<_{\mathcal{O}}$. If $h$
is computable, then an _index_ of $h$ is an index of a function $\hat{h}$ with
these properties.

###  2.2. Preliminaries from computable analysis

Let $r\in\mathbb{R}$, and suppose $\alpha\in\omega_{1}^{\operatorname{CK}}$.
We say $r$ is _left (right) $\Sigma^{0}_{\alpha}$_ if its left (right)
Dedekind cut is $\Sigma^{0}_{\alpha}$. If $r$ is left (right)
$\Sigma^{0}_{\alpha}$, then a _left (right) $\Sigma^{0}_{\alpha}$ index_ of
$r$ is a $\Sigma^{0}_{\alpha}$ index of its left (right) Dedekind cut.

Let $(r_{n})_{n\in\mathbb{N}}$ be a sequence of real numbers, and suppose
$h:\mathbb{N}\rightarrow\omega_{1}^{\operatorname{CK}}$. We say that
$(r_{n})_{n\in\mathbb{N}}$ is _left (right) $\Sigma^{0}_{h}$_ if $r_{n}$ is
left (right) $\Sigma^{0}_{h(n)}$ for every $n\in\mathbb{N}$. Suppose $h$ is
computable, and let $e$ be any index of $h$. If $(r_{n})_{n\in\mathbb{N}}$ is
left (right) $\Sigma^{0}_{h}$, then we say $e$ is a left (right)
$\Sigma^{0}_{h}$ index of $(r_{n})_{n\in\mathbb{N}}$.

###  2.3. Dyadic numerals

We construct here two canonical sets of numerals for the dyadic rationals in
$[0,1]$: $(\nu^{\exists}_{r})_{r\in\mathbb{D}\cap[0,1]}$ and
$(\nu^{\forall}_{r})_{r\in\mathbb{D}\cap[0,1]}$. The latter will be universal,
and the former existential. We make use of the fact that $\mathbb{D}\cap[0,1]$
is the smallest set of real numbers that contains $0$ and $1$ and is closed
under $x\mapsto\frac{1}{2}x$ and $x\mapsto 1-x$. Accordingly,
$\nu^{\exists}_{r}$ and $\nu^{\forall}_{r}$ are defined by the following
equations.

$\displaystyle\nu^{\exists}_{0}$ $\displaystyle=$
$\displaystyle\inf_{x_{0}}\underline{d}(x_{0},x_{0})$
$\displaystyle\nu^{\forall}_{0}$ $\displaystyle=$
$\displaystyle\sup_{x_{0}}\underline{d}(x_{0},x_{0})$
$\displaystyle\nu^{\exists}_{1-r}$ $\displaystyle=$
$\displaystyle\neg\nu^{\forall}_{r}$ $\displaystyle\nu^{\forall}_{1-r}$
$\displaystyle=$ $\displaystyle\neg\nu^{\exists}_{r}$
$\displaystyle\nu_{r/2}^{Q}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\nu_{r}^{Q},\ Q\in\\{\forall,\exists\\}$

##  3\. Proof of main theorem

We divide the main part of the proof of Theorem 1.2 into the following lemmas.

######  Lemma 3.1.

If $r\in[0,1]$ is right $\Sigma^{0}_{2}$, then there is a left
$\Sigma^{0}_{1}$ nonincreasing sequence $(r_{n})_{n\in\mathbb{N}}$ so that
$r=\inf_{n}r_{n}$ and so that $r_{n}\in[0,1]$.

###### Proof.

Suppose $r\in[0,1]$ is right $\Sigma^{0}_{2}$. Then, there is a computable
$R\subseteq\mathbb{N}^{2}\times\mathbb{Q}$ so that for all $q\in\mathbb{Q}$,

$q>r\ \Leftrightarrow\ \exists x_{0}\forall x_{1}R(x_{0},x_{1},q).$

Let

$R_{1}(x_{0},x_{1},q)\ \Leftrightarrow\ (q_{(x_{0})_{1}}\leq q\ \wedge\
R((x_{0})_{0},x_{1},q_{(x_{0})_{1}}).$

Set:

$\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle(-\infty,0)\ \cup\
\\{q\in\mathbb{Q}\cap(-\infty,1)\ :\exists x_{1}\neg R_{1}(x_{0},x_{1},q)\\}$
$\displaystyle s_{n}$ $\displaystyle=$ $\displaystyle\sup S_{n}$
$\displaystyle r_{n}$ $\displaystyle=$
$\displaystyle\min\\{s_{0},\ldots,s_{n}\\}$

Thus, by construction, $(r_{n})_{n\in\mathbb{N}}$ is non-increasing, and
$r_{n}\in[0,1]$.

We now show $r=\inf_{n}r_{n}$. It suffices to show that $r=\inf_{n}s_{n}$. We
first note that for every $q\in\mathbb{Q}$,

$q>r\ \Leftrightarrow\ \exists x_{0}\forall x_{1}R_{1}(x_{0},x_{1},q).$

We now observe that for all $q,q^{\prime}\in\mathbb{Q}$,

$q<q^{\prime}\ \wedge\ R_{1}(x_{0},x_{1},q)\ \Rightarrow\
R_{1}(x_{0},x_{1},q^{\prime}).$

Hence, each $S_{n}$ is closed downward. It now follows that $D^{<}(r)\subseteq
D^{\leq}(\inf_{n}s_{n})$ and that
$D^{<}(\inf_{n}s_{n})\subseteq\bigcap_{n}S_{n}=D^{\leq}(r)$. Thus,
$r=\inf_{n}s_{n}$.

By construction,

$D^{<}(r_{n})=\\{q\in\mathbb{Q}\ :\ \forall k\leq n\ \exists q^{\prime}\in
S_{k}\ q<q^{\prime}\\}.$

Since $S_{n}$ is $\Sigma^{0}_{1}$ uniformly in $n$, $(r_{n})_{n\in\mathbb{N}}$
is left $\Sigma^{0}_{1}$. ∎

We note that Lemma 3.1 uniformly relativizes in that the proof yields a
computable $f:\mathbb{N}\rightarrow\mathbb{N}$ so that for every
$e\in\mathbb{N}$ and $X\subseteq\mathbb{N}$, if $e$ is a right
$\Sigma^{0}_{2}(X)$ index of a number $s\in[0,1]$, then $f(e)$ is a left
$\Sigma^{0}_{1}(X)$ index of a sequence $(r_{n})_{n\in\mathbb{N}}$ so that
$s=\inf_{n}r_{n}$ and so that $r_{n}\in[0,1]$. This uniformity will be heavily
exploited later.

######  Lemma 3.2.

If $r\in[0,1]$ is left $\Sigma^{0}_{2}$, then there is a right
$\Sigma^{0}_{1}$ nondecreasing sequence $(r_{n})_{n\in\mathbb{N}}$ so that
$r=\sup_{n}r_{n}$ and so that $r_{n}\in[0,1]$.

###### Proof sketch.

Suppose $r\in[0,1]$ is left $\Sigma^{0}_{2}$, and let
$R\subseteq\mathbb{N}^{2}\times\mathbb{Q}$ be a computable relation so that

$q<r\ \Leftrightarrow\ \exists x_{0}\ \forall x_{1}\ R(x_{0},x_{1},q)$

for every $q\in\mathbb{Q}$. Let

$R_{1}(x_{0},x_{1},q)\ \Leftrightarrow q\leq q_{(x_{0})_{1}}\ \wedge\
R((x_{0})_{0},x_{1},q_{(x_{0})_{1}}).$

Set:

$\displaystyle S_{n}$ $\displaystyle=$ $\displaystyle(1,\infty)\ \cup\
\\{q\in\mathbb{Q}\cap(0,\infty)\ :\exists x_{1}\neg R_{1}(x_{0},x_{1},q)\\}$
$\displaystyle s_{n}$ $\displaystyle=$ $\displaystyle\inf S_{n}$
$\displaystyle r_{n}$ $\displaystyle=$
$\displaystyle\max\\{s_{0},\ldots,s_{n}\\}$

Thus, by construction, $(r_{n})_{n\in\mathbb{N}}$ is nondecreasing, and
$r_{n}\in[0,1]$. The rest of the proof is a straightforward modification of
the proof of Lemma 3.1. Namely, one flips the order of the inequalities, and
exchanges $\inf$ and $\sup$ as well as ‘upward’ and ‘downward’. ∎

Again, we note that Lemma 3.2 relativizes uniformly.

######  Lemma 3.3.

Let $\alpha<\omega_{1}^{\operatorname{CK}}$. If $r\in[0,1]$ is right
$\Sigma^{0}_{\alpha+1}$, then there is a non-increasing left
$\Sigma^{0}_{\alpha}$ sequence $(r_{n})_{n\in\mathbb{N}}$ so that
$r=\inf_{n}r_{n}$ and so that $r_{n}\in[0,1]$.

###### Proof.

We first consider the case $\alpha=0$. Let $(s_{n})_{n\in\mathbb{N}}$ be an
effective enumeration of $D^{>}(r)$. Set
$r_{n}=\min\\{1,s_{0},\ldots,s_{n}\\}$. Then, $(r_{n})_{n\in\mathbb{N}}$ is a
computable sequence of rational numbers, and $r=\inf_{n}r_{n}$.

The case $\alpha=1$ is handled by Lemma 3.1

Suppose $2\leq\alpha<\omega$. Let $k=\alpha-2$. Then, $D^{>}(r)$ is
$\Sigma^{0}_{2}(\emptyset^{(k+1)})$. By the relativization of Lemma 3.1, there
is a left $\Sigma^{0}_{1}(\emptyset^{(k+1)})$ non-increasing sequence
$(r_{n})_{n\in\mathbb{N}}$ so that $r=\inf_{n}r_{n}$ and so that
$r_{n}\in[0,1]$. Thus, $(r_{n})_{n\in\mathbb{N}}$ is left
$\Sigma^{0}_{\alpha}$.

Finally, suppose $\alpha$ is infinite. Thus, $D^{>}(r)$ is
$\Sigma^{0}_{1}(\emptyset^{(\alpha+1)})=\Sigma^{0}_{1}(\emptyset^{(\alpha)})$.
Again, by the relativization of Lemma 3.1, there is a left
$\Sigma^{0}_{1}(\emptyset^{(\alpha)})=\Sigma^{0}_{\alpha}$ nonincreasing
sequence so that $r=\inf_{n}r_{n}$ and so that $r_{n}\in[0,1]$. ∎

######  Lemma 3.4.

Let $\alpha<\omega_{1}^{\operatorname{CK}}$. Suppose $r\in[0,1]$ is left
$\Sigma^{0}_{\alpha+1}$. Then, there is a right $\Sigma^{0}_{\alpha}$
nondecreasing sequence $(r_{n})_{n\in\mathbb{N}}$ so that $r=\sup_{n}r_{n}$
and so that $r_{n}\in[0,1]$.

###### Proof sketch:.

Similar to proof of Lemma 3.3. ∎

Again, we observe that the above lemmas all uniformly relativize.

######  Lemma 3.5.

Suppose $0<\alpha<\omega_{1}^{\operatorname{CK}}$ is a limit ordinal, and
suppose $r\in[0,1]$ is right $\Sigma^{0}_{\alpha}$. Then, there is a
computable $h:\mathbb{N}\rightarrow\alpha$ and a right $\Sigma^{0}_{h}$ non-
increasing sequence $(r_{n})_{n\in\mathbb{N}}$ so that $r=\inf_{n}r_{n}$ and
so that $r_{n}\in[0,1]$.

###### Proof.

By standard techniques, there is a sequence $(S_{\beta})_{\beta<\alpha}$ so
that $D^{>}(r)=\bigcup_{\beta<\alpha}S_{\beta}$ and so that $S_{\beta}$ is
$\Sigma^{0}_{\beta}$ uniformly in $\beta$. Without loss of generality, we may
assume that $S_{\beta}$ is upwardly closed. Let $s_{\beta}=\inf
S_{\beta}\cup(1,\infty)$. Thus, $s_{\beta}\in[0,1]$ is right
$\Sigma^{0}_{\beta}$ uniformly in $\beta$.

To see that $r=\inf_{\beta}s_{\beta}$, first suppose $q$ is a rational number
so that $q>r$. Then, $q\in S_{\beta}$ for some $\beta<\alpha$, and so
$q\geq\inf S_{\beta}\geq s_{\beta}$. Thus,
$q\geq\inf_{\beta<\alpha}s_{\beta}$. On the other hand, suppose
$q>\inf_{\beta<\alpha}s_{\beta}$. Then, there exists $q^{\prime}\in
S_{\beta}\cup(1,\infty)$ so that $q>q^{\prime}$. If $q^{\prime}\in S_{\beta}$,
then $q\in S_{\beta}$ and so $q>r$. If $q^{\prime}>1$, then since $r\leq 1$,
$q>r$. Thus, $\inf_{\beta<\alpha}s_{\beta}\geq r$.

Finally, let $h$ be any computable surjection of $\mathbb{N}$ onto $\alpha$,
and set

$\displaystyle r_{n}=\min\\{s_{h(0)},\ldots,s_{h(n)}\\}.$

∎

######  Lemma 3.6.

Suppose $0<\alpha<\omega_{1}^{\operatorname{CK}}$ is a limit ordinal, and
suppose $r\in[0,1]$ is left $\Sigma^{0}_{\alpha}$. Then, there is a computable
$h:\mathbb{N}\rightarrow\alpha$ and a left $\Sigma^{0}_{h}$ non-increasing
sequence $(r_{n})_{n\in\mathbb{N}}$ so that $r=\sup_{n}r_{n}$ and so that
$r_{n}\in[0,1]$.

###### Proof sketch.

Similar to proof of 3.5. ∎

###### Proof of Theorem 1.2.

We proceed by effective transfinite recursion. We begin with the case
$\alpha=1$. Suppose $D^{<}(r)$ is $\Sigma^{0}_{1}$. Let $g$ be a computable
surjection of $\mathbb{N}$ onto $D^{<}(r)\cap\mathbb{D}\cap(0,1)\cup\\{1\\}$.
Set $\Phi=\bigcup_{n}\nu^{\exists}_{h(n)}$. Suppose $\mathcal{M}$ is an
interpretation of $L_{M}$ (_i.e._ , a metric space). Then,
$\Phi^{\mathcal{M}}=\inf_{n}r_{n}=r$. by the density of the dyadics, If
$D^{>}(r)$ is $\Sigma^{0}_{1}$, then we take $(r_{n})_{n\in\mathbb{N}}$ to be
an effective enumeration of $D^{>}(r)\cap\mathbb{D}\cap(0,1)\cup\\{0\\}$.

We now handle the recursive steps. Let
$\alpha<\omega_{1}^{\operatorname{CK}}$, and suppose $r$ is left
$\Sigma^{0}_{\alpha+1}$. By Lemma 3.4, there is a right $\Sigma^{0}_{\alpha}$
sequence $(r_{n})_{n\in\mathbb{N}}$ so that $r=\inf_{n}r_{n}$ and so that
$r_{n}\in[0,1]$. From $n\in\mathbb{N}$, it is possible to compute a code of a
$\Sigma^{0}_{\alpha}$ sentence $\Phi_{n}$ of $L_{M}$ so that
$\Phi_{n}^{\mathcal{M}}=r$ for every interpretation $\mathcal{M}$ of $L_{M}$.
We can then compute a code of
$\Phi=\operatorname*{\bigwedge\mkern-15.0mu\bigwedge}_{n}\Phi_{n}$. Hence,
$\Phi^{\mathcal{M}}=r$ for every interpretation $\mathcal{M}$ of $L_{M}$. The
case where $r$ is right $\Sigma^{0}_{\alpha}$ is handled similarly.

Suppose $\alpha>0$ is a limit ordinal. We first consider the case where $r$ is
left $\Sigma^{0}_{\alpha}$. Then, by Lemma 3.6, there is a computable
$h:\mathbb{N}\rightarrow\alpha$ and a left $\Sigma^{0}_{h}$ sequence
$(r_{n})_{n\in\mathbb{N}}$ so that $r=\sup_{n}r_{h(n)}$ and so that
$r_{n}\in[0,1]$. For each $n\in\mathbb{N}$, let $\Phi_{n}$ be a computable
$\Sigma^{0}_{h(n)}$ infinitary formula of $L$ so that
$\Phi^{\mathcal{M}}_{n}=\Phi_{n}$. Set
$\Phi=\operatorname*{\bigvee\mkern-15.0mu\bigvee}_{n}\Phi_{n}$.

The case where $r$ is right $\Sigma^{0}_{\alpha}$ is handled similarly. ∎

##  4\. Conclusion

We conclude with some speculation about the complexity of theories in
continuous logic.

Suppose $\mathcal{M}$ is an interpretation of $L$. By the _computable
infinitary theory of $\mathcal{M}$_ we mean the _function_ that maps each
computable infinitary sentence $\Phi$ of $L$ to $\Phi^{\mathcal{M}}$. We
denote this function $\operatorname{Th}^{C}_{\omega_{1}\omega}(\mathcal{M})$.
Let $X\subseteq\omega$. We say that $X$ _computes_
$\operatorname{Th}^{C}_{\omega_{1}\omega}(\mathcal{M})$ if $X$ computes a
partial function $f:\subseteq\mathbb{N}^{2}\rightarrow\mathbb{Q}$ so that
whenever $e$ is a code of a computable infinitary sentence of $L$ and
$k\in\mathbb{N}$, $|f(e,k)-q|<2^{-k}$.

######  Corollary 4.1.

Suppose $\mathcal{M}$ is an interpretation of $L$, and suppose $X$ computes
$\operatorname{Th}_{\omega_{1}\omega}^{C}(\mathcal{M})$. Then, $X$ computes
every hyperarithmetic set. Thus, no hyperarithmetic set computes
$\operatorname{Th}_{\omega_{1}\omega}^{C}(\mathcal{M})$.

######  Question 4.2.

If $\mathcal{M}$ is an interpretation of $L$, does it follow that there is a
least Turing degree $\mathbf{d}$ so that every $X\in\mathbf{d}$ computes
$\operatorname{Th}^{C}_{\omega_{1}\omega}(\mathcal{M})$?

## References

  * [1] C. J. Ash and J. Knight, _Computable structures and the hyperarithmetical hierarchy_ , Studies in Logic and the Foundations of Mathematics, vol. 144, North-Holland Publishing Co., Amsterdam, 2000.
  * [2] ItaïBen Yaacov, Alexander Berenstein, C. Ward Henson, and Alexander Usvyatsov, _Model theory for metric structures_ , Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 350, Cambridge Univ. Press, Cambridge, 2008, pp. 315–427.
  * [3] Itaï Ben Yaacov and José Iovino, _Model theoretic forcing in analysis_ , Ann. Pure Appl. Logic 158 (2009), no. 3, 163–174.
  * [4] Itaï Ben Yaacov and Arthur Paul Pedersen, _A proof of completeness for continuous first-order logic_ , The Journal of Symbolic Logic 75 (2010), no. 1, 168–190.
  * [5] Itai Ben Yaacov and Alexander Usvyatsov, _Continuous first-order logic and local stability_ , Transactions of the American Mathematical Society 362 (2010), 5213–5259.
  * [6] C. Camrud, I. Goldbring, and T.H. McNicholl, _On the complexity of the theory of a computably presented metric structure_ , Submitted. Preprint available at https://arxiv.org/abs/2106.05372, 2021.
  * [7] Christopher J. Eagle, _Expressive power of infinitary $[0,1]$-logics_, Beyond first order model theory, CRC Press, Boca Raton, FL, 2017, pp. 3–22.
  * [8] Gerald E. Sacks, _Higher recursion theory_ , Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990. MR 1080970 (92a:03062)


# Spin-lattice couplings in two-dimensional CrI3 from first-principles study

Banasree Sadhukhan<EMAIL_ADDRESS>Department of Applied Physics, School of
Engineering Sciences, KTH Royal Institute of Technology, AlbaNova University
Center, SE-10691 Stockholm, Sweden  Anders Bergman  Department of Physics and
Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Yaroslav O.
Kvashnin  Department of Physics and Astronomy, Uppsala University, Box 516,
SE-75120 Uppsala, Sweden  Johan Hellsvik  PDC Center for High Performance
Computing, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Nordita, KTH Royal Institute of Technology and Stockholm University, Hannes
Alfvéns väg 12, SE-106 91 Stockholm, Sweden  Anna Delin  Department of Applied
Physics, School of Engineering Sciences, KTH Royal Institute of Technology,
AlbaNova University Center, SE-10691 Stockholm, Sweden  Swedish e-Science
Research Center (SeRC), KTH Royal Institute of Technology, SE-10044 Stockholm,
Sweden

###### Abstract

Since thermal fluctuations become more important as dimensions shrink, it is
expected that low-dimensional magnets are more sensitive to lattice
distortions and phonons than bulk systems are. Here we present a fully
relativistic first-principles study on the spin-lattice coupling, i.e. how the
magnetic interactions depend on local lattice distortions, of the prototypical
two-dimensional ferromagnet CrI3. We extract an effective measure of the spin-
lattice coupling in CrI3 which is up to ten times larger than what is found
for bcc Fe. The magnetic exchange interactions, including Heisenberg and
relativistic Dzyaloshinskii-Moriya interactions, are sensitive both to the in-
plane motion of Cr atoms and out-of-plane motion of ligand atoms. We find that
significant magnetic pair interactions change sign from ferromagnetic (FM) to
anti-ferromagnetic (AFM) for atomic displacements larger than 0.16 Å. We
explain the observed strong spin-lattice coupling by analyzing the orbital
decomposition of isotropic exchange interactions, involving different crystal-
field-split Cr$-3d$ orbitals. The competition between the AFM t2g \- t2g and
FM t2g \- eg contributions depends on the bond angle formed by Cr and I atoms
as well as Cr-Cr distance. In particular, if a Cr atom is displaced, the FM-
AFM sign change when the I-Cr-I bond angle approaches 90∘. The obtained spin-
lattice coupling constants, along with the microscopic orbital analysis can
act as a guiding principle for further studies of the thermodynamic properties
and combined magnon-phonon excitations in two-dimensional magnets.

##  I Introduction

Until quite recently, no magnetic atomically thin, or two-dimensional (2D),
material was known to exist, despite the rapidly growing number of synthesized
2D materials. Since just a few years this situation has changed due to, e.g.,
the discovery of magnetic long-range order in monolayer CrI3 in 2017 Huang
_et al._ (2017). Understanding the nature of the magnetism in these materials
is connected to fundamental issues in condensed matter physics such as the
relation between dimensionality, thermal fluctuations and critical behavior,
and the onset of topological order in low-dimensional magnetic systems. In
addition, 2D magnetic materials hold promise for several technological
applications, e.g., transistors with magnetic functionality, high-efficiency
spin filters, and ultrathin magnetic sensors Wang  _et al._ (2018); Song  _et
al._ (2018); Xing  _et al._ (2017); Huang  _et al._ (2018); Jimenez  _et al._
(2020). Unsurprisingly, these fascinating materials have quickly become a very
active field of research. In particular, CrI3 has become somewhat of a
canonical system for exploring magnetism in 2D, probably due to that it was
one of the very first 2D magnets to be discovered. In CrI3, the large spin-
orbit coupling (SOC) in the I-ions create a substantial magnetic anisotropy,
stabilizing the 2D magnetic long-range order Huang  _et al._ (2017). In fact,
large enough SOC may also stabilize magnetic long-range order even in atomic
chains, according to some predictions Smogunov  _et al._ (2008). In 2D CrI3,
the magnetic interaction between the individual Cr atoms are of superexchange
type, mediated through the I-ions. The angles in the Cr-I-Cr bonds are close
to 90 degrees, which implies that it is unclear from the Goodenough-Kanamori
rules whether the Cr-Cr nearest-neighbor (NN) interaction will be
ferromagnetic (FM) or antiferromagnetic (AFM). Instead, careful computations
need to be performed. The magnetic Cr ions in CrI3 form a honeycomb lattice,
and topological edge magnons have been predicted to exist in such systems from
general arguments Zhang  _et al._ (2013), a prediction which was subsequently
indirectly confirmed experimentally in CrI3 Chen  _et al._ (2018), using
inelastic neutron scattering to map out the magnon spectrum. To analyze the
physics behind the topological magnon gap in these systems in more detail, the
magnetic interactions and magnetic excitations (magnon spectra) were recently
computed with DFT+$U$ including spin-orbit coupling for bulk, 2D and strips of
CrX3 (X = Cl, I, Br) systems Kvashnin  _et al._ (2020). These calculations
show that a small topological magnon gap is formed, supporting the view that
these systems are indeed topological magnetic insulators (TMI). However, the
obtained gap is minute and much smaller than the experimentally observed gap.
In the same work, it is speculated that the disagreement between previous
calculations and experiment might originate from lattice effects. Supporting
this view is the fact that the phonon modes involving Cr atoms in CrI3 have,
using DFT calculations, earlier been found to be particularly sensitive to the
magnetic ordering, suggesting substantial spin–lattice and spin–phonon
coupling in this system Webster  _et al._ (2018). The effect of lattice
vibrations on these chiral edge magnon modes was recently investigated
theoretically Thingstad  _et al._ (2019), finding that lattice vibrations may
weaken the topological protection, and that magnon polarons many form.
Clearly, the nature and effect of the spin-lattice couplings in CrI3 seem to
be complex issues, warranting careful investigation.

In the present work, we compute the spin-lattice interactions in CrI3,
analysing how both the Heisenberg interaction parameters $J_{ij}$ and the
Dzyaloshinskii-Moriya interaction (DMI) parameters $D_{ij}$ change when the Cr
atoms are displaced from their equilibrium positions. We also present a
detailed orbital analysis of the magnetic interactions using superexchange
theory.

The manuscript is organized as follows: In Sec. II we describe the method and
techniques for the coupled SLD. We present the formalism for computing
magnetic interactions within relativistic limit and give details of the
performed calculations. In Sec. III, we present our main results, namely the
effect of lattice displacements of Cr and I atoms the on magnetic exchange
interactions, DMI. We also present an orbital analysis for deeper microscopic
understanding of how the magnetism can be tailored by the lattice
displacements and estimate the SLD constant for CrI3 monolayer. Section IV
summarizes our conclusions and gives an outlook.

##  II Theory and computational details

The bilinear spin Hamiltonian $\mathcal{H}_{\rm S}$ contains Heisenberg
exchange, Dzyaloshinskii-Moriya interaction (DMI) and symmetric, anisotropic
interactions that in a compact form can be expressed as

$\displaystyle\mathcal{H}_{\rm
S}=-\sum_{ikj}\sum_{\\{\alpha,\beta\\}}e^{\alpha}_{i}J^{\alpha\beta}_{ij}(\\{u_{k}^{\mu}\\})e^{\beta}_{j}$
(1)

where $e^{\alpha}_{i}$ ($e^{\beta}_{j}$) is the $\alpha$ ($\beta$) component
of the unitary vector pointing along the direction of the spin located at the
site $i$ ($j$). The exchange tensor $J_{ij}^{\alpha\beta}$ is a rank 2 tensor
in spin space, with elements that in general have a dependence on the atomic
displacements $\\{u_{k}^{\mu}\\}$ as well as the magnetic configuration. For
clarity, the exchange tensor $J_{ij}^{\alpha\beta}$ explicitly depends on
$\\{u_{k}^{\mu}\\}$. The antisymmetric part of $J^{\alpha\beta}_{ij}$ can be
rewritten in terms of DMI vector, having e.g. an $z$-component

$\displaystyle\vec{D}^{z}_{ij}=(J^{xy}_{ij}-J^{yx}_{ij})/2.$ (2)

The contributions to the mixed spin-lattice Hamiltonian can then be obtained
by expanding the bilinear magnetic Hamiltonian $\mathcal{H}_{\rm S}$ in
displacement. This procedure of coupled SLD has been formulated and applied
successfully to model non-relativistic exhange striction for bcc Fe Hellsvik
_et al._ (2019). In the current manuscript, we generalize the idea by
considering full exchange interaction tensors of $J^{\alpha\beta}_{ij}$ and
focus on the three-body interaction

$\displaystyle\mathcal{H}_{\rm
SL}=-\sum_{ijk}\sum_{\\{\alpha,\beta\\}}\Gamma_{ijk}^{\alpha\beta\mu}e_{i}^{\alpha}e_{j}^{\beta}u_{k}^{\mu},$
(3)

where the coupling tensor is defined as
$\Gamma_{ijk}^{\alpha\beta\mu}=\frac{\partial J_{ij}^{\alpha\beta}}{\partial
u_{k}^{\mu}}$. The so developed approach is used to calculate from microscopic
origins the SLD interaction, and its effect on the magnetism, of CrI3
monolayer.

The exchange interaction tensors $J_{ij}^{\alpha\beta}$ were calculated by
means of magnetic force theorem, as implemented in the full-potential linear
muffin-tin orbital-based code RSPt RSP ; Wills  _et al._ (2010). In this
approach, the exchange interactions are calculated via Green’s functions
within linear response theory by perturbing the spin system by deviating the
initial moments ($\vec{e}_{0}$) with small angles Antropov  _et al._ (1997);
Udvardi  _et al._ (2003); Ebert and Mankovsky (2009); Secchi  _et al._ (2015).
All components of the $J^{\alpha\beta}_{ij}$ tensor are obtained from the
second order in the tilting angles. $\bar{J}$ represents the isotropic
(Heisenberg) part of the interaction, and $\mid\vec{D}\mid$ is the magnitude
of the DMI vector.

The density functional calculations are performed with local spin density
approximation (LSDA). We constructed CrI3 monolayer from bulk structure, with
a vacuum of about 20 Å added between the layers to avoid interactions between
them. Then the crystal structure was relaxed using the projector augmented
wave method (PAW) Kresse and Joubert (1999), as implemented in the VASP code
Kresse and Furthmüller (1996, 1996). The plane-wave energy cutoff was set to
370 eV with a 16$\times$16$\times$1 k-point mesh. The exchange interaction
tensors $J_{ij}^{\alpha\beta}$ were calculated using PBE Perdew  _et al._
(1996) as implemented in the full-potential linear muffin-tin orbital-based
code RSPt relativistic spin polarized toolkit ; energy  _et al._ (2010). To
calculate the spin-lattice coupling constants, we consider 2$\times$2$\times$1
supercells in which one atom is displaced along a specific direction $\mu$.
The magnetic exchange interactions $\bar{J_{ij}}$’s have been calculated
within RSPt RSP ; Wills  _et al._ (2010) for both the unitcell and
2$\times$2$\times$1 supercell with 16$\times$16$\times$1 k-point mesh for
which identical $\bar{J_{ij}}$’s were obtained. In order to calculate spin-
lattice coupling constants, $\bar{J_{ij}}$ have been calculated for the same
2$\times$2$\times$1 cell but now with one atom displaced with a finite
displacement $\Delta{U}$ along $\mu$ direction. Here we consider the
displacement of both the magnetic and ligand atoms along in-plane ($x$, $y$,
$xy$) and out-plane ($z$) directions, respectively, depending on the lattice
geometry of the CrI3 monolayer. Here the $x$, $y$, $xy$, $z$ represent the
[100], [010], [110], [001] directions respectively. From the displaced
supercell, we calculated the spin-lattice coupling constants as given by
$\Gamma_{ijk}^{\alpha\beta\mu}=\frac{\partial J_{ij}^{\alpha\beta}}{\partial
u_{k}^{\mu}}$.

In order to analyze the strength of the spin lattice coupling we define the
quantities

$\displaystyle\Gamma_{ijk}^{\mu}=\frac{\Gamma_{xx}^{\mu}+\Gamma_{yy}^{\mu}+\Gamma_{zz}^{\mu}}{3},$
(4)
$\displaystyle|\Gamma_{ijk}^{\mu}|=\sqrt{(\Gamma_{xx}^{\mu})^{2}+(\Gamma_{yy}^{\mu})^{2}+(\Gamma_{zz}^{\mu})^{2}},$
(5)
$\displaystyle\Gamma_{ijk}=\frac{{\Gamma_{ijk}^{\mu=x}}+{\Gamma_{ijk}^{\mu=y}}+{\Gamma_{ijk}^{\mu=z}}}{3},$
(6)
$\displaystyle|\Gamma_{ijk}|=\frac{|{\Gamma_{ijk}^{\mu=x}}|+|{\Gamma_{ijk}^{\mu=y}}|+|{\Gamma_{ijk}^{\mu=z}}|}{3},$
(7)

where $\Gamma_{ijk}$ and $|\Gamma_{ijk}|$ are an isotropic spin lattice
coupling constant and its absolute value, respectively.

Figure 1: (a) I-Cr-I bond angles in the octahedral environment of Cr atoms.
The change in I-Cr-I bond angles of NN $i$-link, $j$-link, $k$-link with (b)
$\mu_{Cr}=x$ and (c) $\mu_{I}=z$ respectively. Here the displacement magnitude
is chosen as $\Delta{U}=0.25$ Å. The green circle indicates the Cr or I atom
being displaced. Calculated isotropic exchange interaction (${\bar{J}_{ij}}$)
with (d) $\mu_{Cr}=x$ ; (e) $\mu_{I}=z$. The change in (f) Cr-Cr bond distance
and (g) I-Cr-I bond angles for the NN $i$-link, $j$-link, $k$-link with
$\mu_{Cr}=x$ ( $*$ corresponds to $\mu_{I}=z$). Calculated orbital-resolved
exchange interaction of the NN $i$-link ($\bar{J}_{0i}^{mm^{\prime}}$) with
displacement of (h) $\mu_{Cr}=x$ (* corresponds to the NN $j$-link
($\bar{J}_{0j}^{mm^{\prime}}$)) and (i) $\mu_{I}=z$.

##  III Results

###  III.1 Sensitivity of magnetism with displacement of atoms

Bulk CrI3 crystallizes in a layered van der Waals material (low temperature
space group R$\bar{3}$) which can easily be exfoliated to produce 2D
monolayers. The optimized in-plane lattice constant for monolayer CrI3 is
$a=6.817$ Å. Monolayer CrI3 consists of the honeycomb arrangement of the Cr
atoms coordinated by the six I ligands which form a distorted corner-sharing
octahedral environment around each Cr atom. The honeycomb network of Cr ions
is sandwiched by two atomic planes of I atoms. To study the effect of lattice
displacements on the magnetic interactions, we consider a 2$\times$2$\times$1
supercell. In octahedral environment, the I-Cr-I bond angle (same plane of I
atoms) within different octahedra is approximately $90^{\circ}$, whereas the
I-Cr-I the band angle (different plane of I atoms) is less than $180^{\circ}$
within the same octahedra. The I-Cr-I bond angle for the NN links ($i$-link,
$j$-link, $k$-link) of Cr atom connecting the I ligands of opposite planes is
approximately $86^{\circ}$ (see Fig. 1(a)).

Magnetism in CrI3 is associated with the partially filled $d$ orbitals of Cr
atom with an electronic configuration $3s^{0}3d^{3}$. In octahedral
environment of Cr atom, the crystal field interaction with the I ligands
results quenching of orbital moment (L = 0) and splitting of 3$d$ orbitals
into $e_{g}$ ($d_{x^{2}-y^{2}},d_{z^{2}}$) and $t_{2g}$
($d_{xy},d_{yz},d_{zx}$) manifolds. Therefore, the three electrons occupying
the $t_{2g}$ triplet makes CrI3 monolayer almost an ideal realization of a
system with spin $S=3/2$ according to Hund’s rule, and gives an atomic
magnetic moment of $\approx$ 3$\mu_{B}$ per Cr atom.

The exchange parameters, calculated for both the unitcell and the supercell of
CrI3 monolayer, as a function of distance, are found to be in very good
agreement with reported values Kvashnin  _et al._ (2020). Here the isotropic
part of the NN, next NN (NNN) and 3rd NN coupling are contributing in which
$\bar{J_{1}}$ is the dominant one. $\bar{J_{1}}$ consists of two competing
terms originating from $e_{g}$ and $t_{2g}$ orbitals which are strongly
hybridized with $p$ orbitals of the ligands. This suggests a nontrivial role
of the ligand states in the formation of magnetic ordering in CrI3 monolayer.
Later we will show a full multi-orbital analysis of superexchange mechanism as
the origin of ferromagnetism in CrI3 monolayer.

For each Cr atom, there exist three NN, six NNN and again three 3rd NN. As the
isotropic part of the NN, NNN and 3rd NN coupling are the most dominant ones,
we consider 12 NN atoms in total when studying the effect of lattice
displacements on magnetic ordering. For the undisplaced case, all these three
links have identical exchange interactions and DMI obeying the $C_{3}$
symmetry. Lattice displacements break the $C_{3}$ symmetry resulting a lower
symmetry in the obtained set of $\bar{J_{ij}}$ and DMI. However, the exchange
interactions $\bar{J_{ij}}$ for all NN of $i$-link, $j$-link, $k$-link take
different values. The same is also true for the NNN and 3rd NN $i$-link,
$j$-link, and $k$-link exchange interactions and DMI.

Figure 1(d)-(e) show the change in $\bar{J_{ij}}$ for NN $i$-link, $j$-link,
$k$-link for the in-plane displacement of Cr-atom along the $x$, and out-plane
displacement of I-atom along $z$ directions respectively. For the undisplaced
case, the I-Cr-I bond angle for the NN $i$-link, $j$-link, $k$-link is
$\approx$ $86^{\circ}$. The I-Cr-I bond angle changes for the $i,j$-link,
$j,k$-link, $k,i$-link, and $i$-link with the displacements of Cr atom long
$x,y,xy$, and I atom long $z$ respectively as shown in the Fig. 1(b)-(c) where
the green circle indicates the displaced Cr-atom. (see Fig. 6 in Sec. V for
the other two case of displacements $\mu_{Cr}=y,xy$). The strength of FM
exchange interaction for the NN $i$-link decreases with displacements of Cr
atom along $x$, while the NN $j$-link and $k$-link follow the opposite trends.
The strength of the NN $j$-link and $i$-link exchange interaction decrease,
when the displacements are applied along $y$ and $xy$ directions,
respectively. The FM to AFM sign change for a particular NN link
($\bar{J}_{1}$) occur for $\mu_{Cr}^{xy}\geq 0.12$ Å, $\mu_{Cr}^{x/y}\geq
0.16$ Å, $\mu_{I}^{z}\geq 0.18$ Å which is 1.76%, 2.35%, 2.64% of the lattice
constant, respectively.

The transition from the FM to AFM coupling is controlled either by the change
in the I-Cr-I bond angle or by the distance between the corresponding Cr
atoms. Since both parameters change when the atoms are displaced, at this
stage it is hard to identify the main driving force of the sign flip of the
$\bar{J}_{ij}$.

In order to elucidate this, we calculated the Cr-Cr bond length dCr-Cr and
I-Cr-I bond angle for different displacements for the in-plane motion of Cr
atom ($\mu_{Cr}=x$) and out-of-plane motion of ligand atom ($\mu_{I}=z$) as
shown in Fig. 1 (f)-(g) (see also Fig 6 in Sec. V for the other two cases
$\mu_{Cr}=y,xy$). In the undisplaced case, the $d_{\text{Cr-Cr}}$ for the NN
links is 3.938 Å which decreases with the displacement of Cr atom and switches
to AFM, when it reaches the value of 3.796 Å. The bond angle increases with
the displacement of Cr atom and the NN coupling is FM until the I-Cr-I angle
reaches $90^{\circ}$ (see $i$-link $J_{0i}$ for $\mu_{Cr}=x$ in Fig. 1 (g)).
The I-Cr-I bond angles of NN-links increase and decrease for the in-plane
displacement of Cr atom when the bond length decreases and increases
respectively for the corresponding NN-links upon displacement
($\mu_{Cr}=x,y,xy$). The corresponding NN coupling became AFM when I-Cr-I bond
angle exceeds $90^{\circ}$ for the displacement of Cr atom ($\mu_{Cr}=x,y,xy$,
and, see also Fig. 6 in Sec. V). While the Cr displacement entails the changes
of both the Cr-Cr distance and the bond angles, the movement of ligand atom
along “$z$” direction only affect the latter. In this case, we also find that
the displacement of I atom affects the interaction between the Cr atoms, it is
linked to, and can also change the sign of the NN coupling to AFM for
sufficiently large position shift ($\Delta U=0.25$ Å).

The strength of the exchange interactions reduce to NNN-links and always have
the FM sign with the displacements of Cr atom to any directions (see Fig.
7(a)-(e) in Sec. V). The angles between both the NNN-links and 3rd NN are 120∘
and change according to the displacement directions. The $\bar{J}_{ij}$ for
the NNN $i$-link and $j$-link follow the almost opposite trends with
displacements whereas the it is unaffected for the $k$-link with the
$x$-displacement of Cr atom. This is due to fact that the bond angles for NNN
$i,k$-link and $j,k$-link pairs remain same whereas it changes for the other
NNN-link pairs with displacements.

The strength of $\bar{J_{ij}}$ for 3rd NN-links with regard to NN-links
reduces further. The AFM $\bar{J}_{ij}$ for the 3rd NN-links decreases with
the displacements of Cr atom and one of the link ($k$-link) flips its sign at
$\mu=0.25$ Å($\mu=x$) following the angle rules of 3rd NN-link (see Fig.
7(f)-(j) in Sec. V). The 3rd NN links need much larger displacements with
regard to NN links to change its sign from FM to AFM with $\mu_{Cr}^{x}\geq
0.21$ Å which is 3.1% of the lattice constant for the CrI3 monolayer.

###  III.2 Orbital resolved magnetism

Figure 2: Schematic representation of various processes giving rise to the NN
exchange interaction. The color of the orbital denotes the exchange path it
belongs to. Gray arrows indicate the direct hopping process between two
Cr-$t_{2g}$ orbitals.

In order to get an insight into the physical origin behind the observed
changes in the exchange interactions $\bar{J_{ij}}$, we performed their
orbital-by-orbital decomposition.

The sign of the superexchange interaction depends on the symmetry of electron
orbitals arising from the crystal field the Cr atom experiences. In case of
CrI3, the Cr atoms are surrounded by 6 iodine atoms forming a nearly ideal
octahedron. For simplicity, we assign the Cr-$d$ orbitals to $t_{2g}$ and
$e_{g}$ subsets, which would arise in the ideal case. Although the nominal
occupation of the $3d$-states of Cr3+ ions should be roughly three, resulting
in a half-filled $t_{2g}$ and empty $e_{g}$ shells, the electronic structure
calculations reveal a different situation Besbes  _et al._ (2019); Kashin  _et
al._ (2020). There is a finite occupation of the $e_{g}$ orbitals, which have
lobes pointing directly towards iodine atoms thus forming strong $\sigma$
bonds. As in previous reports Besbes  _et al._ (2019); Kashin  _et al._
(2020); Kvashnin  _et al._ (2020), we find that there are two main competing
contributions to the long-range magnetic ordering in CrI3 which originate from
the $t_{2g}-t_{2g}$ and $t_{2g}-e_{g}$ interacting orbital channels. The total
exchange interaction in CrI3 from the multi-orbital approach can be presented
as a sum of two contributions:
$\bar{J_{ij}}=\bar{J_{ij}}^{t_{2g}-t_{2g}}+\bar{J_{ij}}^{t_{2g}-e_{g}}$. The
coupling between nominally empty $e_{g}$ orbitals is negligibly small. The
${t_{2g}-e_{g}}$ contribution is FM involving the transitions between half-
filled $t_{2g}$ and empty $e_{g}$ orbitals via a single intermediate I-$p$
orbital.

Figure 3: Dzyaloshinskii-Moriya Interactions (DMI) terms for the $i$-link,
$j$-link, $k$-link with (a) $\mu_{Cr}$ = $x$ and (b) $\mu_{I}$ = $z$.

Figure 1(h) shows the orbital decomposition of $\bar{J_{ij}}$ for the NN
$i$-link for the displacement of Cr atom along $\mu=x$. For undistorted CrI3,
the FM coupling that is due to $t_{2g}-e_{g}$ hybridization dominates over the
AFM $t_{2g}-t_{2g}$ term, which leads to the stabilization of the FM ground
state. If we displace the Cr atoms along $\mu=x$, both interacting channels
tend to increase with the magnitude of displacements, but the growing rate is
higher for the $t_{2g}-t_{2g}$ term than for its $t_{2g}-e_{g}$ counterpart.
At the displacement magnitude $|\Delta U_{\mathrm{Cr}}|=0.01$ Å, the AFM
$t_{2g}-t_{2g}$ term starts to dominate, and the sign of the total NN coupling
flips. We note that the orbital decomposed $\bar{J_{0i}}$ does not change much
for the displacement of Cr atom along $\mu=y$.

The out-of-plane displacement of iodine ($\mu_{I}=z$), which brings I atom
closer to the Cr-Cr plane, leads to the drastic enhancement of the AFM
$t_{2g}-t_{2g}$ contribution to the NN coupling, while the FM contribution
grows slower, as shown in Fig. 1(i). This only holds true for the Cr-Cr bond,
which is mediated by the displaced iodine atom. The other two NN couplings
remain unaffected, since their exchange paths are intact.

Fig. 2 contains a schematic picture of the orbital contributions to the NN
exchange coupling. The $t_{2g}$-$t_{2g}$ contribution can be characterized by
two distinct processes, which both lead to AFM interaction, as was
demonstrated in Ref. Dupont  _et al._ (2021) for monolayer of CrCl3. One of
them is the Anderson’s superexchange, which involves the $t_{2g}$ orbitals
having distinct symmetries on the two Cr atoms, mediated by the hybridization
with $I-p$ states. Another one is a direct kinetic exchange between the
$t_{2g}$ orbitals of the same symmetry, whose lobes point towards each other.
The FM and AFM contributions are of comparable size and since the I-Cr-I bond
angle is close to 90∘, the balance between them can easily be altered by
lattice distortions.

Figure 4:  The Dzyaloshinskii-Moriya interaction (DMI) for the NN and NNN
links (a) $\mu_{Cr}=x$ and (f) $\mu_{I}=z$ in which the color corresponds to
the sign of the z component of the DMI vectors. The ratio of the DMI and
Heisenberg interaction for the NN and NNN links as a function of distance
along x-axis ($x_{ij}$) and y-axis ($y_{ij}$) with (b)-(e) $\mu_{Cr}=x$ and
(g)-(j) $\mu_{I}=z$. The $\times$ represents the position of the origin i.e
reference Cr atom (i) and the interactions are taken between i and its
neighbors, j with $r_{ij}=(x_{ij},y_{ij})$. The strength of the ratio is given
by the color of the symbols where $\Delta$ and $\circ$ represent the
undisplaced and displaced case respectively in each figure.

In the case of Cr motion, the bond acquiring the shortest Cr-Cr distance is
characterized by the strongest enhancement of the constituent orbital
interactions. Both FM and AFM contributions are expected to increase,
primarily because of increased Cr-I orbital overlaps, which results in larger
hopping amplitudes, generating the superexchange. At the same time, the
$t_{2g}-t_{2g}$ channel also has a contribution stemming from the kinetic
exchange between the orbitals, pointing directly towards each other (see Fig.
2). This latter contribution depends primarily on the Cr-Cr distance, and is
less subject to the changes in the I-Cr-I bond angle. In this case it is also
expected to grow as a function of displacement amplitude and is likely to play
the dominant role in the enhancement of the $t_{2g}-t_{2g}$ term, similarly to
the case of the uniform strain in monolayered CrCl3Dupont  _et al._ (2021).

Here, similarly to the case of Cr displacement, the increase in the orbital
overlap leads to the growth of AFM $t_{2g}-t_{2g}$ term. However, the case of
$\mu_{I}=z$ is different from $\mu_{\text{{Cr}}}=x$, since the direct kinetic
exchange path is not affected by the changing position of I atom. Instead, the
superexchange processes are altered. Inspection of Fig. 2 suggests that by
making the I-Cr-I angle ($\theta$) smaller, one increases the overlap between
the $t_{2g}$ orbitals involved in AFM superexchange (shown with blue color).
At the same time, the displacement $\mu_{I}=z$ (along ”c” direction) does not
substantially increase the overlap between Cr-$e_{g}$ and I-$p$ states. As a
result, the AFM contribution experiences a much stronger increase and thus
starts to dominate at large displacement amplitudes.

To connect the lattice displacements where we see significant effects from the
spin-lattice coupling with a experimentally measurable quantity, we have
calculated the mean square displacement (MSD) at different temperatures using
VASP in combination with PHONOPY Kresse and Joubert (1999); Kresse and
Furthmüller (1996, 1996); Togo and Tanaka (2015); Lane  _et al._ (2012);
Deringer  _et al._ (2014) (see Fig. 8 in Sec. V). The sign change from FM to
AFM for individual NN couplings occur when $\mu_{Cr}^{xy}\geq 0.12$ Å
$\mu_{Cr}^{x/y}\geq 0.16$ Å, or $\mu_{I}^{z}\geq 0.18$ Å which correspond to
the MSD at 132 K, 175 K, and 197 K respectively. All structures are in the low
temperature rhombohedral phase which is consistent with earlier reports
McGuire  _et al._ (2015).

###  III.3 Effect on Dzyaloshinskii-Moriya interactions

CrI3 monolayer is reported as a topological magnetic insulator. The DMI
vectors ideally lie in the plane of Cr network due to the symmetry, but a
small component of the z-component of the DMI vector ($D^{z}$) is responsible
for the opening of the topological gap at the K point. Therefore, any change
in the magnetic interactions, specially, DMI affect the topology of magnon.
Figure 3 depicts the DMI as a function of distance at different displacements
of Cr atom and I atom (see also Figs. 9 and 10 in Sec. V for more details).
The NN $i$-link, $j$-link, $k$-link DMI are forbidden by symmetry of the
systems. Only the NNN DMI’s are finite obeying the $C_{3}$ symmetry of CrI3
monolayer and matches well in the reported values Kvashnin  _et al._ (2020).
The $C_{3}$ symmetry is broken with displacements and DMI terms for NN, NNN
and 3rd NN $i$-link, $j$-link, $k$-link are contributing. The NN DMI terms
increase with displacements and become dominating when ground state changes
FM-AFM ordering in $\bar{J_{1}}$. The dominating DMI increases almost $\sim 6$
times with regard to the undisplaced case at $\Delta{U}=0.25$ Å. The effect on
DMI observed for the displacements of ligand atoms in coupled SLD is almost
double compared to magnetic atoms.

To study the effect on DMI with the displacement of atoms in coupled SLD which
in turn affect the topology of the system, we calculate the ratio of the DMI
and Heisenberg interaction ($D_{ij}^{z}/\bar{J_{ij}}$) for the NN and NNN
links for both in-plane motion of Cr-atom and out-plane motion of I-atom as
shown in Fig. 4. The strength of the ratio determine the domination of the DMI
over the Heisenberg interaction which is the controlling parameter for
magnonic topological transport. Figures 4(a) and 4(f) depict the directions of
the z-component of DMI vectors within a Cr sublattice in coupled SLD where
both the NN and NNN links are contributing. The direction of the DMI vectors
for the NN and NNN links in two adjacent Cr sublattices are opposite with
regard to the other sublattice. The DMI ratio $D_{ij}^{z}/\bar{J_{ij}}$
increases with increasing the displacements of Cr atom and reach a high value
of 1.6 for the NN $i$-links with $\Delta{U}=0.16$ Å. It is because of the
$\bar{J_{0i}}$ switches from FM to AFM. At $\Delta{U}=0.25$ Å,
$D_{ij}^{z}/\bar{J_{ij}}$ decreases (0.1) because the high value of both the
DMI and Heisenberg interaction whereas the NN $k$-links reach a higher value
of 0.6 determined by the coupled effect of both the interactions.

The sensitivity of the DMI ratio is larger for the displacements of ligands
than magnetic atoms. The DMI ratio $D_{ij}^{z}/\bar{J_{ij}}$ got a high value
of 1.5 for the NN i-link at $\Delta{U}=0.25$ Å. The microscopic analysis
presented here shows that the DMI for some pairs is high in magnitude compared
to the Heisenberg exchange in coupled SLD.

Figure 5: (a) Variation of the isotropic exchange interaction ($\bar{J_{ij}}$)
with the displacements of Cr-atom in both in-plane and out-plane directions
for NN. (b) Absolute value of the isotropic spin lattice coupling constants
for the $i$-link, $j$-link, and $k$-link as a function of the distance with
displacements.

###  III.4 Spin-lattice coupling:

The variation of the isotropic exchange interaction ($\Delta{\bar{J_{ij}}}$)
for NN $i$-link, $j$-link, $k$-link with the displacements of Cr-atom in both
in-plane and out-plane directions are shown in Fig. 5(a). The system is
sensitive only to the in-plane motion of the Cr-atom and out-plane motion of
halide atom. The response in the $\Delta{\bar{J_{ij}}}$ for the $i$-link,
$j$-link, $k$-link depends on the change in the I-Cr-I bond angle of the
corresponding links. The response of $\Delta{\bar{J_{ij}}}\sim\Delta{U}$
reduces to ($\sim\frac{1}{10}$) for the NNN links (see Fig. 11(a) in Sec.V).
The sensitivity of CrI3 monolayer in the linear region of coupled SLD i.e.
$\Delta{\bar{J_{ij}}}\sim\Delta{U}$ is up to $|\mu|$ = 0.02 Å and beyond that
limit, it falls into the non-linear region. The dotted (black) line in Fig.
5(a) indicates the ideal variation of $\Delta{\bar{J_{ij}}}\sim\Delta{U}$
which is $<{3\%}$ offset limit of the actual response. The absolute values of
the isotropic spin lattice coupling (SLC) constants for the $i$-link,
$j$-link, $k$-link as a function of the distance with displacements are
depicted in Fig. 5(b). The SLC constants remain almost constants with
displacements within the linear regime of coupled SLD (though it deflects
slightly for smaller displacement $|\mu|$ = 0.005 Å), and suddenly increase
when they fall into the non-linear regime of coupled SLD.

The strength of the spin-lattice coupling in CrI3 can be contrasted to the
strength of the exchange striction in bcc Fe Hellsvik  _et al._ (2019). For
bcc Fe, the ratio of the exchange striction coupling and the Heisenberg
exchange is $|\Gamma_{1}|/|J_{1}|=0.641$ Å-1 and $|\Gamma_{2}|/|J_{2}|=0.481$
Å-1 for the nearest and next nearest neighbor bonds respectively. For CrI3,
the corresponding ratios are $|\Gamma_{1}|/|J_{1}|=6.97$ Å-1 and
$|\Gamma_{2}|/|J_{2}|=2.97$ Å-1, i.e. for the nearest neighbor (next nearest
neighbor) bonds the relative strength of the spin-lattice coupling is a factor
$\sim 10$ ($\sim 6$) stronger in CrI3 than in bcc Fe.

##  IV Conclusion

From fully-relativistic calculations of the magnetic exchange interactions,
including Heisenberg and DMI, when considering finite displacements, we have
found that the spin-lattice coupling is significant in CrI3. In particular it
has been found that dominating exchange interactions can change sign from FM
to AFM coupling when the atomic distance between neighbouring atoms increases.
A microscopic explanation for the strong spin-lattice coupling based on
orbital decomposition has been presented where the angle formed by I-Cr-I, and
the Cr-Cr bond distance affect the magnetic interaction significantly. To this
end, we argue that it is not enough to consider only the isotropic exchange
and anisotropic DMI to study the effect of lattice displacements during
thermal excitations. It needs a orbital resolved analysis of the exchange
interactions to have a complete picture from microscopic origin. For
comparison with three-dimensional ferromagnets we extract an effective measure
of the SLC constants which is ten times larger for CrI3 than for bcc Fe. The
strong spin-lattice coupling in CrI3 and related two-dimensional magnets are
expected to play a significant role for the existence of topological magnons
in these systems and we suggest that coupled spin-lattice dynamics is a
suitable tool for investigating this further.

###### Acknowledgements.

A.D. acknowledges financial support from Vetenskapsrådet (grant numbers VR
2015-04608, VR 2016-05980 and VR 2019-05304), and the Knut and Alice
Wallenberg foundation (grant number 2018.0060). Y.O.K. acknowledges the
financial support from the Swedish Research Council (VR) under the project No.
2019-03569. J.H. acknowledges financial support from the Swedish Research
Council (VR) (neutron project grant BIFROST, Dnr. 2016-06955). A.B.
acknowledges eSSENCE. The computations were enabled by resources provided by
the Swedish National Infrastructure for Computing (SNIC) at PDC and NSC,
partially funded by the Swedish Research Council through grant agreement no.
2018-05973.

## References

  * Huang  _et al._ (2017) B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Nature 546, 270 (2017).
  * Wang  _et al._ (2018) Z. Wang, I. Gutiérrez-Lezama, N. Ubrig, M. Kroner, M. Gibertini, T. Taniguchi, K. Watanabe, A. Imamoğlu, E. Giannini, and A. F. Morpurgo, Nature Communications 9, 2516 (2018).
  * Song  _et al._ (2018) T. Song, X. Cai, M. W.-Y. Tu, X. Zhang, B. Huang, N. P. Wilson, K. L. Seyler, L. Zhu, T. Taniguchi, K. Watanabe, _et al._ , Science 360, 1214 (2018).
  * Xing  _et al._ (2017) W. Xing, Y. Chen, P. M. Odenthal, X. Zhang, W. Yuan, T. Su, Q. Song, T. Wang, J. Zhong, S. Jia, _et al._ , 2D Materials 4, 024009 (2017).
  * Huang  _et al._ (2018) B. Huang, G. Clark, D. R. Klein, D. MacNeill, E. Navarro-Moratalla, K. L. Seyler, N. Wilson, M. A. McGuire, D. H. Cobden, D. Xiao, _et al._ , Nature nanotechnology 13, 544 (2018).
  * Jimenez  _et al._ (2020) V. O. Jimenez, V. Kalappattil, T. Eggers, M. Bonilla, S. Kolekar, P. T. Huy, M. Batzill, and M.-H. Phan, Scientific Reports 10, 4789 (2020).
  * Smogunov  _et al._ (2008) A. Smogunov, A. Dal Corso, A. Delin, R. Weht, and E. Tosatti, Nature Nanotechnology 3, 22 (2008).
  * Zhang  _et al._ (2013) L. Zhang, J. Ren, J.-S. Wang, and B. Li, Phys. Rev. B 87, 144101 (2013).
  * Chen  _et al._ (2018) L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone, A. I. Kolesnikov, Q. Huang, and P. Dai, Phys. Rev. X 8, 041028 (2018).
  * Kvashnin  _et al._ (2020) Y. O. Kvashnin, A. Bergman, A. I. Lichtenstein, and M. I. Katsnelson, Phys. Rev. B 102, 115162 (2020).
  * Webster  _et al._ (2018) L. Webster, L. Liang, and J.-A. Yan, Physical Chemistry Chemical Physics 20, 23546 (2018).
  * Thingstad  _et al._ (2019) E. Thingstad, A. Kamra, A. Brataas, and A. Sudbø, Phys. Rev. Lett. 122, 107201 (2019).
  * Hellsvik  _et al._ (2019) J. Hellsvik, D. Thonig, K. Modin, D. Iuşan, A. Bergman, O. Eriksson, L. Bergqvist, and A. Delin, Phys. Rev. B 99, 104302 (2019).
  * (14) “Relativistic Spin Polarized toolkit (RSPt).” http://fplmto-rspt.org/.
  * Wills  _et al._ (2010) J. M. Wills, O. Eriksson, P. Andersson, A. Delin, O. Grechnyev, and M. Alouani,  _Full-Potential Electronic Structure Method_, Springer Series in Solid-State Sciences, Vol. 167 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2010).
  * Antropov  _et al._ (1997) V. Antropov, M. Katsnelson, and A. Liechtenstein, Physica B: Condensed Matter 237, 336 (1997).
  * Udvardi  _et al._ (2003) L. Udvardi, L. Szunyogh, K. Palotás, and P. Weinberger, Phys. Rev. B 68, 104436 (2003).
  * Ebert and Mankovsky (2009) H. Ebert and S. Mankovsky, Phys. Rev. B 79, 045209 (2009).
  * Secchi  _et al._ (2015) A. Secchi, A. I. Lichtenstein, and M. I. Katsnelson, Annals of Physics 360, 61 (2015).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Computational materials science 6, 15 (1996).
  * Perdew  _et al._ (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * (24) R. relativistic spin polarized toolkit, “http://fplmto-rspt.org/,” .
  * energy  _et al._ (2010) F.-P. E. S. M. energy, force calculations with density functional, J. M. dynamical mean field theory, Wills, M. Alouani, P. Andersson, A. Delin, O. Eriksson, and O. Grechnyev, Vol. 167 (Springer Science & Business Media, 2010).
  * Besbes  _et al._ (2019) O. Besbes, S. Nikolaev, N. Meskini, and I. Solovyev, Phys. Rev. B 99, 104432 (2019).
  * Kashin  _et al._ (2020) I. V. Kashin, V. V. Mazurenko, M. I. Katsnelson, and A. N. Rudenko, 2D Materials 7, 025036 (2020).
  * Dupont  _et al._ (2021) M. Dupont, Y. O. Kvashnin, M. Shiranzaei, J. Fransson, N. Laflorencie, and A. Kantian, Phys. Rev. Lett. 127, 037204 (2021).
  * Togo and Tanaka (2015) A. Togo and I. Tanaka, Scripta Materialia 108, 1 (2015).
  * Lane  _et al._ (2012) N. J. Lane, S. C. Vogel, G. Hug, A. Togo, L. Chaput, L. Hultman, and M. W. Barsoum, Phys. Rev. B 86, 214301 (2012).
  * Deringer  _et al._ (2014) V. L. Deringer, R. P. Stoffel, A. Togo, B. Eck, M. Meven, and R. Dronskowski, CrystEngComm 16, 10907 (2014).
  * McGuire  _et al._ (2015) M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Chemistry of Materials 27, 612 (2015).

##  V Appendix

Figure 6(a) shows the geometry of the undisplaced 2$\times$2$\times$1
supercell, and Fig. 6(b)-(g) show the change in isotropic magnetic exchange
interactions, bond distances and bond angles with the displacement along the
other two in-plane motion of Cr atom $\mu=y,xy$ directions respectively. The
strength of the Heisenberg exchange interactions and bond angles with the in-
plane displacement of Cr atom ($\mu_{Cr}=x,y,xy$) for NNN and 3rd NN are shown
in Fig. 7. The mean square displacement of atoms as a function of temperature
is shown in Fig. 8.

Figure 9 shows the change in the Dzyaloshinskii-Moriya interaction (DMI) for
$i,j,j$-links of NN, NNN and 3rd NN with displacement of Cr atom along
$\mu_{Cr}=y,xy$. To analyse, how the components of DMI ($D_{x},D_{y},D_{z}$)
behave with the displacements, we calculated the $D_{x},D_{y},D_{z}$ for NN
and NNN $i,j,j$-links with $\mu_{Cr}=x$ as shown in Fig. 10(a)-(f). Figure 11
(a)-(b) show the linear regime of coupled spin-lattice dynamics for NNN and
corresponding isotropic spin-lattice coupling constants with displacements
respectively.

Figure 6: (a) 2$\times$2$\times$1 supercell in the $ab$-plane of CrI3. The
shaded region in the supercell corresponds to the unit cell. The change in
I-Cr-I bond angles of NN $i$-link, $j$-link, $k$-link with displacements along
(b) $\mu_{Cr}=y$ and (c) $\mu_{Cr}=xy$ directions respectively. Here the
displacement magnitude is chosen as $\Delta{U}=0.25$ Å. The green circle
indicates the Cr atom being displaced. Calculated isotropic exchange
interaction (${\bar{J}_{ij}}$) with (d) $\mu_{Cr}=y$ ; (e) $\mu_{Cr}=xy$. The
change in (f) Cr-Cr bond distance and (g) I-Cr-I bond angles for the NN
$i$-link, $j$-link, $k$-link with $\mu_{Cr}=xy$ ( $*$ corresponds to
$\mu_{Cr}=y$).  Figure 7:  The exchange interactions and bond angle for
(a)-(e) NNN and (f)-(j) 3rd NN $i$-link, $j$-link, $k$-link without and with
displacement of Cr-atom along $\mu_{Cr}=x,y,xy$ directions respectively . The
displacement magnitude to denote the bond angle in the NNN and 3rd NN links is
chosen as $\Delta{U}$ = 0.25 Å. The green circle indicates the displaced Cr-
atom.  Figure 8: Calculated mean square displacements of CrI3 monolayer with
the temperature. Figure 9: Dzyaloshinskii-Moriya interactions (DMI) for the
$i$-link, $j$-link, $k$-link with (a) $\mu_{Cr}$ = $y$ and (b) $\mu_{Cr}$ =
$xy$. Figure 10: The components of the DMI for (a)-(c) NN links and (d)-(f)
NNN links with $\mu_{Cr}$ = $x$.  Figure 11:  (a) Variation of the isotropic
exchange interaction ($\bar{J_{ij}}$) with the displacements of Cr-atom for
the NNN. (b) Isotropic spin lattice coupling constants for the $i$-link,
$j$-link, $k$-link as a function of the distance with displacements.