# Poisson structures compatible with the cluster algebra structure in
Grassmannians

M. Gekhtman  Department of Mathematics, University of Notre Dame, Notre Dame,
IN 46556<EMAIL_ADDRESS>,  M. Shapiro  Department of Mathematics, Michigan
State University, East Lansing, MI 48823<EMAIL_ADDRESS>,  A. Stolin
Department of Mathematics University of Goteborg 411 24 Goteborg Sweden
<EMAIL_ADDRESS>and  A. Vainshtein  Department of Mathematics &
Department of Computer Science, University of Haifa, Haifa, Mount Carmel
31905, Israel<EMAIL_ADDRESS>
###### Abstract.

We describe all Poisson brackets compatible with the natural cluster algebra
structure in the open Schubert cell of the Grassmannian $G_{k}(n)$ and show
that any such bracket endows $G_{k}(n)$ with a structure of a Poisson
homogeneous space with respect to the natural action of $SL_{n}$ equipped with
an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the
simplest class in the Belavin-Drinfeld classification. Moreover, every
compatible Poisson structure can be obtained this way.

###### Key words and phrases:

Grassmannian, Poisson-Lie group, cluster algebra

###### 1991 Mathematics Subject Classification:

53D17, 14M15

##  1\. Introduction

The notion of a Poisson bracket compatible with a cluster algebra structure
was introduced in [GSV1]. It was used to interpret cluster transformations and
matrix mutations in cluster algebras from a viewpoint of Poisson geometry. In
addition, it was shown in [GSV1] that if a Poisson variety
$\left(\mathcal{M},{\\{\cdot,\cdot\\}}\right)$ possesses a coordinate chart
that consists of regular functions whose logarithms have pairwise constant
Poisson brackets, then one can use this chart to define a cluster algebra
${\mathcal{A}}_{\mathcal{M}}$, which is closely related (and, under rather
mild conditions, isomorphic) to the ring of regular functions on
$\mathcal{M}$, and such that ${\\{\cdot,\cdot\\}}$ is compatible with
${\mathcal{A}}_{\mathcal{M}}$. This construction was applied to an open cell
$G^{0}_{k}(n)$ in the Grassmannian $G_{k}(n)$ viewed as a Poisson homogeneous
space under the action of $SL_{n}$ equipped with the standard Poisson-Lie
structure. The resulting cluster algebra ${\mathcal{A}}_{G^{0}_{k}(n)}$ can be
viewed as a restriction of the cluster algebra structure in the coordinate
ring of $G_{k}(n)$. This “larger” cluster algebra ${\mathcal{A}}_{G_{k}(n)}$
was described in [S] using combinatorial properties of Postnikov’s map from
the space of edge weights of a planar directed network into the Grassmannian
[P]. Poisson geometric properties of Postnikov’s map are studied in [GSV2].
One of the by-products of this study is the existence of a pencil of Poisson
structures on $G_{k}(n)$ compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. It
turnes out that every bracket in the pencil defines a Poisson homogeneous
structure with respect to a Sklyanin Poisson-Lie bracket associated with a
solution of the modified classical Yang-Baxter equation (MCYBE) of the form
$R_{t}=R_{0}+tA\pi_{0}$, where $t$ is a scalar parameter, $A$ is a certain
fixed skew-symmetric $n\times n$ matrix, $R_{0}=\pi_{+}-\pi_{-}$, and
$\pi_{\pm,0}$ are projections onto strictly upper/strictly lower/diagonal part
in $sl_{n}$ (the standard Poisson-Lie structure corresponds to $t=0$).

According to the Belavin-Drinfeld classification [BD], skew-symmetric
solutions of MCYBE are defined by two types of data: discrete data associated
with the Dynkin diagram and called the Belavin-Drinfeld triple and continuous
data associated with the Cartan subalgebra. We will say that two R-matrices
belong to the same class if the corresponding Belavin-Drinfeld triples are the
same. R-matrices $R_{t}$ mentioned above belong to the class associated with
the trivial Belavin-Drinfeld triple. The entire class consists of R-matrices
of the form $R_{S}=R_{0}+S\pi_{0}$, with $S$ arbitrary skew-symmetric. On the
other hand, the Poisson pencil described above does not exhaust all Poisson
structures compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. The main goal of
this paper is to prove

######  Theorem 1.1.

The Poisson homogeneous structure with respect to the Poisson-Lie bracket
associated with any $R_{S}$ is compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$.
Moreover, up to a scalar multiple, all Poisson brackets compatible with
${\mathcal{A}}_{G^{0}_{k}(n)}$ are obtained this way.

It should be noted that Poisson brackets compatible with the “larger” cluster
algebra ${\mathcal{A}}_{G_{k}(n)}$ are naturally associated with Poisson
structures on the Grassmann cone $\mathcal{C}(G_{k}(n))$ that can be realized
as one-dimensional extensions of corresponding structures on $G_{k}(n)$. Both
the formulation and the proof of Theorem 1.1 can be modified in a rather
straightforward way to the case of the cluster algebra
${\mathcal{A}}_{G_{k}(n)}$. A detailed description of the relationship between
${\mathcal{A}}_{G_{k}(n)}$ and ${\mathcal{A}}_{G^{0}_{k}(n)}$ this
modification relies upon is presented in Chapter 4 of the forthcoming book
[GSV3].

The paper is organized as follows. In Section 2 we recall the necessary
information on cluster algebras and compatible Poisson structures and show how
the latter can be completely described via the use of a toric action. Section
3 provides the background on Poisson-Lie groups and Sklyanin brackets.
Finally, in Section 4 we review Poisson homogeneous structures on $G_{k}(n)$
and the construction of ${\mathcal{A}}_{G^{0}_{k}(n)}$, and then proceed to
prove Theorem 1.1, which is re-stated there as Theorem 4.3.

##  2\. Cluster algebras and compatible Poisson brackets

We start with the basics on cluster algebras of geometric type. The definition
that we present below is not the most general one, see, e.g., [FZ3, BFZ] for a
detailed exposition. In what follows, we will use notation $[i,j]$ for an
interval $\\{i,i+1,\ldots,j\\}$ in $\mathbb{N}$, and write $[n]$ instead of
$[1,n]$.

The coefficient group ${\mathfrak{P}}$ is a free multiplicative abelian group
of a finite rank $m$ with generators $g_{1},\dots,g_{m}$. An ambient field is
the field ${\mathfrak{F}}$ of rational functions in $n$ independent variables
with coefficients in the field of fractions of the integer group ring
${\mathbb{Z}}{\mathfrak{P}}={\mathbb{Z}}[g_{1}^{\pm 1},\dots,g_{m}^{\pm 1}]$
(here we write $x^{\pm 1}$ instead of $x,x^{-1}$).

A seed (of geometric type) in ${\mathfrak{F}}$ is a pair $\Sigma=({\bf
x},\widetilde{B})$, where ${\bf x}=(x_{1},\dots,x_{n})$ is a transcendence
basis of ${\mathfrak{F}}$ over the field of fractions of
${\mathbb{Z}}{\mathfrak{P}}$ and $\widetilde{B}$ is an $n\times(n+m)$ integer
matrix whose principal part $B$ (that is, the $n\times n$ submatrix formed by
the columns $1,\dots,n$) is skew-symmetrizable. In this paper, we will only
deal with the case when $B$ is skew-symmetric.

The $n$-tuple ${\bf x}$ is called a cluster, and its elements
$x_{1},\dots,x_{n}$ are called cluster variables. Denote $x_{n+i}=g_{i}$ for
$i\in[m]$. We say that $\widetilde{{\bf x}}=(x_{1},\dots,x_{n+m})$ is an
extended cluster, and $x_{n+1},\dots,x_{n+m}$ are stable variables. It is
convenient to think of ${\mathfrak{F}}$ as of the field of rational functions
in $n+m$ independent variables with rational coefficients.

Given a seed as above, the adjacent cluster in direction $k\in[n]$ is defined
by

${\bf x}_{k}=({\bf x}\setminus\\{x_{k}\\})\cup\\{x^{\prime}_{k}\\},$

where the new cluster variable $x^{\prime}_{k}$ is given by the exchange
relation

$x_{k}x^{\prime}_{k}=\prod_{\begin{subarray}{c}1\leq i\leq n+m\\\
b_{ki}>0\end{subarray}}x_{i}^{b_{ki}}+\prod_{\begin{subarray}{c}1\leq i\leq
n+m\\\ b_{ki}<0\end{subarray}}x_{i}^{-b_{ki}};$

here, as usual, the product over the empty set is assumed to be equal to $1$.

We say that ${\widetilde{B}}^{\prime}$ is obtained from ${\widetilde{B}}$ by a
matrix mutation in direction $k$ and write
${\widetilde{B}}^{\prime}=\mu_{k}({\widetilde{B}})$ if

$b^{\prime}_{ij}=\begin{cases}-b_{ij},&\text{if $i=k$ or $j=k$;}\\\
b_{ij}+\displaystyle\frac{|b_{ik}|b_{kj}+b_{ik}|b_{kj}|}{2},&\text{otherwise.}\end{cases}$

Given a seed $\Sigma=({\bf x},\widetilde{B})$, we say that a seed
$\Sigma^{\prime}=({\bf x}^{\prime},\widetilde{B}^{\prime})$ is adjacent to
$\Sigma$ (in direction $k$) if ${\bf x}^{\prime}$ is adjacent to ${\bf x}$ in
direction $k$ and $\widetilde{B}^{\prime}=\mu_{k}(\widetilde{B})$. Two seeds
are mutation equivalent if they can be connected by a sequence of pairwise
adjacent seeds.

The cluster algebra (of geometric type)
${\mathcal{A}}={\mathcal{A}}(\widetilde{B})$ associated with $\Sigma$ is the
${\mathbb{Z}}{\mathfrak{P}}$-subalgebra of ${\mathfrak{F}}$ generated by all
cluster variables in all seeds mutation equivalent to $\Sigma$.

Let ${\\{\cdot,\cdot\\}}$ be a Poisson bracket on the ambient field
${\mathfrak{F}}$ considered as the field of rational functions in $n+m$
independent variables with rational coefficients. We say that it is compatible
with the cluster algebra ${\mathcal{A}}$ if, for any extended cluster
$\widetilde{{\bf x}}=(x_{1},\dots,x_{n+m})$, one has

$\\{x_{i},x_{j}\\}=\omega_{ij}x_{i}x_{j},$

where $\omega_{ij}\in{\mathbb{Z}}$ are constants for all $i,j\in[n+m]$. The
matrix $\Omega^{\widetilde{\bf x}}=(\omega_{ij})$ is called the coefficient
matrix of ${\\{\cdot,\cdot\\}}$ (in the basis $\widetilde{\bf x}$); clearly,
$\Omega^{\widetilde{\bf x}}$ is skew-symmetric.

In what follows, we denote by $A(I,J)$ the submatrix of a matrix $A$ with a
row set $I$ and a column set $J$. Consider, along with cluster and stable
variables ${\widetilde{\bf x}}$, another $(n+m)$-tuple of rational functions
denoted $\tau=(\tau_{1},\dots,\tau_{n+m})$ and defined by

(2.1) $\tau_{j}=x_{j}^{\varkappa_{j}}\prod_{k=1}^{n+m}x_{k}^{b_{jk}},$

where $\widehat{B}=(b_{jk})_{j,k=1}^{n+m}$ is a fixed skew-symmetric matrix
such that $\widehat{B}([n],[n+m])=\widetilde{B}$, $\varkappa_{j}$ is an
integer, $\varkappa_{j}=0$ for $1\leq j\leq n$. We say that the entries
$\tau_{i}$, $i\in[n+m]$, form a _$\tau$ -cluster_. It is proved in [GSV1],
Lemma 1.1, that $\varkappa_{j}$, $n+1\leq j\leq n+m$, can be selected in such
a way that the transformation ${\widetilde{\bf x}}\mapsto\tau$ is non-
degenerate, provided $\operatorname{rank}{\widetilde{B}}=n$. We denote
$\varkappa=\operatorname{diag}(\varkappa_{i})_{i=1}^{n+m}$ and
$B_{\varkappa}=\widehat{B}+\varkappa$. Nondegeneracy of the transformation
${\widetilde{\bf x}}\mapsto\tau$ is equivalent to nondegeneracy of
$B_{\varkappa}$.

Recall that a square matrix $A$ is reducible if there exists a permutation
matrix $P$ such that $PAP^{T}$ is a block-diagonal matrix, and irreducible
otherwise. The following result is a particular case of Theorem 1.4 in [GSV1].

######  Theorem 2.1.

Assume that $\operatorname{rank}{\widetilde{B}}=n$ and the principal part of
${\widetilde{B}}$ is irreducible. Then a Poisson bracket is compatible with
${\mathcal{A}}({\widetilde{B}})$ if and only if its coefficient matrix
$\Omega^{\tau}$ in the basis $\tau$ has the following property: the submatrix
$\Omega^{\tau}([n],[n+m])$ is proportional to ${\widetilde{B}}$.

Starting with an arbitrary ${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ compatible
with ${\mathcal{A}}$, one can suggest an alternative description of all other
compatible Poisson brackets via the following construction. Let $C=(c_{ij})$
be an integral $(n+m)\times m$ matrix. Define an action of
$({\mathbb{C}}^{*})^{m}=\\{\mathbf{d}=(d_{1},\ldots,d_{m})\ :\ d_{1}\cdots
d_{r}\neq 0\\}$ on ${\widetilde{\bf x}}$ by

(2.2) $\mathbf{d}.{\widetilde{\bf
x}}=\left(x_{i}\prod_{\alpha=1}^{m}d_{\alpha}^{c_{i\alpha}}\right)_{i=1}^{n+m}.$

We say that (2.2) extends to an action of $({\mathbb{C}}^{*})^{m}$ on
${\mathcal{A}}$ if the action induced by it in any cluster in ${\mathcal{A}}$
has a form (2.2) (with possibly different coefficients $c_{i\alpha}$). Lemma
2.3 in [GSV1] claims that (2.2) extends to an action of
$({\mathbb{C}}^{*})^{m}$ on ${\mathcal{A}}$ if and only if
${\widetilde{B}}C=0$. The same condition guarantees that
$\tau_{i}(\mathbf{d}.{\widetilde{\bf x}})=\tau_{i}({\widetilde{\bf x}})$ for
$i\in[n]$. Since $B_{\varkappa}$ is invertible, any such $C$ of full rank has
a form $C=B_{\varkappa}^{-1}([n+m],[n+1,n+m])U$, where $U$ is any invertible
$m\times m$ matrix.

Next, assume that $({\mathbb{C}}^{*})^{m}$ is equipped with a Poisson
structure given by

$\\{d_{i},d_{j}\\}_{V}=v_{ij}d_{i}d_{j},$

where $V=(v_{ij})$ is a fixed skew-symmetric matrix.

######  Proposition 2.2.

Let ${\widetilde{B}}$ satisfy the assumptions of Theorem 2.1. Then for any
skew-symmetric $m\times m$ matrix $V$, there exists a Poisson structure
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ compatible with ${\mathcal{A}}$ such
that the map
$\left(({\mathbb{C}}^{*})^{m}\times{\mathcal{A}},{\\{\cdot,\cdot\\}}_{V}\times{\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}\right)\to\left({\mathcal{A}},{\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}\right)$
extended from the action $(\mathbf{d},{\widetilde{\bf
x}})\mapsto\mathbf{d}.{\widetilde{\bf x}}$ is Poisson. Moreover, every
compatible Poisson bracket on ${\mathcal{A}}$ is a scalar multiple of
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ for some $V$.

###### Proof.

Let $\Omega^{{\widetilde{\bf x}}}$ be the coefficient matrix of
${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ in the basis ${\widetilde{\bf x}}$. It
is easy to see that in the product structure
${\\{\cdot,\cdot\\}}_{V}\times{\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ on
$({\mathbb{C}}^{*})^{m}\times{\mathcal{A}}$,

$\\{(\mathbf{d}.{\widetilde{\bf x}})_{i},(\mathbf{d}.{\widetilde{\bf
x}})_{j}\\}=\left(\Omega^{{\widetilde{\bf
x}}}+CVC^{T}\right)_{ij}(\mathbf{d}.{\widetilde{\bf
x}})_{i}(\mathbf{d}.{\widetilde{\bf x}})_{j}.$

Thus, for the action $(\mathbf{d},{\widetilde{\bf
x}})\mapsto\mathbf{d}.{\widetilde{\bf x}}$ to be Poisson, one must have
$\\{x_{i},x_{j}\\}^{\mathcal{A}}_{V}=\left(\Omega^{\tilde{\bf
x}}+CVC^{T}\right)_{ij}x_{i}x_{j}$ for $i,j\in[n+m]$. Since
$\tau_{i}(\mathbf{d}.{\widetilde{\bf x}})=\tau_{i}({\widetilde{\bf x}})$ if
$i\in[n]$, and $\tau_{i}(\mathbf{d}.{\widetilde{\bf
x}})=\tau_{i}({\widetilde{\bf x}})m_{i}(\mathbf{d})$ for some monomials
$m_{i}(\mathbf{d})$ in $\mathbf{d}$ for $i\in[n+1,n+m]$, we see that
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}=\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{0}$
for $i\in[n]$, $j\in[n+m]$. Since ${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ is a
compatible Poisson bracket, Theorem 2.1 yields that
${\\{\cdot,\cdot\\}}_{V}^{\mathcal{A}}$ is compatible as well.

Now, let $\Omega^{\tau}$ be the coefficient matrix of
${\\{\cdot,\cdot\\}}_{0}^{\mathcal{A}}$ in the basis $\tau$. Denote
$Z_{0}=\Omega^{\tau}([n+1,n+m],[n+1,n+m])$. Consider
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}$ for $i,j\in[n+1,n+m]$:
$\\{\tau_{i},\tau_{j}\\}^{\mathcal{A}}_{V}=z_{ij}\tau_{i}\tau_{j}$. To compute
$Z=(z_{ij})_{i,j=n+1}^{n+m}$, note that the matrix that describes
${\\{\cdot,\cdot\\}}^{\mathcal{A}}_{V}$ in coordinates $\tau$ is
$\Omega_{V}^{\tau}=B_{\varkappa}\left(\Omega^{{\widetilde{\bf
x}}}+CVC^{T}\right)B_{\varkappa}^{T}$, and thus

$Z=\Omega_{V}^{\tau}([n+1,n+m],[n+1,n+m])=Z_{0}+UVU^{-1}.$

It is clear that by varying $V$, one can make $Z$ to be equal to an arbitrary
skew-symmmetric $m\times m$ matrix. Theorem 2.1 implies that up to a scalar
multiple, the matrix block $Z$ determines a compatible Poisson structure
uniquely, and the result follows. ∎

##  3\. Poisson-Lie groups and Sklyanin brackets

We need to recall some facts about Poisson-Lie groups (see, e.g.[ReST]).

Let ${\mathcal{G}}$ be a Lie group equipped with a Poisson bracket
${\\{\cdot,\cdot\\}}$. ${\mathcal{G}}$ is called a Poisson-Lie group if the
multiplication map

${\mathfrak{m}}:{\mathcal{G}}\times{\mathcal{G}}\ni(x,y)\mapsto
xy\in{\mathcal{G}}$

is Poisson. Perhaps, the most important class of Poisson-Lie groups is the one
associated with classical R-matrices.

Let $\mathfrak{g}$ be a Lie algebra of ${\mathcal{G}}$. Assume that
$\mathfrak{g}$ is equipped with a nondegenerate invariant bilinear form $(\ ,\
)$. An element $R\in\operatorname{End}(\mathfrak{g})$ is a classical R-matrix
if it is a skew-symmetric operator that satisfies the modified classical Yang-
Baxter equation (MCYBE)

(3.1) $[R(\xi),R(\eta)]-R\left([R(\xi),\eta]\ +\
[\xi,R(\eta)]\right)=-[\xi,\eta].$

Given a classical R-matrix $R$, ${\mathcal{G}}$ can be endowed with a Poisson-
Lie structure as follows. Let $\nabla f$, $\nabla^{\prime}f$ be the right and
the left gradients for a function $f\in C^{\infty}({\mathcal{G}})$:

(3.2) $(\nabla
f(x),\xi)=\frac{d}{dt}f(\exp{(t\xi)}x)|_{t=0},\qquad(\nabla^{\prime}f(x),\xi)=\frac{d}{dt}f(x\exp{(t\xi)})|_{t=0}.$

Then the bracket given by

(3.3)
$\\{f_{1},f_{2}\\}=\frac{1}{2}(R(\nabla^{\prime}f_{1}),\nabla^{\prime}f_{2})-\frac{1}{2}(R(\nabla
f_{1}),\nabla f_{2})\ $

is a Poisson-Lie bracket on ${\mathcal{G}}$ called the Sklyanin bracket.

We are interested in the case ${\mathcal{G}}=SL_{n}$ and $\mathfrak{g}=sl_{n}$
equipped with the trace-form

$(\xi,\eta)=\operatorname{Tr}(\xi\eta).$

In this case, the right and left gradients (3.2) are

$\nabla f(x)=x\ \mbox{grad}f(x),\qquad\nabla^{\prime}f(x)=\mbox{grad}f(x)\ x,$

where

$\mbox{grad}f(x)=\left(\frac{\partial f}{\partial x_{ji}}\right)_{i,j=1}^{n},$

and the Sklyanin bracket becomes

(3.4) $\\{f_{1},f_{2}\\}_{R}(x)=\\\ \frac{1}{2}(R(\mbox{grad}f_{1}(x)\
x),\mbox{grad}f_{2}(x)\ x)-\frac{1}{2}(R(x\ \mbox{grad}f_{1}(x)),x\
\mbox{grad}f_{2}(x)).$

Every $\xi\in\mathfrak{g}$ can be uniquely decomposed as

$\xi=\pi_{-}(\xi)+\pi_{0}(\xi)+\pi_{+}(\xi),$

where $\pi_{+}(\xi)$ and $\pi_{-}(\xi)$ are strictly upper and lower
triangular and $\pi_{0}(\xi)$ is diagonal. The simplest classical R-matrix on
$sl_{n}$ is given by

(3.5)
$R_{0}(\xi)=\pi_{+}(\xi)-\pi_{-}(\xi)=\left(\operatorname{sign}(j-i)\xi_{ij}\right)_{i,j=1}^{n}.$

Substituting $R=R_{0}$ into (3.4), we find the bracket for a pair of matrix
entries:

(3.6)
$\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{0}}=\frac{1}{2}\left(\operatorname{sign}(i^{\prime}-i)+\operatorname{sign}(j^{\prime}-j)\right)x_{ij^{\prime}}x_{i^{\prime}j}.$

It is known (see [ReST]) that if $R_{0}$ is the standard R-matrix, $S$ is any
linear operator on the space of traceless diagonal matrices that is skew-
symmetric with respect to the trace-form, and $\pi_{0}$ is the natural
projection onto the subspace of diagonal matrices, then

(3.7) $R_{S}=R_{0}+S\pi_{0}$

satisfies MCYBE (3.1), and thus gives rise to a Sklyanin Poisson-Lie bracket.
The operator $S$ can be identified with an $n\times n$ skew-symmetric matrix
whose kernel contains the vector $(1,\ldots,1)$ and thus is uniquely
determined by its $(n-1)\times(n-1)$ submatrix $(s_{ij})_{i,j=1}^{n-1}$, which
we will also denote by $S$. Slightly abusing notation, we denote the remaining
elements of the above $n\times n$ skew-symmetric matrix by

$s_{in}=-\sum_{j=1}^{n-1}s_{ij},\qquad s_{nj}=-\sum_{i=1}^{n-1}s_{ij}.$

The Sklyanin bracket (3.4) that corresponds to (3.7) can be written in terms
of matrix entries as

(3.8)
$\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{S}}=\\{x_{ij},x_{i^{\prime}j^{\prime}}\\}_{R_{0}}+\frac{1}{2}\left(s_{ii^{\prime}}-s_{jj^{\prime}}\right)x_{ij}x_{i^{\prime}j^{\prime}}.$

Let $\mathcal{H}$ denote the subgroup of diagonal matrices in $SL_{n}$:

$\mathcal{H}=\\{\operatorname{diag}(d_{1},\dots,d_{n})\ :d_{1}\cdots
d_{n}=1\\}.$

For any skew-symmetric matrix $V=(v_{ij})_{i,j=1}^{n-1}$, define a Poisson
bracket ${\\{\cdot,\cdot\\}}^{\mathcal{H}}_{V}$ on $\mathcal{H}$ by

(3.9) $\\{d_{i},d_{j}\\}^{\mathcal{H}}_{V}=v_{ij}d_{i}d_{j};$

here $v_{in}$ and $v_{nj}$ have the same meaning as $s_{in}$ and $s_{nj}$
above.

In what follows, we denote the Poisson manifolds
$\left(\mathcal{H},{\\{\cdot,\cdot\\}}^{\mathcal{H}}_{V}\right)$ and
$\left(SL_{n},{\\{\cdot,\cdot\\}}_{R_{S}}\right)$ by $\mathcal{H}^{\\{V\\}}$
and $SL_{n}^{\\{S\\}}$, respectively.

Next, for $S$ defined as in (3.7), consider the direct product of Poisson
manifolds $\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}$; the product
structure we denote below simply by ${\\{\cdot,\cdot\\}}$.

######  Lemma 3.1.

The map $\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to SL_{n}^{\\{S\\}}$
given by $(D_{1},X,D_{2})\mapsto D_{1}XD_{2}$ is Poisson.

###### Proof.

Denote $D_{1}XD_{2}$ by $\widehat{X}=(\hat{x}_{ij})_{i,j=1}^{n}$. Let
$D_{k}=\operatorname{diag}(d_{kl})_{l=1}^{n}$ for $k=1,2$. Then
$\\{\hat{x}_{ij},\hat{x}_{i^{\prime}j^{\prime}}\\}=\\{\hat{x}_{ij},\hat{x}_{i^{\prime}j^{\prime}}\\}_{R_{0}}+x_{ij}x_{i^{\prime}j^{\prime}}\\{d_{1i}d_{2j},d_{1i^{\prime}}d_{2j^{\prime}}\\}$.
The second term is equal to

$\frac{1}{2}(s_{ii^{\prime}}-s_{jj^{\prime}})x_{ij}x_{i^{\prime}j^{\prime}}d_{1i}d_{2j}d_{1i^{\prime}}d_{2j^{\prime}}=\frac{1}{2}(s_{ii^{\prime}}-s_{jj^{\prime}})\hat{x}_{ij}\hat{x}_{i^{\prime}j^{\prime}},$

and the claim follows by (3.8). ∎

##  4\.  Grassmannians

### 4.1.

Let ${\mathcal{P}}$ be a Lie subgroup of a Poisson-Lie group ${\mathcal{G}}$.
A Poisson structure on the homogeneous space
${\mathcal{P}}\backslash{\mathcal{G}}$ is called Poisson homogeneous (with
respect to the Poisson-Lie structure on ${\mathcal{G}}$) [D] if the action map
${\mathcal{P}}\backslash{\mathcal{G}}\times{\mathcal{G}}\to{\mathcal{P}}\backslash{\mathcal{G}}$
is Poisson. In particular, if ${\mathcal{P}}$ is a parabolic subgroup of a
simple Lie group ${\mathcal{G}}$ equipped with the standard Poisson-Lie
structure, then ${\mathcal{P}}\backslash{\mathcal{G}}$ is a Poisson
homogeneous space. We will be interested in the case when
${\mathcal{G}}=SL_{n}$ equipped with the bracket (3.4) and

${\mathcal{P}}={\mathcal{P}}_{k}=\left\\{\begin{pmatrix}A&0\\\
B&C\end{pmatrix}\ :A\in GL_{k},C\in GL_{n-k}\right\\}.$

The resulting homogeneous space is the Grassmannian $G_{k}(n)$ equipped with
what we will call the standard Poisson homogeneous structure
${\\{\cdot,\cdot\\}}_{0}^{Gr}$. We will recall an explicit expression of this
Poisson structure on the open Schubert cell $G^{0}_{k}(n)=\\{X\in
G_{k}(n):x_{[k]}\neq 0\\}$. Here we use the same notation for an element of
the Grassmannian and its matrix representative $X$, and $x_{I}$ denotes the
Plücker coordinate that corresponds to a $k$-element subset $I\subset[n]$.
Elements of $G^{0}_{k}(n)$ can be represented by matrices of the form
$\left[\mathbf{1}_{k}\ Y\right]$, and the entries of the $k\times(n-k)$ matrix
$Y$ serve as coordinates on $G^{0}_{k}(n)$. In terms of matrix elements
$y_{ij}$ of $Y$, the Poisson homogeneous bracket looks as follows [GSV1]:

(4.1)
$\\{y_{ij},y_{\alpha\beta}\\}_{0}^{Gr}=\frac{\operatorname{sign}(\alpha-i)-\operatorname{sign}(\beta-j)}{2}y_{i\beta}y_{\alpha
j}.$

We denote $G_{k}(n)$ equipped with the Poisson bracket (4.1) by
$G_{k}(n)^{\\{0\\}}$.

######  Proposition 4.1.

(i) For an arbitrary skew-symmetric operator $S$, there exists a Poisson
bracket ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ on $G_{k}(n)$, unique up to a scalar
multiple, such that the natural action $(X,D)\mapsto XD$ is a Poisson map from
$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}$ to
$G_{k}(n)^{\\{S\\}}:=\left(G_{k}(n),{\\{\cdot,\cdot\\}}^{Gr}_{S}\right)$.

(ii) The bracket ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ is a Poisson homogeneous
structure on $G_{k}(n)$ with respect to the bracket
${\\{\cdot,\cdot\\}}_{R_{S}}$ on $SL_{n}$ defined by (3.8).

###### Proof.

(i) Let $X=\left[\mathbf{1}_{k}\ Y\right]\in G^{0}_{k}(n)$,
$D=\operatorname{diag}(d_{1},\dots,d_{n})\in\mathcal{H}$ and let
$\left[\mathbf{1}_{k}\ \widetilde{Y}\right]$ be the matrix that represents the
element $XD\in G^{0}_{k}(n)$. Then $\tilde{y}_{ij}=y_{ij}{d_{j+k}}/{d_{i}}$,
and the Poisson bracket of any two Plücker coordinates $\tilde{y}_{ij}$ and
$\tilde{y}_{\alpha\beta}$ in the product structure
${\\{\cdot,\cdot\\}}^{Gr}_{0}\times{\\{\cdot,\cdot\\}}^{\mathcal{H}}_{-\frac{1}{2}S}$
is equal to

$\frac{\operatorname{sign}(\alpha-i)-\operatorname{sign}(\beta-j)}{2}\tilde{y}_{i\beta}\tilde{y}_{\alpha
j}+\frac{s_{i,\beta+k}+s_{j+k,\alpha}-s_{i,\alpha}-s_{j+k,\beta+k}}{2}\tilde{y}_{ij}\tilde{y}_{\alpha\beta}.$

Thus, the bracket defined on $G^{0}_{k}(n)$ by the formula

$\\{y_{ij},y_{\alpha\beta}\\}_{S}^{Gr}=\\{y_{ij},y_{\alpha\beta}\\}_{0}^{Gr}+\frac{s_{i,\beta+k}+s_{j+k,\alpha}-s_{i,\alpha}-s_{j+k,\beta+k}}{2}y_{ij}y_{\alpha\beta}$

is the unique, up to a scalar multiple, Poisson bracket that makes the map
$(X,D)\mapsto XD$ Poisson. Sinse $G^{0}_{k}(n)$ is an open dense subset in
$G_{k}(n)$ the claim follows.

(ii) To see that ${\\{\cdot,\cdot\\}}^{Gr}_{S}$ is Poisson homogeneous with
respect to ${\\{\cdot,\cdot\\}}_{R_{S}}$, we need to check that the natural
action of $SL_{n}$ on $G_{k}(n)$ defines a Poisson map from
$G_{k}(n)^{\\{S\\}}\times SL_{n}^{\\{S\\}}$ to $G_{k}(n)^{\\{S\\}}$. Instead
of a straightforward calculation, we can use the fact that this is true for
$S=0$ and Lemma 3.1. Indeed, it is easy to check that both Poisson maps
$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to
G_{k}(n)^{\\{S\\}}$ given by $(X,D_{1})\mapsto XD_{1}$ and
$\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to SL_{n}^{\\{S\\}}$
given by $(D_{1},X,D_{2})\mapsto D_{1}XD_{2}$ are surjective. Therefore, we
can replace the map $G_{k}(n)^{\\{S\\}}\times SL_{n}^{\\{S\\}}\to
G_{k}(n)^{\\{S\\}}$ by the map

$G_{k}(n)^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\times\mathcal{H}^{\\{\frac{1}{2}S\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to G_{k}(n)^{\\{S\\}}$

given by $(X,D_{1},D_{2},A,D_{3})\mapsto XD_{1}D_{2}AD_{3}$. It is easy to
check that $(D_{1},D_{2})\mapsto D_{1}D_{2}$ is a Poisson map from
$\mathcal{H}^{\\{-\frac{1}{2}S\\}}\times\mathcal{H}^{\\{\frac{1}{2}S\\}}$ onto
$\mathcal{H}^{\\{0\\}}$, which, in turn, is a Poisson-Lie subgroup of
$SL_{n}^{\\{0\\}}$. We thus arrive to the Poisson map
$G_{k}(n)^{\\{0\\}}\times
SL_{n}^{\\{0\\}}\times\mathcal{H}^{\\{-\frac{1}{2}S\\}}\to G_{k}(n)^{\\{S\\}}$
given by $(X,\widetilde{A},D_{3})\mapsto X\widetilde{A}D_{3}$ with
$\widetilde{A}=D_{1}D_{2}A$. The standard Poisson homogeneous structure
ensures that the map $G_{k}(n)^{\\{0\\}}\times SL_{n}^{\\{0\\}}\to
G_{k}(n)^{\\{0\\}}$ is Poisson, and it remains to use part (i) of Proposition
4.1 to complete the proof. ∎

### 4.2.

Now, we recall the construction of the cluster algebra
${\mathcal{A}}_{G^{0}_{k}(n)}$ associated with the open cell $G^{0}_{k}(n)$ as
described in [GSV1, GSV2]. Denote $m=n-k$. For every $i\in[k]$, $j\in[m]$ put

(4.2) $I_{ij}=\begin{cases}[i+1,k]\cup[j+k,i+j+k-1],&\text{if $i\leq
m-j+1$}\\\ \left([k]\setminus[i+j-m,\ i]\right)\cup[j+k,n],&\text{if
$i>m-j+1$}.\end{cases}$

Denote the Plücker coordinate $x_{I_{ij}}$ by $x(i,j)$.

The initial cluster of ${\mathcal{A}}_{G^{0}_{k}(n)}$ is given by

(4.3) $\mathbf{x}=\mathbf{x}(k,n)=\left\\{\frac{x(i,j)}{x_{[k]}}\ :\ i\in[k],\
j\in[m]\right\\}.$

Stable variables are
${\displaystyle\frac{x(1,1)}{x_{[k]}},\ldots,\frac{x(k,1)}{x_{[k]}},\frac{x(k,2)}{x_{[k]}},\ldots,\frac{x(k,m)}{x_{[k]}}}$.
The entries of ${\widetilde{B}}$ that correspond to ${\bf x}$ are all $0$ or
$\pm 1$s. Thus it is convenient to describe ${\widetilde{B}}$ by a directed
graph $\Gamma({\widetilde{B}})$.

Figure 1. Graph $\Gamma({\widetilde{B}})$

The vertices of $\Gamma({\widetilde{B}})$ correspond to all columns of
${\widetilde{B}}$, and, since ${\widetilde{B}}$ is rectangular, the
corresponding edges are either between the cluster variables or between a
cluster variable and a stable variable. In our case, $\Gamma({\widetilde{B}})$
is a directed graph with $km$ vertices labeled by pairs of integers $(i,j)\
i\in[k],j\in[m]$. $\Gamma({\widetilde{B}})$ has edges $(i,j)\to(i,j+1)$,
$(i+1,j)\to(i,j)$ and $(i,j)\to(i+1,j-1)$ whenever both vertices defining an
edge are in the vertex set of $\Gamma({\widetilde{B}})$ (cf. Fig. 1). Each
cluster variable $x(i,j)$ is associated with (placed at) the vertex with
coordinates $(i,j)$ of the grid in Fig. 1 for $i\in[k],\ j\in[m]$. Equation
(2.1) then results in the following formulas for the $\tau$-cluster:

(4.4)
$\tau_{ij}=\frac{x(i+1,j-1)x(i,j+1)x(i-1,j)}{x(i-1,j+1)x(i,j-1)x(i+1,j)},\quad
i\in[k-1],\ j\in[2,m],$

where $x(0,j)=x(i,m+1)=1$.

######  Lemma 4.2.

Functions (4.4) are invariant under the natural action of $\mathcal{H}$ on
$G_{k}(n)$.

###### Proof.

Let $X\in G_{k}(n)$, $D=\operatorname{diag}(d_{1},\dots,d_{n})\in\mathcal{H}$
and $\widetilde{X}=XD$. For any subset $I=\\{i_{1},\ldots,i_{l}\\}\subset[n]$
denote $d^{I}=\prod_{j=1}^{l}d_{i_{j}}$. Then, using (4.2), we obtain

$\tilde{x}(i,j)=x(i,j)d^{I_{ij}}=\begin{cases}{\displaystyle
x(i,j)\frac{d^{[k]}d^{[i+j+k-1]}}{d^{[i]}d^{[j+k-1]}}},&\text{if $i\leq
m-j+1$},\\\\[6.99997pt] {\displaystyle
x(i,j)\frac{d^{[k]}d^{[n]}d^{[i+j-m-1]}}{d^{[i]}d^{[j+k-1]}}},&\text{if
$i>m-j+1$},\end{cases}$

and the equality $\tau_{ij}(\widetilde{X})=\tau_{ij}(X)$ follows from (4.4) by
trivial cancellation. ∎

Now we are ready to prove

######  Theorem 4.3.

A Poisson structure ${\\{\cdot,\cdot\\}}$ on $G_{k}(n)$ is compatible with
${\mathcal{A}}_{G^{0}_{k}(n)}$ if and only if a scalar multiple of
${\\{\cdot,\cdot\\}}$ defines a Poisson homogeneous structure with respect to
${\\{\cdot,\cdot\\}}_{R_{S}}$ for some skew-symmetric operator $S$.

###### Proof.

It follows from Theorem 5.4 in [GSV2] that ${\\{\cdot,\cdot\\}}_{0}^{Gr}$ is
compatible with ${\mathcal{A}}_{G^{0}_{k}(n)}$. The number of stable variables
for ${\mathcal{A}}_{G^{0}_{k}(n)}$ is $n-1$. Since $\mathcal{H}$ is isomorphic
to $({\mathbb{C}}^{*})^{n-1}$, Lemma 4.2 guarantees that the map $(X,D)\mapsto
XD$ translates into an action of $({\mathbb{C}}^{*})^{n-1}$ on
${\mathcal{A}}_{G^{0}_{k}(n)}$ as described in Section 2. Assumptions of
Theorem 2.1 are verified in [GSV1], Section 3.3. Then Proposition 2.2 and
Proposition 4.1 imply that every compatible Poisson bracket on
${\mathcal{A}}_{G^{0}_{k}(n)}$ is a scalar multiple of
${\\{\cdot,\cdot\\}}_{S}^{Gr}$ for some skew-symmetric operator $S$ on the
space of traceless diagonal $n\times n$ matrices. Since
${\\{\cdot,\cdot\\}}_{S}^{Gr}$ is a unique Poisson homogeneous with respect to
${\\{\cdot,\cdot\\}}_{R_{S}}$ (see, e.g. [D]), the claim follows. ∎

## Acknowledgments

M. G. was supported in part by NSF Grant DMS #0801204. M. S. was supported in
part by NSF Grants DMS #0800671 and PHY #0555346. A. S. was supported in part
by KVVA. A. V. was supported in part by ISF Grant #1032/08. The authors are
grateful to A. Zelevinsky for useful comments.

## References

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  * [BFZ] A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells. Duke Math. J. 126 (2005), 1–52.
  * [D] V. G. Drinfeld, On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. and Math. Phys. 95 (1993), no. 2, 524–525
  * [FZ1] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity. J. Amer. Math. Soc. 12 (1999), 335–380.
  * [FZ2] S. Fomin and A. Zelevinsky, Total Positivity: tests and parametrizations., Math. Inteligencer. 22 (2000), 23–33.
  * [FZ3] S. Fomin and A. Zelevinsky, Cluster algebras.I. Foundations. J. Amer. Math. Soc. 15 (2002), 497–529.
  * [GSV1] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry. Mosc. Math. J. 3 (2003), 899–934.
  * [GSV2] M. Gekhtman, M. Shapiro, and A. Vainshtein, Poisson geometry of directed networks in a disk, Selecta Mathematica 15, no. 1, 61-103.
  * [GSV3] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, book in preparation.
  * [P] A. Postnikov,  Total positivity, Grassmannians and networks, arXiv: math/0609764.
  * [ReST] A. Reyman and M. Semenov-Tian-Shansky Group-theoretical methods in the theory of finite-dimensional integrable systems Encyclopaedia of Mathematical Sciences, vol.16, Springer–Verlag, Berlin, 1994 pp. 116–225.
  * [S] J. Scott, Grassmannians and cluster algebras, Proc. London Math. Soc. 92 (2006), 345–380.


††thanks: deceased

# Magnetic phase diagram and possible Kitaev-like behavior of honeycomb-
lattice antimonate Na3Co2SbO6

E. Vavilova  Zavoisky Physical-Technical Institute (ZPhTI) of the Kazan
Scientific Center of the Russian Academy of Sciences, 420029 Kazan, Russia  T.
Vasilchikova  Faculty of Physics, Moscow State University, Moscow 119991,
Russia  A. Vasiliev  Faculty of Physics, Moscow State University, Moscow
119991, Russia  National University of Science and Technology ”MISiS”, Moscow
119049, Russia  Department of theoretical physics and applied mathematics,
Ural Federal University, Ekaterinburg 620002, Russia  D. Mikhailova  Institute
for Complex Materials, Leibniz Institute for Solid State and Materials
Research (IFW) Dresden, 01069 Dresden, Germany  V. Nalbandyan  Faculty of
Chemistry, Southern Federal University, Rostov-on-Don 344090, Russia  E.
Zvereva  Faculty of Physics, Moscow State University, Moscow 119991, Russia
S.V. Streltsov  Institute of Metal Physics, Ural Branch of the Russian Academy
of Sciences, Ekaterinburg 620137, Russia

###### Abstract

Recent theoretical studies have suggested that Kitaev physics and such effects
as formation of a mysterious spin-liquid state can be expected not only in
$\alpha$-RuCl3 and iridates, but also in conventional $3d$ transition metal
compounds. Using DC and AC magnetometry, thermodynamic and 23Na nuclear
magnetic resonance measurements (NMR) we studied such a candidate material
Na3Co2SbO6. A full phase diagram of Na3Co2SbO6 in a wide range of magnetic
fields and temperatures is presented. The results demonstrate transformation
of the magnetic structure realized at low-temperature and suppression of the
AFM order under the external field, the gradual development of the saturation
phase, as well as evidence of gapped spin-liquid-like behavior in certain
parts of the phase diagram.

##  I Introduction

Since proposal of the Kitaev model realization in honeycomb $\alpha$-RuCl3 and
iridates, a lot of efforts had been concentrated on study of these systems [1]
The quest for the spin liquid due to the anisotropic (Kitaev) exchange
interaction had been continued by considering more conventional $3d$ materials
[2]. It has been known for a long time that low-energy excitations in Co2+ can
be modeled by the pseudospin j${}_{\rm eff}=1/2$ [3; 4]. Indeed, in the
octahedral surrounding, $t_{2g}$ states can be described by the effective
orbital moment $l_{\rm eff}=1$, which together with spin $s=1/2$ leads to the
total moment j${}_{\rm eff}=1/2$. Of course, both $t_{2g}$ and $e_{g}$ states
contribute to the exchange coupling, but in a common edge geometry (nearly 90°
metal-ligand-metal bonds), the antiferromagnetic (AFM) $t_{2g}-e_{g}$ exchange
can be largely compensated by the ferromagnetic (FM) $e_{g}-e_{g}$ terms [5],
so that strongly anisotropic Ising-like exchange can dominate. This suggestion
resulted in a flurry of both theoretical and experimental interest to Co2+
compounds with a honeycomb layered geometry as a new playground for the Kitaev
physics.

However, already first ab initio and model calculations have shown that the
situation is more complicated in these systems [6; 7; 8]. In fact, a direct
exchange between $xy$ orbitals for the nearest neighbors leads to a
substantial isotropic exchange, which in case of BaCo2(AsO4)2 turned out to be
an order of magnitude larger than the Kitaev term [7]. Moreover, together with
this, the large exchange coupling between 3rd neighbors strongly questions
possibility of Kitaev model realization in Co2+ compounds. This explains why
most of them orders at very low temperature [9; 10]. Nevertheless, an external
magnetic field was suggested to stabilize an emergent state in such materials
as BaCo2(AsO4)2 [11] and Na2Co2TeO6 [9]. The nature of this field-induced
state is debated now, but it appears to be related to the Kitaev physics.

In the present paper, we study honeycomb antimonate Na3Co2SbO6, that was
recently proposed as a compound which is proximate to a Kitaev spin-liquid
phase [5]. Recent inelastic neutron scattering measurements suggested a
significant AFM Kitaev term in this material [12; 13], while theoretical
calculations in contrast expect large isotropic exchanges for both first and
third nearest neighbors (and small Kitaev term) [8]. Using AC and DC
magnetometry, heat capacity and NMR measurements, we establish a phase diagram
of Na3Co2SbO6 and investigate physical properties of this system. It is shown
that magnetic moments are confined to the honeycomb plane and order below
$T_{\rm N}$ = 7.6 K, while a magnetic field suppresses this ordering already
at $B^{*}\approx 2$ T. Analysis of the experimental data suggests that the
spin gap develops above this field.

The paper is organized as follows. Experimental details are summarized in Sec.
II. Results of 23Na NMR, thermodynamic and magnetometry experiments are
presented in Sec. III. The experimental results and the phase diagram are
given in the same section. The conclusions are summarized in Sec. IV.

##  II Experimental details

Figure 1:  (Color online)X-ray powder diffraction pattern of Na3Co2SbO6 with
the observed (red dots) and calculated (black solid line) curves together with
their difference curves (blue solid line) and Bragg positions given as green
vertical lines.

The polycrystalline Na3Co2SbO6 sample was prepared by a conventional two-step
solid-state synthesis. First, stoichiometric amounts of Sb2O3 (99.6%, Alfa
Aesar), Co3O4 (99.9%, Sigma Aldrich) and 5% excess of Na2CO3 (99.9%, Merck)
were ground together and annealed at 800°C in air for 15 h. After cooling down
to room temperature, the mixture was ground again, pressed into a pellet and
annealed in air at 980°C for 40 h followed by quenching in air. The phase
purity of the sample was verified by powder X-ray diffraction, using STOE
STADI P diffractometer with a Co-Kα1 radiation, $\lambda$ = 1.7889 Å, in a
transmission mode. Rietveld analysis using the Fullprof software [14] (see
Fig. 1) confirmed the structural model based on the neutron diffraction data
[15].

Part of the experimental data was obtained for the sample studied earlier
[16]. Excellent agreement of the lattice parameters (Table 1), together with
some thermodynamic and magnetic characteristics, proves that both samples are
essentially identical.

Heat capacity measurements were carried out by a relaxation method using a
Quantum Design PPMS system. The plate-shaped sample was obtained by cold
pressing of the polycrystalline powder. Data were collected at zero magnetic
field and under an applied field up to 9 T in the temperature range 2–30 K.
The AC magnetic susceptibility data were collected at the magnetic field
$B=0.0001$ T in the temperature range between 2–45 K while varying the
frequency $f$ between 0.5–10 kHz. The static magnetic measurements were
performed by means of a Quantum Design SQUID –magnetometer. The temperature
dependence of magnetic susceptibility was measured at various magnetic fields
0 T $\leq B\leq$ 7 T in the temperature range 1.8–300 K. Nuclear magnetic
resonance (NMR) measurements were performed with a Tecmag Redstone pulsed
spectrometer at fixed Larmor frequencies at several external field ranges. The
spectra were collected by step-by-step sweeping the field and integrating the
echo signal obtained by the standard Hahn echo pulse sequence at each field
step. The longitudinal relaxation rate $T_{1}^{–1}$ of 23Na nuclei was
measured at the maximum of the spectrum magnitude with saturation recovery and
stimulated echo pulse protocols. The evolution of the magnetization after the
pulse sequence is described by the equation for spin 3/2 [17]. Due to the
partial overlap of the main line and quadrupole satellites in the
inhomogeneously broadened powder spectra, the stretch factor $b$ was applied:

$\displaystyle M\propto M_{0}\left(0.1\ {\rm exp}(-\tau/T_{1})^{b}+0.9\ {\rm
exp}(-6\tau/T_{1})^{b}\right),$ (1)

where $\tau$ is a delay between the pulses and T1 is a spin-lattice relaxation
time.

Table 1:  Crystal lattice parameters of Na3Co2SbO6. Present results are given in the last line. Radiation |  $a$, Å |  $b$, Å |  $c$, Å |  $\beta$, ∘
---|---|---|---|---
Neutron [15] | 5.35648(10) | 9.28723(19) | 5.65100(7) | 108.358(1)
Synchrotron [16] | 5.3649(6) | 9.2782(3) | 5.6533(5) | 108.490(8)
Co Kα1 | 5.3640(1) | 9.2774(2) | 5.6545(1) | 108.491(2)

##  III Experimental results and discussion

###  III.1 Specific heat

The specific heat data at various external magnetic fields are presented in
Fig. 2. As it was found earlier [16] the temperature dependence $C_{p}(T)$ in
zero magnetic field demonstrates a distinct $\lambda$-type anomaly, associated
with an onset of the magnetic order at $T_{N}=6.7$ K. Under magnetic field
($B\leq$ 1 T) the singularity at $T_{N}$ somewhat broadens and shifts to lower
temperatures (inset in Fig. 2a). Higher fields lead to progressive weakening
and eventual fading of the anomaly, which finally transforms to a broad hump;
its position shifts to higher temperatures with the field (Fig. 2b).

Figure 2:  (Color online) (a) The temperature dependence of the specific heat
$C_{p}(T)$ in applied magnetic fields for Na3Co2SbO6 (filled circles) and non-
magnetic isostructural analogue Li3Zn2SbO6 (open circles). The inset
represents $C_{p}/T$ at low magnetic fields. (b) The magnetic specific heat
$C_{m}(T)$; the dashed red lines represent fit for a gapped system (Eq. 3); on
the inset $C_{m}(T)$ at $B<1$ T, solid blue and green lines are approximation
in the frame of spin wave theory (Eq. 2).

In order to analyze the nature of the magnetic phase transition in Na3Co2SbO6
at various external fields and evaluate corresponding contribution to the
specific heat we used data for its non-magnetic isostructural analogue
Li3Zn2SbO6. The magnetic contribution to the specific heat was determined by
subtracting the lattice contribution using the data for Li3Zn2SbO6 (Fig. 2b).
Analysis of the low-temperature part of $C_{m}(T)$ was performed in the terms
of the spin-wave (SW) theory [18]

$\displaystyle C_{SW}\propto T^{d/n},$ (2)

where $C{\rm{}_{SW}}$ is low-temperature specific heat due to spin-wave
excitations, $d$ stands for the dimensionality of the magnetic lattice and $n$
is defined as the exponent in the dispersion relation $\omega(q)\sim q^{n}$.
The analysis has shown that below $T_{N}$ the Cm(T) dependence follows
satisfactorily $T^{3}$ – law at low fields $B<$ 1 T (see inset in Fig. 3b).
This result implies a presence of antiferromagnetic magnons consistent with 3D
antiferromagnetic ordering at the low temperatures in this field range. The
same procedure for fields 1 T $<B\leq$ 2 T reveals $d$ = 2, indicating the
predominance of 2D AFM exchanges in the studied compound. In the vicinity of
quantum critical point, temperature dependence of magnetic specific heat obeys
power law $C_{m}$ $\sim$ $T^{2}$ [19]. It is worth mentioning that the low-
temperature magnetic specific heat in Na3Co2SbO6 for fields $B=2$ T and higher
can be described by the same model with the spin gap as was used for
$\alpha$-RuCl3 [19]

$C_{m}\propto\frac{{\rm exp}(-\Delta/k_{B}T)}{T}.$ (3)

In order to estimate the energy gap, $C_{m}(T)$ was fitted by (2), the results
are shown by dashed curves on Fig. 2b.

###  III.2 Static magnetization

In accordance with previously reported data [16], the static susceptibility
$\chi$ = M/B of Na3Co2SbO6 (Fig. 3) demonstrates an antiferromagnetic behavior
with low-field Weiss temperature $\Theta$ $\approx$ -10 K. For low external
fields $B<$ 2 T magnetic susceptibility passes through a maximum with decrease
in temperature and then drops. From the maximum $\chi$(T), one can find the
Néel temperature TN = 7.6 K at 0.05 T and it decreases with magnetic field.
The fit of field dependence of TN (Fig. 3(c)) by expression
$T_{N}\sim(1-B/B_{C})^{\textit{z}\nu}$ with z$\nu$ = 0.25 $\pm$ 0.013 yields
the critical field $B_{C}=1.33\pm 0.002$ T. The critical exponent z$\nu$ is
consistent with the magnetization and neutron scattering data for
$\alpha$-RuCl3 [20]; it can be related to $b$ = 0.25 for 2D chiral XY system
and is often observed in magnetic systems with 2D geometry with coordination
number 3 in the presence of in-plane anisotropy [21]. At the same time, the
magnetization and NMR data (see below) indicate the existence of features
typical for the AFM transition even above 1.3 T. Apparently, the strong
anisotropy of the magnetic properties entails a difference in the field of AFM
order suppression for different orientations of the powder crystallites. Thus,
below we will assume that the AFM order exists in the fields below
$B^{*}\approx$ 2T.

Figure 3:  (Color online) (a) Temperature dependence of magnetic
susceptibility $\chi=M/B$ at various external fields for Na3Co2SbO6 . (b)
Examples of the low temperature part of $d\chi/dT$ curves at two different
fields. (c) Field dependence of the Néel temperature fitted by critical
scenario (see text).

In order to examine the magnetic properties in an external field, we analyzed
the data obtained both for the studied sample [16] and the experimental
results from other publications on this compound [15; 22; 23]. The data on
$M(B)$ show that, firstly, full saturation starts to develop at $B>2.5$ T but
it has not yet been reached in fields 4 - 7 T. Secondly, the field behavior of
magnetization is highly anisotropic. And thirdly, the data from different
experiments demonstrate that the magnetization magnitude varies depending on
the sample. The reason can be either the quality of the sample, or the
difference in the shape of crystallites and the associated demagnetizing
factor.

###  III.3 Nuclear magnetic resonance

To study the local properties of the magnetic spin system, we performed NMR
measurements on the powder sample of Na3Co2SbO6 on the 23Na nuclei which have
a spin $I=3/2$, the gyromagnetic ratio $\gamma_{n}$ = 11.2685 MHz/T, and the
quadruple moment Q = 0.104 10-28 m2. The measurements were carried out at
several frequencies 10.7, 13.512, 22.52, 30.33, 45.04 and 86.714 MHz in the
corresponding field ranges since the effect of an external field on local
magnetic characteristics is of particular interest. Temperature transformation
of NMR spectra in some of field ranges is presented in Fig. 4.

Figure 4: 23Na powder NMR spectra for different frequencies and field ranges.

At temperatures of the order of 100 K and above, a classical powder spectrum
with quadrupole shoulders can be recorded. As the temperature decreases, the
line broadens and shifts towards lower fields. The low temperature spectrum at
$B<B^{*}$ has a flat top but does not have a stepwise rectangular shape that
is typical for antiferromagnets [24]. This can be understood assuming the fact
that the total width of the spectrum at low T is comparable to the quadrupole
splitting and the overlap of the main line and quadrupole satellites smears
the shape of the spectrum creating a bell-like profile. The same reason makes
the modelling of the line powder profile technically impossible and one cannot
resolve the positions of the spectral contributions from the parallel and
perpendicular orientations of the crystallites.

The shift of the NMR line can be defined as $k$ =
(($B_{L}-B_{res})/B_{L})\times$100%, where $B_{res}$ is the field of the
spectral magnitude maximum and $B_{L}$ = $\omega_{Larmor}/\gamma_{n}$. Its
temperature-dependent part corresponds to the local static susceptibility and
in paramagnetic state it is proportional to the bulk susceptibility $k$ =
$A\chi_{bulk}$, where A is the hyperfine tensor. The hyperfine interaction
constants $A_{hf}$ taken from the linear part of the $k$ ($\chi_{bulk})$
dependencies (see Fig. 5) in each external field are found to be almost equal
and the average value $A_{hf}$ is determined as 0.25$\pm$0.014 kOe/$\mu_{B}$.
This value is comparable to that observed in Na2Co2TeO6 [25] and is
significantly smaller than $A_{hf}$ for copper [26] and nickel [27] honeycomb
compounds. The difference in the electronic configurations and orbital
occupancy for cobalt ions seems to be responsible for the smaller orbital
overlap and the transfer of spin density to sodium ions. As a result, despite
the larger susceptibility compared to nickel or copper compounds (see also
[28] for tellurates), the increase in the local hyperfine field on sodium
nuclei is insignificant in cobalt compounds. This leads to a remarkably larger
decoupling of the honeycomb planes and to a more pronounced two-dimensionality
of the strongest exchanges in the cobalt compound as a result.

Figure 5: Dependencies of the local static $k$ (solid triangles) and dynamic
$(T_{1}T)^{-1}$ (open triangles) susceptibility on static bulk susceptibility
at various magnetic fields.

The local static susceptibility $k$ decrease compared to the bulk one at
fields $B<B^{*}$ indicates the development of antiferromagnetic correlations
in the system of cobalt spins at low temperatures suppressing the local field
at the sodium positions located between the cobalt planes. At $B>B^{*}$ the
local static susceptibility does not deviate from the bulk one anymore. In
Kitaev compounds, in the disordered state at high fields, no significant field
dependence in $k$ is observed [29]. For Na3Co2SbO6, the line shift strongly
depends on field below 10 K (Fig. 6) suggesting that the magnetic state is
close to fully polarized at low temperatures above 2.5 T.

Figure 6: Left panel: Temperature dependence of the NMR line shift at
different external fields. Lines are guide for eye. Right panel: Line shift
$k$ and $k\cdot B$ versus external field. Lines are guide for eye.

The dynamics of the electron spin system determines the process of nuclear
spin-lattice relaxation (see Fig. 7, left panels). The relaxation rate at
external fields $B<B^{*}$ exhibits a sharp peak at $T<6$ K indicating the
magnetic phase transition. The temperature of this peak decreases with
increasing field, similarly to the Néel temperature determined from the static
magnetic susceptibility data. Relaxation in the nearest upper vicinity of TN
can be described in terms of the critical exponent

$(T_{1}^{-1})\textfractionsolidus(T_{1\infty}^{-1})\propto((T-T_{N})\textfractionsolidus
T_{N})^{-\textit{p}},$ (4)

where $\textit{p}=\nu(\textit{z}-\eta)$ [30]. The obtained critical parameters
are p = 0.328 $\pm$ 0.013, $T_{N}=5.51\pm 0.08$ K for 1.2 T and p = 0.319
$\pm$ 0.031, $T_{N}=2.57\pm 0.07$ K for 2 T. Such a small value of p is
typical for XY-triangular and honeycomb lattices [31; 27] and can be estimated
as close to the calculated p = 0.32 obtained assuming $\nu\approx$ 0.5 [32]
and z = 0.64 [33]. It confirms the results of paragraph III.1: in-plane 2D
spin correlations at $B>$ 1 T play a key role in the magnetism of Na3Co2SbO6
including onset of a three-dimensional AFM order.

With a further increase of the field, a less pronounced but clearly observed
maximum appears on the temperature dependence of the relaxation rate. Its
position shifts to the higher T with the field and corresponds to the region
where the static susceptibility taken in the same fields changes weakly with
temperature. This allows attributing this maximum $T_{sat}$ to the
establishment of a static quasi-ferromagnetic phase induced by the external
field. On the other hand, in a FM state one can expect a strong increase in
the internal field at sodium positions and, accordingly, a line shift towards
low external fields. Nevertheless, at $B>B^{*}$ the line shift changes very
slightly below the low-temperature maximum compared to the paramagnetic state
(see Fig. 6). This leaves it possible to consider the low-temperature region
in these fields as a spin liquid with slow dynamics. To characterize the spin
dynamics at low temperatures, we tried to estimate the magnon gap $\Delta_{m}$
using a formula for a gapped three-magnon process [34] $T_{1}^{-1}\propto\
T^{n}\exp(-\Delta_{m}/T)$, where $n=2$. At $B>B^{*}$, such an estimation gives
a gap of the order of 4 – 16 K, which grows with the field (see Fig. 9). At
the same time, fitting of the $T_{1}^{-1}$(T) curve below 2.5 T, i.e., in the
AFM state, results in an extremely low value of the gap, and the data can be
described by a power law $T_{1}^{-1}\sim T^{3}$ typical for three-magnon
scattering in the antiferromagnets at temperatures above the energy gap in the
spin wave spectrum. It should be noted that the number of experimental points
does not allow making a reliable quantitative analysis; therefore, the
presented conclusions should be only considered as an estimation.

Above $T_{N}$ or the assumed saturation temperature $T_{sat}$, the temperature
dependence of the relaxation rate in the high fields contains a wide hump. It
is noteworthy that this hump firstly appears at 2 T, which is somewhat lower
than $B^{*}$. This feature can be attributed as the so-called “low-dimensional
maximum”, which characterizes the development of 2D spin correlations in two-
dimensional compounds. Given the possible Kitaev-Heisenberg physics in the
compound under study, we can consider two approaches to describing this
temperature dependence. The Arrhenius plot shown in Fig. 7 (right panel)
allows to evaluate the excitation gap as it was done in [35; 36; 37; 38]. Fit
to the activation law $T_{1}^{-1}\sim\exp(-\Delta_{1}\textfractionsolidus T)$
gives the gap $\Delta_{1}$ values from 2.98 K to 19.69 K in field range 2 –
7.7 T (see Fig. 7). Another approach applied for NMR in $\alpha$-RuCl3
compound [39; 29] describes not just the slowing down of the relaxation below
its maximum but a broad temperature range including the whole hump.
Theoretical model [40] correlates the boundaries of this hump with the
magnitude of the Kitaev exchange and propose the extended exponential function
$T_{1}^{-1}\sim T^{-n}\exp(-\Delta/T)$. We evaluate the spin gap with assuming
$n=1$ (see Fig. 7 left panel) as 8.177 K at 2 T and 44.871 K at 7.7 T (see
Fig. 8) which is comparable to the values obtained for $\alpha$-RuCl3
compound.

Figure 7: Left panel: Temperature dependence of 23Na spin-lattice relaxation
obtained in different external fields. Solid brown lines present fit to
equation $T_{1}^{-1}\sim T^{-n}\exp(-\Delta/T)$, where $n=1$. Red dashed lines
describe a critical behavior of the relaxation 4. Brown dotted lines
correspond to $T_{1}^{-1}\sim T^{2}\exp(-\Delta_{m}/T)$for $B>2.5$ T and
$T_{1}^{-1}\sim T^{n}$ for $B<2.5$ T. Upper panel presents stretched
coefficients from equation 1. Right panel: 23Na spin-lattice relaxation rates
versus inverse temperature. Brown dashed lines correspond to equation
$T_{1}^{-1}\sim\exp(-\Delta_{1}\textfractionsolidus T)$.

And, finally, like the NMR line shift probes the local static susceptibility,
the spin-lattice relaxation can be considered as a direct probe of the local
dynamic susceptibility of the electron spin subsystem [41]

$\frac{1}{T_{1}T}\propto\sum_{\bf q}|A_{\perp}({\bf
q})|^{2}\frac{\chi^{\prime\prime}({\bf q},\omega_{L})}{\omega_{L}},$ (5)

where $A_{\perp}({\bf q})$ is the q-dependent hyperfine constant, q is the
wave vector and $\omega_{L}$ is the Larmor frequency. In the paramagnetic
regime $(T_{1}T)^{-1}$ is proportional to the static bulk susceptibility. The
sharp deviation of the local dynamic susceptibility from the static bulk one
occurs below 10 K in the entire range of fields (see Fig. 5), marking the
critical slowing down of the electron spin fluctuations in the vicinity of the
transition to the static state. In fields above $B^{*}$, the local dynamic
susceptibility deviates from the static one already at 20 - 50 K, marking the
development of dynamic correlations at a relatively low-frequency part of the
spin fluctuation spectra.

###  III.4 Bulk dynamic susceptibility.

The AC susceptibility at zero external field exhibits a sharp peak at the Néel
temperature. Above $B^{*}$, the critical low-temperature anomaly is absent,
but a wide hump is observed, which shifts to higher temperatures with
increasing field. At any fields, the $\chi’(T)$ curves obtained at different
frequencies do not exhibit any frequency dependence (see Fig. 8). This rules
out that the hump is associated with any kind of conventional glassy
transition. The $\chi’$ hump is not accompanied by an anomaly of
$\chi^{\prime\prime}$ and therefore one cannot attribute it to the crossover
to the field-induced ferromagnetic order, since at ferro- or ferrimagnetic
transition or crossover the imaginary component is present [42]. On the other
hand, this broad maximum shifts to higher temperatures with the applied DC
field and it does not allow us to attribute it to the development of two-
dimensional antiferromagnetic correlations in cobalt planes. It is remarkable
that a pronounced anomaly in this temperature range is absent in the
temperature dependence of both the bulk and the local static susceptibility,
but it is observed in the dynamic susceptibility in both the kHz and MHz (NMR
relaxation) ranges indicating the dynamic nature of the emerging magnetic
state. It should be noted that the temperature of the AC susceptibility hump
maximum corresponds in order of magnitude to the values of the Kitaev-
Heisenberg gap obtained from nuclear relaxation analysis.

Figure 8: Left panel: Real part $\chi$’ of the ac magnetic susceptibility at
various external field. Right panel: Real part $\chi$’ (top) and imaginary
part $\chi$” (bottom) of the AC magnetic susceptibility at various AC
frequencies.

###  III.5 Phase diagram.

In the phase diagram at temperatures below 8 K, one can distinguish regions
with the ordered state of the magnetic lattice and several other phases,
characterized by specific fields. Firstly, this is the field $B_{1}$ = 1 T,
where, according to the heat capacity data, the magnon regime changes from 3D
to 2D. It is noteworthy that this approximately corresponds to the field at
which the data of the field dependence of the magnetization [16; 15] register
a metamagnetic transition. The anisotropy gap determined from the field
$B_{1}$ is smaller than 2 K. With a further increase in the external field
above $B_{C}$ = 1.3 T, the antiferromagnetic order is suppressed. In fields
above $\approx$ 2.8 T, both local (NMR line shift) and bulk static
susceptibility indicate the development of saturation at low temperatures. The
characteristic rise in the relaxation rate is observed here, which is not as
sharp as the Néel peak at low fields. Taking the $T_{1}^{-1}(T)$ dependence
below this maximum, one can estimate the magnon gap $\Delta_{m}$. The
longitudinal nuclear relaxation records a transverse dynamics of the magnetic
system. Previous theoretical calculations of XXZ model suggest that these are
the one-magnon excitations, which can have the lowest excitation energy for
parameters typical for Na3Co2SbO6 [43; 22]. NMR, as a local method, fixes
excitations mostly on long wave vectors and therefore one can expect the
observation of a one-magnon process for the saturation region. Indeed, the
slope of the dependence $\Delta_{m}/B=g\beta S/k_{B}$ is about 2.19 K/T (dark
blue circles in Fig. 9). This slope is consistent with the Zeeman energy of a
single spin flip, i.e., a one-magnon excitation, which, with the g-factor
$g=3.3$ [16], would be 2.21 K/T.

Figure 9: The magnetic phase diagram for Na3Co2SbO6

While the ground state of Na3Co2SbO6 is established (zigzag AFM, see
e.g.[22]), nature of the field-induced phase is debated in Kitaev materials.
There were proposed various states to characterize this phase: a chiral Kitaev
spin-liquid, a quantum paramagnet with possible nematic phases or complex
magnetic orders, see e.g. [44; 45; 46; 47] and references therein. Realization
of a particular state depends on details of the exchange interaction, which
have not been established yet for Na3Co2SbO6 (available theoretical data [22;
8] are partially contradicting).

On another hand experimentally the field-induced phase is characterized by a
wide hump on the temperature dependence of the heat capacity, AC
susceptibility, and nuclear relaxation. This feature is typical for low-
dimensional systems in general relating to development of 2D short-range
magnetic correlations. in Na3Co2SbO6 it is remarkable that this pronounced
hump is absent in low fields, where AFM develops at low temperatures. The fits
of these ”hump” dependencies testify to the gap character of spin excitations.
At the same time, some features of the AFM transition are still observed by
the local NMR technique up to 2 T. This is easy to explain taking into account
the powder nature of the sample under study: while a strong anisotropy of
magnetic properties is observed in the Na3Co2SbO6 [48; 22], including the AFM
order suppression field, the result of the measurements is the sum signal of
crystallites of different orientation and the data obtained in a field region
1 – 2 T combine the properties of both the ordered and spin-liquid phases. For
example, the local NMR technique observes both features (AFM transition peak
and high-temperature hump) on the temperature dependence of relaxation at 2T.
The data of bulk heat capacity measurement technique at 2T can be described
both by the gap law and 2D magnon process or a sum of these two contributions
depending on the temperature range of the fit. The observed combination of two
scenarios for the magnetic system behavior at 1T$<B<$ 2T is in good
quantitative agreement with the field values of metamagnetic transitions in
different orientations of the single crystal mentioned in work [48].

At higher temperatures, a comparison of the results of dynamic and static
magnetization indicates the existence of regions with developed magnetic
correlations. The boundary of such a region shifts to low temperatures with an
increase in the external field; then, above 2T, the nature of the correlations
apparently changes and the temperature boundaries of the correlated region
expand again. As it has been mentioned above, as the nuclear spin-lattice
relaxation and heat capacity suggest, spin excitations in this region are
gapped and the temperature behavior of the experimental data in this field and
temperature range are well described by the extended exponential function
characteristic, e.g., to Kitaev model. In this model [49], the Majorana
fermion gap is proportional to $B^{3}$, but other, non-Kitaev interactions
result in the field-induced gap for any field orientation and make it linearly
dependent on the field [50]. A slope of the gap temperature dependence is
about 3.41 K/T for a single exponential model (red stars in Fig. 9) or 6.67
for an extended exponential function which is specific to the Kitaev state
(violet stars in Fig. 9). Note that while there can be many possible reasons
for the gap in the spectrum of spin excitations of frustrated quantum magnets
in an external field, we can confidently say that this gap process is observed
in the temperature range several times higher than the saturation limit. It
formally does not correspond to any single-magnon or two-magnon spin
excitation processes, and the size of the gap significantly exceeds the
obtained value of the exchange anisotropy gap. Moreover, the gap value is
comparable to the energy of the spin wave spectrum peak around $E=1–3$ meV
observed by inelastic neutron scattering experiments and described in
frameworks of Kitaev–Heisenberg model [12]. At the same time, it should be
kept in mind that the gap value is not always the same for all orientations of
the crystal lattice in the magnetic field and, moreover, the gap behavior in
Heisenberg-Kitaev systems is not necessary observed in all directions (see for
example [51]). Studies of a powder sample obviously contain a certain error in
determining the magnitude and field dependence of the Kitaev gap in the
spectrum of spin excitations. It is also necessary to take into account the
possible anisotropy of the $g$-factor used in data analysis; therefore, the
above quantitative parameters should be considered with a caution. While there
is an experimental evidence of Kitaev physics realization in Na3Co2SbO6, we
would like to motivate further researches on single crystals, which could
clarify the boundaries of a possible spin-liquid state around 2 T.

##  IV Conclusions

In conclusion, we performed bulk (heat capacity, DC and AC susceptibility) and
local (NMR) probe measurements of the powder sample of honeycomb $j_{eff}=1/2$
system Na3Co2SbO6 in a wide range of temperatures and magnetic fields. These
experiments reveal regions with a static spontaneous and field-induced
magnetic order, a wide temperature range of spin correlations of various
nature as well as a region of fields, where a gapped quantum spin-liquid can
exist. The parameters of spin excitations in different regions of the phase
diagram were estimated, including the magnitude and temperature dependence of
the field-induced energy gap, which determines the specific shape of the
temperature dependence of bulk and local characteristics well above the
saturation region. Our measurements give some evidence of possible formation
of a spin-liquid-like phase or realizing the Kitaev-Heisenberg scenario in
Na3Co2SbO6 under external magnetic field. In this context, it would be
enlightening to investigate similarly a high-quality single-crystal sample in
order to clarify the details of the model parameters.

##  V Acknowledgments

The authors are grateful to Dr. P. Maksimov, Dr. S. Winter, Dr. H.-J. Grafe,
Prof. B. Büchner, and Dr. V. Kataev for useful discussions. A.V. acknowledges
support by the Megagrant program of Russian Government through the project
075-15-2021-604. The work of D.M. was made in 2021. S.V.S. thanks Russian
Science Foundation for support of theoretical analyses (RSF 20-62-46047). E.V.
would like to thank financial support from the government assignment for FRC
Kazan scientific Center of RAS. .

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Existence and smoothness of the Navier-Stokes equations

and semigroups of linear operators

Yu. N. Kosovtsov

Lviv, Ukraine

email<EMAIL_ADDRESS>
###### Abstract

Based on Leray’s formulation of the Navier-Stokes equations and the conditions
of the exact linear representation of the nonlinear problem found in this
paper, a compact explicit expression for the exact operator solution of the
Navier-Stokes equations is given. It is shown that the introduced linear
operator for Leray’s equations is the generator of one-parameter contraction
semigroup. This semigroup yields the existence of a unique and smooth
classical solution of the associated Cauchy problem of Navier-Stokes equations
in space $\mathbb{R}^{3}$ under smooth initial conditions.

##  1 The Navier-Stokes Equations

Incompressible flows of homogeneous fluids in space $\mathbb{R}^{3}$ are
solutions of the system of equations [1]-[5]

$\frac{\partial{v}}{\partial t}=\nu\Delta
v-\sum_{j=1}^{3}v_{j}\frac{\partial{v}}{\partial x_{j}}-\nabla p+f,$ (1)

$\mathrm{div}\;v\equiv\sum_{j=1}^{3}\frac{\partial{v_{j}}}{\partial
x_{j}}=0,\qquad(x,t)\\\ \in\\\ \mathbb{R}^{3}\times[0,\infty),$ (2)

$v\mid_{t=0}=v_{0},\qquad x\\\ \in\\\ \mathbb{R}^{3},$ (3)

where $v(x,t)\equiv(v_{1},v_{2},v_{3})$ is the fluid velocity, $p(x,t)$ is the
scalar pressure, the gradient operator is $\nabla=(\frac{\partial}{\partial
x_{1}},\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial x_{3}})$ and
the Laplace operator $\Delta\equiv\sum_{j=1}^{3}\frac{\partial^{2}}{\partial
x_{j}^{2}}$, a kinematic constant viscosity $\nu\geq≥0$, initial conditions
$v\mid_{t=0}=v_{0}(x)\equiv(v_{01},v_{02},v_{03})$, $f=f(x,t)$ is given,
externally applied force. Further we will consider Navier-Stokes equations in
the absence of external forces, i.e. with $f=0$.

It is well known [3],[4] that we can find an equation $p=p(v)$ to eliminate
the pressure from (1). If we take the divergence at each side of the momentum
equation (1), which yields, using (2) that

$\Delta\emph{p}=-\sum_{i,j=1}^{3}\frac{\partial{v_{j}}}{\partial
x_{i}}\frac{\partial{v_{i}}}{\partial x_{j}}.$ (4)

The equation (4) with respect to the function $p$ is the well known Poisson
equation, whose solution in $\mathbb{R}^{3}$ [3] has the form (for brevity, we
will sometimes denote this solution as $p_{v}(x,t)\equiv p_{v}$)

$p(x,t)=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}d\xi\sum_{i,j=1}^{3}\frac{\partial{v_{j}(t,\xi)}}{\partial\xi_{i}}\frac{\partial{v_{i}(t,\xi)}}{\partial\xi_{j}}\frac{1}{|x-\xi|},\qquad(\xi\\\
\in\\\ \mathbb{R}^{3}).$ (5)

Substituting the solution (5) into (1), we obtain (Leray’s Formulation of the
Navier-Stokes Equation [3]) _non-linear system of integro-differential
equations_ containing _only_ unknown functions $v_{i}$.

$\frac{\partial{v}}{\partial t}=\nu\Delta
v-\sum_{j=1}^{3}v_{j}\frac{\partial{v}}{\partial x_{j}}-\nabla
p_{v},\qquad(x\\\ \in\\\ \mathbb{R}^{3},t\geq 0).$ (6)

It is easy to see [3] that the systems of the Navier-Stokes equations (1)-(3)
are equivalent to the system of the equations (5),(6), provided that functions
$v_{0}$ in initial conditions are smooth and

$\sum_{i=1}^{3}\frac{\partial{v_{0i}}}{\partial x_{i}}=0,\qquad(x\\\ \in\\\
\mathbb{R}^{3}).$ (7)

Although the Leray’s formulation of the Navier-Stokes equation has been known
for a long time, it has not been widely used to analyze its solutions. The
main theses of this can be traced to the following [3].

The equation (6) is quadratically nonlinear and contains a nonlocal,
quadratically nonlinear operator. These facts make the Navier-Stokes equation
hard to study analytically.

Although (6) constitutes a closed system for $v$, this formulation turns out
to be not very useful for further analysis except by rather crude methods
based on the energy principle. The main reason for this is that this
formulation hides all the properties of vorticity stretching and interaction
between rotation and deformation.

##  2 Exact linear representation of a nonlinear problem

We define for a multi-index $(\alpha_{1},...,\alpha_{1})\in\mathbb{N}^{n}$

$D^{\alpha}=D_{1}^{\alpha_{1}}...D_{3}^{\alpha_{n}},\;\mathrm{where}\;D_{k}=\frac{\partial}{\partial
x_{k}},\;\alpha:=\sum_{i=1}^{n}\alpha_{i}.$

Let us take an arbitrary function $u(x)$, $x\in\mathbb{R}^{n}$ which is
infinitely many times differentiable and decay sufficiently rapidly, i.e.,
$u\in C^{\infty}(\mathbb{R}^{n})$, and

$\lim_{\mid x\mid\rightarrow\infty}\mid
x\mid^{p}D^{\alpha}u(x)=0\qquad\mathrm{for\;some}\;p\in\mathbb{N}\;\mathrm{and\;all}\;\alpha\in\mathbb{N}^{n}.$

Then, using this function, we define the linear space of _analytic_ on all
arguments functions $G(u):\equiv G(u,u_{1},u_{2},...)\in C^{\infty}$, where
$u_{i}:=D_{k}^{i}u(x)$ ($i,k\in\mathbb{N}$) and which decay sufficiently
rapidly with $u$ and all $u_{i}$ provided $\mid x\mid\rightarrow\infty$.

Since we can also consider these functions $G(u(x))$ as functions of
$x\in\mathbb{R}^{n}$ denoting $G(u(x))=\widetilde{G}(x)$. In this case, it is
obvious that the function $\widetilde{G}(x)\in C^{\infty}(\mathbb{R}^{n})$ and
decay sufficiently rapidly when $\mid x\mid\rightarrow\infty$.

When endowed with the usual inner product

$<\widetilde{F}(x),\widetilde{G}(x)>:=\int_{\mathbb{R}^{n}}\widetilde{F}(x)\widetilde{G}(x)\;dx$

the considered space becomes as a complete normed subspace
$L^{2}_{u}(\mathbb{R}^{n})$ of the Hilbert space $L^{2}(\mathbb{R}^{n})$.

Let a differential operator $A$ with domain $D(A)$ on a Banach space
$L^{2}_{u}(\mathbb{R}^{n})$ is of type

$A=\int_{\mathbb{R}^{n}}d\zeta F(u(\zeta))\frac{\delta}{\delta
u(\zeta)},\qquad\zeta\\\ \in\\\ \mathbb{R}^{n},$ (8)

where $u(x)$ is a function of class described above, a _predetermined_
$F(u(x)):\equiv F(u,u_{1},u_{2},...)\in L^{2}_{u}(\mathbb{R}^{n})$ with
$u_{i}:=D_{k}^{i}u(x)$ and we can consider $F(u(x))$ as the linear
multiplication operator, $\frac{\delta}{\delta u(\zeta)}$ is the functional
derivative. We have specially transferred the symbol $d\zeta$ at the beginning
of the formula to emphasize that the right operator is executed first, and
then the integration is performed.

Here by functional derivative we mean the linear mapping with the following
property

$\frac{\delta u(x)}{\delta u(\zeta)}=Dirac(x-\zeta),$

the chain rule is also valid in this context.

Entered operator is a linear operation, i.e., operator $A$ obeys the following
properties:

$AcG(u(x))=cAG(u(x)),\qquad\mathrm{for\;any}\;G\in\\\
L^{2}_{u}(\mathbb{R}^{n})\;\mathrm{and\;any\;scalar}\;c\in\\\ \mathbb{R},$
$A[G(u(x))+H(u(x))]=AG(u(x))+AH(u(x)),\qquad\mathrm{for\;any}\;G,H\in\\\
L^{2}_{u}(\mathbb{R}^{n}).$

It is obvious that $A^{j}u\in L^{2}_{u}(\mathbb{R}^{n})$ for any
$j\in\mathbb{N}$ and operator $A$ is _continuous_ and therefore it is
_bounded_ and _closable_ on the space $\L^{2}_{u}(\mathbb{R}^{n})$.

Let an operator $A$ with some $F(u(x))$ is the _generator of strongly
continuous semigroup_ $(T(t))_{t\geq 0}$ then ([6], p.50)

$\frac{d}{dt}T(t)u(x)=T(t)Au(x)=AT(t)u(x)\qquad\mathrm{for\;all}\;t\geq 0.$
(9)

Note that if we denote

$v(x,t):=T(t)u(x),$ (10)

then from (9) _three_ _equivalent_ equations follows.

First

$\frac{d}{dt}v(x,t)=Av(x,t),$ (11)

the _linear_ one with the _functional derivative_ and expression (10) is its
_general_ solution.

Second

$\frac{d}{dt}v(x,t)=T(t)Au(x),$

but because for considered operator

$Au(x)=F(u(x)),$

then as much as (see [6], p. 58) $T(t)$ is a semigroup of algebra
homomorphisms on $D(A)$, i.e.,

$T(t)(h\cdot g)=T(t)h\cdot T(t)g\qquad\mathrm{for}\;h,g\in
D(A)\;\mathrm{and}\;t\geq 0$

and, since we considering analytic functions $F$, we have

$T(t)Au(x)=F(T(t)u(x))$

and come to

$\frac{d}{dt}v(x,t)=F(v(x,t),v_{x_{1}}(x,t),...,D^{\alpha}v(x,t),...),$ (12)

i.e., to _nonlinear PDE_. The expression (10) is the _general_ solution to
(12) _too_!

This result has parallels with the well known connection between _nonlinear_
ODEs and _linear_ PDEs. There is a third equivalent equation

$Av(x,t)=F(v(x,t),v_{x_{1}}(x,t),...,D^{\alpha}v(x,t),...),$

the _nonlinear_ one with the _functional derivative_ and with the _same_
_general_ solution.

From a comparison of expressions (8) and (12), it is easy to obtain a one-to-
one relationship between the linear operator $A$ and the nonlinear right-hand
side of equation (12).

Note that the strongly continuous semigroup $(T(t))_{t\geq 0}$ generated by
the operator $A$ of the type considered here with some $F(u(x))$ can be given
by [6], [7]

$T(t)=\exp\\{tA\\}=\sum_{n=0}^{\infty}\frac{t^{n}A^{n}}{n!},\qquad t\geq 0,$

where the power series converges for every $t\geq 0$ ([6], pp. 52, 81).

Practically without changing the above reasoning, we can generalize the
results obtained to more complex equations, when

$F(v(t))=F(v(t,x),v_{x_{1}}(t,x),...,D^{\alpha}v(t,x),...,I_{1}v(t,x),I_{2}v(t,x)...),$

where $I_{i}$ are nonlinear _integral_ operators.

Moreover, it is possible to consider in a completely similar way systems of
nonlinear equations

$\frac{d}{dt}v_{i}(x,t)=F_{i}(v_{1}(x,t),...v_{n}(x,t)),\;\;i=1..n,$ (13)

where all $F_{i}$ are functions of the type described above and if we
introduce the following linear operator

$A=\int_{\mathbb{R}^{n}}d\zeta\sum_{i=1}^{n}F_{i}(u_{1}(\zeta),...,u_{n}(\zeta))\frac{\delta}{\delta
u_{i}(\zeta)},\qquad\zeta\\\ \in\\\ \mathbb{R}^{n},$ (14)

where rapidly decreasing $u_{i}\in C^{\infty}(\mathbb{R}^{n})$, and
$F_{i}(u_{1}(x),...u_{n}(x))$ are in the space $\L^{2}_{u}(\mathbb{R}^{n})$
built on vector function $u(x)\equiv(u_{1}(x),...,u_{n}(x))$.

Again assuming that operator $A$ is the _generator of strongly continuous
semigroup_ $(T_{sys}(t))_{t\geq 0}$. In this case the general solutions to the
system (13) is

$v_{i}(x,t)=T_{sys}(t)u_{i}(x),$ (15)

where as before

$T_{sys}(t)=\exp\\{tA\\}=\sum_{n=0}^{\infty}\frac{t^{n}A^{n}}{n!},\qquad t\geq
0.$

Thus we have obtained an important result that, under the above conditions, it
is possible to reduce _exactly_ a _nonlinear_ problem to a _linear_ one, which
allows us to use for solving nonlinear problems the accumulated baggage for
linear problems111Here we consider the case when the operator $A$ does not
depend on $t$. It is possible to generalize the results obtained above for
more general functions $F(t,x,u(x))$ (see the idea in [8])..

Comment. So far as $T(t)\mid_{t=0}=1$ (identity operator) then from (10) or
(15) it follows that function $u(x)$ is directly related to the initial
conditions. However, we need to note one circumstance in previous reasoning.
We use in definition of the operators $A$ the functional derivatives with
respect to the variable $u$ and therefore cannot identify it with everyone
fixed initial condition. That is to obtain the particular solution we have to
substitute $u(x)=v(x,0)$ into the general solution

$v(x,t)=\\{T(t)u(x)\\}\mid_{u(x)=v(x,0)}.$ (16)

##  3 Solution of the Navier-Stokes problem

Turning to the Leray’s formulation of the Navier-Stokes equations (6), we will
consider the vector function $u(x)\equiv(u_{1}(x),u_{2}(x),u_{3}(x))$, where
arbitrary functions $u_{i}(x)$, $x\in\mathbb{R}^{3}$ which, as we discussed in
the previous section, are infinitely many times differentiable and decay
sufficiently rapidly. In accordance with the trick described above let us
introduce the linear differential operator

$\displaystyle{\bf
A}=\int_{\mathbb{R}^{3}}d\zeta\,\sum_{i=1}^{3}{\Large\\{}\nu\Delta_{\zeta}u_{i}(\zeta)-\sum_{j=1}^{3}u_{j}(\zeta)\frac{\partial{u_{i}(\zeta)}}{\partial\zeta_{j}}-$
$\displaystyle\frac{1}{4\pi}\frac{\partial}{\partial\zeta_{i}}{\Large[}\int_{\mathbb{R}^{3}}d\xi\sum_{i,j=1}^{3}\frac{\partial{u_{j}(\xi)}}{\partial\xi_{i}}\frac{\partial{u_{i}(\xi)}}{\partial\xi_{j}}\frac{1}{|\zeta-\xi|}{\Large]}{\Large\\}}\frac{\delta}{\delta
u_{i}(\zeta)}.$ (17)

In short notation

${\bf
A}=\int_{\mathbb{R}^{3}}d\zeta\,\sum_{i=1}^{3}{\Large\\{}\nu\Delta_{\zeta}u_{i}(\zeta)-\sum_{j=1}^{3}u_{j}(\zeta)\frac{\partial{u_{i}(\zeta)}}{\partial\zeta_{j}}-\nabla_{\zeta}\;p_{u}(\zeta){\Large\\}}\frac{\delta}{\delta
u_{i}(\zeta)},$ (18)

where

$p_{u}(\zeta)\equiv\frac{1}{4\pi}{\Large[}\int_{\mathbb{R}^{3}}d\xi\sum_{i,j=1}^{3}\frac{\partial{u_{j}(\xi)}}{\partial\xi_{i}}\frac{\partial{u_{i}(\xi)}}{\partial\xi_{j}}\frac{1}{|\zeta-\xi|}{\Large]}.$

We restrict our attention to Navier-Stokes equations and correspondingly to
operator ${\bf A}$ on all of $\mathbb{R}^{3}$, thus avoiding the delicate
discussion of boundary conditions.

As it follows from the results of the previous section we need to find out
whether the operator is a generator of a semigroup. Let us describe some
properties of the operator ${\bf A}$ introduced above. First of all

${\bf A}\,u_{i}=\nu\Delta
u_{i}-\sum_{j=1}^{3}u_{j}\frac{\partial{u_{i}}}{\partial
x_{j}}-\frac{1}{4\pi}\frac{\partial}{\partial
x_{i}}\int_{\mathbb{R}^{3}}d\xi\sum_{i,j=1}^{3}\frac{\partial{u_{j}(\xi)}}{\partial\xi_{i}}\frac{\partial{u_{i}(\xi)}}{\partial\xi_{j}}\frac{1}{|x-\xi|}$
(19)

and it is obvious that ${\bf A}\;L^{2}_{u}(\mathbb{R}^{3})\mapsto
L^{2}_{u}(\mathbb{R}^{3})$ and ${\bf A}$ is _continuous_ and therefore it is
_bounded_ and _closable_ on the space $\L^{2}_{u}(\mathbb{R}^{n})$ built on
vector function $u(x)\equiv(u_{1}(x),u_{2}(x),u_{3}(x))$. We can rewrite the
expressions (19) in vector form for brevity

${\bf A}u=\nu\Delta u-\sum_{j=1}^{3}u_{j}\frac{\partial{u_{i}}}{\partial
x_{j}}-\nabla p_{u},$ (20)

where

$p_{u}(x)=\frac{1}{4\pi}\int_{\mathbb{R}^{3}}d\xi\sum_{i,j=1}^{3}\frac{\partial{u_{j}(\xi)}}{\partial\xi_{i}}\frac{\partial{u_{i}(\xi)}}{\partial\xi_{j}}\frac{1}{|x-\xi|}.$

Further, by analogy with the proof of the energy principle, we have in view of
(7) and that $u$ is rapidly decreasing and hence $p_{u}$ rapidly decreasing at
infinity too one can obtain the well known result that inner product

$<{\bf
A}u,u>=-\nu\sum_{i,j=1}^{3}\int_{\mathbb{R}^{3}}{\Large(}\frac{\partial{c_{i}}}{\partial
x_{j}}{\Large)}^{2}dx\leq 0.$ (21)

Hence, the linear operator ${\bf A}$ on the Hilbert space
$L^{2}_{u}(\mathbb{R}^{3})$ is _dissipative_ [9]-[13].

By definition [14], the adjoint operator $A^{*}$ to the bounded operator $A$
on the Hilbert space satisfies the identity

$<Au,h>=<u,A^{*}h>,\qquad\mathrm{for\;any}\;u,h\in L^{2}(\mathbb{R}^{3})).$

In the case of the _real_ vector Hilbert space we have for considered operator
${\bf A}$, taking into account (21), that

$<u,{\bf A}^{*}u>=<{\bf A}^{*}u,u>=<{\bf A}u,u>\;\leq 0,$

so adjoint operator ${\bf A}^{*}$ is _dissipative_ too.

Now we use one of the results of semigroup theory ([6], p. 84, 3.17
Corollary).

Proposition 1. _Let_ $(A,D(A))$ _be a densely defined operator on a Banach
space_ $X$. _If both_ $A$ _and its adjoint_ $A^{\prime}$ _are dissipative,
then the closure_ $\overline{A}$ _of_ $A$ _generates a contraction semigroup
on_ $X$.

A contraction semigroup is a special case of strongly continuous semigroup
$(T(t))_{t\geq 0}$ [6], [7]. In each case, the semigroup is given by

$T(t)=\exp\\{tA\\}=\sum_{n=0}^{\infty}\frac{t^{n}A^{n}}{n!},\qquad t\geq 0,$

where the power series converges for every $t\geq 0$.

So the _general_ solution to the Leray’s system (6) is as follows

$v(x,t)=\exp\\{t{\bf A}\\}u(x)=\sum_{n=0}^{\infty}\frac{t^{n}{\bf
A}^{n}}{n!}u(x),\qquad t\geq 0,$ (22)

where $u(x)$ is arbitrary vector function.

And its particular solutions

$v(x,t)=\\{\exp\\{t{\bf
A}\\}u(x)\\}\mid_{u=v_{0}}=\\{\sum_{n=0}^{\infty}\frac{t^{n}{\bf
A}^{n}}{n!}u(x)\\}\mid_{u(x)=v_{0}},\qquad t\geq 0,$ (23)

provided that $v_{0}:=v_{0}(x)$ is divergence-free vector field and the scalar
pressure $p$ restored through known solution $v$ by (5) leads us to the
solution of the Navier-Stokes equations (1)-(3).

Because the operator ${\bf A}$ is the generator of a contraction semigroup, it
follows that the semigroup yields solutions of the associated abstract Cauchy
problem ([6], p.145). Furthemore, if $u(x)\in D({\bf A})$ the map
$v(x,t):=T(t)u(x)$ is the _unique classical_ solution of the X-valued initial
value problem (or abstract Cauchy problem).

Returning to the Navier-Stokes equations, we come to the conclusion that the
operator solution of the system of equations in form (23) provided that
initial condition $v_{0}=v(x,0)$ is divergence-free is the _unique classical_
solution to (6).

That is, we have proved

Proposition 2. _Let initial conditions_ $v(x,0)$ _be any smooth rapidly
decreasing, divergence-free vector field_. _Take_ $f(x,t)$ _to be identically
zero. Then there exist unique smooth rapidly decreasing functions_ $p(x,t)$,
$v_{i}(x,t)$ _on_ $\mathbb{R}^{3}\times[0,\infty)$ _that satisfy_ (1)-(3).

I wish to thank Prof. Pierre Gilles Lemarié-Rieusset for his valuable comments
on this article.

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