# The copolarity of pseudo-cones

Rolf Schneider

###### Abstract

We show that the copolarity of pseudo-cones has analogous properties as the
usual polarity of convex bodies.
Keywords: pseudo-cone, copolarity, support function, gauge function, Gauss
curvature, centro-affine differential geometry
2020 Mathematics Subject Classification: 52A20

##  1 Introduction

For a set $K\subseteq{\mathbb{R}}^{n}$, the polar set is defined by

$K^{\circ}:=\\{u\in{\mathbb{R}}^{n}:\langle u,x\rangle\leq 1\mbox{ for all
}x\in K\\}$

where $\langle\cdot\,,\cdot\rangle$ denotes the scalar product of
${\mathbb{R}}^{n}$. Restricted to the family of compact convex sets $K$
containing the origin $o$ in the interior, polarity is an involution, that is,
it satisfies $K^{\circ\circ}=K$. Polarity of convex bodies has several
interesting and useful properties.

Within their investigation of general classes of dualities, Artstein–Avidan,
Sadovsky and Wyczesany [1] have pointed out that to the duality above there
exists a dual polarity, which has much less been studied. For
$K\subseteq{\mathbb{R}}^{n}$, the copolar set is defined by

$K^{*}:=\\{u\in{\mathbb{R}}^{n}:\langle u,x\rangle\leq-1\mbox{ for all }x\in
K\\}$

(compared to [1], we prefer a reflection in the origin). It satisfies
$K^{**}=K$ if restricted to the so-called pseudo-cones. A pseudo-cone in
${\mathbb{R}}^{n}$ is a nonempty closed convex subset
$K\subset{\mathbb{R}}^{n}$ satisfying $o\notin K$ and $\lambda K\subseteq K$
for $\lambda\geq 1$.

The name ‘copolarity’ was suggested by Rashkovskii [9], while treating a
special case. Parallel to [1, Sect. 5], but independently, a first detailed
study of copolarity (though not under this name) was undertaken by Xu, Li, and
Leng [16], who also coined the name ‘pseudo-cone’ (though with a slightly
different definition). Among several other results, they proved that
copolarity is essentially the only inclusion reversing involution on the set
of pseudo-cones, thus obtaining a counterpart to a characterization of the
classical polarity of convex bodies by Böröczky and Schneider [2].

There are, in fact, more properties of the classical polarity that have
counterparts for copolarity. A systematic investigation seems to be missing.
We mention that in [16] it has been shown that the suitably defined support
function $h_{K}$ and radial function $\varrho_{K}$ of pseudo-cones $K$ satisfy

$\varrho_{K}(x)=\frac{-1}{h_{K^{*}}(x)}$ (1)

for $x$ in the interior of the recession cone of $K$. For a pseudo-cone $K$
whose recession cone is pointed and $n$-dimensional, the exposed faces of
$K^{*}$ in relation to the exposed faces of $K$ were studied in [14]. Here it
turned out that one must distinguish whether faces are bounded or unbounded
and whether or not they are contained in the boundary of the recession cone.

It is the purpose of the following to collect some more properties of
copolarity which are in analogy to known properties of ordinary polarity.

We assume throughout that $K$ is a $C$-pseudo-cone, that is, a pseudo-cone in
${\mathbb{R}}^{n}$ with recession cone $C$, where $C$ is a pointed closed
convex cone with interior points. Then the dual cone $C^{\circ}$ has the same
properties.

##  2 A relation between support functions

We define the support function
$h_{K}:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}={\mathbb{R}}\cup\\{\infty\\}$
of $K$ by

$h_{K}(u):=\sup\\{\langle u,x\rangle:x\in K\\},\quad u\in{\mathbb{R}}^{n}.$

Then $h_{K}$ is positively homogeneous and a closed convex function (see,
e.g., [4, Thm. 2.7]). It is non-positive on $C^{\circ}$ and negative on ${\rm
int}\,C^{\circ}$. If $u\in{\mathbb{R}}^{n}\setminus C^{\circ}$, then there
exists $x\in{\rm int}\,C$ with $\langle u,x\rangle>0$. Since $\lambda x\in K$
for all sufficiently large $\lambda$, it follows that $h_{K}(u)=\infty$.

We note that, for $u\in{\mathbb{R}}^{n}$,

$u\in K^{*}\Leftrightarrow\langle u,x\rangle\leq-1\,\forall x\in
K\Leftrightarrow\sup\\{\langle u,x\rangle:x\in K\\}\leq-1\Leftrightarrow
h_{K}(u)\leq-1.$

For a convex function $f:{\mathbb{R}}^{n}\to\overline{\mathbb{R}}$, the
Legendre transform is defined by

$({\mathcal{L}}f)(u):=\sup\\{\langle
u,x\rangle-f(x):x\in{\mathbb{R}}^{n}\\}\quad\mbox{for }u\in{\mathbb{R}}^{n}.$

We would like to apply this to $\textstyle{\frac{1}{2}}h_{K}^{2}$, but this
function is not convex, and $-\textstyle{\frac{1}{2}}h_{K}^{2}$ is only convex
on $C^{\circ}$. Therefore, we define

$\widetilde{h}_{K}(u):=\left\\{\begin{array}[]{ll}-\textstyle{\frac{1}{2}}h_{K}^{2}(u)&\mbox{if
}u\in C^{\circ},\\\\[5.69054pt] \infty&\mbox{if }u\in{\mathbb{R}}^{n}\setminus
C^{\circ}.\end{array}\right.$

Then $\widetilde{h}_{K}$ is convex. In a similar way, $\widetilde{h}_{K^{*}}$
is defined. With these definitions, we can state that

${\mathcal{L}}\widetilde{h}_{K}=\widetilde{h}_{K^{*}}.$ (2)

This is analogous to a relation for the ordinary polarity (see, e.g., [13,
(1.48)]), with appropriately changed definitions.

For the proof of (2) we note that for $x\in{\mathbb{R}}^{n}$ we have

$\displaystyle h_{K^{*}}(x)$ $\displaystyle=$ $\displaystyle\sup\\{\langle
x,u\rangle:u\in K^{*}\\}=\sup\\{\langle x,u\rangle:h_{K}(u)\leq-1\\}$
$\displaystyle=$ $\displaystyle\sup\\{\langle
x,u\rangle:h_{K}(u)=-1\\}=\sup_{u\in C^{\circ}\setminus\\{o\\}}\\{\langle
x,u\rangle:h_{K}(u)=-1\\},$

since $h_{K}(u)>-1$ if $u\notin C^{\circ}\setminus\\{o\\}$. Thus

$h_{K^{*}}(x)=\sup_{u\in C^{\circ}\setminus\\{o\\}}\frac{\langle
x,u\rangle}{-h_{K}(u)},\quad x\in{\mathbb{R}}^{n}.$

Here we have used that the function $u\mapsto\langle u,x\rangle/(-h_{K}(u))$
is positively homogeneous of degree zero. The latter fact is also used below.
Observing that

$\min_{a\in{\mathbb{R}}}\\{\lambda
a+{\textstyle\frac{1}{2}}a^{2}\\}=-{\textstyle\frac{1}{2}}\lambda^{2}\quad\mbox{for
}\lambda\in{\mathbb{R}},$

we get for $x\in C$ (where $h_{K^{*}}$ is finite)

$\displaystyle\widetilde{h}_{K^{*}}(x)$ $\displaystyle=$
$\displaystyle-{\textstyle\frac{1}{2}}h^{2}_{K^{*}}(x)$ $\displaystyle=$
$\displaystyle\min_{a\in{\mathbb{R}}}\left\\{h_{K^{*}}(x)a+{\textstyle\frac{1}{2}}a^{2}\right\\}=\min_{a\in{\mathbb{R}}}\left\\{\sup_{u\in
C^{\circ}\setminus\\{o\\}}\frac{\langle
x,u\rangle}{-h_{K}(u)}a+{\textstyle\frac{1}{2}}a^{2}\right\\}$
$\displaystyle=$ $\displaystyle\min_{a\in{\mathbb{R}}}\sup_{u\in
C^{\circ}\setminus\\{o\\}}\left\\{\frac{\langle
x,u\rangle}{-h_{K}(u)}a+{\textstyle\frac{1}{2}}a^{2}:-h_{K}(u)=a\right\\}$
$\displaystyle=$ $\displaystyle\sup_{u\in
C^{\circ}\setminus\\{o\\}}\left\\{\frac{\langle
x,u\rangle}{-h_{K}(u)}a+{\textstyle\frac{1}{2}}a^{2}:-h_{K}(u)=a\right\\}$
$\displaystyle=$ $\displaystyle\sup\\{\langle
x,u\rangle+{\textstyle\frac{1}{2}}h_{K}^{2}(u):u\in C^{\circ}\\}$
$\displaystyle=$ $\displaystyle\sup\\{\langle
x,u\rangle-\widetilde{h}_{K}(u):u\in{\mathbb{R}}^{n}\\}$ $\displaystyle=$
$\displaystyle\left({\mathcal{L}}\widetilde{h}_{K}\right)(x).$

For $x\in{\mathbb{R}}^{n}\setminus C$ there exists $u\in{\rm int}\,C^{\circ}$
with $\langle x,u\rangle>0$, hence $\lambda u\in K^{*}$ for all sufficiently
large $\lambda$. This gives

$\left({\mathcal{L}}\widetilde{h}_{K}\right)(x)=\sup\\{\langle
x,u\rangle-\widetilde{h}_{K}(u):u\in{\mathbb{R}}^{n}\\}=\infty.$

Thus, (2) is proved.

##  3 A relation between gauge functions

For a convex body $L\subset{\mathbb{R}}^{n}$, containing the origin in the
interior, one defines the radial function by
$\varrho_{L}(x)=\sup\\{\lambda\geq 0:\lambda x\in L\\}$ and the gauge function
by $\|x\|_{L}=\inf\\{\lambda\geq 0:x\in\lambda L\\}$, thus

$\|x\|_{L}=\frac{1}{\varrho_{L}(x)}=h_{L^{\circ}}(x).$

In calculus of variations and in Finsler geometry (see, e.g., Rund [10]),
where often $L$ figures as the ‘indicatrix’ and $L^{\circ}$ as the
‘figuratrix‘, one prefers the notation $\|\cdot\|_{L}=F$ and
$\|\cdot\|_{L^{\circ}}=H$, and relations between the functions $F$ and $H$ are
of interest. In this section, we study analogous functions for $C$-pseudo-
cones and their relations.

We assume in this section that the $C$-pseudo-cone $K$ is smooth and strictly
convex inside ${\rm int}\,C$. This means that each point
$x\in\partial_{i}K:=\partial K\cap{\rm int}\,C$ is contained in a unique
supporting hyperplane of $K$ and that there is a supporting hyperplane $H$ of
$K$ with $H\cap K=\\{x\\}$. We recall from [14, Def. 4] that a pair
$(x,x^{*})\in{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}$ is called a crucial pair
of $K$ if $x\in\partial K$ and $x^{*}$ is an outer normal vector of $K$ at $x$
such that $\langle x,x^{*}\rangle=-1$. Every point $x\in\partial_{i}K$ belongs
to some crucial pair of $K$. If $(x,x^{*})$ is a crucial pair of $K$ with
$x\in{\rm int}\,C$, then $(x^{*},x)$ is a crucial pair of $K^{*}$, by Lemma 6
of [14]. And if $x\in{\rm int}\,C$, then $x^{*}\in{\rm int}\,C^{\circ}$, since
otherwise $K$ could not be strictly convex. Since we have assumed that $K$ is
smooth and strictly convex inside ${\rm int}\,C$, in a crucial pair
$(x,x^{*})$ of $K$ with $x\in{\rm int}\,C$ each of the vectors determines the
other one uniquely. Therefore, the same holds for $K^{*}$, so that also
$K^{*}$ is smooth and strictly convex inside $C^{\circ}$.

In the present case (and writing $\partial_{i}K^{*}:=\partial K^{*}\cap{\rm
int}\,C^{\circ}$), we can define a map

$f_{K}:\partial_{i}K\to\partial_{i}K^{*}\quad\mbox{by}\quad
f_{K}(x):=x^{*}\quad\mbox{if $(x,x^{*})$ is a crucial pair of $K$}.$

We have $f_{K^{*}}\circ f_{K}={\rm id}$ on $\partial_{i}K$ and
$f_{K}^{-1}(x^{*})=x$. We shall relate this function to the function $F$
defined below.

We define the gauge functions of $K$ and its copolar pseudo-cone $K^{*}$ by

$F(x):=\max\\{\lambda>0:x\in\lambda K\\},\quad x\in{\rm int}\,C,$
$H(x^{*}):=\max\\{\lambda>0:x^{*}\in\lambda K^{*}\\},\quad x^{*}\in{\rm
int}\,C^{\circ}.$

We have

$K\cap{\rm int}\,C=\\{x\in{\rm int}\,C:F(x)\geq 1\\}.$

Defining the radial function $\varrho_{K}$ of $K$ on ${\rm int}\,C$ by

$\varrho_{K}(x):=\min\\{\lambda>0:\lambda x\in K\\},\quad x\in{\rm int}\,C,$

we have by (1)

$F=\frac{1}{\varrho_{K}}=-h_{K^{*}}|_{{\rm int}\,C}.$

Since $K^{*}$ is strictly convex inside ${\rm int}\,C^{\circ}$, the support
function $h_{K^{*}}$ is differentiable on ${\rm int}\,C$, hence $F$ is
differentiable.

Let $X$ be a $C^{1}$ parametrization of $\partial_{i}K$, with local parameters
$u_{\alpha}$. Let $x^{1},\dots,x^{n}$ be the standard coordinates of $X$.
Since $F(X)\equiv 1$, we get

$0=\frac{\partial\frac{1}{2}F^{2}}{\partial
u_{\alpha}}(X)=\sum_{i=1}^{n}F(X)\frac{\partial F}{\partial
x^{i}}(X)\frac{\partial x^{i}}{\partial u_{\alpha}}=\langle\nabla
F(X),X_{\alpha}\rangle,$

where $\nabla$ denotes the gradient and $X_{\alpha}$ is the vector with
coordinates $\frac{\partial x^{i}}{\partial u_{\alpha}}$, $i=1,\dots,n$. Thus,
the gradient $\nabla F(X)$ is orthogonal to the tangent vectors of $X$ and
hence is proportional to the normal vector. Since $F^{2}$ is homogeneous of
degree $2$, Euler’s homogeneity relation gives

$\left\langle\left(\nabla{\textstyle\frac{1}{2}}F^{2}\right)(x),x\right\rangle=F^{2}(x),\quad
x\in{\rm int}\,C,$ (3)

thus

$\left\langle-\left(\nabla{\textstyle\frac{1}{2}}F^{2}\right)(X),X\right\rangle=-1.$

Hence, for $x\in\partial_{i}K$,

$\left(x,-\left(\nabla{\textstyle\frac{1}{2}}F^{2}\right)(x)\right)$

is a crucial pair. It follows that

$f_{K}(x)=-\nabla{\textstyle\frac{1}{2}}F^{2}(x)\quad\mbox{for
}x\in\partial_{i}K.$

Thus, the gradient mapping of $-\frac{1}{2}F^{2}$ maps $\partial_{i}K$ to
$\partial_{i}K^{*}$. Since this mapping is positively homogeneous of degree
one, it maps ${\rm int}\,C$ onto ${\rm int}\,C^{\circ}$.

We state that

$F(x)=H(x^{*})\quad\mbox{if $(x,x^{*})$ is a crucial pair of $K$}.$ (4)

For the proof, we use Remark 1.6.18 of [14] with $f=-\frac{1}{2}F^{2}$ (which
is equal to $-\frac{1}{2}h^{2}_{K^{*}}|_{{\rm int}\,C}$ and hence convex),
$D={\rm int}\,C$ and $U={\rm int}\,C^{\circ}$. Let $(x,x^{*})$ be a crucial
pair of $K$, then $(\nabla f)(x)=x^{*}$, $(\nabla f)^{-1}(x^{*})=x$, and
Euler’s relation (3) reads $\langle x,(\nabla f)(x)\rangle=2f(x)$. We obtain

$({\mathcal{L}}f)(x^{*})=\langle(\nabla f)^{-1}(x^{*}),x^{*}\rangle-f((\nabla
f)^{-1}(x^{*}))=\langle x,(\nabla f)(x)\rangle-f(x)=f(x).$ (5)

By (2) (with $K$ and $K^{*}$ interchanged) we have
${\mathcal{L}}\widetilde{h}_{K^{*}}=\widetilde{h}_{K}$. Here
$\widetilde{h}_{K^{*}}|_{{\rm int}\,C}=-\frac{1}{2}h_{K^{*}}^{2}|_{{\rm
int}\,C}=-\frac{1}{2}F^{2}$, and similarly $\widetilde{h}_{K}|_{{\rm
int}\,C^{\circ}}=-\frac{1}{2}H^{2}$, hence

${\mathcal{L}}(-\textstyle{\frac{1}{2}}F^{2})=-\textstyle{\frac{1}{2}}H^{2}\quad\mbox{on
}C^{\circ}.$

Together with (5) this gives
$-\frac{1}{2}H^{2}(x^{*})={\mathcal{L}}(-\frac{1}{2}F^{2})(x^{*})=f(x)=-\frac{1}{2}F^{2}(x)$
and hence the assertion (4).

More can be said if $F$ is twice continuously differentiable. Then we can
define

$G_{ij}(x):={\textstyle\frac{1}{2}}\frac{\partial^{2}F^{2}}{\partial
x^{i}\partial x^{j}}(x)\quad\mbox{for }x\in{\rm int}\,C.$

Since $(\partial/\partial x^{i})F^{2}$ is homogeneous of degree one, the
homogeneity relations yield

$\sum_{j=1}^{n}{\textstyle\frac{1}{2}}\frac{\partial^{2}F^{2}}{\partial
x^{i}\partial x^{j}}(x)x^{i}={\textstyle\frac{1}{2}}\frac{\partial
F^{2}}{\partial x^{i}}(x).$

Thus

$\sum_{j=1}^{n}G_{ij}x^{i}=\mbox{$i$th coordinate of
$\nabla\left({\textstyle\frac{1}{2}}F^{2}\right)(x)$}.$

In other words, if we denote by $G(x)$ the $n\times n$ matrix with entries
$G_{ij}(x)$, then

$f_{K}(x)=-G(x)x\quad\mbox{for }x\in\partial K.$

##  4 A relation between Gauss curvatures

In this section we assume that the $C$-pseudo-cone $K$ and its copolar pseudo-
cone $K^{*}$ have sufficiently smooth boundaries with positive Gauss
curvatures. Precise assumptions will be given below. For $x\in\partial K$ we
denote by $\nu_{K}(x)$ to outer unit normal vector of $K$ at $x$. If the Gauss
curvature of $\partial K$ at the boundary point $x$ is denoted by
$\kappa(K,x)$, we shall show that for crucial pairs $(x,x^{*})$,

$\frac{\langle
x,\nu_{K}(x)\rangle}{\kappa(K,x)^{\frac{1}{n+1}}}\cdot\frac{\langle
x^{*},\nu_{K^{*}}(x^{*})\rangle}{\kappa(K^{*},x^{*})^{\frac{1}{n+1}}}=1.$ (6)

The term

$\varrho_{\rm aff}(K,x)=\frac{\langle
x,\nu_{K}(x)\rangle}{\kappa(K,x)^{\frac{1}{n+1}}}$ (7)

appearing in (6) expresses in Euclidean terms what in affine differential
geometry is known as the equiaffine distance of $\partial K$ from $o$, or as
the equiaffine support function of $K$, taken at $x$. A hypersurface of class
$C^{2}_{+}$ on which the equiaffine support function is constant is known to
be an affine hypersphere. Thus we can deduce from (6) the following
observation: If $\partial K$ is an affine hypersphere, then also $\partial
K^{*}$ is an affine hypersphere. The corresponding result for ordinary
polarity of convex bodies can be found, e.g., in Salkowski [11, p. 155] (for
$n=3$) or Oliker and Simon [8, p. 107]. About (hyperbolic) affine hyperspheres
in cones and the solution of corresponding conjectures of Calabi, one can read
in Li, Simon and Zhao [7, § 2.7].

For convex bodies with a boundary of class $C^{3}_{+}$ and the ordinary
polarity, relation (6) can be found for $n=3$ in Salkowski [11, § 39, (31),
(32)] and for general $n$ in Kaltenbach [5, Sect. I.3]. We mention that for
general convex bodies and suitable sets of crucial pairs (which may be empty,
as for example in the case of polytopes), a proof has been given by Hug [3].
We can carry over Kaltenbach’s proof to pseudo-cones and copolarity, but since
we use a weaker assumption, the proof is slightly more elaborate. Here we use
the following assumption.

Assumption. The radial function of the $C$-pseudo-cone $K$ is twice
continuously differentiable.

For the proof of (6), we use local parametrizations of $\partial K$ of the
form $X=\varrho_{K}(U)U$, where $U$ is a regular local parametrization of
$\Omega_{C}$ of class $C^{2}$, where

$\Omega_{C}:=C\cap{\rm int}\,C$

is an open subsets of the unit sphere ${\mathbb{S}}^{n-1}$. Let
$N=\nu_{K}(\varrho_{K}(U)U)$, so that $N$ is a $C^{1}$ parametrization of the
outer unit normal vector of the parametrized hypersurface $X$. Denoting the
coefficients of the first and second fundamental forms of $X$ by
$g_{\alpha\beta}=\langle X_{\alpha},X_{\beta}\rangle$,
$b_{\alpha\beta}=-\langle N_{\alpha},X_{\beta}\rangle$ and their matrices by
$(g_{\alpha\beta})_{\alpha,\beta=1}^{n-1}=g$,
$(b_{\alpha\beta})_{\alpha,\beta=1}^{n-1}=b$, we have

$\kappa(K,X)=\frac{\det b}{\det g}.$

From

$\frac{\det(N,N_{1},\dots,N_{n-1})}{\det(N,X_{1},\dots,X_{n-1})}=\frac{\det(N,N_{1},\dots,N_{n-1})\det(N,X_{1},\dots,X_{n-1})}{\det(N,X_{1},\dots,X_{n-1})\det(N,X_{1},\dots,X_{n-1})}=(-1)^{n-1}\frac{\det
b}{\det g}$

we get

$\kappa(K,X)=\left|\frac{\det(N,N_{1},\dots,N_{n-1})}{\det(N,X_{1},\dots,X_{n-1})}\right|.$
(8)

The support function of the copolar pseudo-cone $K^{*}$ is given by

$h_{K^{*}}(u)=\frac{-1}{\varrho_{K}(u)}\quad\mbox{for }u\in{\rm int}\,C$

and hence is of class $C^{2}$. If $u$ is the exterior unit normal vector of
$K^{*}$ at $x^{*}$, then

$x^{*}(u)=\nabla h_{K^{*}}(u),$

by [13, Cor. 1.7.23]. Therefore,

$X^{*}=\nabla h_{K^{*}}(U)$

gives a local $C^{1}$ parametrization of $\partial K^{*}$, with the property
that $X^{*}$ and $X$ have the same local parameters if $(X,X^{*})$ is a
crucial pair of $K$.

We now use some consequences of the fact that $K^{*}$ has a support function
of class $C^{2}$ (here we differ from [5]). We apply [13, pp. 115–117], noting
that the considerations are local, so that the unboundedness of $K^{*}$ does
not make a difference.

First we note that the exterior unit normal vector of $X^{*}$, denoted by
$N^{*}$, is given by $N^{*}=U$, hence it is at least of class $C^{1}$. The
correspondence between $K$ and $K^{*}$ can now be expressed in terms of local
parametrizations, namely

$N^{*}=\frac{X}{\|X\|},\quad X^{*}=\frac{N}{-\langle X,N\rangle},\quad
X=\frac{N^{*}}{-\langle X^{*},N^{*}\rangle},\quad\|X\|=\frac{1}{-\langle
X^{*},N^{*}\rangle}.$ (9)

We denote differentiation with respect to the $\alpha$th local parameter by an
index $\alpha$. By differentiation we get

$N^{*}_{\alpha}=\frac{1}{\|X\|}\left(X_{\alpha}-\frac{\langle
X,X_{\alpha}\rangle}{\|X\|^{2}}X\right),$ (10)
$X^{*}_{\alpha}=\frac{1}{-\langle X,N\rangle}\left(N_{\alpha}-\frac{\langle
X,N_{\alpha}\rangle}{-\langle X,N\rangle}N\right).$ (11)

From [13, (2.46)] and the fact that the reciprocal Gauss curvature is the
product of the principal radii of curvature (taken, respectively, at a
boundary point and the exterior unit normal vector at this point), we obtain
that

$\kappa(K^{*},X^{*})=\left|\frac{\det e^{*}}{\det b^{*}}\right|,$

where $e^{*}$ is the matrix with entries $e^{*}_{\alpha\beta}=\langle
N_{\alpha}^{*},N_{\beta}^{*}\rangle$ and $b^{*}$ has entries $\langle
X^{*}_{\alpha},N^{*}_{\beta}\rangle$. Similarly as (8) was obtained, we deduce
that

$\kappa(K^{*},X^{*})=\left|\frac{\det(N^{*},N^{*}_{1},\dots,N^{*}_{n-1})}{\det(N^{*},X^{*}_{1},\dots,X^{*}_{n-1})}\right|.$
(12)

This looks like (8), but we could not just deduce it “by polarity”, since
$\partial K$ is not assumed to be of class $C^{3}$.

The remaining part follows [5], with the necessary changes. By (10) we obtain

$\displaystyle\det(N^{*},N^{*}_{1},\cdots,N^{*}_{n-1})$
$\displaystyle=\|X\|^{-n}\det\left(X,X_{1}-\frac{\langle
X,X_{1}\rangle}{\|X\|^{2}}X,\dots,X_{n-1}-\frac{\langle
X,X_{n-1}\rangle}{\|X\|^{2}}X\right)$
$\displaystyle=\|X\|^{-n}\det(X,X_{1},\dots,X_{n-1}).$

From this and from

$\frac{\det(X,X_{1},\dots,X_{n-1})}{\det(N,X_{1},\dots,X_{n-1})}=\frac{\det(X,X_{1},\dots,X_{n-1})\det(N,X_{1},\dots,X_{n-1})}{\det(N,X_{1},\dots,X_{n-1})\det(N,X_{1},\dots,X_{n-1})}=\frac{\langle
X,N\rangle}{1}$

we get

$\det(N^{*},N^{*}_{1},\cdots,N^{*}_{n-1})=\|X\|^{-n}\langle
X,N\rangle\det(N,X_{1},\dots,X_{n-1}).$ (13)

In a similar way, we obtain

$\displaystyle\det(N^{*},X_{1}^{*},\dots,X_{n-1}^{*})$ $\displaystyle=$
$\displaystyle\frac{1}{\langle
X^{*},N^{*}\rangle}\det(X^{*},X_{1}^{*},\dots,X_{n-1}^{*})$ $\displaystyle=$
$\displaystyle\frac{-1}{\langle X^{*},N^{*}\rangle\langle
X,N\rangle}\det(N,X_{1}^{*},\dots,X_{n-1}^{*}),$

where (9) was used. From this and (11) we get

$\det(N^{*},X_{1}^{*},\dots,X_{n-1}^{*})=\frac{(-1)^{n}}{\langle
X^{*},N^{*}\rangle\langle X,N\rangle^{n}}\det(N,N_{1},\dots,N_{n-1}).$ (14)

Observing that $1/\|X\|=-\langle X^{*},N^{*}\rangle$ by (9), from (8), (12),
(13) and (14) we obtain

$\kappa(K,X)\kappa(K^{*},X^{*})=\langle X,N\rangle^{n+1}\langle
X^{*},N^{*}\rangle^{n+1},$

which is (6).

##  5 Centro-affine fundamental forms

Under stronger smoothness assumptions on $\partial K$ and $\partial K^{*}$,
some information about the fundamental forms of centro-affine differential
geometry can be carried over from ordinary polarity to copolarity. For
ordinary polarity, these results can be found in Salkowski [11, §§ 37, 38]
(for $n=3$), Laugwitz [6, III.6], Oliker and Simon [8]. For easier comparison,
we use the classical tensor notation employed in [6, 11].

We write crucial pairs of $K$ in the form $(x,x^{*})$ (considering $x^{*}$ as
a vector of ${\mathbb{R}}^{n}$, since we have tacitly used the scalar product
to identify the dual space of ${\mathbb{R}}^{n}$ with ${\mathbb{R}}^{n}$). We
can assume that $\partial K$ is parametrized by parameters $u_{\alpha}$,
$\alpha=1,\dots,n-1$. If the local parametrization of $\partial K$ is given by
$X$, then by demanding that $(X,X^{*})$ is a crucial pair we have
automatically a local parametrization $X^{*}$ of $\partial K^{*}$. In other
words, we have $X^{*}=f_{K}\circ X$, where $f_{K}$ is the map defined in
Section 3. We assume now that both, $X$ and $X^{*}$, are of class $C^{3}$ and
have positive Gauss curvatures.

The centro-affine differential geometry of a parametrized hypersurface is
obtained as its relative differential geometry, if for the relative normal
vector one chooses the negative position vector. For the basics of relative
differential geometry, we refer to [12, Sect. 2], [15, § 1].

We denote differentiation of $X$ or $X^{*}$ with respect to the local
parameter $u_{\alpha}$ by a lower index $\alpha$. By the definition of the
crucial pair $(X,X^{*})$ we have

$\langle X^{*},X_{\alpha}\rangle=0,\quad\langle X^{*},X\rangle=-1.$

Since $-X$ is the relative normal vector, this shows that $X^{*}$ is the
covector of the tangent plane, as defined in [12], [15]. Therefore,

$G_{\alpha\beta}:=\langle X^{*},X_{\alpha\beta}\rangle$

defines the centro-affine metric tensor of $\partial K$. Denoting by ∥α
covariant differentiation with respect to the centro-affine Riemannian metric,
we also have

$G_{\alpha\beta}=\langle X^{*},X_{\alpha\|\beta}\rangle=-\langle
X^{*}_{\alpha},X_{\beta}\rangle=-\langle X^{*}_{\beta},X_{\alpha}\rangle.$

If $\overline{G}_{\alpha\beta}$ denotes the centro-affine metric tensor of
$\partial K^{*}$, then we immediately see that

$G_{\alpha\beta}=\overline{G}_{\alpha\beta}.$ (15)

The cubic fundamental form of $\partial K$ is defined by

$A_{\alpha\beta\gamma}:=\langle X^{*},X_{\alpha\|\beta\|\gamma}\rangle.$

The Ricci relation $G_{\alpha\beta\|\gamma}=0$ gives

$\langle X^{*},X_{\alpha\|\beta}\rangle_{\|\gamma}=0.$

hence

$\langle X^{*}_{\gamma},X_{\alpha\|\beta}\rangle+\langle
X^{*},X_{\alpha\|\beta\|\gamma}\rangle=0.$

Since also $\langle X^{*}_{\gamma},X_{\alpha}\rangle=-G_{\gamma\alpha}$ and
thus $\langle X^{*}_{\gamma},X_{\alpha}\rangle_{\|\beta}=0$, we get

$\langle X^{*}_{\gamma\|\beta},X_{\alpha}\rangle+\langle
X^{*}_{\gamma},X_{\alpha\|\beta}\rangle=0.$

The same relation holds with $X^{*}$ and $X$ interchanged, thus

$\langle X^{*}_{\alpha\|\beta},X_{\gamma}\rangle+\langle
X^{*}_{\alpha\|\beta\|\gamma}\rangle=0,\quad\langle
X^{*}_{\alpha},X_{\gamma\|\beta}\rangle+\langle
X^{*}_{\alpha\|\beta},X_{\gamma}\rangle=0.$

Denoting the cubic fundamental form of $\partial K^{*}$ by
$\overline{A}_{\alpha\beta\gamma}$, we obtain

$A_{\alpha\beta\gamma}=-\langle X_{\gamma},X_{\alpha\|\beta}\rangle=\langle
X^{*}_{\gamma\|\beta},X_{\alpha}\rangle=-\overline{A}_{\gamma\beta\alpha}$

and thus

$A_{\alpha\beta\gamma}=-\overline{A}_{\alpha\beta\gamma},$ (16)

since the cubic fundamental form is symmetric.

Relations (15) and (16) show that the fundamental forms of centro-affine
geometry behave under copolarity in the same way as under the ordinary
polarity.

## References

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  * [9] Rashkovskii, A., Copolar convexity. Ann. Polon. Math. 120 (2017), 83–95.
  * [10] Rund, H., The Differential Geometry of Finsler Spaces. Springer, Berlin, 1959.
  * [11] Salkowski, E., Affine Differentialgeometrie. W. de Gruyter, Berlin, 1934.
  * [12] Schneider, R., Zur affinen Differentialgeometrie im Großen. Math. Z. 101 (1967), 375–406.
  * [13] Schneider, R., Convex Bodies: The Brunn–Minkowski Theory. 2nd edn., Encyclopedia of Mathematics and Its Applications, vol. 151, Cambridge University Press, Cambridge, 2014.
  * [14] Schneider, R., Pseudo-cones. Adv in Appl. Math. 155 (2024), Paper No. 102657, 22pp.
  * [15] Simon, U., Zur Relativgeometrie: Symmetrische Zusammenhänge auf Hyperflächen. Math. Z. 106 (1968), 36–46.
  * [16] Xu, Y., Li, J., Leng, G., Dualities and endomorphisms of pseudo-cones. Adv. in Appl. Math. 142 (2023), Paper No. 102434, 31 pp.


# Quantum Magnetism in Fe2Cu2 Polymeric Branched Chains: Insights from Exactly
Solved Ising-Heisenberg Model

Dávid Sivý 1,†, Katarína Karl’ová 2 and Jozef Strečka 1,†,∗

###### Abstract

The spin-1/2 Ising-Heisenberg branched chain, inspired by the magnetic
structure of three isostructural polymeric coordination compounds
[(Tp)2Fe2(CN)6X(bdmap)Cu2(H2O)] $\cdot$ H2O to be further denoted as Fe2Cu2
(Tp = tris(pyrazolyl)hydroborate, bdmapH = 1,3-bis(dimethylamino)-2-propanol,
HX = acetic acid, propionic acid or trifluoroacetic acid), is rigorously
studied using the transfer-matrix method. The overall ground-state phase
diagram reveals three distinct phases: a quantum antiferromagnetic phase, a
quantum ferrimagnetic phase and a classical ferromagnetic phase. In the zero-
temperature magnetization curve, two quantum ground states are manifested as
intermediate plateaus at zero and half of the saturation magnetization, while
the magnetization reaches its saturated value within the classical
ferromagnetic phase. The bipartite entanglement between nearest-neighbor
Heisenberg spins is more pronounced in the quantum ferrimagnetic phase
compared to the quantum antiferromagnetic phase due to a fully polarized
nature of the Ising spins. A reasonable agreement between theoretical
predictions for the spin-1/2 Ising-Heisenberg branched chain and experimental
data measured for a temperature dependence of the magnetic susceptibility and
a low-temperature magnetization curve suggests strong antiferromagnetic
coupling between nearest-neighbor Cu2+-Cu2+ magnetic ions and moderately
strong ferromagnetic coupling between nearest-neighbor Cu2+-Fe3+ magnetic ions
in the polymeric compounds Fe2Cu2. A thermal entanglement between nearest-
neighbor Cu2+-Cu2+ magnetic ions persists up to a relatively high threshold
temperature $T\approx 224$ K and undergoes a transient magnetic-field-driven
strengthening.

##  1 Keywords:

Ising-Heisenberg Branched Chain, Quantum and Thermal Entanglement, Fractional
Magnetization Plateau, Magnetocaloric Effect, Heterobimetallic Coordination
Polymers, Exact Results

##  2 Introduction

Although one-dimensional (1D) quantum spin models may seem relatively simple
at first glance, achieving an exact solution for these lattice-statistical
models is often accompanied with formidable mathematical difficulties stemming
from noncommuting algebra [1]. As a matter of fact, there exists only a few
paradigmatic examples of exactly solved 1D quantum spin models such as the
quantum spin-1/2 Heisenberg chain solved by the Bethe-ansatz method [2, 3, 4,
5] or the quantum spin-1/2 XY model solved using the Jordan-Wigner
fermionization approach [6, 7]. It should be stressed that the exploration of
1D quantum spin models is of particular importance, because several low-
dimensional magnetic materials may have negligible exchange interactions in
the remaining two spatial dimensions [8]. It has been demonstrated that
several 1D spin chains may display intriguing quantum features such quantized
magnetization plateaus [9] and quantum entanglement [10].

1D Ising-Heisenberg models featuring regularly alternating Ising and
Heisenberg spins are conversely relatively easily exactly solvable using the
transfer-matrix method. In spite of their simplicity, these models still offer
a plausible framework for theoretical modeling of a certain class of
insulating quantum magnetic materials. The first paradigmatic experimental
realization of the spin-1/2 Ising-Heisenberg chain composed from regularly
alternating Ising spins and Heisenberg spin trimers was found by van den
Heuvel and Chibotaru during their analysis of magnetic properties of the
trimetallic 3d-4d-4f coordination polymer
[{(CuL)2Dy}{Mo(CN)8}]$\cdot$CH3CN$\cdot$H2O, where L2- =
N,N’-propylenebis(3-methoxysalicylideneiminato) [11, 12]. A simpler
experimental realization of the spin-1/2 Ising-Heisenberg chain, composed from
regularly alternating Ising and Heisenberg spins, was discovered in the
bimetallic 3d-4f coordination polymer Dy(NO3)(DMSO)2Cu(opba)(DMSO)2 (DMSO =
dimethylsulfoxide, opba = orthophenylenebisoxamato) [13, 14, 15]. The low-
temperature magnetization data of another 3d-4f heterobimetallic coordination
polymer [Dy(hfac)2(CH3OH)2Cu(dmg)(Hdmg)2]n (H2dmg = dimethylglyoxime; Hhfac =
1,1,1,5,5,5-hexafluoropentane-2,4-dione) have been successfully interpreted
within the spin-1/2 Ising-Heisenberg orthogonal-dimer chain [16, 17]. It is
worthwhile to remark that several other exactly solved Ising-Heisenberg spin
chains have provided unbiased descriptions of various intriguing quantum
phenomena including quantized magnetization plateaus [18, 19], enhanced
magnetocaloric effect [19, 20], partition function zeros [21], quantum
teleportation [22], unconventional ‘fire-and-ice’ ground states [23], pseudo-
transitions [24], quantum coherence [25], quantum nonlocality [26], quantum
Fisher information [27] as well as nanostructures [28, 29].

In recent years, considerable attention has been directed towards
investigating magnetic and quantum features of Ising-Heisenberg branched
chains. Notably, the mixed spin-(1/2,5/2,1/2) Ising-Heisenberg branched chain
has demonstrated its utility in theoretical modeling the magnetic
characteristics of the heterotrimetallic coordination polymer
[CuMn(L)][Fe(bpb)(CN)2]ClO4 $\cdot$ H2O [30, 31]. The mixed spin-(1/2,5/2,1/2)
Ising-Heisenberg branched chain provided a satisfactory explanation of the
magnetization and susceptibility behavior of this heterotrimetallic Fe-Mn-Cu
coordination polymer. Furthermore, the Fe-Mn-Cu polymeric branched chain has
been proposed as a promising platform for realizing quantum teleportation [32,
33], while also exhibiting robust quantum coherence [33] and thermal
entanglement [34]. On the other hand, the spin-1/2 Ising-Heisenberg branched
chain inspired by the magnetic structure of the bimetallic coordination
polymer [(Tp)2Fe2(CN)6(OCH3)(bap)Cu2(CH3OH) $\cdot$ 2CH3OH $\cdot$ H2O] (Tp =
tris(pyrazolyl)hydroborate, bapH = 1,3-bis(amino)-2-propanol) theoretically
predicts an intriguing breakdown of the intermediate one-half magnetization
plateau in a low-temperature magnetization curve depending basically on the
interaction anisotropy [35, 36].

Figure 1: A part of the crystal structure of the polymeric coordination
compound [(Tp)2Fe2(CN)6(CH3COO)(bdmap)Cu2(H2O)] depicted according to
crystallographic data reported in Ref. [37]. Tp stands for tris(pyrazolyl)
hydroborate and bdmapH is 1,3-bis(dimetylamino)-2-propanol. Figure 2: A
schematic illustration of the spin-1/2 Ising-Heisenberg branched chain with
the isotropic exchange interaction $J$ between nearest-neighbor Heisenberg
spins and the anisotropic exchange interaction $J_{1}$ between the nearest-
neighbor Heisenberg and Ising spins, respectively. The Heisenberg spins
capture lattice positions of Cu2+ magnetic ions (yellow circles), while the
Ising spins capture lattice positions of Fe3+ magnetic ions (blue circles).

In the present study, we will introduce a similar but structurally distinct
spin-1/2 Ising-Heisenberg branched chain to model the magnetic properties of
three isostructural polymeric coordination compounds
[(Tp)2Fe2(CN)6X(bdmap)Cu2(H2O)] as reported in Ref. [37] [Tp=tris(pyrazolyl)
hydroborate, bdmapH=1,3-bis(dimetylamino)-2-propanol and HX is acetic acid,
propionic acid or trifluoracetic acid]. The magnetic structure of these
polymeric coordination compounds, further abbreviated as Fe2Cu2, is shown in
Fig. 1 and it can be identified as another variant of spin-1/2 Ising-
Heisenberg branched chain. This characterization arises from the presence of
magnetically isotropic spin-1/2 Cu2+ magnetic ions ($S=1/2$) described by the
notion of quantum Heisenberg spins, which is in contrast to highly
magnetically anisotropic spin-1/2 Fe3+ magnetic ions ($\sigma=1/2$) described
by the notion of Ising spins. Consequently, the exchange coupling between
nearest-neighbor Cu2+-Cu2+ magnetic ions will be assumed to be of the
Heisenberg type, whereas the exchange coupling between nearest-neighbor
Fe3+-Cu2+ magnetic ions will be treated as the Ising-type exchange interaction
(see Fig. 2).

The structure of this paper is as follows. In Section 2, we introduce the
spin-1/2 Ising-Heisenberg branched chain along with a brief outline of the
method employed for its calculation. Section 3 is dedicated to discussing the
theoretical findings regarding the ground-state phase diagram, magnetization
curves, entropy, and concurrence. In Section 4, we compare the exact
theoretical results for the spin-1/2 Ising-Heisenberg branched chain with
available experimental data for the magnetization and susceptibility of the
polymeric compound Fe2Cu2 together with the respective theoretical prediction
for the concurrence. Finally, Section 5 provides a summary of the most
important scientific findings obtained in this study.

##  3 Model and method

The spin-1/2 Ising-Heisenberg branched chain schematically illustrated in Fig.
2 can be defined through the following Hamiltonian:

$\displaystyle\hat{\mathcal{H}}\\!=\\!\sum_{i=1}^{N}\\!\Bigl{\\{}J\\!\left[\Delta\\!\left(\\!\hat{S}_{1,i}^{x}\hat{S}_{2,i}^{x}\\!+\\!\hat{S}_{1,i}^{y}\hat{S}_{2,i}^{y}\\!\right)\\!+\\!\hat{S}_{1,i}^{z}\hat{S}_{2,i}^{z}\right]\\!+\\!J_{1}\\!\left[\hat{S}_{2,i}^{z}\hat{\sigma}_{2,i}^{z}\\!+\\!\hat{S}_{1,i}^{z}\\!\left(\hat{\sigma}_{1,i}^{z}\\!+\\!\hat{\sigma}_{1,i+1}^{z}\\!\right)\\!\right]\\!\\!-\\!h_{H}\\!\sum_{j=1}^{2}\\!\hat{S}_{j,i}^{z}\\!-\\!h_{I}\\!\sum_{j=1}^{2}\\!\hat{\sigma}_{j,i}^{z}\\!\Bigr{\\}},\\!\\!\\!$
(1)

where $\hat{\sigma}_{j,i}^{z}$ and $\hat{S}^{\alpha}_{j,i}$ $(\alpha=x,y,z)$
label the Ising and Heisenberg spins ascribed to $\rm{Fe}^{3+}$ and
$\rm{Cu}^{2+}$ magnetic ions, respectively. The coupling constant $J>0$
denotes the antiferromagnetic exchange interaction between the nearest-
neighbor Heisenberg spin pairs capturing lattice positions of the dimeric
$\rm{Cu}^{2+}-\rm{Cu}^{2+}$ units in the polymeric compounds Fe2Cu2, while the
coupling constant $J_{1}>0$ ($J_{1}<0$) stands for the antiferromagnetic
(ferromagnetic) exchange interaction between the nearest-neighbor Ising and
Heisenberg spins capturing lattice positions of $\rm{Fe}^{3+}$ and
$\rm{Cu}^{2+}$ magnetic ions, respectively. Furthermore, the Zeeman’s terms
$h_{H}=g_{\rm Cu}\mu_{\rm B}B$ and $h_{I}=g_{\rm Fe}\mu_{\rm B}B$ account for
the magnetostatic energy of magnetic moments ascribed to the Heisenberg and
Ising spins in an external magnetic field $B$, $g_{\rm Cu}$ and $g_{\rm Fe}$
are the respective Landé g-factors and $\mu_{\rm B}$ is Bohr magneton. The
parameter $\Delta$ measures a degree of the exchange anisotropy in the XXZ
Heisenberg interaction and $N$ denotes the total number of unit cells. For
simplicity, the periodic boundary conditions $\sigma_{1,N+1}=\sigma_{1,1}$ are
assumed.

It is convenient to rewrite the total Hamiltonian (1) as a sum of the cell
Hamiltonians $\hat{\mathcal{H}}=\sum_{i=1}^{N}\hat{\mathcal{H}}_{i}$ with the
following definition of the cell Hamiltonian $\hat{\mathcal{H}}_{i}$:

$\displaystyle\hat{{\mathcal{H}}_{i}}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!J\left[\Delta(\hat{S}_{1,i}^{x}\hat{S}_{2,i}^{x}+\hat{S}_{1,i}^{y}\hat{S}_{2,i}^{y})+\hat{S}_{1,i}^{z}\hat{S}_{2,i}^{z}\right]+J_{1}\left[\hat{S}_{2,i}^{z}\hat{\sigma}_{2,i}^{z}+\hat{S}_{1,i}^{z}(\hat{\sigma}_{1,i}^{z}+\hat{\sigma}_{1,i+1}^{z})\right]$
(2) $\displaystyle-$
$\displaystyle\\!\\!\\!h_{H}(\hat{S}_{1,i}^{z}+\hat{S}_{2,i}^{z})-\frac{h_{I}}{2}(\hat{\sigma}_{1,i}^{z}+\hat{\sigma}_{1,i+1}^{z}+2\hat{\sigma}_{2,i}^{z}).$

Owing to the fact that the cell Hamiltonians (2) commute with each other
$[\hat{\mathcal{H}}_{i},\hat{\mathcal{H}}_{j}]=0$, the partition function of
the spin-1/2 Ising-Heisenberg branched chain can be factorized into the
following product:

$\displaystyle
Z=\sum_{\\{\sigma_{1,i}\\}}\prod_{i=1}^{N}\sum_{\sigma_{2,i}}{\rm
Tr}_{i}~{}\exp\left(-\beta\hat{{\mathcal{H}}_{i}}\right),$ (3)

where $\beta=1/(k_{\rm{B}}T)$, $k_{\rm{B}}$ is the Boltzmann’s constant, $T$
is the absolute temperature, ${\rm Tr}_{i}$ stands for a trace over degrees of
freedom of the Heisenberg dimer $S_{1,i}-S_{2,i}$ from the $i$-th unit cell,
$\sum_{\sigma_{2,i}}$ denotes a summation over two states of the Ising spin
from the branching of the $i$-th unit cell, and the summation
$\sum_{\\{\sigma_{1,i}\\}}$ runs over all possible configurations of the Ising
spins from the main backbone of the branched chain. The partition function (3)
can be rewritten into a more suitable form for further calculations:

$\displaystyle Z=\sum_{\\{\sigma_{1,i}\\}}\prod_{i=1}^{N}\exp\left[\frac{\beta
h_{I}}{2}(\sigma_{1,i}+\sigma_{1,i+1})\right]\sum_{\sigma_{2,i}}\exp\left(\beta
h_{I}\sigma_{2,i}\right){\rm
Tr}_{i}~{}\exp\left(-\beta\hat{{\mathcal{H}}_{i}^{{}^{\prime}}}\right),$ (4)

where the reduced cell Hamiltonian $\hat{{\mathcal{H}}_{i}^{{}^{\prime}}}$ is
defined as:

$\displaystyle\hat{{\mathcal{H}}_{i}^{{}^{\prime}}}=J[\Delta(\hat{S}_{1,i}^{x}\hat{S}_{2,i}^{x}+\hat{S}_{1,i}^{y}\hat{S}_{2,i}^{y})+\hat{S}_{1,i}^{z}\hat{S}_{2,i}^{z}]+\sum_{j=1}^{2}h_{j,i}\hat{S}_{j,i}^{z}.$
(5)

The reduced cell Hamiltonian $\hat{{\mathcal{H}}_{i}^{{}^{\prime}}}$ given by
Eq. (5) can be alternatively viewed as the Hamiltonian of a spin-1/2
Heisenberg dimer in nonuniform effective fields $h_{j,i}$ involving the
interaction with the adjacent Ising spins:

$\displaystyle h_{1,i}=J_{1}(\sigma_{1,i}+\sigma_{1,i+1})-h_{H},\qquad
h_{2,i}=J_{1}\sigma_{2,i}-h_{H}.$ (6)

To perform a trace Tri over degrees of freedom of the spin-1/2 Heisenberg
dimer from the $i$-th unit cell in the relevant expression for the partition
function (4), it is convenient to proceed to a matrix representation of the
reduced cell Hamiltonian (5) in the standard basis:

$\displaystyle\left|\varphi_{1,i}\right\rangle=\left|\uparrow\right\rangle_{1,i}\left|\uparrow\right\rangle_{2,i},\quad\left|\varphi_{2,i}\right\rangle=\left|\uparrow\right\rangle_{1,i}\left|\downarrow\right\rangle_{2,i},\quad\left|\varphi_{3,i}\right\rangle=\left|\downarrow\right\rangle_{1,i}\left|\uparrow\right\rangle_{2,i},\quad\left|\varphi_{4,i}\right\rangle=\left|\downarrow\right\rangle_{1,i}\left|\downarrow\right\rangle_{2,i}.$
(7)

The state vectors $\left|\uparrow\right\rangle_{j,i}$ and
$\left|\downarrow\right\rangle_{j,i}$ are eigenvectors of the $z$-component of
the Heisenberg spin $S_{j,i}^{z}=1/2$ and $-1/2$, respectively. After solving
the relevant eigenvalue problem
$\left|\left\langle\varphi_{k,i}\right|\hat{{\mathcal{H}}_{i}^{{}^{\prime}}}\left|\varphi_{l,i}\right\rangle-
E_{j,i}{\rm\textbf{I}}\right|=0$ one gets four eigenvalues of the reduced cell
Hamiltonian (5):

$\displaystyle E_{1,i}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\frac{J}{4}-\frac{h_{1,i}}{2}-\frac{h_{2,i}}{2},\qquad\qquad\qquad\qquad\quad
E_{2,i}=\frac{J}{4}+\frac{h_{1,i}}{2}+\frac{h_{2,i}}{2},$ $\displaystyle
E_{3,i}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!-\frac{J}{4}+\frac{1}{2}\sqrt{\left(h_{1,i}-h_{2,i}\right)^{2}+\left(J\Delta\right)^{2}},\qquad
E_{4,i}=-\frac{J}{4}-\frac{1}{2}\sqrt{\left(h_{1,i}-h_{2,i}\right)^{2}+\left(J\Delta\right)^{2}},$
(8)

with the corresponding eigenvectors:

$\displaystyle\left|\psi_{1,i}\right\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\left|\uparrow\right\rangle_{1,i}\left|\uparrow\right\rangle_{2,i},\quad\left|\psi_{2,i}\right\rangle=\left|\downarrow\right\rangle_{1,i}\left|\downarrow\right\rangle_{2,i},\quad\left|\psi_{3,i}\right\rangle=A_{i}^{+}\left|\uparrow\right\rangle_{1,i}\left|\downarrow\right\rangle_{2,i}+A_{i}^{-}\left|\downarrow\right\rangle_{1,i}\left|\uparrow\right\rangle_{2,i},$
$\displaystyle\left|\psi_{4,i}\right\rangle\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!A_{i}^{-}\left|\uparrow\right\rangle_{1,i}\left|\downarrow\right\rangle_{2,i}-A_{i}^{+}\left|\downarrow\right\rangle_{1,i}\left|\uparrow\right\rangle_{2,i},$
(9)

where the probability amplitudes $A_{i}^{\mp}$ are given by:

$\displaystyle
A_{i}^{\mp}=\frac{1}{\sqrt{2}}\sqrt{1\\!\mp\\!\frac{J_{1}(\sigma_{1,i}\\!+\\!\sigma_{1,i+1}\\!-\\!\sigma_{2,i})}{\sqrt{J^{2}_{1}(\sigma_{1,i}\\!+\\!\sigma_{1,i+1}\\!-\\!\sigma_{2,i})^{2}\\!+\\!(J\Delta)^{2}}}}.$
(10)

With the use of the eigenvalues (8) one can further rewrite the partition
function (4) into the following simplified form:

$\displaystyle
Z=\sum_{\\{\sigma_{1,i}\\}}\prod_{i=1}^{N}{\rm\textbf{T}}(\sigma_{1,i},\sigma_{1,i+1}),$
(11)

where the effective Boltzmann’s factor
${\rm\textbf{T}}(\sigma_{1,i},\sigma_{1,i+1})$ is defined as follows:

$\displaystyle{\rm\textbf{T}}(\sigma_{1,i},\sigma_{1,i+1})\\!\\!\\!$
$\displaystyle=$ $\displaystyle\\!\\!\\!\exp\left[\frac{\beta
h_{I}}{2}(\sigma_{1,i}+\sigma_{1,i+\\!1})\right]\sum_{\sigma_{2,i}}\exp\left(\beta
h_{I}\sigma_{2,i}\right)\sum_{j=1}^{4}\exp\left(-\beta E_{j,i}\right)$ (12)
$\displaystyle=$ $\displaystyle\\!\\!\\!\exp\left[\frac{\beta
h_{I}}{2}(\sigma_{1,i}\\!+\\!\sigma_{1,i+\\!1})\right]\Biggl{\\{}\\!\exp\left[-\frac{\beta}{4}(J\\!+\\!J_{1})\right]\\!\cosh\left[\beta\left(\frac{J_{1}}{2}(\sigma_{1,i}\\!+\\!\sigma_{1,i+\\!1}\\!)\\!-\\!h_{H}\\!-\\!\frac{h_{I}}{2}\\!\right)\right]\Biggr{.}$
$\displaystyle+$
$\displaystyle\\!\\!\\!\Biggl{.}\exp\left[\frac{\beta}{4}(-J+J_{1})\right]\cosh\left[\beta\left(\frac{J_{1}}{2}(\sigma_{1,i}+\sigma_{1,i+1})-h_{H}+\frac{h_{I}}{2}\right)\right]\Biggr{.}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\Biggl{.}\exp\left[\frac{\beta
J}{4}+\frac{\beta
h_{I}}{2}\right]\cosh\left[\frac{\beta}{2}\sqrt{\left(J_{1}(\sigma_{1,i}+\sigma_{1,i+1})-\frac{J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right]\Biggr{.}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\Biggl{.}\exp\left[\frac{\beta
J}{4}-\frac{\beta
h_{I}}{2}\right]\\!\cosh\left[\frac{\beta}{2}\sqrt{\left(J_{1}(\sigma_{1,i}\\!+\\!\sigma_{1,i+1})\\!+\\!\frac{J_{1}}{2}\right)^{2}\\!\\!+\\!(J\Delta)^{2}}\right]\Biggr{\\}}.$

The effective Boltzmann’s weight (12) depends only on degrees of freedom of
two Ising spins $\sigma_{1,i}$ and $\sigma_{1,i+1}$ and it can be
alternatively viewed as the transfer matrix:

$\displaystyle\\!\\!\\!\\!\rm{\textbf{T}}(\sigma_{1,i},\sigma_{1,i+1})=\begin{pmatrix}{\rm
T}\left(+\frac{1}{2},+\frac{1}{2}\right)\\!&\\!{\rm
T}\left(+\frac{1}{2},-\frac{1}{2}\right)\\\ {\rm
T}\left(-\frac{1}{2},+\frac{1}{2}\right)\\!&\\!{\rm
T}\left(-\frac{1}{2},-\frac{1}{2}\right)\end{pmatrix}=\begin{pmatrix}{\rm
T}_{1}\\!&\\!{\rm T}_{3}\\\ {\rm T}_{3}\\!&\\!{\rm T}_{2}\end{pmatrix},$

where ${\rm T}_{i}$ ($i=1,2,3$) denote three different elements of the
transfer matrix obtained from Eq. (12) by considering all four available
states of two Ising spins $\sigma_{1,i}$ and $\sigma_{1,i+1}$:

$\displaystyle{\rm T}_{1}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!2\exp\left(\frac{\beta h_{I}}{2}\right)\Biggl{\\{}{\rm
e}^{-\frac{\beta}{4}(J\\!+\\!J_{1})}\cosh\left[\beta\left(\frac{J_{1}}{2}\\!-\\!h_{H}\\!-\\!\frac{h_{I}}{2}\right)\right]+{\rm
e}^{\frac{\beta}{4}(-J+J_{1})}\cosh\left[\beta\left(\frac{J_{1}}{2}-h_{H}+\frac{h_{I}}{2}\right)\right]\Biggr{.}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\Biggl{.}\exp\left(\frac{\beta
J}{4}\right)\left[{\rm e}^{\frac{\beta
h_{I}}{2}}\cosh\left(\frac{\beta}{2}\sqrt{\left(\frac{J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right)+{\rm
e}^{-\frac{\beta
h_{I}}{2}}\cosh\left(\frac{\beta}{2}\sqrt{\left(\frac{3J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right)\right]\Biggr{\\}},$
$\displaystyle{\rm T}_{2}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!2\exp\left(-\frac{\beta h_{I}}{2}\right)\Biggl{\\{}{\rm
e}^{-\frac{\beta}{4}(J\\!+\\!J_{1})}\cosh\left[\beta\left(\frac{J_{1}}{2}\\!+\\!h_{H}\\!+\\!\frac{h_{I}}{2}\right)\right]+{\rm
e}^{\frac{\beta}{4}(-J+J_{1})}\cosh\left[\beta\left(-\frac{J_{1}}{2}-h_{H}+\frac{h_{I}}{2}\right)\right]\Biggr{.}$
$\displaystyle+$ $\displaystyle\\!\\!\\!\Biggl{.}\exp\left(\frac{\beta
J}{4}\right)\left[{\rm e}^{\frac{\beta
h_{I}}{2}}\cosh\left(\frac{\beta}{2}\sqrt{\left(\frac{3J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right)+{\rm
e}^{-\frac{\beta
h_{I}}{2}}\cosh\left(\frac{\beta}{2}\sqrt{\left(\frac{J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right)\right]\Biggr{\\}},$
$\displaystyle{\rm T}_{3}\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!2\Biggl{\\{}\exp\left[-\frac{\beta}{4}(J\\!+\\!J_{1})\right]\cosh\left[\beta\left(h_{H}\\!+\\!\frac{h_{I}}{2}\right)\right]+\exp\left[\frac{\beta}{4}(-J+J_{1})\right]\cosh\left[\beta\left(-h_{H}+\frac{h_{I}}{2}\right)\right]\Biggr{.}$
(14) $\displaystyle+$ $\displaystyle\\!\\!\\!\Biggl{.}2\exp\left(\frac{\beta
J}{4}\right)\cosh\left(\frac{\beta}{2}\sqrt{\left(\frac{J_{1}}{2}\right)^{2}+(J\Delta)^{2}}\right)\cosh\left(\frac{\beta
h_{I}}{2}\right)\Biggr{\\}}.$

In the spirit of the transfer-matrix method, the final expression for the
partition function can be finally expressed in terms of the transfer-matrix
eigenvalues:

$\displaystyle
Z=\sum_{\\{\sigma_{1,i}\\}}\prod_{i=1}^{N}{\rm\textbf{T}}(\sigma_{1,i},\sigma_{1,i+1})=\sum_{\sigma_{1,1}=\pm\frac{1}{2}}{\rm\textbf{T}}^{N}(\sigma_{1,1},\sigma_{1,1})={\rm
Tr}~{}{\rm\textbf{T}}^{N}=\rm{\lambda}_{+}^{N}+\rm{\lambda}_{-}^{N}.$ (15)

The transfer matrix (LABEL:TransMatrix) can be readily diagonalized in order
to obtain the eigenvalues $\rm{\lambda}_{\pm}$ defined through the transfer-
matrix elements (14):

$\displaystyle\lambda_{\pm}=\frac{1}{2}\left[{\rm T}_{1}+{\rm
T}_{2}\pm\sqrt{({\rm T}_{1}-{\rm T}_{2})^{2}+4{\rm T}_{3}^{2}}\right].$ (16)

In the thermodynamic limit $N\to\infty$, the Gibbs free energy of the spin-1/2
Ising-Heisenberg branched chain per unit cell is given by the largest
transfer-matrix eigenvalue:

$\displaystyle g=-k_{\rm B}T\lim_{N\to\infty}\frac{1}{N}\ln Z=-k_{\rm
B}T\ln\lambda_{+}.$ (17)

From the Gibbs free energy (17) one can subsequently derive the local
magnetization of the Heisenberg and Ising spins as well as the total
magnetization (per spin) of the spin-1/2 Ising-Heisenberg branched chain:

$\displaystyle
m_{H}=\frac{1}{2}\left\langle\hat{S}_{1,i}^{z}+\hat{S}_{2,i}^{z}\right\rangle=-\frac{1}{2}\left(\frac{\partial
g}{\partial h_{H}}\right)_{T}\\!\\!\\!,\quad
m_{I}=\frac{1}{2}\left\langle\hat{\sigma}_{1,i}^{z}+\hat{\sigma}_{2,i}^{z}\right\rangle=-\frac{1}{2}\left(\frac{\partial
g}{\partial h_{I}}\right)_{T}\\!\\!\\!,\quad m=\frac{m_{H}+m_{I}}{2}.$ (18)

From the exact result (17) for the Gibbs free energy one can eventually derive
other quantities such as magnetic susceptibility, entropy and specific heat
using the standard relations of thermodynamics. Besides, one may also
investigate a strength of the bipartite entanglement inside of the Heisenberg
dimers using the quantity concurrence [38], which can be calculated from the
pair correlation functions and the local magnetization [39, 40]:

$\displaystyle\\!\\!\\!\\!C=\\!\max\\!\left\\{0,4\left|C^{xx}\right|\\!-\\!2\sqrt{\left(\frac{1}{4}\\!+\\!C^{zz}\right)^{2}\\!-\left(m_{H}\right)^{2}}\right\\}.$

Here, $C^{xx}=\left\langle\hat{S}_{1,i}^{x}\hat{S}_{2,i}^{x}\right\rangle$ and
$C^{zz}=\left\langle\hat{S}_{1,i}^{z}\hat{S}_{2,i}^{z}\right\rangle$ denote
two spatial components of the pair correlation function between the nearest-
neighbor Heisenberg spins and $m_{H}$ is the respective local magnetization
given by Eq. (18). The pair correlation functions can be straightforwardly
derived from the Gibbs free energy (17) according to the relations:

$\displaystyle C^{xx}=-\frac{1}{2N}\frac{\partial\ln Z}{\partial(\beta
J\Delta)},\quad C^{zz}=-\frac{1}{N}\frac{\partial\ln Z}{\partial(\beta J)},$
(20)

but their explicit form is too cumbersome to write it down here explicitly.

##  4 Theoretical results

Let us proceed to a discussion of the most interesting results obtained for
the spin-1/2 Ising-Heisenberg branched chain by assuming the isotropic
antiferromagnetic Heisenberg coupling constant $J>0$, while keeping the
anisotropy parameter fixed at the constant value of $\Delta=1$ for
illustrative purposes. Throughout this section, the coupling constant $J>0$
will serve as an energy unit when defining a relative strength of the magnetic
field $h/J$, the dimensionless temperature $k_{\rm B}T/J$ and the interaction
ratio $J_{1}/J$. To reduce the number of free parameters, the Landé
$g$-factors of the Heisenberg and Ising spins assigned to ${\rm Cu}^{2+}$ and
${\rm Fe}^{3+}$ magnetic ions were set equal to each other, thereby ensuring
that the ’local’ magnetic fields exert an equal strength $h\equiv
h_{H}=h_{I}$.

Figure 3: A schematic illustration of typical configurations realized within
all possible ground states of the spin-1/2 Ising-Heisenberg branched chain:
(a)-(b) the quantum antiferromagnetic phases $\left|I\right\rangle$ and
$\left|I^{\prime}\right\rangle$; (c)-(d) the quantum ferrimagnetic phases
$\left|II\right\rangle$ and $\left|II^{\prime}\right\rangle$; (e) the
classical ferromagnetic phase $\left|III\right\rangle$. The pair of arrows
situated at the Heisenberg spins symbolizes a singlet-like quantum
superposition, whereby the size of arrows reflects different magnitudes of the
respective probability amplitudes (24). Figure 4: (a) The ground-state phase
diagram for the spin-1/2 Ising-Heisenberg branched chain in the $J_{1}/J-h/J$
plane by assuming the isotropic value of the parameter $\Delta=1$. The regions
I(I’), II(II’) and III correspond to a quantum antiferromagnetic, a quantum
ferrimagnetic and a classical ferromagnetic phase, respectively; (b)-(d)
Magnetic-field dependencies of the local and total magnetization given by Eq.
(18) for the spin-1/2 Ising-Heisenberg branched chain at three different
temperatures and three selected values of the interaction ratio: (b)
$J_{1}/J=-1.5$; (c) $J_{1}/J=-0.5$; (d) $J_{1}/J=1.5$.

By comparing energies of all available eigenstates of the spin-1/2 Ising-
Heisenberg branched chain one finds just three different ground states
schematically shown in Fig. 3 and mathematically given by the eigenvectors:

$\displaystyle\left|I,I^{\prime}\right\rangle=\prod_{i=1}^{N}\left(a_{-}\left|\uparrow\right\rangle_{S_{1,i}}\left|\downarrow\right\rangle_{S_{2,i}}-a_{+}\left|\downarrow\right\rangle_{S_{1,i}}\left|\uparrow\right\rangle_{S_{2,i}}\right)\left|\uparrow\right\rangle_{\sigma_{1,i}}\left|\downarrow\right\rangle_{\sigma_{2,i}},$
(21)
$\displaystyle\left|II,II^{\prime}\right\rangle=\prod_{i=1}^{N}\left(b_{-}\left|\uparrow\right\rangle_{S_{1,i}}\left|\downarrow\right\rangle_{S_{2,i}}-b_{+}\left|\downarrow\right\rangle_{S_{1,i}}\left|\uparrow\right\rangle_{S_{2,i}}\right)\left|\uparrow\right\rangle_{\sigma_{1,i}}\left|\uparrow\right\rangle_{\sigma_{2,i}},$
(22)
$\displaystyle\left|III\right\rangle=\prod_{i=1}^{N}\left|\uparrow\right\rangle_{S_{1,i}}\left|\uparrow\right\rangle_{S_{2,i}}\left|\uparrow\right\rangle_{\sigma_{1,i}}\left|\uparrow\right\rangle_{\sigma_{2,i}}.$
(23)

While the factorized form of the third ground state (23) is consistent with
the classical nature of the fully polarized ferromagnetic state
$\left|III\right\rangle$, the quantum antiferromagnetic phase
$\left|I,I^{\prime}\right\rangle$ and the quantum ferrimagnetic phase
$\left|II,II^{\prime}\right\rangle$ described by Eqs. (21) and (22) exhibit
bipartite quantum entanglement among the Heisenberg spin pairs
$S_{1,i}$-$S_{2,i}$ unambiguously characterized through the probability
amplitudes $a_{\pm}$ and $b_{\pm}$:

$\displaystyle
a_{\pm}=\frac{1}{\sqrt{2}}\sqrt{1\pm\frac{\frac{3}{2}\frac{J_{1}}{J}}{\sqrt{\left(\frac{3}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}}},\qquad
b_{\pm}=\frac{1}{\sqrt{2}}\sqrt{1\pm\frac{\frac{1}{2}\frac{J_{1}}{J}}{\sqrt{\left(\frac{1}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}}}.$
(24)

From this perspective, it would be insightful to compare the strength of the
bipartite quantum entanglement emerging within the quantum antiferromagnetic
(21) and ferrimagnetic (22) phases using the concept of the concurrence
(LABEL:Concurrence), which acquires within these two ground states the
following values:

$\displaystyle
C_{\left|I,I^{\prime}\right\rangle}=\frac{\Delta}{\sqrt{\left(\frac{3}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}},\qquad
C_{\left|II,II^{\prime}\right\rangle}=\frac{\Delta}{\sqrt{\left(\frac{1}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}}.$
(25)

It is obvious from Eq. (25) that the bipartite quantum entanglement of the
Heisenberg spin pairs is consistently stronger within the quantum
ferrimagnetic phase $\left|II,II^{\prime}\right\rangle$ compared to the
quantum antiferromagnetic phase $\left|I,I^{\prime}\right\rangle$ for a given
set of the interaction ratio $J_{1}/J$ and the exchange anisotropy $\Delta$,
because a fully polarized nature of the enclosing Ising spins tends to bring a
singlet-like state closer to a perfect singlet-dimer state as one of the Bell
states.

For the sake of completeness, let us also quote eigenenergies (per unit cell)
corresponding to the quantum antiferromagnetic phase
$\left|I,I^{\prime}\right\rangle$

$\displaystyle
E_{I/I^{\prime}}=-\frac{J}{4}-\frac{1}{2}\sqrt{\left(\frac{3J_{1}}{2}\right)^{2}+(J\Delta)^{2}},$
(26)

the quantum ferrimagnetic phase $\left|II,II^{\prime}\right\rangle$

$\displaystyle
E_{II/II^{\prime}}=-\frac{J}{4}-\frac{1}{2}\sqrt{\left(\frac{J_{1}}{2}\right)^{2}+(J\Delta)^{2}}-h,$
(27)

the classical ferromagnetic phase $\left|III\right\rangle$

$\displaystyle E_{III}=\frac{J}{4}+\frac{3J_{1}}{4}-2h.$ (28)

It should be stressed that the ground states $\left|I\right\rangle$ and
$\left|I^{\prime}\right\rangle$ have the same energy and they just differ by a
relative orientation of the singlet-like state of the Heisenberg spin pairs
with respect to their neighboring Ising spins [see Fig. 3(a)-(b)] depending on
whether the Ising coupling constant is antiferromagnetic $J_{1}>0$ or
ferromagnetic $J_{1}<0$. The same statement holds true also for the quantum
ferrimagnetic phases $\left|II\right\rangle$ and
$\left|II^{\prime}\right\rangle$ schematically drawn in Fig. 3(c)-(d),
respectively. A direct comparison of all three ground-state energies (26)-(28)
affords exact expressions for the relevant ground-state phase boundaries:

1\. phase boundary $\left|I\right\rangle$/$\left|II\right\rangle$ and
$\left|I^{\prime}\right\rangle$/$\left|II^{\prime}\right\rangle$:

$\displaystyle\frac{h}{J}=\frac{1}{2}\left[\sqrt{\left(\frac{3}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}-\sqrt{\left(\frac{1}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}}\right],$
(29)

2\. phase boundary $\left|I^{\prime}\right\rangle$/$\left|III\right\rangle$:

$\displaystyle\frac{h}{J}=\frac{1}{4}+\frac{3}{8}\frac{J_{1}}{J}+\frac{1}{4}\sqrt{\left(\frac{3}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}},$
(30)

3\. phase boundary $\left|II\right\rangle$/$\left|III\right\rangle$ and
$\left|II^{\prime}\right\rangle$/$\left|III\right\rangle$:

$\displaystyle\frac{h}{J}=\frac{1}{2}+\frac{3}{4}\frac{J_{1}}{J}+\frac{1}{2}\sqrt{\left(\frac{1}{2}\frac{J_{1}}{J}\right)^{2}+\Delta^{2}},$
(31)

4\. phase boundary $\left|II\right\rangle$/$\left|II^{\prime}\right\rangle$:

$\displaystyle J_{1}/J=0.$ (32)

Using these exact expressions, we have plotted in Fig. 4(a) the overall
ground-state phase diagram of the spin-1/2 Ising-Heisenberg branched chain in
the $J_{1}/J-h/J$ plane. The parameter regions denoted in Fig. 4(a) as I(I’),
II(II’) and III correspond to the quantum antiferromagnetic (QAF), the quantum
ferrimagnetic (QFI) and the classical ferromagnetic (CF) phase, respectively.
Note furthermore that the coexistence lines (29)-(31) correspond to
discontinuous magnetic-field-driven phase transitions. All three phases QAF,
QFI and CF coexist together in Fig. 4(a) at a triple coexistence point
emerging for the specific value of the interaction ratio
$J_{1}/J\approx-0.97$. Below this value, the QFI ground state
$\left|II^{\prime}\right\rangle$ is absent and only the two ground states
$\left|I^{\prime}\right\rangle$ and $\left|III\right\rangle$ appear.

To verify this observation, we present magnetic-field dependencies of the
local and total magnetization in Fig. 4(b) for the interaction ratio
$J_{1}/J=-1.5$ and three distinct temperature values. Under this condition,
the local magnetization $m_{H}$ and $m_{I}$ defined as a single-site
magnetization of the Heisenberg and Ising spins are zero until the saturation
field $h_{s}/J\approx 0.3$ is reached. This consequently leads to a zero
magnetization plateau of the total magnetization $m$, which is consistent with
the antiferromagnetic alignment of the Ising spins and the singlet-like
ordering of the Heisenberg spin pairs realized within the QAF ground state
$\left|I^{\prime}\right\rangle$ [see Fig. 3(b)]. The observed single
magnetization jump can thus be attributed to a discontinuous magnetic-field-
driven phase transition from the QAF ground state
$\left|I^{\prime}\right\rangle$ to the CF ground state
$\left|III\right\rangle$. It should be also noted that the true magnetization
jump exists at zero temperature only, which however changes to a steep but
continuous rise upon increasing of temperature.

Next, the local and total magnetization of the spin-1/2 Ising-Heisenberg
branched chain are plotted against the magnetic field in Fig. 4(c) for another
value of the interaction ratio $J_{1}/J=-0.5$ and three distinct temperatures.
The spin-1/2 Ising-Heisenberg branched chain undergoes in this particular case
two magnetic-field-driven phase transitions: initially from the QAF ground
state $\left|I^{\prime}\right\rangle$ towards the QFI ground state
$\left|II^{\prime}\right\rangle$ subsequently followed by a transition from
the QFI ground state $\left|II^{\prime}\right\rangle$ towards the CF ground
state $\left|III\right\rangle$. In agreement with this statement, the local
magnetization of the Ising spins $m_{I}$ does not contribute to the total
magnetization $m$ only at sufficiently low magnetic fields $h/J\lesssim 0.11$
stabilizing the QAF ground state $\left|I^{\prime}\right\rangle$ and then it
undergoes sudden rise towards the saturated value acquired within the QFI
ground state $\left|II^{\prime}\right\rangle$ and the CF ground state
$\left|III\right\rangle$. On the contrary, the local magnetization of the
Heisenberg spins $m_{H}$ does not contribute to the total magnetization $m$
unless the magnetic field exceeds its saturation value $h_{s}/J\approx 0.64$.

Finally, we illustrate in Fig. 4(d) the local and total magnetization of the
spin-1/2 Ising-Heisenberg branched chain for a fixed value of the interaction
ratio $J_{1}/J=1.5$. The behavior observed here closely resembles that of the
previous cases, with two intermediate magnetization plateaus and two magnetic-
field-induced transitions: first from the QAF phase $\left|I\right\rangle$
towards the QFI phase $\left|II\right\rangle$, followed by a transition from
the QFI phase $\left|II\right\rangle$ towards the CF phase
$\left|III\right\rangle$. The primary quantitative difference lies in a
somewhat wider intermediate one-half magnetization plateau corresponding to
the QFI phase $\left|II\right\rangle$, which generally broadens with an
increase in the interaction ratio $J_{1}/J$.

Figure 5: A density plot of the reduced entropy $S/Nk_{\rm B}$ of the spin-1/2
Ising-Heisenberg branched chain in the magnetic field versus temperature plane
$h/J-k_{\rm B}T/J$ for three selected values of the interaction ratio: (a)
$J_{1}/J=-1.5$, (b) $J_{1}/J=-0.5$, and (c) $J_{1}/J=1.5$.

An independent validation of the aforementioned magnetization scenario is
provided through a comprehensive analysis of the magnetic entropy, as depicted
in Fig. 5 in the form of a density plot in the magnetic field versus
temperature plane ($h/J$ \- $k_{\rm B}T/J$). The isoentropy lines shown in
Fig. 5 bring further insight into the adiabatic demagnetization process of the
spin-1/2 Ising-Heisenberg branched chain. It is evident from Fig. 5 that a
sharp decrease (or increase) in temperature is observed upon decreasing (or
increasing) the magnetic field above (below) each magnetic-field-driven phase
transition. Consistent with this observation, the isoentropy lines tend
towards absolute zero temperature at a single magnetic field for the specific
case $J_{1}/J=-1.5$, owing to the absence of the quantum ferrimagnetic phase
$|II^{\prime}\rangle$ at moderate magnetic fields [refer to Fig. 5(a)].
Conversely, for the other two specific cases, $J_{1}/J=-0.5$ and $1.5$, the
isoentropy lines approach absolute zero temperature at two distinct magnetic
fields due to the presence of the quantum ferrimagnetic phase
$|II^{\prime}\rangle$ at moderate magnetic fields.

Figure 6: Temperature dependence of the initial magnetic susceptibility times
the temperature $\chi k_{\rm B}T/J$ of the spin-1/2 Ising-Heisenberg branched
chain for a few different values of the interaction ratio $J_{1}/J$.

The temperature dependence of the initial magnetic susceptibility times
temperature product is illustrated in Fig. 6 for various values of the
interaction ratio $J_{1}/J$. It is noteworthy that the spin-1/2 Ising-
Heisenberg branched chain exhibits the quantum antiferromagnetic (QAF) ground
state in the zero-field limit for any value of the interaction ratio $J_{1}/J$
[refer to Fig. 4(a)]. This aligns with the vanishing behavior of the
susceptibility times temperature product as temperature approaches zero. It is
apparent from Fig. 6 that the product of susceptibility and temperature
gradually tends to zero as temperature decreases for the interaction ratio
$J_{1}/J=1.5$. For the interaction ratio $J_{1}/J=-0.25$, the product $\chi T$
contrarily initially declines, followed by maintaining a nearly constant value
below the temperature $k_{\rm B}T/J\lesssim 0.25$, before rapidly decreasing
to zero as absolute zero temperature is approached. The notable temperature
dependence with a nearly constant value of the susceptibility times
temperature product observed at moderate temperatures is attributed to the
ferromagnetic nature of the Ising coupling constant $J_{1}<0$, which has
opposite tendency of increasing of the product $\chi T$ upon decreasing of
temperature. Indeed, the reinforcement of the ferromagnetic coupling constant
$J_{1}$ leads to an increase in the susceptibility times temperature product
as temperature decreases, as evidenced in Fig. 6 for specific values of the
interaction ratio $J_{1}/J=-1.5$ and $-2.3$, where the product $\chi T$
initially reaches a rounded maximum before ultimately approaching zero as
temperature decreases.

Figure 7: Magnetic-field (a)-(c) and temperature (d)-(f) dependencies of the
concurrence (LABEL:Concurrence) calculated for the Heisenberg spin pairs of
the spin-1/2 Ising-Heisenberg branched chain at three different values of the
interaction ratio: (a) and (d) $J_{1}/J=-1.5$; (b) and (e) $J_{1}/J=-0.5$; (c)
and (f) $J_{1}/J=1.5$.

To better understand the quantum nature of the spin-1/2 Ising-Heisenberg
branched chain, we conduct a detailed analysis of the concurrence
(LABEL:Concurrence), which quantifies the strength of bipartite entanglement
within the Heisenberg spin pairs. In Fig. 7(a), we illustrate the magnetic-
field dependence of the concurrence for a fixed interaction ratio
$J_{1}/J=-1.5$ and three distinct temperatures. At sufficiently low magnetic
fields, the concurrence is non-zero due to the presence of the quantum
antiferromagnetic (QAF) ground state $\left|I^{\prime}\right\rangle$,
characterized by singlet-like states of the Heisenberg dimers. This state
breaks down at the magnetic-field-induced phase transition $h_{c}/J\approx
0.3$, leading to the emergence of the fully polarized classical ferromagnetic
(CF) ground state $\left|III\right\rangle$. Fig. 7(a) demonstrates that
increasing temperature generally smears out the stepwise magnetic-field
dependence of the concurrence. The corresponding thermal variations of the
concurrence are depicted in Fig. 7(d) for three selected magnetic field
values. At zero magnetic field, the concurrence exhibits a standard monotonous
decline with increasing temperature until it completely vanishes at a
threshold temperature. However, a more intricate temperature dependence is
observed for the magnetic field $h/J=0.3$, around which the magnetic-field-
driven phase transition occurs at zero temperature. Under this condition, the
concurrence undergoes an abrupt decrease to a local minimum at the lowest
temperatures, followed by a gradual rise to a local maximum at moderate
temperatures, before eventually tending to zero with further temperature
increase. Moreover, the concurrence initially starts from zero due to the CF
ground state $\left|III\right\rangle$ at the magnetic field $h/J=1$, which
exceeds the saturation value. However, the concurrence exhibits a striking
double-reentrant behavior, developing first at a lower threshold temperature
$k_{\rm{B}}T/J\approx 0.2$, then reaching a rounded maximum at a moderate
temperature, before returning to zero at an upper threshold temperature
$k_{\rm{B}}T/J\approx 0.85$. This double-reentrant behavior is attributed to
the thermal population of the QAF phase $\left|I^{\prime}\right\rangle$.

Next, we depict in Fig. 7(b) the concurrence as a function of the magnetic
field for a fixed interaction ratio $J_{1}/J=-0.5$ and three distinct
temperatures. In contrast to the previous case, the spin-1/2 Ising-Heisenberg
branched chain encompasses at zero temperature all three possible ground
states: QAF, QFI, and CF. Consequently, the concurrence initially attains a
relatively large value ($C\approx 0.8$) due to the QAF ground state
$\left|I^{\prime}\right\rangle$ realized at sufficiently low magnetic fields
($h/J<0.11$). This is followed by an almost maximal value ($C\approx 0.97$)
attributed to the QFI ground state $\left|II^{\prime}\right\rangle$, before
the concurrence abruptly drops to zero near the saturation field
($h_{s}/J\approx 0.64$) at very low temperature ($k_{\rm{B}}T/J=0.01$), as
observed in Fig. 7(b). Furthermore, Fig. 7(b) reveals that the concurrence may
become non-zero above the saturation field ($h_{s}/J\approx 0.64$) at slightly
higher temperatures ($k_{\rm{B}}T/J=0.1$ and 0.3) due to thermal excitations
from the fully polarized CF ground state $\left|III\right\rangle$ to the
quantum states $\left|II^{\prime}\right\rangle$ and
$\left|I^{\prime}\right\rangle$. To gain a comprehensive understanding, Fig.
7(e) illustrates the concurrence against temperature for the same value of the
interaction ratio $J_{1}/J=-0.5$ and three different magnetic field values. At
zero magnetic field, the concurrence starts from a finite value ($C\approx
0.8$) corresponding to the QAF ground state, then steadily rises to reach a
rounded maximum before gradually tending to zero. The prominent rounded peak
of the concurrence can be attributed to thermal excitations from the less
entangled QAF phase to the QFI phase with stronger entanglement. If the
magnetic field $h/J=0.6$ favors the QFI ground state, the concurrence sharply
drops from its almost maximal value within a narrow temperature range
($k_{\rm{B}}T/J\in(0;0.08)$) and then decreases steadily before showing a more
abrupt thermally-induced decline. At a higher magnetic field ($h/J=1$)
exceeding the saturation value, the concurrence exhibits a double-reentrant
behavior quite analogous to the previously described case for $J_{1}/J=-1.5$.

Last but not least, we have depicted in Fig. 7(c) and (f) the magnetic-field
and temperature dependencies of the concurrence for the fixed value of the
interaction ratio $J_{1}/J=1.5$. It is evident that the results shown in Fig.
7(c) and (f) are qualitatively very similar to those displayed in Fig. 7(b)
and (e) for the other specific case $J_{1}/J=-0.5$. The main quantitative
difference lies in the strength of bipartite entanglement, i.e., the magnitude
of concurrence, which generally weakens with an increase in the absolute value
of the interaction ratio $|J_{1}|/J$. Notably, the concurrence attains the
same zero-temperature asymptotic value within the QAF ground states
$\left|I\right\rangle$ and $\left|I^{\prime}\right\rangle$ for two specific
values of the interaction ratio $J_{1}/J=1.5$ and $-1.5$, respectively.

##  5 Magnetic data of polymeric compound Fe2 Cu2:
theory vs. experiment

In this section, we will take advantage of the spin-1/2 Ising-Heisenberg
branched-chain model described by the Hamiltonian (1) to theoretically
interpret available magnetic data experimentally measured for the
heterobimetallic coordination polymer $[(\rm Tp)_{2}\rm Fe_{2}(\rm
CN)_{6}$(CH3COO)$(\rm bdmap)\rm Cu_{2}(\rm H_{2}\rm O)]$ to be further
abbreviated as Fe2Cu2, which serves as its experimental realization [37]. The
experimental data for the temperature dependence of the magnetic
susceptibility times temperature product $\chi_{M}T$ recorded at a magnetic
field of $B=0.2$ T [37] are confronted in Fig. 8(a) with the best theoretical
fit obtained using the spin-1/2 Ising-Heisenberg branched-chain model. The
least-squares fitting procedure yielded a rather strong antiferromagnetic
coupling constant $J/k_{\rm B}=247.8$ K between nearest-neighbor Cu2+-Cu2+
magnetic ions, a moderately strong ferromagnetic coupling constant
$J_{1}/k_{\rm B}=-64.2$ K between nearest-neighbor Cu2+-Fe3+ magnetic ions,
and Landé $g$-factors $g_{\rm Cu}=2.16$ and $g_{\rm Fe}=2.23$ for Cu2+ and
Fe3+ magnetic ions, respectively. A moderate value of the interaction ratio
$J_{1}/J\approx-0.26$ contributes to a relatively rapid decline in the
susceptibility times temperature product as temperature decreases in a high-
temperature regime $T\gtrsim 50$ K. This is successively followed by a less
steep decline observed in a range of moderate temperatures 20 K $\lesssim
T\lesssim$ 50 K before it finally rapidly falls to zero with further
temperature reduction. Note furthermore that the steep reduction of the
susceptibility times temperature product observed in the low-temperature range
is consistent with the quantum antiferromagnetic (QAF) ground state
$|I^{\prime}\rangle$.

Figure 8: Temperature dependence of the molar magnetic susceptibility
$\chi_{M}$ times temperature $T$ data at the magnetic field $B=0.2$ T (a) and
the isothermal magnetization curve of the polymeric coordination compound
$[(\rm Tp)_{2}\rm Fe_{2}(\rm CN)_{6}$(CH3COO)$(\rm bdmap)\rm Cu_{2}(\rm
H_{2}\rm O)]$ recorded at temperature $T=1.8~{}\rm{K}$ (b). Blue squares
represent experimental data reported in Ref. [37], while green curves are the
respective theoretical fits obtained with the help of the spin-1/2 Ising-
Heisenberg branched chain model.

Utilizing the determined set of fitting interaction parameters, we have also
conducted theoretical modeling of the low-temperature magnetization curve of
the polymeric compound Fe2Cu2, as reported in Ref. [37], for sufficiently low
temperatures $T=1.8~{}\rm{K}$ up to a magnetic field of $B=5~{}\rm{T}$. The
magnetization data recorded for the coordination polymer Fe2Cu2 are confronted
in Fig. 8(b) with the corresponding theoretical fit based on the spin-1/2
Ising-Heisenberg branched chain model, calculated up to an ultrahigh magnetic
field of $150~{}\rm{T}$ surpassing the saturation field. The spin-1/2 Ising-
Heisenberg branched chain model provides a relatively satisfactory fit to the
experimental magnetization data. Moreover, an extrapolation to higher magnetic
fields predicts the presence of a sizable intermediate one-half magnetization
plateau in the magnetic field range between $B\approx 10~{}\rm{T}$ and
$140~{}\rm{T}$. This intermediate magnetization plateau reflects the emergence
of the quantum ferrimagnetic phase $|II^{\prime}\rangle$ accompanied by a
stronger bipartite entanglement of the Heisenberg spin pairs. It is worthwhile
to mention that a small deviation between the theoretical and experimental
data can be attributed to the Ising approximation of a highly (but not
infinitely) anisotropic exchange coupling between Cu2+ and Fe3+ magnetic ions.

Figure 9: (a) Temperatures dependencies of the concurrence (LABEL:Concurrence)
calculated for the Cu2+-Cu2+ dimeric units using the spin-1/2 Ising-Heisenberg
branched chain for several values of the magnetic field $B$ (the inset shows a
detail from the low-field region); (b) A density plot of the concurrence
(LABEL:Concurrence) calculated for the Cu2+-Cu2+ dimeric units of the spin-1/2
Ising-Heisenberg branched chain in the magnetic field vs. temperature plane.
Displayed lines correspond to isocontour lines of the concurrence. All
calculations are presented for the set of fitting parameters $J/k_{\rm
B}=247.8~{}K$, $J_{1}/k_{\rm B}=-64.2~{}K$, $g_{Cu}=2.16$ and $g_{Fe}=2.23$
obtained for the polymeric coordination compound $[(\rm Tp)_{2}\rm Fe_{2}(\rm
CN)_{6}$(CH3COO)$(\rm bdmap)\rm Cu_{2}(\rm H_{2}\rm O)]$.

To shed light on the strength of bipartite entanglement within the Cu2+-Cu2+
dimeric units of the polymeric chain Fe2Cu2, we utilized the determined
fitting parameters of the corresponding spin-1/2 Ising-Heisenberg branched
chain to calculate the concurrence (LABEL:Concurrence). Temperature
dependencies of the concurrence (LABEL:Concurrence) calculated for the
Heisenberg spin pairs of the spin-1/2 Ising-Heisenberg branched chain are
illustrated in Fig. 9(a) for a few selected values of the magnetic field $B$.
At zero magnetic field, the concurrence exhibits a striking temperature-
induced rise almost to its maximum possible value, after which it gradually
declines with increasing temperature until it completely vanishes at the
threshold temperature $T\approx 224~{}\rm{K}$. A small magnetic field can be
considered as another driving force that may reinforce the strength of
bipartite entanglement. Indeed, it turns out that the concurrence approaches
almost its maximum value in the zero-temperature asymptotic limit when a small
magnetic field is applied. If the magnetic field exceeds the saturation field,
a double reentrant behavior of the concurrence is observed.

Last but not least, a density plot of the concurrence (LABEL:Concurrence)
calculated for the Heisenberg spin pairs of the spin-1/2 Ising-Heisenberg
branched chain is presented in Fig. 9(b) in the magnetic field versus
temperature plane. This density plot can be interpreted as a phase diagram,
delineating the entangled parameter region from the disentangled one.
Remarkably, the density plot in Fig. 9(b) reveals that a relatively high value
of the concurrence ($C=0.1$) can still be detected even up to relatively high
temperatures of around $T\approx 188~{}\rm{K}$ irrespective of the accessible
magnetic-field strength for experimental testing ($B<10~{}\rm{T}$). This
suggests the persistence of significant bipartite entanglement in the system
over a wide temperature range, which could be of interest for experimental
investigations.

##  6 Conclusion

In this paper, we have investigated the ground-state phase diagram,
magnetization process, magnetocaloric effect, and bipartite entanglement of
the spin-1/2 Ising-Heisenberg branched chain with two different coupling
constants using the transfer-matrix method. Our analysis reveals that the
spin-1/2 Ising-Heisenberg branched chain exhibits five distinct ground states:
two quantum antiferromagnetic phases $\left|I\right\rangle$ and
$\left|I^{\prime}\right\rangle$, two quantum ferrimagnetic phases
$\left|II\right\rangle$ and $\left|II^{\prime}\right\rangle$, and the
classical ferromagnetic phase $\left|III\right\rangle$. These ground states
are separated by discontinuous magnetic-field-driven phase transitions,
whereby the ground states $\left|I,I^{\prime}\right\rangle$ and
$\left|II,II^{\prime}\right\rangle$ are responsible for zero and one-half
magnetization plateaus, respectively. Furthermore, our exact calculation of
the concurrence has confirmed the quantum character of the ground states
$\left|I,I^{\prime}\right\rangle$ and $\left|II,II^{\prime}\right\rangle$. We
find that the strength of bipartite entanglement within the quantum
ferrimagnetic phases $\left|II,II^{\prime}\right\rangle$ is stronger compared
to that within the quantum antiferromagnetic phases
$\left|I,I^{\prime}\right\rangle$. This insight contributes to a deeper
understanding of the entanglement properties of the system and its magnetic
behavior.

It has been demonstrated that the spin-1/2 Ising-Heisenberg branched chain
with two different coupling constants provides a satisfactory description of
the magnetic behavior observed in the heterobimetallic polymeric chain Fe2Cu2.
By fitting available experimental data for the temperature dependence of the
susceptibility times temperature product and the low-temperature magnetization
curve, we have determined a rather strong coupling constant $J/k_{\rm
B}=247.8~{}K$ between the Cu2+-Cu2+ magnetic ions and a moderately strong
coupling constant $J_{1}/k_{\rm B}=-64.2~{}K$ between the Cu2+-Fe3+ magnetic
ions with subtle differences in the Landé $g$-factors $g_{Cu}=2.16$ and
$g_{Fe}=2.23$. Our analysis reveals that the quantum antiferromagnetic phase
is realized at sufficiently low magnetic fields ($B\lesssim 10$ T), while the
quantum ferrimagnetic phase, characterized by a sizable intermediate one-half
magnetization plateau, can be detected in the magnetic-field range between
$B\approx 10~{}\rm{T}$ and $140~{}\rm{T}$. The substantial coupling constant
between the Cu2+-Cu2+ dimeric units further implies a robust bipartite
entanglement between the nearest-neighbor Cu2+-Cu2+ magnetic ions. This
entanglement persists up to a relatively high threshold temperature ($T\approx
224$ K) and undergoes a transient magnetic-field-driven strengthening.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

DS: software, investigation, formal analysis, methodology, data curation,
validation, visualization, and writing–original draft. KK: conceptualization,
investigation, formal analysis, methodology, data curation, validation,
supervision, writing–review and editing. JS: conceptualization, investigation,
formal analysis, funding acquisition, methodology, data curation, validation,
visualization, supervision, project administration, writing–original draft,
writing–review and editing. DS and JS contributed equally to this work. DS and
JS share first and last authorship.

## Funding

This work was financially supported by the grant of the Slovak Research and
Development Agency provided under the contract No. APVV-22-0172 and by the
grant of The Ministry of Education, Science, Research, and Sport of the Slovak
Republic provided under the contract No. VEGA 1/0695/23. This project has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
945380.

## Data Availability Statement

The datasets presented in this article are not readily available. The data are
available upon reasonable request from the corresponding author.

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