let $F:{\mathcal{M}}\to\operatorname{Conv}({\bf R}^{2})$. Given constants
$\lambda_{1},\lambda_{2}\geq 0$, $\lambda_{1}\leq\lambda_{2}$, let us assume
that, for each $x\in{\mathcal{M}}$, the sets $F^{[1]}[x:\lambda_{1}]$ and
$F^{[2]}[x:\lambda_{1},\lambda_{2}]$ are non-empty. Then,

$\displaystyle
F^{[2]}[x:\lambda_{1},\lambda_{2}]=F^{[1]}[x:\lambda_{1}]\cap{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]~{}~{}~{}~{}\text{for
every}~{}~{}~{}x\in{\mathcal{M}}.$ (7.26)

Proof. We note that $F^{[2]}[x:\lambda_{1},\lambda_{2}]\subset
F^{[1]}[x:\lambda_{1}]$, see (7.4). Furthermore,

${\mathcal{T}}_{F,\lambda_{1}}(z)={\mathcal{H}}[F^{[1]}[z:\lambda_{1}]]\supset
F^{[1]}[z:\lambda_{1}],~{}~{}~{}z\in{\mathcal{M}},$

so that

${\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]=\bigcap_{z\in{\mathcal{M}}}\big{\\{}{\mathcal{T}}_{F,\lambda_{1}}(z)+\lambda_{2}\rho(x,z)Q_{0}\big{\\}}\supset\bigcap_{z\in{\mathcal{M}}}\big{\\{}F^{[1]}[z:\lambda_{1}]+\lambda_{2}\rho(x,z)Q_{0}\big{\\}}.$

From this and definition (7.11), we have

$\displaystyle{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]\supset
F^{[2]}[x:\lambda_{1},\lambda_{2}]$ (7.27)

proving that the right hand side of (7.26) contains its left hand side.

Let us prove the converse statement. We will need two auxiliary results. Here
is the first of them: Let $C_{1},C_{2}\in\operatorname{Conv}({\bf R}^{2})$,
and let $r\geq 0$. Suppose that $C_{1}\cap C_{2}\neq\emptyset$. Then

$\displaystyle C_{1}\cap
C_{2}+rQ_{0}=(C_{1}+rQ_{0})\cap(C_{2}+rQ_{0})\cap{\mathcal{H}}[C_{1}\cap
C_{2}+rQ_{0}].$ (7.28)

The second auxiliary result states the following: Let
${\mathcal{K}}\subset\operatorname{Conv}({\bf R}^{2})$ be a finite family of
closed convex sets in ${\bf R}^{2}$ with non-empty intersection, and let
$r\geq 0$. Then

$\displaystyle\left(\,\bigcap_{K\in{\mathcal{K}}}K\right)+rQ_{0}=\bigcap_{K,K^{\prime}\in{\mathcal{K}}}\left\\{(K\cap
K^{\prime})+rQ_{0}\right\\}.$ (7.29)

These result were proved in [34] under the conditions
$C_{1},C_{2}\in{\mathcal{K}}({\bf R}^{2})$ and
${\mathcal{K}}\subset{\mathcal{K}}({\bf R}^{2})$ respectively. (See [34, Lemma
3.4, Lemma 5.4].) Obvious changes in the proofs of these results show that
these property are satisfied for every $C_{1},C_{2}\in\operatorname{Conv}({\bf
R}^{2})$ and any finite family ${\mathcal{K}}\subset\operatorname{Conv}({\bf
R}^{2})$.

Property (7.29) tells us that, for every $z\in{\mathcal{M}}$, we have

$\displaystyle F^{[1]}[z:\lambda_{1}]+\lambda_{2}\rho(x,z)Q_{0}$
$\displaystyle=$
$\displaystyle\left(\bigcap_{z\in{\mathcal{M}}}\big{\\{}F(u)+\lambda_{1}\rho(z,u)Q_{0}\big{\\}}\right)+\lambda_{2}\rho(x,z)Q_{0}$
$\displaystyle=$
$\displaystyle\bigcap_{u,u^{\prime}\in{\mathcal{M}}}\left[\\{F(u)+\lambda_{1}\rho(z,u)Q_{0}\\}\cap\\{F(u^{\prime})+\lambda_{1}\rho(z,u^{\prime})Q_{0}\\}+\lambda_{2}\rho(x,z)Q_{0}\right]$
$\displaystyle=$
$\displaystyle\bigcap_{u,u^{\prime}\in{\mathcal{M}}}A(x:z,u,u^{\prime})$

where

$A(x:z,u,u^{\prime})=\\{F(u)+\lambda_{1}\rho(z,u)Q_{0}\\}\cap\\{F(u^{\prime})+\lambda_{1}\rho(z,u^{\prime})Q_{0}\\}+\lambda_{2}\rho(x,z)Q_{0}.$

Thanks to property (7.28), for every $z,u,u^{\prime}\in{\mathcal{M}}$, we have

$A(x:z,u,u^{\prime})=\\{F(u)+(\lambda_{1}\rho(z,u)+\lambda_{2}\rho(x,z))Q_{0}\\}\cap\\{F(u^{\prime})+(\lambda_{1}\rho(z,u^{\prime})+\lambda_{2}\rho(x,z))Q_{0}\\}\cap{\mathcal{H}}[A(x:z,u,u^{\prime})].$

Recall that $0\leq\lambda_{1}\leq\lambda_{2}$. From this, the triangle
inequality and (7.10), we have

$\displaystyle S_{1}$ $\displaystyle=$ $\displaystyle
F(u)+(\lambda_{1}\rho(z,u)+\lambda_{2}\rho(x,z))Q_{0}$ $\displaystyle\supset$
$\displaystyle F(u)+\lambda_{1}(\rho(z,u)+\rho(x,z))Q_{0}\supset
F(u)+\lambda_{1}\rho(x,u)Q_{0}\supset F^{[1]}[x:\lambda_{1}].$

In the same way we prove that

$S_{2}=F(u^{\prime})+(\lambda_{1}\rho(z,u^{\prime})+\lambda_{2}\rho(x,z))Q_{0}\supset
F^{[1]}[x:\lambda_{1}].$

Let us show that the set ${\mathcal{H}}[A(x:z,u,u^{\prime})]$ contains
${\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]$. Indeed, thanks to
(7.10),

$G=\\{F(u)+\lambda_{1}\rho(z,u)Q_{0}\\}\cap\\{F(u^{\prime})+\lambda_{1}\rho(z,u^{\prime})Q_{0}\\}\supset
F^{[1]}[z:\lambda_{1}],$

so that
${\mathcal{H}}[G]\supset{\mathcal{H}}[F^{[1]}[z:\lambda_{1}]]={\mathcal{T}}_{F,\lambda_{1}}(z)$.
From this and (2.14), we have

$\displaystyle S_{3}$ $\displaystyle=$
$\displaystyle{\mathcal{H}}[A(x:z,u,u^{\prime})]={\mathcal{H}}[G+\lambda_{2}\rho(x,z)Q_{0}]={\mathcal{H}}[G]+\lambda_{2}\rho(x,z)Q_{0}$
$\displaystyle\supset$
$\displaystyle{\mathcal{T}}_{F,\lambda_{1}}(z)+\lambda_{2}\rho(x,z)Q_{0}\supset{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]$

proving the required inclusion
$S_{3}\supset{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]$.

These properties of the sets $S_{1},S_{2}$ and $S_{3}$ imply the following:

$A(x:z,u,u^{\prime})=S_{1}\cap S_{2}\cap S_{3}\supset
F^{[1]}[x:\lambda_{1}]\cap{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]~{}~{}~{}\text{for
every}~{}~{}~{}z,u,u^{\prime}\in{\mathcal{M}}.$

Thus, for every $z\in{\mathcal{M}}$, we have

$F^{[1]}[z:\lambda_{1}]+\lambda_{2}\rho(x,z)Q_{0}=\bigcap_{u,u^{\prime}\in{\mathcal{M}}}A(x:z,u,u^{\prime})\supset
F^{[1]}[x:\lambda_{1}]\cap{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}].$

Thanks to this property and definition (7.2), we get

$F^{[2]}[x:\lambda_{1},\lambda_{2}]=\bigcap_{z\in{\mathcal{M}}}\big{\\{}F^{[1]}[z:\lambda_{1}]+\lambda_{2}\,\rho(x,z)\,Q_{0}\big{\\}}\supset
F^{[1]}[x:\lambda_{1}]\cap{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]$

proving that the left hand side of (7.26) contains its right hand side.

The proof of the lemma is complete. $\blacksquare$

###### Remark 7.10

Let us note the following interesting representation of the mapping
$F^{[2]}[\cdot:\lambda_{1},\lambda_{2}]$: in the settings of Lemma 7.9, for
every $x\in{\mathcal{M}}$, the following is true

$\displaystyle
F^{[2]}[x:\lambda_{1},\lambda_{2}]=F^{[1]}[x:\lambda_{1}]\cap{\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]=F^{[1]}[x:\lambda_{1}]\cap{\mathcal{H}}[F^{[2]}[x:\lambda_{1},\lambda_{2}]].$
(7.30)

This is immediate from Lemma 7.9 and inclusion (7.27). Indeed, because
${\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]$ is a rectangle, from
(7.27) we have
${\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]\supset{\mathcal{H}}[F^{[2]}[x:\lambda_{1},\lambda_{2}]]$.
This and (7.26) imply the second equality in (7.30).

Note that, in general,

${\mathcal{T}}^{[1]}_{F,\lambda_{1}}[x:\lambda_{2}]\supsetneqq{\mathcal{H}}[F^{[2]}[x:\lambda_{1},\lambda_{2}]]~{}~{}~{}\text{on}~{}~{}~{}{\mathcal{M}}.$

However, thanks to (7.30), intersection of these rectangles with
$F^{[1]}[x:\lambda_{1}]$ produces the same set, i.e., the set
$F^{[2]}[x:\lambda_{1},\lambda_{2}]$. $\blacktriangleleft$

Proof of Proposition 7.8. We know that
$\Lambda^{({\mathcal{F}}{\mathcal{P}})}(F)<\infty$ and
$\lambda\in[\Lambda^{({\mathcal{F}}{\mathcal{P}})}(F),+\infty)$. Therefore,
thanks to Theorem 7.5, the $(\lambda;F)$-Iterative Algorithm 7.3 produces the
outcome “Success” and returns the mapping
$f^{[\lambda;F]}:{\mathcal{M}}\to{\bf R}^{2}$ defined by formula (7.14). Its
Lipschitz seminorm is bounded by $\gamma\lambda$ where $\gamma$ is an absolute
constant.

Thanks to part ($\bigstar{\mathscr{A}}$) of Theorem 7.5, in this case, the
sets $F^{[1]}[x:\lambda]$ and $F^{[2]}[x:\lambda,3\lambda]$ are non-empty for
every $x\in{\mathcal{M}}$. Furthermore, Lemma 7.9 tells us that in this case,
for every $x\in{\mathcal{M}}$, the following equality

$\displaystyle
F^{[2]}[x:\lambda,3\lambda]=F^{[1]}[x:\lambda]\cap{\mathcal{T}}^{[1]}_{F,\lambda}[x:3\lambda]$
(7.31)

holds. Recall that, given $x\in{\mathcal{M}}$,

$\displaystyle{\mathcal{T}}^{[1]}_{F,\lambda}[x:3\lambda]=\bigcap_{z\in{\mathcal{M}}}\big{\\{}{\mathcal{T}}_{F,\lambda}(z)+3\lambda\,\rho(x,z)\,Q_{0}\big{\\}}$
(7.32)

and ${\mathcal{T}}_{F,\lambda}(x)={\mathcal{H}}[F^{[1]}[x:\lambda]]$.

Let us see how the mapping $f^{[\lambda;F]}$ can be constructed in $O(N^{2})$
running time.

At the first step of this procedure, for every $x\in{\mathcal{M}}$, we
construct the rectangle ${\mathcal{T}}_{F,\lambda}(x)$. Because
$F:{\mathcal{M}}\to{\mathcal{H}}{\mathcal{P}}({\bf R}^{2})$ and
$\\#{\mathcal{M}}=N$, the convex polygon $F^{[1]}[x:\lambda]$ is determined by
$O(N)$ linear constrains. See (7.10).

Part (i) of Corollary 6.19 tells us that there exists an algorithm which, for
every $x\in{\mathcal{M}}$, constructs the set ${\mathcal{T}}_{F,\lambda}(x)$,
the rectangular hull of $F^{[1]}[x:\lambda]$, in $O(N)$ running time.

Thus, at this step we are able to construct all $N$ rectangles
${\mathcal{T}}_{F,\lambda}(x)$ in $O(N^{2})$ running time.

At the second step of the procedure, following (7.32), we construct each
rectangle ${\mathcal{T}}^{[1]}_{F,\lambda}[x:3\lambda]$, $x\in{\mathcal{M}}$,
in $O(N)$ running time. Thus, the constructing all $N$ rectangles
${\mathcal{T}}^{[1]}_{F,\lambda}[x:3\lambda]$ will require at most $O(N^{2})$
computer operations.

We turn to the third step of the procedure. Thanks to representation (7.31),
each set $F^{[2]}[x:\lambda,3\lambda]$ is a convex polygon determined by at
most $N+4$ linear constraints. Part (ii) of Corollary 6.19 tells us that there
exists an algorithm which, for every $x\in{\mathcal{M}}$, calculates the
distance from $O$ to $F^{[2]}[x:\lambda,3\lambda]$ (i.e., the quantity
$r_{x}$, see (7.13)) using at most $O(N)$ computer operations. Thus,
calculating all $N$ numbers $r_{x}$, $x\in{\mathcal{M}}$, can be proceed in
$O(N^{2})$ running time.

At the fourth step of the procedure, we treat the sets
${\mathscr{F}}[x:\lambda]=F^{[2]}[x:\lambda,3\lambda]\cap Q(0,2r_{x})$,
$x\in{\mathcal{M}}$, see (7.12). Because each $F^{[2]}[x:\lambda,3\lambda]$ is
determined by at most $N+4$ linear constraints, the set
${\mathscr{F}}[x:\lambda]$ is a convex polygon determined by at most $N+8$
linear constraints. Therefore, thanks to part (i) of Corollary 6.19, there
exists an algorithm which, for every $x\in{\mathcal{M}}$, constructs the set
$\Pi_{\lambda,F}(x)$, the rectangular hull of ${\mathscr{F}}[x:\lambda]$ (see
(7.15)), in $O(N)$ running time.

Thus, at this step we construct all $N$ rectangles $\Pi_{\lambda,F}(x)$ using
at most $O(N^{2})$ computer operations.

Finally, at the fifth step of the procedure, for every $x\in{\mathcal{M}}$, we
construct the point $f^{[\lambda;F]}(x)$ as the center of the rectangle
$\Pi_{\lambda,F}(x)$. See formula (7.14). Because the number of all such
rectangles is $N$, this procedure will take $4N$ computer operations.

Thus, we construct the mapping $f^{[\lambda;F]}:{\mathcal{M}}\to{\bf R}^{2}$
in five steps, and each of these steps requires at most $O(N^{2})$ computer
operations. This shows that $f^{[\lambda;F]}$ can be constructed in $O(N^{2})$
running time completing the proof of Proposition 7.8. $\blacksquare$

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Dimensionality significantly affects exciton production and condensation.
Despite the report of excitonic instability in one-dimensional materials, it
remains unclear whether these spontaneously produced excitons can form Bose-
Einstein condensates. In this work, we first prove statistically that one-
dimensional condensation exists when the spontaneously generated excitons are
thought of as an ideal neutral Bose gas, which is quite different from the
inability of free bosons to condense. We then derive a general expression for
the critical temperature in different dimensions and find that the critical
temperature increases with decreasing dimension. We finally predict by first-
principles $GW$-BSE calculations that experimentally accessible single-chain
staircase Scandocene and Chromocene wires are an antiferromagnetic spin-
triplet excitonic insulator and a ferromagnetic half-excitonic insulator,
respectively.

# One-Dimensional Magnetic Excitonic Insulators

Jing Liu  Key Lab of advanced optoelectronic quantum architecture and
measurement (MOE), and Advanced Research Institute of Multidisciplinary
Science, Beijing Institute of Technology, Beijing 100081, China  These authors
contribute equally to this work.  Hongwei Qu  Key Lab of advanced
optoelectronic quantum architecture and measurement (MOE), and Advanced
Research Institute of Multidisciplinary Science, Beijing Institute of
Technology, Beijing 100081, China  These authors contribute equally to this
work.  Yuanchang Li<EMAIL_ADDRESS>Key Lab of advanced optoelectronic
quantum architecture and measurement (MOE), and Advanced Research Institute of
Multidisciplinary Science, Beijing Institute of Technology, Beijing 100081,
China

In the 1960s, theoretical physicists propose a new state of matter called the
excitonic insulator, which has a many-body ground state similar to the
superconducting stateMott ; Knox ; Kohn ; Rice . Its onset is accompanied by
spontaneous symmetry breaking, may leading to interesting physical phenomena
such as superfluidityEisenstein ; Jiangtri  and electron ferroelectricitySham
; Batista . Intertwining with magnetic and topological order also gives rise
to novel quantum states, including single-spin exciton condensatesJianghf ,
spin-triplet excitonic insulatorsJiangtri ; Liufeng ; Mazzone  and topological
excitonic insulatorsDurr ; Daniele ; Wusanfeng ; Cobden ; DongTEI ; LiXZ .
However, to date, there is no universally accepted evidence that excitonic
insulators exist in real-life materials.

The formation of excitonic insulators involves two aspects: (1) excitonic
instability, i.e., spontaneous generation of excitons when the exciton binding
energy exceeds the one-electron excitation gap, and (2) exciton condensation,
i.e., the macroscopic occupation of the ground state by these spontaneously
generated excitons after undergoing a quantum statistical phase transition.
Both aspects are sensitive to dimensionality, but the dependence tends to be
opposite. On the one hand, dimensionality reduction weakens the electron-hole
screening effect and enhances their binding, thus promoting exciton
generation. As such, low-dimensional excitonic insulators have the potential
to outperform the traditional material choices of small-gap semiconductors or
small-band-overlap semimetals. First-principles studies have indeed shown that
excitonic instability is still possible in two-dimensional (2D) semiconductors
with gaps up to $\sim$3 eVJianghf ; Jiangtri . It is for this reason that the
rise of 2D materials in recent years has renewed interest in excitonic
insulatorsusEI ; usDong ; DongAlSb ; Daniele ; Wusanfeng ; Cobden ; LxzNJP ;
DongTEI ; LiXZ . On the other hand, however, exciton condensation in low
dimensions seems to be ruled out by the Hohenberg-Mermin-Wagner
theoremHohenberg ; Mermin , which states the absence of long-range order in 1D
and 2D homogeneous systems at non-zero temperature due to quantum or thermal
fluctuations. Based on a similar application of the Bogoliubov inequality,
this was subsequently extended by Walker to the nonexistence of 1D and 2D
excitonic insulatorsWalker . It is worth noting that all these proofs were
obtained under idealized conditions, e.g., in the thermodynamic limit. With
the continuous progress of low-dimensional physics, there have been different
levels of theoretical models and experimental approaches that go beyond the
limits of the Hohenberg-Mermin-Wagner theorem, as exemplified by the
realization of stable ferromagnetic order in 2D Cr2Ge2Te6Gong  and CrI3Huang .

In this work, we focus on 1D excitonic insulators. Physically, 1D quantum
many-body systems are of great scientific importance because theoretical
descriptions of systems of interacting particles are very difficult and, in
some cases, exact solutions exist only in one dimension. Yet, 1D excitonic
insulators have historically received little attentionRiceJPCM ; Rice9 ;
Rice11 ; VarsanoNC ; Hellgren ; MnCp ; CrBz ; Barborini  compared to 2D ones.
In terms of material platforms, excitonic instabilities have been reported for
conjugated polymerRice9 ; Rice11 , carbon nanotubesVarsanoNC ; Hellgren  and
organometallic wiresMnCp ; CrBz . In terms of Bose-Einstein condensation, by
mapping to the Heisenberg modelBatista , a previous studyBatista4  showed that
the 1D excitonic phase is critical (i.e., there is no long-range order).
Hence, there is a case that 1D exciton condensation is ruled out by the
Hohenberg-Mermin-Wagner theorem. But in practice it is generally accepted that
the finite size and dimension of real-life materials will weaken the statement
of the Hornberger-Mermin-Wagner theorem, and thus the existence of long-range
order (e.g., superconductivity, magnetism, and Bose-Einstein condensation) in
confined geometries remains under debate. To the best of our knowledge, a more
fundamental question of whether 1D condensation can occur when considering
spontaneously generated excitons as an ideal neutral Bose gas has not yet been
addressed. It is well known that ideal free bosons are not allowed to undergo
1D condensation, which makes the study of non-interacting case even more
scientifically interesting.

Consider a free boson gas of mass $m$ and momentum $p$ for a 1D system of
length $L$. The number of quantum states in the energy range $\varepsilon$ to
$\varepsilon$+d$\varepsilon$ is

$D\left(\varepsilon\right){\rm
d}\varepsilon=\frac{{2L}}{h}{\left({\frac{m}{{2\varepsilon}}}\right)^{\frac{1}{2}}}{\rm
d}\varepsilon.$ (1)

Combining Eq. (1) with the Bose-Einstein distribution functions, the total
number $N$ of bosons is given by

$N=\int_{\rm{0}}^{\infty}{\frac{{D(\varepsilon){\rm
d}\varepsilon}}{{{e^{\beta(\varepsilon-\mu)}}-1}}}=\frac{L}{\pi}\int_{0}^{\infty}{\frac{{{\rm
d}k}}{{{e^{\beta(\varepsilon-\mu)}}-1}}},$ (2)

where ${\beta}={\left({{k_{\rm B}}{T}}\right)^{-1}}$. This integral diverges
if chemical potential $\mu$ = 0. Physically, the divergence means that the
continuum states can accommodate infinite particles. Therefore the condensate
is excluded for a 1D free boson gas since its $\mu$ is zero.

However, the situation is different for exciton condensation, either non-
equilibrium or spontaneous. According to statistical mechanics, $\mu$ equals
the change in Gibbs free energy at constant temperature and pressure. The
production of excitons in solids, no matter by photoexcitation or spontaneous
formation, can be approximated as an isothermal and isobaric process.
Therefore, $\mu$ is roughly equivalent to the exciton formation energy. For
non-equilibrium condensation, exciton production is energy-consuming, hence
$\mu>$ 0\. For excitonic insulators, spontaneous production of excitons
implies that the formation energy is negative, hence $\mu<$ 0\. As such, the
divergent behavior of Eq. (2) will be eliminated for exciton condensation.

Because it is the excitons that constitute the ground state of excitonic
insulators, there are discrete exciton bands below the continuous single-
particle states. Without losing generality, let us assume that the lowest
exciton energy is $\varepsilon_{0}<0$. At this point, the upper limit of the
$\mu$ in Eq. (2) is $-\lvert\varepsilon_{0}\rvert$.

Now the maximum particle number $N_{\rm c}$ that can be accommodated in the
continuum spectrum is

${N_{\rm c}}=\frac{L}{\pi}\int_{0}^{\infty}{\frac{{\rm
d}k}{{{e^{\beta\left({\varepsilon+\left|{{\varepsilon_{0}}}\right|}\right)}}-1}}}\\\
=A\sum\limits_{j=1}^{\infty}{\frac{{{{({e^{-\beta|{\varepsilon_{0}}|}})}^{j}}}}{{{j^{\frac{1}{2}}}}}}\\\
<A\sum\limits_{j=1}^{\infty}{({e^{-\beta|{\varepsilon_{0}}|}})}^{j}\\\
=A\frac{1}{e^{\beta|{\varepsilon_{0}}|}-1},$ (3)

where $A=\sqrt{{mL^{2}}/{2\pi{\hbar^{2}}\beta}}$. Definitely, the $N_{\rm c}$
has an upper limit. When the number of preformed excitons $N$ exceeds this
critical value, the excess particles ($N-N_{\rm c}$) can only spill over into
the discrete states. Imagine the extreme case where there is only one discrete
state. Once $N>N_{\rm c}$, the excess excitons will inevitably condense into
this state and form a macroscopic occupation of the ground state. Therefore,
1D excitonic insulators do exist when considering the spontaneously generated
excitons as an ideal neutral Bose gas.

In this sense, one may evaluate the critical temperature $T_{\rm c}$ from the
$N_{\rm c}$ as a function of the 1D exciton density $n_{\rm{1}}$, the exciton
mass $m$, and the lowest exciton level $\varepsilon_{0}$,

${n_{\rm{1}}}=\frac{N_{\rm c}}{L}\\\ =(\frac{mk_{\rm B}T_{\rm
c}}{{2\pi\hbar^{2}}})^{\frac{{\rm{1}}}{2}}\sum\limits_{j=1}^{\infty}{\frac{{{{({e^{-|{\varepsilon_{0}}|/{k_{\rm
B}}{T_{\rm c}}}})}^{j}}}}{{{j^{\frac{1}{2}}}}}}.$ (4)

Likewise, we derive the $T_{\rm c}$ for 2D and 3D excitonic insulators as in
Eqs. (5) and (6)

$\begin{split}{n_{\rm{2}}}=\frac{{mk_{\rm B}T_{\rm
c}}}{{2\pi{\hbar^{2}}}}\sum\limits_{j=1}^{\infty}{\frac{{{{({e^{-|{\varepsilon_{0}}|/k_{\rm
B}T_{\rm c}}})}^{j}}}}{{{j}}}},\end{split}$ (5) ${n_{\rm{3}}}=(\frac{mk_{\rm
B}T_{\rm
c}}{{2\pi\hbar^{2}}})^{\frac{{\rm{3}}}{2}}\sum\limits_{j=1}^{\infty}{\frac{{{{({e^{-|{\varepsilon_{0}}|/k_{\rm
B}T_{\rm c}}})}^{j}}}}{{{j^{\frac{{\rm{3}}}{2}}}}}}.$ (6)

Writing Eqs. (4), (5) and (6) in a unified form gives the $T_{\rm c}$ for
$D$-dimensional excitonic insulators

${n_{D}}=(\frac{mk_{\rm B}T_{\rm
c}}{{2\pi\hbar^{2}}})^{\frac{{D}}{2}}\sum\limits_{j=1}^{\infty}{\frac{{{{({e^{-|{\varepsilon_{0}}|/k_{\rm
B}T_{\rm c}}})}^{j}}}}{{{j^{\frac{{D}}{2}}}}}}.$ (7)

Figure 1:  (Color online) The critical temperature $T_{c}$ in different
dimensions as a function of normalized exciton density $n_{D}$, with the
lowest exciton level $\lvert\varepsilon_{0}\rvert$ and the exciton mass $m$
set to $\lvert\varepsilon_{0}\rvert$ = 0.1 eV and $m$=0.05 $m_{0}$ (Left),
$\lvert\varepsilon_{0}\rvert$ = 0.1 eV and $m$=$m_{0}$ (Middle), and
$\lvert\varepsilon_{0}\rvert$ = 0.01 eV and $m$ = 0.05 $m_{0}$ (Right). Here
$m_{0}$ is the mass of free electrons. Different colours correspond to
different dimensions.

In Fig. 1, we show the $T_{\rm c}$ of different dimensions as a function of
the normalized $n_{D}$, as well as the effects of
$\lvert\varepsilon_{0}\rvert$ and $m$. Three interesting trends are found.
First, under the same conditions, the $T_{\rm c}$ presents the order 1D $>$ 2D
$>$ 3D. Therefore, a decrease in dimensionality favours an increase in the
$T_{\rm c}$. Second, $\lvert\varepsilon_{0}\rvert$ and $m$ have opposite
effects on the $T_{\rm c}$. Increasing the former increases the $T_{\rm c}$,
but increasing the latter decreases the $T_{\rm c}$. Third, the effect of
$\lvert\varepsilon_{0}\rvert$ is more significant in comparison. For example,
at $m$=0.05 $m_{0}$, decreasing $\lvert\varepsilon_{0}\rvert$ by a factor of
ten leads to an order of magnitude decrease in the $T_{\rm c}$. In contrast,
at $\lvert\varepsilon_{0}\rvert$=0.1 eV, a twenty-fold increase in $m$ causes
a much smaller decrease in the $T_{\rm c}$.

Next we turn to consider material realization. Candidates for 1D excitonic
insulators are very rare, and so far only conjugated polymerRice9 ; Rice11 ,
carbon nanotubesVarsanoNC ; Hellgren  and organometallic molecular wiresMnCp ;
CrBz  have been theoretically predicted to exhibit excitonic instabilities.
The recent synthesis of single-chain staircase ferrocene FeCp2 nanowires
(FeNWs) up to 50 nm in length not only breaks the current barrier for the
length of such organometallic wires, but also pave the way for the realization
of similar metallocene wires (TMNWs) with different embedded transition
metalsAFM . In this work, we focus on Scandocene (ScNW) and Chromocene (CrNW)
wires, which can be regarded as ScCp2 and CrCp2 molecules linked by the
dehydrogenation of cyclopentadiene (Cp) rings (See Fig. 2).

Figure 2:  (Color online) (a) Side and (b) top views of a staircase
metallocene nanowire as derived from the experimentAFM . It is composed of
staggered sandwich metallocene molecules (TMCp2) linked by the dehydrogenation
procedure, where one cyclopentadiene (Cp) ring is rotated by $36^{\circ}$ with
respect to the other.

Density-functional-theory calculations were performed within the Perdew-Burke-
Ernzerhof (PBE)PBE  functional as implemented in the Vienna ab initio
Simulation Package (VASP)vasp . The electron-ion interaction was described by
a projector-augmented-wave methodPAW ; PAW2  with an energy cutoff of 450 eV.
A $1\times 1\times 45$ _k_ -mesh was used to sample the Brillouin zone. A
vacuum layer of at least 10 Å was added to separate two neighboring wires. All
atomic positions are fully relaxed without any symmetry constraint until the
residual forces are less than 0.01 eV/Å. The single-shot $G_{0}W_{0}$
methodHybertsen  was employed for the quasi-particle band structures, with a
_k_ -grid of $1\times 1\times 19$ and 240 bands within our computational
capabilities. The Bethe-Salpeter equation (BSE) was solved on top of the $GW$
results for excitonic propertiesRohlfing , including two valence and four
conduction bands to build the Hamiltonian. Spin-orbit coupling was ignored
because our test calculations show that it does not change the conclusion.

Table 1: Optimized lattice constant in Å ($c$), relative formation energy with FeNW as reference zero in eV (${E_{\rm f}}={E_{\rm{TMNW}}}-{E_{\rm{TMCp_{2}}}}+2{E_{\rm H}}$, in which $E_{\rm{TMNW}}$, $E_{\rm{TMCp_{2}}}$ and $E_{\rm H}$ denote the energies of metallocene nanowire, corresponding metallocene molecule and H atom, respectively), energy difference between ferromagnetic and antiferromagnetic configuration in meV ($\Delta E={E_{\rm{FM}}}-{E_{\rm{AFM}}}$), local magnetic moment in $\mu_{\rm B}$ ($M$) on the metal atom, one-electron gap by PBE in eV ($E_{\rm g}$), lowest exciton energy in eV ($\varepsilon_{0}$), corresponding binding energy in eV ($E_{\rm b}$), and excitonic instability or not (EI) for three metallocene nanowires. Spin-resolved results are given for CrNW, and $\uparrow$ and $\downarrow$ denote the results for spin-up and spin-down channels, respectively. | _c_ | $E_{\rm f}$ | $\Delta E$ | $M$ | $E_{\rm g}$ | $\varepsilon_{0}$ | $E_{\rm b}$ | EI
---|---|---|---|---|---|---|---|---
ScNW | 5.58 | -0.05 | 55 | 1 | 0.46 | -0.71 | 2.24 | yes
CrNW | 5.38 | 0.01 | -33 | 2 | 2.23$\uparrow$ 0.48$\downarrow$ | 1.71$\uparrow$ -0.37$\downarrow$ | 2.20$\uparrow$ 3.01$\downarrow$ | no$\uparrow$ yes$\downarrow$
FeNW | 5.17 | 0 | — | 0 | 2.17 | 2.29 | 1.88 | no
Figure 3:  (Color online) (a) Band structures of ScNW, CrNW and FeNW by PBE
(solid and dashed lines) and $G_{0}W_{0}$ (solid and empty squares),
respectively. (b) Projected density of states (PDOS) on transition-metal
$d$-orbitals and C $p$-orbitals for ScNW, CrNW and FeNW. In (a) and (b), the
Fermi levels are set to zero (dashed lines). Bands and PDOS are spin-resolved
for ScNW and CrNW, while not for FeNW. For ScNW, the bands for two spin
channels are exactly degenerate. (c) Characterized transition-metal
$d$-electron configurations for ScNW, CrNW and FeNW. $\uparrow$ and
$\downarrow$ represent an occupied electron in the spin-up and spin-down
channels, respectively.

Table I summarizes the calculated results for the three TMNWs. Structurally,
the lattice constant of TMNWs decreases as the metal atomic number increases
from Sc to Cr to Fe. Energetically, the formation energy of CrNW is comparable
to that of the experimentally synthesised FeNW, while that of ScNW is lower by
0.05 eV per metal atom. Thus, from a thermodynamic point of view, it is also
feasible to fabricate ScNW and CrNW in a similar way to FeNW.

The electronic structures of ScNW, CrNW and FeNW are plotted in Figs. 3(a) and
3(b). All the three features indirect-gap semiconductors by PBE. Both ScNW and
CrNW are spin-polarized, unlike the non-spin-polarized FeNW. Specifically,
ScNW is antiferromagnetic with an energy 55 meV/Sc lower than that of the
ferromagnetic configuration. Instead, CrNW is ferromagnetic with an energy 33
meV/Cr lower than that of the antiferromagnetic configuration.

As shown in Fig. 3(a), ScNW has a minimum PBE gap of 0.46 eV, with the valence
band maximum and conduction band minimum located at $A$ and $\Gamma$
respectively. $G_{0}W_{0}$ increases the gap to 1.43 eV, accompanied by a
slight shift of the valence band maximum from $A$ towards $\Gamma$. CrNW has
PBE gaps of 2.23 eV and 0.48 eV for the spin-up and spin-down channels,
respectively. They are remarkably increased to 3.91 eV and 2.50 eV by
$G_{0}W_{0}$. FeNW shows a PBE gap of 2.17 eV and a $G_{0}W_{0}$ one of 4.17
eV.

From the projected density of states shown in Fig. 3(b), one can see that the
states near the Fermi energy are mainly contributed by transition-metal
$d$-orbitals. It is thus possible to understand the electronic and magnetic
properties of different nanowires in terms of their characteristic
$d$-electron configurationsShen ; LiJPC . According to the Hückel rule, the
Cp, as a five-electron metastable radical, tends to capture an extra electron
to form a stable aromatic configurationShen . Since the unit-cell of TMNWs
contains two Cp rings, this results in the neighbouring metal atom losing its
two 4$s$ electrons. Under an approximately local environment of TMCp2, five TM
3$d$ orbitals split into $a_{1}$ singlet, two-fold degenerate $e_{2}$ and
$e_{1}$ doubletsLiYC . As a result, Sc($d^{1}$), Cr($d^{4}$), and Fe($d^{6}$)
form the characteristic valence-electron populations shown in Fig. 3(c).

With this, it is immediately obvious that ScNW and CrNW are magnetic, with
local magnetic moments on Sc and Cr atoms of 1 and 2 $\mu_{\rm B}$,
respectively, while FeNW is not. These agrees very well with our first-
principles calculations (see Table I) and are also in line with previous
studyDaiJMCC . In addition, the population helps to understand the relative
gap sizes of the different nanowires. Since the splitting of $a_{1}$ and
$e_{2}$ is usually smaller than that of $e_{2}$ and $e_{1}$LiYC , the gap of
ScNW is smaller than that of FeNW. This is also evidenced in CrNW, where the
calculated spin-up gap is much larger than the spin-down one for the same
reason.

Owing to the selection rule, such $d$-$d$ gap nature determines a suppressed
contribution to the system screening from the band-edge statesJianghf ; BuXT .
Combined with their reduced dimension, TMNWs are expected to have very
interesting excitonic effects, not only in terms of large binding energies,
but also in terms of being decoupled from the one-electron gap, all of which
are favourable for the occurrence of excitonic instabilitiesusEI ; linear . In
this regard, we solve the BSE and obtain their low-energy exciton spectra as
presented in Fig. 4. Note that only zero-momentum excitons are considered
here. Despite the presence of an indirect gap, the near-Fermi-level bands are
all relatively flat due to the $d$-orbital nature, so this treatment is
expected to have little effect on the lower exciton energy levels.

Figure 4:  (Color online) Low-energy exciton levels by solving the BSE on top
of $G_{0}W_{0}$ results for (a) ScNW, (b) CrNW and (c) FeNW. For CrNW, only
the spectrum of spin-down channel is given and the lowest-energy level for
spin-up is as high as 1.71 eV (see Table I).

The lowest exciton of ScNW has a negative energy of -0.71 eV, signalling the
presence of excitonic instability. As usual, it is an optically inactive dark
excitonusEI ; usDong . For an antiferromagnet, spin-triplet exciton has lower
energy than spin-singlet one due to the exciton exchange interactionJianghf ;
Mazzone . Accordingly, the ScNW would form a spin-triplet excitonic insulator
when spontaneously generated excitons condense. The two spins of CrNW have
one-electron gaps of very different sizes, and their competition with exciton
binding energies leads to essentially distinct results, yielding spin-resolved
excitonic instabilities. As listed in Table I, the lowest exciton of the spin-
up channel has a positive energy of 1.71 eV; on the contrary, it is negative,
-0.37 eV, for the spin-down channel with a much smaller one-electron gap.
Therefore, the CrNW is a half excitonic insulator with single-spin exciton
condensateJianghf . From our first-principles $GW$-BSE calculations, the
binding energies corresponding to spin-up and spin-down excitons are inferred
to be 2.20 and 3.01 eV, respectively. Whilst, the FeNW is a usual
semiconductor with a lowest exciton energy of 2.29 eV (see Table I and Fig.
4).

Both ScNW and CrNW are magnetic excitonic insulators, which provide an
additional degree of freedom for probing spontaneous condensation of excitons.
For example, spin-triplet exciton condensation can host spin
superfluidityJiangtri . We also anticipate that the interplay between
magnetism and exciton condensation would cause magnetic excitonic insulators
to exhibit some unique features different from those of normal magnetic
insulators, thus providing evidence for identification.

Figure 5:  (Color online) Band structures of (a) ScNW and (b) CrNW by HSE06.
The Fermi levels are set to zero (dashed lines). Gap size as a function of
exact-exchange weights $\alpha$ for (c) ScNW and (d) CrNW. Note that only the
spin-down values are shown for CrNW, since its large spin-up gap makes this
channel independent of excitonic instability at all times. The red horizontal
dashed line indicates the exciton binding energies in Table I, from which the
critical $\alpha$ for excitonic instability is estimated.

In addition, both magnetic ground state and excitonic instability could in
principle be the consequence of a semi-local approximation in the use of PBE
exchange-correlation functionalBarborini . To clarify this pivotal point, we
have further performed calculations using the HSE06 hybrid functional. We
first perform structural relaxations on the supercell containing two
transition-metal atoms. The results show that no Peierls distortion occurs.
Then, we compare the energetics of different magnetic configurations. The
results show that ScNW remains in the antiferromagnetic ground state with an
energy 35 meV/Sc lower than that of the ferromagnetic configuration, while
CrNW remains in the ferromagnetic ground state with an energy 10 meV/Cr lower
than that of the antiferromagnetic configuration. Compared with the results of
the PBE in Table I, HSE06 reduces the energy difference by 20 meV/Sc and 23
meV/Cr, respectively.

Finally, we calculate the one-electron bands at different exact-exchange
weights, i.e., $\alpha$ = 0.125, 0.25, 0.375, and 0.5. We find that these
bands resemble each other except for different gap sizes. In Figs. 5(a) and
5(b) we give typical bands for ScNW and CrNW at $\alpha$ = 0.25. They are
similar to the ones by PBE and GW [see Fig. 3(a)]. Further, we present the gap
size as a function of $\alpha$ in Figs. 5(c) and 5(d), respectively. Note that
only the spin-down values are shown for CrNW, since its large spin-up gap
makes this channel independent of excitonic instability at all times. As can
be seen, the gap approximately exhibits a linear dependence on $\alpha$,
namely, $E_{\rm g}$ = 5.06$\alpha+$ 0.49 and $E_{\rm g}$ = 6.42$\alpha+$0.50
for ScNW and CrNW, respectively. Since solving the BSE on top of HSE06 is very
computationally expensive, we use the exciton binding energies in Table I
[denoted by red horizontal dashed lines in Figs. 5(c) and 5(d)] as estimates.
With this approach, we obtain that the excitonic instability of ScNW and CrNW
remains robust if $\alpha<$ 0.35 and $\alpha<$ 0.39.

In summary, using statistical methods combined with first-principles
calculations, we have investigated spontaneous Bose-Einstein condensation of
excitons in the 1D systems. It is shown that 1D excitonic insulators do exist
and their critical temperatures are even much higher than those of 2D and 3D
excitonic insulators. However, 1D excitonic insulators have rarely been
studied in the past, due in large part to the Hohenberg-Mermin-Wagner theorem
ruling out its existence. We expect that the present study can revive interest
in 1D excitonic insulators, thereby bringing new opportunities for the
identification and future application of the excitonic insulator. Regarding
the material realization, we predict that the single-chain staircase
Scandocene and Chromocene wires are an antiferromagnetic spin-triplet
excitonic insulator and a ferromagnetic half-excitonic insulator,
respectively, providing a platform for exploiting the multifaceted nature of
magnetism and spontaneous exciton condensate.

###### Acknowledgements.

Y.L. thanks Zhirong Liu for fruitful discussions. This work was supported by
the Ministry of Science and Technology of China (Grant Nos. 2023YFA1406400 and
2020YFA0308800) and the National Natural Science Foundation of China (Grant
No. 12074034).

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