# Non-commuting graphs of nilpotent groups

Alireza Abdollahi  Department of Mathematics, University of Isfahan, Isfahan
81746-73441, Iran;
School of Mathematics, Institute for Research in Fundamental Sciences (IPM),
P.O.Box 19395-5746 , Tehran, Iran<EMAIL_ADDRESS>and  Hamid
Shahverdi  School of Mathematics, Institute for research in fundamental
science (IPM), P.O. Box 19395-5746, Tehran, Iran. Iran.
<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-
commuting graph $\Gamma_{G}$ associated to $G$ is the graph whose vertex set
is $G\setminus Z(G)$ and two distinct elements $x,y$ are adjacent if and only
if $xy\neq yx$. We prove that if $G$ and $H$ are non-abelian nilpotent groups
with irregular isomorphic non-commuting graphs, then $|G|=|H|$.

###### Key words and phrases:

Non-commuting graph; nilpotent groups; graph isomorphism; groups with abelian
centralizers

###### 2000 Mathematics Subject Classification:

20D15; 20D60

##  1\. Introduction and results

Let $G$ be a non-abelian group and $Z(G)$ be its center. The non-commuting
graph $\Gamma_{G}$ of $G$ is a graph whose vertex set is $G\setminus Z(G)$ and
two vertices $x$ and $y$ are adjacent if and only if $xy\neq yx$. The non-
commuting graph of a group was first considered by Paul Erdős in 1975 [7].
Many people have studied the non-commuting graph (e.g., [1, 2, 3, 9, 10]). In
[2] the following conjecture was put forward:

######  Conjecture 1.1 (Conjecture 1.1 of [2]).

Let $G$ and $H$ be two finite non-abelian groups such that
$\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.

Conjecture 1.1 was refuted by an example due to Isaacs in [6], however it is
valid whenever one of $G$ or $H$ is a non-abelian finite simple group [3] or
whenever one of $G$ or $H$ has prime power order [1]. The counterexample given
in [6] is a pair $(G,H)$ of nilpotent non-abelian groups with regular non-
commuting graph; recall that a graph is called regular if the degree of all
vertices are the same, otherwise the graph is called irregular. It follows
from a result of Ito [5] that a finite group with a regular non-commuting
graph is a direct product of a non-abelian $p$-group for some prime $p$ and an
abelian group.

Here we study pairs $(G,H)$ of non-abelian finite groups which provide a
counterexample to Conjecture 1.1. It follows from the main result of [1], that
if a pair $(G,H)$ provides a counterexample then none of $G$ and $H$ are of
prime power order. Here we prove that if a pair of non-abelian finite
nilpotent groups provides a counterexample for Conjecture 1.1 then their non-
commuting graphs must be regular.

######  Theorem 1.2.

Let $G$ and $H$ be two finite non-abelian nilpotent groups with irregular non-
commuting graphs such that $\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.

We conjecture that the word “nilpotent” in Theorem 1.2 is sufficient for one
of the groups $G$ and $H$.

##  2\. Non-commuting graphs of nilpotent groups

A non-abelian group is called an $AC$-group if the centralizer of every non-
central element is abelian. For a group $G$ and an element $g\in G$, $g^{G}$
denotes the conjugacy class of $g$ in $G$.

######  Lemma 2.1.

Let $G$ and $H$ be two finite non-abelian groups. If
$\phi:\Gamma_{G}\rightarrow\Gamma_{H}$ is a graph isomorphism and $g$ is a
non-central element of $G$, then the following hold:

  1. (1)

$|G|-|Z(G)|=|H|-|Z(H)|$.

  2. (2)

$|G|-|C_{G}(g)|=|H|-|C_{H}(\phi(g))|$.

  3. (3)

$|C_{G}(g)|-|Z(C_{G}(g))|=|H|-|Z(C_{H}(\phi(g)))|$, where $C_{G}(g)$ is not
abelian.

  4. (4)

If $C_{G}(g)$ is not abelian, then
$\Gamma_{C_{G}(g)}\cong\Gamma_{C_{H}(\phi(g))}$.

  5. (5)

Suppose that $C_{1}=C_{G}(g_{1})$ and $C_{i}=C_{C_{i-1}}(g_{i})$ for $i\geq
2$, where $g_{1}\in G\setminus Z(G)$ and $g_{i}\in C_{i-1}\setminus
Z(C_{i-1})$. Then there exists $k\in\mathbb{N}$ such that $C_{k}$ is an
$AC$-group.

  6. (6)

$|G|=|H|$ if and only if $|C_{G}(g)|=|C_{H}(\phi(g))|$ if and only if
$|Z(G)|=|Z(H)|$.

###### Proof.

It is straightforward. To prove (5), note that if the centralizer $C_{i}$ is
not an $AC$-group, then some proper centralizer in $C_{i}$ is not abelian
guaranteeing the existence of an element $g_{i+1}$. On the other hand, $G$ is
finite so the series $C_{1}>C_{2}>\cdots>C_{i}>\cdots$ will eventually
terminate in an $AC$-group. ∎

######  Lemma 2.2 (see e.g. Theorem 2.1 of [1]).

Let $G$ be a finite non-abelian group and $H$ be a group such that
$\Gamma_{G}\cong\Gamma_{H}$. Then the following hold:

  1. (1)

$|C_{H}(h)|$ divides $(|g^{G}|-1)(|Z(G)|-|Z(H)|)$ for any $g\in G\setminus
Z(G)$ and $h=\phi(g)$, where $\phi$ is any graph isomorphism from $\Gamma_{G}$
to $\Gamma_{H}$.

  2. (2)

If $|Z(G)|\geq|Z(H)|$ and $G$ contains a non-central element $g$ such that
${|C_{G}(g)|}^{2}\geq|G|\cdot|Z(G)|$, then $|G|=|H|$.

We need the following result concerning a number theoretic conjecture due to
Goormaghtigh.

######  Theorem 2.3 (see e.g. Theorem 1.3 of [4]).

Let $x,y,m,n$ be integers such that $y>x>1$ and $m,n>1$. Then the following
equation has at most one pair $(m,n)$ of solution for every fixed pair
$(x,y)$:

$\frac{y^{n}-1}{y-1}=\frac{x^{m}-1}{x-1}.$

######  Theorem 2.4.

Let $G$ be a nilpotent group with at least two distinct non-abelian Sylow
subgroups. Suppose also that $H$ is any non-abelian group such that
$|Z(G)|\geq|Z(H)|$ and $\Gamma_{G}\cong\Gamma_{H}$. Then $|G|=|H|$.

###### Proof.

Suppose $G=P\times Q\times S$, where $P$ and $Q$ are non-abelian Sylow $p$,
$q$-subgroups of $G$ such that $p\neq q$ and $S$ is a subgroup of $G$. If
$x\in P\setminus Z(P)$ and $y\in Q\setminus Z(Q)$, then
$|Z(G)|<|C_{P}(x)||C_{Q}(y)||S|$. Therefore

$|G||Z(G)|<|C_{P}(x)||C_{Q}(y)||P||Q||S|^{2}=|C_{G}(x)||C_{G}(y)|.$

It follows that

$|G||Z(G)|<\max\\{|C_{G}(x)|^{2},|C_{G}(y|^{2}\\}.$

Now, Lemma 2.2(2) completes the proof. ∎

######  Corollary 2.5.

Let $G$ and $H$ be two nilpotent groups each of which have at least two non-
abelian Sylow subgroups. If $\Gamma_{G}\cong\Gamma_{H}$, then $|G|=|H|$.

Both groups $G$ and $H$ in the counterexample of Conjecture 1.1 due to Isaacs
in [6] have the same shape, that is, they are direct products of a non-abelian
group of prime power order $P$ and a non-trivial abelian group $A$ such that
$\gcd(|P|,|A|)=1$ and all non-trivial conjugacy class sizes of $G$ or $H$ have
equal order. The latter property was first studied by Ito [5] and we want to
prove Theorem 1.2 for all nilpotent groups except those satisfying the latter
shape.

##  3\. Proof of Theorem 1.2

Now, we prove Theorem 1.2 in four cases. In this section $G$ and $H$ are
finite non-abelian nilpotent groups with irregular non-commuting graphs and
$\phi:\Gamma_{G}\rightarrow\Gamma_{H}$ is a graph isomorphism. By Corollary
2.5, we may assume that $G$ has exactly one non-abelian Sylow subgroup. If $G$
is of prime power order, the main result of [1] implies that $|G|=|H|$. Thus
we may assume $G=P\times A$, where $P$ is a non-abelian Sylow $p$-subgroup of
$G$ and $A$ is a non-trivial abelian subgroup whose order is prime to $p$.
Also, set $|P|=p^{n}$ and $|Z(P)|=p^{r}$.

Case (a): $H=P_{1}\times B$ for some non-abelian Sylow $p$-subgroup $P_{1}$ of
$H$ and for some non-trivial abelian subgroup $B$ of $H$.
We use the following notation: $|P_{1}|=p^{m}$, $|Z(P_{1})|=p^{s}$ and
$\phi(g_{i})=h_{i}$, where $g_{1},\dots,g_{k}$ are non-central elements of $G$
chosen from conjugacy classes of $G$ with pairwise distinct sizes such that
$|{g_{i}}^{G}|=p^{a_{i}}$ and $|{h_{i}}^{H}|=p^{b_{i}}$ and
$a_{1}<\dots<a_{k}$ and $b_{1}<\dots<b_{k}$. Notice that $k\geq 2$, since
$\Gamma_{G}$ and $\Gamma_{H}$ are irregular.
Since $\Gamma_{G}\cong\Gamma_{H}$,

(1) $|A|p^{r}(p^{n-r}-1)=|B|p^{s}(p^{m-s}-1)$ (2)
$|A|p^{n-a_{i}}(p^{a_{i}}-1)=|B|p^{m-b_{i}}(p^{b_{i}}-1)$

for every $1\leq i\leq k$. Equation (1) implies that $r=s$ and equation (2)
implies that $n-a_{i}=m-b_{i}$. Since $\Gamma_{G}$ is not regular, graph
isomorphism implies that

(3) $|A|(p^{n-a_{1}}-p^{n-a_{2}})=|B|(p^{m-b_{1}}-p^{m-b_{2}}).$

Therefore $|A|=|B|$. Now, equation (2) implies that $a_{1}=b_{1}$. Hence
$|P|=|P_{1}|$.

Case (b): $H=P_{1}\times X$, where $P_{1}$ is a non-abelian Sylow $p$-subgroup
of $H$ and $X$ is an arbitrary group such that $\gcd(p,|X|)=1$.
Suppose $H$ is a minimal counterexample. Also suppose by way of contradiction
that $X$ is a non-abelian group. Then $P_{1}$ and $X$ are $AC$-group. Let
$x\in X\setminus Z(X)$. Then $C_{H}(x)=P_{1}\times B$, where $B\subseteq X$ is
an abelian subgroup of $X$. Therefore Case (a) implies that
$|C_{H}(x)|=|C_{G}(\phi^{-1}(x))|$. Since $\Gamma_{G}\cong\Gamma_{H}$, we have
$|G|=|H|$. Now, $|G|=|H|$ implies that $|P|=|P_{1}|$. Set
$|C_{G}(\phi^{-1}(x))|=p^{n-\alpha}|A|$ for some integer $1<\alpha<n$. By
graph isomorphism, we have

$(p^{n}-p^{n-\alpha})|A|=p^{n}(|X|-|C_{X}(x)|).$

The largest $p$-power dividing the right-hand side of the equation is $\geq
p^{n}$ and the left is $p^{n-\alpha}$. This is a contradiction. Hence $X$ is
abelian and Case (a) completes the proof.

Case (c): $H=Q_{1}\times X$, where $Q_{1}$ is a Sylow $q_{1}$-subgroup of $H$
and $X$ is a non-abelian nilpotent group.
If $p=q_{1}$, then Case (b) completes the proof. We claim that $p\neq q_{1}$
is not possible. Let $H$ be a minimal counterexample. Therefore $Q_{1}$ and
$X$ are $AC$-groups. By the characterization of $AC$-groups [8], a nilpotent
$AC$-group is a direct product of a non-abelian group of prime power order and
an abelian group. Therefore $X=Q_{2}\times B$, where $Q_{2}$ is a non-abelian
$q_{2}$-group for some prime $q_{2}$, $B$ is an abelian group and
$\gcd(|Q_{2}|,|B|)=1$. Let $h_{i}\in Q_{i}\setminus Z(Q_{i})$ for
$i\in\\{1,2\\}$. Also, set $\phi^{-1}(h_{i})=g_{i}$ for $i\in\\{1,2\\}$ and
$|C_{G}(g_{i})|=|A|p^{n-{a_{i}}}$, where $1<a_{i}<n$. If $q_{2}=p$, then again
Case (b) implies that $Q_{1}\times B$ is an abelian group. This is a
contradiction. Therefore $p\neq q_{1},q_{2}$.

We have $C_{H}(h_{1})=C_{Q_{1}}(h_{1})\times Q_{2}\times B$ and
$C_{H}(h_{2})=Q_{1}\times C_{Q_{2}}(h_{2})\times B$ and
$Z(C_{H}(h_{1}))=C_{Q_{1}}(h)\times Z(Q_{2})\times B$ and
$Z(C_{H}(h_{2}))=Z(Q_{1})\times C_{Q_{2}}(h_{1})\times B$. So $Z(H)\subsetneqq
Z(C_{H}(h_{i}))$. Therefore graph isomorphism implies that $Z(G)\subsetneqq
Z(C_{G}(g_{i}))$. Set $|Z(C_{G}(g_{i}))|=|A|p^{d_{i}}$ for $i\in\\{1,2\\}$ and
$|Z(G)|=|A|p^{r}$. It is clear that $d_{i}>r$. Now,
$\Gamma_{G}\cong\Gamma_{H}$ and
$\Gamma_{C_{G}(g_{i})}\cong\Gamma_{C_{H}(h_{i})}$ for $i\in\\{1,2\\}$ imply
that

(4)
$|C_{H}(h_{2})|-|Z(C_{H}(h_{2}))|=(|Q_{1}|-|Z(Q_{1})|)|C_{Q_{2}}(h_{2})||B|=|A|(p^{n-a_{2}}-p^{d_{2}})$
(5)
$|Z(C_{H}(h_{1}))|-|Z(H)|=(|C_{Q_{1}}(h_{1})|-|Z(Q_{1})|)|Z(Q_{2})||B|=|A|(p^{d_{1}}-p^{r})$
(6)
$|H|-|C_{H}(h_{1})|=(|Q_{1}|-|C_{Q_{1}}(h_{1})|)|Q_{2}||B|=|A|(p^{n}-p^{n-a_{1}}).$

Since
$|B|(|Q_{1}|-|Z(Q_{1})|)=|B|(|Q_{1}|-|C_{Q_{1}}(h_{1})|)+|B|(|C_{Q_{1}}(h_{1})|-|Z(Q_{1})|)$,
by equation (5) and (6) the largest $p$-power dividing the right-hand side of
the latter equation is $p^{r}$ and by equation (4) the largest $p$-power
dividing the left hand side is $p^{d_{2}}$. This is a contradiction.

Case (d): $H=Q\times B$, where $Q$ is a non-abelian Sylow $q$-subgroup for
some prime $q\neq p$ and $B$ is a non-trivial abelian subgroup.
Suppose by way of contradiction that $|G|\neq|H|$. Since $\Gamma_{G}$ is not
regular, there exist $g_{1},g_{2}\in G\setminus Z(G)$ such that
$|g_{1}^{G}|=p^{a_{1}}\neq p^{a_{2}}=|g_{2}^{G}|$. Set $|Q|=q^{m}$,
$|Z(Q)|=q^{s}$, $\phi(g_{i})=h_{i}$ for $i\in\\{1,2\\}$ and
$|h_{i}^{H}|=q^{b_{i}}$. Since $\Gamma_{G}\cong\Gamma_{H}$,

(7) $|A|(p^{n}-p^{r})=|B|(q^{m}-q^{s})$ (8)
$|A|(p^{n-a_{i}}-p^{r})=|B|(q^{m-b_{i}}-q^{s}).$

If $u=\gcd(a_{1},a_{2},n-r)$ and $v=\gcd(b_{1},b_{2},m-s)$, by considering
equations (7) and (8) and taking greatest common divisors, we have

(9) $|A|p^{r}(p^{u}-1)=|B|q^{s}(q^{v}-1).$

Now, by dividing equations (7) and (9), we have

(10) $\frac{p^{n-r}-1}{p^{u}-1}=\frac{q^{m-s}-1}{q^{v}-1}$

and by dividing equations (8) and (9), we have

(11) $\frac{p^{n-a_{i}-r}-1}{p^{u}-1}=\frac{q^{m-b_{i}-s}-1}{q^{v}-1}.$

Note that it is not possible that $n-a_{1}-r=u=n-a_{2}-r$, since $a_{1}\neq
a_{2}$. Now, Theorem 2.3 and equation (10) and (11) yield a contradiction.
$\hfill\Box$

Acknowledgments

The authors are grateful to the referee for his/her invaluable comments. The
first author was financially supported by the Center of Excellence for
Mathematics, University of Isfahan. This research was in part supported by
grants IPM (No. 91050219) and IPM (No. 91200045).

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# LaBH8: the first high-$T_{\text{c}}$ low-pressure superhydride

Simone Di Cataldo<EMAIL_ADDRESS>Institute of Theoretical and
Computational Physics, Graz University of Technology, NAWI Graz, 8010 Graz,
Austria  Dipartimento di Fisica, Sapienza Università di Roma, 00185 Roma,
Italy  Christoph Heil  Institute of Theoretical and Computational Physics,
Graz University of Technology, NAWI Graz, 8010 Graz, Austria  Wolfgang von der
Linden  Institute of Theoretical and Computational Physics, Graz University of
Technology, NAWI Graz, 8010 Graz, Austria  Lilia Boeri
<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza Università di Roma,
00185 Roma, Italy

###### Abstract

In the last five years a large number of new high-temperature superconductors
have been predicted and experimentally discovered among hydrogen-rich
crystals, at pressures which are way too high to meet any practical
application. In this work, we report the computational prediction of a hydride
superconductor, LaBH8, with a $T_{\text{c}}$ of 126 K at a pressure of 50 GPa,
thermodynamically stable above 100 GPa, and dynamically stable down to 40 GPa,
an unprecedentedly low pressure for high-$T_{\text{c}}$ hydrides. LaBH8 can be
seen as a ternary sodalite-like hydride, in which a metallic hydrogen
sublattice is stabilized by the chemical pressure exerted by the guest
elements. The combination of two elements with different atomic sizes in LaBH8
realizes a more efficient packing of atoms than in binary sodalite hydrides. A
suitable choice of elements may be exploited to further reduce the
stabilization pressure to ambient conditions.

The discovery of high-temperature superconductivity (HTSC) at 203 K in sulfur
hydride at a pressure of 200 GPa rekindled the dream of achieving room-
temperature superconductivity [1, 2, 3], triggering a hydride rush [4, 5, 6,
7, 8, 9, 10, 11] which culminated in the report of superconductivity with a
critical temperature (Tc) of 287 K (15 ∘C) at 267 GPa in a carbonaceous sulfur
hydride [12]. The first discovery of a room-temperature superconductor set a
major milestone in the history of condensed matter physics, but the
exceptional pressure required to stabilize the superconducting phase thwarts
any practical application.

Obviously, the next challenge for materials research is to find materials
exhibiting comparable superconducting properties at, or close to, ambient
pressures. Also in this case hydrides, which realize the requirements for
conventional HTSC, are a promising hunting ground.

In the five years following the SH3 discovery, all possible combinations of
$X$Hn binary hydrides have been computationally explored in an effort to
achieve room-temperature superconductivity; these studies revealed that the
formation, stability and superconducting properties of these high-pressure
(HP) hydrides strongly depend on the ionic size, electronegativity and
electronic configuration of the $X$ elements. High-Tc superconductors are
found either among covalent hydrides, in which $X$ and H form directional
bonds driven metallic by pressure, [13, 14, 15] or among alkali, alkaline-
earth and rare-earth hydrides, which form sodalite structures, in which the
$X$ atoms do not directly bind to hydrogen but provide a scaffold, that
stabilizes a dense sponge-like hydrogen lattice [16, 17, 18, 19, 20]. Most
HTSC binary hydrides are predicted to be stable above 150-200 GPa; a few are
predicted to survive down to 70 GPa, where they are on the verge of a
dynamical instability [21, 18]; uranium hydrides are stable above 35 GPa, but
do not exhibit HTSC [22, 6].

Having exhausted all possible combinations of binary hydrides, it is natural
to extend the search for HTSC to multinary hydrides, where the addition of a
third element other than hydrogen, enormously expands the phase space [23,
24]. A few works have already tried to exploit this additional flexibility to
to raise the $T_{c}$ of high-pressure hydrides well beyond room temperature.
[25, 26]

In this work, we will demonstrate a strategy to bring the stabilization
pressure of high-$T_{\text{c}}$ superconducting hydride phases close to
ambient pressure in a ternary hydride. In short, it consists of identifying a
suitable combination of elements with different sizes and electronegativity.
Our strategy is demonstrated by the prediction of a new ternary high-
temperature superconductor, identified through a evolutionary search of the
lanthanum-boron-hydrogen (La-B-H) phase diagram [27, 28]. This phase, with
LaBH8 composition, exhibits a superconducting $T_{\text{c}}$ of 126 K at 50
GPa. It is a remarkable example of a ternary sodalite hydride, in which a
cubic La-B scaffolding confines a highly-symmetric, dense hydrogen sublattice,
and makes it stable down to moderate pressures. Ternary sodalite hydride
phases have been predicted only at pressures above 170 GPa [25, 29], while
LaBH8 appears on the ternary La-B-H hull at 110 GPa, and is dynamically stable
down to 40 GPa.

Figure 1: Crystal structure of the $Fm\bar{3}m$ phase of LaBH8 (conventional
unit cell). La, B, and H atoms are shown as green, orange, and blue spheres,
respectively. The Electron Localization Function (ELF) is projected onto the
$\bar{1}00$ plane.

The new LaBH8 phase was identified through a structural search at 100 GPa
using variable-composition evolutionary crystal structure prediction, as
implemented in USPEX [27, 28], by sampling a total of over twelve thousand
structures 111In addition, we re-sampled particularly promising compositions.
Further details are provided in the Supplemental Material. The ternary La-B-H
convex hull was then constructed, including also the zero-point lattice
contribution to the total energy. At 100 GPa four stable compositions lie on
the convex hull: La(BH2)3, La(BH4)3, LaBH5, and LaBH8. The first two exhibit
crystal structures analogous to those observed in other metal borohydrides,
i.e. molecular structures with BH${}_{2}^{-}$ and BH${}_{4}^{-}$ anions
interspaced by La3+ cations [31, 32, 24, 33, 34], and are insulating. LaBH5
and LaBH8 are characterized by the same La-B rocksalt sublattice. In LaBH5,
boron and hydrogen form a BH${}_{4}^{-}$ tetrahedral anion, and an additional
H atom is trapped at the center of a La tetrahedron. For LaBH8 we predict a
$Fm\bar{3}m$ sodalite-like structure. Both LaBH5 and LaBH8 are metallic.
Preliminary $T_{\text{c}}$ calculations showed that the LaBH5 structure
exhibits a $T_{\text{c}}$ of 53 K at 50 GPa, whereas LaBH8 exhibits a
$T_{\text{c}}$ of 126 K at the same pressure 222Additional information on the
crystal structures can be found in the form of Crystallographic Information
File in the Supplemental Material. These preliminary results led us to focus
on the much more promising LaBH8 $Fm\bar{3}m$ sodalite-like structure.

$Fm\bar{3}m$ LaBH8 lies only 23 meV/atom above the hull at 100 GPa, becomes
thermodynamically stable above 110 GPa, and is dynamically stable down to 40
GPa. This implies that this phase can be realistically synthesized by laser
heating at 110 GPa, and quenched at low pressures down to a minimum of 40 GPa.
The crystal structure of $Fm\bar{3}m$-LaBH8 is shown in Fig. 1: La and B
occupy $4b$ and $4a$ Wyckoff positions, respectively, while H atoms sit on
$32f$ sites with $x=0.145$. The hydrogen sites occupy the vertices of a
rhombicuboctahedron centered around La atoms, and vertices of cubes around B
atoms. Interestingly, a structure with an identical $M$-B sublattice was
observed with neutron diffraction on $M$BH4 ($M$ = K, Rb, Cs), which only
differs by the $1/2$ occupancy of the $32f$ site by hydrogen [36].

The Electron Localization Function (ELF) for LaBH8, shown in Fig. 1 along the
$\bar{1}00$ plane, has maxima around the atoms, and along the H-H bonds, but
not between La and H or B and H, indicating that neither La nor B form bonds
with H, but both act as spacers. The absence of a B-H covalent bond is quite
unusual for a borohydride, and implies that this structure is not a covalent
hydride, like SH3. Rather, it is reminiscent sodalite hydrides like LaH10,
where a dense, metallic hydrogen sublattice is stabilized at pressures lower
than the pure hydrogen at metallization pressure, due to the chemical pressure
exerted by the host atoms. In a very simplified picture, one could see LaBH8
as a chemically-precompressed version of LaH10 (for a visual impression see
Fig. S3 of the SM [37]). In fact, the two structures share the same La-La
sublattice, with almost identical lattice parameters at all pressures; LaH10
is stable at the harmonic level only above 200 GPa; in LaBH8, boron atoms fill
the voids between the second-nearest La atoms and provide additional chemical
pressure, making the metallic hydrogen sublattice stable down to 40 GPa. The
analogy of LaBH8 with binary sodalite structures, which is confirmed by the
analysis of the electronic structure and vibrational properties, is ultimately
at the heart of the HTSC at low pressure. We also observe that at all
pressures H-H interatomic distances are 13% larger than sodalite LaH6, and 20%
larger than LaH10 [18] (see Fig. S5 of the SM 333The Supplemental Material is
available at..). In short, both the geometric and bonding properties indicate
that this structure is a natural extension of the concept of sodalite
hydrides, to the case of a ternary hydride [16, 17].

In Fig. 2 we show the electronic band structure, along with the atom-projected
density of states, calculated at 50 GPa. Here and in the following, we will
focus on this pressure, which is sufficiently close to the moderate-pressure
regime, but is about 10 GPa higher than the dynamical instability pressure, so
that predictions are still not dramatically affected by anharmonic effects. A
band structure formation analysis (see SM, Fig. S6 [37]) reveals that the
eight bands in the -15 to 2 eV range from the Fermi level derive from the
eight quasi-free-electron-like bands of the empty H8 sublattice, which are
only weakly perturbed by hybridization with the $2s-2p$ boron states, and more
strongly by hybridization with the three La semi-core bands from -20 to -15
eV. A Bader charge analysis [38] predicts a net charge of +1.46 for La, +0.88
for B, and -0.29 for each H, indicating that both boron and lanthanum donate
charge to the hydrogen sublattice.

Hence, the band structure analysis confirms the bonding picture observed in
real space: H forms a dense sublattice, stabilized by the La-B scaffolding
with which hydrogen forms only weak bonds. The absence of a covalent B-H bond
here is crucial, and explains the free-electron-like behavior of the hydrogen-
derived electronic states. In fact, as a result of this weak hybridization
electronic bands at the Fermi level are highly-dispersive and have an almost
purely (70%) hydrogen character, exactly like binary sodalite hydrides [18].

In the left panel of Fig. 4 we show the Fermi surface decorated with H
character. The Fermi surface is characterized by three sheets i) a large
electron-like sphere around the $\Gamma$ point, which has the greatest weight
in the reciprocal space, ii) a cross-shaped sheet and a small hole pocket
around the X point, mostly of H character, and iii) a small hole pocket around
the L point with mixed B and H character. Overall, the whole Fermi surface
exhibits a strong hydrogen character, i.e. the partial H contribution to the
DOS is never less than 50 %, on average around 70%.

Figure 2: Left panel: electronic band structure of $Fm\bar{3}m$ LaBH8 at 50
GPa, decorated with hydrogen character (blue) vs non-hydrogen character
(gray). Right panel: atom-projected density of states in units of eV-1spin-1.
Projection onto La, B, and H is shown in green, orange, and blue,
respectively. The zero of the energy corresponds to the Fermi level.

In order to compute the superconducting properties of the $Fm\bar{3}m$-LaBH8
phase, we calculated the phonon dispersions and the electron-phonon coupling
using linear response theory within the harmonic approximation, using Wannier
interpolation on very fine $\vec{k}$ and $\vec{q}$ grids, as implemented in
the `EPW` code [39, 40, 41]. In Fig. 3 we show the phonon dispersions
decorated with the partial electron-phonon (e-ph ) coupling coefficients
$\lambda_{\nu\vec{q}}$, together with the atom-projected Eliashberg function
$\alpha^{2}F(\omega)$ and the phonon density of states. The e-ph coupling is
spread rather evenly on all optical branches, and is stronger for modes which
involve vibrations of the hydrogen sublattice, again in close analogy with
other binary sodalite hydrides [42, 18]. The high peak in the Eliashberg
function at 50 meV corresponds to a flat region of the dispersion around the
$\Gamma$ point, which experiences a particularly strong e-ph coupling. This
phonon mode, named $T^{*}_{2g}$ in the following, is characterized by a
$T_{2g}$ symmetry at the $\Gamma$ point and corresponds to a distortion of the
tetrahedra formed by nearest-neighbours H atoms, and carries around 15$\%$ of
the total e-ph coupling (See SM Fig. S13 for more details [37]). In addition,
a triply degenerate branch with $T_{1u}$ symmetry at $\Gamma$, accidentally
quasi-degenerate with the $T_{2g}^{*}$ mode at 50 GPa, is also notable. This
branch, in fact, corresponds to a rattling mode of the boron atoms inside the
cubic hydrogen cages surrounding it, and is mostly dispersionless throughout
the Brillouin zone, coherently with the description of boron as passively
pressurizing the metallic hydrogen sublattice, without bonding to it.

Integrating the Eliashberg function we obtain the two moments: [43, 44]
$\lambda=2\int{\alpha^{2}F(\omega)\omega^{-1}d\omega}=1.54$ and
$\omega_{\text{log}}=\exp\left[2\lambda^{-1}\int
d\omega\alpha^{2}F(\omega)\omega^{-1}\log(\omega)\right]71$ meV. Table 1
reports the critical temperature obtained by a direct solution of ab-initio
Migdal-Eliashberg equations, as implemented in the `EPW` code [41]. Coulomb
effects are included via the Morel-Anderson pseudopotential
$\displaystyle\mu^{*}=\mu/[1+\mu\log(\omega_{\text{el}}/\omega_{\text{ph}})]$
[45], with $\omega_{\text{el}}$ and $\omega_{\text{ph}}$ being characteristic
energies for electrons (band-width of the Fermi surface electrons) and phonons
(highest phonon energy), respectively. The double Fermi-surface average of the
screened Coulomb interaction $\mu$ was evaluated within the random phase
approximation [46, 47, 48, 49]. We find a value of $\mu^{*}$=0.09 at pressures
of 50 and 100 GPa, close to the standard values (0.10-0.14) assumed for most
conventional superconductors [18]. This rules out possible anomalous effects
of Coulomb repulsion which were suggested for yttrium sodalite hydride [5].

The $T_{\text{c}}$ obtained from the fully anisotropic solution
($T_{\text{c}}$=126 K) is extremely close to the isotropic one
($T_{\text{c}}$=122 K). The anisotropy of the superconducting gap is in fact
very limited 444We also checked the dependence of $\mu^{*}$ on Tc and found
that a variation of 0.01 in $\mu^{*}$ changes Tc only by 2-3 K (See SM Fig. S9
for more details) [37]. The distribution of the calculated superconducting gap
on the Fermi surface is shown in the right panel of Fig. 4. Indeed, with the
exception of a hotspot around the X point, which has a negligible weigth in
reciprocal space, the gap is rather uniform, with a deviation of $\pm$ 15%
around its mean value $\Delta_{\text{avg}}$ = 26 meV. The mean value differs
from the isotropic average $\Delta_{\text{iso}}=$ 23.5 meV, as the isotropic
average is affected by the fact that large values of the gap have a small
weight in reciprocal space. The calculated BCS parameter $\displaystyle
2\Delta_{\text{iso}}(0)/T_{c}$ is 4.3, confirming the strong-coupling nature
of superconductivity in LaBH8.

Figure 3: Left panel: phonon dispersions of LaBH8 at 50 GPa (black thin lines), decorated with the e-ph coupling (red thick lines). Center panel: atom-projected (colored filled lines) and total (black line) Eliashberg function, and its first inverse moment $\lambda(\omega)$ (dashed black line). Right panel: atom-projected (colored filled lines) and total (black line) phonon density of states. Projection onto La, B, and H is shown in green, orange, and blue, respectively. P | $N(E_{F})$ | $\lambda$ | $\omega_{\text{log}}$ | T${}_{c}^{\text{AME}}$ | T${}_{c}^{\text{IME}}$ | $\Delta_{\text{iso}}$ |  $\Delta$H
---|---|---|---|---|---|---|---
(GPa) | $(eV^{-1})$ |  | (meV) | (K) | (K) | (meV) | (meV/at)
50 | 0.62 | 1.54 | 71 | 126 | 122 | 23.5 | 125
75 | 0.60 | 1.06 | 91 | 101 | 96 | 16.8 | 71
100 | 0.56 | 0.64 | 88 | 42 | 32 | 5.6 | 23
Table 1: Summary of the main superconducting properties of LaBH8 at 50 and 100
GPa. The DOS at the Fermi level $N(E_{F}$ in the second column is in units of
$eV^{-1}spin^{-1}$. AME and IME correspond to solutions of the anisotropic and
isotropic Migdal-Eliashberg equations, respectively. The $T_{\text{c}}$ is
calculated with $\mu^{*}=0.09$. $\Delta_{\text{iso}}$ represents the isotropic
average of the superconducting gap. The last column describes the enthalpy per
atom (including zero-point energy) above the convex hull for the LaBH8
$Fm\bar{3}m$ phase. Figure 4: Fermi surface of LaBH8 at 50 GPa. Left:
decorated with hydrogen character, right: decorated with the value of the
superconducting gap. The color scale goes from zero to the maximum value of
the H character (0 to 0.75), and the gap (0 to 42 meV), respectively.

Having established that the superconducting properties of LaBH8 at 50 GPa are
extremely promising, we further studied their behavior as a function of
pressure, computing the electron-phonon spectra at 75 and 100 GPa, and solving
the corresponding Eliashberg equations. The main results are summarized in
Tab. 1, and more details are provided in SM Fig. S8 [37]. The main effect of
an increase in pressure is a rather uniform shift of all phonon frequencies to
higher values, which causes a decrease of $\lambda$ and an increase of
$\omega_{log}$. The $T_{2g}^{*}$ mode around $\Gamma$, which at 50 GPa has a
frequency of 50 meV, responds more strongly to pressure than the rest of the
spectrum, causing a small, counterintuitive decrease of $\omega_{\text{log}}$
between 75 and 100 GPa. The same mode drives the system towards a dynamical
instability when pressure is decreased below 50 GPa.

At the harmonic level, the instability occurs at 35 GPa. Vibrations involving
hydrogen and, in general, light elements, exhibit strong anharmonic and
quantum effects [51, 42], which may severely affect the dynamical stability
and/or superconducting properties of hydrides. In LaBH8, the soft $T^{*}_{2g}$
mode is also the only strongly anharmonic one. Hence, in order to estimate the
importance of anharmonic effects in $Fm\bar{3}m$-LaBH8, We recomputed the
frequency of the $T_{2g}^{*}$ mode, solving the Schrödinger equation
numerically as a function of pressure, as described in Ref. [49]. We estimated
that the difference between the harmonic and anharmonic frequencies is almost
constant with pressure, and equal to $\sim 10$ meV. This causes a $\sim 5$ GPa
shift of the stability pressure to 40 GPa (See Figs. S11 and S12 of the SM for
more details [37]), and a negligible effect on the critical temperature (See
Fig. S13 of the SM [37]). The stabilization pressure of LaBH8 represents a new
minimum at which a high-$T_{\text{c}}$ superhydride is predicted to be stable,
beating the previous record of 70 GPa in YbH6 [21]. We believe that the main
reason behind the low stabilization pressure of LaBH8 is chemical pressure.

In conclusion, using a evolutionary crystal structure prediction and ab-initio
Migdal Eliashberg theory we predicted a new ternary hydride phase with LaBH8
stoichiometry and $Fm\bar{3}m$ space group, which is a conventional HTSC at
moderate pressures, with a $T_{\text{c}}$ of 126 K at 50 GPa. According to our
estimate, this structure could be synthesized by means of laser heating at a
pressure around 100 GPa, and remains dynamically stable down to 40 GPa, where
a single zone-center phonon mode drives a structural instability.

LaBH8 is the first conventional superconductor with $T_{\text{c}}$ above
liquid nitrogen boiling point that can be stabilized down to 50 GPa. Its
exceptional superconducting properties can be understood as deriving from a
metallic hydrogen lattice, which is stabilized at low pressures by a boron and
lanthanum scaffolding. The combination of two elements with different atomic
sizes turns out to be a very effective way to boost chemical pressure on the
interstitial hydrogen sublattice. In general, our results demonstrate an
effective new strategy to lower the stabilization pressure of binary hydrides.
It is likely that the $XY$H8 $Fm\bar{3}m$ structure may be tuned to attain
even better performances, through a careful choice of the $X$,$Y$ elements.
The possibility of stabilizing a superhydride to this pressure represents a
giant leap towards hydride-based superconductivity at room pressure.

##  I Acknowledgments

This work was supported by the Austrian Science Fund (FWF) Projects No. P
30269-N36 (Superhydra), the dCluster of the Graz University of Technology and
the VSC3 of the Vienna University of Technology. L.B. acknowledges support
from Fondo Ateneo- Sapienza 2017,2018 and 2019. C. H. acknowledges support
from the Austrian Science Fund (FWF) Project No. P 32144-N36 and the VSC4 of
the Vienna University of Technology. The authors would like to thank Antonio
Sanna for the useful suggestions and for kindly sharing the code to solve the
isotropic Migdal-Eliashberg equations.

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