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##  Appendix A Proofs of approximate factorization of Gibbs states in 1D

This appendix is devoted to the proofs of the new results appearing in Section
2.4, in connection to the approximate factorisation of Gibbs states in 1D.

###  A.1 Proof of Lemma 2.4

Let us recall that we want to prove the following inequality:

$\norm{\tr_{AC}[\rho^{A}\otimes\rho^{C}E_{AB,C}\,E_{A,BC}]^{-1}}_{\infty}\,\leq\,\mathcal{C}\,.$
(69)

Before providing its proof, as a technical tool we introduce the following
measure of locality

$\norm{T}_{I}:=\inf\\{\norm{T-P}_{\infty}:P\in\operatorname{\mathcal{A}}_{I}\\}$

and the following Lemma, whose proof can be found in [15].

######  Lemma A.1.

[15, Lemma 4.1] Let $T,T^{\prime}$ be two observables. Then

$\|TT^{\prime}\|_{I}\leq
2(\|T\|_{I}\|T^{\prime}\|_{\infty}+\|T\|_{\infty}\|T^{\prime}\|_{I})$

and if $T$ is positive definite

$\|T^{-1}\|_{I}\leq 2\|T^{-1}\|_{\infty}^{2}\|T\|_{I}.$

For $T$ an observable on a bipartite system $AB$, $\rho_{A}$ any state on $A$,
$I\subset AB$, and $I^{\prime}=I\cap B$, we also have

$\|\tr_{A}[\rho_{A}T]\|_{I^{\prime}}\leq\|T\|_{I}$

###### Proof.

The proof of the last point has been omitted in the reference, but follows
straightforwardly from the contractiveness of the operator norm. By
compactness, let $P\in\operatorname{\mathcal{A}}_{I}$ be an operator such that
$\|T\|_{I}=\|T-P\|_{\infty}$. Then,

$\|\tr_{A}[\rho_{A}T]\|_{I^{\prime}}=\leq\|\tr_{A}[\rho_{A}T]-\tr_{A}[\rho_{A}P]\|_{\infty}\leq\|T-P\|_{\infty}=\|T\|_{I}.$

∎

###### Proof of Lemma 2.4.

The proof of this lemma closely follows the arguments of the third point of
[15, Corollary 4.4], however, due to subtle differences in the statement we
reprove it here. The original strategy used that the operator $Q$ is almost
local in a small interval (after a partial trace and identification of
neighbouring intervals), a preliminary that we replace with $Q$ being
approximately supported on two small _but distant_ intervals. First, note that

$\displaystyle\tr_{AC}[\rho^{A}\otimes\rho^{C}Q]$
$\displaystyle=\tr_{AC}[\rho^{A}\otimes\rho^{C}e^{-H_{ABC}}e^{H_{A}+H_{B}+H_{C}}]$
$\displaystyle=e^{\frac{1}{2}H_{B}}\tr_{AC}[\rho^{A}\otimes\rho^{C}F_{A,B}F_{AB,C}F_{AB,C}^{*}F_{A,B}^{*}]e^{-\frac{1}{2}H_{B}}$
$\displaystyle=e^{\frac{1}{2}H_{B}}\tr_{AC}[\rho^{A}\otimes\rho^{C}F]e^{-\frac{1}{2}H_{B}}\,,$

with $F_{X,Y}=e^{-\frac{1}{2}H_{XY}}e^{\frac{1}{2}(H_{X}+H_{Y})}$ and
$F=F_{A,B}F_{AB,C}F_{AB,C}^{*}F_{A,B}^{*}\geq 0$. By Proposition 2.1, we know
that each factor $F_{X,Y},F_{X,Y}^{-1}$ is bounded in norm by $\mathcal{G}$
(for $\Phi/2$), and thus
$\norm{F}_{\infty},\norm{F^{-1}}_{\infty}\leq\operatorname{\mathcal{G}}^{4}$.

Let us define the intervals $I_{n}$ as neighbourhoods of the cuts $AB$ and
$BC$ and $I^{\prime}_{n}$ as $I_{n}$ reduced to $B$. More precisely if
$l_{\cdot}$ and $u_{\cdot}$ denote the lower and upper boundaries of an
interval, we define
$I_{n}=\\{u_{A}-n,\ldots,l_{B}+n\\}\cup\\{u_{B}-n,\ldots,l_{C}+n\\}$. Our
objective now is to minimize the impact of $F$ by approximating it solely in
terms of local operators on these sets. Subsequently, we will demonstrate that
by performing a partial trace, these operators are further reduced to act non-
trivially solely on $I^{\prime}_{n}=B\cap I_{n}$.

Using Lemma A.1 and Proposition 2.1 we can derive

$\displaystyle\norm{F}_{I_{n}}$
$\displaystyle=\norm{F_{A,B}F_{AB,C}F_{AB,C}^{*}F_{A,B}^{*}}_{I_{n}}$
$\displaystyle\leq
4\operatorname{\mathcal{G}}^{3}\left(\|F_{A,B}\|_{I_{n}}+\|F_{AB,C}\|_{I_{n}}+\|F_{AB,C}^{*}\|_{I_{n}}+\|F_{A,B}^{*}\|_{I_{n}}\right)$
$\displaystyle\leq
16\operatorname{\mathcal{G}}^{3}\frac{\operatorname{\mathcal{G}}^{n}}{(\lfloor
n/R\rfloor+1)!}.$

Using Lemma A.1 and the fact that the map
$\tr_{AC}[\rho^{A}\otimes\rho^{C}\cdot]$ is positive unital we have

$\displaystyle\|F^{\prime}\|_{I^{\prime}_{n}}$
$\displaystyle\leq\|F\|_{I_{n}}\leq
16\operatorname{\mathcal{G}}^{3}\frac{\operatorname{\mathcal{G}}^{n}}{(\lfloor
n/R\rfloor+1)!}$ $\displaystyle\|F^{\prime-1}\|_{\infty}$
$\displaystyle\leq\|F^{-1}\|_{\infty}\leq\operatorname{\mathcal{G}}^{4},$

where $F^{\prime}=\tr_{AC}[\rho^{A}\otimes\rho^{C}F]$. Employing Lemma A.1
again

$\|F^{\prime-1}\|_{I^{\prime}_{n}}\leq
2\|F^{\prime-1}\|^{2}_{\infty}\|F^{\prime}\|_{I^{\prime}_{n}}\leq
32\operatorname{\mathcal{G}}^{11}\frac{\operatorname{\mathcal{G}}^{n}}{(\lfloor
n/R\rfloor+1)!}.$

The next step is completely analogous to [15, Lemma 4.2] with the only
deviation being that [15, Proposition 3.2] involving uniform estimates for the
complex time evolution of any observable does not apply directly.

Let us denote by $P_{n}$ a collection of $I^{\prime}_{n}$-local operators
fulfilling $\|F^{\prime-1}-P_{n}\|_{\infty}=\|F^{\prime-1}\|_{I^{\prime}_{n}}$
and $S_{1}=P_{1}$, $S_{n}=P_{n}-P_{n-1}$. This implies the bounds

$\|S_{1}\|_{\infty}\leq\|F^{\prime-1}\|_{I^{\prime}_{1}}+\|F^{\prime-1}\|_{\infty}\qquad\textrm{
and
}\qquad\|S_{n}\|_{\infty}\leq\|F^{\prime-1}\|_{I^{\prime}_{n}}+\|F^{\prime-1}\|_{I^{\prime}_{n-1}}\,.$

Note that for $n\geq\lfloor|B|/2\rfloor+1$, $P_{n}=F^{\prime-1}$ and thereby
$S_{n+1}=0$. We start with a decomposition

$F^{\prime-1}=\sum_{n=1}^{\lfloor|B|/2\rfloor+1}S_{n}\,,$

where we stop the local decomposition once the two distinct intervals start
overlapping. Our goal is to bound

$e^{\frac{1}{2}H_{B}}F^{\prime-1}e^{-\frac{1}{2}H_{B}}$

by individually bounding the norms of the imaginary time-evolved $S_{n}$. To
this end, let us note that there exists an orthogonal decomposition for all
$n\leq\lfloor|B|/2\rfloor$

$S_{n}=\sum_{i=1}^{d^{4n}}S^{l}_{n,i}\otimes S^{r}_{n,i}\,.$

Such a decomposition further possesses the property that $\|S^{l}_{n,i}\otimes
S^{r}_{n,i}\|_{\infty}\leq d^{2n}\|S_{n}\|_{\infty}$, which is derived from
the orthogonality of the $S^{l}_{n,i}\otimes S^{r}_{n,i}$ and the inequality:

$\|\cdot\|_{\infty}\leq\|\cdot\|_{2}\leq\sqrt{d^{\prime}}\|\cdot\|_{\infty}\,.$

Here, $d^{\prime}$ is the dimension of the underlying Hilbert space. We can
now use that the time evolution is a group automorphism on each term
individually and that we have [12, Proposition 3.2] allowing us to estimate

$\displaystyle\left\|e^{\frac{1}{2}H_{B}}S^{l}_{n,i}\otimes
S^{r}_{n,i}e^{-\frac{1}{2}H_{B}}\right\|_{\infty}$
$\displaystyle\leq\left\|e^{\frac{1}{2}H_{B}}FS^{l}_{n,i}e^{-\frac{1}{2}H_{B}}\right\|_{\infty}\left\|e^{\frac{1}{2}H_{B}}S^{r}_{n,i}e^{-\frac{1}{2}H_{B}}\right\|_{\infty}$
$\displaystyle\leq\operatorname{\mathcal{G}}^{2n}d^{2n}\|S_{n}\|_{\infty}$

At last for $n=\lfloor|B|/2\rfloor+1$, we simply regard $R_{n}$ as an
$B$-local operator, so the directly applying [12, Proposition 3.2] yields

$\left\|e^{\frac{1}{2}H_{B}}S_{n}e^{-\frac{1}{2}H_{B}}\right\|_{\infty}\leq\operatorname{\mathcal{G}}^{|B|}\|S_{n}\|_{\infty}\,.$

Putting everything together, we have

$\displaystyle\left\|e^{\frac{1}{2}H_{B}}F^{\prime-1}e^{-\frac{1}{2}H_{B}}\right\|_{\infty}$
$\displaystyle\leq
32d^{6}\operatorname{\mathcal{C}}^{2}(\operatorname{\mathcal{G}}^{12}+\operatorname{\mathcal{G}}^{4})+\left(32\operatorname{\mathcal{G}}^{11}\sum_{n=2}^{\lfloor|B/2|\rfloor+1}\operatorname{\mathcal{G}}^{2n}d^{6n}\frac{\operatorname{\mathcal{G}}^{n}}{(\lfloor
n/R\rfloor+1)!}\right)$

and the statement follows by combining constants. ∎

###  A.2 Proof of Lemma 2.5

Next, we prove that, for a Gibbs state $\rho^{A^{\prime}ABCC^{\prime}}$ on a
finite chain $A^{\prime}ABCC^{\prime}$, the following holds.

$\left\|\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}-\operatorname{\mathds{1}}\right\|_{\infty}<\mathcal{C}^{4}\,\left\|\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}-\operatorname{\mathds{1}}\right\|_{\infty}\,,$
(70)

###### Proof.

We absorb $\beta$ in the interaction, so the argument will be made for
$\beta=1$. Let us rewrite

$\displaystyle\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}$
$\displaystyle\hskip
56.9055pt=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}ABCC^{\prime}}\right]^{-1}$
$\displaystyle\hskip
56.9055pt=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}\,\rho_{A^{\prime}ABCC^{\prime}}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}ABCC^{\prime}}\right]^{-1}$
$\displaystyle\hskip
56.9055pt=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}\,e^{-H_{A^{\prime}ABCC^{\prime}}}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[e^{-H_{A^{\prime}ABCC^{\prime}}}\right]^{-1}$
$\displaystyle\hskip
56.9055pt=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}\,e^{-H_{A^{\prime}ABCC^{\prime}}}\,e^{H_{ABC}}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[e^{-H_{A^{\prime}ABCC^{\prime}}}\,e^{H_{ABC}}\right]^{-1}\,.$

Recalling that

$E_{A^{\prime}ABC,C^{\prime}}:=e^{-H_{A^{\prime}ABCC^{\prime}}}\,e^{H_{A^{\prime}ABC}+H_{C^{\prime}}}\;,\quad
E_{A^{\prime},ABC}:=e^{-H_{A^{\prime}ABC}}\,e^{H_{A^{\prime}}+H_{ABC}}\,,$

then, from the above identity, we deduce that

$\displaystyle\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}$
$\displaystyle=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho_{AB}\,\rho_{B}^{-1}\rho_{BCD}\,\rho_{ABCD}^{-1}\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\,e^{-H_{A^{\prime}}-H_{C^{\prime}}}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\,e^{-H_{A^{\prime}}-H_{C^{\prime}}}\right]^{-1}$
$\displaystyle=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho^{A^{\prime}}\otimes\rho^{C^{\prime}}\,\rho_{AB}\,\rho_{B}^{-1}\rho_{BCD}\,\rho_{ABCD}^{-1}\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho^{A^{\prime}}\otimes\rho^{C^{\prime}}\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\right]^{-1}$

Thus

$\displaystyle\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}-\operatorname{\mathds{1}}$
$\displaystyle\hskip
65.44142pt=\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho^{A^{\prime}}\otimes\rho^{C^{\prime}}\,Q\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\right]\,\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho^{A^{\prime}}\otimes\rho^{C^{\prime}}\,\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\right]^{-1}\,,$

where

$Q:=\rho_{A^{\prime}AB}\,\rho_{B}^{-1}\rho_{BCC^{\prime}}\,\rho_{A^{\prime}ABCC^{\prime}}^{-1}-\operatorname{\mathds{1}}\,.$

By Proposition 2.2, we have

$\norm{\operatorname{tr}_{A^{\prime}C^{\prime}}\left[\rho^{A^{\prime}}\otimes\rho^{C^{\prime}}\,Q\,E_{A^{\prime}ABC,C^{\prime}}\,E_{A^{\prime},ABC}\right]}_{\infty}\leq\norm{E_{A^{\prime}ABC,C^{\prime}}}_{\infty}\norm{E_{A^{\prime},ABC}}_{\infty}\norm{Q}_{\infty}\leq\mathcal{C}^{2}\norm{Q}_{\infty}\,,$

and for the term with the inverse we have an upper bound of $\mathcal{C}^{2}$
by Lemma 2.4. Therefore,

$\left\|\rho_{AB}\,\rho_{B}^{-1}\rho_{BC}\,\rho_{ABC}^{-1}-\operatorname{\mathds{1}}\right\|_{\infty}\leq\mathcal{C}^{4}\,\|Q\|_{\infty}\,.$

∎

##  Appendix B Extension of results to exponentially-decaying interactions

In this appendix, we extend the main results of this manuscript to the
framework of exponentially-decaying interactions. All results presented in
Section 2.4 admit extensions to this setting, albeit with some small
modifications in the estimates. Here, we collect the generalisations of all
these technical tools in the context of exponentially-decaying interactions,
referring to the text where these results can be found for some, or proving
them here for others. Next, we use these results to generalise the main
results of this manuscript (Theorem 3.6, Theorem 4.11 and Theorem 4.9) to
exponentially-decaying interactions. To increase readability, we will only
highlight those parts of the proofs that differ from the finite-range
counterpart.

Consider an interaction on $\Sigma\subseteq\mathbb{Z}$. To define
exponentially-decaying interactions, we introduce for each $\lambda,\mu>0$ the
following notation

$\|\Phi\|_{\lambda,\mu}:=\sup_{x\in\Sigma}\sum_{X\Subset\Sigma:x\in
X}\|\Phi({X})\|e^{\lambda|X|+\mu\operatorname{diam}(X)}\in[0,\infty]\,.$ (71)

We say that $\Phi$ is exponentially decaying, if there exist $\mu,\lambda>0$
such that $\norm{\Phi}_{\lambda,\mu}<\infty$. Note that the finite range case
is a particular one of this. Given a finite interval $\Lambda\Subset\Sigma$
split into $X$ and $Y$, the expansional of a Hamiltonian on $X,Y$ is defined
analogously to Eq. (24), and the estimates of Proposition 2.1 extend
similarly, as shown in [64], with the only caveat that point (ii) just
presents exponential decay with $\ell$, rather than superexponential. Here we
recall the simplified formulation of [21, Lemma 31].

######  Proposition B.1 ([64, Theorem 3.1]).

Let $\Phi$ be an exponentially-decaying interaction which is further
translation invariant. Then the following hold:

  1. (i)

There is an absolute constant $\tilde{\mathcal{G}}>1$ depending only on $\mu$,
$\lambda$ and $\beta$ such that, for any finite interval
$\Lambda=XY\Subset\mathbb{Z}$ split into two subintervals $X$ and $Y$, we
have:

$\norm{E_{X,Y}}\,,\,\norm{E_{X,Y}^{-1}}\,\,\leq\,\,\tilde{\mathcal{G}}\,.$

  2. (ii)

There are positive constants $\mathcal{K},\alpha>0$ depending on $\mu$,
$\lambda$ and $\beta$ such that if we add two intervals $\widetilde{X}$ and
$\widetilde{Y}$ adjacent to $X$ and $Y$, respectively, so that we get a larger
interval $\widetilde{X}XY\widetilde{Y}$, then

$\left\|E_{X,Y}^{-1}-E^{-1}_{\widetilde{X}X,Y\widetilde{Y}}\right\|,\left\|E_{X,Y}-E_{\widetilde{X}X,Y\widetilde{Y}}\right\|\,\leq\,\mathcal{K}e^{-\alpha\ell}\,.$

for any $\ell\in\mathbb{N}$ such that $\ell\,\leq\,|X|\,,\,|Y|$.

Next, note that Lemma A.1 is independent of the range of the interactions.
Note further that Proposition 2.2 relies on estimates for expansionals and
certain contractions thereof which allows us to conclude its proof for
exponentially-decaying interactions directly from Lemma A.1 and Point (i) of
Proposition B.1. The statement can be found in Proposition B.2. By similar
arguments, we conclude Lemma 2.4 in the exponentially-decaying setting.

######  Proposition B.2.

In the conditions above, there is an absolute constant $\tilde{\mathcal{C}}>1$
depending only on $\mu,\lambda$ and $\beta$ such that

$\displaystyle\big{\|}\tr_{B}[\rho^{B}Q]\big{\|},\big{\|}\tr_{B}[\rho^{B}Q]^{-1}\big{\|}\,\leq\,\tilde{\mathcal{C}}\,\,,\quad\quad
Q\in\\{E_{B,C}^{*}\,,\,E_{B,C}\,,\,E_{A,B}^{*}\,,\,E_{A,B}\\}\,,$ (72)
$\displaystyle\big{\|}\tr_{AB}[\rho^{AB}Q]\big{\|}\,,\,\big{\|}\tr_{AB}[\rho^{AB}Q]^{-1}\big{\|}\,\leq\,\tilde{\mathcal{C}}\,\,,\quad\quad
Q\in\\{E_{A,B}^{*\,-1}\,,\,E_{A,B}^{\,-1}\\}\,,$ (73)
$\displaystyle\big{\|}\tr_{B}\big{[}\rho^{B}E_{A,B}^{*}E_{AB,C}^{*}\big{]}\big{\|}\,,\,\big{\|}\tr_{B}\big{[}\rho^{B}E_{A,B}^{*}E_{AB,C}^{*}\big{]}^{-1}\big{\|}\,\leq\,\tilde{\mathcal{C}}\,,$
(74)
$\displaystyle\norm{\tr_{AC}[\rho^{A}\otimes\rho^{C}E_{AB,C}\,E_{A,BC}]^{-1}}_{\infty}\,\leq\,\tilde{\mathcal{C}}\,.$
(75)

The next result constitutes a generalisation of [15, Theorem 5.1] to the
setting of exponentially-decaying interactions. Its proof completely follows
that of its finite-range counterpart, and thus we only provide a very brief
sketch of it, highlighting the differences.

######  Theorem B.3.

With $\Phi$ on $Z$ an exponentially-decaying interaction that is further
translation-invariant, there are positive constants
$\tilde{K},\tilde{\alpha}>0$ depending only on $\lambda,\mu$ and $\beta$ such
that for every $\Lambda\Subset\operatorname{\mathbb{Z}}$ split into three
subintervals $\Lambda=ABC$, where $B$ shields $A$ from $C$, for its local
Gibbs state $\rho^{\Lambda}=:\rho_{ABC}$ it holds that

$\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}\leq\tilde{K}e^{-\tilde{\alpha}|B|}\,.$
(76)

###### Sketch of the proof.

From [15, Eq. (19)], we can rewrite

$\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}=E_{A,BC}\tr_{A}[E_{A,BC}\rho^{A}]^{-1}\tr_{A}[\tilde{E}_{A,BC}\rho^{A}]\tilde{E}_{A,BC}\,,$

with

$\tilde{E}_{A,BC}=\tr_{C}[e^{-H_{ABC}}]\tr_{C}[e^{-H_{BC}}]^{-1}e^{H_{A}}\,.$

Now, the conclusion of the proof is a combination of two statements:

  1. 1.

Statement: The four factors in the right-hand side of the previous expression
and their inverses are uniformly bounded. This is a consequence of Proposition
B.1 and Proposition B.2. The universal constant is given by
$\tilde{\mathcal{G}}\,\tilde{\mathcal{C}}^{2}$.

  2. 2.

Statement: The following inequalities hold,

$\displaystyle\norm{\tilde{E}_{A,BC}-E_{A,BC}}$
$\displaystyle\leq\tilde{K}^{\prime}e^{-\tilde{\alpha}^{\prime}|B|}\,,$
$\displaystyle\norm{\tr_{A}[\tilde{E}_{A,BC}\rho^{A}]-\tr_{A}[E_{A,BC}\rho^{A}]}$
$\displaystyle\leq\tilde{K}^{\prime}e^{-\tilde{\alpha}^{\prime}|B|}\,.$

The second inequality follows from the first one by contractivity. For the
first one, we split the terms in the same way as in the proof of [15],
obtaining $\tilde{K}^{\prime}=4\,\mathcal{C}\,\mathcal{K}\,e^{-\alpha}$ and
$\tilde{\alpha}^{\prime}=\alpha/2$.

We conclude by taking $\tilde{\alpha}=\tilde{\alpha}^{\prime}$ and
$\tilde{K}=8\,\tilde{\mathcal{C}}^{7}\tilde{\mathcal{G}}^{3}\mathcal{K}e^{-\alpha}$.
∎

Next, one can immediately conclude with Lemma 2.5 for exponentially-decaying
interactions as of Eq. (75). This, jointly with Theorem B.3, allows us to
deduce a generalisation of Theorem B.3 to the case where the Gibbs state is
given on $\Lambda=A^{\prime}ABCC^{\prime}$ and but we only compare its
marginals on $ABC,AB,B$ and $BC$, respectively. Namely, we can show that, in
this context, the following inequality holds:

$\norm{\rho_{ABC}\rho_{BC}^{-1}\rho_{B}\rho_{AB}^{-1}-\operatorname{\mathds{1}}}_{\infty}\leq\tilde{C}^{4}\tilde{K}e^{-\tilde{\alpha}|B|}\,.$
(77)

To conclude the preliminaries for exponentially-decaying interactions, let us
recall that the exponential decay of correlations and local
indistinguishability hold at any positive temperature for the considered
setting. The proofs of these results can be found in [21, Theorem 26] and [21,
Theorem 28], respectively, as a consequence of [64, Theorem 4.4].

######  Proposition B.4 (Decay of correlations and local
indistinguishability).

For $\Phi$ an exponentially-decaying, translation invariant interaction over
$\operatorname{\mathbb{Z}}$, the following inequalities hold for universal
constants $c,c^{\prime},\gamma,\gamma^{\prime}>0$ depending only on
$\lambda,\mu$ and $\beta$:

$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{A}\,O_{C}]-\Tr_{ABC}[\rho^{ABC}O_{A}]\Tr_{ABC}[\rho^{ABC}O_{C}]\right|$
$\displaystyle\leq\norm{O_{A}}_{\infty}\norm{O_{C}}_{\infty}\,c\,e^{-\alpha|B|}\,,$
$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{A}]-\Tr_{AB}[\rho^{AB}O_{A}]\right|$
$\displaystyle\leq\norm{O_{A}}_{\infty}\,c^{\prime}\,e^{-\gamma^{\prime}|B|}\,,$
$\displaystyle\left|\Tr_{ABC}[\rho^{ABC}O_{C}]-\Tr_{BC}[\rho^{BC}O_{C}]\right|$
$\displaystyle\leq\norm{O_{C}}_{\infty}\,c^{\prime}\,e^{-\gamma^{\prime}|B|}\,.$

###  B.1 Exponential decay of the BS-CMI for exponentially-decaying
interactions

First, note that most estimates from Lemma 3.5 follow straightforwardly in the
context of exponentially-decaying interactions, only the last one differs. For
that, we have as before

$\displaystyle\norm{\rho_{B}^{-1}}_{\infty}$
$\displaystyle=\left\|(\rho^{B})^{-1}\tr_{AC}[E_{A,BC}E_{B,C}\rho^{A}\rho^{C}]\right\|_{\infty}\frac{Z_{ABC}}{Z_{A}Z_{B}Z_{C}}\,,$

with the difference that now

$\displaystyle\norm{(\rho^{B})^{-1}}_{\infty}\leq
e^{2\norm{H_{B}}_{\infty}+\log(d)}\leq
e^{2|B|\norm{\Phi}_{\lambda,\mu}+\log(d)}\,.$

Having this in mind, we can show the following bounds for the decays of the
various versions of the BS-CMI.

######  Theorem B.5.

Let $\Phi$ an exponentially-decaying, translation invariant interaction over
$\operatorname{\mathbb{Z}}$ and $\Lambda\Subset\operatorname{\mathbb{Z}}$ any
interval split into consecutive parts $\Lambda=A^{\prime}ABCC^{\prime}$, with
$A^{\prime}$ and $C^{\prime}$ possibly empty. Then for the marginal on $ABC$
of its local Gibbs state
$\tr_{A^{\prime}C^{\prime}}[\rho^{\Lambda}]=\rho_{ABC}$ there exist universal
constants $c_{\cdot}>0$ and $\alpha_{\cdot}>0$ independent of $\Lambda$ and
only dependent on $\lambda,\mu$ and $\beta$ such that

$\widehat{I}^{\operatorname{os}}_{\rho}(A:C|B)\leq\tilde{c}_{\operatorname{ots}}e^{\tilde{\alpha}_{\operatorname{ots}}(|A|)}e^{(\tilde{\alpha}_{\operatorname{ots}}-\tilde{\alpha})|B|}\,,$
(78)

and

$\widehat{I}^{\operatorname{ts}}_{\rho}(A:C|B)\leq\tilde{c}_{\operatorname{ots}}e^{\tilde{\alpha}_{\operatorname{ots}}(|A|)}e^{(\tilde{\alpha}_{\operatorname{ots}}-\tilde{\alpha})|B|}\,.$
(79)

And further with slightly modified constants

$\widehat{I}_{\rho}^{\operatorname{rev}}(A:C|B)\leq\tilde{c}_{\operatorname{rev}}e^{\tilde{\alpha}_{\operatorname{rev}}(|A|)}e^{(\tilde{\alpha}_{\operatorname{rev}}-\tilde{\alpha})|B|}\,.$
(80)

######  Remark B.6.

Note that, whenever $\tilde{\alpha}>\tilde{\alpha}_{\operatorname{ots}}$,
respectively $\tilde{\alpha}>\tilde{\alpha}_{\operatorname{rev}}$, there is
exponential decay of the corresponding BS-CMI with $|B|$. These constants can
be explicitly estimated in the following way

  1. 1.

For $\tilde{\alpha}_{\operatorname{ots}}$, it follows from the analogue of
Lemma 3.5 for exponentially-decaying interactions. In the latter case, its
finite-range counterpart could be chosen as
${\alpha}_{\operatorname{ots}}=3J\beta+\log d$, where now we are making the
dependence with $\beta$ explicit (see Lemma 3.5). In the current case, we can
analogously take
$\tilde{\alpha}_{\operatorname{ots}}=3\norm{\Phi}_{\lambda,\mu}\equiv
3\beta\norm{\Phi^{0}}_{\lambda,\mu}+\log d$. We use $\Phi^{0}$ to denote the
interaction without dependence on $\beta$, so that we can make the linear
scaling with $\beta$ for $\tilde{\alpha}_{\operatorname{ots}}$ explicit.

  2. 2.

In the case of $\tilde{\alpha}_{\operatorname{rev}}$, we see similarly that
its finite-range counterpart is ${\alpha}_{\operatorname{rev}}=2J\beta+\log
d$, and the exponentially-decaying one thus
$\tilde{\alpha}_{\operatorname{rev}}=2\beta\norm{\Phi^{0}}_{\lambda,\mu}+\log
d$.

  3. 3.

Note now that $\tilde{\alpha}$ results from Eq. (77), and is given by
$\alpha/2$, with $\alpha$ from Point (ii) of Proposition B.1, which in turn is
extracted from [64, Eq. (32)], where it is shown to be
$\alpha=\lambda-\operatorname{\mathcal{O}}(\beta)$.

We leave the explicit description of the relation between $\beta$ and
$\lambda,\mu$ and $d$ for which the latter inequalities between the
$\tilde{\alpha}$’s hold for future work.

###  B.2 Efficient estimation of the global purity for exponentially-decaying
interactions

The objective of this subsection is to expand on the findings about the
efficient evaluation of the overall purity of a Gibbs state on a translation-
invariant spin chain in the context of interactions that decay exponentially.
To achieve this, we will apply each of the necessary technical results to this
context. However, most of the results have similar proofs to the finite-range
case, so we will skip those parts for simplicity and only highlight the
differences.

First, we can prove the following proposition, which is the generalisation of
Proposition 3.9.

######  Proposition B.7.

Let $\Phi$ be an exponentially-decaying, translation-invariant interaction.
Then there exist positive constants $\tilde{c}_{p},\tilde{\alpha}_{p}$
depending only on $\lambda,\mu$ and $\beta$ such that, for every
$\Lambda\Subset\mathbb{Z}$ split as $\Lambda=ABC$ and for
$\rho_{\Lambda}:=\rho^{\Lambda}$ the Gibbs state on $\Lambda$, the following
holds.

$\left|\frac{\Tr_{AB}[\rho_{AB}^{2}]\Tr_{BC}[\rho_{BC}^{2}]}{\Tr_{\Lambda}[\rho_{\Lambda}^{2}]\Tr_{B}[\rho_{B}^{2}]}-1\right|\leq\tilde{c}_{p}e^{-\tilde{\alpha}_{p}|B|}\,.$
(81)

###### Sketch of the proof.

The proof is completely analogous to that of Proposition 3.9. Let us recall
that it follows from the fact that we estimate the left-hand side of Eq. (81)
by:

$\left|\widetilde{\lambda}_{ABC}^{-1}-1\right|+\left|\widetilde{\lambda}_{ABC}^{-1}\right|\left|\chi_{ABC}-1\right|\,,$

where the terms are defined completely analogously to the finite-range case,
but we keep the same notation for simplicity. The estimates of
$\left|\widetilde{\lambda}_{ABC}^{-1}-1\right|$ and
$\left|\widetilde{\lambda}_{ABC}^{-1}\right|$ were shown in [16, Lemma 6.1] to
hold exactly in the same way as for finite range. In the finite range case we
used a combination of estimates on Araki’s expansionals, decay of correlations
and local indistinguishability to conclude the decay of
$\left|\chi_{ABC}-1\right|$. All these properties also hold for exponentially-
decaying interactions as a consequence of Proposition B.1 and Proposition B.4,
hence we can conclude the proof. ∎

Next, note that the global purity of a Gibbs state split into many regions can
be estimated efficiently from this proposition in the same way as in Theorem
4.11. Since the proof is the same as for Theorem 4.11, and follows [79], we
omit it here.

######  Theorem B.8.

Under the conditions of Proposition B.7, there is an algorithm that outputs an
estimate $\hat{P}_{2}(\rho_{1:N})$ such that

$\left|\frac{\hat{P}_{2}(\rho_{1:N})}{\Tr[\rho_{1:N}^{2}]}-1\right|\leq\varepsilon,$
(82)

with a number of samples and classical post-processing cost given by
$\textup{poly}(n/\varepsilon)$.

###  B.3 Learning of Gibbs states via MPO approximations for exponentially-
decaying interactions

In analogy with the previous subsection, the recovery map employed in the
reconstruction of the Gibbs state from its marginals for the MPO approximation
is defined in the same way for exponentially-decaying interactions as for the
finite-range ones. Therefore, the estimate in Theorem 4.4 for the MPO
approximation is still valid, and we omit its explicit formulation in this
context for simplicity. The only change in the learning application would
occur in the analogue of Corollary 4.5, since the bond dimension is inherited
from the decay of the BS-CMI, and thus it is expected to be slightly worse in
this case. Indeed, as a consequence of Theorem B.5, we obtain the following.

######  Corollary B.9.

For a $n$-site marginal of a Gibbs state in one dimension with translation-
invariant, exponentially-decaying interaction $\Phi$ and a given accuracy
$\varepsilon$, whenever $\tilde{\alpha}>2\tilde{\alpha}_{\operatorname{rev}}$
in Theorem B.5, there is an MPO representation of bond dimension

$D=\exp(2\log(d)\tilde{\operatorname{\mathcal{C}}}_{1}\log(n/\varepsilon)),$
(83)

where $\tilde{\operatorname{\mathcal{C}}}_{1}$ is a constant depending on
$\lambda$, $\mu$, and $\beta$. It is given by

$\left\|\left(\bigcirc_{i=1}^{N-1}\operatorname{\mathcal{R}}_{i}\right)(\rho_{1})-\rho_{1:N}\right\|\leq\varepsilon$

choosing the $A_{i}$ as consecutive regions of at least $l$ spins

$|A_{i}|=l\geq\operatorname{\mathcal{C}}_{1}{\log(n/\varepsilon)}\,.$

######  Remark B.10.

Note that the factor $2$ is required in the relation
$\tilde{\alpha}>2\tilde{\alpha}_{\operatorname{rev}}$ because all subregions
$A_{i}$ are chosen above with the same size and we require that the factor
$e^{(\tilde{\alpha}_{\operatorname{rev}}-\tilde{\alpha})|B|}$ outweighs
$e^{\tilde{\alpha}_{\operatorname{rev}}(|A|)}$ in Eq. (80).

##  Appendix C Lifting the upper bound on the DPI of the BS-entropy to
channels

Note that just by restriction of the original Hilbert space to the common
support of $X$ and $Y$, one can obtain the same result as Theorem 3.2 for the
case that $X$ and $Y$ are not full-rank but share the same kernel. In that
case, the inverses are replaced by Moore-Penrose pseudo-inverses. With this at
hand, we can now prove the upper bound on the DPI for the BS-entropy for
quantum channels instead of conditional expectations.

######  Corollary C.1.

For $X,Y\in\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})$ having
the same support and
$\operatorname{\mathcal{T}}:\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{A})\to\operatorname{\mathcal{B}}(\operatorname{\mathcal{H}}_{B})$
a quantum channel, we find that

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{T}}(X)\|\operatorname{\mathcal{T}}(Y))$
$\displaystyle\leq\norm{X^{-1/2}YX^{-1/2}}_{\infty}\norm{X}_{1}\norm{\operatorname{\mathcal{T}}(X)^{1/2}}_{\infty}\norm{\operatorname{\mathcal{T}}(X)^{-1/2}}_{\infty}$
(84)
$\displaystyle\hphantom{\leq}\cdot\norm{\operatorname{\mathcal{T}}(Y)^{-1}\operatorname{\mathcal{T}}(X)}_{\infty}\norm{XY^{-1}\operatorname{\mathcal{T}}^{*}(\operatorname{\mathcal{T}}(Y)\operatorname{\mathcal{T}}(X)^{-1})-\operatorname{\mathds{1}}}_{\infty}$

where inverses are replaced by Moore-Penrose pseudo-inverses in case the
operators are not full-rank.

###### Proof.

Following similar arguments as in [82, 12], we note that by Stinespring’s
dilation theorem, we can write every quantum channel as a composition:

$\operatorname{\mathcal{T}}(\cdot)=\tr_{C}[V\cdot V^{*}]$ (85)

where
$V:\operatorname{\mathcal{H}}_{A}\to\operatorname{\mathcal{H}}_{B}\otimes\operatorname{\mathcal{H}}_{C}$
is an isometry and $\operatorname{\mathcal{H}}_{C}$ an auxiliary space. Using
the isometric invariance of the BS-entropy and its additivity under tensor
products, we get

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{T}}(X)\|\operatorname{\mathcal{T}}(Y))=\widehat{D}(X_{V}\|Y_{V})-\widehat{D}(\pi_{C}\otimes\tr_{C}[X_{V}]\|\pi_{C}\otimes\tr_{C}[Y_{V}])\,,$

where we set $X_{V}=VXV^{*}$ and $Y_{V}=VYV^{*}$. Both of the latter operators
agree in their support, hence with the considerations brought forward at the
beginning of this appendix we utilise Theorem 3.2 to obtain

$\displaystyle\widehat{D}(X\|Y)-\widehat{D}(\operatorname{\mathcal{T}}(X)\|\operatorname{\mathcal{T}}(Y))$
$\displaystyle\leq\norm{X_{V}^{-1/2}Y_{V}X_{V}^{-1/2}}_{\infty}\norm{X_{V}}_{1}\norm{\operatorname{\mathcal{T}}(X)^{1/2}}_{\infty}\norm{\operatorname{\mathcal{T}}(X)^{-1/2}}_{\infty}$
$\displaystyle\hphantom{\leq}\cdot\norm{\operatorname{\mathcal{T}}(Y)^{-1}\operatorname{\mathcal{T}}(X)}_{\infty}\norm{X_{V}Y_{V}^{-1}(\operatorname{\mathcal{T}}(Y)\operatorname{\mathcal{T}}(X)^{-1})\otimes\operatorname{\mathds{1}}_{C}-\operatorname{\mathds{1}}}_{\infty}$

where we already cancelled constants and used isometric invariance of the
Schatten p-norms and Eq. (85) to simplify the expression. Using again
isometric invariance on the remaining terms and identifying
$\operatorname{\mathcal{T}}^{*}$ confirms the claim. ∎


#  High-$T_{c}$ superconducting hydrides formed by LaH24 and YH24 cage
structures as basic blocks.

Peng Song1, Zhufeng Hou2, Pedro Baptista de Castro3,4, Kousuke Nakano1,5,
Kenta Hongo6, Yoshihiko Takano3,4, Ryo Maezono1

1School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292,
Japan

2State Key Laboratory of Structural Chemistry, Fujian Institute of Research on
the Structure of Matter, Chinese Academy of Sciences, Fuzhou 350002, China

3National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki
305-0047, Japan

4University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan

5International School for Advanced Studies (SISSA), Via Bonomea 265, 34136,
Trieste, Italy

6Research Center for Advanced Computing Infrastructure, JAIST, Asahidai 1-1,
Nomi, Ishikawa 923-1292, Japan

###### Abstract

Based on recent studies regarding high-temperature (high-$T_{c}$) La-Y ternary
hydrides (e.g., $P{\bar{1}}$-La2YH12, $Pm{\bar{3}}m$-LaYH12, and
$Pm{\bar{3}}m$-(La,Y)H10 with a maximum $T_{c}\sim 253$ K), we examined the
phase and structural stabilities of the (LaH6)(YH6)y series as high-$T_{c}$
ternary hydride compositions using a genetic algorithm and ab initio
calculations. Our evaluation showed that the $Pm\bar{3}m$-LaYH12 reported in
the previous study was unstable during decomposition into $R\bar{3}c$-LaH6 \+
$Im\bar{3}m$-YH6. We also discovered new crystal structures, namely
$Cmmm$-LaYH12 ($y=1$), $R\bar{3}c$-LaYH12 ($y=1$), $Cmmm$-LaY3H24 ($y=3$), and
$R\bar{3}$-LaY3H24 ($y=3$), showing stability against such decomposition.
While $R\bar{3}c$ ($y=1$) and $R\bar{3}$ ($y=3$) did not exhibit
superconductivity owing to the extremely low density of states at the Fermi
level, $Cmmm$ phases exhibited a $T_{c}$ of approximately 140 K at around 200
GPa owing to the extremely high electron–phonon coupling constant ($\lambda$ =
1.876 for LaYH12). By the twice longer stacking for $Cmmm$-LaY3H24, the
coupling constant increased owing to the chemical pressure of Y, leading to a
slightly increased $T_{c}$.

##  I Introduction

Metal hydrides have always been considered ideal candidates for high-
temperature (high-$T_{c}$) superconductors owing to their ultrahigh phonon
vibration frequency. Ashcroft (2004, 1968) However, realizing
superconductivity by keeping them under the metallicity gradient often
requires extremely high pressures. Ashcroft (2004, 1968); Eremets  _et al._
(2008); Drozdov  _et al._ (2015); Szczȩśniak and Zemła (2015); Snider  _et
al._ (2021) Therefore, theoretical simulations play an extremely important
role in search for novel hydride superconductors. Glass  _et al._ (2006); Wang
_et al._ (2012a); Lonie and Zurek (2011) Through theoretical simulations,
almost all the predictions for binary hydrides have been completed, and
studies have successfully predicted the room-temperature superconductivity of
YH10. Peng  _et al._ (2017); Heil  _et al._ (2019) The successful prediction
of room-temperature superconductivity in ternary Li2MgH16 Sun  _et al._ (2019)
and the experimental discovery of room-temperature superconductivity in the
C–S–H system Snider  _et al._ (2020) have pushed research on superconducting
ternary hydrides system to a climax.

Figure 1:  Clathrate structure of LaYH12 (left panel, at 200 GPa) and LaY3H24
(right panel, at 180 GPa) with $Cmmm$, consisting of La-centered H24 cages and
Y-centered H24 cages. Each H24 cage is surrounded by six squares and eight
hexagons.

However, owing to the diversity of ternary hydrides, the search for new
high-$T_{c}$ superconductors appears very slow, and only $10\sim 20$ potential
superconducting hydrides have been identified. Sun  _et al._ (2019); Snider
_et al._ (2020); Ma  _et al._ (2017a, b); Li  _et al._ (2018); Liang  _et al._
(2019a); Rahm  _et al._ (2017); Liang  _et al._ (2019b); Wei  _et al._ (2020);
Xie  _et al._ (2019); Kokail  _et al._ (2017); Shao  _et al._ (2019, 2019);
Zhang  _et al._ (2020a, b); Di Cataldo  _et al._ (2020); Guo  _et al._ (2020);
Cui  _et al._ (2020); Lv  _et al._ (2020); Yan  _et al._ (2020); Muramatsu
_et al._ (2015); Meng  _et al._ (2019) From the information obtained regarding
binary superconductors, the crystal structure significantly influences the
$T_{c}$. For example, some ultrahigh-$T_{c}$ superconductors often appear in
cubic systems such as ($Fm\bar{3}m$-YH10, $Fm\bar{3}m$-LaH10, and
$Im\bar{3}m$-H3S, among others. Peng  _et al._ (2017); Somayazulu  _et al._
(2019); Errea  _et al._ (2015) However, in ternary hydrides, the stability of
the cubic structure is very poor, and many ternary systems may exhibit phase
decomposition relative to binary hydrides. For example, CSH7, Cui  _et al._
(2020) is stable relative to C, S, and H, but not to binary hydrides CH4 Gao
_et al._ (2010) and H3S Errea  _et al._ (2015), possibly leading to phase
decomposition. Owing to phase decomposition, it is highly unlikely that these
compounds can be synthesized experimentally. Therefore, it is extremely
important to ensure their stability relative to the constituent elements and
binary materials for predicting ternary hydrides.

For searching ternary hydrides, there are two major approaches aiming at
obtaining a high $T_{c}$, one focusing on SHx doped with La, Se, Te, and Cl
etc., Liang  _et al._ (2019b); Liu  _et al._ (2018); Chang  _et al._ (2020);
Nakanishi  _et al._ (2018) and the other on La- or Y-hydrides doped with
metallic elements such as CaYH12, CaMgH12, KScH12, and LaKH12. Liang  _et al._
(2019b); Sukmas  _et al._ (2020); Ishikawa  _et al._ (2019); Song  _et al._
(2021) With no significant improvement in $T_{c}$ observed so far in the
latter approach, the required pressure to realize superconductivity in these
materials is considerably reduced, thereby providing the possibility of their
experimental synthesis. Experiments involving the La–Y–H system report
high-$T_{c}$ superconductivity in (La,Y)H10. Semenok  _et al._ (2020)

Based on these experiments, theoretical predictions of superconductivity in
LaYH12, La2YH18, La4YH30 etc. have been also reported. Semenok  _et al._
(2020) These compositions are referred to as composites of (LaH6)x \+ YH6. The
symmetry of their predicted structures is considerably low mainly being
$P\bar{1}$ with highest mostly $Pm\bar{3}m$ for LaYH12. Examining its
decomposition enthalpy revealed that the structure may exhibit phase
decomposition under high pressure toward $R\bar{3}c$-LaH6 \+ $Im\bar{3}m$-YH6,
being unlikely to be synthesized experimentally. Heil  _et al._ (2019); Peng
_et al._ (2017) We then determined whether LaYH12 has other structures stable
against decomposition based on the USPEX fixed composition method. Glass  _et
al._ (2006) Composition LaYH12 was also found in our machine learning search
based on a gradient boosting tree Song  _et al._ (2021), and it exhibited a
high $T_{c}$. When searching more general forms, namely (LaH6)x(YH6)y, we
restricted the range by fixing $x=1$ because the preceding work has reported
instability along the $x$-axis. Along the $y$-axis, we found $Cmmm$-LaYH12
($y=1$) and $Cmmm$-LaY3H24 ($y=3$) exhibiting a high $T_{c}\sim$ of 140 K.
These compounds are formed through stacking of LaH24 and YH24 cages, as shown
in Fig. 1. A slightly higher $T_{c}$ for $Cmmm$-LaY3H24 compared to that for
LaYH12 can be realized by enhancing the chemical pressure for Y through
further stacking.

##  II Results

Fig. 2 shows the convex hull of our search for LaH6(YH6)x ($x$ = 1-4)
structures at 100, 200, and 300 GPa. LaYH12 ($x$=1) and LaY3H24 ($x$=3) are
stable phases, whereas other LaY2H18 and LaY4H30 structures are likely to be
decomposed at the abovementioned pressures . For the stable compositions,
namely LaYH12 and LaY3H24, we compared their relative enthalpies with those of
the candidate structures over the pressure range, as shown in Fig. 3. The plot
for LaYH12 indicates that the $Pm\bar{3}m$ structure proposed recently Semenok
_et al._ (2020) exhibits instability toward decomposition into binary
compounds, namely YH6($Im\bar{3}m$) and LaH6($R\bar{3}c$).

Figure 2:  Convex hull of La–Y–H at pressures of 100 GPa, 200 GPa, and 300
GPa.

For the entire pressure range, more stable structures are predicted as
$R\bar{3}c\to Cmmm$ with the transition occurring at approximately $P=140$
GPa, without instability toward decomposition. For LaY3H24, stable structures
form as $R\bar{3}\to Cmmm$, preventing its decomposition either to LaYH12 \+
2(YH6) or 3(YH6) + LaH6. As explained later, $R\bar{3}c$-LaYH12 and
$R\bar{3}$-LaY3H24 exhibit an extremely low density of states (DOS) at the
Fermi level, $D\left(\varepsilon_{F}\right)$, leading to a considerably low
$T_{c}$. In contrast, $Cmmm$ compounds are exhibit a higher $T_{c}$, and their
structures are shown in Fig. 1. Detailed geometries for each structure are
provided in Table 2 in the Supplemental Information (S.I.).

Figure 3:  Comparisons of candidate structure enthalpies for LaYH12 [panel
(a)] and LaY3H24 [panel (b)]. Such structures giving higher values than those
of ’YH6 \+ LaH6’ (for the left panel), ’3(YH6) + LaH6’ (for the right), and
’LaYH12 \+ 2(YH6)’ (for the right), are predicted to exhibit instability
toward decomposition into these binary compounds.

As observed in the Fig. 1, the structures are mostly clathrate structures
formed by units with La@H24 and Y@H24 cage structures. Several other clathrate
superconductors have been identified, Peng  _et al._ (2017); Wang  _et al._
(2012b, 2021); Sun  _et al._ (2019); Heil  _et al._ (2019); Salke  _et al._
(2019); Xie  _et al._ (2019) and hence it would be plausible for this
structure to exhibit superconductivity. Different periodicities of the La- and
Y-layers along the $c$-direction correspond to LaYH12 [(La/Y)(La/Y)$\cdots$]
and LaY3H24 [(La/Y/Y/Y)(La/Y/Y/Y)$\cdots$], respectively.

Fig. 4 shows the phonon dispersions evaluated for the clathrate $Cmmm$ LaYH12.
We confirmed no imaginary modes, thus confirming its structural stability. The
phonon dispersions for other structures are given in the S.I., confirming that
all the candidates predicted from Fig. 3, namely $Cmmm$-LaYH12, $Cmmm$-LaY3H24
$R\bar{3}c$-LaYH12, and $R\bar{3}$-LaY3H24, have no imaginary modes. In
contrast, $Pm\bar{3}m$-LaYH12 proposed recently Semenok  _et al._ (2020)
exhibits imaginary modes, as shown in Fig. 5, consistent with the observations
from Fig. 3 in terms of the instability.

Figure 4:  Phonon dispersions, projected phonon densities of states, and
Eliashberg spectral function for $Cmmm$-LaYH12 clathrate at 200 GPa.

Applying the Allen–Dynes formalism, Giannozzi  _et al._ (2009); Nakano  _et
al._ (2017, 2016) we estimated $T_{c}$ as summarized in Table 1. For LaYH12,
$T_{c}$ is in the $130.6\sim 140.55$ [K] range, depending on the choice of
parameter $\mu=0.10\sim 0.13$ for the Coulomb interaction. A slightly higher
$T_{c}$ realized for LaY3H24 is attributed to the higher
$D\left(\varepsilon_{F}\right)$ due to the compressed chemical pressure for
the Y-site with the (La/Y/Y/Y) stacking structure, compared to the
(La/Y)(La/Y) periodicity.

Table 1:  $T_{c}$ (expressed in [K]) estimated using the Allen–Dynes formula through ab initio phonon calculations for LaYH12 and LaY3H24 at each pressure. $\lambda$ and $\omega_{\rm log}$ (expressed in [K]) are the parameters appearing in the formula. The values of $D\left(\varepsilon_{F}\right)$ are are expressed in [states/eV/atom].  |  $P$ [GPa] | $\lambda$ | $\omega_{\rm log}$ | $D\left(\varepsilon_{F}\right)$ |  $T_{c}$ ($\mu$ = 0.1 - 0.13)
---|---|---|---|---|---
LaYH12 | 200 | 1.876 | 1022.54 | 0.097 | 140.55 - 130.61
LaYH12 | 250 | 1.618 | 1051.56 | 0.093 | 128.42 - 117.96
LaY3H24 | 180 | 2.452 | 891.49 | 0.100 | 145.31 - 137.11

##  III Discussions

As observed for $Fd\bar{3}m$-Li2MgH16, a room-temperature superconductor Sun
_et al._ (2019), the key factors for achieving a high $T_{c}$ are its high DOS
and high phonon frequency, which can be measured by $\lambda$ and $\omega_{\rm
log}$, respectively. $\lambda=2$ would be the typical criterion for realizing
superconductivity ($\lambda$ = 3.35 for $Fd\bar{3}m$-Li2MgH16). Sun  _et al._
(2019)

In the S.I., we have provided the band structures and Fermi surfaces of
$R\bar{3}c$-LaYH12 and $R\bar{3}c$-LaYH12 (Figs. 10 and 11). The Fermi
surfaces and the DOS values on these surfaces are extremely small and are
insufficient to achieve superconductivity.

In contrast, for the $Cmmm$ structure, sufficiently large Fermi surfaces and
high DOS values lead to higher $\lambda$, thus resulting in superconductivity.
Examining the phonon dispersions and Eliashberg spectral functions (Figs. 4, 8
and 9) reveals that higher $\lambda$ values can be attributed to the
contribution from the phonon branches appearing at higher ranges ($>$ 10 THz),
amounting to 96.2%. These branches are formed by the vibration of hydrogen
atoms, whereas those at lower ranges ($<$ 10 THz) by La and Y, negligibly
contributing to $\lambda$.

Applying higher pressures (200$\to$250 GPa) to $Cmmm$-LaY3H24 increases its
phonon frequencies for the branches at higher ranges ($>$ 10 THz), as observed
in Figs. 4 and 8. This trend is reflected as the increased $\omega_{\rm log}$
(Table 1), contributing toward increasing the $T_{c}$. However, this trend
disappears when the DOS (around 4.2%) decreases owing to the applied pressure,
leading to a reduced $\lambda$ value. This decrease is more dominant than the
increase in $\omega_{\rm log}$, and hence the applied pressure eventually
lowers the $T_{c}$, as observed in Table 1.

The slightly higher $T_{c}$ determined for LaY3H24 is attributed to the
enhanced $\lambda$, as observed in Table 1. In this case too, changes in
$\omega_{\rm log}$ and $\lambda$ exhibit a cancelling relationship, causing
only a slight increase in the $T_{c}$. Applying further pressure would mean
increasing $\lambda$ further for achieving a higher $T_{c}$; however,
according to the convex hull in our analysis, $Cmmm$-LaY3H24 exhibits
instability toward phase decomposition above 220 GPa.

$Pm\bar{3}m$-LaYH12 exhibits a higher $T_{c}$ (203 K at 180 GPa) Semenok  _et
al._ (2020) However, for this structure, the imaginary phonon modes appeared
as shown in Fig. 5, implying structural instability at least within the extent
of the harmonic approximation. As shown in Fig. 3, another structure,
$Fd\bar{3}m$-LaYH12, is predicted to be more stable than $Pm\bar{3}m$.
However, for both structures, the imaginary modes are found, without vanishing
even on applying further pressure.

Figure 5:  Phonon dispersions and projected phonon densities of states for
$Pm\bar{3}m$-LaYH12 at 180 GPa.

$Pm\bar{3}m$-LaYH12 can be regarded as a compound based on the
$Im\bar{3}m$-YH6 dimer with La substituting for one Y (Fig. 6). The
$Im\bar{3}m$-YH6 actually appears as a stable structure in the convex-hull
analysis given in Fig. 2. Our prediction, $Cmmm$-LaYH12, is actually a
distorted structure of $Pm\bar{3}m$-LaYH12, showing considerably similar X-ray
diffraction (XRD) peak patterns as well as that of $Fd\bar{3}m$, as shown in
Fig. 2. Furthermore, the possibility of the structural transition between
$Cmmm$ and $Pm\bar{3}m$ at higher temperatures is pointed out. Sun  _et al._
(2019)

Figure 6: Structures and the XRD patterns of (a) $Im\bar{3}m$-YH6, (b)
$Pm\bar{3}m$-LaYH12, and (c) $Cmmm$-LaYH1212.

##  IV Conclusion

In summary, we investigated the possible crystal structures for (LaH6)(YH6)y
compounds as candidates for forming ternary hydrides to achieve a high $T_{c}$
using a generic algorithm implemented in USPEX. We determined the thermal
stabilities at $y=1$ (LaYH12) and $y=3$ (LaY3H24). A recently reported
structure, $Pm\bar{3}m$, appeared unstable toward the decomposition, i.e.,
LaYH12 ($Pm\bar{3}m$) $\to$ LaH6($R\bar{3}c$) + YH6($Im\bar{3}m$). However, we
discovered other new stable structures that did not exhibit decomposition,
namely $Cmmm$-LaYH12, $R\bar{3}c$-LaYH12, $Cmmm$-LaY3H24, and
$R\bar{3}$-LaY3H24. Among these, the compounds with $Cmmm$ clathrate
structures exhibited a $T_{c}\sim$ 130 [K] at approximately $P$ = 200 GPa, as
estimated using the Allen–Dynes formalism.

##  V Methods

For the structural predictions, we used the USPEX code Glass  _et al._ (2006)
combined with the ab initio kernel in the Vienna ab initio simulation package
(VASP). Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996a, b) For
example, the LaYH12 composition randomly generated 400 structures ranging from
monomers to tetramers with a LaYH12 unit as the “initial generation.” Each
generation evolved 100 structures with 40% heredity, 40% randomness, 10%
softmutation, and 10% transmutation. When no further evolution occurred for
more than 10 generations, the final structure was identified as the candidate
crystal structure of the composition. For the identified candidates, we
performed ab initio geometrical optimizations using the Perdew–Burke–Ernzerhof
(GGA–PBE) functional for the exchange-correlation functional. Perdew  _et al._
(1996)

For the most stable candidates [$R\bar{3}c$ (100-140 GPa) and $Cmmm$ (140-300
GPa) of LaYH12, for example], we performed ab intio phonon calculations to
evaluate their structural stabilities and $T_{c}$ using the Allen–Dynes
formalism, Giannozzi  _et al._ (2009); Nakano  _et al._ (2017, 2016)
implemented in Quantum Espresso. Giannozzi  _et al._ (2009) Computational
conditions such as the cutoff energy of the plane wave basis set, sizes of the
grid-mesh over the Brillouin zone etc. were determined such that the energies
sufficiently converged in their dependences. We finally used an $8\times
8\times 5$ $k$-mesh used in the self-consistent field convergence assisted by
the Marzari–Vanderbilt smearing scheme. Marzari  _et al._ (1999) The final
energy values were estimated through extrapolation of the smearing parameter
toward zero. Mesh sizes for the phonon calculations were $16\times 16\times
10$ for the $k$\- and $4\times 4\times 2$ for the $q$-mesh.

##  VI Acknowledgments

The computations in this work have been performed using the facilities of
Research Center for Advanced Computing Infrastructure (RCACI) at JAIST. K.H.
is grateful for financial support from the HPCI System Research Project
(Project ID: hp190169) and MEXT-KAKENHI (JP16H06439, JP17K17762, JP19K05029,
and JP19H05169). R.M. is grateful for financial supports from MEXT-KAKENHI
(21K03400 and 19H04692), and the Air Force Office of Scientific Research
(AFOSR-AOARD/FA2386-17-1-4049;FA2386-19-1-4015)

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##  VII Supplemental Information

For each compound reported in the main text, we provide the detailed
geometries [Table 2)], Phonon dispersions and Eliashberg spectral functions
[Figs. 7-9], and electronic band structures and Fermi surfaces [Figs. 10-13].
To show the bonding characteristics of the hydrogen atoms in each compound, we
also provide the results of the electron localization function (ELF) [Figs. 14
and 15], and the crystal orbital Hamilton population (COHP) [Fig. 16].

Table 2:  Crystal structures of LaYH12 and LaY3H24 predicted at each pressure ($P$). Lattice parameters ($a$, $b$, and $c$) are given in units of $\AA$.  |  | $P$ (GPa) | Lattice parameters | Atomic coordinates (fractional)
---|---|---|---|---
|  | Atoms | $x$ | $y$ | $z$
LaYH12 | $R\bar{3}c$ | 100 |  | $a=b=5.389$
---
$c=13.674$
$\alpha=\beta=90^{\circ}$
$\gamma=120^{\circ}$
| La(6$b$)
---
Y(6$a$)
H(36$f$)
H(36$f$)
| 0.00000
---
0.00000
0.00181
0.00492
| 0.00000
---
0.00000
0.23464
0.27691
| 0.00000
---
0.25000
0.36866
0.60940
LaYH12 | $Cmmm$ | 200 |  | $a=3.617$
---
$b=4.956$
$c=5.156$
$\alpha=\beta=\gamma=90^{\circ}$
| La(2$c$)
---
Y(2$a$)
H(8$n$)
H(8$n$)
H(8$m$)
| 0.00000
---
0.00000
0.00000
0.00000
0.25000
| 0.50000
---
0.00000
0.11589
0.38205
0.25000
| 0.50000
---
0.00000
0.35519
0.11708
0.23317
LaY3H24 | $R\bar{3}$ | 100 |  | $a=b=5.324$
---
$c=13.569$
$\alpha=\beta=90^{\circ}$
$\gamma=120^{\circ}$
| La(3$a$)
---
Y(6$c$)
Y(3$b$)
H(18$f$)
H(18$f$)
H(18$f$)
H(18$f$)
| 0.00000
---
0.00000
0.00000
0.00154
0.00245
0.00465
0.01123
| 0.00000
---
0.00000
0.00000
0.26668
0.23293
0.75887
0.72593
| 0.00000
---
0.24782
0.50000
0.60934
0.36741
0.87000
0.11041
LaY3H24 | $Cmmm$ | 180 |  | $a=3.621$
---
$b=5.013$
$c=10.239$
$\alpha=\beta=\gamma=90^{\circ}$
| La(2$a$)
---
Y(4$l$)
Y(2$d$)
H(8$n$)
H(8$n$)
H(8$n$)
H(8$n$)
H(8$m$)
H(8$m$)
| 0.00000
---
0.00000
0.00000
0.00000
0.00000
0.00000
0.00000
0.25000
0.25000
| 0.00000
---
0.50000
0.00000
0.11988
0.12176
0.37658
0.38314
0.25000
0.25000
| 0.00000
---
0.25606
0.50000
0.19483
0.31540
0.43798
0.07100
0.13624
0.37798
Figure 7:  Phonon dispersions and projected phonon densities of states for
$R\bar{3}c$-LaYH12 clathrate at 100 GPa.  Figure 8:  Phonon dispersions,
projected phonon densities of states, and Eliashberg spectral function for
$Cmmm$-LaYH12 clathrate at 250 GPa.  Figure 9:  Phonon dispersions, projected
phonon densities of states, and Eliashberg spectral function for
$Cmmm$-LaY3H24 clathrate at 180 GPa.  Figure 10:  Band structure, DOS, and
Fermi surface of $R\bar{3}c$-LaYH12.  Figure 11:  Band structure, DOS, and
Fermi surface of $R\bar{3}$-LaY3H24.  Figure 12:  Band structure, DOS, and
Fermi surface of $Cmmm$-LaYH12 Figure 13:  Band structure, DOS, and Fermi
surface of $Cmmm$-LaY3H24 Figure 14:  The calculated ELF of (a)
$R\bar{3}c$-LaYH12, (b) $Pm\bar{3}m$-LaYH12, and (c) $Cmmm$-LaYH12 in the
(100) plane.  Figure 15:  The calculated ELF of (a) $R\bar{3}$-LaY3H24 and (b)
$Cmmm$-LaY3H24 in the (100) plane.  Figure 16:  COHP for pairs of H-H in (a)
$R\bar{3}c$-LaYH12 at 100 GPa, (b) $Cmmm$-LaYH12 at 200 GPa, (c)
$R\bar{3}$-LaY3H24 at 100 GPa, and (d) $Cmmm$-LaY3H24 at 180 GPa.