\dot{\mathsf{y}}^{+}\end{bmatrix}=\begin{bmatrix}\Lambda^{-}\\\
&\Lambda^{+}\end{bmatrix}\begin{bmatrix}{\mathsf{y}}^{-}\\\
{\mathsf{y}}^{+}\end{bmatrix},$ (60)

which solution is
${\mathsf{y}}^{\pm}(t)=\exp(t\Lambda^{\pm}){\mathsf{y}}^{\pm}(0)$. One can
readily check that for each couple of real numbers $\alpha^{+}$ and
$\alpha^{-}$ that satisfy

$0<\alpha^{-}<\alpha^{+}<\min\\{\Re\lambda_{i}\,:\,\Re\lambda_{i}>0\\},$ (61)

there exists a so-called exponential dichotomy of the ODE solutions, which
amounts in our case to the existence of two constants $K^{-},K^{+}\geq 1$ such
that

$\displaystyle\|\exp(t\Lambda^{-})\|$ $\displaystyle\leq
K^{-}e^{\alpha^{-}t}\quad\text{for }t\geq 0,$
$\displaystyle\|\exp(t\Lambda^{+})\|$ $\displaystyle\leq
K^{+}e^{\alpha^{+}t}\quad\text{for }t\leq 0,$

see, _e.g._ , [27].

We now consider a non-autonomous perturbation of this ODE, which is
represented in the basis of the columns of $Q$ as

$\dot{\mathsf{y}}(t)=h({\mathsf{y}}(t),t)\quad\text{with}\quad
h(y,t)=\begin{bmatrix}\Lambda^{-}\\\
&\Lambda^{+}\end{bmatrix}y+\varepsilon(y,t),$ (62)

and $\varepsilon:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ is a
continuous function. In the sequel, we shall be interested in the asymptotic
behavior of this equation for the large values of $t$, and therefore, restrict
our study to the interval $\mathbb{I}=[t_{0},\infty)$ for some given
$t_{0}\geq 0$ that we shall fix later. We assume that $\varepsilon(0,\cdot)=0$
on $\mathbb{I}$. We denote as
$\phi:\mathbb{I}\times\mathbb{I}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ the so-
called general solution of (62), which is defined by the fact that
$\phi(\cdot,t,x)$ is the unique noncontinuable solution of (62) such that
$\phi(t,t,x)=x$ for $t\in\mathbb{I}$ and $x\in\mathbb{R}^{d}$, assuming this
solution exists and is unique for each
$(x,t)\in\mathbb{R}^{d}\times\mathbb{I}$.

In the linear autonomous case provided by the ODE (60), the subspace

$\mathcal{G}=\left\\{\left(t,\begin{bmatrix}y^{-}\\\
0\end{bmatrix}\right)\in\mathbb{R}\times\mathbb{R}^{d}\ :\
y^{-}\in\mathbb{R}^{d^{-}}\right\\}$

is trivially invariant in the sense that if $(t,y)\in\mathcal{G}$, then,
$(s,\phi(s,t,y))\in\mathcal{G}$ for each $s\in\mathbb{R}$. This concept can be
generalized to the non-linear and non-autonomous case. We say that the
$\mathcal{C}^{1}$ function
$w:\mathbb{R}^{d^{-}}\times\mathbb{I}\to\mathbb{R}^{d^{+}}$ defines a global
non-autonomous invariant manifold for the ODE (62) if $w(0,t)=0$ for all
$t\in\mathbb{I}$, and, furthermore, if for each $t\in\mathbb{I}$ and each
$y^{-}\in\mathbb{R}^{d^{-}}$, writing $y=(y^{-},w(y^{-},t))$, the general
solution $\phi(s,t,y)=(\phi^{-}(s,t,y),\phi^{+}(s,t,y))$ with
$\phi^{\pm}(s,t,y)\in\mathbb{R}^{d^{\pm}}$ verifies
$\phi^{+}(s,t,y)=w(\phi^{-}(s,t,y),s)$ for each $s\in\mathbb{I}$. The non-
autonomous invariant manifold is the set

$\mathcal{G}=\left\\{\left(t,\begin{bmatrix}y^{-}\\\
w(y^{-},t)\end{bmatrix}\right)\in\mathbb{I}\times\mathbb{R}^{d}\ :\
y^{-}\in\mathbb{R}^{d^{-}}\right\\},$

which obviously satisfies $(t,y)\in\mathcal{G}\ \Rightarrow\
(s,\phi(s,t,y))\in\mathcal{G}\ \text{for each }s\in\mathbb{I}$.

These invariant manifolds are described by the following proposition, which is
a straightforward application of [40, Th. A.1] (see also [31, Th. 6.3 p. 106,
Rem. 6.6 p. 111]). It is useful to note that under the conditions provided in
the statement of this proposition, the existence of the general solution
$\phi$ of the ODE (62) is ensured by Picard’s theorem.

######  Proposition 7.1.

Let $\mathbb{I}=[t_{0},\infty)$ for some $t_{0}\geq 0$. Assume that the
function $\varepsilon(y,t)$ is such that $\varepsilon(0,\cdot)\equiv 0$ on
$\mathbb{I}$, the function $\varepsilon(\cdot,t)$ is continuously
differentiable for each $t\in\mathbb{I}$, and furthermore, the Jacobian matrix
$\partial_{1}\varepsilon(y,t)$ satisfies

$|\varepsilon|_{1}\triangleq\sup_{(y,t)\in\mathbb{R}^{d}\times\mathbb{I}}\|\partial_{1}\varepsilon(y,t)\|<\frac{\alpha^{+}-\alpha^{-}}{4K}$
(63)

with $K=K^{-}+K^{+}+K^{-}K^{+}(K^{-}\vee K^{+})$ and $\alpha^{-},\alpha^{+}$
chosen as in Eq. (61). Then, for each
$\delta\in(2K|\varepsilon|_{1},(\alpha^{+}-\alpha^{-})/2)$ and each
$\gamma\in(\alpha^{-}+\delta,\alpha^{+}-\delta)$, the set

$\mathcal{G}=\left\\{(t,y)\in\mathbb{I}\times\mathbb{R}^{d}\ :\ \sup_{s\geq
t}\|\phi(s,t,y)\|\exp(\gamma(t-s))<\infty\right\\}$

is nonempty, and does not depend on $\gamma$. Moreover, this set is a global
invariant manifold for the ODE (62) that is defined by a continuously
differentiable mapping
$w:\mathbb{R}^{d^{-}}\times\mathbb{I}\to\mathbb{R}^{d^{+}}$. In addition, if
the partial derivatives
$\partial_{1}^{k}\varepsilon:\mathbb{R}^{d}\times\mathbb{I}$ exist and are
continuous for $k\in\\{1,\ldots,m\\}$ with globally bounded partial
derivatives

$|\varepsilon|_{k}\triangleq\sup_{(y,t)\in\mathbb{R}^{d}\times\mathbb{I}}\|\partial_{1}^{k}\varepsilon(y,t)\|<\infty\,,$
(64)

under the gap condition

$m\alpha^{-}<\alpha^{+},\ m\in\mathbb{N}^{*},$ (65)

the partial derivatives $\partial_{1}^{k}w:\mathbb{R}^{d^{-}}\times\mathbb{I}$
exist and are continuous with

$\sup_{(y^{-},t)\in\mathbb{R}^{d^{-}}\times\mathbb{I}}\|\partial_{1}^{k}w(y^{-},t)\|<\infty\quad\text{for
all }k\in\\{1,\ldots,m\\}.$ (66)

Finally, if $\partial_{2}^{n}\partial_{1}^{k}\varepsilon$ exist and are
continuous for $0\leq n<m$ and $0\leq k+n\leq m$, then $w$ is $m$-times
continuously differentiable.

Let us partition the function $h(y,t)$ as

$h(y,t)=\begin{bmatrix}h^{-}(y,t)\\\
h^{+}(y,t)\end{bmatrix}=\begin{bmatrix}\Lambda^{-}y^{-}+\varepsilon^{-}(y,t)\\\
\Lambda^{+}y^{+}+\varepsilon^{+}(y,t)\end{bmatrix},$ (67)

where $h^{\pm}:\mathbb{R}^{d}\times\mathbb{I}\to\mathbb{R}^{d^{\pm}}$,
$y^{\pm}\in\mathbb{R}^{d^{\pm}}$ and
$\varepsilon^{\pm}:\mathbb{R}^{d}\times\mathbb{I}\to\mathbb{R}^{d^{\pm}}$.
With these notations, the previous proposition leads to the following lemma.

######  Lemma 7.2.

In the setting of Prop. 7.1, for each $t$ in the interior of $\mathbb{I}$ and
each vector $y=(y^{-},y^{+})$ such that $y^{\pm}\in\mathbb{R}^{d^{\pm}}$ and
$y^{+}=w(y^{-},t)$, it holds that

$h^{+}(y,t)=\partial_{1}w(y^{-},t)h^{-}(y,t)+\partial_{2}w(y^{-},t)\,.$ (68)

Assume that $\alpha^{-}$ is small enough so that Ineq. (65) and Eq. (64) hold
true with $m=2$. Assume in addition that
$\partial_{2}^{n}\partial_{1}^{k}\varepsilon$ exists and is continuous for
$0\leq n<2$ and $0\leq k+n\leq 2$, and furthermore, that there exists a
bounded neighborhood ${{\mathcal{V}}}\subset\mathbb{R}^{d}$ of zero such that

$\sup_{(y,t)\in{{\mathcal{V}}}\times\mathbb{I}}\left\lVert\partial_{2}\varepsilon(y,t)\right\rVert<+\infty.$
(69)

Then, there exists a neighborhood
${{\mathcal{V}}}^{-}\subset\mathbb{R}^{d^{-}}$ of zero such that

$\displaystyle\sup_{(y^{-},t)\in{{\mathcal{V}}}^{-}\times\mathbb{I}}\left\lVert\partial_{1}\partial_{2}w(y^{-},t)\right\rVert<+\infty\,,$
(70)
$\displaystyle\sup_{(y^{-},t)\in{{\mathcal{V}}}^{-}\times\mathbb{I}}\left\lVert\partial_{2}^{2}w(y^{-},t)\right\rVert<+\infty\,.$
(71)

###### Proof.

By Prop. 7.1, the general solution $\phi(s,t,y)$ of the ODE (62) can be
written as $\phi(s,t,y)=(\phi^{-}(s,t,y),\phi^{+}(s,t,y))$ with
$\phi^{+}(s,t,y)=w(\phi^{-}(s,t,y),s)$ for each $s\in\mathbb{I}$. Equating the
derivatives with respect to $s$ of the two members of this equation and taking
$s=t$, we get the first equation.

Writing
$g:\mathbb{R}^{d}{{}^{-}}\times\mathbb{I}\to\mathbb{R}^{d},(y^{-},t)\mapsto(y^{-},w(y^{-},t))$,
Eq. (68) can be rewritten as

$\partial_{2}w(y^{-},t)=h^{+}(g(y^{-},t),t)-\partial_{1}w(y^{-},t)h^{-}(g(y^{-},t),t).$
(72)

By Prop. 7.1, the function $w$ is twice differentiable, and we can write

$\partial_{2}^{2}w(y^{-},t)=\partial_{1}h^{+}\partial_{2}g+\partial_{2}h^{+}-(\partial_{1}\partial_{2}w)h^{-}-(\partial_{1}w)(\partial_{1}h^{-}\partial_{2}g+\partial_{2}h^{-}),$
(73)

where, _e.g._ , $h^{+}$ is a shorthand notation for $h^{+}(g(y^{-},t),t)$. It
holds from Eq. (67) and the assumptions of Prop. 7.1 that for each
$(y,t)\in\mathbb{R}^{d}\times\mathbb{I}$,

$\|\partial_{1}h(y,t)\|\leq\|\Lambda\|+\|\partial_{1}\varepsilon(y,t)\|\leq
C,$ (74)

where the constant $C>0$ is independent of $(y,t)$ and can change from an
inequality to another in the remainder of the proof. By the mean value
inequality and Prop. 7.1, we also get that

$\|w(y^{-},t)\|=\|w(y^{-},t)-w(0,t)\|\leq\sup_{(u,s)}\|\partial_{1}w(u,s)\|\
\|y^{-} \|\leq C\|y^{-}\|,$

thus, $\|g(y^{-},t)\|\leq C\|y^{-}\|$. By the mean value inequality again,

$\left\lVert h(g(y^{-},t),t)\right\rVert=\left\lVert
h(g(y^{-},t),t)-h(0,t)\right\rVert\leq\sup_{(u,t)}\left\lVert\partial_{1}h(u,t)\right\rVert\left\lVert
g(y^{-},t)\right\rVert\\\ \leq C\left\lVert g(y^{-},t)\right\rVert\leq
C\|y^{-}\|.$

By Eq. (72) and Prop. 7.1, this implies that

$\displaystyle\left\lVert\partial_{2}g(y^{-},t)\right\rVert=\left\lVert\partial_{2}w(y^{-},t)\right\rVert=\left\lVert
h^{+}-(\partial_{1}w)h^{-}\right\rVert\leq C\left\lVert
y^{-}\right\rVert,\quad\text{and}$ (75)
$\displaystyle\left\lVert\partial_{1}\partial_{2}w(y^{-},t)\right\rVert=\left\lVert\partial_{1}h^{+}\partial_{1}g-(\partial_{1}^{2}w)h^{-}-(\partial_{1}w)(\partial_{1}h^{-}\partial_{1}g)\right\rVert\leq
C(\left\lVert y^{-}\right\rVert+1).$ (76)

Let ${{\mathcal{V}}}^{-}\subset\mathbb{R}^{d^{-}}$ be a small enough
neighborhood of zero so that $g(y^{-},t)\in{{\mathcal{V}}}$ for each
$y^{-}\in{{\mathcal{V}}}^{-}$, which is possible by the inequality
$\|g(y^{-},t)\|\leq C\|y^{-}\|$. By the assumption on
$\|\partial_{2}\varepsilon(y,t)\|$ in the statement of Lem. 7.2, we have

$\forall
y^{-}\in{{\mathcal{V}}}^{-},\quad\left\lVert\partial_{2}h(g(y^{-},t),t)\right\rVert=\left\lVert\partial_{2}\varepsilon(g(y^{-},t),t)\right\rVert\leq
C.$ (77)

The bound (70) is an immediate consequence of Eq. (76). Getting back to Eq.
(73), the bound (71) follows from the inequalities (74)–(77). ∎

Prop. 7.1 deals with the case where the function $\varepsilon$ is globally
Lipschitz continuous. In practical cases, such a strong assumption is not
necessarily verified. In particular, for the ODEs we consider for our
application, it is not satisfied (see the function $e$ defined in Subsec.
7.3.1 below). Nonetheless, recall that we only need the existence of a local
non-autonomous invariant manifold, i.e. defined in the vicinity of an
arbitrary solution such as the trivial zero solution (since we suppose here
$\varepsilon(0,\cdot)=0$) whereas the aforementioned strong assumption
provides a global non-autonomous invariant manifold. Indeed, as for the
avoidance of traps result we intend to show, we will only need to look at the
behavior of our ODE in the neighborhood of a trap $z_{\star}$. Therefore, in
prevision of the proof of Th. 4.1, we localize the ODE (62) in the
neighborhood of zero. This is the purpose of the next proposition.

######  Proposition 7.3.

Let $\mathbb{I}=[t_{0},+\infty)$ for some $t_{0}\geq 0$ and let
$h:\mathbb{R}^{d}\times\mathbb{I}\rightarrow\mathbb{R}^{d}$ be defined as in
Eq. (62). Assume that $\varepsilon(0,\cdot)\equiv 0$ on $\mathbb{I}$, that the
function $\varepsilon(\cdot,t)$ is continuously differentiable for every
$t\in\mathbb{I}$ and that

$\lim_{(y,t)\rightarrow(0,+\infty)}\left\lVert\partial_{1}\varepsilon(y,t)\right\rVert=0\,.$
(78)

Then, there exist $\sigma>0,t_{1}>0$, a function
$\tilde{\varepsilon}:\mathbb{R}^{d}\times\mathbb{I}_{1}\to\mathbb{R}^{d}$
where $\mathbb{I}_{1}\triangleq[t_{1},+\infty)$ and a function
$\tilde{h}:\mathbb{R}^{d}\times\mathbb{I}_{1}\rightarrow\mathbb{R}^{d}$
defined for every $y\in\mathbb{R}^{d},t\in\mathbb{I}_{1}$ by
$\tilde{h}(y,t)=\Lambda y+\tilde{\varepsilon}(y,t)$ s.t. $\tilde{h}$ and
$\tilde{\varepsilon}$ verify the assumptions of Prop. 7.1 and for every
$(y,t)\in B(0,\sigma)\times\mathbb{I}_{1}$, we have that
$\tilde{h}(y,t)=h(y,t)$ and $\tilde{\varepsilon}(y,t)=\varepsilon(y,t)$.
Moreover, for any $\delta>0$, we can choose $\sigma,t_{1}$ respectively small
and large enough s.t. the mapping
$w:\mathbb{R}^{d^{-}}\times\mathbb{I}_{1}\rightarrow\mathbb{R}^{d^{+}}$
obtained from Prop. 7.1 (applied to $\tilde{h}$ and $\tilde{\varepsilon}$)
satisfies

$|w|_{1}=\sup_{(y,t)\in\mathbb{R}^{d^{-}}\times\mathbb{I}_{1}}\|\partial_{1}w(y,t)\|<\delta\,.$
(79)

Furthermore, Eq. (68) holds for $\tilde{h}$ and $w$ for all $(y,t)\in
B(0,\sigma)\times\mathbb{I}_{1}$. If, additionally, Eq. (69) holds for
$\varepsilon$, then there exists $\sigma_{1}\leq\sigma$ such that

$\displaystyle\sup_{(y^{-},t)\in
B(0,\sigma_{1})\times\mathbb{I}_{1}}\left\lVert\partial_{1}\partial_{2}w(y^{-},t)\right\rVert<+\infty\,,$
(80) $\displaystyle\sup_{(y^{-},t)\in
B(0,\sigma_{1})\times\mathbb{I}_{1}}\left\lVert\partial_{2}^{2}w(y^{-},t)\right\rVert<+\infty\,.$
(81)

###### Proof.

The idea of the proof is to _localize_ the function $h(y,t)$ to a neighborhood
of zero in the variable $y$ for the purpose of applying Prop. 7.1. This cut-
off technique is known in the non-autonomous ODE literature, see, _e.g._ ,
[31, Th. 6.10]. Let $\psi:\mathbb{R}^{d}\to[0,1]$ be a smooth function such
that $\psi(y)=1$ if $\|y\|\leq 1$, and $\psi(y)=0$ if $\|y\|\geq 2$. Let
$C=\max_{y}\|\nabla\psi(y)\|$ where $\nabla\psi$ is the Jacobian matrix of
$\psi$. Thanks to the convergence (78), we can choose $t_{1}>0$ large enough
and $\sigma>0$ small enough so that

$\sup_{(t,y)\in[t_{1},\infty)\times
B(0,2\sigma)}\|\partial_{1}\varepsilon(y,t)\|<\frac{\alpha^{+}-\alpha^{-}}{4K(1+2C)},$

and we set $\mathbb{I}_{1}=[t_{1},\infty)$. Writing
$\tilde{\varepsilon}(y,t)=\psi(y/\sigma)\varepsilon(y,t)$, it holds that for
each $(t,y)\in\mathbb{I}_{1}\times\mathbb{R}^{d}$,

$\displaystyle\|\partial_{1}\tilde{\varepsilon}(y,t)\|$
$\displaystyle\leq\sigma^{-1}C\mathbbm{1}_{\|y\|\leq
2\sigma}\|\varepsilon(y,t)\|+\mathbbm{1}_{\|y\|\leq
2\sigma}\|\partial_{1}\varepsilon(y,t)\|$
$\displaystyle\leq\left(\max_{\|y\|\leq
2\sigma}\|\partial_{1}\varepsilon(y,t)\|\right)\left(\sigma^{-1}C\|y\|+1\right)\mathbbm{1}_{\|y\|\leq
2\sigma}$ $\displaystyle\leq\frac{\alpha^{+}-\alpha^{-}}{4K},$

where we used the mean value inequality along with $\varepsilon(0,t)=0$ to
obtain the second inequality. Thus, the function $\tilde{h}(y,t)=\Lambda
y+\tilde{\varepsilon}(y,t)$ satisfies all the assumptions of Prop. 7.1. In
addition, the function $\tilde{\varepsilon}$ coincides with the function
$\varepsilon$ on $B(0,\sigma_{1})\times\mathbb{I}_{1}$, and so it is for the
functions $\tilde{h}$ and $h$. Finally, it follows from [31, Th. 6.3] that

$|w|_{1}\leq\frac{2K^{2}}{\alpha_{+}-\alpha_{-}-4K|\tilde{\varepsilon}|_{1}}|\tilde{\varepsilon}|_{1}\,$

(note that $L$ in [31, Th. 6.3] corresponds to $|\tilde{\varepsilon}|_{1}$
with our notations). Using Eq. (78), we can make $|\tilde{\varepsilon}|_{1}$
as small as needed by choosing $\sigma,t_{1}$ respectively small and large
enough, which gives us Eq. (79). The proof of the last two equations follows
from the application of Lemma 7.2 to $\tilde{h}$ and $w$. The result is
immediate after noticing that for
$(y,t)\in\mathbb{R}^{d}\times\mathbb{I}_{1}$, we have
$\left\lVert\partial_{2}\tilde{\varepsilon}(y,t)\right\rVert\leq\left\lVert\partial_{2}\varepsilon(y,t)\right\rVert$.
∎

###  7.2 Proof of Th. 4.1

We shall rely on the following result of Brandière and Duflo. Recall that
$(\Omega,{\mathscr{F}},\mathbb{P})$ is a probability space equipped with a
filtration $({\mathscr{F}}_{n})_{n\in\mathbb{N}}$.

######  Proposition 7.4.

([13, Prop. 4]) Given a sequence $(\gamma_{n})$ of deterministic nonnegative
stepsizes such that $\sum_{k}\gamma_{k}=+\infty$ and
$\sum_{k}\gamma_{k}^{2}<+\infty$, consider the $\mathbb{R}^{d}$–valued
stochastic process $(z_{n})_{n\in\mathbb{N}}$ given by

$z_{n+1}=(I+\gamma_{n+1}H_{n})z_{n}+\gamma_{n+1}\eta_{n+1}+\gamma_{n+1}\rho_{n+1}.$

Assume that $z_{0}$ is ${\mathscr{F}}_{0}$–measurable and that the sequences
$(\eta_{n})$, $(\rho_{n})$ together with the sequence of random matrices
$(H_{n})$ are $({\mathscr{F}}_{n})$–adapted. Moreover, on a given event
$A\in{\mathscr{F}}$, assume the following facts:

  1. )

$\sum_{n}\|\rho_{n}\|^{2}<\infty$.

  2. )

$\limsup\mathbb{E}[\|\eta_{n+1}\|^{2+a}\,|\,{\mathscr{F}}_{n}]<\infty$ for
some $a>0$, and $\mathbb{E}[\eta_{n+1}\,|\,{\mathscr{F}}_{n}]=0$.

  3. )

$\liminf\mathbb{E}[\|\eta_{n+1}\|^{2}\,|\,{\mathscr{F}}_{n}]>0$.

Let $H\in\mathbb{R}^{d\times d}$ be a deterministic matrix such that the real
parts of its eigenvalues are all positive. Then,

$\mathbb{P}\left(A\cap[z_{n}\to 0]\cap[H_{n}\to H]\right)=0.$

We now enter the proof of Th. 4.1. Recall the development (11) of $b(z,t)$
near $z_{\star}$ and the spectral factorization (12) of the matrix $D$. To
begin with, it will be convenient to make the variable change
$y=Q^{-1}(z-z_{\star})$, and set

$h(y,t)=Q^{-1}b(Qy+z_{\star},t)=\Lambda y+\tilde{e}(y,t),$

with $\tilde{e}(y,t)=Q^{-1}e(Qy+z_{\star},t)$, in such a way that our
stochastic algorithm is rewritten as

$y_{n+1}=y_{n}+\gamma_{n+1}h(y_{n},\tau_{n})+\gamma_{n+1}\tilde{\eta}_{n+1}+\gamma_{n+1}\tilde{\rho}_{n+1}\,$

where $\tilde{\eta}_{n}$ is as in the statement of the theorem and
$\tilde{\rho}_{n}=Q^{-1}\rho_{n}$. Observe that the assumptions on the
function $e$ in the statement of the theorem remain true for $\tilde{e}$ with
$z_{\star}$ replaced by zero.

If the matrix $\Lambda$ has only eigenvalues with (strictly) positive real
parts, _i.e._ , $d^{-}=0$, then we can apply Prop. 7.4 to the sequence
$(z_{n})$. Henceforth, we deal with the more complicated case where $d^{-}>0$.

Apply Prop. 7.3 to $h$ to obtain $\tilde{h}$ and $\sigma,t_{1}$ respectively
small and large enough and
$w:\mathbb{R}^{d^{-}}\times\mathbb{I}_{1}\to\mathbb{R}^{d^{+}}$ where
$\mathbb{I}_{1}:=[t_{1},+\infty)$. By Assumption 4 of Th. 4.1 and Prop. 7.3 we
can choose $\sigma_{1}\leq\sigma$ such that Eq. (80) and Eq. (81) hold. Now,
given $p\in\mathbb{N}$, let us define the event

$E_{p}=\left[\forall n\geq p,\
\|y_{n}\|<\sigma_{1},\tau_{n}\in\mathbb{I}_{1}\right].$

On $E_{p}$, it holds that $h(y_{n},\tau_{n})=\tilde{h}(y_{n},\tau_{n})$ and

$\displaystyle\forall n\geq p,\quad y_{n+1}$
$\displaystyle=y_{n}+\gamma_{n+1}h(y_{n},\tau_{n})+\gamma_{n+1}\tilde{\eta}_{n+1}+\gamma_{n+1}\tilde{\rho}_{n+1}$
$\displaystyle=\begin{bmatrix}y^{-}_{n}\\\
y^{+}_{n}\end{bmatrix}+\gamma_{n+1}\begin{bmatrix}h^{-}(y_{n},\tau_{n})\\\
h^{+}(y_{n},\tau_{n})\end{bmatrix}+\gamma_{n+1}\begin{bmatrix}\tilde{\eta}^{-}_{n+1}\\\
\tilde{\eta}^{+}_{n+1}\end{bmatrix}+\gamma_{n+1}\begin{bmatrix}\tilde{\rho}^{-}_{n+1}\\\
\tilde{\rho}^{+}_{n+1}\end{bmatrix}\,$ (82)

where $h$ is partitioned as in (67), and where
$\tilde{\eta}^{\pm}_{n},\tilde{\rho}^{\pm}_{n}\in\mathbb{R}^{d^{\pm}}$. Note
that, by Prop. 7.3 and Assumptions 6 and 7 on the sequence $(\eta_{n})$, we
can choose $\sigma,t_{1}$ respectively small and large enough such that

$\liminf\mathbb{E}[\left\lVert\tilde{\eta}^{+}_{n+1}\right\rVert^{2}|{\mathscr{F}}_{n}]\mathbbm{1}_{E_{p}}(y_{n})\\\
-2\limsup\mathbb{E}[\left\lVert\partial_{1}w(y_{n}^{-},\tau_{n})\tilde{\eta}_{n+1}^{-}\right\rVert^{2}|{\mathscr{F}}_{n}]\mathbbm{1}_{E_{p}}(y_{n})>\frac{c^{2}}{2}\,.$
(83)

This inequality will be important in the end of our proof. Let $t$ be in the
interior of $\mathbb{I}_{1}$, and let $y=(y^{-},y^{+})$ be in a neighborhood
of $0$. Make the variable change $(y^{-},y^{+})\mapsto(u^{-},u^{+})$ with

$\displaystyle u^{+}$ $\displaystyle=y^{+}-w(y^{-},t),$ $\displaystyle u^{-}$
$\displaystyle=y^{-},$

where $w$ is the function defined in the statement of Prop. 7.3, and let

$\displaystyle W(u^{-},u^{+},t)$
$\displaystyle=h^{+}(y,t)-\partial_{1}w(y^{-},t)h^{-}(y,t)-\partial_{2}w(y^{-},t)$
$\displaystyle=h^{+}((u^{-},u^{+}+w(u^{-},t)),t)$
$\displaystyle-\partial_{1}w(u^{-},t)h^{-}((u^{-},u^{+}+w(u^{-},t)),t)-\partial_{2}w(u^{-},t).$

By Prop. 7.3 and Lem. 7.2, it holds that $W(u^{-},0,t)=0$. Moreover,
$W(u^{-},\cdot,t)\in\mathcal{C}^{1}$ by the assumptions on $h$. Therefore,
writing $y(r)=(u^{-},ru^{+}+w(u^{-},t))$ for $r\in[0,1]$, and using the
decomposition (67), we get that

$\displaystyle W(u^{-},u^{+},t)$
$\displaystyle=\int_{0}^{1}\partial_{2}W(u^{-},ru^{+},t)u^{+}\,dr$
$\displaystyle=\Lambda^{+}u^{+}$
$\displaystyle\phantom{=}+\int_{0}^{1}\left(\partial_{1}\varepsilon^{+}(y(r),t)\begin{bmatrix}0\\\
I_{d^{+}}\end{bmatrix}-\partial_{1}w(u^{-},t)\partial_{1}\varepsilon^{-}(y(r),t)\begin{bmatrix}0\\\
I_{d^{+}}\end{bmatrix}\right)u^{+}dr.$

We can also write $y(r)=(y^{-},ry^{+}+(1-r)w(y^{-},t))$. Recalling that
$w(0,t)=0$ and that $\|\partial_{1}w(y^{-},t)\|$ is bounded on
$\mathbb{R}^{d^{-}}\times\mathbb{I}$, we get by the mean value inequality that
$\left\lVert w(y^{-},t)\right\rVert\leq C\left\lVert y^{-}\right\rVert$ where
$C>0$ is a constant. Thus, $\left\lVert y(r)\right\rVert\leq(1+C)\left\lVert
y\right\rVert$. Moreover, $\varepsilon(y,t)=Q^{-1}e(Qy,t)$ for $\|y\|<\sigma$.
Thus, we get by (13) that
$\left\lVert\partial_{1}\varepsilon(y(r),t)\right\rVert\to 0$ as
$(y,t)\to(0,\infty)$ uniformly in $r\in[0,1]$. Using again the boundedness of
$\|\partial_{1}w(\cdot,\cdot)\|$, we eventually obtain that

$W(u^{-},u^{+},t)=\left(\Lambda^{+}+\Delta(y,t)\right)u^{+},\quad\text{with}\quad\lim_{(y,t)\to(0,\infty)}\Delta(y,t)=0.$

On the event $E_{p}$, assume that $n\geq p$, and write

$u^{+}_{n}=y^{+}_{n}-w(y^{-}_{n},\tau_{n}),\quad u^{-}_{n}=y^{-}_{n},$

(see Eq. (82)). Choosing $\alpha_{-}>0$ small enough so that the gap condition
(65) is satisfied with $m=2$, we have by Taylor’s expansion

$\displaystyle w(y^{-}_{n+1},\tau_{n+1})-w(y^{-}_{n},\tau_{n})$
$\displaystyle=w(y^{-}_{n+1},\tau_{n+1})-w(y^{-}_{n},\tau_{n+1})+w(y^{-}_{n},\tau_{n+1})-w(y^{-}_{n},\tau_{n})$
$\displaystyle=\partial_{1}w(y^{-}_{n},\tau_{n+1})(y^{-}_{n+1}-y^{-}_{n})+\gamma_{n+1}\partial_{2}w(y^{-}_{n},\tau_{n})+\epsilon_{n+1}+\epsilon_{n+1}^{\gamma}\,,$
$\displaystyle\text{with}\quad\left\lVert\epsilon_{n+1}\right\rVert$
$\displaystyle\leq\underset{y^{-}\in[y^{-}_{n},y^{-}_{n+1}]}{\sup}\left\lVert\partial_{1}^{2}w(y^{-},\tau_{n+1})\right\rVert\left\lVert
y^{-}_{n+1}-y^{-}_{n}\right\rVert^{2}\,,$
$\displaystyle\text{and}\quad\left\lVert\epsilon^{\gamma}_{n+1}\right\rVert$
$\displaystyle\leq\underset{\tau\in[\tau_{n},\tau_{n+1}]}{\sup}\left\lVert\partial_{2}^{2}w(y^{-}_{n},\tau)\right\rVert\gamma_{n+1}^{2}\,.$

Using this equation, we obtain

$\displaystyle u^{+}_{n+1}-u^{+}_{n}$
$\displaystyle=\gamma_{n+1}W(u^{-}_{n},u^{+}_{n},\tau_{n})+\gamma_{n+1}\left(\tilde{\eta}^{+}_{n+1}-\partial_{1}w(y^{-}_{n},\tau_{n+1})\tilde{\eta}^{-}_{n+1}\right)$
$\displaystyle\phantom{=}+\gamma_{n+1}\left(\tilde{\rho}^{+}_{n+1}-\partial_{1}w(y^{-}_{n},\tau_{n+1})\tilde{\rho}^{-}_{n+1}\right)-\epsilon_{n+1}-\epsilon^{\gamma}_{n+1}$
$\displaystyle\phantom{=}+\gamma_{n+1}\left(\partial_{1}w(y^{-}_{n},\tau_{n})-\partial_{1}w(y^{-}_{n},\tau_{n+1})\right)h^{-}(y_{n},\tau_{n})\,,$

which leads to

$u^{+}_{n+1}=u^{+}_{n}+\gamma_{n+1}\left(\Lambda^{+}+\Delta(y_{n},\tau_{n})\right)u^{+}_{n}+\gamma_{n+1}\bar{\eta}_{n+1}+\gamma_{n+1}\bar{\rho}_{n+1},$
(84)

with
$\bar{\eta}_{n+1}=\tilde{\eta}^{+}_{n+1}-\partial_{1}w(y^{-}_{n},\tau_{n})\tilde{\eta}^{-}_{n+1}$
and

$\bar{\rho}_{n+1}=\tilde{\rho}^{+}_{n+1}-\partial_{1}w(y^{-}_{n},\tau_{n})\tilde{\rho}^{-}_{n+1}-\mathbbm{1}_{\gamma_{n+1}>0}\frac{\epsilon_{n+1}+\epsilon^{\gamma}_{n+1}}{\gamma_{n+1}}\\\
+\left(\partial_{1}w(y^{-}_{n},\tau_{n})-\partial_{1}w(y^{-}_{n},\tau_{n+1})\right)h^{-}(y_{n},\tau_{n})\,.$
(85)

To finish the proof, it remains to check that the noise sequence satisfies the
assumptions of Prop. 7.4 on the event $A_{p}=E_{p}\cap[y_{n}\rightarrow 0]$.
In the remainder, $C^{\prime}$ will indicate some positive constant which can
change from an inequality to another one.

First, we verify that $\sum_{n}\|\bar{\rho}_{n}\|^{2}<\infty$ on $A_{p}$ by
controlling each one of the terms of $\bar{\rho}_{n}$. Combining the
boundedness of $\partial_{1}w(\cdot,\cdot)$ with the summability assumption
$\sum_{n}\|\tilde{\rho}_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}<+\infty$
a.s., we immediately obtain on $A_{p}$ that
$\sum_{n}\|\tilde{\rho}^{+}_{n+1}-\partial_{1}w(y^{-}_{n},\tau_{n})\tilde{\rho}^{-}_{n+1}\|^{2}<+\infty$
given our choice of $\sigma$. Moreover, it holds that
$\left(\|\epsilon^{\gamma}_{n+1}\|/\gamma_{n+1}\right)^{2}\leq
C^{\prime}\gamma^{2}_{n+1}$ by invoking Prop. 7.3. In addition, using the
boundedness of $\partial_{1}^{2}w(\cdot,\cdot)$, we can write

$\mathbbm{1}_{\gamma_{n+1}>0}\left\|\frac{\epsilon_{n+1}}{\gamma_{n+1}}\right\|^{2}\leq\mathbbm{1}_{\gamma_{n+1}>0}\frac{C^{\prime}}{\gamma_{n+1}^{2}}\left\lVert
y_{n+1}-y_{n}\right\rVert^{4}\\\ \leq C^{\prime}\gamma_{n+1}^{2}(\left\lVert
h(y_{n},\tau_{n})\right\rVert^{4}+\left\lVert\tilde{\eta}_{n+1}\right\rVert^{4}+\left\lVert\tilde{\rho}_{n+1}\right\rVert^{4})\,.$

A coupling argument (see [13, p. 401]) shows that we can simplify the
condition
$\limsup\mathbb{E}[\|\eta_{n+1}\|^{4}\,|\,{\mathscr{F}}_{n}]\mathbbm{1}_{z_{n}\in\mathcal{W}}<\infty$
to
$\mathbb{E}[\|\eta_{n+1}\|^{4}\,|\,{\mathscr{F}}_{n}]\mathbbm{1}_{z_{n}\in\mathcal{W}}<C^{\prime}$.
The latter condition implies that
$\mathbb{E}[\mathbbm{1}_{A_{p}}\sum_{n}\gamma_{n+1}^{2}\left\lVert\eta_{n+1}\right\rVert^{4}]\leq\sum_{n}C^{\prime}\gamma_{n+1}^{2}$,
and therefore
$\sum_{n}\gamma_{n+1}^{2}\left\lVert\eta_{n+1}\right\rVert^{4}\mathbbm{1}_{A_{p}}<+\infty$
a.s. As a consequence, noticing also the boundedness of $(h(y_{n},\tau_{n}))$
and $(\tilde{\rho}_{n})$ on $A_{p}$, we deduce that
$\sum_{n}\mathbbm{1}_{\gamma_{n+1}>0}\left\|\frac{\epsilon_{n+1}}{\gamma_{n+1}}\right\|^{2}<+\infty$
on $A_{p}$. We now briefly control the last term of $\bar{\rho}_{n}$. By the
mean value inequality, we obtain that

$\left\|\left(\partial_{1}w(y^{-}_{n},\tau_{n})-\partial_{1}w(y^{-}_{n},\tau_{n+1})\right)h^{-}(y_{n},\tau_{n})\right\|\\\
\leq\gamma_{n+1}\underset{(y^{-},t)}{\sup}\left\lVert\partial_{2}\partial_{1}w(y^{-},t)\right\rVert\|h^{-}(y_{n},\tau_{n})\|\leq
C^{\prime}\gamma_{n+1}\,,$

where the last inequality stems from Prop. 7.3-Eq. (80) together with the
boundedness of the sequence $(h(y_{n},\tau_{n}))$. In view of Eq. (85) and the
above estimates, we deduce that
$\sum_{n}\|\bar{\rho}_{n+1}\|^{2}\mathbbm{1}_{A_{p}}<+\infty$ a.s. on $A_{p}$.

We verify the remaining conditions on the noise sequence $(\bar{\eta}_{n})$.
We can easily remark that $\mathbb{E}[\bar{\eta}_{n+1}|{\mathscr{F}}_{n}]=0$
and $\left\lVert\bar{\eta}_{n+1}\right\rVert\leq
C^{\prime}\left\lVert\eta_{n+1}\right\rVert$ on $A_{p}$. Hence,
$\limsup\mathbb{E}[\|\bar{\eta}_{n+1}\|^{4}\,|\,{\mathscr{F}}_{n}]\mathbbm{1}_{z_{n}\in\mathcal{W}}<\infty$.
The last condition, meaning that the noise is exciting enough, stems from
noting that

$\displaystyle
2\liminf\mathbb{E}[\left\lVert\bar{\eta}_{n+1}\right\rVert^{2}|{\mathscr{F}}_{n}]\mathbbm{1}_{A_{p}}$
$\displaystyle\geq\liminf\mathbb{E}[\left\lVert\tilde{\eta}^{+}_{n+1}\right\rVert^{2}|{\mathscr{F}}_{n}]\mathbbm{1}_{A_{p}}$
$\displaystyle\phantom{=}-2\limsup\mathbb{E}[\left\lVert\partial_{1}w(y_{n}^{-},\tau_{n})\tilde{\eta}_{n+1}^{-}\right\rVert^{2}|{\mathscr{F}}_{n}]\mathbbm{1}_{A_{p}}$
$\displaystyle>\frac{c^{2}}{2}\,,$

where we used our choice of $\sigma,t_{1}$ and Eq. (83).

Noticing that $[y_{n}\to 0]\subset[\Delta(y_{n},\tau_{n})\to 0]$, we can now
apply Prop. 7.4 to the sequence $(u^{+}_{n})$ (see Eq. (84)) with $A=A_{p}$ to
obtain

$\mathbb{P}\left(A_{p}\cap[u^{+}_{n}\to
0]\right)=\mathbb{P}\left(A_{p}\cap[u^{+}_{n}\to
0]\cap[\Delta(y_{n},\tau_{n})\to 0]\right)=0\,.$

We now show that $[y_{n}\to 0]\subset[u^{+}_{n}\to 0]$ , which amounts to
prove that $w(y^{-}_{n},\tau_{n})\to 0$ given $y_{n}\to 0$. To that end, upon
noting that $w(0,\cdot)\equiv 0$ and that $\partial_{1}w(\cdot,\cdot)$ is
bounded, it suffices to apply the mean value inequality, writing :

$\|w(y^{-}_{n},\tau_{n})\|=\|w(y^{-}_{n},\tau_{n})-w(0,\tau_{n})\|\leq\sup_{(y^{-},t)}\|\partial_{1}w(y^{-},t)\|\
\|y^{-}_{n}\|\leq K\|y^{-}_{n}\|\,.$

We have shown so far that $\mathbb{P}(A_{p})=0$. Since $y_{n}=Q^{-1}z_{n}$ and
$[y_{n}\to 0]\subset\bigcup_{p\in\mathbb{N}}E_{p}$, we finally obtain that

$\mathbb{P}[z_{n}\to 0]=\mathbb{P}[y_{n}\to
0]=\mathbb{P}\left(\bigcup_{p\in\mathbb{N}}([y_{n}\to 0]\cap
E_{p})\right)=\mathbb{P}\left(\bigcup_{p\in\mathbb{N}}A_{p}\right)=0.$

Th. 4.1 is proven.

###  7.3 Proofs for Section 4.2.1

####  7.3.1 Proof of Lem. 4.2

The matrix $D$ coincides with $\nabla g_{\infty}(z_{\star})$, where the
function $g_{\infty}$ is defined in (20). As such, its expression is
immediate. Recalling that $p_{\infty}S(x_{\star})-q_{\infty}v_{\star}=0$, we
get

$\displaystyle g(z,t)-D(z-z_{\star})$
$\displaystyle=\begin{bmatrix}{\mathsf{p}}(t)S(x)-{\mathsf{q}}(t)v-p_{\infty}\nabla
S(x_{\star})(x-x_{\star})+q_{\infty}(v-v_{\star})\\\ {\mathsf{h}}(t)\nabla
F(x)-{\mathsf{r}}(t)m-h_{\infty}\nabla^{2}F(x_{\star})(x-x_{\star})+r_{\infty}m\\\
-m\left((v+\varepsilon)^{-\frac{1}{2}}-(v_{\star}+\varepsilon)^{-\frac{1}{2}}\right)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}-{\mathsf{q}}(t)+q_{\infty}&0&({\mathsf{p}}(t)-p_{\infty})\nabla
S(x_{\star})\\\
0&r_{\infty}-{\mathsf{r}}(t)&({\mathsf{h}}(t)-h_{\infty})\nabla^{2}F(x_{\star})\\\
\frac{m}{2(v_{\star}+\varepsilon)^{\frac{3}{2}}}&0&0\end{bmatrix}\begin{bmatrix}v-v_{\star}\\\
m\\\ x-x_{\star}\end{bmatrix}$
$\displaystyle\phantom{=}+\begin{bmatrix}{\mathsf{p}}(t)(S(x)-S(x_{\star})-\nabla
S(x_{\star})(x-x_{\star}))\\\ {\mathsf{h}}(t)(\nabla
F(x)-\nabla^{2}F(x_{\star})(x-x_{\star}))\\\
-m\odot\left(\frac{1}{\sqrt{v+\varepsilon}}-\frac{1}{\sqrt{v_{\star}+\varepsilon}}+\frac{v-v_{\star}}{2(v_{\star}+\varepsilon)^{\frac{3}{2}}}\right)\end{bmatrix}+\begin{bmatrix}{\mathsf{p}}(t)S(x_{\star})-{\mathsf{q}}(t)v_{\star}\\\
0\\\ 0\end{bmatrix}$ $\displaystyle\triangleq e(z,t)+c(t).$

Under the assumptions made, it is easy to see that the function $e(z,t)$ has
the properties required in the statement of Th. 4.1.

####  7.3.2 Proof of Th. 4.3

Consider the matrix $D$ defined in the statement of Lem. 4.2. A spectral
analysis of this matrix as regards its eigenvalues with positive real parts is
done in the following lemma.

######  Lemma 7.5.

Let $D$ be the matrix provided in the statement of Lem. 4.2. Each eigenvalue
$\zeta$ of the matrix $D$ such that $\Re\zeta>0$ is real, and its algebraic
and geometric multiplicities are equal. Moreover, there is a one-to-one
correspondence $\varphi$ between these eigenvalues and the negative
eigenvalues of $V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$. Let
$d^{+}$ be the dimension of the eigenspace of
$V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$ that is associated with
its negative eigenvalues, let

$W=\begin{bmatrix}&w_{1}&\\\ &\vdots\\\
&w_{d^{+}}&\end{bmatrix}\in\mathbb{R}^{d^{+}\times d}$

be a matrix which rows are independent eigenvectors of
$V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$ that generate this
eigenspace, and denote as $\beta_{k}<0$ the eigenvalue associated with
$w_{k}$. Then, the rows of the rank $d^{+}$-matrix

$A^{+}=\begin{bmatrix}0_{d^{+}\times
d},&WV^{\frac{1}{2}},&-\operatorname*{diag}(r_{\infty}+\varphi^{-1}(\beta_{k}))WV^{-\frac{1}{2}}\end{bmatrix}\in\mathbb{R}^{d^{+}\times
3d}$

generate the left eigenspace of $D$, the row $k$ being an eigenvector for the
eigenvalue $\varphi^{-1}(\beta_{k})$.

###### Proof.

It is obvious that the block lower-triangular matrix $D$ has $d$ eigenvalues
equal to $-q_{\infty}$ and $2d$ eigenvalues which are those of the sub-matrix

$\widetilde{D}=\begin{bmatrix}-r_{\infty}I_{d}&h_{\infty}\nabla^{2}F(x_{\star})\\\
-V&0\end{bmatrix}.$

Given $\lambda\in\mathbb{C}$, we obtain by standard manipulations involving
determinants that

$\det(\widetilde{D}-\lambda)=\det(\lambda(r_{\infty}+\lambda)+h_{\infty}V\nabla^{2}F(x_{\star}))=\det(\lambda(r_{\infty}+\lambda)+h_{\infty}V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}).$

Denoting as $\\{\beta_{k}\\}_{k=1}^{d}$ the eigenvalues of
$h_{\infty}V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$ counting the
multiplicities, we obtain from the last equation that the eigenvalues of
$\widetilde{D}$ are the solutions of the second order equations

$\lambda^{2}+r_{\infty}\lambda+\beta_{k}=0,\quad k=1,\ldots,d.$

The product of the roots of such an equation is $\beta_{k}$, and their sum is
$-r_{\infty}\leq 0$. Thus, denoting as $\zeta_{k,1}$ and $\zeta_{k,2}$ these
roots, it is easy to see that if $\beta_{k}\geq 0$, then
$\Re\zeta_{k,1},\Re\zeta_{k,2}\leq 0$, while if $\beta_{k}<0$, then both
$\zeta_{k,i}$ are real, and only one of them is positive. Thus, we have so far
shown that the eigenvalues of $D$ which real parts are positive are themselves
real, and there is a one-to-one map $\varphi$ from the set of positive
eigenvalues of $D$ to the set of negative eigenvalues of
$V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$. Moreover, the
algebraic multiplicity of the eigenvalue $\zeta>0$ of $D$ is equal to the
multiplicity of $\varphi(\zeta)$.

Let us now turn to the left (row) eigenvectors of $D$ that correspond to these
eigenvalues. To that end, we shall solve the equation

$uD=\zeta u\quad\text{with}\ u=[0,u_{1},u_{2}],\quad
u_{1,2}\in\mathbb{R}^{1\times d},$ (86)

for a given eigenvalue $\zeta>0$ of $D$. Developing this equation, we get

$-r_{\infty}u_{1}-u_{2}V=\zeta u_{1},\quad
h_{\infty}u_{1}\nabla^{2}F(x_{\star})=\zeta u_{2}.$

If we now write $\tilde{u}_{1}=u_{1}V^{-\frac{1}{2}}$ and
$\tilde{u}_{2}=u_{2}V^{\frac{1}{2}}$, this system becomes

$-r_{\infty}\tilde{u}_{1}-\tilde{u}_{2}=\zeta\tilde{u}_{1},\quad
h_{\infty}\tilde{u}_{1}V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}=\zeta\tilde{u}_{2},$

or, equivalently,

$\tilde{u}_{2}=-(r_{\infty}+\zeta)\tilde{u}_{1},\quad\tilde{u}_{1}\left(\zeta^{2}+r_{\infty}\zeta+h_{\infty}V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}\right)=0,$

which shows that $\tilde{u}_{1}$ is a left eigenvector of
$V^{\frac{1}{2}}\nabla^{2}F(x_{\star})V^{\frac{1}{2}}$ associated with the
eigenvalue $\varphi(\zeta)$. What’s more, assume that $r$ is the multiplicity
of $\varphi(\zeta)$, and, without generality loss, that the submatrix
$W_{r,\cdot}$ made of the first $r$ rows of $W$ generates the left eigenspace
of $\varphi(\zeta)$. Then, the matrix

$\begin{bmatrix}0_{r\times
d}&W_{r\cdot}V^{\frac{1}{2}}&-(r_{\infty}+\zeta)W_{r\cdot}V^{-\frac{1}{2}}\end{bmatrix}$

is a $r$-rank matrix which rows are independent left eigenvectors that
generate the left eigenspace of $D$ for the eigenvalue $\zeta$. In particular,
the algebraic and geometric multiplicities of this eigenvalue are equal. The
same argument can be applied to the other positive eigenvalues of $D$. ∎

We now have all the elements to prove Th. 4.3. Recall Eq. (14):

$z_{n+1}=z_{n}+\gamma_{n+1}b(z_{n},\tau_{n})+\gamma_{n+1}\eta_{n+1}+\gamma_{n+1}\rho_{n+1},$

where $b(z,t)=g(z,t)-c(t)=D(z-z_{\star})+e(z,t)$ and
$\rho_{n}=c(\tau_{n-1})+\tilde{\rho}_{n}$. With these same notations, we check
that Assumptions 1–6 in the statement of Th. 4.1 are satisfied. The function
$e(z,t)$ satisfies Assumptions 1–4 by Lem. 4.2. We now verify that the
sequence $(\rho_{n})$ fulfills Assumption 5. First, observe that
$\sum_{n}\|c(\tau_{n})\|^{2}<\infty$ under Assumption 4.3-1. Then, we control
the second term $(\tilde{\rho}_{n})$. After straightforward derivations, one
can show the existence of a positive constant $C$ (depending only on
$\varepsilon$ and a neighborhood $\mathcal{W}$ of $z_{\star}$) such that

$\|\tilde{\rho}_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}\leq
C(\|m_{n}-m_{n+1}\|^{2}+\|v_{n+1}-v_{n}\|^{2})\mathbbm{1}_{z_{n}\in\mathcal{W}}\,.$
(87)

Using the boundedness of the sequences $(h_{n})$ and $(r_{n})$ together with
the update rule of $m_{n}$ and Assumption 4.3-3, there exists a positive
constant $C^{\prime}$ independent of $n$ (which may change from an inequality
to another) such that

$\mathbb{E}\left[\|m_{n}-m_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}\right]\leq\gamma_{n+1}^{2}C^{\prime}\mathbb{E}\left[(1+\mathbb{E}_{\xi}\left[\|\nabla
f(x_{n},\xi)\|^{2}\right])\mathbbm{1}_{z_{n}\in\mathcal{W}}\right]\leq
C^{\prime}\gamma_{n+1}^{2}\,.$ (88)

A similar result holds for
$\mathbb{E}\left[\|v_{n}-v_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}\right]$
following the same arguments. In view of Eqs. (87)-(88) and the assumption
$\sum_{n}\gamma_{n+1}^{2}<+\infty$, it holds that
$\mathbb{E}\left[\sum_{n}\|\tilde{\rho}_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}\right]<+\infty$.
Therefore,
$\sum_{n}\|\tilde{\rho}_{n+1}\|^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}<+\infty$
a.s., which completes our verification of condition 5 of Th. 4.1. Assumption 6
follows from condition 4.3-3. Finally, let us make Assumption 7 of Th. 4.1
more explicit. Partitioning the matrix $Q^{-1}$ as
$Q^{-1}=\begin{bmatrix}B^{-}\\\ B^{+}\end{bmatrix}$ where $B^{\pm}$ has
$d^{\pm}$ rows, Lem. 7.5 shows that the row spaces of $B^{+}$ and $A^{+}$ are
the same, which implies that Assumption 7 can be rewritten equivalently as
$\mathbb{E}[\left\lVert
A^{+}\eta_{n+1}\right\rVert^{2}\,|\,{\mathscr{F}}_{n}]\mathbbm{1}_{z_{n}\in\mathcal{W}}\geq
c^{2}\mathbbm{1}_{z_{n}\in\mathcal{W}}$. By inspecting the form of $\eta_{n}$
provided by Eq. (28) (written as a column vector), one can readily check that
Assumption 4.3-4 implies Assumption 7 of Th. 4.1 for a small enough
neighborhood ${{\mathcal{W}}}$, using the continuity of the covariance matrix
$V^{\frac{1}{2}}\mathbb{E}_{\xi}(\nabla f(x,\xi)-\nabla F(x))(\nabla
f(x,\xi)-\nabla F(x))^{\text{T}}V^{\frac{1}{2}}$ when $x$ is near $x_{\star}$.

###  7.4 Proof of Th. 4.4

As mentioned in Section 4.2.2, the proof of Th. 4.4 is almost identical to the
one of Th. 4.3. We point out the main differences here. In Lem. 4.2, replace
$D$ by $\tilde{D}=\begin{bmatrix}0&h_{\infty}\nabla^{2}F(x_{\star})\\\
-I_{d}&0\end{bmatrix}$ and set $c(t)=0$. Then, in Lem. 7.5, replace the matrix
$V^{1/2}\nabla^{2}F(x_{\star})V^{1/2}$ by the Hessian
$\nabla^{2}F(x_{\star})$.

## Acknowledgements

We would like to thank the anonymous reviewers for their outstanding job of
refereeing, especially for their comments on the avoidance of traps results
and the stability of the iterates of the stochastic algorithm which helped us
to improve and clarify our manuscript.
A.B. was supported by the “Futur & Ruptures” research program which is jointly
funded by the IMT, the Mines-Télécom Foundation and the Carnot TSN Institute.
S.S. was supported by the “Région Ile-de-France”.

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# Quantum phases of a frustrated spin-1 system: The 5/7 skewed ladder

Sambunath Das<EMAIL_ADDRESS>Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India  Dayasindhu Dey
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India  Manoranjan Kumar
<EMAIL_ADDRESS>S. N. Bose National Centre for Basic Sciences,
Block - JD, Sector - III, Salt Lake, Kolkata - 700106, India  S. Ramasesha
<EMAIL_ADDRESS>Solid State and Structural Chemistry Unit, Indian
Institute of Science, Bangalore 560012, India

###### Abstract

The quantum phases in a spin-1 skewed ladder system formed by alternately
fusing five- and seven-membered rings are studied numerically using the exact
diagonalization technique up to 16 spins and using the density matrix
renormalization group method for larger system sizes. The ladder has a fixed
isotropic antiferromagnetic (AF) exchange interaction ($J_{2}=1$) between the
nearest-neighbor spins along the legs and a varying isotropic AF exchange
interaction ($J_{1}$) along the rungs. As a function of $J_{1}$, the system
shows many interesting ground states (gs) which vary from different types of
nonmagnetic and ferrimagnetic gs. The study of diverse gs properties such as
spin gap, spin-spin correlations, spin density and bond order reveal that the
system has four distinct phases, namely, the AF phase at small $J_{1}$; the
ferrimagnetic phase with gs spin $S_{G}=n$ for $1.44<J_{1}<4.74$ and with
$S_{G}=2n$ for $J_{1}>5.63$, where $n$ is the number of unit cells; and a
reentrant nonmagnetic phase at $4.74<J_{1}<5.44$. The system also shows the
presence of spin current at specific $J_{1}$ values due to simultaneous
breaking of both reflection and spin parity symmetries.

##  I Introduction

In low-dimensional magnetic systems, confinement leads to strong quantum
fluctuations, and these systems can show many exotic phases in the presence of
frustration induced by the topology of exchange interactions Majumdar and
Ghosh (1969a); *ckm69b; Hamada  _et al._ (1988); Chubukov (1991); Chitra  _et
al._ (1995); White and Affleck (1996); Itoi and Qin (2001); Mahdavifar (2008);
Sirker (2010); Kumar  _et al._ (2015); Soos  _et al._ (2016); Kumar  _et al._
(2010); Chubukov (1991); Kumar and Soos (2012); Kumar  _et al._ (2015); Vekua
_et al._ (2007); Hikihara  _et al._ (2008); Sudan  _et al._ (2009); Dmitriev
and Krivnov (2008); Heidrich-Meisner  _et al._ (2006, 2007, 2009); Parvej and
Kumar (2017); Kecke  _et al._ (2007). Even in a one-dimensional (1D) spin
system, with only a nearest-neighbor Heisenberg antiferromagnetic (HAF)
exchange interaction, the ground state (gs) can be gapped or gapless for
integer or half odd-integer spins, respectively, as pointed out in a seminal
paper by Haldane Haldane (1983a); *haldane83b. The gs of the HAF integer spin
chain can be represented as a valance bond solid (VBS) Affleck  _et al._
(1987, 1988); Schollwöck  _et al._ (1996), and in 1987 Affleck, Kennedy, Lieb,
and Tasaki (AKLT) showed that perfect VBS state may exist on various
geometries with specific spins Affleck  _et al._ (1987, 1988). The AKLT state
still continues to inspire physicists for various reasons; for example, the
AKLT state has led to many recent developments such as the matrix product
states technique Östlund and Rommer (1995); Verstraete  _et al._ (2008);
Schollwöck (2011) which is a form of the density matrix renormalization group
(DMRG) method White (1992, 1993); Schollwöck (2005); Hallberg (2006), the
tensor network method Orús (2014) and the projected entangled pair states
ansatz Verstraete  _et al._ (2008); Schollwöck (2011). AKLT states can also be
represented as cluster states which can be used in measurement-based quantum
computation Verstraete and Cirac (2004); Wei  _et al._ (2011), and recently
these states have been explored in a spin-3/2 on a hexagonal lattice Lemm  _et
al._ (2020); Pomata and Wei (2020).

The HAF spin-1 chain exhibits a topological phase, spin-1/2 edge modes, and
the gs is four fold degenerate in the thermodynamic limit. The correlation
length in the gs of spin-1 is 6.05 lattice units and the eigenvalue spectrum
has large spin gaps Dey  _et al._ (2016); White and Huse (1993). The gs can be
represented as a VBS, which belongs to the same universality class of AKLT
states Affleck  _et al._ (1987, 1988). The two leg HAF spin-1 ladder shows
interesting properties like plaquette-singlet solid state, where two spin-1/2
singlet dimers are sitting at each rung and there is no overlap between the
VBS states in the large rung exchange limit Todo  _et al._ (2001). The AKLT
state in the system breaks down for any finite value of rung exchange
interaction Todo  _et al._ (2001). The spin-1 zigzag ladder shows a transition
from a Haldane phase to a double Haldane phase Hikihara  _et al._ (2002);
Chepiga  _et al._ (2016). In fact the zigzag ladder can be mapped into a chain
system with nearest-neighbor and next nearest-neighbor exchange interactions,
and the gs of the frustrated systems is a singlet. In this work, we explore
the magnetic phases of a spin-1 system on a 5/7-skewed ladder system; it has
been demonstrated that a spin-1/2 system on this lattice shows many exotic
phases Giri  _et al._ (2017).

The 5/7-skewed ladder is inspired by fused Azulene, a ladder like structure
made up of 5- and 7-membered carbon rings alternately fused on a chain,
studied by Thomas _e_ t al. in which they showed that the gs is ferrimagnetic
Thomas  _et al._ (2012). These structures can be mapped to a zigzag like
ladder structure with some missing bonds Thomas  _et al._ (2012); Giri  _et
al._ (2017). The HAF spin-1/2 system on various lattices such as the 5/7, 3/4,
3/5, and 5/5 is studied and it was shown that the gs of these systems exhibits
many interesting magnetic and nonmagnetic gs in their quantum phase diagrams
with strength of the rung exchange interaction as a phase parameter Giri  _et
al._ (2017). In the large rung exchange limit, the gs wavefunction of a 5/7
skewed ladder can be represented as a product of rung singlet dimers and two
ferromagnetically interacting spins per unit cell Dey  _et al._ (2020). In
various parameter regimes this system shows dimer, spiral and chiral vector
phases Giri  _et al._ (2017). In the presence of an axial magnetic field the
HAF spin-1/2 system on the 5/7 skewed ladder exhibits four magnetization
plateau phases Dey  _et al._ (2020).

The structure of zigzag and 5/7 skewed ladder are shown in Figs. 1(a) and 1(b)
and by periodically removing some of the rung bonds, shown in red, from Fig.
1(a) to give 5/7 skewed ladder in Fig. 1(b). In this paper, we are interested
in the gs phases of a spin-1 5/7 skewed ladder as a function of the ratio of
rung-to-leg exchanges $J_{1}$ and $J_{2}$, respectively. We show that this
system is highly frustrated, and in the small rung interaction limit,
$J_{1}/J_{2}<1.06$, singlet dimers along the rung are weak and correlations
along the leg remain short ranged, whereas, for $J_{1}/J_{2}>1.44$, the gs is
magnetic and each unit cell contributes spin-1 to the gs spin $S_{G}$, and
spin densities are distributed over the whole unit cell, with spin density at
sites 3 and 7 being large. For $4.74<J_{1}/J_{2}<5.44$ the system is
nonmagnetic but for $J_{1}>5.63$ gs of the system is magnetic with each unit
cell contributing spin 2 to $S_{G}$ with prominent rung dimers and site spin
densities.

Figure 1: Schematic diagram of (a) a zigzag ladder. The nearest-neighbor or
rung interaction is $J_{1}$ and the next-nearest neighbor (along the leg)
interaction is $J_{2}$. (b) The 5/7 skewed ladder: Some rung bonds shown in
red of the zigzag ladder are periodically removed to give a 5/7 skewed ladder.
Here “$i$” is the index of the unit cell and the numerals 1, 2, …are numbering
of the spins within the unit cell. There are 2 spins per unit cell in the
zigzag ladder while there are 8 spins per unit cell in the 5/7 ladder. The
sites on the top leg are even numbered and on the bottom leg are odd numbered.

This paper is divided into four sections. In Section II we discuss the model
Hamiltonian and the numerical methods. The results are presented and discussed
in Section III under four subsections. Section IV provides a summary of
results and conclusions.

##  II Model and Method

The site numbering used in this paper for the 5/7 skewed ladder is shown in
Fig. 1(b). All nonzero exchange interactions between spins are
antiferromagnetic (AF). The sites are numbered such that odd numbered sites
are on the bottom leg and even numbered sites are on the top leg. Thus the
rung bonds are the nearest-neighbor exchanges $J_{1}$ and the bonds on the
legs are the next-nearest-neighbor exchanges $J_{2}$. The exchange $J_{2}$ is
set to 1 and it defines the energy scale. The model Hamiltonian of the 5/7
skewed ladder can be written as

$\displaystyle H_{5/7}$ $\displaystyle=$ $\displaystyle
J_{1}\sum_{i}\left(\vec{S}_{i,1}\cdot\vec{S}_{i,2}+\vec{S}_{i,4}\cdot\vec{S}_{i,5}\right)$
(1)
$\displaystyle+J_{2}\sum_{i}\bigg{(}\vec{S}_{i,7}\cdot\vec{S}_{i+1,1}+\vec{S}_{i,8}\cdot\vec{S}_{i+1,2}$
$\displaystyle\qquad\qquad+\sum_{k=1}^{6}\vec{S}_{i,k}\cdot\vec{S}_{i,k+2}\bigg{)}.$

where $i$ labels the unit cell and $k$ are the spins within the unit cell
(Fig. 1). The first term denotes the rung exchange terms, and the second term
denotes the exchange interactions along the legs.

We use the exact diagonalization technique for finite ladders with up to 16
spins and impose periodic boundary condition (PBC). There are mirror planes
perpendicular to the ladder, for example, the plane perpendicular to the
ladder and passing through site 3 and the perpendicular bisector of sites 2
and 4, as well as the one passing through site 7 and the perpendicular
bisector of sites 6 and 8, again perpendicular to the ladder. An extra rung is
needed when the open boundary condition (OBC) is used. For larger system sizes
we use the DMRG method White (1992, 1993); Schollwöck (2005); Hallberg (2006)
to handle the large degrees of freedom in the many body Hamiltonian. We retain
up to 500 block states ($m=500$), which are the eigenvectors of the block
density matrix with dominant eigenvalues. The chosen value of “$m$” keeps the
truncation error to less than $\sim 10^{-10}$. We also carry out 6–10 finite
sweeps for improved convergence. The details of building the 5/7 ladder for
the DMRG method is the same as in Ref.[ Giri  _et al._ , 2017]. The largest
system size studied is a system with 130 sites or 16 unit cells with OBC. The
DMRG calculations are carried out for different $S^{z}$ values of ladders. The
gs spin is $S_{G}=l$, for $l$ that satisfies $\Gamma_{l}=0$ and
$\Gamma_{l+1}>0$, where $\Gamma_{l}$ is given by

$\Gamma_{l}=E_{0}(S^{z}=l)-E_{0}(S^{z}=0),$ (2)

with $E_{0}$ being the lowest energy state in the chosen $S^{z}$ sector. The
correlation function and bond orders are computed in the gs, with $S^{z}=S$.

##  III Results and discussions

In the gs, the spin-1 5/7 skewed ladder, like the spin-1/2 system, also shows
many exotic phases like the bond order wave (BOW) phase, chiral points in
parameter space of the Hamiltonian and nonmagnetic to magnetic phase
transition on tuning the value of $J_{1}$. However, there are significant
differences from the spin-1/2 system. To analyze the magnetic transitions in
the quantum phase diagram, various quantities are analyzed as a function of
$J_{1}/J_{2}$ which is the only variable model parameter in this system and
$J_{2}$ is set to 1. Besides the spin gaps $\Gamma_{l}$, we have the computed
correlation function $C(r)=\langle\vec{S}_{i}\cdot\vec{S}_{i+r}\rangle$ to
study the behavior of the spins in the system. The bond order between bonded
neighbors $-\langle\vec{S}_{i}\cdot\vec{S}_{i^{\prime}}\rangle$ where sites
$i$ and $i^{\prime}$ are bonded neighbors, and spin-density $\langle
S^{z}_{i}\rangle$ within a unit cell are also calculated and compared with the
results for a 1D spin-1 chain where appropriate.

###  III.1 Nature of gs

The spin in the gs, $S_{G}$, of the skewed 5/7 ladder systems is obtained from
the magnetic gaps $\Gamma_{l}$ defined in Eq. 2. In Fig. 2(a), we plot the
gaps $\Gamma_{l}$ for different values of “$l$”, as a function of $J_{1}$ for
a system with 24 spins corresponding to 3 unit cells $(n=3)$ under PBC. The
plot shows that there are four distinct regions: In region I with
$0<J_{1}<1.06$, the gs is a singlet and is nonmagnetic; in region II,
$1.06<J_{1}<4.74$, $S_{G}$ is less than or equal to the number of unit cells
in the systems, consequently each unit cell contributes at most spin-1 to
$S_{G}$ and for $1.44<J_{1}<4.74$, the spin $S_{G}$ saturates to the number of
unit cells. We calculate $S_{G}/n$ as a function of $J_{1}$ for systems with
$N=24$ and $N=48$ spins to investigate if the transition to $S_{G}=n$ is
smooth or abrupt. We find that $S_{G}/n$ shows a gradual increase in the
region $1.06<J_{1}<1.44$ for the larger system, shown in Fig. 2(b). The weak
finite size effect in $S_{G}/n$ can be attributed to the short spin-spin
correlation lengths. As shown in Fig. 2(b), the increase in $S_{G}/n$ is
continuous between regions I and II. In region III, $4.74<J_{1}<5.44$, the gs
becomes nonmagnetic for the 24 spin system. The spin-spin correlation function
reveals that in this region the “free” spins in each unit cell align
ferromagnetically while the alignment of the spins across unit cells is AF,
with large periodicity. The transition from region II to region III is abrupt
for $N=24$ spins and we had convergence difficulties even for the $N=48$ spins
and hence cannot comment on the effect of system size.

We investigated the spin gaps at $J_{1}=5.1$ (where there is a peak in the
$\Gamma_{l}$ values for the $N=24$ spin system) as a function of system size
from $N=24$ to $N=96$ spins, retaining 2400 block states in the finite DMRG
calculations for both PBC and OBC. We find that for this $J_{1}$ value, the
gaps $\Gamma_{1}$ to $\Gamma_{n}$ exhibit nonlinear variation with system size
as shown in Figs. 2(c) and 2(d) for PBC and OBC, respectively. The convergence
for higher values of $l$ in $\Gamma_{l}$ for PBC is poor and hence is not
shown in Fig. 2(c). We find that for some system sizes (number of unit cells
$n$) the excitation gaps $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ vanish,
for both PBC and OBC. We surmise that the vanishing of the gaps is because for
these “$n$”, the number of unit cells in the system is an integral multiple of
the periodicity of the spin-spin correlations. In region IV, the spin of the
gs is $2n$, and indicates that all the “free” spins are ferromagnetically
aligned. The transition from region III to region IV was followed for the 24
spin system under PBC, by varying $J_{1}$ in small increments. We find a step
in the gs spin at an intermediate value $1<S_{G}/n<2$ with a step width of
0.18 in $J_{1}$, but this region could not be studied for larger system sizes
due to convergence difficulties. In region IV, with $J_{1}>5.63$, $S_{G}$
corresponds to twice the number of unit cells. We summarize the behavior of
the gs in different regions, in Fig 3

Figure 2: (a) The lowest excitation gaps $\Gamma_{l}$ for different $S^{z}=l$
manifolds are shown as functions of $J_{1}$. For $J_{1}<1.06$, $\Gamma_{1}$ is
nonzero, whereas for $1.06<J_{1}<4.74$, $\Gamma_{l}$ is zero for $l\leq S$
where $S$ is the total spin of the gs. The system exhibits reentrant
nonmagnetic phase for $4.74<J_{1}<5.44$ where $\Gamma_{1}$ is nonzero. For
$1.44<J_{1}<4.74$ $S_{G}\sim n$ and for $J_{1}>5.63$ $S_{G}\sim 2n$. (b) The
variation of the gs spin per unit cell $S_{G}/n$ with $J_{1}$ for system sizes
of 48 $(n=6)$ spins in regions I and II, and of $24$ $(n=3)$ spins in all the
regions. (c) The lowest excitation gaps $\Gamma_{1}$, $\Gamma_{2}$ and
$\Gamma_{3}$ with the inverse of the number of unit cells of a 5/7 with PBC
are shown for $J_{1}=5.1$. (d) The lowest excitation gaps $\Gamma_{1}$ to
$\Gamma_{7}$ with the inverse of the number of unit cells of a 5/7 with OBC
are shown for $J_{1}=5.1$. Figure 3: The variation of the gs spin SG with the
number of unit cells $n$ is shown for $J_{1}=1.0$, 1.9 and 6.0.

###  III.2 Spin correlations

To understand the spin structure in different regions of the parameter space,
we have studied the spin-spin correlations of the total spin
$C(r)=\langle\vec{S}_{k}\cdot\vec{S}_{k+r}\rangle$, where $k$ is the reference
site of spins in the middle of the system. The $z$ component of the spin
correlations, $C^{zz}(r)=\langle S^{z}_{k}S^{z}_{k+r}\rangle-\langle
S^{z}_{k}\rangle\langle S^{z}_{k+r}\rangle$, shows behavior similar to the
total spin correlation. The total spin correlations are shown for a system of
98 spins, which correspond to 12 unit cells for OBC. These are calculated in
the gs with $S^{z}=S_{G}$. There are three different spin correlations that we
have computed. They correspond to the correlation between spins on the lower
leg $C_{1}(49,49+2r)$, $C_{2}(50,50+2r)$ between spins on the upper leg and
$C_{3}(51,51+4r)$ between “free” spins which reside on the lower leg. The
reference site for the correlations is from the middle unit cell which for
$C_{1}$ is site 49, for $C_{2}$ is site 50 and for $C_{3}$ is site 51. For
convenience, we classify the spins on the lower leg as of two types, type 1
“bound” spins, which are bound to three nearest-neighbor spins and type 2 as
“free” spins, which are the middle sites in the five and seven-membered rings.

In Fig. 4, we show the spin correlations in the four different regions of the
parameter space. The correlations $C_{1}(r)$ shown in Fig. 4(a) correspond to
spins in the lower leg. The correlations are given from site 49 which is in
the middle of the system. For $J_{1}=0$, we have the correlation length of the
Haldane system. We also find that as $J_{1}$ is increased, the correlation
length gradually decreases to $\sim 2.2$ for $J_{1}=1.0$. In the transition
region between I and II, the correlations fluctuate rapidly and we can not
extract a correlation length. We note that for $J_{1}=1.0$, the system has a
singlet gs. The correlations fall off rapidly and the correlation length $\xi$
is $\sim 2.2$ sites of the specified kind. In the spin-1 AF chain the
correlation length is longer by almost a factor of three and is approximately
six sites. The shorter correlation length can perhaps be attributed to the
frustration in exchange interactions in the rings. For $J_{1}=2.0$ and 6.0,
the system is in a magnetic state, $C_{1}(r)$ decays very slowly and is AF in
nature. In the reentrant phase the system goes to a nonmagnetic state and the
spin correlations between the spins on the lower leg have long wavelength spin
oscillations whose amplitude shows an exponential decay and corresponds to a
noncollinear spin arrangement. From the spin correlations, it appears that the
magnetic unit cell is tripled in this region. In Fig. 4(b), $C_{2}(r)$ in the
upper leg are shown for the four phases, with the reference spin being the
50th spin in the system. $C_{2}(r)$ for both $J_{1}=1$ and 2 are AF and
exponentially decaying with correlation length, $\xi\approx 3$. For $J_{1}=5$
and 6, $C_{2}(r)$ are vanishingly small, the magnitude is less than $\approx
0.05$ even for nearest-neighbor pair, and show long wavelength behavior.
However, the amplitude of this wave is too small to definitively conclude this
oscillatory behavior. The small correlation in upper leg at high $J_{1}$ is
due to the strong dimer formation along the rungs and between ($i$, 6) and
($i$, 8) sites. The correlations between “free” spins $C_{3}(r)$ shows a rapid
decay in the nonmagnetic state at $J_{1}=1$, while those for $J_{1}=2$ and
$J_{1}=6$, the correlations are ferromagnetic. For the $J_{1}=6$, the spins at
these sites have almost completely aligned ferromagnetically, while for
$J_{1}=2$, the alignment is partially ferromagnetic. This reflects in the net
spin of the gs which is $2n$ for the $J_{1}=6$ case and $n$ for the $J_{1}=2$
case. In the reentrant phase, the free spins in each unit cell are aligned
ferromagnetically while the alignment of these spins across unit cells is AF,
with large periodicity.

Figure 4: The spin-spin correlations between (a) spins in the lower leg, (b)
spins in the upper leg and (c) free spins at sites 3, 7, 11, etc. for a 5/7
ladder with $N=98$ spins with OBC. Four $J_{1}$ values are chosen to represent
different regions of the phase diagram. $J_{1}=1.0$ for the nonmagnetic phase,
$J_{1}=2.0$ for the $S_{G}=n$ phase, $J_{1}=5.0$ for the reentrant phase, and
$J_{1}=6.0$ for the $S_{G}=2n$ phase.

In summary, all the nearest-neighbor spin correlations are always AF, in the
nonmagnetic gs for $J_{1}$ in regime I, and the correlation lengths are much
shorter than the Haldane chain. For $J_{1}$ values in regime-II where
$S_{G}\sim n$, the correlations in the lower leg are AF and very long ranged
while those in the upper leg are AF and fall off rapidly. The “free” spin
correlations are ferromagnetic with an amplitude of about $0.6$ and show very
slow decay. When the $J_{1}$ value is in the reentrant regime III, the lower
leg spin correlations show formation of wave packets over approximately three
unit cells. The “free” spin correlations show a long wavelength oscillatory
behavior with about five unit cell wavelength, which corresponds to a long
period Néel arrangement of “free” spins. While in the classical, frustrated
$J_{1}-J_{2}$ model, the pitch angle is dependent on $J_{1}/J_{2}$, in the
skewed ladder we have not been able to obtain a similar relationship. Besides,
it is unlikely that a classical model will exhibit a reentrant phase. In
regime IV, the “bound” spins on the lower leg are antiferromagnetically
aligned and the correlations fall off very slowly with distance. The
correlations between spins on the upper leg show weak long period Néel
structure. The “free” spins are aligned ferromagnetically with very long
correlation length.

###  III.3 Spin densities and bond order

The correlation lengths in the system are short, often less than distance to a
third equivalent nearest-neighbors. Hence, we can get qualitatively correct
behavior of the system in the thermodynamic limit from high accuracy DMRG
studies on a system with three unit cells. We have carried out studies on a 48
site spin-1 system, with PBC corresponding to six unit cells $(n=6)$. We have
retained 2400 block states for high accuracy in our DMRG computations. We have
computed the spin densities in the gs for $S^{z}=S_{G}$ and the bond orders of
all the nearest-neighbor bonds. The system has reflection symmetry and hence
there are five unique bonds and five unique sites. The spin densities are
computed as the expectation value of $\langle\psi_{gs}\mid
S_{i}^{z}\mid\psi_{gs}\rangle$. They are uniformly zero in the singlet gs. The
bond orders are computed as $b_{i,j}=-\langle\psi_{gs}\mid S_{i}\cdot
S_{j}\mid\psi_{gs}\rangle$, where $i$ and $j$ are nearest-neighbor bonds. When
the gs of the system is a singlet (region I), from the bond orders (Fig. 5) we
can describe the system as weakly coupled spin-1 HAF chains. The upper leg has
a BOW with a periodicity of four bonds, while the lower leg has a BOW with a
periodicity of two bonds. The rung bonds are weak and the leg bond orders vary
between $1.285$ and $1.370$. For comparison, in the spin-1 HAF, all the bond
orders are uniform and have a value of $1.40$. In region II, where $S_{G}=n$,
the upper leg bond between the sites in the pentagon become weak, the rung
bonds become strong and the bonds in the lower leg also become slightly weak.
Besides, the bond on the upper leg between the sites which entirely belongs to
the seven-membered ring also becomes strong. The rung bonds become much
stronger while the ladder bonds become weaker. The spin densities at the
“free” spin sites are nearly equal and there is a net negative spin density on
the rung bonds with the spin density of the sites on the lower leg being large
negative. The “bonded” spin sites on the seven-membered ring on the upper leg
acquire small negative spin densities. In region IV, where the $S_{G}=2n$, the
rung bonds and the bond in the upper leg of the seven-membered ring almost
form singlets, with a bond order close to $2.0$. All other bonds are very
weak. The spin densities of the sites in the seven-membered ring which form
the singlet are very nearly zero, while the rung bonds are qualitatively
different with large negative (on the lower leg) and positive (on the upper
leg) spin densities. The “free” spins are almost completely polarized and have
spin densities that are very nearly unity.

Figure 5:  Bond orders for 5/7 skewed ladder with $N=48$ spins and PBC in the
four different regions: (a) Nonmagnetic at $J_{1}=1.0$, (b) $S_{G}=n$ at
$J_{1}=1.5$, (c) reentrant nonmagnetic at $J_{1}=5.1$, and (d) $S_{G}=2n$ at
$J_{1}=6.0$. The numbers adjacent to the bonds are the bond orders. The width
of the bonds is proportional to the magnitude of the bond order. The site
numbering is given in (a), the zero spin density is represented by open
circles in (a) and (c), the spin density in (b) and (d) is proportional to the
area of the filled circles. Red circles represent positive spin densities and
blue circles represent negative spin densities.

###  III.4 Vector chirality

Broken symmetry states give rise to different quantum phases whose properties
depend on the type of symmetry that is broken in the system. In general,
broken spatial inversion/reflection symmetry gives rise to bond the BOW phase,
whereas broken spin inversion symmetry gives rise to spin density wave (SDW).
If both the spatial and spin inversion symmetries in the system are broken
then the vector chiral phase arises and it leads to a spontaneous spin current
in the system. For these symmetries to break simultaneously, the lowest energy
states in the two subspaces that the symmetry element divides the appropriate
Hilbert space should be degenerate. In this case, any linear combination of
the two low-lying states in the two subspaces which are even (odd) under both
reflection and spin inversion will be degenerate resulting in symmetry
breaking. The symmetry group of the 5/7 skewed ladder system consists of four
elements: $E$, $P$, $\sigma$, and $\sigma P$, where $E$ is identity, $\sigma$
is the reflection symmetry, and $P$ is the spin inversion symmetry and all
these elements commute with each other leading to an Abelian group. The four
irreducible representations correspond to $A^{+}$, $A^{-}$, $B^{+}$, and
$B^{-}$. $A$ ($B$) corresponds to even (odd) under $\sigma$ while ‘$+$’
(‘$-$’) corresponds to even (odd) under $P$. A BOW transition requires a
degeneracy between the lowest states in $A^{+}$ and $B^{+}$ (or $A^{-}$ and
$B^{-}$) subspaces. Similarly an SDW transition requires a degeneracy of the
lowest energy states in $A^{+}$ and $A^{-}$ (or $B^{+}$ and $B^{-}$)
subspaces. For a vector chiral transition, the lowest energy states in $A^{+}$
and $B^{-}$ (or $A^{-}$ and $B^{+}$) subspaces must be degenerate.

Table 1: Two lowest energy levels from different $S^{z}$ sectors at specified $J_{1}$ values (see Fig. 6). $J_{1}$ | $E(S^{z}=0)$ | $E(S^{z}=1)$ | $E(S^{z}=2)$ | $E(S^{z}=3)$ | $E(S^{z}=4)$
---|---|---|---|---|---
1.07 | $-23.8471$ | $-23.8470$ |  |  |
| $-23.8470$ |  |  |  |
1.408 | $-25.0575$ | $-25.0575$ | $-25.0574$ |  |
| $-25.0574$ | $-25.0574$ |  |  |
4.601 | $-44.7871$ | $-44.7871$ | $-44.7871$ | $-44.7871$ |
| $-44.7871$ | $-44.7871$ | $-44.7871$ |  |
5.55 | $-51.7948$ | $-51.7948$ | $-51.7948$ | $-51.7948$ | $-51.7946$
| $-51.7946$ | $-51.7946$ | $-51.7946$ | $-51.7946$ |
Figure 6: The energy gap $\Gamma_{\sigma}$ (see Eq. 3) between two lowest
energy levels belonging to different reflection symmetry subspaces.
$\Gamma_{\sigma}$ vanishes at $J_{1}=1.07$, $1.408$, $4.601$ and $5.55$
indicating degeneracies at these $J_{1}$ values.

Since the spin inversion symmetry, $P$ divides the $S^{z}=0$ subspace into
even and odd total spin ($S$) sectors, the lowest energy states of the odd and
even subspaces under $P$ should be degenerate to break the spin inversion
symmetry. To determine the degeneracy of the lowest energy state in the odd
and even subspaces under spin inversion symmetry, instead of employing ‘$P$’
to divide the Hilbert space with $S^{z}=0$ into even and odd total spin
subspaces, we use the following argument. Whenever there is a degeneracy of
the lowest energy states with odd and even total spin sectors, then the spin
inversion symmetry is broken. We recognize the degeneracy of the gs when two
states in the $S^{z}=0$ sector are degenerate. In this case, we compute the
energies of the lowest states in the higher $S^{z}$ sectors. The spin of the
degenerate spin states is determined by following the degeneracies of the
states in these sectors.

We calculate the energy gap $\Gamma_{\sigma}$, as the modulus of the
difference in energy between the lowest energy states in the $A$ and $B$
subspaces,

$\Gamma_{\sigma}=|E_{0}(\sigma=-1)-E_{0}(\sigma=+1)|,$ (3)

where $E_{0}(\sigma=+1)$ and $E_{0}(\sigma=-1)$ are the lowest energies in the
even and odd subspaces under $\sigma$. In Fig. 6, $\Gamma_{\sigma}$ is shown
as a function of $J_{1}$ for a system size $N=16$ with PBC. We see that it
vanishes at four values of $J_{1}$, namely $J_{1}=1.07$, $J_{1}=1.408$,
$J_{1}=4.601$, and $J_{1}=5.550$. At these values of $J_{1}$, we compute
$\Gamma_{l}$ and find the two degenerate gs have spins $S=0$ and 1, $S=1$ and
2, $S=2$ and 3, and $S=3$ and 4 respectively, at $J_{1}$ 1.07, 1.408, 4.6, and
5.550. In Table 1 we show the lowest energy states in different $S^{z}$
sectors at the four points at which $\Gamma_{\sigma}$ vanishes. The degeneracy
under reflection and spin inversion at a given $J_{1}$ implies a vector chiral
state and nonzero spin currents for these $J_{1}$ values.

Usually, in all known systems, the chiral phase emerges either due to exchange
anisotropy or due to an external magnetic field on a ferrimagnetic gs.
However, in our system, due to the peculiar nature of the frustrated exchange
interactions, accidental degeneracy occurs between the lowest energy states in
the $A^{+}$ and $B^{-}$ symmetries. In the basis of these degenerate states,
the spin chirality operator has nonzero eigenvalues. This leads to nonzero
spin current in a well defined $S_{z}$ state, but the total spin is no longer
conserved in the eigenstate. This implies spontaneous symmetry breaking, i.e.,
the eigenstate does not exhibit the full symmetry of the Hamiltonian.

The magnitude of the $z$ component of the spin current, $\kappa_{z}(j,k)$, is
given by the eigenvalues of the matrix of the spin current operator for the
$j-k$ bond, viz., $(\vec{S}_{j}\times\vec{S}_{k})_{z}$ in the
$\mid\psi_{G}(\pm)\rangle$ basis. The matrix, in this basis, is given by,

$\begin{pmatrix}\langle\psi(+)\mid\hat{\kappa}_{z}\mid\psi(+)\rangle&\langle\psi(+)\mid\hat{\kappa}_{z}\mid\psi(-)\rangle\\\
\langle\psi(-)\mid\hat{\kappa}_{z}\mid\psi(+)\rangle&\langle\psi(-)\mid\hat{\kappa}_{z}\mid\psi(-)\rangle\end{pmatrix},$

where the function $\mid\psi_{G}(+)\rangle$ is the lowest energy state in the
even subspace for reflection and even subspace for spin inversion, similarly,
$\mid\psi_{G}(-)\rangle$ is in the odd supspace under both symmetries. The
matrix elements of the spin current operator can be evaluated easily by using
the operator identity

$\displaystyle\kappa_{z}(j,k)$ $\displaystyle=$
$\displaystyle-i(\vec{S}_{j}\times\vec{S}_{k})_{z}\ $ (4) $\displaystyle=$
$\displaystyle\frac{1}{2}(S^{+}_{j}S_{k}^{-}-S_{j}^{-}S_{k}^{+})$

The diagonal matrix element in the $2\times 2$ matrix is zero, and the
eigenvalues of the spin current are given by
$\pm\frac{1}{2}\left|\langle\psi_{G}(+)\mid(S^{+}_{j}S_{k}^{-}-S_{j}^{-}S_{k}^{+})\mid\psi_{G}(-)\rangle\right|.$

In Fig. 7, we show the spin currents for $J_{1}=1.408$ and $5.550$ for
different degenerate $S^{z}$ values. Spin currents for all $J_{1}$ values, at
which $\Gamma_{\sigma}=0$ and there is a degeneracy between states of odd and
even total spin, are shown in Table 2 for different $S^{z}$ values. We find
that the spin currents are large for $J_{1}=1.408$ compared with
$J_{1}=5.550$. At $J_{1}=1.408$, the spin current is larger in the five-
membered ring compared with that in the seven-membered ring. Also, the
direction of spin currents is opposite in the two rings. The spin currents are
also not uniform for all the bonds, implying the mean angle between the spins
depends upon the bond, as the spin current of a bond is a measure of the angle
between orientations of the spins of the bond. The rung bonds have smaller
currents than the leg bonds and in the five-membered ring, the upper leg has
larger currents than the lower leg, while it is the opposite in the seven-
membered ring. The spin current in the five-membered ring almost vanishes at
$J_{1}=5.550$ while it is much weaker in the seven-membered ring. The spin
currents of the rung bonds are very small and the spin current on the leg
bonds in the seven-membered ring becomes uniform. There is also weak
dependence of the spin current on the $S^{z}$ value of the state for which it
is calculated.

Table 2: Spin currents for different $J_{1}$ values in different $S^{z}$ sectors. At $J_{1}=1.07$, $S=0$ and $S=1$ are degenerate. At $J_{1}=1.408$, $S=1$ and $S=2$ are degenerate. At $J_{1}=4.601$, $S=2$ and $S=3$ are degenerate and finally at $J_{1}=5.5$, $S=4$ and $S=5$ are degenerate. $\kappa_{z}$ is defined in Eq. 4 $J_{1}$ | $S^{z}$ | $\kappa_{z}(1,2)$ | $\kappa_{z}(4,5)$ | $\kappa_{z}(1,3)$ | $\kappa_{z}(5,7)$ | $\kappa_{z}(2,4)$ | $\kappa_{z}(4,6)$
---|---|---|---|---|---|---|---
1.07 | 0 | $-0.302$ | $-0.302$ | $0.165$ | $-0.158$ | $-0.237$ | $0.086$
1.408 | 0 | $-0.316$ | $-0.316$ | $0.225$ | $-0.220$ | $-0.315$ | $0.129$
| 1 | $0.274$ | $0.274$ | $-0.195$ | $0.190$ | $0.273$ | $-0.112$
4.601 | 0 | $-0.045$ | $0.045$ | $0$ | $0.205$ | $0$ | $-0.205$
| 1 | $-0.042$ | $0.042$ | $0$ | $0.193$ | $0$ | $-0.193$
| 2 | $0.033$ | $-0.033$ | $0$ | $-0.153$ | $0$ | $0.153$
5.55 | 0 | $0.027$ | $-0.027$ | $0$ | $-0.147$ | $0$ | $0.147$
| 1 | $-0.026$ | $0.026$ | $0$ | $0.143$ | $0$ | $-0.143$
| 2 | $-0.023$ | $0.023$ | $0$ | $0.127$ | $0$ | $-0.127$
| 3 | $-0.018$ | $0.018$ | $0$ | $0.097$ | $0$ | $-0.097$

Figure 7: Spin currents $\kappa_{z}(j,k)$ of a 5/7 skewed ladder of $N=16$
spins (see Eq. 4) for (a) $J_{1}=1.408$ in the $S^{z}=1$ sector, (b)
$J_{1}=5.55$ in the $S^{z}=2$ sector, and (c) $J_{1}=5.55$ in the $S^{z}=3$
sector. Arrows indicate the direction of the current and the magnitude of the
currents is given adjacent to the arrows. No arrowhead for a bond means the
spin current is zero along that bond.

##  IV Summary and Conclusion

In this paper, we study quantum phases of a spin-1 HAF model on a 5/7 ladder
shown in Fig. 1(b). This system goes from a nonmagnetic state to a partially
magnetized state for $J_{1}>1.06$, and for $J_{1}>1.44$ magnetization per unit
cell is $m=S_{G}/n=1$. The gs again goes to a nonmagnetic state for
$4.74<J_{1}<5.44$, and spins have noncollinear arrangement in this phase. For
large $J_{1}>5.63$, the gs goes to a magnetic state with $m=2$.

The correlation length in the 5/7 ladder decreases monotonically with $J_{1}$
and for $J_{1}=1$ the correlation length $\xi\sim 3$ lattice units in the
singlet gs. The bond order of the rung bonds increase monotonically with
$J_{1}$, and in the large $J_{1}$ limit, the gs is a product of rung dimers
and free spins at 3 and 7 sites in each unit cell. The uniform VBS state of
spin-1 chain disappears even for the small value of $J_{1}$ as spin-1/2 at the
edges of the ladder gets pinned by the rung interaction and forms singlet
pairs. In large $J_{1}$ limit, the spins at sites $8i-2$ and $8i$ ($i$ is the
unit cell index) form a strong singlet dimer, which is comparable to a spin-1
singlet dimer with bond order $\sim 2.0$. Thus, in the large $J_{1}$ limit we
have a VB state with singlets on the rung bonds and between sites $6$ and $8$
in each unit cell. The free spins at sites $8i+3$ and $8i+7$ have
ferromagnetic alignment.

For $J_{1}<1.06$, $\Gamma_{\sigma}$ vanishes and reflection symmetry is broken
leading to dimer order in the system. For larger $J_{1}$ values, the gs is in
ferrimagnetic state and for some $J_{1}$ values the lowest energy states in
the even-even and odd-odd subspace under reflection and spin inversion become
degenerate. This leads to both inversion and spin-parity symmetry being broken
at the degeneracy points; therefore, the gs at this $J_{1}$ value possesses
vector chirality and there is spontaneous spin current in the ladder system.
This is unique as it can have both finite magnetization and spin current in
the absence of an external magnetic field. In this system, maximum gs
magnetization per unit cell, $m=2$, while, for a spin-1/2 system $m=1$ Giri
_et al._ (2017); Dey  _et al._ (2020). In the spin-1/2 system magnetic moments
are localized mostly on ($4i-1$) sites of the system and other sites have
vanishingly small spin density Giri  _et al._ (2017); Dey  _et al._ (2020).
However, in the spin-1 system while magnetization contribution comes mostly
from sites ($4i-1$), there is antiferromagnetically aligned spin density in
the lower leg and ferromagnetically aligned spins in the upper leg due to
strong singlet dimer formation along the rung. The isolated singlet spin-1
dimer between sites $8i-2$ and $8i$ in this extended system is unique.

In conclusion, the HAF spin-1 system on a skewed 5/7 ladder is unique with
different ferrimagnetic gs, and this system exhibits a plethora of exotic
phases in the gs on tuning $J_{1}$. The spin arrangements of the spin-1 system
are vastly different from those on the spin-1/2 on this lattice. This is a
unique ladder system where a singlet spin-1 dimer and BOW can coexist. The
topological phase of the spin-1 chain vanishes for any finite value of
$J_{1}$. The HAF 5/7 ladder system can be mapped onto a spin-1 chain with an
AF nearest- and next-nearest-neighbor interaction $J_{1}$ and $J_{2}$ with
periodically missing $J_{1}$ bonds. This system may be realized in molecular
magnets based on transition metal compounds.

###### Acknowledgements.

MK thanks Department of Science and Technology (DST), India for Ramanujan
fellowship. SR thanks Indian National Science Academy and DST-SERB for
supporting this work. SD and DD contributed equally to this work.

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