# On the Euler-Poincaré characteristics of a simply connected rationally
elliptic CW-complex

Mahmoud Benkhalifa  Department of Mathematics. Faculty of Sciences, University
of Sharjah. Sharjah, United Arab Emirates<EMAIL_ADDRESS>
###### Abstract.

For a simply connected rationally elliptic CW-complex $X$, we show that the
cohomology and the homotopy Euler-Poincaré characteristics are related to two
new numerical invariants namely $\eta_{X}$ and $\rho_{X}$ which we define
using the Whitehead exact sequences of the Quillen and the Sullivan models of
$X$.

###### Key words and phrases:

Rationaly elliptic space, Sullivan model, Quillen model, Euler-Poincaré
characteristic, Whitehead exact sequence

###### 2000 Mathematics Subject Classification:

55P62

##  1\. introduction

A simply connected CW-complex $X$ is called rationally elliptic if both the
graded vector spaces $H^{*}(X;{\mathbb{Q}})$ and
$\pi_{*}(X)\otimes{\mathbb{Q}}$ are finite dimensional. To such a space $X$
are attached two numerical invariants namely the cohomology Euler
characteristic $\chi_{H}$ and the homotopy Euler characteristic $\chi_{\pi}$
defined by

$\chi_{H}=\sum_{i\geq
0}(-1)^{i}\operatorname{dim}H^{i}(X;{\mathbb{Q}})\quad,\quad\chi_{\pi}=\sum_{i\geq
0}(-1)^{i}\operatorname{dim}\pi_{i}(X)\otimes{\mathbb{Q}}.$

Computing $\chi_{H}$ and $\chi_{\pi}$ and studying their properties is a major
task in rational homotopy theory. For instance, it is well-known that (see
[9], Prop. $\left.32.10\right)$

  * •

$\chi_{H}\geq 0$ and $\chi_{\pi}\leq 0.$

  * •

$\chi_{H}>0$ if and only if $\chi_{\pi}=0.$

If $X^{[i]}$ denotes the $i^{\text{th }}$ Postnikov section of $X$ and $X^{i}$
denotes its $i^{\text{th }}$ skeleton, then we introduce two new numerical
homotopy invariants namely

$\rho_{X}=1+\sum_{i\geq
2}(-1)^{i}\operatorname{dim}H^{i}(X^{[i-2]};{\mathbb{Q}})\quad,\quad\eta_{X}=1+\sum_{i\geq
2}(-1)^{i}\operatorname{dim}\Gamma_{i}(X),$

where
$\Gamma_{i}(X)=\operatorname{ker}(\pi_{i}(X^{i})\otimes{\mathbb{Q}}\longrightarrow\pi_{i}(X^{i},X^{i-1})\otimes{\mathbb{Q}})$
for $i\geq 2$.

By exploiting the well-known properties of a rationally elliptic CW-complex
$X$ as well as the virtue of the Whitehead exact sequences associated
respectively to the Sullivan model and the Quillen model of $X$, we prove the
following

######  Theorem 1.

If $X$ is a rationally elliptic CW-complex $X$, then

  1. (1)

$H^{i}(X^{[i-2]};{\mathbb{Q}})\cong\Gamma_{i-1}(X)$ for all $i\geq 2;$

  2. (2)

$\rho_{X}=\eta_{X}>0;$

  3. (3)

Either $\rho_{X}=\chi_{H}$ or $\rho_{X}=-\chi_{\pi}$;

  4. (4)

If $X$ is 2–connected, then $\Gamma_{i}(X)=\pi_{i}(X^{i})\otimes{\mathbb{Q}}$
for all $i\geq 2$.

As a corollary of theorem 1, we deduce the following result

######  Corollary 2.

Let $X$ be a rationally elliptic CW-complex $X$. We have the following

  1. (1)

If $H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0$ for every $i\geq 2$, then $X$ is
rationally an odd-dimensional sphere.

  2. (2)

If $X$ is 2-connected and $\pi_{2i-1}(X^{2i-2})\otimes{\mathbb{Q}}=0$ for
every $i\geq 1$, then $X$ is rationally an odd-dimensional sphere.

A special class of rationally elliptic CW-complexes is formed by the
$F_{0}$-spaces. Such CW-complex is characterised by the fact that its homotopy
Euler characteristics vanishes (see definition 3.14). Thus, from the main
theorem of this paper we show

######  Corollary 3.

If $X$ is a 2-connected $F_{0}$-space, then

$\pi_{2i+1}(X^{2i})\otimes{\mathbb{Q}}=H^{2i+2}(X^{[2i]},{\mathbb{Q}})=0\quad,\quad\forall
i\geq 1.$

We show our results using standard tools of rational homotopy theory by
working algebraically on the models of Quillen and Sullivan of $X$. We refer
to [9] for a general introduction to these techniques. Recall that every
simply connected space of finite type has a corresponding differential
commutative cochain algebra called the Sullivan model of $X$, unique up to
isomorphism, that encodes the rational homotopy of $X$. Dually, every simply
connected space $X$ has a differential graded Lie algebra (DGL for short),
called the Quillen model of $X$ and unique up to isomorphism, which determines
completely the rational homotopy type of $X$.

The paper is organised as follows. In section 2, we recall the definitions of
the Whitehead exact sequences associated with the Quillen and the Sullivan
models of an rationally elliptic CW-complex as well as we formulate and prove
all the results in an algebraic setting. In section 3, we give a mere
transcription of the above results in the topological context.

##  2\. Whitehead exact sequences in rational homotopy theory

###  2.1. Whitehead exact sequence associated with a DGL

Let $W=(W_{\geq 1})$ be a finite dimensional graded vector space over
${\mathbb{Q}}$ and let $(\mathbb{L}(W),\delta)$ be a DGL. We define the linear
maps

$j_{i}:H_{i}(\mathbb{L}(W_{\leq i}))\longrightarrow W_{i}\quad,\quad
b_{i}:W_{i}\longrightarrow H_{i-1}(\mathbb{L}(W_{\leq i-1})),$ (1)

by setting $j_{i}([w+y])=w$ and $b_{i}(w)=[\delta(w)]$, were $[\delta(w)]$
denotes the homology class of $\delta(w)$ in the sub-Lie algebra
$\mathbb{L}_{i-1}(W_{\leq i-1}).$ Recall that if $x\in
H_{i}(\mathbb{L}(W_{\leq i}))$, then $x=[w+y]$, where $w\in
W_{i},y\in\mathbb{L}_{i}(W_{\leq i-1})$ and $\delta(w+y)=0$.

To every DGL $(\mathbb{L}(W),\delta)$ we can assign (see [4, 6, 8] for more
details) the following long exact sequence

$\cdots\rightarrow W_{i+1}\stackrel{{\scriptstyle
b_{i+1}}}{{\longrightarrow}}\Gamma_{i}\rightarrow
H_{i}(\mathbb{L}(W))\rightarrow W_{i}\stackrel{{\scriptstyle
b_{i}}}{{\longrightarrow}}\cdots$ (2)

called the Whitehead exact sequence of $(\mathbb{L}(W),\delta)$, where

$\Gamma_{i}=\operatorname{ker}(j_{i}:H_{i}(\mathbb{L}(W_{\leq i}))\rightarrow
W_{i})\quad,\quad i\geq 2.$ (3)

######  Remark 2.1.

First, it is clear that if $H_{i}(\mathbb{L}(W_{\leq i}))=0$, then
$\Gamma_{i}=0$.

Next, if $(\mathbb{L}(W),\delta)$ satisfies $W_{1}=0$, then

$\Gamma_{i}=H_{i}(\mathbb{L}(W_{\leq i-1}))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,i\geq
2.$ (4)

Indeed; let us consider the following long exact sequence

$\cdots\rightarrow H_{i+1}(\mathbb{L}(W_{\leq i})/\mathbb{L}(W_{\leq
i-1}))\rightarrow H_{i}(\mathbb{L}(W_{\leq i-1}))\rightarrow
H_{i}(\mathbb{L}(W_{\leq i}))\rightarrow\cdots$

which is associated with the following short exact of chain complexes

$(\mathbb{L}(W_{\leq i-1}),\delta)\mapsto(\mathbb{L}(W_{\leq
i}),\delta)\rightarrow(\mathbb{L}(W_{\leq i})/\mathbb{L}(W_{\leq
i-1}),\widetilde{\delta}),$

where $(\mathbb{L}(W_{\leq i})/\mathbb{L}(W_{\leq i-1}),\widetilde{\delta})$
is the quotient chain complex and the differential $\widetilde{\delta}$ is
induced by $\delta.$ Now clearly, we have $(\mathbb{L}(W_{\leq
i})/\mathbb{L}(W_{\leq i-1}))\cong W_{i}$ and as $W_{1}=0$ it follows that
$H_{i+1}(\mathbb{L}(W_{\leq i})/\mathbb{L}(W_{\leq i-1}))=0.$ As a result we
obtain (4).

###  2.2.  Whitehead exact sequence associated with a Sullivan algebra

A Sulivan algebra $(\Lambda V,\partial)$ over ${\mathbb{Q}}$ is simply
connected if $V^{1}=0.$ Let $(\Lambda(V^{\leq i-2}),\partial)$ be the sub
algebra generated by the graded vector apace $V^{\leq i-2}.$ For simplicity
let us write

$L^{i}=H^{i}(\Lambda(V^{\leq i-2}))\quad,\quad i\geq 2.$ (5)

The Whitehead exact sequence of $(\Lambda V,\partial)$ is defined as follows
(see [4, 5, 9] for more details)

$\cdots\rightarrow H^{i}(\Lambda V)\rightarrow V^{i}\stackrel{{\scriptstyle
b^{i}}}{{\longrightarrow}}L^{i+1}\longrightarrow H^{i+1}(\Lambda
V)\longrightarrow V^{i+1}\stackrel{{\scriptstyle
b^{i+1}}}{{\longrightarrow}}\cdots$ (6)

Here $b^{i}(v)=[\partial(v)]$, where $[\partial(v)]$ is the cohomology class
of the cycle $\partial(v)$ in $\Lambda(V^{\leq i-2})$.

###  2.3. Elliptic algebras

A Sullivan algebra $(\Lambda V,\partial)$ is called elliptic if

$\operatorname{dim}H^{*}(\Lambda V)=\sum_{i\geq
0}\operatorname{dim}H^{i}(\Lambda V)<\infty\,\,\,\,\,\,\,\,\,\,\text{ and
}\,\,\,\,\,\,\,\,\,\,\operatorname{dim}V=\sum_{i\geq
2}\operatorname{dim}V^{i}<\infty.$

Let us call $n=\max\left\\{i:H^{i}(\Lambda V)\neq 0\right\\}$ the formal
dimension of $(\Lambda V,\partial)$. For an elliptic Sullivan algebra
$(\Lambda V,\partial)$, we define the following two numbers

$\displaystyle\chi_{\Lambda V}$ $\displaystyle=$ $\displaystyle\sum_{i\geq
0}\operatorname{dim}H^{2i}(\Lambda V)-\sum_{i\geq
0}\operatorname{dim}H^{2i+1}(\Lambda V)=H^{\rm{even}}(\Lambda
V)-H^{\rm{odd}}(\Lambda V),$ $\displaystyle\chi_{V}$ $\displaystyle=$
$\displaystyle\sum_{i\geq 1}\operatorname{dim}V^{2i}-\sum_{i\geq
1}\operatorname{dim}V^{2i+1}=\operatorname{dim}V^{\rm{even}}-\operatorname{dim}V^{\rm{odd}}.$

The following are some important properties of elliptic Sullivan algebras.

######  Theorem 2.2.

([4], §32) Suppose $(\Lambda V,\partial)$ is elliptic of formal dimension $n.$
Then

  1. (1)

$\operatorname{dim}V^{\text{odd }}\geq\operatorname{dim}V^{\text{even }};$

  2. (2)

$V^{i}=0$, for $i\geq 2n$;

  3. (3)

$V^{i}=0$, for $i>n$ and $i$ even;

  4. (4)

There is only one non-trivial $V^{i}$, for $i>n$ and $i$ odd. Necessary, we
have $\operatorname{dim}V^{i}=1;$

  5. (5)

$\chi_{\Lambda V}\geq 0$ and $\chi_{V}\leq 0.$ Moreover $\chi_{\Lambda
V}=0\Longleftrightarrow\chi_{V}<0.$

######  Proposition 2.3.

Suppose $(\Lambda V,\partial)$ is elliptic of formal dimension $n.$

  1. (1)

For every even number i such that $i>n+1$, we have $L^{i}=0$;

  2. (2)

For every odd number i such that i $>n+1$ we have $L^{i}\cong V^{i-1}$;

  3. (3)

For every i such that $i>2n$, we have $L^{i}=0$.

###### Proof.

As $(\Lambda V,\partial)$ has formal dimension $n$, it follows that
$H^{i}(\Lambda V)=0$ for $i>n$. Therefore, by theorem 2.2 the Whitehead exact
sequence of $(\Lambda V,\partial)$ can be written as

$H^{n}(\Lambda V)\rightarrow V^{n}\rightarrow L^{n+1}\rightarrow
0\rightarrow\cdots\to 0\rightarrow V^{2n-1}\to L^{2n}\rightarrow 0\rightarrow
V^{2n}=0\to L^{2n+1}\to 0$

Consequently, the three assertions of proposition 2.3 are easily derived. ∎

######  Definition 2.4.

Let $(\Lambda V,\partial)$ be an elliptic Sullivan algebra. Using (5), we
define

$\rho_{\Lambda V}=1+\operatorname{dim}L^{\text{even
}}-\operatorname{dim}L^{\text{odd }},$

where

$\operatorname{dim}L^{\text{even
}}=\operatorname{dim}L^{4}+\operatorname{dim}L^{6}+\operatorname{dim}L^{8}+\dots$
$\operatorname{dim}L^{\text{odd
}}=\operatorname{dim}L^{5}+\operatorname{dim}L^{7}+\operatorname{dim}L^{9}+\dots$

Note that from (5), we deduce that $L^{i}=0$ for $i=2,3$.

######  Proposition 2.5.

If $(\Lambda V,\partial)$ is elliptic of formal dimension $n$, then we have

$0\leq\rho_{\Lambda V}-\sum_{i=4}^{n+1}(-1)^{i}\operatorname{dim}L^{i}\leq 2.$
(7)

###### Proof.

First by (4) of proposition 2.3, we have

$\rho_{\Lambda
V}=1+\sum_{i=4}^{2n}(-1)^{i}\operatorname{dim}L^{i}=1+\sum_{i=4}^{n+1}(-1)^{i}\operatorname{dim}L^{i}+\sum_{i=n+2}^{2n}(-1)^{i}\operatorname{dim}L^{i}.$

Next, let us focus on the integer
$\sum_{i=n+2}^{2n}(-1)^{i}\operatorname{dim}L^{i}.$ Using the relations (3)
and (4) of theorem 2.2 and (2) of proposition 2.3, we deduce that

$-1\leq\sum_{i=n+2}^{2n}(-1)^{i}\operatorname{dim}L^{i}\leq 1.$

Therefore

$\sum_{i=4}^{n+1}(-1)^{i}\operatorname{dim}L^{i}\leq\rho_{\Lambda
V}\leq\sum_{i=4}^{n+1}(-1)^{i}\operatorname{dim}L^{i}+2,$

as wanted. ∎

The next result shows the relationship between the $\rho_{\Lambda
V},\chi_{\Lambda V}$ and $\chi_{V}.$

######  Theorem 2.6.

If $(\Lambda V,\partial)$ is an elliptic Sullivan algebra, then $\rho_{\Lambda
V}=\chi_{\Lambda V}-\chi_{V}$.

###### Proof.

Let us consider the Whitehead exact sequence of $(\Lambda V,\partial)$ given
in (6)

$\cdots\rightarrow H^{i}(\Lambda V)\rightarrow V^{i}\stackrel{{\scriptstyle
b^{i}}}{{\longrightarrow}}L^{i+1}\longrightarrow H^{i+1}(\Lambda
V)\longrightarrow V^{i+1}\stackrel{{\scriptstyle
b^{i+1}}}{{\longrightarrow}}\cdots$

Let us write $H^{*}=H^{*}(\Lambda V)$. For every $i\geq 2$, let

$V^{i,1}=\operatorname{ker}b^{i}\quad,\quad
V^{i,2}=\operatorname{im}b^{i}\quad,\quad L^{i+1,1}=\operatorname{im}i^{i+1}.$

The exactness of the above sequence implies that

$V^{i}\cong V^{i,1}\oplus V^{i,2}\quad,\quad L^{i+1}\cong V^{i,2}\oplus
L^{i+1,1}\quad,\quad H^{i+1}\cong L^{i+1,1}\oplus V^{i+1,1}.$ (8)

Using the direct summands in (8) we deduce the following

$\displaystyle\operatorname{dim}V^{3}$
$\displaystyle=\operatorname{dim}H^{3}+\operatorname{dim}V^{3,2}$
$\displaystyle=\operatorname{dim}H^{3}+\operatorname{dim}L^{4}-\operatorname{dim}L^{4,1}$
$\displaystyle=\operatorname{dim}H^{3}+\operatorname{dim}L^{4}-\operatorname{dim}H^{4}+\operatorname{dim}V^{4,1}$
$\displaystyle=\operatorname{dim}H^{3}+\operatorname{dim}L^{4}-\operatorname{dim}H^{4}+\operatorname{dim}V^{4}-\operatorname{dim}V^{4,2}.$

Iterating the above process and taking into consideration that $(\Lambda
V,\partial)$ is elliptic which implies that this process must stop, we get the
following formula

$\operatorname{dim}V^{3}=-\sum_{i\geq
3}(-1)^{i}\operatorname{dim}H^{i}+\sum_{i\geq
4}(-1)^{i}\operatorname{dim}L^{i}+\sum_{i\geq
4}(-1)^{i}\operatorname{dim}V^{i},$

which implies

$\sum_{i\geq 3}(-1)^{i}\operatorname{dim}H^{i}-\sum_{i\geq
3}(-1)^{i}\operatorname{dim}V^{i}=\sum_{i\geq
4}(-1)^{i}\operatorname{dim}L^{i}.$ (9)

As $(\Lambda V,\partial)$ is simply connected, it follows that
$H^{0}\cong{\mathbb{Q}}$, $H^{1}=0$ and $H^{2}\cong V^{2}$. Therefore the
relation (9) becomes

$(\operatorname{dim}H^{\text{even }}-\operatorname{dim}H^{\text{odd
}})-(\operatorname{dim}V^{\text{even }}-\operatorname{dim}V^{\text{odd
}})=1+\sum_{i\geq 4}(-1)^{i}\operatorname{dim}L^{i},$

implying $\rho_{\Lambda V}=\chi_{\Lambda V}-\chi_{V}$. ∎

######  Corollary 2.7.

If $(\Lambda V,\partial)$ is an elliptic Sullivan algebra, then $\rho_{\Lambda
V}>0.$ Moreover if $\chi_{\Lambda V}>0$, then $\rho_{\Lambda V}=\chi_{\Lambda
V}$.

###### Proof.

As $\rho_{\Lambda V}=\chi_{\Lambda V}-\chi_{V}$, the relation (5) of theorem
2.2 implies that $\rho_{\Lambda V}\geq 0.$ Now if we assume that
$\rho_{\Lambda V}=0$, then we derive that $\chi_{\Lambda V}=\chi_{V}$. Again
by the relation (5) of theorem 2.2, on one hand we have $\chi_{\Lambda
V}=\chi_{V}=0$ and on the other hand we have $\chi_{V}<0$ which is impossible.
Thus, $\rho_{\Lambda V}>0$.

Now if $\chi_{\Lambda V}>0$, then the relation (4) of theorem 2.2 implies that
$\chi_{V}=0.$ Hence $\rho_{\Lambda V}=\chi_{\Lambda V}-\chi_{V}=\chi_{\Lambda
V}$. ∎

######  Corollary 2.8.

If $(\Lambda V,\partial)$ is an elliptic Sullivan algebra, then
$L^{\rm{even}}\geq L^{\rm{odd}}$. Moreover, if $L^{\rm{even}}=0$, then
$\operatorname{dim}H^{*}(\Lambda V)=\operatorname{dim}V^{*}+1$.

###### Proof.

By combining definition 2.4 and corollary 2.7, we obtain

$\rho_{\Lambda V}=1+\operatorname{dim}L^{\text{even
}}-\operatorname{dim}L^{\text{odd }}>0\Longrightarrow L^{\rm{even}}\geq
L^{\rm{odd}}.$

Thus, if $L^{\rm{even}}=0$, then $L^{\rm{odd}}=0$ implying that
$L^{i}=0\,,\,\forall i\geq 2$. Finally, from the Whitehead exact sequence (6)
of $(\Lambda V,\partial)$ , we deduce that

$H^{i}(\Lambda V)\cong V^{i}\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,\forall
i\geq 2.$ (10)

Consequently, if $n$ is the formal dimension of $(\Lambda V,\partial)$, then
$V^{i}=0\,,\,\forall i>n$ and we obtain

$\displaystyle\operatorname{dim}H^{*}(\Lambda V)$ $\displaystyle=$
$\displaystyle\operatorname{dim}H^{0}(\Lambda
V)+\operatorname{dim}H^{1}(\Lambda V)+\operatorname{dim}H^{2}(\Lambda
V)+\cdots+\operatorname{dim}H^{n}(\Lambda V)$ $\displaystyle=$ $\displaystyle
1+0+\operatorname{dim}V^{2}+\operatorname{dim}V^{3}+\cdots+\operatorname{dim}V^{n}=1+\operatorname{dim}V^{*}$

as desired. ∎

##  3\. Topological applications

Let $X$ be a simply connected CW-complex. For every $i\geq 2$, we define the
vector space

$\Gamma_{i}(X)=\operatorname{ker}(\pi_{i}(X^{i})\otimes{\mathbb{Q}}\longrightarrow\pi_{i}(X^{i},X^{i-1})\otimes{\mathbb{Q}}).$
(11)

Here $X^{i}$ denotes the $i^{\text{th }}$-skeleton of $X$.

Thus, if $(\mathbb{L}(W),\delta)$ denotes the Quillen model of $X$, then by
virtue of the properties of this model, we obtain the following
identifications (valid for any $i\geq 2$)

$\pi_{i}(X)\otimes{\mathbb{Q}}\cong H_{i-1}(\mathbb{L}(W))\quad,\quad
H_{i}(X;{\mathbb{Q}})\cong W_{i-1}\quad,\quad\Gamma_{i}(X)\cong\Gamma_{i-1},$
(12)

where $\Gamma_{i}$ is defined in (3). Therefore the Whitehead exact sequence
(2) of this model can be written as

$\cdots\longrightarrow H_{i+1}(X;{\mathbb{Q}})\stackrel{{\scriptstyle
b_{i+1}}}{{\longrightarrow}}\Gamma_{i}(X)\stackrel{{\scriptstyle}}{{\longrightarrow}}\pi_{i}(X)\otimes{\mathbb{Q}}\stackrel{{\scriptstyle
h_{i}}}{{\longrightarrow}}H_{i}(X;{\mathbb{Q}})\stackrel{{\scriptstyle
b_{i}}}{{\longrightarrow}}\cdots$ (13)

where $h_{i}$ is the Hurewicz homomorphism.

######  Remark 3.1.

Firstly, since $(\mathbb{L}(W_{\leq i-1}),\delta)$ can be chosen as the
Quillen model of the skeleton $X^{i}$ (see [9], pp. 323) implying that

$\pi_{i}(X^{i})\otimes{\mathbb{Q}}\cong H_{i-1}(\mathbb{L}(W_{\leq
i-1}))\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\forall i\geq 2$

and based on remark 2.1 and the identifications (12), it follows that

$\pi_{i}(X^{i})\otimes{\mathbb{Q}}=0\Longrightarrow\Gamma_{i}(X)=0\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\forall
i\geq 2$ (14)

Secondly, if the space $X$ is 2-connected, then by remark 2.1 we deduce that

$\Gamma_{i}\cong H_{i}(\mathbb{L}(W_{\leq
i-1}))\cong\pi_{i+1}(X^{i})\otimes{\mathbb{Q}}\quad,\quad i\geq 2,$

and the Whitehead exact sequence (13) becomes

$\cdots\longrightarrow H_{i+1}(X;{\mathbb{Q}})\stackrel{{\scriptstyle
b_{i+1}}}{{\longrightarrow}}\pi_{i}(X^{i-1})\otimes{\mathbb{Q}}\stackrel{{\scriptstyle
i_{i}}}{{\longrightarrow}}\pi_{i}(X)\otimes{\mathbb{Q}}\longrightarrow
H_{i}(X;{\mathbb{Q}})\stackrel{{\scriptstyle b_{i}}}{{\longrightarrow}}\cdots$

Dually, if $(\Lambda V,\partial)$ is the Sullivan model of $X$, then we have

$H^{i}(X;{\mathbb{Q}})\cong H^{i}(\Lambda
V)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,\,\,\,H^{i+1}(X^{[i-1]};{\mathbb{Q}})\cong
H^{i+1}(\Lambda V^{\leq i-1}),$
$V^{i}\cong\operatorname{Hom}(\pi_{i}(X)\otimes{\mathbb{Q}};{\mathbb{Q}}),$
(15)

where $X^{[i]}$ denotes the $i^{\text{th }}$ Postikov section of $X.$
Therefore, the Whitehead exact sequence of this model can be written as

$\cdots\rightarrow\operatorname{Hom}(\pi_{i}(X)\otimes{\mathbb{Q}},{\mathbb{Q}})\overset{b^{i}}{\rightarrow}H^{i+1}(X^{[i-1]};{\mathbb{Q}})\rightarrow
H^{i+1}(X;{\mathbb{Q}})\rightarrow\operatorname{Hom}(\pi_{i+1}(X)\otimes{\mathbb{Q}},{\mathbb{Q}})\rightarrow\cdots$
(16)

Recall that a simply connected CW-complex $X$ is rationally elliptic if

$\operatorname{dim}H^{*}(X;{\mathbb{Q}})=\sum_{i\geq
0}\operatorname{dim}H^{i}(X;{\mathbb{Q}})<\infty\,\,\,\text{ and
}\,\,\,\,\operatorname{dim}\pi_{*}(X)\otimes{\mathbb{Q}}=\sum_{i\geq
2}\operatorname{dim}\pi_{i}(X)\otimes{\mathbb{Q}}<\infty$

Note that, by virtue of the Sullivan model, $X$ is rationally elliptic if and
only if its Sullivan model is elliptic. In this case the formal dimension of
$X$ is the formal dimension of its Sullivan model.

######  Definition 3.2.

Let $X$ be a rationally elliptic CW-complex. We define

$\eta_{X}=1+\mathrm{dim}\,\Gamma_{\rm{even}}(X)-\mathrm{dim}\,\Gamma_{\rm{odd}}(X)$

where $\Gamma_{*}(X)$ is defined in (11)

######  Definition 3.3.

If $X$ is a rationally elliptic CW-complex and $(\Lambda V,\partial)$ its
Sullivan model, then we define $\rho_{X}=\rho_{\Lambda V}$, where
$\rho_{\Lambda V}$ is given in definition 2.4.

Subsequently, we need the following proposition showing that
$H^{i+1}(X^{[i-1]},{\mathbb{Q}})$ and $\Gamma_{i}(X)$ are isomorphic, as
vector spaces, for every $i\geq 2$.

######  Proposition 3.4.

If $X$ is a rationally elliptic CW-complex, then

$H^{i+1}(X^{[i-1]};{\mathbb{Q}})\cong\text{\rm{Hom}}(\Gamma_{i}(X),{\mathbb{Q}}),\quad
i\geq 2.$ (17)

###### Proof.

Applying the exact functor $\text{\rm{Hom}}(.,{\mathbb{Q}})$ to the exact
sequence (13) we obtain

$\cdots\leftarrow
H^{i+1}(X;{\mathbb{Q}})\leftarrow\text{\rm{Hom}}(\Gamma_{i}(X),{\mathbb{Q}})\leftarrow\text{\rm{Hom}}(\pi_{i}(X)\otimes{\mathbb{Q}},{\mathbb{Q}})\leftarrow
H^{i}(X;{\mathbb{Q}})\leftarrow\cdots$ (18)

Taking in account that

  * •

All the vector spaces involved are finite dimensional.

  * •

The two maps
$H^{i}(X;{\mathbb{Q}})\rightarrow\text{\rm{Hom}}(\pi_{i}(X)\otimes{\mathbb{Q}},{\mathbb{Q}})$
appearing in (16) and (18) are the same morphism for all $i\geq 2$.

  * •

$\text{\rm{Hom}}(H_{*}(X;{\mathbb{Q}});{\mathbb{Q}})\cong
H^{*}(X;{\mathbb{Q}})$.

and by comparing the sequences $(\ref{15}),(\ref{17})$, we get $(\ref{16})$. ∎

######  Remark 3.5.

By virtue of the properties of the Sullivan model $(\Lambda V,\partial)$ of
the space $X$, it is important to mention that for all $i\geq 2$, the linear
map

$b^{i}:V^{i}\longrightarrow H^{i+1}(\Lambda(V^{\leq i-1})),$

induced by the differential $\partial$, is the dual of the $i$-invariant

$k_{i}\in
H^{i+1}(X^{[i-1]};\pi_{i}(X))=\text{\rm{Hom}}(H_{i+1}(X^{[i-1]}),\pi_{i}(X)),$

which is the dual of the map
$\Gamma_{i}(X)\overset{i_{i}}{\longrightarrow}\pi_{i}(X)\otimes{\mathbb{Q}}$
given in (13).

A mere transcription into the topological context of the above results
obtained in section 2 provides the following applications.

######  Corollary 3.6.

If $X$ is a rationally elliptic CW-complex of formal dimension $n$, then

  1. (1)

For every even number i such that $i>n+1$, we have

$H^{i}(X^{[i-2]},{\mathbb{Q}})=\Gamma_{i-1}(X)=0.$

  2. (2)

For every i such that $i>2n$, we have
$H^{i}(X^{[i-2]},{\mathbb{Q}})=\Gamma_{i-1}(X)=0$.

  3. (3)

If $X$ is 2-connected, then
$\Gamma_{i}(X)\cong\pi_{i}(X^{i-1})\otimes{\mathbb{Q}}$ for every $i\geq 2$.

###### Proof.

It follows from remark 2.1, propositions 2.3 and 3.4. ∎

The next result establishes that the two numerical invariants $\eta_{X}$ and
$\rho_{Y}$ are equal although they are defined differently.

######  Theorem 3.7.

If $X$ is a rationally elliptic CW-complex, then $\eta_{X}=\rho_{X}$.

###### Proof.

The result follows from the definitions 2.4 and 3.2 and by applying
proposition 3.4 after taking into account the identifications (12). ∎

######  Corollary 3.8.

If $X$ is a rationally elliptic CW-complex of formal dimension $n$, then

$0\leq\rho_{X}-\sum_{i=4}^{n+1}(-1)^{i}\mathrm{dim}\,H^{i}(X^{[i-2]};{\mathbb{Q}})\leq
2\,\,\,\,\,\,\,\,,\,\,\,\,\,\,0\leq\eta_{X}-\sum_{i=3}^{n}(-1)^{i}\mathrm{dim}\,\Gamma_{i}(X)\leq
2$

###### Proof.

It follows from propositions 2.5, 3.4 and theorem 3.7. ∎

######  Proposition 3.9.

If $X$ is a rationally elliptic CW-complex such that

$H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\forall i\geq
2,$

then we have

  1. (1)

$H^{2i+1}(X^{[2i-1]};{\mathbb{Q}})=0$ for every $i\geq 2$;

  2. (2)

$\Gamma_{i}(X)=0$ for every $i\geq 2$;

  3. (3)

$\operatorname{dim}H^{*}(X;{\mathbb{Q}})=\operatorname{dim}\pi_{i}(X)\otimes{\mathbb{Q}}+1$.

###### Proof.

All the assertions are a consequence of corollary 2.8 and the formula (17). ∎

######  Theorem 3.10.

Let $X$ be a rationally elliptic CW-complex $X$. If
$H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0$ for every $i\geq 2$, then $X$ is
rationally an odd-dimensional sphere.

###### Proof.

Let $(\mathbb{L}(W),\delta)$ denote the Quillen model of $X$. First, if
$H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0$ for every $i\geq 2$, then by (5) and the
identification (3), we deduce that $L^{\text{even }}=0$ implying $L^{\text{odd
}}=0$ according to corollary 3.8. Next, by proposition 17, we get
$\Gamma_{i}=0$ for every $i\geq 2$, where $\Gamma_{i}=0$ is defined in (3).
Therefore, the linear map $b_{i+1}:W_{i+1}\to\Gamma_{i}$, given in (2), is
trivial for $i\geq 2$. Now using the relation (1), it follows that
$b_{i}(w)=[\delta(w)]$, for $i\geq 2$ and every $w\in W_{i+1}$.

Consequently, using the properties of the Quillen model (see [9], pp. 323), it
follows that all the attaching maps of the space $X$ are rationally trivial.
Consequently the Quillen model of $X$ can be chosen as $(\mathbb{L}(W),0)$
(with trivial differential). Moreover, and as the space $X$ is rationally
elliptic, we must have $W=W_{2k}\cong{\mathbb{Q}}$ for a certain $k\geq 1$ .
Thus, the Quillen model of $X$ has the form $(\mathbb{L}(W_{2i}),0)$ implying
that $X$ is rationally an odd-dimensional sphere. ∎

######  Corollary 3.11.

Let $X$ be a rationally elliptic CW-complex $X$. If
$\pi_{2i-1}(X^{2i-1})\otimes{\mathbb{Q}}=0$ for every $i\geq 1$, then $X$ is
rationally an odd-dimensional sphere.

###### Proof.

If $\pi_{2i-1}(X^{2i-1})\otimes{\mathbb{Q}}=0\,,\,\forall i\geq 2$, then by
the implication (14) and corollary 3.6 we deduce that
$H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0\,,\,\forall i\geq 2$. Then we apply theorem
3.10 ∎

######  Corollary 3.12.

Let $X$ be a rationally elliptic CW-complex $X$. If $X$ is 2-connected and
$\pi_{2i-1}(X^{2i-2})\otimes{\mathbb{Q}}=0$ for every $i\geq 1$, then $X$ is
rationally an odd-dimensional sphere

###### Proof.

If $X$ is 2-connected and $\pi_{2i-1}(X^{2i-2})\otimes{\mathbb{Q}}=0$ for
every $i\geq 1$, then by corollary 3.6 we deduce that
$H^{2i}(X^{[2i-2]};{\mathbb{Q}})=0\,,\,\forall i\geq 2$. Then we apply theorem
3.10. ∎

######  Example 3.13.

The purpose of this example is to compute the numerical invariants
$\rho_{\mathbb{C}P_{0}^{n}}$ and $\eta_{\mathbb{C}P_{0}^{n}}$, for the space
$\mathbb{C}P_{0}^{n}$ which is the rationalised complex projective space
$\mathbb{C}P^{n}$. Indeed; the Sullivan model of $\mathbb{C}P_{0}^{n}$ is
$(\Lambda(V^{2}\oplus V^{2n+1}),\partial)$ with $V^{2}=$ $\langle
x\rangle,V^{2n+1}=\langle y\rangle$ and $\partial y=x^{n+1}$ (see [9], example
5, page 333). As we have $L^{i}=$ $H^{i}(\Lambda(V^{\leq i-2}))$, it follows
that $L^{\geq 2n+3}=0.$ Moreover it is easy to see that
$\operatorname{dim}L^{i}=1$ for $i$ even and $\operatorname{dim}L^{i}=0$ for
$i$ odd. Therefore $\rho_{\mathbb{CP}_{0}^{n}}=1+L^{\text{even }}=1+n$ and
since
$\chi_{\pi}(\mathbb{C}P_{0}^{n})=\operatorname{dim}V^{2}-\operatorname{dim}V^{2n+1}=0$,
we deduce that $\rho_{\mathbb{C}P^{n}}=\chi_{\mathbb{CP}_{0}^{n}}=1+n.$ As a
result we get $\eta_{\mathbb{C}P^{n}}=1+n$. Notice that the Quillen model of
$\mathbb{C}P_{0}^{n}$ is given by

$\mathbb{L}(W_{1},\ldots,W_{2n-1})\quad,\quad W_{i}=\left\langle
w_{i}\right\rangle\quad,\quad\delta(w_{k})=\frac{1}{2}\sum_{i+j=k}\left[w_{i},w_{j}\right].$

Hence, for $\Gamma_{i}=H_{i}(\mathbb{L}(W_{\leq i-2}))$, we deduce that
$\eta_{\mathbb{CP}_{0}^{n}}=1+\Gamma_{\text{even }}-\Gamma_{\text{odd }}=1+n$.

###  3.1. $F_{0}$-spaces

######  Definition 3.14.

An $F_{0}$-space is a rationally elliptic CW-complex $X$ whose rational
cohomology algebra is given by
${\mathbb{Q}}[x_{1},\dots,x_{n}]/(P_{1},\dots,P_{n})$, where the polynomials
$P_{1},\dots,P_{n}$ form a regular sequence in
${\mathbb{Q}}[x_{1},\dots,x_{n}]$, i.e., $P_{1}\neq 0$ and for every $i\geq
1$, $P_{i}$ is not a zero divisor in
${\mathbb{Q}}[x_{1},\dots,x_{n}]/(P_{1},\dots,P_{i-1})$.

For instance, products of even spheres, complex Grassmannian manifolds and
homogeneous spaces $G/H$ such that rank $G$ = rank $H$ are $F_{0}$-spaces.

######  Proposition 3.15.

([9] Proposition 32.10). Let $X$ be a rationally elliptic CW-complex and
$(\Lambda V,\partial)$ its Sullivan model. The following statements are
equivalent

  1. (1)

$X$ is an $F_{0}$-space;

  2. (2)

$H^{\rm{odd}}(\Lambda V)=0;$

  3. (3)

$\operatorname{dim}V^{\rm{even}}-\operatorname{dim}V^{\rm{odd}}=0$ and
$(\Lambda V,\partial)$ is pure, i.e., $\partial(V^{\text{even }})=0$ and
$\partial(V^{\text{odd }})\subseteq\Lambda V^{\text{even }}.$

######  Theorem 3.16.

If $X$ is an $F_{0}$-space, then

$\Gamma_{2i+1}(X)=H^{2i+2}(X^{[2i]},{\mathbb{Q}})=0\quad,\quad\forall i\geq
1.$

Moreover, if $X$ is 2-connected, then
$\pi_{2i+1}(X^{2i})\otimes{\mathbb{Q}}=0$ for every $i\geq 1$.

###### Proof.

By (6), the Whitehead exact sequence of $(\Lambda V,\partial)$ can be written
as

$\cdots\rightarrow H^{2i}(\Lambda V)\rightarrow V^{2i}\stackrel{{\scriptstyle
b^{2i}}}{{\longrightarrow}}L^{2i+1}\longrightarrow H^{2i+1}(\Lambda
V)\longrightarrow V^{2i+1}\stackrel{{\scriptstyle
b^{2i+1}}}{{\longrightarrow}}\cdots$

As $X$ is an $F_{0}$-space, then by proposition 3.15, the Sullivan model
$(\Lambda V,\partial)$ of $X$ satisfies $H^{\text{odd }}(\Lambda V)=0$ and
$\partial(V^{\text{even }})=0$, it follows that the maps $b^{\text{even }}=0.$
Consequently, $L^{\text{odd }}=0.$ So, taking into account the identifications
(12) and (3), the result follows from the relation (17). ∎

## Acknowledgments

The author is deeply grateful to the referee for a careful reading of the
paper and valuable suggestions that greatly improved the manuscript.

## References

  * [1] M. Benkhalifa, _On the group of self-homotopy equivalences of an almost formal space_ , Quaestiones Mathematicae, (2022). https://doi.org/10.2989/16073606.2022.2044405.
  * [2] M. Benkhalifa, _Realizability of the group of self-homotopy equivalences and local homotopy theory_ , Homology, Homotopy and Applications, 24 (1), 205-215, 2022
  * [3] M. Benkhalifa, _The group of self-homotopy equivalences of a rational space cannot be a free abelian group_ , to appear in Journal of the Mathematical Society of Japan.
  * [4] M. Benkhalifa, _The effect of cell-attachment on the group of self-equivalences of an elliptic space_ , Michigan Mathematical Journal, (2021). DOI: 10.1307/mmj/20195840
  * [5] M. Benkhalifa, _On the group of self-homotopy equivalences of an elliptic space_ , Proceedings of the American Mathematical Society. Vol.148 (6),2695-2706, 2020
  * [6] M. Benkhalifa and S. B. Smith, The effect of cell attachment on the group of self-equivalences of an $R$-local space, Journal of Homotopy and Related Structures. Vol. 10(3), 549-564, $2015.$
  * [7] M. Benkhalifa, Rational self-homotopy equivalences and Whitehead exact sequence J. Homotopy Relat. Struct. $4(1),135-144,2009$
  * [8] M. Benkhalifa, Whitehead exact sequence and differential graded free Lie algebra, International Journal of Mathematics, Volume 15, N. 12,1-19, 2004 .
  * [9] Y. Félix, S. Halperin, and J.-C. Thomas,Rational homotopy theory, Graduate Texts in Mathematics, Vol. 205, Springer-Verlag, New York, $2001.$


# Improper two-dimensional electron gas formation

Daniel Bennett<EMAIL_ADDRESS>Theory of Condensed Matter,
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge
CB3 0HE, UK  John A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA  Pablo Aguado-Puente
CIC Nanogune and DIPC, Tolosa Hiribidea 76, 20018 San Sebastian, Spain  Emilio
Artacho  Theory of Condensed Matter, Cavendish Laboratory, University of
Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK  CIC Nanogune and DIPC,
Tolosa Hiribidea 76, 20018 San Sebastian, Spain  Ikerbasque, Basque Foundation
for Science, 48011 Bilbao, Spain  Nicholas Bristowe  Centre for Materials
Physics, Durham University, South Road, Durham DH1 3LE, UK

###### Abstract

In spite of the interest in the two-dimensional electron gases (2DEGs)
experimentally found at surfaces and interfaces, important uncertainties
remain about the observed insulator–metal transitions (IMTs). Here we show how
an explicit improper coupling of carrier sources with a relevant soft mode
significantly affects the transition. The analysis presented here for 2DEGs at
polar interfaces is based on group theory, Landau-Ginzburg theory, and
illustrated with first-principles calculations for the prototypical case of
the $\mathrm{LaAlO_{3}/SrTiO_{3}}$ interface, for which such a structural
transition has recently been observed. This direct coupling implies that the
appearance of the soft mode is always accompanied by carriers. For
sufficiently strong coupling an avalanche-like first-order IMT is predicted.

Introduction.—The insulator–metal transition (IMT) is one of the most widely
studied problems in condensed matter physics [1, 2], particularly in oxide
materials [3, 4, 5]. IMTs in two dimensions (2D) have attracted additional
interest [6, 7], in particular the one occurring at polar-nonpolar perovskite
interfaces, first observed for thin films of $\mathrm{LaAlO_{3}}$ (LAO) grown
on $\mathrm{SrTiO_{3}}$ (STO) [8, 9]. For film thickness beyond a critical
value (3 unit cells), a 2D electron gas (2DEG) appears at the interface, or
can be induced by gating [10, 11], see Fig. 1 (a). The gas is associated to Ti
3$d$ states, and has been found to display intriguing magnetic [12] and
superconducting [13] properties, even in co-existence [14, 15], in addition to
non-trivial topological character [16].

If both insulators are centrosymmetric, as is the case in LAO/STO, the polar
discontinuity is quantized [17, 18] in units of $P_{S}=P_{0}/2$, i.e. half a
quantum of polarization, or 0.5 electrons per interface Ti. Although
electronic reconstruction – electrons from the valence band at the surface
becoming more stable at the bottom of the conduction band at the interface –
was initially proposed as the source of carriers, redox defects were soon
recognized as a more likely source [19, 20, 21], most prominently oxygen
vacancies. This has been amply ratified by various alternative experimental
methods to generate very similar 2DEGs, namely, by irradiation [22] and
chemistry (controlling the oxygen partial pressure [23], or depositing oxygen-
hungry metallic layers on STO [24]), in addition to detailed first-principles
calculations [25].

2DEG formation has been also proposed as an alternative screening mechanism to
domains [26, 27] in ferroelectric materials [28, 29], supported by recent
experimental evidence [30, 31, 32]. Similar observations of surface carriers
have been made in stacks of 2D materials [33], which have recently been
predicted [34, 35, 36, 37, 38] and experimentally shown [39, 40] to be
ferroelectric.

Figure 1:  (a) Sketch of 2DEG formation between a polar film and a nonpolar
substrate. (b) Illustration of the possibilities for the onset of interfacial
carriers. The red lines show the 2DEG carrier concentration, for a mean field
theory (solid), beyond mean-field (dashed), and a discontinuous transition
(dot-dashed). The black lines represent the onset of conductivity, which may
differ from the onset of carriers.

Despite considerable interest, some aspects of 2DEG formation and the IMT at
polar–nonpolar interfaces are still unclear. For example, it is not clear
whether the appearance of carriers with film thickness or electric field is
continuous, as predicted in a mean-field approximation [18], or discontinuous,
as either assumed [25], or postulated beyond mean-field [21], see Fig. 1 (b).
Additionally, the distinction between the onset of carriers and the onset of
conduction is not typically made. It may be possible that carriers can appear
but are not conducting. This is supported by experimental evidence: a density
of trapped Ti 3d-like states has been observed in LAO/STO at just 2 unit cells
of LAO, i.e. before the observation of the mobile carriers [41]. For carriers
generated via oxygen vacancies, the charge carriers are trapped below the
surface defects which generated them, and thus will be localized in the plane
of the interface [21]. As the thickness of the film increases, more surface
defects will be created and the associated carriers will become delocalized,
and conductivity will occur at a thickness larger than the one at which
carriers first appear, see Fig. 1 (b). It has also been proposed that both
heavy and light carriers form, with only the lighter carriers contributing to
conduction [42].

It was recently proposed that the coupling between the dielectric properties
and structural instabilities, such as octahedral rotations (tilts, see inset
in Fig. 2), may play a role in the onset of carriers, and possibly the IMT,
and even change the character of the transition, or lead to a sequence of
transitions [43]. Tilting has been experimentally shown to occur in LAO/STO,
where a transition with thickness was recently observed [44].

Such instabilities may play a deeper role in the IMT when considering oxygen
vacancies as the source of carriers. In addition to providing free charge
which can migrate to the interface, removing an oxygen atom will break an
octahedron at the surface, which may strongly affect the tilting and polar
modes. In order to better understand the rich coupling between such soft-mode
instabilities, defects and interfacial carriers, we perform large-scale first-
principles calculations of the LAO/STO interface. From a group theory
analysis, we find that there is an additional direct coupling between
octahedral rotations and oxygen vacancies at the surface, which propagates
into the film and substrate. Using a Ginzburg-Landau (GL) theory, we show that
this direct coupling at the surface can strongly affect the onset of
interfacial carriers, and can drastically change the order of the IMT.

Figure 2:  $\phi_{\parallel}$ tilt profile in LAO/STO from first-principles
calculations. The hollow points show results with no 2DEG
(${\sigma^{\prime}=0}$), the filled points show results with a fully saturated
2DEG (${\sigma^{\prime}=1}$) using VCA (black) and explicit oxygen vacancies
(red). The horizontal line on the LAO side of the interface indicates the
magnitude to the tilts in bulk strained LAO, $\phi^{0}_{\parallel}$. The inset
shows a sketch of an $\mathrm{AlO_{6}}$ octahedron. The axes of the
$\phi_{\parallel}$ and $\phi_{\perp}$ tilts, [100] and [001] respectively, are
shown.

First-principles calculations.—Calculations were performed using the siesta
code [45] on vacuum-terminated $\mathrm{(SrTiO_{3})_{n}/(LaAlO_{3})_{m}}$
slabs, where $n=8$ or 15 unit cells, and $m=3$ unit cells [46]. The vacancies
were explicitly introduced by removing one of the oxygen atoms from the
$\mathrm{AlO_{2}}$ surface; removing one oxygen atom from a $2\times 2$ in-
plane supercell is equivalent to imposing a displacement field of
$D_{\perp}=0.5$ electrons per unit cell surface area. In order to distinguish
between the effect an oxygen vacancy has on just the electrostatic boundary
conditions, compared to the effect of symmetry breaking on structural
instabilities, we redo the calculations using the virtual crystal
approximation (VCA), substituting all of the oxygen atoms on the surface with
‘virtual atoms’ with fractional atomic numbers to impose the same electric
displacement boundary conditions but without surface symmetry breaking [47].

The profiles of in-plane tilts, $\phi_{\parallel}(z)$, are shown in Fig. 2,
for a fully saturated 2DEG introduced via explicit oxygen vacancies (red) and
the VCA (black) and a pristine interface with no 2DEG (hollow points). The STO
substrate also displays tilts around the out of plane axis, but these are
suppressed in the LAO film due to the in-plane expansion imposed by the
clamping to the substrate and are therefore not shown here [46].

For a 2DEG induced via the VCA, there is negligible (although nonzero) tilting
in the LAO film and in STO near the interface, resulting in a tilt profile
virtually identical to the pristine interface. For a 2DEG introduced via
explicit oxygen vacancies, a very large tilt around the [100] axis is induced
at the surface which decays into both LAO and STO (despite the competition
with the out-of-plane tilt considered in STO). The presence of an additional
distortion when explicit vacancies are considered indicates that there is an
additional direct coupling between oxygen vacancies and tilts, separate from
the biquadratic coupling between the 2DEG carrier concentrations and tilts,
which occurs indirectly through the polar mode [44, 43]. We emphasise that the
lack of significant tilting in the VCA case cannot simply be due to the DFT
calculations lying at a saddle-point, since the tilting is small but non-zero.
This implies that the direct coupling with vacancies does not just break the
symmetry to allow for the tilts, but is also sufficiently strong to enforce
the tilting with amplitudes orders of magnitude larger than without the
coupling.

Ginzburg-Landau theory.—Motivated by the first-principles calculations in Fig.
2, we generalize the GL theory developed in Ref. [43] to account for the
additional direct coupling between tilts and oxygen vacancies at the LAO
surface [46]. We write the free energy per unit volume $\mathcal{F}^{\rm GL}$
as

$\mathcal{F}^{\rm GL}=\mathcal{F}^{\rm GL}_{\rm indirect}+\frac{1}{V}\int_{\rm
s}\beta\sigma^{\prime}\phi_{\rm s}^{\prime}\mathop{}\\!\mathrm{d}{S}\;.$ (1)

The indirect terms in $\mathcal{F}^{\rm GL}_{\rm indirect}$ include the
energies associated with the defect formation of surface vacancies and
associated electrostatic screening [18], as well as the tilt double well and
associated Ginzburg term, and finally an indirect coupling between the tilts
and vacancies through the biquadratic coupling between tilts and polar mode
[43]. Interestingly, our symmetry analysis [46] identifies a new direct
coupling term which is bilinear in the surface tilt, $\phi_{\rm s}$, and
carrier density provided by the oxygen vacancies to the 2DEG, $\sigma$, with
coupling coefficient $\beta$, which has units of energy per unit area. $V=Ad$
is the volume of LAO, where $A$ is the in-plane unit cell surface area, and
$d$ is the thickness of the film; while $d$ is a discrete number of unit
cells, we treat it as a continuously varying parameter, since its effect is
equivalent to that of an applied electric field (see a more detailed
discussion in Ref. [43]). Transitions have been predicted and observed for a
given thickness (already close to the transition) under a varying applied
perpendicular field [10, 11] and the results of this paper on the character of
the transition are directly transferable to that situation. In Eq. 1 both
parameters have been renormalized for convenience:
$\phi^{\prime}=\frac{\phi}{\phi_{0}}$, $\phi_{0}$ being the bulk equilibrium
tilt in LAO, and $\sigma^{\prime}=\frac{\sigma}{P_{S}}$, $P_{S}$ being the
polarization associated with the discontinuity at the interface. Being
bilinear, the existence of tilting linearly scales the formation energy of
surface vacancies, and similarly, the existence of vacancies produces forces
on the surface oxygens which automatically induces tilting. In this way,
unlike the even order coupling terms important for triggered phase transitions
[48], the sign of the $\beta$ coefficient is unimportant (simply defining the
sense of the tilting with respect to vacancies on a certain oxygen
sublattice), in connection with the field of improper phase transitions [49,
50, 51]. We note that while a vacancy every $2\times 2$ cells induces tilts
around a [100] axis, in agreement with DFT, our group theory analysis suggests
that alternative vacancy orderings can induce tilts around [110], which is the
expected tilt pattern observed in LAO films [44], and hence the case
considered in the following.

Figure 3:  (a) Tilt profile for three unit cells of LAO for several values of
$\beta$, in units of $\frac{P_{S}^{2}}{\epsilon}$, where $P_{S}$ is half a
polarization quantum of LAO and $\epsilon$ is the dielectric constant of LAO.
The displacement for the tilt mode around [110], a zone-edge nonpolar mode, is
sketched. (b) Tilt profile $\phi^{\prime}(z)$ for LAO films of thickness 3
(black), 4 (red) and 5 (blue) unit cells, with ${\beta=0}$ (dashed) and
${\beta=\frac{1}{8}}$ (solid). In each film the 2DEG is saturated, i.e.
$\sigma^{\prime}=1$.

In order to investigate the role of direct coupling between oxygen vacancies
and surface tilts on the IMT in LAO/STO, Eq. (1) was minimized numerically,
following the methodology in Ref. [43]. For a given thickness of LAO and
sensible material parameters [46], the values of the order parameters
$\phi^{\prime}(z)$ and $\sigma^{\prime}$ which minimize the free energy were
determined. In Fig. 3 (a) we show the tilt profile in 3 u.c. thick LAO for
several values of $\beta$. The magnitude of the the surface tilt which results
from the oxygen vacancies is strongly dependent on $\beta$, and results in a
large tilt profile which decays into the film. For sufficiently strong
$\beta$, the curvature of the tilt profile changes, and the surface tilt can
excede the bulk equilibrium value, i.e. $\phi^{\prime}_{\rm
s}\equiv\phi^{\prime}(d)>1$, which is in agreement with the first-principles
results shown in Fig. (2).

Fig. 3 (b) shows the tilt profile in LAO films consisting of 3–5 unit cells,
with and without direct coupling between oxygen vacancies and surface tilts.
In the absence of the direct coupling term ($\beta=0$), the 3 u.c. film is
predicted to be untilted, with the tilts appearing from 4 u.c. and approaching
the bulk equilibrium value with film thickness. Switching on the direct
coupling always results in a surface tilt when oxygen vacancies are present,
which decays into the film. Thus, direct coupling at the surface results in
the appearance of tilts in the 3 u.c. film, without which no tilts are
predicted, in line with experimental observations [44].

Next, we consider the effect of the direct coupling on the appearance of
carriers as a function of film thickness. Eq. (1) was minimized as a function
of film thickness $d$, which was treated as a smoothly varying parameter, for
several values of $\beta$. The carrier concentration and root mean square tilt
$\phi^{\prime}_{\rm RMS}\equiv\sqrt{\left<\phi^{\prime 2}\right>}$ are shown
in Figs. 4 (a) and (b), respectively. Assuming that in the absence of any
coupling to tilts, the carriers appear at a critical thickness $d_{\text{c}}=$
2 u.c. [41], the biquadratic coupling to homogeneous tilts (described by a
Landau theory) reduces $d_{\text{c}}$ slightly, accompanied by a continuous
increase in tilts. Treating the tilts with a GL theory, but with $\beta=0$,
the tilts are predicted to appear above 3 u.c., with the carriers appearing at
the uncoupled thickness $d_{\text{c}}=$ 2 u.c.. For $\beta\neq 0$, the tilts
always appear simultaneously at $d_{\text{c}}=$ 2 u.c. when the carriers
appear. The rate at which the carrier concentration increases with thickness
is strongly dependent on $\beta$. For sufficiently strong $\beta$, both order
parameters switch on discontinuously at $d_{\text{c}}$, with the
$\sigma^{\prime}=1$ being fully saturated, resembling an avalanche-like phase
transition [52, 53].

Figure 4:  (a) Carrier concentration $\sigma^{\prime}$, and (b) root mean
square of the tilt profile $\phi^{\prime}_{\rm RMS}$, as a function of LAO
film thickness for several value of $\beta$. The dashed line indicates that
the tilts are homogeneous, i.e. described by a Landau theory, and only the
indirect biquadratic coupling between $\sigma^{\prime}$ and $\phi^{\prime}$ is
considered.

Discussion and conclusions.—In this letter, we illustrate with first-
principles calculations that there is a direct coupling between surface tilts
and oxygen vacancies in LAO/STO, going beyond the indirect biquadratic
coupling of the tilts and the carriers, through the polar mode [43]. We
propose that this coupling is described by a bilinear term in the surface tilt
and carrier concentration, supported by group theory analysis. Using a GL
theory, we show that the direct coupling always implies the appearance of
tilts whenever there are oxygen vacancies. This is in agreement with
experimental observations [44], as well as our first-principles calculations,
where thin films of LAO were predicted to be untilted when interface carriers
were artificially introduced using the VCA, but tilted when carriers were
introduced from surface oxygen vacancies.

Our results suggest that the direct coupling between oxygen vacancies and
tilts on the surface of LAO may play an important role in the IMT at the
LAO/STO interface. Our GL theory suggests that this direct coupling strongly
influences the appearance of carriers with film thickness. The direct coupling
between oxygen vacancies and surface tilts may resolve differences between
theory, where a continuous appearance of carriers above 3 u.c. was predicted
[18], and experiment, where a fully saturated 2DEG is observed at 3 u.c. and
above [9].

For sufficiently strong coupling, an avalanche-like phase transition to a
fully saturated 2DEG occurs. To our knowledge, this mechanism of an avalanche-
like IMT driven by the indirect coupling between surface tilts and oxygen
vacancies is unique, adding to the diverse structural mechanisms for IMTs,
such as the triggerd IMT in nickelates, induced by the cooperate coupling
between tilts and a breathing mode [48].

We observed a discrepancy between experimental data [44], where a relatively
small surface tilt was measured in 3 unit cells of LAO, and our DFT results,
where a large surface tilt was calculated. For the same set of material
parameters, the tilts from DFT are well-described by a large value of $\beta$,
Fig. 3 (a), while the experimental measurements of tilts are better described
by a smaller value of $\beta$, Fig. 3 (b). This difference in behaviour may be
due to the different conditions in ideal DFT calculations and experiment,
resulting in very different surface tilts. In Ref. [46] we investigate the
role of the surface tension on the surface tilt and show that both
experimental and DFT results can be described using the same value of $\beta$
but changing the energetics at the surface of LAO, keeping all other material
parameters fixed. Despite the differences between the experimental
measurements and the more ideal situation in the DFT calculations, the direct
coupling between tilts and oxygen vacancies is capable of describing the
behaviour of the tilts in both scenarios, as well as a first-order IMT.

The bilinear coupling proposed here may play a role in IMTs beyond
polar–nonpolar interfaces. A defect ordering described by a non-zone-center
mode will couple to non-zone-center phonon modes (see Fig. 3 (a)), which
typically compete with the polar mode. Thus, for a structural IMT, induced by
the suppression/enhancement of a polar mode, this indirect coupling can play a
strong role, beyond acting as a source of carriers, in inducing an IMT.

In addition to its role in the IMT at polar–nonpolar interfaces, the direct
coupling between soft-mode instabilities and surface defects may prove
interesting and useful in other scenarios. For example, knowledge of this
coupling may make it possible to engineer a regular array of surface defects
by tailoring the soft-mode instabilities in a material, through tolerance
factor, temperature or strain [54], for example.

D.B. thanks M.C. Payne and M. Stengel for interesting discussions. D.B.
acknowledges funding from the EPSRC Centre for Doctoral Training in
Computational Methods for Materials Science under grant number EP/L015552/1,
the US Army Research Office (ARO) MURI project under grant No. W911NF-21-0147
and the National Science Foundation DMREF program under Award No. DMR-1922172.
P.A.P. acknowledges funding from the Diputación Foral de Gipuzkoa through
Grant 2020-FELL-000005-01. Funding from the Spanish MCIN/AEI/
10.13039/501100011033 is also acknowledged, through grants
PID2019-107338RB-C61 and PID2022-139776NB-C65, as well as a María de Maeztu
award to Nanogune, Grant CEX2020-001038-M.

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