# Algebraic localization of Wannier functions implies Chern triviality in non-
periodic insulators

Jianfeng Lu  (JL) Department of Mathematics, Department of Physics, and
Department of Chemistry, Duke University, Box 90320, Durham, NC 27708, USA
<EMAIL_ADDRESS>and  Kevin D. Stubbs  (KDS) Department of Mathematics,
Duke University, Box 90320, Durham, NC 27708, USA<EMAIL_ADDRESS>
###### Abstract.

For gapped periodic systems (insulators), it has been established that the
insulator is topologically trivial (i.e., its Chern number is equal to $0$) if
and only if its Fermi projector admits an orthogonal basis with finite second
moment (i.e., all basis elements satisfy
$\int|\bm{x}|^{2}|w(\bm{x})|^{2}\,\textrm{d}{\bm{x}}<\infty$). In this paper,
we extend one direction of this result to non-periodic gapped systems. In
particular, we show that the existence of an orthogonal basis with slightly
more decay
($\int|\bm{x}|^{2+\epsilon}|w(\bm{x})|^{2}\,\textrm{d}{\bm{x}}<\infty$ for any
$\epsilon>0$) is a sufficient condition to conclude that the Chern marker, the
natural generalization of the Chern number, vanishes.

This work is supported in part by the U.S. National Science Foundation via
grant DMS-2012286 and the U.S. Department of Energy via grant DE-SC0019449.
K.D.S. was supported in part by a National Science Foundation Graduate
Research Fellowship under Grant No. DGE-1644868.

##  1\. Introduction

In electron structure theory, we are often interested in studying the subspace
of low energy states spanned by the range of Fermi projector $P$. For
numerical and theoretical purposes, we are in particular interested in finding
a basis for the occupied space $\operatorname{range}{(P)}$ which is as well
localized in space as possible. The elements of such a basis are known as
Wannier functions or generalized Wannier functions (see review [6] and
references therein). Typically for insulating materials, the Fermi projector
$P$ admits an integral kernel which is exponentially localized in the
following sense (see, for example [5]):

(1) $|P(\bm{x},\bm{y})|\lesssim e^{-c_{gap}|\bm{x}-\bm{y}|}.$

Therefore, we might expect that these insulators admit a basis which decays
exponentially quickly in space. Somewhat surprisingly, even if $P$ satisfies
an estimate like Equation (1), it is not necessarily true that
$\operatorname{range}{(P)}$ admits a basis which decays exponentially quickly
in space due to the existence of so called “topological obstructions”.

In two dimensional periodic insulators, it is now well understood [1, 8, 7]
that the existence of a well localized basis for $\operatorname{range}{(P)}$
is fully characterized by the Chern number which is defined as follows:

$c(P)=\frac{1}{2\pi}\int_{\mathcal{B}}\operatorname{tr}\Bigl{(}P(\bm{k})\bigl{[}\partial_{k_{1}}P(\bm{k}),\partial_{k_{2}}P(\bm{k})\bigr{]}\Bigr{)}dk_{1}\wedge
dk_{2},$

where $\mathcal{B}$ is the first Brillouin zone and $P(\bm{k})$ is the Bloch
decomposition of $P$ (see e.g., [9]).

For periodic systems, $P$ possesses a basis with finite second moment (known
as Wannier functions) if and only if $c(P)=0$, as established in [7].
Furthermore, $c(P)=0$ if and only if there exists a basis of
$\operatorname{range}{(P)}$ which is exponentially localized [1]. These
results, which connect the existence of a basis with finite second moment to
the vanishing of the Chern number and to the existence of an exponentially
localized basis, is known as the localization dichotomy in periodic
insulators.

Since the notion of the Chern number depends on the Bloch decomposition, the
Chern number is no longer well defined for non-periodic systems. For generic
systems, the Chern marker was proposed in [2, 4] as an extension.

######  Definition 1 (Chern Marker).

Let $P$ be a projection on $L^{2}(\mathbb{R}^{2})$ and $\chi_{L}$ be the
indicator function of the set $[-L,L)^{2}$. The Chern marker of $P$ is defined
by

$C(P):=\lim_{L\rightarrow\infty}\frac{2\pi
i}{4L^{2}}\operatorname{tr}{\left(\chi_{L}P\Big{[}[X,P],[Y,P]\Big{]}P\chi_{L}\right)}$

whenever the limit on the right hand side exists.

Note that this generalizes the Chern number as for periodic systems the Chern
number and the Chern marker agree [4, 5]. Therefore, parallel to the periodic
case, it is conjectured that the Chern marker characterizes the existence of
localized Wannier basis for gapped systems [4, 5]. Before continuing to state
the conjecture more precisely and state the main result of this paper, which
confirms the conjecture in one direction, let us start by making some
definitions:

######  Definition 2.

Suppose that $A$ is a bounded linear operator on
$L^{2}(\mathbb{R}^{d})\rightarrow L^{2}(\mathbb{R}^{d})$. We say that $A$
admits an exponentially localized kernel with decay rate $\gamma$, if $A$
admits an integral kernel
$A(\cdot,\cdot):\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{C}$ and
there exists a finite, positive constant $C$ so that:

$|A(\bm{x},\bm{x}^{\prime})|\leq Ce^{-\gamma|\bm{x}-\bm{x}^{\prime}|}\quad
a.e.$

######  Definition 3 ($s$-localized generalized Wannier basis).

Given an orthogonal projector $P$, we say an orthonormal basis
$\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}\subseteq L^{2}(\mathbb{R}^{2})$ is
an $s$-localized generalized Wannier basis for $P$ for some $s>0$ if:

  1. (1)

The collection $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ spans
$\operatorname{range}{(P)}$,

  2. (2)

There exists a finite, positive constant $C$ and a collection of points
$\\{\bm{\mu}_{\alpha}\\}_{\alpha\in\mathcal{I}}\subseteq\mathbb{R}^{2}$ such
that for all $\alpha\in\mathcal{I}$

$\int_{\mathbb{R}^{2}}\langle\bm{x}-\bm{\mu}_{\alpha}\rangle^{2s}|\psi_{\alpha}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}\leq
C,$

where
$\langle\bm{x}-\bm{\mu}_{\alpha}\rangle:=(\lvert\bm{x}-\bm{\mu}_{\alpha}\rvert^{2}+1)^{1/2}$
is the Japanese bracket.

We refer to the collection $\\{\bm{\mu}_{\alpha}\\}_{\alpha\in\mathcal{I}}$ as
the center points of the basis $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$.

With these definitions, the localization dichotomy conjecture for non-periodic
systems is as follows:

######  Conjecture (Localization dichotomy for non-periodic gapped systems).

Let $P$ be an orthogonal projector which admits an exponentially localized
kernel. Then the following statements are equivalent:

  1. (a)

$P$ admits a generalized Wannier basis that is exponentially localized.

  2. (b)

$P$ admits a generalized Wannier basis that is $s$-localized for $s=1$.

  3. (c)

$P$ is topologically trivial in the sense that its Chern marker $C(P)$ exists
and is equal to zero.

Note that obviously (a) implies (b). For the other equivalence, there have
been a few works devoted to the study of non-periodic localization dichotomy.
In particular, recent work [5] has shown that (b) $\Rightarrow$ (c) with
$s>5$. Additionally, our previous work [3] has shown that (b) $\Rightarrow$
(a) (and hence (b) $\Rightarrow$ (c)) with $s>5/2$. In this paper, we improve
upon these previous works by showing that (b) $\Rightarrow$ (c) for $s>1$.
Formally stated, the main result of this paper is the following:

######  Theorem 1.

Suppose that $P$ is an orthogonal projection on $L^{2}(\mathbb{R}^{2})$ which
admits an exponentially localized kernel. If $P$ admits an
$(1+\delta)$-localized generalized Wannier basis for some $\delta>0$, then the
Chern marker $C(P)$ vanishes.

Our proof of Theorem 1 can be extended to $s=1$ by making some additional
technical assumptions on the generalized Wannier basis however a full proof
for the $s=1$ case remains open. We note that Theorem 1 establishes only one
part of the localization dichotomy, while the other direction, $C(P)=0$
implies the existence of localized generalized Wannier basis, is still quite
open.

### Notations

Vectors in $\mathbb{R}^{d}$ will be denoted by bold face with their components
denoted by subscripts. For example,
$\bm{v}=(v_{1},v_{2},v_{3},\cdots,v_{d})\in\mathbb{R}^{d}$. For any
$\bm{v}\in\mathbb{R}^{d}$, we use $|\cdot|$ to denote its $\ell^{2}$-norm and
$|\cdot|_{\infty}$ to denote its $\ell^{\infty}$-norm; that is,
$|\bm{v}|:=\bigl{(}\sum_{i=1}^{d}v_{i}^{2}\bigr{)}^{1/2}$,
$|\bm{v}|_{\infty}:=\max_{i}|v_{i}|$. For any $\bm{x}\in\mathbb{R}^{2}$ and
$a\in\mathbb{R}^{+}$, we define $\chi_{a}$ to be the indicator function of the
set $[-a,a)^{2}$ and $B_{a}(\bm{x})$ be the ball of radius $a$ centered at
$\bm{x}$.

For any $f:\mathbb{R}^{2}\rightarrow\mathbb{C}$, we will use $\|f\|$ to denote
the $L^{2}$-norm. For any bounded linear operator $A$ on
$L^{2}(\mathbb{R}^{2})$, we adopt the following conventions:

  * •

Let $\|A\|$ denote the spectral norm of $A$, $\|A\|:=\sup_{\|f\|=1}\|Af\|$.

  * •

If $A$ is compact, let $\\{\sigma_{n}(A)\\}_{n=1}^{\infty}$ denote the
singular values of $A$ in decreasing order (i.e. if $i<j$ then
$\sigma_{i}(A)\geq\sigma_{j}(A)$).

  * •

If $A$ is compact, let
$\|A\|_{\mathfrak{S}_{p}}=\bigl{(}\sum_{n=1}^{\infty}\sigma_{n}(A)^{p}\bigr{)}^{1/p}$
denote the Schatten $p$-norm for any $p\geq 1$.

Note that with this convention $\|A\|=\|A\|_{\mathfrak{S}_{\infty}}$.

In our estimates, $C$ is used as a generic constants whose value may change
from line to line. We also write $A\lesssim B$ if there exists a constant $C$
such that $A\leq CB$.

### Organization

The remainder of this paper is organized as follows. In Section 2, we outline
the proof of Theorem 1 relying on a number of propositions (Proposition 2.3,
2.4, and 2.5). Next, in Section 3 we state and prove three important technical
estimates which are central to the proofs of these propositions. We provide
proofs of Proposition 2.3 in Section 4, Proposition 2.4 in Section 5, and
Proposition 2.5 in Section 6, respectively.

##  2\. Proof of Main Theorem

We begin our proof by recalling the notion of bounded density which was
introduced in [3] to simplify the analytic estimates. After recalling the
consequences of bounded density (in particular, Lemma 2.2), we will use these
results to prove the main theorem.

###  2.1. Bounded Density

We begin with the definition of bounded density

######  Definition 4.

We say that a collection of points
$\\{\bm{\mu}_{\alpha}\\}_{\alpha\in\mathcal{I}}$ has bounded density if there
exists a constant $M<\infty$ such that for all $\bm{x}\in\mathbb{R}^{2}$ we
have

$\\#\\{\alpha:\bm{\mu}_{\alpha}\in B_{1}(\bm{x})\\}\leq M$

Importantly, if orthogonal projector $P$ has an exponentially localized
kernel, one can show that the center points of every well localized basis must
have bounded density.

######  Lemma 2.1.

Let $P$ be an orthogonal projector which admits an exponentially localized
kernel. If $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ is an $s$-localized
generalized Wannier basis for $P$ for some $s>0$, then the center points for
$\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ have bounded density.

###### Proof.

For this proof, let $\chi_{B_{r}(\bm{a})}$ denote the characteristic function
of the ball $B_{r}(\bm{a})$: $\chi_{B_{r}(\bm{a})}(\bm{x})=1$ if $\bm{x}\in
B_{r}(\bm{a})$ and zero otherwise. We start by observing two important facts.

  1. (i)

If $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ is an $s$-localized basis for
$s>0$ with center points $\\{\bm{\mu}_{\alpha}\\}_{\alpha\in\mathcal{I}}$ then
we have that

$\displaystyle\|(1-\chi_{B_{r}(\bm{\mu}_{\alpha})})\psi_{\alpha}\|^{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}(1-\chi_{B_{r}(\bm{\mu}_{\alpha})}(\bm{x}))\frac{\langle\bm{x}-\bm{\mu_{\alpha}}\rangle^{2s}}{\langle\bm{x}-\bm{\mu_{\alpha}}\rangle^{2s}}|\psi_{\alpha}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle\lesssim
r^{-2s}\int_{\mathbb{R}^{2}}\langle\bm{x}-\bm{\mu_{\alpha}}\rangle^{2s}|\psi_{\alpha}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$

Hence, since the collection $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ is
$s$-localized, there exists a constant $C$, uniform in $\alpha$, so that
$\|(1-\chi_{B_{r}(\bm{\mu}_{\alpha})})\psi_{\alpha}\|^{2}\leq Cr^{-2s}$. Thus
we can find a radius $R>0$ so that for all $\alpha\in\mathcal{I}$ and all
$r\geq R$

(2) $\|(1-\chi_{B_{r}(\bm{\mu}_{\alpha})})\psi_{\alpha}\|^{2}\leq\frac{1}{2}.$

Since
$\|(1-\chi_{B_{r}(\bm{\mu}_{\alpha})})\psi_{\alpha}\|^{2}+\|\chi_{B_{r}(\bm{\mu}_{\alpha})}\psi_{\alpha}\|^{2}=1$,
have that for all $r\geq R$,
$\|\chi_{B_{r}(\bm{\mu}_{\alpha})}\psi_{\alpha}\|^{2}\geq\frac{1}{2}$.

  2. (ii)

Since $P$ admits an exponentially localized kernel, one easily checks that
there exists a constant $K$ so that for all $a\in\mathbb{R}^{2}$:

(3)
$\|\chi_{B_{r}(a)}P\|_{\mathfrak{S}_{2}}^{2}=\int_{\mathbb{R}^{2}}\int_{\mathbb{R}^{2}}\chi_{B_{r}(a)}(\bm{x})|P(\bm{x},\bm{y})|^{2}\,\textrm{d}{\bm{x}}\,\textrm{d}{\bm{y}}\leq
Kr^{2}$

Now let $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ be an $s$-localized basis
for some $s>0$ and towards a contradiction suppose that the center points of
this basis does not have bounded density.

Since the center points for this basis do not have bounded density, we can
find a point $\bm{x}^{*}\in\mathbb{R}^{2}$ so that the ball
$B_{1}(\bm{x}^{*})$ has more than $4K(R+1)^{2}$ center points where the
constant $R$ is from Equation (2) and the constant $K$ is from Equation (3).
Let us denote the set of these center points by
$\mathcal{A}:=\\{\alpha:\bm{\mu}_{\alpha}\in B_{1}(\bm{x}^{*})\\}$.

Due to Equation (2) we have that

$\|\chi_{B_{R+1}(\bm{x}^{*})}P\|_{\mathfrak{S}_{2}}^{2}\leq K(R+1)^{2}$

but on the other hand we have that

$\displaystyle\|\chi_{B_{R+1}(\bm{x}^{*})}P\|_{\mathfrak{S}_{2}}^{2}$
$\displaystyle\geq\sum_{\alpha\in\mathcal{A}}\|\chi_{B_{R+1}(\bm{x}^{*})}\psi_{\alpha}\|^{2}\geq\frac{1}{2}\big{(}\\#\mathcal{A}\big{)}\geq
2K(R+1)^{2}$

where we have used that $\alpha\in\mathcal{A}$ implies that
$B_{R}(\bm{\mu}_{\alpha})\subseteq B_{R+1}(\bm{x}^{*})$ and Equation (2). This
is a contradiction and hence the center points of
$\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ must have bounded density. ∎

The usefulness of the notion of bounded density is that we can effectively
treat any basis with bounded density to have its center points on the integer
lattice.

######  Lemma 2.2.

Let $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ is a $s$-localized basis with
center points $\\{\bm{\mu}_{\alpha}\\}_{\alpha\in\mathcal{I}}$. If we
additionally assume that the center points have bounded density, then we may
find a positive integer $M$ so that we can relabel the basis as
$\\{\psi_{\bm{m}}^{(j)}\\}$ where $\bm{m}\in\mathbb{Z}^{2}$ and
$j\in\\{1,\cdots,M\\}$. Furthermore, the center point of $\psi_{\bm{m}}^{(j)}$
can be taken to be $\bm{m}$ without loss of generality.

###### Proof.

For each $\bm{m}\in\mathbb{Z}^{2}$ let us define the unit square centered at
$\bm{m}$ as follows

$S_{\bm{m}}:=\bigg{[}m_{1}-\frac{1}{2},m_{1}+\frac{1}{2}\bigg{)}\times\bigg{[}m_{2}-\frac{1}{2},m_{2}+\frac{1}{2}\bigg{)}.$

Since the basis $\\{\psi_{\alpha}\\}_{\alpha\in\mathcal{I}}$ has center points
with bounded density, we know that there are at most $M$ center points
contained in the square $S_{\bm{m}}$ (as it is contained in $B_{1}(\bm{m})$).
Because of this, we can relabel this basis as $\\{\psi_{\bm{m}}^{(j)}\\}$
where $\psi_{\bm{m}}^{(j)}$ has its center in $S_{\bm{m}}$ and $j$ is a
degeneracy index which runs from $\\{1,\cdots,M\\}$. If $S_{\bm{m}}$ has fewer
than $M$ center points, say it has $j^{*}$, then we define
$\psi_{\bm{m}}^{(j)}\equiv 0$ for all $j>j^{*}$. Strictly speaking this
enlarged set is no longer a basis, but it does not really matter as we are
only interested in the Chern marker, which is unchanged by this enlargement.

If $\psi_{\bm{m}}^{(j)}$ initially had center point $\bm{\mu}_{\alpha}$, by
construction $|\bm{m}-\bm{\mu}_{\alpha}|_{2}\leq\frac{\sqrt{2}}{2}$.
Therefore, using triangle inequality, it is easy to check that the collection
$\\{\psi_{\bm{m}}^{(j)}\\}$ is $s$-localized if we choose $\bm{m}$ as the
center point of $\psi_{\bm{m}}^{(j)}$ instead. ∎

Throughout our proof, we will assume that $M=1$ to simplify notation.
Considering the case $M>1$ only has the effect of introducing a multiplicative
factor of $M$ to some of our upper bounds and does not change the overall
argument or results.

###  2.2. Proof outline

As discussed in the previous section, as a consequence of bounded density
(with $M=1$), any $s$-localized basis may be written as $\\{\psi_{\bm{m}}\\}$
where $\psi_{\bm{m}}$ has its center point at $\bm{m}$. Given a fixed choice
of basis, we can now define the projector $P_{L}$ which projects onto the
basis functions centered within the box of size $L$:

(4) $P_{L}:=\sum_{|\bm{m}|_{\infty}\leq
L}|\psi_{\bm{m}}\rangle\langle\psi_{\bm{m}}|.$

Throughout the rest of this paper, we will assume that projector $P_{L}$ is
fixed and defined through a basis $\\{\psi_{\bm{m}}\\}$ which is
$(1+\delta)$-localized for some $\delta>0$.

Unlike $\chi_{L}P$ which appears in the definition of the Chern marker, the
projector $P_{L}$ has finite rank and
$\operatorname{range}{(P_{L})}\subseteq\operatorname{range}{(P)}$. In some
sense, the orthogonal projector $P_{L}$ captures the local information of $P$
in more controlled way than multiplying $P$ by the cutoff $\chi_{L}$ as in the
definition of the Chern marker. Importantly, thanks to the decay property of
the basis functions $\\{\psi_{\bm{m}}\\}$, approximating $\chi_{L}P$ with
$P_{L}$ incurs an error which is subleading compared to the area of
$\chi_{L}$:

######  Proposition 2.3.

Suppose that $P$ admits a $(1+\delta)$-localized basis where $\delta>0$. There
exists a constant $C$ such that for all $L\geq 1$:

$\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}\leq CL^{2/3}$

###### Proof.

Proven in Section 4. ∎

As a consequence of this proposition, we can show that replacing $\chi_{L}P$
with $P_{L}$ in the definition of the Chern marker does not change the overall
limit:

######  Proposition 2.4.

If $P$ admits a $(1+\delta)$-localized generalized Wannier basis where
$\delta>0$ then

(5)
$\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\left\|\chi_{L}P\Big{[}[X,P],[Y,P]\Big{]}P\chi_{L}-P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\right\|_{\mathfrak{S}_{1}}=0.$

Hence

(6) $\lim_{L\rightarrow\infty}\frac{2\pi
i}{4L^{2}}\operatorname{tr}{\left(\chi_{L}P\Big{[}[X,P],[Y,P]\Big{]}P\chi_{L}\right)}=\lim_{L\rightarrow\infty}\frac{2\pi
i}{4L^{2}}\operatorname{tr}{\left(P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\right)}$

whenever at least one of the above limits exists.

###### Proof.

Proven in Section 5 ∎

Hence to prove Theorem 1 it suffices to show that if $P$ admits an
$(1+\delta)$-localized generalized Wannier basis then

(7) $\lim_{L\rightarrow\infty}\frac{2\pi
i}{4L^{2}}\operatorname{tr}{\left(P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\right)}=0.$

Towards proving Equation (7), we begin by observing that since $P_{L}$ is
defined through a $(1+\delta)$-localized basis, the position operator $X$ is a
bounded operator on $\operatorname{range}{(P_{L})}$ for each $L$. In
particular, we have that

$\displaystyle\|XP_{L}\|^{2}$ $\displaystyle\leq\sum_{|\bm{m}|_{\infty}\leq
L}\|X\psi_{\bm{m}}\|^{2}$ $\displaystyle\leq\sum_{|\bm{m}|_{\infty}\leq
L}\Big{(}\|(X-m_{1})\psi_{\bm{m}}\|+|m_{1}|\|\psi_{\bm{m}}\|\Big{)}^{2}$
$\displaystyle\leq\sum_{|\bm{m}|_{\infty}\leq
L}\Big{(}\|(X-m_{1})\psi_{\bm{m}}\|+L\Big{)}^{2}$ $\displaystyle\lesssim
L^{4}$

Similarly, it is easily checked that $Y$ is also a bounded operator on
$\operatorname{range}{(P_{L})}$.

We will now use the fact that $X$ and $Y$ are both bounded operators on
$\operatorname{range}{(P_{L})}$ to perform some algebraic manipulations. Using
the fact that $P^{2}=P$ and $[X,Y]=0$, one can verify that (see also [5, 10])

$P\Big{[}[X,P],[Y,P]\Big{]}P=[PXP,PYP].$

Therefore, since $P_{L}=P_{L}P=PP_{L}$, we have the following:

$\displaystyle P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}$
$\displaystyle=P_{L}[PXP,PYP]P_{L}$
$\displaystyle=P_{L}XPYP_{L}-P_{L}YPXP_{L}$
$\displaystyle=P_{L}X(P-P_{L}+P_{L})YP_{L}-P_{L}Y(P-P_{L}+P_{L})XP_{L}$
$\displaystyle=[P_{L}XP_{L},P_{L}YP_{L}]+P_{L}X(P-P_{L})YP_{L}-P_{L}Y(P-P_{L})XP_{L}.$

These manipulations are justified since $X$ and $Y$ are bounded operators on
$\operatorname{range}{(P_{L})}$. Since $P_{L}$ is finite rank,
$[P_{L}XP_{L},P_{L}YP_{L}]$ is traceless and hence

(8)
$\operatorname{tr}\left(P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\right)=\operatorname{tr}\Bigl{(}P_{L}X(P-P_{L})YP_{L}-P_{L}Y(P-P_{L})XP_{L}\Bigr{)}.$

Hence using Hölder’s inequality and $(P-P_{L})=(P-P_{L})^{2}$, we have that

$\displaystyle|\operatorname{tr}\left(P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\right)|$
$\displaystyle\leq\|P_{L}X(P-P_{L})YP_{L}\|_{\mathfrak{S}_{1}}+\|P_{L}Y(P-P_{L})XP_{L}\|_{\mathfrak{S}_{1}}$
(9) $\displaystyle\leq
2\|(P-P_{L})XP_{L}\|_{\mathfrak{S}_{2}}\|(P-P_{L})YP_{L}\|_{\mathfrak{S}_{2}}.{}$

######  Proposition 2.5.

If $P$ admits a $(1+\delta)$-localized generalized Wannier basis where
$\delta>0$ then

(10)
$\displaystyle\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\|(P-P_{L})XP_{L}\|_{\mathfrak{S}_{2}}^{2}=0$
(11)
$\displaystyle\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\|(P-P_{L})YP_{L}\|_{\mathfrak{S}_{2}}^{2}=0.$

Since the mapping $x\mapsto\sqrt{x}$ is continuous for $x>0$, Proposition 2.5
and Equation (9) imply that

$\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\Bigl{\lvert}\operatorname{tr}\Bigl{(}P_{L}\Big{[}[X,P],[Y,P]\Big{]}P_{L}\Bigr{)}\Bigr{\rvert}=0$

which proves Equation (7), completing the proof of Theorem 1.

##  3\. Technical Estimates

In this section we prove two technical estimates (Proposition 3.1 and
Proposition 3.2) which are fundamental in our proofs of Proposition 2.3 and
Proposition 2.5.

######  Proposition 3.1.

If $P$ admits a $(1+\delta)$-localized generalized Wannier basis then for all
$a,b\geq 1$:

$\|(1-\chi_{a+b})P_{a}\|_{\mathfrak{S}_{2}}^{2}\lesssim a^{2}b^{-2(1+\delta)}$

###### Proof.

We can expand the Hilbert-Schmidt norm we want to bound as follows:

(12)
$\|(1-\chi_{a+b})P_{a}\|_{\mathfrak{S}_{2}}^{2}=\sum_{|\bm{m}|_{\infty}\leq
a}\|(1-\chi_{a+b})\psi_{\bm{m}}\|^{2}.$

Because of the separation between the sets
$\\{\bm{m}\in\mathbb{Z}^{2}:|\bm{m}|_{\infty}\leq a\\}$ and
$\operatorname{supp}{(1-\chi_{a+b})}$ we can show that each of the terms in
the above sum are small. In particular

$\displaystyle\|(1-\chi_{a+b})\psi_{\bm{m}}\|^{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}(1-\chi_{a+b}(\bm{x}))|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle=\int_{\mathbb{R}^{2}}(1-\chi_{a+b}(\bm{x}))\frac{(1+|x_{1}-m_{1}|+|x_{2}-m_{2}|)^{2(1+\delta)}}{(1+|x_{1}-m_{1}|+|x_{2}-m_{2}|)^{2(1+\delta)}}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$

Since $|\bm{m}|_{\infty}\leq a$ we have the pointwise bound

$\frac{(1-\chi_{a+b}(\bm{x}))}{(1+|x_{1}-m_{1}|+|x_{2}-m_{2}|)}\leq\frac{1}{1+(a+b)-|\bm{m}|_{\infty}}\leq\frac{1}{1+b}.$

Therefore, for each since $\psi_{\bm{m}}$ is $(1+\delta)$-localized we have
that:

$\displaystyle\|(1-$ $\displaystyle\chi_{a+b})\psi_{\bm{m}}\|^{2}$
$\displaystyle\leq(1+b)^{-2(1+\delta)}\int_{\mathbb{R}^{2}}(1+|x_{1}-m_{1}|+|x_{2}-m_{2}|)^{2(1+\delta)}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle\leq Cb^{-2(1+\delta)}$

for some absolute constant $C$. Using this bound in Equation (12), we conclude
that

$\displaystyle\|(1-\chi_{a+b})P_{a}\|_{\mathfrak{S}_{2}}^{2}$
$\displaystyle\leq\sum_{|\bm{m}|_{\infty}\leq a}Cb^{-2(1+\delta)}$
$\displaystyle\lesssim a^{2}b^{-2(1+\delta)}$

which completes the proof. ∎

######  Proposition 3.2.

If $P$ admits a $(1+\delta)$-localized generalized Wannier basis then for all
$a,b\geq 1$:

$\|\chi_{a}(P-P_{a+b})\|_{\mathfrak{S}_{2}}^{2}\lesssim
b^{-\delta}+ab^{-(1+\delta)}$

We start by stating a lemma which we prove at the end of the section.

######  Lemma 3.3.

Suppose that $\\{\psi_{\bm{m}}\\}$ is a $(1+\delta)$-localized basis. For any
$a\geq 1$ we have the following bounds depending on the location of $\bm{m}$
in relation to $\operatorname{supp}{(\chi_{a})}$:

  1. (i)

If $|m_{1}|>a$ and $|m_{2}|>a$ then

$\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim\langle|m_{1}|-a\rangle^{-(1+\delta)}\langle|m_{2}|-a\rangle^{-(1+\delta)}.$

  2. (ii)

If $|m_{1}|>a$ and $|m_{2}|\leq a$ then

$\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim\langle|m_{1}|-a\rangle^{-2(1+\delta)}$

  3. (iii)

If $|m_{1}|\leq a$ and $|m_{2}|>a$ then

$\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim\langle|m_{2}|-a\rangle^{-2(1+\delta)}$

With this lemma in hand, we can now prove Proposition 3.2.

###### Proof of Proposition 3.2.

By the properties of the Hilbert-Schmidt norm we see that

$\displaystyle\|\chi_{a}(P-P_{a+b})\|_{\mathfrak{S}_{2}}^{2}$
$\displaystyle=\sum_{\|\bm{m}\|>a+b}\|\chi_{a}\psi_{\bm{m}}\|^{2}$

We now split the set $\\{\bm{m}\in\mathbb{Z}^{2}:|\bm{m}|_{\infty}>a+b\\}$
into three parts and bound each part separately

$\begin{split}S_{1}:=\Big{\\{}\bm{m}:|m_{1}|>a+b\text{ and
}|m_{2}|>a+b\Big{\\}}\\\\[4.30554pt] S_{2}:=\Big{\\{}\bm{m}:|m_{1}|>a+b\text{
and }|m_{2}|\leq a+b\Big{\\}}\\\\[4.30554pt]
S_{3}:=\Big{\\{}\bm{m}:|m_{1}|\leq a+b\text{ and }|m_{2}|>a+b\Big{\\}}\\\
\end{split}$

We start with controlling $S_{1}$, by applying Lemma 3.3(1) we have that

$\displaystyle\sum_{\bm{m}\in S_{1}}\|\chi_{a}\psi_{\bm{m}}\|^{2}$
$\displaystyle\leq\sum_{\bm{m}\in
S_{1}}\frac{C}{\langle|m_{1}|-a\rangle^{-(1+\delta)}\langle|m_{2}|-a\rangle^{-(1+\delta)}}$
$\displaystyle\leq Cb^{-\delta}\sum_{\bm{m}\in
S_{1}}\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta/2)}\langle|m_{2}|-a\rangle^{(1+\delta/2)}}$
$\displaystyle\leq
Cb^{-\delta}\sum_{\bm{m}\in\mathbb{Z}^{2}}\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta/2)}\langle|m_{2}|-a\rangle^{(1+\delta/2)}}$

where in the second to last line we have used that since $\bm{m}\in S_{1}$,
$\min\\{\langle|m_{1}|-a\rangle,\langle|m_{2}|-a\rangle\\}>b$. Therefore,

$\sum_{\bm{m}\in S_{1}}\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim b^{-\delta}$

We now turn to bound the sum for $\bm{m}\in S_{2}$. Applying Lemma 3.3(2) we
have that there exists a constant $C$ such that

$\displaystyle\sum_{\bm{m}\in S_{2}}\|\chi_{a}\psi_{\bm{m}}\|^{2}$
$\displaystyle\leq\,\sum_{|m_{1}|>a+b}\sum_{|m_{2}|\leq
a+b}\frac{C}{\langle|m_{1}|-a\rangle^{2(1+\delta)}}$ $\displaystyle\leq
Cb^{-(1+\delta)}\sum_{|m_{1}|>a+b}\sum_{|m_{2}|\leq
a+b}\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta)}}$ $\displaystyle\leq
2C(a+b)b^{-(1+\delta)}\sum_{m_{1}\in\mathbb{Z}}\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta)}}$

where in the second to last line we have used that
$\langle|m_{1}|-a\rangle>b$. Therefore,

$\sum_{\bm{m}\in
S_{2}}\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim(a+b)b^{-(1+\delta)}$

Repeating the same calculation for $S_{3}$ making the obvious changes we have
that

$\sum_{\bm{m}\in
S_{3}}\|\chi_{a}\psi_{\bm{m}}\|^{2}\lesssim(a+b)b^{-(1+\delta)}$

Hence

$\|\chi_{a}(P-P_{a+b})\|_{\mathfrak{S}_{2}}^{2}\leq
C_{1}b^{-\delta}+C_{2}(a+b)b^{-(1+\delta)}+C_{3}(a+b)b^{-(1+\delta)}$

which proves the result. ∎

It remains to prove Lemma 3.3 to finish the proof.

###### Proof of Lemma 3.3.

We will focus on the case when $|m_{1}|>a$ and $|m_{2}|>a$ and note the
changes which must be made for the other cases. For these estimates, we will
introduce the strip characteristic functions $\chi^{\text{strip},X}_{D}$ and
$\chi^{\text{strip},Y}_{D}$ defined as follows

$\chi^{\text{strip},X}_{D}(\bm{x})=\begin{cases}1&|x_{1}|\leq D\\\
0&\text{otherwise}\end{cases}\qquad\qquad\chi^{\text{strip},Y}_{D}(\bm{x})=\begin{cases}1&|x_{2}|\leq
D\\\ 0&\text{otherwise}\end{cases}$

Next, let us define the distances $D_{x}:=|m_{1}|-a$ and $D_{y}:=|m_{2}|-a$.
With these definitions, it is clear that up to a set of measure zero:

$\chi^{\text{strip},X}_{D_{x}}(\bm{x}-\bm{m})\chi_{a}(\bm{x})=0\quad\text{and}\quad\chi^{\text{strip},Y}_{D_{y}}(\bm{x}-\bm{m})\chi_{a}(\bm{x})=0$

Therefore,

$\displaystyle\|\chi_{L}\psi_{\bm{m}}\|^{2}$
$\displaystyle=\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle=\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\Big{(}1-\chi^{\text{strip},X}_{D_{x}}(\bm{x}-\bm{m})\Big{)}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle=\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\Big{(}1-\chi^{\text{strip},X}_{D_{x}}(\bm{x}-\bm{m})\Big{)}\frac{\langle
x_{1}-m_{1}\rangle^{(1+\delta)}}{\langle
x_{1}-m_{1}\rangle^{(1+\delta)}}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$

By definition of $\chi^{\text{strip},X}_{D_{x}}$ we have the pointwise bound:

$\frac{1-\chi^{\text{strip},X}_{D_{x}}(\bm{x}-\bm{m})}{\langle
x_{1}-m_{1}\rangle}\leq\frac{1}{\langle|m_{1}|-a\rangle}$

Therefore,

$\|\chi_{L}\psi_{\bm{m}}\|^{2}\leq\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta)}}\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\langle
x_{1}-m_{1}\rangle^{(1+\delta)}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}.$

By similar logic

$\displaystyle\int_{\mathbb{R}^{2}}$ $\displaystyle\chi_{a}(\bm{x})\langle
x_{1}-m_{1}\rangle|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle=\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\Big{(}1-\chi^{\text{strip},Y}_{D_{y}}(\bm{x}-\bm{m})\Big{)}\frac{\langle
x_{2}-m_{2}\rangle^{(1+\delta)}}{\langle
x_{2}-m_{2}\rangle^{(1+\delta)}}\langle
x_{1}-m_{1}\rangle|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}$
$\displaystyle\leq\frac{1}{\langle|m_{2}|-a\rangle^{(1+\delta)}}\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\langle
x_{1}-m_{1}\rangle^{(1+\delta)}\langle
x_{2}-m_{2}\rangle^{(1+\delta)}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}.$

Hence

$\|\chi_{a}\psi_{\bm{m}}\|^{2}\leq\frac{1}{\langle|m_{1}|-a\rangle^{(1+\delta)}\langle|m_{2}|-a\rangle^{(1+\delta)}}\int_{\mathbb{R}^{2}}\chi_{a}(\bm{x})\langle
x_{1}-m_{1}\rangle^{(1+\delta)}\langle
x_{2}-m_{2}\rangle^{(1+\delta)}|\psi_{\bm{m}}(\bm{x})|^{2}\,\textrm{d}{\bm{x}}.$

Now recall that the geometric mean is bounded by the arithmetic mean so

$\langle x_{1}-m_{1}\rangle^{(1+\delta)}\langle
x_{2}-m_{2}\rangle^{(1+\delta)}\leq\frac{1}{2}\Big{(}\langle
x_{1}-m_{1}\rangle^{2(1+\delta)}+\langle
x_{2}-m_{2}\rangle^{2(1+\delta)}\Big{)}$

Therefore,

$\|\chi_{a}\psi_{\bm{m}}\|^{2}\leq\frac{\|\langle
X-m_{1}\rangle^{(1+\delta)}\psi_{\bm{m}}\|^{2}+\|\langle
Y-m_{2}\rangle^{(1+\delta)}\psi_{\bm{m}}\|^{2}}{2\langle|m_{1}|-a\rangle^{(1+\delta)}\langle|m_{1}|-a\rangle^{(1+\delta)}}.$

which implies the result since $\psi_{\bm{m}}$ is $(1+\delta)$-localized.

The case $|m_{1}|>a$ and $|m_{2}|\leq a$ follows by inserting $\langle
x_{1}-m_{1}\rangle^{2(1+\delta)}\langle x_{1}-m_{1}\rangle^{-2(1+\delta)}$
instead of $\langle x_{1}-m_{1}\rangle^{(1+\delta)}\langle
x_{2}-m_{2}\rangle^{(1+\delta)}\langle x_{1}-m_{1}\rangle^{-(1+\delta)}\langle
x_{2}-m_{2}\rangle^{-(1+\delta)}$; the case $|m_{1}|\leq a$ and $|m_{2}|>a$
follows similarly. ∎

##  4\. Proof of Proposition 2.3

Let us start by fixing some $\ell$ where $\ell\in[1,L)$ to be chosen later. We
can split the quantity we would like to bound into four parts:

$\displaystyle\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}$
$\displaystyle\leq\|\chi_{L}(P-P_{L})\|_{\mathfrak{S}_{2}}+\|(1-\chi_{L})P_{L}\|_{\mathfrak{S}_{2}}$
$\displaystyle\leq\|\chi_{L}(P-P_{L+\ell})\|_{\mathfrak{S}_{2}}+\|\chi_{L}(P_{L+\ell}-P_{L})\|_{\mathfrak{S}_{2}}$
$\displaystyle\hskip
21.52771pt+\|(1-\chi_{L})(P_{L}-P_{L-\ell})\|_{\mathfrak{S}_{2}}+\|(1-\chi_{L})P_{L-\ell}\|_{\mathfrak{S}_{2}}$

The first term is bounded by Proposition 3.2 by letting $a=L$, $b=\ell$ there
exists a constant $C_{1}$ so that:

(13) $\|\chi_{L}(P-P_{L+\ell})\|_{\mathfrak{S}_{2}}\leq
C_{1}(\ell^{-\delta}+L\ell^{-(1+\delta)})^{1/2}.$

The next two terms are bounded by observing that

$\begin{split}&\operatorname{rank}{(P_{L+\ell}-P_{L})}\leq
4((L+\ell)^{2}-L^{2})\leq 12L\ell\\\
&\operatorname{rank}{(P_{L}-P_{L-\ell})}\leq 4(L^{2}-(L-\ell)^{2})\leq
12L\ell\end{split}$

where we have used that $\ell<L$. Hence, there exists a constant $C_{2}$ so
that

(14)
$\|\chi_{L}(P_{L+\ell}-P_{L})\|_{\mathfrak{S}_{2}}+\|(1-\chi_{L})(P_{L}-P_{L-\ell})\|_{\mathfrak{S}_{2}}\leq
C_{2}(L\ell)^{1/2}.$

As for the final term, we can apply Proposition 3.1 with $a=L-\ell$, $b=\ell$
to conclude that there exists a constant $C_{3}$ so that

(15) $\|(1-\chi_{L})P_{L-\ell}\|_{\mathfrak{S}_{2}}\leq
C_{3}(L^{2}\ell^{-2(1+\delta)})^{1/2}$

Combining the bounds in Equations (13), (14), (15), we have that

$\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}\leq
C_{1}(\ell^{-\delta}+L\ell^{-(1+\delta)})^{1/2}+C_{2}(L\ell)^{1/2}+C_{3}(L^{2}\ell^{-2(1+\delta)})^{1/2}.$

Now in the above equation we have four different terms each which have
different big–O as $L\rightarrow\infty$:

$O(\ell^{-\delta})\qquad O(L\ell^{-(1+\delta)})\qquad O(L\ell)\qquad
O(L^{2}\ell^{-2(1+\delta)})$

Since $\ell>1$, it’s clear that the two dominating terms are $O(L\ell)$ and
$O(L^{2}\ell^{-2(1+\delta)})$. Since we are free to choose $\ell$, we will
make a choice of $\ell$ so that these two terms balance. A simple calculation
shows that choosing $\ell=L^{1/(3+2\delta)}$ gives

$\begin{split}&L\ell=L^{2(2+\delta)/(3+2\delta)}\\\
&L^{2}\ell^{-2(1+\delta)}=L^{2(2+\delta)/(3+2\delta)}\\\
&L\ell^{-(1+\delta)}=L^{(2+\delta)/(3+2\delta)}\end{split}$

This valid choice for $\ell$ since $\delta>0$ so $\frac{1}{3+2\delta}<1$. With
this choice of $\ell$, we have that

$\displaystyle\|\chi_{L}(P-P_{L})\|_{2}\leq
C_{1}(L^{-\delta/(3+2\delta)}+2L^{(2+\delta)/(3+2\delta)})^{1/2}+C_{2}L^{(2+\delta)/(3+2\delta)}+C_{3}L^{(2+\delta)/(3+2\delta)}$

Hence, for $L\geq 1$

$\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}\lesssim L^{(2+\delta)/(3+2\delta)}.$

The proof is completed by observing that for all $\delta\geq 0$:

$\frac{2+\delta}{3+2\delta}\leq\frac{2}{3}.$

##  5\. Proof of Proposition 2.4

For this proof, let us abbreviate the commutator in the definition of the
Chern marker as $\mathcal{C}$, that is:

$\mathcal{C}:=\Big{[}[X,P],[Y,P]\Big{]}.$

With this notation, we have that:

$\displaystyle\chi_{L}P\mathcal{C}P\chi_{L}-P_{L}\mathcal{C}P_{L}$
$\displaystyle=(\chi_{L}P-P_{L})\mathcal{C}P\chi_{L}+P_{L}\mathcal{C}(P\chi_{L}-P_{L})$

Applying Hölder’s inequality to the trace norm we want to bound, we have that

$\displaystyle\|\chi_{L}P$
$\displaystyle\mathcal{C}P\chi_{L}-P_{L}\mathcal{C}P_{L}\|_{\mathfrak{S}_{1}}$
$\displaystyle\leq\|(\chi_{L}P-P_{L})\mathcal{C}P\chi_{L}\|_{\mathfrak{S}_{1}}+\|P_{L}\mathcal{C}(P\chi_{L}-P_{L})\|_{\mathfrak{S}_{1}}$
$\displaystyle\leq\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}\|\mathcal{C}\|_{\mathfrak{S}_{\infty}}\|P\chi_{L}\|_{\mathfrak{S}_{2}}+\|P_{L}\|_{\mathfrak{S}_{2}}\|\mathcal{C}\|_{\mathfrak{S}_{\infty}}\|P\chi_{L}-P_{L}\|_{\mathfrak{S}_{2}}.$

The right hand side can be upper bounded by observing that

  1. (i)

Since $P$ admits an exponentially localized kernel,

$\|\mathcal{C}\|_{\mathfrak{S}_{\infty}}=\|\mathcal{C}\|=\|[[X,P],[Y,P]]\|\leq
2\|[X,P]\|\|[Y,P]\|\lesssim 1.$

  2. (ii)

Additionally, since $P$ admits an exponentially localized kernel, one easily
checks that

$\|P\chi_{L}\|_{\mathfrak{S}_{2}}\lesssim L.$

  3. (iii)

Since $\operatorname{rank}{(P_{L})}\leq 4L^{2}$ and $\|P_{L}\|\leq 1$ we have
that

$\|P_{L}\|_{\mathfrak{S}_{2}}\leq 2L.$

  4. (iv)

Proposition 2.3 implies that

$\|\chi_{L}P-P_{L}\|_{\mathfrak{S}_{2}}\lesssim L^{2/3}$

Combining these four bounds, we conclude that

$\|\chi_{L}P\mathcal{C}P\chi_{L}-P_{L}\mathcal{C}P_{L}\|_{\mathfrak{S}_{1}}\lesssim
L^{5/3}$

Hence

$\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\|\chi_{L}P\mathcal{C}P\chi_{L}-P_{L}\mathcal{C}P_{L}\|_{\mathfrak{S}_{1}}=0$

and the proposition is proved.

##  6\. Proof of Proposition 2.5

Our main goal in this section is to show that the following quantity is
$o(L^{2})$:

$\|(P-P_{L})XP_{L}\|_{\mathfrak{S}_{2}}^{2}$

The corresponding bound for $Y$ follows by an analogous argument.

Similar to the proof of Proposition 2.4, our first step will be to introduce a
length parameter $\ell\in[1,\frac{1}{2}L)$ to be fixed later. For any such
choice of $\ell$, by the properties of the Hilbert-Schmidt norm we have that:

(16)
$\|(P-P_{L})XP_{L}\|_{\mathfrak{S}_{2}}^{2}=\|(P-P_{L})XP_{L-2\ell}\|_{\mathfrak{S}_{2}}^{2}+\|(P-P_{L})X(P_{L}-P_{L-2\ell})\|_{\mathfrak{S}_{2}}^{2}$

The second of these terms can be shown to be $O(L\ell)$ using that
$(P-P_{L})(P_{L}-P_{L-2\ell})=0$. In particular:

$\displaystyle\|(P-P_{L})X(P_{L}-P_{L-\ell})\|_{\mathfrak{S}_{2}}^{2}$
$\displaystyle=\sum_{L-2\ell<|\bm{m}|_{\infty}\leq
L}\|(P-P_{L})X\psi_{\bm{m}}\|^{2}$
$\displaystyle=\sum_{L-2\ell<|\bm{m}|_{\infty}\leq
L}\|(P-P_{L})(X-m_{1})\psi_{\bm{m}}\|^{2}$
$\displaystyle\leq\left(\sup_{\bm{m}}\|(X-m_{1})\psi_{\bm{m}}\|^{2}\right)\sum_{L-2\ell<|\bm{m}|_{\infty}\leq
L}1$ $\displaystyle\lesssim L\ell.$

Therefore, this term is $o(L^{2})$ so long as we choose $\ell=o(L)$.

Returning to the first term in Equation (16), using that
$(P-P_{L})P_{L-2\ell}=0$ we have that

$\displaystyle\|(P-P_{L})XP_{L-2\ell}\|_{\mathfrak{S}_{2}}^{2}$
$\displaystyle=\sum_{|\bm{m}|_{\infty}\leq
L-2\ell}\|(P-P_{L})(X-m_{1})\psi_{\bm{m}}\|^{2}$

We will now show that each of the terms in the above sum are small using
Proposition 3.2. In particular, we have the following easy lemma

######  Lemma 6.1.

If $\psi_{\bm{m}}$ is $(1+\delta)$-localized with center point $\bm{m}$ where
$|\bm{m}|_{\infty}\leq L-2\ell$, then

$\|(P-P_{L})(X-m_{1})\psi_{\bm{m}}\|\lesssim\ell^{-\delta}+L\ell^{-(1+\delta)}$

###### Proof.

We start by inserting $\chi_{L-\ell}+(1-\chi_{L-\ell})$ and applying triangle
inequality

$\displaystyle\|(P-P_{L})$ $\displaystyle(X-m_{1})\psi_{\bm{m}}\|$
$\displaystyle\leq\|(P-P_{L})\chi_{L-\ell}(X-m_{1})\psi_{\bm{m}}\|+\|(P-P_{L})(1-\chi_{L-\ell})(X-m_{1})\psi_{\bm{m}}\|$

By Proposition 3.2 we have that

$\|(P-P_{L})\chi_{L-\ell}(X-m_{1})\psi_{\bm{m}}\|\leq\|(P-P_{L})\chi_{L-\ell}\|\|(X-m_{1})\psi_{\bm{m}}\|\lesssim\ell^{-\delta}+L\ell^{-(1+\delta)}.$

By repeating a similar argument as used in the proof of Proposition 3.1 it is
easily verified that

$\displaystyle\|(P-P_{L})(1-\chi_{L-\ell})(X-m_{1})\psi_{\bm{m}}\|\leq\|(1-\chi_{L-\ell})(X-m_{1})\psi_{\bm{m}}\|\lesssim\ell^{-\delta}$

which proves the lemma. ∎

Using Lemma 6.1 and it follows that:

$\|(P-P_{L})XP_{L-\ell}\|_{\mathfrak{S}_{2}}^{2}\lesssim
L^{2}(\ell^{-\delta}+L\ell^{-(1+\delta)})^{2}.$

For our final step, we will choose $\ell=L^{2/(2+\delta)}$. Since $\delta>0$,
note that this choice of $\ell$ is $o(L)$ consistent with our previous
requirement. Furthermore, we have that

$\|(P-P_{L})XP_{L-\ell}\|_{\mathfrak{S}_{2}}^{2}\lesssim
L^{2}(L^{-2\delta/(2+\delta)}+L^{-\delta/(2+\delta)})^{2}.$

Since $\delta>0$, we conclude that

$\lim_{L\rightarrow\infty}\frac{1}{L^{2}}\|(P-P_{L})XP_{L-\ell}\|_{\mathfrak{S}_{2}}^{2}=0$

which completes the proof.

## References

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# Superconducting fluctuations observed far above $\text{T}_{\text{c}}$
in the isotropic superconductor $\text{K}_{3}\text{C}_{60}$

Gregor Jotzu<EMAIL_ADDRESS>Guido Meier  Alice Cantaluppi  Max
Planck Institute for the Structure and Dynamics of Matter, Hamburg, Germany
Andrea Cavalleri  Max Planck Institute for the Structure and Dynamics of
Matter, Hamburg, Germany  Department of Physics, Clarendon Laboratory,
University of Oxford, Oxford, UK  Daniele Pontiroli  Mauro Riccò  Dipartimento
di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di
Parma, Parma, Italy  Arzhang Ardavan<EMAIL_ADDRESS>Moon-
Sun Nam<EMAIL_ADDRESS>Department of Physics, Clarendon
Laboratory, University of Oxford, Oxford, UK

###### Abstract

Alkali-doped fullerides are strongly correlated organic superconductors that
exhibit high transition temperatures, exceptionally large critical magnetic
fields and a number of other unusual properties. The proximity to a Mott
insulating phase is thought to be a crucial ingredient of the underlying
physics, and may also affect precursors of superconductivity in the normal
state above ${T_{c}}$. We report on the observation of a sizeable magneto-
thermoelectric (Nernst) effect in the normal state of
$\text{K}_{3}\text{C}_{60}$, which displays the characteristics of
superconducting fluctuations. The anomalous Nernst effect emerges from an
ordinary quasiparticle background below a temperature of 80K, far above
${T_{c}=20\text{K}}$. At the lowest fields and close to ${T_{c}}$, the scaling
of the effect is captured by a model based on Gaussian fluctuations. The
temperature up to which we observe fluctuations is exceptionally high for a
three-dimensional isotropic system, where fluctuation effects are usually
suppressed.

Recent work suggests that the exceptional properties of alkali-doped fulleride
superconductors, $\text{A}_{3}\text{C}_{60}$, result from an unusual
cooperation between electron-phonon and electron-electron coupling [1, 2, 3].
The former is primarily governed by a dynamical Jahn-Teller distortion of the
$\text{C}_{60}$ molecules leading to an inverted Hund’s coupling between
electrons
[[PhononeffectsmaynotnecessarilyberequiredforaninvertedHund'scoupling,
see][]jiang2016], whilst the latter contributes to a suppression of the
effective bandwidth. With increasing lattice spacing, superconductivity in
$\text{A}_{3}\text{C}_{60}$ acquires a “dome-like” $T_{c}$, eventually
evolving into a Mott insulator with an antiferromagnetic ground state [5, 6].
Unlike other high-temperature superconductors however,
$\text{A}_{3}\text{C}_{60}$ features no anisotropy, and displays the
characteristics of an s-wave superconductor.

$\text{A}_{3}\text{C}_{60}$ also seems to follow the Uemura relation [7, 8],
with a transition temperature $T_{c}$ proportional to the superfluid density,
suggesting that the loss of long-range phase coherence may be responsible for
the disappearance of superconductivity at $T_{c}$. Yet, due to the large
spread of experimentally determined superfluid densities in
$\text{A}_{3}\text{C}_{60}$ [9, 10], some uncertainty remains on this
assignment.

Figure 1:  Probing the Nernst effect in $\text{K}_{3}\text{C}_{60}$. a, The
f.c.c. lattice structure of $\text{K}_{3}\text{C}_{60}$, with potassium atoms
shown in grey and carbon buckyballs in green. b, Schematic of the measurement
configuration. A temperature gradient $\nabla T$ is applied orthogonal to the
external magnetic field, $B$. The voltage, $V$, is then measured orthogonal to
both. Cartoons of various possible contributions to the Nernst signal, such as
quasiparticles (left), short-lived Cooper pairs (centre) and mobile vortices
(right), are included.

In $\text{K}_{3}\text{C}_{60}$ (see Fig. 1a), observables such as the specific
heat and the pressure dependence of $T_{c}$ suggest that the material may be
well described by weak-coupling BCS theory [9, 10], but discrepancies in the
size and temperature dependence of the superconducting gap remain [11]. Very
recent measurements in few-layer thin films of $\text{K}_{3}\text{C}_{60}$
have reported the appearance of a pseudogap up to about twice $T_{c}$ [12].

Finally, upon illumination with mid-infrared laser pulses, optical properties
compatible with superconductivity have been observed in
$\text{K}_{3}\text{C}_{60}$ at temperatures that exceed $T_{c}$ by an order of
magnitude [13, 14, 15, 16], further underscoring a highly unusual normal
state. One of the proposed mechanisms for these phenomena suggests that the
effect of the light field consists in synchronizing pre-existing, but phase-
incoherent, Cooper pairs [15, 17], inspired by the empirical correlation
between materials where light-induced high temperature superconductivity can
be induced and an anomalous Nernst effect above $T_{c}$ exists [17, 18, 19].
Recent experiments have also provided suggestive magnetic anomalies when
$\text{Rb}_{3}\text{C}_{60}$ is made to interact with electromagnetic vacuum
modes in an optical cavity [20].

The Nernst effect describes the appearance of an electric field,
$E_{y}=-e_{\mathrm{N}}\partial_{x}T$, transverse to an applied temperature
gradient, $\partial_{x}T$, and to a magnetic field $B_{z}$ pointing along the
third spatial direction. The Nernst signal $e_{\mathrm{N}}$ is related to the
conductivity and thermoelectric tensors, $\bm{\sigma}$ and $\bm{\alpha}$ via

$e_{\mathrm{N}}=\frac{\alpha_{xy}\sigma_{xx}-\alpha_{xx}\sigma_{xy}}{\sigma_{xx}^{2}+\sigma_{xy}^{2}}\approx\frac{\alpha_{xy}}{\sigma_{xx}}-S\mu_{\mathrm{H}}B_{z}$
(1)

for an isotropic system. $\mu_{\mathrm{H}}=\sigma_{xy}/\sigma_{xx}B_{z}$
denotes the Hall mobility and $S=\alpha_{xx}/\sigma_{xx}$ the Seebeck
coefficient. The approximate equality holds for small Hall angles, and is an
excellent approximation for the parameters used in this work.

For a metal, the effect can be seen as a combination of a flow of charges
carrying entropy along a temperature gradient, the Seebeck effect, and the
deflection of moving charges in the presence of a magnetic field, the Hall
effect (see Fig. 1b). However, with exact particle-hole symmetry, the two
terms in Eq. 1 would cancel exactly [21]. The overall sign and amplitude of
the effect will depend on the details of the quasiparticle band structure,
which the Nernst signal is therefore very sensitive to. In the free-electron
approximation, this signal is linear in temperature.

In a superconductor, a different contribution to the Nernst effect arises from
the movement of superconducting vortices. When these mobile vortices carry
entropy along the applied temperature gradient (see Fig. 1b), they also carry
magnetic flux, which induces a voltage in the transverse direction [22,
[AnadditionalcontributionfromtheMagnusforceisalsopresent, see][]krasnov1997].
This effect is much larger than its metallic counterpart, and shows a highly
non-linear dependence on $B_{z}$, perhaps owing to a competition between
vortex density and vortex mobility.

If, at $T_{c}$, superconducting long-range order breaks down because of
fluctuations of the phase of the order parameter, while a finite Cooper pair
amplitude remains, this vortex Nernst effect would be expected to survive at
temperatures above $T_{c}$. However, even for a transition driven by a thermal
breakdown of Cooper pairing, the thermal diffusion of short-lived Cooper pairs
may also contribute to the Nernst signal above $T_{c}$ [24, 25, 22].

In general, the presence of such precursors of superconductivity is expected
to be suppressed as the dimensionality of the system increases. Indeed, Nernst
signals that could be related to superconducting precursors have been reported
in layered and thin-film samples [26, 25, 18, 27, 22], but we are not aware of
any previous observations in an isotropic three-dimensional system.
Interestingly, $\text{K}_{3}\text{C}_{60}$ and $\text{Rb}_{3}\text{C}_{60}$
were the first fully three-dimensional materials where slight deviations of
the normal state conductivity in the immediate vicinity of $T_{c}$ could be
attributed to paraconductivity [28].

Figure 2:  Nernst signal in the superconducting regime. a, Measured transverse
thermoelectric effect, $e_{\mathrm{N}}$, in the presence of magnetic fields
ranging from 1T (grey line) to 15T (blue line), in steps of 1T. The red dashed
line shows the critical temperature in zero field. Inset: Sample resistance
near the critical temperature, 0T (black) to 15T (blue) range. b, Contour plot
of $e_{\mathrm{N}}$ as a function of temperature and magnetic field. The black
dotted line shows the resistive $T_{c}(B)$. A Gaussian convolution was used
for smoothing, and in the low-T, low-B region the data was sparse and was
interpolated. See Fig. S3b for raw data and the field-normalized
$e_{\mathrm{N}}/B$.

In the experiments reported in this paper, air-sensitive
$\text{K}_{3}\text{C}_{60}$ powders were compressed into pellets and
incorporated into a circuit board printed on an FR4 substrate, which features
low thermal conductivity. Embedded heaters and temperature sensors, as well as
indium-coated contacts with contact resistances below 1 Ohm were used to
optimize these measurements (see Appendix and Fig. S1). Four-probe resistance
measurements (see Appendix) were used to identify the superconducting
transition, as shown in the inset of Fig. 2a. In zero field, we found
$T_{c}(B=0)=19.8$K, in good agreement with previous reports and with magnetic
measurements on the same batch of samples [9, 10, 14]. A Werthamer-Helfand-
Hohenberg theory [29] was used to extrapolate the zero-temperature upper
critical field, $\mu_{0}H_{c2}(0)$, from the field dependence of the resistive
transition via: $\mu_{0}H_{c2}(0)=0.69\mu_{0}T_{c}\left.\frac{\partial
H_{c2}}{\partial T}\right|_{T_{c}}$, where $\mu_{0}$ denotes the vacuum
permeability. This yielded a value of about 39T, corresponding to a zero-
temperature coherence length of
$\xi_{0}=\sqrt{\Phi_{0}/2\pi\mu_{0}H_{c2}(0)}=2.9$nm (here $\Phi_{0}$ is the
magnetic flux quantum). This lies within the range of values extrapolated from
other experiments [9, 10], and closely matches the value recently determined
using pulsed fields in a range exceeding $H_{c2}$ [30].

As expected, no Nernst signal ($e_{\mathrm{N}}$) was observed for temperatures
far below $T_{c}$, likely due to freezing of vortex motion. For higher
temperatures, entering the temperature range where vortices become mobile, the
signal was seen to increase (see Fig. 2a). For a given magnetic field, the
number of vortices in the system remains nearly constant (as all fields in our
measurements were much larger than the lower critical field), but their
mobility increases rapidly. Near $T_{c}$, the Nernst signal reduced again. A
detailed quantitative understanding of this well-known phenomenon is still
lacking – both the increasing vortex-vortex interactions that appear as vortex
length scales increase, as well as a change in the entropy per vortex are
relevant [22, 31]. Although the amplitude of $e_{\mathrm{N}}$ did not saturate
at the largest applied field, the value it approaches agrees with an upper
bound that was found to apply to various superconductors spanning several
orders of magnitude in critical fields and temperatures [32].

The Nernst signal as a function of magnetic field (see Fig. S3a) peaked at a
field $B_{\mathrm{max}}$, that reduces on approaching the critical
temperature. This reduction fits very well to a linear function with a slope
of $-2.27(3)$ T/K, see Fig. 3c. It displays a zero-field intersect at
$T=19.5(1)$ K, close to the resistive critical temperature $T_{c}(B=0)=19.8$K.
$B_{\mathrm{max}}$ has previously been shown to bear some similarity to a
softening mode [33, 22], and its vanishing value when approaching $T_{c}(0)$
suggests that the Nernst effect changes in nature when crossing $T_{c}$.

Figure 3:  Anomalous Nernst effect above $T_{c}$. a, Nernst coefficient
$\mathcal{N}$ close to $T_{c}$ (red dashed line). Open symbols show the raw
data, lines are 10-point boxcar averages for 5T, 10T and 15T (grey to blue).
Inset: Same data re-scaled with the field-dependent $T_{c}(B)$ . b,
$\mathcal{N}$ at higher temperatures, after 10-point boxcar averaging (Note
the reduced $y$-scale). The red curve corresponds to the quasiparticle
contribution $-2.7\mu_{\mathrm{H}}S/T$. c, Characteristic quantities extracted
from the Nernst signal: $B_{\mathrm{max}}$ denotes the fitted peak of
$e_{\mathrm{N}}(B)$ (see Fig. 2b), $T_{0}$ and $T_{\mathrm{min}}$ are the
zero-crossing and minimum of $\mathcal{N}$, respectively. $T_{c}$ denotes the
resistive superconducting transition temperature. All lines are linear fits.
Error bars indicate fit uncertainties.

Above $T_{c}$, precursors of superconductivity may contribute to the Nernst
signal, but quasiparticles can also play a role. The latter contribution is
expected to scale linearly with magnetic field and temperature (at constant
volume) for a simple metallic state. [21] We therefore used the temperature-
and magnetic-field-normalized Nernst coefficient,
$\mathcal{N}=e_{\mathrm{N}}/BT$, in order to detect anomalous behaviour. In
Fig. 3a, $\mathcal{N}$ is shown to evolve smoothly across $T_{c}$, remaining
positive and retaining a strong field dependence, as expected for a signal
that is primarily caused by superconducting fluctuations. The inset in Fig. 3a
shows that $\mathcal{N}$ displays near-universal behaviour upon approaching
the field-dependent critical temperature, $T_{c}(B)$, (see also Fig. S3d).

At higher temperatures, we observed two characteristic features in the data
(see Fig. 3b): First, at $T_{0}(B)\approx 50$K, the signal changed from
positive to negative. The temperature of the zero-crossing, $T_{0}(B)$, shows
a similar field dependence as $T_{c}(B)$, (see Fig. 3c). Second, a minimum in
$\mathcal{N}$ appears at $T_{\mathrm{min}}(B)\approx 80$K, above which
$\mathcal{N}$ shows a linear and positive slope.

In order to disentangle the superconducting contribution to $\mathcal{N}$ we
first focus on the high-temperature limit, where the quasiparticle
contribution should become dominant. We can compare the behaviour of
$\mathcal{N}$ in this regime to the expected signal in a single band free-
electron model of a metal. There, $|\mathcal{N}|\sim\mu_{\mathrm{H}}S/T$, with
a pre-factor of order unity [21]. This relationship implies that the Nernst
effect is proportional to the ratio of the mobility and the Fermi energy in
the metal, which was found to hold in a variety of materials, with
$\mathcal{N}$ ranging from 1mV/$\mathrm{K}^{2}$T down to 1nV/$\mathrm{K}^{2}$T
[22]. We use previously determined values for the $T$-linear $S$ and
$\mu_{\mathrm{H}}$ [34, 35], where the latter shows a linear dependence on
temperature that has been shown to scale with the expansion of the lattice. As
shown in Fig. 3b, we find excellent agreement with our data above
$T_{\mathrm{min}}$, using a quasiparticle contribution of
$-2.7\mu_{\mathrm{H}}S/T$, and no significant field-dependence of
$\mathcal{N}$. Given the complex band structure of
$\text{K}_{3}\text{C}_{60}$, the agreement with such a simple scaling is
remarkable. It extends its range of validity to values of $\mathcal{N}$ which
are two orders of magnitude smaller than those reported so far[22].

The rapid change in the slope of $\mathcal{N}$ signals a significant change in
the electronic properties of the material. Such a change could for example be
caused by the appearance of charge-density wave order [36], and in a number of
cuprate superconductors a very similar feature was found to coincide closely
with the pseudogap temperature $T^{*}$ [27]. In $\text{K}_{3}\text{C}_{60}$, a
transition to a frozen orientational disorder of the $\mathrm{C}_{60}$
molecules is known to occur, but at a temperature very close to 200K [37, 38].
There are no observations pointing at the appearance of competing orders
around 80K, although it is worth noting that a certain deviation from
linearity has been observed in the Seebeck effect, which has been attributed
to electron-phonon coupling or precursors of superconductivity [34, 39, 40].

An effect related to the superconducting ground state seems much more likely,
given that the magnetic field dependence of $\mathcal{N}$ tracks $T_{c}(B)$.
In the following we will consider two scenarios through which precursors of
superconductivity could cause such an upturn in $\mathcal{N}$: a vortex-based
Nernst signal surviving in a phase-fluctuating regime above $T_{c}$, or a
signal caused by short-lived Cooper pairs.

Figure 4:  Scaling of the superconducting Nernst coefficient
$e_{\mathrm{N}}^{S}/B$ above $T_{c}$. a, Quasiparticle-subtracted contribution
to the Nernst coefficient,
$e_{\mathrm{N}}^{S}/B=e_{\mathrm{N}}/B+2.7\mu_{\mathrm{H}}S$, for magnetic
fields between 3T and 6T. Color scale as in Fig. 1. The data are shown as a
function of the distance to the field-dependent critical temperature at zero
field. b, Data for higher fields, up to 15T. The dashed red line shows the
expected low-field value from a model based on Gaussian fluctuations, with a
constant conductivity of 6$(\mathrm{m}\Omega\mathrm{cm})^{-1}$ (see text). The
top axes indicate where the coherence length, $\xi$, becomes equal to the
lattice spacing, $a_{\mathrm{Lattice}}$, or to the magnetic length $l_{B}$ at
the highest field shown in the respective panel.

For the latter scenario, a theory based on Gaussian fluctuations in a
Ginzburg-Landau model [24] predicts a superconducting contribution,
$e_{\mathrm{N}}^{\mathrm{SCG}}$, to the Nernst signal given by:

$\frac{e_{\mathrm{N}}^{\mathrm{SCG}}}{B_{z}}=\frac{\alpha_{xy}^{\mathrm{SCG}}}{\sigma_{xx}B_{z}}=\frac{k_{\mathrm{B}}e^{2}}{12\pi\hbar^{2}}\frac{\xi}{\sigma_{xx}}$
(2)

with

$\xi=\frac{\xi_{0}}{\sqrt{\epsilon}}=\frac{\xi_{0}}{\sqrt{(T-T_{c})/T_{c}}}$
(3)

for a three-dimensional system, where $e$ is the electron charge and $\hbar$
the reduced Planck constant. The two-dimensional version of this theory is in
good agreement with measurements on conventional [25, 22] and some
unconventional [41, 33] superconductors. Its prediction of a field-independent
Nernst coefficient applies to the low-field regime, where the coherence length
$\xi$ is short compared to the magnetic length $l_{B}=\sqrt{\hbar/eB}$, and
$T_{c}(B)\approx T_{c}(0)$. As it is a continuum theory, it may also become
invalid once $\xi$ becomes as short as the lattice spacing, and generally
speaking Ginzburg-Landau theory is only applicable in the vicinity of $T_{c}$.

In Fig. 4a, we compare our measurements for fields up to 6T to this theory by
subtracting the fitted quasiparticle contribution determined above (red line
in Fig. 3b) from the measured Nernst signal. See Fig. S2 for other possible
subtraction schemes. We find that the data is overall well described by the
simple $1/\sqrt{\epsilon}$ scaling predicted by theory if we use a
conductivity of 6$(\mathrm{m}\Omega\mathrm{cm})^{-1}$. Although we expect our
measurement to be sensitive to the intrinsic conductivity, rather than grain
boundary effects [42], this value is still about three times larger than the
conductivity of high-purity single crystals. [9] Interestingly however, a very
similar discrepancy was found in measurements of the paraconductivity [28],
suggesting that the residual conductivity describing the transport properties
of superconducting fluctuations may not be identical to the one extracted from
direct measurements.

At higher temperatures, beyond the expected regime of validity of this model,
deviations from the simple scaling occur, which might be captured by
perturbative expansions [43, 44]. Interestingly, at higher fields (see Fig.
4b), the behaviour does not follow a single power law. This suggests that in
high field, the resistive $T_{c}(B)$ deviates from the thermodynamical
critical point, which is a characteristic of unconventional superconductors
[45].

In order to gauge the plausibility of a phase-fluctuating scenario, we use the
framework proposed by Emery and Kivelson [46] to estimate the temperature
$T_{\theta}$ at which global phase coherence in the superconductor would be
destroyed by thermal fluctuations – even if pairing were to survive up to a
higher “mean field temperature” $T_{\mathrm{MF}}$. Taking the most recent (and
largest) value for the penetration depth in $\text{K}_{3}\text{C}_{60}$,
$\lambda=890$nm [47], we find a temperature $T_{\theta}$ as low as 80K. This
is only 4 times larger than $T_{c}$, whereas for conventional superconductors
$T_{\theta}/T_{\mathrm{MF}}$ can be on the order of $10^{5}$. Additionally, by
taking into account some degree of quantum fluctuations (as expected given the
relative proximity of a Mott-insulating state), it would be possible that the
superconducting transition is somewhat suppressed below $T_{\mathrm{MF}}$.

We are not aware of a quantitative prediction of the vortex Nernst signal
above $T_{c}$ in a three-dimensional system. Results in two dimensions [31]
suggest an important role of the lattice geometry, and can therefore not
simply be extrapolated to our system. As the difference between mean-field
models and fully quantum-mechanical descriptions becomes less pronounced in
higher dimensions, small deviations at high temperatures and magnetic field
may play a key role in distinguishing different scenarios. Our data therefore
also provide an important benchmark for a theoretical framework describing the
appearance of light-induced superconductivity in $\text{K}_{3}\text{C}_{60}$
based on the synchronization of stable but globally phase-incoherent Cooper
pairs, which would also have to correctly describe the initial static state.

It would be highly interesting to extend our work to
$\text{Rb}_{3}\text{C}_{60}$ and especially
$\text{Rb}_{x}\text{Cs}_{3-x}\text{C}_{60}$, where quantum phase fluctuations
caused by the proximity of the Mott-insulating state will be enhanced. For the
latter family, a suppression in $T_{c}$ upon approaching the quantum phase
transition has been observed [5, 6], but could not be reproduced in an
otherwise quantitatively successful theoretical model [1]. Studying the Nernst
effect in this regime, which should be possible using the experimental
framework presented here, would provide new insights concerning the nature of
the superconducting transition in the fullerides, and of phase-incoherent
superconductivity in general.

Acknowledgements We thank Michele Buzzi, Dante Kennes, Daniel Podolsky and
Dharmalingam Prabhakaran for insightful discussions, and Boris Fiedler for
superb technical assistance.

## Appendix

##  Appendix A Sample preparation

The $\text{K}_{3}\text{C}_{60}$ powder used in this work was prepared and
characterized as previously reported in Refs. [13, 14, 15]. In brief, finely
ground C60 powder and metallic potassium were placed in a vessel inside a
Pyrex vial in stoichiometric amounts, evacuated to $10^{-6}$mbar, and sealed.
The two materials were heated at 523K for 72h and then at 623K for 28h, and
kept separated to ensure that the $C_{60}$ was only exposed to clean potassium
vapour. After regrinding and pelletizing in an inert Ar atmosphere, the sample
was annealed at 623K for 5 days. Powder X-ray diffraction measurements
confirmed the purity of $\text{K}_{3}\text{C}_{60}$ and indicated a grain size
between 100nm and 400nm. Magnetic susceptibility measurements yielded a
$T_{c}$ of 19.8K[13]. For the Nernst effect and 4-point resistivity
measurements the sample was handled inside an Ar glove box with $<$0.2ppm
$\text{O}_{2}$ and $\text{H}_{2}$O. It was placed inside an FR4-frame, which
had been glued to the circuit board described below using thermally and
electrically insulating, minimally-outgassing glue (Epo-TeK 301-2FL-T), see
Fig. S1. The powder was then compressed with an FR4 piston, and sealed with
the same glue. The resistance of the sample was monitored to ensure that no
contamination occurred during the sealing process and the subsequent transfer
to the cryostat.

##  Appendix B Nernst effect measurement setup

Figure S1:  Experimental setup. a, The sample (red) is pressed on top of a
circuit board (green) using a PMMA piston (beige). The sample space is
encapsulated using non-conductive epoxy glue. The base temperature is measured
with a sensor (black) pressed against the bottom of the circuit board. The
sample and circuit board are mounted on the cold finger of a cryostat (copper)
using titanium screws (not shown), using spacers made of PMMA (grey). b,
Schematic of the circuit board. The sample compartment is indicated by a grey
rectangle. A temperature sensor is placed on each side of the sample in a
milled pocket, encapsulated by thermally-conductive epoxy glue. Resistive
heaters are placed on each side. The Nernst signal is measured with the
indicated indium-coated contacts in the centre of the sample compartment. The
other contacts are used for four-point resistance measurements. c, Transverse
voltage at +15T (blue) and -15T (grey). For the data shown in the main text,
the Nernst signal was evaluated as half of the difference between signals
measured at opposite fields.

We used a printed copper circuit board on an FR4 substrate (which features a
low thermal conductivity, $\sim 0.1$W/Km at 10K [48]), as shown in Fig. S1b.
Cernox temperature senors were embedded in thermally conductive glue (Stycast
2850FT) in milled pockets on each side of the sample. They were used to
monitor the temperature gradient across the sample, which was induced using a
resistive heater. An additional Cernox sensor was attached to the bottom of
the circuit board to monitor the base temperature. The circuit board was
mounted to the cold finger of a cryostat using non-magnetic (titanium) screws
and spring washers, and PMMA spacers were used for additional thermal
insulation. In the sample compartment, the copper contacts were coated with
indium, yielding contact resistances below 1 Ohm. The transverse voltage was
measured whilst slowly cooling the sample. Data for opposite magnetic fields
(see Fig. S1c), was then subtracted to compute the Nernst signal.

##  Appendix C Resistance measurements

The resistance of the sample was determined using a low-frequency lock-in
measurement in a linear 4-contact configuration, with contacts as shown in
Fig. S1b. Above $T_{c}$, the sample showed an increase in resistance upon
cooling, as previously observed in granular $\text{K}_{3}\text{C}_{60}$
samples [9, 10]. In order to determine $T_{c}$, the point at which the
resistance changes slope was used, which yielded a zero-field $T_{c}$
consistent with magnetic susceptibility measurements on the same batch of
sample. We verified that the width of the transition is not sensitive to
reducing the probe current below the value of 2$\mathrm{\mu}$A which we used.

Figure S2:  Effect of subtracting the quasiparticle signal on the scaling
analysis. a, Same as Fig. 4a, but subtracting the fixed value of
$e_{\mathrm{N}}$/B at 100K instead of a temperature-dependent function. The
grey dotted line shows the theoretical prediction of Eq. 2, but using a
quadratic temperature dependence for the conductivity. b, The Nernst
coefficient $e_{\mathrm{N}}$/B without any subtraction of the quasiparticle
contribution. Note that this results in some negative values for
$e_{\mathrm{N}}$ at higher temperatures, which do not appear in this
logarithmic plot.

##  Appendix D Subtraction of the quasiparticle signal

We verify that the details of how the quasiparticle contribution to
$e_{\mathrm{N}}$ is subtracted do not strongly affect the comparison to the
Gaussian fluctuation model shown in Fig. 3, by comparing different subtraction
schemes: In Fig. S3a, instead of subtracting the temperature-dependent
quasiparticle function ($-2.7\mu_{\mathrm{H}}$S), we subtract its fixed value
at 100K. Here, the simple theoretical model seems to capture the data even at
higher temperatures. In Fig. S3b, we plot $e_{\mathrm{N}}$ without any quasi-
particle subtraction. This leads to a deviation at high temperatures (as
expected given that the signal changes sign there), but within the expected
range of validity of the theoretical model, it still captures the data well.

Figure S3:  B-T maps of the Nernst signal. a, $e_{\mathrm{N}}$ as a function
of magnetic field for different temperatures below $T_{c}(B)$. b, Raw data of
$e_{\mathrm{N}}$ as a function of temperature and magnetic field (see Fig. 2
for a smoothed contour map). Grey dots indicate the B-T values of each
measurements, the data is interpolated to the nearest available point. The
black dotted line shows $T_{c}(B)$. c, Same as (b), but showing the field-
normalized value $e_{\mathrm{N}}/B$. d, Smoothed contour plot of (c).

We have also verified that using a temperature-dependent value of the
conductivity in Eq. 2 (where we have used the quadratic dependence found in
[49] as a comparison), has a negligible effect in the relevant range.

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