# Cyclic inner functions in growth classes and applications to approximation
problems

Bartosz Malman

###### Abstract

It is well-known that for any inner function $\theta$ defined in the unit disk
$\mathbb{D}$ the following two conditons: $(i)$ there exists a sequence of
polynomials $\\{p_{n}\\}_{n}$ such that $\lim_{n\to\infty}\theta(z)p_{n}(z)=1$
for all $z\in\mathbb{D}$, and $(ii)$ $\sup_{n}\|\theta
p_{n}\|_{\infty}<\infty$, are incompatible, i.e., cannot be satisfied
simultaneously. In this note we discuss and apply a consequence of a result by
Thomas Ransford from [13], which shows that if we relax the second condition
to allow for arbitrarily slow growth of the sequence
$\\{\theta(z)p_{n}(z)\\}_{n}$ as $|z|\to 1$, then condition $(i)$ can be met.
In other words, every growth class of analytic functions contains cyclic
singular inner functions. We apply this observation to properties of decay of
Taylor coefficients and moduli of continuity of functions in model spaces
$K_{\theta}$. In particular, we establish a variant of a result of Khavinson
and Dyakonov on non-existence of functions with certain smoothness properties
in $K_{\theta}$, and we show that the classical Aleksandrov theorem on density
of continuous functions in $K_{\theta}$, and its generalization to de Branges-
Rovnyak spaces $\mathcal{H}(b)$, is essentially sharp.

##  1 Background and results

Let $X$ be a topological space consisting of functions which are analytic in
the unit disk $\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\}$ and which satisfies some
customary desirable properties, such as that the evaluation $f\mapsto
f(\lambda)$ is a continuous functional on $X$ for each $\lambda\in\mathbb{D}$,
and that the function $z\mapsto zf(z)$ is contained in the space $X$ whenever
$f\in X$. A function $g\in X$ is said to be cyclic if there exists a sequence
of analytic polynomials $\\{p_{n}\\}_{n}$ for which the polynomial multiples
$\\{gp_{n}\\}_{n}$ converge to the constant function $1$ in the topology of
the space.

The well-known Hardy classes $H^{p}$ are among the very few examples of
analytic function spaces in which the cyclicity phenomenon is completely
understood. The cyclic functions $g$ are of the form

$g(z)=\exp\Big{(}\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}\log(|g(\zeta)|)\,d\textit{m}(\zeta)\Big{)},\quad
z\in\mathbb{D},$ (1)

where $d\textit{m}$ is the (normalized) Lebesgue measure of the unit circle
$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$. Functions as in (1) are called outer
functions. The inner functions are of the form

$\theta(z)=\prod_{n}\frac{\overline{\alpha_{n}}}{|\alpha_{n}|}\frac{z-\alpha_{n}}{1-\overline{\alpha_{n}}z}\cdot\exp\Big{(}-\int_{\mathbb{T}}\frac{\zeta+z}{\zeta-z}d\nu(\zeta)\Big{)},\quad
z\in\mathbb{D},$ (2)

where $\nu$ is a positive finite Borel measure on $\mathbb{T}$ and
$\\{\alpha_{n}\\}_{n}$ is a Blaschke sequence. It is clear that if the
Blaschke product on the left is non-trivial, then $\theta$ vanishes at points
in $\mathbb{D}$ and therefore cannot be cyclic in any reasonable space of
analytic functions $X$. The right factor is a singular inner function, and it
is well-known that if a function $g\in H^{p}$ has a singular inner function as
a factor, then $g$ is not cyclic in $H^{p}$. As a consequence, if
$\\{p_{n}\\}_{n}$ is a sequence of polynomials for which we have

$\lim_{n\to\infty}\theta(z)p_{n}(z)=1,\quad z\in\mathbb{D},$

then necessarily the Hardy class norms of the sequence must explode:

$\lim_{n\to\infty}\,\|\theta
p_{n}\|^{p}_{H^{p}}:=\lim_{n\to\infty}\int_{\mathbb{T}}|\theta
p_{n}|^{p}d\textit{m}=\infty$

for finite $p\geq 1$, or in case $p=\infty$,

$\lim_{n\to\infty}\|\theta
p_{n}\|_{\infty}:=\lim_{n\to\infty}\sup_{z\in\mathbb{D}}|\theta(z)p_{n}(z)|=\infty.$

When other norms are considered, cyclic singular inner functions might exist,
and here the Bergman spaces $L^{p}_{a}(\mathbb{D})$ provide a famous set of
examples. The Bergman norms are of the form

$\|g\|^{p}_{L^{p}(\mathbb{D})}:=\int_{\mathbb{D}}|g(z)|^{p}dA(z),$

where $dA$ is the normalized area measure of $\mathbb{D}$. After a sequence of
partial results by multiple authors, Boris Korenblum in [11] and James Roberts
in [14] independently characterized the cyclic singular inner functions in the
Bergman spaces in terms of the vanishing on certain subsets of $\mathbb{T}$ of
the corresponding singular measure $\nu$ appearing in (2). A construction of a
singular inner function which is cyclic in the classical Bloch space appears
in [3].

The purpose of this note is to discuss and apply a recent theorem of Thomas
Ransford from [13] which deals with singular inner functions which decay
slowly near the boundary of the disk. Here is the statement of Ransford’s
theorem.

######  Theorem 1.1.

pre-ransfordtheorem Let $w:[0,1)\to(0,1)$ be any function satisfying
$\lim_{r\to 1^{-}}w(r)=0$. Then there exists a singular inner function
$\theta$ for which we have

$\min_{|z|<r}|\theta(z)|\geq w(r),\quad r\in(0,1).$ (3)

We refer the reader to the article [13] for the clever proof of the statement,
which relies on an application of Frostman’s classical construction of a
measure with prescribed smoothness properties, and estimates of the Poisson
kernel. We will apply Ransford’s theorem in a slightly different form than it
is stated in [13]. The following consequence of his theorem is the pivotal
point of our discussion.

######  Corollary 1.2.

ransfordtheorem Let $w:[0,1]\to(0,1)$ be any decreasing function satisfying
$\lim_{t\to 1}w(t)=0$. There exists a singular inner function $\theta=S_{\nu}$
and a sequence of analytic polynomials $\\{p_{n}\\}_{n}$ such that

  1. (i)

$\lim_{n\to\infty}\theta(z)p_{n}(z)=1,\quad z\in\mathbb{D},$

  2. (ii)

$\sup_{z\in\mathbb{D}}|\theta(z)p_{n}(z)|w(|z|)\leq 2$.

###### Proof.

Apply Ransford’s pre-ransfordtheorem to the function $w$ to produce a singular
inner function $\theta$ satisfying (3). For integers $n\geq 2$ we set
$r_{n}:=1-1/n$ and $Q_{n}(z):=1/\theta(r_{n}z)$. Then $Q_{n}$ is holomorphic
in a neighbourhood of the closed disk $\overline{\mathbb{D}}$, and because we
are assuming that $w$ is decreasing, we have the estimate

$\sup_{z\in\mathbb{D}}|Q_{n}(z)|w(|z|)\leq\sup_{z\in\mathbb{D}}\frac{w(|z|)}{w(r_{n}|z|)}\leq
1.$

We can approximate $Q_{n}$ by an analytic polynomial $p_{n}$ so that

$\sup_{z\in\mathbb{D}}|Q_{n}(z)-p_{n}(z)|\leq 1/n.$

Then

$\displaystyle\sup_{z\in\mathbb{D}}|\theta(z)p_{n}(z)|w(|z|)\leq\sup_{z\in\mathbb{D}}\Big{(}|\theta(z)Q_{n}(z)|+1/n\Big{)}w(|z|)\leq
2$

It is clear from the construction that $\theta(z)p_{n}(z)\to 1$ as
$n\to\infty$, for any $z\in\mathbb{D}$. ∎

ransfordtheorem says that there exists cyclic singular inner functions in
essentially any space of analytic functions defined in terms of a growth
condition, or in any space in which such a growth space is continuously
embedded.

In the proofs of our main results, which will be stated shortly, we will
concern ourselves with the following weak type of cyclicity of singular inner
functions. Let $Y$ be some linear space of analytic functions which is
contained in $H^{1}$. We want to investigate if there exists a singular inner
function $\theta$ and a sequence of polynomials $\\{p_{n}\\}_{n}$ such that

$f(0)=\int_{\mathbb{T}}f\,d\textit{m}=\lim_{n\to\infty}\int f\overline{\theta
p_{n}}\,d\textit{m}$ (4)

holds for all $f\in Y$. The above situation means that the sequence $\\{\theta
p_{n}\\}_{n}$ converges to the constant $1$, weakly over the space $Y$. Now,
clearly if $Y$ is too large of a space (say, $Y=H^{2}$), then (4) can never
hold for all $f\in Y$. However, if $Y$ is sufficiently small, then the
situation in (4) might occur. For instance, in the extreme case when $Y$ is a
set of analytic polynomials, then any singular inner function $\theta$ and any
sequence of polynomials $\\{p_{n}\\}_{n}$ which satisfies
$\lim_{n\to\infty}p_{n}(z)=1/\theta(z)$ for $z\in\mathbb{D}$, is sufficient to
make (4) hold. Philosophically speaking, it is the uniform smoothness of the
functions in the class $Y$ that allows the existence of singular inner
functions $\theta$ for which the above situation occurs.

Such weak cyclicity of singular inner functions plays a role in the theory of
approximations in model spaces $K_{\theta}$ and the broader class of de
Branges-Rovnyak spaces $\mathcal{H}(b)$. Recall that the space $K_{\theta}$ is
constructed from an inner function $\theta$ by taking the orthogonal
complement of the subspace

$\theta H^{2}:=\\{\theta h:h\in H^{2}\\}$

in the Hardy space $H^{2}$:

$K_{\theta}=H^{2}\ominus\theta H^{2}.$

For background on the spaces $K_{\theta}$ one can consult the books [5] and
[9].

Note that the integrals on the right-hand side in (4) represents the inner
product in $H^{2}$ between a function $f$ and a polynomial multiple of
$\theta$. Under insignificat assumptions on $Y$, a straightforward argument
will show that if $\eqref{weakcyclicity}$ occurs, then the intersection
between $Y$ and $K_{\theta}$ is trivial: $Y\cap K_{\theta}=\\{0\\}$. As a
consequence of these observations, Ransford’s theorem, and some further
analysis, we will be able to construct model spaces $K_{\theta}$ which have
trivial intersections with many examples of spaces $Y$, even ones defined by
very mild smoothness or continuity conditions. Consequently, we will be able
to construct spaces $K_{\theta}$ in which every non-zero function behaves
badly in some precise way.

In one of our main results, we will show that the famous approximation theorem
of Aleksandrov from [1] on density in $K_{\theta}$ of functions which extend
continuously to the boundary, is in fact essentially sharp, as it cannot be
extended to any class of functions satisfying an estimate on their modulus of
continuity. By a modulus of continuity $\omega$ we mean a function
$\omega:[0,\infty)\to[0,\infty)$ which is continuous, increasing, satisfies
$\omega(0)=0$, and for which $\omega(t)/t$ is a decreasing function with

$\lim_{t\to 0^{+}}\omega(t)/t=\infty.$

For such a function $\omega$ we define $\Lambda^{\omega}_{a}$ to be the space
of functions $f$ which are analytic in $\mathbb{D}$, extend continuously to
$\overline{\mathbb{D}}$, and satisfy

$\sup_{z,w\in\overline{\mathbb{D}},z\neq
w}\frac{|f(z)-f(w)|}{\omega(|z-w|)}<\infty.$ (5)

Then $\Lambda^{\omega}_{a}$ is the space of analytic functions on $\mathbb{D}$
which have a modulus of continuity dominated by $\omega$. We make
$\Lambda^{\omega}_{a}$ into a normed space by introducing the quantity

$\|f\|_{\omega}:=\|f\|_{\infty}+\sup_{z,w\in\overline{\mathbb{D}},z\neq
w}\frac{|f(z)-f(w)|}{\omega(|z-w|)}.$

By a theorem of Tamrazov from [16], we could have replaced the supremum over
$\overline{\mathbb{D}}$ by a supremum over $\mathbb{T}$, and obtain the same
space of functions (we remark that a nice proof of this result is contained in
[4, Appendix A]). One of our two main results is the following optimality
statement regarding Aleksandrov’s density theorem.

######  Theorem 1.3.

mocnondensitymodelspaces Let $\omega$ be a modulus of continuity. There exists
a singular inner function $\theta$ such that

$\Lambda^{\omega}_{a}\cap K_{\theta}=\\{0\\}.$

This statement will be proved in Section 3. In fact, we will see that
mocnondensitymodelspaces is a consequence of a variant, and in some directions
a strengthening, of a theorem of Dyakonov and Khavinson from [6]. For a
sequence of positive numbers $\bm{\lambda}=\\{\lambda_{n}\\}_{n=0}^{\infty}$
we define the class

$H^{2}_{\bm{\lambda}}=\Big{\\{}f=\sum_{n=0}^{\infty}f_{n}z^{n}\in
Hol(\mathbb{D}):\sum_{n=0}^{\infty}\lambda_{n}|f_{n}|^{2}<\infty\Big{\\}}.$
(6)

The other of our main results, proved in Section 2, reads as follows.

######  Theorem 1.4.

badmodelspace Let $\bm{\lambda}=\\{\lambda_{n}\\}_{n=0}^{\infty}$ be any
increasing sequence of positive numbers with
$\lim_{n\to\infty}\lambda_{n}=\infty$. Then there exists a singular inner
function $\theta$ such that

$K_{\theta}\cap H^{2}_{\bm{\lambda}}=\\{0\\}.$

The result can be compared to the mentioned result of Dyakonov and Khavinson
in [6], from which the above result can be deduced in the special case
$\bm{\lambda}=\\{(k+1)^{\alpha}\\}_{k=0}^{\infty}$ with any $\alpha>0$.

The theory of de Branges-Rovnyak spaces $\mathcal{H}(b)$ is a well-known
generalization of the theory of model spaces $K_{\theta}$. The symbol of the
space $b$ is now any analytic self-map of the unit disk, and we have
$\mathcal{H}(b)=K_{b}$ whenever $b$ is inner. For background on
$\mathcal{H}(b)$ spaces, see [15], or [7] and [8]. A consequence of the
authors work in collaboration with Alexandru Aleman in [2] is that the above
mentioned density theorem of Aleksandrov generalizes to the broader class of
$\mathcal{H}(b)$ spaces: any such space admits a dense subset of functions
which extend continuously to the boundary. Since mocnondensitymodelspaces
proves optimality of Aleksandrov’s theorem for inner functions $\theta$, one
could ask if at least for outer symbols $b$ any improvement of the density
result in $\mathcal{H}(b)$ from [2] can be obtained. In Section 4 we remark
that this is not the case, and the result in [2] is also essentially optimal,
even for outer symbols $b$. This is shown to be equivalent to a theorem of
Khrushchev from [10].

In the last Section 5 we list a few questions we have not found an answer for,
and some ideas for further research.

##  2 Proof of badmodelspace

In the proof of the theorem we will need to use the following crude
construction of an integrable weight with large moments.

######  Lemma 2.1.

momentlemma Let $\\{\lambda_{n}\\}_{n=0}^{\infty}$ be a decreasing sequence of
positive numbers with $\lim_{n\to\infty}\lambda_{n}=0$. There exists a non-
negative function $\Lambda\in L^{1}([0,1])$ which satisfies

$\lambda_{n}\leq\int_{0}^{1}x^{2n+1}\Lambda(x)\,dx,\quad n\geq 0.$

###### Proof.

Recall that the sequence $(1-1/n)^{n}=\exp(n\log(1-1/n))$ is decreasing and
satisfies

$\lim_{n\to\infty}(1-1/n)^{n}=e^{-1}.$

It follows that

$\inf_{x\in(1-1/n,1)}x^{2n+1}\geq\alpha$

for some constant $\alpha>0$ which is independent of $n$. For $n\geq 1$, we
define the intervals $I_{n}=\big{(}1-1/n,1-1/(n+1)\big{)}$. Our function
$\Lambda$ will be chosen to be of the form

$\Lambda(x)=\sum_{n=0}^{\infty}1_{I_{n}}c_{n},$

where $1_{I_{n}}$ is the indicator function of the interval $I_{n}$ and the
$c_{n}$ are positive constants to be chosen shortly. Note that

$\displaystyle\int_{0}^{1}x^{2N+1}\Lambda(x)\,dx\geq\int_{1-1/N}^{1}x^{2N+1}\Lambda(x)\,dx\geq\alpha\sum_{n=N}^{\infty}|I_{n}|c_{n}.$
(7)

We choose

$c_{n}=\alpha^{-1}|I_{n}|^{-1}(\lambda_{n}-\lambda_{n+1}).$

This choice of coefficients $c_{n}$ makes $\Lambda$ integrable over $[0,1]:$

$\displaystyle\int_{0}^{1}\Lambda(x)dx=\sum_{n=1}^{\infty}|I_{n}|c_{n}=\alpha^{-1}\sum_{n=1}^{\infty}\lambda_{n}-\lambda_{n+1}$
$\displaystyle=\lim_{M\to\infty}\alpha^{-1}\sum_{n=1}^{M}\lambda_{n}-\lambda_{n+1}=\lim_{M\to\infty}\alpha^{-1}(\lambda_{1}-\lambda_{M+1})$
$\displaystyle=\alpha^{-1}\lambda_{1}$

In the last step we used the assumption that the sequence
$\\{\lambda_{n}\\}_{n}$ converges to zero. Moreover, by (7) and the choice of
$c_{n}$ we can estimate

$\displaystyle\int_{0}^{1}x^{2N+1}\Lambda(x)\,dx\geq\alpha\sum_{n=N}^{\infty}|I_{n}|c_{n}$
$\displaystyle=\lim_{M\to\infty}\alpha\sum_{n=N}^{M}|I_{n}|c_{n}=\lim_{M\to\infty}\sum_{n=N}^{\mathcal{M}}\lambda_{n}-\lambda_{n+1}$
$\displaystyle=\lim_{M\to\infty}\lambda_{N}-\lambda_{M+1}=\lambda_{N}.$

The proof is complete. ∎

The significance of the above lemma is the estimate

$\sum_{k=0}^{\infty}\lambda_{k}|f_{k}|^{2}\leq
c\int_{\mathbb{D}}|f(z)|^{2}\Lambda(|z|)dA(z)$ (8)

for some numerical constant $c>0$ and any function $f$ which is holomorphic in
a neighbourhood of the closed disk $\overline{\mathbb{D}}$. The estimate can
be vereified by direct computation of the integral on the right-hand side,
using polar coordinates.

We will also use the following well-known construction.

######  Lemma 2.2.

multiplier For any function $g\in L^{1}([0,1])$ there exists a positive and
increasing function $w:[0,1)\to\mathbb{R}$ which satisfies

$\lim_{t\to 1^{-}}w(t)=\infty$

and

$wg\in L^{1}([0,1]).$

###### Proof.

The integrability condition on $g$ implies that

$\lim_{t\to 1^{-}}\int_{t}^{1}|g(x)|dx=0.$

Thus there exists a sequence of intervals $\\{I_{n}\\}_{n=1}^{\infty}$ which
have $1$ as the right end-point and which satisfy $I_{n+1}\subset I_{n}$ for
all $n\geq 1$, and

$\int_{I_{n}}|g(x)|dx\leq 4^{-n}.$

If we set

$w(t)=1_{[0,1)\setminus I_{1}}+\sum_{n=1}^{\infty}2^{n}1_{I_{n}\setminus
I_{n+1}},$

where $1_{I_{n}\setminus I_{n+1}}$ is the indicator function of the set
difference $I_{n}\setminus I_{n+1}$, then $w$ is increasing, satisfies
$\lim_{t\to 1^{-}}w(t)=\infty$, and

$\int_{I_{n}\setminus I_{n+1}}w(x)|g(x)|dx\leq 2^{-n}$

for all $n\geq 1$. Consequently

$\int_{0}^{1}w(x)|g(x)|dx\leq\int_{0}^{1}|g(x)|dx+\sum_{n}\int_{I_{n}\setminus
I_{n+1}}w(x)|g(x)|dx<\infty.$

∎

###### Proof of badmodelspace.

Let $\Lambda$ be the function in momentlemma which corresponds to the sequence
$\\{1/\lambda_{n}\\}_{n=0}^{\infty}$. That is, $\Lambda$ satisfies

$\frac{1}{\lambda_{n}}\leq\int_{0}^{1}x^{2n+1}\Lambda(x)\,dx,\quad n\geq 0,$

and $\Lambda\in L^{1}[0,1]$. Now, let $w$ be a positive decreasing function
which satisfies $\lim_{x\to 1^{-}}w(x)=0$ and

$\int_{0}^{1}\frac{\Lambda(x)}{w^{2}(x)}\,dx<\infty.$

Existence of such a function follows readily from multiplier. Apply
ransfordtheorem to $w$ and obtain a corresponding inner function $\theta$ and
a sequence of polynomials $\\{p_{n}\\}_{n}$ for which the conclusions $(i)$
and $(ii)$ of ransfordtheorem hold. We will show that for this $\theta$ we
have $K_{\theta}\cap H^{2}_{\bm{\lambda}}=\\{0\\}.$

Indeed, assume $f\in K_{\theta}\cap H^{2}_{\bm{\lambda}}=\\{0\\},$ but that in
fact $f$ is non-zero. Since both $K_{\theta}$ and $H^{2}_{\bm{\lambda}}$ are
invariant for the backward shift operator, we may without loss of generality
assume that $f(0)\neq 0$. Fix an integer $n$ and let

$g(z)=\theta(z)p_{n}(z)-1,z\in\mathbb{D}.$ (9)

Let $\\{f_{k}\\}_{k},\\{g_{k}\\}_{k}$ be the sequences of Taylor coefficients
of $f$ and $g$, respectively. Since $f\in K_{\theta}$, we have

$\displaystyle|f(0)|=\Big{|}\int_{\mathbb{T}}f\,d\textit{m}\Big{|}=\Big{|}\int_{\mathbb{T}}f\overline{\theta
p_{n}-1}\,d\textit{m}\Big{|}=\lim_{r\to
1^{-}}\Big{|}\sum_{k=0}^{\infty}r^{2k}f_{k}\overline{g_{k}}\Big{|}$
$\displaystyle\leq\limsup_{r\to
1^{-}}\Big{(}\sum_{k=0}^{\infty}\lambda_{k}r^{2k}|f_{k}|^{2}\Big{)}^{1/2}\Big{(}\sum_{k=0}^{\infty}\frac{1}{\lambda_{n}}|r^{k}g_{k}|^{2}\Big{)}^{1/2}$

Using inequality (8) on the term on the right-hand side in the last expression
(with $\lambda_{n}$ replaced by $1/\lambda_{n}$), we obtain

$\displaystyle|f(0)|\leq C\limsup_{r\to
1^{-}}\Big{(}\sum_{k=0}^{\infty}\lambda_{k}|f_{k}|^{2}\Big{)}^{1/2}\Big{(}\int_{\mathbb{D}}|g(rz)|^{2}\Lambda(|z|)dA(z)\Big{)}^{1/2}$
$\displaystyle=C\Big{(}\sum_{k=0}^{\infty}\lambda_{k}|f_{k}|^{2}\Big{)}^{1/2}\Big{(}\int_{\mathbb{D}}|g(z)|^{2}\Lambda(|z|)dA(z)\Big{)}^{1/2}.$

By assertion in part $(ii)$ of ransfordtheorem, the function
$|g(z)|^{2}\Lambda(|z|)$ is dominated pointwise in $\mathbb{D}$ by the
integrable function

$\frac{4\Lambda(|z|)}{w^{2}(|z|)},\quad z\in\mathbb{D}$

independently of which polynomial $p_{n}$ is used to defined $g$ in (9). But
if we let $n\to\infty$ in (9), then $|g(z)|^{2}\Lambda(|z|)\to 0$, and so we
infer from the computation above and the dominated convergence theorem that
$f(0)=0$, which is a contradiction. The conclusion is that $K_{\theta}\cap
H^{2}_{\bm{\lambda}}=\\{0\\}$, and the proof of the theorem is complete. ∎

##  3 Proof of mocnondensitymodelspaces

mocnondensitymodelspaces will follow immediately from badmodelspace together
with the following embedding result for the spaces $\Lambda^{\omega}_{a}$.

######  Lemma 3.1.

mocspecestimatelemma Let $\omega$ be a modulus of continuity. There exists an
increasing sequence of positive numbers
$\bm{\alpha}=\\{\alpha_{n}\\}_{n=0}^{\infty}$ satisfying
$\lim_{n\to\infty}\alpha_{n}=\infty$ and such that for any
$f\in\Lambda^{\omega}_{a}$ we have the estimate

$\sum_{n=0}\alpha_{n}|f_{n}|^{2}\leq C\|f\|^{2}_{\omega}$ (10)

where $C>0$ is a numerical constant and $\\{f_{n}\\}_{n}$ is the sequence of
Taylor coefficients of $f$.

###### Proof.

For each $r\in(0,1)$ we have the estimate

$\sum_{n=0}(1-r^{2n})|f_{n}|^{2}=\int_{\mathbb{T}}|f(\zeta)-f(r\zeta)|^{2}d\textit{m}(\zeta)\leq\omega(1-r)^{2}\|f\|^{2}_{\omega}.$
(11)

Since $\lim_{t\to 0}\omega(t)=0$, for each positive integer $N$ there exists a
number $r_{N}\in(0,1)$ such that $\omega(1-r_{N})\leq\frac{1}{2^{N}}$. Since
$\lim_{n\to\infty}r_{N}^{2n}=0$, there exists an integer $K(N)$ such that
$r_{N}^{2n}<1/2$ for $n\geq K(N)$. Then

$\sum_{n=K(N)}^{\infty}\frac{|f_{n}|^{2}}{2}\leq\sum_{n=K(N)}^{\infty}(1-r_{N}^{2n})|f_{n}|^{2}\leq\frac{1}{4^{N}}\|f\|^{2}_{\omega}.$

Consequently

$\sum_{n=K(N)}2^{N}|f_{n}|^{2}\leq\frac{1}{2^{N-1}}\|f\|^{2}_{\omega}.$ (12)

We can clearly choose the sequence of integers $K(N)$ to be increasing with
$N$. If we define the sequence $\bm{\alpha}$ by the equation $\alpha_{n}=1$
for $n\leq K(1)$, and $\alpha_{n}=2^{n}$ for $K(N)\leq n<K(N+1)$, then (10)
follows readily from (12) by summing over all $N\geq 1$. ∎

###### Proof of mocnondensitymodelspaces.

mocspecestimatelemma implies that $\Lambda^{\omega}_{a}$ is contained in some
space of the form $H^{2}_{\bm{\alpha}}$ as defined in (6). If $\theta$ is a
singular inner function given by badmodelspace such that
$H^{2}_{\bm{\alpha}}\cap K_{\theta}=\\{0\\}$, then obviously we also have that
$\Lambda^{\omega}_{a}\cap K_{\theta}=\\{0\\}$, and so the claim follows. ∎

##  4 The case of $\mathcal{H}(b)$ spaces

Here we prove the optimality of the continuous approximation theorem for the
larger class of $\mathcal{H}(b)$ spaces.

######  Proposition 4.1.

Let $\omega$ be a modulus of continuity. There exists an outer function
$b:\mathbb{D}\to\mathbb{D}$ such that

$\Lambda^{\omega}_{a}\cap\mathcal{H}(b)=\\{0\\}.$

###### Proof.

By a result of Khrushchev noted in [10, Theorem 2.4], there exists a closed
subset $E$ of the circle $\mathbb{T}$ with the property that for no non-zero
integrable function $h$ supported on $E$ is the Cauchy integral

$C_{h}(z)=\int_{\mathbb{T}}\frac{h(\zeta)}{1-z\overline{\zeta}}d\textit{m}(\zeta)$

a member of the space $\Lambda^{\omega}_{a}$. It suffices thus to construct an
$\mathcal{H}(b)$ space for which every function can be expressed as such a
Cauchy integral. The simplest choice for the space symbol $b$ is the outer
function with modulus $1$ on $\mathbb{T}\setminus E$ and $1/2$ on $E$. Then
$b$ is invertible in the algebra $H^{\infty}$, and a consequence of the
general theory (see [8, Theorem 20.1 and Theorem 28.1]) is that every function
in the space $\mathcal{H}(b)$ is a Cauchy integral of a function $h$ which is
square-integrable on $\mathbb{T}$ and supported only on $E$. Thus
$\Lambda^{\omega}_{a}\cap\mathcal{H}(b)=\\{0\\}$. ∎

##  5 Some ending questions and remarks

Since Ransford’s theorem seems to be such a powerful tool in establishing
results of the kind mentioned here, we are wondering whether it can be further
applied. In particular, the following questions come to mind.

  1. 1.

Are our methods strong enough to prove that there exists model spaces
$K_{\theta}$ which admit no non-zero functions in the Wiener algebra of
absolutely convergent Fourier series? The result is known, and has been noted
in [12]. However, it was proved as a consequence of a complicated construction
of a cyclic singular inner function in the Bloch space. Is it so that
Ransford’s construction is sufficient to prove the non-density result for the
Wiener algebra in the fashion presented here?

  2. 2.

For $p>2$, the Banach spaces $\ell^{p}_{a}$ consisting of functions $f\in
Hol(\mathbb{D})$ with Taylor series $\\{f_{n}\\}_{n=0}^{\infty}$ satisfying

$\|f\|^{p}_{\ell^{p}_{a}}:=\sum_{n=0}^{\infty}|f_{n}|^{p}<\infty$

are of course larger than the space $H^{2}=\ell^{2}_{a}$. Do there exist
cyclic singular inner functions in these spaces?

### Acknowledgements

The author would like to thank Christopher Felder for his reading of the
manuscript and for his helpful suggestions for improvements. He would also
like to thank Adem Limani for very useful discussions and important comments.

## References

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Bartosz Malman,
KTH Royal Institute of Technology,
Stockholm, Sweden
<EMAIL_ADDRESS>


# Superfluid Density in Conventional Superconductors: From Clean to Strongly
Disordered

Surajit Dutta1, Pratap Raychaudhuri1, Sudhansu S. Mandal2, T.V.Ramakrishnan3
1Tata Institute of Fundamental Research, Mumbai 400005, India
2Department of Physics, Indian Institute of Technology, Kharagpur 721302,
India
3Department of Physics, Indian Institute of Science, Bangalore 560012, India

###### Abstract

The highly convergent form of superfluid density in disordered conventional
superconductors available in the literature and independently obtained by us
following the approach of an earlier paper [Phys. Rev. B $\bm{102}$, 024514
(2020)] has been reformulated to separate out the generally used so-called
‘dirty-limit’ term and an additional term. We use this new expression for
making an extensive comparison with previously published experimental data and
show that the former, generally used, term is not sufficient for analyzing
these results. We point out that consequently, there is a large regime
(disordered superconductors with moderate to no disorder) where theoretical
predictions need to be confronted with experiment.

##  I Introduction

The additional free energy ${\cal F}$ of a superconductor depends on its
nonzero superfluid velocity ${\rm\bf v}_{s}$ as ${\cal F}\sim(\rho_{s}/2)\int
d\bm{r}\,{\rm\bf v}_{s}^{2}$ where $\rho_{s}$ is the superfluid stiffness or
phase rigidity, analogous to the mass (see e.g. Ref.Coleman ). This gauge
invariant superfluid velocity ${\rm\bf v}_{s}$ is related to the phase
$\theta$ of the superconducting order parameter as ${\rm\bf
v}_{s}=(1/m_{e})(\bm{\nabla}\theta-2e\bm{A})$; here $\bm{A}$ is the vector
potential, $e$ and $m_{e}$ are the charge and mass of an electron
respectively, and we set $\hbar=1$. Experimentally, one measures the magnetic
penetration depth $\lambda$ which is related to the superfluid density $n_{s}$
asTinkham  $\lambda^{-2}=\mu_{0}e^{2}n_{s}/m_{e}$. The superfluid density
$n_{s}$ is proportional to the superfluid stiffness;
$n_{s}=(4/m_{e})\rho_{s}$. We use the above relation between the
experimentally measured penetration depth $\lambda$ and the calculated
$\rho_{s}$ to compare in detail theoretical results with experiment, and
suggest that there is a large regime of disorder in relatively clean systems
so that measurements are needed here, to also establish the clean London
limiting value.

The solely diamagnetic response of the electron system to an external magnetic
field leads to $n_{s}^{d}=n$, the electron density. This is the London value
which also follows for the ground state ($T=0$) from Galilean invariance, for
a homogeneous continuum. However, the actual superfluid density is less than
$n_{s}^{d}$ due to the paramagnetic response of the system:
$n_{s}=n_{s}^{d}-n_{s}^{p}$, $n_{s}^{p}$ being the paramagnetic contribution
to the superfluid density. For a clean conventional Bardeen-Cooper-Schrieffer
(BCS) superconductor BCS , $n_{s}^{p}=0$ at zero temperature and is
exponentially small at low temperatures because of the presence of the
quasiparticle gap. However, $n_{s}^{p}$ grows with temperature and eventually
becomes equal to $n_{s}^{d}$ at the superconducting critical temperature
$T_{c}$ where $n_{s}$ vanishes. In disordered superconductors, $n_{s}^{p}\neq
0$ at zero temperature ($T=0$), and the resulting superfluid density is
disorder dependent and is smaller A-G  than the London limiting value at
$T=0$. This, and the temperature dependence of $n_{s}$ have been discussed in
literature AG2 ; AGD ; Weiss ; Nam ; Kogan ; M-R .

The effect of static, short range nonmagnetic disorder on superconductors is
most simply characterized by a broadening $\Gamma\ll\epsilon_{\rm F}$ of the
electron spectral density (here $\epsilon_{\rm F}$ is the Fermi energy Coleman
; A-G . Microscopic calculations generally use on site or zero range disorder
with a Gaussian probability distribution of its strength related to this
broadening. The effect of disorder on electrons is mostly implemented in the
Born approximation, where it leads to a finite lifetime $\tau=(1/\Gamma)$ of
electronic states. Such a treatment neglects Anderson localization effectsMa-
Lee . In this approximation, it is well known that in the so called ‘dirty
limit’, i.e. for $\Delta_{0}/\Gamma<<1$, $n_{s}$ at $T=0$ scales A-G  with the
dc conductivity $\sigma=ne^{2}\tau/m_{e}$ in the normal state, i.e.,
$n_{s}(T=0)=\sigma(\pi m_{e}\Delta_{0}/e^{2})=n\pi\Delta_{0}\tau$, where
$\sigma$ is the electrical conductivity of the system, $n$ is the normal
electron density, and $\Delta_{0}$ is the gap at $T=0$. We note that
$\Delta_{0}$ is independent of disorder, according to Anderson’s
theoremAnderson . A generalized form of this zero-temperature superfluid
density at finite temperatures, namely

$n_{s}(T)=n\pi\tau\Delta(T)\,\tanh\left(\frac{\Delta(T)}{2k_{B}T}\right)\,$
(1)

is often used for analyzing experimental data Lemberger07 ; Mondal11 ;
Mandal20 ; where $\Delta(T)$ is the gap at the temperature $T$. However, this
expression has also been derived AG2 ; Kogan  in the dirty limit. Clearly,
$n_{s}(T)$ in Eq. (1) cannot be valid for all $\tau$ because for $\tau$ large
enough such that $\Delta_{0}\tau>1/\pi$, the superfluid density $n_{s}(T=0)$
exceeds the maximum possible London limiting value $n$.

In this paper, we exhibit the superfluid density as a sum of the commonly used
term (1) and another term in the following way. We reformulate an expression
(2) of superfluid density AGD ; Nam ; Weiss ; Scheffler , which is a
convergent sum of Matsubara frequencies only and which shows explicitly that
$n_{s}$ vanishes when $\Delta$ vanishes. This frequency sum is converted into
a contour integral over complex frequencies, and displays two simple poles at
$\pm\Delta$ and branch cuts for the domains $(\Delta,\infty)$ and
$(-\infty,-\Delta)$. The residue of the simple poles provides the contribution
(1) generally used for the analysis of experimental data. We have derived an
additional contribution arising from the branch cuts; this competes with the
former as they are opposite in sign. We find that the contribution of the
latter is insignificant if $\Delta_{0}\tau\lesssim 10^{-3}$; it begins to be
relevant for $\Delta_{0}\tau\sim 5\times 10^{-3}$. Both the contributions
increase with $\Delta_{0}\tau$, and their difference asymptotically approaches
the London limit at $T=0$ for $\Delta_{0}\tau\rightarrow\infty$. The
contribution of the latter to superfluid density and thus to the measured
absolute value of the penetration depth provides a large regime, which is yet
unexplored, for experimental studies of disorder dependent superfluid density
in relatively clean superconductors over a wide span in $\Delta_{0}\tau$,
namely roughly from $10^{-3}$ to $10$, i.e., from the dirty limit to the clean
limit.

We also find that temperature dependence of the scaled superfluid density
$n_{s}(T)/n_{s}(0)$ is almost independent of disorder; this scaled density
function is easily obtained in the dirty limit as well as in the pure limit,
and is the same. This fact has led to the belief that the dirty limit
expression is appropriate for all disorder, including very weak disorder.

Our finding suggests a disorder dependent study with measurement of the
absolute value of the superfluid density as a function of disorder, and
provides explicit expressions for it at different temperatures and for
different values of disorder. Unfortunately, not much data is available in the
literature where absolute measurement of $n_{s}$ has been performed, so that
our results cannot be easily compared with experiment. In Section III, we
analyze some of the available experimental data in superconductors like Nb-
doped SrTiO${}_{{}_{3}}$, Pb, Sn, Nb, NbN, and a-MoGe. The data for $T_{c}$
and $n$ have been obtained via transport measurements, and the dimensionless
parameter $\delta=\Delta_{0}/(2k_{B}T_{c})$ is obtained from the measurement
of $\Delta_{0}$ in tunneling experiments. We then have just one free parameter
$\Delta_{0}\tau$ which we extract by fitting the above mentioned theoretical
expression where we have explicitly shown also the contributions of both the
terms in the expressions separately. The extracted values of $\Delta_{0}\tau$
range from about $5\times 10^{-5}$ to $0.5$. The ratio $\eta$ of the two
contributions to $n_{s}(T)$ mentioned above, is almost negligible for a-MoGe
and NbN for which $\Delta_{0}\tau$ is very small, but it becomes recognizable
for the Nb sample, it becomes more prominent for Pb and Sn, and for Nb-doped
SrTiO3 it is the largest amongst all the ones analyzed here.

Section IV is devoted to the outlook and discussion where we have pointed out
that many more experiments are needed to be confronted with theoretical
prediction as the highest value of $n_{s}/n$ that has been found in the
earlier experiments is about $0.56$, whereas it can go up to $1.0$ for the
pure limit that may be attained for the samples with $\Delta_{0}\tau\sim 10$.
We also discuss here the physics that cannot be revealed from the theoretical
prediction above.

In appendix A, we have estimated the superfluid density by utilizing the
oscillator sum rule for the real part of optical conductivity. We show that it
reproduces the clean limit exactly and the dirty limit up to a numerical
factor of order unity.

##  II Reformulated Superfluid Density

A highly convergent expression AGD  of super-fluid density (see also Refs.Nam,
; Weiss, ; AG2, ; Kogan, ) at finite temperatures for all disorder (excluding
the localization regime) is given by

$n_{s}(T)=\frac{n\pi}{\beta}\sum_{\omega_{m}}\left[\frac{\tilde{\Delta}^{2}}{(\tilde{\Delta}^{2}+\tilde{\omega}_{m}^{2})^{3/2}}\right]$
(2)

which is obtained also by a series of successive integration by parts for
removing divergences in the approach of Ref.M-R, , where renormalized
frequency, $\tilde{\omega}_{m}$, and gap, $\tilde{\Delta}$, in terms of
Matsubara frequency $\omega_{m}=\pi(2m+1)/\beta$ and the superconducting gap
$\Delta$ can be expressed as

$\frac{\tilde{\omega}_{m}}{\omega_{m}}=\frac{\tilde{\Delta}}{\Delta}=1+\frac{1}{2\tau\sqrt{\Delta^{2}+\omega_{m}^{2}}}.$
(3)

Here $\beta=1/(k_{B}T)$ and the introduction A-G  of finite electronic life-
time $\tau$ in the theory of disordered superconductors through the Nambu-
Green’s functions for Bogoliubov quasiparticles. The superfluid density is
explicitly seen to vanish (2) in the absence of isotropic superconducting gap.
The expression (2) of $n_{s}(T)$ is applicable to both two and three
dimensional superconductors provided the localization effect of disorder does
not set in for very strong disorder. However, the expression (2) has not been
frequently used for analyzing experimental data because it involves
complicated sum over the Matsubara frequency. Here we reformulate Eq.2 below
in terms of a simple term and a simple integral, and analyze available data of
the absolute measurements of superfluid density in the next section.

Figure 1: (Color online) Zero temperature contributions for the expression (5)
of normalized superfluid density, $n_{s}/n$, as a function of dimensionless
disorder $\Delta_{0}\tau$: We show the first term, the second term, and their
difference for several decades of $\Delta_{0}\tau$. The nonzero value of the
difference is apparent on this log-log scale in the separate curve for it.

The frequency sum in Eq. 2 is evaluated in the usual way: we change it into a
contour integration for complex $z$ such that the contour contains only the
poles at $z=i\omega_{m}$,

$n_{s}(T)=n\pi\int_{C}\frac{dz}{2\pi
i}\frac{\Delta^{2}}{(\Delta^{2}-z^{2})(\sqrt{\Delta^{2}-z^{2}}+\frac{1}{2\tau})}\frac{1}{e^{\beta
z}+1}.$ (4)

We now deform the contour to exclude the non-analyticities on the real axis
(energy $\epsilon$), namely the simple poles at $z=\pm\Delta$ as well as the
branch cut from $z=\Delta\to\infty$ and from $-\Delta\to-\infty$ (arising from
the square root term) so that

$\displaystyle\frac{n_{s}}{n}$ $\displaystyle=$
$\displaystyle\pi\Delta\tau\tanh\left(\frac{\beta\Delta}{2}\right)$ (5)
$\displaystyle-$
$\displaystyle\Delta^{2}\int_{\Delta}^{\infty}d\epsilon\frac{\tanh\left(\frac{\beta\epsilon}{2}\right)}{\sqrt{\epsilon^{2}-\Delta^{2}}(\epsilon^{2}-\Delta^{2}+\frac{1}{4\tau^{2}})}\,.$

The first term is due to the contribution from the residues of the poles and
the second term is due to the branch cut. In the dirty limit
$(\Delta_{0}\tau<<1)$ the latter is much smaller than the first and can
therefore be neglected; the contribution of the corresponding branch cut to
the superfluid density is negligible. This fact leads to a considerable
simplification of calculations in the dirty limit. We note that while the
first term in Eq.(5) is present in the superfluid density expression in
Ref.M-R,  as well, the second term differs. This finer difference makes the
expression (5) consistent in temperature dependence at all disorder.

Figure 2: (Color online) Temperature dependence of $n_{s}$ scaled with
electron density for different levels of disorder:
$\Delta_{0}\tau=10^{-3},\,10^{-2},\,10^{-1},\,1,\,10,\,10^{2}$ (in the unit of
$\hbar$). Temperature is scaled with the BCS $T_{c}$. Inset: $n_{s}(T)$ is
scaled with $n_{s0}=n_{s}(T=0)$. Temperature variation of $n_{s}/n_{s0}$ is
almost independent of disorder, although $n_{s0}$ is strongly disorder
dependent.

The zero temperature limit of Eq. (5) yields

$\displaystyle\frac{n_{s}}{n}(T=0)=\pi\Delta_{0}\tau-$
$\displaystyle\left\\{\begin{array}[]{ll}\frac{(2\Delta_{0}\tau)^{2}}{\sqrt{(2\Delta_{0}\tau)^{2}-1}}\tan^{-1}\left(\sqrt{(2\Delta_{0}\tau)^{2}-1}\right)&\text{for}\,\,\,\,2\Delta_{0}\tau>1\\\
\frac{(2\Delta_{0}\tau)^{2}}{\sqrt{1-(2\Delta_{0}\tau)^{2}}}\tanh^{-1}\left(\sqrt{1-(2\Delta_{0}\tau)^{2}}\right)&\text{for}\,\,\,\,2\Delta_{0}\tau\leq
1\end{array}\right.$ (8) (9)

Though superficially different from the well known $T=0$ result A-G ; Nam ;
Kogan ; M-R , this has also the right clean and dirty limits, namely $n$ and
$n\pi\Delta_{0}\tau$. Although the first term in Eq. (5) is sufficient for
extreme dirty limit $(\Delta_{0}\tau<<1)$ as mentioned above, it alone is
incomplete when $\Delta_{0}\tau\sim 1$ as it can exceed the London limit,
namely the electron density $n$! We show variations of the first and second
terms of Eq.(5) and their difference, i.e., $n_{s}/n$ over several decades of
$\Delta_{0}\tau$ at $T=0$ in Fig. 1. Contribution of the second term is
negligible as it is less by 3 orders of magnitude than the first term when
$\Delta_{0}\tau=10^{-4}$. However, the role of the former begins to be
significant even for $\Delta_{0}\tau=5\times 10^{-3}$ when the latter is about
$3$% of the former. While both the terms increase with $\Delta_{0}\tau$, the
difference between them asymptotically becomes unity, namely it approaches the
disorder-free London limit.The zero temperature value of $n_{s}$ depends
strongly on $\Delta_{0}\tau$ and attains the pure limit for
$\Delta_{0}\tau\sim 10$ while it has the dirty limit value for
$\Delta_{0}\tau\lesssim 0.005$.

The temperature dependence of $n_{s}$ is numerically calculated using a
dimensionless form of the variables and parameters of Eq. (5) and reinstating
$\hbar$ as appropriate:

$\displaystyle\frac{n_{s}}{n}$ $\displaystyle=$
$\displaystyle\pi\tilde{\Delta}\left(\frac{\Delta_{0}\tau}{\hbar}\right)\tanh\left(\frac{\delta\tilde{\Delta}}{T/T_{c}}\right)$
(10) $\displaystyle-$
$\displaystyle\tilde{\Delta}^{2}\int_{\tilde{\Delta}}^{\infty}d\tilde{\epsilon}\frac{\tanh\left(\frac{\delta\tilde{\epsilon}}{T/T_{c}}\right)}{\sqrt{\tilde{\epsilon}^{2}-\tilde{\Delta}^{2}}(\tilde{\epsilon}^{2}-\tilde{\Delta}^{2}+\frac{1}{4(\Delta_{0}\tau/\hbar)^{2}})}.$

where $\tilde{\Delta}=\Delta/\Delta_{0}$,
$\tilde{\epsilon}=\epsilon/\Delta_{0}$, and $\delta=\Delta_{0}/(2k_{B}T_{c})$.
Figure 2 shows the temperature dependence of $n_{s}/n$ for a wide range of
$\Delta_{0}\tau$ (in the unit of $\hbar$) and using the BCS value of
$\delta=0.882$. As expected, temperature dependence of $n_{s}$ at low
temperatures is exponentially weak due to the presence of gap $\Delta_{0}$,
but it strongly depends on $T$ beyond a threshold value $T_{\rm th}$ and
eventually vanishes at $T=T_{c}$. Inset of Fig. 2 shows temperature dependence
of scaled $n_{s}(T)$ by its zero-temperature value $n_{s0}$ for wide range of
$\Delta_{0}\tau$ (scaled by $T_{c}$). The unrecognizable differences of
$n_{s}(T)/n_{s}(0)$ with disorder indicates that the experimental techniques
in which absolute value (in lieu of relative value with respect to zero
temperature) of $n_{s}(T)$ is measured is the only one suitable for studying
the disorder dependence of superfluid density.

##  III Comparison with Experiment

Table 1: Experimental data of $T_{c}$, $\Delta(0)$, $\lambda(0)$, $n$, and normal-state resistivity $\rho_{{}_{N}}$, mean free path $\ell$ and effective mas $m^{\ast}$ of an electron obtained from a number of experiments Devlin ; Megerle ; Wyckoff ; Kittel ; Rinderer83 ; Lemberger07 ; Sutton ; Karim ; Mondal11 ; Chand ; Mattheiss ; Chockalingam ; Mandal20 ; Dressel ; Lin  in various samples.  Sample | $T_{c}$ | $\Delta(0)$ | $\lambda(0)$ | $n$ | $\rho_{{}_{N}}$ | $\ell$ | $m^{\ast}/m_{e}$
---|---|---|---|---|---|---|---
| (K) | (meV) | (nm) | ($10^{28}$ m-3) | ($\mu\Omega$-m) | $(A^{o})$ |
Sn | 3.72 Devlin  | 0.555 Megerle  | 42.5 Devlin  | 14.8 Wyckoff  | *** | *** | 1.26 Kittel
Pb | 7.2 Rinderer83  | 1.34 Megerle  | 52.5 Rinderer83  | 13.2 Wyckoff  | *** | *** | 1.97 Kittel
Nb (15.3nm) | 8.17 Lemberger07  | 1.525 Sutton  | 135.08 Lemberger07  | 5.56 Wyckoff  | 0.135 Lemberger07  | 64.6 | 1.81 Karim
NbN-1111$n$ and $\rho_{{}_{N}}$ of NbN is obtained by interpolation using given data set of Ref. Chand, . The three samples correspond to different levels of disorder. | 14.3 Mondal11  | 2.5 Chand  | 358.4 Mondal11  | 16.85 Chand  | 1.14 Chand  | 3.65 | 1.0 Mattheiss ; Chockalingam
NbN-2a | 9.94 Mondal11  | 1.736 Chand  | 583.9 Mondal11  | 11.6 Chand  | 2.22 Chand  | 2.41 | 1.0 Mattheiss ; Chockalingam
NbN-3a | 8.5 Mondal11  | 1.485 Chand  | 759.1 Mondal11  | 11.76 Chand  | 2.41 Chand  | 2.2 | 1.0 Mattheiss ; Chockalingam
MoGe-1 (21 nm)222Three amorphous MoGe thin films with different thickness (within bracket). The carrier density is measured from Hall effect for MoGe-1 and assumed to remain same for other thickness. | 7.56 Mandal20  | 1.28 Mandal20  | 528 Mandal20  | 46 Mandal20  | 1.5 Mandal20  | 1.42 | 1.0
MoGe-2 (11nm)b | 6.62 Mandal20  | 1.25 Mandal20  | 554.6 Mandal20  | 46 Mandal20  | 1.64 Mandal20  | 1.3 | 1.0
MoGe-3 (4.5nm)b | 4.8 Mandal20  | 1.12 Mandal20  | 613.07 Mandal20  | 46 Mandal20  | 1.44 Mandal20  | 1.48 | 1.0
Nb-doped STO | 0.346 Scheffler  | 0.052 Scheffler  | 1349.5 Scheffler  | 0.011 Scheffler  | 0.52 Scheffler  | *** | 4.0 Lin
Figure 3: (Color online) (a) Experimental data (black dots) of $n_{s}/n$ vs.
$T$ for Nb-doped SrTiO3 fitted with Eq.(7); blue and green curves respectively
represent the contributions of 1st and 2nd terms of Eq. (7). (b) Fit of the
same data but with the dirty limit BCS formula which is equivalent to taking
only the first term of Eq.(7); the fit deviates at intermediate temperatures.
(c)–(e) Experimental data for Pb, Sn crystal, and 15.3-nm thick Nb film
respectively; red solid curves are the theoretical fits using Eq.(7). (f) and
(g) respectively correspond to temperature dependence of $n_{s}/n$ for NbN and
MoGe films with various thickness; solid lines are the theoretical fits using
Eq.(7); this fit is indistinguishable from the contribution of the 1st term of
Eq. 71) alone. (h) The ratio of the contributions of the 2nd and 1st terms of
Eq. (7), $\eta$, vs. the parameter $\Delta_{0}\tau$ (in the unit of $\hbar$)
(solid line) and the same extracted from the fits mentioned above for various
samples (dots).

In this section, we analyze some of the published experimental data of
$n_{s}(T)$ which are extracted from the measured penetration depth using
London’s formula Tinkham :

$n_{s}=\frac{m^{\ast}}{\mu_{0}e^{2}}\lambda^{-2}=2.82\times
10^{13}\left(\frac{m^{\ast}}{m_{e}}\right)m^{-1}\lambda^{-2}$ (11)

in the light of the expression (10) derived here, where $m^{\ast}$ is the
effective mass of an electron in a system. One difficulty in comparison
between theory and experiment is that in much of the literature on
conventional superconductors, only the change of penetration depth with
respect to a given temperature rather than the absolute value of $\lambda$ has
been measured in bulk sample. Absolute values have been measured for colloidal
particles Shoenberg ; Lock  and large area thin films on mica, but for those
samples it is difficult to estimate other properties like resistivity and
carrier density which could significantly differ from bulk and have not been
reported. Nevertheless, researchers used indirect schemes to estimate
$\lambda(0)$. For example, in Ref. Rinderer83,  for Pb, $\Delta$ obtained from
tunneling was used as input parameter and $\lambda(0)$ was obtained from
tuning it to the value that consistently reproduced the BCS temperature
dependence $\lambda(T)$ for a set of samples with different amount of
impurity. In some other cases such as in pure Sn crystal Devlin , $\lambda(0)$
was estimated from the normal state properties. More recently, absolute
measurement of $\lambda$ have been performed on a number of superconducting
thin films using two-coil mutual inductance technique Benfatto ; Kamlapure ;
Bose  and on some single crystals using microwave techniques Hafner . Here, we
analyze the data of Nb-doped STO Scheffler  and Sn crystal Devlin ,
polycrystalline Rinderer83  Pb and 15.3 nm thick Nb film Lemberger07 , and
relatively stronger disordered thin films Mondal11 ; Mandal20  of NbN and
a-MoGe. Although Nb-doped SrTiO3 was initially thought to be a multi-band
superconductor Bednorz , the recent data are in favor of a single-band
Thiemann2 ; Hwang ; Eagles  superconductor. Together these systems span a
large range of disorder for which $n_{s}/n\sim 0.6$–$10^{-4}$. In Table 1, we
summarize the properties of these materials. For Sn and Pb, the authors
reported $\lambda(T)$ vs. $\left(1-(T/T_{c})^{4}\right)^{1/2}$; the data was
digitized and converted into $\lambda^{-2}(T)$ vs. $T$. One important
parameter in Table 1 is the effective mass of the electron. This value is
taken either from electronic specific heat (Sn, Pb, NbN) or quantum
oscillations (Nb-doped STO and Nb). For a-MoGe, we did not find an independent
estimate but used the electron mass as has been done in the literature Tashiro
. In figure 3(a)–(g), we show the temperature variation $n_{s}/n$ for
different materials. We first focus on the Nb-doped SrTiO3 crystal which is
the cleanest sample analyzed here. In Fig. 3(a) we fit $n_{s}(T)/n$ using the
full expression in Eq.(10) using the values of $\delta$ as shown in Table 2
and $\alpha$ as the only adjustable parameter. In the same panel we also
separately plot the 1st and 2nd term on the right hand side of Eq. (10). In
Fig. 3(b), we try to fit the same data using only first term which is
equivalent to the dirty limit expression in Eq. (1). As can be seen best fit
curve deviates at high temperature, showing at this level of cleanliness a
small but discernible difference in the T-dependence emerges between the exact
expression and the dirty-limit BCS expression. For Sn, Pb, Nb film (Fig.
3(c)-3(e)) as $n_{s}/n$ decreases, the contribution of the 2nd term in the
overall expression progressively decreases. For the strongly disordered NbN
and a-MoGe films (Fig. 3(f)-3(g)) the contribution of the 2nd term is
negligible and the data can be fitted with the dirty limit BCS expression. The
extracted parameters from the fits are also shown in Table 2. Wherever
resistivity data is available the values of $\tau$ extracted from the present
fits, $\tau_{{}_{P}}$ are consistent with those obtained from resistivity,
$\tau_{{}_{T}}$, using Drude model. In Fig. 3(h), we show the ratio of the
second term to the first term, $\eta$, as a function of $\Delta_{0}\tau$. It
is obvious from the graph that the cleanest superconductor analyzed here, Nb-
doped STO, is far from the BCS clean limit for which $n_{s}(0)/n\sim 1$ and
$\Delta_{0}\tau>>1$. Most studies on pure elemental superconductor show
$n_{s}/n=0.05$–$0.3$ Tai ; Pippard ; Dressel . Surprisingly, there is one
report Strongin  where $n_{s}/n$ values very close to one was reported for
very pure polycrystalline Ta and Nb. However, in that paper $\lambda(0)$
values were obtained from $\lambda(T)$ close to $T_{c}$. However for the same
sample, the low temperature variation of $\lambda(T)$ showed unexpected
distinct deviation from BCS variation, probably from surface contamination.
Similarly it was suggested that Nb-doped SrTiO3 could be in the clean limit
Collignon  but this has been contested from direct measurements of the
penetration depth Scheffler . Therefore there is a need for further
measurements on high purity single crystals to explore if the BCS limit can
indeed be realized.

##  IV Outlook and Conclusion

Our analysis is based on the Born approximation for disorder potential. We
thus have not considered localization effect which plays a major role for
strongly disordered superconductors when $k_{F}\ell\sim 1$ (where $k_{F}$ is
the Fermi wavenumber and $\ell$ is the mean free path of an electron). The
superfluid density presented here is without consideration of higher order
effects due to phase fluctuations which again finds its role for relatively
large disorder when $\alpha=\Delta_{0}\tau/\hbar\lesssim 10^{-5}$, and hence
the physics of pseudogap phase Pratap_jpcm  has also been ignored.

Our study reveals that the absolute measurement of superfluid density at all
temperatures, rather than the relative measurement with respect to a given
$T$, is necessary for determining its dependence on disorder. This is because
$n_{s}(T)/n_{s}(0)$ is weakly disorder dependent while both $n_{s}(T)$ and
$n_{s}(0)$ are disorder dependent. This analysis is based on the assumption
that $\Delta$ is disorder independent, as a consequence of Anderson’s theorem
Anderson .

We find that the estimated relaxation time from the resistivity data and from
the fitted parameter $\alpha$ are in the same ballpark for all the samples
those have been analyzed, excepting purer samples Pb and Sn for which
resistivity data are not available for comparison. One surprising finding in
this study is that most samples on which the temperature dependence of the
superfluid density has been investigated seem to be in the dirty limit where
$n_{s}(0)<<n$. In fact, the paradigmatic BCS clean limit seems to be very
rare. To achieve the clean BCS limit the superconductor needs to have a large
electronic relaxation time, $\tau>\hbar/\Delta_{0}\sim 10^{-11}$–$10^{-12}$s,
which translates into an electronic mean free path, $\ell$, greater than tens
of micrometers. Such a large $\ell$ is indeed very rare and has been realized
in very high purity single crystals of noble metals like Ag and semimetals
like Bi on which electron focusing experiments Benistant83 ; Heil  were
performed. This requirement is even more stringent than the mean free path
required in typical single crystals on which de Haas-van Alphen measurements
are performed at fields of several Tesla. It will be instructive to try to
synthesize superconductors with comparable mean free path to experimentally
verify the temperature variation of $n_{s}/n$ from the clean-limit BCS theory.

Table 2: Parameters calculated using or extracted from the experimental data shown in table 1 for all the samples. Relaxation time calculated using the transport data, $\tau_{{}_{T}}=m^{\ast}/(ne^{2}\rho_{{}_{N}})$, and the same calculated using the parameter $\alpha$ extracted by fitting $n_{s}/n$ with Eq. (10), $\tau_{{}_{P}}$, are in the same ballpark. Sample | $n_{s}(0)$ | $\frac{n_{s}(0)}{n}$ | $\delta=\frac{\Delta(0)}{2k_{B}T_{c}}$ | $\alpha=\frac{\Delta(0)\tau}{\hbar}$ | $\tau_{{}_{T}}=\frac{m^{\ast}}{ne^{2}\rho_{{}_{N}}}$ | $\tau_{{}_{P}}=\frac{\alpha\hbar}{\Delta(0)}$
---|---|---|---|---|---|---
| ($10^{25}$m-3) | $(10^{-3})$ |  | $(10^{-3})$ | ($10^{-17}$ s) | ($10^{-17}$ s)
Sn | 1967.17 | 132.92 | 0.865 | 48.5 | *** | 5751.8
Pb | 2015.56 | 152.69 | 1.082 | 63 | *** | 3094
Nb (15.3nm) | 279.7 | 50.3 | 0.96 | 17.7 | 855 | 763.9
NbN-1 | 21.94 | 1.30 | 1.0135 | 0.415 | 18.4 | 10.9
NbN-2 | 8.27 | 0.713 | 1.0135 | 0.228 | 13.8 | 8.64
NbN-3 | 4.89 | 0.416 | 1.0135 | 0.134 | 12.5 | 5.93
MoGe-1 (21 nm) | 10.11 | 0.219 | 1.06 | 0.0694 | 5.14 | 3.57
MoGe-2 (11nm) | 9.17 | 0.199 | 1.116 | 0.0638 | 4.7 | 3.36
MoGe-3 (4.5nm) | 7.50 | 0.163 | 1.3 | 0.0518 | 5.35 | 3.04
Nb-doped STO | 6.2 | 563.1 | 0.875 | 500 |  $2.5\times 10^{5}$ Scheffler  | $6.3\times 10^{5}$

##  Appendix A Sum Rule for the Suppression of Superfluid Density

While the clean BCS limit can only be reached in specially prepared very clean
single crystals, frequently available polycrystalline and thin film
superconductors are in the opposite limit, i.e., dirty limit where,
$\tau\ll\Delta_{0}/\hbar$. In such a situation, $n_{s}(0)\ll n$. $n_{s}/n$ can
be intuitively estimated based on the oscillator sum rule Kubo ; Glover ;
Ferrell  that gets the result correct within a factor of order unity; here we
outline this derivation and compare with the accurate expression of $n_{s}$
that has already been derived microscopically in this paper (9) and originally
by Abrikosov and Gorkov A-G  in the linear response theory.

The optical conductivity of a metal in Drude theory is given by
$\sigma(\omega)=\sigma^{\prime}(\omega)+i\sigma^{\prime\prime}(\omega)$ where

$\sigma^{\prime}(\omega)=\frac{\sigma_{0}}{1+(\omega\tau)^{2}}\,\,\,;\,\,\,\sigma^{\prime\prime}(\omega)=\frac{\sigma_{0}\omega\tau}{1+(\omega\tau)^{2}}$
(12)

with dc conductivity $\sigma_{0}=ne^{2}\tau/m_{e}$. The well known oscillator
sum rule for $\sigma^{\prime}(\omega)$ is given by

$\int_{0}^{\infty}\sigma^{\prime}(\omega)\,d\omega=\frac{\pi
ne^{2}}{2m_{e}}\,.$ (13)

The sum rule in Eq. (13), however, remains unaltered for finite temperature,
magnetic field, the presence of interaction between electrons, and even when
the metallic system makes a phase transition into the superconducting state.
However, the spectral weight in $\sigma^{\prime}(\omega)$ is redistributed,
depending on the state of the system.

When a metal goes into the superconducting state, a spectral gap opens for the
frequency $\omega<2\Delta_{0}/\hbar$. At a very high frequency
$(\omega>>2\Delta_{0}/\hbar)$, the distribution of spectral weight in the real
part of conductivity in the superconducting state,
$\sigma_{s}^{\prime}(\omega)$, remains unaltered from its metallic
counterpart. $\sigma_{s}^{\prime}(\omega)$ approaches zero as $\omega\to
2\Delta_{0}/\hbar$ from its higher values. However, this depletion of spectral
weight gets accumulated at zero frequency in the form of Dirac delta function:

$\sigma^{\prime}_{s}(\omega)=\frac{\pi n_{s}e^{2}}{m_{e}}\delta(\omega)$ (14)

where the prefactor $\pi n_{s}e^{2}/m_{e}$ is known as Drude weight to the
conductivity that is proportional to the superfluid density. The precise
variation of $\sigma_{s}(\omega)$ for a s-wave superconductor may be obtained
from Mattis-Bardeen theoryBardeen . However for the purpose of an approximate
estimation of $n_{s}$, we consider a discontinuous jump in
$\sigma^{\prime}_{s}(\omega)$ at $\omega=2\Delta_{0}/\hbar$ from its zero
value to normal-metallic value. Following the sum rule (13), we thus write

$\int_{0}^{2\Delta_{0}/\hbar}\sigma^{\prime}(\omega)d\omega\approx\int_{0}^{2\Delta_{0}/\hbar}\sigma^{\prime}_{s}(\omega)d\omega$
(15)

which yields

$\frac{n_{s}}{n}=\frac{2}{\pi}\tan^{-1}\left(\frac{2\Delta_{0}\tau}{\hbar}\right)$
(16)

reproducing the clean limit ($\Delta_{0}\tau\to\infty$), i.e., $n_{s}=n$. In
the dirty limit ( $\Delta_{0}\tau\to 0$), we find
$n_{s}/n=4\Delta_{0}\tau/(\pi\hbar)$ which differs with the microscopic result
only by a numerical factor $\pi^{2}/4$.

It is instructive to write Eq.(16) in terms of the measurable quantities such
as penetration depth and normal state resistivity
$\rho_{{}_{N}}=1/\sigma_{0}$. Substituting $n_{s}$ by
$(m_{e}/\mu_{0}e^{2})\lambda^{-2}(0)$ in Eq.(16) and reinstating the above
mentioned factor $\pi^{2}/4$, we find

$\lambda^{-2}(0)=\frac{\pi\mu_{0}\Delta_{0}}{\hbar\rho_{{}_{N}}}$ (17)

in the dirty limit. The relation (17) is particularly powerful as it relates
three independent measurable quantities $\lambda(0)$, $\Delta_{0}$ and
$\rho_{{}_{N}}$ without any adjustable parameters.

## Acknowledgments

PR would like to thank Mohit Randeria for valuable discussions in 2008 on the
connection between the oscillator sum rule and superfluid density. We thank
Thomas Lemberger and Marc Scheffler for sharing data on Nb and Nb-doped SrTiO3
respectively. We also thank Marc Scheffler and Peter Armitage for valuable
online discussions and feedback after an early draft of this paper was
circulated. We acknowledge financial support by the Department of Atomic
Energy, Govt of India (Project No: RTI4003).

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