# Existence of solutions to a generalized self-dual Chern-Simons equation on
finite graphs

Yuanyang Hu1

###### Abstract.

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions
for the following generalized Chern-Simons equation on $G$

$\Delta
u=\lambda\mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4\pi\sum_{s=1}^{N}\delta_{p_{s}}\quad,$

where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vetex $p_{s}$,
and $p_{1},p_{2},\dots,p_{N}$ are arbitrarily chosen distinct vertices on the
graph. We show that there exists a critial value $\hat{\lambda}$ such that
when $\lambda>\hat{\lambda}$, the generalized Chern-Simons equation has at
least two solutions, when $\lambda=\hat{\lambda}$, the generalized Chern-
Simons equation has a solution, and when $\lambda<\hat{\lambda}$, the
generalized Chern-Simons equation has no solution.

###### Key words and phrases:

Chern-Simons equation, finite graph, existence, uniqueness, the variational
method

###### Key words and phrases:

variational method, mountain-pass theorem, Chern-Simons equation, finite
graph, equation on graphs

1 School of Mathematics and Statistics, Henan University, Kaifeng, Henan
475004, P. R. China.

Emails<EMAIL_ADDRESS>(Y. Hu).

##  1\. Introduction

Abrikosov [2] considered Magnetic vortex configurations in the context of
Ginzburg–Landau theory of superconductivity. Later, Nielsen and Olesen
stressed the relevance to high-energy physics of vortex-line solutions of the
Abelian Higgs model in the context of dual string models [19]. Since then the
interest on vortices has continued to grow both in condensed-matter and
particle physics. A self-dual Chern-Simons model now plays an important role
in various areas of physics, many researchers did a lot of significant work on
the existence of vortices in this Chern-Simons model [6, 8, 12, 21, 22].
Caffarelli and Yang [9] established the existence of condensates or periodic
multivortices in the Abelian Chern-Simons-Higgs model. Bazeia, da Hora, dos
Santos [6], proposed a generalized Chern-Simons model, and obtained a
generalized self-dual Chern-Simons equation. Later, Han [12] established the
existence of doubly periodic multi-vortices solutions to the generalized self-
dual Chern-Simons model.

In recent years, increasing efforts have been devoted to the development of
analysis on graph; see, for example, [3, 6, 16, 13, 17] and the references
therein. Bendito et al. [5] considered the discrete analogues of boundary
value problems for Laplacian operator on Riemannian manifolds. Lin and Wu [18]
established the existence and nonexistence of global solutions for the
following semilinear heat equation on G

$\left\\{\begin{array}[]{lc}u_{t}=\Delta u+u^{1+\alpha}&\text{ in
}(0,+\infty)\times V,\\\ u(0,x)=a(x)&\text{ in }V,\end{array}\right.$

where $G=(V,E)$ is a finite or locally finite connected weighted graph. Bauer
et al. [7] established the Li-Yau gradient estimate for the heat kernel on
graphs. Horn et al. [15] proved Li-Yau-type estimates for bounded and positive
solutions of the heat equation on graphs. Grigor’yan, Lin and Yang [3] studied
Kazdan-Warner equation on a finite graph. Ge, Hua and Jiang [10] studied the
Liouville equation

$-\Delta u=e^{u}$

on a graph $G$ satisfying a certain isoperimetric inequality. In [11], Ge and
Jiang studied the $1$-Yamabe equation

(1.1)
$\Delta_{1}u+g\operatorname{Sgn}(u)=h|u|^{\alpha-1}\operatorname{Sgn}(u),$

on connected finite graph, where $\Delta_{1}$ is the discrete $1$-Laplacian,
$\alpha>1$ and $g,h>0$ are known. In [16], Huang, Lin and Yau studied the mean
field equation

(1.2) $\Delta u+e^{u}=\rho\delta_{0}$

on $V$, where $G=(V,E)$ is a connected finite graph, constant $\rho>0$ and
$\delta_{0}$ is the Dirac mass at the zero point, and proved an existence
result to (1.2), which is different from that on the cotinuous limit-the two
dimensional flat tori, when $\rho=8\pi$.

Let $G=(V,E)$ be a connected finite graph, where $V$ denotes the vetex set and
$E$ denotes the edge set. In this paper, we estabilish the existence of
solutions to following generalized self-dual Chern-Simons equation

(1.3) $\Delta
u=\lambda\mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4\pi\sum_{s=1}^{N}\delta_{p_{s}}\quad\text{
on }\quad V,$

where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vetex $p_{s}$,
and $p_{1},p_{2},\dots,p_{N}$ are arbitrarily chosen distinct vertices on the
graph.

We are now ready to delineate the major result of this paper.

######  Theorem 1.1.

There exists a critical value

$\hat{\lambda}\geq\frac{6^{6}}{5^{5}}\frac{4\pi N}{\text{Vol}(V)}$

such that if $\lambda>\hat{\lambda}$, then (1.3) has at least two solutions,
if $\lambda=\hat{\lambda}$, then (1.3) has a solution, and if
$\lambda<\hat{\lambda}$, then (1.3) admits no solution.

The rest of the paper is arranged as below. In section 2, we present some
results that we will use in the following pages. In section 3, we prove
Theorem 1.1.

##  2\. Preliminary results

For each edge $xy\in E$, We suppose that its weight $w_{xy}>0$ and that
$w_{xy}=w_{yx}$. Set $\mu:V\to(0,+\infty)$ be a finite measure. For any
function $u:V\to\mathbb{R}$, the Laplacian of $u$ is defined by

(2.1) $\Delta u(x)=\frac{1}{\mu(x)}\sum_{y\sim x}w_{yx}(u(y)-u(x)),$

where $y\sim x$ means $xy\in E$. The gradient form of $u$ reads

(2.2) $\Gamma(u,v)(x)=\frac{1}{2\mu(x)}\sum_{y\sim
x}w_{xy}(u(y)-u(x))(v(y)-v(x)).$

We denote the length of the gradient of $u$ by

$|\nabla u|(x)=\sqrt{\Gamma(u)(x)}=\left(\frac{1}{2\mu(x)}\sum_{y\sim
x}w_{xy}(u(y)-u(x))^{2}\right)^{1/2}.$

Denote, for any function $u:V\rightarrow\mathbb{R}$, an integral of $u$ on $V$
by $\int\limits_{V}ud\mu=\sum\limits_{x\in V}\mu(x)u(x)$. Denote
$|V|$=$\text{Vol}(V)=\sum\limits_{x\in V}\mu(x)$ the volume of $V$. For $p\geq
1$, denote $||u||_{p}:=(\int\limits_{V}|u|^{p}d\mu)^{\frac{1}{p}}$. Define a
sobolev space and a norm on it by

$W^{1,2}(V)=\left\\{u:V\rightarrow\mathbb{R}:\int\limits_{V}\left(|\nabla
u|^{2}+u^{2}\right)d\mu<+\infty\right\\},$

and

$\|u\|_{H^{1}(V)}=\|u\|_{W^{1,2}(V)}=\left(\int\limits_{V}\left(|\nabla
u|^{2}+u^{2}\right)d\mu\right)^{1/2}.$

To prove our main result, we need the following Sobolev embedding, Poincaré
inequality and Maximum principle on graphs.

######  Lemma 2.1.

([3, Lemma 5]) Let $G=(V,E)$ be a finite graph. The sobolev space $W^{1,2}(V)$
is precompact. Namely, if ${u_{j}}$ is bounded in $W^{1,2}(V)$, then there
exists some $u\in W^{1,2}(V)$ such that up to a subsequence, $u_{j}\to u$ in
$W^{1,2}(V)$.

######  Lemma 2.2.

([3, Lemma 6]) Let $G=(V,E)$ be a finite graph. For all functions
$u:V\to\mathbb{R}$ with $\int\limits_{V}ud\mu=0$, there exists some constant
$C$ depending only on $G$ such that $\int\limits_{V}u^{2}d\mu\leq
C\int\limits_{V}|\nabla u|^{2}d\mu$.

######  Lemma 2.3.

([16, Lemma 4.1]) Let $G=(V,E)$, where $V$ is a finite set, and $K\geq 0$ is
constant. Suppose a real function $u(x):V\to\mathbb{R}$ satisfies
$(\Delta-K)u(x)\geq 0$ for all $x\in\mathbb{R}$, then $u(x)\leq 0$ for all
$x\in V$.

######  Lemma 2.4.

Let $G=(V,E)$ be a finite graph. For all functions $u:V\to\mathbb{R}$ with
$\int\limits_{V}ud\mu=0$ and $p\geq 1$, there exists a constant $C=C(G,p)$
such that

$||u||_{p}\leq C||\nabla u||_{2}.$

###### Proof.

By Lemma 2.2, there exists $C=C(G)$ such that

$M:=\max\limits_{x\in
V}|u(x)|\leq(\frac{C}{\min\limits_{V}\mu}\int\limits_{V}|\nabla
u|^{2}d\mu)^{\frac{1}{2}}.$

Thus we have $||u||_{p}\leq(\frac{C}{\min\limits_{V}\mu}\int\limits_{V}|\nabla
u|^{2}d\mu)^{\frac{1}{2}}|V|^{1/p}:=C_{2}||\nabla u||_{2}$. ∎

##  3\. The proof of Theorem 1.1

Since $\int\limits_{V}-\frac{4\pi
N}{\text{Vol}(V)}+4\pi\sum\limits_{j=1}^{N}\delta p_{j}d\mu=0$, we can choose
a solution $u_{0}$ of the equation

(3.1) $\Delta u_{0}=-\frac{4\pi
N}{\text{Vol(V)}}+4\pi\sum_{j=1}^{N}\delta_{p_{j}}.$

Letting $u=u_{0}+v$, the equation (1.3) can be reduced to the following
question

(3.2) $\Delta v=\lambda e^{u_{0}+v}\left(e^{u_{0}+v}-1\right)^{5}+\frac{4\pi
N}{\text{Vol}(V)}.$

Set $F(y):=(e^{y}-1)^{5}e^{y}$ on $\mathbb{R}$, it is clear that $F$ has a
unique minimal value $-\frac{5^{5}}{6^{6}}.$ Thus, it follows from (3.2) that

$0=\int\limits_{V}\Delta
vd\mu\geq\lambda\int\limits_{V}-\frac{5^{5}}{6^{6}}d\mu+4\pi
N=-\frac{5^{5}}{6^{6}}\lambda\text{Vol}(V)+4\pi N.$

This implies that

$\lambda\geq\frac{6^{6}}{5^{5}}\frac{4\pi N}{|V|},$

which is a necessary condition for the existence of solutions to (1.3).

To solve (3.2), for a constant $K\geq\lambda$, we define a sequence
$\\{w_{n}\\}$ by a monotone iterative scheme:

(3.3) $\displaystyle(\Delta-K)W_{n}$ $\displaystyle=\lambda
e^{u_{0}+W_{n-1}}\left(e^{u_{0}+W_{n-1}}-1\right)^{5}-KW_{n-1}+\frac{4\pi
N}{\operatorname{Vol}(V)},~{}n=1,2,\ldots$ $\displaystyle W_{0}$
$\displaystyle=-u_{0}.$

Next, we establish a solution to (3.2) by a supersolution and subsolution
method.

######  Definition 3.1.

A function $u$ on $V$ is called a subsolution of (3.2) if

$\Delta u\geq\lambda e^{u_{0}+u}\left(e^{u_{0}+u}-1\right)^{5}+\frac{4\pi
N}{\operatorname{Vol}(V)}.$

######  Lemma 3.1.

Let ${W_{n}}$ be a sequence defined by scheme (3.3) with $K\geq\lambda$. Then

(3.4) $W_{-}\leq\dots\leq W_{n}\leq\dots\leq W_{2}\leq W_{1}\leq W_{0}$

for any subsolution $W_{-}$ of (3.2).

###### Proof.

By (3.1) and (3.3), we have

(3.5) $(\Delta-K)(W_{1}-W_{0})=4\pi\sum_{s=1}^{N}\delta_{p_{s}}>0,x\in V.$

By Lemma 2.3, we deduce that

$(W_{1}-W_{0})(x)\leq 0$

for all $x\in V$. Suppose that

$W_{0}\geq W_{1}\geq\dots\geq W_{k}.$

By (3.3), we conclude that

$\displaystyle(\Delta-K)\left(W_{k+1}-W_{k}\right)$
$\displaystyle=\lambda\mathrm{e}^{u_{0}+W_{k}}\left(\mathrm{e}^{u_{0}+W_{k}}-1\right)^{5}-\lambda\mathrm{e}^{u_{0}+W_{k-1}}\left(\mathrm{e}^{u_{0}+W_{k-1}}-1\right)^{5}-K\left(W_{k}-W_{k-1}\right)$
$\displaystyle=\left[\lambda\mathrm{e}^{u_{0}+\xi}\left(\mathrm{e}^{u_{0}+\xi}-1\right)^{4}\left(6\mathrm{e}^{u_{0}+\xi}-1\right)-K\right]\left(W_{k}-W_{k-1}\right)$
$\displaystyle\geq(\lambda-K)\left(W_{k}-W_{k-1}\right)$ $\displaystyle\geq
0,$

where $W_{k}\leq\xi\leq W_{k-1}$. By Lemma 2.3, we see that $W_{k+1}\leq
W_{k}$ on $V$.

Set $(W_{-}-W_{0})(x_{0})=\max\limits_{x\in V}(W_{-}-W_{0})(x)$, we claim that

$(W_{-}-W_{0})(x_{0})\leq 0.$

Otherwise, $(W_{-}-W_{0})(x_{0})>0$. It follows that

$\Delta(W_{-}-W_{0})(x_{0})\geq\lambda
e^{u_{0}+W_{-}(x_{0})}(e^{u_{0}+W_{-}(x_{0})}-1))^{5}+\frac{4\pi
N}{\text{Vol}(V)}+\Delta u_{0}>0.$

By the definition of Laplace operator, we obtain

$\Delta(W_{-}-W_{0})(x_{0})\leq 0.$

This is a contradicion. Thus we have

$(W_{-}-W_{0})(x_{0})\leq 0,$

which implies that

$W_{-}-W_{0}\leq 0$

on $V$. Assume that $W_{-}-W_{k}\leq 0$ for some integer $k\geq 0$. Thanks to
$W_{-}$ is a subsolution of (3.2) and $K\geq\lambda$, we deduce that

$\displaystyle(\Delta-K)\left(W_{-}-W_{k+1}\right)$
$\displaystyle\geq\lambda\left[\mathrm{e}^{u_{0}+W_{-}}\left(\mathrm{e}^{u_{0}+W_{-}}-1\right)^{5}-\mathrm{e}^{u_{0}+W_{k}}\left(\mathrm{e}^{u_{0}+W_{k}}-1\right)^{5}\right]-K\left(W_{-}-W_{k}\right)$
$\displaystyle\geq\left[\lambda\mathrm{e}^{u_{0}+\eta}\left(\mathrm{e}^{u_{0}+\eta}-1\right)^{4}\left(6\mathrm{e}^{u_{0}+\eta}-1\right)-K\right]\left(W_{-}-W_{k}\right)$
$\displaystyle\geq(\lambda-K)\left(W_{-}-W_{k}\right)$ $\displaystyle\geq 0,$

where $W_{-}\leq\eta\leq W_{k}$. By Lemma 2.3, we have $W_{-}\leq W_{k+1}$ on
$V$.

We now complete the proof. ∎

######  Lemma 3.2.

If $\lambda>0$ is sufficiently large, then there exists a solution of (3.2) on
$V$.

###### Proof.

Assume that $u_{0}$ is a solution of (3.2). Select a constant $Q_{0}$ such
that $u_{0}<Q_{0}$. Let $\hat{W}_{-}\equiv-Q_{0}$ on $V$, then for
sufficiently large $\lambda$, we have

$0=\Delta\hat{W}_{-}>\lambda
e^{u_{0}+\hat{W}_{-}}(e^{u_{0}+\hat{W}_{-}}-1)^{5}+\frac{4\pi
N}{\operatorname{Vol}(V)}.$

Thus $\hat{W}_{-}$ is a subsolution of (3.2). By Lemma 3.1, we get a sequence
$\\{W_{n}\\}$ satisfying

$\hat{W}_{-}\leq\dots\leq W_{n}\leq\dots\leq W_{2}\leq W_{1}\leq-u_{0}.$

Thus we can define $w(x):=\lim\limits_{n\to+\infty}W_{n}(x)$. Letting
$n\to+\infty$ in (3.3), then we know that $w$ is a solution of (3.2). ∎

In order to prove Lemma 3.4, we need the following proposition.

######  Lemma 3.3.

If $u$ is a solution of equation (1.3) in $V$, then $u<0$ on $V$.

###### Proof.

Suppose that

$u(x_{0})=\max\limits_{x\in V}u(x),$

we claim that $u(x_{0})<0$. Suppose by way of contradiction that $u(x_{0})\geq
0$, then

$e^{u(x_{0})}-1\geq 0,$

which implies that $\Delta u(x_{0})>0$. By (2.1), we have

$0\geq\Delta u(x_{0}).$

This is a contradiction. ∎

######  Lemma 3.4.

There exists $\hat{\lambda}\geq\frac{4\pi
N}{\text{Vol}(V)}\frac{6^{6}}{5^{5}}$ such that when
$\lambda\geq\hat{\lambda}$, (1.3) admits a solution, and when
$\lambda<\hat{\lambda}$, (1.3) admits no solutions.

###### Proof.

Denote $A:=\\{\lambda>0|\lambda~{}\text{is such that \eqref{E1} admits a
solution}\\}$. We claim that $A$ is a interval. If $\lambda_{0}\in A$, let
$v^{{}^{\prime}}$ be the solution of (1.3) with $\lambda=\lambda_{0}$. By
Lemma 3.3, we have

$v{{}^{\prime}}<0~{}\text{on}~{}V.$

Set $u^{{}^{\prime}}=v^{{}^{\prime}}-u_{0}$, then

$u^{{}^{\prime}}+u_{0}<0~{}\text{on}~{}V.$

It is easy to check that $u^{{}^{\prime}}$ is a subsolution of (3.2) for
$\lambda\geq\lambda_{0}$. It follows from Lemma 3.1 that $\lambda\in A$ for
$\lambda\geq\lambda_{0}$. Thus $A$ is an interval. Clearly,

$\lambda_{i}:=infA$

is well defined. We can choose a sequence $\\{\lambda_{n}\\}\subset A$ such
that $\lambda_{n}\to\lambda_{i}$. On account of $\lambda_{n}\geq\frac{4\pi
N}{\text{Vol(V)}}\frac{6^{6}}{5^{5}}$, we obtain

$\lambda_{i}\geq\frac{4\pi N}{\text{Vol}(V)}\frac{6^{6}}{5^{5}}.$

For any $\lambda>\hat{\lambda}$, we can find a solution of (3.2) denoted by
$u_{\lambda}(x)$. We next prove that if
$\lambda_{1}>\lambda_{2}>\hat{\lambda}$, then $u_{\lambda_{1}}\geq
u_{\lambda_{2}}$ on $V$. By Lemma 3.3, $u_{0}+u_{\lambda_{2}}<0$. Thus we
deduce that

$\displaystyle\Delta u_{\lambda_{2}}$
$\displaystyle=\lambda_{2}\mathrm{e}^{u_{0}+u_{\lambda_{2}}}\left(\mathrm{e}^{u_{0}+u_{\lambda_{2}}}-1\right)^{5}+\frac{4\pi
N}{\text{Vol}(V)}$
$\displaystyle>\lambda_{1}\mathrm{e}^{u_{0}+u_{\lambda_{2}}}\left(\mathrm{e}^{u_{0}+u_{\lambda_{2}}}-1\right)^{5}+\frac{4\pi
N}{\text{Vol}(V)}.$

and hence that $u_{\lambda_{2}}$ is a subsolution of (3.2) with
$\lambda=\lambda_{1}$. By a similar argument as Lemma 3.1, we can show that

(3.6) $u_{\lambda_{2}}\leq u_{\lambda_{1}}~{}\text{ on }~{}V.$

Thus we can define
$U(x):=\lim\limits_{\lambda\to\hat{\lambda}^{+}}u_{\lambda}(x)\in[-\infty,-u_{0})$.

We claim that

(3.7) $U(x)>-\infty~{}~{}~{}\forall x\in V.$

Suppose that $\lim\limits_{\lambda\to\hat{\lambda}^{+}}u_{\lambda}(x)=-\infty$
for all $x\in V$. Integrating

(3.8) $\Delta u_{\lambda}=\lambda
e^{u_{0}+u_{\lambda}}\left(e^{u_{0}+u_{\lambda}}-1\right)^{5}+\frac{4\pi
N}{\text{Vol}(V)}$

on $V$, we obtain

(3.9) $\displaystyle 0=\int\limits_{V}\Delta u_{\lambda}d\mu$
$\displaystyle=\int\limits_{V}\lambda
e^{u_{0}+u_{\lambda}}\left(e^{u_{0}+u_{\lambda}}-1\right)^{5}d\mu+4\pi N$
$\displaystyle=\lambda\sum_{x\in
V}\mu(x)e^{u_{0}+u_{\lambda}}\left(e^{u_{0}+u_{\lambda}}-1\right)^{5}d\mu+4\pi
N.$

Letting $\lambda\rightarrow\hat{\lambda}^{+}$ in (3.9), we see that $0=4\pi
N$, which is a contradiction. Define

(3.10) $V_{1}=\left\\{x\in
V|\lim_{\lambda\rightarrow\hat{\lambda}^{+}}u_{\lambda}=-\infty\right\\},V_{2}:=\left\\{x\in
V\mid\lim_{\lambda\rightarrow\hat{\lambda}^{+}}u_{\lambda}\text{ exists in
}(-\infty,-u_{0})\right\\}.$

If $V_{1}=\emptyset$, then (3.7) holds. Next, we suppose that
$V_{1}\not=\emptyset$ and $V_{2}\not=\emptyset$. Choose $y_{2}\in V_{2}$, then

$\displaystyle\Delta u_{\lambda}\left(y_{2}\right)$
$\displaystyle=\frac{1}{\mu\left(y_{2}\right)}\sum_{x\sim
y_{2}}w_{xy_{2}}\left(u_{\lambda}(x)-u_{\lambda}\left(y_{2}\right)\right)$
$\displaystyle=\frac{1}{\mu\left(y_{2}\right)}\sum_{y\sim
y_{2},y_{\in}V_{1}}w_{yx_{2}}\left(u_{\lambda}(y)-u_{\lambda}\left(y_{2}\right)\right)+\frac{1}{\mu(y_{2})}\sum_{y\sim
y_{2},y_{\in}V_{2}}w_{yx_{2}}\left(u_{\lambda}(y)-u_{\lambda}\left(y_{2}\right)\right)$
$\displaystyle=:I_{1}(\lambda)+I_{2}(\lambda).$

Clearly,
$\lim\limits_{\lambda\rightarrow\hat{\lambda}^{+}}I_{1}(\lambda)=-\infty$ and
$\lim\limits_{\lambda\rightarrow\hat{\lambda}^{+}}I_{2}(\lambda)$ exists in
$\mathbb{R}$. By (3.8), we have

$\Delta u_{\lambda}(y_{2})\geq\lambda(-\frac{5^{5}}{6^{6}})+\frac{4\pi
N}{\text{Vol}(V)}.$

This is impossible. Thus we have $V_{1}=\emptyset$.

Letting $\lambda\to\hat{\lambda}^{+}$ in (3.8), we can deduce that $U$ is a
solution of (3.2) with $\lambda=\hat{\lambda}$. ∎

Define

(3.11) $I_{\lambda}(v):=\int\limits_{V}\frac{1}{2}|\nabla
v|^{2}+\frac{\lambda}{6}\left(e^{u_{0}+v}-1\right)^{6}+\frac{4\pi
N}{\text{Vol}(V)}vd\mu.$

We may give a sufficient condition under which the problem (3.2) admits a
solution and $I_{\lambda}(v)$ has a minimizer.

######  Lemma 3.5.

If $\lambda>\hat{\lambda}$, then there exists a solution $v_{\lambda}$ of
(3.2) and it is a local minimum of the functional $I_{\lambda}(v)$ defined by
(3.11).

###### Proof.

Thanks to $u_{0}+U(x)<0$, we conclude that $U(x)$ is a subsolution of (3.2)
for $\lambda>\hat{\lambda}$. We define

$A=\left\\{v\in W^{1,2}\mid v\geqslant U\text{ in }V\right\\}\text{. }$

Clearly, $I_{\lambda}$ is bounded from below on $V$. Thus we can define

$\eta_{0}:=\inf\limits_{v\in A}I_{\lambda}(v).$

Set $\\{v_{n}\\}$ be a minimizing sequence and
$v_{n}=v_{n}^{{}^{\prime}}+c_{n}$, $n=1,2,\dots$, where

$c_{n}=\frac{\int\limits_{V}v_{n}d\mu}{\text{Vol(V)}}.$

It is easy to see that

$c_{n}\geq\frac{\int\limits_{V}Ud\mu}{\text{Vol(V)}}.$

Thus, we get

(3.12)
$I_{\lambda}\left(v_{n}\right)\geqslant\int\limits_{V}\frac{1}{2}\left|\nabla
v_{n}\right|^{2}d\mu+\frac{4\pi N}{\text{Vol(V)}}\int\limits_{V}Ud\mu,$

which implies that $\\{\|\nabla v_{n}\|_{2}\\}_{n=1}^{\infty}$ is bounded. By
(3.11), we have

$I_{\lambda}(v_{n})\geq\int\limits_{V}\frac{4\pi N}{\text{Vol(V)}}c_{n}d\mu,$

which implies that

$c_{n}\leq\frac{I_{\lambda}(v_{n})}{4\pi N}.$

Thus, $\\{v_{n}\\}$ is bounded in $W^{1,2}(V)$. Since $V$ is a finite graph,
by passing to a subsequence, there exists $v_{\lambda}(x)$ such that

$v_{n}(x)\to v_{\lambda}(x)$

as $n\to+\infty$ for every $x\in V$. Thus

$I_{\lambda}(v_{\lambda})=\eta_{0}.$

By a similar argument as the appendix of [20], we can deduce that
$v_{\lambda}$ is a solution of (3.2).

We next show that $v_{\lambda}>U$ in $V$. It is easy to check that

(3.13) $\Delta(U-v_{\lambda})>\hat{\lambda}(U-v_{\lambda}).$

By Lemma 2.3, we have $W:=U-v_{\lambda}\leq 0$ on $V$. We claim that

$W(x_{0}):=\max\limits_{V}W<0.$

Otherwise, $W(x_{0})=0$. Clearly, $\Delta W(x_{0})\leq 0$. By (3.13), we
obtain $W(x_{0})<0$, which is a contradiction. Thus we have

$U<v_{\lambda}~{}\text{on}~{}V.$

We claim that $v_{\lambda}$ is a local minimum of $I_{\lambda}(v)$ in $A$. For
any integer $n\geq 1$, we see that

(3.14) $\inf\left\\{I_{\lambda}(w)\mid w\in
W^{1,2}(V),\|w-v_{\lambda}\|_{W^{1,2}(V)}\leq\frac{1}{n}\right\\}=\varepsilon_{n}<I_{\lambda}(v_{\lambda}).$

By a similar argument as above, we can deduce that there exists
$\\{v_{n}\\}\subset W^{1,2}(V)$ satisfying

$\|v_{n}-v_{\lambda}\|_{W^{1,2}(V)}\leq\frac{1}{n}$

and

$I_{\lambda}(v_{n})=\epsilon_{n}.$

Thus we conclude that, by passing to a subsequence, $v_{n}\to v_{\lambda}$ in
$V$ as $n\to+\infty$, and hence that $v_{n}>U$ for sufficiently large $n$.
Therefore, we obtain $I_{\lambda}(v_{n})\geq I_{\lambda}(v_{\lambda})$. This
is a contradiction. ∎

We now prove that $I_{\lambda}(v)$ satisfies the Palais-Smale condition.

######  Lemma 3.6.

Every sequence $\\{v_{n}\\}\subset W^{1,2}(V)$ satisfying

(3.15)
$I_{\lambda}(v_{n})\to\alpha~{}\text{and}~{}\|I^{{}^{\prime}}_{\lambda}(v_{n})\|\to
0~{}\text{as}~{}n\to+\infty$

has a convergent subsequence.

###### Proof.

From (3.15), we have

(3.16) $\frac{1}{2}\left\|\nabla
v_{n}\right\|_{2}^{2}+\frac{\lambda}{6}\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{6}\mathrm{~{}d}x+\frac{4\pi
N}{|V|}\int\limits_{V}v_{n}\mathrm{~{}d}x=\alpha+o(1),\text{as}~{}n\to+\infty$
(3.17)
$\left|\int\limits_{V}\Gamma(v_{n},\varphi)\mathrm{d}x+\lambda\int\limits_{V}e^{u_{0}+v_{n}}\left(e^{u_{0}+v_{n}}-1\right)^{5}\varphi\mathrm{d}x+\frac{4\pi
N}{|V|}\int\limits_{V}\varphi\mathrm{d}x\right|\leq\varepsilon_{n}\|\varphi\|_{W^{1,2}(V)},~{}\epsilon_{n}\to
0$

as $n\to+\infty,$ $\varphi\in H^{1}(V)$. By taking $\varphi=1$ in (3.17), we
have

$\lambda\int\limits_{V}e^{u_{0}+v_{n}}\left(e^{u_{0}+v_{n}}-1\right)^{5}\mathrm{~{}d}x+4\pi
N\leq\varepsilon_{n}|V|^{1/2},$

from which we deduce that

$\displaystyle\frac{\varepsilon_{n}|V|^{1/2}}{\lambda}$
$\displaystyle\geq\frac{4\pi
N}{\lambda}+\int\limits_{V}e^{u_{0}+v_{n}}\left(e^{u_{0}+v_{n}}-1\right)^{5}\mathrm{~{}d}\mu$
$\displaystyle=\frac{4\pi
N}{\lambda}+\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{6}\mathrm{~{}d}\mu+\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{5}\mathrm{~{}d}\mu$
$\displaystyle\geq\frac{4\pi
N}{\lambda}-\frac{1}{6}|V|+\frac{1}{6}\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{6}\mathrm{~{}d}\mu.$

This implies that there exists a constant $C=C(\epsilon_{n},\lambda,|V|)>0$
such that

(3.18) $\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{6}\mathrm{~{}d}\mu\leq
C.$

Hence, we can find $C_{2}>0$ such that

(3.19)
$\int\limits_{V}e^{6\left(u_{0}+v_{n}\right)}\mathrm{d}\mu=\int\limits_{V}\left[\left(e^{u_{0}+v_{n}}-1\right)+1\right]^{6}\mathrm{~{}d}\mu\leq
2^{6}\left[\int\limits_{V}\left(e^{u_{0}+v_{n}}-1\right)^{6}\mathrm{~{}d}\mu+|V|\right]\leq
C_{2}.$

Then by H$\ddot{\text{o}}$lder inequality, there exists $C_{3}>0$ such that

(3.20)
$\int\limits_{V}e^{2\left(u_{0}+v_{n}\right)}\mathrm{d}\mu\leq\left(\int\limits_{V}e^{6\left(u_{0}+v_{n}\right)}\mathrm{d}\mu\right)^{\frac{1}{3}}|V|^{\frac{2}{3}}\leq
C_{3}.$

Similarly, $\int\limits_{V}e^{4\left(u_{0}+v_{n}\right)}\mathrm{d}\mu\leq
C_{4}$ for a suitable constant $C_{4}>0$. Decompose
$v_{n}=v_{n}^{\prime}+c_{n}$, where $\int\limits_{V}v_{n}^{\prime}d\mu=0$ and
$c_{n}\in\mathbb{R}$ for $n=1,2,\dots$. Substituing it in (3.16), we conclude
that

(3.21) $\frac{1}{2}\left\|\nabla
v_{n}^{\prime}\right\|_{2}^{2}+\frac{\lambda}{6}\int\limits_{V}\left(e^{u_{0}+v_{n}^{\prime}+c_{n}}-1\right)^{6}\mathrm{~{}d}\mu+4\pi
Nc_{n}\rightarrow\alpha,$

as $n\to+\infty$, and hence that $c_{n}$ is bounded from above. By (3.16), we
see that there exists an integer $N$ such that

$\alpha-1<I_{\lambda}(v_{n})<\alpha+1$

for $n\geq N$. This implies that

(3.22) $\alpha-1<\frac{1}{2}\left\|\nabla
v_{n}^{\prime}\right\|_{2}^{2}+\frac{\lambda}{6}\int\limits_{V}\left(e^{u_{0}+v_{n}^{\prime}+c_{n}}-1\right)^{6}\mathrm{~{}d}\mu+4\pi
Nc_{n}<\alpha+1.$

From (3.18) and (3.22), we conclude that

(3.23) $\alpha-1+\frac{4\lambda\pi
N}{5}-\left(\frac{\lambda}{6}+\frac{\varepsilon_{n}}{5}\right)|V|<\frac{1}{2}\left\|\nabla
v_{n}^{\prime}\right\|_{2}^{2}+4\pi Nc_{n}<\alpha+1.$

Next we show that $c_{n}$ is bounded from below. Taking $v_{n}^{\prime}$ in
(3.17), by Lemma 2.2, we can find a constant $C_{5}$ such that

(3.24) $\left\|\nabla
v_{n}^{\prime}\right\|_{2}^{2}+\lambda\int\limits_{V}e^{u_{0}+v_{n}}\left(e^{u_{0}+v_{n}}-1\right)^{5}v_{n}^{\prime}\mathrm{d}\mu\leq\varepsilon_{n}\left\|v_{n}^{\prime}\right\|_{W^{1,2}(V)}\leq
C_{5}\varepsilon_{n}\left\|\nabla v_{n}^{\prime}\right\|_{2}.$

This implies that

(3.25) $\displaystyle\|\nabla$ $\displaystyle
v_{n}^{\prime}\|_{2}^{2}+\lambda\int\limits_{V}e^{6\left(u_{0}+c_{n}\right)}\left(e^{6v_{n}^{\prime}}-1\right)v_{n}^{\prime}\mathrm{d}\mu$
$\displaystyle\leq$
$\displaystyle\lambda\int\limits_{V}e^{6\left(u_{0}+c_{n}\right)}v_{n}^{\prime}\mathrm{d}\mu+C_{5}\varepsilon_{n}\left\|\nabla
v_{n}^{\prime}\right\|_{2}$
$\displaystyle+C_{6}\int\limits_{V}e^{u_{0}+v_{n}}\left(e^{4\left(u_{0}+v_{n}\right)}+e^{3\left(u_{0}+v_{n}\right)}+e^{2\left(u_{0}+v_{n}\right)}+e^{u_{0}+v_{n}}+1\right)\left|v_{n}^{\prime}\right|\mathrm{d}\mu.$

By Lemma 2.2, Lemma 2.4 and H$\ddot{\text{o}}$lder inequality, we deduce that

(3.26)
$\int\limits_{V}e^{6\left(u_{0}+c_{n}\right)}v_{n}^{\prime}\mathrm{d}\mu\leq
C_{7}\left\|v_{n}^{\prime}\right\|_{2}\leq C_{8}\left\|\nabla
v_{n}^{\prime}\right\|_{2},$

and

(3.27)
$\int\limits_{V}e^{5\left(u_{0}+v_{n}\right)}\left|v_{n}^{\prime}\right|\mathrm{d}\mu\leq\left(\int\limits_{V}e^{6\left(u_{0}+v_{n}\right)}\mathrm{d}\mu\right)^{\frac{5}{6}}\left(\int\limits_{V}\left|v_{n}^{\prime}\right|^{6}\mathrm{~{}d}\mu\right)^{\frac{1}{6}}\leq
C_{9}\left\|v_{n}^{\prime}\right\|_{6}\leq C_{10}\left\|\nabla
v_{n}^{\prime}\right\|_{2}$

for suitable positive constants $C_{7}-C_{10}$. Similarly, we can get all the
other terms on the right hand side of (3.25) can be bounded by
$\hat{C}||\nabla v_{n}^{\prime}||_{2}$, where $\hat{C}>0$ is a constant. Thus,
there exists constant $C_{11}>0$ such that

(3.28) $\left\|\nabla
v_{n}^{\prime}\right\|_{2}^{2}+\lambda\int\limits_{V}e^{6\left(u_{0}+c_{n}\right)}\left(e^{6v_{n}^{\prime}}-1\right)v_{n}^{\prime}\mathrm{d}\mu\leq
C_{11}\left\|\nabla v_{n}^{\prime}\right\|_{2}.$

Clearly,

(3.29)
$\int\limits_{V}e^{6\left(u_{0}+c_{n}\right)}\left(e^{6v_{n}^{\prime}}-1\right)v_{n}^{\prime}\mathrm{d}\mu\geq
0.$

Hence by (3.28), we have $||\nabla v_{n}^{\prime}||_{2}\leq C_{12}$ for a
suitable constant $C_{12}>0$. Therefore, by (3.23), we deduce that $c_{n}$ is
bounded from below.

Thus $\\{v_{n}\\}$ is bounded in $H^{1}(V)$. Thus, there exists $v\in
H^{1}(V)$ such that, by passing to a subsequence, $v_{n}(x)\to v(x)$ for all
$x\in V$.

Next, we find the second solution of (3.2). From now on, we suppose that
$v_{\lambda}$ is the local minimum as defined by Lemma 3.5(if not, we could
have already found our second solution). Thus there exists $\rho_{0}>0$ such
that

$I_{\lambda}(v_{\lambda})\leq I_{\lambda}(v)$

for all $v:||v-v_{\lambda}||_{H^{1}(V)}\leq\rho_{0}.$ For $c>0$, we have

(3.30) $\displaystyle
I_{\lambda}\left(v_{\lambda}-c\right)-I_{\lambda}\left(v_{\lambda}\right)=$
$\displaystyle\frac{\lambda}{6}\int\limits_{V}\left[\left(e^{u_{0}+v_{\lambda}-c}-1\right)^{6}-\left(e^{u_{0}+v_{\lambda}-1}\right)^{6}\right]d\mu-4\pi
Nc$ $\displaystyle<\frac{\lambda}{6}|V|C_{13}-4\pi Nc\rightarrow-\infty\text{
as }c\rightarrow+\infty.$

There are two possibilities: (I) $v_{\lambda}$ is not a strict local minimum
for $I_{\lambda}$, (II) $v_{\lambda}$ is a strict local minimum for
$I_{\lambda}.$ If case (I) happens, then we deduce that

$\inf\limits_{\left\|v-v_{\lambda}\right\|_{H^{1}(V)}=\rho}I_{\lambda}=I_{\lambda}\left(v_{\lambda}\right)=:\alpha_{\lambda}$

for all $0<\rho<\rho_{0}.$ It follows that there exists a local minimum
$v_{\rho}\in H^{1}(V)$ such that

$||v_{\rho}-v_{\lambda}||=\rho,$

and $I_{\lambda}(v_{\rho})=\alpha_{\lambda}$ for all $\rho\in(0,\rho_{0})$.
Therefore, in this situation, we get a one-parameter family of solutions of
(3.2). If case (II) happens, we can find $\rho_{1}\in(0,\rho_{0})$ such that

(3.31)
$\inf_{\left\|v-v_{\lambda}\right\|_{H^{1}(V)}=\rho_{1}}I_{\lambda}(v)>I_{\lambda}\left(v_{\lambda}\right)=\alpha_{\lambda}.$

By (3.30), we deduce that

$I_{\lambda}\left(u_{\lambda}-c_{0}\right)\leqslant
I_{\lambda}\left(u_{\lambda}\right)-1<I_{\lambda}\left(v_{\lambda}\right)$

for some $c_{0}>|V|^{-\frac{1}{2}}\rho_{1}$.

Define $\mathcal{P}=\left\\{\gamma:[0,1]\rightarrow H^{1}(V)\right.$ $|\gamma$
is continuous and satisfies
$\left.\gamma(0)=v_{\lambda},\gamma(1)=v_{\lambda}-c_{0}\right\\}$ and

$\alpha=\inf\limits_{\gamma\in\mathcal{P}}\sup\limits_{t\in[0,1]}I_{\lambda}(\gamma(t)).$

From (3.31), we conclude that

$\alpha>I_{\lambda}\left(v_{\lambda}\right)\geqslant\max\left\\{I_{\lambda}(\gamma(0)),I_{\lambda}(\gamma(1))\right\\}~{}\forall\gamma\in\mathscr{P}.$

Thus, by Lemma 3.6 , $I_{\lambda}$ satisfies the hypothesis of the mountain-
pass theorem. Thus $\alpha$ is a critical point of $I_{\lambda}$. By virtue of
$\alpha>I_{\lambda}(v_{\lambda})$, we get a second solution of (3.2).

We now complete the proof of Theorem 1.1. ∎

###### Acknowledgements..

The authors would like to thank the anonymous Referees for their valuable
comments which helped to improve the manuscript. ∎

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# Practically universal representation of the Helfand-Werthamer
upper critical field for any transport scattering rate

Ruslan Prozorov  Ames Laboratory, Ames 50011, IA, U.S.A.  Department of
Physics & Astronomy, Iowa State University, Ames 50011, U.S.A.  Vladimir G.
Kogan  Ames Laboratory, Ames 50011, IA, U.S.A.

(20 July 2024)

###### Abstract

The simplified scaling of the orbital upper critical field, $H_{c2}(T)$, for
the isotropic case is discussed. To facilitate the analysis of the
experimental data, we suggest a simple but accurate approximation in the
entire temperature range of the scaled upper critical field valid for any
transport scattering rate
$H_{c2}/H_{c2}(0)\approx(1-t^{2})/(1+0.42\,t^{1.47})$.

##  I Introduction

The upper critical field, $H_{c2}$, is one of the fundamental parameters
characterizing a superconductor. Its temperature dependence is linked to the
anisotropies of the Fermi surface and the order parameter. It is also
sensitive to magnetic (spin-flip) and non-magnetic (potential or transport)
scattering as well as paramagnetic enhancement of the internal magnetic field
and spin-orbit scattering. Analysis of $H_{c2}\left(T\right)$ can provide
important information about these parameters.

The evaluation of $H_{c2}\left(T,\rho\right)$ for isotropic $s-$wave
superconductor with arbitrary transport (non-magnetic) scattering rate,

$\rho=\frac{\hbar}{2\pi T_{c}\tau}\,,$ (1)

where $\tau$ is the mean scattering time, was derived by E. Helfand and N. R.
Werthamer (HW) [1]. The transport dirty limit was discussed by De Gennes [2]
and Maki [3].

In their seminal work [1], HW have shown that the upper critical field
normalized on its slope at $T_{c}$,

$h^{*}=-\frac{H_{c2}}{T_{c}(dH_{c2}/dT)_{T_{c}}}$ (2)

only weakly depends on the scattering rate, $\rho$, as shown in Fig. 1. One
can say that $h^{*}(t,\rho)$ is close to universal for any $\rho$. It reaches
zero-temperature values, $h^{*}(0)=7\zeta(3)e^{2-C}/48\approx 0.7273$ in the
clean case $\rho=0$ and $h^{*}(0)=(\pi^{2}/8)e^{-C}\approx 0.6927$ in the
dirty limit $\rho>>1$ ($C\approx 0.5772$ is the Euler constant). Hence, the
relative difference between the clean and dirty values of $h^{*}(T)$ is less
than $5\%$ at $T=0$, decreases to zero at $T_{c}$, but is still on a $2\%$
level at $t=T/T_{c}=0.9$. This is illustrated in the inset in Fig. 1.

The “near universality” of $h^{*}$ is a useful property commonly used in data
analysis. In this note, we point out that a different normalization, namely
$u(t)=H_{c2}(T,\rho)/H_{c2}(0,\rho)$, practically collapses the upper critical
field curves onto one, much closer to a truly universal curve, compared to the
original HW normalization. Such a degree of scaling behavior is sufficient to
provide a simple formula for this practically universal curve, which may help
the researchers to analyze the experimental data without worrying about the
level of disorder always present in real samples.

The practical usefulness of this scaling representation of the upper critical
field was recognized before. For example, Sun and Maki used it to compare
theoretical $H_{c2}(T)$ in different directions in $p-$wave superconductors
[4]. Godeke _at al._ used it to show on one graph the experimental data of a
variety of 27 Nb-Sn based superconductors and discussed the results in terms
of the dirty-limit Maki - de Gennes model (our Eq. (10)) [5]. However, to the
best of our knowledge, this note is the first in which such scaling behavior
is used to address arbitrary transport scattering. This is practically
relevant since the exact evaluations are time-consuming and, as it turns out,
they do not add new features to the shape of the $H_{c2}(T)$ curve.

Figure 1:  The HW scaling function $h^{*}(t)=-H_{c2}(t)/(dH_{c2}/dt)_{t\to 1}$
for the clean limit, $h^{*}_{cl}$ and the dirty limit, $h^{*}_{d}$, vs. the
reduced temperature $t=T/T_{c}$. Inset: the relative difference between these
two curves, $(h^{*}_{cl}-h^{*}_{d})/h^{*}_{cl}$, showing a difference close to
5% in most of the temperature range, $t<0.7$.

##  II The universal curve

Consider the quantity,

$u(t)=\frac{H_{c2}(T,\rho)}{H_{c2}(0,\rho)}=\frac{h(t,\rho)}{h(0,\rho)}\,,$
(3)

where the reduced field

$h=\frac{H_{c2}}{\phi_{0}}\frac{\hbar^{2}v^{2}}{2\pi T_{c}^{2}}\,.$ (4)

Clearly, $u(0)=1$ is independent of the scattering rate. To see how this
quantity behaves with temperature, let us consider separately the clean and
dirty limits near $T_{c}$. We denote the clean limit by a “cl” subscript, and
the dirty limit by “d” subscript.

The general expression for the slope $(dh/dt)$ at $T_{c}$ is given by HW [1]:

$-\frac{dh}{dt}\Big{|}_{t=1}=3\rho^{2}\left[\psi\left(\frac{1}{2}\right)-\psi\left(\frac{1+\rho}{2}\right)+\frac{\pi^{2}\rho}{4}\right]^{-1}.\qquad$
(5)

In the clean case this gives

$-\frac{dh_{cl}}{dt}\Big{|}_{t=1}=-\frac{24}{\psi^{\prime\prime}(1/2)}=\frac{12}{7\zeta(3)}\,.$
(6)

According to Eq. (3) the slope $u^{\prime}(t)=h^{\prime}(t)/h(0)$. Further,
the clean case $H_{c2}(0)=\pi\phi_{0}T_{c}^{2}e^{2-C}/2\hbar^{2}v^{2}$ (see
e.g. [6]), which gives the reduced value,

$h_{cl}(0)=\frac{e^{2-C}}{4}\,.$ (7)

Hence, we obtain

$-\frac{du_{cl}}{dt}\Big{|}_{t=1}=\frac{48\,e^{C-2}}{7\zeta(3)}\approx
1.3750\,.$ (8)

Note that this number is just the inverse of the HW clean-limit number
$h^{*}_{cl}(0)=0.727$, because in the $h^{*}$ normalization, the slope at
$T_{c}$ is 1.

Figure 2:  The scaled function $u(t)=h(t)/h(0)$ for different types of
scattering. The wide solid green line shows the numerical result in the clean
case; the blue dash-dotted line is the transport dirty limit; the red x-marks
show the finite transport scattering rate case, $\rho=1$, $\rho_{m}=01$. These
three curves are practically collapsed on a single universal curve. In stark
contrast, magnetic scattering results in a more concave curve, shown for pure
magnetic scattering, $\rho_{m}=1$ and $\rho=0$ (violet solid line), and for a
more realistic mixed case, $\rho_{m}=1$ and $\rho=1$ (blue dotted line).
Inset: the relative difference, $(u_{cl}-u_{d})/u_{cl}$, between the two
limiting cases showing about 0.5% difference level for most of the temperature
interval.

In the dirty limit, Eq. (5) yields

$-\frac{dh_{d}}{dt}\Big{|}_{t=1}=\frac{12\rho}{\pi^{2}}\,.$ (9)

The dirty limit self-consistency equation [2, 3], Eq.(40) of the HW paper [1],

$-\ln t=\psi\left(\frac{1}{2}+\frac{h}{6\rho
t}\right)-\psi\left(\frac{1}{2}\right)\,$ (10)

yields

$h_{d}(0)=\frac{3\rho}{2\,e^{C}}\,.$ (11)

Thus, we have,

$\frac{du_{d}}{dt}\Big{|}_{t=1}=\frac{8\,e^{C}}{\pi^{2}}\approx 1.4437\,.$
(12)

Note that this number is just the inverse of the HW number
$h^{*}_{d}(0)=0.693$.

Figure 3: The slopes $du/dt$ near $T_{c}$. The solid red curve shows the clean
limit, the solid blue curve shows the dirty limit, and the dashed black line
shows the derivative of the suggested universal curve, Eq. (14). The dashed
orange lines with numbers show the analytic limits, Eq. (8) and Eq. (12). The
suggested universal approximation, based on Eq. (14), fits in between the
limiting values.

Numerically, one can evaluate $H_{c2}(t,\rho)$ at any temperature using either
the original HW equations or a revised theory [6] better suited for numerical
analysis to obtain $u(t)$ for clean, dirty (and any intermediate) cases. The
latter approach can also be used to compute $H_{c2}$ in the case of finite
magnetic and non-magnetic scattering. The wide solid green line in Fig. 2
shows the numerical result in the clean case; the blue dash-dotted line shows
the transport dirty limit; the red x marks show the case of borderline
transport scattering, $\rho=1$, when the coherence length $\xi$ and the mean
free path, $\ell$, are of the same order. These three curves practically
collapse on a single universal curve and, as shown later, can be well
approximated by a simple equation $u(t)\approx(1-t^{2})/(1+0.42\,t^{1.47})$.
Note that the magnetic scattering results in a concave deviation from the
universal curve. The solid violet line in Fig.,2 shows the case of pure
magnetic scattering, $\rho_{m}=1$ and $\rho=0$, and the blue dotted line shows
a mixed case, $\rho_{m}=1$ and $\rho=1$. This shows that the pair-breaking
scattering will manifest itself in this way regardless of the background
transport disorder, which makes such a simple analysis very useful. The inset
in Fig. 2 shows the relative difference, $(u_{cl}-u_{d})/u_{cl}$, which is
just a downward shift of a similar quantity shown in the inset of Fig.,1, by
approximately 0.048, resulting in about ten times smaller relative difference
in most of the temperature range.

Thus, $u(t)$ is a nearly universal function of $t$. Inspired by the Casimir
and Gorter theory, see Eq.(9) in Ref.[7], we used the following form to fit
the “exact” numerical results,

$u(t)=\frac{1-t^{2}}{1+at^{b}}\,,$ (13)

The fitting yields $a=0.423,\;b=1.468$ in the clean case and
$a=0.416,\;b=1.472$ in the dirty limit. The results are so close, that we
converged on a single function that describes the scaled $u(t)$ curve for any
transport scattering rate with a good accuracy,

$u(t)\approx\frac{1-t^{2}}{1+0.42t^{1.47}}\,,$ (14)

It is worth noting again that, as shown in the inset to Fig. 2, the
“universality parameter” $(u_{cl}-u_{d})/u_{cl}$ for this new scaling is an
order of magnitude smaller than that for original HW, compare with the inset
in Fig. 1.

Figure 4:  Illustration of the use of the suggested scaling for the analysis
of the experimental data. The wide green line shows the numerical clean limit
and solid yellow line shows the universal scaling function, Eq. (14). The data
that follow this scaling very well belong to cubic $s-$wave materials,
represented here by Nb-Sn - based superconductors that include 16 ternary and
11 binary compounds [5] and a very clean niobium, shown by stars [7]. This
contrasts with the data shown by violet rhombi for FeTe0.9Se0.1 films that had
known magnetic scattering [8]. It exhibits a concave curvature, consistent
with Fig.2, where magnetic scattering is considered. Another case, leading to
the convex $u(t)$ curve, is a two-gap $s_{++}$ superconductor, MgB2, shown by
cyan-filled black circles for $H\parallel ab-$plane and by orange-filled black
squares for the $H\parallel c-$axis [9].

To investigate the source of some non-universality of $u$ close to $T_{c}$, we
numerically calculated the slopes $du/dt$. Figure 3 shows that the slopes,
$du/dt$, near $T_{c}$ differ for the clean and dirty cases, approaching
$-1.375$ at $t=1$ in the clean case, Eq. (8) and $-1.444$ in the dirty limit,
Eq. (12). However, this relatively small difference does not show in the plot
of $u(t)$ since near $T_{c}$, $u\to 0$. For comparison, the dashed black line
in Fig. 3 shows the temperature derivative of the suggested universal curve,
Eq. (14). Expectedly, it falls in between the limiting slopes.

To illustrate the practical usability of our results, Fig.4 shows several
cases where $H_{c2}(T)$ agrees with our “universal” curve, Eq. (14), or not
for the reasons provided. Among many reports of the upper critical field, we
sought the most complete data covering the entire temperature range. The
reference lines in Fig.4 are: the wide green line shows the numerical clean
limit (almost invisible under data points), and the solid yellow line shows a
simplified universal $u(t)$ of Eq. (14). The data that follow this scaling
very well belong to cubic $s-$wave materials, represented here by Nb-Sn -
based superconductors that include a rich compilation of 16 ternary and 11
binary compounds [5]. We note that it is likely that these 27 compounds had
different levels of scattering, yet they all collapsed on a single universal
curve. To reinforce this argument, stars in Fig.4 show a very clean niobium
metal [7]. This is contrasted with FeTe0.9Se0.1 data, shown by violet rhombi,
which had known magnetic scattering [8] and exhibits a concave curvature,
consistent with Fig.2. Another case, leading to the convex-type deviation, is
the well-known two-gap superconductor, MgB2, shown by cyan-filled black
circles for $H\parallel ab-$plane and by orange-filled black squares for the
$H\parallel c-$axis [9].

The proposed scaling approach is naturally justified considering that the
error bar in the experimental determination of $H_{c2}$ is usually not small,
so the precision of the proposed scaling function, Eq.(14), is quite
sufficient as can be seen in Fig.4 where real-world experimental data are
shown.

##  III Conclusions

In conclusion, we show that the analysis of the upper critical field in scaled
form $u(t)=H_{c2}(T/T_{c})/H_{c2}(0)$ provides a simple way to check the
compliance of the experimental results with the HW orbital critical field. One
has to use Eq. (14) with a two-parameter fit, $H_{c2}(0)$ and $T_{c}$, to
scale the data irrespective of the transport scattering rate. Significant
deviations will signal the presence of other mechanisms that influence
$H_{c2}(T)$. For example, multi-gap $s_{++}$ superconductivity will show as a
convex curve with a noticeable curvature in the upper half of the temperature
range, whereas magnetic (possibly pairbreaking in general) scattering will
result in a more pronounced concave curve above the universal $u(t)$ line.

###### Acknowledgements.

R.P. acknowledges the useful discussions with A. Gurevich. This research was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering. Ames National Laboratory is
operated for the U.S. Department of Energy by Iowa State University under
Contract No.DE-AC02-07CH11358.

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