# On the strong convergence of partial sums with respect to bounded Vilenkin
systems

G. Tutberidze  G.Tutberidze, The University of Georgia, School of Science and
Technology, 77a Merab Kostava St, Tbilisi 0128, Georgia and Department of
Engineering Sciences and Mathematics, Luleå University of Technology, SE-971
87 Luleå, Sweden and UiT The Arctic University of Norway, P.O. Box 385,
N-8505, Narvik, Norway<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate some strong convergence theorems for partial sums
with respect to Vilenkin system.

The research was supported by a Swedish Institute scholarship and by Shota
Rustaveli National Science Foundation grant YS15-2.1.1-47.

2010 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, partial sums, Fejér means, martingale
Hardy space, strong convergence.

##  1\. Introduction

It is well-known (for details see e.g. [10] and [14]) that Vilenkin system
does not form basis in the space $L_{1}\left(G_{m}\right).$ Moreover, there is
a function in the Hardy space $H_{1}\left(G_{m}\right),$ such that the partial
sums of $f$ are not bounded in $L_{1}$-norm. However, subsequence $S_{M_{n}}$
of partial sums are bounded from the martingale Hardy space
$H_{1}\left(G_{m}\right)$ to the Lebesgue space $L_{1}\left(G_{m}\right):$

(1) $\left\|S_{M_{k}}f\right\|_{H_{1}}\leq c\left\|f\right\|_{H_{1}}\text{ \ \
\ }(k\in\mathbb{N}).$

Moreover, we have the follwing norm equivalence:

(2)
$\left\|f\right\|_{H_{1}}\equiv\left\|\sup_{n}\left|S_{M_{n}}f\right|\right\|_{1}.$

Moreover, Gát [8] proved the following strong convergence result for all $f\in
H_{1}:$

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f-f\right\|_{1}}{k}=0,$

where $S_{k}f$ denotes the $k$-th partial sum of the Vilenkin-Fourier series
of $f.$

It follows that there exists an absolute constant $c,$ such that

(3) $\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{1}}{k}\leq
c\left\|f\right\|_{H_{1}}\text{ \ }\left(n=2,3...\right)$

and

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{1}}{k}=\left\|f\right\|_{H_{1}},$

for all $f\in H_{1}.$

Analogical result for the trigonometric system was proved by Smith [20], for
the Walsh-Paley system by Simon [18].

If partial sums of Vilenkin-Fourier series was bounded from $H_{1}$ to $L_{1}$
we also would have:

(4)
$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}\left\|S_{m}f\right\|_{1}\leq
c\left\|f\right\|_{H_{1}}.$

but as it was present above that boundednes of partial sums does not hold from
$H_{1}$ to $L_{1}$, However, we have inequality (3).

On the other hand, in one-dimensional, Fujji [6] and Simon [17] proved that
maximal operator Fejér means is bounded from $H_{1}$ to $L_{1}$. It follows
that

(5)
$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\left\|\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}S_{m}f\right\|_{1}<c\left\|f\right\|_{H_{1}}.$

So, natural question has arised that if inequality (4) holds true, which would
be generalization of inequality (5) or we have negative answer on this
problem.

In this paper we prove that there exists a function $f\in H_{1}$ such that

$\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n}\underset{m=1}{\overset{n}{\sum}}\left\|S_{m}f\right\|_{1}=\infty.$

This paper is organized as follows: in order not to disturb our discussions
later on some definitions and notations are presented in Section 2. For the
proofs of the main results we need some auxiliary Lemmas. These results are
presented in Section 3. The formulation and detailed proof of main results can
be found in Section 4.

##  2\. Definitions and Notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$

Let $m:=(m_{0,}m_{1},\dots)$ denote a sequence of the positive integers not
less than 2.

Denote by

$Z_{m_{k}}:=\\{0,1,\dots,m_{k}-1\\}$

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group
$Z_{m_{j}}$ with the product of the discrete topologies of $Z_{m_{j}}$ ,s.

The direct product $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \qquad}(j\in Z_{m_{k}})$

is the Haar measure on $G_{m_{\text{ }}}$with $\mu\left(G_{m}\right)=1.$

If $\sup_{n\in\mathbb{N}}m_{n}<\infty$, then we call $G_{m}$ a bounded
Vilenkin group. If the generating sequence $m$ is not bounded then $G_{m}$ is
said to be an unbounded Vilenkin group. In this paper we discuss bounded
Vilenkin groups only.

The elements of $G_{m}$ are represented by sequences

$x:=(x_{0},x_{1},\dots,x_{k},\dots)\qquad\left(\text{ }x_{k}\in
Z_{m_{k}}\right).$

It is easy to give a base for the neighbourhood of $G_{m}$

$I_{0}\left(x\right):=G_{m},$ $I_{n}(x):=\\{y\in G_{m}\mid
y_{0}=x_{0},\dots,y_{n-1}=x_{n-1}\\}\text{ }(x\in G_{m},\text{
}n\in\mathbb{N})$

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$ .

Let

$e_{n}:=\left(0,\dots,0,x_{n}=1,0,\dots\right)\in
G_{m}\qquad\left(n\in\mathbb{N}\right).$

If we define the so-called generalized number system based on $m$ in the
following way:

$M_{0}:=1,\text{ \qquad}M_{k+1}:=m_{k}M_{k\text{ }}\ \qquad(k\in\mathbb{N})$

then every $n\in\mathbb{N}$ can be uniquely expressed as
$n=\sum_{k=0}^{\infty}n_{j}M_{j}$ where $n_{j}\in Z_{m_{j}}$
$~{}(j\in\mathbb{N})$ and only a finite number of $n_{j}`$s differ from zero.
Let $\left|n\right|:=\max$ $\\{j\in\mathbb{N};$ $n_{j}\neq 0\\}.$

For the natural number $n=\sum_{j=1}^{\infty}n_{j}M_{j},$ we define

$\delta_{j}=signn_{j}=sign\left(\ominus n_{j}\right),\text{ \ \ \ \
}\delta_{j}^{\ast}=\left|\ominus n_{j}-1\right|\delta_{j},$

where $\ominus$ is the inverse operation for $a_{k}\oplus
b_{k}=\left(a_{k}+b_{k}\right)$mod$m_{k}.$

We define functions $v$ and $v^{\ast}$ by

$v\left(n\right)=\sum_{j=0}^{\infty}\left|\delta_{j+1}-\delta_{j}\right|+\delta_{0},\text{
\ }v^{\ast}\left(n\right)=\sum_{j=0}^{\infty}\delta_{j}^{\ast},$

Next, we introduce on $G_{m}$ an orthonormal system which is called the
Vilenkin system.

At first define the complex valued function
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions as

$r_{k}\left(x\right):=\exp\left(2\pi\imath x_{k}/m_{k}\right)\text{
\qquad}\left(\imath^{2}=-1,\text{ }x\in G_{m},\text{ }k\in\mathbb{N}\right).$

Now define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
as:

$\psi_{n}\left(x\right):=\prod_{k=0}^{\infty}r_{k}^{n_{k}}\left(x\right)\text{
\qquad}\left(n\in\mathbb{N}\right).$

Specially, we call this system the Walsh-Paley one if $m\equiv 2.$

The norm (or quasi norm) of the space $L_{p}(G_{m})$ is defined by

$\left\|f\right\|_{p}:=\left(\int_{G_{m}}\left|f(x)\right|^{p}d\mu(x)\right)^{1/p}\qquad\left(0<p<\infty\right).$

The Vilenkin system is orthonormal and complete in $L_{2}\left(G_{m}\right)\,$
(for details see e.g. [1, 26]).

If $\ f\in L_{1}\left(G_{m}\right)$ we can establish Fourier coefficients,
partial sums of the Fourier series, Fejér means, Dirichlet kernels with
respect to the Vilenkin system in the usual manner:

$\displaystyle\widehat{f}(k)$ $\displaystyle:$
$\displaystyle=\int_{G_{m}}f\overline{\psi}_{k}d\mu\text{\thinspace\qquad\ \ \
\ }\left(\text{ }k\in\mathbb{N}\text{ }\right)$ $\displaystyle S_{n}f$
$\displaystyle:$
$\displaystyle=\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k}\
\text{\qquad\ \ }\left(\text{ }n\in\mathbb{N}_{+},\text{ }S_{0}f:=0\right)$
$\displaystyle\sigma_{n}f$ $\displaystyle:$
$\displaystyle=\frac{1}{n}\sum_{k=0}^{n-1}S_{k}f\text{ \qquad\ \ \ \ \
}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right)$ $\displaystyle D_{n}$
$\displaystyle:$ $\displaystyle=\sum_{k=0}^{n-1}\psi_{k\text{ }}\text{ \qquad\
\ \qquad}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right).$

Recall that

(6) $\quad D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{l}\text{
}M_{n}\text{ \ \ \ }x\in I_{n}\\\ \text{ }0\text{ \qquad}x\notin
I_{n}\end{array}\right.$

and

(7)
$D_{s_{n}M_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}\psi_{kM_{n}}=D_{M_{n}}\sum_{k=0}^{s_{n}-1}r_{n}^{k}\text{
\qquad}1\leq s_{n}\leq m_{n}-1.$

The $n$-th Lebesgue constant is defined in the following way

$L_{n}=\left\|D_{n}\right\|_{1}.$

If $f\in L_{1}(G_{m}),$ the maximal functions are also be given by

$f^{*}\left(x\right)=\sup_{n\in\mathbb{N}}\frac{1}{\left|I_{n}\left(x\right)\right|}\left|\int_{I_{n}\left(x\right)}f\left(u\right)\mu\left(u\right)\right|$

Hardy martingale space $H_{1}\left(G_{m}\right)$ consist of all martingales
for which (for details see e.g. [27, 28])

$\left\|f\right\|_{H_{1}}:=\left\|f^{*}\right\|_{1}<\infty.$

##  3\. Auxiliary results

######  Lemma 1.

[11] Let $n\in\mathbb{N}$. Then

$\frac{1}{4\lambda}v\left(n\right)+\frac{1}{\lambda}v^{\ast}\left(n\right)+\frac{1}{2\lambda}\leq
L_{n}\leq\frac{3}{2}v\left(n\right)+4v^{\ast}\left(n\right)-1,$

where $\lambda=\sup_{n\in\mathbb{N}}m_{n}.$

######  Lemma 2.

[12] Let $n\in\mathbb{N}$. Then there exists an ansolute constant $c,$ such
that

$\frac{1}{nM_{n}}\underset{k=1}{\overset{M_{n}-1}{\sum}}v\left(k\right)\geq
c>0.$

##  4\. Main Result

######  Theorem 1.

There exists a martingale $f\in H_{1},$ such that

$\sup_{n\in\mathbb{N}}\frac{1}{n}\overset{n}{\underset{k=1}{\sum}}\left\|S_{k}f\right\|_{1}=\infty.$

##  5\. Proof of the Theorem

###### Proof of Theorem 1.

Let $\left\\{\alpha_{k}:k\in\mathbb{N}\right\\}$ be an increasing sequence of
the positive integers such that

(8) $\sum_{k=0}^{\infty}\frac{1}{\alpha_{k}^{1/2}}<c<\infty.$

Let

$f=\sum_{k=1}^{\infty}\frac{a_{k}}{\alpha_{k}^{1/2}},$

where

$a_{k}=D_{M_{\alpha_{k}+1}}-D_{M_{{}_{\alpha_{k}}}}.$

It is evident that

$S_{M_{n}}f=\sum_{\left\\{k;\text{
}\alpha_{k}<n\right\\}}\frac{a_{k}}{\alpha_{k}^{1/2}},$

and

$\left|S_{M_{n}}f\right|\leq\sum_{\left\\{k;\text{
}\alpha_{k}<n\right\\}}\frac{\left|a_{k}\right|}{\alpha_{k}^{1/2}}\leq\sum_{k=1}^{\infty}\frac{\left|a_{k}\right|}{\alpha_{k}^{1/2}},$

It follows that

$\sup_{n\in\mathbb{N}}\left|S_{M_{n}}f\right|\leq\sum_{k=1}^{\infty}\frac{\left|a_{k}\right|}{\alpha_{k}^{1/2}}.$

Since (see equality (6))

$\left\|a_{k}\right\|\leq 2,\ \ \text{fol all}\ \ k\in\mathbb{N}.$

by combining (2) and (8) we get that

$\displaystyle\left\|f\right\|_{H_{1}}\leq
c\left\|\sup_{k\in\mathbb{N}}\left|S_{M_{k}}f\right|\right\|_{1}$
$\displaystyle\leq$ $\displaystyle
c\left\|\sum_{k=1}^{\infty}\frac{\left|a_{k}\right|}{\alpha_{k}^{1/2}}\right\|\leq
c\sum_{k=1}^{\infty}\frac{\left\|a_{k}\right\|}{\alpha_{k}^{1/2}}\leq
2c\sum_{k=1}^{\infty}\frac{1}{\alpha_{k}^{1/2}}\leq c<\infty.$

Moreover,

(9)
$\widehat{f}(j)=\left\\{\begin{array}[]{l}\frac{1}{\alpha_{k}^{1/2}},\,\,\text{\
\ \ \
\thinspace\thinspace}j\in\left\\{M_{\alpha_{k}},...,M_{\alpha_{k}+1}-1\right\\},\text{
}k\in\mathbb{N}\\\ 0\text{ },\text{
\thinspace\qquad\thinspace\thinspace\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{k=1}^{\infty}\left\\{M_{\alpha_{k}},...,M_{\alpha_{k}+1}-1\right\\}.\text{
}\end{array}\right.$

Let $M_{\alpha_{k}}\leq j<M_{\alpha_{k+1}}.$ Since

$D_{j+M_{\alpha_{k}}}=D_{M_{\alpha_{k}}}+\psi_{{}_{M_{\alpha_{k}}}}D_{j},\text{
\qquad when \thinspace\thinspace}j<M_{\alpha_{k}},$

if we apply (9) we obtain that

$\displaystyle S_{j}f$ $\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\sum_{v=M_{\alpha_{k}}}^{j-1}\widehat{f}(v)\psi_{v}$
$\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\sum_{v=M_{\alpha_{k}}}^{j-1}\widehat{f}(v)\psi_{v}$ (11)
$\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}^{1/2}}\sum_{v=M_{\alpha_{k}}}^{j-1}\psi_{v}$
(12) $\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}^{1/2}}\left(D_{j}-D_{M_{\alpha_{k}}}\right)$
(13) $\displaystyle=$ $\displaystyle
S_{M_{\alpha_{k}}}f+\frac{M_{\alpha_{k}}^{1/p-1}}{\alpha_{k}^{1/2}}\psi_{M_{\alpha_{k}}}D_{j-M_{\alpha_{k}}}$
(14) $\displaystyle=$ $\displaystyle III_{1}+III_{2}.$

In view of (1) we can write that

(15)
$\displaystyle\left\|III_{1}\right\|_{1}\leq\left\|S_{M_{\alpha_{k}}}f\right\|_{1}\leq
c\left\|f\right\|_{H_{1}}.$

By combining Lemma 1 and (15) we get that

$\displaystyle\left\|S_{n}f\right\|_{1}\geq\left\|III_{2}\right\|_{1}-\left\|III_{1}\right\|_{1}\geq\frac{cv\left({n-M_{\alpha_{k}}}\right)}{\alpha_{k}^{1/2}}-c\left\|f\right\|_{H_{1}}.$

Hence, according to Lemma 2 we can conclude that

$\displaystyle\underset{n\in\mathbf{\mathbb{N}}_{+}}{\sup}\frac{1}{n}\underset{k=1}{\overset{n}{\sum}}\left\|S_{k}f\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{M_{\alpha_{k}+1}}\underset{\left\\{M_{\alpha_{k}}\leq
l\leq 2M_{\alpha_{k}}\right\\}}{\sum}\left\|S_{l}f\right\|_{1}$
$\displaystyle\geq$
$\displaystyle\frac{1}{M_{\alpha_{k}+1}}\underset{\left\\{M_{\alpha_{k}}\leq
l\leq
2M_{\alpha_{k}}\right\\}}{\sum}\left(\frac{v\left(l-M_{\alpha_{k}}\right)}{\alpha_{k}^{1/2}}-c\left\|f\right\|_{H_{1}}\right)$
$\displaystyle\geq$
$\displaystyle\frac{c}{\alpha_{k}^{1/2}M_{\alpha_{k}}}\underset{l=1}{\overset{M_{\alpha_{k}}-1}{\sum}}v\left(l\right)-c\left\|f\right\|_{H_{1/2}}^{1/2}$
$\displaystyle\geq$ $\displaystyle c\alpha_{k}^{1/2}\rightarrow\infty,\text{
as \ }k\rightarrow\infty.$

The proof is complete. ∎

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# Signatures of topological phase transitions in the s-wave superconductor
at finite temperature

Stefan Gorol  Florian Loder  Daniel Braak  Arno P. Kampf  Thilo Kopp  Center
for Electronic Correlations and Magnetism, Institute of Physics, University of
Augsburg, 86135 Augsburg, Germany

(August 27, 2024)

###### Abstract

In two dimensions, an s-wave superconductor in the presence of Rashba spin-
orbit coupling possesses distinct topologically non-trivial ground state
phases controlled by Zeeman splitting and band filling. These phases can be
characterized in terms of spin textures in momentum space. Although the spin
texture becomes topologically trivial at finite temperatures, we identify
thermodynamic signatures that are directly related to the topological phase
transitions of the ground state. In particular, relative maxima in the entropy
as a function of the magnetic field in the vicinity of topological phase
transitions emerge and are attributed to a sign change in the derivative of
the magnetization with respect to temperature.

##  I Introduction

Topologically non-trivial electronic systems in two dimensions (2d) are often
characterized by a non-zero Chern number [1, 2, 3] which reflects a property
of the ground state wave function. This topological invariant is essential for
the quantum Hall effect [4] and determines the quantized Hall conductivity in
the zero temperature limit, or the thermal Hall effect [5, 6, 7] for which the
finite Chern number dictates the term in the thermal Hall conductivity linear
in temperature. The 2d s-wave superconductor can be topological—in the sense
that the Chern number may differ from zero—if spin-orbit coupling (SOC) and
Zeeman spin splitting are present [8, 9, 10, 11].

In these systems, thermodynamic properties of Lifshitz transitions concomitant
with topological phase transitions have been investigated [12, 13, 14, 15].
Those transitions are accompanied by a kink in the first derivative of the
thermodynamic potential with respect to the chemical potential at zero
temperature [16, 17]. At finite temperatures, the kink is broadened but may
still be detected as a peak in the third derivative of the thermodynamic
potential with respect to the chemical potential [18, 14, 19, 20]. On the
other hand the bulk topology can be related to topological edge states by the
bulk-boundary correspondence [1, 21, 22]. By using Hill thermodynamics the
presence of the edge states in finite-size systems can be taken into account
[12, 13], resulting in a linear contribution to the heat capacity.

In addition to the Chern number, non-trivial bulk topology may be
characterized by a non-zero skyrmion number for spin textures in reciprocal
space [23, 9, 24, 8, 25]. At zero temperature, the skyrmion number in a
topological s-wave superconductor is related to the Chern number [9], but any
finite temperature destroys its topological character as will be discussed in
more detail in section II. While the Lifshitz transition is related to the
Fermi surface structure of the corresponding normal conducting phase, we study
here thermodynamic signatures of the spin texture and predict a maximum of the
entropy as function of magnetic field at constant temperature in the vicinity
of the topological phase transition. This entropy maximum at non-zero
temperatures may serve as an experimentally accessible way to detect the
transition into a topologically non-trivial ground state of the system.

The realization of a topological 2d s-wave superconductor is, however, not
straightforward, because the necessary magnetic fields—if applied
perpendicular to the plane—are usually larger than the upper critical magnetic
field $h_{\mathrm{c2}}$. It has been argued that topologically non-trivial
phases in the s-wave superconductor may be accessible by magnetic field
rotation towards in-plane orientations [26], which preserves the non-trivial
topology and avoids exceeding $h_{\mathrm{c2}}$ [27]. In this way, the
topological phase transition may be observed in spin-orbit coupled s-wave
superconductors with sufficiently strong Zeeman splitting [8].

(a)

(b)

Figure 1: (a) Phase diagram of topological ground states specified by the
Chern numbers $N_{\mathrm{C}}$ for $\alpha/t=0.5$ and $V/t=0.75$. (b) Filling
$n^{*}$ as a function of $\alpha/t$, defined as the band filling at which
$h_{\mathrm{t},1}=\Delta_{\mathrm{OP}}$ (see text).

##  II Topological Characterization

First, we briefly review the model of an topological s-wave superconductor and
its topological phases. We exclusively concentrate on the situation of intra-
band pairing discussed in Reference [28]. The Bogoliubov-de Gennes Hamiltonian
for the 2D superconducting system under consideration is diagonal in momentum
space and given by the 4x4 matrix

$\mathcal{H}(\boldsymbol{k})=\left(\begin{array}[]{cc}\mathcal{H}_{0}(\boldsymbol{k})&\mathbb{D}\\\
\mathbb{D}^{\dagger}&-\mathcal{H}_{0}^{\dagger}(\boldsymbol{k})\end{array}\right),$
(1)

where the Nambu spinor basis
$\boldsymbol{\psi}_{\boldsymbol{k}}=\left(\hat{c}_{\boldsymbol{k},\uparrow},\hat{c}_{\boldsymbol{k},\downarrow},\hat{c}^{\dagger}_{-\boldsymbol{k},\uparrow},\hat{c}^{\dagger}_{-\boldsymbol{k},\downarrow}\right)^{\dagger}$
is used with $\boldsymbol{k}=(k_{\mathrm{x}},k_{\mathrm{y}})^{\mathrm{T}}$.
The 2x2 matrix $\mathcal{H}_{0}$ reads

$\mathcal{H}_{0}=\left(\begin{array}[]{cc}\epsilon(\boldsymbol{k})-\mu+h_{\mathrm{z}}&\alpha(\boldsymbol{k})+h_{\mathrm{x}}-\mathrm{i}h_{\mathrm{y}}\\\
{\alpha}^{*}(\boldsymbol{k})+h_{\mathrm{x}}+\mathrm{i}h_{\mathrm{y}}&\epsilon(\boldsymbol{k})-\mu-
h_{\mathrm{z}}\end{array}\right)$ (2)

while $\epsilon(\boldsymbol{k})=-2t\,(\cos k_{\mathrm{x}}+\cos
k_{\mathrm{y}})$ with $t$ and $\mu$ denoting the usual tight binding hopping
energy and the chemical potential, respectively. The Rashba SOC is
$\alpha(\boldsymbol{k})=\alpha\left(\sin k_{\mathrm{y}}+i\sin
k_{\mathrm{x}}\right)$. The coupling of the magnetic field to the spin by the
Zeeman term is included as
$\mu_{\mathrm{B}}\boldsymbol{h}\cdot\boldsymbol{\sigma}$ where
$h_{\mathrm{x}}$, $h_{\mathrm{y}}$ and $h_{\mathrm{z}}$ are the magnetic field
components in x-,y- and z-direction, respectively. We use units such that
$k_{\mathrm{B}},\mu_{\mathrm{B}},\hbar=1$. Regular s-wave pairing is assumed

$\mathbb{D}=\left(\begin{array}[]{cc}0&\Delta\\\ -\Delta&0\end{array}\right)$
(3)

with $\Delta$ being the order parameter. The grand canonical potential reads

$\Omega=-\frac{1}{\beta}\sum_{\boldsymbol{k}}\sum_{\nu}\ln\left[2\cosh\left(\frac{\beta\lambda_{\nu}(\boldsymbol{k},\mu)}{2}\right)\right]-2N\mu+\frac{N|\Delta|^{2}}{V}.$
(4)

where $\lambda_{\nu}$ denotes the eigenvalues of the Hamiltonian (1) and $N$
is the number of lattice points. The attractive on-site interaction is $V>0$
and $\beta=1/T$. From the gap equation $\partial\Omega/\partial\Delta=0$ the
$\boldsymbol{k}$-independent mean-field order parameter $\Delta_{\mathrm{OP}}$
is obtained and the particle number density is determined from
$n=-\frac{1}{N}{\partial\Omega}/{\partial\mu}$.

In this system, a relevant topological invariant is the Chern number

$N_{\mathrm{C}}=\frac{1}{2\pi}\sum_{\nu,\mathrm{occupied}}\int\mathrm{d}^{2}k\,\Omega^{\nu}_{\mathrm{B}}(\boldsymbol{k});\qquad
N_{\mathrm{C}}\in\mathbb{Z}$ (5)

with the Berry curvature $\Omega^{\nu}_{\mathrm{B}}(\boldsymbol{k})$ given by

$\Omega^{\nu}_{\mathrm{B}}(\boldsymbol{k})=\left.\mathrm{i}\nabla\times\langle\nu,\boldsymbol{k}|\nabla_{\boldsymbol{k}}|\nu,\boldsymbol{k}\rangle\right|_{z},$
(6)

where $\nu$ runs over all occupied bands and $|\nu,\boldsymbol{k}\rangle$
refers to an eigenstate of the band $\nu$. For
$\boldsymbol{h}\neq\boldsymbol{0}$ the four eigenbands $\lambda_{\nu}$ with
$\nu\in\\{1,2,3,4\\}$ are non-degenerate. The bands are in the following
indexed by increasing energy. The Nambu-space bands are related by
$\lambda_{1}(\boldsymbol{k},\mu)=-\lambda_{4}(\boldsymbol{k},\mu)$ and
$\lambda_{2}(\boldsymbol{k},\mu)=-\lambda_{3}(\boldsymbol{k},\mu)$.

A further topological invariant is the skyrmion number $N_{\mathrm{S}}$,
defined by

$N_{\mathrm{S}}=\frac{1}{4\pi}\int
d^{2}k\,\boldsymbol{S}(\boldsymbol{k})\cdot\left(\partial_{k_{x}}\boldsymbol{S}(\boldsymbol{k})\times\partial_{k_{y}}\boldsymbol{S}(\boldsymbol{k})\right),$
(7)

where $\boldsymbol{S}(\boldsymbol{k})$ is the normalized spin vector

$\boldsymbol{S}(\boldsymbol{k})=\boldsymbol{s}(\boldsymbol{k})/|\boldsymbol{s}(\boldsymbol{k})|$
(8)

with
$\boldsymbol{s}(\boldsymbol{k})=\left(\langle\hat{s}_{x}(\boldsymbol{k})\rangle,\langle\hat{s}_{y}(\boldsymbol{k})\rangle,\langle\hat{s}_{z}(\boldsymbol{k})\rangle\right)$.
We obtain the spin expectation values at each $\boldsymbol{k}$-point for a
given temperature $T$ with

$\displaystyle\langle\boldsymbol{s}(\boldsymbol{k})\rangle$
$\displaystyle=\frac{1}{2}\sum_{\nu=1,2}\tanh\left(\frac{\beta\lambda_{\nu}(\boldsymbol{k},\mu)}{2}\right)\langle\nu,\boldsymbol{k}|\left(c^{\dagger}_{\boldsymbol{k},\uparrow}c^{\dagger}_{\boldsymbol{k},\downarrow}\right)\boldsymbol{\sigma}\left(c_{\boldsymbol{k},\uparrow}c_{\boldsymbol{k},\downarrow}\right)^{\mathrm{T}}|\nu,\boldsymbol{k}\rangle.$
(9)

It was shown previously [9] that in a system described by (1) the skyrmion
number and the Chern number are related by $N_{\mathrm{C}}=-2N_{\mathrm{S}}$
at zero temperature. The skyrmion number takes half-integer values because the
spin texture is actually of meron character [29, 30, 31]. In the following,
for simplicity, we will omit expectation-value brackets and the hat
identifying operators.

In order to enter any non-trivial topological phase it is necessary to apply
magnetic field energies larger than $\Delta_{\mathrm{OP}}$ but the upper
critical magnetic field energy set by $h^{2}_{\mathrm{c}2}$ is usually smaller
than $\Delta_{\mathrm{OP}}$. If the magnetic field is rotated into an in-plane
orientation, the topological transition fields $h_{\mathrm{t},1}$ and
$h_{\mathrm{t},2}$ are decreased.

For $h_{\mathrm{x}}$, $h_{\mathrm{y}}\ll\alpha$

$h_{\mathrm{t},1,2}(h_{\mathrm{x},\mathrm{y}}\neq 0)\approx
h_{\mathrm{t},1,2}(h_{\mathrm{x},\mathrm{y}}=0)-t\frac{h_{\mathrm{x}}^{2}+h_{\mathrm{y}}^{2}}{\alpha^{2}}$
(10)

is obtained. Concomitantly, orbital depairing is suppressed in the rotated
field setup such that $h_{\mathrm{c,2}}$ is increased in layered
superconductors [27, 32]. It may hence be possible to realize the situation
where $h_{\mathrm{t},1}<h_{\mathrm{c,2}}$. The required field rotation into an
in-plane orientation leads to pairing with finite center-of-mass momentum [33,
34, 35, 36, 37, 34, 35], however, this does not destroy the inherent
topological character of the considered phases [9] and, consequently, the
finite temperature signatures addressed in the following are not affected
qualitatively. Therefore, without loss of generality, we discuss here the
situation for $h_{\mathrm{x}}=h_{\mathrm{y}}=0$. The topological phase-
transition fields are defined by two magnetic fields $h_{\mathrm{t,1}}(n)$ and
$h_{\mathrm{t,2}}(n)$, at which the band gap closes, given by [38, 39]

$\displaystyle h_{\mathrm{t},1}(h_{\mathrm{x},\mathrm{y}}=0)$
$\displaystyle=\sqrt{\left({\Delta_{\mathrm{OP}}}\right)^{2}+(\epsilon(\boldsymbol{0})-\mu(n))^{2}},$
(11) $\displaystyle h_{\mathrm{t},2}(h_{\mathrm{x},\mathrm{y}}=0)$
$\displaystyle=\sqrt{\left({\Delta_{\mathrm{OP}}}\right)^{2}+\mu^{2}(n)}.$
(12)

Figure 2: Dispersion of the normal conducting bands $\xi_{\boldsymbol{k}}^{+}$
and $\xi_{\boldsymbol{k}}^{-}$ for the cases $h_{\mathrm{z}}<h_{\mathrm{t},1}$
(left figure), $h_{\mathrm{z}}>h_{\mathrm{t},1}$ (right figure). The arrows
indicate the spin direction related to each band. The situation around
$h_{\mathrm{t},2}$ is similar.

The Chern number as a function of $h_{\mathrm{z}}$ and the band filling $n$ is
calculated and the result is shown in Figure 1 indicating four topologically
distinct phases with $N_{\mathrm{C}}\in\left\\{0,-1,1,2\right\\}$. Due to
Rashba SOC, the superconducting order parameter is finite even for large
magnetic fields, if orbital depairing is not included, and it vanishes only in
the $h_{\mathrm{z}}/t\to\infty$ limit [40].

For low band fillings $n\ll 1$ one has $\mu(n)\approx\epsilon(\boldsymbol{0})$
and hence $h_{\mathrm{t},1}<h_{\mathrm{t},2}$. However, around half filling
$n\approx 1$ one finds $\mu(n)\approx 0$ and thus
$h_{\mathrm{t},1}>h_{\mathrm{t},2}$. In between these cases the crossing point
of both transition-field curves is found at around quarter filling $n\approx
1/2$. At this point all topological phases merge as shown in Figure 1a.

Gap closing points emerge whenever one band of the normal conducting state is
depleted — related to $h_{\mathrm{t},1}$ and $h_{\mathrm{t},2}$ of Equations
(11) and (12) in the $\Delta_{\textrm{OP}}\to 0$ limit — as illustrated in
Figure 2. There, $\xi^{+}_{\boldsymbol{k}}$ and $\xi_{\boldsymbol{k}}^{-}$ are
the helical spin-split eigenbands of $\mathcal{H}_{0}$. The transition fields
depend on $\alpha$ through $\mu(n)$ obtained by solving the particle-number
equation.

The non-trivial topological phase is characterized by $N_{\mathrm{C}}=-1$ in
the low filling regime. Around half filling non-trivial topological phases
with $N_{\mathrm{C}}=2$ and $N_{\mathrm{C}}=1$ are possible. The latter
requires large magnetic fields with
$h_{\mathrm{z}}\geq\epsilon(\boldsymbol{0})/2=h_{0}$ where $h_{0}$ is the
minimal magnetic field for which
$h_{\mathrm{z}}>h_{\mathrm{t},1},h_{\mathrm{t},2}$ is fulfilled.

In the low filling regime, the minimum transition field
$\min(h_{\mathrm{t},1}(n))=\Delta_{\textrm{OP}}$ is obtained for a filling
$n^{*}$ for which $\mu(n^{*})=\epsilon(\boldsymbol{0})$; $n^{*}$ depends on
the Rashba SOC as shown in Figure 1b. Here it should be noted that
$\epsilon(\boldsymbol{0})$ is not the lowest band energy on account of the
finite Rashba SOC. Since $N_{\mathrm{C}}=-2N_{\mathrm{S}}$ at zero temperature
[9], the ground state phase diagram in Figure 1a also amounts to four
topologically distinct spin textures in reciprocal space. At
$h_{\mathrm{x},\mathrm{y}}=0$ and $T=0$, the spin texture for the
topologically trivial phase for low band filling is depicted in Figure 3 a.
There are vortex patterns at the time-reversal invariant momentum (TRIM)
points $(0,0)$ and $(\pi,\pi)$ and antivortices at the TRIM points $(0,\pi)$
and $(\pi,0)$. The vortex (antivortex) center points are denoted as
$\boldsymbol{k}_{\mathrm{VC}}$ in the following. For
$h_{\mathrm{x},\mathrm{y}}\neq 0$, the $\boldsymbol{k}_{\mathrm{VC}}$ are
shifted away from the TRIM points. At each center of a vortex or antivortex
the spin normalization must be defined by the limit
$\boldsymbol{k}\rightarrow\boldsymbol{k}_{\mathrm{VC}}$ which may be singular.
For $N_{\mathrm{C}}=0$, it is singular at all four
$\boldsymbol{k}_{\mathrm{VC}}$. The normalized spin textures for the phases
with $N_{\mathrm{C}}=-1,2,1$ are shown in Figures 3 b - d. In each of the
these phases, the spin points upwards, $\boldsymbol{S}=(0,0,1)^{T}$, at one or
more of the $\boldsymbol{k}_{\mathrm{VC}}$ as depicted in Figure 3.

As the normalized spin $\boldsymbol{S}(\boldsymbol{k})$ can be viewed as a map
from the torus (the 2d Brillouin zone) to the upper hemisphere of the sphere
S2, the number of the $\boldsymbol{k}_{\mathrm{VC}}$ mapped onto the “north
pole” are less then or equal to four, whereas the remaining points are mapped
to the equator, where the map becomes singular. The manifold of the spin
expectation values can be compactified to the unit sphere such that the
equator is mapped to the “south pole” of the sphere proving the spin texture’s
topological nature. In Figure 3, the color of the arrows indicates on which
latitude of the sphere the normalized spin expectation value is positioned
after compactification. Dark red corresponds to the covering of the north pole
while dark blue implies the mapping onto the south pole. Reference [9]
describes further details of this mapping.

The number of north pole coverings is one, zero, two, and three for the phases
$N_{\mathrm{C}}=-1,0,1,2$, respectively. At any finite temperature $T$, the
$s_{z}$ spin components are finite everywhere in the Brillouin zone. The north
pole is then covered four times since the in-plane spin components
$s_{\mathrm{x}}=s_{\mathrm{y}}=0$ by symmetry. Hence, the number ob
$\boldsymbol{k}_{\mathrm{VC}}$ mapped onto the equator is zero and the
compactified map does not yield a full covering of the S2. Because the map is
still smooth, the associated skyrmion number vanishes for any $T>0$.

Figure 3: Zero temperature spin textures for (a) the topologically trivial
phase with $N_{\mathrm{S}}=N_{\mathrm{C}}=0$, $n=0.19$,
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}=-0.5$, (b) for the
topologically non-trivial phases with $N_{\mathrm{S}}=1/2$,
$N_{\mathrm{C}}=-1$, $n=0.12$, (c)
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}=0.18$,
$N_{\mathrm{S}}=-1$, $N_{\mathrm{C}}=2$, $n=0.9$,
$(h_{\mathrm{z}}-h_{\mathrm{t},2})/h_{\mathrm{t},2}=0.3$ , and (d)
$N_{\mathrm{S}}=-1/2$, $N_{\mathrm{C}}=1$, $n=0.54$,
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}=0.05$. The arrows indicate
the direction of the normalized spin vectors. The color codes the
$z$-component of the normalized spin, where red corresponds to
$S_{\mathrm{z}}=1$ whereas blue corresponds to $S_{\mathrm{z}}=0$. The
positions of the $\boldsymbol{k}_{\mathrm{VC},i}$, which are defined in the
main text, are marked with red and green symbols for vortices and
antivortices, respectively.

In order to characterize the spin structure beyond the skyrmion number for
finite temperature, the quantity

$\Sigma_{\mathrm{L}}=\frac{1}{2\pi}\int
d^{2}k\,\boldsymbol{S}_{\mathrm{L}}(\boldsymbol{k})\cdot\left(\partial_{k_{x}}\boldsymbol{S}_{\mathrm{L}}(\boldsymbol{k})\times\partial_{k_{y}}\boldsymbol{S}_{\mathrm{L}}(\boldsymbol{k})\right),$
(13)

was introduced in Reference [9] with the in-plane normalized spin vector

$\boldsymbol{S}_{\mathrm{L}}=\left(\frac{s_{\mathrm{x}}}{\sqrt{s_{\mathrm{x}}^{2}+s_{\mathrm{y}}^{2}}},\frac{s_{\mathrm{y}}}{\sqrt{s_{\mathrm{x}}^{2}+s_{\mathrm{y}}^{2}}},s_{\mathrm{z}}\right)=\left(S_{\mathrm{L}}^{\mathrm{x}},S_{\mathrm{L}}^{\mathrm{y}},s_{\mathrm{z}}\right).$
(14)

$\Sigma_{\mathrm{L}}$ is not a topological invariant, and not confined to
integer values. In fact, $\boldsymbol{S}_{\mathrm{L}}(\boldsymbol{k})$ is a
map from the torus to the open unit cylinder, which is not compact. However,
in the zero-temperature limit $\Sigma_{\mathrm{L}}\to N_{\mathrm{C}}$, because
the map becomes smooth after the cylinder is compactified similar to the
hemisphere above and $s_{z}$ is either zero or 1 at all vortex (antivortex)
center points $\boldsymbol{k}_{\mathrm{VC}}$.

By partial integration of (13) we can rewrite $\Sigma_{\mathrm{L}}$ as

$\displaystyle\Sigma_{\mathrm{L}}$
$\displaystyle=\sum_{i=1}^{4}s_{\mathrm{z}}(\boldsymbol{k}_{\mathrm{VC},i})\lim_{\epsilon\to
0}\oint\limits_{C(\boldsymbol{k}_{\mathrm{VC},i},\epsilon)}\frac{\mathrm{d}\boldsymbol{k}}{2\pi}\,\left(S_{\mathrm{L}}^{\mathrm{x}}\nabla_{\boldsymbol{k}}S_{\mathrm{L}}^{\mathrm{y}}-S_{\mathrm{L}}^{\mathrm{y}}\nabla_{\boldsymbol{k}}S_{\mathrm{L}}^{\mathrm{x}}\right)$
$\displaystyle=\sum_{i=1}^{4}s_{\mathrm{z}}(\boldsymbol{k}_{\mathrm{VC},i})\mathcal{V}(\boldsymbol{k}_{\mathrm{VC},i}).$
(15)

The index $i$ runs over all vortex-center points where $\langle
s_{\mathrm{x}}(\boldsymbol{k}_{\mathrm{VC},i})\rangle=\langle
s_{\mathrm{y}}(\boldsymbol{k}_{\mathrm{VC},i})\rangle=0$, and the transverse
components of $\boldsymbol{S}_{\mathrm{L}}$ cannot be normalized. The quantity
$\mathcal{V}(\boldsymbol{k})$ denotes the winding number (vorticity) of the
in-plane spin around the momentum $\boldsymbol{k}$. The spin winding results
from Rashba SOC. Whereas the number of points $\boldsymbol{k}_{\mathrm{VC}}$
with $\boldsymbol{S}(\boldsymbol{k}_{\mathrm{VC}})=\boldsymbol{0}$ changes at
the topological phase transition, their vortex or anti-vortex character is
preserved. $C(\boldsymbol{k}_{\mathrm{VC},i},\epsilon)$ denotes a
counterclockwise circular path around $\boldsymbol{k}_{\mathrm{VC},i}$ with
radius $\epsilon$. $\Sigma_{\mathrm{L}}$ is integer valued, if
$s_{\mathrm{z}}$ takes the extremal values 0 or 1 at the vortex points, as is
the case for $T=0$. As all $\mathcal{V}(\boldsymbol{k}_{\mathrm{VC},i})$ are
unchanged at the topological phase transition $\Sigma_{\mathrm{L}}$ is
exclusively determined by the values of
$s_{\mathrm{z}}(\boldsymbol{k}_{\mathrm{VC},i})$ and these change only at the
gap closing points.

(a)

(b)

(c)

(d)

Figure 4: Thermodynamic signatures of topological spin-texture changes in the
ground state close to the transition field $h_{\mathrm{t},1}$. (a)
$\partial_{T}M_{\mathrm{z}}\cdot
T_{\mathrm{c}}/(M_{\mathrm{z}}(T_{\mathrm{c}})-M_{\mathrm{z}}(0))$ while
$(M_{\mathrm{z}}(T_{\mathrm{c}})-M_{\mathrm{z}}(0))$ is always positive and
(b)
$(\Delta_{\mathrm{OP}}(T)-\Delta_{\mathrm{OP}}(0))/\Delta_{\mathrm{OP}}(0)$ as
functions of $T/T_{\mathrm{c}}$ for
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}=0.19$,
$\Delta_{\mathrm{OP}}(T=0)=0.05t$ (blue) and
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}=-0.19$,
$\Delta_{\mathrm{OP}}(T=0)=0.06t$ (grey); s-wave superconductor without SOC
for $\Delta_{\mathrm{OP}}=0.05t$ and $h_{\mathrm{z}}=0.6\Delta_{\mathrm{OP}}$
(dashed grey). (c) Entropy $S(h_{\mathrm{z}})$ for
$T/T_{\mathrm{c}}=0.11,0.21,0.23,0.27$ from bottom to top. The blue dashed
line connects the local maxima of $S(h_{\mathrm{z}},T)$. (d) $\partial
M_{\mathrm{z}}/\partial T<0$ (blue area); $\partial M_{\mathrm{z}}/\partial
T>0$ (white area) and minimum gap $\Delta_{\mathrm{min}}$ as a function of
$(h_{\mathrm{z}}-h_{\mathrm{t},1})/h_{\mathrm{t},1}$ (solid black). We used
$V/t=1.5$ corresponding to a $T_{\mathrm{c}}\approx 3\cdot 10^{-2}t$,
$n=0.028$ and $\alpha/t=0.5$.

##  III Thermodynamics

The values of $s_{\mathrm{z}}(\boldsymbol{k}_{\mathrm{VC},i})$ at isolated
points of the BZ are not accessible by measurements of thermodynamic
quantities. However, the modification of the spin-structure in a sizable
region of the Brillouin zone can affect thermodynamic properties and thereby
exhibit signatures of the topological ground-state transition even at non-zero
temperatures.

The energy gap $\Delta_{\mathrm{min}}$ depends on $\boldsymbol{k}$ due to the
Rashba SOC which is discussed in more detail in Reference [28]. At low
temperatures $T\ll\Delta_{\mathrm{min}}$ thermal excitations are restricted to
a small neighborhood of the energy gap minima. Thus, information about
variations in the $s_{\mathrm{z}}(\boldsymbol{k})$-values in the vicinity of
the gap closing points can be obtained. Taking the partial derivative of the
magnetization in z-direction with respect to temperature, and neglecting
contributions further away from the Fermi-level yields

$-\left.\frac{\partial}{\partial T}\left(\left.\frac{\partial\Omega}{\partial
h_{\mathrm{z}}}\right|_{T,\mu}\right)\right|_{h_{\mathrm{z}},n}=\left.\frac{\partial
M_{\mathrm{z}}}{\partial T}\right|_{n,h_{\mathrm{z}}}=\left.\frac{\partial
S}{\partial
h_{\mathrm{z}}}\right|_{n,T}=\frac{1}{N}\\!\\!\int\mathrm{d}^{2}k\,\frac{\partial\lambda_{3}(\boldsymbol{k},\mu)}{\partial
h_{\mathrm{z}}}\frac{\lambda_{3}(\boldsymbol{k},\mu)}{T^{2}}\;\mathrm{sech}^{2}\left(\frac{\lambda_{3}(\boldsymbol{k},\mu)}{2T}\right).$
(16)

We omitted the contributions of two eigenbands $\lambda_{1}$ and $\lambda_{4}$
which are far away from the Fermi level and used the symmetry
$\lambda_{2}(\boldsymbol{k},\mu)=-\lambda_{3}(\boldsymbol{k},\mu)$.
$T\ll\Delta_{\mathrm{min}}$ is reflected in a peak in
$T^{-2}\mathrm{sech}^{2}(\lambda_{3}(\boldsymbol{k},\mu)/2T)$ at the gap
minimum $\boldsymbol{k}_{\mathrm{min}}$ where
$\lambda_{3}(\boldsymbol{k},\mu)$ is minimal. In Equation (16), $S$ is the
entropy and a Maxwell relation is used to relate the derivate of the
magnetization with respect to $T$ to the derivative of the entropy with
respect to the magnetic field.

In the following, we discuss the thermodynamic signatures of a topological
ground-state phase transition exemplarily in the low filling regime. Figure 4a
shows $\partial_{T}M_{\mathrm{z}}$ for $h_{\mathrm{x}}=h_{\mathrm{y}}=0$ and
$h_{\mathrm{z}}$ above and below $h_{t,1}$, see blue and grey line,
respectively. For magnetic fields below the topological phase transition
$\partial_{T}M_{\mathrm{z}}>0$ for $T<T_{\mathrm{c}}$ similar to the case of
an s-wave superconductor without SOC except in a narrow region around the
topological ground state transition field $h_{\mathrm{t},1}$ as shown in
Figure 4d. In contrast, the figure shows $\partial_{T}M_{\mathrm{z}}$ being
non-monotonous and even negative as a function of $T$ for
$h_{\mathrm{z}}>h_{\mathrm{t},1}$ and $T\ll T_{\mathrm{c}}$. The temperature
scale for non-vanishingly small $\partial_{T}M_{\mathrm{z}}$ is set by the
band gap minimum which is proportional to $h_{\mathrm{z}}-h_{\mathrm{t},1}$ as
shown in Figure 4d. The dependence of $\partial_{T}M_{\mathrm{z}}$ on $T$ for
a superconductor without SOC is given by the dashed grey curve where
$\partial_{T}M_{\mathrm{z}}$ is exponentially increasing with
$\Delta_{\mathrm{OP}}$. The temperature dependence of the order parameter
$\Delta_{\mathrm{OP}}$ is qualitatively similar to the dependence of
$\partial_{T}M_{\mathrm{z}}$ as depicted in Figure, 4b.

At $T=0$, $s_{\mathrm{z}}(\boldsymbol{0})=0$ for magnetic fields
$h_{\mathrm{z}}<h_{\mathrm{t},1}$ because the spin expectation values of the
chiral spin split bands compensate each other. However, $s_{\mathrm{z}}$ turns
into a maximum if $h_{\mathrm{z}}$ is tuned through the topological phase
transition (the image in the spin map of the momentum
$\boldsymbol{k}=\boldsymbol{0}$ switches from the south to the north pole).
Finite temperature excitations now have a quantitatively different effect on
the $s_{\mathrm{z}}(\boldsymbol{k}=0)$-values in the trivial and the
topological phase where $s_{\mathrm{z}}(\boldsymbol{k}=0,T)$ is increasing and
decreasing, respectively.

This qualitative difference extends to a finite region around the
$\Gamma$-point such that the change from the minimum in $s_{\mathrm{z}}$ to
the maximum is visible as a sign change in $\partial_{T}M_{\mathrm{z}}$. At
$T\ll\Delta_{\mathrm{min}}$ the integral in (16) can be approximated by the
value of the integrand at the band gap minimum due to the sharp peak in
$T^{-2}\mathrm{sech}^{2}(\lambda_{3}/2T)$. The sign of
$\partial_{T}M_{\mathrm{z}}$ is therefore determined by the sign of
$\partial_{h_{\mathrm{z}}}\lambda_{3}$ at the band gap minimum.

At finite temperatures in the regime $T\ll T_{\mathrm{c}}$, the sign change of
$\partial_{T}M_{\mathrm{z}}$ does not occur exactly at the transition field
$h_{\mathrm{t},1}$ but in a magnetic field range where the two normal
conducting bands are still filled in the vicinity of the $\Gamma$-point. The
points at which $\partial_{T}M_{\mathrm{z}}=0$ (as a function of
$h_{\mathrm{z}}$ and $T$) are given by the blue dashed line in Figure 4d. They
correspond to the positions of the relative maxima in the entropy in the
region with $T\ll\Delta_{\mathrm{min}}$ around $h_{\mathrm{t},1}$ as shown by
the blue dashed line in Figure 4c.

##  IV Conclusion and final remarks

Topologically non-trivial states can be realized in s-wave superconductors in
the presence of Zeeman splitting and Rashba spin-orbit coupling [26]. The
topologically non-trivial phases in the system may be described by either the
Chern number or the skyrmion number for the spin-texture in momentum space.
The Chern number description is valid only for the ground state while the
skyrmion number characterization yields trivial topology at any finite
temperature. Nevertheless, there exist signatures of the topological phase
transition at finite temperature related to the profound change in the spin
texture signified in certain thermodynamic quantities.

A topological superconductor with spin-orbit coupling shows a characteristic
vortex structure of the in-plane spin components at the time-reversal
invariant momenta in the Brillouin zone. The out-of-plane spin component at
distinct vortex centers flips from zero to one at a critical Zeeman splitting
for $T=0$ and attains a maximum in the topologically non-trivial phase even
for $T>0$. This sudden change manifests itself in a maximum of the entropy as
function of magnetic field at constant temperature. Equivalently, it
corresponds to a sign change of the derivative of the magnetization with
respect to temperature. We propose a suitable generalization of the skyrmion
number, $\Sigma_{\mathrm{L}}$, which measures the singular character of the
spin texture and is not confined to the ground state. In this way,
$\Sigma_{\mathrm{L}}$ captures the topological nature of the phase transition
even at non-zero temperature although it is not a topological invariant.

Other two-dimensional systems with broken time-reversal symmetry, whose
topological ground state depends on the magnetic field e. g. the topological
mixed-parity superconductor [41, 42], are candidates for similar
investigations close to topological phase transitions. Thermodynamic
signatures of topological phase transitions are not restricted to topological
superconductors but expected as well for topological insulators. Replacing the
magnetization by a pseudospin polarization, the Chern insulators described in
terms of topologically non-trivial pseudospin textures may be analyzed at
finite temperatures in a similar fashion [25].

###### Acknowledgements.

Acknowledgements
Financial support by the Deutsche Forschungsgemeinschaft (project number
107745057, TRR 80) is gratefully acknowledged.

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# Optimization of Magnetized Electron Cooling with JSPEC

Stephen J. Coleman<EMAIL_ADDRESS>RadiaSoft, LLC, Boulder, CO  David L.
Bruhwiler  RadiaSoft, LLC, Boulder, CO  Dan T. Abell  RadiaSoft, LLC, Boulder,
CO  Boaz Nash  RadiaSoft, LLC, Boulder, CO  Ilya Pogorelov  RadiaSoft, LLC,
Boulder, CO  He Zhang  Thomas Jefferson National Accelerator Laboratory,
Newport News, VA

###### Abstract

The Electron-Ion-Collider (EIC) will be a next-generation facility located at
Brookhaven National Laboratory (BNL), built with the goal of accelerating
heavy ions up to 275 GeV. To prevent ion beam size growth during the
acceleration phase, cooling techniques will be required to keep the beam size
from growing due to intra-beam scattering. The JSPEC (JLab Simulation Package
for Electron Cooling) C ++ package is a tool designed to numerically model
magnetized and unmagnetized cooling through friction forces between co-
propagating electron and ion bunches. Here we describe a feature that has been
added to the JSPEC package, which implements a Nelder-Mead Simplex
optimization algorithm to allow a user to optimize certain beam parameters in
order to achieve a target cooling time.

##  1 Introduction

The Electron-Ion-Collider (EIC), the layout of which is shown in figure 1,
will be a next-generation facility located at Brookhaven National Laboratory
(BNL), built with the goal of accelerating heavy ions up to
$275\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ and luminosities up to
${10}^{34}\text{\,}\mathrm{c}\mathrm{m}^{-2}\mathrm{s}^{-1}$ [1]. To achieve
these luminosities, ion beams will need to have high intensity and low
momentum spread. Ion beam size growth arises due to stochastic interactions
associated with Intra-Beam Scattering (IBS). The growth can be slowed or
balanced by introducing some form of cooling tuned to the characteristics of
the beam. Initial design studies plan for strong cooling of the ion beam using
a coherent electron cooling technique, but the implementation of this
technique depend on future R&D efforts. Magnetized electron cooling could be
an alternative or a backup plan for controlling growth should the primary
effort fall short [1, 2].

In magnetized electron cooling, stochastic Coulomb collisions of individual
ions in an ion bunch occur with co-propagating electrons that are confined in
a solenoidal section. Interactions with electrons following ‘frozen’ Larmor
trajectories cause the ensemble of ions in the bunch to feel a drag, reducing
the momentum spread through dynamical friction. As this friction force is
reducing the average velocity of the ion beam in the beam rest frame, it can
be considered as ‘cooling’ the ion beam. The possible benefits of magnetized
electron cooling are reviewed in [3, 4, 5].

Figure 1: Current overview of the electron-ion collider concept[1]. The
electron cooler is at 3 o’clock.

##  2 JSPEC

The JLab Simulation Package for Electron Cooling (JSPEC) package is an open-
source C ++ package originally developed at the Thomas Jefferson National
Accelerator Facility [6, 7]. In this package, friction force models and IBS
models are applied to a model beam orbiting an accelerator with MAD-X lattice
and other properties supplied by the user. JSPEC was extensively benchmarked
against the Betacool electron cooling simulation package [8], an older package
based on MAD8 that also models IBS and cooling.

The JSPEC dynamic calculation models the evolution of an ion bunch over time,
after many passes in the cooler. The user has the option to produce dynamic
simulations in one of two ways, either propagating with moments of
distributions or with individual macro-particles. In the former, the
properties of representative ions are drawn randomly from the initial moments
of each distribution. At the conclusion of each step in the dynamic
calculation, new moments are calculated after perturbations from IBS and
cooling kicks have been applied to the ensemble of representative ions. At the
beginning of the subsequent step, new representative ions are initialized with
properties drawn from those perturbed moments. In this way, the distributions
are guaranteed to stay Gaussian. In the macro-particle method, initial
properties of macro-particles are drawn from Gaussian distributions as before,
but in the series of dynamic simulation steps each macro-particle experiences
its own history of cooling and IBS kicks. The dynamics and the summary
statistics are drawn from the ensemble of independent macro-particles. This is
a more accurate representation of the physics, and non-Gaussian distributions
are possible. For the first time step, where initial cooling rates are
calculated and reported, these two methods yield identical results.

####  2.0.1 Rate Calculation

JSPEC reports the rate of change of the emittance in horizontal, vertical, and
longitudinal directions in units of 1/sec. The initial rate of change is also
broken down into components caused by IBS and by electron cooling. Initial IBS
rates are calculated after initializing an ion bunch with known emittance
values, propagating that bunch forward one step to induce IBS kicks (but not
cooling kicks) to a set of macro-particles. The emittances are then re-
calculated from the perturbed distributions. The IBS rate is simply the
difference in the emittances divided by the simulation time step. Two IBS
models are available in JSPEC, the Bjorken-Mtingwa model [9] and the Martini
model [10]. The initial cooling rate calculation is performed in a similar way
to the IBS rate calculation. The rate is calculated by determining the
difference in the emittance before and after cooling kicks are applied in the
absence of IBS kicks, and dividing that difference by the time step.

Several new friction force models were added to the JSPEC package, bringing it
in line with the set of models available in Betacool. This set includes the
Derbenev & Skrinsky model [11], the Meshkov asymptotic model [12], the Budker
un-magnetized model [13], and two other numerical approximations of the first-
principles unmagnetized model [8]. Implementation of these models required
numerical evaluation of indefinite integrals. These were solved through
numerical routines provided by the GSL library [14], and these calculations
were parallelized through use of the OpenMP package [15]. These friction force
formulas in JSPEC have been benchmarked with BETACOOL as shown in figure 2.

Figure 2: Friction force curves as modeled in JSPEC and in Betacool.

###  2.1 Optimization

There are many parameters that may affect the cooling rate calculated for a
proposed magnetized electron cooler. We have implemented a Nelder-Mead simplex
optimization algorithm in JSPEC, which allows users to search a multi-
dimensional parameter space for a set of values that meets their design needs.
A full listing of parameters that may be varied within the optimization
algorithm is shown in Table 1.

Parameter | Variable |  i values
---|---|---
Ion Twiss $\alpha_{i}$ | alpha_i | v,h
Ion Twiss $\beta_{i}$ | beta_i | v,h
$e^{-}$ bunch RMS width $\sigma_{i}$ | sigma_i | x,y,s
Dispersion | disp_i | v,h
Dispersion Derviative | disp_der_i | v,h
Cooler Magnetic Field $B$ | bfield | —
$e^{-}$ Temperature | temp_i | tr,long
# of electrons | n_electron | —
Table 1: A full listing of variables that users may make available to the
optimizer. Variables that are not initialized within the optimize section of
the input file are held fixed. Here replace i by v or h for vertical or
horizontal variables respectively.

The Nelder-Mead simplex algorithm [16], available within the GSL library [14],
was implemented because it is a gradient-free method, which makes it
insensitive to the statistical noise encountered when cooling rates are
repeatedly calculated with multiple independent simulations using a finite
number of macro-particles. Each of the macro-particle parameters are Gaussian
distributed and are independently drawn at each optimization step. While the
parameter space may be smooth in the limit of an infinite number of simulated
macro-particles, the statistical noise associated with smaller numbers of
macro-particles can lead to problems when evaluating the gradient. The impact
of this statistical noise is further mitigated by sampling a large number of
macro particles, at the cost of longer simulation times (the default value is
${10}^{7}$).

###  2.2 Cost function

The cost function for this optimization approach uses existing JSPEC cooling
rate calculations. The cooling rate is then converted into an approximate
cooling time $T_{i}$ in a particular direction, $i\in x,y,s$. In units of
minutes, this time is given by

(1) $T_{i}=60\frac{1}{R_{i}}$

where $R_{i}$ is the initial cooling rate in the $i$ direction. The value
being minimized within the cost function $C$ is then

(2) $C_{i}=|T_{0}-T_{i}|$

where the target cooling time is $T_{0}$. Note that cooling time and thus
optimization can only be performed in one direction, so users must decide to
prioritize cooling in a longitudinal or transverse direction based on their
design constraints. Without a defined target time, the cost function would
effectively behave with $T_{0}=0$, leading the algorithm to select undesirable
or extreme values for the free parameters, particularly for parameters that
have approximately linear relationships with cooling time. In that case, the
optimization algorithm finds reductions to the cost function by repeatedly
varying a single parameter, ignoring all others.

###  2.3 Uniqueness of results

For an optimization problem with a large number of free parameters, there
exist an infinite number of possible solutions along a locus in the multi-
dimensional parameter space. As a trivial example, consider an optimization
problem with three free parameters, with selected values that satisfy the
optimization condition at $\alpha=\alpha_{0}$, $\beta=\beta_{0}$, and
$\gamma=\gamma_{0}$. Now assume the two parameters $\alpha$ and $\beta$ are
anti-correlated, meaning that an infinitesimal increase in one parameter
coupled with an infinitesimal decrease in the other will yield the same
result. While this may remain true for perceptible deviations, a significant
change (say, $\alpha\rightarrow\alpha_{1}$ and $\beta\rightarrow\beta_{1}$)
may cause the third free parameter to compensate as
$\gamma\rightarrow\gamma_{1}$. Now, at this new point, infinitesimal changes
about $\alpha_{1}$ can be compensated with anti-correlated changes about
$\beta_{1}$. Thus, many equivalent solutions exist and repeated attempts at
optimization with identical initial conditions may not yield the same sets of
optimal parameters.

###  2.4 Searching beyond local minima

Nelder-Mead Simplex optimization procedures may sometimes fall into local
minima and fail to emerge. For this reason, the optimization procedure is
attempted many times, each with slightly different initial conditions. The
initial conditions for each attempt are drawn randomly from a Gaussian
distribution, using the user-provided starting point as the mean of the
distribution. For each attempt, initial conditions are then drawn at random
and used to initialize the Nelder-Mead Simplex algorithm. After a number of
attempts (15 by default) the optimal set of parameters producing the minimum
value of the cost function $C$ from any of these attempts is stored in a
plaintext output file, best.txt.

If the Nelder-Mead Simplex suggests a parameter that is unphysical (e.g. a
negative bunch length, or a negative electron density), the optimization
routine will alert the user that an unphysical attempt has been made and will
discard the suggestion.

###  2.5 Parameter Scans

Users may hold $N-1$ parameter values constant and scan values of the holdout
parameter in order to see its effect on cooling rates. Betacool does not have
a built-in support for parameter scans. A comparison of scanned values for
matching configurations in JSPEC and in Betacool is shown in Figure 3.

Figure 3: Parameter scan of the horizontal dispersion parameter performed in
JSPEC compared to single-point sampled values in Betacool.

A reasonable strategy for optimization would be to allow the optimization
algorithm to find a suitable set of parameters, then examine the 1-d parameter
scans of each free parameter in order to understand the nature of the
dependence of cooling time on each parameter in the local region of parameter
space. This can also be useful for practical purposes. For example, an optimum
set of parameters may include a solenoid magnetic field of
$>$10\text{\,}\mathrm{T}$$, a difficult and expensive element to produce. A
parameter scan might reveal that a lower solenoid field strength will suffice
to satisfy the design objectives of the cooler. The syntax for configuring a
JSPEC parameter scan are shown in Appendix B.

##  3 Examples for EIC

The EIC will accelerate ions, from protons ($Z=1$) to Au+ ($Z=79$) to
collision energies ranging from 41–275 GeV. We will construct examples at both
ends of the ion mass spectrum and use them to demonstrate the optimization and
parameter scan features of JSPEC. We will also construct an example to
demonstrate magnetized electron cooling at pre-injection. We will select the
Parkhomchuk [17] friction force model for all cases.

###  3.1 Cooling at pre-injection

One concept being considered for cooling at EIC is to cool protons at pre-
injection, with ion beam energies of 23.8 GeV. This is favorable for
magnetized electron cooling, because the friction force scales with
$1/\gamma^{2}$. Cooling at lower energies is generally easier. The parameters
being considered for this concept are shown in Table 2.

Parameter Name | value
---|---
Species | Proton
Ion Energy [GeV] | 23.8
Bunch Intensity [$10^{10}$] | 2.6
Beam Current [A] | 0.69
$\beta$ horizontal [cm] | 150-200
$\beta$ vertical [cm] | 250-200
$\epsilon_{\mathrm{Horiz}}$ (Norm.) [$\mu$m] | 2.7
$\epsilon_{\mathrm{Vert}}$ (Norm.) [$\mu$m] | 0.25
$\Delta p/p$ [$10^{-4}$] | 10.3
Bunch Length [cm] | 60
Length of cooling section [m] | 130
Table 2: Table of low-energy cooling parameters. Values from [2].

####  3.1.1 Preliminary test

We start the optimization procedure by allowing only a few parameters to
float: the transverse beam sizes $\sigma_{x}$ and $\sigma_{y}$, the solenoid
field strength $B$, and the number of electrons in a bunch. We set the target
cooling time to $T_{0}=20$ minutes. The initialization code for this is shown
in Appendix A and the results are listed in Table 3. We see that the
optimization has yielded a cooling time within a second of 20 minutes. While
this satisfies the target time, we notice that the electron density is quite
high from an operational standpoint.

Parameter Name | optimization result
---|---
$\sigma_{x}$ [$\mu$ m] | 48.67
$\sigma_{y}$ [$\mu$ m] | 317.94
B field [T] | 1.93
$N_{e}$ [$10^{10}$] | 0.94
Cost function $C$ [min] | $4.29\text{\times}{10}^{-4}$
Table 3: Table of optimized parameters after the preliminary optimization for
20 minute cooling times for beam conditions shown in Table 2.

####  3.1.2 Alternate optimizer parameterization

Examining the 1-dimensional dependence of the cooling time on a particular
parameter may inform excursions away from optimum values based on such
practicalities. While the friction force models may inform the dependence on
some parameters, for example a linear dependence on $n_{e}$, non-linear
relationships may be observed with sampling in a parameter scan. In Figure 4
increasing $B$-field values, holding all other parameters fixed, yields
shorter cooling times, but the relationship is asymptotic. A user may judge
whether a 2 Tesla solenoid is sufficient for the cooling goals.

Figure 4: Parameter scan of the electron bunch length parameter around the
central value determined from the optimization algorithm. The grey vertical
line marks the value suggested by the optimizer, while the dashed horizontal
line denotes the boundary between ion beam heating (positive values) or
cooling (negative) values.

We can now fix the B-field value and run the optimization algorithm again,
with a different set of floating parameters. The free parameters were the
electron bunch length $\sigma_{s}$, the number of electrons in the bunch
$n_{e}$, the transverse electron temperature $T_{\perp}$, the longitudinal
electron temperature $T_{\parallel}$, and the $\beta_{v}$ and $\beta_{h}$ of
the cooler. The target was a 20 minute cooling time in the longitudinal
direction, and then separately in the vertical direction, to match the EIC
magnetized electron cooling design study [2]. Results are summarized in Table
4.

Parameter Name |  $s$ optimization |  $y$ optimization
---|---|---
$\beta_{y}$ [m] | 99.8 | 100.1
$\beta_{y}$ [m] | 98.8 | 102.0
$N_{e}$ [$10^{10}$] | 1.06 | 1.04
$T_{\perp}$ [eV] | 0.014 | 0.011
$T_{\parallel}$ [eV] | 0.01 | 0.013
$\sigma_{s}$ [cm] | 4.48 | 4.90
Cost function $C$ [min] | $1.12\text{\times}{10}^{-4}$ | $1.96\text{\times}{10}^{-4}$
Table 4: Table of optimized parameters for 20 minute cooling times in the
longitudinal direction and the vertical direction for beam conditions shown in
Table 2.

This satisfies the target cooling time with the inverse of the initial rate,
but it’s worth looking at the long-term dynamic behavior of an ion bunch that
passes through the cooler on multiple orbits. An approximate bunch of macro-
particles representing the whole ion bunch may be simulated with JSPEC. Each
of the macro-particles evolves independently, and summary statistics and
emittance are then calculated from the ensemble.

###  3.2 Cooling for EIC

We can use the optimization algorithm to explore the use of magnetized
electron cooling within the EIC hadron storage ring. Here we simulate the
lowest operating energy range within the full range of ion masses. The
optimization algorithm was initialized with the fixed values shown in Table 5,
with either protons or gold ions. The cooling target time was set to be 20
minutes in the longitudinal direction in both cases. The optimized parameters
are presented in Table 6.

Parameter Name | proton |  Au+
---|---|---
Ion Energy [GeV] | 41 | 41
Bunch Intensity [${10}^{10}$] | 2.6 | 0.036
Beam Current [A] | 0.69 | 0.41
$\beta$ horizontal [cm] | 90 | 90
$\beta$ vertical [cm] | 7.1 | 4
$\epsilon_{\text{Horiz}}$ (Norm.) [$\mu$m] | 2.7 | 3.0
$\epsilon_{\text{Vert}}$ (Norm.) [$\mu$m] | 0.25 | 0.3
$\Delta p/p$ [${10}^{-4}$] | 10.3 | 10.0
Bunch Length [cm] | 7.5 | 11.6
Table 5: Table of EIC parameters. Proton and gold ion values are taken from Table 3.3 and Table 3.5 in [1] respectively. Parameter Name | proton |  Au+
---|---|---
Ion Energy [GeV] | 41 | 41
$B$-field [T] | 2.07 | 2.35
$N_{e}$ [$10^{10}$] | 1.22 | 0.156
$T_{\perp}$ [eV] | 0.011 | 0.013
$T_{\parallel}$ [eV] | 0.009 | 0.008
$\sigma_{s}$ [cm] | 5.8 | 5.8
Cost function $C$ [min] | $4.8\times 16^{-5}$ | $1.26\times 16^{-4}$
Table 6: Table of optimized parameters for 20 minute cooling times in the
longitudinal direction for beam conditions shown in Table 5.

##  4 Conclusion

We have shown techniques that can be used to optimize the design of an
arbitrary magnetized electron cooler. A Nelder-Mead Simplex optimization
algorithm was introduced to the JSPEC magnetized electron cooling simulation
code in order to sample many possible cooler configurations. We then
demonstrated these techniques with examples relevant to the proposed electron
ion collider. The optimized sets of parameters were then validated through
dynamic simulations, also generated with JSPEC. Readers may simulate the
optimized sets of parameters for themselves using the cloud-based Sirepo
interface for JSPEC available at https://sirepo.com/jspec.

##  Appendix A Code for defining a parameter scan

Much of the previous appendix conveys for a parameter scan. The changes which
initialize a parameter scan, in this case for the magnetic field strength $B$
(bfield), are shown here:

... section_optimization bfield = 1.0e-4 bfield = 10.0 steps = 100 section_run
total_expansion_rate optimize_cooling Here the repeated definition of the
variable initializes the parameter scan and activates the steps variable,
which defines the granularity of the scan. The output is stored in a plaintext
scan.txt file showing the cooling rate for $x,y$, and $s$ directions at each
step.

##  Appendix B Input code for pre-injection cooling optimization

The full plaintext input file parsed by JSPEC is shown below, for an
optimization problem. This input file requires the lattice file to be in Mad-X
Twiss parameter format (.tfs file type), and that the file be present in the
working directory. Values defined in section_optimization over-ride values for
the same parameter defined in earlier sections. All parameters that are not
defined in the section_optimization are fixed, and are not allowed to vary
with the optimization algorithm.

section_scratch ion_mass = 938.272 ion_ke = 23800.0 ion_gamma = 1 +
ion_ke/ion_mass section_ion charge_number = 1 mass = ion_mass kinetic_energy =
ion_ke norm_emit_x = 2.5e-06 norm_emit_y = 2.5e-06 momentum_spread = 0.001
particle_number = 2.6e10 rms_bunch_length = 0.6 section_ring lattice =
Lattice.tfs section_ibs nu = 100 nv = 100 log_c = 20.6 coupling = 0.0
section_cooler length = 130.0 section_number = 1 magnetic_field = 5.06 bet_x =
150 bet_y = 150 disp_x = 0.0 disp_y = 0.0 alpha_x = 0.0 alpha_y = 0.0 disp_dx
= 0.0 disp_dy = 0.0 section_e_beam gamma = ion_gamma tmp_tr = 0.0001 tmp_l =
0.01 shape = bunched_gaussian radius = 0.009 current = 4.0 sigma_x = 0.0002
sigma_y = 0.0002 sigma_z = 0.07 length = 0.05 e_number = 5e10 section_ecool
sample_number = 100000.0 ion_sample = MONTE_CARLO force_formula = PARKHOMCHUK
section_run create_ion_beam create_ring create_e_beam create_cooler
section_simulation ibs = on e_cool = on time = 2000.0 step_number = 100
output_file = JSPECdump.SDDS model = RMS ref_bet_x = 10.0 ref_bet_y = 10.0
ref_alf_x = 0.0 ref_alf_y = 0.0 ref_disp_x = 0.0 ref_disp_y = 0.0 ref_disp_dx
= 0.0 ref_disp_dy = 0.0 section_optimization sigma_x = 2e-4 sigma_y = 2e-4
bfield = 2.0 n_electron = 1.5 axis = s time = 20 section_run
total_expansion_rate optimize_cooling The best fit parameters will be stored
in a plaintext file in the working directory called bests.txt.

## References

  * [1] Ferdinand Willeke and J. Beebe-Wang.  Electron ion collider conceptual design report 2021.  2 2021.
  * [2] A. Fedotov et al.  Low energy cooling for electron ion collider.  Technical Report BNL-220686-2020-TECH, Brookhaven National Laboratory, 12 2020.
  * [3] V V Parkhomchuk and A N Skrinsky.  Electron cooling: physics and prospective applications.  Reports on Progress in Physics, 54(7):919–947, jul 1991.
  * [4] M Larsson.  Atomic and molecular physics with ion storage rings.  Reports on Progress in Physics, 58(10):1267–1319, oct 1995.
  * [5] I.N. Meshkov.  Electron cooling — the first 30 years and thereafter.  Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 391(1):1–11, 1997.
  * [6] JSPEC source code.
  * [7] H. Zhang et al.  The latest code development progress of JSPEC.  In Proc. of NAPAC2019, pages 539–541, Lansing, MI, 2019.
  * [8] I. Meshkov et al.  BETACOOL Physics Guide.  2007\.
  * [9] James D. Bjorken and Sekazi K. Mtingwa.  Intrabeam Scattering.  Part. Accel., 13:115–143, 1983.
  * [10] M. Martini.  Intrabeam scattering in the ACOOL-AA machines.  Technical Report CERN PS/84-9 AA, CERN, May 1984.
  * [11] Ya.S. Derbenev and A.N. Skrinsky.  Fyzika plasmy, 4:492, 1978.
  * [12] I. N. Meshkov.  Electron cooling: Status and perspectives.  Phys. Part. Nucl., 25:631–661, 1994.
  * [13] G. Budker.  In Proc. Intern. Symp. Electron and Positron Storage Rings, pages 11–1–1, Saclay, 1966.
  * [14] M. Galassi et al.  GNU Scientific Library Reference Manual (3rd Ed.).  Network Theory Ltd., 2009.
  * [15] Leonardo Dagum and Ramesh Menon.  OpenMP: An industry-standard API for shared-memory programming.  IEEE Comput. Sci. Eng., 5:46–55, January 1998.
  * [16] J.A. Nelder and R. Mead.  A simplex method for function minimization.  Computer Journal, 7:308–313, 1965.
  * [17] V. V. Parkhomchuk.  New insights in the theory of electron cooling.  Nucl. Instrum. Meth. A, 441:9–17, 2000.


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