ORLICZ SPACES ASSOCIATED WITH A SEMI-FINITE VON NEUMANN ALGEBRA

Sh. A. Ayupov${}^{1},$ V.I. Chilin${}^{2},$ R. Z. Abdullaev3

1 _Institute of Mathematics and Information Technologies, Uzbekistan Academy
of Science, Dormon yoli, 29, 100125, Tashkent. Uzbekistan_

and

_The Abdus Salam International Centre for Theoretical Physics. Triest. Italy._

Coresponding author. e-mail<EMAIL_ADDRESS>
2 _National University of Uzbekistan,_ e-mail<EMAIL_ADDRESS>
3 _Tashkent State Pedagogical University,_ e-mail<EMAIL_ADDRESS>
Introduction

Construction and investigation of various classes of symmetric spaces of
measurable operators affiliated with a von Neumann algebra $M$ is one of
important applications of the non commutative integration theory for a
faithful normal semi-finite trace on the von Neumann algebra $M.$ Examples of
such spaces are given, in particular by non commutative $L_{p}$-spaces
$L_{p}(m,\tau)$ [16] and by Orlicz spaces $L_{\Phi}(M,\tau)$ associated with
an $N$-function $\Phi$ [5],[6],[7]. All these spaces are realized as ideal
subspaces of the $*-$algebra $S(M)$ of measurable operators affiliated with
$M.$

Investigations based on the modular theory for von Neumann algebras enable to
consider non commutative versions of $L_{p}$-spaces associated with states and
weights (see e.g. the survey [14]).But in these cases in general
$L_{p}$-spaces can not be realizes as ideal subspaces of $S(M).$ This fact
explains in particular why in their attempt to introduce non commutative
Orlicz spaces for states in [1] as a subspaces of $S(M),$ the authors we
unable to prove the completeness of these spaces with respect to the Luxemburg
norm.

In the present paper we introduce a certain class of non commutative Orlicz
spaces, associated with arbitrary faithful normal locally-finite weights on a
semi-finite von Neumann algebra $M.$ We describe the dual spaces for such
Orlicz spaces and, in the case of regular weights , we show that they can be
realized as linear subspaces of the algebra of $LS(M)$ of locally measurable
operators affiliated with $M.$

For the terminology and notations from the von Neumann algebras theory we
refer to [10] and from theory of measurable and locally measurable operators
refer to [10],[15].

Preliminaries

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$ with
$\mathbf{1}$-the identity operator on $H,$ and let $P(M)=\\{p\in
M:p=p^{2}=p^{*}\\}$ be the lattice of all projection from $M$. Denote by
$S(M)$ (respectively by $LS(M)$) the ∗-algebra of all measurable
(respectively, locally-measurable) operators affiliated with $M$. It is well-
known that $S(M)$ is a ∗-subalgebra in $LS(M)$, and $M$ is a ∗-subalgebra of
$S(M)$ ([8], Ch.2).

If $x\in LS(M)$ and $x=u|x|$ is its polar decomposition, where
$|x|=(x^{*}x)^{1/2}$ and $u$ is a partial isometry, then we have that $u\in M$
and $|x|\in LS(M).$ It is also known that the spectral family of projections
$e_{\lambda}(x)$ for a self adjoint operator $x\in LS(M)$, always belongs to
$P(M).$

Given a subset $A\subset LS(M)$, put

$A_{h}=\\{x\in A:x=x^{*}\\},$ and

$A_{+}=\\{x\in A:(x\xi,\xi)\geq 0\\}$ for all $\xi\in D(x),$

where $D(x)$ is the domain of the operator $x\in LS(M),$ and $(\cdot,\cdot)$
is the inner product in the Hilbert space $H.$

Let $\tau$ be a faithful normal semi-finite trace on $M.$ For each real number
$p\geq 1$ consider the set

$L_{p}(M,\tau)=\left\\{x\in
S(M):\int\limits_{0}^{\infty}\lambda^{p}d\tau(e_{\lambda}(|x|))<\infty\right\\}.$

It known [16] that $L_{p}(M,\tau)$ is a linear subspace in $S(M)$ and the
function
$\|x\|_{p}=\left(\int\limits_{0}^{\infty}\lambda^{p}d(\tau(e_{\lambda}(|x|))\right)^{1/p}$
is a norm, which turns $L_{p}(M,\tau)$ into a Banach space.

A map $\varphi:M_{+}\rightarrow[0,\infty]$ is said to be _a weight_ if

$\varphi(x+y)=\varphi(x)+\varphi(y),\ \ \varphi(\lambda x)=\lambda\varphi(x),\
\ (x,y\in M_{+},\lambda\geq 0,\textrm{where, }0\cdot\infty=0).$

A weight $\varphi$ is said to be

— _normal_ , if $\varphi(x)=\sup\varphi(x_{i})(x_{i}\nearrow x;x_{i},x\in
M_{+});$

— _faithful_ , if $\varphi(x)=0,$ $x\in M_{+}$ implies that $x=0;$

— _semi-finite_ , if the linear span $m_{\varphi}$ of the cone
$m_{\varphi}^{+}=\\{x\in M_{+}:\varphi(x)<\infty\\}$ is dense in $M$ with
respect to the ultra-weak topology;

— _locally finite_ , if

$\forall x\in M_{+}\ \ (x\neq 0)\ \ \exists y\in M_{+}:y\leq
x,0<\varphi(y)<\infty;$

— _regular_ , if

$\forall\omega\in(M_{*})_{+}\,\,(\omega\neq 0)\
\exists\omega^{\prime}\in(M_{*})_{+}\,\,(\omega^{\prime}\neq
0):\omega^{\prime}\leq\omega,\omega^{\prime}\leq\varphi,$

where $(M_{*})_{+}$ is the set of all positive ultra-weakly continuous linear
functionals on $M.$

If the weight $\varphi$ is a trace, i.e. when
$\varphi(x^{*}x)=\varphi(xx^{*})$ for all $x\in M,$ the properties of semi-
finiteness and locally finiteness (and respectively of faithfulness and
regularity) of $\varphi$ coincide with each other [13].

For a faithful normal semi-finite weight $\varphi$ on $M$ there exists a
uniquely defined non singular self-adjoint positive operator $h$, affiliated
with $M$ such that $\varphi(\cdot)=\tau(h\cdot)$ , and which is called the
Radon-Nikodym derivative of the weight $\varphi$ with respect is the trace
$\tau$ [9].

Recall the following result

Theorem 1.[13] _Let $\tau$ be a faithful normal semi-finite trace on $M$ and
let $\varphi=\tau(h\cdot)$ be a faithful normal semi-finite weight on $M,$
where $h$ is the Radon-Nikodym derivative of $\varphi$ with respect to $\tau.$
Then_

$(i)$ _the weight $\varphi$ is locally finite if and only if the operator $h$
is locally measurable;_

$(ii)$ _the weight $\varphi$ is regular if and only if the operator $h^{-1}$
is locally measurable._

Now let $\varphi(\cdot)=\tau(h\cdot)$ be a faithful normal locally finite
weight on $M.$ For real numbers $p\geq 1$ and $\alpha\in[0,1]$ put

$m_{\alpha}^{1/p}=\\{x\in M:h^{\alpha/p}xh^{(1-\alpha)/p}\in
L_{p}(M,\tau)\\};$ $\|x\|_{p,\alpha}=\|h^{\alpha/p}xh^{(1-p)/p}\|_{p}.$

In [12] it has been proved that $m_{\alpha}^{1/p}$ is a linear subspace in
$M,$ and $\|\cdot\|_{p,\alpha}$ is a norm on $m_{\alpha}^{1/p}.$ The
completion of the normed space $(m_{\alpha}^{1/p},\|\cdot\|_{p,\alpha})$ is
denoted by $L_{p}(M,\varphi).$ In [12] it is proved that the Banach space
$(L_{p}(M,\varphi),\|\cdot\|_{p,\alpha})$ is isometrically isomorphic to the
space $(L_{p}(M,\tau),\|\cdot\|_{p})$ for all $\alpha\in[0,1].$

In order to define the Orlicz space associated with a weight, we need the
notion of $N$-function.

A continuous non-negative convex monotone increasing function $\Phi$ on the
set of real numbers $\mathbb{R}$ is called $N$-_function_ [4], if

$\Phi(t)=\int\limits_{0}^{|t|}p(s)ds,$

where $p(s)$ is a non-decreasing function, positive for $s>0$ and right
continuous for $s\geq 0$ , which satisfies the conditions

$p(0)=0,\ \ p(\infty)=\lim\limits_{s\rightarrow\infty}p(s)=\infty.$

For each $N$-function $\Phi(t)$ a complementary $N$-function $\Psi(t)$ is
defined as

$\Psi(t)=\int\limits_{0}^{|t|}q(s)ds,$

where $q(s)=\sup\\{t\geq 0:p(t)\leq s\\}.$ It is clear that the complementary
$N$-function for the $N$-function $\Psi(t)$ coincides with the initial
function $\Phi(t),$ and moreover the following Young inequality is valid

$ts\leq\Phi(t)+\Psi(s)\ \ \textrm{for all}\ \ t,s\geq 0.$

We say that an $N$-function $\Phi(t)$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, if given any real $k>0$ there exists a
positive number $r(k)$ such that $\Phi(kt)\leq r(k)\Phi(t)$ for all $t\geq 0.$
Examples of $N$-function which satisfy the $(\delta_{2},\Delta_{2})$-condition
are given by the function $\Phi(t)=\frac{1}{p}|t|^{p},$ $p>1.$

Let $\Phi(t)$ be an $N$-function and let $x\in LS_{h}(M),$
$x=\int\limits_{-\infty}^{\infty}\lambda de_{\lambda}(x).$ It is known ([8],
§2.3) that one can define a self-adjoint operator
$\Phi(x)=\int\limits_{-\infty}^{\infty}\Phi(\lambda)de_{\lambda}(x),$ and
moreover $\Phi(x)\in LS(M).$

Let us extend the faithful normal semi-finite trace $\tau$ from $M_{+}$ to
operators from $LS_{+}(M)$ as

$\tau(x)=\sup\limits_{t\geq 1}\tau\left(\int\limits_{0}^{t}\lambda
de_{\lambda}(x)\right)=\int\limits_{0}^{\infty}\lambda d\tau(e_{\lambda}(x)).$

It is known (e.g. [8], §4.1), that

$\tau(x)=\sup\\{\tau(y):y\in M_{+},y\leq x\\}$

for all $x\in LS_{+}(M).$

It is clear that $\tau(|x|)<\infty$ for $x\in LS(M)$ if and only if $x\in
L_{1}(M,\tau);$ in this case $\tau(\mathbf{1}-e_{\lambda}(|x|))<\infty$ for
all $\lambda>0.$ Further we shall need the following result.

Proposition 1. [3] _If $x,y\in LS_{+}(M),$ then_

$(i)$ $\tau(f(x))\leq\tau(f(y))$ for $x\leq y$ _for each continuous monotone
increasing function $f:[0,\infty)\rightarrow\mathbb{R}$ with $f(0)=0;$_

$(ii)$ $\tau(f(\lambda
x+(1-\lambda)y))\leq\lambda\tau(f(x))+(1-\lambda)\tau(f(y))$ _for all
$\lambda\in[0,1]$ and each convex monotone increasing function $f$ with
$f(0)=0.$_

Let $\Phi$ be an $N$-function. The set

$K_{\Phi}=\\{x\in S(M):\tau(\Phi(|x|))\leq 1\\}$

is an absolutely convex subset in $S(M)$ [5]. The linear subspace
$L_{\Phi}(M,\tau)=\bigcup\limits_{n=1}^{\infty}nK_{\Phi}$ equipped with the
norm

$\|x\|_{\Phi}=\inf\left\\{\lambda>0:\frac{x}{\lambda}\in K_{\Phi}\right\\},$
$None$

is a Banach space [5] which is called the Orlicz space associated with
$M,\tau$ and $\Phi$. If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then

$L_{\Phi}(M,\tau)=\\{x\in S(M):\tau(\Phi(|x|))<\infty\\},$

moreover the linear subspace

$m_{\Phi}^{\tau}=\\{x\in M:\tau(\Phi(|x|))<\infty\\}$

is dense in $\left(L_{\Phi}(M,\tau),\|\cdot\|_{\Phi}\right).$

Note that

$m_{\tau}=\\{x\in M:\tau(|x|)<\infty\\}\subset m_{\Phi}^{\tau}.$ $None$

Indeed, from the equalities

$\lim\limits_{t\downarrow 0}\frac{\Phi(t)}{t}=\lim\limits_{t\downarrow
0}p(t)=0$

it follows that $\Phi(t)\leq t$ for sufficiently small $t>0.$

Therefore for $x\in m_{\tau}$ there exists $t_{0}>0$ such that

$\tau(\Phi(|x|e_{t_{0}}(|x|)))=\int\limits_{0}^{t_{0}}\Phi(\lambda)d\tau(e_{\lambda}(|x|))\leq\int\limits_{0}^{t_{0}}\lambda
d\tau(e_{\lambda}(|x|))=\tau(|x|e_{t_{0}}(|x|))<\infty.$

Since $\tau(\mathbf{1}-e_{t_{0}}(|x|))<\infty,$ we have that

$\tau(\Phi(|x|(\mathbf{1}-e_{t_{0}})))\leq\Phi(\|x\|_{M})\tau(\mathbf{1}-e_{t_{0}}(|x|))<\infty,$

where $\|\cdot\|_{M}$ is the $C^{*}$-norm on $M.$ Therefore
$\tau(\Phi(|x|))<\infty,$ i.e. $x\in m_{\Phi}^{\tau}.$

Proposition 2. _If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then $m_{\tau}$ is dense in
_$L_{\Phi}(M,\tau).$

_Proof._ Since $m_{\tau}\subset m_{\Phi}^{\tau}$ (see (2)) and
$m_{\Phi}^{\tau}$ is dense in $L_{\Phi}(M,\tau),$ it sufficient to prove that
$m_{\tau}$ is dense in $m_{\Phi}^{\tau}.$ Moreover since each element of
$m_{\phi}^{\tau}$ is a finite linear combination of positive elements from
$m_{\Phi}^{\tau}$ it sufficient to show that every element from
$x\in\left(m_{\Phi}^{\tau}\right)_{+}$ belongs to the closure of $m_{\tau}$ in
$L_{\Phi}(M,\tau).$ First, let us show that

$x_{n}=x(\textbf{1}-e_{\frac{1}{n}})\in m_{\tau},$

where $e_{\lambda}=e_{\lambda}(x),$ $\lambda>0$ is the spectral family of $x.$
From

$\Phi\left(\frac{1}{n}\right)\tau\left(\textbf{1}-e_{\frac{1}{n}}\right)=\tau\left(\Phi\left(\frac{1}{n}\left(\textbf{1}-e_{\frac{1}{n}}\right)\right)\right)\leq\tau\left(\Phi\left(x\left(\textbf{1}-e_{\frac{1}{n}}\right)\right)\right)\leq\tau(\Phi(x))<\infty,$

it follows that $\tau\left(\textbf{1}-e_{\frac{1}{n}}\right)<\infty$ and the
inequality $0\leq
x\left(\textbf{1}-e_{\frac{1}{n}}\right)\leq\|x\|_{M}\left(\textbf{1}-e_{\frac{1}{n}}\right)$
implies that

$x_{n}=x\left(\textbf{1}-e_{\frac{1}{n}}\right)\in m_{\tau}.$

Since $0\leq xe_{\frac{1}{n}}\downarrow 0$ when $n\rightarrow\infty,$ it
follows that
$\tau\left(\Phi\left(\frac{1}{\varepsilon}xe_{\frac{1}{n}}\right)\right)\downarrow
0$ for any $\varepsilon>0.$ In particular, there exists $n(\varepsilon)$ such
that
$\tau\left(\Phi\left(\frac{1}{\varepsilon}xe_{\frac{1}{n}}\right)\right)<1$
for $n\geq n(\varepsilon),$ i.e.
$\left\|xe_{\frac{1}{n}}\right\|_{\Phi}<\varepsilon.$ This means that
$\|x-x_{n}\|_{\Phi}\rightarrow 0$, i.e. $m_{\tau}$ is dense $m_{\Phi}^{\tau}.$

The proof is complete. $\Box$

Let $\Psi$ be the complementary $N$-function for the $N$-function $\Phi$
satisfying the $(\delta_{2},\Delta_{2})$-condition. In this case given any
$y\in L_{\Psi}(M,\tau)$ the function $f_{y}(x)=\tau(xy),x\in
L_{\Phi}(M,\tau),$ defines the general form of continuous linear functionals
on $L_{\Phi}(M,\tau)$ [5], moreover

$\|f_{y}\|=\sup\\{|\tau(xy)|:x\in L_{\Phi}(M,\tau),\|x\|_{\Phi}\leq
1\\}=\|y\|_{\Psi}.$

Further we shall need also two inequalities from the following proposition.

Proposition 3. _Let $\tau$ be a faithful normal semi-finite trace on a von
Neumann algebra $M$. Then_

$(i)$([8], §3.4). _Given any $x,y\in LS(M)$ there exist two partial isometries
$u,v\in M$ such that_

$|x+y|\leq u^{*}|x|u+v^{*}|y|v.$

$(ii)$ [2]. _For every $N$-function $\Phi$, arbitrary operator $z\in M$ with
$\|z\|_{M}\leq 1$, and for each $x\in LS_{+}(M)$ we have the following
inequality_

$\tau(\Phi(z^{*}xz))\leq\tau(z^{*}\Phi(x)z).$

Orlicz spaces associated with a weight

In this section an approach is suggested for the construction of Orlicz spaces
associated with a faithful normal locally finite weight on a semi-finite von
Neumann algebra for an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition. For these spaces the dual spaces are
described. In the case of regular locally finite normal weights the
constructed Orlicz spaces are represented as spaces of locally measurable
operators.

Let $\tau$ be a faithful normal semi-finite trace on a von Neumann algebra
$M$. From now on $\varphi$ denotes a faithful normal locally finite weight on
$M$. Therefore the Radon-Nikodym derivative $h$ of the weight $\varphi$ with
respect to $\tau$ is a positive locally measurable non-singular operator.

Given an $N$-function $\Phi$ and a real number $\alpha\in[0,1]$ put

$U(x)=U_{\Phi,\alpha}^{\varphi,\tau}(x)=(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha},\
\ x\in LS(M).$

It is clear that $U(x)\in LS(M)$ and $\Phi(|U(x)|)\in LS(M).$

Consider the functional on $LS(M)$ defined by

$O_{\Phi,\alpha}^{\varphi,\tau}(x)=\tau(\Phi(|U(x)|)),$

and put

$m_{\Phi,\alpha}^{\varphi,\tau}=\left\\{x\in
M:O_{\Phi,\alpha}^{\varphi,\tau}(x)<\infty\right\\}.$

Consider on the set $m_{\Phi,\alpha}^{\varphi,\tau}$ the functional

$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda>0:O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq
1\right\\}.$

Theorem 2. _If the $N$-function $\Phi$ satisfies the
$(\delta_{2},\Delta_{2})$-condition, then the set
$m_{\Phi,\alpha}^{\varphi,\tau}$ is a linear subspace in $M$._

In order to prove this theorem we need the following inequality.

Lemma 1. _For the $N$-function $\Phi$ and real number $\lambda\in[0,1]$ the
following inequality is valid_

$\tau(\Phi(|U(\lambda x)|))\leq\lambda\tau(\Phi(|U(x)|))$ $None$

_for all_ $x\in m_{\Phi,\alpha}^{\Phi,\tau}.$

_Proof._ By the linearity of the map $U$ we have

$\tau(\Phi(|U(\lambda x)|))=\tau(\Phi(\lambda|U(x)|)).$

From the inequality $(ii)$ in Proposition 1 with $y=0$, we obtain

$\tau(\Phi(|U(\lambda x)|))\leq\lambda\tau(\Phi(|U(x)|)).$

The proof of lemma is complete. $\Box$

_Proof of the theorem 2._ The inequality (3) above implies that
$m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under the multiplication by complex
number $k$ with $|k|\leq 1.$ Let us show that for $x\in
m_{\Phi,\alpha}^{\varphi,\tau}$ and any complex number $k$ with $|k|>1$ we
have that $kx\in m_{\Phi,\alpha}^{\varphi,\tau},$ i.e.
$\tau(\Phi(|U(kx)|))<\infty.$

Since $\Phi$ satisfies the $(\delta_{2},\Delta_{2})$-condition, given any
positive number $|k|$ there exists a positive number $r(|k|)$ such that
$\Phi(|k|t)\leq r(|k|)\Phi(t)$ for all $t\geq 0$. Therefore
$(\delta_{2},\Delta_{2})$-condition implies that

$\tau(\Phi(|U(kx)|))=\tau(\Phi(|k||U(x)|))\leq$ $\leq
r(|k|)\tau(\Phi(|U(x)|))<\infty,$

i.e. the set $m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under multiplication
by any complex number.

Now let us prove that the sum of any two operators from
$m_{\Phi,\alpha}^{\varphi,\tau}$ also belongs to
$m_{\Phi,\alpha}^{\varphi,\tau}$. Let $x,y\in m_{\Phi,\alpha}^{\varphi,\tau},$
i.e. $\tau(\Phi(|U(x)|))<\infty$ and $\tau(\Phi(|U(y)|))<\infty.$ The
inequalities $(i)$ and $(ii)$ from proposition 3, the linearity of the
operator $U$, the convexity of $\Phi$, the tracial property of $\tau$ and the
fact that $m_{\Phi,\alpha}^{\varphi,\tau}$ is closed under the multiplication
by complex numbers imply:

$\tau(\Phi(|U(x+y)|))=\tau(\Phi(|U(x)+U(y)|))\leq\tau(\Phi(u^{*}|U(x)|u+v^{*}|U(y)|v))\leq$
$\leq\tau\left(\frac{1}{2}(\Phi(2u^{*}|U(x)|u)+\Phi(2v^{*}|U(y)|v))\right)=$
$=\frac{1}{2}(\tau\left(\Phi(u^{*}2|U(x)|u))+\tau(\Phi(v^{*}2|U(y)|v)\right)\leq\frac{1}{2}\left(\tau(u^{*}\Phi(2|U(x)|)u)+\tau(v^{*}\Phi(2|U(y)|)v)\right)\leq$
$\leq\frac{1}{2}(\tau(\Phi(2|U(x)|))+\tau(\Phi(2|U(y)|)))=\frac{1}{2}(\tau(\Phi|U(2x)|)+\tau(\Phi|U(2y)|))<\infty,$

i.e. $x+y\in m_{\Phi,\alpha}^{\varphi,\tau}.$

The proof is complete. $\Box$

Theorem 3. _The set_

$K_{\Phi,\alpha}^{\varphi,\tau}=\\{x\in
M:O_{\Phi,\alpha}^{\varphi,\tau}(x)\leq 1\\}$

_is absolutely convex and absorbing in $m_{\Phi,\alpha}^{\varphi,\tau}.$_

_Proof._ Let us prove the convexity of $K_{\Phi,\alpha}^{\varphi,\tau}$. Let
$x,y\in K_{\Phi,\alpha}^{\varphi,\tau}$ and $\lambda\in[0,1].$ In view of
Proposition 3$(i)$ there exist partial isometries $u$ and $v$ in $M$ such that

$|\lambda U(x)+(1-\lambda)U(y)|\leq\lambda
u^{*}|U(x)|u+(1-\lambda)v^{*}|U(y)|v.$

From the inequalities of Proposition 1 and 3 and from the tracial property of
$\tau$ we obtain

$\tau(\Phi|\lambda
U(x)+(1-\lambda)U(y)|)\leq\lambda\tau(\Phi(u^{*}|U(x)|u))+(1-\lambda)\tau(\Phi(v^{*}|U(y)|v))\leq$
$\leq\lambda\tau(u^{*}\Phi(|U(x)|u))+(1-\lambda)\tau(v^{*}\Phi(|U(y)|)v)\leq\lambda\tau(\Phi(|U(x)|))+(1-\lambda)\tau(\Phi(|U(y)|)),$

i.e.

$O_{\Phi,\alpha}^{\varphi,\tau}(\lambda x+(1-\lambda)y)\leq\lambda
O_{\Phi,\alpha}^{\varphi,\tau}(x)+(1-\lambda)O_{\Phi,\alpha}^{\varphi,\tau}(y),$

which implies the convexity of $K_{\Phi,\alpha}^{\varphi,\tau}$.

The inequality (3) shows that the set $K_{\Phi,\alpha}^{\varphi,\tau}$ is
balanced, and hence is absolutely convex.

Finally let us move that $K_{\Phi,\alpha}^{\varphi,\tau}$ is absorbing in
$m_{\Phi,\alpha}^{\varphi,\tau}$.

If $x\in m_{\Phi,\alpha}^{\varphi,\tau},$ then there exists $t>1$ such that
$O_{\Phi,\alpha}^{\varphi,\tau}(x)<t.$ Let $\lambda\in\mathbb{C}$ and
$|\lambda|\geq t$ By Lemma 1 we have that

$O_{\Phi,\alpha}^{\varphi,\tau}(\frac{x}{\lambda})\leq\frac{1}{|\lambda|}O_{\Phi,\alpha}^{\varphi,\tau}(x)\leq\frac{1}{t}O_{\Phi,\alpha}^{\varphi,\tau}(x)<1,$

i.e. $\frac{x}{\lambda}\in K_{\Phi,\alpha}^{\varphi,\tau}(x).$ The proof is
complete. $\Box$

Corollary 1. _The Minkovsky functional of the set
$K_{\Phi,\alpha}^{\varphi,\tau}$ defined as_

$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda>0:\frac{x}{\lambda}\in
K_{\Phi,\alpha}^{\varphi,\tau}\right\\},$ $None$

_is a norm on the linear space $m_{\Phi,\alpha}^{\varphi,\tau}$._

_Proof._ It is sufficient to prove that $\|x\|_{\Phi,\alpha}^{\varphi,\tau}=0$
implies that $x=0$. Indeed, if $\|x\|_{\Phi,\alpha}^{\varphi,\tau}=0$ then
$O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq 1$ for all
$\lambda\in(0,1)$. By Lemma 1 we obtain that
$\frac{1}{\lambda}O_{\Phi,\alpha}^{\varphi,\tau}\left(x\right)\leq
O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq 1$ for all
$\lambda\in(0,1),$ i.e. $O_{\Phi,\alpha}^{\varphi,\tau}(x)=0.$ Faithfulness of
$\tau$ then implies that $\Phi^{-1}(h)^{\alpha}x\Phi^{-1}(h)^{1-\alpha}=0$.
Since $h\in LS_{+}(M)$ (see theorem 1$(i)$) and $h$ is a non singular
operator, we have that $\Phi^{-1}(h)^{\alpha},$ $\Phi^{-1}(h)^{1-\alpha}\in
LS_{+}(M)$ and $\Phi^{-1}(h)^{\alpha},$ $\Phi^{-1}(h)^{1-\alpha}$ are non
singular operators too. Let

$\Phi^{-1}(h)=\int\limits_{0}^{\infty}\lambda de_{\lambda},$
$x_{n}=\int\limits_{\frac{1}{n}}^{n}\left(\frac{1}{\lambda}\right)^{\alpha}\lambda
de_{\lambda},$
$y_{n}=\int\limits_{\frac{1}{n}}^{n}\left(\frac{1}{\lambda}\right)^{1-\alpha}\lambda
de_{\lambda}.$

Using
$x_{n}\Phi^{-1}(h)^{\alpha}=e_{n}-e_{\frac{1}{n}}=\Phi^{-1}(h)^{1-\alpha}y_{n}$
we see that
$\left(e_{n}-e_{\frac{1}{n}}\right)x\left(e_{n}-\frac{1}{n}\right)=0.$ Since
$\Phi^{-1}(h)$ is a non singular operator, it follows that
$\left(e_{n}-e_{\frac{1}{n}}\right)\uparrow\mathbf{1}$ for all $n=1,2,\ldots$
when $n\rightarrow\infty.$ Consequently, $x=0.$ The proof is complete. $\Box$

Denote by $L_{\Phi,\alpha}(M,\varphi,\tau)$ the Banach space obtained as the
completion of $m_{\Phi,\alpha}^{\varphi,\tau}$ in the norm
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and call this completion _the Orlicz
space_ constructed by the $N$-function $\Phi$ on the von Neumann algebra $M$
with respect to the faithful normal locally finite weight $\varphi$. It is
clear that if $\varphi$ is a trace or $M$ is a commutative von Neumann
algebra, then the norm $\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ do not depend on $\alpha\in[0,1].$

Note also that in the case where $\Phi(t)=\frac{1}{t}|t|^{p},\ p>1,$ the norm
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ and the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ do not depend on the choice of the faithful
normal semi-finite trace $\tau$ and of $\alpha\in[0,1]$ [12].

For general $N$-functions $\Phi$ this is not true even in the commutative
case.

Example. Take $M=l_{\infty},$ $f_{i}=\\{0,...,0,1,0...\\},$ where 1 is on the
$i$-th position, and put $\Phi(t)=|t|^{\beta}(\ln|t|+1),\ t\neq 0,\ \beta>1,\
\Phi(0)=0.$ In ([4], Ch. I, §4) it is proved that $\Phi$ is an $N$-function
satisfying the $(\delta_{2},\Delta_{2})$-condition. Consider the trace $\nu$
on $l_{\infty}$ defined as
$\nu(f_{i})=\frac{1}{i^{2}}((e^{i^{2}})^{2\beta}(2i^{2}+1))^{-1}.$ Put

$h=\\{e^{\beta i^{2}}+(i^{2}+1)\\}_{i=1}^{\infty},$
$f=\Phi^{-1}(h)=\\{e^{i^{2}}\\}_{i=1}^{\infty}.$

Now define the trace $\mu$ on $l_{\infty}$ as $\mu(\cdot)=\nu(h\cdot).$

Let us show that in this case the norms $\|\cdot\|_{\Phi,1}^{\mu,\nu}$ and
$\|\cdot\|_{\Phi,1}^{\mu,\mu}$ are not equivalent on the ideal $E$ of all
finite sequences from $l_{\infty}$ (it is clear that $E\subset
m_{\Phi,\alpha}^{\mu,\nu}$ and $E\subset m_{\Phi,\alpha}^{\mu,\mu}$). For this
it is sufficient to find a sequence $\\{x_{n}\\}$ of elements from
$(K_{\Phi,1}^{\mu,\nu})\cap E$ such that
$\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda K_{\Phi,1}^{\mu,\mu}$ for all
$\lambda>0.$ Let $x_{n}=\sum\limits_{i=2}^{n}e^{i^{2}}f_{i}.$ It is clear that
for commutative algebras one has

$O_{\Phi,1}^{\mu,\nu}(x)=\nu(\Phi(|\Phi^{-1}(h)x|))$

and

$O_{\Phi,1}^{\mu,\mu}(x)=\mu(\Phi(|x|)).$ $None$

Therefore

$O_{\Phi,1}^{\mu,\nu}(x_{n}f_{i})=\nu(\Phi(fx_{n}f_{i}))=\nu(\Phi((e^{2i^{2}}f_{i})^{2}))=(e^{i^{2}})^{2\beta}(2i^{2}+1)\nu(f_{i})=\frac{1}{i^{2}}.$

Hence

$O_{\Phi,1}^{\mu,\nu}(x_{n})=\sum\limits_{i=2}^{n}\frac{1}{i^{2}}<1,\textrm{i.e.}\,x_{n}\in
K_{\Phi,1}^{\mu,\nu},$ $None$

for all $n.$ Let us show that $\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda
K_{\Phi,1}^{\mu,\mu}$ for all positive real $\lambda.$ From (5) we have

$O_{\Phi,1}^{\mu,\mu}(x_{n}f_{i})=\mu(\Phi(x_{n}f_{i}))=\nu(h\Phi(x_{n})f_{i})=\nu(\Phi(\Phi^{-1}(h))\Phi(x_{n})f_{i})=\nu(\Phi(f)\Phi(x_{n})f_{i})=$
$=(e^{i^{2}})^{2\beta}(i^{4}+2i^{2}+1)\nu(f_{i})>(e^{i^{2}})^{2\beta}i(2i^{2}+1)\nu(f_{i})=\frac{1}{i}.$

Therefore $O_{\Phi,1}^{\mu,\mu}(x_{n})>\sum\limits_{i=2}^{n}\frac{1}{n},$ and
hence

$\\{x_{n}\\}\subset\\!\\!\\!\\!\\!/\lambda K_{\Phi,1}^{\mu,\mu}$ $None$

for all positive $\lambda.$ From (6) and (7) it follows that the norms
$\|\cdot\|_{\Phi,1}^{\mu,\nu}$ and $\|\cdot\|_{\Phi,1}^{\mu,\mu}$ are not
equivalent on $E.$ In particular the identity mapping from $E$ into $E$ can
not be extended to an isomorphism between
$L_{\Phi,\alpha}(l_{\infty},\mu,\nu)$ and
$L_{\Phi,\alpha}(l_{\infty},\mu,\mu).$

At the same time by following theorem the Orlicz spaces
$L_{\Phi,\alpha}(l_{\infty},\mu,\nu)$ and
$L_{\Phi,\alpha}(l_{\infty},\mu,\mu)$ are isometrically isomorphic.

Theorem 4. _Let the $N$-function $\Phi$ satisfy the
$(\delta_{2},\Delta_{2})$-condition, $\alpha\in[0,1].$ Then the Banach space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is isometrically isomorphic to the Banach
space_ $L_{\Phi}(M,\tau)=L_{\Phi,1}(M,\tau,\tau)$.

_Proof._ For every $x\in m_{\Phi,\alpha}^{\varphi,\tau}$ we have

$U(x)=(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\in L_{\Phi}(M,\tau).$

Therefore from definitions (1) and (4) of the norms we obtain

$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\|(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\|_{\Phi}.$

This means that the map $U$ defined as

$m_{\Phi,\alpha}^{\varphi,\tau}\ni
x\longrightarrow^{\\!\\!\\!\\!\\!\\!\\!\\!\\!U}\
(\Phi^{-1}(h))^{\alpha}x(\Phi^{-1}(h))^{1-\alpha}\in L_{\Phi}(M,\tau)$ $None$

is a linear isometry. Let us show that the
$U(m_{\Phi,\alpha}^{\varphi,\tau})=(\Phi^{-1}(h))^{\alpha}m_{\Phi,\alpha}^{\varphi,\tau}(\Phi^{-1}(h))^{1-\alpha}$
is dense in $L_{\Phi}(M,\tau)$.

Let $h=\int\limits_{0}^{\infty}\lambda de_{\lambda}$ and
$q_{n}=\int\limits_{\frac{1}{n}}^{n}de_{\lambda}\,\,,(n=1,2,...)$. Consider
the set

$\mathcal{F}=\bigcup\limits_{m,n=1}^{\infty}q_{m}m_{\tau}q_{n}.$

Since $q_{n}\leq q_{n+1},$ it follows that $\mathcal{F}$ is a linear subspace
in $m_{\tau}$ and by (2) $\mathcal{F}\subset L_{\Phi}(M,\tau).$

First. Let us prove that $\mathcal{F}$ is dense in $L_{\Phi}(M,\tau)$. From
the $(\delta_{2},\Delta_{2})$-condition it follows that for $y\in
L_{\Psi}(M,\tau)$ (where $\Psi$ is the complementary $N$-function for $\Phi$)
the functional $f(x)=\tau(xy),$ $x\in L_{\Phi}(M,\tau),$ defines the general
form of continuous linear functional on $L_{\Phi}(M,\tau)$.

Let $y\in L_{\Psi}(M,\tau)$ and suppose that
$f(q_{m}xq_{n})=\tau((q_{m}xq_{n})y)=0$ for all $x\in m_{\tau}$ and
$m,n=1,2,....$ In order to prove that $\mathcal{F}$ is dense in
$L_{\Phi}(M,\tau)$ it is sufficient to show that $y=0.$

From the tracial property of $\tau$ we have that $\tau(xq_{n}yq_{m})=0$ for
all $x\in m_{\tau}.$ By proposition 2 $m_{\tau}$ is dense in
$L_{\Phi}(M,\tau)$ and hence $q_{n}yq_{m}=0$ for all $m,n=1,2,...$. Since
$q_{n}\nearrow\textbf{1}$ as $n\rightarrow\infty,$ this implies that $y=0$.
Therefore $\mathcal{F}$ is dense $L_{\Phi}(M,\tau)$.

Now let us show that $\mathcal{F}\subset U(m_{\Phi,\alpha}^{\varphi,\tau}).$
For this it is sufficient to prove that given any $x\in m_{\tau}$ and
$m,n=1,2,...,$ there exists $y\in m_{\Phi,\alpha}^{\varphi,\tau}$ such that
$q_{m}xq_{n}=U(y).$

Since the operators $(\Phi^{-1}(h))^{-\alpha}q_{m}$ and
$(\Phi^{-1}(h))^{\alpha-1}q_{n}$ belong to $M,$ the operator
$y=U^{-1}(q_{m}xq_{n})=(\Phi^{-1}(h))^{-\alpha}(q_{m}xq_{n})(\Phi^{-1}(h))^{\alpha-1}$
also belongs to $M.$ From (2) and from $\tau(|q_{m}xq_{n}|)<\infty$ we obtain
that $\tau(\Phi(|U(y)|))=\tau(\Phi(|q_{m}xq_{n}|))<\infty,$ i.e. $y\in
m_{\Phi,\alpha}^{\varphi,\tau}.$ This implies that $\mathcal{F}\subset
U(m_{\Phi,\alpha}^{\varphi,\tau}).$

Now since $m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(L_{\Phi,\alpha}(M,\varphi,\tau),\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau})$ and
$U\left(m_{\Phi,\alpha}^{\varphi,\tau}\right)$ is dense in
$(L_{\Phi}(M,\tau),\|\cdot\|_{\Phi})$ the isometry
$U:m_{\Phi,\alpha}^{\varphi,\tau}\rightarrow L_{\Phi}(M,\tau)$ defined in (8)
can be uniquely extended to an isometric isomorphism between
$L_{\Phi,\alpha}(M,\varphi,\tau)$ and $L_{\Phi}(M,\tau).$ The proof is
complete. $\Box$

Since every faithful normal semi-finite trace $\tau_{1}$ on $M$ is a locally
finite weight [13] the theorem 4 implies the following

Corollary 2._If $\tau_{1}$ and $\tau$ are faithful normal semi-finite traces
on a von Neumann algebra $M,$ $\Phi$ is an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition, then the Orlicz spaces
$L_{\Phi}(M,\tau_{1})$ and $L_{\Phi}(M,\tau)$ are isometrically isomorphic._

Theorem 4 and Corollary 2 together imply the following theorem

Theorem 5._Let $\tau_{1}$ and $\tau$ be faithful normal traces on a von
Neumann algebra $M,$ and let $\varphi_{1},\varphi_{2}$ be faithful normal
locally finite weights on $M.$ Suppose that $\Phi$ is an $N$-function
satisfying the $(\delta_{2},\Delta_{2})$-condition, $\alpha,\beta\in[0,1].$
Then the Orlicz spaces $L_{\Phi,\alpha}(M,\varphi_{1},\tau_{1})$ and
$L_{\Phi,\beta}(M,\varphi_{2},\tau_{2})$ are isometrically isomorphic._

Theorem 4 implies also the following

Corollary 3. _Let $\Phi$ be an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition and let $\Psi$ be the complementary
$N$-function for $\Phi$, and $\alpha,\beta\in[0,1].$ Then the dual space
$(L_{\Phi,\alpha}(M,\varphi,\tau))^{*}$ for the Orlicz space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is isometrically isomorphic to the space
$L_{\Psi}(M,\tau).$ If moreover $\Psi$ also satisfies the
$(\delta_{2},\Delta_{2})$-condition then
$(L_{\Phi,\alpha}(M,\varphi,\tau))^{*}$ is isometrically isomorphic to
$L_{\Psi,\beta}(M,\varphi,\tau)$ and the Banach space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ is reflexive._

Now let us give a representation of the space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ by locally measurable operators in the case
where $\varphi$ is a regular locally finite weight, and the $N$-function
$\Phi$ satisfies $(\delta_{2},\Delta_{2})$-condition.

Consider the following subset in the algebra $LS(M)$ of locally measurable
operators affiliated with the von Neumann algebra $M$:

$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)=\\{x\in
LS(M):O_{\Phi,\alpha}^{\varphi,\tau}(x)<\infty\\},$

and for each $x\in\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)$ put

$\|x\|_{\Phi,\alpha}^{\varphi,\tau}=\inf\left\\{\lambda\geq
0:O_{\Phi,\alpha}^{\varphi,\tau}\left(\frac{x}{\lambda}\right)\leq
1\right\\}.$

It is clear that

$m_{\Phi,\alpha}^{\varphi,\tau}=M\bigcap\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau).$

Repeating the proof of the Theorems 2 and 3 and of Corollary 1 we obtain that
$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)$ is a linear subspace of $LS(M)$
and that $\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}$ is a norm on
$\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau).$

Theorem 6._Let $\varphi$ be a regular locally finite normal weight on $M$ and
suppose that $\Phi$ is an $N$-function satisfying the
$(\delta_{2},\Delta_{2})$-condition and $\alpha\in[0,1].$ Then
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau})$ is a Banach space and
$m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$_

_Proof._ Let $h$ be the Radon-Nikodym derivative of the weight $\varphi$ with
respect to the trace $\tau$, and suppose that
$h=\int\limits_{0}^{\infty}\lambda de_{\lambda}$ is the spectral resolution of
the operator $h$. From Theorem 1 it follows that the operators $h$ and
$h^{-1}$ are locally measurable. Therefore the operators
$\Phi^{-1}(h)=\int\limits_{0}^{\infty}\Phi^{-1}(\lambda)de_{\lambda}$ and
$(\Phi^{-1}(h))^{-1}=\int\limits_{0}^{\infty}((\Phi^{-1}(\lambda)))^{-1}de_{\lambda}$
are also locally measurable. There exists a linear isometry $U$ from
$m_{\Phi,\alpha}^{\varphi,\tau}$ into $L_{\Phi}(M,\tau)$ (see (8)), in
particular $U$ is injective. Thus there exists the converse map $U^{-1}$ for
the map $U.$ Since $U(m_{\Phi,\alpha}^{\varphi,\tau})$ is dense in
$L_{\Phi}(M,\tau)$ (see the proof of theorem 4), the converse map $U^{-1}$ can
be extended to a linear isometry from $L_{\Phi}(M,\tau)$ to the closure of
$m_{\Phi,\alpha}^{\varphi,\tau}$ in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$

Similar to the proof of Theorem 4 we obtain that
$m_{\Phi,\alpha}^{\varphi,\tau}$ is dense in
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}).$ Therefore $U^{-1}$ can be extended
to a linear isometry from $(L_{\Phi}(M,\tau),$ $\|\cdot\|_{\Phi})$ onto
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}),$ i.e.
$(\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau),$
$\|\cdot\|_{\Phi,\alpha}^{\varphi,\tau}),$ is a Banach space. The proof is
complete. $\Box$

Theorem 6 implies that in the case where $h$ and $h^{-1}$ are locally
measurable operators and the $N$-function $\Phi$ satisfies
$(\delta_{2},\Delta_{2})$-condition, the Orlicz space
$L_{\Phi,\alpha}(M,\varphi,\tau)$ can be described by locally measurable
operators in the following form

$L_{\Phi,\alpha}(M,\varphi,\tau)=\mathcal{L}_{\Phi,\alpha}(M,\varphi,\tau)=(\Phi^{-1}(h))^{-\alpha}L_{\Phi}(M,\tau)(\Phi^{-1}(h))^{\alpha-1}\subset
LS(M).$

Acknowledgements

_The final version of this work was done within the framework of the
Associateship Scheme of the Abdus Salam International Centre for Theoretical
Physics (ICTP), Triest. Italy. The first author thanks ICTP for providing
financial support and all facilities during his stay in ICTP (July-August,
2011). This work is supported in part by the DFG AL 214 136-1 project
(Germany). The first and the third authors would like to thank the Institute
of Applied Mathematics of the Bonn University for hospitality (April-May,
2011)._

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# Geometry-induced topological superconductivity

Po-Hao Chou1, Chia-Hsin Chen2, Shih-Wei Liu2, Chung-Hou Chung3, and Chung-Yu
Mou1,2,4 1Physics Division, National Center for Theoretical Sciences, P.O.Box
2-131, Hsinchu, Taiwan, R.O.C.  2Center for Quantum Technology and Department
of Physics, National Tsing Hua University, Hsinchu,Taiwan 300, R.O.C.
3Electrophysics Department, National Chiao-Tung University, Hsinchu,Taiwan
300, R.O.C.  4Institute of Physics, Academia Sinica, Nankang, Taiwan 115,
R.O.C.

###### Abstract

Intrinsic topological superconductors with $p$-wave pairing are rare in
nature. Its underlying reason is due to the fact that it is usually difficult
to change the relative strength between the singlet and triplet channels for
the electron-electron interaction in material. Here we show that by
considering superconductivity occurring on surfaces of topological insulators
(TIs), the relative strength between the singlet and triplet channels can be
changed by geometry and sizes of TIs. Specifically, we show that pairing of
electrons at different locations on the surface of a topological insulator
generally tends to favor the triplet pairing and can induce topological
superconductivity by controlling the surface curvature and size of the
topological insulator. We illustrate the effects in two configurations, thin
film geometry and the spherical geometry with a sphere or a hemisphere, and
find that topological superconductivity arises with the $p\pm ip$ pairing
symmetry dominated in nanoscale size of the TI. As a consequence, vortices can
spontaneously form on surfaces of topological insulators with roughness of
appropriate curvature. These vortices support a Majorana zero mode inside each
core and can be used as a platform to host Majorana zero modes without
invoking real magnetic fields. Our theoretical discovery opens a new route to
realize topological superconductivity in material.

##  I Introduction

The discovery of the topological insulators (TIs) has naturally led to
consider superconductors with nontrivial topologyKanerev ; Zhangrev ;
Zhang2009 ; Xue2011 . Due to additional particle-hole symmetry imposed by
superconductivity, quasi-particles in topological superconductors can realize
the Majorana fermion which is its own anti-particle with quasi-particle
operators satisfying $\gamma^{\dagger}=\gamma$. In particular, the localized
Majorana zero mode is considered as an important building block for
constructing topological quantum computersquantumcomp .

To realize topological superconductivity, a number of proposals have been put
forth. Fu and Kane proposed to realize topological superconductivity through
the proximity of the surface states in TIs to s-wave superconductors, where in
the presence of magnetic fields, the surface Dirac cone can be used to
simulate the chiral $p_{x}+ip_{y}$ pairingFuKane . This idea was further
developed to use other materials with spin-orbit interactions in proximity to
both ferromagnetic insulators and s-wave superconductorsSarma . While much
experimental progress has been made along this approachXu2015 ; William2012 ;
Wang2012 ; Sun2016 , the confinement of Majorana zero modes to the interface
of the TI and a superconductor limits experimental access for manipulations.
On the other hand, materials that develop intrinsic topological
superconductivity generally requires the presence of the odd-parity pairing
symmetrySato2017 . In real materials, singlet and triplet pairing channels are
governed by the same electron-electron interactions but projected to different
channels. For the singlet pairing channel, the wavefuction of the Cooper pair
needs to be symmetric in space; while for the triplet pairing channel, the
wavefuction for the Cooper pair has to be anti-symmetric in space. The Coulomb
interaction energy is reduced when the orbital wavefunction is antisymmetric
in space; while the energy for the attractive interaction between electrons
due to the electron-phonon interaction is reduced when the orbital
wavefucntion is symmetric in space. For typical superconductors, the
attractive interaction between electrons dominates so that the summation of
electron-phonon interaction and the Coulomb interaction yields a net
attraction between electrons near the Fermi surface. As a result, the
condition for the odd-parity superconductivity to emerge is usually difficult
to be satisfied. Hence natural topological superconductors are rare. The well-
known old candidates are superfluid 3HeHe3  and Sr2RuO4SRO . Recent intensive
search has led to the discovery of a number of new possible candidates such as
CuxBi2Se3, InxSn1-xTe, PrOs4Sb12, $\beta$-PdBi2, UTe2CuBiSe1 ; CuBiSe2 ;
Bi2Se3pressure ; Sb2Se3pressure ; InSnTe ; CuPbSeBiSe ; SrBi2Se3 ; Tl5Te3 ;
PrOs4Sb12 ; betaPdBi2 ; UTe2 , and etc. Although there are experimental
evidences of unconventional superconductivity in these materials, due to
possible alternative scenario to account for experimental observationsLee2012
; SRO1 , unambiguous signature of topological superconductivity has not been
firmly established.

In this paper, by considering the surface superconductivity that was recently
found on TIsSb2Te3 , we would like to explore an alternative way to realize
the topological superconductivity. By exploiting the competition of the
Coulomb interaction and the electron-electron attraction due to phonons on
surfaces, we show that the relative strength of the Coulomb interaction and
the electron-electron attraction due to phonons can be tuned by curvatures and
sizes of TI. Specifically, we show that through pairing of electrons at
different locations on surfaces of a TI, surfaces of TIs can be turned into
topological superconductors by controlling surface curvature and sizes of TIs.
We illustrate the mechanism by first considering competition of pairing of
electrons on the same surface and paring between top and down surfaces
(interlayer) in the thin film geometry, and we then extend the illustration to
the same competition on a curve surface by using the spherical geometry with a
sphere and a hemisphere. In the thin film geometry, we find that the inter-
layer pairing with $p\pm ip$ pairing symmetry emerges when the thickness
decreases to the nanoscale. In the spherical geometry, we show that $p\pm ip$
pairing dominates and vortices can spontaneously form without invoking real
magnetic fields. These vortices generally supports a Majorana zero mode inside
each vortex core. Our results thus indicate that TIs with appropriate geometry
(such as surface roughness) can be used as a platform to host Majorana zero
modes on their surfaces.

##  II Theoretical model

We start by considering electrons in a general manifold $M$ occupied by a
topological insulator. The low energy physics is characterized by interacting
Dirac Hamiltonian

$\displaystyle H=$ $\displaystyle\int_{M}d\vec{r}\
\psi^{{\dagger}}({\vec{r}})\begin{pmatrix}(m_{\vec{r}}-\mu)I&\lambda_{so}\vec{\sigma}\cdot\vec{p}\\\
\lambda_{so}\vec{\sigma}\cdot\vec{p}&(-m_{\vec{r}}-\mu)I\end{pmatrix}\psi({\vec{r}})$
$\displaystyle+\frac{1}{2}\int_{M}d\vec{r}d\vec{r^{\prime}}V(\vec{r},\vec{r}^{\prime})\psi^{{\dagger}}(\vec{r})\psi(\vec{r})\psi^{{\dagger}}(\vec{r}^{\prime})\psi(\vec{r}^{\prime}).$

Here
$\psi^{{\dagger}}(\vec{r})=(C_{\vec{r},A\uparrow}^{{\dagger}},C_{\vec{r},A\downarrow}^{{\dagger}},C_{\vec{r},B\uparrow}^{{\dagger}},C_{\vec{r},B\downarrow}^{{\dagger}})$
with $C_{\vec{r},\tau\sigma}^{{\dagger}}$ being the creation operator of
conduction electrons with orbital $\tau=A,B$Zhang2010  and spin
$\sigma=\uparrow,\downarrow$ at position $\vec{r}$. $\mu$ is the chemical
potential. $m_{\vec{r}}=m_{0}+m_{2}\vec{p}^{2}$ is the effective Dirac mass
with $\vec{p}$ being the momentum operator, $m_{0}>0$, and $m_{2}<0$. The
interaction $V(\vec{r},\vec{r}^{\prime})$ includes the phonon-mediated
attractive interaction $V^{ph}(\vec{r},\vec{r}^{\prime})$ and the repulsive
Coulomb interaction $V^{c}(\vec{r},\vec{r}^{\prime})$. We shall assume that
the superconductivity occurs in topological protected surface states so that
$H$ will be projected onto the curved surface of $M$ and become an effective
surface Hamiltonian $H_{eff}$. The energy dispersion of $H_{eff}$ includes
Dirac cones with particle-hole symmetry. Here the chemical potential $\mu$ and
the superconducting pairing energy cutoff will be set to lie in lower Dirac
cone.

##  III Topological $p$-wave pairing induced in thin film geometry

Figure 1: (a) Schematic plot of Dirac cone states on top (t) and bottom (b)
surfaces in a thin film of thickness L. (b) Pairing of electrons in the same
surface, $\Delta_{s}$ (top surface) and $-\Delta_{s}$ (bottom surfaces), is
singlet. (c) Pairing of electrons between top and bottom surfaces,
$\Delta_{t}$, is triplet.

We first analyze superconductivity due to the competition between $V^{ph}$ and
$V^{c}$ in the thin film geometry. Let the thickness of the thin film be $L$.
As shown in Fig. 1(a), electronic states at top and down surfaces are
described by the Dirac cone states when $L$ is large. The possible pairing of
electrons are intra-surface and inter-surface pairings as indicated in 1(b)
and (c). Clearly, the combined intra-surface pairing of top ($t$) and bottom
($b$) surfaces is singlet-pairing, represented by $\Delta_{s}$; while the
inter-surface pairing is triplet-pairing, represented by $\Delta_{t}$. The
Hamiltonian for describing the surface superconductivity is the specialization
of Eq.(II) to the thin film geometry and can be written as

$\displaystyle
H=\sum_{\mathbf{k}z}\Psi^{\dagger}_{{\mathbf{k}}z}(H_{\mathbf{k}z}-\varepsilon_{F})\Psi_{{\mathbf{k}}z}+$
$\displaystyle\frac{1}{2\Omega}\sum_{\mathbf{k}\mathbf{k}^{\prime}\mathbf{q},zz^{\prime}}V_{\mathbf{q},zz^{\prime}}(\Psi^{\dagger}_{{\mathbf{k}}-{\mathbf{q}}z}\Psi_{{\mathbf{k}}^{\prime}z})(\Psi^{\dagger}_{{\mathbf{k}}^{\prime}+{\mathbf{q}}z^{\prime}}\Psi_{{\mathbf{k}}z^{\prime}}),$
(1)

where $\varepsilon_{F}$ is the Fermi energy, $\Omega$ is the surface area, A
and B are indices for atomic orbits, $z$ is the index for quintuple layers
along z-axis, ${\mathbf{k}}=(k_{x},k_{y})$ is the in-plane momentum along the
surface, and
$\Psi^{\dagger}_{{\mathbf{k}}z}=(C^{\dagger}_{{\mathbf{k}}z,A\uparrow},C^{\dagger}_{{\mathbf{k}}z,A\downarrow},C^{\dagger}_{{\mathbf{k}}z,B\uparrow},C^{\dagger}_{{\mathbf{k}}z,B\downarrow})$
is the creation operator for surface electrons. The Hamiltonian matrix
$H_{\mathbf{k}z}$ is given by the following formZhang2010

$\displaystyle H_{\mathbf{k}z}=(m_{0}+m_{1}p^{2}_{z}+m_{2}k^{2})\tau_{z}+$
$\displaystyle\left[\lambda_{so}^{z}p_{z}\sigma_{z}+\lambda_{so}\left(k_{x}\sigma_{x}+k_{y}\sigma_{y}\right)\right]\tau_{x},$
(2)

where $\tau_{\alpha}$ ($\alpha=x,y,z$) are Pauli matrices in the pseudo space
formed by atomic orbits A and B and parameters are given as follows:
$m_{0}=-175$ (meV), $m_{1}=200$ (meV$\cdot\mbox{nm}^{2}$), $m_{2}=500$
(meV$\cdot\mbox{nm}^{2}$), $\lambda_{so}=330$ (meV$\cdot\mbox{nm}$) and
$\lambda_{so}^{z}=330$ (meV$\cdot\mbox{nm}$). The interaction
$V_{\mathbf{q},zz^{\prime}}$ is a summation of bare Coulomb interaction and
phonon-mediated
interaction,$V_{\mathbf{q},zz^{\prime}}=V^{c}_{\mathbf{q},zz^{\prime}}+V^{ph}_{\mathbf{q},zz^{\prime}}$,
with

$\displaystyle V^{c}_{\mathbf{q},zz^{\prime}}=\frac{4\pi
e^{2}}{\kappa}\frac{e^{-q|z-z^{\prime}|}}{2q},$ $\displaystyle
V^{ph}_{\mathbf{q},zz^{\prime}}=-\frac{(Z\hbar
q\overline{V^{c}_{q}})^{2}}{Ma^{2}}\sum_{\varepsilon}\frac{\phi_{\varepsilon}(z)\phi_{\varepsilon}(z^{\prime})}{(v_{p}q)^{2}+\varepsilon^{2}}.$
(3)

Here $\overline{V^{c}_{q}}$ is the average of
$V^{c}_{\mathbf{q},zz^{\prime}}$, $Z=3.53e$ is the renormalized ion charge,
$M=124.7u$ is the average ion mass, $a=0.4$ nm is the lattice constant along
the surface, $v_{p}=2$ km/s is the speed of sound, $\varepsilon$ is the phonon
dispersion calculated by 1D phonon model with open boundary condition along
z-axis, and $\phi_{\varepsilon}(z)$ is the corresponding wave function of
phonon.

There are four surface energy eigenstates for each Fourier mode
${\mathbf{k}}$, which are related to the real space basis by a unitary
transformation $U$

$C^{\dagger}_{{\mathbf{k}},v\alpha}=\sum_{z\tau\sigma}U^{\tau\sigma*}_{{\mathbf{k}},v\alpha}(z)C^{\dagger}_{{\mathbf{k}}z,\tau\sigma}.$
(4)

Here $v=\pm$ specifies the spin rotation direction of the surface state, i.e.
the chirality of the surface state, $\alpha=p,n$ specifies sign of the eigen-
energy(positive or negative), $\tau=A,B$ specifies atomic orbits, and $\sigma$
is the index for spin. We shall assume that only the negative energy surface
states lie on the Fermi level, hence only the $\alpha=n$ channel is kept and
we shall drop the index for $\alpha$. The screened Coulomb interaction and
phonon-mediated interaction, $V^{c}_{\mathbf{q},zz^{\prime}}$ and
$V^{ph}_{\mathbf{q},zz^{\prime}}$, can be obtained from the z-axis charge
screening as done in Ref[screened, ]. The resulting BCS Hamiltonian is then
given by

$\displaystyle
H_{eff}=\sum_{\mathbf{k}}\sum_{v=\pm}(\varepsilon_{k}-\mu)C^{{\dagger}}_{\mathbf{k},v}C_{\mathbf{k},v}+\frac{1}{2\Omega}\times$
$\displaystyle\sum_{\begin{subarray}{c}\mathbf{k}\mathbf{k}^{\prime}\\\
v_{1},v_{2},v_{3},v_{4}=+,-\end{subarray}}\Gamma^{v_{1}v_{2}v_{3}v_{4}}_{\mathbf{k}\mathbf{k}^{\prime}}C^{{\dagger}}_{\mathbf{k},v_{1}}C^{{\dagger}}_{-\mathbf{k},v_{2}}C_{-\mathbf{k}^{\prime},v_{3}}C_{\mathbf{k}^{\prime},v_{4}}.\quad$
(5)

Here $\Omega$ is the surface area of the film,
$\varepsilon_{k}=-\sqrt{\lambda_{so}^{2}k^{2}+\Delta^{2}_{g,k}}$ with
$\Delta_{g,k}$ being the hybridized gap of top and bottom surface states,
$v=\pm$ is the index for chirality, and
$\Gamma^{v_{1}v_{2}v_{3}v_{4}}_{\mathbf{k}\mathbf{k}^{\prime}}$ is the
projected effective interaction between quasi-particles of different
chiralities.

Figure 2: Thickness dependence of electron-phonon coupling $\lambda_{\alpha}$,
strength of Coulomb interaction $\mu^{*}_{\alpha}$, and $T_{c}$ for three
different Fermi energiesparameter1 : (a) $0.5\varepsilon_{F}$ (b)
$0.84\varepsilon_{F}$ (c) $\varepsilon_{F}$, where $\varepsilon_{F}$ is the
bulk Fermi energy of $\mbox{Sb}_{2}\mbox{Te}_{3}$Sb2Te3 . Here $L$ is number
of quintuple layers(QL, 1 QL $\sim$ 1nm). Subscripts $s$ and $t$ denote intra-
surface and inter-surface.

The pairing amplitudes, $\Delta_{s}$ and $\Delta_{t}$, are determined by the
effective potential by
$g_{s,\mathbf{k}\mathbf{k}^{\prime}}=2(\Gamma^{++++}_{\mathbf{k}\mathbf{k}^{\prime}}-\Gamma^{++--}_{\mathbf{k}\mathbf{k}^{\prime}})$
and
$g_{t,\mathbf{k}\mathbf{k}^{\prime}}=2(\Gamma^{+--+}_{\mathbf{k}\mathbf{k}^{\prime}}+\Gamma^{+-+-}_{\mathbf{k}\mathbf{k}^{\prime}})$.
In the mean-field approximation, the pairing term in $H_{eff}$ becomes

$\displaystyle
H_{\Delta}=\sum_{\mathbf{k}}\left\\{-i\Delta_{s}(C^{{\dagger}}_{\mathbf{k}\uparrow}C^{{\dagger}}_{-\mathbf{k}\downarrow}-C^{{\dagger}}_{\mathbf{k}\downarrow}C^{{\dagger}}_{-\mathbf{k}\uparrow})+\Delta_{t}(k_{x}-ik_{y})\right.$
$\displaystyle\left.C^{{\dagger}}_{\mathbf{k}\uparrow}C^{{\dagger}}_{-\mathbf{k}\uparrow}-\Delta_{t}(k_{x}+ik_{y})C^{{\dagger}}_{\mathbf{k}\downarrow}C^{{\dagger}}_{-\mathbf{k}\downarrow}+h.c.\right\\}.$
(6)

As a result, the energy of quasi-particles is given by
$E_{k}=\sqrt{\xi^{2}_{k}+\Delta^{2}_{s}+\Delta^{2}_{t}}$ with
$\xi_{k}=\varepsilon_{k}-\mu$ and the gap equations are given by
$\Delta_{\alpha}=-\sum_{\mathbf{k}^{\prime}}g_{\alpha,\mathbf{k}\mathbf{k}^{\prime}}\frac{\Delta_{\alpha}}{2E_{k^{\prime}}}\tanh{\frac{\beta
E_{k^{\prime}}}{2}},$ where $\alpha$= $t$ or $s$ and $\beta=1/k_{B}T$.
Clearly, the gap equation cannot be simultaneously satisfied for both
$\Delta_{t}$ and $\Delta_{s}$. Hence $\Delta_{t}$ and $\Delta_{s}$ cannot
coexist for a given thickness $L$. Since interactions between quasi-particles
crucially depend on $L$, their competition may lead to the change of pairing
symmetry as the thickness changes. An estimation of the transition temperature
can be found based on the empirical McMillan formulaMcMillan

$T_{c}=\frac{\Theta_{D}}{1.45}\exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}(1+0.62\lambda)}\right].$
(7)

Figure 3: Phase diagram of superconducting orders for thin film geometry with
the Fermi energy being in the lower surface Dirac cone. Here $L$ is the
thickness of the thin film in the unit of number of quintuple layers, $n_{h}$
is the surface hole density, and $n_{h,0}$ is the bulk hole density of
$\mbox{Sb}_{2}\mbox{Te}_{3}$.

[hbtp] Here $\Theta_{D}$ is the Debye temperature, $\lambda$ is the electron-
phonon coupling constant, and $\mu^{*}$ is the Coulomb pseudo-potential.
$\lambda$ and $\mu^{*}$ are obtained by setting $k,k^{\prime}\sim k_{F}$ and
performing the average over the angle of $k^{\prime}$,
$\phi_{{\mathbf{k}}^{\prime}}$ ,for the effective potential due tothe
electron-phonon coupling
$g^{ph}_{\alpha,{\mathbf{k}}{\mathbf{k}}^{\prime}}$and the Coulomb interaction
$g^{c}_{\alpha,{\mathbf{k}}{\mathbf{k}}^{\prime}}$ respectively

$\displaystyle\lambda_{\alpha}=\int_{0}^{2\pi}d\phi_{{\mathbf{k}}^{\prime}}g^{ph}_{\alpha,{\mathbf{k}}_{F}{\mathbf{k}}_{F}^{\prime}},$
$\displaystyle\mu_{\alpha}=\int_{0}^{2\pi}d\phi_{{\mathbf{k}}^{\prime}}g^{c}_{\alpha,{\mathbf{k}}_{F}{\mathbf{k}}_{F}^{\prime}},$
$\displaystyle\mu^{*}_{\alpha}=\frac{\mu_{\alpha}}{1+\mu_{\alpha}\ln{(W/\Theta_{D})}},$
(8)

where $\alpha=s$ or $t$, and $W\sim\varepsilon_{F}$ is the Coulomb interaction
cut off . Fig. 2(a) shows $\lambda_{\alpha}$ and $\mu^{*}_{\alpha}$ versus $L$
with values $\lambda_{s}$ at large $L$ being consistent with experimentslambda
. In Fig. 2(b), we find that the triplet pairing wins over for $L\sim 5$-$10$
nm for larger $\varepsilon_{F}$phase , indicating that pairing between
parallel surfaces may enhance the triplet pairing and turn a TI into a surface
topological superconductor in thin film geometry. Finally, the resulting phase
diagram of superconducting orders in thin films with the Fermi energy being in
the lower surface Dirac cone is given in Fig. 3.

##  IV Effective magnetic field induced by curvature

The results of thin film geometry imply that pairing of electrons at different
locations on a curved surface may also enhance the triplet pairing. Hence we
consider superconductivity on a general surface. It is well known that the
Dirac equation on a curved space acquires the spin connection as a gauge
fieldNakahara . In particular, it is found that a spin-1/2 Fermion on a sphere
in the local frame will see an effective magnetic monopole $\pm 1/2$ for the
local spin direction being $\mp$ respectivelymonopole . We have generalized
this result to a general surface $S$ and proved that the effective local
magnetic field $\vec{B}_{eff}$ is proportional to the Gauss curvature $K$ as
general_monopole

$\vec{B}_{eff}=\pm\frac{1}{2}K\hat{n},$ (9)

Figure 4: The total magnetic flux due to the spin-connection gauge field that
passes through a surface $S$ is a topological invariant, which depends on the
genus $g$ of the surface as $\oint_{S}\vec{B}_{eff}\cdot d\vec{a}=\pm
2\pi(1-g)$, where $\pm$ are the local spin directions (down and up) of
electrons.

where $\hat{n}$ is the unit vector normal to the surface and $\pm$ are the
local spin directions of electrons. As a result, according to the Gauss-Bonnet
theorem, the total magnetic flux through a surface $S$,
$\oint_{S}\vec{B}_{eff}\cdot d\vec{a}$, is a topological invariant that
depends only the genus $g$ of $S$ as illustrated in Fig. 4. For a general
surface with genus $g$, considering possible formation of vortices, the
Poincaré-Hopf theorempoincare  further implies that the minimum number of
vortices on the surface is $2(1-g)$. The formation of vortices is determined
by $\vec{B}_{eff}$ and energetics with the configuration of the lowest free
energy being realized.

##  V Curvature induced superconducting vortex states

To consider effects of effective magnetic fields on surface superconductivity,
we shall consider the simplest surface with $g=0$. In particularly, we shall
consider the surface on a sphere or a semi-sphere with different curvature. As
we shall see, the results allow one to further estimate the formation of
vortices on general surfaces. In the presence of effective magnetic fields
induced by curvature, since electrons with different spin directions see
opposite monopole charges, we expect that only Cooper pairs of the triplet
pairing, $++$ and $--$, are affected by non-vanishing $B_{eff}$, while the
singlet pairing would not be affected by $B_{eff}$. The effective Hamiltonian
is given by

$\displaystyle H_{eff}=\int
d\Omega\Psi^{\dagger}_{\Omega}(\mathcal{H}_{surf}-\mu)\Psi_{\Omega}+$
$\displaystyle\frac{1}{2}\int\int d\Omega
d\Omega^{\prime}V(\Omega,\Omega^{\prime})(\Psi^{\dagger}_{\Omega}\Psi_{\Omega})(\Psi^{\dagger}_{\Omega^{\prime}}\Psi_{\Omega^{\prime}}).$
(10)

Here
$\Psi^{\dagger}_{\Omega}\equiv(C^{\dagger}_{\Omega,+},C^{\dagger}_{\Omega,-})$
is the creation operator for the local spinor at the solid angle $\Omega$.
$\mathcal{H}_{surf}$ is the projective Hamiltonian describing the free local
surface states

$\displaystyle\mathcal{H}_{surf}=\frac{\lambda_{so}}{R}\Bigg{(}\begin{array}[]{cccc}0&D_{-}\\\
D_{+}&0\end{array}\Bigg{)},$ (13)

where $R$ is the radius of sphere and $D_{\pm}=e^{\pm
i\phi}(\pm\partial_{\theta}+\frac{i}{\sin\theta}\partial_{\phi}\mp\frac{1}{2}\tan\frac{\theta}{2})$local
. $V(\Omega,\Omega^{\prime})$ is the effective interaction that includes the
screened Coulomb interaction adapted to the sphere,
$V^{c}(\Omega,\Omega^{\prime})=\frac{V_{C}}{2R\sin\frac{\Delta\Omega}{2}}\exp(-\alpha_{c}|\sin(\frac{\Delta\Omega}{2})|)$
and the attractive phonon-mediated interaction
$V^{ph}(\Omega,\Omega^{\prime})=-V_{ph}\exp(-\alpha_{ph}(\Delta\Omega)^{2})$
with $\Delta\Omega=\Omega-\Omega^{\prime}$. Here the interactions are modeled
by a Gaussian formSC_sphere  and the Yukawa potential form with
$\alpha_{ph}=\frac{R}{\xi_{ph}}$ and $\alpha_{c}=\frac{2R}{\xi_{C}}$
($\xi_{ph}$ and $\xi_{C}$ are the corresponding decay length and screening
length). Finally, as the attractive phonon interaction is cutoff in energy, we
assume that $\mathcal{H}_{surf}$ is cutoff in energy near the chemical
potential with the cutoff $\Lambda$.

The mean-field pairing amplitude is given by
$\Delta^{s,s^{\prime}}_{\Omega,\Omega^{\prime}}=V(\Omega,\Omega^{\prime})\langle
C_{\Omega,s}C_{\Omega^{\prime},s^{\prime}}\rangle$ so that the pairing
Hamiltonian is $H_{\Delta}=\frac{1}{2}\int\int d\Omega
d\Omega^{\prime}\sum\limits_{(s,s^{\prime})}\Delta^{s,s^{\prime}}_{\Omega,\Omega^{\prime}}C^{\dagger}_{\Omega,s}C^{\dagger}_{\Omega^{\prime},s^{\prime}}+h.c.$.
To solve the self-consistent gap equation on the sphere, we need to go to the
eigen-basis of ${H}_{surf}$ with the eigen-energies being
$E_{\pm}=\pm(\lambda_{so}/R)(j+1/2)$ with $j=0,1,2,\cdots$. The corresponding
annihilation operators are denoted by $C_{jm,E_{\pm}}$, which are related to
the local spinor $C_{\Omega\pm}$ by $C_{jm,E_{\pm}}=\int
d\Omega(Y^{m*}_{-1/2,j}C_{\Omega+}\pm Y^{m*}_{1/2,j}C_{\Omega-})$. Here
$Y^{m}_{q,j}$ are the monopole harmonics with $q$ being the monopole charge
and $(j,m)$ being the angular momentum quantum number. By keeping the $E_{-}$
operator and dropping the subindex $E_{\pm}$, we obtain

$\displaystyle\begin{array}[]{l}C_{jm}^{\dagger}=\frac{1}{{\sqrt{2}}}\int{d\Omega(Y_{-1/2,j}^{m}(\Omega)C_{\Omega,+}^{\dagger}-Y_{1/2,j}^{m}(\Omega)C_{\Omega,-}^{\dagger})},\\\
C_{\Omega,+}^{\dagger}=\frac{1}{{\sqrt{2}}}\sum\limits_{jm}{C_{jm}^{\dagger}Y_{-1/2,j}^{m*}(\Omega)},\\\
C_{\Omega,-}^{\dagger}=-\frac{1}{{\sqrt{2}}}\sum\limits_{jm}{C_{jm}^{\dagger}Y_{1/2,j}^{m*}(\Omega)}.\end{array}$
(17)

The interaction $V(\Omega,\Omega^{\prime})$ can be re-expressed in this basis.
First, we set $\gamma=\Omega-\Omega^{\prime}$ and apply the addition theorem
of spherical harmonics to write $V(\Omega,\Omega^{\prime})$ as

$\displaystyle V(\Omega,\Omega^{\prime})=\sum\limits_{l}{\frac{{4\pi
V_{l}}}{{2l+1}}\sum\limits_{\bar{m}}{{{(-1)}^{\bar{m}}}Y_{l}^{-\bar{m}}(\Omega)Y_{l}^{\bar{m}}(\Omega^{\prime})}},$
(18)

where

$\displaystyle{V_{l}}=\frac{{2l+1}}{2}\int_{-1}^{1}{V(\gamma){P_{l}}(\cos{\gamma})d\cos{\gamma}},$
(19)

$P_{l}(x)$ is the $l_{th}$ Legendre polynomial, and $Y_{l}^{\bar{m}}$ is the
spherical harmonics. By using Eq.(17), the interaction term in Eq.(V) can be
expressed as

$\displaystyle\int\int{d\Omega
d\Omega^{\prime}V(\Omega,\Omega^{\prime})(\Psi_{\Omega}^{\dagger}{{\Psi}_{\Omega}})(\Psi_{\Omega^{\prime}}^{\dagger}{{\Psi}_{\Omega^{\prime}}})}$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int\int{d\Omega
d\Omega^{\prime}V(\Omega,\Omega^{\prime})(C_{\Omega,+}^{\dagger}{C_{\Omega,+}}{\rm{+}}C_{\Omega,-}^{\dagger}{C_{\Omega,-}})}\times$
$\displaystyle(C_{\Omega^{\prime},+}^{\dagger}{C_{\Omega^{\prime},+}}{\rm{+}}C_{\Omega^{\prime},-}^{\dagger}{C_{\Omega^{\prime},-}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\begin{subarray}{c}j_{1}j_{2}j_{3}j_{4}\\\
m_{1}m_{2}m_{3}m_{4}\end{subarray}}V_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}C^{{\dagger}}_{j_{1}m_{1}}C^{{\dagger}}_{j_{2}m_{2}}C_{j_{3}m_{3}}C_{j_{4}m_{4}},$

where

$\displaystyle V_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}=$
$\displaystyle\int\int d\Omega d\Omega^{\prime}\sum_{l}\frac{\pi
V_{l}}{(2l+1)}\sum_{\bar{m}}(-1)^{\bar{m}}Y_{l}^{-\bar{m}}(\Omega)Y_{l}^{\bar{m}}(\Omega^{\prime})$
$\displaystyle\times\sum_{q,q^{\prime}=\pm
1/2}Y_{q,j_{1}}^{m_{1}*}(\Omega)Y_{q,j_{4}}^{m_{4}}(\Omega)Y_{q^{\prime},j_{2}}^{m_{2}*}(\Omega^{\prime})Y_{q^{\prime},j_{3}}^{m_{3}}(\Omega^{\prime}).$
(21)

The integrals involved between three harmonics are given bysup4

$\displaystyle\int{d\Omega}{Y_{l}^{\bar{m}}(\Omega)Y_{q,j}^{m*}(\Omega)Y_{q,j^{\prime}}^{m^{\prime}}(\Omega)}$
$\displaystyle=$
$\displaystyle{(-1)^{l+j+j^{\prime}-q-m}}{\left[{\frac{{(2l+1)(2j+1)(2j^{\prime}+1)}}{4\pi}}\right]^{1/2}}$
(26) $\displaystyle\times\left({\begin{array}[]{*{20}{c}}l&j&{j^{\prime}}\\\
{\bar{m}}&{-m}&{m^{\prime}}\end{array}}\right)\left({\begin{array}[]{*{20}{c}}l&j&{j^{\prime}}\\\
0&{-q}&q\end{array}}\right),$

where $\left({\begin{array}[]{*{20}{c}}l&j&{j^{\prime}}\\\
{\bar{m}}&{-m}&{m^{\prime}}\end{array}}\right)$ is the $3-j$ symbol.
Furthermore, to ensure that the anti-commuting properties of fermion operator
are preserved in the interaction, we define

$\displaystyle\Gamma_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}=\frac{1}{4}(V_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}-V_{j_{2}j_{1}j_{3}j_{4}}^{m_{2}m_{1}m_{3}m_{4}}$
$\displaystyle-
V_{j_{1}j_{2}j_{4}j_{3}}^{m_{1}m_{2}m_{4}m_{3}}+V_{j_{2}j_{1}j_{4}j_{3}}^{m_{2}m_{1}m_{4}m_{3}}).$
(27)

Eq.(V) is then turned into the following form

$\displaystyle
H^{surf}_{eff}=\sum_{jm}\left[-\lambda_{so}/R(j+1/2)-\mu\right]C_{jm}^{{\dagger}}C_{jm}$
$\displaystyle+\frac{1}{2}\sum_{\begin{subarray}{c}j_{1}j_{2}j_{3}j_{4}\\\
m_{1}m_{2}m_{3}m_{4}\end{subarray}}\Gamma_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}C^{{\dagger}}_{j_{1}m_{1}}C^{{\dagger}}_{j_{2}m_{2}}C_{j_{3}m_{3}}C_{j_{4}m_{4}},\quad$

while the corresponding mean-field equations are given by

$\displaystyle\Delta^{mm^{\prime}}_{jj^{\prime}}=\sum_{\begin{subarray}{c}j_{3}j_{4}\\\
m_{3}m_{4}\end{subarray}}\Gamma_{jj^{\prime}j_{3}j_{4}}^{mm^{\prime}m_{3}m_{4}}\langle
C_{j_{3}m_{3}}C_{j_{4}m_{4}}\rangle,$ (29)

where $\Gamma_{j_{1}j_{2}j_{3}j_{4}}^{m_{1}m_{2}m_{3}m_{4}}$ is the
interaction $V(\Omega,\Omega^{\prime})$ expressed in the $(j,m)$ basis.

To find the superconducting state, we first solve
$\Delta^{mm^{\prime}}_{jj^{\prime}}$. The solution is then converted to the
solid-angle basis, $\Delta^{s,s^{\prime}}_{\Omega,\Omega^{\prime}}$. The gap
function generally depends on solid angles $\Omega$ and $\Omega^{\prime}$ of
two electrons in a Cooper pair and can be decomposed into the superposition of
products of the amplitude at the center of mass point
$\bar{\Omega}=(\Omega+\Omega^{\prime})/2$ and the amplitude for the relative
motion that presents different pairing symmetries as
$\Delta^{s,s^{\prime}}_{\Omega,\Omega^{\prime}}=\Delta_{S}(\bar{\Omega})S(\Omega,\Omega^{\prime})+\Delta_{P}(\bar{\Omega})P(\Omega,\Omega^{\prime})+\cdots,$
where $S(\Omega,\Omega^{\prime}),P(\Omega,\Omega^{\prime}),\cdots$ denote
pairing symmetries of s-wave, p-wave, $\cdots$ with the corresponding
amplitudes $\Delta_{s}(\bar{\Omega}),\Delta_{p}(\bar{\Omega}),\cdots$ at the
center of mass point. The s-wave, $S(\Omega,\Omega^{\prime})$, is a constant.
The local pairing symmetry, $p_{x}\pm ip_{y}$, on a sphere is taken to be in
the form and their higher harmonics in solid anglesSC_sphere :
$(P+iP)(\Omega,\Omega^{\prime})=\alpha\beta^{\prime}-\beta\alpha^{\prime}$ and
$(P-iP)(\Omega,\Omega^{\prime})=\alpha\beta^{\prime*}-\beta\alpha^{\prime*}$,
where ($\alpha$, $\beta$) denotes the local spinorlocal ; local1 . Numerical
values of the pairing amplitude can be obtained by first taking the limit
$\Omega-\Omega^{\prime}\rightarrow 0$ to find amplitudes of relative motion
using the different form of pairing symmetries SC_sphere_2 . The coefficients
that go with different pairing symmetries are the amplitudes at the center of
mass point when $\Omega=\Omega^{\prime}$.

Figure 5: Self-consistent vortex solution on the surface of a spherical TI and
a semi-spherical TI. Here $\lambda_{so}=35$meV$\cdot$nm, $\xi_{C}=4$nm,
$\xi_{ph}=50$nm, $R=50$ nm, $V_{ph}=7.76$meV, $V_{C}=50$ meV$\cdot$nm,
$\mu=-9.5$meV, the cutoff $\Lambda=2.1$meV, and the electron density is
$2.7\times 10^{-3}$nm-2parameter2 . (a) The superconducting amplitude,
$\Delta^{++}$ (meV) (pairing symmetry $p_{x}$+$ip_{y}$), on a sphere for the
vortex solution with vortex charge $Q=1$ in the vortex phase. Here the color-
scale decreases from 0.001meV(yellow) to 0(blue). Contours and arrows mark
directions of supercurrents. (b) The superconducting amplitude, $\Delta^{++}$
(meV), for a vortex solution on a semi-sphere. (c) The quasi-particle energy
spectrum (in the unit of meV) inside the vortex core exhibits the Majorana
zero mode (indicated by the arrow, tolerance is 0.02meV). Here $m$ and $n$ are
z-component of angular momentum of the quasi-particle. (d) Relative strength
of the singlet pairing to the triplet pairing ($\Delta^{++}_{-}$) versus
$\Omega-\Omega^{\prime}$ in the Meissner phase. Here $V_{C}=70$ meV and
$T=1.24$meV.(e) and (f): Pairing components in $\Delta^{++}$ and $\Delta^{--}$
for vortex phase (e) and vortex* phase (f). Here $\theta$ is the azimuthal
angle relative to the north pole, red circles represent $\Delta^{++}_{+}$,
blue squares represent $\Delta^{--}_{-}$, and black dots represent
$\Delta^{++}_{-}$ (=$\Delta^{--}_{+}$).

In Fig. 5, we show numerical solutions of generic superconducting states found
on spheres and semi-spheres. It is found that vortices of two different
charges ($Q=1$ and $Q=2$) are spontaneously formed in different regimesvortex
. Fig. 5(a) shows $\Delta^{++}$ with the pairing symmetry $p_{x}+ip_{y}$ in
color-scale for the vortex solution with one vortex of charge $Q=1$ at the
north pole and the other vortex of charge $Q=1$ at the south pole. Phase with
such vortices is termed as the vortex phase. A similar solution with minor
component exhibiting the vortex solution of charge $Q=2$ is also found. Phase
with such vortices is termed as the vortex* phase. This is consistent with the
Poincaré-Hopf theorem which states that the total winding number (= $Q$) of a
vector field on a sphere should be two. In Fig. 5(b), we show that the
spontaneously-formed vortex solution survives on a semi-sphere, which
simulates a bump on the surface. This solution implies that vortices can also
form on bumps with appropriate curvature on surfaces of TIs. Fig. 5(c) shows
that each vortex in the vortex phase support a Majorana zero mode in the core.
For the vortex* phase, since the vortex solution only exists in the minor
component ($\Delta^{++}_{-}$, see Fig. 5(f)), the vortex core does not host
Majorana zero modeSato . Fig. 5(d) shows the relative strength of singlet
pairing to triplet pairing versus $\Omega-\Omega^{\prime}$ in the Meissner
phase in which there is no vortex. Here only $\Delta^{++}$ and $\Delta^{--}$
($=\Delta^{++}$) are non-vanishing triplet components in the Meissner phase.
Clearly, due to the momentum-locking of Dirac fermions at different solid
angles, the triplet pairing has higher chance of occurrence and dominates.
This results in the topological superconductivity. The triplet component for
the vortex phase can be generally expressed as
$\Delta^{++}=\Delta^{++}_{-}(p_{x}-ip_{y})+\Delta^{++}_{+}(p_{x}+ip_{y})$ and
$\Delta^{--}=\Delta^{--}_{-}(p_{x}-ip_{y})+\Delta^{--}_{+}(p_{x}+ip_{y})$. The
components for different pairing symmetries are shown in Fig. 5(d) and (e).
These components $\Delta^{ss}_{s^{\prime}}$ can be expressed in terms of
monopole harmonics.

As the local magnetic field is determined by $1/R^{2}$, the interplay between
the Meissner phase and vortex phases on a sphere is generally controlled by
the radius and the strength of Coulomb interaction. Fig. 6 shows typical phase
diagrams we find for superconducting states on a sphere. It is seen that the
vortex phase generally exists at low temperature on surfaces with radius of
curvature being around the nanoscale (smaller than 50 nm).

Figure 6: (a) The phase diagram of superconducting states on a sphere:
$1/R^{2}$ versus temperature $T$. Here $1/R^{2}$ act similar as the local
magnetic field. (b) The phase diagram of superconducting states on a sphere:
$V_{C}$ versus $T$. Here $\xi_{C}=4$nm, $\xi_{ph}=50$nm, $V_{ph}=7.76$meV,
$\mu=-9.5$meV, and the cutoff $\Lambda=2.1$meV. Note that vortex phases
diminish as $R\rightarrow\infty$, which can be accessed in (b) with
$V_{C}\rightarrow 0$.

##  VI Discussion and conclusion

Although the emergence of vortices with Majorana zero mode is mainly
illustrated on a perfect sphere, our results are not restricted to a perfect
sphere. This is illustrated in Fig. 5(b) by a semi-sphere, which simulates a
bump on a plane. The solution also contains a Majorana zero mode inside the
vortex core. It implies that vortices can also spontaneously form on bumps
with appropriate curvature on surfaces of TIs. On general surfaces, the vortex
state is competing with the Meissner phase in energy. At the critical
curvature when the vortex state with a single vortex wins over the Meissner
phase, the free energy for the vortex state is equal to that of the Meissner
phase. Similar to lower critical field$H_{c1}$ for conventional
superconductors, this determines the critical curvature $K_{c}$ for the
formation of a single vortex as

$K_{c}=\pm\frac{8\pi\epsilon}{\Phi_{0}},$ (30)

where $\epsilon$ is the free energy per unit length of the vortex and
$\Phi_{0}$ is the flux quantum. Hence there is a critic radius of curvature
for forming a vortex. Below the critical radius of curvature, the Meissner
phase wins. The numerical values for $K_{c}$ can be estimated by using the
phase digram shown in Fig. 5(a), where at different temperatures, the critical
curvature for a vortex to form is determined by the boundary between the
Meissner phase and the vortex phase. Our results can be thus applied to an
ellipsoid or bumps on surface of topological insulator in which vortices with
Majorana modes could emerge at places where the local curvature ($~{}1/R^{2}$)
exceeds the threshold as marked in Fig. 5(a).

The connection of the thin-film geometry to the spherical geometry is in the
fact that they all belong to the $g=0$ surfaces. In this case, the Poincaré-
Hopf theorem implies that the minimum number of vortices is two, but
generally, number of vortices in a vortex state with charge 1 can be 2, 4,6,
$\dots$. The extra vortices that deviate from 2 contain vortex pairs of $\pm
1$ charges so that the Poincaré-Hopf theorem is satisfied. For the thin-film
(slab-like) geometry, it also has the genus number, $g=0$. Depending on
curvatures of the corners, if they exceed the threshold curvature, vortices
can form. The formation of vortices will appear at corners. However, minimum
number of vortices that can appear is still two. In this case, one still has
to control the curvature of the corner to enable the formation of vortices.

In conclusion, we have shown that pairing of electrons on different locations
of a TI can generally induce topological superconductivity through the control
of surface curvature and size of the TI. These are illustrated both in the
thin film geometry and in the spherical geometry. The generic phase diagram of
topological superconductivity on a sphere is constructed. For general
surfaces, our results imply that vortices can spontaneously formed on bumps
with appropriate curvature on surfaces of TIs. These vortices generally host a
Majorana zero mode inside each core, which is shown to be robust against the
presence of the disorder potential. We expect that the experimental detection
can be performed on surfaces of Sb2Te3Sb2Te3  by using STM and one looks for
surface roughness of size around 50 nm or less at temperatures below 9K. The
surface of a TI can be thus used as a platform to host Majorana zero modes
without invoking real magnetic fields.

###### Acknowledgements.

This work was supported by the Ministry of Science and Technology (MoST),
Taiwan. We also acknowledge support from the Center for Quantum Technology
within the framework of the Higher Education Sprout Project by the Ministry of
Education (MOE) in Taiwan.

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