# A note on the local Weyl formula on compact Lie groups

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Julio Delgado  Julio
Delgado: Departamento de Matemáticas Universidad del Valle Cali-Colombia
E-mail address<EMAIL_ADDRESS>and  Michael Ruzhansky
Michael Ruzhansky: Department of Mathematics: Analysis, Logic and Discrete
Mathematics Ghent University, Belgium and School of Mathematical Sciences
Queen Mary University of London United Kingdom E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this note we investigate a local Weyl type formula for an arbitrary pseudo-
differential operator on a compact Lie group in terms of its matrix-valued
symbol. We link our formula to the principal symbol of the operator, to the
representation theory of the group, and to the problem of the quantum unique
ergodicity.

###### Key words and phrases:

Weyl formula, eigenfunctions, quantum limits, pseudo-differential operators,
spectrum

###### 2020 Mathematics Subject Classification:

35S30, 42B20; Secondary 42B37, 42B35

The authors were supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis
and Partial Differential Equations, by the Methusalem programme of the Ghent
University Special Research Fund (BOF) (Grant number 01M01021) and by the
DyCon Project 2015 H2020-694126.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivations: the quantum ergodicity problem
    2. 1.2 Contributions of the note
  2. 2 Preliminaries
    1. 2.1 The Fourier analysis of a compact Lie group
    2. 2.2 The quantisation formula
    3. 2.3 Global Hörmander classes on compact Lie groups
  3. 3 Local Weyl formula on compact Lie groups
    1. 3.1 Notations
    2. 3.2 Local Weyl formula for zero order operators
    3. 3.3 Local Weyl formula for operators of positive order
    4. 3.4 Expected value of multiplication operators in the energy states $\phi_{j}$

##  1\. Introduction

As it was pointed out in [17] by Robert Strichartz, what is “spectral
asymptotics” if not the study of the asymptotic behavior as
$\lambda\rightarrow\infty$ of the eigenvalue counting function

$N(\lambda)=|\\{j:|\lambda_{j}|\leq\lambda\\}|$

of some operator. So, without the usual assumptions of ellipticity, given a
pseudo-differential operator $A$ of order zero on a compact Lie group $G,$ in
this note we address the problem of computing its local Weyl formula in terms
of its matrix-valued symbol (which in this setting is equivalent to computing
it in terms of the representation theory of these groups). Similar formulae
are obtained for operators of positive order. Formulae of this type involving
the representation theory of the group are only known for the case of the
torus $G=\mathbb{T}^{n}$ (where the symbols are scalar-valued), see e.g.
Zelditch [20] and here we deal with its extension to the non-commutative
setting.

###  1.1. Motivations: the quantum ergodicity problem

On a compact Riemannian manifold $(M,g)$, the local Weyl law of Zelditch
establishes that for a pseudo-differential operator $A\in\Psi^{0}(M),$ of
order $0,$ in terms of the normalised spectral data
$\\{\phi_{j},\lambda_{j}^{2}\\}$ of the positive Laplacian $-\Delta_{g}$ on
$(M,g)$ (that are, its system of eigenfunctions $\phi_{j}$ with corresponding
eigenvalues $\lambda_{j}^{2}$) the following asymptotic expansion is valid
(see Zelditch [20])

$\sum_{\lambda_{j}\leq\lambda}(A\phi_{j},\phi_{j})=\left(\smallint_{T^{*}\mathbb{S}(M)}a_{0}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1}),$
(1.1)

for $\lambda$ big enough where $a_{0}\in\Gamma(T^{*}M)$ is the (section of the
cotangent bundle $T^{*}M$ that provides the) principal symbol of the operator
$A,$ and $d\mu_{L}(x,\xi)$ is the Liouville measure on the spherical cotangent
bundle $T^{*}\mathbb{S}(M)=\\{(x,\xi)\in T^{*}M:\|\xi\|_{g}=1\\}.$

In this note, when $M=G$ is a compact Lie group, we address the problem of
computing the left-hand side of (1.1) in terms of the representation theory of
the group. To do this (in our main Theorem 1.1), and by following the modern
techniques of the theory of pseudo-differential operators, we will use the
formalism of a matrix-valued global symbol as developed in [11]. Indeed,
according to such a matrix-valued quantisation, to a pseudo-differential
operator, one can associate a unique and globally defined symbol on the non-
commutative phase space $G\times\widehat{G},$ with $\widehat{G}$ being the
unitary dual of the group. Thus, the main problem in computing the left-hand
side of (1.1) is to translate the spectral information $(A\phi_{j},\phi_{j})$
in terms of the algebraic one that provides the unitary dual $\widehat{G},$
that encodes the representation theory of the group.

Before presenting the contribution of this note in the form of Theorem 1.1, we
present our main motivation, which arose from the understanding of the local
Weyl formula and its relation with the quantum ergodicity problem on compact
manifolds.

Indeed, let us explain the connection of (1.1) with the quantum ergodicity
problem. We have denoted the eigenvalues of the Laplacian by
$\lambda_{j}^{2}.$ They are usually understood as different levels of energies
$E_{j}=\lambda_{j}^{2}.$ Their square roots are the frequencies
$\lambda_{j}\geq 0,$ and the most basic quantities testing the asymptotics of
eigenfunctions are the matrix elements $(A\phi_{j},\phi_{j}).$ These matrix
elements measure the expected value of the observable $A$ (when the order of
this operator is zero) in the energy state $\phi_{j}$. Much of the work in
quantum ergodicity since the pioneering work of Schnirelman [15] is devoted to
the study of the limits of $(A\phi_{j},\phi_{j}).$ Although difficult to
determine, these limits are still the most accessible aspects of
eigenfunctions. Indeed, a real number $Q$ for which there exists a subsequence
of $(A\phi_{j},\phi_{j})$ with $Q$ as its limit point is called a quantum
limit of the system $(\phi_{j})$ with respect to the observable $A$. A
priority problem in quantum mechanics (and then, in spectral geometry) is to
determine whether or not, the sequence $(A\phi_{j},\phi_{j})$ has a unique
quantum limit. Even, it is also important to answer whether there exists a
sparse exceptional sequence that has a singular concentration. This is known
as the quantum unique ergodicity problem.

The local Weyl formula is connected with the quantum unique ergodicity
problem. Indeed, if $N(\lambda)=\\{j:\lambda_{j}\leq\lambda\\}$ is the
eigenvalue counting function of $\sqrt{-\Delta_{g}},$ then (1.1) implies that

$\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{\lambda_{j}\leq\lambda}(A\phi_{j},\phi_{j})=\smallint_{T^{*}\mathbb{S}(M)}a_{0}(x,\xi^{\prime})d\mu_{L}(x,\xi).$
(1.2)

So, if the observable $A$ is a self-adjoint operator of order zero, there is a
sub-sequence of matrix elements $(A\phi_{j},\phi_{j})$ which converges to
$\textnormal{QL}:=\smallint_{T^{*}\mathbb{S}(M)}a_{0}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$
This proves that the quantity in the right hand side of (1.2) is a quantum
limit of the sequence $(A\phi_{j},\phi_{j}).$ In particular, a sequence of
eigenfunctions $\phi_{j}$ is ergodic if the limit of $(A\phi_{j},\phi_{j})$ is
the quantity
$\smallint_{T^{*}\mathbb{S}(M)}a_{0}(x,\xi^{\prime})d\mu_{L}(x,\xi)$.
Applications of quantum ergodicity to problems in nodal geometry can be found
in Zelditch [19]. In this context, it is natural for analysts to ask how one
can use such matrix elements to study classical problems on eigenfunctions
such as growth and distributions of zeros and critical points. Numerical
evidence allows one to deduce that ergodicity causes maximal oscillation in
the real and complex domain of the eigenfunctions and gives rise to special
distributions of zeros and (conjecturally) to critical points.

Although we do not discuss it here, a good portion of the literature on
quantum chaos is dedicated to numerics and physicists’ heuristics. Some
classics are Berry [2] and Heller [8, 9]. The article of A. Barnett [1] is a
relatively recent discussion of the numerical results in quantum ergodicity.
There is a wealth of phenomenology standing behind the rigorous results and
heuristic proofs in this field. Much of it still lies far beyond the scope of
the current mathematical techniques. For an excellent expository work on the
subject we refer the reader to Zelditch [19].

###  1.2. Contributions of the note

We refer the reader to Subsection 3.1 for the notations used here. In
particular, in the case of a compact Lie group $G,$ we denote by
$\mathcal{L}_{G}$ the positive Laplace-Beltrami operator. Its discrete
spectrum $\\{\mu_{\xi}:[\xi]\in\widehat{G}\\}$ can be enumerated with the
unitary dual $\widehat{G}.$ Then, we use the notations
$|\xi|:=\sqrt{{\mu}}_{\xi}$ and
$N(\lambda)=\\{[\xi]\in\widehat{G}:|\xi|\leq\lambda\\}$ for the eigenvalues of
$\sqrt{\mathcal{L}}_{G}$ as well as for its Weyl eigenvalue counting formula,
respectively. The following theorem is our main result.

######  Theorem 1.1.

Let $G$ be a compact Lie group of dimension $n,$ and let $A\in\Psi^{0}(G)$ be
a pseudo-differential operator of order zero. In terms of the following data:

  * •

first, with $\sigma_{\textnormal{loc}}$ being the (Hörmander) principal symbol
of $A$ (which is a section of the cotangent bundle $T^{*}G$),

  * •

denoting by $\sigma_{\textnormal{glob}}$ the matrix-valued symbol of $A$
defined on $G\times\widehat{G},$

  * •

and with $\mu_{L}$ denoting the Liouville measure on the spherical vector
bundle $T^{*}\mathbb{S}(G),$

for any $\lambda>0,$ the partial trace of $A$ admits the asymptotic expansion

$\sum_{[\xi]\in\widehat{G}:|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1}).$
(1.3)

Moreover, the convergence at infinity of its average with respect to the
eigenvalue counting function
$N(\lambda):=\\{[\xi]\in\widehat{G}:|\xi|\leq\lambda\\}$ is given by the
quantum limit

$\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{[\xi]\in\widehat{G}:|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=C_{n}\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$
(1.4)

Now, we present the following application of Theorem 1.1 to quantum limits.

######  Corollary 1.2.

Let $G$ be a compact Lie group of dimension $n,$ and let
$A\in\Psi^{0}(G,\textnormal{loc})$ be a pseudo-differential operator of order
zero. Assuming that $A$ is self-adjoint (an observable) on $L^{2}(G),$ there
exists a sequence of unitary representations $\xi_{j}$ of the group such that
the sequence

$\\{d_{\xi_{j}}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi_{j}))dx\\}_{j}$

converges to the quantum limit
$\textnormal{QL}:=\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$

The following Theorem 1.3 is an extension of the local Weyl formula for
pseudo-differential operators of positive order. There we relax the
ellipticity condition on the operator that one usually requires in the
standard Weyl type formulae (see e.g. the discussion in Strichartz [17]) by
imposing the positivity of its matrix-valued symbol.

######  Theorem 1.3.

Let $A\in\Psi^{m}(G)$ be a pseudo-differential operator of order $m>0$ and
assume that on every representation space the trace of its matrix-valued
symbol is positive, i.e.

$\forall(x,[\xi])\in
G\times\widehat{G},\,\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,[\xi]))\geq
0.$ (1.5)

In terms of the following data:

  * •

first, with $\sigma_{\textnormal{loc}}$ being the (Hörmander) principal symbol
of $A$ (which is a section of the cotangent bundle $T^{*}G$),

  * •

and with $\mu_{L}$ denoting the Liouville measure on the spherical vector
bundle $T^{*}\mathbb{S}(G),$

for any $\lambda>0,$ the partial trace of $A$ satisfies the growth estimate

$\sum_{[\xi]\in\widehat{G}:0<|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx\leq
C_{n,A}\lambda^{n+m}+O(\lambda^{n+m-1}),$ (1.6)

where

$C_{n,A}:=(2\pi)^{-n}\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$
(1.7)

######  Remark 1.4.

In Theorem 1.3 the matrix-valued symbol of the operator $A\in\Psi^{m}(G)$
satisfies the trace condition (1.5) if e.g. on every representation space the
matrix-valued symbol is a positive definite matrix i.e. if
$\forall(x,[\xi])\in G\times\widehat{G},$
$\sigma_{\textnormal{glob}}(x,[\xi])\geq 0.$

######  Remark 1.5.

In [4], [5], [6], and [7], the second and the third author have studied the
$L^{p}$-nuclearity and the membership in Schatten-von-Neumann classes of
Fourier multipliers and of pseudo-differential operators on compact Lie
groups. It follows from the analysis in these references that any pseudo-
differential operator $A\in\Psi^{m}(G)$ belongs to the Schatten-von-Neumann
class $S_{r}(L^{2}(G))$ on $L^{2}(G),$ with $r>0,$ provided that $m<-n/r$. In
particular, $S_{1}(L^{2}(G))$ is the ideal of the trace class operators on
$L^{2}(G),$ and the order condition $m<-n$ implies that $A\in
S_{1}(L^{2}(G)).$

In this setting, for any $A\in\Psi^{m}(G),$ $m<-n,$ the trace formula

$\textnormal{Tr}(A)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=\lim_{\lambda\rightarrow\infty}\sum_{[\xi]\in\widehat{G}:|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx$
(1.8)

holds and it agrees with the sum of the eigenvalues $\lambda_{n}(A)$ of the
trace class operator $A.$ Since the order condition $m<-n$ is sharp, the trace
formula in (1.8) is not valid if
$A\in\Psi^{0}(G)\setminus\Psi^{-\varepsilon}(G),$ where $0<\varepsilon\leq n.$
That is the case if e.g. $A\in\Psi^{0}$ is an elliptic operator. However, the
local Weyl formula in Theorem 1.1 gives us information about the growth of the
partial sums

$S_{n}(A):=\sum_{[\xi]\in\widehat{G}:|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx,$

as well as the growth estimate (1.6) in Theorem 1.3.

On the other hand, from the proof of (1.4) (see Section 3) and from the Weyl
eigenvalue counting formula for the operator $\sqrt{\mathcal{L}}_{G}$ one has
that (c.f. [17]) $N(\lambda)\sim\textnormal{Vol}(G)\lambda^{n},$ so that the
constant $C_{n}$ in (1.4) is proportional to $(2\pi)^{-n}/\textnormal{Vol}(G)$
where $n$ is the dimension of the group $G.$

##  2\. Preliminaries

###  2.1. The Fourier analysis of a compact Lie group

Let $dx$ be the Haar measure on a compact Lie group $G.$ The Hilbert space
$L^{2}(G)$ will be endowed with the inner product

$(f,g)=\smallint\limits_{G}f(x)\overline{g(x)}dx.$

According to the Peter-Weyl theorem the spectral decomposition of $L^{2}(G)$
can be done in terms of the entries of unitary representations on a compact
Lie group $G$. To present such a theorem we will give some preliminaries.

######  Definition 2.1 (Unitary representation of a compact Lie group).

A continuous and unitary representation of $G$ on $\mathbb{C}^{\ell}$ is any
continuous mapping $\xi\in\textnormal{Hom}(G,\textnormal{U}(\ell)),$ where
$\textnormal{U}(\ell)$ is the Lie group of unitary matrices of order
$\ell\times\ell.$ The integer number $\ell=\dim_{\xi}$ is called the dimension
of the representation $\xi$ since it is the dimension of the representation
space $\mathbb{C}^{\ell}.$

######  Remark 2.2 (Irreducible representations).

A subspace $W\subset\mathbb{C}^{d_{\xi}}$ is called $\xi$-invariant if for any
$x\in G,$ $\xi(x)(W)\subset W,$ where $\xi(x)(W):=\\{\xi(x)v:v\in W\\}.$ The
representation $\xi$ is unitary if its only invariant subspaces are
$W=\emptyset$ and $W=\mathbb{C}^{d_{\xi}},$ the trivial ones. On the other
hand, any unitary representation $\xi$ is a direct sum of unitary irreducible
representations. We denote it by $\xi=\xi_{1}\otimes\cdots\otimes\xi_{j},$
with $\xi_{i}$ being irreducible representations on factors
$\mathbb{C}^{d_{\xi_{i}}}$ that decompose the representation space

$\mathbb{C}^{d_{\xi}}=\mathbb{C}^{d_{\xi_{1}}}\otimes\cdots\otimes\mathbb{C}^{d_{\xi_{j}}}.$

######  Definition 2.3 (Equivalent representations).

Two unitary representations

$\xi\in\textnormal{Hom}(G,\textnormal{U}(d_{\xi}))\textnormal{ and
}\eta\in\textnormal{Hom}(G,\textnormal{U}(d_{\eta}))$

are equivalent if there exists a linear mapping
$S:\mathbb{C}^{d_{\xi}}\rightarrow\mathbb{C}^{d_{\eta}}$ such that for any
$x\in G,$ $S\xi(x)=\eta(x)S.$ The mapping $S$ is called an intertwining
operator between $\xi$ and $\eta.$ The set of all the intertwining operators
between $\xi$ and $\eta$ is denoted by $\textnormal{Hom}(\xi,\eta).$

######  Remark 2.4 (Schur Lemma).

In view of the 1905’s Schur lemma, if
$\xi\in\textnormal{Hom}(G,\textnormal{U}(d_{\xi}))$ is irreducible, then
$\textnormal{Hom}(\xi,\xi)=\mathbb{C}I_{d_{\xi}}$ is formed by scalar
multiples of the identity matrix $I_{d_{\xi}}$ of order $d_{\xi}.$

######  Definition 2.5 (The unitary dual).

The relation $\sim$ on the set of unitary representations
$\textnormal{Rep}(G)$ defined by: $\xi\sim\eta$ if and only if $\xi$ and
$\eta$ are equivalent representations, is an equivalence relation. The
quotient

$\widehat{G}:={\textnormal{Rep}(G)}/{\sim}$

is called the unitary dual of $G.$

The unitary dual encodes all the Fourier analysis on the group. The Fourier
transform is defined as follows.

######  Definition 2.6 (Group Fourier transform).

If $\xi\in\textnormal{Rep}(G),$ the Fourier transform $\mathscr{F}_{G}$
associates to any $f\in C^{\infty}(G)$ a matrix-valued function
$\mathscr{F}_{G}f$ defined on $\textnormal{Rep}(G)$ as follows

$(\mathscr{F}_{G}f)(\xi)\equiv\widehat{f}(\xi)=\int\limits_{G}f(x)\xi(x)^{*}dx,\,\,\xi\in\textnormal{Rep}(G).$

######  Remark 2.7 (The Fourier inversion formula on a compact Lie group).

The discrete Schwartz space
$\mathscr{S}(\widehat{G}):=\mathscr{F}_{G}(C^{\infty}(G))$ is the image of the
Fourier transform on the class of smooth functions. This operator admits a
unitary extension from $L^{2}(G)$ into $\ell^{2}(\widehat{G}),$ with

$\ell^{2}(\widehat{G})=\left\\{\phi:\forall[\xi]\in\widehat{G},\,\phi(\xi)\in\mathbb{C}^{d_{\xi}\times
d_{\xi}}\textnormal{ and
}\|\phi\|_{\ell^{2}(\widehat{G})}:=\left(\sum_{[\xi]\in\widehat{G}}d_{\xi}\|\phi(\xi)\|_{\textnormal{HS}}^{2}<\infty\right)^{\frac{1}{2}}\right\\}.$
(2.1)

The norm $\|\phi(\xi)\|_{\textnormal{HS}}$ is the standard Hilbert-Schmidt
norm of matrices. The Fourier inversion formula takes the form

$f(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\xi(x)\widehat{f}(\xi)],\,f\in
L^{1}(G),$ (2.2)

where the summation is understood in the sense that from any equivalence class
$[\xi]$ we choose randomly a unitary representation.

###  2.2. The quantisation formula

Let $A:C^{\infty}(G)\rightarrow C^{\infty}(G)$ be a continuous linear operator
with respect to the standard Fréchet structure on $C^{\infty}(G).$ There is a
way of associating to the operator $A$ a matrix-valued function $\sigma_{A}$
defined on the non-commutative phase space $G\times\widehat{G}$ to rewrite the
operator $A$ in terms of the Fourier inversion formula and in terms of the
Fourier transform. Such a expression is called the quantisation formula. To
introduce it we require the following definition.

######  Definition 2.8 (Right convolution kernel of an operator).

The Schwartz kernel theorem associates to $A$ a kernel
$K_{A}\in(C^{\infty}(G),\mathscr{D}^{\prime}(G))$ such that

$Af(x)=\int\limits_{G}K_{A}(x,y)f(y)dy,\,\,f\in C^{\infty}(G).$

The distribution defined via $R_{A}(x,xy^{-1}):=K_{A}(x,y)$ that provides the
convolution identity

$Af(x)=\int\limits_{G}R_{A}(x,xy^{-1})f(y)dy,\,\,f\in C^{\infty}(G),$

is called the right-convolution kernel of $A.$

######  Remark 2.9 (The quantisation formula).

Now, we will associate a global symbol
$\sigma_{A}:G\times\widehat{G}\rightarrow\cup_{\ell\in\mathbb{N}}\mathbb{C}^{\ell\times\ell}$
to $A.$ Indeed, in view of the identity $Af(x)=(f\ast R_{A}(x,\cdot))(x),$ and
after taking the Fourier transform with respect to $x\in G,$ we get

$\widehat{Af}(\xi)=\widehat{R}_{A}(x,\xi)\widehat{f}(\xi).$

Then, the Fourier inversion formula gives the following representation of the
operator $A$ in terms of the Fourier transform,

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\xi(x)\widehat{R}_{A}(x,\xi)\widehat{f}(\xi)],\,f\in
C^{\infty}(G).$ (2.3)

In view of the identity (2.3), from any equivalence class
$[\xi]\in\widehat{G},$ we can choose one and only one irreducible unitary
representation $\xi_{0}\in[\xi],$ such that the matrix-valued function

$\sigma_{A}(x,[\xi])\equiv\sigma_{A}(x,\xi_{0}):=\widehat{R}_{A}(x,\xi_{0}),\,(x,[\xi])\in
G\times\widehat{G},$ (2.4)

such that

$Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\xi_{0}(x)\sigma_{A}(x,[\xi])\widehat{f}(\xi_{0})],\,f\in
C^{\infty}(G).$ (2.5)

The representation in (2.5) is independent of the choice of the representation
$\xi_{0}$ from any equivalent class $[\xi]\in\widehat{G}.$ This is a
consequence of the Fourier inversion formula.

In the following quantisation theorem we observe that the distribution
$\sigma_{A}$ in (2.5) is unique and can be written in terms of the operator
$A,$ see Theorems 10.4.4 and 10.4.6 of [11, Pages 552-553].

######  Theorem 2.10.

Let $A:C^{\infty}(G)\rightarrow C^{\infty}(G)$ be a continuous linear
operator. The following statements are equivalent.

  * •

The distribution
$\sigma_{A}(x,[\xi]):G\times\widehat{G}\rightarrow\cup_{\ell\in\mathbb{N}}\mathbb{C}^{\ell\times\ell}$
satisfies the quantisation formula

$\forall f\in C^{\infty}(G),\,\forall x\in
G,\,\,Af(x)=\sum_{[\xi]\in\widehat{G}}d_{\xi}\textnormal{Tr}[\xi(x)\sigma_{A}(x,[\xi])\widehat{f}(\xi)].$
(2.6)

  * •

$\forall(x,[\xi])\in
G\times\widehat{G},\,\sigma_{A}(x,\xi)=\widehat{R}_{A}(x,\xi).$

  * •

$\forall(x,[\xi]),\,\sigma_{A}(x,\xi)=\xi(x)^{*}A\xi(x),$ where
$A\xi(x):=(A\xi_{ij}(x))_{i,j=1}^{d_{\xi}}.$

######  Remark 2.11.

In view of the quantisations formulae (2.5) and (2.6), a symbol $\sigma_{A}$
can be considered as a mapping defined on $G\times\widehat{G}$ or as a mapping
defined on

$G\times\textnormal{Rep}(G)$

by identifying all the values
$\sigma_{A}(x,\xi)=\sigma_{A}(x,\xi^{\prime})=\sigma(x,[\xi])$ when
$\xi^{\prime},\xi\in[\xi].$

######  Example 2.12 (The symbol of a Borel function of the Laplacian).

Let $\mathbb{X}=\\{X_{1},\cdots,X_{n}\\}$ be an orthonormal basis of the Lie
algebra $\mathfrak{g}.$ The positive Laplacian on $G$ is the second order
differential operator

$\mathcal{L}_{G}=-\sum_{j=1}^{n}X_{j}^{2}.$ (2.7)

The operator $\mathcal{L}_{G}$ is independent of the choice of the orthonormal
basis $\mathbb{X}$ of $\mathfrak{g}.$ The $L^{2}$-spectrum of
$\mathcal{L}_{G}$ is a discrete set that can be enumerated in terms of the
unitary dual $\widehat{G}$

$\textnormal{Spect}(\mathcal{L}_{G})=\\{\mu_{[\xi]}:[\xi]\in\widehat{G}\\}.$
(2.8)

For a Borel function $f:\mathbb{R}^{+}_{0}\rightarrow\mathbb{C},$ the right-
convolution kernel $R_{f(\mathcal{L}_{G})}$ of the operator
$f(\mathcal{L}_{G})$ (defined by the spectral calculus) is determined by the
identity

$f(\mathcal{L}_{G})\phi(x)=\phi\ast R_{f(\mathcal{L}_{G})}(x),\,x\in G.$ (2.9)

This kernel satisfies the identity

$\widehat{R}_{f(\mathcal{L}_{G})}([\xi])=f(\mu_{[\xi]})I_{d_{\xi}}.$ (2.10)

Then the matrix-valued symbol of $f(\mathcal{L}_{G})$ can be determined e.g.
using Theorem 2.10 as follows

$\sigma_{f(\mathcal{L}_{G})}(x,\xi)=\widehat{R}_{f(\mathcal{L}_{G})}([\xi]).$
(2.11)

Since the operator $f(\mathcal{L}_{G})$ is central the symbol
$\sigma_{f(\mathcal{L}_{G})}(\xi)=\sigma_{f(\mathcal{L}_{G})}(x,\xi)$ does not
depend of the spatial variable $x\in G.$ Of particular interest for us will be
the Japanese bracket function

$\langle t\rangle:=(1+t)^{\frac{1}{2}},\,t\geq-1.$ (2.12)

In particular the symbol of the operator $\langle\mathcal{L}_{G}\rangle$ is
given by

$\sigma_{\langle\mathcal{L}_{G}\rangle}([\xi]):=\langle\xi\rangle
I_{d_{\xi}},\,\,\,\langle\xi\rangle:=\langle\mu_{\xi}\rangle.$ (2.13)

###  2.3. Global Hörmander classes on compact Lie groups

In this section we denote for any linear mapping $T$ on $\mathbb{C}^{\ell}$ by
$\|T\|_{\textnormal{op}}$ the standard operator norm

$\|T\|_{\textnormal{op}}=\|T\|_{\textnormal{End}(\mathbb{C}^{\ell})}:=\sup_{v\neq
0}\|Tv\|_{e}/\|v\|_{e},$

where $\|\cdot\|_{e}$ is the Euclidean norm.

For introducing the Hörmander classes on compact Lie groups we have to measure
the growth of derivatives of symbols in the group variable, for this we use
vector fields $X\in T(G).$ To derivate symbols with respect to the discrete
variable $[\xi]\in\widehat{G}$ we use difference operators. Before introducing
the Hörmander classes on compact Lie groups we have to define these
differential/difference operators.

######  Definition 2.13 (Left-invariant canonical differential operators).

If $\\{X_{1},\cdots,X_{j}\\}$ is an arbitrary family of left-invariant vector
fields, we will denote by

$X_{x}^{\alpha}:=X_{1,x}^{\alpha_{1}}\cdots X_{n,x}^{\alpha_{n}}$

an arbitrary canonical differential operator of order $m=|\alpha|.$

Also, we have to take derivatives with respect to the “discrete” frequency
variable $\xi\in\textnormal{Rep}(G).$ To do this, we will use the notion of
difference operators introduced in [13]. Indeed, the frequency variable in the
symbol $\sigma_{A}(x,[\xi])$ of a continuous and linear operator $A$ on
$C^{\infty}(G)$ is discrete. This is since $\widehat{G}$ is a discrete space.

######  Definition 2.14 (Canonical difference operators on the dual
$\widehat{G}$).

If $\xi_{1},\xi_{2},\cdots,\xi_{k},$ are fixed irreducible and unitary
representation of $G$, which not necessarily belong to the same equivalence
class, then each coefficient of the matrix

$\xi_{\ell}(g)-I_{d_{\xi_{\ell}}}=[\xi_{\ell}(g)_{ij}-\delta_{ij}]_{i,j=1}^{d_{\xi_{\ell}}},\,\quad
g\in G,\,\,1\leq\ell\leq k,$ (2.14)

that is each function $q^{\ell}_{ij}(g):=\xi_{\ell}(g)_{ij}-\delta_{ij}$,
$g\in G,$ defines a difference operator

$\mathbb{D}_{\xi_{\ell},i,j}:=\mathscr{F}_{G}(\xi_{\ell}(g)_{ij}-\delta_{ij})\mathscr{F}^{-1}_{G}.$
(2.15)

We can fix $k\geq\mathrm{dim}(G)$ of these representations in such a way that
the corresponding family of difference operators is admissible, that is,

$\textnormal{rank}\\{\nabla q^{\ell}_{i,j}(e):1\leqslant\ell\leqslant
k\\}=\textnormal{dim}(G).$

To define higher order difference operators of this kind, let us fix a unitary
irreducible representation $\xi_{\ell}$. Since the representation is fixed we
omit the index $\ell$ of the representations $\xi_{\ell}$ in the notation that
will follow. Then, for any given multi-index
$\alpha\in\mathbb{N}_{0}^{d_{\xi_{\ell}}^{2}}$, with
$|\alpha|=\sum_{i,j=1}^{d_{\xi_{\ell}}}\alpha_{i,j}$, we write

$\mathbb{D}^{\alpha}:=\mathbb{D}_{1,1}^{\alpha_{11}}\cdots\mathbb{D}^{\alpha_{d_{\xi_{\ell}},d_{\xi_{\ell}}}}_{d_{\xi_{\ell}}d_{\xi_{\ell}}}$

for a difference operator of order $m=|\alpha|$.

Now, we are ready for introducing the global Hörmander classes on compact Lie
groups.

######  Definition 2.15 (Global $(\rho,\delta)$-Hörmander classes in the whole
range $0\leq\delta,\rho\leq 1$).

We say that $\sigma\in{S}^{m}_{\rho,\delta}(G\times\widehat{G})$ if the
following symbol inequalities

$\|{X}^{\beta}_{x}\mathbb{D}^{\alpha}\sigma(x,\xi)\|_{\textnormal{op}}\leqslant
C_{\alpha,\beta}\langle\xi\rangle^{m-\rho|\gamma|+\delta|\beta|},$ (2.16)

are satisfied for all $\beta$ and $\gamma$ multi-indices and for all
$(x,[\xi])\in G\times\widehat{G},$ where $\langle\xi\rangle$ denotes the
Japanese bracket function at $\lambda_{\xi}$ defined in 2.13.

The class
$\Psi^{m}_{\rho,\delta}(G\times\widehat{G})\equiv\textnormal{Op}({S}^{m}_{\rho,\delta}(G\times\widehat{G}))$
is defined by those continuous and linear operators on $C^{\infty}(G)$ such
that $\sigma_{A}\in{S}^{m}_{\rho,\delta}(G\times\widehat{G}).$

In the next theorem we describe some fundamental properties of the global
Hörmander classes of pseudo-differential operators [11].

######  Theorem 2.16.

Let $\rho,\delta\in[0,1]$ be such that $0\leqslant\delta\leqslant\rho\leqslant
1,$ $\rho\neq 1.$ Then

$\Psi^{\infty}_{\rho,\delta}(G):=\cup_{m\in\mathbb{R}}\Psi^{m}_{\rho,\delta}(G)$

is an algebra of operators stable under compositions and adjoints, that is:

  * -

the mapping $A\mapsto
A^{*}:\Psi^{m}_{\rho,\delta}(G\times\widehat{G})\rightarrow\Psi^{m}_{\rho,\delta}(G\times\widehat{G})$
is a continuous linear mapping between Fréchet spaces.

  * -

The mapping $(A_{1},A_{2})\mapsto A_{1}\circ
A_{2}:\Psi^{m_{1}}_{\rho,\delta}(G\times\widehat{G})\times\Psi^{m_{2}}_{\rho,\delta}(G\times\widehat{G})\rightarrow\Psi^{m_{1}+m_{2}}_{\rho,\delta}(G\times\widehat{G})$
is a continuous bilinear mapping between Fréchet spaces.

Moreover, any operator in the class
$\Psi^{0}_{\rho,\delta}(G\times\widehat{G})$ admits a bounded extension from
$L^{2}(G)$ to $L^{2}(G).$

######  Remark 2.17.

With $0\leqslant\delta<\rho\leqslant 1$ such that $\rho\geq 1-\delta,$ the
condition $A\in\Psi^{m}_{\rho,\delta}(G\times\widehat{G})$ where
$m\in\mathbb{R},$ is equivalent to the fact that, when microlocalising the
operator $A$ into a local coordinate system $U,$ the operator $A$ takes the
form

$\forall f\in C^{\infty}_{0}(U),\,\forall
x\in\mathbb{R}^{n},\,\,Af(x)=\smallint\limits_{\mathbb{R}^{n}}\smallint\limits_{\mathbb{R}^{n}}e^{2\pi
i(x-y)\cdot\xi}a(x,\xi)f(y)dy(\xi)d\xi,$

where the function $a=a_{U},$ is such that for every compact subset $K\subset
U$ and for all $\alpha,\beta\in\mathbb{N}_{0}^{n},$ the inequalities

$|\partial_{x}^{\beta}\partial_{\xi}^{\alpha}a(x,\xi)|\leqslant
C_{\alpha,\beta,K}(1+|\xi|)^{m-\rho|\alpha|+\delta|\beta|},$ (2.17)

hold uniformly in $(x,\xi)\in K\times\mathbb{R}^{n}.$ This characterisation of
the Hörmander classes on $G$ was proved in [12]. So, for any compact Lie group
$G,$ the classes $\Psi^{m}_{\rho,\delta}(G\times\widehat{G})$ agree with the
ones introduced by Hörmander [10] when $0\leqslant\delta<\rho\leqslant 1$ and
$\rho\geq 1-\delta.$

##  3\. Local Weyl formula on compact Lie groups

###  3.1. Notations

We will use the following notations taken from [11, 20].

  * •

$(M,g)$ is a compact Riemannian manifold without boundary. We are interested
when $M=G$ is a compact Lie group and $g$ is an Ad-invariant Riemannian
metric. We will say in this case that $g(X,Y):=(X,Y)_{g}$ is $G$-invariant.

  * •

For all $m\in\mathbb{R},$ $\Psi^{m}(G):=\Psi^{m}_{1,0}(G)$ denotes the Kohn-
Nirenberg class of pseudo-differential operators of order $m.$

  * •

$T^{*}G$ is the cotangent bundle of $G$, and $T^{*}G\backslash\\{0\\}$ is the
cotangent bundle where the zero section is removed.

  * •

$g_{ij}=g(\frac{\partial}{\partial x_{i}},\frac{\partial}{\partial x_{j}})$,
$[g^{ij}]$ is the inverse matrix to $[g_{ij}]$. Then,

$\|\xi\|_{g}=\sqrt{\sum_{i,j=1}^{n}g^{ij}(x)\xi_{i}\xi_{j}}:T^{*}G\backslash\\{0\\}\rightarrow\mathbb{R}^{+}$

denotes the length of a (co)-vector.

  * •

The quantization of the Hamiltonian $\|\xi\|_{g}$ is the square root
$\sqrt{-\Delta_{g}}$ of the positive Laplacian $-\Delta_{g}$ of $(G,g)$. It
generates the wave operator

$U_{t}=e^{it\sqrt{\Delta}},$

which is a group of Fourier integral operators which propagate singularities
along geodesics. We denote by $|\xi|:=\sqrt{\mu}_{\xi},$
$[\xi]\in\widehat{G},$ the system of eigenvalues of $\sqrt{-\Delta_{g}}.$

  * •

The unit (co-) ball bundle is denoted by
$T^{*}\mathbb{B}(G)=\\{(x,\xi):\|\xi\|_{g}\leq 1\\}$. Its boundary
$T^{*}\mathbb{S}(G)=\\{\|\xi\|_{g}=1\\}$ is the spherical vector bundle.

  * •

$d\mu_{L}$ is the Liouville measure on $T^{*}\mathbb{S}(G)$, i.e. the surface
measure $d\mu=\frac{dxd\xi}{dH}$ induced by the Hamiltonian $H=\|\xi\|_{g}$
and by the symplectic volume measure $dxd\xi$ on $T^{*}G$.

  * •

$\widehat{G}$ denotes the unitary dual of a compact Lie group, that is formed
of all equivalence classes of unitary, irreducible and continuous
representations $\xi:G\rightarrow\textnormal{End}(\mathbb{C}^{\ell}),$ where
$\ell:=\dim(\xi)$ is the dimension of the representation space.

  * •

We denote by
$\sigma_{\textnormal{glob}}:G\times\widehat{G}\rightarrow\bigcup_{\ell\times\ell\in\widehat{G}}\textnormal{End}(\mathbb{C}^{\ell})$
the matrix-valued global symbol of a pseudo-differential operator
$A\in\Psi^{0}(G,\textnormal{loc})$ in the standard Hörmander class of oder
zero. Then, by Theorem 2.10

$\forall[\xi]\in\widehat{G},\,\forall x\in
G,\,\sigma_{\textnormal{glob}}(x,\xi)=\xi(x)^{*}A\xi(x).$ (3.1)

###  3.2. Local Weyl formula for zero order operators

Now we prove our main theorem. In this subsection $A\in\Psi^{0}(G)$ denotes a
pseudo-differential operator of order zero.

###### Proof of Theorem 1.1.

Let us choose one and only one unitary and irreducible representation $\xi$
from each equivalence class $[\xi]\in\widehat{G}.$ Then, any
$\xi:G\rightarrow\textnormal{End}(\mathbb{C}^{\ell})$ is a continuous mapping
and $\ell=\dim(\xi)$ is the dimension of its representation space. Moreover,
for any $x\in G,$ $\xi(x)$ is a unitary linear operator. For any $x\in G,$ and
in terms of a basis $B_{\xi}:=\\{e_{i}^{\xi}:1\leq i\leq\ell=\dim(\xi)\\}$ of
the representation space,
$\xi(x):\mathbb{C}^{\ell}\rightarrow\mathbb{C}^{\ell},$ can be identified with
a unitary matrix $\xi(x)\cong(\xi_{ij}(x))_{i,j=1}^{d_{\xi}},$ where, for any
$(i,j),$

$\xi_{ij}(x):=(e_{i}^{\xi},\xi(x)e^{\xi}_{j}),$

is the $(i,j)$-entry of the unitary operator $\xi(x)$ with respect to the
matrix $B_{\xi},$ and $d_{\xi}=\dim(\xi).$ We have denoted by $(\cdot,\cdot)$
the standard inner product on $\mathbb{C}^{\ell}.$

Let us consider the system of functions

$B:=\\{d_{\xi}^{\frac{1}{2}}\xi_{ij}:1\leq i,j\leq
d_{\xi},\,[\xi]\in\widehat{G}\\}$ (3.2)

formed by $d_{\xi}^{\frac{1}{2}}$ times the matrix entries $\xi_{ij}$ of the
unitary representations $\xi.$ By the Peter-Weyl theorem, $B$ is a system of
eigenfunctions of the positive Laplacian that provides an orthonormal basis of
$L^{2}(G).$ In view of the local Weyl formula, see Zelditch [20], we have the
asymptotic formula

$\sum_{|\xi|\leq\lambda}\sum_{i,j=1}^{d_{\xi}}(A(d_{\xi}^{\frac{1}{2}}\xi_{ij}),d_{\xi}^{\frac{1}{2}}\xi_{ij})=(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1}),$
(3.3)

for any $\lambda>0,$ where $\sigma_{\textnormal{loc}}\in\Gamma(T^{*}G)$ is the
(Hörmander) principal symbol of $A,$ and $\mu_{L}$ denotes the Liouville
measure on the spherical vector bundle $T^{*}\mathbb{S}(G).$

Now, we want to prove that the left-hand side of (3.3) agrees with the left-
hand side of (1.3). So, let us use (3.1)

$\sigma_{\textnormal{glob}}(x,\xi)=\xi(x)^{*}A\xi(x),\,x\in
G,\,[\xi]\in\widehat{G}.$ (3.4)

In consequence,

$A\xi(x)=\xi(x)\sigma_{\textnormal{glob}}(x,\xi),\,x\in
G,\,[\xi]\in\widehat{G}.$ (3.5)

Let us re-write this algebraic information. Note that for any $1\leq i,j\leq
d_{\xi},$

$A\xi_{ij}(x)=(\xi(x)\sigma_{\textnormal{glob}}(x,\xi))_{ij}=\sum_{k=1}^{d_{\xi}}\xi(x)_{ik}\sigma_{\textnormal{glob}}(x,\xi)_{kj}.$
(3.6)

The inner product can be written as

$(A(d_{\xi}^{\frac{1}{2}}\xi_{ij}),d_{\xi}^{\frac{1}{2}}\xi_{ij})=d_{\xi}(A\xi_{ij},\xi_{ij})=d_{\xi}\smallint\limits_{G}A\xi_{ij}(x)\overline{\xi(x)}_{ij}dx.$
(3.7)

Plugging (3.6) into (3.7) we obtain that

$\sum_{|\xi|\leq\lambda}\sum_{i,j=1}^{d_{\xi}}d_{\xi}(A\xi_{ij},\xi_{ij})=\sum_{|\xi|\leq\lambda}d_{\xi}\sum_{i,j,k=1}^{d_{\xi}}\smallint\limits_{G}\xi(x)_{ik}\sigma_{\textnormal{glob}}(x,\xi)_{kj}\overline{\xi(x)}_{ji}^{t}dx$
$=\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}[\xi(x)\sigma_{\textnormal{glob}}(x,\xi){\xi(x)}^{*}]dx.$

Now, taking into account that the spectral trace is invariant under unitary
transformations, we have that

$\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}[\xi(x)\sigma_{\textnormal{glob}}(x,\xi){\xi(x)}^{*}]dx=\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}[\sigma_{\textnormal{glob}}(x,\xi)]dx.$

The previous analysis shows the validity of (1.3). To prove the equality

$\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}),$
(3.8)

where $N(\lambda):=\\{[\xi]:|\xi|\leq\lambda\\}$ is the spectral function of
the Bessel potential $(1+\mathcal{L})^{\frac{1}{2}}$, just note that

$\frac{1}{N(\lambda)}\smallint\limits_{G}d_{\xi}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=\frac{1}{(2\pi)^{n}}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\frac{\lambda^{n}}{N(\lambda)}+O(\lambda^{-1}).$
(3.9)

Since

$\lambda^{n}\times N(\lambda)^{-1}\rightarrow
c_{n}>0,\,\lambda\rightarrow\infty,$

the term of the right hand side can be neglected since it goes to zero when
$\lambda\rightarrow\infty$ proving (1.4). ∎

###  3.3. Local Weyl formula for operators of positive order

In this subsection $A\in\Psi^{m}(G)$ denotes a pseudo-differential operator of
order $m>0$.

###### Proof of Theorem 1.3.

Let $(\cdot,\cdot)_{g}$ be a $G$-invariant inner product on $\mathfrak{g}.$ If
$X_{1},\cdots,X_{n}$ is an orthonormal basis of $\mathfrak{g}$ with respect to
the inner product $(\cdot,\cdot)_{g},$ then the positive Laplacian on $G$ is
given by $\mathcal{L}_{G}=-\sum_{i=1}^{n}X_{i}^{2}.$ Note that
$(\cdot,\cdot)_{g}$ induces an inner product on the dual $\mathfrak{g}^{*}.$
Below we continue with the notation in Subsection 3.1.

Since $T^{*}G$ is parallelizable we can make the identification $T^{*}G\cong
G\times\mathfrak{g}^{*}.$ Then, the (Hörmander) principal symbol of
$\mathcal{L}_{G}$ is given by (see e.g. Wallach [18, Page 16])

$\forall(x,\theta)\in(T^{*}G\setminus\\{0\\})\cong(G\times\mathfrak{g}^{*}\setminus\\{0\\}),\,\sigma_{\textnormal{loc},\mathcal{L}_{G}}(x,\theta)=(\theta,\theta)_{g}=:\|\theta\|_{g}^{2}.$
(3.10)

Moreover, using Seeley’s functional calculus [14], we have that for any
$\alpha\in\mathbb{R},$

$\sqrt{\mathcal{L}}_{G}^{\alpha}\in\Psi^{\alpha}(G)$

is a pseudo-differential operator of order $\alpha\in\mathbb{R}.$ Moreover,
for any $z\in\mathbb{C}$ with $\textnormal{Re}(z)<0,$ one has that (see Stein
[16])

$\sqrt{\mathcal{L}}_{G}^{z}\in\Psi^{\textnormal{Re}(z)}(G),\textnormal{ and
}\sqrt{\mathcal{L}}_{G}^{z}=\left(\sqrt{\mathcal{L}}_{G}+\textnormal{Proj}_{\textnormal{Ker}(\sqrt{\mathcal{L}}_{G})}\right)^{z}-\textnormal{Proj}_{\textnormal{Ker}(\sqrt{\mathcal{L}}_{G})}.$
(3.11)

Define the pseudo-differential operator

$\tilde{A}=A\sqrt{\mathcal{L}}_{G}^{-m}\in\Psi^{0}(G).$ (3.12)

In view of Theorem 1.1 we have the local Weyl formula

$\sum_{[\xi]\in\widehat{G}:|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}({\tilde{\sigma}}_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}{\tilde{\sigma}}_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1}),$
(3.13)

where $\tilde{\sigma}_{\textnormal{glob}}$ and
$\tilde{\sigma}_{\textnormal{loc}}$ denote the matrix-valued symbol and the
(Hörmander) local principal symbol of $\tilde{A},$ respectively. Note that on
the co-sphere, $\|\theta\|_{g}=1$ for all
$\theta\in\mathfrak{g}^{*}\setminus\\{0\\},$ and

$\forall(x,\theta)\in(T^{*}G\setminus\\{0\\})\cong
G\times\mathfrak{g}^{*}\setminus\\{0\\},\,\,\tilde{\sigma}_{\textnormal{loc}}(x,\theta)=\sigma_{\textnormal{loc}}(x,\theta)\|\theta\|_{g}^{-m}=\sigma_{\textnormal{glob}}(x,\theta).$
(3.14)

On the other hand, note that the matrix-valued symbol of $\tilde{A}$ on every
representation space is given by

$\forall(x,\xi)\in
G\times(\widehat{G}\setminus\\{1_{\widehat{G}}\\}),\,\,\tilde{\sigma}_{\textnormal{glob}}(x,\xi)=\sigma_{\textnormal{glob}}(x,\xi)|\xi|^{-m}$
(3.15)

where $1_{\widehat{G}}$ denotes the irreducible trivial representation (which
is one-dimensional, and therefore unique up to isomorphism). Indeed, since
$|\xi|=\sqrt{\mu}_{\xi},$ only for the trivial representation
$\xi_{0}=1_{\widehat{G}}$ we have $|\xi_{0}|=0.$ Then, we have

$\displaystyle\sum_{[\xi]\in\widehat{G}:0<|\xi|\leq\lambda}d_{\xi}|\xi|^{-m}\smallint\limits_{G}\textnormal{Tr}({\sigma}_{\textnormal{glob}}(x,\xi))dx$
$\displaystyle=(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}{\tilde{\sigma}}_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1})$
$\displaystyle-\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]dx$
$\displaystyle=(2\pi)^{-n}\left(\smallint\limits_{\\{(x,\xi^{\prime}):\|\xi^{\prime}\|_{g}=1\\}}{\sigma}_{\textnormal{loc}}(x,\xi^{\prime})\|\xi^{\prime}\|_{g}^{-m}d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}$
$\displaystyle+O(\lambda^{n-1})-\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]dx$
$\displaystyle=(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}{\sigma}_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1})$
$\displaystyle-\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]dx.$

For all $0<|\xi|\leq\lambda,$ we have that $\lambda^{-m}\leq|\xi|^{-m}.$
Because of the trace condition

$\forall(x,[\xi])\in
G\times\widehat{G},\,\textnormal{Tr}[\sigma_{\textnormal{glob}}(x,[\xi])]\geq
0,$

we can write the inequality

$\displaystyle\lambda^{-m}\sum_{[\xi]\in\widehat{G}:0<|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}({\sigma}_{\textnormal{glob}}(x,\xi))dx$
$\displaystyle\leq(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}{\sigma}_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n}+O(\lambda^{n-1})$
$\displaystyle-\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]dx.$

Consequently,

$\displaystyle\sum_{[\xi]\in\widehat{G}:0<|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}({\sigma}_{\textnormal{glob}}(x,\xi))dx$
$\displaystyle\leq(2\pi)^{-n}\left(\smallint\limits_{T^{*}\mathbb{S}(G)}{\sigma}_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime})\right)\lambda^{n+m}+O(\lambda^{n+m-1})$
$\displaystyle-\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]dx\lambda^{m}.$

The analysis above proves (1.6). Indeed, note that
$\smallint_{G}\textnormal{Tr}[\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})]=0.$
To see this note that $1\in\textnormal{Ker}(\mathcal{L}_{G}),$ and the
functional calculus implies that for any function
$f\in\textnormal{Ker}(\mathcal{L}_{G}),$

$\sqrt{\mathcal{L}_{G}}^{-m}f\equiv 0.$

Then, since $1_{\widehat{G}}(x)=1$ for any $x\in G,$ we have that

$\displaystyle\tilde{\sigma}_{\textnormal{glob}}(x,1_{\widehat{G}})=1_{\widehat{G}}(x)^{*}A\sqrt{\mathcal{L}_{G}}^{-m}1_{\widehat{G}}(x)=A\sqrt{\mathcal{L}_{G}}^{-m}1=0.$

The proof of Theorem 1.3 is complete. ∎

###  3.4. Expected value of multiplication operators in the energy states
$\phi_{j}$

Now, let us analyse a simple prototype of a pseudo-differential operator.
Consider $\varkappa\in C^{\infty}(G)$ be a smooth function. Let

$A_{\varkappa}:L^{2}(G)\mapsto L^{2}(G),\,A_{\varkappa}f:=\varkappa f,$ (3.16)

be the corresponding multiplication operator by $\varkappa.$ In general, to
analyse the behaviour of the matrix-coefficients
$(A_{\varkappa}\phi_{j},\phi_{j})=({\varkappa}\phi_{j},\phi_{j})$ for an
arbitrary normalised system of eigenfunctions $\phi_{j}$ of the positive
Laplacian $-\Delta_{g}$ is a difficult task. Let us analyse the behaviour of
the expected values

$(A(d_{\xi}^{\frac{1}{2}}\xi_{ij}),d_{\xi}^{\frac{1}{2}}\xi_{ij})=d_{\xi}(A\xi_{ij},\xi_{ij})=d_{\xi}({\varkappa}\xi_{ij},\xi_{ij})$
(3.17)

where any $\phi_{ij,[\xi]}:=d_{\xi}^{\frac{1}{2}}\xi_{ij}$ is an energy state
that belongs to the basis $B$ defined in (3.2). To analyse the matrix-
coefficients in (3.17) we consider the convergence of its average, that
according with the proof of Theorem 1.1 satisfies the identity

$\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}\sum_{i,j=1}^{d_{\xi}}d_{\xi}(A\xi_{ij},\xi_{ij})=\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx,$
(3.18)

where $\sigma_{\textnormal{glob}}(x,\xi)$ is the global symbol of
$A_{\varkappa}.$ Now, by using the global quantisation formula, note that

$\sigma_{\textnormal{glob}}(x,\xi)=\xi(x)^{*}\varkappa(x)\xi(x)=\varkappa(x)I_{d_{\xi}},\,[\xi]\in\widehat{G},\,x\in
G,$

where $I_{d_{\xi}}$ is the identity operator in $\mathbb{C}^{d_{\xi}}.$
Consequently,

$\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}d_{\xi}\smallint\limits_{G}\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=\lim_{\lambda\rightarrow\infty}\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}d_{\xi}^{2}\smallint\limits_{G}\varkappa(x)dx=\smallint\limits_{G}\varkappa(x)dx.$
(3.19)

In view of Theorem 1.1 we deduce that following simplified version of (1.4)

$C_{n}\smallint\limits_{T^{*}\mathbb{S}(G)}\varkappa(x)d\mu_{L}(x,\xi^{\prime})=\smallint\limits_{G}\varkappa(x)dx.$
(3.20)

Note that, if the function $\varkappa\in C^{\infty}(G)$ is real-valued, the
convergence of the average
$\frac{1}{N(\lambda)}\sum_{|\xi|\leq\lambda}\sum_{i,j=1}^{d_{\xi}}d_{\xi}(A\xi_{ij},\xi_{ij})$
when $\lambda\rightarrow\infty,$ implies the existence of a sub-sequence
$d_{\xi}^{\frac{1}{2}}\xi_{i_{k},j_{k}}$ of energy states in $B$ such that

$\lim_{k\rightarrow\infty}d_{\xi}(\varkappa\xi_{i_{k},j_{k}},\xi_{i_{k},j_{k}})=\smallint\limits_{G}\varkappa(x)dx,$

which shows that the Haar integral $\smallint\limits_{G}\varkappa(x)dx$ is a
quantum limit of the sequence $B$ with respect to the multiplication operator
$A_{\varkappa}$.

###### Proof of Corollary 1.2.

Let us use a numerable correspondence between $\mathbb{N}$ and the unitary
dual $\widehat{G}=\\{[\xi_{j}]:j\in\mathbb{N}\\}.$ Let us define the sequence

$A_{n}:=d_{\xi_{n}}\smallint\limits_{G}\textnormal{Tr}[\sigma_{\textnormal{op}}(x,\xi_{n})]dx.$

Let $S_{n}$ be the average sums of the sequence
$\\{A_{n}\\}_{n\in\mathbb{N}}.$ Note that from Theorem 1.1 we have the
identity

$\lim_{n\rightarrow\infty}\frac{1}{N(n)}\sum_{[\xi_{j}]:\langle\xi_{j}\rangle\leq
n}A_{j}=(2\pi)^{-n}=\lim_{n\rightarrow\infty}\frac{1}{N(n)}S_{N(n)}=\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$
(3.21)

Then, there exists a sub-sequence $\\{A_{n_{j}}\\}$ such that when
$j\rightarrow\infty,$

$A_{n_{j}}\rightarrow\smallint\limits_{T^{*}\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi^{\prime})d\mu_{L}(x,\xi^{\prime}).$
(3.22)

The proof of Corollary 1.2 is complete. ∎

## References

  * [1] Barnett, A. Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards, Comm. Pure Appl. Math. 59(10), 1457–1488, (2006).
  * [2] Berry, M. Regular and irregular semiclassical wavefunctions, J. Phys. A 10, 2083-2091, (1977).
  * [3] Bott, R. The index theorem for homogeneous differential operators. Differential and Combinatorial Topology. A Symposion in Honour of Marston Morse, 167–186, 1965.
  * [4] Delgado, J., Ruzhansky, M. Fourier multipliers, symbols, and nuclearity on compact manifolds. J. Anal. Math. 135, no. 2, pp. 757–800, (2018).
  * [5] Delgado, J., Ruzhansky, M. Schatten classes and traces on compact groups. Math. Res. Lett. 24, no. 4, pp. 979–1003, (2017).
  * [6] Delgado, J., Ruzhansky, M. Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds. C. R. Math. Acad. Sci. Paris 352, no. 10, pp. 779–784, (2014).
  * [7] Delgado, J., Ruzhansky, M. $L^{p}$-nuclearity, traces, and Grothendieck-Lidskii formula on compact Lie groups. J. Math. Pures Appl. (9) 102, no. 1, pp. 153–172, (2014).
  * [8] Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems: scars of periodic orbits. Phys. Rev. Lett. 53(16), 1515–1518, (1984).
  * [9] Heller, E. J. Wavepacket dynamics and quantum chaology. Chaos et physique quantique (Les Houches, 1989), 547–664, North-Holland, Amsterdam, 1991.
  * [10] Hörmander, L. The analysis of the linear partial differential operators, Vol. III-IV. Springer-Verlag, (1985).
  * [11] Ruzhansky, M., Turunen, V. Pseudo-differential Operators and Symmetries: Background Analysis and Advanced Topics Birkhäuser-Verlag, Basel, 2010.
  * [12] Ruzhansky, M., Turunen, V., Wirth, J. Hörmander class of pseudo-differential operators on compact Lie groups and global hypoellipticity, J. Fourier Anal. Appl. 20 (2014), 476–499.
  * [13] Ruzhansky, M., Wirth, J. $L^{p}$ Fourier multipliers on compact Lie groups, Math. Z., 280, pp. 621–642, (2015).
  * [14] Seeley, R. T. Complex powers of an elliptic operator. 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288–307 Amer. Math. Soc., Providence, R.I.
  * [15] Shnirelman, A. I. Ergodic properties of eigenfunctions, Usp. Math. Nauk. 29(6), 181-182, (1974).
  * [16] Stein, E. M. Topics in harmonic analysis related to the Littlewood-Paley theory, Vol. 63 of Annals of Mathematics Studies, Princeton University Press, 1970.
  * [17] Strichartz, R. S. Spectral asymptotics revisited. J. Fourier Anal. Appl. 18(3), 626—659 (2012)
  * [18] Wallach, N.  Harmonic analysis on homogeneous spaces.  Pure and Applied Mathematics, No. 19. Marcel Dekker, Inc., New York, 1973. xv+361 pp.
  * [19] Zelditch, S. Recent developments in mathematical quantum chaos. in Current Developments in Mathematics, 2009, D. Jerison, B. Mazur, T. Mrowka, W. Schmid, R. Stanley, S-T Yau (eds.), International Press, 2009.
  * [20] Zelditch, S. Eigenfunctions of the Laplacian on a Riemannian Manifold. CBMS Regional Conference Series in Mathematics. Conference Board of the Mathematical Sciences, 2017.


# Room temperature ferromagnetic semiconductors through metal-semiconductor
transition in monolayer MnSe2

Jia-Wen Li  Kavli Institute for Theoretical Sciences, University of Chinese
Academy of Sciences, Beijng 100049, China  Gang Su<EMAIL_ADDRESS>Kavli
Institute for Theoretical Sciences, University of Chinese Academy of Sciences,
Beijng 100049, China  CAS Center for Excellence in Topological Quantum
Computation, University of Chinese Academy of Sciences, Beijng 100190, China
Physical Science Laboratory, Huairou National Comprehensive Science Center,
Beijing 101400, China  School of Physical Sciences, University of Chinese
Academy of Sciences, Beijng 100049, China  Bo Gu<EMAIL_ADDRESS>Kavli
Institute for Theoretical Sciences, University of Chinese Academy of Sciences,
Beijng 100049, China  CAS Center for Excellence in Topological Quantum
Computation, University of Chinese Academy of Sciences, Beijng 100190, China
Physical Science Laboratory, Huairou National Comprehensive Science Center,
Beijing 101400, China

###### Abstract

To realize room temperature ferromagnetic semiconductors is still a challenge
in spintronics. Recent experiments have obtained two-dimensional (2D) room
temperature ferromagnetic metals, such as monolayer MnSe2. In this paper, we
proposed a way to obtain room temperature ferromagnetic semiconductors through
metal-semiconductor transition. By the density functional theory calculations,
a room temperature ferromagnetic semiconductor is obtained in monolayer MnSe2
with a few percent tensile strains, where a metal-semiconductor transition
occurs with 2.2% tensile stain. The tensile stains raise the energy of d
orbitals of Mn atoms and p orbitals of Se atoms near the Fermi level, making
the Fermi level sets in the energy gap of bonding and antibonding states of
these p and d orbitals, and opening a small band gap. The room temperature
ferromagnetic semiconductors are also obtained in the heterostructures MnSe2/X
(X = Al2Se3, GaSe, SiH, and GaP), where metal-semiconductor transition happens
due to the tensile strains by interface of heterostructures. In addition, a
large magneto-optical Kerr effect (MOKE) is obtained in monolayer MnSe2 with
tensile strain and MnSe2-based heterostructures. Our theoretical results pave
a way to obtain room temperature magnetic semiconductors from experimentally
obtained 2D room temperature ferromagnetic metals through metal-semiconductor
transitions.

##  I Introduction

In spintronics, it is still a challenge in experiments to realize room
temperature ferromagnetic semiconductors. In 2017, the successful synthesis of
two-dimensional (2D) van der Waals ferromagnetic semiconductors CrI3 [1] and
Cr2Ge2Te6 [2] in experiments has attracted extensive attention to 2D
ferromagnetic semiconductors. According to Mermin-Wagner theorem [3], the
magnetic anisotropy is essential to produce long-range magnetic order in 2D
systems. Recently, more 2D ferromagnetic semiconductors have been obtained in
experiments, such as Cr2S3 [4], CrCl3 [5], CrBr3 [6], CrSiTe3 [7], CrSBr [8],
where the Curie temperatures $T_{C}$ are far below room temperature. On the
other hand, the 2D van der Waals ferromagnetic metals with high $T_{C}$ have
been obtained in recent experiments. For example, $T_{C}$ = 140 K in CrTe [9],
300 K in CrTe2 [10], 344 K in Cr3Te6 [11], 160 K in Cr3Te4 [12], 280 K in CrSe
[13], 300 K in Fe3GeTe2 [14, 15], 270 K in Fe4GeTe2 [16], 229 K in Fe5GeTe2
[17], 380 K in Fe3GaTe2 [18], 300 K in MnSe2 [19], etc.

Recent studies have shown that the physical properties of 2D materials are
sensitive to external regulations, such as electric filed [14, 20, 21, 22,
23], doping [24, 25, 26, 27, 28], surface functionalization [29, 30],
intercalation [31, 32], heterostructure [33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48], strain [49, 50, 51, 52, 34, 53], etc. Among them,
strain engineering is an effective technique to change the lattice and
electronic structures and thus to control various properties of 2D materials.
In contrast to bulk materials, 2D materials have stronger deformation capacity
and thus can withstand greater elastic strain without fracture, which shows
great advantages in strain engineering. For example, monolayer MoS2 can
sustain strains as large as 11% [54], monolayer FeSe can sustain strains up to
6% [55, 56], and single-layer graphene can even withstand 25% elastic strain
[57]. It has been observed that in few-layer black phosphorus that biaxial
tensile strain can increases the band gap, while biaxial compressive strain
can decrease the band gap [49]. Under uniaxial strain up to 1.7%, a band gap
reduction of 0.3 eV was observed in monolayer MoS2 [50].

In addition, heterostructures can have a significant impact on the magnetism
of magnetic materials [38, 39, 40, 45, 46, 47]. Heterostructures also have a
great impact on the band structure. Especially, due to the effect of
interlayer interaction and strain, metal-semiconductor transition happens in
some semimetal/semiconductor heterostructures, and a band gap $E_{g}$ can be
opened, such as $E_{g}$ = 0.44 eV in silicene/GaP [41], 0.22 eV in
silicene/Ga2SeS [42], 0.67 eV in CSe/BP [58], 13 meV in graphene/MoSi2N4 [43]
and 17 meV in Ge/SMoSe [44].

As a room temperature ferromagnetic material, MnSe2 has attracted a lot of
attentions due to its interesting properties [59, 36, 37, 60, 61, 62, 63, 64,
65, 66]. The structure of MnSe2 is shown in Fig. 1(a) with space group of
P$\overline{3}$m1 (164). The band structure from density functional theory
(DFT) with Perdew-Burke-Ernzerhof (PBE) pseudopotential shows semimetal
properties [62, 63, 64, 65]. Because PBE pseudopotential always underestimated
the band gap, Heyd-Scuseria-Ernzerhof (HSE) hybrid functional approach is
believed to give a better description of the band structures [67]. By the DFT
with HSE, MnSe2 was calculated to be a semimetal [63, 45] or semiconductor
with small band gap of 0.01 eV [65]. Experimental results suggest that MnSe2
at Bi2Se3 substrate has a small band gap [60]. On the other hand, the
electronic properties of monolayer MnSe2 are still unclear. It was reported
that tensile strain could enhance the ferromagnetic properties of monolayer
MnSe2 and weaken the metallicity [63, 65]. It has been discussed that $T_{C}$
can be enhanced by tensile strain in some 2D ferromagnetic semiconductors [68,
38, 69, 70] The evolution of band structures of monolayer MnSe2 was calculated
by DFT with PBE, giving no band gap with tensile strain up to 8% [65]. Is it
possible to open a band gap in these room temperature ferromagnetic metals by
some methods, and obtain room temperature ferromagnetic semiconductors?

In this paper, we propose a way to obtain room temperature ferromagnetic
semiconductor by metal-semiconductor transition in monolayer MnSe2 and
MnSe2-based heterostructures. The DFT calculation with HSE hybrid functional
shows that monolayer MnSe2 is room temperature ferromagnetic semimetal. A 2.2%
tensile strain could open a small band gap in monolayer MnSe2. An in-plane to
out-of-plane magnetic anisotropy transition happens with 1.7% tensile strain,
and $T_{C}$ increases with tensile strain. In addition, the heterostructures
MnSe2/X with X= GaP [71, 72, 73], GaSe [74], SiH [75], and Al2Se3 [40] were
studied. The room temperature ferromagnetic semiconductors are also obtained
in these heterostructures, where metal-semiconductor transition occurs due to
tensile stain by interfaces. Large magneto-optical Kerr effect is found in
monolayer MnSe2 with a percent tensile strain and MnSe2-based
heterostructures. Our theoretical results propose a way to obtain room
temperature ferromagnetic semiconductors by metal-semiconductor transition in
monolayer MnSe2 by applying tensile strain or building heterostructures.

##  II Method

All calculations were based on the DFT as implemented in the Vienna ab initio
simulation package (VASP) [76]. The exchange-correlation potential is
described by the PBE form with the generalized gradient approximation (GGA)
[77]. The electron-ion potential is described by the projector-augmented wave
(PAW) method [78]. We carried out the calculation of PBE + U with U = 4 eV for
3d electrons in Mn [62, 63, 64, 65]. The band structures were calculated in
HSE06 hybrid functional [67]. The plane-wave cutoff energy is set to be 650
eV. The 9$\times$9$\times$1 $\Gamma$ center K-point was used for the Brillouin
zone (BZ) sampling. To obtain accurate results of magnetic anisotropy energy
(MAE), K-points were chosen as $\Gamma$-centered 25$\times$25$\times$1\. The
density of states (DOS) were obtained from HSE and K-points of
$\Gamma$-centered 18$\times$18$\times$1\. The structures of all materials were
fully relaxed, where the convergence precision of energy and force was
$10^{-6}$ eV and $10^{-2}$ eV/Å, respectively. The van der Waals effect is
included with DFT-D3 method [79]. The Wannier90 code was used to construct a
tight-binding Hamiltonian [80, 81]. The WannierTools code was used to obtain a
tight-binding band structure from tight-binding Hamiltonian [82]. The
Heisenberg-type Monte Carlo simulation was performed on a
50$\times$50$\times$1 lattice with 2500 magnetic points for MnSe2.
1$\times$105 steps were carried for each temperature, and the last one-thirds
steps were used to calculate the temperature-dependent physical quantities.

##  III RESULTS AND DISCUSSION

###  III.1 Monolayer MnSe2

The crystal structure of monolayer MnSe2 is shown in Fig. 1(a), which shows a
triangle lattice. The calculated in-plane lattice constant is $a_{0}$ = 3.658
Å, in agreement with previous calculations [62, 63, 64, 65]. The band
structure of monolayer MnSe2 with HSE hybrid functional is shown in Fig. 1(b).
The lowest band above Fermi level and the highest band below Fermi level
slightly overlap, showing the semimetal behavior. The bands near Fermi level
contain only one component of spins, showing the half-metal behavior. In
addition, the Monte Carlo results of magnetization and susceptibility as a
function of temperature for monolayer MnSe2 is shown in Fig. 1(c), giving a
high $T_{C}$ = 354 K.

Figure 1:  (a) Top and side views of crystal structure of monolayer MnSe2. (b)
Spin polarized band structure of MnSe2, obtained by the DFT calculation with
HSE hybrid functional. (c) Magnetization and susceptibility of MnSe2 as a
function of temperature, obtained by the Monte Carlo simulation based on a 2D
Heisenberg model. The Curie temperature of monolayer MnSe2 is calculated as
$T_{C}$ = 354 K, being consistent with the experiment [19].

###  III.2 Monolayer MnSe2 with strain

Figure 2:  For monolayer MnSe2, tensile strain dependence of (a) band gap, (b)
Curie temperature $T_{C}$, and (c) magnetic anisotropy energy (MAE), obtained
by the DFT and Monte Carlo calculations. Numerical results of four MnSe2-based
heterostructures are also included, where strain comes from the interface of
heterostructures. Experimental $T_{C}$ and MAE of heterostructure MnSe2/GaSe
[19] are included for comparison.

To study the effect of strain on electronic properties, we applied biaxial
strain. The strain ratio is defined as $\epsilon=(a-a_{0})/{a_{0}}$, where $a$
and $a_{0}$ are the in-plane lattice constants with and without strain,
respectively. The variation of band gap of monolayer MnSe2 with tensile strain
calculated by DFT with HSE is shown in Fig. 2 (a). A metal-semiconductor
transition happens with a tensile strain of 2.2%. Fig. 2 (b) shows $T_{C}$ of
monolayer MnSe2 as a function with tensile strain. $T_{C}$ increase with
tensile strain, in agreement with previous report [68, 38, 69, 70]. The MAE is
defined as $(E_{\perp}-E_{\parallel})/N_{Mn}$, where E⟂ and E∥ are energies of
MnSe2 with out-of-plane and in-plane magnetic polarization, respectively.
$N_{Mn}$ is the number of Mn atoms in a unit cell. For monolayer MnSe2 without
strain, as shown in Fig. 2(c), the in-plane MAE of 0.75 meV/Mn is obtained, in
agreement with the previous report [63]. By applying tensile strain, an in-
plane to out-plane MAE transition is obtained at 1.7% tensile strain. Thus,
monolayer MnSe2 will become a room temperature ferromagnetic semiconductor
with out-of-plane MAE by applying tensile strains above 2.2%.

Figure 3:  Metal-semiconductor transition of monolayer MnSe2 due to strain.
DFT results of partial density of state (PDOS) of (a) d orbitals of Mn and (b)
p orbitals of Se without stain. PDOS of (c) d orbitals of Mn and (d) p
orbitals of Se with 4% tensile stain.

In order to analyze the metal-semiconductor transition of monolayer MnSe2
under strain, the partial density of states (PDOS) were calculated, as shown
in Fig. 3. PDOS of d orbitals of Mn and p orbitals of Se without strain is
shown in Figs. 3(a) and 3(b), respectively. PDOS of Mn and Se with 4% tensile
strain is shown in Figs. 3(c) and 3(d), respectively, with a small gap. The
tensile stains raise the energy of d orbitals of Mn atoms and p orbitals of Se
atoms near the Fermi level, making the Fermi level lie in the energy gap of
bonding and antibonding states of these p and d orbitals, and opening a small
band gap.

###  III.3 Heterostructures MnSe2/X (X = GaP, GaSe, SiH, and Al2Se3)

Considering the mismatch from substrate is an efficient way to provide strain
for 2D material in experiments [83], we constructed MnSe2-based
heterostructures with 2D semiconductors to provide strain. GaP [71, 72], GaSe
[74], SiH [75], and Al2Se3 [40] are nonmagnetic semiconductors. The calculated
results with PBE at monolayer limit give band gaps of 1.22, 1.80, 2.18 and
1.69 eV, respectively, and lattice constants of 3.916, 3.814, 3.888 and 3.788
Å, respectively. According to the lattice mismatch
$\delta=2\times(a_{1}-a_{2})/(a_{1}+a_{2})\times 100\%$, the lattice mismatch
of GaP, GaSe, SiH, and Al2Se3 with MnSe2 are 6.8%, 4.2%, 6.1%, and 3.5%,
respectively. The detailed results are given in Supplemental Material [84].

Figure 4:  Structure and spin polarized band structure of MnSe2-based
heterostructures, (a, e) MnSe2/GaP, (b, f) MnSe2/GaSe, (c, g) MnSe2/SiH, and
(d, h) MnSe2/Al2Se3. The band structures are obtained by the DFT calculation
with HSE hybrid functional.

We consider different stacking models of heterostructures, and the detailed
data are given in Supplemental Material [84]. As shown in Fig. 4, all
heterostructures are ferromagnetic semiconductors. The optimised lattice
constants of MnSe2/X with X = GaP, GaSe, SiH, and Al2Se3 are about 3.83, 3.76,
3.81, and 3.75 Å, respectively, with an effective tensile strain of 4.7%,
2.8%, 4.2%, and 2.5%, respectively. The binding energy $E_{b}$ for MnSe2/X is
defined as $E_{b}=(E_{MnSe_{2}/X}-E_{MnSe_{2}}-E_{X})/N_{tot}$, where
$E_{MnSe_{2}/X}$, $E_{MnSe_{2}}$, and $E_{X}$ represent the total energies of
heterostructure MnSe2/X, monolayer MnSe2, and monolayer X, respectively, and
$N_{tot}$ is the total atom number in a unitcell. The calculated results are
-0.24, -0.17, -0.10, and -0.20 eV/atom for X = GaP, GaSe, SiH, and Al2Se3,
respectively, indicating their stability. $T_{C}$ of heterostructures were
obtained through DFT calculations and Monte Carlo simulation. As shown in Fig.
2, the calculation results predict that MnSe2/X with X = GaP, GaSe, SiH, and
Al2Se3 are room temperature magnetic semiconductors with out-of-plane MAE. It
is noted that for the heterostructure MnSe2/GaSe, Tc above room temperature
and out-of-plane magnetization were observed in the experiment [19]. Thus, our
calculation results in Fig. 2 are consistent with the experiment of MnSe2/GaSe
[19]. The detailed calculation results are given in Supplemental Material
[84].

Heterostructures not only provide effective strain, but also induce interlayer
interaction. The calculated results of the most stable stacks of
heterostructures is shown in Fig. 2. The calculated results of
heterostructures are different with those of monolayer MnSe2 with the same
tensile strain. For example, heterostructure MnSe2/GaSe has a band gap of 0.18
eV, while monolayer MnSe2 with same lattice constants has a band gap of 0.08
eV. Heterostructure MnSe2/Al2Se3 give a MAE of -1.11 meV/Mn, while monolayer
MnSe2 with same lattice constants give a MAE of -0.36 meV/Mn. Therefore, the
interlayer interaction may play an important role in properties of
heterostructures.

###  III.4 MOKE in monolayer MnSe2 and heterostructures MnSe2/X

Figure 5:  DFT results of Kerr rotation angle for MnSe2 with 0%, 2%, 4%
tensile strain and heterostructures MnSe2/X (X = SiH, Al2Se3, GaSe, and GaP).
Experimental and numerical results of Fe [85] are also included for
comparison.

We investigated the magneto-optical Kerr effect for MnSe2 based structures.
The Kerr rotation angle is given by:

$\displaystyle\begin{aligned}
\theta_{K}(\omega)=Re\frac{\varepsilon_{xy}}{(1-\varepsilon_{xx})\sqrt{\varepsilon_{xx}}},\\\
\end{aligned}$ (1)

where $\varepsilon_{xx}$ and $\varepsilon_{xy}$ are the diagonal and off-
diagonal components of the dielectric tensor $\varepsilon$, and $\omega$ is
the frequency of incident light. The dielectric tensor $\varepsilon$ can be
obtained by the optical conductivity tensor $\sigma$ as
$\varepsilon(\omega)=\frac{4{\pi}i}{\omega}\sigma(\omega)+I$, where I is the
unit tensor. The calculated Kerr rotation angle as a function of photon energy
for monolayer MnSe2 with strain is shown in Fig. 5 (a), the result for
heterostructures MnSe2/X with X = GaP, GaSe, SiH and Al2Se3 is shown in Fig. 5
(b). According to our calculation results, there exist large Kerr rotation
angles with out-of-plane magnetization, such as monolayer MnSe2 with 2% and 4%
tensile strain. On the contrary, Kerr rotation angles with in-plane
magnetization are small, such as monolayer MnSe2 without strain. In addition,
heterostructures MnSe2/X with out-of-plane magnetization also have a large
Kerr rotation angle. The experimental result for Fe [85] and our DFT result
for Fe bulk are also included for comparison. The Kerr rotation angles for
stretched MnSe2 are about 4 times bigger than that of bcc Fe. Detailed results
of Kerr rotation angles are given in Supplemental Material [84].

##  IV Conclusion

Based on the DFT calculations, we studied the properties of monolayer MnSe2
with strain and MnSe2-based heterostructures. For monolayer MnSe2, a metal-
semiconductor transition happens at 2.2% tensile strain, and an in-plane to
out-of-plane MAE transition happens at 1.7% tensile strain. In addition,
$T_{C}$ of monolayer MnSe2 increases with tensile strain. The heterostructures
MnSe2/X with X = GaP, GaSe, SiH, and Al2Se3 were studied, and the DFT
calculation results show that they are room temperature ferromagnetic
semiconductors with out-of-plane MAE. Large magneto-optical Kerr effect was
found in monolayer MnSe2 with a few percent tensile strain and MnSe2-based
heterostructures. Our results propose a way to obtain room temperature
ferromagnetic semiconductors by metal-semicondutor transition in monolayer
MnSe2 by applying a few percent tensile stain or building heterostructures.

##  V Acknowledgements

This work is supported by National Key R&D Program of China (Grant No.
2022YFA1405100), National Natural Science Foundation of China (Grant No.
12074378), Chinese Academy of Sciences (Grants No. YSBR-030, No.
JZHKYPT-2021-08, No. XDB33000000), Beijing Municipal Science and Technology
Commission (Grant No. Z191100007219013).

## References

  * Huang  _et al._ [2017] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A. McGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-Herrero, and X. Xu, Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit, Nat. 546, 270 (2017).
  * Gong  _et al._ [2017] C. Gong, L. Li, Z. Li, H. Ji, Alex Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals, Nat. 546, 265 (2017).
  * Mermin and Wagner [1966] N. D. Mermin and H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models, Phys. Rev. Lett. 17, 1307 (1966).
  * Chu  _et al._ [2019] J. Chu, Y. Zhang, Y. Wen, R. Qiao, C. Wu, P. He, L. Yin, R. Cheng, F. Wang, Z. Wang, J. Xiong, Y. Li, and J. He, Sub-millimeter-Scale Growth of One-Unit-Cell-Thick Ferrimagnetic Cr2S3 Nanosheets, Nano. Lett. 19, 2154 (2019).
  * Cai  _et al._ [2019] X. Cai, T. Song, N. P. Wilson, G. Clark, M. He, X. Zhang, T. Taniguchi, K. Watanabe, W. Yao, D. Xiao, M. A. McGuire, D. H. Cobden, and X. Xu, Atomically Thin CrCl3: An In-Plane Layered Antiferromagnetic Insulator, Nano. Lett. 19, 3993 (2019).
  * Zhang  _et al._ [2019a] Z. Zhang, J. Shang, C. Jiang, A. Rasmita, W. Gao, and T. Yu, Direct Photoluminescence Probing of Ferromagnetism in Monolayer Two-Dimensional CrBr3, Nano. Lett. 19, 3138 (2019a).
  * Achinuq  _et al._ [2021] B. Achinuq, R. Fujita, W. Xia, Y. Guo, P. Bencok, G. van der Laan, and T. Hesjedal, Covalent Mixing in the 2D Ferromagnet CrSiTe3 Evidenced by Magnetic X-Ray Circular Dichroism, Phys. Status. Solidi. 16, 2100566 (2021).
  * Lee  _et al._ [2021] K. Lee, A. H. Dismukes, E. J. Telford, R. A. Wiscons, J. Wang, X. Xu, C. Nuckolls, C. R. Dean, X. Roy, and X. Zhu, Magnetic Order and Symmetry in the 2D Semiconductor CrSBr, Nano. Lett. 21, 3511 (2021).
  * Sun  _et al._ [2022] S. Sun, J. Liang, R. Liu, W. Shen, H. Wu, M. Tian, L. Cao, Y. Yang, Z. Huang, W. Lin, J. Du, Z. Ni, Y. Xu, Q. Chen, and Y. Zhai, Anisotropic magnetoresistance in room temperature ferromagnetic single crystal CrTe flake, J. Alloy. Compd. 890, 161818 (2022).
  * Meng  _et al._ [2021] L. Meng, Z. Zhou, M. Xu, S. Yang, K. Si, L. Liu, X. Wang, H. Jiang, B. Li, P. Qin, P. Zhang, J. Wang, Z. Liu, P. Tang, Y. Ye, W. Zhou, L. Bao, H.-J. Gao, and Y. Gong, Anomalous thickness dependence of Curie temperature in air-stable two-dimensional ferromagnetic 1T-CrTe2 grown by chemical vapor deposition, Nat. Commun. 12, 94 (2021).
  * Chua  _et al._ [2021] R. Chua, J. Zhou, X. Yu, W. Yu, J. Gou, R. Zhu, L. Zhang, M. Liu, M. B. H. Breese, W. Chen, K. P. Loh, Y. P. Feng, M. Yang, Y. L. Huang, and A. T. S. Wee, Room Temperature Ferromagnetism of Monolayer Chromium Telluride with Perpendicular Magnetic Anisotropy, Adv. Mater. 33, 2103360 (2021).
  * Li  _et al._ [2022a] B. Li, X. Deng, W. Shu, X. Cheng, Q. Qian, Z. Wan, B. Zhao, X. Shen, R. Wu, S. Shi, H. Zhang, Z. Zhang, X. Yang, J. Zhang, M. Zhong, Q. Xia, J. Li, Y. Liu, L. Liao, Y. Ye, L. Dai, Y. Peng, B. Li, and X. Duan, Air-stable ultrathin Cr3Te4 nanosheets with thickness-dependent magnetic biskyrmions, Mater. Today 57, 66 (2022a).
  * Zhang  _et al._ [2019b] Y. Zhang, J. Chu, L. Yin, T. Shifa, Z. Cheng, R. Cheng, F. Wang, Y. Wen, X. Zhan, Z. Wang, and J. He, Ultrathin Magnetic 2D Single‐Crystal CrSe, Adv. Mater. 31, 1900056 (2019b).
  * Deng  _et al._ [2018] Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2, Nat. 563, 94 (2018).
  * Fei  _et al._ [2018] Z. Fei, B. Huang, P. Malinowski, W. Wang, T. Song, J. Sanchez, W. Yao, D. Xiao, X. Zhu, A. F. May, W. Wu, D. H. Cobden, J.-H. Chu, and X. Xu, Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2, Nat. Mater. 17, 778 (2018).
  * Seo  _et al._ [2020] J. Seo, D. Kim, E. An, K. Kim, G.-Y. Kim, S.-Y. Hwang, D. Kim, B. Jang, H. Kim, G. Eom, S. Seo, R. Stania, M. Muntwiler, J. Lee, K. Watanabe, T. Taniguchi, Y. Jo, J. Lee, B. Min, M. Jo, H. Yeom, S.-Y. Choi, J. Shim, and J. Kim, Nearly room temperature ferromagnetism in a magnetic metal-rich van der Waals metal, Sci. Adv. 6, eaay8912 (2020).
  * May  _et al._ [2019] A. May, D. Ovchinnikov, Q. Zheng, R. Hermann, S. Calder, B. Huang, Z. Fei, Y. Liu, X. Xu, and M. Mcguire, Ferromagnetism Near Room Temperature in the Cleavable van der Waals Crystal Fe5GeTe2, ACS Nano 13, 4436 (2019).
  * Zhang  _et al._ [2022] G. Zhang, F. Guo, H. Wu, X. Wen, L. Yang, W. Jin, W. Zhang, and H. Chang, Above-room-temperature strong intrinsic ferromagnetism in 2D van der Waals Fe3GaTe2 with large perpendicular magnetic anisotropy, Nat. Commun. 13, 5067 (2022).
  * O’hara  _et al._ [2018] D. O’hara, T. Zhu, A. Trout, A. Ahmed, Y. Luo, C. Lee, M. Brenner, S. Rajan, J. Gupta, D. Mccomb, and R. Kawakami, Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit, Nano. Lett. 18, 3125 (2018).
  * Jiang  _et al._ [2018] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Controlling magnetism in 2D CrI3 by electrostatic doping, Nat. Nanotechnol. 13, 549 (2018).
  * You  _et al._ [2021] J.-Y. You, X.-J. Dong, B. Gu, and G. Su, Electric field induced topological phase transition and large enhancements of spin-orbit coupling and Curie temperature in two-dimensional ferromagnetic semiconductors, Phys. Rev. B 103, 104403 (2021).
  * Zhao  _et al._ [2021] F. Zhao, T. Cao, and S. G. Louie, Topological Phases in Graphene Nanoribbons Tuned by Electric Fields, Phys. Rev. Lett. 127, 166401 (2021).
  * Shi  _et al._ [2019] J. Shi, Y. Ou, M. A. Migliorato, H. Wang, H. Li, Y. Zhang, Y. Gu, and M. Zou, Tuning the electronic structure of GeC/WS2 van der Waals heterostructure by electric field and strain: A first principles study, Comput. Mater. Sci. 160, 301 (2019).
  * Hao  _et al._ [2019] Q. Hao, H. Yi, H. Su, B. Wei, Z. Wang, Z. Lao, Y. Chai, Z. Wang, C. Jin, J. Dai, and W. Zhang, Phase Identification and Strong Second Harmonic Generation in Pure $\epsilon$-InSe and Its Alloys, Nano Lett. 19, 2634 (2019).
  * Wang  _et al._ [2018] B. Wang, Q. Wu, Y. Zhang, Y. Guo, X. Zhang, Q. Zhou, S. Dong, and J. Wang, High Curie-temperature intrinsic ferromagnetism and hole doping-induced half-metallicity in two-dimensional scandium chlorine monolayers, Nanoscale Horiz. 3, 551 (2018).
  * Feng  _et al._ [2017] S. Feng, Z. Lin, X. Gan, R. Lv, and M. Terrones, Doping two-dimensional materials: ultra-sensitive sensors, band gap tuning and ferromagnetic monolayers, Nanoscale Horiz. 2, 72 (2017).
  * Fang  _et al._ [2022] J. Fang, H. Song, B. Li, Z. Zhou, J. Yang, B. Lin, Z. Liao, and Z. Wei, Large unsaturated magnetoresistance of 2D magnetic semiconductor Fe-SnS2 homojunction, J. Semicond. 43, 092501 (2022).
  * Zhao  _et al._ [2022] D. Zhao, C. Zhang, C. Zhang, W. Ji, S. Li, and P. Wang, Magnetic tuning in a novel half-metallic Ir2TeI2 monolayer, J. Semicond. 43, 052001 (2022).
  * González-Herrero  _et al._ [2016] H. González-Herrero, J. M. Gómez-Rodríguez, P. Mallet, M. Moaied, J. J. Palacios, C. Salgado, M. M. Ugeda, J.-Y. Veuillen, F. Yndurain, and I. Brihuega, Atomic-scale control of graphene magnetism by using hydrogen atoms, Sci. 352, 437 (2016).
  * Datta  _et al._ [2017] S. Datta, Y. Cai, I. Yudhistira, Z. Zeng, Y.-W. Zhang, H. Zhang, S. Adam, J. Wu, and K. P. Loh, Tuning magnetoresistance in molybdenum disulphide and graphene using a molecular spin transition, Nat. Commun. 8, 94 (2017).
  * Pathirage  _et al._ [2023] V. Pathirage, S. Khatun, S. Lisenkov, K. Lasek, J. Li, S. Kolekar, M. Valvidares, P. Gargiani, Y. Xin, I. Ponomareva, and M. Batzill, 2D materials by Design: Intercalation of Cr or Mn between two VSe2 van der Waals Layers, Nano Lett. , e2004557 (2023).
  * Zhou  _et al._ [2021] J. Zhou, Z. Lin, H. Ren, X. Duan, I. Shakir, Y. Huang, and X. Duan, Layered Intercalation Materials, Adv. Mater. 33, 9579 (2021).
  * Chen  _et al._ [2019a] W. Chen, Z. Sun, Z. Wang, L. Gu, X. Xu, S. Wu, and C. Gao, Direct observation of van der Waals stacking-dependent interlayer magnetism, Sci. 366, 983 (2019a).
  * Zhang  _et al._ [2021a] X. Zhang, Q. Lu, W. Liu, W. Niu, J. Sun, J. CoOk, M. Vaninger, P. F. Miceli, D. J. Singh, S.-W. Lian, T.-R. Chang, X. He, J. Du, L. He, R. Zhang, G. Bian, and Y. Xu, Room-temperature intrinsic ferromagnetism in epitaxial CrTe2 ultrathin films, Nat. Commun. 12, 2492 (2021a).
  * Kumar  _et al._ [2016] P. Kumar, Y. S. Chauhan, A. Agarwal, and S. Bhowmick, Thickness and Stacking Dependent Polarizability and Dielectric Constant of Graphene-Hexagonal Boron Nitride Composite Stacks, J. Phys. Chem. C 120, 17620 (2016).
  * Chen  _et al._ [2023] Z. Chen, X. Liu, X. Li, P. Gao, Z. Li, W. Zhu, H. Wang, and X. Li, Large tunneling magnetoresistance in spin-filtering 1T-MnSe2/h-BN van der Waals magnetic tunnel junction, Nanoscale 15, 8447 (2023).
  * Wu  _et al._ [2023] B. Wu, J. Yang, S. Liu, S. Fang, Z. Liu, Z. Lin, J. Shi, W. Yang, Z. Luo, C. Wang, H. Du, J. Yang, and J. Lu, Layer-Dependent Magnetoresistance and Spin-Transfer Torque in MnSe2-Based Magnetic Tunnel Junctions, Phys. Rev. Applied 19, 064008 (2023).
  * Dong  _et al._ [2020] X.-J. Dong, J.-Y. You, Z. Zhang, B. Gu, and G. Su, Great enhancement of Curie temperature and magnetic anisotropy in two-dimensional van der Waals magnetic semiconductor heterostructures, Phys. Rev. B 102, 144443 (2020).
  * Chen  _et al._ [2019b] S. Chen, C. Huang, H. Sun, J. Ding, P. Jena, and E. Kan, Boosting the Curie Temperature of Two-Dimensional Semiconducting CrI3 Monolayer through van der Waals Heterostructures, J. Phys. Chem. C 123, 17987 (2019b).
  * Li  _et al._ [2023a] Y. Li, J. Deng, Y.-F. Zhang, X. Jin, W.-H. Dong, J.-T. Sun, J. Pan, and S. Du, Nonvolatile electrical control of spin polarization in the 2D bipolar magnetic semiconductor VSeF, npj Comput. Mater. 9, 50 (2023a).
  * Ren  _et al._ [2016] M. Ren, M. Li, C. Zhang, M. Yuan, P. Li, F. Li, W. Ji, and X. Chen, Band structures in silicene on monolayer gallium phosphide substrate, Solid State Commun. 239, 32 (2016).
  * Caglayan  _et al._ [2022] R. Caglayan, H. E. Guler, and Y. Mogulkoc, An analysis of Schottky barrier in silicene/Ga2SeS heterostructures by employing electric field and strain, Phys. Chem. Chem. Phys. 24, 10210 (2022).
  * Yuan  _et al._ [2022] G. Yuan, Z. Cheng, Y. Cheng, W. Duan, H. Lv, Z. Liu, C. Han, and X. Ma, Highly Sensitive Band Alignment of the Graphene/MoSi2N4 Heterojunction via an External Electric Field, ACS Appl. Electron. Mater. 4, 2897 (2022).
  * Yuan  _et al._ [2023] J. Yuan, F. Wang, Z. Zhang, B. Song, S. Yan, M.-H. Shang, C. Tong, and J. Zhou, Effects of electric field and interlayer coupling on Schottky barrier of germanene/MoSSe vertical heterojunction, Phys. Rev. B 108, 125404 (2023).
  * Hong  _et al._ [2022] J. Hong, C.-J. Kang, and J. Kim, Role of electronic correlations in room-temperature ferromagnetism of monolayer MnSe2, Phys. Rev. B 106, 195428 (2022).
  * Yang  _et al._ [2023] J. Yang, X. Wang, S. Li, X. Wang, M. Pan, M. Ai, H. Yuan, X. Peng, R. Wang, Q. Li, F. Zheng, and P. Zhang, Robust Two-Dimensional Ferromagnetism in Cr5Te8/CrTe2 Heterostructure with Curie Temperature above 400 K, ACS Nano , 23160 (2023).
  * Zhu  _et al._ [2023] W. Zhu, C. Song, L. Han, H. Bai, C. Chen, and F. Pan, Interface-enhanced room-temperature Curie temperature in CrPS4/graphene van der Waals heterostructure, Phys. Rev. B 108, l100406 (2023).
  * Lu  _et al._ [2023] X. Lu, Y. Deng, Y. Pei, Z. Chen, and G. Wang, Recent advances in NiO/Ga2O3 heterojunctions for power electronics, J. Semicond. 44, 061802 (2023).
  * Huang  _et al._ [2019] S. Huang, G. Zhang, F. Fan, C. Song, F. Wang, Q. Xing, C. Wang, H. Wu, and H. Yan, Strain-tunable van der Waals interactions in few-layer black phosphorus, Nat. Commun. 10, 94 (2019).
  * Li  _et al._ [2020] Z. Li, Y. Lv, L. Ren, J. Li, L. Kong, Y. Zeng, Q. Tao, R. Wu, H. Ma, B. Zhao, D. Wang, W. Dang, K. Chen, L. Liao, X. Duan, X. Duan, and Y. Liu, Efficient strain modulation of 2D materials via polymer encapsulation, Nat. Commun. 11, 94 (2020).
  * Xu and Zhu [2018] Z. Xu and H. Zhu, Two-Dimensional Manganese Nitride Monolayer with Room Temperature Rigid Ferromagnetism under Strain, J. Phys. Chem. C 122, 14918 (2018).
  * Wu  _et al._ [2019] Z. Wu, J. Yu, and S. Yuan, Strain-tunable magnetic and electronic properties of monolayer CrI3, Phys. Chem. Chem. Phys. 21, 7750 (2019).
  * Wei [2023] D. Wei, The room temperature ferromagnetism in highly strained two-dimensional magnetic semiconductors, J. Semicond. 44, 040401 (2023).
  * Bertolazzi  _et al._ [2011] S. Bertolazzi, J. Brivio, and A. Kis, Stretching and Breaking of Ultrathin MoS2, ACS Nano 5, 9703 (2011).
  * Peng  _et al._ [2014] R. Peng, H. C. Xu, S. Y. Tan, H. Y. Cao, M. Xia, X. P. Shen, Z. C. Huang, C. Wen, Q. Song, T. Zhang, B. P. Xie, X. G. Gong, and D. L. Feng, Tuning the band structure and superconductivity in single-layer FeSe by interface engineering, Nat. Commun. 5, 94 (2014).
  * Zhang  _et al._ [2016] P. Zhang, X.-L. Peng, T. Qian, P. Richard, X. Shi, J.-Z. Ma, B. B. Fu, Y.-L. Guo, Z. Q. Han, S. C. Wang, L. L. Wang, Q.-K. Xue, J. P. Hu, Y.-J. Sun, and H. Ding, Observation of high-TC superconductivity in rectangular FeSe/SrTiO3(110) monolayers, Phys. Rev. B 94, 104510 (2016).
  * Cocco  _et al._ [2010] G. Cocco, E. Cadelano, and L. Colombo, Gap opening in graphene by shear strain, Phys. Rev. B 81, 241412 (2010).
  * Idrees  _et al._ [2023] M. Idrees, M. Khurami, B. Amin, Y. Chen, and X. Yan, Interfacial characteristics, metal-semiconductor contact and optical properties of CSe and BX (X=B, As and Sb) monolayers, Mat. Sci. Semicon. Proc. 163, 107593 (2023).
  * Hu and Ouyang [2021] H. Hu and G. Ouyang, First-principles calculations of interface engineering for 2D $alpha$-In2Se3-based van der Waals multiferroic heterojunctions, Appl. Surf. Sci. 545, 149024 (2021).
  * Noesges  _et al._ [2020] B. A. Noesges, T. Zhu, J. J. Repicky, S. Yu, F. Yang, J. A. Gupta, R. K. Kawakami, and L. J. Brillson, Chemical migration and dipole formation at van der Waals interfaces between magnetic transition metal chalcogenides and topological insulators, Phys. Rev. Materials 4, 054001 (2020).
  * Xiao  _et al._ [2023] C. Xiao, W. Wu, H. Wang, Y.-X. Huang, X. Feng, H. Liu, G.-Y. Guo, Q. Niu, and S. A. Yang, Time-Reversal-Even Nonlinear Current Induced Spin Polarization, Phys. Rev. Lett. 130, 166302 (2023).
  * Eren  _et al._ [2019] I. Eren, F. Iyikanat, and H. Sahin, Defect tolerant and dimension dependent ferromagnetism in MnSe2, Phys. Chem. Chem. Phys. 21, 16718 (2019).
  * Xie  _et al._ [2021] W.-Q. Xie, Z.-W. Lu, C.-C. He, X.-B. Yang, and Y.-J. Zhao, Theoretical study of tunable magnetism of two-dimensional MnSe2 through strain, charge, and defect, J. Phys.: Condens. Matter 33, 215803 (2021).
  * Li  _et al._ [2022b] Q. Li, C. xin Zhang, D. Wang, K.-Q. Chen, and L.-M. Tang, Giant valley splitting in a MoTe2/MnSe2 van der Waals heterostructure with room-temperature ferromagnetism, Mater. Adv. 3, 2927 (2022b).
  * Kan  _et al._ [2014] M. Kan, S. Adhikari, and Q. Sun, Ferromagnetism in MnX2 (X = S, Se) monolayers, Phys. Chem. Chem. Phys. 16, 4990 (2014).
  * He  _et al._ [2022] J. He, S. Li, L. Zhou, and T. Frauenheim, Ultrafast Light-Induced Ferromagnetic State in Transition Metal Dichalcogenides Monolayers, J. Phys. Chem. Lett. 13, 2765 (2022).
  * Heyd  _et al._ [2003] J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003).
  * Dong  _et al._ [2019] X.-J. Dong, J.-Y. You, B. Gu, and G. Su, Strain-Induced Room-Temperature Ferromagnetic Semiconductors with Large Anomalous Hall Conductivity in Two-Dimensional Cr2Ge2Se6, Phys. Rev. Appl. 12, 014020 (2019).
  * Zhang  _et al._ [2021b] Z. Zhang, J.-Y. You, X.-Y. Ma, B. Gu, and G. Su, Kagome quantum anomalous Hall effect with high Chern number and large band gap, Phys. Rev. B 103, 014410 (2021b).
  * O’Neill  _et al._ [2022] A. O’Neill, S. Rahman, Z. Zhang, P. Schoenherr, T. Yildirim, B. Gu, G. Su, Y. Lu, and J. Seidel, Enhanced Room Temperature Ferromagnetism in Highly Strained 2D Semiconductor Cr2Ge2Te6, ACS Nano 17, 735 (2022).
  * Li  _et al._ [2022c] H. Li, Y. Liu, Z. Bai, J. Xiong, F. Liu, G. Zhou, T. Qing, S. Zhang, and J. Lu, Ohmic contact in graphene and hexagonal III-V monolayer (GaP, GaAs, InP, and InAs) van der Waals heterostructures: Role of electric field, Phys. Lett. A 433, 128029 (2022c).
  * Parmar  _et al._ [2022] P. Parmar, S. Khengar, and P. Thakor, Structural And Electronic Properties of PtSe2/GaP Heterostructure, Mater. Today:. Proc. 67, 161 (2022).
  * Wei  _et al._ [2022] T. Wei, X. Li, Z. Li, W. Yang, Y. Wu, Z. Xing, and S. Lu, Interfacial dynamics of GaP/Si(100) heterostructure grown by molecular beam epitaxy, J. Semicond. 43, 122101 (2022).
  * Li  _et al._ [2023b] W. Li, X. Zhang, J. Yang, S. Zhou, C. Song, P. Cheng, Y.-Q. Zhang, B. Feng, Z. Wang, Y. Lu, K. Wu, and L. Chen, Emergence of ferroelectricity in a nonferroelectric monolayer, Nat. Commun. 14, 2757 (2023b).
  * Wani  _et al._ [2022] A. F. Wani, B. Rani, S. Dhiman, U. B. Sharopov, and K. Kaur, Sih monolayer: A promising two-dimensional thermoelectric material, Int. J. Energ. Res. 46, 10885 (2022).
  * Kresse and Furthmüller [1996] G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  * Perdew  _et al._ [1996] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).
  * Blöchl [1994] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  * Grimme  _et al._ [2010] S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, A consistent and accurate ab initio parametrization of density functional dispersion correction DFT-D for the 94 elements H-Pu, J. Chem. Phys. 132, 154104 (2010).
  * Mostofi  _et al._ [2008] A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, and Nicola Marzari, wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178, 685 (2008).
  * Mostofi  _et al._ [2014] A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza, D. Vanderbilt, and Nicola Marzari, An updated version of wannier90: A tool for obtaining maximally-localised wannier functions, Comput. Phys. Commun. 185, 2309 (2014).
  * Wu  _et al._ [2018] Q. Wu, S. Zhang, H.-F. Song, M. Troyer, and Alexey A. Soluyanov, Wanniertools: An open-source software package for novel topological materials, Comput. Phys. Commun. 224, 405 (2018).
  * Qi  _et al._ [2023] Y. Qi, M. A. Sadi, D. Hu, M. Zheng, Z. Wu, Y. Jiang, and Y. P. Chen, Recent Progress in Strain Engineering on Van der Waals 2D Materials: Tunable Electrical, Electrochemical, Magnetic, and Optical Properties, Adv. Mater. 35, 2205714 (2023).
  * [84] See supplemental material .
  * Oppeneer  _et al._ [1992] P. M. Oppeneer, T. Maurer, J. Sticht, and J. Kübler, Ab initio calculated magneto-optical Kerr effect of ferromagnetic metals: Fe and Ni, Phys. Rev. B 45, 10924 (1992).