# Cycle integrals of meromorphic modular forms and Siegel theta functions

Markus Schwagenscheidt

###### Abstract.

We study meromorphic modular forms associated with positive definite binary
quadratic forms and their cycle integrals along closed geodesics in the
modular curve. We show that suitable linear combinations of these meromorphic
modular forms have rational cycle integrals. Along the way, we evaluate the
cycle integrals of the Siegel theta function associated with an even lattice
of signature $(1,2)$ in terms of Hecke’s indefinite theta functions, which is
of independent interest.

##  1\. Introduction

Let $\mathcal{Q}_{d}$ be the set of all (positive definite if $d<0$) integral
binary quadratic forms of discriminant $d$. For $k\in\mathbb{N}$ with $k\geq
2$ we consider the functions

$f_{k,d}(z)=\frac{|d|^{k-1/2}}{\pi}\sum_{Q\in\mathcal{Q}_{d}}\frac{1}{Q(z,1)^{k}},$

which transform like modular forms of weight $2k$ for
$\Gamma=\operatorname{PSL}_{2}(\mathbb{Z})$. These functions were first
studied by Zagier [12] for $d>0$, in which case they are cusp forms, and by
Bengoechea for $d<0$, in which case they are meromorphic modular forms. We
obtain a refinement $f_{k,P}$ of $f_{k,d}$ by summing only over quadratic
forms in the equivalence class of a fixed form $P$.

In [1] we computed the cycle integrals of the meromorphic modular forms
$f_{k,P}$ associated with positive definite quadratic forms $P$. These cycle
integrals are defined by111If $f_{k,P}$ has poles on $S_{A}$ then the cycle
integral can be defined using the Cauchy principal value as in [11].

$\mathcal{C}_{A}(f_{k,P})=\int_{\Gamma_{A}\backslash
S_{A}}f_{k,P}(z)A(z,1)^{k-1}dz,$

where $A=[a,b,c]$ is an indefinite integral binary quadratic form of positive
non-square discriminant $D>0$, $\Gamma_{A}$ denotes the stabilizer of $A$, and
$S_{A}=\\{z\in\mathbb{H}\,:\,a|z|^{2}+b\Re(z)+c=0\\}$ is the geodesic
corresponding to $A$. We let $\mathcal{C}_{D}(f_{k,P})$ be the $D$-th trace of
cycle integrals, that is, the sum of $\mathcal{C}_{A}(f_{k,P})$ over all
classes $A\in\mathcal{Q}_{D}/\Gamma$, and we let $M_{3/2-k}^{!}$ denote the
space of weakly holomorphic modular forms of weight $3/2-k$ for
$\Gamma_{0}(4)$ satisfying Kohnen’s plus space condition $a_{g}(n)=0$ unless
$(-1)^{k+1}n\equiv 0,1\ \,\left(\mathrm{mod}\,4\right)$. Then the main result
of [1] is as follows.

######  Theorem 1.1 (Theorem 1.1 in [1]).

Let $k\geq 2$ be even and let $g\in M_{3/2-k}^{!}$. Suppose that the
coefficients $a_{g}(-D)$ are rational for $D>0$ and vanish if $D$ is a square.
Then the linear combinations of traces of cycle integrals

$\sum_{D>0}a_{g}(-D)\mathcal{C}_{D}(f_{k,P})$

of the individual meromomorphic modular forms $f_{k,P}$ are rational.

In [11] we gave some refinements of this result and made a conjecture
concerning the rationality of the individual cycle integrals of linear
combinations of various $f_{k,d}$’s, which can be viewed as a dual version of
Theorem 1.1 for odd $k$. Our goal is to prove this conjecture.

######  Theorem 1.2 (Conjecture 2.5 in [11]).

Let $k\geq 3$ be odd and let $g\in M_{3/2-k}^{!}$. Suppose that the
coefficients $a_{g}(d)$ for $d<0$ are rational. Then the individual cycle
integrals

$\mathcal{C}_{A}\left(\sum_{d<0}a_{g}(d)f_{k,d}(z)\right)$

of linear combinations of the meromorphic $f_{k,d}$ are rational.

In [11] this result was stated as a conjecture for higher level, even and odd
$k$, and twisted versions of $f_{k,d}$. Our arguments also work in this
general setup, see Section 5.

The proof of Theorem 1.2 follows a method of Bruinier, Ehlen, and Yang (see
the proof of [8, Theorem 5.4]). We use the well-known fact that $f_{k,d}$ can
be written as a regularized theta lift. Then we interchange the cycle integral
with the regularized integral, which leads us to study the cycle integrals of
the Siegel theta function $\Theta_{L}(\tau,z)$ associated with an even lattice
$L$ of signature $(1,2)$. The evaluation of these cycle integrals is the main
novelty of this work. Let us describe the result in more detail.

The indefinite quadratic form $A$ defines a vector of negative length in the
lattice $L$, and hence yields sublattices $I=L\cap(\mathbb{Q}A)^{\perp}$ and
$N=L\cap(\mathbb{Q}A)$ of signature $(1,1)$ and $(0,1)$, respectively. Hecke
[9] constructed an indefinite theta series $\vartheta_{I}(\tau)$ associated
with the lattice $I$, which is a cusp form of weight $1$ for the Weil
representation $\rho_{I}$, see (3.3). Moreover, there is a classical
holomorphic weight $3/2$ theta series $\Theta_{3/2,N}(\tau)$ associated to the
unary lattice $N$, see (3.1).

######  Theorem 1.3.

Suppose that $L=I\oplus N$. Then we have

$\mathcal{C}_{A}\left(\frac{\partial}{\partial
z}\Theta_{L}(\tau,z)\right)\stackrel{{\scriptstyle\cdot}}{{=}}\vartheta_{I}(\tau)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}.$

Here $\stackrel{{\scriptstyle\cdot}}{{=}}$ means equality up to an explicit
non-zero constant factor. For the precise version of this result we refer to
Theorem 3.1. The idea of the proof is that $\Theta_{L}(\tau,z)$ essentially
splits as a product of two Siegel theta functions $\Theta_{I}(\tau,z)$ and
$\Theta_{N}(\tau)$ along the geodesic $S_{A}$, and the cycle integral of
$\Theta_{I}(\tau,z)$ yields the Hecke theta series $\vartheta_{I}(\tau)$.

Combining the above evaluation of the cycle integral of the Siegel theta
function with the theta lift realization of $f_{k,d}$, we obtain a
representation of $\mathcal{C}_{A}(f_{k,d})$ as a regularized Petersson inner
product, which can in turn be evaluated using the theory of harmonic Maass
forms. This gives an explicit formula for the cycle integrals appearing in
Theorem 1.2 (see Theorem 4.1), from which their rationality can easily be
deduced.

This paper is organized as follows. In Section 2 we recall the theta lift
realization of $f_{k,d}$. In Section 3 we evaluate the cycle integrals of the
Siegel theta function, which is the technical heart of this work. In Section 4
we put everything together and deduce an explicit rational formula for the
cycle integrals of $f_{k,d}$, which also implies Theorem 1.2. Finally, in
Section 5 we sketch the proof of a higher level version of Theorem 1.2.

### Acknowledments

The author was supported by SNF project PZ00P2_202210.

##  2\. Meromorphic modular forms as theta lifts

In this section we recall the realization of the meromophic modular form
$f_{k,d}$ as a theta lift of a harmonic Maass form, following [3]. We consider
the even lattice

$L=\left\\{X=\begin{pmatrix}b&c\\\
-a&-b\end{pmatrix}:a,b,c\in\mathbb{Z}\right\\}$

with the quadratic form $q(X)=\det(X)$. It has signature $(1,2)$, and its dual
lattice is given

$L^{\prime}=\left\\{X=\begin{pmatrix}b/2&c\\\
-a&-b/2\end{pmatrix}:a,b,c\in\mathbb{Z}\right\\}.$

The group $\Gamma=\operatorname{PSL}_{2}(\mathbb{Z})$ acts on $L$ via
$g.X=gXg^{-1}$. We can view $L^{\prime}$ as the set of integral binary
quadratic forms $[a,b,c]$, where $-4q(X)=b^{2}-4ac$ corresponds to the
discriminant.

Let $\operatorname{Gr}(L)$ be the Grassmannian of positive definite lines in
$V(\mathbb{R})=L\otimes\mathbb{R}$. We can identify $\operatorname{Gr}(L)$
with $\mathbb{H}$ by sending $z=x+iy\in\mathbb{H}$ to the positive line
generated by

$X(z)=\frac{1}{\sqrt{2}y}\begin{pmatrix}-x&|z|^{2}\\\ -1&x\end{pmatrix}.$

This bijection is compatible with the actions of
$\operatorname{PSL}_{2}(\mathbb{R})$, in the sense that $g.X(z)=X(gz)$ for any
$g\in\operatorname{PSL}_{2}(\mathbb{R})$. We also consider the vectors

$U_{1}(z)=\frac{1}{\sqrt{2}y}\begin{pmatrix}x&-x^{2}+y^{2}\\\
1&-x\end{pmatrix},\qquad U_{2}(z)=\frac{1}{\sqrt{2}y}\begin{pmatrix}y&-2xy\\\
0&-y\end{pmatrix},$

which, together with $X(z)$, form an orthogonal basis of $V(\mathbb{R})$. For
each fixed $z=x+iy\in\mathbb{H}$ we define the polynomials

$\displaystyle Q_{X}(z)$
$\displaystyle=-\sqrt{2}y(X,U_{1}(z)+iU_{2}(z))=az^{2}+bz+c,$ $\displaystyle
p_{X}(z)$ $\displaystyle=\sqrt{2}(X,X(z))=\frac{1}{y}(a|z|^{2}+bx+c),$

on $V(\mathbb{R})$. They satisfy the invariances

$\displaystyle p_{X}(gz)=p_{g^{-1}.X}(z),\qquad
Q_{X}(gz)=(cz+d)^{-2}Q_{g^{-1}.X}(z),$ (2.1)

for each $g=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{PSL}_{2}(\mathbb{R})$. For brevity
we write $X_{z}$ and $X_{z^{\perp}}$ for the orthogonal projection of $X$ to
the positive line generated by $X(z)$ and to the negative plane $X(z)^{\perp}$
spanned by $U_{1}(z),U_{2}(z)$, respectively. We have the useful relations

$\displaystyle q(X_{z})=\frac{1}{4}p_{X}(z)^{2},\qquad
q(X_{z^{\perp}})=-\frac{1}{4}y^{-2}|Q_{X}(z)|^{2}.$ (2.2)

Since $L$ has signature $(1,2)$, the Siegel theta function

$\Theta_{L}(\tau,z)=v\sum_{X\in
L^{\prime}}e\left(q(X_{z})\tau+q(X_{z^{\perp}})\overline{\tau}\right)\operatorname{\mathfrak{e}}_{X}$

transforms like a modular form of weight $-1/2$ in $\tau$ for the Weil
representation $\rho_{L}$ associated with $L$. Moreover, it is
$\Gamma$-invariant in $z$. Here
$\operatorname{\mathfrak{e}}_{X}=\operatorname{\mathfrak{e}}_{X+L}$ are the
standard basis vectors of the group ring $\mathbb{C}[L^{\prime}/L]$. The
derivative $R_{0}\Theta_{L}(\tau,z)=2i\frac{\partial}{\partial
z}\Theta_{L}(\tau,z)$ transforms like a modular form of weight $2$ in $z$. It
is explicitly given by

$\displaystyle R_{0}\Theta_{L}(\tau,z)=2\pi v^{2}\sum_{X\in
L^{\prime}}p_{X}(z)y^{-2}\overline{Q_{X}(z)}e\left(q(X_{z})\tau+q(X_{z^{\perp}})\overline{\tau}\right)\operatorname{\mathfrak{e}}_{X}.$
(2.3)

Let $A_{k,L}$ denote the space of all $\mathbb{C}[L^{\prime}/L]$-valued
functions on $\mathbb{H}$ which transform like modular forms of weight $k$ for
$\rho_{L}$. If $M\subset L$ is a sublattice of finite index, we have natural
maps $A_{k,L}\to A_{k,M},f\mapsto f_{M}$, and $A_{k,M}\to A_{k,L},g\mapsto
g^{L}$, given by

$\displaystyle(f_{M})_{\gamma}=\begin{cases}f_{\gamma}&\text{if $\gamma\in
L^{\prime}/M$,}\\\ 0,&\text{if $\gamma\notin
L^{\prime}/M$,}\end{cases}\qquad\text{and}\qquad(g^{L})_{\gamma}=\sum_{\beta\in
L/M}g_{\beta+\gamma}.$ (2.4)

These maps are adjoint with respect to the bilinear pairings on
$\mathbb{C}[L^{\prime}/L]$ and $\mathbb{C}[M^{\prime}/M]$. Moreover, the
Siegel theta functions for $M$ and $L$ are related by
$\Theta_{L}(\tau,z)=\Theta_{M}(\tau,z)^{L}$.

For a smooth modular form $f$ of weight $1/2$ for $\overline{\rho}_{L}$ the
regularized theta lift is defined by

$\Phi_{L}(f,z)=\int_{\mathcal{F}}^{\operatorname{reg}}f(\tau)\cdot\Theta_{L}(\tau,z)d\mu(\tau),$

where $\mathcal{F}$ is a fundamental domain for $\Gamma\backslash\mathbb{H}$,
the dot denotes the natural bilinear pairing on $\mathbb{C}[L^{\prime}/L]$,
and $d\mu(\tau)$ is the invariant measure. The integral is regularized as
explained in [7].

We let $P_{3/2-k,d}(\tau)$ be the unique harmonic Maass form of weight $3/2-k$
for the dual Weil representation $\overline{\rho}_{L}$ with principal part
$q^{d}\operatorname{\mathfrak{e}}_{d}+O(1)$.

######  Proposition 2.1 (Proposition 4.1 in [3]).

For odd $k\geq 3$ we have

$f_{k,d}(z)=c_{k}\cdot
R_{0}^{k}\Phi_{L}\left(R_{3/2-k}^{(k-1)/2}P_{3/2-k,d},z\right)$

with the constant $c_{k}=(2^{k+1}\pi^{(k+1)/2}(k-1)!)^{-1}$. Here
$R_{\kappa}^{n}=R_{\kappa+2n-2}\circ\dots\circ R_{\kappa}$ is an iterated
version of the raising operator $R_{\kappa}=2i\frac{\partial}{\partial
z}+\kappa y^{-1}$.

##  3\. Cycle integrals of Siegel theta functions

In this section we compute the cycle integrals of the derivative of the Siegel
theta function $\Theta_{L}(\tau,z)$. Throughout this section we let

$A=\begin{pmatrix}b/2&c\\\ -a&-b/2\end{pmatrix}\in L^{\prime},\qquad
D=-4q(A)=b^{2}-4ac>0,$

be an indefinite integral binary quadratic form of positive non-square
discriminant $D$. We let $S_{A}=\\{z\in\mathbb{H}\,:\,a|z|^{2}+bx+c=0\\}$ be
the corresponding geodesic semi-circle in $\mathbb{H}$. Changing $A$ to $-A$
only changes the sign of the cycle integral, so we will assume $a>0$ from now
on. Then $S_{A}$ is oriented counter-clockwise. Since $A$ is as a vector of
negative norm in $L^{\prime}$, we can define two sublattices

$I=L\cap(\mathbb{Q}A)^{\perp},\qquad N=L\cap(\mathbb{Q}A),$

of signature $(1,1)$ and $(0,1)$, respectively. Note that $I$ is anisotropic
since $A$ has non-square discriminant, and that $N^{-}=(N,-q)$ is a positive
definite one-dimensional lattice. We define the weight $3/2$ unary theta
function corresponding to $N$ (or rather $N^{-}$) by

$\displaystyle\Theta_{3/2,N}(\tau)=\sum_{W\in
N^{\prime}}(W,A)e(-q(W)\tau)\operatorname{\mathfrak{e}}_{W}.$ (3.1)

It is a cusp form of weight $3/2$ for the dual Weil representation
$\overline{\rho}_{N}$.

Let $w=\frac{-b-\sqrt{D}}{2a}$ be the real endpoint of the geodesic $S_{A}$,
and consider the vector

$\displaystyle Y_{0}=\begin{pmatrix}-w&w^{2}\\\ -1&w\end{pmatrix}\in
I\otimes\mathbb{R}.$ (3.2)

We define the _Hecke theta series_ associated with the signature $(1,1)$
lattice $I$ by

$\displaystyle\vartheta_{I}(\tau)=\sum_{\begin{subarray}{c}Y\in\Gamma_{I}\backslash
I^{\prime}\\\
q(Y)>0\end{subarray}}\frac{\operatorname{sgn}(Y,Y_{0})}{|\Gamma_{Y}|}e(q(Y)\tau)\operatorname{\mathfrak{e}}_{Y}.$
(3.3)

It is a cusp form of weight $1$ for $\rho_{I}$, which was first constructed by
Hecke [9] as a theta lift.

######  Theorem 3.1.

We have

$\mathcal{C}_{A}\big{(}R_{0}\Theta_{L}(\tau,\,\cdot\,)\big{)}=-\frac{4\pi}{\sqrt{D}}\left(\vartheta_{I}(\tau)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}\right)^{L},$

with the map $f^{L}$ defined in (2.4).

###### Proof.

Note that $I\oplus N$ is a sublattice of finite index in $L$, and we have

$\mathcal{C}_{A}(R_{0}\Theta_{L})=\mathcal{C}_{A}(R_{0}\Theta_{I\oplus
N})^{L},$

so we can assume $L=I\oplus N$ for now. We first split $R_{0}\Theta_{I\oplus
N}$ along the geodesic $S_{A}$ into tensor products of theta functions
associated with $I$ and $N$ . We can write $X\in(I\oplus N)^{\prime}$ uniquely
as $X=Y+W$ with $Y\in I^{\prime}$ and $W\in N^{\prime}$. Note that
$W=\frac{(W,A)}{(A,A)}A$. Hence, for $z\in S_{A}$ we have

$p_{W}(z)=0,\qquad W_{z}=0,\qquad
y^{-2}\overline{Q_{W}(z)}=-2(W,A)Q_{A}(z)^{-1}.$

Plugging this into the explicit formula (2.3) for $R_{0}\Theta_{I\oplus N}$,
we obtain for $z\in S_{A}$ the splitting

$\displaystyle R_{0}\Theta_{I\oplus N}(\tau,z)$
$\displaystyle=2\pi\left(\Theta_{I}^{*}(\tau,z)\otimes\overline{\Theta_{1/2,N}(\tau)}v^{1/2}\right)-4\pi\left(Q_{A}(z)^{-1}\Theta_{I}(\tau,z)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}\right),$

with the Siegel theta functions

$\displaystyle\Theta_{I}^{*}(\tau,z)$ $\displaystyle=v^{3/2}\sum_{Y\in
I^{\prime}}p_{Y}(z)y^{-2}\overline{Q_{Y}(z)}e(q(Y_{z})\tau+q(Y_{z^{\perp}})\overline{\tau})\operatorname{\mathfrak{e}}_{Y},$
$\displaystyle\Theta_{I}(\tau,z)$ $\displaystyle=v^{1/2}\sum_{Y\in
I^{\prime}}p_{Y}(z)e(q(Y_{z})\tau+q(Y_{z^{\perp}})\overline{\tau})\operatorname{\mathfrak{e}}_{Y},$

and the holomorphic weight $1/2$ and weight $3/2$ theta functions associated
with $N$.

It remains to compute the cycle integrals of $\Theta_{I}^{*}(\tau,z)$ and
$Q_{A}(z)^{-1}\Theta_{I}(\tau,z)$. Note that the geodesic $S_{A}$ can be
identified with the Grassmannian $\operatorname{Gr}(I)$ of positive lines in
$I\otimes\mathbb{R}$. Hence, the cycle integrals can be computed as done by
Hecke [9]. We give the details for $\Theta_{I}(\tau,z)$.

Recall that we assume $a>0$. The two real endpoints $w<w^{\prime}$ of $S_{A}$
are given by $w=\frac{-b-\sqrt{D}}{2a}$ and
$w^{\prime}=\frac{-b+\sqrt{D}}{2a}$. The matrix

$\sigma=\frac{a^{\frac{1}{2}}}{D^{\frac{1}{4}}}\begin{pmatrix}w^{\prime}&w\\\
1&1\end{pmatrix}\in\operatorname{PSL}_{2}(\mathbb{R})$

maps $0$ to $w$ and $i\infty$ to $w^{\prime}$ and hence maps the imaginary
axis (oriented from $i\infty$ to $0$) to $S_{A}$ (oriented counter-clockwise).
Note that $\sigma^{-1}.A=[0,-\sqrt{D},0]$. Modding out the stabilizer
$\Gamma_{A}=\Gamma_{I}$ in the sum over $Y$, and using the parametrization
$z=\sigma.it$ of $S_{A}$, we obtain

$\int_{\Gamma_{A}\backslash
S_{A}}\Theta_{I}(\tau,z)\frac{dz}{Q_{A}(z)}=\frac{v^{1/2}}{\sqrt{D}}\int_{0}^{\infty}\sum_{Y\in\Gamma_{I}\backslash
I^{\prime}}\frac{1}{|\Gamma_{Y}|}p_{Y}(\sigma.it)e(q(Y_{\sigma.it})\tau+q(Y_{\sigma.it^{\perp}})\overline{\tau})\operatorname{\mathfrak{e}}_{Y}\frac{dt}{t}.$

If we write $Y=\left(\begin{smallmatrix}\beta/2&\gamma\\\
-\alpha&-\beta/2\end{smallmatrix}\right)\in I^{\prime}$, a short compuation
shows that

$\sigma^{-1}.Y=\begin{pmatrix}0&\lambda\\\
\lambda^{\prime}&0\end{pmatrix},\qquad\lambda=\frac{a}{\sqrt{D}}(\alpha
w^{2}+\beta w+\gamma)\in\mathbb{Q}\big{(}\sqrt{D}\big{)},$

where $\lambda^{\prime}$ is the conjugate of $\lambda$ in
$\mathbb{Q}(\sqrt{D})$. Using (2.1) and (2.2) we find

$\displaystyle p_{Y}(\sigma.it)$ $\displaystyle=p_{\sigma^{-1}.Y}(it)=\lambda
t^{-1}-\lambda^{\prime}t,$ $\displaystyle q(Y_{\sigma.it^{\perp}})$
$\displaystyle=q((\sigma^{-1}.Y)_{it^{\perp}})=-\frac{1}{4}(\lambda
t^{-1}+\lambda^{\prime}t)^{2}.$

Hence, using
$q(Y_{z})\tau+q(Y_{z^{\perp}})\overline{\tau}=q(Y)\tau-2ivq(Y_{z^{\perp}})$,
we get

$\displaystyle\int_{\Gamma_{A}\backslash
S_{A}}\Theta_{I}(\tau,z)\frac{dz}{Q_{A}(z)}=\frac{v^{1/2}}{\sqrt{D}}\sum_{Y\in\Gamma_{I}\backslash
I^{\prime}}\frac{1}{|\Gamma_{Y}|}\left(\int_{0}^{\infty}(\lambda
t^{-1}-\lambda^{\prime}t)e^{-\pi v(\lambda
t^{-1}+\lambda^{\prime}t)^{2}}\frac{dt}{t}\right)e(q(Y)\tau)\operatorname{\mathfrak{e}}_{Y}.$

A standard computation222For example, one can use the $K$-Bessel function
$\sqrt{\frac{\pi}{2}}\frac{e^{-x}}{\sqrt{x}}=K_{1/2}(x)=\frac{1}{2}\sqrt{\frac{1}{2}z}\int_{0}^{\infty}\exp\left(-t-\frac{x^{2}}{4t}\right)\frac{dt}{t^{3/2}}$.
shows that the integral is given by

$\int_{0}^{\infty}(\lambda t^{-1}-\lambda^{\prime}t)e^{-\pi v(\lambda
t^{-1}+\lambda^{\prime}t)^{2}}\frac{dt}{t}=\frac{1}{2}v^{-1/2}(\operatorname{sgn}(\lambda)-\operatorname{sgn}(\lambda^{\prime}))e^{-2\pi
v|\lambda\lambda^{\prime}|}e^{-2\pi v\lambda\lambda^{\prime}},$

which equals $v^{-1/2}\operatorname{sgn}(\lambda)$ if
$q(Y)=-\lambda\lambda^{\prime}>0$ and vanishes otherwise. Using
$\sqrt{D}\lambda=a(\alpha w^{2}+\beta w+\gamma)=a(Y,Y_{0})$, with $Y_{0}$ as
in (3.2), we get $\operatorname{sgn}(\lambda)=\operatorname{sgn}(Y,Y_{0})$.
Summarizing, we obtain

$\displaystyle\int_{\Gamma_{A}\backslash
S_{A}}\Theta_{I}(\tau,z)\frac{dz}{Q_{A}(z)}$
$\displaystyle=\frac{1}{\sqrt{D}}\sum_{\begin{subarray}{c}Y\in\Gamma_{I}\backslash
I^{\prime}\\\
q(Y)>0\end{subarray}}\frac{\operatorname{sgn}(Y,Y_{0})}{|\Gamma_{Y}|}e(q(Y)\tau)\operatorname{\mathfrak{e}}_{Y}=\frac{1}{\sqrt{D}}\vartheta_{I}(\tau).$

The computation of the cycle integral of $\Theta_{I}^{*}(\tau,z)$ is very
similar. Using

$\int_{0}^{\infty}(\lambda^{2}t^{-2}-\lambda^{\prime 2}t^{2})e^{-\pi v(\lambda
t^{-1}+\lambda^{\prime}t)^{2}}\frac{dt}{t}=\left[\frac{1}{2\pi v}e^{-\pi
v(\lambda t^{-1}+\lambda^{\prime}t)^{2}}\right]_{0}^{\infty}=0,$

we see that we in fact have $\mathcal{C}_{A}(\Theta_{I}^{*})=0$. This finishes
the proof. ∎

######  Remark 3.2.

One can compute the cycle integrals of $R_{0}\Theta_{L}(\tau,z)$ along
infinite geodesics (if $D$ is a square) in a similar way. In this case, the
lattice $I$ is isotropic, and $\vartheta_{I}$ has to be replaced with a weight
$1$ Hecke Eisenstein series.

##  4\. Cycle integrals of meromorphic modular forms: The proof of Theorem 1.2

We are now ready to give an explicit evaluation of the cycle integrals
appearing in Theorem 1.2. As before, we let $A\in L^{\prime}$ be an indefinite
integral binary quadratic form of non-square discriminant $D=-4q(A)>0$ with
$a>0$, and we let $I=L\cap(\mathbb{Q}A)^{\perp}$ and $N=L\cap(\mathbb{Q}A)$ be
the corresponding indefinite and negative definite sublattices of $L$.
Moreover, we let $\vartheta_{I}$ be the Hecke theta series of weight $1$ for
$\rho_{I}$ defined in (3.3) and $\Theta_{3/2,N}$ the holomorphic theta
function of weight $3/2$ for $\overline{\rho}_{N}$ as in (3.1).

We choose a harmonic Maass form $\widetilde{\Theta}_{3/2,N}$ of weight $1/2$
for $\rho_{N}$ with

$\xi_{1/2}\widetilde{\Theta}_{3/2,N}=\frac{1}{\sqrt{D}}\Theta_{3/2,N},$

and we let $\widetilde{\Theta}_{3/2,N}^{+}$ be its holomorphic part. Moreover,
we require the Rankin-Cohen bracket

$\left[f,g\right]_{n}=\sum_{s=0}^{n}(-1)^{s}\binom{\kappa+n-1}{s}\binom{\ell+n-1}{n-s}f^{(n-s)}g^{(s)},\qquad\left(f^{(s)}=\frac{1}{(2\pi
i)^{s}}\frac{\partial^{s}}{\partial\tau^{s}}f\right),$

which maps modular forms $f$ of weight $\kappa$ and $g$ of weight $\ell$ to a
form of weight $\kappa+\ell+2n$.

######  Theorem 4.1.

Let $k\geq 3$ be odd and let $g\in M_{3/2-k,\overline{\rho}_{L}}^{!}$ be a
weakly holomorphic modular form of weight $3/2-k$ for $\overline{\rho}_{L}$.
Then we have

$\mathcal{C}_{A}\left(\sum_{d<0}a_{g}(d)f_{k,d}\right)=-(4D)^{(k-1)/2}\operatorname{CT}\left(g_{I\oplus
N}\cdot\left[\vartheta_{I},\widetilde{\Theta}_{3/2,N}^{+}\right]_{(k-1)/2}\right),$

with the Rankin-Cohen bracket in weights $\kappa=1$ and $\ell=1/2$, and
$g_{I\oplus N}$ defined in (2.4).

###### Proof.

We follow the idea of the proof of [8, Theorem 5.4]. For brevity we put
$f_{k,g}=\sum_{d<0}a_{g}(d)f_{k,d}$. Using Proposition 2.1 we can write
$f_{k,g}$ as a theta lift,

$f_{k,g}(z)=c_{k}\cdot R_{0}^{k}\Phi_{L}\left(R_{3/2-k}^{(k-1)/2}g,z\right),$

with the constant $c_{k}=(2^{k+1}\pi^{(k+1)/2}(k-1)!)^{-1}$. Now we take the
cycle integral $\mathcal{C}_{A}$ on both sides. As a preliminary step, we will
reduce the power of the outer raising operator $R_{0}^{k}$. It is well-known
that $\Phi_{L}=\Phi_{L}(R_{3/2-k}^{(k-1)/2}g,z)$ is an eigenform of the
Laplacian $\Delta_{0}$ with eigenvalue $k(1-k)$, see [7, Theorem 4.6]. Hence,
a repeated application of [5, Theorem 1.1] gives

$\mathcal{C}_{A}\big{(}R_{0}^{k}\Phi_{L}\big{)}=c_{k}^{\prime}\,\mathcal{C}_{A}\left(R_{0}\Phi_{L}\right),$

with the constant $c_{k}^{\prime}=D^{(k-1)/2}(k-1)!^{2}/(\frac{k-1}{2})!^{2}$.
Interchanging the cycle integral and the regularized integral gives

$\mathcal{C}_{A}\big{(}f_{k,g}\big{)}=c_{k}c_{k}^{\prime}\int_{\mathcal{F}}^{\operatorname{reg}}R_{3/2-k}^{(k-1)/2}g(\tau)\cdot\mathcal{C}_{A}\big{(}R_{0}\Theta_{L}(\tau,\,\cdot\,)\big{)}d\mu(\tau).$

Now we plug in the evaluation of the cycle integral from Theorem 3.1 to get

$\mathcal{C}_{A}\big{(}f_{k,g}\big{)}=c_{k}c_{k}^{\prime}c_{k}^{\prime\prime}\int_{\mathcal{F}}^{\operatorname{reg}}R_{3/2-k}^{(k-1)/2}g_{I\oplus
N}(\tau)\cdot\vartheta_{I}(\tau)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}d\mu(\tau),$

with the constant $c_{k}^{\prime\prime}=-4\pi/\sqrt{D}$. Here we used that the
maps $f^{L}$ and $g_{I\oplus N}$ are adjoint to each other. Next, we use the
“self-adjointness” of the raising operator to compute

$\displaystyle\mathcal{C}_{A}\big{(}f_{k,g}\big{)}$
$\displaystyle=(-1)^{(k-1)/2}c_{k}c_{k}^{\prime}c_{k}^{\prime\prime}\int_{\mathcal{F}}^{\operatorname{reg}}g_{I\oplus
N}(\tau)\cdot
R_{-1/2}^{(k-1)/2}\left(\vartheta_{I}(\tau)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}\right)d\mu(\tau)$
$\displaystyle=(-1)^{(k-1)/2}c_{k}c_{k}^{\prime}c_{k}^{\prime\prime}\int_{\mathcal{F}}^{\operatorname{reg}}g_{I\oplus
N}(\tau)\cdot\left(R_{1}^{(k-1)/2}\vartheta_{I}(\tau)\right)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}d\mu(\tau).$

In the last step we used that $\Theta_{3/2,N}$ is holomorphic. By [8,
Proposition 3.6] we have

$\left(R_{1}^{(k-1)/2}\vartheta_{I}(\tau)\right)\otimes\overline{\Theta_{3/2,N}(\tau)}v^{3/2}=c_{k}^{\prime\prime\prime}L_{k+1/2}\left[\vartheta_{I},\widetilde{\Theta}_{3/2,N}\right]_{(k-1)/2},$

with the constant
$c_{k}^{\prime\prime\prime}=4^{k-1}(-\pi)^{(k-1)/2}\sqrt{D}(\frac{k-1}{2})!^{2}/(k-1)!$.
We arrive at

$\mathcal{C}_{A}\big{(}f_{k,g}\big{)}=(-1)^{(k-1)/2}c_{k}c_{k}^{\prime}c_{k}^{\prime\prime}c_{k}^{\prime\prime\prime}\int_{\mathcal{F}}^{\operatorname{reg}}g_{I\oplus
N}(\tau)\cdot
L_{k+1/2}\left[\vartheta_{I},\widetilde{\Theta}_{3/2,N}\right]_{(k-1)/2}d\mu(\tau).$

Note that the constants in front simplify to
$(-1)^{(k-1)/2}c_{k}c_{k}^{\prime}c_{k}^{\prime\prime}c_{k}^{\prime\prime\prime}=-(4D)^{(k-1)/2}$.
Now a standard application of Stokes’ Theorem as in [8, Theorem 5.4] finishes
the proof. ∎

We remark that there is a similar formula for the linear combination of cycle
integrals of $f_{k,P}$ from Theorem 1.1, see [2, Theorem 1.1]. We can now
prove Theorem 1.2.

###### Proof of Theorem 1.2.

We show that the right-hand side in Theorem 4.1 is rational. Note that $g$ and
$\vartheta_{I}$ have rational Fourier coefficients. Moreover, by [10, Theorem
1.1] we can choose $\widetilde{\Theta}_{3/2,N}$ such that its holomorphic part
has rational Fourier coefficients. Here we use that the lattice $N$ is of the
form $(\mathbb{Z},q(x)=-Dr^{2}x^{2})$ for some rational number $r$. It remains
to note that the Rankin-Cohen brackets preserve the rationality of the Fourier
coefficients. ∎

##  5\. Generalization to higher level

In [11, Conjecture 2.5], Theorem 1.2 was stated as a conjecture for level
$\Gamma_{0}(N)$ and twisted versions of $f_{k,d}$. Here we sketch the proof of
this general conjecture.

Let $N\in\mathbb{N}$. We let $\Delta\in\mathbb{Z}$ be a fundamental
discriminant (possibly $1$) and $\rho\in\mathbb{Z}/2N\mathbb{Z}$ such that
$\Delta\equiv\rho^{2}\ \,\left(\mathrm{mod}\,2N\right)$. Moreover, we let
$d<0$ be a negative integer and $r\in\mathbb{Z}/2N\mathbb{Z}$ with
$d\equiv\operatorname{sgn}(\Delta)r^{2}\ \,\left(\mathrm{mod}\,2N\right)$. We
consider the set $\mathcal{Q}_{N,d|\Delta|,r\rho}$ of (not necessarily
positive definite) integral binary quadratic forms $Q=[aN,b,c]$ of
discriminant $d|\Delta|$ with $b\equiv r\rho\
\,\left(\mathrm{mod}\,2N\right)$. We let $\chi_{\Delta}$ be the usual
generalized genus character on $\mathcal{Q}_{N,d|\Delta|,r\rho}$. For
$k\in\mathbb{N}$ with $k\geq 2$ we define the function

$f_{k,d,r,\Delta,\rho}(z)=\frac{|d\Delta|^{k-1/2}}{2\pi}\sum_{Q\in\mathcal{Q}_{N,d|\Delta|,r\rho}}\frac{\operatorname{sgn}(Q)\chi_{\Delta}(Q)}{Q(z,1)^{k}},$

where we put $\operatorname{sgn}(Q)=\operatorname{sgn}(a)$. It defines a
meromorphic modular form of weight $2k$ for $\Gamma_{0}(N)$. For $N=1$ and odd
$k$ we recover $f_{k,d}$ as $f_{k,d,d,1,1}$.

We consider the signature $(1,2)$ lattice

$L_{N}=\left\\{X=\begin{pmatrix}b&c/N\\\
-a&-b\end{pmatrix}:a,b,c\in\mathbb{Z}\right\\}$

with the quadratic form $q_{N}(X)=N\det(X)$. We have
$L_{N}^{\prime}/L_{N}\cong\mathbb{Z}/2N\mathbb{Z}$, so we will write
$\operatorname{\mathfrak{e}}_{r}$ with $r\in\mathbb{Z}/2N\mathbb{Z}$ for the
standard basis elements of $\mathbb{C}[L_{N}^{\prime}/L_{N}]$. Note that the
elements of $L_{N}^{\prime}$ correspond to binary quadratic forms $[aN,b,c]$,
with the discriminant being given by $-4Nq_{N}(X)$. We let
$\widetilde{\rho}_{L_{N}}$ be equal to $\overline{\rho}_{L_{N}}$ if
$\Delta>0$, and to $\rho_{L_{N}}$ if $\Delta<0$.

We have the following higher level version of Theorem 1.2.

######  Theorem 5.1.

Let $k\in\mathbb{N}$ with $k\geq 2$ and let $g$ be a weakly holomorphic
modular form of weight $3/2-k$ for $\widetilde{\rho}_{L_{N}}$. Suppose that
the Fourier coefficients $a_{g}(d,r)$ for $d<0$ and
$r\in\mathbb{Z}/2N\mathbb{Z}$ are rational. Then the cycle integrals

$\mathcal{C}_{A}\left(\sum_{r\in\mathbb{Z}/2N\mathbb{Z}}\sum_{d<0}a_{g}(d,r)f_{k,d,r,\Delta,\rho}\right)$

are rational.

###### Proof.

This can be proved analogously to Theorem 1.2, so we only give a sketch.
First, by [3, Proposition 4.1] the meromorphic modular form
$f_{k,d,r,\Delta,\rho}$ can be written as a twisted theta lift on the lattice
$L_{N}$. We can reduce to the non-twisted case using an intertwining operator
for the involved Weil representations as explained in [4]. Then it remains to
evaluate the cycle integrals of certain Siegel theta functions associated with
a general even lattice $L$ of signature $(1,2)$. However, note that we did not
use the precise shape of $L$ in the proof of Theorem 3.1, so the arguments
work exactly the same for arbitrary $L$, at least for odd $k$. For even $k$
the theta lift involves a Siegel theta function of the shape

$\Theta_{L}^{*}(\tau,z)=v\sum_{X\in
L^{\prime}}p_{z}(X)e\left(q(X_{z})\tau+q(X_{z^{\perp}})\overline{\tau}\right)\operatorname{\mathfrak{e}}_{X}.$

A computation similar to the proof of Theorem 3.1 shows that we have

$\mathcal{C}_{A}\big{(}\Theta_{L}^{*}(\tau,z)\big{)}\stackrel{{\scriptstyle\cdot}}{{=}}\left(\vartheta_{I}(\tau)\otimes\overline{\Theta_{1/2,N}(\tau)}v^{1/2}\right)^{L},$

with the holomorphic weight $1/2$ theta function
$\Theta_{1/2,N}(\tau)=\sum_{X\in
N^{\prime}}e(-q(X)\tau)\operatorname{\mathfrak{e}}_{X}$. Apart from this, the
proofs for even and odd $k$ are analogous. ∎

Finally, one might speculate that the method from Theorem 3.1 could be used to
compute cycle integrals of Siegel theta functions (and the corresponding theta
lifts) on lattices of other signatures, e.g. signature $(1,n)$ or $(2,n)$. We
hope to come back to this problem in the future.

## References

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#  Optical study on electronic structure of the locally non-centrosymmetric
CeRh2As2

Shin-ichi Kimura<EMAIL_ADDRESS>FBS and Department of Physics,
Osaka University, Suita, Osaka 565-0871, Japan  Institute for Molecular
Science, Okazaki, Aichi 444-8585, Japan  Jörg Sichelschmidt  Max Planck
Institute for Chemical Physics of Solids, 01187 Dresden, Germany  Seunghyun
Khim  Max Planck Institute for Chemical Physics of Solids, 01187 Dresden,
Germany

###### Abstract

The electronic structures of the heavy-fermion superconductor CeRh2As2 with
the local inversion symmetry breaking and the reference material LaRh2As2 have
been investigated by using experimental optical conductivity
($\sigma_{1}(\omega)$) spectra and first-principal DFT calculations. In the
low-temperature $\sigma_{1}(\omega)$ spectra of CeRh2As2, a $4f$–conduction
electron hybridization and heavy quasiparticles are clearly indicated by a
mid-infrared peak and a narrow Drude peak. In LaRh2As2, these features are
absent in the $\sigma_{1}(\omega)$ spectrum, however, it can nicely be
reproduced by DFT calculations. For both compounds, the combination between a
local inversion symmetry breaking and a large spin-orbit (SO) interaction
plays an important role for the electronic structure, however, the SO
splitting bands could not be resolved in the $\sigma_{1}(\omega)$ spectra due
to the small SO splitting size.

##  I Introduction

Materials with non-centrosymmetric crystal structure and crystal surfaces with
inversion symmetry breaking have recently attracted attention for novel
physical properties combined with spin-orbit interaction (SOI) [1]. By using
the SOI and the electric field gradient generated by the spatial symmetry
breaking, a spin-polarized current originating from spin-polarized bands is
generated and is regarded to be useful for spintronics applications. A
breaking of local inversion symmetry also generates a toroidal current owing
to the Dzyaloshinsky-Moriya interaction [2] and a Cooper pair with a helical
spin structure, which produces a superconducting state with a high critical
field [3]. Actually, an extremely high upper critical field $H_{c2}$ as high
as $5~{}{\rm T}$ has been observed in the heavy-fermion superconductor CePt3Si
($T_{c}$ $\sim 0.75~{}{\rm K}$) having a globally non-centrosymmetric crystal
structure [4].

Figure 1:  Crystal structure of $R$Rh2As2 ($R=$ Ce, La) with the space group
of $P4/nmm$ depited by using VESTA [5]. Rh 1 and Rh 2 are located at different
sites and therefore have different effects on the electronic structure (see
Figs. 4 and 6).

CeRh2As2 has recently been discovered as a novel heavy-fermion superconductor
($T_{c}$ = $0.26~{}{\rm K}$) [6]. It crystallizes in the CaBe2Ge2-type
structure ($P4/nmm$), which lacks a local inversion symmetry of the Ce and one
of the Rh and As sites while maintaining a global inversion center (see Fig.
1). Noteworthy, the majority of $RM_{2}X_{2}$ ($R$ = rare earth, $M$ =
transition metal, $X$ = Si, Ge) compounds including the first heavy-fermion
superconductor CeCu2Si2 [7] have a tetragonal ThCr2Si2-type crystal structure
with inversion symmetry ($I4/mmm$).

The fundamental physical properties of CeRh2As2 including superconductivity
have recently been reported [6]. The Kondo temperature ($T_{\rm K}$) and the
electronic specific heat coefficient ($\gamma$) have been evaluated as $20-40$
K and $\approx 1$ J/mol K2 [6], respectively, which are similar to those of
CeCu2Si2 [8, 9]. The origin of the superconducting property of locally non-
centrosymmetric heavy fermions including CeRh2As2 has been investigated
theoretically [10, 11, 12, 13, 14, 15, 16] and experimentally [17]. However,
the electronic structure of the material, which is the most basic information
for the discussion of the physical properties, has not been investigated
experimentally so far, although the electronic structure using DFT
calculations has been reported [13, 14, 15].

A locally non-centrosymmetric crystal structure can produce a local electric
field to the electron orbitals. With a large SOI, the local electric field
induces the band splitting such as the Rashba effect [18, 19], which can be
directly observed by an angle-resolved photoelectron spectroscopy (ARPES) [20]
and (magneto-optical) infrared spectroscopies [21, 22, 23]. ARPES studies on
non-centrosymmetric heavy fermions have been performed on CeIrSi3 [24] and UIr
[25]. However, so far, no studies of optical conductivity were reported for
this type of materials.

We investigated the fundamental electronic structure of the locally non-
centrosymmetric heavy-fermion material CeRh2As2 by measuring the optical
conductivity ($\sigma_{1}(\omega)$) spectra and comparing them with first-
principal DFT calculations. As a reference material without $4f$ electrons, we
also investigated LaRh2As2 in order to discuss the electronic structure
without the effect of the hybridization between conduction and $4f$ electrons
($c$-$f$ hybridization). Firstly, the obtained $\sigma_{1}(\omega)$ spectrum
of LaRh2As2 is compared with the DFT calculations for the fundamental
electronic structure without $4f$ electrons. Next, the $\sigma_{1}(\omega)$
spectrum of CeRh2As2 is compared with the corresponding calculation results.
Finally, the temperature-dependent $\sigma_{1}(\omega)$ spectra of CeRh2As2 is
compared with other heavy fermion materials to discuss the evolution of the
$c$-$f$ hybridization.

##  II Experiment and calculation methods

Figure 2:  Temperature-dependent reflectivity ($R(\omega)$) spectra of the as-
grown $(001)$ surfaces of CeRh2As2 and LaRh2As2 in the photon energy
$\hbar\omega$ range of $0.01-1.2$ eV. Fine structures of CeRh2As2 at the
photon energies $\hbar\omega$ of about 15, 23, and 32 meV originate from TO
phonons (not discussed here). (Inset) Wide-energy-range $R(\omega)$ spectra of
CeRh2As2 and LaRh2As2 in the $\hbar\omega$ range of $0.1-30$ eV at 300 K. Both
spectra are almost identical suggesting the similar electronic structure at
room temperature.

Single-crystalline CeRh2As2 and LaRh2As2 samples were synthesized by the Bi-
flux method [6]. The optical reflectivity $R(\omega)$ measurements have been
performed using the as-grown (001) plane. Near-normal-incident $R(\omega)$
spectra were acquired in a wide photon-energy range of 8 meV – 30 eV to ensure
accurate Kramers-Kronig analysis (KKA) [26]. Infrared (IR) and terahertz (THz)
measurements at the photon energy $\hbar\omega$ regions of 8 meV–1.5 eV have
been performed using conventional near-normal reflectivity measurement setups
to obtain absolute $R(\omega)$ at an accuracy of $\pm 0.3$ % with a feed-back
positioning system in the temperature range of 10–300 K [27]. To obtain the
absolute $R(\omega)$ values, the in-situ gold evaporation method was adopted.
Obtained $R(\omega)$ spectra of CeRh2As2 and LaRh2As2 are shown in Fig. 2. In
the photon energy range of 1.5–30 eV, the $R(\omega)$ spectrum was measured
only at 300 K by using the synchrotron radiation setup at the beamline 3B of
UVSOR-III Synchrotron [28], and connected to the spectra for $\hbar\omega$
$\leq 1.5$ eV for KKA. In order to obtain $\sigma_{1}(\omega)$ via KKA of
$R(\omega)$, the spectra were extrapolated below 8 meV with a Hagen-Rubens
function [$R(\omega)=1-\\{2\omega/(\pi\sigma_{DC})\\}^{1/2}$] due to the
metallic $R(\omega)$ spectra, and above 30 eV with a free-electron
approximation $R(\omega)\propto\omega^{-4}$ [29]. Here, the values of the
direct current conductivity ($\sigma_{DC}$) were adopted from the experimental
values [6]. The extrapolations were confirmed not to severely affect to the
$\sigma_{1}(\omega)$ spectra at around 100 meV, which are the main part in
this paper.

Table 1:  Lattice parameters of CeRh2As2 and LaRh2As2 used for the DFT calculations. The parameters were obtained by x-ray diffraction methods.  Sample |  | CeRh2As2 |  |  | LaRh2As2 |
---|---|---|---|---|---|---
Crystal structure | CaBe2Ge2-type
Space group |  $P_{4}/nmm$ (No. 129)
Lattice constant |  |  |  |  |  |
a (Å), b (Å), c (Å) | 4.283(1), | 4.283(1), | 9.865(2) | 4.3137(2), | 4.3137(2), | 9.8803(4)
Position | X | Y | Z | X | Y | Z
Ce, La | 0.25 | 0.25 | 0.254686(3) | 0.25 | 0.25 | 0.2543(2)
Rh 1 | 0.75 | 0.25 | 0 | 0.75 | 0.25 | 0
Rh 2 | 0.25 | 0.25 | 0.617418(4) | 0.25 | 0.25 | 0.6159(2)
As 1 | 0.75 | 0.25 | 0.5 | 0.75 | 0.25 | 0.5
As 2 | 0.25 | 0.25 | 0.864075(7) | 0.25 | 0.25 | 0.871(9)

First-principal DFT calculations have been performed by using the Wien2k code
including SOI [30] to explain the experimentally obtained $\sigma_{1}(\omega)$
spectra. Lattice parameters obtained from room-temperature x-ray diffraction
measurements shown in Table I were adopted to the calculations. The obtained
band structure of CeRh2As2 was consistent with the recent report [13, 15].
$\sigma_{1}(\omega)$ spectra of the interband transitions have also been
calculated by using the Wien2k code. For the discussion of LaRh2As2, we also
performed DFT calculations on the basis of a hyphothetical ThCr2Si2-type
crystal structure, assuming the same lattice constants as for the original
CaBe2Ge2-type structure.

##  III Results and Discussion

Figure 3:  Temperature-dependent optical conductivity ($\sigma_{1}(\omega)$)
spectra of CeRh2As2 and LaRh2As2.

The measured $R(\omega)$ spectra of CeRh2As2 and LaRh2As2, shown in Fig. 2, at
various temperatures, were used for KKA in order to obtain their
$\sigma_{1}(\omega)$ spectra as shown in Fig. 3. At 300 K,
$\sigma_{1}(\omega)$ of both materials monotonically increase with decreasing
$\hbar\omega$ suggesting a typical metallic character. The spectrum of
CeRh2As2 at 300 K is very similar to that of LaRh2As2, as demonstrated by
almost identical $R(\omega)$ spectra in the inset of Fig. 2. This suggests
fully localized Ce $4f$ states at 300 K. With decreasing temperature, the
spectra strongly change and new features emerge: A double-peak structure
(“mid-IR peak”) appears at 0.12 and 0.4 eV in CeRh2As2, whereas in LaRh2As2, a
weak single peak becomes visible at 0.45 eV. The mid-IR peak is usually
observed in many Ce compounds and suggests the emergence of the $c$-$f$
hybridization [31]. The peak appearing in LaRh2As2 does not obviously
originate from the $c$-$f$ hybridization, but can be explained by the
electronic structure of LaRh2As2.

###  III.1 Electronic structure of LaRh2As2

Figure 4:  Calculated band structure of LaRh2As2 without (left) SOI and with
(center) SOI. Spin-orbit splitted bands are red encircled. Inset depicts
symmetry points in the Brillouin zone. (Right) Total density of states and
partial density of states of La, Rh 1 and Rh 2 (see Fig. 1) with SOI.

Figure 4 shows the band calculation results of LaRh2As2 near the Fermi energy
($E_{\rm F}$) with and without SOI. The main effect of including the SOI
appears as a spin-orbit (SO) splitting near $E_{\rm F}$ along the $M-X$ and
$A-R$ lines (see the red encircled regions in Fig. 4). The bands mainly
originate from the $4d$ states of Rh 1 as shown in the partial density of
states depicted in Fig. 4. The SO splitting is much smaller than that of
LaPt3Si [32] and BiTeI [23]. The different SO splitting size probably
originate from the different orbital moments ($4d$ for Rh, $5d$ for Pt, and
$6p$ for Bi), locally/globally non-centrosymmetric crystal structure, and/or
the different dimensionality, three-dimensional LaRh2As2 and LaPt3Si, and two-
dimensional BiTeBr.

Figure 5:  Optical conductivity ($\sigma_{1}(\omega)$) spectrum of LaRh2As2 at
$T=10$ K (Experiment, dotted line). Thick line denotes the interband
transition part obtained after subtracting the Drude part (grey line) from the
$\sigma_{1}(\omega)$ spectrum. Calculated $\sigma_{1}(\omega)$ spectra with
and without SOI are shown by red and blue lines, respectively. The calculated
spectrum of a centrosymmetric ThCr2Si2-type crystal structure ($I4/mmm$) is
also plotted for comparison (dashed green line).

Figure 5 shows the experimental $\sigma_{1}(\omega)$ spectrum of LaRh2As2 at
$T$ = 10 K together with spectra obtained from the DFT calculation results
(Fig. 4) either with SOI (red solid line) or without SOI (blue solid line).
The free charge carrier response (Drude peak, grey solid line) evaluated from
$\sigma_{DC}$ of LaRh2As2 is also shown. The experimental $\sigma_{1}(\omega)$
spectrum (denoted by “Experiment” in Fig. 5) is much larger than the expected
Drude curve suggesting the existence of other components due to interband
transitions overlapping on the Drude curve. These interband transitions are
indicated by the thick solid line (denoted by “Experiment - Drude” in Fig. 5),
which was derived by subtraction of the Drude curve from the
$\sigma_{1}(\omega)$ spectrum. The interband transition spectrum has a broad
peak at around $\sim 0.1$ eV and a sharp peak at $\sim 0.5$ eV.

Both calculated $\sigma_{1}(\omega)$ spectra with and without SOI are almost
identical suggesting a weak effect of the SOI on $\sigma_{1}(\omega)$. This is
consistent with the small SOI intensity of LaRh2As2. The two significant peaks
at $\hbar\omega$ $\sim 0.2$ eV and $0.45$ eV can be attributed to the
experimentally observed peaks at $\sim 0.1$ eV and $\sim 0.5$ eV. They both
originate from the bands near the $\Gamma-M$ and $Z-A$ lines in the Brillouin
zone (see Fig. 4). The background intensity of the calculated spectra is
consistent with that of the experimental spectrum. Therefore, the experimental
$\sigma_{1}(\omega)$ spectrum can be explained well by the DFT calculations.

In order to investigate the effect of the crystal structure, Fig. 5 also shows
a $\sigma_{1}(\omega)$ spectrum calculated by using the centrosymmetric
ThCr2Si2-type crystal structure ($I4/mmm$) while keeping the lattice
parameters as in the CaBe2Ge2-type structure. The large peak at $\sim 0.5$ eV
does not appear and the background spectral intensity is much lower than that
of the experimental spectrum. This suggests that both the 0.5 eV peak and the
high background intensity are characteristic properties of LaRh2As2 and the
locally non-centrosymmetric CaBe2Ge2-type crystal structure.

###  III.2 Electronic structure of CeRh2As2

Figure 6:  Calculated band structure of CeRh2As2 without (left) SOI and with
(center) SOI. Spin-orbit splitted bands are red encircled. Inset depicts
symmetry points in the Brillouin zone. (Right) Total density of states and
partial density of states of Ce, Rh 1 and Rh 2 (see Fig. 1) with SOI.

Figure 6 shows the calculated band structures of CeRh2As2 with and without
SOI. An important effect of the SOI is a splitting of the Ce $4f$ bands from
almost flat bands at $\sim 0.3$ eV into two parts, namely $4f_{5/2}$ bands at
$\sim 0.2$ eV and $4f_{7/2}$ bands at $\sim 0.5$ eV. The splitting energy is
roughly $0.25-0.3$ eV, which is a characteristic value for various Ce-based
compounds as observed in $\sigma_{1}(\omega)$ spectra [31, 33] and
photoelectron spectra [8, 34].

As in LaRh2As2, the SO split bands appear near $E_{\rm F}$ along the $M-X$ and
$A-R$ lines. The size of SO splitting is larger than that of LaRh2As2. Since
the bands originate from the hybridization between the Rh 1 $4d$ and Ce $4f$
bands (see the partial densities of states of Ce and Rh 1 in the right figure
of Fig. 6), the $4f$ state is considered to be relevant for the SO splitting.
Moreover, as flat bands are claimed to be important for the superconductivity
[13], the SO splitting may play an important role for the exotic
superconductivity in CeRh2As2.

Figure 7:  Optical conductivity ($\sigma_{1}(\omega)$) spectrum of CeRh2As2 at
10 K (Experiment, magenta solid line). The Drude curve expected by the
experimental electrical resistivity is shown by a gray solid line, and
calculated $\sigma_{1}(\omega)$ spectra with and without SOI are shown by a
red marked line and a blue solid line, respectively. The total and partial $f$
density of states above $E_{\rm F}$ are also plotted in the lower figure for
the comparison with the $\sigma_{1}(\omega)$ spectrum. The $E_{\rm F}$ of the
density of states is set to $\hbar\omega$ $=0$ eV of $\sigma_{1}(\omega)$
spectra. The peaks at about 0.2 and 0.5 eV originate from the unoccupied Ce
$4f_{5/2}$ and $4f_{7/2}$ states.

Figure 7 shows the experimentally obtained $\sigma_{1}(\omega)$ spectrum at
$T=$ 10 K together with the calculated $\sigma_{1}(\omega)$ spectra with and
without SOI. In comparison with the $\sigma_{1}(\omega)$ spectra of LaRh2As2,
the calculated spectra cannot reproduce the experimental spectrum well. The
main reason is that the mid-IR peak below $\hbar\omega$ $\sim 0.6$ eV does not
appear in the calculation. The experimental mid-IR peak can be attributed to
the SO splitting of the Ce $4f$ states [35], which appears in the unoccupied
density of states above $E_{\rm F}$ as shown in the lower frame of Fig. 7.
However, the corresponding mid-IR peak structure does not clearly appear in
the $\sigma_{1}(\omega)$ calculation shown in the upper frame of Fig. 7. The
inconsistency suggests that the calculated $c$-$f$ hybridization intensity in
the DFT calculations is much smaller than the experimental value because the
Kondo interaction between conduction and localized $4f$ electrons is not
included in the calculation.

According to the band calculation in Fig. 6, the signature of the SO splitting
along the $\Gamma-M$ line may appear at $\hbar\omega$ $\leq 0.1$ eV. In this
energy region, there are three phonon peaks at $\hbar\omega$ = 15.2, 22.8, and
31.7 meV (see Fig. 2), but no other significant structures except for the
Drude peak. Therefore, we conclude that, in the $\sigma_{1}(\omega)$ spectra,
there are no visible signatures for a SO splitting expected in the band
structure.

###  III.3 $c$-$f$ hybridization of CeRh2As2

Figure 8:  Relative temperature dependence of the change of the center of
gravity ($\Delta\langle\omega\rangle$, left axis) and the spectral weight
($\Delta SW$, right axis) of CeRh2As2 and reference materials, CeCu2Si2 and
CeNi2Ge2 [31]. These values were normalized to their room-temperature values.

The $c$-$f$ hybridization in CeRh2As2 can be characterized by using the
temperature dependence of the mid-IR peak. Figure 8 shows the temperature
dependencies of the center of gravity [$\Delta\langle\omega\rangle$
$=\left[\langle\omega(T)\rangle-\langle\omega({\rm
300~{}K})\rangle\right]/\langle\omega({\rm 300~{}K})\rangle$], where
$\langle\omega(T)\rangle=\int_{\omega_{1}}^{\omega_{2}}\omega\sigma_{1}(\omega)d\omega/\int_{\omega_{1}}^{\omega_{2}}\sigma_{1}(\omega)d\omega$,
and the spectral weight transfer [$\Delta SW$
$=\int_{\omega_{1}}^{\omega_{2}}[|\sigma_{1}(\omega,T)-\sigma_{1}(\omega,{\rm
300~{}K})|/\sigma_{1}(\omega,{\rm 300~{}K})]d\omega$] of the mid-IR peak of
CeRh2As2 relative to those at $T=$ 300 K. The integration range was set as
$\omega_{1}=0~{}{\rm eV}\leq\omega\leq\omega_{2}=0.8~{}{\rm eV}$, where the
spectral change in the lower energy region is almost recovered. In the figure,
the evaluated $\Delta\langle\omega\rangle$ and $\Delta SW$ of two heavy-
fermion compounds CeCu2Si2 ($T_{\rm K}$ $\sim 10$ K) [36] and CeNi2Ge2
($T_{\rm K}$ $\sim$ several K) [33] are also plotted for comparison. The
temperature dependencies of $\Delta\langle\omega\rangle$ and $\Delta SW$
correspond to the evolution of the $c$-$f$ hybridization with temperature
[31], i.e., an increasing itinerancy corresponds to increasing values of
$\Delta\langle\omega\rangle$ and $\Delta SW$ whereas constant values suggest a
localization of the $4f$ state.

In CeRh2As2, $\Delta\langle\omega\rangle$ is constant near 300 K and increases
toward low temperatures below $\sim 150$ K, suggesting a rapid development of
the $c$-$f$ hybridization below the temperature. This is consistent with the
$c$-$f$ hybridization intensity being almost negligible at 300 K, which is
demonstrated by almost identical $R(\omega)$ spectra of CeRh2As2 and LaRh2As2
at 300 K (see inset of Fig. 2).

The $\Delta SW$ values monotonically increase with decreasing temperature
below 300 K. As was shown previously [31], an increase from 300 K to $\sim
100$ K originates from electron-phonon interactions, and the $c$-$f$
hybridization effect appears below the temperature of $\sim 100$ K. In
CeRh2As2, the $\Delta SW$ increases at low temperatures, which is consistent
with $\Delta\langle\omega\rangle$. The values of $\Delta\langle\omega\rangle$
and $\Delta SW$ of CeRh2As2 at the lowest accessible temperature are located
in between those of CeCu2Si2 and CeNi2Ge2 suggesting an intermediate $c$-$f$
hybridization intensity$\tilde{V}$, which is included in the Kondo temperature
expression

$T_{\rm K}\propto\exp[-\tilde{V}^{-2}D_{c}(E_{\rm F})^{-1}],$ (1)

where $D_{c}(E_{\rm F})$ is the density of states of the conduction band at
$E_{\rm F}$ [37]. Hence, compared to CeCu2Si2, the higher $T_{\rm K}\sim
20-40$ K and smaller $\tilde{V}$ of CeRh2As2 should be related to a larger
$D_{c}(E_{\rm F})$. Indeed, such conclusion is consistent with the
$\sigma_{1}(\omega)$ spectral intensity of LaRh2As2 for $\hbar\omega$ $\leq
0.8$ eV, which reflects the density of states near the $E_{\rm F}$. In this
region, $\sigma_{1}(\omega)$ is enhanced compared to the calculated one using
the ThCr2Si2-type structure as shown in Fig. 5, which is also due to a flat Rh
$4d$ band near the $E_{\rm F}$ as shown in Fig. 4. Therefore, the locally non-
centrosymmetric CaBe2Ge2-type crystal structure of CeRh2As2 supports a higher
$T_{\rm K}$ than the ThCr2Si2-type structure of CeCu2Si2.

##  IV Conclusion

To summarize, optical conductivity $\sigma_{1}(\omega)$ spectra of a locally
non-centrosymmetric heavy fermion superconductor CeRh2As2 and LaRh2As2 as a
reference material without $4f$ electrons were measured and compared with the
corresponding DFT calculations. The experimentally obtained
$\sigma_{1}(\omega)$ spectrum of LaRh2As2 can be explained well by the DFT
calculations. Besides, the experimental $\sigma_{1}(\omega)$ spectrum of
CeRh2As2 at low temperatures has a stronger $c$-$f$ hybridization intensity
than the DFT calculation because the Kondo interaction is effective at low
temperatures, which is also seen in the temperature dependence of the
$\sigma_{1}(\omega)$ spectrum. The evidence of the SO splitting due to the
locally non-centrosymmetric crystal structure, unfortunately, could not be
observed in the $\sigma_{1}(\omega)$ spectra because of the SO splitting being
too small for resolvable spectral features.

## Acknowledgments

We would like to thank Profs. Noriaki Kimura, Takahiro Ito, Hiroshi Watanabe,
and Yoshiyuki Ohtsubo for their fruitful discussion and UVSOR Synchrotron
staff members for their support during synchrotron radiation experiments. Part
of this work was performed under the Use-of-UVSOR Synchrotron Facility Program
(Proposals No. 20-735) of the Institute for Molecular Science, National
Institutes of Natural Sciences. This work was partly supported by JSPS KAKENHI
(Grant No. 20H04453).

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