# Surface density of states and tunneling spectroscopy of a spin-3/2
superconductor with Bogoliubov Fermi Surfaces

Ryoi Ohashi1, Shingo Kobayashi2, Shotaro Kanazawa3, Yukio Tanaka3,4, and Yuki
Kawaguchi3,4 1Department of Materials Engineering Science, Osaka University,
Toyonaka 560-8531, Japan
2RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan
3Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan
4Research Center for Crystalline Materials Engineering, Nagoya University,
Nagoya 464-8603, Japan

###### Abstract

Bogoliubov Fermi surfaces of superconducting states arise from point or line
nodes by breaking time-reversal symmetry. Because line and point nodes often
accompany topologically protected zero-energy surface Andreev bound states
(ASBSs) and thereby lead to a characteristic zero-bias conductance peak (ZBCP)
in tunneling spectroscopy, we investigate how these properties change when the
line and point nodes deform into BFSs. In this paper, we consider spin-quintet
$J_{\rm pair}=2$ pairing states of spin-3/2 electrons with BFSs and calculate
the surface density of states and the charge conductance. Comparing the
obtained results with the cases of spin-singlet d-wave pairing states having
the same symmetry, we find that the ZBCP associated with point and/or line
nodes is blunted or split in accordance with the appearance of the BFSs. On
the other hand, when the spin-singlet d-wave state has point nodes but does
not have SABS on the surface, we obtain a nonzero small electron conductivity
at zero bias through the zero-energy states on the BFSs.

††preprint: APS/123-QED

##  I Introduction

Nodal structures of a superconducting pair potential lead to the
characteristic properties of unconventional superconductors. An example
includes the power-law temperature dependence of bulk physical quantities at a
low temperature, such as specific heat, spin relaxation rate, thermal
conductivity, and London penetration depth [1]. These signatures have been
indeed observed in high-temperature superconductors and heavy fermion
superconductors [2, 3]. Another characteristic originating from nodal
structures is the appearance of surface Andreev bound states (SABSs) at zero
energy due to the change of the internal phase degrees of freedoms of pair
potentials [4, 5, 6, 7, 8, 9]. The existence of zero-energy SABSs is related
to the bulk topological properties of superconductors [10, 11, 12, 13, 14,
15]. Namely, there is a one-to-one correspondence between the number of zero-
energy SABSs and the bulk topological invariant via the bulk-boundary
correspondence. When the superconducting pair potential has line or point
nodes, a topological invariant takes a nonzero value in a subspace of the
Brillouin zone, leading to SABSs with dispersion or zero-energy flat-band
SABSs in the surface Brillouin zone [16, 17, 18, 19, 20, 21, 22].

Experimentally, SABSs manifest themselves in the low voltage profile of the
tunneling spectroscopy of quasiparticles. For instance, in the case of spin-
singlet $d$-wave superconductors with line nodes, the zero-energy flat-band
SABSs lead to a zero bias conductance peak (ZBCP) in normal
metal/superconductor (N/S) junctions [23, 24, 25], which was observed in
tunneling spectroscopy experiments on high $T_{\rm c}$ cuprate superconductors
[26, 27, 28, 25]. On the other hand, in the case of 3D chiral superconductors,
which has a pair of point nodes in addition to a line node [29, 30], the
dispersive SABSs originated from the point nodes also lead to a ZBCP, but the
height of the ZBCP is strongly suppressed as compared to the case of the zero-
energy flat-band SABSs [31, 32, 33, 34, 35, 36, 37, 38].

Recently, a new type of nodal structure, called the Bogoliubov Fermi surface
(BFS), is predicted for time-reversal symmetry (TRS) breaking
superconductivity in a multi-band system [39, 40, 41, 42, 43, 44], where zero-
energy quasiparticle states appear on a surface in the 3D bulk Brillouin zone.
The unique feature of this superconducting state is that the BFSs arise from
the inflation of point or line nodes due to splitting of superconducting
energy bands around the nodes enforced by a TRS breaking effect [40, 45]. This
implies that the BFSs are topological objects characterized by multiple
topological invariants [46]; namely, the one characterizes the BFS and the
others are associated with the underlying point or line nodes. It is expected
that the multiple topological natures come together in tunneling spectroscopy
and bring about unique low voltage profile distinguished from the conventional
ZBCP attributed to line or point nodes. Although tunnelling spectroscopy in a
superconductor with the BFSs has been studied in Refs. [47, 48, 42, 49], the
influence of multiple topological natures on tunneling conductance has yet to
be clarified.

In addition, recent experiments on the iron-based superconductor Fe(Se,S) have
indicated the existence of Fermi surface in the superconducting states. In
these experiments, the signature of the BFS has been detected as a nonzero
residual density of states at zero energy at a low temperature in specific
heat, thermal conductivity, and tunneling spectroscopy [50, 51, 52]. More
recently, the direct evidence of the BFS has been reported as an unusual
superconducting gap anisotropy with zero gap regions using high-energy
resolution laser-based angle-resolved photoemission spectroscopy [53]. Thus, a
detailed study on the influence of the BFSs on experimental observables is
called for.

In this paper, we study the SABSs and tunneling spectroscopy of spin-quintet
$J_{\textrm{pair}}=2$ pairing states of spin-3/2 electrons. We calculate the
surface density of states (SDOS) and the charge conductance in N/S junctions
utilizing the recursive Green’s function method [54] and the generalized Lee-
Fisher formula [55, 56] which are available for the lattice models of
Bogoliubov-de Gennes (BdG) Hamiltonian [57, 58, 59]. Here, we consider the
three $J_{\textrm{pair}}=2$ states, $T_{2g}(0,i,1)$,
$T_{2g}(1,\omega,\omega^{2})$ and $E_{g}(1,i)$, all of which possess the BFSs.
Because these states can be mapped to spin-singlet d-wave pairing states with
line and/or point nodes, we compare the obtained results for the
$J_{\textrm{pair}}=2$ pairing states with those for the corresponding spin-
singlet d-wave pairing states of spin-1/2 electrons. The effects of the BFSs
on the SABSs and the charge conductance are summarized as follows. When the
spin-singlet d-wave state exhibits the zero-energy peak in the SDOS and the
ZBCP due to the existence of the topologically protected zero-energy surface
flat bands or surface arc states, the corresponding $J_{\textrm{pair}}=2$
pairing state has blunted or split peaks. This is because the broken TRS in
the $J_{\textrm{pair}}=2$ states violates the topological protection for the
zero-energy SABSs in the d-wave pairing states and makes the SABSs more
dispersive. On the other hand, in the case when the d-wave state has no SABS,
and hence, has zero SDOS and zero charge conductance at zero energy, the small
SDOS at zero energy coming from the bulk BFSs and the charge conductance
through the zero-energy states become visible in the $J_{\textrm{pair}}=2$
pairing state. These features will serve as a guide to detect the BFSs by
tunneling spectroscopy.

The organization of this paper is as follows. In Sec. II, we review how the
BFSs appear in a two-band system. In Sec. III, we introduce the Luttinger-Kohn
model of spin-$3/2$ electrons with cubic symmetry and describe the pair
potentials for spin-singlet $J_{\textrm{pair}}=0$ and spin-quintet
$J_{\textrm{pair}}=2$ pairing states. Projection to the spin-singlet d-wave
pairing states of spin-1/2 electrons is also discussed here. In Sec. IV, we
explain the detailed setup for our calculation and the calculation method. In
Sec. V, we discuss the behavior of the SABSs in the presence of the BFSs with
showing the numerical results of the SDOS and the charge conductance for
$T_{2g}(1,i,0)$, $T_{2g}(1,\omega,\omega^{2})$, and $E_{g}(1,i)$ states. In
Sec. VI, we conclude our results.

##  II Bogoliubov Fermi Surface

We briefly review the underlying physics of the appearance of BFSs [40]. BFSs
can appear in even-parity superconducting states of inversion symmetric
systems with broken TRS. Examples include spin-singlet superconductors under
an external magnetic field [60, 42, 61] and multi-band superconductors with
TRS-breaking inter-band pairings. Here, we consider the latter case, i.e., the
case when TRS is preserved in a normal state and spontaneously broken in a
superconducting state. A minimal model for such systems is a two-band model.

We start from a normal-state Hamiltonian given in the band basis as

$\displaystyle\mathcal{H}_{\rm N}$
$\displaystyle=\sum_{{\bm{k}}}(\bm{c}_{\bm{k}}^{\dagger})^{\rm
T}H_{0}({\bm{k}})\bm{c}_{\bm{k}},$ (1) $\displaystyle H_{0}({\bm{k}})$
$\displaystyle=\operatorname{Diag}[E_{{\bm{k}},+},E_{{\bm{k}},+},E_{{\bm{k}},-},E_{{\bm{k}},-}]-\mu\mathbbm{1}_{4},$
(2)

where $E_{{\bm{k}},+}$ and $E_{{\bm{k}},-}$ are the energy dispersion of the
two bands, each of which is doubly degenerate due to the TRS, $\mu$ is the
chemical potential, $\mathbbm{1}_{n}$ denotes the $n\times n$ identity matrix,
and
$\bm{c}_{\bm{k}}=(c_{{\bm{k}}+\uparrow},c_{{\bm{k}}+\downarrow},c_{{\bm{k}}-\uparrow},c_{{\bm{k}}-\downarrow})^{\rm
T}$ and
$\bm{c}^{\dagger}_{\bm{k}}=(c^{\dagger}_{{\bm{k}}+\uparrow},c^{\dagger}_{{\bm{k}}+\downarrow},c^{\dagger}_{{\bm{k}}-\uparrow},c^{\dagger}_{{\bm{k}}-\downarrow})^{\rm
T}$ with $c_{{\bm{k}}\pm s}$ being the annihilation operator of an electron in
the Bloch state $|{\bm{k}},\pm,s\rangle$ in the $\pm$ band with momentum
${\bm{k}}$ and pseudospin $s=\uparrow,\downarrow$. From the assumptions of
inversion symmetry and TRS, the space inversion operator $P$ and the time-
reversal operator $T$ transforms the Bloch states as

$\displaystyle P|{\bm{k}},\pm,s\rangle$
$\displaystyle=|-{\bm{k}},\pm,s\rangle,$ (3) $\displaystyle
T|{\bm{k}},\pm,s\rangle$
$\displaystyle=-\operatorname{sign}(s)|-{\bm{k}},\pm,-s\rangle,$ (4)

where $\operatorname{sign}(\uparrow)=1$ and
$\operatorname{sign}(\downarrow)=-1$. Due to the inversion symmetry, the
energy dispersion satisfies $E_{{\bm{k}},\pm}=E_{-{\bm{k}},\pm}$.

The superconducting state is described by the Bogoliubov-de Gennes (BdG)
Hamiltonian given by

$\displaystyle\mathcal{H}$
$\displaystyle=\frac{1}{2}\sum_{{\bm{k}}}\begin{pmatrix}(\bm{c}_{\bm{k}}^{\dagger})^{\rm
T}&\bm{c}_{-{\bm{k}}}^{\rm
T}\end{pmatrix}H_{\bm{k}}\begin{pmatrix}\bm{c}_{\bm{k}}\\\
\bm{c}_{-{\bm{k}}}^{\dagger}\end{pmatrix},$ (5) $\displaystyle H_{\bm{k}}$
$\displaystyle=\begin{pmatrix}H_{0}({\bm{k}})&\Delta({\bm{k}})\\\
\Delta^{\dagger}({\bm{k}})&-H_{0}^{\rm T}(-{\bm{k}})\end{pmatrix},$ (6)

where the pairing term $\Delta({\bm{k}})$ should have even or odd parity. For
the case of even-parity pairing, which is our interest, the general form of
$\Delta({\bm{k}})$ is given by

$\displaystyle\Delta({\bm{k}})=\begin{pmatrix}\psi_{{\bm{k}},+}i\sigma_{y}&(\psi_{{\bm{k}},I}\sigma_{0}+\bm{d}_{\bm{k}}\cdot\bm{\sigma})i\sigma_{y}\\\
(\psi_{{\bm{k}},I}\sigma_{0}-\bm{d}_{\bm{k}}\cdot\bm{\sigma})i\sigma_{y}&\psi_{{\bm{k}},-}i\sigma_{y}\end{pmatrix},$
(7)

where $\bm{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ and $\sigma_{0}$ are
the vector of Pauli matrices and the identity matrix in the pseudospin space,
respectively, and all functions in the matrix are even functions of $\bm{k}$.
Here, $\psi_{{\bm{k}},\pm}$ denotes the pseudospin-singlet pair in each band,
whereas $\psi_{{\bm{k}},I}$ and ${\bm{d}}_{\bm{k}}$ are the pseudospin-singlet
and triplet components, respectively, in the inter-band pairing. Note that the
pseudospin-triplet pairing should be band singlet due to the Fermi-Dirac
statistics, which is the reason for the different signs in front of
${\bm{d}}_{\bm{k}}$ in the off-diagonal terms. By defining the pseudospin
operator for $+$ and $-$ bands as

$\displaystyle\bm{S}_{+}=\begin{pmatrix}\bm{\sigma}&0\\\ 0&0\end{pmatrix},\ \
\bm{S}_{-}=\begin{pmatrix}0&0\\\ 0&\bm{\sigma}\end{pmatrix},$ (8)

the polarization in the pairing state at momentum ${\bm{k}}$ is given by

$\displaystyle\operatorname{Tr}[\Delta^{\dagger}({\bm{k}})\bm{S}_{\pm}\Delta({\bm{k}})]=2i{\bm{d}}_{\bm{k}}\times{\bm{d}}_{\bm{k}}^{*}\pm
4{\rm Re}(\psi_{{\bm{k}},I}^{*}{\bm{d}}_{\bm{k}}),$ (9)

which means that TRS is spontaneously broken when ${\bm{d}}_{\bm{k}}$ is a
complex vector such that ${\bm{d}}_{\bm{k}}^{*}\neq{\bm{d}}_{\bm{k}}$ or when
the pseudospin-singlet and triplet components are mixed in the inter-band
pairing.

BFS is a topological object defined for an inversion-symmetric BdG Hamiltonian
with even-parity pairing. As a general property, the BdG Hamiltonian possesses
particle-hole symmetry:

$\displaystyle CH_{-{\bm{k}}}C^{-1}=-H_{\bm{k}}$ (10)

where $C=\mathcal{K}\tau_{x}$ with $\tau_{i=x,y,z}$ being the Pauli matrices
in the Nambu space and $\mathcal{K}$ being the complex conjugation operator.
Under the assumption $H_{0}({\bm{k}})=H_{0}(-{\bm{k}})$, the Hamiltonian is
also symmetric under inversion:

$\displaystyle PH_{-{\bm{k}}}P^{-1}=H_{\bm{k}},$ (11)

where $P=\tau_{0}$ ($P=\tau_{z}$) for even-parity (odd-parity) pairing states.
Thus, the product of these symmetries satisfies

$\displaystyle(CP)H_{\bm{k}}(CP)^{-1}=-H_{\bm{k}}.$ (12)

It is generally proved that any $CP$ symmetric Hamiltonian $H_{\bm{k}}$ can be
unitarily transformed into an antisymmetric matrix $\tilde{H}_{\bm{k}}$, if
the square of $CP$ is unity: $(CP)^{2}=+1$, which is the case for even-parity
pairing states [39]. To be more concrete, we obtain the antisymmetric matrix,
$\tilde{H}_{\bm{k}}=-\tilde{H}_{\bm{k}}^{\rm T}$, by the unitary
transformation $\tilde{H}_{\bm{k}}=\Omega H_{\bm{k}}\Omega^{\dagger}$, where

$\displaystyle\Omega=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\
-i&i\end{pmatrix}$ (13)

is a unitary matrix acting on the Nambu space. It follows that the
eigenspectrum of $H_{\bm{k}}$ is characterized by the Pfaffian of
$\tilde{H}_{\bm{k}}$, ${\rm Pf}(\tilde{H}_{\bm{k}})$, which is always real
owing to the relation ${\rm Pf}(\tilde{H}_{\bm{k}})^{2}={\rm det}H_{\bm{k}}>0$
for two-band systems. In general, when the number of bands is even (odd),
${\rm Pf}(\tilde{H}_{\bm{k}})$ is always real (purely imaginary), because
$H_{\bm{k}}$ always has pairs of positive and negative eigenvalues due to
particle-hole symmetry. Thus, the Brillouin zone is divided into two regions
according to the signs of ${\rm Pf}(\tilde{H}_{\bm{k}})$, and zero-energy
states appear at the boundary between the two regions. In the case of a three-
dimensional system, zero-energy states appear on a surface. That is the BFS
[39, 40].

The discussion above does not require the breaking of TRS. Below, we show that
BFSs indeed appear when (i) the intra-band pair potential has a node, and (ii)
the inter-band pairing breaks TRS. For the even-parity pairing state given by
Eq. (7), the Pfaffian is calculated as

$\displaystyle P({\bm{k}})\equiv$
$\displaystyle\operatorname{Pf}(\tilde{H}_{\bm{k}})$ $\displaystyle=$
$\displaystyle\left|\psi_{{\bm{k}},+}\psi_{{\bm{k}},-}-\psi_{{\bm{k}},I}^{2}+{\bm{d}}_{\bm{k}}\cdot{\bm{d}}_{\bm{k}}^{*}\right|^{2}$
$\displaystyle+\left(\xi_{{\bm{k}},+}\xi_{{\bm{k}},-}+{\bm{d}}_{\bm{k}}\cdot{\bm{d}}_{\bm{k}}^{*}+|\psi_{{\bm{k}},I}|^{2}\right)^{2}$
$\displaystyle+\xi_{{\bm{k}},+}^{2}|\psi_{{\bm{k}},-}|^{2}+\xi_{{\bm{k}},-}^{2}|\psi_{{\bm{k}},+}|^{2}$
$\displaystyle-({\bm{d}}_{\bm{k}}\cdot{\bm{d}}_{\bm{k}}^{*}+|\psi_{{\bm{k}},I}|^{2})^{2},$
(14)

where $\xi_{{\bm{k}},\pm}=E_{{\bm{k}},\pm}-\mu$. Suppose that the $+$ band has
a normal-state Fermi surface, ${\rm
FS}_{+}\equiv\\{{\bm{k}}|\xi_{{\bm{k}},+}=0\\}$, and its intra-band pair
potential $\psi_{\bm{k},+}$ has a node along a certain direction. Along the
nodal direction, the Pfaffian (14) reduces to

$\displaystyle\left.P({\bm{k}})\right|_{\psi_{\bm{k},+}=0}=$
$\displaystyle\left(\xi_{{\bm{k}},+}\xi_{{\bm{k}},-}+{\bm{d}}_{\bm{k}}\cdot{\bm{d}}_{\bm{k}}^{*}+|\psi_{{\bm{k}},I}|^{2}\right)^{2}$
$\displaystyle-\left|{\bm{d}}_{\bm{k}}\times{\bm{d}}_{\bm{k}}^{*}\right|^{2}-4\left|{\rm
Re}(\psi_{{\bm{k}},I}^{*}{\bm{d}}_{\bm{k}})\right|^{2}$
$\displaystyle+\xi_{{\bm{k}},+}^{2}|\psi_{{\bm{k}},-}|^{2}.$ (15)

We further assume that the energy scales of the inter-band and intra-band
pairings are much smaller than the energy splitting
$|E_{{\bm{k}},+}-E_{{\bm{k}},-}|$ in the vicinity of ${\rm FS}_{+}$. Then,
there should exist a point ${\bm{k}}_{0}$ in the vicinity of ${\rm FS}_{+}$
such that the first term of the right hand side of Eq. (15) disappears. The
Pfaffian at ${\bm{k}}={\bm{k}}_{0}$ is then given by

$\displaystyle\left.P({\bm{k}})\right|_{{\bm{k}}={\bm{k}}_{0}}=$
$\displaystyle-\left|{\bm{d}}_{\bm{k}}\times{\bm{d}}_{\bm{k}}^{*}\right|^{2}-4\left|{\rm
Re}(\psi_{{\bm{k}},I}^{*}{\bm{d}}_{\bm{k}})\right|^{2}$
$\displaystyle+\frac{|\psi_{{\bm{k}},-}|^{2}({\bm{d}}_{\bm{k}}\cdot{\bm{d}}_{\bm{k}}^{*}+|\psi_{{\bm{k}},I}|^{2})^{2}}{\xi_{{\bm{k}},-}^{2}}.$
(16)

From the assumption that
$|\xi_{{\bm{k}},-}|\simeq|E_{{\bm{k}},+}-E_{{\bm{k}},-}|$ is much larger than
the pair potentials, the last term of Eq. (16) is negligible, and the right-
hand-side of Eq. (16) becomes negative if the inter-band pairing breaks TRS
[see Eq. (9)]. On the other hand, far from the normal state Fermi surfaces of
both $+$ and $-$ bands, the Pfaffian (14) becomes positive because of the
$(\xi_{{\bm{k}},+}\xi_{{\bm{k}},-})^{2}$ term. Thus, the sign of the Pfaffian
changes in the Brillouin zone and BFSs appear.

##  III Spin-$\bm{3/2}$ System with cubic symmetry

As a concrete model for a two-band system with TRS, we consider spin-3/2
electrons in the Luttinger-Kohn model [62], where four internal degrees of
freedom reflect the total angular momentum $j=3/2$ combined from spin angular
momentum $s=1/2$ and orbital angular momentum $l=1$ of the $p$-band. Spin-3/2
electrons are known to appear, for example, in the $\Gamma_{8}$ band of cubic
crystals with strong spin-orbit interactions.

In this section, we introduce an $O_{h}$-symmetric BdG Hamiltonian in the
$j=3/2$ basis. When we describe the internal degrees of freedom using the
$j=3/2$ basis, we use quantities with bars, such as
$\bar{\bm{c}}_{\bm{k}}=(\bar{c}_{{\bm{k}},3/2},\bar{c}_{{\bm{k}},1/2},\bar{c}_{{\bm{k}},-1/2},\bar{c}_{{\bm{k}},-3/2})^{\rm
T}$, $\bar{H}_{0}({\bm{k}})$, and $\bar{\Delta}({\bm{k}})$, which are related
to the ones in the band basis via a unitary transformation.

###  III.1 Normal-state Hamiltonian

The normal part of an $O_{h}$-symmetric Hamiltonian in a continuum up to the
second order of momentum is given by [62]

$\displaystyle\bar{H}_{0}({\bm{k}})=$
$\displaystyle\left(\frac{\hbar^{2}k^{2}}{2m_{0}}-\mu\right)\mathbbm{1}_{4}$
$\displaystyle+\frac{3\hbar^{2}}{4m_{1}}\sum_{\nu=1,2,3}\mathscr{Y}_{\nu}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}})$
$\displaystyle+\frac{3\hbar^{2}}{4m_{2}}\sum_{\nu=4,5}\mathscr{Y}_{\nu}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}}),$
(17)

where $\mathscr{Y}_{\nu}(\bm{A})\ (\nu=1,2,3,4,5)$ are the rank-2 irreducible
tensors constitute of a three-dimensional vector
${\bm{A}}=(A_{x},A_{y},A_{z})$:

$\displaystyle\mathscr{Y}_{1}(\bm{A})$
$\displaystyle=\frac{1}{\sqrt{3}}(A_{x}A_{y}+A_{y}A_{x}),$ (18a)
$\displaystyle\mathscr{Y}_{2}(\bm{A})$
$\displaystyle=\frac{1}{\sqrt{3}}(A_{y}A_{z}+A_{z}A_{y}),$ (18b)
$\displaystyle\mathscr{Y}_{3}(\bm{A})$
$\displaystyle=\frac{1}{\sqrt{3}}(A_{z}A_{x}+A_{x}A_{z}),$ (18c)
$\displaystyle\mathscr{Y}_{4}(\bm{A})$
$\displaystyle=\frac{1}{\sqrt{3}}(A_{x}^{2}-A_{y}^{2}),$ (18d)
$\displaystyle\mathscr{Y}_{5}(\bm{A})$
$\displaystyle=\frac{1}{3}(2A_{z}^{2}-A_{x}^{2}-A_{y}^{2}),$ (18e)

and ${\bm{J}}=(J_{x},J_{y},J_{z})$ is the vector of the spin-3/2 matrices. A
set of $\gamma^{\nu}\equiv\mathscr{Y}_{\nu}({\bm{J}})$ corresponds to the
anticommuting Dirac matrices [40]. The specific forms of the matrices
$J_{x,y,z}$ and $\gamma^{\nu}$ will be given in Appendix A. Since
$\mathscr{Y}_{1,2,3}(\bm{A})$ and $\mathscr{Y}_{4,5}(\bm{A})$ are the basis of
irreducible representations $T_{2g}$ and $E_{g}$, respectively, of the point
group $O_{h}$, the effective masses $m_{1}$ and $m_{2}$ are independent
constants. For the sake of simplicity, we choose $m_{1}=m_{2}$ in the
following calculations, with which the Hamiltonian (17) becomes invariant
under $O(3)$. Then, the eigenvalue of the Hamiltonian (17) is given by

$\displaystyle\xi_{{\bm{k}},\pm}$
$\displaystyle=\frac{\hbar^{2}k^{2}}{2M_{\pm}}-\mu,$ (19) $\displaystyle
M_{\pm}$ $\displaystyle=\frac{1}{1/m_{0}\pm 1/m_{1}},$ (20)

which gives spherically symmetric normal-state Fermi surfaces. We further
choose $0<m_{1}<|m_{0}|$ and $\mu>0$ so that only the $+$ band has a Fermi
surface as shown in Fig. 1.

$E$$\bm{k}$$-\mu$$\xi_{\bm{k},-}$$\xi_{\bm{k},+}$$-k_{\mathrm{F}}$$k_{\mathrm{F}}$$\bm{0}$
Figure 1:  Schematic energy dispersion of $j=3/2$ electrons in the Luttinger-
Kohn model around ${\bm{k}}={\bm{0}}$ for $0<m_{1}=m_{2}<|m_{0}|$ and $\mu>0$.
There are two doubly-degenerate bands, $\xi_{\bm{k},+}$ and $\xi_{\bm{k},-}$
[Eq. (19)], and only the $+$ bands have Fermi surfaces.

###  III.2 Pairing Hamiltonian

The pair potential for a spin-3/2 system is distinct from that for a simple
two-band system in that the pairing states can be classified according to the
total angular momentum of a Cooper pair $J_{\textrm{pair}}$. For the case of
even-parity pairing, the internal degrees of freedom of a Cooper pair can be
singlet ($J_{\textrm{pair}}=0$), quintet ($J_{\textrm{pair}}=2$), or a
superposition of them [63]. By using the unitary part of the time-reversal
operator

$\displaystyle U_{T}=\exp\left(-i\pi J_{y}\right)=\begin{pmatrix}0&0&0&-1\\\
0&0&1&0\\\ 0&-1&0&0\\\ 1&0&0&0\end{pmatrix}$ (21)

and the rank-2 irreducible tensors $\mathscr{Y}_{j}({\bm{J}})$, we write a
general form of the pairing part of the Hamiltonian as

$\displaystyle\bar{\Delta}({\bm{k}})=\eta_{0}({\bm{k}})U_{T}+\sum_{\nu=1}^{5}\eta_{\nu}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}})U_{T},$
(22)

where the first and second terms corresponds to the $J_{\textrm{pair}}=0$ and
$J_{\textrm{pair}}=2$ pair potentials, respectively, and $\eta_{0}({\bm{k}})$
and $\eta_{\nu}({\bm{k}})$ are even functions of ${\bm{k}}$.

Under the cubic symmetry, the pair potential is further divided into three
irreducible representations $A_{1g}$, $T_{2g}$, and $E_{g}$, respectively
given by

$\displaystyle A_{1g}:\ \ \bar{\Delta}({\bm{k}})$
$\displaystyle=\eta_{0}({\bm{k}})U_{T},$ (23a) $\displaystyle T_{2g}:\ \
\bar{\Delta}({\bm{k}})$
$\displaystyle=\sum_{\nu=1,2,3}\eta_{\nu}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}})U_{T},$
(23b) $\displaystyle E_{g}:\ \ \bar{\Delta}({\bm{k}})$
$\displaystyle=\sum_{\nu=4,5}\eta_{\nu}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}})U_{T}.$
(23c)

Because we are interested in TRS-broken states, we consider the following
three cases of $J_{\textrm{pair}}=2$ pairs:

$\displaystyle T_{2g}(0,i,1):$ $\displaystyle\
\bm{\eta}=\Delta_{0}(0,0,i,1,0,0),$ (24a) $\displaystyle
T_{2g}(1,\omega,\omega^{2}):$ $\displaystyle\
\bm{\eta}=\Delta_{0}(0,1,\omega,\omega^{2},0,0),$ (24b) $\displaystyle
E_{g}(1,i):$ $\displaystyle\ \bm{\eta}=\Delta_{0}(0,0,0,0,1,i),$ (24c)

where
$\bm{\eta}({\bm{k}})\equiv(\eta_{0}({\bm{k}}),\eta_{1}({\bm{k}}),\cdots,\eta_{5}({\bm{k}}))$,
$\omega\equiv e^{i2\pi/3}$, and we consider only the simplest cases of
${\bm{k}}$-independent gap functions.

###  III.3 Description in the band basis

We introduce a unitary transformation $U(\bm{k})$ that diagonalizes the normal
Hamiltonian

$\displaystyle
U^{\dagger}({\bm{k}})\bar{H}_{0}({\bm{k}})U({\bm{k}})=\operatorname{Diag}[\xi_{{\bm{k}}+},\xi_{{\bm{k}}+},\xi_{{\bm{k}}-},\xi_{{\bm{k}}-}],$
(25)

and describe the system in the band basis. Because the band basis
$\bm{c}_{\bm{k}}$ is related to the $j=3/2$ bases $\bar{\bm{c}}_{\bm{k}}$ as
$\bar{\bm{c}}_{\bm{k}}=U({\bm{k}})\bm{c}_{\bm{k}}$ and
$\bar{\bm{c}}^{\dagger}_{\bm{k}}=U^{*}({\bm{k}})\bm{c}_{\bm{k}}^{\dagger}$,
the pairing part of the Hamiltonian is written in the band basis as

$\displaystyle\Delta({\bm{k}})=U^{\dagger}({\bm{k}})\bar{\Delta}({\bm{k}})U^{*}({\bm{k}}).$
(26)

According to the discussion in Sec. II, this $\Delta({\bm{k}})$ is written in
the form of Eq. (7).

We note here that because the unitary transformation $U({\bm{k}})$ depends on
${\bm{k}}$, $\Delta({\bm{k}})$ in the band basis becomes ${\bm{k}}$-dependent
even when $\bar{\Delta}({\bm{k}})$ is ${\bm{k}}$-independent. In particular,
in the present cases of $J_{\textrm{pair}}=2$ under the $O(3)$ symmetric
normal-state Hamiltonian, the effective intra-band pair potential
$\psi_{{\bm{k}},+}$ has $d$-wave symmetry. This can be proved as follows.
First, $U({\bm{k}})$ for the present system is given by

$\displaystyle
U({\bm{k}})=e^{i\pi}e^{-iJ_{z}\phi}e^{-iJ_{y}\theta}\begin{pmatrix}1&0&0&0\\\
0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\end{pmatrix}$ (27)

where $\theta=\arccos{k_{z}/k}$ and $\phi=\arg(k_{x}+ik_{y})$. This unitary
operator transforms $\mathscr{Y}_{\nu}({\bm{J}})U_{T}$ in the pair potential
as

$\displaystyle\mathcal{P}_{+}[U^{\dagger}({\bm{k}})\mathscr{Y}_{\nu}({\bm{J}})U_{T}U^{*}({\bm{k}})]=\frac{3}{2}\mathscr{Y}_{\nu}({\bm{k}})i\sigma_{y}$
(28)

where $\mathcal{P}_{+}$ means the projection onto the $+$ band, i.e., taking
the upper left $2\times 2$ matrix of a $4\times 4$ matrix. Equation (28)
clearly shows that the anisotropic pairing in the spin space of the
$J_{\textrm{pair}}=2$ system is transcribed to the anisotropic
${\bm{k}}$-dependence of the intra-band pairing. The spin-singlet pair
potentials corresponding to the states in Eq. (24) are respectively given by

$\displaystyle T_{2g}(0,i,1):$ $\displaystyle\
\psi_{{\bm{k}},+}=\Delta_{0}\sqrt{3}k_{z}(k_{x}+ik_{y}),$ (29a) $\displaystyle
T_{2g}(1,\omega,\omega^{2}):$ $\displaystyle\
\psi_{{\bm{k}},+}=\Delta_{0}\sqrt{3}(k_{x}k_{y}+\omega
k_{y}k_{z}+\omega^{2}k_{z}k_{x}),$ (29b) $\displaystyle E_{g}(1,i):$
$\displaystyle\ \psi_{{\bm{k}},+}=\Delta_{0}e^{i\pi/6}(k_{x}^{2}+\omega
k_{y}^{2}+\omega^{2}k_{z}^{2}).$ (29c)

Here, the intra-band pair potential for $T_{2g}(0,i,1)$ is the 3D chiral
$d$-wave state, which is a possible candidate for the superconducting state in
Sr2RuO4 [64], whereas that for $T_{2g}(1,\omega,\omega^{2})$ is the 3D cyclic
$d$-wave state, which is predicted to appear in neutron stars [65].

All of the three pairing states in Eq. (29) have nodal gap structures,
indicating the appearance of the BFSs in the original spin-3/2 system. In
addition, because the d-wave pairing states in Eq. (29) are topological, zero-
energy surface flat bands and surface arc states appear in a particular
direction of the interface, which resonantly increases the tunnel conductance
at zero bias [29, 30]. We, therefore, focus on how such a behavior of the
tunnel conductance around zero bias changes when the line nodes and point
nodes deform to the BFSs by comparing the $J_{\rm pair}=2$ pairing states in
Eq. (24) and the spin-singlet d-wave pairing states in Eq. (29).

##  IV Setup and methods

We consider a normal metal/superconductor (N/S) junction as schematically
shown in Fig. 2 and compare the results for $j=3/2$ and $j=1/2$ systems having
the same symmetry. For the case of the $j=3/2$ system, we use the same normal-
state Hamiltonian $\bar{H}_{0}({\bm{k}})$ given by Eq. (17) both in the normal
metal and the superconductor. The pair potential is zero in the normal metal,
whereas that in the superconductor is one of Eq. (24). For the case of the
$j=1/2$ system, we use the normal state Hamiltonian
$\xi_{{\bm{k}},+}\otimes\mathbbm{1}_{2}$, where $\xi_{{\bm{k}},+}$ is given by
Eq. (19) with $M_{+}$ and $\mu$ being the same as those in the $j=3/2$ model.
The pair potential is zero in the normal metal, whereas that in the
superconductor has d-wave symmetry given by one of Eq. (29).

Figure 2: Schematic illustration of a normal metal/superconductor (N/S)
junction. We use the same normal state Hamiltonian for both sides and
introduce superconducting pair potential in the $z>0$ side. The transfer
matrix at the interface is multiplied by $1/Z$ with $Z$ being the barrier
parameter.

To calculate the Green’s function at the interface, we use the recursive
Green’s function method [54, 59], which is applicable for a tight-binding
model along the perpendicular direction to the interface. We choose the $z$
axis perpendicular to the interface and replace the terms including $k_{z}$ in
the BdG Hamiltonian as

$\displaystyle k_{x,y}k_{z}$ $\displaystyle\to\frac{1}{a}k_{x,y}\sin k_{z}a,$
(30a) $\displaystyle k_{z}^{2}$ $\displaystyle\to\frac{2}{a^{2}}(1-\cos
k_{z}a),$ (30b)

with $a$ being the lattice spacing along the $z$ axis, so that the Hamiltonian
is rewritten in terms of the Fourier-transformed operators with respect to
$k_{z}$,
${\bm{f}}_{l,{\bm{k}}_{\perp}}=\sum_{k_{z}}e^{ik_{z}al}\bar{\bm{c}}_{{\bm{k}}}$
with ${\bm{k}}_{\perp}=(k_{x},k_{y})$. The resulting Hamiltonian with an
interface between $l=0$ and $1$ is given by
$\mathcal{H}=\sum_{{\bm{k}}_{\perp}}\mathcal{H}({\bm{k}}_{\perp})$ with

$\displaystyle\mathcal{H}({\bm{k}}_{\perp})=$
$\displaystyle\sum_{l\leq-1}h^{\rm N}_{\rm L}(l,{\bm{k}}_{\perp})+h_{\rm
I}({\bm{k}}_{\perp})$ $\displaystyle+\sum_{l\geq 1}\left[h^{\rm N}_{\rm
R}(l,{\bm{k}}_{\perp})+h^{\rm S}_{\rm R}(l,{\bm{k}}_{\perp})\right],$ (31)

where $h^{\rm N}_{\rm L(R)}(l,{\bm{k}}_{\perp})$, $h^{\rm S}_{\rm
R}(l,{\bm{k}}_{\perp})$, $h_{\rm I}({\bm{k}}_{\perp})$ are the normal-state
Hamiltonian in the left-hand (right-hand) side of the interface, the pairing
Hamiltonian in the right-hand side, and the Hamiltonian at the interface,
respectively. We use the same normal-state Hamiltonian:

$\displaystyle h_{\rm L}^{\rm N}(l,{\bm{k}}_{\perp})=$ $\displaystyle h_{\rm
R}^{\rm N}(l,{\bm{k}}_{\perp})$ $\displaystyle=$
$\displaystyle-(\bm{f}_{l+1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}t({\bm{k}}_{\perp})\bm{f}_{l,{\bm{k}}_{\perp}}$
$\displaystyle-(\bm{f}_{l,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}t^{\dagger}({\bm{k}}_{\perp})\bm{f}_{l+1,{\bm{k}}_{\perp}}$
$\displaystyle+(\bm{f}_{l,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}u({\bm{k}}_{\perp})\bm{f}_{l,{\bm{k}}_{\perp}},$ (32)

where $t({\bm{k}}_{\perp})$ and $u({\bm{k}}_{\perp})$ are the
${\bm{k}}_{\perp}$-dependent hopping matrix and the on-site potential matrix,
respectively, derived from the continuum model with the replacement in Eq.
(30). Here, $t({\bm{k}}_{\perp})$ and $u({\bm{k}}_{\perp})$ are $4\times 4$
($2\times 2$) matrices for the $j=3/2$ ($j=1/2$) system, whose detailed forms
are given in Appendix B. We introduce barrier parameter $Z$ at the interface
and use the following interface Hamiltonian:

$\displaystyle h_{\rm I}({\bm{k}}_{\perp})=$
$\displaystyle-\frac{1}{Z}\left[(\bm{f}_{1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}t({\bm{k}}_{\perp})\bm{f}_{0,{\bm{k}}_{\perp}}+(\bm{f}_{0,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}t^{\dagger}({\bm{k}}_{\perp})\bm{f}_{1,{\bm{k}}_{\perp}}\right]$
$\displaystyle+(\bm{f}_{0,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}u({\bm{k}}_{\perp})\bm{f}_{0,{\bm{k}}_{\perp}}.$ (33)

The pairing Hamiltonian consists of the on-site and off-site pairing terms:

$\displaystyle h_{\rm R}^{\rm S}(l,{\bm{k}}_{\perp})=$
$\displaystyle(\bm{f}_{l,\bm{k}_{\perp}}^{\dagger})^{\rm T}\bar{\Delta}_{\rm
on}({\bm{k}}_{\perp})\bm{f}_{l,-\bm{k}_{\perp}}^{\dagger}$
$\displaystyle+(\bm{f}_{l,\bm{k}_{\perp}}^{\dagger})^{\rm T}\bar{\Delta}_{\rm
off}({\bm{k}}_{\perp})\bm{f}_{l+1,-\bm{k}_{\perp}}^{\dagger}$
$\displaystyle+\textrm{H. c.},$ (34)

where H. c. stands for the Hermitian conjugate. In the case of the $j=3/2$
system with the $\bm{k}$-independent pair potentials in Eq. (24), only the on-
site pair potential exists, i.e.,

$\displaystyle\bar{\Delta}_{\rm on}({\bm{k}}_{\perp})=\bar{\Delta},\ \
\bar{\Delta}_{\rm off}=0,$ (35)

where $\bar{\Delta}$ is given by Eq. (22) with ${\bm{k}}$-independent
$\eta_{j}$’s in Eq. (24). On the other hand, in the $j=1/2$ system, the pair
potential has $d$-wave symmetry, and $h_{\rm R}^{\rm S}$ includes both the on-
site and off-site pairing terms. The concrete forms of $\bar{\Delta}_{\rm
on}({\bm{k}}_{\perp})$ and $\bar{\Delta}_{\rm off}({\bm{k}}_{\perp})$
corresponding to the pair potentials in Eq. (29) is given in Appendix B.

The retarded (advanced) Green’s function for the BdG Hamiltonian is written as

$\displaystyle\mathcal{G}^{\rm R(A)}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})$
$\displaystyle=\left[\epsilon+(-)i\delta_{\epsilon}-\mathcal{H}({\bm{k}}_{\perp})\right]^{-1}_{ll^{\prime}}$
(36) $\displaystyle=\begin{bmatrix}G^{\rm
R(A)}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})&F^{\rm
R(A)}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})\\\\[2.84526pt] \tilde{F}^{\rm
R(A)}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})&\tilde{G}^{\rm
R(A)}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})\end{bmatrix}.$ (37)

Here, $G^{\rm R(A)}_{ll^{\prime}},\tilde{G}^{\rm R(A)}_{ll^{\prime}},F^{\rm
R(A)}_{ll^{\prime}}$ and $\tilde{F}^{\rm R(A)}_{ll^{\prime}}$ are $4\times 4$
($2\times 2$) matrices for the $j=3/2$ ($j=1/2$) system. We calculate the
Green’s function at the interface by using the recursive Green’s function
method.

To see the effects of the BFSs and topological surface states on the charge
conductance, we first calculate the SDOSs of the $j=1/2$ and $j=3/2$
superconductors and compare them with the LDOSs deep inside the
superconductors. Here, SDOS, $\rho_{\rm S}(\epsilon)$, and momentum-resolved
SDOS, $\rho_{\rm S}(\epsilon,{\bm{k}}_{\perp})$, are calculated via

$\displaystyle\rho_{\rm S}(\epsilon)$
$\displaystyle=\sum_{{\bm{k}}_{\perp}}\rho_{\rm
S}(\epsilon,{\bm{k}}_{\perp}),$ (38a) $\displaystyle\rho_{\rm
S}(\epsilon,{\bm{k}}_{\perp})$ $\displaystyle=-\frac{1}{\pi}{\rm Im}[G^{\rm
R}_{11}(\epsilon,{\bm{k}}_{\perp})],$ (38b)

under the Hamiltonian with $h_{\rm L}^{\rm N}(l,{\bm{k}}_{\perp})=h_{\rm
I}({\bm{k}}_{\perp})=0$. Similarly, bulk LDOS,
$\rho_{\textrm{S}}^{\textrm{(bulk)}}(\epsilon)$, inside the superconductor is
given by the same form as Eq. (38) but with $G_{11}^{\rm
R}(\epsilon,{\bm{k}}_{\perp})$ calculated by adding the same pairing
Hamiltonian in both sides and choosing $Z=1$. We also define the normal-state
version of these quantities, $\rho_{\textrm{N}}(\epsilon)$ and
$\rho_{\textrm{N}}^{\textrm{(bulk)}}(\epsilon)$, which are calculated using
the same code but with choosing $\Delta_{0}=0$.

Next, we calculate the charge conductance by utilizing the Lee-Fisher formula
[55]. Here, we use the generalized one for the case with internal degrees of
freedom [56]. We describe the formula in the Nambu space (see Appendix C for
details) and obtain

$\displaystyle\Gamma(\epsilon)=\frac{e^{2}}{2h}$
$\displaystyle\operatorname{Tr}\sum_{{\bm{k}}_{\perp}}$
$\displaystyle\bigg{[}$
$\displaystyle-\bar{\mathcal{G}}_{01}(\epsilon,{\bm{k}}_{\perp}){T}({\bm{k}}_{\perp})\bar{\mathcal{G}}_{01}(\epsilon,{\bm{k}}_{\perp}){T}({\bm{k}}_{\perp})$
$\displaystyle-\bar{\mathcal{G}}_{10}(\epsilon,{\bm{k}}_{\perp}){T}^{\dagger}({\bm{k}}_{\perp})\bar{\mathcal{G}}_{10}(\epsilon,{\bm{k}}_{\perp}){T}^{\dagger}({\bm{k}}_{\perp})$
$\displaystyle+\bar{\mathcal{G}}_{11}(\epsilon,{\bm{k}}_{\perp}){T}({\bm{k}}_{\perp})\bar{\mathcal{G}}_{00}(\epsilon,{\bm{k}}_{\perp}){T}^{\dagger}({\bm{k}}_{\perp})$
$\displaystyle+\bar{\mathcal{G}}_{00}(\epsilon,{\bm{k}}_{\perp}){T}^{\dagger}({\bm{k}}_{\perp})\bar{\mathcal{G}}_{11}(\epsilon,{\bm{k}}_{\perp}){T}({\bm{k}}_{\perp})\bigg{]},$
(39)

where

$\displaystyle\bar{\mathcal{G}}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})$
$\displaystyle=\frac{1}{2i}[\mathcal{G}^{\rm
A}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})-\mathcal{G}^{\rm
R}_{ll^{\prime}}(\epsilon,{\bm{k}}_{\perp})],$ (40) $\displaystyle
T({\bm{k}}_{\perp})$
$\displaystyle=\begin{bmatrix}\bm{t}({\bm{k}}_{\perp})&\bf 0\\\ \bf
0&\bm{t}^{*}(-{\bm{k}}_{\perp})\end{bmatrix}.$ (41)

To see the effect of the BFSs on the tunneling spectroscopy, we calculate the
conductance $\Gamma_{\rm S}(\epsilon)$ for the N/S junction and normalize it
with that for the normal metal/normal metal junction, $\Gamma_{\rm
N}(\epsilon)$, which is calculated by choosing $\Delta_{0}=0$.

In the following numerical calculations, we use $\mu$ as the unit of energy
and choose

$\displaystyle\frac{\hbar^{2}}{m_{0}a^{2}}$ $\displaystyle=\frac{\mu}{6},$
(42a) $\displaystyle\frac{\hbar^{2}}{m_{1}a^{2}}$
$\displaystyle=\frac{\hbar^{2}}{m_{2}a^{2}}=4\mu.$ (42b)

The corresponding Fermi wave number for the $+$ band in the continuum model is
$k_{\rm F}a=\sqrt{2M_{+}\mu}/\hbar=2\sqrt{3}/5\sim 0.69$. We choose the
infinitesimal value $\delta_{\epsilon}$ as $\delta_{\epsilon}/\mu=5.0\times
10^{-5}$ for LDOS and SDOS calculations and $\delta_{\epsilon}/\mu=1.0\times
10^{-8}$ for conductance calculations.

##  V Results

In this section, we show the results of the SDOS and the charge conductance
for each pairing state of Eqs. (24) and (29), which are briefly summarized as
follows. The $d$-wave pairing states in Eqs. (29a), (29b), and (29c) have
topological zero-energy flat-band SABSs, topological SABSs with surface arc
states, and no SABS, respectively, on the (001) surface. When the point nodes
and line nodes deform to BFSs in the $J_{\textrm{pair}}=2$ pairing states, the
topological SABSs shift from zero energy or move in the surface Brillouin
Zone. As a result, the zero-energy peak of the SDOS and the ZBCP arising in
the $d$-wave states are blunted or split in the corresponding
$J_{\textrm{pair}}=2$ states [$T_{2g}(0,i,1)$ and
$T_{2g}(1,\omega,\omega^{2})$ states in Eqs. (24a) and (24b), respectively].
On the other hand, the states on the BFSs have a nonzero contribution to the
charge conductance at zero energy, as in the case of normal state conduction.
This contribution is visible in the absence of SABSs [$E_{g}(1,i)$ state in
Eq. (24c)].

In the following sections, the results for d-wave pairing states are
independent of $\Delta_{0}/\mu$ when we scale the energy by $\Delta_{0}$. We
therefore show the numerical results only for $\Delta_{0}/\mu=0.2$ unless
otherwise noted. On the other hand, the results for the $J_{\textrm{pair}}=2$
pairing states in spin-3/2 systems exhibit a clear dependence on
$\Delta_{0}/\mu$, which will be discussed in detail.

###  V.1 $\bm{T_{2g}\;(0,i,1)}$

First, we consider the $T_{2g}(0,i,1)$ state given by Eqs. (24a) and (29a).
Here, the $j=1/2$ superconducting state is the 3D chiral d-wave state which
features one line node on the $k_{x}k_{y}$ plane and two point nodes on the
$k_{z}$ axis as illustrated in Fig. 3(a). The nodal regions in the $j=3/2$
system are extended to the BFSs as shown in Fig. 3(b), which is calculated by
using Eq. (14). The BFSs extend alongside the Fermi surface [39]: the line
node expands into a circular ribbon shape, while the two point nodes stretch
into a disc-like geometry.

Figure 3: (a) Point and line nodes in 3D chiral $d$-wave superconducting state
[Eq. (29a)] and (b) Bogoliubov Fermi surfaces (BFSs) in spin-3/2
$T_{2g}(0,i,1)$ state [Eq. (24a)] with $\Delta_{0}/\mu=0.2$.  Figure 4:
Momentum-resolved SDOS at $k_{y}=0$ for 3D chiral $d$-wave state (a) and
spin-3/2 $T_{2g}(0,i,1)$ state with $\Delta_{0}/\mu=0.05$ (b) and $0.1$ (c).
Panel (a) does not depend on the value of $\Delta_{0}/\mu.$ In this and the
following figures, the results for the d-wave case are calculated for
$\Delta_{0}/\mu=0.2$ unless otherwise noted. The doubly-degenerate zero-energy
flat bands in (a) split in the presence of BFSs in (b) and (c), where the
energy splitting increases in proportion to $(\Delta_{0}/\mu)^{2}$.  Figure 5:
Momentum-resolved zero-energy SDOS for 3D chiral d-wave state (a) and spin-3/2
$T_{2g}(0,i,1)$ state with $\Delta_{0}/\mu=0.05$ (b) and $0.1$ (c). Panel (a)
does not depend on the value of $\Delta_{0}/\mu.$ In panels (b) and (c), the
zero-energy states remain on a ring, protected by the $CC_{2z}$ symmetry (see
text).

Figures 4 and 5 show the numerically obtained momentum-resolved SDOS at
$k_{y}=0$ and $\epsilon=0$, respectively. In the case of the 3D chiral
$d$-wave state [Figs. 4(a) and 5(a)], doubly-degenerate zero-energy flat bands
emerge, which are protected by the 1D winding number [29]. On the other hand,
in the case of the spin-3/2 $T_{2g}(0,i,1)$ state, the TRS-breaking pair
potential lifts the degeneracy, and the surface states become dispersive as
shown in Figs. 4(b), 4(c), 5(b), and 5(c). Because the spin polarization Eq.
(9), which is proportional to $\Delta_{0}^{2}$, works as a pseudo-magnetic
field, the maximum energy splitting at $k_{x}=k_{y}=0$ increases in proportion
to $(\Delta_{0}/\mu)^{2}$ [40]. Here, we note that the zero-energy states
always remain on a ring on the surface Brillouin zone as shown in Figs. 5(b)
and 5(c) due to the ${\bm{k}}$-dependence of the spin polarization. This is a
distinctive difference from the case when we apply a uniform magnetic field to
the 3D chiral d-wave state, where the flat bands remain flat and shift in
opposite directions. The existence of the ring-shaped zero-energy states is
topologically protected by the Pfaffian associated with the $CC_{2z}$ symmetry
of the BdG Hamiltonian with a boundary [66, 67], where $C_{2z}$ is the
rotation by $\pi$ about the $z$ axis.

Figure 6:  Bulk LDOS (a) and SDOS (b) of $T_{2g}(0,i,1)$ states normalized by
their values at $\Delta_{0}=0$. Inset in (a) is a magnified view around
$\epsilon=0$, and inset in (b) is the SDOS of 3D chiral d-wave state with an
expanded vertical axis. The results for the d-wave state are independent of
the value of $\Delta_{0}/\mu.$ (a) The BFSs contribute to the nonzero bulk
LDOS at $\epsilon=0$. (b) The appearance of the BFSs blunts the sharp ZEP in
the d-wave case.

To clarify the contributions of BFSs, we compare the SDOS with the bulk LDOS
in Fig. 6. As known in the previous studies [29, 30, 68, 69], the bulk LDOS in
the 3D chiral d-wave superconductor exhibits a V-shaped structure in the
vicinity of $\epsilon=0$ because of the line node. By contrast, the bulk LDOS
in the spin-3/2 $T_{2g}(0,i,1)$ superconductor takes a constant value around
$\epsilon=0$ proportional to $\Delta_{0}/\mu$. This increase in the LDOS at
$\epsilon=0$ is indeed attributed to the zero-energy states on the BFSs, whose
area in the momentum space is proportional to $\Delta_{0}/\mu$ [39].
Furthermore, the energy range for the constant LDOS is proportional to
$(\Delta_{0}/\mu)^{2}$, being consistent with the shift in the dispersion due
to the spin-polarization. The splitting of the coherence peak at
$|\epsilon/\Delta_{0}|\sim 0.7$ is also explained by the dispersion shift due
to the pseudo-magnetic field, and the width of the splitting is proportional
to $(\Delta_{0}/\mu)^{2}$. When we terminate the system at the (001) surface,
a sharp zero-energy peak emerges in the SDOS of the 3D chiral d-wave state due
to the zero-energy surface flat bands [Fig. 6(b)]. Though the SDOS of the
spin-3/2 $T_{2g}(0,i,1)$ state also has a zero-energy peak, it becomes less
pronounced or blunted. This blunting effect is a consequence of the appearance
of the BFSs: The peak width is determined by the dispersion shift, being in
proportion to $(\Delta_{0}/\mu)^{2}$. Here, each of SDOS of the spin-3/2
$T_{2g}(0,i,1)$ state has several small peaks, which is attributed to the
contribution of the surface state that originally exists in the normal state
(see the discussion around Fig. 12).

Figure 7:  Normalized charge conductance
$\Gamma_{\textrm{S}}(\epsilon)/\Gamma_{\textrm{N}}(\epsilon)$ at N/S junction
with $Z=10$ for 3D chiral d-wave and spin-3/2 $T_{2g}(0,i,1)$ states, where
$\Gamma_{\textrm{N}}(\epsilon)$ is calculated by using the same code as
$\Gamma_{\textrm{S}}(\epsilon)$ but with $\Delta_{0}=0$. Inset is the result
for 3D chiral d-wave state with an expanded vertical axis, which is
independent of the value of $\Delta_{0}/\mu.$ In accordance with the change in
SDOS in Fig. 6(b), the appearance of the BFSs blunts the ZBCP.

Figure 7 displays the results of the normalized charge conductance
$\Gamma_{\textrm{S}}(\epsilon)/\Gamma_{\textrm{N}}(\epsilon)$ in the N/S
junction with barrier parameter $Z=10$. In the case of the 3D chiral d-wave
superconductor, the ZBCP associated with the zero-energy peak in SDOS emerges
as known in the previous studies [29, 30, 70]. The charge conductance for the
spin-3/2 $T_{2g}$(0,i,1) state also reflects the structure in SDOS: Similarly
to the zero-energy peak in Fig. 6(b), the ZBCP is blunted as a consequence of
the appearance of the BFSs. This blunting of the ZECP is an observable
signature of the BFSs.

Figure 8:  The transmittance dependence of ZBCP for 3D chiral d-wave and
spin-3/2 $T_{2g}(0,i,1)$ states at $Z=10$. Inset is the result in 3D chiral
d-wave SC state with an expanded vertical axis, which is independent of the
value of $\Delta_{0}/\mu$. In the case of spin-3/2 $T_{2g}(0,i,1)$ state, the
ZBCP is more suppressed for a larger $\Delta_{0}/\mu$ and remains finite even
at low transmittance.

At the end of this subsection, we discuss the normalized charge conductance at
zero energy. Figure 8 shows the relationship between the normal conductance
and the normalized charge conductance. It is known that the zero-energy
conductance of the 3D chiral d-wave superconductor, regardless of the
magnitude of the pair amplitude, diverges as the normal state conductance
decreases (or barrier parameter $Z$ increases), according to the relation
$\Gamma_{\mathrm{S}}/\Gamma_{\mathrm{N}}\propto 1/\Gamma_{\mathrm{N}}$ [23].
By contrast, the zero-energy conductance of the spin-3/2 $T_{2g}(0,i,1)$ state
takes a finite value in the limit of $\Gamma_{\mathrm{N}}\to 0$, which
decreases as $\Delta_{0}/\mu$ increases. This result can be explained by
noting the fact that $\Gamma_{\textrm{S}}(\epsilon=0)$ in the presence of BFSs
consists of two contributions: One is the Andreev reflection via the zero-
energy SABSs; The other is the residual electrical conduction via the zero-
energy states on the BFSs. In the present case, the former contribution is
strongly suppressed because most of the SABSs shift from zero energy due to
the existence of the BFSs [see the difference between Figs. 5(a) and 5(b)(c)]
[29]. On the other hand, the latter behaves similarly to the normal-state
conduction and goes zero as $\Gamma_{\textrm{N}}\to 0$. The combined effect of
these two results in a finite value of
$\Gamma_{\textrm{S}}/\Gamma_{\textrm{N}}$ at $\Gamma_{\textrm{N}}\to 0$.

Short summary for $T_{2g}(0,i,1)$ state: When the line node in the 3D chiral
d-wave state changes into the BFS, the topologically protected zero-energy
surface flat bands split and become dispersive. This change suppresses the
sharp zero-energy peak in the SDOS and the ZBCP, and the peak width becomes
roughly in proportion to the square of the area of the BFS in the momentum
space. In addition to the conventional Andreev reflection, the zero-energy
states on the BFSs contribute to the charge conductance at $\epsilon=0$,
leading to the unusual transmittance dependence of the normalized conductance.

###  V.2 $\bm{T_{2g}\;(1,\omega,\omega^{2})}$

Figure 9:  (a) Point nodes in 3D cyclic $d$-wave superconducting state [Eq.
(29b)] and (b) BFSs in spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ state [Eq.
(24b)] with $\Delta_{0}/\mu=0.2$. Red (blue) color on the point nodes and BFSs
indicates that the Chern number defined on a surface enclosing the point node
or the BFS is $+2$ ($-2$).

Next, we consider the $T_{2g}(1,\omega,\omega^{2})$ state whose pairing
potentials are given by Eqs. (24b) and (29b) for spin-3/2 and spin-1/2
superconductors, respectively. The spin-1/2 pairing state is the 3D cyclic
d-wave superconducting state, which has a pair of point nodes along each of
the $k_{x}=k_{y}=k_{z}$ line and the $x$, $y$, and $z$ axes, and has eight
point nodes in total [Fig. 9(a)]. Correspondingly, the spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ state has eight BFSs extend alongside the Fermi
surface as illustrated in Fig. 9(b), which is calculated by using Eq. (14).
The color of each node and each BFS in Fig. 9 indicates the sign of Chern
number $\operatorname{Ch}$ at each node, where the red (blue) color denotes
$\operatorname{Ch}=+2(-2)$. Here, Chern number is defined by [40, 29]:

$\displaystyle\operatorname{Ch}=\frac{i}{2\pi}\sum_{n\in\mathrm{occ}}\oint_{\mathcal{S}}d^{2}\bm{s}({\bm{k}})\cdot\left[\nabla_{{\bm{k}}}\times\braket{u_{n}({\bm{k}})}{\nabla_{{\bm{k}}}}{u_{n}({\bm{k}})}\right],$
(43)

where $\mathcal{S}$ is a closed surface surrounding the node or the BFS,
$d^{2}\bm{s}({\bm{k}})$ is a vectorial surface element in the momentum space,
$\ket{u_{n}({\bm{k}})}$ is an eigenstate of the BdG Hamiltonian $H_{\bm{k}}$,
and the summation is taken over all of the occupied states. The BFSs spread
alongside the Fermi surface with the increase of $\Delta_{0}$, and eventually,
four of them merge into one when $\Delta_{0}$ exceeds a critical value
($\Delta_{0}/\mu\sim 0.2168$ for our setup). The Chern number for the combined
BFS is the summation of those for each BFS before the merge.

Figure 10:  Momentum-resolved zero-energy SDOS for 3D cyclic $d$-wave (a) and
spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ states with $\Delta_{0}/\mu=0.05$ (b)
and $0.1$ (c). Panel (a) is independent of the value of $\Delta_{0}/\mu$. Two
arcs come out of each of the point nodes (a) or BFSs (b), (c) projected onto
the surface Brillouin zone. The BFSs split the doubly-degenerate arc states in
(a).

Figure 10 illustrates the momentum-resolved SDOS at $\epsilon=0$ for the 3D
cyclic d-wave superconductor (a), and for the spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ superconductor at $\Delta_{0}/\mu=0.05$ (b) and
$0.1$ (c). In Fig. 10(a), arc states arise on the $k_{x}=0$, $k_{y}=0$, and
$k_{x}=k_{y}$ lines, connecting the nodal points projected onto the surface
Brillouin zone. The location of the arcs is protected by the 1D winding number
associated with the chiral symmetry: In addition to the inversion symmetry,
the Hamiltonian with pair potential Eq. (29b) has the pseudo-time-reversal
symmetry on these lines, i.e., $H_{\bm{k}}$ satisfies
$\tilde{T}^{\dagger}H_{{\bm{k}}}\tilde{T}=H_{-{\bm{k}}}$ where
$\tilde{T}=U^{\dagger}TU$ with $T=i\sigma_{y}\tau_{0}\mathcal{K}$ being the
conventional time-reversal operator and
$U=e^{-i(\pi/3)\sigma_{0}\tau_{z}},e^{-i(2\pi/3)\sigma_{0}\tau_{z}}$ and $1$
for $H_{\bm{k}}$ on the lines $k_{x}=0,k_{y}=0$, and $k_{x}=k_{y}$,
respectively [29]. The winding number further reveals that the arc states on
these lines are doubly degenerate. In the spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ state, however, the pseudo-time-reversal
symmetry on these lines is no longer preserved, and the location of the arcs
changes. Nevertheless, the total number of arcs does not change because each
of the point nodes or BFSs projected onto the surface Brillouin zone is
attached to two arcs, reflecting their winding number of
$|\operatorname{Ch}|=2$. The doubly-degenerate arc states in the 3D cyclic
d-wave state split into two, and the splitting becomes larger as $\Delta_{0}$
increases.

Figure 11:  Bulk LDOS (a) and SDOS (b) of $T_{2g}(1,\omega,\omega^{2})$ states
normalized by their values at $\Delta_{0}=0$. Inset in (a) is a magnified view
around $\epsilon=0$, and inset in (b) is the SDOS of 3D cyclic d-wave state
with an expanded vertical axis. The results for the d-wave state are
independent of the value of $\Delta_{0}/\mu.$ (a) The BFSs contribute to the
nonzero bulk LDOS at $\epsilon=0$. (b) The appearance of the BFSs splits the
ZEP in the d-wave case.  Figure 12:  Relationship between SABSs and SDOS in
spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ state. (a)-(e) Momentum-resolved SDOS
at different energy levels indicated by arrows. (f) SDOS of spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ state at $\Delta_{0}/\mu=0.2$. The data in (f)
is the same as that shown in Fig. 11(b).

We show the bulk LDOS and the SDOS in Figs. 11(a) and 11(b), respectively. In
the case of the 3D cyclic d-wave superconductor with the point nodes shown in
Fig. 9(a), the bulk LDOS exhibits a characteristic U-shaped structure, with a
zero value at zero energy [71]. The bulk LDOS in the spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ state also has the same U-shaped structure.
However, as in the case of $T_{2g}(0,i,1)$ state, there are nonzero
$\rho_{\mathrm{S}}^{\textrm{(bulk)}}$ values at zero energy. This increment at
zero energy is nothing but the contributions from the states on the BFSs. On
the other hand, more drastic change happens in the SDOS: Whereas the SDOS of
the 3D cyclic d-wave state has a zero-energy peak with finite width, that of
the spin-3/2 $T_{2g}(1\omega,\omega^{2})$ state exhibits distinct dip and peak
structures. The disappearance of the zero-energy peak is the consequence of
the vanishing crossing point of arcs at $k_{x}=k_{y}=0$ in Fig. 10(a).

We can further understand the dip and peak structures of the SDOS by analyzing
the momentum-resolved SDOS with changing $\epsilon$. In Fig. 12, we show the
magnified view of the result for $\Delta_{0}/\mu=0.2$ in Fig. 11(b) together
with the momentum-resolved SDOS at each $\epsilon$ indicated by arrows. By
comparing the SDOS in Figs. 12(a) and 12(b), in particular inside the pink
solid circles and pink dashed circles, one can see that the small dip at
$\epsilon/\mu\sim 0.04$ in Fig. 12(f) is associated with the vanishing of two
SABSs encircled by pink solid curves in Fig. 12(a). On the other hand, the
peak at $\epsilon/\mu\sim 0.3$ in Fig. 12(f) is due to the reconnection of the
SABSs as seen in Figs. 12(c)-(e): In the course of the reconnection, the SABSs
have a crossed structure [Fig. 12(d)] as in the 3D cyclic d-wave case [Fig.
10(a)], leading to a peak in the SDOS. In general, a crossing of surface
states leads to a peak in SDOS [38]. In our model, the normal-state
Hamiltonian has a topological surface state with sharp dispersion, whose
contribution to the momentum-resolved SDOS is too narrow to be recognized. The
small peaks of SDOS in Figs. 6(b) and 17(b) are attributed to the reconnection
between SABSs and the surface state in normal state.

Figure 13:  Normalized charge conductance
$\Gamma_{\textrm{S}}(\epsilon)/\Gamma_{\textrm{N}}(\epsilon)$ at N/S junction
with $Z=10$ for 3D cyclic d-wave and spin-3/2 $T_{2g}(1,\omega,\omega^{2})$
states. Inset is the result for 3D chiral d-wave state with an expanded
vertical axis, which is independent of the value of $\Delta_{0}/\mu.$ In
accordance with the change in SDOS in Fig. 11(b), the appearance of the BFSs
splits the ZBCP.

Figure 13 shows the normalized charge conductance at the N/S junction with the
barrier parameter $Z=10$. In both cases of the 3D cyclic d-wave state and the
spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ state, the normalized charge
conductance has a similar structure as the SDOS in Fig. 11(b). Namely, the
ZBCP in the 3D cyclic d-wave state splits into two small peaks at nonzero
energy with the appearance of the BFSs.

Short summary for $T_{2g}(1,\omega,\omega^{2})$ state: In both the 3D cyclic
d-wave state and the spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ state, topological
surface arc states arise because the point nodes and BFSs projected onto the
surface Brillouin zone have nonzero Chern numbers. In the 3D cyclic d-wave
state, the location of the arcs is topologically restricted on the $k_{x}=0$,
$k_{y}=0$, and $k_{x}=k_{y}$ lines, and hence, the arcs cross at
$k_{x}=k_{y}=0$, whereas such a constraint is removed in the spin-3/2
$T_{2g}(1,\omega,\omega^{2})$ state, resulting in the disappearance of the
crossing point of the arcs. This change splits the zero-energy peak in SDOS
and the ZBCP in the 3D cyclic d-wave state and two small peaks appear at
nonzero energies in the spin-3/2 $T_{2g}(1,\omega,\omega^{2})$ state.

###  V.3 $\bm{E_{g}\;(1,i)}$

As a third case, we consider the $E_{g}(1,i)$ state, whose pair potential is
given by Eqs. (29c) and (24c) for spin-$1/2$ and spin-$3/2$ superconductors,
respectively. The spin-1/2 $E_{g}(1,i)$ state has eight point nodes along the
$(\pm 1,\pm 1,\pm 1)$ directions as shown in Fig. 14(a). Correspiondingly, the
spin-3/2 $E_{g}(1,i)$ state has eight BFSs that extend alongside the Fermi
surface as illustrated in Fig. 14(b). As in the case of the
$T_{2g}(1,\omega,\omega^{2})$ state, the point nodes and BFSs in the
$E_{g}(1,i)$ state have nonzero Chern numbers. The red (blue) color on the
nodes and BFSs indicates that the Chern number [Eq. (43)] of the node and the
BFS is $+2$ ($-2$).

Figure 14:  (a) Point nodes in spin-1/2 $E_{g}(1,i)$ state [Eq. (29c)] and (b)
BFSs in spin-3/2 $E_{g}(1,i)$ state [Eq. (24c)] with $\Delta_{0}/\mu=0.2$. Red
(blue) color on the point nodes and BFSs indicates that the Chern number
defined on a surface enclosing the point node or the BFS is $+2$ ($-2$).
Figure 15:  Momentum-resolved zero-energy SDOS for spin-1/2 $E_{g}(1,i)$ state
(a) and spin-3/2 $E_{g}(1,i)$ state with $\Delta_{0}/\mu=0.05$ (b) and $0.2$
(c). Panel (a) is independent of the value of $\Delta_{0}/\mu$. In (a),
$\rho_{\textrm{S}}(0)$ is nonzero only on the point nodes, which are invisible
in this graph. The area of nonzero $\rho_{\textrm{S}}(0)$ increases as
$\Delta_{0}$ increases in (b) and (c). In all cases, no SABS appears.

We show the momentum-resolved SDOS at $\epsilon=0$ and at $k_{x}=k_{y}$ in
Figs. 15 and 16, respectively. One can see from Fig. 15 that the BFSs expand
as $\Delta_{0}$ increases. However, no SABS is observed on the (001) surface.
The absence of SABS is also confirmed from the momentum-resolved SDOS on the
$k_{\parallel}-\epsilon$ space shown Fig. 16, where $k_{\parallel}$ is the
momentum along the $\hat{k}_{\parallel}$ direction depicted in Fig. 15(c). The
absence of SABS is because the summation of the Chern numbers of the BFSs
projected onto the same place in the surface Brillouin Zone is zero. Thus, in
this case, we can investigate the transport through the zero-energy states on
the BFSs without being buried by that via SABSs. We comment that the ABSs
appear when we create a surface in other directions, such as the (1,1,1)
surface [40].

Figure 16:  Momentum-resolved SDOS along the $\hat{k}_{\parallel}$ direction
shown in Fig. 15(c) for spin-1/2 $E_{g}(1,i)$ state (a) and spin-3/2
$E_{g}(1,i)$ states with $\Delta_{0}/\mu=0.05$ (b), and $0.2$ (c). Panel (a)
is independent of the value of $\Delta_{0}/\mu$.

Figures 17(a) and 17(b) show the bulk LDOS and the SDOS, respectively. As in
the case of $T_{2g}(1,\omega,\omega^{2})$ states, the bulk LDOS has a U-shaped
structure, a characteristic of superconductors with point nodes, and the value
at $\epsilon=0$ increases as $\Delta_{0}$, or equivalently, the area of the
BFSs increases. On the other hand, because of the lack of SABSs, the SDOS has
a similar structure as the bulk LDOS, differently from the case of
$T_{2g}(1,\omega,\omega^{2})$ states. The residual SDOS at $\epsilon=0$ is
again due to the BFSs, although its magnitude is smaller than that in bulk
LDOS.

Figure 17:  Bulk LDOS (a) and SDOS (b) of $E_{g}(1,i)$ states normalized by
their values at $\Delta_{0}=0$. Insets in (a) and (b) are magnified views
around $\epsilon=0$. The legend ”d-wave” denotes the result for the spin-1/2
$E_{g}(1,i)$ state, which is independent of the value of $\Delta_{0}/\mu$.
There is no zero-energy peak in SDOS because of the absence of SABSs. The
nonzero bulk LDOS (a) and SDOS (b) at $\epsilon=0$ are the contributions from
the BFSs.

The normalized charge conductance
$\Gamma_{\textrm{S}}(\epsilon)/\Gamma_{\textrm{N}}(\epsilon)$ at the N/S
junction with the barrier parameter $Z=10$ is shown in Fig. 18. Reflecting the
$\epsilon$ and $\Delta_{0}$ dependence of the SDOS in Fig. 17(b), the
normalized conductance has a dip instead of a peak at $\epsilon=0$. The
minimum conductance at $\epsilon=0$ in the spin-3/2 $E_{g}(1,i)$ state
increases as $\Delta_{0}$ increases. This is the residual electrical
conduction via the zero-energy states on the BFSs. We comment here that the
normalized conductance at $\epsilon=0$ does not fall to zero even in the
spin-1/2 $E_{g}(1,i)$ state due to the finite value of the barrier parameter
$Z$.

Figure 18:  Normalized charge conductance
$\Gamma_{\textrm{S}}(\epsilon)/\Gamma_{\textrm{N}}(\epsilon)$ at N/S junction
with $Z=10$ for $E_{g}(1,i)$ states. Inset is the magnified view around
$\epsilon=0$. The legend ”d-wave” denotes the result for the spin-1/2
$E_{g}(1,i)$ state, which is independent of the value of $\Delta_{0}/\mu$.
Because of the absence of the ZBCP, the increase in the conductance at zero
bias due to the transport through the states on the BFSs becomes visible.

Short summary for $E_{g}(1,i)$ state: In the absence of the topological SABSs,
no peak structure appears around zero energy in the SDOS nor the charge
conductance. In accordance with the deformation from point nodes to BFSs,
nonzero SDOS arises at $\epsilon=0$, which leads to a nonzero electron
conduction in the N/S junction at zero bias.

##  VI Conclusion

In this paper, we have studied surface Andreev bound states (SABSs) of
superconductors having Bogoliubov Fermi surfaces (BFSs) and their
contributions to the charge conductance at normal metal/superconductor (N/S)
junctions. We consider spin quintet $J_{\textrm{pair}}=2$ pairings of spin-3/2
electrons, which exhibit anisotropic behavior due to the asymmetry in the spin
space even when the pair potential is isotropic in the momentum space. When
the system is projected to an effective pseudo-spin-1/2 system, the
$J_{\textrm{pair}}=2$ pairs are mapped to spin-singlet $d$-wave pairs
belonging to the same irreducible representation as the pristine
$J_{\textrm{pair}}=2$ pair. In particular, the $J_{\textrm{pair}}=2$ pairs
possessing BFSs, which are our interest, are mapped to the $d$-wave pair
potentials with line nodes and/or point nodes, which accompany topological
zero-energy surface flat bands and surface arc states on specific surfaces.
Since such topological surface states are featured by the zero-bias
conductance peak (ZBCP) at an N/S junction, we have investigated how the
situation changes in the spin-3/2 $J_{\textrm{pair}}=2$ pairs with BFSs having
the same symmetry.

We have calculated surface density of states (SDOS) and quasiparticle
tunneling conductance in N/S junctions based on the recursive Green’s function
method and a generalized Lee-Fisher formula. We have discussed three spin-3/2
$J_{\textrm{pair}}=2$ pairing states:
$T_{2g}(0,i,1),T_{2g}(1,\omega,\omega^{2})$, and $E_{g}(1,i)$ states. The
corresponding spin-1/2 $d$-wave superconducting states have zero-energy
surface flat bands, surface arc states, and no SABS, respectively, on the
(001) surface. In the first case, $T_{2g}(0,i,1)$, the doubly-degenerate zero-
energy surface flat bands in the $d$-wave pairing state split due to the
appearance of the BFSs. Thus, the zero-energy peak of the SDOS and the ZBCP
observed in the spin-1/2 $d$-wave pairing case are blunted, although the zero-
energy surface states remain on a ring on the surface Brillouin Zone. In the
second case, $T_{2g}(1,\omega,\omega^{2})$, the location of the surface arc
states, which is topologically restricted in the case of the spin-1/2 $d$-wave
pairing state, changes in the spin-3/2 system. Accordingly, the crossing of
the arcs disappears, leading to the splitting of the zero-energy peak of the
SDOS and the ZBCP into two small peaks at nonzero energies. In the third case,
$E_{g}(1,i)$, there is no SABS on the (001) surface for both cases of spin-1/2
and spin-3/2 systems. Because there is no zero-energy peak of the SDOS, we can
investigate the electronic transport though the zero-energy states on the
BFSs, and indeed, we obtained nonzero charge conductance at zero bias, which
increases as the BFSs become larger. Summarising above, the topological
protections existing in the spin-1/2 $d$-wave pairings are easily violated in
the corresponding spin-3/2 $J_{\textrm{pair}}=2$ pairings, leading to blunting
or splitting of ZBCPs. At the same time, zero-energy states on BFSs directly
contribute to the electron transport at zero bias, which is visible in the
absence of ZBCP.

There are several remaining problems. Recently, the relevance of odd-frequency
pairing and BFSs are discussed [72, 73, 74]. It is reasonable that the odd-
frequency pair amplitude exists even in the bulk state since gapless
superconducting state is realized. On the other hand, it is known that an
even-frequency superconductor has a nonzero odd-frequency pairing amplitude at
an interface or a surface due to the breaking of the translational symmetry
[75, 76, 77, 12]. Especially, in the presence of zero energy SABS, the
magnitude of the odd frequency pairing at the interface is amplified. It is an
interesting future work to analyze the obtained results focusing on the
symmetry of the Cooper pair including odd-frequency pairings and resulting
anomalous proximity efect [75, 78, 79].

Another possible direction is to study the effect of a magnetic field on the
SABSs. Under an applied magnetic field, the Doppler shift of quasiparticle
energy spectra occurs, resulting in the splitting of the ZBCP in spin-singlet
d-wave superconductors [80, 81] and the enhancement of the ZBCP in 3D chiral
superconductors for a certain surface [30]. Since the magnetic field only
changes the volume of the BFSs, tunneling spectroscopy under a magnetic field
may provide a way to distinguish the SABSs from the BFSs.

###### Acknowledgements.

This work was supported by JSPS KAKENHI (Grant Numbers JP19H01824, JP19K14612,
JP20H00131, JP21H01039, JP22K03478, and JP23K17668) and JST CREST (Grant No.
JPMJCR19T2).

##  Appendix A $\bm{J=3/2}$ representation of the rank-2 tensor

Using the $4\times 4$ matrix form of the angular momentum operator $\bm{J}$,

$\displaystyle J_{x}$ $\displaystyle=\begin{pmatrix}0&\sqrt{3}/2&0&0\\\
\sqrt{3}/2&0&1&0\\\ 0&1&0&\sqrt{3}/2\\\ 0&0&\sqrt{3}/2&0\end{pmatrix},$ (44)
$\displaystyle J_{y}$ $\displaystyle=\begin{pmatrix}0&-i\sqrt{3}/2&0&0\\\
i\sqrt{3}/2&0&-i&0\\\ 0&i&0&-i\sqrt{3}/2\\\ 0&0&i\sqrt{3}/2&0\end{pmatrix},$
(45) $\displaystyle J_{z}$ $\displaystyle=\begin{pmatrix}3/2&0&0&0\\\
0&1/2&0&0\\\ 0&0&-1/2&0\\\ 0&0&0&-3/2\end{pmatrix},$ (46)

the the rank-2 tensor $\mathscr{Y}_{j}(\bm{J})\ (j=1,2,3,4,5)$ is given by

$\displaystyle\mathscr{Y}_{1}(\bm{J})$
$\displaystyle=\begin{pmatrix}0&0&-i&0\\\ 0&0&0&-i\\\ i&0&0&0\\\
0&i&0&0\end{pmatrix}=\tau_{y}\otimes\sigma_{0},$ (47)
$\displaystyle\mathscr{Y}_{2}(\bm{J})$
$\displaystyle=\begin{pmatrix}0&-i&0&0\\\ i&0&0&0\\\ 0&0&0&i\\\
0&0&-i&0\end{pmatrix}=\tau_{z}\otimes\sigma_{y},$ (48)
$\displaystyle\mathscr{Y}_{3}(\bm{J})$
$\displaystyle=\begin{pmatrix}0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&-1\\\
0&0&-1&0\end{pmatrix}=\tau_{z}\otimes\sigma_{x},$ (49)
$\displaystyle\mathscr{Y}_{4}(\bm{J})$
$\displaystyle=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{pmatrix}=\tau_{x}\otimes\sigma_{0},$ (50)
$\displaystyle\mathscr{Y}_{5}(\bm{J})$
$\displaystyle=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&1\end{pmatrix}=\tau_{z}\otimes\sigma_{z}.$ (51)

##  Appendix B Tight-binding Hamiltonian

We summarize the matrix in the tight-binding model along the $z$ direction.

###  B.1 spin-1/2 system

The hopping matrix $t({\bm{k}}_{\perp})$ and on-site potential matrix
$u({\bm{k}}_{\perp})$ in the normal-state Hamiltonian is derived from
$H_{0}({\bm{k}})=\xi_{{\bm{k}},+}\otimes\mathbbm{1}_{2}$ as

$\displaystyle\bm{t}(\bm{k}_{\perp})$
$\displaystyle=\frac{\hbar^{2}}{2M_{+}a^{2}}\mathbbm{1}_{2},$ (52)
$\displaystyle\bm{u}(\bm{k}_{\perp})$
$\displaystyle=\left[\frac{\hbar^{2}}{2M_{+}}\left(k_{x}^{2}+k_{y}^{2}+\frac{2}{a^{2}}\right)-\mu\right]\mathbbm{1}_{2}.$
(53)

The pair potential with the d-wave symmetry given by Eq. (29) includes both
the on-site and off-site pairing terms. For each state in Eq. (29), the pair
potentials are given as follows:

$\displaystyle T_{2g}(0,i,1):$ $\displaystyle\hskip 28.45274pt\Delta_{\rm
on}({\bm{k}}_{\perp})=0,$ (54) $\displaystyle\hskip 28.45274pt\Delta_{\rm
off}({\bm{k}}_{\perp})=\frac{\sqrt{3}\Delta_{0}}{2a}(-ik_{x}+k_{y})i\sigma_{y},$
(55) $\displaystyle T_{2g}(1,\omega,\omega^{2}):$ $\displaystyle\hskip
28.45274pt\Delta_{\rm
on}({\bm{k}}_{\perp})=\frac{\sqrt{3}\Delta_{0}}{2}k_{x}k_{y}i\sigma_{y},$ (56)
$\displaystyle\hskip 28.45274pt\Delta_{\rm
off}({\bm{k}}_{\perp})=-i\frac{\sqrt{3}\Delta_{0}}{2a}(\omega
k_{x}+\omega^{2}k_{x})i\sigma_{y},$ (57) $\displaystyle E_{g}(1,i):$
$\displaystyle\hskip 28.45274pt\Delta_{\rm
on}({\bm{k}}_{\perp})=\frac{\Delta_{0}e^{i\pi/6}}{2}\left(k_{x}^{2}+\omega
k_{y}^{2}+\frac{2\omega^{2}}{a^{2}}\right)i\sigma_{y},$ (58)
$\displaystyle\hskip 28.45274pt\Delta_{\rm
off}({\bm{k}}_{\perp})=-\frac{\omega^{2}\Delta_{0}e^{i\pi/6}}{a^{2}}i\sigma_{y}.$
(59)

###  B.2 spin-3/2 system

The hopping matrix $t({\bm{k}}_{\perp})$ and on-site potential matrix
$u({\bm{k}}_{\perp})$ in the normal-state Hamiltonian is derived from Eq. (17)
with $m_{1}=m_{2}$ as

$\displaystyle\bm{t}(\bm{k}_{\perp})=$
$\displaystyle\frac{\hbar^{2}}{2m_{0}a^{2}}\mathbbm{1}_{4}+\frac{\hbar^{2}}{2m_{1}a^{2}}\mathscr{Y}_{5}(\bm{J})$
$\displaystyle-i\frac{\sqrt{3}\hbar^{2}}{2m_{1}a}\left[k_{x}\mathscr{Y}_{1}(\bm{J})+k_{y}\mathscr{Y}_{2}(\bm{J})\right],$
(60) $\displaystyle\bm{u}(\bm{k}_{\perp})=$
$\displaystyle\left[\frac{\hbar^{2}}{2m_{0}}\left(k_{x}^{2}+k_{y}^{2}+\frac{2}{a^{2}}\right)-\mu\right]\mathbbm{1}_{4}$
$\displaystyle+\frac{\sqrt{3}\hbar^{2}}{2m_{1}}k_{x}k_{y}\mathscr{Y}_{1}(\bm{J})$
$\displaystyle+\frac{\sqrt{3}\hbar^{2}}{4m_{1}}\left(k_{x}^{2}-k_{y}^{2}\right)\mathscr{Y}_{4}(\bm{J})$
$\displaystyle+\frac{\hbar^{2}}{4m_{1}}\left(\frac{4}{a^{2}}-k_{x}^{2}-k_{y}^{2}\right)\mathscr{Y}_{5}(\bm{J}).$
(61)

We consider ${\bm{k}}$-independent $J_{\rm pair}=2$ pair potentials, which
leads to $\Delta_{\rm off}=0.$ The on-site pair potentials corresponding to
Eqs. (24a)-(24c) are given by

$\displaystyle T_{2g}(0,i,1):$ $\displaystyle\hskip 28.45274pt\Delta_{\rm
on}=\Delta_{0}\left[i\mathscr{Y}_{2}(\bm{J})+\mathscr{Y}_{3}(\bm{J})\right],$
(62) $\displaystyle T_{2g}(1,\omega,\omega^{2}):$ $\displaystyle\hskip
28.45274pt\Delta_{\rm
on}=\Delta_{0}\left[\mathscr{Y}_{1}(\bm{J})+\omega\mathscr{Y}_{2}(\bm{J})+\omega^{2}\mathscr{Y}_{3}(\bm{J})\right],$
(63) $\displaystyle E_{g}(1,i):$ $\displaystyle\hskip 28.45274pt\Delta_{\rm
on}=\Delta_{0}\left[\mathscr{Y}_{4}(\bm{J})+i\mathscr{Y}_{5}(\bm{J})\right].$
(64)

##  Appendix C Generalized Lee-Fisher formula in Nambu space

We derive the Lee-Fisher formula [55] generalized for the case with internal
degrees of freedom in Nambu space. We consider a normal-state Hamiltonian with
generic spin-dependent hopping terms

$\displaystyle\mathcal{H}_{\rm hop}$
$\displaystyle=-\sum_{l,{\bm{k}}_{\perp}}\sum_{\lambda,\lambda^{\prime}}\left[f_{l+1,{\bm{k}}_{\perp},\lambda}^{\dagger}t_{\lambda\lambda^{\prime}}({\bm{k}}_{\perp})f_{l,{\bm{k}}_{\perp},\lambda^{\prime}}+\textrm{H.
c.}\right],$ (65)
$\displaystyle=-\sum_{l,{\bm{k}}_{\perp}}\left[(\bm{f}_{l+1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}\bm{t}({\bm{k}}_{\perp})\bm{f}_{l,{\bm{k}}_{\perp}}+\textrm{H. c.}\right],$
(66)

where $t_{\lambda\lambda^{\prime}}({\bm{k}}_{\perp})$ is an element of the
hopping matrix with $\lambda$ and $\lambda^{\prime}$ being the indices for the
internal degrees of freedom. The second line is just rewriting the first line
using the matrix forms. We can rewrite $\mathcal{H}_{\rm hop}$ using the Nambu
representation as

$\displaystyle\mathcal{H}_{\rm
hop}=-\frac{1}{2}\sum_{l,{\bm{k}}_{\perp}}\begin{pmatrix}(\bm{f}_{l+1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}&(\bm{f}_{l+1,-{\bm{k}}_{\perp}})^{\rm
T}&(\bm{f}_{l,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}&(\bm{f}_{l,-{\bm{k}}_{\perp}})^{\rm
T}\end{pmatrix}\begin{pmatrix}\bm{0}&\tau_{3}\bm{T}({\bm{k}}_{\perp})\\\\[5.69054pt]
\bm{\tau}_{3}\bm{T}^{\dagger}({\bm{k}}_{\perp})&\bm{0}\end{pmatrix}\begin{pmatrix}\bm{f}_{l+1,{\bm{k}}_{\perp}}\\\
\bm{f}_{l+1,-{\bm{k}}_{\perp}}^{\dagger}\\\ \bm{f}_{l,{\bm{k}}_{\perp}}\\\
\bm{f}_{l,-{\bm{k}}_{\perp}}^{\dagger}\end{pmatrix},$ (67)

where $\bm{T}({\bm{k}}_{\perp})$ is given by Eq. (41). Here, $\tau_{3}$ in
front of $\bm{T}(\bm{k}_{\perp})$ and $\bm{T}^{\dagger}(\bm{k}_{\perp})$ comes
from the Fermi statistics. The current operator $J$ is defined from the
equation of continuity:

$\displaystyle-|e|\frac{\partial}{\partial
t}\sum_{{\bm{k}}_{\perp}}\left(\bm{f}_{l,{\bm{k}}_{\perp}}^{\dagger}\right)^{\rm
T}\bm{f}_{l,{\bm{k}}_{\perp}}+\frac{J(l)-J(l-1)}{a}=0,$ (68)

where $e<0$ is the elementary charge and $a$ is the lattice constant.
Considering the fact that the current comes from the hopping Hamiltonian, the
current operator at site $l$ is given by

$\displaystyle J(l)=$
$\displaystyle\frac{|e|a}{i\hbar}\sum_{{\bm{k}}_{\perp}}\left[(\bm{f}_{l+1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}\bm{t}({\bm{k}}_{\perp})\bm{f}_{l,{\bm{k}}_{\perp}}-\textrm{H. c.}\right]$
$\displaystyle=$
$\displaystyle\frac{|e|a}{i2\hbar}\sum_{{\bm{k}}_{\perp}}\begin{pmatrix}(\bm{f}_{l+1,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}&(\bm{f}_{l+1,-{\bm{k}}_{\perp}})^{\rm
T}&(\bm{f}_{l,{\bm{k}}_{\perp}}^{\dagger})^{\rm
T}&(\bm{f}_{l,-{\bm{k}}_{\perp}})^{\rm
T}\end{pmatrix}\begin{pmatrix}\bm{0}&\bm{T}({\bm{k}}_{\perp})\\\\[5.69054pt]
-\bm{T}^{\dagger}({\bm{k}}_{\perp})&\bm{0}\end{pmatrix}\begin{pmatrix}\bm{f}_{l+1,{\bm{k}}_{\perp}}\\\
\bm{f}_{l+1,-{\bm{k}}_{\perp}}^{\dagger}\\\ \bm{f}_{l,{\bm{k}}_{\perp}}\\\
\bm{f}_{l,-{\bm{k}}_{\perp}}^{\dagger}\end{pmatrix},$ (69)

which indicates that $\bm{T}({\bm{k}}_{\perp})$ is the hopping matrix for the
current operator in the Nambu description. Using this
$\bm{T}({\bm{k}}_{\perp})$ and following the procedure to derive the Lee-
Fisher formula [55], see, e.g., Appendix 1 of Ref. [56] for details, we obtain
Eq. (39).

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