# Doping induced singlet to triplet superconducting transition in Ba2CuO3+δ

Priyo Adhikary  Center for Atomistic Modelling and Materials Design, Indian
Institute of Technology Madras, Chennai, 600036, India  Department of Physics,
Indian Institute Of Technology Madras, Chennai, 600036, India  Mayank Gupta
Center for Atomistic Modelling and Materials Design, Indian Institute of
Technology Madras, Chennai, 600036, India  Condensed Matter Theory and
Computational Lab, Department of Physics, Indian Institute Of Technology
Madras, Chennai, 600036, India  B. R. K. Nanda<EMAIL_ADDRESS>Center for
Atomistic Modelling and Materials Design, Indian Institute of Technology
Madras, Chennai, 600036, India  Condensed Matter Theory and Computational Lab,
Department of Physics, Indian Institute Of Technology Madras, Chennai, 600036,
India  Shantanu Mukherjee<EMAIL_ADDRESS>Center for Atomistic Modelling
and Materials Design, Indian Institute of Technology Madras, Chennai, 600036,
India  Department of Physics, Indian Institute Of Technology Madras, Chennai,
600036, India

###### Abstract

In this study, we perform a numerical simulation on the recently discovered
high-temperature superconductor ($T_{c}$= 73K) Ba2CuO3.2 [1] while focusing on
doping dependence of alternating CuO6 octahedra and CuO chain-like states.
Employing the multiband random-phase approximation, we compute the spin-
fluctuation mediated pairing interaction, subsequently determining its pairing
eigenvalues and eigenfunctions relative to oxygen-doping levels. We find that,
for the certain range of hole doping in Ba2CuO3+δ, a singlet
$d_{x^{2}-y^{2}}$-wave pairing symmetry emerges as long as we keep the doping
below the critical value $x_{c}$. Interestingly upon hole doping, the dominant
pairing symmetry undergoes a transition to a triplet (odd paring) type from
the singlet state. This change in pairing is driven by the competition between
the nesting vectors coming from the Fermi surface of $d_{z^{2}}$ and
$d_{x^{2}-y^{2}}$ orbitals within the CuO6 octahedra. This triplet state is
attainable through hole doping, while supressing inter-layer self-doping
effects. Furthermore, we present the density of states within the
superconducting phase, offering a potential comparison with tunnelling spectra
in Ba2CuO3+δ. Our research provides novel insights into the intricate pairing
symmetries in Ba2CuO3+δ and their underlying pairing mechanisms.

##  I Introduction

One of the intriguing features of cuprate superconductors is the possibility
of $d$-wave pairing symmetry, that leads to a superconducting gap with nodes
on the Fermi surface (FS)[2, 3, 4]. However, recent experiments have pointed
towards the possibility of a nodeless pairing state in certain electron and
hole doped regions of the cuprate high-Tc superconductors [5, 6, 7, 8, 9, 10,
11, 12, 13, 14]. This deviation from a $d$-wave nature of the superconducting
gap suggests a more complex phase diagram of cuprates than previously
considered[15]. Theoretical studies have also shown that a triplet state can
emerge in cuprates within both one-band and three-band models[16, 17],
depending on the doping level and the interaction parameters. However, there
is no direct experimental evidence for a triplet state in cuprates so far. It
is within this context that the recent discovery of Superconductivity (SC) in
Ba2CuO3+δ (BCO)[1] becomes particularly significant, as it may provide a new
platform to explore the nature of the pairing symmetry in cuprates.

The Unit cell of Ba2CuO3.25 has two primary layers: layer-I and layer-II.
Layer-I exhibits octahedral and square planar Cu-O complexes alternately
stacked along the b-direction and layer-II has only square planar complexes
along the b-axis, see Fig. 1(a). This two layer feature originates from the
missing oxygen atoms in Ba2CuO4 lattice. The oxygen k-edge x-ray absorption
spectra[1, 18] have provided evidence that points towards such a layered
arrangement. Electronic structure calculations show that the interlayer
hybridization has a pronounced effect on the effective band structure, leading
to a shift in the Van Hove singularity (VHS) and $d_{x^{2}-y^{2}}$ orbital
moves below the Fermi level[19, 20]. However, as we increase the hole doping,
the $d_{x^{2}-y^{2}}$ orbital crosses the Fermi level and contributes
significantly to the VHS. This is crucial for determining the pairing symmetry
in BCO.

In this paper, we investigate the variations in superconducting pairing
symmetry and strength as a function of hole doping. We consider a 14-orbital
basis, which is subsequently downfolded to a five-orbital basis[20]. This
includes the $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals from the CuO6
octahedra of the Cu(1) atom in layer-I, and the $d_{b^{2}-c^{2}}$ orbital from
the chain state of the Cu(2) atom in layer-I and the Cu(3)/Cu(4) atoms in
layer-II. This Hamiltonian accurately replicates the low-energy bands observed
in density functional theory (DFT) calculations. Notably, due to octahedral
compression, the Cu-$d_{x^{2}-y^{2}}$ orbital is positioned below the Fermi
level, see 1(b). As we increase hole doping, this orbital becomes crucial in
determining the superconducting pairing symmetry. Our model assumes that
superconducting pairing interactions arise from spin-fluctuation mechanisms,
incorporating multiband Hubbard interactions within the framework of the weak
coupling random phase approximation (RPA). The most robust solution of this
pairing interaction indicates towards dominant superconducting gap. The
dominant pairing strength in the singlet channel originates from a nesting
vector of $\pm(\pi$-$\delta$,$\pi$-$\delta$). Consequently, the
superconducting gap function changes sign between momentum vectors connected
by this nesting vector, leading to a $d_{x^{2}-y^{2}}$ pairing symmetry.

Prompted by the recent discovery of SC in BCO at high doping level (40%), we
aim to elucidate the superconducting pairing symmetry across the entire range
of hole doping. We ascertain that below a critical doping threshold ($x_{c}$),
the superconducting pairing symmetry predominantly exhibits the conventional
$d_{x^{2}-y^{2}}$ type. However, with increased doping, the superconducting
pairing symmetry transitions to a multi-band triplet solution. Notably, this
feature demonstrates a significant robustness against Hund’s coupling. This
change in pairing symmetry we attribute to the inter-orbital nesting vectors
that connect the $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ FSs. While such triplet
symmetry has not yet been observed in Ba2CuO3.2, our findings suggest that
through chemical doping or the application of pressure, this type of
superconducting gap structure could be realized in future experiments.

The rest of the paper is organised as follows: Section II we provide a summary
of the tight-binding model, along with the mechanisms underlying the multiband
RPA spin fluctuation approach. In Section III, we delve into the FS topology,
examining the nesting profile, RPA spin susceptibility, and pairing symmetry.
This section also explores the variation in pairing strength as a function of
doping and investigates the density of states in the SC state. Finally in
section IV we give a comprehensive discussion and conclusion of our findings.

##  II Method and Model

###  II.1 Tight-binding model

We consider a 14-orbital, tight-binding (TB) Hamiltonian that effectively
replicates the low energy density functional theory (DFT) band structure [20].
In our model, the four Cu atoms in the unit cell contribute five Cu-$d$
orbitals, while the adjacent oxygen atoms provide nine $p$ orbitals. The
orbital weights contribution to the DFT results shows that oxygen bands lie
deep inside the Fermi level. Using Löwdin downfolding procedure we integrate
out the oxygen bands and obtain an effective five-band model Hamiltonian. A
more comprehensive analysis of this contribution is available in the appendix.
The Hamiltonian is expressed as follows,

$H({\bf k})=\sum_{\alpha\beta}\sum_{\begin{subarray}{c}{\bf k}\\\
\sigma\in(\uparrow,\downarrow)\end{subarray}}\Big{[}\xi_{\alpha\beta}({\bf
k})+\mu_{\alpha}\delta_{\alpha\beta}\Big{]}c_{{\bf
k},\alpha,\sigma}^{\dagger}c_{{\bf k},\beta,\sigma}.$ (1)

$\xi_{\alpha\beta}({\bf k})$ is the TB matrix element fitted with the DFT
bands. Fermion creation and annihilation operator are denoted by
$c^{\dagger}_{{\bf k},\alpha,\sigma}$ and $c_{{\bf k},\beta,\sigma}$
respectively. The onsite energy for orbital $\alpha$ is $\mu_{\alpha}$

###  II.2 Multiband RPA susceptibility

We use the multi-band Hubbard model to study the topology of FS and
corresponding spin-fluctuation potential. The Hamiltonian of the Hubbard model
is given by [22],

$\displaystyle H_{\rm int}$ $\displaystyle=$ $\displaystyle\sum_{\alpha,{\bf
q}}Un_{\alpha\uparrow}({\bf q})n_{\alpha\downarrow}(-{\bf
q})+\sum_{\alpha\neq\beta}\sum_{{\bf q}}\frac{V}{2}n_{\alpha}({\bf
q})n_{\beta}(-{\bf q})$ (2) $\displaystyle-\sum_{\alpha\neq\beta}\sum_{{\bf
q}}\frac{J}{2}{\bf S}_{\alpha}({\bf q})\cdot{\bf S}_{\beta}(-{\bf q})$

Where $U$ and $V$ are the intra-orbital and inter-orbital Hubbard interaction
between Cu-$d$ orbitals and $J_{H}$ is the Hund coupling.

Using perturbative expansion of the spin density and charge-density
correlation function we obtain random-phase approximation (RPA) spin and
charge susceptibilities,

$\displaystyle\tilde{\chi}_{\rm s/c}({\bf q})=\tilde{\chi}_{0}({\bf
q})\left(\tilde{\mathbb{I}}\mp\tilde{U}_{s/c}\tilde{\chi}_{0}({\bf
q})\right)^{-1},$ (3)

The nonzero components of onsite Hubbard interactions for spin and charge
fluctuation are $\tilde{U}_{s}$ and $\tilde{U}_{c}$[17, 22]. The bare
susceptibility is enhanced at the nesting wave vector which leads to a
corresponding enhanced peak in the RPA spin susceptibility. The overall
momentum space structure of the susceptibility can in general be more complex
in multi orbital systems owing to the presence of matrix elements. In general,
due to the presence of $(1-\tilde{U}\chi_{0})$ in the denominator for the RPA
spin susceptibility, the contribution from the spin channel is enhanced
compared to the charge channel that contains a $(1+\tilde{U}\chi_{0})$
contribution.

###  II.3 Superconducting pairing symmetry

Superconducting pairing in Cu -d electrons is mediated via spin
fluctuations[20]. We calculate the spin-fluctuation pairing potential by
expanding the $H_{\rm int}$ from Eq. (2) into a perturbation series and
collecting bubble and ladder diagrams. The effective Hamiltonian we obtain as
[17, 23],

$\displaystyle H_{\rm eff}=\sum_{\alpha\beta\gamma\delta}\sum_{{\bf
kq},\sigma\sigma^{\prime}}\Gamma_{\alpha\beta}^{\gamma\delta}({\bf
q})c_{\alpha\sigma}^{\dagger}({\bf k})c_{\beta\sigma^{\prime}}^{\dagger}(-{\bf
k})$ $\displaystyle c_{\gamma\sigma^{\prime}}({\bf-k-q})$ (4) $\displaystyle
c_{\delta\sigma}({\bf k+q}).$

Here, the pairing potential is a tensor of four orbital indices. For singlet
and triplet channels the spin-fluctuation pairing potential is given by [25,
24, 26, 27, 28, 29],

$\displaystyle\tilde{\Gamma}_{S}({\bf q})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}3{\tilde{U}}_{s}{\tilde{\chi}}_{s}({\bf
q}){\tilde{U}}_{s}-{\tilde{U}}_{c}{\tilde{\chi}}_{c}({\bf
q}){\tilde{U}}_{c}+{\tilde{U}}_{s}+{\tilde{U}}_{c}\big{]},$ (5a)
$\displaystyle\tilde{\Gamma}_{T}({\bf q})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\big{[}{\tilde{U}}_{s}{\tilde{\chi}}_{s}({\bf
q}){\tilde{U}}_{s}+{\tilde{U}}_{c}{\tilde{\chi}}_{c}({\bf
q}){\tilde{U}}_{c}\big{]}.$ (5b)

Using a unitary transformation we obtain pairing potential in the band basis.

$\displaystyle\tilde{\Gamma}_{\mu\nu}({\bf
k,q})=\sum_{\alpha\beta\gamma\delta}\Gamma_{\alpha\beta}^{\gamma\delta}({\bf
q})\psi^{\mu\dagger}_{\alpha}({\bf k})\psi^{\mu\dagger}_{\beta}(-{\bf k})$
$\displaystyle\psi^{\nu}_{\gamma}({\bf-k-q})$
$\displaystyle\psi^{\nu}_{\delta}({\bf k+q})$

Where $(\mu,\nu)$ represent the band indices, and $\psi^{\mu}_{\alpha}({\bf
k})$ is the eigenvector component corresponding to orbital $\alpha$, band
$\mu$ and calculated at the wave vector ${\bf k}$. We obtain superconducting
pairing symmetry by solving the linearized gap equation,

$\displaystyle\Delta_{\mu}({\bf k})=-\lambda\frac{1}{N}\sum_{\nu,{\bf
q}}\tilde{\Gamma}_{\mu\nu}({\bf k,q})\Delta_{\nu}({\bf k+q}).$ (7)

$\lambda$ is known as a superconducting coupling constant. By solving Eq. (7)
we obtain pairing eigenfunction for largest eigenvalue. This largest
eigenvalue determines the stability of superconducting gap function
$\Delta({\rm\bf k})$[25].

The unconventional SC within spin fluctuation theory originates from a
repulsive interaction that usually favours a sign change of the
superconducting gap over the FS. Since $\chi_{s}$ (see Eq. (3)) is positive
and larger than $\chi_{c}$, pairing potential Eq. (5a) is repulsive. The only
possible solution of the gap equation for repulsive interaction is when
$\Delta$ (see Eq. (7) ) changes sign between momentum vectors ${\bf k}$ and
${\bf k+Q}$. This leads to an anisotropic solution of the gap function in the
momentum space whose underlying symmetry transforms according to the
irreducible representation of the crystals point group symmetry.

##  III Results

###  III.1 Electronic structure

Figure 1: (a) Crystal structure of Ba2CuO3.25. Layer-I consists of copper in
octahedral coordination (Cu(1)) and square planar complexes (Cu(2)), while
Layer-II features Cu(3)/Cu(4) in square planar complexes, aligned along the
$b$-axis. (b) Tight binding model fitted with DFT band structure [20]. (c)
Orbital resolved tight binding bands of BCO. The red and blue dashed lines
indicate two specific doping levels at $E_{F}$ of 0.0778 eV and -0.0778 eV,
based on rigid band approximations. These correspond to doping levels of $x$ =
0.24 and $x$ = 0.37, respectively. The critical doping level, where SC pairing
symmetry transitions from singlet to triplet, is shown in Fig. 5. The critical
values are $x_{c}$=0.29, 0.33 for $J_{H}=0$ and $J_{H}\neq 0$ respectively.
Fermi surfaces are shown at two different $k_{z}$ cuts (0 and $\pi$) for two
different doping levels, $x<x_{c}$ in (d)-(e) and $x>x_{c}$ in (f)-(g)
respectively. The colour scheme for different orbitals are, $d_{z^{2}}$ (brick
red), $d_{x^{2}-y^{2}}$ (orange), Cu(2)-$d_{b^{2}-c^{2}}$(green), Cu(3)/Cu(4)-
$d_{b^{2}-c^{2}}$ (navy blue).

In this study, we begin by analyzing the electronic structure and FS topology
at two distinct doping levels. We choose two representative doping levels,
$x$=0.24 and $x$ = 0.37, corresponding to a Fermi energy ($E_{F}$) of 0.0778
eV and -0.0778 eV, respectively. The electronic band structures are depicted
in Fig. 1(c). The red and blue horizontal dashed line in 1(c) denotes the
low($x$=0.24) and high( $x$ = 0.37) doping values. Our findings reveal that
increasing hole doping leads to a reduction in the electron filling of the
$d_{x^{2}-y^{2}}$ orbital. This creates additional hole pockets reminiscent of
cuprate superconductors[17]. Notably, when the $d_{x^{2}-y^{2}}$ hole pocket
becomes fully depleted, the FS is predominantly characterized by open
electron-like pockets of $d_{z^{2}}$ at $k_{z}$=0 and closed electron pockets
at $k_{z}$=$\pi$, with an enhanced contribution from $d_{b^{2}-c^{2}}$
orbitals on the open electron pockets persisting across the entire range of
hole doping. Experimentally, the FS results can be corroborated by ARPES
measurements conducted on BCO samples. Intriguingly, the presence of the hole
pocket introduces additional nesting vectors, absent in the 40% hole doping
regime. In the following sections we will demonstrate that these alterations
in the $d_{x^{2}-y^{2}}$ FS significantly impact the overall pairing potential
and, consequently, the pairing symmetry within the BCO system.

###  III.2 Evolution of FS nesting with doping

Figure 2:  (a)-(d) Physical RPA spin susceptibilities for two representation
dopings. (a-b) is for $x<x_{c}$ with $J_{H}$ = 0 and $\frac{U}{3}$
repectively. (c-d) is for $x>x_{c}$ with $J_{H}$ = 0 and $\frac{U}{3}$
repectively.

In the following section, we delve into the evolution of the FS nesting
profile as a function of hole doping concentration. For this purpose, we
compute the RPA spin-susceptibility, as shown in Fig. 2 for $q_{z}$=0, and
orbital-resolved components in Fig. 3. At low doping, FS has a mixed character
of $d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals near (0, $\pi$) region. RPA
susceptibility of the $d_{x^{2}-y^{2}}$ orbital is much lower than the inter-
orbital contribution, as evident from Fig. 3(b)-(c). We have found a
pronounced peak at the physical spin susceptibility near the
$\pm(\pi-\delta$,$\pi-\delta$) wave-vector, in Fig. 2, which we posit to be a
precursor to a $d_{x^{2}-y^{2}}$-type superconducting gap.

Figure 3:  (a)-(l) Orbital resolved RPA spin susceptibilities for two
representation dopings. Column 1 and 2 belong to intra-orbital (
$\chi_{11}^{11}$, $\chi_{22}^{22}$ ) and column 3 is for inter-orbital
($\chi_{12}^{12}$ ) contributions. Row 1 is for $x<x_{c}$ with $J_{H}$ = 0 ,
row 2 for $x<x_{c}$ with $J_{H}=U/3$, Row 3 is for $x>x_{c}$ with $J_{H}$ = 0
, row 4 for $x>x_{c}$ with $J_{H}=U/3$.

The FS displays a $k_{z}$ dependence, as depicted in Fig. 1(d)-(g). However,
the distribution of orbital weight across the Fermi level is such that the RPA
spin susceptibility exhibits negligible $q_{z}$ dependence. Consequently, we
have focused solely on the $q_{z}$=0 components in our susceptibility
calculations. With increasing hole doping, in addition to the smaller
electron-like pocket at the center of the Brillouin zone (BZ), we have an
additional hole pocket with $d_{x^{2}-y^{2}}$ character emerges near the
corners of BZ [see Fig. 1. This new hole pocket, reminiscent of typical
cuprate superconductors which gives a nesting vector of ($\pi$, $\pi$) to the
CuO4 plane. However, in BCO, because of the structural anisotropy (relative to
Ba2CuO4 ) the hole pocket significantly alters the nesting vector, as
demonstrated in the susceptibility plot in Fig. 2(c)-(d). Moreover, there is a
notable reversal in the contributions of the $d_{x^{2}-y^{2}}$ and $d_{z^{2}}$
orbitals to the intra-orbital RPA susceptibility components in Fig. 3(g)-(h).
The contribution of the intraorbital $d_{x^{2}-y^{2}}$ is marginally more
significant than the inter-orbital susceptibility, which in turn correlates
with a weaker $d_{z^{2}}$ intraorbital susceptibility in Fig. 3(g)-(i).

Fig. 2(b) and(d) demonstrate the influence of Hund’s coupling in diminishing
the effectiveness of charge screening, together with their orbital resolved
components in Fig. 3(d)-(f) and (j)-(l). It is noteworthy that in the absence
of Hund’s coupling, the peak value of the RPA spin susceptibility is larger,
yet the overall nesting vector remains unchanged. Hence we expect that the
nodal structure of the SC gap does not change. Indeed, as will be demonstrated
in Fig. 4, the symmetry of the superconducting gap does not vary for a given
doping level. However, since the superconducting pairing potential is directly
proportional to the strength of the RPA spin susceptibility, the strength of
the pairing symmetry is attenuated in the presence of Hund’s coupling.

Figure 4:  (a)-(h)Computed pairing eigenfunction $\Delta$(k) for the leading
eigenvalue, plotted on the corresponding FSs, for two representative values of
doping ($x$). $x_{c}$ is the critical value of the doping at which leading SC
gap changes from singlet to triplet pairing.

###  III.3 Superconducting properties

We now turn our attention to the doping dependence of SC in BCO. We include
doping levels comparable to those considered in our susceptibility
discussions. In Fig. 4, we present a plot of the superconducting gap on the
FS, highlighting the largest pairing eigenvalue. The colorbar in this figure
illustrates the sign change in the superconducting gap function. Two distinct
solutions emerge at varying doping values. Our results indicate that the
singlet potential solution of Eq. (5a) exhibits the highest pairing eigenvalue
($\lambda$) at lower doping levels. Conversely, at higher doping levels, the
triplet channel, Eq. (5b), gives the largest pairing strength.

As discussed earlier in the context of Fig. 2, at low doping levels, the FS
nesting at $(\pi-\delta,\pi-\delta)$ significantly contributes to the RPA spin
susceptibility. The FS exhibits $C_{2}$ symmetry, which is mirrored in the
solution of the superconducting pairing potential. For doping levels below
40%, the nesting condition within the CuO6 octahedra fosters a pairing
symmetry that fulfils the relation $\Delta(k+Q_{1})=-\Delta(k)$ for the SC gap
function, where $Q_{1}=(\pi-\delta,\pi-\delta)$. At higher doping values
$(x>x_{c})$, the uniaxial nesting condition arising from the $d_{x^{2}-y^{2}}$
orbital yields a nesting vector $Q_{2}=(\pi-\delta,q_{y}\approx\text{small})$,
affecting all Fermi momentum vectors. In this scenario, the FS leads to a gap
function with a $\sin(k_{x})$ structure. This corresponds to the triplet
solution that emerges when the hole pocket crosses the van Hove singularity
point from below. Consequently, in the overdoped region of BCO, we observe a
triplet $p$-wave solution.

Next, we show the highest pairing eigenvalue across the entire hole-doped
region, as depicted in Fig. 5. The selection of interaction parameters is
chosen from the Stoner criterion, which is essential for fulfilling the normal
state paramagnetic solution. Superconducting pairing strength, $\lambda$,
decreases with increased doping. This behaviour remains robust against Hund’s
coupling. Hence the doping dependence of $\lambda$ is fundamentally linked to
the FS nesting properties. The observed decrease in $\lambda$ with hole doping
is elucidated by examining the orbital-resolved DOS for BCO [20]. With hole
doping, the DOS of the $d_{x^{2}-y^{2}}$ orbital increases, approaching the
VHS, while the DOS of the $d_{z^{2}}$ orbital diminishes. Additionally, a
comparative analysis of the FS volumes in two doping regions indicates an
increase in FS volume with enhanced electron filling. This contributes
significantly to the calculations of $\lambda$ using Eq. 7. The evolution of
FS with doping is non monotonic, marked by the emergence of new orbitals
($d_{x^{2}-y^{2}}$). Within the framework of weak-coupling theory, orbital
resolved SC gap is also expected to form in these new orbitals.

###  III.4 Superconducting spectral function

We consider the Nambu spinor basis, $\Psi_{\bf k}=\Big{(}\phi_{\bf
k\sigma},\phi^{\dagger}_{-\bf k-\sigma}\Big{)}^{T}$, to calculate the spectral
function. Eigenstates of the non-interacting Hamiltonian($H({\bf k})$), see
Eq. (1), are denoted by $\phi_{\bf k\sigma}$. Using this, we construct mean-
field Hamiltonian,

$\displaystyle H_{SC}=\begin{pmatrix}H({\bf k})&\tilde{\Delta}({\bf k})\\\
\tilde{\Delta}^{\dagger}(-{\bf k})&-H^{\dagger}(-{\bf k})\end{pmatrix}.$ (8)

Where $\tilde{\Delta}({\bf k})$ is the SC gap functions we obtain from the
self-consistent solution of Eq. (7) for band $\nu$. We have adopted tilde
notation to indicate that, the SC gap exhibits multi-band components. To
elucidate this aspect, the off-diagonal terms are explicitly written as
follows,

$\displaystyle\tilde{\Delta}({\bf k})=\left(\begin{matrix}&\Delta_{1}({\bf
k})&.&.&.&.\\\ &.&\Delta_{2}({\bf k})&.&.&.\\\ &.&.&\Delta_{3}({\bf k})&.&.\\\
&.&.&.&\Delta_{4}({\bf k})&.\\\ &.&.&.&.&\Delta_{5}({\bf k})\\\
\end{matrix}\right).$ (9)

The spectral functions we obtain from, $\tilde{A}({\bf
k},\omega)=-\frac{1}{\pi}{\rm Im}\Big{[}\frac{1}{(\omega+i\delta){{\bf{\it
I}}}-H_{SC}}\Big{]}$. We have shown in Fig. 6. the imginary part of the
spectral function summed over all momentum.

Figure 5:  (a)-(b) Doping-dependent SC coupling constant $\lambda$ for BCO for
a choice of $J_{H}$ = 0 eV and $J_{H}=\frac{U}{3}$ eV.

Figure 6:  (a)-(d) Total DOS for the SC state plotted for two representative
values of doping ($x$). (a-b) is for $x<x_{c}$ with $J_{H}$ = 0 and
$\frac{U}{3}$ repectively. (c-d) is for $x>x_{c}$ with $J_{H}$ = 0 and
$\frac{U}{3}$ repectively.

The SC gap function we obtain from the spin-fluctuation calculation at the
Fermi momentum points. To extend these results across the entire BZ, we employ
a Radial Basis Function (RBF)[30] for the extrapolation of two-dimensional
data points for different $k_{z}$ values. Using this extrapolated gap function
we obtain the spectral function and density of states.

In the low hole doping regime ($x<x_{c}$), the electron pockets at the Fermi
level predominantly arise from the Cu(1) $d_{z^{2}}$ and Cu(3)
$d_{b^{2}-c^{2}}$ orbitals in the singlet channel, whereas the Cu(1)
$d_{x^{2}-y^{2}}$ and Cu(2)/Cu(4) planar atoms exhibit negligible
contributions to superconductivity. Conversely, in the high hole doping regime
($x>x_{c}$), the emergence of an additional hole pocket attributable to the
Cu(1) $d_{x^{2}-y^{2}}$ orbital facilitates the triplet channel. The SC gap
structure, as previously discussed, varies significantly between these two
channels. In Fig. 6 we illustrates DOS in the SC state, highlighting the
contributions from the Cu(1) $d_{z^{2}}$ and Cu(3) $d_{b^{2}-c^{2}}$ orbitals
in (a) and (b) respectively, and an additional Cu(1) $d_{x^{2}-y^{2}}$ orbital
contribution in (c). In Fig. 6(b), (d) we have shown DOS when Hund’s coupling
is present.

In presence of Hund’s coupling the V-shape feature is prominent. Otherwise DOS
have a mixed V and U shape character. This DOS feature’s robustness against
variations in Hund’s coupling is significant. However, the DOS differences
between singlet and triplet states are primarily attributed to the multi-band
structure of the SC-gap function. The observed V-shaped feature suggests a
nodal gap structure. The residual DOS is attributable to the SC originating
predominantly from two (or three) orbitals in the singlet (or triplet)
channel, additionally the hopping parameter substantially larger than the SC
gap’s maximum across the BZ. The SC gap diminishes exponentially faster
towards the BZ boundary compared to the hopping parameter. Our findings offer
a theoretical framework that can be empirically validated through scanning
tunneling microscopy (STM) measurements on BCO at different doping conditions.

##  IV Discussions and Conclusions

We have identified a triplet p-wave pairing in BCO, occurring at hole doping
levels significantly above the conventional optimal doping range for cuprate
superconductors. This contrasts with other cuprates, where the
Cu-$d_{x^{2}-y^{2}}$ orbital typically leads to d-wave pairing due to the
($\pi,\pi$) nesting vector. The key difference in BCO lies in the octahedral
compression, which suppresses the $d_{x^{2}-y^{2}}$ orbital relative to the
$d_{z^{2}}$ orbital. The absence of $d_{x^{2}-y^{2}}$ changes the AFM nesting
vector different than $(\pi,\pi)$. The nesting vector is changed by shifts in
the Fermi level, enabling the emergence of $d_{x^{2}-y^{2}}$ SC at lower
doping levels. The absence of the $d_{x^{2}-y^{2}}$ orbital due to octahedral
compression can be compensated by shifting above the VHS point through a rigid
band shift. This adjustment introduces an additional hole pocket, which in
turn suppresses the RPA susceptibility of the $d_{z^{2}}$ orbital, previously
dominant in the absence of $d_{x^{2}-y^{2}}$. Since the spin-fluctuation
pairing potential is directly proportional to the RPA spin susceptibility,
there is now a substantial contribution at the
$Q_{2}=(\pi-\delta,q_{y}\approx\text{small})$ nesting vector. Consequently,
the SC gap function exhibits a change in sign corresponding to the reversal of
the momentum vector.

Triplet odd parity SC exhibits a range of fascinating applications, including
in superconductor/ferromagnet heterostructures and topological SC, leading to
the Majorana modes[31], etc. Notably, triplet odd parity SC has been recently
observed in the heavy fermion superconductor CeRh2As2[32]. Additionally, a
handful of studies on cuprates have indicated the presence of p-wave SC [33,
34, 35]. However similar results in cuprates is still lacking. In this
context, our results could serve as a motivation for experimental
investigations into the signatures of triplet SC in doped BCO. Such
investigations might include field angle magnetic field measurements, nuclear
magnetic resonance (NMR), Knight shift measurements, and Angle-Resolved
Photoemission Spectroscopy (ARPES).

##  Appendix A Details of the tight-binding Hamiltonian

The matrix representation of the SK-TB model Hamiltonian of BCO3.25 is shown
in Eq. 2 of the main text. Here, the sub-matrix $H$ of the layer-I in basis
set order $d_{z^{2}}$, $d_{x^{2}-y^{2}}$, $d_{x^{2}-y^{2}}$, $p_{x}$, $p_{z}$,
$p_{z}$, $p_{y}$, and $p_{y}$ is:

$H_{l1}=\left(\begin{array}[]{cccccccc}\xi_{1}&\xi_{1,2}&\xi_{1,3}&\xi_{1,4}&\xi_{1,5}&0&\xi_{1,7}&0\\\
\cline{1-1}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\xi_{2}&\xi_{2,3}&\xi_{2,4}&\xi_{2,5}&\xi_{2,5}^{*}&0&0\\\
\cline{2-2}\cr&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\xi_{3}&0&\xi_{2,5}^{*}&\xi_{2,5}&0&0\\\
\cline{3-3}\cr&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(3)}&0&0&0&0\\\
\cline{4-4}\cr&{\rm
h.c.}&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(1)}&0&0&0\\\
\cline{5-5}\cr&&&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(1)}&0&0\\\
\cline{6-6}\cr&&&&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(1)}&0\\\
\cline{7-7}\cr&&&&&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(1)}\\\
\cline{8-8}\cr\end{array}\right)$ (10)

h.c. denotes the Hermitian conjugate of the upper-triangular matrix.

The Hamiltonian sub-matrix for layer-II in orbital basis set of
$d_{x^{2}-y^{2}}$, $d_{x^{2}-y^{2}}$, $p_{z}$, $p_{z}$, $p_{y}$ and $p_{y}$ is
given as

$H_{l2}=\left(\begin{array}[]{cccccc}\xi_{4}&\xi_{4,5}&\xi_{4,6}&0&0&0\\\
\cline{1-1}\cr\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\xi_{5}&0&\xi_{4,6}&0&0\\\
\cline{2-2}\cr&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(7)}&0&0&0\\\
\cline{3-3}\cr&{\rm
h.c.}&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(7)}&0&0\\\
\cline{4-4}\cr&&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(7)}&0\\\
\cline{5-5}\cr&&&&\lx@intercol\hfil\hfil\lx@intercol\vrule\lx@intercol&\mu^{(7)}\\\
\cline{6-6}\cr\end{array}\right)$ (11)

Further, the Hamiltonian sub-matrix containing the interaction between layer-I
and layer-II is

$H_{l1-l2}=\left(\begin{array}[]{cccccc}0&0&0&0&\xi_{2}^{12}&-(\xi_{2}^{12})^{*}\\\
0&0&0&0&0&0\\\
\xi_{1}^{12}&(\xi_{1}^{12})^{*}&0&0&\xi_{3}^{12}&-(\xi_{3}^{12})^{*}\\\
0&0&0&0&0&0\\\ :&&&:&&:\\\ 0&0&0&0&0&0\\\ \end{array}\right)$ (12)

The components of the Hamiltonian matrices are found to be,

$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
t^{(5)}\cos(k_{x})+\mu^{(4)}+t^{(15)}\cos(k_{y})-0.1\cos(2k_{x})$ (13)
$\displaystyle\xi_{2}$ $\displaystyle=$ $\displaystyle
t^{(10)}\cos(ky)+t^{(11)}\cos(k_{x})+\mu^{(2)}$ (14) $\displaystyle\xi_{3}$
$\displaystyle=$ $\displaystyle\xi_{2}+\mu^{(5)}$ (15) $\displaystyle\xi_{4}$
$\displaystyle=$ $\displaystyle t^{(14)}\cos(k_{x})+\mu^{(6)}$ (16)
$\displaystyle\xi_{5}$ $\displaystyle=$ $\displaystyle\xi_{4}+0.02$ (17)
$\displaystyle\xi_{1,2}$ $\displaystyle=$ $\displaystyle
t^{(3)}\cos(k_{x})+t^{(9)}\cos(k_{y})$ (18) $\displaystyle\xi_{1,3}$
$\displaystyle=$ $\displaystyle t^{(4)}\exp(ik_{y}/4)$ (19)
$\displaystyle\xi_{2,3}$ $\displaystyle=$ $\displaystyle t^{(1)}\cos(k_{y}/2)$
(20) $\displaystyle\xi_{1,4}$ $\displaystyle=$ $\displaystyle
it^{(8)}\sin(k_{x}/2)$ (21) $\displaystyle\xi_{1,5}$ $\displaystyle=$
$\displaystyle it^{(8)}\sin(k_{y}/4)$ (22) $\displaystyle\xi_{1,7}$
$\displaystyle=$ $\displaystyle it^{(6)}\sin(0.156k_{z})$ (23)
$\displaystyle\xi_{2,4}$ $\displaystyle=$ $\displaystyle
it^{(7)}\sin(k_{x}/2)$ (24) $\displaystyle\xi_{2,5}$ $\displaystyle=$
$\displaystyle t^{(4)}\exp(ik_{y}/4)$ (25) $\displaystyle\xi_{4,5}$
$\displaystyle=$ $\displaystyle t^{(12)}\cos(k_{y}/2)$ (26)
$\displaystyle\xi_{4,6}$ $\displaystyle=$ $\displaystyle
it^{(13)}\sin(k_{y}/4)$ (27) $\displaystyle\xi_{1}^{12}$ $\displaystyle=$
$\displaystyle 2t_{12}^{(1)}\exp(-ik_{y}/4)\cos(k_{z}/2)\cos(k_{x}/2)$ (28)
$\displaystyle\xi_{2}^{12}$ $\displaystyle=$ $\displaystyle
it_{12}^{(2)}\exp(ik_{y}/4)\sin(k_{z}/3)\cos(kx/2)$ (29)
$\displaystyle\xi_{3}^{12}$ $\displaystyle=$ $\displaystyle
it_{12}^{(2)}\exp(-ik_{y}/4)\sin(k_{z}/3)\cos(k_{x}/2)$ (30)

The tight-binding parameters are, $t^{(1-14)}$ = $\big{[}$0.9775, 0.0237,
0.3396, 0.005, -0.78, 0.2076, -0.46 , 0.0, 0.035, -0.119, -0.074, 0.91,
0.6928, -0.054, -0.06.$\big{]}$ $\mu^{(1-7)}$ = $\big{[}$ -0.74, -0.31, -1.24,
0.5, 1.18, 0.28, -1.64 $\big{]}$ $t_{12}^{(1-2)}$ = $\big{[}$ -0.02,
-0.3$\big{]}$ The Löwdin method used for the downfolding mechanism can be
explained by,

$H^{\rm
downfold}_{\alpha,\beta}=H_{\alpha,\beta}+\sum_{\gamma\neq\alpha}^{\prime}\frac{H_{\alpha,\gamma}(H_{\beta,\gamma})^{*}}{H_{\alpha,\alpha}-H_{\gamma,\gamma}}$
(31)

Here, $H^{\rm downfold}_{\alpha,\beta}$ is final 5$\times$5 downfolded
Hamiltonian matrix, $\gamma$ contains O-$p$ – O-$p$, Cu-$d$ – O-$p$ orbitals
interaction respectively which are projected on A.

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[a,b]Maksym Ovchynnikov

#  Sensitivities to feebly interacting particles: public and unified
calculations

Jean-Loup Tastet  Oleksii Mikulenko  Kyrylo Bondarenko

###### Abstract

The increasing interest in Long-Lived Particles (LLPs) has led to numerous
proposed experiments in order to search for them. However, the sensitivity
estimates published by these experiments tend to rely on disparate
assumptions. To ensure an accurate comparison of their potential to find LLPs,
a unified estimation of their sensitivity is therefore required. In this
contribution, we introduce SensCalc, a Mathematica-based code that uses a
semi-analytic approach to calculate the event rate of GeV-scale LLPs, and we
present several case studies.

##  1 The need for a unified public sensitivity evaluator

Recently, a lot of experiments have been proposed to search for LLPs in the
GeV-scale range [1]. Most of them are at an early stage of research and
development, meaning in particular that the setup is constantly changing. To
understand the relative potential of each experiment to explore LLPs, the
Physics Beyond Colliders (PBC) working group proposed to compare their
sensitivities in the LLP parameter space for several benchmark models of
Higgs-like scalars, axion-like particles, heavy neutral leptons, dark photons,
millicharged particles, and light dark matter [2].

This myriad of experiments with changing designs, combined with different
descriptions of the LLP phenomenology existing in the literature, have made it
particularly difficult to conduct a fair and accurate comparison. Notable
difficulties include the different assumptions made about the LLP
phenomenology in the calculations done by different experiments, the slow
updates of sensitivities when changing the setup, and, last but not least, the
lack of access to the tools used to perform the sensitivity calculations by
the broader community (outside the collaborations), which affects the
transparency of the calculations. For some LLPs, the calculations have been
performed by groups outside the collaborations, making the picture even more
inhomogeneous.

This lack of consistency poses the need for a public sensitivity evaluator
that can calculate the event rates for LLPs at different experiments in a
unified way. The requirements for such an evaluator are: i) a good accuracy,
comparable with the Monte-Carlo evaluators, ii) its relative simplicity,
allowing people to use it without any efforts for already-implemented LLPs and
experiments, and also implement new ones easily, and iii) a short runtime when
calculating the sensitivity from scratch.

##  2 SensCalc

Semi-analytic calculations [3] provide a good starting point for such a
unified approach. The idea is to represent the number of events as the
integral of the product of several pre-computed quantities: the LLP angle-
energy distribution, the geometric acceptance of the experiment, the LLP decay
probability, and the acceptance for the decay products. This method has been
successfully used for a number of experiments and LLPs [4, 5, 6] (see also [7,
8]).

In ref. [9], we have systematically implemented this approach by developing
the Mathematica-based code SensCalc.111Available at
https://doi.org/10.5281/zenodo.7957784 and also at
https://github.com/maksymovchynnikov/SensCalc . It is made of four modules:
one calculating the tabulated geometric acceptance of the experiment and the
decay products’ acceptance, one calculating the LLP angle-energy distribution
function, one that combines these quantities to compute the tabulated number
of events and the sensitivity; and finally one which produces plots.

The implemented LLPs includes heavy neutral leptons with various mixing
patterns, Higgs-like scalars with mixing and trilinear coupling to the Higgs
boson, dark photons, ALPs coupled to fermions, gluons, or photons, and
anomaly-free $U(1)$ mediators coupled to the baryon and lepton currents. Many
other models will be implemented in the future. The included experiments are
those currently running or proposed at Fermilab and at the SPS, LHC, and FCC-
hh (see Fig. 1 for some examples).

Figure 1: Some of the implemented experiments: the Downstream setup at LHCb
[10] (top left), SHiP [11] (top right), MATHUSLA [12] (bottom left) and the
ceiling configuration of ANUBIS [13] (bottom right). The black dot shows the
location of proton-proton or proton-target interaction; the blue domains
indicate the decay volumes and the red domains are detectors.

SensCalc features a lot of options, including: specifying the visible LLP
decay channels; imposing pre-selection criteria for the decay products, e.g.,
on the invariant mass of particle pairs, their energy, the spatial separation
between tracks, and the impact parameter; estimating the effect of the
detector’s dipole magnet on the trajectories of the decay products; studying
explicitly the impact of various factors entering the number of events (such
as the geometric acceptance and decay probability) on the event rate;
producing the event rate distribution in terms of the LLP’s angle and energy;
producing the event density in the plane of the LLP’s mass and coupling.
SensCalc’s main limitation comes from the fact that it does not produce a
detailed event record, so it is not suited for a detailed event analysis or
further processing with the full detector simulation. An advanced version of
SensCalc that produces the event data is in preparation.

SensCalc has been cross-checked for various LLPs and geometries against
several Monte-Carlo codes (more details can be found in ref. [9]).

##  3 Case studies

Let us illustrate some interesting applications of SensCalc.

###  3.1 Qualitative comparison of the sensitivities of different experiments

Let us consider a few experiments that differ significantly in their housing
facility, geometric placement, and detector equipment. As an example, we will
use the SHiP, FASER2, and MATHUSLA experiments, which would be located at the
SPS (SHiP) and the LHC (MATHUSLA, FASER2), on-axis (SHiP, FASER2) and off-axis
(MATHUSLA). In addition, SHiP and FASER2 are equipped with electromagnetic and
hadronic calorimeters, unlike MATHUSLA, which lacks those. The goal is to
understand qualitatively the impact of these factors on the sensitivity.

For this purpose, we will consider the regime of large LLP lifetimes, such
that the decay probability is in the linear regime for all of the mentioned
experiments,

$P_{\text{decay}}\approx\frac{\Delta L_{\text{decay
volume}}}{c\tau_{\text{LLP}}}m_{\text{LLP}}\langle p_{\text{LLP}}^{-1}\rangle$
(1)

and study the mass dependence of the total number of produced LLPs (denoted by
$\mathcal{I}_{0}$), the fraction of LLPs intersecting the decay volume
($\mathcal{I}_{1}$), the fraction of those decaying inside
($\mathcal{I}_{2}$), and the subset of these events for which the decay
products meet the decay acceptance criteria ($\mathcal{I}_{3}$), which
corresponds to the total number of events (more details on the procedure may
be found in Sec. IV.A of ref. [9]). The ratios of these quantities between the
three considered experiments, for the models of Heavy Neutral Leptons and
Higgs-like scalars, are shown in Fig. 2.

Figure 2: The _ratio_ of the $\mathcal{I}_{i}$ quantities (see text for
details) for SHiP$/$FASER2 (solid lines) or SHiP$/$MATHUSLA (dashed lines) for
Heavy Neutral Leptons coupled solely to the electron neutrino (left) and
Higgs-like scalars mixing with the Higgs boson (right). Note that the blue
lines overlap because of the same luminosity to be collected during the
runtime of FASER2 and MATHUSLA.

From the figure, we see that the total number of produced LLPs (blue lines)
from $B$ mesons (above $2\,\mathrm{GeV}$ for HNLs, everywhere for Higgs-like
scalars) is larger at the LHC, whereas the amount of the LLPs produced from
$D$ mesons (below $2\,\mathrm{GeV}$ for HNLs) is much larger at the SPS. The
situation changes once we require (red lines) the LLPs to pass through the
decay volumes of SHiP, FASER2, and MATHUSLA, as FASER2 has a much smaller
geometric acceptance than SHiP, while MATHUSLA is located off-axis where the
flux of $B$ is suppressed. Next, taking also into account the decay
probability (1) (green lines), we see that the ratio SHiP$/$FASER2 further
increases as SHiP has a much longer decay volume and LLPs at the SPS have a
much larger $\langle p_{\text{LLP}}^{-1}\rangle$, while SHiP$/$MATHUSLA
decreases, since $\langle p_{\text{LLP}}^{-1}\rangle$ at MATHUSLA is much
larger due to the off-axis placement. Considering finally the decay acceptance
(black lines), the ratio SHiP$/$FASER2 decreases (since decay products at
FASER2 always point to the detector), while SHiP$/$MATHUSLA increases (as
decay products at MATHUSLA are low-energetic and may easily miss the
detector).

###  3.2 ALPs coupled to fermions and Higgs-like scalars

Two examples of long-lived particles for which the published sensitivities [2,
1] have been computed under disparate assumptions are ALPs coupled to fermions
(denoted by BC10) and Higgs-like scalars (BC4). In particular, some of
experiments use the inclusive description of the dark scalar production from
$B$ mesons (which may underestimate the number of events by a factor of 10 or
higher in the domain of large masses) and are missing production and decay
channels for the ALPs coupled to fermions (the interested reader is referred
to refs. [9, 14]). Finally, some of the setups used to compute the sensitivity
are already outdated, or even disagree with all published descriptions of the
experiment.

Figure 3: Sensitivities of the SHiP, FASER2, and MATHUSLA experiments to ALPs
that couple universally to fermions (left) and to dark scalars mixing with the
Higgs boson (right). The solid lines denote the phenomenology newly derived in
refs. [14, 15], while the dashed lines assume the inclusive production of dark
scalars and, for ALPs, the absence of hadronic decays and the production only
through the channels $B\to K+a$ and $B\to K^{*}+a$, similarly to what was
previously used in ref. [2]. For SHiP, we adopted the decay products selection
criteria specified in ref. [11] and the setup from ref. [16]. For FASER2 and
MATHUSLA, we did not assume any selection on the decay products beyond the
geometric acceptance itself, and we used the setups from ref. [12, 17].

In Fig. 3, we show the sensitivities of the SHiP, MATHUSLA, and FASER2
experiments to the scalars and ALPs under different sets of assumptions about
their phenomenology, in order to demonstrate the impact of assumptions on the
parameter space probed. We notably observe that the boundary of the probed
region may change by orders of magnitude.

In conclusion, we have presented a public and unified sensitivity evaluator —
SensCalc — that performs sensitivity calculations in a transparent and robust
way and is made available to the whole scientific community. Its aim is to
qualitatively improve the exploration of the FIP parameter space, by ensuring
greater consistency between the sensitivity studies.

## References

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