# Unique $d_{xy}$ Superconducting State in the Cuprate Member Ba2CuO3.25

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  Amit Chauhan  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  Sashi Satpathy
Condensed Matter Theory and Computational Lab, Department of Physics, Indian
Institute Of Technology Madras, Chennai, 600036, India  Department of Physics
& Astronomy, University of Missouri, Columbia, MO 65211, USA  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  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

###### Abstract

Recent discovery of superconductivity at a transition temperature of $73$K in
the doped layered compound Ba2CuO3+x for $x\sim 0.2$ has generated a lot of
interest. Experiments in this alternately stacked oxygen octahedral and chain
layered structure reveal that a compression of the octahedra causes the Cu-
$d_{z^{2}}$ orbital to lie above the Cu- $d_{x^{2}-y^{2}}$ orbital unlike in
the well-known cuprate superconducting materials. Our first-principle
calculations and low-energy Hamiltonian studies on the $x$ = 0.25 system
reveal that this energy ordering results in formation of $d_{z^{2}}$ dominated
electron pockets. The strong nesting in the Fermi pockets leads to an AFM spin
fluctuation mediated $d_{xy}$ wave superconducting state dominated by pairing
among the $d_{z^{2}}$ orbitals. This is in contrast to the cuprate
superconductors (e.g. YBCO) where both electron and hole pockets exist and the
superconducting state with B1g symmetry are formed by the $d_{x^{2}-y^{2}}$
orbital electrons. Unlike the earlier reports we find the inter-layer
hybridization has an important contribution to the low energy band structure
and formation of the unconventional superconducting state.

A large class of cuprate compounds shows a high-temperature superconducting
phase at moderate carrier doping, where Cu $d_{x^{2}-y^{2}}$ orbital electrons
are responsible for the formation of the cooper pair condensate. Recently a
new class of overdoped cuprate material Liu  _et al._ (2006); Gauzzi  _et al._
(2016) has emerged, which exhibits higher superconducting transition
temperature than the typical cuprates at similar carrier doping Damascelli
_et al._ (2003); Scalapino (2012); Bennemann and Ketterson (2008). Among them
is the orthorhombic compound Ba2CuO3+x Li  _et al._ (2019) with a
superconducting transition temperature of $T_{c}\sim 73$K. In this material,
an octahedral distortion breaks the degeneracy of the $e_{g}$ orbitals,
leading to a partially filled $d_{z^{2}}$ orbital and a fully occupied
$d_{x^{2}-y^{2}}$ orbital. The presence of high-temperature superconductivity
in this material in spite of significantly higher doping levels and low energy
physics that is dominated by $d_{z^{2}}$ orbitals provides a new channel for
understanding high-temperature superconductivity.

Figure 1: (a) Crystal structure (Pmmm) of Ba2CuO3.25 (BCO). Layer-I has
alternate stacking of octahedral and square planar Cu-O complexes (along $b$).
The octahedra form a corner-share network along $a$-direction. In layer-II,
due to missing oxygen atoms when compared to Ba2CuO4, only square planar
complexes exist and they form a chain along b-direction (b) The Fermi surface
of BCO. Center elliptical pocket with strong nesting is formed by
Cu(1)-$d_{z^{2}}$ orbital of layer-I while the open electron sheets are formed
by Cu(3, 4)-$d_{b^{2}-c^{2}}$ layer-II.

Recent experimental studies on Ba2CuO3+x with $x$ = 0.2 Li  _et al._ (2019);
Fumagalli  _et al._ (2021) measuring oxygen K-edge X-ray absorption spectra
(XAS) estimate 40% doping, that is significantly higher than doping in
overdoped cuprates like YBCO. The Zhang-Rice singlet state is observed at this
oxygen doping with dominant pre-peak at 528 eV photon energy Li  _et al._
(2019). The XAS measurements Li  _et al._ (2019) on the Ba2CuO3.2 suggested a
compressed octahedral structure. This prompted Maier $et.\hskip 2.84544ptal.$
Maier  _et al._ (2019) to suggest a simplified two-band model using
$d_{z^{2}}$ and $d_{x^{2}-y^{2}}$ orbitals within the 214 structure. They
propose two dome superconductivity one at low doping and the other at high
oxygen doping. Similar calculations based on spin-fluctuation theory on Lieb
lattice Yamazaki  _et al._ (2020) structure give $s\pm$ wave
superconductivity.

The unit cell of Ba2CuO3.25 (BCO) has two layers, layer-I and layer-II (see
Fig. 1). The DFT+DMFT based calculations Worm  _et al._ (2022) proposed the
presence of a charge transfer between layer-I and layer-II and proposed that
due to the presence of a quasi-1D band, antiferromagnetic spin-fluctuation may
occur to give rise to superconductivity. The specific-heat measurement Li  _et
al._ (2019) on BCO indicates that superconductivity is very anisotropic in
contrast to the exponential jump at $T_{c}$ of conventional electron-phonon
superconductivity.

In this letter, using a combination of first-principles DFT calculations and a
spin fluctuation mediated superconducting pairing mechanism Scalapino (1995),
we find a crucial inter-layer hybridization present in BCO which not only
helps to stabilize a novel $d_{xy}$ symmetry superconducting state but also
leads to a weak 3-dimensional character to the superconducting gap function.
Experimentally, in the XAS spectra, a transition $2p^{6}3d^{9}L\rightarrow
2p^{5}3d^{10}L$ is seen Li  _et al._ (2019) which we attribute to this
hybridization and is associated with hopping between Cu-$d$ and
O-$p_{y}/p_{z}$ orbitals. We identify the ground state superconducting gap
function for decoupled individual layers and the bulk BCO that includes inter-
layer hybridization. The analysis provides a comparative study between a layer
decoupled and hybridized low energy Hamiltonian to elucidate the role of the
latter towards superconductivity. Despite the larger electronic doping, we
find that the Fermi surface (FS) remains significantly nested, and it leads to
a large paramagnetic susceptibility and superconducting pairing potential. The
obtained superconducting gap function belongs to pairing between electrons
predominantly in $d_{z^{2}}$ orbitals.

We find that the gap symmetry and the strength of the pairing of the bulk are
different from a model incorporating decoupled layers. Furthermore, due to the
presence of inter-layer hybridization, the magnitude of pairing eigenfunction
has a weak $k_{z}$ dependence, which is otherwise absent in the case of
decoupled layers. The pairing symmetry of the bulk has a gap function with
$d_{xy}$ symmetry and the pairing symmetry of individual layer-I is $s+$ type
and layer-II is $s-$ type with additional nodes at the FS. Further, the broken
$C_{4}$ rotational symmetry due to structural distortion results in a pairing
symmetry belonging to the $D_{2h}$ point group.

Electronic structure - Employing the DFT+DMFT method, Worm et al. Worm  _et
al._ (2022) have examined the electronic structure of BCO to make a broad
prediction of the presence of an almost half-filled, strongly nested, quasi-1D
$d_{b^{2}-c^{2}}$ band which is probable cause of superconductivity. Here, we
would like to make a comprehensive analysis of the electronic structure using
DFT and to develop a tight-binding (TB) model that examines the inter-layer
coupling and its effect on FS of the BCO. As discussed later and in the
Supplementary Material (SM), the minimal basis set TB model is developed by
both the Slater-Koster formulation and the Loẃdin downfolding technique.

The DFT-derived bands are shown in blue in Fig. 2 (a), along with the orbital
resolved density of states shown in Fig. 2 (b). Details of the DFT
calculations and orbital resolved band structure are provided in the SM. Below
we mention the main findings of DFT results. (I) $d_{z^{2}}$ orbital of Cu(1)
is within the range of -0.56 eV to 1.13 eV w.r.t. the Fermi level (EF). The
$d_{x^{2}-y^{2}}$ orbital is completely occupied and lies in the range -1.57
eV to -0.09 eV. This highlights the role of $d_{z^{2}}$ orbitals in the
typical energy scales associated with the superconducting transition. (II)
$d_{b^{2}-c^{2}}$ orbital of Cu(2) is about 0.89 eV above EF (0.89 eV to 1.56
eV). (III) The O-$p$ states are extended in valence bands and lie up to -0.27
eV below EF.

To gain further insight into the electronic structure, specifically to obtain
the eigenvectors of the states occupying the Fermi level, we developed a low
energy TB Hamiltonian initially with a 14 orbital basis and later downfolded
to a five orbital basis. The Hamiltonian reproduces well the DFT band
structure in the corresponding energy range (see Fig. 2 (a)) and thereby
enables us to calculate the random phase approximation (RPA) spin
susceptibility.

In this model, four Cu atoms in the unit cell contribute five Cu-$d$ orbitals,
and the nearest neighbor O atoms contribute to the nine O-$p$ orbitals near
the Fermi level (See SM for detailed analysis). We write the Hamiltonian in
the following form,

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

Here, $c^{\dagger}_{{\bf k},\alpha,\sigma}$ ($c_{{\bf k},\alpha,\sigma}$ ) is
the Fermion creation (annihilation) operator for orbital $\alpha$ with spin
$\sigma=\uparrow,\downarrow$. $\xi_{\alpha\beta}({\bf k})$ is the kinetic
energy term containing hopping parameters and $\mu_{\alpha}$ is the onsite
energy.

Figure 2:  (a) The five-band TB model fitted with DFT bands with and without
inter-layer coupling. The inter-layer hybridization shifts the Van Hove
singularities near Y and T points at the high symmetry $k$-path of the
Brillouin zone. The energy gap between Van Hove singularities is absent for
the decoupled individual layers. The high symmetry $k$-path used to plot the
band structure is provided in the Fig. S1 of the SM. (b) DFT obtained partial
density of states.

Since there are two weakly coupled inequivalent layers in the system, the TB
Hamiltonian contains intra-layer and inter-layer Hamiltonian contributions. In
a matrix form, the Hamiltonian can be expressed as:

$H=\left(\begin{array}[]{cc}H_{l1}&H_{l1-l2}\\\ H_{l1-l2}^{\dagger}&H_{l2}\\\
\end{array}\right)$ (2)

Here, $H_{l1}$ and $H_{l2}$ are the intra-layer Hamiltonian submatrices for
layer-I and layer-II respectively, and $H_{l1-l2}$ accounts for the inter-
layer hybridization. Using Loẅdin downfolding Löwdin (1951) procedure, we
obtain an effective Hamiltonian by integrating out the oxygen subspace while
keeping only the Cu-$d$ orbitals in the Hamiltonian. The matrix elements and
the downfolding formalism are provided in the SM.

Figure 3: (a), (b) Fermi surfaces of decoupled individual layer-I and layer-II
in ${k_{x}}-{k_{y}}$ plane. The Fermi pockets are electron-like, coming from
the $d_{z^{2}}$ orbital of Cu(1) atom and $d_{b^{2}-c^{2}}$ orbital of Cu(3)
atom. (c), (d) RPA spin susceptibility [$\rm Tr[\chi_{s}]$] of decoupled
individual layer-I and layer-II within the ${q_{x}}-{q_{y}}$ plane. The
dominant nesting vectors are denoted by ${\bf Q_{1}}$, ${\bf Q_{2}}$. ${\bf
G}$ is the reciprocal lattice vector.

The TB bands with and without inter-layer hybridization are shown in Fig. 2
(a) and are compared with the DFT obtained band structure. The lack of inter-
layer coupling does not reproduce the band structure well with the subtle
differences can be seen in Fig. 2 (a) when momenta changes along
$T(0,\pi,\pi)$ to $Z(0,0,\pi)$ direction. The distinction between them comes
from inter-layer hybridizations Cu(1)-$d$ – Cu(3, 4)-$d$ of strength
$t_{12}^{(1)}$ and between Cu(1)-$d$ – O-$p$ of strength $t_{12}^{(2)}$. We
find $t_{12}^{(1)}$ to be one order of magnitude higher than $t_{12}^{(2)}$
(see SM). Most importantly, the inter-layer hopping pushes the Van Hove
singularity slightly above the Fermi level. The $d_{z^{2}}$ orbital of Cu(1)
and $d_{b^{2}-c^{2}}$ orbital of Cu(3, 4) cross Fermi level and form electron
like pockets. Both electron pockets hybridize near $Y(0,\pi,0)$ point. The
$d_{x^{2}-y^{2}}$ orbital of Cu(1) octahedra lies slightly below the Fermi
level. This happens because of octahedral distortion in the BCO structure at
very high doping. We can see in Fig. 2 (b) that Cu(1)-$d_{z^{2}}$ and Cu(3,
4)-$d_{b^{2}-c^{2}}$ have larger DOS at the EF than Cu(2)-$d$ orbitals. The FS
topology of the bulk is shown in Fig. 1 (b).

Superconducting state - The multi-orbital superconducting pairing Kernel is
derived from a spin fluctuation pairing mechanism. Scalapino (2012); Bennemann
and Ketterson (2008); Scalapino  _et al._ (1986); Rice and Ueda (1986); effer
(1964); Schrieffer  _et al._ (1989); Monthoux  _et al._ (1991); Sigrist and
Ueda (1991); Davis and Lee (2013); Das  _et al._ (2014); Mazin  _et al._
(2008); Graser  _et al._ (2009); Yao  _et al._ (2009); Das and Balatsky
(2011); Chubukov (2012); Takimoto  _et al._ (2004a, b); Das  _et al._ (2015);
Ikeda  _et al._ (2015); Nomoto and Ikeda (2016, 2017); Schmalian (1998); Saito
and Yoshida (2011) The fluctuation exchange approximation (FLEX) that has been
successfully utilized to extract the ground state superconducting states of
both cuprate Scalapino (2012); Bennemann and Ketterson (2008); Scalapino  _et
al._ (1986); Rice and Ueda (1986); effer (1964); Schrieffer  _et al._ (1989);
Monthoux  _et al._ (1991); Sigrist and Ueda (1991); Davis and Lee (2013); Das
_et al._ (2014) and iron based superconductors Scalapino (2012); Bennemann and
Ketterson (2008); Mazin  _et al._ (2008); Graser  _et al._ (2009); Yao  _et
al._ (2009); Das and Balatsky (2011); Chubukov (2012). The pairing Kernel
involves contribution from paramagnetic and charge susceptibilities that are
calculated from a Hubbard-Hund Hamiltonian within the RPA (See SM for detailed
analysis). Finally, the pairing interaction is included in the self-consistent
linearized gap equation in order to extract the ground-state superconducting
gap functions. The gap equation reads,

$\displaystyle\Delta_{\nu}({\bf k})$ $\displaystyle=$
$\displaystyle-\lambda\frac{1}{\Omega_{\rm BZ}}\sum_{\nu^{\prime},{\bf
q}}\Gamma^{\prime}_{\nu\nu^{\prime}}({\bf k,q})\Delta_{\nu^{\prime}}({\bf
k+q}).$ (3)

Where $\Gamma^{\prime}_{\nu\nu^{\prime}}({\bf k,q})$ is SC pairing potential
and $\lambda$ is the pairing strength. We obtain SC pairing potential by
expanding the interaction term of the Hubbard Hamiltonian in a perturbation
series and collecting the bubble and ladder diagrams,

$\displaystyle\tilde{\Gamma}_{s}({\bf q})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}3{\tilde{U}}_{s}{\tilde{\chi}}_{s}({\bf
q}){\tilde{U}}_{s}-{\tilde{U}}_{c}{\tilde{\chi}}_{c}({\bf
q}){\tilde{U}}_{c}+{\tilde{U}}_{s}+{\tilde{U}}_{c}\big{]}.$ (4)

Here, $\chi_{s}(\chi_{c})$ is the RPA spin (charge) susceptibility.

Figure 4: (a), (b) The superconducting pairing eigenfunction for the largest
eigenvalue on the Fermi surface for decoupled individual layer-I and layer-II
respectively. The pairing symmetry of layer-I and layer-II belongs to A1g
irreducible representation of the D2h point group.

In BCO, the effect of a broken $C_{4}$ rotational symmetry can be seen on the
FS and corresponding spin susceptibility calculations. We first explore the
Hamiltonian in the limit of no inter-layer hybridization. In Figs. 3 (a) and
(b) we show the FS topology of layer-I and layer-II respectively at $k_{z}=0$.
We find that layer-I shows a stronger 2D dispersion as compared to layer-II.
The effect of the quasi-1D nature of layer-II shows up in the 1D
susceptibility peaks observed from its FS nesting. The dominant FS nesting
vectors for individual layers are also shown in Figs. 3 (a) and (b). We show
the spin susceptibility of layer-I and layer-II in Figs. 3 (c) and (d)
respectively. This chain-like 1D FS enhances spin susceptibility in each layer
for small values for Hubbard interactions and leads to a dominant nesting for
layer-II at the incommensurate wavevector ${\bf Q_{2}}=(\pm 0.96\pi,\pm
0.68\pi)$. Similarly, the spin susceptibility result of layer-I leads to a
corresponding peak at ${\bf Q_{1}}=(\pm 0.66\pi,0)$. We find that $\chi_{s}$
at $\bf Q_{2}$ is larger than the corresponding maximum for $\bf Q_{1}$.

When inter-layer hybridization is absent, SC of each layer is decoupled from
the bulk BCO. Peaks of spin susceptibility will determine the maximum pairing
potential when nesting condition ($\xi_{{\bf k+Q}}=-\xi_{{\bf k}}$ ) is
satisfied by the momentum ${\bf q=Q}$ at the FS. The strong deviation of the
dominant susceptibility from the $C_{4}$ susceptibility of the well-known
cuprates can lead to corresponding deviations in the superconducting state. We
plot the superconducting gap function for layer-I and layer-II for the largest
pairing eigenvalue in Figs. 4 (a) and (b) respectively. Colormap blue to red
denotes the sign of the pairing symmetry. The pairing symmetry of layer-I and
layer-II leads to a dominant spin singlet superconducting gap that would
transform as an A1g irreducible representation of the D2h point group
symmetry. Whereas for layer-I the gap is only anisotropic near the region of
large curvature of the FS, the gap on layer-II shows the presence of
accidental nodes. We also find that the superconducting gaps have a 2D
structure over the FS within negligible $k_{z}$ dispersion.

The formation of a SC gap with A1g symmetry despite of a repulsive pairing
interaction is explained by the dominance of the inter-orbital pairing channel
over the intra-orbital pairing contribution (see the discussion in Section IV
of SM). This large off-diagonal contribution leads to an attractive pairing.
Such a scenario can also be induced by Hund’s interaction roig , although for
BCO it is already present at the non-interacting level.

Figure 5: (a), (b) The Fermi surface of BCO in the presence of inter-layer
hybridization at two different ${k_{z}}$ values. The weak 3D dispersion
induced by the inter-layer hybridization causes ellipticity of the electron
pocket at $k_{z}=\pi$ as compared to the pocket at $k_{z}=0$. (c), (d) RPA
spin susceptibility, $\rm Tr[\chi_{s}]$ for two ${q_{z}}$ values. The dominant
nesting vectors are denoted by ${\bf Q_{3}^{0}}$, ${\bf Q_{3}^{\pi}}$.

.

We next include inter-layer hybridization in the non-interacting Hamiltonian.
In Figs. 5 (a) and (b) we show the FS of bulk BCO at $k_{z}=0$ and $k_{z}=\pi$
respectively. Interestingly, the inter-layer hybridization not only enhances
the dispersion of the electronic bands along the $k_{z}$ direction, it also
leads to a significant shift of the dominant susceptibility peak. The RPA spin
susceptibility at $q_{z}=0$, and $q_{z}=\pi$ are shown in Figs. 5 (c) and (d).
As shown in Fig. 5 (c), the dominant nesting vector is still in the
$k_{x}$-$k_{y}$ plane but leads to a susceptibility peak at the wave vector
${\bf Q_{3}^{0}}=(\pm 0.48\pi,\pm 0.52\pi)$ for $k_{z}=0$. The larger
ellipticity of the electron pocket at $k_{z}=\pi$ as compared to the pocket at
$k_{z}=0$ shows the weak 3D dispersion induced by the inter-layer
hybridization, and as shown in Fig. 5 (d) leads to a susceptibility peak at
around ${\bf Q_{3}^{\pi}}=(\pm 0.44\pi,\pm 0.56\pi)$ wavevector.

The effect of inter-layer hybridization is even more significant for the
ground-state superconducting gap functions. In Figs. 6 (a) and (b) we plot the
superconducting gap function of bulk BCO for the largest pairing eigenvalues
at two different $k_{z}$ values. The pairing symmetry of bulk BCO on the
elliptical hole pocket can be expressed in the form
$\Delta(\bf{k})=\Delta_{0}\sin\it(k_{x})\sin(k_{y})$ with line nodes along the
$k_{x}=0$ and $k_{y}=0$ lines on the FS. This is similar to the cuprate B1g
superconducting basis function on a $\pi/4$ rotated axis. We find that
although the sign of the gap remains unchanged along $k_{z}$, the gap function
magnitude ($\Delta_{0}$) gets enhanced with increasing $k_{z}$. In bulk BCO,
the pairing eigenvalue $\lambda$ seems to closely track the transition of
unhybridized layer-I model (See SM Fig. S4). This feature is likely due to the
dominance of the $d_{z^{2}}$ electrons and corresponding orbital resolved
pairing interaction for the layer-I model. Our predictions can be probed by
experimental techniques such as ARPES, STM among a variety of techniques that
have been successfully utilized to understand the superconducting state in the
cuprates.

Figure 6:  (a), (b) The solution of the superconducting gap equation on the
Fermi surface of the bulk BCO for two representatives ${k_{z}}$ values. The
pairing symmetry in presence of inter-layer hybridization is $d_{xy}$ type
with nodes along the $k_{x}=0$ and $k_{y}=0$ lines on the Fermi surface.

Conclusions - The recent discovery of superconductivity in BCO at
T${}_{c}=73$K with high hole doping levels places this material in a new
parameter regime among the various classes of cuprate high-temperature
superconductors. From the DFT calculations, we find that the $d_{z^{2}}$
orbital of the Cu(1) atom belonging to the octahedra lies at the Fermi level,
and $d_{x^{2}-y^{2}}$ orbital is fully occupied. This makes BCO different from
usual cuprate superconductors. We propose an effective 5-orbital tight-binding
model consisting of the selective $d$\- orbitals of the Cu atoms that shows
excellent agreement with the DFT band structure when an inter-layer
hybridization is included in the model. From the tight-binding analysis, we
find the hybridization between the BCO layers, albeit weak, significantly
influences the band structure along $Y\Gamma$ and $TZ$ where the van Hove
singularities exist. The $d$-orbitals coming from Cu(3)(/Cu(4)) and Cu(1)
atoms form electron like pocket at the Fermi level. Similar to the YBCO, the
planar layer of bulk BCO forms quasi-1D chain states. However, the hole pocket
coming from $d_{x^{2}-y^{2}}$ orbital in YBCOAdhikary and Das (2020) and
infinite layer nickelatesAdhikary  _et al._ (2020) are absent in BCO.

In BCO, the presence of inter-layer hybridization plays a pivotal role in
reshaping the Fermi surface. It removes the parallel Fermi pocket regions
connected by dominant inter-orbital contributions. This leads to the dominance
of intra-orbital nesting and susceptibility along the $(\pi-\delta,0)$ wave-
vector [See SM Section IV for orbital resolved susceptibility contributions].
This diagonal intra-orbital pairing contribution will support an
unconventional superconducting order. Additionally, the $(\pi-\delta,0)$
nesting wave-vector found in our susceptibility calculations would support
$d_{xy}$ symmetry superconducting order. We find an $s$-wave gap on Fermi
pocket (without sign change over) for layer-I, if we ignore the inter-layer
hybridization. However, for layer-II, we do find a sign change of the
superconducting gap over the Fermi pocket that is expected from repulsive
interaction. Therefore, the gap over the entire Fermi pocket should not be
considered as a conventional $s$-wave gap but belonging to A1g symmetry with
higher harmonic contributions that can lead to a sign change of gap on layer-
II. The signatures of the nodal sign changing $d_{xy}$ gap can be probed in
future thermodynamic measurements like low-temperature specific heat and
thermal conductivity measurements. With synthesis of good quality single
crystals, the gap should also be directly observable in ARPES and scanning
tunneling spectroscopic experiments.

Acknowledgements: This work was funded by the Department of Science and
Technology, India, through Grant No. CRG/2020/004330. S.S. thanks SERB India
for the Visiting Advanced Joint Research (VAJRA) program of the Science and
Engineering Research Board, Department of Science and Technology (SERB-DST),
Government of India.

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Supplementary Materials for “ Unique $d_{xy}$ Superconducting State in the
Cuprate Member Ba2CuO3.25”

##  I Computational and Structural details

Density functional theory (DFT) calculations were performed using the Vienna
ab-initio Simulation Package (VASP) vasp  code with projector-augmented-wave
(PAW) PAW1 ; PAW2  pseudopotentials. Perdew–Burke–Ernzerhof (PBE) PBE
exchange-correlation functional scheme of generalized gradient approximation
(GGA) was considered to take care of exchange and correlation functional. We
have employed a kinetic energy cutoff of 500 eV and $\Gamma$-centred
16$\times$12$\times$4 $k$-mesh which yields 189 irreducible points for
Brillouin zone (BZ) sampling. A Slater-Koster (SK) Slater  based minimal basis
set tight-binding (TB) formalism is used to design the model Hamiltonian.

Figure S1:  First Brillouin zone (BZ) of BCO unitcell crystal structure along
with the high symmetry k-points.

The crystal structure of Ba2CuO3.25 (BCO) is obtained from Ba2CuO4 by removing
a few O-atoms as shown in Fig. S2 (a). The structure has been proposed in
previous studies bco_dft1 . The primitive unitcell of BCO contains two
inequivalent layers. As shown in Fig. S2 (a), layer-I forms an alternate CuO6
octahedron, and CuO4 square plane connected through the corner sharing O-atoms
along the b-direction, and layer-II contains a one-dimensional chain of CuO4
square plane extended along b-direction. Thus the unitcell has four
inequivalent Cu atoms ( Cu(1), Cu(2) from layer-I and Cu(3), Cu(4) from layer-
II ) and 13 O-atoms (seven in layer-I and six in layer-II).

Figure S2:  (a) Unitcell crystal structure of BCO. (b) DFT obtained orbital
resolved band structure of BCO. The bands are plotted along the high symmetry
k-path provided in the BZ.

In order to get the correct orbital basis set required for constructing the
model Hamiltonian, we have further performed the DFT calculations and studied
the electronic structure properties of the compound. Fig. S2(b) shows the
orbital resolved band structure of BCO, and it suggests that bands near the
Fermi are contributed by the covalent hybridization of Cu-O atomic orbitals.
However, not all hybridized orbitals contributions are present in the bands
near the Fermi level, the exact molecular orbitals are different for different
Cu atoms. For example, {Cu(2), Cu(3), and Cu(4)}-$x^{2}-y^{2}$ orbital
hybridized with nearest-neighbor in-plane O-{$p_{x}$, $p_{y}$} orbitals,
Cu(1)-$x^{2}-y^{2}$ hybridized with nearest-neighbor in-plane O-{$p_{x}$,
$p_{y}$}, and Cu(1)-$z^{2}$ hybridized with nearest-neighbor out-of-plane
O-$p_{z}$ orbital are contributing to the bands near the Fermi. Thus a total
of 14 orbital basis-set with five Cu-$d$, and nine O-$p$ orbitals are
sufficient to design the TB model Hamiltonian.

##  II Details of the tight-binding Hamiltonian

The matrix representation of the SK-TB model Hamiltonian of BCO is shown in
Eq. 2 of the main text. Here, the sub-matrix $H$ of the layer-I in basis set
order Cu(1) { $d_{z^{2}}$, $d_{x^{2}-y^{2}}$}, Cu(2) {$d_{x^{2}-y^{2}}$},
$p_{x}$, $p_{z}$, $p_{z}$, $p_{y}$, and $p_{y}$ is:

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

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

The Hamiltonian sub-matrix for layer-II in orbital basis set of Cu(3)/Cu(4)
{$d_{x^{2}-y^{2}}$}, $p_{z}$, $p_{z}$, $p_{y}$ and $p_{y}$ is given as

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

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

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

Figure S3: (Color online) (a) TB (red) fitted DFT (blue) band structure of
layer-I of BCO crystal. (b) Same for layer-II of BCO crystal. We plot bands
along the high symmetry k-path in the BZ as shown in Fig. S1. The dotted
circles in (a) show the disagreement between DFT and TB bands at those regions
when inter-layer interactions are ignored.

The components of the Hamiltonian matrices are found to be,

$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
t^{(5)}\cos(k_{x})+\mu^{(4)}+t^{(15)}\cos(k_{y})-0.1\cos(2k_{x})$
$\displaystyle\xi_{2}$ $\displaystyle=$ $\displaystyle
t^{(10)}\cos(ky)+t^{(11)}\cos(k_{x})+\mu^{(2)}$ (4a) $\displaystyle\xi_{3}$
$\displaystyle=$ $\displaystyle\xi_{2}+\mu^{(5)}$ (4b) $\displaystyle\xi_{4}$
$\displaystyle=$ $\displaystyle t^{(14)}\cos(k_{x})+\mu^{(6)}$ (4c)
$\displaystyle\xi_{5}$ $\displaystyle=$ $\displaystyle\xi_{4}+0.02$ (4d)
$\displaystyle\xi_{1,2}$ $\displaystyle=$ $\displaystyle
t^{(3)}\cos(k_{x})+t^{(9)}\cos(k_{y})$ (4e) $\displaystyle\xi_{1,3}$
$\displaystyle=$ $\displaystyle t^{(4)}\exp(ik_{y}/4)$ (4f)
$\displaystyle\xi_{2,3}$ $\displaystyle=$ $\displaystyle t^{(1)}\cos(k_{y}/2)$
(4g) $\displaystyle\xi_{1,4}$ $\displaystyle=$ $\displaystyle
it^{(8)}\sin(k_{x}/2)$ (4h) $\displaystyle\xi_{1,5}$ $\displaystyle=$
$\displaystyle it^{(8)}\sin(k_{y}/4)$ (4i) $\displaystyle\xi_{1,7}$
$\displaystyle=$ $\displaystyle it^{(6)}\sin(0.156k_{z})$ (4j)
$\displaystyle\xi_{2,4}$ $\displaystyle=$ $\displaystyle
it^{(7)}\sin(k_{x}/2)$ (4k) $\displaystyle\xi_{2,5}$ $\displaystyle=$
$\displaystyle t^{(4)}\exp(ik_{y}/4)$ (4l) $\displaystyle\xi_{4,5}$
$\displaystyle=$ $\displaystyle t^{(12)}\cos(k_{y}/2)$ (4m)
$\displaystyle\xi_{4,6}$ $\displaystyle=$ $\displaystyle
it^{(13)}\sin(k_{y}/4)$ (4n) $\displaystyle\xi_{12}^{1}$ $\displaystyle=$
$\displaystyle 2t_{12}^{(1)}\exp(-ik_{y}/4)\cos(k_{z}/2)\cos(k_{x}/2)$ (4o)
$\displaystyle\xi_{12}^{2}$ $\displaystyle=$ $\displaystyle
it_{12}^{(2)}\exp(ik_{y}/4)\sin(k_{z}/3)\cos(kx/2)$ (4p)

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$\big{]}$. We further use Loẅdin downfolding mechanism to reduce the
Hamiltonian size. The Loẅdin method used for the downfolding mechanism can be
explained by,

$H^{\rm
downfold}_{\alpha,\beta}=H_{\alpha,\beta}+\sum_{\gamma\neq\alpha}^{\prime}\frac{H_{\alpha,\gamma}(H_{\beta,\gamma})^{*}}{H_{\alpha,\alpha}-H_{\gamma,\gamma}}$
(5)

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. We have numerically
downfolded the full Hamiltonian and the resulting band structures are shown in
Fig. 2 (a) of main text which shows an excellent agreement around the Fermi
level with the all-electron band structure obtained from density functional
theory (DFT) calculations.

##  III Superconductivity

RPA spin susceptibility-  We use the multi-band Hubbard model to study the
topology of Fermi surface (FS) and corresponding spin-fluctuation potential.
The Hamiltonian of the Hubbard model is,

$\displaystyle H_{\rm int}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}\sum_{{\bf
k,k^{\prime},q}}U_{\alpha}c_{\alpha\uparrow}^{\dagger}({\bf
k})c_{\alpha\downarrow}^{\dagger}({\bf k}^{\prime})c_{\alpha\downarrow}({\bf
k^{\prime}-q})c_{\alpha\uparrow}({\bf
k+q})\sum_{\begin{subarray}{c}\alpha\neq\beta\\\
\sigma\sigma^{\prime}\in(\uparrow,\downarrow)\end{subarray}}\sum_{{\bf
k,k^{\prime},q}}\Big{[}V_{\alpha,\beta}c_{{\bf
k},\alpha,\sigma}^{\dagger}c_{{\bf
k}^{\prime},\beta,\tilde{\sigma}}^{\dagger}c_{{\bf
k^{\prime}-q},\beta,\tilde{\sigma}}c_{{\bf k+q},\alpha,\sigma}$
$\displaystyle+\Big{(}V_{\alpha,\beta}-J_{H}\Big{)}c_{{\bf
k},\alpha,\sigma}^{\dagger}c_{{\bf
k}^{\prime},\beta,{\sigma}}^{\dagger}c_{{\bf
k^{\prime}-q},\beta,{\sigma}}c_{{\bf k+q},\alpha,\sigma}\Big{]}.$

Here $c^{\dagger}_{{\bf k},\alpha,\sigma}$ and $c_{{\bf k},\beta,\sigma}$ are
fermion creation and annihilation operator in orbital $\alpha$ and $\beta$ and
$\tilde{\sigma}=-\sigma$. Where $U_{\alpha}$ and $V_{\alpha,\beta}$ are the
intra-orbital and inter-orbital Hubbard interaction between Cu-$d$ orbitals
and $J_{H}$ is the Hund’s coupling.

Non-interacting electron-hole density-density correlation function in the
orbital basis is given by fwave ; nickelates1  ,

$\displaystyle[\chi_{0}(\textbf{q})]_{\alpha\beta}^{\gamma\delta}$
$\displaystyle=$ $\displaystyle-\frac{1}{N}\sum_{{\bf
k},\nu\nu^{\prime}}\psi^{\nu}_{\beta}({\bf k})\psi^{\nu\dagger}_{\alpha}({\bf
k})\psi^{\nu^{\prime}}_{\delta}({\bf
k+q})\psi^{\nu^{\prime}\dagger}_{\gamma}({\bf
k+q})\frac{f(\xi_{\nu^{\prime}}({{\bf k+q}}))-f(\xi_{\nu}({{\bf
k}}))}{\xi_{\nu^{\prime}}({\bf k+q})-\xi_{\nu}({\bf k})+i\eta}.$ (7)

N is the volume of the phase space. Using S-matrix expansion of the spin
density and charge-density correlation function we obtain random-phase
approximation (RPA) spin and charge susceptibilities,

$\displaystyle\tilde{\chi}_{\rm s/c}({\bf q})=\tilde{\chi}_{0}({\bf
q})\left(\tilde{\mathbb{I}}\mp\tilde{U}_{s/c}\tilde{\chi}_{0}({\bf
q})\right)^{-1},$ (8)

The nonzero components of onsite Hubbard interactions for spin and charge
fluctuation are $\tilde{U}_{s}$ and $\tilde{U}_{c}$graser . Nesting of the FS
is captured in the Lindhard function. At the nesting vector, bare
susceptibility shows a strong peak which leads to a large peak in the RPA spin
susceptibility. The presence of $(1-\tilde{U}\chi_{0})$ in the denominator,
contribution from the spin channel enhanced whereas the charge channel is
suppressed due to $(1+\tilde{U}\chi_{0})$.

Spin-fluctuation theory-  We assume that the superconducting pairing in Cu-$d$
electrons is mediated via spin fluctuations. We calculate the spin-fluctuation
pairing potential by expanding the $H_{\rm int}$ from Eq. (LABEL:Hint) into a
perturbation series and collecting bubble and ladder diagrams. The effective
Hamiltonian we obtain as fwave ; nickelates1 ,

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{\alpha\beta\gamma\delta}\sum_{{\bf
kq},\sigma\sigma^{\prime}}\Gamma_{\alpha\beta}^{\gamma\delta}({\bf
q})c_{\alpha\sigma}^{\dagger}({\bf k})c_{\beta\sigma^{\prime}}^{\dagger}(-{\bf
k})c_{\gamma\sigma^{\prime}}({\bf-k-q})c_{\delta\sigma}({\bf k+q}).$ (9)

Here, the pairing potential is a tensor of four orbital indices. For singlet
and triplet channels the spin-fluctuation pairing potential is given by
SCrepulsive ; SCcuprates ; SCpnictides ; SCHF ; SCorganics ,

$\displaystyle\tilde{\Gamma}_{S}({\bf q})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}3{\tilde{U}}_{s}{\tilde{\chi}}_{s}({\bf
q}){\tilde{U}}_{s}-{\tilde{U}}_{c}{\tilde{\chi}}_{c}({\bf
q}){\tilde{U}}_{c}+{\tilde{U}}_{s}+{\tilde{U}}_{c}\big{]},$ (10a)
$\displaystyle\tilde{\Gamma}_{T}({\bf q})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\big{[}{\tilde{U}}_{s}{\tilde{\chi}}_{s}({\bf
q}){\tilde{U}}_{s}+{\tilde{U}}_{c}{\tilde{\chi}}_{c}({\bf
q}){\tilde{U}}_{c}\big{]}.$ (10b)

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

Figure S4: We plot superconducting pairing eigenvalue as a function onsite
Hubbard interaction for layer-I, layer-II, and bulk BCO.

Figure S5:  First column shows orbital weights on the bands of Layer-I in the
absence of inter-layer hybridization, Second column shows orbital weights at
the Fermi surface. Third and fourth column shows inter-orbital and intra-
orbital Lindhard susceptibilities in absence of the inter-layer hybridization.

Figure S6:  First column shows different orbital weights on BCO bands in the
presence of inter-layer hybridization. Second column shows orbital weights at
the Fermi surface at $k_{z}=0$. Third and fourth col. shows inter and inter-
orbital Lindhard susceptibilities respectively at $q_{z}=0$.

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

We obtain superconducting pairing symmetry by solving the linearized gap
equation,

$\displaystyle\Delta_{\mu}({\bf k})=-\lambda\frac{1}{N}\sum_{\nu,{\bf
q}}\tilde{\Gamma}_{\mu\nu}({\bf k,q})\Delta_{\nu}({\bf k+q}).$ (12)

$\lambda$ is known as a superconducting coupling constant. By solving Eq. (12)
we obtain pairing eigenfunction for largest eigenvalue. This largest
eigenvalue determines the stability of superconducting gap function
$\Delta({\rm\bf k})$SCrepulsive .

The unconventional SC within spin fluctuation theory originates from the
nesting at the FS. Since $\chi_{s}$ (see Eq. (8)) is positive and larger than
$\chi_{c}$, pairing potential Eq. (10a) is repulsive. The only possible
solution of the gap equation for repulsive interaction is when $\Delta$ (see
Eq. (12) ) changes sign between momentum vectors ${\bf k}$ and ${\bf k+Q}$,
where ${\bf Q}$ is the nesting vector that connects the Cooper pairs. This
leads to an anisotropic solution of the gap function in the momentum space
which should reflect underlying point group symmetry.

We study the effect of Hubbard interaction on the superconducting pairing
eigenvalue (see Fig. S4 ). This result shows a comparison of dominant pairing
eigenvalue for layer-I, and layer-II for unhybridized bands and bulk BCO with
hybridized bands as a function of onsite Hubbard interaction. We see that
pairing eigenvalue $\lambda$ extracted by solving the linearized gap equation
is slightly larger on layer-II in comparison to layer-I for the unhybridized
bands for $U>0.4$. However, the decoupled quasi-1D layer would generically be
more susceptible to quantum fluctuations that are likely to suppress the
transition temperature strongly.

##  IV Orbital resolved susceptibilities- The role of interlayer hybridization

### Absence of inter-layer hybridization

In Fig. S5, we have illustrated the orbital resolved susceptibilities and
electronic structures when inter-layer hybridization is absent. The key
insight provided by Fig. S5 lies in the comparison between the inter and
intra-orbital susceptibilities shown in the third and fourth columns.
Specifically, we have demonstrated that the inter-orbital susceptibility is
dominant over the intra-orbital susceptibility. This observation is essential
in understanding how attractive pairing interactions can arise without Hund’s
coupling.

The physical susceptibility in the main manuscript suggests that intra-orbital
susceptibility might dominate due to $(\pi-\delta,0)$ nesting, but a closer
look (Fig. S5) reveals that inter-orbital susceptibility is actually dominant.
This inter-orbital susceptibility peaks at a different nesting vector and is
smaller at $(\pi-\delta,0)$. This finding explains why layer-I can create an
effective attractive pairing channel.

### Presence of inter-layer hybridization

In Fig. S6, we have shown the orbital weights across different bands and the
Fermi surface of BCO under the influence of inter-layer hybridization. As
expected, the presence of inter-layer hybridization significantly alters the
Fermi surface and orbital weights near the $(0,\pi)$ region, thereby makes a
significant effect on the pairing interactions.

Specifically, as can be seen from the calculated orbital susceptibility plots,
the introduction of inter-layer hybridization results in a reduction of the
contribution from inter-orbital susceptibility. This change in susceptibility
is a direct consequence of the modified Fermi surface and orbital weights. In
particular, in the presence of hybridization, the elliptical Fermi pocket for
layer-I is now completely dominated by the $d_{z^{2}}$ orbital that will now
support susceptibility in intra-orbital channel. Secondly, the modification of
the Fermi pocket shape near the hybridization region suppresses the parallel
Fermi pocket regions that previously supported the inter-orbital nesting in
the absence of inter-layer hybridization. As is well known, such parallel
regions are required to have sufficient weight from the $\bf k$-sum at a given
q wave-vector in susceptibility calculations. The above scenario therefore not
only will support a nesting vector along the $(\pi-\delta,0)$ wave-vector, but
the susceptibility will be driven by the dominant intra-orbital contributions.
This will naturally lead to a dominant intra-orbital pairing (or dominant
diagonal contribution to the pairing matrix) and the corresponding repulsive
pairing interaction plays a pivotal role in the emergence of the observed
$d_{xy}$ pairing symmetry in BCO.

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