# Role of crystal-field-splitting and longe-range-hoppings on superconducting
pairing symmetry of La3Ni2O7

Hongquan Liu1,2111These authors contributed equally to this work., Chengliang
Xia1††footnotemark: , Shengjie Zhou1 and Hanghui
<EMAIL_ADDRESS>1NYU-ECNU Institute of Physics, NYU Shanghai,
Shanghai 200122, China
2Department of Physics, Brown University, 182 Hope Street, Providence, RI
02912, USA
3Department of Physics, New York University, New York, New York 10012, USA

###### Abstract

We study the bilayer two-orbital model for superconducting pairing symmetry of
La3Ni2O7 under pressure. By combining density-functional-theory (DFT),
maximally-localized-Wannier-function, and linearized Eliashberg equation with
random-phase-approximation, we find that the superconducting pairing symmetry
of La3Ni2O7 is robustly $d_{xy}$ if its DFT band structure is accurately
reproduced in the downfolded model. We further show that fine-tuning of
crystal-field-splitting between two Ni-$e_{g}$ orbitals qualitatively affects
superconducting pairing symmetry of the bilayer two-orbital model, which
changes from $d_{xy}$ to $s_{\pm}$ as the crystal-field-splitting exceeds a
critical value. When the model only includes nearest-neighbor and second-
nearest-neighbor hoppings, the crystal-field-splitting obtained by fitting to
the DFT band structure is larger than the critical value and thus leads to
$s_{\pm}$ superconducting pairing symmetry. When all nonzero long-range-
hoppings are also included in the model, the fitted crystal-field-splitting is
reduced and smaller than the critical value, which makes $d_{xy}$
superconducting pairing symmetry more favorable than $s_{\pm}$ symmetry. Our
work demonstrates that in downfolded effective models, the details of band
structure can play a crucial role in determining pairing symmetry in multi-
orbital unconventional superconductors (such as La3Ni2O7).

The recent discovery of superconductivity in La3Ni2O7 under pressure has drawn
great attention Sun et al. (2023); Hou et al. (2023); Qin and Yang (2023);
Shilenko and Leonov (2023); Luo et al. (2023a); Huang et al. (2023); Zhang et
al. (2023a); Yang et al. (2023a); Gu et al. (2023); Zhang et al. (2023b); Luo
et al. (2023b); Oh and Zhang (2023); Geisler et al. (2023); Lange et al.
(2023); LaBollita et al. (2023); Zhang et al. (2023c); Yang et al. (2023b);
Pan et al. (2023); Tian et al. (2023); Zhang et al. (2023d); Ryee et al.
(2023); Lu et al. (2023a); Jiang et al. (2023); Wang et al. (2023); Yang et
al. (2023c); Kaneko et al. (2023); Liao et al. (2023); Rhodes and Wahl (2023);
Zhang et al. (2023e); Qu et al. (2023); Lu et al. (2023b); Chen et al.
(2023a); Cao and feng Yang (2023); Christiansson et al. (2023); Shen et al.
(2023); Lechermann et al. (2023); Li et al. (2023); Chen et al. (2023b). It
not only adds a new member to the family of nickelate superconductors Li et
al. (2019, 2020); Zeng et al. (2020); Gu et al. (2020a, b); Zeng et al.
(2022); Osada et al. (2021); Xia et al. (2022); Chen et al. (2023c), but also
substantially increases the transition temperature to near 80 K Sun et al.
(2023). A bilayer two-orbital model that consists of Ni-$e_{g}$ orbitals has
been widely used as the minimum model to study La3Ni2O7 under pressure Luo et
al. (2023a); Yang et al. (2023a); Huang et al. (2023); Luo et al. (2023b); Gu
et al. (2023); Zhang et al. (2023b, a). The prototype is from Ref. Luo et al.
(2023a) (referred to as Luo’s model in this work). Based on the bilayer two-
orbital model, a number of studies use different many-body techniques and find
that the leading superconducting instability of La3Ni2O7 has $s_{\pm}$ pairing
symmetry Luo et al. (2023a); Yang et al. (2023a); Huang et al. (2023); Luo et
al. (2023b); Gu et al. (2023); Zhang et al. (2023b, a), distinct from the
pairing symmetry of well-known cuprate superconductors Van Harlingen (1995);
Tsuei and Kirtley (2000); Taillefer (2010) but reminiscent of iron-based
superconductors Chubukov et al. (2008); Hirschfeld et al. (2011); Stewart
(2011). On the other hand, Ref. Lechermann et al. (2023) finds $d$-wave
pairing symmetry using the same effective model and weak-coupling theory
calculations.

In this work, we combine density-functional-theory (DFT) Hohenberg and Kohn
(1964); Kohn and Sham (1965), maximally-localized-Wannier-function (MLWF)
Marzari et al. (2012); Mostofi et al. (2008), and linearized Eliashberg
equation with random-phase-approximation (RPA) Hirsch (1985); Yanase et al.
(2003); Graser et al. (2009); Scalapino (2012); Wu et al. (2020) to study the
bilayer two-orbital model for superconducting pairing symmetry of La3Ni2O7. We
find that if the downfolded model accurately reproduces the DFT band structure
of La3Ni2O7, the leading superconducting instability has a robust $d_{xy}$
symmetry. We refer to the bilayer two-orbital model with this type of “exact”
downfolding as Wannier’s model. We further show that in both Luo’s and
Wannier’s models, fine-tuning of crystal-field-splitting between two
Ni-$e_{g}$ orbitals qualitatively affects superconducting pairing symmetry,
which changes from $d_{xy}$ to $s_{\pm}$ when the crystal-field-splitting
exceeds a critical value. In Luo’s model which only includes nearest-neighbor
and second-nearest-neighbor hoppings, its crystal-field-splitting obtained
from fitting to the DFT band structure of La3Ni2O7 is larger than the critical
value, thereby leading to a $s_{\pm}$ superconducting pairing symmetry. By
contrast, in Wannier’s model in which all nonzero long-range-hoppings are also
included, its crystal-field-splitting obtained from Wannier fitting is reduced
and substantially smaller than the critical value. Thus the pairing symmetry
of leading superconducting instability in Wannier’s model is robustly
$d_{xy}$. Our work highlights the sensitivity of superconducting pairing
symmetry to the details of band structure in the bilayer two-orbital model
when it is applied to study La3Ni2O7 under pressure.

We perform DFT calculations, as implemented in the Vienna ab initio simulation
package (VASP) Payne et al. (1992); Kresse and Furthmüller (1996). We use the
MLWF to do the downfolding from DFT band structure Chen et al. (2022) and
obtain a bilayer two-orbital model for linearized Eliashberg equation
calculations. We use RPA to treat the local Hubbard interaction on Ni-$e_{g}$
orbitals and obtain the effective pairing potential. The basis of the bilayer
two-orbital model in this study is:
(Ni1-$d_{3z^{2}-r^{2}}$,Ni1-$d_{x^{2}-y^{2}}$, Ni2-$d_{3z^{2}-r^{2}}$,
Ni2-$d_{x^{2}-y^{2}}$). The dispersion parameters of Luo’s model and Wannier’s
model, the details of DFT, RPA and linearized Eliashberg equation
calculations, as well as other computational specifics are found in the
Supplementary Materials.

Figure 1:  (a) Crytal structure of La3Ni2O7 under high pressure. (b)
Comparison of DFT band structure of La3Ni2O7 under 30 GPa (blue solid curves)
to the band structure of Luo’s model (red dashed curves). (c) The Fermi
surface of Luo’s model. (d) Comparison of DFT band structure of La3Ni2O7 under
30 GPa (blue solid curves) to the band structure of Wannier’s model (red
dashed curves). (e) The Fermi surface of Wannier’s model.

Fig. 1(a) shows the crystal structure of La3Ni2O7 under high pressure. In DFT
calculations, we find that after atomic relaxation, La3Ni2O7 under 30 GPa
crystallizes in a $Fmmm$ structure (space group no.69), consistent with the
experiment Sun et al. (2023). However, the in-plane lattice constants $a$ and
$b$ are very close, almost leading to a $C_{4}$ symmetry about $z$ axis
Geisler et al. (2023). Following the previous studies Luo et al. (2023a); Yang
et al. (2023a); Huang et al. (2023); Luo et al. (2023b); Gu et al. (2023);
Zhang et al. (2023b, a), because the inter-block interaction is weak, we focus
on one-block of La3Ni2O7, whose DFT band structure is used to do the
downfolding to a bilayer two-orbital model. Fig. 1(b) shows the comparison of
DFT band structure of La3Ni2O7 to the band structure of Luo’s model, which
reasonably reproduces the four Ni-$e_{g}$-derived bands around the Fermi
level. Fig. 1(c) shows the Fermi surface of Luo’s model, which has three
sheets in the two-dimensional (2D) Brillouin zone, labeled as $\alpha$,
$\beta$ and $\gamma$. Fig. 1(d) shows the comparison of DFT band structure of
La3Ni2O7 and the band structure from Wannier’s model. By additionally
including all nonzero long-range-hoppings, Wannier’s model exactly reproduces
the four Ni-$e_{g}$-derived bands. The resulting Fermi surface also has three
sheets, which is very similar to that of Luo’s model, as shown in Fig. 1(e).
However, as we show below, the two seemingly identical Fermi surfaces have
different pairing symmetries of leading superconducting instability.

Figure 2:  Bare susceptibility $\chi_{0}(\textbf{q})$ at $T=116$ K. (a) Total
and orbital-projected $\chi_{0}(\textbf{q})$ of Luo’s model along the high-
symmetry q-path. (b) Total $\chi_{0}(\textbf{q})$ of Luo’s model in the 2D
Brillouin zone. (c) Total and orbital-projected $\chi_{0}(\textbf{q})$ of
Wannier’s model along the high-symmetry q-path. (d) Total
$\chi_{0}(\textbf{q})$ of Wannier’s model in the 2D Brillouin zone.

Next we compare total and orbital-projected bare susceptibility
$\chi_{0}(\textbf{q})$ of the two models (the definition of different kinds of
susceptibility is found in the Supplementary Materials). Fig. 2(a) and (b)
show $\chi_{0}(\textbf{q})$ of Luo’s model on a high-symmetry q-path and in
the 2D Brillouin zone. In panel (a), both total and orbital-projected
$\chi_{0}(\textbf{q})$ are shown, while in panel (b), we only show total
$\chi_{0}(\textbf{q})$. In Luo’s model, Ni-$d_{3z^{2}-r^{2}}$ orbital overall
has a larger susceptibility than Ni-$d_{x^{2}-y^{2}}$ orbital. The highest
peak value of total $\chi_{0}(\textbf{q})$ is along $\Gamma$ to $M$, close to
$\left(\frac{\pi}{2},\frac{\pi}{2}\right)$. This peak value would be
substantially enhanced when local Hubbard interactions are added on Ni
orbitals in RPA calculations (see the Supplementary Materials). Then we
perform the same calculations on Wannier’s model and show the results in Fig.
2(c) and (d). We find that while $\chi_{0}(\textbf{q})$ of Wannier’s model is
qualitatively similar to that of Luo’s model, there are some notable
differences. In Wannier’s model, the $d_{3z^{2}-r^{2}}$-projected
$\chi_{0}(\textbf{q})$ is increased, in particular at $M$ point. This suggests
that $d$-wave pairing might be favored in Wannier’s model.

Figure 3:  (a) The eigenvalues $\lambda$ of linearized Eliashberg’s equation
with RPA method, using Luo’s model. The first three leading superconducting
instabilities have $s_{\pm}$ (blue circles), $d_{xy}$ (red squares), and
$d_{x^{2}-y^{2}}$ (green triangles) symmetries, respectively. (b) Gap function
for the leading superconducting instability of Luo’s model calculated at
$U=1.2$ eV and $J=0.15U=0.18$ eV, which exhibits $s_{\pm}$ symmetry. (c) The
eigenvalues $\lambda$ of linearized Eliashberg’s equation with RPA method,
using Wannier’s model. The first three leading superconducting instabilities
have $d_{xy}$ (red squares), $d_{x^{2}-y^{2}}$ (green triangles), and
$s_{\pm}$ (blue circles) symmetries, respectively. (d) Gap function for the
leading superconducting instability of Wannier’s model calculated at $U=1.2$
eV and $J=0.15U=0.18$ eV, which exhibits $d_{xy}$ symmetry.

Now we study superconducting pairing symmetry of the two models, which have
qualitative differences, despite the fact that the two models have very
similar band structure and Fermi surface. We use linearized Eliashberg
equation with RPA method to treat the local Hubbard interaction for the
effective pairing potential. In Fig. 3(a), we show the superconducting pairing
symmetry of Luo’s model. We find that the first three leading superconducting
instabilities have $s_{\pm}$, $d_{xy}$, and $d_{x^{2}-y^{2}}$ pairing
symmetries, respectively. This result does not change as interaction strength
$U$ varies in a reasonable range. Fig. 3(b) shows the gap function for the
leading eigenvalue of linearized Eliashberg equation, using Luo’s model. In
this gap function, $\alpha$ and $\gamma$ sheets are of the same sign, while
$\beta$ sheet is of the opposite sign, which is consistent with the previous
studies Gu et al. (2023); Zhang et al. (2023b, a); Yang et al. (2023a). In
Fig. 3(c), we show the superconducting pairing symmetry of Wannier’s model. By
contrast, we find that the first three leading superconducting instabilities
have $d_{xy}$, $d_{x^{2}-y^{2}}$, and $s_{\pm}$ pairing symmetries,
respectively. This result is qualitatively different from Luo’s model and does
not change with interaction strength, either. Fig. 3(d) shows the gap function
for the leading eigenvalue of linearized Eliashberg equation, using Wannier’s
model. This gap function clearly exhibits $d_{xy}$ symmetry, consistent with
Ref. Lechermann et al. (2023).

Figure 4:  (a) Comparison of DFT band structure of La3Ni2O7 under 30 GPa
(gray) to the band structure of Luo’s model at a few different crystal-field-
splitting $\Delta$. (b) The eigenvalues $\lambda$ of linearized Eliashberg
equation with RPA method for the first three superconducting instabilities in
Luo’s model, calculated at $U=1.2$ eV and $J=0.15U=0.18$ eV. The leading
superconducting pairing symmetry is $d_{xy}$ ($s_{\pm}$) when $\Delta$ is
smaller (larger) than 0.30 eV. The dashed brown line highlights the optimal
value of $\Delta$ = 0.37 eV in the original Luo’s model. (c) Comparison of DFT
band structure of La3Ni2O7 under 30 GPa (gray) to the band structure of
Wannier’s model at a few different crystal-field-splitting $\Delta$. (d) The
eigenvalues $\lambda$ of linearized Eliashberg equation with RPA method for
the first three superconducting instability in Wannier’s model. The leading
superconducting pairing symmetry is $d_{xy}$ ($s_{\pm}$) when $\Delta$ is
smaller (larger) than 0.37 eV. The dashed brown line highlights the optimal
value of $\Delta$ = 0.22 eV in the original Wannier’s model. The purple dotted
line shows that spin-density-wave instability occurs at this value of $\Delta$
in the RPA calculations.

To reveal the key difference between the two models and what is the control
parameter for superconducting pairing symmetry, we perform a “thought-
experiment”. For each model, we manually change the onsite energy difference
$\Delta$ between Ni-$d_{3z^{2}-r^{2}}$ and Ni-$d_{x^{2}-y^{2}}$ orbitals,
which is a measure of crystal-field-splitting between the two Ni-$e_{g}$
orbitals. For a given value of $\Delta$, we re-calculate the band structure
and the superconducting pairing symmetry using linearized Eliashberg equation
with RPA method. We use $U=1.2$ eV and $J=0.15U=0.18$ eV. The following
results do not qualitatively change if the interaction strength takes other
reasonable values (see the Supplementary Materials). In Fig. 4(a), we compare
the band structure of Luo’s model to the DFT band structure of La3Ni2O7 under
30 GPa, as $\Delta$ is fine-tuned in a small range from 0.20 eV to 0.40 eV.
The optimal value of $\Delta$ in Luo’s model is 0.37 eV, based on the fitting.
We find that tuning $\Delta$ in this small range yields marginal effects on
the overall band structure, since the total bandwidth is about 4 eV. In Fig.
4(b), we show the eigenvalues for the first three leading superconducting
instabilities, using Luo’s model, as $\Delta$ varies. We find that the leading
superconducting instability has $d_{xy}$ symmetry ($s_{\pm}$ symmetry) if
$\Delta$ is smaller (larger) than 0.30 eV. The brown dashed line highlights
the optimal value of $\Delta$ in Luo’s model, at which the leading
superconducting pairing symmetry is $s_{\pm}$, consistent with the previous
studies Luo et al. (2023a); Yang et al. (2023a); Huang et al. (2023); Luo et
al. (2023b); Gu et al. (2023); Zhang et al. (2023b, a). In Fig. 4(c), we
compare the band structure of Wannier’s model to the DFT band structure of
La3Ni2O7 under 30 GPa, as $\Delta$ is fine-tuned in the same range. The
optimal value of $\Delta$ in Wannier’s model is 0.22 eV, smaller than that in
Luo’s model due to the inclusion of all nonzero long-range-hoppings. Similar
to Luo’s model, we find insignificant changes in the band structure of
Wannier’s model by such fine-tuning of $\Delta$. Then we re-calculate the
superconducting pairing symmetry at the same $U$ and $J$ values, using
Wannier’s model, as $\Delta$ varies. Fig. 4(d) shows the eigenvalues for the
three leading superconducting instabilities. We find a similar trend that a
smaller (larger) $\Delta$ favors $d_{xy}$ symmetry ($s_{\pm}$ symmetry) with
the critical point at 0.37 eV. The dashed brown line highlights the optimal
value of $\Delta$ in Wannier’s model, which is substantially smaller than the
critical value. Thus the leading superconducting pairing symmetry of Wannier’s
model is $d_{xy}$, consistent with Ref. Lechermann et al. (2023). In passing,
we mention that when the crystal-field-splitting $\Delta$ of Wannier’s model
is 0.39 eV (purple dotted line in panel (d)), spin-density-wave instability
occurs in the RPA calculations.

Experimentally it is challenging to directly probe superconducting pairing
symmetry in La3Ni2O7 because so far La3Ni2O7 only exhibits superconductivity
under pressure Sun et al. (2023). The take-home message from this study is
that superconducting pairing symmetry of La3Ni2O7 is very sensitive to the
details of band structure in downfolded models, such as crystal-field-
splitting and long-range-hoppings. A small increase in Ni-$e_{g}$ crystal-
field-splitting qualitatively changes the superconducting pairing symmetry of
the bilayer two-orbital model from $d_{xy}$ to $s_{\pm}$, even though the
modification to band structure and Fermi surface is marginal. When the model
only includes short-range-hoppings up to second-nearest-neighbors, the
resulting crystal-field-splitting from fitting to DFT band structure of
La3Ni2O7 is larger than the critical value, which stabilizes $s_{\pm}$
superconducting pairing symmetry. When the model includes all nonzero
hoppings, which exactly reproduces the DFT band structure of La3Ni2O7, the
crystal-field-splitting obtained from the Wannier fitting gets reduced and is
substantially smaller than the critical value. This leads to a robust $d_{xy}$
superconducting pairing symmetry in RPA calculations. We conjecture that such
sensitivity of superconducting pairing symmetry to band structure may exist in
other multi-orbital unconventional superconductors and thus care is needed in
the downfolding procedure when studying unconventional superconductivity in
realistic materials.

###### Acknowledgements.

We are grateful to Xianxin Wu, Yi-feng Yang and Dao-Xin Yao for useful
discussions. This project was financially supported by the National Key R&D
Program of China under project number 2021YFE0107900, and Science and
Technology Commission of Shanghai Municipality under grant number 23ZR1445400.
NYU High-Performance-Computing (HPC) provides computational resources.

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