# Optical conductivity signatures of open Dirac nodal lines

David Santos-Cottin<EMAIL_ADDRESS>LPEM, ESPCI Paris; CNRS; PSL
University, 10 rue Vauquelin, F-75005 Paris, France  Sorbonne Université,
CNRS, LPEM, F-75005 Paris, France  Department of Physics, University of
Fribourg, 1700 Fribourg, Switzerland  Michele Casula  IMPMC-Sorbonne
Université, CNRS and MNHN, 4, place Jussieu, 75005 Paris, France  Luca de’
Medici  LPEM, ESPCI Paris; CNRS; PSL University, 10 rue Vauquelin, F-75005
Paris, France  Sorbonne Université, CNRS, LPEM, F-75005 Paris, France  F. Le
Mardelé  Department of Physics, University of Fribourg, 1700 Fribourg,
Switzerland  J. Wyzula  LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes, Univ.
Toulouse, Univ. Toulouse 3, INSA-T, Grenoble and Toulouse, France  M. Orlita
LNCMI-EMFL, CNRS UPR3228, Univ. Grenoble Alpes, Univ. Toulouse, Univ. Toulouse
3, INSA-T, Grenoble and Toulouse, France  Institute of Physics, Charles
University in Prague, CZ-12116 Prague, Czech Republic  Yannick Klein  IMPMC-
Sorbonne Université, CNRS and MNHN, 4, place Jussieu, 75005 Paris, France
Andrea Gauzzi  IMPMC-Sorbonne Université, CNRS and MNHN, 4, place Jussieu,
75005 Paris, France  Ana Akrap  Department of Physics, University of Fribourg,
1700 Fribourg, Switzerland  R. P. S. M. Lobo<EMAIL_ADDRESS>LPEM, ESPCI Paris;
CNRS; PSL University, 10 rue Vauquelin, F-75005 Paris, France  Sorbonne
Université, CNRS, LPEM, F-75005 Paris, France

(August 27, 2024)

###### Abstract

We investigate the optical conductivity and far-infrared magneto-optical
response of BaNiS2, a simple square-lattice semimetal characterized by Dirac
nodal lines that disperse exclusively along the out-of-plane direction. With
the magnetic field aligned along the nodal line the in-plane Landau level
spectra show a nearly $\sqrt{B}$ behavior, the hallmark of a conical-band
dispersion with a small spin-orbit coupling gap. The optical conductivity
exhibits an unusual temperature-independent isosbestic line, ending at a Van
Hove singularity. First-principles calculations unambiguously assign the
isosbestic line to transitions across Dirac nodal states. Our work suggests a
universal topology of the electronic structure of Dirac nodal lines.

Certain Dirac semimetals belonging to a class of nonsymmorphic square-lattice
system Schoop  _et al._ (2016); Chen  _et al._ (2017); Yang  _et al._ (2018);
Takane  _et al._ (2016) outshine the numerous topological systems known today.
They have Dirac points located on in-plane symmetry lines, in contrast to
high-symmetry points of the Brillouin zone as in graphene. This is a promising
property for novel concepts of electronic devices, as it makes the topological
properties of two-dimensional (2D) materials tunable. Indeed, it has been
shown that electronic correlations effectively modify the dispersion of the
Dirac nodes Li  _et al._ (2008); Shao  _et al._ (2020); Nilforoushan  _et al._
(2020); Gatti  _et al._ (2020). This dispersion typically follows either an
open line or a close loop, thus reducing the effective dimension from three to
two Carbotte (2016). The challenge is that the most studied Dirac nodal line
(DNL) materials have intricate Fermi surfaces with Dirac lines dispersing in
different reciprocal space directions. It is therefore important to identify
model systems where the physics of DNLs is as simple as possible.

A key finding in the physics of these materials happened long before any
topological effects were known, when Hoffmann and Tremel Tremel and Hoffmann
(1987) showed that the nonsymmorphic tetragonal $P4/nmm$ space group can lead
to dispersive DNLs. Only a handful of such compounds are actively explored
Takane  _et al._ (2016); Ebad-Allah  _et al._ (2019). Among the less explored
compounds, we focus on BaNiS2 Grey and Steinfink (1970) whose DNLs have been
discovered and studied only recently Santos-Cottin  _et al._ (2016a, b);
Nilforoushan  _et al._ (2021, 2020). BaNiS2 is formed by square-lattice layers
of puckered NiS5 pyramids shown in the inset of Fig. 1(a). Remarkable features
of the band structure are fourfold-symmetry DNLs within the $k_{x}-k_{y}$
plane, with Dirac nodes pinned at the Fermi energy at $k_{z}=0$. These nodes
slightly disperse along $k_{z}$ and raise above the Fermi level as $k_{z}$
increases (Fig. 3(a)). The resulting nodal lines are open, wrapping the first
Brillouin zone (IBZ) torus in the $k_{z}$ direction. Their in-plane momentum
position is tunable along the $\Gamma$–M path (Fig. 2(a)) through chemical
substitution Nilforoushan  _et al._ (2020), and their Fermi velocities can be
modulated by external perturbations, such as light pulses Nilforoushan  _et
al._ (2021).

Figure 1: (a) Temperature dependent in-plane reflectivity of BaNiS2 below 0.8
eV. Inset: Crystal structure of BaNiS2. (b) Temperature dependence of the in-
plane real part of the optical conductivity. The grey areas show the SW
transfer from high to low energy between 300 K and 5 K. The Inset shows the
differential spectral weight $\Delta SW(T)=SW(T)-SW_{300K}$, for different
upper cut-off energies, as a function of temperature. (c) Color-plot of the
relative magneto-reflectance, $R_{B}/R_{0}$. Lighter areas correspond to
inter-LL excitations, which were assigned to transitions: $-n\rightarrow n+1$
and $-n-1\rightarrow n$ and fitted using the LL spectrum in Eq. 1.

This ensemble of favorable properties makes BaNiS2 a model system for probing
open DNLs through dynamic charge transport. In this Letter we report its first
magneto-optical conductivity study combined with first-principles
calculations. We find that the nodal lines’ contribution to the spectrum has
universal optical conductivity fingerprints: the spectrum has a linear energy
dispersion; and it is temperature ($T$-) independent over an “isosbestic”
range of frequencies, ending at high energy peaks, which do have a
$T$-dependence. The linear dispersion and $T$-independent conductivity is
similar to that found in graphene, confirming the effective two-dimensionality
of the conical dispersion in BaNiS2 Falkovsky and Varlamov (2007); Falkovsky
and Pershoguba (2007); Kuzmenko  _et al._ (2008); Mak  _et al._ (2008, 2011);
Schilling  _et al._ (2017). The $T$-dependent peaks cannot be dissociated from
the lower-energy isosbestic range, because the bands generating the Dirac
cones are continuous and IBZ-periodic. According to the Morse theorem Morse
(1925), these bands must show saddle points, yielding Van Hove singularities
(VHSs) with a logarithmic divergence in the 2D density of states Van Hove
(1953), and leading to $T$-dependent peaks in the optical conductivity once
finite lifetime and many-body effects are taken into account Mak  _et al._
(2011, 2014). Considering that the above connection between Dirac bands and
VHSs is a topological property of a quasi-2D electronic structure, it follows
that the features observed in BaNiS2 are universal fingerprints of open DNL
semimetals, expected in layered beyond-graphene materials.

We measured the in-plane reflectance of 0.5-1 mm-sized BaNiS2 single crystal
platelets Santos-Cottin  _et al._ (2016a); Shamoto  _et al._ (1995) from 3 meV
to 5 eV. We also measured their far-infrared response in fields up to 34 T SM
. First-principles calculations were carried out within density functional
theory in the Quantum Espresso plane-wave implementation Giannozzi  _et al._
(2009, 2017). We employed a modified hybrid Heyd-Scuseria-Ernzerhof (HSE)
functional Heyd  _et al._ (2003, 2006); Krukau  _et al._ (2006), able to
accurately reproduce the fermiology of BaNiS2. From the band structure, we
computed the optical conductivity in the Kubo formalism, by explicitly
including thermal scattering effects via a Fermi-liquid self-energy
$\Sigma=i\Gamma=i\pi(k_{B}T)^{2}/(k_{B}T_{0})$, with $T_{0}=1000$ K SM ;
Mostofi  _et al._ (2014); Kubo (1957); Pruschke  _et al._ (1993); Tomczak and
Biermann (2009).

Figure 2: (a) First Brillouin zone and the HSE-calculated Fermi surface. Hole
and electron pockets are indicated in yellow and purple, respectively. The red
cylinder represents the constrained momentum space used to isolate the
contribution of the nodal line. (b) HSE band structure, and DOS at zero and
300 K. (c) Low-energy optical conductivity calculated at different
temperatures for the full IBZ (solid lines). Dashed lines show
$\sigma_{1}(\omega)$ for transitions in the restricted $k$-space, defined by
the shaded volume in Fig. 2(a). (d) Full optical conductivity at $100$ K
decomposed into different $k$-space contributions, for various cylinders in
IBZ. Green, blue, and red lines shows $\sigma_{1}(\omega)$ for a cylinder
centered at $k_{\textrm{Dirac}}$, $\Gamma$, and X, respectively. The radius
R${}_{\textrm{sel}}$ of each cylinder is given in crystal units.

To access the complex optical conductivity, we measured the frequency-
dependent reflectance, shown in Fig. 1(a) Homes  _et al._ (1993); SM . We
calculated the optical conductivity from the reflectance employing Kramers-
Kronig relations Tanner (2015). Its real part $\sigma_{1}(\omega)$, shown in
Fig. 1(b), contains a highly unusual isosbestic range where it remains
$T$-independent from 80 to 250 meV. This isosbestic line is in contrast with
the isosbestic point commonly observed in gapped systems. Such an isosbestic
line means that the spectral weight (SW), defined as the area underneath
$\sigma_{1}(\omega)$, depleted from low frequencies at low temperatures gets
transferred to high energies, skipping over the 80 to 250 meV range. The inset
of Fig. 1(b) illustrates that the SW lost from dc to about 100 meV is the same
as the one lost from dc to about 250 meV. Only then does it start to get
compensated until it is fully recovered by 600 meV.

Focusing on the 5 K curve, we see that $\sigma_{1}(\omega)$ contains several
contributions. A very narrow Drude component is followed by a roughly linear
increase up to 100 meV. At this point, another linear dispersion begins with a
smaller slope and a finite zero-energy intercept. This second linear
dispersion is the isosbestic line where $\sigma_{1}(\omega)$ at all
temperatures overlap. It terminates in a pair of broad, partly overlapping
peaks, at 0.3 and 0.4 eV.

Applying the magnetic field along the nodal-lines direction (001), a series of
excitations is observed in the relative magneto-reflectivity spectra,
$R_{B}/R_{0}$, Fig. 1(c) 111The reason we plot the reflectivity maxima is
because $\epsilon_{2}$ is larger than $\epsilon_{1}$, and therefore dominates
the reflectivity response. SM . The maxima correspond to the transition
energies and are electric-dipole excitations, with selection rules
$\Delta|n|=\pm 1$, between pairs of Landau levels (LLs) of 2D massive Dirac
electrons:

$\varepsilon_{\pm n}(B)=\pm\sqrt{2\hbar
ev_{1}v_{2}Bn+\Delta^{2}},\;\;n=0,1,2\ldots.$ (1)

The most pronounced line corresponds to the lowest interband excitations
$-1\rightarrow 0$ and $0\rightarrow 1$. The latter dominates in the hole doped
parts of the nodal lines (away from $k_{z}=0$). At lower energies, intraband
(cyclotron-resonance-like) inter-LL excitations appear. The extracted gap is
$2\Delta=16$ meV, and the asymptotic velocity parameter,
$v=\sqrt{v_{1}v_{2}}=1.55\times 10^{5}$ m/s, where $v_{1}$ and $v_{2}$ are
velocities for the two in-plane directions. BaNiS2 is the nodal line system
having the smallest gap, 3 times smaller than the one in ZrSiSe Shao  _et al._
(2020), twice smaller than that of ZrSiS Schoop  _et al._ (2016), and 7 times
smaller than the gap in NbAs2 Shao  _et al._ (2019). The velocity parameter
$v$ is 1.6 times lower than in ZrSiSe Shao  _et al._ (2020).

The optical conductivity is dominated by interband contributions, and these
can be best understood by knowing the exact band structure. Figures 2(a) and
2(b) display the calculated Fermi surface, band structure and the density of
states (DOS) of BaNiS2 around the Fermi level. The computed band structure
yields two pockets at the Fermi energy ($E_{F}$), in agreement with previous
results (Santos-Cottin  _et al._ , 2016a; Klein  _et al._ , 2018). Halfway
through the $\Gamma$–M line, there are two spin-degenerate linear bands. These
bands cross at $E_{F}$ and $k_{\textrm{Dirac}}$ at $k_{z}=0$, forming a well-
defined Dirac cone, a common signature of the square-net family Tremel and
Hoffmann (1987); Yang  _et al._ (2018). Because of the SOC, the Dirac cone is
slightly gapped, 2$\Delta_{calc}=$ $18\text{\,}\mathrm{meV}$, in agreement
with magneto-optics. The Dirac point progressively shifts to higher energies
as $k_{z}$ increases, leading to a hole-like Fermi pocket along the Z–A line,
compensated by an electron pocket centered at $\Gamma$. In contrast to other
square net lattice members, BaNiS2 shows no additional low-energy Dirac
structures in the $\Gamma$–X or Z–R directions at $E_{F}$. This makes BaNiS2 a
cleaner case, and a good candidate to isolate the signature of the dispersive
nodal line.

Figure 3:  (a) 3D view of the connection between the two DNLs dispersing along
$k_{z}$ obtained from HSE calculations. (b) Comparison between the measured
$\sigma_{1}(\omega)$ at $10\text{\,}\mathrm{K}$ and $100\text{\,}\mathrm{K}$
(thick lines), and the analytic form of Eq. 3 (dashed line) with an electric
field perpendicular to the nodal line dispersion $k_{z}$. The thin green line
shows the $k$-space restricted theoretical $\sigma_{1}(\omega)$ at 100 K. (c)
Dispersion of a 2D model on a square lattice. (d) $\sigma_{1}(\omega)$ of the
model in panel (c) at different temperatures. The peak corresponds to
interband transitions between two saddle points (VHSs) of the model. The black
dashed line is the universal value of $\sigma_{1}(\omega)$ for a perfect
conical model with 4 valleys and 1/2-spin fermions.

Full lines in Fig. 2(c) display the calculated optical conductivity. For
$T<$200\text{\,}\mathrm{K}$$, the Drude width is small enough to reveal a
minimum around $80\text{\,}\mathrm{meV}$, followed by a linear increase of
$\sigma_{1}(\omega)$ up to $\sim$ $120\text{\,}\mathrm{meV}$. In the
200–$450\text{\,}\mathrm{meV}$ range, $\sigma_{1}(\omega)$ collapses into a
linear-in-energy response, roughly independent of temperature. Two peaks
appear around 550 and $650\text{\,}\mathrm{meV}$. These features are enhanced
at lower temperatures, and they appear at energies $\sim 1.5$ times higher
than those experimentally determined. This discrepancy points to sizable band
renormalization effects, driven by electronic correlations, which are not
perfectly captured by the modified HSE functional, and to exitonic many-body
effects beyond single-particle description Mak  _et al._ (2011, 2014).
Nevertheless, all the important characteristics of the data are reproduced by
the first-principles results: (i) a Drude peak and a small gap at low
temperatures; (ii) a $T$-independent isosbestic line with linear energy
dependence; (iii) SW transfer from below to above this line; and (iv) similar
absolute values for $\sigma_{1}(\omega)$ in theory and experiment.

While the match between our first-principles calculations and the experiment
is very good, the question remains whether we can (i) isolate the contribution
of the DNL and (ii) explain such an unusual isosbestic line. To answer these
two questions, we constrain the $k$-integration range in the current-current
response function calculation to a cylinder in the IBZ (shaded volume in Fig.
2(a)). This cylinder selects intra and interband transitions involving states
that mainly belong to the Dirac cone centered at $k_{\textrm{Dirac}}$. When
the radius R${}_{\textrm{sel}}$ of the cylinder is set to 1/4 in crystal units
(Figs. 2(c–d)), $\sigma_{1}(\omega)$ becomes flat between 0.2 and 0.4 eV, and
its second peak at $650\text{\,}\mathrm{meV}$ disappears. By further shrinking
R${}_{\textrm{sel}}$ to 0.2, to better isolate the DNLs around
$k_{\textrm{Dirac}}$, the peak at $550\text{\,}\mathrm{meV}$ is suppressed as
well, while the plateau in the 200–$400\text{\,}\mathrm{meV}$ range remains
(Fig. 2(d)). The flat and $T$-independent response of $\sigma_{1}(\omega)$
(Fig. 2(c)) is characteristic of a 2D conical dispersion Gusynin  _et al._
(2006); Kuzmenko  _et al._ (2008); Mak  _et al._ (2008); Raveendran‐Nair  _et
al._ (2008). This is a strong first-principles evidence that the isosbestic
character of $\sigma_{1}(\omega)$ from 0.2 to 0.4 eV is caused by the
electronic states involved in the dispersive nodal line. The same Dirac states
reveal their conical in-plane dispersion in our magneto-optic measurements.

However, this does not yet explain the actual slope seen in both experiment
and full, unrestricted $\sigma_{1}(\omega)$ calculations over that energy
window. To solve this mismatch, we also analyze the $\sigma_{1}(\omega)$
contributions coming from transitions around $\Gamma$ and X, excluded in the
previous $k$-selection in Fig. 2(c). We plot the results of $k$-selected
transitions delimited by cylinders of radius $R_{\textrm{sel}}=0.2$ centered
at $\Gamma$ and X in Fig. 2(d). The calculated $\sigma_{1}$ with $k$-points
from $\Gamma$ perfectly covers the 200–$450\text{\,}\mathrm{meV}$ frequency
range, with a slope very close to full $\sigma_{1}$. This linear
$\sigma_{1}(\omega)$ contribution can be explained in the band structure by
the presence of a second conical shape centered at $\Gamma$, which is
connected to the bands of the DNL at the $\Gamma$–M line mid-point, Fig. 3(a).
The contribution of the $k$-points around X starts at
$300\text{\,}\mathrm{meV}$. By dissecting the IBZ into distinct patches, we
prove that the isosbestic line is made out of a flat and strictly
$T$-independent part coming from the response of the DNL at the $\Gamma$–M
line mid-point; and a higher-energy linear contribution from conical states
near $\Gamma$. The former contribution is typical of the 2D and 3D conical
model, whose optical response becomes temperature independent at an energy far
enough from the optical gap Falkovsky and Varlamov (2007); Falkovsky and
Pershoguba (2007); Stauber  _et al._ (2008); Tabert  _et al._ (2016) The
isosbestic line is then the optical signature of purely conical excitations.
In BaNiS2 the isosbestic energy range becomes clearly visible because of the
quasi-2D nature of the material. This leads to open Dirac lines developing
along $k_{z}$ such that, in a first approximation, one can see the system as
made of infinite copies (in $k_{z}$) of a 2D Dirac system.

We will now focus on the low-temperature optical conductivity at energies
below the isosbestic line. An ideal 2D Dirac cone with non-$k_{z}$-dispersing
line nodes would have a constant $\sigma_{1}(\omega)$ Kuzmenko  _et al._
(2008), and would not explain the dip and the steep slope of
$\sigma_{1}(\omega)$ below the nearly flat plateau. These features come from
the $k_{z}$-dispersing nodal lines in BaNiS2 as shown in Fig. 3(a). The
plotted gapped Dirac nodes projected in the $\Gamma$–M direction as a function
of kz are obtained from the HSE band structure in Fig. 2(b). This can be
modeled by the energy dispersion Shao  _et al._ (2019):

$\epsilon_{\pm}=\pm\sqrt{\Delta^{2}+v_{1}^{2}k_{1}^{2}+v_{2}^{2}k_{2}^{2}}+v_{z}k_{z},$
(2)

where $k_{1}$ and $k_{2}$ are momenta perpendicular to the nodal line,
$\Delta$ is the SOC-induced gap, and $v_{i}$ the in-plane asymptotic
velocities that describe the slope of the Dirac cone. The HSE band structure
gives the asymptotic velocities $v_{1}=2.61$ $\text{\,}\mathrm{eV}$Å and
$v_{2}=0.94$ $\text{\,}\mathrm{eV}$Å. Their average is 1.57
$\text{\,}\mathrm{eV}$Å, fairly close to the experimental $v=1.02$
$\text{\,}\mathrm{eV}$Å.

$\sigma_{1}(\omega)$ associated with the dispersion in Eq. 2 is Shao  _et al._
(2019):

$\sigma_{\it
NL}^{i}(\omega)=\dfrac{N}{8}\dfrac{e^{2}}{h}k_{0}(\omega)\dfrac{v_{i}^{2}}{v_{1}v_{2}}\left(1+\dfrac{4\Delta^{2}}{\omega^{2}}\right)\Theta\left(\omega-2\Delta\right),$
(3)

where $N$ is the number of nodal line (including the spin multiplicity) in the
IBZ. The electric field is polarized along the $i$ direction. The length
$k_{0}(\omega)$ is the chunk of the nodal line accessible at a given $\omega$,
see the supplementary for its photon energy dependence SM . We take Eq. 3 with
the electric field averaged in plane, and plug in the magneto-optical
parameters. We also fit the energy shift of the nodal line along $k_{z}$, so
as to match the experimental onset of interband excitations. All of this gives
us a $\sigma_{\it NL}(\omega)$ dependence, shown in Fig. 3(b) as a dashed
black line. This curve follows the experimental data well up to
$80\text{\,}\mathrm{meV}$, but fails to reproduce the steeper isosbestic slope
at higher energies. In contrast, $\sigma_{\it NL}(\omega)$ behaves much more
like the _ab initio_ calculations restricted around $k_{\textrm{Dirac}}$. This
confirms that the dip, the slope and the isosbestic flat region of
$\sigma_{1}(\omega)$ come from DNLs dispersing along $k_{z}$.

Finally, we want to explain the experimental peak in $\sigma_{1}$ at
$350\text{\,}\mathrm{meV}$, captured by the _ab initio_ peak at
$550\text{\,}\mathrm{meV}$. We developed a simplified 2D tight-binding model
on a square lattice SM ; Berthod  _et al._ (2013); Ando  _et al._ (2002); Min
and MacDonald (2009); Mukherjee and Carbotte (2017), in line with the quasi-2D
nature of BaNiS2. The electronic structure of this model is plotted in Fig.
3(c). The fourfold Dirac cones, like in BaNiS2, are connected by arches whose
topmost part leads to VHSs in the DOS, and to the peak found in the computed
$\sigma_{1}(\omega)$ at $\approx$ 1.1 eV, Fig. 3(d). This toy model shows that
the higher energy peaks of $\sigma_{1}(\omega)$, looming over the flatter
isosbestic regions are caused by the connecting Dirac cones within the IBZ.
The cones’ straight walls eventually bend to merge with one another. These
VHSs are thus universal in a system of Dirac cone networks Mak  _et al._
(2011, 2014); Santos-Cottin  _et al._ (2020); Ebad-Allah  _et al._ (2019);
Martino  _et al._ (2019); Le Mardelé  _et al._ (2020).

In conclusion, we measured the optical conductivity signatures of the open
dispersive nodal lines dominating the low-energy spectrum of BaNiS2 and a VHS
at a higher energy, with an isosbestic line connecting both regions. The
magneto-optical response indicates a slightly gapped in-plane linear
dispersion. Using first-principles calculations we fully reproduce the
experimental optical spectra and we reverse engineer them. We show that BaNiS2
is an exemplary nodal line Dirac semimetal, whose band structure is simpler
than the more investigated counterparts, and with strong 2D features, despite
being a 3D layered material. This simplicity allowed us to uncover the optical
signatures of conical dispersions, and the impact of their $k_{z}$ and
temperature dependence. The observed isosbestic line and the VHS are universal
fingerprints of untainted quasi-2D Dirac excitations in the low-energy range.
The VHS also marks the conical model breakdown when the Dirac cones connect in
reciprocal space. This peculiar optical response is thus an experimental proxy
which can unequivocally identify new beyond-graphene materials.

M.C. acknowledges GENCI allocations for computer resources under the project
number 0906493. L.d.M is supported by the European Commission through the ERC-
CoG2016, StrongCoPhy4Energy, GA No 724177. A. A. acknowledges funding from the
Swiss National Science Foundation through project PP00P2_170544. This work was
supported by the ANR DIRAC3D project (ANR-17-CE30-0023).

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