# On the Klein-Gordon Gürses-oscillators and pseudo-Gürses-oscillators:
vorticity-energy correlations and spacetime associated degeneracies

Omar Mustafa<EMAIL_ADDRESS>Department of Physics, Eastern
Mediterranean University, G. Magusa, north Cyprus, Mersin 10 - Turkey.

###### Abstract

Abstract: We discuss KG-oscillators in the (1+2)-dimensional Gürses spacetime
and under position-dependent mass (PDM) settings. We observe that the KG-
Gürses oscillators are introduced as a byproduct of the very nature of the
Gürses spacetime structure. We report that the energy levels of such KG-Gürses
oscillators admit vorticity-energy correlations as well as spacetime
associated degeneracies (STAD). We discuss KG-Gürses oscillators’ results
reported by Ahmed Ahmed1 2019  and pinpoint his improper treatment of this
model so that his results should be redirected to those reported in this
study. Moreover, we introduce a new set of KG pseudo-Gürses oscillators that
admits isospectrality and invariance with the KG-Gürses oscillators and
inherits the same vorticity-energy correlations as well as STADs.

PACS numbers: 05.45.-a, 03.50.Kk, 03.65.-w

Keywords: Klein-Gordon oscillators, Gürses spacetime, position-dependent mass,
vorticity-energy correlations, spacetime associated degeneracies.

##  I Introduction

Klein-Gordon (KG) and Dirac oscillators Moshinsky 1989 ; Bruce 1993 ; Dvoeg
1994 ; Mirza 2004  have received much attention over the years. KG-oscillators
in Gödel-type spacetime (e.g., Moshinsky 1989 ; Bruce 1993 ; Dvoeg 1994 ; Das
2008 ; Carvalho 2016 ; Garcia 2017 ; Vitoria 2016 ), in cosmic string
spacetime and Kaluza-Klein theory backgrounds (e.g., Ahmed1 2021 ; Boumal 2014
), in Minkowski spacetime with space-like dislocation Mustafa1 2022 , in Som-
Raychaudhuri Wang 2015 , in (1+2)-dimensional Gürses space-time backgrounds
(e.g., Gurses 1994 ; Ahmed1 2019 ; Mustafa2 2022 ). The KG-oscillators in a
(1+2)-dimensional Gürses spacetime described by the metric

$ds^{2}=-dt^{2}+dr^{2}-2\Omega
r^{2}dtd\theta+r^{2}\left(1-\Omega^{2}r^{2}\right)d\theta^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu};\text{
}\mu,\nu=0,1,2,$ (1)

were investigated investigated by Ahmed Ahmed1 2019 ., using
$a_{{}_{0}}=b_{{}_{0}}=e_{{}_{0}}=1$,
$b_{{}_{1}}=c_{{}_{0}}=\lambda_{{}_{0}}=0$, and vorticity $\Omega=-\mu/3$, in
the Gürses metric

$ds^{2}=-\phi
dt^{2}+2qdtd\theta+\frac{h^{2}\psi-q^{2}}{a_{{}_{0}}}d\theta^{2}+\frac{1}{\psi}dr^{2}$
(2)

(i.e., as in Eq.(5) of Gurses 1994 ) where

$\phi=a_{{}_{0}},\,\psi=b_{{}_{0}}+\frac{b_{{}_{1}}}{r^{2}}+\frac{3\lambda_{{}_{0}}}{4}r^{2},\,q=c_{{}_{0}}+\frac{e_{{}_{0}}\mu}{3}r^{2},\,h=e_{{}_{0}}r,\,\lambda_{{}_{0}}=\lambda+\frac{\mu^{2}}{27}.$
(3)

In this note, we shall show that there are more quantum mechanical features
indulged in the spectroscopic structure of the KG-oscillators in the
background of such a Gürses spacetime metric (1) than those reported by Ahmed
Ahmed1 2019 , should this model be properly addressed. Throughout this note,
such KG-oscillators shall be called KG-Gürses oscillators.

We organize the current note in the following manner. In section 2, we revisit
KG-oscillators in the (1+2)-dimensional Gürses spacetime of (1) and present
them in a more general form, that includes position-dependent mass (PDM, which
is a metaphoric notion) settings along with Mirza-Mohadesi’s KG-oscillators
Mirza 2004  recipe. We observe that the KG-Gürses oscillators are introduced
as a byproduct of the very nature of the Gürses spacetime structure. This
motivates us to first elaborate and discuss, in section 3, the effects of
Gürses spacetime on the energy levels of the KG-Gürses oscillators, without
the KG-oscillator prescription of Mirza-Mohadesi Mirza 2004 . Therein, we
report that such KG-Gürses oscillators admit vorticity-energy correlations as
well as spacetime associated degeneracies (STADs). In section 4, we discuss
Ahmed’s model Ahmed1 2019  that includes Mirza-Mohadesi Mirza 2004  recipe and
pinpoint Ahmed’s Ahmed1 2019  improper treatment of the model at hand. We
consider the PDM KG-Gürses oscillators in section 5. We discuss and report KG
pseudo-Gürses oscillators in section 6, where we observe that they admit
isospectrality and invariance with the KG Gürses-oscillators and inherit the
same vorticity-energy correlations as well as STADs. Our concluding remarks
are given in section 7.

##  II KG-Gürses oscillators and PDM settings

The covariant and contravariant metric tensors corresponding to the
(1+2)-dimensional Gürses spacetime of (1), respectively, read

$g_{\mu\nu}=\left(\begin{tabular}[]{ccc}$-1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$0$&$-\Omega r^{2}$\\\ $0$&$1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$0$\\\ $-\Omega
r^{2}$&$\,0$&$\,r^{2}\left(1-\Omega^{2}r^{2}\right)$\end{tabular}\right)\Longleftrightarrow
g^{\mu\nu}=\left(\begin{tabular}[]{ccc}$\left(\Omega^{2}r^{2}-1\right)$&$0\vskip
3.0pt plus 1.0pt minus 1.0pt$&$-\Omega$\\\ $0$&$1\vskip 3.0pt plus 1.0pt minus
1.0pt$&$0$\\\ $-\Omega$&$\,0$&$\,1/r^{2}$\end{tabular}\right)\text{ };\text{ \
}\det\left(g_{\mu\nu}\right)=-r^{2}.$ (4)

Then the corresponding KG-equation is given by

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi\right)=m^{2}\Psi.$
(5)

However, we shall now use the momentum operator

$p_{\mu}\longrightarrow p_{\mu}+i\mathcal{F}_{\mu},$ (6)

so that it incorporates the KG-oscillator prescription of Mirza-Mohadesi Mirza
2004  as well as position-dependent mass (PDM) settings proposed by Mustafa
Mustafa1 2022 . Where $\mathcal{F}_{\mu}=\left(0,\mathcal{F}_{r},0\right)$ and
our $\mathcal{F}_{r}=\eta r;$ $\eta=m\omega,$ of Ahmed1 2019  and
$\mathcal{F}_{r}=\eta r+g^{\prime}\left(r\right)/4g\left(r\right)$ to also
include PDM settings as in Mustafa Mustafa1 2022 . This would suggest that
Ahmed’s model is retrieved when the positive-valued scalar multiplier
$g\left(r\right)=1$. Nevertheless, the reader should be aware that the regular
momentum operator $p_{\mu}$ is replaced by the PDM-momentum operator
$p_{\mu}+i\mathcal{F}_{\mu}$ to describe PDM KG-particles in general (for more
details on this issue the reader is advised to refer to Mustafa1 2022 ).

Under such assumptions, the KG-equation (5) would transform into

$\frac{1}{\sqrt{-g}}\left(\partial_{\mu}+\mathcal{F}_{\mu}\right)\left[\sqrt{-g}g^{\mu\nu}\left(\partial_{\nu}-\mathcal{F}_{\nu}\right)\Psi\right]=m^{2}\Psi,$
(7)

which consequently yields

$\left\\{-\partial_{t}^{2}+\left(\Omega\,r\,\partial_{t}-\frac{1}{r}\partial_{\theta}\right)^{2}+\partial_{r}^{2}+\frac{1}{r}\partial_{r}-M\left(r\right)-m^{2}\right\\}\Psi=0,$
(8)

where

$M\left(r\right)=\frac{\mathcal{F}_{r}}{r}+\mathcal{F}_{r}^{\prime}+\mathcal{F}_{r}^{2}.$
(9)

We now substitute

$\Psi\left(t,r,\theta\right)=\exp\left(i\left[\ell\theta-
Et\right]\right)\psi\left(r\right)=\exp\left(-i\left[\ell\theta-
Et\right]\right)\frac{R\left(r\right)}{\sqrt{r}}$ (10)

to imply

$R^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\ell^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}r^{2}-\tilde{M}\left(r\right)\right]R\left(r\right)=0,$
(11)

where $\ell=0,\pm 1,\pm 2,\cdots$ is the magnetic quantum number,

$\tilde{M}\left(r\right)=-\frac{3}{16}\left(\frac{g^{\prime}\left(r\right)}{g\left(r\right)}\right)^{2}+\frac{1}{4}\frac{g^{{}^{\prime\prime}}\left(r\right)}{g\left(r\right)}+\frac{1}{4}\frac{g^{\prime}\left(r\right)}{rg\left(r\right)}+\frac{1}{2}\frac{g^{\prime}\left(r\right)}{g\left(r\right)}\eta
r,$ (12)

and

$\lambda=E^{2}-2\,\Omega\,\ell\,E-2\eta-m^{2}\text{ ; \
}\tilde{\omega}^{2}=\Omega^{2}E^{2}+\eta^{2}.$ (13)

It is obvious that we retrieve Ahmed’s model Ahmed1 2019  when
$g\left(r\right)=1$. Moreover, we observe that the KG-Gürses oscillators are
introduced as a byproduct of the very nature of the Gürses spacetime
structure. This motivates us to first elaborate and discuss the effects of
Gürses spacetime on the energy levels of the KG-Gürses oscillators, without
the KG-oscillator prescription of Mirza-Mohadesi Mirza 2004  (i.e., with
$\eta=0$).

##  III KG-Gürses oscillators: vorticity-energy correlations and spacetime
associated degeneracies

It is obvious that KG-Gürses oscillators are introduced by the very structure
of Gürses spacetime. That is, for $\eta=0$, and $g\left(r\right)=1$ our KG-
equation (11) collapses into the two-dimensional Schrödinger oscillator

$R^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\ell^{2}-1/4\right)}{r^{2}}-\Omega^{2}E^{2}r^{2}\right]R\left(r\right)=0,$
(14)

which admits exact textbook solvability so that the eigenvalues and radial
eigenfunctions, respectively, read

$\lambda=2\left|\Omega E\right|\left(2n_{r}+\left|\ell\right|+1\right)$ (15)

and

$R\left(r\right)\sim r^{\left|\ell\right|+1/2}\exp\left(-\frac{\left|\Omega
E\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\Omega
E\right|r^{2}\right)\Longleftrightarrow\psi\left(r\right)\sim
r^{\left|\ell\right|}\exp\left(-\frac{\left|\Omega
E\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\Omega
E\right|r^{2}\right).$ (16)

where $L_{n_{n}}^{\left|\ell\right|}\left(\left|\Omega E\right|r^{2}\right)$
are the generalized Laguerre polynomials. Now with the help of (13) and (15)
we obtain

$E^{2}-2\,\Omega\,\ell\,E-m^{2}=2\left|\Omega
E\right|\left(2n_{r}+\left|\ell\right|+1\right).$ (17)

Figure 1:  The energy levels for KG-Gürses oscillators of (20) and (21) are
plotted $m=1$ (a) for $n_{r}=0,\,\ell=0,\pm 1,\pm 2$, and (b) for $n_{r}=3$,
$\ell=0,\pm 1,\pm 2,\pm 3$.

This result should be dealt with diligently and rigorously, as mandated by the
very nature of $\left|\Omega E\right|=\Omega_{\pm}E_{\pm}\geq 0$ or
$\left|\Omega E\right|=-\Omega_{\mp}E_{\pm}\geq 0$ (that secures the
finiteness and square integrability of the radial wavefunction (16)), where
$\Omega_{\pm}=\pm\left|\Omega\right|$ and $E_{\pm}=\pm\left|E\right|$, That
is, for $\left|\Omega E\right|=\Omega_{\pm}E_{\pm}$ in (17) we obtain

$E_{\pm}^{2}-2\,\Omega_{\pm}E_{\pm}\,\tilde{n}_{+}-m^{2}=0;\;\tilde{n}_{+}=2n_{r}+\left|\ell\right|+\ell\,+1,$
(18)

and for $\left|\Omega E\right|=-\Omega_{\mp}E_{\pm}$ we get

$E_{\pm}^{2}+2\,\Omega_{\mp}E_{\pm}\,\tilde{n}_{-}\,-m^{2}=0;\;\tilde{n}_{-}=2n_{r}+\left|\ell\right|-\ell\,+1.$
(19)

Which would allow us to cast

$E_{\pm}=\Omega_{\pm}\,\tilde{n}_{+}\pm\sqrt{\Omega^{2}\tilde{n}_{+}^{2}+m^{2}}\Rightarrow\left\\{\begin{tabular}[]{l}$E_{+}=\Omega_{\pm}\,\tilde{n}_{+}+\sqrt{\Omega^{2}\tilde{n}_{+}^{2}+m^{2}}$\\\
$E_{-}=\Omega_{\pm}\,\tilde{n}_{+}-\sqrt{\Omega^{2}\tilde{n}_{+}^{2}+m^{2}}$\end{tabular}\right.,$
(20)

for $\left|\Omega E\right|=\Omega_{\pm}E_{\pm}$ and

$E_{\pm}=-\Omega_{\mp\,}\tilde{n}_{-}\,\pm\sqrt{\Omega^{2}\tilde{n}_{-}^{2}+m^{2}}\Rightarrow\left\\{\begin{tabular}[]{l}$E_{+}=-\Omega_{-\,}\tilde{n}_{-}\,+\sqrt{\Omega^{2}\tilde{n}_{-}^{2}+m^{2}}$\\\
$E_{-}=-\Omega_{+\,}\tilde{n}_{-}\,-\sqrt{\Omega^{2}\tilde{n}_{-}^{2}+m^{2}}$\end{tabular}\right..$
(21)

Consequently, one may rearrange such energy levels and cast them so that

$E_{\pm}^{\left(\Omega_{+}\right)}=\pm\left|\Omega\right|\,\tilde{n}_{\pm}\pm\sqrt{\Omega^{2}\tilde{n}_{\pm}^{2}+m^{2}},$
(22)

for positive vorticity, and

$E_{\pm}^{\left(\Omega_{-}\right)}=\pm\left|\Omega\right|\,\tilde{n}_{\mp}\pm\sqrt{\Omega^{2}\tilde{n}_{\mp}^{2}+m^{2}}.$
(23)

for negative vorticity. Notably, we observe that
$\tilde{n}_{\pm}\left(\ell=\pm\ell\right)=\tilde{n}_{\mp}\left(\ell=\mp\ell\right)$
which would in effect introduce the so called vorticity-energy correlations so
that
$E_{\pm}^{\left(\Omega_{+}\right)}\left(\ell=\pm\ell\right)=E_{\pm}^{\left(\Omega_{-}\right)}\left(\ell=\mp\ell\right)$.
We have, therefore, four branches of energy levels so that the upper half
(above $E=0$ line) is represented by $E_{+}$ and the lower half (below $E=0$
line) is represented by $E_{-}$ in the correlations mentioned above. Yet for
massless KG-Gürses oscillators we obtain
$E_{\pm}^{\left(\Omega_{+}\right)}=\pm 2\left|\Omega\right|\,\tilde{n}_{\pm}$
and $E_{\pm}^{\left(\Omega_{-}\right)}=\pm
2\left|\Omega\right|\,\tilde{n}_{\mp}$.

Moreover, in Figures 1(a) and 1(b) we observe yet a new type of degeneracies
in each branch of the energy levels (i.e., in each quarter of the figures).
That is, states with the irrational quantum number
$\tilde{n}_{+}=2n_{r}+\left|\ell\right|+\ell\,+1$ collapse into $\ell=0$ state
for $\forall\ell=-\left|\ell\right|$ and states with
$\tilde{n}_{-}=2n_{r}+\left|\ell\right|-\ell\,+1$ collapse into $\ell=0$ state
for $\forall\ell=+\left|\ell\right|$. This type of degeneracies is introduced
by the structure of spacetime (Gürses spacetime is used here) and therefore
should be called, hereinafter, spacetime associated degeneracies (STADs).

##  IV KG-Gürses plus Mirza-Mohadesi’s oscillators

We now consider KG-Gürses plus Mirza-Mohadesi’s oscillators with $\eta\neq 0$,
and $g\left(r\right)=1$. In this case, our KG-equation (11) collapses again
into the two-dimensional Schrödinger oscillator

$R^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\ell^{2}-1/4\right)}{r^{2}}-\tilde{\omega}^{2}r^{2}\right]R\left(r\right)=0,$
(24)

which admits exact textbook solvability so that the eigenvalues and radial
eigenfunctions, respectively, read

$\lambda=2\left|\tilde{\omega}\right|\left(2n_{r}+\left|\ell\right|+1\right)=2\left|\Omega
E\right|\sqrt{1+\frac{\eta^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\ell\right|+1\right)$
(25)

and

$R\left(r\right)\sim
r^{\left|\ell\right|+1/2}\exp\left(-\frac{\left|\tilde{\omega}\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\tilde{\omega}\right|r^{2}\right)\Longleftrightarrow\psi\left(r\right)\sim
r^{\left|\ell\right|}\exp\left(-\frac{\left|\tilde{\omega}\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\tilde{\omega}\right|r^{2}\right).$
(26)

Then, equation (13) along with (25) imply

$E^{2}-2\Omega E\ell-2\left|\Omega
E\right|\sqrt{1+\frac{\eta^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\ell\right|+1\right)-\left(m^{2}+2\eta\right)=0.$
(27)

It is obvious that for $\eta=0$ in (27) one would exactly obtain the results
for the KG-Gürses oscillators discussed above. In Figure 2(a), we notice that
the vorticity-energy correlations as well as STADs are now only partially
valid because of the energy shifts introduced by Mirza-Mohadesi’s Mirza 2004
parameter $\eta$. In Figures 2(b) and 2(c) we can clearly observe such shifts
in each quarter of the figures. That is, quarters 1 and 2 are for
$\Omega=\Omega_{+}=+\left|\Omega\right|$ (i.e., for
$E_{\pm}^{\left(\Omega_{+}\right)}$), and 3 and 4 are for
$\Omega=\Omega_{-}=-\left|\Omega\right|$ (i.e., for
$E_{\pm}^{\left(\Omega_{-}\right)}$).

At this point, it should be pointed out that this equation was improperly
treated by Ahmed Ahmed1 2019 , as he expressed the energies in terms of
$\tilde{\omega}$ where $\tilde{\omega}=\sqrt{\Omega^{2}E^{2}+\eta^{2}}$ (see
(16) vs (21) with (22) and (16) vs (35) with (36) of Ahmed1 2019 ). That is,
the energies are given in terms of the energies and his results starting form
his equation (21) to the end of his paper are rendered misleading, and are
incorrect. His results should be redirected to the results reported in current
note, therefore.

Figure 2:  The energy levels for KG-Gürses oscillators of (27) are plotted
with $m=1$ (a) for $\eta=5$, $n_{r}=1$, $\ell=0,\pm 1,\pm 2$, (b) for
$n_{r}=2$, $\ell=1$, $\eta=0,1,3,6,9$ and (c) for $n_{r}=2$, $\ell=-2$,
$\eta=0,1,3,6,9$.

##  V PDM KG-Gürses oscillators

In this section we consider PDM settings for KG-Gürses oscillators, where
$g\left(r\right)=\exp\left(2\beta r^{2}\right);\;\beta\geq 0$. Under such
settings, KG-equation (11) reads

$R^{\prime\prime}\left(r\right)+\left[\lambda-\frac{\left(\ell^{2}-1/4\right)}{r^{2}}-\tilde{\Omega}^{2}r^{2}\right]R\left(r\right)=0,$
(28)

with

$\lambda=E^{2}-2\,\Omega\,\ell\,E-2\beta-m^{2}\text{ ; \
}\tilde{\Omega}^{2}=\Omega^{2}E^{2}+\beta^{2}.$ (29)

In this case, the eigenvalues and radial wavefunctions, respectively, read

$\lambda=2\left|\tilde{\Omega}\right|\left(2n_{r}+\left|\ell\right|+1\right)=2\left|\Omega
E\right|\sqrt{1+\frac{\beta^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\ell\right|+1\right),$
(30)

and

$R\left(r\right)\sim
r^{\left|\ell\right|+1/2}\exp\left(-\frac{\left|\tilde{\Omega}\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\tilde{\Omega}\right|r^{2}\right)\Longleftrightarrow\psi\left(r\right)\sim
r^{\left|\ell\right|}\exp\left(-\frac{\left|\tilde{\Omega}\right|r^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\tilde{\Omega}\right|r^{2}\right).$
(31)

Consequently, the energies are given by

$E^{2}-2\,\Omega\,\ell\,E-2\beta-m^{2}\text{ }=2\left|\Omega
E\right|\sqrt{1+\frac{\beta^{2}}{\Omega^{2}E^{2}}}\left(2n_{r}+\left|\ell\right|+1\right).$
(32)

Obviously, the effect of $\beta$ on the energy levels is the same as that of
the Mirza-Mohadesi’s oscillators Mirza 2004  parameter $\eta$. This would
suggest that Mirza-Mohadesi’s oscillators Mirza 2004  may very well be
considered as a special case of PDM KG-oscillators.

##  VI KG pseudo-Gürses oscillators: vorticity-energy correlations and
spacetime associated degeneracies

We now consider a spacetime described by the metric

$ds^{2}=-dt^{2}+g\left(r\right)\,dr^{2}-2\Omega
Q\left(r\right)r^{2}dtd\theta+Q\left(r\right)r^{2}\left(1-\Omega^{2}Q\left(r\right)r^{2}\right)d\theta^{2}.$
(33)

Next, let us introduce a transformation of the radial part so that

$\rho=\sqrt{Q\left(r\right)}r=\int\sqrt{g\left(r\right)}dr\Rightarrow\sqrt{g\left(r\right)}=\sqrt{Q\left(r\right)}\left[1+\frac{Q^{\prime}\left(r\right)}{2Q\left(r\right)}r\right],$
(34)

where $\mathbb{R}\ni\left(\rho,r\right)\in\left[0,\infty\right]$, and hence
$Q\left(r\right)\in\mathbb{R}$ is a positive-valued dimensionless scalar
multiplier (so is $g\left(r\right)$). In this case, our spacetime metric (33)
now reads

$ds^{2}=-dt^{2}+\,d\rho^{2}-2\Omega\rho^{2}dtd\theta+\rho^{2}\left(1-\Omega^{2}\rho^{2}\right)d\theta^{2}.$
(35)

This metric looks very much like that of Gürses (1) and consequently the KG-
equation (14) that describes KG-Gürses oscillators is indeed invariant and
isospectral with the corresponding KG pseudo-Gürses oscillators equation

$R^{\prime\prime}\left(\rho\right)+\left[\lambda-\frac{\left(\ell^{2}-1/4\right)}{\rho^{2}}-\Omega^{2}E^{2}\rho^{2}\right]R\left(\rho\right)=0.$
(36)

Hence, our KG pseudo-Gürses oscillators would copy the same energies for the
KG-Gürses oscillators of (22) and (23) (discussed in section 3) so that

$E_{\pm}^{\left(\Omega_{+}\right)}=\pm\left|\Omega\right|\,\tilde{n}_{\pm}\pm\sqrt{\Omega^{2}\tilde{n}_{\pm}^{2}+m^{2}},$
(37)

for positive vorticity, and

$E_{\pm}^{\left(\Omega_{-}\right)}=\pm\left|\Omega\right|\,\tilde{n}_{\mp}\pm\sqrt{\Omega^{2}\tilde{n}_{\mp}^{2}+m^{2}}.$
(38)

for negative vorticity. However, the radial wavefunctions are now given by

$R\left(\rho\right)\sim\rho^{\left|\ell\right|+1/2}\exp\left(-\frac{\left|\Omega
E\right|\rho^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\Omega
E\right|\rho^{2}\right)\Longleftrightarrow\psi\left(\rho\right)\sim\rho^{\left|\ell\right|}\exp\left(-\frac{\left|\Omega
E\right|\rho^{2}}{2}\right)L_{n_{n}}^{\left|\ell\right|}\left(\left|\Omega
E\right|\rho^{2}\right).$ (39)

The following notes on our spacetime metric (33) are unavoidable.

(a)

The spacetime metric (35) looks very much like Gürses spacetime one of (1) and
should be called, hereinafter, pseudo-Gürses spacetime, therefore.

(b)

If we set $\Omega=-\mu/3$, $a_{{}_{0}}=1$ in

$\phi=a_{{}_{0}},\,\psi=b_{{}_{0}}+\frac{b_{{}_{1}}}{\rho^{2}}+\frac{3\lambda_{{}_{0}}}{4}\rho^{2},\,q=c_{{}_{0}}+\frac{e_{{}_{0}}\mu}{3}\rho^{2},\,h=e_{{}_{0}}\rho,\,\lambda_{{}_{0}}=\lambda+\frac{\mu^{2}}{27},$
(40)

of (3) and use

$Q\left(r\right)=e_{{}_{0}}+\frac{3c_{{}_{0}}}{\mu r^{2}}\Longleftrightarrow
g\left(r\right)=\frac{\mu e_{{}_{0}}^{2}r^{2}}{\mu
e_{{}_{0}}r^{2}+3c_{{}_{0}}},$ (41)

(where the parametric values are adjusted so that
$\left(Q\left(r\right),g\left(r\right)\right)\in\mathbb{R}$ are positive-
valued functions, i.e., $c_{{}_{0}}<0$ ) we obtain

$q=c_{{}_{0}}+\frac{e_{{}_{0}}\mu}{3}r^{2},\;\psi=\frac{1}{e_{{}_{0}}}+\frac{3c_{{}_{0}}}{\mu
e_{{}_{0}}^{2}r^{2}}.$ (42)

Which is yet another feasible structure for the Gürses spacetime of (2) and
(3) with

$b_{{}_{0}}=\frac{1}{e_{{}_{0}}},\,b_{{}_{1}}=\frac{3c_{{}_{0}}}{\mu
e_{{}_{0}}^{2}},\,\lambda_{{}_{0}}=0,\text{ }h=e_{{}_{0}}r.$ (43)

(c)

As long as condition (34) is satisfied, all KG-pseudo-Gürses oscillators
(including the one in (b) above) in the spacetime model of (35) admit
isospectrality and invariance with the KG-Gürses oscillators (14) and inherit
the same vorticity-energy correlations so that
$E_{\pm}^{\left(\Omega_{+}\right)}\left(\ell=\pm\ell\right)=E_{\pm}^{\left(\Omega_{-}\right)}\left(\ell=\mp\ell\right)$
as well as they inherit the spacetime associated degeneracies, discussed in
section 3.

##  VII Concluding remarks

In the current proposal, we revisited KG-oscillators in the (1+2)-dimensional
Gürses spacetime of (1) so that PDM settings and Mirza-Mohadesi’s KG-
oscillators Mirza 2004  are included. We have observed that KG-Gürses
oscillators are introduced as a byproduct of the very nature of the Gürses
spacetime structure. This has, in turn, motivated us to first elaborate and
discuss the effects of Gürses spacetime on the energy levels of the KG-Gürses
oscillators. We have found that such KG-Gürses oscillators admit vorticity-
energy correlations as well as spacetime associated degeneracies (STADs)
(documented in Figures 1(a) and 1(b)).. However, for KG-Gürses plus Mirza-
Mohadesi’s oscillators we have observed that the vorticity-energy correlations
as well as STADs are only partially valid because of the energy shifts
introduced by Mirza-Mohadesi’s Mirza 2004  parameter $\eta$ (documented in
Figures 2(a), 2(b), and 2(c)). Nevertheless, this model was studied by Ahmed
Ahmed1 2019  who has reported improper treatment and incorrect results.
Consequently, his reported results (starting from his equation (21) to the end
of his paper) should be redirected to the ones reported in the current study.
Moreover, we have shown that PDM setting may very well have the same effect on
the spectrum as that reported for KG-Gürses plus Mirza-Mohadesi’s oscillators.
Yet, a new set of the so called KG pseudo-Gürses oscillators is introduced and
is shown to be invariant and isospectral with KG-Gürses oscillators.
Therefore, such KG pseudo-Gürses-oscillators would inherit the vorticity-
energy correlations as well as STADs of the KG-Gürses oscillators.

Data Availability Statement Authors can confirm that all relevant data are
included in the article and/or its supplementary information files. The author
confirms that there are no online supplementary files (for web only
publication) in this article.

## References

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# Multiple magnetic transitions, metamagnetism and large magnetoresistance in
GdAuGe single crystals

D. Ram  Department of Physics, Indian Institute of Technology, Kanpur 208016,
India  J. Singh  Department of Physics, Indian Institute of Technology
Hyderabad, Kandi, Medak 502 285, Telangana, India  M. K. Hooda  Department of
Physics, Indian Institute of Technology, Kanpur 208016, India  K. Singh
Institute of Low Temperature and Structure Research, Polish Academy of
Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland V. Kanchana
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology
Hyderabad, Kandi, Medak 502 285, Telangana, India  D. Kaczorowski
<EMAIL_ADDRESS>Institute of Low Temperature and Structure Research,
Polish Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland Z. Hossain
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Kanpur
208016, India

###### Abstract

We report the physical properties of GdAuGe single crystals, which were grown
using Bi flux. The powder x-ray diffraction data shows that the compound
crystallizes in hexagonal NdPtSb-type structure (space group P63mc).
Magnetization measurements performed for field configuration H $\parallel$ c
and H $\perp$ c show that GdAuGe orders antiferromagnetically at the Néel
temperature, TN = 17.2 K. Around this temperature, heat capacity and
electrical resistivity data exhibit prominent anomaly due to the
antiferromagnetic (AFM) transition. In addition to an AFM phase transition,
the magnetization data for H $\parallel$ c display the signature of field-
induced metamagnetic (MM) transitions below TN. The critical field range for
these transitions vary from 0.2 to 6.2 T. The critical fields for the MM
transitions decrease with increasing temperature and approach to zero value
for temperature approaching TN. For instance, in high field MM transition,
critical field changes from 6.2 T at 1.7 K to 1.8 T at 16 K. Interestingly,
the magnetoresistance (MR) data (for H $\parallel$ c) record a sharp increase
in values at the critical fields that coincide with those seen in
magnetization data, tracking the presence of MM transitions. MR is positive
and large ($\approx$ 169% at 9 T and 2 K) at low temperatures. Above TN, MR
becomes small and switches to negative values. Hall resistivity data reveal
the predominance of hole charge carriers in the system. In addition, we
observe an emergence of step-like feature in the Hall resistivity data within
the field range of second MM, and a significantly large anomalous Hall
conductivity of $\sim$ 1270 $\Omega$-1 cm-1 at 2 K. The H$-$T phase diagram
constructed from our detailed magnetization and magnetotransport measurements
reveals multiple intricate magnetic phase transitions. The electronic and
magnetic structure of GdAuGe are also thoroughly investigated using first-
principles methods. The electronic band structure calculations reveal that
GdAuGe is a Dirac nodal-line semimetal.

##  I Introduction

Ternary rare-earth intermetallic compounds continue to receive the attention
of scientific community because of their complex relationships between
composition and structure, interesting magnetic, thermodynamic and transport
properties [1, 2, 3, 4, 5, 6]. These compounds possess a wide range of
magnetic characteristics, ranging from simple diamagnetic behavior to very
complex magnetic phases, depending upon the degree of hybridization between 4f
and conduction electrons [5]. The majority of these compounds either display
antiferromagnetic (AFM) or ferromagnetic (FM) ordering due to the long-range
nature of dominant Ruderman-Kittel-Kasuya-Yosida interactions present in them
[5, 6, 7, 8, 9]. The application of an external magnetic field to the AFM
ground states of some rare-earth compounds is observed to disrupt their low
magnetization state, leading to metamagnetic (MM) transitions [5, 10, 11].
These transitions occur in both strongly and weakly anisotropic magnetic
structures and are very sensitive to the crystalline electric field (CEF)
effects, which constrain the magnetic moments along a specific axis [5, 7, 10,
11, 12]. For example, GdAgSi exhibits one MM transition at 4 K with critical
field of $\sim$ 0.29 T [11].

Furthermore, these intermetallic compounds become more interesting, when
interplay between magnetism and novel electronic states generates new exotic
quantum states and intriguing physical properties such as quantum critical
behavior [13, 14], and unconventional superconductivity [15, 16]. Among them,
Eu- and Gd-based intermetallic compounds attract more attention due to their
oxidation state Gd3+ and Eu2+ (most stable oxidation state) having the
electron configuration 4f7 with a half-filled f shell, resulting in a quenched
orbital momentum and very weak spin-orbit coupling (SOC). The f-electron
systems having magnetic frustration can also give rise to a skyrmion phase or
noncollinear spin texture with nonzero scalar spin chirality [$\chi_{s}$ =
Si·(Sj$\times$Sk) $\neq$ 0, where Si, Sj, and Sk are the three nearest spins].
These can act as a fictitious magnetic field on the conduction electrons,
giving rise to the topological Hall effect (THE) [17, 18, 19]. For example,
Gd2PdSi3 is a centrosymmetric triangular lattice with AFM ordering, it
exhibits an intrinsic THE arising from a skyrmion phase under magnetic field
[20]. It is further interesting to note that GdAgGe single crystals previously
investigated by us do not show any MM character up to field of 7 T, but it is
a topological nodal line containing the drumhead surface states in kz = 0
plane, protected by the inversion symmetry [21]. Thus, it becomes important to
examine whether it is possible to induce MM transitions and novel electronic
state in rare-earth based germanide systems by substituting transition metals.

In this context, equiatomic rare-earth gold-germanide (RAuGe) series can be
interesting. The compounds of this series crystallize in the non-
centrosymmetric hexagonal crystal structure with space group P63mc, where two-
dimensional infinite chains of [AuGe] polyanions are separated by rare earth
element ions [3]. The detail studies on the magnetic and physical properties
of polycrystalline RAuGe compounds have been already reported in the
literature [3, 4, 22, 23, 24, 25]. It is observed that HoAuGe and NdAuGe in
RAuGe series display the MM transitions at 2 K with the critical fields of 0.4
and 3 T, respectively [4, 22]. Here, we focus on another member of RAuGe
series i.e. GdAuGe. Polycrystalline GdAuGe was reported to order
antiferromagnetically at 16.9 K [25, 26].

In this report, we study the anisotropic magnetic and electronic transport
properties of GdAuGe single crystals with high magnetic fields up to 9 T, as
well as detailed electronic structure using first-principles calculations. Our
study on GdAuGe single crystals shows the AFM ground state at 17.2 K for
fields perpendicular and parallel to the crystallographic c axis. The field
applied parallel to the c axis of crystal induces MM transitions below the AFM
ordering temperature, which correlate well with the magnetotransport
properties of the compound. Furthermore, we present a first-principles
calculations of the magnetic ground state and topological character of the
compound.

##  II Experimental details

The single crystals of GdAuGe were synthesized using Bi as an external flux.
Starting elements Gd (99.9%, Alfa Aesar), Au (99.99%, Alfa Aesar), Ge
(99.999%, Alfa Aesar) and Bi (99.999%, Alfa Aesar) were taken in a molar ratio
of 1:1:1:10. The constituent elements were put into an alumina crucible, which
was then transferred to a silica quartz tube. The tube was sealed under
partial pressure of argon gas. In the next step, the sealed assembly was
heated to temperature of 1050 ∘C, where it was held for 24 h in order to
obtain homogeneous solution. Subsequently, the slow cooling to 680 ∘C at the
rate of 2.5 ∘C/h produced very shiny plate-like single crystals with a typical
size of 5 $\times$ 3 $\times$ 0.4 mm3 (as shown in bottom inset of Fig. 1(b)),
which were separated from the Bi flux by centrifuging.

Figure 1: (a) Rietveld refined powder XRD patterns of crushed single crystals
of GdAuGe at room temperature. The blue line represents the difference between
the observed intensity (red solid circles) and the calculated intensity (solid
black line). The olive vertical lines represent the position of Bragg peaks.
(b) The single-crystal XRD pattern of a GdAuGe, showing only (00$l$)
reflections. Upper inset shows the rocking curve of peak (004). Lower inset
shows a optical image of crystals.

The phase purity and orientation of as grown crystals were analyzed by x-ray
diffraction (XRD) using a PANalytical X’Pert PRO diffractometer with Cu Kα1
radiation. The XRD pattern of powdered crystals and a representative single
crystal recorded at room temperature is shown in Figs. 1(a) and 1(b),
respectively. It confirms the single phase growth of the compound
crystallizing in the hexagonal crystal structure with space group P63mc (No.
186). The lattice parameters ($a$ = $b$ = 4.4281 Å, and $c$ = 7.4262 Å)
obtained from Rietveld refinement are in good agreement with previously
reported data in the literature [25, 26, 27]. The presence of (00$l$) peaks in
the single crystal diffraction pattern shows that the crystallographic $c$
axis of crystal is perpendicular to its flat plane. The upper inset of Fig.
1(b) presents the rocking curve of (004) peak with a full width at half
maximum (FWHM) $\Delta$$\theta$ = 0.024∘, indicating a high quality of the
single crystal used. The desired chemical composition of crystals was further
confirmed by energy-dispersive x-ray spectroscopy using a JEOL JSM-6010LA
scanning electron microscope. Electrical resistivity and magnetoresistance
measurements were performed using a Quantum Design physical property
measurement system (PPMS) by the standard four-probe method. Heat capacity
measurements were performed by the conventional relaxation method in the same
PPMS platform. The magnetic susceptibility and magnetization were measured
down to 1.7 K using a Quantum Design magnetic property measurement system.

Based on the density functional theory (DFT) [28, 29], the first-principles
calculations were carried out using the projector augmented wave [30] approach
as implemented in the Vienna ab initio simulation package [31, 32]. The
generalised gradient approximation (GGA) with the Perdew-Burke-Ernzerhof [33]
parametrization was utilised to account for exchange-correlation effects. A
Hubbard U parameter (GGA+U) of 6 eV was used to address the correlation
effects of Gd-$f$ states [34, 35]. The calculations were done with a plane
wave energy cutoff of 600 eV, and the energy convergence criterion was chosen
to be 10-8 eV. The geometry optimization was performed with 2 $\times$ 2
$\times$ 2 supercell using a 16 $\times$ 16 $\times$ 8 k-mesh as per the
Monkhorst-Pack method [36].

Figure 2: (a) Temperature-dependent magnetic susceptibility of GdAuGe measured
under an applied magnetic field of $\mu_{0}$H = 0.1 T for H $\parallel$ c and
H $\perp$ c in ZFC and FC modes. (b) The inverse magnetic susceptibility as a
function of temperature for H $\parallel$ c and H $\perp$ c. The solid orange
lines show the Curie-Weiss fit above 50 K.

##  III Results and discussion

###  III.1 Magnetic properties

Figure 2(a) presents the temperature (T) dependence of magnetic susceptibility
($\chi$) measured under zero-field cooling (ZFC) and field cooling (FC)
conditions at the constant magnetic field of 0.1 T applied perpendicular and
parallel to the crystallographic c axis. A maximum of $\chi$(T) is visible at
TN = 17.2 K for both field configurations, which is indicative of an AFM
ordering in the compound and marks the boundary between AFM and paramagnetic
(PM) phase. This value is very close to the previously reported TN for GdAuGe
[25, 26, 27]. It is to be noted that $\chi$(T) shows bifurcation in ZFC$-$FC
measurements below 15 K for field configuration H $\perp$ c. It points out
spin reorientations in the compound below 15 K, which also corroborates with
second anomaly in heat capacity data. The magnetic anisotropy of the system in
the AFM region is low as evident from Fig. 2(b). It becomes insignificant in
the PM region (above TN). Above 50 K, the data plotted as inverse magnetic
susceptibility ($\chi^{-1}$) vs. T in Fig. 2(b) fit to the Curie-Weiss formula
$\chi(T)=C/(T-\Theta)$, where $C$ and $\Theta$ are the Curie constant and
Curie-Weiss temperature, respectively. The least-square fitting yields the
value of $\Theta$ $\approx$ -4.7 and -6.9 K for H $\perp$ c and H $\parallel$
c, respectively. The estimated effective magnetic moment of $\mu_{eff}$ = 7.76
(for H $\perp$ c) and 7.80 $\mu_{B}$/Gd (for H $\parallel$ c) is in close
agreement with the theoretical value expected for Gd3+ ion.

Figure 3: (a) Temperature-dependent magnetic susceptibility of GdAuGe measured
under different applied magnetic fields of $\mu_{0}$H = 0.01, 0.5, 1, 2, 3, 4,
5, 5.5, 6.0, 6.5, and 7.0 T for H $\parallel$ c. (b) Magnified view of
magnetic susceptibility curves at higher fields of $\mu_{0}$H = 5.0, 5.5, 6.0,
6.5, and 7.0 T. The arrows are guide to various magnetic transitions such as
AFM TN (magenta) and field-induced anomalies Tm1 (blue) and Tm2 (red). Figure
4: (a) Isothermal magnetization of GdAuGe at several different temperatures of
T = 1.7, 4, 6, 8, 10, 12, 14, and 16 K for H $\parallel$ c. Top inset of (a)
shows the magnetic isotherms in temperature range of T = 18$-$30 K. Bottom
inset of (a) presents the magnetic field dependence of magnetization measured
at temperature T = 1.7 K for H $\perp$ c. (b) Magnetic field dependence of
differential magnetization at the various temperatures for H $\parallel$ c.
The dotted arrows show the critical fields Hc1 and Hc2 at various temperatures
corresponding to two MM transitions observed in GdAuGe. Inset of (b) shows a
zoom view of low-field magnetizations at 1.7 K in both field-directions.

Next, we study the $\chi_{c}$(T) behavior under different applied magnetic
fields (0.01$-$7 T) along the c axis. The data plotted under various magnetic
fields are shown in Figs. 3(a) and 3(b). At very low field, $\mu_{0}$Hc $\sim$
0.01 T, a very sharp peak, which is a typical characteristics of an AFM
ordering can be observed at TN $\sim$ 17.2 K (Fig. 3(a)). With further
increase in field, this peak gets suppressed in magnitude and becomes broad
along with shift towards low temperatures. Above 0.5 T, we observe the onset
of field-induced anomalies in addition to an AFM transition. These anomalies
shift towards low temperatures with an increase in the field. Fig. 3(b)
presents the magnified view of these anomalies along with AFM transition at
higher fields. The peaks of anomalies are marked by arrows to facilitate the
view. It is interesting to note that AFM peak becomes almost flat at
$\mu_{0}$Hc $\sim$ 7 T, and the curvature of $\chi$ vs. T tends to approach
the FM state through these field-induced anomalies in the system.

Further, we measured isothermal magnetization for H $\parallel$ c between 1.7
and 30 K with magnetic fields up to 7 T, as shown in Fig. 4(a). They show
monotonic increase in magnetization values with no sign of saturation.
Magnetization value reaches $\sim$ 2.02 $\mu_{B}$/Gd at 7 T, which is much
smaller than the value expected for free Gd3+ ion. Magnetic isotherms further
reveal the sudden change in slope at two critical fields below 18 K,
indicating the emergence of two MM transitions in GdAuGe. It is noteworthy to
mention here that we also measured in-plane magnetization (H $\perp$ c). But
it did not reveal any MM transition besides the negligible hysteresis at low
magnetic fields (see insets of Figs. 4(a) and 4(b)). Next, we calculate the MM
critical fields at various temperatures using the maxima of field-dependent
differential magnetization curves as shown in Fig. 4(b). At 1.7 K, the
critical fields are $\mu_{0}$Hc1 $\sim$ 0.8 and $\mu_{0}$Hc2 $\sim$ 6.2 T,
which decrease with increasing temperature. This trend is in accordance with
spin-flop transition expected in antiferromagnets [37] and has been observed
in other Gd-based compounds such as Gd2Te3 [38]. Above 16 K, MM transitions
completely disappear and the compound enters into PM state.

Figure 5: (a) Temperature-dependent heat capacity (Cp) of GdAuGe single
crystal. The solid blue line represents the Debye-Einstein Model fit to the
experimental data. The inset shows a magnified view of the Cp behavior at low
temperatures. (b) Magnetic entropy of GdAuGe as a function of temperature.
Figure 6: Temperature-dependent electrical resistivity of GdAuGe single
crystal in (a) without and (b) with magnetic field. The current is applied
along ab plane of crystal. Transverse magnetoresistance as a function of
magnetic field at different temperatures (c) below TN and inset above TN for H
$\parallel$ c. The arrows and dotted line mark different phase transitions.

The origin of MM transitions in our data remains unclear like other rare-earth
silver and gold germanides systems. Although, it is observed that a number of
factors such as strong magnetocrystalline anisotropy, CEF effects, and
competition between long-range FM and AFM interactions contribute to the MM
transitions observed in rare earth compounds [5, 7, 10]. In the present case,
magnetocrystalline anisotropy is small and CEF effects are minimal considering
the fact that Gd3+ ions are in symmetric 8S7/2 state [7]. Recently, multiple
MM transitions observed in CeRh3Si2 were explained using the Ising model,
which generates the series of commensurate and incommensurate phases, leading
to the metamagnetism like features [39]. The transition from a commensurate to
incommensurate phase was reported in isostructural HoAuGe, which shows MM
transition $\sim$ 0.4 T at 2 K [22]. Such possibility in GdAuGe is subject of
future investigations.

###  III.2 Heat capacity and entropy

Figure 5(a) shows the heat capacity (Cp) of GdAuGe single crystal measured in
T range 2$-$300 K. A broad peak feature in low temperature Cp data near TN is
consistent with an AFM ordering observed from the magnetic measurements. The
size of this peak ($\Delta$Cp) is $\sim$ 7.2 J/mol K, which is almost half of
the value predicted by mean-field theory for amplitude-modulated magnetic
structure. A close zoom-in view of the peak shows that it is split into two
parts at TN = 16.8 and 14.9 K (see inset of Fig. 5(a)). Two magnetic
transitions in GdAuGe based on Cp(T) data were reported earlier in Ref. [25,
27]. It is suggested to be associated with spin-reorientation processes in the
compound [25]. Furthermore, we observe a broad hump around 6 K in low
temperature Cp data. This kind of broad hump has been reported in other Gd
based compounds, for example, in GdCu2Si2, it is observed at $\sim$ 3 K and is
associated with emergence of (2J+1)-fold degeneracy of multiplet in the
ordered system [40, 41].

Figure 7: (a) The Hall resistivity measured at various temperatures from 2 to
100 K. The inset displays the magnitude of anomalous Hall resistivity at 2 K.
In panels (b) and (c), the solid black line represents the fit of Eq. (4) to
the experimental data at 2 and 4 K. Panel (d) shows the temperature variation
of ordinary and anomalous Hall coefficient. (e) The anomalous Hall resistivity
$\rho_{xy}^{A}$ ($\displaystyle\star$), and conductivity $\sigma_{xy}^{A}$
($\displaystyle\blacksquare$) and $\sigma_{xy}^{A^{\prime}}$
($\displaystyle\blacktriangleright$) as a function of temperature.

At 300 K, Cp value approaches $\sim$ 73.74 J/mol K, which is within the
Dulong-Petit limit. Attempts to determine the electronic specific-heat
coefficient, $\gamma$ from low temperature Cp data fail due to the
nonlinearity caused by the magnetic anomalies. Cp data above TN can be well
described by the following expression

$C_{p}(T)=\gamma T+qC_{D}(T)+(1-q)C_{E}(T)$ (1)

where $q$ is the weight factor, and $C_{D}$(T) and $C_{E}$(T) are Debye and
Einstein Cp contributions, respectively, defined as

$C_{D}(T)=9nR\left(\frac{T}{\Theta_{D}}\right)^{3}\int_{0}^{\Theta_{D}/T}\frac{x^{4}e^{x}}{(e^{x}-1)^{2}}dx$
(2)

and

$C_{E}(T)=3nR\left(\frac{\Theta_{E}}{T}\right)^{2}\frac{e^{\Theta_{E}/T}}{(e^{\Theta_{E}/T}-1)^{2}}$
(3)

where $\Theta_{D}$ and $\Theta_{E}$ are the Debye and Einstein temperatures,
respectively. The values of various fitting parameters follow as, $\gamma$ =
3.9 mJ/mol K2, $\Theta_{D}$ = 307, $\Theta_{E}$ = 90 K, and $q$ = 0.58. The
magnetic entropy (shown in Fig. 5(b)) is calculated using the formula,
S${}_{m}=\int\frac{C_{m}}{T}dT$, where magnetic contribution, Cm is obtained
by subtracting lattice part from the experimental data using Eq. (1). The
entropy Sm released at TN is slightly lower than the theoretical value S =
$R$ln(2J+1) = 17.3 J/mol K for Gd3+ with J = 7/2. The Sm(T) reaches $R$ln8 at
24 K and then saturates above 28 K. A slightly higher value of saturation
entropy is due to the partial subtraction of phonon contribution [42].

###  III.3 Magnetotransport

The electrical resistivity, $\rho$ as a function of T measured along the ab
plane of crystal is shown in Fig 6(a). The investigated crystal shows the room
temperature resistivity value of around 206 $\mu$$\Omega$ cm, and residual
resistivity ratio ($\rho_{300K}$/$\rho_{2K}$) $\approx$ 11.33. It is
comparable to the values reported in the literature for other Gd based ternary
intermetallic compounds [43, 44]. The $\rho$(T) exhibits a typical metal like
behavior. It decreases systematically with decreasing T until it registers a
sharp drop in value near the magnetic transition temperature. The sharp drop
at TN = 17.2 K is the result of substantial reduction in the spin-disorder
scattering and corroborates the results of magnetic and heat capacity
measurements. Furthermore, we also measured the temperature-dependent
electrical resistivity with magnetic field H $\parallel$ c, as shown in Fig.
6(b). As field strength increases, the AFM transition TN shifts to lower
temperatures, as marked with black arrows. Above 1 T, a second anomaly appears
at Tm1, which does not change so much with magnetic field. The values of TN
and Tm1 are consistent with the $\chi_{c}$(T) data as discussed above.

The transverse magnetoresistance (MR) measured for field configuration H
$\parallel$ c in T range 2$-$40 K, are shown in Fig. 6(c). The MR is positive
for T $\leq$ 14 K and becomes negative at T $\geq$ 16 K, as observed in AFM
systems. In weak fields, it follows H1.3 field dependence and its value
reaches only about 7% at 0.8 T. For fields higher than 0.8 T, a weak anomaly
is visible in MR data, thereafter, MR increases sublinearly. It reaches up to
$\approx$ 123% at 2 K until an onset of another anomaly at 6.2 T. In the
vicinity of this anomaly, MR value suddenly jumps to $\approx$ 160% and tends
to saturate above 7 T. The anomalies observed in MR data in the vicinity of
critical fields, where we observed the MM transitions in magnetic isotherms,
clearly indicate that they are related to the MM transitions observed in the
compound. The positive MR below TN for an AFM phase is quite naturally
expected, however large and positive MR due to the MM transitions in GdAuGe is
in contrast to small and negative MR observed in several rare-earth compounds
[5, 45]. Ideally, application of magnetic field reduces the electrical
resistivity of ferromagnet and paramagnet, leading to negative MR. However, in
the literature, numerous cases of this type of sudden enhancement in MR have
been observed [5, 46, 47, 48, 49]. The MR value of our crystal is almost twice
the value of $\sim$ 82% observed for TbAgGe crystals [5] and comparable to the
value reported for EuAg4As2 single crystals [49]. With increasing T, sharp
steps of increase in MR observed due to MM transitions gradually decreases and
completely disappear above TN, leading to the negative values of MR.

Figure 8: (a) The crystal structure of GdAuGe. (b) The irreducible Brillouin
zone of the bulk along with the (001) projected surface. Figure 9: (a)$-$(h)
FM and AFM configurations for 2 $\times$ 2 $\times$ 2 supercell with Gd spins.
Here, AFM1, AFM3, AFM4 are A-, C-, G-type, respectively, whereas other
configurations are stripe-type AFM. Red and green arrows denote the spin-up
and spin-down, respectively.

Figure 7(a) displays the Hall resistivity ($\rho_{xy}$) of single-crystalline
GdAuGe, measured within the ab plane over a T range of 2 to 100 K. The
$\rho_{xy}$ increases continuously with an increasing magnetic field in a
slightly nonlinear manner and its value remains positive throughout the
temperature range, indicating that holes are the majority charge carriers.
Moreover, we estimate the carrier concentration and mobility to be
approximately 2.69 $\times$ 1020 cm-3 and 167 cm2 V-1 s-1, respectively, by
obtaining the slope from the linear fit of the 100 K dataset. At low
temperatures (below 16 K), the $\rho_{xy}$ exhibits a step-like increase
around the critical magnetic field range, where we observed the signature of
MM transition in magnetization and MR data. Considering that the step-like
feature in $\rho_{xy}$ is a part of the anomalous Hall resistivity
($\rho_{xy}^{A}$). To calculate the magnitude of $\rho_{xy}^{A}$, we adopt the
method used in Ref. [50], as shown in the inset of Fig. 7(a). The
$\rho_{xy}^{A}$ at 2 K is around 3.61 $\mu\Omega$ cm, and its magnitude
decreases with increasing T, reaching $\sim$ 1.2 $\mu\Omega$ cm at 12 K (see
the Fig. 7(e)). In general, the total Hall resistivity including the
$\rho_{xy}^{A}$ term is given by the following expression

$\rho_{xy}=\rho_{xy}^{O}+\rho_{xy}^{A}=R_{0}H+R_{s}\mu_{0}M,$ (4)

where R0 and Rs are the ordinary and anomalous Hall coefficients, respectively
and M(H) is isothermal magnetization data as a function of field. In our data,
it is difficult to separate $\rho_{xy}^{O}$ and $\rho_{xy}^{A}$ contributions
as the magnetic moments do not saturate up to field strength 7 T. Therefore,
we have simulated the experimental data up to $\mu_{0}H=$ 7 T using Eq. (4).
The results for 2 and 4 K data set are displayed in Figs. 7(b) and 7(c),
respectively. The estimated values of $R_{0}$ and $R_{s}$ from simulated data
are presented in Fig. 7(d). The values of $R_{s}$ are significantly larger
than $R_{0}$. Next, we present the anomalous Hall conductivity (AHC),
$\sigma_{xy}^{A}$ = $\rho_{xy}^{A}$/($\rho_{xx}^{2}+\rho_{xy}^{2}$), in Fig.
7(e), and the AHC decreases with the increasing temperatures. At 2 K, its
value is about 1270 $\Omega$-1 cm-1, which is of the same order as reported
for AFM topological systems DyPtBi [51] and TbPtBi [52]. To further check the
consistency of calculated AHC, we have estimated the AHC
($\sigma_{xy}^{A^{\prime}}$) using the $R_{s}$ and change in magnetization
value around the MM transition. The obtained values of
$\sigma_{xy}^{A^{\prime}}$ are quite close to that directly calculated from
$\rho_{xy}$(H) curves (see Fig. 7(e)).

###  III.4 Electronic structure

Figure 10: (a) Total and projected density of states of GdAuGe. (b) Electronic
band structure along $\Gamma$-$M$-$K$-$\Gamma$-$A$-$L$-$H$-$A$ path without
SOC. (c) The orbital-decomposed electronic band structure without SOC. (d) The
electronic band structure with SOC. Inset shows the Dirac points DP1, DP2 and
DP3. Figure 11: (a) and (c) The illustration of the nodal line, where $a$,
$b$, $c$, $d$ are equally spaced points between $M$ and $K$ along kz = 0
plane, and $e$, $f$, $g$, $h$ between $L$ and $H$ along kz = 0.5 plane,
respectively. (b) and (d) Electronic band structures along the k-paths as
indicated in (a) and (c), respectively. Iso-energy Fermi contours along (e) kz
= 0 and (f) kz = 0.5 planes, which show the nodal lines.

The unit cell of GdAuGe consists of six atoms, with Gd, Au, and Ge atoms
occupying Wyckoff positions 2a, 2b, and 2b, respectively. As shown in Fig.
8(a), it crystallizes in a hexagonal structure with the space group
$P$63${mc}$ (186). Along with three vertical mirror planes
$\widetilde{M}$${}_{x\bar{y}}$ = {M${}_{x\bar{y}}$$|$00$\frac{1}{2}$},
$\widetilde{M}$2xy = {M2xy$|$00$\frac{1}{2}$}, and My, the structure has
threefold rotational symmetry, C3z and twofold screw rotational symmetry, S2z
= {C2z$|$00$\frac{1}{2}$}. In Fig. 8(b), we display the (001) surface
Brillouin zone (BZ) beside the bulk BZ. In order to investigate the possible
magnetic configurations, we have examined the FM and seven AFM (including A-,
C-, G\- and stripe-type) spin configurations with 2 $\times$ 2 $\times$ 2
supercell. The possible magnetic configurations are shown in Fig. 9. The
calculated ground state energy for each configuration is presented in Table 1.
From the Table 1, it can be seen that the AFM5 configuration yields the lowest
energy. Here, AFM5 configuration exhibits the AFM coupling along the a axis,
whereas FM coupling along the b and c axes. To further confirm the spin
orientations in the AFM5 case, we have calculated the ground state energies
along different spin alignments such as [001], [010], [100], [011], [101],
[110] and [111]. The computed ground state energy differences are given in
Table 2. The minimum ground state energy is observed for the [100] spin
configuration. A similar AFM5 magnetic structure has also been reported in the
isostructural compounds RAuGe (R = Tb$-$Er), where magnetic moments are
inclined with respect to the c axis [22, 23, 3]. This inclination angle of
magnetic moment decreases with increase in number of 4f electrons. For
example, TbAuGe magnetic moment is inclined at an angle of 65∘ to the c axis,
while ErAuGe magnetic moment is along the c axis. Following the trend of the
magnetic structure of isostructural RAuGe compounds, the magnetic moments of
GdAuGe are likely to be aligned along the ab plane, as suggested by our DFT
calculations. However, it cannot be completely ascertained, as our
experimental data indicate that moments are preferably aligned along the c
axis. Further, the CEF effects are absent in GdAuGe, unlike isostructural
RAuGe compounds, which could affect the orientation of magnetic moments [23].
Our DFT results are valid for T = 0 K. Furthermore, we have used a fixed value
of U (= 6 eV) in the absence of experimentally determined U value. Thus,
correlation effects are not taken care of appropriately. These inherent
limitations might be responsible for the difference between the theoretically
predicted magnetic structure and the magnetic measurements. To determine the
precise orientation of Gd spins within the GdAuGe and to resolve the
discrepancy between our theoretical calculations and experimental
observations, further investigations are required, especially using
microscopic techniques.

Table 1: Calculated energies of different magnetic configurations (in meV) with the reference energy considered to be 0 meV. Configuration | Energy (meV) | Configuration | Energy (meV)
---|---|---|---
FM | 11.93 | AFM4 | 3.59
AFM1 | 5.72 | AFM5 | 0.00
AFM2 | 5.71 | AFM6 | 3.59
AFM3 | 0.02 | AFM7 | 2.79
Table 2: Calculated energies of different spin configurations in AFM5 case with the reference energy considered to be 0 $\mu$eV. Configuration | [001] |  [010] |  [100] |  [011] |  [101] |  [110] |  [111]
---|---|---|---|---|---|---|---
Energy ($\mu$eV) | 49.79 | 21.56 | 0.00 | 34.79 | 24.12 | 20.11 | 15.09
Figure 12: The H-T phase diagram of the GdAuGe when magnetic field is applied
along the c axis. Denotations are mentioned in main text. The dark cyan dot
line illustrates the fit of molecular field theory equation
H=H0[1$-$TN(H)/TN(H = 0)]1/2 to the experimental data. Dotted gray lines are
just guides to the eyes.

Furthermore, the total density of states (DOS) and projected density of states
(PDOS) were calculated for AFM5 case, to illustrate the behavior of Gd, Au and
Ge elements, and the results are displayed in Fig. 10(a). The valence band
region is equally contributed by Gd, Au and Ge atoms, whereas the conduction
region is dominated by Gd in both spin channels. Moreover, GdAuGe has a small
DOS at the Fermi level, which confirms the semimetallic nature of the
compound. We have also investigated the electronic band structure properties.
Figure 10(b) shows the electronic band structure with the spin-up (in red
color) and the spin-down (in blue color) channels. The electronic band
structure exhibits some band crossing points near the Fermi level along the kz
= 0 as well as kz = 0.5 plane, which might lead to the nodal line. To
determine the non-trivial nature of these bands, we calculated the orbital
decomposed band structure (Fig. 10(c)) and it infers that Gd-$d$ and Ge-$p$
states are main contributors to the band crossing points. From the Fig. 10(c),
we observe two crossing points along kz = 0 plane. The band inversion between
Gd-$d$ and Ge-$p$ in one crossing point can be seen, which reflects the non-
trivial nature of crossing points, whereas another crossing point lacks the
band inversion and shows the trivial nature of crossings. Similarly, the band
inversion can also be seen along kz = 0.5 plane. Notably, each band along the
kz = 0.5 plane is twofold degenerate due to the anticommutation relation
between My and S2z symmetries [53], which show the four-fold degeneracy in
bands at the crossing point and hints towards the presence of a Dirac nodal
line. To analyze these band crossings, we have performed a detailed
calculation of the band structures along $\Gamma$-$M/a/b/c/d/K$ paths (see
Fig. 11(a)) as well as $A$-$L/e/f/g/h/H$ paths (see Fig. 11(c)) and found that
the Dirac-type band crossings appeared in all the above-mentioned paths (see
Figs. 11(b) and 11(d)) reflecting the occurrence of two $\Gamma$-centered
nodal rings protected by My symmetry, and one $A$-centered Dirac nodal ring
protected by My and S2z symmetries. Furthermore, we have confirmed the
presence of nodal lines through iso-energy Fermi contours and shown in Figs.
11(e) and 11(f). With the inclusion of SOC, we can see the band opening at the
crossing points, which is shown in Fig. 10(d). In addition, there exist
multiple Dirac points (DP1, DP2, and DP3) along $A$-$L$ path, which is shown
in the inset of Fig. 10(d). The Dirac points DP1, DP2, and DP3 are generated
by My and non-symmorphic (S2z) symmetries.

###  III.5 H-T phase diagram

Based on the experimental data presented above, we have constructed the H$-$T
phase diagram for H $\parallel$ c, which is depicted in Fig. 12. The phase
line boundaries are calculated using the peak positions of derivatives of the
$\chi_{c}$(T), M(H), $\rho_{xx}$(H), and $\rho_{xx}$(T) data. The resulting
phase diagram shows four distinct regions in the magnetically ordered state.
The first region, labeled AFM I, corresponds to AFM phase. The magnetic
structure in this region below TN is collinear AFM at low fields as evidenced
by our electronic band structure calculations (see Fig. 9(f)) and
experimentally observed value of $\chi_{c}$(1.7 K)/$\chi_{c}$(TN) $\approx$
0.5 (refer to Fig. 2(a)). As the strength of field increases, we move into
region AFM II. In this region, AFM spins tend to align along the direction of
external magnetic field, and get partially flopped above the critical field
Hc1, which is deduced from the M(H) measurements. At Hc1, the first spin-flop
transition occurs, followed by the second spin-flop transition at the critical
field Hc2. After Hc2 and below the mean field fitting line, the system is in
region AFM III, which likely corresponds to an incommensurate magnetic
structure. Both Hc1 and Hc2 decrease with increasing T. The phase boundary of
Hc2 nearly overlap with that of TN in the T range 12$-$16 K. However, below 12
K, it is well separated from TN and its values, derived from the magnetization
and transport measurements, are in a good agreement. Such complex magnetic
phases are also observed in the isostructural RAuGe (Tb$-$Er) [23, 22, 3]. We
further note that TN shifts towards low temperatures with increasing field
like Hc1 and Hc2. Its trend is in well accord with that predicted by the
molecular field theory equation H = H0[1$-$TN(H)/TN(H = 0)]1/2, where H0 is
the critical field strength required to completely destroy the AFM phase
transition [54]. The fit of this equation yields H0 = 8.57 T and TN(H = 0) =
17.15 K. Further increase in field drives the system from region AFM III to
PPM, which we refer to as the polarized paramagnetic (PPM) phase, where a
large part of the magnetic moments is aligned along the field. It displays two
field-induced anomalies, Tm1 and Tm2, in the vicinity of TN. These anomalies
show small variation in their temperatures with increasing fields. As the
temperature surpasses Tm1, the compound transitions to the PM state.

##  IV Conclusions

We have investigated the magnetic, thermodynamic and magnetotransport
properties of GdAuGe single crystals grown using Bi flux. The magnetic
susceptibility measurements for field configuration H $\parallel$ c and H
$\perp$ c revealed the AFM ground state in the compound with TN = 17.2 K. The
anomalies observed near TN in the heat capacity data and a sharp drop in
electrical resistivity data below TN further confirmed the AFM ordering in the
compound. The magnetization data for H $\parallel$ c showed two successive MM
transitions at T = 1.7 K with critical fields of $\sim$ 0.82 and 6.2 T. The
magnetotransport data recorded for H $\parallel$ c near the critical fields of
MM transitions was observed to show unexpectedly positive and large values of
transverse MR (169% at 9 T and 2 K) for temperatures less than TN. At higher
temperatures, the MR decreases and becomes negative in the PM regime. A large
anomalous Hall conductivity $\sim$ 1270 $\Omega$-1 cm-1 was observed near Hc2
at 2 K. The phase-diagram in H $\parallel$ c vs. T plane was constructed from
the magnetization and magnetotransport measurements, which unveiled multiple
magnetic phase transitions including a collinear AFM ground state, two
successive spin-flop transitions and a polarized paramagnetic state
corresponding to the two field-induced magnetic anomalies. The electronic band
structure analysis shows the presence of two nodal rings along kz = 0 plane
and a Dirac nodal ring along kz = 0.5 plane, which makes GdAuGe a Dirac nodal-
line semimetal.

##  V Acknowledgment

We acknowledge IIT Kanpur and the Department of Science and Technology, India,
[Order No. DST/NM/TUE/QM-06/2019 (G)] for financial support. J.S and V.K.
acknowledge the National Supercomputing Mission (NSM) for providing computing
resources of ‘PARAM SEVA’ at IIT, Hyderabad. V.K. would like to acknowledge
DST-FIST (SR/FST/PSI-215/2016) for the financial support. J.S. was supported
through a CSIR scholarship. K.S. and D.K. acknowledge financial support from
the National Science Centre (Poland) under Research Grant No.
2021/41/B/ST3/01141.

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