# Emergence of highly bond-dependent anisotropic magnetic interactions in
Sr4RhO6: a theoretical study

Shishir Kumar Pandey<EMAIL_ADDRESS>International Center for
Quantum Materials, School of Physics, Peking University, Beijing 100871, China
AI for Science Institute, Beijing, China  Qiangqiang Gu  School of
Mathematical Science, Peking University, Beijing 100871, China  AI for Science
Institute, Beijing, China  Rajarshi Tiwari  School of Physics, AMBER and CRANN
Institute, Trinity College Dublin, Dublin 2, Ireland

###### Abstract

The quantum spin liquid states as a natural ground state of the Kitaev model
have led to a quest for new materials candidates hosting Kitaev physics. Yet,
there are very few materials hosting Kitaev physics. Using a combination of
$ab$ $initio$ and model Hamiltonian methods, we propose that Sr4RhO6 belongs
to this category. With a tight-binding model and exact diagonalization
approach, we show that despite substantial trigonal-like distortion, the
electronic and magnetic properties of Sr4RhO6 can be well described in terms
of pseudo-spin = 1/2 states. Magnetic interactions among this pseudo-spins,
estimated using the second-order perturbation method are highly bond-dependent
anisotropic in nature with two particularly noticeable features,
antiferromagnetic Kitaev and Dzyaloshinskii-Moriya interactions. The gaped
spin-wave spectra of Sr4RhO6 obtained with linear spin-wave theory are
consistent with the underlying magnetic frustration. Further spectra analysis
reveals a crucial role of these interactions in Sr4RhO6. In the end, we
briefly discuss the enhanced magnetic interactions in a similar material
Ca4RhO6 and propose it be another magnetic candidate belonging to the 4$d$
class.

Suggested keywords

##  I Introduction

Orbital and spin angular momentum of an electron are coupled through a
relativistic effect called spin-orbit coupling (SOC). Many interesting
phenomena such as the anomalous Hall effect, manipulation of spin currents,
the emergence of topological properties in weakly correlated systems has been
extensively studied [1, 2, 3] . However, the strongly correlated materials
provide host even richer physics because of the presence of additional
interactions such as crystal field splitting ($\Delta^{\text{\tiny CF}}$) and
on-site Hubbard ($U$), often competing with SOC [4, 5]. This competition gives
rise to exotic phenomenon like realization of unconventional superconductivity
[6, 7, 8], emergence of topological phases [9] and Kitaev physics [10]. Among
these examples, Kitaev physics [11] particularly has recently got a lot of
attention as a driving mechanism in realization of quantum spin liquid states
[12, 13].

The work of Jackeli and Khaliullin [10] accelerated the progress towards the
realization of Kitaev physics in real materials. Their proposal was based on
magnetic interactions between pseudo-spins on a honeycomb lattice of
transition metal ions originating from the interplay of strong electrostatic
crystal field (CF) of anions and SOC at transition metal sites. The five
degenerate $d$ orbitals of transition metal atom split into triply degenerate
$t_{2g}$ and doubly degenerate $e_{g}$ orbitals due to $\Delta^{\text{\tiny
CF}}$ (see Fig. 1(a)). Energetically lower $t_{2g}$ manifold further splits in
the presence of SOC to form the half-filled pseudo-spin $J_{\text{eff}}$
$=\frac{1}{2}$ states dominating low energy space of materials. Magnetic
interactions between these $J_{\text{eff}}$ $=\frac{1}{2}$ pseudo-spin states
was proposed to be dominantly Kitaev-type. Cobaltates [14, 15, 16, 17, 18, 19,
20, 21, 22, 23], iridates [24, 25, 26, 27, 28, 29, 30, 31] and $\alpha$-RuCl3
[32, 33, 34, 35, 36] are some of the examples falling in this category. Recent
studies on Ir-based double perovskite compounds have further widen the horizon
of Kitaev physics on frustrated fcc lattice formed by magnetic ions with
spatially separated octahedral environment [37, 38, 39, 40, 41].

These pseudo-spin $J_{\text{eff}}$ =$\frac{1}{2}$ behave as Kramers’s doublet
(i.e. $J_{\text{eff}}$γ, where $\gamma$ = $x$, $y$, $z$ operators follow the
fermionic anti-commutation relation) only in the limits, $\Delta^{\text{\tiny
CF}}$ $\to$ $\infty$ and when the splitting among the $t_{2g}$ manifold due to
additional trigonal(tetragonal) distortions $\Delta^{\text{\tiny
CF}}_{\text{\tiny tri.}}$($\Delta^{\text{\tiny CF}}_{\text{\tiny tet.}}$)
$\to$ 0.

However, the real materials mentioned above are far from these ideal limits
making the situation even more complex. Such complexities are inevitable when
a minor change in details of these interactions may have dramatic effects on
the macroscopic behavior of the material. For example, in iridates despite the
presence of additional $\Delta^{\text{\tiny CF}}_{\text{\tiny
tri.}}$($\Delta^{\text{\tiny CF}}_{\text{\tiny tet.}}$) distortions which are
responsible for mixing between $J_{\text{eff}}$ =$\frac{1}{2}$ and
$\frac{3}{2}$ states [27, 31], the large SOC of Ir 5$d$ orbitals still allow
$J_{\text{eff}}$$=\frac{1}{2}$ description of the magnetic properties.
However, same cannot be pre-assumed for a 4$d$ transition metal compound where
SOC strength is nearly half of its 5$d$ counterpart and $\Delta^{\text{\tiny
CF}}_{\text{\tiny tri.}}$($\Delta^{\text{\tiny CF}}_{\text{\tiny tet.}}$)
distortions of octahedra might be comparable to the SOC strength. This
inhibits any generic rule for behavior prediction of such materials and hence,
a case to case study is often required.

The scarcity of 4$d$ magnetic compounds with $J_{\text{eff}}=\frac{1}{2}$
behavior makes it even more difficult to obtain any comprehensive
understanding coming from experiments. To the best of our knowledge, the only
example of magnetic material in this category is $\alpha$-RuCl3 and has been
the subject of extensive theoretical and experimental investigations [32, 33,
34, 35, 36]. Other 4$d$ materials such as Li2RhO3, Sr2RhO4, and some
theoretically predicted Rh and Ir-based fluorides are either non-magnetic
(Li2RhO3 shows spin-glass behavior) or paramagnetic in nature [42, 43, 44]. In
the quest of new Kitaev candidates, Sr4RhO6 is another possible example of a
4$d$ oxide [45, 46]. Materials like Sr4RhO6 and some Ir-based double
perovskites [37, 38, 39, 40, 41]) with isolated metal-anion octahedra (as
shown in Fig. 1(b)) may possess an advantage over materials with edge-shared
geometry because the larger spatial separation between the magnetic ions in
the former can minimize the direct overlap of $d$ orbitals as compared to edge
shared geometry. This in turn may result in suppression of additional
$undesirable$ nearest-neighbor as well as farther neighbor Heisenberg-like
isotropic coupling. Sr4RhO6is believed to exhibit ideal cubic octahedral
environment on Rh-sites [46] in a centrosymmetric crystal structure. Such a
distinctive feature may lead to the realization of $pure$ $J_{\text{eff}}$
=1/2 states, a feature not realized in any of the previously mentioned Kitaev
candidate materials. Despite purportedly having such lucrative features with
the possibility of hosting rich physics, it is surprising to find scarcity of
theoretical study dedicated to this material and hence is the focus of our
study in this article.

In this study, using a combination of first-principles calculations and a
tight-binding model, we first show that contrary to the earlier belief [46],
the Rh-O6 octahedra in Sr4RhO6 is not perfect and the octahedral crystal field
at Rh-sites has additional trigonal-like distortions originating from the
influence of the extended environment of Sr atoms. Using the exact
diagonalization (ED) technique, we show that despite such a distortion, mixing
between $J_{\text{eff}}$ = 1/2-3/2 states is small and description of low-
energy space in terms of $J_{\text{eff}}$ = 1/2 states is still valid in this
material. Magnetic interaction among these pseudo-spins estimated using
second-order perturbation theory show highly bond-dependent anisotropic
behavior with additional diagonal/off-diagonal terms appearing alongside two
particularly noticeable features, antiferromagnetic Kitaev and Dzyaloshinskii-
Moriya (DMI) interactions on some of the first nearest-neighbor (1NN) Rh-Rh
bonds. We attribute the appearance of DMI to the local inversion symmetry
breaking due to the extended environment of Sr+2 ions. The second and third
nearest-neighbor interactions are found to be negligibly small. The
classically optimized magnetic ground state brings an antiferromagnetic
configuration which is energetically close to the previously proposed magnetic
structure. Spin wave spectra calculated using linear spin-wave theory is found
to be gaped throughout the Brillouin zone, consistent with underlying
frustrated magnetic interactions. The origin of various features of the
spectra is analyzed separately by examining the role of various magnetic
interaction terms in the spin Hamiltonian. In the end, we also show that the
replacement of Sr with Ca atoms can substantially enhance the magnetic
interactions and may be a better magnetic candidate in the category of 4$d$
oxides.

##  II Methods

Density-functional theory calculations have been performed using projector-
augmented wave method [47, 48], implemented within Vienna $ab$ $initio$
simulation package(VASP) [49]. The Perdew-Burke-Ernzerhof functional [50] is
used for the exchange-correlation functional within the GGA formalism. We
start with the experimental lattice parameters of trigonal crystal system of
Sr4RhO6 with centrosymmetric space group space group $R\bar{3}c$ (No. 167)
which are $|$a$|$ = $|$b$|$ = 9.740 Å, $|$c$|$ = 11.840 Å; $\alpha$ = $\beta$
= 90° and $\gamma$ = 120° [46]. Using plane wave cutoff energy 550 eV,
$2\times 2\times 4$ $\Gamma$-centered $k$-mesh and energy convergence criteria
of 10-5 eV, we optimize the lattice parameters with experimentally proposed
magnetic ground (accommodated within 24 Rh atoms $2\times 2\times 1$
supercell) considering SOC effect at the self-consistent level. A DFT+$U$
approach employing Liechtenstein [51] scheme with on-site Coulomb interaction
$U$ = 2.5 eV and exchange interaction $J_{\text{H}}$ = 0.9 eV was used. The
values of $U$ and $J_{\text{H}}$ parameters are consistent with the previous
study [46]. Optimized $a$ and $b$ lattice constants were found to be
overestimated by $\sim$ 3.1 % while $c$ remains the same. Since this change in
lattice constants is significant, we have used the optimized structure in
further calculations.

Non-spin polarised tight-binding (TB) Hamiltonian ($H_{\text{TB}}$) was
calculated by projecting onto all the five Rh-$d$ orbitals using the
Wannierization procedure [52] and is shown in Fig. 2(a). Crystal field matrix
on site $i$ ($\Delta^{\text{\tiny CF}}_{i}$) is extracted from the onsite part
of $H_{\text{TB}}$. We estimated the Coulomb matrix elements
$U_{ijkl}$($\omega$ = 0) within the constrained random phase approximation
(cRPA) by neglecting the screening effects for all the five Rh $d$ orbitals,
states corresponding to which are energetically well-separated form other
states [53, 54, 55]. This provides $U$ = 2.474 eV and $J_{\text{H}}$ = 0.106
eV which was further used in our multi-band Hubbard model.

##  III Results

###  III.1 Structural analysis and electronic properties

Figure 1: (a) $J_{\text{eff}}$ picture for a $d^{5}$ system arising from
octahedral crystal field ($\Delta^{\text{\tiny CF}}$) and spin-orbit coupling
($\lambda$). Additional splitting of $J_{\text{eff}}$ =3/2 states due to
trigonal like distortions ($\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$)
into two doublets $a_{1g}$ and $e_{g}^{\pi}$. (b) Side view of Sr4RhO6 crystal
structure. Spatially separated octahedrons are evident. Local $x$, $y$, and
$z$ axes on two of the octahedron are shown. $a$, $b$ and $c$ are the global
crystallographic axes. Two kinds of color-coded Rh-Sr bonds along with O-Rh-O
bond angles obtained after optimization of the crystal structure are shown.
(c) Extended local environment (including Sr atoms) around an Rh atom in
Sr4RhO6. Four types of Rh-Rh nearest neighbors A, B, C, and D-bonds with Rh-O
Octahedra on these bonds are also shown.

Under a large $\Delta^{\text{\tiny CF}}_{i}$, the low energy space Rh-4$d^{5}$
ions can be described by a single hole within the $t_{2g}$ manifold with
effective spin moment s = 1/2 and effective orbital angular moment L′ = 1. The
spin-orbit coupling then leads to an effective total angular momentum of
J${}_{\text{eff}}$ = s \- L′ resulting in doubly degenerate pseudospin-1/2
states forming low-energy space in this material. This is schematically shown
in Fig. 1(a). However, the lowering of cubic $O_{h}$ symmetry of octahedron
due additional $\Delta^{\text{\tiny CF}}_{\text{ \tiny
tri.}}$($\Delta^{\text{\tiny CF}}_{\text{\tiny tet.}}$) terms can invalidate
this picture. Hence it is important to first examine whether Rh-O6 octahedra
in Sr4RhO6 retains the $O_{h}$ symmetry as was proposed earlier in Ref [46].
In this experimental crystal structure, all the six Rh-O bond lengths are
$\sim$ 2.044 Å while O-Rh-O bond angles are quite close to the ideal 90° with
the largest deviation being 0.1°. However, full structural optimization with
the magnetic ground state in our DFT calculation brings substantial changes in
$a$ and $b$ lattice constants along with the changes in the local octahedral
environment. The optimization enhanced $a$ and $b$ lattice constants to 10.046
Å and also all the six Rh-O bond lengths elongated to 2.109 Å. The structural
optimization also alters the O-Rh-O bond angles to 92.28 and 87.72°(see Fig.
1(b)). Also, out of eight Sr neighbors in the extended environment of Rh
atoms, two “apical” Sr atoms along $c$ axis are at 2.952 Å distance while
other “non-apical” six are at 3.340 Å in the optimized structure. This is
shown in Fig. 1(b) (short Rh-Sr distances are along A-bonds and long ones are
along B/C/D-bonds). These two kinds of Rh-Sr distances were 2.960 and 3.238 Å
in the starting structure. Almost similar Rh-Rh 1NN distances $\sim$ 5.98/6.0
Å of all the 8 bonds before optimization has changed substantially now to
$\sim$ 5.9 and 6.13 Å for two A-bonds, and six B/C/D bonds respectively. Thus
optimization of structure results in substantial changes in Rh-O, non-apical
Rh-Sr, and overall Rh-Rh bond lengths and is consistent with enhancement of
$a$ and $b$ lattice constants.

In order to understand how these changes in crystal structure affect the CF,
we set up a TB model with $d$ orbital basis of $\psi^{\dagger}$ =
$[d^{\dagger}_{z^{2}}$, $d^{\dagger}_{x^{2}-y^{2}}$, $d^{\dagger}_{xz}$,
$d^{\dagger}_{yz}$, $d^{\dagger}_{xy}]$ using Wannierization procedure as
mentioned in methods section and the band structures obtained from both, DFT
and TB are shown in Fig 2(a). The CF matrix $\Delta^{\text{\tiny CF}}_{i}$
extracted from the onsite part of $H_{\text{TB}}$ is given in Eq. 1. Entries
in the matrix are in the unit of eV.

$\Delta^{\text{CF}}_{i}=\begin{bmatrix}[r]2.5385&-0.0034&0.0936&-0.1324&-0.0511\\\
-0.0034&2.5447&0.0033&-0.0111&-0.117\\\ 0.0936&0.0033&0.1102&0.0504&-0.0533\\\
-0.1324&-0.0111&0.0504&0.1094&0.0567\\\
-0.0511&-0.1170&-0.0533&0.0567&0.1179\\\ \end{bmatrix}$ (1)

By diagonalizing this matrix one can find that the $t_{2g}$-$e_{g}$ crystal
field splitting ($\Delta^{t_{2g}-e_{g}}_{i}$) is $\sim$ 2.630 eV while the
triply degenerate $t_{2g}$ splits into a $a_{1g}$ singlet and $e_{g}^{\pi}$
doublet by $\Delta^{\text{\tiny CF}}_{\text{\tiny tri}}$ $\sim$ 160 meV with
doublet being higher in energy than the singlet. The corresponding
eigenvectors are graphically represented in Fig. 2(b), where each column in
the 5 $\times$ 5 graph represents an eigenvector with row representing
absolute weight of individual orbitals. This representation clearly highlights
the nature of $\Delta^{\text{\tiny CF}}_{i}$ in Sr4RhO6 which has been
depicted in Fig. 1(a).

Figure 2: (a) Band structure plot from the $ab$ $initio$ and Wannier-based TB
model calculations considering all the Rh $d$ orbitals in the basis. (b)
Graphical representation of eigenvectors of matrix in Eq. 1. In column-wise
representation of eigenvectors each row represents absolute weight of
individual orbitals. Labeling of eigenstates is done in accordance of Fig.
1(a).

This particular form of $\Delta^{\text{\tiny CF}}_{i}$ can be understood as
follows. The shorter “apical” Sr-Rh bond passes through the center of two
triangular faces of Rh-O6 octahedra as shown in Fig. 1(b)-(c). This bond is
also one of the four three-fold rotational symmetry ($C_{3}$) axis of the
Rh-O6 octahedra. The electrostatic repulsion along these shorter bonds behaves
as compressing strain causing changes in Rh-O bond lengths and O-Rh-O bond
angles. This is analogous to the case of trigonal distortions where bond
distortions take place along one of the four $C_{3}$ axes of the octahedra.
Thus, in Sr4RhO6, an extended anisotropic environment of Sr atoms produces a
non-spherical crystalline potential responsible for additional
$\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ of Rh-O6 octahedra. The cubic
$O_{h}$ symmetry then lowers to $D_{3d}$ in this case.

Figure 3: $Ab$ $initio$ band structure plots near the Fermi level for the
cases of, (a) non-magnetic (NM), (b) SOC and (c) SOC + $U$ (on Rh $d$
orbitals). Red dashed line at 0 eV indicates the Fermi level. The fined dashed
box in (b) shows the energy window of where separation of $J_{\text{eff}}$
=1/2 bands from other $J_{\text{eff}}$ =3/2 bands takes place with inclusion
of SOC.

Distortions like $\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ tend to lower
the energy separation between $J_{\text{eff}}$ =1/2 and 3/2 states. This
brings a genuine concern about whether the strength of SOC in Sr4RhO6 is
sufficient enough to separate out these two states. To examine this point, we
calculated the $ab$ $initio$ band structures for three cases, (i) non-
magnetic, (ii) with SOC, and (iii) with SOC + $U$. SOC was included at the
self-consistent level in these calculations.

In Fig. 3, we only show the bands near the Fermi level which are dominantly
contributed by $t_{2g}$ orbitals, linear combinations of which form the
$J_{\text{eff}}$ =1/2 and 3/2 states. One can see that the inclusion of SOC
leads to substantial changes in the bands structure around Fermi level in Fig.
3(b) in comparison with non-magnetic case in Fig. 3(a). In particular, SOC
leads to separation of bands near -0.1 eV (red box in Fig. 3(b)) from the
other bands ($J_{\text{eff}}$ = 3/2 bands) near -0.1 eV. Inclusion of $U$ on
Re $d$ states further contributes to this band separation as shown in Fig.
3(c). Imposition of the magnetic ground state in band structure calculation
(not shown) fully opens the gap at the Fermi level making it insulating. This
is similar to the case of $\alpha$-RuCl3 [35] and hence $\lambda$ in Sr4RhO6
is expected to be similar of $\alpha$-RuCl3 i.e. $\lambda$ $\sim$ 140 meV
[36]. One can conclude that the electronic structure of Sr4RhO6 is the
combined efforts of $U$, SOC, and magnetism. Having examined the role of SOC,
one can further quantify the $J_{\text{eff}}$ =1/2 and 3/2 states admixture
due to $\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ by considering a multi-
band Hubbard model for an isolated Rh+4 ion. This is discussed in the next
section.

###  III.2 Onsite Hamiltonian and the atomic features

One way to estimate the extent of mixing between the $J_{\text{eff}}$ =1/2 and
3/2 states is by calculating the projection of a “pure” $J_{\text{eff}}$ =1/2
and 3/2 states for the case when $\Delta^{\text{\tiny CF}}_{\text{\tiny
tri.}}$ = 0 onto the “true” $J_{\text{eff}}$ =1/2 obtained with Eq. 1.These
states are atomic features and hence can be described in an isolated atom
limit. In this limit, a multiband Hubbard Hamiltonian at site, $i$ in the
five-orbital basis reads as,

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle
H_{\text{cf}}+H_{\text{soc}}+H_{\text{int}}$ (2) $\displaystyle=$
$\displaystyle\sum_{i,\sigma}\psi_{i\sigma}^{\dagger}{\Delta^{\text{CF}}_{i}}\psi_{i\sigma}+\sum_{i}\lambda\bm{L}_{i}\cdot\bm{s}_{i}$
$\displaystyle+$
$\displaystyle\frac{U}{2}\displaystyle\sum_{i,\alpha}n_{i\alpha\sigma}n_{i\alpha\sigma^{\prime}}+\frac{U^{\prime}}{2}\displaystyle\sum_{i,\alpha\neq\beta}n_{i\alpha}n_{i\beta}$
$\displaystyle-$
$\displaystyle\frac{J_{\text{H}}}{2}\displaystyle\sum_{i,\sigma,\sigma^{\prime},\alpha\neq\beta}\psi^{\dagger}_{i\alpha\sigma}\psi_{i\alpha\sigma^{\prime}}\psi^{\dagger}_{i\beta\sigma^{\prime}}\psi_{i\beta\sigma}$
$\displaystyle-$
$\displaystyle\frac{J^{\prime}}{2}\displaystyle\sum_{i,\sigma\neq\sigma^{\prime},\alpha\neq\beta}\psi^{\dagger}_{i\alpha\sigma}\psi_{i\beta\sigma^{\prime}}\psi^{\dagger}_{i\alpha\sigma^{\prime}}\psi_{i\beta\sigma}$

In above expression, $U$/$U^{\prime}$ are intraorbital/interorbital Hartree
energies; and $J_{\text{H}}$ and $J^{\prime}$ are Hund’s coupling and pair
hopping interaction, respectively. Rotational invariance in the isolated atom
limit dictates the relationships: $U^{\prime}$ = $U$ \- 2$J_{\text{H}}$ and
$J_{\text{H}}$ = $J^{\prime}$. We use $U$ = 2.474 eV and $J_{\text{H}}$ =
0.106 eV which are estimated from cRPA as mentioned in the methods section. We
consider $\lambda$ =140 meV which is similar to $\alpha$-RuCl3 [56]. However,
on a later stage we also show that $\lambda$ = 195 meV does not qualitatively
influence the magnetic interactions in Sr4RhO6. We diagonalize the above
Hamiltonian considering five electrons of Rh+4 ions which give a total of 252
eigenstates, the lowest two and the next four of which are the
$J_{\text{eff}}$ =1/2 states and $J_{\text{eff}}$ =3/2 states, respectively.

| $\braket{\phi^{\prime}_{\alpha}}{\phi_{\beta}}$
---|---
$\bra{\phi^{\prime}_{\alpha}}$ | $\ket{\phi_{1}}$ | $\ket{\phi_{2}}$
1 | 0.919 | 0.297
2 | 0.297 | 0.919
3 | 0.053 | 0.158
4 | 0.110 | 0.122
5 | 0.070 | 0.150
6 | 0.162 | 0.041
Table 1: Projections of $J_{\text{eff}}$ =1/2, 3/2 states obtained when
$\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ = 0, onto $J_{\text{eff}}$ =1/2
states with $true$ CF from Eq. 1.

For $\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ = 0, $t_{2g}$-$e_{g}$
splitting was fixed at 2.790 eV and all the off-diagonal matrix elements were
zeroed in Eq. 1. The lowest six states in this case are represented by
$\\{\phi^{\prime}_{\alpha}\\}$, $\alpha$ = 1, 6 while lowest two states
obtained using $true$ CF from Eq. 1 are labelled as $\\{\phi_{\beta}\\}$,
$\beta$ = 1-2. The projections $\braket{\phi^{\prime}_{\alpha}}{\phi_{\beta}}$
are listed in Table 1. From the table, since
$\braket{\phi^{\prime}_{\alpha}}{\phi_{\beta}}$ = 0.919 for $\alpha/\beta$ =
1, 2, one can conclude that the $J_{\text{eff}}$ =1/2 states retain their
major weight despite a substantial $\Delta^{\text{\tiny CF}}_{\text{\tiny
tri.}}$, validating applicability of $J_{\text{eff}}$ = 1/2 picture in
Sr4RhO6. The non-zero value of projections
$\braket{\phi^{\prime}_{\alpha}}{\phi_{\beta}}$ ($\sim$ 0.16) for $\alpha$ =
3-6, $\beta$ = 1-2 indicates a small admixture of $J_{\text{eff}}$ =1/2 and
3/2 states.

One of the quantities which can be measured from the resonant inelastic X-ray
scattering experiments are the single-point excitations represented by sharp
peaks in the scattering intensity in the relevant energy range. It can be a
direct probe for cubic symmetry lowering of the Rh-O6 octahedra in Sr4RhO6.
Theoretically, such a low-lying crystal field-assisted many-body excitations
bear a close resemblance to the eigenvalues obtained from diagonalization of
many-body Hamiltonian in Eq. 2. For Sr4RhO6, analysis of eigenvalue reveals
that the $J_{\text{eff}}$ = 3/2 states split into two doublets by $E_{2}$ =
0.133 eV (see Fig. 1(a)) which would otherwise be four-fold degenerate if
$\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ = 0. Energy separation of the
$J_{\text{eff}}$ = 1/2 doublet with the lower $J_{\text{eff}}$ = 3/2 doublet
is $E_{1}$ = 0.181 eV. It can also be observed that $E_{1}$ is $\sim$ 30 meV
smaller than the expected value of $\frac{3}{2}\lambda$ due to finite
$\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$. From the higher
$J_{\text{eff}}$ = 3/2 doublets, the next single ion excitation is at $\sim$
1.695 eV. From this point, a broad continuum of states with energy separations
of few meV in the window of $\sim$ 165 meV are found in our calculations.
Having investigated the electronic properties of Sr4RhO6, we now discuss its
magnetic properties in the next section.

###  III.3 Magnetism

We start by projecting the Hamiltonian in Eq. 2 to the pseudo-spins $J_{1/2}$
subspace and introduce hoping ($H_{\text{hop}}$) as perturbation. The hopping
amplitudes are extracted from $H_{\text{TB}}$ and are listed in Appendix
Appendix A: First neighbors Rh-Rh hopping amplitudes in Sr4RhO6 expressed in
the basis ($d^{\dagger}_{z^{2}}$, $d^{\dagger}_{x^{2}-y^{2}}$,
$d^{\dagger}_{xz}$, $d^{\dagger}_{yz}$, $d^{\dagger}_{xy}$).. In the limit
$U\gg t$, the second-order perturbation term brings,

$\displaystyle H^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{ij}\sum_{\alpha\beta\alpha^{\prime}\beta^{\prime}}\mathcal{H}(i,j)_{\alpha\beta\alpha^{\prime}\beta^{\prime}}|i\alpha,j\beta\rangle\,\langle
i\alpha^{\prime},j\beta^{\prime}|,$
$\displaystyle\mathcal{H}(i,j)_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$
$\displaystyle=$ $\displaystyle\sum_{kl}\sum_{\gamma\lambda}\frac{1}{\Delta
E}\langle i\alpha,j\beta|H_{\text{hop}}|k\gamma,l\lambda\rangle\,\langle
k\gamma,l\lambda|H_{\text{hop}}|i\alpha^{\prime},j\beta^{\prime}\rangle,$ (3)

where $1/\Delta
E=\frac{1}{2}[{1}/(E_{i\alpha}+E_{j\beta}-E_{k\lambda}-E_{l\gamma})$. Here,
$|i\alpha,j\beta\rangle$ and $|i\alpha^{\prime},j\beta^{\prime}\rangle$ are
two-site states made of $J_{1/2}$ doublets, and $|k\lambda,l\gamma\rangle$ are
two-site excited states with $d^{6}$ and $d^{4}$ configurations with Hilbert
space dimensions of 210 for both. $H_{\text{hop}}$ connects a two-site ground
state to these excited states. The eigenstates of isolated Rh ions with 4 and
6 $d$ electrons are obtained again by exact diagonalization.

One can represnt the pseudo-spins $J_{1/2}$ as
$S^{\mu}=\braket{i\alpha}{\bm{J^{\mu}_{i,\text{ \bf eff}}}}{i\beta}$ which are
the expectation values of pseudospin $J^{\mu}_{\text{eff}}$ operators with
$\mu=0,x,y,z$. Here, $J^{0}_{\text{\bf eff}}$ = $\mathds{1}_{2\times 2}$ is
the matrix representation of operator $J^{0}_{\text{\bf eff}}$. Using it, Eq.
(3) can be mapped to a spin Hamiltonian of the form,

$\displaystyle H_{\text{spin}}$ $\displaystyle=$ $\displaystyle
S_{i}^{\mu}\Gamma(i,j)^{\mu\nu}S_{j}^{\nu}$ $\displaystyle=$
$\displaystyle\Gamma(i,j)^{\mu\nu}\phi_{i\alpha}^{\dagger}S_{\alpha\alpha^{\prime}}^{\mu}\phi_{i\alpha^{\prime}}\phi_{j\beta}S_{\beta\beta^{\prime}}^{\nu}\phi_{j\beta}^{\dagger},$

In the above expression, summation over all repeated indexes is implied. The
map can be achieved by solving the linear equations,

$-S_{\alpha\alpha^{\prime}}^{\mu}S_{\beta\beta^{\prime}}^{\nu}\Gamma(i,j)^{\mu\nu}=\mathcal{H}(i,j)_{\alpha\beta\alpha^{\prime}\beta^{\prime}}$
(4)

Here, degeneracy of the Kramers doublet leads $\Gamma^{0\mu}$ = $\Gamma^{\mu
0}$ = 0. Thus, the most general form of exchange interaction matrix on an Rh-
Rh bond $l\in$ $(i,j)$ is defined as,

$\Gamma_{l}=\begin{pmatrix}J+\zeta&\eta+D&\eta^{\prime}-D^{\prime}\\\
\eta-D&J-\zeta&\eta^{\prime\prime}+D^{\prime\prime}\\\
\eta^{\prime}+D^{\prime}&\eta^{\prime\prime}-D^{\prime\prime}&J+K\end{pmatrix}$
(5)

In the above expression, $J$, $K$, and $\eta$/$\eta^{\prime}$ are the
Heisenberg, Kitaev and off-diagonal interaction terms between the
pseudospins-1/2, while $\zeta$ is the diagonal anisotropic term. DMI is
represented by ($D$, $D^{\prime}$, $D^{\prime\prime}$) vector.

| Bond-type
---|---
Term | A | B | C
$J$ | $0.291$ | $-1.062$ | $3.260$
$K$ | $0.040$ | $1.360$ | $0.292$
$\zeta$ | $0.000$ | $0.464$ | $-0.121$
$\eta$ | $0.020$ | $-0.867$ | $0.463$
$\eta^{\prime}$ | $0.000$ | $-1.802$ | $0.354$
$\eta^{\prime\prime}$ | $-0.020$ | $1.047$ | $-0.274$
$D$ | $0.000$ | $1.512$ | $0.000$
$D^{\prime}$ | $0.000$ | $0.830$ | $0.000$
$D^{\prime\prime}$ | $0.000$ | $-1.343$ | $0.000$
Table 2:  Estimated first neighbor (NN) Heisenberg $J$, Kitaev $K$ and
diagonal $\zeta$ and off-diagonal $\eta$, $\eta^{\prime}$,
$\eta^{\prime\prime}$ anisotropic terms for Sr4RhO6 given in meV. The second
nearest neighbor interactions were found to be negligibly small ($<$ 0.1 meV).
Parameters used are $U$ = 2.474 eV, $J_{\text{H}}$ = 0.106 eV and $\lambda$ =
140 meV.

The Rh atoms forms a body-centered cubic lattice in Sr4RhO6 and thus each Rh
atom has eight 1NNs. Based on the nature of magnetic interactions between
different 1NNs, we subdivide the Rh-Rh bonds into three distinct categories
which are indicated as A/B/C/D bonds in Fig. 1(c). Values of magnetic
interactions are listed in Table 2. For bond A and C, the $\Gamma_{l}$ matrix
acquire a more symmetric form since on these bonds $\zeta$ = $\eta$ =
$\eta^{\prime}$ = $\eta^{\prime\prime}$ = $D$ = $D^{\prime}$
=$D^{\prime\prime}$ = 0. However, the magnetic interactions on these two bonds
differ in their strengths. On B-bond, $\Gamma_{\text{B}}$ takes the general
form of Eq. 5 and $\Gamma_{\text{D}}$ can be obtained by simply taking the
transpose of $\Gamma_{\text{B}}$.

Several remarks are in order. First, one can see that the strength, as well as
signs of interactions, differ for different bonds. For example, for A and C
bonds $J$, $\eta$, and $\eta^{\prime}$ are antiferromagnetic while for B-bond
they are ferromagnetic and the antiferro Kitaev coupling is stronger on B-bond
than the others. We emphasize that the antiferromagnetic Kitaev coupling in
Sr4RhO6, although smaller, distinctly differs from the previous reports on
iridates and $\alpha$-RuCl3 [36, 56]. Second, quite interestingly, DMI appears
on B and D bonds in the centrosymmetric structure of Sr4RhO6. However, $D$,
$D^{\prime}$, and $D^{\prime\prime}$ have opposite signs on these two bonds.
We attribute this to the local inversion symmetry breaking due to anisotropic
crystalline potential produced by Sr atoms in the extended environment around
Rh atoms. In the bipartite bcc lattice of Rh atoms, the first neighbors of an
Rh atom are all symmetry inequivalent Rh atoms and are not related by
inversion. For A-bond, the apical Sr atoms situated at the center of the bonds
act as inversion center (see Fig. 1(c)) and hence the hopping matrix on this
bond in appendix Appendix A: First neighbors Rh-Rh hopping amplitudes in
Sr4RhO6 expressed in the basis ($d^{\dagger}_{z^{2}}$,
$d^{\dagger}_{x^{2}-y^{2}}$, $d^{\dagger}_{xz}$, $d^{\dagger}_{yz}$,
$d^{\dagger}_{xy}$). is symmetric, i.e. $T_{ij}^{t}=T_{ij}$, resulting in the
suppression of DMI. However, this arrangement is not available for B and D
bonds leading to the form $T_{ij}^{t}\neq T_{ij}$ of hopping matrix, resulting
in the appearance of finite DMI on these bonds. Third, one may think that the
Sr+2 ions on A-bond may mediate superexchange interaction between Rh atoms
through their $s$ orbitals. However, on the contrary, we find highly
suppressed interactions on this bond suggesting a destructive role of the
anisotropic crystalline potential of the Sr+2 on magnetic interactions.
Fourth, we found large off-diagonal terms on some of the Rh-Rh bonds. This is
similar to the case of iridates and $\alpha$-RuCl3 [56] resulting from
substantial $\Delta^{\text{\tiny CF}}_{\text{\tiny tri.}}$ distortions present
in all these materials. Based on the two particularly noticeable features in
the first two points, $viz-a-viz$ antiferromagnetic Kitaev terms and
appearence of DMI, one may consider Sr4RhO6 a special 4$d$ magnetic material.

Figure 4: (a) Experimentally proposed magnetic ground state of Sr4RhO6. (b)
Classical ground state obtained from optimization of classical ground state
using exchange interactions of Table 2. Color coded spin orientation of only
Rh lattice is shown here.

Increasing the magnitude of SOC strength $\lambda$ in our model does not
change the interactions at a qualitative level. For $\lambda$ = 195 meV, we
find that on A-bond, the magnetic interactions remain almost similar. However,
on C-bond, the magnitude of $J$ is reduced by $\sim$ 13% while other
interactions reduced to 48 % of their values from $\lambda$ = 140 meV case. On
B-bond, while $J$, $K$, and $\zeta$, remain more or less the same, the
magnitude of all other terms is reduced by 24 % with the increase of
$\lambda$. The sign of all the magnetic terms remains the same. The findings
of this analysis are consistent with the previous study on $\alpha$-RuCl3
[36]. Further analysis in coming sections is based on $\lambda$ = 140 meV
case, which is expected to be closer to the real value in Sr4RhO6.

Magnetic interaction of Table 2 are used to optimize the classical magnetic
state using SpinW package [57]. The obtained magnetic ground state,
represented by ordering vector $\sim$(1.0 0.5 0), is shown in Fig. 4(b) along
with the experimentally proposed one in Fig. 4(a). The antiferromagnetic state
obtained in our calculations successfully captures most of the experimental
features. In the experimental magnetic structure, the spin arrangement on Rh-
Rh bonds(Fig. 1(c)), A and B are antiferromagnetic while on C and D it is
ferromagnetic. Optimized magnetic state in Fig. 4(b) from our calculations
retains antiferromagnetic coupling on A and ferromagnetic coupling at D bonds.
However, this configuration differs from the one shown in Fig. 4(a) on bonds B
and C where the spin arrangement in the two cases are just opposite to each
other i.e. on B-bond the coupling is ferro while on C-bond it is
antiferromagnetic in our optimized structure. Swapping the interactions at
bonds B and C does not bring the experimentally observed ground state
indicating a joint meticulous effort of all the magnetic interactions to bring
the ground state. We find a slight deviation of magnetic moments from the $ac$
plane mainly due to the presence of off-diagonal terms like
$\eta$/$\eta^{\prime}$/$\eta^{\prime\prime}$ and DMI on some of the Rh-Rh
bonds. This is consistent with the experimental finding of small tilting from
$c$ axis [46]. Our optimized magnetic configuration is energetically close to
the experimentally proposed one with the former stabilized by 1.552 meV/spin.
The second and third neighbor magnetic interactions are found to be negligibly
small in Sr4RhO6 and do not bring any distinguishable change in the
optimization of the magnetic ground state. Thus we ignore them in further
calculations of spin-wave spectra.

Here, we would like to comment on the scale of magnetic ordering temperature
of a material depends on various parameters like the strength of exchange
interactions, number of neighbors, their corresponding exchange contributions,
and spatial dimensions of the magnetic lattice. Although, the magnetic lattice
of Rh atoms in Sr4RhO6 form a three-dimensional bulk structure with eight
first magnetic neighbors, the strongly frustrated anisotropic nature of bond-
dependent magnetic interactions might be the reason behind its experimentally
observed low $T_{\text{N}}$ of $\sim$ 7.5 K.

###  III.4 Spin-wave spectra

We further use the magnetic interactions listed in Table 2 in linear spin-wave
theory to obtain the spin-wave spectra using SpinW package [57]. Obtained
spectra along various reciprocal space directions is shown in Fig. 5.

Figure 5: Spin wave spectra of Sr4RhO6 obtained within linear spin wave theory
considering magnetic interactions of Table 2.

Several points are to be noted about the spectra. First, one can see that the
spectra have gaped along all directions in reciprocal space with a Goldstone
gap of $\sim$ 2 meV. This feature of spin-wave spectra may be caused by the
breakdown of $SU(2)$ symmetry of the isotropic Heisenberg Hamiltonian. Such a
symmetry breaking can be a result of additional Ising like Kitaev terms and/or
diagonal/off-diagonal anisotropic terms like $\zeta$, $\eta$, $\eta^{\prime}$
and $\eta^{\prime\prime}$. Second, one branch $\sim$ 8 meV in the spectra
appears to be dispersion-less. It is separated from the dispersing branch by
$\sim$ 0.5 meV. Such a feature has previously been observed from the inelastic
neutron scattering experiments on some of the cobaltates [16], pertinent
material candidates for Kitaev physics [23]. Third, it can observe that the
spin-wave spectra near ${\bf\Gamma}$ point is quadratic in nature. This is in
contradiction to the expected linear dispersion of spin-wave dispersion for an
antiferromagnetic ground state.

Figure 6: Breakdown of spin-wave spectra shown in Fig. 3 to individual
contributions of combination of various magnetic interactions. Spectra from,
(a) $J+K$ terms, (b) only diagonal anisotropic term $\zeta$, (c) $J+K+\zeta$
terms and (d) $D+D^{\prime}+D"+\eta+\eta^{\prime}+\eta^{\prime\prime}$ terms
from Table 2.

In order to investigate the origin of previously mentioned features of spin-
wave spectra of Sr4RhO6, we break it down to the contribution of either
individual or a specific combination of magnetic interactions. The plots are
shown in Fig. 6. Considering $J$ and $K$ terms together, we immediately obtain
both branches with a lower branch, at ${\bf\Gamma}$, showing the linear
dispersion behavior of an antiferromagnet. The plot is shown in Fig. 6(a).
However, the spectra are barely gaped in this case due to dominant $J$ over
$K$ and which is also responsible for the dispersion width of $\sim$ 5.5 meV
of the lower branch. Consideration of $K$-only term in the Hamiltonian
produces a completely flat branch at $\sim$ 4 meV (not shown) consistent with
the previous theoretical study on Kitev model [58]. $\zeta$-only term indeed
causes the gap opening along with deviation towards a quadratic dispersion at
${\bf\Gamma}$ of the lower branch and as shown in Fig. 6(b). However, the
energy scale, in this case, is smaller than that of the original spectra in
Fig. 5. A combination of $J+K+\zeta$ (Fig. 6(c)) reproduces some of the
features in the more or less similar spectral windows as that of the original
spectra. However, the dispersion width and nature of the lower branch, in this
case, are inconsistent with the original one in Fig 5. Additionally, near
${\bf\Gamma}$, dispersion of the lower branch appears to be further deviating
from quadratic to higher powers of $\bm{k}$. The terms
$D+D^{\prime}+D^{\prime\prime}+\eta+\eta^{\prime}+\eta^{\prime\prime}$
produces similar but relatively flatter branches than the $\zeta$ terms and is
shown Fig. 6(d). The spectral energy window, in this case, is similar to that
of $J+K+\zeta$ term. Thus one can say conclusively that the dominant off-
diagonal terms are mainly responsible for the gap in spin-wave spectra of
Sr4RhO6 while the diagonal anisotropic term decides the nature of dispersion
near ${\bf\Gamma}$ point in spin-wave spectra of Sr4RhO6. The overall spectra
which resemble a typical magnetic system with strong frustration is a joint
effort of all the terms of magnetic Hamiltonian.

##  IV Case of Ca4RhO6

The crystal structure of Ca4RhO6 by replacing Sr with Ca atoms in Sr4RhO6 and
performing a full structural optimization retains the original trigonal
symmetry of crystal structure. However, $|\bf{a}|$=$|\bf{b}|$ reduces by
$\sim$ 4 % while $|\bf{c}|$ reduces by $\sim$ 5.1 %. This reduction is in
accordance with the smaller +2 ionic radius of Ca atoms when compared to Sr
atoms. We estimate the magnetic interactions in Ca4RhO6 in a similar fashion
as done for Sr4RhO6 and obtained values are listed in Table 3 below.

| Bond-type
---|---
Term | A | B | C
$J$ | $0.447$ | $-1.944$ | $5.119$
$K$ | $0.038$ | $3.104$ | $0.460$
$\zeta$ | $0.000$ | $0.483$ | $-0.180$
$\eta$ | $0.052$ | $-1.362$ | $0.694$
$\eta^{\prime}$ | $0.000$ | $-2.832$ | $0.560$
$\eta^{\prime\prime}$ | $-0.040$ | $1.989$ | $-0.460$
$D$ | $0.000$ | $2.911$ | $0.000$
$D^{\prime}$ | $0.000$ | $1.565$ | $0.000$
$D^{\prime\prime}$ | $0.000$ | $-2.176$ | $0.000$
Table 3:  Estimated first neighbor (NN) Heisenberg $J$, Kitaev $K$ and
diagonal $\zeta$ and off-diagonal $\eta$, $\eta^{\prime}$ and
$\eta^{\prime\prime}$ anisotropic terms for Ca4RhO6 given in meV. Parameters
used are $U$ = 2.474 eV, $J_{\text{H}}$ = 0.106 eV and $\lambda$ = 140 meV.

As one can see from the table, most of the magnetic interactions are
substantially enhanced when compared to their Sr4RhO6 counterparts, except on
the A-bond. This is due to enhanced hopping amplitudes among $d$ orbitals of
Rh atoms. Such an enhancement is consistent with the reduced volume in the
current case due to the reduction of lattice parameters. In Ca4RhO6, the Rh-Rh
1NN bondlengths were found to be substantially smaller than that of Sr4RhO6
and are $\sim$ 5.62 (for A-bonds) and 5.71 Å (other six B/C/D bonds),
resulting in enhanced hopping amplitudes of Rh-$d$ orbitals. Thus, Ca4RhO6
maybe a better material candidate belonging to the rare class of
4$d$-transition metal-based magnetic compound and calls for an experimental
exploration.

##  V Conclusion

In the quest for new Kitaev candidates, in this work, we have investigated the
electronic and magnetic properties of Sr4RhO6. Through $ab$ $initio$
calculations and a TB model, we show the lowering of cubic symmetry of Rh-O6
octahedra due to additional trigonal-like distortions which are in
contradiction to the previous experimental proposal. Using the exact
diagonalization technique, we show that despite such a distortion, electronic
and magnetic properties of Sr4RhO6 can be well described with the pseudo-spin
1/2 framework. The magnetic interactions between these pseudo-spins were found
to be highly bond-dependent anisotropic in nature. We found two particularly
noticeable features of the 1NN magnetic interactions in Sr4RhO6 which are,
appearance of antiferromagnetic Kitaev term and DMI. This may place Sr4RhO6 in
a distinct class of materials as previusly proposed Kitaev candidates shown to
have ferromagnetic Kitaev couplings and DMI appears on the 2nd neighbor bonds
[56]. The analysis of spin-wave spectra obtained employing linear spin-wave
theory considering these interactions reveals the crucial role of diagonal and
off-diagonal magnetic interactions in producing a gaped spectrum of Sr4RhO6.
In the end, we also propose Ca4RhO6 in the 4$d$ category of transition metal
compounds as a better magnetic candidate with larger magnetic interaction than
Sr4RhO6. Our theoretical study provides deeper insights of structural,
electronic and magnetic properties of these compounds and calls for further
experimental investigations of these compounds.

## Appendix A: First neighbors Rh-Rh hopping amplitudes in Sr4RhO6 expressed
in the basis ($d^{\dagger}_{z^{2}}$, $d^{\dagger}_{x^{2}-y^{2}}$,
$d^{\dagger}_{xz}$, $d^{\dagger}_{yz}$, $d^{\dagger}_{xy}$).

A-bond | B-bond | C-bond
---|---|---
$\begin{pmatrix}[r]-0.0251&-0.0015&0.0144&-0.0042&0.0217\\\ -0.0015&-0.0288&0.0240&0.0182&-0.0319\\\ 0.0144&0.0240&-0.0369&0.0131&0.0263\\\ -0.0042&0.0182&0.0131&0.0270&-0.0094\\\ 0.0217&-0.0319&0.0263&-0.0094&-0.0199\end{pmatrix}$ | $\begin{pmatrix}[r]0.0245&-0.0302&0.0517&0.0006&0.0141\\\ -0.0334&-0.0100&0.0009&-0.0077&-0.0192\\\ -0.0791&0.0633&0.0175&-0.0004&0.0087\\\ 0.0368&-0.0626&0.0285&-0.0040&0.0339\\\ 0.0476&0.0256&0.0130&-0.0294&0.0033\ \end{pmatrix}$ | $\begin{pmatrix}[r]-0.0083&-0.0001&-0.0217&0.0134&-0.0419\\\ -0.0001&-0.0173&-0.0057&0.0618&0.0128\\\ -0.0217&-0.0057&-0.0194&0.0233&0.0103\\\ 0.0134&0.0618&0.0233&-0.0245&0.0622\\\ -0.0419&0.0128&0.0103&0.0622&-0.0684\\\ \end{pmatrix}$

Table A-1: First neighbor Rh-Rh hopping amplitudes on different types of bonds
shown in Fig. 1(c)

.

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# How to delay death and look further into the future if you fall into a black
hole

A.V. Toporensky  S.B. Popov

###### Abstract

In this note, we present a pedagogical illustration of peculiar properties of
motion in the vicinity and inside black holes. We discuss how a momentary
impulse can modify the lifetime of an object radially falling into a
Schwarzschild black hole down to singularity. The well known upper limit for a
proper time spent within a horizon, in fact, requires an infinitely powerful
kick. We calculate the proper time interval (perceived as personal lifetime of
a falling observer) till the contact with the singularity, as well as the time
interval in the Lemaître frame (which reflects how far into the future of the
outer world a falling observer can look), for different values of the kick
received by the falling body. We discuss the ideal strategy to increase both
time intervals by the engine with a finite power. This example is suitable for
university seminars for undergraduate students specializing in General
Relativity and related astrophysical subjects.

###### keywords:

general relativity, black holes

Dr. Alexei Toporensky is working at the Sternberg Astronomical Institute and
also is giving seminars at the HSE University, Moscow. His main research
interests are in cosmology of the early Universe and theories of gravity.

Prof. Sergei Popov is working as the Sternberg Astronomical Institute and
lecturing at the Lomonosov Moscow State University. Main scientific interests
are related to evolution of compact objects, especially — neutron stars. He
also actively participates in popular science activity and public outreach.

GENERAL ARTICLE

##  1 Introduction

It is well known that even simple processes involving black holes (BHs) can
produce apparent paradoxes. Many of them are discussed in the literature,
others still wait for a detailed explanation. Analysis of such interesting
problems which involve non-trivial aspects of General relativity (GR) effects
can be very useful in a pedagogical practice. Bright examples of interesting
phenomena with clear physical explanation can help students to understand BH
properties and motivate them for further studies of GR.

In this note we apply and develop some results from [1] to illustrate how a
fall time into a Schwarzschild BH (till the singularity is reached) is
modified by a momentary kick. We used this example on seminars for masters-
level students at the Lomonosov Moscow State University and HSE University.
Discussions during these seminars demonstrated that the chosen topic helps to
shed light on important aspects which are not well-understood by many
students.

The task is to derive and discuss necessary conditions to make the fall time
as long as possible. For a Schwarzschild black hole the ideal strategy for
maximizing the proper time can be easily derived from equations of motion. The
metric in static coordinates $(t,r,\phi,\theta)$ has the form (we set the
units so that $c=1$)

$ds^{2}=-fdt^{2}+\frac{dr^{2}}{f}+r^{2}d\omega^{2}\text{,}$ (1)

where $f=1-r_{g}/r$, $r_{g}$ is the Schwarzschild radius $(r_{g}=2GM)$ and the
angular part of the metric is
$d\omega^{2}=d\theta^{2}+d\phi^{2}\sin^{2}\theta$.

We write equations of motion for a particle in the space-time defined by eq.
(1) in such a form that the proper time $\tau$ is present explicitly (the
angular coordinates are chosen so that the motion to occur within the plane
$\theta=\frac{\mathrm{\pi}}{2}$). This form of the equations of motion can be
obtained by using the existence of two integrals of motion: energy $E$ (since
the metric is static) and angular momentum $L$ (since the metric does not
depend on $\phi$), see [1] for details.

$\dot{r}=\pm Z\text{,}$ (2) $\dot{t}=\frac{\varepsilon}{f}\text{,}$ (3)
$m\dot{\phi}=\frac{L}{r^{2}}\text{.}$ (4)

In the equations above a dot denotes differentiation with respect to the
proper time $\tau$. We denote $\varepsilon=E/m$ and

$Z=\sqrt{\varepsilon^{2}-f\left(1+\frac{L^{2}}{m^{2}r^{2}}\right)}\text{.}$
(5)

The $`+^{\prime}$ sign in eq. (2) indicates outward motion, while the
$`-^{\prime}$ sign corresponds to inward motion. Since we are interested in a
motion under horizon we consider only the latter possibility in this paper.

From eq. (2), it follows that a proper time required for a travel from the
horizon $r_{g}$ to $r_{1}<r_{g}$ is equal to

$\tau=\int_{r_{1}}^{r_{g}}\frac{dr}{Z}=-\int_{r_{g}}^{r_{1}}\frac{dr}{\sqrt{\varepsilon^{2}-f\left(1+\frac{L^{2}}{m^{2}r^{2}}\right)}}.$
(6)

For an object which is radially free falling into a black hole with zero
velocity at infinity we have $\varepsilon=1$ and $L=0$, so the free-fall time
from the Schwarzschild radius $r_{g}$ to singularity is:

$\tau_{\text{ff}}=-\int_{r_{g}}^{0}\frac{dr}{\sqrt{r_{g}/r}}=\frac{2}{3}r_{g}.$
(7)

Is it possible to increase this time for an object equipped with a rocket
engine? The known answer is: ‘yes, but not much’. Since under the horizon both
integrals of motion enter equations in the quadratic form with positive sign,
the integral in eq. (6) has its maximum value for $\varepsilon=0$, $L=0$. This
maximum value is equal to $(\mathrm{\pi}/2)r_{g}$ for $r_{1}=0$. So, the best
strategy for maximizing the proper time under a horizon is to reach this
optimal trajectory with two integrals of motion being equal to zero. Note,
that this trajectory is a geodesic one, since the motion is two-dimensional
and the two above mentioned integrals of motion represent a full set of
integrals fixing a particular geodesic.

The fact that the life time of an object with a jet engine is maximal at a
geodesic trajectory (i.e., at a trajectory which requires the engine to be
turned off after the ideal trajectory is reached) is in some sense a counter-
intuitive statement. Sometimes it is claimed that this result easily follows
from the fact that the interval $ds$ along a curve is just a proper time
$d\tau$ of an observer following this curve, so as a geodesic maximises $s$,
it also maximises $\tau$. However, as it have been already discussed in [2]
this is correct only if we consider a motion between two fixed space-time
points. Only under this condition a geodesic is unique and maximises $\tau$.
However, a singularity is not a space-time point! Its appearance differs for
different coordinate systems, for the Gullstrand-Painlevé coordinates which we
use below it is simply the line $r=0$. So that, we can not apply the above
argument directly. Indeed, both $\varepsilon=1$ and $\varepsilon=0$
trajectories are geodesics, but the latter works obviously better.

However, we will see soon that if the problem is considered in a bit more
realistic situation — with an engine of a finite power, — the final answer
appears to be closer to intuitive feeling.

##  2 Maximization of the proper fall time using a single-pulse engine

We consider a free-fall which is perturbed by a single engine thrust. It is
assumed that this event momentary gives a peculiar velocity $V_{p}$ to the
falling body. Afterwards, a free-fall continues till the object reaches the
singularity.

Velocity $V_{p}$ can be illustrated in the following simple way. The ship with
an engine is free falling and just near it along the whole trajectory till the
engine is turned on there is another free falling object. $V_{p}$ is their
relative velocity immediately after the momentary engine thrust.

The thrust can be given just once. We consider only situations when the
thrusting happens below the horizon ( for a more realistic modelling of a
rocket behaviour near a BH horizon see, for example, [3]). The task is to
figure out the best strategy to maximize the time (in the falling body frame)
till the singularity is reached for a given $V_{p}$. We assume $\varepsilon=1$
which is approximately true for an object with its kinetic energy being small
in comparison with $mc^{2}$ at a large distance from the BH. We also assume a
radial fall.

Kinematics of the motion with respect to the frame with $\varepsilon=1$, $L=0$
(usually called the Lemaître frame) have been considered recently in detail
[4, 5]. The natural coordinate system associated with such a fall is the
Lemaître one:

$ds^{2}=-d\tilde{t}^{2}+(r_{g}/r)d\rho^{2}+r^{2}(\rho,\tilde{t})d\omega^{2},$
(8)

with timelike coordinate $\tilde{t}$ and spacelike coordinate $\rho$, so that
the metric in these coordinates (in contrast to static coordinates $t$ and
$r$) is regular at a horizon and everywhere else except for the singular point
$r=0$. The Lemaître time is defined as $d\tilde{t}=dt+dr\sqrt{1-f}/f$ and
approximately coincides with the static time $t$ in the large distance limit
where $f\sim 1$. On the contrary, its behavior is very different for small
$r$, in particular it is finite at a horizon and inside a horizon. The former
static coordinate $r$ is now a function which for the Schwarzschild metric is
known to be $r=r_{g}^{1/3}((3/2)(\rho-\tilde{t}))^{2/3}$. This system is a
synchronous one, and the coordinate $\rho$ is constant during the free fall
for particles with $\varepsilon=1$ and $L=0$. However, sometimes it is better
to keep the coordinate $r$ and use the metric in the following form
(Gullstrand-Painlevé metric):

$ds^{2}=-d\tilde{t}^{2}+(dr+vd\tilde{t})^{2}+r^{2}d\omega^{2},$ (9)

with $v=\sqrt{1-f}$. This form of metric is not so common in pedagogical
literature, though the famous textbook [6] is based mostly on it, where it is
called as a ‘rain frame’. The price of using $r$ is the appearance of the off-
diagonal term. The coefficient of this term is easily recognizable as a free
fall velocity with respect to the stationary frame $v=\sqrt{r_{g}/r}$. Unlike
a coordinate velocity, the velocity with respect to a frame has a direct
physical meaning since it is by definition the velocity which is measured by a
local observer belonging to the considered frame. In particular, this velocity
is always subluminal. Under a horizon a stationary frame does not exist
anymore, so $v$ looses its direct physical meaning becoming superluminal.
Note, that $v$ characterises the ‘rain frame’ itself. A velocity of a particle
with respect to this frame will be introduced in the next paragraph.

The metric (9) has rather interesting properties. We mention here that the
proper distance between two points at the same radius measured at $\tilde{t}=$
const hypersurface is simply the difference in $r$ coordinate [7]. Moreover,
$\tilde{t}=$ const sections are flat. This property is rather useful when we
try to visualize the structure of a BH demonstrating together regions outside
and inside the horizon in a single picture. What is also useful, is that the
rate of change of the radial coordinate $r$ of any object with respect to the
Lemaître time $\tilde{t}$ can be decomposed as

$\frac{dr}{d\tilde{t}}=v-v_{p},$ (10)

where $v_{p}$ is the velocity of an object with respect to the Lemaître frame.
So that, the second term in the right hand side has a local meaning, while two
other terms do not. It is important, that this relation (which resembles the
Galilean summation rule for classical velocities) is valid independently of
the values of $v$ and $v_{p}$ everywhere, even under the horizon where $v$
exceeds the speed of light. Since the left hand side of this equation is a
coordinate (i.e., not physical!) velocity, this does not contradict Special
Relativity. The same situation is known for cosmology where the velocity of a
distant object (which is a non-local entity, and, so, is not bounded from
above) is decomposed into a sum of the Hubble and peculiar velocities (see,
for example, an excellent discussion in [8]). To avoid confusion, we repeat
that in our notation the value $V_{p}$ indicates the velocity caused by the
engine, and, so, represents its power capability (i.e., it is a constant in a
given set of conditions), while $v_{p}$ indicates a changing with time
velocity of any object with respect to the Lemaître frame and so can be
considered as a variable peculiar velocity.

We need also to take into account the fact that for a radially falling object
with the velocity $v_{p}$ with respect to the Lemaître frame its integral of
motion can be written as:

$\varepsilon=\frac{1-vv_{p}}{\sqrt{1-v_{p}^{2}}}.$ (11)

The equation above also determines the integral of motion $\varepsilon$ of the
spaceship after the engine have been used. Both $v$ and $v_{p}$ should be
taken at the point of the thrust. Since we assume that initially the spaceship
have followed trajectory with $\varepsilon=1$, being at rest with respect to
the Lemaître frame, the value $v_{p}$ immediately after the thrust equals
$V_{p}$.

Now we return to the question of the proper time. We know that the proper time
spent inside the horizon is maximized for $\varepsilon=0$. However, we see
from the above equation that to reach such a trajectory is not an easy task!
It can be done only well inside a horizon since $\varepsilon=0$ corresponds to
$v_{p}=1/v$, so that $r$ should be smaller than $r_{g}V_{p}^{2}$ to accomplish
this maneuver. In particular, it can not be made exactly at a horizon since it
requires an infinitely powerful engine ($\epsilon=0$ corresponds to $v_{p}=1$
at a horizon), so that the well known upper limit for a proper time inside a
horizon equal to $(\mathrm{\pi}/2)r_{g}$ is an unreachable limit, indeed.

The natural goal for the captain of a spaceship would be to maximize the
proper time since the fate of the ship is understood. The captain can decide
to wait till $r_{c}=r_{g}V_{p}^{2}$ is reached and then switch to the
$\epsilon=0$ trajectory. The price needed to be paid is the time interval from
the initial point (which we assume to be at the horizon) till the critical
point which located inside the horizon. During this period the ship is falling
along the trajectory with $\epsilon=1$, i.e. far from the optimal one.
Instead, the captain could decide to turn the engine on immediately, reaching
the minimal possible value of $\varepsilon$ and then to fall along this still
not optimized trajectory. Finally, the jet firing can be executed at some
$r_{\mathrm{on}}$ in between $r_{g}$ and $r_{g}V_{p}^{2}$. To understand the
best strategy in this setting we need to make corresponding numerical
calculations.

Figure 1: Falling from the horizon. A radial coordinate at the horizontal axis
corresponds to the point where the engine is turned on. The vertical axis
gives the time of falling from the horizon to the center of the BH. Different
curves correspond to different values of $V_{p}$.

We start calculations at the horizon. Then the time of the fall is written as
a sum of two integrals:

$\tau=-\left(\int_{r_{0}}^{r_{\text{on}}}\frac{dr}{\sqrt{r_{g}/r}}+\int_{r_{\text{on}}}^{0}\frac{dr}{\sqrt{r_{g}/r+(\varepsilon^{2}-1)}}\right).$
(12)

In this exercise we neglect tidal effects which would cause an actual death of
an observer before reaching the singularity (for stellar mass BHs even before
horizon crossing!). So, to be realistic we think about supermassive BHs where
tidal effects close to the horizon are negligible. Of course, tidal distortion
is always strong near the singularity, but it is easy to demonstrate, see e.g.
[6], that effects are terribly strong just for a fraction of second, which is
much shorter than the considered time of a free fall into a supermassive BH.
Thus, we safely integrate down to $r=0$.

We can see that the earlier the engine is activated — the longer proper time
is reached (see Fig. 1). So that, it is better not to wait until the optimal
trajectory becomes available, but to react to the situation immediately. If
the spaceship captain recognizes the bad luck before the point
$r=r_{g}V_{p}^{2}$, then the best strategy (i.e., switching on the engine for
the full power as early as possible) can be considered as matching our
‘general’ intuition. Only if the critical point is already passed, the captain
should deliberately switch off the engine at the trajectory with
$\varepsilon=0$ despite some fuel is still unused.

In principle, the curves in Fig. 1 can be prolonged even further to the left,
i.e. to the region which corresponds to a free fall from outside of the
horizon. Our calculations show that curves there are still monotonic: the
earlier is the kick given — the larger is the proper time of a free fall.
However, a radial kick outside the horizon is not the best option since the
angular momentum enters eq. (12) with the negative sign for $r>r_{g}$ — so, it
is better to deflect the trajectory than to slow down the free fall. That is
why we do not show this part of curves in our Fig. 1.

##  3 Maximization of the Lemaître time

Though maximization of the proper time has its obvious motivation, this is not
the unique reasonable strategy for a falling observer. The other option is to
maximize the Lemaître time. The reason for this strategy is to observe as much
as possible events in the ‘outer’ world. Since the Lemaître time coincides
with the static time $t$ for infinitely large $r$, the larger is the Lemaître
time needed to reach the singularity — the larger part of the history of the
Universe surrounding the black hole in question can be witnessed.

Properties of the Lemaître free fall time are quite different from those
considered in the previous section. To see this we can calculate the interval
of $\tilde{t}$ for a radial trajectory with $\varepsilon=0$. It is possible to
obtain an analytical result using the fact that at such trajectory
$v_{p}=-1/v$. Then the integral

$\tilde{t}=\int^{r_{1}}_{0}\frac{dr}{v-v_{p}}$ (13)

which is giving us the free fall time from $r=r_{1}$ to singularity, can be
expressed through $v$ only. After substitution $v=\sqrt{r_{g}/r}$ we get:

$\tilde{t}=r_{g}\ln\left(\frac{{\sqrt{r_{g}}+\sqrt{r_{1}}}}{{\sqrt{r_{g}}-\sqrt{r_{1}}}}\right)-2\sqrt{r_{g}r_{1}}.$
(14)

The equation evidently diverges if $r_{1}\to r_{g}$ — there is no upper limit
for the Lemaître free fall time! Again, this infinite limit is unreachable
since it is impossible to switch to the trajectory with $\varepsilon=0$
exactly at the horizon. Note, however, that the ability to see a very remote
future is limited only by the power of the observer’s engine.

We can stop here for a moment and remind the reader that the ability to see
infinitely remote future during a free fall into a black hole is one of the
most popular misconceptions in the black hole physics which is often
considered in teaching of GR (see, for example, [9]). However, while crossing
a horizon along a geodesics can not help to witness a remote future, the
powerful engine ignited at a horizon can help! An infinitely remote future
still can not be seen, but the more power engine is available — the longer is
the period of history of the outer Universe which observed during the fall.

Figure 2:  Fall time from the horizon in the Lemaitre frame. Different curves
correspond to different values of $V_{p}$.

Eq. (13) tells us also that the larger is $v_{p}$ — the larger is the Lemaître
free fall time, so the seemingly paradoxical situation when in some cases it
is better to switch the engine off, never occurs while maximizing $\tilde{t}$.
The fact that the working engine can increase the coordinate time while
decreasing the proper time have already been mentioned in [2] where the
Eddington-Finkelstein time have been considered. The unreachable upper limit
for the Lemaître time evidently corresponds to $v_{p}=1$, so it is given by
the integral

$\tilde{t}_{max}=\int^{r_{1}}_{0}\frac{dr}{\sqrt{r_{g}/r}-1}.$ (15)

Note, that the maximum possible proper time is given by

$\tau_{max}=\int^{r_{1}}_{0}\frac{dr}{\sqrt{(r_{g}/r)-1}},$ (16)

so for any $r_{1}$ the maximum Lemaitre time is larger that the maximum proper
time left to a singularity. For a finite power engine with $V_{p}<1$ we need
the evolution equation for $v_{p}$ which can be obtained by reversing eq.
(11):

$v_{p}=\frac{v-\varepsilon Z}{v^{2}+\varepsilon^{2}}.$ (17)

Substituting eq. (17) into eq. (13) we can obtain the Lemaitre time in the
same way as we got the proper time in the previous section. The numerical
results plotted in Fig. 2 show that in order to maximize $\tilde{t}$ we should
(as for maximizing $\tau$) switch on the engine the sooner the better. This
again matches our intuitive feeling that it is better to start to struggle
without any delay. Moreover, now this struggle should use all available fuel
independently of the observer’s position.

##  4 Conclusions

Our teaching experience shows that problems like the one about maximization of
the fall time serve as a good motivator for students. However, during our
seminars students never provided the standard (correct) answer to the question
about the ideal strategy to maximize the proper time (‘to reach the optimal
trajectory and to turn-off the engine’) before this problem was analyzed at a
blackboard in details. Indeed, it can be hard to accept the counter-intuitive
approach when the usage of the engine after some point might be considered as
an act of sabotage. Luckily, the detailed analyzes (see above) shows that it
is necessary to have an unrealistically powerful engine to reach the optimal
trajectory close to the horizon. However, a realistic strategy for a finite
power engine is more intuitive and students typically provide such an answer:
to use all the power as soon as possible. In this note we provided a detailed
consideration, that this is the best approach when the optimal trajectory
cannot be reached and the falling body did not pass the critical point
($r=v_{p}^{2}$), yet.

In this note we also demonstrated that maximization of the proper and Lemaître
time might be done following different procedures. If we already have
$r<r_{g}V_{p}^{2}$ then the curiosity to know a remote future (i.e.,
maximizing the Lemaître time) has its price in decreasing the proper life
time. Note, however, that since the integral defining the Lemaître time
diverges at a horizon, any maneuver deep inside the horizon can not increase
it significantly. For example, consider a situation when a spaceship with the
engine giving at most $V_{p}=0.9$ is located already at $r=0.49r_{g}$. The
proper time maximization strategy requires to use the boost only to
$v_{p}=0.7$ which gives the proper time equal to $0.275(r_{g}/c)$. So, using
the full power a hypothetical observer would sacrifice $0.024(r_{g}/c)$ of his
own proper time, instead increasing the Lemaitre time for about $23\%$ from
$0.335(r_{g}/c)$ to $0.413(r_{g}/c)$.

The considerations of the present paper assumes that the initial kinetic
energy of the spaceship is small with respect to its rest energy, so we use
the Lemaître frame to describe the motion. However, in our science-fiction set
up we can easily imagine a situation when this condition is not satisfied. To
deal with such a situation we need to use the kinematic with respect to a
general radial free falling frame. We leave this question for a future work.

## Acknowledgements

We thank Dr. Oleg Zaslavsky from Karazin National University (Kharkiv,
Ukraine) and Dr. Tarun Deep Saini from Indian Institute of Science (Bangalore,
India) for discussions and useful comments. We also thank the referee for
notes and suggestions which help to improve the paper.

SP was supported by the Ministry of science and higher education of Russian
Federation under the contract 075-15-2020-778 in the framework of the Large
scientific projects program within the national project ‘Science’.

## References

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