# Interplay of the Jahn–Teller effect and spin-orbit coupling: The case of
trigonal vibrations

Sergey V. Streltsov∗ Institute of Metal Physics, S. Kovalevskoy St. 18, 620990
Ekaterinburg, Russia  Department of Theoretical Physics and Applied
Mathematics, Ural Federal University, Mira St. 19, 620002 Ekaterinburg, Russia
<EMAIL_ADDRESS>Fedor V. Temnikov  Institute of Metal Physics, S.
Kovalevskoy St. 18, 620990 Ekaterinburg, Russia  Kliment I. Kugel  Institute
for Theoretical and Applied Electrodynamics, Russian Academy of Sciences,
Izhorskaya str. 13, 125412 Moscow, Russia  National Research University Higher
School of Economics, 101000 Moscow, Russia  Daniel I. Khomskii  II.
Physikalisches Institut, Universit$\ddot{a}$t zu K$\ddot{o}$ln,
Z$\ddot{u}$lpicher Stra$\ss$e 77, D-50937 K$\ddot{o}$ln, Germany

(August 27, 2024)

###### Abstract

We study an interplay between the orbital degeneracy and the spin-orbit
coupling (SOC) giving rise to spin-orbital entangled states in concentrated
systems (cooperative Jahn-Teller (JT) effect). As a specific example, we
analyze the interaction of electrons occupying triply degenerate single-ion
$t_{2g}$ levels with trigonal vibrations (the $t\otimes T$ problem). A more
general problem of the electron–lattice interaction involving both tetragonal
and trigonal vibrations is also considered. It is shown that the result of
such interaction crucially depends on the occupation of $t_{2g}$ levels
leading to either the suppression or the enhancement of the JT effect by the
SOC.

DOI: 10.1103/PhysRevB.105.205142

##  I Introduction

The effects related to spin-orbit coupling (SOC) have recently become quite
topical especially due to their decisive role in the physics of topological
insulators and other topological materials. These effects are also important
in such strongly correlated electron systems as 4$d$ and 5$d$ transition metal
compounds. In contrast with 3$d$ compounds, the large SOC characteristic of
4$d$ and 5$d$ transition metal ions can play a dominant role in the formation
of electron structure determining the sequence and multiplet characteristics
of the energy levels. Therefore, in such systems, we are dealing with the
spin-orbit entangled electron states Takayama et al. (2021). This means that
the spin and orbital degrees of freedom become intermixed leading to a more
pronounced contribution of magnetism to the orbital characteristics.

Indeed, the orbital degeneracy, leading in particular to the Jahn–Teller (JT)
effect, is quite common in many transition metal compounds. Until recently, it
was predominantly studied in 3$d$ systems containing such well-known JT ions
as Mn3+ and Cu2+. Currently, however, the attention is gradually shifting to
the study of 4$d$ and 5$d$ compounds. In this case, the SOC starts to play a
more and more important role. Therefore a question arises: what is the
concerted outcome of the JT effect and strong SOC? The most natural
expectation is that SOC would suppress the JT effect. Indeed, due to JT
distortions, the orbital degeneracy is lifted, and it becomes favorable to put
an electron at the state with a real wave function, with a particular
quadrupole moment. At the same time, the SOC rather prefers the states with
complex wave functions. Note here that even the first, rather old treatment
Öpik and Pryce (1957) has revealed that in the simplest case of one electron
per site the JT effect is gradually suppressed with increasing SOC
(characterized by the SOC constant $\lambda$).

In 3$d$ systems we usually deal with the high-spin state (satisfying the first
Hund’s rule stabilizing the state with maximum possible spin), in which case
we often have a situation with partially filled $e_{g}$ states (like those in
Mn3+ or Cu2+). However, for $e_{g}$ electrons, the SOC is in the first
approximation quenched. In contrast, for 4$d$ and 5$d$ systems, we typically
have low-spin states with very often partial filling of triply-degenerate
$t_{2g}$ orbitals, but for these, the SOC is not quenched, and just in this
case, the most realistic for 4$d$ and 5$d$ systems, one should expect an
important role of SOC.

At the same time, in many cases, there still remains an orbital degeneracy
even if the SOC is very strong. The orbital degeneracy typically manifests
itself in the involvement of the crystal lattice occurring in the form of
vibronic interactions, i.e. those related to the JT effect Bersuker and
Polinger (1989); Bersuker (2006); Khomskii (2014); Streltsov and Khomskii
(2017); Khomskii and Streltsov (2020).

Such a strong interplay of electronic and lattice characteristics in the
systems with spin-orbit entangled states should lead to a plethora of novel
quantum phenomena, the analysis of which now seems to be only at the initial
stage. In this connection, let us note some early Öpik and Pryce (1957);
Moffitt and Thorson (1957); O’Brien (1969); Bacci et al. (1975); Bates (1978);
Judd (1984) and several recent Chen and Balents (2011); Plotnikova et al.
(2016); Liu and Khaliullin (2019); Nikolaev et al. (2018); Ishikawa et al.
(2019); Paramekanti et al. (2020); Khaliullin et al. (2021); Mosca et al.
(2021) papers but particularly mention an unduly rarely cited paper by K.D.
Warren Warren (1982). Using the so-called angular overlap model, Warren was
able to treat limiting situations of small and very large SOC for all possible
occupations of $d$ electrons. It was shown that this interaction may
substantially modify JT coupling constants. In a recent study a more general
situation of the SOC of arbitrary strength was considered by a very different
approach Streltsov and Khomskii (2020) for $E=\\{Q_{2},Q_{3}\\}$ type
distortions in case of a static JT problem.

The main results of Streltsov and Khomskii (2020) can be summarized as
following: Vibronic and spin-orbit coupling interactions can either enhance or
suppress each other depending on a particular situation, first of all, on the
number of electrons per site. For one electron at triply degenerate $t_{2g}$
states, for which the SOC is not quenched, an increase in SOC gradually
suppresses the JT effect, which, however, remains nonzero even for very strong
SOC. For the $d^{2}$ configuration, the JT effect is also suppressed by SOC.
In the case of $d^{4}$ and $d^{5}$ configurations (with all electrons in
$t_{2g}$ states, which is the case for the low-spin states typical of 4$d$ and
5$d$ systems) the JT effect also vanishes as the strength of SOC grows.
However, in contrast with the $d^{2}$ case, the JT effect disappears at finite
value of $\lambda_{cr}$ in an almost abrupt way, and it is strictly zero above
$\lambda_{cr}$. Nevertheless, there may also be an opposite effect: the SOC
may enhance rather than suppress the JT distortions. This is the situation for
the $d^{3}$ configuration, for which the SOC does not impede but activates the
JT effect, which for this configuration is absent for $\lambda=0$.

These results were obtained in Ref. Streltsov and Khomskii (2020) by
considering the JT coupling of $t_{2g}$ electrons with doubly degenerate $E$
(tetragonal and orthorhombic) distortions – the so called $t\otimes E$
problem. However, the $t_{2g}$ states can be also split by trigonal
distortions – the $t\otimes T$ problem and moreover JT coupling constant for
$T$ vibrations can be as large as for $E$ phonons Iwahara et al. (2018).

It is both interesting scientifically and important practically to know how
the SOC would affect such trigonal distortions, which are often present in
real situations. Theoretically it is even more interesting: for very strong
SOC the states of $t_{2g}$ electrons are split into $j=3/2$ quartet and
$j=1/2$ doublet, with $j=3/2$ states lying lower. Such quartet is actually
formed by two Kramers doublets, i.e. the situation in this sense resembles
that of the usual $e_{g}$ states and sometimes indeed these doublets are
regarded as an effective $e_{g}$ orbitals, see, e.g., Ref. Takayama et al.
(2021), but how far does this analogy go? In particular, $e_{g}$ levels are
not split by the trigonal deformation, i.e. there is no interaction with
trigonal $T$-phonons. However, the situation with $t_{2g}$ electrons in case
of strong SOC might be very different from the $e\otimes E$ case, just because
of a strong spin-orbit entanglement introduced by SOC. And indeed, it is known
that for $d^{1}$ configuration in the case of infinitely strong SOC, where we
are dealing with the $j=3/2$ quartet, JT coupling to $T_{2}$ vibrations still
survives Moffitt and Thorson (1957); Judd (1984). The case of intermediate JT
coupling, not considered in the previous literature, present special interest
because real $4d$ and $5d$ systems usually belong to this category. This is
what is done in the present paper. Another question we concentrate on is what
the situation is with the $t\otimes T$ problem in case of strong SOC for other
electron configurations - $d^{2}$, $d^{3}$ etc. In these cases, besides purely
JT electron–phonon interaction, an electron–electron interaction, especially
the Hund’s rule exchange, play crucial role and can strongly modify the
behavior of a system, in particular the manifestations of JT effect in those.

Finally, one important comment has to be made. In Ref. Streltsov and Khomskii
(2020) and in the present paper we mainly have in mind concentrated systems,
especially those with $4d$ and $5d$ transition metal ions, such as Kitaev
magnets containing e.g. Ru3+ (RuCl3) or Ir4+ (Li2IrO3), or double perovskites
like Na2BaOsO6 or Sr2CaIrO6. Interplay of strong SOC and JT effect in these
systems present an interesting and important problem, both theoretically and
experimentally Ishikawa et al. (2019); Streltsov and Khomskii (2020); Kloß et
al. (2021); Kim et al. (2019); Huang et al. (2022); Prodan et al. (2021); Weng
and Dong (2021); Zhang et al. (2022). To treat it one has to start from the
case of single transition metal ions, as we do in the present paper. In
treating the single site case some extra complications can enter the game,
such as vibronic effects connected with treating not only electrons but also
the lattice (nucleus) quantum-mechanically. These effects can lead to certain
modifications, for example to the well-known Ham’s reduction of different
nondiagonal matrix elements Ham (1965). In concentrated system such as those
we have in mind, the vibronic effects are practically always neglected, and
the lattice is treated (quasi)classically, see e.g. Englman (1972); Keimer et
al. (2000). Keeping in mind this situation and aiming at application to
concentrated system, we in our calculations do not also take into account
vibronic effects leaving the study of their possible effects for future.

##  II Model

The model Hamiltonian used in the present paper includes three components

$\displaystyle\hat{H}=\hat{H}_{SOC}+\hat{H}_{JT}+\hat{H}_{U},$ (1)

where the first, second, and third terms correspond to the SOC, the JT
electron-lattice coupling, and the Hubbard on-site electron–electron
interaction, respectively. The SOC is taken in a full vector form

$\displaystyle\hat{H}_{SOC}=-\zeta\sum_{\alpha}\hat{\bf
l}_{\alpha}\cdot\hat{\bf s}_{\alpha},$ (2)

where $\hat{\bf l}_{\alpha}$ and $\hat{\bf s}_{\alpha}$ are orbital and spin
operators of the $\alpha$th electron, $\zeta$ is the SOC constant, and the
minus sign appears because we deal with the $t_{2g}$ orbitals with effective
orbital moment $l_{\it eff}=1$ Abragam and Bleaney (1970). In the LS coupling
scheme (for SOC weaker than the Hund’s rule coupling), one can also write this
part of the Hamiltonian as $H_{SOC}=-\lambda\hat{\bf L}\cdot\hat{\bf S}$,
where $\hat{\bf L}=\sum_{\alpha}\hat{\bf l}_{\alpha}$, $\hat{\bf
S}=\sum_{\alpha}\hat{\bf s}_{\alpha}$ are the total orbital and spin moments
of a particular configuration, and $\lambda=\zeta/2S$.

The interaction part is written in the standard rotationally invariant form
Georges et al. (2013)

$\displaystyle\hat{H}_{U}=(U-3J_{H})\frac{\hat{N}(\hat{N}-1)}{2}-2J_{H}{\hat{S}}^{2}-\frac{J_{H}}{2}{\hat{L}}^{2}+\frac{5}{2}J_{H}\hat{N},$
(3)

where $U$ is the Hubbard repulsion (not important here since we consider a
single site), $J_{H}$ is the Hund’s rule intraatomic exchange, and $\hat{N}$
is the operator for the total number of electrons.

The JT term includes the elastic energy contribution and the linear coupling
of the electron subsystem with the corresponding vibrations. In Sec. III,
where the $t\otimes T$ problem is considered, we use the following form of the
JT Hamiltonian

$\displaystyle\hat{H}_{JT}^{T}$ $\displaystyle=$
$\displaystyle-g\Big{(}(\hat{l}_{y}\hat{l}_{z}+\hat{l}_{z}\hat{l}_{y})Q_{4}+(\hat{l}_{x}\hat{l}_{z}+\hat{l}_{z}\hat{l}_{x})Q_{5}$
(4) $\displaystyle+$
$\displaystyle(\hat{l}_{y}\hat{l}_{x}+\hat{l}_{x}\hat{l}_{y})Q_{6}\Big{)}+\frac{B^{2}}{2}\left(Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}\right).$

where $Q_{4}$, $Q_{5}$, and $Q_{6}$ are the phonon modes, illustrated in Fig.
1, with the corresponding coefficients $g$ and $B$ Bates (1978). For
simplicity, in most of numerical calculations, we assume that $g=B=1$.
Positive $Q_{4}$, $Q_{5}$ and $Q_{6}$ in the combination $Q_{4}+Q_{5}+Q_{6}$
would give trigonal distortion corresponding to the elongation of octahedron
in the [111] direction, see Fig. 2. A more general form the JT term used in
Sec. IV is presented in Eq. (IV).

Figure 1:  Sketch demonstrating distortions of metal-ligand octahedron by
trigonal $Q_{4}$, $Q_{5}$ and $Q_{6}$ modes.

It has to be stressed that here we consider not a lattice of JT ions, but a
single JT center. Nevertheless, we keep in mind the situation of concentrated
systems.

Figure 2:  Energy level splitting and distortions in the $t\otimes T$ problem
without the SOC in the case of $t_{2g}^{1}$ (a) and $t_{2g}^{2}$ (b)
configurations.

It has to be noted that the present approach differs from conventional ones
used to treat the JT effect. It is not perturbative, but it is based on
numerically exact solution of the many-electron problem including all the
necessary interactions (electron–lattice, Hund’s rule exchange, and SOC) for
an arbitrary distortion with subsequent global minimization of the total
energy with respect to all possible phonon modes. In this scheme, different
specific electronic states correspond to each particular nuclear
configuration, i.e. all vibronic effects, such as the Ham’s reduction Ham
(1965); Abragam and Bleaney (1970), can be included if one would use this
scheme for calculating dynamical properties of isolated JT impurities as
investigated e.g. by paramagnetic resonance spectroscopy.

##  III $t\otimes T$ problem

Figure 3:  Amplitude of the JT distortion (compression) defined as
$Q=\sqrt{Q^{2}_{4}+Q^{2}_{5}+Q^{2}_{6}}$ as a function of the SOC constant
$\zeta$ in the case of $d^{1}$ electronic configuration for different ratios
of the $g$ and $B$ parameters. The inset shows the constant energy surface for
$\lambda=0$ corresponding to
$E(Q_{4},Q_{5},Q_{6})\approx-1.3\frac{g^{2}}{2B}$, which is close to the
absolute energy minimum $E=-\frac{4}{3}\frac{g^{2}}{2B}$.

In this section, we not only study how the SOC affects the JT effect in the
case of the $t\otimes T$ problem, but also pay attention to the role of
intraatomic Hund’s rule exchange. It is assumed here that the $t_{2g}-e_{g}$
crystal-field splitting, $10Dq$, is very large (always larger than the SOC
constant $\lambda$).

###  III.1 $d^{1}$ configuration

The situation in the case of $d^{1}$ configuration without SOC is well
documented, and the JT effect results in the trigonal compression, i.e.
distortion along one of four possible vectors: $[-1,-1,-1]$, $[-1,1,1]$,
$[1,-1,1]$, or $[1,1,-1]$ in the $Q_{4},Q_{5},Q_{6}$ space. This leads to the
level splitting such that the $a_{1g}$ orbital turns out to be lower than
$e_{g}^{\pi}$ and a single electron occupies this $a_{1g}$ orbital, see Fig.
2(a). These four minima are clearly seen in the inset of Fig. 3, where the
constant energy surface $E(Q_{4},Q_{5},Q_{6})$ corresponding to 99% of global
energy minimum $E=-\frac{4}{3}\frac{g^{2}}{2B}$ is presented. In the
($Q_{4},Q_{5},Q_{6}$) space, these minima are located along four [111]
directions with the total tetrahedral symmetry. These minima would be located
at other ends of these [111] axes for the opposite sign of the coupling
constant $g$ in Hamiltonian (4).

Figure 4:  Evolution of the energy isosurface $E(Q_{4},Q_{5},Q_{6})$ as a
function of the SOC constant $\lambda$ in the case of $d^{1}$ configurations.
a) $\lambda=0$ b) $\lambda=5g^{2}/B$ c) $\lambda=50g^{2}/B$. To calculate the
isosurface the Hamiltonian (4) is rewritten in a spherical coordinates
$Q_{4}=r\sin{\theta}\cos{\phi}$, $Q_{5}=r\sin{\theta}\sin{\phi}$,
$Q_{6}=r\cos{\theta}$ and then it is minimized in $r$ for each $(\theta,\phi)$
point.

The account taken of the SOC results in the gradual suppression of the JT
distortions as shown in Fig. 3. In agreement with previous studies the system
retains trigonal minima in the presence of not very strong SOCÖpik and Pryce
(1957); Bacci et al. (1975). However, one might see that the SOC tends to
stabilize an electron at very different orbitals (as compared to those
favorable with respect to the JT effect) and this results in the suppression
of the amplitude of the distortion and JT coupling constant as was noted in
Warren (1982). However, the SOC cannot lift the degeneracy completely – we
still have an electron at the doubly degenerate (without taking into account
the Kramers degeneracy) $j=3/2$ subshell. Therefore, the JT effect will never
be suppressed completely.

The results of these calculations also answer the question formulated in the
Introduction: : To which extent does the ground-state quartet $j=3/2$ (two
Kramers doublets), reached for strong SOC, resembles the $e_{g}$ quartet (also
two Kramers doublets) for the usual $d$ electrons in cubic crystal field
without SOC? We remind that for the usual $d$ electrons with one electron (as,
e.g., in Mn3+) or one hole (as in Cu2+) at $e_{g}$ levels, the JT effect
leading to the lifting of this degeneracy exists for tetragonal and
orthorhombic distortions but not for trigonal ones. In our case, however, for
one electron at the $j=3/2$ quartet, not only tetragonal Streltsov and
Khomskii (2020) but also trigonal distortions lead to the JT effect, see Fig.
3. Thus we see that the $j=3/2$ quartet is in this sense not equivalent to the
usual $e_{g}$ case. A different character of the corresponding wave functions
in this case, with the strong spin-orbit entanglement, leads to different
characteristics of the JT effect for strong SOC. The study of the coupling to
trigonal modes (the $t\otimes T$ problem) thus allows us to reveal the role of
spin-orbit entanglement for the JT effect.

Figure 5:  Cuts of the “Mexican globe” along the $\phi=45^{\circ}$ meridian in
the case of $d^{1}$ configurations at the SOC constant $\lambda=0$ (a),
$\lambda=5g^{2}/B$ (b) and $\lambda=50g^{2}/B$ (c). $Q=(Q_{4}+Q_{5})/\sqrt{2}$
is a normalized distortion along the $\phi=45^{\circ}$ meridian. Figure 6:
Corrugation energy $\delta\varepsilon_{\text{corrug}}$ as a function of the
SOC constant $\lambda$ in the case of $d^{1}$ configurations.
$\delta\varepsilon_{\text{corrug}}$ is the absolute difference of the energy
minimum and a saddle point.

In fact, we see that for the $d^{1}$ configuration, the situation for trigonal
distortions is actually similar to that for tetragonal ones Streltsov and
Khomskii (2020): the strong SOC reduces JT distortions but not completely.
Similarly to the $t\otimes E$ case, for the strong SOC, different trigonal
distortions (elongation and compression along different [111] axes) become
equivalent, so that any linear combination thereof has the same energy, which
is the situation of the “Mexican hat” (continuous manifold of degenerate
states, which for the $e\otimes E$ problem indeed has a form of a “Mexican
hat”, see, e.g., Refs. Bersuker (2006); Khomskii (2014); Khomskii and
Streltsov (2020)). Thus, in this case for very strong SOC, we also have the
manifold of degenerate states, forming “Mexican hat”, but in the 4D space, see
Appendix A. The fact that in the limit of $\lambda\to\infty$ the JT effect for
the case of one electron at the $j=3/2$ quartet (in the Bethe notation, the
$\Gamma_{8}$ quartet) leads to the continuum of degenerate states (1D
manifold, the trough in the Mexican hat in the $t\otimes E$ problem, 2D
manifold – the “Mexican globe” for the $t\otimes T$ case) is well-known in the
JT literature Moffitt and Thorson (1957); Bates (1978); Judd (1984). We
demonstrated how this state is reached with increasing $\lambda$, i.e., how
the energy surface evolves from that of $\lambda=0$ to the limiting solution
of the Mexican hat in the $t\otimes E$ or the Mexican globe for the $t\otimes
T$ case for infinite SOC.

Figure 7:  Amplitude of the JT distortion defined as
$Q=\sqrt{Q^{2}_{4}+Q^{2}_{5}+Q^{2}_{6}}$ as a function of the SOC constant
$\lambda$ in the case of $d^{2}$ and $d^{3}$ electronic configurations for
$g=B=1$.

Analysis of the energy surface of the $t\otimes T$ problem is quite difficult
compared to that for the $t\otimes E$ and $e\otimes E$ problems, since
$E(Q_{4},Q_{5},Q_{6})$ is a 4D function. The energy isosurfaces
$E(Q_{4},Q_{5},Q_{6})=\text{const}$ can be plotted (e.g., see Fig. 4) but they
are difficult to compare with the Mexican hats in the $t\otimes E$ and
$e\otimes E$ problems. However, one can decrease dimensionality of
$E(Q_{4},Q_{5},Q_{6})$ if we make some cut of the “Mexican globe” by combining
two phonon modes (e.g. $Q_{4}$ and $Q_{5}$) into one
$Q=(Q_{4}+Q_{5})/\sqrt{2}$ (where $1/\sqrt{2}$ is normalization factor). It
corresponds to cutting the “globe” by the corresponding meridian. Then we can
plot the energy surfaces of the trigonal and tetragonal modes.

The cuts of the “Mexican globe” along the $\phi=45^{\circ}$ meridian at the
various values of the SOC constant $\lambda$ are shown in Fig. 5. Two global
minima of the energy cut corresponds to the $[-1,-1,-1$] and $[1,1,-1]$ minima
in the $Q_{4},Q_{5},Q_{6}$ space. Other minima of the Mexican globe (Fig. 4)
can be obtained if the cut is made along other directions. The last (local)
minimum is in fact a saddle point in the $Q_{4},Q_{5},Q_{6}$ space; its energy
is equal to the energy of the saddle point between the global minima. The
energy difference between the global minima and the saddle point (with the
local minima) becomes smaller with increasing SOC (see Fig. 6). Thus the cut
of the Mexican globe turns into the well-known Mexican hat at $\lambda\gg
E_{JT}$ (Fig. 5c).

The cuts in Fig. 5 are generally similar to the Mexican hats of the $t\otimes
E$ and $e\otimes E$ problems. Both such “hats” have conical points at $(0,0)$.
The cuts have three minima in Fig. 5(a) and Fig. 5(b); however, whereas in the
$t\otimes E$ and $e\otimes E$ problems all three minima are equal, here, in
this cross-section, we get two global and one local minima. Also we obtain the
continuous set of minimum points in Fig. 5(c). Except for the presence of the
local minimum, the evolution of the cuts of Mexican globe in the $t\otimes T$
problem and the Mexican hat of the $t\otimes E$ problem as the function of
$\lambda$ very similar. Note that in the $e\otimes E$ problem the Mexican hat
gets corrugated due to higher order JT coupling. Here, however, we get some
corrugation already for the linear JT coupling but for finite SOC; so, instead
of the continuous set of minimum points, only four minima exist for finite
$\lambda$.

###  III.2 $d^{2}$ configuration

In the case of $d^{2}$ electronic configuration, one needs to take into
account the intraatomic exchange interaction, $J_{H}$. Here, we assume that
the energy gain due to the JT distortions is always smaller than $J_{H}$, but
the strength of the SOC, $\lambda$, can be larger or smaller than the Hund’s
rule energy $J_{H}$ and $g^{2}/{2B}$.

Having two electrons in the absence of the SOC, we gain more energy by
elongating the metal–ligand octahedron in [111] direction and by putting two
electrons with parallel spins onto $e_{g}^{\pi}$ orbitals, as shown in Fig.
2(b). This results in the trigonal elongation along one of four possible [111]
directions discussed above (in solids, the exchange interaction or
electron–phonon coupling could choose one of these directions). The SOC in
turn favors the occupation of very different orbitals. These are $j=3/2$ spin-
orbitals, see Eq. (B) in Appendix B.

Therefore, by increasing the strength of SOC, we reduce the maximum possible
energy gain (and the distortion as a result) due to the vibronic coupling, see
the upper panel of Fig. 7. This is exactly what is observed in our numerical
calculations – the JT distortion amplitude decreases with $\lambda$.

Moreover, formally as $\lambda\to\infty$, the JT distortions asymptotically
vanish. This is in contrast with the situation with $d^{1}$ configuration. One
can easily understand this by noting that also at very strong SOC, the intra-
atomic exchange makes two electrons to occupy $j^{z}_{3/2}$ and $j^{z}_{1/2}$,
or $j^{z}_{-3/2}$ and $j^{z}_{-1/2}$, orbitals (to have maximal spin
projections, see Eq. (B)). However, the distortions induced by such occupation
compensate each other exactly: Using Eqs. (4) and (B), it can be readily shown
that e.g. $j^{z}_{3/2}$ gives $Q_{JT}^{3/2}=-\frac{1}{3}\frac{g}{B}$, while
$j^{z}_{1/2}$ results in $Q_{JT}^{1/2}=\frac{1}{3}\frac{g}{B}$.

These results were obtained taking into consideration only the $t_{2g}$
manifold, assuming that the cubic crystal field leading to splitting ($10Dq$)
of $t_{2g}$ and $e_{g}$ levels is the largest parameter in the system.
Admixture of $e_{g}$ states in case of finite $10Dq$ can bring about some
modifications, largely of numerical character. This problem, especially
important for $d^{2}$ configuration, will be considered separately.

###  III.3 $d^{3}$ configuration

In the case of three electrons and zero SOC, we fill three $t_{2g}$ levels by
electrons, which have the same spin projection due to the strong intraatomic
Hund’s rule exchange. Such a state does not exhibit any orbital degeneracy and
therefore it is inactive for the JT effect.

Figure 8:  Amplitude of the JT distortion defined as
$Q=\sqrt{Q^{2}_{4}+Q^{2}_{5}+Q^{2}_{6}}$ as a function of the SOC constant
$\lambda$ in the case of $d^{4}$ electronic configuration for $g=B=1$.

The SOC acts against the Hund’s rule exchange and redistributes electrons in
such a way as to make them occupy $j=3/2$ states. This results in orbital
degeneracy (three electrons at the fourfold degenerate $j=3/2$ states) and
activates the JT effect as was pointed out by Warren Warren (1982). The
distortion amplitude as a function of SOC $\lambda$ is shown in Fig. 7. One
might expect that, similarly to the $t\otimes E$ problem Streltsov and
Khomskii (2020), here one would also have the Mexican hat geometry of the
adiabatic potential energy surface in the formal limit of $\lambda\to\infty$.
Indeed, as in the case of the $d^{1}$ configuration, we have here a single “JT
active” particle at the $j_{3/2}$ levels, but it is a hole in the case of
$d^{3}$.

It is interesting to study the effect of intra-atomic exchange on JT
distortions. First, one may see in Fig. 7 that it is ratio $\lambda/J_{H}$,
which plays a crucial role. A half maximum possible JT distortion is achieved
at $\lambda/J_{H}\sim 0.7-1$. On the other hand, indeed the Hund’s rule and
SOC favor very different occupations of the spin-orbitals by electrons:
eigenfunctions (19) of spin-orbit operator (2) are obviously not optimum from
the viewpoint of intra-atomic exchange, which favors having as many as
possible electrons with the same spin projection. Therefore, by increasing
$J_{H}$, we suppress the JT distortions induced by the SOC, see Fig. 7.

###  III.4 $d^{4}$ and $d^{5}$ configurations

These two configurations demonstrate very similar behavior in the case of the
$t\otimes E$ problem Streltsov and Khomskii (2020). For $T$ vibrations, this
result remains the same. The corresponding plots of distortion amplitude are
summarized in Figs. 8 and 9.

Figure 9:  Amplitude of the JT distortion defined as
$Q=\sqrt{Q^{2}_{4}+Q^{2}_{5}+Q^{2}_{6}}$ as a function of the SOC constant
$\lambda$ in the case of $d^{5}$ configuration. This function is the same for
different $g$ and $B$, if $\lambda$ is measured in the units of
$\frac{g^{2}}{B}$ and $Q$ is measured in the units of $\frac{g}{B}$. Insets
demonstrate the constant energy surfaces $E(Q_{4},Q_{5},Q_{6})=E_{iso}$, which
are close to the global energy minima, at $\lambda=0<\lambda_{c}$ (left) and
$\lambda=1.2>\lambda_{c}$ (right).

The case of $d^{5}$ configuration without SOC can be described in terms of one
hole at $t_{2g}$ levels, so it can be derived from the $d^{1}$ case by
replacing $g$ with $-g$. Then, the points characterizing absolute energy
minima are of the opposite signs as compared to the $d^{1}$ case: $[1,1,1]$,
$[1,-1,-1]$, $[-1,1,-1]$, or $[-1,-1,1]$ in $Q_{4}Q_{5}Q_{6}$ space (Fig. 9,
left inset). The $ML_{6}$ octahedron is elongated in one of these directions,
and we put five electrons at the crystal-field levels of Fig. 2(b). This is
the typical situation for such ions as Ir4+ and Ru3+, important, e.g., for the
Kitaev materials.

As $\lambda$ increases, the JT distortions decrease and near the critical
value $\lambda_{c}$ abruptly disappear (Fig. 9). The absence of distortions at
large $\lambda$ values can be explained in the $jj$ coupling scheme, which is
relevant in this case. In this scheme, a single hole occupies the upper
Kramers doublet $j=\frac{1}{2}$, which does not have any orbital degeneracy,
so the JT effect does not work in this situation.

The same picture also explains similar behavior for the $d^{4}$ configuration,
Fig. 9: in the $jj$ scheme, four electrons completely fill four $j=3/2$ states
leading to the nondegenerate and nonmagnetic $J=0$ state .

##  IV Full $t\otimes(T+E)$ problem

Figure 10:  Energy as a function of $\lambda$ for the $t\otimes T$ (black
solid line) and $t\otimes E$ (red circles and blue solid line) problems in the
case of $d^{1}$ configuration. The coefficient $g_{e}=\sqrt{3}$ is chosen to
have $E^{e}=E^{t}$ at $\lambda=0$ and e.g. for $g_{e}=0.9\sqrt{3}$:
$|E^{e}|<|E^{t}|$.

Typically, for the description of specific materials it is enough to treat the
coupling of electrons with $E$ or $T$ vibrations. Nevertheless, for
completeness, below we consider the general $t\otimes(T+E)$ problem, which
includes both tetragonal ($Q_{2}$, $Q_{3}$) and trigonal ($Q_{4}$, $Q_{5}$,
$Q_{6}$) displacements. In this situation, the JT term is written in the
following form

$\displaystyle\hat{H}_{JT}^{TE}$ $\displaystyle=$
$\displaystyle\frac{B_{e}}{2}(Q_{2}^{2}+Q_{3}^{2})+\frac{B_{t}}{2}(Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2})$
$\displaystyle-$ $\displaystyle
g_{e}\Big{(}\frac{1}{\sqrt{3}}(\hat{l}^{2}_{x}-\hat{l}^{2}_{y})Q_{2}+\Big{(}\hat{l}^{2}_{z}-\frac{2}{3}\Big{)}Q_{3}\Big{)}$
$\displaystyle-$ $\displaystyle
g_{t}\Big{(}(\hat{l}_{y}\hat{l}_{z}+\hat{l}_{z}\hat{l}_{y})Q_{4}+(\hat{l}_{x}\hat{l}_{z}+\hat{l}_{z}\hat{l}_{x})Q_{5}+$
$\displaystyle+$
$\displaystyle(\hat{l}_{x}\hat{l}_{y}+\hat{l}_{y}\hat{l}_{x})Q_{6}\Big{)}.$

Here, $B_{e}$ and $g_{e}$ ($B_{t}$ and $g_{t}$) are constants corresponding to
$E$ ($T$) distortions.

The solution of Eq. (IV) is well known for the case of zero SOC, $\zeta=0$.
There are three types of extremum points: three correspond to tetragonal
minima with $Q_{4}=Q_{5}=Q_{6}=0$, four are trigonal minima with
$Q_{2}=Q_{3}=0$, and six are orthorhombic points O’Brien (1969). The
difference between the energies $E^{e}=-2g_{e}^{2}/9B_{e}$ (the coupling to
$E$ modes) and $E^{t}=-2g_{t}^{2}/3B_{t}$ ($T$ modes) is crucial for the
$t\otimes(T+E)$ problem. If $E^{e}<E^{t}$, the tetragonal extremum points are
absolute minima and the others are saddle points. Conversely, if
$E^{t}_{JT}<E^{e}_{JT}$, then the trigonal points correspond to global minima,
and again the others are saddle points. Orthorhombic points always remain to
be saddle points.

The $E^{e}=E^{t}$ case is more complicated. All types of extrema become
minimum points. Moreover, there is a continuous subset of minima. For a
special case $B_{e}=B_{t}$ and $g_{e}=g_{t}/\sqrt{3}$ (the so-called $t\otimes
D$ problem) all minima obey the relationship
$Q_{2}^{2}+Q_{3}^{2}+Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}=Q_{0}^{2}=g_{t}^{2}/3B_{t}$,
so they can be parameterized as

$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle-\sqrt{3}Q_{0}\sin^{2}\theta\cos 2\phi,$ $\displaystyle Q_{3}$
$\displaystyle=$ $\displaystyle-Q_{0}(3\cos^{2}\theta-1),$ $\displaystyle
Q_{4}$ $\displaystyle=$ $\displaystyle-\sqrt{3}Q_{0}\sin 2\theta\sin\phi,$ (6)
$\displaystyle Q_{5}$ $\displaystyle=$ $\displaystyle-\sqrt{3}Q_{0}\sin
2\theta\cos\phi,$ $\displaystyle Q_{6}$ $\displaystyle=$
$\displaystyle-\sqrt{3}Q_{0}\sin^{2}\theta\sin 2\phi.$

Let us consider how the situation changes with the account taken of the SOC.
First, if we compare the results of the $t\otimes T$ and $t\otimes E$
problems, one can notice that all modes have similar dependence on $\lambda$.

Therefore, one might expect that the ground state energies of the $t\otimes E$
and $t\otimes T$ problems have the same dependence on $\lambda$. Direct
numerical calculations of $E^{e}$ with $B_{t}=g_{t}=0$ and $E^{t}$ with
$B_{e}=g_{e}=0$ (using Eq. (IV)) show that this is indeed the case (see Fig.
10). If $E^{e}$ is equal (larger or less than) $E^{t}$ at $\lambda=0$, then
$E^{e}$ is equal (larger or less than) $E^{t}$ at any $\lambda$. Consequently,
all conclusions derived for the $t\otimes(T+E)$ problem without SOC remain the
same in the case of nonzero SOC.

Moreover, parametrization (6) can also be used for the case of $B_{e}=B_{t}$,
$g_{e}=g_{t}/\sqrt{3}$, but now all modes (including $Q_{0}$) become functions
of $\lambda$, which are similar to the $Q$ function (Fig. 3). Direct
calculations with the JT term described by Eq. (IV) with substitution from Eq.
(6) for some values of $\lambda$ show that the ground state energy of the
system is the same for any $\theta$ and $\phi$. Thus, SOC does not destroy the
continuous set of minimum points in the $t\otimes D$ problem.

The $\lambda\to\infty$ case is considered separately. We use the same
algorithm as described in Appendix A, but with an additional step. Exact
expression for the total energy is rather cumbersome, but one can expand this
into the Laurent series at $\lambda\to\infty$ and take the leading terms
($\lambda^{1}$ and $\lambda^{0}$). After these transformations, the ground
energy takes the form

$\begin{gathered}E(Q_{2},Q_{3},Q_{4},Q_{5},Q_{6})=-\frac{\lambda}{2}+\frac{B_{e}}{2}(Q^{2}_{2}+Q^{2}_{3})+\frac{B_{t}}{2}(Q^{2}_{4}+\\\
+Q^{2}_{5}+Q^{2}_{6})-\frac{1}{3}\sqrt{g_{e}^{2}(Q_{2}^{2}+Q_{3}^{2})+3g_{t}^{2}(Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2})}.\end{gathered}$
(7)

If we take $B_{e}=B_{t}=B$ and $g_{e}=g_{t}/\sqrt{3}=g$, Eq.(7) becomes
$E=B\sum_{i}Q_{i}^{2}-\frac{1}{3}g\sqrt{\sum Q_{i}^{2}}-\frac{\lambda}{2}$.
Now we can again use Eq. (6) and obtain
$E=5BQ_{0}^{2}/2-\frac{\sqrt{5}}{3}gQ_{0}-\frac{\lambda}{2}$. The last
expression has the minimum at $Q_{0}^{2}=g^{2}/45B^{2}$. This is the “Mexican
hat” again, but in the 6D space.

Consider the case $E^{e}\neq E^{t}$. Now Eq. (7) depends on two sums: the sum
of $E_{g}$ modes $Q_{2}^{2}+Q_{3}^{2}$ and the sum of $T_{2g}$ modes
$Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}$. One can transform $E_{g}$ modes to
cylindrical coordinates and $T_{2g}$ modes to spherical coordinates

$\displaystyle Q_{2}$ $\displaystyle=$ $\displaystyle Q_{e}\sin\phi,$
$\displaystyle Q_{3}$ $\displaystyle=$ $\displaystyle Q_{e}\cos\phi,$
$\displaystyle Q_{4}$ $\displaystyle=$ $\displaystyle Q_{t}\sin\theta\cos\phi$
(8) $\displaystyle Q_{5}$ $\displaystyle=$ $\displaystyle
Q_{t}\sin\theta\sin\phi,$ $\displaystyle Q_{6}$ $\displaystyle=$
$\displaystyle Q_{t}\cos\theta.$

Then

$E(Q_{e},Q_{t})=\frac{B_{e}}{2}Q_{e}^{2}+\frac{B_{t}}{2}Q_{t}^{2}-\frac{1}{3}\sqrt{g_{e}^{2}Q_{e}^{2}+3g_{t}^{2}Q_{t}^{2}}-\frac{\lambda}{2}.$
(9)

This energy equation is a 3D one, so it can be easily treated analytically.
Using the first derivatives, we obtain stationary points (0, $\pm
g_{t}/\sqrt{3}B_{t}$) (with $E^{t}_{JT}=-g_{t}^{2}/18B_{t}$) and ($\pm
g_{e}/3B_{e}$, 0) (with $E^{e}_{JT}=-g_{e}^{2}/6B_{e}$) in $(Q_{e},Q_{t})$
coordinates. Then, we calculate the Hessians and find that the point (0, $\pm
g_{t}/\sqrt{3}B_{t}$) is the absolute minimum if
$3g_{t}^{2}B_{t}>g_{e}^{2}/B_{e}$ (or $E^{t}_{JT}<E^{e}_{JT}$), and the point
($\pm g_{e}/3B_{e}$, 0) is the absolute minimum if
$3g_{t}^{2}B_{t}<g_{e}^{2}/B_{e}$ (or $E^{e}_{JT}<E^{t}_{JT}$).

Hence, in the $\lambda\to\infty$ limit, we have three “Mexican hats”: the
first is 4D with only $T_{2g}$ modes
($Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}=Q_{t}^{2}=g_{t}^{2}/3B_{t}$) for
$E^{t}_{JT}<E^{e}_{JT}$. The second 3D hat includes only $E_{g}$ modes
$Q_{2}^{2}+Q_{3}^{2}=Q_{e}^{2}=g_{e}^{2}/9B_{e}^{2}$ for
$E^{e}_{JT}<E^{t}_{JT}$. Finally, the last “Mexican hat” is a 6D in $T_{2g}$
and $E_{g}$ modes space that was considered at the beginning of this section.

Here we have considered the static $t\otimes(T+E)$ problem of the linear JT
coupling at arbitrary SOC strength. Previously, the dynamical properties for
the strong JT coupling O’Brien (1969) and coexistence of tetragonal,
orthorhombic and trigonal distortions for the quadratic JT interaction Bacci
et al. (1975) was studied for this problem. However, the SOC was treated only
for special cases, such as $g$-value O’Brien (1969) and ${}^{3}T$ term Bacci
et al. (1975).

##  V Conclusions

In this paper, we analyzed an interplay between SOC and vibronic interactions
in ions with partially occupied $t_{2g}$ levels. A special emphasis was put on
the $t\otimes T$ problem, i.e. on the interactions of $t_{2g}$ electrons with
trigonal vibrational modes $Q_{4}$, $Q_{5}$, and $Q_{6}$ of a metal–ligand
octahedron.

In the case of the $d^{1}$ configuration, an increase in the SOC leads to a
gradual decay (but not vanishing) of the characteristic JT distortions. At a
strong SOC, we obtain a 4D analog of the “Mexican hat” adiabatic potential
energy surface with the potential for concomitant quantum effects.

For the $d^{2}$ configuration, the SOC also suppresses the JT distortions.
However, in contrast with the $d^{1}$ case, these distortions can vanish due
to such additional factor as the Hund’s rule intraatomic exchange, $J_{H}$.
Quite an unusual situation arises for the $d^{3}$ configuration, for which in
the absence of SOC, owing to the strong Hund’s rule exchange, three electrons
with parallel spins occupy three $t_{2g}$ levels, thus removing orbital
degeneracy. The SOC redistributes such electrons favoring the occupation of
the $j=3/2$ state. That is why, the orbital degeneracy is restored, and the JT
effect begins to work, i.e. in this case the SOC does not suppress but
activates JT effect.

The $d^{4}$ and $d^{5}$ cases turn out to be quite similar in their behavior.
In both cases, the JT distortions abruptly vanish at a sufficiently strong SOC
since the latter favors the formation of the $j=1/2$ doublet for the $d^{4}$
and a singlet $J=0$ state for $d^{5}$ configurations, which do not exhibit the
orbital degeneracy, thus removing the JT effect.

The results, in a nutshell, are that the qualitative behavior of JT effect for
trigonal distortions (the $t\otimes T$ problem) for the strong SOC coupling is
qualitatively similar to that for coupling to tetragonal distortions (the
$t\otimes E$ problem) considered earlier Streltsov and Khomskii (2020). This
agrees with previous results, where limiting situations of a very large and
small SOC were considered Warren (1982). In particular, the JT effect for
trigonal distortions can survive even for very strong SOC, when we can
describe the situation by the $j=3/2$ quartet. In this sense, the situation in
such a limit is not identical to the actual $e_{g}$ case with two Kramers
doublets. The more complicated nature of the SOC-stabilized states with strong
entanglement of spins and orbitals, changes the situation drastically and
makes it quite nontrivial.

An interesting conclusion is that the very strong SOC leads to a continuous
degeneracy of the ground states (”Mexican hat”, in this case ”Mexican globe”).
In contrast with the usual situation without SOC, where this continuous
degeneracy is lifted by higher-order effects, here it is destroyed already for
linear JT coupling but for finite SOC. This can lead to interesting effects in
particular in the dynamics of such systems.

It is worthwhile to note that features of local distortions of the ligand
octahedra in $4d$ and $5d$ transition metal compounds have become a subject of
many recent studies. These are e.g. local point symmetry breaking in Ba2NaOsO6
seen by local methods such as NMR Lu et al. (2017), while diffraction does not
detect any deviations from the cubic symmetry Erickson et al. (2007) or
noncubic crystal-field seen by the resonant inelastic x-ray scattering in
various iridates expected to be in undistorted octachedra in the limit of
large SOC Sala et al. (2014); Liu et al. (2012); Revelli et al. (2019). One
might also mention unexpected elongation of the octahedra in Ba2SmMoO6
Mclaughlin (2008), Ba2NdMoO6 Cussen et al. (2006), Sr2MgReO6 Sarapulova et al.
(2015), Sr2LiOsO6 da Cruz Pinha Barbosa et al. (2022), and K2TaCl6 Ishikawa et
al. (2019), which sometimes is accompanied by even further lowering of the
symmetry and thus might involve $T$ modes. The coupling to the trigonal
vibrations should be especially relevant for systems containing corresponding
transition metal ions with face-sharing octahedra – such as, for example,
systems of the type of Ba3TMRu2O9 or Ba3TMIr2O9, with TM =Na, Ca, Y, Ce etc.
Detailed study of these materials is an important, but at the same time
complicated problem, since there are many other factors affecting lattice
distortions in addition to the conventional JT effect such as purely steric
factors defined by the Goldschmidt tolerance factor, or possible high-order
multipolar orderings Mosca et al. (2021); Pourovskii et al. (2021).

Thus, we see that even a single-site problem involving the SOC provides a real
cornucopia of interesting new physics. Taking into consideration the
interactions between JT sites in a lattice results in an interplay among
orbital, spin, and lattice degrees of freedom. It has been shown on example of
the $E$ distortions that the spin-orbit and vibronic interactions also compete
in this case as well and, e.g., for $d^{1}$ configuration suppression of the
JT distortions by the SOC also occurs Streltsov and Khomskii (2020). taking
into account the interactions of the JT ion with the lattice may bring even a
richer physical content. Indeed, these are not simple electronic orbitals, but
spin-orbitals, which are now coupled with lattice distortions and therefore
one might expect other novel, e.g. magneto-elastic, effects in this case, but
details depend on a particular occupation of $d$ orbitals, lattice
connectivity and of course the strength of the SOC. We believe that the
results of this work should create a good basis for a further study of these
effects.

## Acknowledgements

S.V.S., F.T., and K.I.K. acknowledge the support of the Russian Science
Foundation (project No. 20-62-46047) in the part concerning the numerical
calculations. The work of D.I.Kh. is supported by the Deutsche
Forschungsgemeinschaft (project No. 277146847-CRC 1238).

##  Appendix A Goldstone modes for $d^{1}$ in the case of $\lambda\to\infty$

In this Appendix, we will show analytically that the adiabatic potential
energy surface for electronic configuration $d^{1}$ in the limit of
$\lambda\to\infty$ is similar to the “Mexican hat” in the space of 4D
($Q_{4}$, $Q_{5}$, $Q_{6}$, $E$), where $E$ is the energy.

For the sake of simplicity, we first perform the derivation for the $t\otimes
E$ problem, which was considered in detail in Ref. Streltsov and Khomskii
(2020). In this case, instead of $Q_{4}$, $Q_{5}$, and $Q_{6}$ one has only
two phonon modes, $Q_{2}$ and $Q_{3}$. As the first step, we transform full
Hamiltonian (3) of Ref. Streltsov and Khomskii (2020) including both SOC and
JT terms to the basis, which is diagonal in the space of $j_{1/2}$ and
$j_{3/2}$ states. In the limit of $\lambda\to\infty$, the splitting $j_{1/2}$
and $j_{3/2}$ becomes infinitely large and one can work only with $4\times 4$
Hamiltonian for $j_{3/2}$ states. Its diagonalization gives the spectrum with
the lowest in energy eigenvalue

$\displaystyle
E(Q_{2},Q_{3})=-\frac{\lambda}{2}-\frac{g}{3}\sqrt{Q_{2}^{2}+Q_{3}^{2}}+\frac{B}{2}(Q_{2}^{2}+Q_{3}^{2}).$
(10)

There are two types of extrema – the first one at $(Q_{2}=0,Q_{3}=0)$ is
absolutely unstable and the second one corresponding to the absolute minimum
is parametrized by the equation

$\displaystyle Q_{2}^{2}+Q_{3}^{2}=\frac{4}{9}\frac{g^{2}}{B^{2}}.$ (11)

This is nothing else, but the equation describing the trough of the “Mexican
hat”. We see that the ground state of our problem is highly degenerate and it
is described by the rotation in the $Q_{2}Q_{3}$ space, i.e. by the Goldstone
mode.

Now, one can repeat the same calculations for the $t\otimes T$ problem. Then,
we obtain

$\displaystyle E(Q_{4},Q_{5},Q_{6})$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{2}-\frac{g}{\sqrt{3}}\sqrt{Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}}$
(12) $\displaystyle+$
$\displaystyle\frac{B}{2}(Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}),$

i.e. the same quadratic form characteristic for the Goldstone modes, which
again gives equation for the trough of the “Mexican hat”, but now in the 4D
space

$\displaystyle Q_{4}^{2}+Q_{5}^{2}+Q_{6}^{2}=\frac{4}{3}\frac{g^{2}}{B^{2}}.$
(13)

##  Appendix B  Wave functions

If one considers a metal-ligand octahedron with the axes directed to the
metal-ligand bonds, then the trigonal orbitals are

$\displaystyle|a_{1g}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{3}}\left(|xy\rangle+|xz\rangle+|yz\rangle\right),$
(14) $\displaystyle|e_{g}^{\pi}\rangle$ $\displaystyle=$
$\displaystyle\pm\frac{1}{\sqrt{3}}\left(|xy\rangle+{\rm e}^{\pm 2\pi{\rm
i}/3}|xz\rangle+{\rm e}^{\mp 2\pi{\rm i}/3}|yz\rangle\right).$ (15)

However, if $z$ axis is chosen along the trigonal $[1,1,1]$ direction, they
can be written in a more suitable form Khomskii et al. (2016):

$\displaystyle|a_{1g}\rangle$ $\displaystyle=$
$\displaystyle|3z^{2}-r^{2}\rangle,$ (16) $\displaystyle|e_{g,1}^{\pi}\rangle$
$\displaystyle=$
$\displaystyle-\frac{2}{\sqrt{6}}|xy\rangle+\frac{1}{\sqrt{3}}|yz\rangle,$
$\displaystyle|e_{g,2}^{\pi}\rangle$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{6}}|x^{2}-y^{2}\rangle+\frac{1}{\sqrt{3}}|xz\rangle.$
(17)

Then, one may construct $l^{z}=\pm 1$ states from the $e_{g}^{\pi}$ orbitals

$\displaystyle|l^{z}_{\pm 1}\rangle=|e_{g,1}^{\pi}\pm{\rm
i}e_{g,2}^{\pi}\rangle,$ (18)

while $|l^{z}_{0}\rangle=|a_{1g}\rangle$.

Finally the $j=3/2$ wave-functions are:

$\displaystyle|j_{3/2},j^{z}_{3/2}\rangle$ $\displaystyle=$
$\displaystyle|l^{z}_{1},\uparrow\rangle,$
$\displaystyle|j_{3/2},j^{z}_{-3/2}\rangle$ $\displaystyle=$
$\displaystyle|l^{z}_{-1},\downarrow\rangle,$
$\displaystyle|j_{3/2},j^{z}_{1/2}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}|l^{z}_{0},\uparrow\rangle+\frac{1}{\sqrt{3}}|l^{z}_{1},\downarrow\rangle$
$\displaystyle|j_{3/2},j^{z}_{-1/2}\rangle$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2}{3}}|l^{z}_{0},\downarrow\rangle+\frac{1}{\sqrt{3}}|l^{z}_{-1},\uparrow\rangle$
(19)

## References

  * Takayama et al. (2021) T. Takayama, J. Chaloupka, A. Smerald, G. Khaliullin, and H. Takagi, “Spin-orbit-entangled electronic phases in 4$d$ and 5$d$ transition-metal compounds,” J. Phys. Soc. Japan 90, 062001 (2021).
  * Öpik and Pryce (1957) U. Öpik and M. H. L. Pryce, “Studies of the Jahn–Teller effect. I. A survey of the static problem,” Proc. R. Soc. A 238, 425 (1957).
  * Bersuker and Polinger (1989) I. B. Bersuker and V. Z. Polinger, _Vibronic Interactions in Molecules and Crystals_ (Springer-Verlag, 1989).
  * Bersuker (2006) I. B. Bersuker, _The Jahn–Teller Effect_ (Cambridge University Press, 2006).
  * Khomskii (2014) D. I. Khomskii, _Transition Metal Compounds_ (Cambridge University Press, 2014).
  * Streltsov and Khomskii (2017) S. Streltsov and D. Khomskii, “Orbital physics in transition metal compounds: New trends,” Phys.-Usp. 60, 1121 (2017).
  * Khomskii and Streltsov (2020) D. Khomskii and S. Streltsov, “Orbital effects in solids: Basics, recent progress, and opportunities,” Chem. Rev. 121, 2992 (2020).
  * Moffitt and Thorson (1957) W. Moffitt and W. Thorson, “Vibronic states of octahedral complexes,” Phys. Rev. 108, 1251 (1957).
  * O’Brien (1969) M. C. O’Brien, “Dynamic Jahn-Teller Effect in an Orbital Triplet State Coupled to Both $E_{g}$ and $T_{2\sigma}$ Vibrations,” Physical Review 187, 407 (1969).
  * Bacci et al. (1975) M. Bacci, A. Ranfagni, M. Fontana, and G. Viliani, “Coexistence of tetragonal with orthorhombic or trigonal Jahn-Teller distortions in an $O_{h}$ complex: A plausible interpretation of alkali-halide phosphors luminescence,” Physical Review B 11, 3052 (1975).
  * Bates (1978) C. A. Bates, “Jahn–Teller effects in paramagnetic crystals,” Phys. Rep. 35, 187 (1978).
  * Judd (1984) B. R. Judd, “Jahn–Teller trajectories,” Adv. Chem. Phys. LVII, 247 (1984).
  * Chen and Balents (2011) G. Chen and L. Balents, “Spin-orbit coupling in $d^{2}$ ordered double perovskites,” Phys. Rev. B 84, 094420 (2011).
  * Plotnikova et al. (2016) E. M. Plotnikova, M. Daghofer, J. van den Brink, and K. Wohlfeld, “Jahn–Teller effect in systems with strong on-site spin-orbit coupling,” Phys. Rev. Lett. 116, 106401 (2016).
  * Liu and Khaliullin (2019) H. Liu and G. Khaliullin, “Pseudo-Jahn–Teller effect and magnetoelastic coupling in spin-orbit Mott insulators,” Phys. Rev. Lett. 122, 57203 (2019).
  * Nikolaev et al. (2018) S. Nikolaev, I. Solovyev, A. Ignatenko, V. Irkhin, and S. V. Streltsov, “Realization of the anisotropic compass model on the diamond lattice of Cu2+ in CuAl2O4,” Phys. Rev. B 98, 201106 (2018).
  * Ishikawa et al. (2019) H. Ishikawa, T. Takayama, R. K. Kremer, J. Nuss, R. Dinnebier, K. Kitagawa, K. Ishii, and H. Takagi, “Ordering of hidden multipoles in spin-orbit entangled $5d^{1}$ Ta chlorides,” Phys. Rev. B 100, 045142 (2019).
  * Paramekanti et al. (2020) A. Paramekanti, D. D. Maharaj, and B. D. Gaulin, “Octupolar order in $d$-orbital Mott insulators,” Phys. Rev. B 101, 054439 (2020).
  * Khaliullin et al. (2021) G. Khaliullin, D. Churchill, P. P. Stavropoulos, and H.-Y. Kee, “Exchange interactions, Jahn–Teller coupling, and multipole orders in pseudospin one-half $5{d}^{2}$ Mott insulators,” Phys. Rev. Research 3, 033163 (2021).
  * Mosca et al. (2021) D. F. Mosca, L. V. Pourovskii, B. H. Kim, P. Liu, S. Sanna, F. Boscherini, S. Khmelevskyi, and C. Franchini, “Interplay between multipolar spin interactions, Jahn–Teller effect and electronic insulator,” Phys. Rev. B 103, 104401 (2021).
  * Warren (1982) K. D. Warren, in _Complex Chemistry. Structure and Bonding._ (Springer, 1982), vol. 57, p. 119.
  * Streltsov and Khomskii (2020) S. Streltsov and D. Khomskii, “Jahn–Teller effect and spin-orbit coupling: Friends or foes?,” Phys. Rev. X 10, 031043 (2020).
  * Iwahara et al. (2018) N. Iwahara, V. Vieru, and L. F. Chibotaru, “Spin-orbital-lattice entangled states in cubic $d^{1}$ double perovskites,” Phys. Rev. B 98, 075138 (2018).
  * Kloß et al. (2021) S. D. Kloß, M. L. Weidemann, and J. P. Attfield, “Preparation of Bulk-Phase Nitride Perovskite LaReN3 and Topotactic Reduction to LaNiO2-Type LaReN2,” Angewandte Chemie - International Edition 60, 22260 (2021).
  * Kim et al. (2019) C. H. Kim, S. Baidya, H. Cho, V. V. Gapontsev, S. V. Streltsov, D. I. Khomskii, J.-g. Park, A. Go, and H. Jin, “Theoretical Evidence of Spin-Orbital-Entangled J=1/2 State in the 3d Transition Metal Oxide CuAl2O4,” Phys. Rev. B 100, 161104 (2019).
  * Huang et al. (2022) H. Y. Huang, A. Singh, C. I. Wu, J. D. Xie, J. Okamoto, A. A. Belik, E. Kurmaev, A. Fujimori, C. T. Chen, S. V. Streltsov, et al., “Resonant inelastic X-ray scattering as a probe of J${}_{\text{eff}}$=1/2 state in 3d transition-metal oxide,” npj Quantum Materials 7, 33 (2022).
  * Prodan et al. (2021) L. Prodan, S. Yasin, A. Jesche, J. Deisenhofer, H.-A. K. von Nidda, F. Mayr, S. Zherlitsyn, J. Wosnitza, A. Loidl, and V. Tsurkan, “Unusual field-induced spin reorientation in ${\mathrm{FeCr}}_{2}{\mathrm{S}}_{4}$: Field tuning of the Jahn-Teller state,” Phys. Rev. B 104, L020410 (2021).
  * Weng and Dong (2021) Y. Weng and S. Dong, “Manipulation of ${J}_{\mathrm{eff}}\phantom{\rule{4.0pt}{0.0pt}}=\phantom{\rule{4.0pt}{0.0pt}}\frac{3}{2}$ states by tuning the tetragonal distortion,” Phys. Rev. B 104, 165150 (2021).
  * Zhang et al. (2022) Y. Zhang, L. F. Lin, A. Moreo, and E. Dagotto, “Electronic and magnetic properties of quasi-one-dimensional osmium halide OsCl4,” Applied Physics Letters 120, 023101 (2022).
  * Ham (1965) F. S. Ham, “Dynamical Jahn-Teller Effect in Paramagnetic Resonance Spectra: Orbital Reduction Factors and Partial Quenching of Spin-Orbit Interaction,” Phys. Rev. 138, A1727 (1965).
  * Englman (1972) R. Englman, _The Jahn-Teller Effect in Molecules and Crystals_ , Interscience Monographs and Texts in Physics and Astronomy (Wiley-Interscience, 1972), ISBN 9780471241683\.
  * Keimer et al. (2000) B. Keimer, D. Casa, A. Ivanov, J. W. Lynn, M. V. Zimmermann, J. P. Hill, D. Gibbs, Y. Taguchi, and Y. Tokura, “Spin Dynamics and Orbital State in LaTiO3,” Physical Review Letters 85, 3946 (2000).
  * Abragam and Bleaney (1970) A. Abragam and B. Bleaney, _Electron Paramagnetic Resonance of Transition Ions_ (Clarendon press, Oxford, 1970).
  * Georges et al. (2013) A. Georges, L. D. Medici, and J. Mravlje, “Strong correlations from Hund’s coupling,” Annu. Rev. Condens. Matter Phys. 4, 137 (2013).
  * Lu et al. (2017) L. Lu, M. Song, W. Liu, A. P. Reyes, P. Kuhns, H. O. Lee, I. R. Fisher, and V. F. Mitrović, “Magnetism and local symmetry breaking in a Mott insulator with strong spin orbit interactions,” Nat. Comm. 8, 14407 (2017), eprint 1701.06117.
  * Erickson et al. (2007) A. S. Erickson, S. Misra, G. J. Miller, R. R. Gupta, Z. Schlesinger, W. A. Harrison, J. M. Kim, and I. R. Fisher, “Ferromagnetism in the Mott insulator Ba2NaOsO6,” Phys. Rev. Lett. 99, 016404 (2007).
  * Sala et al. (2014) M. M. Sala, K. Ohgushi, A. Al-Zein, Y. Hirata, G. Monaco, and M. Krisch, “CaIrO3: A Spin-Orbit Mott Insulator Beyond the j${}_{\text{eff}}=1/2$ Ground State,” Phys. Rev. Lett. 112, 176402 (2014).
  * Liu et al. (2012) X. Liu, V. M. Katukuri, L. Hozoi, W.-G. Yin, M. P. M. Dean, M. H. Upton, J. Kim, D. Casa, A. Said, T. Gog, et al., “Testing the Validity of the Strong Spin-Orbit-Coupling Limit for Octahedrally Coordinated Iridate Compounds in a Model System Sr3CuIrO6,” Phys. Rev. Lett. 109, 157401 (2012).
  * Revelli et al. (2019) A. Revelli, C. C. Loo, D. Kiese, P. Becker, T. Fröhlich, T. Lorenz, M. Moretti Sala, G. Monaco, F. L. Buessen, J. Attig, et al., “Spin-orbit entangled $j=1/2$ moments in Ba2CeIrO6: A frustrated fcc quantum magnet,” Phys. Rev. B 100, 085139 (2019).
  * Mclaughlin (2008) A. C. Mclaughlin, “Simultaneous Jahn-Teller distortion and magnetic order in the double perovskite Ba${}_{2}^{154}$SmMoO6,” Phys. Rev. B 78, 132404 (2008).
  * Cussen et al. (2006) E. J. Cussen, D. R. Lynham, and J. Rogers, “Magnetic Order Arising from Structural Distortion: Structure and Magnetic Properties of Ba2LnMoO6,” Chem. Mater. 18, 2855 (2006).
  * Sarapulova et al. (2015) A. Sarapulova, P. Adler, W. Schnelle, D. Mikhailova, C. Felser, L. H. Tjeng, and M. Jansen, “Sr2MgOsO6: A Frustrated Os6+ (5$d^{2}$) Double Perovskite with Strong Antiferromagnetic Interactions,” Z. Anorg. Allg. Chem. 641, 769 (2015).
  * da Cruz Pinha Barbosa et al. (2022) V. da Cruz Pinha Barbosa, J. Xiong, P. M. Tran, M. A. McGuire, J. Yan, M. T. Warren, R. V. Aguilar, W. Zhang, M. Randeria, N. Trivedi, et al., “The Impact of Structural Distortions on the Magnetism of Double Perovskites Containing $5d^{1}$ Transition-Metal Ions,” Chem. Mater. 34, 1098–1109 (2022).
  * Pourovskii et al. (2021) L. V. Pourovskii, D. F. Mosca, and C. Franchini, “Ferro-octupolar order and low-energy excitations in $d^{2}$ double perovskites of osmium,” Phys. Rev. Lett. 127, 237201 (2021).
  * Khomskii et al. (2016) D. I. Khomskii, K. I. Kugel, A. O. Sboychakov, and S. V. Streltsov, “Role of local geometry in the spin and orbital structure of transition metal compounds,” JETP 122, 484 (2016).


# Horizon Thermodynamics and Gravitational Field Equations in Hořava-Lifshitz
Gravity

Rong-Gen Cai<EMAIL_ADDRESS>Key Laboratory of Frontiers in Theoretical
Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China  Department
of Physics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan  Nobuyoshi
Ohta<EMAIL_ADDRESS>Department of Physics, Kinki University,
Higashi-Osaka, Osaka 577-8502, Japan

###### Abstract

We explore the relationship between the first law of thermodynamics and
gravitational field equation at a static, spherically symmetric black hole
horizon in Hořava-Lifshtiz theory with/without detailed balance. It turns out
that as in the cases of Einstein gravity and Lovelock gravity, the
gravitational field equation can be cast to a form of the first law of
thermodynamics at the black hole horizon. This way we obtain the expressions
for entropy and mass in terms of black hole horizon, consistent with those
from other approaches. We also define a generalized Misner-Sharp energy for
static, spherically symmetric spacetimes in Hořava-Lifshtiz theory. The
generalized Misner-Sharp energy is conserved in the case without matter field,
and its variation gives the first law of black hole thermodynamics at the
black hole horizon.

††preprint: KU-TP 035

##  I Introduction

The holographic principle might be one of the principles of nature, which
states that a theory with gravity could be equivalent to a theory without
gravity in one less dimension. The well-known AdS/CFT correspondence Mald  is
a realization of the holographic principle, while the latter is motivated by
black hole thermodynamics. The black hole thermodynamics says that a black
hole behaves as an ordinary thermodynamic system with temperature and entropy.
The temperature of a black hole is proportional to surface gravity at its
horizon, while the entropy of the black hole is measured by its horizon area.
Black hole mass, temperature and entropy satisfy the first law of
thermodynamics. These results come from a combination of quantum mechanics,
black hole geometry and general relativity. This implies that there might
exist a deep connection between thermodynamics and gravity theory.

Indeed some pieces of evidence have been accumulated for the connection
between thermodynamics and gravity theory in the literature. Assuming there is
a proportionality between entropy and horizon area, Jacobson Jac  derived the
Einstein field equation by using the fundamental Clausius relation, $\delta
Q=TdS$, connecting heat, temperature and entropy. The key idea is to demand
that this relation holds for all the local Rindler causal horizon through each
spacetime point, with $\delta Q$ and $T$ interpreted as the energy flux and
Unruh temperature seen by an accelerated observer just inside the horizon. In
this way, the Einstein field equation is nothing but an equation of state of
spacetime. More recently, Jacobson’s argument has been generalized to all
diffeomorphism-invariant theories of gravity Brus (however, see also Parikh ).
For $f(R)$ theory and scalar-tensor theory, see also ES ; WuYZ . In fact,
investigating the thermodynamics of spacetime for $f(R)$ theory Jac1 ; AC1 ;
AC  and scalar-tensor theory AC1 ; CC1 , it is found that a nonequilibrium
thermodynamic setup has to be employed. Further, it is argued that if shear of
spacetime is not assumed to vanish, the nonequilibrium thermodynamic setting
is required even for the Einstein general relativity Eling ; CL . There an
internal entropy production term has to be introduced to balance energy
conservation. The internal entropy production term $dS_{i}$ is proportional to
the squared shear of the horizon and the ratio of the proportionality constant
to the area entropy density is $1/4\pi$. The latter is a universal value for
many kinds of conformal field theories with AdS duals KSS .

There exists another route in exploring the relationship between
thermodynamics and gravity theory. Padmanabhan Pad1  first noticed that the
gravitational field equation in a static, spherically symmetric spacetime can
be rewritten as a form of the ordinary first law of thermodynamics at a black
hole horizon. This indicates that Einstein’s equation is nothing but a
thermodynamic identity. For a recent review on this, see Pad2 . This
observation was then extended to the cases of stationary axisymmetric horizons
and evolving spherically symmetric horizons in the Einstein gravity KSP ,
static spherically symmetric horizons KP  and dynamical apparent horizons CCHK
in Lovelock gravity, and three-dimensional Banados-Teitelboim-Zanelli black
hole horizons Akbar1 . On the other hand, the relationship between the first
law of thermodynamics and dynamical equation of spacetime has been intensively
investigated in a Friedmann-Robertson-Walker (FRW) cosmological setup in
various gravity theories CK ; others ; AC07 ; CC2 ; AC1 ; AC ; CC1 ; GW ;
Others1 ; it is shown that (modified) Friedmann equations can be cast to a
form of the first law of thermodynamics, and there exists a Hawking radiation
associated with apparent horizon in a FRW universe CCH .

Recently a field theory model for a UV complete theory of gravity was proposed
by Hořava Hor , which is a nonrelativistic renormalizable theory of gravity
and is expected to recover Einstein’s general relativity at large scales. This
theory is named Hořava-Lifshitz theory in the literature since at the UV fixed
point of the theory space and time have different scalings. Since then a lot
of work has been done in exploring various aspects of the theory; for a more
or less complete list of references, see, for example, WDR .

In this paper we discuss the relationship between the first law of
thermodynamics and gravitational field equation in Hořava-Lifshitz theory. In
static spherically symmetric black hole spacetimes, we show that the
gravitational field equation can be rewritten as $dE-TdS=PdV$ at the black
hole horizon. Note that in Hořava-Lifshitz theory the full diffeomorphism
invariance is broken to the “foliation-preserving” diffeomorphism. Therefore
our result is a nontrivial generalization of Padmanabhan’s observation. In
addition, we discuss the question of whether one can define a generalized
Misner-Sharp quasilocal energy in Hořava-Lifshitz theory. The answer is
positive. We define a generalized Misner-Sharp energy. It is a conserved
charge when the matter field is absent, while its variation at a black hole
horizon gives the first law of black hole thermodynamics.

This paper is organized as follows. In the next section we review the
Padmanabhan’s observation by extending his discussion to a more general
spherically symmetric spacetime. In Sec. III we consider black hole spacetimes
in Hořava-Lifshitz theory. In Sec. IV the case of IR modified Hořava-Lifshitz
theory is discussed. In Sec. V we define a generalized Misner-Sharp quasilocal
energy for static, spherically symmetric spacetimes in Hořava-Lifshitz theory
and discuss its properties. The conclusion is given in Sec. VI.

##  II Black Holes in Einstein Gravity

As a warm-up exercise, in this section, we will briefly review the observation
made by Padmanabhan Pad1  by generalizing his discussion to a more general
spherically symmetric case. In Einstein’s general relativity, the
gravitational field equations are

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi GT_{\mu\nu},$ (1)

where $G_{\mu\nu}$ is Einstein tensor and $T_{\mu\nu}$ is the energy-momentum
tensor of matter field. On the other hand, for a general static, spherically
symmetric spacetime, its metric can be written down as

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+b^{2}(r)(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (2)

where $f(r)$ and $b(r)$ are two functions of the radius coordinate $r$. (Note
that in the Padmanabhan’s discussion Pad1 , the metric is assumed in a form
(2) with $b(r)=r$; when matter is present, however, such a metric form is not
always satisfied. See also KSP .) Suppose the metric (2) describes a
nonextremal black hole with horizon at $r_{+}$, then the function $f(r)$ has a
simple zero at $r=r_{+}$. Namely, $f^{\prime}(r)|_{r=r_{+}}=0$, but
$f^{\prime\prime}(r)|_{r=r_{+}}\neq 0$. It is easy to show that the Hawking
temperature of the black hole associated with the horizon $r_{+}$ is

$T=\frac{1}{4\pi}f^{\prime}(r)|_{r=r_{+}}\equiv\frac{1}{4\pi}f^{\prime}(r_{+}),$
(3)

where a prime stands for the derivative with respective to $r$. Einstein’s
equations in the metric (2) have the components

$\displaystyle G^{t}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{b^{2}}(-1+fb^{\prime
2}+b(f^{\prime}b^{\prime}+2fb^{\prime\prime})),$ $\displaystyle G^{r}_{r}$
$\displaystyle=$
$\displaystyle\frac{1}{b^{2}}(-1+bf^{\prime}b^{\prime}+fb^{\prime 2}).$ (4)

Note that at the horizon, one has $f(r)=0$, and then

$G^{t}_{t}|_{r=r_{+}}=G^{r}_{r}|_{r=r_{+}}=\frac{1}{b^{2}}(-1+bf^{\prime}b^{\prime})|_{r=r_{+}}.$
(5)

Therefore at the horizon, the $t-t$ component of Einstein’s equations can be
expressed as

$-1+bf^{\prime}b^{\prime}=8\pi Gb^{2}P,$ (6)

where $P=T^{r}_{r}|_{r=r_{+}}$ is the radial pressure of matter at the
horizon. Note that here (5) guarantees $T^{t}_{t}=T^{r}_{r}$ at the horizon.
Now we multiply $dr_{+}$ on both sides of (6) and rewrite this equation as

$\frac{1}{2G}bf^{\prime}b^{\prime}dr_{+}-\frac{1}{2G}dr_{+}=4\pi
b^{2}Pdr_{+}.$ (7)

Note that $b$ is a function of $r$ only and $f^{\prime}$ has a relation to the
Hawking temperature as (3). One then can rewrite the above equation as

$Td\left(\frac{4\pi b^{2}}{4G}\right)-d\left(\frac{r_{+}}{2G}\right)=PdV,$ (8)

where $dV=4\pi b^{2}dr_{+}$. Therefore $V$ is just the volume of the black
hole with horizon radius $r_{+}$ in the coordinate (2). The equation (8) can
be further rewritten as

$TdS-dE=PdV,$ (9)

with identifications

$S=\frac{4\pi b^{2}}{4G}=\frac{A}{4G},\ \ \ E=\frac{r_{+}}{2G}.$ (10)

Clearly here $S$ is precisely the entropy of the black hole, while $E$ is the
Misner-Sharp energy MS  at the horizon. Thus we have shown in general that at
black hole horizon, Einstein’s equations can be cast into the form of the
first law of thermodynamics.

##  III Black Holes in Hořava-Lifshitz Gravity

In the $(3+1)$-dimensional Arnowitt-Deser-Misner formalism, where the metric
can be written as

$ds^{2}=-N^{2}dt^{2}+g_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt),$ (11)

and for a spacelike hypersurface with a fixed time, its extrinsic curvature
$K_{ij}$ is

$K_{ij}=\frac{1}{2N}(\dot{g}_{ij}-\nabla_{i}N_{j}-\nabla_{j}N_{i}),$ (12)

where a dot denotes a derivative with respect to $t$ and covariant derivatives
defined with respect to the spatial metric $g_{ij}$.

The action of Hořava-Lifshitz theory is Hor ; LMP

$\displaystyle I$ $\displaystyle=$ $\displaystyle\int dtd^{3}x({\cal
L}_{0}+{\cal L}_{1}+{\cal L}_{m}),$ (13) $\displaystyle{\cal L}_{0}$
$\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{2}{\kappa^{2}}(K_{ij}K^{ij}-\lambda
K^{2})+\frac{\kappa^{2}\mu^{2}(\Lambda
R-3\Lambda^{2})}{8(1-3\lambda)}\right\\},$ $\displaystyle{\cal L}_{1}$
$\displaystyle=$
$\displaystyle\sqrt{g}N\left\\{\frac{\kappa^{2}\mu^{2}(1-4\lambda)}{32(1-3\lambda)}R^{2}-\frac{\kappa^{2}}{2\omega^{4}}\left(C_{ij}-\frac{\mu\omega^{2}}{2}R_{ij}\right)\left(C^{ij}-\frac{\mu\omega^{2}}{2}R^{ij}\right)\right\\}.$

where $\kappa^{2}$, $\lambda$, $\mu$, $\omega$ and $\Lambda$ are constant
parameters and the Cotten tensor, $C_{ij}$, is defined by

$C^{ij}=\epsilon^{ikl}\nabla_{k}\left(R^{j}_{\
l}-\frac{1}{4}R\delta^{j}_{l}\right)=\epsilon^{ikl}\nabla_{k}R^{j}_{\
l}-\frac{1}{4}\epsilon^{ikj}\partial_{k}R.$ (14)

The first two terms in ${\cal L}_{0}$ are the kinetic terms, others in $({\cal
L}_{0}+{\cal L}_{1})$ give the potential of the theory in the so-called
“detailed-balance” form, and ${\cal L}_{m}$ stands for the Lagrangian of other
matter field.

Comparing the action to that of general relativity, one can see that the speed
of light, Newton’s constant and the cosmological constant are

$c=\frac{\kappa^{2}\mu}{4}\sqrt{\frac{\Lambda}{1-3\lambda}},\ \
G=\frac{\kappa^{2}c}{32\pi},\ \ \tilde{\Lambda}=\frac{3}{2}\Lambda,$ (15)

respectively. Let us notice that when $\lambda=1$, ${\cal L}_{0}$ could be
reduced to the usual Lagrangian of Einstein’s general relativity. Therefore it
is expected that general relativity could be approximately recovered at large
distances when $\lambda=1$. Here we will mainly consider the case of
$\lambda=1$, but will also discuss the $\lambda\neq 1$ case briefly at the end
of this paper.

Now we consider black hole spacetime with metric ansatz LMP ; CCO

$ds^{2}=-\tilde{N}^{2}(r)f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega_{k}^{2},$
(16)

where $d\Omega_{k}^{2}$ denotes the line element for a two-dimensional
Einstein space with constant scalar curvature $2k$. Without loss of
generality, one may take $k=0$, $\pm 1$, respectively. Substituting the metric
(16) into (13), we have

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Lambda\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{-3\Lambda r^{2}-2(f-k)-2r(f-k)^{\prime}\right.$ (17)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.+\frac{(\lambda-1)f^{\prime
2}}{2\Lambda}+\frac{(2\lambda-1)(f-k)^{2}}{\Lambda
r^{2}}-\frac{2\lambda(f-k)}{\Lambda r}f^{\prime}+\alpha r^{2}{\cal
L}_{m}\right\\},$

where a prime denotes the derivative with respect to $r$, $\Omega_{k}$ is the
volume of the two-dimensional Einstein space and the constant
$\alpha=8(1-3\lambda)/\kappa^{2}\mu^{2}\Lambda$. In the case of $\lambda=1$ we
can rewrite the action as

$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left\\{\left(x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\right)^{\prime}+x^{2}(\frac{\alpha}{-\Lambda}){\cal
L}_{m}\right\\}.$ (18)

Note that here $x=\sqrt{-\Lambda}r$, a prime becomes the derivative with
respect to $x$. Varying the action with $\tilde{N}$, we obtain the equations
of motion

$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}-2(f-k)-\frac{(f-k)^{2}}{x^{2}}-2xf^{\prime}+\frac{2(f-k)f^{\prime}}{x}\right)=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}.$ (19)

Suppose the nonextremal black hole (16) has a horizon radius $r_{+}$, namely
$x_{+}=\sqrt{-\Lambda}r_{+}$. Then the Hawking temperature of the black hole
is

$T=\frac{1}{4\pi}\tilde{N}(r)\left.\frac{df}{dr}\right|_{r=r_{+}}=\frac{\sqrt{-\Lambda}}{4\pi}\tilde{N}(x)f^{\prime}|_{x=x_{+}}.$
(20)

Now we consider a class of solutions with $\tilde{N}(r)=$ const. For example,
the charged black hole solution discussed in CCO  belongs to this class of
solutions. In this case one can set $\tilde{N}=1$ by rescaling the time
coordinate $t$. Note that here not all solutions with matter field have the
form $\tilde{N}=1$. At the horizon $x_{+}$, Eq. (19) is reduced to

$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}_{+}+2k-\frac{k^{2}}{x^{2}_{+}}-2x_{+}f^{\prime}-\frac{2kf^{\prime}}{x_{+}}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}.$ (21)

Multiplying both sides with $dx_{+}$, a variation of the horizon radius, we
have

$\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(2(x_{+}+\frac{k}{x_{+}})f^{\prime}dx_{+}-(3x^{2}_{+}+2k-\frac{k^{2}}{x^{2}_{+}})dx_{+}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}Pdx_{+},$
(22)

where $P=\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}$. Note that the Hawking temperature
turns to be $T=f^{\prime}\sqrt{-\Lambda}/4\pi$ when $\tilde{N}=1$. The above
equation then can be rewritten as

$TdS-dE=PdV,$ (23)

where

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k\ln
x_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16x_{+}}\left(x_{+}^{2}+k\right)^{2},$
(24)

$V=\frac{\Omega_{k}}{3}r_{+}^{3}$, and $S_{0}$ is an undetermined constant.
Clearly $V$ is the volume of black hole with radius $r_{+}$. Comparing (III)
with black hole entropy and mass defined through a Hamiltonian approach in our
previous papers CCO , we see that $S$ and $E$ are just black hole entropy and
mass in terms of horizon radius $x_{+}$, and the gravitational field equation
at the black hole horizon can be cast to the form of the first law of
thermodynamics. Note that here we have obtained expressions for black hole
entropy and mass, but have not used any explicit black hole solutions. In
other words, the above way provides a universal method to derive black hole
entropy and mass.

Now we turn to the case without the detailed-balance condition by considering
the action as LMP ; CCO

$I=\int dtd^{3}x({\cal L}_{0}+(1-\epsilon^{2}){\cal L}_{1}+{\cal L}_{m})$ (25)

where the parameter $\epsilon^{2}\neq 0$. In this case, instead of (18), we
have

$I=\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\int
dtdx\tilde{N}\left\\{\left(x^{3}-2x(f-k)+(1-\epsilon^{2})\frac{(f-k)^{2}}{x}\right)^{\prime}+x^{2}(\frac{\alpha}{-\Lambda}){\cal
L}_{m}\right\\}.$ (26)

Varying the action with respect to $\tilde{N}$ yields

$\displaystyle-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}-2(f-k)-(1-\epsilon^{2})\frac{(f-k)^{2}}{x^{2}}\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\left.2xf^{\prime}+(1-\epsilon^{2})\frac{2(f-k)f^{\prime}}{x}\right)=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}.$ (27)

At the black hole horizon where $f=0$, the equation reduces to

$-\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(3x^{2}_{+}+2k-(1-\epsilon^{2})\frac{k^{2}}{x^{2}_{+}}-2x_{+}f^{\prime}-(1-\epsilon^{2})\frac{2kf^{\prime}}{x_{+}}\right)=x^{2}_{+}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}.$ (28)

Multiplying $dx_{+}$ on both sides, one can express this equation as the form
(23) with the condition $\tilde{N}=1$, again. But this time we have

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k(1-\epsilon^{2})\ln
x_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16x_{+}}\left(x_{+}^{4}+2kx_{+}+(1-\epsilon^{2})k^{2}\right).$
(29)

These are nothing but the entropy and mass, expressed in terms of horizon
radius $x_{+}$, of the black hole solutions found in CCO .

Now we turn to the case with $z=4$ terms, where $z$ is the dynamical critical
exponent. Such terms are super-renormalizable ones. The vacuum black hole
solution for this case has been discussed in CLS . Including $z=4$ terms
changes ${\cal L}_{1}$ in (13) to

$\displaystyle{\cal{L}}_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{g}N\frac{\kappa^{2}}{8}\Big{\\{}\frac{4}{\omega^{4}}C^{ij}C_{ij}-\frac{4\mu}{\omega^{2}}C^{ij}R_{ij}-\frac{4}{\omega^{2}M}C^{ij}L_{ij}+\mu^{2}G_{ij}G^{ij}+\frac{2\mu}{M}G^{ij}L_{ij}$
(30) $\displaystyle~{}+\frac{2\mu}{M}\Lambda
L+\frac{1}{M^{2}}L^{ij}L_{ij}-\tilde{\lambda}\big{(}\frac{L^{2}}{M^{2}}-\frac{\mu
L}{M}(R-6\Lambda)+\frac{\mu^{2}}{4}R^{2}\big{)}\Big{\\}},$

where

$\displaystyle G^{ij}$ $\displaystyle=$ $\displaystyle
R^{ij}-\frac{1}{2}g^{ij}R$ $\displaystyle L^{ij}$ $\displaystyle=$
$\displaystyle(1+2\beta)(g^{ij}\nabla^{2}-\nabla^{i}\nabla^{j})R+\nabla^{2}G^{ij}$
$\displaystyle~{}~{}+2\beta
R(R^{ij}-\frac{1}{4}g^{ij}R)+2(R^{imjn}-\frac{1}{4}g^{ij}R^{mn})R_{mn},$
$\displaystyle L$ $\displaystyle\equiv$ $\displaystyle
g^{ij}L_{ij}=\bigg{(}\frac{3}{2}+4\beta\bigg{)}\nabla^{2}R+\frac{\beta}{2}R^{2}+\frac{1}{2}R_{ij}R^{ij},$
(31)

and $\tilde{\lambda}=\lambda/(3\lambda-1)$, $\beta$ and $M$ are two new
parameters. When $\beta=-3/8$ and $\lambda=1$, the action in the metric (16)
reduces to

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{{\kappa}^{2}\Omega_{k}}{16\sqrt{-\Lambda^{3}}}\int
dtdx\tilde{N}\left\\{\Big{[}\tilde{\mu}^{2}\Big{(}x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}\Big{)}\right.$
(32)
$\displaystyle\left.~{}~{}-2\tilde{\beta}\tilde{\mu}\Big{(}\frac{(f-k)^{3}}{x^{3}}-\frac{(f-k)^{2}}{x}\Big{)}+\tilde{\beta}^{2}\frac{(f-k)^{4}}{x^{5}}\Big{]}^{{}^{\prime}}+x^{2}\tilde{\alpha}{\cal
L}_{m}\right\\},$

where we define
$\tilde{\mu}=-\mu\Lambda,\tilde{\beta}=\frac{\Lambda^{2}}{4M}$,
$\tilde{\alpha}=16/\kappa^{2}$, and the prime is still the derivative with
respect to $x$. Varying the action with respect to $\tilde{N}$ yields

$\displaystyle-\frac{{\kappa}^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\Big{(}x^{3}-2x(f-k)+\frac{(f-k)^{2}}{x}$
$\displaystyle~{}~{}~{}~{}~{}~{}-2\frac{\tilde{\beta}}{\tilde{\mu}}(\frac{(f-k)^{3}}{x^{3}}-\frac{(f-k)^{2}}{x})+\frac{\tilde{\beta}^{2}}{\tilde{\mu}^{2}}\frac{(f-k)^{4}}{x^{5}}\Big{)}^{{}^{\prime}}=x^{2}\frac{\Omega_{k}}{(-\Lambda)^{3/2}}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}$ (33)

Again, we take the values of all quantities at the black hole horizon and then
multiply $dx_{+}$ on both sides, the above equation turns to be

$TdS-dE=PdV,$ (34)

where $P=\left.\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}\right|_{x=x_{+}}$, $V=\Omega_{k}r_{+}^{3}/3$ is the
volume of the black hole, and

$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\sqrt{-\Lambda}\Omega_{k}}{16}\left(x^{3}_{+}+2kx_{+}+\frac{k^{2}}{x_{+}}+2\frac{\tilde{\beta}}{\tilde{\mu}}(\frac{k^{3}}{x_{+}^{3}}+\frac{k^{2}}{x})+\frac{\tilde{\beta}^{2}}{\tilde{\mu}^{2}}\frac{k^{4}}{x_{+}^{5}}\right)$
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{4}\left(x_{+}^{2}+2k\ln
x_{+}-3\frac{\tilde{\beta}k^{2}}{\tilde{\mu}x_{+}^{2}}-\frac{\tilde{\beta}^{2}k^{3}}{\tilde{\mu}^{2}x_{+}^{4}}+4\frac{\tilde{\beta}k}{\tilde{\mu}}\ln
x_{+}\right)+S_{0}$ (35)

This way we have obtained entropy and mass of black hole solutions CLS ,
again, and shown that at the black hole horizon, gravitational field equation
can be cast into the form of the first law of thermodynamics.

##  IV Black Holes in IR Modified Hořava-Lifshitz Gravity

In this section we consider the case with broken detailed-balance by
introducing a term $\mu^{4}R$ to the action KS . Such theory is called IR
modified Hořava-Lifshitz theory. In this way, it is found that one can get
asymptotically flat solutions. In fact, introducing the parameter
$\epsilon^{2}$ to the original action of Hořava-Lifshitz theory with detailed-
balance can also lead to asymptotically flat solutions LMP ; CCO . Here we
show that for the IR modified Hořava-Lifshitz theory, gravitational field
equations at the black hole horizon can also be cast into a form of the first
law of thermodynamics.

Now we add a new term

${\cal L}_{3}=\sqrt{g}N\frac{\kappa^{2}\mu^{2}\nu}{8(3\lambda-1)}R,$ (36)

to the action (13). Here $\nu$ is a new parameter. The term (36) “softly”
violates the so-called detailed-balance. The action in the metric (16) changes
to Park

$\displaystyle I$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega}{8(1-3\lambda)}\int
dtdr\tilde{N}\left((2\lambda-1)\frac{(f-1)^{2}}{r^{2}}-2\lambda\frac{f-1}{r}f^{\prime}+\frac{\lambda-1}{2}f^{\prime
2}\right.$ (37)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-2(\nu-\Lambda)(1-f-rf^{\prime})-3\Lambda^{2}r^{2}+\alpha\Lambda
r^{2}{\cal L}_{m}),$

where we have restricted to the case with $k=1$ and a prime stands for the
derivative with respect to $r$. Considering the case with $\lambda=1$, and
varying the action with respect to $\tilde{N}$, one has the equation of motion

$\frac{\kappa^{2}\mu^{2}\Omega}{16}\left(\frac{(f-1)^{2}}{r^{2}}-2\frac{f-1}{r}f^{\prime}-2(\nu-\Lambda)(1-f-rf^{\prime})-3\Lambda^{2}r^{2}\right)=r^{2}\Omega\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}},$ (38)

At a black hole horizon where $f=0$ and $f^{\prime}|_{r=r_{+}}=4\pi T$, by the
same approach, we can rewrite the equation as

$TdS-dE=PdV,$ (39)

where

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\kappa^{2}\mu^{2}\Omega}{4}\left((\nu-\Lambda)r_{+}^{2}+2\ln
r_{+}\right)+S_{0},$ $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega}{16}\left(\Lambda^{2}r_{+}^{3}+2(\nu-\Lambda)r_{+}+\frac{1}{r_{+}}\right).$
(40)

This energy is the same as that given in Park , up to a factor, which is not
figured out there. The entropy is given for the first time, although related
results on thermodynamics of black holes in the modified Hořava-Lifshitz
theory have been discussed in Myung .

##  V Generalized Misner-Sharp energy and the first law of black hole
thermodynamics

Quasilocal energy is an important concept in general relativity. In
particular, the so-called Misner-Sharp energy is intensively discussed in the
literature. In a spherically symmetric spacetime with metric

$ds^{2}=h_{ab}dx^{a}dx^{b}+r^{2}(x)(d\theta^{2}+\sin^{2}\theta d\varphi^{2}),$
(41)

where $a=0$ and $1$, the Misner-Sharp energy is defined as MS

$M(r)=\frac{r}{2G}\left(1-h^{ab}\partial_{a}r\partial_{b}r\right),$ (42)

which is valid for general relativity in four dimensions. For Schwarzschild
solution, (42) just gives the Schwarzschild mass, while it gives the effective
Schwarzschild mass $m(r)$ at $r$ for a static spherically symmetric spacetime
(16) with $f(r)=1-\frac{2Gm(r)}{r}$. Therefore at a black hole horizon
$r_{+}$, the Misner-Sharp energy (42) gives us the energy of gravitational
field at the horizon $r_{+}$: $M=r_{+}/2G$.

In the previous sections, we have shown that the gravitational field equations
at a black hole horizon can be cast to a form of the first law of
thermodynamics in Hořava-Lifshtiz theory. In this section, we show that the
form of action for the Hořava-Lifshtiz theory allows us to give a generalized
Misner-Sharp quasilocal energy in the case of static, spherically symmetric
spacetime (16).

Let us start with the action (13) with the detailed-balance. In this case, the
gravitational part of the action can be rewritten in a derivative form (18),
which enables us to define a generalized Misner-Sharp energy as

$M(r)=\frac{\kappa^{2}\mu^{2}\Omega_{k}}{16\ r}\left(\Lambda^{2}r^{4}-2\Lambda
r^{2}(k-f)+(k-f)^{2}\right).$ (43)

It is easy to see that at a black hole horizon $r_{+}$, this quasilocal energy
$E(r)$ gives the mass (III) of the black hole solution. The variation of the
generalized Misner-Sharp energy with respect to $r$ gives

$dM(r)=-r^{2}\Omega_{k}\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}dr,$ (44)

from which one can see clearly that $-\frac{\delta(\tilde{N}{\cal
L}_{m})}{\delta\tilde{N}}$ is the energy density of matter field. In the case
without matter field, the generalized Misner-Sharp energy is conserved,
$dM(r)=0$. At the horizon we have

$dM(r)|_{r=r_{+}}=dE-TdS,$ (45)

where $E$ and $S$ are just mass and entropy of the black hole, given in (III).
When the matter field is absent, it gives us $dE=TdS$, which is the first law
of black hole thermodynamics. In CCO , we have used the first law to derive
the black entropy.

In the case including the $z=4$ term, the generalized Misner-Sharp energy can
be read down from the action (32)

$\displaystyle M(r)$ $\displaystyle=$
$\displaystyle\frac{\kappa^{2}\mu^{2}\Omega_{k}}{16\
r}\left(\Lambda^{2}r^{4}-2\Lambda r^{2}(k-f)+(k-f)^{2}\right.$ (46)
$\displaystyle\left.-2\frac{\tilde{\beta}}{\Lambda\mu}\left((k-f)^{2}-\frac{(k-f)^{3}}{\Lambda
r^{2}}\right)+\frac{\tilde{\beta}^{2}}{\Lambda^{2}\mu^{2}}\frac{(k-f)^{4}}{\Lambda^{2}r^{4}}\right),$

while for the IR modified Hořava-Lifshtiz theory, we have, from (37),

$M(r)=\frac{\kappa^{2}\mu^{2}\Omega}{16\
r}\left(\Lambda^{2}r^{4}+2(\nu-\Lambda)r^{2}(1-f)+(1-f)^{2}\right).$ (47)

It is easy to show that, when the matter field is absent, these two
generalized Misner-Sharp energies are conserved, while when the matter field
appears, their variation satisfies (44). At the black hole horizon, the
variation of the generalized Misner-Sharp energy obeys (45), which is closely
related to the first law of black hole thermodynamics.

##  VI Conclusions and Discussions

The black hole thermodynamics implies that there might exist a deep connection
between thermodynamics and gravity theory, although they are quite different
subjects. Such a connection must be closely related to the holographic
properties of gravity. The holography is an essential feature of gravity. In
this paper we investigated the relationship between the first law of
thermodynamics and gravitational field equation at a static, spherically
symmetric black hole horizon in Hořava-Lifshtiz theory with/without detailed-
balance. It turns out that, as in the cases of Einstein gravity and Lovelock
gravity, the gravitational field equation can be cast to a form of the first
law of thermodynamics at the black hole horizon. This way we obtained entropy
and mass expressions in terms of black hole horizon, and they are exactly the
same as those resulting from the integration method for black hole entropy and
the Hamiltonian approach for black hole mass CCO .

Note that Hořava-Lifshtiz theory, different from general relativity, is not
fully diffeomorphism invariant and only keeps the “foliation-preserving”
diffeomorphism. Our results on the relation between the first law of
thermodynamics and gravity field equation in the Hořava-Lifshtiz theory
indicate that this relation is a robust one, and is of some universality. In
addition, unlike the case in general relativity, the first law of black hole
mechanics has not yet been established so far in Hořava-Lifshtiz theory. Our
result is a first step towards that goal. Furthermore, let us stress that in
the process to derive the entropy and mass of black holes in Hořava-Lifshtiz
theory, we have not employed an explicit solution of the theory. This is quite
different from the previous works in the literature. This manifests that the
relation between the first law and gravity field equation has a deep
implication.

We also defined generalized Misner-Sharp energy for static, spherically
symmetric spacetimes in Hořava-Lifshtiz theory. The generalized Misner-Sharp
energy is conserved in the case without matter field, and its variation gives
the first law of black hole thermodynamics at the black hole horizon.

Note that we have restricted ourselves to the case with $\lambda=1$ in Sec.
III. Here let us make a simple discussion of the case with $\lambda\neq 1$. In
this case, the reduced action can be expressed as CCO

$I=\frac{\kappa^{2}\mu^{2}\Omega_{k}}{8(1-3\lambda)}\int
dtdr\tilde{N}\left\\{\frac{(\lambda-1)}{2}F^{\prime
2}-\frac{2\lambda}{r}FF^{\prime}+\frac{(2\lambda-1)}{r^{2}}F^{2}\right\\},$
(48)

where $F(r)=k-\Lambda r^{2}-f(r)$. Varying the action with respect to $F$ and
$\tilde{N}$ yields the equations of motion

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(\frac{2\lambda}{r}F-(\lambda-1)F^{\prime}\right)\tilde{N}^{\prime}+(\lambda-1)\left(\frac{2}{r^{2}}F-F^{\prime\prime}\right)\tilde{N},$
(49) $\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{(\lambda-1)}{2}F^{\prime
2}-\frac{2\lambda}{r}FF^{\prime}+\frac{(2\lambda-1)}{r^{2}}F^{2}.$ (50)

These equations have the solution with LMP ; CCO

$F(r)=\alpha r^{s},\ \ \ \tilde{N}(r)=\gamma r^{1-2s},$ (51)

where $\alpha$ and $\gamma$ are both integration constants and

$s=\frac{2\lambda\pm\sqrt{2(3\lambda-1)}}{\lambda-1}.$

As discussed in the second reference of CCO , to have a well-defined physical
quantities and well-behaved asymptotical behavior for the solution, we have to
take the negative branch in $s$ and $s$ is in the range $s\in[-1,2)$. The
temperature of the black hole in this case is

$T=\frac{1}{4\pi}\tilde{N}(r)f^{\prime}(r)|_{r=r_{+}}=\frac{\gamma}{4\pi
r_{+}^{2s}}\left(-\Lambda r_{+}^{2}(2-s)-sk\right).$ (52)

We can rewrite Eq. (50) as

$\frac{2\lambda}{r}(k-\Lambda r^{2}-f)f^{\prime}+4\lambda\Lambda(k-\Lambda
r^{2}-f)+\frac{(\lambda-1)}{2}F^{\prime 2}+\frac{(2\lambda-1)}{r^{2}}F^{2}=0.$
(53)

On the black hole horizon where $f(r_{+})=0$, the above equation reduces to

$\frac{2\lambda}{r_{+}}(k-\Lambda
r^{2}_{+})f^{\prime}+4\lambda\Lambda(k-\Lambda
r^{2}_{+})+\frac{(\lambda-1)}{2}F^{\prime
2}(r_{+})+\frac{(2\lambda-1)}{r^{2}}F^{2}(r_{+})=0.$ (54)

Multiplying Eq. (54) by

$\frac{\sqrt{2}\kappa^{2}\mu^{2}\Omega_{k}\tilde{N}(r_{+})}{16\lambda\sqrt{3\lambda-1}}dr_{+},$

and considering the expression of the temperature (52), we find that the first
term in (54) can be expressed as $TdS$, where

$S=\frac{\pi\kappa^{2}\mu^{2}\Omega_{k}}{\sqrt{2(3\lambda-1)}}\left(k\ln(\sqrt{-\Lambda}r_{+})+\frac{1}{2}(\sqrt{-\Lambda}r_{+})^{2}\right)+S_{0},$
(55)

where $S_{0}$ is an integration constant. On the other hand, with the solution
(51), the other three terms in (54) can be expressed as $-dM$, where $M$ is

$M=\frac{\sqrt{2}\kappa^{2}\mu^{2}\gamma\Omega_{k}}{16\sqrt{3\lambda-1}}\frac{(k-\Lambda
r_{+}^{2})^{2}}{r_{+}^{2s}}.$ (56)

Thus we have shown that on the black hole horizon, the equation of motion (50)
can be expressed as $TdS-dM=0$, where $S$ and $M$ are just the black hole
entropy and mass, as found in the second reference of CCO . Here we would like
to mention that unlike the case of $\lambda=1$, due to the presence of $F^{2}$
and $F^{\prime 2}$ in the equation of motion, we have to use the black hole
solution (51) in order to express the equation of motion in the form of the
first law of thermodynamics. In addition, we here have discussed the case with
the vacuum solution.

## Acknowledgments

We thank T. Padmanabhan for useful correspondences. RGC is supported partially
by grants from NSFC, China (No. 10525060, No. 108215504 and No. 10975168) and
a grant from MSTC, China (No. 2010CB833004). NO was supported in part by the
Grant-in-Aid for Scientific Research Fund of the JSPS No. 20540283, and also
by the Japan-U.K. Research Cooperative Program. This work is completed during
RGC’s visit to Kinki University, Japan with the support of JSPS invitation
fund, the warm hospitality extended to him is greatly appreciated.

## References

  * (1) J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113 (1999)] [arXiv:hep-th/9711200].
  * (2) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995) [arXiv:gr-qc/9504004].
  * (3) R. Brustein and M. Hadad, Phys. Rev. Lett. 103, 101301 (2009) [arXiv:0903.0823 [hep-th]].
  * (4) M. K. Parikh and S. Sarkar, arXiv:0903.1176 [hep-th]; T. Padmanabhan, arXiv:0903.1254 [hep-th].
  * (5) E. Elizalde and P. J. Silva, Phys. Rev. D 78, 061501 (2008) [arXiv:0804.3721 [hep-th]].
  * (6) S. F. Wu, G. H. Yang and P. M. Zhang, Prog. Theor. Phys. 120, 615 (2008) [arXiv:0805.4044 [hep-th]].
  * (7) C. Eling, R. Guedens and T. Jacobson, Phys. Rev. Lett. 96, 121301 (2006) [arXiv:gr-qc/0602001].
  * (8) M. Akbar and R. G. Cai, Phys. Lett. B 635, 7 (2006) [arXiv:hep-th/0602156].
  * (9) M. Akbar and R. G. Cai, Phys. Lett. B 648, 243 (2007) [arXiv:gr-qc/0612089].
  * (10) R. G. Cai and L. M. Cao, Phys. Rev. D 75, 064008 (2007) [arXiv:gr-qc/0611071].
  * (11) C. Eling, JHEP 0811, 048 (2008) [arXiv:0806.3165 [hep-th]].
  * (12) G. Chirco and S. Liberati, arXiv:0909.4194 [gr-qc].
  * (13) P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. 94, 111601 (2005) [arXiv:hep-th/0405231].
  * (14) T. Padmanabhan, Class. Quant. Grav. 19, 5387 (2002) [arXiv:gr-qc/0204019].
  * (15) T. Padmanabhan, arXiv:0910.0839 [gr-qc].
  * (16) D. Kothawala, S. Sarkar and T. Padmanabhan, Phys. Lett. B 652, 338 (2007) [arXiv:gr-qc/0701002].
  * (17) A. Paranjape, S. Sarkar and T. Padmanabhan, Phys. Rev. D 74, 104015 (2006) [arXiv:hep-th/0607240]; D. Kothawala and T. Padmanabhan, Phys. Rev. D 79, 104020 (2009) [arXiv:0904.0215 [gr-qc]].
  * (18) R. G. Cai, L. M. Cao, Y. P. Hu and S. P. Kim, Phys. Rev. D 78, 124012 (2008) [arXiv:0810.2610 [hep-th]].
  * (19) M. Akbar, Chin. Phys. Lett. 24, 1158 (2007) [arXiv:hep-th/0702029]; M. Akbar and A. A. Siddiqui, Phys. Lett. B 656, 217 (2007).
  * (20) R. G. Cai and S. P. Kim, JHEP 0502, 050 (2005) [arXiv:hep-th/0501055]; R. G. Cai, L. M. Cao and Y. P. Hu, arXiv:0807.1232 [hep-th];
  * (21) A. V. Frolov and L. Kofman, JCAP 0305, 009 (2003) [arXiv:hep-th/0212327]; U. H. Danielsson, Phys. Rev. D 71, 023516 (2005) [arXiv:hep-th/0411172]; R. Bousso, Phys. Rev. D 71, 064024 (2005) [arXiv:hep-th/0412197].
  * (22) M. Akbar and R. G. Cai, Phys. Rev. D 75, 084003 (2007) [arXiv:hep-th/0609128].
  * (23) R. G. Cai and L. M. Cao, Nucl. Phys. B 785, 135 (2007) [arXiv:hep-th/0612144]; A. Sheykhi, B. Wang and R. G. Cai, Nucl. Phys. B 779, 1 (2007) [arXiv:hep-th/0701198]; A. Sheykhi, B. Wang and R. G. Cai, Phys. Rev. D 76, 023515 (2007) [arXiv:hep-th/0701261]; R. G. Cai, Prog. Theor. Phys. Suppl. 172, 100 (2008) [arXiv:0712.2142 [hep-th]]; X. H. Ge, Phys. Lett. B 651, 49 (2007) [arXiv:hep-th/0703253]
  * (24) Y. Gong and A. Wang, Phys. Rev. Lett. 99, 211301 (2007) [arXiv:0704.0793 [hep-th]].
  * (25) S. F. Wu, G. H. Yang and P. M. Zhang, arXiv:0710.5394 [hep-th]. S. F. Wu, B. Wang and G. H. Yang, Nucl. Phys. B 799, 330 (2008) [arXiv:0711.1209 [hep-th]; S. F. Wu, B. Wang, G. H. Yang and P. M. Zhang, arXiv:0801.2688 [hep-th]; T. Zhu, J. R. Ren and S. F. Mo, arXiv:0805.1162 [gr-qc]; M. Akbar, arXiv:0808.0169 [gr-qc].
  * (26) R. G. Cai, L. M. Cao and Y. P. Hu, Class. Quant. Grav. 26, 155018 (2009) [arXiv:0809.1554 [hep-th]].
  * (27) P. Horava, Phys. Rev. D 79, 084008 (2009) [arXiv:0901.3775 [hep-th]]; JHEP 0903, 020 (2009) [arXiv:0812.4287 [hep-th]].
  * (28) A. Wang, D. Wands and R. Maartens, arXiv:0909.5167 [hep-th].
  * (29) C. W. Misner and D. H. Sharp, Phys. Rev. 136, B571 (1964).
  * (30) H. Lu, J. Mei and C. N. Pope, Phys. Rev. Lett. 103, 091301 (2009) [arXiv:0904.1595 [hep-th]].
  * (31) R. G. Cai, L. M. Cao and N. Ohta, Phys. Rev. D 80, 024003 (2009) [arXiv:0904.3670 [hep-th]]; R. G. Cai, L. M. Cao and N. Ohta, Phys. Lett. B 679, 504 (2009) [arXiv:0905.0751 [hep-th]].
  * (32) R. G. Cai, Y. Liu and Y. W. Sun, JHEP 0906, 010 (2009) [arXiv:0904.4104 [hep-th]].
  * (33) A. Kehagias and K. Sfetsos, Phys. Lett. B 678, 123 (2009) [arXiv:0905.0477 [hep-th]].
  * (34) M. i. Park, JHEP 0909, 123 (2009) [arXiv:0905.4480 [hep-th]].
  * (35) Y. S. Myung, arXiv:0908.4132 [hep-th]; Y. S. Myung, Phys. Lett. B 678, 127 (2009) [arXiv:0905.0957 [hep-th]].