# Probing Palatini-type gravity theories through gravitational wave detections via quasinormal modes

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## Abstract

The possibility of testing gravity theories with the help of gravitational wave detections has become an interesting arena of recent research. In this paper, we follow this direction by investigating the quasinormal modes (QNMs) of the axial perturbations for charged black holes in the Palatini-type theories of gravity, specifically (1) the Palatini *f*(*R*) gravity coupled with Born-Infeld nonlinear electrodynamics and (2) the Eddington-inspired-Born-Infeld gravity (EiBI) coupled with Maxwell electromagnetic fields. The coupled master equations describing perturbations of charged black holes in these theories are obtained with the tetrad formalism. By using the Wentzel-Kramers-Brillouin (WKB) method up to 6th order, we calculate the QNM frequencies of the EiBI charged black holes, the Einstein-Born-Infeld black holes, and the Born-Infeld charged black holes within the Palatini \(R+\alpha R^2\) gravity. The QNM spectra of these black holes would deviate from those of the Reissner-Nordström black hole. In addition, we study the QNMs in the eikonal limit and find that for the axial perturbations of the EiBI charged black holes, the link between the eikonal QNMs and the unstable null circular orbit around the black hole is violated.

## 1 Introduction

It is quite fair to say that one of the most enchanting events of recent discovery in modern physics is the direct detection of gravitational waves (GWs) from the coalescence of binary black holes [1, 2]. The reason why the direct detections of GW signals are so appealing is that they not only confirm the predictions of Einstein’s general relativity (GR) once again, but also render GWs spectacularly a suitable tool for human being to *hear* deep into the sky far beyond the reach of electromagnetic signals. Not long after the first detection of GWs, the LIGO-VIRGO collaboration succeeded in detecting the GWs emitted from the merger of binary neutron stars, with an accurate localization of the source [3]. The prompt and accompanied electromagnetic signals emitted from the source were also detected. This outstanding accomplishment has initiated a new era of multi-messenger astronomy.

In addition, the direct detections of GWs could help us to test other gravitational theories, or even to falsify some extended theories of gravity [4]. In fact, one of the reasons to consider extended theories of gravity is that GR inevitability predicts the existence of spacetime singularities like the big bang singularity and the black hole singularity. To ameliorate these spacetime singularities, one may resort to some extended theories of gravity which modify Einstein equation at the large curvature limits, but reduce to GR when curvature becomes small. Within the new era of GW astronomy, one plausible way to test these extended theories of gravity, for instance, is via the speed of GWs, as was done in Refs. [5, 6, 7, 8, 9].

Another interesting aspect regarding GW detections could be the ringdown signals in the final stage of a merger event. Essentially, the final product of a merger event, no matter if seeded from binary black holes or from binary neutron stars, is usually a black hole. Before the final black hole settles itself, there is an intermediate stage where the distortion of the black hole is gradually relieved, with the emission of GWs. In practice, the ringdown stage can be described by the theory of black hole perturbations and the frequencies of the GWs are featured by quasinormal modes (QNMs). In this stage, the distorted black hole can be regarded as a dissipative system. The perturbations have a discrete spectrum and the QNM frequencies are complex numbers. The real part of the frequencies describes the oscillations of the perturbations and the imaginary part corresponds to the decay of the amplitude. Interestingly, these QNM frequencies merely depend on the parameters characterizing the black holes, such as the mass, the charge, and the spin. If there are additional parameters appearing in the underlying theory, these parameters should manifest themselves in the QNM spectra. Along this direction of research, the QNMs of black holes in several gravity theories have been investigated, such as in the Horndeski gravity [10, 11, 12, 13, 14], metric *f*(*R*) gravity [15, 16, 17], massive gravity [18, 19], Einstein-dilaton-Gauss-Bonnet gravity [20, 21, 22, 23], the Randall-Sundrum braneworld model [24], Hořava-Lifshitz gravity [25], higher dimensional black holes [26, 27, 28], and Einstein-aether theory [29], etc. Furthermore, the QNMs of some regular black holes [30, 31] and the black holes with non-commutative geometry [32, 33] have been analyzed in the literature. In addition, probing signatures of the black hole phase transitions in modified theories of gravity via QNMs has been shown to be possible [34]. See Refs. [35, 36, 37, 38] for nice reviews on the latest progress of the field.

In this paper, as a further extension of our previous work [39] in which the QNMs of massless scalar field perturbations were studied, we consider the QNMs of the axial perturbations for the charged black holes in two Palatini-type gravity theories: (1) the Palatini *f*(*R*) gravity coupled with Born-Infeld nonlinear electrodynamics (NED) and (2) the Eddington-inspired-Born-Infeld (EiBI) gravity coupled with linear electromagnetic fields. To calculate the QNM frequencies, we use the WKB method up to 6th order [40, 41, 42, 43]. We also calculate the QNMs in the eikonal limit in which the multipole number \(l\rightarrow \infty \). Furthermore, the QNM frequencies will be compared with those of the Reissner-Nordström (RN) black hole in GR. Note as well that for the merger events of binary neutron stars, the ringdown timescale is usually shorter than the timescale of charge neutralization of the black hole [44]. This justifies to some extent the validity of studying QNMs of charged black holes from the astrophysical point of view.

The charged black holes in the Palatini *f*(*R*) gravity coupled with Born-Infeld NED have been studied in detail in Ref. [45]. The black hole solutions are very close to the RN black hole at the exterior spacetime, while deviate from it inside the event horizon because of the Born-Infeld NED source and the nonlinear function *f*(*R*). It has been shown that there exist some regions in the parameter space where the singularity inside the event horizon is replaced with a finite size wormhole structure [46]. Moreover, one can construct the Einstein-Born-Infeld (EBI) black hole by choosing \(f(R)=R\). The properties of this charged black hole have been widely studied in the literature [47, 48, 49, 50]. Again, due to the Born-Infeld corrections from the NED source, the interior structure of the black hole would change significantly as compared with that of the RN black hole in GR.

The EiBI gravity was formally proposed in Ref. [51] to resolve the initial big bang singularity [52]. This theory reduces to GR in vacuum but differs from it when matter is included. The exact expressions and some interesting properties of the charged black holes in the EiBI theory were studied in Refs. [53, 54] (see Ref. [55] for a review on the EiBI gravity). Due to the Born-Infeld corrections from the gravity sector, the interior structure of the black hole could deviate from that of the RN black hole notably [56, 57, 58]. One can then compare the QNM frequencies of the EBI black holes and those of the EiBI charged black holes to see how the Born-Infeld structure from the matter and the gravitational sector affects the QNMs.

This paper is outlined as follows. In Sect. 2, we briefly review the tetrad formalism which will be used later to derive the master equations describing the axial perturbations of the black holes. In Sect. 3, the perturbed Maxwell equation for NED is obtained for the sake of later convenience. In Sect. 4, we derive sequentially the coupled master equations of the axial perturbations for charged black holes in the Palatini *f*(*R*) gravity coupled with Born-Infeld NED and in the EiBI gravity with linear electromagnetic fields. In Sect. 5, we calculate the QNM frequencies of these charged black holes by using the WKB semi-analytic method. The QNMs in the eikonal limit are discussed as well. We finally conclude in Sect. 6.

## 2 Tetrad formalism

*t*(\(t=x^0\)), radial coordinate

*r*(\(r=x^2\)), and polar angle \(\theta \) (\(\theta =x^3\)). Because the system is axisymmetric, the metric functions are independent of the azimuthal angle \(\phi \) (\(\phi =x^1\)). In this work, the notation used in Ref. [59] is strictly followed. The only difference is that the metric function \(\omega \) used in Ref. [59] is replaced with \(\sigma \) in Eq. (2.1) since we will use \(\omega \) to denote the frequency of the perturbations later. Note that in the background spacetime which is static and spherically symmetric, we have \(\sigma =q_2=q_3=0\).

## 3 Perturbed Maxwell equation for NED

*t*,

*r*) and (

*r*,

*t*) components, i.e. the (0, 2) and (2, 0) components of the field strength \(F_{\mu \nu }\) appear at the background level. In the tetrad frame, the field strength \(F_{(a)(b)}\) at the background level satisfies [45]

*t*,

*r*, and \(\theta \). In this case, the Bianchi identity of the field strength \(F_{[(a)(b)|(c)]}=0\) leads to

*X*should be decomposed into the background and the 1st order parts:

^{1}Eq. (3.4). Note that the quantity

*X*at the background level is given by \(X=F_{(0)(2)}^2\).

*B*(or the 1st order field strength \(F_{(0)(1)}\)) on the left hand side. In order to deduce the other coupled equation, the perturbed gravitational equation should be taken into account.

## 4 The master equations

As we have mentioned previously, the master equations describing the gravitational perturbations of a charged black hole are two coupled equations. This is because of the coupling between the gravitational field and the electromagnetic field in the system. So far we have derived one of the coupled equations, i.e., Eq. (3.22), from the Maxwell equation of the NED source. In this section, we will carry out the derivation of the other coupled equation from the gravitational equation of the theory. We will first consider the Palatini *f*(*R*) gravity coupled with NED and obtain the master equations in this theory. After that we will turn to deduce the master equations of the EiBI gravity coupled with linear electromagnetic fields.

### 4.1 Axial perturbations of charged black holes in Palatini *f*(*R*) with NED

*f*(

*R*) theory coupled with NED. The action of the theory reads [45]

*f*of the Ricci scalar \(R\equiv g^{\mu \nu } R_{\mu \nu }(\Gamma )\), the equations of motion would be different from those in the metric

*f*(

*R*) theory.

*f*(

*R*) gravity coupled with the Born-Infeld NED Lagrangian (4.2) have been studied in Ref. [45]. The QNMs of a massless scalar field of such black holes have been discussed in Ref. [39]. The most general form of the metric functions of these black holes have been derived in Ref. [45] as well (see also Eqs. (3.24) and (3.25) in Ref. [39]).

*F*(.., ..; ..; ..) is the hypergeometric function [60] and \(r_m\) is defined as \(r_m\equiv \sqrt{Q_*/\beta _m}\). Note that we have used the following dimensionless rescalings:

As mentioned in Ref. [39], it is interesting to compare the QNMs of the EBI charged black hole with those of the charged black holes within the EiBI gravity coupled with Maxwell electromagnetic fields. One can then compare directly the QNMs of charged black holes within two theories, one with the Born-Infeld correction from the matter sector and the other one with this kind of modification from the gravitational sector.

#### 4.1.1 Perturbed field equations

*f*(

*R*) gravity, the auxiliary metric and the physical metric are conformally related \(q_{\mu \nu }=f_Rg_{\mu \nu }\), where \(f_R=df/dR\). Therefore, their metric functions are related as follows

*R*can be written as

*f*(

*R*) gravity coupled with NED.

*B*and

*Q*, which correspond to the perturbations of the matter field and the gravitational field, respectively.

#### 4.1.2 Effective potentials

*l*is the multipole number. From Eq. (4.22), it can be proven that

- 1.
When \(f_R=1\) and the NED model is assumed to be the Born-Infeld NED given by Eq. (4.2), the master equations reduce to those of the EBI black hole given in Ref. [61].

- 2.
When \(\phi =X\), it can be proven that the Ricci scalar

*R*, the function*f*, and its derivative \(f_R\) are just constants at the background level. They manifest themselves as an effective cosmological constant \(\Lambda _{eff}\equiv f/(2f_R)\). Furthermore, it can be shown that, after a constant rescaling of \(H_i^{(-)}\), the master equations reduce to those of the RN-dS(AdS) spacetime given in Refs. [62, 63, 64, 65, 66]. - 3.If \(\phi =X\), and \(f_R=1\), we have$$\begin{aligned} V_{12}&=V_{21}=-\frac{2Q_*\mu }{r^3}e^{2\nu }\,, \end{aligned}$$(4.31)$$\begin{aligned} V_{11}&=\frac{e^{2\nu }}{r^3}\left[ (\mu ^2+2)r+\frac{4Q_*^2}{r}\right] \,, \end{aligned}$$(4.32)The master equations turn out to be those of the RN black hole [59].$$\begin{aligned} V_{22}&=\frac{e^{2\nu }}{r^3}\left[ (\mu ^2+2)r-3+\frac{4Q_*^2}{r}\right] . \end{aligned}$$(4.33)
- (iv)If \(\phi =X\), \(Q_*=0\), and \(f_R=1\), we have \(V_{12}=V_{21}=0\), and$$\begin{aligned} V_{11}&=\frac{e^{2\nu }}{r^2}l(l+1), \end{aligned}$$(4.34)Therefore, the potential for pure electromagnetic perturbations and for pure axial gravitational perturbations of the Schwarzschild black hole (the Regge-Wheeler equation [67]) are recovered, respectively.$$\begin{aligned} V_{22}&=\frac{e^{2\nu }}{r^2}\left[ l(l+1)-\frac{3}{r}\right] . \end{aligned}$$(4.35)

*f*(

*R*), the potentials would change significantly and, consequently, alter the QNM frequencies. We will discuss this issue later in Sect. 5.

### 4.2 Axial perturbations for EiBI charged black holes

#### 4.2.1 Perturbed field equations

*f*(

*R*) gravity, we have constructed another tetrad basis \(\tilde{e}^{(a)}_\mu \) to map the auxiliary metric \(q_{\mu \nu }\) onto the tetrad frame.

*f*(

*R*) gravity, that is Eq. (4.10), in the EiBI theory these two bases are still related implicitly via Eqs. (4.45) and (4.47). We will immediately show how to derive the master equations of the axial perturbations by using these two equations.

#### 4.2.2 Effective potentials

In addition, we should highlight a crucial result following from the master equations (4.72) and (4.73). It can be seen that in the second term on the right hand side of Eq. (4.72), i.e., the term containing \(\mu ^2\), there is a factor \(\sigma _+/\sigma _-\). On the other hand, in the second term on the right hand side of Eq. (4.73), there is a factor \(\sigma _-/\sigma _+\). Actually, these factors play a crucial role when the QNMs in the eikonal limit (\(l\rightarrow \infty \)) are considered. In that limit, we will show later in Sect. 5 that, because of these factors, the QNMs cannot be calculated directly from the associated quantities of the unstable photon sphere of the black hole and the correspondence proposed in Ref. [68] is not satisfied for the EiBI charged black holes (for more fundamental illustration on the photon sphere, see Ref. [69]).

- 1.If \(\epsilon =+1\), the metric functions read [39, 53, 54]where \(r_g\equiv \sqrt{Q_*/\beta _g}\) and we have used the rescaling \(\beta _gr_s\rightarrow \beta _g\).$$\begin{aligned} e^{2(\nu +\mu _2)}=&\frac{r^4}{r^4+r_g^4},\nonumber \\ e^{-2\mu _2}=&\frac{r^4+r_g^4}{r^4-r_g^4}\nonumber \\&\left[ 1\!-\frac{r}{\sqrt{r^4+r_g^4}} \left( 1\!-\frac{4r_g^4\beta _g^2}{3r}F \left( \frac{1}{4},\frac{1}{2};\frac{5}{4}; -\frac{r_g^4}{r^4}\right) \right) -\frac{r_g^4\beta _g^2}{3r^2}\right] ,\nonumber \\ e^{2\mu _3}=&r^2\,,\qquad e^{2\psi }=r^2\sin ^2\theta , \end{aligned}$$(4.74)
- 2.If \(\epsilon =-1\), the metric functions read [39, 53, 54]where$$\begin{aligned} e^{2(\nu +\mu _2)}=&\frac{r^4}{r^4-r_g^4}\,,\nonumber \\ e^{-2\mu _2}=&\frac{r^4-r_g^4}{r^4+r_g^4}\nonumber \\&\left[ 1-\frac{r}{\sqrt{r^4-r_g^4}} \left( 1-\frac{r_g^3\beta _g^2}{3}B \left( \frac{1}{4},\frac{1}{2}\right) \right. \right. \nonumber \\&\left. \left. +\frac{2\sqrt{2}r_g^3\beta _g^2}{3}F \left( \cos ^{-1}\frac{r_g}{r},\frac{1}{\sqrt{2}}\right) \right) \!-\!\frac{r_g^4\beta _g^2}{3r^2}\right] \,,\nonumber \\ e^{2\mu _3}=&r^2\,,\qquad e^{2\psi }=r^2\sin ^2\theta , \end{aligned}$$(4.75)
*B*(.., ..) is the Beta function and*F*(.., ..) is the elliptic function of the first kind, respectively [60].

## 5 QNM frequencies: the 6th order WKB method

The evaluation of the QNM frequencies is essentially based on treating the master equations of the perturbations as an eigenvalue problem with proper boundary conditions. In the literature, there have been several methods to calculate the QNMs, ranging from numerical approaches [70, 71] to semi-analytic methods (see Refs. [35, 36, 37, 38] and references therein). In this paper, we will use a semi-analytical approach firstly formulated in the seminal paper [40]. This approach is based on the WKB approximation and the QNMs can be calculated by just using a simple formula once the potential terms in the master equations are given. In Refs. [41, 42], the 1st order WKB method was extended to the 3rd and 6th orders WKB approximation, respectively. Recently, a further extension of the WKB method up to the 13th order has been proposed with the help of the Padé transforms [43]. The WKB method is expected to be accurate as long as the multipole number *l* is larger than the overtone *n* [36]. On the other hand, for astrophysical black holes, the fundamental mode \(n=0\) has the longest decay time and therefore dominates the late time signal of the ringdown stage. At this regard, we will mainly focus on the fundamental mode.

The formulation of the WKB method to calculate the QNMs is essentially based on the fact that the master equations can be written like a Schrödinger wave equation in quantum mechanics. The potential term, in most cases (including ours), has a finite value when \(r_*\rightarrow \infty \) (spatial infinity) and \(r_*\rightarrow -\infty \) (at the event horizon). Furthermore, the potential has a peak at some finite \(r_*\). One can then treat the problem as a quantum scattering process through a potential barrier after suitable boundary conditions for the problems are imposed. At spatial infinity, only outgoing waves moving away from the black hole exist. On the other hand, there can only exist ingoing waves moving toward the black hole at the event horizon.

The idea of the WKB method to encompass the aforementioned boundary conditions is to consider a quantum scattering process without incident waves, while the reflected and the transmitted waves have comparable amounts of amplitudes. The peak value of the effective potential \(V_{\text {eff}}(r_*)\equiv -\omega ^2+V\) is required to be slightly larger than zero in the sense that there are two classical turning points near the peak. The solutions far away from the turning points (\(r_*\rightarrow \pm \infty \)) are solved by using the WKB approximation up to a desired order and the boundary conditions should be taken into account. At the vicinity of the peak, the potential is expanded into a Taylor series up to a given order, and one uses a series expansion method to solve the differential equation. Finally, the numerical values of the QNM frequencies \(\omega \) can be obtained by matching the solution near the peak with the solutions deduced from the WKB approximation simultaneously at the two turning points.

*m*denotes the quantities evaluated at the peak of the potential. \(V_m''\) is the second order derivative of the potential with respect to \(r_*\), calculated at the peak. \(\Lambda _i\) are constant coefficients resulting from higher order WKB corrections. These coefficients contain the value and derivatives (up to the 12th order) of the potential at the peak.

^{2}

### 5.1 Fundamental QNMs

### 5.2 Eikonal QNMs

*f*(

*R*) gravity coupled with NED satisfy the approximation (5.2). The same is also valid for the massless scalar field perturbations of an EiBI charged black hole [39].

*p*denotes the quantities calculated at the peak of the potentials. Note that at the location of the peak, we have

In Fig. 4, we exhibit the eikonal QNM frequencies of the EiBI charged black holes in terms of \(1/\beta _g\). The charge is fixed to \(Q_*=0.4\) (we choose this value to amplify the effects of the charge on the QNMs). The blue (red) curves correspond to a positive (negative) EiBI coupling constant. The dashed and the dotted curves are the eikonal QNMs described by the potential \(V_1\) and \(V_2\), respectively. We also present the eikonal QNMs for the massless scalar field perturbations which can be described by the potential (5.3) in the sense that \(V_s=V\) (see Ref. [39]). It can be seen that the eikonal QNMs of the axial perturbations for the EiBI charged black holes (dashed and dotted curves) deviate from those corresponding to the unstable null circular orbit (solid curves).

Before closing this subsection, we would like to mention that the violation of Eq. (5.2) for the axial perturbations of the EiBI charged black holes could be due to the non-trivial coupling between the electromagnetic and the gravitational fields in this theory. On the other hand, if we assume that the electromagnetic perturbations do not alter the spacetime geometry, the electromagnetic perturbations will be described by the master equation (4.38) without the metric perturbation terms on the right hand side. In this regard, the potential describing the electromagnetic perturbations in the eikonal limit can be approximated as Eq. (5.3), and the correspondence proposed in Ref. [68], i.e., Eq. (5.2) is satisfied.

## 6 Conclusions

In this paper, we consider specifically two gravitational theories within the Palatini formulation and study the QNMs of the axial perturbations for the charged black holes in these theories. These theories of gravity are, respectively, the Palatini *f*(*R*) gravity coupled with Born-Infeld NED and the EiBI gravity coupled with linear electromagnetic fields. One of our goals is to see how the Born-Infeld structures from the gravitational sector and from the matter sector change differently the QNM frequencies. Therefore, we pay special attention to the comparison between the QNMs of the EBI black holes and the EiBI charged black holes. The QNMs of the Born-Infeld charged black holes in the Palatini \(R+\alpha R^2\) gravity are also discussed. In fact, our paper can be regarded as a further extension of our previous work [39] in which we studied the QNMs of the massless scalar field perturbations to these different charged black holes.

By using the tetrad formalism, we have derived the coupled master equations describing the axial perturbations of the charged black holes. In the two theories that we are considering, the coupled equations reduce to those of the RN black holes when the ratio of the charge and the Born-Infeld coupling constant \(Q_*/\beta _m\) (or \(Q_*/\beta _g\)) is small. The QNM frequencies of the charged black holes are evaluated by using the WKB method up to the 6th order, which is accurate for modes whose multiple is larger than the overtone \(l>n\). In this paper, we mainly focus on the QNMs of the fundamental modes (\(n=0\)), since these modes have the longest decay time and would dominate the late time ringdown signals from an astrophysical perspective. Our results indicate that the charged black holes are all stable against the axial perturbations. Besides, the QNM frequencies would deviate from those of the RN black hole when nonlinearity of matter fields (Born-Infeld NED) or modification of the gravitational theory (EiBI or *f*(*R*)) are considered. For instance, both the absolute value of the real part and the imaginary part of the QNM frequencies for the EBI charged black holes would increase with the value of \(1/\beta _m\). On the other hand, we show that by increasing the value of \(1/\beta _g\), the real part of the QNM frequencies and the decay time (\(\propto 1/|\text {Im}\,\omega |\)) would increase (decrease) for the EiBI charged black holes with \(\epsilon =+1\) (\(\epsilon =-1\)).

Furthermore, we study the QNMs of these black holes in the eikonal limit (\(l\rightarrow \infty \)). Interestingly, we find that the QNM frequencies in this limit for the EiBI charged black holes cannot be described by the properties of the unstable null circular orbit around the black hole. In other words, the QNM formula (5.2) proposed in Ref. [68] is not valid for the EiBI charged black holes. This violation could be an extra possible imprint from the EiBI corrections on the QNMs, aside from the QNM spectra, that may be detectable in the future.

In addition to the axial perturbations, it is necessary to investigate the QNMs of the polar perturbations (even parity perturbations) for the charged black holes considered in this work. For the Schwarzschild [72] and the RN charged black holes [59] in GR, it is well-known that their axial and polar perturbations are isospectral. This means that the potential terms in their master equations satisfy a certain relation in such a way that the QNMs of the axial and polar perturbations have identical spectra. The isospectrality could be violated in the presence of, for instance, nonlinearity in the matter source [73, 74], or modifications of the Einstein-Hilbert action [16], and so on. The violation/fulfillment of the isospectrality for the charged black holes in the Palatini-type gravity theories could be an additional tool to test the underlying theories and we shall leave this interesting issue for a coming work.

## Footnotes

- 1.
We only use a delta into the linear order perturbations of quantities whose values at the background level do not vanish. The quantities that vanish at the background level, such as the metric functions \(\sigma \), \(q_2\), and \(q_3\), and the Maxwell tensor components \(F_{(i)(j)}\) (\(ij\ne 02\) or 20), shall be regarded as linear order perturbation quantities directly.

- 2.

## Notes

### Acknowledgements

CYC would like to thank R. A. Konoplya for providing the WKB approximation. CYC and PC are supported by Taiwan National Science Council under Project No. NSC 97-2112-M-002-026-MY3, Leung Center for Cosmology and Particle Astrophysics (LeCosPA) of National Taiwan University, and Taiwan National Center for Theoretical Sciences (NCTS). MBL is supported by the Basque Foundation of Science Ikerbasque. She also would like to acknowledge the partial support from the Basque government Grant No. IT956-16 (Spain) and from the project FIS2017-85076-P (MINECO/AEI/FEDER, UE). PC is in addition supported by US Department of Energy under Contract No. DE-AC03-76SF00515. MBL is also thankful to LeCosPA (National Taiwan University, Taipei) for kind hospitality while part of this work was done.

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