Hall coefficient signals orbital differentiation in the Hund's metal Sr$_2$RuO$_4$

TL;DR

This paper explains the non-monotonic temperature dependence of the Hall coefficient in Sr$_2$RuO$_4$, linking it to orbital differentiation and scattering crossovers, supported by theoretical calculations.

Contribution

It reveals the connection between Hall coefficient sign changes and orbital differentiation in Hund's metals through combined Boltzmann and dynamical mean-field theory.

Findings

Hall coefficient shows two sign reversals with temperature.

Sign changes correspond to crossovers in scattering regimes.

Orbital differentiation influences transport properties in Hund's metals.

Abstract

The Hall coefficient $R_H$ of Sr$_2$RuO$_4$ exhibits a non-monotonic temperature dependence with two sign reversals. We show that this puzzling behavior is the signature of two crossovers which are key to the physics of this material. The increase of $R_H$ and the first sign change upon cooling are associated with a crossover into a regime of coherent quasiparticles with strong orbital differentiation of the inelastic scattering rates. The eventual decrease and the second sign change at lower temperature is driven by the crossover from inelastic to impurity-dominated scattering. This qualitative picture is supported by quantitative calculations of $R_H(T)$ using Boltzmann transport theory in combination with dynamical mean-field theory, taking into account the effect of spin-orbit coupling. Our insights shed new light on the temperature dependence of the Hall coefficient in materials with strong orbital differentiation, as observed in Hund's metals.

Equations

$$R_H=\frac{{\sigma }_e^2{R}_{H,e}+{\sigma }_h^2{R}_{H,h}}{{\left({\sigma }_e+{\sigma }_h\right)}^2},$$

$${\eta }_{\nu }(\mathbf{k})=\sum _m|⟨{\psi }_{\mathbf{k}}^{\nu }|{\chi }_m⟩{|}^2{\eta }_m\text{with}m=\{xy,xz,yz\}.$$

Full text

Introduction

Measuring the Hall coefficient $R_{H}$ is a standard way of characterizing charge carriers in quantum materials. For free carriers of a single type the Hall coefficient $R_{H}$ is simply given by the inverse of the density of carriers $n$ and their charge $e$. However, in complex materials with a Fermi surface (FS) composed of multiple sheets, interpreting $R_{H}$ can be more complicated and also provides richer information when both electron-like and hole-like carriers are present. For instance, in the case of one hole-like and one electron-like FS sheet, the corresponding Hall coefficient is given by an average of $R_{H,e}<0$ and $R_{H,h}>0$:

[TABLE]

weighted by the squares of the individual hole and electron conductivities, $\sigma_{h}$ and $\sigma_{e}$ respectively. Hence, the ratio of scattering rates between the two types of carriers enters in a key manner to determine both the overall sign and magnitude of the Hall coefficient.

The 4d transition metal oxide \ceSr2RuO4 is such a complex material: with low-energy bands built out of three Ru-$t_{2g}$ orbitals ($d_{xy},d_{yz},d_{xz}$) hybridized with O-$2p$ states, it has a FS comprising two electron-like sheets, $\beta$ and $\gamma$, and one hole-pocket, $\alpha$ Mackenzie1996b ; Bergemann2000 ; Damascelli2000 . And indeed, experiments Shirakawa1995 ; Mackenzie1996 ; Galvin2001 have observed a particularly intriguing temperature dependence of $R_{H}$ in \ceSr2RuO4, as depicted on Fig. 1. $R_{H}$ increases from a negative value of about $-1\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{C}}^{-1}$ at low temperatures (values between $-1.37\text{\times}{10}^{-10}$ and $-0.7\text{\times}{10}^{-10}\text{\,}{\mathrm{m}}^{3}\text{\,}{\mathrm{C}}^{-1}$ for $T\rightarrow 0$ have been reported Shirakawa1995 ; Mackenzie1996 ; Galvin2001 ; Kikugawa2004 ), exhibits a sign change at $T_{1}=$30\text{,}\mathrm{K} (in the cleanest samples), reaches a positive maximum at about $80\text{\,}\mathrm{K}$, changes sign a second time around $T_{2}=$120\text{\,}\mathrm{K} and eventually saturates to a slightly negative value for $T>$200\text{,}\mathrm{K}$$.

Obviously, this rich temperature dependence is a signature of the multi-carrier nature of \ceSr2RuO4, as realized early on in Ref. Shirakawa1995 , which considered a Drude model with two types of carriers. This was refined later in several works Mazin2000 ; Noce2000 ; Noce1999 using Boltzmann transport theory calculations for tight-binding models assuming scattering rates $1/\tau_{\nu}=A_{\nu}+B_{\nu}\,T^{2}$ for the different FS sheets $\nu=\{\alpha,\beta,\gamma\}$, with adjustable parameters $A_{\nu}$ and $B_{\nu}$. The overall take-home message of these phenomenological models is that $R_{H}$ is highly sensitive to the precise details of the FS sheets and also to the temperature and sheet dependence of the scattering rates.

Another remarkable experimental finding provides insight in interpreting the temperature dependence of $R_{H}$ Galvin2001 : Adding small amounts of Al impurities has a drastic impact on the intermediate temperature regime such that $R_{H}$ no longer turns positive and instead increases monotonically from the low-T to the high-T limit, as indicated by dotted lines in Fig. 1. Arguably, the similarity of the low-T values of $R_{H}$ for different impurity concentrations provides evidence that the elastic-scattering regime has been reached where $R_{H}$ is mainly determined by FS properties (see also Ref. Mackenzie1996 ). In contrast, the temperature dependence itself must be due to inelastic scattering, possibly associated with electronic correlations Galvin2001 .

In this work, we address this rich temperature dependence of $R_{H}$ in \ceSr2RuO4 and provide a clear interpretation of its physical meaning. We show that the two sign changes of $R_{H}(T)$ in clean samples are the signatures of two important crossovers in the physics of this material. The increase of $R_{H}$ upon cooling from high temperature signals the gradual formation of coherent quasiparticles, which is associated with a strong temperature dependence of the ratio of inelastic scattering rates between the $xy$ and $xz/yz$ orbitals. At low temperatures the decrease of $R_{H}$ is due to the crossover from inelastic to impurity-dominated scattering. These qualitative insights have relevance to a wide class of materials with orbital differentiation.

Our qualitative picture is supported by a quantitative calculation of $R_{H}(T)$ using Boltzmann transport theory in combination with dynamical mean-field theory (DMFT) Georges1996 , taking into account the electronic structure of the material. The spin-orbit coupling (SOC) is found to play a key role Haverkort2008 ; Veenstra2014 , because it has a strong influence on the shape of the FS and also controls the manner in which the scattering rates associated with the different orbitals combine into ${\bm{k}}$-dependent quasiparticle scattering rates at a given point on the FS Haverkort2008 ; Veenstra2014 ; Tamai2018 .

Results

Dependence on scattering rate ratios

The orbital dependence of scattering rates is crucial for the understanding of the Hall effect. Therefore, we begin by assigning scattering rates $\eta_{xy}$, $\eta_{xz}$ and $\eta_{yz}$ (due to crystal symmetries $\eta_{xz}=\eta_{yz}$) to each orbital, irrespective of the microscopic details of the underlying scattering mechanisms, which will be addressed at a later stage. Then, these scattering rates are converted into band and ${\bm{k}}$-dependent scattering rates using the overlap of the orbital wave-function with the Bloch wave-function:

[TABLE]

We calculate $R_{H}$ with Boltzmann transport theory using a realistic Wannier-Hamiltonian for the effective low-energy \ceRu-$t_{2g}$ subspace (see Methods). In Boltzmann transport theory $R_{H}$ only depends on the scattering rates through their ratio $\xi=\eta_{xy}/\eta_{xz/yz}$ and not their absolute magnitude; a point we verified in our calculations. This also implies that within the constant isotropic scattering rate approximation, i.e. $\xi=1$, the full temperature dependence of $R_{H}$ cannot be explained.

The calculated $R_{H}$ as a function of the scattering rate ratio $\xi$ is displayed in Fig. 2. We compare calculations with and without SOC, and note that in the following all results labeled with ‘SOC’ actually take the correlation-enhancement of the effective SOC into account Liu2008 ; Zhang2016 ; Kim2018 ; Tamai2018 (see Methods). Without SOC $R_{H}$ remains negative for all values of $\xi$ and approaches zero as $\xi\gg 1$. In this limit the $\gamma$ sheet drops out and the contributions of the hole-like $\alpha$ sheet and electron-like $\beta$ sheet compensate each other. This means that it is not possible to explain the positive value of $R_{H}$ observed experimentally in clean samples for $T_{1}<T<T_{2}$ (Fig. 1) without taking SOC into account. With SOC we observe a very different behavior of $R_{H}(\xi)$; it turns from negative to positive at $\xi\simeq 2.6$. This is a result of two effects Haverkort2008 ; Veenstra2014 ; Tamai2018 (see Fig. 2, inset): First, SOC changes the shape and size of the FS sheets, and secondly, it induces a mixing between different orbital characters, which varies for each point on the FS. Thus, the manner in which the scattering rates associated with the different orbitals combine into ${\bm{k}}$-dependent quasiparticle scattering rates (Eq. 2) is controlled by the SOC. From the calculated dependence of $R_{H}(\xi)$ in the presence of SOC we deduce that agreement with experiments would require $\xi$ to be smaller than $2.6$ at high temperatures, increase above this value at $\sim T_{2}$ and then decrease again to reach a value close to unity at low temperatures.

Inelastic electron-electron scattering

We turn now to microscopic calculations by first considering inelastic electron-electron scattering ratios calculated with DMFT (see Methods). These calculations consider the $t_{2g}$ subspace of states with Hubbard-Kanamori interactions of $U=$2.3\text{,}\mathrm{e}\mathrm{V} and $J=$0.4\text{\,}\mathrm{e}\mathrm{V} Mravlje2011 . The calculated $\xi(T)$ from inelastic scattering only is displayed in Fig. 3 (a). In agreement with previous studies Mravlje2011 ; Deng2016 , we find that the $xy$ orbital is less coherent than $xz/yz$ at all temperatures and $\eta_{xy}>\eta_{xz/yz}$. In \ceSr2RuO4 the crossover from the low-T coherent Fermi liquid regime with $\eta\sim T^{2}$ to an incoherent regime with a quasilinear temperature dependence of the scattering rate is well-documented Mravlje2011 ; Stricker2014 and also manifested in deviations of the resistivity from a low-temperature quadratic behavior to a linear one Hussey1998 . Importantly, this coherence-to-incoherence crossover as well as the corresponding coherence scales are strongly orbital dependent. When approaching the Fermi liquid regime ($T_{FL}\approx$25\text{,}\mathrm{K}$$ Hussey1998 ; Maeno1997 ; Mackenzie2003 ) the scattering rate ratio reaches a value as large as $\xi^{FL}\sim 3$ (Fig. 3 (a)), but decreases rapidly upon heating with $\xi=1.8$ at $300\text{\,}\mathrm{K}$. We do not find a substantial change for even higher temperatures; at 500 K the scattering rate ratio is $\xi=1.6$.

Connecting these results to the discussion of Fig. 2 above, the temperature dependence of $\xi$ directly translates into that of $R_{H}$, as shown in Fig. 3 (b). Like in experiments, $R_{H}$ is negative at high temperatures, but when the temperature is lowered it increases and crosses zero at $110\text{\,}\mathrm{K}$. This demonstrates that electronic correlations are indeed able to turn $R_{H}$ positive and suggests the following physical picture: The electronic transport in \ceSr2RuO4 crosses from a regime governed by incoherent electrons at high temperatures, connected to a weaker orbital differentiation of scattering rates and a negative $R_{H}$, over to a coherent Fermi liquid regime, with a stronger orbital differentiation and positive $R_{H}$. The resulting sign change at $110\text{\,}\mathrm{K}$ can be seen as a direct consequence of this coherence-to-incoherence crossover. We emphasize that this sign change is only observed when SOC is taken into account. Without SOC $R_{H}$ is purely negative and shows only a weak temperature dependence (Fig. 3 (b), dashed line).

When moving along the FS from $\Gamma$-M ($\theta=0^{\circ}$) to $\Gamma$-X ($\theta=45^{\circ}$), the mixing of the orbital character induced by SOC (Eq. 2) leads to angular-dependent scattering rates $\eta_{\nu}(\theta)$ (Fig. 3 (c)). At $\theta=0^{\circ}$ the ratio of scattering rates between the $\gamma$ and $\beta$ sheets is large, because these bands still have mainly $xy$ and $xz/yz$ character, respectively (Fig. 2, inset). As expected from Fig. 3 (a), this sheet dependence decreases with increasing temperature. On the other hand, at $\theta=45^{\circ}$ the ratio is small, due to a very similar orbital composition of the $\gamma$ and $\beta$ sheets. The $\alpha$ pocket (being almost entirely $xz/yz$) has the lowest scattering rate and turns $R_{H}$ positive when $\xi$ becomes large enough at low temperatures. To shed more light on the interplay of the individual FS sheets we can phenomenologically assign constant scattering rates to each FS sheet, as shown in Fig. 3 (d). We see that for $R_{H}$ to be positive a necessary condition is $\eta_{\beta}>\eta_{\alpha}$. This again highlights the importance of SOC, because without SOC the $\alpha$ and $\beta$ sheets have entirely $xz/yz$ orbital character, and thus $\eta_{\alpha}=\eta_{\beta}$. Should one make this assumption also in the presence of SOC, it would not result in $R_{H}>0$ for any ratio $\eta_{\gamma}/\eta_{\alpha}$ (Fig. 3 (d), dashed line).

Impurity-dominated scattering

Considering inelastic scattering only would yield a positive $R_{H}$ at even lower temperatures deep in the Fermi liquid regime. However, at such low temperatures elastic scattering is expected to dominate over inelastic scattering. The extracted DMFT scattering rates at $29\text{\,}\mathrm{K}$ with $5.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for the $xy$ and $1.9\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ for the $xz/yz$ orbitals are of the order of the impurity scattering for ‘clean’ samples with residual resistivities of $\sim$0.5\text{,}\mathrm{\SIUnitSymbolMicro}\Omega\mathrm{c}\mathrm{m}$$. Therefore, we add a constant elastic scattering $\eta^{\text{el.}}$ to the orbital-dependent inelastic scattering $\eta^{\text{inel.}}_{m}$. This elastic term is assumed to be isotropic: $\eta^{\text{el.}}_{xy}=\eta^{\text{el.}}_{xz/yz}$. The resulting temperature dependence of $R_{H}$ for values of $\eta^{\text{el.}}$ ranging from $0.1$ to $10\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ is shown in Fig. 4. The dashed lines are calculated with the Fermi liquid form $\eta^{\text{inel.}}_{m}=A_{m}T^{2}$ and parameters $A_{m}$ determined from the calculated inelastic scattering rates at $29\text{\,}\mathrm{K}$.

For small enough $\eta^{\text{el.}}$ we observe a second zero crossing of $R_{H}(T)$ and a regime with $R_{H}<0$ at low temperatures, which is consistent with $R_{H}(T)$ depicted in Fig. 1. For $T\rightarrow 0$ the fully elastic scattering regime is reached, and thus $R_{H}$ is not influenced by the magnitude of the (isotropic) scattering rate, but rather by the shape of the FS only. This regime corresponds to $\xi=1$ in Fig. 2, for which we obtain $R_{H}=$-0.94\text{\times}{10}^{-10}\text{,}{\mathrm{m}}^{3}\text{,}{\mathrm{C}}^{-1}$$, in good quantitative agreement with experiments Shirakawa1995 ; Mackenzie1996 ; Galvin2001 ; Kikugawa2004 . With increasing temperature the influence of elastic scattering fades away and the precise interplay with inelastic scattering shapes the overall temperature dependence of $R_{H}$. Hence, we see that also the low-temperature zero crossing has a simple physical interpretation: it signals the crossover between the regime dominated by elastic scattering at low temperatures and the regime dominated by inelastic scattering at higher temperatures. Matching the two terms in the scattering rate, a simple estimate of the corresponding crossover scale is $T_{1}\sim\sqrt{\eta^{\text{el.}}/A_{xy}}\sim\sqrt{\eta^{\text{el.}}T_{\mathrm{FL}}}$. This scale obviously depends on the elastic scattering rate, and coincides approximately with the Fermi liquid coherence scale $T_{\mathrm{FL}}$ only for the cleanest samples reported in which $\eta^{\text{el.}}\sim T_{\mathrm{FL}}$. For even cleaner samples we predict $T_{1}<T_{\mathrm{FL}}$.

On the contrary, for larger $\eta^{\text{el.}}$ we find that $R_{H}(T)$ ceases to exhibit any zero crossing and is negative in the whole temperature range. Only in very clean samples can the inelastic scattering rate sufficiently exceed the elastic one for the sign changes of $R_{H}$ to occur. This is further substantiated by experimental Hall measurements for samples where the residual resistivity was altered by introducing different amounts of Al impurities, cf. the dependence of $R_{H}$ on $\eta^{\text{el.}}$ in Fig. 4 and the inset with experimental data from Ref. Galvin2001 .

In the high-T limit we obtain a value of $R_{H}$ which is more negative than the one reported in experiments Shirakawa1995 ; Mackenzie1996 ; Galvin2001 . Within Boltzmann transport theory this would imply that a larger ratio $\eta_{xy}/\eta_{xz/yz}$ is needed. Likewise, resistivities are significantly underestimated in DMFT transport calculations for $T>$300\text{,}\mathrm{K}$$ in this material Deng2016 . A possible explanation is that other sources of inelastic scattering, e.g. electron-phonon scattering, could play an important role in the high-T regime. We emphasize, however that all experimental evidence points towards negligible magnetic contribution (due to processes like skew scattering) and a standard orbital-dominated Hall effect in \ceSr2RuO4 Mackenzie1996 ; Noce2000 ; Kikugawa2004 .

Discussion

In summary, our quantitative calculations and qualitative interpretations explain the highly unusual temperature dependence of the Hall coefficient of \ceSr2RuO4. The high-T sign change of $R_{H}(T)$ in clean samples is the direct consequence of the crossover from a high-T incoherent regime to a coherent regime with orbital differentiation. The orbital composition of each quasiparticle state on the Fermi surface, as well as the distinct scattering rates of the different orbitals, are crucial to this phenomenon and are properly captured by DMFT. This is in line with recent insights from angle-resolved photoemission spectroscopy Tamai2018 . In turn, the low-T sign change is due to the crossover from inelastic to impurity-dominated scattering, which is further substantiated by comparing our results to experimental data on samples with a higher impurity concentration. Because it directly affects the shape of the Fermi surface sheets and strongly mixes their orbital character, spin-orbit coupling is found to be essential in explaining $R_{H}(T)$.

Orbital differentiation is actually a general feature common to Hund’s metals Aichhorn2010 ; Mravlje2011 ; Lanata2013 ; Miao2016 ; Medici2017 ; Kostin2018 , a broad class of materials in which the electronic correlations are governed by the Hund’s coupling, comprising for example transition metal oxides of the 4d series as well as iron based superconductors Werner2008 ; Haule2009 ; Yin2011 ; Georges2013 ; Medici2017 . We note that a non-monotonic temperature dependence of the Hall coefficient has also been reported for \ceSr3Ru2O7 Perry2000 . Beyond ruthenates, \ceLiFeAs and \ceFeSe are two compounds without Fermi surface reconstruction due to long-range magnetic order, which display striking similarities to \ceSr2RuO4 in many regards. The Fermi surface of these superconductors is also composed of multiple electron- and hole-like sheets with distinct orbital composition and strong orbital differentiation Miao2016 ; Kostin2018 . And indeed, the Hall coefficient of \ceLiFeAs has a strong temperature dependence Heyer2011 and that of \ceFeSe displays two sign changes in the tetragonal phase Watson2015 ; Sun2017 . These examples show that strongly-correlated materials with multiple Fermi surface sheets of different or mixed orbital character and a orbital-differentiated coherence-to-incoherence crossover are expected to show a pronounced temperature dependence of the Hall coefficient. Sign-changes then emerge in materials with balanced electron and hole-like contributions. These observations point to a wide relevance of our findings beyond the specific case of \ceSr2RuO4.

Method

Hamiltonian and SOC

We use a maximally-localized Wannier function construction MLWF1 ; MLWF2 to obtain an effective low-energy Hamiltonian for the three $t_{2g}$-like orbitals centered on the \ceRu atoms. This construction is based on a non-SOC density functional theory calculation (DFT), using the software packages WIEN2K Wien2k with GGA-PBE PBE , wien2wannier wien2wannier and Wannier90 wannier90 . We incorporate the SOC as an additional local term, where we neglect the coupling to $e_{g}$ orbitals, as these are well separated in energy. It has been shown that electronic correlations lead to an effective enhancement of the SOC in \ceSr2RuO4 by nearly a factor of 2 Liu2008 ; Zhang2016 ; Kim2018 . As the corresponding off-diagonal elements of the self-energy (in the cubic basis) are approximately frequency-independent Kim2018 , we model the effect of correlations by using a static effective SOC strength of $\lambda=$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V}, instead of the DFT value of about $\lambda^{DFT}=$100\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}. This is crucial to obtain precise agreement with the Fermi surface recently measured with photoemission experiments Tamai2018 . We refer to Ref. Tamai2018 for further details on the Hamiltonian.

DMFT

Due to the fermionic sign-problem, low temperatures are only accessible without SOC. However, in \ceSr2RuO4 the diagonal parts of the self-energy are, to a good approximation, unchanged upon the inclusion of SOC Kim2018 ; Linden2018 . Therefore, we perform DMFT calculation using the Hamiltonian without SOC and Hubbard-Kanamori interactions of $U=$2.3\text{,}\mathrm{e}\mathrm{V} and $J=$0.4\text{\,}\mathrm{e}\mathrm{V} obtained with cRPA Mravlje2011 . We use the TRIQS library TRIQS in combination with DFTTools TRIQS/DFTTOOLS and the CT-HYB TRIQS ; TRIQS/CTHYB impurity solver ($3.85\text{\times}{10}^{9}$ measurements). Inelastic scattering rates are extracted from non-SOC self-energies $\Sigma(i\omega_{n})$ by fitting a polynomial of $4^{\text{th}}$ order to the lowest 6 Matsubara points and extrapolating Im$\left[\Sigma\left(i\omega_{n}\rightarrow 0\right)\right]$, a procedure used in Ref. Mravlje2011 . We calculate the standard deviation with 9 consecutive DMFT iterations to obtain errorbars for the inelastic scattering rate ratios.

Transport calculations

We calculate $R_{H}$ using Boltzmann transport theory as implemented in the BoltzTraP package and described in the corresponding Refs. Madsen2006 ; Madsen2018 . We use a $46\times 46\times 46$ input ${\bm{k}}$-grid, which is interpolated on a 5 times denser grid with BoltzTraP Madsen2006 ; Madsen2018 . From Eq. 2 we obtain a scattering rate for each band and ${\bm{k}}$-point. Off-diagonal elements of $\eta_{\nu\nu^{\prime}}\left({\bm{k}}\right)$ and possible inter-band transitions are not considered in BoltzTraP, but we verified with Kubo transport calculations (TRIQS/DFTTools TRIQS/DFTTOOLS ) that these are negligible for the ordinary conductivity $\sigma_{xx}$. For all transport calculations labeled with ‘SOC’ we use the effective one-particle SOC term with a coupling strength of $\lambda=$200\text{,}\mathrm{m}\mathrm{e}\mathrm{V}$$, however even with $\lambda^{DFT}$ we find the same qualitative conclusions. We note that recently interest has been devoted to the theoretical descriptions of the Hall effect in strongly-correlated systems beyond Boltzmann transport theory Auerbach2018 ; Nourafkan2018 ; Miterscherling2018 .

Data Availability

All data generated and analyzed during this study are available from the corresponding author upon reasonable request.

Acknowledgment

We gratefully acknowledge useful discussions with Gabriel Kotliar, Andrew Mackenzie, Hugo Strand, Andrea Damascelli, Reza Nourafkan, André-Marie Tremblay. JM is supported by Slovenian Research Agency (ARRS) under Program P1-0044. MA acknowledges support from the Austrian Science Fund (FWF), project Y746, and NAWI Graz. This work was supported in part by the European Research Council grant ERC-319286-QMAC. The Flatiron Institute is a division of the Simons Foundation.

Author Contribution

M.Z. performed all calculations and the results were analyzed by M.Z. and A.G. All authors discussed and interpreted the results at different stages. The whole project was initiated by A.G. The manuscript was written by M.Z with the help of all authors.

Additional Information

Competing interests: The Authors declare no Competing Financial or Non-Financial Interests

Corresponding Authors: Manuel Zingl (mzingl@flatironinstitute.org) or Antoine Georges (ageorges@flatironinstitute.org)