Liangcai Xu

Xiaokang Li

Xiufang Lu

Clément Collignon

Huixia Fu

Jahyun Koo

Benoît Fauqué

Binghai Yan
email: binghai.yan@weizmann.ac.il

Zengwei Zhu
email: zengwei.zhu@hust.edu.cn

Kamran Behnia
email: kamran.behnia@espci.fr

Finite-temperature violation of the anomalous transverse Wiedemann-Franz law

The Wiedemann-Franz (WF) law links the ratio of electronic charge and heat conductivity to fundamental constants. It has been tested in numerous solids, but the extent of its relevance to the anomalous transverse transport, which represents the topological nature of the wave function, remains an open question. Here we present a study of anomalous transverse response in the noncollinear antiferromagnet Mn3Ge extended from room temperature down to sub-Kelvin temperature and find that the anomalous Lorenz ratio remains close to the Sommerfeld value up to 100 K, but not above. The finite-temperature violation of the WF correlation is caused by a mismatch between the thermal and electrical summations of the Berry curvature, rather than the inelastic scattering as observed in ordinary metals. This interpretation is backed by our theoretical calculations, which reveals a competition between the temperature and the Berry curvature distribution. The accuracy of the experiment is supported by the verification of the Bridgman relation between the anomalous Ettingshausen and Nernst effects. Our results identify the anomalous Lorenz ratio as an extremely sensitive probe of Berry spectrum near the chemical potential.

INTRODUCTION

The Berry curvature of electrons can give rise to the anomalous Hall effect (AHE) [1,2]. This happens if the host solid lacks time-reversal symmetry, which impedes cancellation after integration over the whole Fermi surface. Explored less frequently [3][4][5], the thermoelectric and thermal counterparts of the AHE (the anomalous Nernst and anomalous Righi-Leduc effects) arise also by the same fictitious magnetic field [6][START_REF] Xiao | Berry-Phase Effect in Anomalous Thermoelectric Transport[END_REF][START_REF] Onoda | Quantum transport theory of anomalous electric, thermoelectric, and thermal Hall effects in ferromagnets[END_REF]. How do the magnitudes of these anomalous off-diagonal coefficients correlate with each other? Do the established correlations between the ordinary transport coefficients hold? Satisfactory answers to these questions are still missing. A semiclassical formulation of anomalous Hall effect [START_REF] Sinitsyn | Semiclassical theories of the anomalous Hall effect[END_REF] is laborious, because the concept of Berry connection (or the 'anomalous velocity' [START_REF] Karplus | Effect in Ferromagnetics[END_REF]) is based on off-diagonal matrix elements linking adjacent Bloch functions and not wave packets of semi-classical transport theory [START_REF] Sinitsyn | Semiclassical theories of the anomalous Hall effect[END_REF]. This makes any intuitive picture of how Berry curvature combined to a longitudinal thermal gradient can produce a transverse electric field (an anomalous Nernst response) [START_REF] Xiao | Berry-Phase Effect in Anomalous Thermoelectric Transport[END_REF][START_REF] Mccormick | Semiclassical theory of anomalous transport in type-II topological Weyl semimetals[END_REF] or a transverse thermal gradient (an anomalous thermal Hall response) [START_REF] Matsumoto | Rotational motion of magnons and the thermal Hall effect[END_REF][START_REF] Qin | Energy Magnetization and the Thermal Hall Effect[END_REF] even more challenging.

Here, we present a study of correlations between the anomalous off-diagonal transport coefficients of a magnetic solid, with a focus on the relation between anoma-lous electrical, σ A ij , and thermal, κ A ij , Hall conductivities. The anomalous Lorenz ratio is defined as:

L A ij = κ A ij T σ A ij . (1)
We track L A ij in order to compare it with the Sommerfeld value:

L 0 = π 2 3 k B e 2 . (2)
We find that over a wide temperature range (0.5 K < T < 100 K), L A ij and L 0 remain close to each other, but a deviation starts above 100 K. We argue that this observation implies a hitherto unnoticed mechanism for finitetemperature violation of the Wiedemann-Franz (WF) law and points to a small (10 meV) energy scale in the Berry spectrum of Mn 3 Ge that is absent in Mn 3 Sn. This experimental observation is backed by theoretical calculations identifying the source of the Berry curvature in this family. By directly measuring the anomalous Ettingshausen and anomalous Nernst effects, we verify the validity of the Bridgman relation connecting the two transverse thermoelectric coefficients to each other. This confirms that the Bridgman relation, a consequence of Onsager reciprocity and based on thermodynamics of irreversible processes [START_REF] Callen | The Application of Onsager's Reciprocal Relations to Thermoelectric, Thermomagnetic, and Galvanomagnetic Effects[END_REF], holds regardless of microscopic details. Finally, we quantify the anomalous transverse thermoelectric response α A ij and find that the ratio of α A ij /σ A ij tends to saturate towards a value close to k B /e in the high-temperature limit. Following theoretical propositions [START_REF] Chen | Anomalous Hall Effect Arising from Noncollinear Antiferromagnetism[END_REF][START_REF] Kübler | Non-collinear antiferromagnets and the anomalous Hall effect[END_REF], a large AHE was found in Mn 3 X (X = Sn, Ge) family of non-collinear antiferromagnets [START_REF] Nakatsuji | Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature[END_REF][START_REF] Nayak | Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3[END_REF][START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF] below a remarkably high Néel temperature [START_REF] Zimmer | Investigation of the Magnetic Phase Transformation in Mn3Sn[END_REF][START_REF] Tomiyoshi | Polarized neutron diffraction study of Mn3Sn[END_REF][START_REF] Tomiyoshi | Magnetic Structure and Weak Ferromagnetism of Mn3Sn Studied by Polarized Neutron Diffraction[END_REF]. These newcomers to the emerging field of antiferromagnetic spintronics [START_REF] Smejkal | Topological antiferromagnetic spintronics[END_REF], present a distinct profile of the Hall resistivity in which the extraction of the anomalous Hall conductivity becomes straightforward. An anomalous thermoelectric (Nernst) [START_REF] Ikhlas | Large anomalous Nernst effect at room temperature in a chiral antiferromagnet[END_REF][START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF] and Righi-Leduc [START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF], counterparts of AHE, were also observed in Mn 3 Sn. In the case of Mn 3 Sn the triangular order is destroyed at finite temperature [START_REF] Nakatsuji | Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature[END_REF][START_REF] Ikhlas | Large anomalous Nernst effect at room temperature in a chiral antiferromagnet[END_REF][START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF]. This is not the case of Mn 3 Ge where the fate of these signals can be followed down to sub-Kelvin temperatures.

BASIC PROPERTIES

The room-temperature field-dependence of the three transport properties in Mn 3 Ge is shown in Fig. 1. Like in Mn 3 Sn, a hysteretic jump is triggered at a welldefined magnetic field, marking the nucleation of domains of opposite polarity induced by magnetic field [START_REF] Li | Momentum-space and real-space Berry curvatures in Mn3Sn[END_REF]. The large jump, the small magnetic field required for inverting polarity and the weakness of the ordinary Hall response lead to step-like profiles contrasting with other topological solids exhibiting AHE [START_REF] Liang | Anomalous Nernst Effect in the Dirac Semimetal Cd3As2[END_REF][START_REF] Liang | Anomalous Hall effect in ZrTe5[END_REF][START_REF] Shekhar | Anomalous Hall effect in Weyl semimetal half-Heusler compounds RPtBi (R = Gd and Nd)[END_REF]. A steplike profile of anomalous transverse response (for other

T ( K ) ؒ x x ؒ z z z [ 0 0 0 1 ] c ) -S x x ( µV / K ) T ( K ) B | | x [ 2 1 1 0 ] B = 0 . 1 T M ( e m u / g ) T ( K ) f ) a ) C / T ( J / m o l K 2 ) T 2 ( K 2 ) M n S i M n 3 S n U R u 2 S i 2 ( E T ) 2 C u ( N S C ) 2 ( E T ) 2 B r M n 3 G e Y b A l 3 Y b C u 4 . 5 U B e 1 3 Y b C u 4 A g 2 Y b C u 2 S i 2 B i -2 2 0 1 P d -S / T ( µV / K 2 ) ؇ | (m J / m o l K 2 ) C u FIG. 2.
Antiferromagnetic, dirty and correlated: a) A sketch of the magnetic texture of Mn3Ge showing the orientation of spins of Mn atoms. Red and blue represent two adjacent planes. b) Temperature dependence of the magnetization with Néel temperature visible at 370 K. c) Temperature dependence of resistivity along two orientations. d) The Seebeck coefficient, S, as a function of temperature. e) Low temperature specific heat C/T , as a function of T 2 . Extrapolation to T = 0 yields γ = 24.3 mJ mol -1 K -2 . f) Plot of the absolute value of S/T vs. γ for a number of correlated metals including Mn3X and MnSi [START_REF] Stishov | Heat capacity and thermal expansion of the itinerant helimagnet MnSi[END_REF][START_REF] Arsenijević | Manifestation of the spin textures in the thermopower of MnSi[END_REF].

varieties [START_REF] Sakai | Giant anomalous Nernst effect and quantum-critical scaling in a ferromagnetic semimetal[END_REF][START_REF] Liu | Giant anomalous Hall effect in a ferromagnetic kagome-lattice semimetal[END_REF]) makes the extraction of the anomalous component straightforward. Panels of Fig. 1 show the measured Hall resistivity, Nernst signal and thermal Hall resistivity, which were used to extract electric, thermoelectric and thermal Hall conductivities.

Figure 2 presents a number of basic properties of the system under study. The spin texture [START_REF] Nakatsuji | Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature[END_REF][START_REF] Zimmer | Investigation of the Magnetic Phase Transformation in Mn3Sn[END_REF][START_REF] Tomiyoshi | Polarized neutron diffraction study of Mn3Sn[END_REF] is shown in Fig. 2a. This magnetic order is stabilized thanks to the combination of Heisenberg and Dzyaloshinskii-Moriya interactions [START_REF] Liu | Anomalous Hall Effect and Topological Defects in Antiferromagnetic Weyl Semimetals: Mn3Sn/Ge[END_REF]. As seen in Fig. 2b, which shows the magnetization, it emerges below T N = 370 K. The small residual ferromagnetism has been attributed to the residual magnetic moment of octupole clusters of Mn atoms [START_REF] Suzuki | Cluster multipole theory for anomalous Hall effect in antiferromagnets[END_REF] in this pseudo-Kagome lattice.

A carrier density of n = 3.1 × 10 22 cm -3 is extracted from the magnitude of the ordinary Hall number[34] in agreement with a previous report [START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF]. The electrical resistivity shows little variation with temperature (Fig. 2c). and its magnitude of 150 µΩcm implies a mean free path as short as 0.9 nm, compatible with the fact that Mn 3 X crystals are not stoichiometric [START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF][START_REF] Tomiyoshi | Polarized neutron diffraction study of Mn3Sn[END_REF]. In our crystals, we found the Mn:Ge ratio ranges from 3.32:1 to 3.35:1 [34]. Since one tenth of Ge sites are occupied by Mn atoms, the average distance between these defects is 3 √ 10 times the average lattice parameter (a = 0.53 and c = 0.43 nm) and comparable to the short electronic mean free path.

The Seebeck coefficient (Fig. 2d) has a nonmonotonic temperature dependence with a peak around 60 K, and a sign change above 200 K, and a large low-temperature slope indicative of electronic correlations. The T -linear electronic specific heat (Fig. 2e), is as large as γ = 24.3 mJ mol -1 K -2 , thirty times larger than copper and five times larger than iron [START_REF] Kittel | Introduction to Solid state Physics[END_REF]. Assuming a single spherical Fermi surface corresponding to the known carrier density, such a γ, implies an effective mass as large as m * = 14.5 m e , which should not be taken literally given that the system is multiband. The slope of the Seebeck coefficient at low-temperature (S/T -0.2 µVK -2 ) correlates with γ yielding q = SN Av e T γ 1, as observed in other correlated systems [START_REF] Behnia | On the thermoelectricity of correlated electrons in the zero-temperature limit[END_REF] (Fig. 2f). On the other hand, the low temperature resistivity presents a weak upturn impeding the detection of any T -square term and the Wilson ratio (of the specific heat and magnetic susceptibility [START_REF] Coleman | Heavy fermions: Electrons at the edge of magnetism[END_REF]) implies that mobile electrons do not play any significant role in the magnetic response [34].

THE ANOMALOUS TRANSVERSE WF LAW

For each temperature, we measured σ A zx and κ A zx , identified as jumps in σ zx (B) and κ zx (B). This led to the determination of L A zx at each temperature and a comparison with L 0 to check the WF law.

Our main finding is presented in Fig. 3. Below 100 K, the anomalous Lorenz ratio, L A zx was found to be flat with a magnitude slightly larger than the Sommerfeld value, equal to it within the experimental margin. The results were reproduced by two different measuring methods (with resistive sensors and thermocouples [34]) and in two different samples. In one of them, we checked the persistent validity of this equality below 1 K down to 0.3 K. As seen also in the figure ,L A zx begins a steady downward deviation from L 0 above 100 K. Interestingly, σ A zx begins a steady decrease itself above 100 K. The temperature dependence of σ A zx is similar to what was reported in previous reports [START_REF] Nayak | Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3[END_REF][START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF] and its zero-temperature magnitude (which depends on stoichiometry [START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF]) is in agreement with was reported for a similar Mn content [START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF]34]. As seen in Fig. 3d, the anomalous Lorenz ratio in Mn 3 Ge and in Mn 3 Sn behave very differently, in spite of the fact that resistivity in both shows only a slight change with temperature (Fig. 3e), in contrast to elemental ferromagnets.

THE BRIDGMAN RELATION

Several previous reports of the violation of WF law have been refuted afterwards. One may therefore wonder if our data can be validated by an independent criteria. The answer is affirmative. Their validity is supported by the verification of the Kelvin relation (for normal longitudinal transport coefficients) and the Bridgman relation (for anomalous transverse coefficients). Their violation counter the thermodynamics of irreversible processes.

To check the Bridgman relation, we directly measured both the Nernst (Fig. 4a) and the Ettingshausen (Fig. 4b) effects. The former is the transverse electric field generated by a longitudinal thermal gradient, S zx = Ez ∇xT and the latter is the transverse thermal gradient produced by a longitudinal charge current, A zx = ∇zT Jx . The anomalous Ettingshausen effect (Fig. 4b) is easily invertible by a small magnetic field, like other anomalous transverse responses. We measured S A zx and A zx at several different temperatures. Combined with longitudinal thermal conductivity data κ xx (T ), this allowed us to check the Bridgman relation [START_REF] Bridgman | The Connections between the Four Transverse Galvanomagnetic and Thermomagnetic Phenomena[END_REF], which links these three quantities:

A zx = T S A zx κ xx . (3)
As seen in Fig. 4c, the two sides of the equation remain close to each other in the whole temperature range. The Bridgman relation, derivable by a thermodynamic argument [START_REF] Sommerfeld | The Statistical theory of thermoelectric, galvano-and thermomagnetic phenomena in metals[END_REF] is based on Onsager reciprocity [START_REF] Callen | The Application of Onsager's Reciprocal Relations to Thermoelectric, Thermomagnetic, and Galvanomagnetic Effects[END_REF]. Its experimental validity has been confirmed in semiconductors [START_REF] Delves | Thermomagnetic effects in semiconductors and semimetals[END_REF] and in superconductors hosting mobile vortices [START_REF] Huebener | Superconductors in a temperature gradient[END_REF]. While there is a previous report on simultaneous measurements of anomalous Nernst and Ettingshausen coefficients [START_REF] Seki | Relationship between anomalous Ettingshausen effect and anomalous Nernst effect in an FePt thin film[END_REF], the present study is the first experimental confirmation of the validity of Bridgman relation in the context of topological transverse response. We also verified the Kelvin relation linking the Seebeck and Peltier coefficients [34].

The temperature dependence of the anomalous transverse thermoelectric conductivity, α A zx , extracted from anomalous Nernst coefficient, is presented in Fig. 4d. As expected, it vanishes in the zero-temperature limit, but is remarkably large at room temperature. This can be seen by noting that the ratio of α A zx to σ A zx is close to k B /e at room temperature. This contrasts with the ordinary longitudinal counterpart of this ratio which includes a T /T F damping factor, inversely scaling with the Fermi temperature, T F [START_REF] Behnia | Nernst effect in metals and superconductors[END_REF].

ORIGIN OF THE FINITE-TEMPERATURE VIOLATION

Having discussed the thermoelectric response, let us now turn back to the thermal transport. The zerotemperature validity of the WF law implies that the transverse flow of charge and entropy caused by Berry

1 2 0 2 4 0 1 1 0 1 0 0 0 . 2 3 0 0 1 2 3 a ) -σ A z x ( 1 0 2 Ω -1 c m -1 ) b )  ؋ A z x / T ( 1 0 -4 W K -2 m -1 ) e ) d ) c ) L A z x ( 1 0 -8 V 2 K -2 ) L 0 0 1 2 3 L 0 M n 3 G e M n 3 S n ( L i 2 0 1 7 )
M n 3 S n ( S u g i i 2 0 1 9 )

L A i j ( 1 0 -8 V 2 K -2
) curvature conforms to a ratio of π 2 3 k B e 2 . This confirms Haldane's argument [START_REF] Haldane | Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological Fermi-Liquid Property[END_REF] that AHE is a property of the topological quasi-particles of the Fermi surface. As argued previously [START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF], only the states within the thermal window of the Fermi surface can be affected by a temperature gradient and give rise to a finite κ A zx . However, even in the case of ordinary longitudinal transport [START_REF] Ziman | Principles of the Theory of Solids[END_REF], the WF law ceases to be valid in presence of inelastic scattering. This is because small-angle inelastic collisions decay the momentum flow less efficiently than the energy flow both for electron-phonon [START_REF] Ziman | Principles of the Theory of Solids[END_REF] and electron-electron [START_REF] Jaoui | Departure from the Wiedemann-Franz Law in WP2 Driven by Mismatch in T-square Resistivity Prefactors[END_REF] scattering. However, if inelastic scattering played a significant role, one would have not observed such a drastic difference between Mn 3 Sn and Mn 3 Ge. Moreover, since the violation of the WF law is not correlated with any significant change in the magnitude of resistivity, it cannot be attributed to the gradual emergence of an extrinsic source of AHE [6]. Therefore, one should look for a hitherto unidentified route to the finite-temperature violation of the anomalous transverse WF law.

0
In the semi-classic picture of electronic transport, charge and heat conductivity are set by the mean-freepath and the Fermi radius averaged over the whole Fermi surface with a pondering factor, which is [START_REF] Ziman | Principles of the Theory of Solids[END_REF][START_REF] Van Houten | Thermo-electric properties of quantum point contacts[END_REF][START_REF] Behnia | Fundamentals of Thermoelectricity[END_REF]:

F n ( k ) = ( k -µ) n ∂f ( k ) ∂ k . ( 4
)
This expression for the pondering factor has been obtained for both Boltzmann [START_REF] Ziman | Principles of the Theory of Solids[END_REF] and Landauer [START_REF] Van Houten | Thermo-electric properties of quantum point contacts[END_REF] formalisms with n = 0 for charge transport, n = 1 for thermoelectric transport and n = 2 for thermal transport. Therefore, the main source of each transport coefficient is not located at the same place in the k-space. The Berry curvature summed over the Fermi sheets with these pondering factors can potentially generate a mismatch between σ A ij and κ A ij . As the temperature raises, the summations extend over a broader interval in the kspace inversely proportional to the thermal de Broglie length of electrons (Λ = h √ 2πm * k B T ) [START_REF] Behnia | Fundamentals of Thermoelectricity[END_REF]. The electrical

µ 0 H (mT) T*|S A zx |/κ T*|S A zx | /κ (µK m/A) |ε A zx | |ε A zx | (µK m/A) T*S A /κε A T (K) #2-5 #10 σ A zx -σ A zx (10 2 Ω -1 cm -1
) and thermal summations of the Berry curvature depend on these two length scales as well as the fine details of the spectrum, i.e. the size of the Berry curvature sources (and sinks) as well as their relative distance to the Fermi level (See Fig. 5). One can see that the validity of the WF correlation becomes harder to satisfy with warming. The higher the temperature, the larger the k-space area swept by the two different pondering functions and the easier to find a mismatch between σ A ij and κ A ij if the Berry curvature changes significantly across Λ -1 . In this context, one can conceive a mismatch between thermal and electrical summations of the Berry curvature (whose typical scale in the momentum space is an order of magnitude smaller).

T (K) α A zx -α A zx (A/Km) 𝑆𝑆 𝑧𝑧𝑧𝑧 = 𝐸𝐸 𝑧𝑧 𝛻𝛻 𝑧𝑧 𝑇𝑇 ε 𝑧𝑧𝑧𝑧 = 𝛻𝛻 𝑧𝑧

THEORETICAL CALCULATIONS OF THE ANOMALOUS LORENZ RATIO

The anomalous Hall conductivity σ A zx and anomalous thermal Hall conductivity κ A zx are expressed in the form of the Berry curvature Ω n zx (k) [START_REF] Vafek | Quasiparticle Hall transport of d-wave superconductors in the vortex state[END_REF],

σ A zx (µ) = e 2 ∞ -∞ dξ - ∂f (ξ -µ) ∂ξ σzx (ξ), (5)
κ A zx (µ) = 1 T ∞ -∞ dξ -(ξ -µ) 2 ∂f (ξ -µ) ∂ξ σzx (ξ), (6)
σzx (ξ) = BZ dk (2π) 3 n <ξ Ω n zx (k) (7) Ω n zx (k) = 1 i m =n n|v z |m n|v x |m -(x ↔ z) ( n -m ) 2 . ( 8
)
Here f (ξ -µ) = 1/(e ξ-µ k B T + 1) is the Fermi-Dirac function and vx/z is the velocity operator,and σzx (ξ) is actually the zero-temperature anomalous Hall conductivity (divided by e 2

). We point out that the electric and thermal coefficients are mainly different in the pondering factors for σzx (ξ). As shown in Fig. 5, the pondering factor -∂f (ξ-µ) ∂ξ in Eq. 5 is a δ-function while The total Berry curvature σzx (ξ) is determined by the intrinsic band structure of a give material. It is not surprising that L A zx depends sensitively on the position of µ because κ A zx (µ) and σ A zx (µ) do. In experiment, Mn 3 Ge(Sn) is usually off-stoichiometric in the form of Mn 3+x Ge(Sn) 1-x [START_REF] Nakatsuji | Large anomalous Hall effect in a non-collinear antiferromagnet at room temperature[END_REF][START_REF] Nayak | Large anomalous Hall effect driven by a nonvanishing Berry curvature in the noncolinear antiferromagnet Mn3[END_REF][START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF][START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF], where there are usually excess of Mn but deficiency of Ge(Sn). Such an offstoichiometry can shift the chemical potential up in energy compared to the charge neutral point.

-(ξ -µ) 2 ∂f (ξ-µ)
By shifting µ slightly above, we can successfully reproduce the general trend of T -dependence of L A zx for both Ge and Sn compounds in Fig. 6. For Mn 3 Ge at µ = 180 meV above the charge neutral point, L A zx drops quickly from 100 ∼ 250 K and goes up again after 250 K, qualitatively consistent with the experiment. The first drop is induced by dips of σ A zx at ∼0.1 and ∼0.3 eV, because the κ A zx (T ) sums smaller Berry curvature than σ A zx (T ) does in this case (see Eqs.5-6). We note that the dip at ∼0.1 eV is induced by an anti-crossing gap while the dip at ∼0.3 eV is caused by a Weyl point. (See the supplementary Fig. S11) The following up-turn is related to the increase of σ A zx after these dips. We also note that the theoretical violation is smaller than the experimental one. This may point to a significant role played by electronic correlations, which is neglected in the density-functional theory employed here. For Mn 3 Sn at µ = 140 meV, however, L A zx remains close to L 0 in a large temperature range. Different behaviors between two materials originate in their different Berry curvature σzx (ξ), as shown in Fig. 6(a) and (c). For Mn 3 Sn, σzx (ξ) is approximately anti-symmetric with respect to µ = 140 meV in an energy window of nearly 200 meV, inducing L A zx ≈ L 0 up to 400 K and even above. However, µ may vary from such an ideal anti-symmetric point in different samples. This explains the observed deviation of of L A zx in a different Mn 3 Sn sample, as shown in Fig. S9. To demonstrate the sensitive role of the chemical potential, we show L A zx -T curves for different µ in Fig. 6. Because it depends on the competition of the Berry curvature [σ zx (ξ)] profile and the temperature (5k B T ), L A zx ≈ L 0 at low T for different µ. In addition, there are interesting topological features near the chose chemical potential (see Fig. 6e-f). Along the Γ -M line in the Brillouin zone, there are two Weyl points induced by the crossing between the lowest and second lowest conduction bands. These Weyl points were not revealed in previous study which usually focuses on the crossing between the lowest conduction band and the highest valence band. For Mn 3 Sn at µ = 140 meV, a Weyl point exists and contributes large Berry curvature to the anomalous Hall conductivity. For Mn 3 Ge at µ = 180 meV, however, the Weyl cone is strongly tilted so that negative and positive Berry curvatures nearly cancel each other near the Weyl point (see more information in the supplementary Fig. S10).

CONCLUDING REMARKS

To summarize, we measured counterparts of the anomalous Hall effect associated with the flow of entropy and found that the Wiedemann-Franz law linking the magnitude of the thermal and electrical Hall effects is valid at zero-temperature, but a finite deviation emerges above 100 K. This deviation is not caused by inelastic scattering, but arises because of a mismatch in thermal and electrical summations of Berry curvature over the Fermi surface. The Bridgman relation, which links anomalous Nernst and Ettingshausen coefficients is satisfied over the whole temperature range. Finally, we observed that the room-temperature α A zx /σ A zx ratio is close to k B /e.

METHODS

Sample preparation and transport measurement:

Single crystals of Mn 3 Ge were grown from polycrystalline samples, using Bridgman-Stockbarger technique. The raw materials, Mn (99.99% purity) and Ge (99.999% purity), were weighed and mixed in an Argon glove box with a molar ratio of 3.3:1, loaded in an alumina crucible then sealed in a vacuum quartz ampule. The mixture was heated up to 1050 o C, remained for 2 hours to ensure homogeneity of melt, then was cooled slowly down to 800 o C to obtain polycrystalline samples. The polycrystalline Mn 3 Ge were ground, loaded in an alumina crucible and sealed into another vacuum quartz ampule. The growth temperature was controlled at 980 o C and 800 o C for high-temperature and low-temperature end, respectively. Finally, to obtain high temperature hexagonal phase, the quartz ampule was quenched with water. The single crystals were cut by a wire saw into typical di-mensions of 0.3×1.5×2 mm 3 for transport measurements. The stoichiometry was found to be Mn 3.08 Ge 0.92 (Mn:Ge = 3.32-3.35:1) using energy dispersive X-ray spectroscopy (EDX), This is close to the ratio of the raw materials and comparable to previous reports [START_REF] Kiyohara | Giant Anomalous Hall Effect in the Chiral Antiferromagnet Mn3Ge[END_REF].

Longitudinal and Hall resistivity were measured by the standard four-probe method using a current source (Keithley 6221) with a DC-Nanovoltmeter (Keithley 2182A) in a commercial measurement system (PPMS, Quantum Design). The thermal conductivity and thermal Hall effect were performed using a heater and two pairs of thermocouples in the PPMS in a high-vacuum environment [START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF]. For temperatures below 4.2 K, the measurements were performed in a dilution refrigerator inserted in a 14 T superconducting magnet using oneheater-three-thermometers set-up, allowing to measure longitudinal and transverse transport coefficients with the same contacts.

Theoretical calculations: The band structure was calculated with the density-functional theory in the framework of the generalized-gradient approximation [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF]. The Bloch wave functions were projected to atomic-orbital-like Wannier functions [START_REF] Mostofi | Wannier90: A Tool for Obtaining Maximally-Localised Wannier Functions[END_REF]. Based on the Wannier-projected tight-binding Hamiltonian, we calculated the Berry curvature and the anomalous Hall conductivity in the clean limit. More details can be found in Ref. [START_REF] Zhang | Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn3X (X = Ge, Sn, Ga, Ir, Rh, and Pt)[END_REF]. As shown in Fig. 6(a) and (c), the σzx (ξ) data are analytically fitted so that integrals in Eqs. 5 and 6 can be evaluated accurately with dense energy steps.

Data availability: The data that support the findings of this study are available upon reasonable request.

Supplemental Material for "Finitetemperature violation of the anomalous transverse Wiedemann-Franz law in absence of inelastic scattering" 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0 5 0

1 0 0 1 5 0 2 0 0 ؓ A H ( Ω -1 c m -1 ) T ( K ) # 1 B | | y # 9 B | | x # 1 0 B | | x

S1. MAGNITUDE OF THE ANOMALOUS HALL CONDUCTIVITY

The zero-temperature anomalous Hall conductivity found here (190 Ω -1 cm -1 ) is to be compared with previous reports of 150-450 Ω -1 cm -1 in Ref. [1] and 320-380 Ω -1 cm -1 in Ref. [2]. The difference in stoichiometry between the present sample (Mn 3.1 Ge 0.9 ) and the one studied in Ref. [2] (Mn 3.05 Ge 0.95 ) combined to the fact that the magnitude of the anomalous Hall resistivity depends

1 0 -3 1 0 -1 1 0 1 1 1 0 1 0 0 0 . 2 1 0 -3 1 0 -1 ؋ A z x ( # 1 ) ؋ A z x ( # 2 ) ؋ ( W / K m ) ؋ x x ؋ z z a ) b ) W A z x ( K m / W ) T ( K ) W A z x ( # 1 ) W A z x ( # 2 )
FIG. S3. a) Temperature dependent longitudinal thermal conductivity κxx and κzz and anomalous transverse thermal conductivity κ A zx , the diamond and star denote the data measured in #1 and #2, respectively. b) The anomalous transverse thermal resistance W A zx with the function of temperature.

on Mn content [2], provide an explanation for this apparent discrepancy. The temperature dependent anomalous Hall conductivity in different samples cut from the same batch are similar (see Fig. S1).

S2. CARRIER DENSITY

The field dependence of Hall resistivity along two different directions, yield the same value for ∂ρ H /∂B at high magnetic field at 2 K (see Fig. S2). The slope was found to be R H = ∂ρ H /∂B = -0.02 µΩ cm/T. Using a single-band model (n = 1 eR H ), we found a carrier density of 3.1×10 22 cm -3 , which is comparable to Ref. [1]. A small anomalous Hall resistivity observed previously [1] in the B z configuration is also observed in our sample.

L 0 L x x / L 0 T ( K ) L x x / L 0 ( M n 3 G e ) FIG.
S4. The Lorenz ratio for longitudinal thermal and electrical transport in Mn3Ge.

S3. LONGITUDINAL THERMAL TRANSPORT

Fig. S3 shows the temperature dependence of thermal conductivity along x axis, κ xx , and along z axis, κ zz as well as the anomalous transverse thermal conductivity, κ A zx . The latter was extracted using the anomalous thermal resistivity, W A zx and the longitudinal conductivities. Fig. S4 shows Lorenz ratio L xx for longitudinal transport in Mn 3 Ge. At finite temperature, L xx /L 0 exceeds unity indicating a large contribution from phonons. As expected, the ratio tends to unity in the zero temperature limit.

S4. THE KELVIN RELATION

The Peltier effect is the heat current produced by a charge current. The Kelvin relation relates the Peltier and Seebeck coefficients: Π xx = T S xx . Fig. S5b shows Π xx and T S xx in Mn 3 Ge. The ratio of Π xx /T S xx is close to 1 in the whole temperature range. The Peltier coefficient Π xx and the Ettingshausen coefficient zx were obtained by symmetrizing the signals obtained by injecting positive and negative currents through the sample in order to eliminate the Joule heating effect. Comparing to Ettingshausen effect, we can measure the heat absorb/release from the sample. We measured the temperature gradient along the sample to get the heat combining with the thermal conductivity [3].

S5. THERMAL EVOLUTION OF ANOMALOUS TRANSVERSE RESPONSE

Fig. S6 displays the field dependence of the three transverse coefficients at different temperatures. The temperature dependence of the anomalous transverse transport coefficients presented in main text were deduced from these curves.

S6. SPECIFIC HEAT CP AND MAGNETIC SUSCEPTIBILITY χ

The Fig. S7 a) and b) show the specific heat data for Mn 3 Ge and Mn 3 Sn. As seen in the inset, the electronic contribution in Mn 3 Sn is 19.1 mJ mol -1 K -2 . From the low-temperature magnetization curve (Fig. S7c), the magnetic susceptibility χ is deduced to be 6.2 × 10 -3 emu Oe -1 mol -1 . As seen in Fig. S8, the magnetic susceptibility is much larger than what is expected for the Pauli susceptibility of mobile electrons according to the Wilson ratio. This implies that most of the susceptibility is due to Mn spins.

S7. THE DEPENDENCE OF LORENZ RATIO IN

Mn3Sn ON STOICHIOMETRY

The focus of this paper is the validity and/or the violation of the anomalous transverse WF law in Mn 3 Ge. Nevertheless, let us notice that in the case of Mn 3 Sn,

0 1 0 0 2 0 0 3 0 0 0 1 -1 0 1 2 b ) װ x x /T S x x T ( K )  װ x x | & -T S x x ( m V ) װ x x T S x
0 0 K 2 0 0 K 3 0 0 K ؒ z x ( µΩc m ) a ) b )  ؓ z x ( 10 2 Ω -1 c m -1 ) c ) -S z x ( µV / K ) d )   z x ( Α/ K m ) e ) W z x (1 0 -4 K m / W ) ؍ 0 H ( m T ) f )  ؋ z x (1 0 -4 W / K m ) FIG. S6.
Field dependence of transverse transport coefficients at different temperatures.

the published data suggests that the the precise chemical composition of the compound can effect validity or verification of the WF law with little effect on the temperature dependence of resistivity. The precise location of the chemical potential depends on the the Mn. A violation of the WF law in Mn 3 Sn is conceivable if the The electronic specific heat vs. magnetic susceptibility χ ; the red rectangle refers to Mn3Ge, the others are from Ref. [4], the solid line represent ) FIG. S9. The Lorentz ratio in Mn3Sn with different content of Mn.(digitized from [5,6]) chemical potential shifts away from the µ = 140 meV (See Fig. 6).

S8. THE BAND STRUCTURE AND BERRY CURVATURE

FIG. 1 .

 1 FIG.1. Anomalous transverse coefficients, definitions and profiles: Off-diagonal components of the three conductivity tensors (electric, σ, thermoelectric, ᾱ, and thermal, κ) can be finite in the absence of magnetic field. As shown in the three left panels, they link to four vectors, which are charge density current, J, electric field, E, thermal gradient, ∇T and heat density current, JQ. a) Hall resistivity, ρzx; b) Hall conductivity, σzx, extracted from ρzx, ρxx and ρzz; c) Nernst signal, Szx; d) Transverse thermoelectric conductivity, αzx, extracted from Szx, Sxx, ρxx, ρzz and ρzx; e) thermal Hall resistivity, Wzx; f) thermal Hall conductivity, or the Righi-Leduc coefficient, κzx extracted from off-diagonal and diagonal thermal resistivities.

FIG. 3 .

 3 FIG. 3. Anomalous transverse Wiedemann-Franz law: Temperature dependence of the anomalous Hall conductivity σ A zx (a) ; the anomalous thermal Hall conductivity divided by temperature κ A zx /T (b); c) the anomalous Lorenz ratio κ A zx /σ A zx T . Different symbols are used for data obtained with two different set-ups: resistive thermometers (diamonds) and thermocouples (circles). Star symbols refer to a third set of data obtained on another sample measured down to sub-Kevin temperatures. The horizontal solid line marks L0 = 2.44 × 10 -8 V 2 K -2 . The deviation between L and L0 starts at T > 100 K and is concomitant with the decrease in σ A zx . d) Temperature dependence of the anomalous Lorenz ratio in Mn3Ge and in Mn3Sn[START_REF] Li | Anomalous Nernst and Righi-Leduc effects in Mn3Sn: Berry curvature and entropy flow[END_REF][START_REF] Sugii | Anomalous thermal Hall effect in Anomalous thermal Hall effect the topological antiferromagnetic state[END_REF]. e) Comparison of their in-plane resistivity. The large deviation from the WF law in Mn3Ge occurs in spite of the fact that the temperature dependence of its resistivity is even more modest than that in Mn3Sn.

FIG. 4 .FIG. 5 .

 45 FIG. 4. Anomalous Nernst and Ettingshausen effects and the Bridgman relation: a) The transverse electric field created by a finite longitudinal temperature gradient as a function of magnetic field (the Nernst effect); b) the transverse thermal gradient produced by a finite longitudinal charge current (the Ettingshausen effect) at the same temperature. Insets show experimental configurations. c) The temperature dependence of the anomalous Nernst (S A zx ) and anomalous Ettingshausen ( A zx ) coefficients. A zx and S A zx T /κxx remain equal as expected by the Bridgman relation. d) Temperature dependence of α A zx extracted from the Nernst signal S A zx . Also shown is the temperature dependence of Anomalous Hall conductivity σ A zx .

FIG. 6 .

 6 FIG. 6. The zero-temperature Berry curvature σ A zx (µ) (a) (c) and the anomalous Lorenz ratio L A zx (c) (d). The charge neutral point is set to zero. The green, red and blue lines represent µ = 0, 140 and 180 meV, respectively. The dashed horizontal black lines represents L0 in (c) and (d). In the band structure (e) and (f), the color indicates the Berry curvature value. The blue arrows point out two Weyl points between the lowest and second lowest conduction bands.

  FIG. S1. Temperature dependence of anomalous Hall conductivity in three samples from the same batch in two configurations, B y and B x.

B

  FIG. S2. Hall resistivity ρH vs. B for B y (black line) and B z (red line) at 2 K. The blue lines are linear fits to quantify the ordinary Hall resistivity.

  FIG. S5. Kelvin relation: a) The comparison of Peltier coefficient, Πxx and Seebeck, Sxx times temperature, T at different temperatures. b) The ratio of Πxx/T Sxx as a function of temperature.

2 )T 2 ( K 2 )

 222 FIG. S7. Temperature dependence of specific heat of Mn3Ge, a) and Mn3Sn ,b). Inset show the C/T vs. T 2 of Mn3Sn . c) Magnetization curve of Mn3Ge measured at 2 K, the susceptibility χ = 6.2 × 10 -3 emu.Oe -1 mol -1 is extracted from the high-field slope (black dashed line).

2 )

 2 FIG. S8.The electronic specific heat vs. magnetic susceptibility χ ; the red rectangle refers to Mn3Ge, the others are from Ref.[4], the solid line represent

π 2 κ 2 B 3µ 2 B

 22 χ. The large magnetic susceptibility of Mn3Ge is not caused by correlated electrons but by ordered spins.

  FIG. S10. Contrasting Mn3Ge and Mn3Sn in the band structure and Berry curvature. The upper and lower panels are for Mn3Ge and Mn3Sn, respectively. (a) and (b) The Berry curvature Ωxz distribution in the kxky plane integrated along kz for Mn3Ge and Mn3Sn, respectively. The color bar of Ωxz is in arbitrary unit. Corresponding noncollinear AFM spin structure is shown in (i). There is a My mirror symmetry to constrain that only Ωxz is even to the mirror plane, i.e. only the anomalous Hall conductivity σxz is nonzero. The dashed circle in (a) indicates two hot spots near the M -point in the Brillouin zone (the black hexagon). (c)-(d) The band structure along kz through the center of a hot spot for Mn3Ge and Mn3Sn, respectively. The Fermi energy is set to zero. The color code indicates the Berry curvature distribution in the band structure. At the charge neutral point, the Berry curvature is dominantly contributed by an anti-crossing gap in along the kz axis for Mn3Ge, while such a gap is missing for Mn3Sn. (e)-(f) The band structure along Γ -M . When it lies at 180 meV, the chemical potential crosses a type-II Weyl point for Mn3Ge. Because of the strong tilting of the Weyl cone, its net contribution to the total Berry curvature (sum over all occupied bands) is small. In contrast, he chemical potential of 140 meV crosses a weakly-tilted Weyl point for Mn3Sn, inducing large Berry curvature.

Acknowledgements We thank Alaska Subedi for numerous discussions. This work was supported by the National Science Foundation of China (Grant No. 51861135104 and No. 11574097) and The National Key Research and Development Program of China(Grant No.2016YFA0401704). Z. Z. was supported by the 1000 Youth Talents Plan. K. B. was supported by China Highend foreign expert program and Fonds-ESPCI-Paris and by the KITP through the National Science Foundation Grant No. NSF PHY17-48958. B.Y. acknowledges the financial support by the Willner Family Leadership Institute for the Weizmann Institute of Science, the Benoziyo Endowment Fund for the Advancement of Science, Ruth and Herman Albert Scholars Program for New Scientists, and the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 815869). B.Y. thanks the fruitful discussion with Yang Zhang at the Max Planck Institute in Dresden.

Supplementary


Dante Fedele
email: dante.fedele@univ-lille.fr

Grotius and Late Medieval Ius Commune on Rebellion and Civil War

Keywords: Grotius, Ius commune, Rebellion, Civil War, Use of force

This paper explores the presence of late medieval ius commune in Grotius's thought on the use of force in internal strife and war, based on De iure belli ac pacis (1625). To this end, it examines Grotius's use of ius commune sources, and considers some similar sources, which he does not actually cite, but which relate to his discussion. By clarifying Grotius's selection and use of ius commune sources, the paper intends to contribute to the achievement of a double aim: firstly, to determine the place of rebellion and civil war in De iure belli ac pacis, especially in relation to (external) war; and, secondly, to assess Grotius's approach to the two former issues, particularly with regard to the criteria by which a distinction between rebellion and civil war can be drawn, and to the effects of this distinction.

This paper explores the presence of late medieval ius commune in Grotius's thought on the use of force in internal strife and war, based on De iure belli ac pacis (1625). To this end, it examines Grotius's use of ius commune sources, and considers some similar sources, which he does not actually cite, but which relate to his discussion and can be fruitfully studied to broaden the scope of the analysis. By clarifying Grotius's selection and use of ius commune sources, the paper intends to contribute to the achievement of a double aim: firstly, to determine the place of rebellion and civil war in De iure belli ac pacis, especially in relation to (external) war; and, secondly, to assess Grotius's approach to the two former issues, particularly with regard to the criteria by which a distinction between rebellion and civil war can be drawn, and to the effects of this distinction.

One obvious starting point for this kind of investigation is Robert Feenstra's annotation of De iure belli ac pacis, which guides us neatly through the numerous sources cited by Grotius in his magnum opus, and enables the reader to appreciate the strong presence of ius commune jurisprudence. 1 Feenstra also provided an overview of this subject in an article which appeared in 1992, in which he demonstrates that Grotius sometimes had direct knowledge of the works he cited (although when writing De iure belli ac pacis he had no longer access to his personal library), and sometimes relied on second-hand references to medieval jurists (in the works of early modern scholars like Francisco de Vitoria, Alberico Gentili, Diego de Covarrubias and Fernando Vázquez de Menchaca, in particular). 2 A ground-breaking study in the field -predating Feenstra -was Peter Haggenmacher's Grotius et la doctrine de la guerre juste, which is especially relevant to our analysis insofar as it focuses on war and the use of force, and -through careful analysis of both De iure praedae and De iure belli ac pacis -assesses the presence of ius commune authorities in these works. 3 Subsequently, James Muldoon studied Grotius's use of medieval canon law in Mare liberum (1609), in order to show how the Dutch jurist, although he used the same source material as his Spanish predecessors, actually elaborated a different conception of the international order, criticising the idea of a trade monopoly based on papal donation, and asserting a basis for trade freedom in natural law. 4 More recently, Robert Fredona has investigated the presence of late medieval jurisprudence in Mare liberum, insisting on the 'technical legal character' of this work and its scholarly apparatus, which includes 'an average of 2.25 mentions' of Roman, canon, and feudal law texts or commentaries per page, and features Bartolus de Sassoferrato and Baldus de Ubaldis as the most cited authors of that tradition. 5 IBP. Translations are from H. Grotius, De jure belli ac pacis libri tres, transl. F.W. Kelsey (Oxford: Clarendon Press, 1925).

Grotius's categorisation of war in De iure belli ac pacis is well known. 6 He starts from a distinction between 'private war' (fought by private persons), 'public war' (fought by people invested with jurisdiction) and 'mixed war' (which is private on the one side, and public on the other). Public war is then divided into two sub-categories, i.e. 'formal (solenne)' on the one hand (which requires two conditions: a formal declaration of war, and the sovereignty of both belligerent parties), 7 and 'less formal (minus solenne)' on the other (which is defined by the absence of either of these conditions). Internal strife and war fall under the category of 'mixed war' -they are, in fact, the only instance of mixed war that Grotius discusses (I.4). However, he also examines them in his analysis of 'less formal public war' (I.3). Actually, these two chapters seem to reflect different perspectives: in chapter 3 Grotius considers rebellion from the viewpoint of the public official (who has the authority to repress rebellion), whereas in chapter 4 he considers the same issue from the viewpoint of private persons (who normally enjoy no right of resistance). Internal strife and war are also touched upon in other passages of De iure belli ac pacis, including those on the right to send ambassadors (II.18.2.3), the right of burial (II.19.4), going to war on behalf of others (II.25.8), formal war (III.3.1), the right to appropriate objects captured in war (III.6.27), the rights over prisoners of war (III.7.5.3 and III.7.7), and the keeping of good faith with enemies (III.19.6-10).

This paper is divided into two parts. In the first, it deals with rebellion and its relation to the notion of war (bellum); in the second, it concentrates on civil war and -in particularsome medieval antecedents of Grotius's conceptualisation of the two belligerent parties as two distinct gentes.

Rebellion

When he discusses less formal public war, in I.3.4, Grotius begins by presenting the opinion that not every public official has the authority to 'wage war for the protection of the people entrusted to his charge', since war can only be waged under the authority of the holder of the 'sovereign power in the state (summa in civitate potestatem)'. 8 He refers to Plato's Laws (12.7), and to both Roman and canon law: lex Iulia maiestatis is cited, 9 as are a constitution by Valentinian and Valens (from 364), and an excerpt from Augustin included in Gratian's Decretum. 10 All of these sources state that the order to take up arms can only be issued by the princeps -a principle which is duly emphasised by the glosses to all these Roman and praescriptione, Fredona uses this as a case-study about Grotius's 'second-hand erudition', see ibid., p. 62, note 23.

6 See Haggenmacher, Grotius, pp. 529-37.

7 On Grotius's notion of sovereignty, see ibid., pp. 537-47. canon law passages.11 Two exceptions, however, are established: firstly, Grotius says, 'it is permissible for a public official, who has proper authority over a district, [...] to restrain by force a few that are disobedient, whenever there is no need of larger forces for the purpose, and danger does not threaten the state'. Note that a distinction between rebellion and civil war (the 'danger' which threatens the state) seems to be implied here, and based on a quantitative criterion; this point will be discussed later. Secondly, necessity is another exception to the laws requiring an order by the princeps to declare war: 'if the danger is so pressing that time does not permit consultation with him who has the supreme authority in the state, in that case also necessity will make an exception'. 12 In order to support this assertion about the authority of subordinate public officials to put down an insurrection, Grotius cites Bartolus de Sassoferrato, Martinus Garatus Laudensis, Luis de Molina and Francisco de Vitoria. 13 However, the only sources that specifically touch on this issue are actually Bartolus's comment on lex Ex hoc iure (Dig. 1.1.5), and a quaestio of Martinus Garatus's De iure belli. Bartolus attributed to Pierre de Belleperche and Guillaume de Cun the opinion that subordinate public officials can legitimately take up arms against their subjects without authorisation by the princeps both in a defensive war, and (if there is no princeps, or he has no de facto power) in an offensive war. 14 Martinus Garatus, on the other hand -writing approximately a century after Bartolus -, stated that 'the princes and cities of Italy', since they recognised no superior, could wage war against either their subjects or external enemies, and went on to extend this principle to the situations in which the superior authority was absent. 15 After establishing the right of subordinate officials to take up arms, Grotius wonders (in I.3.5) whether or not such war should properly be deemed 'public'. In this section he seems to adopt a more restrictive view than in the previous section, and to identify 'public war' with 'formal war', since -arguing against 16 th -century authorities like Balthasar Ayala and Silvestro Mazzolini 16 -he relies on late medieval jurists in declaring that, if the word 'public' is understood 'as characterizing that which is done with due formality, [...] such wars are not public, for the reason that both the decision of the sovereign power and other conditions are necessary for the fulfillment of the legal requirements involved'. 17 Indeed, Bartolus had denied the 'public' character of the 'wars' waged by private persons to defend themselves or recover their properties, while Innocent IV had called them 'defense (defensio)' rather than 'war (bellum)', and had understood military actions against rebellious subjects as 'executio iurisdictionis vel iustitia' or 'executio iuris' rather than war. Nicolaus de Tudeschis (Panormitanus) had then talked of the war waged by subjects (including subordinate officials) without the authorisation of their superior as 'bellum improprie sumptum': this could be lawful or unlawful depending on whether or not there was a 'just cause' -the need to put down an insurrection, for example. 18 It is noteworthy that, in chapter 6 of De iure praedae (1604-1608), Grotius had criticised these very medieval jurists when arguing that even (external) wars declared by subordinate officials -in cases of absence or negligence of the supreme magistrate -had to be termed 'public': he was then assuming that the requisite authority had been tacitly delegated to the subordinate officials concerned. 19 In De iure belli ac pacis I.3.5 he no longer assumes this, and puts forward a stricter notion of 'public' war. , pp. 175-6. A few lines below, Grotius also makes reference to a consilium by Franciscus Aretinus about the status of property taken in war, stating that the deprivation of property is not limited to formal war but also occurs in rebellions: see Franciscus de Aretio, Consilia (Ludguni: Iacobus Giunta, 1546), consilium 14, fol. 12 rb , n. 9.

19 See Haggenmacher, Grotius, pp. 145-6, and Hugo Grotius, De iure praedae commentarius, ms Leiden, University Library, BPL 917, chap. 6, fol. 28 v -29 r (hereafter cited as IPC; a critical edition of this manuscript by Jan Waszink is available at http://arkyves.org/view/DeIurePraedae, accessed 30.07.2020). This argument is restated in IPC, chap. 13, fol. 128 v -129 r , where Grotius maintains that the States Assembly of Holland had a right to wage war on the Portuguese, before going on to consider the justness of the Santa Catarina's capture as an act of 'public' war carried out by the Dutch East India Company (VOC). However, ibid., fol. 141 r Grotius says that 'even if no order specifically concerned with prize had been issued, nevertheless, owing to the fact that both the Admiral of the fleet and the captains of the individual vessels had been granted jurisdiction by the state, these commanders would have been empowered -in the absence of other judges, and in defence of the rights of subjects as well as of their own authority -to impose punishment upon Portuguese offenders against that authority, and to seize the property of such offenders'. Grotius then refers again to the abovementioned comments by Bartolus, Innocent IV and Panormitanus to argue that 'this conclusion is especially tenable in the event that it is supported by a decree issued in advance; and in the present case such a decree was issued by the Admiral [dated December 4, 1602 marginal note] and rendered duly effective by the supreme naval council which assisted him in an assessorial capacity' (I used the translation by Gwladys L. Williams published in H. Grotius, Commentary on the Law of Prize and Booty, edited and with an Introduction by M.J. van Ittersum (Indianapolis: Liberty Fund, 2006), p. 429).

The discussion on rebellion continues in chapter I.4 (on 'mixed war'), where Grotius excludes resistance against both the ruler who actually exercises his power in a despotic way, and the ruler who has no lawful title and is therefore a usurper (i.e. the two kinds of tyrannus described by Bartolus de Sassoferrato in his treatise De tyranno). 20 There are some exceptions to this principle, the most interesting of which is that of the king who sets out to 'destroy' his people and, since he has thus proven himself to be 'an enemy of the whole people (hostis totius populi)', is considered to have renounced his kingdom. Grotius specifically refers to the king who rules over several peoples, and who 'may wish to have one people destroyed for the sake of another, in order that he may colonize the territory thus made vacant (ut colonias ibi faciat)'. 21 Drawing on a note by Jean Barbeyrac, Jules Basdevant and Peter Haggenmacher have pointed out that Grotius very likely considered this description to fit Philip II's attitude towards the United Provinces perfectly. 22 A distinction thus emerges between individuals' right of resistance (which Grotius does not admit) and that of an entire people (which, under some circumstances, he does admit), suggesting again that mere rebellion and civil war can be quantitatively distinguished. Moreover, the fact that the king who sets out to destroy his people is called hostis totius populi seems to imply that there is little difference between collective resistance and a state of external public war. This argument appears to be in line with the view already expounded in De iure praedae -despite the much greater emphasis given to the justification of resistance in the latter work -, where Grotius attributes the right to resist the unjust ruler (i.e. Philip II) to magistrates that are vested with the 'supreme authority' (i.e. the Dutch States General), and ultimately considers war against Spain to be an instance of 'external war (bellum [...] externum)' rather than rebellion or civil war 23 . 20 Grotius, IBP, I.4.1.3, p. 138, and I.4.15, pp. 159-60. See Bartolus de Sassoferrato, Tractatus de tyranno, ed. D. Quaglioni, Politica e diritto nel Trecento italiano. Il "De tyranno" di Bartolo da Sassoferrato (1314-1357) (Florence: Leo S. Olschki, 1983), quaestio VI, p. 185, l. 206-9, and quaestio VIII, p. 196, l. 447-9. Reference to Bartolus with regard to this twofold classification of tyranny is found in Grotius, IPC, chap. 13, fol. 135 r (the passage including this reference is a later addition by Grotius). 21 Grotius, IBP, I.4.11, p. 158, transl. Kelsey, pp. 157-8. For other exceptions, see Grotius, IBP, I.4.8-14.

22 See J. Barbeyrac, in H. Grotius, Le droit de la guerre et de la paix, trad. J. Barbeyrac (Amsterdam: chez Pierre de Coup, 1729), p. 233, note 3, commenting on the passage quoted in the previous footnote: 'On a attribué ce dessein à Philippe II. Roi d'Espagne, par rapport aux Pais-Bas' (emphasis in original). This note is referred to by J. Basdevant, 'Hugo Grotius', in Les fondateurs du droit international, avec une introduction de A. Pillet (Paris: V. Giard In De iure belli ac pacis I.4, no specific reference is made to medieval ius commune authorities. That said, near the end of it, Grotius cites a passage from Thomas Aquinas's Summa theologiae -which he had already used in De iure praedae -in making yet another argument against the right of resistance. 24 It is worth remarking that Bartolus, in his treatise De Guelphis et Gebellinis, had used the very same passage by Aquinas to argue in favour of rebellion -in the absence of a superior authority to which to resort -for the sake of public good. 25 'A tyrannical government -Aquinas said -is not just, because it is directed, not to the common good, but to the private good of the ruler, as the Philosopher states (Pol. 3.5; Eth. Nic. 8.10). Consequently there is no sedition in disturbing a government of this kind, unless indeed the tyrant's rule be disturbed so inordinately (sic inordinate), that his subjects suffer greater harm (maius detrimentum) from the consequent disturbance than from the tyrant's government. Indeed it is the tyrant rather that is guilty of sedition, since he encourages discord and sedition among his subjects, that he may lord over them more securely; for this is tyranny, being conducive to the private good of the ruler, and to the injury of the multitude'. 26 While in De iure praedae Grotius had fully accepted Aquinas's line of argument, in De iure belli ac pacis he does not consider the possibility of rebellion for the public good, and simply states that, for Aquinas, 'the destruction of a government even though tyrannical is sometimes (interdum) an act of sedition'. 27 This different use of Aquinas's passage in the two works provides a glimpse into Grotius's attitude towards sources and authorities: in fact, rather than being interested in interpreting them in their own context, or building up a consistent philosophical system, he appears to be guided by the need imperio summarum potestatum circa sacra (1614-1618, but published posthumously in 1648): see H. Grotius, De imperio summarum potestatum circa sacra, critical edition with Introduction, English translation and commentary by H.-J. van Dam (Leiden/Boston: Brill, 2001), III.6-8 and 14, pp. 212-7 and 227-8. Although only very few references to legal sources are found here, many theological references are the same which are found in IBP: see the editor's remarks, ibid., pp. 670-1 and 681-2. It is well known that Grotius wrote IBP in France, and dedicated it to Louis XIII, who had welcomed him after exile: along with his wish to maintain good relations with James I, these were probably reasons that helped to make Grotius avoid extending the possibilities of (individual) resistance against sovereigns: see recently P.E. Vauthier Borges de Macedo, Catholic and Reformed Traditions in International Law. A Comparison Between the Suarezian and the Grotian Concept of Ius Gentium (Cham: Springer, 2017), pp. 121 and 273-82. The right of resistance is also denied in IBP I.3.8 (where Grotius criticises the notion of popular sovereignty), II.1.9, II.22.1 and III.7.7. 24 Thomas Aquinas discussed the issue of rebellion (seditio) in S.Th. II a -II ae , quaestio 42. 25 See Bartolus de Sassoferrato, Tractatus de Guelphis et Gebellinis, ed. Quaglioni, Politica, p. 139, l. 157-64.

26 S.Th. II a -II ae , quaestio 42, articulus 2, ad tertium, transl. the Fathers of the English Dominican Province (New York: Benzinger, 1947-1948), available at https://www.documentacatholicaomnia.eu/03d/1225-1274,_Thomas_Aquinas,_Summa_Theologiae_%5B1%5D,_EN.pdf (accessed 30.07.2020). In his previous commentary on Peter Lombard's Sententiae, dealing with the case of a tyrant who has acquired political authority by unlawful means, Aquinas had authorised rebellion without even referring to the 'greater harm' limit: see Scriptum super Libros Sententiarum, liber II, dist. 44, quaestio 2, articulus 2 ('Utrum Christiani teneantur obedire potestatibus saecularibus, et maxime tyrannis') and, on the issue, Reichberg, Thomas Aquinas, 123-7. to argue a specific point, or support a particular view, if necessary disregarding those sources' original meaning [START_REF] Van Dam | Introduction[END_REF] .

The fact that Grotius -in general -denies the right of resistance to subjects does not stop him from allowing, in some cases, a third party to intervene in order to rescue them from the tyrannical acts of their ruler. 29 In II.25.8 Grotius discusses the right to undertake war 'on behalf of the subjects of another ruler, in order to protect them from wrong at his hands', and he states that if some ruler 'should inflict upon his subjects such treatment as no one is warranted in inflicting, the exercise of the right vested in human society (ius humanae societatis) is not precluded': in this case, even though 'subjects cannot justifiably take up arms', it does not follow that 'others may not take up arms on their behalf [...]. The restriction, in fact, which prevents a subject from resisting, does not arise from a cause which is identical in the case of a subject and of one who is not a subject, but from the personal condition which is not transferred to others'. [START_REF] Grotius | [END_REF] If one excepts classical sources, only two references to Vitoria's De Indis can be found in this passage, [START_REF] De Vitoria | De Indis et de iure belli relectiones[END_REF] but elsewhere Grotius expounds a similar view relying on some medieval authorities. In II.20.40 he explicitly endorses Innocent IV's opinion that 'war may be waged upon those who sin against nature (qui in natura delinquunt)'; [START_REF] Grotius | See Innocent IV, Super libros[END_REF] and in I.5 he cites Innocent IV, Bartolus, Paulus Castrensis, Panormitanus, Jason de Mayno and Silvestro Mazzolini to argue in favour of service rendered to other men, if not due to other ties, at least by virtue of the 'common bond of human nature (humanae naturae communio)'. [START_REF] Grotius | [END_REF] It is worth pointing out that, among them, only Innocent IV had briefly touched on war waged 'causa pietatis vel humanitatis'; 34 Bartolus, Paulus, Panormitanus, and Jason de Mayno, on the other hand, had admitted the use of force in defence of someone else in the context of a discussion which concerned neither public war in particular, nor the rebellion of subjects against their ruler, but rather people who generically suffered injustice. 35 This principle had been applied to war by Johannes de Legnano and, based on the latter, by Silvestro Mazzolini, under the entry on 'war (bellum)' of his Summa, which gave it a particular significance in the context of the discussion of war. 36 While he discusses the right to undertake war to support the rebellious subjects of another ruler, Grotius does not tackle the issue of whether or not someone can intervene in favour of another ruler and against the insurgents in a rebellion. This issue had certainly not been ignored in the ius commune tradition: Bartolus had raised it in his commentary on the constitutions Ad reprimenda and Quoniam nuper, possibly the most renowned late medieval legal source on resistance. 37 Ad reprimenda and Quoniam nuper had been enacted by Henry VII in 1313 to provide a legal basis for his conviction of King Robert of Naples for rebellion against the empire. The emperor had sentenced Robert in absentia to be beheaded, and his land confiscated; pope Clement V had reacted by declaring the conviction invalid, both because Robert -who was domiciled in the terrae ecclesiae -could not be summoned from outside the borders of the empire, and because he had been stripped of his right to defend himself. 38 In his commentary, Bartolus pondered on the legitimacy of rebellion and its consequences. Although in other writings he had admitted resistance against tyrants, he said that no cause could justify resistance against the emperor. 39 In his view, the emperor was the supreme authority who legitimised all other powers, a principle which led to two consequences. Firstly, by definition the emperor could not himself be a tyrant. Secondly, as regards the consequences of rebellion, rebels against the empire (either private persons or 35 See Bartolus de Sassoferrato, In Primam Digesti Veteris Partem, ad Dig. 1.1.3, fol. 7 ra , n. 7-8: drawing on the Ordinary gloss, Bartolus first held that it was licit to defend 'consanguinei' based on 'affectio'. Then he admitted that other people ('ali[i] extrane[i]') could also be succoured, but suggested that before setting out to help them a request from their part ('mandatum') should be awaited. 39 See Bartolus de Sassoferrato, Consilia, quaestiones, et tractatus (Venetiis: [Iunta], 1596], Super constitutione 'Qui sint rebelles' [= 'Quoniam nuper'], ad v. rebellando, fol. 105 ra , n. 1: 'Et advertendum est, quod in illo, qui rebellat contra principem, haec constitutio loquitur simpliciter, quasi contra eum non possit esse aliqua iusta causa resistendi'. Bartolus had admitted resistance against the tyrant, for example, in the passage cited supra, note 25. See Quaglioni, Politica. cities) lost ipso iure all rights derived from ius civile, and could be punished by anyone: if they were private persons, they could even be killed or enslaved, whereas if they were cities, they could be taken and plundered -although they could neither be destroyed, nor their population enslaved, without the consent of the emperor, to whom they belonged. 40 Grotius did not raise the issue, nor did he cite Bartolus's commentary (in either De iure praedae or De iure belli ac pacis), although he must have been familiar with it, since it circulated both in early modern editions of Bartolus' work, and in those of the Corpus iuris civilis, where it was printed as the Ordinary gloss to the two constitutions. 41 One may speculate on what Grotius might have written had he decided to elaborate on the issue. However, it is highly unlikely that he would have referred to Bartolus's commentary anyway, so much was the latter grounded in the notion of the emperor's universal sovereignty. Grotius, as is well known, strongly opposed this notion, discarding it as 'absurd' and drawing on Aristotle and jurists like Panormitanus, Jason de Mayno, Charles Dumoulin and Jacobus Menochius to oppose the universalism of Dante and Bartolus. 42 In light of the above, it may be said that, in De iure belli ac pacis, Grotius chose to use only the best known ius commune sources on war, paying much less attention to the rich body of reflection on rebellion developed by both civil and canon lawyers. This drastic selection of his sources reveals Grotius's real interest, which lay in war, and the link between rebellion and the notion of war, rather than in rebellion per se. 43

Civil War

We have already referred to certain passages in which Grotius draws a distinction between the rebellion of individuals and the uprising of entire populations. Civil war -unlike rebellion -is not simply an internal matter: it bears some similarity to external war, although it does not lead to all the latter's consequences. This limitation is made apparent when Grotius discusses the right to appropriate objects captured in war, and the right to enslave prisoners of war. 44 In fact, in his view, the former right is 'so peculiar to a war that is formal (solennis) according to the law of nations that in other wars it finds no place', so that 'in wars between citizens, whether these be great or small, no change of ownership is made except by the authority of a judge'. 45 As for the rights over prisoners of war, Grotius observes -relying on 40 Bartolus, Consilia, fol. 105 va , n. 13, and fol. 105 vb -106 ra , n. 23-4. 41 In the fifth volume, called Volumen or Volumen parvum. 42 Grotius, IBP, II.22.13, pp. 560-1. For the authorities to which Grotius referred in this passage, see Feenstra's notes 561e-561l, ibid., p. 999. 43 The argument that Grotius's primary interest in De iure belli ac pacis was war has already been developed by Haggenmacher, Grotius, and A. Wijffels, 'De iure belli ac pacis libri tres. In quibus ius naturae & Gentium: item iuris publici praecipua explicantur [...] 1625. Hugo Grotius', in The Formation and Transmission of Western Legal Culture. 150 Books that Made the Law in the Age of Printing, ed. S. Dauchy et al. (Cham: Springer, 2016), pp. 173-7: 175. 44 Grotius, IBP, II.19.4.2, p. 456 can also be considered, where Grotius observes that 'legimus quidem in historiis exempla eorum qui insepulti abiecti sunt, frequentiora civilibus quam externis bellis'.

Plutarch and Tacitus -that in civil wars 'on many occasions captives have been killed because they could not be reduced to slavery'. 46 This approach to civil war was rooted in Roman law 47 . In the second half of the 14 th century, Baldus de Ubaldis had defined 'civil war (bellum civile)' as the war 'which a people begins against itself'. He had identified its causes in factional divisions within cities, quoting a 'wise Genoese' according to whom 'division in the city is the entry of the worm into the cheese'. Baldus had also distinguished this sort of war from external war ('bellum de gente ad gentem', as he called it elsewhere) by explaining that in a civil war captives do not become the slaves of their captors, nor do the postliminium or the fictio legis Corneliae apply, because this is not a genuine 'war' in the proper sense of the word. [START_REF] Baldus De Ubaldis | Sed nunquid capti in isto bello fiunt servi capientium? Respondeo non: quia non sunt proprie hostes, sed partiales. Quia civitas divisa est in partes, ut ff. de cap. l. si quis ingenuam[END_REF] In the theological tradition, Isidore of Seville had distinguished external war (either just or unjust) from 'civil (civile)' and 'more than civil (plusquam civile)' war: 'civil war -he stated -occurs when factions arise among fellow-citizens and hostilities are stirred up, as between Sulla and Marius', whereas 'a more than civil war is where not only fellow-citizens, but also kinfolk fight -this was done by Caesar and Pompey'. 49 However, he had not drawn a clear distinction between civil (and more than civil) war and insurrection, or sedition; on the contrary, he seems to have equated them when he wrote that 'as what is waged against external enemies is called "war", so "insurrection (tumultus)" is what is stirred up by civil sedition. Sedition (seditio) is a dissension (dissensio) of citizens, so called because they separate (seorsum ire) into different factions'. 50 bottom of the page Grotius had divided the bellum publicum into bellum civile and bellum externum, as opposed to bellum privatum.

Although he touches on civil war in several passages of De iure belli ac pacis, Grotius never really dwells on this issue. Probably the most interesting insight on this matter is found in his chapter on the law of embassies. Here, after saying that the right to send and receive ambassadors in principle belongs only to sovereigns (because ius gentium applies between different nations: 'inter gentes est diversas'), [START_REF] Grotius | Sed primum sciendum est, qualecunque est hoc ius gentium de quo videbimus, ad eos legatos pertinere quos mittunt qui summi imperii sunt compotes inter se. nam qui extra hos legati sunt provinciales, municipales atque alii, non iure gentium, quod inter gentes est diversas, sed iure civili reguntur[END_REF] Grotius makes an exception, and explains that 'in civil wars, however, necessity sometimes opens the way for the exercise of this right, though in an irregular fashion. Such a case will arise when a people has been divided into parts so nearly equal that it is doubtful which of the two sides possesses sovereignty [...]. Under such circumstances a single people is considered for the time being as two peoples (quasi duae gentes)'. [START_REF] Grotius | In bellis vero civilibus necessitas interdum locum huic iuri facit, extra regulam, puta cum ita divisus est populus in partes quasi aequales, ut dubium sit ab utra parte stet ius imperii: aut cum iure admodum controverso de regni successione duo decertant. nam hoc eventu gens una pro tempore quasi duae gentes habetur[END_REF] Grotius just discusses a couple of examples heretaken from Roman historical sources -and no legal authorities are mentioned. However, this specific issue had been debated in legal literature since the second half of the 16 th century, as demonstrated by the discussion engaged in by François Hotman, Pierre Ayrault, Alberico Gentili, and Jean Hotman (François's son) on the right of each faction, during civil war, to send ambassadors (i.e. agents enjoying full diplomatic status, including inviolability). All these scholars had admitted that factions in a civil war (not in a mere rebellion) enjoyed the rights established by ius gentium, and ius legationis in particular. [START_REF] Fedele | Ius gentium. The Metamorphoses of a Legal Concept (Ancient Rome to Early Modern Europe)[END_REF] Moreover, ways in which civil war contributed to the shaping of the conceptualisation of internal factions as independent universitates were already being recognised -and analysed -in the Middle Ages, with the struggles between rival factions in Italian cities from the 13 th century onwards. This analysis took place in the context of a broader debate (which culminated in Bartolus' treatise De Guelphis et Gebellinis) on the status of city-parties, in times of both peace and internal disorder. [START_REF] Costa | Bonum commune e partialitates: il problema del conflitto nella cultura politico-giuridica medievale[END_REF] A quaestio discussed around 1273 by the Bolognese jurist Thomas de Piperata is a remarkable example of this debate, and is of particular relevance to any discussion of late medieval ius commune on internal strife and its legal consequences, in light of the considerable analysis it devotes to the issue. At the time, Bologna was riven by factional discord, and Piperata -together with his fellow members of the Lambertazzi (i.e. Ghibellines) -would himself be exiled from the city in 1274. [START_REF]The quaestio is preserved in two manuscripts in the Biblioteca Apostolica Vaticana[END_REF] The text of his quaestio was partially edited (or rather summarised), in the 14 th century, by Albericus de Rosciate, who used it in his comment on Dig. 49.15.21.1. This excerpt from Ulpian deals with civil dissensions (civiles dissensiones), and says that in such situations, since 'the contest is not fought for the destruction of the respublica', 'those who go off to join one or other side are not counted as true enemies among whom the rights of captivity and postliminium apply'. 56 This passage, with some others, is quoted in its entirety by Grotius in III.3.1.1 to argue that a formal war according to ius gentium can only take place between different peoples. [START_REF] Grotius | [END_REF] At the beginning of his comment, Albericus de Rosciate remarked on the damage caused by civil strife, and defined the lex as 'golden and singular for the matter, which often occurs in miserable Italy, where constantly one party is expelling the other from cities and fortified places'. [START_REF] De Rosciate | Secundam ff. Novi partem Commentarij (Venetiis[END_REF] Then he summarised Piperata's quaestio, which deals with three issues: first, whether an expelled party can appoint a syndic (i.e. a representative), which would imply that it enjoys the status of a fully-fledged corporate body (universitas). Second, whether the members of an expelled party can be summoned by the city's podestà and whether or not he, if they ignore such a summons, can convict and punish them. And third, whether the intrinseci, if they are a minority of the people, still have the authority to represent the whole city, i.e. to alienate its property, or enter treaties on its behalf. To Piperata, the solution to all three issues depends on whether or not the expulsion was just. If an expulsion is motivated by a 'just cause (iusta causa)', the members of the exiled party are considered to be rebels who cannot constitute a separate universitas: they have no power to appoint a syndic and can be summoned, convicted and punished by the city's podestà; the intrinseci, on the other hand, retain the power to act on behalf of the whole city although they are a minority. If, however, the expulsion is not motivated by a 'just cause', the pars extrinseca constitutes a genuine universitas, the city's podestà is not entitled to summon its members, and the pars intrinseca (as it is a minority of the people) has no authority to bind the whole city. So, the criterion by which mere rebellion and civil war are distinguished is the just cause of the expulsion: if the expulsion is just, it is mere rebellion; if, on the contrary, it is unjust, Piperata says, 'it seems that two corporate bodies are made out of one (de una universitate videntur facte due)'. 59 While the criterion of the just cause evidently differs from that which is used by Grotius and other early modern scholars, the latter expression used by Piperata is remarkably close to the 'quasi duae gentes' mentioned by the Dutch jurist, and provides us with a significant antecedent of Grotius's conceptualisation of the two parties in a civil war as two distinct gentes.

The 'just cause' criterion appears in the comments on lex Omnes populi made by Bartolus de Sassoferrato, Raynerius de Arsendis and Baldus de Ubaldis. In their discussion, they ask whether, after one party has been expelled from a city, the other can pass a statute that is binding on the entire people (including the members of the expelled party); and their answer depends on the justice, or otherwise, of the expulsion: if the expulsion was not motivated by a iusta causa, the new statute will only be binding on the party which remained in the city. 60 Baldus adds another element to the discussion: if an expulsion is just, he says, 'jurisdiction remains in the hands of those inside the city because it adheres to the territory, and those inside possess the territory. [...] Those inside are a corporate body (universitas), and those expelled are separate individuals (singuli)'. 61 The just cause criterion is combined here with the idea that jurisdiction adheres to the territory, and the intrinseci are still in control of the city's territory.

Elsewhere, Baldus reports an opinion of Raynerius de Arsendis about the capacity of the expelled party to acts that are binding on the whole city. The case discussed by Raynerius had taken place in Todi, and concerned the expelled Guelph party's recognition of the authority held by the Roman Church over the city. 62 The jurist had considered the recognition to be invalid, since if the Guelphs had acted 'as separate individuals (ut singuli)' their action had no consequence for the city, and if they had acted 'collectively (ut universi)' their act was invalid, because their universitas was not licit. Baldus concurs with Raynerius, but makes an exception for the case in which 'the major part of the city were expelled, because then -he clarifies -it would make a licit corporation under the name of the majority and not that of a sect'. This passage is significant for our purposes, since here Baldus seems to rely on a quantitative criterion, not the just cause: in his view, the majority is held to embody the universitas and thus to represent the city as a whole. 63 In arguing this thesis, Baldus is drawing on Innocent IV and objecting to Bartolus's opinion: the latter had maintained that neither the expelled party nor the party within the city could alone submit the whole city to someone else, since neither had the power to represent it. 64 Baldus argues to the contrary: the surrender 'is valid if it is done by the majority' albeit only 'to the prejudice of those making the surrender, [...] not of their superior' (meaning that if the city has a superior no surrender can be made at the expense of his rights). 65 In these comments, moreover, both Bartolus and Baldus refer to another quaestio discussed in Bologna in the second half of the 13 th century. Its author was Lambertinus de Ramponibus who, unlike Thomas de Piperata, was a member of the Geremei (i.e. Guelph) party of Bologna; he nonetheless actively promoted peace with the Lambertazzi and in 1285, as a city councillor, endeavoured to have them return to Bologna. 66 Despite the opposition of those who, like Dynus de Mugello, believed that parties did not have the status of an 'approved guild', Lambertinus stated that both Guelphs and Ghibellines were entitled to elect a syndic to make peace with one another. The issue under discussion here was therefore not the extent to which parties could legitimately act as representative of the whole city, but the possibility for each to elect a representative in order to make peace with the other within the same city. Bartolus remarks that this solution was actually observed in every-day practice. 67 These examples show that interest in the nature and consequences of internal strife and war had already developed in the late medieval ius commune, which clearly distinguished between civil war (implying the recognition of both parties' legal personality) and rebellion (resulting in mere executio iurisdictionis against the insurgents, as stated by Innocent IV). Admittedly, the approach of late medieval jurists differed from that later adopted by Grotius, as did the criteria they proposed to distinguish the two situationswhether they relied on the just cause, the adherence of jurisdiction to territory, or (like Grotius himself) on a quantitative criterion. Given the scant attention paid by Grotius and his almost complete avoidance of previous scholarship on the issue -again, his real interest seems to have been external war -, it is hard to say whether these discussions ever attracted his attention. Nevertheless, just as in the case of rebellion, his remarks on civil war can confidently be placed in the broader context of the ius commune tradition, and thus contribute to further our understanding of premodern and early modern scholarship on the use of force in internal strife and war.

67

  See the references to Bartolus and Baldus provided supra, notes 64 and 65. I have been unable to find the original opinion of Lambertinus. As for Dynus de Mugello, a group of quaestiones authored by him are preserved in ms Vatican City, Biblioteca Apolistolica Vaticana, Borgh. 274; on fol. 86 va there is a passage related to Dig. 30.32.2: 'In l. si quis a filio § fi. [Dig. 30.32.2], questio fuit de facto an pars lambertinorum aut gibilinorum Bononie esset licita ad credendum seu constituendum syndicum, et dicebatur quod sic per hanc l[egem]. Sed valde parum facit, Dy[nus]'.

24 and executio iurisdictionis, see F.H. Russell, The Just War in the Middle Ages (Cambridge et al.: Cambridge University Press, 1975

  Philippum Iuntam, 1580), vol. III, consilium 68, fol. 98 ra -98 rb , n. 69, and 78-80; fol. 98 va , n. 82; fol. 99 ra , n. 94; and fol. 100 vb , n. 140-1. See also [Silvestro Mazzolini da Prierio], Sylvestrinae summae, quae summa summarum merito nuncupatur (Antverpiae: Ex officina Christophori Plantini, 1569), pars prima, ad v. Bellum, p. 90 b , n. 2.

	17 Grotius, IBP, I.3.5.1, p. 97, transl. Kelsey pp. 98-9.
	18 See Bartolus de Sassoferrato, In Secundam Digesti Novi Partem (Venetiis: Iunta, 1596), ad Dig. 49.15.24,
	fol. 215 ra , n. 9; Innocent IV, Super libros quinque Decretalium (Francofurti ad Moenum: per Martinum Lechler,
	1570; repr. Frankfurt am Main: Minerva, 1968), ad c. 12, X 2.13, fol. 231 va-b , n. 8; ibid. ad c. 29, X 2.24, fol.
	288 va , n. 5; and Abbas Panormitanus, Commentaria Secundae Partis in Secundum Decretalium Librum
	(Venetiis: [Società dell'Aquila che si rinnova], 1591), ad c. 29, X 2.24, fol. 161 va , n. 9. On Innocent IV's
	comment ad c. 29, X 2.

  & E. Brière, 1904), p. 190, note 1, who in turn is cited by Haggenmacher, Grotius, p. 535. IPC, XIII, fol. 141 r . See also Haggenmacher, Grotius, p. 535-6 and -for De iure praedae -p. 147. In addition to IPC, there are other works previous to IBP in which Grotius dealt with resistance: see H. Grotius, Commentarius in theses XI: An early treatise on sovereignty, the just war, and the legitimacy of the Dutch revolt, ed. P. Borschberg (Bern et al.: Lang, 1994), which M.J. van Ittersum, 'Dating the manuscripts of De Jure Praedae (1604-1608): What watermarks, foliation and quire divisions can tell us about Hugo Grotius' development as a natural rights and natural law theorist', History of European Ideas 35 (2) (2009), 125-93: 143 considers to be a spin-off of IPC, XIII, dating probably to 1607-1609. See also Grotius's Theses LVI, ms Leiden, University Library, BPL 922, fol. 287 r -290 v , which P. Borschberg, 'Grotius, the Social Contract and Political Resistance. A Study of the Unpublished Theses LVI', IILJ Working Paper 2006/7 (www.iilj.org) dates to either 1602-1605 or 1613-1615. Ibid., p. 62, note 181 Borschberg quotes Grotius's letter to Georg Michael Lingelsheim of 8 September 1617, which is also relevant to the issue of resistance, and on p. 64, note 187 he cites another draft treatise by Grotius dealing with resistance against tyrants and preserved in ms BPL 922, De bello ob libertatem eligendo ex thesibus politicis M. Tullii, Lib. 9 ad Attic. Ep. 3. Grotius also strongly rejected individuals' and lower officials' right of resistance (except by mere disobedience to unjust laws) in his De

23 See Grotius,

27

  See Grotius, IPC, XIII, fol. 134 v -135 r (Grotius does not cite Bartolus' De Guelphis et Gebellinis here, although he refers to it previously in the same chapter, fol. 129 v ), and Grotius, IBP, I.4.19.3, p. 162, transl. Kelsey p. 162.

  See also Paulus Castrensis, In Primam Digesti Veteris partem Commentaria (Lugduni: [excudit Ant. Blanc], 1585), ad Dig. 1.1.1.4, fol. 3 vb , n. 10-12; Abbas Panormitanus, Commentaria Primae Partis in Secundum Decretalium Librum (Venetiis: [Società dell'Aquila che si rinnova], 1591), ad c. 12, X 2.13, fol. 184 va , n. 18; Jason de Mayno, In Primam Digesti Veteris Partem Commentaria (Lugduni: [excudit Ant. Blanc], 1581), ad Dig. 1.1.3, fol. 6 vb , n. 28-29 ('ubicunque in aliqua lege [...] fit mentio de se, vel sua persona, vel sim, talia verba comprehendunt non solum propriam personam, sed etiam alios, quos circa se necessarios habet: puta uxorem, filios, & similes [...]. Sed pulchrum dubium est, quid si sit extraneus, an liceat pro eo defendendo alium occidere? Bart[olus] & communiter alij hoc concludunt quod sic, praesertim eo petente auxilium'). 36 See Johannes de Legnano, Tractatus de bello de represaliis et de duello, ed. Th.E. Holland ([n.p.]: Printed for the Carnegie Institution of Washington at the Oxford University Press, 1917), chap. 42, pp. 115-16, and [Silvestro Mazzolini da Prierio], Sylvestrinae summae, pars prima, ad v. Bellum, pp. 92 b -93 a , n. 8. 37 See D. Quaglioni, '"Fidelitas habet duas habenas". Il fondamento dell'obbligazione politica nelle glosse di Bartolo alle costituzioni pisane di Enrico VII', in Origini dello Stato. Processi di formazione statale in Italia fra medioevo ed età moderna, ed. G. Chittolini, A. Molho and P. Schiera (Bologna: Il Mulino, 1994), pp. 381-96. 38 See W.M. Bowsky, Henry VII in Italy. The Conflict of Empire and City-State 1310-1313 (Lincoln: University of Nebraska Press, 1960), and K. Pennington, The Prince and the Law, 1200-1600. Sovereignty and Rights in the Western Legal Tradition (Berkeley et al.: University of California Press, 1993), pp. 165-201.

59

  Ibid., fol. 222 vb -223 rb , n. 2-10 (for references to the original quaestio of Piperata, see supra, note 55). 60 See Bartolus de Sassoferrato, In Primam Digesti Veteris Partem, ad Dig. 1.1.9, fol. 10 ra , n. 17; Raynerius Arsendis ad Dig. 1.1.9, in Albericus de Rosciate, In Primam ff. Veter. Part. Commentarij (Venetiis: [Società dell'Aquila che si rinnova], 1585; repr. Bologna: Forni, 1974), fol. 18 rb -18 va , n. 40; and Baldus de Ubaldis, In primam Digesti Veteris partem Commentaria (Venetiis: [Società dell'Aquila che si rinnova], 1599; repr. Goldbach, Keip Verlag, 2004), ad Dig. 1.1.9, fol. 13 va , n. 34-6. 61 Baldus de Ubaldis, In primam Digesti Veteris partem Commentaria, ad Dig. 1.1.9, fol. 13 va , n. 35, transl. J. Canning, The Political Thought of Baldus de Ubaldis (Cambridge et al.: Cambridge University Press, 1987), p. 130. In primam Digesti Veteris partem Commentaria, ad Dig. 3.4.1, fol. 193vb, n. 5. I have been unable to find Raynerius' original consilium.

62

See Baldus de Ubaldis,

  to civil war, 63 See Baldus de Ubaldis, In primam Digesti Veteris partem Commentaria, ad Dig. 3.4.1, fol. 193vb, n. 5 ('[...] nisi forte esset expulsa maior pars civitatis, quia tunc sub nomine maioris partis, non sub nomine sectae faceret collegium licitum iuxta no[tata] per In[nocentium IV] extra, de no. ope. nun. c. cum ex iniuncto [c. 2 X 5.32]'), transl. Canning, The Political Thought, p. 154. See Innocent IV, Super libros quinque Decretalium, ad c. 2, X 5.32, fol. 529 va , n. 2. 64 See Bartolus de Sassoferrato, In Secundam Infortiati Partem (Venetiis: [Iunta], 1596), ad Dig. 30.32.2, fol. 10 ra . 65 Baldus de Ubaldis, In Primam et Secundam Infortiati Partem Commentaria (Venetiis: [Società dell'Aquila che si rinnova], 1599; repr. Goldbach: Keip Verlag, 2004), ad Dig. 30.32.2, fol. 130 ra -130 rb , transl. Canning, The Political Thought, p. 154. 66 See G. Mazzanti, s.v. 'Ramponi, Lambertino', Dizionario Biografico degli Italiani, vol. 86 (Rome: Istituto

	dell'Enciclopedia	Italiana,	2016),	available	at	http://www.treccani.it/enciclopedia/lambertino-
	ramponi_(Dizionario-Biografico)/ (accessed 30.07.2020).

See R. Feenstra, 'Ius commune et droit comparé chez Grotius: nouvelles remarques sur les sources citées dans ses ouvrages juridiques, à propos d'une réimpression du De iure belli ac pacis' (1992), in Id., Legal Scholarship

See Ordinary gloss to Dig. 48.4.3, ad v. aut castra concesserit, ad v. delectumve, and ad v. comparuerit; Ordinary gloss to Cod. 11.47.1, ad v. movendorum; and Ordinary gloss to c. 4, C. 23, q. 1, casus and gloss ad

See B. Ayala, De iure et officiis bellicis, et disciplina miliari, (Duaci: Ex officina Ioannis Bogardi Typogr., 1582), I.2, fol. 8 r , n. 9, who draws on M. Socini [junior], Consiliorum sive malis responsorum (Venetiis: apud

'public'). This had already been a relevant issue in Grotius, IPC, fol. 141 r ; ibid., fol. 14 r , in a note added at the

Grotius, IBP, III.7.5.3, pp. 708-9, transl. Kelsey p. 692 (the references to Plutarch and Tacitus were first added in the 1631 edition).