Dirk Saurenz

Funda Demirhan

Philippe Richard

Rinaldo Poli
email: rinaldo.poli@u-bourgogne.fr

Helmut Sitzmann
email: sitzmann@chemie.uni-kl.de

European Journal of Inorganic Chemistry Cyclopentadienylmolybdenum(VI) and Molybdenum(V) Oxo Chemistry: New Synthetic and Structural Features

published or not. The documents may come

Introduction

High oxidation state organometallic chemistry has experienced a rapid development in the last 20 years, mostly justified by the search for efficient oxidation catalysts. The half-sandwich oxo derivatives of rhenium, [1][2][3] displaying a very rich chemical reactivity and catalytic activity, are an example of a well investigated class of compounds. Some of the earliest high oxidation state organometallics, however, are the cyclopentadienyl oxo derivatives of molybdenum, first developed in the laboratory of M. L. H. Green. [4,5] The stability of high oxidation state complexes relies on the presence of electronegative and donating ligands such as the halides or negatively charged oxygen (oxo, alkoxo) or nitrogen (nitrido, imido, amido) based ligands. The organic fragments that are most compatible with this ligand environment are odd-electron ones such as alkyls (1-electron), allyls (3-electron) dienyls (5-electron), etc., while neutral organic fragments (olefins, arenes, etc.) necessitate electron-richer, low-valent metal centers to optimize bonding. These conditions confer a high degree of covalency to the resulting metal-carbon bonds, which consequently become quite resistance to hydrolytic conditions.

It is therefore somewhat surprising that the physical behavior and chemical reactivity of high oxidation state organometallics is not systematically investigated in water, although aqueous reagents are sometimes used for their syntheses. This contrasts with the well developed aqueous chemistry of high oxidation state oxomolybdenum complexes. [6,[START_REF] Richens | The Chemistry of Aqua Ions[END_REF] On the other hand, the rich electrochemical behavior displayed by the concerned metals, notably molybdenum, makes these molecules quite interesting for potential electrocatalytic applications. For these reasons, we have initiated a research effort aimed at increasing our basic knowledge of the physical properties and chemical reactivity of organomolybdenum compounds in a variety of high oxidation states in water.

Although several investigations of cyclopentadienyl oxo complexes of molybdenum in high oxidation states have been reported, [8,[START_REF] Abel | Comprehensive Organometallic Chemistry II[END_REF] the access routes to these material cannot be considered as completely satisfactory. In this first contribution, we shall focus on the improvements of synthetic procedures to access compounds Cp2Mo2O5 and Cp*2Mo2O5, which we plan to use as starting materials for our studies. In addition, we shall report an extension of the synthetic procedure to analogues containing differently substituted cyclopentadienyl rings. We are interested in cyclopentadienyl rings with a high degree of substitution as they are known to afford more inert metal-cyclopentadienyl bonds by way of disfavoring access to the metal center, while at the same time rendering the other ligands more labile. Interesting reactivity may then be expected. Some basic aqueous chemistry which is related to the optimization of the syntheses reported in this paper will also be discussed. Detailed studies of the behavior of these compounds in water as a function of the pH [START_REF] Collange | [END_REF] and of the electrode potential [START_REF] Gun | [END_REF] have also been completed and are being reported separately.

Results

1. The cyclopentadienyl oxo molybdenum(VI) system.

(a) Discussion of previously reported access routes.

As mentioned in the Introduction, compound Cp2Mo2O5 has been known since 1964, but a reliable high yield and practical synthesis does not seem to be yet available. It was first obtained in 20% yield by the photolytic degradation of [CpMo(CO)3]2 in chloroform, together with several other products. [4] It also forms in small yields by adventitious oxygenation of solutions of various CpMo derivatives. [4,5] A higher yield synthesis (79%) was reported by oxidation of the dinuclear Mo(V) compound [CpMoO2]2. This, however, requires use of Ag2O as the oxygen transfer agent and warming in CH2Cl2 at 55° for 1.5 days. [START_REF] Bunker | [END_REF] The precursor complex, in turn, can be obtained by hydrolysis of CpMoX4 (X = Cl, Br) with excess water. A red isomer, formulated as a tetranuclear compound, is obtained immediately while a longer exposure to water or dissolution of the red isomer in acetone or chloroform slowly afford a dinuclear orange isomer. [5,[START_REF] Bunker | [END_REF] A similar evolution was described for analogous compounds with monosubstituted cyclopentadienyl ligands. [START_REF] Bunker | [END_REF] Since the reported behavior of the red tetrameric form suggests that it is a chemically more labile form, we wondered whether the oxidation of this red isomer by sufficiently rapid oxygen transfer agents might provide an easier and high yield access to the desired Mo(VI) product before its conversion to the less reactive orange isomer.

(b) New synthetic strategy

Several effective reagents are commonly used for oxygen transfer chemistry, the most easily accessible one being H2O2. We were hesitant to use this reagent because the formation of peroxo compounds of Mo(VI) or W(VI) such as Cp*W(O2)(O)R, [13][14][15] or Cp*M(O2) (O)Cl (M = Mo, W) [16,17] by treatment of suitable precursors with either H2O2 or hydroperoxides had previously been described. In fact, oxidation tests with H2O2 gave unsatisfactory results. The use of PhIO, [18] on the other hand, provided a rapid and convenient synthesis in two steps (>50% overall yield) from CpMoCl4, which is itself easily prepared from commercially available Mo(CO)6 (Equations 1 and 2). [START_REF] Gordon | Synthetic Methods of Organometallic and Inorganic Chemistry[END_REF] The second step is complete within 15 min at room temperature and the best results are obtained when the solution is immediately worked up. If the product is allowed to sit in the dichloromethane solution, darker insoluble by-products start to form. The intermediate red Mo(V) oxo complex must be isolated and removed from water. An attempt to carry through the synthetic procedure without this isolation step gave unsatisfactory results. This procedure, however, constitutes a significant improvement over the previously available ones, especially in terms of time efficiency as both steps can be carried out at room temperature and the overall procedure may be accomplished in a half day. The procedure described below for the Cp* and related analogues, via the hitherto unknown

[CpMoO3] -anion, [START_REF]anion are currently underway in our laboratory[END_REF] has not given satisfactory results for the Cp system.

The pentamethylcyclopentadienyl oxo molybdenum(VI) system.

(a) Discussion of previously reported access routes.

The Cp* compound was first reported in 1985 by Herberhold et al. as the product of the aerial oxidation of Cp*Mo(CO)2(NO) [START_REF] Herberhold | [END_REF] and was subsequently shown to form as a small yield by-product from a variety of other reactions. [22][23][24][25] A high yield synthesis was reported by Faller and Ma by aerial oxidation of [Cp*Mo(CO)2]2 in chloroform, but the product is said to be contaminated by ca. 15% of the chloride complex Cp*MoO2Cl and needs further exposure to acetone/water for purification, [26] while the synthesis of the starting compound from commercially available Mo(CO)6 is not itself free from pitfalls. [27,28] This reaction was re-examined later by Bottomley and shown to yield a host of other incompletely oxidized products under a variety of conditions. [29,30] The preparation in 70-75% yield from Cp*MoCl4 reported by Umakoshi and Isobe [31] is related to the synthetic procedure reported by us here, but was not optimized for large scale syntheses and uses the exotic sodium oxometallate salts Na2MoO4, Na2WO4 or NaVO3 as oxidants. Another reported high yield syntesis is by protonolysis of the unusual starting compound Cp*Mo(NBu t )2Cl with aqueous HCl. [32] Finally, work carried out in Geoffroy's laboratory has shown that the hydrolysis of Cp*MCl4 (M = Mo, W) in the presence of an amine (Bu t NH2) yields Bu t NH3 + [Cp*MO3] -in good yields, [33] while Sundermeyer has shown that the same anions are also formed upon treatment of the oxo-bridged dimers Cp*2M2O5 with tetrabutylammonium hydroxide. [32] The acidification of the anion solution, studied in the Geoffroy study for the tungsten compound, gives back the dinuclear complex via the presumed condensation of initially formed Cp*WO2(OH). [33] A detailed discussion of this reversible anion/dimer transformation and the aqueous chemistry of the tetrachloride compounds is essential, as it forms the basis of our improved synthetic strategy.

Geoffroy et al. have reported that the reaction of Cp*MoCl4 with 2 equiv of NaOH in acetone-water in an open flask for more than 30 min gives Cp*MoO2Cl, while use of an unspecified excess of NaOH affords the dinuclear Cp*2Mo2O5. [34] The reaction between Cp*MoO2Cl and excess (3 equivalents)

of NaOH was also reported to yield the dinuclear compound. These observations seem odd to us in view of the above mentioned Sundermeyer result, which we have confirmed (see next section). [START_REF]On the other hand, it is interesting to note that the use of excess aqueous NaOH in the hydrolysis reaction of Cp*WCl4 did[END_REF] In contrast with the apparent lack of reactivity between Cp*2Mo2O5 and excess aqueous NaOH, excess aqueous Bu t NH2 (a weaker base than NaOH) yields Bu t NH3 + [Cp*MoO3] -in good yields under analogous conditions. [34] Furthermore, it was shown that the acidification of dichloromethane solutions of the Bu t NH3 + [Cp*MoO3] -salt with acetic acid affords Cp*2Mo2O5, which was recovered in 58% yields. From these observations, it would seem logical to imagine a pH-dependent equilibrium between the anionic mononuclear and the neutral dinuclear forms (Equation 3) in water. [36] The anion should therefore be the ultimate product when excess strong base (e.g. NaOH) is present, in contrast to Geoffroy's report. On the basis of this hypothesis, we have developed the new synthetic procedure reported in the following section. Preliminary investigations show that, indeed, yellow Cp*2Mo2O5 solutions in MeOH-H2O become nearly colorless upon treatment with NaOH, and the resulting basic solutions turn back yellow upon lowering the pH by addition of strong acids.

Cp* 2 Mo 2 O 5 + 2 OH - 2 [Cp*MoO 3 ] -+ H 2 O (3)

(b) New synthetic strategy

The Cp*MoCl4 compound is conveniently prepared in high yields and purity [START_REF] Keogh | Synthetic Methods of Organometallic and Inorganic Chemistry[END_REF] from commercially available Mo(CO)6 and is therefore the choice starting compounds for our synthetic work. As mentioned in the previous section, it seemed to us that excess NaOH under aerobic conditions should lead directly to the cyclopentadienyltrioxomolybdate anion, according to the stoichiometry of Equation 4.

2 Cp*MoCl 4 + 10 OH -+ ½ O 2 2 [Cp*MoO 3 ] -+ 8 Cl -+ 5 H 2 O (4)
Treatment of acetone solutions of Cp*MoCl4 with a slight excess (ca. 6 equivalents) of aqueous NaOH in air for over a day at room temperature yielded a suspension of the pale yellow Cp*MoO3 - Na + precipitate but also minor quantities of a soluble colored by-product (identified as [Cp*MoO2]2, vide infra). Separation of this by-product is accomplished by complete evaporation of the solvent mixture and redissolution in pure water, which dissolves the sodium salt but not the by-product. Since the sodium organomolybdate salt may be difficult to separate by fractional crystallization from NaCl and excess NaOH, we have proceeded directly to its conversion to the neutral dimer, which is insoluble in pure water. Dropwise addition of acetic acid from a burette leads, after neutralization of the excess base, to the instantaneous precipitation of the yellow product. The acid addition is stopped when the administration of one further drop of the acid solution does not cause any further precipitation. The synthetic procedure can therefore be handled as a simple titration reaction allowing the use of close to the stoichiometric amount of acid. This is important because the excess acetic acid is also partially extracted into the ether layer and large quantities of this compound may interfere with the crystallization procedure or even contaminate the organomolybdenum product. This procedure consistently gives a pure product with yields in excess of 60 % on a 2 g scale. The synthesis can also be accomplished stepwise by reaction of Cp*MoCl4 with 3 equivalents of NaOH for 1 day, followed by the addition of 3 more equivalents of NaOH. In this fashion, the formation of the Mo(V) byproduct is avoided (see discussion). The isolated yield of the desired product, however, is not increased by this modified procedure. plane, syn as in I [START_REF] Arzoumanian | [END_REF] (like the analogous Cp system) [39] or anti as in II. [40] We have redetermined the structure of the syn form at lower temperature with a better precision. The results have been deposited

with the CCDC but, since they do not significantly differ from those of the higher temperature structure, they are not explicitly reported here. The metal-metal distance in the syn polymorph [2.587(1) Å] is significantly shorter than in the anti polymorph [2.647(3) Å], [40] certainly because of the butterfly arrangement which allows the metals to approach each other relative to the planar arrangement of the anti isomer. The same situation is found for the [(L)2Mo2O4] 2+ (L = 1,5,9triazacyclononane). [41,42] (a) Syntheses.

The procedure described above for the Cp* system was applied in a straightforward manner to the hydrolysis of 4 CpMoCl4 and Cp'''MoCl4. Again, the aerial hydrolysis affords in both cases solutions containing the water soluble Na + [(Ring)MoO3] -salt. This was not isolated, but its presence is indicated by the identical behavior to the corresponding Cp* system described above upon acidification. Compounds 4 Cp2Mo2O5 and Cp'''2Mo2O5 were obtained as yellow or yellow-orange crystalline solids that are insoluble in water but soluble in ether. Both were crystallized and their structure was confirmed by X-ray diffraction methods (see next section). They display two IR vibrations that are typical of the Mo=O functionalities (symmetric and asymmetric at ca. 885 and 920 cm -1 , respectively) and one due to the asymmetric Mo-O-Mo stretch (at ca. 760 cm -1 ), like their Cp and Cp* analogues. [4,22] While the 4 Cp system afforded the expected dinuclear Mo(VI) product with no indication of incomplete oxidation, the Cp''' system behaved like the Cp* system described above and gave also a water insoluble fraction, from which the dinuclear Mo(V) complex [Cp'''MoO2]2 was recovered. This compound was also characterized by X-ray diffraction methods (see below).

(b) X-ray structures of the (Ring)2Mo2O5 compounds.

Several X-ray structural determinations have been reported over the years for various polymorphs of Cp*2Mo2O5 [23][24][25][26] and a discussion of the similarities between them is already available. [25] It is interesting to see whether the steric bulk of the cyclopentadienyl ligand plays any role on the other structural parameters. The geometries of compounds 4 Cp2Mo2O5 and Cp'''2Mo2O5 are shown in Figures 1 and2, respectively, and the relevant metric data are compared in Table 1. Mo( 1)

Mo(1)# O(1) O(2) O(3) CNT(1)
Mo( 1)

Mo(2) O(1) O(2) O(3) O(4) O(5) CNT(1) CNT(2) Figure 1. (c) X-ray structure of compound [Cp'''MoO2]2.
The geometry of compound [Cp'''MoO2]2 is shown in Figure 3 and selected bonding parameters are reported in Table 2. The immediate striking feature is the relative anti arrangement of the two Cp''' groups, while they are syn for the Cp analogue and both arrangements are possible for the Cp* molecule. Although an unambiguous conclusion cannot be drawn, it seems reasonable to imagine that the steric bulk of the Cp''' ligand enforces the anti arrangement, while the compound might prefer the syn arrangement for electronic reason. Therefore, the syn arrangement will be observed in the less crowded cases (e.g. Cp). It is easy to imagine how the placement of two Cp''' ligands into the syn structure (see drawing I) would introduce a severe strain in the molecule.

Figure 3.

An ORTEP view of compound [Cp'''MoO2]2. Ellipsoids are drawn at the 50% probability level. [40] This parameter is therefore much more sensitive to the geometry of the central Mo2O2 moiety (bent for the syn and planar for the anti) than to the type of cyclopentadienyl ring. The Mo=O and Mo-(-O) distances do not seem sensitive to either effect, being essentially identical with those of all other previously described molecules with this stoichiometry, whether of syn or anti type.

4. The 4 CpMoO2Cl system.

According to the previous Cp* literature, [34] the aerobic hydrolysis of the tetrachloro compound with 2 equivalents of aqueous NaOH leads directly to the corresponding dioxochloro derivative. For the less electron-rich Cp system, on the other hand, excess water yields quite rapidly the dioxo system as seen above even in the absence of base. [5] The 4 Cp system behaves similarly to the Cp* with which it is sterically and electronically more similar. Identical yields of 4 CpMoO2Cl were obtained when 4 CpMoCl4 was hydrolyzed in the presence of either 3 or 2 equivalents of NaOH. The compound shows essentially identical Mo=O stretching vibrations in the IR spectrum as the related oxo-bridged dinuclear complex and no band in the Mo-O-Mo stretching region, as expected. The structure of this product has also been confirmed by X-ray crystallography (see Figure 4). The relevant bond distances and angles are reported in Table 3. While no structure for a (Ring)MoO2X (X = Cl or other halogen) seem to be reported, related structures are those of {C5Ph4[C6H3(OMe)2-2,5]}MoO2(OMe) [43] and Cp*MoO2[(2,6-Pr i 2C6H3)N(SiMe3)]2Si2O2(OH)3. [44] The Mo=O distances are quite comparable in all these compounds. The Mo-Cl distance compares with related Mo(V)-Cl distances and appears to be affected mostly by the cyclopentadienyl ring steric pressure, as suggested by the comparison of the following examples: [Cp*MoOCl]2O: 2.368(2) Å; [31] CpMoOCl2: 2.301(4) Å (average); [45] Cp*MoOCl2: 2.245(9) and 2.330(5) Å (averages) in two independent reports. [30,46] Mo O

O

Cl

Discussion

From the previous literature reports and from our results, the following simplified picture emerges for the aerobic hydrolysis of (Ring)MoCl4 compounds (see Scheme 1). Hydrolysis of the first two Mo-Cl bonds leads to (Ring)MoOCl2, which can proceed either to further hydrolysis leading to [(Ring)MoO2]2 or it may be oxidized to (Ring)MoO2Cl. Finally, the dioxo Mo(V) dimer can be oxidized and the chlorodioxo Mo(VI) complex can be hydrolyzed, in both cases leading to (Ring)2Mo2O5. Under basic conditions, however, the final product converts to the [(Ring)MoO3] - anion. Of course, other intermediates must also form during these transformations, notably hydroxo complexes whose relative stability is supported by some evidence. [START_REF] Bunker | [END_REF]31,34] H

Scheme 1

If hydrolysis prevails over oxidation at the level of the first dichlorooxo intermediate leading to the dinuclear, metal-metal bonded dioxo compound [(Ring)MoO2]2, further oxidization by dioxygen to the Mo(VI) dimer is rather difficult. This is what happens for the simple Cp system. In fact, even oxygen delivering agents such as Ag2O react with it slowly. [START_REF] Bunker | [END_REF] However, the kinetic red form is more reactive and a successful strategy for the formation of Cp2Mo2O5 via this route by using PhIO has been presented in this paper.

The rate of hydrolysis seems to decrease as the steric bulk of the cyclopentadienyl ring increases.

The greater donating power of the bulkier rings may also retard the complete hydrolysis for electronic reasons. The oxidation of the dichlorooxo Mo(V) species, on the other hand, should be made more favourable by electron-richer ligands. At any rate, the oxidation step may then prevail over hydrolysis leading to the chlorodioxo Mo(VI) species provided there is dioxygen available. This is the situation encountered for the Cp* system, for which the aerobic hydrolysis in the presence of 2 equiv of NaOH was reported to give a good yield of Cp*MoO2Cl. [34] The same occurs also for the 4 Cp system reported here with either 2 or 3 equivalents of NaOH. The use of neutral aqueous conditions should produce 3 acid equivalents as shown in Equation 5. Evidently, the last chloride ligand is not hydrolyzed under neutral conditions, whereas at high pH (excess NaOH) the only recovered product (after acification) is 4 Cp2Mo2O5. The relative rate of hydrolysis and oxidation for (Ring)MoOCl2 should depend also on other factors such as the concentration of water, the nature of the organic solvent, and the pH. Although we have not carried out a comprehensive study of these effects, we do have qualitative indications that more Mo(V) dimer complex is obtained when using a larger amount of water for the Cp* and Cp''' systems. If Scheme 1 is valid, one would expect that a greater concentration of OH -(larger NaOH excess) should favor the hydrolysis pathway leading to the [(Ring)MoO2]2 system. Indeed, we do obtain a significant amount of this compound as a by-product when the hydrolysis is carried out in the presence of >5 equivalents of NaOH, whereas a good yield of Cp*MoO2Cl was obtained by Geoffroy et al. by using only 2 equivalents, with no mention of the formation of the Mo(V)

product. [34] Indeed, repeating the synthesis of Cp*2Mo2O5 by stepwise addition of 3 equivalents of NaOH, followed by three more equivalents a day later (without isolation of the intermediate Cp*MoO2Cl complex) completely avoided the formation of the [Cp*MoO2]2 by-product. Some subtleties, however, are not yet completely understood. For instance, the Mo(V) dimer was obtained for the Cp''' system like for the Cp* system, but an analogous product was not obtained for the 4 Cp system, even when using 5 equiv of NaOH. The use of excess NaOH, however, provided that the oxidation route is followed, allows access to the water soluble sodium salts of (Ring)MoO3 -.

Although we have not isolated and characterized the new anions of this type for the 4 Cp and Cp''' systems, their formation is indicated by the protonation reactivity which is identical with that of the Cp* system: acidification of the water solutions yields the yellow water insoluble (Ring)2Mo2O5

products. [36]

Conclusion

High oxidation state monocyclopentadienyl oxo derivatives of molybdenum are often the end products of oxidative decompositions of lower valent derivatives and were previously available only in low yields or by using impractical starting materials and/or reagents. The hydrolytic route from the easily accessible cyclopentadienylmolybdenum(V) tetrachlorides has been optimized and extended to the preparation of new compounds with sterically more encumbering rings. The stability of these compounds in air and aqueous media makes the exploration of their aqueous chemistry and electrochemistry possible. Investigations in those directions are ongoing in our laboratory.

Experimental Section

General Remarks: All reactions were carried out in a Jacomex glove box or by the use of standard Schlenk techniques under an argon atmosphere. The solvents were dried by conventional methods (THF, Et2O, heptane and pentane from sodium benzophenone ketyl and CH2Cl2 from P4O10) and distilled under argon prior to use. EPR measurements were carried out at the X-band microwave frequency on a Bruker ESP300 spectrometer. The spectrometer frequency was calibrated with DPPH (g= 2.0037). 1 H NMR spectra were recorded using a Bruker AC-200 and referenced to the residual solvent protons of the deuterated solvent. The elemental analyses were carried out by the analytical service of the Laboratoire de Synthèse et d'Electrosynthèse Organométallique with a Fisons EA 1108 apparatus. Compounds (Ring)MoCl4 (Ring = Cp, Cp* and 4 Cp) were prepared as described in the literature. [START_REF] Gordon | Synthetic Methods of Organometallic and Inorganic Chemistry[END_REF][START_REF] Keogh | Synthetic Methods of Organometallic and Inorganic Chemistry[END_REF]47] Compound Cp'''MoCl4 was prepared by a straightforward adaptation of those procedures as described below.

Tetrachloro(1,2,4-tri-tert-butylcyclopentadienyl)molybdenum(V), Cp'''MoCl4 : Mo(CO)6 (2.64 g, 10.0 mmol) and Cp'''Na (2.66 g , 10.3 mmol) were mixed as solids and suspended in 40 ml THF. The suspension turned to an orange solution during a 10 min period. It was then refluxed for 18 h. An IR monitoring indicated the formation of Cp'''Mo(CO)3 -Na + (strong peaks at 1885, 1784 and 1736 cm -1 , comparable with literature values for the Cp * analogue).

After cooling to room temperature, MeI (700 µl, 11.25 mmol) was added to the deep orange-red solution causing darkening to brown. After an additional reflux for 3 h the solvent was removed in vacuo leaving a yellow brown residue which was extracted in 60 ml CH2Cl2. After filtration, the deep orange solution was evaporated to dryness yielding 4. Tetraoxo--oxodicyclopentadienyldimolybdenum(VI), Cp2Mo2O5: Compound CpMoCl4 (0.555 g, 1.83 mmol) was added to a flask containing distilled water (10 ml). Shaking of the resulting suspension for a few minutes gave rise to the formation of fine deep red microcrystals. The product ({CpMoO2}4) [START_REF] Bunker | [END_REF] was separated by decanting off the mother liquor, washed with water (3x10 mL) and quickly with acetone (5 mL), and dried in vacuo for 2 h (yield 0.28 g, 79.5%). This material (1.45 mmol of Mo) was then dissolved in dichloromethane (ca. 100 mL) and to the resulting solution was added solid PhIO (0.22 g, 1.0 mmol). The mixture was stirred at room temperature for 15 m, resulting in a color change from red to yellow. A longer reaction time should be avoided as it affords larger amounts of insoluble by-products. The solution was filtered and the filtrate was evaporated to give yellow microcrystals. The product was purified by recrystallization from CH2Cl2/pentane, and dried in vacuo. Yield 0.207 g, 71.1% (56.5% from CpMoCl4). The analytical (IR, NMR) properties of the compound correspond to those reported in the literature [4] and confirm its purity. resulting dark brown suspension was stirred for 18 h causing a lightening of the colour to orange and the precipitation of a pale yellow solid. After evaporation to dryness the brownish residue was extracted in 75 ml of distilled water and filtered. The pale yellow filtrate was acidified with diluted glacial acetic acid causing precipitation of a yellow solid at pH 3.8 -4.0. The acetic acid addition was stopped when the addition of one further drop did not cause any additional precipitation. The precipitate was extracted with 4x100 ml portions of ether, the last extract being colourless. The combined organic layers were washed with distilled water and dried over MgSO4 overnight. Evaporation of the solvent yielded 1.55 g (2.86 mmol; 61.3 %) of [(Cp * MoO2)2O] as a yellow solid. The 1 H NMR spectrum of this compound (singlet at 2.01 ppm in CDCl3) corresponds with that described in the literature. [26] From another experiment starting from 2.03 g of Cp*MoCl4, the water insoluble orange-red solid remaining from the water extraction was also worked up. It was washed with 20 ml ether then with 20 ml pentane and dried in vacuo affording the dinuclear Mo(V) compound [(Cp * MoO2)2] as a orange crystalline solid (192 mg, 0.37 mmol; 13.6 %).

Tetraoxo

Elem. Anal.: Calcd for C10H15MoO2:C, 45.64; H, 5.74 %. Found: C, 45.36; H, 5.71 %. The 1 H NMR spectrum (1.98 ppm in CDCl3) corresponds with that reported in the literature. [START_REF] Arzoumanian | [END_REF] Suitable single crystals for x-ray diffraction could be grown by layering a saturated THF solution with heptane at room temperature. Normally up to ca. 20 % of this byproduct was recovered depending on the reaction conditions.

Tetraoxo--oxobis(tetraisopropylcyclopentadienyl)dimolybdenum(VI), 4 Cp2Mo2O5. To a stirred mixture of acetone (75 mL), THF (75 mL) and aqueous NaOH (2.6 mL, 3 M, 7.8 mmol) in an open beaker was quickly added deep purple 4 CpMoCl4 as a solid (609 mg, 1.29 mmol). A brown suspension immediately formed which lightened to orange yellow over a period of 2 h. After stirring overnight the solvents were evaporated and the resulting yellow residue was extracted with 100 ml of warm water (50 °C). After filtration the aqueous phase was acidified with diluted glacial acetic acid as described in the previous section causing the precipitation of a yellow solid at pH 4. Extraction of this solid with 3x100 ml portions of ether, followed by separation of the organic layer, washing with water and drying over MgSO4 yielded a yellow ether solution, which was allowed to stand overnight and then was evaporated to dryness to The orange-yellow filtrate was collected and evaporated to dryness. The resulting yellow residue was extracted with 70 ml of distilled water and the extract was acidified with diluted glacial acetic acid to pH 4, as described above for the Cp* analogue. The yellow solid which precipitated was extracted with several portions of ether and the combined organic layers (300 ml) were washed with 200 ml of distilled water and dried over MgSO4. The shiny yellow ether phase was allowed to stay overnight and was then evaporated to dryness, yielding 120 mg (0,16 mmol. analysis was grown by slowly cooling a hot saturated heptane solution to room temperature. The same reaction was found to take place with just two equivalents of NaOH used, yielding the same product in an identical overall yield.

X-ray Crystallographic Study:

The crystal and refinement parameters for all structures are collected in Table 4. All set of data were collected on an Enraf-Nonius KappaCCD diffractometer at 110 K using Mo K radiation. The structures were solved via a Patterson search program [START_REF] Sheldrick | SHELXS97. Program Structure solution[END_REF] and refined with full-matrix least-squares methods [START_REF] Sheldrick | SHELXL97. Program for Crystal Structure refinement[END_REF] based on |F 2 | with the aid of the WINGX program. [START_REF] Farrugia | [END_REF] All non-hydrogen atoms were refined with anisotropic thermal parameters. Hydrogen atoms were located on final Fourier difference maps and freely refined with isotropic temperature factors fixed to 1.2 or 1.5 (for CH3) times those of the corresponding parent atoms. Selected bond distances and angles are listed in Table 1 for both independent molecules of 4 Cp2Mo2O5 and for Cp'''2Mo2O5, in Table 2 for for [Cp'''MoO2]2, and in Table 3

( c )

 c Isolation and structure of the [Cp*MoO2]2 by-product.The water-insoluble crop generated during the synthetic procedure described above contains orangered crystalline [Cp*MoO2]2, which can be recovered in yields up to 20% depending on the operating conditions. It seems that not only the time of exposure to air but also the relative amount of water and acetone is important to determine the ratio between the Mo(V) and Mo(VI) products (see Discussion). Smaller amounts of water seem to favor a greater extent of oxidation. Compound [Cp*MoO2]2 has previously been reported and shown to adopt two possible structures which differ by the relative arrangement of the Cp* and terminal oxo ligands across the central Mo(-O)2Mo

  Figures1 and 2, respectively, and the relevant metric data are compared in Table1. The 4 Cp compound has a symmetry-imposed inversion center on the bridging oxygen atom, but two independent half molecules are contained in the asymmetric unit. The Cp''' compound, on the other hand, has a single entire molecule in the asymmetric unit. While the Mo-O-Mo angle is forced to be linear in the first case, it deviates quite substantially from linearity in the second one. This deviation is larger than the largest one previously observed for the related Cp* polymorphs [172.7(3)°]. The Mo=O and Mo-(-O) distances for the Cp* structures were found to be in the 1.67-1.72 Å and 1.85-1.90 Å ranges, respectively. The same parameters are within these ranges for both4 Cp and Cp''' structures, the Mo-(-O) distances being around the long end of this range in both cases.

Figure 1 .

 1 Figure 1. An ORTEP view for one of the two independent molecules of compound 4 Cp2Mo2O5. Ellipsoids are drawn at the 50% probability level. The atom numbering correspondence for the other molecule is as follows: Mo(2) for Mo(1), O(4) for O(1), O(5) for O(2), O(6) for O(3), and CNT(2) for CNT(1).

Figure 2 .

 2 Figure 2.An ORTEP view of compound Cp'''2Mo2O5. Ellipsoids are drawn at the 50% probability level.

Figure 4 .

 4 Figure 4.An ORTEP view of compound 4 CpMoO2Cl. Ellipsoids are drawn at the 50% probability level.

2 .

 2 --oxobis(pentamethylcyclopentadienyl)dimolybdenum(VI), Cp*2Mo2O5 and Dioxodi-oxobis(pentamethylcyclopentadienyl)dimolybdenum(V), [Cp*MoO2]Deep violet Cp * MoCl4 (3.48 g; 9.33 mmol) was added to a stirred solution of aqueous NaOH (19 mL; 3 M; 57 mmol) in acetone (200 mL) in an open beaker. The

  ||Fo|-|Fc||)/|Fo|. b wR2=[w(Fo 2 -Fc 2 ) 2 /[w(Fo 2 ) 2 ] 1/2 where w=1/[ 2 (Fo 2 +(aP) 2 +bP] and P=(Max(Fo 2 ,0)+2*Fc 2 )/3 where (a, b) = (0.028, 0.0) for 1, (0.023, 1.69) for 2, (0.26, 0.0) for 3, (0.018, 0.79) for 4 and (0.024, 2.22) for 5.

Table 1 .

 1 Selected bond distances (Å) and angles (°) for compounds (Ring)2Mo2O5 (Ring =4 Cp,

	Cp''').
		4 Cp2Mo2O5 a	Cp'''2Mo2O5
	Mo(1)-O(1)	1.89560(18) 1.89985(17) Mo(1)-O(3)	1.8950(12)
		Mo(2)-O(3)	1.8978(11)

Mo(1)-O(2)

1.7092(15) 1.7071(

15

) Mo(1)-O(1) 1.7088

(12)

Table 2 .

 2 Selected bond distances (Å) and angles (°) for compound [Cp'''MoO]2. Mo distance in compound [Cp'''MoO2]2 compares quite well with that in the anti form of [Cp*MoO2]2 [2.647(3) Å].

	Mo(1)-Mo(2)	2.65115(19)	CNT(2)-Mo(2)-Mo(1)	127.7(1)

Table 3 .

 3 Selected bond distances (Å) and angles (°) for compound 4 CpMoO2Cl.

	Mo-Cl	2.3251(7)	Cl-Mo-O(2)	101.54(8)
	Mo-O(1)	1.720(2)	Cl-Mo-CNT	108.8(2)
	Mo-O(2)	1.700(2)	O(1)-Mo-O(2)	106.83(11)
	Mo-CNT	2.113(3)	O(1)-Mo-CNT	118.1(2)
	Cl-Mo-O(1)	102.15(8)	O(2)-Mo-CNT	117.2(3)

  Hz, CH(CH3)2). The 1 H NMR spectrum did not show any detectable Mo(V) byproduct. Yellow-orange single crystals suitable for x-ray diffraction were grown by slowly cooling a hot saturated heptane solution to room temperature.Cp2Mo2O5 as the only NMR active species. The final mixture was filtered and the solid was washed with water and redissolved in pentane. After evaporation of the solvent, the residue was dried in vacuo leaving 240 mg (0.33 mmol; 46.8 %) of 4 Cp2Mo2O5 as a orange powder (confirmed by 1 H NMR). Orange-red crystals could be grown by slowly cooling a hot saturated heptane solution to room temperature or by cooling a pentane solution to -25 °C. A single crystal from this batch showed the same unit cell as that of the yellow-orange crystals obtained from the procedure in the previous section.

	1.19 ppm (d, 6H, JHH = 6.7 Hz, CH(CH3)2). 1 H NMR (C6D6): 6.14 ppm (s, 1H, ring-H), 3.14 ppm (m, 4H, CHMe2),
	1.47 ppm (d, 6H, JHH = 6.7 Hz, CH(CH3)2), 1.44 ppm (d, 6H, JHH = 7.0 Hz, CH(CH3)2), 1.23 ppm (d, 6H, JHH = 7.0 Hz,
	CH(CH3)2), 1.02 ppm (d, 6H, JHH = 7.0 Attempted synthesis of Dioxodi--oxobis(tetraisopropylcyclopentadienyl)dimolybdenum(V), [ 4 CpMoO2]2.
	Compound 4 CpMoCl4 (670 mg, 1.42 mmol) was added to 150 ml of a stirred 1:1 mixture of acetone and distilled water,
	yielding immediately a brown suspension. The mixture was stirred for 40 hours, during which aliquots were
	withdrawn, the solid was separated, washed with water, and investigated by 1 H NMR, showing the spectrum of
	4
	Tetraoxo--oxobis(1,2,4-tri-tert-butylcyclopentadienyl)dimolybdenum(VI), Cp'''2Mo2O5 and Dioxodi--
	oxobis(pentamethylcyclopentadienyl)dimolybdenum(V), [Cp'''MoO2]2. To a stirred mixture of THF (75 mL),
	acetone (75 mL), water (10 mL) and aqueous NaOH (5.7 mL, 3 M, 17.1 mmol) in an open beaker was added purple
	Cp'''MoCl4 (1.36 g , 2.89 mmol). Some foaming occurred and a bordeaux-red solution formed immediately. No
	precipitate was detected during the following 24 hours of stirring at room temperature. Subsequent gentle heating (40-
	50 °C) of the mixture for two hours caused the precipitation of a red-violet product, which was collected on a filter,
	washed with water, redissolved in THF and evaporated to dryness leaving the Mo(V) dimer [(Cp'''MoO2)2] as the main
	product. Yield: 380 mg (0.53 mmol, 36.7 %). Elem. anal. Calcd for C17H29MoO2: C, 56.51; H, 8.09 %. Found: C,
	56.72; H, 8.12 %. IR (Nujol mull): 922s, 885s, 764vs cm -1 . 1 H NMR (C6D6): 6.43 ppm (s, 2H, ring-H), 1.46 ppm (s,
	18H, CMe3 1,2-positions), 1.37 ppm (s, 9H, CMe3 4-position). Suitable single crystals were grown by slowly cooling a
	hot saturated heptane solution to room temperature.
	yield 222 mg (0,3 mmol; 46.6 %) of yellow [( 4 CpMoO2)2O]. Elem. Anal.: Calcd for C34H58Mo2O5: C, 55.28; H, 7.91
	%. Found: C, 54.69; H, 8.25 %. IR (Nujol mull): 919s, 889s, 760s cm -1 . 1 H NMR (CDCl3): 6.01 ppm (s, 1H, ring-H),
	3.10 ppm (m, 4H, CHMe2), 1.39 ppm (d, 6H, JHH = 7.3 Hz, CH(CH3)2), 1,30 ppm (d, 12H, JHH = 7.3 Hz, CH(CH3)2),

  The ether layer remained colorless until the aqueous one was acidified with diluted acetic acid, causing the formation of a shiny yellow precipitate, which was extracted by the ether. After addition of the last traces of acid, which led to further precipitation, the aqueous phase was clear and colorless. The ether phases was washed three times with distilled water and dried over MgSO4 overnight. Evaporation to dryness gave 890 mg (1.20 mmol, 73.2 %) of orange-yellow Cp'''2Mo2O5 without any detectable Mo(V) by-product.

	Chlorodioxo(tetraisopropylcyclopentadienyl)molybdenum(VI), 4 CpMoO2Cl. Aqueous NaOH (2.89 ml, 3M, 8.67
	mmol) was added to 250 ml of acetone. After addition of solid 4 CpMoCl4 (1.362 g, 2.89 mmol) the solution turned red
	forming a white precipitate. Stirring overnight changed the color of the supernatant to a bright yellow-orange. All solvents
	were removed in vacuo leaving a yellow residue which was extracted with 200 ml of pentane. This pentane solution was
	dried over Na2SO4, filtered and evaporated to dryness leaving 1.0 g of a yellow-orange solid (2.52 mmol, 87 %), which
	was then recrystallized from a saturated heptane solution at -30 °C. Elem. anal. Calcd for C17H29ClMoO2: C, 51.45; H,
	7.36 %. Found: C, 51.04; H, 7.87 %. IR (Nujol mull): 920s, 890s cm -1 . 1 H NMR (CDCl3): 6.06 ppm (s, 1H, ring-H), 3.14
	ppm (m, 4H, CHMe2), 1.40 ppm (d, 6H, JHH = 7.4 Hz, CH(CH3)2), 1.35 ppm (d, 6H, JHH = 7.3 Hz, CH(CH3)2), 1.31 ppm
	(d, 6H, JHH = 6.7 Hz, CH(CH3)2), 1.25 ppm (d, 6H, JHH = 7.0 Hz, CH(CH3)2). A suitable single crystal for the X-ray
	11.2 %) of
	yellow [(Cp'''MoO2)2O]. Elem. anal. Calcd for C34H58Mo2O5: C, 55.28; H, 7.91 %. Found: C, 55.04; H, 7.94 %. 1 H
	NMR (C6D6): 5.63 ppm (s, 2H, ring-H), 1.58 ppm (18H, CMe3 1,2-position), 1.48 ppm (9H, CMe3 4-position). Suitable
	single crystals were grown by slowly cooling a hot saturated heptane solution to room temperature.
	Alternative route to pure Cp'''2Mo2O5. Acetone (120 mL) was mixed with an aqueous NaOH solution (6.6 mL, 3 M,

19.8 mmol). Purple Cp'''MoCl4 (1.55 g, 3.29 mmol) was added causing a deep red solution followed by slow turning of the supernatant to orange with formation of a white yellow precipitate within a 3 h period. After stirring overnight all solvents were evaporated under reduced pressure and the yellowish residue was extracted with 100 mL of water and layered with 100 mL of ether.

  Crystallographic data (excluding structure factors) for the structures reported in this paper have been deposited with the Cambridge Crystallographic Data Centre as supplementary publication no. CCDC-173139-173143.Copies of the data can be obtained free of charge on application to CCDC, 12 Union Road, Cambridge CB2 1EZ, UK [Fax: (internat.) + 44-1223/336-033; Email: deposit@ccdc.cam.ac.uk].

for 4 CpMoO2Cl.

Table 4 .

 4 Crystal data and structure refinement for all compounds.

	Compound	4 Cp2Mo2O5, 1	Cp ''' 2Mo2O5, 2	Cp * 2Mo2O4, 3	[Cp ''' MoO2]2, 4 4 CpMoO2Cl, 5
	Formula	C34H58Mo2O5	C34H58Mo2O5	C20H30Mo2O4	C34H58Mo2O4	C17H29ClMoO2
	M	738.68	738.68	526.32	722.68	396.79
	T; K	110(2)	110(2)	110(2)	110(2)	110(2)
	Crystal system	monoclinic	monoclinic	tetragonal	triclinic	monoclinic
	Space group	P21/c	P21/c	P-421c	P-1	P21/c
	a; Å	11.8993(2)	14.8025(2)	11.5418(9)	10.3976(1)	8.8290(2)
	b; Å	20.0429(3)	10.3088(1)	11.5418(9)	11.9801(2)	12.3811(3)
	c; Å	14.7150(2)	23.6064(3)	15.3875(3)	14.0109(2)	17.1372(6)
	; °	90	90	90	84.658(1)	90
	; °	95.597(1)	107.226(1)	90	88.904(1)	103.690(1)
	, °	90	90	90	88.898(1)	90
	V; Å 3	3492.74(9)	3440.66(7)	2049.8(2)	1737.09(4)	1820.09(9)
	Z	4	4	4	2	4
	F(000)	1544	1544	1064	756	824
	Dcalc; g/cm 3	1.405	1.426	1.705	1.382	1.448
	Diffract., scan type		Kappa CCD, mixture of  rotations and  scans
	; Å			0.71073
	; mm -1	0.755	0.766	1.245	0.755	0.870
	Crystal size; mm 3	0.30x0.25x0.13	0.43x0.25x0.25	0.20x0.20x0.12 0.32x0.25x0.20 0.25x0.15x0.08
	sin()/ max; Å -1	0.65	0.65	0.65	0.65	0.69
	Index ranges	h: -15; 15	h: -19; 19	h: -12; 14	h: -12; 13	h: -12; 6
		k: -25; 26	k: -13; 12	k: -14; 13	k: -15; 15	k: -16; 15
		l: -19; 19	l: -30; 30	l: -19; 18	l: -18; 18	l: -21; 23
	RC = Refl. Collected	25939	22209	10012	13541	8953
	IRC = unique RC / R(int)	7990 / 0.0403	7828 / 0.0243	2317 / 0.0284	7847 / 0.0141	4530 / 0.0190
	IRCGT = IRC and [I>2(I)]	5940	6728	2198	7073	3952
	Refinement method			Full-matrix L.S. on F 2
	Data / restr. / param.	7990 / 0 / 545	7828 / 0 / 545	2317 / 0 / 164 7847 / 0 / 535 4530 / 0 / 277
	R for IRCGT

Acknowledgments

We are grateful to the Ministère de la Recherche and the CNRS for general support through UMR 5632 and the French and German Ministeries of Foreign Affairs for a bilateral Procope grant. DS thanks the Conseil Régional de Bourgogne for a post-doctoral fellowship and FD thanks the Scientific and Technical Research Council of Turkey (TUBITAK) for a NATO Science Fellowship Programme scholarship, and Celal Bayar University for a leave of absence.

Graphical Abstract

Improved syntheses are reported for compounds Cp2Mo2O5 and Cp*2Mo2O5. Extension of these procedures have yielded homologues with sterically more encumbering cyclopentadienyl rings and related compounds, some of which have been structurally characterized.


Tsvi Tlusty

Exceptional Topology in Ordinary Soft Matter

Hydrodynamics is shown to induce non-Hermitian topological phenomena in ordinary, passive soft matter. This is demonstrated for the first time by subjecting a 2D elastic lattice to a low-Reynolds viscous flow. The interplay of hydrodynamics and elasticity splits Dirac cones into bulk Fermi arcs, pairing exceptional points with opposite half-integer topological charges. The bulk Fermi arc is a generic hallmark of the system exhibited in all lattice and flow symmetries. Analytic model and simulations explain how the emergent singularities shape the spectral bands and give rise to a web of van Hove singularity lines in the density of states. The present findings suggest that non-Hermitian physics can be explored in a broad class of ordinary soft matter, living and artificial alike, opening avenues for topology-based technology in this regime.

INTRODUCTION

The conservation of energy in isolated Hermitian systems is a basic tenet of physics, but in practice, most systems are open, exchanging energy and information with the external world. This inherent non-Hermiticity is traditionally seen as an inevitable imperfection of realistic systems, yet recent studies revealed that it gives rise to distinctive phenomena unmatched in Hermitian physics [START_REF] Bender | Real spectra in nonhermitian hamiltonians having pt symmetry[END_REF]-most notably skewed spectral bands prone to symmetry breaking when exceptional points emerge [START_REF] Doppler | Dynamically encircling an exceptional point for asymmetric mode switching[END_REF][START_REF] El-Ganainy | Nonhermitian physics and pt symmetry[END_REF][START_REF] Shen | Topological band theory for non-hermitian hamiltonians[END_REF][START_REF] Moiseyev | Non-Hermitian quantum mechanics[END_REF][START_REF] Bergholtz | Exceptional topology of non-hermitian systems[END_REF]. And this discovery kicked off intensive efforts to engineer non-Hermitian systems in diverse classical and quantum settings, ranging from photonics [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF][START_REF] Feng | Non-hermitian photonics based on paritytime symmetry[END_REF][START_REF] Miri | Exceptional points in optics and photonics[END_REF][START_REF] Weidemann | Topological funneling of light[END_REF], phononics [START_REF] Rivet | Constant-pressure sound waves in non-hermitian disordered media[END_REF][START_REF] Zhu | Simultaneous observation of a topological edge state and exceptional point in an open and non-hermitian acoustic system[END_REF][START_REF] Ma | Topological phases in acoustic and mechanical systems[END_REF], and optomechanics [START_REF] Xu | Topological energy transfer in an optomechanical system with exceptional points[END_REF] to electronics [START_REF] Schindler | Experimental study of active lrc circuits with PT symmetries[END_REF] and atomic lattices [START_REF] Zhang | Observation of parity-time symmetry in optically induced atomic lattices[END_REF]. The potential prowess of non-Hermitian technology has been demonstrated in developing new meta-materials and devices [START_REF] Guo | Observation of PT -symmetry breaking in complex optical potentials[END_REF][START_REF] Regensburger | Paritytime synthetic photonic lattices[END_REF][START_REF] Wiersig | Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection[END_REF][START_REF] Hodaei | Enhanced sensitivity at higher-order exceptional points[END_REF].

But an open question remains: besides sophisticated engineered systems, can one observe and utilize these so-called exotic topological phenomena in more common and natural settings?-After all, we are immersed in a dissipative, non-Hermitian world, and living systems are immanently open to exchange with the surrounding environment at all scales [START_REF] Libchaber | Walking droplets, swimming microbes: on memory in physics and life[END_REF]. The present study offers a clear positive answer: simple model and simulations demonstrate for the first time non-Hermitian topological hallmarks in standard, passive elastic networks subject to ordinary viscous flow [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Shani | Longrange orientational order in two-dimensional microfluidic dipoles[END_REF]. Such settings are omnipresent in the overdamped low-Reynolds regime, typical to cells, macromolecules and simple soft matter systems, suggesting that non-Hermitian topology is much more common than previously realized.

Among the exotic phenomena observed in non-Hermitian materials, bulk Fermi arcs [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF] hold a special place. In contrast to the ingrained intuition that frequency levels are closed curves, each Fermi arc is an open isofrequency curve ending at two exceptional points. * tsvitlusty@gmail.com These endpoints are defects with opposite half-integer topological charges. It is important to note that bulk Fermi arcs are topological hallmarks of non-Hermiticity in the bulk spectrum of the lattice, unlike the more familiar surface Fermi arcs induced by Weyl points in 3D Hermitian systems [START_REF] Wan | Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Xu | Observation of fermi arc surface states in a topological metal[END_REF][START_REF] Yang | Ideal weyl points and helicoid surface states in artificial photonic crystal structures[END_REF][START_REF] Morali | Fermi-arc diversity on surface terminations of the magnetic weyl semimetal Co3Sn2S2[END_REF]. Bulk arcs have been so far elusive and were empirically observed in one photonic crystal [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF]. Recent theoretical studies predict bulk arcs in the spectra of heavy fermions [START_REF] Nagai | Dmft reveals the non-hermitian topology and fermi arcs in heavyfermion systems[END_REF] and spin liquids [START_REF] Yang | Exceptional spin liquids from couplings to the environment[END_REF]. Strikingly, the present work finds that the elusive bulk arcs are a generic phenomenon of driven viscoelastic matter [START_REF] Beatus | The physics of 2d microfluidic droplet ensembles[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF][START_REF] Baron | Hydrodynamic crystals: Collective dynamics of regular arrays of spherical particles in a parallel-wall channel[END_REF][START_REF] Beatus | Anomalous microfluidic phonons induced by the interplay of hydrodynamic screening and incompressibility[END_REF][START_REF] Beatus | One-dimensional microfluidic crystals far from equilibrium: Acoustic phonons, instabilities and confinement[END_REF] whose emergence does not require fine tuning and stems directly from the symmetry of the interactions.

An important step into the dissipative regime are recent theoretical studies [START_REF] Scheibner | Odd elasticity[END_REF][START_REF] Scheibner | Nonhermitian band topology and skin modes in active elastic media[END_REF] demonstrating non-Hermitian "odd" elastodynamics [START_REF] Shmuel | Linking scalar elastodynamics and non-hermitian quantum mechanics[END_REF] in a network of active units specifically designed to exert circular forces. Likewise, non-Hermitian viscosity waves were shown to emerge in a system with "odd viscosity" [START_REF] Avron | Odd viscosity[END_REF]. Common to these "odd" systems are nonreciprocal elasticity or viscosity coefficients, originating from active modules that are hard to realize in microscopic settings. In sharp contrast, the present system requires no active parts, but merely standard passive soft matter in a laminar background flow, where non-reciprocity is a direct outcome of the hydrodynamic interaction [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF]. Thus, linking hydrodynamics and topology in soft matter can expand the realm of non-Hermitian physics to yet further fields, beyond the already rich plethora of applications developed in recent years.

In the following Results section, we derive the dynamic equations of the hydro-elastic lattice and map its spectrum to a standard non-Hermitian form in 2D momentum space. Then, we demonstrate how the interplay of Hermitian and skew-Hermitian interactions generates bulk Fermi arcs, which are shown to be a generic hallmark of the flow-induced dynamics. Next, we discuss the topological features of these singularities and their links to the density of states. Finally, in the Discussion, we suggest possible experimental realizations and examine potential implications and future directions.

RESULTS

The motion of the hydro-elastic lattice. To see how non-Hermitian topology arises in ordinary elastic matter at low-Reynolds, consider the following model system (Fig. 1A). A two-dimensional triangular lattice made of spherical particles of size -joined by thin elastic struts of length a and spring constant κ-is submerged in a viscous fluid of viscosity η and is moving at a velocity u relative to the fluid. The resulting viscous drag on each particle is γu, where γ is the friction coefficient. Such relative motion can be obtained in the lab by holding the lattice in a flow or by or by driving the lattice with various forces (see Discussion).

Perturbing the surrounding fluid, the driven particles induce long-range interactions throughout the lattice [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Beatus | The physics of 2d microfluidic droplet ensembles[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF][START_REF] Baron | Hydrodynamic crystals: Collective dynamics of regular arrays of spherical particles in a parallel-wall channel[END_REF] (Fig. 1B). When the flow is limited to a thin sheet of viscous fluid between solid floor and ceiling, these hydrodynamics forces are dipolar [START_REF] Liron | Stokes flow for a stokeslet between two parallel flat plates[END_REF][START_REF] Cui | Anomalous hydrodynamic interaction in a quasi-two-dimensional suspension[END_REF], and the force exerted by particle j on particle i is

f hyd ij = γ u 2 R -2 ij cos 2θ ij sin 2θ ij , (1)
where R i are the particle positions and the distance vectors are

R ij = R i -R j = R ij (cos θ ij , sin θ ij ).
The magnitude of the hydrodynamic dipole in (1) scales as ∼u 2 .

The Hookean elastic forces are proportional to the change of the length, ∆R ij = R ij -Rij , where Rij the equilibrium length of the springs (a in the lattice),

f elas ij = κ ∆R ij cos θ ij sin θ ij . (2)
Owing to the dipolar symmetry of (1), the hydrodynamic forces a pair of particles exert on each other are equal, f hyd ij = f hyd ji (because θ ji = π + θ ij ). Thus, the dipolar forces break Newton's third law of momentum conservation (Fig. 1B). This is because viscous flow is an inherently open system, an effective representation of energy and momentum transfer from hydrodynamic degrees-of-freedom to microscopic ones. To steadily move the lattice, the momentum leakage ('loss') needs to be constantly compensated by the driving force ('gain'). In contrast, the elastic forces (2) are opposite and conserve linear momentum, f elas ij = -f elas ji . As shown below, the interplay of conservative and non-conservative forces give rise to skewed non-Hermitian topology.

Two timescales govern the dynamics of the lattice: the hydrodynamic timescale, τ hyd = a 3 /(u 2 ), and the elastic relaxation time, τ elas = γ/κ. Their ratio is the hydroelastic number,

≡ τ -1 elas τ -1 hyd = κa 3 γu 2 , (3)
which controls the system's behavior: when 1, it is dominated by hydrodynamics, and when 1 by elasticity. Hereafter, we measure times in units of the hydrodynamic timescale τ hyd and frequencies in τ -1 hyd . The dynamical equations combine Stokes flow with Hookean elasticity, a linear regime where the correspondence between experiment and theory is well-established. In this overdamped regime, the friction force on each particle is counterbalanced by the driving force, F = F x, and by the hydrodynamic and elastic interactions in the lattice,

γ Ṙi = F + j =i (f hyd ij + f elas ij ) , ( 4
)
where Ṙi is the i th particle's velocity (see App. A 1).

In equation ( 4), the long-range hydrodynamic forces [START_REF] Bender | Real spectra in nonhermitian hamiltonians having pt symmetry[END_REF] are summed over all particles, while elastic interactions (2) are summed only among neighbors connected by struts. Owing to the lattice parity symmetry, the sums of interactions vanish at steady-state, when the lattice traverses uniformly at a velocity u = F/γ.

Dynamics in momentum space. The coaction of elastic and hydrodynamic forces excites collective motion in the lattice. Expanding the dynamics (4) in small x + Ω 2 y . The spectrum exhibits six Dirac points (green, one denoted as D) on the boundary of the Brillouin zone (black hexagon). Middle: At each Dirac point, the bands merge, forming a graphene-like double-cone, "diabolo" shape. Right: the 3D shape of the frequency bands in the first Brillouin zone (grey hexagon), showing the double-cones, which are halved by the zone's boundary. (B) At = 0, the symmetry is broken when H includes a skew-Hermitian component, H elas . Left: The real part of the bands, Re(ω+) = -Re(ω-), drawn for = π, exhibits six bulk Fermi arcs (white lines) in the first Brillouin zone (the whole spectrum is shown in Fig. 7). Each arc splits from a Dirac point (green) and joins two Exceptional Points (ExPs) with topological charges ± 1 2 (orange and light blue, E+ and E-are two ExPs that split from D). Middle: the real parts of the bands merge along the Fermi arc, forming a double-wedge shape. Right: 3D shape of the the frequency bands in the first Brillouin zone (grey hexagon), showing the double-wedges (halved by the zone's boundary).

deviations r j of the particles from their steady-state positions in the moving lattice Rj , we find that the collective modes are plane waves r j = e k exp i(k • Rj -ωt) (App. A 2-A 3). The 2D polarization of the wave, e k , is an eigenstate of a Schrödinger-like equation with an eigenfrequency ω,

H e k = ω e k , (5)
where the operator H is a momentum-space representation of the forces in the lattice. The "Hamiltonian" H is a 2×2-matrix, which sums the hydrodynamic and elastic interactions with a relative weight . In the basis of left-and right-circular polarizations, we obtain

H = H hyd + H elas , with H hyd = Ω x σ x + Ω y σ y , (6) and
H elas = -i (ω x σ x + ω y σ y + ω 1 1) ,
where σ x , σ y and 1 are Pauli's and the unity matrices. The frequencies Ω x , Ω y , ω x , ω y and ω 1 , are the Fourier sums of the interactions-all real by the parity symmetry of the lattice (see App. A 3). The hydroelastic operator H (6) is analogous to the Hamiltonian of spin-1 2 particles in a complex magnetic field with damping [START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF], and this spinor-like nature shows in the spectrum, as discussed below. Equation ( 5) with its non-Hermitian Hamiltonian H (6) is standard and appears in numerous 2D topological systems. However, the symmetry of H in momentum space is non-trivial and leads to the formation of bulk Fermi arcs.

Symmetry: Parity and Hermiticity. The hydrodynamic operator H hyd in ( 6) is Hermitian, a sum of products of Hermitian Pauli matrices and real numbers. Likewise, the elastic operator H elas is skew-Hermitian (i.e., a Hermitian operator multiplied by i),

H hyd = H † hyd , H elas = -H † elas .
Note that the hydrodynamic forces do not conserve momentum, but the effective hydrodynamic operator is Hermitian. This is because, in the low-Reynolds regime, the sum of the forces is proportional to the velocity (the drag force in ( 4)). Mathematically speaking, the imaginary unit factors of the time derivative (iω) and the spatial derivative (ik) cancel each other. For the same reason, the effective elastic operator is skew-Hermitian, reflecting the overdamped nature of elastic modes in this regime.

As for parity symmetry in k-space, the hydrodynamic part is odd and the elastic part is even,

H hyd (-k) = -H hyd (k) , H elas (-k) = +H elas (k) .
This follows from the parity of the interactions (1,2): Ω x and Ω y are odd functions of k whereas ω 1 , ω x and ω y are even (see App. A 3). It is important to note that unlike "odd" systems [START_REF] Scheibner | Odd elasticity[END_REF][START_REF] Avron | Odd viscosity[END_REF], H does not introduce any nonreciprocal coefficients of elasticity or viscosity. Here, non-Hermiticity is simply the outcome of hydrodynamics.

The spectrum: Dirac cones and Fermi arcs. The interplay of odd, Hermitian hydrodynamics and even, skew-Hermitian elasticity brings about distinctive topological signatures (Fig. 2). The spectrum of equations [START_REF] Moiseyev | Non-Hermitian quantum mechanics[END_REF][START_REF] Bergholtz | Exceptional topology of non-hermitian systems[END_REF] exhibits two eigenfrequency bands,

ω ± = -i ω 1 ± √ ν + ν -, (7)
where

ν ± = (Ω x -i ω x ) ± i(Ω y -i ω y )
, and all frequencies are measured in units of τ -1 hyd . The corresponding polarization eigenstates are (App. A 4)

e ± k = 1 |ν + | + |ν -| √ ν ± √ ν ∓ . ( 8
)
Without elastic forces (Fig. 2A), a purely hydrodynamic system ( = 0) exhibits real spectrum of propagating phonon-like waves [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Baron | Hydrodynamic crystals: Collective dynamics of regular arrays of spherical particles in a parallel-wall channel[END_REF], ω + = -ω -= (Ω 2

x + Ω 2 y ) 1/2 . On the edge of the Brillouin zone there are six Dirac points where the hydrodynamic interaction vanishes, Ω x = Ω y = 0. At a Dirac point, negative and positive bands kiss, ω + = ω -= 0, forming a graphenelike double cone [START_REF] Yarkony | Diabolical conical intersections[END_REF] (App. A 6). The introduction of skew-Hermitian elasticity at non-zero breaks the symmetry. Fig. 2B shows the real part of the bands, Re(ω + ) = -Re(ω -), for a hydroelastic number = π (the whole spectrum is shown in Fig. 7). Six bulk Fermi arcs-four S-shaped and two vertical lines-emerge from the Dirac points. Along the arcs, the real parts of the bands merge at Re(ω + ) = Re(ω -) = 0. Each arc is therefore an openended isofrequency contour joining a pair of isolated exceptional points (ExPs). As shown below, these points are topological defects of opposite ± 1 2 charges (App. A 7).

Singularities and bifurcations. At the ExPs, the spectral bands [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF] and their corresponding eigenstates [START_REF] Feng | Non-hermitian photonics based on paritytime symmetry[END_REF] simultaneously coalesce (Fig. 2B): the bands are purely imaginary, ω + = ω -= -i ω 1 , and the coalescing eigenstates, e + k = e - k , are either right-or left-circular polarizations, reflecting the chirality of the topological charges (see App. A 7). ExPs occur where the determinant in [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF] vanishes, (ν + = 0 or ν -= 0) at

Ω y ω x = - Ω x ω y = ± . (9)
At these double branching singularities, of both Re(ω) and Im(ω), the spectrum becomes gapless. In contrast to the Dirac points whose eigenspaces are two-dimensional, the eigenstates at the ExPs are parallel, signifying a reduction of the eigenspace dimension to one.

Tuning the hydroelastic number advances the ExPs along 1D trajectories, from the Dirac points at = 0 to the corners or the center of the Brillouin zone at = ∞ (App. A 7). Along these trajectories, the bands exhibit square-root singularities (Fig. 3),

ω ± = -i ω 1 ± ω 2 x + ω 2 y (s 2 -2 ) , (10)
where s ≡ Ω x /ω y = -Ω x /ω y is a 1D coordinate defined by [START_REF] Miri | Exceptional points in optics and photonics[END_REF]. The Fermi arc is the branch-cut of the square root [START_REF] Weidemann | Topological funneling of light[END_REF] stretching between the bifurcation transitions at the ExPs, s = ± . The square-root singularity reflects strong level repulsion between the bands, compared to the linear opening of the gap at the Dirac cone [START_REF] Miri | Exceptional points in optics and photonics[END_REF].

Bulk Fermi Arcs are generic.

As mentioned, the bulk Fermi arcs are topological signature of non-Hermiticity [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF] in the driven elastic lattice-unrelated to the more common surface Fermi arcs induced by Weyl points in 3D Hermitian systems [START_REF] Wan | Topological semimetal and fermi-arc surface states in the electronic structure of pyrochlore iridates[END_REF][START_REF] Xu | Observation of fermi arc surface states in a topological metal[END_REF][START_REF] Yang | Ideal weyl points and helicoid surface states in artificial photonic crystal structures[END_REF][START_REF] Morali | Fermi-arc diversity on surface terminations of the magnetic weyl semimetal Co3Sn2S2[END_REF]. The bulk arcs are a direct outcome of the broken symmetry, at = 0, when hydrodynamics and elasticity with opposite symmetries are mixed. Thus, the bulk arcs are a generic feature of the system, which are indeed observed in other lattice symmetries (Fig. 9) and for all directions of the driving flow. This is formally shown by a small-expansion around the Dirac cone of a purely hydrodynamic system (App. A 8), demonstrating that solutions to the ExP double condition (9) always exist. Thus, the pair of ExPs and the bulk arc that stretches between them are generic. Topological charges. The ExPs are topological defects: a closed eigenfrequency loop encircling an ExP cannot shrink to a point without passing through the ExP. The corresponding topological charge can be found from the vorticity of the band-gap [START_REF] Shen | Topological band theory for non-hermitian hamiltonians[END_REF]

, ∆ω = ω + -ω - (Fig 4), V = - 1 2π dk • ∇ k (arg ∆ω) . (11)
Since the band-gap is ∆ω = 2 √ ν + ν - [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF], the vorticity is

V = - 1 4π dk • ∇ k (arg ν + + arg ν -) = ± 1 2 , (12)
with a sign corresponding to the left-or right-handed chiralities of the gradient field (App. A 10). The vorticity is determined by the branch-cuts, where the jumps of arg ν ± by ±2π yield the topological charges q ± = ± 1 2 . The charges and their opposite chiralities originate from the square-root singularity of the Riemann surface [START_REF] Weidemann | Topological funneling of light[END_REF], and reflect the spinor-like nature of the polarization eigenstates, which accumulate a ±π phase when circling the ExP and passing through the branch-cut of the arcs.

Likewise, the charges can be computed from the integral of the Berry phase [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF] (see App. A 9). Berry's connections are the vectors

A ± (k) = i e ± * k ∇ k e ± k ,
and Berry's phases γ ± are the loop integrals

γ ± = A ± (k) • dk = ±π . ( 13
)
The corresponding charges, q ± = γ ± /(2π) = ± 1 2 , are determined by the jump of the phase (13) at the branch-cut, as in the case of the vorticity [START_REF] Rivet | Constant-pressure sound waves in non-hermitian disordered media[END_REF].

The density of states and its singularities. Projecting the Riemann surfaces of the spectral bands ω ± onto the complex frequency-plane reveals the density of states g(ω) (Fig. 5),

g(ω) = a 2π 2 d 2 k δ (ω -ω(k)) . ( 14
)
The Dirac points (green) split between the two banks of the branch cut. Notable are sharp-edged ridges of the density merging at logarithmically diverging summits, akin to van Hove singularities in Hermitian systems [START_REF] Van Hove | The occurrence of singularities in the elastic frequency distribution of a crystal[END_REF].

Level repulsion shows in low density of states around the ExP singularities in Fig. 5. The analytic density of states g(ω) is similar to the one obtained from spectra of simulated lattices, with deviations owing to finite size and boundary conditions (Fig. 10).

DISCUSSION

Realizations and physical limitations. Four physical conditions determine the experimentally accessible regime:

I. The topology is most notable when the elastic and hydrodynamic forces are comparable, around ∼ 1. Using (3), this condition can be written as κ ∼ ηu( /a) 3 (with Stokes law γ ∼ η ).

II. The Reynolds number Re must be kept in the overdamped non-inertial regime, Re = ρu /η ≤ Re * , with an upper bound Re * 10 -2 -10 -1 . Using condition I, this amounts to κ ≤ Re * [η 2 /(ρ )]( /a) 3 . III. To avoid lattice "melting", the thermal fluctuations should be kept small. A Lindemann-like criterion implies that the fluctuations are smaller than the particle size,

r 2 i ∼ (k B T /κ) ln(L/a) ≤ 2 ,
where k B T is the thermal energy scale and the fluctuations of the 2D elastic lattice increase logarithmically with size L [START_REF] Safran | Statistical thermodynamics of surfaces, interfaces, and membranes[END_REF]. This condition is recast as a lower bound, κ ≥ (k B T / 2 ) ln(L/a).

IV. The dynamics should be fast enough to allow adequate collection of statistics, η /κ ∼ τ elas ∼ τ hyd ≤ τ * , with an upper bound taken as τ * ∼ 10 s [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF]. This sets another lower bound on the elasticity, κ ≥ η /τ * .

Examination of the bounds in the -κ plane (Fig. 6) shows that the regime they define is easily accessible in standard soft matter settings. The limits are drawn for the parameters of a prototypical microfluidic system [START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF].

We see that all conditions I-IV are met by the optimal spring constant κ ∼ 10 2 -10 3 k B T/µm 2 (from condition I), corresponding to interaction energies of order 10 2 -10 4 k B T between micron-sized particles. Physical settings of microscopic bodies whose interactions are of order ∼10 3 k B T are characteristic to various soft matter systems, in particular colloids [START_REF] Safran | Statistical thermodynamics of surfaces, interfaces, and membranes[END_REF]. For example, spring constants of about ∼10 2 -10 3 k B T/µm 2 were measured in colloidal gels [START_REF] Dinsmore | Microscopic structure and elasticity of weakly aggregated colloidal gels[END_REF], small lattices [START_REF] Park | Pairwise interactions of colloids in two-dimensional geometric confinement[END_REF] and 2D colloidal crystals [START_REF] Buttinoni | Mechanical properties of colloidal crystals at fluid interfaces[END_REF][START_REF] Zhang | Self-assembly of patchy particles[END_REF][START_REF] Chen | Directed selfassembly of a colloidal kagome lattice[END_REF][START_REF] Mao | Entropy favours open colloidal lattices[END_REF]. Note that elasticity can be replaced by any other conservative central force, such as magnetic [START_REF] Keim | Harmonic lattice behavior of two-dimensional colloidal crystals[END_REF] and electrostatic [START_REF] Kelleher | Phase behavior of charged colloids at a fluid interface[END_REF] interactionsall retain the Hermiticity and parity symmetries of elasticity and enter the equations of motion as an effective spring constant. Other relevant systems include lattices of DNA-coated colloids [START_REF] Wang | Crystallization of dna-coated colloids[END_REF][START_REF] Rogers | Using dna to program the self-assembly of colloidal nanoparticles and microparticles[END_REF], colloidal lattices embedded in gels, and self-assembling microfluidic crystals [START_REF] Lee | Dynamic self-assembly and control of microfluidic particle crystals[END_REF][START_REF] Uspal | Self-organizing microfluidic crystals[END_REF].

To observe the exceptional topology, such soft matter crystals should be subjected to viscous flow, for example in a quasi-2D microfluidic channel. The lattices can be driven by the surrounding flow, slowed down by friction [START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF] or held in place by boundary conditions or external forces, for example, by dynamic holographic optical tweezers [START_REF] Curtis | Dynamic holographic optical tweezers[END_REF]. What matters is the velocity of the lattice relative to the embedding fluid, which induces the hydrodynamic interactions. The geometry of the lattice can be easily adapted. In our model system (Fig. 1A), we assumed for simplicity that the particles are joined by thin struts. However, the struts can be replaced by other particles, forming for example kagome, honeycomb or square-kagome lattices. This would augment the non-Hermitian spectrum with additional optical branches.

Moreover, long-range magnetic or electrostatic interactions obviate the need for direct mechanical connection among the particles. So one can leave out the struts and form, for instance, large 2D lattices of magnetic colloids [START_REF] Zahn | Dynamic criteria for melting in two dimensions[END_REF], disks [START_REF] Grzybowski | Dynamics of self assembly of magnetized disks rotating at the liquidair interface[END_REF], hydrogel particles [START_REF] Hwang | Microfluidic-based synthesis of non-spherical magnetic hydrogel microparticles[END_REF], or droplets [START_REF] Katsikis | Synchronous universal droplet logic and control[END_REF], with effective springs induced by magnetic dipole-dipole interactions [START_REF] Keim | Harmonic lattice behavior of two-dimensional colloidal crystals[END_REF]. The steady state configuration of such a lattice is governed by the confining boundaries, which can be crafted to allow the passage of viscous flow.

Outlook and summary. The proposed non-Hermitian model system can also be realized also in the sub-micron regime. Shrinking the system's dimensions (Fig. 6) tightens the thermal stability bound on κ (condition III), which approaches the optimal value (condition I) at a particle size of about ∼ 100 nm. But the bounds may be relaxed by increasing ηu (shifting up the dashed line), for example using higher pressures in hardened microfluidic chambers [START_REF] Shani | Longrange orientational order in two-dimensional microfluidic dipoles[END_REF][START_REF] Beatus | The physics of 2d microfluidic droplet ensembles[END_REF]. Tentative candidate systems in this regime include lattices made of DNA [START_REF] Rogers | Using dna to program the self-assembly of colloidal nanoparticles and microparticles[END_REF][START_REF] Thaner | Entropy-driven crystallization behavior in dna-mediated nanoparticle assembly[END_REF], protein [START_REF] Sinclair | Generation of protein lattices by fusing proteins with matching rotational symmetry[END_REF][START_REF] Ben-Sasson | Design of biologically active binary protein 2d materials[END_REF][START_REF] Zhang | Assembly of a patchy protein into variable 2d lattices via tunable multiscale interactions[END_REF] and patchy nano-particles [START_REF] Zhang | Self-assembly of patchy particles[END_REF][START_REF] Liljeström | Electrostatic self-assembly of soft matter nanoparticle cocrystals with tunable lattice parameters[END_REF][START_REF] Li | Kinetics-controlled design principles for two-dimensional open lattices using atom-mimicking patchy particles[END_REF]. At the other extreme, the system can be expanded up to the sub-millimetric regime, as long as the effective springs remain soft. Minding the omnipresence of both periodic lattices [START_REF] Messner | Occurrence, structure, chemistry, genetics, morphogenesis, and functions of s-layers[END_REF][START_REF] Charrier | Engineering the s-layer of caulobacter crescentus as a foundation for stable, high-density, 2d living materials[END_REF] and low-Reynolds flows [START_REF] Purcell | Life at low reynolds number[END_REF][START_REF] Brody | Biotechnology at low reynolds numbers[END_REF][START_REF] Lauga | The hydrodynamics of swimming microorganisms[END_REF] in biology and biotechnology, one may speculate that non-Hermitian topological phenomena are also widespread in these realms.

To sum, the reported skewed topology with bulk Fermi arcs ending at isolated exceptional points was recently observed in a photonic crystal with radiation loss [START_REF] Zhou | Observation of bulk fermi arc and polarization half charge from paired exceptional points[END_REF]. Here, we proposed a blueprint a controllable realization of this exotic topology in ordinary, passive soft matter, where the analogue of radiation loss is viscoelastic dissipation. The performed analyses showed the ubiquity of exceptional points and bulk Fermi arcs in generic elastic lattices driven by viscous flow. This suggests that non-Hermitian topology-which is robust against defects and noise-can be used to develop devices in the low-Reynolds regime. While these directions are intriguing, our main objective here was to demonstrate that simple viscoelastic systems can form an easily-accessible playground to investigate fundamental features of topological matter in the overdamped low-Reynolds regime, typical to soft and living matter. Future directions include examination of skin modes [START_REF] Bergholtz | Exceptional topology of non-hermitian systems[END_REF] and 3D geometries, where the long-range hydrodynamic interaction is much stronger than in 2D. In addition, one could explore composite architectures combining passive and active soft matter, also at the nanoscale [START_REF] Jee | Enzyme leaps fuel antichemotaxis[END_REF][START_REF] Jee | Catalytic enzymes are active matter[END_REF]. components),

f hyd ij, x = γΛ j =i X 2 ij -Y 2 ij X 2 ij + Y 2 ij 2 = γΛ j =i cos 2θ ij R 2 ij , (A1)
f hyd ij, y = γΛ j =i 2X ij Y ij X 2 ij + Y 2 ij 2 = γΛ j =i sin 2θ ij R 2 ij ,
where the positions of the dipoles are R i = (X i , Y i ), and

R ij = R i -R j = (X ij , Y ij ) are the distance vectors. In polar coordinates, R ij = (R ij , θ ij ), where R ij = |R ij |
and θ ij the angle. Equation ( 1) is a compact form of (A1).

The coupling constant Λ scales as the strength of the dipoles, Λ ∼ u 2 , where is the size of the particle, and u its velocity relative to the fluid. This defines the typical timescale τ hyd of the hydrodynamic interaction,

τ hyd ≡ a 3 Λ = a 3 u 2 , (A2)
the time it takes a perturbation to traverse a distance a at a sound velocity c s ∼ Λ/a 2 (in the continuum long wavelength limit). The timescale τ hyd depends on the physical forces driving the hydrodynamic interaction. For example, in a lattice of particles sedimenting in a quasi-2D fluid, the relative velocity scales as u ∼ ∆ρ g 2 /η, where ∆ρ is the density difference and η the fluid's viscosity.

In the quasi-2D flow of squeezed droplets, the coupling is Λ = 2 Ku, where K = u particle /u fluid ≤ 1 is the ratio of the velocities of particle and the surrounding fluid that drags it in the channel [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF].

The elastic Hookean forces are described by harmonic springs with constant k and equilibrium length Rij , the lattice constant a. The elastic forces are proportional to the change of the length, ∆R ij = R ij -Rij ,

f elas i = κ∆R ij n ij , (A3)
where n ij is a unit vector in the direction of the distance Rij . Equation ( 2) is (A3) expressed in polar coordinates.

In the overdamped regime, the elastic relaxation time is

τ elas = γ k . ( A4
)
This is the typical decay time of vibrational modes in the absence of hydrodynamic driving force. The hydroelastic number measures the relative significance of elastic and hydrodynamic interactions (3),

≡ 1/τ elas 1/τ hyd = κa 3 γu 2 .
Unlike odd viscosity [START_REF] Avron | Odd viscosity[END_REF] or odd elasticity [START_REF] Scheibner | Odd elasticity[END_REF], both hydrodynamic and elastic forces are non-circular, ∇ Rij × f ij = 0, as they conserve angular momentum.

Linear Expansion

In the overdamped low-Reynolds regime, the equations of motion are (4). At steady-state, the lattice interactions vanish by symmetry, j =i (f hyd ij + f elas ij ) = 0, and the lattice moves uniformly at a velocity u = F/γ. Expansion of the equations of motion (4) in small deviations of the lattice positions around the steady-state positions, r j = R j -Rj , yields a linear dynamic equation,

ṙ = H r , ( A5
)
where r is 2N -vector of the N particle deviations, r i .

The tensor H = H hyd + H elas , which is analogous to a Hamiltonian, combines contributions from hydrodynamics and elasticity. H is a 2N × 2N -matrix, composed of 2 × 2 blocks H ij that account for interactions between the i th and j th particles,

H ij = H ij hyd + H ij elas , (A6)
where the hydrodynamic term is

H ij hyd = 2 a Rij 3 cos 3θ ij sin 3θ ij sin 3θ ij -cos 3θ ij , (A7)
and the elastic one

H ij elas = 1 2 1 + cos 2θ ij sin 2θ ij sin 2θ ij 1 -cos 2θ ij , (A8)
which are conveniently expressed in terms of Pauli's matrices σ x , σ y , σ z (and 1 the unity matrix),

H ij hyd = 2 a/ Rij 3 (sin 3θ ij σ x + cos 3θ ij σ z ) , (A9) H ij elas = 1 2 (1 + sin 2θ ij σ x + cos 2θ ij σ z ) .
The diagonal terms ensure zero sums, H ii =j =i H ij . Since the angles obey θ ji = π + θ ij , it follows from (A9) that H ji hyd is odd with respect to particle exchange (i↔j), while H ij elas is even,

H ji hyd = -H ij hyd , H ji elas = +H ij elas .
The mutual forces between the i th and the j th particles are f ij = H ij (r j -r i ) and f ji = H ji (r i -r j ). Thus, we verify that the hydrodynamic forces violate Newton's third law of momentum conservation, whereas the lastic forces obey it (see Fig. 1B),

f hyd ji = +f hyd ij , f elas ji = -f elas ij .
While the microscopic molecular forces in the fluid obey Newton's law, the hydrodynamic interactions are effective macroscopic forces that do not conserve momentum. The momentum is leaking through the walls and is compensated by the driving force (e.g., gravitation or pressure gradient).

Momentum space

To exploit the crystal symmetry, one represents the dynamics in the momentum space of the wave-vectors k = (k x , k y ) by Fourier transform. The linearized dynamical equations (A5,A9) are expanded in plane waves, such that the deviation of each particle from its mechanical equilibrium position Rj is

r j (t) = e k e ik• Rj = e k e i(k• Rj -ωt) , (A10)
where e k (t) = e k exp(-iωt) is a 2D polarization vector in k-space.

The equation of motion of e k (t) is a non-unitary Schrödinger-like equation,

i ∂ ∂t e k (t) = He k (t) , ( A11
)
whose eigenvector is e k with eigenfrequency ω (5),

He k = ωe k .
The "Hamiltonian" H is a 2×2-matrix expressed in terms of Pauli's matrices as

H = H hyd + H elas (A12) = (Ω x σ z + Ω y σ x ) -i (ω 1 1 + ω x σ z + ω y σ x ) .
The contributions of the long-range hydrodynamic to H (A12) are Fourier sums,

Ω x = -2i j =0 a Rj 3 cos 3θ j e ik• Rj , Ω y = -2i j =0 a Rj 3 sin 3θ j e ik• Rj ,
where Rj = Rj (cos θ j , sin θ j ) are the distances of the steady-state lattice positions from an arbitrary origin particle 0. Owing to the crystal's parity symmetry, we can rearrange the summation to be over pairs at inverse positions, ± Rj , demonstrating that Ω x and Ω y are always real in a crystal,

Ω x Ω y = 2 j =0
a Rj

3 cos 3θ j sin 3θ j sin k • Rj . (A13)
The elastic contributions in (A12) are sums over the neighbors connected by springs to particle 0,

ω x = 1 2 j =0 cos 2θ j 1 -e ik• Rj , ω y = 1 2 j =0 sin 2θ j 1 -e ik• Rj , ω 1 = 1 2 j =0
1 -e ik• Rj .

Again, owing to the crystal parity symmetry, the sums can be rearranged, demonstrating that ω x , ω y and ω 1 are all real,

  ω 1 ω x ω y   = j =0   1 cos 2θ j sin 2θ j   sin 2 1 2 k • Rj . (A14)
The hydrodynamic and elastic interaction in a triangular lattice are shown in Fig. 8.

Spectra

The eigenfrequencies ω are found by solving the secular equation corresponding to (A11,A12). There are two eigenfrequency bands,

ω ± = -i ω 1 ± (Ω x -i ω x ) 2 + (Ω y -i ω y ) 2 = -i ω 1 ± √ ν + ν -, (A15)
where ν ± are defined as

ν ± ≡ (Ω x -i ω x ) ± i (Ω y -i ω y ) .
In the hydrodynamics-dominated regime, 1, the spectrum (A15) is purely real,

ω hyd ± ± Ω 2 y + Ω 2 x 1/2 , (A16)
while in elasticity-dominated regime 1 the spectrum is purely imaginary,

ω elas ± -i ω 1 ± (ω 2 x + ω 2 y ) 1/2 . (A17)
The polarization eignevectors, e + k and e - k , are

e ± k = Ω x -i ω x + ω ± + i ω 1 Ω y -i ω y (A18) = Ω x -i ω x ± √ ν + ν - Ω y -i ω y ,
where the eigenvalues ω ± are given in (A15). When normalized, the eigenvectors become In this basis, the Hamiltonian (A12) becomes equation ( 6),

e ± k = 1 √ 2 |ν + | + |ν -| √ ν + ± √ ν - i √ ν + ∓ √ ν - . (
H = H hyd + H elas = (Ω x σ x + Ω y σ y ) -i (ω 1 1 + ω x σ x + ω y σ y ) .
The eigenvectors in this representation take the simple form

e + k = 1 |ν + | + |ν -| √ ν + √ ν - , (A20) e
- k = 1 |ν + | + |ν -| √ ν - √ ν + .

Dirac points and cones

Dirac points occur in the purely hydrodynamic system ( = 0), at wavevectors k D for which the hydrodynamic interaction vanishes, Ω x = Ω y = 0 . From (A13), one sees that this happens when sin k • Rj = 0, that is for wavevectors that are halves of the reciprocal lattice base vectors, 1 2 b 1 , 1 2 b 2 , and their combinations, k D = 1 2 β 1 b 1 + 1 2 β 2 b 2 (where β 1 , β 2 ∈ {-1, 0, 1}). In the triangular lattice, the six Dirac points are (β 1 , β 2 ) = (0, ±1), (±1, 0), (±1, ±1). Note that these are the midpoints of the Brillouin zone edges and not the corners as in graphene.

The expansion of (A13) around the Dirac point is linear

in dk = k -k D , Ω x ∼ ∇ k Ω x • dk , Ω y ∼ ∇ k Ω y • dk ,
where the gradients are

∇ k Ω x = 2a j =0
(-1) β1α j 1 +β2α j 2 cos θ j sin θ j cos 3θ j Rj /a 2 , (A21)

∇ k Ω y = 2a j =0
(-1)

β1α j 1 +β2α j 2 cos θ j sin θ j sin 3θ j Rj /a 2 .

The α j 1 and α j 2 in (A21) are the indices of the lattice points, Rj = α j 1 a 1 + α j 2 a 2 , with the basis vectors, a 1 and a 2 . It is straightforward to verify that the gradients at the Dirac point are orthogonal,

∇ k Ω x • ∇ k Ω y = 0 , (A22)
resulting in an elliptic cone (Fig. 2A). It is also useful to note that, at the Dirac points, the elastic interactions, ω x , ω y and ω 1 , have extrema (minima, maxima, and saddles), and are therefore constant to first order in dk (Fig. 8).

Exceptional points and Fermi arcs

Exceptional points (ExPs) occur when both eigenfrequencies (A15) and eigenvectors (A19) coincide,

ω + = ω -= -i ω 1 , (A23) e + k = e - k = 1 √ 2 1 ±i ,
where the ± signs correspond to the paired ExPs. In the circular polarization basis, the eignevectors at the ExPs are the unit vectors,

e + k = 1 0 , e - k = 0 1 .
ExPs are positioned exactly where the determinant, ν + ν -, in (A15) vanishes. In other words, when ν + = 0 or ν -= 0, which amounts to the double condition [START_REF] Miri | Exceptional points in optics and photonics[END_REF],

Ω y ω x = - Ω x ω y = ± .
As one varies , the ExPs move along the 1D curve Ω x ω x + Ω y ω y = 0 , (A24) from the Dirac cones at = 0 to the Brillouin zone corners (or its center) at = ∞. The stretch of the trajectory between the two ExPs is the Fermi arc. Along these trajectories, the eigenfrequencies are The correspondence between the simulation and the analytic solution is excellent, owing to the commensurable four-fold symmetry of the lattice and the periodic boundary conditions, with some coarseness due to the finite size of the simulated system.

ω ± = -i ω 1 ± Ω 2 x + Ω 2 y -2 ω 2 x + ω 2

FIG. 1 .

 1 FIG. 1. (A) A triangular lattice made of spheres connected by elastic struts is immersed in a thin layer of viscous fluid between two walls in the z direction (perpendicular to the page). The lattice is driven at a velocity u relative to the fluid. (B) A pair of moving particles induce dipolar flow fields (grey streamlines), thereby exerting on each other equal hydrodynamic forces, f hyd ij = f hyd ji (blue arrows) in the direction of the induced flow. The elastic forces along the elastic strut are opposite, f elas ij = -f elas ji (red arrows), thus conserving linear momentum.

FIG. 2 .

 2 FIG. 2. (A) A purely hydrodynamic system, = 0. Left: The operator H = H hyd is Hermitian with real frequency bands ω+ = -ω-= Ω 2x + Ω 2 y . The spectrum exhibits six Dirac points (green, one denoted as D) on the boundary of the Brillouin zone (black hexagon). Middle: At each Dirac point, the bands merge, forming a graphene-like double-cone, "diabolo" shape. Right: the 3D shape of the frequency bands in the first Brillouin zone (grey hexagon), showing the double-cones, which are halved by the zone's boundary. (B) At = 0, the symmetry is broken when H includes a skew-Hermitian component, H elas . Left: The real part of the bands, Re(ω+) = -Re(ω-), drawn for = π, exhibits six bulk Fermi arcs (white lines) in the first Brillouin zone (the whole spectrum is shown in Fig.7). Each arc splits from a Dirac point (green) and joins two Exceptional Points (ExPs) with topological charges ± 1 2 (orange and light blue, E+ and E-are two ExPs that split from D). Middle: the real parts of the bands merge along the Fermi arc, forming a double-wedge shape. Right: 3D shape of the the frequency bands in the first Brillouin zone (grey hexagon), showing the double-wedges (halved by the zone's boundary).

FIG. 3 .

 3 FIG.3. The double branching transition in the real (blue) and imaginary (red) parts of the bands at the exceptional points[START_REF] Weidemann | Topological funneling of light[END_REF]. Zoom on the upper right bulk Fermi arc in Fig.2B.

FIG. 4 .

 4 FIG. 4. The vorticity of the band-gap, ∆ω = ω+-ω-(11), for = π. The argument of the band-gap, arg ∆ω, is color-coded and arrows denote its gradient field, ∇ k (arg ∆ω). Hexagonal black line shows the Brillouin zone boundary, with Dirac points (green) and exceptional points with ± 1 2 charges (orange and blue).

FIG. 5 .

 5 FIG. 5. Left: The density of states g(ω) plotted in log-scale, showing the Dirac points (green circles) and the exceptional points (blue/orange circles), all connected by bulk Fermi arcs (white line) at Re ω = 0. Right: 3D representation of g(ω).

FIG. 6 .

 6 FIG. 6. The physical limitations I-IV in the -κ plane, drawn for for typical values of a microfluidic system [31]: velocity u = 500 µm/s, viscosity η = 30 cP, geometric factor ( /a) 3 = 0.1, and lattice size (L/a) 2 = 10 4 . The regions excluded by bounds II-VI are shaded, and condition I ( ∼ 1) is the dashed line. The accessible region is in the remaining white domain.

FIG. 7 . 2 (

 72 FIG. 7. The spectrum of the triangular lattice. The real (top) and imaginary (bottom) components of the eigenfrequencies ω+ (left) and ω-(right) of a triangular lattice at = π (Fig. 2B). The black hexagon is the boundary of the Brillouin zone. Shown are the Dirac points (green) at the midpoints of the Brillouin zone edges, and the ExPs with their + 1 2 (red) and -1 2

A19) 5 .

 5 Circular polarization basisOne can represent the Hamiltonian in the basis of left and right circularly polarized unit vectors (which are the coalescing eigenstates at the ExPs),

FIG. 8 .

 8 FIG. 8. Hydrodynamic and elastic interactions and frequency bands. The black hexagon is the boundary of the Brillouin zone. The eigenfrequencies of the purely hydrodynamic systems with negligible elasticity, ω hyd ± (A16), are real because the effective Hamiltonian is Hermitian, H = H hyd . (top-right and center-right).In the absence of flow, the elastic system is purely damping. H = H elas is skew-Hermitian, and therefore its spectrum, ω elas ± (A17) is purely imaginary (top-center and center). The elastic interactions, Ωx and Ωy vanish at the Dirac points, the middles of Brillouin zone edges (bottom left and center), while the elastic interactions, ωx, ωy and ω1, exhibit extrema (left column).

FIG. 9 .

 9 FIG. 9. The spectrum of a square lattice. The real (top) and imaginary (bottom) components of the eigenfrequencies, ω+ (left) and ω-(right), of a square lattice at = π. The black square is the boundary of the Brillouin zone. Shown are the Dirac points (green) and the ExPs with their + 1 2 (red) and -1 2 (blue) charges. Bulk Fermi arcs are grey curves.

y 1 / 2 (ω 2 y 1 / 2 ,FIG. 10 .

 1221210 FIG.10. Density of states g(ω): analytic solution vs. simulation. (A-B): g(ω) for triangular lattice, analytic (A) and simulation (B), for = π. The simulated system is a 121 × 121 triangular lattice with periodic boundary conditions (a torus). The spectrum is obtained by Fourier analysis of the motion computed from the dynamical equation (4). The simulated g(ω) exhibits general similarity to the analytic solution. The incommensurately of the four-fold symmetry of the torus and the six-fold symmetry of the lattice results (C-D): g(ω) for square lattice, analytic (A) and simulation of a 121 × 121 lattice (B), for = π. The correspondence between the simulation and the analytic solution is excellent, owing to the commensurable four-fold symmetry of the lattice and the periodic boundary conditions, with some coarseness due to the finite size of the simulated system.

115, E10812 (2018).

ACKNOWLEDGMENTS

This work was supported by the Institute for Basic Science, Project Code IBS-R020. I thank Tsevi Beatus, Roy Bar-Ziv, Sam Safran, Hyuk Kyu Pak, Vincenzo Vitelli, William Irvine and their coworkers for essential comments.

Appendix A: Derivation of the model

Dynamics of hydroelastic lattices

The following is a brief description of the derivation. An elastic lattice is moving in the x-y plan of a thin 2D fluid layer at a velocity u relative to the fluid. In this quasi-2D geometry, the narrow dimension is z (perpendicular to the page in Fig. 1). The lattice is made of particles of size joined by thin elastic rods of average length a and spring constant κ. The viscous drag on each particle is γu, where γ is the friction coefficient (inverse of the mobility). Therefore, this non-equilibrium steadystate of uniformly moving lattice requires driving forces F (force per particle) that inject momentum and energy to compensate for the dissipative friction forces.

The particles' motion with respect to the fluid induces dipolar perturbations with a velocity field decaying as the inverse square of the distance, ∼u ( /r) 2 , and this dipolar flow fields give rise to collective hydrodynamic interactions [START_REF] Beatus | Phonons in a one-dimensional microfluidic crystal[END_REF][START_REF] Beatus | The physics of 2d microfluidic droplet ensembles[END_REF][START_REF] Beatus | Twodimensional flow of driven particles: a microfluidic pathway to the non-equilibrium frontier[END_REF][START_REF] Baron | Hydrodynamic crystals: Collective dynamics of regular arrays of spherical particles in a parallel-wall channel[END_REF]. The hydrodynamic force f hyd ij exerted by the j th particle on the i th particle is (in x, y where s ≡ Ω x /ω y = -Ω x /ω y is the coordinate along the trajectory, from s = 0, the Dirac point, through s = , the ExP, to the corner of the Brillouin zone s = ∞ (the four S-shaped arcs), or its center (the two straight arcs). The exceptional point is the position of a bifurcation transition where the branch-cut of the square root stretches along the Fermi arc.

Bulk Fermi arcs are generic

The bulk Fermi arcs observed in the triangular (Figs. 2,7) or the square (Fig. 9) lattices are a generic phenomenon of the hydroelastic lattice. They emerge as soon as symmetry is broken, at = 0, when the skew-Hermitian elasticity is introduced. To see this formally, one may examine the emergence of arcs for small . Then, the exceptional point condition ( 9) can be linearly expended in small deviations from the Dirac point, dk = kk D (A21). The double condition (9) becomes a linear equation in dk,

where the ± signs correspond to the two ExPs. Since the gradients are linearly-independent (A22), solutions always exist, and the positions of the ExPs are

Hence, the arcs and the ExPs are generic. The bulk Fermi arc stretches between the two solutions, and its size grows linearly with .

Berry's phase and the topological charges

In the momentum representation, Berry's connections are the vectors [START_REF] Berry | Quantal phase factors accompanying adiabatic changes[END_REF],

corresponding to the eigenvectors e ± k . For convenience, we write the normalized eigenvectors as Jones polarization vectors,

In this notation, we find that the Berry connections are simply

For the eignevectors (A18), the Jones parameters are

The Berry phases γ ± are the path integrals

To find the Berry phase of the ExPs, we integrate over a small circular counterclockwise path around each point. For example, consider the point where ν + = 0. In the vicinity of an ExP, c 1 2 , and the connections are therefore,

Along the circular integration path, the phase arg ν + passes all quadrants. In particular, it passes through the branch cut, arg ν + = ±π, where it jumps by ±2π, depending on the gradients of ν + . Thus, we find that Berry's phase is γ ± = ±π. Likewise, at the other ExP, Berry's phase takes the opposite sign, γ ± = ∓π. Altogether, we find that the corresponding topological charges are q ± = γ ± /(2π) = ± 1 2 .

Vorticity and Topological Charges

Another way to find the topological charges is through the vorticity of the eigenfrequency band-gap, ∆ω = ω +ω -, defined as [START_REF] Shen | Topological band theory for non-hermitian hamiltonians[END_REF]

where ω ± are the given in (A13). Since, ∆ω = ω + -ω -= 2 √ ν + ν -, we find that the vorticity is

As in the case of the Berry phase, the vorticity around the ExPs is determined by their branch-cuts. Namely, the jumps of arg ν ± by ±2π when passing through the branch cuts yield the topological charges q ± = ± 1 2 .