Claire Le Paven-Thivet

A Ziani

Florent Marlec

Laurent Le Gendre

Franck Tessier

Mohamad Haydoura

Ratiba Benzerga

François Cheviré

K Takanabe

Ala Sharaiha

C Le Paven

Structural and photoelectrochemical properties of SrTaO2N oxynitride thin films deposited by reactive magnetron sputtering

Introduction

Conversion of solar energy is attractive because it corresponds to an inexhaustible source of energy on a human scale. Over a year, the amount of solar energy absorbed by the Earth is 3,850 ZJ (ZJ, 10 21 Joules) and the human energy requirement is 0.5 ZJ. The challenge now remains to convert a large fraction of the received energy into storable and reusable energy.

Many projects using hydrogen as a new energy carrier have emerged in recent years [START_REF] Fujishima | Electrochemical Photolysis of Water at a Semiconductor Electrode[END_REF][START_REF] Rao | Solar photochemical and thermochemical splitting of water[END_REF][START_REF] Takanabe | Photocatalytic water splitting: quantitative approaches toward photocatalyst by design[END_REF][START_REF] Sivula | Semiconducting materials for photoelectrochemical energy conversion[END_REF]. An ecological and almost inexhaustible source of hydrogen atoms is at our disposal: water. It is, therefore, planned to convert and store solar energy in a chemical form (i.e., in binding energies) by splitting water into hydrogen and oxygen through a catalytic process involving a photoactive material. Among the possible processes, photoelectrochemical water splitting can achieve production of hydrogen in a single reactor. This process involves several reaction steps, including the absorption of photons, generation of electron/hole pairs, and the use of these charge carriers in the chemical reactions producing H2 and O2 [START_REF] Takanabe | Photocatalytic water splitting: quantitative approaches toward photocatalyst by design[END_REF]. To utilize photon energies in the visible-light range, it is necessary for the photoresponsive materials to absorb less than 3.1 eV of photon energy, corresponding to the typical bandgap of between 2 and 3 eV [START_REF] Sivula | Semiconducting materials for photoelectrochemical energy conversion[END_REF][START_REF] Bahers | Combined theoretical and experimental characterizations of semiconductors for photoelectrocatalytic applications[END_REF], and to have conduction and valence bands well-positioned relative to the redox potentials for hydrogen and oxygen evolutions.

J o u r n a l P r e -p r o o f

The material studied in the present contribution is SrTaO2N (abbreviated as STON hereafter), a perovskite oxynitride compound. SrTaO2N has an electronic band structure comparable to TaON [START_REF] Xiao | Tantalum (oxy)nitride: narrow bandgap photocatalysts for solar hydrogen generation[END_REF] and LaTiO2N [START_REF] Le Paven-Thivet | Photoelectrochemical properties of crystalline perovskite lanthanum titanium J o u r n a l P r e -p r o o f oxynitride films under visible light[END_REF] and, based on its bandgap and band positions, has been identified as a relevant photocatalyst candidate for the water-splitting reactions [START_REF] Ziani | Photophysical properties of SrTaO2N thin films and influence of the anion ordering: a joint theoretical and experimental investigation[END_REF][START_REF] Higashi | Photocatalytic overall water splitting under visible light using ATaO2N (A = Ca, Sr, Ba) and WO3 in a IO3 -/Ishuttle redox mediated system[END_REF]. In our previous study of STON films deposited on Nb:STO substrates [START_REF] Ziani | Photophysical properties of SrTaO2N thin films and influence of the anion ordering: a joint theoretical and experimental investigation[END_REF], we identified the influence of the stoichiometry, namely anion ordering (O/N). Here we aim to generate new insight that will improve our understanding of the physicochemical phenomena responsible for the photoelectrochemical response. This study reports extended experimental data by varying crystallization of films, type of crystallization, impact of different substrates, and thickness of films, and to study their impact on the photoelectrochemical activity of STON thin films. The objective of the study is the determination of intrinsic properties and identification of limiting parameters of STON. In this sense, our study is in line with that of Pichler et al. on the photoelectrochemical study of perovskite oxynitride LaTiO2N films [START_REF] Pichler | LaTiOxNy Thin Film Model Systems for Photocatalytic Water Splitting: Physicochemical Evolution of the Solid-Liquid Interface and the Role of the Crystallographic Orientation[END_REF]. The present research presents the deposition of three series of STON films by means of reactive radiofrequency (rf) magnetron sputtering (substrate temperature upon deposition, film thickness, and type of substrate), which were varied in order to study their influence on the optoelectronic, structural, and photoelectrochemical properties.

Experimental

Oxynitride STON sputtering target

The sputtering target was made via multi-step processes involving the synthesis of the STON powder and its subsequent shaping as a cylindrical target disc for the deposition of thin films by rf sputtering. For the powder, the first step is to make the Sr2Ta2O7 oxide precursor compound by a solid-state chemistry route. Stoichiometric amounts of high purity powders of SrCO3 and Ta2O5 were homogenized in isopropanol for 1 h in an agate mortar and dried in an The oxynitride target (75 mm in diameter) was separately made using the STON powder, compacted by uniaxial pressing at 15 tons. No sintering processes were carried out in air due to the susceptibility of oxynitrides to oxidation at moderate and high temperature [START_REF] Rachel | Tantalum and niobium perovskite oxynitrides: synthesis and analysis of the thermal behaviour[END_REF][START_REF] Aguiar | Thermal oxidation of oxynitride perovskites in different atmospheres[END_REF]. This limitation has, therefore, made it impossible to use an organic binder to ensure a minimal mechanical strength of the target. The latter is, therefore, very brittle, as shown by the slight cracks observed on the photo shown in Supplementary Figure S1. Moreover, the tested substrates had specific roles in our study: FTO and FG substrates served for the UV-visible transmittance; the FTO, Nb:STO (niobium-doped Nb:SrTiO3), Au(200 nm)/Ti(10 nm)/FG, and Ta (0.5 mm thick tantalum foil) substrates were employed for the photoelectrochemical study. In that case, a small area of the FTO and Au/Ti/FG substrates were covered by a mask during the deposition in order to maintain a conductive area at the front of the sample for the photo-electrical measurements because, unlike the conductive substrates (Nb:STO, Ta), their backside is not conductive.

Thin-film deposition

Characterization

X-ray diffraction patterns of films were obtained using a Rigaku SmartLab diffractometer (Cu K 1 radiation). Conventional  diagrams were recorded at 0.01° intervals with a 2 s count time at each step. The Lotgering factor (FL) was used to quantify the film orientation. The Lotgering factor compares the orientation of a film versus a randomly oriented material (i.e., powder) and can vary from 0 for a non-oriented layer to 1 for a fully oriented layer. The Lotgering factor is calculated using the following relations, considering a (00l) orientation of films [START_REF] Lotgering | Topotactical reactions with ferrimagnetic oxides having hexagonal crystal structures[END_REF]:

𝐹 𝐿 = 𝑃 00𝑙 -𝑃 0(00𝑙) 1-𝑃 0(00𝑙) with 𝑃 (00𝑙) = ∑ 𝐼 (00𝑙)
∑ 𝐼 (ℎ𝑘𝑙)

J o u r n a l P r e -p r o o f where (hkl) is used for all families of planes, I(00l) and I(hkl) are the intensities of the (00l) and

(hkl) indexed peaks, respectively, and the subscript 0 is relative to the powder reference (JCPDS data).

UV-visible transmittance spectra were recorded using a Perkin-Elmer Lambda 20 spectrometer in the 200-1100 nm wavelengths range. For each sample, the absorption coefficient  is calculated from the transmittance (T) using the relation  = (1/t)(ln(1/T)), where t is the thickness of the film. The bandgap of the material (Eg) is calculated assuming a direct allowed transition and using the relation (h  = h -Eg, where h is the photon energy. In the plot of (h  as a function of h, the extrapolation of the linear region of the plot at zero gives the value of Eg [START_REF] Kubelka | Ein beitrag zur optik der farbanstriche[END_REF]. Additional measurements for the determination of the absorption coefficient have been performed using a JASCO J670 spectrophotometer with an integrating sphere that simultaneously measures the transmittance, absorbance and reflectance of the sample.

Energy dispersive spectrometry (EDS) was used to investigate the cations and nitrogen relative compositions of films; analyses were performed using the Aztek OXFORD software through a JEOL JSM IT100 scanning electron microscope (SEM) operating at 10 kV. Surface and cross-section morphologies of films were characterized by the above SEM; the error on the thickness values from the cross-section observations is estimated as being ± 10 nm.

Transient absorption (TA) measurements were performed using the pulse energy of the fundamental output of a Ti:sapphire fs regenerative amplifier operating at 800 nm with 35 fs pulses and a repetition rate of 1 kHz. The pump pulses at 320 nm were created from fs pulses generated in an optical parametric amplifier (TOPAS Prime, Spectra-Physics). To generate the probe pulses (white light), another fraction of the 800 nm amplified pulses was focused onto the 2-mm thick continuously moving CaF2 crystal. The pump and probe were spatially overlapped on the STON thin film surface after passing through a synchronized mechanical J o u r n a l P r e -p r o o f chopper (500 Hz) that blocked an alternative pump pulse. The transmitted probe pulses were directed to the optical fiber. Depending on the path length of the probe beam, the excited state lifetime of the photoexcited charge carrier was measured to be up to 3.5 ns with respect to the pump pulses. To achieve better signal to noise ratios, the resulting white light was split into two channels named as probe and reference, respectively, and focused on the two different fiber optics. The change in absorption (ΔA) of the excited state is calculated by subtracting absorption of the excited and unexcited sample. A is defined as the following:

A = APump -ANoPump = log( 𝐼 𝑁𝑜𝑃𝑢𝑚𝑝 𝐼 𝑃𝑢𝑚𝑝 )
where A is the absorbance defined by the Beer-Lambert law [A = log(I/I0) with

I(t) = I0.e -.t ,
where I0 is the incident intensity, I is the intensity of the transmitted probe (without (INoPump)

and with (IPump) the pump),  is the absorption coefficient and t is the thickness of film]. Spectra were averaged over a time period of 2 s for each time delay, yielding every time-resolved spectrum.

Linear scanning voltammetry (LSV) measurements for the photoelectrochemical characterization were performed on a photoelectrochemical (PEC) system. Three electrodes were used: a working electrode (WE), which is the studied STON film and acts as the anode on which the oxidation of water occurs; a counter-electrode (CE) in platinum that acts as the cathode with the reduction of protons and the production of H2; and a reference electrode (Ag/AgCl Sat. with KCl) with a standard potential of 0.198 V vs. RHE (reversible hydrogen electrode). A NaOH electrolyte (0.1 M, pH = 13) was used with an Ar bubbling. The measuring cell (Supplementary Figure S2) has a capacity of 100 mL and a flat Pyrex window that allows irradiation. The latter is performed under solar simulator irradiation adjusted to 100 mW cm -2

(1 sun). The irradiation is chopped with a mechanical shutter with a frequency of 0.11 Hz (open for 4.5 s, closed for 4.5 s), in order to reveal the influence of irradiation on the sample.

J o u r n a l P r e -p r o o f

Results and Discussion

Effects of the FTO substrate temperature during sputtering

This series of experiments concerns films deposited on FTO conducting substrate to examine the synthesis of oxynitride films at different substrate temperatures, from 300 to 650°C (the latter is the highest possible temperature on this substrate due to its degradation beyond).

The film thicknesses were all around 300 nm. XRD  diagrams of the samples are presented in Figure 2. At TS = 350°C, diffraction peaks for SrTaO2N (JCPDS 79-1311) are identified and exhibit a polycrystalline growth. It is quite remarkable that TS as low as 350°C led to the crystallization of the oxynitride phase in contrast to the high synthesis temperatures (~1000°C)

generally used for the synthesis of oxynitride powders [START_REF] Chen | Direct synthesis of nearly single phase SrTaO2N from SrCO3/TaN[END_REF][START_REF] Fuertes | Synthetic approaches in oxynitride chemistry[END_REF]. Figure 2 also shows that the increase of the substrate temperature induces an increase of the 𝑐 ⃗ axis orientation of films as reflected by the intense (00l) peaks and by the evolution of the Lotgering factor values (Table 1, Supplementary Figure S3).

Figure 3 presents the evolution of the transmittance of films as a function of the wavelength. As can be seen in Table 1, a fall of the bandgap (from 5.30 to 2.50 eV) is observed between the substrate temperatures TS = 350 and 400°C. Above 400°C (Supplementary Figure S4), bandgaps show a shift towards lower values when the substrate temperature increases. The lowest measured bandgap (2.35 eV) approaches those previously reported on powders (2.1 eV) [START_REF] Zhong | Enhanced water-splitting performance of perovskite SrTaO2N photoanode film through ameliorating interparticle charge transport[END_REF] and thin films (2.27 eV) [START_REF] Oka | Possible ferroelectricity in perovskite oxynitride SrTaO2N epitaxial thin films[END_REF].

Concerning the chemical composition, the Sr/Ta ratios and nitrogen contents of films deposited at various substrate temperatures are given in Table 1, and their evolution as a function of TS is depicted in Figure 4. It is found that the Sr/Ta ratio decreases as the substrate temperature increases; the ratio Sr/Ta = 1.05 is obtained at TS = 650°C. Note that the Sr/Ta = 1.00 value can be obtained but at a higher substrate temperature and so, on another substrate (on Ta as seen in Figure 4). For the nitrogen content, except the value obtained at TS = 300°C

J o u r n a l P r e -p r o o f for which the nitrogen content is high (26 at.%), we can assume that all other data are similar and that the observed differences are within the experimental uncertainty of the EDS analysis.

The variation of the cationic composition, evidenced here as a function of the deposition temperature, can be explained by the flexibility of the perovskite unit-cell which can account relatively large deviations of the chemical composition compared to the strict stoichiometry. This is in line with the findings of Rodriguez et al. [START_REF] Rodriguez | A new metastable thin-film strontium tantalate perovskite[END_REF] who pointed out a perovskite signature in XRD patterns of SrxTaO3 films for strontium contents (x) in the range [0.8-1.2]. A similar feature was also observed by Regnery et al. in their study of Sr-Ta-O films [START_REF] Regnery | SrTa2O6 thin films for high-K dielectric applications grown by chemical vapor deposition on different substrates[END_REF].

The surface morphologies of the STON films deposited on FTO at different substrate temperatures are shown in Figure 5. A finely granular and very dense structure is seen regardless of the deposition temperature. The roughness associated with a polycrystalline nature seems apparent as the substrate temperature increases. The size of the grains is typically in the range of 100-200 nm.

Overall, using the FTO substrate, the stoichiometry becomes closer to its theoretical stoichiometry at higher Ts, up to 650°C, but at temperatures beyond this, FTO loses its conductivity. At 650°C, the bandgap approached the reported values on the STON material. A summary of the samples characteristics is presented in Table 1.

Effects of film thickness on FG substrate

This series of experiments uses films deposited on FG substrates, a transparent insulating substrate that can be heated at high temperature for optical measurement purposes.

First, two samples were prepared at TS = 650°C and TS = 800°C with a film thickness of 300 nm. Results of the EDS analysis (Table 1) show that the Sr/Ta ratio and the nitrogen content of films approach that of the stoichiometric STON compound of when the substrate temperature is increased from TS = 650°C to 800°C (for the latter, Sr/Ta = 1.03 and at.%N = 20.5). The

J o u r n a l P r e -p r o o f

XRD patterns of the STON films are ascribed to STON without any impurity phase (Supplementary Figure S5). Maintaining TS = 800°C, the film thickness was controlled by regulating sputtering time to generate thicknesses of 150 and 600 nm in addition to 300 nm.

The UV-visible transmittance, reflectance, and absorbance spectra of the samples as a function of the incident wavelength are presented in Supplementary Figure S6. Figure 6 shows the absorption coefficient  as a function of the wavelength. Comparing the bandgap values of the two samples with a thickness of 300 nm (Supplementary Figure S7), we observe that the film deposited at TS = 650°C has a slightly larger bandgap (2.39 eV) than that deposited at TS = 800°C (2.33 eV). The latter value is close to those already reported on the STON thin films [START_REF] Oka | Possible ferroelectricity in perovskite oxynitride SrTaO2N epitaxial thin films[END_REF]. Moreover, we notice that the bandgap slightly decreases with decreasing thickness (Supplementary Figure S8). Concerning the absorption (Figure 6), we observe that the absorption coefficient values of the different samples are in the same range in the visible domain and are slightly lower than 100,000 cm -1 at 400 nm. These values are similar to those obtained in one of our previous works on STON films deposited on Nb:STO with different thickness [START_REF] Ziani | Photophysical properties of SrTaO2N thin films and influence of the anion ordering: a joint theoretical and experimental investigation[END_REF].

These values are also comparable to those obtained by Trapalis et al. on the Zn3N2 nitride material [START_REF] Trapalis | Structural, electrical, and optical characterization of as grown and oxidized zinc nitride thin films[END_REF]. These values are somewhat lower than those obtained by Ziani et al. on Ta3N5 thin films, which were in the range 8 × 10 5 to 1 × 10 4 cm -1 from 300 to 600 nm [START_REF] Ziani | Carrier dynamics of a visible-light-responsive Ta3N5 photoanode for water oxidation[END_REF]. Moreover, here for the wavelength value of interest at 585 nm in the next transient absorption measurements, low values close to 3000 cm -1 were measured.

Dielectric characteristics of the STON films in the UV-Visible range can be calculated from the above measurements using the relations explained in the reference [START_REF] Ziani | Carrier dynamics of a visible-light-responsive Ta3N5 photoanode for water oxidation[END_REF]. Permittivities (Figure 7) shows a fairly large fluctuation as a function of the energy of the incident photon in the range 1 -3 eV due to interference fringes in the reflectance spectra (Figure S6). In the range 1-3 eV, average permittivities are moderate, between 4.5 and 8, with dielectric loss tangents between 0.2 and 0.7. The highest permittivity is obtained for the 150-nm thin film. These values J o u r n a l P r e -p r o o f are consistent with those obtained in our previous work on STON films deposited on Nb:STO [START_REF] Ziani | Photophysical properties of SrTaO2N thin films and influence of the anion ordering: a joint theoretical and experimental investigation[END_REF] and are comparable to those of Ta2O5 showing an optical permittivity of 4.4 [START_REF] Bécu | Etude des non-linéarités de permittivité de diélectriques utilisés en microélectronique[END_REF]. In contrast, Ta3N5 films were reported to exhibit a slightly higher permittivity value [START_REF] Zhong | Enhanced water-splitting performance of perovskite SrTaO2N photoanode film through ameliorating interparticle charge transport[END_REF] with lower dielectric losses (0.03) from 10 14 to 10 15 Hz [START_REF] Ziani | Carrier dynamics of a visible-light-responsive Ta3N5 photoanode for water oxidation[END_REF].

To evaluate the lifetime kinetics of the photo-charge carriers, preliminary measurements of transient absorption (TA) by means of femtosecond pump-probe spectroscopy were performed on the above STON-FG films. TA spectra as a function of time delay from 1 ps to 250 ps in the [420 -620 nm] range for the 150 nm and 600 nm STON-FG thin films are presented in Figures 8a and8b. To highlight differences, a comparison of the TA at time delay +1 ps for the two samples is also presented at Figure 8c. Three regions are observed in the transient absorption spectra. The first signal is a ground state bleaching GSB (negative) signal around 450 nm, a second GSB signal at 500 nm, and a broad third signal is an induced absorption (positive TA signal) between 510 to 620 nm.

The first GSB signal at 450 nm is due to the depopulation of the ground state due to bandgap transition and is correlated to the VB free hole dynamics [START_REF] Maity | Study of the bulk charge carrier dynamics in anatase and rutile TiO2 single crystals by femtosecond time-resolved spectroscopy[END_REF]. Both the STON-150

and STON-600 sample present this transition. The kinetics of this transition is rapid; it happens at almost 9 ps (Figure 9a). As a comparison, a career lifetime of a similar material, Ta3N5 thin films synthesized by reactive sputtering [START_REF] Ziani | Carrier dynamics of a visible-light-responsive Ta3N5 photoanode for water oxidation[END_REF], was measured to be also very short with a career trapping occurring in less than 10 ps. The presence of the second GSB signal at 500 nm can be correlated to the formation of hole trapping nitrogen vacancies (VN) above the valence band (VB). This process was recently reported on Ta3N5 material by Murthy et al. [28] and Ziani et al. [START_REF] Ziani | Carrier dynamics of a visible-light-responsive Ta3N5 photoanode for water oxidation[END_REF]. Two bleaching signals were observed at 554 and 590 nm; the first one was assigned to the bandgap transition and the second was assigned to a VN to CB transition. Here, by correlating the energetic positions of bleaching at 450 nm and 500 nm, we can potentially assign the presence of the GSB signal at 500 nm to the formation of hole trapping VN states situated J o u r n a l P r e -p r o o f 0.27 eV above the VB maximum. Only the STON-600nm presents the two transitions. The STON-150nm looks less defected and presents mainly the bandgap transition. Finally, the broad signal between 510-620 nm can be assigned to be shallowly trapped electrons [28]. As a summary, characteristics of films of this series are presented in Table 1 and underline that the substrate temperature can be further increased beyond TS = 650°C, up to TS = 800°C, leading to films very close to the STON stoichiometry. The different thicknesses of the STON films consistently give similar absorption coefficients, while low coefficients of absorption were obtained in the 500-600 nm range; these could represent a first factor explaining the low measured photocurrents on the STON films as seen after. A second factor is also evidenced by TA measurements, the low lifetimes of photo-charge carriers on the order of a few picoseconds to a few tens of picoseconds.

Effects of the substrate identity

J o u r n a l P r e -p r o o f

This series of experiments makes use of deposition on different types of substrates, FG, Nb:STO, Au/Ti/FG, and Ta, at TS = 800°C. Film thicknesses are fixed to be around 300 nm.

The film deposited on the FG substrate was the only one measured by UV-visible transmittance because the other substrates (metalized or conductive) are non-transparent; the extracted bandgap is 2.33 eV. Photos of the prepared films are presented in Supplementary Figure S9.

XRD measurements of the films were carried out in θ-2θ measurements while keeping the Kβ line of copper to obtain higher peak intensities and thus to detect the possible existence of parasitic phases in very small quantities. Bare substrates were also analyzed to locate their diffraction peaks. Patterns of films and substrates are shown in Figure 10.

All deposited films are identified as STON but have different crystalline states, from polycrystalline to totally oriented growth. To quantify the orientation of the films along with their 𝑐 ⃗ axis, the Lotgering factors (FL) were calculated (Table 1). The film deposited on the amorphous FG substrate has an FL = 0.17; this film is polycrystalline, as can also be seen in Figure 10a. Contrarily, the {00l} family peaks are the only visible peaks for the film deposited on the Nb:STO substrate (Figure 10b), the film is therefore fully 𝑐 ⃗-oriented with a Lotgering factor of 1. This is explained by the weakest mismatch (m) within this series between the SrTaO2N material and the Nb:STO single-crystalline substrate (m = + 3.3%); this is the substrate that promotes the best crystal orientation of STON. Several (hkl) peaks are found for the other films deposited on Au/Ti/FG and Ta substrates (Figures 10c and10d); the films are polycrystalline. Nevertheless, they display a significant degree of 𝑐 ⃗-axis orientation with FL = 0.33 and FL = 0.58 on Au/Ti/FG and Ta, respectively. One can also notice that the peaks are of very low intensity on the Au/Ti/FG substrate, denoting a poor crystallization. For the film deposited on Ta, a fairly intense peak is observed at  ~ 60.6°, which might correspond to the TaN tantalum nitride (JCPDS 49-1283) with a (220) orientation.

J o u r n a l P r e -p r o o f

By the EDS analysis (Table 1), all Sr/Ta ratios are similar and close to 1, regardless of the substrate; the amount of nitrogen was consistently higher (marginally) than 20 at.%, except for the film on Nb:STO substrate (N = 24.0 at.%).

Photoelectrochemical measurements on STON films

Photoelectrochemical measurements under visible light were performed by linear sweep voltammetry. As reported by many authors, for instance, Pichler et al. in their photoelectrochemical study of LaTiO2N thin films [START_REF] Pichler | LaTiOxNy Thin Film Model Systems for Photocatalytic Water Splitting: Physicochemical Evolution of the Solid-Liquid Interface and the Role of the Crystallographic Orientation[END_REF], oxynitride materials undergo a selfoxidation by consuming the created photo-holes. This process, in addition to degrading the photoanode material, produces a photocurrent that adds to the O2 evolution. As an example, Supplementary Figure S10 presents the first four measurements done on a STON film deposited on Nb:STO substrate deposited at TS = 800°C. The results show that the photocurrent density was more or less stabilized from the second measurement. Consequently, for the rest of the study, we report the photocurrent density values obtained for the second measurement on each sample.

Supplementary Figure S11 shows the photoresponse of STON films with the same thickness (300 nm) deposited on different substrates (FTO, Nb:STO, Au/Ti/FG and Ta). First, it can be seen that all samples exhibit a photocurrent when irradiated, which confirms that the material responds to visible light. The low photoresponses of the STON-FTO and STON-Au/Ti/FG films can be explained first by their polycrystalline and weak crystallization, respectively, evidenced in the XRD analysis. Photoresponses of the well-crystallized oriented STON-Nb:STO and STON-Ta films are actually higher and demonstrates, as already reported [START_REF] Le Paven-Thivet | Photoelectrochemical properties of crystalline perovskite lanthanum titanium J o u r n a l P r e -p r o o f oxynitride films under visible light[END_REF], that the crystallinity of films is a key to exhibit a good photoresponse. The photocurrent (19 A cm -2 at 1 V vs. RHE) measured for the STON-Ta film is consistent with those already reported on similar oxynitride films in the same conditions, in particular in the absence of cocatalysts or electron donor in the reaction medium [START_REF] Le Paven-Thivet | Photoelectrochemical properties of crystalline perovskite lanthanum titanium J o u r n a l P r e -p r o o f oxynitride films under visible light[END_REF][START_REF] Higashi | Photocatalytic overall water splitting under visible light using ATaO2N (A = Ca, Sr, Ba) and WO3 in a IO3 -/Ishuttle redox mediated system[END_REF][START_REF] Nishimura | Effect of TiCl4 treatment on the photoelectrochemical properties of LaTiO2N electrodes for water splitting under visible light[END_REF][START_REF] Menon | Development of Zn1-x-yGaxCoyO1-zNz as a non-oxide semiconductor material with visible light photoelectrochemical activity[END_REF]. For example, Pischler et al.

J o u r n a l P r e -p r o o f report photocurrent densities close to 10 A cm -2 at 1.5 V vs. RHE for LaTiO2N films deposited on TiN/MgO substrates [START_REF] Higashi | Photocatalytic overall water splitting under visible light using ATaO2N (A = Ca, Sr, Ba) and WO3 in a IO3 -/Ishuttle redox mediated system[END_REF]. Also, a LaTiO2N thin film deposited on Nb:STO substrate exhibited a photocurrent density of 5 A cm -2 [START_REF] Le Paven-Thivet | Photoelectrochemical properties of crystalline perovskite lanthanum titanium J o u r n a l P r e -p r o o f oxynitride films under visible light[END_REF], which was increased by a factor of 16 to 80 A cm -2 when iridium oxide (IrO2) was deposited on the LaTiO2N film. Moreover, here, no influence of the cationic composition on the photoresponse can be underlined. On the one hand, the Sr/Ta ratios of the STON films on Nb:STO and Ta are in the range 0.99 -1.02 and we can assume that they correspond to the stoichiometric value taking into account the EDS uncertainty. On the other hand, only one sample (TS = 650°C) of the series deposited on FTO (showing a significant variation of the Sr/Ta ratio as a function of the deposition temperature) has been used for the photoelectrochemical experiments. Anyway, for this sample, its weak polycrystalline character is a more prominent factor influencing the photoresponse than any other of its characteristics.

Figure 11 presents the responses of the STON films deposited on Ta substrates with three thicknesses, 150, 300, and 600 nm. The highest photocurrent density is reached on the thickest studied film (600 nm). All the measured films are polycrystalline, with similar

Lotgering factors (Table 1) regardless of the thickness, so the observed evolution is not primarily due to the crystallization. From Figure 6, the absorption coefficient showed notable low values in the 400-600 nm range (for example,  ~ 3000 cm -1 at  = 585 nm). This could be one of the key factors explaining the low measured photocurrent densities: despite having the proper bandgap (~2.35 eV) for absorption in the visible domain, the STON films exhibit a low absorbance in this range of interest, meaning a non-optimal capture of photons for the photochemical reactions. From this point, the higher photocurrent density obtained for the 600 nm-thick STON film could be explained by the effect of the absorbance that increases with the thickness of films. For higher thicknesses, one can assume a limit for which the recombination of the photoelectrons within the material occurs, resulting in a global reduction of the J o u r n a l P r e -p r o o f photoelectrochemical response. Finally, it is noted that the photoresponses start at more negative than 0 V vs. RHE, suggesting that the conduction band position is located more negative than hydrogen evolution potential, proving that STON is a potential candidate to achieve overall water splitting with a single semiconductor. This result is promising, and performance can likely be further improved by decorating electrocatalyst on its surface.

Conclusion

The present study of SrTaO2N (STON) films deposited by reactive magnetron rf sputtering identified some key optoelectronic characters for O2-evolving photoanode. Films approach stoichiometry when the substrate temperature increases to TS = 800°C, with a bandgap of 2.33 eV, a cation ratio Sr/Ta = 1.03, and a nitrogen content of 19.5 at % on FG substrate.

However, on FG and Au/Ti/FG, the crystallization of STON is weak with a polycrystalline nature. In contrast, c-axis oriented films are obtained on Ta and Nb:STO substrates. The PEC measurements show a promising very negative onset potential for photoresponse, suggesting that STON is a potential candidate to achieve overall water splitting using a single semiconductor. Three limiting factors can then be highlighted here: a low absorption coefficient, especially in the visible domain from 500 to 600 nm, the short lifetime of photoexcited charge carriers, and the moderate permittivity values (< 10) in the visible domain.

Among the deposited SrTaO2N, the highest photoresponse for O2 evolution under visible light was achieved for the 600 nm-thick sample deposited on Ta because of the dual effects of an oriented growth and a higher absorbance. J o u r n a l P r e -p r o o f

J o u r

  n a l P r e -p r o o f oven at 110°C. The obtained powder was pressed into a pellet using a uniaxial press, then calcined in air for 15 h at 1000°C, milled, and calcined again at 1400°C for 15 h, and checked for phase identification by X-Ray Diffraction (XRD, PANalytical X'Pert Powder). Once the Sr2Ta2O7 compound is synthesized, the powder is placed in a tubular oven under ammonia at 1000°C. The complete nitridation of the powder was conducted through a series of thermal treatments of 40 h each, controlled each time by XRD analysis to check if the STON compound is formed. If the nitrided powder still contains traces of the Sr2Ta2O7 oxide phase, it is nitrided for an additional 40 h. To nitride 40 g of powder necessary for the realization of the sputtering target, a total of 370 h of heat treatment has been necessary, as seen in Figure1giving the patterns after various nitridation steps. The elemental composition of the final powder was checked by energy dispersive spectroscopy (EDS, Oxford Aztek in a JEOL JSM IT100 SEM): this returned a Sr/Ta ratio of 1.01 and a nitrogen content of 20.5 at.%, which are close to expected values and, thus, confirms the synthesis of the SrTaO2N compound.

Films

  were deposited by (Ar + N2) reactive rf magnetron sputtering (Plassys MP450S) using the prepared STON target. Deposition conditions were the following: rf power PRF = 60 W, total pressure pT = 13 mTorr, and 25 vol% N2. Thicknesses were controlled by the sputtering time of the deposition. Three series of samples were deposited: the first series on FTO substrate J o u r n a l P r e -p r o o f (fluorine-doped tin oxide (F:SnO2) layer on soda-lime glass) by varying the substrate temperature TS from 300 to 650°C in order to analyze the formation of the oxynitride compound at a moderate temperature; the second series on fused glass (FG) substrate with different thicknesses to analyze the influence of the thickness on the optical, dielectric properties and lifetimes of photo-carriers; and the third series at a fixed substrate temperature (TS = 800°C) on different type of substrates in order to study the impact of the nature of the substrate (amorphous, single-crystal, metallic) on the properties of the deposited films, especially their crystallization and photoresponse.

Figure 9a depicts the

  Figure 9a depicts the absorbance change evolution as a function of the wavelength for

Figure 1 .

 1 Figure 1.  X-ray diffraction patterns for Sr2Ta2O7 powder transformed to SrTaO2N under

Figure 1 .

 1 Figure 1.  X-ray diffraction patterns for Sr2Ta2O7 powder transformed to SrTaO2N under nitridation in flowing NH3 at 1000°C (circled peaks belong to the Sr2Ta2O7 phase (JCPDS 72-0921); the top index corresponds to the SrTaO2N compound (JCPDS 79-1311)).

Figure 2 .

 2 Figure 2. X-ray diffraction patterns of STON films deposited on FTO substrates prepared

Figure 2 .

 2 Figure 2. X-ray diffraction patterns of STON films deposited on FTO substrates prepared at different substrate temperatures by reactive rf magnetron sputtering. Indexation is made according to the SrTaO2N compound. Asterisks indicate peaks belonging to the FTO/SG substrate.

Figure 3 .

 3 Figure 3. UV-visible transmittance spectra of STON films deposited on FTO substrates at

  r n a l P r e -p r o o f

Figure 3 .

 3 Figure 3. UV-visible transmittance spectra of STON films deposited on FTO substrates at different substrate temperatures by reactive rf magnetron sputtering.

Figure 4 .

 4 Figure 4. Evolution of the ratio Sr/Ta and the nitrogen content (N at.%) as a function of the

  r n a l P r e -p r o o f

Figure 4 .

 4 Figure 4. Evolution of the ratio Sr/Ta and the nitrogen content (N at.%) as a function of the substrate temperature for the STON films deposited on FTO substrates prepared by reactive rf magnetron sputtering (as comparison, is given in blue the data for the STON film deposited on Ta substrate at TS = 800°C).

Figure 5 .

 5 Figure 5. SEM surface morphologies of STON films deposited on FTO substrates at different

Figure 5 .

 5 Figure 5. SEM surface morphologies of STON films deposited on FTO substrates different substrate temperatures by reactive rf magnetron sputtering (substrate temperature is indicated in the photos).

Figure 6 .Figure 6 .

 66 Figure 6. Absorption coefficient of STON films with different thicknesses deposited on FG

Figure 7 .

 7 Figure 7. Permittivity (') and dielectric loss (tan) as a function of the photon energy of STON

Figure 7 .

 7 Figure 7. Permittivity (') and dielectric loss (tan) as a function of the photon energy of STON films deposited on FG substrate by reactive rf magnetron sputtering.

Figure 8 .

 8 Figure 8. (a) Absorbance change A for an excitation at 330 nm as a function of time delay for

Figure 8 .

 8 Figure 8. (a) Absorbance change A for an excitation at 330 nm as a function of time delay for the STON-150nm film on FG substrate, and (b) for STON-600nm film deposited on FG substrate. (c) Comparison of the TA of STON-150nm and STON-600nm at time delay +1ps.

J o u rFigure 9 .

 9 Figure 9. (a) Kinetics at 450 nm as a function of the delay and the thickness of STON films

Figure 9 .

 9 Figure 9. (a) Kinetics at 450 nm as a function of the delay and the thickness of STON films deposited on FG substrate and (b) kinetics at 550nm determined by femto pump-probe spectroscopy for an excitation at 330 nm.

Figure 10 .

 10 Figure 10.X-ray diffraction patterns of STON films deposited on different types of

J o u rFigure 10 .

 10 Figure 10. X-ray diffraction patterns of STON films deposited at TS = 800°C by reactive rf magnetron sputtering on different types of substrate: (a) FG, (b) Nb:SrTiO3, (c) Au/Ti/FG, (d) Ta. The pattern of the bare substrate is shown below each film diagram. Indexation of films is made according to the SrTaO2N compound. Asterisks indicate peaks belonging to the substrate.

Figure 11 .

 11 Figure 11. Linear scanning voltammetry measurements of STON films deposited on Ta

Figure 11 .

 11 Figure 11. Linear scanning voltammetry measurements of STON films deposited on Ta substrates with different thicknesses. Electrolyte: NaOH (pH 13), Reference Electrode: Ag/AgCl, Counter Electrode: Platinum, Light: 100 mW cm -2 .

Table 1 .

 1 List of STON film properties deposited on various substrates at different substrate temperatures by reactive rf magnetron sputtering.

	Substrate TS (°C) Thickness (nm)	Eg (eV)	Sr/Ta	(at.%) N	factor F L Lotgering	Crystallization
		300		5.10 1.13	26.0	/	Amorphous
		350		5.30 1.11	23.5	0.15	Polycrystalline
		400		2.50 1.10	23.0	0.17	Polycrystalline
	FTO	500	300	2.45 1.08	22.0	0.38	Polycrystalline
		550		2.40 1.06	23.5	0.41	Polycrystalline
		600		2.35 1.04	23.0	0.43	Polycrystalline
		650		2.35 1.05	23.5	0.47	Polycrystalline
		650	300	2.39 1.19	19.5	0.11	Polycrystalline
	FG	800 800	150 300	2.31 2.33 1.03 -a	-a 19.5	0.23 0.17	Polycrystalline Polycrystalline
		800	600	2.34 0.95	20.5	0.01	Polycrystalline
	Nb:STO 800	300	-	1.02	24.0	1.00	Fully textured
	Au/Ti/FG	800	300	-	1.01	21.0	0.33	Poorly crystallized
		800	150		-a	-a	0.44	Polycrystalline
	Ta	800	300	-	1.00	20.5	0.58	Polycrystalline
		800	600	-	1.01	20.5	0.57	Polycrystalline

J o u r n a l P r e -p r o o f

Acknowledgments

This work was supported by the European Union through the European Regional Development Fund (ERDF) and by the Ministry of Higher Education and Research, Brittany Région and Côtes d'Armor Département through the CPER Projects MATECOM and SOPHIE-STICC. This publication work was also supported by the Conseil Départemental des Côtes d'Armor, Saint Brieuc Armor Agglomération, and the Syndicat de Gestion du Pole Universitaire de Saint Brieuc (France). The authors thank Prof. Hicham Idriss and Dr. Habib Katsiev at SABIC corporation for providing the fs pump-probe spectrometer utilities and for the support in the measurements. J o u r n a l P r e -p r o o f

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

J o u r n a l P r e -p r o o f


Jean Zinn-Justin

RANDOM VECTOR AND MATRIX THEORIES: A RENORMALIZATION GROUP APPROACH

The study of the statistical properties of random matrices of large size has a long history, dating back to Wigner, who suggested using Gaussian ensembles to give a statistical description of the spectrum of complex Hamiltonians and derived the famous semi-circle law, the work of Dyson and many others. Later, 't Hooft noticed that, in SU (N ) non-Abelian gauge theories, tessalated surfaces can be associated to Feynman diagrams and that the large N expansion corresponds to an expansion in successive topologies.

Following this observation, some times later, it was realized that some ensembles of random matrices in the large N expansion and the so-called double scaling limit could be used as toy models for quantum gravity: 2D quantum gravity coupled to conformal matter. This has resulted in a tremendous expansion of random matrix theory, tackled with increasingly sophisticated mathematical methods and number of matrix models have been solved exactly. However, the somewhat paradoxical situation is that either models can be solved exactly or little can be said.

Since the solved models display critical points and universal properties, it is tempting to use renormalization group (RG) ideas to determine universal properties, without solving models explicitly. Initiated by Brézin and Zinn-Justin, the approach has led to encouraging results, first for matrix integrals and then quantum mechanics with matrices, but has not yet become a universal tool as initially hoped. In particular, general quantum field theories with matrix fields require more detailed investigations.

To better understand some of the encountered difficulties, we first apply analogous ideas to the simpler O(N ) symmetric vector models, models that can be solved quite generally in the large N limit. Unlike other attempts, our method is a close extension of Brézin and Zinn-Justin. Discussing vector and matrix models with similar approximation scheme, we notice that in all cases (vector and matrix integrals, vector and matrix path integrals in the local approximation), at leading order, non-trivial fixed points satisfy the same universal algebraic equation, and this is the main result of this work. However, its precise meaning and role have still to be better understood.

1 O(N ) symmetric vector models: simple integrals O(N ) symmetric vector models can be solved, quite generally, in the large N limit [START_REF] Moshe | Quantum field theory in the large N limit: a review[END_REF]. The large N behaviour and critical properties of O(N )-symmetric vector models can then be studied in the same spirit as matrix models but the analysis can be extended also to path and field integrals and analogous double scaling limits can then be exhibited [START_REF] Vecchia | Double scaling limit in O(N) vector models[END_REF][START_REF] Zinn-Justin | O(N) vector field theories in the double scaling limit[END_REF].

In the case of simple integrals, the geometric interpretation of critical points is related to the statistical properties of a class of continuous chains (obtained as limits of chains of bubble diagrams, see figure 1), of the form of branched polymers or filamentary surfaces, classified by the amount of stretching and the number of vertices (see figure 2).

Fig. 1 Vector models: The dominant diagrams in the large N limit. Fig. 2 Vector models: Vertices in the large N limit.

Here, we first recall how various O(N ) symmetric vector integrals can be calculated in the large N limit and exhibit multicritical points and scaling relations. We then try to recover the results by RG methods extending those used for matrix models [START_REF] Brézin | Renormalization group approach to matrix models[END_REF], in order to better understand the difficulties encountered in the latter case.

Zero dimension or simple integrals

We consider the integral

e Z N = N 2π N/2 d N x e -NV (x 2 ) , (1.1)
where x is an N -component vector and we first choose the normalization V (ρ) = ρ/2 + O(ρ 2 ). We assume that the function V (ρ) is analytic in a neighbourhood of the real positive axis, the simplest example being a polynomial.

To evaluate the integral, we first integrate over angles, which reduces the integral to

(ρ = x 2 ) e Z N = N dρ ρ e -Nσ(ρ) (1.2) with σ(ρ) = V (ρ) -1 2 ln ρ and N = N 2π N/2 π N/2 Γ(N/2) ∼ N→∞ N 4π e N/2 .
The large N limit can be determined by using the steepest descent method. The saddle point equation ρ c is given by

σ ′ (ρ c ) = 0 ⇔ 2V ′ (ρ c )ρ c = 1 . (1.3)
Expanding to σ(ρ) up order (ρρ c ) 2 and performing the Gaussian integration, one finds

Z N = N 1 2 -V (ρ c ) + 1 2 ln ρ c -1 2 ln 2ρ 2 c V ′′ (ρ c ) + 1 + O(1/N ).
For later purpose, it is convenient to introduce the function

R(ρ) = 1/2V ′ (ρ). (1.4)
The saddle point equation then reduces to

R(ρ) = ρ . (1.5)

Critical and multicritical points

Critical points [START_REF] Vecchia | Double scaling limit in O(N) vector models[END_REF][START_REF] Zinn-Justin | O(N) vector field theories in the double scaling limit[END_REF] correspond to situations where a number of derivatives of 2V ′ (ρ) -1/ρ vanish at the saddle point ρ = ρ c . At a (multi)critical point,

σ(ρ) -σ(ρ c ) ∝ (ρ -ρ c ) m , m ≥ 2 ,
and relevant values of ρρ c are of order N -1/m , which leads to non-trivial scaling properties. For m > 2, the fluctuations at the saddle point are no longer Gaussian.

Note that for m odd, the saddle point is only relevant if the integral is defined as a contour integral after some appropriate analytic continuation. A similar situation is found in matrix models. The problem will arise again in quantum mechanics and field theory and thus an analytic continuation will be, when necessary, implicitly assumed.

We normalize ρ such that the saddle point is located at ρ = m -1, m integer with m ≥ 2, and assume that the critical potential V c satisfies

2V ′ c (ρ)ρ -1 = -1 - ρ m -1 m-1 + O ((ρ -m + 1) m ) .
Then,

σ c (ρ) ≡ V c (ρ) -1 2 ln ρ = 1 2m 1 - ρ m -1 m + O (ρ -m + 1) m+1 .
Note that a strict equality is impossible since V (ρ) is regular at ρ = 0. Returning to the integral and setting z = 1ρ/(m -1), we obtain

e Z N ∝ dz exp -N z m 2m + O(z m+1 ) .
We see that the values of z = 1ρ/(m -1) contributing to the integral are of order N 1/m . A general relevant (in the RG sense) perturbation to the critical function V c has then the form

V q (ρ) = v q z q + O(z q+1 ), v q = 0 , 1 < q < m -1 , (1.6)
(q = m -1 corresponds to a translation of z) and the partition function Z N is given for N large and v q small by e

Z N ∝ dz exp -N z m 2m + N m-2 q=1 v q z q . (1.7)
Rescaling z into zN -1/m we see that the scaling region corresponds to take v q vanishing with N like N q/m-1 . For q > m, the perturbations are irrelevant and, at v q fixed, their contributions vanish like N q/m-1 . For m = 2, that is, for the Gaussian integral, all perturbations are irrelevant.

A renormalization group (RG) inspired strategy

Studying some specific properties of O(N ) vector models in the large N limit, we have exhibited critical points and scaling laws. We are thus reminded of the general theory of critical phenomena [START_REF] Wilson | The renormalization group and the ǫ expansion[END_REF], where critical properties can be derived using RG arguments without solving models explicitly. Even though in the large N limit, most vector models can be solved exactly, having in mind an application to matrix models, we try to recover the vector model results by simple-minded renormalization group considerations.

We expect that a change N → N + δN can be compensated by a change of the parameters of the model with the same continuum physics. We show here that this is the case, at least in some approximation scheme, at the expense of enlarging the space of coupling constants very much as in the Wilson's scheme of integration over the momenta in the shell Λ -dΛ < |p| < Λ and, which in the Wegner-Wilson form, leads to functional RG equations [START_REF] Wilson | The renormalization group and the ǫ expansion[END_REF]. Some aspects of this problem for simple integrals have been investigated previously, but in a different spirit in [START_REF] Higuchi | Exact beta functions in the vector model and renormalization group approach[END_REF]. Here, we are only interested in methods that can readily be generalized to path integrals and corresponding matrix integrals [START_REF] Brézin | Renormalization group approach to matrix models[END_REF], and this requires some approximation scheme.

For this purpose, we consider the O(N + 1) integral and integrate over only one variable:

e Z N +1 = N + 1 2π (N+1)/2 d N x dy e -(N+1)V (x 2 +y 2 ) .
For N → ∞, we can expand in powers of y. Keeping the leading term and integrating, we find

e Z N +1 = N + 1 2π N/2 d N x e -(N+1)V (x 2 )-1 2 ln 2V ′ (x 2 ) (1 + O(1/N )) ,
the leading relative correction being

- 3 8N V ′′ (x 2 ) V ′2 (x 2 ) .
This confirms that for this method to work V ′ (ρ) ∝ R -1 (ρ) must be a strictly positive function, a condition that we assume from now on.

In the spirit of the RG, we then renormalize the integration vector,

x → x 1 - 1 2N (1 + γ) ,
where γ is a free parameter. We infer,

Z N+1 (V ) = -1 2 γ + Z N (V + δV ) + O(N -1 ) with N δV (ρ) = V (ρ) -(1 + γ)ρV ′ (ρ) + 1 2 ln 2V ′ (ρ) + O(N -1
).

(2.1)

General flow equation

From the variation of the function V (ρ), we infer, for N → ∞,

N [Z N+1 (V ) -Z N (V )] ∼ N ∂ ∂N Z N (V ) = -N γ + dρ V (ρ) -(1 + γ)ρV ′ (ρ) + 1 2 ln 2V ′ (ρ) δ δV (ρ) Z N (V ), (2.2)
where for N large we have treated 1/N as a continuous parameter and γ is chosen such that

V (ρ) = 1 2 ρ + O(ρ 2 ) ⇒ γ = 2V ′′ (0).
Correspondingly, after a rescaling N → λN , we find for V the general flow equation

λ d dλ V (ρ, λ) = V (ρ, λ) -1 + γ(λ) ρV ′ (ρ, λ) + 1 2 ln 2V ′ (ρ, λ), (2.3)
where N → ∞ corresponds to λ → ∞. It is convenient to differentiate the flow equation with respect to ρ. In terms of the function

R(ρ, λ) = 1/2V ′ (ρ, λ), (2.4)
one obtains

λ d dλ R(ρ, λ) = γ(λ)R(ρ, λ) -1 + γ(λ) ρR ′ (ρ, λ) + R ′ (ρ, λ)R(ρ, λ) (2.5)
with R(0, λ) = 1 and, thus, γ(λ) = -R ′ (0, λ). The equation that will again appear in all more general problems.

Fixed points

To equation (2.3), corresponds the fixed point equation

V (ρ) -(1 + γ)ρV ′ (ρ) + 1 2 ln 2V ′ (ρ) = 0 .
We first discuss the equation in a perturbative and then linear approximation.

We then consider the full equation but conveniently expressed in terms of R = 1/2V ′ .

Critical point: Perturbative approximation

We first consider the simple example

V (ρ) = 1 2 ρ + 1 4 gρ 2 .
Since the contribution 1 2 ln 2V ′ (ρ) is no longer a polynomial, we assume that we can use a small ρ approximation and expand up to order ρ 2 :

1 2 ln 2V ′ (ρ) = gρ -1 2 g 2 ρ 2 + O(ρ 3 ).
The variation of V then reduces to a variation of g. Taking into account that γ = g, one finds

λ ∂g ∂λ = -β(g) with β(g) = g (1 + 3g) .
This equation determines two fixed points g * = 0 with β ′ (g * ) = 1, the attractive Gaussian fixed point, and a non-trivial repulsive fixed point g * = -1 3 with β(g * ) = -1. The exact critical point is g * = -1 4 and the exact scaling variable for m = 3 and q = 1 is N v 3/2 1 instead of here v 1 N , where v 1 ∝ gg * , in semi-quantitative agreement.

General fixed points: Linear perturbative approximation

We first examine the RG flow in a perturbative approximation: we assume that V is close to the Gaussian term and linearize the log term, substituting

ln 2V ′ (ρ) → 2V ′ (ρ) -1 .
In this approximation, the flow equation reduces to the linear equation

λ d dλ V (ρ, λ) = V (ρ, λ) + 1 -1 + γ(λ) ρ V ′ (ρ, λ) -1 2 ,
or, after differentiating with respect to ρ (R = 1/2V ′ ),

λ d dλ R(ρ, λ) = γ(λ)R(ρ, λ) + 1 -1 + γ(λ) ρ R ′ (ρ, λ).
The fixed point equation is simply

V (ρ) + [1 -(1 + γ)ρ]V ′ (ρ) -1 2 = 0
and the solution is

V (ρ) = 1 2 -1 2 [1 -(1 + γ)ρ] 1/(1+γ) .
Since V (ρ) must be a regular function, we conclude

γ = 1 m -1
with m a strictly positive integer and thus

V (ρ) = 1 2 -1 2 (1 -ρ/m) m .
The stability properties of the fixed point are determined by the eigenvalues of the operator obtained by varying the fixed point equation with respect to V . Defining then

Ω = 1 + (1 -ρ/m) d dρ ,
we can write the equation for an eigenvalue κ and eigenvector h as

Ωh = κh + 1 2 δγρ[1 -ρ/m] m-1 ,
where δγ is determined by the condition h ′ (0) = 0. Since the global normalization of h is arbitrary, we can choose δγ = 2/m. A special solution of the equation then is

h 0 = 1 κ (1 -ρ/m) m + m 1 -mκ (1 -ρ/m) m-1 .
The solution h 1 of the homogeneous equation is

h 1 = 1 -κ κ(1 -mκ) (1 -ρ/m) m(1-κ) .
The solution h = h 0 + h 1 must be regular at ρ = m and this implies

κ = 1 -p/m
with p a strictly positive integer. However, no regular solution exists for κ = 0 and κ = 1/m. The spectrum can be compared to the exact spectrum.

The value m = 1 corresponds to the Gaussian fixed point. All eigenvalues are negative and the fixed point is stable. Moreover, the approximation corresponds to linearizing the flow equation near the fixed point and, thus, the result is also valid for the full equation.

The value m = 2 corresponds to the first critical point. Then, κ = 1 is the only positive eigenvalue, corresponding to a direction of instability and all others κ = -1 2 , -1, -3 2 ... are negative. More generally, for generic m, there exists (m -1) positive eigenvalues corresponding to (m -1) directions of instability.

Saddle point evolution: A source of difficulties

We now return to the general flow equation (2.5). We assume that for the saddle point corresponds to the finite value ρ = ρ c (λ) and for ρ -

ρ c (λ) → 0, R(ρ, λ) = ρ + r p (λ) ρ -ρ c (λ) p + O ρ -ρ c (λ) p+1 . (3.1)
We set R = ρ + S, which leads to the equation

λ d dλ S(ρ, λ) = 1 + γ(λ) + S ′ (ρ, λ) S(ρ, λ) -γ(λ)ρS ′ (ρ, λ). (3.2)
We then set (p ≥ 1)

S(ρ, λ) = ∞ q=p r q (ρ -ρ c ) q .
Identifying the term of order (ρρ c ) p-1 in the flow equation, one obtains

λ d dλ ρ c = γρ c . Thus, ln ρ c (λ) = ln ρ c (1) + λ 1 dλ ′ γ(λ ′ ) λ ′ .
First, we conclude that, as expected, the critical behaviour is invariant under the RG flow. However, the location ρ c can have a finite non-vanishing limit only if γ(λ) → 0 for λ → ∞ and, thus as we will show, if the flow converges toward the Gaussian fixed point. By contrast if γ(λ) has a negative limit, then ρ c (λ) → 0, which is a boundary value; if it has a positive limit, ρ c (λ) → +∞. We show later that the limit of γ(λ) must be negative.

Full flow equation: Fixed points

In terms of the function R, the full fixed point equation reads

γR + RR ′ -(1 + γ)ρR ′ = 0 . (3.3)
Then, there are two cases: (i) γ = 0 and the equation reduces to

R ′ (R -ρ) = 0 .
It has two solutions R(ρ) ≡ const. = R(0), which corresponds to the Gaussian fixed point and R(ρ) = ρ, which is a trivial fixed point since in radial coordinates the integrand is then N -independent. The Gaussian fixed point has already been discussed in the framework of the linearized equation.

(ii) γ = 0. We then multiply the equation by R -1/γ-2 and it becomes a total derivative and can be integrated. For R(0) = 0 (the other case is trivial), normalizing R and ρ such that R(0) = 1, we obtain the integrated equation

R 1+1/γ (ρ) = R(ρ) -ρ . (3.4)
The function R has square root singularities. The location is obtained by imposing that R also satisfies the equation obtained by differentiating with respect to R:

(1 + 1/γ)R 1/γ = 1
and, thus,

R = γ 1 + γ γ , ρ = γ γ (1 + γ) 1+γ .
The condition that solutions V ′ (ρ) should be regular on the real positive axis implies γ < 0 and thus, according to the discussion of section 3.3, ρ c (λ) → 0.

Duality relation and series expansion

One can rewrite the integrated fixed point equation as

R 1/γ + ρR -1 = 1 .
Denoting by R(ρ, γ) the solution of the equation, one verifies the relation

R(ρ, γ) = ρR -γ (ρ 1/γ , 1/γ).
In particular, if R(ρ, γ) is positive and regular on the real positive axis, the same applies to R(ρ, 1/γ). Moreover, it is sufficient to study the domain γ ≤ -1 or -1 ≤ γ < 0.

The expansion of the solution in powers of ρ is

R(ρ) = -γ n=0 ρ n n! Γ n + γ(n -1) Γ 1 + γ(n -1) ⇒ R α = -αγ n=0 ρ n n! Γ n + γ(n -α) Γ 1 + γ(n -α) .
The behaviour of the series is consistent with the location and nature of the singularities already found.

Discussion

One verifies easily that the fixed point solution is such that the saddle point equation R(ρ) = ρ has no solution for ρ finite, a result that is consistent with the analysis of the flow of the saddle point. Indeed, since the function R has been assumed to be positive, R(ρ) > ρ, and the only possible saddle points are ρ infinite or ρ = 0. Since ρ can be eliminated, the only remaining possibility is ρ infinite. Asymptotically, R(ρ) ∼ ρ, the leading term of V thus cancelling the contribution of the ρ-measure and the ρ-integral does not converge.

Moreover, the leading correction is

R(ρ) -ρ ∼ ρ→∞ ρ 1/γ+1 .
The order of the fixed point is governed by 1/γ + 1. We seem to find a continuous set of critical points, while we know that the set is discrete. However, the interpretation of ρ = ∞ as a saddle point implies that the solution should be a regular function of 1/ρ at ρ = ∞. Only γ = -1/n satisfies the condition and the fixed point equation becomes polynomial:

R n (ρ) -ρR n-1 (ρ) -1 = 0 .

One can then linearize the flow equation at the fixed point and determine the corresponding eigenvalues.

A few special solutions

With the ansatz γ = -1/n, the location of singularities is given by

ρ n = - n n (n -1) n-1 .
The solutions are monotonously increasing for ρ > 0. For complex solutions, if R(ρ) is a solution, ωR(ρ/ω) with ω n = 1 is a solution. For the first values of n, the equation can be solved explicitly. For example, for n = 1, 2, 3, respectively,

R = 1 + ρ , R = 1 2 ρ + ρ 2 + 4 , R = 1 27 ρ 3 + 1 2 + 1 2 1 + 4 27 ρ 3 1/3 + ρ 2 9 1 27 ρ 3 + 1 2 + 1 2 1 + 4 27 ρ 3 1/3 + ρ 3 .

Non-linear renormalization

With a simple rescaling of the integration variable ρ, it is impossible to generate a non-Gaussian fixed point corresponding to a saddle point at a finite value of ρ. Non-linear changes of variables are required:

ρ → ρ - 1 N h(ρ), h(0) = 0 .
The variation (up to a possible constant) of the function V is then

N δV (ρ) = V (ρ) -h(ρ)V ′ (ρ) + 1 2 ln 2V ′ (ρ) + O(N -2 ). (3.5)
The fixed point equation for

R(ρ) = 1/2V ′ (ρ) then reads 1 -h ′ (ρ) R(ρ) -R ′ (ρ) R(ρ) -h(ρ) = 0 .
If we impose the saddle point position ρ = 1, we have the constraint

1 -h ′ (1) -R ′ (1) 1 -h(1) = 0 .
We can then invert the process and, given a fixed point candidate, determine from the equation a function h(ρ). It is obtained by solving

-Rh ′ + R ′ h + R -RR ′ = 0 .
which integrated yields

h(ρ) = R(ρ) ρ 0 dρ ′ R(ρ ′ ) -ln R(ρ) + const. ,
where the integration constant is determined by the condition h(0) = 0. Therefore, provided one chooses as a fixed point function R(ρ), a function that does not vanish for ρ ≥ 0, one can find a a suitable change of variables. However, it is not clear how to guess a suitable non-linear change of variables a priori in the more general situation we examine later.

Quantum mechanics

We now generalize the preceding analysis to path integrals. We consider the free energy given by the path integral

e Z N = [dx(t)] e -S(x) , (4.1)
where S is an O(N ) invariant Euclidean action of the form

S(x) = N dt 1 2 ẋ2 (t) + V x 2 (t) , (4.2)
for example,

V (x) = 1 2 x 2 + g 4 x 2 2 .
We could discuss this problem using radial coordinates from the Schrödinger equation point of view. Because eventually we want to extend the analysis to field theory we use a more general method. The large N expansion can be obtained from standard techniques (for a review see, for example, Ref. [START_REF] Moshe | Quantum field theory in the large N limit: a review[END_REF]). To determine the large N limit, one introduces two paths λ(t) and ρ(t) and adds to the action

1 2 N dt λ(t) x 2 (t) -ρ(t) .
The integration over the Lagrange multiplier λ(t) imposes the constraint ρ(t) = x 2 (t) and one can replace V (x 2 ) by V (ρ). The action then takes the form

S(x, λ, ρ) = N dt 1 2 ẋ2 (t) + V ρ(t) + 1 2 λ(t) x 2 (t) -ρ(t) .
After the Gaussian integration over x, the path integral becomes e Z N ∝ [dλ(t)dρ(t)] e -S(λ,ρ)

with

S(λ, ρ) = N 2 dt -λ(t)ρ(t) + 2V ρ(t) + tr ln -d 2 t + λ . (4.3)
The dependence on N of the partition function is now explicit. In the large N limit the path integral can be calculated by the steepest descent method. We look for two constants λ, ρ solutions of

2V ′ (ρ) -λ = 0 , ρ = 1 2π dω ω 2 + λ = 1 2 √ λ .
Eliminating λ between the two equations, we obtain

8ρ 2 V ′ (ρ) = 1 ,
an equation identical to the zero dimension equation, R(ρ) = ρ, if we now set

V ′ (ρ) = 1 8R 2 (ρ) . (4.4)
We expect a critical point when the equation has a double root. The condition is R ′ (ρ) = 1 .

One verifies that the equation expresses that the determinant of second partial derivatives of the action for constant paths vanishes. Critical points are then associated with the vanishing of the determinant of the second functional derivatives at the saddle point. The matrix of second derivatives in the Fourier representation is

M = N 2 2V ′′ (ρ) -1 -1 -B where B(ω) = 1 2π dω ′ (ω ′2 + λ) (ω -ω ′ ) 2 + λ .
In particular, B(0

) = 1 4λ 3/2 .
Then, using the saddle point equations, one finds

det M(ω = 0) = -1 4 N 2 1 + 2V ′′ (ρ) B(0) = -1 4 N 2 [1 -R ′ (ρ)] .
Diagonalizing the matrix M, we infer that for a linear combination µ of λ and ρ, the potential vanishes in the N → ∞ limit. The path corresponding to the second eigenvector is not critical in this sense (in one dimension a phase transition is impossible) and can be integrated out.

The µ-path effective action

The resulting action is non-local but can be expanded in powers of µ and derivatives. Rescaling time and path,

t → αt , µ → const. × (α/N ) 1/2 µ ,
we can then look for a scaling limit [START_REF] Zinn-Justin | O(N) vector field theories in the double scaling limit[END_REF]. If we relate α and N by α = (N z) 1/5 , z fixed we find that all terms in the action with more than two derivatives or powers of µ higher than three vanish with N . The action then reduces to

S(µ) = 1 2 dt μ2 (t) + √ zµ 3 (t) .
More generally, we can adjust the potential V in such a way that the coefficients of all interactions up to the µ m-1 vanish. The scaling limit corresponds then to choose α = (N z) (m-2)/(m+2) and thus µ → µN -2/(m+2) .

Correspondingly, the effective action in the large N limit becomes

S(µ) = 1 2 dt μ2 (t) + z (m-2)/2 µ m (t) .
Relevant perturbations correspond to adding to the action in the initial normalization a term N v q dt µ q (t).

A non-trivial scaling limit is then obtained by choosing v q ∝ N -2(m-q)/(m+2) .

By contrast, note that in matrix models the situation is more complicate because logarithmic deviations from a simple scaling law are found.

RG inspired method

Like in the case of the simple integral, we now integrate over only one component of the path. We start from the action with (N + 1) components (x, y),

S N+1 (x, y) = (N + 1) dt 1 2 ẋ2 + V (x 2 ) + 1 2 ẏ2 + V ′ (x 2 )y 2 + O(y 4 ) ,
where we have assumed

V (ρ) = 1 2 ρ + O(ρ 2
). In the large N limit, the integral over y reduces to a Gaussian approximation and (with a suitable normalization of the path integral) one obtains

S N+1 (x) = (1 + 1/N )S N (x) + 1 2 W + O(1/N ) with W = tr ln -d 2 t + 2V ′ (x 2 ) .
Immediately we face a new problem, the contribution W to the action is not local. Then we rescale x,

x → x(1 -(1 + γ)/2N ),
even though we have seen in the zero-dimensional example that a linear rescaling may lead to difficulties. We then obtain the variation of the action

N δS N (x) = dt -1 2 γ ẋ2 + V (x 2 ) -(1 + γ)x 2 V ′ (x 2 ) + 1 2 W .
We still have to perform a time renormalization to normalize the coefficient of ẋ2 to 1 2 :

t → t(1 -γ/N ).
This yields an additional contribution -γV :

N δS N (x) = dt (1 -γ)V (x 2 ) -(1 + γ)x 2 V ′ (x 2 ) + 1 2 W .
We can then determine γ by the condition δV ′ (0) = 0. A first way to solve the locality issue is to expand W at leading order in V ′ -1,

W = W 0 + 1 2 dt V ′ (x 2 ) -1 2 + O (V ′ -1 2 ) 2 ,
where W 0 is a constant but the next term in such an expansion breaks locality. The local approximation. More globally, from the preceding analysis, we expect that, for N large, a local expansion (which is also a large V expansion) should be meaningful because the fluctuations of x 2 are small. Then,

tr ln -d 2 t + 2V ′ (x 2 ) ∼ dt 2V ′ (x 2 ), (4.5)
the leading correction being

dt (x • ẋ) 2 V ′′2 (x 2 ) 2 [2V ′ (x 2 )] 5/2 .
The flow equation then becomes a simple scalar equation for the function V (ρ) of the form

N ∂ ∂N V (ρ) = (1 -γ)V (ρ) -(1 + γ)ρV ′ (ρ) + 1 2 2V ′ (ρ) -1 , (4.6)
where the constant is adjusted to enforce consistency for ρ = 0. Differentiating with respect to ρ and setting V ′ (ρ) = 1/8R 2 , one obtains

N ∂ ∂N R(ρ) = γR(ρ) -(1 + γ)ρR ′ (ρ) + R(ρ)R ′ (ρ), (4.7)
an equation identical to equation (2.5) after the change N → N λ.

Fixed point equation

Like in the example of the simple integral, we first consider the perturbative and also linear approximation. Alternatively, we then use the first term of the local expansion of W .

The perturbative approximation: Fixed points

The fixed point equation then reads

(ρ = x 2 ) 2(1 -γ)V (ρ) + [1 -2(1 + γ)ρ]V ′ (ρ) -1 2 = 0 . The solution is V = 1 4(1 -γ) 1 -1 -2(1 + γ)ρ (1-γ)/(1+γ) .
The regularity condition implies

γ = 1 -m 1 + m ⇒ V = 1 + m 8m 1 -1 - 4ρ m + 1 m ,
where m is a strictly positive integer.

The perturbative approximation: Eigenvalues

We define the operator

Ω = 4m m + 1 + 1 - 4 m + 1 ρ d dρ .
Denoting by κ the eigenvalue and h the eigenvector, we find the eigenvalue equation

Ωh = κh + 2δγ 1 + m 8m 1 + (m -1) 1 - 4ρ m + 1 m + m 1 - 4ρ m + 1 m-1
, where δγ is determined by the condition h ′ (0) = 0. The solution h 1 of the homogeneous equation is

h 1 = 1 - 4ρ m + 1 m-κ(m+1)/4
.

The regularity condition implies κ = 4 mq m + 1 .

The local expansion

In the local approximation, we reduce the problem essentially to the example of the simple integral. The fixed point equation now reads

(1 -γ)V (x 2 ) -(1 + γ)x 2 V ′ (x 2 ) + 1 2 2V ′ (x 2 ) = 0 .
Setting ρ = x 2 and differentiating the fixed point equation with respect to ρ, we obtain

-2γV ′ (ρ) -(1 + γ)ρV ′′ (ρ) + V ′′ (ρ) 2 2V ′ (ρ) = 0 .
We then set V ′ = 8/R 2 and the equation becomes indeed identical to the equation for the simple integral:

γR(ρ) + R ′ (ρ)R(ρ) -(1 + γ)ρR ′ (ρ) = 0 .
We do not have to repeat the discussion of the simple integral. The only modification is the implication of the fixed point solution for V . For example, since for ρ → ∞, R(ρ) ∼ ρ then V (ρ) goes to a constant with as a leading correction -1/8ρ.

Statistical field theory

The analysis leading to the double scaling limit can be generalized to statistical field theories in two dimensions with generic potentials and in three dimensions to theories with a sextic potential [START_REF] Zinn-Justin | O(N) vector field theories in the double scaling limit[END_REF].

From the discussion of the quantum mechanical case, we have already some intuition about the main difference with d = 1. The critical limit corresponds to having one bound state associated with the composite φ 2 field becoming massless, (the φ-field itself remaining non-critical) that is, to a phase transition of an Ising-like system. For d = 1 a phase transition is impossible while it is possible in higher dimensions. We now show how this difference shows up technically, first in the special case of the (φ 2 ) 2 interaction. We consider the partition function

Z N = [dφ(x)] e -S(φ) ,
where S(φ) is a O(N ) symmetric action:

S(φ) = N 1 2 [∂ µ φ(x)] 2 + V φ 2 (x) d d x (6.1)
with as simplest example

V (ρ) = 1 2 rρ + 1 4 gρ 2 .
A cut-off of order 1, consistent with the symmetry, is implied. More precisely we have written explicitly in action (6.1) only the two first terms of the inverse propagator in a local expansion (in Fourier space small momentum) expansion.

In particular, the parameter r is defined as the value of the inverse propagator at zero momentum. Following the method explained in section 4, we introduce two fields λ(x), ρ(x), add to the action (6.1),

1 2 N d d x λ(x) φ 2 (x) -ρ(x)
and, accordingly, replace V (φ 2 ) by V (ρ). We then perform the Gaussian integration over φ and find

Z = [dλ(x)dρ(x)] e -S(λ,ρ) with S(λ, ρ) = N 2 d d x -λ(x)ρ(x) + 2V ρ(x) + tr ln -∆ x + λ .
In the large N limit the field integral can be calculated by the steepest descent method.

The large N limit

We look for a uniform saddle point λ(x) = λ, ρ(x) = ρ. The saddle point equations are

-λ + 2V ′ (ρ) = 0 , -ρ + 1 (2π) d Λ d d p p 2 + λ = 0 . (6.2)
where we have introduced a cut-off Λ since otherwise the integral is divergent for d ≥ 2. With a cut-off and for d < 4, the second equation becomes

-ρ + 1 (4π) d/2 Γ(1 -d/2)λ d/2-1 + Λ d-2 C(d) = 0 ,
where the pole at d = 2 of C(d) cancels the pole of the Γ-function. We call ρ 0 (d) the cut-off dependent term, set

K(d) = Γ(1 -d/2) (4π) d/2 , (6.3)
and introduce the function

R(ρ) = K(d)[2V ′ (ρ)] d/2-1 . (6.4)
Using the first equation (6.2), the equation can then be rewritten as

ρ -ρ 0 = R(ρ). (6.5)
The critical point is defined by

0 = 1 -R ′ (ρ) = 1 + 2V ′′ (ρ) 1 (2π) d d d q (q 2 + λ) 2 . (6.6)
The determinant of the matrix of the second derivatives of the action at the saddle point,

M = N 2 -∆(p) -1 -1 2V ′′ (ρ) with ∆(p) = 1 (2π) d d d q (q 2 + λ) (p + q) 2 + λ , is proportional to 1 + 2V ′′ (ρ) ∆(p).
The criticality condition (6.6) thus implies that the determinant vanishes at p = 0 and the propagator of a linear combination µ of the ρ and λ fields,

µ(x) = ρ(x) -ρ + 2V ′′ (ρ) λ(x) -λ ,
has a pole at zero momentum. The field µ(x) becomes massless while the other component and also the φ-field remain massive.

The scaling limit in field theory

We have determined the partition function in the large N limit. We now look for a scaling limit [START_REF] Zinn-Justin | O(N) vector field theories in the double scaling limit[END_REF]. Since the µ field is a one-component field, it can remain critical for d > 1 even in presence of interactions. We have to examine the most IR divergent terms in perturbation theory. We face a standard problem in the theory of critical phenomena. The deviation ε of a parameter in V from its critical value plays exactly the role of a deviation from the critical temperature.

Two dimensions. We first examine dimension 2. The effective action for the µ-field is non-local and contains arbitrary powers of the field. However, because the other fields are not critical, we can again make a local expansion. Standard arguments of the theory of critical phenomena tell us that the most IR divergent terms come from interactions without derivatives and with the lowest power of the field. Here the leading interaction is proportional to µ 3 . To characterize the IR divergences of the perturbative expansion in powers in 1/N we rescale distances and field µ,

µ(x) → µ(x)N -1/2 , x → Λx,
where Λ plays the role of a cut-off. The effective action at leading order, after some additional finite renormalizations, is

S eff. (µ) = d 2 x 1 2 (∂ µ µ) 2 + 1 2 εΛ 2 µ 2 + Λ 2 6 √ N µ 3 ,
where a cut-off Λ is again implied. If d < 4 the theory is super-renormalizable. We want the coefficient of µ 3 to have a limit and we thus set

Λ 2 / √ N = u.
In contrast with quantum mechanics, we cannot just fix the product εΛ 2 = uε √ N because the perturbative expansion is not finite in the large N , large cut-off Λ limit. Instead, we have to introduce a counter-term which renders µ(x) finite. A short calculation yields the relation

ε = 1 u √ N 1 2 µ 4 + u 2 16π ln N u 2 /m 4 ,
in which m is a renormalized mass parameter. We have shown that a scaling limit exists which leads to a renormalized µ 3 field theory. However, the relation between ε and N has itself no longer a simple scaling form.

Higher dimensions. The situation in higher dimensions is similar, though slightly more complicate, as long as the theory is super-renormalizable, because more counter-terms are required. Naive scaling would predict a scaling variable N ε (6-d)/2 . However, due to counter-terms, ε does not go to zero as fast as naively expected. We perform here the general analysis not for the µ 3 theory but rather for the more interesting µ 4 theory, which can only been obtained from φ interactions depending on more parameters.

Multicritical points. The problem can be dealt with by the method explained in the case of quantum mechanics. The results are very similar. Because only one mode is critical, one can introduce additional parameters in the effective interaction of the critical mode in such a way that the most IR relevant terms can be cancelled. In the language of critical phenomena we reach multicritical points. We generate then renormalized λ p interactions provided we again choose the relation between ε and N to cancel the UV divergences of perturbation theory. In two dimensions the relation, with standard normalizations, is

Λ 2 = N u, ε = 1 N u m 2 + 1 8π ln(N u/m 2 ) .
In dimension 2 < d < 4 the relation becomes more complicate because more counter-terms are needed. Note that at leading order for N large

ε ∼ Λ d-2 N ∼ N -(4-d)/2 ,
while naive scaling would indicate

ε ∼ N -2/(4-d) .
In dimensions d > 4 perturbation theory is no longer IR divergent and, therefore, no scaling limit can be found.

RG inspired strategy

We now generalize the strategy used in the quantum mechanics case (d = 1) to field theory. Again we evaluate the determinant generated by the integration over one component in the local approximation. We start from the action (6.1) for (N + 1) components and integrate over only one component in the large N limit:

S N+1 (φ, χ) = (N + 1) d d x 1 2 [∂ µ φ(x)] 2 + V φ 2 (x) + d d x 1 2 ∂ µ χ(x) 2 + χ 2 (x)V ′ φ 2 (x) + • • • (6.7)
Performing the integration over χ in the Gaussian approximation, one finds the action

(1 + 1/N )S N (φ) + 1 2 tr ln[-∆ x + 2V ′ (φ 2 )] + O(1/N ).
In the local approximation, the determinant can be evaluated and one finds

tr ln[-∆ x + 2V ′ (φ 2 )] ∼ 2K(d) d d d x 2V ′ φ 2 (x) d/2 , (6.8)
where K(d) has been defined in equation (6.3). In even dimensions, an additional cut-off dependent contribution cancels the pole of the Γ-function. For d < 4, it is proportional to V ′ (φ 2 ) and for d = 4 it is quadratic in V ′ (φ 2 ). Adding to expression (6.8) the divergent part, one obtains

d d x K(d) d 2V ′ φ 2 (x) d/2 + ρ 0 (d)V ′ φ 2 (x) .
We now introduce a field renormalization

φ(x) → φ(x) 1 -(1 + γ)/N .
Omitting the cut-off dependent term, the variation δS N of the action then becomes

N δS N (φ) = d d x -1 2 γ(∂ µ φ) 2 + V (φ 2 ) -(1 + γ)φ 2 V ′ (φ 2 ) + K(d) d 2V ′ (φ 2 ) d/2 .
A space rescaling is then required to normalize the coefficient of the kinetic term to 1 2 :

x → x 1 + γ (d -2)N ,
a form that shows that d = 2 has to be examined separately. The final form of the variation of the potential then reads

N δV (ρ) = (1 + γd/(d -2))V (ρ) -(1 + γ)ρV ′ (ρ) + K(d) d [2V ′ (ρ)] d/2 + ρ 0 V ′ (ρ).
Differentiating with respect to ρ, one obtains

N δV ′ (ρ) = 2γ d -2 V ′ (ρ) -(1 + γ)ρV ′′ (ρ) + V ′′ (ρ) K(d)(2V ′ (ρ)) d/2-1 + ρ 0 .
In terms of the function (6.4),

δR R = (d -2) δV ′ 2V ′ and V ′′ V ′ = 2R ′ (d -2)R .
The variation equation for R then reads

δR = γR -(1 + γ)ρR ′ + (R + ρ 0 )R ′ , (6.9)
which leads to an equation identical to equation (2.5) up the divergent addition. Fixed point equation. For γ = 0, we then multiply the fixed point equation

γR -(1 + γ)ρR ′ + (R + ρ 0 )R ′ = 0
by R -1/γ-2 and integrate:

ρ - ρ 0 1 + γ -R(ρ) = R 1/γ+1 (ρ)R -1/γ-1 (0).
We find the same equation as in previous examples, up to a simple shift of ρ.

The equation is no longer singular at d = 2 but a more detailed study of the d = 2 limit is required.

The Hermitian one-matrix model and random surfaces

The matrix model representation of two-dimensional quantum gravity [START_REF] David | Planar diagrams, two-dimensional lattice gravity and surface models[END_REF] has led to explicit solutions for minimal conformal fields coupled to gravity [START_REF] Brézin | Exactly solvable field theories of closed strings[END_REF][START_REF] Brézin | Scaling violation in a field theory of closed strings in one physical dimension[END_REF][START_REF] Ph | For a review on early developments in random matrix theory and 2D quantum gravity see e[END_REF]. However, this approach has not been of real help for understanding the difficulties which arise when the central charge c of the matter field is larger than one although it is very easy to write matrix models for those cases as well.

We note that the central result of matrix models is the existence of a 'double scaling limit' that is, a continuum limit with critical exponents that describe how the coupling constants of the theory have to be tuned to reach this limit. These simple scaling laws for c < 1, with logarithmic deviations at c = 1 are quite reminiscent of the theory of continuous phase transitions for space dimensions d ≤ 4. There, information about the critical behaviour is obtained without solving models explicitly but constructing a renormalization group [START_REF] Wilson | The renormalization group and the ǫ expansion[END_REF]. This is a strategy we want to use again here, extending the approach of reference [START_REF] Brézin | Renormalization group approach to matrix models[END_REF] and in the spirit of the method we have explained in the vector model example in section 2.

The one-matrix models

We consider an integral over N × N Hermitian matrices M of the form e Z(N) = dM e -N tr V (M ) , where V is a polynomial, for example,

V (M ) = 1 2 M 2 + 1 3 gM 3 .
The formal expansion of Z(N, g) in powers of g generates a sum of connected fat Feynman diagrams (see figure 3) whose dual are connected triangulated surface (see figure 4). Giving a unit area to all triangles, one concludes that the term of order g n is the sum of surfaces of area n. Moreover, Z(N, g) has the large N expansion

Z(N, g) = N 2 Z 0 (g) + Z 1 (g) + 1 N 2 Z 2 (g) + • • • = h N 2-2h Z h (g),
where Z h (g) is the sum of all surfaces of genus h.

The continuum limit

The quantity Z 0 (g) can be calculated by the steepest descent method. One then finds a critical value g c of g where Z 0 (g) is singular,

Z 0,sing. (g) ∝ (g -g c ) 2-γ .
The specific heat exponent γ is called string susceptibility in the string terminology. At this point the average surface area A diverges:

A = n = ∂ ln Z 0,sing. (g) ∂g ∝ 2 -γ g -g c ,
allowing to define a continuum limit by shrinking the size of the polygons to zero.

The double scaling limit

More generally, it can be shown that Z h (g) diverges at the same critical value like

Z h (g) ∝ (g c -g) (2-γ)(1-h) .
This allows defining a double scaling limit in which N → ∞ in a correlated way with g → g c . Setting

κ -1 = N (g -g c ) (2-γ)/2
one finds, in the double scaling limit, a sum over continuum surfaces with increasing genus h:

Z(N, g) ∼ Z(κ) = h=0 κ 2-2h f h . (7.1)
In the double scaling limit, the partition function can be calculated explicitly by various techniques, for example, using orthogonal polynomials [START_REF] Ph | For a review on early developments in random matrix theory and 2D quantum gravity see e[END_REF].

Multicritical points and central charge

Up to now we have assumed a simple model in which V (M ) depends only on one parameter. If V (M ) depends on several parameters, like in the theory of critical phenomena one can tune some of them to reach multicritical points with different values of γ and scaling partition functions. The continuum Liouville theory predicts a relation between γ and the central charge c that labels the unitary discrete series of conformal 2D field theories, c = 1 -6/m(m + 1):

γ = 1 12 c -1 -(1 -c)(25 -c) = -1 m .

The renormalization group approach

Again, instead of trying to solving models exactly, we want to construct a renormalization group. If we are able to construct such a renormalization group, we know that the scaling laws and the exponents, which characterize the double scaling limit, will follow automatically. Therefore, we have to understand how the coupling constants of the theory, -in the simplest example the string coupling constant and the cosmological constant mapped, respectively, onto the size N of the matrices and some matrix parameter,-evolve under a rescaling of the regularization length introduced in the triangulation of the world sheet.

We expect that a change N → N + δN can be compensated by a change of the matrix parameter with the same continuum physics. We show that for matrix models this is indeed the case [START_REF] Brézin | Renormalization group approach to matrix models[END_REF] (see also Ref. [START_REF] Higuchi | Renormalization group approach to multiple-arc random matrix models[END_REF]), at least in some approximation scheme.

Construction of flow equation: Gaussian integration

In order to study the existence at once of critical and multicritical points within one-matrix models, we immediately allow for an arbitrary analytic even potential depending on an N × N matrix φ N matrix,

V (φ N , g k ) = N tr V (φ N ) ≡ N ∞ 1 g k 2k tr φ 2k N (7.2)
with g 1 = V ′′ (0) = 1 and we set g 2 = V (4) (0)/6 = g. We then consider the matrix integral

I N (g) = dφ N e -N tr V (φ N ) .
To calculate the integral over (N + 1) × (N + 1) matrices φ N+1 , we parametrize the matrix φ N+1 in terms of an N × N submatrix φ N , a complex N -component vector v a , and a real number α:

φ N+1 = φ N v a v * a α . (7.3)
One verifies easily that all the terms involving α are of relative order 1/N and can be dropped at leading order; we thus can set α = 0. Expanding the polynomial V in powers of v, we obtain

tr V (φ N+1 ) = tr V (φ N ) + v * V ′ (φ N )φ -1 N v + O |v| 4 .
In the large N limit, at leading order we truncate the expansion at the quadratic order in v, integrate over v to find the new action

V (φ N ) + δV (φ N ) = (N + 1) tr V (φ N ) + tr ln V ′ (φ N )φ -1 N .
After rescaling of the matrix to enforce the condition δV ′′ (0) = 0,

φ N = ζφ ′ N with ζ = 1 - g + 1/2 N + O 1 N 2 ,
the variation of the function V becomes (µ is a real variable and 1/N is considered as a continuous variable)

N ∂ ∂N V (µ) ≡ N δV (µ) = V (µ) -g + 1 2 µV ′ (µ) + ln V ′ (µ)/µ . (7.4)

Perturbative approximation: the gravity exponent

We know from the exact solution that the one-matrix φ 4 model is sufficient to describe, near its critical point, pure gravity (c = 0). It consists of the integral over an N × N Hermitian matrix φ N ,

I N (g) = dφ N exp [-N tr V (φ N , g)]
with an 'action'

V (φ N , g) = tr 1 2 φ 2 N + 1 4 gφ 4 N . (7.5)
The double scaling limit is then reached in the limit g → g * with

g * = -1/12 and thus V * (µ) = 1 -(1 -µ 2 /2m) m .
First, m = 1 corresponds to the Gaussian fixed point. For m = 2 one finds µ 2 /2-µ 4 /16, instead of the exact µ 2 /2-µ 4 /48; for m = 3, µ 2 /2-µ 4 /12+µ 6 /216 instead of µ 2 /2µ 4 /12 + µ 6 /180. With this simple approximation, we find the right sequence of multicritical points with their property of being alternatively unbounded below for even m's and bounded for odd m's.

For the m-th multicritical point, critical exponents are related to the eigenvalues at the fixed point of the operator obtained by varying δV with respect to V . We define

Ω = 1 + 1 µ - µ 2m d dµ .
Denoting by κ an eigenvalue, we have to solve the eigenvalue equation

Ωh = κh + δgµ 2 (1 -µ 2 /2m) m-1 ,
where δg is determined by the condition that the eigenvectors h satisfy h ′′ (0) = 0. A special solution to the equation is

h 0 (µ) = 2mδg 1 κ (1 -µ 2 /2m) m + m 1 -mκ (1 -µ 2 /2m) m-1 .
The solution h 1 of the homogeneous equation such that h 1 (0) + h 0 (0) = 0 is

h 1 (µ) = - 2mδg κ(1 -mκ)
(1µ 2 /2m) -m(κ-1) .

The function h 1 must be regular for µ 2 = 2m and thus

κ = 1 - p m ,
where p is a positive integer, p ≥ 1. In general the values κ = 1/m and κ = 0 are excluded because the regular solution h 1 + h 0 vanishes. For m = 1, κ < 0 and the Gaussian fixed point is stable. For m = 2, the only value is κ = 1 and we recover the direction of instability. In general, one finds (m -1) eigenvalues corresponding to relevant perturbations. By tuning (m -2) parameters, one can select the eigenvector corresponding to the smallest eigenvalue 2/m. For the m-th multicritical point, instead of the exact value 3/2m, one thus finds γ = 2 + 2/(-2/m) = 2m, in good qualitative and semi-quantitative agreement with the exact answer (here we have calculated the opposite of the eigenvalues of the usually defined β(g)-functions).

General calculation

We now keep the exact fixed point equation as obtained after Gaussian integration: V (µ) -(g + 1 2 )µV ′ (µ) + ln V ′ (µ)/µ = 0 . More generally, for ρ → ∞ and g < 0, one finds f ∼ 1/ρ ⇒ V (µ) ∼ 2 ln µ. A first problem with this solution, is that it seems to depend on a continuous parameter g while we expect a discrete spectrum of multicritical points.

Moreover, the behaviour for large matrices is such that the matrix integral no longer converges, potential and measure being of same order.

We also set f = 1/R. We then find the fixed point equation

RR ′ + 2gR -(1 + 2g)ρR ′ = 0 . (7.8)
Remarkably enough, we find an equation that is identical to the equation (2.5) that appears in vector models and seems to be a universal feature of this kind of approximations when combined with a local approximation in higher dimensions. We do not need to discuss it again here.

In the case of matrix models, for a still improved determination of critical and multicritical points, one has presumably to introduce more general potentials, that is, with products of traces, and special parametrizations or non-linear matrix transformations as the study of vector models as indicated.

Conclusion

From the analysis of the vector model by RG methods, it seems that at least two problems have to be handled: non-linear transformations of the integration variables to generate fixed points with suitable properties and, for path or field integrals, local expansion of the contribution to the action of the determinant generated by the partial Gaussian integration. These problems seem to arise also in matrix models. But in matrix models, potentials with product of traces will have to be introduced and this will generate additional difficulties.

Finally, it is quite remarkable that in this RG inspired method, in the large N limit and within the local approximation scheme, only one (one-parameter) fixed point equation arises both in matrix and different vector models and this point, together with the meaning of the equation, remains to be better understood.

Fig. 3

 3 Fig. 3 Hermitian matrix propagator. Hermitian matrix three-point vertex.

Fig. 4 tr

 4 Fig. 4 tr M 3 vertex and triangulated surface.

(7. 6 ) 1 .

 61 Fixed point solution. Setting (this assumes V is even)f (ρ = µ 2 /2) = V ′ (µ)/µand differentiating the fixed point equation, one findsf ′ (ρ) = 2gf 2 (ρ) + (1 + 2g)ρf (ρ)f ′ (ρ) (7.7) with f (0) =Multiplying the equation by f 1/2g-1 , one can integrate and obtains the algebraic equationf 1/2g = ρf 1+1/2g + 1 .Example: for g = -1/2, the equation has the simple solution f = 1/(1 + ρ) ⇒ V (µ) = ln(1 + µ /2).

and the susceptibility exponent is γ = -1/2 .

In the RG approach, to reduce the flow equation to a one parameter flow, we expand the equation in powers of µ, a purely numerical approximation. To order µ 4 , the equation reduces to

The β-function has two zeros corresponding to two fixed points: g * = 0 is the attractive Gaussian fixed point and g * = -1/6 is the non-trivial repulsive fixed point since

Indeed, the pure gravity exponents are obtained only when one 'tunes' the cosmological constant g near its critical value g * . In the exact theory the value of g * is -1/12, in contrast to our first approximation -1/6. Since at this order β ′ (g * ) is equal to -1, the calculation yields for the string susceptibility exponent [START_REF] Brézin | Renormalization group approach to matrix models[END_REF]

to be compared with the exact value -1/2 .

Linear approximation

We give here only a brief discussion of multicritical points, in this RG approach, based on assuming a small deviation from the Gaussian model.

We use an approximation of small deviation from the Gaussian model replacing the determinant contribution by (this is at variance with has been done in the first example) by the linear deviation:

The fixed point equation corresponding to equation (7.4) reduces to

Integrating we find

Since V * must be regular function (and assuming g + 1 2 > 0), we infer

Path integral

We consider now the Euclidean action

where V (φ) has the form (7.2).

The problem of quantum mechanics with large size matrices has also been discussed within other RG approaches, in [START_REF] Alfaro | The D = 1 matrix model and the renormalization group[END_REF][START_REF] Dasgupta | Renormalization Group Approach to c=1 Matrix Model on a circle and D-brane Decay[END_REF]. Field theory has also been considered [START_REF] Nishigaki | Wilsonian approximated renormalization group for matrix and vector models in 2 < d < 4[END_REF].

The discussion now combines the arguments given for the vector path integral with the matrix integral.

The action S N+1 expressed in terms the parametrization (7.3), becomes

After integration over the vector v in the Gaussian approximation, we find

We then rescale φ and t:

in such a way that the coefficient of φ2 becomes again 1 2 . In the local approximation (4.5), the variation of the action takes the form

This leads to a scalar flow equation involving the function V (µ):

Differentiating with respect to µ, we obtain

2V ′ (µ) .

and setting V ′ (x) = 1/2R 2 , one recovers the universal equation (2.5).