Manju Ananya Chattaraj

Mishra Patidar

V Ganesan

Sébastien Joulie

Virginie Serin

Alain Claverie

Vijay Kumar

Aloke Kanjilal
email: aloke.kanjilal@snu.edu.in

Manju Mishra Patidar

Atomic scale microstructural insights of superconducting β-tungsten thin films

Keywords: β-phase tungsten, thin films, microstructure, disorder, superconductivity

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INTRODUCTION

The α phase of tungsten (W) with body-centered cubic (BCC) structure is a well-known type-I elemental superconductor with a critical superconducting transition temperature (Tc) of 0.01 K. [START_REF] Gibson | Superconductivity of tungsten[END_REF] But its allotrope β-W with A15 structure has received overwhelming interest because of its exotic superconducting behavior that includes a higher Tc of ~ 3-4 K and its sustainability at higher upper critical magnetic fields (Hc2). [START_REF] Sengupta | Superconducting nanowires by electron-beam-induced deposition[END_REF][START_REF] Kondo | Superconducting characteristics and the thermal stability of tungsten-based amorphous thin films[END_REF][START_REF] Sun | Voltage-current properties of superconducting amorphous tungsten nanostrips[END_REF][START_REF] Charlton | Allotropes of tungsten[END_REF][START_REF] Basavaiah | Superconductivity in β-Tungsten Films[END_REF] There is a long-standing interest in materials with the A15 structure such as Nb3Ge for high magnetic field dependent applications and especially for their usage in superconducting magnets. [START_REF] Xu | A review and prospects for Nb3Sn superconductor development[END_REF][START_REF] Devantay | The physical and structural properties of superconducting A15-type Nb-Sn alloys[END_REF] Given the A15 structure, β-W is also promising, although its structural stability is considered to be dictated by the presence of impurities such as oxygen. [START_REF] Vink | Stress, strain, and microstructure in thin tungsten films deposited by dc magnetron sputtering[END_REF][START_REF] O'keefe | Phase transformation of sputter deposited tungsten thin films with A-15 structure[END_REF][START_REF] Shen | Influences of oxygen on the formation and stability of A15 β-W thin films[END_REF][START_REF] Lee | Growth and characterization of α and β-phase tungsten films on various substrates[END_REF][START_REF] Choi | Phase, grain structure, stress, and resistivity of sputter-deposited tungsten films[END_REF] Earlier, superconductivity has been observed in β-W films by tunneling method and the structure has been analyzed by X-ray diffraction (XRD). [START_REF] Basavaiah | Superconductivity in β-Tungsten Films[END_REF] In another study an enhancement in the value of Tc was reported in W-based amorphous thin films by incorporating 5 to 70 at.% metalloids, with a parallel verification of their thermal stability. [START_REF] Kondo | Superconducting characteristics and the thermal stability of tungsten-based amorphous thin films[END_REF] Attempts have also been made to understand the effects of structural instability on superconductivity due to the exposure of A15 superconducting compounds to high-energy laser. [START_REF] Testardi | Destruction of superconductivity by laser light[END_REF] Kammerer and Strongin [START_REF] Kammerer | Superconductivity in tungsten films[END_REF] have also studied superconductivity in W films prepared by (i) electron beam (at a pressure of 2×10 - 6 Torr) and (ii) sputter (in argon atmosphere) deposition techniques. They found a maximum Tc of ~4.1 K in a film of about 200 nm thickness and also measured the critical magnetic field.

The structure of these films was examined by both XRD and electron microscopy. From these measurements they separated out the effects of strain and particle size. However, no conclusion was established on the observed line broadening. Two possible reasons were proposed for the higher Tc in these films: (i) the presence of surface oxygen and the existence of oxide metal interface, and (ii) the presence of disorder, [START_REF] Kammerer | Superconductivity in tungsten films[END_REF] although these were not so clearly manifested to understand the underlying superconductivity. In another study, an enhancement of superconductivity related parameters in W films was also found and it was discussed in the light of structural changes [START_REF] Chopra | Enhancement of superconductivity in tungsten films[END_REF] Superconductivity has also been reported 2 in low dimensional structures, such as electron beam grown W nanowires, though it is vulnerable to ion beamproduced defects. Majkova et al. [START_REF] Majkova | Superconductivity and critical fields in amorphous tungsten/silicon multilayers[END_REF] further studied the superconducting properties of amorphous W/Si multilayers, where they demonstrated an almost linear rise in Tc with the increasing number of layers. There are some other studies based on the superconducting fluctuations in amorphous WSi. [START_REF] Surick | Growth of amorphous tungsten silicide and study of the proximity effect at low dimensions for superconducting applications[END_REF][START_REF] Zhang | Superconducting fluctuations and characteristic time scales in amorphous WSi[END_REF] As superconductivity in low dimensional systems is believed to be somewhat ruled by the growth conditions, a clear understanding of the microstructure including the local atomic structure is very essential not only for a fundamental understanding but also for future applications, especially for oxygen-induced β-W. Although, the discovery of β-W was made a long time ago, [START_REF] Charlton | Allotropes of tungsten[END_REF][START_REF] Hägg | β-Tungsten'as a tungsten oxide[END_REF][START_REF] Morcom | The preparation and characterization of beta-tungsten, a metastable tungsten phase[END_REF] an understanding of the formation and stability of this phase has been quite challenging. It has been found that oxygen plays a crucial role in stabilizing β-W. [START_REF] Sun | Voltage-current properties of superconducting amorphous tungsten nanostrips[END_REF][START_REF] Shen | Influences of oxygen on the formation and stability of A15 β-W thin films[END_REF][START_REF] Lee | Growth and characterization of α and β-phase tungsten films on various substrates[END_REF] The role of oxygen in this phase formation has been recently discussed in detail by a combined experimental and theoretical study [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF] . The solution energy of O in β is significantly higher compared to α and this leads to the stabilization of the β phase with oxygen doping. [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF] The presence of oxygen leads to disorder in the local atomic structure and this is further enhanced with an increasing number of dopants (>20 at.%). [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF][START_REF] Chattaraj | Crucial role of oxygen on the bulk and surface electronic properties of stable β phase of tungsten[END_REF] Therefore, it is difficult to recognize and explain the observed superconductivity in β-W truly without having a detailed understanding of the elemental distribution and their influence on the atomic structure.

In the present study, we have measured the exotic superconductivity of β-W thin films and analysed it in the light of the microstructure of the films for the purpose of not only applications but also to achieve a fundamental understanding. Several films were grown by electron beam deposition technique with varying film thickness. Among them, two best quality films have been used for further low-temperature measurements, where it has been found that Tc and Hc2(0) vary with film thickness. Also, based on the atomic structure we found a Tc (1.38 K) which does not belong to either α-W or β-W, and has not been found earlier. This value of Tc is very useful for developing photon number resolving transition-edge sensors for astronomical and quantum information applications as well as for usage in a cryogenic particle detectors. [START_REF] Lita | Tuning of tungsten thin film superconducting transition temperature for fabrication of photon number resolving detectors[END_REF][START_REF] Irwin | Tungsten thin films for use in cryogenic particle detectors[END_REF] The phase formation and corresponding structures of the deposited films with two different thicknesses have been studied in detail through grazing incidence XRD (GIXRD) and transmission electron microscopy (TEM) investigations. The structures become disordered due to the presence of O which is revealed from the energy dispersive X-ray spectroscopy (EDX), [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF][START_REF] Chattaraj | Crucial role of oxygen on the bulk and surface electronic properties of stable β phase of tungsten[END_REF] and further confirmed from ab initio calculations. Using these results and observations, the effect of the atomic structure in the underlying superconductivity of β-W is well recognized.

EXPERIMENTAL METHODS

A 500 µm thick Si (100) wafer was diced into pieces of area 1×1 cm 2 and cleaned by dipping in trichloroethylene, acetone, isopropanol, and deionized water followed by heating for 2 min in each step. After chemical cleaning of the Si substrate, W films were grown at room temperature (RT) by electron beam deposition technique with a deposition rate of 0.01 nm/s. The chamber base pressure was ~5 ×10 -7 Torr and it was maintained at ~1×10 -6 Torr during deposition. Around 99.95% pure W powder (Alfa Aesar) was used to deposit W thin films. The thickness of the deposited films was estimated by a surface profilometer (DektakXT, Bruker).

The evolution of the β-W phase was confirmed by GIXRD using Cu-Kα radiation (λ = 0.154 nm) from a D8-Discover (Bruker) instrument. The detailed microstructural investigation was carried out by TEM in both plan-view and cross-sectional geometries using a Cs corrected Tecnai-FEI system operated with an acceleration voltage of 200 kV. The electrical transport measurements down to 2K and 14T magnetic fields were carried out by conventional four probe measurements with a dc current of 100 μA using 14T/2K PPMS system from Quantum Design USA. For measurements below 2K, a dilution refrigerator insert from the same Quantum Design was used. This was equipped with an ac resistance measurement configuration. A drive frequency of 12.2 Hz with the same current level in four probe configuration was employed. Typical current densities used are 280 A/cm 2 for a low thickness (~35 nm) film and 114 A/cm 2 for higher thickness (~60 nm) film.

CALCULATION DETAILS

The atomic structure of β-W with and without oxygen was studied from ab initio calculations.

We performed simulated annealing with ab initio molecular dynamics on bulk β-W having different concentrations of oxygen. The atomic structures were fully optimized. The details of the ab-initio calculations have been given in Ref. [22]. The electronic densities of states, which are influenced by the atomic structure have been also calculated and analyzed with and without the doping of oxygen in β-W. [START_REF] Chattaraj | Crucial role of oxygen on the bulk and surface electronic properties of stable β phase of tungsten[END_REF]

RESULTS

Atomic Structure

We considered two film thicknesses of ~35 nm (W35) and ~60 nm (W60) in this study. The GIXRD patterns of W35 and W60 films in the range of 2 from 10 o to 80 o are shown in Fig. 1. As discerned, the broad peak centered at ~40 o in W35 creates an ambiguity in A15 -W phase identification (JCPDS file #00-047-1319) as it signifies the presence of a disordered structure. Such a broad reflection is associated with the formation of a disordered -W phase when the oxygen concentration is increased above 25 at.%. [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF] However, the characteristic reflections from the (200), (210), and (211) planes of -W (A15 structure) are identified in W60 film having lower (~16 at.%) oxygen concentration [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF] compared with the W35 film. It should be noted here that the -W(110) reflection (JCPDS#04-0806) is expected to overlap with the (210) peak due to their close proximity. Another weak and broad peak with its maximum at ~70 o is found in W35 film, but three different reflections can be distinguished in W60 film, namely (222), (320), and (321) (JCPDS file #00-047-1319). It can be seen in Fig. 1 that the peak corresponding to (210) for W35 film shifts to slightly lower 2θ value compared to the one in W60. This is in agreement with the results from simulated oxygen doped W bulk which show an increase in the lattice parameter with an increase in oxygen concentration. The sharp peak at ~52 o (indicated by *) is the reflection from the Si substrate. The strain and average crystallite size in the film have been calculated from the GIXRD data by using Williamson-Hall (W-H) equation,

b cos= 0.9λ/D + 4ε sinθ, ( 1
)
where b is the full-width at half maximum (FWHM) of a diffraction peak, ε is the strain, [START_REF] Chattaraj | Probing the impact of surface reactivity on charge transport in dimensional phase changed tungsten films[END_REF][START_REF] Purushotham | X-ray determination of crystallite size and effect of lattice strain on Debye-Waller factors of platinum nano powders[END_REF] and D is the average crystallite size. The W-H equation basically gives a straight line, from where we can extract the average crystallite size and strain from the intercept and slope, respectively. The diffraction peaks for both W35 and W60 are fitted in a standard software and their FWHMs are extracted. Using six distinct diffraction peaks in W60, the W-H equation is employed for determining  and D (see Fig. S1 in Supporting Information). Because of the availability of only a couple of broad peaks (absence of sufficient numbers of reflections), the W-H equation is not suitable for W35 film. Instead, we have used the Scherer equation for determining the average crystallite size,

D = 0.9λ/(bcosθ) (2)
and the associated strain ε = b/(4 tanθ). [START_REF] Ooi | X-ray line profile analysis of BaTiO3 thin film prepared by sol-gel deposition[END_REF] The results are summarized in Table 1. In this case, the instrumental broadening is corrected by using the formula b = [(bM) 2 -(bI) 2 ] 0.5 where bM and bI are the FWHMs of the measured and standard peaks, respectively. Here, the strain is found to be increased with decreasing crystallite size. Therefore, the observed broadening in the XRD peaks is due to the size effect of the crystallites, especially the absence of long-range order with small (~1.5 nm) crystallites in W35 film.

Table 1. Average crystallite size (D) and strain () determined from the recorded GIXRD patterns of W35 and W60 films. spacing is marked by yellow lines. As -W phase is found to be stable in the presence of oxygen, the films were further diagnosed by EDX in cross-sectional mode. The measurements have been done in both ways; along the surface (in different depths) and perpendicular to the surface. [START_REF] Chattaraj | Unravelling oxygen driven α to β phase transformation in tungsten[END_REF][START_REF] Chattaraj | Crucial role of oxygen on the bulk and surface electronic properties of stable β phase of tungsten[END_REF] In any case, a nearly uniform distribution of O throughout the film was manifested, and a typical EDX image in Fig. 2(e) shows the superimposition of the signals from W, O, and Si elements across the film. One can see from Fig. 2(e) that Si did not diffuse in the W film, but oxygen is found to be distributed all over. The average atomic concentration of oxygen across the film is found to be reduced from ~20 at.% to ~16 at.% by increasing film thickness from 35 nm to 60 nm. atomic ordering is also noticed in W35, as can be realized from the inset in Fig. 3(b). In contrary, big crystallites (> 20 nm) are formed in W60 film, which is also in good agreement with our GIXRD result (Fig. 1 and Table 1). A drastic change in atomic ordering is also observed from the inset of the lower panel of Fig. 3(b). The increase in the average crystallite size in W60 film is further reflected from the selected area electron diffraction (SAED) patterns in Fig. 3(c). The bright spots in the polycrystalline rings of the recorded SAED pattern of W60 are the signature of the formation of relatively bigger crystallites. Moreover, the rings are fitting well with the -W phase according to JCPDS file #03-065-6453. [START_REF] Petroff | Microstructure, growth, resistivity, and stresses in thin tungsten films deposited by rf sputtering[END_REF] On the other side, the SAED of W35 film exhibits a bright ring surrounded by several diffused rings. Here, the arrays of bright spots superimposed on these rings show the reflections from the Si substrate.

Considering the bright ring and its adjacent diffused rings into account, the calculated interplanar spacing (d) values of ~0.201 nm, 0.229 nm, and 0.247 nm suggest reflections from the (211), (210), and (200) planes of the partially crystalline -W grains. Since the d spacing values of (110) and (210) are ~0.223 nm and 0.225 nm, it is impossible to discard the existence of the former one. [START_REF] Girault | Controlled nanostructuration of polycrystalline tungsten thin films[END_REF] The analysis of the remaining rings is, however, not straightforward because of their diffusive nature arising from poor atomic ordering in small grains. However, the approximate d spacings for the remaining rings has been estimated to be ~0.142 nm, 0.129 nm, and 0.114 nm, which can be related to the reflections from the (222)/(320), (321)/(400), and (420)/(421) planes, respectively. Similarly, reflection from the (211) planes is also expected with  (321)/(400). All the rings are indexed accordingly for clarity. The evolution of -W phase was also examined by ab initio molecular dynamics calculations based on density functional theory. We studied doping of 10 to 40 at.% oxygen in the supercells of αand -W. The lattice parameters as well as the ionic positions were fully relaxed after simulated annealing run. It was found that the favorable position of oxygen is the tetrahedral interstitial site (TIS) to sit in the A15 structure. The corresponding cohesive energy, lattice parameters, bond lengths, etc. have been optimized and discussed in detail in Ref. [22]. It is revealed that the lattice parameters are slightly stretched by increasing oxygen doping concentration and this lead to a disordered structure. However, the structure remains intact for comparatively low O doping cases. From the calculations, we found that the degree of disorder depends not only on the O doping concentration, but also on the distribution of oxygen in the lattice. Clustering of oxygen was found to be favorable and therefore in a sample with less than ~15 at.% O concentration, there is likely to be a coexistence of α and β phases. This can be also recognized from the GIXRD and TEM results of film W60.

Low temperature resistivity

To verify the superconducting behavior of the W films, low temperature electrical resistivity (ρ) was measured with and without a magnetic field. The data in the absence of the external magnetic field for both W35 and W60 films are shown in Fig. 5. The superconducting transition temperatures are found to be ~3.10 K and ~1.40 K for W35 and W60 films, respectively. This thickness dependent transition in the resistivity is the major finding in the present study. In order to understand this intriguing phenomenon, we further explored the behavior of resistivity under a magnetic field.

Resistivity under magnetic field

The resistivity has also been measured with an applied magnetic field and the corresponding dρ/dT has been also calculated for further analysis. These are shown in Figs. 6( In the case of W35, one can see from Fig. 6(a) that ρ starts to fall from 3.9 K (TPP) itself while it drastically drops below 3.33 K (Ton, the onset of superconductivity) and it completely vanishes below 2.92 K (Tc0). The corresponding mid-point of the transition or mean field transition temperature (Tm) is ~3.14 K. We should note here that the Tm is also denoted by the peak in the dρ/dT vs T plot [see Fig. 6(b)]. As the transition in the critical region is almost sharp, therefore here it is considered that Tm ~ Tc, superconducting critical transition temperature. It is also clear from Fig. 6(a) that the Tc gradually shifts to lower values with increasing magnetic field. As discerned from Fig. 6(b), a sharp peak is visible at ~3.14 K for zero magnetic field, while a gradual enhancement in transition width is found with increasing magnetic field. Moreover, the critical magnetic field with respect to temperature is also displayed in the inset and the observed linear behavior will be discussed in the following. As in the case of the W35 film, Tc is found to shift towards lower values with increasing magnetic field in W60 film also [see Fig. 7(a)]. Moreover, a sharp peak is visible at ~1.38 K in Fig. 7(b) for zero magnetic field. This is the Tc for the W60 film. The corresponding transition points as well as the temperatures related to various stages of transitions for the W35 and W60 films are listed in Table 2. The superconducting transition width taken from dρ/dT is found to be almost the same (see Table 2), signifying a sharp transition as in conventional superconductors. However, more broadening is seen in W35 film as compared to W60 due to the disordered structure. Likely in W35 film, the critical magnetic field is also found to be following a linear trend with temperature for the W60 film [inset, Fig. 7(b)].

Critical magnetic field

The Hc2(0) value determined from the variation of Hc2 as a function of T [inset, Fig. 6(b)] for the W35 film is estimated to be ~10.72 T by extrapolating the fitted data. Interestingly, the temperature variation of Hc2 is found to be a linear function rather than the one expected from the conventional BCS theory,

𝐻 𝑐2 = 𝐻 𝑐2 (0) [1 -( 𝑇 𝑇 𝑐 ) 𝑡 ] (3)
Using this equation, the fitting process gives the value of t as 1.07. Using the Werthamer-Helfand-Hohenberg (WHH) equation

𝐻 𝑐2 (0) = -0.693𝑇 𝑐 ( 𝑑𝐻 𝑐2 𝑑𝑇 ) 𝑇 𝑐 (4)
we find the value of Hc2(0) to be ~8.35 T for the W35 film. The temperature variation of Hc2 is also plotted in the inset of Fig. 7(b) for the W60 film, where Hc2(0) is determined to be ~2.34

T by extrapolating the fitted data. In this case the WHH estimate yields the value of Hc2(0) to be ~1.64 T.

Coherence length

The coherence length  of W35, a type-II superconductor, is estimated to be ~5.3 nm from

𝜉 2 = 𝜙 0 2𝜋𝐻 𝑐2 (0) (6)
where the quanta of flux 𝜙 0 = ℎ𝑐 𝑒  2.067 × 10 -15 Wb. [START_REF] Dodaro | Generalization of Anderson's theorem for disordered superconductors[END_REF] Here, the calculated  is found to be higher than the average crystallite size of ~1.47 nm in the W35 film as listed in Table 1.

However, an estimate of the coherence length  for the W60 film turns out to be 11.83 nm, which is less compared to the projected average crystalline grain size of ~22.84 nm.

DISCUSSION

Our results show that the W35 film has a long-range disordered structure yet organized microstructure with a higher internal strain and Tc0 ~ 2.92 K. However, comparatively ordered β phase with lower strain provides a Tc0 ~1.18 K. These results point towards the involvement of microstructure and associated strain in controlling Tc, in accordance with the prior studies. [START_REF] Chen | Strain-Controlled Superconductivity in Few-Layer NbSe2[END_REF][START_REF] Pogrebnyakov | Enhancement of the Superconducting Transition Temperature of MgB2 by a Strain-Induced Bond-Stretching Mode Softening[END_REF][START_REF] Chandra | Internal strain induced superconductivity in arc melted Ti0.97Fe0.03 alloy[END_REF][START_REF] Ahadi | Enhancing superconductivity in SrTiO3 films with strain[END_REF][START_REF] Herrera | Strainengineered interaction of quantum polar and superconducting phases[END_REF] In fact, Pogrebnyakov et al. [START_REF] Pogrebnyakov | Enhancement of the Superconducting Transition Temperature of MgB2 by a Strain-Induced Bond-Stretching Mode Softening[END_REF] reported an increase in Tc by introducing epitaxial tensile strain in MgB2. Precipitation induced appearance of internal strain was also proposed to justify the increase in Tc. [START_REF] Chandra | Internal strain induced superconductivity in arc melted Ti0.97Fe0.03 alloy[END_REF] Chloe Herrera et al. [START_REF] Herrera | Strainengineered interaction of quantum polar and superconducting phases[END_REF] further discussed the strain engineered appearance of superconducting phases in SrTiO3. However, in most of the cases, an epitaxial layer was only considered to explain the role of strain on superconductivity. In the present case also, a higher Tc (~3 K) has been detected in the W35 film in the presence of a relatively higher strain (see Table 1) in granular structure. Since the strain can affect lattice vibrations and heat capacity, [START_REF] Ma | Strain effect on lattice vibration, heat capacity, and thermal conductivity of graphene[END_REF][START_REF] Wu | Strain engineering and lattice vibration manipulation of atomically thin TaS2 films[END_REF] it is expected to modulate phonon dispersion and hence the phonon density of states. As a matter of fact, it has a direct impact on Tc according to the standard Bardeen-Cooper-Schrieffer (BCS) theory. [START_REF] Choi | Enhancing the critical temperature of strained Niobium films[END_REF][START_REF] Mata-Pinzón | Superconductivity in bismuth. A new look at an old problem[END_REF] However, the atomic structures in the present case, especially the atomic ordering and average grain size and shape are the decisive factors for controlling the superconducting phenomenon in the W35 and W60 films. The effect of disorder on Tc has also been reported in a few theoretical studies. [START_REF] Gastiasoro | Enhancing superconductivity by disorder[END_REF][START_REF] Dodaro | Generalization of Anderson's theorem for disordered superconductors[END_REF] One such study emphasized that the anisotropy in the order parameter of a superconductor plays an important role in its response to impurities, where both the inter-band and intra-band impurity scattering processes are considered. 41` The grains are quite small in the case of the W35 film (see Table 1). They are especially smaller than the obtained coherence length (5.3 nm) and therefore the volume of a grain is lesser than the coherence volume. It characterizes the existence of a conventional superconductivity. However, the number of Cooper pairs is not so large like in a conventional BCS superconductor as the number of grains is small (~2) within a coherence length in this case. Under this circumstance, two consecutive grains are separated by an insulating medium, mostly due to the involvement of oxygen at the grain boundaries (see Fig. 3). This intermediate thin oxide medium ultimately participates in electron tunnelling from one grain to another and in turn takes part in changing the physical property of the entire system. Even, depending on this narrow tunnelling medium the electronic coupling would be either strong or weak and subsequently control Tc. It is somewhat similar to the Mott-type metal-insulator transition. [START_REF] Deutscher | The Role of the Short Coherence Length in Unconventional Superconductors[END_REF] A similar explanation was proposed by Kammerer and Strongin in an earlier study. [START_REF] Kammerer | Superconductivity in tungsten films[END_REF] However, due to the lack of the structural analysis, it was not established there.

Additionally, a long-range disorder also exists in the W35 film. It is well known that the disordered type superconductors follow the Anderson type transition. [START_REF] Gastiasoro | Enhancing superconductivity by disorder[END_REF][START_REF] Dodaro | Generalization of Anderson's theorem for disordered superconductors[END_REF][START_REF] Deutscher | The Role of the Short Coherence Length in Unconventional Superconductors[END_REF] However, in this case the disorder is not homogeneous as the presence of short-range crystallites cannot be ignored. Based on the microstructural analysis, we can instead infer that the granular superconductivity is dominated in the W35 film as compared to only disorder-based transition. This is quite similar to the case superconductivity in granular Al. [START_REF] Moshe | From orbital to Pauli-limited critical fields in granular aluminum films[END_REF] Whereas in the W60 film, the volume of the average crystalline grain is higher than the coherence volume, even of the order of 2𝜉. Therefore, this can be considered as an unconventional superconductor. In fact,

26

 26

Figure 1 .

 1 Figure 1. Typical GIXRD patterns of W35 and W60 films. Different peaks in the case of W60

  achieve a comprehensive knowledge of the structural evolution with increasing W film thickness, further investigations have been made by TEM. A typical bright-field cross-sectional TEM (XTEM) image of W35 is displayed in Fig.2(a), showing the formation of self-assembled columnar structure. A close inspection indicates the appearance of a very thin natural oxide (SiOx) layer at the W/Si interface, while high-resolution (HR) XTEM image in Fig.2(b) further confirms its thickness of ~3 nm. One can further distinguish the formation of a disordered W film, and this continues up to the surface [see Fig.2(c)]. However, it shows some dark patches (highlighted by yellow dashed curves) along with short-range atomic ordering in the adjacent regions (marked by downward arrows). This is again reflected in the plan-view TEM (PVTEM) image [see Fig.2(d)]. Here, few (200) planes are evidenced, where a typical inter-planar

Figure 2

 2 Figure 2 (a) The bright-field XTEM image of W35 in low magnification, showing the formation

Figure 3

 3 Figure3shows a comparison of the microstructure features in W35 and W60 films. In

Figure 3 .

 3 Figure 3. (a) High-resolution PVTEM and (b) XTEM images of W35 (top row) and W60

Figure 4 (

 4 Figure 4(a) displays a perfectly aligned plane in -W along the [001] zone axis with a

Figure 4 .

 4 Figure 4. (a) A square lattice structure is found in W60 in HRTEM image, whereas (b) shows

Figure 5 .

 5 Figure 5. Temperature variation of resistivity for β-W films at zero magnetic field.

  a) and (b), respectively, for the W35 film while the same for the W60 film are presented in Figs. 7(a) and (b).

Figure 6 .

 6 Figure 6. (a) Temperature variation of  of β-W35 film under an applied magnetic field up to

Figure 7 .

 7 Figure 7. (a) Temperature variation of  for W60 film for applied magnetic fields up to 16 T.

Table 2 :

 2 Various indicators of the transition temperatures for the W35 and W60 films.

	Sample	TPP (K)	Ton (K)	Tm (K)	Tc0 (K)
	W35	3.90	3.33	3.14	2.92
	W60	2.09	1.53	1.38	1.18

due to the short coherence length, tunnelling of electrons between the grains is difficult in this case, and in turn Tc gets reduced. The upper critical magnetic field has also been found to be anisotropic and is dependent on film thickness. This signifies the deviation from the standard 3D BCS formalism. It even also deviates from the predictions for the temperature dependence of Hc2 calculated by WHH. This is due to the corrections needed in the standard BCS formalism to describe superconductivity in a two-dimensional superconductor, as shown by Tinkham, especially for thin films. [START_REF] Tinkham | Effect of fluxoid quantization on transitions of superconducting films[END_REF] Tinkham 45 also discussed the importance of inhomogeneities in the superconducting films for achieving unusual behaviour in the presence of impurities by impeding the free movement of flux quanta. Considering these facts, we believe from the present experimental work that the structural modification and the associated strain, and the corresponding effect in microstructures with varying oxygen concentration as a function of film thickness led to the observed superconductivity.

SUMMARY

In summary, our results show that the Tc is higher (~3.14 K) for the low thickness (W35) film, whereas it reduces to ~1.38 K in the thicker (W60) film. In the former case there is a longrange disorder associated with a higher strain. The orderliness in the structure is basically controlled by the oxygen concentration as supported by our ab initio calculations. Though there is no significant change in the transition width, a slight increase in field induced resistive broadening in the W35 film is due to the presence of disorder. The analysis of the results led to two main understanding: (i) the impact of internal strain on the critical transition temperature Tc and superconductivity in β-W thin films in the absence of external magnetic field. In particular, the strain has been shown to be released by reducing the mean oxygen concentration from ~20 to ~16 at.% with increasing film thickness from 35 nm (W35) to 60 nm (W60) (ii)

The average grain size and the grain boundaries along with the involved atomic structures are found to be responsible for the origin of type-II superconductivity. However, it requires further theoretical analysis to understand the underlying superconductivity in β-W films.


Xhek Turkeshi

Marcello Dalmonte

Rosario Fazio

Marco Schirò

Entanglement Transitions from Stochastic Resetting of Non-Hermitian Quasiparticles

We put forward a phenomenological theory for entanglement dynamics in monitored quantum many-body systems with well-defined quasiparticles. Within this theory entanglement is carried by ballistically propagating non-Hermitian quasiparticles which are stochastically reset by the measurement protocol with rate given by their finite inverse lifetime. We write down a renewal equation for the statistics of the entanglement entropy and show that depending on the spectrum of quasiparticle decay rates different entanglement scaling can arise and even sharp entanglement phase transitions. When applied to a Quantum Ising chain where the transverse magnetization is measured by quantum jumps, our theory predicts a critical phase with logarithmic scaling of the entanglement, an area law phase and a continuous phase transition between them, with an effective central charge vanishing as a square root at the transition point. We compare these predictions with with exact numerical calculations on the same model and find an excellent agreement.

Entanglement is a fundamental property of quantum mechanics, a key resource for emerging quantum technologies and a powerful tool to characterize quantum phases of matter, in and out of equilibrium [START_REF] Amico | Entanglement in many-body systems[END_REF][START_REF] Horodecki | Quantum entanglement[END_REF][START_REF] Laflorencie | Quantum entanglement in condensed matter systems[END_REF]. For ground-state critical systems in one dimension it is by now understood that entanglement entropy scales logarithmically with system size, with a prefactor given by the central charge of the associated conformal field theory [START_REF] Calabrese | Entanglement entropy and quantum field theory[END_REF]. Gapped systems on the other hand satisfy an area law [START_REF] Hastings | An area law for one-dimensional quantum systems[END_REF]. For translation-invariant systems evolving under unitary dynamics the entanglement growth is generally expected to be linear in time and to lead to a volume law scaling, both for integrable systems [START_REF] Calabrese | Evolution of entanglement entropy in one-dimensional systems[END_REF][START_REF] Alba | Entanglement and thermodynamics after a quantum quench in integrable systems[END_REF] as well as for ergodic quantum many-body systems [START_REF] Kim | Ballistic spreading of entanglement in a diffusive nonintegrable system[END_REF][START_REF]Entanglement growth in diffusive systems[END_REF]. For random unitary circuits it was shown that fluctuations around this linear growth contain important information on the nature of the many-body dynamics and an effective mapping to models of stochastic interface growth was discovered [START_REF] Nahum | Quantum entanglement growth under random unitary dynamics[END_REF]. Furthermore, the slow, logarithmic, growth of entanglement entropy in highly excited disordered quantum many-body systems is considered among the key signatures of Many-Body Localizaton [START_REF] Abanin | Colloquium: Many-body localization, thermalization, and entanglement[END_REF].

Recently, a new class of phase transitions characterized by a qualitative change of the entanglement properties has been identified for quantum many-body systems under the effect of stochastic measurements [START_REF] Li | Quantum zeno effect and the many-body entanglement transition[END_REF][START_REF] Li | Measurementdriven entanglement transition in hybrid quantum circuits[END_REF][START_REF] Skinner | Measurementinduced phase transitions in the dynamics of entanglement[END_REF][START_REF] Chan | Unitary-projective entanglement dynamics[END_REF]. For hybrid quantum circuits with random unitary gates and projective measurements this protocol leads to an entanglement transition between an error correcting/volume law phase and a Zeno phase with area-law scaling [START_REF] Li | Quantum zeno effect and the many-body entanglement transition[END_REF][START_REF] Li | Measurementdriven entanglement transition in hybrid quantum circuits[END_REF][START_REF] Skinner | Measurementinduced phase transitions in the dynamics of entanglement[END_REF][START_REF] Nahum | Measurement and entanglement phase transitions in allto-all quantum circuits, on quantum trees, and in landau-ginsburg theory[END_REF][START_REF] Bao | Theory of the phase transition in random unitary circuits with measurements[END_REF][START_REF] Gullans | Dynamical purification phase transition induced by quantum measurements[END_REF][START_REF] Gullans | Scalable probes of measurement-induced criticality[END_REF][START_REF] Medina | Entanglement transitions from restricted boltzmann machines[END_REF][START_REF] Sierant | Dissipative floquet dynamics: from steady state to measurement induced criticality in trapped-ion chains[END_REF][START_REF] Choi | Quantum error correction in scrambling dynamics and measurementinduced phase transition[END_REF][START_REF] Hashizume | Measurement-induced phase transitions in sparse nonlocal scramblers[END_REF]. Several aspects of this transition have been discussed, including its critical properties [START_REF] Vasseur | Entanglement transitions from holographic random tensor networks[END_REF][START_REF] Jian | Measurement-induced criticality in random quantum circuits[END_REF][START_REF] Lopez-Piqueres | Mean-field entanglement transitions in random tree tensor networks[END_REF][27][START_REF] Zabalo | Critical properties of the measurement-induced transition in random quantum circuits[END_REF][START_REF] Zabalo | Operator scaling dimensions and multifractality at measurement-induced transitions[END_REF][START_REF] Agrawal | Entanglement and charge-sharpening transitions in u(1) symmetric monitored quantum circuits[END_REF][START_REF] Li | Statistical mechanics model for clifford random tensor networks and monitored quantum circuits[END_REF][START_REF] Sierant | Universal behavior beyond multifractality of wave-functions at measurementinduced phase transitions[END_REF], the role of symmetries [START_REF] Bao | Symmetry enriched phases of quantum circuits[END_REF] and dimensionality [START_REF] Lunt | Measurementinduced criticality and entanglement clusters: A study of one-dimensional and two-dimensional clifford circuits[END_REF][START_REF] Turkeshi | Measurementinduced criticality in (2+1)-dimensional hybrid quantum circuits[END_REF][START_REF] Block | The measurement-induced transition in long-range interacting quantum circuits[END_REF][START_REF] Sharma | Measurement-induced criticality in extended and longrange unitary circuits[END_REF]. Recently an experimental implementation with trapped ions has been achieved [START_REF] Noel | Observation of measurement-induced quantum phases in a trapped-ion quantum computer[END_REF]. When the unitary dynamics is generated by a Hamiltonian one can expect a richer pattern of entanglement properties, which has just started to be unveiled. In the case of free fermions it was shown that the volume law is unstable for any value of the measurement strength to a subextensive entanglement content [START_REF] Cao | Entanglement in a fermion chain under continuous monitoring[END_REF][START_REF] Fidkowski | How Dynamical Quantum Memories Forget[END_REF][START_REF] Coppola | Growth of entanglement entropy under local projective measurements[END_REF][START_REF] Botzung | Engineered dissi-pation induced entanglement transition in quantum spin chains: from logarithmic growth to area law[END_REF][START_REF] Tang | Quantum criticality in the nonunitary dynamics of (2 + 1)-dimensional free fermions[END_REF]. Nonetheless an entanglement transition between a logarithmic and area law phase at a critical measurement strength was found in several translationally invariant free-fermionic models under different measurement protocols corresponding to the continuous moni-toring limit [START_REF] Alberton | Entanglement transition in a monitored free-fermion chain: From extended criticality to area law[END_REF][START_REF] Turkeshi | Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks[END_REF][START_REF] Buchhold | Effective theory for the measurement-induced phase transition of dirac fermions[END_REF][START_REF] Müller | Measurementinduced dark state phase transitions in long-ranged fermion systems[END_REF]. These results raise the question of whether a simple picture for entanglement transitions in these settings might exist, generalizing the quasi-particle picture available for integrable systems under unitary evolution [START_REF] Calabrese | Evolution of entanglement entropy in one-dimensional systems[END_REF][START_REF] Alba | Entanglement and thermodynamics after a quantum quench in integrable systems[END_REF]. A first attempt in this direction lead to the collapsed quasiparticle ansatz of Ref. [START_REF] Cao | Entanglement in a fermion chain under continuous monitoring[END_REF], predicting an area law scaling for any finite measurement rate, that appears at odds with the above results for monitored free-fermions.

In this Letter we put forward a phenomenological theory for entanglement dynamics in continuously monitored quantum many-body systems with well-defined quasi-particles and we test it against exact numerical simulations on the quantum Ising chain evolving under quantum jumps. We assume that due to the measurement process the ballistically propagating quasiparticles, responsible for entanglement growth in the unitary case, acquire now a finite lifetime given by the associated non-Hermitian Hamiltonian [START_REF] Wiseman | Quantum Measurement and Control[END_REF][START_REF] Ashida | Non-hermitian physics[END_REF]. We then describe the entanglement dynamics as a stochastic resetting process [START_REF] Evans | Diffusion with stochastic resetting[END_REF]51] , where those non-Hermitian quasiparticles are randomly reset with a rate given by their inverse lifetime. We write down a renewal equation for the entanglement statistics and compute its average value, from which we obtain that entanglement growth is directly connected to the spectrum of decay rates of those non-Hermitian quasiparticles. We show that our phenomenological picture captures all the key features of the monitored Ising chain, including the critical logarithmic entanglement phase, related to the existence of slowly decaying quasiparticles, its transition into an area law phase and an effective central charge vanishing continuously at the transition.

Unitary Dynamics and Measurement Protocol -We consider a quantum many-body system evolving under the combined effect of a unitary evolution generated by a Hamiltonian H and a measurement apparatus acting occasionally but abruptly on the quantum state to measure a local operator L i , leading to Quantum Jump (QJ) trajectories described by the following stochastic Schrödinger equation [START_REF] Wiseman | Quantum Measurement and Control[END_REF][START_REF] Dalibard | Wave-function approach to dissipative processes in quantum optics[END_REF][START_REF] Dum | Monte carlo simulation of the atomic master equation for spontaneous emission[END_REF][START_REF] Carmichael | Quantum trajectory theory for cascaded open systems[END_REF][START_REF] Plenio | The quantum-jump approach to dissipative dynamics in quantum optics[END_REF][START_REF] Daley | Quantum trajectories and open many-body quantum systems[END_REF][START_REF] Fuji | Measurement-induced quantum criticality under continuous monitoring[END_REF]

d|ψ(N t ) = -iH - γ 2 i L † i L i -L † i L i t dt|ψ(N t ) + i   L i L † i L i t -1   dN i t |ψ(N t ) , (1)
where

• t ≡ ψ(N t )| • |ψ(N t )
, γ is the measurement rate, and N t = {N i,t } are Poisson processes dN i,t = 0, 1, statistically independent dN i,t dN j ,t = δ i,j dN i,t , and with average value dN i,t = γdt L † i L i t . We assume the unitary evolution, the first term in (1), to be described by a Hamiltonian containing well-defined quasiparticle degrees of freedom and the operator being measured L i to be local in these quasiparticles -see below for a specific example.

In the following we will be mainly interested in the (von Neumann) entanglement entropy, defined as [START_REF] Amico | Entanglement in many-body systems[END_REF][START_REF] Horodecki | Quantum entanglement[END_REF][START_REF] Calabrese | Entanglement entropy and quantum field theory[END_REF]]

S(N t ) = -tr A [ρ A (N t ) ln ρ A (N t )] , (2)
where we have introduced a partition A ∪ B and the re-

duced density matrix ρ A (N t ) = tr B |ψ(N t ) ψ(N t )|.
In particular we will focus on the conditional average entanglement entropy, given by S t = DN t P (N t )S(N t ).

Stochastic Resetting of Non-Hermitian Quasiparticles -We now present our phenomenological theory which takes the form of an effective stochastic process for the entanglement dynamics. In absence of any measurement, it is well known that entanglement is carried by pairs of quasiparticles with momenta k, -k moving along the light-cone with velocity v k . The measurement protocol in (1) gives rise to two effects, namely a (i) nonunitary evolution described by a non-Hermitian Hamiltonian [START_REF] Wiseman | Quantum Measurement and Control[END_REF]

H eff = H -i γ 2 i L † i L i and
(ii) stochastic quantum jumps which tend to suppress entanglement. We assume that the non-Hermitian Hamiltonian can be still written in terms of quasiparticles with momentum k which propagate ballistically but acquire a finite decay rate Γ k given by the imaginary part of their complex energy Λ k = E k + iΓ k . Drawing from the literature on stochastic resetting [START_REF] Evans | Diffusion with stochastic resetting[END_REF]51,[START_REF] Gupta | Fluctuating interfaces subject to stochastic resetting[END_REF][START_REF] Majumdar | Dynamical transition in the temporal relaxation of stochastic processes under resetting[END_REF][START_REF] Mukherjee | Quantum dynamics with stochastic reset[END_REF][START_REF] Magoni | Ising model with stochastic resetting[END_REF][START_REF] Perfetto | Designing nonequilibrium states of quantum matter through stochastic resetting[END_REF] we postulate that the combined effect of measurements is to randomly reset these non-Hermitian quasiparticles, according to a Poisson process with a rate that we identify with their inverse lifetime Γ k and that therefore strongly depends on the competition between unitary dynamics and local measurements.

The key quantity in our theory is therefore the probability P k (s, t) of having at time t a contribution s to the entanglement entropy due to quasiparticles at momenta k, -k, propagating with velocity v k and being reset with a rate Γ k . An infinite number of resetting events contribute to this probability, which can be resummed into a renewal equation of the form [63-66]

P k (s, t) = e -Γ k t p 0 k (s, t) + t 0 dτ Γ k e -Γ k τ P k (s, t -τ ) (3) with p 0 k (s, t) = δ(s -s * k (t)) with s * k (t) = s 0 min (2|v k |t, )
describing the ballistic contribution of the isolated system with s 0 the thermodynamic entropy density [START_REF] Alba | Entanglement and thermodynamics after a quantum quench in integrable systems[END_REF][START_REF] Fagotti | Evolution of entanglement entropy following a quantum quench: Analytic results for the xy chain in a transverse magnetic field[END_REF][START_REF] Calabrese | Entanglement spreading in non-equilibrium integrable systems[END_REF]. We further assume the initial state of the dynamics to be completely uncorrelated and that the resetting process restarts the entanglement trajectory s * k (t) from its product state value.

The two terms in (3) have a clear meaning as renewal processes, where the first term describes a situation where the no resetting has taken place up to time t, whose probability is e -Γ k t and the entanglement statistics is given by the unitary evolution, while the second describes the situation in which the first resetting event occurs at time τ and the following evolution during t -τ accounts for all possible resetting events. An equivalent formulation in terms of the last resetting time can be also obtained [63]. To solve the renewal equation it is convenient to introduce the characteristic function F k (q, t) defined as

F k (q, t) = dse -iqs P k (s, t) . (4)
A simple calculation gives [63]

F k (q, t) = Γ k + iqB k (t)e -iq Bk min(t,t * ) Γ k + iq Bk , (5)
with

Bk = 2s 0 |v k |, B k (t) = Bk e -Γ k min(t,t * ) and t * = /2|v k |.
Let us now discuss the consequences of this ansatz for the average entanglement entropy, which can be written after simple manipulations as [63]

S t ( ) = dk 2π 2|v k | Γ k 1 -e -Γ k min(t,t * ) . (6)
Taking the long-time limit we obtain an expression for the stationary state entanglement entropy which reads

S ∞ ( ) = dk 2π 2|v k | Γ k 1 -e -Γ k /2|v k | . (7)
We see that for a resetting rate given by the bare measurement rate, i.e. Γ k ≡ γ, our theory reduces to the collapsed quasi-particle ansatz of Ref. [START_REF] Cao | Entanglement in a fermion chain under continuous monitoring[END_REF] leading, for any measurement rate, to an exponential saturation of the entanglement in time and a stationary state with area law. As we are going to show, this scenario can dramatically change if the decay rate Γ k vanishes for certain k points, leading to a gapless dispersion of decay modes. In this case, the integral in (6) might diverge with either time or system size and different scaling of the entanglement entropy can be obtained depending on the nature of the decay rate.

Application: Quantum Ising chain with Quantum Jumps -To test our quasiparticle picture we consider a quantum Ising chain, where the transverse magnetization is measured through a QJ protocol. Specifically, we take in (1) the Hamiltonian of the form

H = -J L-1 i=1 σ x i σ x i+1 -h i σ z i (8)
with open boundary conditions, where σ α are Pauli matrices. We note that with respect to Ref. [START_REF] Turkeshi | Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks[END_REF] here we include also a transverse field h > 0 in the unitary part of the evolution. For what concerns the measuring process we choose the local operator in [START_REF] Amico | Entanglement in many-body systems[END_REF] as we start at t = 0 from the ground-state of H with J = 1 and h = h 0 1 [63]. To compute the entanglement entropy we take a cut of length = L/4. We first discuss the predictions from our phenomenological theory. To this extent we consider the non-Hermitian Hamiltonian associated with the QJ protocol which in the case under consideration takes the form of a quantum Ising chain with complex transverse field [START_REF] Hickey | Time-integrated observables as order parameters for full counting statistics transitions in closed quantum systems[END_REF][START_REF] Lee | Heralded magnetism in non-hermitian atomic systems[END_REF][START_REF] Biella | Many-body quantum zeno effect and measurement-induced subradiance transition[END_REF], H eff = H -iγ/4 i σ z i . This model can be diagonalized exactly in terms of non-Hermitian quasiparticles with complex spectrum Λ k

L i = (1+σ z i )/2 and
Λ k = ε 2 k -γ 2 /4 + 2iγ (h -cos k) ≡ E k + iΓ k (9)
and finite lifetime Γ k . In ( 9)

ε k = 4 (h 2 + 1 -2h cos k)
is the dispersion relation of the Hermitian Ising chain, featuring a quantum critical point at h = h c = 1 for ordering at k = 0. It is easy to see [63] that the nonhermitian quasiparticle spectrum in (9) features a spectral (subradiant) transition [START_REF] Biella | Many-body quantum zeno effect and measurement-induced subradiance transition[END_REF][START_REF] Gopalakrishnan | Entanglement and purification transitions in non-hermitian quantum mechanics[END_REF], at a critical value of the measurement strength where the properties of the decay rates Γ k change qualitatively. As we are going to show now within our phenomenological theory, this directly affects the scaling of the entanglement entropy and leads to a sharp entanglement transition. To present our results it is useful to distinguish two cases, namely h < h c and h > h c . In the first case, we have that for γ < γ c (h) = 4 √ 1 -h 2 the real-part of the spectrum is gapped while the imaginary part, controlling the long- 6) we obtain a logarithmic growth of the entanglement either in time , S t ∼ c eff log t, or as a function of the size of the cut , S ∞ ∼ c eff log with an effective central charge [START_REF] Alberton | Entanglement transition in a monitored free-fermion chain: From extended criticality to area law[END_REF][START_REF] Turkeshi | Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks[END_REF] that we can obtain analytically from [START_REF] Calabrese | Evolution of entanglement entropy in one-dimensional systems[END_REF]. Using as velocity v k the group velocity of quasi-particles in the initial state we obtain c eff ∼ √ γ c -γ, which vanishes as a square root at γ = γ c (h) [63]. Right at the critical point, γ = γ c (h) we have that α * → ∞ and the decay is non-analytic Γ k ∼ |k -k * | 1/2 which gives a convergent contribution to the entanglement entropy and only leaves a smooth exponential component. On the other hand, for γ > γ c (h) a gap opens up in the imaginary part of the quasiparticle spectrum, while the real part is gapless. In this regime, (6) leads to an exponential dynamics of the entanglement and to an area law-phase, independent from the size of the cut. As the transverse field is increased above h c = 1 the spectrum changes structure. Specifically, the imaginary part remains always gapped for any γ and we, therefore, we do not expect any entanglement transition, a result that we can qualitatively understand by noting that for large h the system remains always very close to an exact eigenstate of the measurement operator and therefore is not able to generate entanglement. We now present the numerical results obtained by evolving the system under the stochastic dynamics in (1) and compare them with our predictions. In Fig 2 we plot the average entanglement as a function of time for different values of the transverse field h, the measuring strength γ, and system size L [63]. We see that for small h and γ (top left panel) the entanglement entropy shows the expected logarithmic growth in time, which eventually saturates to a stationary value on time scales that depend on system size. Upon increasing γ at a fixed value of h = 0.4 (top right panel) we see a clear transition in the dynamics of the entanglement which for large γ shows an exponentially fast approach to the stationary value and a weak dependence on system size as expected for a Zeno phase. A similar transition in the entanglement dynamics is found for fixed γ = 1.5 upon increasing h (bottom right panel). Finally, for large values of the transverse field, h = 1.5, and γ = 1 the entanglement entropy reaches a stationary value exponentially fast and no signature of the logarithmic growth phase or transition is found, again in agreement with our phenomenological theory.

In Fig. 3 we plot (right-panels) the dependence of the stationary entanglement from the system size L for a few representative values of h, γ, which confirms for h < h c = 1 the existence of a logarithmic phase for small γ, where S ∞ ∼ c eff log L, and an area law-phase for large γ, separated by an entanglement transition [START_REF]= a log L + b neglecting small system sizes and identifying c eff ≡ a[END_REF], while for h > h c = 1 (bottom panel) only an area law scaling is found, as predicted by [START_REF] Calabrese | Evolution of entanglement entropy in one-dimensional systems[END_REF]. We note that for certain values of γ the stationary entanglement entropy features a crossover with system size, from an area law at small L to a logarithmic scaling at large L, missed by our theory which only predicts the large L behavior. A similar crossover was found for the Ising chain in the no-click limit [START_REF] Turkeshi | Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks[END_REF] suggesting that the origin of this effect, very different from what usually observed in ground-state critical systems, might be related to the entanglement dynamics of non-Hermitian systems, which is largely unex-plored [START_REF] Bácsi | Dynamics of entanglement after exceptional quantum quench[END_REF]. Finally, in the left panel we draw a numerical phase diagram in the (γ, h) plane by plotting the numerical value of the effective central charge (in the inset we present three representative values of h), and compare it with the predicted critical coupling γ c (h) (dashed line) showing a remarkable qualitative agreement.

Overall these results show that our phenomenological picture captures the essence of the measurement-induced transition in the Quantum Ising model. We further note that our results for the QJ protocol are qualitatively very similar to those obtained for the same model evolving under continuous monitoring or in the no-click limit [START_REF] Turkeshi | Measurement-induced entanglement transitions in the quantum ising chain: From infinite to zero clicks[END_REF], suggesting that our phenomenology might apply more broadly [75].

Conclusions -In this paper we have introduced a phenomenological theory for entanglement transitions in quantum many-body systems with well-defined quasiparticles evolving under the effect of continuous monitoring. The idea is that entanglement is carried by non-Hermitian quasiparticles which propagate ballistically and are reset stochastically with a rate given by their inverse lifetime. We have written down a renewal equation for the entanglement statistics from which one can obtain the average entanglement in terms of only two parameters, the quasiparticle velocity and their lifetime. We have applied our theory to the Quantum Ising chain measured through quantum jumps. We have shown that our phenomenological theory captures all the key features of this problem, including the logarithmic growth phase, an effective central charge vanishing continuously at the entanglement transition into an area law, and the overall phase diagram. Natural extensions of this work could include monitored free fermionic systems in d ≥ 2, where one could expect different entanglement scaling to emerge in presence of a gapless spectrum of decay rates, or even interacting integrable quantum many-body systems for which the structure of the effective non-Hermitian Hamiltonian preserve its integrability [START_REF] Ashida | Non-hermitian physics[END_REF]. This work was supported by the ANR grant "NonEQuMat"(ANR-19-CE47-0001). We acknowledge computational resources on the Collége de France IPH cluster.

The logarithm of the characteristic function Φ k (q, t) = logF k (q, t) generates all the connected moments of the entanglement entropy contribution of each quasiparticle. We can define therefore the characteristic function of the full entanglement distribution

Φ(q, t) = dk 2π Φ k (q, t) (S29)
which by construction generates the connected moments of the entanglement entropy, for example for the average we have

∂Φ/∂q| q=0 = -iS(t) (S30)
We can now compute explicitly the characteristic function F k (q, t). Plugging the renewal equation into the definition we have

F k (q, t) = e -Γ k t e -iqS * k (t) + Γ k t 0 dτ e -Γ k τ e -iqS * k (τ ) . (S31)
In order to evaluate the integral on the right hand side we have to distinguish whether t is smaller or greater than the time scale t * = l/2|v k |. Specifically for t < t * we have that also τ < t * and we obtain

F k (q, t) = e -Γ k t e -iqs02|v k |t + Γ k t 0 dτ e -Γ k τ e -iqs02|v k |τ (S32)
while for t > t * we have two different contributions to the integral

F k (q, t) = e -Γ k t e -iqs0l + Γ k t * 0 dτ e -Γ k τ e -iqs02|v k |τ + Γ k t t * dτ e -Γ k τ e -iqs0l . (S33)
Performing the integrals and after simple manipulation we end up with the following result for the characteristic function

F k (q, t) = Γ k + iqB k (t)e -iq Bk min(t,t * ) Γ k + iq Bk (S34)
where we have defined

Bk = 2s 0 |v k | (S35) B k (t) = Bk e -Γ k min(t,t * ) . (S36)
We can now evaluate the average entanglement entropy as a function of time. To this extent we use Eqns. (S29), (S30), and take the derivative with respect to q to get

S = i∂Φ/∂q| q=0 = dk 2π 1 F k (q = 0) i∂F k ∂q | q=0 (S37)
A simple calculation for the first derivative evaluated at q = 0 gives the result

∂F k ∂q | q=0 = i B k (t) -Bk Γ k (S38)
from which we immediately recover Eq. ( 6) of the main text, i.e. In the main text we consider the following non-Hermitian Ising chain

S t ( ) = dk 2π 2|v k | Γ k 1 -e -Γ k min(t,t * ) . (
H eff = -J i σ x i σ x i+1 -h i σ z i -i γ 4 i σ z i (S40)
which is obtained in the no-click limit of the QJ protocol discussed in the main text, using the fact that L i = (1 + σ z i )/2 ≡ L 2 i and dropping the constant factor. Using a Jordan-Wigner transformation of spins into fermions this model can be diagonalised exactly in terms of new fermionic degrees of freedoms γk , γk [START_REF] Hickey | Time-integrated observables as order parameters for full counting statistics transitions in closed quantum systems[END_REF][START_REF] Lee | Heralded magnetism in non-hermitian atomic systems[END_REF][START_REF] Biella | Many-body quantum zeno effect and measurement-induced subradiance transition[END_REF] and brought into a form

H eff = k>0 Λ k γk γk + γ-k γ-k -1 (S41) the complex quasiparticle spectrum Λ k reads Λ k = ε 2 k -γ 2 /4 + 2iγ (h -cos k) (S42)
where ε k = 4 (h 2 + 1 -2h cos k) is the dispersion of the Hermitian Ising chain and we fixed J = 1. We define the real and imaginary part of the complex eigenvalues Λ k in such a way that the imaginary part is always positive (decays modes).

Λ k = E k + iΓ k . (S43)
As discussed in the main text, the scaling of the average entanglement entropy predicted from the stochastic resetting theory is directly controlled by the spectral property of the decay modes, encoded in the quasiparticle lifetime Γ k . By inspecting Eq. (S42) we immediately see that two cases have to be distinguished, depending on the value of h. For h < h c = 1 the imaginary part of the argument of the square root in Eq. (S42) vanishes at a pair of k points given by k = k * (h) = ± arccos(h). Then we see that at these values of k the spectrum reads Λ k * = 4(1 -h 2 ) -γ 2 /4 and it is therefore either purely real, when Near k * we can linearize the spectrum and find that for γ < γ c (h) the decay rate Γ k behaves as

Γ k α * |k -k * | (S45)
where α * can be obtained from the derivative of Λ k and reads

α * = Im dΛ dk | k=k * = Im 1 2Λ k * 2ε k * dε dk | k=k * + 2iγ sin k * = 2γ 1 -h 2 / γ 2 c -γ 2
As the critical point is approached, γ → γ c , we have that α * → ∞ and the decay rate is non analytic, Γ k ∼ |k -k * | 1/2 . On the other hand for h > h c = 1 the imaginary part of the argument of the square root in Eq. (S42) is always different from zero and as such there is always a finite decay rate in the system. Using the results above and Eq.(S39) we can obtain the predictions for the average entanglement entropy discussed in the main text. In particular for γ < γ c , when Eq. (S45) holds, we have

S t ( ) = dk 2π 2|v k | Γ k 1 -e -Γ

Figure 1 .

 1 Figure 1. (Top) A quantum many-body system evolves under the action of its own Hamiltonian and local quantum jump measurements with rate γ. Its entanglement entropy of a partition A is obtained from the reduced density matrix ρA. (Bottom) Cartoon of the phenomenological picture for the entanglement dynamics where non-Hermitian quasiparticles with velocity v k are stochastically reset by the measurement with a rate given by their finite inverse lifetime Γ k .

Figure 2 .

 2 Figure2. Average entanglement entropy dynamics for different values of transverse field h and measuring strength γ, obtained from the QJ protocol. Top panels: logarithmic growth of the entanglement for γ < γc and h < hc (left), which turns into an exponential approach to a stationary value upon increasing γ above γc(h) (right). Bottom panels: Entanglement dynamics for h > hc shows a weak dependence from system size (left), while upon decreasing h a transition into a logarithmic growth phase emerges.

Figure 3 .

 3 Figure 3. Steady-state entanglement transition in the quantum Ising chain with QJ measurements. Left-panel: Phase diagram in the (γ, h) plane featuring a critical logarithmic growth phase for γ < γc(h) and an area-law (Zeno) phase elsewhere. Rightpanels: Stationary entanglement versus system size for different values of γ, h showing the logarithmic growth phase and the area-law. Inset: Effective central charge c eff (γ) for three values of h (see vertical cuts on the phase diagram).

  S39) S5 SPECTRUM OF NON-HERMITIAN ISING MODEL AND SCALING OF ENTANGLEMENT ENTROPY

4 1 - 1 -h 2 .

 112 h 2 > γ 2 /4 (S44)or purely imaginary otherwise. The condition above defines in fact a spectral phase transition in the non-Hermitian Hamiltonian for measurement strength γ equal to the critical value γ c (h) = 4 √ For γ < γ c (h) we haveE k * = γ 2 c /4 -γ 2 /4 Γ k * = 0 while for γ > γ c (h) we have E k * = 0 Γ k * = γ 2 /4 -γ 2 c /4

Supplemental Material to 'Entanglement Transitions from Stochastic Resetting of Non-Hermitian Quasiparticles'

In this Supplemental Material, we provide details on (i) the derivation of the renewal equation for the entanglement probabilty, (ii) an alternative formulation in terms of last-resetting renewal equation, (iii) the explicit solution of the renewal equation, (iv) the spectrum of the non-Hermitian Ising chain and the scaling of the average entanglement (v) the quantum jumps protocol for the quantum Ising chain and its numerical implementation.

RENEWAL EQUATION FOR ENTANGLEMENT PROBABILITY

Here we show how to derive the renewal equation given in Eq. (3) of the main text from a resummation of resetting events [51]. For the sake of generality we consider resetting events to be distributed according to a waiting-time probability ϕ k (t) which gives the probability of having a resetting event in the interval t, t + dt. By construction, this has to be normalized

and for a Poisson process, considered in the main text, this reduces to ϕ k (t) = Γ k e -Γ k t . As a consequence the probability of no resetting up to time t is

where

is the probability of having at least one resetting in the time interval (0, t). In terms of these quantities the renewal equation given in the main text, also called first-resetting renewal equation, reads

where we recall that p 0 k (s, t), given in the main text, is the probability of an entanglement contribution s in absence of any resetting.

In order to derive Eq. (S3) we write the probability P k (s, t) as a series expansion over an arbitrary number n of resettings events, i.e.

k (s, t) is the probability of having an entanglement contribution s at time t in presence of n resettings. We start to evaluate explicitly the first terms of this series. For n = 0, corresponding to no resettings up to time t, we have by definition

For n = 1 we have one resetting at time t 1 ∈ (0, t), followed by no resetting in a time interval of duration t -t 1 with free propagation starting from s * = 0 for a time t -t 1 . This gives a contribution

Similarly, for n = 2 we can have one resetting at time t 1 ∈ (0, t) and a second after a time t 2 ∈ (0, t -t 1 ), followed by no resetting in a time interval of duration t -t 1 -t 2 with free propagation starting from s * = 0 for a time t -t 1 -t 2 , with contribution

This procedure can be straightforwardly iterated to any order n to obtain the entanglement probability in presence of n resettings. It is now easy to see that the renewal equation given in Eq. (S3) generates the very same expansion.

In fact we take Eq. (S3) for P k (s, t) and we use it to express the probability at time t -τ , i.e. P k (s, t -τ ) entering the right hand side, we obtain

Plugging this result in Eq. (S3) we obtain

If we stop at this order and identify the distribution P k (s, t -τ -τ 1 ) appearing on the right hand side as the bare one, i.e.

we obtain the resummation of resetting events up to n = 2 that we discussed above. However we can continue this process, using Eq. ( S3) to generate a full series of nested integrals which reproduces the series expansion in resetting events to all orders. We conclude therefore that the renewal equation provides a formal resummation of resetting events of arbitrary number.

ALTERNATIVE FORMULATION: LAST-RESETTING RENEWAL EQUATION

Here we show that the series expansion in Eq. (S4), or equivalently the first-resetting renewal equation in Eq. ( S3), can be in fact resummed analytically in a closed form known as last-resetting renewal equation [51], which reads

To see this it is convenient to express the series in Eq. (S4) in Laplace domain. Specifically we can introduce the Laplace transform

and evaluate each term in the expansion. For the n = 1 term in Laplace space we get

where φk (z) is the Laplace transform of ϕ k (t) and we have defined Fk (s, z)

With similar manipulations we can get for n = 2

If we iterate this procedure and resum the geometric series we obtain

From the solution in Laplace space we can go back to the real-time. It is easy to see that the function

gives back Eq. (S16) under Laplace transform, provided the Laplace transform of the function Υ k (t) is such that

In fact we have

which coincides with Eq. (S16) provided that Eq. (S18) holds. For Poissonian resetting we can further simplify Eq. (S17) since we have

and in Laplace domain

from which we obtain Υk (z) = Γ k /z which in the time domain gives Υ k (t) = Γ k θ(t) and allow us to rewrite Eq. (S17) as

which coincides with Eq. (S11) which therefore provides a closed form solution to the first-time renewal equation.

SOLUTION OF RENEWAL EQUATION AND AVERAGE ENTANGLEMENT

Once the probability of the entanglement contribution P k (s, t) is known from Eq. (S11) we can obtain all the moments of the entanglement contribution of each momentum k

as well as the associated moments of the entanglement entropy

In order to obtain an explicit expression for P k (s, t) it is useful to introduce its characteristic function F k (q, t) defined as

which we see is infrared divergent in the large , t limit and leads to a logarithmic scaling of the entanglement entropy, S t ( ) ∼ c eff log with a prefactor

where we have used as quasiparticle velocity v k the group velocity of quasi-particles in the initial state, v k = dε k /dk = 4h 0 sin k/ε k (h 0 ) ∼ 2 sin k for h 0 1.

DETAILS ON NUMERICAL SIMULATIONS: QUANTUM JUMP EQUATION AND ENTANGLEMENT EVOLUTION

In this section we consider the evolution of the TFIM subject to quantum jumps. We first introduce the equation of motion for the quantum state in the Hilbert space, which we then recast in an evolution for the two-point correlation function. Throughout this section we simplify the notation introducing |ψ t ≡ |ψ(N t ) .

Stochastic Schrödinger equation for the quantum state -The state dynamics is given by

where H =i (Jσ x i σ x i+1 +hσ z i ) is the Ising Hamiltonian, σ α l are Pauli matrices acting on site l, n l = (1+σ z l )/2 = c † l c l is the number operator acting on site l, and O t ≡ ψ t |O|ψ t is the expectation value at time t. The trajectory nature of Eq. ( S46) is fully captured in the Poisson process dN l,t = 0, 1 with dN l,t dN l ,t = δ l,l dN l,t and disorder average value dN l,t = γdt n l t .

Eq. (S46) preserves the Gaussianity: given an initial state |ψ 0 in the fermionic operators c and c † , the state state at time t is also Gaussian. A major consequence is that the state is fully encoded in the two-body correlation functions of the fermionic operators, i.e. in the 2L × 2L matrix

with the submatrices given by

Furthermore, the entropy can be easily derived from Eq. (S47), as detailed in Ref. [76]. Introducing the matrix

where 1 L×L is the L × L identity matrix, we define the matrix (en passant, we note this is the correlation matrix for Majorana fermions [76])

The entanglement entropy of a connected partition

The matrix A is real and anti-symmetric and can be reduced to canonical form through Schur's decomposition [76]

with

, the entanglement entropy is given the Yang-Yang entropy [77]

Stochastic Schrödinger equation for the correlation functions -In this paragraph we derive the equation of motion for G(t). We preliminary note the stochastic Schrödinger equation Eq. (S46) can be divided in two parts: a deterministic evolution, generated by the non-Hermitian Hamiltonian

and by quantum jumps

Since the measurement nature is strong (dN = 1 dt and n l is a projection), we can separately consider a non-Hermitian evolution for G interspersed with stochastic measurements (we closely follow the discussion in Ref. [START_REF] Breuer | The theory of open quantum systems[END_REF] and Ref. [START_REF] Daley | Quantum trajectories and open many-body quantum systems[END_REF]).

The evolution under the non-Hermitian part is given by

having non-zero elements A m,m = h, A m,m+1 = A m+1,m = -J/2, B j,j+1 = -B j+1,j = -J/2. Using the above prescription, and implementing the Wick theorem on the four-body terms (e.g. {n l , c m c † n } t ), a simple but lengthly computation gives

where we introduced the 2L × 2L matrices

where 0 L×L is the zero matrix and 1 L×L is the L × L identity matrix. Quantum jumps are instead taken into account by

and by

where we have used Wick theorem and simple algebraic manipulations. Similarly the remaining matrix elements can be computed via the algebraic relationship explicit in Eq. (S47). We formally denote the action of the jumps Eqs. (S62) and (S63) as

With this notation, we can formally write the stochastic equation for the correlation matrix

Numerical Implementation -We numerically implement Eq. (S65) following the prescription in Ref. [START_REF] Daley | Quantum trajectories and open many-body quantum systems[END_REF]. The transition probability in a small time-step δt is given by the decay of the norm δp = δt ψ t |i(H eff -H † eff )|ψ t = δt l γ n l ≡ l δp l . (S66)

We can interpret δp l as the probability that the action of n l will occur during this particular time step. We draw a uniform random number r 1 ∈ [0, 1], and compare it with δp. If r 1 > δp, no jump occurs, and the evolution is given by Eq. (S60). Instead, if r 1 < δp a jump occur. To identify the site where the jump occurs, we draw a random number r For the numerical simulations we fix J = 1, δt = 0.01. We consider the values of transverse field in the range h = 0 ÷ 1.5 and the values of the measurement rate γ = 0.5 ÷ 4.8. We consider system sizes in the range L = 8 ÷ 128.