Matthew Bristow
email: matthew.bristow@physics.ox.ac.uk

William Knafo

Pascal Reiss

William Meier

Paul C Canfield

Stephen J Blundell

Amalia I Coldea
email: amalia.coldea@physics.ox.ac.uk

Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor CaKFe 4 As 4

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I. INTRODUCTION

The upper critical field, H c2 , is an important property of superconductors that defines their limit for practical applications. It also describes the complex interplay between different pairing gaps and symmetry and can shed light on the nature of the superconducting mechanism. Furthermore, the temperature dependence of the upper critical field can also provide evidence for the presence of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF][START_REF] Song | Quantum FFLO State Iclean Layered Superconductors[END_REF], in which the order parameter varies in space. Iron-based superconductors have unusually large values of the upper critical field which reveal exotic effects caused by the interplay of orbital and paramagnetic pairbreaking in multiband superconductors with unconventional pairing symmetry [START_REF] Hirschfeld | Gap symmetry and structure of Fe-based superconductors[END_REF]. They also provide the right conditions for the FFLO state to develop in clean materials due to the likely presence of shallow bands [START_REF] Mou | Enhancement of the Superconducting Gap by Nesting in CaKFe 4 As 4 : A New High Temperature Superconductor[END_REF] and very large Pauli paramagnetic effects [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF].

CaKFe 4 As 4 is a clean and stoichiometric superconductor with a relatively high T c = 35 K and it belongs to a family of 1144 iron-based superconductors [START_REF] Iyo | New-structure-type Fe-based superconductors: CaAFe 4 As 4 (A = K, Rb, Cs) and SrAFe 4 As 4 (A = Rb, Cs)[END_REF]. This system lacks longranged magnetic order or a nematic electronic state at low temperatures [START_REF] Iyo | New-structure-type Fe-based superconductors: CaAFe 4 As 4 (A = K, Rb, Cs) and SrAFe 4 As 4 (A = Rb, Cs)[END_REF][START_REF] Iida | Spin resonance in the new-structure-type iron-based superconductor CaKFe 4 As 4[END_REF][START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF][START_REF] Bud'ko | 57Fe Mössbauer study of stoichiometric iron-based superconductor CaKFe 4 As 4 : A comparison to KFe 2 As 2 and CaFe 2 As 2[END_REF][START_REF] Bud'ko | Coexistence of superconductivity and magnetism in CaK(Fe 1-x Ni x ) 4 As 4 as probed by 57 Fe Mössbauer spectroscopy[END_REF][START_REF] Cui | Magnetic fluctuations and superconducting properties of CaKFe 4 As 4 studied by 75 As NMR[END_REF][START_REF] Zhang | High-T c superconductivity in CaKFe 4 As 4 in absence of nematic fluctuations[END_REF] but, upon doping with Ni, a hedgehog magnetic structure is stabilized [START_REF] Bud'ko | Coexistence of superconductivity and magnetism in CaK(Fe 1-x Ni x ) 4 As 4 as probed by 57 Fe Mössbauer spectroscopy[END_REF][START_REF] Ding | NMR study of the new magnetic superconductor CaK(Fe 0.951 Ni 0.049 ) 4 As 4 : Microscopic coexistence of the hedgehog spin-vortex crystal and superconductivity[END_REF][START_REF] Ding | Hedgehog Spin-Vortex Crystal Antiferromagnetic Quantum Criticality in CaK(Fe 1-x Ni x ) 4 As 4 Revealed by NMR[END_REF]. CaKFe 4 As 4 has an exceptionally large critical current density due to strong pointlike defects caused by local structural site effects as well as surface pinning [START_REF] Singh | Ultrahigh critical current densities, the vortex phase diagram, and the effect of granularity of the stoichiometric high-T c superconductor CaKFe 4 As 4[END_REF][START_REF] Haberkorn | Effect of Ni doping on vortex pinning in CaK(Fe 1-x Ni x ) 4 As 4 single crystals[END_REF][START_REF] Ishida | Unique defect structure and advantageous vortex pinning properties in superconducting CaKFe4As4[END_REF]. Due to reduced symmetry compared with the 122 family of iron-based superconductors, CaKFe 4 As 4 is predicted to have up to ten different bands [Figs. 1(e) and 4]. However, angle-resolved photoemission spectroscopy (ARPES) detects a Fermi surface composed of three hole pockets and two electron pockets. The superconducting gaps are nearly isotropic and different for each of the Fermi surface sheets [START_REF] Mou | Enhancement of the Superconducting Gap by Nesting in CaKFe 4 As 4 : A New High Temperature Superconductor[END_REF]. The presence of electron and hole sheets supports a spin resonance corresponding to the (π, π) nesting wave vector detected by neutron diffraction [START_REF] Iida | Spin resonance in the new-structure-type iron-based superconductor CaKFe 4 As 4[END_REF] and promotes a s ± superconducting pairing symmetry in CaKFe 4 As 4 . Thus, this clean system is a model system for understanding the effect of pairing on its upper critical field.

To understand the superconducting properties of CaKFe 4 As 4 we have measured the upper critical fields for two orientations in magnetic fields up to 90 T using electrical transport measurements. These experimental studies provide a complete H c2 (T ) phase diagram and allow us to model the entire temperature dependence, as previous work in magnetic fields up to 60 T could not reach the low-temperature region [START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF]. We find that CaKFe 4 As 4 is highly isotropic at the lowest temperature. A two-band model describes the dependence of the upper critical fields for both directions and the band-coupling parameters indicate the presence of different pairing channels. At low temperatures, the upper critical field does not saturate but shows an upturn, consistent with the emergence of a FFLO state in CaKFe 4 As 4 . Pulsed magnetic-field measurements were performed at LNCMI Toulouse, using a 70 T single coil and a 90 T dual coil at constant temperatures below 35 K for both field orientations (an example is shown in Fig. 6). Due to the high crystallinity of our samples that display sharp superconducting transitions, we find that the critical tem-perature from magnetization measurements is similar to the offset temperature determined from transport measurements [Fig. 5(a)]. To completely suppress the superconductivity of CaKFe 4 As 4 , we have used pulsed magnetic fields up to 90 T. Resistivity data against magnetic fields up to 90 T measured at fixed temperature are shown in Figs. 1(c) and 1(d) for the two different orientations. Despite the strong disparity in the degree of suppression of superconductivity between the two field directions close to T c , at the lowest measurable temperature of 4.2 K, the normal state is reached at a similar field ∼85 T for both orientations.

III. UPPER CRITICAL FIELD FROM TRANSPORT MEASUREMENTS

IV. UPPER-CRITICAL-FIELD PHASE DIAGRAM

Based on these experimental data, we have constructed the complete upper-critical-field phase diagram of CaKFe 4 As 4 down to 4.2 K as shown in Fig. 2(a) for the two orientations. Our results are in good agreement with previous studies up to 60 T for both offset and onset critical fields [START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF] [see Figs. 9(a) and 9(b)] and reveal extremely large upper critical fields, reaching almost ∼3 T c at the lowest temperatures. These values are above the Pauli paramagnetic limit, ∼1.85 T c , estimated to be ∼65 T for a single-band superconductor and assuming g = 2 and the weak-coupling limit. The anisotropy of the upper critical field, defined as the ratio of the upper critical field for different orientations, = H ab c2 /H c c2 , drops dramatically with decreasing temperature from ∼4 to 1, as shown in Fig. 2(b). Interestingly, the upper critical fields for the two orientations cross at T ∼ 4.2 K, leading to isotropic superconducting behavior in the low-temperature limit. This phenomenon has been found in optimally doped iron-based superconductors, such as FeSe 0.5 Te 0.5 [START_REF] Serafin | Anisotropic fluctuations and quasiparticle excitations in FeSe 0.5 Te 0.5[END_REF][START_REF] Braithwaite | Evidence for anisotropic vortex dynamics and Pauli limitation in the upper critical field of FeSe 1-x Te x[END_REF] and (Ba,K)Fe 2 As 2 [START_REF] Yuan | Nearly isotropic superconductivity in (Ba,K)Fe 2 As 2[END_REF]. This behavior reflects the large Pauli paramagnetic effects in iron-based superconductors and the influence of Fermi surface details on limiting the orbital effects [START_REF] Yuan | Nearly isotropic superconductivity in (Ba,K)Fe 2 As 2[END_REF]. The Fermi surface of CaKFe 4 As 4 has significant warping for the outer electron and hole band that can potentially allow circulating currents out of plane [see Fig. 4(d) in the Appendix]. Furthermore, the calculated anisotropy of the penetration depth based on plasma frequencies (as in Ref. [START_REF] Hashimoto | Evidence for superconducting gap nodes in the zone-centered hole bands of KFe 2 As 2 from magnetic penetration-depth measurements[END_REF]) is = λ c /λ ab ∼ 4.5, (see Fig. 4) similar to the measured anisotropy close to T c [Fig. 2(b)], suggesting that the Fermi surface details play an important role in understanding its superconducting properties.

Having experimentally determined the upper critical fields for different orientations in magnetic field, we can extract the associated coherence lengths in different temperature regimes, as detailed in the Appendix. In the vicinity of T c , the Ginzburg-Landau coherence lengths shown in Fig. 2(b) are ξ GL ab = 1.66(4) nm and ξ GL c = 0.53(2) nm for CaKFe 4 As 4 , similar to previously reported values [START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF]. In the low-temperature limit, we find coherence lengths of ξ ab = 1.83(3) nm and ξ c = 1.87(6) nm, as shown in Fig. 7. This demonstrates the presence of an isotropic superconducting state at lowest temperature in CaKFe 4 As 4 . The coherence lengths approach the 3D to 2D crossover close to c/ √ 2 ∼ 0.911 nm, as shown in Fig. 2(b) [START_REF] Iyo | New-structure-type Fe-based superconductors: CaAFe 4 As 4 (A = K, Rb, Cs) and SrAFe 4 As 4 (A = Rb, Cs)[END_REF][START_REF] Suetin | Electronic properties and Fermi surface for new layered high-temperature superconductors CaAFe4As4 (A = K, Rb, and Cs): FLAPW-GGA calculations[END_REF]. The low-temperature extracted ξ values are larger than ∼0.7 nm reported from scanning tunnelling microscopy (STM) measurements, which probe locally the vortex lattice rather than an overall averaged effect [START_REF] Fente | Influence of multiband sign-changing superconductivity on vortex cores and vortex pinning in stoichiometric high-T c CaKFe 4 As 4[END_REF]. Knowing the coherence length allows us to estimate the depairing current density, J d , as described in the Appendix [START_REF] Tinkham | Introduction to Superconductivity[END_REF]. Previous magnetization data reported a value of μ 0 H c1 = 22(1) mT when H||c at low temperatures which, combined with H c2 reported here, give κ ∼ 99(2) and λ = 183(6) nm. Using this value for the penetration depth and ξ = 1.86(3) nm gives J d = 1.61(9) × 10 8 A/cm 2 , which is one of the largest among iron-based superconductors. This supports the extremely large critical current densities, reported previously for CaKFe 4 As 4 in Ref. [START_REF] Singh | Ultrahigh critical current densities, the vortex phase diagram, and the effect of granularity of the stoichiometric high-T c superconductor CaKFe 4 As 4[END_REF]. The small coherence length of CaKFe 4 As 4 is compatible with the presence of the large upper critical field, consistent with small Fermi velocities and high T c values of CaKFe 4 As 4 (ξ ∼ hv F /(2π k B T c ) [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF]). Surface superconductivity can survive in a thin layer of thickness ∼ξ in systems with clean surfaces and it can lead to a critical field larger than H c2 . To establish the importance of those effects, future angular-dependent studies would be necessary to identify the role played by surface superconductivity in CaKFe 4 As 4 close to T c . Using the extrapolated zero-temperature normal state resistivity ρ 0 ∼ 9.4 μ cm in Fig. 1(f) and carrier concentrations from Ref. [START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF], we can estimate a mean-free path of = 26.6 nm. Since the mean-free path due to elastic scattering from impurities is far larger than the coherence length, ξ (0)

, CaKFe 4 As 4 can be described as being in the clean limit. Resistivity data in Fig. 1(f) also show a sharp superconducting transition indicating a highquality single crystal. Furthermore, the extrapolated high-field resistivity at low temperatures [see Fig. 4(d)] displays a T 2 dependence indicative of a Fermi-liquid behavior that extends up to ∼55 K.

To assess the role of orbital and Pauli paramagnetic effects on the upper critical field of CaKFe 4 As 4 , we first describe the temperature dependence of the upper critical field using the three-dimensional Werthamer-Helfand-Hohenberg (WHH) model [START_REF] Werthamer | Temperature and purity dependence of the superconducting critical field, H c2 . III. electron spin and spin-orbit effects[END_REF], with the inclusion of spin paramagnetism effects. The slope close to T c (H c2 = -|dH c2 /dT | T =T c ) is used to estimate the zero-temperature orbital upper critical field and H orb c2 = 0.73T c H c2 for the clean limit (0.69T c H c2 in the dirty limit) for a single-band weak-coupling superconductor with ellipsoidal Fermi surface [START_REF] Helfand | Temperature and purity dependence of the superconducting critical field, H c2 . II[END_REF]. We find that the orbital pair breaking dominates the temperature dependence for H||c down to 10 K, below which it deviates, as shown in Fig. 2(c). However, when the magnetic field is aligned along the conducting (ab) plane, a Pauli pair-breaking contribution has to be included which reduces the orbital-limited critical field by

μ 0 H P = μ 0 H orb c2 / √ 1 + α 2 ,
where α is the Maki parameter. The extracted Maki parameter α is small where the orbital effects dominate (α ∼ 0.25 for H||c), but it becomes significant reaching α = 4.2 for H||(ab) [see Fig. 2(c)]. This value is close to that of FeSe 0.6 Te 0.6 single crystals where α ∼ 5.5, suggesting that the upper critical field is dominated by Pauli paramagnetic effects [START_REF] Khim | Evidence for dominant Pauli paramagnetic effect in the upper critical field of single-crystalline FeTe 0.6 Se 0.4[END_REF]. Strong paramagnetic effects are an important signature of optimally doped ironbased superconductors [START_REF] Tarantini | Significant enhancement of upper critical fields by doping and strain in iron-based superconductors[END_REF][START_REF] Cho | Anisotropic upper critical field and possible Fulde-Ferrell-Larkin-Ovchinnikov state in the stoichiometric pnictide superconductor LiFeAs[END_REF][START_REF] Gurevich | Iron-based superconductors at high magnetic fields[END_REF][START_REF] Maiorov | Inversion of the upper critical field anisotropy in FeTeS films[END_REF][START_REF] Fuchs | Orbital and spin effects for the upper critical field in As-deficient disordered Fe pnictide superconductors[END_REF]. For a clean isotropic single band, the Maki parameter is given by α = π 2 /(4E F ). The values of the measured superconducting gap varies between 2.4 -13 meV whereas the Fermi energies vary significantly for different bands, being smallest for the inner hole band, α (∼3 meV) and the shallow electron band δ (∼10 meV) [START_REF] Mou | Enhancement of the Superconducting Gap by Nesting in CaKFe 4 As 4 : A New High Temperature Superconductor[END_REF][START_REF] Khasanov | In-plane magnetic penetration depth of superconducting CaKFe 4 As 4[END_REF]. The band-structure calculation predicts four electron pockets centered at the M point, as shown in Fig. 4. Experimentally, only one electron pocket can be resolved in experiments due to a significant intrinsic linewidth and the fact that the bottom of these bands is located very close to Fermi level [START_REF] Mou | Enhancement of the Superconducting Gap by Nesting in CaKFe 4 As 4 : A New High Temperature Superconductor[END_REF]. Thus, the presence of the shallow bands, together with the small Fermi energies E F and large superconducting gap create conditions for large α and Pauli pair-breaking.

V. UPPER CRITICAL FIELD DESCRIBED BY A TWO-BAND MODEL

To describe the complete temperature dependence of H c2 (T ) for CaKFe 4 As 4 , including the upturn below ∼10 K for both orientations, a two-band model in the clean limit is considered, as detailed in Ref. [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF] and in the Appendix. This model accounts for the presence of two different bands with interband scattering (λ 11 , λ 22 ) and intraband effects (λ 12 , λ 21 ) and includes paramagnetic effects and allows for the presence of an FFLO inhomogeneous state at high fields and low temperatures [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF]. An FFLO state is characterized by a real-space modulation of the superconducting order parameter either in amplitude or phase such that the system energy is minimized under the constraints of a large Zeeman energy and superconducting condensation energy [START_REF] Song | Quantum FFLO State Iclean Layered Superconductors[END_REF]. As the FFLO wave vector Q appears spontaneously below a certain temperature, T FFLO , the spinodal instability line in H c2 (T ) at a finite Q acquires the characteristic upturn [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF].

In most iron-based superconductors, the pairing is expected to be mediated by spin fluctuations leading to a sign changing s ± order parameter. In this case, upper critical fields are generally described by dominant interband coupling parameters with λ 11 = λ 22 = 0 and λ 12 = λ 21 ∼ 0.5 (λ 11 λ 22 λ 12 λ 21 ) [START_REF] Hirschfeld | Gap symmetry and structure of Fe-based superconductors[END_REF][START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF][START_REF] Tarantini | Significant enhancement of upper critical fields by doping and strain in iron-based superconductors[END_REF][START_REF] Cho | Anisotropic upper critical field and possible Fulde-Ferrell-Larkin-Ovchinnikov state in the stoichiometric pnictide superconductor LiFeAs[END_REF][START_REF] Kano | Anisotropy of the upper critical field in a Co-doped BaFe 2 As 2 single crystal[END_REF][START_REF] Bristow | Upper critical fields across the nematic end point in FeSe 1-x S x[END_REF]. Furthermore, in these multiband systems, orbital fluctuations can dominate the pairing interactions in certain conditions favoring a gap with equal sign on each pocket, resulting in the s ++ pairing [START_REF] Kontani | Orbital-Fluctuation-Mediated Superconductivity in Iron Pnictides: Analysis of the Five-Orbital Hubbard-Holstein Model[END_REF] (λ 11 λ 22 λ 12 λ 21 ). To model the upper critical fields for both field orientations, we have considered different input parameters from previous experiments on CaKFe 4 As 4 . For example, the ratio between the extreme velocities of the Fermi surfaces is taken from ARPES experiments [START_REF] Khasanov | In-plane magnetic penetration depth of superconducting CaKFe 4 As 4[END_REF] and gives a starting value of η c ∼ 0.02, which is the squared velocity ratio between the two bands. Upper critical field simulations have a strong sensitivity to η values, as shown in Fig. 8 I). The values of v 2 are similar in both orientations, indicative of a strongly warped sheet (see Fig. 4). On the other hand, v 1 is larger for H||c and is close to ∼360 meVÅ, reported for the α hole band from ARPES measurements [START_REF] Khasanov | In-plane magnetic penetration depth of superconducting CaKFe 4 As 4[END_REF]. The band-coupling constants, λ i,j , that describe our data, suggest that the interband scattering due to spin fluctuations is dominated by the intraband effects in CaKFe 4 As 4 , which can occur in the presence of strong orbital fluctuations [START_REF] Kontani | Orbital-Fluctuation-Mediated Superconductivity in Iron Pnictides: Analysis of the Five-Orbital Hubbard-Holstein Model[END_REF]. For s ++ pairing to overcome s ± pairing normally requires the presence of strong disorder, which leads to similar superconducting gaps on different Fermi surfaces [START_REF] Efremov | Disorder-induced transition between s ± and s ++ states in two-band superconductors[END_REF]. Furthermore, the band coupling constants for the s ++ case resemble those used to describe a two-band superconductor in the dirty limit [START_REF] Golubov | Specific heat of MgB 2 in a one-and a two-band model from first-principles calculations[END_REF]. However, CaKFe 4 As 4 is clean, suggesting that disorder effects are negligible. Thus, based on the band-coupling parameters, we conclude that orbital fluctuations may dominate over spin fluctuations.

When the magnetic field lies along the conducting planes, the low-temperature upturn of H c2 (T ) in Fig. 3(b) cannot be fully captured by only considering the two-band model with large Pauli paramagnetic effects, α ∼ 3.2. The formation of the FFLO state in a system with a cylindrical Fermi surface requires a large Zeeman energy and a critical Maki's parameter of α c = 4.76 [START_REF] Song | Quantum FFLO State Iclean Layered Superconductors[END_REF], compared to α c = 1.8 [START_REF] Gruenberg | Fulde-Ferrell Effect in Type-II Superconductors[END_REF] for a threedimensional Fermi surface. CaKFe 4 As 4 has a complex Fermi surface with two-dimensional cylindrical and highly warped sheets and together with a large value of α creates the conditions for the emergence of an FFLO state [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF]. A FFLO state can be realized in very clean materials with weak scattering of quasiparticles and it generally manifests as change in slope in the upper critical field at low temperatures [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF]. We find indeed that to describe the upper critical field data of CaKFe 4 As 4 over the entire temperature range, a FFLO state could be stabilized below T FFLO ∼ 14 K, as shown in Fig. 3(c). Possible contenders to support such an effect are the shallow electron Fermi surface pocket, δ, in the zone corner which is very close to the Fermi level (∼10 meV) as well as the inner hole band α (∼3 meV), as detected by ARPES measurements [START_REF] Mou | Enhancement of the Superconducting Gap by Nesting in CaKFe 4 As 4 : A New High Temperature Superconductor[END_REF].

Q / Q(0) H||(ab) Q/Q(0) 0 H c2 (T) s s ++ (a) (b) (c)
The temperature dependence of the upper critical field data in CaKFe 4 As 4 implies that the intraband scattering is likely to dominate the interband scattering, the latter of which is promoted by spin fluctuations. In the presence of spin-orbit coupling, the orbital fluctuations can lead to s ++ pairing, as suggested for LiFeAs [START_REF] Saito | Revisiting orbital-fluctuation-mediated superconductivity in LiFeAs: Nontrivial spin-orbit interaction effects on the band structure and superconducting gap function[END_REF]. Usually, s ++ pairing results in far lower critical fields than s ± pairing [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF], so our results for CaKFe 4 As 4 are somehow unusual as H c2 is almost 3 T c at the lowest temperature. The presence of several scattering channels in a multiband system like CaKFe 4 As 4 can increase the upper critical field to a much greater extent than in single gap superconductors, caused by the relative weight of different scattering channels. CaKFe 4 As 4 theoretically has up to six large cylindrical hole bands, (with the equivalent of 50% hole doping in a 122 iron-based superconductor) providing a large density of states (see Figs. 1(e) and 4 and Ref. [START_REF] Khasanov | In-plane magnetic penetration depth of superconducting CaKFe 4 As 4[END_REF]) and it can also promote intraband scattering driven by orbital or electron-phonon couplings. On the other hand, the shallow bands in CaKFe 4 As 4 are likely to be involved in the stabilization of the FFLO state. Thus, the temperature dependence of the upper critical field of CaKFe 4 As 4 can reflect the behavior of two dominant superconducting gaps that reside on different sheets of the cylindrical Fermi surface, possibly one on a large hole band and another on a shallow small band. A theoretical work studying the superconducting instabilities in CaKFe 4 As 4 also found two different symmetry representations of the superconducting gap suggesting different pairing interactions, involving both conventional s± wave but also pairing that would lead to additional sign change within the hole and electron pockets [START_REF] Lochner | Electronic properties, low-energy Hamiltonian, and superconducting instabilities in CaKFe 4 As 4[END_REF] In summary, we have experimentally mapped the upper critical fields of CaKFe 4 As 4 up to 90 T and down to 4.2 K, providing a complete phase diagram for this stoichiometric superconductor. The anisotropy decreases dramatically with temperature, the system becoming essentially isotropic near 4 K. Upper critical fields are extremely large in this system, reaching close to ∼3 T c at the lowest temperature, well above the expectation based on conventional single-band superconductivity. Instead, the temperature dependence of the upper critical field can be described using by a two-band model in the clean limit. The band-coupling constants suggest a dominant orbital pairing over spin fluctuations pairing in CaKFe 4 As 4 . Furthermore, for magnetic fields aligned in the conducting plane and due to large Pauli paramagnetic effects, we find that the temperature dependence of the upper critical field is consistent with the emergence of an FFLO state at low temperatures.

In accordance with the EPSRC policy framework on research data, access to the data will be made available [42].

for the technical support during the pulsed field experiments.

APPENDIX

In this Appendix, we present additional data to support the findings presented in the main paper.

Experimental details.

Several high-quality single crystals were used in this study with large residual resistivity ratios RRR = ρ(300 K)/ρ(36 K) ∼ 14.5, small ρ 0 values of ∼9.2 μ cm and sharp superconducting transitions with T c ∼ 0.1 K, as shown in Figs. 5(c) and 1(f). Temperature sweeps were performed at constant magnetic field for different orientations, H||c and H||(ab) to build the high-temperature part of the H c2 (T ) phase diagram. Magnetic fields up to 90 T were produced by using current pulses through two different solenoid coils. An example of the magnetic field produced as a function of time is shown in Fig. 6.

Determining superconducting coherence lengths

The superconducting coherence lengths can be found in both field orientations near T c using the Ginzburg-Landau (GL) approach from the equations

ξ GL ab = 0 2πμ 0 dH c c2 dT T c T c , ξ GL c = 0 2πμ 0 dH ab c2 dT T c T c ξ GL ab , ( A1
)
where 0 is the magnetic flux quanta, μ 0 |dH c c2 /dT | T c is the slope near T c when H||c, μ 0 |dH ab c2 /dT | T c is the slope near T c when H||(ab), and ξ GL ab and ξ GL c are the respective GL coherence lengths.

At the lowest temperature, the coherence lengths are extracted using μ 0 H c2 (T → 0) and the following = λ c /λ ab = 313.68/70 = 4.5, using a similar approached used in Ref. [START_REF] Hashimoto | Evidence for superconducting gap nodes in the zone-centered hole bands of KFe 2 As 2 from magnetic penetration-depth measurements[END_REF]. These calculations are in agreement with previous results reported in Ref. [START_REF] Suetin | Electronic properties and Fermi surface for new layered high-temperature superconductors CaAFe4As4 (A = K, Rb, and Cs): FLAPW-GGA calculations[END_REF]. equations:

ξ ab = 0 2πμ 0 H c c2 (T → 0) , ξ c = 0 2πμ 0 H ab c2 (T → 0)ξ ab , ( A2
)
where ξ ab and ξ c are the coherence lengths in the (ab) plane and along the c axis, and μ 0 H c c2 (T → 0) and μ 0 H ab c2 (T → 0) are the critical fields at the lowest temperature when H||c and H||(ab), respectively (see Fig. 7). Using Eqs. (A2) at different constant temperatures, one can estimate ξ i → ξ i (T ) and H c2 → H c2 (T ), and the anisotropy ratio, , as shown in Fig. 2(b) in the main text and Fig. 7.

FIG. 5. Transport data of CaKFe 4 As 4 as a function of temperature and magnetic field. (a) Estimating the critical temperature, T c , from resistivity against temperature in zero magnetic field (left axis) and from magnetic susceptibility in a small applied field of 5 mT (right axis). The midpoint of the transition, T mid , is identified as the peak in the derivative dρ/dT . Linear fits are performed above and below the midpoint transition and the intercepts are defined as T off and T on , as labeled in the figure. T c from susceptibility is defined as the point at which diamagnetism occurs and deviates from the high-temperature background and it is closest to the offset temperature in transport. The critical temperatures used to build the phase diagrams in magnetic field were extracted using T off from the transport data. (b) The estimation of the upper critical field, H off , from resistivity in pulsed magnetic field data measured at constant temperature using the same method as in (a). (c) Resistivity against temperature for different samples of CaKFe 4 As 4 between 300 K and 2 K. The superconducting transition is very sharp, with T c,0 = 35.0 K and width T c = 0.15 K, values typical for all measured single crystals indicating their high quality. (d) Resistivity data as a function of magnetic field for the sample S1 when H ||(ab) at 18 K. The dashed black line shows the linear extrapolation used to obtain the zero-field resistivity at 18 K and the intercept, ρ H →0 (T ), and gradient are shown. Using this method, the extracted values of zero-field resistivity measured at constant temperatures are shown in Fig. 1(f).

Depairing current density

Using the coherence length, one can also determine the critical depairing current density of a superconductor, J d . As a function of temperature and magnetic field, this is expressed as [START_REF] Tinkham | Introduction to Superconductivity[END_REF]

J d (T, H ) = 2 3μ 0 λ 2 (T ) μ 0 H 2 c2 (T ) -H 2 1/2 0 6π , ( A3
)
where λ is the superconducting penetration depth and 0 is the magnetic flux quantum. At zero temperature and in no magnetic field, this gives

J d (0, 0) = 2 3μ 0 λ 2 (0) μ 0 H c2 (0) 0 6π . ( A4
)
Substituting μ 0 H c2 = 0 /2πξ 2 , this equation can be rearranged to give

J d (0, 0) = 0 3 √ 3πμ 0 1 λ 2 (0)ξ (0) . ( A5
)
Previous magnetization reported μ 0 H c1 = 22(1) mT [START_REF] Singh | Ultrahigh critical current densities, the vortex phase diagram, and the effect of granularity of the stoichiometric high-T c superconductor CaKFe 4 As 4[END_REF] when H||c at low temperature which, combined with H c2 values reported here, give κ ∼ 99(2) and λ = 183(6) nm. Using ξ = 1.86 nm, we find that the depairing current density is J d ∼ 1.62 × 10 8 A/cm 2 , which is among the largest of any known iron-based superconductor.

The upper critical fields using a two-band model

To describe the entire temperature dependence of H c2 (T ) for CaKFe 4 As 4 , including the upturn below ∼10 K for both orientations, we use a two-band model in the clean limit. This models includes paramagnetic effects and allows the emergence of a FFLO state, as detailed in Ref. [START_REF] Gurevich | Upper critical field and the Fulde-Ferrell-Larkin-Ovchinnikov transition in multiband superconductors[END_REF]. The values of the upper critical field for different temperatures are found from estimating the following expressions: where a 1 = (λ 0 + λ -)/2w, a 2 = (λ 0 -λ -)/2w, λ ± = λ 11 ± λ 22 , w = λ 11 λ 22 -λ 12 λ 21 , and λ 0 = (λ 2 -+ 4λ 12 λ21) 1/2 . Here λ i j represents a coupling constant between bands i and j, and as the cross terms λ 12 and λ 21 only appear together, we set λ 21 = λ 12 for simplicity. The values of the coupling constants allows us to establish the dominant pairing. For s ++ pairing, for strong intraband coupling w > 0 [w ∼ 0.248 for the case presented in Figs. 3(c Intraband pairing could indicate a nonsign changing order parameter (s ++ ) between different Fermi surfaces. However, many iron-based materials are dominated by strong interband pairing, suggestive of a sign changing order parameter (s ± ).

a 1 [ln t + U 1 ] + a 2 [ln t + U 2 ] + [ln t + U 1 ][ln t + U 2 ] = 0, ( A6
U 1 and U 2 are defined as

U 1 = 2e q 2 Re ∞ n=0 ∞ q due -u 2 × u n + 1/2 - t √ b tan -1 u √ b t (n + 1/2) + iαb , (A7) U 2 = 2e q 2 s Re ∞ n=0 ∞ q √ s due -u 2 × u n + 1/2 - t √ bη tan -1 u √ bη t (n + 1/2) + iαb (A8)
The variables used for this two-band model, b, α, q, η, and s, are defined as

b = h2 v 2 1 H c2 8πφ 0 k B 2 T 2 c , α = 4μφ 0 k B T c h2 v 2 1 , q 2 = Q 2 φ 0 1 2π H c2 , η = v 2 2 v 2 1 , s = 2 1 . ( A9
)
v i is the in-plane Fermi velocity of band i, i is the mass anisotropy ratio, i = m ⊥ i /m i is related to the ratio between the gradients near T c in different field orientations, s is band mass anisotropy between the two bands, and γ = -1/2 . Various simulations of upper critical fields using this two-band model are shown in Fig. 8. The slope of H c2 near T c of CaKFe 4 As 4 , is -3.3 for H||c and -10.8 for H||(ab), giving an ∼ 1/10. Q is the magnitude of the FFLO modulation and it is found for a given temperature when H c2 is maximal (dH c2 /dQ = 0). The simulations of H c2 for H||(ab) assumed that the bands have the same anisotropy parameter 1 = 2 = . Other parameters are estimated based on the fitted values using a single-band model and the velocities are minimized locally to find an optimum solution. This approach was used to infer the presence of either interband or intraband dominated pairings, as described in the main body.

A rescaling of the upper critical field has been performed previously mapping

H c c2 → H ab c2 using q ab → q c -3/4 , α ab → α c -1/2 , b 1/2 ab → b 1/2 c 1/4 in U 1 , (η ab b ab ) 1/2 → (η c b 1/2 c ) 1/4 in U 2 .
The values of the α parameters change as α ab,1 → α c -1/2 and α ab,2 → α/(η 1/2 ). However, this rescaling has not been used in this study, as the ratio between the Fermi velocities η is a variable parameter that can be changed between the two field orientations.

Previous studies on CaKFe 4 As 4 have described the onset upper critical field data up to 60 T using the parameters λ 11 = λ 22 = 0, λ 12 λ 21 = 0.25, η = 0.2, α = 0.5, = 1/6, and the above rescaling [START_REF] Meier | Anisotropic thermodynamic and transport properties of single crystalline CaKFe 4 As 4[END_REF]. Using these parameters, we find that the onset upper critical field data (open blue and red symbols) clearly fail to describe the full temperature dependence of the upper critical field of CaKFe 4 As 4 , as shown in Figs. 9(a) and 9(b) (blue and red curves). As discussed in the main body of text, we can reasonably describe the upper critical field of CaKFe 4 As 4 using a two-band model with band parameters for the s ± and s ++ pairing (see Table I), as shown in Figs. 9(c) and 9(e), respectively. We note here that an FFLO state can be stabilized at low temperatures for H||(ab) [Figs. 9(c) and 9(e), right axis].

To further test the obtained parameters, we have modelled both the offset and onset critical fields for both field orientations assuming the s ± and s ++ pairing. The parameters for the two cases are listed in Table I and the simulations of the upper critical field are shown in Fig. 9. We find that the parameters describing the offset and onset upper critical fields are similar, with small differences found for the α values, which are slightly larger as the curvature for H on is greater as compared with the H off case. The velocities for H on are also slightly smaller due to the larger slopes near T c , which scales as |dH c2 /dT | T c ∝ 1/v 2 . Using the values for H off and H on , we expected a ∼20% difference in velocities when H||c ) for H ||(ab), and it is scaled to the zero-temperature value Q(0). We note that the FFLO state emerges when H ||(ab), regardless of the pairing limit or which critical field criteria is chosen. This reinforces the robustness of this result. and a ∼10% difference when H||(ab), as shown in Table I. However, independent of the definition of the upper critical field, the stabilization of an FFLO state in CaKFe 4 As 4 for H||(ab) is realized in both cases.

FIG. 1 .

 1 FIG. 1. Resistivity versus temperature for CaKFe 4 As 4 (S1) at different constant magnetic fields for (a) H ||c and (b) H ||(ab). Resistivity versus magnetic field at constant temperatures measured concomitantly in pulsed fields up to 90 T, for (c) H ||c (S2) and (b) H ||(ab) (S1). (e) Fermi surface of CaKFe 4 As 4 using experimental lattice parameters (as detailed in Fig. 4). The colors reflect the variation of the Fermi velocity. (f) Resistivity against temperature for sample S1 together with the zero-field extrapolated values from high fields shown in (c) and (d) [see also Fig. 5(d) in the Appendix]. The zero-temperature resistivity ρ 0 is estimated as 9.4(2) μ cm using a Fermi-liquid T 2 behavior [red line in (f)].contacts were achieved using indium solder, giving contact resistances of less than 0.5 . Samples were measured using the standard four-and five-point AC lock-in techniques and a current of 1 mA. We investigated several high-quality single crystals, with large residual resistivity ratios ∼14.5 [RRR = ρ(300 K)/ρ(36 K)], small residual resistivity, ρ 0 ∼ 9.4 μ cm, and sharp superconducting transitions, T c ∼ 0.1 K [see Figs. 1(f) and 5(c)]. Transport measurements were performed in Oxford up to 16 T in a Quantum Design PPMS in constant magnetic fields and for different magnetic field orientations [H||c and H||(ab)]. Pulsed magnetic-field measurements were performed at LNCMI Toulouse, using a 70 T single coil and a 90 T dual coil at constant temperatures below 35 K for both field orientations (an example is shown in Fig.6).

Figure 1 (

 1 Figure 1(a)-1(b) shows resistivity against temperature for different fixed magnetic fields for orientations of the sample in relation to the applied magnetic field. As the magnetic field increases, the superconducting transition becomes broader and suppressed faster for H||c by 5 K in 16 T [Fig. 1(a)], as compared with the H||(ab) case for which H c2 only changes by 2 K in 16 T [Fig. 1(b)], as the orbital effects are less effective in suppressing superconductivity in this orientation. These data are used to extract the upper critical field near T c , defined as the offset field, as shown in Figs. 5(a) and 5(b).Due to the high crystallinity of our samples that display sharp superconducting transitions, we find that the critical tem-

FIG. 2 . 2 .

 22 FIG. 2. (a) Upper critical fields as a function of temperature in CaKFe 4 As 4 when H ||c (solid squares) and H ||(ab) (open triangles), together with previous reported data measured up to 60 T from Ref. [7]. (b) The temperature dependence of coherence length extracted close to T c , as described in the Supplemental Material. The H c2 (T = 0) values were used to find the zero-temperature coherence lengths. The horizontal dashed line represents the 3D-2D crossover when coherence length ∼c/ √ 2. The anisotropy = H ab c2 /H c c2 on the right axis approaches = 1 at ∼4.2 K. Solid lines are guides to the eye. (c) Upper critical fields for H ||c (black squares) and H ||(ab) (black triangles) scaled by the superconducting transition temperature, T c , and the slope near T c from the WHH model against reduced temperature T /T c . The dashed blue and red lines are fits to the WHH model for H ||c and H ||(ab) using different values of α. Deviation from the WHH model occurs at low temperatures for both field orientations.

  in the Appendix, and it can change for different field orientations due to differences in the in-plane velocities on the Fermi surface [see Figs. 1(e) and 4]. We assume that s = 1 [defined by Eqs. (A9) in the Appendix)] for both field orientations and the starting values of the Maki parameters are those from the single band model of H c2 (T ) in Fig. 2(c). First, the H c2 (T ) data are simulated for CaKFe 4 As 4 using parameters corresponding to the s ± case [dashed lines Figs. 3(a) and 3(b)], which provide a reasonable representation of the observed behavior for both orientations. However, we find that H c2 (T ) is best described using λ 11 = 0.81, λ 22 = 0.29, and λ 12 = λ 21 = 0.1 [solid lines in Figs. 3(a) and 3(b)], which is consistent with s ++ pairing. For the later model, the constrained Fermi velocities are v 1 ∼ 380(20) meVÅ and v 2 ∼ 54(4) meVÅ when H||c, and v 1 ∼ 195(20) meVÅ and v 2 ∼ 41(4) meVÅ when H||(ab) (see Table

FIG. 3 .

 3 FIG. 3. Upper critical fields for (a) H ||c (solid squares) and (b), (c) H ||(ab) (open triangles) described by the two-band model. The dotted lines correspond to s ± pairing [λ 11 = λ 22 = 0, λ 12 = λ 21 ∼ 0.5, α 1 = 0.95, α 2 = 0 for H ||c and α 1 = 3.1 and α 2 = 0.7 for H ||(ab)]. Solid lines represent s ++ pairing [λ 11 = 0.81, λ 22 = 0.29 and λ 12 = λ 21 = 0.1, α 1 = 0.5 for H ||c and α 1 = 3.2, α 2 = 0 for H ||(ab)]. The values of η vary between ∼0.02 in (a) and ∼0.04 in (b). (c) The temperature dependence of the upper critical field including the emergence of the FFLO state for the two different pairing symmetries with the same parameters as in (b). The lower dashed and red lines represent the scaled FFLO Q/Q(0) modulation for H ||(ab) for the respective pairing (right axis).

FIG. 4 .

 4 FIG. 4. Fermi surface of CaKFe 4 As 4 .(a) Fermi surface with velocity color. Slices at the (b) and (c) Z point, indicating the size of the different sheets. Solid lines indicate the Brillouin zone. (d) Fermi surface slice along the [110] diagonal of the Brillouin zone indicated by the solid lines. The calculations were performed using WIEN2K and GGA approximation and the experimental lattice parameters described by the P4/mmm symmetry group were a = b = 3.86590 Å, c = 12.88400 Å [5]. The calculated value of anisotropy,= λ c /λ ab = 313.68/70 = 4.5, using a similar approached used in Ref.[START_REF] Hashimoto | Evidence for superconducting gap nodes in the zone-centered hole bands of KFe 2 As 2 from magnetic penetration-depth measurements[END_REF]. These calculations are in agreement with previous results reported in Ref.[START_REF] Suetin | Electronic properties and Fermi surface for new layered high-temperature superconductors CaAFe4As4 (A = K, Rb, and Cs): FLAPW-GGA calculations[END_REF].

FIG. 6 .

 6 FIG.6. Generation of the high-magnetic field pulse. Applied magnetic field as a function of time from the 90 T magnet in Toulouse. The peak of this pulse was ∼88 T and is generated by overlaying pulsed current though two different coils. The pulse on the inner coil which generates the extremely high fields lasts ∼30 ms.

FIG. 7 .

 7 FIG. 7. Coherence length extraction. Upper critical fields in CaKFe 4 As 4 as a function of temperature. The dashed blue and red lines are linear fits near T c when H ||c and H ||(ab), respectively. The gradients of these fits (shown in the figure) were used to calculate the coherence lengths using the Ginzburg-Landau theory [Eq. (A1)]. The solid blue and red lines are linear fits to the low-temperature region when H ||c and H ||(ab), respectively. The zero-temperature extrapolation of H c2 (T → 0) for the two orientations were used to find the zero-temperature coherence lengths, as shown in the figure.

  ) and 9(c)], whereas for strong interband coupling w < 0 [Figs. 3(a), 3(b), 9(a), and 9(b)].

FIG. 8 .

 8 FIG. 8. Two-band upper-critical-field simulations for different band-coupling parameters and values of η. Simulations of the reduced upper critical field h using the two-band model based on Eqs. (A6)-(A9). The parameters listed in each panel correspond to s ± pairing in (a) and (b) with λ 11 = λ 22 = 0, λ 12 = λ 21 = 0.5, s ± pairing in (c) and (d) with λ 11 = λ 22 = 0.81, λ 12 = λ 21 = 0.5, and s ++ pairing in (e) and (f) with λ 11 = 0.81, λ 22 = 0.29, and λ 12 = λ 21 = 0.1, α 1 = 0.95 for H ||c. For panels (b), (d), and (f) the lower curves (left corners) represent the magnitude of the FFLO modulation, Q/(2Q(0)).

FIG. 9 .

 9 FIG. 9. Two-band upper-critical-field models for CaKFe 4 As 4 using different band-coupling parameters. (a), (b) λ 11 = λ 22 = 0, λ 12 λ 21 = 0.25, η = 0.2, α = 0.5, and = 1/6 used previously to describe the onset upper critical field of CaKFe 4 As 4 up to 60 T [7] when H ||c and H ||(ab), respectively. (c) s ± pairing with λ 11 = λ 22 = 0, λ 12 = λ 21 = 0.5. Here α 1 = 0.95, α 2 = 0, and η ∼ 0.006 for H ||c, and α 1 = 3.1, α 2 = 0.7, and η ∼ 0.04 for H ||(ab). (d) s ± pairing for the onset critical field using the same parameters as in (c), but with η ∼ 0.008 and α 1 = 1.5 when H ||c and α 1 = 4.2 when H ||(ab). (e) s ++ pairing with λ 11 = 0.81, λ 22 = 0.29, and λ 12 = λ 21 = 0.1. Here α 1 = 0.95, α 2 = 0, and η ∼ 0.02 for H ||c, and α 1 = 3.1, α 2 = 0.0 and η ∼ 0.04 for H ||(ab). (f) s ++ pairing for the onset critical field using the same parameters as in (e), but with α 1 = 1.0 when H ||c, and α 1 = 3.5 when H ||(ab). In (c)-(f), blue lines show the two-band model (with no FFLO state present) when H ||c, dashed black lines show the two-band model with no FFLO state when H ||(ab), and the red lines show the two-band model with an FFLO state when H ||(ab). The temperature dependence of the FFLO Q vector is shown by the dashed red lines (the right axis in (c)-(f)) for H ||(ab), and it is scaled to the zero-temperature value Q(0). We note that the FFLO state emerges when H ||(ab), regardless of the pairing limit or which critical field criteria is chosen. This reinforces the robustness of this result.

  This work was mainly supported by Oxford Centre for Applied Superconductivity and EPSRC Grants (No. EP/I004475/1 and No. EP/EP/I017836/1). P.R. acknowledges the support of the Oxford Quantum Materials Platform (Grant No. EP/M020517/1). Part of this work was supported HFML-RU/FOM and LNCMI-CNRS, members of the European Magnetic Field Laboratory (EMFL) and by EPSRC (UK) via its membership to the EMFL (Grant No. EP/N01085X/1). Part of this work at the LNCMI was supported by Programme Investissements d'Avenir under Program No. ANR-11-IDEX-0002-02, Reference No. ANR-10-LABX-0037-NEXT. Work at Ames Laboratory was supported by the U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames Laboratory is operated by the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358. W.M. was supported by the Gordon and Betty Moore Foundation's EPiQS Initiative through Grant No. GBMF4411. We also acknowledge financial support of the John Fell Fund of the Oxford University. A.I.C. acknowledges an EPSRC Career Acceleration Fellowship (No. EP/I004475/1) and Oxford Centre for Applied Superconductivity [43].

TABLE I .

 I The different parameters used for the simulations of the upper critical field (defined either using an offset or onset magnetic field), using a two-band model and considering different pairing symmetries and magnetic field orientations in CaKFe 4 As 4 .

			μ 0 H off				μ 0 H on
		s ±		s ++		s ±		s ++
		H ||c	H ||(ab)	H ||c	H ||(ab)	H ||c	H ||(ab)	H ||c	H ||(ab)
	λ 11	0	0	0.81	0.81	0	0	0.81	0.81
	λ 22	0	0	0.29	0.29	0	0	0.29	0.29
	λ 12	0.5	0.5	0.1	0.1	0.5	0.5	0.1	0.1
	α 1	0.95	3.1	0.5	3.2	1.5	4.2	1.0	3.5
	α 2	0	0 . 7	0	0	0	0	0	0
	v 1 (meVÅ)	510(20)	253(20)	380(20)	195(20)	400(20)	215(20)	312(20)	181(20)
	v 2 (meVÅ)	40(4)	51(5)	54(4)	41(4)	36(3)	43(4)	44(4)	38(4)
	η	∼0.006	∼0.04	∼0.02	∼0.04	∼0.008	∼0.04	∼0.02	∼0.045

ACKNOWLEDGMENTS

We thank Alex Gurevich for helpful discussions related to the two-band modeling of the upper critical field and Jérôme Béard, Marc Nardone, and Abdelaziz Zitouni

Competing pairing interactions responsible for the large upper critical field in a stoichiometric iron-based superconductor CaKFe 4 As 4


Pascal Reiss

David Graf

Amir A Haghighirad

William Knafo

Loïc Drigo

Matthew Bristow

Andrew J Schofield

Amalia I Coldea

Quenched nematic criticality separating two superconducting domes in an iron-based superconductor under pressure

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

An electronic nematic ordered state can occur in a strongly interacting electronic system in which a Fermi surface undergoes a spontaneous distortion to a shape with lower symmetry compared to the underlying crystal lattice. The observation of electronic nematic order in different families of hightemperature superconductors imply that the same interactions may be involved in stabilizing both the nematic and superconducting states [START_REF] Fernandes | What drives nematic order in iron-based superconductors?[END_REF][START_REF] Lederer | Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point[END_REF][START_REF] Sprau | Discovery of Orbital-Selective Cooper Pairing in FeSe[END_REF]. However, the presence of other competing electronic phases, such as spin or charge-density waves neighbouring superconductivity can obscure the relevance of the nematic fluctuations in superconducting pairing. Hydrostatic pressure is a powerful method not only to fine-tune across different electronic phase transitions and disentangle competing electronic phases, but also to access new regimes, as often found in organic superconductors and iron-based superconductors [START_REF] Furukawa | Quantum criticality of Mott transition in organic materials[END_REF][START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. FeSe is a striking example of a nematic superconductor in which applied pressure leads to a four-fold increase in its bulk superconductivity (from 9 K towards a high-critical temperature, T c ∼ 40 K) [START_REF] Mizuguchi | Superconductivity at 27 K in tetragonal FeSe under high pressure[END_REF][START_REF] Medvedev | Electronic and magnetic phase diagram of β-Fe1.01Se with superconductivity at 36.7 K under pressure[END_REF]. In the normal state, the nematic phase transition of FeSe is suppressed with increasing pressure, but the quantum phase transition is masked by an emerging magnetic ordering stabilized under high pressures [START_REF] Terashima | Pressure-Induced Antiferromagnetic Transition and Phase Diagram in FeSe[END_REF][START_REF] Terashima | Fermi surface reconstruction in FeSe under high pressure[END_REF][START_REF] Wang | Nematicity and quantum paramagnetism in FeSe[END_REF][START_REF] Wang | Strong interplay between stripe spin fluctuations, nematicity and superconductivity in FeSe[END_REF][START_REF] Sun | High-Tc Superconductivity in FeSe at High Pressure: Dominant Hole Carriers and Enhanced Spin Fluctuations[END_REF][START_REF] Kothapalli | Strong cooperative coupling of pressure-induced magnetic order and nematicity in FeSe[END_REF][START_REF] Imai | Why does undoped FeSe become a high-Tc superconductor under pressure?[END_REF][START_REF] Bendele | Pressure Induced Static Magnetic Order in Superconducting FeSe1-x[END_REF]. As the nematic and magnetic phases of FeSe are intertwined in the pressure-temperature phase diagram, it is difficult to establish the roles played by nematic or spin fluctuations for pairing and stabilizing a high-T c state. Similarly to applied external pressure, isoelectronic sulfur substitution in FeSe 1-x S x , equivalent to internal positive chemical pressure, suppresses the ne-matic order, but does not stabilize a magnetically ordered state [START_REF] Watson | Emergence of the nematic electronic state in FeSe[END_REF][START_REF] Watson | Suppression of orbital ordering by chemical pressure in FeSe1-xSx[END_REF][START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF][START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF][START_REF] Hosoi | Nematic quantum critical point without magnetism in FeSe1-xSx superconductors[END_REF]. Consequently, by combining chemical pressure and hydrostatic pressure, the nematic quantum critical point is unmasked, as the magnetic order is shifted to higher pressures with increasing sulfur concentration [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF][START_REF] Hosoi | Nematic quantum critical point without magnetism in FeSe1-xSx superconductors[END_REF][START_REF] Xiang | Dome of magnetic order inside the nematic phase of sulfur-substituted FeSe under pressure[END_REF][START_REF] Yip | Weakening of the diamagnetic shielding in FeSe1-xSx at high pressures[END_REF]. This opens a unique path for studying the nature of nematic criticality in detail using hydrostatic pressure as a clean tuning parameter and to probe the role of nematic fluctuations in stabilizing superconductivity.

As superconductivity depends not only on the origin of the attractive pairing interaction but also on the details of the Fermi surface from which it emerges [START_REF] Coldea | The key ingredients of the electronic structure of FeSe[END_REF], understanding the changes in the electronic structure as a function of different tuning parameters is paramount. Furthermore, the size of the Fermi energy triggers different electronic instabilities either a nematic order at small values, superconductivity in an intermediate regime or a magnetically-ordered state for large energies [START_REF] Chubukov | Magnetism, superconductivity, and spontaneous orbital order in iron-based superconductors: Which comes first and why?[END_REF]. To test these different regimes in FeSe 1-x S x , one requires experimental probes which give access to the Fermi surface under applied pressures. Quantum oscillations are a powerful technique to access the evolution of the electronic structure and correlations under extreme conditions. In FeSe, quantum oscillations showed that the extremal areas of the Fermi surface increase in the nematic phase, and a sudden reconstruction was interpreted to occur in the magnetic phase at higher pressures [START_REF] Terashima | Pressure-Induced Antiferromagnetic Transition and Phase Diagram in FeSe[END_REF][START_REF] Terashima | Fermi surface reconstruction in FeSe under high pressure[END_REF]. On the other hand, quantum oscillations and ARPES studies using chemical pressure detected a continuous growth of the Fermi surface of FeSe 1-x S x , as the highly in-plane-distorted Fermi surface arXiv:1902.11276v1 [cond-mat.supr-con] 28 Feb 2019 of FeSe becomes isotropic in the tetragonal phase [START_REF] Watson | Suppression of orbital ordering by chemical pressure in FeSe1-xSx[END_REF][START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF][START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF]. Additionally, quantum oscillations detect an unusual Lifshitz transition in FeSe 1-x S x , which coincides with the disappearance of the nematic state [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF] and raises the prospect that these effects may be relevant for stabilising superconductivity.

In this Article we explore the electronic behaviour across the pressure-temperature phase diagram of FeSe 0.89 S 0.11 , in the absence of long-range magnetic order. Using external pressure, we finely tune the system across it critical nematic region to understand the role played by the electronic structure, scattering and correlations in stabilizing superconductivity. We identify different regimes of scattering in the proximity of the nematic and the high-T c phases from detailed resistivity measurements under pressure. The nematic phase is surrounded by an extended region with a ∼ T 3/2 resistivity, different from a prototypical quantum critical behaviour. Remarkably, at low temperatures a Fermi liquid behaviour is recovered across the nematic critical region. Quantum oscillations identify a Lifshitz transition at the nematic quantum phase transition, that separates two types of superconducting domes having different Fermi surface sizes and different strength of electronic correlations. Importantly, the cyclotron effective masses are significantly suppressed across the nematic phase transition, as well as superconductivity. We interpret this behaviour as a signature of quenched critical fluctuations due to a strong nematoelastic coupling to the lattice [START_REF] Paul | Lattice Effects on Nematic Quantum Criticality in Metals[END_REF].

Temperature dependence of resistivity with applied pressure. Figure 1A shows the evolution of the temperature dependence of the resistivity as a function of applied pressure below 100 K (the full range up to 300 K is shown in Fig. S1). With increasing pressure, the nematic transition temperature T s is quickly suppressed from 60 K at ambient pressure until it becomes unobservable around p c (Fig. S1). By extrapolating the pressure evolution of T s to zero, we can locate the position of the nematic quantum phase transition at p c ∼ 5.8(5) kbar in Fig. 1E. Within the nematic phase, there is only a weak pressure dependence of the superconducting transition temperature T c (defined as the zero resistivity temperature) similar to previous findings using chemical pressure [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF][START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF]. Outside the nematic phase, superconductivity is suppressed rather than enhanced in the proximity of the nematic quantum phase transition and reaches a minimum T c ∼ 6.4 K (at ∼ 7 kbar) close to p c . However, superconductivity is enhanced significantly towards 22 K at 20 kbar, and T c was reported to reach 30 K at higher pressures at around 30 kbar (see Fig. 4A) [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. The superconducting transition at high pressures broadens significantly compared with the low-pressure region (by a factor ≈ 10), suggesting that the superconducting phase becomes rather inhomogeneous at high pressures or that it coexists with another electronic phase. In contrast to FeSe where nematicity, magnetism and superconductivity may coexist under pressure [START_REF] Terashima | Pressure-Induced Antiferromagnetic Transition and Phase Diagram in FeSe[END_REF][START_REF] Sun | High-Tc Superconductivity in FeSe at High Pressure: Dominant Hole Carriers and Enhanced Spin Fluctuations[END_REF][START_REF] Xiang | Dome of magnetic order inside the nematic phase of sulfur-substituted FeSe under pressure[END_REF], magnetic order in our composition is expected to be stabilized only at high pressures exceeding 40 kbar [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF], and a reduction of the superconducting fraction was detected well-beyond 30 kbar in FeSe 0.88 S 0.12 [START_REF] Yip | Weakening of the diamagnetic shielding in FeSe1-xSx at high pressures[END_REF]. To elucidate the nature of quasi-particle scattering in the proximity of the nematic quantum phase transition, we investigate the resistivity exponent n, defined as ρ(T ) = ρ 0 +AT n , as a function of pressure and temperature. For a non-or weakly interacting Fermi liquid, one expects n = 2. In contrast, for systems in the proximity of the quantum critical points, enhanced order parameter fluctuations can lead to non-Fermi liquid behaviour with an exponent n < 2 and additional scattering [START_REF] Lederer | Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point[END_REF][START_REF] Rosch | Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point[END_REF][START_REF] Dmitrii | Resistivity of a Non-Galilean-Invariant Fermi Liq-uid near Pomeranchuk Quantum Criticality[END_REF][START_REF] Barber | Resistivity in the Vicinity of a van Hove Singularity: Sr2RuO4 under Uniaxial Pressure[END_REF]. The precise power law n value is influenced by the nature of the critical fluctuations (their wave-vector and dynamical exponent), by dimensionality [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF] by the amount of disorder present [START_REF] Rosch | Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point[END_REF] and by nearby quantum critical points [START_REF] Thomas | Quantum multicriticality[END_REF]. At low temperatures away form the critical points in a clean system, Fermi-liquid like behaviour is often recovered [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]. In this regime, the A coefficient, which is a measure of the strength of the electronic correlations as A ∼ (m * /m e ) 2 , diverges upon approaching the critical region as fluctuations become critical (m * is the quasiparticle effective mass) [START_REF] Kasahara | Evolution from non-Fermi-to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx) 2 superconductors[END_REF][START_REF] Analytis | Transport near a quantum critical point in BaFe2(As1xPx)2[END_REF].

Figure 1E shows the temperature and pressure dependence of n as a color map (defined as n = ∂ log(ρ -ρ 0 )/∂ log T ) with ρ 0 extrapolated from low temperatures) from which several different non-Fermi liquid scattering regimes can be identified. Firstly, inside the nematic phase, the resistivity follows a sublinear dependence, with n ranging from ∼ 0.4 -1, similar to previous observations as a function of chemical pressure [START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF][START_REF] Bristow | The anomalous high-magnetic-field electronic state of the nematic superconductors Fe(Se1-xSx)[END_REF][START_REF] Urata | Non-Fermi liquid behavior of electrical resistivity close to the nematic critical point in Fe1-xCoxSe and FeSe1-ySy[END_REF] or in FeSe under applied pressure [START_REF] Knöner | Resistivity and magnetoresistance of FeSe single crystals under helium-gas pressure[END_REF]. Secondly, in the tetragonal phase surrounding the nematic phase above p c , the resistivity displays a dominant n ∼ 3/2 dependence over a wide temperature and pressure range (Figs. 1B andC); small deviations towards a n = 4/3 power law are found for intermediate temperatures (T < 50 K) for pressures close to p c (Fig. 1B). Thirdly, in the high-pressure region (close to 20 kbar), the resistivity has an extended region of almost linear temperature dependence (Fig. 1A), similar to related pressure studies in the vicinity of the maximum T c for this composition [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. The observation of an (almost) linear dependence of the resistivity is unusual and is often associated with antiferromagnetic fluctuations close to a magnetic quantum critical point which, in this case, is away from the expected nematic critical point around p c [START_REF] Kasahara | Evolution from non-Fermi-to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx) 2 superconductors[END_REF].

Next, we comment on the possible origin of the non-Fermi liquid behaviour outside the nematic phase, where critical fluctuations are expected to affect scattering and the relevant resistivity exponents. Above p c , a prominent exponent n = 3/2 can be assigned to strong antiferromagnetic fluctuations in the dirty limit [START_REF] Rosch | Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point[END_REF]. However, an NMR study under pressure on a compound with similar sulfur concentration (x ∼ 0.12) suggested that spin fluctuations are suppressed outside the nematic phase above p c and a regime of nematic fluctuations extends towards ∼ 20 kbar [START_REF] Kuwayama | Two different superconducting states and possible antiferromagnetic quantum critical points in S-doped FeSe under pressure[END_REF]. This would imply that the n ∼ 3/2 region of non-Fermi liquid behaviour in resistivity can be caused by a region of strong nematic fluctuations. Theoretically, the nematic critical fluctuations can be exceedingly effective in destroying quasiparticles and can produce a striking nodal-antinodal dichotomy generating non-Fermi liquid-like behaviour over much of the Fermi surface enhancing resistivity [START_REF] Lederer | Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point[END_REF]. Furthermore, the proximity to a nematic quantum critical behaviour in certain quasi-2D systems was associated to a resistivity exponent of n = 4/3 [START_REF] Dmitrii | Resistivity of a Non-Galilean-Invariant Fermi Liq-uid near Pomeranchuk Quantum Criticality[END_REF] and the nematoelastic coupling with the lattice can also generate non-Fermi liquid-like behaviour due to the presence of robust hot spots on the electron pockets [START_REF] Paul | Lattice Effects on Nematic Quantum Criticality in Metals[END_REF].

Near the critical pressure, p c , the resistivity in our single crystal of FeSe 0.89 S 0.11 tuned by applied pressure does not follow a linear T dependence, as found near magnetic critical points [START_REF] Kasahara | Evolution from non-Fermi-to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx) 2 superconductors[END_REF] nor does it display the expected quantum critical fan with a constant exponent [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF]. Instead, the nematic critical region is best represented by a temperature-dependent resistivity exponent n, shown in Fig. 1B, that evolves from a value around n = 3/2 at high temperatures towards the n = 2 value in the low temperature regime. In fact, n shows a marked upturn below 30 K for pressures close to p c that can be linearly extrapolate towards n = 2, as shown by dashed lines in Fig. 1B. To further demonstrate this result, we can use magnetic fields to suppress superconductivity, and by performing a two-band model fit to the magnetotransport data [START_REF] Proust | Fermi liquid behavior of the in-plane resistivity in the pseudogap state of YBa2Cu4O8[END_REF], we can extrapolate the zero-field resistivity in the absence of superconductivity (Fig. S2). At the lowest temperatures, we find a Fermi-liquid behaviour for all pressures across p c , as shown in Fig. 1D (see also Figs. S1 andS3). Furthermore, the corresponding A coefficient as well as the zero-temperature resistivity, ρ 0 , are continuously suppressed with increasing pressure across the transition (Figs. 4B andS1). This strongly suggests that the nematic fluctuations do not become critical at p c .

The evolution of the Fermi surface with pressure. Next we use quantum oscillations measurements under pressure to follow the evolution of the Fermi surface and to assess the strength of electronic correlations across the phase diagram of FeSe 0.89 S 0.11 under pressure. Figures 2A andB show the field dependence of the resistance and TDO data for pressures up to 17 kbar and up to 45 T, and the ambient pressure measurements up to 80 T for three different samples. Results on additional samples and pressures are shown in Figs. S4-S8. There are different field regimes seen in the raw data: a) the superconducting state with zero resistance at low magnetic fields, b) finite resistance that increases strongly with magnetic field in the crossover vortex-liquid region to the normal state at higher fields and c) normal magnetoresistance accompanied by quantum oscillations in high magnetic fields.

At low pressures in the nematic state, the magnetotransport data is dominated by a low-frequency oscillation (Fig. 2A). With increasing pressure, the frequency is reduced and the oscillation disappears beyond 4.7 kbar, in the proximity of p c (Figs. 2A, B andC). This behaviour is remarkably similar to the Lifshitz transition observed on the border of the nematic phase transition, also identified in quantum oscillations as a function of chemical pressure in FeSe 1-x S x [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF]. At high pressures and in high fields, a high-frequency oscillation is visible in the raw data (see Figs. 2A andB), associated with a large Fermi surface sheet. This rules out that a reconstruc- tion of the Fermi surface occurs at high pressures in this compound, similar to to the tetragonal phases of FeSe 1-x S x [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF], but in contrast to the magnetic phase in FeSe under pressure [START_REF] Terashima | Fermi surface reconstruction in FeSe under high pressure[END_REF].

To quantify the complex oscillatory spectra of quantum oscillations, we use both a fast Fourier transform in Fig. 2D (after removing a smooth and monotonic polynomial background from the raw data) or directly fitting the Lifshitz-Kosevich formalism to the raw data in field (with a low-pass filter applied), as shown in Figs. 2C, S4 and S6. This analysis reveals several distinct peaks at ambient pressure, consistent with a complex multi-band electronic structure. Based on ARPES data and previous quantum oscillation measurements, the ambient pressure Fermi surface is formed of two concentric electronlike and one outer hole-like quasi-two dimensional sheets (β and δ orbits) as well as a small inner 3D hole pocket centered at the Z-point (χ), shown in Fig. 3D [START_REF] Watson | Suppression of orbital ordering by chemical pressure in FeSe1-xSx[END_REF][START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF][START_REF] Coldea | The key ingredients of the electronic structure of FeSe[END_REF]. As a function of applied pressure, the cylindrical Fermi surfaces can become warped along the k z direction, either due to changes in the degree of interlayer hopping term and/or electronic correlations leading to a Lifshitz transition and the disappearance of the neck orbits (like β or α 1 in Fig. 3D). The multi-band structure of FeSe 0.89 S 0.11 could give rise up to a maximum of seven frequencies depending on pressure assuming that all the cross-sectional areas are dissimilar for different Fermi surface sheets. In our quantum oscillations data, we can distinguish fewer frequencies but we observe clearly the high frequency region, dominated by the largest orbit of the hole band, δ, as shown in Fig. 2D. Weaker features associated to the largest orbit of the electron band, γ, can be also detected at some pressures and in the TDO signal, as shown in Fig. S7.

The dominant low frequency, λ 1 , can be extracted by directly fitting of the data in Fig. S6, which confirms that this low frequency disappears close to p c (see λ 1 in Fig. 3). Since this observation is in agreement with previous studies using sulfur substitution as the tuning parameter, we attribute this Lifshitz transition, caused by the change in topology of the Fermi surface, as a common feature of the electronic structure at the nematic phase transition tuned either by chemical or applied pressures. Furthermore, the TDO data over the entire available magnetic field range seem to indicate the presence of an additional low-frequency oscillation beyond the critical nematic phase (see Figs. S4 andS6).

With increasing pressure all frequencies increase in size, with the exception of the lowest two frequencies, as summarized in Fig. 3A. At the highest measured pressure, the Fermi surface area of the δ pocket, A k , given by the Onsager relation [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF], F = A k /(2πe), has expanded almost by a factor two, as compared with its size at the ambient pressure. This is much more than the expected growth of the Brillouin zone (less than 4%), assuming a simple contraction of the in-plane unit cell [START_REF] Kothapalli | Strong cooperative coupling of pressure-induced magnetic order and nematicity in FeSe[END_REF][START_REF] Tomita | Correlation between Tc and crystal structure in S-Doped FeSe superconductors under pressure: Studied by x-ray diffraction of FeSe0.8S0.2 at Low Temperatures[END_REF]. This implies significant pressure-induced topological changes of the Fermi surface, similar to those observed as a function of chemical pressure [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF].

Quasiparticle masses. We now turn to the evolution of the electronic correlations across the phase diagram. The evolution of the cyclotron effective mass, m * , is extracted from the temperature dependence of the quantum oscillation amplitudes, analyzed quantitatively within the Lifshitz-Kosevich formalism (Fig. 2E) [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF], and summarized in Fig. 3B. The cyclotron mass of the highest frequency, δ, (as well as χ and γ in Fig. S7), decreases significantly with increasing pressure from 5 m e to 3 m e , signifying a strong reduction in electronic correlations away from the nematic phase transition. This shows that the quasiparticle mass divergence is avoided in the vicinity of the nematic quantum phase transition around p c . We find similar trends in the evolution of the A coefficient and the quasiparticle mass of the δ orbit (m * δ ) across the nematic quantum phase transition, as shown in Fig. 4B. In the high pressure regime, a slight increase in the effective masses of the δ and ξ orbits is observed, but their values remain smaller than in the nematic phase, despite the fact that superconductivity has increased twice.

Quantum oscillation amplitude variation. An unusual observation of our study is the significant variation of the quantum oscillations amplitude of the highest frequency δ across the nematic phase transition, shown in Fig. 2F. According to the Lifshitz-Kosevich formalism, the amplitude of quantum oscillations is affected by different damping terms directly linked to changes of the quasiparticle masses, impurity scattering, spin-splitting effects, as well as the curvature factor of the Fermi surface [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]. As the same single crystal is pressurized, the level of impurity scattering is not expected to change over the entire pressure range. This is supported by the Dingle analysis that shows a similar impurity scattering rate at all pressures in Fig. S8. Furthermore, spin-splitting effects are likely to be negligible for large Fermi surface sheets in a non-magnetic system [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF]. As the amplitude variation strongly exceeds the expectation linked to the changes in the quasiparticle masses, as shown in Fig. 2F, other effects must play an important role. The curvature factor is the largest for a two-dimensional Fermi surface [START_REF] Shoenberg | Magnetic Oscillations in Metals[END_REF] but at the Lifshitz transition an increase in the warping and three-dimensionality of the Fermi surface with pressure would generate a smaller curvature factor [START_REF] Subhasish Mandal | Strong pressure-dependent electron-phonon coupling in FeSe[END_REF]. Lastly, NMR studies have revealed strong antiferromagnetic fluctuations within the nematic phase, before being suppressed close to the nematic phase transition [START_REF] Kuwayama | Two different superconducting states and possible antiferromagnetic quantum critical points in S-doped FeSe under pressure[END_REF]. Based on this, the unusual suppression of the amplitude of the high frequency quantum oscillation, δ, within the nematic phase may be linked to smearing of the Fermi surface caused by the strength of the antiferromagnetic fluctuations within the nematic phase. Resistivity data within the nematic phase has an almost linear dependence (see Fig. 1) above T c suggesting that antiferromagnetic fluctuations may be relevant deep inside the nematic phase. At the lowest temperatures below p c , a resistivity exponent n is harder to estimate reliably due to the larger upper critical fields and the drastic changes in magnetoresistance caused by the low frequency quantum oscillations (see Fig. S1). Discussion. The size of the Fermi energy, E F , in a multiband superconductor is a parameter that provides important clues about the stabilization of different ground states competing with superconductivity [START_REF] Chubukov | Magnetism, superconductivity, and spontaneous orbital order in iron-based superconductors: Which comes first and why?[END_REF]. In the case of a small Fermi energy, a Pomeranchuck instability is favoured, whereas a spin-density wave is stabilized for large Fermi energies. From our quantum oscillation study, we find that the largest Fermi energies increase significantly outside the nematic phase as a function of pressure in FeSe 0.89 S 0.11 , as shown in Fig. 3C. Large hole and electron Fermi surface sheets at high pressures are expected to enhance nesting, promoting the stabilization of a magnetically ordered-state, such as the spin-density wave (SDW in Fig. 4A), identified by other pressure studies [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. The small pockets in Fig. 3C have Fermi energies of the order 3-5 meV, comparable to the superconducting gap, ∆ (Fig. S4); under such conditions a BCS-BEC crossover can partially occur [START_REF] Kasahara | Fieldinduced superconducting phase of FeSe in the BCS-BEC crossover[END_REF].

A 1/2 T c (0T) m* δ m* δ nematic T (K) A 1/2 T c (8T) m* (m e ) 2
Based on our experimental findings, the superconducting phase of FeSe 0.89 S 0.11 under pressure is composed of two dif-ferent superconducting domes, separated by a Lifshitz transition. The first one at low pressure emerges from the nematic phase (SC1 in Fig. 4B) and the second one at high pressure approaching the spin-density wave phase (SC2 in Figs. 4A,B). The two domes can also be visualized in the pressuretemperature phase diagram in magnetic fields of 8 T in Fig. 4B. Remarkably, the nematic quantum phase transition is located in the normal state between the two superconducting domes.

The cyclotron effective masses measure that the strength of electronic correlations varies with pressure, being in general larger inside the nematic phase compared with those in the tetragonal phase. Interestingly, we find a quantitative link between the cyclotron masses of the δ orbit, the A coefficient from the low temperature resistivity data, and the value of T c in the nematic phase and in the low-T c tetragonal phase. This supports that the hole bands are closely involved in the pairing mechanism (Fig. 4B). This correlation between the band renormalisation and T c was also captured by quantum oscillations experiments and ARPES studies on FeSe 1-x S x [START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF][START_REF] Reiss | Suppression of electronic correlations by chemical pressure from FeSe to FeS[END_REF]. Furthermore, NMR studies show that the strength of antiferromagnetic fluctuations also correlates with T c within the nematic phase, being suppressed together with the nematicity [START_REF] Kuwayama | Two different superconducting states and possible antiferromagnetic quantum critical points in S-doped FeSe under pressure[END_REF][START_REF] Wiecki | Persistent correlation between superconductivity and antiferromagnetic fluctuations near a nematic quantum critical point in FeSe1-xSx[END_REF]. For higher pressures, the correlation between the hole-like band masses and T c is lost, suggesting changes in the pairing mechanism towards the high-T c phase.

Upon approaching a critical region as a function of the tuning parameter, one would expect the divergence of the quasiparticle masses, A coefficient and an enhanced superconducting phase [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF][START_REF] Analytis | Transport near a quantum critical point in BaFe2(As1xPx)2[END_REF][START_REF] Shishido | Evolution of the Fermi Surface of BaFe2(As1-xPx) 2 on Entering the Superconducting Dome[END_REF]. Here, near the nematic critical point at p c , in the absence of magnetic order, we find a smoothly evolving A coefficient and non-divergent changes in the effective masses for the large, well-defined orbit (δ). This suggests that the nematic fluctuations are finite and not critical at p c . Strikingly, the superconducting transition temperatures are minimal in the vicinity of the nematic quantum phase transition (Fig. 4). Therefore, superconductivity is not enhanced by non-critical nematic fluctuations in the absence of the magnetic order. This is in contrast to other iron-based superconductors where critical nematic and/or magnetic fluctuations enhance superconductivity approaching a spin-density wave state [START_REF] Lederer | Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point[END_REF][START_REF] Kasahara | Evolution from non-Fermi-to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx) 2 superconductors[END_REF]. Based on these observations, we interpret our results in the context of quenched nematic criticality due to a strong nematoelastic coupling to the lattice [START_REF] Paul | Lattice Effects on Nematic Quantum Criticality in Metals[END_REF][START_REF] Labat | Pairing instability near a latticeinfluenced nematic quantum critical point[END_REF]. This coupling is responsible for the suppressed superconductivity [START_REF] Labat | Pairing instability near a latticeinfluenced nematic quantum critical point[END_REF] and the finite quasiparticle mass in the proximity of the nematic quantum critical point (compared with data in Fig. S7).

The transport across the two superconducting domes tuned by pressure is dominated by different types of scattering. These processes can be either completely independent of each other or they can overlap in certain pressure regions. A linear resistivity is observed over a large temperature range in the high pressure regime in the vicinity of a antiferromagnetic quantum critical point, expected around 30 kbar [START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. On the other hand, outside nematic phase boundaries, we find a non-Fermi liquid behaviour, with a dominant T 3/2 resistivity surrounding the nematic phase, consistent with the region domi-nated by strong nematic fluctuations, as identified by an NMR study [START_REF] Kuwayama | Two different superconducting states and possible antiferromagnetic quantum critical points in S-doped FeSe under pressure[END_REF]. In reality, the resistivity exponent n close to p c in our study is not constant and it has a strong temperature dependence (Figs. 1B). The exponent changes from n = 3/2 at high temperatures towards the Fermi liquid regime with n = 2 at low temperature for all pressures across the nematic quantum phase transition. This is consistent with the predictions of the model assuming a strong nematoelastic coupling that can generate non-Fermi liquid behaviour at high temperatures and the transformation into a Fermi liquid below T F L . We can estimated that for our system T F L ∼ 13 K (from ARPES and nematic susceptibility measurements for similar compositions [START_REF] Watson | Suppression of orbital ordering by chemical pressure in FeSe1-xSx[END_REF][START_REF] Hosoi | Nematic quantum critical point without magnetism in FeSe1-xSx superconductors[END_REF]), in good agreement with our experiments (Fig. 1). Thus, the transport behaviour near the nematic critical point revealed by our study is very different from the linear Tbehaviour in the proximity of a magnetic critical point [START_REF] Kasahara | Evolution from non-Fermi-to Fermi-liquid transport via isovalent doping in BaFe2(As1-xPx) 2 superconductors[END_REF].

The resistivity behaviour in the vicinity of the nematic critical point tuned using applied hydrostatic pressure seems to be in contrast to studies using chemical pressure in FeSe 1-x S x , which identified a linear-T region at the nematic quantum phase transition and equally no evidence of an antiferromagnetic phase for the highest available doping was found [START_REF] Urata | Non-Fermi liquid behavior of electrical resistivity close to the nematic critical point in Fe1-xCoxSe and FeSe1-ySy[END_REF][START_REF] Licciardello | Electrical resistivity across a nematic quantum critical point[END_REF]. While in our study a single high quality single crystal is tuned as a function of pressure across different phases, studies using chemical substitution require different samples to be measured and the varying impurity scattering between samples needs to be taken into account. In fact, the role played by impurities on quantum critical behaviour can have drastic effects on scattering. For example, critical antiferromagnetic fluctuations cause strong scattering only in the vicinity of hotspots on the Fermi surface which can be short-circuited by other parts of the Fermi surface in the ultra-clean limit giving a Fermi-liquid T 2 resistivity [START_REF] Löhneysen | Fermi-liquid instabilities at magnetic quantum phase transitions[END_REF][START_REF] Thomas | Quantum multicriticality[END_REF]. Impurities or multiple scattering modes can make the scattering more isotropic and could produce a T -linear resistivity close to an antiferromagnetic critical point in a two-dimensional weakly disorder metal [START_REF] Rosch | Interplay of disorder and spin fluctuations in the resistivity near a quantum critical point[END_REF]. Similar arguments can be applied to nematic critical systems where nematoelastic coupling generate isolated hot spots on the Fermi surface [START_REF] Lederer | Superconductivity and non-Fermi liquid behavior near a nematic quantum critical point[END_REF][START_REF] Paul | Lattice Effects on Nematic Quantum Criticality in Metals[END_REF][START_REF] Wang | Scattering mechanisms and electrical transport near an ising nematic quantum critical point[END_REF].

Remarkably, the fine tuning using hydrostatic pressure in FeSe 0.89 S 0.11 provides an unique insight into the behaviour of a pristine nematic quantum critical point. The electronic signature of this point is substantially different from other critical points in iron-based superconductors, where both magnetic and nematic phases closely coexist. Superconductivity is not enhanced by non-critical nematic fluctuations, in the absence of magnetic order, as they are cut off along certain directions by the noncritical shear strains that are invariably present in real materials [START_REF] Labat | Pairing instability near a latticeinfluenced nematic quantum critical point[END_REF]. Our findings should have profound implications on understanding the superconducting mechanisms in different classes of superconductors where nematic phases are present.

Methods

Sample characterization. Single crystals of FeSe 0.89 S 0.11 were grown by the KCl/AlCl 3 chemical vapour transport method, as reported previously [START_REF] Böhmer | Lack of coupling between superconductivity and orthorhombic distortion in stoichiometric single-crystalline FeSe[END_REF][START_REF] Böhmer | Variation of transition temperatures and residual resistivity ratio in vapor-grown FeSe[END_REF]. More than 20 crystals were screened at ambient pressure and they showed sharp superconducting transition regions of ∼ 0.1 K, and large residual resistivity ratios up to 17 between room temperature and the onset of superconductivity. Crystals from the same batch were previously used in quantum oscillations and ARPES studies [START_REF] Watson | Emergence of the nematic electronic state in FeSe[END_REF][START_REF] Coldea | Evolution of the Fermi surface of the nematic superconductors FeSe1-xSx[END_REF].

Magnetotransport measurements. High magnetic field measurements up to 45 T at ambient pressure and under hydrostatic pressure were performed using the hybrid magnet dc facility at the NHMFL in Tallahassee, FL, USA. Pressures up to 17 kbar were generated using a NiCoCr piston cylinder cell, using Daphne Oil 7575 as pressure medium. The pressure inside the cell was determined by means of ruby fluorescence at low temperatures where quantum oscillations were observed. Different samples were measured simultaneously under pressure: sample B was used for transport measurements, and sample D-TDO was positioned inside a coil and the resonant frequency of an LC tank circuit driven by a tunnel diode was recorded (TDO). Transport sample E was measured at ambient pressure at the same time. Magnetotransport measurements were performed using the standard ac technique. Pulsed-field measurements up to 80 T at ambient pressure were carried out in Toulouse, using the same samples (transport measurements on sample B, TDO measurements on sample D). Magnetotransport and Hall effect measurements under pressure using a 5-contact configuration were carried out on sample A in low fields up to 16 T in an Oxford Quantum Design PPMS and an ElectroLab High Pressure Cell, using Daphne Oil 7373 which ensures hydrostatic conditions up to about 23 kbar. The pressure inside this cell was determined via the superconducting transition temperature of Sn after cancelling the remanent field in the magnet. The magnetic field was applied along the crystallographic c axis for all samples. A maximum current of up to 2 mA flowing in the conducting tetragonal ab plane was used.

FIG. 1 .

 1 FIG. 1. Transport under pressure in FeSe0.89S0.11. (A) Temperature dependence of resistivity for different pressures. The transition temperature into the nematic state occurs at Ts and into the superconducting phase at Tc. The onset and offset of superconductivity are indicated by crosses and solid circles, respectively in E. (B) Local resistivity exponent n for pressures close to pc. Dashed lines are linear fits at low temperatures above the onset of superconductivity. (C) High temperature resistivity plotted against T 3/2 . Dashed lines are linear fits. Data in panels A and C are shifted vertically for clarity. (D) Low-temperature resistivity (solid lines) and normal state resistivity extrapolated from symmetrized magnetic field measurements (points) plotted against T 2 (Fig.S2). Dashed lines are fits to Fermi liquid behaviour, ρ = ρ0 + AT 2 , the slope gives the A coefficient and ρ0 is the zero-temperature residual resistivity. Fit residuals are shown in Fig. S3. (E) Pressure-temperature phase diagram. The color map represents the local resistivity exponent n (ρ ∼ T n ), see color scale in inset. Symbols indicate the transition temperatures defined in panel (A). The dashed line is a guide to the eye. The solid line is a fit of Ts ∼ (p -pc) giving ∼ 0.4 and pc = 5.8 kbar.

FIG. 2 .

 2 FIG. 2. Evolution of quantum oscillations with pressure in FeSe0.89S0.11. (A) Magnetotransport and (B) TDO resonant frequency variation, ∆f (MHz), for several different samples as a function of applied pressures (see also Figs. S4-S5). Horizontal dashed lines in A indicate the sample resistance in zero field at the onset of superconductivity. (C) The oscillatory component of the magnetoresistance (after of a smooth background with a low-pass filter applied) shows the evolution of the dominant low-frequency oscillation. Dashed lines here indicate the base line. The horizontal line indicates the period of the dominant low frequency (λ1). (D) Fourier transform of the oscillatory part of the magnetotransport data in (A) after the background subtraction. The distinct peaks identified correspond to extremal cross-sections of the Fermi surface. (E) The temperature dependence of the amplitude gives the cyclotron effective band mass of the δ orbit for different constant pressures. Lines are fits to the Lifshitz-Kosevich mass damping term [38]. (F) Low-temperature amplitude variation with pressure of the δ orbit (solid symbols) and the expected amplitude variation due to the mass damping term RT (cross symbols). Solid lines are guides to the eye. Data in panels C and D are shifted vertically for clarity.

2 FIG. 3 .

 23 FIG. 3. Pressure tuning of Fermi surface and electronic correlations. (A) Quantum oscillation frequencies, and (B) the cyclotron masses as a function of pressure for samples B, D and E. The masses associated to χ and δ orbits show a clear decrease upon crossing the nematic phase transition. A topological Lifshitz transition occurs at pc and one small orbit in the kz=0 plane disappears (either β or α1 can be assigned to λ1). (C) The Fermi energy EF assuming parabolic band dispersions, based on Fermi surface crosssections and quasiparticle masses (panels A and B). Lines are guides to the eye. The splitting of the γ and δ frequencies is summarized in Fig. S7. Shaded areas indicate the nematic phase (D) Sketch of the Fermi surface in the nematic phase and the different two-dimensional orbits, based on ARPES and quantum oscillation measurements [17-19].

FIG. 4 .

 4 FIG.4. Pressure-temperature phase diagrams. (A) Extended phase diagram of FeSe0.89S0.11 based on our work and previous reports[START_REF] Matsuura | Maximizing T c by tuning nematicity and magnetism in FeSe1-xSx superconductors[END_REF]. (B) Comparison of the band masses of the δ orbit with Tc and the Fermi liquid coefficient, A. A striking similar pressure dependence of electronic correlations and superconductivity is maintained across the nematic phase transition, but it is lost towards the high pressure high-Tc state. The temperature-pressure phase diagram in a static magnetic field of 8 T shows the separation of the two superconducting domes unmasking the nematic quantum phase transition at pc. Symbols for the masses of the δ orbits are identical to Fig.3. Solid lines are guides to the eye and dashed lines are extrapolations.

Acknowledgements We thank Penglin Cai for technical support provided with setting up the PPMS pressure cell and Andrey Chubukov, Erez Berg, Raphael Fernandes, Indranil Paul, Roser Valenti, Ilya Vekhter, Matthew Watson, Zachary Zajicek for useful discussions and comments. This work was mainly supported by EPSRC (EP/I004475/1, EP/I017836/1). AAH acknowledges the financial support of the Oxford Quantum Materials Platform Grant (EP/M020517/1). A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-1157490 and the State of Florida. Part of this work was supported by HFML-RU/FOM and LNCMI-CNRS, members of the European Magnetic Field Laboratory (EMFL) and by EPSRC (UK) via its membership to the EMFL (grant no. EP/N01085X/1). Part of this work at the LNCMI was supported by Programme Investissements d'Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT. AIC thanks the hospitality of KITP supported by the National Science Foundation under Grant No. NSF PHY-1125915. We also acknowledge financial support of the John Fell Fund of the Oxford University. AIC acknowledges an EPSRC Career Acceleration Fellowship (EP/I004475/1).