Anomalous high-magnetic field electronic state of the nematic superconductors FeSe$_{1-x}$S$_x$

Understanding superconductivity requires detailed knowledge of the normal electronic state from which it emerges. A nematic electronic state that breaks the rotational symmetry of the lattice can potentially promote unique scattering relevant for superconductivity. Here, we investigate the normal transport of superconducting FeSe$_{1-x}$S$_x$ across a nematic phase transition using high magnetic fields up to 69 T to establish the temperature and field-dependencies. We find that the nematic state is an anomalous non-Fermi liquid, dominated by a linear resistivity at low temperatures that can transform into a Fermi liquid, depending on the composition $x$ and the impurity level. Near the nematic end point, we find an extended temperature regime with $T^{1.5}$ resistivity. The transverse magnetoresistance inside the nematic phase has as a $H^{1.55}$ dependence over a large magnetic field range and it displays an unusual peak at low temperatures inside the nematic phase. Our study reveals anomalous transport inside the nematic phase, driven by the subtle interplay between the changes in the electronic structure of a multi-band system and the unusual scattering processes affected by large magnetic fields and disorder

Understanding superconductivity requires detailed knowledge of the normal electronic state from which it emerges. A nematic electronic state that breaks the rotational symmetry of the lattice can potentially promote unique scattering relevant for superconductivity. Here, we investigate the normal transport of superconducting FeSe1−xSx across a nematic phase transition using high magnetic fields up to 69 T to establish the temperature and field-dependencies. We find that the nematic state is an anomalous non-Fermi liquid, dominated by a linear resistivity at low temperatures that can transform into a Fermi liquid, depending on the composition x and the impurity level. Near the nematic end point, we find an extended temperature regime with ∼ T 1.5 resistivity. The transverse magnetoresistance inside the nematic phase has as a ∼ H 1.55 dependence over a large magnetic field range and it displays an unusual peak at low temperatures inside the nematic phase. Our study reveals anomalous transport inside the nematic phase, driven by the subtle interplay between the changes in the electronic structure of a multi-band system and the unusual scattering processes affected by large magnetic fields and disorder.
Magnetic field is a unique tuning parameter that can suppress superconductivity to reveal the normal low-temperature electronic behavior of many unconventional superconductors [1,2]. High-magnetic fields can also induce new phases of matter, probe Fermi surfaces and determine the quasi-particle masses from quantum oscillations in the proximity of quantum critical points [1,3]. In unconventional superconductors, close to antiferromagnetic critical regions, an unusual scaling between a linear resistivity in temperature and magnetic fields was found [4,5]. Magnetic fields can also induce metal-toinsulator transitions, as in hole-doped cuprates, where superconductivity emerges from an exotic electronic ground state [2].
FeSe is a unique bulk superconductor with T c ∼ 9 K which displays a variety of complex and competing electronic phases [6]. FeSe is a bad metal at room temperature and it enters a nematic electronic state below T s ∼ 87 K. This nematic phase is characterized by multi-band shifts driven by orbital ordering that lead to Fermi surface distortions [6,7]. Furthermore, the electronic ground state is that of a strongly correlated system and the quasiparticle masses display orbital-dependent enhancements [7,8]. FeSe shows no long-range magnetic order at ambient pressure, but complex magnetic fluctuations are present at high energies over a large temperature range [9]. Below T s , the spin-lattice relaxation rate from NMR experiments is enhanced as it captures the low-energy tail of the stripe spin-fluctuations [10,11]. Furthermore, recent µSR studies invoke the close proximity of FeSe to a magnetic quantum critical point as the muon relaxation rate shows unusual temperature dependence inside the nematic state [12].
The changes in the electronic structure and magnetic fluctuations of FeSe can have profound implication on its transport and superconducting properties. STM reveals a highly anisotropic superconducting gap driven by orbital-selective Cooper pairing [13]. Due to the the presence of the small inner bands, whose Fermi energies are comparable to the superconducting gap, FeSe was placed inside the BCS-BEC crossover regime [14]. In large magnetic fields, when the Zeeman energy is comparable to the gap and Fermi energies, a peculiar highly-polarized superconducting state may occur [14].
To establish the role played by different competing interactions on nematicity and superconductivity, an ideal route is provided by the isoelectronic substitution of selenium by sulphur ions in FeSe 1−x S x [15]. This tuning parameter suppresses nematicity and it leads to changes in the electronic structure, similar to the temperature effects, with the Fermi surface becoming isotropic in the tetragonal phase and the electronic correlations becoming weaker [3,6,15,16]. As nematicity is suppressed, it creates ideal conditions to explore a potential nematic critical point [17] in the absence of magnetism. The superconducting dome extends outside the nematic state but anisotropic pairing remains robust [18] and a different superconducting state was suggested to be stabilized in the tetragonal phase [19].
In this paper we study the normal electronic state across the nematic transition in FeSe 1−x S x using magnetotransport studies in high-magnetic fields up to 69 T. We find that the nematic state has a non-Fermi-liquid behaviour with an unusual transverse magnetoresistance (∼ H 1.55 ), reflecting an unconventional scattering mechanism. Just outside the nematic phase, resistivity is dominated by a ∼ T 1.5 dependence, similar to studies under pressure [20]. The transverse magnetoresistance is significant inside the nematic phase and it shows an unusual change in slope at low temperatures. Inside the nematic phase at low temperatures, we find linear resistivity followed by Fermi-liquid behaviour for certain x and   impurity levels. Our study reveals anomalous transport in the nematic state due to the subtle changes in the electronic structure and/or scattering, which are also influenced by impurity levels.

RESULTS AND DISCUSSION
Figs. 1a-e show the transverse magnetoresistance, ρ xx , of different single crystals of Fe(Se 1−x S x ) up to 35 T at various fixed temperatures inside the nematic phase and up to 69 T for x ∼ 0.25 in the tetragonal phase. From these constant temperature runs, we can extract the magnetoresistance at fixed fields for each composition x, as shown in Fig. 1f-j, which reveals several striking features. Firstly, the magnetoresistance increases significantly once a system enters the nematic state at T s , and its magnitude dependents on the concentration x, being largest for FeSe, just above T c . Secondly, in the vicinity of T c in magnetic fields much larger than the upper critical field, the magnetoresistance shows an unusual temperature dependence that varies strongly with x across the phase diagram, as shown in Fig. 1(f-g). The resistivity slope dρ xx /dT in 34 T of FeSe changes sign around a crossover temperature, T * ∼ 14 K, as shown in Fig.1f (also in the colour plot of the slope in Fig. 3d). With increasing sulphur substitution from FeSe towards x ∼ 0.07 (defined as the nematic A region), the position of T * shifts to a slightly higher temperature of ∼ 20 K, and the peak in magnetoresistance is much smaller than for FeSe. For higher concentrations, approaching the nematic phase boundary, (x ∼ 0.11 − 0.17 defined as the nematic B region), there is a small peak at T * but the negative slope dρ xx /dT in 34 T is strongly enhanced at low temperatures, different from the nematic A phase (see Fig. 1(h,i) and Fig. 3(d)). Lastly, in the tetragonal phase, the magnetoresistance shows a conventional behaviour and increases quadratically in magnetic fields ( Fig. 1(e) and (j)).
The unusual downturn in resistivity in high-field fields below T * inside the nematic A phase was previously assigned to large superconducting fluctuations in FeSe in magnetic fields up to 16 T [10,11]. We find that this behaviour remains robust in magnetic fields at least a factor of 2 higher than the upper critical field of ∼16 T for H||c [10]. Furthermore, it also manifests in x ∼ 0.07 inside the nematic A phase but it disappears for higher x 0.1. As T c and the upper critical field inside the nematic phase for different x remain close to that of FeSe [3,21], the changes in the resistivity slope in high magnetic fields are likely driven by field-induced effects that influence scattering and/or the electronic structure.
The Hall coefficient, R H = ρ xy /µ 0 H, extrapolated in the low-field limit (below 1 T) for FeSe 1−x S x has an unusual temperature dependence, as shown in Fig.2b. For a compensated metal, the sign of the Hall coefficient depends on the difference between the hole and electron mobilities [22]. In the tetragonal phase above T s and for x 0.18, R H is close to zero (Fig.2b), as expected for a two-band compensated metal. On the other hand, in the low-temperature nematic A phase the sign of R H is negative suggesting that transport is dominated by a highly mobile electron band [15,23]. It becomes positive inside the nematic B phase, dominated by a hole-like band ( Fig. 2(a)). It is worth mentioning that inside the nematic B phase the quantum oscillations are dominated by a low-frequency pocket with light-mass that disappears at the nematic end point [3]. Thus, the behaviour of R H is linked to the disappearance of a small 3D hole pocket center at the Z-point in FeSe below T s and its re-emergence in the nematic B phase with x substitution around x ∼ 0.11, as found in ARPES studies [15] and sketched in Fig.1(m). Interestingly, the subtle changes in the electronic structure in FeSe 1−x S x seem to correlate with the different features observed both in magnetoresistance ( Fig.1(f-i)) and in the Hall coefficient |R H | that shows a maximum near T * (Fig.2(b)). In a high magnetic field, the Hall component of FeSe is complex, changing sign and being non-linear [15,21]. A magnetic field can induce changes in scattering and/or field-induced Fermi-surface effects in the limit when the cyclotron energy is close to the Zeeman energy. The smallest inner bands of FeSe 1−x S x shift in energy as a function of composition x (and temperature [3]), as shown in Figs.1(k-o). Furthermore, Hall effect in ironbased superconductors can be affected by the spin fluctuations that induce mixing of the electron and hole currents [24].
Next, we attempt to quantify the magnetoresistance across the phase diagram and in the vicinity of the nematic end point in FeSe 1−x S x , as shown in Fig.1(a-e). At the lowest temperature, inside the nematic phase, the transverse magnetoresistance of most samples is dominated by quantum oscillations [3] making difficult to quantify its dependence. A near-linear magnetoresistance is detected for x ∼ 0.07 in Fig. 1b and for a dirty sample (with low residual resistivity ratio ∼ 8.5) in Fig. S9. The quasi-linear field magnetoresistance at low temperature can arise from squeezed trajectories of carriers in semiclassically large magnetic fields in case of small Fermi surfaces (ω c τ 1) [25,26]. Another explanation for an almost linear magnetoresistance is the presence of mobility fluctuations caused by spatial inhomogeneities, as found in low carrier density systems [26][27][28].
Classical magnetoresistance in systems with a single dominant scattering time is expected to follow a H 2 dependence [25]. This results in Kohler's rule, which is violated in FeSe 1−x S x suggesting that the magnetoresistance is not dominated by a single scattering time, as shown in Fig. S2(a-c). Magnetoresistance is quadratic in magnetic fields up to 69 T in the tetragonal phase (x ≥ 0.19) (see Fig.1e and Fig.S4(e-f)) but not inside the nematic phase. FeSe 1−x S x are compensated multi-band systems [6] where the high-field magnetoresistance is expected to be very large and dependent on scattering times of electron and hole bands [22]. Magnetoresistance has a complex form and instead simpler scaling have been sought to reveal its importance, in particular in the vicinity of critical points [4,5]. For example, in BaFe 2 (As 1−x P x ) for x ∼ 0.33 at the antiferromagnetic critical point, a universal H − T scaling was empirically found between the linear resistivity in temperature and magnetic field [4]. For FeSe 1−x S x near the nematic end point at x ∼ 0.17 we find that a H − T dependence collapses onto a single curve, as shown in Fig. S2(e). Despite this, the energy scaling of magnetoresistance used to described the antiferromagnetic critical point in Ref. [4] is not obeyed in the vicinity of the nematic end point in FeSe 1−x S x , as detailed in Fig. S2(g-i). This could be due to additional constrains to be included either to account for the nematoelastic coupling [29] and/or the effect of impurities. For example, a very dirty sample of FeSe 1−x S x close to x nom ∼ 0.18 was recently suggested to obey H − T scaling [30].
For reasons described above, we propose a different approach to model the magnetoresistance data in the nematic state of FeSe 1−x S x , using a power law in magnetic fields given by ρ xx (H) = ρ 0 (H) + bH δ . Strikingly, we find that all the magnetoresistance data inside the nematic phase can be described by a unique exponent δ ∼ 1.55(5) over a large field window, as shown by the colour plot in Fig.3(c) as well as in Figs.2(c) and S4(a-d). A detailed method of the extraction of δ and its stability over a large temperature and field window is shown in Fig.S3. Furthermore, this gives δ ∼ 2 for samples in the tetragonal phase (see Fig.3(c)). Inside the nematic phase, the Fermi surface of FeSe 1−x S x distorts anisotropically [6,7] and an unusual type of scattering could become operational due to presence of hot and cold spots along certain directions [31].
In the absence of magnetic field the transport behaviour can also be described by a power law, ρ(T ) = ρ 0 + AT γ . Fig. 3a shows a colour plot of the exponent γ, which is close to unity at low temperatures inside the nematic phase and becomes sublinear close to the nematic phase boundary, indicating a significant deviation from Fermi-liquid behaviour (a value of γ=1.1(2) was previously reported for FeSe [32]). Outside the nematic phase a T 1.5 dependence of resistivity describes the data well over a large temperature range up to 120 K (see Fig. 2(a) and Fig.3(a)), in agreement with previous studies of FeSe 1−x S x under pressure [20]. Using the high-magnetic field data below T c , we extract the low-temperature resistivity in the absence of superconductivity, ρ H→0 (T). Fig. 2(df) shows resistivity against temperature for different values of x, together with the extrapolated high-field points, using longitudinal magnetoresistance when H||(ab) plane, shown in Fig.S5. We also use transverse magnetoresistance data to extract the zero-field resistivity, using the established power law H 1.55 , as shown in Fig. S7. From both measurements, we find strong evidence for a linear resistivity in the low temperature regime, below T * , inside the nematic phase. Linear resistivity was also detected from the 35 T temperature dependence of the longitudinal magnetoresistance in Ref. [33], however, it was assumed to occur near the nematic critical point defined as x nom ∼ 0.16, which corresponds to x ∼ 0.13 in our phase diagrams in Fig.3 and Fig.S1(b) (as the resistivity derivative in Ref. [33] show a T s ∼ 51 K). At low temperatures, we observe that Fermi-liquid behaviour recovers in the tetragonal phase (see also Refs. [33,34]) and inside the nematic phase, below T F L (see Figs. 2(d-f) and 3(b)). This is strongly dependent on composition and impurity level, even in the vicinity of the nematic end point (see Figs. S8 and S9). We find that T F L is highest for the samples with the largest residual resistivity ratio (above ∼ 16) (see Figs.S1(c) and S6). Theoretical models suggest that the temperature exponent, γ, in vicinity of critical points is highly dependent on the presence of cold spots on different Fermi surfaces, due to the symmetry of the nematic order parameter [31,35]. On the other hand, near a antiferromagnetic critical point in the presence of spin fluctuations the impurity level also affects the temperature exponent [36]. Furthermore, the scale at which the crossover to Fermi liquid behavior occurs at T F L in nematic critical systems could depend on the strength of the coupling to the lattice [29].
An overall representation of the resistivity slope dρ xx (34 T)/dT in 34 T for FeSe 1−x S x as a function of temperature is shown in the phase diagram in Fig. 3d. The low-temperature manifestation of the nematic A and B phases is clearly different below T * . In order to identify possible sources of scattering responsible for these changes, we consider the role of spin fluctuations. Recent NMR data found that anti-ferromagnetic spin fluctuations are present inside the nematic phase of FeSe 1−x S x , being strongest around x ∼ 0.1 [37]. In FeSe, spin fluctuations are rather anisotropic [37,38] and strongly field-dependent below 15 K [11]. Interestingly, the spin-fluctuations relaxation rate is enhanced below T * (Fig. 3(d)), suggesting a correlation between spin-dependent scattering, the high-field magnetoresistance and the low-temperature transport inside the nematic state. High-magnetic fields are expected to align magnetic spins and could affect the energy dispersion of low-energy spin excitations and spin-dependent scattering in magnetic fields. In FeSe, the spin-relaxation rate in different magnetic fields up to 19 T deviates at T * [11] but it remains relatively constant in 19 T at the lowest temperatures. This may suggest the variation in magnetoresistance in high magnetic fields at low temperatures in FeSe 1−x S x is more sensitive to the changes in the electronic behaviour rather to the spin fluctuations across the nematic phase. The low-temperature regime below T * displays linear resistivity, which is a potential manifestation of scattering induced by critical spin-fluctuations in clean systems [36]. µSR studies place FeSe near an itinerant antiferromagnetic quantum critical point at very low temperatures [12] and spinfluctuations are only found inside the nematic state [11,37]. On the other hand, close to the nematic end point in FeSe 1−x S x we find that resistivity is not linear in temperature but is dominated by a T 1.5 dependence. This is contrast to the linear resistivity found near a antiferromagnetic critical point in BaFe 2 (As 1−x P x ) [32]. Theoretically, γ = 3/2 could describe the resistivity caused by strong antiferromagnetic critical fluctuations in the dirty limit [36,39]. However, in FeSe 1−x S x the spin fluctuations are suppressed and a Lifshitz transition was detected at the nematic end point [3]. At a nematic critical point the divergent fluctuations for different Fermi surfaces could display unusual power laws in resistivity, as discussed in Refs. [31,35,40]. To asses the critical behaviour, it is worth emphasizing that the effective masses associated to the outer hole bands do not show any divergence close to the nematic end point x ∼ 0.18 [3]. This agrees with the variation of the A 1/2 coefficient (see Fig. S11) and previous studies under pressure [20], suggesting the critical nematic fluctuations could be quenched by the coupling to the lattice along certain directions in FeSe 1−x S x .
The striking difference in magnetotransport behaviour between the nematic and tetragonal phase in FeSe 1−x S x can have significant implications on what kind of superconductivity is stabilized inside and outside the nematic phase as different pairing channels may be dominant in different regions, as found experimentally [18,19]. Linear resistivity found at low temperatures inside the nematic state is present in the region where spin-fluctuations are likely to be present. Furthermore, the absence of superconductivity enhancement at the nematic end point in FeSe 1−x S x is supported by the lack of divergent critical fluctuations, found both with chemical pressure [3] and applied pressure [20]. It is expected that the coupling to the relevant lattice strain restricts criticality in nematic systems only to certain high symmetry directions [29,41].
In conclusion, we have studied the evolution of the lowtemperature magnetotransport behaviour in FeSe 1−x S x in high-magnetic fields up to 69 T. We find that the nematic state has non-Fermi liquid behaviour and displays unconventional power laws in magnetic field, reflecting the dominant anomalous scattering inside the nematic phase. In high magnetic fields, well-above the upper critical fields, the transverse magnetoresistance shows a change in slope that reflects the changes in the spin-fluctuations and/or the electronic structure. In the low-temperature limit, high magnetic field suppresses superconductivity and it reveals an extended linear resistivity in temperature followed by a Fermi-liquid like dependence, highly dependent on the composition and impurity level. Our study reveals the anomalous transport behaviour of the nematic state, strikingly different from the tetragonal phase, that influences how superconductivity is stabilized in different phases.

MATERIALS AND METHODS
Single crystals of FeSe 1−x S x were grown by the KCl/AlCl 3 chemical vapor transport method [42]. The composition for samples from the same batch were checked using EDX as reported previously in Ref. [3]. Note that in Refs. [30,33] the nominal, x nom were can be at least 80% less than the real x (see also Ref. [3,17,37]). The structural transition at T s also provides useful information about the expected x value, as shown in Fig.S1. More than 30 samples were screened for high magnetic field studies to test their physical properties. Residual resistivity ratio varies between 15-44, as shown in Fig.S1c. We observed the variation within the same batch due to the inhomogeneous distribution of sulfur with increasing x (see Figs.S1 and S8). We estimate that the nematic end point is located close to x ∼ 0.180(5) (see Figs.S1) and S11).
In-plane transport measurements (I||(ab)) were performed in a variable temperature cryostat in dc fields up to 38 T at HFML, Nijmegen and up to 70 T at LNCMI, Toulouse with the magnetic field applied mainly along the c-axis (transverse magnetoresistance) but also in the (ab) conducting plane (longitudinal magnetoresistance) at constant temperatures. Lowfield measurements were performed in a 16 T Quantum Design PPMS. The resistivity ρ xx and Hall ρ xy components were measured using a low-frequency five-probe technique and were separated by (anti)symmetrizing data measured in positive and negative magnetic fields. Good electrical contacts were achieved by In soldering along the long edge of the single crystals and electrical currents up to 3 mA were used to avoid heating. Magnetic fields along the c-axis suppress superconductivity in fields higher than 20 T for all x values [3].  (5), which agrees with previous reports [3,17]. This value however differs from that reported in Ref. [33], where the nominal concentrations have been used. For example, in Ref [33] xnom = 0.16 has Ts ∼ 51 K, which would correspond to x ∼ 0.13, based on our phase diagram and previous reports [3,17]. The two x ∼ 0.17 and the x ∼ 0.18 samples come from the same batch and their differences reflect the sulphur variation and the degree of disorder (x ∼ 0.18 is cleaner with an RRR of ∼ 24 compared with ∼ 16 for the two x ∼ 0.17 samples). For x ∼ 0.18 the derivative in (b) evolves more gradually, without a well-defined structural transition as compared to the others, and we believe that this sample is the closest to the nematic end point (just inside the nematic state).     Kohler's rule is clearly violated for all samples over a large temperature range, providing evidence that electrical transport in these systems is not governed by a single scattering time.
(d-f) H − T scaling of ∆ρxx/ρ0 ∼ µ0H /T , where ∆ρxx = ρxx − ρ0 and ρ0 is the zero-temperature zero-field resistivity. There is clearly no scaling for x = 0 and 0.25; for the x = 0.17 sample in the vicinity of the nematic end point, data are dominated by a low-frequency quantum oscillation and collapse onto a single curve at low temperature, as shown in (e), but we have not identified yet an appropriate scaling law for it. However, in Ref. [30] for a particularly dirty sample of xnom = 0.18 with RRR ∼ 5 and weak magnetoresistance (a factor ∼ 100 smaller than our x ∼ 0.17) H/T scaling was proposed suggesting that disorder may play an important role in this type of scaling in FeSe1−xSx. (g-i) Energy scaling of resistivity as Γ, where Γ = αkBT 1 + (β/α) 2 (µBµ0H /(kBT )) 2 using α = 1 and β = 1, for x = 0, 0.17 and 0.19, respectively. Our data do not follow the proposed energy scaling for any reasonable value of α/β and for any sulphur concentration in FeSe1−xSx. This magnetoresistance scaling was used to describe the antiferromagnetic critical region in BaFe2(As1−xPx)2 [4].   (a-f) Temperature dependence of resistivity at low temperatures for different compositions used to build the low-temperature phase diagram in Fig. 3(b). Solid squares show the extrapolated normal state resistivity, ρH→0 from (g-l), and the solid triangles in (e) are the resistivity data at 35 T from (k), when the magnetic field is along the conducting (ab) plane. Solid lines are the zero-field resistivity for each sample. Fermi-liquid like behaviour is observed in certain samples (with the largest resistivity ratio in Fig.S1(c)) below TFL, as indicated by arrows. (g-l) Magnetic field-dependence of the resistivity at the lowest temperature when H ||(ab). Dashed lines are the linear extrapolation towards H → 0 . For x = 0.11 and 0.17, the upper critical field is too large to reach the normal state in magnetic fields up to 38 T in this orientation.  Fig. S5. The red dashed lines are fits of resistivity to a quadratic temperature dependence below TFL, and the blue dashed lines show a linear dependence between ∼ TFL and T * . Resistivity data taken at 16 T for x ∼ 0.18 is also shown in (e) and follows the zero field curve at high temperatures as expected for longitudinal magnetoresistance. (b, d, f, h) The temperature dependence of resistivity against T 2 illustrating the Fermi-liquid behaviour, given by ρ = ρ0 + AT 2 . Here the dashed red lines are linear fits in T 2 and the zero-temperature resistivity values, ρ0, and A parameters are listed in each panel. We find that the samples with the larger RRR also display larger TFL.   [31,35,40]. The γ exponent depends on the value of ρ0, which was extracted at the lowest temperature from longitudinal magnetoresistance in Fig. S5.