Unveiling the hidden nematicity and spin subsystem in FeSe

Unveiling the hidden nematicity and spin subsystem in FeSe Chih-Wei Luo , Po Chung Cheng, Shun-Hung Wang, Jen-Che Chiang, Jiunn-Yuan Lin, Kaung-Hsiung Wu, Jenh-Yih Juang, Dmitry A. Chareev , Olga S. Volkova 6, 7 and Alexander N. Vasiliev 6, 7 1. Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan 2. Taiwan Consortium of Emergent Crystalline Materials, Ministry of Science and Technology, Taipei 10601, Taiwan 3. Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan 4. Institute of Experimental Mineralogy, Russian Academy of Sciences, 142432, Chernogolovka, Moscow District, Russia 5. Physics Faculty, Moscow State University, Moscow 119991, Russia 6. Institute of Physics and Technology, Ural Federal University, Mira st. 19, 620002 Ekaterinburg, Russia 7. National University of Science and Technology "MISiS", Moscow 119049, Russia


INTRODUCTION
The progress in understanding Fe-based superconductors has formed a most intriguing chapter in modern condensed matter physics [1][2][3] . The existence of nematic order has become well established in Fe-based superconductors and is considered an essential ingredient to understand the mechanism of Fe-based superconductivity [4][5][6] .
The nematic order breaks the rotational symmetry by making the x and y directions in the plane non-equivalent, while preserving the time-reversal symmetry. The chalcogenide FeSe has a superconducting transition temperature Tc ~ 8.5 K; its tetragonal structure undergoes a transition to orthorhombic below Ts = 90 K. No longrange magnetic order has ever been detected in FeSe down to the lowest temperatures [7][8][9][10] . In this respect, as the structurally simplest Fe-based superconductor, FeSe has unexpectedly emerged in the frontier of Fe-based superconductivity research [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] . Up to date, few consensuses have been reached on either the electronic structure of FeSe or its nematic and superconducting mechanisms. For example, very recent angleresolved photoemission spectroscopy (ARPES) has indicated a small Fermi surface (FS) above Ts that cannot be reproduced by density functional theory calculations 10,[14][15][16][26][27][28] . Furthermore, various ARPES groups concur on an even smaller FS below Ts 10, [14][15][16][26][27][28][29][30] , which is in general consistent with the Sommerfeld coefficient observed from the specific heat 31,32 . Nevertheless, how and why a FS is reconstructed in FeSe through Ts is poorly characterized. As for the nematic order under Ts, almost everyone agrees on an electronic origin, as a 0.2% orthorhombic distortion is unlikely to lead to the observed FS elongation and the shift of band energy at the M point 15 . Nevertheless, whether the nematicity in FeSe is magnetically or orbitally driven is under current fierce debate, whereas it is generally considered to be driven by magnetism in pnictides 6 .
This controversy occurs largely due to the absence of the magnetic order in FeSe that remains an unsolved puzzle. The existence of nematic fluctuations above Ts is, likewise, not entirely clear in the literature.
In the present work, we utilized the polarized femtosecond pump-probe spectroscopy of FeSe to elucidate the above issues. This probe is relevant to both the charge and spin channels, and is sensitive to fluctuations or the short-range order. For example, the wavelength-dependent femtosecond spectroscopy clearly revealed the magnetic fluctuations at T = 170 K in HoMnO3, far above the long-range antiferromagnetic TN = 76 K 33 . A similar technique has been applied to detect the nematic fluctuations above Ts in pnictides [34][35][36] . Here we employed polarized ultrafast spectroscopy to elucidate the detailed orientation and temperature dependence of the quasiparticle dynamics in FeSe. As a results of this comprehensive survey, the hidden nematic fluctuations and spin subsystem in FeSe is unveiled. Figure 1 shows the typical polarization-dependent photoinduced reflectivity (ΔR/R) transients on the (001) plane of an FeSe single crystal at various temperatures. At T = 60 K, below Ts, the ΔR/R transients demonstrate clear nematicity (Fig. 1a) in this phase, as also indicated in other experiments 10,[14][15][16][26][27][28] . We show below that this nematicity in dynamics reveals information of both the quasiparticle and magnetic channels.

RESULTS
Astonishingly, ΔR/R shows profound nematic fluctuations even at T = 150 K, far above Ts (Fig. 1b); this two-fold symmetry persists up to at least 200 K (Fig. 2c). Overall, the raw data in Fig. 1 indicate clear nematic signals in ultrafast dynamics at the highest temperatures unprecedented in preceding reports. Furthermore, the two-fold symmetry pattern shifts by 90 when the temperature passes through Ts as shown in Fig. 1. To depict the context of Fig. 1 more clearly, Fig. 2 shows the typical ΔR/R transients with the electric field E along  = 0 and  = 90 at the temperatures associated with those in Fig. 1. (Angles  = 0 and  = 90 were chosen to represent the largest nematic signals, which are corresponding to a-axis and b-axis of orthorhombic structure, respectively.) Both the sign and the amplitude of ΔR/R transients show clear nematicity between  = 0 and  = 90 at 60 K, as shown in Fig. 2a. With T increasing to 150 K, the sign of ΔR/R transients along  = 90 dramatically reverses from negative to positive.
Although this pattern shift was unexpected, it manifests a valuable clue to the coupling between magnetism and the FS topology in FeSe, as discussed below.
The relaxation processes (t > 0) of ΔR/R transients in FeSe single crystals are described phenomenologically with The first term in the right side of Eq. (1) is the decay of the photoexcited electrons (or quasiparticles, QPs) with an initial population number A1, through phonon coupling with a relaxation time τ1. The second term pertains to the decay of QPs with an initial population number A2, through spin coupling with a corresponding decay time τ2. The third term describes the energy loss from the hot spot to the ambient environment on a time scale of microsecond, which is much longer than the period of the measurement (~50 ps) and is hence taken as a constant. The ascriptions of the first and the second terms are due mainly to the time and energy scales of τ1 and τ2. (See the sections S2 of  S4 Supplementary Information.) To depict better the temperature dependence of nematic ultrafast dynamics, we undertook another thorough run of ΔR/R transient measurements with E along both  = 0 and  = 90. According to Eq. (1), each component was extracted from 290 K to 30 K, as shown in Fig. 3a-d. We discuss first the results for T  Ts; this nematic phase of FeSe has been defined better than the state of T > Ts. For the fast component in R/R, a remarkable difference in the amplitude A1 was observed between  = 0 and  = 90 in the low-temperature regime, shown in Fig. 3a. For  = 0, the sign of A1,0 below Ts is positive; in contrast, that of A1,90 below Ts is negative for  = 90. In the literature, this difference is known to manifest the nematicity of the electronic structure. For example, the anisotropic single-particle and collective excitations in the quasi-1D charge-density wave semiconductor K0.3MoO3 37 and the d-wave symmetry of the superconducting gap in cuprate superconductors YBCO 38-41 have been unambiguously revealed by polarized pump-probe spectroscopy. As intriguingly, the orientation anisotropy is shown also in τ1 (Fig. 3c). τ1,90 for  = 90 (red solid circles) shows a notable divergence near Ts; this divergence in the rate of QP relaxation indicates a gap opening, at least on some part of the FS. The presence of a gap in the QP density of states gives rise to a bottleneck for carrier relaxation. The mechanism of the bottleneck is described by the Rothwarf-Taylor model 42 ; indeed, the temperature dependence of  Fig. 3e. However, there is no such signature of divergence for τ1,0 near Ts, which implies a major difference in carrier dynamics and in the band structure along various k orientations in the electronic structure. This discrepancy between τ1,0 and τ1,90 seems puzzling; but it actually fits well into the fascinating ARPES observation that, for T < Ts at M point, a gap is opened along ky, whereas there is no gap opening along kx (see the illustration of the band structure in Fig. 3f) 27 . It is therefore plausible to assign the directions of 0 and 90 as x and y, respectively. The abrupt decrease in τ1,0 at 90 K, i.e., the relaxation of QPs becoming efficient, probably indicates that an increased density of states is involved in the relaxation processes along kx 43,44 . In this scenario, the results of Figs. 3a and 3c also imply that the reconstruction of FS at the M point occurs mainly near 90 K, with no significant fluctuation of electronic nematicity at the M point above 100 K. As ultrafast spectroscopy is a bulk probe, the present results provide bulk evidence to support the electronic structure according to the surface-sensitive ARPES.
We turn to the slow component in Eq.

DISCUSSION
The state of T > Ts in FeSe has been much less revealed than in the nematic phase, partly due to lack of tools appropriate to investigate the nematic fluctuations. In the following, we show that nematic ultrafast dynamics above Ts elucidates surprising details of this largely uncharted territory. As shown in Figs. 1, 2 and 3, when T increases beyond Ts, the nematic signatures persist until at least 200 K. In comparison, the magnitude of ΔR/R for T > Ts is smaller than that for T < Ts, but two features of the fast

DATA AVAILABILITY
The authors declare that the data supporting the findings of this study are available within the paper and its Supplementary Information files.

COMPETING INTERESTS
The authors declare no competing financial interests.     The error bars are the standard deviations estimated from several measurements.

S2. Ascriptions of the first and second terms in Eq. (1)
In FeSe, the electronic excitations generated by the pump pulses result in a rapid rise of ΔR/R at zero time delay, as shown in Fig. 2 Subsequently, the disordered spins would reorder on the timescale of tens of picoseconds 2,3 , which is expressed by the A2 component with two-fold symmetry below T * in Fig. 4. This spin-related process is described with the second term in Eq. (1) and the green dashed lines in Fig. S2(a)-(f).  (Fig. S2(f)) also show two relaxation channels as we observed at  = 0. The difference between these two ΔR/R transients at  = 90 and 0 is only the amplitude, which is caused by the Fermi surface (FS) distortion at  point as shown in the inset of Fig. 4. Above argument is also applied to both fast and slow relaxation channels at T > Ts and even T = Ts (= 90 K). However, when temperature is lower than Ts (= 90 K), the amplitude of fast relaxation channel at  = 0 is larger than that at  = 90, which is opposite to the cases at T  Ts. This is because that the electronic relaxation (fast relaxation channel) is dominated by the FS distortion at M point, whose distortion is much serious than that at  point and rotated by 90, as shown in the inset of Fig. 4. Additionally, when T < Ts, the slow relaxation channel, which pertains to spin reordering through the spin-phonon coupling, totally disappears in the ΔR/R transient at  = 90. This means that the relaxation channel through spin-phonon coupling is broken along  = 90, but it is still surviving along  = 0. If the phonon is isotropy, this huge anisotropy observed in the slow relaxation channel at T < Ts should come from the spin subsystem, which is not the necessary consequences of FS distortion.  opening at the M point due to band splitting; therefore it might be slightly lower than the real structure transition temperature at which FS reconstruction just begins to emerge. All of above parameters were also applied to the fitting of τ1,90 in Fig. 3c. The fitting result is in accord with the energy splitting between dyz and dxz near the M point in the Brillouin zone revealed from ARPES 6,7 . Very recently, this gap was assigned to the splitting (M) between dxz/yz and dxy bands at the M point 8,9 . However, the gap amplitude M between dxz/yz and dxy bands, and its temperature evolution are also consistent with temperature-dependent |A1,90| in Fig. 3e and

S4. The energy scale of magnetic fluctuations
In order to estimate the energy scale of magnetic fluctuations, we fitted the temperature-dependent A2 by the Rothwarf-Taylor model (for the details, please see section S3), which has been used for the temperature-dependent A1,90 in Fig. 3a. For the temperature-dependent A2,0 in Fig. 3b, the fitting with Rothwarf-Taylor model leads to an energy scale of 2m(0) = 72 meV and x = 0.216; The characteristic temperature Tm is fixed at 90 K as required by the experimental data points. The energy scale of 72 meV from the fitting is consistent with the value obtained from inelastic neutron scattering 11 . Tm is associated with the spin subsystem and is not necessary identical to Ts in S3. Figure S3a shows the temperature-dependent (|A1,0|-|A1,90|) and (A2,0-A2,90) above

S5. Onset temperatures of nematic and magnetic fluctuations above Ts
Ts. The onset temperature of non-zero (|A1,0|-|A1,90|), which is associated with the charge subsystem, is 200 K. The temperature-dependent (A2,0-A2,90) associated with the spin subsystem in the high-temperature region reveals the onset (marked by an arrow in Fig.   S3a) about 230 K, which is completely consistent with the onset temperature of slope change in the temperature-dependent magnetic moment of Fig. S3b. This feature at 230 K was revealed in our previous work by non-polarization-resolved pump-probe spectroscopy 12 , which further infers the opening of a spin gap in FeSe (see S4).
Solid lines are guides to the eyes. The arrows indicate the onset temperature of nonzero (A2,0-A2,90) and the slope change in (T).

S6. Orientation dependence of the relaxation times τ1 and τ2 at various temperatures
To reveal the overall anisotropic dynamics of electronic structure and the spin subsystem in FeSe, we plot the orientation-dependent relaxation times extracted from shows a strong anisotropy at 60 K on (001) plane of FeSe. The significantly enhanced τ1 is caused by the gap opening along ky at the M point as illustrated in Fig. 3f 6 . In contrast, the relaxation of excitation energy through coupling with spins is also strongly orientation-dependent at 60 K, indicating that the spins mainly align along kx (see the relaxation time drastically shrinks. For τ2, its anisotropy persists to temperatures between 200 K and 250 K to indicate that the spin nematicity in FeSe can be observed above 200 K, i.e., the temperature marked by the arrows in Fig. S3. The anisotropy of τ1 is observable only until 200 K. This comparison indicates that the nematicity of spin subsystem appears at temperatures higher than that of the electronic nematicity concurring with Fig. S3a.