Magnetic field-induced quantum phase transitions in a van der Waals magnet

Magnetic field-induced quantum phase transitions in a van der Waals magnet Siwen Li1, +, Zhipeng Ye2, +, Xiangpeng Luo1, +, Gaihua Ye2, Hyun Ho Kim3, Bowen Yang3, Shangjie Tian4, Chenghe Li4, Hechang Lei4, Adam W. Tsen3, Kai Sun1, Rui He2, *, and Liuyan Zhao1, * 1Department of Physics, University of Michigan, 450 Church Street, Ann Arbor, Michigan 48109, USA 2Department of Electrical and Computer Engineering, 910 Boston Avenue, Texas Tech University, Lubbock, Texas 79409, USA

Chromium triiodide (CrI3) stands out amongst the large family of van der Waals (vdW) ferromagnets (FMs) investigated so far, because its isolated atomic crystal is one of the first two to realize twodimensional (2D) ferromagnetic long-range order 1,2 in the monolayer limit, and more importantly, is the first-ever interlayer antiferromagnetic (AFM) semiconductor 8,9,17,18 in its multilayer form. This also makes CrI3 distinct from the 2D intralayer AFMs [19][20][21] . From an application perspective, such unique magnetic properties of CrI3 have already drawn enormous interest in developing novel device functionalities that are tunable by either magnetic 8,9,17,18 or electric 7 fields. From a fundamental science point of view, extensive static and dynamic measurements in both three-dimensional (3D) bulk 22,23 and 2D film 8,9,17,18,24,25 CrI3 have been performed to understand the mechanism of the magnetism that has remained elusive until now. Our study here approaches the nature of CrI3 magnetism through exploring the magnetic field dependence of spin wave excitations in CrI3 and interrogating the interplay between the magnetic order and the crystalline structure.
In its bulk form, CrI3 goes through a monoclinic (!2/$) to rhombohedral (%3 ' ) structural phase transition at ( ) = 220 K and develops a FM long range order at ( ,-= 61 K 22 . Across ( ) , the major structural change involves the shearing of the vdW bonded CrI3 layers of the honeycomb lattice from a tilted to an aligned ABC stacking sequence 22 . Below ( ,-, all the spin moments within and between layers align along the out-of-plane direction 22 and a spin wave gap of ~ 1 meV emerges at the Brillouin zone center Γ point as a result of the Ising-type exchange anisotropy 23 . The magnetic and crystallographic degrees of freedom couple strongly in bulk CrI3 as evidenced by the enhanced reduction of interlayer spacing below ( ,- 22 . In its thin layer form, the CrI3 magnetism is known to survive below ( 1,-= 45 K, however, in a unique layered AFM order in which spins align along the same out-of-plane direction within each layer and alternate to the opposite orientation between adjacent layers 8,9,17,18 , while its crystal structure is much less studied experimentally. The origin for this crossover from the FM in bulk to the layered AFM in thin films, as well as the role of the crystallographic structure in this transition, remains as outstanding open questions in the field of 2D magnetism. In order to probe both the magnetic and crystallographic degrees of freedom in CrI3, we carried out polarized Raman spectroscopy to detect the symmetry-resolved collective excitations of spin precessions (i.e., magnons) 26 and lattice vibrations (i.e., phonons) 27 , respectively. To further reveal the interplay between these two degrees of freedom, we performed both temperature and magnetic field dependent Raman measurements covering a temperature (T) range from room temperature down to 10 K and a magnetic field (B) range from 0 up to 7 T. Because the stray magnetic fields cause the Faraday rotation of linearly polarized light transmitted through the objective in close proximity to the magnet 28 , we chose circularly polarized light to perform reliable selection rule measurements (see Supplementary Information section 1 for Raman selection rules in the circular polarization basis for CrI3, and see Methods for details of the Raman measurements).
We start by showing Raman spectra in both parallel and crossed circularly polarized channels, labeled as LL and LR in Fig. 1a, respectively, taken on a freshly cleaved 3D CrI3 crystal at T = 10 K and B = 0 T, where LL(R) stands for incident and scattered light being left and left (right) circularly polarized, respectively. We categorize all the observable Raman modes into three categories based on their symmetry properties. The first category contains phonon modes of either Ag or Eg symmetry of the C3i point group as reported before using linearly polarized light 24,29,30 . Under the circular polarization basis in Fig. 1a, the Ag phonon modes (labeled P(Ag)) only appear in the LL channel and the Eg ones (P(Eg)) are solely in the LR channel, as expected for the rhombohedral crystal structure. The second group consists of two modes, M1 and M2, that were previously shown to be antisymmetric and attributed to surface magnetic excitations 24 , showing up in the LL channel here. The third kind is exclusively very low frequency modes, M0, that, however, have neither been detected experimentally 24,29,30 nor been predicted theoretically 31,32 before in Raman studies and are present in both LL and LR channels. Symmetry-wise, these M0-type modes violate the selection rules for the rhombohedral crystal structure and the FM order. Energy-wise, their frequencies of ~ 4 cm -1 (0.49 meV) are close to the reported bulk spin wave gap, which is on the order of 1 meV 23 . We note that the M0 intensity is stronger in the anti-Stokes than the Stokes channel, possibly due to the resonance excitation to the charge transfer transition in CrI3 and the broken time reversal symmetry of M0.
To explore the nature of these potentially magnetism-related Raman modes, M0-2, and investigate the magneto-elastic coupling between magnetic and phononic modes, we carried out careful out-of-plane magnetic field (B ⊥ 67) dependent Raman measurements at 10 K and show the results in the LL channel in Fig. 1b  independence of M1 and M2 before 2 T immediately rules out the possibility of them being conventional spin wave excitations. More insights about their potential nature will be discussed later on in Fig. 3. Third, the phonons exhibit contrasting selection rules below and above 2 T while their frequencies remain more or less independent of magnetic field. For example, the Eg mode (at ~ 109 cm -1 ) leaks into the LL channel ( Fig. 1b) and the Ag modes (at ~ 79 and 129 cm -1 ) show up in the LR channel above 2 T (see Supplementary Information section 2).
Having described the Raman spectra evolution with increasing magnetic field, we show in Fig. 1c both Raman spectra in the LL and LR channels at our highest available magnetic field of 7 T. Comparing to the spectra taken at 0 T in Fig. 1a, one of the M0 modes shifts up to nearly 9 cm -1 with its spectral intensity primarily in the LR channel while neither M1 nor M2 are present in either the LR or LL channel. In addition, observable fractions of phonon intensities leak into the corresponding orthogonal channels, suggesting that the crystalline symmetry is lowered from the C3i point group at 0 T. So far, we have established that both the magnetic order and the crystalline structure change across a critical magnetic field ; 8 of about 2 T. To gain more insights into these magnetic field-induced phase transitions and their relationships to one the other, in the following, we first discuss the magnetic phase transition from the magnetic field dependence of M0 and then address the structural phase transition from the combination of the magnetic field dependence of the M1-2, Ag and Eg phonons.
To provide a comprehensive picture of the magnetic field dependence of M0, we summarize in Fig. 2a the key experimental result of the M0 frequencies shifting as a function of the external magnetic field and the corresponding field-dependent spin wave calculations in the left and right panels, respectively. Here, we show the average M0-type mode frequencies from the Lorentzian fit of the Raman spectra for both Stokes and anti-Stokes shift in both the LL and LR channels. Strikingly, despite the fact that bulk CrI3 is considered as a simple Ising ferromagnet 22 , we observe that three spin wave branches at magnetic fields lower than 2 T collapse into one across ; 8 = 2 T. In particular, a pair of the three branches below 2 T start with close frequencies of ~ 3.4 and 3.9 cm -1 at 0 T and evolve in opposite trends at increasing magnetic field (M0a and M0b), while the third one increases linearly since its appearance at ~ 1 T and continues after ; 8 of 2 T with a weak discontinuity of frequency redshift (M0c  . 6 @ to the > 6 . . 7 @ form, suggesting the loss of the three-fold rotational symmetry and thus the breaking of the rhombohedral crystal symmetry. We propose the shearing of vdW layers away from the aligned ABC stacking order (Fig. 3e), which is indeed a structural instability for CrI3 bulk 22,36 , as one means to transit from the C3i rhombohedral (nearly D3d because of weak interlayer interactions) to the C2h monoclinic crystal symmetry (see Supplementary Information section 1 for C2h Raman selection rules and Supplementary Information section 7 for its comparison to the high temperature monoclinic structure).
Interestingly, this magnetic field-induced monoclinic phase mimics a 3D nematic long-range order with a director as its order parameter (indicated as elongated ellipses in Fig. 3e). It is known that such a 3D nematic order must emerge through the first-order phase transition 37 as is indeed our case here.  Fig. 3b and c). These two cases are consistent with the Eg(C3i) phonons transforming into the Ag(C2h) and Bg(C2h) phonons, respectively, which further corroborates the proposed structural phase transition at ; 8 . Third, the antisymmetric M1, 2 modes stay nearly constant until ; 8 and disappear right above ; 8 (see M2 in Fig. 3d and M1 in Supplementary Information section 8). Both modes were initially interpreted as surface magnetic excitations based on their broken time reversal symmetry and thickness independence 24 . Here, we can immediately rule out the possibility that they are conventional bulk spin waves based on the magnetic field independence of their frequencies below ; 8 , and can confidently associate them with the sAFM because of their disappearance above ; 8 . Based on this, we propose one possible origin of M1, 2 to be a collective excitation made of two parts, one being the c-axis zone boundary phonon of the non-magnetic lattice A(C D⃗ F , H), and the other being the layered AFM order JK−C D⃗ F , 0L, where C D⃗ F is the out-of-plane wavevector for the constituent phonon (A) and magnetism (J) at frequencies of H and 0 (i.e., elastic scattering off the layered AFM order), respectively. Such a collective excitation breaks the time reversal symmetry, possesses a total momentum of zero and becomes inaccessible in the FM phase above ; F due to the finite momenta (see detailed analysis in Supplementary Information section 9). In other words, M1,2 corresponds to a special zone-folding by a single copy (or more precisely, odd number of copies) of the magnetic order that breaks the time reversal symmetry. However, this proposed origin cannot account for the thickness independence reported in Ref. 24 . Further experiments are needed to pin down the exact nature of both M1 and M2 modes. Nevertheless, they are good indicators for the sAFM state and the rhombohedral crystal lattice ( Fig. 3d and Fig. 1d). In contrast to conventional structural transitions, the emergence of this 3D nematic order is driven by an external magnetic field that has a much stronger coupling to electrons than to ions, supporting an electronic origin for this structural transition. One natural mechanism for it could be that this interlayer shear deformation increases the distance between the nearest interlayer spins and thereby reduces the exchange energy penalty for the fieldinduced layered AFM to FM transition. Magnetic field dependent Raman data on CrI3 flakes show consistent results (see Supplementary Information section 10).
We then proceed to construct the temperature versus magnetic field phase diagram for bulk CrI3 by performing temperature (magnetic field) dependent Raman measurements at multiple magnetic fields (temperatures). We show in Fig. 4a the temperature dependence of the M0c FM frequency at a series of magnetic fields, and we clearly observe a crossover between FM and paramagnetic (PM) states at ~ 65 K at all applied magnetic fields 37,38 . This crossover is represented by the striped line with the experimental data points from this work in Fig. 4c, and its extrapolation to zero magnetic field indeed corroborates the FM transition temperature of 61 K obtained from the magnetization measurements in 3D CrI3 bulk 22 .
Meanwhile, we display in Fig. 4b the magnetic field dependence of the M2 intensity at several temperatures, and we discover a decreasing trend of the critical magnetic field ; 8 with increasing temperatures. This shows that the novel mixed state of sAFM and bFM in bulk CrI3 is bounded by a line of phase transitions of both the layered AFM to FM and the rhombohedral to monoclinic structures, which is highlighted by a solid brown line in Fig. 4c. Furthermore, based on the fact that the crystal structure of 3D CrI3 bulk transits from the rhombohedral to monoclinic symmetry across ~ 220 K from below at 0 T 22 , we propose the presence of a phase boundary for the paramagnetic rhombohedral phase so as to connect the two regions of monoclinic structures above 220 K at ; = 0 T and below 45 K at ; > ; 8 . Please note that while the starting point of this phase boundary is determined to be at ( =~ 220 K, ; = 0 T, its end point is unknown due to a lack of experimental data, and thus the gray dashed line in Fig. 4c only presents one possibility.
Our findings in 3D CrI3 bulk of a novel mixed state of the surface layered AFM and the bulk FM and a magnetic field-induced first-order structural phase transition reveal a rich phase diagram for this vdW magnetic semiconductor, which unambiguously resolves the puzzling evolution from the FM in 3D to the layered AFM in 2D CrI3. Furthermore, controlling this unique magnetism in vdW magnets and its interplay with crystalline structures opens up new possibilities for the realization of novel 2D magnetic phases and the applications in modern spintronics. While a broader class of vdW magnets with such unique magnetism and strong magneto-elastic coupling are to be identified, CrI3 serves as an ideal platform to explore a variety of external controls, such as electric field, strain, and charge carrier doping, on the magnetism and its interplay with other degrees of freedoms.

Growth of CrI3 single crystals
Single crystals of CrI3 were grown by the chemical vapor transport method. Chromium power (99.99% purity) and iodine flakes (99.999%) in a 1:3 molar ratio were put into a silicon tube with a length of 200 nm and an inner diameter of 14 mm. The tube was pumped down to 0.01 Pa and sealed under vacuum, and then placed in a two-zone horizontal tube furnace. The two growth zones were raised up slowly to 903 K and 823 K for 2 days, and were then held there for another 7 days. Shiny, black, plate-like crystals with lateral dimensions of up to several millimeters can be obtained from the growth. In order to avoid degradation, the CrI3 crystals were stored in a nitrogen-filled glovebox and exfoliated in dark to expose fresh surfaces right before the experiment (the air exposure time in a dark environment was less than 10s before the sample was sealed in the cryostat vacuum chamber). used to achieve the variable out-of-plane magnetic field from 0 T to 7 T. The cryostat cold finger, on which the samples were mounted, was inserted into the center of the room-temperature-bore of the magnet. In this work, the selection rule measurements were performed mainly under the circularly polarized basis to eliminate any artifacts of the Faraday effect from the microscope objective subject to the strong stray magnetic field. All thermal cycles were performed at a base pressure lower than 7 × 10 -7 torr.

Competing financial interests
The authors declare no competing financial interests.

Data availability
The data that support the plots of this paper are available from the corresponding authors (R. H. and L. Z.) upon reasonable request.

S1. Raman selection rules for CrI 3 with the circular polarization basis
Under magnetic fields, the quartz-based optical components manifest the magneto-optical Faraday effect by rotating the electric field polarization of linearly polarized light, as is the case for a transmission objective in proximity to a strong magnet in magnetic field dependent micro-Raman measurements.  Table. S1. Note that in the rhombohedral phase A g and E g modes are expected to appear exclusively in parallel and crossed channels, respectively, whereas in the monoclinic phase A g modes show up in both channels and the B g modes only in the crossed channel.
As elaborated in Section S8, the antiferromagnetism (AFM) related Raman modes M 1,2 in the rhombohedral phase are anti-symmetric, with their Raman tensors being of the form Such modes are non-vanishing only in the LL channel.

LL LR
Rhombohedral/C 3i A g , M 1, 2 E g Monoclinic/C 2h A g A g , B g Table. S1. Raman selection rules for CrI 3 in the rhombohedral (C 3i ) and monoclinic (C 2h ) structures in circularly parallel (LL) and crossed (LR) channels.

S2.
Extended data of magnetic field dependent Raman spectra on 3D CrI 3 bulk

S2.1
Magnetic field dependent Raman spectra of 3D CrI 3 bulk in the LR channel at 10 K.

Fig. S1
Magnetic field dependent Raman spectra of 3D CrI 3 bulk in the LR channel at 10 K.

S2.2 Magnetic field dependent spin wave branches in 3D CrI 3 bulk in both Stokes and anti-Stokes sides and in both the LL and LR channels
The false color maps of magnetic field dependent Raman spectra in Fig. 1b in the main text and Fig. S1 above were plotted to have individual panels normalized to their own maxima, in order to make weak features more visible within each panel. However, we cannot make a direct comparison of Raman intensities across different panels. In Fig. S2 below, we provide color maps of magnetic field dependent spin wave modes in Fig. 1b and Fig. S1 using the same color scale so as to make a fair comparison between the spin wave branch intensities of Stokes and anti-Stokes sides in the LL and LR channels.

Fig. S2
Color maps for the magnetic field-dependent Raman spectra. The data were acquired in (a) the LL, and (b) the LR channel at 11 K. The color scale is set the same for the anti-Stokes and Stokes sides for both the LL and LR channels.

S2.3
Magnetic field dependent resonant Raman spectra on three freshly cleaved 3D CrI 3 bulk

S2.4
Comparison of the magnetic field dependent Raman spectra between the increasing and decreasing magnetic field.

Fig. S4
Comparison of the magnetic field dependent Raman spectra between increasing and decreasing magnetic field. Data taken with (a) the increasing magnetic field, and (b) the decreasing magnetic field.

Fig. S5
Resonant Raman spectroscopy data on two CrI 3 flakes with different thicknesses, together with their magneto-tunneling resistivity spectra. Raw magnetic field-dependent magnon Raman spectra maps, the plots of the fitted magnon frequencies v.s. magnetic fields, and magentotunneling resistivity spectra for (a) 8-layer-thick CrI 3 flake, (b) 20-layer-thick CrI 3 flake. Below B c , we can see the interlayer AFM spin wave branches (yellow and orange) in both samples, but the interlayer FM spin wave branch is missing in both samples. Above B c , there are two FM spin wave branches in both flake samples, in contrast to the single FM spin wave branch in the 3D CrI 3 bulk. This is because the thin flakes break translational symmetry along the c-axis (the out-of-plane direction) and therefore result in satellite modes near the primary one, which can be seen for phonon modes as well and have been discussed in Nature Communications 9, 5122 (2018). The magneto-tunneling resistivity measurements for both thin layer samples show a critical magnetic field of B c ~ 1.7 T, which is consistent with that found in resonant Raman spectroscopy. To test whether Option 2 is plausible in 3D CrI 3 , we therefore performed a calculation on the long-range magnetic dipole-dipole interaction for the CrI 3 with the rhombohedral crystal structure both at the surface and in the bulk with the lattice constant adopted from literature. We calculated the magnetic energy of a single magnetic moment at the surface interacting with all the rest of magnetic dipole moments within a cylinder (whose radius R is 3 Em, a typical in-plane ferromagnetic domain size, and depth is labeled by the number of layers N) through dipole-dipole interactions. There is no simple analytical solution to this dipole-dipole interaction, and therefore, we performed computational calculations. The result is plotted in Fig. R1 below. We can see that the magnetic energy per single Cr 3+ ion from long-range dipole-dipole interactions is only < 1 EeV for 20 layers of CrI 3 (20 layers are the estimated thickness of the surface AFM phase). This magnetic energy scale is over two orders of magnitude weaker than the interlayer AFM exchange couple of 150 E eV. Therefore, the long-range magnetic dipole interaction is not sufficient to overcome the short-range interlayer AFM exchange coupling so as to achieve the interlayer FM phase in regions of 3D CrI 3 about 20 layers from the surface. This finding is consistent with the literature on dipole-dipole interaction dominated magnets (whose leading energy scale is typically in order of EeV and transition temperature is sub-K) Our calculations rule out Option 2, and therefore suggest Option 1 (surface reconstruction) as a likely reason for achieving surface AFM and bulk FM.

Fig. S7
The plot of the magnetization v.s. out-of-plane magnetic field data taken on 3D CrI 3 bulk. The main panel shows the clear signature of the poling field of the bFM domain states, B p ~ 0.1 T. The two insets show the close-look around B c = ± 2T, the critical field for the sAFM to FM transition, in which a weak but observable anomaly in the magnetization is seen.
With these additional results from S3 and S4 sections above, we provide an in-depth analysis that confirms the mixed state of sAFM and bFM in 3D CrI 3 .
First, we establish the interlayer AFM state in addition to the FM state in 3D CrI 3 .
We summarize the experimental findings point-by-point as follows: 1. Through the magneto-tunneling resistance measurements, we find that the CrI 3 flakes of the thickness used in our experiment have interlayer AFM ordering, as evidenced by the extremely low magnetotunneling resistance at low magnetic fields. Across the critical field B c , about 1.7 T, the interlayer AFM transits to the FM phase, which is supported by the significant reduction of magneto-tunneling resistance above B c .
2. Through the magnetization measurements, we find that the bulk crystal is primarily in the FM order, and the saturation poling field B p is as low as ~ 0.1 T. This is to say that the magnetic field required to polarize the FM domains is as low as ~ 0.1 T. In addition, there is a very small but observable signature of the interlayer AFM to FM transition at ~ ± 2 T (Fig. S7).
3. Through the magnetic field dependent Raman spectroscopy measurements of CrI 3 flakes, both samples that are interlayer AFMs show a) a magnetic phase transition at a critical field B c ~ 1.7 T, which is consistent with the interlayer AFM to FM transition detected by the magneto-tunneling resistance measurements; b) two spin wave branches with opposite magnetic field dependence below B c , which corroborates with the interlayer AFM phase; c) the absence of the third spin wave branch below B c that is present in 3D CrI 3 , which is again consistent with the surface interlayer AFM phase.
Comparing the three experiments above in both bulk and multilayer CrI 3 , we can confidently conclude: We find in bulk CrI 3 , the FM phase for a great majority of the bulk volume, clear signatures of the interlayer AFM phase, and its phase transition into the FM phase at B c . These layered AFM-related behaviors are exactly the same as those in CrI 3 flakes known to be interlayer AFMs. At the same time, we also exclude the possibilities of spin waves from FM domains with opposite magnetization as it only takes ~ 0.1 T to polarize the FM domains whereas we see the three spin wave branches in 3D CrI 3 surviving up to B c ~ 2 T.
Second, we answer the question where the interlayer AFM happens in 3D CrI 3 bulk.
Resonant Raman spectroscopy measurements are subject to finite penetration depth, and therefore only survey a finite number of layers from the top surface. We have measured more than ten bulk CrI 3 samples with freshly cleaved surfaces and have observed the very same signatures of interlayer AFM and magnetic transitions in every individual sample. Fig. S5 shows the results from two flake examples. Such a consistent observation, together with the very small volume of interlayer AFM in 3D CrI 3 bulk, suggests that the interlayer AFM has to happen at the top layers, rather than deep in the bulk.

S6.
Spin wave calculations for the sAFM and bFM mixed state and its magnetic field dependence We have performed standard spin wave calculations for a 3D magnet made of ABC stacked 2D honeycomb magnetic lattices.
To calculate the spin wave dispersion for the sAFM state, we simulate with an AFM interlayer exchange coupling and FM intralayer exchange couplings, and the spin Hamiltonian can be written as In the mixed sAFM and bFM state of 3D CrI 3 , the bFM provides an effective magnetic field l G to the sAFM even when there is no external magnetic field. This results in the slight splitting of the two magnons per branch at 0 T. In our experimental data, the magnon branch we observed is the acoustic branch ; B whose field dependence is shown in Fig. 2a and labeled with the sAFM.
In the case of bFM below 2 T or FM above 2 T, we simulate the effective interlayer coupling with a FM one, and there are only 2 different spin sites because of two sublattices per honeycomb layer and one layer per FM unit cell. They correspond to one acoustic and one optical branch at 0 T, but are both spin -1 without any degeneracy. Experimentally, we observed the acoustic branch labeled as bFM in Fig. 2a. The interlayer exchange coupling `] is no longer the same as that in Eq. 2, but this has no impact on the magnetic field dependence of the acoustic branch which takes the following form of Comparing ; B t and ; B , the magnon with spin -1 of bFM has smaller energy than that of sAFM that is indeed consistent with our experimental observation. Given the fact that only the acoustic magnon branch is probed experimentally, we can only fit out the intralayer exchange anisotropy `a −`d, but not the individual values of `d and `a.
It is worth mentioning that, apart from studying magnetic excitations, we can also estimate the interlayer [^= −jE k lb The phase transition happens when the total energy of FM becomes lower than that of the sAFM, that is jE k l v b = b @`] . With l v = 2 T, we get `] = 0.15 meV, which corroborates the previous result of `] = 0.15 meV obtained using magnon energies.

S7. Comparison between the magnetic field-and temperature-induced monoclinic structure
We compare the Raman spectra between two cases, the magnetic field-induced monoclinic phase below T N above B c and the temperature-induced monoclinic phase above T s and at B = 0 T, as shown in Fig. S8 below. For the magnetic field-induced monoclinic phase, we can clearly observe the relaxation of selection rules from the rhombohedral point group, as the leakage of A g and E g modes appear in the LR and LL channel, respectively ( Fig. S8(a)). In contrast, for the temperatureinduced monoclinic phase, we barely find the selection rules relaxation from the rhombohedral C 3i to the monoclinic C 2h point group, because the A g and E g modes remain nearly absent in the LR and LL channel, respectively ( Fig. S8(b)). It could be the weaker interlayer van der Waals coupling between layers in the temperature-induced monoclinic phase than in the magnetic field-induced monoclinic structure, that makes the in-plane phonon modes (e.g., the A g and E g in the main text figure Fig. 3) hardly feel the symmetry reduction from the shearing between layers.

Fig. S8
Comparison between the Raman spectra taken on the magnetic field-and temperatureinduced monoclinic structure. Raman spectra in both LL and LR channels taken on (a) magnetic fieldinduced monoclinic phase, T = 11 K and B = 3 T, where a clear relaxation of the selection rule from rhombohedral structure is observed (the leakage of A g from LL into LR channel, and the leakage of E g from LR to LL channel), and (b) temperature-induced monoclinic phase, T = 290 K and B = 0 T, where the selection rule relaxation is hardly observable.

S8.
Magnetic field dependence of M 1 intensity As shown in Eq. 9, Δ1 KL is antisymmetric, so that the mode for such a composite object only shows up in the crossed channel with the linear polarization basis or the LL channel with the circular polarization basis.