Re-order parameter of interacting thermodynamic magnets

Abstract

Phase diagrams of materials are typically based on a static order parameter, but it faces challenges when distinguishing subtle phase changes, such as re-ordering. Here, we report a dynamic nonequilibrium order parameter termed re-order parameter to determine subtle phases and their transitions in interacting magnets. The dynamical precession of magnetization, so-called magnon, premises as a reliable re-order parameter of strong spin-orbit coupled magnets. We employ orthoferrites YFeO₃ and its Mn-doped variations, where diverse magnetic phases, including canted antiferromagnetic (Γ₄) and collinear antiferromagnetic (Γ₁) states, have been well-established. Low-energy magnon uncovers the spin-orbit coupling-induced subtle magnetic structures, resulting in distinct terahertz emissions. The temporal and spectral parameters of magnon emission exhibit characteristics akin to BCS-type order parameters, constructing the magnetic phase diagram of Mn-doped YFeO₃. This approach further reveals a concealed ferrimagnetic phase within the Γ₁ state, underscoring its potential to search for hidden phases of materials, completing their phase diagrams.

Introduction

A phase diagram represents diverse states of a substance, based on external factors such as temperature and pressure^1,2. This is vital for decision-making, materials design, and matter understanding. An order parameter, a measure of the degree of order across the boundaries in a phase transition system^1,2, serves as a comprehensive determinant of matter phases in classical thermodynamics, such as magnetization in ferromagnets and polarization in ferroelectrics^3,4. They effectively distinguish the ordered state, characterized by a maximum saturated value, from the disordered state, where the value is zero^3,4. Complex interactions within ordered states give rise to multiple distinct phases. Then, classical order parameters face challenges in distinguishing these phases with subtle changes, such as re-ordering. This necessitates the development of novel parameters to classify the nuanced phases emerging from interactions like electron correlation and spin-orbit coupling.

Magnetic materials primarily divided into two categories: ferromagnets (FMs), where aligned spins create net magnetization (M), and antiferromagnets (AFMs), where antiparallel spin alignment yields zero M (refs. ^3,4). This M has served as the equilibrium order parameter that distinguishes these core magnetic phases. Temperature-dependent M(T) unveils ferromagnetic transitions with rotational symmetry breaking—M sharply rises from null at the transition temperature T_c, showing the first-order transition (Fig. 1a) (refs. ^3,4). However, due to spin-orbit coupling (SOC) that causes spin canting, AFMs also can exhibit net M (refs. ^5,6), introducing intricacy to the phase diagram and casting doubt on the classical order parameter. Indeed, the conventional equilibrium parameter M, acquired through static intensity measurements, such as magnetometry faces challenges in categorizing subtle magnetic structures, e.g., between canted AFM and FM.

Fig. 1: Introduction of re-order parameter.

a Classical order parameter in thermal equilibrium. As temperature (T) decreases, the disordered state transitions into an ordered state through the alignment of the thermodynamic order parameter, namely magnetization M(T) for ferromagnets. The order parameter exhibits a first-order-like transition at the critical temperature T_c, which can be effectively described by a BCS-type transition curve. b Dynamic re-order parameter in nonequilibrium. As T decreases, the ordered state undergoes re-ordering at the re-ordering temperature T_RO, accompanied by a first-order-like transition in the re-order parameter, denoted as dM(T)/dt. This re-order parameter, responsible for the precession of magnetization in magnets, demonstrates a high sensitivity to subtle re-ordered phases.

This necessitates the development of a new order parameter extending beyond static equilibrium, offering a credible alternative not only suitable for classical magnets but also capable of accommodating interaction effects. SOC induces subtle changes in spin ordering, such as the emergence of canted AFM phases with net M (refs. ^5,6), resulting in distinct magnetization dynamics. Interestingly, the observation of the magnon within various magnetic phases suggests their potential utility in distinguishing these subtle magnetic states^7,8,9,10,11. Despite their clear association with both static spin arrangement and spin canting, magnons have yet to be recognized as the order parameter.

We propose that magnons, representing the precession of M(T), assume the role of a re-order parameter (Fig. 1b), effectively delineating first-order-like magnetic transitions from the ordered state to the re-ordered state. An exceptional feature of this re-order parameter lies in its ability to directly excite magnons, excluding other spin effects stemming from inevitable sample imperfections, such as magnetic boundaries and clusters, aiding magnetic phase diagram studies, even in polycrystalline magnets. The counterintuitive notion that magnons, as dynamical modulation of the conventional order parameter M, determine magnetic phases, is remarkably unexpected.

Results

Orthoferrite YFeO₃, characterized by an orthorhombic perovskite structure (space group Pbnm)^12,13,14,15, can serve as a testbed for validating magnons as the re-order parameter. Strong SOC in these materials guarantees rich magnetic phases across the canted AFM (Γ₄) and collinear AFM (Γ₁) phases^12,13,14,15. Figure 2a depicts a canted AFM spin ordering of YFeO₃ below Néel temperature T_N = 645 K (refs. ^12,13,14,15). The complex spin ordering can be simplified with two magnetizations M₁ and M₂. The Γ₄ phase manifests a weak FM moment from canting of M₁ and M₂ along the c-axis due to SOC, which persists below T_N without applying an external magnetic field along the a-axis that incudes spin-reorientation transition¹⁶.

Fig. 2: Spin waves as the re-order parameter in canted antiferromagnetic (AFM) YFeO₃.

a Magnetic ordering of YFeO₃ below T_N. The magnetic ordering can be simplified by sublattice spins (S₁, S₂, S₃, S₄) and further by magnetization M₁( = S₁ + S₃) and M₂( = S₂ + S₄), resulting in the net M( = M₁ + M₂) and Néel vector A( = M₁‒M₂). b Two types of spin waves in YFeO₃. The quasi-FM spin wave (left) is a precession of net M, represented as a sum of M₁ and M₂ with a phase difference of π/2. The quasi-AFM spin wave (right) is a precession of net M with in-phase M₁ and M₂. ω_FM and ω_AFM are the precession frequencies of quasi-FM and quasi-AFM spin waves, respectively. c Terahertz (THz) field-excitation of spin waves: (I) M is rotated by a torque induced by the THz magnetic field (H_THz), resulting in the excitation of spin waves. (II) The excited spin wave radiates THz E-field (E_THz) during relaxation, captured by a THz detector. d Time-domain signal and its Fourier-transformed spectrum. In the time domain, the sample signal (orange) transmitted through the sample exhibits a delay relative to the reference signal (black) transmitted through the blank holder. The sample signal (top panel) consists of the primary pulsed E_THz-field and the oscillatory E_THz-field from the magnon emission. The inset shows the magnified magnon emission signal. The transmission spectrum (red curve) of the entire time-domain signal and the emission spectrum (blue curve) of the oscillatory spin wave emission reveal a quasi-ferromagnetic spin wave at 10 cm^‒1 ( ≈ 1.24 meV ≈ 0.3 THz) and a quasi-AFM spin wave at 18 cm^‒1 (≈ 2.23 meV ≈ 0.54 THz). e Magnon density (ω_p²) serving as the re-order parameter. The ω_p² of the quasi-FM magnon mode is acquired from the Lorentzian model fitting to the quasi-FM magnon conductivity spectrum (left inset). Dots represent experimental data, while solid lines represent fitting lines. No phase transition occurs for low-doped YFe_0.97Mn_0.03O₃ (black), while a first-order-like transition appears for high-doped YFe_0.85Mn_0.15O₃ (red), as described by a BCS-type transition curve (red line). The auxiliary re-order parameter, scattering rate (γ, inverse lifetime), shows a peak at the re-ordering temperature (right inset).

The presence of spin canting in orthoferrites inevitably results in the coexistence of quasi-FM and quasi-AFM magnons, as illustrated in Fig. 2b. The quasi-FM mode arises from the precession of M₁ and M₂ with a phase difference of π/2, while the quasi-AFM mode is attributed to precession without any phase difference. The introduction of transition metal ions into the Fe³⁺ sites is known to induce spin-reoriented phases, exemplified by the substitution of Mn³⁺ causing the Morin-type spin-reorientation transition from the Γ₄ phase to the collinear AFM phase Γ₁ (refs. ^17,18,19,20). Naturally, this phase transition is supposed to change the dynamics of magnons.

By circumventing other charge effects, terahertz (THz) excitation facilitates low-energy magnon excitation in semiconducting orthoferrites. Notably, their substantial band gap and lack of free carriers or low-energy phonons underscore THz field generation of magnon that serves as the re-order parameter. As described in Fig. 2c, the magnetic field component (H_THz) of the THz transient applies torque to the net moment, inducing precession and resulting in THz electromagnetic field emission during relaxation, governed by the Landau–Lifshitz–Gilbert equation²¹:

$$\frac{{{\mbox{d}}}{{{{{\bf{M}}}}}}}{{{\mbox{d}}}t}=-\left|\gamma \right|\left({{{{{\bf{M}}}}}}\times {{{{{{\bf{H}}}}}}}_{{{{{\rm{eff}}}}}}\right)+\frac{\alpha }{{{{{{\bf{M}}}}}}}\left({{{{{\bf{M}}}}}}\times \frac{{{\mbox{d}}}{{{{{\bf{M}}}}}}}{{{\mbox{d}}}t}\right)$$

(1)

with gyromagnetic ratio |γ| and Gilbert damping α. Featuring SOC-induced spin canting, orthoferrites possess both THz quasi-FM and quasi-AFM magnon modes, distinct from collinear AFMs.

Time-resolved THz spectroscopy accurately captures magnon dynamics in picoseconds. Figure 2d (top panel) presents THz electric (E_THz) fields—a primary pulse remains after magnon generation and is followed by magnon emission. Although the H_THz-field component triggers magnon excitation, the resulting emission registers as an E_THz-field. In the case of the parent compound YFeO₃, two distinctive magnons associated with the canted AFM are observed in the time-domain as a superposition of two oscillatory signals with distinct frequencies (inset of Fig. 2d (top panel)).

Next, the transmission spectrum is obtained through the Fourier transformation of the complete time-domain signal (comprising the primary pulsed E_THz-field and the oscillatory magnon emission), whereas the emission spectrum exclusively utilizes the oscillatory magnon emission that follows the primary pulsed E_THz-field. These spectra consistently reveal a quasi-FM magnon at 10 cm^‒1 ( ≈ 1.24 meV ≈ 0.3 THz) and a quasi-AFM magnon at 18 cm^‒1 (≈ 2.23 meV ≈ 0.54 THz) as consistent with the previous works^7,8,10,20. We emphasize that the nature of the sample, whether it is a single crystal or polycrystal, does not affect magnon observation.

The spectral analysis (Fig. 2e) unveils key magnon parameters, including the precession frequency (ω₀), magnon density (ω_p²), and scattering rate (γ) (or equivalently, inverse lifetime γ = 1/τ). These parameters are acquired from the Lorentzian model fitting to the quasi-FM magnon conductivity spectrum (left inset of Fig. 2e) (see Supplementary Figs. 1‒3 for fitting details). For low-doped YFe_0.97Mn_0.03O₃, ω_p² presents no clear change over the temperatures (4 K ≤ T ≤ 295 K). In contrast, such temperature dependence of ω_p² distinctly depicts a first-order-like transition for high-doped YFe_0.85Mn_0.15O₃, which is reminiscent of order parameter M described by BCS-type transition (Methods). The magnetic transition consistently adheres to the BCS model (Methods) as well as the power law in mean-field theory (order parameter ∝ (1‒T/T_c)^β). However, considering the significance of the saturation value reflecting the magnon density, we opted for the BCS model. This choice not only provides the transition temperature but also furnishes the saturation value, in contrast to the power law, which only identifies the transition temperature.

The transition temperature of ω_p² closely coincides with the spin reorientation temperature of YFe_0.85Mn_0.15O₃ (ref. ¹⁸), highlighting the potential of ω_p² as a reliable re-order parameter. Indeed, E_THz ∝ d²M/dt² for FM magnons²² guarantees dM/dt as an integration of E_THz proportional to the strength of the magnon emission peak (∝ ω_p²). Note that the quasi-AFM magnon does not exhibit any order parameter-like features (See Supplementary Fig. 4). Moreover, γ also exhibits a distinct peak at the re-ordering temperature (right inset of Fig. 2e), suggesting an auxiliary re-order parameter complementing the primary re-order parameter ω_p².

We present a comprehensive phase diagram encompassing both the parent compound YFeO₃ and its Mn-doped variants (YFe_1‒xMn_xO₃), utilizing the re-order parameter ω_p² (Fig. 3a). The auxiliary re-order parameter γ (Fig. 3b), as a complementary of re-order parameter, aids in comprehending the re-ordering process and the subtle variations in magnetization induced by Mn doping. Within the diagram, the phase boundaries are determined by tracking the transition temperatures of quasi-FM magnons, with respect to varying doping levels (x). The comprehensive phase diagram (Fig. 3c) reveals numerous intricate phases, encompassing the Mn fluctuation regime, Mn ordering regime, magnetic compensation boundary, and ferrimagnetic (FerriM) phase. These findings significantly expand upon previous research, which primarily focused on the simplistic depiction of two phases: canted AFM and collinear AFM, surrounding the spin reorientation boundary (white circles in Fig. 3c, ref. ¹⁸).

Fig. 3: Establishing a phase diagram using the re-order parameter for Mn-Doped YFeO₃.

a Temperature-dependent re-order parameter ω_p² for Mn-doped YFeO₃ (YFe_1‒xMn_xO₃). Experimental data are represented by dots, while the BCS model fitting results are shown as lines. For 0.1 ≤ x ≤ 0.4, red stars indicate the first-order-like transition temperatures, and blue stars correspond to temperatures where the experimental data begin to deviate from BCS traits. For x = 0.45, black and orange stars indicate indistinguishable points. Above 180 K, no experimental data are available due to significant broadening of magnon peak, and a grey line represents the experimental data for x = 0.4. b Temperature-dependent auxiliary re-order parameter γ. The stars correspond to those in a. c Magnetic phase diagram, constructed using our re-order parameter data (stars). Results from magnetometry in reference (white circles) are also plotted for comparison. The phase diagram encompasses the canted antiferromagnetic, collinear antiferromagnetic, and ferrimagnetic phases. The analysis of the re-order parameter reveals spin reorientation, magnetization fluctuation, and ordering due to Mn doping, as well as magnetization compensation between phases. The figure illustrates the spin arrangement of each phase, with a blue arrow indicating the Fe moment, a grey arrow representing the Mn moment, and a red arrow denoting the net moment. d Temperature-dependent magnetic susceptibility data for the Mn-doped YFeO₃ samples identical to those used in THz experiments. Shaded regions indicate the temperature of phase boundaries, determined by magnetic susceptibility. A constant external magnetic field (H) of 100 Oe is applied in this magnetometry experiment.

First, the undoped and lightly doped samples (up to x = 0.08) exhibit no discernible phase transitions in the re-order parameter ω_p² (See Supplementary Fig. 1). This suggests that undoped YFeO₃ and lightly doped YFe_1‒xMn_xO₃ exhibit no discernible alterations in magnetic ordering without spin reorientation, spin fluctuations, or magnetic compensation. In contrast, highly doped YFe_1‒xMn_xO₃ (0.1 ≤ x ≤ 0.45) reveal phase transitions in the re-order parameter ω_p² (See Supplementary Fig. 2). Additional noteworthy observation is that the crossover temperatures tend to increase with higher values of x. For the range 0.1 ≤ x ≤ 0.4, first-order-like transitions of re-order parameter ω_p² are evident at re-ordering temperatures (red stars in Fig. 3), all of which are well-described by the BCS model fitting (Fig. 3a).

For highly doped YFe_1‒xMn_xO₃, these re-ordering temperatures closely coincide with the Morin-type spin reorientation temperatures (denoted by the white circles in Fig. 3c, ref. ¹⁸), signifying the transition of AFM ordering from the a-axis to the b-axis and the elimination of canting, as reported in previous studies^17,18,19,20. This consistency serves to validate the robustness and reliability of the re-order parameter ω_p². Moreover, the scattering rate γ of magnons (Fig. 3b) experiences an increase of more than twofold at the re-ordering temperatures. This elevated γ implies the disruption of magnon coherence, which could be associated with the fluctuation in the magnetization of Mn (S_Mn) on the background of the magnetization of Fe (S_Fe). Therefore, the γ can serve as an auxiliary re-order parameter.

At lower temperatures (marked by blue stars in Fig. 3a), a significant deviation from the BCS characteristics induced by spin reorientation occurs. These previously unreported transitions raise intriguing questions. It is plausible that this deviation results from the influence of Mn doping, which alters the Mn-Mn interaction, ultimately leading to Mn ordering at lower temperatures. As Mn ordering takes place, the concurrent increase in both ω_p² and γ signifies an augmentation in magnon density but with a diminished level of magnon coherence. We also find an occurrence of crossing near Mn ordering temperatures (blue stars) in the two-dimensional plot of magnon emission with the x-axis (temperature) and y-axis (time), signaling the change in the relaxation time τ (See Supplementary Fig. 5).

In the case of x = 0.45, distinctive transitions (marked by black and orange stars in Fig. 3a,b) become apparent along with Mn ordering at lower temperatures. However, experimental data above 180 K are unobtainable due to the substantial peak broadening (i.e., the substantial magnon scattering rate γ). It is worth noting that the re-order parameter ω_p² exhibits a sudden decrease below 100 K, signifying a reduction in magnon density due to magnetization compensation (black stars), where S_Fe and S_Mn offset each other. This compensation can be interpreted as an intermediate state in the Mn ordering process, transitioning towards a ferrimagnetic state (orange star) characterized by perfect collinearity between S_Fe and S_Mn.

Indeed, our investigation unveils the presence of the FerriM phase deep within the collinear AFM phase for x = 0.45, as indicated by the subsequent rise in the re-order parameter ω_p². As the magnetization (M) saturates, the magnon density (∝ ω_p²) experiences a sharp increase below the compensation temperature, aligning with expectations. Notably, the distinct characteristics of magnon emissions, as revealed by the (auxiliary) re-order parameters, offer precise identification of the FerriM phase. This is particularly important since the FerriM phase may appear somewhat ambiguous in magnetometry results, which typically exhibit negative magnetic susceptibility^18,23.

To highlight the strength of (auxiliary) re-order parameters, we compare the phase identification based on the re-order parameter with that based on the conventional order parameter. Figure 3d presents temperature-dependent magnetic susceptibility data, where an external magnetic field (H) of 100 Oe is applied to the Mn-doped YFeO₃ samples used in the THz experiments.

Foremost, the alignment between THz results and magnetometry data reinforces the credibility of the re-order parameter. Around the spin reorientation temperature (marked by the red stars), both the magnetic susceptibility and THz data consistently exhibit a pronounced decrease with decreasing temperature. Additionally, a modest rise in magnetic susceptibility can be interpreted as indicative of Mn ordering (indicated by blue stars), whereas a null magnetic susceptibility points to magnetization compensation (black stars). While negative susceptibility alone may not serve as direct evidence for the FerriM phase (noted by the orange star), its combined observation alongside the sharp increase in the re-order parameter ω_p² provides compelling support for the FerriM phase.

The pivotal distinction lies in the precision of the re-order parameter in discerning phases and identifying phase boundaries, as it provides sharp and well-defined transition temperatures. In comparison, magnetometry data displays a broad transition temperature range (shaded regions in Fig. 3d) with considerable error bars, often exceeding 50 K. Magnetic susceptibility involves extensive trial and error with varying magnetic fields to achieve accurate measurements, whereas re-order parameters do not require the application of magnetic fields. Correcting issues like the splitting of M during field cooling and heating cycles (Fig. 3d) and accounting for magnetic domain boundaries and clusters can be challenging. Relying solely on magnetic susceptibility makes it challenging to identify complex magnetic phases beyond the spin reorientation boundary, such as the Mn fluctuation regime, Mn ordering regime, compensation boundary, and FerriM phase.

Discussion

We conclude this report by offering a perspective on dynamic order parameters, as depicted in Fig. 4. Dynamic order parameters, exemplified by dM/dt, serve a dual purpose, functioning as both classical order parameters and re-order parameters. Beyond the re-order parameter (main text), dM/dt for FM magnons can also act as an order parameter: it remains at dM/dt = 0 in the disordered state (M = 0) and exhibits a sharp increase as M increases in the ordered state. The phases arising from ordering and re-ordering transitions can be understood through Landau free energy landscapes, where the minimum position dictates the dynamic order parameter value. With time-resolved measurements, time-domain oscillatory signals are responsible for the notable changes in dM/dt across the ordering and re-ordering temperatures. The spectral area, proportional to the dynamic order parameter (e.g., ω_p² ∝ dM/dt), can identify the nuanced phases in interacting thermodynamic magnets. In addition to Γ₄ to Γ₁ transition in Mn-doped YFeO₃, we also validate the applicability of the re-order parameter for Γ₄ to Γ₂ transition in HoFeO₃ (See Supplementary Fig. 6).

Fig. 4: Generalization to dynamic order parameter: serving as both order parameter and re-order parameter.

This figure illustrates the concept of a dynamic order parameter, which serves as both an order parameter and a re-order parameter Main panel: The figure demonstrates the revelation of ordering at T_c (critical temperature) or re-ordering at T_RO (re-ordering temperature) through the abrupt change in the dynamic order parameter, such as dM/dt. First layer scheme: The corresponding Landau free energy landscape, representing the potential structure of the system. Second layer: Real-space arrangement of the order parameter, depicted as arrows on a square lattice, providing insight into the spatial distribution of the ordered state. Third layer: Time-domain signal, featuring oscillatory signals that are responsible for the dynamics of the order parameter, for example, the precession of magnetization. Fourth layer: The spectrum, showing Lorentzian peaks obtained through the Fourier transformation of the time-domain signal. The area of the spectrum is directly proportional to the dynamic order parameter, such as ω_p² ∝ dM/dt.

Our study highlights the utility of dynamic order parameters in uncovering uncharted phases and their significance in understanding physical phenomena in complex thermodynamic systems. Indeed, dynamic order parameters recently applied to phenomena such as charge density waves²⁴ and colloidal glasses²⁵.

Methods

Cryogenic terahertz (THz) spectroscopy

THz spectroscopic measurements were conducted on a Teraview TPS3000 (Teraview Ltd., UK) over the frequency range of 0.1–3 THz with controlling the temperature over the range of 4–300 K by a helium-free optical cryostat (Cryostation, Montana Instruments, Ltd., USA). All measurements were carried out in vacuum to remove the water-vapor absorption. The samples were attached to a holder with a 3-mm hole by using silver paste. The THz-TDS technique yielded time-dependent waveforms of electric fields as raw data, and these were converted into complex-valued functions in the frequency domain through the fast Fourier transform (FFT).

Sample preparation

Polycrystalline YFe_1‒xMn_xO₃ (x = 0–0.4) compounds were synthesized by a conventional solid state reaction method. A stoichiometric mixture of Y₂O₃ (99.98% Alfa Aesar), Fe₂O₃ (99.998% Alfa Aesar), and MnO₂ (99.997% Alfa Aesar) powders was ground by using a pestle in a corundum mortar, followed by pelletizing and calcining at 1100°C for 5 h. The calcined pellet was reground and sintered at 1200°C for 12 h. The compound was again finely reground and then sintered at 1300°C for 30 h and cooled to room temperature at a rate of 100°C/h. The crystallographic structure of the YFe_1‒xMn_xO₃ samples was confirmed by using an X-ray diffractometer (Ultima IV, Rigaku) with Cu-Ka radiation. The temperature and magnetic field dependences of DC magnetization were examined by a vibrating sample magnetometer for temperatures of 2–300 K and magnetic fields of ‒9 T to 9 T by using a physical properties measurement system (PPMS, Quantum Design, Inc.).

Spectral analysis

We conducted the fitting of conductivity spectra using Lorentz models. (i) For lowly Mn-doped YFe_1‒xMn_xO₃ samples over 0 ≤ x ≤ 0.08, we adopted four Lorentzian peaks, where ω_p is the oscillation strength, ω₀ is the center frequency, and γ is the broadening parameter.

$${\sigma }_{{{{{{\rm{total}}}}}}}\left(\omega \right)= \frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{FM}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{FM}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{FM}}}}}}}\omega }\left({{{{{\rm{FM\; magnon}}}}}}\right) \\ +\frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{AFM}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{AFM}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{AFM}}}}}}}\omega }\left({{{{{\rm{AFM\; magnon}}}}}}\right) \\ +\frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{bg}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{bg}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{bg}}}}}}}\omega }\left({{{{{\rm{Background}}}}}}\right) \\ +\frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{imp}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{imp}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{imp}}}}}}}\omega }\left({{{{{\rm{Impurity}}}}}}\right).$$

(ii) For highly Mn-doped YFe_1‒xMn_xO₃ samples over 0.1 ≤ x ≤ 0.45, we adopted three Lorentzian peaks, where ω_p is the oscillation strength, ω₀ is the center frequency, and γ is the broadening parameter.

$${\sigma }_{{{{{{\rm{total}}}}}}}\left(\omega \right)= \frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{FM}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{FM}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{FM}}}}}}}\omega }\left({{{{{\rm{FM\; magnon}}}}}}\right) \\ +\frac{\omega }{4{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}}\frac{{\omega }_{{{{{{\rm{p}}}}}},{{{{{\rm{bg}}}}}}}^{2}}{{\omega }_{0,{{{{{\rm{bg}}}}}}}^{2}-{\omega }^{2}-{{{{{\rm{i}}}}}}{\gamma }_{{{{{{\rm{bg}}}}}}}\omega }\left({{{{{\rm{Background}}}}}}\right).$$

BCS model fitting

The re-order parameter ω_p² is fitted to the first-order-like transition with a BCS-type transition curve as below:

$${{{\omega }}_{{{{{{\rm{p}}}}}}}}^{2}(T)={A}*{\tanh} ((({{T}_{{{{{\rm{RO}}}}}}}/T)-1)^{B})+C$$

where T_RO is the re-ordering temperature. A is a free parameter, B is fixed to 0.5 or 0.8, and C is a constant offset (ref. ²⁶).