Introduction

Phase transitions and associated critical phenomena near thermodynamic critical points are fundamental research themes in modern condensed matter physics1,2,3. Pioneering studies since the 1960s4,5,6 established that phase transitions are governed by divergent correlation lengths and emergent scale invariance, resulting in critical phenomena characterized by power-law singularities in thermodynamic quantities. These singularities manifest as distinctive scaling behaviors in macroscopic quantities such as magnetization, susceptibility, and specific heat. Their critical exponents, classified through the universality principle, form the cornerstone of critical phenomena characterization7. Experimentally, critical exponents are typically determined by fitting the divergent behaviors of these quantities8,9,10,11. Scaling laws establish fundamental relationships between critical exponents, demonstrating that singularities in macroscopic quantities originate from microscopic long-range correlations. While critical exponents associated with correlation length and time provide more direct insights into critical dynamics, their experimental determination remains challenging due to the simultaneous requirements of spatial and temporal resolution. Consequently, experimental characterization of phase transitions has historically lacked a perspective rooted in these intrinsic parameters.

In magnetic systems, the extensively studied paramagnetic-ferromagnetic phase transition serves as a vital platform for investigating critical behaviors. The magnetic noise generated by thermally driven spin (cluster) fluctuations during phase transitions encodes information about long-range correlations12,13,14. Consider an experiment where a sensor placed above a magnetic material measures magnetic perturbations caused by these fluctuations. This naturally raises two questions: (1) what quantities can be extracted from these measurements, and (2) how they reflect the critical physics. The power spectral density SB(ω) of the measured magnetic field noise is directly related to the spectral density of magnetization SM(ω). Crucially, as shown in ref. 15, the noise spectra exhibit a strong dependence on the reduced temperature (T − Tc)/Tc, where Tc is the critical temperature. Thus, noise spectrum measurements could provide essential insights into critical phenomena, including long-range correlations and critical exponents15,16,17.

The nitrogen-vacancy (NV) centers18,19,20 in diamond are ideal quantum sensors for this purpose. Here, we employ NV centers to measure critical spin fluctuations in the van der Waals magnet Fe3GeTe2, yielding two key findings. First, the NV centers’ decoherence rate shows a pronounced peak at Tc, accompanied by a change in its temperature dependence. Second, we derive a theoretical model that quantitatively links the decoherence rate to both the noise spectra and the sensor-sample separation distance. Importantly, our experiments explicitly reveal the crossover from white noise to 1/f noise as the system approaches criticality. These results establish a framework for probing phase transitions through critical fluctuations across diverse quantum materials.

Results

We investigate the critical fluctuations in Fe3GeTe2 (hereafter abbreviated as FGT), a van der Waals magnet exhibiting a paramagnetic-ferromagnetic phase transition. FGT is an exfoliable magnet that retains robust ferromagnetism with strong out-of-plane anisotropy even at the monolayer limit9,21,22. For bulk FGT, the Tc is approximately 210 K8,23. When thinned down to a monolayer, Tc is significantly suppressed due to enhanced fluctuations24, accompanied by a transition in magnetic properties from 3D to 2D Ising ferromagnetism22,23,25,26.

The investigation of critical fluctuations near the critical point constitutes the primary focus of this work. To achieve this, we developed a cryogenic wide-field microscopy system utilizing ensemble NV centers27, with the experimental configuration depicted in Fig. 1a. The system employs a linearly polarized 532 nm laser to initialize and read out the state of NV centers, while microwave radiation delivered through an antenna enables quantum state manipulation. An in-plane magnetic field was applied to lift the degeneracy of the \(\left\vert \pm 1\right\rangle\) of NV centers and to avoid perturbing the intrinsic anisotropy of the FGT sample. Hexagonal boron nitride (hBN) substrates were transferred above and below the FGT flakes to prevent oxidation and to adjust the distance d between the FGT and NV centers, respectively. More details about the setup are presented in Supplementary Section 1.

Fig. 1: Experiment setup and pulse sequences.
Fig. 1: Experiment setup and pulse sequences.
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a Exfoliated FGT flakes are transferred to a [100] oriented diamond chip with single-layer ensemble NV centers. Black arrows in the diamond chip indicate the four possible orientations of the NV centers. Microwaves are radiated through an Ω-shaped wave guide, and the magnetic field B0 is applied in-plane. hBN substrate layers are placed above and below the FGT flakes. Inset: Crystal structure of FGT and NV centers, where d denotes the NV-FGT distance. The crystal data were obtained at SpringerMaterials59. b Hahn-echo pulse sequences for wide-field coherence detection. Laser pulse initializes the state of NV centers to \(\left\vert 0\right\rangle\) and reads out the final state. The first π/2 microwave pulse drives the system to a superposition state. After free evolution for time τ, a π-pulse decouples noise components with frequencies beyond the 1/2τ band. Final π/2 or 3π/2 pulses convert coherence to \(\left\vert 0\right\rangle\) or \(\left\vert 1\right\rangle\) populations. A CMOS camera records photoluminescence from these populations as signal and reference frames, respectively. Each subsequence executes N times per frame. The pulse sequence is averaged over M ~100–200 times for each τ.

NV centers, serving as quantum sensors, have become indispensable tools in condensed matter physics, materials science, and quantum sensing28,29,30,31. The optically detected magnetic resonance technique enables static magnetic field sensing through continuous wave (CW) spectroscopy32, and dynamic decoupling protocols33,34,35,36 establish frequency domain detection channels, particularly effective for resolving varying magnetic signals through spectral engineering37,38. Notably, these methodologies have unveiled critical fluctuations in van der Waals magnets, including Fe3GeTe239, \({{{{\rm{MnBi}}}}}_{2}{{{{\rm{Te}}}}}_{4}{({{{{\rm{Bi}}}}}_{2}{{{{\rm{Te}}}}}_{3})}_{{{{\rm{n}}}}}\)40, CrPS441, CrSBr42, and twisted double trilayer CrI343, primarily through relaxometry near phase transitions. The relaxation time of NV centers is predominantly governed by noise components at frequencies resonant with their energy levels, typically centered around 2.87 GHz. The observed increase in relaxation rate reflects enhanced spin fluctuations near the Tc. Meanwhile, subHertz magnetic domains’ reverse has been observed in our previous work near the Tc44. These results demonstrate NV centers’ capability to detect spin fluctuations across different time scales. Notably, the aforementioned works and ref. 45 have merely reported the existence of critical fluctuations, without exploring their underlying mechanisms, particularly failing to establish connections with scaling theory.

In contrast to previous experiments, our work leverages the quantum coherence of NV centers to probe critical fluctuations. The details of the NV centers and FGT samples are presented in Supplementary Sections 2 and  3. The Hahn-echo pulse sequence (Fig. 1b) exhibits a narrow-band spectral weight function, selectively coupling to noise components whose frequencies match the pulse sequence periodicity46. This frequency-selective coupling modulates the coherence time of NV centers, enabling spectral mapping of magnetic fluctuations.

Multi-mode Imaging for static and fluctuating magnetic fields

Firstly, we perform wide-field imaging measurements to characterize the coherence time of NV centers adjacent to the FGT sample. When T > Tc, although the FGT is paramagnetic and lacks long-range magnetic order, local magnetic fluctuations still suppress the coherence of NV centers. The results are presented in Fig. 2a. The decoherence of NV centers was measured through Hahn-echo pulse sequences, as shown in Fig. 1b. The contrast is defined as the normalized difference between the two frames: C = (Isig − Iref)/Iref, where Isig(Iref) is the gray value of the signal (reference) frame. The decoherence curves are depicted in Fig. 1a, which are obtained by recording contrast as a function of the pulse interval τ. By fitting the decoherence profile using

$$C(t)={C}_{0}\exp (-\chi (t))={C}_{0}\exp (-{(t/{T}_{2})}^{\alpha }),$$
(1)

where C0 is a fitting parameter related to contrast; T2 is the coherence time, determined by χ(t) = 1; and α is the stretch exponent related to the properties of noise. The spatial variation of T2 is presented in the inset. The sample comprises three regions. Region ‘a’: without FGT flakes and exhibiting the longest T2; Region ‘b’: with a 70 nm-thick FGT flake and exhibiting intermediate T2; Region ‘c’: with a 90 nm-thick FGT sample and exhibiting the shortest T2. The three curves in Fig. 2a demonstrate that magnetic noise originating from the FGT affects both the T2 and the stretch exponent α. In region ‘a’, we observe α ≈ 1.5, consistent with previous reports in ref. 47. This value decreases significantly in FGT-covered regions ‘b’ and ‘c’. The stretch exponent α is a rather subtle issue, which is affected by the inhomogeneity of the sample47,48,49.

Fig. 2: Dynamic and static magnetometry in paramagnetic and ferromagnetic states.
Fig. 2: Dynamic and static magnetometry in paramagnetic and ferromagnetic states.
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a Decoherence curves of NV centers when FGT is in the paramagnetic phase. Solid lines represent fits using Eq. (1), yielding stretch exponents α = 1.306 ± 0.113 (green point), 0.772 ± 0.089 (red square), 0.698 ± 0.098 (blue rhombus). The inset shows the T2 spatial map at 211 K. The colors of the lettered markers in each region correspond to those of the decoherence curves. Statistical distributions of α and T2 are provided in Supplementary Section 6. b CW spectrum of NV centers when FGT is in the ferromagnetic phase. The inset shows the FWHM map of the CW spectrum at 188 K. The data in both figures are measured from sample #6. Scale bar: 5 μm.

In Fig. 2b, with T < Tc, spontaneous magnetization emerges in the FGT sample. We employ CW spectroscopy (see Supplementary Section 4) to map the static magnetic field distribution. A clear spectral dip at about 2.79 GHz is observed in region ‘a’, while in regions ‘b’ and ‘c’ the stray magnetic field from the FGT broadens the spectral lines and reduces the contrast. In Supplementary Section 5, static magnetic field imaging of Sample #9 was performed at a temperature far below Tc, revealing distinct CW spectral characteristics between domain walls and interiors. Near domain walls, spectral lines are significantly broadened with nearly undetectable optical contrast, while in domain interiors, spectral lines maintain their intrinsic width. As shown in the inset of Fig. 2b, the spectral broadening observed across region ‘b’ originates from reduced magnetic domain sizes near Tc. Quasi-static noise arising from domain reversion may also contribute to this contrast reduction.

Huge critical magnetic fluctuations near the T c

Next, we investigate the coherence dynamics of NV centers near the critical regime of FGT. As shown in Fig. 3a, the T2 distribution exhibits two dominant peaks: a shorter value of approximately 3.2 μs in regions covered by FGT films and a longer value of 4.7 μs in FGT-free regions. We define the distribution peak position as the characteristic T2 with the FGT sample. Further analysis in Fig. 3b represents the characteristic T2 as a function of temperature for various samples with thicknesses ranging from 10 to 90 nm. All measured T2 decreases monotonically as temperature approaches Tc from above. Although Tc varies with thickness (see Supplementary Table S1) of various FGT sample, the decrease of T2 is consistent. The intrinsic coherence time of the NV centers (bare diamond) is approximately 4.5 μs in our diamond chips47. Thus, the decrease of T2 originates from the magnetic noise generated by critical fluctuations. Further details are provided in Supplementary Section 2.

Fig. 3: Distribution of T2 and its temperature dependence.
Fig. 3: Distribution of T2 and its temperature dependence.
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a Statistical distribution of T2 across the field of view, measured from Sample #1 at 175 K. The two peaks in the histogram correspond to pixels inside and outside the FGT sample, respectively. The inset shows the corresponding T2 map. The distribution of α is presented in Supplementary Section 6. b Temperature-dependent T2 measured from eight FGT samples (Sample #1–#8) above Tc. Data points with T2 ≤ 500 ns are excluded due to insufficient fitting reliability. By adjusting the NV-FGT distance, this can be resolved and Fig. 4b shows the complete measurement across Tc. On sample #6, two independent measurements were performed. Data marked with empty squares highlight that T2 is unaffected when far from Tc. Scale bar: 5 μm.

Spin fluctuation behavior throughout the complete phase transition

In the preceding experiments, the distance d from the FGT layer to the ensemble NV centers layer is about 60 nm50 without inserting hBN flakes. As T → Tc, critical fluctuations intensify dramatically, leading to complete suppression of coherence. The statistical method illustrated in Fig. 3a fails to yield reliable results when T2 is extremely short, so the critical fluctuations at phase transitions can not be detected. We can mitigate this difficulty by inserting the hBN flakes with finite thickness between the diamond chip and the FGT sample. This configuration increases the distance d, thereby reducing the magnetic noise coupling to the NV centers. This protocol enables us to investigate thickness-dependent decoherence rates of NV centers with an individual FGT sample (see Supplementary Section 7). Preliminary estimation indicates magnetic fluctuations decay scaling as \(\langle \delta {B}^{2}\rangle \sim {d}^{-2.5}\).

Therefore, by increasing the distance d, the measured spin noise decreases, enabling the measurement of the T2 of NV centers across Tc. Figure 4a shows three decoherence curves at 190 K, 180 K, and 162 K, corresponding to temperatures above, near, and below Tc. Obviously, the shortest T2 occurs at 180 K, closest to Tc. In the ferromagnetic state (T < Tc), significant magnetic noise variations emerge within the FGT sample, as shown in the inset of Fig. 4a. The data processing strategy illustrated in Fig. 3a becomes inadequate. Instead, we integrate data from all pixels within the region of interest to extract the characteristic coherence time. For T > Tc, both data processing methods yield consistent results, as demonstrated in Supplementary Section 8. This consistency confirms the reliability of our experimental data and reflects global properties rather than localized features.

Fig. 4: Coherence of NV centers across the critical point.
Fig. 4: Coherence of NV centers across the critical point.
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a Decoherence curves for the entire region delineated by the green dashed line in the inset at three temperatures: above Tc (190 K, green point), near Tc (180 K, red square), and below Tc (162 K, blue rhombus). Solid lines represent fits using Eq. (1), yielding α = 1.183 ± 0.055 (green), 1.187 ± 0.098 (red), 0.845 ± 0.028 (blue). The inset shows the T2 map measured from Sample #12 at these three temperatures. b Decoherence rate Γ of NV centers as a function of temperature across Tc. Data are shown for two samples with distinct thicknesses and distances d: Sample #11 (47 nm thickness, d = 310 nm) and Sample #12 (16 nm thickness, d = 245 nm). Vertical dashed lines indicate the Tc determined from the maximum. Error bars represent the standard error of Γ for pixels within the dashed boundaries of Supplementary Figs. S8 and S9. Scale bar: 5 μm.

Analogous to the relaxation rate39,40,41, the decoherence rate is defined as Γ = 1/T2 − 1/T2,0, where T2,0 denotes the intrinsic coherence time of NV centers in the absence of FGT sample. The decoherence rates of NV centers measured with FGT samples #11 and #12 are presented in Fig. 4b with corresponding T2 maps provided in Supplementary Figs. S8 and S9. Despite differences in sample thickness and NV-FGT separation distances, the experimental results for both samples exhibit consistent behavior. The decoherence rate is observed to peak at the critical point, coinciding with the strongest fluctuations. This finding aligns with theoretical predictions from ref. 15. Remarkably, even at NV-FGT separations of hundreds of nanometers, magnetic fluctuations in the MHz frequency range exhibit intensities of hundreds of kHz-far exceeding those reported in prior relaxation-based measurements. This contrast in noise intensity between MHz and GHz frequencies directly reflects the spectral structure of critical fluctuations.

Theoretical model based on scaling theory

Building on these observations, we can now elucidate the mechanisms underlying experimental results. The two questions posed in the Introduction can be resolved. The effective Hamiltonian of the NV centers should be written as18

$$H={\Delta }_{0}\left({\hat{S}}_{{{{\rm{z}}}}}^{2}-\frac{1}{3}\right)+\gamma \left({B}_{0}+\delta B\right){\hat{S}}_{{{{\rm{z}}}}},$$
(2)

where Δ0 = 2.87 GHz is the zero-field splitting level, \({\hat{S}}_{{{{\rm{z}}}}}\) is the Pauli operator, B0 is the bias field, and δB is the fluctuation field generated by FGT flakes. The coefficient γ = 28 MHz mT−1 is the electron gyromagnetic ratio. This noise δB induces a decoherence process determined by \(\langle \exp (i\phi (t))\rangle=\exp (-\langle \phi {(t)}^{2}\rangle /2)\), where \(\phi (t)=\gamma \int_{0}^{t}\delta B({t}^{{\prime} })d{t}^{{\prime} }\) is the accumulated phase. According to the fluctuation-dissipation theorem51,52, \(\langle \delta B({t}_{1})\delta B({t}_{2})\rangle=\frac{1}{2\pi }\int\,d\omega \exp (-i\omega ({t}_{1}-{t}_{2}))S(\omega )\), where S(ω) is the power spectral density. Then we have (see Eq. (1))

$$\chi (t) \sim \frac{1}{2}\langle {\phi }^{2}(t)\rangle=\frac{1}{4\pi }\int\,d\omega S(\omega ){W}_{t}(\omega ),$$
(3)

with a frequency filter function for the Hahn-echo pulse sequence by \({W}_{t}(\omega )=8\sin {(\omega t/4)}^{4}/{\omega }^{2}\) 46. Thus for noise with the S(ω) ~ 1/ω μ, we have

$$\chi (t) \sim \int\,d\omega \frac{\sin {(\omega t/4)}^{4}}{{\omega }^{2}}\frac{1}{{\omega }^{\mu }} \sim {t}^{1+\mu }.$$
(4)

Inspired by the numerical results from Monte Carlo simulations in ref. 15,17, we hypothesize that the magnetic noise spectrum of critical fluctuations can be approximated by

$$S(\omega )=A/({\omega }_{0}^{\,\mu }+{\omega }^{\,\mu }),$$
(5)

where ω0 ~ ∣(T − Tc)/Tc∣zν is the reciprocal of the relaxation time53, an intrinsic property of the sample. z is the dynamic critical exponent associated with the relaxation time, and ν is the critical exponent associated with the divergence of the correlation length. As shown in Fig. 5a, this formula matches the characteristics across both low-frequency and high-frequency asymptotic regimes perfectly. The characteristic frequency ω0 marks the crossover between white noise and 1/f noise regimes. This noise spectrum is dominated by white noise for ω ≪ ω0, whereas it exhibits 1/f-type scaling when ω ≫ ω0.

Fig. 5: Theoretical interpretation of enhanced decoherence rates via noise spectral analysis.
Fig. 5: Theoretical interpretation of enhanced decoherence rates via noise spectral analysis.
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a Red lines: temperature-dependent magnetic noise spectral density S(ω) of spin fluctuations; blue line: filter function Wt(ω) of the Hahn-echo pulse sequence. b Contour maps of parameter \({{{\mathcal{A}}}}\) versus Γ and ω0 for \({{{\mathcal{A}}}}\) = 0.1, 0.5, and 1. Red arrows denote parameter evolution driven by increasing distance d and approaching Tc, respectively. The blue point corresponds to the experiment results, and the black dashed line marks the condition Γ = ω0, dividing the parameter space into two regimes: Γ > ω0 and Γ < ω0. c Fitted Γ (see Eq. (6)) versus (T − Tc)/Tc measured from multiple samples. d Double-logarithmic representation of (c); Arrow marks the crossover from white-noise plateau to 1/f-slope region.

In the following, we bridge the experimental results with theoretical models. As shown in Fig. 5a, the filter function Wt(ω) peaks at ωt ≃1, where t represents the interval of pulse sequences. Only noise components near ω ≃1/t contribute to the decoherence, while other frequency components are filtered out. The noise spectrum S(ω) can be reconstructed from the decoherence profile of a single NV center54,55,56. However, the decoherence profile of ensemble NV centers constitutes a superposition of decoherence curves of spatially indistinguishable NV centers, inherently complex and lacking direct physical interpretation47. Usually, the stretch exponents α qualitatively reflect spatial inhomogeneity of fluctuation field57,58, consistent with our results (see Supplementary Section 6).

Unlike the stretch exponent α, the decoherence rate Γ is independent of the specific decoherence profiles. When T2 ≪ T2,0, Γ can be approximated as 1/T2, thereby yields χ(1/Γ ) = 1. Since Wt(ω) is narrow compared with S(ω) in the frequency range, substituting t with 1/Γ into Eq. (3) yields 1 = S(Γ )∫W1/Γ (ω)dω = πS(Γ )/2Γ. This establishes a quantitative correspondence between experimental observations and the theoretical framework:

$$S(\Gamma )=\frac{{{{\mathcal{A}}}}}{{\omega }_{0}^{\, \mu }+{\Gamma }^{\mu }}=\Gamma,$$
(6)

where \({{{\mathcal{A}}}}\) is a distance-dependent parameter determined by the NV-FGT separation d. The interplay between Γ, ω0 and \({{{\mathcal{A}}}}\) is illustrated in Fig. 5b. For a fixed \({{{\mathcal{A}}}}\) (thus fixed d ), the asymptotic behavior ω0 → 0 as T → Tc reflects critical slowing down. Moreover, increasing d reduces \({{{\mathcal{A}}}}\) (since \({\lim }_{d\to \infty }{{{\mathcal{A}}}}=0\)), thereby reducing Γ. Employing the scaling relation ω0 = m∣(T − Tc)/Tc∣z ν, where m is a fitting parameter, the temperature dependence of Γ is obtained. As shown in Fig. 5c, the experimentally measured Γ values align well with theoretical predictions derived from Eq. (6), where m and \({{{\mathcal{A}}}}\) are determined via global fitting of the data. This analysis adopts critical exponents (ν = 1 and z = 2.17, yielding μ = 1 + γ/zν = 1.81) from the 2D Ising model to describe the magnetism of FGT15,22,23,25,26. Crucially, the developed theoretical framework here is universal, independent to the specific spin Hamiltonian.

There are two intriguing limits,

$$\Gamma \sim \left\{\begin{array}{ll}{{{\rm{constant}}}}\quad &{\omega }_{0}\ll \Gamma \\ {\left| \frac{T-{T}_{c}}{{T}_{c}}\right| }^{-z\nu \mu }\quad &{\omega }_{0}\gg \Gamma .\end{array}\right.$$
(7)

Both limits and their crossover are distinctly evident in the log–log representation. As illustrated in Fig. 5d, all four independent measurements exhibit nearly identical behavior. The crossover is given by Γ = ω0. Notably, in the vicinity of (T − Tc)/Tc ≈0.05, the relaxation time of critical fluctuations reaches approximately 1/ω0 ~2.5 μs. This value represents an intrinsic property of the FGT material and remains independent of experimental configurations. By adjusting the NV-FGT separation distance, additional spin correlation times can be probed at distinct reduced temperatures. To our knowledge, this work constitutes the first experimental determination of spin-correlation times in real physical systems undergoing phase transitions.

Discussion

In summary, this work reports an experimental demonstration that links critical fluctuation phenomena to scaling theory. A noise crossover signature emerges from the temperature-dependent evolution of NV centers’ coherence times. Inspired by numerical results, we propose a phenomenological noise spectrum formula that exhibits excellent agreement with measurements. Wide-field magnetic imaging provides robust global properties of the sample but neglects local inhomogeneities. Notably, our research unveils a paradigm for characterizing criticality through microscopic dynamical quantities, advancing systematic methodologies for phase transition analysis. We anticipate that in the future, the correlation time, correlation length, and their corresponding critical exponents will be determined through single NV center-based experiments.

Methods

Extended information on used methods is provided in Supplementary information.