Main

Magnetism has been studied since antiquity, yet understanding how magnetic order emerges in complex, interacting systems remains a challenge. In solids, magnetism generally stems from electron spin or orbital motion. Although spin moments dominate in most materials, recent studies on two-dimensional (2D) van der Waals systems reveal a leading role for orbital magnetism. Unlike spin-based magnetism, orbital magnetism reflects the distribution of electron wavefunctions and encodes the band topology and Berry curvature. In addition to spin, electrons in graphene-like 2D materials exhibit a valley degree of freedom, which strongly influences their behaviour. Crucially, spontaneous magnetization is forbidden if time-reversal or full spin-valley isospin symmetry is preserved. Thus, magnetism in these systems offers key insights into symmetry breaking and electronic interactions in strongly interacting 2D materials1,2,3.

An important characteristic of atomic-layer materials is their highly tunable electronic structure, which, together with a high density of states (DOS) and non-trivial band geometry, can drive spontaneous symmetry breaking and the formation of magnetic ground states even at zero magnetic field4,5,6,7,8. The resulting isospin symmetry breaking produces a rich array of textures, including spin and valley polarization, intervalley coherence (IVC), spin-valley locking via spin–orbit coupling (SOC), and skyrmion-like patterns9,10,11,12,13,14. Rhombohedral multilayer graphene (RMG) has recently emerged as a low-disorder platform for realizing such a phase15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30, including correlated insulators16,21,23,31, Chern insulators31,32, ferromagnets16,19,22,24,28,29,30,31,32,33,34,35,36, ferroelectrics19, superconductors25,33,37,38,39,40,41,42,43 and multiferroics28. Furthermore, proximity-induced SOC can stabilize certain phases34,35,37,38,39,40,41,42,43, whereas moiré engineering enables exotic fractional quantum states24,29,30,32,44,45.

In the absence of interactions, electronic bands with different valley and spin polarizations are degenerate. A key factor determining the energetic hierarchy of isospin symmetry-broken states is the Hund’s exchange interaction, which describes interactions between electrons in the different isospin bands, comprising intravalley and intervalley components. Although intravalley exchange between opposite spins is usually the dominant energy due to the long-range nature of Coulomb interactions, intervalley exchange crucially shapes the material’s low-energy physics. For example, in the absence of SOC, intervalley Hund’s coupling favours a spin-ferromagnetic state over spin-antiferromagnetic states. SOC, by contrast, tends to stabilize intervalley spin-antiferromagnetic order in the absence of Hund’s coupling. Thus, observing ferromagnetism indicates dominant intervalley Hund’s interaction. It also permits spin-ferromagnetic IVC (a charge density wave), in contrast to antiferromagnetic IVC (a spin density wave). Despite its fundamental importance, a direct measurement of the Hund’s interaction energy UHu has been lacking. In RMG, UHu is believed to be ferromagnetic, but directly probing its strength is challenging because it requires a perturbation that couples oppositely to spins in different valleys—unavailable experimentally. However, intrinsic Ising SOC (λSOC) acts exactly as such a perturbation, generating a small spin anisotropy of the order of \({\lambda }_{\rm{SOC}}^{2}/{U}_{\rm{Hu}}\). By rotating a small external magnetic field, we probe this anisotropy, enabling a quantitative measurement of UHu.

Previous studies on isospin ordering in van der Waals materials have largely relied on global measurements and theoretical models22,24,28,29,30,31,32,34,35,39,42,46. Here we introduce a nanoscale magnetic imaging technique to directly visualize local isospin textures in the symmetry-broken phases of crystalline ABCA graphene. We further explore how magnetic fields affect isospin state stability, particularly via interactions with Berry curvature and orbital magnetization. The field influences both spin moments through the Zeeman effect and orbital moments via a valley Zeeman effect, resulting in band shifts15,47. Our results show that the combination of momentum-dependent Berry curvature and unique band structure induces non-trivial magnetic band reconstruction, establishing RMG as a versatile platform for exploring the interplay of topology and interaction-driven physics12.

Transport and topological magnetic band reconstruction

Three dual-gated ABCA graphene devices were studied, showing qualitatively similar behaviour (Fig. 1 and Extended Data Figs. 1 and 5). The d.c. top- and bottom-gate voltages \({V}_{\rm{tg}}^{\rm{dc}}\) and \({V}_{\rm{bg}}^{\rm{dc}}\), respectively, control the carrier density n and out-of-plane displacement field D. Figure 1d presents the longitudinal resistance Rxx(n, D) at zero out-of-plane magnetic field (Bz = 0) in device A, which displays metallic behaviour with Rxx < 1 kΩ for |n| ≳ 0.5 × 1012 cm−2. Several ridge-like features partition the phase diagram. We focus on the weak ridge, na(D) (Fig. 1d, white dashed line), which traces a kink in the quantum oscillations (QOs) when Bz is raised to 3 T (Fig. 1e). To its left, vertical QOs reflect a simple band structure with a single Fermi surface (Fig. 1c, left inset) and fourfold-degenerate Landau levels (LLs). Our single-particle band structure calculations (Extended Data Fig. 3) accurately reproduce both transition line na(D) and QOs in regions A and B (Fig. 1f). The na(D) line reflects a Lifshitz transition—annulus opening in the valence band—producing a sharp step in the DOS (Fig. 1c, black arrow). Since at |n| > |na(D)|, the band structure is simple and no substantial interaction effects are expected due to low DOS, no additional transitions at higher densities are anticipated, as confirmed at zero field (Fig. 1d). Unexpectedly, in this full metal (M) phase, the QOs exhibit a break along the nπ(D) line (Fig. 1e, cyan dashed line), across which the QOs display a π shift. Such a shift has been observed previously in rhombohedral structures, but remains unexplained1. The high-resolution measurement and low disorder allow resolving the π shift as splitting of fourfold LLs and their recombination with neighbouring LLs. As described in the Supplementary Information, this is a result of the topological magnetic band reconstruction (TMBR) due to large Berry-curvature-induced orbital magnetization \({{\mathscr{M}}}_{{SR}}\) in the annular Fermi surface (Fig. 1a and Extended Data Fig. 2), which in the presence of Bz, gives rise to pronounced splitting in the DOS between the K and K′ valleys (Fig. 1c) and in band distortion, \(\varepsilon (k)={\varepsilon }_{0}(k)-{{\mathscr{M}}}_{{SR}}(k){B}_{z}\) (Fig. 1b).

Fig. 1: TMBR and transport measurements in ABCA graphene.
Fig. 1: TMBR and transport measurements in ABCA graphene.
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a, Surface plot of the calculated low-energy band structure of the valence band ε(k) with overlaid colour-coded self-rotation magnetization \({{\mathscr{M}}}_{{SR}}(k)\) of the K′ valley for displacement potential Δ1 = 10 meV at Bz = 0 T (Supplementary Information). b, Energy dispersion, \(\varepsilon (k)={\varepsilon }_{0}(k)-{{\mathscr{M}}}_{{SR}}(k){B}_{z}\), along (0, ky) for Δ1 = 10 meV at Bz = 0 T (dashed) and Bz = 3 T, resulting in TMBR and degeneracy lifting between K (red) and K′ (blue) valleys. c, Corresponding valence band DOS, ρ, versus n without (black) and with valley degeneracy lifting by TMBR for K (red) and K′ (blue) valleys. A full valley polarization occurs at |n| < 0.76 × 1012 cm−2 at Bz = 3 T. The insets show the Fermi surface structure at the red points and the black arrow indicates the sharp step in DOS at annulus opening na(D). d, Longitudinal resistance Rxx(n, D) of device A at Bz = 0 T at T = 500 mK. The white dashed line is the theoretical fit of the Lifshitz transition na(D). e, Same data as d, but for Bz = 3 T, showing different patterns of QOs in the various labelled regions. Region A, full metal with single Fermi surface (s-M); region B, a-M with annular Fermi surface and partial TMBR-induced valley polarization; region C, full metal with spontaneous partial spin polarization and TMBR-induced valley polarization; region D, valley-degenerate spin-polarized 1/2M with single Fermi surface (s-1/2M); region E, a-1/2M with annular Fermi surface and possible partial valley polarization; region F, 1/4M with single Fermi surface; region G, 1/4M with annular Fermi surface. The cyan dashed curve in region A is the π-shift nπ(D) line in full metal derived from the TMBR calculations with no fitting parameters. The cyan dotted curves in region D mark equivalent π-shift lines in the 1/2M phase. f, Calculated DOS showing oscillations due to LLs at Bz = 3 T in the presence of TMBR. The single-particle DOS effectively reproduces the QOs in regions A and B in e, where interaction effects are negligible (D = 1 V nm−1 corresponds to Δ1 = 97 meV).

Spontaneous symmetry breaking

The rich QOs in Rxx (Fig. 1e) reveal several symmetry-broken states. The vertical QO lines in region A correspond to fourfold-degenerate LLs in a full metal with a single Fermi surface. The annulus opening at na(D) introduces two Fermi surfaces in region B, with LLs from each. The TMBR further lifts valley degeneracy, yielding a full metal with four sets of intersecting LLs (Extended Data Fig. 3). At lower |n|, interaction-driven isospin symmetry breaking emerges. Regions D and F exhibit vertical QOs like region A, but with periods \(\Delta n=\frac{Nh}{e{B}_{z}}\), corresponding to LL degeneracies of N = 2 and N = 1, revealing half-metal (1/2M) and quarter-metal (1/4M) phases, respectively, with simple Fermi surfaces (h, Planck’s constant; e, elementary charge; Extended Data Fig. 4).

Imaging isospin magnetism

To probe the nature of the symmetry-broken states, we performed nanoscale magnetic imaging on device B utilizing a superconducting quantum interference device (SQUID) on the apex of a sharp pipette (SQUID on tip (SOT))48,49. A small a.c. voltage \({V}_{\rm{tg}}^{{ac}}\) applied to the top gate modulates n by nac (Fig. 2d), inducing a differential change in the local magnetization m(x, y) and corresponding stray field \({B}_{z}^{{ac}}\left(x,y\right)\), detected by the SOT. Figure 2a shows \({B}_{z}^{{ac}}(n,D)\) at a fixed location above the sample at low Bz = 10 mT. Two sharp peak lines indicated an abrupt increase in magnetization \({\mathscr{M}}\) at M-to-1/2M and 1/2M-to-1/4M phase transitions, consistent with the transport data (Extended Data Fig. 5). In a state that preserves time-reversal symmetry, the net magnetization vanishes. Time-reversal symmetry breaking induces magnetization via charge imbalance in spin or valley sectors. The 2D graphene structure dictates a highly anisotropic orbital magnetic moment (schematic of the current loops is shown in Fig. 2f–h) with an easy \(\hat{z}\) axis. By contrast, our understanding of spin anisotropy is lacking due to limited ability for direct experimental visualization of the spin properties. Contrary to orbital moment, spin anisotropy (Fig. 2f–h, spheres with arrows) is governed by the interplay between SOC and Hund’s interactions18 and can, therefore, vary considerably between the different states. To investigate isospin anisotropy, we image \({B}_{z}^{{ac}}\left(x,y\right)\) as Ba is rotated.

Fig. 2: Symmetry-broken isospin phase diagram in ABCA graphene.
Fig. 2: Symmetry-broken isospin phase diagram in ABCA graphene.
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a, \({B}_{z}^{{ac}}(n,D)\) measured in the bulk of the sample at T = 20 mK and Bz = 10 mT, showing a sharp differential magnetic signal along the M-to-1/2M and 1/2M-to-1/4M phase transition lines. b, Schematic of the experimentally derived phase diagram with labelled states. s-M, fourfold-degenerate full metal with a simple Fermi surface; a-M, full metal with an annular Fermi surface; PSP-M, partially spin polarized metal with TMBR-induced valley polarization; s-1/2M, twofold-degenerate fully spin polarized 1/2M with a simple Fermi surface; PVP-1/2M, partially valley polarized annular 1/2M; s-1/4M, onefold-degenerate simple 1/4M; a-1/4M, annular 1/4M. Solid lines indicate sharp magnetic transitions and dashed lines represent continuous transitions. The red–orange striped regions schematically show the expansion of the s-1/4M and a-1/4M phases on increasing Bz (Extended Data Figs. 5 and 6). c, Calculated scHF phase diagram, with the colour bar indicating FD. The squares show the corresponding derived ground-state fermiology, with the coloured areas representing hole-occupied states of the four flavours. In the grey area, at low D, the numerical calculations are unreliable due to the overlap of conduction and valence bands. d, Schematic of the sample layout with scanning SOT revealing the local magnetization pattern (left). Schematic of the cross-section of the encapsulated ABCA graphene with the electrical diagram (right). e, Optical image of the ABCA graphene device B patterned into Hall bar geometry, with the scan window indicated by the red dotted rectangle. The dark circular patches in the central region are bubbles. f–h, Isospin schematics with orbital (circular trajectories) and spin (balls with arrows) sectors. f, Fourfold-degenerate metal with zero magnetization. g, Twofold valley-degenerate spin-polarized 1/2M in out-of-plane (top) and tilted (bottom) Ba, with isotropic magnetization and spin direction following the Ba orientation. h, Onefold spin- and valley-polarized 1/4M in out-of-plane (top) and tilted (bottom) Ba. The isospin magnetization is highly anisotropic with easy \(\hat{z}\) axis and spin locked to the orbital moment.

Isospin anisotropy in 1/2M

Figure 3g shows \({B}_{z}^{{ac}}\left(x,y\right)\) measured in Bz = 15 mT (Bx = 0 mT, tanθB = Bx/Bz = 0) across the M-to-1/2M phase transition. Using an inversion procedure (Methods), we reconstruct the local differential magnetization m(x, y) (Fig. 3j). Since nac modulation induces a.c. switching between adjacent phases, the differential magnetization, \({\bf{m}}={{\mathscr{M}}}_{1/2M}-{{\mathscr{M}}}_{M}\), reflects the difference in local magnetization between the 1/2M and M phases (for clarity, the differential m is normalized by n in units of µB/e). Since the M phase has no magnetization, the measured m reflects the magnetization of the 1/2M phase.

Fig. 3: Isospin texture in the 1/2M phase.
Fig. 3: Isospin texture in the 1/2M phase.
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a, Schematic of the cross-section of graphene (orange) with isospin magnetic moments (red) oriented along \(\hat{z}\) (θs = 0) with the corresponding calculated stray magnetic field lines. b,c, Same data as a, but for isospins tilted at θs = 45° (b) and θs = –45° (c). d, Calculated out-of-plane stray magnetic field Bz(x) at a height of 200 nm above graphene (dashed line in a) for uniform magnetization of 0.63 × 1012 µB cm−2. The vertical dotted lines correspond to graphene edges. e, Same data as d, but for θs = 45°. The positive Bz peak resides on the inner side of the right graphene edge, whereas the negative peak resides on the outer side of the left edge. f, Same data as e, but for θs = –45°. g, Measured out-of-plane stray field \({B}_{z}^{{ac}}(x,y)\) at the M-to-1/2M transition (black star in Fig. 2a) for Ba at θB = 0. The blue circular regions correspond to the locations of the bubbles (Fig. 2e). h, Same data as g, but for θB = 45°. The negative \({B}_{z}^{{ac}}\) is enhanced outside the left edge, whereas the positive \({B}_{z}^{{ac}}\) is enhanced on the inner side of the right edge. i, Same data as h, but for θB = –45°. j, Local differential magnetization m(x, y) reconstructed from g. k, Calculated \({B}_{z}^{{ac}}(x,y)\) from m(x, y) in j tilted to θs = 45°, reproducing the measured \({B}_{z}^{{ac}}\left(x,y\right)\) in h. l, Same data as k, but for θs = –45°.

Next, applying Bz = 15 mT rotates the field to θB = 45°, with the resulting \({B}_{z}^{{ac}}(x,y)\) shown in Fig. 3h. Although the overall \({B}_{z}^{{ac}}(x,y)\) looks similar, the stray field along the inner right edge is enhanced (red) and the outer edge field vanishes (green), rather than being negative (blue), as shown in Fig. 3g. By contrast, on the left edge, the outer field becomes more negative and the inner field is reduced. Additionally, bubble domains with suppressed \({B}_{z}^{{ac}}\) become more oval and shift rightwards. For θB = –45°, similar effects but with a reversed orientation appear (Fig. 3i). These changes are consistent with the calculated stray fields for a uniform sample with tilted magnetization (Fig. 3b,c,e,f).

The data show that the isospin orientation in the 1/2M phase depends on the direction of Ba (Fig. 2g). To quantify the magnetization angle, we begin with the reconstructed \({\bf{m}}\left(x,y\right)=m\left(x,y\right)\hat{z}\) at θB = 0 (Fig. 3j), rotate it by an angle θs such that \({\bf{m}}\left(x,y\right)\)\(=m\left(x,y\right)(\cos {\theta }_{s}\hat{z}+\sin {\theta }_{s}\hat{x})\), and compute the resulting \({B}_{z}^{{ac}}(x,y)\). The actual isospin magnetization angle θs is then extracted by best fitting the calculated and measured \({B}_{z}^{{ac}}(x,y)\) (Extended Data Fig. 8). The calculated \({B}_{z}^{{ac}}(x,y)\) (Fig. 3k) effectively reproduces the measured one (Fig. 3h), including the negative \({B}_{z}^{{ac}}\) along the left edge, the enhanced field at the right edge and bubble distortions. A similarly accurate fit for θB = –45° is shown in Fig. 3l. This approach enables the extraction of both magnitude and orientation of the isospin magnetic moment (Methods).

Figure 4f shows θs versus θB, revealing that in the 1/2M phase, isospin magnetism is nearly isotropic and aligns closely with Ba. Given the strong \(\hat{z}\)-axis anisotropy of orbital magnetism, this confirms that 1/2M in tetralayer rhombohedral graphene is a valley-degenerate, spin-polarized ferromagnet, consistent with prior RMG studies21,24,31. However, full spin isotropy in the presence of SOC implies a large intervalley Hund’s coupling, as discussed below.

Fig. 4: Isospin texture in the 1/4M phase.
Fig. 4: Isospin texture in the 1/4M phase.
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a, Measured \({B}_{z}^{{ac}}(x,y)\) at the 1/2M-to-1/4M transition (yellow star in Fig. 2a) for Ba along θB = 0. b, Same data as a, but for θB = 83°. In contrast to Fig. 3h, the negative \({B}_{z}^{{ac}}\) occurs at the right graphene edge due to spin tilting in the 1/2M phase. c, Same data as b, but for θB = –83°. d, Local differential magnetization m(x, y) reconstructed from a. e, \({B}_{z}^{{ac}}(x,y)\) calculated using a superposition of the orbital m(x, y) in the 1/4M shown in d and of a tilted spin m(x, y) in the 1/2M phase based on Fig. 3j, qualitatively reproducing the measured \({B}_{z}^{{ac}}\left(x,y\right)\) in b. f, Derived spin tilt angle |θs| versus the applied field tilt angle |θB| for positive (black squares) and negative (blue diamonds) angles across the M-to-1/2M and 1/2M-to-1/4M (solid magenta triangles) transitions. The magenta error bars at 1/2M-to-1/4M transitions are estimated by comparing \({B}_{z}^{{ac}}\) for |θB| = 78° and |θB| = 83° (Methods). The green open circles represent the average spin tilt angle, \({\bar{\theta }}_{s}={(|\theta }_{s}\left( > 0\right)|+{|\theta }_{s}\left( < 0\right)|)/2\), and the corresponding black error bars are defined as (|θs(>0)| – |θs(<0)|). The solid lines show the calculated θs versus θB (Methods) for various indicated values of λSOC in 1/4M and α in 1/2M for Bz = 15 mT and variable Bx = BztanθB. The 1/4M lines set a lower bound on Ising SOC of λSOC ≳ 120 µeV, substantially higher than the previously evaluated λSOC = 50 µeV (yellow curve) in ABC graphene18. The magnetism in the 1/2M phase is essentially fully isotropic with the calculated solid lines setting an upper bound on spin anisotropy of |α| < 0.25 µeV, which sets a lower bound on UHu ≳ 6.5 meV (Methods). The red dashed line shows the calculated θs versus θB for α = 0.4 µeV based on previously assumed UHu = 4.2 meV (refs. 41,51), which is inconsistent with the data.

For Ba along \(\hat{z}\) (θB = 0), the symmetric negative stray field \({B}_{z}^{{ac}}\) outside the right and left edges (Fig. 3g) is symmetric, which matches the calculations for m(x, y) aligned along \(\hat{z}\) (Fig. 3a,d and Methods). The reconstructed m(x, y) is rather uniform in the sample bulk (Fig. 3g,j, red–yellow), except for circular patches (blue) in which the magnetization vanishes. These are the bubbles formed during fabrication (Fig. 2e), which induce strain and disorder that destroy the symmetry-broken phases, as seen in magic-angle bilayer graphene50.

Anisotropy in 1/4M

Figure 2a reveals a sharp peak in differential magnetization at the 1/2M-to-1/4M transition, indicating greater magnetization in the 1/4M phase. Since 1/2M is already spin polarized, this increase must stem from additional valley polarization. To probe the isospin structure, we image \({B}_{z}^{{ac}}(x,y)\) across the transition. Figure 4a shows the data at θB = 0°, with the reconstructed m(x, y) shown in Fig. 4d. The m(x, y) pattern resembles that of the M-to-1/2M transition, but with over threefold-higher amplitude, exceeding 1 µB/e. Since for θB = 0°, the spin is aligned along \(\hat{z}\) with the same contribution to magnetization in both 1/2M and 1/4M phases, the observed differential m(x, y) across the transition reflects only the Berry-curvature-induced orbital magnetization in 1/4M, emphasizing its dominant contribution.

Rotating Ba to θB = 83°, the resulting \({B}_{z}^{{ac}}(x,y)\) in Fig. 4b shows the negative stray field shifting to the right edge—opposite to its position in Fig. 3e,h—suggesting that the differential magnetization behaves as if tilted to negative θs (Fig. 3f,i). This counterintuitive behaviour reveals a strong SOC in the 1/4M phase, aligning both spin and orbital moments along \(\hat{z}\). As a result, the difference between the out-of-plane magnetization in 1/4M and the θs = 83° spin tilt in 1/2M results in differential magnetization that appears to a negative θs. Similarly, for θB = –83°, the differential magnetization appears to be oriented in the positive direction with negative \({B}_{z}^{{ac}}\) on the left edge (Fig. 4c). As discussed in the Methods, we can qualitatively reproduce the measured \({B}_{z}^{{ac}}(x,y)\) in Fig. 4b by the proper superposition of orbital and spin-tilted m(x, y) patterns from Figs. 4d and 3j, as shown in Fig. 4e. These observations show that, unlike the nearly isotropic 1/2M phase, the 1/4M phase exhibits strong isospin anisotropy with an easy \(\hat{z}\) axis (Fig. 4f (magenta triangles) and Extended Data Fig. 9).

Analysis of isospin anisotropy and Hund’s and SOC energies

The stark anisotropy difference between the 1/2M and 1/4M phases can be qualitatively understood as follows. In 1/4M, the spin anisotropy is dictated by SOC energy λSOC, which tends to align the spin and valley moments. Considering SOC and Zeeman effects under tilted fields, the spin tilt angle is given by \(\tan {\theta }_{s}=\frac{{B}_{x}}{{B}_{z}+{\lambda }_{\rm{SOC}}/(g{\mu }_{B})}\) (Methods), with g being the gyromagnetic ratio. The three rightmost curves in Fig. 4f show the calculated θs versus θB = tan−1(Bx/Bz) for different values of λSOC at Bz = 15 mT and variable Bx, which allows us to set a lower bound on Ising λSOC ≳ 120 µeV. This experimental estimate of the SOC strength in ABCA is more than twice the previously evaluated λSOC = 50 µeV in ABC graphene18.

The extracted λSOC raises a question regarding the isotropic spin moment measured in the 1/2M phase. A finite SOC means that full spin rotation is not a symmetry of the system, and hence, we expect the spin ferromagnetic phase to be anisotropic (either easy axis or easy plane). However, the spin anisotropy α in 1/2M is strongly suppressed by Hund’s coupling \({\alpha \cong \lambda }_{\rm{SOC}}^{2}/8{U}_{\rm{Hu}}\), where UHu = nJH is the intervalley Hund’s coupling energy (Methods). Therefore, to explain the seemingly isotropic spin moment, UHu should be sufficiently large. The 1/2M phase (Fig. 4f, solid lines) shows the calculated θs versus θB dependence for various values of α (Extended Data Fig. 7), from which we can set an upper bound on |α| < 0.25 µeV. Using the attained λSOC > 120 µeV, this α value leads to a lower bound on the Hund’s coupling of UHu > 6.5 meV = 75.4 K, and of the corresponding JH > 3.1 × 10−12 meV cm2. This measurement of the Hund’s-type intervalley exchange energy is substantially larger than the values of JH ≅ 2 × 10−12 meV cm2 previously used in the literature for modelling rhombohedral graphene18,41,51. Such a large Hund’s energy scale means that relative spin fluctuations between the two valleys are suppressed. For temperatures below this energy scale, the total spin moment for the two valleys should be treated as a single degree of freedom, whereas at higher temperatures, the spin moments associated with each valley should be considered as independent degrees of freedom.

Phase diagram

Low-field magnetization data (Fig. 2a) reveal three main phases—M, 1/2M and 1/4M—separated by sharp phase transitions. Examination of the QOs at high magnetic fields (Fig. 1e and Extended Data Fig. 4), by contrast, discloses additional symmetry-broken states within the main phases (Fig. 2b). The M phase, in which all four flavours are occupied, is subdivided into three states: simple s-M with a single Fermi surface, a-M with an annular Fermi surface, and spontaneously symmetry-broken partially spin-polarized PSP-M phase. Since the partially spin-polarized state is partially polarized, the sharp transition from PSP-M to fully spin-polarized 1/2M should be accompanied by a reduced jump in magnetization, in accordance with the observed m ≅ 0.3 µB/e at the M-to-1/2M transition. The 1/2M phase is subdivided into valley-symmetric s-1/2M with a simple Fermi surface and a-1/2M with an annular Fermi surface, which is possibly also a partially valley-polarized PVP-1/2M phase. Note that similar to full metal, the TMBR-induced π shifts are also observed in s-1/2M (Fig. 1e, cyan dotted lines), revealing magnetic-field-induced partial valley polarization. Finally, the 1/4M phase is also subdivided into a single Fermi surface (s-1/4M) and annular (a-1/4M) states. Since 1/4M comprises only one valley and one spin, no π shifts should occur. In particular, the 1/4M region expands markedly with the field (Fig. 2b, striped regions), as observed by comparing Fig. 1d,e and described in Extended Data Fig. 6. This expansion is a result of the magnetic energy gain due to orbital magnetic moment in 1/4M, which is absent in the adjacent 1/2M phase (Supplementary Information). These conclusions are corroborated by self-consistent Hartree–Fock (scHF) calculations with a gate-screened Coulomb interaction (Supplementary Information). Figure 2c shows the derived phase diagram with colour-coded flavour degeneracy: FD = n/max(ni), where ni is the carrier density in each isospin flavour, showing a sequence of symmetry-breaking transitions from M to 1/4M, which broadly agrees with the measured diagram in Fig. 2b.

Discussion

A comparison of isospin transitions in ABCA and ABC graphene18 shows that in both systems, full spin polarization does not occur immediately. Instead, there is a gradual polarization, too weak to be detected locally, followed by an abrupt transition to a fully spin-polarized 1/2M phase. Similarly, in the presence of a magnetic field, the spin-polarized 1/2M phase apparently undergoes gradual valley polarization, before an abrupt transition to a 1/4M phase. In particular, although ABC graphene exhibits IVC ordering within 1/4M, ABCA shows no such signatures, either in measurements or scHF simulations. This is particularly interesting since hole-doped ABC graphene has been found to become superconducting at subkelvin temperatures25, whereas, so far, no superconducting phase has been reported in a pure hole-doped tetralayer43. The ABCA phase diagram, thus, presents a valuable comparison for narrowing down the superconducting mechanism. Importantly, the hole-doped ABCA seems to be lacking IVC phases and its superconductivity is apparently suppressed, whereas in ABC, both phases are present, which is consistent with recent suggestions of a pairing mechanism mediated by IVC fluctuations10,11,14,27. Since our findings clearly indicate that the isospin anisotropy throughout the phase diagram depends subtly on small differences in microscopic properties, a better understanding of the underlying electronic correlations is necessary.

The findings not only shed light on the microscopic aspects of isospin ordering in fine detail but also demonstrate that RMG is a highly tunable and increasingly well-understood system with superior device quality compared with twisted stackings. Recent work on spin–orbit-proximitized RMG has uncovered superconducting phases absent without a strong SOC37,38,39,40,41,42, potentially originating from a spin-canted in-plane ferromagnetic state17,41. Our experimental estimate of Hund’s coupling is crucial for a quantitative understanding of spin-canted ferromagnetism and possibly related superconducting phases. We conclude that screening effects on the Hund’s coupling are weak, as the attained UHu > 6.5 meV is suppressed by less than a factor of two relative to the maximum value of intervalley Coulomb energy in a vacuum (Methods). Comparable Hund’s coupling is, thus, expected across different rhombohedral graphene systems.

Methods

Device fabrication

The hexagonal boron nitride (hBN)-encapsulated tetralayer graphene heterostructures were fabricated using the dry-transfer method. The hBN and graphene flakes were exfoliated onto a Si/SiO2 (285 nm) substrate and picked up using a polycarbonate on a polydimethylsiloxane dome stamp. The number of layers of the graphene flakes was determined via optical contrast analysis against the Si substrate (using the G values in the RGB triplet) and further confirmed through step strength analysis in the dark-field mode. The stacking order of the graphene flakes was determined by the shape of the Raman 2D peak (Extended Data Fig. 1e), and its rhombohedral-stacked regions were isolated using high-power laser cutting. The Raman analysis was subsequently repeated on the final stack (Extended Data Fig. 1f) to rule out the possible relaxation of graphene to Bernal stacking.

To increase the yield of the rhombohedral structure, a single-pick strategy was used to avoid multiple pick-ups of graphene flakes during the stacking procedure, which enhanced relaxation to Bernal stacking. This involved pre-preparation of the lower part of the stack, comprising a backgate (annealed patterned metal or graphite stripe) covered with an hBN flake, followed by annealing at 500 °C under a vacuum. For the same reason, the graphene flake manipulations were done faster, at lower temperatures (<70 °C) than usual, and along the zigzag direction of graphene52. This modified stacking protocol can, in turn, encourage bubble formation (Fig. 2e). Unlike in other graphene stacks, the bubbles cannot be removed by thermal annealing or subsequent pick-ups since the graphene stacking configuration is susceptible to these procedures. The chirality of the edges (zigzag/armchair) was first determined from the dependence of the second-harmonic generation on the polarization angle in the hBN flakes53,54 and in Bernal-stacked trilayer region within the same flake as the tetralayer graphene of interest.

Subsequently, the top hBN and graphene flakes were transferred onto the pre-prepared bottom part. During these processes, the crystal axes (discerned from the straight edges) of the hBN and tetralayer graphene were intentionally misaligned using a mechanical rotation stage to avoid moiré correlations. A Ti (2 nm)/Au (10 nm) top gate was deposited onto the finalized stacks. Then, the devices were etched into a Hall bar geometry, and one-dimensional contacts were established through SF6 and O2 plasma etching, followed by Cr (4 nm)/Au (60–90 nm) evaporation. The Raman, second-harmonic generation measurements and laser cutting were performed using a WITec alpha300 R Raman imaging microscope using 532-nm and 1,064-nm wavelengths.

Device summary:

Devices A and C: Ti (2 nm)/Au (10 nm) top gate and graphite bottom gate. The top and bottom hBN thicknesses are ~17 nm and ~31 nm, respectively.

Device B: Ti (2 nm)/Au (10 nm) top gate and Ti (2 nm)/Pt (10 nm) bottom gate. The top and bottom hBN thicknesses are ~18 nm and ~29 nm, respectively.

SOT measurements

The local magnetic imaging was performed using a custom-built scanning SOT microscope in a dilution refrigerator. All the measurements were conducted at a nominal base temperature of T = 20 mK. For detecting the local magnetic fields, we used an indium SQUID fabricated on the tip of a pulled quartz pipette with an integrated shunt resistor55. The effective diameter of the In SOT was 194 nm, with a field sensitivity down to 10 nT Hz−1/2 and operating fields up to 0.4 T. A cryogenic-series SQUID array amplifier was implemented to read-out the SOT signal56,57. To control the scan height during imaging, the SOT was attached to a quartz tuning fork vibrating at a resonance frequency of around 32 kHz. A small a.c. voltage \({V}_{\rm{tg}}^{{ac}}\) was applied to the top gate with an amplitude of 100 mV to 200 mV (r.m.s.) and frequency of 0.7 kHz to 2 kHz, modulating the carrier density by nac, and the corresponding a.c. change in the local magnetic field \({B}_{z}^{{ac}}\left(x,y\right)\) was acquired by the SOT.

All the presented \({B}_{z}^{{ac}}\left(x,y\right)\) images were measured on device B to avoid magnetic signals originating from the graphite bottom gate. Single-point measurements (like Fig. 2b) were recorded at a height of 70 nm above the surface of the top metal gate. The 2D spatial images (Figs. 3 and 4) were scanned at a constant height of about 150 nm above the top-gate surface, with a pixel size of 50 nm and integration time of 1.2 s per pixel. Figures 3g–i and 4a–c were acquired at (D, n) values of (0.29 V nm−1, –2.1 × 1012 cm−2), (0.31 V nm−1, 0.97 × 1012 cm−2) and (0.31 V nm−1, 0.9 × 1012 cm−2), respectively.

The angle-dependent measurements were performed at fixed Bz = 15 mT or Bz = 85 mT and variable Bx. All the data points in Fig. 4f were acquired at Bz = 15 mT, except for θB < 30° points, which were measured also at Bz = 85 mT.

Single-particle band structure calculations

To calculate the low-energy single-particle band structure of rhombohedral tetralayer graphene, we adopted the Slonczewski–Weiss–McClure parameterized tight-binding model12. Using the sublattice states {A1, B1, A2, B2, A3, B3, A4, B4}, the low-energy Hamiltonian can be written as,

$${H}_{0}=\left[\begin{array}{cccccccc}{\Delta }_{1}+{\Delta }_{2} & {v}_{0}{\pi }^{\dagger } & {v}_{4}{\pi }^{\dagger } & {v}_{3}\pi & 0 & \frac{1}{2}{\gamma }_{2} & 0 & 0\\ {v}_{0}\pi & {\Delta }_{1}+{\Delta }_{2}+\delta & {\gamma }_{1} & {v}_{4}{\pi }^{\dagger } & 0 & 0 & 0 & 0\\ {v}_{4}\pi & {\gamma }_{1} & \frac{{\Delta }_{1}}{3}-{\Delta }_{2}+\delta & {v}_{0}{\pi }^{\dagger } & {v}_{4}{\pi }^{\dagger } & {v}_{3}\pi & 0 & \frac{1}{2}{\gamma }_{2}\\ {v}_{3}{\pi }^{\dagger } & {v}_{4}\pi & {v}_{0}\pi & \frac{{\Delta }_{1}}{3}-{\Delta }_{2}+\delta & {\gamma }_{1} & {v}_{4}{\pi }^{\dagger } & 0 & 0\\ 0 & 0 & {v}_{4}\pi & {\gamma }_{1} & -\frac{{\Delta }_{1}}{3}-{\Delta }_{2}+\delta & {v}_{0}{\pi }^{\dagger } & {v}_{4}{\pi }^{\dagger } & {v}_{3}\pi \\ \frac{1}{2}{\gamma }_{2} & 0 & {v}_{3}{\pi }^{\dagger } & {v}_{4}\pi & {v}_{0}\pi & -\frac{{\Delta }_{1}}{3}-{\Delta }_{2}+\delta & {\gamma }_{1} & {v}_{4}{\pi }^{\dagger }\\ 0 & 0 & 0 & 0 & {v}_{4}\pi & {\gamma }_{1} & {-\Delta }_{1}+{\Delta }_{2}+\delta & {v}_{0}{\pi }^{\dagger }\\ 0 & 0 & \frac{1}{2}{\gamma }_{2} & 0 & {v}_{3}{\pi }^{\dagger } & {v}_{4}\pi & {v}_{0}\pi & {-\Delta }_{1}+{\Delta }_{2}\end{array}\right],$$

where π = τkx + iky, \({v}_{i}=\frac{\sqrt{3}}{2\hslash }{a}_{0}{\gamma }_{i}\) (i = 0, 3, 4) is the hopping velocity, τ is the valley index (+1 for K valley and –1 for K′ valley) and a0 = 2.46 Å is the graphene lattice constant. The parameter δ corresponds to the on-site potential for the sublattices, which have the nearest neighbour on the adjacent layer. Δ determines the potential difference between the outer layers and is approximately proportional to the applied displacement field D, and Δ2 encodes the potential difference between the outer and middle layers. The hopping parameters were chosen by fitting the annulus opening line na(D) in the full metal in Fig. 1d, which can be achieved with the same intra- and interlayer hopping values reported previously for rhombohedral trilayer graphene24, leading us to adjust only the value of the on-site energy Δ2 compared with the trilayer. This results in the following values:

Parameter

γ0

γ1

γ2

γ3

γ4

δ

Δ2

Value (in eV)

3.1

0.38

−0.015

−0.29

−0.141

0.0105

0.002

From the fit to the π-shift line (Fig. 1), we find that the outer-layer potential difference Δ1 = 97 meV corresponds to the applied displacement field D = 1 V nm−1.

Isospin anisotropy analysis and derivation of spin–orbit and Hund’s coupling bounds

To understand the interplay between valley polarization and spin-moment anisotropy, we use a mean-field-inspired model assuming that isospin polarization is momentum independent, which allows us to parameterize the system as a density matrix, using two spinors (assuming no intervalley coherence):

$$\begin{array}{c}\hat{\rho }=\frac{{n}_{K}}{n}\left|{\bf{\Psi }}_{{\rm{k}}}\right\rangle {{\langle }}{\bf{\Psi} }_{{\rm{k}}}|+\frac{{n}_{K^{\prime} }}{n}\left|{\bf{\Psi }}_{{\rm{k}}^{\prime} }\right\rangle {{\langle }}{\bf{\Psi} }_{{\rm{k}}^{\prime} }|\\ \left|{\bf{\Psi}}_{{\rm{k}}}\right\rangle ={a}_{1}\left|\uparrow K\;\right\rangle +{a}_{2}\left|\downarrow K\right\rangle ,\qquad\,\left|{\bf{\Psi}}_{{\rm{k}}{\prime} }\right\rangle ={a}_{3}\left|\uparrow K^{\prime} \right\rangle +{a}_{4}\left|\downarrow {K}^{{\prime} }\right\rangle ,\end{array}$$
(1)

where n = nK + nK′, and both |Ψk〉 and |Ψk′〉 are normalized to 1. The energy per carrier is given by

$$E=\frac{1}{2}{\chi }_{\tau }^{-1}{\left\langle {\tau }_{z}\right\rangle }^{2}-{U}_{\rm{Hu}}\left\langle {\bar{S}}_{+}\right\rangle \cdot \left\langle {\bar{S}}_{-}\right\rangle -\frac{{\lambda }_{\rm{SOC}}}{2}\left\langle {\tau }_{z}{S}_{z}\right\rangle -\frac{g}{2}{\mu }_{B}\bar{B}\cdot \left\langle \bar{S}\right\rangle -{\mu }_{v}{B}_{z}\left\langle {\tau }_{z}\right\rangle ,$$
(2)

where \(\left\langle {\bar{S}}_{\pm }\right\rangle \equiv \frac{1}{2}\left\langle \left(1\pm {\tau }_{z}\right)\bar{S}\right\rangle\) are valley-spin polarization vectors, χτ is the valley susceptibility, μv is the valley orbital magnetic moment, the spin and valley operators are taken to have eigenvalues of ±1, and we take g = 2 for the g-factor. The interaction energy UHu = nJH is the Hund’s coupling energy scale, determined by the carrier density times the intrinsic Hund’s coupling, and we assume that it is ferromagnetic, as there is plenty of evidence that the 1/2M phase is a spin ferromagnet. The Ising SOC (\(\frac{{\lambda }_{\rm{SOC}}}{2}\left\langle {\tau }_{z}{S}_{z}\right\rangle\)) can be related to a Kane–Mele-type SOC in monolayer graphene. In monolayer graphene, the Ising SOC vanishes due to either inversion or C2 rotation symmetry and does not result in any observable spin anisotropy. However, as rhombohedral graphene under the displacement field breaks the inversion symmetry and does not have C2 rotation symmetry to begin with, it develops a finite sublattice polarization, and the Kane–Mele-type SOC of the graphene monolayer translates to an Ising-type SOC with magnitude proportional to the amount of sublattice polarization of the Bloch wavefunctions. In general, for a fixed displacement field, the amount of sublattice polarization increases with the number of layers, as it is related to the potential difference between the top and bottom layers.

We use the following parameterization:

$$\begin{array}{c}{\hat{\rho }}=\frac{1+\delta n}{2}\left|{\bf{\Psi} }_{{\rm{k}}}\right\rangle {{\langle }}{\bf{\Psi} }_{{\rm{k}}}|+\frac{1-\delta n}{2}\left|{\bf{\Psi} }_{{\rm{k}}^{\prime} }\right\rangle {{\langle }}{\bf{\Psi} }_{{\rm{k}}^{\prime} }|\\ \left|{\bf{\Psi} }_{{\rm{K}}}\right\rangle =\left(\cos {\theta }_{-}\left|\uparrow \right\rangle +\sin {\theta }_{-}\left|\downarrow \right\rangle \right)\otimes \left|K\;\right\rangle \left|{\bf{\Psi} }_{{{\rm{K}}}^{{\prime} }}\right\rangle =\left(\cos {\theta }_{+}\left|\uparrow \right\rangle +\sin {\theta }_{+}\left|\downarrow \right\rangle \right)\otimes \left|{K}^{{\prime} }\right\rangle ,\end{array}$$
(3)

where θ± = θs ± δθ; therefore, we find

$$\begin{array}{l}\begin{array}{l}E=\frac{1}{2}{\chi }_{\tau }^{-1}{\left(\delta n-{\chi }_{\tau }{\mu }_{v}{B}_{z}\right)}^{2}-{U}_{\rm{Hu}}\frac{1-\delta {n}^{2}}{4}\cos \left(2\delta \theta \right)-\frac{{\lambda }_{\rm{SOC}}}{2}\left(\sin {\theta }_{s}\sin \delta \theta\right. \\\quad+\left.\delta n\cos {\theta }_{s}\cos \delta \theta \right)\\\quad-\frac{g}{2}{\mu }_{B}\left[{B}_{x}\left(\sin {\theta }_{s}\cos \delta \theta -\delta n\cos {\theta }_{s}\sin \delta \theta \right)\right.\\\quad+\left.{B}_{z}\left(\cos {\theta }_{s}\cos \delta \theta +\delta n\sin {\theta }_{s}\sin \delta \theta \right)\right].\,\end{array}\end{array}$$
(4)

Physically, θs is the average spin angle, δθ is the relative spin-canting angle between the valleys, and δn is the relative valley density imbalance normalized by the total density (Extended Data Fig. 7). Assuming λSOC to be a small energy scale relative to \({\chi }_{\tau }^{-1}\) and UHu, we find the ground state to be 1/4M (〈τz〉 = ±1) for \({\chi }_{\tau }^{-1}+\frac{\left|{U}_{\rm{Hu}}\right|}{2} < 0\), and 1/2M (〈τz〉 = 0) otherwise.

1/4M

As valley polarization breaks the time-reversal symmetry, in the 1/4M phase, there is an effective spontaneous magnetic field acting on the spin degree of freedom, originating from the SOC. We assume full valley polarization δn = ±1, and find

$${E}_{{{\rm{QM}}}}=-\frac{{\lambda }_{\rm{SOC}}}{2}\left\langle {\tau }_{z}{S}_{z}\right\rangle -\frac{g}{2}{\mu }_{B}\bar{B}\cdot \left\langle \bar{S}\right\rangle =\mp \frac{{\lambda }_{\rm{SOC}}}{2}\left\langle {S}_{z}\right\rangle -\frac{g}{2}{\mu }_{B}\bar{B}\cdot \left\langle \bar{S}\right\rangle .$$
(5)

The sign of valley polarization 〈τz〉 = δn can be trained by an external field.

We derive the angle of spin moment from the z axis for a positive value of Bz:

$$\tan {\theta }_{s}=\frac{{B}_{x}}{{B}_{z}+\frac{{\lambda }_{\rm{SOC}}}{g{\mu }_{B}}}.$$
(6)

Figure 4f shows the calculated θs versus θB from equation (6) for fixed Bz = 15 mT and varying Bx for two values of λSOC in the 1/4M phase. Compared with the 1/4M data, we can set a lower bound on SOC of λSOC ≳ 120 µeV.

1/2M

In 1/2M, the total magnetic moment includes both valley and spin moments. It is defined as

$$\bar{{\mathscr{M}}}=-{\bar{\nabla }}_{B}{E}_{0},$$

where E0 is the ground-state energy expressed as

$${E}_{0}=\min \left\{E\left({\theta }_{s},\delta \theta ,\delta n\right)|{\theta }_{s}\in \left[0,\uppi \right],\delta \theta \in \left[-\frac{\uppi }{2},\frac{\uppi }{2}\right],\delta n\in \left[-1,1\right]\right\}.$$

We expect both δθ and δn to be perturbatively small in λSOC/UHu, as indeed confirmed numerically by minimizing the energy E in equation (4). Thus, the energy of 1/2M can be expressed as a function of θs as the only variable. We do this by minimizing the expression for E over δθ and δn, and find

$${E}_{{{\rm{eff}}}}\approx \alpha {\left\langle {S}_{z}\right\rangle }^{2}-\frac{g}{2}{\mu }_{B}\bar{B}\cdot \left\langle \bar{S}\right\rangle =\alpha {\cos }^{2}\left({\theta }_{s}\right)-\frac{g}{2}{\mu }_{B}B\cos \left({\theta }_{s}-{\theta }_{B}\right),$$
(7)

where

$$\alpha ={\left(\frac{{\lambda }_{\rm{SOC}}}{2}\right)}^{2}\left(\frac{1}{2{U}_{\rm{Hu}}}-\frac{1}{{U}_{\rm{Hu}}+2{\chi }_{\tau }^{-1}}\right)=\frac{{\lambda }_{\rm{SOC}}^{2}}{8{U}_{\rm{Hu}}}\left(\frac{2{\chi }_{\tau }^{-1}-{U}_{\rm{Hu}}}{2{\chi }_{\tau }^{-1}+{U}_{\rm{Hu}}}\right)\mathop{\approx }\limits_{{\chi }_{\tau }^{-1}\gg {U}_{\rm{Hu}}}\frac{{\lambda }_{\rm{SOC}}^{2}}{8{U}_{\rm{Hu}}},$$
(8)

with easy-plane ferromagnet for α > 0, and an easy-axis ferromagnet otherwise. Intuitively, the first term of the expression for α comes from the fact that easy-plane ferromagnet gains energy from SOC by canting the spin in each valley in the opposite direction (towards \(\left\langle {\tau }_{z}\right\rangle \cdot \hat{z}\)), which comes at the cost of intervalley exchange energy, because the spin moments of the two valleys are not perfectly aligned. The second term is the energy gained by an easy-axis ferromagnet due to SOC. In this case, the system gains energy by generating a finite valley polarization (〈τz〉 ≠ 0). From scHF numerical calculations, we derive χτ = 15.8 eV−1. Since \({\chi }_{\tau }^{-1}=\) 63.3 meV ≫ UHu, the second term is small and, hence, we expect positive α and correspondingly easy-plane spin ferromagnet.

The 1/2M phase (Fig. 4f, solid lines) shows the calculated θs versus θB by minimizing Eeff in equation (7) for various indicated positive and negative values of α for fixed Bz = 15 mT and varying Bx. From comparison with the data, we can set an upper bound on |α| < 0.25 µeV. Combining with the derived λSOC ≳ 120 µeV and the numerically attained \({\chi }_{\tau }^{-1}\), we can set a lower bound on the interaction energy as

$${U}_{\rm{Hu}} > 6.5\,{\rm{meV}}\Rightarrow {J}_{\text{H}} > 3.1\times {10}^{-12}\,{\rm{meV}}\,{{\rm{cm}}}^{2},$$

where we used the experimental carrier density, n = 2.1 × 1012 cm−2. Crucially, this bound is not very sensitive to the numerically extracted value of χτ as long as \({\chi }_{\tau }^{-1}\) \(\gg {U}_{\rm{Hu}}\) (equation (8)).

With these parameters, δn is vanishingly small, and hence, the orbital magnetization is negligible. As a result, the magnetization in the 1/2M phase is given by the spin magnetization. Similarly, δθ is very small, and hence, the spin canting is negligible. Note that the attained value of UHu > 6.5 meV is not much lower than the maximum theoretical value of UHu = Vq=K(ϵ = 1) = 11.2 meV, which is the intervalley Coulomb energy in a vacuum (\(\bar{K}=\) (\(0,\frac{4\pi }{3}\)) is the intervalley wavevector). Note that the dielectric constant at wavevector \(\bar{K}\) can be substantially different from the dielectric constant at small wavevectors, which enters the screened Coulomb interaction at the distance between electrons.

In the above discussion, we have assumed momentum-independent Hund’s coupling. In the presence of a k-dependent coupling, the anisotropy expression \({\alpha \cong \lambda }_{\rm{SOC}}^{2}/8{U}_{\rm{Hu}}\) in the limit of λSOC ≪ UHu will remain valid, where UHu should be thought of as momentum-averaged Hund’s energy (that is, it remains the average Hund’s interaction energy per electron). Our reasoning for the Hund’s origin of spin isotropy in 1/2M is as follows: given the fact that the 1/4M phase is fully out-of-plane polarized, we infer that the spin orientation in each valley is dominated by an Ising-like SOC term. In this case, the only way to get an isotropic non-zero spin magnetism in the 1/2M phase is by coupling the spins between the valleys such that they will have a common orientation and the effect of the SOC on each valley will cancel out. The simplest term that has this effect and is allowed by the symmetries of the problem is an intervalley Hund’s exchange coupling. We are not aware of any additional symmetry-allowed term that, under reasonable assumptions, can explain the observed spin isotropy in the 1/2M phase.

Experimental determination of isospin tilt angle in 1/2M

The tilt angle of the isospin moment at the M-to-1/2M transition was determined as follows. The magnetic moment distribution \({\bf{m}}\left({\bf{r}}\right)=m\left({\bf{r}}\right)\hat{z}\) attained in out-of-plane Ba (θB = 0, Extended Data Fig. 7a) is rotated by a trial angle θtr, \({\bf{m}}\left({\bf{r}}\right)=m\left({\bf{r}}\right)\cos {\theta }_{\rm{tr}}\hat{z}+m\left({\bf{r}}\right)\sin {\theta }_{\rm{tr}}\hat{x}\), and the corresponding \({B}_{z}^{\rm{tr}}\left({\bf{r}},{\theta }_{\rm{tr}}\right)\) is calculated:

$${B}_{z}^{{\rm{tr}}}\left({\bf{r}},{\theta }_{{\rm{tr}}}\right)=\frac{{\mu }_{0}}{4\pi }\iint \left[\frac{3{\bf{m}}({\bf{r}}^{\prime} )\cdot \left({\bf{r}}-{\bf{r}}^{\prime} \right)}{{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|}^{5}}\left({\bf{r}}-{\bf{r}}^{\prime} \right)\cdot {\hat{z}}-\frac{{\bf{m}}({\bf{r}}^{\prime} )\cdot {\hat{z}}}{{\left|{\bf{r}}-{\bf{r}}^{\prime} \right|}^{3}}\right]{\text{d}}{A}^{{\prime} }.$$

We then calculate the mean squared deviation between the measured \({B}_{z}^{{ac}}\left({\bf{r}}\right)\) in the presence of Ba along a given θB (Extended Data Fig. 8b) and the calculated \({B}_{z}^{\rm{tr}}\left({\bf{r}},{\theta }_{\rm{tr}}\right)\) (Extended Data Fig. 8c), quantified by \({\rm{Error}}=\langle {|{B}_{z}^{{ac}}\left({\bf{r}},{\theta }_{B}\right)-{B}_{z}^{\rm{tr}}\left({\bf{r}},{\theta }_{{\rm{tr}}}\right)|}^{2}\rangle\). We identify the isospin tilt angle θs with θtr that results in minimal error (Extended Data Fig. 8d). The resulting θs for various positive and negative θB values is presented in Fig. 4f.

In the above procedure, we have made a simplifying assumption that the spin tilt angle θs is position independent, whereas disorder and local strain may result in position-dependent magnetic anisotropy. We note that the inversion problem with arbitrary spin orientation has no single-valued mathematical solution, and hence, some constraints must be imposed. We have, therefore, applied the simplest assumption of uniform θs that shows a good fit to the experimental data.

To provide an additional assessment of the applicability of this assumption and to test for spatial variations in θs, we have repeated the analysis over several narrow strips of the sample (Extended Data Fig. 8e, coloured dashed rectangles). To improve the signal-to-noise ratio, we integrated \({B}_{z}^{{ac}}(x,y)\) over y within each strip. Extended Data Fig. 8f (red curves) shows the averaged \({B}_{z}^{{ac}}(x)\) profiles in the three strips attained from Fig. 3g, as compared with the calculated ones shown in Extended Data Fig. 8e (blue) for the case of θB = 0. Extended Data Fig. 8g shows the results of the same procedure for θB = –45°. The corresponding error function for the three strips for θB = –45° is shown in Extended Data Fig. 8h. The quality of the fits and the derived uncertainty of the tilt angles θs in the three strips is consistent with global values (Extended Data Fig. 8d) within our uncertainty of 5°, suggesting that the variations in the local anisotropy are weak, and the orientation of magnetization can be considered homogeneous.

Our understanding of how strain affects the electronic properties is generally limited, and its impact on magnetic anisotropy is even less understood. In the case of rhombohedral tetralayer graphene, we believe that the strain’s effect on local magnetization orientation should be weak. This conclusion is based on our finding that 1/2M is highly isotropic, whereas 1/4M is strongly easy-axis anisotropic. The theoretical curves in Fig. 4f show that substantial variations in α in 1/2M and in λSOC in 1/4M result in relatively small deviations in the resulting spin orientation θs. Therefore, we believe that, in practice, possible strain-induced local variations in the SOC or anisotropy parameter will result in only small local variations in θs, unresolvable within our experimental resolution.

Evaluation of isospin texture in the 1/4M phase

Extended Data Fig. 9a shows a schematic of the different components of magnetization vectors at the 1/2M-to-1/4M transition. The measured differential magnetization across the transition, m (magenta vector), is given by the difference between the magnetization in 1/4M, \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}\) (blue vector), and 1/2M, \({{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}\) (purple): \({{\bf{m}}={\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}-{{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}={{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{o}}}+{{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}-{{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}\). Here \({{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}\) is the spin magnetization in 1/2M (purple) oriented along the applied field direction θB, and \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}\) (blue) is given by the sum of the orbital magnetization, \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{o}}}\) (brown), oriented along \(\hat{z}\), and the spin magnetization, \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}\) (green), with a tilt angle θs that we aim to determine as follows.

Since the carrier density at the two sides of the 1/2M-to-1/4M transition is the same, the spin magnetization should have the same magnitude (\(|{{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}|=|{{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}|\); the green and purple vectors are of the same length), and should equal \(\left|{{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}\right|\) at the M-to-1/2M transition (Fig. 3j), rescaled by the relative carrier densities at the two transitions and by the partial polarization in the PSP-M region. For out-of-plane Ba (θB = 0), \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}={{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}\) (the green and purple vectors are parallel), and hence, \({\bf{m}}={{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{o}}}\). The brown vector is, thus, known and is given by Fig. 4d. We can then evaluate θs by comparing the measured \({B}_{z}^{{ac}}(x,y)\) corresponding to m (magenta), with \({B}_{z}^{{ac}}(x,y,{\theta }_{\rm{tr}})\) derived from the summation of the green, brown and purple vectors, \({\bf{m}}={{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{o}}}+{{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}({\theta }_{\rm{tr}})-{{\mathscr{M}}}_{{\bf{1}}/{\bf{2}}{\bf{M}}}^{{\bf{s}}}({\theta }_{B})\), where θtr is a trial angle for the spin orientation θs in 1/4M (green).

Extended Data Fig. 9b,c shows the measured \({B}_{z}^{{ac}}\left(x,y\right)\) at θB = 78° and the calculated \({B}_{z}^{\rm{tr}}(x,y,{\theta }_{\rm{tr}})\) using the above procedure, whereas Fig. 4e shows the calculated \({B}_{z}^{\rm{tr}}(x,y,{\theta }_{\rm{tr}})\) at θB = 83°. The corresponding mean square error \({\rm{Error}}=\langle {|{B}_{z}^{{ac}}\left(x,y\right)-{B}_{z}^{\rm{tr}}\left(x,y,{\theta }_{\rm{tr}}\right)|}^{2}\rangle\) at θB = 78° is plotted as a function of θtr (Extended Data Fig. 9d). The minimal error at θtr = 1.4° designates the spin tilt angle (θtr = θs) of \({{\mathscr{M}}}_{{\bf{1}}/{\bf{4}}{\bf{M}}}^{{\bf{s}}}\) (green vector) and the resulting \({B}_{z}^{{ac}}(x,y,{{\theta }_{\rm{tr}}=\theta }_{s})\) is shown in Extended Data Fig. 9c. The magenta triangles in Fig. 4f show the average |θs| derived from the above procedure at θB = ±78° and θB = ±83°.

We can attain an additional bound on θs in 1/4M as follows. Because the isospin magnetization is isotropic in 1/2M, the spin will be essentially in-plane polarized at large in-plane field Bx. Further increase in Bx will hardly affect the magnetization in 1/2M, but may tilt the spin orientation in 1/4M away from the easy axis. We first apply a small Bz = 15 mT and a large Bx = 70 mT (θB = 78°) and acquire the corresponding \({B}_{z}^{{ac}+}(x,y)\) across the 1/2M-to-1/4M transition. The in-plane field is then set to Bx = –70 mT and the resulting \({B}_{z}^{{ac}-}(x,y)\) is measured. The two sets of data are subtracted, with the difference, \({B}_{z}^{{ac}\pm }\left(x,y\right)={B}_{z}^{{ac}+}\left(x,y\right)-{B}_{z}^{{ac}-}(x,y)\), presented in Extended Data Fig. 9e. This procedure eliminates the out-of-plane component of the differential magnetization, which is independent of the Bx direction, with \({B}_{z}^{{ac}\pm }\left(x,y\right)\) reflecting only the stray field due to the in-plane component of differential magnetization across the transition. The same procedure is repeated with substantially larger Bx = ±120 mT (θB = 83°), with the resulting \({B}_{z}^{{ac}\pm }\left(x,y\right)\) presented in Extended Data Fig. 9f. Extended Data Fig. 9g presents the difference \(\Delta {B}_{z}^{{ac}}\left(x,y\right)={{B}_{z}^{{ac}\pm }\left(x,y\right)}_{120\,{\rm{mT}}}-{{B}_{z}^{{ac}\pm }\left(x,y\right)}_{70\,{\rm{mT}}}\), which shows that the change in differential in-plane magnetization across the transition, and hence the possible change in the isospin orientation in the 1/4M, are unresolvable between the two Bx values within our resolution. By averaging over the y axis for improved signal-to-noise ratio of the \({{B}_{z}^{{ac}\pm }\left(x,y\right)}_{120\,{\rm{mT}}}\) data (red) and the \(\Delta {B}_{z}^{{ac}}\left(x,y\right)\) data (black) in Extended Data Fig. 9h, we can estimate that the change in in-plane magnetization is less than 9%, placing an upper bound on possible spin tilt angle change in 1/4M between Bx = 70 mT and Bx = 120 mT of about 5°, which we use as the error bar for θs in the 1/4M phase (Fig. 4f).

Reconstruction of 2D isospin magnetization

To reconstruct the out-of-plane magnetization, instead of the Tikhonov-regularized Fourier-transform-based approach58, we use a numerical method based on a deep neural network, similar to that in ref. 59. In this method, we consider a three-layer network architecture, consisting of the input layer, deep layer and recovered image layer. Both deep layer and recovered image layer utilize the rectified linear unit activation function. The recovered image layer represents the magnetization m. The convolution of the recovered image layer with the Biot–Savart kernel yields \({B}_{z,{{\rm{sim}}}}^{{ac}}\), the magnetic field predicted by the network. To achieve faster convergence, the input layer is initialized with the measured values of \({B}_{z}^{{ac}}(x,y)\). The network’s weights are optimized by minimizing the merit function as

$${f}_{{\rm{M}}}={\Vert {B}_{z,{\rm{sim}}}^{ac}-{B}_{z}^{ac}\Vert }^{2}+{\lambda }_{\alpha }{\Vert {\nabla }^{2}m\Vert }^{2},$$

which ensures that the theoretical magnetic field closely matches the measured field and maintains smoothness in magnetization. Here λα is the Tikhonov regularization parameter.