Pressure-correlation phase diagram unraveling quantum phase transitions in altermagnetic $ε$-FeOOH and realization of piezo-anomalous Hall effect

Correlated systems, due to their rich electronic and magnetic phase diagram, have always remained at the forefront for exploring emerging quantum phases and insights in condensed matter physics [3, 4, 5, 6, 7, 8] [1, 2]. Oxide minerals are often considered the appropriate class of materials for investigating these phases and their transitions, as they can withstand high pressures and temperatures. In recent years, due to the existence of large interstitial space in the form of tunnels and vander Waals separations, encouraging quantum confinement and strong octahedral crystal fields, the iron oxyhydroxide polymorphs [9, 10, 11, 12, 13, 14, 15, 16, 17], namely goethite, akaganeite, and lepidocrocite, have gained renewed attention in the context of magnetism, structural breakdown and transitions, and desirable electrochemical activities.

Particularly, the quenched goethite, $\epsilon$-FeOOH, which can endure phenomenally high pressure (around 100 GPa) and can shrink volume by 50%, has emerged as a reference system as it demonstrates switchable spin states, tunable spin anisotropy, and structural transitions [12, 18, 19, 20].

A brief review of the literature provides the following information about the critical pressures across which the phase transitions occur. From the experimental perspective, the structure has a second-order transition ($P2_{1}nm \rightarrow Pnnm$ phase) around 18 GPa and a first-order phase transition around 45 GPa [12, 19, 20]. The theoretical studies propose the structural transition pressure to be in the range from 37-45 GPa [18, 20]. Concerning the high spin (HS) to low spin (LS) crossover, both from a theoretical and experimental perspective, there is

a lack of clarity, and it lies somewhere in between 45 and 70 GPa [18, 19, 20]. In addition, a recent experimental studies by Ikeda et al. propose a possible change in the spin quantization axis in the system at 8 GPa [12]. In the absence of a complete electronic and magnetic phase diagram in the space spanned by pressure and correlation, the understanding of the underlying mechanism driving these transitions is missing. Also, with several competing interactions at play, there can be other missing quantum transitions in this class of minerals.

Furthermore, with the recent discovery of altermagnetism, identified with non-relativistic spin-splitting of the bands of a real space antiferromagnet [21, 22], where fundamental understanding is still emerging, and nontrivial quantum transports are being envisaged [23, 24, 25, 26, 27, 28], the FeOOH mineral class, with its spin anisotropy and pressure tunable symmetries, can serve as a promising family to investigate.

In this work, we have carried out electronic and magnetic structure calculations, and analyzed the results from the perspective of a strong correlation effect (i) to establish a complete phase diagram, (ii) to present the underlying mechanism responsible for the reported quantum phase transitions, and (iii) to unravel yet to be reported phase transitions and phases including altermagnetism.

The critical findings from this work, summarized in Figure 1, are as follows. There are three correlation-tunable critical pressures across which the phase transitions occur. Validating the majority of the experimental observation, the structural transition from $P2_{1}nm$ (non-centrosymmetric) to $Pnnm$ (centrosymmetric), at a critical pressure $P_{c2}$, occurs while the system is in the HS phase (intermediate pressure), and the crossover to the LS state occurs at a higher critical pressure ($P_{c1}$). Interestingly, $P_{c2}$ is almost independent of the correlation effect, while $P_{c1}$ is sensitive to it. Consequently, concurrent with the HS-LS crossover, from the electronic structure perspective, the system changes from a charge transfer insulator (CTI) to a Mott insulator (MI) type, which is an outcome of competition between Hund’s coupling and crystal field. Symmetry compatibility ensures altermagnetism in both the structural phases, and therefore, it makes this system unique in achieving pressure switchable non-centrosymmetric to centrosymmetric altermagnet. The spin quantization axis alters its direction from $\hat{z}$ to $\hat{y}$ to $\hat{x}$ at two critical pressures $P_{c3}$ and $P_{c1}$, respectively; one at the HS regime and one at the crossover of HS to LS, which enables the system to exhibit piezo-anomalous Hall effect, a phenomenon yet to be reported.

Results

Pressure-correlation phase diagram

The phase diagram is computed by performing DFT+$U$ calculations on relaxed structures by varying the onsite Coulomb repulsion ($U$) and the volume of the $\epsilon$-FeOOH unit cell. The structural and computational details, as well as the relation between pressure and volume, are provided in the appendix. The phase diagram is shown in Figure 2(a). The three distinct critical pressure curves, $P_{c1}(U) > P_{c2}(U) > P_{c3}(U)$ identifying the multiple phase transitions are shown.

We will first discuss the case of $P_{c1}(U)$, which exhibits the HS-LS crossover as well as the CTI to MI transition by mapping our DFT results to the following multiorbital Hubbard Hamiltonian [29, 30, 31]:

where $U' = U - 2J$ and $n_{i\mu} = n_{i\mu\uparrow} + n_{i\mu\downarrow}$. Here, the first and second terms represent the onsite energy and electron hopping, respectively. The third and fourth terms (i.e., $\mathcal{H}_{\mathrm{corr}}$ and $\mathcal{H}_{\mathrm{mag}}$) consist of Hubbard interactions among electrons in the same orbital, different orbitals, and Hund’s coupling energy, respectively. The mean-field expansion of the Hubbard terms, involving orbital and spin space occupancies, is provided in the appendix.

With 3+ charge state of Fe in $\epsilon$-FeOOH, the $3d^{5}4s^{0}$ electronic configuration, under the influence of octahedral crystal field $\Delta$ and Hund’s coupling $J$, governs the electronic and magnetic structure of the system. $J$ is primarily an atomistic phenomenon, while $\Delta$, which gets incorporated in the first term of Eq. (1), is driven by the volume and shape of the octahedron. In addition to this, the electron hopping $t$ and repulsion $U$, both of which depend on the atomistic characteristics as well as the crystal structure, complicate the electron filling scenarios [32]. The strength of $\Delta$ that splits the triply degenerate $t_{2g}$ manifold and doubly degenerate $e_g$ manifold increases with compression. In a nonmagnetic state, the splitting leads to a five-sixth occupied $t_{2g}$ state and an unoccupied $e_g$ state, and they are well-separated in the equilibrium structure as shown in the schematic Figure 2(a-i). The large DOS at the Fermi level ($E_F$) compels the system to lower its energy through spin polarization with the help of Hund’s exchange splitting $J$. As the system is half-filled, a strong $J$ ($> \Delta$), driven by the last term of Eq. (1), helps the Fe to achieve the HS state with $S = 5/2$. The Fe-O-Fe superexchange stabilizes the antiferromagnetic state, and it results in a Slater insulator as demonstrated in Figure 2(a-ii).

Upon application of Coulomb repulsion, the first term of $\mathcal{H}_{\mathrm{corr}}$ in Eq. (1) dominates to push the occupied $t_{2g}$ and $e_g$ spin majority states below the O-$p$ spectrum and to push the unoccupied spin minority states above in energy to stabilize the system in a CTI phase. The process of achieving CTI is shown through Figure 2(a-ii - a-iv). When the system is compressed, the octahedra are shrunk to yield a large $\Delta$ separating the $t_{2g}$ and $e_g$ (see Figure 2(a-vi)). Simultaneously, with a reduction in the bond length, the kinetic hopping $t$ increases to reduce the effective Hund’s coupling $J_{\mathrm{eff}}$. Therefore, the competing $\Delta$ and $J_{\mathrm{eff}}$ determine the spin occupancy of $t_{2g}$ and $e_g$ states. For $\Delta$ larger than $J_{\mathrm{eff}}$, which occurs beyond a critical pressure $P_{c1}$, the $e_g$ states remain unoccupied and do not contribute to $\mathcal{H}_{\mathrm{mag}}$ and $\mathcal{H}_{\mathrm{corr}}$. This leads to a LS state ($S = 1/2$; $t_{2g}^{3\uparrow,2\downarrow}e_g^0$). In the absence of strong correlation, the partially occupied $t_{2g}\downarrow$ sub-band yields a metallic state as shown in Figure 2(a-vii). With the in-

clusion of $\mathcal{H}_{\mathrm{corr}}$, the strength of the first term, involving the Coulomb repulsion of the opposite spins, is effectively reduced to one-third (= $\langle n_{i\mu\uparrow}\rangle- \langle n_{i\mu\downarrow}\rangle$), and as a consequence, the $t_{2g}$ spin majority bands lie above the $O$-$p$ spectrum. Hence, CTI ceases to exist. However, the second term, involving the inter-orbital Coulomb repulsion, separates the $t_{2g}$ spin minority bands into two subbands and creates a Mott gap at the Fermi (see Figure 2a-viii). The Mott gap increases with an increase in $U$ (see Figure 2(a-ix)). From our first-principles calculation, we found that $P_{c1}$ increases with $U$, which is also reflected in the local spin moment (LSM) plot in Figure 2(a-x).

To provide a quantitative explanation of the stability as well as transition of the phases, in Figure 2(c) and (d), we have plotted the relative energies as a function of volume for $U = 3$ and $5$ eV, respectively. For the pressure-free system, the $P2_1nm$ HS phase (green curve) has lower energy as this curve lies distinctly below the blue curve indicating the $Pnnm$ HS phase (blue curve). The latter phase stabilizes over the former beyond $P_{c2}$. However, with further application of pressure, the energy difference between HS and LS (magenta curve) reduces, and the latter wins over the former at a critical volume $P_{c1}$. Unlike $P_{c1}$, $P_{c2}$ ($\approx45$ GPa) is less sensitive to the Hubbard parameter $U$ as can be seen in Figure 2(a). The value of 45 GPa is in reasonable agreement with the one reported by Gleason et al. (43 GPa) [20] and Insixiengmay et al. (37 GPa) [18]. However, the experimental studies predict a second-order phase transition at 18 GPa and a first-order phase transition between 45 - 60 GPa [19]. The former is attributed to hydrogen bond symmetrization, and the latter is attributed to possible HS-LS spin crossover. If we go by the observation of Ikeda et al. [12], at 18 GPa, even though the system makes a transition to the $Pnnm$ phase, the hydrogen bond symmetrization is not complete, as the H atom still has the fractional occupancy at two different Wyckoff positions, which escapes the scrutiny by DFT methods designed to study the periodic systems. However, we find that at $P_{c2}$, a perfect hydrogen bond symmetrization occurs in agreement with both theory and experiment.

Altermagnetism and its characterization

Stabilization A-type Néel ordering in $\epsilon$-FeOOH, encourages us to examine the symmetry compatibility of this compound to exhibit altermagnetism. In the $P2_1nm$ phase, the three symmetry operations: one two fold screw, one mirror, and one glide along with identity $\mathcal{E}$ form the crystallographic Laue group, i.e., $G = \{\mathcal{E}, C_{2x}^{\frac{1}{2},0,\frac{1}{2}}, M_z, M_y^{\frac{1}{2},0,\frac{1}{2}}\}$. The elements of $G$ can be divided into two spin halving subgroups $H_{31} = \{\mathcal{E}, M_z\}$ and $(G-H) = \{C_{2x}^{\frac{1}{2},0,\frac{1}{2}}, M_y^{\frac{1}{2},0,\frac{1}{2}}\}$ such that they form type III Laue spin group $R_s = [\mathcal{E} \Vert H] + [C_2 \Vert(G-H)]$. For the case of the $Pnnm$ phase, the hydrogen bond symmetrization, in addition to the existing symmetries, introduces two more symmetry operations, namely, inversion ($\mathcal{I}$) and proper rotation. They reconstruct the crystallographic Laue group $G = \{\mathcal{E}, \mathcal{I}, C_{2x}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, C_{2y}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, C_{2z}, M_x^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, M_y^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, M_z\}$, where $H = \{\mathcal{E}, \mathcal{I}, C_{2z}, M_z\}$ and $(G-H) = \{C_{2x}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, C_{2y}^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, M_x^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}, M_y^{\frac{1}{2},\frac{1}{2},\frac{1}{2}}\}$. A compound lying in the type III Laue spin group exhibits altermagnetism if there exist no inversion or translation symmetries in the halving subgroup $G-H$, but there must exist a proper or improper rotational symmetry $A$ ($\in G-H$) such that $G-H = AH$. As is the case for both $P2_1nm$ and $Pnnm$ phases, $\epsilon$-FeOOH is an altermagnet in both phases.

The $Pnnm$ phase of $\epsilon$-FeOOH is centrosymmetric, and the inversion symmetry connects the same spin sublattices as it belongs to the halving subgroup $H$. As a result,

the antiferromagnetic spin degeneracy in the band structure is lifted to induce altermagnetism. The presence of $[\mathcal{T}_{\sigma} \Vert\mathcal{M}_x]$ and $[\mathcal{T}_{\sigma} \Vert\mathcal{M}_y]$ construct the $k_x-k_z$ and $k_y-k_z$ planes as nodal planes (see appendix; Eq. (14) and D8), thereby yielding an centrosymmetric $d$-wave altermagnet. Furthermore, since the $\epsilon$-FeOOH manifes colossal magnetism, in the non-relativistic limit, even though the system is noncentrosymmetric as in the $P2_1nm$ phase, the bands undergo $[\mathcal{E}_{\sigma} \Vert\mathcal{T}_L]$ operation (identity operation on spin space and time-reversal operation on lattice space), which ensures AMSS $\Delta E(\mathbf{k}) = E(\sigma,\mathbf{k}) - E(-\sigma,\mathbf{k}) = \Delta E(-\mathbf{k})$ [21, 28]. Interestingly, in the $P2_1nm$ phase, $[\mathcal{T}_{\sigma} \Vert\mathcal{M}_y]$, along with $[\mathcal{E}_{\sigma} \Vert\mathcal{T}_L]$, again construct the $k_x-k_z$ and $k_y-k_z$ planar nodal planes (see appendix; Eq. (17) and D12) to yield a rare noncentrosymmetric $d$-wave altermagnet. The other space groups that exhibits 3D collinear noncentrosymmetric altermagnets reported so far are $I4_1md$ (GdAlSi [33], GdAlGe [34]), $P\bar{6}2c$ (FeS [35]), $P\bar{6}m2$ (doped CrSb [28]), and $P2_1$ (RbMnPO$_4$ [28]).

We will now validate the existence of altermagnetism through band structure, which is shown in Figure 3, and examine the impact of several phase transitions on altermagnetism. The upper (Figure 3(a,Figure 3b)) and middle panels (Figure 3(c,Figure 3d)) show the band structure for the HS phase with the $P2_1nm$ space group and the LS phase with the $Pnnm$ space group, respectively. The band structure on the left demonstrates antiferromagnetic band degeneracy as the $k$-path considered lies on the nodal plane. The band structures on the right show the AMSS to reflect the altermagnetism in both HS and LS phases.

When we compare the band structure of HS and LS phases, we notice two striking differences. Firstly, the valence band maximum (VBM) and conduction band minimum (CBM) shift their position. In the case of LS, the CBM lies at $\Gamma$ with a well-dispersive parabolic band (see Figure 3(c)), and the VBM lies at Z, which also has large dispersion. In the case of HS, the bands are weakly dispersed. There is a near absence of CBM as the conduction band edge is nearly flat across $\Gamma-X-S-Y$ (see Figure 3(a)), while the VBM shifts from Z to X. The alteration in the behavior of VBM and CBM can have consequences on the opto-electronics behavior of the system.

Secondly, the altermagnetic spin splitting (AMSS) is stronger for LS and weaker (almost reduced by half) for HS. This can be explained as follows. For the LS, the reduction in the bond length enhances the covalent hybridization, and therefore, the bands are expected to be more dispersed. On top of it, since the LS phase is MI, the spin carrying Fe-$d$ bands occupy the Fermi, while the O-$p$ bands lie below, which is substantiated through the atom resolved DOS (see Figure 2(b)). The spin-carrying bands are expected to show larger AMSS as compared to the O-$p$ bands, which are at best spin-induced due to hybridization. As the HS phase is a CTI, the localized Fe-$d$ bands lie far away from the Fermi energy $E_\mathrm{F}$, while the spin-induced O-$p$ bands lie close to the $E_\mathrm{F}$. These weakly spin-polarized bands generate weak AMSS.

Magnetic anisotropy

Concerning the quantum transport, one of the expectations to make altermagnetic materials useful for spintronics applications is to demonstrate anomalous Hall conductivity (AHC) in them. The basic criterion here is to see that they have a preferred spin quantization axis that is symmetry compatible to exhibit AHC. For $\epsilon$-FeOOH, if the preferred spin orientation is along $\hat{x}$ or $\hat{y}$, then AHC is expected. As we will demonstrate here, the applied pressure, in addition to the aforementioned structural and magnetic phase transition, alters the spin quantization axis.

For the equilibrium configuration, the $\hat{x}$ axis remains as hard axis of the spin orientation, while both $\hat{y}$ and $\hat{z}$ axes become energetically favorable as energy difference

of $\Delta E_{yz}$ ($= E_y - E_z$) tends to zero ($\approx-0.05$ meV). If the external pressure is applied or the correlation strength is reduced, the $\hat{y}$ quantization becomes distinctly energetically favorable beyond the critical pressure $P_{c3}$, as shown in Figure 2(a), with $\Delta E_{yz}$ reaching its peak value of $-0.3$ meV. Aligning with the above observation, the recent experiment by Ikeda et al. [12] reported the spin quantization axis as $\hat{z}$ for the equilibrium configuration and switches to $\hat{y}$ with pressure [12]. Interestingly, at the critical pressure $P_{c1}$, where the HS-LS spin crossover occurs, $\hat{x}$ quantization becomes favorable with $\Delta E_{yx}$ more than 2 meV. This observation opens up scopes for the experimenter to validate and establish spin quantization switching.

Direction-dependent piezo-anomalous Hall effect

The role of spin quantization on the anomalous Hall transport in the $\epsilon$-FeOOH is discussed next. This is premised on the fact that the existence of AHC and the allowed components of its tensor depend on the Néel vector orientation and the corresponding magnetic space group (MSG). The symmetry-allowed AHC tensor components for different magnetic configurations are summarized in Tables III and IV of the appendix. For the $P2_1nm$ phase in equilibrium condition, the Néel vector is oriented along the $\hat{z}$ direction, as discussed earlier. The corresponding magnetic symmetry forbids all components of the AHC tensor, leading to a vanishing intrinsic AHC. Beyond the critical pressure $P_{c3}$, the Néel vector changes to the $\hat{y}$ direction, which in turn, changes the MSG. The new symmetry elements allow the $\sigma_{yz}$ component of the AHC, while all other tensor components remain forbidden. The same is the case for the $Pnnm$ phase with HS state and spin quantization along $\hat{y}$. Upon further application of pressure beyond $P_{c1}$, where the spin crossover occurs, the Néel vector reorients along the $\hat{x}$ axis, which further modifies the corresponding symmetries. In the LS state, only the $\sigma_{xz}$ component of AHC is allowed. In Figure 4, we have presented the numerical results for various components of the AHC using Berry curvature in different situations allowed by the corresponding symmetries. The magnitude and direction switching of AHC with pressure are clearly seen in the figure. Furthermore, from the figure, we observe that the magnitude of the AHC increases with pressure. The details of the numerical calculation are provided in the appendix. Therefore, ε-FeOOH is an ideal compound to establish piezo-anomalous Hall response, where AHC can be switched on and off or its direction can be changed by tuning the pressure.

Discussion

Necessitated by experimentally observed pressure-tunable electronic and magnetic structures in ε-FeOOH, in this work, we have established a pressure-correlation phase diagram for it. Through this phase diagram, we reaffirmed the existence of two critical pressures driving the structural transition ($P_{c2}$) and spin crossover ($P_{c1}$). While $P_{c2}$ is the least influenced by the correlation, $P_{c1}$ is sensitive to it. In addition, we identified a third critical pressure $P_{c3}$ across which the spin quantization axis changes from $\hat{z}$ to $\hat{y}$, which is broadly consistent with the very recent experimental observation by Ikeda et al. [12]. $P_{c3}$ is found to be highly dependent on the electron-electron correlation. We established another change in the spin quantization axis (from $\hat{y}$ to $\hat{x}$) at $P_{c1}$ that was missing in earlier studies. The present work, through a thorough analysis of electronic structure and mapping that to a multiorbital Hubbard Hamiltonian, provides the underlying mechanisms governing the mechanism of HS and LS states and their transitions. It demonstrates that due to competing the crystal field and effective Hund’s coupling, the HS state is synonymous with the CTI electronic state, and the LS state is synonymous with the MI electronic state. Our group theoretical analysis, along with the band structure studies, infer that the compound is altermagnetic in both $P2_1nm$ (non-centrosymmetric) and $Pnnm$ (centrosymmetric) phases. Therefore, ε-FeOOH is a unique system to exhibit pressure switchable centrosymmetric to non-centrosymmetric altermagnet. Furthermore, we find that the system exhibits finite AHC provided the spin quantization axis is either $\hat{y}$ or $\hat{x}$. Therefore, the critical pressures $P_{c1}$ and $P_{c3}$ imply the realization of piezo-anomalous Hall effect, which is promising for spintronics applications.

Methods

In this study, DFT calculations are performed using the plane-wave projector augmented-wave (PAW) method [36, 37], as implemented in the Vienna $Ab$ initio Simulation Package (VASP) [38], to investigate the electronic and magnetic properties of the compound. The Perdew-Burke-Ernzerhof generalized gradient approximation is chosen for the exchange correlation functional. The Brillouin zone is sampled using a $\Gamma$-centered $6 \times6 \times8$ $k$-point mesh, which is decided from a convergence test. The plane-wave energy cutoff of 600 eV is adopted. To account for the effects of electron correlations, the DFT+$U$ approach is employed within the rotationally invariant formalism proposed by Dudarev [39], using the effective Hubbard parameter $U_{\mathrm{eff}} = U - J$. To evaluate the AHC, a tight-binding Hamiltonian was constructed using maximally localized Wannier functions constructed using the Wannier90 code [40]. The AHC was subsequently calculated from this Wannier-based Hamiltonian using the WannierBerri code [41].

Acknowledgement

AM thanks MoE India for the PMRF fellowship. The authors acknowledge IIT Madras for the computational resources. The authors acknowledge Babuji Dandigunta for useful discussions.

Structural details of ε-FeOOH

The ε phase of FeOOH, a high-pressure quenchable polymorph of its α phase (goethite), was first synthesized by Chenevas et al. in 1973, and has an InOOH-type crystal structure with $P2_1nm$ space group [42, 43]. As shown in Figure 5, each Fe atom is surrounded by six oxygen atoms forming a FeO$_6$ stretched octahedron. The octahedra are edge-shared along $c$. Along $a$, the octahedra are separated each other through a void. Furthermore, the octahedra are layer stacked along $b$, and the stretched axis of the octahedra along the diagonal of $ab$ plane are zig-zagged to constitute a cooperative distortion.

Oxygen from every alternate layer forms hydroxy bonds, while oxygen from the other alternate layers forms hydrogen bonds in such a way that there is an O−H$\cdots$O bond lying diagonally in the void. The lattice parameters of the unit cell are optimized to be $a = 4.99$ Å, $b = 4.45$ Å, $c = 3.02$ Å, which are in close agreement with experimental reported values of $a = 4.93918$ Å, $b = 4.43693$ Å, $c = 2.99433$ Å[12].

Upon applying hydrostatic pressure to this compound, a structural phase transition occurs. The $P2_{1}nm$ phase changes to the $Pnnm$ phase, which is further verified by estimating the relative energy difference obtained from the $ab$-initio calculations. The structural transition from $P2_{1}nm$ to $Pnnm$ phase is concurrent with the vanishing of the hydrogen bond (i.e., H$\cdots$O). With an intermediate pressure ($\approx18$ GPa) [12], the O−H$\cdots$O is now replaced with a single O−H bond. However, there is a 50% probability that the H atom is attached to the lower oxygen layer and another 50% probability is that it is connected to the upper oxygen layer. As the pressure increases, the O−H$\cdots$O bond becomes an O−H−O bond, where H is rigidly connected to both the oxygen atoms instead of flipping from one to the other. As the DFT calculations are done on a static equilibrium configuration, the fractional occupancies cannot be accommodated, and we have considered the O−H−O bond both for the intermediate and high-pressure $Pnnm$ phase.

Pressure vs Volume curve

The volume of the unit cell for a given pressure and correlation strength is obtained by relaxing the atomic positions and cell volume in DFT calculations. The results are shown in Figure 6. From the figure, we see a sudden reduction in volume at the critical pressure $P_{c1}$, where HS-LS crossover occurs. This discontinuity was reported in Ref. [19] as a first-order phase transition. Furthermore, we observe that $P_{c1}$ increases with $U$, which suggests that HS-LS crossover not only occurs due to an increase in the crystal field as reported by Inxixieng-may et al. [18], but also that a competition among the crystal field, Hund’s coupling, and electron correlation is involved in it.

Multi-orbital Hubbard Model

The multi-orbital Hubbard Hamiltonian is written as

where $U' = U - 2J$, and,

Since $\epsilon$-FeOOH stabilizes in the $A$-type antiferromagnetic spin ordering, the two Fe atoms exhibit opposite spins. Here, we analyze the occupation of different orbitals of the Fe atom, which has the spin-up majority. The same analysis applies to the other Fe atom. The electron occupation numbers for the HS and LS states are tabulated in Table I. From the table and the above-mentioned equations, we see that in the HS state, the Hamiltonian $H_2$ does not create an energy gap between the spin-up and spin-down states. For $U=0$, Hamiltonian $H_1$ also has no role in the formation of the energy gap. Therefore, the Hund’s coupling energy can be estimated from Hamiltonian $H_3$. The energy gap for $e_g$ orbitals from the DFT-produced DOS is obtained as $J_{\mathrm{eff}}=3$ eV. Hence, $3=-\frac{J}{2}(1+1)$ (see Figure 6). From here, we can estimate $J\approx3$ eV.

Spin state	$d_1 \uparrow$	$d_2 \uparrow$	$d_3 \uparrow$	$d_4 \uparrow$	$d_5 \uparrow$	$d_1 \downarrow$	$d_2 \downarrow$	$d_3 \downarrow$	$d_4 \downarrow$	$d_5 \downarrow$
HS	1	1	1	1	1	0	0	0	0	0
LS	1	1	1	0	0	$2/3$	$2/3$	$2/3$	0	0

If the external pressure is applied to it, the volume of the unit cell decreases, which results in an increase in crystal field splitting $\Delta$. At the moment $\Delta$ exceeds the $J$, the system jumps from the HS state to the LS state. From the nonmagnetic DOS, we find $\Delta\approx4$ eV at 100 GPa pressure. Furthermore, the gap due to Hund’s coupling here should be approximately $3\times(2/3)=2$ eV, which matches with the DFT obtained DOS with $U=0$ eV (see Figure 6).

Altermagnetic symmetry and nodal planes

For a collinear magnet, as is the case of $\epsilon$-FeOOH, under the non-relativistic limit the bands always have the $[\mathcal{E}_{\sigma}\Vert\mathcal{T}_{L}]$ symmetry (where, $\mathcal{E}_{\sigma}$ is identity operation on spin space while $\mathcal{T}_{L}$ is the time-reveral operation on lattice space), which mimics the effect of the inversion symmetry in the band structure [21, 28], i.e.,

Therefore, the nonrelativistic bands of collinear spin systems always have the inversion symmetry irrespective of their centrosymmetricity. This ensures $\Delta E(k)=\Delta E(-k)$, where $\Delta E=E(\sigma)-E(-\sigma)$.

In the centrosymmetric $Pnnm$ phase of $\epsilon$-FeOOH, the crystal symmetries impose the following constraints on the band structure.

Here we see that the results in Eqs. (8) and (9) are identical as discussed earlier. $[\mathcal{T}_{\sigma}\Vert M_x]$ and $[\mathcal{T}_{\sigma}\Vert M_y]$ make the $k_x-k_z$ and $k_y-k_z$ plane spin degenerate, respectively, while $[\mathcal{T}_{\sigma}\Vert C_{2x}]$ and $[\mathcal{T}_{\sigma}\Vert C_{2y}]$ make the $k_x$ and $k_y$ axis spin degenerate, respectively. This makes the system a centrosymmetric $d$ wave altermagnet as shown in Figure 3(a) in the main text.

In the $P2_1nm$ phase, the crystal symmetry-driven constraints on the band structure are given by,

Note that inversion symmetry is broken in this compound. Here, only $[\mathcal{T}_{\sigma}\Vert M_y]$ and $[\mathcal{T}_{\sigma}\Vert C_{2x}]$ can establish a relation between the band energies of two opposite spins, as evident from Eqs. (17) and (18). Out of this two $[\mathcal{T}_{\sigma}\Vert M_y]$ makes the $k_x-k_z$ plane spin degenerate, while $[\mathcal{T}_{\sigma}\Vert C_{2x}]$ makes the $k_x$ axis spin degenerate. Moreover, the above-mentioned symmetries followed by $[\mathcal{E}_{\sigma}\Vert\mathcal{T}_{L}]$ provide a similar effect of $[\mathcal{T}_{\sigma}\Vert C_{2y}]$ and $[\mathcal{T}_{\sigma}\Vert M_x]$, which results in a spin band degenerate $k_y$ axis and $k_y-k_z$ plane, respectively, as shown below.

As a whole, the $P2_1nm$ phase of $\epsilon$-FeOOH also has two distinct spin band degenerate planes, namely $k_x-k_z$ and $k_y-k_z$ (the degenerate axes are included in the degenerate planes), which are known as the altermagnetic nodal plane. Therefore, the $P2_1nm$ phase is a noncentrosymmetric $d$ wave altermagnet.

Anomalous Hall effect

As FeOOH is an altermagnet whose magnetic orientation changes with pressure, we examined the possibility of pressure-induced AHE using magnetic symmetry analysis and first-principles calculations. The symmetry compatibility for the existence of finite AHC for different spin quantization axes is provided in Table II and Table III. The Berry curvature, $\Omega_n^\nu(\mathbf{k})$, was calculated over the Brillouin zone using [44].

Symmetry	$\sigma_{xy}$	$\sigma_{xz}$	$\sigma_{yz}$
$P2_1nm$ with spin direction $\hat{z}$
$M_z$	$\checkmark$	$\times$	$\times$
$M_y t_{\frac{1}{2},0,\frac{1}{2}}$	$\times$	$\checkmark$	$\times$
$C_{2x}t_{\frac{1}{2},0,\frac{1}{2}}$	$\times$	$\times$	$\checkmark$
$P2_1nm$ with spin direction $\hat{y}$
$\tau M_z$	$\times$	$\checkmark$	$\checkmark$
$\tau M_y t_{\frac{1}{2},0,\frac{1}{2}}$	$\checkmark$	$\times$	$\checkmark$
$C_{2x}t_{\frac{1}{2},0,\frac{1}{2}}$	$\times$	$\times$	$\checkmark$

Symmetry	$\sigma_{xy}$	$\sigma_{xz}$	$\sigma_{yz}$
$Pnnm$ with spin direction $\hat{x}$
$\tau C_{2x}t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\checkmark$	$\checkmark$	$\times$
$C_{2y}t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\times$	$\checkmark$	$\times$
$\tau C_{2z}$	$\times$	$\checkmark$	$\checkmark$
$\tau M_x t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\checkmark$	$\checkmark$	$\times$
$M_y t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\times$	$\checkmark$	$\times$
$\tau M_z$	$\times$	$\checkmark$	$\checkmark$
$Pnnm$ with spin direction $\hat{y}$
$C_{2x}t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\times$	$\times$	$\checkmark$
$\tau C_{2y}t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\checkmark$	$\times$	$\checkmark$
$\tau C_{2z}$	$\times$	$\checkmark$	$\checkmark$
$M_x t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\times$	$\times$	$\checkmark$
$\tau M_y t_{\frac{1}{2},\frac{1}{2},\frac{1}{2}}$	$\checkmark$	$\times$	$\checkmark$
$\tau M_z$	$\times$	$\checkmark$	$\checkmark$

Here, $v_\gamma$ is the velocity operator, and $E_n(\mathbf{k})$ is the energy eigenvalue of the $n$th band. The AHC was then obtained using the Kubo formula [45],

where $e$, $\hbar$, $\epsilon_{\gamma\delta\nu}$, and $f(\epsilon_n(\mathbf{k}))$ denote the electron charge, reduced Planck’s constant, the Levi-Civita symbol, and the Fermi–Dirac distribution function, respectively.

FeOOH crystallizes in the $P2_1nm$ and $Pnnm$ space groups in different pressure ranges.