Continuous PT-Symmetry Breaking as a Design Variable for Giant Altermagnetic Spin Splitting

Kichan Chun; Gunn Kim

cond-mat.mtrl-sciApr 11, 2026

Continuous PT-Symmetry Breaking as a Design Variable for Giant Altermagnetic Spin Splitting

Authors: Kichan Chun, Gunn Kim

Organizations: Department of Physics, Sejong University, Seoul 05006, Korea

Abstract

Magnetic point-group analysis classifies altermagnets but returns only a binary symmetry verdict, leaving spin-splitting energy (SSE) inaccessible without spin-polarized density functional theory (DFT). This binary ceiling is not fundamental. Sublattice symmetry breaking is promoted here to a continuous, DFT-free scalar -- the Motif Symmetry-Breaking Index (MSBI) -- that quantifies $\mathcal{PT}$ -symmetry breaking between antiparallel magnetic motifs directly from crystal coordinates. SHAP analysis of an XGBoost surrogate trained on 3,851 DFT-labeled binary structures identifies three dominant descriptors: MSBI (symmetry-breaking axis), motif packing fraction MPF (superexchange axis), and the $p/d$ electron ratio (covalency axis), each mapping onto a directly tunable experimental handle. A controlled VO--CrSb comparison within the same P $6_3$ /mmc host lattice demonstrates that composition alone boosts SSE sevenfold. Bayesian optimization over this three-axis space, followed by independent DFT validation, recovers $α$ -NiS (SSE $= 0.823$ ,eV) as cross-validation against an independent symmetry-based prediction and identifies three previously unrecognized high-SSE candidates -- square-planar FeS (1.297,eV), octahedral CoS (1.103,eV), and FeAs (1.089,eV) -- all matching or exceeding CrSb. Square-planar Fe--S is proposed as a transferable coordination motif for giant altermagnetic spin splitting, advancing altermagnet design from symmetry classification to continuous quantitative optimization.

Figures & tables

Figure 1: Overview of the inverse-design workflow for high-SSE altermagnetic materials. (Left) Candidate structures are generated with MatterGen under an energy-above-hull constraint and labeled with DFT-computed SSE, defined as the maximum spin splitting within $[E_{F}-2~\text{eV},\,E_{F}]$ . After binary-altermagnet screening ( $|\boldsymbol{M}|\leq 0.005~\mu_{\text{B}}$ , $\mathrm{SSE}>1~\mathrm{meV}$ ) and strain sampling ( $N=3{,}851$ ), 52 motif-level descriptors are constructed entirely from crystal structure and composition, without any electronic-structure input. An XGBoost surrogate trained on these descriptors predicts SSE, and SHAP analysis identifies MSBI, MPF, and the $p/d$ electron ratio as the three leading contributors. (Right) Descriptor-space Bayesian optimization then maximizes the predicted SSE; top candidates are stratified by coordination number (CN $\in\{4,6\}$ ) and mapped onto periodic structures through prototype matching for independent DFT validation.

	Sc	Ti	V	Cr	Mn	Fe	Co	Ni	Cu	Zn
$U_{\mathrm{eff}}$	3	3	4	4	4	4	3	7	7	7
	Y	Zr	Nb	Mo	Tc	Ru	Rh	Pd	Ag	Cd
$U_{\mathrm{eff}}$	3	3	3	4	–	4	4	4	0	0

Table 1:

U_{\mathrm{eff}}

values (eV) for transition-metal

d

orbitals. A dash indicates an element not included in the present candidate set (Tc).

Figure 2: Three descriptors dominate SSE prediction. (a) Top-12 descriptors ranked by mean absolute SHAP value $\bar{S}_{i}$ (eV), averaged over 100 independently trained XGBoost models. MSBI, MPF, and the $p/d$ ratio together account for 37.4% of total SHAP attribution. Feature labels are shared with panel (b). (b) SHAP beeswarm plot for the same 12 descriptors. Each point represents one structure; horizontal position shows the signed SHAP value (eV). Color indicates normalized feature value (red = high, blue = low). Positive SHAP values indicate that the feature increases predicted SSE.

Feature (Symbol)	Mean $\|\text{SHAP}\|$ (eV)	Ratio $R_{i}$	Cumulative
MSBI	0.1122	0.2320	0.2320
MPF	0.0359	0.0743	0.3063
$p/d$ electron ratio	0.0328	0.0679	0.3742
$\sigma_{\text{inhom}}$	0.0256	0.0529	0.4270
$d_{CC}^{(1)}$	0.0255	0.0526	0.4797
$n_{p,X}$	0.0191	0.0395	0.5192

Table 2: Top-12 descriptors ranked by mean absolute SHAP value

\bar{S}_{i}

(eV). The normalized contribution ratio

R_{i}=\bar{S}_{i}/\sum_{k}\bar{S}_{k}

expresses each descriptor’s share of the model’s total explanatory budget; the cumulative fraction is the running sum of

R_{i}

. Both are computed relative to the total SHAP magnitude (

\sum_{k}\bar{S}_{k}=0.4836

eV). The top three descriptors account for 37.4% of the model’s total attribution.

Figure 3: MSBI controls symmetry breaking and SSE. (a–c) SSE versus (a) MSBI, (b) MPF, and (c) $p/d$ electron ratio, with points colored by mean SHAP value (eV) for the corresponding descriptor. All three panels use logarithmic $x$ -axes. MSBI exhibits a sharp threshold near 0.5, above which SSE systematically exceeds 0.4 eV. CuO and FeSi are annotated in (a) as extreme-MSBI case studies. (d) CuO (MSBI = 1.018, SSE = 1.634 eV): crystal structure with magnetic atoms (red) and ligands (cyan), motif projection showing coordination geometry and spin orientations, and spin-resolved band structure near $E_{F}$ (red = spin-up, blue = spin-down). (e) FeSi (MSBI = 0.0003, SSE = 0.003 eV): crystal structure, motif projection, and band structure. Bands are plotted in black as spin-up and spin-down channels are visually indistinguishable.

Figure 4: Packing and covalency jointly govern SSE magnitude. (a) SSE versus nearest motif-centroid distance $d_{CC}^{(1)}$ , colored by MPF. Large SSE ( $>0.8$ eV) requires high packing (MPF $>0.20$ ) and short inter-motif separations ( $d_{CC}^{(1)}<3.5$ Å). (b) Hybridization spin asymmetry $\Delta H=|H_{\uparrow}-H_{\downarrow}|$ versus maximum spin-resolved $p$ – $d$ hybridization $\max(H_{\uparrow},H_{\downarrow})$ , colored by SSE. Black circles denote $p/d<1$ ; red triangles denote $p/d\geq 1$ . Large SSE concentrates in the upper-right region (strong hybridization and large spin asymmetry, predominantly $p/d<1$ ). VO and CrSb are annotated as case studies. (c) VO ( $p/d=1$ , SSE = 0.165 eV): crystal structure in the P $6_{3}$ /mmc prototype with spin orientations indicated (top), and spin-resolved $p$ – $d$ hybridization fatbands along $L$ – $\Gamma$ – $L^{\prime}$ (bottom), showing weak, spin-symmetric hybridization. (d) CrSb ( $p/d=0.667$ , SSE = 1.194 eV): crystal structure (top) and fatbands (bottom), displaying strong, spin-asymmetric hybridization. In (c) and (d), band opacity scales with $H_{\sigma}$ ; separate colorbars indicate spin-up (red) and spin-down (blue) channels. The band pair for which SSE is evaluated is highlighted with a thicker linewidth.

Coordination ( $N_{X}$ )	Rank	$M$ – $X$ composition	Prototype	Pred. SSE (eV)	DFT SSE (eV)	Cos. sim. $c$	Distance $d$
4	1	Fe–S a	Square-planar CuO	1.061	1.297	0.694	9.17
4	3	Ni–S b	Square-planar CuO	1.042	–	0.555	12.93
4	4	Co–O	Square-planar CuO	1.033	0.972	0.634	10.95
4	5	Cr–S	Square-planar CuO	1.009	0.552	0.662	9.75
6	1	Ni–S	NiAs	0.893	0.823	0.897	20.72
6	2	Co–S	NiAs	0.874	1.103	0.905	24.35

Table 3: Top BO candidates for each coordination class, their nearest structural prototypes, and independent DFT validation results. Structural similarity is quantified by cosine similarity

c

and Euclidean distance

d

in the normalized structural-descriptor space. For reference, the median pairwise Euclidean distance among the 323 parent prototypes in the dataset is

d\approx 7.8

in this space.

Figure 5: DFT validation of BO-selected candidates. (a) Square-planar FeS and (b) NiAs-type NiS. Top panels show the relaxed crystal structures with spin order, and bottom panels show spin-resolved $p$ – $d$ hybridization fatbands along $T_{2}$ – $\Gamma$ – $U_{2}$ for FeS and $L$ – $\Gamma$ – $L^{\prime}$ for NiS. The DFT-relaxed descriptors are FeS (MSBI = 0.871, MPF = 0.533, $p/d$ = 0.571) and NiS (MSBI = 0.681, MPF = 0.333, $p/d$ = 0.445), and both lie in the favorable regime identified by the SHAP analysis. Band opacity scales with $H_{\sigma}$ , and the band pair used for SSE evaluation is highlighted.

Materials Science

Materials science studies how composition, structure, processing, and defects determine material properties and how to design useful materials.

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Predicting magnetism with first-principles AI

Max Geier, Liang Fu

Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Materials Property Prediction Materials Science Quantum Many-Body Physics Computational Materials Science

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Computational discovery of magnetic materials remains challenging because magnetism arises from the competition between kinetic energy and Coulomb interaction that is often beyond the reach of standard electronic-structure methods. Here we tackle this challenge by directly solving the many-electron Schrödinger equation with neural-network variational Monte Carlo, which provides a highly expressive variational wavefunction for strongly correlated systems. Applying this technique to transition metal dichalcogenide moiré semicondutors, we predict itinerant ferromagnetism in WSe

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/WS

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S_z=0

sector, removing the need to compute and compare multiple

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Abstract

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