Machine-learning force-field models for dynamical simulations of metallic magnets

Mac hine-learning force-ﬁeld mo dels for dynamical sim ulations of metallic magnets Gia-W ei Chern, 1 , ∗ Y unhao F an, 1 Sheng Zhang, 1 and Puhan Zhang 1 1 Dep artment of Physics, University of Vir ginia, Charlottesvil le, V A 22904, USA W e review recent adv ances in mac hine-learning (ML) force-ﬁeld metho ds for Landau-Lifshitz- Gilb ert (LLG) simulations of itinerant electron magnets, fo cusing on their scalability and transfer- abilit y . Built on the principle of locality , a deep neural-netw ork mo del is dev elop ed to eﬃcien tly and accurately predict electron-mediated forces gov erning spin dynamics. Symmetry-aw are descrip- tors constructed through a group-theoretical approach ensure rigorous incorporation of both lattice and spin-rotation symmetries. The framew ork is demonstrated using the prototypical s-d exchange mo del widely employ ed in spintronics. ML-enabled large-scale simulations reveal no vel nonequilib- rium phenomena, including anomalous coarsening of tetrahedral spin order on the triangular lattice and the freezing of phase-separation dynamics in lightly hole-doped, strong-coupling square-lattice systems. These results establish ML force-ﬁeld frameworks as scalable, accurate, and v ersatile to ols for mo deling nonequilibrium spin dynamics in itineran t magnets. I. INTR ODUCTION Itineran t frustrated magnets host a rich v ariet y of spin and electronic textures stabilized b y electron-mediated in teractions. Among them, magnetic v ortices and skyrmions ha ve attracted particular atten tion for their top ological stabilit y and p oten tial spintronic applica- tions [ 1 – 7 ]. The Berry phases asso ciated with these textures act as emergen t magnetic ﬁelds, giving rise to no vel transport phenomena such as the anomalous Hall eﬀect. Another striking example is the mixed-phase state in colossal magnetoresistance (CMR) manganites, where ferromagnetic metallic and an tiferromagnetic or c harge-ordered insulating regions coexist [ 8 – 16 ]. These nanoscale textures are highly resp onsiv e to external stim- uli—magnetic ﬁelds, pressure, or doping—leading to dra- matic transp ort changes. Mo deling the real-time dynamics of such complex spin textures remains computationally challenging. While classical spins evolv e according to the Landau-Lifshitz- Gilb ert (LLG) equation, the eﬀectiv e ﬁelds driving the dynamics originate from quan tum electron–spin ex- c hange interactions. Direct sim ulations are therefore prohibitiv ely expensive, as each LLG step requires solv- ing a man y-b o dy electronic problem. Mac hine learning (ML) oﬀers a p o w erful alternative. As emphasized b y Kohn [ 17 , 18 ], linear-scaling electronic algorithms hinge on the principle of lo calit y—a concept naturally real- ized in Behler-P arrinello (BP)-t ype ML force-ﬁeld frame- w orks dev elop ed for ab initio molecular dynamics [ 19 – 33 ] and extended to adiabatic lattice dynamics in condensed- matter systems [ 34 – 37 ]. Here, w e present a scalable ML force-ﬁeld framework for the adiabatic dynamics of itinerant magnets, gener- alizing the BP architecture [ 38 – 42 ]. A central feature is the use of symmetry-preserving magnetic descriptors that remain in v arian t under lattice point-group and spin- rotation op erations while b eing diﬀerentiable with re- ∗ gchern@virginia.edu ; Corresp onding author sp ect to spin orientations. W e demonstrate the method for the one-band s-d exchange mo del, widely used in spin tronics. ML-trained force ﬁelds repro duce noncopla- nar tetrahedral spin order and its anomalous coarsen- ing dynamics on the triangular lattice [ 42 ], and reveal a correlation-induced freezing phenomenon in the phase- separated state of a lightly hole-dop ed, strong-coupling square-lattice system [ 38 ]. I I. BEHLER-P ARRINELLO FOR CE-FIELD AR CHITECTURE W e consider a one-band s-d mo del, as a represen tative system for itinerant magnets, describ ed by the following Hamiltonian ˆ H = − X ij,α t ij  ˆ c † iα ˆ c j α + h.c.  − J X i,αβ S i · ˆ c † iα σ αβ ˆ c iβ . (1) Here ˆ c † iα (ˆ c iα ) is the creation (annihilation) op erators of an electron with spin α = ↑ , ↓ at site- i . The ﬁrst term de- scrib es the Hamiltonian for the s -electrons, while J in the second term represents the exchange coupling b et ween lo- cal magnetic momen t S i from the d -electrons and spin of s -electrons. Here the local momen ts are appro ximated b y classical spins with a ﬁxed unit length. The dynamics of the classical local spins is gov erned b y the Landau-Lifshitz-Gilb ert equation d S i dt = γ S i × ( H i + η i ) − γ λ S i × [ S i × ( H i + η i )] . (2) Here, γ denotes the gyromagnetic ratio, λ is a dimen- sionless damping parameter, H i is the local eﬀective ﬁeld arising from the s-d exchange in teraction with itinerant electrons, and η i represen ts the sto c hastic magnetic ﬁeld asso ciated with thermal ﬂuctuations mo deled as a Gaus- sian random pro cess. Within the adiabatic appro ximation–analogous to the Born-Opp enheimer scheme in quan tum MD [ 43 ], the lo- cal exchange force is given by the deriv ative of an eﬀectiv e 2 2 AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E3gnoM5OIxonlAsoTZyWwyZHZ2mekVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYkUBl3328ltbG5t7+R3C3v7B4dHxeOTlolTzXiTxTLWnYAaLoXiTRQoeSfRnEaB5O1gXJ/77SeujYjVI04S7kd0qEQoGEUrPZTZZb9YcivuAmSdeBkpQYZGv/jVG8QsjbhCJqkxXc9N0J9SjYJJPiv0UsMTysZ0yLuWKhpx408Xp87IhVUGJIy1LYVkof6emNLImEkU2M6I4sisenPxP6+bYnjrT4VKUuSKLReFqSQYk/nfZCA0ZygnllCmhb2VsBHVlKFNp2BD8FZfXietasW7rlzdV0u1ehZHHs7gHMrgwQ3U4A4a0AQGQ3iGV3hzpPPivDsfy9ack82cwh84nz+N/o1T (c ) 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AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN6CevAYwTwgWcLsZDYZM49lZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSjgz1ve/vcLK6tr6RnGztLW9s7tX3j9oGpVqQhtEcaXbETaUM0kblllO24mmWESctqLRzdRvPVFtmJIPdpzQUOCBZDEj2Dqp2SVD1rvtlSt+1Z8BLZMgJxXIUe+Vv7p9RVJBpSUcG9MJ/MSGGdaWEU4npW5qaILJCA9ox1GJBTVhNrt2gk6c0kex0q6kRTP190SGhTFjEblOge3QLHpT8T+vk9r4MsyYTFJLJZkvilOOrELT11GfaUosHzuCiWbuVkSGWGNiXUAlF0Kw+PIyaZ5Vg/Pq1f15pXadx1GEIziGUwjgAmpwB3VoAIFHeIZXePOU9+K9ex/z1oKXzxzCH3ifP0gKjvU= b E AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FETxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RRWVtfWN4qbpa3tnd298v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWjm6nfekKleSwfzThBP6IDyUPOqLHSQ9C77ZUrbtWdgSwTLycVyFHvlb+6/ZilEUrDBNW647mJ8TOqDGcCJ6VuqjGhbEQH2LFU0gi1n81OnZATq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwks/4zJJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0SjYEb/HlZdI8q3rn1av780rtOo+jCEdwDKfgwQXU4A7q0AAGA3iGV3hzhPPivDsf89aCk88cwh84nz8MXI2q b F AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FQTxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RRWVtfWN4qbpa3tnd298v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWjm6nfekKleSwfzThBP6IDyUPOqLHSQ9C77ZUrbtWdgSwTLycVyFHvlb+6/ZilEUrDBNW647mJ8TOqDGcCJ6VuqjGhbEQH2LFU0gi1n81OnZATq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwks/4zJJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0SjYEb/HlZdI8q3rn1av780rtOo+jCEdwDKfgwQXU4A7q0AAGA3iGV3hzhPPivDsf89aCk88cwh84nz8N4I2r b G AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FD3qsaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrK6tbxQ3S1vbO7t75f2Dpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YRK81g+mnGCfkQHkoecUWOlh6B32ytX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhJd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXOq1f355XadR5HEY7gGE7BgwuowR3UoQEMBvAMr/DmCOfFeXc+5q0FJ585hD9wPn8AD2SNrA== b H AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLz1WtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju5nffkKleSwfzSRBP6JDyUPOqLHSQ9Cv98sVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14bWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmldVL3L6s39ZaV2m8dRhBM4hXPw4ApqUIcGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8Q6I2t b A AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9VLx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7dRvPaHSPJaPZpygH9GB5CFn1FjpIehd98oVt+rOQJaJl5MK5Kj3yl/dfszSCKVhgmrd8dzE+BlVhjOBk1I31ZhQNqID7FgqaYTaz2anTsiJVfokjJUtachM/T2R0UjrcRTYzoiaoV70puJ/Xic14aWfcZmkBiWbLwpTQUxMpn+TPlfIjBhbQpni9lbChlRRZmw6JRuCt/jyMmmeVb3z6tX9eaV2k8dRhCM4hlPw4AJqcAd1aACDATzDK7w5wnlx3p2PeWvByWcO4Q+czx8GTI2m b B AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG+lXjxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSQ9Cv98sVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14bWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmldVL3L6s39ZaVWz+Mowgmcwjl4cAU1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8H0I2n b C AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj+sxvP6HSPJaPZpKgH9Gh5CFn1FjpIejX++WKW3XnIKvEy0kFcjT65a/eIGZphNIwQbXuem5i/Iwqw5nAaamXakwoG9Mhdi2VNELtZ/NTp+TMKgMSxsqWNGSu/p7IaKT1JApsZ0TNSC97M/E/r5ua8NrPuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuqt5l9eb+slK7zeMowgmcwjl4cAU1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8JVI2o b D AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FPXisaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrK6tbxQ3S1vbO7t75f2Dpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YRK81g+mnGCfkQHkoecUWOlh6B32ytX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhJd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXOq1f355XadR5HEY7gGE7BgwuowR3UoQEMBvAMr/DmCOfFeXc+5q0FJ585hD9wPn8ACtiNqQ== F I G . 1 . S y m m e t ry - i n v a ri a n t d e s c ri p t o r fo r t h e n e i g h b o rh o o d s p i n c o n ﬁg u ra t i o n . ( a ) T h e fo u r b o n d v a ri a b l e s b A ,b B ,b C ,b D , c o n s t ru c t e d fro m t h e i n n e r p ro d u c t o f t h e fo u r n e i g h b o ri n g s p i n s ( w i t h e q u a l d i s t a n c e t o t h e c e n t e r s i t e - i )a n d t h e c e n t e rs p i n s , fo rm t h e b a s i s o f a fo u r- d i m e n s i o n a l re d u c i b l e re p re s e n t a t i o n o f t h e D 4 p o i n t g ro u p . ( b ) S i m i l a rl y , t h e fo u r s c a l a r c h i ra l i t y  A ,  B ,  C ,  D a l s o fo rm t h e b a s i s o f a re d u c i b l e 4 - d i m re p re s e n t a t i o n o f D 4 . H e re  A = S i · S j ⇥ S k and s o on. (c) T he e igh t b o n d v a ri a b l e s b A ,b B , ·· · ,b H a re t h e b a s i s o f a n 8 - d i m e n s i o n a l re d u c i b l e re p re s e n t a t i o n . i l ar l y b e d e c om p os e d i n t o I R s w i t h t h e b as i s : f A 1 =  A +  B +  C +  D f B 1 =  A   B +  C   D f E =(  A   C ,  B   D ) F i n al l y , w e gi v e an e x am p l e of t h e 8- d i m e n s i on al b l o c k , s h o w n i n F i g. 1 ( c ) . T h e c or r e s p on d i n g b as i s ar e t h e 8 b on d v ar i ab l e s { b A ,b B , ·· · ,b H } w h i c h d o n ot i n v ol v e t h e c e n t e r s p i n . T h e d e c om p os i t i on of t h i s 8- d i m e n s i on al r e p - r e s e n t at i on i s 8 = A 1  A 2  B 1  B 2  2 E , an d t h e b as i s of t h e c or r e s p on d i n g I R s ar e [ 4 ] f A 1 = b A + b B + b C + b D + b E + b F + b G + b H f A 2 = b A  b B + b C  b D + b E  b F + b G  b H f B 1 = b A  b B  b C + b D + b E  b F  b G + b H f B 2 = b A + b B  b C  b D + b E + b F  b G  b H f E 1 =( b A  b E ,b C  b G ) f E 2 =( b B  b F ,b D  b H ) B y r e p e at i n g t h e s am e p r o c e d u r e s f or al l i n v ar i an t b l o c k s , on e ob t ai n s al l t h e I R s of t h e b on d /c h i r al i t y v ar i ab l e s i n t h e n e i gh b or h o o d C i . F or c on v e n i e n c e , w e ar r an ge t h e b as i s f u n c t i on s of a gi v e n I R i n t h e d e c om p on s i t i on i n t o a v e c t or f  r =( f  r, 1 ,f  r, 2 , ·· · ,f  r, D  )w h e r e  l ab e l s t h e IR, r e n u m e r at e s t h e m u l t i p l e o c c u r r e n c e of I R  in the d e c om p os i t i on of t h e f - e l e c t r on c on ﬁ gu r at i on , an d D  is t h e d i m e n s i on of t h e I R . G i v e n t h e s e b as i s f u n c t i on s , on e c an i m m e d i at e l y ob t ai n a s e t of i n v ar i an t s c al l e d p o w e r sp ect rum { p  r } , w h i c h ar e t h e am p l i t u d e s of e ac h i n d i - v i d u al I R c o e ﬃ cien t s, i.e. p  r =   f  r   2 . Ho w e v e r , f e at u r e v ar i ab l e s b as e d on l y on p o w e r s p e c t r u m ar e i n c om p l e t e i n t h e s e n s e t h at t h e r e l at i v e p h as e s b e t w e e n d i ↵ eren t IRs ar e i gn or e d . F or e x a m p l e , t h e r e l at i v e “an gl e ” b e t w e e n t w o I R s of t h e s am e t y p e : c os ✓ =( f  r 1 · f  r 2 ) / | f  r 1 || f  r 2 | is al s o an i n v ar i an t of t h e s y m m e t r y gr ou p . W i t h ou t s u c h p h as e i n f or m at i on , t h e NN m o d e l m i gh t s u ↵ e r f r om ad - d i t i on al e r r o r d u e t o t h e s p u r i ou s s y m m e t r y , n am e l y t w o I R s c an f r e e l y r ot at e i n d e p e n d e n t of e ac h ot h e r . As d i s c u s s e d ab o v e , a s y s t e m at i c ap p r oac h t o i n c l u d e al l r e l e v an t i n v ar i an t s , i n c l u d i n g b ot h am p l i t u d e s a n d r e l - at i v e p h as e s , i s t h e b i s p e c t r u m m e t h o d [ 2 , 3 ]. Giv en the co e ﬃ cien t s f  r of al l I R s , t h e b i s p e c t r u m c o e ﬃ c i e n t s ar e gi v e n b y B  ,  1 ,  2 r, r 1 ,r 2 = C  ;  1 ,  2 ↵ ,  ,  f  r, ↵ f  1 r 1 ,  f  2 r 2 ,  ( 4) where C  ;  1 ,  2 ar e t h e C l e b s c h - G or d an c o e ﬃ c i e n t s of t h e p oi n t gr ou p [ 2 ]. T he bisp ectrum co e ﬃ c i e n t s ar e t r i p l e p r o d u c t of c o e ﬃ c i e n t s of d i ↵ e r e n t I R s , s i m i l ar t o t h e s c al ar p r o d u c t a · b ⇥ c of t h r e e v e c t or s . W h e n on e of t h e thre e I R , s a y  , i s t h e t r i v i al i d e n t i t y I R , t h e r e s u l t an t bisp ectrum c o e ﬃ c i e n t i s r e d u c e d t o t h e i n n e r p r o d u c t of t w o I R s of t h e s am e t y p e , i . e . t h e y t r an s f or m i n e x ac t l y t h e s am e w a y u n d e r t h e s y m m e t r y op e r at i on s . T h e r e - s u l t an t i n v ar i an t s ar e s i m i l ar t o t h e i n n e r p r o d u c t a · b of t w o v e c t or s . Ho w e v e r , t h e n u m b e r of b i s p e c t r u m c o e ﬃ c i e n t s i s of - t e n t o o l ar ge f or p r ac t i c al ap p l i c at i on s , an d s om e of t h e m ar e r e d u n d an t . He r e w e h a v e i m p l e m e n t e d a d e s c r i p t or t h at i s m o d i ﬁ e d f r om t h e b i s p e c t r u m m e t h o d [ 1 ]. W e i n t r o d u c e t h e r e f e r e n c e b as i s f u n c t i on s f  re f f or e ac h i r - r e p of t h e p oi n t gr ou p . T h e s e r e f e r e n c e b as i s ar e c om - p u t e d b y a v e r agi n g l ar ge b l o c k s of b on d an d c h i r al - i t y v ar i ab l e s , s u c h t h at t h e y ar e l e s s s e n s i t i v e t o s m al l c h an ge s i n t h e n e i gh b or h o o d s p i n c on ﬁ gu r at i on s . W e t h e n d e ﬁ n e t h e r e l at i v e “p h as e ” of a i r r e p as t h e p r o- j e c t i on of i t s b as i s f u n c t i on s on t o t h e r e f e r e n c e b as i s : ⌘  r ⌘ f  r · f  re f / | f  r || f  re f | .T h e e ↵ e c t i v e c o or d i n at e s ar e t h e n t h e c ol l e c t i on of p o w e r s p e c t r u m c o e ﬃ c i e n t s an d t h e r e l at i v e p h as e s : { G ` } = { p  r , ⌘  r } . T h e v ar i ou s s t e p s of t h e d e s c r i p t or ar e s u m m ar i z e d i n t h e f ol l o w i n g C i ! { b jk ,  jm n } ! { f  r } ! { p  r , ⌘  r } T h e ge n e r al i z e d c o or d i n at e s { G ` } , or f e at u r e v ar i ab l e s c h ar ac t e r i z i n g t h e n e i gh b or h o o d s p i n s , ar e t h e n f or - w ar d e d t o t h e n e u r al n e t w or k w h i c h p r o d u c e s t h e l o c al e n e r gi e s at i t s ou t p u t n o d e . F or t h e c u t o ↵ r ad i u s R c =5 a u s e d i n t h i s w or k , t h e r e i s a t ot al of 539 b on d /c h i r al i t y v ar i ab l e s i n e ac h n e i gh b or h o o d . + AAAB6HicbVDLSgNBEOyNrxhfUY9eBoMgCGFXAupFAl48JmAekCxhdtKbjJmdXWZmhRDyBV48KOLVT/Lm3zhJ9qCJBQ1FVTfdXUEiuDau++3k1tY3Nrfy24Wd3b39g+LhUVPHqWLYYLGIVTugGgWX2DDcCGwnCmkUCGwFo7uZ33pCpXksH8w4QT+iA8lDzqixUv2iVyy5ZXcOskq8jJQgQ61X/Or2Y5ZGKA0TVOuO5ybGn1BlOBM4LXRTjQllIzrAjqWSRqj9yfzQKTmzSp+EsbIlDZmrvycmNNJ6HAW2M6JmqJe9mfif10lNeO1PuExSg5ItFoWpICYms69JnytkRowtoUxxeythQ6ooMzabgg3BW355lTQvy16lfFOvlKq3WRx5OIFTOAcPrqAK91CDBjBAeIZXeHMenRfn3flYtOacbOYY/sD5/AF0MYy3 H i =  @ E @ S i 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AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexKQb2VevFYwdZCu5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YSK4sZ73jQpr6xubW8Xt0s7u3v5B+fCobVSqKWtRJZTuhMQwwSVrWW4F6ySakTgU7CEc38z8hyemDVfy3k4SFsRkKHnEKbFOavfoiPcb/XLFq3pz4FXi56QCOZr98ldvoGgaM2mpIMZ0fS+xQUa05VSwaamXGpYQOiZD1nVUkpiZIJtfO8VnThngSGlX0uK5+nsiI7Exkzh0nTGxI7PszcT/vG5qo6sg4zJJLZN0sShKBbYKz17HA64ZtWLiCKGau1sxHRFNqHUBlVwI/vLLq6R9UfVr1eu7WqXeyOMowgmcwjn4cAl1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwBFAo7z  C AAAB7XicbVBNSwMxEJ3Ur1q/qh69BIvgqexKQb0Ve/FYwdZCu5Rsmm1js8mSZIWy9D948aCIV/+PN/+NabsHbX0w8Hhvhpl5YSK4sZ73jQpr6xubW8Xt0s7u3v5B+fCobVSqKWtRJZTuhMQwwSVrWW4F6ySakTgU7CEcN2b+wxPThit5bycJC2IylDzilFgntXt0xPuNfrniVb058Crxc1KBHM1++as3UDSNmbRUEGO6vpfYICPacirYtNRLDUsIHZMh6zoqScxMkM2vneIzpwxwpLQrafFc/T2RkdiYSRy6zpjYkVn2ZuJ/Xje10VWQcZmklkm6WBSlAluFZ6/jAdeMWjFxhFDN3a2Yjogm1LqASi4Ef/nlVdK+qPq16vVdrVK/yeMowgmcwjn4cAl1uIUmtIDCIzzDK7whhV7QO/pYtBZQPnMMf4A+fwBGho70  D AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKeyKoN6CevAYwTwgWcLsZDYZM49lZlYIS/7BiwdFvPo/3vwbJ8keNLGgoajqprsrSjgz1ve/vcLK6tr6RnGztLW9s7tX3j9oGpVqQhtEcaXbETaUM0kblllO24mmWESctqLRzdRvPVFtmJIPdpzQUOCBZDEj2Dqp2SVD1rvtlSt+1Z8BLZMgJxXIUe+Vv7p9RVJBpSUcG9MJ/MSGGdaWEU4npW5qaILJCA9ox1GJBTVhNrt2gk6c0kex0q6kRTP190SGhTFjEblOge3QLHpT8T+vk9r4MsyYTFJLJZkvilOOrELT11GfaUosHzuCiWbuVkSGWGNiXUAlF0Kw+PIyaZ5Vg/Pq1f15pXadx1GEIziGUwjgAmpwB3VoAIFHeIZXePOU9+K9ex/z1oKXzxzCH3ifP0gKjvU= b E AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FETxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RRWVtfWN4qbpa3tnd298v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWjm6nfekKleSwfzThBP6IDyUPOqLHSQ9C77ZUrbtWdgSwTLycVyFHvlb+6/ZilEUrDBNW647mJ8TOqDGcCJ6VuqjGhbEQH2LFU0gi1n81OnZATq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwks/4zJJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0SjYEb/HlZdI8q3rn1av780rtOo+jCEdwDKfgwQXU4A7q0AAGA3iGV3hzhPPivDsf89aCk88cwh84nz8MXI2q b F AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FQTxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RRWVtfWN4qbpa3tnd298v5BU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWjm6nfekKleSwfzThBP6IDyUPOqLHSQ9C77ZUrbtWdgSwTLycVyFHvlb+6/ZilEUrDBNW647mJ8TOqDGcCJ6VuqjGhbEQH2LFU0gi1n81OnZATq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwks/4zJJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0SjYEb/HlZdI8q3rn1av780rtOo+jCEdwDKfgwQXU4A7q0AAGA3iGV3hzhPPivDsf89aCk88cwh84nz8N4I2r b G AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FD3qsaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrK6tbxQ3S1vbO7t75f2Dpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YRK81g+mnGCfkQHkoecUWOlh6B32ytX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhJd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXOq1f355XadR5HEY7gGE7BgwuowR3UoQEMBvAMr/DmCOfFeXc+5q0FJ585hD9wPn8AD2SNrA== b H AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLz1WtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfju5nffkKleSwfzSRBP6JDyUPOqLHSQ9Cv98sVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14bWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmldVL3L6s39ZaV2m8dRhBM4hXPw4ApqUIcGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8Q6I2t b A AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9VLx4r2g9oQ9lsJ+3SzSbsboQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR7dRvPaHSPJaPZpygH9GB5CFn1FjpIehd98oVt+rOQJaJl5MK5Kj3yl/dfszSCKVhgmrd8dzE+BlVhjOBk1I31ZhQNqID7FgqaYTaz2anTsiJVfokjJUtachM/T2R0UjrcRTYzoiaoV70puJ/Xic14aWfcZmkBiWbLwpTQUxMpn+TPlfIjBhbQpni9lbChlRRZmw6JRuCt/jyMmmeVb3z6tX9eaV2k8dRhCM4hlPw4AJqcAd1aACDATzDK7w5wnlx3p2PeWvByWcO4Q+czx8GTI2m b B AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG+lXjxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj25nffkKleSwfzSRBP6JDyUPOqLHSQ9Cv98sVt+rOQVaJl5MK5Gj0y1+9QczSCKVhgmrd9dzE+BlVhjOB01Iv1ZhQNqZD7FoqaYTaz+anTsmZVQYkjJUtachc/T2R0UjrSRTYzoiakV72ZuJ/Xjc14bWfcZmkBiVbLApTQUxMZn+TAVfIjJhYQpni9lbCRlRRZmw6JRuCt/zyKmldVL3L6s39ZaVWz+Mowgmcwjl4cAU1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8H0I2n b C AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG/FXjxWtB/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RTW1jc2t4rbpZ3dvf2D8uFRS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj+sxvP6HSPJaPZpKgH9Gh5CFn1FjpIejX++WKW3XnIKvEy0kFcjT65a/eIGZphNIwQbXuem5i/Iwqw5nAaamXakwoG9Mhdi2VNELtZ/NTp+TMKgMSxsqWNGSu/p7IaKT1JApsZ0TNSC97M/E/r5ua8NrPuExSg5ItFoWpICYms7/JgCtkRkwsoUxxeythI6ooMzadkg3BW355lbQuqt5l9eb+slK7zeMowgmcwjl4cAU1uIMGNIHBEJ7hFd4c4bw4787HorXg5DPH8AfO5w8JVI2o b D AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FPXisaD+gDWWznbRLN5uwuxFK6E/w4kERr/4ib/4bt20O2vpg4PHeDDPzgkRwbVz32ymsrK6tbxQ3S1vbO7t75f2Dpo5TxbDBYhGrdkA1Ci6xYbgR2E4U0igQ2ApGN1O/9YRK81g+mnGCfkQHkoecUWOlh6B32ytX3Ko7A1kmXk4qkKPeK391+zFLI5SGCap1x3MT42dUGc4ETkrdVGNC2YgOsGOppBFqP5udOiEnVumTMFa2pCEz9fdERiOtx1FgOyNqhnrRm4r/eZ3UhJd+xmWSGpRsvihMBTExmf5N+lwhM2JsCWWK21sJG1JFmbHplGwI3uLLy6R5VvXOq1f355XadR5HEY7gGE7BgwuowR3UoQEMBvAMr/DmCOfFeXc+5q0FJ585hD9wPn8ACtiNqQ== F I G . 1 . S y m m e t ry - i n v a ri a n t d e s c ri p t o r fo r t h e n e i g h b o rh o o d s p i n c o n ﬁg u ra t i o n . ( a ) T h e fo u r b o n d v a ri a b l e s b A ,b B ,b C ,b D , c o n s t ru c t e d fro m t h e i n n e r p ro d u c t o f t h e fo u r n e i g h b o ri n g s p i n s ( w i t h e q u a l d i s t a n c e t o t h e c e n t e r s i t e - i )a n d t h e c e n t e rs p i n s , fo rm t h e b a s i s o f a fo u r- d i m e n s i o n a l re d u c i b l e re p re s e n t a t i o n o f t h e D 4 p o i n t g ro u p . ( b ) S i m i l a rl y , t h e fo u r s c a l a r c h i ra l i t y  A ,  B ,  C ,  D a l s o fo rm t h e b a s i s o f a re d u c i b l e 4 - d i m re p re s e n t a t i o n o f D 4 . H e re  A = S i · S j ⇥ S k and s o on. (c) T he e igh t b o n d v a ri a b l e s b A ,b B , ·· · ,b H a re t h e b a s i s o f a n 8 - d i m e n s i o n a l re d u c i b l e re p re s e n t a t i o n . i l ar l y b e d e c om p os e d i n t o I R s w i t h t h e b as i s : f A 1 =  A +  B +  C +  D f B 1 =  A   B +  C   D f E =(  A   C ,  B   D ) F i n al l y , w e gi v e an e x am p l e of t h e 8- d i m e n s i on al b l o c k , s h o w n i n F i g. 1 ( c ) . T h e c or r e s p on d i n g b as i s ar e t h e 8 b on d v ar i ab l e s { b A ,b B , ·· · ,b H } w h i c h d o n ot i n v ol v e t h e c e n t e r s p i n . T h e d e c om p os i t i on of t h i s 8- d i m e n s i on al r e p - r e s e n t at i on i s 8 = A 1  A 2  B 1  B 2  2 E , an d t h e b as i s of t h e c or r e s p on d i n g I R s ar e [ 4 ] f A 1 = b A + b B + b C + b D + b E + b F + b G + b H f A 2 = b A  b B + b C  b D + b E  b F + b G  b H f B 1 = b A  b B  b C + b D + b E  b F  b G + b H f B 2 = b A + b B  b C  b D + b E + b F  b G  b H f E 1 =( b A  b E ,b C  b G ) f E 2 =( b B  b F ,b D  b H ) B y r e p e at i n g t h e s am e p r o c e d u r e s f or al l i n v ar i an t b l o c k s , on e ob t ai n s al l t h e I R s of t h e b on d /c h i r al i t y v ar i ab l e s i n t h e n e i gh b or h o o d C i . F or c on v e n i e n c e , w e ar r an ge t h e b as i s f u n c t i on s of a gi v e n I R i n t h e d e c om p on s i t i on i n t o a v e c t or f  r =( f  r, 1 ,f  r, 2 , ·· · ,f  r, D  )w h e r e  l ab e l s t h e IR, r e n u m e r at e s t h e m u l t i p l e o c c u r r e n c e of I R  in the d e c om p os i t i on of t h e f - e l e c t r on c on ﬁ gu r at i on , an d D  is t h e d i m e n s i on of t h e I R . G i v e n t h e s e b as i s f u n c t i on s , on e c an i m m e d i at e l y ob t ai n a s e t of i n v ar i an t s c al l e d p o w e r sp ect rum { p  r } , w h i c h ar e t h e am p l i t u d e s of e ac h i n d i - v i d u al I R c o e ﬃ cien t s, i.e. p  r =   f  r   2 . Ho w e v e r , f e at u r e v ar i ab l e s b as e d on l y on p o w e r s p e c t r u m ar e i n c om p l e t e i n t h e s e n s e t h at t h e r e l at i v e p h as e s b e t w e e n d i ↵ eren t IRs ar e i gn or e d . F or e x a m p l e , t h e r e l at i v e “an gl e ” b e t w e e n t w o I R s of t h e s am e t y p e : c os ✓ =( f  r 1 · f  r 2 ) / | f  r 1 || f  r 2 | is al s o an i n v ar i an t of t h e s y m m e t r y gr ou p . W i t h ou t s u c h p h as e i n f or m at i on , t h e NN m o d e l m i gh t s u ↵ e r f r om ad - d i t i on al e r r o r d u e t o t h e s p u r i ou s s y m m e t r y , n am e l y t w o I R s c an f r e e l y r ot at e i n d e p e n d e n t of e ac h ot h e r . As d i s c u s s e d ab o v e , a s y s t e m at i c ap p r oac h t o i n c l u d e al l r e l e v an t i n v ar i an t s , i n c l u d i n g b ot h am p l i t u d e s a n d r e l - at i v e p h as e s , i s t h e b i s p e c t r u m m e t h o d [ 2 , 3 ]. Giv en the co e ﬃ cien t s f  r of al l I R s , t h e b i s p e c t r u m c o e ﬃ c i e n t s ar e gi v e n b y B  ,  1 ,  2 r, r 1 ,r 2 = C  ;  1 ,  2 ↵ ,  ,  f  r, ↵ f  1 r 1 ,  f  2 r 2 ,  ( 4) where C  ;  1 ,  2 ar e t h e C l e b s c h - G or d an c o e ﬃ c i e n t s of t h e p oi n t gr ou p [ 2 ]. T he bisp ectrum co e ﬃ c i e n t s ar e t r i p l e p r o d u c t of c o e ﬃ c i e n t s of d i ↵ e r e n t I R s , s i m i l ar t o t h e s c al ar p r o d u c t a · b ⇥ c of t h r e e v e c t or s . W h e n on e of t h e thre e I R , s a y  , i s t h e t r i v i al i d e n t i t y I R , t h e r e s u l t an t bisp ectrum c o e ﬃ c i e n t i s r e d u c e d t o t h e i n n e r p r o d u c t of t w o I R s of t h e s am e t y p e , i . e . t h e y t r an s f or m i n e x ac t l y t h e s am e w a y u n d e r t h e s y m m e t r y op e r at i on s . T h e r e - s u l t an t i n v ar i an t s ar e s i m i l ar t o t h e i n n e r p r o d u c t a · b of t w o v e c t or s . Ho w e v e r , t h e n u m b e r of b i s p e c t r u m c o e ﬃ c i e n t s i s of - t e n t o o l ar ge f or p r ac t i c al ap p l i c at i on s , an d s om e of t h e m ar e r e d u n d an t . He r e w e h a v e i m p l e m e n t e d a d e s c r i p t or t h at i s m o d i ﬁ e d f r om t h e b i s p e c t r u m m e t h o d [ 1 ]. W e i n t r o d u c e t h e r e f e r e n c e b as i s f u n c t i on s f  re f f or e ac h i r - r e p of t h e p oi n t gr ou p . T h e s e r e f e r e n c e b as i s ar e c om - p u t e d b y a v e r agi n g l ar ge b l o c k s of b on d an d c h i r al - i t y v ar i ab l e s , s u c h t h at t h e y ar e l e s s s e n s i t i v e t o s m al l c h an ge s i n t h e n e i gh b or h o o d s p i n c on ﬁ gu r at i on s . W e t h e n d e ﬁ n e t h e r e l at i v e “p h as e ” of a i r r e p as t h e p r o- j e c t i on of i t s b as i s f u n c t i on s on t o t h e r e f e r e n c e b as i s : ⌘  r ⌘ f  r · f  re f / | f  r || f  re f | .T h e e ↵ e c t i v e c o or d i n at e s ar e t h e n t h e c ol l e c t i on of p o w e r s p e c t r u m c o e ﬃ c i e n t s an d t h e r e l at i v e p h as e s : { G ` } = { p  r , ⌘  r } . T h e v ar i ou s s t e p s of t h e d e s c r i p t or ar e s u m m ar i z e d i n t h e f ol l o w i n g C i ! { b jk ,  jm n } ! { f  r } ! { p  r , ⌘  r } T h e ge n e r al i z e d c o or d i n at e s { G ` } , or f e at u r e v ar i ab l e s c h ar ac t e r i z i n g t h e n e i gh b or h o o d s p i n s , ar e t h e n f or - w ar d e d t o t h e n e u r al n e t w or k w h i c h p r o d u c e s t h e l o c al e n e r gi e s at i t s ou t p u t n o d e . F or t h e c u t o ↵ r ad i u s R c =5 a u s e d i n t h i s w or k , t h e r e i s a t ot al of 539 b on d /c h i r al i t y v ar i ab l e s i n e ac h n e i gh b or h o o d . AAACJnicbVDLSgMxFM3UV62vUZdugkWooGVGpLop1FbBZQX7gLYMmUymDc1MhiQjlKFf48ZfceOiIuLOTzHTVtDqgXAP59zLzT1uxKhUlvVhZJaWV1bXsuu5jc2t7R1zd68peSwwaWDOuGi7SBJGQ9JQVDHSjgRBgctIyx3WUr/1QISkPLxXo4j0AtQPqU8xUlpyzHLX5cyTo0CXxB87N7AMC108oM4VPIVTUjuZ1eq3cH2cKh5X0jHzVtGaAv4l9pzkwRx1x5x0PY7jgIQKMyRlx7Yi1UuQUBQzMs51Y0kihIeoTzqahiggspdMzxzDI6140OdCv1DBqfpzIkGBTC/RnQFSA7nopeJ/XidW/mUvoWEUKxLi2SI/ZlBxmGYGPSoIVmykCcKC6r9CPEACYaWTzekQ7MWT/5LmWdEuFUt35/lKdR5HFhyAQ1AANrgAFXAL6qABMHgEz2ACXo0n48V4M95nrRljPrMPfsH4/AJHtaMs f E =(  A   C ,  B   D ) , ··· FIG. 1. Sc hematic diagram of ML force-ﬁeld mo del for itinerant electron magnets. A descriptor transforms the neighborho o d spin conﬁguration C i to eﬀective coordinates { p Γ r , η Γ r } which are then fed into a neural netw ork (NN). The output no de of the NN corresp onds to the lo cal energy ϵ i = ε ( C i ) asso ciated with spin S i . The corresp onding total p otential energy E is obtained from the summation of these lo cal energies. Automatic diﬀerentiation is emplo yed to compute the deriv ativ es ∂ E /∂ S i from whic h the lo cal exchange ﬁelds H i are obtained. spin-dep enden t energy E = E ( { S i } ): H i = − ∂ E ∂ S i (3) F ormally , this eﬀective energy can b e obtained by in- tegrating out electrons giv en a frozen spin conﬁgura- tion. F or example, in the weak-coupling limit ( J ≪ t ij ), second-order p erturbation theory yields an RKKY-type long-range spin-spin in teraction [ 44 – 46 ]. At in termediate to strong coupling, a systematic expansion is, ho wev er, exceedingly cum b ersome and often impractical. The ef- fectiv e energy can b e computed by numerically in tegrat- ing out electrons on-the-ﬂy , E =  ˆ H ( { S i } )  = T r  ˆ ρ e ˆ H  , (4) with ˆ ρ e = e − β ˆ H / Z the equilibrium density matrix for a ﬁxed spin conﬁguration and Z = T r e − β ˆ H the partition function. Since the Hamiltonian is quadratic in fermionic op erators, exact diagonalization (ED) can be employ ed, but its O ( N 3 ) scaling makes large-scale LLG sim ulations prohibitiv e. More eﬃcient schemes, such as the linear- scaling kernel p olynomial metho d (KPM), are a v ailable, though their implemen tation is in tricate and they cannot treat systems with electron-electron interactions. Mac hine learning (ML) metho ds pro vide a general linear-scaling approac h to computing eﬀective lo cal ﬁelds H i . As p oin ted out b y W. Kohn, such scaling is en- abled b y the lo calit y principle, or the “nearsigh tedness” of many-electron systems [ 17 , 18 ]. This principle im- plies that extensiv e quantities can b e ev aluated through a divide-and-conquer strategy . Indeed, the lo cality princi- ple underlies mo dern ML-based force-ﬁeld metho ds that enable large-scale MD sim ulations with quantum accu- racy [ 19 – 32 ]. A linear-scaling framework exploiting this principle w as pioneered by Behler-Parrinello (BP) [ 19 ] and Bart´ ok et al. [ 20 ]. Here, we extend the BP archi- tecture to predict lo cal exchange ﬁelds in itinerant spin systems. A schematic of the ML force-ﬁeld mo del is sho wn in Fig. 1 . The eﬀective energy E , deﬁned in Eq. ( 4 ), is de- comp osed in to lo cal con tribution ϵ i asso ciated with lat- tice site- i , E = X i ϵ i = X i ε ( C i ) . (5) In the second expression, we in vok e the lo calit y princi- ple, assuming that the site energy ϵ i dep ends only on the local magnetic environmen t, denoted as C i , through a universal function ε ( · ) whic h is determined b y the un- derlying electron Hamiltonian. In practical implemen- tations, the neighborho o d is deﬁned as the collection of spins within a cutoﬀ radius r c around the i -th site: C i = { S j   | r j − r i | < r c } . The complex functional dep en- dence of the lo cal energy on the magnetic environmen t is to b e approximated by a deep-learning neural netw ork. The eﬀective lo cal ﬁeld H i , deﬁned as the deriv ativ e of the total energy [Eq. ( 3 )], can b e eﬃciently computed using automatic diﬀerentiation [ 47 , 48 ]. A k ey component of the ML force-ﬁeld model is the represen tation, or descriptor, of lo cal spin conﬁgura- tions C i . In BP-t ype schemes, the output energy ϵ i is a scalar and m ust remain in v arian t under the system’s symmetry op erations. Thus, descriptors must both dis- tinguish distinct spin conﬁgurations and preserv e the un- derlying symmetries of the electron mo del. Descriptors pla y a similarly central role in BP-t yp e force ﬁelds for quan tum MD, where they are required to respect sym- metry op erations, such as translation, rotation, and re- ﬂection, of the 3D Euclidean group E (3), as well as p er- 3 m utation of identical atoms [ 19 – 21 , 49 – 54 ]. Here we employ group-theoretical metho ds to con- struct magnetic descriptors for the s-d model. In con trast to atomic descriptors in MD systems, where the relev an t symmetry is the full E (3) group, the lattice-based s-d mo del reduces this to the discrete translation group and the on-site p oin t group [ 34 , 55 , 56 ]. In addition, the mo del p ossesses a con tinuous symmetry: the s-d Hamil- tonian is in v arian t under global SO(3)/SU(2) spin rota- tions [ 39 ]. The local energy function ε ( C i ) m ust respect b oth spin-rotation and lattice symmetries. T o this end, w e ﬁrst note that spin-rotation inv ariance is ensured b y expressing ε ( C i ) in terms of b ond v ariables b j k and scalar c hiralities χ j mn , b j k = S j · S k , χ j mn = S j · S m × S n , (6) whic h corresp ond to tw o- and three-spin correlations. F or lattice symmetries, we ﬁrst decomp ose the set of b ond/c hirality v ariables { b j k , χ j mn } around site- i , whic h forms a basis of a high-dimensional representa- tion of the on-site p oin t group in to irreducible represen- tations (IRs). F or instance, consider the four c hiralit y v ariables χ A , χ B , χ C , χ D in Fig. 1 for a square lattice with on-site symmetry group D 4 . The decomp osition is 4 = A 1 + B 1 + E , with corresp onding IR co eﬃcients: f A 1 = χ A + χ B + χ C + χ D , f B 1 = χ A − χ B + χ C − χ D , and f E = ( χ A − χ C , χ B − χ D ). F or each fundamen- tal IR, we collect its comp onent co eﬃcien ts in to a vector f (Γ ,r ) = ( f (Γ ,r ) 1 , · · · , f (Γ ,r ) n Γ ), where Γ denotes the IR, n Γ its dimension, and r the m ultiplicity . A basic class of in v arian ts is the p o wer sp ectrum, p Γ r = | f Γ r | 2 . (7) While the p o wer sp ectrum ignores relative phases b e- t ween IRs, these can b e incorp orated through bisp ec- trum co eﬃcien ts b Γ , Γ 1 , Γ 2 r,r 1 ,r 2 [ 51 , 57 ], constructed from triple pro ducts of IR co eﬃcien ts and Clebsc h-Gordon coe ﬃ- cien ts. Because IRs often hav e large m ultiplicities, bis- p ectrum descriptors can b e n umerous and redundant. T o reduce complexity , w e introduce reference IR co eﬃcients f Γ ref [ 34 , 39 , 55 ], obtained from the IR decomp osition of b ond and c hirality v ariables coarse-grained o ver large blo c ks. These deﬁne a phase v ariable, η Γ r ≡ f Γ r · f Γ ref / | f Γ r || f Γ ref | , (8) allo wing relativ e phases to b e inferred consistently . The ﬁnal feature set th us consists of amplitudes p Γ r , phases η Γ r , and a selected set of bisp ectrum co eﬃcients obtained from the reference IR. The site energy ϵ i , produced by the NN, depends only on these in v arian ts, thereb y preserving the full symmetry of the s-d mo del. As discussed ab o ve, one of the ma jor adv an tages of the proposed scalable ML framew ork is its dramatically enhanced computational eﬃciency–b oth in terms of al- gorithmic complexit y and actual run time. F or instance, AAAB8nicbVBNSwMxEJ2tX7V+VT16CRbBU9kVqR6LXgQRKvQLtkvJptk2NMkuSVYoS3+GFw+KePXXePPfmLZ70NYHA4/3ZpiZFyacaeO6305hbX1jc6u4XdrZ3ds/KB8etXWcKkJbJOax6oZYU84kbRlmOO0mimIRctoJx7czv/NElWaxbJpJQgOBh5JFjGBjJb/Zz3pKoPvGw7RfrrhVdw60SrycVCBHo1/+6g1ikgoqDeFYa99zExNkWBlGOJ2WeqmmCSZjPKS+pRILqoNsfvIUnVllgKJY2ZIGzdXfExkWWk9EaDsFNiO97M3E/zw/NdF1kDGZpIZKslgUpRyZGM3+RwOmKDF8YgkmitlbERlhhYmxKZVsCN7yy6ukfVH1atXa42WlfpPHUYQTOIVz8OAK6nAHDWgBgRie4RXeHOO8OO/Ox6K14OQzx/AHzucPrmSQ4Q== T KPM AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDabTbt2sxt2J0Ip/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemApu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVpyppUCaU7ITFMcMmayFGwTqoZSULB2uHodua3n5g2XMkHHKcsSMhA8phTglZq9SImkPTLFa/qzeGuEj8nFcjR6Je/epGiWcIkUkGM6fpeisGEaORUsGmplxmWEjoiA9a1VJKEmWAyv3bqnlklcmOlbUl05+rviQlJjBknoe1MCA7NsjcT//O6GcbXwYTLNEMm6WJRnAkXlTt73Y24ZhTF2BJCNbe3unRINKFoAyrZEPzll1dJ66Lq16q1+8tK/SaPowgncArn4MMV1OEOGtAECo/wDK/w5ijnxXl3PhatBSefOYY/cD5/AJUejyc=  AAAB8HicbVDLSgNBEOyNrxhfUY9eBoPgKezmED0GvXiMYB6SLGF2MpsMmccyMyuEJV/hxYMiXv0cb/6Nk2QPmljQUFR1090VJZwZ6/vfXmFjc2t7p7hb2ts/ODwqH5+0jUo1oS2iuNLdCBvKmaQtyyyn3URTLCJOO9Hkdu53nqg2TMkHO01oKPBIspgRbJ30OHYb1EhjMShX/Kq/AFonQU4qkKM5KH/1h4qkgkpLODamF/iJDTOsLSOczkr91NAEkwke0Z6jEgtqwmxx8AxdOGWIYqVdSYsW6u+JDAtjpiJynQLbsVn15uJ/Xi+18XWYMZmklkqyXBSnHFmF5t+jIdOUWD51BBPN3K2IjLHGxLqMSi6EYPXlddKuVYN6tX5fqzRu8jiKcAbncAkBXEED7qAJLSAg4Ble4c3T3ov37n0sWwtePnMKf+B9/gAuj5Cr histogram AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCXRxhKjfCRwIXvLHGzY27vs7pkQwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzSRBP6JDyUPOqLHSQ5le9oslt+IuQNaJl5ESZGj0i1+9QczSCKVhgmrd9dzE+FOqDGcCZ4VeqjGhbEyH2LVU0gi1P12cOiMXVhmQMFa2pCEL9ffElEZaT6LAdkbUjPSqNxf/87qpCa/9KZdJalCy5aIwFcTEZP43GXCFzIiJJZQpbm8lbEQVZcamU7AheKsvr5NWteLVKrX7aql+k8WRhzM4hzJ4cAV1uIMGNIHBEJ7hFd4c4bw4787HsjXnZDOn8AfO5w+LnY1T (a) AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCXRxhKjfCRwIXvLHGzY27vs7pkQwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzSRBP6JDyUPOqLHSQzm47BdLbsVdgKwTLyMlyNDoF796g5ilEUrDBNW667mJ8adUGc4Ezgq9VGNC2ZgOsWuppBFqf7o4dUYurDIgYaxsSUMW6u+JKY20nkSB7YyoGelVby7+53VTE177Uy6T1KBky0VhKoiJyfxvMuAKmRETSyhT3N5K2IgqyoxNp2BD8FZfXietasWrVWr31VL9JosjD2dwDmXw4ArqcAcNaAKDITzDK7w5wnlx3p2PZWvOyWZO4Q+czx+NIo1U (b) AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoMQm3CXIloGbSwjmg9IjrC3t5cs2d07dveEcOQn2FgoYusvsvPfuEmu0MQHA4/3ZpiZFyScaeO6305hY3Nre6e4W9rbPzg8Kh+fdHScKkLbJOax6gVYU84kbRtmOO0limIRcNoNJrdzv/tElWaxfDTThPoCjySLGMHGSg/V8HJYrrg1dwG0TrycVCBHa1j+GoQxSQWVhnCsdd9zE+NnWBlGOJ2VBqmmCSYTPKJ9SyUWVPvZ4tQZurBKiKJY2ZIGLdTfExkWWk9FYDsFNmO96s3F/7x+aqJrP2MySQ2VZLkoSjkyMZr/jUKmKDF8agkmitlbERljhYmx6ZRsCN7qy+ukU695jVrjvl5p3uRxFOEMzqEKHlxBE+6gBW0gMIJneIU3hzsvzrvzsWwtOPnMKfyB8/kDkCyNVg== (d) AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCXRxhKjfCRwIXvLHGzY27vs7pkQwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzSRBP6JDyUPOqLHSQ5ld9oslt+IuQNaJl5ESZGj0i1+9QczSCKVhgmrd9dzE+FOqDGcCZ4VeqjGhbEyH2LVU0gi1P12cOiMXVhmQMFa2pCEL9ffElEZaT6LAdkbUjPSqNxf/87qpCa/9KZdJalCy5aIwFcTEZP43GXCFzIiJJZQpbm8lbEQVZcamU7AheKsvr5NWteLVKrX7aql+k8WRhzM4hzJ4cAV1uIMGNIHBEJ7hFd4c4bw4787HsjXnZDOn8AfO5w+Op41V (c) AAAB8XicbVBNSwMxEJ2tX7V+VT16CRbBU9kVaT0WvXhQqNAvbJeSTbNtaJJdkqxQlv4LLx4U8eq/8ea/MW33oK0PBh7vzTAzL4g508Z1v53c2vrG5lZ+u7Czu7d/UDw8aukoUYQ2ScQj1QmwppxJ2jTMcNqJFcUi4LQdjG9mfvuJKs0i2TCTmPoCDyULGcHGSo+NftpTAt3fTfvFklt250CrxMtICTLU+8Wv3iAiiaDSEI617npubPwUK8MIp9NCL9E0xmSMh7RrqcSCaj+dXzxFZ1YZoDBStqRBc/X3RIqF1hMR2E6BzUgvezPxP6+bmPDKT5mME0MlWSwKE45MhGbvowFTlBg+sQQTxeytiIywwsTYkAo2BG/55VXSuih7lXLl4bJUu87iyMMJnMI5eFCFGtxCHZpAQMIzvMKbo50X5935WLTmnGzmGP7A+fwBDkGQiA== T ML FIG. 2. (a) T etrahedral spin order on a triangular lattice [Eq. ( 9 )]. (b) Two inequiv alent spins in a quadrupled unit cell with opp osite chiralit y . (c) Comparison of ML-predicted torque T ML with KPM results T KPM , where T i = S i × H i . (d) Histogram of prediction error δ = T KPM − T ML . a 10,000-time-step simulation of a 50 × 50 system us- ing the exact-diagonalization (ED) metho d requires ap- pro ximately 20 CPU hours. While limited paralleliza- tion on GPU can oﬀer a mo dest improv emen t, the ED calculation based on the LAP ACK pac k age is inher- en tly non-parallelizable. In contrast, the same sim u- lation p erformed with the ML force-ﬁeld mo del takes only ab out 5 minutes, representing nearly a three-order- of-magnitude ( ∼ 1000-fold) sp eedup. This dramatic im- pro vemen t stems from the diﬀerence in computational scaling— O ( N 3 ) for the ED approach v ersus linear O ( N ) scaling for the ML mo del. Even when compared with the KPM, which also exhibits linear scaling and can b e eﬃcien tly implemented on GPU, the ML framework re- mains sup erior. A 500-step sim ulation of a 96 × 96 system tak es ab out 100 min utes using KPM, whereas the same run with the ML mo del requires only around 20 min utes, demonstrating an additional ∼ 5-fold gain in eﬃciency . I II. CHIRAL DOMAIN CO ARSENING OF TETRAHEDRAL SPIN ORDER W e ﬁrst apply the ab o ve ML framew ork to the nearest- neigh b or triangular-lattice s-d mo del at electron ﬁlling fraction f ∼ 1 / 4. Notably , the ground state at this ﬁlling fraction as w ell as f = 3 / 4 is a non-coplanar tetrahedral order with a quadrupled unit cell [ 58 – 62 ]. In this or- der, the lo cal spins point tow ard the corners of a regular tetrahedron; see Fig. 2 (a) and (b). More generally , mag- netic orders with a quadrupled unit cell on the triangular 4 AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCXRxhKjfCRwIXvLHGzY27vs7pkQwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzSRBP6JDyUPOqLHSQ5le9oslt+IuQNaJl5ESZGj0i1+9QczSCKVhgmrd9dzE+FOqDGcCZ4VeqjGhbEyH2LVU0gi1P12cOiMXVhmQMFa2pCEL9ffElEZaT6LAdkbUjPSqNxf/87qpCa/9KZdJalCy5aIwFcTEZP43GXCFzIiJJZQpbm8lbEQVZcamU7AheKsvr5NWteLVKrX7aql+k8WRhzM4hzJ4cAV1uIMGNIHBEJ7hFd4c4bw4787HsjXnZDOn8AfO5w+LnY1T (a) AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4Kolo9SIUvXisYD+gDWWz3bRLN5uwOxFK6I/w4kERr/4eb/4bt20O2vpg4PHeDDPzgkQKg6777aysrq1vbBa2its7u3v7pYPDpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDf1W09cGxGrRxwn3I/oQIlQMIpWaiG5IZeu2yuV3Yo7A1kmXk7KkKPeK311+zFLI66QSWpMx3MT9DOqUTDJJ8VuanhC2YgOeMdSRSNu/Gx27oScWqVPwljbUkhm6u+JjEbGjKPAdkYUh2bRm4r/eZ0Uw2s/EypJkSs2XxSmkmBMpr+TvtCcoRxbQpkW9lbChlRThjahog3BW3x5mTTPK161Un24KNdu8zgKcAwncAYeXEEN7qEODWAwgmd4hTcncV6cd+dj3rri5DNH8AfO5w9l7I5R t = 500 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(a) Snapshots of lo cal scalar c hiralit y χ ij k = S i · S j × S k , with ( ij k ) denoting sites of an elemen tary triangle, at diﬀerent times after a thermal quench of the triangular s–d mo del near n = 1 / 4. (b) Characteristic length L versus time, extracted from the c hirality structure factor (linear scales on b oth axes). Y ellow triangles and green circles sho w data from LLG quenc hes using KPM and ML-calculated ﬁelds, resp ectively . The ML mo del was trained only on KPM–LLG data up to t training = 800, mark ed by the yello w shaded region. lattice can b e describ ed by a triple- Q structure, S i = ∆ 1 e i M 1 · r i + ∆ 2 e i M 2 · r i + ∆ 3 e i M 3 · r i , (9) with ordering vectors M 1 = (2 π /a, 0) and M 2 , 3 = ( − π /a, ± √ 3 π /a ) at the midp oints of the hexagonal Bril- louin zone (BZ) edge, and v ector order parameters ∆ η . The symmetric tetrahedral state corresp onds to ∆ η = ∆ ˆ e η , where ∆ is the amplitude and ˆ e η ( η = 1 , 2 , 3) are three orthogonal unit vectors, yielding spins at the cor- ners of a regular tetrahedron. The tetrahedral order is further c haracterized by a dis- crete Z 2 symmetry associated with the handedness of the spin texture. The scalar chiralit y on a triangular plaque- tte is deﬁned as χ ij k = S i · S j × S k , which tak es the uniform v alue χ ij k = ± 4 S 3 / 3 √ 3 in the p erfect tetra- hedral state. The tw o signs corresp ond to an Ising-like Z 2 symmetry spontaneously brok en at lo w temperatures. The associated electronic state exhibits striking trans- p ort and magneto electric prop erties. In particular, the c hiral spin order pro duces a quantized Hall conductivity σ xy = ± e 2 /h without external magnetic ﬁelds [ 58 , 63 ]. Although long-range tetrahedral order is destroy ed by thermal ﬂuctuations in 2D, the Z 2 c hirality order survives at ﬁnite temp erature. Monte Carlo sim ulations of the tri- angular s-d mo del at n ≈ 1 / 4 [ 60 ] revealed a c hiral phase transition that, despite its Ising-lik e order parameter, is strongly ﬁrst order. This deviation from the exp ected 2D Ising univ ersality suggests that the coarsening dynamics of c hiral domains may also diﬀer from the Allen-Cahn gro wth la w expected for 2D Ising model. T o in vestigate this, w e built a ML force ﬁeld model for the triangular- lattice s-d mo del with Hund’s rule coupling J = 3, and a chemical p oten tial µ = − 3 . 2, corresp onding to a ﬁll- ing fraction n ≈ 1 / 4; here b oth energies are measured in units of the nearest-neighbor hopping t nn . The training dataset w ere obtained from KPM metho d [ 64 – 66 ]. The neural netw ork was implemented in PyT orch [ 67 ] and consists of eigh t hidden lay ers with 2048 × 1024 × 512 × 256 × 128 × 64 × 64 × 64 neurons. The input la yer con tains 1806 no des, corresp onding to the feature v ariables G Γ r , while the single output neuron predicts the lo cal energy . Rectiﬁed Linear Unit (ReLU) acti- v ations [ 68 ] are used b etw een la yers, and the netw ork is optimized using the Adam algorithm [ 69 ]. Since the torque T i = S i × H i is the primary driving force in LLG spin dynamics, the loss function incorporates the mean- square errors (MSE) of b oth the lo cal torques and total energy: L = P N i =1    T i − ˆ T i    2 + η E    E − P N i =1 ˆ ϵ i    2 , where quan tities with hats denote ML predictions and η E con- trols the relativ e weigh t of the energy term. Two mo dels w ere trained: one using only the torque loss ( η E = 0) and another with η E = 0 . 01. The b enchmark of the η E = 0 mo del is shown in Fig. 2 (c,d), the predicted forces agree well with the exact results, ac hieving an MSE of 1 . 64 × 10 − 5 p er spin without signs of o v erﬁtting. Addi- tional dynamical b enchmarks are provided in Ref. [ 42 ]. W e also note that although b oth achiev e similar MSE v al- ues, including the energy term generally makes training more diﬃcult. Since LLG simulations dep end directly on the predicted torques, it is practical to b egin training with the torque-only loss and gradually in troduce the en- ergy term by increasing η E in later ep o c hs. The training of the the ab ov e NN mo dels with 8 hidden lay ers to ok ab out ∼ 5 da ys to reach the rep orted MSE. Next we applied the trained ML mo del to p erform large-scale thermal quenc h simulations of the s-d mo del. An initial state with random spins was suddenly coupled to a thermal bath at low temp erature T = 0 . 01. Fig. 3 (a) sho ws snapshots of the lo cal scalar c hiralit y χ △ ( r ) at suc- cessiv e times following a thermal quench at t = 0, with color indicating the chiralit y magnitude. Initially , the random spin conﬁguration pro duces a disordered c hiral- it y pattern. As relaxation pro ceeds, domains of uniform c hirality emerge and gro w. These domains are separated b y sharp interfaces, only a few lattice constants wide, across whic h the chiralit y v anishes. The characteristic size L ( t ) of chiral domains is ob- tained from the inv erse width ∆ q − 1 of the q = 0 p eak in the ensemble-a v eraged chiralit y structure factor. Fig. 3 (b) compares L ( t ) from LLG simulations using ML- and KPM-calculated ﬁelds, sho wing excellent agreement and conﬁrming that the ML force ﬁeld accurately repro- duces the phase-ordering dynamics. Notably , although the ML mo del was trained only on KPM-LLG data up to t training = 800, it con tinues to predict L ( t ) reliably at 5 m uch longer times, demonstrating strong extrap olation b ey ond the training window. More imp ortan tly , we ﬁnd that the chiral domain size gro ws linearly with time, L ( t ) ∼ L 0 + a ( t − t 0 ), in sharp contrast to the Allen–Cahn law L ∼ √ t ex- p ected for a non-conserved Ising order parameter. Since the Allen–Cahn b ehavior arises from curv ature-driv en domain motion, the observed linear growth likely re- ﬂects the distinctiv e domain morphology . As sho wn in Fig. 3 (a), late-stage c hiral domains exhibit nearly straigh t interfaces aligned with the principal symmetry directions of the triangular lattice. The zero curv ature of these straigh t b oundaries implies v anishing domain-w all motion under the Allen–Cahn mechanism. In this case, domain gro wth is most lik ely gov erned by the dynamics of sharp corners, suggesting the need for a theoretical framew ork of chiral-domain coarsening based on corner motion as p oin t defects. IV. PHASE SEP ARA TION D YNAMICS OF DOUBLE-EX CHANGE MODELS In this section, we apply the ML force-ﬁeld frame- w ork to study phase-separation dynamics in the strong- coupling, or double-exchange [ 70 – 72 ], regime of the s- d mo del. Phase separation phenomena play a cen tral role in the b eha vior and functionality of many strongly correlated electron systems [ 8 – 13 , 73 – 76 ]. A promi- nen t example is the complex inhomogeneous states ob- serv ed in manganites and magnetic semiconductors ex- hibiting the colossal magnetoresistance (CMR) eﬀect [ 8 – 13 ]. These nanoscale textures arise from the segregation of hole-rich ferromagnetic clusters within half-ﬁlled an- tiferromagnetic domains [ 14 – 16 ]. Because the n umber of dop ed carriers is conserved, the segregation dynamics follo w the classic theory of Lifshitz, Slyozo v, and W ag- ner (LSW) [ 77 , 78 ], whic h predicts a coarsening exp onent α = 1 / 3. The square-lattice double-exchange mo del has been ex- tensiv ely studied theoretically [ 79 – 83 ]. In the regime of sligh t hole doping from half-ﬁlling, a mixed-phase state – consisting of hole-rich ferromagnetic puddles embed- ded in a half-ﬁlled an tiferromagnetic insulator – emerges as a stable thermo dynamic phase at strong Hund’s cou- pling [ 80 – 82 ]. Despite extensive researc h on the static prop erties of suc h mixed-phase states in CMR materials, the kinetics of electron-correlation-driv en phase separa- tion remain largely unexplored. F undamental questions, suc h as whether the process exhibits dynamical scaling and whether late-stage domain growth follo ws the LSW p o w er law, are still op en. The sup erior scalability of the ML force-ﬁeld framework pro vides a p ow erful means to address these questions through large-scale simulations of phase separation in this mo del. W e trained a six-lay er neural net w ork (NN) using Py- T orc h [ 67 , 68 ] on 3500 snapshots of spin conﬁgurations and local exchange forces from ED–LLG simulations on - 0. 5 0. 0 0. 5 1. 0 0. 0 0. 5 1. 0 1. 5 2. 0  AAAB7XicbVBNS8NAEJ34WetX1aOXxSJ4KkkV9Fjw4rGC/YA2lM1m0q7dbMLuRiih/8GLB0W8+n+8+W/ctjlo64OBx3szzMwLUsG1cd1vZ219Y3Nru7RT3t3bPzisHB23dZIphi2WiER1A6pRcIktw43AbqqQxoHATjC+nfmdJ1SaJ/LBTFL0YzqUPOKMGiu1+yEKQweVqltz5yCrxCtIFQo0B5WvfpiwLEZpmKBa9zw3NX5OleFM4LTczzSmlI3pEHuWShqj9vP5tVNybpWQRImyJQ2Zq78nchprPYkD2xlTM9LL3kz8z+tlJrrxcy7TzKBki0VRJohJyOx1EnKFzIiJJZQpbm8lbEQVZcYGVLYheMsvr5J2veZd1ur3V9VGo4ijBKdwBhfgwTU04A6a0AIGj/AMr/DmJM6L8+58LFrXnGLmBP7A+fwBkkCPHg== 0 . 5 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(a) ML-predicted exchange ﬁelds versus exact results from the test dataset. (b) The distribution of the force diﬀer- ence δ = H ML − H exact , w ell ﬁtted b y a normal distribution (red line) with v ariance σ 2 = 0 . 035. (c) Spin-spin correlation ⟨ S i · S j ⟩ as a function of distance r ij = | r j − r i | along the x direction at ﬁlling f = 0 . 485. Red dots show LLG results using NN mo dels without Langevin noise, while blue lines cor- resp ond to ED-LLG simulations at T = 0 . 022. a 30 × 30 lattice. Fig. 4 (a) compares NN-predicted ex- c hange ﬁelds H i with exact results, while the deviation δ = H ML − H exact follo ws a Gaussian distribution with v ariance σ 2 = 0 . 035 [Fig. 4 (b)]. The normal distribu- tion of δ suggests that the ML uncertain t y may act as an eﬀective temperature in Langevin dynamics. W e then com bined the trained NN with LLG dynamics to simu- late a thermal quench from random spins at T = 0 . 022 on a 30 × 30 lattice, using iden tical parameters as in the ED reference. The resulting spin–spin correlations, sho wn in Fig. 4 (c) for ﬁlling f = 0 . 485 (1.5% hole doping), agree closely with ED-LLG results, conﬁrming the accuracy and dynamical transferability of the ML force ﬁeld. Large-scale quench simulations were p erformed using the trained ML mo del on a 100 × 100 lattice. Fig- ure 5 (a) shows snapshots of the lo cal spin correlation, deﬁned as the av erage ov er four nearest-neigh b or b onds, b i = ( S i · S i + x + S i · S i − x + S i · S i + y + S i · S i − y ) / 4. Positiv e (negativ e) b i corresp onds to ferromagnetic (FM) [an tifer- romagnetic (AFM)] regions. The ML–LLG sim ulations rev eal relaxation into an inhomogeneous state compris- ing extended AFM domains interspersed with small FM clusters. W e next analyze the kinetics of FM-domain gro wth. A FM cluster is deﬁned as a connected region where all nearest-neighbor bonds exceed b th = 0 . 5. The time ev olution of the c haracteristic length L is shown in Fig. 5 (b). Because the n umber of dop ed holes is con- serv ed, the coarsening of such conserv ed ﬁelds follo ws the 6 AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCXRxhKjfCRwIXvLHGzY27vs7pkQwk+wsdAYW3+Rnf/GBa5Q8CWTvLw3k5l5QSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzSRBP6JDyUPOqLHSQ5le9oslt+IuQNaJl5ESZGj0i1+9QczSCKVhgmrd9dzE+FOqDGcCZ4VeqjGhbEyH2LVU0gi1P12cOiMXVhmQMFa2pCEL9ffElEZaT6LAdkbUjPSqNxf/87qpCa/9KZdJalCy5aIwFcTEZP43GXCFzIiJJZQpbm8lbEQVZcamU7AheKsvr5NWteLVKrX7aql+k8WRhzM4hzJ4cAV1uIMGNIHBEJ7hFd4c4bw4787HsjXnZDOn8AfO5w+LnY1T (a) 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AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2m3bpZhN2J0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemEph0PO+ncLa+sbmVnG7tLO7t39QPjxqmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gpHtzO/9cS1EYl6xHHKg5gOlIgEo2ilB9/1euWK53pzkFXi56QCOeq98le3n7As5gqZpMZ0fC/FYEI1Cib5tNTNDE8pG9EB71iqaMxNMJmfOiVnVumTKNG2FJK5+ntiQmNjxnFoO2OKQ7PszcT/vE6G0XUwESrNkCu2WBRlkmBCZn+TvtCcoRxbQpkW9lbChlRThjadkg3BX355lTSrrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c2Rzovz7nwsWgtOPnMMf+B8/gBUzI0r 0 . 6 AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4CkkV9Vj04rGi/YA2lM120i7dbMLuRiilP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLU8G18bxvZ2V1bX1js7BV3N7Z3dsvHRw2dJIphnWWiES1QqpRcIl1w43AVqqQxqHAZji8nfrNJ1SaJ/LRjFIMYtqXPOKMGis9eO5lt1T2XG8Gskz8nJQhR61b+ur0EpbFKA0TVOu276UmGFNlOBM4KXYyjSllQ9rHtqWSxqiD8ezUCTm1So9EibIlDZmpvyfGNNZ6FIe2M6ZmoBe9qfif185MdB2MuUwzg5LNF0WZICYh079JjytkRowsoUxxeythA6ooMzadog3BX3x5mTQqrn/uVu4vytWbPI4CHMMJnIEPV1CFO6hBHRj04Rle4c0Rzovz7nzMW1ecfOYI/sD5/AFcXo0w 0 . 2 AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2m3bpZhN2J0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemEph0PO+ncLa+sbmVnG7tLO7t39QPjxqmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gpHtzO/9cS1EYl6xHHKg5gOlIgEo2ilB8+t9soVz/XmIKvEz0kFctR75a9uP2FZzBUySY3p+F6KwYRqFEzyaambGZ5SNqID3rFU0ZibYDI/dUrOrNInUaJtKSRz9ffEhMbGjOPQdsYUh2bZm4n/eZ0Mo+tgIlSaIVdssSjKJMGEzP4mfaE5Qzm2hDIt7K2EDammDG06JRuCv/zyKmlWXf/Crd5fVmo3eRxFOIFTOAcfrqAGd1CHBjAYwDO8wpsjnRfn3flYtBacfOYY/sD5/AFWTo0s  0 . 2 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBiyGpgh6LXjxWsB/QhrLZbtqlu5uwuxFK6F/w4kERr/4hb/4bN20O2vpg4PHeDDPzwoQzbTzv2ymtrW9sbpW3Kzu7e/sH1cOjto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5C73O09UaRbLRzNNaCDwSLKIEWxy6cJz64NqzXO9OdAq8QtSgwLNQfWrP4xJKqg0hGOte76XmCDDyjDC6azSTzVNMJngEe1ZKrGgOsjmt87QmVWGKIqVLWnQXP09kWGh9VSEtlNgM9bLXi7+5/VSE90EGZNJaqgki0VRypGJUf44GjJFieFTSzBRzN6KyBgrTIyNp2JD8JdfXiXtuutfuvWHq1rjtoijDCdwCufgwzU04B6a0AICY3iGV3hzhPPivDsfi9aSU8wcwx84nz+/8Y1j  0 . 6 AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4MSRV1GPRi8cK9gPaUDbbTbt0dxN2N0IJ/QtePCji1T/kzX/jps1BWx8MPN6bYWZemHCmjed9Oyura+sbm6Wt8vbO7t5+5eCwpeNUEdokMY9VJ8SaciZp0zDDaSdRFIuQ03Y4vsv99hNVmsXy0UwSGgg8lCxiBJtcOvfcq36l6rneDGiZ+AWpQoFGv/LVG8QkFVQawrHWXd9LTJBhZRjhdFrupZommIzxkHYtlVhQHWSzW6fo1CoDFMXKljRopv6eyLDQeiJC2ymwGelFLxf/87qpiW6CjMkkNVSS+aIo5cjEKH8cDZiixPCJJZgoZm9FZIQVJsbGU7Yh+IsvL5NWzfUv3NrDZbV+W8RRgmM4gTPw4RrqcA8NaAKBETzDK7w5wnlx3p2PeeuKU8wcwR84nz/GAY1n  1 . 0 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBiyGpgh6LXjxWsB/QhrLZbtqlu5uwuxFK6F/w4kERr/4hb/4bN20O2vpg4PHeDDPzwoQzbTzv2ymtrW9sbpW3Kzu7e/sH1cOjto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5C73O09UaRbLRzNNaCDwSLKIEWxy6cJ3vUG15rneHGiV+AWpQYHmoPrVH8YkFVQawrHWPd9LTJBhZRjhdFbpp5ommEzwiPYslVhQHWTzW2fozCpDFMXKljRorv6eyLDQeipC2ymwGetlLxf/83qpiW6CjMkkNVSSxaIo5cjEKH8cDZmixPCpJZgoZm9FZIwVJsbGU7Eh+Msvr5J23fUv3frDVa1xW8RRhhM4hXPw4RoacA9NaAGBMTzDK7w5wnlx3p2PRWvJKWaO4Q+czx++b41i ( d ) t i m e = 100 AAAB+nicbVC7TsMwFHXKq5RXCiOLRYtUliopAyxIFSyMRaIPqYkqx3Faq7YT2Q6oCv0UFgYQYuVL2Pgb3DYDtBzpSkfn3Kt77wkSRpV2nG+rsLa+sblV3C7t7O7tH9jlw46KU4lJG8cslr0AKcKoIG1NNSO9RBLEA0a6wfhm5ncfiFQ0Fvd6khCfo6GgEcVIG2lgl2vhmQc15cSD1SvXcaoDu+LUnTngKnFzUgE5WgP7ywtjnHIiNGZIqb7rJNrPkNQUMzIteakiCcJjNCR9QwXiRPnZ/PQpPDVKCKNYmhIaztXfExniSk14YDo50iO17M3E/7x+qqNLP6MiSTUReLEoShnUMZzlAEMqCdZsYgjCkppbIR4hibA2aZVMCO7yy6uk06i75/XGXaPSvM7jKIJjcAJqwAUXoAluQQu0AQaP4Bm8gjfryXqx3q2PRWvBymeOwB9Ynz/cYZHJ ( b ) t i m e = 20 AAAB+XicbVBNS8NAEN3Ur1q/oh69LLZCvZQkHvQiFL14rGA/oAlls522S3eTsLsplNB/4sWDIl79J978N27bHLT1wcDjvRlm5oUJZ0o7zrdV2Njc2t4p7pb29g8Oj+zjk5aKU0mhSWMey05IFHAWQVMzzaGTSCAi5NAOx/dzvz0BqVgcPelpAoEgw4gNGCXaSD3broaXPtZMgI8rt55T6dllp+YsgNeJm5MyytHo2V9+P6apgEhTTpTquk6ig4xIzSiHWclPFSSEjskQuoZGRIAKssXlM3xhlD4exNJUpPFC/T2REaHUVISmUxA9UqveXPzP66Z6cBNkLEpSDRFdLhqkHOsYz2PAfSaBaj41hFDJzK2YjogkVJuwSiYEd/XlddLyau5VzXv0yvW7PI4iOkPnqIpcdI3q6AE1UBNRNEHP6BW9WZn1Yr1bH8vWgpXPnKI/sD5/AGiUkY4= ( a) t i m e = 0 AAACBHicbZC7SgNBFIZn4y3GW9QyzWAQYhN2o6CNENDCMoK5QDaE2clJMmT2wsxZMSwpbHwVGwtFbH0IO9/GSbKFJv4w8PGfczhzfi+SQqNtf1uZldW19Y3sZm5re2d3L79/0NBhrDjUeShD1fKYBikCqKNACa1IAfM9CU1vdDWtN+9BaREGdziOoOOzQSD6gjM0VjdfKLkID5iwyYlL54jChwm9pHY3X7TL9kx0GZwUiiRVrZv/cnshj30IkEumdduxI+wkTKHgEiY5N9YQMT5iA2gbDJgPupPMjpjQY+P0aD9U5gVIZ+7viYT5Wo99z3T6DId6sTY1/6u1Y+xfdBIRRDFCwOeL+rGkGNJpIrQnFHCUYwOMK2H+SvmQKcbR5JYzITiLJy9Do1J2TsuV27Ni9TqNI0sK5IiUiEPOSZXckBqpE04eyTN5JW/Wk/VivVsf89aMlc4ckj+yPn8As42Xew== ( c ) t i m e = 60 AAACBXicbZC7SgNBFIZnvcZ4i1pqMRiE2ITdKGojBLSwjGAukA1hdnKSDJm9MHNWDMs2Nr6KjYUitr6DnW/j5FJo4g8DH/85hzPn9yIpNNr2t7WwuLS8sppZy65vbG5t53Z2azqMFYcqD2WoGh7TIEUAVRQooREpYL4noe4Nrkb1+j0oLcLgDocRtHzWC0RXcIbGaucOCi7CAyY8PXbpBFH4kNJLema3c3m7aI9F58GZQp5MVWnnvtxOyGMfAuSSad107AhbCVMouIQ068YaIsYHrAdNgwHzQbeS8RUpPTJOh3ZDZV6AdOz+nkiYr/XQ90ynz7CvZ2sj879aM8buRSsRQRQjBHyyqBtLiiEdRUI7QgFHOTTAuBLmr5T3mWIcTXBZE4Ize/I81EpF56RYuj3Nl6+ncWTIPjkkBeKQc1ImN6RCqoSTR/JMXsmb9WS9WO/Wx6R1wZrO7JE/sj5/ADedl70= 0 . 35 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0haRY9FLx4r2FpoQ9lsN+3S3U3Y3Qgl9C948aCIV/+QN/+NmzYHbX0w8Hhvhpl5YcKZNp737ZTW1jc2t8rblZ3dvf2D6uFRR8epIrRNYh6rbog15UzStmGG026iKBYhp4/h5Db3H5+o0iyWD2aa0EDgkWQRI9jkkuc2LgfVmud6c6BV4hekBgVag+pXfxiTVFBpCMda93wvMUGGlWGE01mln2qaYDLBI9qzVGJBdZDNb52hM6sMURQrW9Kgufp7IsNC66kIbafAZqyXvVz8z+ulJroOMiaT1FBJFouilCMTo/xxNGSKEsOnlmCimL0VkTFWmBgbT8WG4C+/vEo6dddvuPX7i1rzpoijDCdwCufgwxU04Q5a0AYCY3iGV3hzhPPivDsfi9aSU8wcwx84nz/Nn41s 0 . 4 AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4Ckkt6LHoxWNF+wFtKJvtpl262YTdiVBKf4IXD4p49Rd589+4bXPQ1gcDj/dmmJkXplIY9LxvZ219Y3Nru7BT3N3bPzgsHR03TZJpxhsskYluh9RwKRRvoEDJ26nmNA4lb4Wj25nfeuLaiEQ94jjlQUwHSkSCUbTSg+dWe6Wy53pzkFXi56QMOeq90le3n7As5gqZpMZ0fC/FYEI1Cib5tNjNDE8pG9EB71iqaMxNMJmfOiXnVumTKNG2FJK5+ntiQmNjxnFoO2OKQ7PszcT/vE6G0XUwESrNkCu2WBRlkmBCZn+TvtCcoRxbQpkW9lbChlRThjadog3BX355lTQrrn/pVu6r5dpNHkcBTuEMLsCHK6jBHdShAQwG8Ayv8OZI58V5dz4WrWtOPnMCf+B8/gBZVo0u 0 . 45 AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4Ckmt6LHoxWMF+wFtKJvtpl26uwm7G6GE/gUvHhTx6h/y5r9x0+agrQ8GHu/NMDMvTDjTxvO+nbX1jc2t7dJOeXdv/+CwcnTc1nGqCG2RmMeqG2JNOZO0ZZjhtJsoikXIaSec3OV+54kqzWL5aKYJDQQeSRYxgk0ueW79alCpeq43B1olfkGqUKA5qHz1hzFJBZWGcKx1z/cSE2RYGUY4nZX7qaYJJhM8oj1LJRZUB9n81hk6t8oQRbGyJQ2aq78nMiy0norQdgpsxnrZy8X/vF5qopsgYzJJDZVksShKOTIxyh9HQ6YoMXxqCSaK2VsRGWOFibHxlG0I/vLLq6Rdc/1Lt/ZQrzZuizhKcApncAE+XEMD7qEJLSAwhmd4hTdHOC/Ou/OxaF1zipkT+APn8wfPJI1t 0 . 5 AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4CklV9Fj04rGi/YA2lM120i7dbMLuRiilP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLU8G18bxvZ2V1bX1js7BV3N7Z3dsvHRw2dJIphnWWiES1QqpRcIl1w43AVqqQxqHAZji8nfrNJ1SaJ/LRjFIMYtqXPOKMGis9eO5lt1T2XG8Gskz8nJQhR61b+ur0EpbFKA0TVOu276UmGFNlOBM4KXYyjSllQ9rHtqWSxqiD8ezUCTm1So9EibIlDZmpvyfGNNZ6FIe2M6ZmoBe9qfif185MdB2MuUwzg5LNF0WZICYh079JjytkRowsoUxxeythA6ooMzadog3BX3x5mTQqrn/uVu4vytWbPI4CHMMJnIEPV1CFO6hBHRj04Rle4c0Rzovz7nzMW1ecfOYI/sD5/AFa2o0v 1 . 0 AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2m3bpZhN2J0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemEph0PO+ncLa+sbmVnG7tLO7t39QPjxqmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gpHtzO/9cS1EYl6xHHKg5gOlIgEo2ilB9/1euWK53pzkFXi56QCOeq98le3n7As5gqZpMZ0fC/FYEI1Cib5tNTNDE8pG9EB71iqaMxNMJmfOiVnVumTKNG2FJK5+ntiQmNjxnFoO2OKQ7PszcT/vE6G0XUwESrNkCu2WBRlkmBCZn+TvtCcoRxbQpkW9lbChlRThjadkg3BX355lTSrrn/hVu8vK7WbPI4inMApnIMPV1CDO6hDAxgM4Ble4c2Rzovz7nwsWgtOPnMMf+B8/gBUzI0r 0 . 6 AAAB6nicbVBNS8NAEJ34WetX1aOXxSJ4CkkV9Vj04rGi/YA2lM120i7dbMLuRiilP8GLB0W8+ou8+W/ctjlo64OBx3szzMwLU8G18bxvZ2V1bX1js7BV3N7Z3dsvHRw2dJIphnWWiES1QqpRcIl1w43AVqqQxqHAZji8nfrNJ1SaJ/LRjFIMYtqXPOKMGis9eO5lt1T2XG8Gskz8nJQhR61b+ur0EpbFKA0TVOu276UmGFNlOBM4KXYyjSllQ9rHtqWSxqiD8ezUCTm1So9EibIlDZmpvyfGNNZ6FIe2M6ZmoBe9qfif185MdB2MuUwzg5LNF0WZICYh079JjytkRowsoUxxeythA6ooMzadog3BX3x5mTQqrn/uVu4vytWbPI4CHMMJnIEPV1CFO6hBHRj04Rle4c0Rzovz7nzMW1ecfOYI/sD5/AFcXo0w 0 . 2 AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0iqoMeiF48V7Qe0oWy2m3bpZhN2J0Ip/QlePCji1V/kzX/jts1BWx8MPN6bYWZemEph0PO+ncLa+sbmVnG7tLO7t39QPjxqmiTTjDdYIhPdDqnhUijeQIGSt1PNaRxK3gpHtzO/9cS1EYl6xHHKg5gOlIgEo2ilB8+t9soVz/XmIKvEz0kFctR75a9uP2FZzBUySY3p+F6KwYRqFEzyaambGZ5SNqID3rFU0ZibYDI/dUrOrNInUaJtKSRz9ffEhMbGjOPQdsYUh2bZm4n/eZ0Mo+tgIlSaIVdssSjKJMGEzP4mfaE5Qzm2hDIt7K2EDammDG06JRuCv/zyKmlWXf/Crd5fVmo3eRxFOIFTOAcfrqAGd1CHBjAYwDO8wpsjnRfn3flYtBacfOYY/sD5/AFWTo0s  0 . 2 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBiyGpgh6LXjxWsB/QhrLZbtqlu5uwuxFK6F/w4kERr/4hb/4bN20O2vpg4PHeDDPzwoQzbTzv2ymtrW9sbpW3Kzu7e/sH1cOjto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5C73O09UaRbLRzNNaCDwSLKIEWxy6cJz64NqzXO9OdAq8QtSgwLNQfWrP4xJKqg0hGOte76XmCDDyjDC6azSTzVNMJngEe1ZKrGgOsjmt87QmVWGKIqVLWnQXP09kWGh9VSEtlNgM9bLXi7+5/VSE90EGZNJaqgki0VRypGJUf44GjJFieFTSzBRzN6KyBgrTIyNp2JD8JdfXiXtuutfuvWHq1rjtoijDCdwCufgwzU04B6a0AICY3iGV3hzhPPivDsfi9aSU8wcwx84nz+/8Y1j  0 . 6 AAAB63icbVBNS8NAEJ34WetX1aOXxSJ4MSRV1GPRi8cK9gPaUDbbTbt0dxN2N0IJ/QtePCji1T/kzX/jps1BWx8MPN6bYWZemHCmjed9Oyura+sbm6Wt8vbO7t5+5eCwpeNUEdokMY9VJ8SaciZp0zDDaSdRFIuQ03Y4vsv99hNVmsXy0UwSGgg8lCxiBJtcOvfcq36l6rneDGiZ+AWpQoFGv/LVG8QkFVQawrHWXd9LTJBhZRjhdFrupZommIzxkHYtlVhQHWSzW6fo1CoDFMXKljRopv6eyLDQeiJC2ymwGelFLxf/87qpiW6CjMkkNVSS+aIo5cjEKH8cDZiixPCJJZgoZm9FZIQVJsbGU7Yh+IsvL5NWzfUv3NrDZbV+W8RRgmM4gTPw4RrqcA8NaAKBETzDK7w5wnlx3p2PeeuKU8wcwR84nz/GAY1n  1 . 0 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBiyGpgh6LXjxWsB/QhrLZbtqlu5uwuxFK6F/w4kERr/4hb/4bN20O2vpg4PHeDDPzwoQzbTzv2ymtrW9sbpW3Kzu7e/sH1cOjto5TRWiLxDxW3RBrypmkLcMMp91EUSxCTjvh5C73O09UaRbLRzNNaCDwSLKIEWxy6cJ3vUG15rneHGiV+AWpQYHmoPrVH8YkFVQawrHWPd9LTJBhZRjhdFbpp5ommEzwiPYslVhQHWTzW2fozCpDFMXKljRorv6eyLDQeipC2ymwGetlLxf/83qpiW6CjMkkNVSSxaIo5cjEKH8cDZmixPCpJZgoZm9FZIwVJsbGU7Eh+Msvr5J23fUv3frDVa1xW8RRhhM4hXPw4RoacA9NaAGBMTzDK7w5wnlx3p2PRWvJKWaO4Q+czx++b41i ( d ) t i m e = 100 AAAB+nicbVC7TsMwFHXKq5RXCiOLRYtUliopAyxIFSyMRaIPqYkqx3Faq7YT2Q6oCv0UFgYQYuVL2Pgb3DYDtBzpSkfn3Kt77wkSRpV2nG+rsLa+sblV3C7t7O7tH9jlw46KU4lJG8cslr0AKcKoIG1NNSO9RBLEA0a6wfhm5ncfiFQ0Fvd6khCfo6GgEcVIG2lgl2vhmQc15cSD1SvXcaoDu+LUnTngKnFzUgE5WgP7ywtjnHIiNGZIqb7rJNrPkNQUMzIteakiCcJjNCR9QwXiRPnZ/PQpPDVKCKNYmhIaztXfExniSk14YDo50iO17M3E/7x+qqNLP6MiSTUReLEoShnUMZzlAEMqCdZsYgjCkppbIR4hibA2aZVMCO7yy6uk06i75/XGXaPSvM7jKIJjcAJqwAUXoAluQQu0AQaP4Bm8gjfryXqx3q2PRWvBymeOwB9Ynz/cYZHJ ( b ) t i m e = 20 AAAB+XicbVBNS8NAEN3Ur1q/oh69LLZCvZQkHvQiFL14rGA/oAlls522S3eTsLsplNB/4sWDIl79J978N27bHLT1wcDjvRlm5oUJZ0o7zrdV2Njc2t4p7pb29g8Oj+zjk5aKU0mhSWMey05IFHAWQVMzzaGTSCAi5NAOx/dzvz0BqVgcPelpAoEgw4gNGCXaSD3broaXPtZMgI8rt55T6dllp+YsgNeJm5MyytHo2V9+P6apgEhTTpTquk6ig4xIzSiHWclPFSSEjskQuoZGRIAKssXlM3xhlD4exNJUpPFC/T2REaHUVISmUxA9UqveXPzP66Z6cBNkLEpSDRFdLhqkHOsYz2PAfSaBaj41hFDJzK2YjogkVJuwSiYEd/XlddLyau5VzXv0yvW7PI4iOkPnqIpcdI3q6AE1UBNRNEHP6BW9WZn1Yr1bH8vWgpXPnKI/sD5/AGiUkY4= AAAB63icbVBNSwMxEJ2tX7V+VT16CRbBU9kt0noRil48VrAf0C4lm2bb0CS7JFmhLP0LXjwo4tU/5M1/Y7rdg7Y+GHi8N8PMvCDmTBvX/XYKG5tb2zvF3dLe/sHhUfn4pKOjRBHaJhGPVC/AmnImadsww2kvVhSLgNNuML1b+N0nqjSL5KOZxdQXeCxZyAg2mXRTc4flilt1M6B14uWkAjlaw/LXYBSRRFBpCMda9z03Nn6KlWGE03lpkGgaYzLFY9q3VGJBtZ9mt87RhVVGKIyULWlQpv6eSLHQeiYC2ymwmehVbyH+5/UTE177KZNxYqgky0VhwpGJ0OJxNGKKEsNnlmCimL0VkQlWmBgbT8mG4K2+vE46tapXr9YfrirN2zyOIpzBOVyCBw1owj20oA0EJvAMr/DmCOfFeXc+lq0FJ585hT9wPn8ARY+NwA== t = 20 AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEoheh6MVjBfsBbSib7aZdursJuxuhhP4FLx4U8eof8ua/cZvmoK0PBh7vzTAzL0w408Z1v53S2vrG5lZ5u7Kzu7d/UD08aus4VYS2SMxj1Q2xppxJ2jLMcNpNFMUi5LQTTu7mfueJKs1i+WimCQ0EHkkWMYJNLt347qBac+tuDrRKvILUoEBzUP3qD2OSCioN4VjrnucmJsiwMoxwOqv0U00TTCZ4RHuWSiyoDrL81hk6s8oQRbGyJQ3K1d8TGRZaT0VoOwU2Y73szcX/vF5qousgYzJJDZVksShKOTIxmj+OhkxRYvjUEkwUs7ciMsYKE2PjqdgQvOWXV0n7ou75df/hsta4LeIowwmcwjl4cAUNuIcmtIDAGJ7hFd4c4bw4787HorXkFDPH8AfO5w9Lo43E t = 60 AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqheh6MVjBdMW2lA22027dLMJuxOhlP4GLx4U8eoP8ua/cdvmoK0PBh7vzTAzL0ylMOi6305hbX1jc6u4XdrZ3ds/KB8eNU2SacZ9lshEt0NquBSK+yhQ8naqOY1DyVvh6G7mt564NiJRjzhOeRDTgRKRYBSt5OON57q9csWtunOQVeLlpAI5Gr3yV7efsCzmCpmkxnQ8N8VgQjUKJvm01M0MTykb0QHvWKpozE0wmR87JWdW6ZMo0bYUkrn6e2JCY2PGcWg7Y4pDs+zNxP+8TobRdTARKs2QK7ZYFGWSYEJmn5O+0JyhHFtCmRb2VsKGVFOGNp+SDcFbfnmVNC+qXq1ae7is1G/zOIpwAqdwDh5cQR3uoQE+MBDwDK/w5ijnxXl3PhatBSefOYY/cD5/ALLGjfk= t = 100 AAAB73icbVBNS8NAEJ34WetX1aOXxSJ4KolI9Vj04rGC/YA2lM12067dbOLuRCihf8KLB0W8+ne8+W/ctjlo64OBx3szzMwLEikMuu63s7K6tr6xWdgqbu/s7u2XDg6bJk414w0Wy1i3A2q4FIo3UKDk7URzGgWSt4LRzdRvPXFtRKzucZxwP6IDJULBKFqpHfQy8TAhpFcquxV3BrJMvJyUIUe9V/rq9mOWRlwhk9SYjucm6GdUo2CST4rd1PCEshEd8I6likbc+Nns3gk5tUqfhLG2pZDM1N8TGY2MGUeB7YwoDs2iNxX/8zophld+JlSSIldsvihMJcGYTJ8nfaE5Qzm2hDIt7K2EDammDG1ERRuCt/jyMmmeV7xqpXp3Ua5d53EU4BhO4Aw8uIQa3EIdGsBAwjO8wpvz6Lw4787HvHXFyWeO4A+czx9/F4+h b ij FIG. 5. (a) Density maps of lo cal b ond v ariables b ( r i ) at four diﬀeren t times during the ML–LLG simulation on a 100 × 100 lattice with 1 . 5% hole doping. (b) Av erage linear size L = ⟨ s ⟩ 1 / 2 of FM clusters v ersus time after a thermal quenc h. The dash-dotted line denotes the t 1 / 3 p o w er-law growth, while the dashed line sho ws a sublogarithmic dep endence L ( t ) ∼ (log t ) β with β = 0 . 11. The inset plots the time evolution of the av erage cluster size ⟨ s ⟩ . Cahn–Hilliard equation [ 84 , 85 ] or LSW theory [ 77 , 78 ], predicting L ( t ) ∼ t 1 / 3 . How ever, as sho wn in Fig. 5 (b), this scaling holds only at early times; at later stages, the domain size L gro ws muc h more slo wly than predicted b y LSW theory . The LSW theory describ es diﬀusive interactions among minorit y-phase domains, where larger clusters gro w b y absorbing material from smaller ones via an ev apora- tion–condensation mec hanism. In our case, this corre- sp onds to the migration of dop ed holes from small to large FM clusters. The early-stage t 1 / 3 scaling lik ely reﬂects this pro cess. As AFM correlations develop in the half-ﬁlled bac kground, how ev er, eac h dop ed hole b e- comes surrounded b y parallel spins through the double- exc hange mechanism, forming a self-trapp ed FM cloud. This lo calization suppresses hole ev ap oration and halts the coarsening pro cess, leading to a breakdown of the LSW scenario. V. CONCLUSION AND OUTLOOK In this pap er, we hav e presen ted a scalable ML force- ﬁeld framework for LLG simulations of itineran t electron magnets. Based on the principle of lo calit y , the metho d expresses eﬀective spin-dependent forces as functions of lo cal spin environmen ts using group-theoretical descrip- tors that rigorously preserv e b oth lattice and spin sym- metries. Analogous to the BP approach for ab initio MD sim ulations, the neural-net w ork mo del predicts local ex- c hange ﬁelds as energy gradients, ac hieving linear-scaling eﬃciency and high transferability across spin conﬁgu- rations. Applications to the protot ypical s–d exch ange mo del demonstrate the versatilit y of this framework and rev eal nonequilibrium dynamical phenomena b ey ond the reac h of conv en tional metho ds. While this study focuses on the BP-type form ulation, other ML arc hitectures provide complementary and of- ten more ﬂexible approaches. Conv olutional neural net- w orks (CNNs), with their intrinsic lo cal connectivity , naturally implement the principle of lo calit y and en- able scalable ML force ﬁelds for spin systems [ 86 , 87 ]. Recen t adv ances further in tro duce elegant wa ys to em- b ed b oth symmetry and scalabilit y: equiv arian t neural net works enforce symmetry constraints directly within the architecture [ 88 – 90 ], while graph neural netw orks (GNNs) ac hiev e linear scaling via localized message pass- ing and symmetry-preserving transformations of equiv a- len t graph elemen ts [ 91 – 94 ]. Extending these adv anced arc hitectures to mo del spin dynamics in itineran t mag- nets represen ts a promising av en ue for future work. Data a v ailability The data that supp ort the ﬁndings of this study are op enly av ailable in https://github.com/cherngroupUVA/ML double exchange where one can ﬁnd C and Python co des, trained mo del samples and data samples to successfully run the mac hine learning spin dynamics and re produce results discussed in the main text. A CKNOWLEDGMENTS This work was supp orted b y the US Department of Energy Basic Energy Sciences under Contract No. DE- SC0020330. 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Machine-learning force-field models for dynamical simulations of metallic magnets

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