Êîìïüþòåðíîå ìîäåëèðîâàíèå ìîäåëè Èçèíãà ñ äàëüíîäåéñòâóþùèì âçàèìîäåéñòâèåì Ñ.Â. Áåëèì È.Á. Ëàðèîíîâ belimsv@omsu.ru me@g0gi.ch Îìñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò èì. Ô.Ì. Äîñòîåâñêîãî, Îìñê, Ðîññèÿ Àííîòàöèÿ  ñòàòüå ìîäåëèðóåòñÿ êðèòè÷åñêîå ïîâåäåíèå òðåõìåðíîé ìîäåëè Èçèíãà ñ äàëüíîäåéñòâóþùèì âçàèìîäåéñòâèåì. Ðàññìîòðåí ñëó- ÷àé äàëüíîäåéñòâóþùèõ ñèë, óáûâàþùèõ ïî ñòåïåííîìó çàêîíó. Âû÷èñëåíû çàâèñèìîñòè êðèòè÷åñêîé òåìïåðàòóðû è êðèòè÷åñêèõ èíäåêñîâ îò ïàðàìåòðîâ äàëüíîäåéñòâèÿ. Êðèòè÷åñêàÿ òåìïåðàòóðà ïîä÷èíÿåòñÿ ëèíåéíîìó çàêîíó. Ïîêàçàíî, ÷òî äàëüíîäåéñòâóþùèå ýôôåêòû äîìèíèðóþò äëÿ áîëüøèõ ñèñòåì. Ââåäåíèå Ïðè èññëåäîâàíèè êëàññè÷åñêîé òðåõìåðíîé ìîäåëè Èçèíãà ó÷èòûâàåòñÿ âçàèìîäåéñòâèå òîëüêî ìåæäó áëèæàéøèìè ñîñåäÿìè, ïðè ýòîì îáìåííûé èíòåãðàë ñ÷èòàåòñÿ ïîñòîÿííîé âåëè÷èíîé. Äàííîå ïðèáëè- æåíèå îïðàâäàíî â ñèëó òîãî, ÷òî â ðåàëüíûõ ôåððîìàãíèòíûõ ñèñòåìàõ îáìåííûé èíòåãðàë óáûâàåò ïî ýêñïîíåíöèàëüíîìó çàêîíó. Îäíàêî â ðÿäå âåùåñòâ âáëèçè ëèíèè ôàçîâîãî ïåðåõîäà ýêñïåðèìåíòàëüíî îáíàðóæåíî ïîâåäåíèå òåðìîäèíàìè÷åñêèõ ôóíêöèé, ñóùåñòâåííî îòêëîíÿþùååñÿ îò ïðåäñêàçûâàåìîãî íà îñíîâå ìîäåëè Èçèíãà. Ýòî îòêëîíåíèå ìîæåò áûòü îáúÿñíåíî áîëåå ìåäëåííûì óáûâàíèåì îáìåííîãî èíòåãðàëà ñ ðàññòîÿíèåì [1, 2, 3, 4, 5]. Ïðè ýòîì íåîáõîäèìî ó÷èòûâàòü íå òîëüêî âçàèìîäåéñòâèå ìåæäó áëèæàéøèìè ñïèíàìè, íî è ñî ñëåäóþùèìè çà áëèæàéøèìè [2, 6]. Âçàèìîäåéñòâèå, ñâÿçàííîå ñ ìåäëåííûì óáûâàíèåì îáìåííîãî èíòåãðàëà ñ ðàññòîÿíèåì, ïðèíÿòî íàçûâàòü äàëüíîäåéñòâóþùèì.  ðàìêàõ ìîäåëè Èçèíãà äàëüíîäåéñòâóþùåå âçàèìîäåéñòâèå ìîæåò áûòü àïïðîêñèìèðîâàíî ñòåïåííîé ôóíêöèåé [7]: 1 J(r) ∼ , rD+σ ãäå D  ðàçìåðíîñòü ñèñòåìû, σ  ïàðàìåòð äàëüíîäåéñòâèÿ.  ñòàòüå [2] èññëåäîâàíî ìàãíèòíîå êðèòè÷åñêîå ïîâåäåíèå EuO. Ýêñïåðèìåíòàëüíî ïîëó÷åíû êðèòè- ÷åñêèå èíäåêñû γ = 1.29 ± 0.01 è β = 0.368 ± 0.005. Òàêæå â äàííîé ñòàòüå âû÷èñëåíî îòíîøåíèå îáìåí- íîãî èíòåãðàëà ñîñåäåé, ñëåäóþùèõ çà áëèæàéøèìè J2 , ê îáìåííîìó èíòåãðàëó áëèæàéøèõ ñîñåäåé J1 : J2 = (0.5 ± 0.2)J1 .  ñòàòüå [1] èçìåðåíû êðèòè÷åñêèå èíäåêñû ôåððîìàãíèòíîãî ôàçîâîãî ïåðåõîäà äëÿ ñïëàâà La0.5 Sr0.5 CoO3 (γ = 1.351±0.009, β = 0.321±0.002). Àíàëîãè÷íûå ýêñïåðèìåíòû äëÿ La1−x Srx CoO3 (0.2 ≤ x ≤ 0.3) [3] ïðèâåëè ê çíà÷åíèÿì êðèòè÷åñêèõ èíäåêñîâ 0.43 ≤ β ≤ 0.46, 1.39 ≤ γ ≤ 1.43. Ýô- ôåêòû äàëüíîäåéñòâèÿ òàêæå ýêñïåðèìåíòàëüíî áûëè îáíàðóæåíû â La0.1 Ba0.9 V S3 [4] è F e90−x M nx Zr10 (0 ≤ x ≤ 16) [5].  îáîèõ ñëó÷àÿõ êðèòè÷åñêèå èíäåêñû èìåþò èìåþò áëèçêèå çíà÷åíèÿ γ = 1.366, β = 0.501. Copyright c by the paper's authors. Copying permitted for private and academic purposes. In: Sergey V. Belim, Nadezda F. Bogachenko (eds.): Proceedings of the Workshop on Data Analysis and Modelling (DAM 2016), Omsk, Russia, October 2016, published at http://ceur-ws.org Êðèòè÷åñêîå ïîâåäåíèå òðåõìåðíîé ìîäåëè Èçèíãà ñ ýôôåêòàìè äàëüíîäåéñòâèÿ áûëî èññëåäîâàíî ìå- òîäàìè ðåíîðì-ãðóïïû â ðàìêàõ ε-ðàçëîæåíèÿ [7, 8, 9]. Òàêæå áûëî îñóùåñòâëåíî êîìïüþòåðíîå ìîäåëè- ðîâàíèå îäíîìåðíûõ è äâóìåðíûõ ñèñòåì [10, 11, 12]. Âî âñåõ ýòèõ ðàáîòàõ áûëî ïîêàçàíî, ÷òî ýôôåêòû äàëüíîäåéñòâèÿ îêàçûâàþò ñóùåñòâåííîå âëèÿíèå ïðè çíà÷åíèÿõ ïàðàìåòðà σ < 2. Ìîäåëü Èçèíãà ñ ýôôåêòàìè äàëüíîäåéñòâèÿ èññëåäîâàíà íåïîñðåäñòâåííî â òðåõìåðíîì ïðîñòðàíñòâå â ðàìêàõ òåîðåòèêî-ïîëåâîãî ïîäõîäà â ñòàòüÿõ [13, 14, 15, 16, 17].  ðàáîòàõ [18, 19] îñóùåñòâëåíî êîì- ïüþòåðíîå ìîäåëèðîâàíèå äâóìåðíîé ìîäåëè Èçèíãà ñ ýôôåêòàìè äàëüíîäåéñòâèÿ. Àâòîðû äåëàþò âûâîä îá ýêâèâàëåíòíîñòè êðèòè÷åñêîãî ïîâåäåíèÿ ðàññìàòðèâàåìîé ñèñòåìû êëàññè÷åñêîé ìîäåëè Èçèíãà ðàç- ìåðíîñòè d = 4 + D − 2σ .  ñòàòüå [20] ïðîâåäåíî êîìïüþòåðíîå ìîäåëèðîâàíèå äâóìåðíîé ìîäåëè Èçèíãà ñî ñëó÷àéíûìè äàëüíîäåéñòâóþùèìè ñâÿçÿìè. Öåëüþ äàííîé ðàáîòû ñòàâèòñÿ êîìïüþòåðíîå ìîäåëèðîâàíèå êðèòè÷åñêîãî ïîâåäåíèÿ âáëèçè ëèíèè ôàçîâîãî ïåðåõîäà âòîðîãî ðîäà òðåõìåðíîé ìîäåëè Èçèíãà ñ äàëüíîäåéñòâóþùèì âçàèìîäåéñòâèåì. 1 Îïèñàíèå ñèñòåìû Äëÿ èññëåäîâàíèÿ ìîäåëè Èçèíãà ñ ýôôåêòàìè äàëüíîäåéñòâèÿ çàïèøåì åå ãàìèëüòîíèàí: X bJ X H=J Si Sj + D+σ Si Sj n r nn Çäåñü Si  çíà÷åíèå ñïèíà (+1/2 èëè -1/2), J  çíà÷åíèå îáìåííîãî èíòåãðàëà, b  îòíîñèòåëüíàÿ èíòåí- ñèâíîñòü äàëüíîäåéñòâèÿ, σ  ïàðàìåòð äàëüíîäåéñòâèÿ, õàðàêòåðèçóþùèé áûñòðîòó óáûâàíèÿ ýíåðãèè âçàèìîäåéñòâèÿ ñ ðàññòîÿíèåì, D  ðàçìåðíîñòü ñèñòåìû, â äàëüíåéøåì D = 3.  ïåðâîì ñëàãàåìîì ñóì- ìèðîâàíèå îñóùåñòâëÿåòñÿ òîëüêî ïî áëèæàéøèì ñîñåäÿì (n), âî âòîðîì ñëàãàåìîì, êðîìå áëèæàéøèé ñîñåäåé, ó÷èòûâàþòñÿ òàêæå ñïèíû, ðàñïîëîæåííûå âíóòðè ñôåðû ðàäèóñîì 2a (nn), ãäå a  ïîñòîÿííàÿ ðåøåòêè. Êîìïüþòåðíîå ìîäåëèðîâàíèå îñóùåñòâëÿëîñü c ïîìîùüþ àëãîðèòìà Ìåòðîïîëèñà. Ðàññìàòðèâàëèñü ñèñòåìû ñ ïðîñòîé êóáè÷åñêîé ðåøåòêîé ðàçìåðîì L × L × L. Íàêëàäûâàëèñü ñòàíäàðòíûå ïåðèîäè÷åñêèå ãðàíè÷íûå óñëîâèÿ. Äëÿ îïðåäåëåíèÿ òåìïåðàòóðû ôàçîâîãî ïåðåõîäà èñïîëüçîâàëèñü êóìóëÿíòû Áèíäåðà ÷åòâåðòîãî ïî- ðÿäêà [21]: hm4 i UL = 1 − . (3hm2 i2 ) Óãëîâûå ñêîáêè èñïîëüçîâàíû äëÿ îáîçíà÷åíèÿ ñðåäíåé âåëè÷èíû ïî ðàçëè÷íûì êîíôèãóðàöèÿì ñèñòåìû, m  ìàãíèòíûé ìîìåíò ñèñòåìû. Ñîãëàñíî òåîðèè êîíå÷íî ðàçìåðíîãî ñêåéëèíãà [17] âñå êóììóëÿíòû ñèñòåì ñ ðàçëè÷íûìè ëèíåéíûìè ðàçìåðàìè L ïåðåñåêàþòñÿ â îäíîé òî÷êå, ñîîòâåòñòâóþùåé êðèòè÷åñêîé òåìïåðàòóðå Tc . Íàìàãíè÷åííîñòü ñèñòåìû ìîæåò áûòü îïðåäåëåíà êàê ìàãíèòíûé ìîìåíò, ïðèõîäÿùèéñÿ íà îäèí ñïèí ñèñòåìû: hmi M= , N ãäå N = L3  êîëè÷åñòâî ñïèíîâ. Äëÿ îïðåäåëåíèÿ âîñïðèèì÷èâîñòè ñèñòåìû èñïîëüçîâàëîñü ñîîòíîøåíèå: χ = N K(hm2 i − hmi2 ), ãäå K = |J|/kB T , N = L3  ÷èñëî óçëîâ, m  íàìàãíè÷åííîñòü ñèñòåìû, óãëîâûå ñêîáêè èñïîëüçîâàíû äëÿ îáîçíà÷åíèÿ óñðåäíåíèÿ ïî ðàçëè÷íûì êîíôèãóðàöèÿì. Ñîãëàñíî òåîðèè êîíå÷íî ðàçìåðíîãî ñêåéëèíãà ïîâåäåíèå âîñïðèèì÷èâîñòü âáëèçè êðèòè÷åñêîé òåìïå- ðàòóðû óäîâëåòâîðÿþò ñîîòíîøåíèþ: Èç ýòîãî ñîîòíîøåíèÿ ìîæåò áûòü îïðåäåëåíî îòíîøåíèå êðèòè÷åñêèõ èíäåêñîâ γ/ν . Êðèòè÷åñêèé èí- äåêñ ν ìîæåò áûòü âû÷èñëåí èç ñîîòíîøåíèÿ: dU4 ∼ L−1/ν . dT Îñòàëüíûå êðèòè÷åñêèå èíäåêñû âû÷èñëÿþòñÿ èç ñêåéëèíãîâûõ ñîîòíîøåíèé: γ ν η = σ − , β = (D − σ + η), α = σ − Dν. ν 2 2 Ðåçóëüòàòû êîìïüþòåðíîãî ìîäåëèðîâàíèÿ Êîìïüþòåðíûé ìîäåëèðîâàíèå êðèòè÷åñêîãî ïîâåäåíèÿ îñóùåñòâëÿëîñü äëÿ ñèñòåì ñ ëèíåéíûìè ðàçìå- ðàìè îò L = 25 äî L = 50 ñ øàãîì L=5. Êîëè÷åñòâî øàãîâ Ìîíòå-Êàðëî íà ñïèí 3 · 105 . Äëÿ óòî÷íåíèÿ ðåçóëüòàòîâ â ðÿäå ñëó÷àåâ ìîäåëèðîâàëèñü ñèñòåìû ðàçìåðîì äî L = 90. Äëÿ èññëåäîâàíèÿ ñâîéñòâ ìî- äåëè â âû÷èñëèòåëüíîì ýêñïåðèìåíòå âàðüèðîâàëîñü äâà ïàðàìåòðà  b è σ . Ïàðàìåòð σ èçìåíÿëñÿ â èíòåðâàëå îò 1.5 äî 2.0 ñ øàãîì 0.1. Ïàðàìåòð b ïðèíèìàë çíà÷åíèÿ îò 0.1 äî 0.9 ñ øàãîì 0.1. Âû÷èñëåíèÿ âûïîëíÿëèñü íà ãðàôè÷åñêîì ïðîöåññîðå ñ èñïîëüçîâàíèåì òåõíîëîãèè CUDA. Ïåðâûé íà- áîð èòåðàöèé îñóùåñòâëÿëñÿ íà îáû÷íîì ïðîöåññîðå äëÿ ïðèâåäåíèÿ ñèñòåìû â ðàâíîâåñíîå ñîñòîÿíèå. Ïîñëå ýòîãî ñèñòåìà ðàçáèâàëàñü íà áëîêè, êàæäûé èç êîòîðûõ îáðàáàòûâàëñÿ â ñâîåì ïîòîêå íà ãðà- ôè÷åñêîì ïðîöåññîðå. Îáùåå êîëè÷åñòâî ïîòîêîâ äîñòèãàëî 445. Äàííûå îò âñåõ ïîòîêîâ ñîáèðàëèñü è îáðàáàòûâàëèñü ñîâìåñòíî äëÿ ïîëó÷åíèÿ ñòàòèñòè÷åñêèõ çàêîíîìåðíîñòåé. Êîìïüþòåðíûé ýêñïåðèìåíò ïîêàçàë, ÷òî äàëüíîäåéñòâóþùèå ñèëû îêàçûâàþò ñóùåñòâåííîå âëèÿíèå íà êðèòè÷åñêóþ òåìïåðàòóðó ôàçîâîãî ïåðåõîäà. Çàâèñèìîñòü êðèòè÷åñêîé òåìïåðàòóðû îò îòíîñèòåëüíîé èíòåíñèâíîñòè äàëüíîäåéñòâóþùåãî âçàèìîäåéñòâèÿ b ïðåäñòàâëåíà íà ðèñóíêå 1. Ðèñ. 1: Ãðàôèêè çàâèñèìîñòè êðèòè÷åñêîé òåìïåðàòóðû îò ïàðàìåòðà èíòåíñèâíîñòè ýôôåêòîâ äàëüíî- äåéñòâèÿ b ïðè ðàçëè÷íûõ çíà÷åíèÿõ ïàðàìåòðà σ . Ëèíåéíàÿ çàâèñèìîñòü êðèòè÷åñêîé òåìïåðàòóðû îò ïàðàìåòðà b ïðè âñåõ çíà÷åíèÿõ σ ìîæåò áûòü îáú- ÿñíåíà ðîñòîì ýíåðãèè íåîáõîäèìîé äëÿ ïåðåâîðîòà îäíîãî ñïèíà çà ñ÷åò âëèÿíèÿ ñîñåäåé, ñëåäóþùèõ çà áëèæàéøèìè. Èç ãðàôèêîâ òàêæå âèäíî, ÷òî çíà÷åíèÿ êðèòè÷åñêîé òåìïåðàòóðû óáûâàþò ñ ðîñòîì σ ïðè îäíîì è òîì æå b. Äàííûé ýôôåêò îáúÿñíÿåòñÿ, ÷òî ñ óâåëè÷åíèåì σ äàëüíîäåéñòâóþùåå âçàèìîäåéñòâèå áûñòðåå óáûâàåò ñ ðàññòîÿíèåì è âëèÿíèå ñïèíîâ, ñëåäóþùèõ çà áëèæàéøèìè, ñòàíîâèòñÿ ìåíüøå. Íà ðèñóíêå 2 ïðåäñòàâëåíà çàâèñèìîñòü òàíãåíñà óãëà íàêëîíà t ãðàôèêîâ êðèòè÷åñêèõ òåìïåðàòóð. Ãðàôèê, ïðåäñòàâëåííûé íà ðèñóíêå 2 ìîæåò áûòü ñ âûñîêîé òî÷íîñòüþ àïïðîêñèìèðîâàí ñîîòíîøåíè- åì: t = −(1.29 ± 0.02)σ + (9.93 ± 0.04). Ñëåäîâàòåëüíî çàâèñèìîñòü êðèòè÷åñêîé òåìïåðàòóðû îò ïàðàìåòðîâ äàëüíîäåéñòâèÿ èìååò ñëåäóþùèé âèä: Tc (b, σ) = (−(1.29 ± 0.02)σ + (9.93 ± 0.04))b + (4.52 ± 0.01). Ñëåäóåò îòìåòèòü, ÷òî çà ïðåäåëàìè èññëåäóåìûõ èíòåðâàëîâ b è σ âîçìîæíû îòêëîíåíèÿ îò ïîëó÷åííîé çàâèñèìîñòè. 3 Çàêëþ÷åíèå Òàêèì îáðàçîì äàëüíîäåéñòâóþùèå ñèëû, óáûâàþùèå ïî ñòåïåííîìó çàêîíó, ñóùåñòâåííî âëèÿþò íà òåì- ïåðàòóðó ôàçîâîãî ïåðåõîäà. Çíà÷åíèå êðèòè÷åñêîé òåìïåðàòóðû îò îáîèõ ïàðàìåòðîâ äàëüíîäåéñòâèÿ íîñèò ëèíåéíûé õàðàêòåð. Ïðè÷åì ïàðàìåòð b îáåñïå÷èâàåò ëèíåéíûé ðîñò êðèòè÷åñêîé òåìïåðàòóðû, à ïàðàìåòð σ ïðèâîäèò ê ëèíåéíîìó óáûâàíèþ òåìïåðàòóðû. Ðèñ. 2: Ãðàôèê çàâèñèìîñòè òàíãåíñà óãëà íàêëîíà ïðÿìîé ðîñòà êðèòè÷åñêîé òåìïåðàòóðû t îò ïàðàìåòðà äàëüíîäåéñòâèÿ σ . Ñïèñîê ëèòåðàòóðû [1] S. Mukherjee, P. Raychaudhuri, A.K. Nigman Critical behavior in La0.5 Sr0.5 CoO3 . Phys. Rev. B., 61:86518653, 2000. [2] N. Menyuk, K. Dwight, T.B. Reed Critical Magnetic Properties and Exchange Interactions in EuO. Phys.Rev. B., 3:16891698, 1971. [3] J. Mira, J. Rivas, M. Vazquez, J.M. Garcia-Beneytez, J. Arcas, R.D. Sanchez, M.A. Senaris-Rodriguez Critical exponents of the ferromagnetic-paramagnetic phase transition of La1−x Srx CoO3 (0.20 < x < 0.30). Phys.Rev. B., 59:123126, 1999. [4] R. Cabassi, F. Bolzoni, A. Gauzzi, F. Licci Critical exponents and amplitudes of the ferromagnetic transition in La0.1 Ba0.9 V S3 . Phys. Rev. B., 74:184425184430, 2006. [5] A. Perumal, V. Srinivas Critical behavior of weak itinerant ferromagnet F e90−x M nx Zr10 (0 < x < 16) alloys. Phys. Rev. B., 67:094418094423, 2003. [6] J.C. Le Guillou, J. Zinn-Justin Critical exponents from eld theory. Phys. Rev. B., 21:39763998, 1980. [7] M.E. Fisher, S.-K. Ma, B.G. Nickel Critical Exponents for Long-Range Interactions. Phys. Rev. Lett., 29:917920, 1972. [8] J. Honkonen Critical behaviour of the long-range (phi2 )2 model in the short-range limit. J. Phys. A., 23(5):825831, 1990. [9] E. Luijten, H. Mebingfeld Criticality in One Dimension with Inverse Square-Law Potentials. Phys. Rev. Lett., 86:53055308, 2001. [10] E. Bayong, H.T. Diep Eect of long-range intaraction on the critical behavior of the continuous Ising model. Phys. Rev. B., 59:1191911924, 1999. [11] E. Luijten Test of renormalization predictions for universal nite-size scaling functions. Phys. Rev. E. 60:75587561, 1999. [12] E. Luijten, H.W.J. Blote Classical critical behavior of spin models with long-range interactions. Phys. Rev. B., 56:8945-–8958, 1997. [13] S.V. Belim Inuence of long-range eects on the critical behavior of three-dimensional systems. Jetp Lett., 77:112114, 2003. [14] S.V. Belim Eect of long-range interactions on the critical behavior of three-dimensional disordered systems. Jetp Lett., 77: 434437, 2003. [15] S.V. Belim Critical dynamics of three-dimensional spin systems with long-range interactions. J. Exp. Theor. Phys. 98:745749, 2004. [16] S.V. Belim Eect of elastic deformations on the critical behavior of disordered systems with long-range interactions. J. Exp. Theor. Phys., 98:316321, 2004. [17] S.V. Belim Eect of long-range interactions on the multicritical behavior of homogeneous systems. J. Exp. Theor. Phys., 98:338441, 2004. [18] T. Blanchard, M. Picco, M.A. Rajabpour Inuence of long-range interactions on the critical behavior of the Ising model. Europhysics Letters, 101(5):56003(5), 2013. [19] M. Picco Critical behavior of the Ising model with long range interactions. arXiv:1207.1018v1 . [20] X. Zhang, M.A. Novotny Critical behavior of ising models with random long-range (small-world) interactions. Braz. J. Phys., 36(3):664671, 2006. [21] Binder K. Critical Properties from Monte-Carlo Coarse-Graining and Renormalization. Phys. Rev. Lett., 47:693696, 1981. Computer Simulation of Ising Model with Long-Range Interaction Sergey V. Belim, Igor B. Larionov Critical behavior of 3D Ising model with long-range interaction is simulated. The power law case for long-range forces is considered. Dependences of critical temperature and critical exponents from long-range parameters are calculated. Critical temperature changes under the linear law. Long-range eects dominate for big systems.