Êîìïüþòåðíîå ìîäåëèðîâàíèå ìîäåëè Èçèíãà ñ äàëüíîäåéñòâóþùèì âçàèìîäåéñòâèåì
Ñ.Â. Áåëèì
belimsv@omsu.ru È.Á. Ëàðèîíîâ 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]:
J(r)∼ 1 rD+σ, ãäåD ðàçìåðíîñòü ñèñòåìû,σ ïàðàìåòð äàëüíîäåéñòâèÿ.
 ñòàòüå [2] èññëåäîâàíî ìàãíèòíîå êðèòè÷åñêîå ïîâåäåíèå EuO. Ýêñïåðèìåíòàëüíî ïîëó÷åíû êðèòè-
÷åñêèå èíäåêñû γ = 1.29±0.01è 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 ñïëàâàLa0.5Sr0.5CoO3(γ= 1.351±0.009,β = 0.321±0.002). Àíàëîãè÷íûå ýêñïåðèìåíòû äëÿLa1−xSrxCoO3
(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 âLa0.1Ba0.9V S3 [4] èF e90−xM nxZr10
(0≤x≤16) [5].  îáîèõ ñëó÷àÿõ êðèòè÷åñêèå èíäåêñû èìåþò èìåþò áëèçêèå çíà÷åíèÿγ= 1.366,β= 0.501.
Copyright cby 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 ðàìêàõε-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 ïàðàìåòðàσ <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- ìåðíîñòèd= 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:
H =JX
n
SiSj+ bJ rD+σ
X
nn
SiSj
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 äàëüíåéøåìD= 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), ãäåa ïîñòîÿííàÿ ðåøåòêè.
Êîìïüþòåðíîå ìîäåëèðîâàíèå îñóùåñòâëÿëîñü c ïîìîùüþ àëãîðèòìà Ìåòðîïîëèñà. Ðàññìàòðèâàëèñü ñèñòåìû ñ ïðîñòîé êóáè÷åñêîé ðåøåòêîé ðàçìåðîìL×L×L. Íàêëàäûâàëèñü ñòàíäàðòíûå ïåðèîäè÷åñêèå ãðàíè÷íûå óñëîâèÿ.
Äëÿ îïðåäåëåíèÿ òåìïåðàòóðû ôàçîâîãî ïåðåõîäà èñïîëüçîâàëèñü êóìóëÿíòû Áèíäåðà ÷åòâåðòîãî ïî- ðÿäêà [21]:
UL= 1− hm4i (3hm2i2).
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 ðàçìåðàìèLïåðåñåêàþòñÿ â îäíîé òî÷êå, ñîîòâåòñòâóþùåé êðèòè÷åñêîé òåìïåðàòóðåTc.
Íàìàãíè÷åííîñòü ñèñòåìû ìîæåò áûòü îïðåäåëåíà êàê ìàãíèòíûé ìîìåíò, ïðèõîäÿùèéñÿ íà îäèí ñïèí ñèñòåìû:
M =hmi N , ãäåN =L3 êîëè÷åñòâî ñïèíîâ.
Äëÿ îïðåäåëåíèÿ âîñïðèèì÷èâîñòè ñèñòåìû èñïîëüçîâàëîñü ñîîòíîøåíèå:
χ=N K(hm2i − hmi2),
ãäåK=|J|/kBT,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
dT ∼L−1/ν.
Îñòàëüíûå êðèòè÷åñêèå èíäåêñû âû÷èñëÿþòñÿ èç ñêåéëèíãîâûõ ñîîòíîøåíèé:
η=σ−γ
ν, β=ν
2(D−σ+η), α=σ−Dν.
2 Ðåçóëüòàòû êîìïüþòåðíîãî ìîäåëèðîâàíèÿ
Êîìïüþòåðíûé ìîäåëèðîâàíèå êðèòè÷åñêîãî ïîâåäåíèÿ îñóùåñòâëÿëîñü äëÿ ñèñòåì ñ ëèíåéíûìè ðàçìå- ðàìè îòL = 25äîL = 50ñ øàãîì L=5. Êîëè÷åñòâî øàãîâ Ìîíòå-Êàðëî íà ñïèí3·105. 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. Ïàðàìåòðb 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 âçàèìîäåéñòâèÿb ïðåäñòàâëåíà íà ðèñóíêå 1.
Ðèñ. 1: Ãðàôèêè çàâèñèìîñòè êðèòè÷åñêîé òåìïåðàòóðû îò ïàðàìåòðà èíòåíñèâíîñòè ýôôåêòîâ äàëüíî- äåéñòâèÿb ïðè ðàçëè÷íûõ çíà÷åíèÿõ ïàðàìåòðàσ.
Ëèíåéíàÿ çàâèñèìîñòü êðèòè÷åñêîé òåìïåðàòóðû îò ïàðàìåòðàbïðè âñåõ çíà÷åíèÿõσìîæåò 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 ðîñòîìσïðè îäíîì è òîì æåb. Äàííûé ýôôåêò îáúÿñíÿåòñÿ, ÷òî ñ óâåëè÷åíèåìσäàëüíîäåéñòâóþùåå âçàèìîäåéñòâèå áûñòðåå óáûâàåò ñ ðàññòîÿíèåì è âëèÿíèå ñïèíîâ, ñëåäóþùèõ çà áëèæàéøèìè, ñòàíîâèòñÿ ìåíüøå. Íà ðèñóíêå 2 ïðåäñòàâëåíà çàâèñèìîñòü òàíãåíñà óãëà íàêëîíàt 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 èíòåðâàëîâbè σâîçìîæíû 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 ïàðàìåòðb îáåñïå÷èâàåò ëèíåéíûé ðîñò êðèòè÷åñêîé òåìïåðàòóðû, à ïàðàìåòðσïðèâîäèò ê ëèíåéíîìó óáûâàíèþ òåìïåðàòóðû.
Ðèñ. 2: Ãðàôèê çàâèñèìîñòè òàíãåíñà óãëà íàêëîíà ïðÿìîé ðîñòà êðèòè÷åñêîé òåìïåðàòóðûtîò ïàðàìåòðà äàëüíîäåéñòâèÿσ.
Ñïèñîê ëèòåðàòóðû
[1] S. Mukherjee, P. Raychaudhuri, A.K. Nigman Critical behavior in La0.5Sr0.5CoO3. Phys. Rev. B., 61:86518653, 2000.
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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.
