=Paper=
{{Paper
|id=Vol-1732/paper9
|storemode=property
|title=
Компьютерное
моделирование
модели Изинга с
дальнодействующим
взаимодействием
(Computer Simulation of Ising Model with Long-Range Interaction)
|pdfUrl=https://ceur-ws.org/Vol-1732/paper9.pdf
|volume=Vol-1732
|authors=Sergey Belim,Igor Larionov
}}
==
Компьютерное
моделирование
модели Изинга с
дальнодействующим
взаимодействием
(Computer Simulation of Ising Model with Long-Range Interaction)
==
https://ceur-ws.org/Vol-1732/paper9.pdf
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. Â 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]. Â 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 . Â 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. Â 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 .
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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.