D:\sbornik\...\Hakob2012.DVI


Mathematical Problems of Computer Science 36, 17{27, 2012.

Classical Spin Glasses with Consider ation of

Relaxation E ®ects

A s h o t S . Ge vo r kya n a n d H a ko b G. A b a jya n

Institute for Informatics and Automation Problems of NAS of RA

e-mail g ashot@sci.am, habajyan@ipia.sci.am

Abstract

The complex-classical short-range interaction Hamiltonian is used for the ¯rst time
for solving spin glasses with consideration of relaxation e®ects. A system of recurrent
equations is obtained on the nodes of the 1D lattice. An e±cient mathematical al-
gorithm is developed on the basis of these equations with consideration of extended
Sylvester conditions which allows node-by-node construct a huge number of stable
spin chains in parallel. As a result of the simulation, distribution functions of di®er-
ent parameters of a spin glass are constructed from the ¯rst principles of complex-
classical mechanics. Also, the critical properties of spin glass such as catastrophes in
the Clausius-Mossotti equation are studied depending on the external ¯eld. It is shown
that the developed approach excludes these catastrophes, which allows to organize con-
tinuous parallel computation based on the whole-range values of the external ¯eld. A
new representation of the partition function is suggested which, opposite to the usual
de¯nition, is a complex function with the derivatives de¯ned everywhere, including at
critical points.

Refer ences

[1 ] K . B in d e r , A . P . Y o u n g , Spin glasses: E xperimental facts, theoretical concepts, and
open questions, R e vie ws o f Mo d e r n P h ys ic s , vo l. 5 8 , n o . 4 , p p . 8 0 1 -9 7 6 , 1 9 8 6 .

[2 ] M. M¶e z a r d , G. P a r is i, M. A . V ir a s o r o , Spin Glass Theory and B eyond, W o r ld S c ie n t i¯ c ,
vo l. 9 , 1 9 8 7 .

[3 ] A . P . Y o u n g ( e d .) , Spin Glasses and R andom F ields, W o r ld S c ie n t ī c , 1 9 9 8 .

[4 ] S . F. E d wa r d s , P . W . A n d e r s o n , Theory of spin glasses, Jo u r n a l o f P h ys ic s F, vo l. 9 ,
p . 9 6 5 , 1 9 7 5 .

[5 ] R . Fis c h , A . B . H a r r is , Spin-glass model in continuous dimensionality, P h ys ic a l R e vie w
L e t t e r s , vo l. 4 7 , N o . 8 , p . 6 2 0 , 1 9 8 1 .

[6 ] A . B o vie r , Statistical M echanics of D isordered Systems: A M athematical P erspective,
Ca m b r id g e S e r ie s in S t a t is t ic a l a n d P r o b a b ilis t ic Ma t h e m a t ic s , n o . 1 8 , p . 3 0 8 , 2 0 0 6 .

[7 ] Y . Tu , J. Te r s o ®, G. Gr in s t e in , P roperties of a Continuous-R andom-Network M odel
for Amorphous Systems, P h ys ic a l R e vie w L e t t e r s , vo l. 8 1 , n o . 2 2 , p p . 4 8 9 9 -4 9 0 2 , 1 9 9 8 .

[8 ] K . V . R . Ch a r y, G. Go vil, NM R in B iological Systems: F rom M olecules to Human,
S p r in g e r , vo l. 6 , p . 5 1 1 , 2 0 0 8 .

1 7



1 8 Classical Spin Glasses with Consideration of Relaxation E®ects

[9 ] E . B a a ke , M. B a a ke , H . W a g n e r , Ising Quantum Chain is a E quivalent to a M odel of
B iological E volution, P h ys ic a l R e vie w L e t t e r s , vo l. 7 8 , n o . 3 , p p . 5 5 9 -5 6 2 , 1 9 9 7 .

[1 0 ] D . S h e r r in g t o n , S . K ir kp a t r ic k, A Solvable M odel of a Spin-Glass, P h ys ic a l R e vie w
L e t t e r s , vo l. 3 5 . p . 1 9 7 2 .

[1 1 ] B . D e r r id a , R andom-E nergy M odel: An E xactly Solvable M odel of D isordered Systems,
P h ys ic a l R e vie w B , vo l. 2 4 , p p . 2 6 1 3 -2 6 2 6 , 1 9 8 1 .

[1 2 ] G. P a r is i, In¯nite Number of Order P arameters for Spin-Glasses, P h ys ic a l R e vie w
L e t t e r s , vo l.4 3 . p p .1 7 5 4 -1 7 5 6 , 1 9 7 9 .

[1 3 ] A . J. B r a y, M. A . Mo o r e , R eplica-Symmetry B reaking in Spin-Glass Theories, P h ys ic a l
R e vie w L e t t e r s , vo l. 4 1 , p p . 1 0 6 8 -1 0 7 2 , 1 9 7 8 .

[1 4 ] J. F. Fe r n a n d e z , D . S h e r r in g t o n , R andomly L ocated Spins with Oscillatory Interactions,
P h ys ic a l R e vie w B , vo l. 1 8 , p p . 6 2 7 0 -6 2 7 4 , 1 9 7 8 .

[1 5 ] F. B e n a m ir a , J. P . P r o vo s t , G. J. V a ll¶e e , Separable and Non-Separable Spin Glass
M odels, Jo u r n a l d e P h ys iqu e , vo l.4 6 , n o .8 , p p .1 2 6 9 -1 2 7 5 , 1 9 8 5 .

[1 6 ] D . Gr e n s in g , R . K Äu h n , On Classical Spin-Glass M odels, Jo u r n a l d e P h ys iqu e , vo l. 4 8 ,
n o .5 , p p . 7 1 3 -7 2 1 , 1 9 8 7 .

[1 7 ] A . S Ge vo r kya n , Quantum 3D Spin-Glass System on the Scales of Space-Time P eriods
of E xternal E lectromagnetic F ields, in p r e s s , P h ys ic s o f A t o m ic N u c le i.

[1 8 ] C. M. B e n d e r , J. H . Ch e n , D . W . D a r g , K . A . Milt o n , Classical Trajectories for Complex
Hamiltonians, Jo r n a l o f P h ys ic s A : Ma t h e m a t ic a l a n d Ge n e r a l, vo l. 3 9 , p . 4 2 1 9 , 2 0 0 6 .

[1 9 ] C. M. B e n d e r , D . W . D a r g , Spontaneous B reaking of Classical P T Symmetry, Jo u r n a l
o f Ma t h e m a t ic a l P h ys ic s , vo l. 4 8 , p . 2 7 0 3 , 2 0 0 7 .

[2 0 ] C. M. B e n d e r , D . W . H o o k, E xact isospectral pairs of P T symmetric Hamiltonians,
Jo u r n a l o f P h ys ic s A : Ma t h e m a t ic a l a n d Th e o r e t ic a l, vo l. 4 1 , p p . 1 7 5 1 -8 1 1 3 , 2 0 0 8 .

[2 1 ] A . S . Ge vo r kya n , e t a l., R egular and chaotic quantum dynamic in atom-diatom reactive
collisions, P h ys ic s o f A t o m ic N u c le i, vo l.7 1 , p p . 8 7 6 -8 8 3 , 2 0 0 8 .

[2 2 ] A . V . J. S m ilg a , Cryptogauge symmetry and cryptoghosts for crypto-Hermitian Hamil-
tonians, Jo u r n a l o f P h ys ic s A : Ma t h e m a t ic a l a n d Th e o r e t ic a l, vo l. 4 1 , p . 4 0 2 6 , 2 0 0 8 .

[2 3 ] C. It z yks o n , J. M. D r o u ®e , Statistical F ield Theory: F rom B rownian motion to renor-
malization and lattice gauge theory, Ca m b r id g e U n ive r s it y P r e s s , vo l. 2 , p . 4 2 8 , 1 9 9 1 .

[2 4 ] A . S . Ge vo r kya n , H . G. A b a jya n , H . S . S u kia s ya n , A new parallel algorithm for simu-
lation of spin-glass systems on scales of space-time periods of an external ¯eld, Jo u r n a l
o f Mo d e r n P h ys ic s , vo l.2 , p p . 4 8 8 -4 9 7 , 2 0 1 1 .

[2 5 ] A . V . B o g d a n o v, A . S . Ge vo r kya n , G. V . D u b r o vs kiy, On mechanisms of proton-
hydrogen resonance recharge at moderate energies, P is m a v Zh .T.F., vo l. 9 , p p . 3 4 3 -3 4 8 ,
1 9 8 3 .

[2 6 ] I. Ib r a g im o v, Y u . L in n ik, Independent and Stationary Sequences of R andom Variebles,
W o lt e r s -N o o r d h o ® P u b lis h in g Gr o n in g e n , vo l. 4 8 , p p . 1 2 8 7 -1 7 3 0 , 1 9 7 1 .

[2 7 ] E . B o lt h a u s e n , A . B o vie r ( e d s .) , Spin glasses, S p r in g e r , vo l. 1 6 3 , p p . 1 9 0 0 -2 0 7 5 , 2 0 0 7 .

[2 8 ] G. W a n n ie r , Statistical P hysics, D o ve r P u b lic a t io n s , p . 5 3 2 , 1 9 8 7 .



A. Gevorkyan and H. Abajyan 1 9

¸³ë³Ï³Ý ëåÇݳÛÇÝ ³å³ÏÇÝ»ñÁ ѳßíÇ ³éÝí³Í
é»É³ùë³óÇáÝ »ñ¨áõÛÃÝ»ñÁ

². ¶¨áñ·Û³Ý ¨ Ð. ²µ³çÛ³Ý

²Ù÷á÷áõÙ

²ß˳ï³ÝùáõÙ áõëáõÙݳëÇñí³Í ¿ ³ñï³ùÇÝ ¹³ßïÇ ³éϳÛáõÃÛ³Ùµ ï³ñµñ
»ñϳñáõÃÛ³Ùµ 1 D ãϳñ·³íáñí³Í ï³ñ³Í³Ï³Ý ëåÇݳÛÇÝ ßÕóݻñÇ (îêÞ) ѳÙáõÛÃÇ
íÇ׳ϳ·ñ³Ï³Ý ѳïÏáõÃÛáõÝÝ»ñÁ‘ ѳßíÇ ³éÝ»Éáí é»É³ùë³óÇáÝ »ñ¨áõÛÃÝ»ñÁ: ²é³çÇÝ
³Ý·³Ù û·ï³·áñÍí»É ¿ ÏáÙåÉ»ùë-¹³ë³Ï³Ý гÙÇÉïáÝdzÝÁ: ä³ñµ»ñ³Ï³Ý 1 D
ó³ÝóÇ Ñ³Ý·áõÛóÝ»ñáõÙ ëï³óí»É ¿ é»Ïáõñ»Ýï »é³ÝÏÛáõݳã³÷³Ï³Ý ѳí³ë³ñáõÙÝ»ñÇ
ѳٳϳñ·Á, áñáÝù êÇÉí»ëïñÇ å³ÛÙ³ÝÝ»ñÇ Ñ»ï ÙdzëÇÝ ³Ý³ÉÇïÇÏáñ»Ý ß³ñáõݳÏíáõÙ
»Ý ÏáÙåÉ»ùë ï³ñ³ÍáõÃÛ³Ý Ù»ç ¨ Ñݳñ³íáñáõÃÛáõÝ »Ý ï³ÉÇë ѳݷáõÛó ³é ѳݷáõÛó
ѳßí»É ëåÇÝÇ áõÕÕáñ¹í³ÍáõÃÛáõÝÁ‘ ѳßíÇ ³éÝ»Éáí ëåÇݳÛÇÝ ßÕóݻñáõÙ é»É³ùë³óÇáÝ
»ñ¨áõÛÃÝ»ñÁ:

àõëáõÙݳëÇñí³Í »Ý ݳ¨ ëåÇݳÛÇÝ Ñ³ÙáõÛÃáõÙ ï»ÕÇ áõÝ»óáÕ áñáß³ÏÇ ÏñÇïÇϳϳÝ
»ñ¨áõÛÃÝ»ñ, ÇÝãåÇëÇù »Ý Îɳáõ½Çáõë-ØáëëáïÇÇ (Î-Ø) ѳí³ë³ñÙ³Ý Ù»ç ³Õ»ïÝ»ñÁ‘
ϳËí³Í ³ñï³ùÇÝ ¹³ßïÇ Ù»ÍáõÃÛáõÝÇó:

²é³ç³ñÏí³Í ¿ íÇ׳ϳ·ñ³Ï³Ý ·áõÙ³ñÇ Ýáñ Ý»ñϳ۳óáõÙ í»ñç³íáñ Ãíáí
ÇÝï»·ñ³É³ÛÇÝ ³ñï³Ñ³ÛïáõÃÛ³Ý ï»ëùáí‘ ¿Ý»ñ·Ç³ÛÇ ¨ µ¨»é³óí³ÍáõÃÛ³Ý ï³-
ñ³ÍáõÃÛáõÝáõÙ:

Êëàññè÷åñêèå ñïèíîâûå ñòåêëà ñ ó÷åòîì
ðåëàêñàöèîííûõ ýôôåêòîâ

À. Ñ. Ãåâîðêÿí À. Ã. Àáàäæÿí

Àííîòàöèÿ

 äàííîé ðàáîòå èññëåäîâàíû ñòàòèñòè÷åñêèå ñâîéñòâà àíñàìáëÿ íåóïîðÿ-
äî÷åííûõ 1 D ïðîñòðàíñòâåííûõ ñïèí öåïî÷åê (ÏÑÖ) ñ îïðåäåëåííîé äëèíîé âî
âíåøíåì ïîëå ñ ó÷åòîì ðåëàêñàöèîííûõ ýôôåêòîâ. Äëÿ ðåøåíèÿ ýòîé ïðîáëåìû
âïåðâûå áûë èñïîëüçîâàí êîðîòêîäåéñòâóþùèé êîìïëåêñíî-êëàññè÷åñêèé
Ãàìèëüòîíèàí è ðàçðàáîòàí ýôôåêòèâíûé ìàòåìàòè÷åñêèé àëãîðèòì, êîòîðûé,
ñ ó÷åòîì ðàñøèðåííûõ óñëîâèé Ñèëüâåñòðà, ïîçâîëÿåò ïàðàëëåëüíî, øàã
çà øàãîì ïîñòðîèòü áîëüøîå êîëè÷åñòâî ñòàáèëüíûõ 1 D ÏÑÖ. Ôóíêöèè
ðàñïðåäåëåíèÿ ðàçëè÷íûõ ïàðàìåòðîâ ñïèíîâîãî ñòåêëà ïîñòðîåíû íà îñíîâå
àíàëèçà ðåçóëüòàòîâ ðàñ÷åòà 1 D ÏÑÖàíñàìáëÿ. Ïîêàçàíî, ÷òî ðàñïðåäåëåíèÿ
ðàçíûõ ïàðàìåòðîâ ñïèíîâîãî ñòåêëà ïî-ðàçíîìó âåäóò ñåáÿ â çàâèñèìîñòè îò
âíåøíåãî ïîëÿ. Ïîêàçàíî, ÷òî îáîáùåííûé êîìïëåêñíî-êëàññè÷åñêèé ïîäõîä
èñêëþ÷àåò âîçìîæíîñòü âîçíèêíîâåíèÿ êàòàñòðîô â óðàâíåíèè Êëàóçèóñà-
Ìîññîòòè, ÷òî ïîçâîëÿåò îðãàíèçîâàòü íåïðåðûâíûå âû÷èñëåíèÿ íà âñåì
èíòåðâàëå çíà÷åíèé âíåøíåãî ïîëÿ, âêëþ÷àÿ êðèòè÷åñêèå òî÷êè. Íà îñíîâå
ïðîâåäåííûõ èññëåäîâàíèé ïðåäëîæåí íîâûé, áîëåå òî÷íûé ñïîñîá ïîñòðîåíèÿ
ñòàòèñòè÷åñêîé ñóììû ñèñòåìû, êîòîðàÿ â îòëè÷èå îò îáû÷íûõ ïðåäñòàâëåíèé,
ÿâëÿåòñÿ êîìïëåêñíîé ôóíêöèåé. Ñòàòèñòè÷åñêàÿ ñóììà, è åå ïðîèçâîäíûå
àíàëèòè÷íû ïîâñþäó âêëþ÷àÿ êðèòè÷åñêèå òî÷êè.