Orientational phase transitions in topologically

Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 45–66

An integral equation approach to orientational phase transitions in two and three dimensional disordered systems E.Lomba 1 , F.Lado 2 , J.J.Weis 3 1 2 3

Instituto de Qu´ımica F´ısica Rocasolano, CSIC, Serrano 119, E-28006 Madrid, Spain Department of Physics, North Carolina State University, Raleigh, North Carolina 27695-8202, USA ´ ˆ Laboratoire de Physique Theorique, Batiment 210, Universite´ de Paris-Sud, 91405 Orsay Cedex France

Received August 1, 2000 The use of inhomogeneous Ornstein-Zernike equations to analyze phase transitions and ordered phases in magnetic systems is explored both in bulk three dimensional disordered Heisenberg systems and in a simple model for a two dimensional ferrofluid monolayer. In addition to closures like the Mean Spherical Approximation, Hypernetted Chain and Zerah-Hansen approximation, the inhomogeneous Ornstein-Zernike equation must be complemented by a one-body closure, for which the Born-Green equation has been used in this paper. The results obtained prove that the proposed approach can furnish accurate estimates for the paramagneticferromagnetic transition in the three dimensional Heisenberg spin fluid, reproducing reliably the structure of the isotropic and ordered phases. In two dimensions, the results are fairly accurate as well, both for the dipolar film alone and in the presence of external perpendicular fields. At high densities/dipole moments the equation seems to predict a transition to a phase in which the dipoles lie mostly in the plane and are aligned into vortex-like structures. Evidence of this new phase is found in the simulation at somewhat higher couplings. Key words: integral equations, ferromagnetic transitions, antiferromagnetism, dipolar films PACS: 61.20.Gy, 68.15.+e, 75.10.-b, 75.30.-m

1. Introduction


   
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Integral equations in magnetic systems

KMLN
OPNRQTUW S VYXZ S1[\ X]Z [ S_^` O Zbadc `efhgTij c `jj N c1kml `j K lnk L k LN ePoe c N
p kml#ql#r#lnkml N e k LN
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l k N x K lnk L k LN N
OPO v lnkml l O l e N k ` k O suje yN
O e N€ o c1k ozs kml `je
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 KML l#c L„OPN1€zN c1kƒl j…s xl‡† yN
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O e N v su~j N kml#c ePoe c N
p kml#ql#r#lnk }gi k L sue q N
N j pO ` yN xk L s ktˆ ` yN kPkI‰ e N
Š ozs kml `jsuo k ` v s kml#c s r#r } OPN
pO ` x o c N e k LN8pOPN e N j c N `u^‹‚_` r#x e k `j N v ` x N e l jE` O l N j k s kml `j † s }$` O N
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51

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n ^+ao[]^+b_.^+a]` [ k aj^ , & \Gfpd\G^+a (+*k _GmpfpeGd ,rq []\`es[]\G^t[]\G^ui)^+aja * bc`deG^2[]f ( []a]`e , fP[]f * e FJv \Gf ,wn ` , _`ax[]f (+k my`ajmPz^2{Ef g | ^+eQ[6dfP{^+es[]\G^ | fP} (+k mP[]fp^ , ^+e (+*k eQ[]^+aj^ |B~ z[]\G^ , fpb k my` []f * e[ * ` [j[r`fpe€`eQ[]fPi)^+aja * bc`deG^2[ g f (c* a | ^+ajfpeGdG n \Gf ( \‚`_G_`aj^+eQ[]mPzƒ[r`„^ , b kG( \m * eGd^+al[ * ~ k fpm | F%v \Gf , f , (+* eGeG^ ( []^ | n fP[]\ []\G^…i0` ( [[]\` [ n \G^+aj^` , []\G^B_Ga * _`dQ` []f * e * iui)^+aja * bc`deG^2[]f (…* a | ^+ajfpeGd†f , ^+eG^+ajd^2[]f ( `mpmPz i0`{ *k a]` ~ mp^ & `mpmmy`z^+a , * i_`a]`mpmp^+m‡_`ax[]f ( mp^ , []^+e | [ * m *
n ^+al[]\G^c^+eG^+ajdz * iw[]\G^ , z , []^+b q fpes[]\G^ ( ` , ^ * i‡`eQ[]fPi)^+aja * bc`deG^2[]f ( fpeQ[]^+a]` ( []f * e ,ˆ,jkG(+( ^ ,j, fP{^my`z^+a ,6* i * a | ^+aj^ | , _Gfpe , \`{^ * _G_ *, fpeGd (+* eQ[]ajf ~ k []f * e , ‰f F ^ F ?Ša , [leG^+fpd\ ~ *k a , `aj^`eQ[]fp_`a]`mpmp^+m & m *
n ^+ajfpeGds[]\G^^+eG^+ajdz q n \G^+aj^` ,‹, ^ (+* e | eG^+fpd\ ~ *k a , ~ ^+fpeGdŒ_`a]`mpmp^+m[]^+e | [ * fpe ( aj^` , ^[]\G^s^+eG^+ajdz‡`e | ,j** e F v \G^ (+* e | fP[]f * e , i * aŽ[]\G^=i * ajbc` []f * e * i%`eQ[]fPi)^+aja * bc`deG^2[]f ( _G\` , ^ , `aj^=[]\G^+es[ *E*, ^2{^+aj^t[ * ~ ^t^` , fpmPz\`e | mp^ |…~ z[]\G^ , fpb k my` []f * e_Ga *E( ^ | k aj^ , `e | \Gfpe | ^+aw[]\G^ (+* eQ{^+ajd^+e ( ^ * ifpeQ[]^ g da]`m^+‘ k ` []f * e , F >6*
n ^2{^+a+ n ^u\`{^=i *k e | []\` [ | ^ ( aj^` , fpeGd []\G^=a]`eGd^ * i‰[]\G^=fpeQ[]^+a]` ( []f * e []a]`e , my` []^ , fpeQ[ * ` (+* e , f | ^+a]` ~ mp^m *
n ^+ajfpeGd * i[]\G^ ( ajfP[]f ( `m ‹’ Fv \ kG,  n \G^+aj^` , f ,j* []a * _Gf ( ;6>=“uA”( `m (+k my` []f * e , fpe | f ( ` []^ []\` [•` [ 7Q– ?[]\G^™[]a]`e , fP[]f * e *E(+(+k a , i * a ‹’tš L
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n []^+b_.^+a]` [ k aj^ (+* b ~ fpeG^ | n fP[]\™[]\G^ , \ * ax[ a]`eGd^[]a]`e , my` []^ , fpeQ[ * ` ( ^+ax[r`fpesž , []f ( „Qz.Ÿ ( \`a]` ( []^+a[]\` [fpe | kG( ^ , ` (+* e , f | ^+a]` ~ mp^6`b *k eQ[ * i ( m kG, []^+ajfpeGd FJv \G^+ajb * | zEe`bf (+, `e | * a | ^+aw_`a]`b^2[]^+a ,* ~ [r`fpeG^ | i * aŽ[]\Gf , b * | ^+m kG, fpeGd ;= `e | , fpb k my` []f * ei * a{`ajf *kG,, [r` []^ , `aj^ (+* mpmp^ ( []^ | fpe‹[r` ~ mp^6  FXv \G^ , ^uaj^ ,jk mP[ , `aj^ , []fpmpm {^+axz†_Gaj^+mpfpbfpe`axz ~ k [fpe | f ( ` []^‹[]\` [ | ^ , _GfP[]^[]\G^ | fP} (+k mP[]fp^ , []\G^cb^2[]\ * | ( `e , []fpmpm ~ ^ `_G_Gmpfp^ | [ * `eQ[]fPi)^+aja * bc`deG^2[]f (, z , []^+b , F  ¡[[ k aje ,w*k [[]\` [Ž[]\G^ * eG^=_`ax[]f ( mp^ | f , []ajf ~ k []f * e i k e ( []f * e€fpe[]\G^ , ^ , z , []^+b , i * mpm *
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Integral equations in magnetic systems

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