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Condensation of vortices and disorder parameter in 3d Heisenberg model
UCLEAR PHYSIC~
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
ELSEVIER
Condensation
of vortices
and disorder parameter
in 3d Heisenberg
model
A. Di Giacomo a *, D. Martelli b, G. Paffuti ~ aDipartimento di Fisica Universit~ and INFN Pisa (Italy) DISAS, Trieste (Italy) The 3d Heisenberg model is studied from a dual point of view in terms of 2d solitons (vortices). It is shown that the disordered phase corresponds to condensation of vortices in the vacuum, and the critical indices are computed from the corresponding disorder parameter.
. Introduction
2. T h e H e i s e n b e r g
model
as a fiber bundle.
The Heisenberg ferromagnet is defined by the partition function
Usually the colour frame to which the direction of ~ is referred is a fixed frame, independent of x
Z[#] = f I I [ d a ( x ) e x p ( - S )
~(i=1,2,3)
where 1 S = ~# Z
[A~ F(x)]2
g2(x) = 1
(1)
(2)
~A~=~ai
°
~.~=~,#
A body fixed frame can be defined[5] by three unit vectors ~ ( z ) (i = 1, 2, 3)
D,X
and df't(x) is the element of solid angle for the orientation of F in colour space. The model presents a 2nd order phase transition at/~c ~- 0.7[1]. For # > #c there is an ordered phase, with order parameter the magnetization (F) # 0; for/3
and (3(x) - if(x). The frame is defined up to an arbitrary rotation around if(x) = (3 (x). Since ~ = 1
a.Ei(x) = ~ A Ei(x) or
D~,~(x) D.
=
(0. -
= 0
(3)
(Ta)ij = i~iaj are the generators of the 0(3) symmetry group. Eq.(3) is nothing but the definition of parallel transport. From eq.(3) it follows = 0 or, by completeness of [nuln,, ] =
~F. F ~ ( w ) = 0
(4)
o~ is a pure gauge, apart from singularities. The general solution of eq.(3) is then, for ~(x) = (3(x) ~(x) = P exp
0920-5632/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved. PII S0920-5632(99)00060-2
i
,c
T~,(x')dx
4o
746
A. Di Giacomo et aL /Nuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
where g0 is the value of ~(x) at infinity, and the dependence on the path C is trivial, because F ~ is a pure gauge, eq.(4). This is true apart from singularities. We will show that such singularities exist. The current J~ = 1 ~ . ( 0 ~ A 0 ~ ) is identically conserved
We measure the correlator ~(X0) =
<~--q(O,x
)~q(0,0))
By cluster property
l)(Xo)
~--
Ae -Mlz°l -4- <#q)2
(8)
(5)
0.J. = 0
If we look at the theory as the euclidean version of a field theory, with euclidean time on the 3 axis, the corresponding conserved quantity is
(#q) # 0 signals spontaneous breaking of the U(1) symmetry (5), or condensation of vortices. By use of the definition (7) it is easy to see that
Q = ~
l)(zo) - z [ s + a s ]
1/
d2z~7 • (01~ ^ 02~)
z[s] which is nothing but the topological charge of the 2 dimensional configurations of the theory. Q can assume positive and negative integer values. By use of eq.(3) and eq.(4) it is easy to show that
where S + a s replacement
is obtained from S, eq.(2) by the
[aoS(~,o)] 2 -~ [ n ; ' ( ~ , o ) n ( ~ , O = ~
a2z,~. ( ~ ^ ~ ) = -
( ~ , . , ~ ) d x ~ (6)
[aon(~,xo)] ~ ~
where the path C is the contour of the region in the 2 dimensional space (x3 = const.) where eq.(4) holds. Since Q = + n Eq.(6) shows that a~, is not always a pure gauge. This can be explicitely checked on a configuration corresponding to a static 2 dimensional instanton propagating in time (vortex). The conserved current J , identifies a U(1) symmetry. We will show that this symmetry is Wigner in the ordered phase t > tic, and is spontaneously broken in the disordered phase.
+ 1) - n(~,~o)] 2
(#) = exp I f : p ( t ' ) d t q , p is easier to measure and contains all the information on the transition. The behaviour of p is shown in fig.1. For t < tc, P -+ finite limit consistent with zero, or (#) ~ 0, which means condensation of vortices. For/~ > t¢ P can be evaluated in perturbation theory and behaves as J
Let Rq (~, y-') be a £ dependent singular rotation creating a vortex of charge q at the site ~7 in a 2 dimensional configuration.
p = - C l L + c2
The creation operator of a vortex #q at site if, time t, will be defined as
#q(ff, t) = e x p { - t ~ [ ( n ~ -1 (£, y-')~(Z, t + 1) - ~(£, t)) 2 t + 1) - ~(~, t)) ~] }
[R;'(~,0)n(~,xo
and that this really amounts to have a vortex propagating from (6, 0) to (0,xo)[2]. Instead of :D(x0) itself it proves convenient[2,3] to study the quantity p(xo) = ½dlnl)(Xo)/dt. As Ix01 --+ c~ from eq.(8) we have for p p(xo = c~), p "" dln(#>/dt. Since (#)~=o = 1,
3. D i s o r d e r p a r a m e t e r
-(,~(~,
1) - s ( ~ , o ) ] 2
(7)
cl > 0
(9)
L is the lattice size. Eq.(9) implies that (#) = 0 for t > t c in the thermodynamical limit V --+ c~. Around tc a finite size scaling analysis can be performed
<,>
,4. Di Giacomo et al.INuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
747
Conclusions ,
.
,
,
•
,
•
•
•
o
o
.
,
.
,
.
,
.
The phase transition to disorder in 3d Heisenberg model is produced by condensation of topological solitons. A disorder parameter can be defined and out of it the critical indices can be determined.
,
-10.0 o o
~ouv o. v ~0.0
~
v
•
Acknowledgement -50.0
This work has partially supported by MURST. A. Di Giacomo is grateful to EC, TMR project, ERBFMX-CT97-0122 for financing the partecipation to the conference.
~ •
•
L = 6
L = 8
* L=12
-70.0
s
REFERENCES -9°°0'o ,:o 2'0 a'0 ,:o's'.0
e'o 7'0 0'o
'
9:0
'
,o.o
Fig.1 and since 1/~ _~ (tic -fl)v, we get the scaling law[2]
P
= f(LZ/~(~ c - ~ ) )
The scaling law is verified, fig.2, and allows to extract/3c and u /3c = u
0.695 =E0.003 [0.6028]
---- 0.70 4- 0.02 [0.698]
T h e y agree with the values d e t e r m i n e d from (~), which are indicated in parentheses[i,6]. 0.0
-0.5
• L=6 "7 Q..
:'
~ L=8 o L=12
-1.5
~----~.695v=0.70
-2.5
-3"00.2
0.7
1.2
1.7
2.2 Uv
({~-~)L Fig.2
2.7
3.2
4.2
1. E. Brezin, J. Zinn-Justin, Nucl. Phys. B257, 867, (1985). 2. A. Di Giacomo, G. Paffuti, Phys. Rev. D 5 6 , 6816, (1997). 3. L.Del Debbio, A.Di Giacomo, G.Paffuti and P.Pieri, Phys. Lett.B 355 (1995) 255. 4. G. Di Cecio, A. Di Giacomo, G. Pa~uti, M. Trigia.nte, Nucl. Phys. B489, 739, (1997). 5. A. Di Giacomo, M. Mathur, Phys. Lett. B400, 129, (1997). 6. P. Peczak, A.L. Ferrenberg, D.P. Landau, Phys. Rev. 4 3 , 6087, (1991).
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
ELSEVIER
Condensation
of vortices
and disorder parameter
in 3d Heisenberg
model
A. Di Giacomo a *, D. Martelli b, G. Paffuti ~ aDipartimento di Fisica Universit~ and INFN Pisa (Italy) DISAS, Trieste (Italy) The 3d Heisenberg model is studied from a dual point of view in terms of 2d solitons (vortices). It is shown that the disordered phase corresponds to condensation of vortices in the vacuum, and the critical indices are computed from the corresponding disorder parameter.
. Introduction
2. T h e H e i s e n b e r g
model
as a fiber bundle.
The Heisenberg ferromagnet is defined by the partition function
Usually the colour frame to which the direction of ~ is referred is a fixed frame, independent of x
Z[#] = f I I [ d a ( x ) e x p ( - S )
~(i=1,2,3)
where 1 S = ~# Z
[A~ F(x)]2
g2(x) = 1
(1)
(2)
~A~=~ai
°
~.~=~,#
A body fixed frame can be defined[5] by three unit vectors ~ ( z ) (i = 1, 2, 3)
D,X
and df't(x) is the element of solid angle for the orientation of F in colour space. The model presents a 2nd order phase transition at/~c ~- 0.7[1]. For # > #c there is an ordered phase, with order parameter the magnetization (F) # 0; for/3
and (3(x) - if(x). The frame is defined up to an arbitrary rotation around if(x) = (3 (x). Since ~ = 1
a.Ei(x) = ~ A Ei(x) or
D~,~(x) D.
=
(0. -
= 0
(3)
(Ta)ij = i~iaj are the generators of the 0(3) symmetry group. Eq.(3) is nothing but the definition of parallel transport. From eq.(3) it follows = 0 or, by completeness of [nuln,, ] =
~F. F ~ ( w ) = 0
(4)
o~ is a pure gauge, apart from singularities. The general solution of eq.(3) is then, for ~(x) = (3(x) ~(x) = P exp
0920-5632/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved. PII S0920-5632(99)00060-2
i
,c
T~,(x')dx
4o
746
A. Di Giacomo et aL /Nuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
where g0 is the value of ~(x) at infinity, and the dependence on the path C is trivial, because F ~ is a pure gauge, eq.(4). This is true apart from singularities. We will show that such singularities exist. The current J~ = 1 ~ . ( 0 ~ A 0 ~ ) is identically conserved
We measure the correlator ~(X0) =
<~--q(O,x
)~q(0,0))
By cluster property
l)(Xo)
~--
Ae -Mlz°l -4- <#q)2
(8)
(5)
0.J. = 0
If we look at the theory as the euclidean version of a field theory, with euclidean time on the 3 axis, the corresponding conserved quantity is
(#q) # 0 signals spontaneous breaking of the U(1) symmetry (5), or condensation of vortices. By use of the definition (7) it is easy to see that
Q = ~
l)(zo) - z [ s + a s ]
1/
d2z~7 • (01~ ^ 02~)
z[s] which is nothing but the topological charge of the 2 dimensional configurations of the theory. Q can assume positive and negative integer values. By use of eq.(3) and eq.(4) it is easy to show that
where S + a s replacement
is obtained from S, eq.(2) by the
[aoS(~,o)] 2 -~ [ n ; ' ( ~ , o ) n ( ~ , O = ~
a2z,~. ( ~ ^ ~ ) = -
( ~ , . , ~ ) d x ~ (6)
[aon(~,xo)] ~ ~
where the path C is the contour of the region in the 2 dimensional space (x3 = const.) where eq.(4) holds. Since Q = + n Eq.(6) shows that a~, is not always a pure gauge. This can be explicitely checked on a configuration corresponding to a static 2 dimensional instanton propagating in time (vortex). The conserved current J , identifies a U(1) symmetry. We will show that this symmetry is Wigner in the ordered phase t > tic, and is spontaneously broken in the disordered phase.
+ 1) - n(~,~o)] 2
(#) = exp I f : p ( t ' ) d t q , p is easier to measure and contains all the information on the transition. The behaviour of p is shown in fig.1. For t < tc, P -+ finite limit consistent with zero, or (#) ~ 0, which means condensation of vortices. For/~ > t¢ P can be evaluated in perturbation theory and behaves as J
Let Rq (~, y-') be a £ dependent singular rotation creating a vortex of charge q at the site ~7 in a 2 dimensional configuration.
p = - C l L + c2
The creation operator of a vortex #q at site if, time t, will be defined as
#q(ff, t) = e x p { - t ~ [ ( n ~ -1 (£, y-')~(Z, t + 1) - ~(£, t)) 2 t + 1) - ~(~, t)) ~] }
[R;'(~,0)n(~,xo
and that this really amounts to have a vortex propagating from (6, 0) to (0,xo)[2]. Instead of :D(x0) itself it proves convenient[2,3] to study the quantity p(xo) = ½dlnl)(Xo)/dt. As Ix01 --+ c~ from eq.(8) we have for p p(xo = c~), p "" dln(#>/dt. Since (#)~=o = 1,
3. D i s o r d e r p a r a m e t e r
-(,~(~,
1) - s ( ~ , o ) ] 2
(7)
cl > 0
(9)
L is the lattice size. Eq.(9) implies that (#) = 0 for t > t c in the thermodynamical limit V --+ c~. Around tc a finite size scaling analysis can be performed
<,>
,4. Di Giacomo et al.INuclear Physics B (Proc. Suppl.) 73 (1999) 745-747
747
Conclusions ,
.
,
,
•
,
•
•
•
o
o
.
,
.
,
.
,
.
The phase transition to disorder in 3d Heisenberg model is produced by condensation of topological solitons. A disorder parameter can be defined and out of it the critical indices can be determined.
,
-10.0 o o
~ouv o. v ~0.0
~
v
•
Acknowledgement -50.0
This work has partially supported by MURST. A. Di Giacomo is grateful to EC, TMR project, ERBFMX-CT97-0122 for financing the partecipation to the conference.
~ •
•
L = 6
L = 8
* L=12
-70.0
s
REFERENCES -9°°0'o ,:o 2'0 a'0 ,:o's'.0
e'o 7'0 0'o
'
9:0
'
,o.o
Fig.1 and since 1/~ _~ (tic -fl)v, we get the scaling law[2]
P
= f(LZ/~(~ c - ~ ) )
The scaling law is verified, fig.2, and allows to extract/3c and u /3c = u
0.695 =E0.003 [0.6028]
---- 0.70 4- 0.02 [0.698]
T h e y agree with the values d e t e r m i n e d from (~), which are indicated in parentheses[i,6]. 0.0
-0.5
• L=6 "7 Q..
:'
~ L=8 o L=12
-1.5
~----~.695v=0.70
-2.5
-3"00.2
0.7
1.2
1.7
2.2 Uv
({~-~)L Fig.2
2.7
3.2
4.2
1. E. Brezin, J. Zinn-Justin, Nucl. Phys. B257, 867, (1985). 2. A. Di Giacomo, G. Paffuti, Phys. Rev. D 5 6 , 6816, (1997). 3. L.Del Debbio, A.Di Giacomo, G.Paffuti and P.Pieri, Phys. Lett.B 355 (1995) 255. 4. G. Di Cecio, A. Di Giacomo, G. Pa~uti, M. Trigia.nte, Nucl. Phys. B489, 739, (1997). 5. A. Di Giacomo, M. Mathur, Phys. Lett. B400, 129, (1997). 6. P. Peczak, A.L. Ferrenberg, D.P. Landau, Phys. Rev. 4 3 , 6087, (1991).