- Email: [email protected]
Condensation of vortices in the XY model in 3D: A disorder parameter
ELSEVIER
Nuclear Physics B 489 [FS] (1997) 739-747
Condensation of vortices in the XY model in 3D: a disorder parameter* G. Di C e c i o a, A. Di G i a c o m o b, G. Paffuti b'l, M. Trigiante c a Department of Physics and Astronomy, Louisiana State University, 102 Nicholson Hall, Baton Rouge, LA 70803-4001, USA b Dipartimento di Fisica and INFN, 2 Piazza Torricelli, 56100 Pisa, Italy c ISAS, Via Beirut 2, 34014 Trieste, Italy
Received 31 May 1996; revised 20 September 1996; accepted 20 December 1996
Abstract
A disorder parameter is constructed which signals the condensation of vortices. The construction is tested by numerical simulations. (~) 1997 Elsevier Science B.V. PACS: 75.40.Cx; 05.70.Jk; 64.60.Cn
1. Introduction
The XY model in three dimensions is an interesting system in statistical mechanics. Physically it describes the critical behaviour of superfluid He4 [ 1]. From the theoretical point of view it provides a relatively simple example of an order-disorder phase transition in which the condensation of solitons plays an essential role [2-5]. It is fairly well established that the transition is continuous, and the basic critical indices are known with good precision [6-8]. A Ginzburg-Landau kind of description of the system has been given, describing phenomenologically the condensation of vortices in the high temperature phase [9]. In this paper we shall study the system from a slightly different point of view. We shall look at it as the lattice formulation of a 2 + 1-dimensional quantum field theory. * Partially supported by MURST (Italian Ministry of the University and of Scientific and Technological Research) and by the EC contract CHEX-CT92-0051. l E-mail: [email protected] 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PH S 0 5 5 0 - 3 2 1 3 ( 9 7 ) 0 0 0 1 9 - 9
740
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747
We shall then show that the (two-dimensional) ground state of the theory spontaneously breaks the conservation of the number of vortices at high temperature; vortices condense in the vacuum in the same way as Cooper pairs do in a superconductor. In that phase the number of vortices is not defined in the vacuum, which is a superposition of states with different vorticity. We shall directly exhibit this condensation by showing that the vacuum expectation value (vev) of an operator which changes the number of vortices is non-zero. An analogous technique has been used to detect monopole condensation in the ground state of U(1) gauge theory [10] and in SU(2) gauge theory, as a mechanism for confinement of colour [ 11 ]. In Section 2 we shall briefly introduce the model and fix the notation. We shall also define a creation operator for a vortex. In Section 3 we shall present and discuss the results of numerical simulations where the vev of the above operator is measured. The result will be a direct detection of the spontaneous breaking of the conservation of the number of vortices in the disordered phase and a direct evidence of vortex condensation. We shall also obtain an alternative determination of known critical indices, which agrees with existing results, and measure the critical index related to our disorder parameter.
2. The creation operator of vortices The lattice action of the system is S = f l ~ (1 - - c o s ( A ~ O ( i ) ) ) . /z,i
(1)
The field variable is the angle 0 in the site i:/z runs from 0 to 2, 0 being the time axis. In the limit of zero lattice spacing (a --, 0) S '~ /3 ~ (A#0)2 a 2 q- O(a4) . 2 /z,i
(2)
At high fl's the system describes a massless free scalar particle. The couplings of higher dimension appearing in the series expansion of the cosine become important at lower /3. At /3c ~-- 0.454 the system undergoes a phase transition, which is known to be of second order [6]. The system has soliton configurations with the geometry of vortices: numerical simulations show that vortices play an important role in the phase transition [3-5]. A vector field
A~ = a~,o
(3)
can be defined, and a current j~ = e~,~#a,~A# associated with it, which is trivially conserved in smooth configurations:
a~ j~ = 31ze.g,~#3'~A # = O,
(4)
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747
741
because of the antisymmetry of the Ricci tensor. Eq. (4) is the analog of the Bianchi identity in QED. The constant of the motion is
q~= f d2xj°(x,x°) = / d 2 x (V AA) .
(5)
On the other hand, from Eq. (3)
~P = ~ A .
dx = n - 2 ~
(6)
,/
C
with C any closed path. A vortex is defined as a configuration for which n is non-zero,
O(x
( x - y)2 - y) = arctan - -
(7)
(x-y)1
is an example of such a configuration with n = 1. It has a singularity at x = y. Eqs. ( 3 ) - ( 6 ) tell that the number of vortices is a constant of motion, and define a U( 1 ) symmetry. We shall show that, for fl > tic, this symmetry is realized ~ la Wigner: for/3
eipalx) = Ix + a) can be used. The translation operator
ke(x,t)=expfi/d3yaoO(y,t)O(x-y)] indeed, when applied to the state
IO(x, t)},
(8) gives
tz(x,t)lO(y,t) ) = ]O(y,t) + O(x - y)).
(9)
The operator number of vortices
V(t) = / d 2 x j ° ( x , t)
(10)
has a commutation rule with/z, I v ( t ) , / x ( x , t)] = where A = Vt~.
f d 2 z • A A(z,
t ) / z ( x , t) = 2~'be(x, t ) ,
( 11 )
742
G. Di Cecio et aL/Nuclear Physics B 489 [FS] (1997) 739-747
If the U(1) symmetry, which counts the vortices, is realized ~ la Wigner, then the ground state must have a definite number of vortices and (/x) = 0. Since/z changes the number of vortices a, vev (/z) :# 0 signals their condensation. The naive translation of Eq. (8) on the lattice is [ 10]
Iz(n'n°)=exp[-flZsin(d°O(n"n°))O(n'-n'n°)] ' n '
(12)
where the site n t runs on the sites of a two-dimensional slice of the lattice at constant no, except the position of the vortex. Instead of Eq. (12) we shall use a compactified version of it. By this we mean /.t = exp - f l Z
(cos(d00 + 0) - cos(~00))]
/
I1 t
( 13)
which coincides with Eq. (12) at first order in 0. With the definition (13) (/z) is independent of the choice of the gauge for i.e. of the zero for the angle 0: a redefinition of it is reabsorbed in the integration variable. Two remarks are in order. (1) In principle one expects (/.t) -- 0 in the ordered phase fl > tic. In fact, (/z) is an analytic function of fl if the number of degrees of freedom is finite, and therefore, if it were zero for fl > it would be identically zero for all fl's. Only in the limit of infinite volume (thermodynamic limit) singularities develop, and (/z) can be identically zero above without being zero everywhere. Indeed a simple computation at large fl's, where the integral is approximately gaussian, gives
Au,
tic
tic,
(tz) ~ exp I-fl (cl Vl/3 + c2 + O( ~ ) ] .
(14)
(2) Due to periodic boundary conditions the total number of vortices must be zero as easily seen from Eq. (6) with the boundary C at spatial infinity. In principle then the correct way to determine (/z) is to observe a vortex-antivortex pair correlation
( ~ ( x ) ~ ( 0 ) ) x ~ o (~) 2 .
(15)
By the cluster property and C invariance it tends exponentially to (/x) 2 at large distances. In practice, since in the thermodynamical limit physics becomes independent of the boundary conditions and decorrelated from them at distances larger than the correlation length, what happens is that also a single vortex can be created, and a direct measurement of (/~) can be done. (/~) determined in this way is indeed independent of the boundary conditions within errors. When a single vortex is put on the lattice by our operator, a dislocation appears at the boundary with opposite vorticity to correct for the boundary conditions, and, if the lattice is big compared to the correlation length, what is obtained is the same (p)2 as with a vortex-antivortex pair at large relative distance.
G. Di Cecio et aL/Nuclear Physics B 489 [FS] (1997) 739-747
743
3. Numerical simulation and results
Simulations have been done on a 202 x 40 and 302 x 60 lattices to measure the (/2 At) correlation. Single vortices (At) have also been measured on the same lattices and on 203, 303 and 403 lattices. Instead of directly determining (At) as a function o f / 3 it is more convenient to measure [ 10] 1 d p = ~ - ~ ln(At) 2
and to
(At)2 as
reconstruct
(At)2 =exp
(16)
is } 2
p(fl) dfl
•
(17)
0
From the definition of (At)z, (At)
exp
=
[-fl(S + S~)]
,
(18)
with S defined by Eq. (1), z --
t,2
) exp
[-ns]
and
[cos( 0e(n',no)
"F nv(no)Z ~ ( n t --
no n' 4=nv(no)
nv(no) ) ) - cos( doO(n',no) )] (19)
it follows that [ 10] p = (S)s - ( s + S ' ) s + s , .
(20)
The subscripts S and S+S p in Eq. (20) indicate the action used to perform the average. In Eq. (19) the sum runs over the spatial locations n ~ different from the locations of vortices nv (no), in the time slices no where vortices are created or destroyed. In principle one could compute (At) directly. However, since the operator to be averaged in Eq. (18), exp(-flSP), is the exponential of a quantity which is roughly proportional to the spatial volume, V, and fluctuates as v/-V, its values fluctuate over many orders of magnitude and do not seem to obey a gaussian distribution. In fact the relative uncertainty does not decrease with increasing statistics, p does not suffers this disease. Determining p, instead of (At)2, is analogous to determine the internal energy of a system instead of its partition function. The observed shape of p is shown in Fig. 1. The sharp negative peak at tic indicates a steep drop of (At) toward zero at tic. At low /3's p is compatible with zero and
744
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 500.0
0.0
........ Ii
•
•
a
o
- - - • - -
- - _ • _ _
P
-500.0
Lattice 203 o Lattice 30 s • Lattice 40 a
-I~0.0
•
i
-1500.0.0
I
0.5
I
1.0
1.5
Fig. l. p as a function of/3. The dashed lines at high values of/3 are the perturbative estimates. ,
10.0
- -
0.0
P
-10.0 • ~---0.15
~-0.30 -20.0
"30"00.000
* IL-0.40
I
I
I
I
t
0.010
0.020
0.030 1/L
0.040
0.050
0.060
Fig. 2. p as a function of 1/L (L = size of the lattice) in the condensed phase, p converges to a value consistent with zero. The abscissae for different betas are slightly displaced for graphical reasons.
independent of V (Fig. 2). This means by Eq. (17) that limv_o~(/z) ~- 1 for fl < tic, and thus that vortices do condense in the vacuum. An estimate of (/z) by strong coupling expansion at low fl gives p '~ - k f l 3
(21)
with k a positive constant: numerically for the configurations we have studied k ___0.30.5 so that for fl < 0.4 the estimate (21) is smaller than the errors in Fig. 2, which are typically of the order of a few units, and consistent with our numerical determinations. At fl > tic the behaviour of Eq. (14) can be checked by numerical calculation of the gaussian integral. For a cubic lattice with a vortex in the centre the result is
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 0.0
,
,
-1.0
++
-2.0
~
745
++
÷+
~
+
!
" Lattice 202x40 o Lattice 302x60
-3.0
"4'%.5
0.0
0.5
1.0
1.5
(rJ:{)L '~' Fig. 3. Quality of the finite size scaling analysis. The figure corresponds to optimal values of tic and u.
p = -11.332. L + 72.669+0(1/fl).
(22)
The comparison with numerical simulations is shown in Fig. 1. Eq. (22) means that (#z)2 is exactly zero in the limit V -+ co. Eq. (22) agrees with the observed values of p at large fl's (Fig. 1). Near tic, where the correlation length becomes large, a finite size analysis can be done to explore the limit of infinite volume and determine the critical index 77of #z, tic, and the critical index ~, of the correlation length. As ( f l - f l ~ ) - ~ 0 - and V ~ c~
(Iz) ~" (tic - fl)~.
(23)
At finite L, (#z)=(/x)(~,~)
.
(24)
As fl --+ tic, ~ diverges, ~ ,-~ (tic - f l ) - ~ and the dependence on a l ~ can be neglected, \
/
(25)
or
c/v (c/'(nc- n))
(26)
p l L l i " in the scaling region must be a universal function of L l l u ( f l c - f l ) . The quality
of this scaling is shown in Fig. 3. The critical index u and the critical temperature tic are determined by enforcing this scaling behaviour. Fig. 4 shows the data for a vortex-antivortex pair at large distance, compared to the single vortex data. In the condensed phase the single vortex behaves in fact as a pair
746
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 200.0
0.0
t
i
........| o
- - - t ~ - -
-200.0 Q
-400.0
• Lattice 202x40 vortex
o Lattice 202x40 pair d=l 6
l -600"00.0
e
i
i
0.5
1.0
L
1.5
Fig. 4. Comparisonof p for single vortex to p for vortex-antivortexpair at large distance. The two curves coincide within errors in the condensed phase. They differ at high fl and agree with the perturbative determination (dashed lines on the right side). at large distance, confirming that there is no problem in the thermodynamical limit. At /3 >/3c the two quantities are different and at/3 sufficiently large are well described by Eq. (14) with the different appropriate values of Cl and c2, as shown in the figure. A more precise finite size scaling analysis assuming a specific form for f for/3 < tic,
f(LU~(/3c -/3)) =
¢5
L~/~ (/3c -/3) + v,
(27)
+ v~
gives a precise determination of the critical indices and of tic, 6 = 0 . 7 4 0 :i: 0.029,
(28)
/3c = 0.4538 ± 0.0003,
(29)
v = 0 . 6 6 9 i 0.065,
(30)
with x Z / d o f = 1.07. For comparison the determination of Ref. [8] is v = 0.670(7) and /3c = 0.45419(2). The behaviour of the correlator (/.t(x)/2(0)) as a function of the distance gives an estimate of the correlation length scM(/3), (/.t(x)/.t(0)) - (/.t) z ~ exp [ - x / ( M ( / 3 ) ] .
(31)
~:M is of the order of a few lattice spacings (see Fig. 5). s~M(/3) should scale as (/3c - / 3 ) - ~ as (/3 - / 3 c ) --~ 0 - . With our relatively large statistical errors we were not able to check this behaviour.
747
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 0.0
-50.0
-100.0
• 13=0.448 • [~--0.440 "150"05.0
.
.
.
.
i
10.0
i
i
15.0 d
20.0
25.0
Fig. 5. p as a function of distance at two values of ft. The estimated correlation length is few lattice spacings. 4. C o n c l u d i n g r e m a r k s We have constructed a disorder parameter for the ,~ transition in the 3D XY model which is the vev o f an operator with non-trivial vortex number. It is different from zero at high temperatures ( f l < t i c ) in the disordered phase, thus demonstrating condensation of vortices. In the ordered phase it tends to zero exponentially with the linear size of the lattice. tic and the critical index ~, of the correlation length can be determined with good precision by a finite size scaling analysis o f our disorder parameter (/x). As ( f l - tic) O - tz ,,~ ( t i c - f l ) 8 with 8 = 0.74 + 0.03.
A similar construction also works for other models u n d e r g o i n g an o r d e r - d i s o r d e r transition produced by condensation o f solitons. A n example is the condensation o f m o n o p o l e s in gauge theories [10,11].
References [11 R.L. Onsager, Nuovo Cimento Suppl. 6 (1949) 249; R.P. Feynman, in Progress in Low Temperature Physics, ed. C.J. Gorter, Vol. 1 (North-Holland, Amsterdam, 1955) p. 17. [2] T. Banks, R. Meyerson and J.B. Kogut, Nucl. Phys B 129 (1977) 493. [3] G. Kohring, R.E. Shrock and E Wills, Phys. Rev. Lett. 57 (1986) 1358; Nucl. Phys. B 288 (1987) 397. [4] S.R. Shenoy, Phys. Rev. B 40 (1989) 5056. [5] M. Ferer, M.A. Mosre and M. Wortiz, Phys. Rev. B 8 (1973) 5205. [6] W. Janke and H. Kleinert, Nucl. Phys. B 270 (1986) 399. [7] W. Janke, Phys. Lett. A 148 (1990) 306. [8] A.E Gottlob and M. Hasenbusch, CERN TH 6885-93. [9] H. Kleinert, Phys. Lett. A 93 (1982) 86. [10] L. Del Debbio, A. Di Giacomo and G.Paffuti, Phys. Lett. B 349 (1995) 513. [11] L. Del Debbio, A. Di Giacomo, G. Paffuti and P. Pied, Phys. Lett. B 355 (1995) 255.
Nuclear Physics B 489 [FS] (1997) 739-747
Condensation of vortices in the XY model in 3D: a disorder parameter* G. Di C e c i o a, A. Di G i a c o m o b, G. Paffuti b'l, M. Trigiante c a Department of Physics and Astronomy, Louisiana State University, 102 Nicholson Hall, Baton Rouge, LA 70803-4001, USA b Dipartimento di Fisica and INFN, 2 Piazza Torricelli, 56100 Pisa, Italy c ISAS, Via Beirut 2, 34014 Trieste, Italy
Received 31 May 1996; revised 20 September 1996; accepted 20 December 1996
Abstract
A disorder parameter is constructed which signals the condensation of vortices. The construction is tested by numerical simulations. (~) 1997 Elsevier Science B.V. PACS: 75.40.Cx; 05.70.Jk; 64.60.Cn
1. Introduction
The XY model in three dimensions is an interesting system in statistical mechanics. Physically it describes the critical behaviour of superfluid He4 [ 1]. From the theoretical point of view it provides a relatively simple example of an order-disorder phase transition in which the condensation of solitons plays an essential role [2-5]. It is fairly well established that the transition is continuous, and the basic critical indices are known with good precision [6-8]. A Ginzburg-Landau kind of description of the system has been given, describing phenomenologically the condensation of vortices in the high temperature phase [9]. In this paper we shall study the system from a slightly different point of view. We shall look at it as the lattice formulation of a 2 + 1-dimensional quantum field theory. * Partially supported by MURST (Italian Ministry of the University and of Scientific and Technological Research) and by the EC contract CHEX-CT92-0051. l E-mail: [email protected] 0550-3213/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PH S 0 5 5 0 - 3 2 1 3 ( 9 7 ) 0 0 0 1 9 - 9
740
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747
We shall then show that the (two-dimensional) ground state of the theory spontaneously breaks the conservation of the number of vortices at high temperature; vortices condense in the vacuum in the same way as Cooper pairs do in a superconductor. In that phase the number of vortices is not defined in the vacuum, which is a superposition of states with different vorticity. We shall directly exhibit this condensation by showing that the vacuum expectation value (vev) of an operator which changes the number of vortices is non-zero. An analogous technique has been used to detect monopole condensation in the ground state of U(1) gauge theory [10] and in SU(2) gauge theory, as a mechanism for confinement of colour [ 11 ]. In Section 2 we shall briefly introduce the model and fix the notation. We shall also define a creation operator for a vortex. In Section 3 we shall present and discuss the results of numerical simulations where the vev of the above operator is measured. The result will be a direct detection of the spontaneous breaking of the conservation of the number of vortices in the disordered phase and a direct evidence of vortex condensation. We shall also obtain an alternative determination of known critical indices, which agrees with existing results, and measure the critical index related to our disorder parameter.
2. The creation operator of vortices The lattice action of the system is S = f l ~ (1 - - c o s ( A ~ O ( i ) ) ) . /z,i
(1)
The field variable is the angle 0 in the site i:/z runs from 0 to 2, 0 being the time axis. In the limit of zero lattice spacing (a --, 0) S '~ /3 ~ (A#0)2 a 2 q- O(a4) . 2 /z,i
(2)
At high fl's the system describes a massless free scalar particle. The couplings of higher dimension appearing in the series expansion of the cosine become important at lower /3. At /3c ~-- 0.454 the system undergoes a phase transition, which is known to be of second order [6]. The system has soliton configurations with the geometry of vortices: numerical simulations show that vortices play an important role in the phase transition [3-5]. A vector field
A~ = a~,o
(3)
can be defined, and a current j~ = e~,~#a,~A# associated with it, which is trivially conserved in smooth configurations:
a~ j~ = 31ze.g,~#3'~A # = O,
(4)
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747
741
because of the antisymmetry of the Ricci tensor. Eq. (4) is the analog of the Bianchi identity in QED. The constant of the motion is
q~= f d2xj°(x,x°) = / d 2 x (V AA) .
(5)
On the other hand, from Eq. (3)
~P = ~ A .
dx = n - 2 ~
(6)
,/
C
with C any closed path. A vortex is defined as a configuration for which n is non-zero,
O(x
( x - y)2 - y) = arctan - -
(7)
(x-y)1
is an example of such a configuration with n = 1. It has a singularity at x = y. Eqs. ( 3 ) - ( 6 ) tell that the number of vortices is a constant of motion, and define a U( 1 ) symmetry. We shall show that, for fl > tic, this symmetry is realized ~ la Wigner: for/3
eipalx) = Ix + a) can be used. The translation operator
ke(x,t)=expfi/d3yaoO(y,t)O(x-y)] indeed, when applied to the state
IO(x, t)},
(8) gives
tz(x,t)lO(y,t) ) = ]O(y,t) + O(x - y)).
(9)
The operator number of vortices
V(t) = / d 2 x j ° ( x , t)
(10)
has a commutation rule with/z, I v ( t ) , / x ( x , t)] = where A = Vt~.
f d 2 z • A A(z,
t ) / z ( x , t) = 2~'be(x, t ) ,
( 11 )
742
G. Di Cecio et aL/Nuclear Physics B 489 [FS] (1997) 739-747
If the U(1) symmetry, which counts the vortices, is realized ~ la Wigner, then the ground state must have a definite number of vortices and (/x) = 0. Since/z changes the number of vortices a, vev (/z) :# 0 signals their condensation. The naive translation of Eq. (8) on the lattice is [ 10]
Iz(n'n°)=exp[-flZsin(d°O(n"n°))O(n'-n'n°)] ' n '
(12)
where the site n t runs on the sites of a two-dimensional slice of the lattice at constant no, except the position of the vortex. Instead of Eq. (12) we shall use a compactified version of it. By this we mean /.t = exp - f l Z
(cos(d00 + 0) - cos(~00))]
/
I1 t
( 13)
which coincides with Eq. (12) at first order in 0. With the definition (13) (/z) is independent of the choice of the gauge for i.e. of the zero for the angle 0: a redefinition of it is reabsorbed in the integration variable. Two remarks are in order. (1) In principle one expects (/.t) -- 0 in the ordered phase fl > tic. In fact, (/z) is an analytic function of fl if the number of degrees of freedom is finite, and therefore, if it were zero for fl > it would be identically zero for all fl's. Only in the limit of infinite volume (thermodynamic limit) singularities develop, and (/z) can be identically zero above without being zero everywhere. Indeed a simple computation at large fl's, where the integral is approximately gaussian, gives
Au,
tic
tic,
(tz) ~ exp I-fl (cl Vl/3 + c2 + O( ~ ) ] .
(14)
(2) Due to periodic boundary conditions the total number of vortices must be zero as easily seen from Eq. (6) with the boundary C at spatial infinity. In principle then the correct way to determine (/z) is to observe a vortex-antivortex pair correlation
( ~ ( x ) ~ ( 0 ) ) x ~ o (~) 2 .
(15)
By the cluster property and C invariance it tends exponentially to (/x) 2 at large distances. In practice, since in the thermodynamical limit physics becomes independent of the boundary conditions and decorrelated from them at distances larger than the correlation length, what happens is that also a single vortex can be created, and a direct measurement of (/~) can be done. (/~) determined in this way is indeed independent of the boundary conditions within errors. When a single vortex is put on the lattice by our operator, a dislocation appears at the boundary with opposite vorticity to correct for the boundary conditions, and, if the lattice is big compared to the correlation length, what is obtained is the same (p)2 as with a vortex-antivortex pair at large relative distance.
G. Di Cecio et aL/Nuclear Physics B 489 [FS] (1997) 739-747
743
3. Numerical simulation and results
Simulations have been done on a 202 x 40 and 302 x 60 lattices to measure the (/2 At) correlation. Single vortices (At) have also been measured on the same lattices and on 203, 303 and 403 lattices. Instead of directly determining (At) as a function o f / 3 it is more convenient to measure [ 10] 1 d p = ~ - ~ ln(At) 2
and to
(At)2 as
reconstruct
(At)2 =exp
(16)
is } 2
p(fl) dfl
•
(17)
0
From the definition of (At)z, (At)
exp
=
[-fl(S + S~)]
,
(18)
with S defined by Eq. (1), z --
t,2
) exp
[-ns]
and
[cos( 0e(n',no)
"F nv(no)Z ~ ( n t --
no n' 4=nv(no)
nv(no) ) ) - cos( doO(n',no) )] (19)
it follows that [ 10] p = (S)s - ( s + S ' ) s + s , .
(20)
The subscripts S and S+S p in Eq. (20) indicate the action used to perform the average. In Eq. (19) the sum runs over the spatial locations n ~ different from the locations of vortices nv (no), in the time slices no where vortices are created or destroyed. In principle one could compute (At) directly. However, since the operator to be averaged in Eq. (18), exp(-flSP), is the exponential of a quantity which is roughly proportional to the spatial volume, V, and fluctuates as v/-V, its values fluctuate over many orders of magnitude and do not seem to obey a gaussian distribution. In fact the relative uncertainty does not decrease with increasing statistics, p does not suffers this disease. Determining p, instead of (At)2, is analogous to determine the internal energy of a system instead of its partition function. The observed shape of p is shown in Fig. 1. The sharp negative peak at tic indicates a steep drop of (At) toward zero at tic. At low /3's p is compatible with zero and
744
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 500.0
0.0
........ Ii
•
•
a
o
- - - • - -
- - _ • _ _
P
-500.0
Lattice 203 o Lattice 30 s • Lattice 40 a
-I~0.0
•
i
-1500.0.0
I
0.5
I
1.0
1.5
Fig. l. p as a function of/3. The dashed lines at high values of/3 are the perturbative estimates. ,
10.0
- -
0.0
P
-10.0 • ~---0.15
~-0.30 -20.0
"30"00.000
* IL-0.40
I
I
I
I
t
0.010
0.020
0.030 1/L
0.040
0.050
0.060
Fig. 2. p as a function of 1/L (L = size of the lattice) in the condensed phase, p converges to a value consistent with zero. The abscissae for different betas are slightly displaced for graphical reasons.
independent of V (Fig. 2). This means by Eq. (17) that limv_o~(/z) ~- 1 for fl < tic, and thus that vortices do condense in the vacuum. An estimate of (/z) by strong coupling expansion at low fl gives p '~ - k f l 3
(21)
with k a positive constant: numerically for the configurations we have studied k ___0.30.5 so that for fl < 0.4 the estimate (21) is smaller than the errors in Fig. 2, which are typically of the order of a few units, and consistent with our numerical determinations. At fl > tic the behaviour of Eq. (14) can be checked by numerical calculation of the gaussian integral. For a cubic lattice with a vortex in the centre the result is
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 0.0
,
,
-1.0
++
-2.0
~
745
++
÷+
~
+
!
" Lattice 202x40 o Lattice 302x60
-3.0
"4'%.5
0.0
0.5
1.0
1.5
(rJ:{)L '~' Fig. 3. Quality of the finite size scaling analysis. The figure corresponds to optimal values of tic and u.
p = -11.332. L + 72.669+0(1/fl).
(22)
The comparison with numerical simulations is shown in Fig. 1. Eq. (22) means that (#z)2 is exactly zero in the limit V -+ co. Eq. (22) agrees with the observed values of p at large fl's (Fig. 1). Near tic, where the correlation length becomes large, a finite size analysis can be done to explore the limit of infinite volume and determine the critical index 77of #z, tic, and the critical index ~, of the correlation length. As ( f l - f l ~ ) - ~ 0 - and V ~ c~
(Iz) ~" (tic - fl)~.
(23)
At finite L, (#z)=(/x)(~,~)
.
(24)
As fl --+ tic, ~ diverges, ~ ,-~ (tic - f l ) - ~ and the dependence on a l ~ can be neglected, \
/
(25)
or
c/v (c/'(nc- n))
(26)
p l L l i " in the scaling region must be a universal function of L l l u ( f l c - f l ) . The quality
of this scaling is shown in Fig. 3. The critical index u and the critical temperature tic are determined by enforcing this scaling behaviour. Fig. 4 shows the data for a vortex-antivortex pair at large distance, compared to the single vortex data. In the condensed phase the single vortex behaves in fact as a pair
746
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 200.0
0.0
t
i
........| o
- - - t ~ - -
-200.0 Q
-400.0
• Lattice 202x40 vortex
o Lattice 202x40 pair d=l 6
l -600"00.0
e
i
i
0.5
1.0
L
1.5
Fig. 4. Comparisonof p for single vortex to p for vortex-antivortexpair at large distance. The two curves coincide within errors in the condensed phase. They differ at high fl and agree with the perturbative determination (dashed lines on the right side). at large distance, confirming that there is no problem in the thermodynamical limit. At /3 >/3c the two quantities are different and at/3 sufficiently large are well described by Eq. (14) with the different appropriate values of Cl and c2, as shown in the figure. A more precise finite size scaling analysis assuming a specific form for f for/3 < tic,
f(LU~(/3c -/3)) =
¢5
L~/~ (/3c -/3) + v,
(27)
+ v~
gives a precise determination of the critical indices and of tic, 6 = 0 . 7 4 0 :i: 0.029,
(28)
/3c = 0.4538 ± 0.0003,
(29)
v = 0 . 6 6 9 i 0.065,
(30)
with x Z / d o f = 1.07. For comparison the determination of Ref. [8] is v = 0.670(7) and /3c = 0.45419(2). The behaviour of the correlator (/.t(x)/2(0)) as a function of the distance gives an estimate of the correlation length scM(/3), (/.t(x)/.t(0)) - (/.t) z ~ exp [ - x / ( M ( / 3 ) ] .
(31)
~:M is of the order of a few lattice spacings (see Fig. 5). s~M(/3) should scale as (/3c - / 3 ) - ~ as (/3 - / 3 c ) --~ 0 - . With our relatively large statistical errors we were not able to check this behaviour.
747
G. Di Cecio et al./Nuclear Physics B 489 [FS] (1997) 739-747 0.0
-50.0
-100.0
• 13=0.448 • [~--0.440 "150"05.0
.
.
.
.
i
10.0
i
i
15.0 d
20.0
25.0
Fig. 5. p as a function of distance at two values of ft. The estimated correlation length is few lattice spacings. 4. C o n c l u d i n g r e m a r k s We have constructed a disorder parameter for the ,~ transition in the 3D XY model which is the vev o f an operator with non-trivial vortex number. It is different from zero at high temperatures ( f l < t i c ) in the disordered phase, thus demonstrating condensation of vortices. In the ordered phase it tends to zero exponentially with the linear size of the lattice. tic and the critical index ~, of the correlation length can be determined with good precision by a finite size scaling analysis o f our disorder parameter (/x). As ( f l - tic) O - tz ,,~ ( t i c - f l ) 8 with 8 = 0.74 + 0.03.
A similar construction also works for other models u n d e r g o i n g an o r d e r - d i s o r d e r transition produced by condensation o f solitons. A n example is the condensation o f m o n o p o l e s in gauge theories [10,11].
References [11 R.L. Onsager, Nuovo Cimento Suppl. 6 (1949) 249; R.P. Feynman, in Progress in Low Temperature Physics, ed. C.J. Gorter, Vol. 1 (North-Holland, Amsterdam, 1955) p. 17. [2] T. Banks, R. Meyerson and J.B. Kogut, Nucl. Phys B 129 (1977) 493. [3] G. Kohring, R.E. Shrock and E Wills, Phys. Rev. Lett. 57 (1986) 1358; Nucl. Phys. B 288 (1987) 397. [4] S.R. Shenoy, Phys. Rev. B 40 (1989) 5056. [5] M. Ferer, M.A. Mosre and M. Wortiz, Phys. Rev. B 8 (1973) 5205. [6] W. Janke and H. Kleinert, Nucl. Phys. B 270 (1986) 399. [7] W. Janke, Phys. Lett. A 148 (1990) 306. [8] A.E Gottlob and M. Hasenbusch, CERN TH 6885-93. [9] H. Kleinert, Phys. Lett. A 93 (1982) 86. [10] L. Del Debbio, A. Di Giacomo and G.Paffuti, Phys. Lett. B 349 (1995) 513. [11] L. Del Debbio, A. Di Giacomo, G. Paffuti and P. Pied, Phys. Lett. B 355 (1995) 255.