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A Monte Carlo study of the XY model
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
A MONTE CARLO STUDY OF THE X Y MODEL F. FUCITO 1 Istituto di Fisica "G. Marconi", Universithdi Roma, Rome, Italy INFN Sezione di Roma, Rome, Italy and California Institute of Technology, Pasadena, CA 91125, USA Received 11 October 1982
Spin systems which have a phase transition without an order parameter, like the X Y model, have great importance in physics. Here we show how it is possible to study the behaviour of such a system by computer simulation without studying the spin-spin correlation functions.
Planar spin systems are believed not to have a longrange order associated to a low-temperature phase. One can show, using spin wave theory, that no spontaneous magnetization exists. Nevertheless, computer simulations [1] and high-temperature series expansions [2], suggest that a phase transition does occur. Kosterlitz and Thouless [3] have proposed a theoretical model for such systems. The physical scenario of this model is the following: a standard "phase transition" does not occur in the sense that there is no order parameter. The spin-spin correlation function has a power-law behaviour under a "critical" temperature Tc, and an exponential decay for T > T c. The behaviour of the spin-spin correlation functions is explained by the existence of pairs of topological defects. In a magnet (for the sake of simplicity, from now on we will consider only these systems) these topological defects are spin vortices. For T < T c they are bound in pairs while for T > T c they unbind disordering the system. The purpose of this work is to show how it is possible to check this kind of behaviour without directly measure spin-spin correlation functions which are highly fluctuating quantities. We will perform our simulation on the simple X Y model because this model has been studied in great detail both theoretically and with Monte Carlo methods [4] ; so we can compare 1 Weingart Fellow in Theoretical Physics.
our findings with the ones in literature. Anyway, apart from these reasons, this method can be used for any system that has an XY-type behaviour (i.e. recently [5], it has been suggested that the q = 3 , D = 2 antiferromagnetic Potts model may exhibit such a behaviour). These models have a great physical relevance not only because the problem is interesting in itself but also because of wide analogies with four-dimensional abelian lattice gauge theories and the problem of confmement [6]. The hamiltonian of our model is H = - T -1 ~ cos(O i - 0/), ~ij>
(1)
where T stands for k B T / J , J being the coupling strength, 0 i is the angle between the ith spin and some fixed axis in the plane, and the sum runs over nearest-neighbours spin pairs (each pair is counted once).
The spins are placed on a square lattice with periodic boundary conditions just in one direction. In the other direction the angles 0 i have been forced to be zero at the boundaries. This corresponds to the effects of a magnetic field acting at the boundaries which forces these spins to be oriented along a fixed direction. An approach similar to this, but in a different context, has been studied in ref. [7]. The simulation has been performed using the tradi-
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
99
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
B
5
8
0.832
A
~. 0.77E
0.72C
A
A
0.664
B 0.60E , 0.0
I
2.5
i
I
5.0
~
I
7.5
, "'.J
A ~ I
10.0
,
12.5
I
15.0
,
I
17.5
~
l
20.0
L
I
22.5
BISTANCE
Fig. 1. Power-law fit to our data at T = 0.7 after averagingthe magnetization over all our measures. The values of the x ~ test is 2.9 X 10 -2 . (A) Experimental curve; (B) fit curve. tional Metropolis Monte Carlo algorithm [8]. The change of the angle Oi was adjusted to give an acceptance ratio between 0.4 and 0.5. Our system contains 62 × 62 spins. We gathered results for temperature 0.7 ~< T~< 1.3 spaced by 0.1. We did this choice considering previous Monte Carlo work [4] that suggests the critical temperature lying between T = 0.85 and T = 0.95. We collected our first results starting, for T = 0.7 (T = 1.3), from a low- (high-) temperature configuration and waiting the system to reach equilibrium. For 0.7 < T < 1.3 we brought the system to equilibrium starting from the equilibrated configuration reached by the preceding temperature. As we have a magnetic field at the boundaries we look at the decay of the magnetization as a function of the distance at fbxed temperature. For the sake of simplicity, we chose the component cos 0 i of the magnetization. The expected behaviour of m (x) is: rn(x)cc l / i x l ~,
T
m ( x ) ~ x e -~xl/~,
T>T c .
(2)
For each distance in the x direction, we averaged all the spins with the same x and then we grouped them in subsets. The original lattice had 62 distances out of which we obtained 20 subsets. As we have a magnetic field at both boundaries of our lattice we 100
then performed a left-right average of these values, to force them to be symmetric with respect to the center of the lattice. At last we fit the functional forms of eq. (2). To respect the periodicity of the lattice, for T < T c, we fit a functional form of the type A / I x l ~ + B / I L - x[ ~ ,
(3)
where L is the total length of the lattice. In figs. 1 and 2 two typical fits for the functional forms of eq. (2) are shown 4-1. In our system the energy reaches equilibrium very early while the magnetization is much slower. For this reason we created, for each value of the temperature, a configuration thermalized with 12500 Monte Carlo steps. Then we took separate sets of measures every 3000 Monte Carlo steps. We checked that these different sets of measures were independent one from the other. So it is possible to give an estimate of the statistical error. For T = 0.8, 0.9 we had to take more than five measures because these points are very close to the transition and this causes a further instability. So, in the whole, we did 27500 Monte Carlo steps for T = 0.7
4-1 We would like to draw the reader's attention to the fact that the values of magnetizations in fig. i are such that the asymptotic behaviour of m (x) [true for small m (x)] given in eq. (2), could be spoiled. For higher T, anyway, we get smaller magnetization values.
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
Table 2 Values of 8 as they came out of our powerqaw fit to our lowtemperature data. The measures with a star have a large value
g
of x 2 .
Measure
Z
T
0.7 0 1 2 3 4 5 6 7 8
10-I
I
I
i
I
I
I
I
I
i
I
1
2
3
4
5
6
7
8
9
10
0.248 0.229 0.337 0.396* 0.224
0.8
0.9
1.0
1.1
0.215 0.141 0.832* 0.726* 0.549* 0.363 0.250 0.157 0.149
0.424* 0.280 0.182 0.215 0.296 0.473
0.421 0.578 0.327* 0.371 0.514
0.980* 0.976* 0.933* 1" 1"
8 is obtained by the power-law fit to our low-temperature data. We used the fit program o f the C E R N library called MINUIT. In table 4 we show the values o f 8 and obtained fitting our data w i t h the functional forms of eq. (2), after averaging the magnetization at each disDISTANCE
Fig. 2, Linear fit to our data at T = 1.2 alter averaging all magnetizations over all measures. The line is drawn to guide the eye. and 1.0 ~< T ~< 1.3, 3 0 5 0 0 Monte Carlo steps for T = 0.9 and 39500 Monte Carlo steps for T = 0.8. Notice that most o f the measures for 0.8 ~< T~< 1.0 fit b o t h the power-law behaviour and the e x p o n e n tial decay. The criterium with which we have judged the goodness o f the power-law fit is the standard X2. In table 1 we give the values o f the m e a n energy for the various temperatures, averaged over the last 3 0 0 0 Monte Carlo steps. In tables 2 and 3 we give the various measures for 8 and ~. ~ is extracted as the slope o f the linear fit to our high-temperature data;
Table 3 Values of ~ as the slope of our linear fit to our high-temperature data. The data with a star are fit very badly by a straight line. Measure
T O.8
0 1 2 3 4 5 6 7 8
35.2* 11" 9.2 6.1 12 12.2 14.7" 11.0" 14.7"
0.9 15.2 13.2* 13.2" 22* 17.6 • 10.4
1.0
1.1
1.2
1.3
44* 9.8* 5.8 5.5 5.5' 8.0
4.5 2.9 3.9 2.9 2.5
1.6 0.98 1.4 1.3 1.3
1.2 0.97 1.1 1.3 0.8
Table 1 Mean values of the internal energy. T 0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.5985-+0.0002
1.5266-+0.0002
1.4421-+0.0007
1.3358-+0.0007
1.2017-+0.0032
1.0732-+0.0006
0.9634-+0.0008
101
PHYSICS LETTERS
Volume 94A, number 2
Table 4 Values of 8 and ~ from fits of the magnetizations averaged over all measures. T
~
8
0.7 0.8 0.9
13.2 9.4
0.29 0.35 0.31
1.0
5.6
0.44
1.1
3.2
1
1.2 1,3
1.3 1.0
tance over all our measures. These values are almost the same which could be obtained averaging the values o f tables 2 and 3. All fits for the averaged magnetizations at 0.7 <~ T ~< 1.0 have a good X2 . Linear fits for the averaged magnetizations at 0.8 < T ~< 1.0 seem to be affected by a large systematic error, opposite to the majority o f those from which we extracted in table 3. This suggests that this is the region o f the transition. An estimate o f the statistical error is given by the mean square deviation of measures in tables 2 and 3. It ranges from 20 to 100%. Now, at last, we can draw some conclusions: looking at the magnetization in a square lattice, with a magnetic field at the boundaries, it is possible to study planar systems which have no order parameter. With this method we can easily extract qualitative informations even if for a quantitative type of information a very high statistics is required. F r o m the experience we gained in this work it does not seem justified to let the system internal energy thermalize with only few thousands of Monte Carlo steps and then fit the theoretical prediction to extract reliable measures. In fact it is enough to take another measure after a few thousands o f Monte Carlo steps to get a result
102
28 February 1983
which is usually quite far from the preceding one. This because the thermalization o f the internal energy of the system is much faster than the thermalization of the magnetization or of the correlation functions. This work required 30 min of A.P. FS 190L and around 50 h o f V A X 11/780. The author is grateful to G. Parisi for pointing out the problem to him and for many useful conversations. He would like to thank also K. Wilson and the Theory Group of the Newman Laboratory at CorneU University for warm hospitality and for the use of the A.P. FS 190-L. His stay at Cornell University has been possible due to support o f the Foundation "A. Della Riccia", Florence, Italy.
References [1] D.A. Young and B.J. Alder, J. Chem. Phys. 60 (1974) 1254; R.C. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B20 (1975) 326. [2] H.E. Stanley and T.A. Kaplan, Phys. Rev. Lett. 17 (1966) 913. [3] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [4] J. Tobochnik and G.V. Chester, Phys. Rev. B20 (1979) 3761; S. Miyashita, H. Nishimori, A. Kuroda and M. Suzuki, Prog. Theor. Phys. 60 (1978) 1669; S. Miyashita, Prog. Theor. Phys. 63 (1978) 797; 65 (1980) 1595; [5] A.N. Berker and L.P. Kadanoff, J. Phys. A13 (1980) L259; G.C.Grest and J.R. Banavar, Phys. Rev. Lett. 46 (1981) 1458. [6] J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659. [7] K.H. MIlller and K. Schilling, Nucl. Phys. B200 (1982) 362. [8] K. Binder, in: Monte Carlo methods, ed. K. Binder (Springer, Berlin, 1979).
PHYSICS LETTERS
28 February 1983
A MONTE CARLO STUDY OF THE X Y MODEL F. FUCITO 1 Istituto di Fisica "G. Marconi", Universithdi Roma, Rome, Italy INFN Sezione di Roma, Rome, Italy and California Institute of Technology, Pasadena, CA 91125, USA Received 11 October 1982
Spin systems which have a phase transition without an order parameter, like the X Y model, have great importance in physics. Here we show how it is possible to study the behaviour of such a system by computer simulation without studying the spin-spin correlation functions.
Planar spin systems are believed not to have a longrange order associated to a low-temperature phase. One can show, using spin wave theory, that no spontaneous magnetization exists. Nevertheless, computer simulations [1] and high-temperature series expansions [2], suggest that a phase transition does occur. Kosterlitz and Thouless [3] have proposed a theoretical model for such systems. The physical scenario of this model is the following: a standard "phase transition" does not occur in the sense that there is no order parameter. The spin-spin correlation function has a power-law behaviour under a "critical" temperature Tc, and an exponential decay for T > T c. The behaviour of the spin-spin correlation functions is explained by the existence of pairs of topological defects. In a magnet (for the sake of simplicity, from now on we will consider only these systems) these topological defects are spin vortices. For T < T c they are bound in pairs while for T > T c they unbind disordering the system. The purpose of this work is to show how it is possible to check this kind of behaviour without directly measure spin-spin correlation functions which are highly fluctuating quantities. We will perform our simulation on the simple X Y model because this model has been studied in great detail both theoretically and with Monte Carlo methods [4] ; so we can compare 1 Weingart Fellow in Theoretical Physics.
our findings with the ones in literature. Anyway, apart from these reasons, this method can be used for any system that has an XY-type behaviour (i.e. recently [5], it has been suggested that the q = 3 , D = 2 antiferromagnetic Potts model may exhibit such a behaviour). These models have a great physical relevance not only because the problem is interesting in itself but also because of wide analogies with four-dimensional abelian lattice gauge theories and the problem of confmement [6]. The hamiltonian of our model is H = - T -1 ~ cos(O i - 0/), ~ij>
(1)
where T stands for k B T / J , J being the coupling strength, 0 i is the angle between the ith spin and some fixed axis in the plane, and the sum runs over nearest-neighbours spin pairs (each pair is counted once).
The spins are placed on a square lattice with periodic boundary conditions just in one direction. In the other direction the angles 0 i have been forced to be zero at the boundaries. This corresponds to the effects of a magnetic field acting at the boundaries which forces these spins to be oriented along a fixed direction. An approach similar to this, but in a different context, has been studied in ref. [7]. The simulation has been performed using the tradi-
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
99
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
B
5
8
0.832
A
~. 0.77E
0.72C
A
A
0.664
B 0.60E , 0.0
I
2.5
i
I
5.0
~
I
7.5
, "'.J
A ~ I
10.0
,
12.5
I
15.0
,
I
17.5
~
l
20.0
L
I
22.5
BISTANCE
Fig. 1. Power-law fit to our data at T = 0.7 after averagingthe magnetization over all our measures. The values of the x ~ test is 2.9 X 10 -2 . (A) Experimental curve; (B) fit curve. tional Metropolis Monte Carlo algorithm [8]. The change of the angle Oi was adjusted to give an acceptance ratio between 0.4 and 0.5. Our system contains 62 × 62 spins. We gathered results for temperature 0.7 ~< T~< 1.3 spaced by 0.1. We did this choice considering previous Monte Carlo work [4] that suggests the critical temperature lying between T = 0.85 and T = 0.95. We collected our first results starting, for T = 0.7 (T = 1.3), from a low- (high-) temperature configuration and waiting the system to reach equilibrium. For 0.7 < T < 1.3 we brought the system to equilibrium starting from the equilibrated configuration reached by the preceding temperature. As we have a magnetic field at the boundaries we look at the decay of the magnetization as a function of the distance at fbxed temperature. For the sake of simplicity, we chose the component cos 0 i of the magnetization. The expected behaviour of m (x) is: rn(x)cc l / i x l ~,
T
m ( x ) ~ x e -~xl/~,
T>T c .
(2)
For each distance in the x direction, we averaged all the spins with the same x and then we grouped them in subsets. The original lattice had 62 distances out of which we obtained 20 subsets. As we have a magnetic field at both boundaries of our lattice we 100
then performed a left-right average of these values, to force them to be symmetric with respect to the center of the lattice. At last we fit the functional forms of eq. (2). To respect the periodicity of the lattice, for T < T c, we fit a functional form of the type A / I x l ~ + B / I L - x[ ~ ,
(3)
where L is the total length of the lattice. In figs. 1 and 2 two typical fits for the functional forms of eq. (2) are shown 4-1. In our system the energy reaches equilibrium very early while the magnetization is much slower. For this reason we created, for each value of the temperature, a configuration thermalized with 12500 Monte Carlo steps. Then we took separate sets of measures every 3000 Monte Carlo steps. We checked that these different sets of measures were independent one from the other. So it is possible to give an estimate of the statistical error. For T = 0.8, 0.9 we had to take more than five measures because these points are very close to the transition and this causes a further instability. So, in the whole, we did 27500 Monte Carlo steps for T = 0.7
4-1 We would like to draw the reader's attention to the fact that the values of magnetizations in fig. i are such that the asymptotic behaviour of m (x) [true for small m (x)] given in eq. (2), could be spoiled. For higher T, anyway, we get smaller magnetization values.
Volume 94A, number 2
PHYSICS LETTERS
28 February 1983
Table 2 Values of 8 as they came out of our powerqaw fit to our lowtemperature data. The measures with a star have a large value
g
of x 2 .
Measure
Z
T
0.7 0 1 2 3 4 5 6 7 8
10-I
I
I
i
I
I
I
I
I
i
I
1
2
3
4
5
6
7
8
9
10
0.248 0.229 0.337 0.396* 0.224
0.8
0.9
1.0
1.1
0.215 0.141 0.832* 0.726* 0.549* 0.363 0.250 0.157 0.149
0.424* 0.280 0.182 0.215 0.296 0.473
0.421 0.578 0.327* 0.371 0.514
0.980* 0.976* 0.933* 1" 1"
8 is obtained by the power-law fit to our low-temperature data. We used the fit program o f the C E R N library called MINUIT. In table 4 we show the values o f 8 and obtained fitting our data w i t h the functional forms of eq. (2), after averaging the magnetization at each disDISTANCE
Fig. 2, Linear fit to our data at T = 1.2 alter averaging all magnetizations over all measures. The line is drawn to guide the eye. and 1.0 ~< T ~< 1.3, 3 0 5 0 0 Monte Carlo steps for T = 0.9 and 39500 Monte Carlo steps for T = 0.8. Notice that most o f the measures for 0.8 ~< T~< 1.0 fit b o t h the power-law behaviour and the e x p o n e n tial decay. The criterium with which we have judged the goodness o f the power-law fit is the standard X2. In table 1 we give the values o f the m e a n energy for the various temperatures, averaged over the last 3 0 0 0 Monte Carlo steps. In tables 2 and 3 we give the various measures for 8 and ~. ~ is extracted as the slope o f the linear fit to our high-temperature data;
Table 3 Values of ~ as the slope of our linear fit to our high-temperature data. The data with a star are fit very badly by a straight line. Measure
T O.8
0 1 2 3 4 5 6 7 8
35.2* 11" 9.2 6.1 12 12.2 14.7" 11.0" 14.7"
0.9 15.2 13.2* 13.2" 22* 17.6 • 10.4
1.0
1.1
1.2
1.3
44* 9.8* 5.8 5.5 5.5' 8.0
4.5 2.9 3.9 2.9 2.5
1.6 0.98 1.4 1.3 1.3
1.2 0.97 1.1 1.3 0.8
Table 1 Mean values of the internal energy. T 0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.5985-+0.0002
1.5266-+0.0002
1.4421-+0.0007
1.3358-+0.0007
1.2017-+0.0032
1.0732-+0.0006
0.9634-+0.0008
101
PHYSICS LETTERS
Volume 94A, number 2
Table 4 Values of 8 and ~ from fits of the magnetizations averaged over all measures. T
~
8
0.7 0.8 0.9
13.2 9.4
0.29 0.35 0.31
1.0
5.6
0.44
1.1
3.2
1
1.2 1,3
1.3 1.0
tance over all our measures. These values are almost the same which could be obtained averaging the values o f tables 2 and 3. All fits for the averaged magnetizations at 0.7 <~ T ~< 1.0 have a good X2 . Linear fits for the averaged magnetizations at 0.8 < T ~< 1.0 seem to be affected by a large systematic error, opposite to the majority o f those from which we extracted in table 3. This suggests that this is the region o f the transition. An estimate o f the statistical error is given by the mean square deviation of measures in tables 2 and 3. It ranges from 20 to 100%. Now, at last, we can draw some conclusions: looking at the magnetization in a square lattice, with a magnetic field at the boundaries, it is possible to study planar systems which have no order parameter. With this method we can easily extract qualitative informations even if for a quantitative type of information a very high statistics is required. F r o m the experience we gained in this work it does not seem justified to let the system internal energy thermalize with only few thousands of Monte Carlo steps and then fit the theoretical prediction to extract reliable measures. In fact it is enough to take another measure after a few thousands o f Monte Carlo steps to get a result
102
28 February 1983
which is usually quite far from the preceding one. This because the thermalization o f the internal energy of the system is much faster than the thermalization of the magnetization or of the correlation functions. This work required 30 min of A.P. FS 190L and around 50 h o f V A X 11/780. The author is grateful to G. Parisi for pointing out the problem to him and for many useful conversations. He would like to thank also K. Wilson and the Theory Group of the Newman Laboratory at CorneU University for warm hospitality and for the use of the A.P. FS 190-L. His stay at Cornell University has been possible due to support o f the Foundation "A. Della Riccia", Florence, Italy.
References [1] D.A. Young and B.J. Alder, J. Chem. Phys. 60 (1974) 1254; R.C. Gann, S. Chakravarty and G.V. Chester, Phys. Rev. B20 (1975) 326. [2] H.E. Stanley and T.A. Kaplan, Phys. Rev. Lett. 17 (1966) 913. [3] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181; J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [4] J. Tobochnik and G.V. Chester, Phys. Rev. B20 (1979) 3761; S. Miyashita, H. Nishimori, A. Kuroda and M. Suzuki, Prog. Theor. Phys. 60 (1978) 1669; S. Miyashita, Prog. Theor. Phys. 63 (1978) 797; 65 (1980) 1595; [5] A.N. Berker and L.P. Kadanoff, J. Phys. A13 (1980) L259; G.C.Grest and J.R. Banavar, Phys. Rev. Lett. 46 (1981) 1458. [6] J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659. [7] K.H. MIlller and K. Schilling, Nucl. Phys. B200 (1982) 362. [8] K. Binder, in: Monte Carlo methods, ed. K. Binder (Springer, Berlin, 1979).