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Multigrid method for the 2-d XY model
Nuclear Physics B (Proc . Suppl.) 17 (1990) 354-358 North-Holland
354
MULTIGRID METHOD FOR THE 2-D XY MODEL Arjan Hulsebos, Jan Smit, Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands and Jeroen C. Vink, Center for High Energy Astrophysics, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands . We describe a multigrid method for spin systems and we test it on the two dimensional XY model using the Langevin algorithm . The method appears to reduce z to somewhere between 0.5 and 1 .0.
In performing simulations near a critical point one is confronted with the problemofcritical slowing down: near a critical point the autocorrelation time -r grows as -r oc CZ , where C is the correlation length of the system . For conventional local updating algorithms, z >_ 2. The use of overrelaxation will reduce z to about 1 i .2, although being a local updating algorithm . The cluster Swendsen-Wang-Wolff algorithm 3 results in a z which appears to be consistent with zero. Multigrid methods are also very promising, for a review, see ref. 4. We describe a multigrid method which we apply to the 2-d XY model using the Langevin algorithm. This method is also presented in ref. 4. We consider a spin system invariant under the symmetry group G x G. Its partition function is given by 2
= f DV
exp S(V)
c where DV is the Haar measure of the group G, and the action S can be written as: S
=E x,'a
Re Tr ( Vtp.,AV.+a) .
(2)
The models are defined by f=,w = Q independent of x and A, but in (2) we generalized ~Q to a matrix which may depend on x and A, for reasons which will become clear at short notice. *Presented by A. Hulsebos. 0920-5632/90/$3 .50 © Elsevier Science Publishers B.V. North-Holland
We now divide the lattice into 2 x 2 blocks By, labelled by Y (see fig . l) and we make a transformation of variables Vx = -r.Vff For instance, take V to be equal to Vl , and ir_ = V Vt , i = 2, 3, 4. The Haar measure is invariant under this transformation, dV1dV2dV3dV4 = dV dir2dw3d7r4 (cf. fig. 1). Inserting (3) into (2) leads to S
x,ß
E
Re Tr
(VE ,~Oz,,a
(4)
Vx+,a
-I- ( terms independent of V))
,
where XEBy x+AEB=+j ,
7r.t
N.,a
7rx+K
(5)
When updating Vx --> WVF, effectively each Vx in a block BE gets multiplied by the same group element W. _ As the action in terms of V has the same form as the action in terms of V, we can repeat the transformation (3), but now in terms of V's and L"s . In this way we can produce a collection of effective actions on
A. Rulsebos et al./ Multigrid method for the 2-D XY model
N 8
0
S
y
8
.. ... ... .. .
........... . i
z
i
8 8 16
.. ..... ..
~1
2~ '
.. ........ .... . ,
16
V
16
S
' ...... ...
. . .. .... ..
16 32 32 32
Figure 1: The blocks B.
32
2n-1 x 2n-! lattices, l = 1, . . . , n, if we start from a 2n x 2n lattice. All these effective actions are of the form (4) . For the model we consider here, the XY model, the group G is the group U(1) and Vx = exp (i9. ) while on the finest level P is a positive real number, independent of x and A. In our multigrid method s, we used W-cycles and the following first order Langevin algorithm,
64
9 x+1 = 6 -f. e F.' =
fla
Fxr, + 2e rl'
Im [exp ( -i0x )ß.,g exp (ie+, a)
where 77 is the standard noise term. One update consisted of n Langevin steps with stepsize e, chosen in such a way that ne = 0.5, i.e. the Langevin time tL = 0.5. Notice that on a 1 x 1 lattice an update reduces to just the addition of a random phase to the spin V on that lattice. In order to check the algorithm, we first made runs of 1100 half W-cycles, while discarding the first 100 measurements for thermalization . Each time the finest lattice was visited, we measured the following observables m = 1 2 Re
(magnetization)
64
ß
0.78
e
0.01 0.005 0.82 0.01 0.005 0.86 0.01 0.005 0.90 0.01 0.005 0.78 0.01 ' 0.005 0.82 0.01 0.002 0.86 0.01 0.005 0.90 0.01 0.002 0.82 0.01 0.005 0.86 0.01 0.005 0.90 0.01 0.005 0.92 0.01 0.005 1/1.07 0.01 0 .005 1/1.055 0.01 0.0025
355 E
0.992(5) 1 .007(5) 1 .079(5) 1.088(5) 1 .154(5) 1 .158(5) 1.233(5) 1.234(5) 0.947(3) 0.951(3) 1.012(3) 1.031(3) 1.081(3) 1.097(3) 1.162(3) 1.182(2) 1.001(1) 1.012(1) 1.065(1) 1.079(1) 1.134(1) 1.151(1) 1.174(1) 1.186(1) 1 .1953(6) 1.2085(7) 1 .2213(6) 1.2361(6)
X(e) 17.2(3) 17 .6(3) 21 .3(3) 21.3(3) 24 .7(3) 24 .4(3) 28.7(3] 28 .4(3) 22 .1(6) 22 .1(6) 29 .8(7) 34 .1(7) 41 .3(8) 46 .6(9) 61(1) 67 .4(9) 29 .7(6) 33 .2(9) 45(2) 50(1) 77(2) 84(2) 127(3) 124(3) 143(4) 182(5) 221(5) 271(6)
X(0), X 18 .0(3) 23 .9(1) 21 .3(3) 36 .3(2) 24 .1(3) 60 .0(3) c8 .1(3) 111.43(44) 22 .1(6) 23 .9(1) 35 .2(7) 36 .3(2) 51.9(1 .0) 60 .0(3) 69.0(9) 111.43(44) 36 .7(1.2) 36.3(3) 55(1) 60.0(3) 91(2) 111.43(44) 121(3) 162.68(61) 221(6) 220.8(9) 288(6) 302.2(9)
Table 1 : The results for the energy and the magnetic susceptibility. =-2
E
(magnetic susceptibility) N2ß (energy)
The results are presented in table 1. In the last column X(0) (upper value) is the susceptibility obtained after malting a linear extrapolation to e = 0, while X (lower value) is taken from refs. 2 and 3, for which finite size effects are presumably negligible . The results for m are within errors equal to zero. In fig. 2, we have plotted X(0)/X versus t/N, where we also took C from refs . 2 and 3. For small l; I N we find consistency with refs . 2,3, which indicates that the simple linear extrapolation in e works. The deviations for t/N>0.25 indicate finite size scaling. To study critical slowing down, we then made runs of 2050 full W-cycles, again discarding 100 measure-
356
A. Hulse6os et a1./Multigrid method for the 2-D XY model
f_Tht~_li_i_______
N 8 16 16 32 32
ß 0.70 0.78 0.82 0.86 0 .90
= 0.01) 12.0(1) 22.5(3) 31.5(4) 45(1) 77(1)
X(E
T
2.8(1.0) 3.4(6) 5.8(2.0) 4.4(1.2) 8(2)
~(X)T~--~ 2.1 3.2 3.8 5.0 6.5
2.24 3.36 4.32 5.843 8.476
TFyble 2: The results for the autocorrelation time of the susceptibility.
1
slowing down. Note that for m this result is trivial, because of the random phase update for the 1 x 1 lattice. The results for X are given in table 2. In tiûs table l;(X) is the correlation length corresponding to the X(e = 0.01) value. The relation between X and ~ was taken from refs . 2,3 by linear interpolation . .2
.3
.
.5 .5 .7 .8 .9
Figure 2: Finite size eflFects for the magnetic susceptibility X .
We expect that ~(X) is a reasonable estimate of the actual correlation length at e = 0.01 . The errors on ~ were obtained from binning . In order to find the critical exponent x we made a log(T)-log(1;) and a log(r)-1og(~(x)) plot (See fig . 3). We found x = 0.6fi(34) and z(X) = 0.73(43) .
meats, on (N,ß) _ (8,0.70), (16,0.78), (10,0.82), (32,0.86) slid (32,0 .90) for e = 0.01. For these values ~(Q)lN < 0.25 and there is little dependence on the ~/N . For the observables, we also calculated the autocorrelation function A and the integrated auto~orrelation function C, defined as ( ) - B(t) A t , B(0) 7ma: -t B(t) . O(j + t)O(j) ( 10) j+nas - t ~ j-1
;~na:_t
f 9max
O(.7)lt -t ~ jmax -
jmas t
i_~1
~(~)~
C(t) _ Thc irtegrat~ed autocorrelation tlme
T
is defin~ed as:
where T is the value of t at the first intersection of C(t) with the line C = t/6 s . From this, we saw that only X remains as an observable still suf%ring from critical slowing down. The other observables, m and E, show no signs of critical
Figure 3: T vs. ~ (solid dots and solid line) and ~(X) (open dots and dashed line).
T
vs.
A. Hulsebos et al./Multigrid method for the 2-D XY model
We have described a multigrid method for the XY model which can be used for other spin systems in any dimension. The method appears to reduce the critical exponent z below 1. Furthermore, the algorithm vectorizes well. Our results seem to be more favorable than those mentioned in ref. 4. Perhaps this is due to the combination with the Langevin algorithm. ACKNOWLEDGEMENTS We would like to thank our colleagues at NIKHEF-H for generous support and computer timA on the Gould NP1 computer. This research was partly supported by the `Stichting voor Fundamenteel Onderzoek der Materie (FOM)' . AH is grateful to Pierre van Baal for a beer in La Residenza. REFERENCES 1. S.L. Adler, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 437 (Fermilab 1988)
357
2. R. Gupta, J. DeLapp, G.G. Batrouni, G.C. Fox, C.F. Baillie, J. Apostolakis, Phys . Rev. Lett . 61 (1988) 1996. 3. R.H. Swendsen and J.-S. Wang, Phys . Rev. Lett. 58 (1987) 86. U. Wolff, Phys . Rev. Lett. 60 (1988) 1461; Nucl. Phys. B322 (1989) 759. R.G. Edwards and A.D. Sokal, Phys. Rev . D40 (1989) 1374 . 4. J. Goodman and A.D. Sokal, Phys. Rev. D40
(1989) 2035.
5. A. Hulsebos, J. Smit and J.C. Vink, Nucl. Phys . B (Proc. Suppl.) 9 (1989) 512 (Fermilab 1988) ; preprint ITFA-89-16, to appear in Nucl. Phys. B 6. N. Madras and A.D. Sokal, J . Stat. Phys. 50 (1988) 109.
354
MULTIGRID METHOD FOR THE 2-D XY MODEL Arjan Hulsebos, Jan Smit, Institute of Theoretical Physics, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands and Jeroen C. Vink, Center for High Energy Astrophysics, P.O. Box 41882, 1009 DB Amsterdam, The Netherlands . We describe a multigrid method for spin systems and we test it on the two dimensional XY model using the Langevin algorithm . The method appears to reduce z to somewhere between 0.5 and 1 .0.
In performing simulations near a critical point one is confronted with the problemofcritical slowing down: near a critical point the autocorrelation time -r grows as -r oc CZ , where C is the correlation length of the system . For conventional local updating algorithms, z >_ 2. The use of overrelaxation will reduce z to about 1 i .2, although being a local updating algorithm . The cluster Swendsen-Wang-Wolff algorithm 3 results in a z which appears to be consistent with zero. Multigrid methods are also very promising, for a review, see ref. 4. We describe a multigrid method which we apply to the 2-d XY model using the Langevin algorithm. This method is also presented in ref. 4. We consider a spin system invariant under the symmetry group G x G. Its partition function is given by 2
= f DV
exp S(V)
c where DV is the Haar measure of the group G, and the action S can be written as: S
=E x,'a
Re Tr ( Vtp.,AV.+a) .
(2)
The models are defined by f=,w = Q independent of x and A, but in (2) we generalized ~Q to a matrix which may depend on x and A, for reasons which will become clear at short notice. *Presented by A. Hulsebos. 0920-5632/90/$3 .50 © Elsevier Science Publishers B.V. North-Holland
We now divide the lattice into 2 x 2 blocks By, labelled by Y (see fig . l) and we make a transformation of variables Vx = -r.Vff For instance, take V to be equal to Vl , and ir_ = V Vt , i = 2, 3, 4. The Haar measure is invariant under this transformation, dV1dV2dV3dV4 = dV dir2dw3d7r4 (cf. fig. 1). Inserting (3) into (2) leads to S
x,ß
E
Re Tr
(VE ,~Oz,,a
(4)
Vx+,a
-I- ( terms independent of V))
,
where XEBy x+AEB=+j ,
7r.t
N.,a
7rx+K
(5)
When updating Vx --> WVF, effectively each Vx in a block BE gets multiplied by the same group element W. _ As the action in terms of V has the same form as the action in terms of V, we can repeat the transformation (3), but now in terms of V's and L"s . In this way we can produce a collection of effective actions on
A. Rulsebos et al./ Multigrid method for the 2-D XY model
N 8
0
S
y
8
.. ... ... .. .
........... . i
z
i
8 8 16
.. ..... ..
~1
2~ '
.. ........ .... . ,
16
V
16
S
' ...... ...
. . .. .... ..
16 32 32 32
Figure 1: The blocks B.
32
2n-1 x 2n-! lattices, l = 1, . . . , n, if we start from a 2n x 2n lattice. All these effective actions are of the form (4) . For the model we consider here, the XY model, the group G is the group U(1) and Vx = exp (i9. ) while on the finest level P is a positive real number, independent of x and A. In our multigrid method s, we used W-cycles and the following first order Langevin algorithm,
64
9 x+1 = 6 -f. e F.' =
fla
Fxr, + 2e rl'
Im [exp ( -i0x )ß.,g exp (ie+, a)
where 77 is the standard noise term. One update consisted of n Langevin steps with stepsize e, chosen in such a way that ne = 0.5, i.e. the Langevin time tL = 0.5. Notice that on a 1 x 1 lattice an update reduces to just the addition of a random phase to the spin V on that lattice. In order to check the algorithm, we first made runs of 1100 half W-cycles, while discarding the first 100 measurements for thermalization . Each time the finest lattice was visited, we measured the following observables m = 1 2 Re
(magnetization)
64
ß
0.78
e
0.01 0.005 0.82 0.01 0.005 0.86 0.01 0.005 0.90 0.01 0.005 0.78 0.01 ' 0.005 0.82 0.01 0.002 0.86 0.01 0.005 0.90 0.01 0.002 0.82 0.01 0.005 0.86 0.01 0.005 0.90 0.01 0.005 0.92 0.01 0.005 1/1.07 0.01 0 .005 1/1.055 0.01 0.0025
355 E
0.992(5) 1 .007(5) 1 .079(5) 1.088(5) 1 .154(5) 1 .158(5) 1.233(5) 1.234(5) 0.947(3) 0.951(3) 1.012(3) 1.031(3) 1.081(3) 1.097(3) 1.162(3) 1.182(2) 1.001(1) 1.012(1) 1.065(1) 1.079(1) 1.134(1) 1.151(1) 1.174(1) 1.186(1) 1 .1953(6) 1.2085(7) 1 .2213(6) 1.2361(6)
X(e) 17.2(3) 17 .6(3) 21 .3(3) 21.3(3) 24 .7(3) 24 .4(3) 28.7(3] 28 .4(3) 22 .1(6) 22 .1(6) 29 .8(7) 34 .1(7) 41 .3(8) 46 .6(9) 61(1) 67 .4(9) 29 .7(6) 33 .2(9) 45(2) 50(1) 77(2) 84(2) 127(3) 124(3) 143(4) 182(5) 221(5) 271(6)
X(0), X 18 .0(3) 23 .9(1) 21 .3(3) 36 .3(2) 24 .1(3) 60 .0(3) c8 .1(3) 111.43(44) 22 .1(6) 23 .9(1) 35 .2(7) 36 .3(2) 51.9(1 .0) 60 .0(3) 69.0(9) 111.43(44) 36 .7(1.2) 36.3(3) 55(1) 60.0(3) 91(2) 111.43(44) 121(3) 162.68(61) 221(6) 220.8(9) 288(6) 302.2(9)
Table 1 : The results for the energy and the magnetic susceptibility. =
E
(magnetic susceptibility)
The results are presented in table 1. In the last column X(0) (upper value) is the susceptibility obtained after malting a linear extrapolation to e = 0, while X (lower value) is taken from refs. 2 and 3, for which finite size effects are presumably negligible . The results for m are within errors equal to zero. In fig. 2, we have plotted X(0)/X versus t/N, where we also took C from refs . 2 and 3. For small l; I N we find consistency with refs . 2,3, which indicates that the simple linear extrapolation in e works. The deviations for t/N>0.25 indicate finite size scaling. To study critical slowing down, we then made runs of 2050 full W-cycles, again discarding 100 measure-
356
A. Hulse6os et a1./Multigrid method for the 2-D XY model
f_Tht~_li_i_______
N 8 16 16 32 32
ß 0.70 0.78 0.82 0.86 0 .90
= 0.01) 12.0(1) 22.5(3) 31.5(4) 45(1) 77(1)
X(E
T
2.8(1.0) 3.4(6) 5.8(2.0) 4.4(1.2) 8(2)
~(X)T~--~ 2.1 3.2 3.8 5.0 6.5
2.24 3.36 4.32 5.843 8.476
TFyble 2: The results for the autocorrelation time of the susceptibility.
1
slowing down. Note that for m this result is trivial, because of the random phase update for the 1 x 1 lattice. The results for X are given in table 2. In tiûs table l;(X) is the correlation length corresponding to the X(e = 0.01) value. The relation between X and ~ was taken from refs . 2,3 by linear interpolation . .2
.3
.
.5 .5 .7 .8 .9
Figure 2: Finite size eflFects for the magnetic susceptibility X .
We expect that ~(X) is a reasonable estimate of the actual correlation length at e = 0.01 . The errors on ~ were obtained from binning . In order to find the critical exponent x we made a log(T)-log(1;) and a log(r)-1og(~(x)) plot (See fig . 3). We found x = 0.6fi(34) and z(X) = 0.73(43) .
meats, on (N,ß) _ (8,0.70), (16,0.78), (10,0.82), (32,0.86) slid (32,0 .90) for e = 0.01. For these values ~(Q)lN < 0.25 and there is little dependence on the ~/N . For the observables, we also calculated the autocorrelation function A and the integrated auto~orrelation function C, defined as ( ) - B(t) A t , B(0) 7ma: -t B(t) . O(j + t)O(j) ( 10) j+nas - t ~ j-1
;~na:_t
f 9max
O(.7)lt -t ~ jmax -
jmas t
i_~1
~(~)~
C(t) _ Thc irtegrat~ed autocorrelation tlme
T
is defin~ed as:
where T is the value of t at the first intersection of C(t) with the line C = t/6 s . From this, we saw that only X remains as an observable still suf%ring from critical slowing down. The other observables, m and E, show no signs of critical
Figure 3: T vs. ~ (solid dots and solid line) and ~(X) (open dots and dashed line).
T
vs.
A. Hulsebos et al./Multigrid method for the 2-D XY model
We have described a multigrid method for the XY model which can be used for other spin systems in any dimension. The method appears to reduce the critical exponent z below 1. Furthermore, the algorithm vectorizes well. Our results seem to be more favorable than those mentioned in ref. 4. Perhaps this is due to the combination with the Langevin algorithm. ACKNOWLEDGEMENTS We would like to thank our colleagues at NIKHEF-H for generous support and computer timA on the Gould NP1 computer. This research was partly supported by the `Stichting voor Fundamenteel Onderzoek der Materie (FOM)' . AH is grateful to Pierre van Baal for a beer in La Residenza. REFERENCES 1. S.L. Adler, Nucl. Phys. B (Proc. Suppl.) 9 (1989) 437 (Fermilab 1988)
357
2. R. Gupta, J. DeLapp, G.G. Batrouni, G.C. Fox, C.F. Baillie, J. Apostolakis, Phys . Rev. Lett . 61 (1988) 1996. 3. R.H. Swendsen and J.-S. Wang, Phys . Rev. Lett. 58 (1987) 86. U. Wolff, Phys . Rev. Lett. 60 (1988) 1461; Nucl. Phys. B322 (1989) 759. R.G. Edwards and A.D. Sokal, Phys. Rev . D40 (1989) 1374 . 4. J. Goodman and A.D. Sokal, Phys. Rev. D40
(1989) 2035.
5. A. Hulsebos, J. Smit and J.C. Vink, Nucl. Phys . B (Proc. Suppl.) 9 (1989) 512 (Fermilab 1988) ; preprint ITFA-89-16, to appear in Nucl. Phys. B 6. N. Madras and A.D. Sokal, J . Stat. Phys. 50 (1988) 109.