- Email: [email protected]
Numerical simulations of some first order phase transitions
Nuclear Physics B (Proc. Suppl.) 20 (1991) 661-664
661
North-Holland
NUMERICAL SIMULATIONS OF SOME FIRST ORDER PHASE TRANSITIONS Alain BILLOIRE. Robert LACAZE and Andrd MOREL Service de Physique Thdorique de Saclay. 91191 Gif-sur-Yvette Cedex. FRANCE" We present simulations of the 2d Ports model with q = 10 and q = 7. Results are compared with the predictions of the Bindsr Landau two-peak formalism. 1.
Introduction Numerical simulations have proven to be a very efficient method to locate phase transitions, and to determine whether the transition is first order or second order. There are difficult cases, however. namely first order transitions with a very small latent heat and/or a very large correlation length (by correlation length we mean the "physical". pure phase, correlation length). Correlation lengths and (relative) latent heats of some models of interest can be found in Tab.1. Model
Ref.
2d q=10
z4
~o
2d q=7
14
~ 30
3d q=3 " / = 0. 3d q=3 " / = -.2 SU(3) Art = 4
17
11 (I)
s.9
(~)
~
AE/E
= F~
s3
~o
.26
(,)
.249 (2)
=
2.
Formalism in the vicinity of a first order phase transition. the energy probabiFrb/density ~s the sum of two gauseians7.89. one for each of the coexisting phases: L~(~
PL(E) -
5 - 10
5 - ZO
.0054
- ~O[oo e
-
Eo(~)) •
2Co
aa
~.048
16
3
~ating study is needed in order to conclude about the order of the transkion. We have chosen the Ports model6 to show this method at work. After reviewing the predictions of the two-peak formalism, we present our data~ and compare them with the theory.
(2.1)
2G~
(2) with.
Table 1: Correlation lengths at the transition point in the ordered (~o) and disordered (~g) phases, and relative latent heats A E / E of the q=10 2d Ports model, the q=7 model, the pure q=3 3d model. the q=3 model with "7 = - . 2 anti-ferromagnetic admixture, and Nt = 4 pure gauge SU(3). A E / E is 2(Ea - Eo)/(tEd + Eol) for the 2d models, it is computed from BL=~,, for the others.
Eo(fl) = Eo -- 6T Co
(2.2)
Ed(,6) = Ed -F 6 T C,~
Eo and Co (E,~ and Cd) are the infinite volume energy and specific heat in the pure ordered (resp. disordered) phase, at the transition point ~¢. £T = (fie - f l ) [ ~ . The coefficients ao and ad. the total weight of each component, are given by
Among those is the famous case of the deconfinement phase transition of pure gauge SU(3) lattice theory with Art = 4 1,2,3,4,5. a very "weak first order" transition. In such cases, a finite size
ao = ad
z
qe -z~ eA
*Laboratoire de la Direction des Sciences de la Mati~re du Commissariat h I'Energie Atomique. 0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-ttolland)
(2.3)
662
A. Billoire et al./ Some first order phase transitions ~00
where q is the multiplicity of the ordered phase. and
2, =
=
La -~fl(Fo(fi) - Fd(fl))
Ld ~(~
....
I ....
I ....
CVmax
(2.4)
600
1 C - Z o ) ( E o - E . + -~( o - C~)6T). 400
Fo(~) and Fd(~) are the infinite volume free energies of the two phases. All energy moments are obtained, up to order 1 / L d. in terms of the infinite volume (finite volume corrections to the E~s and C~s are exponentially small ~ e-L/~) internal energies and specific heats of the pure phases, at /Y,. The specific heat. computed with Eq.2.1. has a maximum at
~( c v . ~ ) = #c
In q
1
~(~) ~-cv + 0 (1/La~).
Ed _ Eo L~ + - ~
200
1000
8000
Figure 1: The specific heat. as function o f L 2. for the q = 10 2-d Potts model. -0.10
(2.5) The height o f this maximum increases linearly with Ld
2000
....
I ....
I ....
I ....
-0.12
-0.14 (2.6)
The coefficients ~{c2v } and C V (2} carl be found in Ref.10. they depends on both Ei's and Ci's. Another very interesting observable is the ratio8:
1 ( 1 - < E* > ) B L = -~ < E 2 >2 "
(E~ - E~) 2 BLC2) 12(EoEd) ~ F ~ ÷ O ( 1 / L 2d)
(2.8) at a point
:~(2)
-0.18
(2.7)
The only case where it does not vanish in the infinite volume Emit. is at a first order transition point. In all other situations, the fluctuations disappear in this Emit. Calculating with Fq.2..1, one finds that B L reaches a minimum equal to l l
BL,.,u. =
-0.16
~ ( B L , ~ . ) = ~o In(q(E°/~d)~) 1 ._~_ : ~d). Ed -- Eo L d + L zd + O ' l ' L (2.9) Expressions for the coefficients ~(2) ~BL and BL (2) can be found in Ref.lO.
-O.ZO
....
0.000
I .... 0.002
I .... 0.004
I , , . , 0.006
0.008
Figure 2: BL,~I,. as function o f 1/L 2. for the q = 10 2-d Potts model. The three curves are drawn using the central value and the one standard deviation estimates for Cd. 3.
Numerical results for the ten states 2d model We have investigated 12 the q = 10. 2d Potts model as an example of a model with a clear first order transition. We have made runs on lattice sizes L -- 12, 16, 20, 24, 36 and 50. Our program simulates the trajectories o f sixteen equivalent systems (same size. same fl value) using the Metropolis algorithm. We have performed from 106 to 20.4 106 iterations (during one iteration, all 16L 2
A. Billoire et al./ Some first o r d e r phase transitions
laa~,,,,l 0.000
.... I .... t,,,~1 0.00"2 0 . 0 0 4 0 . 0 0 6 0.008
Figure 3: fl(CV__~._~)and fl(BL,~,~), as function of 1/L 2. for the q = 10 2-d Potts model.
-
O
-O.t5
.
....
0.000
l
O
~
I .... I .... I .... I,., 0.001 0.002 0.003 0.004:
663
3, together with the theoretical prediction of Eqs. 2.6, 2.8, 2.5. and 2.g. Among the five parameters needed, four are known exactly13, namely/3~. Eo, Ed and the difference Co - Ca. We use the value Cd = 12.2 ± .2 as obtained from the (perfect) fit of our specific heat data of Fig.l. The statistical errors on the data points, and the uncertainties on Ca are not visible. The agreement between theory and data is less ideal for BL,=,=. Small deviations from the predicted behavior of BL,~,~ are seen in Fig.2 for L < 24. For some reasons, the corrections to the two peak model (contributions from mixed phases, residual size dependence of the parameters, neglected powers of 1/L a) are larger in this case. Nevertheless. this figure shows unmistakably that the transition is first order. Significant, although smaller, deviations from the predicted behavior are seen in Fig3 where fl(G-~==) and ~(BL,~.,~) are plotted. Note that the I [ L ~ contributions are amazingly small, and that the uncertainLies on Ca are not visible, in conclusion our data for the q = 10 case are in excellent agreement with the two gaussian description of a first order phase transition. This gives us good confidence in this model. 4.
°
.005
Figure 4: BL,~,~. as function of 1/L 2. for the q = 7 2-d Potts model. The three curves are drawn using the central value and the one standard deviation estimates for Ud. spins are updated once). Very long runs are needed because of extreme slowing down close to tic- On the 50 ~ lattices at Bit. we see in average one flip between an ordered phase and the disordered phase every 4 106 iterations. It would be prohibitively time consuming to simulate larger systems. Our results for the maximum of the specific heat. the minimum of BL and the corresponding effective/~c's can be found in Figs. 1. 2 and
The seven states model Fig.4 shows the results for BL,,~,= of our simulation12 of the q ~ 7 model on 163. 24s and 32s lattices. Deviations from the 1/L a behavior are quite large. The curves are drawn with the educated guess Ca = 37 _ 5 from the specific heat data. Nevertheless. one would correctly deduce from Fig.4 alone that the transition is first order. Note that we do expect larger deviations from the predicted behavior since the correlation length is five times larger/4in the q = 7 case than in the q -- 10 case. 5.
The three states 3d Ports model with antiferromagnetic admixture The claim has been made15 that the three states 3d model with antiferromagnetic next to nearest neighbor admixture (3' ~ -.2) has a second order phase transition. Our results 16 have shown that the transition is indeed first order.
664
A. Billoire et al./ Some first order phase transitions
However we did not observe, on the lattice used, the behavior predicted by the two-peak formalism. The reason for this discrepancy can be found in Fig.5 where ]/La Iog(Pr(E)) is plotted as function of E for L = 24, 32 and 48 lattices. -''1
....
I .... 24
0,00016
x
........ 48 o I
I 8 -~
+
0,00015
(1983)
0.00014
0.00013
0.00012
2. S. Gottlieb. in Proceedings of the International Symposium Lattice 89 . (Isola di Capri} edited by N. Cabibbo et al (North Holi~nd 1990). 3. M. Fukugita. M. Okawa and A. Ukawa. Nucl. Phy:, 337B (1990) 181. 4. G. Bhanot and S. Sanielevici. Phys. Rev. D40 (1989) 3454. 5. N.A. Alves, B.A. Berg and S. Sanielevici. Phys. Lett. B241 (1990) 557: B.A. Berg. R. Villanova and C. Vohwinkel. Phys. Rev. Lett. 62 (1989) 2433. 6. For a general review on the Ports model, see F.Y. Wu. Rev. Mod. Phys. 54 (1982): 55 (E)
,,,.V..,, 0.42
0.43
.
0.44
.
.
.
0.45
0.46
Figure 5: 1/Ldlog(Pz(E)) as a function of - E with L = 24,36 and 48 for the 3-d Potts model with "7 = -.2. In order for the two peaks to have the same heights, the curves have been drawn for slightly different fl values, namely 1.19009 (L = 24) 1.19008 (L = 32) and 1.19012 (L = 48). and suitable vertical shifts of the curves have been performed. The left hand side of the ordered peak moves with the lattice size, in contradiction with the model. This can be explained by a residual size dependence of Eo. A lattice as large as 323 is not yet asymptotic for this model. Note the flattening of the curves between the two peaks, for increasing lattice sizes. This is what is expected from the Maxwell construction (mixed ordered-disorderedstates dominates the region between the two peaks, in the large volume limit). In particular, the difference between the peak and dip values scales like 1/L. REFERENCES 1. F,R, Brown in Proceedings of the International Symposium Lattice 89 . (Isola di Capri) edited by N. Cabibbo et al (North Holland 1990),
7. K. Binder and D.P. Landau. Phys. Rev. B30 (!984) 1477. 8. M.S. Challa. D.P. Landau and K. Binder. Phys. Rev. B34 (1986) 1841. 9. C. Borgs and R. Koteck~. preprint HUTMP 90/B258:B259 C. Borgs, R. Kotecky and S. Miracle-Sol~. preprint HUTMP 90/B265. 10. J. Lee and J.M. Kosterlitz. Brown Preprint (June lgg0). 11. A. Billoire. S. Gupta. A. Irb~ick. R. Lacaze. A. Morel and B. Petersson. Phys. Roy. B42 (1990) 6743. 12. A. Billoire, R. Lacaze and A. Morel. Saclay Preprint. to appear. 13. R.J. Baxter. J. Phys. A15, (1982) 3329; R.J. Baxter, Ezaetly solved models in statistical mechanics. Academic Press {1982). 14. P. Peczak and D.P. Landau. Phys. Rev. B39 (1989) 11932. 15. L.A. Fernandez. E. Marinari. G. Parisi. S. Roncolini and A. Taranc6n. Phys. Lett. B 217 (1989) 309: L.A. Fernandez. U. Marini Bettolo Marconi and A. Taran¢6n. Phys. Lett. B217 (1989) 314; Physica A161 (1989) 284. 16. A. Billoire. R. Lacaze and A. Morel. Nucl. Phys. B340 ~(1990) 542. 17. M, Fukugita and M. Okawa. Phys. Rev. Lett. 63 (1989} 13: M. Fukugita, H. Mino, M. Okawa and A. Ukawa, J. Stat. Phys. 59 (1990) 1397.
661
North-Holland
NUMERICAL SIMULATIONS OF SOME FIRST ORDER PHASE TRANSITIONS Alain BILLOIRE. Robert LACAZE and Andrd MOREL Service de Physique Thdorique de Saclay. 91191 Gif-sur-Yvette Cedex. FRANCE" We present simulations of the 2d Ports model with q = 10 and q = 7. Results are compared with the predictions of the Bindsr Landau two-peak formalism. 1.
Introduction Numerical simulations have proven to be a very efficient method to locate phase transitions, and to determine whether the transition is first order or second order. There are difficult cases, however. namely first order transitions with a very small latent heat and/or a very large correlation length (by correlation length we mean the "physical". pure phase, correlation length). Correlation lengths and (relative) latent heats of some models of interest can be found in Tab.1. Model
Ref.
2d q=10
z4
~o
2d q=7
14
~ 30
3d q=3 " / = 0. 3d q=3 " / = -.2 SU(3) Art = 4
17
11 (I)
s.9
(~)
~
AE/E
= F~
s3
~o
.26
(,)
.249 (2)
=
2.
Formalism in the vicinity of a first order phase transition. the energy probabiFrb/density ~s the sum of two gauseians7.89. one for each of the coexisting phases: L~(~
PL(E) -
5 - 10
5 - ZO
.0054
- ~O[oo e
-
Eo(~)) •
2Co
aa
~.048
16
3
~ating study is needed in order to conclude about the order of the transkion. We have chosen the Ports model6 to show this method at work. After reviewing the predictions of the two-peak formalism, we present our data~ and compare them with the theory.
(2.1)
2G~
(2) with.
Table 1: Correlation lengths at the transition point in the ordered (~o) and disordered (~g) phases, and relative latent heats A E / E of the q=10 2d Ports model, the q=7 model, the pure q=3 3d model. the q=3 model with "7 = - . 2 anti-ferromagnetic admixture, and Nt = 4 pure gauge SU(3). A E / E is 2(Ea - Eo)/(tEd + Eol) for the 2d models, it is computed from BL=~,, for the others.
Eo(fl) = Eo -- 6T Co
(2.2)
Ed(,6) = Ed -F 6 T C,~
Eo and Co (E,~ and Cd) are the infinite volume energy and specific heat in the pure ordered (resp. disordered) phase, at the transition point ~¢. £T = (fie - f l ) [ ~ . The coefficients ao and ad. the total weight of each component, are given by
Among those is the famous case of the deconfinement phase transition of pure gauge SU(3) lattice theory with Art = 4 1,2,3,4,5. a very "weak first order" transition. In such cases, a finite size
ao = ad
z
qe -z~ eA
*Laboratoire de la Direction des Sciences de la Mati~re du Commissariat h I'Energie Atomique. 0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-ttolland)
(2.3)
662
A. Billoire et al./ Some first order phase transitions ~00
where q is the multiplicity of the ordered phase. and
2, =
=
La -~fl(Fo(fi) - Fd(fl))
Ld ~(~
....
I ....
I ....
CVmax
(2.4)
600
1 C - Z o ) ( E o - E . + -~( o - C~)6T). 400
Fo(~) and Fd(~) are the infinite volume free energies of the two phases. All energy moments are obtained, up to order 1 / L d. in terms of the infinite volume (finite volume corrections to the E~s and C~s are exponentially small ~ e-L/~) internal energies and specific heats of the pure phases, at /Y,. The specific heat. computed with Eq.2.1. has a maximum at
~( c v . ~ ) = #c
In q
1
~(~) ~-cv + 0 (1/La~).
Ed _ Eo L~ + - ~
200
1000
8000
Figure 1: The specific heat. as function o f L 2. for the q = 10 2-d Potts model. -0.10
(2.5) The height o f this maximum increases linearly with Ld
2000
....
I ....
I ....
I ....
-0.12
-0.14 (2.6)
The coefficients ~{c2v } and C V (2} carl be found in Ref.10. they depends on both Ei's and Ci's. Another very interesting observable is the ratio8:
1 ( 1 - < E* > ) B L = -~ < E 2 >2 "
(E~ - E~) 2 BLC2) 12(EoEd) ~ F ~ ÷ O ( 1 / L 2d)
(2.8) at a point
:~(2)
-0.18
(2.7)
The only case where it does not vanish in the infinite volume Emit. is at a first order transition point. In all other situations, the fluctuations disappear in this Emit. Calculating with Fq.2..1, one finds that B L reaches a minimum equal to l l
BL,.,u. =
-0.16
~ ( B L , ~ . ) = ~o In(q(E°/~d)~) 1 ._~_ : ~d). Ed -- Eo L d + L zd + O ' l ' L (2.9) Expressions for the coefficients ~(2) ~BL and BL (2) can be found in Ref.lO.
-O.ZO
....
0.000
I .... 0.002
I .... 0.004
I , , . , 0.006
0.008
Figure 2: BL,~I,. as function o f 1/L 2. for the q = 10 2-d Potts model. The three curves are drawn using the central value and the one standard deviation estimates for Cd. 3.
Numerical results for the ten states 2d model We have investigated 12 the q = 10. 2d Potts model as an example of a model with a clear first order transition. We have made runs on lattice sizes L -- 12, 16, 20, 24, 36 and 50. Our program simulates the trajectories o f sixteen equivalent systems (same size. same fl value) using the Metropolis algorithm. We have performed from 106 to 20.4 106 iterations (during one iteration, all 16L 2
A. Billoire et al./ Some first o r d e r phase transitions
laa~,,,,l 0.000
.... I .... t,,,~1 0.00"2 0 . 0 0 4 0 . 0 0 6 0.008
Figure 3: fl(CV__~._~)and fl(BL,~,~), as function of 1/L 2. for the q = 10 2-d Potts model.
-
O
-O.t5
.
....
0.000
l
O
~
I .... I .... I .... I,., 0.001 0.002 0.003 0.004:
663
3, together with the theoretical prediction of Eqs. 2.6, 2.8, 2.5. and 2.g. Among the five parameters needed, four are known exactly13, namely/3~. Eo, Ed and the difference Co - Ca. We use the value Cd = 12.2 ± .2 as obtained from the (perfect) fit of our specific heat data of Fig.l. The statistical errors on the data points, and the uncertainties on Ca are not visible. The agreement between theory and data is less ideal for BL,=,=. Small deviations from the predicted behavior of BL,~,~ are seen in Fig.2 for L < 24. For some reasons, the corrections to the two peak model (contributions from mixed phases, residual size dependence of the parameters, neglected powers of 1/L a) are larger in this case. Nevertheless. this figure shows unmistakably that the transition is first order. Significant, although smaller, deviations from the predicted behavior are seen in Fig3 where fl(G-~==) and ~(BL,~.,~) are plotted. Note that the I [ L ~ contributions are amazingly small, and that the uncertainLies on Ca are not visible, in conclusion our data for the q = 10 case are in excellent agreement with the two gaussian description of a first order phase transition. This gives us good confidence in this model. 4.
°
.005
Figure 4: BL,~,~. as function of 1/L 2. for the q = 7 2-d Potts model. The three curves are drawn using the central value and the one standard deviation estimates for Ud. spins are updated once). Very long runs are needed because of extreme slowing down close to tic- On the 50 ~ lattices at Bit. we see in average one flip between an ordered phase and the disordered phase every 4 106 iterations. It would be prohibitively time consuming to simulate larger systems. Our results for the maximum of the specific heat. the minimum of BL and the corresponding effective/~c's can be found in Figs. 1. 2 and
The seven states model Fig.4 shows the results for BL,,~,= of our simulation12 of the q ~ 7 model on 163. 24s and 32s lattices. Deviations from the 1/L a behavior are quite large. The curves are drawn with the educated guess Ca = 37 _ 5 from the specific heat data. Nevertheless. one would correctly deduce from Fig.4 alone that the transition is first order. Note that we do expect larger deviations from the predicted behavior since the correlation length is five times larger/4in the q = 7 case than in the q -- 10 case. 5.
The three states 3d Ports model with antiferromagnetic admixture The claim has been made15 that the three states 3d model with antiferromagnetic next to nearest neighbor admixture (3' ~ -.2) has a second order phase transition. Our results 16 have shown that the transition is indeed first order.
664
A. Billoire et al./ Some first order phase transitions
However we did not observe, on the lattice used, the behavior predicted by the two-peak formalism. The reason for this discrepancy can be found in Fig.5 where ]/La Iog(Pr(E)) is plotted as function of E for L = 24, 32 and 48 lattices. -''1
....
I .... 24
0,00016
x
........ 48 o I
I 8 -~
+
0,00015
(1983)
0.00014
0.00013
0.00012
2. S. Gottlieb. in Proceedings of the International Symposium Lattice 89 . (Isola di Capri} edited by N. Cabibbo et al (North Holi~nd 1990). 3. M. Fukugita. M. Okawa and A. Ukawa. Nucl. Phy:, 337B (1990) 181. 4. G. Bhanot and S. Sanielevici. Phys. Rev. D40 (1989) 3454. 5. N.A. Alves, B.A. Berg and S. Sanielevici. Phys. Lett. B241 (1990) 557: B.A. Berg. R. Villanova and C. Vohwinkel. Phys. Rev. Lett. 62 (1989) 2433. 6. For a general review on the Ports model, see F.Y. Wu. Rev. Mod. Phys. 54 (1982): 55 (E)
,,,.V..,, 0.42
0.43
.
0.44
.
.
.
0.45
0.46
Figure 5: 1/Ldlog(Pz(E)) as a function of - E with L = 24,36 and 48 for the 3-d Potts model with "7 = -.2. In order for the two peaks to have the same heights, the curves have been drawn for slightly different fl values, namely 1.19009 (L = 24) 1.19008 (L = 32) and 1.19012 (L = 48). and suitable vertical shifts of the curves have been performed. The left hand side of the ordered peak moves with the lattice size, in contradiction with the model. This can be explained by a residual size dependence of Eo. A lattice as large as 323 is not yet asymptotic for this model. Note the flattening of the curves between the two peaks, for increasing lattice sizes. This is what is expected from the Maxwell construction (mixed ordered-disorderedstates dominates the region between the two peaks, in the large volume limit). In particular, the difference between the peak and dip values scales like 1/L. REFERENCES 1. F,R, Brown in Proceedings of the International Symposium Lattice 89 . (Isola di Capri) edited by N. Cabibbo et al (North Holland 1990),
7. K. Binder and D.P. Landau. Phys. Rev. B30 (!984) 1477. 8. M.S. Challa. D.P. Landau and K. Binder. Phys. Rev. B34 (1986) 1841. 9. C. Borgs and R. Koteck~. preprint HUTMP 90/B258:B259 C. Borgs, R. Kotecky and S. Miracle-Sol~. preprint HUTMP 90/B265. 10. J. Lee and J.M. Kosterlitz. Brown Preprint (June lgg0). 11. A. Billoire. S. Gupta. A. Irb~ick. R. Lacaze. A. Morel and B. Petersson. Phys. Roy. B42 (1990) 6743. 12. A. Billoire, R. Lacaze and A. Morel. Saclay Preprint. to appear. 13. R.J. Baxter. J. Phys. A15, (1982) 3329; R.J. Baxter, Ezaetly solved models in statistical mechanics. Academic Press {1982). 14. P. Peczak and D.P. Landau. Phys. Rev. B39 (1989) 11932. 15. L.A. Fernandez. E. Marinari. G. Parisi. S. Roncolini and A. Taranc6n. Phys. Lett. B 217 (1989) 309: L.A. Fernandez. U. Marini Bettolo Marconi and A. Taran¢6n. Phys. Lett. B217 (1989) 314; Physica A161 (1989) 284. 16. A. Billoire. R. Lacaze and A. Morel. Nucl. Phys. B340 ~(1990) 542. 17. M, Fukugita and M. Okawa. Phys. Rev. Lett. 63 (1989} 13: M. Fukugita, H. Mino, M. Okawa and A. Ukawa, J. Stat. Phys. 59 (1990) 1397.