- Email: [email protected]
The free energy of the Potts model
I ~ll!ll I I"-I ,'i ',]-"t'k'! [4kl ~t
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 671-673 North-Holland
The free energy of the Potts model T. Bhattacharya a , R. Lacaze b and A. Morel b "MS B285, Group T-8, Los Alamos National Laboratory, NM 87544, U.S.A. bService de Physique Thdorique de Saclay 91191 Gif-sur-Yvette Cedex, France We present a large q expansion of the 2d q-states Potts model free energies up to order 10 in 1/V ~. Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q -- 4, and predicts, for q > 4, the n th energy cumulant scales as the power (3n/2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q -- 10 with lattices of linear dimensions up to L = 50. The pure phase specific heats are predicted to be much larger, at q < 10, than the values extracted from current finite size scaling analysis of extrema.
Much effort has been recently devoted to the 2-d Potts model, both with numerical and analytical techniques. In the former case, the goal was either to test numerical algorithms and criteria for distinguishing first-order from continuous transitions, or to learn how to extract previously unknown quantities such as the interface tension. Although much progress was made, there remain some unsatisfactory issues such as, for example, slight inconsistencies in finite size scaling analysis of the energy cumulants close to the transition temperature fiE-1, and discrepancies between exact results and numerical simulations for the interface tension. Many properties of the model are known exactly [1]. The value of/3t is ln(v/~ + 1), and duality relates the ordered and disordered free energies. The transition is second-order for q _< 4, and its critical properties are described, e.g., by the a and u indices, which at q = 4 take the common value 2/3. Accordingly, the correlation length and the specific heat there diverge as Cq=4 -,, q=4 "l Z - A 1-2/3 .
(i)
C/~
Hence in the vicinity of 13 = f3t, the ratio remains finite at q = 4_. The first-order transition region is q > 4. There the pure phase internal energies are known at fit [2], where the correlation lengths as q ---* 4+ be-
have in both phases like [3,4] 7r2
~ x = exP(21n½(v~+V~:_~).
(2)
rapidly diverges as q ~ 4+ and the leading behavior given by Eq. 2 is accurate over a very wide range of q values. The pure phase specific heats Co and Cd are unknown, but their known differencevanishes when q ---*4+ as x-½. In order to get analyticalinformation on these quantities and higher energy cumulants, we compute a large q expansion of the ordered free energy Fo (that for Fd followsfrom duality) obtained through the Fortuin-Kasteleyn [5] representation of the Potts model partitionfunction Z = E ( e ~ - 1)lq n ,
(3)
X
where X is any configuration of bonds on a square lattice, l its number of bonds and n its number of connected components, or clusters of sites ( two sites bound to each other belong to the same cluster, an isolated site is a cluster). The completely ordered configuration corresponds to n = 1 and I = 2V. So the partition function can be reorganized as an expansion in q-½ about this configuration:
Zo = q(e ~ - 1 ) ~v ~
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00369-7
pjn
Np,n(v)(eZ~nl)-pqn-},(4) V~
672
T. Bhattacharya et al. ~Thefree energy of the Potts model
where Np,n(V) is the number of configurations in a volume V with p removed bonds and n + 1 dusters. The enumeration of all the Np,n(V) such that (p - 2n) g M yields an expansion of Zo to order M. We have computed up to order M = 10, including terms up to N60,25(V). This series exponentiates in V to give Fo (up to order M + 1) and provides us with similar series for the k th derivative with respect to/3, F (k). At/3 =/3t the k = 0 (free energy) and k = 1 (internal energy) series match the exact results [2] up to M = 10. The k = 2 and 3 cases give F(2) = 16 +
34
114
254
882
T 1944
+
6128
13550
+-F- +
36980
+-F-
+. •
(5)
is not the case away from/3t where F ( Z ) / V terms must be kept ( F (s) ~ - 1 8 0 0 at q = 10 [ ). The result for F(2) and F(S)/F(2) as functions of x is well reproduced over a.n astonishingly large range by the simple forms F (2) ~_ Cst x and F(S)/F (2) ~_ Cst x s/~ corresponding to the behavior expected from Eqs. 1 and 2. We then propose the following ansatz: There exists a region of q > 4 where the free energies around /3 = /3t reflect accurately the scaling properties associated with the second-order point lying at q = 4,/3 = ln3 and characterized by the corre. sponding critical indices a and v. Specifically, according to the known value 2/3 of a, we parametrize F(2)(/3) at q = 4 and for/3 --* (/st)+ as Fo(2)(/3) = A (/3 - / 3 0 - m
F~3) =
64 q
224732 qr/2
430 q3/2 888024 q4
2654 q2
12186 qS/2
3164682 qg/~
57018 q3 11243178 (6) q~
The coefficients are fastly increasing with the order, the more so for larger values of k. We thus conjecture for the k th cumulant at low q - 4 a divergence which becomes the more violent as k increases. We then assume that, as ~, F (2) and F (s) have an essential singularity at q = 4 and construct Pad~ approximants for the series of In F (k) instead of F (k), i.e. take as estimates of F(k): F(~)e.t = exp[ Pad~(InF(k))]. The values we predict for the ordered phase specific heat are Co = 5.362(3), 18.06(4), 37.5(4), 71.3(1.0) respectively at q = 20, 10, 8 and 7. At q = 20 our prediction is in good agreement with the numerical estimate[6,7]. It strongly disagrees at q _< 10 with values obtained from a finite size analysis of the maximum of the specific heat measured in the coexistence regime. At q = 7 for example our value 71.3(1.0) has to be compared with 47.5(2.5) [8], 50.(10.) [9] or 44.4(2.2) [10]; at q = 10 our value 18.06(4) has to be compared with 10.6(1.1) [11] and 12.7(3) [9]. But we support the values Co = 17.97(23) [7] and ..~ 18 [9] obtained from q = 10 data analysed at /3 = /3t. There the total specific heat receives no contribution from k > 2 pure phase cumulants [12], which
The higher derivatives are trivially deduced and in their expressions we replace ( / 3 - fit) by ( Cst ~-s/2 ) from Eq. 1 and, boldly continuing above q = 4 at /3 =/3t, reexpress Fo(p+2) as a function of z via Eq. 2 to get • ,p r(2/3+p) Fo(P+2)(/3t) = A ( - ) r(2/3) Bx
l+sp/2
'
(7)
'
where the constant B takes into account proportionality constants. It is easy to sum the expansion of Fo(/3). For later convenience, we introduce scaled temperature (v), energy (e) and length ($) variables v =
e=
E ( B x ) 1/2
's2 =
(Bz) 2
'
and end up with the following compact result
$2"
Fo(/3) = F(flt)+ 1 [ - ~3-
(co+
Fd(/3) = Fo(v ---+- v , eo ~ --ed).
1)v+~_(1+v)4/s] (8)
The latter equation follows from duality. These expressions summarize accurately, over a wide range of q > 4 values, all the properties of the model. Let us justify this statement by comparing the predictions of Eq. 8 for the energy distribution to data [9] taken at q = 10 with various L values. This distribution is obtained by inverse Laplace transform of the partition function,
T. Bhattacharya et al. ~Thefree energy of the Potts model
673
doubt on attempts to determine a [8,10] unless a plateau in E is seen [14]. It is a pleasure to thank R. Balian and A. Billoire for illuminating discussions. We also acknowledge useful conversations with B. Grossmann and T. Neuhaus.
1
REFERENCES
1. F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235. 2. R.J. Baxter, J. Phys. C6 (1973) L445. 3. A. Klfimper, A. Schadschneider and J. Zittartz, Z Phys. B76 (1989) 247; E. Buffenoir and S. Wallon, J. Phys. A26 (1993) 3045. 4. C. Borgs and W. Janke,J. Phys. I (France) 2
(1992) 649.
-2
-1.5 -1 Energy / Site
Figure 1. Energy densities for L = 16, 20, 24, 36, 44 and 50. Curves are parameter free predictions.
Z = q exp[VFo(fl)] + exp[VFd(fl)], computed numerically. The results are shown together with the data of [9] on Fig. 1 for L = 16, 20, 24, 36, 44, 50. Remembering that the continuous curves are absolute predictions without any free parameter, it is quite striking to see how such a simple ansatz as Eq. 8 yields good results. Because ( L / S ) is not large, they are very different from what an asymptotic expansion would give. For example E p e a k - - Eo behaves effectively as ,,, 1/L over a wide range of L values, whereas the asymptotic expectation is Eo-Epeak - - F ( S ) / ( 2 F (2)) 1/L 2. We also determine a pseudo interface tension from the size dependence of the dip to the peak ratio in the energy densities [13]. Although this quantity diverges asymptotically as L (no interface in our construction), we find numerically a value of order 0.11 for all L < 50, unexpectedly close to the exact value[4] ¢r = 1/~d = 0.095 ! This casts some
5. P.W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Japan 26 (Suppl.), 11 (1969). 6. A. Billoire, T. Neuhaus and B. Berg, Nucl. Phys. B396 (1993) 779. 7. W. Janke and S. Kappler, these proceedings. 8. W. Janke, B. Berg and M. Katoot, Nucl. Phys. B382 (1992) 649. 9. A. Billoire, R. Lacase and A. Morel, Nucl. Phys. B 370 (1992) 773. 10. K. Rummukainen, Nucl. Phys. B390 (1993) 621. 11. J. Lee and J. M. Kosterlitz, Phys. Rev. B43 (1991) 3265. 12. C. Borgs and R. Koteck~, J. Stat. Phys. 61 (1990) 79; C. Borgs, R. Koteck~ and S. Miracle-Sol~, J. Stat. Phys. 62 (1991) 529. 13. K. Binder, Phys. Rev A25 (1982) 1699; Z. Phys. B43 (1981) 119. 14. B. Berg, U. Hansmann and T. Neuhaus, to appear in proc. "Computer Simulations Studies in Condensed Matter Physics", Athens 1992; B. Grossmann and M. L. Laursen, Preprint HLRZ-93-7; A. Billoire, T. Neuhaus and B. Berg, in preparation.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 34 (1994) 671-673 North-Holland
The free energy of the Potts model T. Bhattacharya a , R. Lacaze b and A. Morel b "MS B285, Group T-8, Los Alamos National Laboratory, NM 87544, U.S.A. bService de Physique Thdorique de Saclay 91191 Gif-sur-Yvette Cedex, France We present a large q expansion of the 2d q-states Potts model free energies up to order 10 in 1/V ~. Its analysis leads us to an ansatz which, in the first-order region, incorporates properties inferred from the known critical regime at q -- 4, and predicts, for q > 4, the n th energy cumulant scales as the power (3n/2 - 2) of the correlation length. The parameter-free energy distributions reproduce accurately, without reference to any interface effect, the numerical data obtained in a simulation for q -- 10 with lattices of linear dimensions up to L = 50. The pure phase specific heats are predicted to be much larger, at q < 10, than the values extracted from current finite size scaling analysis of extrema.
Much effort has been recently devoted to the 2-d Potts model, both with numerical and analytical techniques. In the former case, the goal was either to test numerical algorithms and criteria for distinguishing first-order from continuous transitions, or to learn how to extract previously unknown quantities such as the interface tension. Although much progress was made, there remain some unsatisfactory issues such as, for example, slight inconsistencies in finite size scaling analysis of the energy cumulants close to the transition temperature fiE-1, and discrepancies between exact results and numerical simulations for the interface tension. Many properties of the model are known exactly [1]. The value of/3t is ln(v/~ + 1), and duality relates the ordered and disordered free energies. The transition is second-order for q _< 4, and its critical properties are described, e.g., by the a and u indices, which at q = 4 take the common value 2/3. Accordingly, the correlation length and the specific heat there diverge as Cq=4 -,, q=4 "l Z - A 1-2/3 .
(i)
C/~
Hence in the vicinity of 13 = f3t, the ratio remains finite at q = 4_. The first-order transition region is q > 4. There the pure phase internal energies are known at fit [2], where the correlation lengths as q ---* 4+ be-
have in both phases like [3,4] 7r2
~ x = exP(21n½(v~+V~:_~).
(2)
rapidly diverges as q ~ 4+ and the leading behavior given by Eq. 2 is accurate over a very wide range of q values. The pure phase specific heats Co and Cd are unknown, but their known differencevanishes when q ---*4+ as x-½. In order to get analyticalinformation on these quantities and higher energy cumulants, we compute a large q expansion of the ordered free energy Fo (that for Fd followsfrom duality) obtained through the Fortuin-Kasteleyn [5] representation of the Potts model partitionfunction Z = E ( e ~ - 1)lq n ,
(3)
X
where X is any configuration of bonds on a square lattice, l its number of bonds and n its number of connected components, or clusters of sites ( two sites bound to each other belong to the same cluster, an isolated site is a cluster). The completely ordered configuration corresponds to n = 1 and I = 2V. So the partition function can be reorganized as an expansion in q-½ about this configuration:
Zo = q(e ~ - 1 ) ~v ~
0920-5632/94/$07.00 © 1994 - Elsevier Science B.V. All rights reserved. SSDI 0920-5632(94)00369-7
pjn
Np,n(v)(eZ~nl)-pqn-},(4) V~
672
T. Bhattacharya et al. ~Thefree energy of the Potts model
where Np,n(V) is the number of configurations in a volume V with p removed bonds and n + 1 dusters. The enumeration of all the Np,n(V) such that (p - 2n) g M yields an expansion of Zo to order M. We have computed up to order M = 10, including terms up to N60,25(V). This series exponentiates in V to give Fo (up to order M + 1) and provides us with similar series for the k th derivative with respect to/3, F (k). At/3 =/3t the k = 0 (free energy) and k = 1 (internal energy) series match the exact results [2] up to M = 10. The k = 2 and 3 cases give F(2) = 16 +
34
114
254
882
T 1944
+
6128
13550
+-F- +
36980
+-F-
+. •
(5)
is not the case away from/3t where F ( Z ) / V terms must be kept ( F (s) ~ - 1 8 0 0 at q = 10 [ ). The result for F(2) and F(S)/F(2) as functions of x is well reproduced over a.n astonishingly large range by the simple forms F (2) ~_ Cst x and F(S)/F (2) ~_ Cst x s/~ corresponding to the behavior expected from Eqs. 1 and 2. We then propose the following ansatz: There exists a region of q > 4 where the free energies around /3 = /3t reflect accurately the scaling properties associated with the second-order point lying at q = 4,/3 = ln3 and characterized by the corre. sponding critical indices a and v. Specifically, according to the known value 2/3 of a, we parametrize F(2)(/3) at q = 4 and for/3 --* (/st)+ as Fo(2)(/3) = A (/3 - / 3 0 - m
F~3) =
64 q
224732 qr/2
430 q3/2 888024 q4
2654 q2
12186 qS/2
3164682 qg/~
57018 q3 11243178 (6) q~
The coefficients are fastly increasing with the order, the more so for larger values of k. We thus conjecture for the k th cumulant at low q - 4 a divergence which becomes the more violent as k increases. We then assume that, as ~, F (2) and F (s) have an essential singularity at q = 4 and construct Pad~ approximants for the series of In F (k) instead of F (k), i.e. take as estimates of F(k): F(~)e.t = exp[ Pad~(InF(k))]. The values we predict for the ordered phase specific heat are Co = 5.362(3), 18.06(4), 37.5(4), 71.3(1.0) respectively at q = 20, 10, 8 and 7. At q = 20 our prediction is in good agreement with the numerical estimate[6,7]. It strongly disagrees at q _< 10 with values obtained from a finite size analysis of the maximum of the specific heat measured in the coexistence regime. At q = 7 for example our value 71.3(1.0) has to be compared with 47.5(2.5) [8], 50.(10.) [9] or 44.4(2.2) [10]; at q = 10 our value 18.06(4) has to be compared with 10.6(1.1) [11] and 12.7(3) [9]. But we support the values Co = 17.97(23) [7] and ..~ 18 [9] obtained from q = 10 data analysed at /3 = /3t. There the total specific heat receives no contribution from k > 2 pure phase cumulants [12], which
The higher derivatives are trivially deduced and in their expressions we replace ( / 3 - fit) by ( Cst ~-s/2 ) from Eq. 1 and, boldly continuing above q = 4 at /3 =/3t, reexpress Fo(p+2) as a function of z via Eq. 2 to get • ,p r(2/3+p) Fo(P+2)(/3t) = A ( - ) r(2/3) Bx
l+sp/2
'
(7)
'
where the constant B takes into account proportionality constants. It is easy to sum the expansion of Fo(/3). For later convenience, we introduce scaled temperature (v), energy (e) and length ($) variables v =
e=
E ( B x ) 1/2
's2 =
(Bz) 2
'
and end up with the following compact result
$2"
Fo(/3) = F(flt)+ 1 [ - ~3-
(co+
Fd(/3) = Fo(v ---+- v , eo ~ --ed).
1)v+~_(1+v)4/s] (8)
The latter equation follows from duality. These expressions summarize accurately, over a wide range of q > 4 values, all the properties of the model. Let us justify this statement by comparing the predictions of Eq. 8 for the energy distribution to data [9] taken at q = 10 with various L values. This distribution is obtained by inverse Laplace transform of the partition function,
T. Bhattacharya et al. ~Thefree energy of the Potts model
673
doubt on attempts to determine a [8,10] unless a plateau in E is seen [14]. It is a pleasure to thank R. Balian and A. Billoire for illuminating discussions. We also acknowledge useful conversations with B. Grossmann and T. Neuhaus.
1
REFERENCES
1. F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235. 2. R.J. Baxter, J. Phys. C6 (1973) L445. 3. A. Klfimper, A. Schadschneider and J. Zittartz, Z Phys. B76 (1989) 247; E. Buffenoir and S. Wallon, J. Phys. A26 (1993) 3045. 4. C. Borgs and W. Janke,J. Phys. I (France) 2
(1992) 649.
-2
-1.5 -1 Energy / Site
Figure 1. Energy densities for L = 16, 20, 24, 36, 44 and 50. Curves are parameter free predictions.
Z = q exp[VFo(fl)] + exp[VFd(fl)], computed numerically. The results are shown together with the data of [9] on Fig. 1 for L = 16, 20, 24, 36, 44, 50. Remembering that the continuous curves are absolute predictions without any free parameter, it is quite striking to see how such a simple ansatz as Eq. 8 yields good results. Because ( L / S ) is not large, they are very different from what an asymptotic expansion would give. For example E p e a k - - Eo behaves effectively as ,,, 1/L over a wide range of L values, whereas the asymptotic expectation is Eo-Epeak - - F ( S ) / ( 2 F (2)) 1/L 2. We also determine a pseudo interface tension from the size dependence of the dip to the peak ratio in the energy densities [13]. Although this quantity diverges asymptotically as L (no interface in our construction), we find numerically a value of order 0.11 for all L < 50, unexpectedly close to the exact value[4] ¢r = 1/~d = 0.095 ! This casts some
5. P.W. Kasteleyn and C. M. Fortuin, J. Phys. Soc. Japan 26 (Suppl.), 11 (1969). 6. A. Billoire, T. Neuhaus and B. Berg, Nucl. Phys. B396 (1993) 779. 7. W. Janke and S. Kappler, these proceedings. 8. W. Janke, B. Berg and M. Katoot, Nucl. Phys. B382 (1992) 649. 9. A. Billoire, R. Lacase and A. Morel, Nucl. Phys. B 370 (1992) 773. 10. K. Rummukainen, Nucl. Phys. B390 (1993) 621. 11. J. Lee and J. M. Kosterlitz, Phys. Rev. B43 (1991) 3265. 12. C. Borgs and R. Koteck~, J. Stat. Phys. 61 (1990) 79; C. Borgs, R. Koteck~ and S. Miracle-Sol~, J. Stat. Phys. 62 (1991) 529. 13. K. Binder, Phys. Rev A25 (1982) 1699; Z. Phys. B43 (1981) 119. 14. B. Berg, U. Hansmann and T. Neuhaus, to appear in proc. "Computer Simulations Studies in Condensed Matter Physics", Athens 1992; B. Grossmann and M. L. Laursen, Preprint HLRZ-93-7; A. Billoire, T. Neuhaus and B. Berg, in preparation.