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Correlation length exponent of the 2-d Z(4) model using an exact method
Volume 146, number 5
PHYSICS LETTERSA
28 May 1990
CORRELATION LENGTH EXPONENT OF THE 2-d Z(4) MODEL USING AN EXACT METHOD Peter WILLIAM Supercomputer Computations Research Institute and Department of Physics, Florida State University, Tallahassee, FL 32306, USA Received 1 March 1990; revised manuscript received 20 March 1990; accepted for publication 21 March 1990 Communicated by A.A. Maradudin
Using a recently suggested exact method to determine the partition function ofa discrete model, the correlation length exponent v for the two-dimensional Z(4) spin model is estimated. This estimation is made from a study of the zeros of the partition function on finite lattices.
1. Introduction
where S and S’ indicate configurations ofthe one dimensional spin system given by
In a recent paper a new method has been suggested to compute the density of states exactly for a set of discrete models [1]. The idea is an extension of the method given in an earlier work [2] to compute the partition function. Following the lines of ref. [1], discrete Z( n) models in two dimensions and, in particular, the Z( 4) model is considered. We give a brief discussion of the method used. The details can be found in ref. [1]. Consider an example of Z( 2) symmetry (the Ising model). Boltzmann weights of u = e~or u = 1 are assigned to a bond depending on whether the spins are aligned antiparallel or parallel. In orderto obtain the density ofstates one has to store the weights ofthe bonds ofeach configuration, hence two arrays W0 and W,, are constructed, which contam information regarding the weights of the vertical bonds generated in each configuration. Each time a spin is added the whole array is updated. The specific updating procedure is described below. This process makes the updates highly nonlocal but results in the algorithm being efficient. The arrays are defined as old and new, and the basic transformation of the weight array with the addition of a spin at the lattice site labelled i, is given as
Wa(S)
=
W0(S) +uW0(S’)
,
(1)
S= (s1, s2, ~
—
...,
s1,
(
...,
s1)
~ 2,
— ~‘~‘
,
:,
,
,
(2) (3)
/
The bar over s, just switches the spin value at that location from 0 to 1 and vice versa. With each added spin, the weight arrays are updated until the whole row is completed. After the row is completed, one just multiplies all the arrays by the Boltzmann weight ofthe spins in the row. This takes care ofall the bonds up to the particular layer being presently considered. The whole lattice is thus constructed layer by layer. From this process one could compute the partition function of the system at a particular temperature. However, one would, in general, be more interested in computing the density of states, and thus avoid the problem of performing simulations at numerous values of temperatures. With the choice of u given above the partition function can be expressed as a polynomial of finite degree in u, em
Z(u)
~ G(k)uk.
=
(4)
k0
Here em is the maximum possible value of the energy. If we choose the value of u such that u=cVm,
0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland)
(5) 261
Volume 146, number 5
PHYSICS LETTERS A
where c1 and rn are non-negative integers, then we observe that (6)
Z(u)= k~O
This implies that for multiplication by a factor u, we have —
uZ(u)= k=O ~
41 Crn(k)U’<
=Cm(0)u+Crn(l)u2+...+Crn(ml)u’~.
(7)
28 May 1990
ory, and this limits the method from being used in very large volumes. The memory that is required in d dimensions is roughly given by mem~2.Onhd_em. (13) Another limitation results from the accuracy to which the coefficients can be computed. Since one is basically integerthat arithmetic, one is restricted byperforming the largest integer can be represented on a machine in maximum precision. The accuracy inbitsisroughlygivenby
Considering a new set of coefficients C’ ‘s which are
acc l’~’log
themselves functions of the variable u, we also have that 1’ rn—I uZ(u) k=o ~ C’m(k)U
2n (14) The generalisation for considering Z(n) models follows quite naturally with a few redefinitions. In one phase factors which take values isfrom 0 tobyn—I. dimension then,Z(n) configuration labelled the The bond energy, in the case of even n, between a .
=C~,n(0)+C~rn(l)u+...+C~rn(rnl)um~(8) .
pair of spins labelled by values n, and n 1 is given by
Comparing terms in the two previous forms given above, we have the following, C’n(O)ciCm(ml)
(9)
E(n,, n1) =d1 d
—n— n—n , j
—
for d< ~n, cor A> I .-2n,
,
—
A
I,
‘15
.
(10)
Similar equations hold in the case of odd values of
We thus see how the important step of the updating procedure as given by (1) transforms in terms of Cm’S.
n with n replaced by n 1 in the inequalities. The generalisation ofthe updating procedure given by (1) is now given by
One can treat each of the weight arrays W(S) as a partial partition function since
W~(S)= ~ exp[ —/3E(n1, n1)] W0(S’)
J=1
C~nU)=Crn(jl),
rn—i.
—
.
(16)
ni = 0
Z(u)=~W(S).
(11) where the states S and S’ are defined by
The rule used in performing the multiplication by u on the whole partition function can be applied to each of the weight arrays separately. At each intermediate step to obtain W~,we add up the respective values of Cm’5 from W0(S) and uW0(S’). Finally, for a particular m we add up the coefficients to get the Cm of the whole partition function. Comparing forms (4) and (6) we deduce that if rn = em + 1 then the spectral coefficients are given by G(k)—C (k) k—0 rn—i (12) —
m
—
This provides a method for obtaining the density of states without having to resort to repeated simulations at different temperatures. However, this process has resulted in a considerable increase of mem262
S = (n1, s = (n I,
n~,
...,
...)
..
,
...)
(17) (18)
.~
2. Scaling of zeros and critical exponents Having obtained the density of states, we can now examine the behaviour of the Lee—Yang [3] zeros of the partition function in the complex plane, to look for phase transitions. Using the results ofref. [4J one can obtain quantitative predictions for the correlation length critical exponent v. The zero closest to the real axis scales like P on an li lattice. Hence the real and imaginary parts of
Volume 146, number 5
PHYSICS LETTERS A
the zeros obey the following relations given by [4] u, (1)
—
u~= 1- l/vf
(19)
1(0),
Re[u
1(1)] —u~=L”f~’(0)
(20)
,
(21)
Im(u1)=l’~f~(0) .
Here u, refers to the ith root, u,, to the infinite volume critical point and I is in general a complex number. From these relations one finds estimates for the critical exponent v, f
V
—
~Imu1(1+l) =log~,, Imu1(1)
)[
(1)]~ log
(22)
In order to obtain the critical exponent v and the critical temperature, we find estimates for the exponent from (22) above and use an extrapolation scheme to get the infinite volume values. Depending on the geometry being considered two different extrapolation schemes were used.
28 May 1990
Approximation scherne. The VBS approximation scheme is described in the following paragraph. A table of approximants is constructed from a given sequence of values, to which the limiting value is to be obtained. Let the given sequence be denoted by [N, 0] (23) Then successive columns are generated by the formula AN
.
[N,M+l]—[N,M] + [NM—l]—[NM]
=
[N+l,M]—[N,M] _________________
+ [N— 1, M]
—
[N, M]
(24)
An auxiliary condition [N, 1] =cx is also imposed. The value of aM chosen, for M= 0, 1, 2, is given by aM=—~[l—(—i)]. (25) —
...
3.Results We used the method to study the Z(4) spin model. For this model one expects a single phase transition from results obtained by Elitzur et al. in considering the clock models first introduced by Jose et al. [6]. More recently Catterall [7] has examined the scaling of the free energy to obtain the exponent of the correlation length in the case of strip geometries. For the case of symmetric lattices, the Bulirsch and Stoer (BS) [8] extrapolation scheme was used and found to work quite satisfactorily. However, the infinite volume estimates, using the Vanden Broeck and Schwartz (YBS) [9] approximation scheme used by Hamer and Barber [10] in their finite size scaling analysis, were found to work better in the case of strip geometries. [51
Successive columns then generate the required limiting value. The displayed tables show the results for the location ofthe closest zeros in the complex plane, for the different lattices considered. Applying the BS extrapolation scheme to the data obtained in table 1, we get an infinite volume estimate for the exponent v to be 1.061(6). Considering data in table 2, v is 1.029(7). However the VBS approximation scheme produces an asymptotic value of 0.989(2). From the strip geometry data given in table 3, the VBS approximation scheme gives the value of v to be 1.077(4).
Table 1 Location of the closest zero with free boundary conditions. L
Re(u)
Im(u)
v
2 3 4 5 6
0.000000000000000 0.201248632385597 0.269826888749186 0.304402531897320 0.325340085360403
0.414213562328067 0.305193065395396 0.238324399472003 0.195707107877198 0.166226294340994
1.327491190300671 1.163236294918317 1.132630261103010 1.116693629114001
263
Volume 146, number 5
PHYSICS LETTERS A
28 May 1990
Table 2 Closest zero with periodic boundary conditions in the L direction.
L
Re(u)
Im(u)
p
2 3 4 5 6
0.257065866406596 0.348326604256138 0.372604293780331 0.383701687949191 0.390063758089077
0.529085514236190 0.304776453932598 0.220254659216910 0.173904945415597 0.144181649043851
0.7351088249307292 0.8857366163862909 0.9444207297869311 0.9727194764403912
Table 3 Closest zero with periodic boundary conditions in the L direction.
L
L
Re(u)
Im(u)
p
2 3 4 5 6
10 10 10 10 10
0.3226097680521 0.4235834643656 0.5400899140669 0.4796154047344 0.4493215213282
0.4751100854037 0.1545557866117 0.09056826396000 0.06604828388895 0.05423491799773
0.3610580355155123 0.5382756144486865 0.7067815795067042 0.9252032837678587
4. Conclusion The results obtained from this simulation agree well with those found previously and this method provides an efficient method to estimate the exponent with considerably lesser requirements on cornputer time, the limitation being on the size of lattices that could be considered due to large memory requirements. However, one does not really require to simulate the system at very large lattices as the sealing of the zeros is a finite size effect and good results are already obtained with the small lattices considered. The generation of the partition function took less than 30 minutes on the SCRI VAX 8700 for the largest system considered.
Acknowledgement I am deeply indebted to Dr. Gyan Bhanot for many patient explanations during the early stages of the project. I also thank Dr. D. Duke, Dr. U. Heller, R.
264
Villanova, R. Bertram for help and encouragement. Thanks are due to Dr. K.M. Bittar for clarifications and criticisms regarding this work. This work was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the U.S. Department of Energy through Contract No. DE-FCO5-85ER250000.
References [I] G. Bhanot Preprint FSU-SCRI-89-90 (1989). [2] K. Binder, Physica 62 (1972)508. [3] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404. [4] C. Itzykson, RB. Pearson and B. Zuber, NucI. Phys. B 220 (1983) 415. [5] S. Elitzur, R.B. Pearson and J. Shigemitsu, Phys. Rev. 19 (1979) 3698. [6] J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys.Rev.B 16(1977)1217. [7] S.M. Catterall, Phys. Lett. B 231 (1989)141. [8]R. Bulirsch and J. Stoer, N. Math. 6 (1964) 413. [9]J.M. Vanden Broeck and L.W. Schwartz, SIAM J. Math. Anal. 10 (1979) 658. [10]C.J. Hamer and M.N. Barber, J. Phys. A 14 (1981) 2009.
PHYSICS LETTERSA
28 May 1990
CORRELATION LENGTH EXPONENT OF THE 2-d Z(4) MODEL USING AN EXACT METHOD Peter WILLIAM Supercomputer Computations Research Institute and Department of Physics, Florida State University, Tallahassee, FL 32306, USA Received 1 March 1990; revised manuscript received 20 March 1990; accepted for publication 21 March 1990 Communicated by A.A. Maradudin
Using a recently suggested exact method to determine the partition function ofa discrete model, the correlation length exponent v for the two-dimensional Z(4) spin model is estimated. This estimation is made from a study of the zeros of the partition function on finite lattices.
1. Introduction
where S and S’ indicate configurations ofthe one dimensional spin system given by
In a recent paper a new method has been suggested to compute the density of states exactly for a set of discrete models [1]. The idea is an extension of the method given in an earlier work [2] to compute the partition function. Following the lines of ref. [1], discrete Z( n) models in two dimensions and, in particular, the Z( 4) model is considered. We give a brief discussion of the method used. The details can be found in ref. [1]. Consider an example of Z( 2) symmetry (the Ising model). Boltzmann weights of u = e~or u = 1 are assigned to a bond depending on whether the spins are aligned antiparallel or parallel. In orderto obtain the density ofstates one has to store the weights ofthe bonds ofeach configuration, hence two arrays W0 and W,, are constructed, which contam information regarding the weights of the vertical bonds generated in each configuration. Each time a spin is added the whole array is updated. The specific updating procedure is described below. This process makes the updates highly nonlocal but results in the algorithm being efficient. The arrays are defined as old and new, and the basic transformation of the weight array with the addition of a spin at the lattice site labelled i, is given as
Wa(S)
=
W0(S) +uW0(S’)
,
(1)
S= (s1, s2, ~
—
...,
s1,
(
...,
s1)
~ 2,
— ~‘~‘
,
:,
,
,
(2) (3)
/
The bar over s, just switches the spin value at that location from 0 to 1 and vice versa. With each added spin, the weight arrays are updated until the whole row is completed. After the row is completed, one just multiplies all the arrays by the Boltzmann weight ofthe spins in the row. This takes care ofall the bonds up to the particular layer being presently considered. The whole lattice is thus constructed layer by layer. From this process one could compute the partition function of the system at a particular temperature. However, one would, in general, be more interested in computing the density of states, and thus avoid the problem of performing simulations at numerous values of temperatures. With the choice of u given above the partition function can be expressed as a polynomial of finite degree in u, em
Z(u)
~ G(k)uk.
=
(4)
k0
Here em is the maximum possible value of the energy. If we choose the value of u such that u=cVm,
0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland)
(5) 261
Volume 146, number 5
PHYSICS LETTERS A
where c1 and rn are non-negative integers, then we observe that (6)
Z(u)= k~O
This implies that for multiplication by a factor u, we have —
uZ(u)= k=O ~
41 Crn(k)U’<
=Cm(0)u+Crn(l)u2+...+Crn(ml)u’~.
(7)
28 May 1990
ory, and this limits the method from being used in very large volumes. The memory that is required in d dimensions is roughly given by mem~2.Onhd_em. (13) Another limitation results from the accuracy to which the coefficients can be computed. Since one is basically integerthat arithmetic, one is restricted byperforming the largest integer can be represented on a machine in maximum precision. The accuracy inbitsisroughlygivenby
Considering a new set of coefficients C’ ‘s which are
acc l’~’log
themselves functions of the variable u, we also have that 1’ rn—I uZ(u) k=o ~ C’m(k)U
2n (14) The generalisation for considering Z(n) models follows quite naturally with a few redefinitions. In one phase factors which take values isfrom 0 tobyn—I. dimension then,Z(n) configuration labelled the The bond energy, in the case of even n, between a .
=C~,n(0)+C~rn(l)u+...+C~rn(rnl)um~(8) .
pair of spins labelled by values n, and n 1 is given by
Comparing terms in the two previous forms given above, we have the following, C’n(O)ciCm(ml)
(9)
E(n,, n1) =d1 d
—n— n—n , j
—
for d< ~n, cor A> I .-2n,
,
—
A
I,
‘15
.
(10)
Similar equations hold in the case of odd values of
We thus see how the important step of the updating procedure as given by (1) transforms in terms of Cm’S.
n with n replaced by n 1 in the inequalities. The generalisation ofthe updating procedure given by (1) is now given by
One can treat each of the weight arrays W(S) as a partial partition function since
W~(S)= ~ exp[ —/3E(n1, n1)] W0(S’)
J=1
C~nU)=Crn(jl),
rn—i.
—
.
(16)
ni = 0
Z(u)=~W(S).
(11) where the states S and S’ are defined by
The rule used in performing the multiplication by u on the whole partition function can be applied to each of the weight arrays separately. At each intermediate step to obtain W~,we add up the respective values of Cm’5 from W0(S) and uW0(S’). Finally, for a particular m we add up the coefficients to get the Cm of the whole partition function. Comparing forms (4) and (6) we deduce that if rn = em + 1 then the spectral coefficients are given by G(k)—C (k) k—0 rn—i (12) —
m
—
This provides a method for obtaining the density of states without having to resort to repeated simulations at different temperatures. However, this process has resulted in a considerable increase of mem262
S = (n1, s = (n I,
n~,
...,
...)
..
,
...)
(17) (18)
.~
2. Scaling of zeros and critical exponents Having obtained the density of states, we can now examine the behaviour of the Lee—Yang [3] zeros of the partition function in the complex plane, to look for phase transitions. Using the results ofref. [4J one can obtain quantitative predictions for the correlation length critical exponent v. The zero closest to the real axis scales like P on an li lattice. Hence the real and imaginary parts of
Volume 146, number 5
PHYSICS LETTERS A
the zeros obey the following relations given by [4] u, (1)
—
u~= 1- l/vf
(19)
1(0),
Re[u
1(1)] —u~=L”f~’(0)
(20)
,
(21)
Im(u1)=l’~f~(0) .
Here u, refers to the ith root, u,, to the infinite volume critical point and I is in general a complex number. From these relations one finds estimates for the critical exponent v, f
V
—
~Imu1(1+l) =log~,, Imu1(1)
)[
(1)]~ log
(22)
In order to obtain the critical exponent v and the critical temperature, we find estimates for the exponent from (22) above and use an extrapolation scheme to get the infinite volume values. Depending on the geometry being considered two different extrapolation schemes were used.
28 May 1990
Approximation scherne. The VBS approximation scheme is described in the following paragraph. A table of approximants is constructed from a given sequence of values, to which the limiting value is to be obtained. Let the given sequence be denoted by [N, 0] (23) Then successive columns are generated by the formula AN
.
[N,M+l]—[N,M] + [NM—l]—[NM]
=
[N+l,M]—[N,M] _________________
+ [N— 1, M]
—
[N, M]
(24)
An auxiliary condition [N, 1] =cx is also imposed. The value of aM chosen, for M= 0, 1, 2, is given by aM=—~[l—(—i)]. (25) —
...
3.Results We used the method to study the Z(4) spin model. For this model one expects a single phase transition from results obtained by Elitzur et al. in considering the clock models first introduced by Jose et al. [6]. More recently Catterall [7] has examined the scaling of the free energy to obtain the exponent of the correlation length in the case of strip geometries. For the case of symmetric lattices, the Bulirsch and Stoer (BS) [8] extrapolation scheme was used and found to work quite satisfactorily. However, the infinite volume estimates, using the Vanden Broeck and Schwartz (YBS) [9] approximation scheme used by Hamer and Barber [10] in their finite size scaling analysis, were found to work better in the case of strip geometries. [51
Successive columns then generate the required limiting value. The displayed tables show the results for the location ofthe closest zeros in the complex plane, for the different lattices considered. Applying the BS extrapolation scheme to the data obtained in table 1, we get an infinite volume estimate for the exponent v to be 1.061(6). Considering data in table 2, v is 1.029(7). However the VBS approximation scheme produces an asymptotic value of 0.989(2). From the strip geometry data given in table 3, the VBS approximation scheme gives the value of v to be 1.077(4).
Table 1 Location of the closest zero with free boundary conditions. L
Re(u)
Im(u)
v
2 3 4 5 6
0.000000000000000 0.201248632385597 0.269826888749186 0.304402531897320 0.325340085360403
0.414213562328067 0.305193065395396 0.238324399472003 0.195707107877198 0.166226294340994
1.327491190300671 1.163236294918317 1.132630261103010 1.116693629114001
263
Volume 146, number 5
PHYSICS LETTERS A
28 May 1990
Table 2 Closest zero with periodic boundary conditions in the L direction.
L
Re(u)
Im(u)
p
2 3 4 5 6
0.257065866406596 0.348326604256138 0.372604293780331 0.383701687949191 0.390063758089077
0.529085514236190 0.304776453932598 0.220254659216910 0.173904945415597 0.144181649043851
0.7351088249307292 0.8857366163862909 0.9444207297869311 0.9727194764403912
Table 3 Closest zero with periodic boundary conditions in the L direction.
L
L
Re(u)
Im(u)
p
2 3 4 5 6
10 10 10 10 10
0.3226097680521 0.4235834643656 0.5400899140669 0.4796154047344 0.4493215213282
0.4751100854037 0.1545557866117 0.09056826396000 0.06604828388895 0.05423491799773
0.3610580355155123 0.5382756144486865 0.7067815795067042 0.9252032837678587
4. Conclusion The results obtained from this simulation agree well with those found previously and this method provides an efficient method to estimate the exponent with considerably lesser requirements on cornputer time, the limitation being on the size of lattices that could be considered due to large memory requirements. However, one does not really require to simulate the system at very large lattices as the sealing of the zeros is a finite size effect and good results are already obtained with the small lattices considered. The generation of the partition function took less than 30 minutes on the SCRI VAX 8700 for the largest system considered.
Acknowledgement I am deeply indebted to Dr. Gyan Bhanot for many patient explanations during the early stages of the project. I also thank Dr. D. Duke, Dr. U. Heller, R.
264
Villanova, R. Bertram for help and encouragement. Thanks are due to Dr. K.M. Bittar for clarifications and criticisms regarding this work. This work was supported by the Florida State University Supercomputer Computations Research Institute which is partially funded by the U.S. Department of Energy through Contract No. DE-FCO5-85ER250000.
References [I] G. Bhanot Preprint FSU-SCRI-89-90 (1989). [2] K. Binder, Physica 62 (1972)508. [3] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404. [4] C. Itzykson, RB. Pearson and B. Zuber, NucI. Phys. B 220 (1983) 415. [5] S. Elitzur, R.B. Pearson and J. Shigemitsu, Phys. Rev. 19 (1979) 3698. [6] J.V. Jose, L.P. Kadanoff, S. Kirkpatrick and D.R. Nelson, Phys.Rev.B 16(1977)1217. [7] S.M. Catterall, Phys. Lett. B 231 (1989)141. [8]R. Bulirsch and J. Stoer, N. Math. 6 (1964) 413. [9]J.M. Vanden Broeck and L.W. Schwartz, SIAM J. Math. Anal. 10 (1979) 658. [10]C.J. Hamer and M.N. Barber, J. Phys. A 14 (1981) 2009.