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On the dynamics of the Ashkin-Teller-Potts model
Volume 58A, number 5
PHYSICS IETTERS
20 September 1976
ON THE DYNAMICS OF THE ASHKIN—TELLER—POTTS MODEL S. TRIMPER Sckfion Physik, Karl-Marx-Universitat, Leipzig, DDR Received 14 July 1976 The dynamical critical exponent of the s-states Ashkin--Teller -Potts model is found with the help of the linear response theory in the c = d~— d expansion (d~= 4 and 6).
In the present letter we investigate the dynamic (critical) behaviour of a system which exhibits cubic and quartic invariants of the order parameter field p(x) in the effective Hamiltonian. Such a case is realized in the continuum version of the s-states Ashkin--Teller—Potts (AlP) model [I]. The static properties of the ATP model are clarified with the help of renormalization group approach [1,2] or with a parquet graph calculation [3,4] in the framework of e-expansion, e d~ d. For the restricted ATP model (without the cubic invariant) the “critical” dimension d~=4 and otherwise dc= 6. For an investigation of the dynamics near the phase transition there are no such general and closed methods like in the static case. However, a simple possibility is to start with a kinetic equation following from the theory of irreversible processes [5] ~ t) (1)
1, ...n (number of components),
i
=
~•
p
~ a-4
...
e7 P
The special symmetry of the ATP model is “hidden” in the algebraic properties of the state vectors e~,which obey the rule [1] (repeated indices are summed) e~ef= —~--~ ö~— —j
(3)
.
The are the bare coupling constants of the model. Let us seek the dynamic exponent z by solving eq. (1) with the help of linear response theory [6, 71. Firstly, we consider the cubic part of the interacting Hamiltonian only, because it dominates the critical behaviour for d~6. Eqs. (I) and (2) yield after including an external field h; 3(s—l)3/~2
L~
1p1—— ~
73VjJk~J~,Ok—hj,
F represents the bare kinetic coefficient, H is the effective Hamiltonian of the ATP model. Repeating the analysis made by Zia and Wallace [1] and taking into account more carefully the s dependence of the prefactors in front of the cubic and quartic invariants of the order parameter field we find (s_l)312 13H=fddx
ry1 + [~
+rov~÷4(~~~2— ~ 0 ik0ii + 5i10jk) (~ij~A~i+
(s — 1)21 + ‘f2u,!kl ~ ]~i~°/’Pk~’l ‘
290
with o =
~1
.
—-
1/
We are interested in the full susceptibility G(x, t) of the system with respect to a small change of the external field h~-÷h~+ oh1. We seek the solution of eq. (1) perturbationally in the forn~~= )+ ~co)+ + where ~p(e)is the random equilibrium value of~x, t)and ~m)are corrections, which are proportional to 6h~(for all m, linear response) and totheory. in the corresponding order m of the perturbation Besides, in the framework of the linear response theory non-equilibrium properties are expressed by averages with respect to equilibrium states denoted by (...): ...
73Vijk~Oi~tPk
(2)
PHYSISCS LETTERS
Volume 58A, number 5
20 September 1976
Eq. (5) is analyzed for small w, k0 and T T~.As the results one obtains after inserting eqs. (6)—(8) (a)
(b)
iw
G~~0 \ ,w \ ‘—-—--~1—3’2 1 /r~ = 31l0—3s~ 1 ~ / i~nw, ~ Immediately the dynamical exponent follows: z 2+ c~,c = 2. __________
Fig. 1. Leading diagrams: AlP model (a), restricted model (b).
(/p~(x,t)) = fG~
1(x x’, —
t — t’)
0h1(x’, t’) d’~x’d
The exponent has the same structure as the conti-
t’ .
fig. la. The straight line represents G°whereas the broken line stands for ~°(x, t) = (~(e)(Xt)~(e)(0,0)). The point corresponds to the bare coupling constants 73 in fig. la. The full susceptibility is found in the simplest approximation by summing up all contributions coming from diagram la. Instead of the bare coupling constants had to the points renormalized ones, which are one replaced byinsert its fixed value y~(linear in e = 6—d):
nuum version of the kinetic Ising model [8], however there are different values of the constant c. The results is applicabel to the limiting cases s=0, the physical oncases i~<0, e.g. the[3],and critical for exponent gin In ofboth which is given recently s = 1. z tends toward the value of the conventional theory; for s =2, ,~= 0 and z = 2, the classical value. Let us remark, that for s = 3 the renormalized vertex part F 3 is a constant [4] and the normal scaling picture breaks down. Besides, one finds, that for 10/3 > s > 2 the static exponent of the susceptibility ~ < 1 in contradiction to the stability requirement 5 ~‘ 1. For s ~ 3 ~ is imaginary. This is a reference to a first order transition [9]. Depending on the sign of i~one obtains different be1+), haviour of the kinetic (T— T~)( = 1/2—5/4i~. Whereascoefficient for i~< 0,FF~diverges at the transition point, F tends to zero if r~ >0.
G 1 (k, w)
Tothe complete ourmodel investigation z for restricted (73 = 0, we d~give 4). the Withexponent the
In zeroth order one obtains G~,(k,w) =
2 + iwF
_____________
+ k
= G°(k,w)O~,.
(4)
In second order (first order calculation yields for T~ Tc no contributions) we find a contribution shown in
G’(O, 0) + k~F—1+k2
dtG°(x,t) ~°(x, t)exp [i(kx
The fixed point value y~is *2
36(s 2)7~2 (5)
x fddx
73
—
—
wt) —1]
.
[2—41
— —
same technique one finds the two exponents z according with the two fixed points of the restricted model obtained with a renormalization group approach [1] or a skeleton graph summation [10]. Taking into account the leading diagram (fig. lb) the result is = 2+ /‘~1
—
18K
~
6k”-’
~O)
jS1
Remark, additional factor thenot nominator of eq. (6)thatasan pointed out in [2] s in can be confirmed. The analytical form of G°(x,t) is found for d~6and T T~ —
G°(x,t)
—
0(t) exp(—x2/4rt). 32ir F t
ç~° and G°are connected by
(7)
the classical relation
1.~ —
—
C1
A /~
1
C 1121.~,w1uiC—usn-t1j—i an 2 — 8 s + 18) (s — 2) (s — (s 54(s2 — 7 s + 14)2 1 = e~
3)
s(si-6)e2 e=4—d. 54(s+2)2 The kinetic coefficient of this model tends to zero at the transition point as expected.
~ (k, ci,) = 2T/w Im G(’)(k, to). It follows
~°(x, t) =
T
[1—exp(—x2/4Ft) (1 +x2/4Ft)], (8)
am indebted to Professor Kühnel for many helpful Idiscussions.
2in2x4 in accordance with the general relation ç(x)cxx—(d—2). 291
Volume 58A, number 5
PHYSICS LETTERS
References [1] R.K.P. Zia and D.J. Wallace, J. Phys. A: General Physics A8 (1975) 1495. (2] A.B. Harris et al., Phys. Rev. Lett. 35 (1975) 327. [3] M.J. Stephens, Phys. Lett. 56A (1976) 149. [4] Th. Nattermann, submitted to Phys. Lett. [51 L.D. Landau, E.M. Lifshitz, Statistical physics (Pergamon, New York, 1968).
292
20 September 1976
[6] S. Ma and G.F. Mazenko, Phys. Rev. Bi 1(1975)4077. [7] A.S. Patashinskii and W.L. Pokrovskii, Fluctuation theory of phase transitions (Moscow, Nauka, 1975, russ.). (81 B.J. Halperin, P.C. I-lohenberg and S. Ma, Phys. Rev. BlO (1974) 139. 191 Th. Nattermann and S. Trimper, J. Phys. A: General Physics A8 (1975) 2000. [101 S. Trimper, unpublished.
PHYSICS IETTERS
20 September 1976
ON THE DYNAMICS OF THE ASHKIN—TELLER—POTTS MODEL S. TRIMPER Sckfion Physik, Karl-Marx-Universitat, Leipzig, DDR Received 14 July 1976 The dynamical critical exponent of the s-states Ashkin--Teller -Potts model is found with the help of the linear response theory in the c = d~— d expansion (d~= 4 and 6).
In the present letter we investigate the dynamic (critical) behaviour of a system which exhibits cubic and quartic invariants of the order parameter field p(x) in the effective Hamiltonian. Such a case is realized in the continuum version of the s-states Ashkin--Teller—Potts (AlP) model [I]. The static properties of the ATP model are clarified with the help of renormalization group approach [1,2] or with a parquet graph calculation [3,4] in the framework of e-expansion, e d~ d. For the restricted ATP model (without the cubic invariant) the “critical” dimension d~=4 and otherwise dc= 6. For an investigation of the dynamics near the phase transition there are no such general and closed methods like in the static case. However, a simple possibility is to start with a kinetic equation following from the theory of irreversible processes [5] ~ t) (1)
1, ...n (number of components),
i
=
~•
p
~ a-4
...
e7 P
The special symmetry of the ATP model is “hidden” in the algebraic properties of the state vectors e~,which obey the rule [1] (repeated indices are summed) e~ef= —~--~ ö~— —j
(3)
.
The are the bare coupling constants of the model. Let us seek the dynamic exponent z by solving eq. (1) with the help of linear response theory [6, 71. Firstly, we consider the cubic part of the interacting Hamiltonian only, because it dominates the critical behaviour for d~6. Eqs. (I) and (2) yield after including an external field h; 3(s—l)3/~2
L~
1p1—— ~
73VjJk~J~,Ok—hj,
F represents the bare kinetic coefficient, H is the effective Hamiltonian of the ATP model. Repeating the analysis made by Zia and Wallace [1] and taking into account more carefully the s dependence of the prefactors in front of the cubic and quartic invariants of the order parameter field we find (s_l)312 13H=fddx
ry1 + [~
+rov~÷4(~~~2— ~ 0 ik0ii + 5i10jk) (~ij~A~i+
(s — 1)21 + ‘f2u,!kl ~ ]~i~°/’Pk~’l ‘
290
with o =
~1
.
—-
1/
We are interested in the full susceptibility G(x, t) of the system with respect to a small change of the external field h~-÷h~+ oh1. We seek the solution of eq. (1) perturbationally in the forn~~= )+ ~co)+ + where ~p(e)is the random equilibrium value of~x, t)and ~m)are corrections, which are proportional to 6h~(for all m, linear response) and totheory. in the corresponding order m of the perturbation Besides, in the framework of the linear response theory non-equilibrium properties are expressed by averages with respect to equilibrium states denoted by (...): ...
73Vijk~Oi~tPk
(2)
PHYSISCS LETTERS
Volume 58A, number 5
20 September 1976
Eq. (5) is analyzed for small w, k0 and T T~.As the results one obtains after inserting eqs. (6)—(8) (a)
(b)
iw
G~~0 \ ,w \ ‘—-—--~1—3’2 1 /r~ = 31l0—3s~ 1 ~ / i~nw, ~ Immediately the dynamical exponent follows: z 2+ c~,c = 2. __________
Fig. 1. Leading diagrams: AlP model (a), restricted model (b).
(/p~(x,t)) = fG~
1(x x’, —
t — t’)
0h1(x’, t’) d’~x’d
The exponent has the same structure as the conti-
t’ .
fig. la. The straight line represents G°whereas the broken line stands for ~°(x, t) = (~(e)(Xt)~(e)(0,0)). The point corresponds to the bare coupling constants 73 in fig. la. The full susceptibility is found in the simplest approximation by summing up all contributions coming from diagram la. Instead of the bare coupling constants had to the points renormalized ones, which are one replaced byinsert its fixed value y~(linear in e = 6—d):
nuum version of the kinetic Ising model [8], however there are different values of the constant c. The results is applicabel to the limiting cases s=0, the physical oncases i~<0, e.g. the[3],and critical for exponent gin In ofboth which is given recently s = 1. z tends toward the value of the conventional theory; for s =2, ,~= 0 and z = 2, the classical value. Let us remark, that for s = 3 the renormalized vertex part F 3 is a constant [4] and the normal scaling picture breaks down. Besides, one finds, that for 10/3 > s > 2 the static exponent of the susceptibility ~ < 1 in contradiction to the stability requirement 5 ~‘ 1. For s ~ 3 ~ is imaginary. This is a reference to a first order transition [9]. Depending on the sign of i~one obtains different be1+), haviour of the kinetic (T— T~)( = 1/2—5/4i~. Whereascoefficient for i~< 0,FF~diverges at the transition point, F tends to zero if r~ >0.
G 1 (k, w)
Tothe complete ourmodel investigation z for restricted (73 = 0, we d~give 4). the Withexponent the
In zeroth order one obtains G~,(k,w) =
2 + iwF
_____________
+ k
= G°(k,w)O~,.
(4)
In second order (first order calculation yields for T~ Tc no contributions) we find a contribution shown in
G’(O, 0) + k~F—1+k2
dtG°(x,t) ~°(x, t)exp [i(kx
The fixed point value y~is *2
36(s 2)7~2 (5)
x fddx
73
—
—
wt) —1]
.
[2—41
— —
same technique one finds the two exponents z according with the two fixed points of the restricted model obtained with a renormalization group approach [1] or a skeleton graph summation [10]. Taking into account the leading diagram (fig. lb) the result is = 2+ /‘~1
—
18K
~
6k”-’
~O)
jS1
Remark, additional factor thenot nominator of eq. (6)thatasan pointed out in [2] s in can be confirmed. The analytical form of G°(x,t) is found for d~6and T T~ —
G°(x,t)
—
0(t) exp(—x2/4rt). 32ir F t
ç~° and G°are connected by
(7)
the classical relation
1.~ —
—
C1
A /~
1
C 1121.~,w1uiC—usn-t1j—i an 2 — 8 s + 18) (s — 2) (s — (s 54(s2 — 7 s + 14)2 1 = e~
3)
s(si-6)e2 e=4—d. 54(s+2)2 The kinetic coefficient of this model tends to zero at the transition point as expected.
~ (k, ci,) = 2T/w Im G(’)(k, to). It follows
~°(x, t) =
T
[1—exp(—x2/4Ft) (1 +x2/4Ft)], (8)
am indebted to Professor Kühnel for many helpful Idiscussions.
2in2x4 in accordance with the general relation ç(x)cxx—(d—2). 291
Volume 58A, number 5
PHYSICS LETTERS
References [1] R.K.P. Zia and D.J. Wallace, J. Phys. A: General Physics A8 (1975) 1495. (2] A.B. Harris et al., Phys. Rev. Lett. 35 (1975) 327. [3] M.J. Stephens, Phys. Lett. 56A (1976) 149. [4] Th. Nattermann, submitted to Phys. Lett. [51 L.D. Landau, E.M. Lifshitz, Statistical physics (Pergamon, New York, 1968).
292
20 September 1976
[6] S. Ma and G.F. Mazenko, Phys. Rev. Bi 1(1975)4077. [7] A.S. Patashinskii and W.L. Pokrovskii, Fluctuation theory of phase transitions (Moscow, Nauka, 1975, russ.). (81 B.J. Halperin, P.C. I-lohenberg and S. Ma, Phys. Rev. BlO (1974) 139. 191 Th. Nattermann and S. Trimper, J. Phys. A: General Physics A8 (1975) 2000. [101 S. Trimper, unpublished.