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Free energy for the Lifshitz point in the disordered phase
Volume 79A, number 1
PHYSICS LETTERS
15 September 1980
FREE ENERGY FOR THE LIFSHITZ POINT IN THE DISORDERED PHASE ~ S. TALUKDAR Serin Physics Laboratory, Rutgers University, Piscata way, NJ 08854, USA Received 23 June 1980
Th~free energy for the biaxial Lifshitz point is calculated to order e integral technique.
=
5
—
d in the disordered phase using the trajectory
The Lifshitz point was introduced by Hornreich et al. [1] for the following model: 1q 1 dd 5—q ddq1 Sq ddq2 d’ 3 -_H/T=~f__~v(q)Sq 2)(Sq3 Sq1_q2_q3), 2
~m q~, q~
(la)
Ld
q~. (Ib) v(q) = r + q~+ 6q~+ (q~) The ordered state of the magnetic crystal may have uniform magnetization or a lower symmetry, e.g. a periodic structure characterized by a wave number ~ If thermodynamic parameters such as pressure and temperature are varied ordering occurs into a state of uniform magnetization in one range of pressure while outside this range a periodic ordering may be formed. Moreover, the wavenumber of the ordered phase may be a continuous function of pressure and temperature so that q~= 0 is a line separating uniformly and periodically magnetized states. The point where this line intersects the line of the phase transition is the Lifshitz point. In the model described by eqs. (1), r, ~ and u are thermodynamic fields. When ö < 0, and r is lowered the inverse propagator r + q~+ + q~’ vanishes at a value q 11 ~ 0 and the ordering occurs in the periodic phase. For ~ > 0 the uniformly magnetized state is obtained. The upper critical dimension for the given model is dc(m) = 4 + m/2, m ‘s~8. The critical exponents have been calculated by Hornreich et al. [1] and Sak and Grest [2] for them = 2,d = 5 e and m = 6,d = 7 e cases. In this note we calculate the free energy and the associated scaling function for the disordered phase. It is well known that the perturbative calculation of the susceptibility using Feynman graph techniques [3] exhibits a series of logarithms of renormalized temperatures. These can be exponentiated only when the coupling constant is at its fixed point value. This method prevents calculation of the scaling function in terms of arbitrary values of irrelevant fields. Using Wegner—Houghton [4] type differential renormalization group equations, the parameters can be integrated out until the correlation length in the renormalized system is of order unity. This is the idea that Rudnick and Nelson [5] followed to calculate scaling functions for multicritical points. After mapping the initial system onto one with r = 1, they employed a fluctuation corrected Landau theory to relate the noncritical free energy, specific heat, susceptibility, etc., to corresponding quantities close to criticality. These relations together with solutions of recursion relations, give the scaling functions directly. Ford = 5 e, m = 2, the differential equations for r, u and ~ to leading order in e are ,
=
z1
i=m+1
—
—
—
*
This work was supported in part by the National Science Foundation under Grant No. DMR 79-2 1 360.
107
Volume 79A, number I
PHYSICS LETTERS
dr(1)/dl
2r(1) +Au(l) -Au(1) 6(1)
-
du(l)/dl
eu(l) —Bu2(l)/[1
,
+
r(l)]2
(4/3ir)Au(l)6(l) d6(t)/dl
=
15 September 1980
u(1)
-
dO cos 02 ~11~_~_,
6(1),
(2a) (2b,c)
where B = (ii + 8)/16ir2, A (n + 2)/16ir2 and tan 0 = q~/q~, Z [(q~)2 + q~]1/2= e112. The momentum shell has been integrated between e~’2<(q~+q~)1I2< I which corresponds to 1/a < ~ < 1, 1/b
=
re~ Q(f)A/B
~Au(f)
—
Q(i)A/B
—~
n12
+
e~ ~Au(1) QQ}4/B
— ~
4
6(1)
dO cos2O Q( [(l)+6(l) sin 012 ln[l +r(1) +6(1) sin 0]
1)i_AIB
,
1, u(1) = ued/Q(1), where Q(l) = I
+
6(1)
(Bu/e)(e~ 1) —
(3a) (3b)
6e (3c)
.
Defining another variable t(l) ~r(l)
+ ~Au(l) + (4/3ir)Au(l)
21
21
ir/2
f
+ ~ Q(l)A/B
6(l) 2
dO cos20 Q(l)1_A/B [r(l)+6(T)smO]
ln[l
+
r(l) + 6(l) sin 0],
a simpler scaling form is obtained apart from the weak dependence of “l” in t(l)
=
[r + ~Au
+
(4/3ir)A6u] e2h/Q(1y’h/B
(4)
QQ):
te2~’/Q(l’y4/B
.
(5)
The phase space shell is integrated out till the renormalized temperature is of order unity, i.e., r(l*),r~te2l*/[l +(Bu/e)(ed*_ i)1A/B ~* =t—1I~[(l—Bu/e)t~/2+Bu/e]
=
1
(6) (7)
,
,
where ~ = 2 —[(n + 2)/(n + 8)] e. The free energy of the system can be expressed [5] as ~=
f e—@-~I2)”9(1’) dl’
+ e-(d—m/2)1F(r(l),
u(l), 6(1)),
(8)
where the first term on the right-hand side of eq. (8) is the trajectory integral, which is generated by the renormalization group transformations of the constant part in free energy. The kernel of the trajectory integral is 3
9(1)~(~—)
ir/2
f
dO cos20{ln[l +rQ)+6(1)cos0] —2/(d.—m/2)}.
In the second term in eq. (8), F(r(1), u(1), 6(l)) is the free energy of the system after the parameters r, u, 6 have been mapped to r(l), u(T), 6(l). Diagrammatically, it comes from a single closed loop and is given by 108
(9)
Volume 79A, number I
PHYSICS LETTERS
15 September 1980
~gt
Fig. 1. The shaded area denotes the region of validity of 9~.
F(r(l),u(l), 6(l))~~f 4ir 0
dO cos2O
f dzz(~m/2—Oin[r(1)+z2+6(OzsinO].
(10)
0
From eqs. (9) and (10) the singular part in the free energy ~F(r,u, 6) turns out to be 2)1 t2(l) + ~2 = —
~2
d
m/2
d
rn/2 ~
dJ e~(~~mI
f
dI e~m/2)l t(1) 62(1).
(II)
The region of validity of eq. (11) is shown in fig. I. In order to express ~ in terms of scaling functions scaling fields must be introduced. These are gt(l) = g~e~’it, g~(l)= g 0 eXu~and g8 (1) = g8 eXS 1, where =
1 —e/Bu
,
g~ 6,
X~2—[(n+2)/(n+8)]e,
X~—e,
X6
=
1
Using these nonlinear scaling fields in eq. (11) 2 g~ [(1 ~ 1 28ir which, with the introduction of =
(
—
x—1 _g 0g~øu
Y=g~gi’~
g~g~2)(4 —n)/(n+ 8)
Ø~=—e/2,
—
~iLi~g~11+(n+2)/(n+ 8)le g~(1
Ø~=1/2-i-
—
g~g~I2)6I(fl4 8)]
(12)
[(n+2)/(n+8)]e/2=vX6
yields ~ =~(x~Y)~ where
(13)
4—~)/(~~8) — y~(1 — x)61(fl+8)] (l4) e(4—n) 48 —n(n + 8) 128ir2 1 [(1 — ~)( and a = ~ [(4 — n)/(n + 8)] e. Corresponding to the integration of the Brfflioun zone shell 1/a <(q~+ q~~2)hI2< 1, a e~ a parallel calculation similar to that above follows. The form of the scaling function remains the same except for a multiplicative factor of 1/4. The origin of this numerical constant owes to the fact that the shell 1/a <(q4~+ q~2)1/2< 1 is 1/2 smaller radially than the shell 1/b <(q~+ q~)1I2 < 1 when a2 = b. ~
,
c1(x,y)
,
The author gratefully thanks Professor J.V. Sak for suggesting the problem and J. Bruno for many useful discussions. 109
Volume 79A, number 1
PHYSICS LETTERS
References [1] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35(1975)1678. [2]J. Sak and G.S. Grest, Phys. Rev. B17 (1978) 3602. L3] E. Brézin, D.J. Wallace and K.G. Wilson, Phys. Rev. B7 (1973) 232. [4] K.!. Wegner and A. Houghton, Phys. Rev. A8 (1973) 401. [5] J. Rudnick and D.R. Nelson, Phys. Rev. B13 (1976) 2208; D.R. Nelson and J. Rudnick, Phys. Rev. Lett. 35(1975)178. [61S.-K. Ma, Modern theory of critical phenomena (Benjamin, Reading, MA, 1976); D.R. Nelson, Phys. Rev. Bli (1975) 3504.
110
15 September 1980
PHYSICS LETTERS
15 September 1980
FREE ENERGY FOR THE LIFSHITZ POINT IN THE DISORDERED PHASE ~ S. TALUKDAR Serin Physics Laboratory, Rutgers University, Piscata way, NJ 08854, USA Received 23 June 1980
Th~free energy for the biaxial Lifshitz point is calculated to order e integral technique.
=
5
—
d in the disordered phase using the trajectory
The Lifshitz point was introduced by Hornreich et al. [1] for the following model: 1q 1 dd 5—q ddq1 Sq ddq2 d’ 3 -_H/T=~f__~v(q)Sq 2)(Sq3 Sq1_q2_q3), 2
~m q~, q~
(la)
Ld
q~. (Ib) v(q) = r + q~+ 6q~+ (q~) The ordered state of the magnetic crystal may have uniform magnetization or a lower symmetry, e.g. a periodic structure characterized by a wave number ~ If thermodynamic parameters such as pressure and temperature are varied ordering occurs into a state of uniform magnetization in one range of pressure while outside this range a periodic ordering may be formed. Moreover, the wavenumber of the ordered phase may be a continuous function of pressure and temperature so that q~= 0 is a line separating uniformly and periodically magnetized states. The point where this line intersects the line of the phase transition is the Lifshitz point. In the model described by eqs. (1), r, ~ and u are thermodynamic fields. When ö < 0, and r is lowered the inverse propagator r + q~+ + q~’ vanishes at a value q 11 ~ 0 and the ordering occurs in the periodic phase. For ~ > 0 the uniformly magnetized state is obtained. The upper critical dimension for the given model is dc(m) = 4 + m/2, m ‘s~8. The critical exponents have been calculated by Hornreich et al. [1] and Sak and Grest [2] for them = 2,d = 5 e and m = 6,d = 7 e cases. In this note we calculate the free energy and the associated scaling function for the disordered phase. It is well known that the perturbative calculation of the susceptibility using Feynman graph techniques [3] exhibits a series of logarithms of renormalized temperatures. These can be exponentiated only when the coupling constant is at its fixed point value. This method prevents calculation of the scaling function in terms of arbitrary values of irrelevant fields. Using Wegner—Houghton [4] type differential renormalization group equations, the parameters can be integrated out until the correlation length in the renormalized system is of order unity. This is the idea that Rudnick and Nelson [5] followed to calculate scaling functions for multicritical points. After mapping the initial system onto one with r = 1, they employed a fluctuation corrected Landau theory to relate the noncritical free energy, specific heat, susceptibility, etc., to corresponding quantities close to criticality. These relations together with solutions of recursion relations, give the scaling functions directly. Ford = 5 e, m = 2, the differential equations for r, u and ~ to leading order in e are ,
=
z1
i=m+1
—
—
—
*
This work was supported in part by the National Science Foundation under Grant No. DMR 79-2 1 360.
107
Volume 79A, number I
PHYSICS LETTERS
dr(1)/dl
2r(1) +Au(l) -Au(1) 6(1)
-
du(l)/dl
eu(l) —Bu2(l)/[1
,
+
r(l)]2
(4/3ir)Au(l)6(l) d6(t)/dl
=
15 September 1980
u(1)
-
dO cos 02 ~11~_~_,
6(1),
(2a) (2b,c)
where B = (ii + 8)/16ir2, A (n + 2)/16ir2 and tan 0 = q~/q~, Z [(q~)2 + q~]1/2= e112. The momentum shell has been integrated between e~’2<(q~+q~)1I2< I which corresponds to 1/a < ~ < 1, 1/b
=
re~ Q(f)A/B
~Au(f)
—
Q(i)A/B
—~
n12
+
e~ ~Au(1) QQ}4/B
— ~
4
6(1)
dO cos2O Q( [(l)+6(l) sin 012 ln[l +r(1) +6(1) sin 0]
1)i_AIB
,
1, u(1) = ued/Q(1), where Q(l) = I
+
6(1)
(Bu/e)(e~ 1) —
(3a) (3b)
6e (3c)
.
Defining another variable t(l) ~r(l)
+ ~Au(l) + (4/3ir)Au(l)
21
21
ir/2
f
+ ~ Q(l)A/B
6(l) 2
dO cos20 Q(l)1_A/B [r(l)+6(T)smO]
ln[l
+
r(l) + 6(l) sin 0],
a simpler scaling form is obtained apart from the weak dependence of “l” in t(l)
=
[r + ~Au
+
(4/3ir)A6u] e2h/Q(1y’h/B
(4)
QQ):
te2~’/Q(l’y4/B
.
(5)
The phase space shell is integrated out till the renormalized temperature is of order unity, i.e., r(l*),r~te2l*/[l +(Bu/e)(ed*_ i)1A/B ~* =t—1I~[(l—Bu/e)t~/2+Bu/e]
=
1
(6) (7)
,
,
where ~ = 2 —[(n + 2)/(n + 8)] e. The free energy of the system can be expressed [5] as ~=
f e—@-~I2)”9(1’) dl’
+ e-(d—m/2)1F(r(l),
u(l), 6(1)),
(8)
where the first term on the right-hand side of eq. (8) is the trajectory integral, which is generated by the renormalization group transformations of the constant part in free energy. The kernel of the trajectory integral is 3
9(1)~(~—)
ir/2
f
dO cos20{ln[l +rQ)+6(1)cos0] —2/(d.—m/2)}.
In the second term in eq. (8), F(r(1), u(1), 6(l)) is the free energy of the system after the parameters r, u, 6 have been mapped to r(l), u(T), 6(l). Diagrammatically, it comes from a single closed loop and is given by 108
(9)
Volume 79A, number I
PHYSICS LETTERS
15 September 1980
~gt
Fig. 1. The shaded area denotes the region of validity of 9~.
F(r(l),u(l), 6(l))~~f 4ir 0
dO cos2O
f dzz(~m/2—Oin[r(1)+z2+6(OzsinO].
(10)
0
From eqs. (9) and (10) the singular part in the free energy ~F(r,u, 6) turns out to be 2)1 t2(l) + ~2 = —
~2
d
m/2
d
rn/2 ~
dJ e~(~~mI
f
dI e~m/2)l t(1) 62(1).
(II)
The region of validity of eq. (11) is shown in fig. I. In order to express ~ in terms of scaling functions scaling fields must be introduced. These are gt(l) = g~e~’it, g~(l)= g 0 eXu~and g8 (1) = g8 eXS 1, where =
1 —e/Bu
,
g~ 6,
X~2—[(n+2)/(n+8)]e,
X~—e,
X6
=
1
Using these nonlinear scaling fields in eq. (11) 2 g~ [(1 ~ 1 28ir which, with the introduction of =
(
—
x—1 _g 0g~øu
Y=g~gi’~
g~g~2)(4 —n)/(n+ 8)
Ø~=—e/2,
—
~iLi~g~11+(n+2)/(n+ 8)le g~(1
Ø~=1/2-i-
—
g~g~I2)6I(fl4 8)]
(12)
[(n+2)/(n+8)]e/2=vX6
yields ~ =~(x~Y)~ where
(13)
4—~)/(~~8) — y~(1 — x)61(fl+8)] (l4) e(4—n) 48 —n(n + 8) 128ir2 1 [(1 — ~)( and a = ~ [(4 — n)/(n + 8)] e. Corresponding to the integration of the Brfflioun zone shell 1/a <(q~+ q~~2)hI2< 1, a e~ a parallel calculation similar to that above follows. The form of the scaling function remains the same except for a multiplicative factor of 1/4. The origin of this numerical constant owes to the fact that the shell 1/a <(q4~+ q~2)1/2< 1 is 1/2 smaller radially than the shell 1/b <(q~+ q~)1I2 < 1 when a2 = b. ~
,
c1(x,y)
,
The author gratefully thanks Professor J.V. Sak for suggesting the problem and J. Bruno for many useful discussions. 109
Volume 79A, number 1
PHYSICS LETTERS
References [1] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35(1975)1678. [2]J. Sak and G.S. Grest, Phys. Rev. B17 (1978) 3602. L3] E. Brézin, D.J. Wallace and K.G. Wilson, Phys. Rev. B7 (1973) 232. [4] K.!. Wegner and A. Houghton, Phys. Rev. A8 (1973) 401. [5] J. Rudnick and D.R. Nelson, Phys. Rev. B13 (1976) 2208; D.R. Nelson and J. Rudnick, Phys. Rev. Lett. 35(1975)178. [61S.-K. Ma, Modern theory of critical phenomena (Benjamin, Reading, MA, 1976); D.R. Nelson, Phys. Rev. Bli (1975) 3504.
110
15 September 1980