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Curvilinear coordinates in the scaling theory of tricritical points
Solid State Communications,
Vol. 13, Pp. 1665—1668, 1973.
Pergamon Press.
Printed in Great Britain
CURVILINEAR COORDINATES IN THE SCALING THEORY OF TRICRITICAL POINTS David J. Bergman, Yoseph Imry and Ora Entin-Woh.lman Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel (Received 25 April 1973; in revised form 3 September 1973 by P. G. de Gennes)
We demonstrate the existence of homogeneity properties of the singular part of the free energy at the tricritical point of a soluble magneto—elastic model. The free energy is only homogeneous when viewed as a function of the appropriate curvilinear coordinates in the space of independent thermodynamic variables. It is not homogeneous as a function of the linear coordinates of either Griffiths or Riedel. The implications of this result for the general scaling theory of tricritical points are discussed.
MUCH OF the recent interest in tricritical points, initiated by Griffiths,1 has been concentrated in the attempt to construct a scaling theory for tricritical behavior.~5A crucial step in this attempt is the assumption that the singular part of the free energy (and consequently, the singular part of any thermodynamic quantity) is a homogeneous function of some appropriate thermodynamic variables, called scaling fields,1 2 in the neighborhood of the tricritical point. A proof of this assertion, using renormalization group ideas, has been given by Wegner.6 Unfortunately, an explicit calculation by Reatto,7 using the droplet model to calculate the free energy of a metamagnetic transition near its tricritical point, seems to show that this is not true.
,
9c ~
,•.
AP
P~CT)
CT P)
9/ i
p ~.
/
-m
T
FIG. 1. Schematic graph of a line of phase transitions P~(T)[or T~(P)Jin the T—Pplane and its tangent at the tricritical point (T~,P~). The graph illustrates the various coordinates that can be used to describe an arbitrary pomt (7’, F) m the plane.
In this letter we try to resolve this paradox by pointing out that the homogeneity property can be very sensitive to the proper choice of scaling fields. Two different particular choices were suggested by Riedel2 and by Griffiths,’ ~ and these are shown in Fig. 1. Both the pair p~,P2, proposed by Riedel,2 and the pair g, AP,8 proposed by Griffiths,4 are simple linear combinations of ~ T T 7’~and AP P P~,and the appropriate coordinate axes are straight lines in the T—Pplane. It is easy to show that, if the Gibbs free energy G(T, F) is homogeneous in one of these pairs, it is also homogeneous in the
other pair. Hence they are equally useful in the description of tricritical behavior. By considering a soluble model that has a tncritical point (7,P~)we will show that neither of these choices is an appropriate one for the model.
—
—
We would like to stress that references 3 and 7 and the present work deal with three different models. We show the need for curvilinear coordinates only in 1665
1666
SCALING THEORY OF TRICRITICAL POINTS
a special but exactly soluble model. Wegner and Riedel3 have found the scaling fields in their model by employing approximate renormalization group recursion formulae. The question of whether the linear approximation to the scaling fields (equation 12 of reference 3) is accurate enough to display the homogeneous form of the free energy is not discussed in reference 3. The same type of linear approximation did not lead to homogeneity in Reatto’s model.7 Our model is the Baker—Essam model9 for a compressible Ising lattice, whose Statistical Mechanics can be reduced rigorously to that of a rigid Ising lattice. This model has been shown~°’~ to have a magnetic transition line T~(F)at H 0 which corresponds to 2nd order phase transitions for F> 0, first order phase transitions for P < 0 and a tricritical point at P = 0. While it is true that this model has some unrealistic aspects, we are only using it to investigate some of the general assumptions made in the scaling theory of tricritical points. As far as we know, this is the only known case of a soluble model that has a tricritical point, Since the model is only soluble in a special ensemble, the X-ensemble, all thermodynamic quantities were originally calculated as functions of the independent variables T and X.’°” The variable A is the external force per particle exerted on particles at the surface of the system. It is connected to the pressure P and the lattice spacing a by (the model assumes a simple cubic lattice)
Vol. 13, No. 10
very similar to those of the rigid Ismg model, except that T— T~is replaced by 7’— T~(X).But we want to express these functions in terms of T— TC(P), since the pressure is the external parameter that we can usually control experimentally. In order to express T —T~(X)in terms of 7’— T~(P)we differentiate 7’— T~[X(T, F)] with respect to 7’. Using reference 11, it can be shown that
d [T
—
i~ (X)J
dT
C
14
flI
~
‘
=
Pad_l,
=
S
=
1
J\1
(‘~\
c~~”) )
We integrate the reciprocal of this equation from T~(P)to T. For P>0, T~(P)= T~(X(T~(P), F)), so that 7’— T~(X)vanishes at the lower limit of integration. But for P < 0, there are two different values of X(T~ (F), F) corresponding to the two sides of the first order transition.10’11 In that case we use the fact that T~(P)= T~{X~[7’~(F)]}, where X~(T)is the inverse function of T~(X),to write’1 T~(P) ~{X [T~(P),P] —
}
X~[T~(P)I
—
X ET~(P),P] (4)
IF I on both sides of the transition. Using this result for P < 0, we can write the integral of (3) as follows
(r— ~(X)\
T— T~(P)
~
~11~’
)‘
(5)
(1)
where’2
where d is the dimensionality. In terms of these variables we can write the following differential equation for the Gibbs free energy
_~(~)
‘7’ —
(3)
F A
‘7’
~
F1
\ =
ign x
—.
-
~ [B;(lSi~P)÷AxI+
~a~~IxI~1
.
(6)
Sc(X)~S;IT?c(X)I’~’+...,
(2) where the two signs refer to the ordered and disordered regimes. A similar expression can be written for the volume per particle ad. The first term on the R.H.S. is analytic in A, while the second term describes the Ising-type singularity that occurs at the line 7~(X). In fact, all of the thermodynamic functions of the Baker—Essam model have singular parts which are
B~is a positive constant that generally has different values on the two sides of the first order transition. Equation (5) can be solved to yield 7’— T~(A) IFI”~ = F
(r— r~(P)\ ~
~
).
(7)
Vol. 13, No. 10
SCALING THEORY OF TRICRITICAL POINTS
Substituting this result in equation (2) we find’2
is an ordinary critical exponent. If we now use equations (13), [14(a)] and [14(b)] to write G~in terms ofg and P, we find
IT— T (P)\
S
=
S~(X)+IFtO_a1~ct1S
).
1~ PIi~~
(8)
2I:
From equation (1) and the form ofa one can easily see that the second term on the R.H.S. is the leading singular term in S(T, P).’3 By integrating equation (8) over Twe find the following form for the leading singular part of G:~’14
=
IF
2_&t)G 1
I
—a)
G,
=
IPI2(2t)Fi(Jj
for ~Pj~1g—A~P [16(a)] for ~FI0~ Ig—A~P2I and F> 0:
~, G
GS(T,P)
1667
—
~pj2(2.~)+4)(O~at)~
.,,
[16(b)]
IL
(9) These expressions indicate that G8(g, F) has different homogeneity properties in the two regions
\IP I where we have defined
04Ig—A~F2I, and
g
T— T~(F) = g—g~(P)
~n
1/as
—
(10) (11) 1”
at =
at and 0 are tricritical scaling exponents, defined after 4 The function g~(F) has the form Gnffiths.
g~(F)=
(13)
ACP2
for small F, i.e. in the vicinity of the tricritical point. This results from the fact that T~(P)is an analytic function. G5 in equation (9) has the same homogeneous 4 except that that Griffiths assumed, gform has been replaced would by the have curvilinear coordinate We will now show that it is impossible to express G 8 by a homogeneous function of g and P. The function G1 has the usual asymptotic forms
~.
G
2 1(x)
Gj(x)
forx~’l
[14(a)]
x x2 + const.x2~forx
‘~
I andP> 0, [14(b)]
where —
1
—
a (15)
~P(~- Ig—A~F2~
(PI
(17) for P>0. We believe that these are the regions one should properly designate as the tricritical region and the critical region, respectively. In order for G 5(g, F) homogeneous neighborhood gto=beP truly = 0, we would haveintothe have either 0 2 of or a = a~.Clearly, neither of these conditions hold in the Baker—Essam model. It is straightforward to show that G 5 is not a homogeneous function of Riedel’s variables Ui ,P~either. Our result makes it clear that, at least sometimes. the free energy must be written as a function of curvilinear coordinates in the space of the independent thermodynamic variablesassociated in order to bring out the homogeneity properties with a tricritical point. We would like to go further and conjecture that this is perhaps true in general. In order to investigate this conjecture one ought to try and decipher the nature of the thatIn appear in Wegner’s 6 that wasscaling quotedfields earlier. this connection theorem it would also be very interesting to see whether the fact that Reatto s calculation does not produce a homogeneous form for the singular part of the free energy7 is not due to an improper choice of the scaling fields. A detailed discussion of the tricritical point in the Baker—Essam model as well as in more general types of systems with a renormalized second order phase transition will be given elsewhere.
1668
SCALING THEORY OF TRICRITICAL POINTS
Acknowledgements
—
Vol. 13, No. 10
We would like to thank B.I. Halperin for extremely helpful discussions.
REFERENCES 1. 2.
GRIFFITHS R.B.,Phys. Rev. Lett. 24,715 (1970). RIEDELE.K.,Phys. Rev. Len. 28, 675 (1972).
3.
RIEDEL E.K. and WEGNER FJ., Fhys. Rev. Lett. 29, 349 (1972).
4.
GRIFFITHS R.B.,Fhys. Rev. B7, 545 (1973).
5.
CHANG TS., HANKEY A. and STANLEY H.E., To appear as article no BaEl in the Proc. 13th mt. Conf Low Temperature Physics, Boulder, Colorado, August (1972).
6.
WEGNER FJ.,Phys. Rev. 85,4529 (1972).
7. 8.
REATTO L.,Phys. Rev. BS, 204 (1972). Griffiths used the symbol A instead of ~.P.We use a different notation in order to avoid confusion because we have a different use for A [see equation (I)]. BAKER Jr., G.A. and ESSAM J.W.,Fhys. Rev. Lett. 24,447 (1970).
9. 10. 11.
GUNTHER L., BERGMAN D.J. and IMRY Y.,Phys. Rev. Len. 27, 558 (1971). BERGMAN D.J., IMRY Y. and GUNTHER L., J. Stat. Phys. 7(4), 337 (April 1973).
12.
It should be kept in mind that the function F(x), as weLl as the functions I~,S~,and G1 defined in equations (7—9), obviously depend on Sign F. For P <0, they also depend on which side of the first order transition we are on: In equation (6) the first term has different values B,., B_ in the two phases. We have not denoted these facts explicitly in order to keep the equations from becoming too cumbersome. While S~(X)also contributes some singular terms when we substitute A = X(T, P) as solved from equation (1), these are negligible near the tricritical point.
13. 14.
One might wonder why we did not substitute equation (7) directly into G kTlog Z (13, A) (d 1)NPad (see references 10 and 11). While this is possible, it would then be more difficult to separate out the leading singular part of G. ~—
—
—
Nous démontrons l’existence de 1’homog~neitéde la partie singuliêre de l’énergie libre au voisinage du point tricritique d’un modêle magneto— élastique soluble. L’énergie libre est homogène seulement comme fonction des coordonnées curvilineai.res appropriées, dans l’espace de variables thermodynamiques independentes. Elle n’est pas homogène comme fonction des coordonnées linéaires ni de Griffiths ni de Riedel. Nous discutons l’implication de ces resultats sur Ia théorie générale de scaling en points tricritiques.
Note added in proof It has been pointed out to us by Profs. T.S. Chang and H.E. Stanley that the importance of curvilinear coordinates near tricritical points can also be inferred from their formulation of tricritical scaling (see reference 5, Phys. Rev. Lett. 29, 278 (1972) and Phys. Rev. B, the issue of May 1973). We would like to thank Profs. Chang and Stanley for their comments.
Vol. 13, Pp. 1665—1668, 1973.
Pergamon Press.
Printed in Great Britain
CURVILINEAR COORDINATES IN THE SCALING THEORY OF TRICRITICAL POINTS David J. Bergman, Yoseph Imry and Ora Entin-Woh.lman Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel (Received 25 April 1973; in revised form 3 September 1973 by P. G. de Gennes)
We demonstrate the existence of homogeneity properties of the singular part of the free energy at the tricritical point of a soluble magneto—elastic model. The free energy is only homogeneous when viewed as a function of the appropriate curvilinear coordinates in the space of independent thermodynamic variables. It is not homogeneous as a function of the linear coordinates of either Griffiths or Riedel. The implications of this result for the general scaling theory of tricritical points are discussed.
MUCH OF the recent interest in tricritical points, initiated by Griffiths,1 has been concentrated in the attempt to construct a scaling theory for tricritical behavior.~5A crucial step in this attempt is the assumption that the singular part of the free energy (and consequently, the singular part of any thermodynamic quantity) is a homogeneous function of some appropriate thermodynamic variables, called scaling fields,1 2 in the neighborhood of the tricritical point. A proof of this assertion, using renormalization group ideas, has been given by Wegner.6 Unfortunately, an explicit calculation by Reatto,7 using the droplet model to calculate the free energy of a metamagnetic transition near its tricritical point, seems to show that this is not true.
,
9c ~
,•.
AP
P~CT)
CT P)
9/ i
p ~.
/
-m
T
FIG. 1. Schematic graph of a line of phase transitions P~(T)[or T~(P)Jin the T—Pplane and its tangent at the tricritical point (T~,P~). The graph illustrates the various coordinates that can be used to describe an arbitrary pomt (7’, F) m the plane.
In this letter we try to resolve this paradox by pointing out that the homogeneity property can be very sensitive to the proper choice of scaling fields. Two different particular choices were suggested by Riedel2 and by Griffiths,’ ~ and these are shown in Fig. 1. Both the pair p~,P2, proposed by Riedel,2 and the pair g, AP,8 proposed by Griffiths,4 are simple linear combinations of ~ T T 7’~and AP P P~,and the appropriate coordinate axes are straight lines in the T—Pplane. It is easy to show that, if the Gibbs free energy G(T, F) is homogeneous in one of these pairs, it is also homogeneous in the
other pair. Hence they are equally useful in the description of tricritical behavior. By considering a soluble model that has a tncritical point (7,P~)we will show that neither of these choices is an appropriate one for the model.
—
—
We would like to stress that references 3 and 7 and the present work deal with three different models. We show the need for curvilinear coordinates only in 1665
1666
SCALING THEORY OF TRICRITICAL POINTS
a special but exactly soluble model. Wegner and Riedel3 have found the scaling fields in their model by employing approximate renormalization group recursion formulae. The question of whether the linear approximation to the scaling fields (equation 12 of reference 3) is accurate enough to display the homogeneous form of the free energy is not discussed in reference 3. The same type of linear approximation did not lead to homogeneity in Reatto’s model.7 Our model is the Baker—Essam model9 for a compressible Ising lattice, whose Statistical Mechanics can be reduced rigorously to that of a rigid Ising lattice. This model has been shown~°’~ to have a magnetic transition line T~(F)at H 0 which corresponds to 2nd order phase transitions for F> 0, first order phase transitions for P < 0 and a tricritical point at P = 0. While it is true that this model has some unrealistic aspects, we are only using it to investigate some of the general assumptions made in the scaling theory of tricritical points. As far as we know, this is the only known case of a soluble model that has a tricritical point, Since the model is only soluble in a special ensemble, the X-ensemble, all thermodynamic quantities were originally calculated as functions of the independent variables T and X.’°” The variable A is the external force per particle exerted on particles at the surface of the system. It is connected to the pressure P and the lattice spacing a by (the model assumes a simple cubic lattice)
Vol. 13, No. 10
very similar to those of the rigid Ismg model, except that T— T~is replaced by 7’— T~(X).But we want to express these functions in terms of T— TC(P), since the pressure is the external parameter that we can usually control experimentally. In order to express T —T~(X)in terms of 7’— T~(P)we differentiate 7’— T~[X(T, F)] with respect to 7’. Using reference 11, it can be shown that
d [T
—
i~ (X)J
dT
C
14
flI
~
‘
=
Pad_l,
=
S
=
1
J\1
(‘~\
c~~”) )
We integrate the reciprocal of this equation from T~(P)to T. For P>0, T~(P)= T~(X(T~(P), F)), so that 7’— T~(X)vanishes at the lower limit of integration. But for P < 0, there are two different values of X(T~ (F), F) corresponding to the two sides of the first order transition.10’11 In that case we use the fact that T~(P)= T~{X~[7’~(F)]}, where X~(T)is the inverse function of T~(X),to write’1 T~(P) ~{X [T~(P),P] —
}
X~[T~(P)I
—
X ET~(P),P] (4)
IF I on both sides of the transition. Using this result for P < 0, we can write the integral of (3) as follows
(r— ~(X)\
T— T~(P)
~
~11~’
)‘
(5)
(1)
where’2
where d is the dimensionality. In terms of these variables we can write the following differential equation for the Gibbs free energy
_~(~)
‘7’ —
(3)
F A
‘7’
~
F1
\ =
ign x
—.
-
~ [B;(lSi~P)÷AxI+
~a~~IxI~1
.
(6)
Sc(X)~S;IT?c(X)I’~’+...,
(2) where the two signs refer to the ordered and disordered regimes. A similar expression can be written for the volume per particle ad. The first term on the R.H.S. is analytic in A, while the second term describes the Ising-type singularity that occurs at the line 7~(X). In fact, all of the thermodynamic functions of the Baker—Essam model have singular parts which are
B~is a positive constant that generally has different values on the two sides of the first order transition. Equation (5) can be solved to yield 7’— T~(A) IFI”~ = F
(r— r~(P)\ ~
~
).
(7)
Vol. 13, No. 10
SCALING THEORY OF TRICRITICAL POINTS
Substituting this result in equation (2) we find’2
is an ordinary critical exponent. If we now use equations (13), [14(a)] and [14(b)] to write G~in terms ofg and P, we find
IT— T (P)\
S
=
S~(X)+IFtO_a1~ct1S
).
1~ PIi~~
(8)
2I:
From equation (1) and the form ofa one can easily see that the second term on the R.H.S. is the leading singular term in S(T, P).’3 By integrating equation (8) over Twe find the following form for the leading singular part of G:~’14
=
IF
2_&t)G 1
I
—a)
G,
=
IPI2(2t)Fi(Jj
for ~Pj~1g—A~P [16(a)] for ~FI0~ Ig—A~P2I and F> 0:
~, G
GS(T,P)
1667
—
~pj2(2.~)+4)(O~at)~
.,,
[16(b)]
IL
(9) These expressions indicate that G8(g, F) has different homogeneity properties in the two regions
\IP I where we have defined
04Ig—A~F2I, and
g
T— T~(F) = g—g~(P)
~n
1/as
—
(10) (11) 1”
at =
at and 0 are tricritical scaling exponents, defined after 4 The function g~(F) has the form Gnffiths.
g~(F)=
(13)
ACP2
for small F, i.e. in the vicinity of the tricritical point. This results from the fact that T~(P)is an analytic function. G5 in equation (9) has the same homogeneous 4 except that that Griffiths assumed, gform has been replaced would by the have curvilinear coordinate We will now show that it is impossible to express G 8 by a homogeneous function of g and P. The function G1 has the usual asymptotic forms
~.
G
2 1(x)
Gj(x)
forx~’l
[14(a)]
x x2 + const.x2~forx
‘~
I andP> 0, [14(b)]
where —
1
—
a (15)
~P(~- Ig—A~F2~
(PI
(17) for P>0. We believe that these are the regions one should properly designate as the tricritical region and the critical region, respectively. In order for G 5(g, F) homogeneous neighborhood gto=beP truly = 0, we would haveintothe have either 0 2 of or a = a~.Clearly, neither of these conditions hold in the Baker—Essam model. It is straightforward to show that G 5 is not a homogeneous function of Riedel’s variables Ui ,P~either. Our result makes it clear that, at least sometimes. the free energy must be written as a function of curvilinear coordinates in the space of the independent thermodynamic variablesassociated in order to bring out the homogeneity properties with a tricritical point. We would like to go further and conjecture that this is perhaps true in general. In order to investigate this conjecture one ought to try and decipher the nature of the thatIn appear in Wegner’s 6 that wasscaling quotedfields earlier. this connection theorem it would also be very interesting to see whether the fact that Reatto s calculation does not produce a homogeneous form for the singular part of the free energy7 is not due to an improper choice of the scaling fields. A detailed discussion of the tricritical point in the Baker—Essam model as well as in more general types of systems with a renormalized second order phase transition will be given elsewhere.
1668
SCALING THEORY OF TRICRITICAL POINTS
Acknowledgements
—
Vol. 13, No. 10
We would like to thank B.I. Halperin for extremely helpful discussions.
REFERENCES 1. 2.
GRIFFITHS R.B.,Phys. Rev. Lett. 24,715 (1970). RIEDELE.K.,Phys. Rev. Len. 28, 675 (1972).
3.
RIEDEL E.K. and WEGNER FJ., Fhys. Rev. Lett. 29, 349 (1972).
4.
GRIFFITHS R.B.,Fhys. Rev. B7, 545 (1973).
5.
CHANG TS., HANKEY A. and STANLEY H.E., To appear as article no BaEl in the Proc. 13th mt. Conf Low Temperature Physics, Boulder, Colorado, August (1972).
6.
WEGNER FJ.,Phys. Rev. 85,4529 (1972).
7. 8.
REATTO L.,Phys. Rev. BS, 204 (1972). Griffiths used the symbol A instead of ~.P.We use a different notation in order to avoid confusion because we have a different use for A [see equation (I)]. BAKER Jr., G.A. and ESSAM J.W.,Fhys. Rev. Lett. 24,447 (1970).
9. 10. 11.
GUNTHER L., BERGMAN D.J. and IMRY Y.,Phys. Rev. Len. 27, 558 (1971). BERGMAN D.J., IMRY Y. and GUNTHER L., J. Stat. Phys. 7(4), 337 (April 1973).
12.
It should be kept in mind that the function F(x), as weLl as the functions I~,S~,and G1 defined in equations (7—9), obviously depend on Sign F. For P <0, they also depend on which side of the first order transition we are on: In equation (6) the first term has different values B,., B_ in the two phases. We have not denoted these facts explicitly in order to keep the equations from becoming too cumbersome. While S~(X)also contributes some singular terms when we substitute A = X(T, P) as solved from equation (1), these are negligible near the tricritical point.
13. 14.
One might wonder why we did not substitute equation (7) directly into G kTlog Z (13, A) (d 1)NPad (see references 10 and 11). While this is possible, it would then be more difficult to separate out the leading singular part of G. ~—
—
—
Nous démontrons l’existence de 1’homog~neitéde la partie singuliêre de l’énergie libre au voisinage du point tricritique d’un modêle magneto— élastique soluble. L’énergie libre est homogène seulement comme fonction des coordonnées curvilineai.res appropriées, dans l’espace de variables thermodynamiques independentes. Elle n’est pas homogène comme fonction des coordonnées linéaires ni de Griffiths ni de Riedel. Nous discutons l’implication de ces resultats sur Ia théorie générale de scaling en points tricritiques.
Note added in proof It has been pointed out to us by Profs. T.S. Chang and H.E. Stanley that the importance of curvilinear coordinates near tricritical points can also be inferred from their formulation of tricritical scaling (see reference 5, Phys. Rev. Lett. 29, 278 (1972) and Phys. Rev. B, the issue of May 1973). We would like to thank Profs. Chang and Stanley for their comments.