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Continuous phase transitions which should be first order
Solid State Communications,
Vol. 14, pp. 1069—1071, 1974.
Pergamon Press.
Printed in Great Britain
CONTINUOUS PHASE TRANSITIONS WHtCH SHOULD BE FIRST ORDER S. Alexander Israel Defense Forces and The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel (Received 21 September 1973; in revised form 5 November 1973 by P.G. de Gennes)
It is demonstrated that phase transitions predicted to be first order by the Landau symmetry arguments can be continuous for a fmite range of the physical parameters. For transitions with a rank-two tensor order parameter it is shown that fluctuations in a phase associated with the anisotropy of the tensor result in a renormalization of the cubic term. This leads to continuous transitions when the coefficient of the bare cubic term is small compared to the coefficient of the quartic term. It is shown that the argument can be extended to more general cases when the order parameter belongs to a multidimensional representation of the broken symmetry group.
LANDAU’ type symmetry arguments are usually considered reliable in predicting the order of phase transitions with a symmetry change. When cubic terms in the order parameter are allowed in the Landau expansion of the free energy the transition is expected to be first order, except for isolated points where the cubic term vanishes accidentally. There is reason to doubt the general validity of these arguments. It has been known for a long time2 that the isotropic to nematic transition of liquid crystals shows very large, almost critical, fluctuations in the disordered phase. Straley and Fisher3 have recently shown that the transition of the 3 state Potts model on the square lattice is probably continuous. Both transitions have rank-2 tensor order parameters4 and are predicted to be first order.4 From a theoretical point of view it also seems surprising that the predictions of the Landau theory do not distinguish transitions with a symmetry change from transitions without such a change when cubic terms are allowed. *
The suggestion of Straley and Fisher that their model has some ‘hidden’ symmetries is in fact erroneous. The symmetries in their problem are exactly of the type envisaged by Landau. It is also fairly easy to construct a molecular field theory retaining the full symmetry. This predicts a i st order transition.5
-
Our purpose here is to demonstrate that these predictions are in fact erroneous. We will show that for transitions with a symmetry change there are important renormalization effects on the cubic term which were neglected in previous discussions. As a result there is a range of values of the parameters for which the transitions should be continuous and show critical behaviour. For clarity we restrict our discussion to situations where the order parameter is a rank two tensor, and the broken symmetry is spherical (as for nematics) or cubic4 (as for the 3 state Potts model). The generalization is, in principle, fairly straightforward, if one uses the Landau Lifshitz technique of associating the order parameter with the irreducible representations of the (broken) high symmetry group.1 We start by constructing the expansion for the free energy. The five components of the order parameter (Q,~) always define a principal axis system (x, y, z). The energy is obviously independent of the orientation of this system and the angular fluctuations have been considered in detail by de Gennes.2 Since the energy is obviously invariant under permutations .
.
.
of the three axis (i.e. has cubic symmetry in this frame) we can always write for the ‘bare’ free energy density (in the Wilson sense)
1069
1070
CONTINUOUS PHASE TRANSITIONS F
=
~ [Aq~,—Bq~+ Cq~+ K Vq~121
[1(a)]
with the constraint = 0 where the ~ (-~(3cos2O~ I)) are the components of Q2 along the three axis, and we have used the constraint [equation 1(b)] to eliminate all mixed terms in F. Equation (1) is thus quite general to this order. It is convenient to eliminate the constraint and rewrite Fin the form: —
F
Ap2 —Bp3cos3O + Cp4 + K( lVpI2 + P2 1V012) (2) where p and 0 are defined by: pe’0 = q~+ e2fl~/3q~ + e2~”3 =
Thus p is the magnitude of Q and the phase 0 measures the anisotropy of Q. This expression for F can also be derived directly from the group theoretical arguments of the Landau Lifthitz. Their procedure6 is now to start by minimizing F with respect to 0 at constant p. The rest of their well known argument then follows. l’his procedure amounts to neglecting fluctuations in 0. Ii is however clear from equation 2 that such fluctuations will be large for small p and this will tend to decrease the effective assymmetry. We will use the most straightforward procedure to estiniate the effect of this renormalization of the cubic term. We first calculate the average (cos3O> at
Vol. 14, No. 11
(cos3O)
exp —(R ~p)/pä)
=
where the function R (p) and the leading exponent (in l/p) ô both depend on the constants B, K and T but are independent of C(and A). This should be substituted for cos3O in equation 2 giving the effective expression F = Ap2 —Bp~exp—(R(p)/p6)+ Cp4 + K~Vpi2 (7) The symmetry change occurs as a first order transition if and only if F becomes negative at some finite p f~rpositive A. This requires a range of p for which: —Bexp—-(R(p)/p6) + Cp <0 and is thus possible only when: C/B < ~(R, ô) = i~(B,K, 7) (8) When C/B is larger than this maximum value the transition is necessarily continuous with the usual type of critical behaviour. The evaluation of the integrals in equation (5) is straightforward. It is however important to remember that the cutoff dependent terms which are usually neglected, are divergent and therefore important in this case (because of the factor p2 in front). The most reasonable choice seems to be a quantum mech~cal cutoff Kp2k~ 0= kB T.
constant p and then use the result to eliminate 0 from equation (2).
(cos30)~,r= exp—(~(O2>~,r) where (02)
=
(4)
kc.o ~~0k0-k)
-c
=
p2
I
(9)
Defining 9B ci
Assuming small fluctuations, we can expand cos 30 to lowest order in 0. This gives from the relevant (0 dependent) part ofF 2~[(9B/2)-p+Kk2]0k0_k (3) Fo~p where the 0k are the fourier transforms of 0(r). Treating the 0~as statistically independent we then have:
(6)
=
=
2kB T
(J~P2
one finds for one dimension (d = 1) 2tan~(ap3~”2~ R1(p) = (2~0)’a” =15/2 [10(a)] for d = 2 R 2 (p)
2in I + (ap3)’ ; &~= 2 =
(2ir~o)
[10(b)] for d = 3 R 3(l (ap3)”2 tan~(cip3)~2; 3 3(p) = 3(2ir~o~ [10(c)] —
dk1~P+ Kk2J
-
(5) where T is the absolute temperature, c a normalization constant, and k~ 0a suitably chosen cutoff. Since the integral equation possibly 2 oneinalways gets5 a cannot result of the form vanish as fast as p
and ford=4) R 3lnIi + (cip3)~I); ~ 4(p) = 2(2ir~~)~(1c~p
=
4
—
[10(d)] where we have retained the full expressions. Since the form ofR and therefore also ‘q, depends on three
Vol. 14, No. 11
CONTINUOUS PHASE TRANSITiONS
independent physical parameters a more detailed discussion does not seem useful. It is obvious that R increases as K and B decrease. To conclude this discussion we would like to point out that our expression for (02) [equation (5)J certainlyunderestimates the true value. Even the bare potential (cos30) is always weaker than the value we have retained (~ 82) and proper renormalization would make it even weaker. We believe we have shown that transitions of the -
.
.
type considered can indeed by continuous for a fimte range of the parameters contrary to the predictions of the Landau theory. A properly renormalized evalu-
1071
ation is of course desirable but should not change the essential features of our results. The present results can be generalized in a straightforward way to the general case of phase transitions with a symmetry change when the order parameter belongs to a multidimensional irreducible representation of the broken symmetry group.’ In all such cases the free energy expansion involves the magnitude of the order parameter and phase angles analogous to 0. The results should therefore be similar. Acknowledgements The author would like to thank D. Bergman for a very helpful discussion in the early stages of this work, H. Gutfreund, G. Horwitz, I. Imry and G. Yuval for many discussions and P.G. de Gennes for introducing him to many of the problems involved. —
REFERENCES 1. (a)LANDAU L.D., Z. Eksp. i Teor. Fiz. 7 627 (1937); translation in Collected Papers of LANDAU L.D., p. 209, Pergamon (1965). (b)See also LANDAU L.D. and LIFSHITZ E.M.,StarisricalPhysics second edition 137—139, pp. 424—445, Pergamon (1968).
2. 3. 4. 5.
6.
DE GENNES P.G.,Physics Lett. A30 454 (1969); Mo!. Cryst. 12 193 (1971). STRALEY JP. and FISHER M.E.,J. Phys. C (in press). ALEXANDER S. and YUVAL G., J. Phys. C (to be published) (1973). ALEXANDER S., unpublished (1973). e.g. p. 438 of reference 1(b).
On démontre que des transitions de phase, qui devraient etre du premier ordre selon les arguments de symétrie de Landau, peuvent etre continues pour un domaine fini des parametres physiques. Pour des transitions avec un parametre d’ordre tensoriel de rang 2, on montre que les fluctuations dans une phase associée avec l’anisotropie du tenseur a pour consequence une renormalisation du terme cubique’ Ccci conduit a une transition continue quand le coefficient du terme cubique (isolé) est petit devant le coefficient du terme quartique. On montre que cette discussion peut etre etendue a des cas plus généraux ou Ic parametre d’ordre appartient a une representation multidimensionnelle du groupe de symétrie bnsée.
Vol. 14, pp. 1069—1071, 1974.
Pergamon Press.
Printed in Great Britain
CONTINUOUS PHASE TRANSITIONS WHtCH SHOULD BE FIRST ORDER S. Alexander Israel Defense Forces and The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel (Received 21 September 1973; in revised form 5 November 1973 by P.G. de Gennes)
It is demonstrated that phase transitions predicted to be first order by the Landau symmetry arguments can be continuous for a fmite range of the physical parameters. For transitions with a rank-two tensor order parameter it is shown that fluctuations in a phase associated with the anisotropy of the tensor result in a renormalization of the cubic term. This leads to continuous transitions when the coefficient of the bare cubic term is small compared to the coefficient of the quartic term. It is shown that the argument can be extended to more general cases when the order parameter belongs to a multidimensional representation of the broken symmetry group.
LANDAU’ type symmetry arguments are usually considered reliable in predicting the order of phase transitions with a symmetry change. When cubic terms in the order parameter are allowed in the Landau expansion of the free energy the transition is expected to be first order, except for isolated points where the cubic term vanishes accidentally. There is reason to doubt the general validity of these arguments. It has been known for a long time2 that the isotropic to nematic transition of liquid crystals shows very large, almost critical, fluctuations in the disordered phase. Straley and Fisher3 have recently shown that the transition of the 3 state Potts model on the square lattice is probably continuous. Both transitions have rank-2 tensor order parameters4 and are predicted to be first order.4 From a theoretical point of view it also seems surprising that the predictions of the Landau theory do not distinguish transitions with a symmetry change from transitions without such a change when cubic terms are allowed. *
The suggestion of Straley and Fisher that their model has some ‘hidden’ symmetries is in fact erroneous. The symmetries in their problem are exactly of the type envisaged by Landau. It is also fairly easy to construct a molecular field theory retaining the full symmetry. This predicts a i st order transition.5
-
Our purpose here is to demonstrate that these predictions are in fact erroneous. We will show that for transitions with a symmetry change there are important renormalization effects on the cubic term which were neglected in previous discussions. As a result there is a range of values of the parameters for which the transitions should be continuous and show critical behaviour. For clarity we restrict our discussion to situations where the order parameter is a rank two tensor, and the broken symmetry is spherical (as for nematics) or cubic4 (as for the 3 state Potts model). The generalization is, in principle, fairly straightforward, if one uses the Landau Lifshitz technique of associating the order parameter with the irreducible representations of the (broken) high symmetry group.1 We start by constructing the expansion for the free energy. The five components of the order parameter (Q,~) always define a principal axis system (x, y, z). The energy is obviously independent of the orientation of this system and the angular fluctuations have been considered in detail by de Gennes.2 Since the energy is obviously invariant under permutations .
.
.
of the three axis (i.e. has cubic symmetry in this frame) we can always write for the ‘bare’ free energy density (in the Wilson sense)
1069
1070
CONTINUOUS PHASE TRANSITIONS F
=
~ [Aq~,—Bq~+ Cq~+ K Vq~121
[1(a)]
with the constraint = 0 where the ~ (-~(3cos2O~ I)) are the components of Q2 along the three axis, and we have used the constraint [equation 1(b)] to eliminate all mixed terms in F. Equation (1) is thus quite general to this order. It is convenient to eliminate the constraint and rewrite Fin the form: —
F
Ap2 —Bp3cos3O + Cp4 + K( lVpI2 + P2 1V012) (2) where p and 0 are defined by: pe’0 = q~+ e2fl~/3q~ + e2~”3 =
Thus p is the magnitude of Q and the phase 0 measures the anisotropy of Q. This expression for F can also be derived directly from the group theoretical arguments of the Landau Lifthitz. Their procedure6 is now to start by minimizing F with respect to 0 at constant p. The rest of their well known argument then follows. l’his procedure amounts to neglecting fluctuations in 0. Ii is however clear from equation 2 that such fluctuations will be large for small p and this will tend to decrease the effective assymmetry. We will use the most straightforward procedure to estiniate the effect of this renormalization of the cubic term. We first calculate the average (cos3O> at
Vol. 14, No. 11
(cos3O)
exp —(R ~p)/pä)
=
where the function R (p) and the leading exponent (in l/p) ô both depend on the constants B, K and T but are independent of C(and A). This should be substituted for cos3O in equation 2 giving the effective expression F = Ap2 —Bp~exp—(R(p)/p6)+ Cp4 + K~Vpi2 (7) The symmetry change occurs as a first order transition if and only if F becomes negative at some finite p f~rpositive A. This requires a range of p for which: —Bexp—-(R(p)/p6) + Cp <0 and is thus possible only when: C/B < ~(R, ô) = i~(B,K, 7) (8) When C/B is larger than this maximum value the transition is necessarily continuous with the usual type of critical behaviour. The evaluation of the integrals in equation (5) is straightforward. It is however important to remember that the cutoff dependent terms which are usually neglected, are divergent and therefore important in this case (because of the factor p2 in front). The most reasonable choice seems to be a quantum mech~cal cutoff Kp2k~ 0= kB T.
constant p and then use the result to eliminate 0 from equation (2).
(cos30)~,r= exp—(~(O2>~,r) where (02)
=
(4)
kc.o ~~0k0-k)
-c
=
p2
I
(9)
Defining 9B ci
Assuming small fluctuations, we can expand cos 30 to lowest order in 0. This gives from the relevant (0 dependent) part ofF 2~[(9B/2)-p+Kk2]0k0_k (3) Fo~p where the 0k are the fourier transforms of 0(r). Treating the 0~as statistically independent we then have:
(6)
=
=
2kB T
(J~P2
one finds for one dimension (d = 1) 2tan~(ap3~”2~ R1(p) = (2~0)’a” =15/2 [10(a)] for d = 2 R 2 (p)
2in I + (ap3)’ ; &~= 2 =
(2ir~o)
[10(b)] for d = 3 R 3(l (ap3)”2 tan~(cip3)~2; 3 3(p) = 3(2ir~o~ [10(c)] —
dk1~P+ Kk2J
-
(5) where T is the absolute temperature, c a normalization constant, and k~ 0a suitably chosen cutoff. Since the integral equation possibly 2 oneinalways gets5 a cannot result of the form vanish as fast as p
and ford=4) R 3lnIi + (cip3)~I); ~ 4(p) = 2(2ir~~)~(1c~p
=
4
—
[10(d)] where we have retained the full expressions. Since the form ofR and therefore also ‘q, depends on three
Vol. 14, No. 11
CONTINUOUS PHASE TRANSITiONS
independent physical parameters a more detailed discussion does not seem useful. It is obvious that R increases as K and B decrease. To conclude this discussion we would like to point out that our expression for (02) [equation (5)J certainlyunderestimates the true value. Even the bare potential (cos30) is always weaker than the value we have retained (~ 82) and proper renormalization would make it even weaker. We believe we have shown that transitions of the -
.
.
type considered can indeed by continuous for a fimte range of the parameters contrary to the predictions of the Landau theory. A properly renormalized evalu-
1071
ation is of course desirable but should not change the essential features of our results. The present results can be generalized in a straightforward way to the general case of phase transitions with a symmetry change when the order parameter belongs to a multidimensional irreducible representation of the broken symmetry group.’ In all such cases the free energy expansion involves the magnitude of the order parameter and phase angles analogous to 0. The results should therefore be similar. Acknowledgements The author would like to thank D. Bergman for a very helpful discussion in the early stages of this work, H. Gutfreund, G. Horwitz, I. Imry and G. Yuval for many discussions and P.G. de Gennes for introducing him to many of the problems involved. —
REFERENCES 1. (a)LANDAU L.D., Z. Eksp. i Teor. Fiz. 7 627 (1937); translation in Collected Papers of LANDAU L.D., p. 209, Pergamon (1965). (b)See also LANDAU L.D. and LIFSHITZ E.M.,StarisricalPhysics second edition 137—139, pp. 424—445, Pergamon (1968).
2. 3. 4. 5.
6.
DE GENNES P.G.,Physics Lett. A30 454 (1969); Mo!. Cryst. 12 193 (1971). STRALEY JP. and FISHER M.E.,J. Phys. C (in press). ALEXANDER S. and YUVAL G., J. Phys. C (to be published) (1973). ALEXANDER S., unpublished (1973). e.g. p. 438 of reference 1(b).
On démontre que des transitions de phase, qui devraient etre du premier ordre selon les arguments de symétrie de Landau, peuvent etre continues pour un domaine fini des parametres physiques. Pour des transitions avec un parametre d’ordre tensoriel de rang 2, on montre que les fluctuations dans une phase associée avec l’anisotropie du tenseur a pour consequence une renormalisation du terme cubique’ Ccci conduit a une transition continue quand le coefficient du terme cubique (isolé) est petit devant le coefficient du terme quartique. On montre que cette discussion peut etre etendue a des cas plus généraux ou Ic parametre d’ordre appartient a une representation multidimensionnelle du groupe de symétrie bnsée.