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Critical amplitudes and “cosine of the angle” between order parameters
Volume 83A, number 7
PHYSICS LETTERS
15 June 1981
CRITICAL AMPLITUDES AND “COSINE OF THE ANGLE” BETWEEN ORDER PARAMETERS V.N. PLECHKO Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received 22 January 1981
For any two order parameters A,B we introduce a function that can be called “cosine of the angle between A and B” and discuss its utility for critical-point studies, in particular for the interpretation of experimental data.
When studying critical phenomena one usually deals with the simultaneous behaviour of several order parameters and the corresponding susceptibilities (see, e.g., ref. [11). Here we introduce the “cosine of the angle” between any two order parameters and briefly discuss its utility for critical-point analyses. Suppose an equilibrium system is given with hamiltonian F, temperature 0 = kT and number of partides N, which we denote by rio; and also operator order parameters A and B which we suppose to be hermitean, A = A+ ,B = B+. Introduce the generalized susceptibility XA B(FiO) [(a/~y) (A )~—yNB/O ] y =0 where (A means the equilibrium Gibbs average. Let In) be the quantum-mechanical representation in which F is diagonal, F~the eigenvalues of F, and Amn, ~nm the matrix elements of the operators A = A (A ~ B = B
‘
—
N
XAB(F/O)= Q
1
~
e_I’flIO
—
—
mn AmnBnm
e_I’nIO — e_FmtO Fm — ~
,(2)
.
(
0 for A
xAB(F1O) = XBA(FiO)
= 0),
(3a) (3b)
,
IxAB(FiO)12 ~xAA(F/O)xBB(F/O)
.
(3c)
Here (3c) is an inequality of the Cauchy—Schwarz type~ (strictly speaking it follows from other properties). Inequalities of this type have first been introduced in statistical mechanics by Bogolubov in 1961 [4] ; one can consider (3) to be a particular case of some general relations discussed in ref. [4] The relations (3) make it possible to introduce “the squared cosine of the angie between the order parameters A and B with respect to the system rio”: 2 2 cosAB(FiO) = IxAB(riO)I 1XAA (~i°)xBB(~I°); (4) independently of the system F/U 2 O~cosAB(FIO,~l
~4a
.
Remark 1. In a special case of the critical system F0/00 (which we are just interested in) some of the susceptibilities may appear to be infinite (if, as usual, N= oo). In that case one should first take some noncritical system rio in the neighbourhood of F0/00, construct cosAB(I’/O) and define cosAB(Fo/O0) by the limit procedure F/U r’01o0. We must stress that the result in general will depend upon the “trajectory” of this limit F/U [‘o/Uo[that is why the alternative definition: “calculate’ cosAB(r’O/UO)for N ~ 00 and then put N co” seems not to be so acceptable]. -÷
It is easy to see now that, independently of rio, x.4B(F/U) is a bilinear form in the operators A, B satisfying all the
XAA (F/U) ~a ~
axioms of
the scalar products
-+
-‘~
0 031—9163/81/0000--0000/$ 02.50 © North-Holland Publishing Company
357
Volume 83A, number 7
PHYSICS LETTERS
In order to illustrate the utility of dO5~B for critical behaviour studies consider some simple ferromagnetic system with a fixed hamiltonian H, critical tempera-
ci + 2~+ y = 2 P0
+~
ture O~(HIOc is the critical system) and a one-compo-
nent order parameter keep in mind, e.g., the Ising(magnetization)S; model. Define alsoone the may “temperature order parameter” L = —H/N. Consider a system near criticality F/O H hNS/O~(1 s), h >0. E >0, h 0, c -~-0, where h >0 is the magnetic field, and 0 = °~(‘ s) is the temperature (below °~). For this system one can represent the specific heat Ch (for fixed field), with the error vanishing when h 0, L 0, as —
15 June 1981
=
(6a)
,
2P~,
2B2/A ~F_
0 <0 and if cosSL(L
=
=
cosSL(E
+0)
=
=
+0) ~ 1
,
(6b) (6c)
0 then
—
(7)
-~
—
in particular, if ci + 20 + y = 2 then P 0
c17(F/O)
O’XLL(F/O), where XLL is the standard form of the susceptibility, see (1). Also note that a(S)~16iao= 0—lxsj (F/U). So, instead ofGh and we shall consider below XLL and XLS• Consider the typical situation of power law asymp=
totics with possible logarithmic corrections for order
parameters and susceptibilities. the short-hand notation F(c) F[H/0~(l — E)], Using F(h) ~EF[H—hNS/O~], F = S, L, XSL,Ctc., and keeping only the main terms in the asymptotic forms, we write: S(E)=BtlncV’~,
(Sa)
XSS(L)= FE~IlflEI~, _0I In EI’°°
XLL (~) = A
S(h) = (h/D)~ Iln h P5 L(h)
=
(Sb)
,
,
ZhtIln h I~t+ const.,
XLLQ1) = Eh~I1nh ~
(9a)
,
P5 + P~ = 2Pt ~ IE = cossL(h = 0) ~ 1 2z2D1 0
—
1/5 >1;
in particular, if 2~+ p 5 +P>2P~.
.
(9b) (9c)
(10) —
1/5
=
1 then (11)
(Sd)
The well-known equality (6a) has been first proposed by Essan1 and Fisher [5] . The general inequalities
(Se)
(7) and (10) have been derived by Rushbrooke [6] and Coopersmith [7] on the basis of the well-known ther-
(Sf)
modynamic relation between the specific heats Ch and C~(see also the discussion of (6a) in ref. [8]); in our notation Cm(F/O) = Ch(F/O) [1 cos~LF(F/0)],L~ —
(Sg)
Xs1~(L)= 0Bs~’Iln 6I~P
(Sh)
1 in h
(5i)
XSL(h) = ~Zh~
where a, ~, are critical indices, P p~ p logarithmic indices,A , B F critical amplitudes, Constructing by the asymptotic expressions (5) dO5SL (E) dO5SL [H/Uc(l s)] , passing to the limit +0 and taking into account (4a), we get restrictions on the critical parameters. If co5~L(c= +0)> 0 then ...,
—
358
2~+e—l/5l
The possibilities considered cover all the relations between critical parameters allowed by (4a).
~
necessarily
2P~. (8) Quite analogously, considering cosSL(h) ash ~ 0, we conclude that if cos~L(h= 0)> 0, then
(Sc)
and hence
_
+ P~>
—
caF/N
So, we here have a new proof of the inequalities due Rushbrooke and Coopersmith and amplitudes, also some new to relations for logarithmic indices and rectly based on the i.e. onbeing the propwhich does not deal “first with principles”, thermodynamics, dierties of the Gibbs canonical distribution contained in the inequality (4a). This new proof is sufficiently more general than previous ones. The relations (6)—(8) remain valid if instead of the temperature variation one considers any arbitrary variation of the hamiltonian H -+ H + V(c), V(c) -÷0 as c -÷+0, and if one assumes for Sand L two
Volume 83A, number 7
PHYSICS LETTERS
arbitrary order parameters [with asymptotic expressions (5) for xss(c), XLL(), XSL(E)] for which there is no relation like for Ch and Cm.
Let c and h again be temperature and field parameters. Note that for the interpretation of experimental data it has been proposed to use the dimensionless am-
2 plitude ratiosA÷/A,F,~/F andR =A÷F~/B(here A~and F~are analogs ofA and F for 0 > 0c) [9, 10] The proposition has been put forward, in particular, that these amplitude ratios may demonstrate universality (like critical indices), i.e., depend only upon the dimensionality and symmetry of the system [9,10] (for detailed discussion see ref. [10]). The relations (6)_(8) show that in the case of onecomponent order parameters, instead of R a more nat—
.
ural parameter is the ratio
2B2 R*
=
02
(12)
.
A.F, If (6a) and (6b) hold, then necessarily 0
June
1981
table for R * * and compare the values of cos~L(h
= 0) for different systems and with dO5~.L(L= +0) for the same systems. However, due to lack of necessary data we are unable to do this. As one can see from table 1, do5~L(c + 0) monotonically decreases with decreasing dirnensionality of the system, i.e., the “angle” between the order param-
eter and the hamiltonian increases. This permits a physi-
cal interpretation that will be discussed in more detail elsewhere. Let us now formulate some concluding remarks. A. The natural dimensionless amplitude ratios for one-component order parameters are R* and R** appearing in (6c) and (9c). Our suggestion is to include
R* ,R** and/or cO5SL(E
=
+0), cossL(h = 0) in the
tables of critical amplitudes.
A~ F÷ =--
15
—
~.
Remark 2. For isotropic ferromagnets (n ~ 2) F and formally R* = 0 (if A 0).
= +00
Values of R* and CosSL( = + 0) for Ising lattices of different dimensionality d are presented in table I (for all the necessary data and references to the original papers see e.g. ref. [10] ; for the d = 4 Ising model see also ref. [11]). Here “yes” and “no” indicate whether (6a) and (6b) are valid or not, X means absence of logarithmic corrections, P 0 = P~= P7 = 0. Note that for the d = 4 Ising model P0 = P~= F,.1, = 1/3 and (6b) holds. It seems to be highly desirable to form an analogous
B. If there are logarithmic corrections one should test the relations (6b), (8), (9b) and (11). C. It seems highly desirable to have a complete set
of data for the “field” critical parameters (including that for L(h) [or xSL(h)] and C(h)= 0 ‘XLL(h), see (5)), at least for the simplest models. D. In general, we should like to note, that the critical amplitudes and logarithmic indices are of considerable interest, perhaps not less, if not more, than the critical indices themselves since they are more “delicate” characteristics than critical indices. The author would like to thank Professor N.N. Bogolubov Jr. for valuable remarks and support,
and also the members of the Professor V.K. Fedyanin seminar for discussions. References [1] L.P. Kadanoff, in: Phase transitions and critical phenomena,eds. C. Domb and M.S. Green, Vol. 5A (AlP, 1976) [2] R.M. Wilcox,
Ta le 1 * 2 . The values of R and coseL(s = +0) for the Ising model for ‘ different lattice dimensionahties. 2 cosSL(E = +0)
Model
(6a)
(6b)
R. *
molecular field (d =°°) Ising d = 4 Isingd = 3 Ising d = 2
yes yes yes yes
X
1
yes
0.75
0.75
X no
0.53
0.53 0
1
1.85
J. Math. Phys. 8 (1966) 962. [3] Physica 32 D-781 (1966) (Dubna, 933. [4] N.N. N.N. Bogolubov Bogolubov,Jr., preprint JINR 1969); see also: (Gordon Lectures and on quantum statistics, 2, Quasiaverages Breach, New York,Vol. 1971). [5] J.W. 802. Essam and M.E. Fisher J. Chem. Phys. 38 (1963) [6] G.S. Rushbrooke, J. Chem. Phys. 39 (1963) 842. [7] M.S. Coopersmith, Phys. Rev. 167 (1968) 478. [8] G.S. Rushbrooke, J. Chem. Phys. 43 (1965) 3439. [9] H.C. Bauer and G.R. Brown, Phys. Lett. 51A (1975) 68. [10] A. Aharony and P.C. Hohenberg, Phys. Rev. 13B (1976) 3081. [11] E.Brézin,J.dePhys.Lett.36(1975)L51.
359
PHYSICS LETTERS
15 June 1981
CRITICAL AMPLITUDES AND “COSINE OF THE ANGLE” BETWEEN ORDER PARAMETERS V.N. PLECHKO Joint Institute for Nuclear Research, Laboratory of Theoretical Physics, Dubna, USSR Received 22 January 1981
For any two order parameters A,B we introduce a function that can be called “cosine of the angle between A and B” and discuss its utility for critical-point studies, in particular for the interpretation of experimental data.
When studying critical phenomena one usually deals with the simultaneous behaviour of several order parameters and the corresponding susceptibilities (see, e.g., ref. [11). Here we introduce the “cosine of the angle” between any two order parameters and briefly discuss its utility for critical-point analyses. Suppose an equilibrium system is given with hamiltonian F, temperature 0 = kT and number of partides N, which we denote by rio; and also operator order parameters A and B which we suppose to be hermitean, A = A+ ,B = B+. Introduce the generalized susceptibility XA B(FiO) [(a/~y) (A )~—yNB/O ] y =0 where (A means the equilibrium Gibbs average. Let In) be the quantum-mechanical representation in which F is diagonal, F~the eigenvalues of F, and Amn, ~nm the matrix elements of the operators A = A (A ~ B = B
‘
—
N
XAB(F/O)= Q
1
~
e_I’flIO
—
—
mn AmnBnm
e_I’nIO — e_FmtO Fm — ~
,(2)
.
(
0 for A
xAB(F1O) = XBA(FiO)
= 0),
(3a) (3b)
,
IxAB(FiO)12 ~xAA(F/O)xBB(F/O)
.
(3c)
Here (3c) is an inequality of the Cauchy—Schwarz type~ (strictly speaking it follows from other properties). Inequalities of this type have first been introduced in statistical mechanics by Bogolubov in 1961 [4] ; one can consider (3) to be a particular case of some general relations discussed in ref. [4] The relations (3) make it possible to introduce “the squared cosine of the angie between the order parameters A and B with respect to the system rio”: 2 2 cosAB(FiO) = IxAB(riO)I 1XAA (~i°)xBB(~I°); (4) independently of the system F/U 2 O~cosAB(FIO,~l
~4a
.
Remark 1. In a special case of the critical system F0/00 (which we are just interested in) some of the susceptibilities may appear to be infinite (if, as usual, N= oo). In that case one should first take some noncritical system rio in the neighbourhood of F0/00, construct cosAB(I’/O) and define cosAB(Fo/O0) by the limit procedure F/U r’01o0. We must stress that the result in general will depend upon the “trajectory” of this limit F/U [‘o/Uo[that is why the alternative definition: “calculate’ cosAB(r’O/UO)for N ~ 00 and then put N co” seems not to be so acceptable]. -÷
It is easy to see now that, independently of rio, x.4B(F/U) is a bilinear form in the operators A, B satisfying all the
XAA (F/U) ~a ~
axioms of
the scalar products
-+
-‘~
0 031—9163/81/0000--0000/$ 02.50 © North-Holland Publishing Company
357
Volume 83A, number 7
PHYSICS LETTERS
In order to illustrate the utility of dO5~B for critical behaviour studies consider some simple ferromagnetic system with a fixed hamiltonian H, critical tempera-
ci + 2~+ y = 2 P0
+~
ture O~(HIOc is the critical system) and a one-compo-
nent order parameter keep in mind, e.g., the Ising(magnetization)S; model. Define alsoone the may “temperature order parameter” L = —H/N. Consider a system near criticality F/O H hNS/O~(1 s), h >0. E >0, h 0, c -~-0, where h >0 is the magnetic field, and 0 = °~(‘ s) is the temperature (below °~). For this system one can represent the specific heat Ch (for fixed field), with the error vanishing when h 0, L 0, as —
15 June 1981
=
(6a)
,
2P~,
2B2/A ~F_
0 <0 and if cosSL(L
=
=
cosSL(E
+0)
=
=
+0) ~ 1
,
(6b) (6c)
0 then
—
(7)
-~
—
in particular, if ci + 20 + y = 2 then P 0
c17(F/O)
O’XLL(F/O), where XLL is the standard form of the susceptibility, see (1). Also note that a(S)~16iao= 0—lxsj (F/U). So, instead ofGh and we shall consider below XLL and XLS• Consider the typical situation of power law asymp=
totics with possible logarithmic corrections for order
parameters and susceptibilities. the short-hand notation F(c) F[H/0~(l — E)], Using F(h) ~EF[H—hNS/O~], F = S, L, XSL,Ctc., and keeping only the main terms in the asymptotic forms, we write: S(E)=BtlncV’~,
(Sa)
XSS(L)= FE~IlflEI~, _0I In EI’°°
XLL (~) = A
S(h) = (h/D)~ Iln h P5 L(h)
=
(Sb)
,
,
ZhtIln h I~t+ const.,
XLLQ1) = Eh~I1nh ~
(9a)
,
P5 + P~ = 2Pt ~ IE = cossL(h = 0) ~ 1 2z2D1 0
—
1/5 >1;
in particular, if 2~+ p 5 +P>2P~.
.
(9b) (9c)
(10) —
1/5
=
1 then (11)
(Sd)
The well-known equality (6a) has been first proposed by Essan1 and Fisher [5] . The general inequalities
(Se)
(7) and (10) have been derived by Rushbrooke [6] and Coopersmith [7] on the basis of the well-known ther-
(Sf)
modynamic relation between the specific heats Ch and C~(see also the discussion of (6a) in ref. [8]); in our notation Cm(F/O) = Ch(F/O) [1 cos~LF(F/0)],L~ —
(Sg)
Xs1~(L)= 0Bs~’Iln 6I~P
(Sh)
1 in h
(5i)
XSL(h) = ~Zh~
where a, ~, are critical indices, P p~ p logarithmic indices,A , B F critical amplitudes, Constructing by the asymptotic expressions (5) dO5SL (E) dO5SL [H/Uc(l s)] , passing to the limit +0 and taking into account (4a), we get restrictions on the critical parameters. If co5~L(c= +0)> 0 then ...,
—
358
2~+e—l/5l
The possibilities considered cover all the relations between critical parameters allowed by (4a).
~
necessarily
2P~. (8) Quite analogously, considering cosSL(h) ash ~ 0, we conclude that if cos~L(h= 0)> 0, then
(Sc)
and hence
_
+ P~>
—
caF/N
So, we here have a new proof of the inequalities due Rushbrooke and Coopersmith and amplitudes, also some new to relations for logarithmic indices and rectly based on the i.e. onbeing the propwhich does not deal “first with principles”, thermodynamics, dierties of the Gibbs canonical distribution contained in the inequality (4a). This new proof is sufficiently more general than previous ones. The relations (6)—(8) remain valid if instead of the temperature variation one considers any arbitrary variation of the hamiltonian H -+ H + V(c), V(c) -÷0 as c -÷+0, and if one assumes for Sand L two
Volume 83A, number 7
PHYSICS LETTERS
arbitrary order parameters [with asymptotic expressions (5) for xss(c), XLL(), XSL(E)] for which there is no relation like for Ch and Cm.
Let c and h again be temperature and field parameters. Note that for the interpretation of experimental data it has been proposed to use the dimensionless am-
2 plitude ratiosA÷/A,F,~/F andR =A÷F~/B(here A~and F~are analogs ofA and F for 0 > 0c) [9, 10] The proposition has been put forward, in particular, that these amplitude ratios may demonstrate universality (like critical indices), i.e., depend only upon the dimensionality and symmetry of the system [9,10] (for detailed discussion see ref. [10]). The relations (6)_(8) show that in the case of onecomponent order parameters, instead of R a more nat—
.
ural parameter is the ratio
2B2 R*
=
02
(12)
.
A.F, If (6a) and (6b) hold, then necessarily 0
June
1981
table for R * * and compare the values of cos~L(h
= 0) for different systems and with dO5~.L(L= +0) for the same systems. However, due to lack of necessary data we are unable to do this. As one can see from table 1, do5~L(c + 0) monotonically decreases with decreasing dirnensionality of the system, i.e., the “angle” between the order param-
eter and the hamiltonian increases. This permits a physi-
cal interpretation that will be discussed in more detail elsewhere. Let us now formulate some concluding remarks. A. The natural dimensionless amplitude ratios for one-component order parameters are R* and R** appearing in (6c) and (9c). Our suggestion is to include
R* ,R** and/or cO5SL(E
=
+0), cossL(h = 0) in the
tables of critical amplitudes.
A~ F÷ =--
15
—
~.
Remark 2. For isotropic ferromagnets (n ~ 2) F and formally R* = 0 (if A 0).
= +00
Values of R* and CosSL( = + 0) for Ising lattices of different dimensionality d are presented in table I (for all the necessary data and references to the original papers see e.g. ref. [10] ; for the d = 4 Ising model see also ref. [11]). Here “yes” and “no” indicate whether (6a) and (6b) are valid or not, X means absence of logarithmic corrections, P 0 = P~= P7 = 0. Note that for the d = 4 Ising model P0 = P~= F,.1, = 1/3 and (6b) holds. It seems to be highly desirable to form an analogous
B. If there are logarithmic corrections one should test the relations (6b), (8), (9b) and (11). C. It seems highly desirable to have a complete set
of data for the “field” critical parameters (including that for L(h) [or xSL(h)] and C(h)= 0 ‘XLL(h), see (5)), at least for the simplest models. D. In general, we should like to note, that the critical amplitudes and logarithmic indices are of considerable interest, perhaps not less, if not more, than the critical indices themselves since they are more “delicate” characteristics than critical indices. The author would like to thank Professor N.N. Bogolubov Jr. for valuable remarks and support,
and also the members of the Professor V.K. Fedyanin seminar for discussions. References [1] L.P. Kadanoff, in: Phase transitions and critical phenomena,eds. C. Domb and M.S. Green, Vol. 5A (AlP, 1976) [2] R.M. Wilcox,
Ta le 1 * 2 . The values of R and coseL(s = +0) for the Ising model for ‘ different lattice dimensionahties. 2 cosSL(E = +0)
Model
(6a)
(6b)
R. *
molecular field (d =°°) Ising d = 4 Isingd = 3 Ising d = 2
yes yes yes yes
X
1
yes
0.75
0.75
X no
0.53
0.53 0
1
1.85
J. Math. Phys. 8 (1966) 962. [3] Physica 32 D-781 (1966) (Dubna, 933. [4] N.N. N.N. Bogolubov Bogolubov,Jr., preprint JINR 1969); see also: (Gordon Lectures and on quantum statistics, 2, Quasiaverages Breach, New York,Vol. 1971). [5] J.W. 802. Essam and M.E. Fisher J. Chem. Phys. 38 (1963) [6] G.S. Rushbrooke, J. Chem. Phys. 39 (1963) 842. [7] M.S. Coopersmith, Phys. Rev. 167 (1968) 478. [8] G.S. Rushbrooke, J. Chem. Phys. 43 (1965) 3439. [9] H.C. Bauer and G.R. Brown, Phys. Lett. 51A (1975) 68. [10] A. Aharony and P.C. Hohenberg, Phys. Rev. 13B (1976) 3081. [11] E.Brézin,J.dePhys.Lett.36(1975)L51.
359