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Free energies of the spherical model of a spin glass
0038-1098/84 $3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 52, No. 12, pp. 1003-1006, 1984. Printed in Great Britain.
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS K. Nemoto and H. Takayama Department of Physics, Hokkaido University, Sapporo 060, Japan
(Received 26 July 1984 by J. Kanamori) Free energies g(m, ms) and f(m, q) of the spherical spin glass (SG) model due to Kosterlitz et aL are calculated explicitly as functions of the uniform magnetization m, and SG order parameter m s and the EdwardsAnderson order parameter q. It is shown that g(0, m,) and f ( 0 , q) below the transition temperature Tit are constant in the ranges 0 g m s < and 0 __gq ~_ qo respectively, where qo = (1 -- T/T#) = m~. The_proper m,o equilibrium values of ms(= mso) and q(= qo) are then f'lxeg from the inspection of their behaviors under infinitesimal uniform field proproportional to N- a(a 2_ 0). THE MEAN FIELD THEORY of spin glasses has been considerably developed by making use of the Sherrington -Kirkpatrick model of the infinite-ranged Ising spin glass (SG) system [1 ]. However an intuitive interpretation of the theory is still not transparent due to various reasons. One of them is the lack of an explicit free energy expression below the transition temperature Tg as a function of certain order parameters in their whole range. Particularly that of the paramagnetic state has not yet been obtained. Another reason may lie on the process to take the thermodynamic limit (N ~ o~). According to the recent interpretations [2-5 ] on Parisi's replica-symmetry breaking solution [6], it contains properly informations on the overlap between many states with free-energy local minima which are separated by energy barriers of the order ofNa(a <~1/4). However it is by no means easy to understand how such a subtle feature in the large N limit can be incorporated in Parisi's solution, since the latter is a solution of a saddle-point equation which itself is derived in the thermodynamic limit. Keeping these difficulties in mind we have reexamined the spherical version of the SK model [7]. In this communication we will report the following now results, which, we believe, offer certain guides to our original problems in the Ising SK model. (i) The explicit free energy forms g(m, ms) and f(m, q) are derived in the entire region of the uniform
= 1 -- T/Tg = qo.
The spherical SG model in the limit of the infiniteranged interactions Jo with a Gaussian probability distribution was def'med and solved by Kosteflitz et al. [7], whose notations we follow unless specified explicitly. In the present work we restrict ourselves to a system with Jo = 0. Then in the limit N ~ 00 the eigenvalue density ofJ 0 obeys the familiar semi-circular law with the largest eigenvalue J,x = 2 ] (hereafter the index A denotes this largest eigenvalue mode). The order parameters m, m a and q are defined as m = N Z~m~, rn, =N-I/2mA and q = N -1 ~im~ = N- *~x m~, respectively, where m~mx) is the thermal average
(). Let us first investigate g(m, ms) which is derived from the following microcanonical partition function
E(m, ms)= ~D[s] 6(N-- ~s2)5 (Nm-- ~si) 6 (Nm s -- N 1/2SA) e- I3H
(2)
where ID [s] = fIIids i = fIIxds x. By making use of the standard manipulation of the spherical model, we obtain g(m, ms) as T
g(m, ms) = -- lim - - [ l n . ( m , ms)]a
magnetization rn, the SG order parameter (or the staggered magnetization)ms, and the EdwardsAnderson order parameter q. Particularly we find that g(0, ms) and f(0, q) become constant below 7"# in the ranges ms ~- mso and q g_qo respectively, where ms2o
proper equilibrium values of m, and q in the thermodynamic limit are fixed, which are given by equation (1l
Nooo N
= [~' (m, mg;z)]a
= --z+ T[~ y~X~(Z) 1
+ -~ X-l(z)m 2 +
(1)
(ii) From the inspection of ms and q under infinitesimal uniform field h proportional to N - a (a ~ 0), the 1003
1 x-,' (z)ml
(3
1004
Vol. 52, No. 12
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS
g,
lim
f
(8a)
lim ms = mso
hs~O
h~O
lim
(8b)
lim ma = 0.
h ~ O hs-~O
j~
--
./
' ms
mso
~q
0 //,
.,., , / "
q0
a
b
Fig. 1. Free energies g(m, ms) vs ms(a),f(m, q) vs q (b). The dashed line represents the unphysical extrapolation of the expressions which are only valid in the region m ~_ mso or q g qo-
Before going to discussions on equation (8), let us examine the free energy f(m, q), which is evaluated from the spherical version of the Thouless-AndersonPalmer (TAP) free energy f~sd, {mx} [9]. For this purpose we briefly describe the TAP theory of the spherical SG. The Helmholtz free energyfn{hx}, h x = Ei(Xl i)hi and h i being the external field at the ith site, is calculated as
fH = -- lim
J
[sl
where [ . . . ] j indicates the random average over JU, and 1[~ x(z) =
1
(IIA) 2 ] z-S ,12
=
y2)1/2 }
Xs(z) -
1
In
{
z
l_3`2X~(Z)
Z k
rr
/
1 v
h~.
(9)
4 N ~ z -- Jx]2'
where z is a solution of the following SP equation 1 = (6)
It is well-established that in case (6) has no solution, the saddle point sticks at z =3` [8] andg(m, ms) is simply given by [ ~e(m, ms; z = 3`)] a. The expression g(m, m,) in a whole range ofm and m s is thus derived. In the present problem the SP sticking occurs below Tg(= 3`) ifm = 0 and 1
T 1 1 hi 2N ~ -z - --J x / 2 + ~-~ ~ X ( z - J x / 2 ) 2"
(1 -- /33`)2msZ + ¼(1 --/323`2)m~ + ~m, I > /3J(1--ms2)
(7a)
1 < /3.](1 -- m~).
(7b)
The free energy g(m, ms) below Tg is shown in Fig. l(a). The dashed line in Fig. l(a) represents the simple extrapolation of equation (7a) to its invalid region, and has no physical meaning. An equilibrium value of m, can not be fixed since any m, in the range 0 ~ ms g m,o yields the same minimum g(0, ms). One way to get rid of this uncertainty may be to examine m s in the limit h, hs ~ 0, where h = ag/Om and hs = Og/Oms. However, reflecting the fact that the SP sticking condition depends crucially on whether m = 0 or m 4: 0, we obtain [see also Fig. 1(a)]
(10)
The TAP free energy is given by the Legendre transformation f~AP = fH + Zxhxmx with mx = -- NbfH/ ah x. Taking the random average of the terms which depend on m x only through q, we obtain
f~aI" --
1 2N ~ Jxm~ --z(1 --q) k
+-2
/3g(0, ms) --/3g(0, 0)
=
/3(z-A/2)
(5)
In equation (4) above (1 IA) = 1IN in denotes the uniform mode and z in equation (3) is the solution of the saddle point (SP) equation
3`X(z) = /33` l - - m ,
= - z+~
(4)
1 - 2(z--Y)"
2 Z--JA/2
-- /3 H - - Z hx sx h
a 1
exp
{z-(z
2
In
(11) '
with the SP condition/3(1 -- q) = X(z), where X(z) is given by equation (4). Again f ~ u , below Tg is divided into two branches depending on whether this SP equation has a solution or not:
8 _ fTAP
1
2N
• Jxm~ + y (q), x
(12)
where
f(q)
/3y2 -
T (1 _ q ) 2 _ 5 [1 + In 27r(1 - - q ) ] , (13a)
for 1 > / 3 ] ( 1 -- q) [with ] X = / 3 ] ( 1 -- q)] and
f ( q ) = -- ] ( 1 - - q) -- -~ ln13]
'
(13b)
Vol. 52, No. 12
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS
for 1
f ( m , q ) = ½,~-1 [m 2 --(1
+Y2£:)q I +/*(q),
(14)
where J~(q) is given by equation (13) and ~ ( = X) is now a function o f m and q, j 2 ~2 = 1 -- m2[q. In Fig. l(b) f(m, q) below Tg is drawn, andf(O, q) is expanded as
t3f(0, q) - ~f(0, 0) ½(1 _ # y ) 2 q + ¼(1-/~2)'2)q2 + ~q3 + . . . = 0
1 > /3J(1 -- q)
(15a)
1 < /3av(1-- q),
(15b)
which is, with q = me2, identical to equation (7). However, as seen in Fig. l(b) the limiting behaviour o f q under the symmetry breaking field h is different from equation (8): lim q = q o = m,o. 2
h--*0
(16)
Finally we examine how these two results (8) and (16) are consistently understood. For this purpose we consider a system with a large but finite N, and investigate a SP solution of equation (10) below 7"# under h proportional to N - a (a _2 0). The solution has to be very close to the branch point z = Y. Therefore we p u t z = J ( 1 + 6) with6 " N - b ( b _2 0) and extract the terms due to the A-mode from the sums in equation (10) and evaluate the rest approximately by making use of the semi-circular law. Then equation (10) reduces to
1 - T/Tg
T h2 h2 2Na~ 6 + 2~2(26)1/2 + 4N.~,26--------------~
'~ N -l+b + N -2a+b/2 + N -l+2b-2a,
(17)
where the first and third terms in r.h.s, are the contributions from the A-mode, and only leading order terms in N x are written for the remaining contributions. The requirement that r.h.s, of equation (17) has to be of the order o f N ° determines b as a function ofa. Substituting the results into m = xh, ms = bfn/ahA and q = Y-,x(afH/ahh) 2 with h x = (hi 1)h ~ h/N 1/2 , we obtain explicit dependence of m, ms and q on h. When h = N o , m and q(--- m 2 ) are finite but m s ~ N -v2 ,
1005
which corresponds to equation (8b). With decreasing h, m decreases in proportion to h while m s increases as h- a. In the range N - 1/2 < h 5 N - 1/6 ms and q (-~ m~) are t'mite, which corresponds to equations (8a) and (16). When h <~N-x/2, q (ms ) decreases in proportion to h2(h ), which simply means that at least the uniform field of the order o f N-x/2 is required to break full symmetry of the phase space. The above argument demonstrates clearly a subtle feature in a large but finite system, which is completely wiped out in the thermodynamic limit. From a mathematical point of view, the occurrence of a fiat region in the free energies is entirely due to the SP sticking in evaluating the partition functions. A similar result is obtained in the spherical model of shortrange ferromagnets [10]. One may therefore think that the results in the present work are the properties peculiar to the spherical model but not necessarily to the SG system. But we emphasize the close similarities between the present results and those of the Ising SK model, such as the TAP theories on these two SG's. Particularly the equation of state for q derived from equation (15a) is almost identical to equation (20) in [9], and their compatibility condition corresponds to that of the existence of a solution for the SP equations in the present work. On the boundary of that condition the inverse susceptibility a2f,rAp/amt~ml becomes semipositive definite for the spherical SG as well as for the Ising SG [11 ]. In this sense, this boundary corresponds to the Almeida-Thouless (AT) line [12]. This semipositivity holds even in the SP sticking region and the order parameter susceptibility always diverges. Also marginal stability of Parisi's solution is confirmed near Tg [13, 14]. Therefore we think it plausible that the coincidence of the free energies of the SG and paramagnetic states below Tg may be also a common property of the two SG's. This coincidence in the limit N -+ oo does not necessarily mean the nonexistence of a phase transition, as demonstrated below equation (16). There a subtle feature in the process of the limit N ~ oo has been clarified by reinterpreting approximately the saddle-point equation (10) as the one for a system with a large but finite N. We believe much subtlety in the large N limit are hidden behind Parisi's solution for the Ising SK model. There are, of course, differences between the SG orders in the two models. In the spherical SG it is at once destroyed, as demonstrated above, by infinitesimal h because it is the condensation of the (single) A-mode, while in the Ising SG it survives up to a certain value of h specified by the AT line because it is the condensation of infinite modes [15, 16]. It may be of interest to examine how the SP sticking in the spherical limit is modified when the Ising nature of spines is introduced, which is now in progress.
1006
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS REFERENCES
1. 2. 3. 4. 5.
6. 7.
D. Sherrington & S. Kirkpatrick, Phys. Rev. Lett. 35, 1972 (1975). C. de Dominicis & A.P. Young, J. Phys. A16, 2063 (1983). A.P. Young, Phys. Rev. Lett. 51, 1206 (1983). G. Parisi,Phys. Rev. Lett. 50, 1946 (1983). N.D. Mackenzie & A.P. Young, Z Phys. C16, 5321 (1983). G. Parisi,Phys. Rev. Lett. 43, 1754 (1979);Phys. Lett. 73A, 203 (1979);J. Phys. A13, 1101 (1980). J.M. Kosterlitz, D.J. Thouless & R.C. Jones, Phys. Rev. Lett. 36, 1217 (1976).
8. 9. 10. 11. 12. 13. 14. 15. 16.
Vol. 52, No. 12
T.H. Berlin & M. Kac, Phys. Rev. 86,821 (1952). D.J. Thouless, P.W. Anderson & R.J. Palmer, Phil. Mag. 35,593 (1977). J.S. Langer, Phys. Rev. 137, A1531 (1965). T. Plefka, J. Phys. AIS, 1971 (1982). J.R.L. de Almeida & D.J. Thouless, J. Phys. A11, 983 (1978). D.J. Thouless, J.R.L. de Almeida & J.M. Kosterlitz, J. Phys. C13, 3271 (1980). C. de Dominicis & I. Kondor, Phys. Rev. B27,606 (1983). H. Sompolinsky, Phys. Rev. B23, 1371 (1981). C. Dasgupta & H. Somplinsky, Phys. Rev. B27, 4511 (1983).
Solid State Communications, Vol. 52, No. 12, pp. 1003-1006, 1984. Printed in Great Britain.
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS K. Nemoto and H. Takayama Department of Physics, Hokkaido University, Sapporo 060, Japan
(Received 26 July 1984 by J. Kanamori) Free energies g(m, ms) and f(m, q) of the spherical spin glass (SG) model due to Kosterlitz et aL are calculated explicitly as functions of the uniform magnetization m, and SG order parameter m s and the EdwardsAnderson order parameter q. It is shown that g(0, m,) and f ( 0 , q) below the transition temperature Tit are constant in the ranges 0 g m s < and 0 __gq ~_ qo respectively, where qo = (1 -- T/T#) = m~. The_proper m,o equilibrium values of ms(= mso) and q(= qo) are then f'lxeg from the inspection of their behaviors under infinitesimal uniform field proproportional to N- a(a 2_ 0). THE MEAN FIELD THEORY of spin glasses has been considerably developed by making use of the Sherrington -Kirkpatrick model of the infinite-ranged Ising spin glass (SG) system [1 ]. However an intuitive interpretation of the theory is still not transparent due to various reasons. One of them is the lack of an explicit free energy expression below the transition temperature Tg as a function of certain order parameters in their whole range. Particularly that of the paramagnetic state has not yet been obtained. Another reason may lie on the process to take the thermodynamic limit (N ~ o~). According to the recent interpretations [2-5 ] on Parisi's replica-symmetry breaking solution [6], it contains properly informations on the overlap between many states with free-energy local minima which are separated by energy barriers of the order ofNa(a <~1/4). However it is by no means easy to understand how such a subtle feature in the large N limit can be incorporated in Parisi's solution, since the latter is a solution of a saddle-point equation which itself is derived in the thermodynamic limit. Keeping these difficulties in mind we have reexamined the spherical version of the SK model [7]. In this communication we will report the following now results, which, we believe, offer certain guides to our original problems in the Ising SK model. (i) The explicit free energy forms g(m, ms) and f(m, q) are derived in the entire region of the uniform
= 1 -- T/Tg = qo.
The spherical SG model in the limit of the infiniteranged interactions Jo with a Gaussian probability distribution was def'med and solved by Kosteflitz et al. [7], whose notations we follow unless specified explicitly. In the present work we restrict ourselves to a system with Jo = 0. Then in the limit N ~ 00 the eigenvalue density ofJ 0 obeys the familiar semi-circular law with the largest eigenvalue J,x = 2 ] (hereafter the index A denotes this largest eigenvalue mode). The order parameters m, m a and q are defined as m = N Z~m~, rn, =N-I/2mA and q = N -1 ~im~ = N- *~x m~, respectively, where m~mx) is the thermal average
(
E(m, ms)= ~D[s] 6(N-- ~s2)5 (Nm-- ~si) 6 (Nm s -- N 1/2SA) e- I3H
(2)
where ID [s] = fIIids i = fIIxds x. By making use of the standard manipulation of the spherical model, we obtain g(m, ms) as T
g(m, ms) = -- lim - - [ l n . ( m , ms)]a
magnetization rn, the SG order parameter (or the staggered magnetization)ms, and the EdwardsAnderson order parameter q. Particularly we find that g(0, ms) and f(0, q) become constant below 7"# in the ranges ms ~- mso and q g_qo respectively, where ms2o
proper equilibrium values of m, and q in the thermodynamic limit are fixed, which are given by equation (1l
Nooo N
= [~' (m, mg;z)]a
= --z+ T[~ y~X~(Z) 1
+ -~ X-l(z)m 2 +
(1)
(ii) From the inspection of ms and q under infinitesimal uniform field h proportional to N - a (a ~ 0), the 1003
1 x-,' (z)ml
(3
1004
Vol. 52, No. 12
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS
g,
lim
f
(8a)
lim ms = mso
hs~O
h~O
lim
(8b)
lim ma = 0.
h ~ O hs-~O
j~
--
./
' ms
mso
~q
0 //,
.,., , / "
q0
a
b
Fig. 1. Free energies g(m, ms) vs ms(a),f(m, q) vs q (b). The dashed line represents the unphysical extrapolation of the expressions which are only valid in the region m ~_ mso or q g qo-
Before going to discussions on equation (8), let us examine the free energy f(m, q), which is evaluated from the spherical version of the Thouless-AndersonPalmer (TAP) free energy f~sd, {mx} [9]. For this purpose we briefly describe the TAP theory of the spherical SG. The Helmholtz free energyfn{hx}, h x = Ei(Xl i)hi and h i being the external field at the ith site, is calculated as
fH = -- lim
J
[sl
where [ . . . ] j indicates the random average over JU, and 1[~ x(z) =
1
(IIA) 2 ] z-S ,12
=
y2)1/2 }
Xs(z) -
1
In
{
z
l_3`2X~(Z)
Z k
rr
/
1 v
h~.
(9)
4 N ~ z -- Jx]2'
where z is a solution of the following SP equation 1 = (6)
It is well-established that in case (6) has no solution, the saddle point sticks at z =3` [8] andg(m, ms) is simply given by [ ~e(m, ms; z = 3`)] a. The expression g(m, m,) in a whole range ofm and m s is thus derived. In the present problem the SP sticking occurs below Tg(= 3`) ifm = 0 and 1
T 1 1 hi 2N ~ -z - --J x / 2 + ~-~ ~ X ( z - J x / 2 ) 2"
(1 -- /33`)2msZ + ¼(1 --/323`2)m~ + ~m, I > /3J(1--ms2)
(7a)
1 < /3.](1 -- m~).
(7b)
The free energy g(m, ms) below Tg is shown in Fig. l(a). The dashed line in Fig. l(a) represents the simple extrapolation of equation (7a) to its invalid region, and has no physical meaning. An equilibrium value of m, can not be fixed since any m, in the range 0 ~ ms g m,o yields the same minimum g(0, ms). One way to get rid of this uncertainty may be to examine m s in the limit h, hs ~ 0, where h = ag/Om and hs = Og/Oms. However, reflecting the fact that the SP sticking condition depends crucially on whether m = 0 or m 4: 0, we obtain [see also Fig. 1(a)]
(10)
The TAP free energy is given by the Legendre transformation f~AP = fH + Zxhxmx with mx = -- NbfH/ ah x. Taking the random average of the terms which depend on m x only through q, we obtain
f~aI" --
1 2N ~ Jxm~ --z(1 --q) k
+-2
/3g(0, ms) --/3g(0, 0)
=
/3(z-A/2)
(5)
In equation (4) above (1 IA) = 1IN in denotes the uniform mode and z in equation (3) is the solution of the saddle point (SP) equation
3`X(z) = /33` l - - m ,
= - z+~
(4)
1 - 2(z--Y)"
2 Z--JA/2
-- /3 H - - Z hx sx h
a 1
exp
{z-(z
2
In
(11) '
with the SP condition/3(1 -- q) = X(z), where X(z) is given by equation (4). Again f ~ u , below Tg is divided into two branches depending on whether this SP equation has a solution or not:
8 _ fTAP
1
2N
• Jxm~ + y (q), x
(12)
where
f(q)
/3y2 -
T (1 _ q ) 2 _ 5 [1 + In 27r(1 - - q ) ] , (13a)
for 1 > / 3 ] ( 1 -- q) [with ] X = / 3 ] ( 1 -- q)] and
f ( q ) = -- ] ( 1 - - q) -- -~ ln13]
'
(13b)
Vol. 52, No. 12
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS
for 1
f ( m , q ) = ½,~-1 [m 2 --(1
+Y2£:)q I +/*(q),
(14)
where J~(q) is given by equation (13) and ~ ( = X) is now a function o f m and q, j 2 ~2 = 1 -- m2[q. In Fig. l(b) f(m, q) below Tg is drawn, andf(O, q) is expanded as
t3f(0, q) - ~f(0, 0) ½(1 _ # y ) 2 q + ¼(1-/~2)'2)q2 + ~q3 + . . . = 0
1 > /3J(1 -- q)
(15a)
1 < /3av(1-- q),
(15b)
which is, with q = me2, identical to equation (7). However, as seen in Fig. l(b) the limiting behaviour o f q under the symmetry breaking field h is different from equation (8): lim q = q o = m,o. 2
h--*0
(16)
Finally we examine how these two results (8) and (16) are consistently understood. For this purpose we consider a system with a large but finite N, and investigate a SP solution of equation (10) below 7"# under h proportional to N - a (a _2 0). The solution has to be very close to the branch point z = Y. Therefore we p u t z = J ( 1 + 6) with6 " N - b ( b _2 0) and extract the terms due to the A-mode from the sums in equation (10) and evaluate the rest approximately by making use of the semi-circular law. Then equation (10) reduces to
1 - T/Tg
T h2 h2 2Na~ 6 + 2~2(26)1/2 + 4N.~,26--------------~
'~ N -l+b + N -2a+b/2 + N -l+2b-2a,
(17)
where the first and third terms in r.h.s, are the contributions from the A-mode, and only leading order terms in N x are written for the remaining contributions. The requirement that r.h.s, of equation (17) has to be of the order o f N ° determines b as a function ofa. Substituting the results into m = xh, ms = bfn/ahA and q = Y-,x(afH/ahh) 2 with h x = (hi 1)h ~ h/N 1/2 , we obtain explicit dependence of m, ms and q on h. When h = N o , m and q(--- m 2 ) are finite but m s ~ N -v2 ,
1005
which corresponds to equation (8b). With decreasing h, m decreases in proportion to h while m s increases as h- a. In the range N - 1/2 < h 5 N - 1/6 ms and q (-~ m~) are t'mite, which corresponds to equations (8a) and (16). When h <~N-x/2, q (ms ) decreases in proportion to h2(h ), which simply means that at least the uniform field of the order o f N-x/2 is required to break full symmetry of the phase space. The above argument demonstrates clearly a subtle feature in a large but finite system, which is completely wiped out in the thermodynamic limit. From a mathematical point of view, the occurrence of a fiat region in the free energies is entirely due to the SP sticking in evaluating the partition functions. A similar result is obtained in the spherical model of shortrange ferromagnets [10]. One may therefore think that the results in the present work are the properties peculiar to the spherical model but not necessarily to the SG system. But we emphasize the close similarities between the present results and those of the Ising SK model, such as the TAP theories on these two SG's. Particularly the equation of state for q derived from equation (15a) is almost identical to equation (20) in [9], and their compatibility condition corresponds to that of the existence of a solution for the SP equations in the present work. On the boundary of that condition the inverse susceptibility a2f,rAp/amt~ml becomes semipositive definite for the spherical SG as well as for the Ising SG [11 ]. In this sense, this boundary corresponds to the Almeida-Thouless (AT) line [12]. This semipositivity holds even in the SP sticking region and the order parameter susceptibility always diverges. Also marginal stability of Parisi's solution is confirmed near Tg [13, 14]. Therefore we think it plausible that the coincidence of the free energies of the SG and paramagnetic states below Tg may be also a common property of the two SG's. This coincidence in the limit N -+ oo does not necessarily mean the nonexistence of a phase transition, as demonstrated below equation (16). There a subtle feature in the process of the limit N ~ oo has been clarified by reinterpreting approximately the saddle-point equation (10) as the one for a system with a large but finite N. We believe much subtlety in the large N limit are hidden behind Parisi's solution for the Ising SK model. There are, of course, differences between the SG orders in the two models. In the spherical SG it is at once destroyed, as demonstrated above, by infinitesimal h because it is the condensation of the (single) A-mode, while in the Ising SG it survives up to a certain value of h specified by the AT line because it is the condensation of infinite modes [15, 16]. It may be of interest to examine how the SP sticking in the spherical limit is modified when the Ising nature of spines is introduced, which is now in progress.
1006
FREE ENERGIES OF THE SPHERICAL MODEL OF A SPIN GLASS REFERENCES
1. 2. 3. 4. 5.
6. 7.
D. Sherrington & S. Kirkpatrick, Phys. Rev. Lett. 35, 1972 (1975). C. de Dominicis & A.P. Young, J. Phys. A16, 2063 (1983). A.P. Young, Phys. Rev. Lett. 51, 1206 (1983). G. Parisi,Phys. Rev. Lett. 50, 1946 (1983). N.D. Mackenzie & A.P. Young, Z Phys. C16, 5321 (1983). G. Parisi,Phys. Rev. Lett. 43, 1754 (1979);Phys. Lett. 73A, 203 (1979);J. Phys. A13, 1101 (1980). J.M. Kosterlitz, D.J. Thouless & R.C. Jones, Phys. Rev. Lett. 36, 1217 (1976).
8. 9. 10. 11. 12. 13. 14. 15. 16.
Vol. 52, No. 12
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