- Email: [email protected]
Free energy of a spin glass
Volume lOBA, number 2
PHYSICS LETTERS
18 March 1985
FREE ENERGY OF A SPIN GLASS
Daniel C. MATTIS and Antonio CALIRI Physics Department, Universityof Utah, Salt Luke city, UT 84112, USA Received 7 February 1985
We give the theory of a model spin glass of the Sherrington-Kirkpatrick type, Determining that the free energy is given by the potential function of a two-dimensional electrostatic medium, we find exact expressions for this quantity in terms of a multipole expansion of the charge distribution. We also obtain the internal energy, entropy, and specific heat in the form of explicit integrals over the multipole distributions. Pending the outcome of a quantitative investigation into the structure of these functions, here we discuss their properties in a qualitative way.
In a recent paper we evaluated the free energy and other relevant thermodynamic functions of an Ising model system described by a hamiltonian : H = -i
c”cJijSiSj i
(each Si = *l) ,
j
(1)
in which the bonds were constant, all Jij = J/N, with J > 0 or < 0 [ 11. An analysis of the zeros of the partition function in the complex-temperature plane yielded the usual mean-field behavior for the ferromagnet (J > 0), while demonstrating the absence of any phase transition in the fully-frustrated case (J < 0). In the present paper we extend our studies to a model spin glass, characterized by random bonds of the type: Jii = eijJlN’t2
,
with eij = +l , random variables “frozen-in” with equal probability. We have found this version of the “canonical” mean-field spin glass [2] to be relatively tractable to our methods. Here we shall outline the theory in terms of a set of functions, with the help of which the thermodynamic properties of the model may be obtained. In a companion paper under preparation 131, denoted II henceforth, we shall provide quantitative estimates of these functions, based upon our study of the exact free energy of a (manageably) small sample. We start by writing down the partition function associated with H for a given configuration 0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(cll)of the random variables, Z, = 2N[cosh@J/N1/2)]N(N - l)i2 X (01
’ij
(1 + fijsixsi”)lO>,
(3)
where 10)is the state of all spins “up”, SF = ST t S,T is a Pauli spin operator, and tii E tanh(eijflJ/Nl i2). WewriteF,=-TInZ,(kg=l,T=/3-l)asthesum of two contributions: F = F(O) t F(l), (Y with F(O) = -NT In 2 - NJ2i/4
(4)
the contribution of the first two factors in the limit N + 00. Defining the third factor (01 ... 10)as P,(t), a polynomial of degree G N(N - 1)/2 in t = tanh@J/ N1j2), we obtain F(,‘) = -TlnP,(t)
= -T C ln(l -
t/Zj>,
(5)
i
where the Zj are the zeros of P,,
i.e. Pol(Zj>= 0. These occur either as complex pairs, Zj and Zj* (xj + iyj), or on the real axis, outside the “physical cut” (0 < Xi < 1, y. = 0). These conditions are required to maintain P,(t 5 real and nonnegative over the physical range of r *l . 111
We next define F(I) as the configurational average over the ensemble of (11,F( ‘) = tiaI)). It is a true free energy in the thermodynamic sense; with fl”) in eq. (4), the thermodynamic properties of the model are given precisely by F = F(O) + F(l) and its derivatives. For the purpose of evaluating it, we introduce the density function p(z), p(z) =;
(c 6(z - Zj))
i
cv
It is convenient to express z in polar coordinates, I exp i0, and thus to obtain F(1) in the form
X ln[l t (t/r)* - 2(t/r) cos 01 .
z =
(8)
This integral has the appearance of a potential in twodimensional electrostatics, with p the charge distribution. Clearly, a solution of the problem requires knowledge of this function. Let us start with some general properties. Unlike Lee-Yang [4] zeros in the complex magnetic field plane, the zeros in the complex T plane do not necessarily coalesce on lines, but may fill areas. (This point was recently noted in connection with entirely different applications of the Ising model, by van Saarloos and Kurtze [S] and, independently, by the present authors [l] .) Nevertheless, in the study of any particular random canfiguration (a) of the bonds connecting a finite number of spins, we *l Other restrictions on the distribution of zeros will appear as the consequence of further restrictions on the model. For example, under the symmetry J- -J we can map xi + -xi and therefore when this symmetry applies, there can be no zeros on the negative segment - 1 Q xi =Z0 either. (Note, in spin glasses with ferromagnetic or antiferromagnetic bias, there is no such symmetry.) We have additionally considered “special random” (SR) distributions, such that at each spin half the bonds have E = - 1. (This requires that N - 1 be divisible by 4; for 9 spins, there are only 16 SR’s as out of a total of 236 configurations cy,thus in numerical investigations it is expeditious to study them.) The SR distributions have the additional symmetry that for each pair of zeros at zi and Zi* there exist conjugate zeros at l/q and l/q*, as if the z’s represented point charges in the presence of a perfect conductor wrapped on the unit circle. In the present paper we make no use of the special symmetries described in this footnote, although they will play a role in our paper II.
112
18 March 1985
PHYSICS LETTERS
Volume 108A, number 2
consistently
find the zeros to lie on fairly complicated
curves which
avoid the real axis, filling areas only in
the neighborhood
of the imaginary
aging over o! the situation
changes
cause the curves for different
axis. Upon averdramatically;
configurations
bedo not
coincide, the resulting density p(r, 0) becomes strictly two-dimensional, filling the complex plane from a small circle of radius rc < 1 (r, corresponds approximately to fl,J = 1 according to our preliminary calculations, in accord with original estimates [2]), to a second circle of radius l/r,. (When the SR model is applicable, the density functions inside the unit circle and outside of it are necessarily related by rp(r, 0) = r-3p( l/r, 71t 0); this causes us to look for densities of the form p = r-*R(r, l/r; 0) in the ranger < 1, with R symmetric in r, l/r: R(l/r, r) = R(r, l/r).) We also note that as r is increased from rc the density rises quickly and that p is non-negative always. Already we can comment on a well-known aspect of the long-ranged spin glasses, i.e. the slow relaxation rates to equilibrium, and the “trapping” in metastable states. A given configuration cy, corresponding to a large region in an infinite (thermodynamic) sample, has thermodynamic properties dominated by curvilinear trajectories of zeros near the physical cut, which are different from the trajectories of a different a’; in fact, as the two are practically disjoint, it is difficult to see how the system can make the transition from LYto (Y’.In calculating the thermodynamic average in eq. (8) we sidestep these questions of ergodicity (which, anyhow, lie outside the purview of any strictly Ising model without dynamics,) but remark that, aside from causing p to become twodimensional, the randomness is an important element in the analysis. The evaluation of (8) proceeds after Fourier transformation:
p(r,e)=r-*(RO(r)f~~IR,(r)cos(“8)).
(9)
Expanding F(l) = To + Z ‘3,,, as in (9), we have the contributions of each multipole (harmonic) as follows: P
To = -NT =
0,
s 0,
dr r-lRo(r)
ln@/r),
for p > 0, , (loa)
forb<&,
(lob)
PHYSICS LETTERS
Volume 108A, number 2
andform>
1, P
3,,, = +NTm-'
(s
dr r-l R,(r)(r/fl)”
0,
R,(r)@/r)m
drr-l P
P>P, 2 (114
PC’
= +NTn-l
s
drr-l R,(r)@/rY,
P
(1lb)
PC
Note the change in overall sign. For reasons which will become apparent, the distribution outside the unit circle (in complex t space) does not contribute to the 9’s, therefore the upper limits 0;’ (which came from r;’ on the integrals in (1 la, b) must be replaced by 00. This remark follows from the fact that Fill is identically zero at all 0 < &, because F(O) [eq. (4)] is the exact free energy at high temperature *2. Thus, the charge distribution lying between r, and 1, within the unit circle, minus the compensating uniform background distribution, is identically zero. By inversion symmetry (z + l/z) the charges outside the unit circle also add up to zero. As we cannot penetrate the region r > 1 (it lies outside the physical cut) the outer distribution is thus unobservable at all T in the range 0 < T < 00, and can be discarded. With the modified limits, eqs. (10) and (11) are the principal results of this paper. It is easy enough to differentiate these expressions with respect to T to obtain the entropy S = -aFlaT, the internal energy U = apF/ap, or the heat capacity c = iW/aT, so we shall not take the space to write the formulas here. The low-temperature properties come from the behavior of the R, near the unit circle, while critical properties at T, are related to the critical exponents R,
=a,,,,(r -r,)v(m),
r>r,.
18 March 1985
A knowledge of these quantities can only come from careful examination of the Y,, which we are in the process of doing and will publish in II. In conclusion: In the spin glass problem, the highT region is trivial while the low-T region (both the critical region just below T, and the ultra-low T region) has heretofore defied exact analysis! *3 We have now shown that it, too, can be fully understood, through a mapping onto a reasonably standard problem in 2D electrostatics. We have found it possible to identify certain symmetries. For example, in the $R model (and, conceivably, in general whenever the distribution of bonds is even) the symmetry J cf J implies that all odd multipole distributions vanish identically (RI = R3 = . .. E 0;) while in a spin glass with ferromagnetic (or antiferromagnetic) tendencies, the odd muitipoles will contribute - most especially, to the magnetic susceptibility (which we have not discussed above), but also importantly to the thermodynamics. It is tempting to compare the present problem with the “spherical-model spin glass” which was given an explicit, exact solution by Kosterlitz, Thouless and Jones [8]. In both cases, the free energy is an explicit integral over two factors: one involving the random elements, the other a property of the spin system. In the spherical model, the function of the random elements is the density-of-states function of a random matrix, a well-understood and easily calculated quantity. In the present case, it is a number of functions (the R,) of the properties of a random polynomial, which has not been studied heretofore. Nevertheless, we are confident that after sufficient analysis, this model will become as thoroughly understood as the other. This work was partially supported by an NSF grant, DMR81-06223 and by a grant of the CNPq (A.C.). It contains material which will be submitted in partial fulfiment of the requirements of the Ph.D. degree (A.C.).
(12)
*2 The proof of this statement comes from the original (replica-trick) treatment of this problem, which is certainly exact at high temperature [ 21, or from the cumulant expansion for the free energy [6], which also yields’precisely eq. (4) in the absence of a symmetry-breaking field.
*3 This problem has been amazingly intractable - a recent review [7] Lists over 700 recent papers dealing with meanfield spin glasses. Although some are experimental, many of these papers involve the theory.
113
Volume
108A, number
2
PHYSICS
References (11 A. Caliri and D.C. Mattis, Phys. Lett. 106A (1984) 74. and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792; Phys. Rev. B17 (1978) 4384. [3] A. Caliri and D.C. Mattis, (1985), to be submitted. [4] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404.
[ 21 D. Sherrington
1985
[5] W. van Saarloos and D.A. Kurtze, J. Phys. Al7 (1984) 1301. [6] D. Mattis, The theory of magnetism,Vol. 2, Thennodynamics and statistical mechanics (Springer, Berlin, 1985), to be published. [7] D. Chowdhury and A. Mookerjee, Phys. Rep. 114 (1984)
1
.181_ ;: Kosterlitz, 36 (1976)
114
18 March
LETTERS
D. Thouless 1217.
and R.C. Jones.
Phys. Rev. Lett.
PHYSICS LETTERS
18 March 1985
FREE ENERGY OF A SPIN GLASS
Daniel C. MATTIS and Antonio CALIRI Physics Department, Universityof Utah, Salt Luke city, UT 84112, USA Received 7 February 1985
We give the theory of a model spin glass of the Sherrington-Kirkpatrick type, Determining that the free energy is given by the potential function of a two-dimensional electrostatic medium, we find exact expressions for this quantity in terms of a multipole expansion of the charge distribution. We also obtain the internal energy, entropy, and specific heat in the form of explicit integrals over the multipole distributions. Pending the outcome of a quantitative investigation into the structure of these functions, here we discuss their properties in a qualitative way.
In a recent paper we evaluated the free energy and other relevant thermodynamic functions of an Ising model system described by a hamiltonian : H = -i
c”cJijSiSj i
(each Si = *l) ,
j
(1)
in which the bonds were constant, all Jij = J/N, with J > 0 or < 0 [ 11. An analysis of the zeros of the partition function in the complex-temperature plane yielded the usual mean-field behavior for the ferromagnet (J > 0), while demonstrating the absence of any phase transition in the fully-frustrated case (J < 0). In the present paper we extend our studies to a model spin glass, characterized by random bonds of the type: Jii = eijJlN’t2
,
with eij = +l , random variables “frozen-in” with equal probability. We have found this version of the “canonical” mean-field spin glass [2] to be relatively tractable to our methods. Here we shall outline the theory in terms of a set of functions, with the help of which the thermodynamic properties of the model may be obtained. In a companion paper under preparation 131, denoted II henceforth, we shall provide quantitative estimates of these functions, based upon our study of the exact free energy of a (manageably) small sample. We start by writing down the partition function associated with H for a given configuration 0.3759601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(cll)of the random variables, Z, = 2N[cosh@J/N1/2)]N(N - l)i2 X (01
’ij
(1 + fijsixsi”)lO>,
(3)
where 10)is the state of all spins “up”, SF = ST t S,T is a Pauli spin operator, and tii E tanh(eijflJ/Nl i2). WewriteF,=-TInZ,(kg=l,T=/3-l)asthesum of two contributions: F = F(O) t F(l), (Y with F(O) = -NT In 2 - NJ2i/4
(4)
the contribution of the first two factors in the limit N + 00. Defining the third factor (01 ... 10)as P,(t), a polynomial of degree G N(N - 1)/2 in t = tanh@J/ N1j2), we obtain F(,‘) = -TlnP,(t)
= -T C ln(l -
t/Zj>,
(5)
i
where the Zj are the zeros of P,,
i.e. Pol(Zj>= 0. These occur either as complex pairs, Zj and Zj* (xj + iyj), or on the real axis, outside the “physical cut” (0 < Xi < 1, y. = 0). These conditions are required to maintain P,(t 5 real and nonnegative over the physical range of r *l . 111
We next define F(I) as the configurational average over the ensemble of (11,F( ‘) = tiaI)). It is a true free energy in the thermodynamic sense; with fl”) in eq. (4), the thermodynamic properties of the model are given precisely by F = F(O) + F(l) and its derivatives. For the purpose of evaluating it, we introduce the density function p(z), p(z) =;
(c 6(z - Zj))
i
cv
It is convenient to express z in polar coordinates, I exp i0, and thus to obtain F(1) in the form
X ln[l t (t/r)* - 2(t/r) cos 01 .
z =
(8)
This integral has the appearance of a potential in twodimensional electrostatics, with p the charge distribution. Clearly, a solution of the problem requires knowledge of this function. Let us start with some general properties. Unlike Lee-Yang [4] zeros in the complex magnetic field plane, the zeros in the complex T plane do not necessarily coalesce on lines, but may fill areas. (This point was recently noted in connection with entirely different applications of the Ising model, by van Saarloos and Kurtze [S] and, independently, by the present authors [l] .) Nevertheless, in the study of any particular random canfiguration (a) of the bonds connecting a finite number of spins, we *l Other restrictions on the distribution of zeros will appear as the consequence of further restrictions on the model. For example, under the symmetry J- -J we can map xi + -xi and therefore when this symmetry applies, there can be no zeros on the negative segment - 1 Q xi =Z0 either. (Note, in spin glasses with ferromagnetic or antiferromagnetic bias, there is no such symmetry.) We have additionally considered “special random” (SR) distributions, such that at each spin half the bonds have E = - 1. (This requires that N - 1 be divisible by 4; for 9 spins, there are only 16 SR’s as out of a total of 236 configurations cy,thus in numerical investigations it is expeditious to study them.) The SR distributions have the additional symmetry that for each pair of zeros at zi and Zi* there exist conjugate zeros at l/q and l/q*, as if the z’s represented point charges in the presence of a perfect conductor wrapped on the unit circle. In the present paper we make no use of the special symmetries described in this footnote, although they will play a role in our paper II.
112
18 March 1985
PHYSICS LETTERS
Volume 108A, number 2
consistently
find the zeros to lie on fairly complicated
curves which
avoid the real axis, filling areas only in
the neighborhood
of the imaginary
aging over o! the situation
changes
cause the curves for different
axis. Upon averdramatically;
configurations
bedo not
coincide, the resulting density p(r, 0) becomes strictly two-dimensional, filling the complex plane from a small circle of radius rc < 1 (r, corresponds approximately to fl,J = 1 according to our preliminary calculations, in accord with original estimates [2]), to a second circle of radius l/r,. (When the SR model is applicable, the density functions inside the unit circle and outside of it are necessarily related by rp(r, 0) = r-3p( l/r, 71t 0); this causes us to look for densities of the form p = r-*R(r, l/r; 0) in the ranger < 1, with R symmetric in r, l/r: R(l/r, r) = R(r, l/r).) We also note that as r is increased from rc the density rises quickly and that p is non-negative always. Already we can comment on a well-known aspect of the long-ranged spin glasses, i.e. the slow relaxation rates to equilibrium, and the “trapping” in metastable states. A given configuration cy, corresponding to a large region in an infinite (thermodynamic) sample, has thermodynamic properties dominated by curvilinear trajectories of zeros near the physical cut, which are different from the trajectories of a different a’; in fact, as the two are practically disjoint, it is difficult to see how the system can make the transition from LYto (Y’.In calculating the thermodynamic average in eq. (8) we sidestep these questions of ergodicity (which, anyhow, lie outside the purview of any strictly Ising model without dynamics,) but remark that, aside from causing p to become twodimensional, the randomness is an important element in the analysis. The evaluation of (8) proceeds after Fourier transformation:
p(r,e)=r-*(RO(r)f~~IR,(r)cos(“8)).
(9)
Expanding F(l) = To + Z ‘3,,, as in (9), we have the contributions of each multipole (harmonic) as follows: P
To = -NT =
0,
s 0,
dr r-lRo(r)
ln@/r),
for p > 0, , (loa)
forb<&,
(lob)
PHYSICS LETTERS
Volume 108A, number 2
andform>
1, P
3,,, = +NTm-'
(s
dr r-l R,(r)(r/fl)”
0,
R,(r)@/r)m
drr-l P
P>P, 2 (114
PC’
= +NTn-l
s
drr-l R,(r)@/rY,
P
(1lb)
PC
Note the change in overall sign. For reasons which will become apparent, the distribution outside the unit circle (in complex t space) does not contribute to the 9’s, therefore the upper limits 0;’ (which came from r;’ on the integrals in (1 la, b) must be replaced by 00. This remark follows from the fact that Fill is identically zero at all 0 < &, because F(O) [eq. (4)] is the exact free energy at high temperature *2. Thus, the charge distribution lying between r, and 1, within the unit circle, minus the compensating uniform background distribution, is identically zero. By inversion symmetry (z + l/z) the charges outside the unit circle also add up to zero. As we cannot penetrate the region r > 1 (it lies outside the physical cut) the outer distribution is thus unobservable at all T in the range 0 < T < 00, and can be discarded. With the modified limits, eqs. (10) and (11) are the principal results of this paper. It is easy enough to differentiate these expressions with respect to T to obtain the entropy S = -aFlaT, the internal energy U = apF/ap, or the heat capacity c = iW/aT, so we shall not take the space to write the formulas here. The low-temperature properties come from the behavior of the R, near the unit circle, while critical properties at T, are related to the critical exponents R,
=a,,,,(r -r,)v(m),
r>r,.
18 March 1985
A knowledge of these quantities can only come from careful examination of the Y,, which we are in the process of doing and will publish in II. In conclusion: In the spin glass problem, the highT region is trivial while the low-T region (both the critical region just below T, and the ultra-low T region) has heretofore defied exact analysis! *3 We have now shown that it, too, can be fully understood, through a mapping onto a reasonably standard problem in 2D electrostatics. We have found it possible to identify certain symmetries. For example, in the $R model (and, conceivably, in general whenever the distribution of bonds is even) the symmetry J cf J implies that all odd multipole distributions vanish identically (RI = R3 = . .. E 0;) while in a spin glass with ferromagnetic (or antiferromagnetic) tendencies, the odd muitipoles will contribute - most especially, to the magnetic susceptibility (which we have not discussed above), but also importantly to the thermodynamics. It is tempting to compare the present problem with the “spherical-model spin glass” which was given an explicit, exact solution by Kosterlitz, Thouless and Jones [8]. In both cases, the free energy is an explicit integral over two factors: one involving the random elements, the other a property of the spin system. In the spherical model, the function of the random elements is the density-of-states function of a random matrix, a well-understood and easily calculated quantity. In the present case, it is a number of functions (the R,) of the properties of a random polynomial, which has not been studied heretofore. Nevertheless, we are confident that after sufficient analysis, this model will become as thoroughly understood as the other. This work was partially supported by an NSF grant, DMR81-06223 and by a grant of the CNPq (A.C.). It contains material which will be submitted in partial fulfiment of the requirements of the Ph.D. degree (A.C.).
(12)
*2 The proof of this statement comes from the original (replica-trick) treatment of this problem, which is certainly exact at high temperature [ 21, or from the cumulant expansion for the free energy [6], which also yields’precisely eq. (4) in the absence of a symmetry-breaking field.
*3 This problem has been amazingly intractable - a recent review [7] Lists over 700 recent papers dealing with meanfield spin glasses. Although some are experimental, many of these papers involve the theory.
113
Volume
108A, number
2
PHYSICS
References (11 A. Caliri and D.C. Mattis, Phys. Lett. 106A (1984) 74. and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792; Phys. Rev. B17 (1978) 4384. [3] A. Caliri and D.C. Mattis, (1985), to be submitted. [4] C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) 404.
[ 21 D. Sherrington
1985
[5] W. van Saarloos and D.A. Kurtze, J. Phys. Al7 (1984) 1301. [6] D. Mattis, The theory of magnetism,Vol. 2, Thennodynamics and statistical mechanics (Springer, Berlin, 1985), to be published. [7] D. Chowdhury and A. Mookerjee, Phys. Rep. 114 (1984)
1
.181_ ;: Kosterlitz, 36 (1976)
114
18 March
LETTERS
D. Thouless 1217.
and R.C. Jones.
Phys. Rev. Lett.