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Saddle points stability in the replica approach off equilibrium
M. Ferrero, M. Virasoro
To cite this version:
M. Ferrero, M. Virasoro. Saddle points stability in the replica approach off equilibrium. Journal de Physique I, EDP Sciences, 1994, 4 (12), pp.1819-1827. �10.1051/jp1:1994109�. �jpa-00247036�
Classification Physics Abstracts
05.20 75 10H 75.10N
Saddle points stability in the replica approach off equilibrium
M.E. Ferrero and M.A. Virasoro
Dipartimento di Fisica, Università di Rcma, La Sapienza, 1-00185 Rcma, Italy
INFN Sezione di Roma I, Roma, Italy
(Received 28 April1994, received in final form 5 August 1994, accepted 12 August 1994)
Abstract. We study trie replica free energy surface for a spin glass model near the glassy temperature. In this model the simplicity of the equilibrium solution hides non trivial metastable saddle points. By means of the stability analysis performed for one and two real replicas con- strained, an interpretation for some of them
is achieved
1. Introduction.
In spin glass models, replica symmetry breaking implies trie existence of many pure states of equilibrium. Their hierarchical organization is described by trie order parameter Q
" (qab)
iii. In trie SK model qab bas a continuum form and it is a marginally stable saddle point of the replica free energy surface in trie limit n - 0 [2]. In other mortels, like the p-spin spherical mortel, trie stable solution is just one-step, yielding a simpler eqmlibrium state organization Î31.
In this paper we investigate this different feature from trie stability point of view. We use the generalization of the replica method recefitly developed to study non equihbrium states [4, Si. The general idea is to consider R identical copies of the model (real replicas) constrained
to have specific mutual overlaps. Then, by forcing the overlaps eut of equilibrium, one cari probe the phase-space structure of the original model and obtain information on some of its
non eqmlibrium states.
By performing trie stability analysis in trie replica approach off equilibrium, we uncover trie physical meaning for some saddle points other than trie one yielding trie Gibbs-measure results.
2. Trie unconstrained mortel and its saddle points.
2.1 THE MODEL. Let us consider trie following truncated replica free energy density
~~~~ Î~~~~~~Î~~~~Χ~~~~Î2§~~~~ ~~~
Q Les Editions de Physique 1994
1820 JOURNAL DE PHYSIQUE I N°12
where we consider t
r~J Tc T to be small. Trie cubic term £q(~, absent in trie SK-model,
gives a y independent replica symmetry breaking. A term of this kind appears in trie p-spin spherical model and in trie Potts model [6j. Trie y parameter will play trie rote of a control parameter. We will assume it to be small (y « t) but positive or negative (in trie SK model it is positive).
In [6j it was shown that for y < 0 trie stable solution is simply a matrix with one replica symmetry breaking (IRSB). Trie probability distribution of trie overlap between equilibrium configurations is then
~eq(~) " ~lô(~) + (~ ~l)~(~ ~EA) (~)
where qEA " 2t + (10/3)yt~ and mi
" (1/2) + vi are obtained by stationarity conditions.
In the solution we will consistently truncate trie O(y~) terms. As a consequence of the
stationarity conditions, the O(y~) terms in the solution give O(y~) contribution to free energy.
In the following we will then consider free energy up to O(y3)
2.2 UNSTABLE SADDLE POINTS. For positive y we know that a better approximation to
trie fuit solution is given by trie 2RSB saddle point
q~ = 0 0 < x < mi
q(X) " ~~ ~
Î~~~
~~ ~ ~ ~ ~~
q~ = 2t + ~yt~ m2 < x <
1
mi " j + (Ut
~~ =
~
+ ~yt (3)
2 2
alla free ellergy F2RSB > FIRSB (F2RSB FIRSB
~ à~).
To study fluctuations around this 2RSB S-P- we consider trie eigenvalue equation
(2t + qab + yqlb + À)rab + jQ> Rab " o (4)
where f = Q QSP. This equation, analysed in details in trie context of trie SK model [7], con
be solved through trie expirait construction of trie eigenvectors exploiting trie symmetry of trie equation. In trie notation of [7] trie eigenvectors are classified as longitudinal rab), anomalous
f$~) and replicon f$~~). For positive y trie minimum eigenvalue is = -yt~ /6 and it is well known that unstable directions belong also to trie rephcon family.
Let us discuss trie analytic continuation to negative y of this solution. It leads to m2 < mi, a result which does not allow the usual probabilistic interpretation. In this case we expect new
unstable directions.
We found the minimum eigenvalue to be negative (À = syt~/6). The unstable eigenvectors belong to trie longitudinal and anomalous familles. They are
fo(0) = 0 if a n b
= 0
longitudinal - rab " Îo(1) # 0 if a n b = 1
fo(2) # 0 if a n b
= 2
fi(0) = 0 if a n b = 0
fi(1) # 0 if a n b
= 1 max(a n ~t,b n ~t) = 0
anomalous - f$~ = fi(2) # 0 if an b
= 2 max(a n ~t,bn ~t) = 0
fi (1)(1 n/mi if a n b
= max(a n ~t,bn ~t) > 0
fi (2)(1 n/mi if a n b
= 2 max(a n ~t,b n ~t) > 0
(5) where, as in [7], a n b = k if qab
= qk.
For future reference, we anticipate that by considering additional RSB trie new unstable S.P.
will have m~+i m~
r~J y. There is, of course, the analytic continuation (to negative y) of the continuum solution (slope
= 1fg), which is the only one marginally stable for positive y. It has obviously a free energy greater than the other solutions.
3. Trie constrained mortel.
Let us consider R real replicas and introduce in the partition function R(R -1) /2 complex La- grangean multipliers (e~~) to keep the mutual overlaps constrained (q[~). The effective averaged
free energy has two contributions [4j
~R(Q,~Î~)
" ~(Q) + ~constr(fÎ, ~Î~) (6)
The functional F(Q) has the same form as for a single real replica with Q a Rn x Rn matrix
oti p12 plR
p12~ Q22 p2R
Q " (~)
plR~ p2R~ QRR
where (PrS)aa
" e[~. The "interaction" term Fconstr(e[~,q[S) reads
Fconstr(~7>~l~) " ~ifl2 £(~l~ ~7)~ (8)
r,s,a
The stationarity conditions are then
In trie following we consider trie particular case R
= 2 and we make for Q and P a Parisi ansatz. Before analyzing trie solution, let us explain how we studied trie stability.
3.1 2RR STABILITY. From (9) we see that trie saddle point for fa, which originally is
integrated along trie imaginary axis, lies on trie real axis. Trie integration patin can be deformed to go through trie saddle point but trie integration patin will be perpendicular to trie real axis.
As usual this S-P- bas to be a maximum when compared to values along trie integration path
and therefore a minimum when compared to values along the real axis. Notice that this
distinguishes fa from ail other matrix terms which are integrated along trie real axis [8j. Trie stability with respect to fa fluctuations around the saddle point depends on the quadratic term
(8) that is not proportional to t. Therefore these fluctuations are much smaller and we can substitute everywhere fa by its S-P- value which we will denote pà.
1822 JOURNAL DE PHYSIQUE I N°12
The equation for trie hessian of F2 is then
(2t + qap + vq$p + À)fap + lQ> flap
= ° (i°)
where off
= 1,..., 2n but a fl # n. In order to reduce this problem to an ultrametric one
we can observe that the symmetry group of the Hamiltonian without constraint is S2n (ail rephcas are equivalent). After imposing the constraint it becomes Sn 4§ (52)~" (to permute
simultaneously replicas in bath systems or to permute equivalent replicas between systems).
Since ail solutions found are symmetrical under the action of 52 on ail the replicas, we divide the eigenvectors in two superfamilies, the symmetrical and the antisymmetrical one. We then
parametrise the fluctuation matrix f around the Qs.P in terms of
+f~ +f~ +f~ +f~
f
= + (ii)
+f~ +f~ -f~ -f~
where the f~i~>~ are n * n symmetrical matrices and f~ antisymmetrical.
The eigenvalue equations are now decoupled and read, for a # b,
(2~ + ~ab + Yqlb + À)flb + lQ, f~ lab + if~ Piab
" o
(2~ + Pab + YPÎb + À)f]b + lP>f~ lab + if~>Qlab
" o (12)
and
(2t + gab + yglb + À)flb + (Q, Î~)ab + f~,Pjab
" 0
(2~ + Pab + YPÎb + À)flb ~ lQ>f~lab + lf~,Plab
" o (13)
The eigenvectors of (12) and (13) are then respectively of trie form
+f~ +f~ +f) +f( +f)~ +f(~
f = f~ = f~~ = (14)
+Î~ +Î~ +Î( +ÎÎ +Î(v +ÎÎV
+f~ +f~ +f( +f( +f(~ +f(~
f = f~ = f~~ = (15)
IX fZ f§ fZ f§ fZ
M M MU MU
where f~, f( and fJ~ are antisymmetrical n * n matrices breaking trie hierarchical subgroups
of permutation group (as the IRR ones).
3.2 2RR CONSTRAINED SADDLE POINTS. We investigate the possible 2RSB solutions in
the range 0 < pà < qEA. We try to follow the solution by continuity when pà varies. At different points in the interval a stable solution becomes unstable and we have to search for the new stable solution.
Let us consider g < 0. We find four critical pd values 0 < p~ < p~ < p~ < p~ < qEA. At each critical pd value, two different solutions coïncide and zero modes appear to modify their
stability.
We start at pd CÎ 0. The stable solution is
10
0<x<m~
q(x) = 10
~~~~~~~jY~~ ~2<~<~
Ù<X<1112
~~ Îpd m2<x<1
1112 " -+Yt (ifi)
2 and the free energy density is
F°A Fjtjjnstr
=
aF°A
= ((-2yt2p( pi) + o(y4) (17)
If pd
" 0 the free energy is the same as the equilibrium one and ôF°~ /ôpd
"
dF°~ /dpà
= 0.
For small pd the free energy is an increasing function of pd (Fig. l). The minimum eigenvalue
is = -(2/3)yt~ pd. Thus when pd > p~
= -(2/3)yt2 this solution becomes unstable and trie new stable solution is:
qo" 0 0<x<mi
~~~~ Ql Pd )àl(i'd l) llll < X < 1112
~~ ~~ ~ Î~~~ YPd(pd 4t) m~ < ~ ~
po = 0 0 < x < mi
~~~~ ~~ ~~ ~~~~~
~~ ~~ ~ ~ ~ ~~
p~= pd m2<x<1
1 pd
~~ 4 ~ ~
4
m~ = + y(t + ~~ (18)
The free energy is
zlF~~
=
~ (2pdt~ 2p(t3 + p(t~ pli) + O(y~) (19)
This solution crosses, at pd " p~, trie previous one. For pd
'~~ -y trie minimum eigenvalue
is = (pd/2) + (yt~/3). As can be seen in figures and 2, zlF~~ is an increasing function until pd
" prnax " t + (23/12)yt~, where dF~ /dpd
"
ôF~B /ôpd
" 0. For future reference we
1824 JOURNAL DE PHYSIQUE I N°12
AF
o
o pA
Fig 1 AF as a function of pd plotted for 0 < pd < 2p~, t
= 0.05 and y = -0.01
AF
op~ P'~~~ P~ P~ P~ qEA
Fig 2 AF as a function of pd plotted for 0 < pd < qEA. t = 0.05 and y
= -0 01.
rewrite (18) on this specific value
qo" 0 0<x<mi
q(x) = qi
" t + ~~yt~ mi < x < m2
q2 = 2t + ~~yt~ m2 < x < 1
lPo
" 0 0 < x < mi
~~~~
~~ ~
Î~~~
~~ ~ ~ ~
t
~~ 4 ~ ~4
m2 =
j+y) (20)
The minimum eigenvalue is
= -yt~ /6. As pà increases further, zIF~B begins to decrease while ail eigenvalues read positive until trie second critical value p~. Trie minimum eigenvalue
for pà ct p~ is
= -y(p( 8pàt + 8t~)/6 and it corresponds to fluctuations of qi and pi p~
is defined by À(~B = 0.
For pà greater than p~ the stable solution becomes
Q° ° 0 < x < mi
'
q(x) = Qi " Pd jv(-P( + i4Pàt 14t2) mi < z < m~
q2 " 2t + Ît~ ~(~(Pà 4t) m~ < z < i
Po " 0 0 <x < mi
~~~~ Pi Pd jY(PÎ ~lld~ + ~~~) llll < X <1112
i'2" i'd 1112<X<Î
pà
mi " j+Yj
m2 " + y(t + ~~ (21)
The free energy of this solution (zlF~~), which differs from (18) by terms of O(y~), is
decreasing in pà (Fig. 2). Again at pd " P~ the two solutions crosses each other. The minimum
eigenvalue is
= y(p( 8pdt+8t~) /6. We have pi qi
= -2y(p( -8pdt+8t~) /9
r~J and the free
energy difference between (18) and (21), expressed in terms of (21), reads
r~J
(pi qi)~ '~~ -Y~.
We found (21) to be stable for pB < pd < p~ m 3/2t. In the range p~ < pà < p~ < qEA we
didn't find any 2RSB stable solution. In particular (21) is unstable with respect to replicon fluctuations. We expect that additional RSB lead to the stable solution.
Thus, let us consider pd > p~
= 2t + 23yt~/6. The stable solution is
10
~ ~ ~ ~ ~~
q(x) = 2
(p~ + t) + y~(Pd + t)~ mi < x <
3
P~~~ ~~~
mi " j + j(Pd + ~) (22)
The free energy, expressed in term of à
= pd qEA '~~ Y, reads (Fig. 3) zIF~QEA
= (-26~t~y ô~) + O(y~) (23)
and the minimum eigenvalue is
= (46 2yt~)/3.
Clearly for positive y these solutions are just approximations to the fuit solution. In particular
we found (20) to have the same instability than the 2RSB free solution (À = -yt~/6).
1826 JOURNAL DE PHYSIQUE I N°12
AF
P~ ~~~
Fig. 3. AF as a function of pd plotted for pd # qEA, t = 0.05 and y
= -0.01.
4. Discussions.
Let us analyse the previous results for y < 0. For pà
= 0 the two real replicas lie on config-
urations belonging to different equihbrium states. As the multiplicity of pure states give no
contributions to the entropy, the free energy per replica is equal to that of the unconstrained system and q(x)
= q(x)~~~°~~~r and p(x)
= 0. If pà increases we force trie overlap between couple of replicas to be out of equilibrium. Trie free energy increases until trie maximum is
reached at p~'~~ while if pà > p~'~~ trie free energy faits until pà of order of qEA, where we
are choosing configurations inside trie same state. Since trie overwhelming majority of pairs of
configurations inside a state are at distance qEA from each other, if pà
= qEA we have twice trie energy and twice trie intra-state entropy.
At pd " P~~~ we bave ôF~B /ôpd
"
dF~B /dpd
" 0. Trie constraining force is then zero (on
trie S-P-) and trie 2 real replicas are unconstrained. In fact trie solution (20) is related through
a replica permutation to trie solution (3), because m)~~
=
m(~~/2. Then F$ff
= Fj([.
However trie constraint still acts on trie fluctuations and prevents trie 2 replicas from sliding along trie unstable direction.
It is remarkable that in this way we bave been able to interpret an unstable 2RSB. We conjecture that with more constraints one will be able to stabilise solutions with more levels of
breaking. Therefore trie linear solution, which is trie limit for an infinite number of breakings,
will reappear.
Trie Parisi parameter q(x) plays two different rotes in trie SK mortel. On trie one nana it appears in trie Gibbs measure and determines trie P~q(q). In this case q(x) must be necessarily
monotonous. On trie other hand x(q) also appears in the Cavity Method where monotonicity
does not seem to be required. It remains as an intriguing open question whether in systems as
trie present a non monotonous x(q) could be relevant. Perhaps solving trie R real replicas by
a generalization of trie Cavity Method could elucidate this issue.
Acknowledgments.
We are grateful to S. Franz, I. Kondor, J. Kurchan and T. Temesvari for useful discussions.
References
iii For a recent exposition on mean field theory of Spin Glasses see for example: Mézard M., Parisi G. and Virasoro M.A., "Spin Glass Theory and Beyond" (Singapore: World Scientific, 1987).
[2] de Dominicis C. and Kondor I., Phys. Reu. 27 (1982) 606-608.
[3] Crisanti A. and Sommers H-J-, Z. Phys. 887 (1992) 341-354.
[4] Franz S
,
Pansi G. and Virasoro M.A., J Phys I Fronce 2 (1992) 1869.
[5] Kurchan J., Pansi G. and Virasoro M.A., J Phys. I France 3 (1993) 1819.
[fil Gross D.J., Kanter I. and Sompohnsky H., Phys. Reu. 55 (1985) 304-307.
[7] de Domimcis C and Kondor I., Europhys. Lent 2 (1986) 617-624.
[8] We thank I. Kondor and T Temesvan for pointing ont to us that there was an apparent incon- cistency m reference [4] because this distinction m the path of integration was overlooked.
