Stability of the Mezard-Parisi solution for random manifolds. D. M. Carlucci. Scuola Normale Superiore di Pisa. Pisa 56126, Italy. C.

Stability of the Mezard-Parisi solution for random manifolds

D. M. Carlucci

Scuola Normale Superiore di Pisa Piazza dei Cavalieri

Pisa 56126, Italy

Internet: CARLUCCI@UX2SNS.SNS.IT

C. De Dominicis S.Ph.T., CE Saclay 1991 Gif sur Yvette , France Internet: CIRANO@AMOCO.SACLAY.CEA.FR

T. Temesvari

Institute for Theoretical Physics Eotvos University

1088 Budapest, Hungary Internet: TEMTAM@HAL9000.ELTE.HU

(February 7, 1996)

Abstract

The eigenvalues of the Hessian associated with random manifolds are con- structed for the general case of^Rsteps of replica symmetry breaking. For the Parisi limit^R!1 (continuum replica symmetry breaking) which is relevant for the manifold dimension ^D<2, they are shown to be non negative.

1

Resume

Les valeurs propres de la Hessienne, associee avec une variete aleatoire, sont construites dans le cas general de R etapes de brisure de la symmetrie des repliques. Dans la limite de Parisi, R ^!1 (brisure continue de la symmetrie des repliques) qui est pertinente pour la dimension de la variete D <2, on montre qu'elles sont non negative.

PACS

numbers: 05.20, 05.50, 02.50

The behaviour of uctuating manifolds pinned by quenched random impurities encom- passes a rich variety of physical situations, from the directed polymers in a random potential (with a manifold dimension D = 1) to interfacial situations (D = d^?1, d being the total dimension of the space). The directed polymer problem, for example, is itself related to sur- face growth, turbulence of the Burgers equation and spin glass. The literature concerning various aspects is wide and orishing, a good deal of it being quoted in some recent reviews [1,2]. The approaches taken have been quite diverse and it is not the purpose of this paper to describe them. This paper is a rst step in an approach to random manifolds that uses eld theoretic techniques in the replica formalism. Field theoretic techniques have been used with success (in their functional renormalization group variety) by Balents and Fisher [3]

to describe the large N (N = d^?D is the transverse dimension) for the 4^? dimensional manifold. In the context of directed polymers Hwa and Fisher [4], using more conventional eld theory, have emphasized the role of large but rare uctuations. Alternately a unied de- scription of the general random manifold has been developed by Mezard and Parisi [6](MP), where, ^via the use of the replica technique, they have laid down the rst step (mean eld and self-consistent Hartree-Fock level) of a systematic eld theory 1=N expansion¹. This is the approach we want to pursue here and this paper may be considered as a rst sequel to MP.

In this paper we study the eigenvalues of the Hessian associated with the 1=N expansion as laid down by MP. We do it in a general form by keeping discrete the R steps of replica symmetry breaking, and we show that the MP continuous solution (R=¹, as is appropriate for D < 2) does not generate negative eigenvalues.

The general discrete expression derived here will be of use in discussing solutions for the case D >2, which is unclear to date.

1The intricacies of the rst loop correction were brie y considered in further work [7]

3

In section 2 we summarize the MP approach down to the quadratic uctuation terms dening the Hessian (see also [8]). Section 3 is devoted to the introduction of a discrete Fourier-like transform on the replica overlaps. In section 4 we brie y discuss the equation of state at mean eld level. In section 5 we discuss and exhibit the replicon eigenvalues.

Section 6-8 are devoted to the tougher case of the longitudinal-anomalous eigenvalues.

I I. SUMMARIZINGMP APPROACH

We thus consider the MP random manifold model dened by

?H(~!) =^?1 2

Z dx^D

2

4 D

X

=1

@~!(x)

@x^

!2

+ 2V(x;~!(x)) +~!²(x)

3

5 (1)

whered=N +D is the dimension of the space, N the dimension of ~!, ^i.e. of the tranverse space, and D the dimension of the vector x, the longitudinal directions. The quenched random potential V has a gaussian distribution of zero mean and correlation

V(x;~!)V(x⁰;~!⁰) =^?^(D)(x^?x⁰)Nf

([~!^?~!⁰]² N

)

(2) with

f(y) = g

2(1^? ) (+y)^1? (3)

where the function is regularized at small arguments by the cut-o .

Averaging over the random quenched variable V via replicas replaces (1) by

?H

r ep(~!^a) =^?1 2

Z dx^d

2

4 X

a 8

<

: D

X

=1

@~!^a(x)

@x^

!2

+~!^a2(x)

9

=

;+^X

a;b

Nf

([~!^a(x)^?~!^b(x)]² N

) 3

5

(4) In studying (4) MP have used rst a variational approach (identical to the Hartree-Fock approximation), which becomes exact in the large N limit. Then they introduced auxiliary elds with the advantage that it generates correction to Hartree-Fock, namely

(i) The zero loop term embodying the eect of quadratic uctuation, ^i.e. the logarithm of the determinant of the Hessian matrix

4

(ii) ^N1 corrections in a systematic way.

In this work we are interested in (i) studying the Hessian and its eigenvalues around the mean eld (Hartree-Fock or variational) solutions. We leave the study of (ii) for a separate publication [9].

Following MP we introduce the n(n+ 1)=2 component auxiliary elds

r^ab(x) = 1N ~!^a(x)~!^b(x) (5) and its associated Lagrange multipliers s^ab(x). We need to compute the generating function Zⁿ =^{Z Yn}

a=1d[~!^a]^{Z Yn}

ab

d[r^ab]d[s^ab]^

exp

(

?

2

X

ab

Z dxs^ab(x)[Nr^ab(x)^?~!^a(x)~!^b(x)]

)

 (6)

exp

8

<

:

?

2

Z dx^Xn

a=1

2

4 D

X

=1

@~!^a(x)

@x^

!2

+~!^2a(x)

3

5

?

² 2

Z dx^X

ab

Nf(r^aa(x) +r^bb(x)^?2r^ab(x))

9

=

;

now quadratic in ~!'s, hence after integrating over ~!^a Zⁿ=^{Z Y}

ab

d[r^ab]d[s^ab]exp[N^Gfr;s^g]

Gfr;s^g=^? 2

X

ab

Z dx[s^ab(x)r^ab(x) +f(r^aa(x) +r^bb(x)^?2r^ab(x))] +^S[s] (7) exp^S[s] =^{Z Y}

a

d[~!^a]exp

(

?

2

X

ab

Z dx^h~!^a(x)^(^?r2+)^ab?s^ab(x)^~!^b(x)ⁱ

)

Assuming an uniform saddle point (for large N) with r^ab(x) =^ab+r^ab(x)

s^ab(x) =^ab+s^ab(x) (8) one can expand around it. MP thus obtain

5

(i) The mean eld contibution ^G(0)fr;s^g

(ii) The stationarity condition (vanishing of linear terms) that denes  and  (identical to the variational answer²)

 Stationarity with respect to s^ab(x):

^ab = 1

X

p

G^ab(p) (9)

where

G^?1ab = (p²+)^ab?^ab (10)

 Stationarity with respect to r^ab(x):

^ab = 2f⁰(^aa+^bb?2^ab) (11)

^aa=^{? X}

b(⁶=^a)^ab (12)

(iii) The Hessian matrix of the quadratic form in s^ab (after integration of the quadratic terms in r^ab which in particular says that s^aa=^Pa6=^bs^ab, ^i.e.

Z^nexp^hN^G0f;^{giZ Y}

a<b

d[s^ab]exp^?1 2

X

a<b

c<d X

p

s^ab(p)M^ab;cd(p)s^cd(^?p) (13)

M^ab;cd(p) =^? ^ab;cd

2f⁰⁰(^aa+^bb?2^ab) ^?

X

q

fG^ac(q) +G^bd(q)^?G^ad(q)^?G^bc(q)^g

fG^ac(p^?q) +G^bd(p^?q)^?G^ad(p^?q)^?G^bc(p^?q)^g (14)

2Only in the large ^N limit. If ^N is nite, in the variational answer^f(^y) is replaced by ^^f(^y) =

1

?(N=2) R

1

0 d

N=2?1

e

?

f(2^y=N) with ^^{f !f} as^{N !1}. 6

We shall work in a formalism where the level R of replica symmetry breaking is kept discrete (R = 0 replica symmetric,R = 1 one step of breaking, R =¹ Parisi breaking, ^i.e.

continuous overlaps). In the end, if necessary, we shall let R ^!1.

For an observablea, witha =a^r for an overlap^\ =r, dene the discrete Fourier transform as

^a^k =^RX+1

r=^k p^r(a^r?a^{r ?}1) (15) and its inverse

a^r =^Xr

k=0

p1^{k (^}a^{k ?}a^^k+1) (16) Here the p^r's are the size of the successive Parisi boxes, p0 ^ n, p^R+1 ^ 1. The above transform has the properties

a^r?a^{r ?}1 = 1p^{r( ^}a^r?a^^r+1) (17)

X

a b  =c (18)

^a^{k^}b^k = ^c^k (19)

In particular

(ad^?1)^k = 1^a^{k (20)}

All this is but a discrete version of the operations introduced in MP for the continuum (R^!1). Note that in the above all quantities with indices outside the range (0;1;:::;R+1) are to be taken identically null. If the limit R ^!1 is understood as

^a^r!a^(u)

a^r!a(u) (21)

7

then the connexion with MP is, in their notations,

^a(u) = ~a^?ha^i?[a](u)

[a](u) =^?Z₀^udta(t) +ua(u) (22)

IV.THE EQUATION OF STATE

With the above denitions we have

^r = 1

X

p

G^r(p) (23)

G^^k(p) = 1

p²+^?^^{k (24)}

and the equation of state

^r= 2f^{0 2}

X

q

[G^R+1(q)^?G^r(q)]

!

r= 0;1;:::;R (25) 0 = ^?RX+1

t=0 p^t(^t?^t?1)r=R+ 1 (26) The last equation, identical to (12), also writes

^0 =^RX+1

r=0p^t(^t?^t?1) = 0 (27)

To solve for the equation of state one uses the transform that allows to write [G^r(q)^?G^{r ?}1(q)] = 1p^r

hG^^r(q)^?G^^r+1(q)ⁱ (28) and

y^r = 2

X

q

[G^R+1(q)^?G^r(q)] = 2

X

q 8

<

: R

X

k=^r+1

p1^{k(^}^k?^^k+1) ^G^k(q) ^G^k+1(q) + ^G^R+1(q)

9

=

; (29) The equation of state then writes, with

f⁰(y^r) = g

2 [+y^r]^? ; 8

^r = 2f⁰(y^r) (30)

i.e.

"

g1

r

X

k=0

p1^{k[^}^k?^^k+1]

#

? 1

=+ 2

X

q 8

<

: R

X

r=^k+1

p1^{k(^}^{k ?}^^k+1) ^G^k(q) ^G^k+1(q) + ^G^R+1(q)

9

=

;: (31) Here we have

G^^k(q) = 1

q²+^?^^{k (32)}

where  is an infrared cut-o, which can be set to zero in appropriate situations. In the continuum limit

?^^{k !}[](x) p^{k !}x

(in the Parisi gauge) easily restituting the MP solution as given for the case of noise with long range correlations.

For later use, let us take a one step dierence in the equation of state

^r?^{r ?}1 = 1p^{r (^}^r?^^r+1) = 2[f⁰(y^r)^?f⁰(y^{r ?}1)] (33) with

y^{r ?1?}y^r = 2

X

q

[G^r(q)^?G^{r ?1}(q)] = 2

X

q

p1^r

hG^^r(q)^?G^^r+1(q)ⁱ (34)

i.e.

= 2 ^{^}^r?^^r+1

p^r

!

X

q

G^^r(q) ^G^r+1(q) (35) For small y^{r ?}1^?y^r (innitesimal in the continuum) one gets

1 = ^?4^X

q

G^^r(q) ^G^r+1(q)f⁰⁰(y^r) (36) a formula used below.

9

The Hessian (14) written in terms of overlaps becomes

M^ab;cd(p) =M^ur;^;vs(p) (37) with

r =a^\b;s=c^\d

u= max(a^\c;a^\d) (38)

v= max(b^\c;b^\d)

In ultrametric space, only three of the four overlaps are distinct. In the replicon subspace r = s, and we have two independent overlaps u;v ^ r+ 1. In the longitudinal anomalous sector, if r⁶=s, one can keep max(u;v) to parametrizeM.

For the replicon sector, and on general ground [10{12], the eigenvalues are given by the double discrete Fourier transform

^p(r;k;l) =^RX+1

u=^k p^uRX+1

v=^kp^vM^ur;^;vr?M^u?r;r1;^{v ?}M^ur;^{;rv ?}1+M^u?r;r1;^{v ?}1

 (39)

for k;l^r+ 1.

Using the Hessian expression M^ur;;^vr =^?^uKr;^R+1^vKr;^R+1

2f^{r00 ?}

X

q

[G^u(q) +G^v(q)^?2G^r(q)][G^u(p^?q) +G^v(p^?q)^?2G^r(p^?q)]

(40) Hence, for the ^double transform

^p(r;k;l) =^? 1 2f^{r00 ?}

X

q

hG^^k(q) ^G^l(p^?q) + ^G^l(q) ^G^k(p^?q)ⁱ (41) and using (36), valid in particular in the continuum,

^p(r;k;l) =^X

q

n2 ^G^r(q) ^G^r+1(q)^?hG^^k(q) ^G^l(p^?q) + ^G^l(q) ^G^k(p^?q)^io

l;k^r+ 1 (42)

10

The propagators ^G^r(q) are decreasing functions of ^jq^j and of r (since ^?^^r is an increasing function). Hence the most ^dangerous eigenvalue is

^p(r;r+ 1;r+ 1) = 2^X

q

nG^^r(q) ^G^r+1(q)^?G^^r+1(q) ^G^r+1(p^?q)^o (43) or in the continuum limit

^p(x;x;x) = 2 1(2)^D

Z dq^D

( 1

q²+^?^(x) ^? 1

(p^?q)²+^?^(x)

) 1

q²+^?^(x) (44) ForD <4, one can set the ultraviolet cut-o to innity and rewrite the dangerous replicon eigenvalues, in the continuum limit, as

^p(x;x;x) =^X

q

hG^^x(q)^?G^^x(p^?q)ⁱ² (45) where the marginal stability is obviously displayed.

VI. HESSIAN EIGENVALUES: THE LONGITUDINAL ANOMALOUS SECTOR

The analysis of longitudinal anomalous sector is notoriously more dicult. It has been considerably simplied by the introduction [13,11,12] of ^kernels, ^i.e. by going to a block- diagonal representation of the ultrametric matrices. Take in (37) a component of the longi- tudinal anomalous sectorM^ur;^;sv, r⁶=s, which, given that only three overlaps can be distinct, can be written³

M^tr;s t = max(u;v)

ToM^tr;sis associated a kernel (in fact a ^Fouriertransform) K^kr;s, wherek = 0;1;2;:::;R+1, k = 0 corresponding to the longitudinal sector. Likewise for the inverse M^?1 of the mass operator matrix we have a kernel F^kr;s. The relationship between them is the Dyson's equation [11,12]

3Except for the components^{Mu;vr ;r} that also incorporate a replicon contribution.

11

F^^kr;s=K^{kr;s? 1}₂^RX+1

t=0K^kr;t
k(t)

^k(t)F^^kt;s (46) F^^kr;s=^?^k(r)F^kr;s^k(s) (47) Here the ^k(r) are special replicon eigenvalues dened by

^k(r) =

8

>

>

>

>

<

>

>

>

>

:

(r;k;r+ 1) k ^r+ 1

(r;r+ 1;r+ 1) k < r+ 1 (48) with  as in (39) and

^k(t) =

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

12(p^t?p^t+1) t < k^?1

12(p^t?2p^t+1) t =k^?1 (p^t?p^t+1) t > k^?1

(49)

The kernel K^kr;s itself is a transformed of the mass operator matrix M^tr;s. It writes as a generalized discrete Fourier transform

K^kr;s =

0

@ r

X

t=^k+2 ^Xs

t=^r+1+4 ^RX+1

t=^s+1

1

Ap^t(M^tr;s?M^t?r;s1) (50) where we have displayed the case k ^r ^s (for dierent orderings the summation ^Pt=^k is only kept in the allowed summands). Knowing K^kr;s, one then computes F^{kr;s via} (46, 47) and (M^?1)^{rt;s via} an inverse transform. That is if one is able to solve the integral equation (46) which is only possible with especially simple kernels K^kr;s. This turned out to be the case when studying the bare propagator of the standard spin glass [11] where

14K^kr;s=A^k(min(r;s)) (51) The same functional form also appears for the spin glass model introduced by Th.Nieuwenhuizen [14]. For the system studied here, let us construct the kernel K^kr;s with the matrix M^tr;s as given by the Hessian (14) manifestly symmetric in r;s. Let us choose r < s and we then nd

12

14K^kr;s(p) =^?X

q

R^X+1

t=max(^s+1^;k)p^tf[G^t(q)G^t(p^?q)^?G^t?1(q)G^t?1(p^?q)]^?

?G^s(q)[G^t(p^?q)^?G^t?1(p^?q)]^? [G^t(q)^?G^t?1(q)]G^s(p^?q)^g (52) which also rewrite as

14K^kr;s =

8

>

>

>

>

<

>

>

>

>

:

B^k(s) r < s < k^?1

B^s+1(s) s^k^?1 (53)

where

B^k(s) =^?X

q n

G(q)Gd(p^?q)^

k

?G^s(q) ^G^k(p^?q)^?G^^k(q)G^s(p^?q)^o (54) The important point is that the kernel only depends on max(r;s)

14K^kr;s =B^k(max(r;s)) (55) thus rendering soluble the integral equation (46).

VI I.SOLUTION OF THE DYSON EQUATION: THE PROPAGATOR KERNEL

In the case of ¹₄K^kr;s = A^k(min(r;s)) the solution of the Dyson equation (46) has been given (in the continuum limit) in [11]. We wish to extend it here to the case ¹₄K^kr;s = B^k(max(r;s)). In this section we work for a given value of k, ^i.e. we remain in a sub-block of the longitudinal anomalous sector (k = 0 for the longitudinal).

The propagator kernel of eqs (46, 47) is obtained as 14F^kr;s = ^+k(min(r;s))^?k(max(r;s))

^k(r)
^k^k(s) (56)

where
^k is a ^Wronskian and ^k the regular and irregular solution of a discrete Sturm- Liouville equation and ^k the special replicon eigenvalues of (48). Namely we have for the Wronskian

13

^k = ^?k(s)^Ds+k(s)^?+k(s)^Ds?k(s)

D

sB^k(s) (57)

with

D

sf(s)^f^s+1 ^?f^s (58)

The functions ^ are given by 12^+k(s) = 1 +^Xs

t=0[B^k(t)^?B^k(s)]
^k(t)

^k(t)^+k(t) (59) 12^?k(s) =^?XR

t=^s[B^k(t)^?B^k(s)]
^k(t)

^k(t)^?k(t)^?B^k(s)C^k (60) with B^k(s) as in (53). Note that, contrary to the solution obtained for (51), where C^k = 1, here, ^?k and ^+k remain coupled ^via C^k

C^{k  XR}

t=0

^k(t)

^k(t)^+k(t) In terms of ^+k the Wronskian can also be written as

^k = 4C^k

"

1 +^XR

s=0B^k(s)
^k(s)

^k(s)^+k(s)

#

: (61)

VI I I. STABILITYDISCUSSION IN THE LA SECTOR

Given the explicit expressions (53) ofB^k(s) in terms of ^G^t(orG^t) itself a positive denite function, decreasing with increasing t

D

tG^t= 1p^t+1

^^t+1^?^^t+2

G^^?t+1¹ G^^?t+2¹ (62) one can easily show that

(i)B(r)<0 (63)

(ii)^DrB(r)^B(r+ 1)^?B(r)>0 (64) 14

Under these conditions we can now give a^{lower bound}for the longitudinal anomalous eigen- values. We can write out the determinant of the eigenvalues restricted to the sub-block k :

det^M^{r s(k)?}I^= ^YR

r=0[^k(r)^?]

"

1 +^XR

t=0B^k(t)
^k(t)

^k(t)^?^+k(t)

#

(65) which is directly related to the Wronskian (apart from the C^k factor).

Note that, from equation (59), one also obtains

D

s^+k(s) =^?2^DsB^k(s)^Xs

t=0

^k(t)

^k(t)^?^+k(t): (66) With the boundary value ^+k(0) = 2 and ^DsB(s)>0, one infers that, in case of  < ^k(t) for all t,

^+k(s)>0 (67)

D

s^+k(s)>0 (68)

Consider now in equation (65) a small value of  such that  < ^k(t) for all t . For that value  the determinant of (65) ^{cannot vanish}, it is always positive on account of (63, 67).

Hence the spectrum of the longitudinal anomalous sector is certainly bounded by the bottom values of the replicon spectrum ^p=0(r;r+ 1;r+ 1). This establishes stability, a marginal stability as in the continuum limit where ^p=0(x;x;x) = 0.

IX. CONCLUSIONS

We thus have set up for random manifolds the formalism to discuss the stability of mean eld solutions. For the well established continuous (R =¹) solution, adequate for D <2, we are able to prove its marginal stability. The formalism used here contains a discussion of any type (discrete, continuous or mixed) mean eld solutions.

It is worth noticing that, when discussing the stability, the only place where properties of the noise correlation function may play a direct role (outside of the fully continuousR =¹

15

solutions) is in (33-36). There the equation of state is used to nd the alternative analytical form for f⁰⁰, that cast in evidence the non-negative character of the replicon eigenvalues.

ACKNOWLEDGEMENTS

One of us (CD) gladly acknowledges useful discussions with M.Mezard.

16

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11

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4

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27

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17