Analytic determination of the complex field zeros of REM

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Analytic determination of the complex field zeros of REM

C. Moukarzel, N. Parga

To cite this version:

C. Moukarzel, N. Parga. Analytic determination of the complex field zeros of REM. Journal de

Physique I, EDP Sciences, 1992, 2 (3), pp.251-261. �10.1051/jp1:1992141�. �jpa-00246480�

Classification Physics Abstracts

75.10N 05.20

Analytic deterndnation of the complex field

_zeros

of REM

C.

Moukarzel(*)

^{and N.}

Parga

Centro At6mico Bariloche, 8400 S-C- de

Bariloche(**),

Rio Negro, Argentina

(Received

5 September 1991, accepted in final form 21

November1991)

Abstract The complex field _zeros of the Random Energy Model

are analytically determined.

For T _< Tc they _are distributed in the whole complex plane with _a density that decays _very fast with the real component ^{of H.} For T _> Tc _a region is found which is free of _zeros and is

associated _to the paramagnetic phase of the system. ^This region is separated from the former

by _a line of _zeros. The real axis -is found to be enclosed in _a cloud of _zeros in the whole frozen

phase of the system.

1 Intro duction.

The determination of the _zeros of the

partition

function allows much

insight

into the behavior of _a model. Since their relation with

phase

transitions _were clarified

by

the works of

Yang

and Lee

[lj

_2] and Fisher [3], many

interesting properties

of _zeros distributions have been studied in

homogeneous

systems

[4-9].

More

limitedj

_on the other

hand,

_are the results

concerning

_zeros distributions of disordered models.

Taking

into account that all the information about _a

given

model is contained in

the _zeros of its

partition function,

it is reasonable to expect ^that _many ^{of the}

interesting

properties

[10] ^{of disordered} systems will be somehow reflected

by

them.

An

example

of this _can be found in the

complex magnetic

field _zeros of

spin glasses

j

which _are dense _near the real axis. This property has been

suggested by

Parisi

[I

I] as

being

_a consequence of the

multiplicity

of states and _was

recently

illustrated [17]

by

a numerical calculation of the

complex

field _zeros distribution in the Random

Ehergy

Model

(REM) [12].

Here _an

analytical

calculation of the

complex

field _zeros of REM is

reported.

The

technique

is the _{same as}

recently employed by

Derrida [13] ^{to calculate the}

complex fl

_zeros of the _same

modelj

which had been also

numerically

estimated [14, 15].

In section 2 _we describe the basis of the REM and

give

_an account of the ideas that allow the calculation of its _zeros. Section 3 is devoted to the REM in _a

complex

field and the calculation

(*

Fellowship

Holder of the Consejo Nacional de

Investigaciones

Cientificas _y T4cnicas.

(**) ^{Comis16n NacionaJ de} Energia At6mica.

JOURNAL DE PHYSIQUEI T 2, N' 3, MARCH 1992

of the dominant contribution _to its

partition

function in each

region

of the

complex

h ₌

flH plane.

In section 4 these results _are used to obtain the

density

of _zeros of REM in the

complex

field

plane,

^while section 5 contains _our conclusions.

2. The random _energy model.

The REM is defined [12] to have 2/~

independent

random _energy levels

E;, I=1,..2/~

distributed

according

to

P(E)

₌

(xN)~~/~ exp(-E~ IN).

The

partition

function of _a

sample (a sample

is

a set of 2/~ numbers

E;),

Z ₌

£(~~ ^e~P~<

_can be written _as Z ₌

£~ n(E) ^e~P~

where

n(E)

is the number of levels between E and _{E +} AE. This

density

of states

n(E)

if found to

satisfy

(n(E))

^{~- exP}

^N(log

²

^(()~)

~~~

(n(E)~) (n(E))~

~-

(n(E))

so two different _energy

regions

_can be identified:

If

(E(

<

Nec

,

with _{cc =}

(log 2)~/~,

^then the number of states

n(E)

is much greater than _one and its fluctuations _are

small,

_so for _a

typical sample

_we will have [13]

n~~~(E)

₌

In(E))

₊

nE(n(E))~/~ (2)

where qE is _a random number of order I.

On the other

hand,

if

(E(

> Nec

,

then the number of states is

exponentially

small

meaning

that _a

typical sample

will have

no levels in this

region

+~YP(E)

₌ °

(3)

The

partition

function for _a

typical sample

will then be

ztyp

= A + B

(4)

with

A ₌

£ (n(E))e~~~ (5)

jEj<Ne<

and

B"

L QE(n(E))~~~e~~~ (6)

jEj<Ne<

The calculation of A _may be done

by

the steepest ^{descent method} [16] ^for

complex fl

₌

pi +ifl2.

+ec A

~

2~

_de

exp

-N(e~

₊

_{fle) (7)}

e<

The

integration

contour is stretched to

infinity through paths

of steepest ^{descent for the real}

part of the exponent ^{of the}

integrand.

When the

integration

limits

lay

_on

opposite

sides of the saddle

point

in _{e =}

-fl/2.we

_can write

/+ec

-co +co +ec

# + +

(8)

-ec

~ec ~co ~co

and A has contributions both from the saddle

point

and from the

integration

limits for

large

N. This

happens

if

[pi

< flc ₌ 2ec _"

2(log2)~/~

and _we get

A ~- exP

lNflecl

_{+ exP}

lN(log

2 +

(fl/2)~) l (9)

whereas if

)pi

> flc the two steepest descent

paths

_run from the

integration

limits down _to

infinity

_on the _same side of the saddle

point

_so it does _not contribute and _we have

/+ec

+co _~E<

" +

(1°)

-ec

~ec ~cc

so

A

~

exp(Nflec) ₍₁₁₎

It has to be mentioned [13] that the lowest

lying

states in REM have order _one fluctuations from

sample

to

sample

_{[12] so} the contributions

coming

from the

integration limits,

both in

(9)

and

(11),

_are known _up to _a

fluctuating complex (if fl

is

complex)

factor of order _one. We will return to this

point

later _on because it results crucial for _our calculation.

The contribution B of the fluctuations _can be calculated for

large

N.

Owing

to the _ran- domness of _{qE, B} is not the

integral

of _an

analytic

function _so the steepest ^{descent method} is

no

longer applicable.

On the other

hand,

the _{qE are} uncorrelated for different values of E _so the term with the

largest

modulus will dominate the _sum in

(6).

We _can then estimate the

modulus of B _as

(B( _~

_~~fl~~~~ ll~(E))~/~~~~~ (12)

It is

easily

seen that when

lpi

<

flc/2

the maximum is in between the limits _so

lBl

_{~- exP}

I(('Og

2 +

Pi)) ₍₁₃₎

while if

lpi

>

flc/2

one of the limits dominates and

(B(

_~- exP

IN _{(Pi lee)1} (14)

We will _{now use} these results to calculate Z for

a

typical sample

of REM when _a

complex

field H acts on it. We will take

fl

real from _{now onj as} well _as hi and h2

non-negative

(h

=

flH

₌ hi ₊

ih2)

which _can be done without _any loss of

generality.

3. REM i11 _a

complex

field.

When _a

magnetic

field H acts on the system, ^the

energies

E; of the states _now

have,

in addition to the internal

energies (coming

from interactions _among

spins)

_a contribution from the interaction with the

field, E;

_{= U;}

HM;j

where the

M;

_are the

magnetizations

of the

states. The

partition

function Z ₌

£;

^e~P~< is then

~N

z(p h) £ ~-pU,+hM, (~~)

,

i=I

The internal

energies

U; have _a Gaussian distribution which is

independent

of the

field,

_so

taking

into account that among the

2~

_{states of the system,}

gm = gM =

N~M

^have

magnetization

M ₌

Nm,

_{we can} write 2

Z(fl, h)

₌

£ e~~'~

_Zm

(fl) (16)

-iimii

where

gm

Zm(fl)

=

L e~~~' (17)

is the

partition

function of _a

sample

of REM with _gm instead of

2~

_{states. If}

we write gm =

(qm)~j

then all the results of last section

concerning

Z are valid for Zm with the sole

condition of

replacing log

2

by

log

_{qm =}

log

2 [(1 ⁺

m) log(I

+

m)

+

(I m) log(I m)]

(for large N).

These restricted

partition

functions Zm can be written _as

Zm(fl)

₌

L _{n(tL, m)} e~~~" (18)

where

n(u,m)

_are the numbers of states with internal energy NV and

magnetization

Nm.

They

can be _seen to

satisfy

(n(lt, ^m))

^{~- exP}

^N('°g

^qm

^tt~)

(19) In(U, m)~) ln(U, m))~

~-

ln(U, m))

Then for _a

typical sample

we will have

(YP(U, ml

~-

(n(Ui ml

+ n(tL,

m)(n(tL, m)1~/~

,

if Iv <

tLc(m)

~~o~

n YP(tL,

m)

_~- °

,

if (U( ^>

tLc(m)

where the

q(u,m)

_are random numbers

describing

the

typical fluctuations,

and the energy threshold

uc(m)

is _now

m-dependent.

uc(m)

₌

(log qm)~/~ (21)

The

partition

^function of _a

typical sample

under field will then be

Z~~~(fl, h)

_"

Le~~~Z$~~(fl) (22)

m

We _{now use} the results of last section and write

zjYP(p)

=

A~(p)

₊

B~( p) (23)

with

Am

^{= pm exp}

^(Nfluc(m))

j

if

fl

>

2uc(m)

(24) Am

_{= pm exp}

(Nfluc(m))

+ _exp

(N(log

_qm +

(fl/2)~))

,

if

fl

<

2uc(m)

and

Bm

^{= nm exp}

lNfltLc(m)I

i

if

fl

>

tLc(m)

Bm _{= nm exp}

I)('Og

qm +

fl~)I

,

if

fl

<

tLc(m)

~~~~

Here qm is

a random number of order _one which describes the fluctuations in the

density

of states _as

already

mentioned. The

origin

^of _pm is somewhat different.

They

_are due to

fluctuations in the values of the lowest

lying

states: The

ground

state energy

(internal

_energy

in this

case)

Uo has order _one

sample

to

sample

fluctuations

[12,

13] so

U)YP(m)

_~

N(uc(m)

+

(IN)

with

(

_an order _one random

numb<r

_so

eP~°

~

pe~P"<(~')

j

where _p is

apositive fluctuating

number.

We _see in this _way that both in A and B the

ground

state contributions _are

non-analytic (due

to

disorder),

and therefore

non-integrable by

the steepest ^{descent method. The}

only

analytic

contribution is the

saddle-point

term in

(24).

Let _{us now} turn to the evaluation of the _sum in

(22).

We define

flc(m)

_as

flc(m)

₌

21tc(m) (26)

and its inverse

mc(fl)

such that satisfies

i 1/2

fl/2

₌

log

2

[(1+ mc) log(I

+

mc)

₊

(I _mc) log(I

-m

)]j (27)

2

The relation

hc(fl)

₌

atanh(mc(fl))

defines the critical field

hc

which is the limit between the frozen and

paramagnetic phases

of REM with field [12]. ^{This critical field satisfies}

fl/2

=

Jog

2 +

log(cash(hc)) hc tanh(hc)]~/~ (28) Using (26)

_{we can} rewrite the condition

fl

_>

2uc(m)

_{as (m( >}

mc(fl)

and the condition

fl

_>

uc(m)

_{as (m(} >

mc(2fl).

The _sum in

(22)

can now be written

Z~YP

= A+ B

(29)

with

A =

£

_Am

_{(fl)e~~'~ (30)}

m

and

B ₌

£ Bm(fl)e~~'~ (31)

We then have

A = A~ + A~ ₌

£ p~eNlPu<(m)+mhl

+

£ eNj(p/2)?+iogq~+mhj

~~~~

-ismsi jmj<m~(p)

B ₌ B~ +

B~

₌

£ ~~eNlPu<(m)+mhl

+

£ ~~e+lP~+'Ogqm+2mhj

_~~~~

lml>mc(2p) jmj<m~(2p)

The

only analytic

contribution is

A2,

^which will be estimated

by

_means of the steepest ^descent method. We will

ignore

Bi ^{because it} is

already

^included in

AI

Let _us first discuss the

evaluation of the

fluctuating

contributions Al and 82. The

non-analyticity

of the

integrand

makes it

impossible

to _use the steepest descent method. On the other

hand,

_as

already

discussed in section II, the term with the greatest ^{modulus will dominate the} sums so we have

(AI

~-

_n~~~

^exP

lNlfl(logqm)~/~

+

mhill (34)

and

(82

_'~ mm exp

( ) ~d~(log qm)

+

2mhi] ) (35)

(m( <m<(2p) It is worth

noticing

that

(82(

₌ 0 if

fl

_>

flc/2 (36)

On the other

hand,

if

fl

<

flc/2,then mc(2fl)

_> 0 and there exist two ranges of

hi

in which 82

gives

different contributions:

The maximum of

[82(

is at iii

=

tanh(2hi)

when

(l(

_<

mc(2fl)

^and at

+mc(2fl)

when [iii) >

mc(2fl).

This _can be

writtenj using (27)

^and

(28)

_as

exp ⁽

~

~d~ +

log

2 + lo

g(cosh(2hi))])

,

if hi <

hc(2fl) [82(

'~

~ ~

(37)

exp

(N

_§Y~ +

[hi (mc(2fl)])

,

if

hi

>

hc (2fl)

Now _we discuss the evaluation of

AI

The maximization condition is satisfied _at fh ₌

tanh(I)

where

I

_is

a function of

hi

and

fl

that satisfies

(

=

(log

^{2 +}

log cosh(I) ^I tanh(I)j

^~~~

⁽³⁸⁾

If _we put

I

=

hc($)

with

$

=

all

,

now the unknown is

a(fl, hi)- Replacing

this in

(38)

_we find

I

=

ahi,

_so the

following

set of

equations:

1#/2 _#h

⁼

^Jog

^{2 +}

^{log cosh(I) I tanh(I)]}

^{~/~ ~~g~}

=

PI

is

equivalent

to

(38).

It is _easy to _see that when hi ₌

hc(fl)

_we obtain _a

= I while _{a <} I if

hi

>

hc(fl)

and _{a >} I if

hi

_<

hc(fl),

_so

⁽

^will

always

be between

hi

and

hc(fl).

The term

AI

will then have the

following

form

jAi1

_~ eXp

jNj(

_{+ hi}

_{tanh(i)jj (40)}

The calculation of A2 _may ^{be done}

by

the steepest ^{descent method}

(see

the

appendix

for

details)

for

large

N. It follows that

again

two different

expressions

_are

found,

_now

depending

on the values of h and

fl.

A smooth _arc is found in the

complex

h

plane

which divides these two

possibilities.

This _arc touches the real axis at

hc(fl)

and the

imaginary

^axis at h ₌

ix/2.

In the inner part of this _{arc we} find

A2

'~ exp

N( ^~

+

hmc(fl))

_{+ exp}

N( ^~

^{~ ~~} +

log(cosh(h))) (41)

and in the _outer side

A2

~ eXp

Nj(

+

hmc(fl)j (42)

It is clear that Z will in

general

be the _sum of several _terms of the form e~W with

~9 ^a

complex function,

_so for

large

N

just

that term with the greatest ^{modulus will be relevant. We then} have to

determine,

in each

region

of the

complex

h

plane,

the term with the greatest ^real part of _~9. This will

give

rise to the appearance of _a

phase diagram

for the model in

complex

h.

In what follows _we will be

only

interested in the determination of the modulus of Z,

which,

_as

we shall _see, is

enough

to calculate the

density

of _zeros of the model. If _we write Z

[=

_exp

N#,

the different contributions _we have to compare are

11

_" ^~~

~

+

log cosh(h)

^for

fl

< flc ^and

[hi

<

h(~~(h2, fl)

4~

₌ +

[hi jm~(p)

for

p

< p~ and

vhi

#~

= +

hi tanh(I)

_{Vfl and}

_Vhi

#4

_" +

)

+

log(cash(2hi))

for

fl

< and

[hi

<

(hc(2fl)

#5

= fl~ +

[hi tanh(hc(2fl))

for

fl

<

'

and

[hi

>

h~(2fl)

Three different _ranges of temperature can be identified:

a)

If

fl

> flc then

only

#3 exists _so ~°~ ~

=

#3 ,Vh.

b)

If flc >

fl

_>

flc/2

_we get contributionsN

from11, 42

and

#3.

In this _case two different

regions

are found in the

complex

h

plane (Fig, I,

_upper

portion), according

to what term is the most

important.

These

regions

are

separated by

an arc which _goes from

(hc(fl),0)

_on the real axis to

(0, h[(fl))

_on the

imaginary

axis. In the inside of this _arc

ii

dominates while in the outside it is

#3

which is the most

important. #2

is _never relevant.

c)

^If

^fl

<

flc/2

then also

#4

and

#5 contribute,

and

they

have to be

compared

to

ii

and

#3

in their

respective

_zones. It is found that

#5

is _never

relevant,

I-e-

(11 45)

and

(#3 Is)

are

never

negative.

On the other band

#4

is

greater

than

#3 (for hi

_<

hc(2fl)

and it is also

2

greater

^than

ii

in the _upper

portion

of the _zone

where11

>

#3,

_so the line where

#4 equals ii

is lower than that where

#3 equals ii

The situation in _case

c)

^is that

depicted

in the lower

portion

of

figure

I. The _arc

separating

#i

and

#4

_now touches the

imaginary

axis at a

point (0, h((fl))

which results from

equating #4

to

ii

for

hi

_" 0 and satisfies

p2 ~p2

CDS(hl(fl))

= e~~ ^~ 8

(43)

so we can see that

h)(fl

=

0)

₌

x/4.

w/2

~~

fl = 0.75 flc

«/4 ~

@~

w/2

4,

_fl

= o.30 j

4l3

«/4

#~ hi

o-o

o-o 1.o 2.o 3.o

Fig. 1. Shown _are the phases of the model in the complex field plane for _two different values of the temperature. For Tc _< T _< 2Tc

(upper figure)

two distinct _{zones appear} separated by an arc of _zeros.

Above this _arc

(phase 3)

densely distributed _{zeros are} found whfle the region below it

(phase I)

is free

of _zeros. For T _> 2Tc the outer part of the _{arc appears} divided in two _zones by _a vertical line, which

is not _a line of

zeros. The densities of _{zeros are} different to the left

(phase

4) ^and

right (phase

3) of

this vertical line.

4.

Density

of

complex

field _zeros of REM.

Having

^obtained

#

₌ ^{~°~ ~} in the various

phases

of the

model,

_{we can} calculate the

density

N

of _zeros

p(h)

of Z

by

_means of the formula [13]

1

32 32

~~~~

2x~~~

2x

3h]

~

3hj) ^~~~~'~~~

^~~~~

which is _a consequence of the electrostatic

analogy

_{[2] that} identifies

p(h)

with _a

charge density

and

#

with the

resulting

electrostatic

potential.

The

density

of _zeros

associated

to

ii

is null

(because

the real part of _an

analytic

function has _a

vanishing Laplacian).

The densities of _zeros

corresponding

to

#3

and

#4

_are

respectively

p2

p~ =

-i7~#3

=

a(I tanh~(ahi))

~ ~

(45)

2x fl2 +

2hi(1

tanh

(ahi))

p4 "

i7~#4

_"

2(1- tanh~(2hi)) (46)

where _a

=

ilhi

as

already

defined.

In addition to these distributed _zerosj there exist other _zeros located

right

_on the lines divid-

ing

the different _zones, and which

density

is

proportional

to the

discontinuity

in the normal component of the '~electric field" _across that line. On _a line

separating

two

generic phases

_a

and b _we then have

Pt~

₌

) _{it(4a -16) (47)}

The

density

of _{zeros on} line 1-3 is found _to be

~~~

x~~~~~~~~~ osh(i~~~~~s(2h2)~~

~

cosh(i~~~~~s(2h2)~~~

~~~~

On line 1-4 the

expression

is the _same with the

only

difference that

I (I

=

ahi)

is

replaced by 2hi

Their zero-densities _are

equal

where lines 1-3 and 1-4 meet.

The line

3-4,

which is the vertical at

hi

_"

jhc(2fl)

^{is not a line of zeros, I.e. the electric}

field is found to be continuous there. On the _other band the

density

of distributed _zeros

jumps

as one crosses line 3-4

by

_a factor

(1+ 2(hi/fl)~(l tanh~(2hi))).

The

density

of _{zeros on} the line 1-3 is found to be null at the

point

where that line touches the real h

axis,

in

agreement

^{with the second order character of the} field-driven transition of the model.

5 Conclusions.

We have estimated the behavior of the

partition

function of REM in

complex

field. Three different

phases

_appear:

For T _< Tc there is

only

_one

phase

which _we called

phase

3 and is the continuation to

complex

field of the frozen

phase

of the model. For Tc < T the

paramagnetic phase

is also present, ^which we call

phase

I

(Fig. I,

_upper

portion). Finally

for T _> 2Tc what _we called

phase

4 also _{appears at}

complex

values of h

(Fig. I,

^lower

portion).

There is _no

physical (I.e.

for real values of

h)

counterpart ^{for this}

phase.

We have also calculated the densities of _zeros in the different

phases.

A

particular

structure has been revealed: The _{zeros are} dense _near the real axis in the frozen

phase.

This characteristic

of the _zeros distributions is

intimately

related to the

properties

of

spin glasses

^{and has} been

discussed

by

Parisi

Ill], although

_no

explicit

calculation had been done before. A numerical estimate of the

complex

field _zeros of REM has been done [17] ^{and shows}

good

coincidence [18] ^{with the results here}

reported.

The existence of distributed _zeros is in this model related to disorder in

a very clear _way:

The

fluctuating

factors _pm and _{qm are}

responsible

for the appearance, in

Z,

of

nonanalyticities

of the sort which _{are necessary} to obtain _{a non-zero}

Laplacian

and

consequently

distributed

zeros.

Acknowledgements.

We thank B. Derrida for

sending

_us ^his results

prior

to

publication.

Appendix.

Here _we describe the evaluation ofA2

by

_means of the steepest ^{descent method} [16].

Starting

from

A2

_"

~~~~~

dm

e~'(~') (Al)

-mc(P) with

~(m)

= mh +

log

2 [(1 +

m) log(I

+

m)

+

(I m) log(I m)]

+

~ (A2)

2

~

we see that

A2

is _non-zero

only

for

fl

<

flc,

^otherwise

mc(fl)

₌ 0 and this _term does _not

contribute.

Following

the usual

procedure

for

e~,aluating

this kind of

integrals

in the

complex plane,

_we

stretch the

integration

contour to

infinity following paths

of steepest ^{descent for the real} part of it The

imaginary

part ^{of the}

integrand

will in this _way be constant

along

the whole

pathj

and _may be taken _out from the

integral.

The level _curves of it _are

topologically

similar to those of z~. _Two steepest ^{descent lines} cross each other _at the saddle

point

which is located at

mo "

tanh(h):

_one of them _goes down to the

valleys

to the left and

right

of the

saddle-point

and will be called AA'while the other climbs towards the mountain tops ^and is the line LL'.

The way in which the

original integration path

has to be deformed

depends

_on the relative

positions

of +mc and LL'.

Depending

_on

fl

and

h,

two situations _are

possible:

a)

^When the

integration

limits

+mc(fl) lay

_on

opposite

sides of

LL',

there

are contributions from three

paths:

the _one that _runs

through

the

saddle-point (path AA')

j

and those

ending

at

+m~(fl).

b)

When both

integration

limits _{are on} the _same side of

LL',

the

path

that

joins

the

valleys through

^the

saddle-point

does not contribute _to the

integral.

The limit condition for this two _cases is that

mc(fl) lays exactly

_on

LL',

which in turn is

equivalent

to the condition that

Imag(it(mc)) =Imag(it(mo)).

This last

identity

results from the fact that

LL', being

_a steepest ^descent for the real part ^of

it,

is _a level _curve for the

imaginary

part of it. This limit condition is then written

t~nh(hi)

_"

~~~~~~~~~~ _~~~~~~~~~

~~~~

and for

fl

_{< flc} it defines _a smooth _arc in the

complex

h

plane (with hi

and h2

Positive)

that

joins

the

points (hc(fl), ₀₎

and

(0, x/2).

The

shape

and size of this _arc

depend

on

fl.

In its

inner part ^situation

a)

is valid _so for

large

hi

2 +~ ex

~ ~

~ _~ ~ ~

~'~C(fl))

+ _exp

Nj ^fl

⁺

fll

4 ~

'°g(cosh(h)))

while in the outer side of this _arc

b)

holds and

A2

~ exp

N( ^~~

₊

hmc(fl)) (A5)

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