We discuss the utility of analytical and numerical investigation of spin
models, in particular spin glasses, on ordinary ``thin'' random graphs (in
effect Feynman diagrams) using methods borrowed from the ``fat'' graphs of two
dimensional gravity. We highlight the similarity with Bethe lattice
calculations and the advantages of the thin graph approach both analytically
and numerically for investigating mean field results.Comment: Contribution to Parallel Session at Lattice95, 4 pages. Dodgy
  compressed ps file replaced with uuencoded LaTex original + ps figure

Ambjorn

Bachas

Baillie

Bethe

Brezin

C.F. Baillie

D.A. Johnston

de Dominicis

Eggarter

Goldschmidt

Inawashiro

Kanter

Mehta

Mezard

Mottishaw

Muller-Hartmann

Penney

Sherrington

Thouless

Viana

Wong

English

arXiv

We discuss the utility of analytical and numerical investigation of spin models, in particular spin glasses, on ordinary "thin" random graphs (in effect Feynman diagrams) using methods borrowed from the "fat" graphs of two dimensional gravity. We highlight the similarity with Bethe lattice calculations and the advantages of the thin graph approach both analytically and numerically for investigating mean field results.</p

Baillie, C. F.

Johnston, D. A.

Heriot Watt Pure

Spin models on thin graphs

We discuss the utility of analytical and numerical investigations of spin models, in particular spin glasses, on ordinary ``thin'' random graphs (in effect Feynman diagrams) using methods borrowed from the ``fat'' graphs of two dimensional gravity. We highlight the similarity with Bethe lattice calculations and the advantages of the thin graph approach both analytically and numerically for investigating mean field results

Baillie, C F

Johnston, D A

CERN Document Server

1Spin Models on Thin Graphs
C. F. Baillie
a
and D. A. Johnston
b
a
Computer Science Department,
Campus Box 430, University of Colorado,
Boulder CO80309, USA
b
Mathematics Department, Heriot-Watt University,
Edinburgh, EH14 4AS, United Kingdom
We discuss the utility of analytical and numerical investigation of spin models, in particular spin glasses, on
ordinary \thin" random graphs (in eect Feynman diagrams) using methods borrowed from the \fat" graphs of
two dimensional gravity. We highlight the similarity with Bethe lattice calculations and the advantages of the
thin graph approach both analytically and numerically for investigating mean eld results.
1. INTRODUCTION
The analytical investigation of spin glasses on
random graphs of various sorts has a long and
honourable history [1,2], though there has been
little in the way of numerical simulations. Ran-
dom graphs with a xed or xed average con-
nectivity have a locally tree like structure, which
means that loops in the graph are predominantly
large, so Bethe-lattice-like [3] (ie mean eld) criti-
cal behaviour is expected for spin models on such
lattices. Given this, the analytical solution for
a spin model or, in particular, a spin glass on
a Bethe lattice [4,5] can be translated across to
the appropriate xed connectivity random lattice.
Alternatively, a replica calculation can be carried
out directly in some cases for spin glasses on var-
ious sorts of random lattices.
A rather dierent way of looking at the problem
of spin models on random graphs was put forward
in [6], where it was observed that the requisite
ensemble of random graphs could be generated
by considering the Feynman diagram expansion
for the partition function of the model. For an
Ising ferromagnet with Hamiltonian
H = 
X
<ij>

i

j
; (1)
where the sum is over nearest neighbours on
three-regular random graphs (ie 
3
Feynman di-
agrams), the partition function is given by
Z
n
()N
n
=
1
2i
I
d

2n+1
Z
d
+
d
 
2
p
detK
exp( S)
where N
n
is the number of undecorated graphs
with 2n points, K is dened by
K
ab
=
 
p
g
1
p
g
1
p
g
p
g
!
(2)
and the action itself is
S =
1
2
X
a;b

a
K
 1
ab

b
 
1
3
(
3
+
+ 
3
 
): (3)
where the sum runs over  indices. The coupling
in the above is g = exp(2J) where J = 1 for
the ferromagnet and the 
+
eld can be thought
of as representing \up" spins with the 
 
eld
representing \down" spins. An ensemble of z-
regular random graphs would simply require re-
placing the 
3
terms with 
z
and a xed average
connectivity could also be implemented with the
appropriate choice of potential.
This approach was inspired by the considerable
amount of work that has been done in recent years
onNN matrix
1
versions of such integrals which
generate \fat" or ribbon graphs graphs with suf-
cient structure to carry out a topological expan-
sion [7] because of the matrix index structure.
1
N , the size of the matrix is not to be confused with n,
the number of vertices in the graph!
2The natural interpretation of such fat graphs as
the duals of triangulations, quadrangulations etc.
of surfaces has led to much interesting work in
string theory and particle physics [8]. The parti-
tion function here is a poor, \thin" (no indices,
so no ribbons), scalar cousin of these, lacking the
structure to give a surface interpretation to the
graph. Such scalar integrals have been used in
the past to extract the large n behaviour of vari-
ous eld theories [9] again essentially as a means
of generating the appropriate Feynman diagrams,
so a lot is known about handling their quirks.
2. (ANTI)FERROMAGNETS AND SPIN
GLASSES
For the Ising ferromagnet on three-regular (
3
)
graphs, solving the saddle point equations at large
n

+
=
p
g
2
+
+
1
p
g

2
 

 
=
p
g
2
 
+
1
p
g

2
+
(4)
shows that the critical behaviour appears as an
exchange of dominant saddle point solutions to
the saddle point equations. The high and low
temperature solutions respectively are

+
; 
 
=
p
g
g + 1

+
; 
 
=
p
g
2(g   1)

1
r
g   3
g + 1

(5)
which give a low temperature magnetized phase.
The critical exponents for the transition can also
be calculated in this formalism and, as expected,
are mean eld. In general a mean eld transition
appears at
exp(2
FM
) = z=(z   2) (6)
on 
z
graphs, which is the value predicted by the
standard approaches. Simulations nicely conrm
this mean eld picture for the ferromagnet [10].
Analysis of the Binder's cumulant for the magne-
tization also shows that the critical temperatures
are identical to the corresponding Bethe lattices
(ie g = 3 for 
3
graphs). The specic heat is
shown in Figure.1 for various sizes of 
3
graphs.
Figure 1. The specic Heat for the Ising ferro-
magnet on 
3
graphs of various size.
There are various possibilities for addressing
spin glass order in the Feynman diagram ap-
proach. In [6] the entropy per spin was calculated
for the Ising anti-ferromagnet on 
3
graphs and it
was found to become negative for suciently neg-
ative , which is often indicative of a spin glass
transition. Simulations again conrm the picture.
Taking a quenched distribution of couplings of the
form
P (J) = p (J   1) + (1  p) (J + 1); (7)
which gives the antiferromagnet for p = 0, pro-
duces results for the spin glass order parameter,
the overlap, that are very similar to the in-
nite range mean eld (Sherrington-Kirkpatrick)
model. Dening the overlap as
q =
1
n
X
i

i

i
: (8)
with two Ising replicas on each graph, 
i
; 
i
and
histogramming
P
n
(q) =
h
h(q   1=n
X

i

i
)i
i
; (9)
where [ ] denotes the quenched disorder average,
we get the distribution shown in Figure.2 [10,11]
3in the putative spin glass phase at low tempera-
ture. The long tail stretching down to q = 0 is
characteristic of the mean-eld spin glass picture
of many inequivalent states.
Figure 2. P (q) at  = 1:2 for various p on 
3
graphs of size 500.
It is possible to make some analytical inroads
as well by looking at the solutions to the saddle
point equations for k Ising replicas [11]
~
 =
Z
P (J)

k
K dJ
~

2
(10)
where we have denoted the 2
k
elds that now ap-
pear as
~
. The Hessian for these equations is
analytically calculable for any k
k
Y
m=0

2
Z
P (J) tanh(J)
m
dJ   1

(
k
m
)
(11)
and its zeroes show that the k = 0 transition tem-
perature observed in simulations or calculated by
using the analogy with the Bethe lattice is identi-
cal to the k = 2 transition temperature. This
also occurs in the nite replica version of the
Sherrington-Kirkpatrick model [12], so yet again
the thin graph results are resolutely mean eld.
For three or more replicas one does not see the
continuous transition because a rst order transi-
tion occurs at higher temperature to a replica-
symmetric state. The situation appears to be
rather similar for Q > 2 state Potts models where
the saddle point calculation nds a continuous
transition at one of the spinodal points and misses
a rst order transition occurring at higher tem-
perature. The 3-state Potts model, for instance,
with action
S =
1
2
(
2
a
+ 
2
b
+ 
2
c
)  c(
a

b
+ 
a

c
+ 
b

c
)
 
1
3
(
3
a
+ 
3
b
+ 
3
c
); (12)
gives high and low temperature solutions

a;b;c
= 1  2c ;

a;b
=
1 +
p
1  4c  4c
2
2
;

c
=
1 + 2c 
p
1  4c  4c
2
2
(13)
where c = 1=(g + 1).
Figure 3. P (q) at  = 2:2 for the 3-state Potts
model for 
3
graphs of size 1000 (labelled) and
500 (unlabelled).
4Calculations and simulations for Potts glasses
are just as easy as for the Ising spin glass. In
Figure.3 overleaf we have plotted the distribution
of overlaps for the three state Potts model at low
temperature. The overlap for a Q state Potts
model is now dened as
q =
1
n
n
X
i=1
(Q

i
;
i
  1) (14)
and the lack of an inversion symmetry in the
spins gives a dierent pattern of replica symmetry
breaking in mean-eld theory to the Ising model.
The numerical results are still consistent with a
mean eld picture.
3. Conclusions
In summary, spin models on thin graphs oer a
promising arena for the application of ideas from
matrix models, large-n calculations in eld the-
ory and bifurcation theory. In the spin glass case
the tensor (or near-tensor) product structure of
the inverse propagator allows some quite general
expressions to be derived for the Hessian in the
saddle point equations and oers a powerful line
of attack on questions such as replica symmetry
breaking. As a subject for numerical simulations
they oer the great advantage of mean eld re-
sults with no innite range interactions and no
boundary problems.
The bulk of the simulations were carried out
on the Front Range Consortium's 208-node Intel
Paragon located at NOAA/FSL in Boulder. CFB
is supported by DOE under contract DE-FG02-
91ER40672, by NSF Grand Challenge Applica-
tions Group Grant ASC-9217394 and by NASA
HPCC Group Grant NAG5-2218. CFB and
DAJ were partially supported by NATO grant
CRG910091.
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Spin Models on Thin Graphs

Spin Models on Thin Graphs

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