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Simple frustrated systems : chains, strips and squares
Bernard Derrida, J. Vannimenus, Y. Pomeau
To cite this version:
Bernard Derrida, J. Vannimenus, Y. Pomeau. Simple frustrated systems : chains, strips and squares.
Journal of Physics C: Solid State Physics, Institute of Physics (IOP), 1978, 11 (23), pp.4749-4765.
�10.1088/0022-3719/11/23/019�. �hal-03285942�
J.
Phys.
C:
Solid State Phys
,
Vol
11,
1978. Printed in Great Britain.
0
1978
Simple
frustrated systems
:
Chains, strips and squares
B
Derridaf,
J
Vannimenus: and
Y
Pomeauf
+
Institut Laue-Langevin,
BP
156. 38042 Grenoble Cedex, France
1
Laboratoire de Physique de 1’Ecole Normale Superieure, 75231 Paris Cedex 05, France
5
DPpartement de Physique Theorique,
CEN
Saclay, BP
2,
91 190 Gif sur Yvette, France
Received
8
May 1978
Abstract. The ground-state properties of several systems showing the Toulouse frustration
effect are investigated analytically. First. the energy and entropy of the random-bond king
chain in a uniform field are obtained for all concentrations of antiferromagnetic bonds, using
the transfer-matrix method. The susceptibility has discontinuities for an infinite number of
critical values
of
the field. where the entropy shows spikes in addition
to
discontinuities. We
relate these effects to the physics of frustration. The second system studied consists of
frustrated strips, which we argue are the proper one-dimensional limit of spin glasses. Two
kinds
of
strips are considered and the results are compared with recent numerical works
on
two-dimensional spin glasses and with the exact results for random (3
x
3) squares.
1.
Introduction
A
new fundamental concept, frustration, was recently introduced to interpret the
remarkable properties of such disordered systems
as
spin glasses (Toulouse 1977).
As
emphasised by Toulouse, it is necessary to study the frustration effect in its own right, at
first on simple cases unobscured by all the intricacies of real spin glasses. Like percolation,
its older counterpart, the frustration effect raises a large number of questions: What are
the numerical values for the critical concentration of competing bonds? How can one
define critical exponents? What scaling laws do they obey? How do they depend
011
spin
and lattice dimensionality? Up to now, most work on the problem has relied on numerical
methods (Bray
et
~111978, Kirkpatrick 1977, Vannimenus and Toulouse 1977). It appears
very difficult to find solvable models other than the completely frustrated ‘odd model’
(Villain 1977).
The natural idea is to look at one-dimensional models, but
a
chain of spins with
randomly ferromagnetic and antiferromagnetic bonds
is
not frustrated.
A
simple change
of variables makes the problem trivial and this approach looks unprofitable at first sight.
The same situation exists in percolation theory for a chain with randomly cut bonds,
since one cut suffices to disconnect the chain. To obtain an interesting one-dimensional
problem, one has to modify the original problem, for instance by introducing
a
‘ghost site’
(Reynolds
et
a1
1.977). In this way one obtains critical exponents in one dimension, which
verify the scaling laws for percolation. In the present paper we follow a similar line
of
thought, and study some models which are essentially one-dimensional and still exhibit
non-trivial frustration effects.
0022-3719/78/0023-4749
$03.00
@
1978 The Institute
of
Physics
4749
4750
B
Derrida,
J
Vannimenus and
Y
Pomeau
The first model we consider is the random-bond chain of Ising spins in a uniform
magnetic field, at zero temperature and for an arbitrary concentration of antiferromagnetic
bonds. At a series
of
field strengths, it becomes favourable to flip well-defined spin
clusters and the magnetisation jumps suddenly. In general, there are several equivalent
choices for the spins to flip, because it is not possible to satisfy ev.ery bond at the same time.
This is a typical frustration effect and
it
gives rise to jumps in the entropy of the ground
state. The same qualitative features appear in another related model we have studied, the
ferromagnetic chain in a random magnetic field.
Next, we consider two random chains of Ising spins coupled by random bonds, with
periodic boundary conditions. One can analyse this system in terms ofa strip
of
frustrated
plaquettes, following Toulouse (1977). Indeed, such a strip is the real one-dimensional
version of the frustration problem, and we compute its energy and entropy for two
different assumptions on the bond distribution. Actual calculations are rather intricate
and we have only performed them at zero temperature and without external field.
It is tempting to study several coupled chains and see whether any indication of a
singularity in the entropy of the
2D
frustrated model shows up for
a
small system. Unfor-
tunately, our methods lead to prohibitively long calculations,
so
we turned to the simpler
case
of
periodic frustrated squares. We present the results for the ground-state energy
and entropy of the
3
by
3
square, which give some insight into the behaviour of the
2D
model. Here again a lot of work is needed to proceed to the 4 by 4 square, and these
difficulties seem inherent to disordered frustrated systems.
2.
Ising
chains
2.1.
Thc random chain in a uniformfield
The model is defined by the Hamiltonian:
x
=-
pi,i+lSiSi+l
-
HISi
where the exchange integral
Ji,
i+,
equals
J
with probability
(1
-
x)
and
-
J
with prob-
ability
x,
and the bond disorder is quenched. The problem has already been studied
by
several authors, mostly by integral equations or numerical methods (Matsubara 1974,
Landau and Blume 1976, Fernandez 1977), but to our knowledge the analytical expres-
sion for the magnetisation at arbitrary
x
and
H
is not available. The behaviour of the
entropy, which is important to understand frustration effects, has not been investigated.
To get some physical understanding of what happens, let us examine
a
group
of
three
successive antiferromagnetic bonds in an otherwise ferromagnetic chain.
If
H
is larger
than
25,
all spins must point up.
If
it is smaller, either
of
the spins surrounded by two
negative bonds flips in order to minimise the total energy, but it is not favourable to flip
both spins,
so
the degeneracy is
2.
To
compute the total entropy
of
the disordered chain,
one must take into account all possible configurations. The direct approach is not practi-
cal, however, and one has to use the transfer-matrix method. The difficulty then comes
from the fact that one has to consider several non-commuting matrices, and study the
product
of
a large number of randomly chosen matrices. For the present problem, the
two transfer matrices are:
/,-l+a ,l+a
\
Ml=(;l-a
z-
'
1
-a
)with probability
x
Simple frustrated systems
:
Chains, strips and squares
475
1
with probability
(1
-
x)
zl+a
Z-l+z
M2
=(
z-l-a
Z1-a
where
z
=
exp(J/kT) and
H
=
aJ.
For very low temperatures, it is possible to follow the
evolution of the dominant term (in powers of
z)
under successive matrix products; this
yields the free energy directly. Since the calculations are lengthy, we defer details of the
solution
to
Appendix
1,
and only present the results here. The ground-state energy per
spin is given by:
(3)
r2x2
+
rx(2
-
x)
+
(2x
-
I)
(x
-
I)
+
a(I
-
x)
(2rx
+
I)
(1
+
rx)(l
-
x
+
rx)
E=
-J
and its entropy per spin is
_-
s
-
x(l
-
x)2
2
(
rx2
)"-'inn.
k,
(1
-
x
+
rx)'
1
-
x
+
rx
(4)
In these formulae the integer
r
is defined by
r<2/a<r+1
Both magnetisation and entropy are constant in the intervals corresponding to successive
values
of
r,
and they jump when
H
=
2J/r.
At these critical values of the field, the
arrangement of spins changes and the degeneracy of the ground state is much 1arger.The
entropy is then:
I-x
1+(r-2)x
S/k,
=
1
R,(t)
In
t.
(5)
The summation bears over all points
t
that can be reached from
to
=
2
by the following
non-commuting operations
:
t
+
T,(t)
=
1
+
t
t
+
T2(t)
=
1
+
l/t
with
~~(2)
=
xz(l
-
x)/[l
+
(r
-
1)x]
R,[T,(~)]
=
~,(t)
x2(r
-
1)/[1
+
(r
-
2) x]
R,[T,(t)]
=
R,(t)x(l
-
x)/[l
+
(r
-
2)x].
For
H
=
2J
(I'
=
I),
only
T2
contributes and formula
(5)
reduces to a simple series: its
value is
0.1428
for
x
=
0.5,
much larger than the value obtained from equation
4.
The same kind of generalised series also appears in the strip problem and it seems
likely that this is a general feature
of
problems involving products of random non-
commuting matrices. Necessary details
on
their derivation are given in the Appendix and
here we only present some comments on the results displayed in figures
1
to
4
for
a
few
selected values of the parameters:
(i) All properties are continuous with respect to the concentration
x
ofnegative bonds,
and for
x
=
0.5
the magnetisation
is
in agreement with the low-temperature numerical
results (Fsrnandez
1977),
except in the very low-field region which the numerical method
could not study in detail.
4152
B
Drrriclu,
-1.
Vannimenus
and
Y
Pomeau
X
Figure
1.
Magnetisation of the random-bond Ising chain in a uniform field
H
as a function
of
the concentration
x
of antiferromagnetic bonds at zero temperature. The different curves
correspond to a range
of
values of
H:
2/r
>
H/J
>
2/(r
+
1)
for
r
=
0
(a),
1
(b),
2
(c),
10
(d),
100
(e).
1r
I
M
0.5
ll
A
I
0
1
2
H/J
Figure
2.
Magnetisation as a function
of
H
for different concentrations
x
=
0.2
(ti),
0.5
(b),
0.8(~.).
When
H
tends to zero, the magnetisation
of
the chain shows an infinite number of
discontinuities but the susceptibility has a finite limit (equation
6).
r
X
Figure3.
Entropy
of
the random-bond chain as a function ofx for two ranges
of
values of the
field:
2
z
H/J
>
1
(a);
1
>
H/J
>
5
(b).
The slope dS/dx is infinite for
x
=
1
(equation
7).
Note that at low
x,
the entropy is smaller in a larger field.