Random-field p-spin glass model on regular random graphs

0.99 MB·English

by  Yoshiki Matsuda

Random-field p-spin glass model on regular random
1 graphs
1
0
2
Yoshiki Matsuda1,4, Hidetoshi Nishimori1, Lenka Zdeborov´a2
n
a and Florent Krzakala3
J
1Department of Physics, Tokyo Institute of Technology, Oh-okayama,Meguro-ku,
1
3 Tokyo 152-8551,Japan
2 Institue de Physique Th´eorique, IPhT, CEA Saclay, and URA 2306, CNRS, 91191
]
n Gif-sur-Yvette cedex, France
n 3 CNRS and ESPCI ParisTech,10 rue Vauquelin, UMR 7083 Gulliver, Paris 75005
-
s France
i
d E-mail: [email protected]
.
t
a
m Abstract. Weinvestigateindetailthephasediagramsofthep-body J Isingmodel
±
- withandwithout randomfields onrandomgraphswith fixedconnectivity. One ofour
d
most interesting findings is that a thermodynamic spin glass phase is present in the
n
o three-body purely ferromagneticmodel in random fields, unlike for the canonicaltwo-
c body interaction random-field Ising model. We also discuss the location of the phase
[
boundarybetweentheparamagneticandspinglassphasesthatdoesnotdependonthe

1 changeoftheferromagneticbias. Thisbehaviorisexplainedbyagaugetransformation,
v
whichshowsthatgauge-invariantpropertiesgenericallydonotdependonthestrength
3
6 of the ferromagnetic bias for the J Ising model on regular random graphs.
±
8
5
.
1
PACS numbers: 05.50.+q, 75.50.Lk
0
1
1
:
v
i
X
r
a
Random-field p-spin glass model on regular random graphs 2
1. Introduction
Two typical examples of disordered system in statistical physics are the spin glass and
the random-field Ising model. The basic model of the former is the Edwards-Anderson
model [1] with quenched randomness in interactions. The Sherrington-Kirkpatrick
model [2] is the infinite-range (fully-connected) version of the Edwards-Anderson model
andisconsideredtoconstitutethemean-fieldparadigmofthetheoryofspinglasses. The
exact solution of the Sherrington-Kirkpatrick model is now established and is known to
possess quite unusual properties represented by the symmetry breaking in the abstract
space of replicas [3, 4, 5, 6, 7]. The spin glass theory has also been applied to a variety
of problem in other disciplines including information theory [6, 7].
The Edwards-Anderson model on the random graph is closer to real physical
systems than the Sherrington-Kirkpatrick model because the former has a finite
connectivity whereas the latter is fully connected although both are of mean-field
nature. Random systems on the tree-like lattice can be analyzed by the cavity
method [3]. Stability of the replica-symmetric (RS) state as formulated by de Almeida
and Thouless for the Sherrington-Kirkpatrick model [8] can also be analyzed on the
Bethe lattice [9, 10, 11], and the effects of replica-symmetry breaking (RSB) have been
studied by a generalization of the cavity method[12]. The problem ofspin glasses on the
finitely-connected sparse graphs in general has also been a useful platform to formulate
random optimization problems [7].
As for the random-field Ising model (RFIM), the infinite-range version has been
solved by Aharony [13]. This model has ferromagnetic and paramagnetic phases but
no spin-glass phase. The boundary between the two phases represents a second-order
phase transitionforsmall values ofthe fieldstrength andafirst-order transitionforlarge
values if the distribution of random fields is bimodal. The RFIM on a finite connectivity
graph also shows a first-order phase transition for larger values of connectivity than
three [14, 15]. Forthe three-dimensional RFIMthe replica field theory hassuggested the
existence of a replica-symmetry-broken (RSB) phase (spin-glass phase) in the vicinity
of the phase boundary [5, 16, 17, 18, 19, 20, 21, 22, 23]. However, nonexistence of the
spin-glass phaseintheRFIMwith two-bodyinteractions hasbeen proved recently forall
latticesandfielddistributions [24], andtheproblemoftheexistence ofaspin-glassphase
in the RFIM has thus been solved negatively as long as the interactions are two-body
and purely ferromagnetic.
It is therefore interesting to study in more detail the interplay of randomness in
interactions and fields on the random graph, in particular in the presence of many-
body interactions. Our numerical results show that the phase boundary between the
spin glass and paramagnetic phases does not change if we vary the ferromagnetic
bias ρ, the probability that an interaction on a bond is ferromagnetic in the J
±
model. This has been known for the J Ising model without field on the regular
±
random graph as a horizontal paramagnetic-spin glass boundary on the ρ-T phase
diagram. Our result generalizes this knowledge to phase diagrams on the H-T
Random-field p-spin glass model on regular random graphs 3
plane and the ρ-H plane, where H is the magnitude (strength) of symmetrically
distributed random fields. This ρ-independence can be understood by a simple gauge
transformation. We also show that the RFIM has a thermodynamic spin-glass phase
if it has three-body even purely ferromagnetic interactions. Thus the proof for the
absence of a spin glass phase in the RFIM [24] cannot be generalized to more than
two-body interactions. This result reinforces previous findings on the glassy behavior of
ferromagnetic systems [25, 26, 27, 28].
2. Formulation
In this section, we formulate the J Ising model in random fields on the regular random
±
graph. We review the cavity method, which enables us to study infinite-size systems
numerically and analytically.
2.1. Cavity method
Weconsiderthe J Isingmodelinrandomfieldswithp-bodyinteractionsontherandom
±
lattice having connectivity c (figure 1). The Hamiltonian is
N
= J S S S H σ S , (1)
H − ij1···jp−1 i j1··· jp−1 − i i
hi,j1X,···,jp−1i Xi=1
where σ is randomly distributed as
i
1 1
P (σ ) = δ(σ 1)+ δ(σ +1) (2)
σ i i i
2 − 2
and J follows
ij1···jp−1
P J = ρδ J 1 +(1 ρ)δ J +1 , (3)
J ij1···jp−1 ij1···jp−1 − − ij1···jp−1
where 1/2 ρ (cid:0)1. (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
≤ ≤
k
1
u
k
u
j
j
k 1
2
l i
1
j
2
u
l
l
2
Figure 1. The local structure of the three-body (p=3) Ising model on the tree with
the connectivity c=2.
Random-field p-spin glass model on regular random graphs 4
Our definition of the regular random graph is a randomly connected graph with
fixed connectivity c [12, 29, 30]. In the thermodynamic limit, the regular random graph
can be identified with the Bethe lattice with proper boundary conditions, because the
local structure is tree-like (figure 1) and the typical length of the loop on the lattice
diverges as logN, where N is number of sites. In this situation we can apply the cavity
method. To fix the notation as well as for completeness of presentation, we present a
short review of the cavity method for the generalized Ising model on a locally tree-like
lattice.
The effective field at site i is calculated iteratively by a partial trace from c 1
−
spins connected to the current site. The independent effects of c 1 interactions have
−
been passed to the current site i in terms of the cavity field h and cavity biases u ,
i j
{ }
the definitions of which are given by
c−1 p−1
1
h = σ H + tanh−1 tanh βJ tanh(βh ) (4)
i i β ij1···jp−1 jk
!
j=1 k=1
X (cid:0) (cid:1)Y
c−1
= σ H + u (J ,h ), (5)
i j ij1···jp−1 jk
j=1
X
and
p−1
βu = tanh−1 tanh βJ tanh(βh ) , (6)
j ij1···jp−1 jk
!
k=1
(cid:0) (cid:1)Y
where β is the inverse temperature, j labels c 1 interactions and k labels p 1 spins
− −
per interaction on the layer previous to the current site i (figure 1).
After very many steps of these update rules, the effective field is obtained as
c
h(c) = σ H + u (J ,h ), (7)
i j ij1···jp−1 jk
j=1
X
and the marginal probability that site i takes S = 1 is
i
±
eβh(c)Si
χSi = . (8)
2coshβh(c)
The Bethe free energy f is calculated from the sum of the site and the bond terms
[31, 32, 33] as
βf = (c 1)log 2coshβh(c)
− − −
c
+ log Trexp(cid:2)βJ S S(cid:3) S + ph S . (9)
p (cid:20)Si i1···ip i1 i2··· ip i i i (cid:21)
(cid:0) P (cid:1)
If we neglect the possibility of RSB for the regular random graph, the distribution
of the cavity field at the lth step Pl(h) is updated as [7]
c−1 c−1p−1
Pl+1(h) = δ(h σ H u J ,h ) Pl(h )dh ,(10)
− i − j ij1···jp−1 jk jk jk
" #
Z Xj=1 (cid:0) (cid:1) J,H Yj=1kY=1
Random-field p-spin glass model on regular random graphs 5
where [ ] denotes the average over the random interactions J and fields σ .
··· J,H ij1···jp−1 i
If the distribution of the cavity field converges, the P (h) = lim Pl(h) satisfies the
l→∞
following self-consistent equation,
c−1 c−1p−1
P (h) = δ(h σ H u J ,h ) P (h )dh ,(11)
− i − j ij1···jp−1 jk jk jk
" #
Z Xj=1 (cid:0) (cid:1) J,H Yj=1Yk=1
and the distribution of the effective field is derived from
c c p−1
P(c)(h) = δ(h σ H u J ,h ) P (h )dh .(12)
− i − j ij1···jp−1 jk jk jk
" #
Z Xj=1 (cid:0) (cid:1) J,H Yj=1kY=1
Sometimes the RS solution is not correct and we need an RSB ansatz, which is
much more complicated even for the one-step RSB (1RSB) solution. In order to draw
phase diagrams of the present model, we use the 1RSB cavity method with the Parisi
parameter m = 1 (to be distinguished from magnetization) [34, 35]. To this end, we
define the message ψSi(h ) = eβhiSi/2coshβh which denotes the marginal probability
i i
of S = 1 in the absence of the spin of the next step. The joint probability distribution
i
±
of messages of the RS and 1RSB (m = 1) ansatz is updated as [34, 35]
P ψSi′ ,ψSi′ = δ ψSi′ F ψSj′k δ ψSi′ F ψSj′k
Si 1RSB RS 1RSB − { 1RSB} RS − { RS }
(cid:16) (cid:17) {XSjk}Z h (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
c−1p−1
S′ S′ S′ S′
w( S S ) P ψ jk ,ψ jk dψ jk dψ jk,
· { jk}| i J,H Sjk 1RSB RS 1RSB RS
i Yj=1kY=1 (cid:16) (cid:17)
where
exp βS J S ψSj′k=Sjk
i j ij1···jp−1 k jk j k RS
w( S S ) = , (13)
{ jk}| i e(cid:16)xp βPS J Q (cid:17)SQ Q ψSj′k=Sjk
{Sjk} i j ij1···jp−1 k jk j k RS
(cid:16) (cid:17)
F ψSjk = P {Sjk}exp βSPi j Jij1···jp−1Q kSjk +QβQHσiSi j kψSjk .
(cid:0){ }(cid:1) PSi {Sjk}ex(cid:16)p βSPi jJij1···jp−Q1 kSjk +βHσi(cid:17)SiQ Qj kψSjk
(cid:16) (cid:17)
The initial condition for distPributPion P is P Q Q Q
Si
PSiniit ψ1SRi′SB,ψRSi′S = P ψRSi′S δ ψ1SRi′SB −δSi′,Si (14)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
In order to identify the 1RSB spin glass phase boundary, we define the complexity
function Σ as the difference between the free energies calculated with ψ and ψ ,
RS 1RSB
Σ = βf +βf m , (15)
RS 1RSB( =1)
−
where f is obtained from equation (9). On the other hand, f m is calculated
RS 1RSB( =1)
from [34, 35]
βf m = βfsite βflink , (16)
− 1RSB( =1) − 1RSB − 1RSB
Random-field p-spin glass model on regular random graphs 6
(c) S′ S′
where, by using the probability distribution of the marginal P χ i ,χ i ,
Si 1RSB RS
(cid:16) (cid:17)
βfsite = (c 1) χSi′=Silog 2coshβh(c) P(c) χSi′ ,χSi′ dχSi′ dχSi′
− 1RSB − − RS Si Si 1RSB RS 1RSB RS
XSi Z h i (cid:16) (cid:17)
(17)
c
βflink = log Trexp(βJ S′′S′′ S′′ +β ph S′′) w′( S )
− 1RSB p S′′ i1···ip i1 i2 ··· ip i Si i { i}
X{Si}Z (cid:20) h i P i (cid:21)J
p
S′ S′ S′ S′
P ψ i ,ψ i dψ i dψ i (18)
· Si 1RSB RS 1RSB RS
Yi=1 (cid:16) (cid:17)
βh(c) = S′tanh−1 2χSi′ 1 , βh = S′tanh−1 2ψSi′ 1 (19)
Si i 1RSB − Si i 1RSB −
exp(cid:16) βJ S (cid:17)S S +β ph S(cid:16) ψSi′=Si(cid:17)
w′( S ) = i1···ip i1 i2··· ip i Si i i RS . (20)
{ i} {Si}e(cid:0)xp βJi1···ipSi1Si2···Sip +Pβ pi hS(cid:1)iSQi iψRSi′S=Si
Actual calculations tPo identify(cid:0)the phase boundaries aPre descri(cid:1)bQed in the next
subsection by using the cavity and effective field distributions.
2.2. Numerical implementation
The cavity methodisimplemented numerically intermsof thepopulationmethod which
realizes the probability distributions, P(c)(h), P (u) and P (ψSi′ ,ψSi′ ) by a large
{ Si 1RSB RS }
amount ofrepresentative samples. Thenumber ofvariablesrepresenting thedistribution
we use is N = 106.
pop
There is an ambiguity in the initial condition of the RS iteration equation. As
typical initial conditions, we have used the following two conditions for the RS cavity
method.
(i) Ferromagnetic: all surface spins being in the up-state, which corresponds to
uinit = J .
j ij1···jp−1
(ii) Free: all surface spins being indeterminate as uinit = ǫ where ǫ 0. This initial
j ± →
condition induces the paramagnetic solution of equation (5).
Theseconditionspossiblyderivedifferentconvergentdistributions. Anexampleisafirst-
order phasetransition. Inthiscase, weshould chooseanappropriateinitial conditionfor
the iterative solution of equation (11) to find the correct solution having the lowest free
energy among all solutions of equation (11). Then the other solutions are metastable
states due to the nature of the first-order phase transition.
We have performed at least 10,000 updates of the cavity iteration until the
distributions converge. After that, we have measured the physical values from the
average of additional 20,000 iterations.
Physical values characterizing the ferromagnetic and spin-glass phases are
numericallycalculatedusingthepopulationmethod. Thedisorderaveragedspontaneous
Random-field p-spin glass model on regular random graphs 7
magnetization is given by
∂f
m = lim = dhP(c)(h)tanh(βh), (21)
−H0→0(cid:20)∂H0(cid:21)J,H Z
where H is an additional uniform external field. The staggered magnetization is also
0
calculated as
∂f
(c) (c)
mˆ = = dh P (h) P (h) tanh(βh), (22)
− ∂H + − −
(cid:20) (cid:21)J,H Z (cid:16) (cid:17)
(c)
where P denotes the distributions of the effective field having positive and negative
±
signs of external field, respectively. The distribution of the cavity field satisfies
P(c)(h) = P(c)(h)+P(c)(h). (23)
+ −
The onset of a spin-glass phase based on the local stability of the RS ansatz is
signaled by the divergence of the spin-glass susceptibility defined as
2 2 2
∂ S ∂ S ∂u ∂u
χ = h i0 = h i0 G 0 , (24)
SG
∂h ∂u ∂h ∂u
G "(cid:18) G (cid:19) #J,H G "(cid:18) 0 G(cid:19) (cid:18) G(cid:19) #J,H
X X
where the subscript G denotes the index from a fixed site 0 of the graph to site G along
the unique path. The divergence of χ is caused when
SG
2 2
∂ S ∂u ∂u
lim (c 1)G h i0 G 0 = 0. (25)
G→∞ − "(cid:18) ∂u0 ∂hG(cid:19) (cid:18)∂uG(cid:19) #J,H 6
The essential part of thedivergence is (∂u /∂u )2 . This is implemented asaneffect
0 G J,H
of a small perturbation to the cavity bias in numerical calculations [36, 37, 38, 24, 30].
(cid:2) (cid:3)
The numerical evaluation of the factor ∂u /∂u is straightforward,
0 G
∂u u (u +∆u ) u (u )
0 0 G G 0 G
− , (26)
∂u ≈ ∆u
G G
where ∆u is a small perturbation which we set to 10−4. We prepare two replicas
G
of identical convergent population of u and introduce a uniform perturbation
i
{ }
into only one of the two replicas. Then we observe the square average of the
variation, (1/N ) Npop(u (u +∆u ) u (u ))2, after 10,000updates of the iteration
pop i=1 i G G − i G
equation (6) with the same bond and site randomness for the two replicas.
P
The 1RSB spin-glass phase is signaled by the complexity function defined in
equation (15). With temperature decreasing, the complexity function Σ would jump
from zero to a positive value at the dynamical 1RSB transition temperature and become
zero at the equilibrium 1RSB transition temperature [34, 35]. We have calculated this
value for three-body (p = 3) systems as shown in the next section.
3. Phase diagrams
The results of the numerical implementation described in the previous section are shown
here in terms of the phase diagrams under various conditions.
Random-field p-spin glass model on regular random graphs 8
1.0 1.0
0.8 0.8
0.6 0.6
^m ^m
0.4 0.4
p = 2, c = 3 p = 2, c = 4
T=0 T=0
0.2 T=0.1 0.2 T=0.1
T=0.3 T=0.3
0.0 T=0.5 0.0 T=0.5
1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0
H H
Figure 2. (Colour online) Staggered magnetization for the random field Ising model
(ρ = 1) on the regular random graph with two-body interactions (p = 2) and
connectivityc=3(left)andc=4(right). Solidanddashedlinesdenotethenumerical
resultswithferromagneticallyfixedandfreeinitialconditions,respectively. Thesystem
has singularities at H = (c 2)+2/M (M = 1,2, ) at zero temperature, and the
spinodal point of the ferrom−agnetic phase is given b·y··HRF =c 2.
F
−
3.1. The random-field Ising model with two-body purely ferromagnetic interactions
Firstly we show our investigation for the purely ferromagnetic random-field Ising model
(ρ = 1) with two-body interactions (p = 2) and connectivities c = 3 and 4. The absence
of a spin-glass phase has been proved for this case (ρ = 1, p = 2) irrespective of the
connectivity [24].
At zero temperature, the spinodal point of the ferromagnetic phase (i.e. the
largest H admitting a ferromagnetic fixed point) has been solved as HRF =
F
lim ((c 2)+2/M) = c 2, where M is interpreted as the number of spins
M→∞
− −
in ferromagnetic clusters at zero temperature [14]. A ferromagnetic phase transition
is observed when the size of the cluster M is infinity, but also there are Griffiths
singularities at H = (c 2) + 2/M for M = 1,2, only at zero temperature. For
− ···
finite temperatures the phase boundary has been evaluated approximately by several
techniques [14, 15, 39, 40, 41, 42, 43, 24], and with the cavity method in [24]. Here we
give more details for finite-temperature case.
When p = 2 and c 3, the transition is second order. Otherwise it is first order
≤
where hysteresis is observed in the magnetization [14, 15, 44]. Figure 2 shows the
staggered magnetization for p = 2, c = 3 and 4 from T = 0 to 0.5. Only at zero
temperature, the staggered magnetization jumps at H = (c 2)+2/M for both cases
−
of c = 3 and 4 but does not for finite temperature except at transition points.
Forc = 4,theconvergent distributionP(c)(h)hastwosolutions. Themagnetization
exhibits different results depending on the initial condition for equation (11) at low
temperature, while there is a unique solution for c = 3. Thus, for the systems with
c = 4, thetransitionisfirst orderatlowtemperature. Thehighertransitiontemperature
Random-field p-spin glass model on regular random graphs 9
2.0
3.0
1.5 P 2.5 P
2.0
T1.0 T
1.5
F 1.0 F TCP
0.5
0.5
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0
H H
Figure 3. (Colouronline)Phasediagramsofthe randomfieldIsingmodelwithp=2
ontheregularrandomgraphwithc=3(left)andc=4(right). Thesymbolsstandfor
theparamagnetic(P)andferromagnetic(F)phases,andTCPisforthetricriticalpoint.
Zero field transition points are given by TRF(H =0)=1/tanh−1(1/(c 1)) 1.820
F
− ≈
(c = 3) and 2.885 (c = 4) for the ferromagnetic phase. In the right panel, spinodal
lines are drawn dashed for paramagnetic (blue) and ferromagnetic (black) phases.
represents the limit of stability of the ferromagnetic solution and the lower is for the
paramagnetic solution. The two transition points may be identified by changing the
field H, i.e. increasing or decreasing H as indicated in arrows in figure 2.
The equilibrium phase boundary lies between these two critical-field values, which
correspond to the metastability limits of respective phases. As shown on the right panel
of figure 3, these two metastability limits merge at a tricritical point, and a second-order
phase transition takes over beyond the tricritical point for lower H and higher T. The
equilibrium phase boundary is calculated as the points where free energies for the two
solutions have the same value by using equation (9).
3.2. The random-field J Ising model with two-body interactions
±
No spin-glass phase is observed under the two-body pure-ferromagnetic condition as
shown in the previous subsection but it readily appears by the introduction of a small
amount of bond randomness. In fact, the zero-temperature para-ferro transition point
of the pure ferromagnetic case, as shown in figure 3, turns out to be a multicritical
point where paramagnetic, ferromagnetic and spin-glass phases coexist as will be shown
below.
Figure 4 shows the zero temperature (T = 0) phase diagram of the J Ising model
±
in random fields with p = 2 and c = 3 (left) and c = 4 (center and right). We see that
the transition point at ρ = 1 and T = 0 behaves as a multicritical point. This means
that the introduction of any small amount of bond randomness produces a spin-glass
phase as an equilibrium state (c = 3) and a metastable state (c = 4). The spin-glass
has ferromagnetically ordered state in the mixed phase.
Random-field p-spin glass model on regular random graphs 10
2.0
1.0 P 1.0 2.0 P HcF 2.0 1.8
0.8 0.8 Hc HcP 1.6
1.5 1.5
TCP MCP 1.4
0.6 0.6
SG SG 2.0
H H1.0 1.0
0.4 F 0.4 1.8
F
0.2 M 0.2 0.5 M 0.5 MCP 1.6
TCP
1.4
0.0 7/8 11/12 0.0 0.0 0.0
0.5 0.6 0.7 0.8 0.9 1.0 0.5 0.6 0.7 0.8 0.9 1.0 0.96 0.98 1.00
Figure 4. (Colour online) Zero temperature (T = 0) phase diagram of the J Ising
±
modelinrandomfields withtwo-bodyinteractionswithconnectivityc=3(leftpanel)
and c = 4 (center and right two panels). The red lines denote the boundaries of the
spinglassphase. Thesymbolsstandforthespinglass(SG)andmixed(M)phasesand
the tricritical (TCP) and multicritical (MCP) points. The mixed phase is the region
surrounded by the spin glass and ferromagnetic phase boundaries. For c =3 the zero
field critical probability is analytically known: one has ρSG =11/12 for the spin-glass
c
phase[45,46]whileρF(RS)=7/8attheRSlevel[45,46]andρF(1RSB) 0.857atthe
1RSB level. In the limc it of H 0 and T 0, our value of ρFcis a bit l≈ower than the
→ → c
RS one of references [45, 46] since we work with real cavity fields whereas references
[45, 46] have integer ones. It is still, however, larger than the 1RSB value which is
usually very close from the exact value. For c=4 (center and right panels) the phase
diagramhasaphasecoexistenceregion. Therightpanelshowsablowupofthe center
panel in the vicinity of ρ = 1 with the ferromagnetically-fixed initial condition (i) in
Section 2.2 (upper) and the free initial condition (ii) (lower). The equilibrium phase
boundaries are shown by solid lines and spinodal lines are by dashed lines. The spin
glass-paramagneticboundariesat H =1.0 (c=3)and H =1.5275(5)(c=4) showno
dependence on ρ.
For c > 3, a first-order ferromagnetic transition has been found at ρ = 1. The
first-order transition line extends from ρ = 1 to some ρ < 1, and a locally stable
TCP
ferromagnetic solution exists in this area. The phase diagram (right panel of figure 4)
is thus more complex than the case c = 3 due to the nature of the first-order phase
transition.
Thephaseboundarybetweenthespinglassandparamagneticphasesisindependent
of the ferromagnetic bias ρ. This behaviour is clearly shown in the finite temperature
phase diagram: Figure 5 depicts phase diagrams for ρ < ρMCP = 2+√2 /4(= 0.8536)
(ρMCP here is defined such that at this density TF = TSG) and ρ = 0.90, ρSG(= 11/12 =
c c (cid:0) (cid:1)c
0.9167) and 0.93 with c = 3. The ferromagnetic phase vanishes for ρ < ρMCP, and
ρSG = 11/12 is defined as the limit below which (ρ < ρSG) the H = 0 system has a finite
c c
temperature spin glass phase. For ρ < ρMCP, the spin-glass phase boundary shows no
dependence on ρ down to 1/2. On the other hand, the ferromagnetic phase exists for
ρ > ρMCP, and phase boundaries of ferromagnetic and mixed phases depend on ρ, which
shrinks with decreasing ρ.