Behavior of Ising Spin Glasses in a Magnetic Field

Abstract

We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed along a nontrivial path in the field-temperature plane that must cross any putative de Almeida-Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice where the instability line separating the spin glass from the paramagnetic state is also computed analytically. While the instability line is reproduced by our simulations on the mean-field Bethe lattice, no such instability line can be found numerically for the short-range three-dimensional model.

Full text

arXiv:0712.2009v2 [cond-mat.dis-nn] 13 May 2008
Behavior of Ising Spin Glasses in a Magnetic Field
Thomas J¨org,1, 2 Helmut G. Katzgraber,3and Florent Krz¸aka la4
1LPTMS, UMR 8626 CNRS et Universit´e Paris-Sud, 91405 Orsay CEDEX, France
2´
Equipe TAO - INRIA Futurs, 91405 Orsay CEDEX, France
3Theoretische Physik, ETH Z¨urich, CH-8093 Z¨urich, Switzerland
4Laboratoire PCT, UMR Gul liver CNRS-ESPCI 7083, 10 rue Vauquelin, 75231 Paris, France
We study the existence of a spin-glass phase in a field using Monte Carlo simulations performed
along a nontrivial path in the field–temperature plane that must cross any putative de Almeida-
Thouless instability line. The method is first tested on the Ising spin glass on a Bethe lattice
where the instability line separating the spin glass from the paramagnetic state is also computed
analytically. While the instability line is reproduced by our simulations on the mean-field Bethe
lattice, no such instability line can be found numerically for the short-range three-dimensional model.
PACS numbers: 75.50.Lk, 75.40.Mg, 05.50.+q, 64.60.-i
Since its proposal in the mid-70’s, the Edwards and
Anderson Ising spin-glass Hamiltonian has become a
source of inspiration in statistical physics, especially in
the context of mean field theory [1, 2, 3, 4], and has been
applied to a wide variety of problems across scientific
disciplines. However, basic yet simple questions about
the very nature of the spin-glass state in (experimentally
relevant) finite space dimensions are still subject of con-
troversy. The most prominent of such open questions is
the existence of spin-glass ordering in a magnetic field.
The fully-connected mean-field version of the EA
model, called the Sherrington-Kirkpatrick (SK) model
[2], was solved using the replica method by Parisi [3, 4];
the obtained free energy recently proven to be rigor-
ously exact by Talagrand [5]. In this model, the low-
temperature spin-glass phase is characterized by a com-
plex free energy landscape made of many different val-
leys. This phase also exists for low externally-applied
magnetic fields and the so-called de Almeida-Thouless
(AT) instability line [6] separates the spin-glass from the
paramagnetic phase at finite fields/temperatures. A sim-
ilar scenario arises in the Bethe lattice approximation
(see Fig. 1). However, within the more phenomenological
description of spin glasses known as the droplet picture
[7] any infinitesimal field destroys the spin-glass order, in
stark disagreement with the aforementioned mean-field
description. Numerical evidence favoring the absence of
a spin-glass state in a field for short-range spin glasses
below the upper critical dimension have become stronger
[8, 9, 10, 11, 12, 13, 14], although different opinions re-
main [15, 16, 17, 18].
Capitalizing on the success of studying the suscepti-
bility and the finite-size correlation length [20, 21] to
probe the spin-glass phase, we study the problem using
a novel numerical approach backed up with analytic cal-
culations on the Bethe lattice. On the numerical side,
we use a multi-spin coded version of exchange (parallel
tempering) Monte Carlo [22] with a new twist where the
replicas “live” in the H–Tplane along a nontrivial path
that guarantees a crossing with a potential AT line (see
T
h(T)
PM
c
H
T
1.5 2
0
0.5
1
2
1.5
0.5 0
SG
1
FIG. 1: (Color online) Magnetic field Hversus temperature
Tphase diagram of the Ising spin glass on a mean-field ran-
dom regular graph with connectivity c= 6 with Gaussian
interactions computed using the cavity method. The para-
magnetic (PM) phase is separated from the spin-glass (SG)
phase by the de Almeida-Thouless line (light red curve). The
diagonal (dark blue) line h(T) represents the simulation path
followed in the Monte Carlo simulations. Because the path
starts at T > Tcfor H= 0 and then increases along a diag-
onal h(T) = r(T−Tmin) in the H–Tplane above Tmin ≪Tc
up to a value beyond Tc(dashed vertical line), an intersection
with a putative AT line is guaranteed. The same approach is
used for the diluted 3D simulations where Tc= 0.663(6) [19].
Fig. 1) and that in contrast to Refs. [23, 24] has both
ends in the high temperature region where decorrelation
is fast. We solve the thermalization problems on diago-
nal paths found in Ref. [10] by studying the link-diluted
version of the model [19] where the cluster moves intro-
duced in Refs. [25] and [19] can be used. In this model,
the cluster updates allow the thermalization times to be
decreased by a factor of at least 103, allowing us to probe
relatively large system sizes down to very low tempera-
tures and large magnetic fields. We first demonstrate the
efficiency of our strategy by applying it on the model de-
fined on a regular random graph—which corresponds to a
Bethe lattice in this context—for which we compute the
AT line analytically, generalizing the original result by de
Almeida and Thouless for the SK model [6]. The analyt-
ical and numerical results for the Bethe lattice agree to
high precision. However, results on the 3D model show

2
no sign of an AT line. The fact that we do observe an
AT line for the mean-field Bethe lattice, exactly where
it is predicted, and do not for the short-range 3D Ising
spin glass is convincing evidence that the phase diagram
is indeed trivial and there is no spin-glass state in a field
in three dimensions (3D).
Models — The spin-glass Hamiltonian is given by
H=−X
i,j
Jij SiSj−HX
i
Si.(1)
The Ising spins Si∈ {±1}have nearest-neighbor interac-
tions. We study the mean-field case where Nspins lie on
the vertices of a regular random graph, and the EA model
on a cubic lattice of size N=L3with periodic boundary
conditions, where the interactions Jij ∈ {−1,0,1}are
chosen from a link-diluted bimodal distribution with a
link occupation probability of 45% [19].
Theoretical predictions — Following recent progress
in the study of finite-connectivity mean-field systems, it
is now possible to study (quite) precisely spin-glass mod-
els on Bethe lattices using the cavity method [26]. Within
this formalism, first steps in computing the phase dia-
gram in the field–temperature plane have been achieved
(see Refs. [27] and [14]). In order to determine the AT
line, one needs to compute the onset of divergence of the
spin-glass susceptibility χin the paramagnetic phase in
which case, thanks to the finite correlation length, the
model is equivalent to a spin glass on an infinite tree
with random boundary conditions [26] (this is reminis-
cent of the exact analysis presented in Ref. [28]). We
thus consider here the simplest Bethe-Peierls cavity ap-
proach for the problem and compute the AT line in this
context. We refer the reader to Refs. [26], [27], [14], and
[29] for a description of the method. For H= 0, we re-
cover the well-known result first found by Thouless [30]:
k[tanh(βcJ)2]av = 1, where βc= 1/Tc,k=c−1 (cthe
connectivity), and [···]av is a disorder average. For finite
external field, one has to perform a numerical solution of
the cavity equations to solve the cavity recursion and to
compute the point where the susceptibility diverges [29].
Using this approach, we compute points along the AT
line for different values of the field Hwith Gaussian
and bimodal disorder and connectivities c= 3 and 6.
We have also checked that the large-connectivity limit
of our computation yields the SK result. We find that,
close to Tc, the data scale as hAT(T)∼(Tc−T)3/2—
as first predicted in Ref. [30] and checked in Ref. [27]—
and the instability line is then approximately linear close
to zero temperature [31]. A fit to hAT(T) = a1(Tc−
T)3/2exp(a2T+a3T2) gives very accurate results whose
precision is within the error of our numerical evalua-
tion of the cavity recursion. For c= 6, we obtain
a1= 0.786 (0.875), a2= 0.111 (0.221), and a3=−0.054
(−0.127) and Tc= 1.807 (2.078) for Gaussian (bimodal)
distributed disorder. For c= 3, we find a1= 0.785
(0.827), a2= 0.251 (0.413), and a3= 0.117 (0.083) and
Tc= 0.748 (1.135). The Gaussian case with c= 6 is
shown in Fig. 1 (light red curve).
The presence of the instability line in finite-
dimensional systems has been criticized before, especially
in the context of the droplet model [7]. Other pictures
were proposed where no such line is present [32, 33, 34].
It has been also suggested that the line disappears below
the upper critical dimension du= 6 [12, 35, 36, 37], and
even more complex scenarios are possible [38, 39, 40]. We
now attempt to clarify this issue in the 3D case.
Numerical method — The simulations are performed
using exchange Monte Carlo (EMC) [22]. Traditionally,
the field is fixed at a constant value and replicas at differ-
ent temperatures perform a Markov chain in temperature
space. There have been alternate approaches where the
temperature Tis fixed to T < Tcand the replicas per-
form a Markov chain in field space. This method has not
proven to be efficient because tunneling across H= 0
requires special moves [23, 24] and especially because in
this case no replica has the chance to reach the zero-field
high-Tphase where relaxation is fast. In order to solve
this problem, we propose an EMC method in the H–
Tplane (see Fig. 1) at which part of the replicas have
labels (T , H = 0) for temperatures values in the range
Tmin ≤T≤Tmax with Tmax ≫Tc(in 3D Tmin = 0.30).
We also couple the replicas in the EMC scheme to a sec-
ond set of replicas along a diagonal path in the H–T
plane with labels (T , H) and h(T) = r(T−Tmin ), rcon-
stant. It is important not to choose the slope rof h(T)
too steep because EMC is least efficient for large fields
[23, 24, 41]. In addition, Tmax(H > 0) > Tc(H= 0)
has to be chosen to ensure that the a potential AT line is
crossed. The method has the advantage to deliver data at
zero (horizontal blue path in Fig. 1), as well as finite field
(diagonal blue path in Fig. 1). Finally, it is advantageous
to choose the slope of the path in the H–Tplane such
that it crosses the phase boundary orthogonally (cleaner
signal of the transition).
We test equilibration by a logarithmic binning of the
data until the data in the last three bins agree within
error bars. To access relatively large system sizes in
3D, we use a multi-spin coded version of the program
that updates 32 copies of the system in parallel. In or-
der to obtain thermalization in reasonable time, it is
necessary to apply the cluster algorithm introduced in
Refs. [19] and [25]. The speedup obtained by this ap-
proach is of three orders of magnitude over conventional
approaches, even for the smallest sizes simulated. Note
that the speedup increases with increasing system size;
our simulations would not have been possible otherwise.
Simulation parameters are listed in Table I.
Results on the Bethe lattice — We have com-
puted the connected spin-glass susceptibility using χ=
Nhq2iT− hqi2
Tav where q= 1/N PN
i=1 qiwith qi=
S1
iS2
ibeing the overlap at site ibetween two indepen-
dent replicas of the system. Here h· · · iTrepresents

3
TABLE I: Simulation parameters: Nsa is the number of sam-
ples, Nsw is the number of Monte Carlo sweeps for one sample,
Tmin is the lowest temperature simulated, Hmax is the max-
imum field studied and Nris the number of replicas used in
the EMC method. hBL (T) = 1.0(T−0.5) for the Bethe lattice
(top set ); in 3D h3D(T) = 0.7(T−0.3) (bottom set).
N Nsa Nsw Tmin Hmax Nr
64 12834 100000 0.5 2.0 25
128 12689 100000 0.5 2.0 25
256 5107 200000 0.5 2.0 25
512 2546 1000000 0.5 2.0 49
L Nsa Nsw Tmin Hmax Nr
4 20000 200000 0.3 0.49 47
5 20000 400000 0.3 0.49 47
6 20000 400000 0.3 0.49 47
8 10528 1000000 0.3 0.49 51
10 6080 2000000 0.3 0.49 51
12 2944 4000000 0.3 0.49 51
FIG. 2: (Color online) Rescaled susceptibility χ/N1/3from
Monte Carlo data of the regular random graph model with
Gaussian disorder and connectivity c= 6. Left: Data for zero
field cross at the analytically predicted critical temperature
Tc= 1.807. Right: Data along hBL(T) = 1.0(T−0.5). The
crossing point agrees very well with the values in Fig. 1.
a thermal average. The susceptibility of the mean-
field model has a finite-size scaling form χ(T, H ) =
N1/3e
GN1/3[T−Tc(H)][23, 24] hence data for differ-
ent Nshould cross at Tcwhen plotted as χ/N1/3versus
T. This is shown in Fig. 2 for c= 6 and a Gaussian
distribution of the interactions: The left panel is at zero
field, whereas the right panel is along the diagonal path
hBL(T) = 1.0(T−0.5). In both cases, the data cross
in agreement with the analytical results (Fig. 1). Hence
our numerical approach allows for a precise detection and
location of any putative AT line.
Results in three dimensions — In the finite-
dimensional system the scaling of the susceptibility re-
FIG. 3: (Color online) Finite-size correlation length as a func-
tion of T /Tcfor the 3D Ising spin glass at zero field with bi-
modal interactions. The data cleanly cross at Tc= 0.663(6).
quires an additional parameter ηwhose putative in-field
value is unknown, as opposed to the mean-field case. We
thus prefer to study the transition via the scaling of the
finite-size two-point correlation length [20, 21] given by
ξL=1
2 sin(|kmin|/2) χ(0)
χ(kmin)−11/2
,(2)
where kmin = (2π/L, 0,0) is the smallest nonzero wave
vector and χ(k) the wave-vector-dependent spin-glass
susceptibility defined as
χ(k) = 1
NX
i,j
[hqiqjiT− hqiiThqjiT]av eik·(Ri−Rj).(3)
Note that we used a slightly different definition of the
susceptibility than in Ref. [10]. However, the two ex-
pressions are equivalent, except that the definition used
here requires only two replicas (unlike the four needed
in Ref. [10]) thus saving computational resources. The
correlation length divided by the system size Lhas the
scaling relation ξL/L =e
XL1/ν[T−Tc(H)], where νis
the usual critical exponent. When T=Tc(H) data for
different Lcross signaling the existence of a transition.
Data for the 3D model in zero field are shown in Fig. 3.
A clear crossing point at Tc= 0.663(6) is observed, thus
signaling the presence of a spin-glass state for system
sizes up to L= 16 and in agreement with previous re-
sults [19]. In contrast, data along the finite-field branch
[h3D(T) = 0.7(T−0.3)] (see Fig. 1) show no sign of a
transition (see Fig. 4). The inset shows a zoom of the
data for small fields where the crossing points (arrows)
between the different lines wander towards zero for in-
creasing L. This shows that at temperature T= 0.3, the
instability is below h < 0.006. This should be compared

4
FIG. 4: (Color online) Finite-size correlation length as a func-
tion of h3D(T) = 0.7(T−0.3) (finite-field branch in Fig. 1)
for the 3D Ising spin glass. The apparent crossing of the data
wanders towards zero for increasing system size (see inset for a
zoom into the low-field region), i.e., there is no sign of a tran-
sition for H > 0 (arrows mark crossings between L-pairs).
with the mean field case where, e.g., for c= 3 with Gaus-
sian couplings (having a similar Tcas the 3D model), one
observes h(0.3) ≈0.26, i.e., a value 40 times larger. Per-
forming a one-parameter scaling with fixed volume ra-
tio hc(L)∼h∞
c+b/L2gives h∞
c=−0.00004(14), i.e.,
compatible with zero with a Q-factor of 94.9%. Fixing
h∞
c= 0 gives Q= 99.7%; i.e., the data suggest the ab-
sence of a spin-glass state in a field.
Summary and discussion — We have presented cal-
culations of the AT line of a spin glass on a Bethe lattice
using the cavity method. These results for the mean-field
model on the Bethe lattice allowed the validation of our
Monte Carlo simulations where we have observed an AT
line close to its predicted value. This is in stark contrast
to the 3D Ising spin glass with bimodally-distributed
bonds where data for a wide range of fields and tem-
peratures clearly show a lack of ordering in a field.
It would be interesting to study higher-dimensional
systems using the method presented here to verify
whether or not an AT line is observed above the upper
critical dimension [12] and we hope our results will also
spark new theoretical developments in this direction.
We thank A. P. Young for helpful discussions. The sim-
ulations have been performed on the ETH Z¨urich Hreidar
cluster. H.G.K. acknowledges support from the Swiss
National Science Foundation under Grant No. PP002-
114713. T.J. acknowledges support from EEC’s HPP
HPRN-CT-2002-00307 (DYGLAGEMEM) and FP6 IST
contracts under IST-034952 (GENNETEC).
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