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Our scheme is illustrated in Fig. 1, where we plot the specific heat of a specific random-field realization for a lattice size
L = 8 . The main simulation was performed at h
RF
= 2.25 and several extrapolations, from h
RF
= 2.1 to h
RF
= 2.4 , are
shown together with results obtained independently via direct Wang-Landau simulation. It is clear that for values of h RF
very close to 2.25 , i.e. h
RF
= 2.2 and h
RF
= 2.3 , the results practically coincide, while for the values h
RF
= 2.1 and
h
RF
= 2.4 , there is only a small deviation mainly around the peak. However in that case too, the location of the pseudocritical
temperature, i.e. the temperature where the specific heat attains its maximum, is accurately determined.
In Fig. 2 we present finite-size phase diagrams for systems of L × L× L spins with L ∈ {4,8,12,16,20} using the results
from an ensemble of 20 disorder realizations simulated at each lattice size. The simulation was performed at h
RF
= 2.25 and
extrapolated to h
RF
= 2.1,2.15,2.2,2.3,2.35, and
3.0
2.4 , using the above described scheme. For each
3.0
L=4-20
value of the disorder strength we have estimated
2.9
the temperature where the average specific heat
2.8
M
1
curve [ C] av
= ∑ C( T)
, where M = 20 counts
2.6
2.8
M i = 1
2.4
the number of realizations, attains its maximum.
2.29(2)
For a given size L , we applied an elliptical fit of
2.7
2.2
x
*
⎛ T ⎞
2.0
the form hRF
( T) = hRF
1−⎜ * ⎟ ,
2.6
0.0 0.1 0.2 0.3 0.4
⎝T
( hRF
= 0) ⎠
L -1/ν
where T * ( h
RF
= 0) is the pseudocritical
2.5
temperature of the corresponding zero-field 3d
Ising model taken from Ref. [6], for the chosen
2.4
L . Thus, we obtained the values for the
4 8 12 16 20 pseudocritical disorder strengths h
* RF
( L ) , as
L
shown in Fig. 2. (Note that the exponent x in the
elliptical form of the phase boundary approaches
Fig. 3: A plot of the pseudocritical disorder strength h
* the value 2 with increasing lattice size, indicating
RF
( L ) versus
L for the range L = 4 − 20 . The inset shows the same data as a
that the sequel of finite-size phase diagrams
1/
approaches a normal ellipse in the limit L →∞).
function of L
− ν
c
. The arrow marks the value for h
RF
.
In Fig. 3 we plot the data for h
* RF
( L ) as a
function of the lattice size. The shift of these pseudocritical disorder strengths is used to estimate the infinite-size critical
c
strength of the random-field, h
RF
and also the correlation length’s exponentν . The solid line shows a very good fit to the
* c 1/
function hRF
( L)
= hRF
+ bL − ν
c
2 2 −7
giving h
RF
= 2.29(2) and ν = 1.40(4) , with a very small value of χ ( χ = 1.4× 10 ) . The
1/
inset shows the same data (but without error bars) as a function of L
− ν , marking with an arrow the value of the critical
randomness. Note that, some recent numerical studies for the Gaussian RFIM using graph theoretical algorithms at zero
c
c
temperature gave the values h
RF
= 2.28(1) and ν = 1.32(7) [3] and h
RF
= 2.270(4) and ν = 1.37(9) [4].
In conclusion, we have studied, via a new simulation approach and a reliable extrapolation scheme, the low-T part of the
c
phase diagram of the bimodal 3d RFIM. We have estimated the critical value of the disorder strength h
RF
= 2.29(2) and the
correlation length’s exponent ν = 1.40(4) . Further research in this area is in progress, mainly focusing on the existence, or
not, of a tricritical point in the phase diagram of the model, as predicted by mean-field theoretical calculations [8].
h * RF (L)
h * RF (L)
[1] Y. Imry, S.-K. Ma, Phys. Rev. Lett. 35 (1975) 1399.
[2] See, e.g. the articles by D.P. Belanger and T. Nattermann in Spin Glasses and Random Fields, edited by A.P. Young,
World Scientific, Singapore, 1998.
[3] A.K. Hartmann, A.P.Young, Phys. Rev. B 64 (2001) 214419.
[4] A.A. Middleton, D.S. Fisher, Phys. Rev. B 65 (2002) 134411.
[5] A. Malakis, N.G. Fytas, Eur. Phys. J. B 51 (2006) 257.
[6] A. Malakis, A. Peratzakis, N.G. Fytas, Phys. Rev. E. 70 (2004) 066128.
[7] F. Wang, D. P. Landau, Phys. Rev. Lett. 86 2050 (2001).
[8] A. Aharony, Phys. Rev. B 18 (1978) 3328.
N.G. Fytas acknowledges financial support by the Alexander S. Onassis Public Benefit Foundation.
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