# Effect of platykurtic and leptokurtic distributions in the random-field Ising model: mean-field approach.

**ABSTRACT** The influence of the tail features of the local magnetic field probability density function (PDF) on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically, we assign a quenched random field whose value is in accordance with a generic distribution that bears platykurtic and leptokurtic distributions depending on a single parameter tau<3 to each site. For tau<5/3, such distributions, which are basically Student-t and r distribution extended for all plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got the same asymptotic power-law behavior as a alpha-stable Lévy distribution with alpha=(3-tau)/(tau-1). For every value of tau, at specific temperature and width of the distribution, the system undergoes a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz (tau=2) which is accompanied with a divergent free energy per spin (at zero temperature).

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**ABSTRACT:**In this paper, we develop a stochastic process rules (SPR) based Markov chain method to calculate the degree distributions of evolving networks. This new approach overcomes two shortcomings of Shi, Chen and Liu’s use of the Markov chain method (Shi et al. 2005 [21]). In addition we show how an SPR-based Markov chain method can be effectively used to calculate degree distributions of random birth-and-death networks, which we believe to be novel. First SPR are introduced to replace traditional evolving rules (TR), making it possible to compute degree distributions in one sample space. Then the SPR-based Markov chain method is introduced and tested by using it to calculate two kinds of evolving network. Finally and most importantly, the SPR-based method is applied to the problem of calculating the degree distributions of random birth-and-death networks.Physica A: Statistical Mechanics and its Applications 06/2012; 391(11):3350–3358. · 1.68 Impact Factor - SourceAvailable from: Nuno Crokidakis[Show abstract] [Hide abstract]

**ABSTRACT:**Using a single functional form which is able to represent several different classes of statistical distributions, we introduce a preliminary study of the ferromagnetic Ising model on the cubic lattices under the influence of non-Gaussian local external magnetic field. Specifically, depending on the value of the tail parameter, $\tau $ ($\tau < 3$), we assign a quenched random field that can be platykurtic (sub-Gaussian) or leptokurtic (fat-tailed) form. For $\tau< 5/3$, such distributions have finite standard deviation and they are either the Student-$t$ ($1< \tau< 5/3$) or the $r$-distribution ($\tau< 1$) extended to all plausible real degrees of freedom with the Gaussian being retrieved in the limit $\tau \rightarrow 1$. Otherwise, the distribution has got the same asymptotic power-law behaviour as the $\alpha$-stable L\'{e}vy distribution with $\alpha = (3 - \tau)/(\tau - 1)$. The uniform distribution is achieved in the limit $\tau \rightarrow \infty$. Our results purport the existence of ferromagnetic order at finite temperatures for all the studied values of $\tau$ with some mean-field predictions surviving in the three-dimensional case. Comment: Report of the first results of an ongoing project. 12 pages and 8 figures. Comments welcome11/2010;

Page 1

arXiv:0906.5602v1 [cond-mat.stat-mech] 30 Jun 2009

Effect of platy- and leptokurtic distributions in the random-field Ising model: Mean

field approach

S´ ılvio M. Duarte Queir´ os1,∗Nuno Crokidakis2,†and Diogo O. Soares-Pinto3‡

1Unilever R&D Port Sunlight

Quarry Road East Wirral, CH63 3JW UK

2Instituto de F´ ısica - Universidade Federal Fluminense

Av.Litorˆ anea s/n

24210-340Niter´ oi - RJ

3Centro Brasileiro de Pesquisas F´ ısicas

Rua Dr Xavier Sigaud 150

22290-180 Rio de Janeiro - RJ

(Dated: 30th June 2009)

Brazil

Brazil

The influence of the tail features of the local magnetic field probability density function (PDF)

on the ferromagnetic Ising model is studied in the limit of infinite range interactions. Specifically,

we assign a quenched random field whose value is in accordance with a generic distribution that

bears platykurtic and leptokurtic distributions depending on a single parameter τ < 3 to each site.

For τ < 5/3, such distributions, which are basically Student-t and r-distribution extended for all

plausible real degrees of freedom, present a finite standard deviation, if not the distribution has got

the same asymptotic power-law behavior as a α-stable L´ evy distribution with α = (3 − τ)/(τ − 1).

For every value of τ, at specific temperature and width of the distribution, the system undergoes

a continuous phase transition. Strikingly, we impart the emergence of an inflexion point in the

temperature-PDF width phase diagrams for distributions broader than the Cauchy-Lorentz (τ = 2)

which is accompanied with a divergent free energy per spin (at zero temperature).

I.INTRODUCTION

Disorder is ubiquitous in Nature. Regarding materials and their statistical properties, disordered magnetic systems

have been systematically studied in condensed matter and statistical physics. From a theoretical point of view, the

most studied case has certainly been the Random Field Ising Model (RFIM) [1, 2], because of its simplicity as a

frustrated system and relevancy to experiments [3, 4] which has been quite boosted after the identification of the

RFIM with diluted antiferromagnets in the presence of a uniform magnetic field [3, 5, 6, 7] and several ferromagnetic

compounds as well [3, 4, 8].

In order to generate the local random field, both the Gaussian and the bimodal probability density function (PDF)

have intensively been used [9, 10, 11]. Nevertheless, controversy over the order of the low-temperature phase transition

has still been at the helm of several discussions. On one hand, a high temperature series expansion up to 15th order

showed a continuous phase transition for both the Gaussian and the bimodal PDF [12]. On the other hand, from an

exact determination of the ground states in higher dimensions (d = 4), Swift et al [13] found a discontinuous phase

transition for the bimodal random field, whereas for d = 3 dimensions and the Gaussian distribution the transition is

continuous. By applying the Wang-Landau algorithm [14], recent simulations on 3D lattices claimed the discovery of

first-order-like features in the strongly disordered regime for both those PDFs [15, 16].

As an alternative to the above mentioned approaches, there is the mean field theory which can present a good

qualitative agreement with some short-range interaction models and experiments. Once more, the Gaussian and the

bimodal PDF have been widely investigated [17, 18] as well as related distributions such as the trimodal [19, 20] and

the double-Gaussian [21] or the treble-Gaussian [22]. In the Gaussian RFIM case, the phase diagram only presents

continuous phase transitions [17], whereas in the bimodal case the phase diagram presents a continuous phase transition

for high temperatures and low random-field intensities and for low temperatures and high random-field intensities a

first-order transition arises therefrom [18]. In other more elaborated cases a rich critical behavior can be found for

finite temperatures as it has been recently conveyed in [21, 22]. Accordingly, we can understand that the choice of the

local random field PDF is of crucial importance for a good theoretical description of real systems. In this particular

context and based on the identification of the RFIM with diluted antiferromagnets in a uniform field, for which the

∗Corresponding author: sdqueiro@gmail.com

†nuno@if.uff.br

‡dosp@cbpf.br

Page 2

local random fields are expressed in terms of quantities that vary in both signal and magnitude [5, 7], the use of

continuous PDFs has demonstrated to be a very promising approach [21, 22].

The utilization of Dirac Delta and Gaussian related distributions is much supported on the easiness of the analytical

treatment of the subsequent equations as well as the pervasiveness of the Gaussian distribution. Although the Gaus-

sian was assumed for many generations as the “natural distribution”, in the last decades the concept of (asymptotic)

scale-invariance of probability density functions have abundantly emerged [23]. In the realm of disordered systems,

PDFs different to the n-Gaussian or the n-Dirac Delta were used to explain the critical behavior of several com-

pounds. For instance, PDFs with very fat tails were introduced to analyze organic charge-transfer compounds like:

N-methyl-phenazium tetra-cyanoquinodimethanide (NMP-TCNQ), quinolinium-(TCNQ)2, acridinium-(TCNQ)2and

phenazine-TCNQ, as first reported in Refs. [24]. Conversely, a sub-Gaussian distribution was used to account for the

magnetic properties and the critical behavior of poly(metal phosphinates) [25]. Last but not least, as was proven

by Gosset [26], asymptotic scale invariant distributions can be derived from the Gaussian distribution when finite

elements are taken into account so that finite and scale-dependent systems can be treated as infinite and (asymp-

totically) scale-independent. Therefore, the study of more general continuous PDFs turns up very interesting as it

furnishes a more widespread picture of disordered magnetic systems than the distributions used up to now. With such

a goal in mind, we study herein the aftermath of applying a more general family of continuous PDFs in the mean field

RFIM. Explicitly, our PDF reproduces the r- and t-distributions for real degrees of freedom. For specific values of

the triplet composed of the degree of freedom, the temperature and the PDF width, our results show that the system

experiences a continuous phase transition that does not dependent on the finiteness of the standard deviation and the

scale behavior (dependence or independence) of the random field. Moreover, for PDFs fatter than the Cauchy-Lorentz,

we determine the emergence of an inflexion point in the temperature versus PDF width phase diagrams that coexists

with a divergence at zero temperature of the free energy per spin.

II.THE MODEL

The infinite-range-interaction Ising model in the presence of an external random magnetic field is defined in terms

of the Hamiltonian,

H = −J

N

?

(i,j)

SiSj−

?

i

HiSi, (1)

where the sum?

(i,j)runs over all distinct pairs of spins Si = ±1 (i = 1,2,...,N). The random fields {Hi} are

quenched variables and ruled by a PDF that is defined by a parameter τ (generic degree of freedom). For τ < 1,

Pi(Hi) =

?

1 − τ

π

Bτ

Γ

?

5−3τ

2(1−τ)

?

?

Γ

2−τ

1−τ

? [1 − Bτ(1 − τ)H2

i]

1

1−τ,(2)

(with |H| ≤ [Bτ(1 − τ)]−1/2) which is the generalized r-distribution, and for τ > 1, we have

Ps(Hi) =

?τ − 1

π

Bτ

Γ

?

2(τ−1)

1

τ−1

3−τ

?

Γ

?

?[1 − Bτ(1 − τ)H2

i]

1

1−τ,(3)

which is the generalized Student-t distribution. By generalized we mean that the degrees of freedom, m and n, of t- and

r-distributions are extended to the entire domain of feasible real values according to the relations τ = (m + 3)/(m + 1)

[m ≥ 0] and τ = (n − 4)/(n − 2) [n ≥ 2], respectively. In Eqs. (2) and (3), Γ[.] is the Gamma function and Bτ is

given by

Bτ=

1

(3 − τ)ω2,(4)

where ω is the width of the PDF. For τ < 5/3 the width and the standard deviation, σ, are related by

(5 − 3τ)σ2= (3 − τ)ω2.(5)

Alternatively, the functional form of Eqs. (2) and (3) can be obtained by optimizing the entropic form presented in

[27] by applying the concept of escort distribution, p(H) ≡ Pτ(H)/?Pτ(H) dH [28, 29], and for that is many times

Page 3

?4

?2024

0.0

0.1

0.2

0.3

0.4

0.5

H

P?H?

Τ?5?2

Τ?3?2

Τ?1

Τ?1?2

Τ??20

0.11101001000

104

10?22

10?18

10?14

10?10

10?6

0.01

H

P?H?

FIG. 1: (Color online) Random-field probability distributions for some values of the parameter τ (from botton to top: τ =

5/2,3/2,1,1/2 and −20), in the normal (left panel) and log-log scale (right panel). We have used ω = 1 in all cases.

called q-Gaussian. In this case, ω2plays the role of the constraint,?H2p(H) dH = ω2, which is always finite for

of the escort distribution and it is finite even when the distribution per se has got a divergent standard deviation,

?H2P (H) dH = σ2. Therefore, it represents a way of appraizing the broadness of the distribution and this is the

acknowledge both nomenclatures we use the traditional terminology of r− and t− distributions that is quite well

established in the Statistics community since a long time. The PDF defined in Eqs. (2) and (3) is symmetrical

around H = 0 and represents a family of continuous distributions that recovers some well-known distributions using

appropriate limits, namely:

τ < 3 with the corresponding Lagrange multiplier given by Eq. (4). Expressly, ω2represents the standard deviation

reason why we named ω width. Recently, Pi(s)(H) has also been coined generalized Lorentzian [30]. Although we

• the uniform distribution, for τ → −∞;

• compact support distributions (limited), for τ < 1;

• the Gaussian distribution, for τ → 1;

• the Cauchy-Lorentz distribution, for τ = 2.

• Dirac Delta, for every τ < 3 and ω → 0.

To boot, the functional form (3) is an asymptotic power-law decaying PDF with finite standard deviation for

1 < τ < 5/3 and an asymptotic power-law decaying PDF, but with infinite standard deviation instead. In both cases

the decay exponent is equal to 2/(τ − 1). The latter case is also capable of reproducing the tail behavior of α-stable

L´ evy distributions

Lα(H) =

?∞

−∞

exp[−a |k|α+ ikH] dk,

with α = (3 − τ)/(τ − 1) and broadness a, whose escort-distribution has got a finite width as well. For the case of

the Cauchy-Lorentz, α = 1 (τ = 2) [the only case for which L´ evy distributions are explicitly defined in real space],

the parameter a is equal to width ω. Accordingly, if we bear in mind the previous work by Aharony [18], we can

hold that our enquiry also sheds light on the low temperature behavior of the random-field Ising model with the local

magnetic field associated with a α-stable L´ evy distribution. In Fig. 1, we depict PDFs (2) and (3) for some values of

τ. Regarding the kurtosis,

κ ≡

?H4?

?H2?2,(6)

Page 4

the distribution is platykurtic, κ < 3, for τ < 1 or leptokurtic, κ > 3, for τ > 1. At this point it is important to stress

that, as it has been made until now, in spite of being able to present non-mesokurtic distributions the combination

of Gaussians results in asymptotic scale-dependent distributions.

From the free energy, F({Hi}), associated with a given realization of site fields, {Hi}, we calculate the quenched

average, [F({Hi})]H,

??

The general mean field result of the free energy per spin, in terms of any PDF of the random fields, is well-known

[17, 18], and is given by

[F({Hi})]H=

i

[dHiP(Hi)]F({Hi}) .(7)

f =J2

2m2−1

β?log[2coshβ(Jm + H)]?H

(8)

and the magnetization is given by,

m = ?tanh[β(Jm + H)]?H,(9)

where ?...?H stands for averages over realizations of the disorder, i.e.,

?...?H=

?+∞

−∞

dH P(H) (...) .

Close to a continuous transition between ordered and disordered phases, the magnetization m is small. So, we can

expand Eq. (9) in powers of m,

m = Am + Bm3+ Cm5+ O(m7), (10)

where the coefficients are given by

A = βJ{1 − ρ1},

B = −(βJ)3

(βJ)5

15

(11)

3

{1 − 4ρ1+ 3ρ2},(12)

C =

{2 − 17ρ1+ 30ρ2− 15ρ3},(13)

with

ρk= ?tanh2k(βH)?H .

With the aim of finding the continuous critical frontier we set A = 1, provided that B < 0. If a first-order critical

frontier also occurs, the continuous line must end when B = 0; in such cases, the continuous and the first-order critical

frontiers converge at a tricritical point, whose coordinates are obtained by solving the equations A = 1 and B = 0,

on condition that C < 0. Thus, for A = 1, we obtain

kT

J

= 1 − ?tanh2(βH)?H . (14)

In the following section, we discuss the role of PDFs (2) and (3) when they are considered in the formulae presented

in this section. Our survey includes the analysis of the phase diagrams for the whole domain of τ.

III.FINITE TEMPERATURE ANALYSIS

Following the above presented results, we proceed by calculating the critical frontiers of the model when the

temperature is different from zero. In the RFIM, we have a single transition between the two possible phases of the

magnetization: the ferromagnetic phase (m ?= 0) and the paramagnetic phase (m = 0). The critical frontier separating

these two phases is found by solving Eq. (14). On account of the fact that Eq. (14) is analytically unsolvable, we have

been compelled to solve it by numerical means using the Global Adaptative Strategy algorithm [31] that has been

proven as the best (i.e., fast and accurate) numerical integration procedure for smooth integrands [32].

Page 5

0.00.2 0.40.6 0.81.0

0.0

0.2

0.4

0.6

0.8

1.0

Ω ? J

k T ? J

P

F

FIG. 2: (Color online) Phase diagram of the model, in the plane temperature vs ω (in units of J), for some values of the

parameter τ < 1. The grey dotted line is for τ = −∞

ω (T = 0) = 0.9831...J; the dot-dashed orange line is for τ = 0 and ω (T = 0) = 0.8660...J; the dotted green line is for

τ = 1/2 and ω (T = 0) = 3?

onto the ω/J axis were exactly calculated through a zero-temperature analysis [section IV.A] where from we can see a good

agreement between the analytical and numerical results which by interpolation indicates discrepancies never greater than 1%.

and ω (T = 0) = J; the brown dashed line is for τ = −20 and

5/64J and the black full line is the Gaussian case with ω (T = 0) =

continuous phase transitions between the Ferromagnetic (F) and the Paramagnetic (P) phases for all values of τ. The points

?

2/πJ. We can observe

A. Platykurtic case: τ < 1

Let us denote fiand mias the free energy and the magnetization for this regime of τ, respectively. Thus, Eqs. (8)

and (9) become,

fi=J2

2m2

i−1

β

?

1

√

Bτ(1−τ)

−

1

√

Bτ(1−τ)

dH Pi(H)log[2coshβ(Jmi+ H)],(15)

and

mi=

?

1

√

Bτ(1−τ)

−

1

√

Bτ(1−τ)

Pi(H)tanhβ(Jmi+ H) ,(16)

where Pi(H) is given by the PDF in Eq. (2). The continuous critical frontier has been found when we have solved

Eq. (14). For all solutions obtained, we have calculated a negative value of B, Eq. (12), which has confirmed the

continuous character of the phase transition.

If a first-order transition existed as well, the critical frontier would be found by equalizing the free energy at each

side of this line, i.e., f(m = 0) = f(m ?= 0). Using this procedure, we have numerically determined the critical

frontiers separating the paramagnetic and ferromagnetic phases, for typical values of τ < 1. We have confirmed that

the above coefficient B, Eq. (12), is always negative. The phase diagram is shown in Fig. 2, on the plane defined

by the temperature, T, and the PDF width, ω (both in units of J), for some typical values of τ < 1. In that

figure, the lines represent the numerical solution of Eq. (14), whereas the points were analytically obtained through

a zero-temperature analysis, which is going to be discussed in the next section. Notice that the ferromagnetic phase

is reduced by increasing the parameter τ from τ = −∞ to τ = 1 as shown in Fig. 2, and for the maximum value for

r-distributions, τ = 1, we recover the simple phase diagram of the Gaussian distribution [17].

Page 6

0.00.20.4 0.60.81.0

0.0

0.2

0.4

0.6

0.8

1.0

Ω ? J

k T ? J

P

F

FIG. 3: (Color online) Phase diagram of the model, in the plane temperature versus ω (in units of J), for some values of

the parameter τ > 1. The black full line is the Gaussian case with ω (T = 0) =

ω (T = 0) =

√3πJ; the purple dotted line is for τ = 2 and ω (T = 0) =

ω (T = 0) = 0.4754 ...J. We can observe continuous phase transitions between the Ferromagnetic (F) and the Paramagnetic

(P) phases for all values of τ. The points represent the results obtained by the zero-temperature analysis. Notice the change

in the concavity of the critical frontier for large values of τ (> 2.0). The vertical dashed line is ω = 0.275J which is close to

the inflexion point of the critical line for τ = 5/2. In this figure, we have distinguished the points with finite free energy per

spin from the points with a divergent free energy per spin representing the latter by empty circles. Again, we can observe a

good agreement between the numerics and the expansion at T = 0. The difference between the analytical approximation and

interpolation is again never greater than 1%.

?2

πJ; red dashed line is for τ = 3/2 and

πJ; the dot-dashed blue line is for τ = 5/2 with

4

2

B.Leptokurtic case: τ > 1

Analogously to the platykurtic case, we denote fsand msas the free energy and the magnetization per spin for

this regime of τ. The expansion of the magnetization Eq. (10) is valid for this case as well, but the averages over the

disorder ?...?H must be made according to PDF (3),

J2

2m2

β

−∞

?+∞

where in this case the integration limits are taken in the range (−∞,+∞).

By considering PDF (3), the above presented procedure for the determination of the critical frontiers can be

employed once more. In other words, Eq. (14) provide the continuous critical line of the phase diagram. Using this

procedure, we have numerically evaluated the critical frontiers separating the paramagnetic and ferromagnetic phases

for typical values of τ > 1. Like the platykurtic case, the leptokurtic case has only given negative values of B, i.e.,

no other than continuous phase transition occurs. The phase diagram is shown in Fig. 3, on the plane formed by the

temperature and the PDF width ω (in units of J), for some specific values of τ > 1. Still, the lines represent numerical

solutions of Eq. (14), while at the same time the points were analytically obtained through a zero-temperature analysis,

which is going to be discussed shortly. As we have perceived in the platykurtic case, the ferromagnetic phase is reduced

by augmenting τ. Similar behavior was found in the Gaussian [17] and the double-Gaussian RFIM [21] by increasing

the standard deviation of such PDFs. However, a chief difference emerges. For distributions with fatter tails than the

Cauchy-Lorentz PDF, the concavity of the critical frontier changes in the high-temperature region. So far as we are

aware, this is the first time that such a change is observed in the mean-field RFIM phase diagram.

fs =

s−1

?+∞

dH Ps(H)log[2coshβ(Jms+ H)],(17)

ms =

−∞

Ps(H)tanhβ(Jms+ H) ,(18)

Page 7

IV.ZERO TEMPERATURE ANALYSIS

Moving forwards, we now consider the phase diagram of the model at zero temperature. As in the finite-temperature

case, we evolve twofold: the platykurtic case and the leptokurtic case, τ < 1 and τ > 1, respectively.

A.Platykurtic case: τ < 1

In the limit T → 0, the free energy and magnetization become1, respectively,

Γ

?

2(2 − τ)Γ

fi =

5−3τ

2(1−τ)

?

?

2−τ

1−τ

?

?

?

1 − τ

(5 − 3τ)π

4

(5 − 3τ)σ3

(2 − τ)2F1

1

2,

1

τ−1;3

2;

(1−τ)J2

(5−3τ)σ2m2

i

?

J2

σ

m2

i+

?

2

?

1 −(1 − τ)J2

(5 − 3τ)σ4m2

?1

i

?1+

2;1

1

1−τ

− 1 −?σ2?σ2− 1??1+

(1 − J mi)

1

1−τ

?

+

2(2 − τ)

?5 − 3τ

1 − τ

2F1

2,

1

τ − 1;3σ2

?

?

,(19)

and

mi= 2

?

1 − τ

(5 − 3τ)π

Γ

?

5−3τ

2(1−τ)

?

?

Γ

2−τ

1−τ

?

?J

σ

?

2F1

?1

2,

1

τ − 1;32;

(1 − τ)J2

(5 − 3τ)σ2m2

i

?

mi, (20)

where2F1[.,.;.;.] is the Gauss hypergeometric function [34]. In the same way as in the finite-temperature analysis,

we expand the above magnetization (20) in powers of mi, so that

mi= aimi+ bim3

i+ cim5

i+ O(m7

i),(21)

where

ai = −2

?J

σ

??

(1 − τ)3

(5 − 3τ)π

Γ

?

5−3τ

2(1−τ)

?

?3/2Γ

?

Γ

1

1−τ

? ,

?

Γ

?

5−3q

2(1−q)

?

(22)

bi = −

2

3√π

?J

σ

?3?1 − τ

5 − 3τ

5−3τ

2(1−τ)

1

1−τ

?

? ,

?

(23)

ci =

1

5√π

?J

σ

?5?

1 − τ

(5 − 3τ)5τ

Γ

?

Γ

2−τ

1−τ

? .(24)

The continuous critical frontier at zero temperature is obtained for ai= 1,

σ

J=2(1 − τ)

√π

?1 − τ

5 − 3τ

?1/2Γ

?

5−3τ

2(1−τ)

?

?

Γ

1

1−τ

? ,(25)

providing that bi< 0, which occurs for all τ < 1. The last-mentioned equation allows determining the exact point

at which the critical frontiers obtained in section 3.A reach the zero-temperature axis (the circles in Fig. 2). The

zero-temperature phase diagram is shown in Fig. 4.

1For the purpose of obtaining the following expressions we made use of the integrals presented in Ref. [33].

Page 8

B.Leptokurtic case: τ > 1

In this regime, PDF (3) presents a distinct behavior for 1 < τ < 5/3 and τ > 5/3. Explicitly, the former case

corresponds to the case in which the standard deviation is finite and the latter to the case for which the distribution

has the same asymptotic behavior as the L´ evy distribution.

1.Finite standard deviation: 1 < τ < 5/3

For this range of τ, the free energy and the magnetization become,

fs =

?

5 − 3τ

(τ − 1)π

Γ

?

2(τ−1)

2−τ

τ−1

3−τ

?

Γ

?

? σ

1

2,

τ − 1;3

?

1 −

1 − τ

5 − 3τ

?J

σ

?2

m2

s

?

1

1−τ+1

+

2(2 − τ)

?J

σ

?2

2F1

?

1

2;−1 − τ

5 − 3τ

?J

σ

?2

m2

s

?

m2

s

?

,(26)

and

ms = 2ms

?J

σ

??

τ − 1

(5 − 3τ)π

Γ

?

2(τ−1)

?J

1

τ−1

3−τ

?

Γ

?

? ×

m2

(27)

2F1

?

1

2,

1

τ − 1;32;−1 − τ

5 − 3τσ

?2

s

?

,

respectively. Similarly to the τ < 1 analysis, we can expand the magnetization ms, Eq. (27), in powers of ms,

ms= asms+ bsm3

s+ csm5

s+ O(m7

s),(28)

where,

as = 2

?J

σ

??

τ − 1

(5 − 3τ)π

Γ

?

2(τ−1)

Γ

1

τ−1

3−τ

?

Γ

?

?,

1

τ−1

5−3τ

2(τ−1)

1

τ−1

3−τ

2(τ−1)

(29)

bs = −4

3

?J

σ

?3?

(τ − 1)3

(5 − 3τ)5π

?

?

Γ

?

?,(30)

cs =

?J

σ

?5τ

5

?

τ − 1

(5 − 3τ)5π

Γ

?

?

Γ

?

?.(31)

The continuous critical frontier at zero temperature is obtained for as= 1,

σ

J=

2

√π

?

τ − 1

(5 − 3τ)

Γ

?

2(τ−1)

1

τ−1

3−τ

?

Γ

?

? , (32)

as long as bs < 0. In the range 1 < τ < 5/3, we notice that the coefficient bs is always negative, indicating the

occurrence of continuous phase transitions for all values of τ. This expression permit us to determine the values of

σ/J, or equivalently, the values of ω/J [see Eq. (5)] at T = 0 of the phase diagrams depicted in Fig. 3. In Fig. 4, we

show the zero-temperature phase diagram, on the plane ω/J vs τ.

Page 9

?10123

0.0

0.2

0.4

0.6

0.8

1.0

Τ

Ω?J

Platykurtic Leptokurtic

P

F

FIG. 4: (Color online) Zero-temperature phase diagram separating the Ferromagnetic (F) and the Paramagnetic (P) phases

for platykurtic (τ < 1) and leptokurtic (τ > 1) distributions. The horizontal dotted line represents the limiting case τ → −∞,

i.e., the uniform distribution (ω/J = 1), the dashed vertical line represents the limit for finite standard deviation (τ = 5/3)

and the dot-dashed line the limit for finite average (τ = 2). The points emphasize the intersection between vertical lines and

the critical line. They correspond to values of ω/J equal to?

2/π, 1/√2 and 2/π, respectively.

2.Finite width: 5/3 < τ < 3

Mark that in this range we must use ω. Thus, as previously, the free energy and magnetization are respectively,

fs =

?

τ − 1

(3 − τ)π

?2

Γ

?

2(τ−1)

1

2,

1

τ−1

3−τ

?

Γ

?

? ω

2;−1 − τ

3 − τ

(3 − τ)

(2 − τ)

?

1 −1 − τ

3 − τ

?2

?J

ω

?2

m2

s

?

1

1−τ+1

+

2

?J

ω

2F1

?

1

τ − 1;3

?J

ω

m2

s

?

m2

s

?

,(33)

and

ms= 2ms

?J

ω

??

τ − 1

(3 − τ)π

Γ

?

2(τ−1)

1

τ−1

3−τ

?

Γ

?

?2F1

?1

2,

1

τ − 1;32;(τ − 1)J2

(τ − 3)ω2m2

s

?

.(34)

Analogously to the above cases, we expand the magnetization ms, Eq. (34), in powers of ms,

ms= asms+ bsm3

s+ csm5

s+ O(m7

s), (35)

with the coefficients,

as = 2

?J

ω

??

τ − 1

(3 − τ)π

Γ

?

2(τ−1)

1

τ−1

3−τ

?

Γ

?

?, (36)

Page 10

FIG. 5: (Color online) Tri-dimensional phase diagram of the model in the axis temperature, τ and ω/J, separating the

Ferromagnetic (F) and the Paramagnetic (P) phases. We have used a darker color to represent the regime of τ (τ > 2) in

which we have determined a divergent free energy per spin analytically found at T = 0.

bs = −2

3

?J

ω

?3?

1

π [(4 − τ)τ − 3]

Γ

?

2(τ−1)

Γ

?

Γ

?

1

τ−1

3−τ

?

Γ

?

?,

?

(37)

cs =

2

45

?J

ω

?5?

τ − 1

(3 − τ)π

(5 + 9τ)

(3 − τ)2

1

τ−1

3−τ

2(τ−1)

?,(38)

Thus, the continuous critical frontier is given by

ω

J=

2

√π

?τ − 1

3 − τ

?1/2 Γ

?

2(τ−1)

1

τ−1

3−τ

?

Γ

?

? ,(39)

and we have again verified that bs< 0 for all values 5/3 < τ < 3. We can see in Fig. 4 the zero-temperature phase

diagram in the plane containing the width ω and the generalized degree of freedom τ.

In this case, it is worth noticing an important result. In the free energy per spin (33), the integrals are finite only

for τ < 2, i.e., the free energy at temperature equal to zero is not finite for probability density functions broader than

the Cauchy-Lorentz. Although we do not have an unequivocal physical account for this phenomenon, we introduce

some insight into this result with the help of the statistical meaning of our distributions. As mentioned in section 2,

for τ > 1, distribution (3) is understood as a generalization of Student-t for real degrees of freedom according to the

relation,

τ =3 + m

1 + m.

(40)

The Cauchy-Lorentz distribution, Eq. (3) with τ = 2, corresponds to the case for which the distribution presents

a divergence in the average but a null average value of the corresponding escort distribution. The divergence of the

mean value of the free energy for τ > 2 emerges from that feature of the property of Eq. (3). Moreover, this divergence

was experimentally observed in organic charge-transfer compounds [24].

In order to summarize the results presented in the manuscript, we show in Fig. 5 a tridimensional phase diagram

separating the ferromagnetic (F) and the paramagnetic (P) phases defined by the axis temperature (in units of J),

Page 11

τ and ω (also in unit of J). We observe a contraction of the ferromagnetic phase for increasing values of τ. We have

spotted the above-described change in the concavity of the critical frontier for τ > 2, as well as the dwindling of the

ferromagnetic phase (for increasing values of τ) which in limit τ → 3 turns into the point ω = 0.

V. CONCLUDING REMARKS

In this work we have investigated the infinite-range-interaction Ising model in the presence of a random magnetic

field following a family of continuous probability density functions, defined by a parameter τ comprising the r-

distribution, for τ < 1, and the Student-t, for τ > 1, which have already found their statistical relevance within

other contexts of disordered systems. Moreover, specific PDFs like the Gaussian (τ → 1), the uniform (τ → −∞)

and the Cauchy-Lorentz (τ = 2) are obtained thereof. Independently of τ, we have observed a continuous phase

transition with the lessening of the ferromagnetic phase in thek T

by 0 ≤k T

noted the appearance of an inflexion point for finite

Jand

energy per spin at null temperature for which we have provided with an explanation based on the statistical nature

of distributions that are fatter than the Cauchy-Lorentz.

As an extension of this work, a numerical study by means of Monte Carlo simulations of the model defined by

Eqs. (1), (2) and (3) in the case of nearest-neighbors interactions is though to bring a better understanding of the

physical properties of the Ising model in the presence of random magnetic fields that follow continuous probability

distributions [35].

J

vsω

Jplane that corresponds to the region defined

ω

J=k T

J

= 0 for τ → 3. For τ ≥ 2, we have

that is also associated with a divergence of the free

J

≤ 1 and 0 ≤ω

J≤ 1 in the uniform case and to the point

ω

k T

J

Acknowledgments

The authors acknowledge F. D. Nobre for discussions on several aspects of disordered magnetic systems. SMDQ

benefits from financial support from the European Union’s Marie Curie Fellowship Programme and NC and DOSP

thank the financial support from the Brazilian agency CNPq.

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