The Anisotropic van Hemmen model with a random field in a random network

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The Anisotropic van Hemmen model with a random field in a random network Alexandre Silveira, S.G. Magalhaes, R. Erichsen Jr.

∗

Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil

article

info

Article history: Received 29 January 2019 Received in revised form 26 September 2019 Available online xxxx Keywords: Ferromagnetism Disordered systems Random networks Random field van Hemmen model

a b s t r a c t In this paper we investigate the three-state spins van Hemmen model with a crystalline field in the random network. The van Hemmen model is known for being treated without the use of replicas. The same method used by van Hemmen is utilized here to average over disordered exchange interactions. To deal with averages over the realizations of the lattice we utilized the replica symmetry formalism of order parameter functions. The order parameters calculated to detect which phase the system encounters are calculated numerically by means of a population dynamics algorithm. Firstly we obtained phase diagrams in low fixed temperature in the diagram crystalline field versus random field, we observe higher connectivities to give rise to segregation between high activity and low activity spin glass (SG) phases, also we verify the appearance of a tricritical point in the low crystalline field region. Finally to account the effects of high thermal fluctuations we have drawn phase diagrams in the temperature versus random field plane for fixed values of crystalline field. Here we observe an important modification of the phase plane topology by the increment of network connectivity. It is also notable the presence of a reentrant behavior in first and second order phase transitions when the crystalline field is sufficiently large. © 2019 Elsevier B.V. All rights reserved.

0. Introduction The two manifestations of disorder in magnetism, random exchange interactions and random fields, are usually studied independently. However, there are many examples in real physical systems of the simultaneous presence of two forms of disorder [1]. In terms of spin models, it is known that the presence of only one of these forms of disorder is enough to produce drastic changes concerning their phase transitions. Therefore, one can expect that the simultaneous presence of both forms of disorder can produce even more drastic changes in its phase diagrams. The effects of adding disorder in spins models can be well illustrated in the Blume–Capel [2,3] and the Ghatak– Sherrington ones [4]. In fact, the Ghatak–Sherrington model can be considered a type of disordered counterpart of the Blume–Capel model since the uniform exchange interaction is replaced by a random one which provides not only disorder but also frustration. Furthermore, both models have important ingredients that leads to a first order phase transition: (i) 3-state spins and, (ii) the presence of a crystal field term. The presence of a bimodal random field in the Blume–Capel model leads to the unfolding of the original ferromagnetic phase in two distinct ferromagnetic phases with the appearance of several multicritical points [5]. In the Ghatak–Sherrington model there is no unfolding of the ferromagnetic phase, but this phase is replaced by a SG one with a tricritical point in the transition spin glass-paramagnetism [4]. ∗ Corresponding author. E-mail address: [email protected] (R. Erichsen). https://doi.org/10.1016/j.physa.2019.123267 0378-4371/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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It should be remarked that the results mentioned above about the Blume–Capel with random field and the Ghatak– Sherrington models have one common aspect. They were obtained at mean field level. This approximation can be described as a lattice model having infinite connectivity among the spin sites. Nevertheless, it has been demonstrated recently that many of results concerning the phase transitions in the fully connected are substantially modified when this model is placed in a random network where the site connectivity is finite. Particularly, that is the case of random field Blume–Capel model [6]. It should be also remarked that the mean field results have to be recovered when the connectivity is large enough. Therefore, one can ask what happens with the phase transitions in a model combining random exchange interaction and random field if the connectivity is finite. Are the unfolding of phases and the existence and multiplication of multicritical points robust? The goal of the present work is to study an anisotropic 3-state spins model containing a random exchange interaction and a bimodal random field, using a method which allows to control the lattice connectivity. However, there is one important technical issue related to the way the disorder is treated. For instance, in the case of the Ghatak–Sherrington model, it is used the replica method with replica symmetry and replica symmetry breaking (RSB) solutions [7]. It is known that the theoretical study of a spin glass (SG) with random field using replicas imposes a major difficulty. The reason is that the random field couples with the local magnetization, precluding the SG order parameter to assume a zero value. This makes highly non-trivial to localize the glassy transition without a replica symmetry breaking calculation which multiplies the number of order parameters [8–11]. In addition, the precise location of first order transitions spin glassparamagnetism is also extremely complicated when the problem is treated within the replica method. The reason is that there is no more a clear criteria of stability guiding the location of this type of phase transition [12]. One interesting alternative model which allows to overcome the previously mentioned problems has been proposed by van Hemmen [13,14]. The fully connected van Hemmen model is differentiated in relation to the Sherrington–Kirkpatrick model [15] in one very important aspect: the replica symmetry (RS) solution is already exact. That is directly related with the specific form of the Jij in that model which is given as a product of random variables [16,17]. As a consequence, it is not necessary any investigation on the stability of the RS solution for the order parameters nor the use replica symmetry breaking. On the other hand, the infinite range interaction or, the fully connected van Hemmen model does not have a complex free energy landscape. Despite of that, the van Hemmen model is the simplest on-site disorder model for SG which does have nontrivial disorder in the sense that the frustration cannot be eliminated by a local gauge transformation. In the presence of a random field, the van Hemmen SG parameter filters the random field effects, and the glassy transition becomes clear. Other important advantage of the van Hemmen model is to locate easily the first order phase transitions. Since there is no problem with the replica method, the free energy can be used again. Indeed, our study offers a novel alternative analytic route to both, numerical simulations and standard mean field approaches. A recent study using both forms of disorder with infinite connectivity has demonstrated not only the unfolding of ferromagnetic phase but also the unfolding of the SG phase and mixed ones with a multiplication of multicritical points [18]. Our results reveal a rich thermodynamic scenario where these unfoldings of phases and multicritical points are strongly dependent on the average network connectivity. It is important to remark that we use the van Hemmen rule to model the exchange interactions. However, we have changed the network architecture. Instead of the fully connected network, we changed to a random network with finite connectivity. To deal with the averages over Jij (ξi , ηj ) we use a method that was developed by van Hemmen in a work about nonlinear neural network [19]. The procedure to deal with realizations of the network matrix cij was previously utilized in the context of satisfiability problems [20] which introduces order parameter functions. We use a combination of the two methods which have already been applied by Coolen [21] to study the Hopfield model in a random network. We restrict our study to the RS solution. The issue of RS solution stability in the present model will be discussed in the Conclusions section. This paper is organized as follows: the model and the replica procedure for van Hemmen interactions and random networks are presented in Section 1, followed by the results and discussion in Section 2. Conclusions and further remarks can be found in Section 3. 1. Model and replica procedure We consider a system of N → ∞ spins assuming states σi = 0, ±1, where i = 1 . . . N is the spin index, with the Hamiltonian given by H(σ ) = −

1∑ c

cij Jij σi σj +

(i,j)

∑

hi σi + D

i

∑

σi2 .

(1)

i

The cij are independent, identically distributed random variables (i.i.d.r.v.) indicating if the spins i and j are interacting (cij = 1) or not (cij = 0), chosen according the probability distribution

( )

p cij =

c

( c) δcij ,1 + 1 − δcij ,0

N N being the average connectivity c a finite constant. The Jij are given by Jij = J(ξi ηj + ξj ηi ) .

(2)

(3)

Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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The variables ξi and ηi are independent, identically distributed random variables (i.i.d.r.v.), each one assuming values +1 or −1 with equal probability. This form of coupling, proposed some time ago by van Hemmen [13,14], avoids the use of replicas when averaging over the disorder on fully connected networks calculations. The local fields hi are i.i.d.r.v. assuming values +h0 and −h0 with equal probability. Here, we use the replica method to average over the cij , but keep the van Hemmen couplings in order to well localize the glassy transition in the presence of a random field, in conjunction with a finite connectivity network. In order to obtain the thermodynamic properties, the disorder-averaged free energy density is calculated through the standard replica procedure, 1

f (β ) = − lim

N →∞

βN

lim

n→0

1 n

log⟨Z n ⟩ ,

(4)

where Z =

e−β H(σ )

∑

(5)

σ1 ...σN

is the partition function. The angle brackets in Eq. (4) stand for the disorder average. The natural order parameter for a random finite connectivity model is the distribution of local fields Wξ η (x, y) [20], where x and y are fields that couple with the local magnetization σ and the local activity σ 2 , respectively. The index ξ η refers to the four sub-networks in which the network is divided according the local specification of random variables ξ and η [21]. A detailed development of the finite connectivity replica symmetric (RS) theory is presented in the Appendix. The main outcome is the saddle-point equation for the local field distribution, Wξ η (x, y) =

k ∑ e−c c k ⟨⟨ ∫ ∏

k!

k

dxl dyl Wξl ηl (xl , yl )

(6)

l=1

k k ( ) ( )⟩ ⟩ 1 ∑ 1 ∑ × δ x−h− φ1 (xl , yl ) δ y − D + φ2 (xl , yl ) . {ξl ηl } h β β l=1

(7)

l=1

where

χ+1 (xl , yl ) , χ−1 (xl , yl ) χ+1 (xl , yl )χ−1 (xl , yl ) φ2 (xl , yl ) = log 2 χ0 (xl , yl ) φ1 (xl , yl ) =

1

2 1

log

(8) (9)

and

χτ (xl , yl ) =

∑

[

exp β xl σ − β yl σ 2 +

σ

β ( c

]

J ξ ηl + ξl η τ σ ,

)

(10)

where the summation is over σ = {−1, 0, +1}. The sub-networks are introduced in order to average over the ξi and ηi randomness, such that: Iξ ,η = {i|ξi = ξ , ηi = η}. The method to solve Eq. (7) will be explained in Section 2. Since we are interested in the spin-glass phase, the order parameters are the van Hemmen spin-glass order parameter [13]

∫

q=

1 ⟨⟨ 2

dx dy Wξ η (x, y)(ξ + η)⟨σ ⟩

⟩ ⟩ ξη h

,

(11)

which evaluates the global order of the system and the activity order parameter, which gives the average magnetic occupation with σ = 0 or ±1, as Q =

⟨⟨∫

dx dy Wξ η (x, y)⟨σ 2 ⟩

⟩ ⟩ ξη h

,

(12)

where

⟨σ ⟩ =

2e−β y sinh(β x)

(13)

1 + 2e−β y cosh(β x)

and

⟨σ 2 ⟩ =

2e−β y cosh(β x) 1 + 2e−β y cosh(β x)

.

(14)

The thermodynamic phases studied are characterized by the order parameters, such that: when the system is in a SG phase, q > 0 and Q > 0. For the paramagnetic phase (P), q = 0 and Q > 0 and the non-magnetic phase (NM) q = 0 and Q = 0. Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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The RS free-energy is given by f (β ) =

⟨∫

c

dx dy dx′ dy′ Wξ η (x, y)Wξ ′ η′ (x′ , y′ )

2β

⟩ × log GE (x, y, x , y ) ′

′

− ξ η,ξ ′ η′

1

⟨⟨

∑ e−c c k

β

⟨∫

k!

k

⟩

k ∏

dxl dyl

l=1

⟩⟩

× Wξl ηl (xl , yl ) log GS ({xl }, {yl })

,

(15)

ξ l ηl h ξ η

where

χ (x, y, x′ y′ ) , χ0 (x, y)χ0 (x′ , y′ ) [ ∑ ) ] β ( χ (x, y, x′ ,′ ) = exp β xσ − β yσ 2 + β x′ τ − β y′ τ 2 + J ξ η′ + ξ ′ η σ τ ,

GE (x, y, x′ , y′ ) =

c

στ

(16) (17)

and GS ({xl }, {yl }) = 1 + eβ h−β D

∏ χ+1 (xl , yl ) l

χ0 (xl , yl )

+ e−β h−β D

∏ χ−1 (xl , yl ) l

χ0 (xl , yl )

.

(18)

2. Results and discussions Before presenting our results, it worths to briefly discuss the numerical procedure. All the order parameters and the free-energy can be calculated under the solution of Eq. (7) by a population dynamic algorithm like in [22]. First, a population of N two coordinate vector fields is created for each sub-cell. There are four sub-cells, since ξ = ±1 and η = ±1. Then, the following iterative procedure is applied: in each step a number k is chosen according to a Poissonian distribution and a random field h is picked out from the random field distribution. Then, k local fields (xl , yl ) and their respective sub-cells are randomly picked out from the population. Finally, another local field, randomly chosen, x, y is updated according to x=h+

k 1 ∑

β

φ1 (xl , yl )

(19)

φ2 (xl , yl ) .

(20)

l=1

and y=D−

k 1 ∑

β

l=1

This procedure is repeated till the local field distribution converges in all sub-cells. In order to visualize the local field distribution, we define the one-dimensional local field distribution by

wξ η (x) =

∫

dy Wξ η (x, y) .

(21)

Fig. 1 shows examples of low temperature (T = 0.03) one-dimensional local field distributions for h0 = 0.2, D = 0.464 and c = 12, where h0 stands for the random field. These coordinates are over the coexistence line between the low and high activity SG phases. See Fig. 1. Fig. 2 shows examples of order parameters q and Q and the free-energy f curves in function of the crystal field D, for the random field amplitude h0 and temperature T and average connectivity c as in Fig. 1. This set of parameters was chosen because it allows to illustrate the localization of two first order transitions. To discuss the role played by the crystal field D and the random field amplitude h0 , we start by presenting the ground state phase diagrams for some representative values of the average connectivity c, in Fig. 3. The results show the evolution of the SG phase as the average connectivity increases. For small connectivities, like c = 4 that is shown in the top left panel of Fig. 3, the SG phase is restricted to small values of h0 and D. This phase undergoes a continuous transition to the paramagnetic phase P as h0 increases. At large D, SG undergoes a discontinuous transition to the NM phase. This can be understood by the fact that D favors the non-magnetic local state σi = 0, while h0 favors the active states σi = ±1, although in a disordered way. The figure still shows a continuous transition between P and NM. The open circle signals the bicritical point located at the encounter of a discontinuous transition line between SG and NM and two continuous transition lines, between SG and P and between P and NM. As the average connectivity increases, the SG phase is reinforced. The top right panel of Fig. 3, for c = 6, shows the appearing of a SG phase like a strip between phases NM and P. This can be devoted to the increase of the network Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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Fig. 1. Two co-existing solutions for the one-dimensional local field distributions for T = 0.03, h0 = 0.2, D = 0.464: the low activity solution (left panel) and the high activity solution (right panel). See the text.

Fig. 2. Order parameters and free-energy as a function of the crystal field for h0 = 0.2, T = 0.03 and c = 12. Solid (dashed) line is the high (low) activity SGH (SGL ). Thin solid line is the NM solution. The left arrow in the free-energy curve signals the transition between the two SG phases, while the right arrow signals the SG to NM transition.

Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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Fig. 3. Ground state phase diagram D vs. h0 for c = 4 (top left); c = 6 (top right); c = 8 (bottom left); c = 12 (bottom right). Solid (dashed) lines correspond to continuous (discontinuous) transitions. Open (closed) circle signals a bicritical (tricritical) point.

frustration, due to the increase in c and due to the competition between the random and crystal field. As the connectivity increases to c = 8 and c = 12, shown, respectively, at the bottom left and bottom right panels of Fig. 3, the width of the strip increases. The bottom right panel, for c = 12, deserves a deeper discussion. There appears a tricritical point over the SG to P transition, such that this transition becomes discontinuous at low D values. Furthermore, the SG phase unfolds in two, one with low activity (SGL ) and one with high activity (SGH ). Both phases have the same SG order parameters, but differ in the parameter activity, as shown in Fig. 2. The left arrow in the free-energy signals the discontinuous transition between the two SG phases, and the right arrow signals the discontinuous transition to the NM phase. A more detailed picture of the coexisting phases is shown in Fig. 1. This figure shows the local field distribution for the four sub-lattices, for the two coexisting phases over the transition line. As it can be seen, only the W1−1 (x) and W−11 (x) differ. The high activity solution having wider distributions than the lower one. The one dimensional y distributions shows a similar behavior, with W1−1 (y) and W−11 (y) being enlarged in the high activity solution. The unfolding of SG phase in (SGL ) and (SGH ) for c = 12 indicates that this value of connectivity already approaches the fully connected c → ∞ three-state van Hemmen model under a random field, discussed in Ref. [18]. Temperature effects, when superimposed to average connectivity variation, produces a remarkable richness of behaviors. In order to make the discussion more comprehensive, we present the temperature vs. random field phase diagrams for some representative crystal field and average connectivity values. We start with the low average connectivity regime c = 4. Fig. 3 (top left) suggests that , for c = 4, the most interesting range to investigate is where the crystal field varies in the range 0.6 to 0.75. For D ≤ 0.6 the SG phase is confined to the low h0 , low T region of the T vs. h0 phase diagram, as shown in the left panel of Fig. 4. This kind of phase diagram is found in general for D ≤ 0.6 and c ≤ 12. For D = 0.72, shown in the middle panel of Fig. 4, there is a strong reentrant effect, with the appearing of a NM phase in the low h0 , low T region. This occurs because the crystal field favors the inactive state, while the random field favors the active states, as do the temperature. Then the SG phase appears as a compromise between h0 and T . At large h0 and T the high noise level favors the P phase. For D = 0.72 the transition between NM and SG is discontinuous. For D = 0.75, shown in the right panel of Fig. 4 we found a similar T vs h0 phase diagram, but with a continuous transition between NM and SG. We guess that there is a short interval in the range 0.72 ≤ D ≤ 0.75 where there is a tricritical point in the transition between or NM and SG. It still worths to remark that the continuous transitions from SG to P are markedly reentrant for D = 0.72 and D = 0.75. We will return to this point later. Next we show the phase diagrams for c = 8. For D ≤ 0.5 there is nothing new to remark about the T vs. h0 , since the phase diagrams are similar to that shown in the left panel of Fig. 4, except that the SG to P transition is no longer reentrant. The left panel of Fig. 5, for D = 0.6, shows a strong reentrance with a discontinuous NM to SG transition, like that observed for c = 4 and D = 0.72. A remarkable changes is found for D = 0.7, shown in Fig. 5, right panel. The SG phase is displaced for higher values of h0 , reflecting the fact that it results as a compromise between the crystalline and the random fields. The higher the D the higher h0 range where SG is present. This can be already previewed in the ground state phase diagram, where SG occupies a diagonal strip region. As the average connectivity increases, the general results approaches those obtained in mean field calculations, that corresponds to c → ∞, and were discussed in Ref. [18]. We show in Fig. 6 some representative phase diagrams for c = 12. For D = 0, shown in the left panel, the transition SG to P becomes first order at low temperature, with a tricritical point Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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Fig. 4. T vs. h0 phase diagrams for c = 4 for some representative values of the crystal field: D = 0.6 (left); D = 0.72 (middle); D = 0.75 (right). Solid (dashed) lines correspond to continuous (discontinuous) transitions.

Fig. 5. T vs. h0 phase diagrams for c = 8 for two representative values of the crystal field: D = 0.6 (left); D = 0.7 (right). Solid (dashed) lines correspond to continuous (discontinuous) transitions.

Fig. 6. T vs. h0 phase diagrams for c = 12 for two representative values of the crystal field: D = 0 (left); D = 0.6 (right). Solid (dashed) lines correspond to continuous (discontinuous) transitions.

at T ≃ 0.3 and no reentrance. It worths to remark that reentrance is observed in the NM to SG transitions, even when they are second order, but not in the SG to P transitions. This point deserves a further comment, see below. The right panel shows the phase diagram for D = 0.6, where the SG to P transition remains second order. To close this section we present an argument to justify the reentrant behavior observed in the NM to SG transitions. This argument is an extension to the NM to SG transition of one argument that was used to justify the reentrance in the NM to Ferromagnetic (F) phase in Refs. [6,23]. A large value of the crystal field D favors the inactive states. Then, consider a small set of three sites, the center site being of coordination two, and being the only connection path between the lateral sites. When the center site is in the inactive state the lateral sites are disconnected. Then we calculate the absolute value of the effective coupling between the lateral sites by summing over all the states of the center site, at a given temperature. This results

|Jeff | =

c 2β

⟨ log

(

1 + 2e−β D cosh

4β J c

− βh ( )

1 + 2e−β D cosh

4β J c

)⟩ .

(22)

h

The effective coupling is positive in the NM to F transition, and can be positive or negative in the NM to SG transition, that is the present case. Curves of |Jeff | vs T , for some values of D, for c = 4 and h0 = 0.2 are shown in Fig. 7, where we can observe that the effective coupling becomes an increasing function of the temperature for sufficiently large D values. This way, it seems reasonable that in some specific circumstances a certain thermal noise level favors the active states Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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Fig. 7. Effective coupling as a function of the temperature for c = 4, D = 0.25 (solid line), D = 0.5 (dashed line) and D = 0.75 (dash-dotted line).

and, consequently, the SG phase. It worths to stress that the entropy has the correct behavior, since the NM state is a highly ordered phase where all the sites are in the inactive state at T = 0. This simple analysis also allows to understand why the NM to SG is reentrant while the SG to P is not, except for small c values. For large D the first one is observed in the small random field side of the SG phase where the inactive states are favored, and the second one is observed in the large random field side of SG, where the spins are already activated by the random field and the temperature effects are less important. For sufficiently small c, like in Fig. 4, even the SG to P transition becomes reentrant. For such low c, small random field amplitudes already destabilizes the SG phase, displacing the transition to low h0 values, where temperature effects become relevant. 3. Concluding remarks Random networks of finite average coordination number can be analytically investigated by an improved replica method. In this paper this method was combined with the van Hemmen model for three state spins to produce an original SG model. As a further ingredient, a zero average, unitary variance, binary random field was included. It worths to stress that the van Hemmen model allows to study the glassy transition in the presence of a random field. We have shown that the average connectivity c has strong effects on the general model behavior. For small c the SG phase is confined to low random field amplitude h0 and crystal field D. For simplicity, we have restricted our analysis to the range 0 ≤ h0 ≤ 1, 0 ≤ D ≤ 1. For increasing c the SG phase appears as a competition between h0 , that favors randomly the active states and D, that favors the inactive states. Temperature effects were also investigated, with the production of T vs h0 phase diagrams for several D and c combinations. In general lines, we want to remark the appearing of strong reentrance effects mainly at small c, to which an explanation was addressed based upon the temperature dependent effective coupling. Also deserves attention the appearing of tricritical points on the NM to SG and the SG to P transitions. As one could expect, the general behavior approaches that of the infinite range mean field theory when c increases. We point out that the changes in the phase diagrams obtained in the present work could also be interpreted as a result of the changing from infinite range interaction (infinite connectivity) to finite range interaction (finite connectivity). The results presented in this work refers to disordered magnetic systems with no net magnetization. Nevertheless, by adding a positive, constant term to the coupling (Eq. (3)), a similar setup may be employed to investigate the interplay between spin-glass and ferromagnetic phases in the presence of random and crystalline fields. Further developments may include more than one set of random variables, e.g. {ξα , ηα }, and letting α run from 1 to p, where p is a natural. We restricted ourselves to the study of the RS solution. Although we believe that solution is probably stable due to the low level of disorder, RSB effects cannot be completely ruled out. Nevertheless, already the first step RSB scheme implies to deal with populations of fields [20] and this is beyond the scope of the present work. However, if there is a RSB phase in our modified van Hemmen model, one could expect important differences regarding the RS solution [24]. An alternative that we are currently investigating is to check the stability of the RS solution using the two replica method as introduced by Kwon and Thouless [25] in the Bethe lattice. To conclude, although our results refer to a particular magnetic model and, therefore, it refers to problems containing the interplay among random interaction, random field, ferromagnetic interaction and crystal field, we expect that our results help also to clarify problems of broader interest as, martensitic alloys. In these systems where it appears strain glasses which is the analogue of SG concerning lattice distortions [26,27]. Acknowledgments The present study was supported by the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil and CAPES, Brazil. We also acknowledge the support from UAM-Santander, Spain CEALAL/2017-08. Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

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Appendix The replicated partition function is given by

⟨ n⟩ Z

=

⟨ ∑

{

exp

n ∑∑

β

σ 1 ...σ n

− βD

n ∑∑

σi2α +

α=1

i

hi σiα

(23)

α=1

i

β∑ c

cij Jij

i
n ∑

σiα σjα

}⟩ {ξi }{ηi }{cij }{hi }

α=1

,

where σ α is a vector with components σ1α . . . σN α representing the state of the replica α . Averaging over {cij } and introducing Eq. (3) this can be rewritten, in the limit N → ∞,

⟨ n⟩ Z

=

⟨ ∑

{

exp

∑

β

σ 1 ...σ n

N

∑

i,α

c ∑⟨

+

hi σiα − β D

e

σi2α

(24)

i,α

βJ

c (ξi ηj +ξj ηi )

∑

α σiα σjα

−1

⟩

}⟩

.

{hi }

{ξi }{ηi }

i
The partition function becomes

⟨ n⟩ Z

=

⟨ ∑

{

exp

β

∑

σ 1 ...σ n

hi σiα − β D

∑

i,α

+

i,α

c ∑ ∑ [ 2N

−1

σi2α

e

βJ

′ ′ c (ξ η +ξ η)

′ α σα σα

∑

σσ ′ ξ η,ξ ′ η′

]∑ ∑

}⟩

δσσ i δσ ′ σ j

.

{h}

i∈Iξ η j∈Iξ ′ η′

(25)

Next we introduce the sub-lattice spin distribution Pξ η (σ ), with σ = {σ1 . . . σn }, through the identity

∫ ∏

1=

{

1

[

dPˆ ξ η (σ )dPξ η exp iPˆ ξ η (σ ) Pξ η (σ ) −

ξ η,σ

| Iξ η |

∑

]}

δσσ i

,

(26)

i∈Iξ η

where |Iξ η | is the number of sites that belong to sub-lattice ξ η. Introducing this in Eq. (25) and after some standard manipulations and changing variables Pˆ ξ η (σ ) → |Iξ η |Pˆ ξ η (σ ), we have

⟨ n⟩ Z

=

∫ ∏

|Iξ η |dPˆ ξ η (σ )dPξ η (σ )exp

{

iN

∑

ξ η,σ

+

Nc ∑ 2

+N

ξ η,σ

∑

pξ η pξ ′ η′

ξ η,ξ ′ η′

∑

pξ η Pˆ ξ η (σ )Pξ η (σ )

[

Pξ η (σ )Pξ ′ η′ (σ ) e ′

βJ

′ ′ c (ξ η +ξ η)

′ α σα σα

∑

] −1

σσ ′

pξ η log

⟨∑

(

exp −iPˆ ξ η (σ ) + β h

∑

σ

ξη

σα − β D

∑

α

σα2

) ⟩}

,

(27)

α

where pξ η is the fraction of sites that belong to sub-lattice ξ η. In the limit N → ∞, the integral can be evaluated by the saddle-point method. The consequent free-energy becomes 1

f (β ) = − lim

βn ⟨ ∑ c

n→0

+

2

⟨

extrPˆ

ξ η (σ )Pξ η (σ )

{ ∑⟨ i

Pˆ ξ η (σ )Pξ η (σ )

⟩

σ

[

Pξ η (σ )Pξ ′ η′ (σ ′ ) e

βJ

′ ′ c (ξ η +ξ η)

′ α σα σα

∑

ξη

]⟩ −1

ξ η,ξ ′ η′

σσ ′

+ log

⟨∑

( ∑ ∑ )⟩ ⟩ } exp −iPˆ ξ η (σ ) + β h σα − β D σα2 .

σ

α

h ξη

α

(28)

Eliminating Pˆ ξ η (σ ′ ) through the saddle point equations we obtain, unless for a normalization factor,

⟨

( ∑⟨

Pξ η (σ ) = exp c

[

P ξ ′ η ′ (σ ′ ) e

βJ

′ ′ c (ξ η +ξ η)

′ α σα σα

∑

]⟩ −1

ξ ′ η′

σ′

+ βh

∑ α

σα − β D

∑ α

σα2

)⟩ h

.

(29)

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10

A. Silveira, S.G. Magalhaes and R. Erichsen, Jr. / Physica A xxx (xxxx) xxx

and the free-energy becomes 1

f (β ) = − lim

extrPξ η (σ )

c ∑⟨

−

Pξ η (σ )Pξ ′ η′ (σ ′ )

[

e

βJ

′ ′ c (ξ η +ξ η)

′ α σα σα

∑

βn 2 ′ σσ ( ∑⟨ ⟨ ⟨∑ [ βJ ′ ′ ∑ ]⟩ ′ exp c + log Pξ ′ η′ (σ ′ ) e c (ξ η +ξ η) α σα σα − 1

n→0

−1

{

]⟩ ξ η,ξ ′ η′

σ

∑

+ βh

ξ ′ η′

σ′

∑

σα − β D

α

σα2

)⟩ ⟩ }

.

h ξη

α

(30)

All the observables can be obtained by solving Eq. (29). To do this we assume the replica symmetric ‘‘Ansatz’’ for three states spins [28], P ξ η (σ ) =

eβ x

∫

α σα −β y

∑

dx dy Wξ η (x, y) (∑

σ

2 α σα

∑

eβ xσ −β yσ

) .

(31)

2 n

Expanding the exponential and introducing the ‘‘Ansatz’’, Eq. (29) can be rewritten as k ⟨ ∑ ∫ dxl dyl Wξ η (xl , yl ) ∑ e− c c k ∏ l l [∑ ] β xl σ −β yl σ 2 n k! e σ σl k l=1 ]⟩ ⟩ [ ∑ ∑ ∑ βJ σα σα l . × exp β xl σα l − β y l σα2l + (ξ ηl + ξl η)

⟨

P ξ η ( σ ) = eβ h

α σα −β D

∑

∑

2 α σα

α

c

α

(32)

ξl ηl h

α

After some arrangements we have

∫

βx

dx dy Wξ η (x, y)e

α σα −β y

∑

α σα

∑

∫ dx dy

=

∑ e−c c k k!

k k ⟨⟨ ∫ ∏

(

dxl dyl Wξl ηl (xl , yl ) × δ x − h −

l=1

k 1 ∑

β

φ1 (xl , yl )

)

l=1

k

) ∑ ( ⟩ ∑ 2 ⟩ 1 ∑ φ2 (xl , yl ) eβ x α σα −β y α σα ×δ y−D+ . {ξl ηl } h β

(33)

l=1

Comparing both sides of Eq. (33) we obtain Eq. (7) The free-energy has energetic and entropic contributions. The energetic contribution is the first term of the right-hand side of Eq. (30), that can be written, in the RS solution, as f E (β ) =

(∑ ×

(⟨ ∫

c

dx dy dx′ dy′ Wξ η (x, y)Wξ ′ η′ (x′ , y′ )

2β n

2 +β x′ σ ′ −β y′ (σ ′ )2 + β J (ξ η′ +ξ ′ η)σ σ ′ c

eβ xσ −β y(σ )

σσ′

)n ⟩

′

][

) .

−1

1 + 2e−β y cosh(β x) 1 + 2e−β y cosh(β x′ )

[

(34)

] ξ η,ξ ′ η′

In the limit n → 0, this reads f E (β ) =

⟨∫

c 2β

⟩ dx dy dx dy Wξ η (x, y)Wξ ′ η′ (x , y ) log GE (x, y, x , y ) ′

′

′

′

′

,

′

(35)

ξ η,ξ ′ η′

where GE (x, y, x′ , y′ ) is given by Eq. (16) The entropic contribution is the second term of the right-hand side of Eq. (30). It can be written as f S (β ) = −

1

βn

⟨

⟨ ∑

log

e

α σα −β D

∑

2 α σα

∑

σ

∏ α

k ∑ e−c c k ∏ k

∑ × Wξl ηl (xl , yl ) ×

βh

σα l

eβ xl σαl −β yl σαl + 2

βJ

k!

dxl dyl

(36)

l=1

c (ξ ηl +ξl η)σα σα l

1 + 2e−β yl cosh(β xl )

⟨∫

⟩

⟩⟩ . ξl ηl h ξ η

Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.

A. Silveira, S.G. Magalhaes and R. Erichsen, Jr. / Physica A xxx (xxxx) xxx

11

Summing over the σα l and rearranging terms using that the replicas are non-interacting, this becomes fS (β ) = −

∏ 1 + 2e

⟨ log

βn

k

×

⟨

1

∑ e−c c k k

−β yl

1+

l=1

k!

cosh(β xl + 2e−β yl

⟨∫

k ∏

[ ∑

dxl dyl Wξl ηl (xl , yl )

c

(ξ ηl + ξl η)σ )

cosh(β xl )

2

σ

l=1

βJ

eβ hσ −β Dσ

] ⟩

⟩⟩ .

n

(37)

ξ l ηl h ξ η

Summing over σ , taking the limit n → 0, f S (β ) = −

1

β

⟨⟨

∑ e−c c k k

k!

⟨∫

k ∏

dxl dyl

(38)

l=1

⟩

⟩⟩

× Wξl ηl (xl , yl ) log GS ({xl }, {yl })

, ξl ηl h ξ η

where GS ({xl }, {yl }) is given by Eq. (18). Summing Eqs. (35) and (38) we obtain Eq. (15). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

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Please cite this article as: A. Silveira, S.G. Magalhaes, R. Erichsen Jr., The Anisotropic van Hemmen model with a random field in a random network, Physica A (2019) 123267, https://doi.org/10.1016/j.physa.2019.123267.