Diluted antiferromagnet in a ferromagnetic enviroment
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Diluted antiferromagnet in a ferromagneticenviroment
M O Hase and J F F Mendes
Departamento de F´ısica da Universidade de Aveiro, 3810-193 Aveiro, PortugalE-mail: [email protected]
PACS numbers: 89.20.-a, 05.50.+q, 64.60.Cn
Abstract.
The question of robustness of a network under random “attacks” is treatedin the framework of critical phenomena. The persistence of spontaneous magnetizationof a ferromagnetic system to the random inclusion of antiferromagnetic interactions isinvestigated. After examing the static properties of the quenched version (in respectto the random antiferromagnetic interactions) of the model, the persistence of themagnetization is analysed also in the annealed approximation, and the difference inthe results are discussed. iluted antiferromagnet in a ferromagnetic enviroment
1. Introduction
The investigation on resilience of networks, which examines the persistence/breakdownof some global properties of a graph under, for instance, removal of vertices or edges, isknown to have practical importance. Many “real networks” (Internet, highways, manybiological systems, et cetera ) depend on the fact that there exist links between the nodesto ensure their functionality. A damage that breaks the interconnection between verticescan trigger a profound impact on the network[1]. The connection of these questions tothe percolation theory is clear, which has been a powerful tool to tackle these problems[2](see also [3, 4]).In this work, the question of resilience of networks is analysed in the frameworkof critical phenomena of magnetic systems, and is not directly related to the classicalproblems concerning the existence of giant components in a graph[5, 6].Critical phenomena have been exhaustively exploited in networks[7, 8], and itwas immediately noticed that it displays a quite rich thermodynamical behaviourwhen compared to regular lattice structures[11, 12, 13, 14] or gives place to nontrivialeffects[15, 16]. This shows the role of the topology of the underlying graph on criticalbehaviour. Moreover, networks constitute a path toward a more “realistic” system ofmean - field models, the main cause being the introduction of (finite) connectivity as aparameter[17, 18, 19, 20, 21, 22].This work will focus on the persistence of a order imposed by a backgroundenviroment to random inclusion of opposite effects on the network. More precisely,starting from a ferromagnetic system, some of its interactions are weakened (or evenchanged) by antiferromagnetic ones. The criterion for the “functionality” of the networkwill be the ferromagnetic order of the model, and by “resilience” it means how strongis the spontaneous magnetization against random introduction of antiferromagneticinteraction that contributes to the disorder. The model is detailed in section II, andthe thermodynamical analysis of its quenched version is given in section III, followedby a characterization of the order - disorder critical line in the annealed approximationin section IV. The critical line that determines the breakdown of the ordered phase isdiscussed in section V, and the last section is devoted to conclusions.
2. Model
Consider a (finite) graph Λ N ( | Λ N | = N ) where each vertex x ∈ Λ N allocates an Isingspin σ x ∈ {− , } . The Hamiltonian is given by H ( { σ x } , { a xy } ) = − J N X x ∈ Λ N σ x ! + J A X x,y ∈ Λ N x
0) between everypair of spins and the second sum is the antiferromagnetic interaction ( − J A <
0) betweensome pair of spins. Therefore, the network is a complete graph where the edges represent iluted antiferromagnet in a ferromagnetic enviroment { a xy } decides if two vertices, x and y , have antiferromagneticinteraction ( a xy = 1) or not ( a xy = 0), and the couplings { a xy } are independent andidentically distributed random variables that obey the distribution P ( a xy ) = (cid:16) − pN (cid:17) δ a xy , + pN δ a xy , . (2)It is known that the quenched version of model (1) without the ferromagneticterm (the first sum) is a frustrated system with a spin glass phase at lowtemperatures[23](since the model has a single sublattice). Therefore, between the twoeffects of antiferromagnetic interactions, namely the contribution to disorder and acontribution to ferromagnetic order (due to the effect of frustration arisen in some loopsof the graph), the former dominates over the later.As usual, the partition function is Z ( β, { a xy } ) := Tr { σ x } e − βH ( { σ x } , { a xy } ) , (3)where β is the inverse of the temperature T ( k B = 1) and the trace indicates thesum over 2 N states. This partition function also depends on the configuration of theantiferromagnetic interactions. Finally, given a function g = g ( { a xy } ) of the set ofrandom variables { a xy } , the average of g over the configuration { a xy } will be denotedas h g i := Z g ( { a xy } ) Y x 3. Thermodynamics The (quenched) free energy f q of the model is evaluated through the replica trick f q ( β ) = − lim N →∞ βN h ln Z ( β ) i = − lim n → lim N →∞ βN n ln h Z n ( β ) i , (5)where besides the analytic continuation in n ∈ N → n ∈ R , the order of the limits N → ∞ and n → h Z n ( β ) i , whichcan be casted as h Z n ( β ) i = (cid:18) N βJ π (cid:19) n n Y α =1 Z R dλ α Tr { σ αx } exp (cid:20) − N p − N βJ n X α =1 ( λ α ) ++ βJ n X α =1 λ α X x ∈ Λ N σ αx + p N X x,y ∈ Λ N e − βJ A P nα =1 σ αx σ αy + O (1) (cid:21) . (6)Throughout this work, upper and lower indices at spin variables indicate replica index(greek letter) and site position (roman letter), respectively.Introducing the order parameter[24] ψ ( µ ) := 1 N X x ∈ Λ N δ { σ αx } , { µ α } , (7) iluted antiferromagnet in a ferromagnetic enviroment δ { σ αx } , { µ α } := Q nα =1 δ σ αx ,µ α , the equation (6) can be written as h Z n ( β ) i ∼ (cid:18) N βJ π (cid:19) n n Y α =1 dλ α ! Z D ψ D ˆ ψe − Nφ q [ ψ, ˆ ψ ]( β, { λ α } ) , (8)where φ q [ ψ, ˆ ψ ]( β, { λ α } ) := p βJ n X α =1 ( λ α ) + Tr { µ α } ψ ( µ ) ˆ ψ ( µ ) −− p { µ α } Tr { τ α } ψ ( µ ) ψ ( τ ) e − βJ A P nα =1 µ α τ α − ln ζ [ ˆ ψ ]( β, { λ α } ) (9)and ζ [ ˆ ψ ]( β, { λ α } ) := Tr { σ α } exp " βJ n X α =1 λ α σ α + ˆ ψ ( σ ) . (10)Apart from a factor β , φ q [ ψ, ˆ ψ ]( β, { λ α } ) is just the variational free energy. Theequation (8) suggests that one should invoke the saddle - point method to determinethe stationary free energy. The extremum conditions, necessary to ensure the infimumof φ q over the suitable functions ( ψ and ˆ ψ ) and variables ( { λ α } ), are ψ ( µ ) = 1 ζ [ ˆ ψ ]( β, { λ α } ) exp " βJ n X α =1 λ α µ α + ˆ ψ ( µ ) ˆ ψ ( µ ) = p Tr { τ α } ψ ( τ ) exp " − βJ A n X α =1 µ α τ α λ α = 1 ζ [ ˆ ψ ]( β, { λ α } ) Tr { σ α } σ α exp " βJ n X α =1 λ α σ α + ˆ ψ ( σ ) . (11)In an attempt to solve the above equations, one should cast the replica symmetric Ansatz ψ ( µ ) = ψ ( P nα =1 µ α ) = Z R dhP ( h ) e βh P nα =1 µ α [2 cosh( βh )] n ˆ ψ ( µ ) = ˆ ψ ( P nα =1 µ α ) = p Z R dyQ ( y ) e βy P nα =1 µ α [2 cosh( βy )] n λ α = λ , ∀ α ∈ { , · · · , n } , (12)where P and Q are probability distributions. iluted antiferromagnet in a ferromagnetic enviroment n → ζ [ ˆ ψ ]( β, { λ α } ) = e p ψ ( x ) = e − p ∞ X r =0 p r r ! r Y j =1 Z R dy j Q ( y j ) exp " βx J λ + r X k =1 y k ! ˆ ψ ( x ) = p Z R dhP ( h ) (cid:20) cosh ( βJ A − βh )cosh ( βJ A + βh ) (cid:21) x λ = e − p ∞ X r =0 p r r ! r Y j =1 Z R dy j Q ( y j ) tanh " β J λ + r X k =1 y k ! , (13)the (quenched) free energy f q is evaluated as βf ( β ) = βJ λ p p Z R dhP ( h ) Z R dyQ ( y ) ln cosh [ β ( h + y )] −− p Z R dh P ( h ) Z R dh P ( h ) ln n e βh cosh [ β ( h − J A )] + e − βh cosh [ β ( h + J A )] o −− ln 2 − e − p ∞ X r =0 p r r ! r Y s =1 Z R dy s Q ( y s ) ln cosh " β J λ + r X j =1 y j ! , (14)the extremum condition leads P and Q to satisfy P ( h ) = e − p ∞ X r =0 p r r ! r Y j =1 Z R dy j Q ( y j ) δ h − " J λ + r X k =1 y k Q ( y ) = Z R dhP ( h ) δ (cid:18) y + 1 β tanh − [tanh( βJ A ) tanh( βh )] (cid:19) , (15)and λ is calculated through the last equation of (13). The above set of equations (15)can be unified as P ( h ) = e − p ∞ X r =0 p r r ! r Y j =1 Z R dh j P ( h j ) ×× δ h − " J λ − β r X k =1 tanh − [tanh( βJ A ) tanh( βh k )] . (16)In the context of replica method, the formulas for the magnetization, m , and thespin - glass order parameter, q , are given by m = lim N →∞ N X x ∈ Λ N lim n → n n X α =1 Tr { σ ηz } σ αx * n Y γ =1 e − βH ( { σ γu } , { a uv } ) + (17)and q = lim N →∞ N X x ∈ Λ N lim n → n ( n − n X α = θ Tr { σ ηz } σ αx σ θx * n Y γ =1 e − βH ( { σ γu } , { a uv } ) + . (18) iluted antiferromagnet in a ferromagnetic enviroment λ = m , m = Z R dhP ( h ) tanh( βh ) and q = Z R dhP ( h ) tanh ( βh ) , (19)as usual. The critical line is determined in the neighborhood of m ∼ q ∼ 0. Inthis regime, the field h is expected to be narrowly distributed around h = 0. Then, the Ansatz ǫ k := Z R dhP ( h ) h k = O ( ǫ k ) , | ǫ | ≪ , (20)is introduced in the equation (16) to evaluate the transition lines[13]. This Ansatz leadsto m = O ( ǫ ) and q = O ( ǫ ), which means that the line of transition from ferromagneticto disordered phase is governed by O ( ǫ ) in the equation (16) and the transition fromparamagnetic to spin - glass phase by the order O ( ǫ ) ( ǫ = 0 is assumed in this case).As a result of these calculations, one has β c J = 1 + p tanh( β c J A ) (order - disorder transition line) , (21)and the transition line from spin - glass to paramagnetic phase is evaluated as β c J = 12 ( J A /J ) ln (cid:18) √ p + 1 √ p − (cid:19) , p > . (22)The phase diagram of the model, generated by the equations (21) and (22), ispresented in FIG 1. The “ m = 0” phase displays spontaneous magnetization, and itmay be a combination of a ferromagnetic and a mixed phases. The exact scenario ofthe “ m = 0” phase can be established from a stability analysis[25] ‡ , which will not beprovided here, since hereafter this work will focus on the critical line that separates theordered phase and disordered one in comparison with the annealed version of the model. 4. Annealed average In the annealed approximation, the free energy f a is written as f a ( β ) = − lim N →∞ βN ln h Z ( β ) i . (23)The problem can be solved in the annealed approximation in a simpler way. Althoughthe interesting case is the quenched one (which was provided in the previous section),the results will be derived for comparison.It is straighforward to show that h Z ( β ) i = Tr { σ x } exp (cid:26) − N p N p βJ A ) + 12 N h βJ − p sinh( βJ A ) i(cid:16) X x ∈ Λ N σ x (cid:17) ++ O (1) (cid:27) , (24) ‡ To be more precise, the stability analysis can also change the exact location of the line that separatesthe spin - glass phase and the paramagnetic phase; however, the phase diagram showed in FIG 1 isbelieved to be qualitatively correct. iluted antiferromagnet in a ferromagnetic enviroment p T / J J A /J = 1 P SGm = 0 / Figure 1. Phase diagram with J A /J = 1. and introducing the order parameter m N := 1 N X x ∈ Λ N σ x , (25)it is easy to show that h Z ( β ) i ∼ Z R dm N e − Nφ a ( β,m N ) (26)for sufficiently large N , where φ a ( β, m N ) := p − p cosh( βJ A )2 − βJ m N + p sinh( βJ A )2 m N + m N (cid:18) m N − m N (cid:19) ++ 12 ln (cid:0) − m N (cid:1) − ln 2 . (27)Therefore, in the thermodynamic limit, with m := lim N →∞ m N , the free energy canbe written as f a ( β ) = 1 β inf m { φ a ( β, m ) } , (28)where the infimum of φ a is achieved from the solutions of the extremum condition m = tanh h βJ m − p sinh( βJ A ) m i . (29)This equation allows one to obtain the critical temperature β c , which then obeys β c J = 1 + p sinh( β c J A ) , (30)and is cleary different from the correspondent expression (21) from the quenchedsituation. iluted antiferromagnet in a ferromagnetic enviroment 5. Breakdown of the spontaneous magnetization This section will analyse the different behaviour of the critical line evaluated in thequenched and annealed approaches, which are β c J = 1 + p tanh( β c J A ) (quenched case) (31)and β c J = 1 + p sinh( β c J A ) (annealed case) . (32)Firstly, it is easy to see that if p = 0, one recovers the ferromagnetic mean - fieldresult β c J = 1 in both cases, as it should be. Now, let p assume nonzero values. Actually,given p and J A /J , the equation (32) may yield two roots, but only the physicallyreasonable one for the order - disorder transition is chosen. The numerical solutionsof the equations are plotted in FIG 2 as a function of p . As one can see in the figure,for sufficiently small J A (in the sense that β c J A ≪ T c /J ∼ − p ( J A /J ) in both cases. p T c / J From left to right:J A /J = 1.0J A /J = 0.5J A /J = 0.25 p T c / J From left to right:J A /J = 1.0J A /J = 0.5J A /J = 0.25 Figure 2. Dependence of the critical temperature on the mean connectivity p (left:quenched case; right: annealed case). However, for a fixed value of J A /J , one can see the difference from the quenched andannealed cases as the mean connectivity p increases. The spontaneous magnetization isbroken for sufficiently large values of p in an abrupt way in the annealed case. Onthe other hand, the critical temperature for the quenched case decays slowly, andreaches T c = 0 for p → ∞ only (see (31)). One should remember, initially, that eachvertex links to another N − J/N = O ( N − ) and an iluted antiferromagnet in a ferromagnetic enviroment J A = O (1)) withprobability p/N = O ( N − ).In the quenched case, where the configuration { a xy } remains frozen during anobservational time, only an infinitesimal fraction (of O ( N − )) of the links haveantiferromagnetic interactions. Suppose that the edge xy (shared between the vertices x and y ) is one of them; this edge has a ferromagnetic component of intensity J/N andan opposite effect of J A ( ≫ J/N ). Although the antiferromagnetic part exceeds theferromagnetic one, the solely effect is make just the spins σ x and σ y having oppositesigns. Therefore, despite the fact that all the antiferromagnetic interactions dominatesover the ferromagnetic interactions in the links where a xy = 1, the total number of suchedges is much smaller than the total number of edges of the network (the thermodynamiclimit is taken for a fixed value of p ), which then becomes predominantely ferromagnetic.This is the cause of persistence of spontaneous magnetization for any finite p , as shownin FIG 2 (left).On the other hand, in the annealing approximation, where the antiferromagneticlinks fluctuate during the observation time, one sees an “averaged” antiferromagneticinteraction between spins (vertices). This means that, although the mean connectivity p is fixed, the effective number of edges with an antiferromagnetic interaction is muchlarger (and the effective intensity is also smaller than J A ). Heuristically speaking, theintensity J A is better distributed over the edges (differently from the quenched case),and the antiferromagnetic effect is better exploited in the annealed case, which makesthe magnetization vanishes even for finite values of p . 6. Conclusions Throughout this work, the static properties of a diluted antiferromagnet on aferromagnetic background was examined, with particular emphasis on the breakdownof the spontaneous magnetization of the system. The phase diagram displayed anonzero magnetization at low temperature regime for any finite mean connectivity p ofantiferromagnetic interactions. The disordered phase is constituted by a paramagneticand a spin - glass phase.The critical temperature of the order - disorder transition, which indicates thebreakdown of the magnetic order of the model, was determined from both the quenchedand annealed approaches. The main difference, noticed from FIG 2, relies on thefact that the spontaneous magnetization vanishes for finite values of p in the annealedapproximation. This phenomena is observed due to the rapid fluctuation of the randomvariables { a xy } , which distributes more efficiently the antiferromagnetic interactions overthe whole graph. This means that in the present work, networks are resilient to non -fluctuating random “attacks” (even strong ones), while they are weaker to “annealingattacks”, which turns the antiferromagnetic interaction more accessible to the edges,although weakening its mean strength. iluted antiferromagnet in a ferromagnetic enviroment Acknowledgements The authors thank A. V. Goltsev. This work was supported by the project DYSONET. References [1] Albert R, Jeong H and Barab´asi A -L 2000 Nature Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Random Graphs (New York: Academic Press)[6] Molloy M and Reed B 1995 Random Struct. 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