Ising spin glass in a random network with a gaussian random field
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Ising spin glass in a random network with a gaussian random field
R. Erichsen Jr, https://pt.overleaf.com/project/5f7f0df369a1bf00014a6d7dA.Silveira and S. G. Magalhaes
Instituto de F´ısica, Universidade Federal do Rio Grande do Sul,Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil (Dated: February 24, 2021)
Abstract
We investigate the thermodynamic phase transitions of the joint presence of Spin Glass (SG) andRandom Field (RF) using a random graph model that allows to deal with the quenched disorder.Therefore, the connectivity becomes a controllable parameter in our theory allowing to answerwhether and which are the differences between this description and the mean field theory i. e.,the fully connected theory. We have considered the Random Network Random Field Ising Model(RNRFIM) where the spin exchange interaction, as well as the RF, are random variables followinga gaussian distribution. The results were found within the replica symmetric (RS) approximationwhose stability is obtained using the Two-replica method. This also puts our work in the contextof a broader discussion which is the RS stability as a function of the connectivity. In particular,our results show that for small connectivity there is a region at zero temperature where the RSsolution remains stable above a given value of the magnetic field no matter the strength of RF.Consequently, our results show important differences with the crossover between the RF and SGregimes predicted by the fully connected theory.
PACS numbers: 64.60.De,87.19.lj,87.19.lgKeywords: Disordered systems, Random field Ising model, Finite connectivity . INTRODUCTION The issue of disorder in spin systems is an inexhaustible source of problems. Two manifes-tations of disorder, spin glass (SG) and random fields (RFs), illustrate how rich this researcharea can be [1]. Undeniably, the corresponding theory has been recognized for its conceptualrichness, awakening interest and providing knowledge not only in physics, but also in otherfields such as information theory and computer science, among others. Therefore, one canexpect that the joint presence of SG and RFs can bring plenty of fascinating possibilities.Most importantly, it is not only a theoretical possibility. In fact, the joint presence of SGand RF has been suggested in physical systems as distinct as ferro and antiferroelectriccrystals such as Rb − x (NH ) x H PO [2], the diluted antiferromagnet Fe x Zn − x F [3] andthe diluted ferromagnet LiHo x Y − x F [4, 5]. The first case being the realization of theelectrical equivalent of a SG with pseudospins degree of freedom. In the three cases, theapplied magnetic field couples with Ising spins (or pseudospins). For Rb − x (NH ) x H PO and LiHo x Y − x F , the field transverse to the Ising direction leads to a quantum phase tran-sition [6–8]. The diversity of systems and scenarios to describe them can anticipate that thetheoretical description of the joint presence of SG and RF might be a ground favoring therise of conceptual and methodological novelties.A pertinent question is the extent to which the mean field theory can provide a realisticdescription of the joint presence of SG and RF. As an example but which may allow moregeneral conclusions, one can mention the mean field description of the Fe x Zn − x F . AlthoughFe x Zn − x F has short-range interactions, the mean field description [9], i.e., the infinite-rangeSherrington-Kirkpatrick (SK) model [10] with a gaussian distribution for the RF to describesome aspects of the behavior of the the mentioned system depending on the parameter ∆ /J where ∆ and J are the RF and random spin exchange interaction variances, respectively.Interestingly, it was proposed that there is a crossover between the SG and RF regimes byvarying ∆ /J or by varying T /J ( T is the temperature) for a fixed ∆ /J . This crossover isdescribed by τ ≡ T − T ∼ h /φ ( T is the freezing temperature without field). The crossoverhas the exponent φ = 1 when the RF regime dominate and φ = 3 as given by the Almeida-Thouless (AT) line [11], i.e., the line that signals the limit of stability of the replica symmetric(RS) solution. In addition, the mean field theory predicts that the typical behavior of theAT line is robust for any value of ∆ /J as h increases, i.e., there is an exponentially small2egion with the SG non-trivial ergodicity breaking even for ∆ ≫ J . However, one canasks whether this crossover description is robust. In this direction of investigation, a morespecific question can be raised. For instance, what happens in the limit of high magneticfields? The mean field description in Ref. [9] predicts that the behavior in that limit is givenby the AT line. This scenario is particularly not easy to reconcile with the direction thatthe debate on the existence of the AT line for disordered spins SG systems with short-rangeinteraction has taken (see, for instance, [12–16]). Nevertheless, the difficulties in providinganswers to these questions are in fact the difficulties in describing disordered spins systemswith short-range interactions. This puts the need for alternative approaches that can bringsubstantial improvements over the mean field description.Our proposal is to use random networks. The main reason is that these networks doallow that the coordination number, i.e., the network connectivity becomes a controllableparameter of the theory [17–20]. Thereby, one can interpolate between the limit of highconnectivity that would be closer to the usual mean field theory (called from now on, thefully connected theory) until the situation with a spin with very few connections to otherspins. Although this limit is not equivalent to treating the problem with short-range inter-actions, it can certainly highlight, at least, the limitations of the fully connected description.Actually, this approach has been already used in Ising spin system with a RF. The RandomNetwork Random Field Ising Model (RNRFIM) was developed in Ref. [21] to study theferromagnetic (FM) to paramagnetic (PM) transition in networks with non-uniform, finiteaveraged connectivity. There, the existing couplings were uniformly ferromagnetic and thedisorder was restricted to the existence or not of a bond between two given sites. The resultsthere showed that the existence of a tricritical point, when the RF distribution is discrete,is very dependent on the connectivity. In fact, the tricritical point tends to disappear whenconnectivity is very small. This result is in evident contrast with the fully connected theoryfor the RF Ising model. [22].In the present work, we use the RNRFIM where, besides the presence of a RF witha gaussian distribution, the spin couplings are also disordered following the same kind ofdistribution. This type of choices allows us to investigate not only the SG to PM phase tran-sition but also the SG to FM phase transition in the presence of a RF. Since the RF coupleswith the local magnetic moments, the replica symmetric (RS) Edwards-Anderson SG orderparameter will be induced whenever a RF is applied, turning out to be not useful to localize3he SG transition. Therefore, it is inescapable to test the RS stability to locate the onset ofnon-trivial ergodicity associated to the SG transition [23]. In order to accomplish that, weuse the Two-replica method [24] which is quite suitable to our approach, since it allows toobtain the limits of stability of the RS approximation using the RS calculations themselves.In particular, having obtained the limits of RS stability , i.e., the AT line, and counting con-nectivity and RF variance as controllable parameters, we can check any crossover betweenSG and RF regimes in low and high connectivity scenarios when a magnetic field is applied.For completeness, we also investigate effects of connectivity on the non-linear susceptibility χ . This quantity is a well established fingerprint of the SG transition [25]. It is known that χ is strongly affected by the RF in the limit of the fully connected random network [7, 8].Therefore, it is also an interesting issue how the RF affects in the χ at low connectivity.Lastly, we remark that there exist other approaches to deal with finite connectivity inspins disordered problems, such as the cavity method (see, for instance, [26, 27]). However,we focus mainly in the RS approximation for which the random network is quite suitable.The development of a replica symmetry breaking theory for the random network with finiteconnectivity for the SG problem with RF is beyond the objective of this paper.The paper is organized as follows: in Sec. II, the free energy and order parameter areobtained using finite connectivity within the RS scheme. The two-replica method, employedto localize the AT line is explained in this section. Sections III and IV present the theoreticalresults and the results obtained from numerical simulations, respectively. Section V offersconcluding remarks. II. THE MODEL
The hamiltonian is an extension of RNRFIM which has two-sites disordered interactionand local random field to single site interaction terms, H = − X i,j
The saddle-point equation for the local field distribution, Eq. (7), is solved by the pop-ulation dynamics method [20]. It starts with a randomly chosen population of local fields.Typically, the size of the population is 100,000 fields. The method is iterative. Each it-eration, a number k ∈ N is sorted according to the poissonian distribution with average c . Then, k fields are randomly chosen from the field population and the summation of theargument of the δ -function in Eq. (7) is evaluated. The result is assigned to another fieldrandomly chosen from the same population. This recipe is applied till W ( x ) converges. Ittakes, on average, 100 iterations per field to converge. The distribution W ( x, y ) is calculatedsimilarly. Examples of joint distributions are shown in Figure 1.6 x -6 -4 -2 0 2 4 6 y W ( x , y ) -6 -4 -2 0 2 4 6 x -6 -4 -2 0 2 4 6 y W ( x , y ) FIG. 1: Joint distributions for c = 4, h /J = 0 . /J = 0 .
0. Top:
T /J = 0 .
8, PM phase, RS-stable diagonal distribution. Bottom:
T /J = 0 .
6, SG phase, RS-unstable non-diagonal distribution.
In the PM phase the system is ergodic, and the RS solution is stable. As mentioned above,the correspondent joint distribution is diagonal, as is shown in the top panel of Figure 1.Conversely, the ergodicity is broken in the SG phase, i.e., this phase is RS unstable, and thejoint distribution is no longer diagonal, as can be seen in the bottom panel of Figure 1. Weproceed by considering that the SG to PM transition coincides with the AT line.Representative examples of how the order parameters q and q ′ behaves as the temperaturevaries is shown in Figure 2, for c = 4 and two sets of parameters ( h , ∆ , J ). The set(0 . , , . , . , .2 0.4 0.6 T/J q,q’
FIG. 2: q (thick lines) and q ′ (thin lines) vs. T /J for c = 4, h = 0 .
15, ∆ = 0, J = 0 . h = 0, ∆ = 0 . J = 0 (dashed lines). being broken at low temperature.A more complete view of the role played by the random field and average connectivityon finite connectivity spin-glasses is revealed through the phase diagrams. In Figure 3, T /J vs. J /J phase diagrams for c = 4, c = 8 and the fully connected network, with randomfield and without random field are presented. The SG to PM transition, as well as thetransition from the mixed phase FM’ to FM are AT lines, i.e., lines that signal the locuswhere the ergodicity is broken, with q and q ′ becoming different. The FM’ to SG and F toPM transitions are signaled by the magnetization m going to zero. All the transitions arecontinuous. The mixed FM’ phase is a non-ergodig ferromagnetic phase, where m = 0, q > q ′ = 0 (it is the correlation between two distinctreplicas), and q = q ′ . As a general remark, when increasing the random field the transitionlines are displaced in a way that the surface occupied by more entropic phases increases,and the less entropic phases are reduced. The differences between averages connectivies c = 4 and c = 8 are mainly quantitative. When increasing c the transition lines are againdisplaced, this time in a way the area occupied by the less entropic phases increases, whilethe more entropic are reduced. So speaking, the SG to PM transition line displaces upwards,the SG to FM’ displaces to the left and FM to PM displaces to the left and FM to FM’displaces downwards.It should be remarked that, by comparing finite and fully connectivity, the relevantqualitative difference is that, in the fully connected case, the transition line FM’ to FM goes8symptotically to T = 0 when J /J increases, unlike the finite case c , where the transitionline intersects the T zero axis. This means that a finite connectivity favors the ergodicityat zero temperature.Next, we analyze the SG to PM transition through T /J vs. h /J phase diagrams. Figure4 shows the SG to PM transitions for connectivity c = 4 and c = 8, for four representativevalues of the field disorder. Although the pictures for different values of c are qualitativelysimilar, with both the uniform field component h and the random field component ∆, bothgiven in units of J , acting to suppress the SG phase in favor of the PM phase, there aresome aspects to consider. First, ∆ is much more effective in suppressing the SG at smallthan at a large h . Second, it is also more effective the smaller the c . This, even consideringthat both the mean value and the variance of the couplings scale with c (see Eq. 3). Thismeans that a more connected network with weaker couplings produces a more robust SGphase than a less connected network with stronger couplings one. Other aspect that mustbe very stressed is that the SG phase is suppressed completely above a certain h for anyvalue of ∆. Moreover, for the same fixed value of ∆, this suppression is much more effectivefor c=4 than for c=8.We remark in Figure 4 that the convexity of the curves at small h /J values changes from∆ /J = 0 to ∆ /J >
0. To investigate this in detail, in Figure 5 we plot, in logarithmic scale,the reduced temperature τ = ( T − T ) /J vs. h /J , for small h /J , where T = T ( h /J = 0),for ∆ /J = 0 .
00 and ∆ = 0 .
01, with connectivity ranging from c = 4 to c = 16. If thereduced temperature is expressed as a power law τ ∼ ( h /J ) /φ as h /J →
0, the slopes infigure indicate that φ = 2 for ∆ /J = 0 .
00 and φ = 1 for ∆ /J as small as 0.01. Here, wecompare our results with those obtained for the fully connected network [9], where φ = 3 for∆ /J = 0 .
00, identified as SG regime, and φ = 1 for finite ∆ /J , identified as the RF regime.We guess that the φ = 3 in the SG regime is a particularity of c → ∞ , since the curvesfor increasing c in Figure 5 superimpose, suggesting that φ = 2 in the SG regime is robustfor all finite c . To resume, the results for both finite and fully connected networks indicatesthat a crossover from SG to RF regime takes place as ∆ /J becomes non zero.The non-linear susceptibility χ = ∂ m/∂h | T/J,h /J =0 diverges at the SG to PM transition(AT line) in the SG regime (zero ∆ /J ) [25]. The non-linear susceptibility as a function of T /J is shown in Figure 6, for some small values of ∆ /J . For ∆ /J = 0 indeed there is adivergence at the same T /J where the two-replica method localizes the SG to PM transition,9 .0 1.0 2.0 3.0 J /J T/J
SG PM FMFM’0.0 1.0 2.0 3.0 J /J T/J
FM’SG PM FM0.0 1.0 2.0 3.0 J /J T/J
SGPM FMFM’
FIG. 3:
T /J vs. J /J phase diagrams for h /J = 0 .
0, ∆ /J = 0 . /J = 0 . c = 4 (top), c = 8 (middle) and fully connected (bottom). .0 0.5 1.0 h /J T/J ∆ /J= ∆ /J =0.2 ∆ /J= ∆ /J= h /J T/J ∆ /J= ∆ /J= ∆ /J= ∆ /J= FIG. 4: Top:
T /J vs. h /J phase diagrams for c = 4 and J = 0 for different values of ∆ /J .Bottom: the same, but for c = 8. for both c = 4 and c = 8. For ∆ /J > /J increases, the peak becomes quickly lesspronounced and moves to higher temperature values, as can be seen in Figure 6.To complete the description, T /J vs. ∆ /J phase diagrams for h /J = 0 and J /J constant are present in Figure 7. In the top left panel we have J /J = 0, that is a prototypefor all phase diagrams where the uniform part of the coupling constant is to weak to allowthe appearing of ferromagnetic phases. In the top right panel, a constant J /J in the re-entrant region was chosen. Here, the uniform coupling becomes sufficiently strong to allow11 ,001 0,010 h / J τ c =4, ∆ = 0.00 c =4, ∆=0.01 c =8, ∆ =0.00 c = 8, ∆ =0.01 c =16, ∆ =0.00 c =16, ∆ =0.01 FIG. 5: Reduced temperature vs. RF amplitude h /J for ∆ /J = 0 .
00 and ∆ /J = 0 . c = 4, c = 8 and c = 16. The solid lines are only guidelines. Shown in dashed, slope 1 and slope 2 straightlines. the appearing of FM and FM’ phases, although the SG remains as the most ordered phaseat zero temperature. For c = 4 and c = 8 this takes place, e.g., in the neighborhood of J /J = 1 .
22 and J /J = 1 .
15, respectively. In the bottom panel the
T /J vs. ∆ /J phasediagram for a constant J /J = 1 . c = 4 and c = 8 is shown. Here, the uniformcoupling becomes stronger, and the mixed phase FM’ can be found at zero temperature.The phase diagrams are qualitatively similar for both values of the connectivity. All thetransitions are continuous. As ∆ /J increases, FM and FM’ phases are the first to besuppressed. Then, the SG to PM transition line decreases monotonically to T /J = 0 andPM is the only remaining phase at large ∆.The last comment concerns the comparison with the fully connected network. Hereagain, the FM’ to FM transition intercepts the zero temperature axis, contrary to the fullyconnected network. If the two-replica method to localize the AT line is correct, the finiteconnectivity results should approach the fully connected ones as c increases. This seemsto be the case for the general aspects of the phase diagrams shown above, except in thehigh field and high coupling constant regimes. In the fully connected network there is noparamagnetic phase at zero temperature, in contrast to the finite connectivity case. Notethat this does not mean that there is no magnetization at zero T : there is, indeed a field-12 ,0 1,5 2,0 T/J χ ∆ /J= ∆ /J =0.01 ∆ /J= T/J χ ∆ /J= ∆ /J =0.01 ∆ /J= FIG. 6: Top: nonlinear susceptibility vs. temperature for c = 4 and several values of ∆ /J . Bottom:the same, but for c = 8. induced magnetization. The main outcome is that we can find the finite connectivity networkergodic at zero T , contrary to the fully connectivity network. To illustrate how the zero T ergodicity region evolves as a function of the connectivity is shown in Figure 8. The figureshows the limiting h /J of the SG phase, at T = 0. As one should expect, this limit increasesslowly but monotonically with c . III. CONCLUSIONS
In this work we have investigated the problem of the joint presence of SG and RF usingrandom network. In order to accomplish that, we have considered the Random Network13 .0 0.5 1.0 1.5 ∆/ J T / J PMSG ∆/ J T/J
PMSGFMFM’ ∆/ J T / J PMFM’ SGFM
FIG. 7:
T /J vs. ∆ /J phase diagram for h = 0. Top left: J /J = 0; c = 4 (solid lines) and c = 8(dashed lines). Top right: J /J = 1 .
22 and c = 4 (solid lines); J /J = 1 .
15 and c = 8 (dashedlines). Bottom: J /J = 1 . c = 4 (solid lines) and c = 8 (dashed lines). All the transitions arecontinuous. Random Field Ising Model [21] where the spin exchange interaction as well as the RF arerandom variables following a gaussian distribution. Our goal has been, using the connectivityas a control parameter in the theory, to verify whether and which are the differences withthe mean field theory i.e., the fully connected theory [9]. Particularly, in the presence of amagnetic field. As a methodological novelty in the problem, we performed the check of the14
20 40 60 80 100 c h /J ∆ /J =0.0 ∆ /J =0.2SGP FIG. 8: Limiting h /J of the SG phase vs. c at T = 0, for ∆ /J = 0 . /J = 0 .
2. Below andabove the lines are the loci of the SG (non-ergodic) and PM (ergodic), respectively. stability of the RS solution using the Two-replica method. This procedure, which gives theAT line, has been used for the SG problem without RF. Thus, in fact, it can be consideredthat our work also belongs to a more general discussion concerning the description of the SGnon-trivial ergodicity breaking using the AT line when the network connectivity can vary.Following a population dynamics algorithm, the effective distribution of local fields wasdetermined, allowing to the calculation of relevant order parameters. Then, we obtainedphase diagrams temperature versus ferromagnetic exchange interaction J (see Eq. (3)) andtemperature versus the magnetic field h (see Eq. (4)) for several values of the random fieldvariance ∆ and for two values of the average connectivity, namely c = 4 and c = 8. Allenergy scales in the problem are given in units of the variance J of the random exchangespin interaction. The differences in the phase diagrams with the two values of c are mainlyquantitative. Nevertheless, the results have shown that the less entropic phases occupygrowing areas to the detriment of the more entropic ones as c increases. This means thatthe connectivity favors the ordered phases, even considering that the coupling constant iscorrectly normalized with c (see Eq. 3). Moreover, we do remark that the AT line interceptsboth the J and h axis at zero temperature, contrary to the observed in the fully connectedtheory. In other words, the SG ground state prevails only within a certain interval of J and h . This also means that, even with quenched disorder, for finite connectivity there isa region at zero temperature where the ergodicity remains unbroken above a given value15f the magnetic field, no matter the strength of the RF gaussian variance ∆. We noticethat, in the limit of large values of c , there are indications that the fully connected theoryis recovered, particularly with regard to the AT line.To conclude, one of the main outcomes of the present investigation concerns the crossoverbetween the RF and the SG regime. Within the fully connected theory, the crossover betweenthe RF and SG regimes was described by τ ≡ T − T ∼ h /φ ( T is the freezing temperaturewithout field). In the fully connected theory, the values φ = 1 and 3 corresponds to RFand SG regimes, respectively. We found, in this work, that at small h , τ ∼ h at ∆ = 0and τ ∼ h for any finite ∆. In other words, φ = 1 and φ = 2 in the RF and SG regimes,respectively. Acknowledgments
The authors acknowledge F. L. Metz and F. D. Nobre for fruitfull discussions. Thepresent work was supported, in part, by the Brazilian agency CNPq.
Appendix
The finite connectivity replica method has become standard. We rewrite here only somekey points and refer to [17, 18] for details. After averaging over c ij we obtain, in the c/N → h Z n i { J ij ,h i ,c ij } = X σ ··· σ n D exp h β X i,α h i σ αi + c N X i,j = i (cid:16) e βJ ij σ i · σ j − (cid:17)iE { J ij ,h i } . (12)Next, we introduce the fraction P ( σ ) of sites where the replica configuration σ is realizedand the auxiliary variables ˆ P ( σ ) and evaluate the trace over the spin variables. This reducesto the problem of one site, and the replicated partition function can be rewritten as h Z n i { J ij ,h i ,c ij } = Z Y σ dP ( σ ) d ˆ P ( σ ) exp n N log X σ D exp h βh X α σ α − ˆ P ( σ ) iE h + N X σ ˆ P ( σ ) P ( σ ) + N c X σσ ′ P ( σ ) P ( σ ′ ) D(cid:16) e βJ σ · σ ′ − (cid:17)E J o . (13)16n the limit N → ∞ , the saddle-point method applies, and the free-energy becomes f ( β ) = − lim n → βn Extr P ( σ ) n − c X σσ ′ P ( σ ) P ( σ ′ ) D(cid:16) e βJ σ · σ ′ − (cid:17)E J + log X σ D exp h βh X α σ α + c X σ ′ P ( σ ′ ) D(cid:16) e βJ σ · σ ′ − (cid:17)E J iE h o , (14)where the auxiliary variables ˆ P ( σ ) were eliminated by the saddle-point equations ∂f ( β ) /∂P ( σ ) = 0. The variables P ( σ ) must satisfy the remaining saddle-point equations, P ( σ ) = D exp h βh P α σ α + c P σ ′ P ( σ ′ ) D(cid:16) e βJ σ · σ ′ − (cid:17)E J iE h P σ ′ D exp h βh P α σ ′ α + c P σ ′′ P ( σ ′′ ) D(cid:16) e βJ σ ′ · σ ′′ − (cid:17)E J iE h . (15)We are interested in those solutions satisfying the RS “ansatz”, P ( σ ) = Z dx W ( x ) e βx P α σ α [2 cosh( βx )] n . (16)This expression is equivalent under permutation of replicas. Introducing the RS ansatz inEq. (15), we obtain a recursive equation for the distribution of effective local fields W ( x ),Eq. (7). [1] A. P. Young (ed), Spin Glasses and Random Fields (Singapore: World Scientific), (1998).[2] J. Slak, R. Kind, R. Blinc, E. Courtens, S. Zumer, Phys. Rev. B , 85 (1984).[3] D. P. Belanger, H. Yoshizawa, Phys. Rev. B , 5051 (1993).[4] W. Wu, D. Bitko, T. F. Rosenbaum, G. Aeppli, Phys. Rev. Lett. 71, 1919 (1993).[5] S. M. A. Tabei, M. P. J. Gingras, Y.-J. Kao, P. Satsiak, J.-Y. Fortin, Phys. Rev. Lett. 95,237203 (2006).[6] R. Pirc, B. Tadic, R. Blinc, Phys. Rev. B , 8607 (1987).[7] C. A. Morais, F. M. Zimmer, M. J. Lazo, S. G. Magalhaes, F. D. Nobre, Phys. Rev. B ,224206 (2016).[8] S. G. Magalhaes, C. A. Morais, F. M. Zimmer, M. J. Lazo, F. D. Nobre, Phys. Rev. B ,064201 (2017).[9] R. F. Soares, F. D. Nobre and J. R. L. de Almeida, Phys. Rev. B , 6151 (1994).[10] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. , 1792 (1975).
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