The phase diagram of the mean-field Ising model in a random field obeying a symmetric three-peak distribution is determined. This distribution is relevant to diluted antiferromagnets in a uniform magnetic field. The phase diagram includes a fourth-order point, tricritical points, ordered critical points, critical end points, and a double critical end point. An ordered phase persists for arbitrarily large random fields at low temperatures

Kaufman, Miron

Khurana, A.

Klunzinger, Philip E.

English

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Multicritical Points in an Ising Random-Field Model

Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUPhysics Faculty Publications Physics Department10-1-1986Multicritical Points in an Ising Random-FieldModelMiron KaufmanCleveland State University, m.kaufman@csuohio.eduPhilip E. KlunzingerCleveland State UniversityA. KhuranaPrinceton UniversityFollow this and additional works at: https://engagedscholarship.csuohio.edu/sciphysics_facpubPart of the Physics CommonsHow does access to this work benefit you? Let us know!Publisher's StatementCopyright 1986 American Physical Society. Available on publisher's site at http://link.aps.org/doi/10.1103/PhysRevB.34.4766.Repository CitationKaufman, Miron; Klunzinger, Philip E.; and Khurana, A., "Multicritical Points in an Ising Random-Field Model" (1986). Physics Faculty Publications.100.https://engagedscholarship.csuohio.edu/sciphysics_facpub/100This Article is brought to you for free and open access by the Physics Department at EngagedScholarship@CSU. It has been accepted for inclusion inPhysics Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contactlibrary.es@csuohio.edu.Original CitationKaufman, Miron, Philip E. Klunzinger, and A. Khurana. "Multicritical Points in an Ising Random-Field Model." Physical Review B 34(1986): 4766-4770.PHYSICAL REVIEWER 8 VOLUME 34, NUMBER 7Multicritical points in an Ising random-field model1 OCTOBER 1986Miron Kaufman and Philip E. KlunzingerDepartment ofPhysics, Cleueland State Uniuersity, Cleue!and, Ohio 44115A. KhuranaDepartment ofPhysics, Princeton Uniuersity, Princeton, New Jersey (%5444'Received 27 May 1986)The phase diagram of the mean-field Ising model in a random field obeying a symmetric three-peak distribution is determined. This distribution is relevant to diluted antiferromagnets in a uni-form magnetic field. The phase diagram includes a fourth-order point, tricritical points, orderedcritical points, critical end points, and a double critical end point. An ordered phase persists for ar-bitrarily large random fields at low temperatures.I. INTRODUCTIONExperimental' and theoretical ' studies of Ising sys-tems subjected to random magnetic fields have concen-trated mainly on the question of what is the lower criticaldimension. The consensus opinion is that the lowest spa-tial dimension (lower critical dimension) above whichlang-range order can be established, is two. However,strong hysteretic effects' prevent or make it difficult toobserve the ordered phase in equilibrium. Still unresolvedis the issue" of what crossover is actually observed in someexperiments, whether random field to random exchangeor random field to classical criticality.This paper addresses the effect of different field distri-butions on the topology of the equilibrium phase diagram.Most of the theoretical work has focused on the Gaussianand two-peak distributions. We consider here randomfields which can take three values +ho, —ho, or 0, withprobabilities p/2, p/2, or 1 —p, respectively. This three-peak symmetric distribution is relevant to diluted antifer-romagnets, such as Fe,Zni, F2, in a uniform field ho. Inthe Zeeman Hamiltonian' the field conjugated to the anti-ferromagnetic order parameter (one-half of the differencebetween neighboring spins on opposite sublattices} takesthree values: h o, —h o, and 0 with probabilitiesp/2=x(1 —x}, p/2=x(1 —x), and 1 —p=x +(1—x),where 1 —x is the concentration of nonmagnetic impuri-ties. This distribution has also been considered byMattis, who calls it the trimodal distribution. Grinsteinand Mukamel have studied the case: ho~ ao for a one-dimensiona1 system.We determine the complete phase diagram of theequivalent-neighbor (mean-field) Ising model with randomfields obeying the three-peak symmetric distribution. Wefind a variety of multicritical points: fourth-order, tricrit-ical, ordered critical, critical end, and double critical end.With the exception of the tricritical point, none of theothers have been detected in a previous study of themode1.For p &1, an ordered (ferromagnetic) phase exists forarbitrarily large values of the random-field strength, ho.In the context of the equivalent-neighbor model, this out-come is a consequence of the fact that a fraction 1 —p ofthe spins are subjected to no magnetic field, and are suffi-ciently interconnected by the equivalent-neighbor bonds toorder at low temperatures. We believe that the orderedphase will also be formed on a finite-dimensional lattice,provided p is sufficiently small. Indeed, if p &1—p,(where p, is the site-percolation threshold} there is a finiteprobability for the formation of an infinite cluster of spinssubjected to zero magnetic field If th.is cluster is not tree-like, we expect the ordered phase, which forms at zerotemperature to persist at low but finite temperatures, forarbitrarily large ho.For p = —,, the three-peak distribution was argued to bea good approximation of the Gaussian distribution, be-cause the low-order inomenta (h2} and (h ) are thesame, respectively, for both distributions. Indeed, no tri-critical point occurs for p= —, , which is also true for theGaussian distribution. On the other hand, the Gaussiandistribution phase diagram has no ordered phase for largerandom fields, while the three-peak distribution exhibitsan ordered phase for arbitrarily large ho at low tempera-tures. Therefore, different field distributions will result,in general, in topologically different phase diagrams. Thematching of the low-order momenta does not ensure theequivalence of the phase diagrams except at low values ofthe random-field strength.The remainder of this article contains a description ofthe model and its solution, in Sec. II, and our concludingremarks in Sec. III.II. MODEL AND SOLUTIONIn the equivalent-neighbor Ising model, each pair of theN spins tr; =+1 is connected by a bond of strength 1/N.Each spin is subjected to a random inagnetic field h ac-cording to the following distribution:P(h) = —,p[5(h —hp)+5(h+ho)]+(1—p)5(h),where 0 &p & 1, ho &0, and 5 is the Dirac delta function.The energy A associated with the X spins is34 4766 1986 The American Physical Society34 Mor.TreRrTrcm polrrr3 rw ~N rsrNo rUorooM-FrELo rrooar.(2)The quenched averaged free energy isf= —kr r (—InZ), (3)where Z is the partition function for a given configura-tion of the magnetic fields. In the thermodynamic limit:if=ks T min t(r(rri ), (4)Q(m) = —,' Jmi —(ln2coshl(rri+h ) ) .In Eq. (5): 7=1/krr T, rri is the average site magnetiza-tion, and ( ) denotes an average over the magnetic fields,using the distribution of Eq. (1).The detailed analysis of this model involves a numericalsolution of Eqs. (4) and (5), which is presented in subsec-tion C below. Certain features of the phase diagratn aredetermined analytically; in subsection A, zero tempera-ture, and in subsection B, multicritical points.A. Zero temperatureAt zero temperature, T=O or J= oo, Eq. (5) reduces toJ 2= —,' m~ —(~rri+h ( ) . (6)B. Molticritical pointsClose to a continuous transition between ordered anddisordered phases, rn is small, and we can expand P inpowers of m:p(m) i'(0) =a2m —+agm 4+a6m 6+asm' . (7)The a coefficients are the following:The values of m which minimize 1(r are the following: forsmall hp, m=+1, and for large hp, m=+(1 —p). Afirst-order transition between the two ordered phasesoccurs at hp —1 —p/2. The transition point, countingboth positive and negative rri's, is actually a point wherefour phases coexist.The physical origin of the two ordered phases is as fol-lows. For small hp the bonds are strong enough to allowall spins to order, i.e., m =+1. For large hp, a fraction pof the spins randomly points up and down, aligned withthe local fields, while a fraction 1 —p of the spins, subject-ed to no field, is in order, i.e., rri =+(1—p). The essentialfeature is that in the latter group the spins are intercon-nected by the equivalent-neighbor bonds, forming an in-finite cluster which can order at low temperatures.This condition should also be satisfied for a finite di-mensional lattice, if 1 —p is larger than p„ the site-percolation threshold. Hence, for p&1 —p, an infinitecluster is formed. Provided this cluster has loops, i.e., it isnot treelike, the spins on the percolating cluster will orderat low temperatures even for large fields. In the one-dimensional version of the model with /lp~00 rn is zerofor all p gO, which is consistent with our argument be-cause the one-dimensional threshold is p, = l.a4 ——,', J (1—4pu+3pu ),a6=~J ( —2+17pu —30pu +15pu ),as —z,ip J (17—248pu +756pu —840pu 3+315pu 4),(8)is the critical line; then he looked for the "vestigial" tri-critical point (the D point). His estimate is p=0.75,krr T=0.44, and hp —0.57. The discrepancy betweenMattis's and our results is small for p and hp, less than2%, but rather substantial for the temperature, about20%. The source of this discrepancy is the fact that theline of tricritical points is exactly located at a2 —a4 —0and a6 &0, while Mattis's requirement dhpldT =0 is nota necessary condition for a tricritical point to occur. Thelatter provides, however, a lower bound on the correct tri-critical temperature, and an upper bound on the correcttricritical hp, if no reentrance of the disordered phaseoccurs (i.e., dh p/d T always negative). The 20%discrepancy on the temperature estimate indicates thatMattis's method to determine tricritical coordinates, firstused in the high-temperature analyses' of random-fieldsystems, ls not always accurate. To obtain another esti-mate of accuracy, we looked at the tricritical point atp =1, exactly locateds at ks T=0.67 and hp =0.44.Mattis's approximation locates the tricritical point atkrr T=0.58 and lip =0.45. The percentage errors are 13%for temperature and 2% for hp.The ordered critical point B, terminates a line of first-order transition points A, in the midst of the orderedphase A . In the (ai,a4,a6) space the B points form aline which, close to the fourth-order point, isa2 l a6 a4 3 a6as 16 as ' as 8 a8 , a,&0, a8&0. (9)where J= 1/kq T and u =tanh (Jh ).Following Griffiths, we use the following symbols todenote the various entities on the phase diagram: A—one-phase point (disordered phase); A —two-phase point(ordered phase); A —three-phase point (coexistence of or-dered A and disordered A phases); A —four-phase point(coexistence of two ordered A phases); B—critical point;BA —critical end point (coexistence of critical B and or-dered A phases); B —isolated critical point in the midstof an ordered phase (coexistence of two critical B phases),which we shall call ordered critical point; C—tricriticalpoint; and D—fourth-order point.The fourth-order point D occurs at az ——a4 —a6 —0,and as &0. From Eq. (8) we compute the coordinates ofthis point: p = » =0.76, krr T= —,', =—0.55, and hp-=0. 56.This int lies at the intersection' of a line of tricriti-cal"' points C, a line of ordered critical points B,and aline of critical end points BA 2. It can therefore be viewedas the last tricritical point, i.e., for p &0.76 there are notricritical points. Mattis has estimated the location ofthis point by first determining the line of tricritical pointsfrom a2 —0 and dhpldT=O, wherekrrT=1 —p tanh (hp/kgT)MIRON KAUFlNAN, PHILIP E. KLUNZINGER, AND A. KHURANA 34l.o- l.O-- 0.585I0.2)0.6 I. O 0.210.4 k~T I. OFIG. 1. Phase diagram for p =1: solid line, critical points;dashed line, first-order transitions. A tricritical point C alsooccurs.FIG. 3. Phase diagram for p=0.74: solid line, critical points;dashed line, first-order transitions. Two critical end points BAand an ordered critical point 8 also occur. The inset shows acloser view of the region in the oval.04 j Q6Q8 4 QsI,a6&0, a8~0. (10)The critical points 8 form a surface in the (ai,a4,as)space, or in the ( T,ho,p) space which, when not preeinpt-ed by first-order transitions, is given byai —0 or kqT=1 —p tanh (ho/EsT) .The critical end point BA occurs at the intersection ofa line of critical points 8 and a line of first-order transi-tion points A' and A . Close to the fourth-order point,BA coordinates are the following:C. Phase diagramWe determine numerically the phase diagrams in theplane T, ho for different values of p. There are two spe-cial p values: p = —",, —=0.76 and p=0.73, which divide theinterval [0,1] in three subintervals corresponding to threetopologically different types of phase diagrams.For p =1, we obtain Aharony's phase diagram, Fig. 1,with a tricritical point at kqT=0. 67 and ho —0.44. Anew ordered phase appears for any p & 1, at large ho. Oilthis phase the magnetization is smaller than that of theordered phase at weak ho. The two ordered phases areseparated, at low temperatures, by a first-order line of Apoints, starting at ho —1 —pl2, at T=O. The small-mordered phase terminates at a line of critical points which,l.O- l.0-I0.210.6 1.0i0.2i0.6 l.OFIG. 2. Phase diagram for @=0.8: solid line, critical points;dashed line, first-order transitions. A critical end point BAand a tricritical point C also occur.FIG. 4. Phase diagram for p=0.67: solid line, critical points;dashed hne, first-order transitions. An ordered critical point 8also occurs.MULTICRITICAL POINTS IN AN ISING RAPlDOM-FIELD MODEL 4769l.0-IO.Z(0.6 l. 0 O.Zl0.4 I.OFIG. 5. Phase diagram for p=0.33: solid bne, critical points;dashed line, first-order transitions. An ordered critical point 8also occurs. Contrary to a suggestion in Ref. 7, this phase dia-gram quahtatively differs from the phase diagram for theGaussian distribution,for hp ~ oo, is at k~ T= 1 —p. As p is lowered, the tricrit-ical point persists for p y» ~0.76, Fig. 2. The "vestigi-al" tricritical point, at p= », is the fourth-order pointoccurring at AT= —,', -=0.55 and hp=0. 56. A criticalend point, BA, occurs at the intersection of a critical lineterminating the small m phase and the first-order lineseparating the small m and the disordered phases fromthe large m phase.For 0.73 &p &0.76, a new type of phase diagram ap-pears (Fig. 3), in which the tricritical point is replaced byan ordered critical point 8 and a new critical end pointBA, i.e., there are now two critical end points. Asp ~0.73, the two critical end points coalesce. Thiscoalescing point is not a new multicritical point. Viewedin the three-dimensional parameter space (T,hp, p) it lieson a smooth line of BA points, at an extremum. It couldbe called a double critical end point, by analogy with thedouble critical point (see Appendix A of Ref. 11) whichplays a similar role on a line of critical points.For 0&p &0.73, we obtain a third type of phase dia-gram (Figs. 4 and 5), with the critical line uninterruptedfrom hp ——0, kiiT=1, to hp~ao, kgT=1 —p. In the or-dered phase there is a first-order line of A points endingat an ordered critical point 8 . This first-order lineshrinks to zero as p goes to zero. At p =0 the modelreduces to the pure Ising model in zero magnetic field,and hp is a redundant variable which does not affect thefree energy. The phase diagram (Fig. 6) has a critical lineat k~T=1, for any ho.FIG. 6. Phase diagram for p =0. The vertical solid line is aline of critical points.III. CONCLUDING REMARKSWe have determined the complete phase diagram of theequivalent-neighbor (mean-field) Ising model in a randomfield obeying a three-peak symmetric distribution. Themain conclusion of our study is that the topology of thephase diagram depends strongly on the particularrandom-field distribution used. Previously studied phasediagrams for the Gaussian and two-peak distributionsresemble each other topologically, even though the orderof the transition differs at low temperatures. In particu-lar, for both distributions, there is no ordered phase forlarge random fields at low temperatures. The phase dia-gram for the symmetric three-peak distribution is qualita-tively different: the ordered phase persists for arbitrarilylarge random fields at low temperatures. We believe thisordered phase occurs for realistic short-range interactionsin firiite dimensions.Diluted antiferromagnets, e.g., Fe„Zni „F2, in a uni-form field are realizations of the random-field Isingmodel with random fields obeying a three-peak symmetricdistribution with p=2x(1 —x) (see Sec. I). In spite ofdifferences between the model and its experimental reali-zations, such as the random character of the exchange in-teractions in the experimental systems, it is conceivablethat an ordered phase will form even for large magneticfields if 1 —p is larger than the site-percolation thresholdp, . In terms of the magnetic concentration x this meansx & [1—(2p, —1)'~ ]/2 and x) [(1+(2p —1)'~ ]/2 if p,is larger than —,'. If p, is smaller than —, the long-rangeorder should be established at large fields for any concen-tration x. Since p=2x(1 —x)&0.5, our computationssuggest the absence of a tricritical point for diluted anti-ferromagnets in a uniform field.R. J. Birgeneau, Y. Shapira, G. Shirane, R. A. Cowley, and H.Yoshizawa, Physica 8 137, 83 (1986), and references therein;Po-zen %'ong, S. von Molnar, and P. Dimon, J. Appl. Phys.53, 7954 (1982).~Y. Imry and S.-K. Ma, Phys. Rev. Lett. 35, 1399 (1975); J. Z.Imbrie, ibid. 53, 1747 (1984).~A. Aharony, Y. Imry, and S.-K. Ma, Phys. Rev. Lett. 37, 1364,(1976).4M. Kaufman and M. Kardar, Phys. Rev. 8 31, 2913 (1985);M.Kaufman, Superlattices and Microstructures 1, 511 (1985),and references therein.5T. Schneider and E. Pytte, Phys. Rev. 8 15, 1519 (1977).MIRON KAUFMAN, PHILIF E. KLUNZINGER, AND A. KHURANA 346A. Aharony, Phys. Rev. B 18, 3318 (1978).7D. C. Mattis, Phys. Rev. Lett. 55, 3009 (1985).Q. Qrinstein and D. Mukamel, Phys. Rev. 8 27, 4503 (1983}.9R. B.Griffiths, Phys. Rev. B 12, 345 (1975).A similar topology was observed in the two-dimensional BEGmodel; A. N. Berker and M. %ortis, Phys. Rev. B 14, 4946(1976).~ M. Kaufman, R. B. Griffiths, J. M. Yeomans, and M. E.Fisher, Phys. Rev. B 23, 3448 (1981).2S. Galam, Phys. Rev. B 31, 7274 (1985).3A. Houghton, A. Khurana, and F. J. Seco, Phys. Rev. B 34,1700 (1986);Phys. Rev. Lett. 55, 856 (1985).

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Multicritical Points in an Ising Random-Field Model

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