The global phase diagram of the Blume-Capel model in a random field obeying the bimodal symmetric distribution is determined by using the mean-field method. The phase diagram includes an isolated ordered critical end point and two lines of tricritical points. A new phase emerges for strong enough random fields: the ferromagnetic-nonmagnetic phase. It is argued that such a phase occurs in three dimensions

Kanner, M.

Kaufman, Miron

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Cleveland-Marshall College of Law

Cleveland State UniversityEngagedScholarship@CSUPhysics Faculty Publications Physics Department8-1-1990Random-Field Blume-Capel Model: Mean-FieldTheoryMiron KaufmanCleveland State University, m.kaufman@csuohio.eduM. KannerFollow 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 1990 American Physical Society. Available on publisher's site at http://link.aps.org/doi/10.1103/PhysRevB.42.2378.Repository CitationKaufman, Miron and Kanner, M., "Random-Field Blume-Capel Model: Mean-Field Theory" (1990). Physics Faculty Publications. 16.https://engagedscholarship.csuohio.edu/sciphysics_facpub/16This 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 and M. Kanner. "Random-Field Blume-Capel Model: Mean-Field Theory." Physical Review B 42 (1990): 2378-2382.PHYSICAL REVIE% 8 VOLUME 42, NUMBER 4Random-field Blume-Capel model: Mean-field theory1 AUGUST 1990Miron Kaufrnan and Michael KannerDepartment ofPhysics, Cleveland State University, Cleveland, Ohio 44115(Received 27 February 1990)The global phase diagram of the Blume-Capel model in a random field obeying the bimodal sym-metric distribution is determined by using the mean-field method. The phase diagram includes anisolated ordered critical end point and two lines of tricritical points. A new phase emerges forstrong enough random fields: the ferromagnetic-nonmagnetic phase. It is argued that such a phaseoccurs in three dimensions.I. INTRODUCTIONQuenched randomness causes first-order transitions tobe replaced by continuous transitions in two dimensions,and the tricritical temperature to be lowered in higher di-mensions. This was recently shown by Hui and Berker'and Aizenman and Wehr by using respectively position-space renormalization-group techniques and rigorousmethods. It is our goal in this paper to further study theeffect of random fields on a tricritical phase diagram byemploying the mean-field approximation, which is areasonable approximation for high dimensions, and it isexact for the equivalent-neighbor lattice.The simplest spin model exhibiting a tricritical phasediagram in the absence of randomness is the Blume-Capelmodel. This model and its generalizations were used tosimulate the thermodynamics of a variety of systems suchas UO2 and 'He-"He mixtures, ' and it has been studiedextensively with a variety of techniques: mean-field, 'position-space renormalization group, ' and othermethods. In this paper we study the Blume-Capel rnod-el in the presence of a random magnetic field, which takestwo values +H with equal probability.The Blume-Capel model includes two thermodynamicfields: the temperature T and the crystal field D conju-gated to —s, where s is the spin, which can take threevalues, +1 and 0. At zero temperature there are twophases: the ferromagnetic phase (trt =—(s ) =+1,Q = (s ) =1) for small D, and the nonmagnetic phase(m =0, Q =0) for large D. These phases coexist at someintermediate value of D. When the bimodal random field+H is turned on, a novel phase emerges: the mixedferromagnetic-nonmagnetic phase, m =+—,', Q =—,'. Inthe plane H, D, for intermediate values of H, this phaseoccupies a buffer separating the ferromagnetic and non-magnetic phases. Thus the emergence of the mixed phasecan be viewed as a new manifestation of the weakening ofthe first-order transitions by the randomness: the jump inm from +1 to 0 is replaced by two smaller jumps: +1 to+—,' and +—,' to 0.We believe that the emergence of the mixed phase isnot a mere artifact of the mean-field approximation, andshould also occur in three dimensions. At low tempera-tures and large D the nonmagnetic phase (m =0, Q =0)is stable, while at low temperatures and large H theparamagnetic phase (rrt =0, Q=1) is stable. A directtransition between these planes, for large values of bothH and D, is unlikely because it will involve a discontinui-ty in Q but no discontinuity in m. A continuous transi-tion (Q decreases from unity to zero in the paramagneticphase} could hold in two dimensions. In three dimen-sions however, we expect the emergence of the new phasebetween the paramagnetic and nonmagnetic phases to al-low for jumps in both m and Q at the two phase boun-daries.The mixed ferromagnetic-nonmagnetic phase persistsfor arbitrarily large H provided the crystal field D is alsolarge. The existence of long-range order (~m~)0) for un-bounded values of the random-field strength is also ob-served in the Ising model in a trimodal random-field, 'but not for the bimodal" and Gaussian distributions.We find in the three-dimensional parameter spaceT, H, D that there are two noncontiguous tricritical lines.The first starts at the pure Blume-Capel tricriticalpoint ' and shows a monotonic decrease of the tricriticaltemperature when H is increased from zero. This is amanifestation of the weakening of first-order transitionsby random fields. ' The second line which starts at theAharony tricritical point of the Ising model in a binaryrandom field, D = —ac, does not show this simple behav-ior but exhibits an extremurn corresponding to a doubletricritical point. Thus moderate to large randomness hasa subtler and more complicated effect than weak random-ness on the phase diagram. In the same context we notethe occurrence of reentrance phenomena and of non-monotonic dependence of the densities m and Q on thethermodynamic fields.Besides the tricritical points the global phase diagramalso includes the following multicritical points: a line ofordered critical points' (two coexisting critical phases)and an isolated ordered critical end point (coexistence oftwo critical phases and a disordered phase).The remainder of this article contains the solution ofthe Blurne-Capel model with a bimodal random field inSec. II, and our conclusions in Sec. III.II. MODEL AND SOLUTIONAt each site of the equivalent-neighbor lattice (themean-field model} there is a spin s, =+1,0. The Hamil-42 2378 1990 The American Physical Society42 RANDOM-FIELD BLUME-CAPEL MODEL: MEAN-FIELD THEORY 2379P(H; )= —,'[5(H; H)—+5(H, +H)], (2)where H ~0 measures the strength of the random fieldand 5 is the Dirac 5 function.In the thermodynamic limit the quenched averagedfree energy per spin is1f=—min 4(m), (3)where4=—,'Jm2 —(lnIl+e I 2cosh[J(m+H;)]I ) . (4)The quenched average ( . ) in Eq. (4) is performed byusing the distribution in Eq. (2). The value of m whichminimizes 4 is the average site magnetization (s; ) whilethe average of s, ise 2cosh[J(m +H;)]1+e 2 cosh[J(m +H; )]tonian associated with the W spins isW—/T= ps;s, J—D gs; +J QH s;,Jt, J twhere J=1/T. The magnetic field H; is distributed ac-cording to the bimodal distribution:the ferromagnetic phase (m =+1, Q =1,f =D ——,'); thenonmagnetic phase (m =0, Q =0, f =0); the paramag-netic phase (m =0, Q =1, f =D H—); theferromagnetic-nonmagnetic phase [m =+—,', Q = —,',f =,'(D H———,' )]. By comparing the energies we deter-mine the phase diagrams shown in Fig. 1.In the ferromagnetic-nonmagnetic phase half of thespins are equal to zero and the other half are equal to + 1{or —1). The emergence of this phase can be viewed as amanifestation of the weakening effect of randomness onfirst-order transitions: the jump, m =+1 to m =0, atsmall H &—,' is replaced by two smaller jumps, m =+1 tom =+—,' and m =+—,' to m=0, at—,' &0 &—,'. For evenstronger randomness H &—,' the ferromagnetic phase isreplaced by the paramagnetic phase. A reentrancephenomenon, or nonmonotonic dependence of densities,takes place in this regime: lowering D the magnetizationis first 0, then +—,', then 0 again.A. Zero temperatureThe ground-state energies determine the phase dia-gram at zero temperature or J = ~. We find four phases: 0.5-0-0.5-0.5FIG. 1. The zero-temperature phase diagram. The fourphases: ferromagnetic (F), nonmagnetic (NM), paramagnetic(P), and ferromagnetic-nonmagnetic (F-NM) are separated bylines of first-order transitions.FIG. 2. Projections of the two tricritical lines along the Daxis in (a) and along the T axis in (b).2380 MIRON KAUFMAN AND MICHAEL KANNER 420+ 0750.375-0IQ.5 0.25FIG. 3. Phase diagram for D = —5 with a tricritical point C.The dashed line represents first-order transitions and the solidone is a line of critical points.FIG. 4. Phase diagram for D =0.4 with a tricritical point Cand an ordered critical point B . The dashed lines representfirst-order transitions and the solid one is a line of criticalpoints.B. Tricritical pointsWe list the entities on the phase diagram by usingGriffiths' notation system A—one-phase point (disor-dered phase, m =0), A —two-phase point (orderedphase, (s;) =+~m~), A —three-phase point (coexistenceof ordered and disordered phases), A —four-phase point(coexistence of two ordered phases), A —five-phasepoint (coexistence of two ordered phases and the disor-dered phase), B—critical point, B —ordered-criticalpoint (coexistence of two critical phases), C—tricriticalpoints, B A —ordered-critical end point (coexistence oftwo critical phases and the disordered phase).The location of the tricritical points is determined byexpanding 4(m) in powers of m:0.750.3?5-0I0.2 OA4(m) —4(0)=a2m +a„m +a6m (6)The fields az, a4, and a6 depend on 1, H, and D accord-ing toTJ 1a= —+Nu —m2FIG. 5. Phase diagram for D =0.47 with two tricriticalpoints —C and an ordered critical point B . The dashed linesrepresent first-order transitions and the solid one is a line ofcritical points.g4a~=, [6w u —12w u+w~(4u+3) —w],g6a6=, [ 120w u —360w u +w (120u +270u)—w (150u +30)+w (16u+15)—w],where(7)0.375u =tanh (JH),w =e 2 cosh( JH) [1+e 2 cosh( JH) ]and J =1/T. The tricritical points C occur at a2 =a4=0and a6 & 0. We checked' that a6 & 0 in the entire T,H, Dspace and thus there is no fourth-order point in this rnod-el.In the T,H, D space the tricritical manifold containstwo noncontiguous lines. The first starts at the tricriticalpoint of the pure Blurne-Capel model: 8=0,D =(ln4) l3, T = —,'. The second line starts at the tricriti-0 015I03'FIG. 6. Phase diagram for D =0.492 with two tricriticalpoints C and an ordered critical point B'. The dashed linesrepresent first-order transitions and the solid one is a line ofcritical points. The gap between 8 and the first-order line A 'is about to be closed.42 RANDOM-FIELD BLUME-CAPEL MODEL: MEAN-FIELD THEORY 2381SL?50.375-1i a015I0.3 yFIG. 7. Phase diagram for D =0.493 with two tricriticalpoints C and an A ' point. The dashed lines represent first-ordertransitions and the solid one is a line of critical points.cal point of the Ising model in a birnodal random field:"D = —~, H =[ln(2+ v 3)]/3, T = —,'. Projections ofthese lines along D and T directions are shown in Fig. 2.The first tricritical line exhibits the expected monotonicdecrease of T with H. However the second tricritical lineexhibits a monotonic dependence of T and D on H. Ithas an extremum at T=—0.46, H—=0.41, D=—0, corre-sponding to a double tricritical point (see Appendix A ofRef. 7).50I0% 03'FIG. 8. Phase diagram for D =0.75 with two tricriticalpoints C. The dashed lines represent first-order transitions andthe solid one is a line of critical points.C. Global phase diagramWe determine the phase diagram by numericallyminimizing 4 given in Eq. (4). We choose to presenttwo-dimensional slices, at fixed D, of the three-dimensional parameter space. There are five topological-ly different such two-dimensional phase diagrams.For D &—,' the phase diagram, Fig. 3, is topologicallyequivalent to Aharony's phase diagram" for the Isingmodel in a birnodal random field, which includes a tricrit-ical point.For—,' ~ D ((ln4)/3 there is a line of 3 points endingat a B point inside the ordered phase (see Fig. 4). Alongthis line the ferromagnetic and ferro-nonmagnetic phasescoexist. At the ordered critical point the two phasescoalesce.For D =(ln4)/3 a second tricritical point emerges asH=O, T=—,'. Thus for (ln4)/3 ~D &0.493 there are twotricritical C points and one ordered critical B point, seeFigs. 5 and 6.At D —=0.493 (slightly below this value), T=0. 1—4,H=—0.24, there is an ordered-critical end point B A,which viewed in the three-dimensional parameter space islocated at the intersection of the B line and the A sur-face. To our knowledge this multicritical point has notbeen observed in any previous theoretical or experimentalstudy. For 0.493 D &—,' this rnulticritical point is re-placed by an A' point, see Fig. 7. As D approaches —,',the A point and the segment of A points below it ap-proach the T=O axis.For D—,' the ferromagnetic-nonmagnetic phase isseparated from the disordered phase (m=O) by a phaseboundary consisting of two first-order transitions seg-ments and a segment of critical points. There are two tri-critical points in this case, see Fig. 8.III. CONCLUSIONS AND DISCUSSIONWe have determined the global phase diagram of theBlume-Capel model in a bimodally distributed randomfield by using the the mean-field approximation. Thehighest-order rnulticritical entity is the tricritical point.An ordered critical end point is also included in the phasediagramA novel phase, the ferromagnetic nonma-gnetic phaseemerges at sufticiently strong random fields. We believethat this phase will also occur for realistic short-range in-teractions in three dimensions. Indeed at low tempera-tures and for large H the magnetic spins are favored overthe nonmagnetic ones and simultaneously are random-ized by the random field: m=O, Q=1. At large D, onthe other hand, the nonmagnetic spins are favored:m =0, Q=O. What happens in the intermediate regionH =D? We envision three scenarios: (i) the two phasesare separated by a special first-order transition where Qjumps from 1 to 0 but m stays equal to 0; this dichotomyin the behavior of the two densities makes this scenarioless likely; (ii) there is a continuous transition; Q variesfrom 1 to 0 in the paramagnetic phase and m stays equalto 0; this scenario could work in two dimensions whererandomness causes first-order transitions to be replacedby continuous transitions; ' (iii) the ferromagnetic-nonmagnetic phase emerges as a buffer between theparamagnetic and the nonmagnetic phases and is separat-ed from them by first-order transitions; this is the mean-field behavior and we expect it holds in three dimensions.It will be very interesting to check whether this phaseindeed emerges in three dimensions by using an alterna-tive technique (Monte Carlo, renormalization group).The effect of randomness on the tricritical points ismore complicated than anticipated. There are two non-contiguous tricritical lines. On one of them the tricriticaltemperature decreases monotonically with the strength of2382 MIRON KAUFMAN AND MICHAEL KANNER 42the random 6eld, as expected. On the second 1ine howev-er, there is a double tricritical point located at an ex-tremum on the tricritica1 line, i.e., nonmonotonic depen-dence.ACKNOWLEDGMENTSWe are grateful to P. E. Klunzinger for help with thecomputer programming.'K. Hui and A. N. Berker, Phys. Rev. Lett. 62, 2507 (1989).M. Aizenman and J. Wehr, Phys. Rev. Lett. 62, 2503 (1989).T. Schneider and E. Pytte, Phys. Rev. 8 15, 1519 (1977).4M. Blume, Phys. Rev. 141, 517 (1966); H. W. Capel, Physica32, 966 (1966).5M. Blume, V. J. Emery, and R. B. Griffiths, Phys. Rev. A 4,1071(1971).A. N. Berker and M. Wortis, Phys. Rev. B 14, 4946 (1976).7M. Kaufman, R. B. Griffiths, J. M. Yeomans, and M. E. Fish-er, Phys. Rev. B 23, 3448 (1981).For a review of tricritical models, see I. D. Lawrie and S. Sar-bach, in Phase Transitions and Critical Phenomena, edited byC. Domb and J. L. Lebowitz (Academic, New York, 1984),Vol. 9, p. 1.Transitions in which only one density undergoes a jump whileother densities vary continuously are rare but not unheard of,see the following: D. J. Thouless, Phys. Rev. 187, 732 (1969);M. Kaufman and R. B. Griffiths, Phys. Rev. 8 26, 5282(1982).0M. Kaufman, P. E. Klunzinger, and A. Khurana, Phys. Rev.8 34, 4766 (1986).'A. Aharony, Phys. Rev. 8 18, 3318 (1978).' R. B.Griffiths, Phys. Rev. 12, 345 (1975).'~The stability condition a& )0 was not checked in R. M. Sebas-tianes and V. K. Saxena, Phys. Rev. 8 35, 2058 (1987) thus re-sulting in a partially erroneous phase diagram for the Isingmodel in a trimodal random field. The correct phase diagramof this model was first presented in Ref. 10.

Random-Field Blume-Capel Model: Mean-Field Theory

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Random-Field Blume-Capel Model: Mean-Field Theory

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