Random-field Blume-Capel model: Mean-field theory

Cleveland State University EngagedScholarship@CSU Physics Faculty Publications Physics Department 8-1-1990 Random-Field Blume-Capel Model: Mean-Field heory Miron Kaufman Cleveland State University, [email protected] M. Kanner Publisher's Statement Copyright 1990 American Physical Society. Available on publisher's site at htp://link.aps.org/doi/10.1103/ PhysRevB.42.2378. Original Citation Kaufman, Miron and M. Kanner. "Random-Field Blume-Capel Model: Mean-Field heory." Physical Review B 42 (1990): 2378-2382. Repository Citation Kaufman, Miron and Kanner, M., "Random-Field Blume-Capel Model: Mean-Field heory" (1990). Physics Faculty Publications. Paper 16. htp://engagedscholarship.csuohio.edu/sciphysics_facpub/16 his Article is brought to you for free and open access by the Physics Department at EngagedScholarship@CSU. It has been accepted for inclusion in Physics Faculty Publications by an authorized administrator of EngagedScholarship@CSU. For more information, please contact [email protected]. PHYSICAL REVIE% 8 VOLUME 42, NUMBER 4 Random-field 1 AUGUST 1990 Blume-Capel model: Mean-field theory Miron Kaufrnan and Michael Kanner Department of Physics, 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 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 phase. It is argued that such a phase strong enough random fields: the ferromagnetic-nonmagnetic occurs in three dimensions. I. INTRODUCTION Quenched randomness causes first-order transitions to be replaced by continuous transitions in two dimensions, and the tricritical temperature to be lowered in higher dimensions. This was recently shown by Hui and Berker' and Aizenman and Wehr by using respectively positionspace renormalization-group techniques and rigorous methods. It is our goal in this paper to further study the effect of random fields on a tricritical phase diagram by which is a employing the mean-field approximation, reasonable approximation for high dimensions, and it is exact for the equivalent-neighbor lattice. The simplest spin model exhibiting a tricritical phase diagram in the absence of randomness is the Blume-Capel model. This model and its generalizations were used to simulate the thermodynamics of a variety of systems such as UO2 and 'He-"He mixtures, ' and it has been studied extensively with a variety of techniques: mean-field, ' renormalization position-space and other group, ' methods. In this paper we study the Blume-Capel rnodel in the presence of a random magnetic field, which takes two values +H with equal probability. The Blume-Capel model includes two thermodynamic fields: the temperature T and the crystal field D conjugated to — where s is the spin, which can take three values, +1 and 0. At zero temperature there are two — the phases: ferromagnetic (trt = (s ) =+1, phase D, and the nonmagnetic phase Q = (s ) =1) for small (m =0, Q =0) for large D. These phases coexist at some intermediate value of D. When the bimodal random field +H is turned on, a novel phase emerges: the mixed ', Q = —,'. In ferromagnetic-nonmagnetic phase, m —, the plane H, D, for intermediate values of H, this phase occupies a buffer separating the ferromagnetic and nonmagnetic phases. Thus the emergence of the mixed phase can be viewed as a new manifestation of the weakening of the first-order transitions by the randomness: the jump in m from +1 to 0 is replaced by two smaller jumps: +1 to + —,' and + —,' to 0. We believe that the emergence of the mixed phase is not a mere artifact of the mean-field approximation, and should also occur in three dimensions. At low temperatures and large D the nonmagnetic phase (m =0, Q =0) is stable, while at low temperatures and large H the s, =+ 42 paramagnetic phase (rrt =0, Q=1) is stable. A direct transition between these planes, for large values of both H and D, is unlikely because it will involve a discontinuity in Q but no discontinuity in m. A continuous transition (Q decreases from unity to zero in the paramagnetic phase} could hold in two dimensions. In three dimensions however, we expect the emergence of the new phase between the paramagnetic and nonmagnetic phases to allow for jumps in both m and Q at the two phase boundaries. The mixed ferromagnetic-nonmagnetic phase persists for arbitrarily large H provided the crystal field D is also 0) for unlarge. The existence of long-range order ( m bounded values of the random-field strength is also observed in the Ising model in a trimodal random-field, ' but not for the bimodal" and Gaussian distributions. We find in the three-dimensional parameter space T, H, D that there are two noncontiguous tricritical lines. The first starts at the pure Blume-Capel tricritical point ' and shows a monotonic decrease of the tricritical temperature when H is increased from zero. This is a manifestation of the weakening of first-order transitions ' The second line which starts at the by random fields. Aharony tricritical point of the Ising model in a binary random field, D = —ac, does not show this simple behavior but exhibits an extremurn corresponding to a double tricritical point. Thus moderate to large randomness has a subtler and more complicated effect than weak randomness on the phase diagram. In the same context we note the occurrence of reentrance phenomena and of nonmonotonic dependence of the densities m and Q on the thermodynamic fields. Besides the tricritical points the global phase diagram also includes the following multicritical points: a line of ordered critical points' (two coexisting critical phases) and an isolated ordered critical end point (coexistence of two critical phases and a disordered phase). The remainder of this article contains the solution of the Blurne-Capel model with a bimodal random field in Sec. II, and our conclusions in Sec. III. ~ II. MODEL ~ ) AND SOLUTION At each site of the equivalent-neighbor lattice (the model} there is a spin s, =+1,0. The Hamil- mean-field 2378 1990 The American Physical Society RANDOM-FIELD BLUME-CAPEL MODEL: MEAN-FIELD THEORY 42 tonian associated with the W spins is J ps;s, J—D gs; +J QH s;, W—/T= J J =1/T. The magnetic field H; is distributed t, P(H; )= —,'[5(H; H)— +5(H, +H)], ac(2) where H ~0 measures the strength of the random field and 5 is the Dirac 5 function. In the thermodynamic limit the quenched averaged free energy per spin is 1 min 4(m), f=— (3) where 4= 'Jm2 —(lnIl+e I —, 2cosh[J(m+H;)]I ) . '); the the ferromagnetic phase (m =+1, Q =1, =D ——, nonmagnetic phase (m =0, Q =0, =0); the paramag=D H—); the netic phase (m =0, Q =1, ', Q = —,', ferromagnetic-nonmagnetic phase —, [m ' — — '(D H —, )]. By comparing the energies we determine the phase diagrams shown in Fig. 1. In the ferromagnetic-nonmagnetic phase half of the spins are equal to zero and the other half are equal to + 1 {or —1). The emergence of this phase can be viewed as a manifestation of the weakening effect of randomness on first-order transitions: the jump, m =+1 to m =0, at ' is replaced by two smaller jumps, m =+1 to small H & —, ' ' ' and m m For even —, to m=0, at —, &0 & —, —, ' the & randomness H ferromagnetic phase is stronger —, replaced by the paramagnetic phase. A reentrance phenomenon, or nonmonotonic dependence of densities, takes place in this regime: lowering D the magnetization is first 0, then + —,', then 0 again. f t where cording to the bimodal distribution: 2379 (4) f =, f =+ '. =+ =+ f The quenched average ( . ) in Eq. (4) is performed by using the distribution in Eq. (2). The value of m which minimizes 4 is the average site magnetization (s; ) while the average of s, is 2cosh[J(m +H;)] 1+e 2 cosh[J(m +H; )] e A. Zero temperature The ground-state energies gram at zero temperature or determine the phase diafour phases: J = ~. We find 0.5- 0 -0.50.5 FIG. 1. The zero-temperature phase diagram. The four phases: ferromagnetic (F), nonmagnetic (NM), paramagnetic (F-NM) are separated by (P), and ferromagnetic-nonmagnetic lines of first-order transitions. FIG. 2. Projections of the two tricritical lines along the D axis in (a) and along the T axis in (b). 42 MIRON KAUFMAN AND MICHAEL KANNER 2380 0+ 075 0.375- I 0 0.25 Q.5 FIG. 3. Phase diagram for D = —5 with a tricritical point C. The dashed line represents first-order transitions and the solid one is a line of critical points. FIG. 4. Phase diagram for D =0.4 with a tricritical point C and an ordered critical point B . The dashed lines represent first-order transitions and the solid one is a line of critical points. B. Tricritical points We list the entities on the phase diagram by using Griffiths' notation system A one-phase point (disortwo-phase point (ordered dered phase, m =0), A three-phase point (coexistence phase, (s;) =+~m~), A four-phase point of ordered and disordered phases), A five-phase (coexistence of two ordered phases), A point (coexistence of two ordered phases and the disorordered-critical dered phase), B critical point, B point (coexistence of two critical phases), C tricritical ordered-critical end point (coexistence of points, B A two critical phases and the disordered phase). The location of the tricritical points is determined by expanding 4(m) in powers of m: — — — — — — — — 0.75 0.3?5- — 4(m) —4(0)=a2m +a„m +a6m (6) The fields az, a4, and a6 depend on 1, H, and D according to T a= J —+Nu —m I 0 0.2 OA FIG. 5. Phase diagram for D =0.47 with two tricritical C and an ordered critical point B . The dashed lines points represent first-order transitions and the solid one is a line of critical points. — 1 2 g4 a~=, [6w —12w u+w~(4u+3) —w], u g6 a6=, [ 120w —w u —360w (150u u (7) +w (120u +270u) +30)+w (16u+15) —w], 0.375 where u =tanh (JH), w =e 2 cosh( JH) [1+e 2 cosh( JH) ] J and =1/T. The tricritical points C occur at a2 =a4=0 and a6 & 0. We checked' that a6 & 0 in the entire T, H, D space and thus there is no fourth-order point in this rnod- el. In the T, H, D space the tricritical manifold contains two noncontiguous lines. The first starts at the tricritical 8=0, model: of the pure Blurne-Capel point D =(ln4) l3, T = —,'. The second line starts at the tricriti- I 0 015 03' FIG. 6. Phase diagram for D =0.492 with two tricritical points C and an ordered critical point B'. The dashed lines represent first-order transitions and the solid one is a line of critical points. The gap between 8 and the first-order line A ' is about to be closed. RANDOM-FIELD BLUME-CAPEL MODEL: MEAN-FIELD THEORY 42 ?5 at a B point inside the ordered phase (see Fig. 4). Along this line the ferromagnetic and ferro-nonmagnetic phases coexist. At the ordered critical point the two phases SL 0.375- i 1 I a 0.3 015 y FIG. 7. Phase diagram for D =0.493 with two tricritical points C and an A ' point. The dashed lines represent first-order transitions 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 of these lines along D and T directions are shown in Fig. 2. The first tricritical line exhibits the expected monotonic decrease of T with H. However the second tricritical line exhibits a monotonic dependence of T and D on H. It = 0.41, D= —0, corre—0.46, H— has an extremum at T= sponding to a double tricritical point (see Appendix A of Ref. 7). '. C. Global phase diagram We determine the phase diagram by numerically minimizing given in Eq. (4). We choose to present two-dimensional slices, at fixed D, of the threedimensional parameter space. There are five topologically different such two-dimensional phase diagrams. For D & —,' the phase diagram, Fig. 3, is topologically equivalent to Aharony's phase diagram" for the Ising model in a birnodal random field, which includes a tricrit- 4 ical point. For ' —, ~ D ((ln4)/3 there is a line of 3 points ending coalesce. For D =(ln4)/3 a second tricritical point emerges as H=O, T= —,'. Thus for (ln4)/3 ~D &0.493 there are two tricritical C points and one ordered critical B point, see Figs. 5 and 6. = 0. 493 (slightly below this value), T= 0. 1— At D — 4, H=—0. 24, there is an ordered-critical end point B A, which viewed in the three-dimensional parameter space is located at the intersection of the B line and the A surface. To our knowledge this multicritical point has not been observed in any previous theoretical or experimental study. For 0. 493 D & —,' this rnulticritical point is replaced by an A' point, see Fig. 7. As D approaches —,', the A point and the segment of A points below it approach the T=O axis. For D —,' the ferromagnetic-nonmagnetic phase is separated from the disordered phase (m=O) by a phase boundary consisting of two first-order transitions segments and a segment of critical points. There are two tricritical points in this case, see Fig. 8. III. CONCLUSIONS AND DISCUSSION We have determined the global phase diagram of the Blume-Capel model in a bimodally distributed random field by using the the mean-field approximation. The highest-order rnulticritical entity is the tricritical point. An ordered critical end point is also included in the phase diagram A novel phase, the ferromagnetic nonma-gnetic phase emerges at sufticiently strong random fields. We believe that this phase will also occur for realistic short-range interactions in three dimensions. Indeed at low temperatures and for large H the magnetic spins are favored over the nonmagnetic ones and simultaneously are randomized by the random field: m=O, Q=1. At large D, on the other hand, the nonmagnetic spins are favored: m =0, Q=O. What happens in the intermediate region H =D? We envision three scenarios: (i) the two phases are separated by a special first-order transition where Q jumps from 1 to 0 but m stays equal to 0; this dichotomy in the behavior of the two densities makes this scenario less likely; (ii) there is a continuous transition; Q varies from 1 to 0 in the paramagnetic phase and m stays equal to 0; this scenario could work in two dimensions where randomness causes first-order transitions to be replaced transitions; (iii) the ferromagneticby continuous nonmagnetic phase emerges as a buffer between the paramagnetic and the nonmagnetic phases and is separated from them by first-order transitions; this is the meanfield behavior and we expect it holds in three dimensions. It will be very interesting to check whether this phase indeed emerges in three dimensions by using an alternative technique (Monte Carlo, renormalization group). The effect of randomness on the tricritical points is more complicated than anticipated. There are two noncontiguous tricritical lines. On one of them the tricritical temperature decreases monotonically with the strength of ' 5 I 0 2381 0% 03' FIG. 8. Phase diagram for D =0.75 with two tricritical points C. The dashed lines represent first-order transitions and the solid one is a line of critical points. 2382 MIRON KAUFMAN AND MICHAEL KANNER the random 6eld, as expected. On the second 1ine however, there is a double tricritical point located at an extremum on the tricritica1 line, i.e., nonmonotonic dependence. '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, Physica 32, 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. Fisher, Phys. Rev. B 23, 3448 (1981). For a review of tricritical models, see I. D. Lawrie and S. Sarbach, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz (Academic, New York, 1984), Vol. 9, p. 1. View publication stats 42 ACKNOWLEDGMENTS We are grateful to P. computer programming. E. Klunzinger for help with the Transitions in which only one density undergoes a jump while other 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. Sebastianes and V. K. Saxena, Phys. Rev. 8 35, 2058 (1987) thus resulting in a partially erroneous phase diagram for the Ising model in a trimodal random field. The correct phase diagram of this model was first presented in Ref. 10. )