Equilibrium, Metastability, and Hysteresis in a Model Spin-crossover Material with Nearest-neighbor Antiferromagnetic-like and Long-range Ferromagnetic-like Interactions
aa r X i v : . [ c ond - m a t . m t r l - s c i ] F e b Equilibrium, Metastability, and Hysteresis in a Model Spin-crossover Material withNearest-neighbor Antiferromagnetic-like and Long-range Ferromagnetic-likeInteractions
Per Arne Rikvold, Gregory Brown, Seiji Miyashita, , Conor Omand, , Masamichi Nishino Department of Physics, Florida State University, Tallahassee, FL 32306-4350,USA Computational Science and Mathematics Division,Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, USA Department of Physics, Graduate School of Science,The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada Computational Materials Science Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan
Phase diagrams and hysteresis loops were obtained by Monte Carlo simulations and a mean-field method for a simplified model of a spin-crossover material with a two-step transition betweenthe high-spin and low-spin states. This model is a mapping onto a square-lattice S = 1 / PACS numbers: 75.30.Wx,64.60.My,64.60.Kw,75.60.-d
I. INTRODUCTION
In many materials, local elastic interactions induce effective long-range interactions via the macroscopic strainfield.
In addition to elastic crystals, physical realizations of long-range interactions include systems as diverse asearthquake faults and long-chain polymers. Phase transitions in such systems belong to the mean-field universalityclass , which has some unusual properties. In particular, critical clusters can be strongly suppressed compared totransitions caused by purely short-range interactions. This effect should be experimentally observable as an absenceof critical opalescence.
A particular class of systems that exemplify these interesting properties are spin-crossover (SC) materials.
These are molecular crystals in which the individual organic molecules contain transition metal ions, such as Fe(II),Fe(III), or Co(II), that can exist in two different spin states: a low-spin ground state (LS) and a high-spin excited state(HS). Molecules in the HS state have higher effective degeneracy and larger volume than those in the LS state. Dueto the higher degeneracy of the excited HS state, crystals of such molecules can be brought into a majority HS stateby increasing temperature, changing pressure or magnetic field, or by exposure to light.
If the intermolecularinteractions are sufficiently strong, this change of state can become a discontinuous phase transition such that theHS phase becomes metastable and hysteresis occurs.
In the case of optical excitation into the metastable phase,this effect is known as light-induced excited spin-state trapping (LIESST).
The different magnetic and opticalproperties of the two phases make such cooperative SC materials promising candidates for applications such asswitches, displays, memory devices, sensors, and activators.
Various experimental results over the last decade have led to the suggestion that the dominant interaction mechanismin SC materials is elastic and therefore effectively long-range, due to the larger size of the molecule in its HS state.
Such systems can be modeled by a pseudo-spin Hamiltonian of the form H = − J X h i,j i s i s j −
12 ( k B T ln g − D ) X i s i + H LR . (1.1)The first two terms constitute the Wajnflasz-Pick Ising-like model, in which the pseudo-spin variables s i denote thetwo spin states ( − J is a nearest-neighbor interaction. The effective field term, H = ( k B T ln g − D ) / , (1.2)contains D >
0, which is the energy difference between the HS and LS states, g , which is the ratio between the HSand LS degeneracies, and k B T , the absolute temperature in energy units. The long-range interactions are representedby H LR . The usual order parameter for SC materials is the proportion of HS molecules, n HS , which is related tothe pseudo-spin variables as n HS = ( m + 1) /
2, where m = P i s i /N (with N the total number of molecules) is thepseudo-magnetization.While the long-range term H LR is most often considered as a true elastic interaction, many aspects of themodel can be obtained at much lower computational cost by replacing this by a Husimi-Temperley (a.k.a. equivalent-neighbor) effective Hamiltonian. The latter is the approach we follow in the present paper.In previous work we have considered models of SC materials that show a direct, or one-step, transition betweenthe LS and HS phases. In these cases, the short-range interactions favor configurations in which nearest-neighbormolecules are in the same state (LS-LS or HS-HS). This corresponds to a positive short-range interaction constant J in Eq. (1.1). In the pseudo-spin language often used in the SC literature, this case is called ferromagnetic-like, orsimply ferromagnetic. We emphasize that this is only an analogy and does not imply a magnetic character of theinteractions. In the remainder of this paper, we will use the simplified terms, ferromagnetic and antiferromagnetic,for interactions that favor uniform and checkerboard spin-state arrangements, respectively.When both the short-range and long-range interactions are ferromagnetic, any nonzero long-range interaction hasthe effect of changing the universality class of the critical point of the LS/HS phase transition caused by the short-range interactions from the Ising to the mean-field universality class. As a result, critical clusters are suppressed,and the system develops true metastable phases limited by sharp spinodal lines in the phase diagram.There also exist SC materials, in which the transition between the LS and HS phases proceeds as a two-steptransition via an intermediate phase.
For some materials, such as Fe(II)[2-picolylamine] Cl · Ethanol, it hasbeen shown by x-ray diffraction that spontaneous symmetry breaking induces an intermediate phase, characterized bylong-range order on two interpenetrating sublattices with nearest-neighbor molecules in different states (HS-LS). This situation can be modeled by the Ising-like model of Eq. (1.1) with antiferromagnetic nearest-neighbor interactions(
J < and without along-range ferromagnetic term.In a recent work, Nishino and Miyashita studied such a model using an elastic long-range interaction. Theyfound that, for weak applied field, the antiferromagnetic phase transition remains in the Ising universality class. Atlow temperatures they observed tricritical points separating lines of second-order and first-order phase transitions.However, much of their high-temperature analysis replaced the short-range interactions by a two-sublattice mean-fieldmodel, which neglects the effects of local fluctuations. The aim of the present paper is to investigate the effects of suchfluctuations in an Ising-like model with nearest-neighbor antiferromagnetic interactions and a long-range ferromagneticinteraction of the Husimi-Temperley form. This enables us to obtain excellent long-time statistics for systems withover one million individual pseudo-spins, and for several different values of the long-range interaction strength. Somepreliminary results were included in Ref. 43.The organization of the rest of the paper is as follows. In Sec. II we present the Ising Hamiltonian with antiferromag-netic nearest-neighbor interactions and a ferromagnetic Husimi-Temperley type long-range interaction of adjustablestrength. Here we also obtain ground-state diagrams (zero-temperature phase diagrams) including phase coexistencepoints and spinodal points for different strengths of the long-range interaction term. In Sec. III we present results for amean-field approximation to this model, in which the nearest-neighbor antiferromagnetic interactions are replaced by atwo-sublattice mean-field approximation. Two different strengths of the long-range ferromagnetic interactions, whichyield qualitatively different phase diagrams, are considered. In Sec. IV we return to our original, nearest-neighborantiferromagnetic interactions, investigating the resulting phase diagrams by Metropolis importance-sampling MonteCarlo (MC) simulations. We find that, in a certain range of the long-range interaction strength, the resulting phasediagrams are qualitatively different from those obtained by the mean-field approximations. In particular, tricriticalpoints predicted by the mean-field approximations are found by MC to decompose into pairs of critical endpoints andmean-field critical points surrounded by horn-shaped regions of metastability. Here we also present hysteresis curvesthat are particularly relevant to the system’s interpretation as a model of two-step transitions in SC materials. Ourconclusions and suggestions for future work are presented in Sec. V.
II. HAMILTONIAN AND GROUND-STATE ANALYSIS
The square-lattice Ising antiferromagnet with weak, long-range (Husimi-Temperley) ferromagnetic interactions isdefined by the Hamiltonian H = − J X h i,j i s i s j − N (cid:18) H + A m (cid:19) m (2.1)with J <
A >
0. Here, H is the applied field, s i = ±
1, and m = N − P i s i . For a square lattice of side L , N = L . The strength of the long-range interaction is A . It is defined such that the critical temperature of the purelong-range ferromagnet ( J = 0) equals A/k B , where k B is Boltzmann’s constant. For convenience we will hereafteruse the dimensionless variables, h = H/ | J | , a = A/ | J | , and t = k B T / | J | .The model’s equilibrium and metastable phases at zero temperature are found by a simple ground-state analysis.The per-site energies of the fully ordered antiferromagnetic (AFM) and field-induced ferromagnetic (FM) phases aregiven by E AFM / ( N | J | ) = − E + / ( N | J | ) = 2 − a/ − h , and E − / ( N | J | ) = 2 − a/ h , respectively. Equating theAFM and FM energies yields the zero-temperature transition values of h as h AFM / + = 4 − a/ h AFM / − = − a/ . (2.3)For a >
8, the AFM ground state disappears, and the system in equilibrium has a direct transition between the m = − m = +1 FM ground states at h = 0. (For a = 8, the AFM and both FM ground states are degenerateat h = 0.)The limits of local stability of metastable phases at t = 0 (zero-temperature spinodal fields) are the field values atwhich the energy change due to a flip of a single spin in the metastable phase becomes negative. For metastability,a positive energy change is required. The spinodal fields are determined as follows. A. a < Decreasing h , attempting to nucleate a transition from metastable m = +1 to the AFM or m = − → + − ++ + + + + + , the energy change is ∆ E/ | J | = − h + 2 a (1 − /N ) > ⇒ h > − a (1 − /N ) . (2.4)Thus, (in the limit N → ∞ ) the m = +1 phase is metastable for 4 − a < h < − a/
2. By symmetry, the m = − − a/ < h < − a .Increasing h , attempting to nucleate a transition from metastable AFM to the m = +1 ground state by a singlespin flip, − + − − + − + − + → + + + − + − − + − , the energy change is ∆ E/ | J | = 8 − h − a/N > ⇒ h < − a/N . (2.5)Thus, (in the limit N → ∞ ) the AFM phase is metastable against decay to m = +1 for 4 − a/ < h <
4. Bysymmetry, the AFM phase is metastable against decay to m = − − < h < − a/
2. Four mean-field sharpspinodal lines extend from these zero-temperature spinodal points toward higher t . The zero-temperature limits ofthe phase diagrams shown in Figs. 1(a), 4(a), and 5(a) illustrate the positions of coexistence and spinodal points forvalues of a < B. a > The only ground states are m = +1 for h ≥ m = − h ≤
0. Increasing h , attempting to nucleate atransition from metastable m = − m = +1 ground state by a single spin flip, − − − − − −− − − → − + −− − − − − − , the energy change is ∆ E/ | J | = − − h + 2 a (1 − /N ) > ⇒ h < a (1 − /N ) − . (2.6)Thus, (in the limit N → ∞ ) m = − < h < a −
4. By symmetry, m = +1 is metastable for − a + 4 < h <
0. Using Eq. (2.5) and symmetry, we find that the AFM state, while never a ground state, is metastablefor − < h <
4. The zero-temperature limits of the phase diagrams shown in Figs. 3(a) and 9(a) illustrate thepositions of coexistence and spinodal points for a > III. MEAN-FIELD APPROXIMATION
In order to obtain an approximate picture of the behavior of the model at finite t , we employ a simple, two-sublatticeBragg-Williams mean-field approximation with sublattice magnetizations m A = 2 N − P i ∈ A s i on sublat-tice A ( m A ∈ [ − , +1]) and analogously for m B on sublattice B. The magnetization and staggered magnetization aregiven by m = ( m A + m B ) / m Stag = ( m A − m B ) /
2, respectively. The approximation is defined by the Hamiltonian H MFA N | J | = 42 m A m B − a m − hm = 12 (cid:16) − a (cid:17) m A m B − a m + m ) − h m A + m B ) . (3.1)Note that the effective interaction between the two sublattice magnetizations includes the long-range interactionstrength a and changes sign from AFM for weak a to FM for strong a at a = 8. For a = 8, the mean-fieldapproximation describes two independent, ferromagnetic sublattices.The system entropy in the mean-field approximation is the sum of the two sublattice entropies, S MFA = − N B X X=A (cid:18) m X m X − m X − m X (cid:19) , (3.2)and the resulting free energy is F MFA = H MFA − k B T S
MFA . (3.3)The coupled self-consistency equations, ∂F MFA /∂m A = 0 and ∂F MFA /∂m B = 0, become m A = tanh " (cid:0) a − (cid:1) m B + a m A + ht (3.4)and equivalent for m B with A and B interchanged on the right-hand side.For a < h = 0, the global free-energy minima lie along the AFM axis ( m B = − m A ) in the order-parameterplane, and the self-consistency equation for the staggered magnetization becomes m Stag = tanh 4 m Stag t (3.5)with the N´eel temperature t N = 4, independent of a . For a > h = 0, the global free-energy minima lie alongthe FM axis ( m B = m A ), and the self-consistency equation for the magnetization becomes m = tanh ( a − mt (3.6)with the a -dependent Curie-Weiss critical temperature t C = ( a − m A , m B ) plane were obtained numerically using Mathematica. Resulting phase diagramsfor a = 7 and a = 10 are shown in Figs. 1 and 3, respectively. A. a = 7 Typical mean-field phase diagrams for the model with a < a = 7. In Fig. 1(a),the line of N´eel critical points, marking the continuous phase transition between the stable AFM phase and the high-temperature disordered phase, extends from t N = 4 to two tricritical points at a lower temperature, which decreaseswith decreasing a . (For a = 7, the tricritical points are found at t ≈ .
75 and h ≈ ± . h = ± (4 − a/
2) = ± .
5. Sharp spinodal field lines marking the limits ofmetastability for the AFM phase extend from the tricritical points to the zero-temperature spinodal points at h = ± − phases extend to their zero-temperaturetermination points at h = 4 − a = − h = − a = +3, respectively. The corresponding zero-field orderparameters, the equilibrium AFM m Stag and the metastable FM m , which follow Eqs. (3.5) and (3.6), respectively,are shown in Fig. 1(b). However, the free-energy barriers that prevent the decay of the metastable FM phases (whichare possible at h = 0 only for 4 < a <
8) into an AFM phase vanish at a spinodal temperature t sFM ≈ . m to zero. This temperature corresponds to the crossing of the two FM spinodals inFig. 1(a). Above it, the FM solutions of the self-consistency equations become saddle points. Samples of free-energycontour plots in the m A , m B plane, based on Eq. (3.3), overlaid with curves representing the individual solutions ofthe two self-consistency equations, Eq. (3.4), are shown in Fig. 2.Mean-field models of two-step crossover have previously been considered by Zelentsov et al., Bousseksou et al., and Bolvin, with Bragg-Williams approaches similar to ours, and by Chernyshov et al. using Landau theory.However, Bolvin does not include any FM interaction, and consequently the resulting mean-field phase diagrams arethose of a pure Ising antiferromagnet without any first-order transitions (Figs. 5 and 6 of Ref. 33). Zelentsov et al.and Bousseksou et al. confine the FM interactions to each sublattice separately, and so these interactions do not affectthe effective inter-sublattice interaction as they do in our model [see Eq. (3.1)]. Neither paper contains explicit phasediagrams. However, their plots of HS fraction vs temperature indicate that phase diagrams for the case of identicalsublattices (their intra-sublattice interactions J A = J B ) should be similar to ours, including tricritical points and first-order transitions at low temperatures. However, their intra-sublattice FM interactions lower the energies of the FMand AFM ground states by the same amount, with the result that they, in contrast to the corresponding term in ourmodel, do not influence the ground-state diagram of the pseudo-spin model. In that respect, their approaches wouldcorrespond to a mean-field approximation to a square-lattice Ising model with AFM nearest-neighbor (inter-sublattice)and FM next-nearest neighbor (intra-sublattice) interactions. In the Landau-theory approach of Chernyshov etal., the ferromagnetic interactions are not limited to the separate sublattices. For not too negative values of theirtemperature-like parameter α , the phase diagram in Fig. 5 of Ref. 34 is quite similar to our Fig. 1(a). We believean approach, in which the ferromagnetic interactions are not limited to the individual sublattices, provides a betterapproximation for the effects of a long-range elastic interaction. B. a = 10 Figure 3 shows typical mean-field phase diagrams for the model with a >
8, here using a = 10. There are onlyFM equilibrium phases, which coexist at h = 0 up to the Curie temperature, t C = a − − phases extend from the critical point totheir zero-temperature termination points at h = 4 −
10 = − h = − t and h , and the spinodals marking their limits of local stability arealso shown. The corresponding zero-field order parameters, the equilibrium FM m and the metastable AFM m Stag ,which follow Eqs. (3.6) and (3.5), respectively, are shown in Fig. 3(b). However, the metastable AFM phases becomeunstable toward decay into a FM phase at a spinodal temperature t s AFM ≈ . a , near the critical, tricritical, or spinodal temperature thespinodal fields obey the power law, | h Spin − h Coex | ∼ ( t c − t ) / , (3.7)where h Spin and h Coex are the spinodal and coexistence fields, respectively, and t c represents the appropriate temper-ature where they meet. This is shown in the insets in Figs. 1(a) and 3(a). IV. MONTE CARLO SIMULATIONS
The standard mean-field approximation for the short-range interactions, discussed in Sec. III, does not properlydescribe the microscopic fluctuations that are important in low-dimensional systems, especially near critical andmulticritical points.
We therefore return to the full model described by Eq. (2.1) to further investigate its phasediagrams and dynamics using importance-sampling Metropolis MC simulations. We consider L × L square latticeswith L = 64, ..., 1024 and periodic boundary conditions. Results are extrapolated to the thermodynamic limit usingknown finite-size scaling relations.Critical points are located by crossings of fourth-order Binder cumulants for the antiferromagnetic order parameter m Stag , U L = 1 − h m i L h m i L . (4.1)(In general, the moments included in this equation are central moments , but since the model contains no staggeredfield, this is automatically satisfied for the moments of m Stag .) This method significantly reduces finite-size effects, andthe results are further linearly extrapolated to 1 /L = 0. For isotropic interactions and periodic boundary conditionson a square lattice, as used here, the Ising fixed-point value of the cumulant is U ∗ = 0 . ... . Coexistence lines represent first-order phase transitions between stable equilibrium phases, i.e., equality of the corre-sponding bulk free energies. Here, we locate the coexistence lines by starting simulations from an initial configurationconsisting of two slabs, one in the AFM ground state and one in the FM ground state corresponding to the sign of h (mixed start method), and searching for the field at which the final state would be either with approximately 50%probability.The long-range, ferromagnetic interactions produce a finite barrier in the free-energy density , separating themetastable and stable phases. As a result, the total free-energy barrier increases linearly with the system size, N = L , leading to an exponential size divergence for the lifetime of such a “true” metastable phase. (We notethat this situation is radically different from the case of metastable decay in systems with only local interactions. Inthat case, the decay occurs through nucleation and growth of compact droplets of the stable phase, and the metastablelifetime becomes system-size independent in the thermodynamic limit. ) The sharp spinodal lines at which the free-energy barrier vanishes are located by starting an L × L system in the equilibrium phase and slowly scanning h past the coexistence line (where the initial phase becomes metastable) until the order parameters undergo simulta-neous, discontinuous jumps denoting the limit of metastability. The field h Spin , corresponding to the instability, wasextrapolated to the thermodynamic limit according to the finite-size scaling relation | h Spin − h L | ∼ L − / , (4.2)where h L is the field at which the metastable phase becomes unstable for the given value of L . A. a = 4 The h, t phase diagram for a = 4 is shown in Fig. 4. Except for the absence of metastable FM phases at h = 0,which is due to the lower value of a used here, the phase diagram is topologically identical to the mean-field phasediagram for a = 7, shown in Fig. 1. However, the line of critical points belongs to the Ising universality class, asevidenced by cumulant crossing values U ∗ ≈ .
61. At h = 0 the critical temperature is near the exact Ising value, t c ( h = 0) = 2 / ln(1 + √ ≈ . L as large as 1024 were used. To estimate the position of the tricritical point, we extrapolated theseparation between the L -extrapolated spinodal fields to zero according to Eq. (3.7) to find the tricritical temperatureand from it the corresponding field values. The result is t ≈ .
914 and h ≈ ± .
383 The coexistence line was obtainedby the mixed start method with L = 512. No significant differences were observed with larger L .We note that, for this relatively weak long-range interaction, our MC phase diagram shown in Fig. 4 is qualitativelysimilar to mean-field phase diagrams, both our Fig. 1 and Fig. 5 of Ref. 34. However, for stronger long-rangeinteractions, complex features that are not seen in the mean-field approximations are revealed by our MC simulations.These are discussed in Secs. IV B and IV C below. B. a = 7
1. Phase diagram
The MC phase diagram in the h, t plane for a = 7 is shown in Fig. 5. Because of the stronger long-range interactions,metastable FM phases are possible for weak fields and low temperatures. However, the main difference from the caseof a = 4 (Fig. 4) is that the tricritical points have been transformed into critical endpoints, where the line of Isingcritical points meets the coexistence lines at a large angle (light gray squares, magenta online). Above the temperatureof the critical endpoints, the coexistence lines continue toward higher temperatures, each eventually terminating ata mean-field critical point (large, black circles). Below the critical line and between the two coexistence lines, thestable phase is AFM. On the positive (negative) side of the right-hand (left-hand) coexistence line, the stable phaseis FM+ (FM − ). Above the critical line and between the coexistence lines, the stable phase is disordered with localfluctuations of AFM symmetry and a small magnetization in the direction of the applied field. For clarity, the insetin Fig. 5(a) shows the phase diagram with only the stable phases and corresponding phase transition lines and pointsincluded. The critical and coexistence lines were obtained as described above, with the coexistence lines calculatedwith L = 1024 to minimize the uncertainty. Our best estimates for the positions of the critical endpoints and mean-field critical points, based on simulations up to L = 1024 and finite-size scaling extrapolations, are t = 2 . h = ± . t = 2 . h = ± . − phases cross the line of critical points at fields significantly weaker than those of the critical endpoints. Eachof the FM spinodals continues on to meet the corresponding disorder spinodal at a mean-field critical point, forminga horn-like region of metastability. (As discussed below, the disorder spinodal and the disorder/FM coexistence linecoincide within our numerical accuracy for t & . h at constant t acrossthe coexistence line and monitoring the maximum of the susceptibility, χ max . At the critical point χ max ∼ L γ/ν eff ,with the mean-field exponents γ = 1 and ν eff = 2 /d = 1. Above the critical temperature, the scaling is sublinearin L , and below it is superlinear. The gray (orange online), diagonal line corresponds to the path for the hysteresisloops shown in Fig. 7.The main part of Fig. 5(b) shows a magnified image of the horn region. At this level of detail, two interestingphenomena become apparent. The first is that the Ising critical line (obtained from the crossings of fourth-ordercumulants and linearly extrapolated to 1 /L = 0) continues beyond the critical endpoint where it meets the coexistenceline, until it meets the spinodal line at h ≈ . L up to 1024 with a completely random spin configurationand equilibrating at t = 2 .
15 and h = 0 .
645 [marked by an up triangle in Fig. 5(c)] for 10 MCSS before measuringthe order parameters. The metastability of the disordered phase in the same region was confirmed by scanning h inthe positive direction across the coexistence line until it decayed discontinuously to FM+ at the spinodal. At thepoint where the extended critical line meets the spinodal line, the interpretation of the latter changes from the limitof metastability of the AFM phase at lower t to being the limit of metastability of the disordered phase at higher t .The line was determined by the same method in both temperature regions.The second phenomenon observed in Figs. 5(b) and (c) is that, above t ≈ .
20, the coexistence line and the spinodalline for the disordered phase coincide within our numerical accuracy. The inset in Fig. 5(b) demonstrates that theseparation of the spinodals near the tip of each horn obeys Eq. (3.7). At h = 0 the critical temperature is again nearthe exact Ising value, t c ( h = 0) ≈ .
2. Hysteresis loops
The phase diagram for a = 7, shown in Fig. 5, suggests the existence of complex hysteresis loops. Constant-temperature hysteresis loops for two different temperatures in the horn region are shown in Fig. 7. In Fig. 7(a) we use t = 2 .
18, which lies between the temperature of the critical endpoint and the temperature at which the FM spinodallines cross the critical line. Following the curves in the negative direction from h = +0 .
8, the system starts in the stableFM+ phase, which becomes metastable as the phase point crosses the coexistence line at h ≈ +0 .
63. (Note that thedisorder/FM coexistence lines are at only slightly weaker fields than the disorder spinodal lines at this temperature.See Figs. 5(b) and (c).) Crossing the FM+ spinodal line at h ≈ +0 .
33, the system changes discontinuously to theequilibrium AFM phase. It remains in this phase until it crosses the critical line into the equilibrium disordered phaseat h ≈ − .
52. Finally there is a discontinuous change to the FM − phase across the disorder spinodal at h ≈ − . h as the field is reversed from h = − . . t = 2 .
25 (which lies between the temperature at which the FM spinodals cross the criticalline and the zero-field critical temperature) in Fig. 7(b), the main difference is that the system changes discontinuouslyfrom the FM+ phase to the equilibrium disordered phase at h ≈ +0 .
38, only passing into the equilibrium AFM phaseas it crosses the critical line at h ≈ +0 .
24. At h ≈ − .
24 it again crosses the critical line into the disordered phase,which it leaves through a discontinuous jump as it crosses out of the horn region at the negative disorder spinodal at h ≈ − .
60. The path is again symmetric during the field reversal. At both temperatures the nonzero values of thestaggered magnetization in the disordered equilibrium phase are a finite-size effect.When using the model studied here to represent phase transitions in SC materials, the magnetic field is replacedby the temperature-dependent effective field of Eq. (1.2). A path for temperature driven hysteresis within thisinterpretation of the model is represented by the diagonal line segment in Fig. 5(a) (degeneracy ratio ln g = 20 / D = 18 | J | ). The corresponding phase transitions and hysteresis loops are shown in Fig. 8. The loopsare asymmetric. The narrow loop above the critical temperature corresponds to passage across the positive- h horn,while the wider loop below the critical temperature lies between the negative AFM spinodal and the FM − spinodal.The nonzero values of m Stag in the disordered phase region are again a finite-size effect.We note that the pattern of transitions and hysteresis loops shown in Fig. 8 closely resembles recent experimentalresults for thermal two-step transitions with hysteresis in several different SC materials.
Some of these experimentsare also discussed in two recent reviews of this rapidly developing field. C. a = 10 For a > h, t phase diagram with a = 10 is shown for h ≥ h = 0. The black curve with data points in the main figure shows theFM − spinodal. For weak fields and temperatures below the Ising critical temperature ( t c ( h = 0) ≈ . h = 0, the disordered phase is metastable between the zero-fieldIsing critical temperature and t ≈ .
68. Like the metastable AFM phase, it is separated from the equilibrium FMphases by sharp mean-field spinodals. A magnified view of the region containing the metastable disordered phaseis shown in the inset in Fig. 9(a). The spinodals and the metastable critical line shown in Fig. 9(a) were obtainedfrom finite-size scaling extrapolations of MC data up to L = 1024. These features do not appear in the simple mean-field approximation shown in Fig. 3(a). Stable and metastable order parameters at h = 0 are shown in Fig. 9(b),corresponding to the mean-field results shown in Fig. 3(b). The MC simulations for the metastable order parameterare seen to be in excellent agreement with the Onsager-Yang exact order parameter for a square-lattice Ising modelin zero field, m Stag ( t,
0) = (cid:8) − (cid:2) sinh(2 /t ) − (cid:3)(cid:9) / . The metastable disordered phase is characterized by values ofthe simulated m Stag that decrease linearly with L and go discontinuously to zero at a sharp spinodal temperature.This phase lies between the Ising critical temperature and the spinodal temperature, whose L -extrapolated value ismarked by a vertical, dashed line. If the system is heated in the metastable phases, this is the temperature at whichthe stable order parameter m will jump discontinuously from near zero to its equilibrium value. The observation of acritical line separating the metastable AFM and disordered phases in this large- a regime supports our interpretationof the critical line in the small part of the horn region for a = 7, shown in Fig. 5(c) as a transition line separating twometastable phases.The phase transition of the stable FM phase at t ≈ .
98 involves a small discontinuity (seen only for L = 1024) andnegative values of the Binder cumulants above the transition temperature (seen for L = 1024 and 512, not shown).These features suggest that this transition is weakly first-order. Exploratory simulations for a = 8 . a = 20at h = 0 indicate a strongly first-order transition in the former case, and a continuous transition in the mean-fielduniversality class in the latter. Further investigation of the strong long-range interaction regime of a ≥ V. CONCLUSIONS
In this paper we present a detailed investigation of the phase diagrams of a simplified model of an SC materialwith a two-step transition as a square-lattice Ising model with AFM nearest-neighbor interactions and FM long-rangeinteractions of the Husimi-Temperley (equivalent-neighbor) kind. An AFM equilibrium phase for weak applied fieldsis replaced by field-induced FM phases at stronger fields. These phases are separated by coexistence lines surroundedby sharp spinodal lines representing limits of metastability.In a range of intermediate-strength long-range interactions, we find significant differences between the phase dia-grams of this model, calculated by importance-sampling MC simulations, and those of a model in which fluctuationshave been neglected by replacing the nearest-neighbor interactions by a two-sublattice mean-field approximation. Thedifference consists in the replacement of each tricritical point in the mean-field model with a pair consisting of a crit-ical endpoint and a mean-field critical point in the h, t phase diagram, surrounded by horns representing metastablephases. For even stronger long-range interactions, the AFM equilibrium phase disappears. However, metastable AFMand disordered phases can still be observed. These complex phase diagrams give rise to hysteresis loops reminiscentof two-step transitions observed in several SC materials.
We find it likely that the horn type phase diagrams and related two-step hysteresis loops revealed in our modelby MC simulations may be observed in more realistic models of SC materials with elastic interactions, and alsoin future experiments. An investigation into the former possibility is in progress. Other interesting avenues offurther research include cluster mean-field approximations for the short-range AFM interactions and calculation offree-energy surfaces in the m A , m B plane by Wang-Landau MC simulations. Acknowledgments
P.A.R. and C.O. gratefully acknowledge hospitality by the Department of Physics, University of Tokyo. P.A.R.thanks H.R.D. Barclay for useful suggestions. Work at Florida State University was supported in part by U.S. NSFGrant No. DMR-1104829. G.B. is supported by Oak Ridge National Laboratory, which is managed by UT-Battelle,LLC. The work was also supported by Grants-in-Aid for Scientific Research C (Nos. 26400324 and 25400391) fromMEXT of Japan, and the Elements Strategy Initiative Center for Magnetic Materials under the outsourcing project ofMEXT. The numerical calculations were supported by the supercomputer center of ISSP of The University of Tokyoand the Florida State University Research Computing Center. C. Teodosiu,
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Statistical Mechanics, Third Edition (Butterworth-Heinemann, Oxford, 2011), Ch. 13, andreferences cited therein. M. Nishino, S. Miyashita, and P. A. Rikvold, in preparation. C. H. Chan, G. Brown, P. A. Rikvold, and S. Miyashita, in preparation. -4 -3 -2 -1 0 1 2 3 4h00.511.522.533.54 t AFM/PM critical lineTricritical pointsAFM/FM coex. linesAFM spinodal linesFM+ spinodal lineFM- spinodal line ( ∆ h) t FM+ stableFM- stable stable
FM- , FM+ metastable
AFM metastable AFM metastable
FM- metastableFM+ metastable (a)
AFM m , m S t a g Stable AFM m
Stag , T N = 4Metatable FM m, T C = 3 (b) FIG. 1: (Color online) Mean-field phase diagrams for a = 7 <
8. (a) In the h, t plane, showing a line of equilibrium criticalpoints terminated by two tricritical points, equilibrium coexistence lines, and sharp spinodals. The FM+ phase is stableeverywhere on the positive- h side of the right-hand coexistence line, and analogously for FM − on the negative- h side. Theinset demonstrates the ( t c − t ) / behavior of the spinodal fields, as given by Eq. (3.7). Here, ∆ h is the difference betweenthe FM and corresponding AFM spinodals. (b) Showing the stable AFM order parameter m Stag and the metastable FM orderparameter m vs t for h = 0. The latter terminates at the spinodal temperature t sFM ≈ . FIG. 2: (Color online) Contour plots in the m A , m B plane of the mean-field free energy for a = 7. The black and light gray(yellow online) curves represent the solutions of ∂F MFA /∂m A = 0 and ∂F MFA /∂m B = 0, respectively. Crossings of the curvescorrespond to extrema and saddle points of the free-energy surfaces. (a) t = 2 . h = 0. Global minima in the secondand fourth quadrant represent the degenerate AFM stable phases. Local minima in the first and third quadrants represent thedegenerate FM metastable phases. (b) t = 2 . h = 0 . t = 3 . h = 0 . m B = m A ) axis. -6 -4 -2 0 2 4 6h0123456 t FM- spinodalFM+ spinodalFM-/FM+ coexistenceAFM spinodals h t FM- stable FM+ stableFM- metastableAFM metastableFM+ metastable (a) m , m S t a g Stable FM m, T C = 6Metastable AFM m Stag , T N = 4 (b) FIG. 3: (Color online) Mean-field phase diagrams for a = 10 >
8. (a) In the h , t plane, showing the equilibrium coexistenceline at h = 0 and the spinodal field curves. The inset demonstrates the ( t c − t ) / behavior of the spinodal fields, as givenby Eq. (3.7). (b) Showing the stable FM order parameter m and the metastable AFM order parameter m Stag vs t at h = 0.The latter terminates at the spinodal temperature t sAFM ≈ . t Critical lineAFM spinodalFM+ spinodalAFM/FM+ coexistenceTricritical point ∆ h t FM+ stableAFM stable
AFM metastableFM+ metastable
FIG. 4: (Color online) (a) MC phase diagram for the full model with a = 4. Except for the absence of metastable FM phasesat h = 0, the phase diagram resembles the mean-field phase diagram for a = 7, shown in Fig. 1. However, in contrast withthe mean-field model, the line of critical points belongs to the two-dimensional Ising universality class. At h = 0 the criticaltemperature is near the exact Ising value, t c ( h = 0) ≈ . t ≈ . h ≈ ± . h is defined as in Fig. 1(a). See further discussion in the text. -4 -3 -2 -1 0 1 2 3 4h00.20.40.60.811.21.41.61.822.22.42.6 t Ising critical lineCoexistence linesAFM/Dis. spinodalsFM+ spinodalFM- spinodalCritical endpointsMF critical pointsHysteresis path -1 -0.5 0 0.5 1 h t AFM stable
FM- stable metastable AFM and FM-
FM+ stable
AFM AFM AFM
Both FM metastable stable metastablemetastablemetastableAFM and FM+ (a) stableDis. FM- FM+AFMDisordered t Ising critical lineCoexistence lineAFM and Disordered spinodalsFM+ spinodalCritical endpointMF critical pointSnapshot location ( ∆ h) t FM+ metastable
FM+ stable
AFM
FM+ stableAFM stable metastable
FM+ metastable (b)
Disordered phase stable t Ising critical lineCoexistence lineAFM and Disordered spinodalsCritical endpointSnapshot locationFM+ stablestable FM+ stableDisordered Disordered
AFM stable
FM+ metastable (c)
FM+ metastable metastable
AFM metastable
FIG. 5: (Color online) (a) MC phase diagram in the h, t plane for the full model with a = 7. In contrast with the mean-fieldmodel, the critical line is in the two-dimensional Ising universality class. Regions of phase stability and metastability aremarked with text. The inset shows the phase diagram including only the stable phases. The diagonal line marks the path forthe hysteresis loops in Fig. 7. (b) Detail of the horn region of the phase diagram. Our estimates of the positions of the criticalendpoint and the mean-field critical points are t = 2 . h = ± . t = 2 . h = ± . h = 0 the critical temperature is near the exact Ising value. The diamond marks the position of the snapshots in Fig. 6. Theinset demonstrates that the width of the horn region, ∆ h , obeys Eq. (3.7). The straight line is a guide to the eye. (c) Furthermagnified detail of the triangular region between the coexistence line, the critical line, and the spinodal line for the disorderedphase. Here, the stable phase is FM+ (confirmed by simulations at the point marked with a triangle) and the disordered phaseis metastable. See further discussion in the text. FIG. 6: (Color online) Snapshots for a = 7 at t = 2 .
25 and h = 0 .
50 in the horn region (marked by a diamond in Fig. 5(b)). (a)and (b) show the equilibrium disordered phase with AFM fluctuations. The system was initiated in an uncorrelated, randomconfiguration and equilibrated for 10 MCSS before the image was recorded. (c) and (d) show the metastable FM+ phase.The system was initiated in the fully ordered FM+ configuration and “equilibrated” for 10 MCSS. In the “straight” images,(a) and (c), up and down spins are colored dark gray (blue online) and light gray (yellow online), respectively. The “masked”images (b) and (d) emphasize AFM domains by coloring up spins on the A sublattice and down spins on the B sublattice darkgray (magenta online) and down on A, up on B light gray (cyan online). -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8h-1-0.8-0.6-0.4-0.200.20.40.60.81
Crit h Coex --> --><--<----> <----><-- (a) -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8h-1-0.8-0.6-0.4-0.200.20.40.60.81
Stag |>h
Crit --> --><--<----> <----><-- (b)
FIG. 7: (Color online) Constant-temperature hysteresis loops for a = 7. The system size was L = 512, except near wherethe hysteresis path crosses the critical line (marked with light gray (green online) vertical, dashed lines) and in the disorderedphase, where L = 1024 was used. The nonzero values of m Stag in the disordered phase are a finite-size effect. (a) At t = 2 . h = +0 .
34. Theloops are symmetric under reversal of h and m . The dark, vertical, short-dashed lines mark the coexistence lines between thedisordered and FM phases. At this temperature they lie very close to the disorder spinodals. (b) At t = 2 .
25. The metastableFM+ phase decays discontinuously into the disordered phase at the FM+ spinodal near h = +0 .
39. The loops are symmetricunder reversal of h and m . At this temperature, the disorder/FM coexistence lines and the disorder spinodals coincide withinour numerical accuracy. See further discussion in the text.
See further discussion in the text. t FM- spinodalAFM and Disorder spinodalsAFM/Disorder critical line h t Dis.metastableAFMmetastable (a) m , m S t a g Exact m
Stag a=0<|m
Stag |> L = 128<|m
Stag |> L = 256t spin
L extrapolated<|m|> L=16 <|m|> L=32<|m|> L=64<|m|> L=128<|m|> L=256<|m|> L=512<|m|> L=1024 (b)