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A mean field criterion for identifying a reentrant phase transition
Journal of Magnetism and Magnetic North-Holland, Amsterdam
Materials
A MEAN FIELD CRITERION K. KORNIK, Departmeni
Received
R.M. ROSHKO
81 (1989) 323-330
323
FOR IDENTIFYING and Gwyn
of Physics, University of Manitoba,
A REENTRANT
PHASE TRANSITION
WILLIAMS
Winnipeg, Canada
20 April 1989
We present numerical simulations of the differential susceptibility in the reentrant regime of a simple Ising mean field model which yield a singular anomaly in the nonlinear components of the susceptibility in the vicinity of the reentrant transition temperature, where asymptotic expansions are invalid. These simulations bear a striking resemblance to an anomaly which has recently been observed in measurements of the nonlinear frequency dependent susceptibility of a reentrant (EFe)Mn alloy, and reproduce many of the systematics of the experimental data, over a comparable range of reduced fields and temperatures. The calculations provide theoretical justification for regarding the experimental anomaly as a manifestation of critical behaviour associated with a reentrant F-SG phase transition.
1. Introduction The magnetic phase diagram of systems characterized by a spatially random arrangement of magnetic ions connected by exchange bonds Jii which assume both positive and negative values according to some probability distribution P( $), and which are thus in conflict with each other, frequently exhibits both ferromagnetic and spin glass ground states. A subset of these systems, notably (Pd, -,,Fe,,)i -Mn, [l], Fe,Cr, --x [2], Au I_XFeX [3] and Eu,Sr,_,S [4], also appear to exhibit a reentrant sequence of phase transitions over a restricted portion of their magnetic phase diagrams: upon cooling at fixed concentration, the system passes from a paramagnet (P) to a ferromagnet (F) and then to a spin glass (SG) phase. Such a sequence is predicted by the SherringtonKirkpatrick (SK) model [5]; however, the reentrant F-SG phase boundary lies below the de Almeida-Thouless (AT) line [6] in temperature, and hence in the region where the replica-syrmnetric solution of the SK model is unstable and is superceded by the apparently “exact” Parisi solution [7]. Instead of a SG ground state, the ferromagnetic phase evolves into a modified (often 0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
called a “mixed”) ferromagnetic phase with irreversibility. (In vector spin models, the “mixed” phase occurs below the Gabay-Toulouse (GT) instability line [8], and is characterized by the coexistence of a spontaneous (ferromagnetic) magnetization and a spin glass ordering of the transverse components of the spins, with a crossover from weak to strong longitudinal magnetic irreversibility at a lower temperature.) However, the experimental manifestations of these theoretical instability lines are difficult to extract from the theory and are frequently a product of “folklore mapping” (such as the association of the AT line with the onset of irreversibility); in particular, broken-symmetry models are currently incapable of reproducing the double-peaked structure observed in the temperature dependence of the reversible susceptibility of nominally reentrant systems measured in fixed applied fields. Consequently, the interpretation of the anomalies observed in susceptibility and inelastic neutron scattering data has remained highly controversial, due primarily to the absence of a rigorous, experimentally accessible, criterion for identifying a genuine thermodynamic F-SG phase transition. One of the distinctive features of the “pure” B.V.
324
K. Kornik et al. / Identifying a reentrant phase transition
P-SG transition, which furnishes convincing evidence for the existence of genuine critical behaviour, is the observation of apparently critical “singularities” in the temperature dependent coefficients of the nonlinear field-dependent components of the magnetization (or susceptibility). Recently, a critical analysis [9] of the nonlinear magnetic response of the reentrant system (Pd,,,,,was performed in the vicinity Fe,.,,, )o.ossMn0.05 of both the P-F critical temperature T, and the proposed reentrant F-SG temperature TSG (the latter, for the first time in a reentrant system). Two anomalies were observed in the temperature dependence of the leading nonlinear term in the frequency dependent susceptibility ( xNL( w) = a(t)H when Hi,, is the internal applied field, corrected for demagnetizing effects): the expected ferromagnetic critical singularity at T,, and another distinct, although substantially weaker, anomaly coincident with the estimated TsG, and highly reminiscent of that associated with the direct P-SG transition. In this paper, we present numerical simulations based on a simple Ising mean field model with a reentrant phase boundary, which replicate the anomalous behaviour observed in (PdFe)Mn, and which suggest a potentially valuablh?criterion for recognizing critically reentrant systems.
2. The model and numerical techniques The model [lo] is a generalization to arbitrary spin quantum number S of an effective field model for spin systems with quenched random spatial disorder, developed by Kaneyoshi [ll] and Southern [12], and is based on the following Ising Hamiltonian H=
- ~JjSiS,-h,~Si, i
with -S
+S.
i (1)
The exchange bonds Jlj are treated as independent random variables, and the effects of disorder are incorporated through various approximation schemes for the moments of the effective field distribution. One such approximation scheme [12]
yields a Gaussian distribution of internal fields, and the following set of coupled equations for the magnetization m = ((S,),), and order parameter 4=
CCsi>‘,>J:
X
epa2j2 da,
[
S2B,2 fiS( .&m + yql”oL
X
eea2/’ da,
+
ha)]
(2)
where & and .? are related to the mean value and variance of the exchange bond distribution. The model does not employ the replica method and consequently avoids the spurious behaviour associated with the solutions of the SK model. Hence, while the coupled equations (2) and the zero field magnetic phase diagram reduce to those of the SK model when S = l/2, the corresponding expressions for the free energy and entropy are different. The thermodynamic properties are well-behaved at low temperatures and, in particular, the third law is not violated. Thus, within the limitations of this effective field approach, the solutions of the coupled equations (2) are stable everywhere and the reentrant phase boundary is well defined. These features make the model uniquely suited to the present calculation. (The absence of instabilities such as those exhibited by the SK solution, is a direct consequence of treating the thermal averages in the Weiss molecular field approximation [12]; the implications of an AT-like instability line for the present effective field calculations will be discussed in a later section.) The coupled equations (2) were differentiated with respect to h to obtain an expression for the susceptibility x = am/ah, and then solved numerically using Newton’s method. The integrals over (Y were performed by replacing the infinite integration limits with finite limits of +lO, dividing the domain of integration into 50 equal intervals, and applying a ten-point Gauss-Legendre quadrature over each interval.
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K. Komik et al. / Identifying a reentrant phase transition
3. Results T>Tc
All of the numerical simulations presented here were performed for spin S = 5/2 and for a single value of - --7/J0 = 0.9 within the reentrant regime (0.8 I J/J, I 1.0). The reentrant F-SG transition temperature was determined numerically by establishing the temperature at which the spontaneous ferromagnetic magnetization vanished, and this procedure yielded a value of T,,/T, = 0.269285, where T, = (S(S + 1)/3)&/k, is the ferromagnetic critical temperature. The temperature dependence of the model susceptibility generated in a sequence of fixed reduced fields h = h,/k,T, exhibits features which are superficially identical to those obtained from the spin-l/2 SK model [13] and, as pointed out previously, bears a remarkable qualitative resemblance to the experimental frequency-dependent susceptibility of reentrant systems measured in finite static biasing fields (in zero applied field, the measured susceptibility possesses an essentially temperature-independent plateau between T, and TSG related to domain wall dynamics, which the model is incapable of reproducing): the susceptibility consists of two peaks which are suppressed in amplitude and driven apart in temperature by increasing the applied field h, with the reentrant peak amplitude > the ferromagnetic critical peak amplitude in a fixed field h. The zero field model susceptibility diverges in the vicinity of both transitions as It *-1 1-y with y = 1, where t* = T/T, for the P-F transition and t* = T/T,, for the reentrant F-SG transition. As mentioned earlier the “pure” P-SG transition is characterized by an order parameter which couples to the nonlinear components of the susceptibility, and expansions [14] of the coupled equations (2) in the “pure” spin glass regime (0 I &/.? -C1.0) for small values of the argument of the Brillouin function (that is, in the neighbourhood of the P-SG phase boundary), confirm that critical singularities are indeed present only in the nonlinear terms and, furthermore, that they are symmetric with respect to TsG, as observed in the PdMn system [15] (there is some evidence for symmetric critical behaviour in AgMn as well [16]). The focus of the present investigation was to
J/&=0.9
II OO
1.0
h2(1c?0) Fig. 1. Numerical susceptibility isotherms in the paramagnetic phase above T, for the following reduced temperatures (in order of decreasing vertical intercept): t = T/T, = 1.0004, 1.0010, 1.0014, 1.0020, 1.0030, 1.0050. The crossing of the isotherms is a consequence of the peak in the temperature dependence of the susceptibility above T,.
identify analogous critical anomalies in the leading field dependence of the model susceptibility in the reentrant regime, particularly in the neighbourhood of the reentrant phase boundary, and to compare these with the experimental anomalies observed in (PdFe)Mn. In order toreplicate the experimental investigation, the model susceptibility was generated over a comparable domain of reduced fields h and reduced temperatures C.Although the ferromagnetic critical analysis is relatively standard, some of the specific predictions of the model regarding the amplitude and range of validity of the various critical terms are relevant to subsequent comparisons with the experimental (PdFe)Mn data and are not available from a gene&scaling formalism. Figs. 1 and 2 show some typical susceptibility isotherms in the neighbourhood of the ferromagnetic critical temperature T,; in each temperature regime (T >< T,), the numerical susceptibility data has been plotted as a function of its dominant (leading) field dependence in that regime as pre-
K. Kornik et al. / Identifying
326
I
3.2
f
T < T,
h (lo-5) Fig. 2. Numerical susceptibility isotherms in the ferromagnetic phase below T, for the following reduced temperatures (top to t = 0.9995, 0.9992, 0.9988, 0.9984, 0.9919, 0.9965, bottom): 0.9950.
dieted by expansion of the coupled equations, that is, quadratic ( h2) above T, and linear (h) below T, (the latter is a consequence of the broken symmetry induced by a nonvanishing spontaneous magnetization). As t + l*, both sets of isotherms develop increasingly pronounced curvature and the range of applied fields over which the leading term dominates the magnetic response progressively diminishes, while the zero field slope of the isotherms, which measures the temperature dependence of the leading term, increases dramatically. The form of this temperature dependence was established by constructing double logarithmic plots of the zero field slope above and below T, as a function of the reduced temperature 1t - 11, as shown in fig. 3; this analysis yielded a divergent power law of the form 1t - 11 -y’ in both regimes, with y’ = 4 for T > T, (this is the expected mean field exponent y’ = 3-r + 2j3 = 4) and y’ = 5/2 for T-c T,. The weak deviations from a strict power law dependence which gradually appear at higher reduced temperatures reflect the presence of nonsingular (noncritical) components in the susceptibility, which become increasingly significant far - from T,, particularly as J/J, + 1 [17].
a reentrant phase transition
In contrast to other mean field models which exhibit sequential transitions, usually in the form of instability boundaries, the present model yields detailed, quantitative predictions concerning the field and temperature dependence of the magnetic response at the reentrant phase boundary, and thus permits specific comparisons with experimental measurements. Expansions of the coupled equations are, however, no longer valid in this regime since the order parameter q is large, and consequently a numerical determination of the leading field dependence of the susceptibility above and below TSG was necessary. A detailed examination of the very low field behaviour of the model susceptibility established that the initial response was linear in h above TSG and quadratic in h below TsG, and figs. 4 and 5 show typical numerical susceptibility isotherms in each of these temperature regimes plotted as a function of the appropriate leading field dependence. These iso-
T c Tc
i'
Fig. 3. Double logarithmic plots of the zero field slope of the P-F susceptibility isotherms above and below T, as a function of reduced temperature 1t - 1I.
327
K. Komik et al. / Ia’entifVing a reentrant phase transition
I -0
1.0
h (IO-~)
Fig. 4. Numerical susceptibility isotherms in the ferromagnetic phase above Ts, for the following reduced temperatures (top t* = T/T,, = 1.0015, 1.0027, 1.0041, 1.0060, to bottom): 1.0101. 1.0212.
therms clearly possess many features in common with those generated in the vicinity of the P-F phase boundary and also bear a striking similarity to their experimental counterparts in the (PdFe) Mn system, near the onset of the apparentreentrant phase. Figs. 4 and 5 show that the reentrant phase boundary is indeed characterized by anomalous behaviour in the initial slope of the model isotherms, similar to that associated with the ferromagnetic transition. Below TsG, the behaviour is typical of a “pure” P-SG transition in the sense that the anomaly occurs in the nonlinear susceptibility (actually, in the quadratic component) while, above TsG, the presence of a spontaneous magnetization leads to a singular behaviour which manifests itself initially in the linear term. (The experimental (PdFe)Mn data appears to be compatible with an h* analysis above, as well as below, Tso, Possibly because the ferromagnetic domain structure between T,, and T, conceals the
J/J,=
0.9
1 T
J/J,= 0.9
I
OO
1.0
Fig. 5. Numerical susceptibility isotherms in the spin glass phase below T,, for the following reduced temperatures (in order of decreasing vertical intercept): t* = 0.9971, 0.9952, 0.9934, 0.9915, 0.9878, 0.9841,0.9804, 0.9581.
Fig. 6. Double logarithmic plots of the zero field slope of the F-SG reentrant susceptibility isotherms above and below TsG as a function of reduced temperature 1t * - 1 I.
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K. Kornik et al. / Identifying a reentrant phase transition
existence of the spontaneous magnetization to a large extent, with the result that the linear term is suppressed. Of course, the model also yields a singular h2 term above TsG, but this is not the dominant term in this regime.) As before, the values of the critical exponent y’ were determined from the double logarithmic plots shown in fig. 6; the straight line below TSG yields an exponent y’ = 4 in the reentrant phase, comparable to the experimental (PdFe)Mn value of y’ = 3.6 + 0.6, while above Tz y’ = 5/2 (the temperature dependence of the experimental h2 coefficient in this regime was too weak to establish a critical exponent [9]). Comparison of fig. 6 with fig. 3 shows that the h2 coefficient below TSG is several orders of magnitude larger than its ferromagnetic counterpart above T,, in agreement with the experimental data analysis.
4. Discussion The numerical simulations presented here suggest a valuable criterion for experimentally identifying a sequence of genuine phase transitions from P to F to SG, and lend support to the contention that the anomalous behaviour observed in the (PdFe)Mn system is indeed a manifestation of critical fluctuations. The calculations were based on a mean field model with thermodynamically well-behaved solutions at low temperatures, and, a well-defined reentrant phase consequently, boundary. As pointed out previously, the model is characterized by a particular hierarchy of mean field approximations which essentially amounts to neglecting the Onsager reaction term in the local mean field equations; hence, with the exception of the generalization to arbitrary spin S, the coupled equations (2) are identical to those of the SK model, but avoid the spurious behaviour associated with replica symmetry breaking. Moreover, other considerations suggest that these instabilities may not play a significant role here and, furthermore, that the Parisi solution may not be the relevant one in the present context: (1) There is some evidence to suggest that, in spite of the AT instabilities, numerical calculations of the differential susceptibility and magneti-
zation based on the replica symmetric SK model, or the present effective field version, may well possess physical relevance as the theoretical equivalent of dynamic probes such as the frequency dependent (ac) susceptibility and the zero field cooled (ZFC) magnetization. The peaks observed in frequency dependent measurements of the susceptibility in the “pure” spin glass regime of PdMn below TSG display systematics as a functionof applied field and temperature which are replicated remarkably well by the model calculations [18], while the symmetry of the critical behaviour observed in the nonlinear components of the experimental SG susceptibility with respect to TSG in PdMn [15] is also a characteristic of these mean fia models. Further evidence for such critical symmetry is provided by a recent dynamical study at ultralow frequencies of the nonlinear susceptibility of a very dilute AgMn spin glass [16]: in a static field of 90 G anGust below TsG, the system appears to approach quasistatic thermodynamic equilibrium for applied frequencies less than about 10m3 Hz, with an effective nonlinear critical exponent y which is close to its value above T,, and hence consistent with the predictions of the effective field model, but not with the Parisi solution which does not yield singular behaviour below TSG. By contrast, neither the ac susceptibility nor the ZFC magnetization in the “pure” SG regime ever seem to exhibit the temperature independent plateau predicted by the Parisi solution, even in the ultralow frequency limit [16]; this feature appears to be uniquely characteristic of the field cooled magnetization, which measures the total (equilibrium?) response. Recent numerical calculations indicate that these correlations extend into the reentrant regime of the magnetic phase diagram as well: measurements of the ZFC magnetization of very dilute -PdFe and PdMn alloys [19,20] with concentrations intermediate between the “pure” ferromagnetic and spin glass phases have been fitted successfully with numerical magnetization curves generated in the vicinity of the tricritical point of the effective field- model, with reentrant values for the parameter J/JO. (2) Although the zero field AT instability line in the ferromagnetic regime of the magnetic phase
K. Komik et al. / Identifying a reentrant phase transition
diagram lies above the reentrant phase boundary in temperature, the curvature of the instability surface, plotted as a function of h and .?,,/.? as independent variables, is strongly concave up, particularly in the vicinity of the tricritical point (z/J= 1) where the slope aT,,/ah approaches infinity. This means that the instability temperature in the reentrant regime is rapidly depressed by the application of a field h so that, while the zero field boundary itself may not be visible, some of the manifestations of the transition in finite field, both above and below the reentrant transition temperature, will penetrate the surface and survive in the regime where the solutions are considered to be stable. (An analogous situation occurs in the pure SG regime, where the model susceptibility peaks in fixed field lie above the AT instability line, but below the zero field critical temperature TsG, and furnish a valuable anomaly for comparing the systematics of the numerical and experimental data [18].) Thus, while the actual singularity in the nonlinear susceptibility may be obscured close to the reentrant temperature TsG, evidence for critical behaviour in the nonlinear’ components will nevertheless still be identifiable farther from TsG. (3) The effective field model (and its replicasymmetric SK equivalent) is currently the only model which is capable of simulating the anomaly in the nonlinear susceptibility observed in the vicinity of the reentrant transition in the (PdFe)Mn system; not only are the model calculations able to imitate the systematics of the field and temperature dependence of xNL, such as the progressively diminishing range of applied fields over which the leading term dominates the nonlinear response as T + TsG, as well as the relative magnitudes of the h* coefficients in the spin glass (T < T,.) and paramagnetic (T > T,) phases, but, perhaps more importantly, they allow these comparisons to be pursued below TSG. This is particularly significant from an experimental point of view since the analysis of the upper side of the transition is normally complicated considerably by interference from domain wall dynamics, and hence only the low temperature side is available for critical analysis in practice. This limitation is clearly illustrated by the (PdFe)Mn investigation: -
329
while the anomaly in the nonlinear susceptibility is distinct, it is asymmetric in the sense that the variation in the temperature dependent coefficient of the quadratic term is considerably weaker above TSG, so that power law behaviour and a corresponding critical exponent can be identified only below TSG. It should be emphasized that, while instability lines in both Ising and Heisenberg models are also accompanied by singularities in the “staggered” susceptibility xso, the implications of these instabilities for the nonlinear susceptibility in finite fields h have thus far remained unexplored. In summary, we have presented the first clear evidence of model calculations which predict an anomaly in the nonlinear susceptibility in the vicinity of the reentrant F-SG transition, with systematics which are highly reminiscent of those observed experimentally in the canonical reentrant (PdFe)Mn system. This anomaly may provide a valuable criterion for identifying critical reentrant behaviour.
Acknowledgement This work has been supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
References 111B.H. Verbeek, G.J. Nieuwenhuys,
H. Stocker and J.A. Mydosh, Phys. Rev. Lett. 40 (1978) 586. 121S.M. Shapiro, C.R. F&her, A.C. Palumbo and R.D. Parks, Phys. Rev. B 24 (1981) 6661. 131 B.R. Coles, B.V.B. Sarkissian and R.H. Taylor, Phil. Mag. 37 (1978) 489. 141 H. Maletta and W. Felsch, Z. Phys. B 37 (1980) 55. PI D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1972. WI J.R.L. de Almeida and D.J. Thouless, J. Phys. A 11 (1978) 983. 171 G. Parisi and G. Toulouse, J. de Phys. 41 (1980) L361. 181 M. Gabay and G. Toulouse, Phys. Rev. Lett. 47 (1981) 201. [91 H. Kunkel and G. Williams, J. Magn. Magn. Mat. 75 (1988) 98. WI R.M. Roshko and G. Williams, J. Phys. F 14 (1984) 703. 1111 T. Kaneyoshi, J. Phys. C 8 (1975) 3415.
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K. Kornik et al. / Identifying a reentrant phase transition
[12] B.W. Southern, J. Phys. C 9 (1976) 4011. [13] G.J. Nieuwenhuys, H. Stocker, B.H. Verbeek and J.A. Mydosh, Solid State Commun. 27 (1978) 197. [14] R.M. Roshko and G. Williams, J. Magn. Magn. Mat. 50 (1985) 311. [15] E. Zastre, R.M. Roshko and G. Williams, Phys. Rev. B 32 (1985) 7597. [16] L.P. L&y, Phys. Rev. B 38 (1988) 4963.
[17] I. Yeung, R.M. Roshko and G. Williams, J. Magn. Magn. Mat. 68 (1987) 39. [18] P. Gash, R.M. Roshko and G. Williams, J. Phys. F 14 (1984) 1501. [19] K. Kornik, H. Kunkel and R.M. Roshko, J. Appl. Phys. 64 (1988) 5619. [20] K. Komik, H. Kunkel and R.M. Roshko, J. de Phys. Colloq. 49 (1988) C8-1139.
Materials
A MEAN FIELD CRITERION K. KORNIK, Departmeni
Received
R.M. ROSHKO
81 (1989) 323-330
323
FOR IDENTIFYING and Gwyn
of Physics, University of Manitoba,
A REENTRANT
PHASE TRANSITION
WILLIAMS
Winnipeg, Canada
20 April 1989
We present numerical simulations of the differential susceptibility in the reentrant regime of a simple Ising mean field model which yield a singular anomaly in the nonlinear components of the susceptibility in the vicinity of the reentrant transition temperature, where asymptotic expansions are invalid. These simulations bear a striking resemblance to an anomaly which has recently been observed in measurements of the nonlinear frequency dependent susceptibility of a reentrant (EFe)Mn alloy, and reproduce many of the systematics of the experimental data, over a comparable range of reduced fields and temperatures. The calculations provide theoretical justification for regarding the experimental anomaly as a manifestation of critical behaviour associated with a reentrant F-SG phase transition.
1. Introduction The magnetic phase diagram of systems characterized by a spatially random arrangement of magnetic ions connected by exchange bonds Jii which assume both positive and negative values according to some probability distribution P( $), and which are thus in conflict with each other, frequently exhibits both ferromagnetic and spin glass ground states. A subset of these systems, notably (Pd, -,,Fe,,)i -Mn, [l], Fe,Cr, --x [2], Au I_XFeX [3] and Eu,Sr,_,S [4], also appear to exhibit a reentrant sequence of phase transitions over a restricted portion of their magnetic phase diagrams: upon cooling at fixed concentration, the system passes from a paramagnet (P) to a ferromagnet (F) and then to a spin glass (SG) phase. Such a sequence is predicted by the SherringtonKirkpatrick (SK) model [5]; however, the reentrant F-SG phase boundary lies below the de Almeida-Thouless (AT) line [6] in temperature, and hence in the region where the replica-syrmnetric solution of the SK model is unstable and is superceded by the apparently “exact” Parisi solution [7]. Instead of a SG ground state, the ferromagnetic phase evolves into a modified (often 0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
called a “mixed”) ferromagnetic phase with irreversibility. (In vector spin models, the “mixed” phase occurs below the Gabay-Toulouse (GT) instability line [8], and is characterized by the coexistence of a spontaneous (ferromagnetic) magnetization and a spin glass ordering of the transverse components of the spins, with a crossover from weak to strong longitudinal magnetic irreversibility at a lower temperature.) However, the experimental manifestations of these theoretical instability lines are difficult to extract from the theory and are frequently a product of “folklore mapping” (such as the association of the AT line with the onset of irreversibility); in particular, broken-symmetry models are currently incapable of reproducing the double-peaked structure observed in the temperature dependence of the reversible susceptibility of nominally reentrant systems measured in fixed applied fields. Consequently, the interpretation of the anomalies observed in susceptibility and inelastic neutron scattering data has remained highly controversial, due primarily to the absence of a rigorous, experimentally accessible, criterion for identifying a genuine thermodynamic F-SG phase transition. One of the distinctive features of the “pure” B.V.
324
K. Kornik et al. / Identifying a reentrant phase transition
P-SG transition, which furnishes convincing evidence for the existence of genuine critical behaviour, is the observation of apparently critical “singularities” in the temperature dependent coefficients of the nonlinear field-dependent components of the magnetization (or susceptibility). Recently, a critical analysis [9] of the nonlinear magnetic response of the reentrant system (Pd,,,,,was performed in the vicinity Fe,.,,, )o.ossMn0.05 of both the P-F critical temperature T, and the proposed reentrant F-SG temperature TSG (the latter, for the first time in a reentrant system). Two anomalies were observed in the temperature dependence of the leading nonlinear term in the frequency dependent susceptibility ( xNL( w) = a(t)H when Hi,, is the internal applied field, corrected for demagnetizing effects): the expected ferromagnetic critical singularity at T,, and another distinct, although substantially weaker, anomaly coincident with the estimated TsG, and highly reminiscent of that associated with the direct P-SG transition. In this paper, we present numerical simulations based on a simple Ising mean field model with a reentrant phase boundary, which replicate the anomalous behaviour observed in (PdFe)Mn, and which suggest a potentially valuablh?criterion for recognizing critically reentrant systems.
2. The model and numerical techniques The model [lo] is a generalization to arbitrary spin quantum number S of an effective field model for spin systems with quenched random spatial disorder, developed by Kaneyoshi [ll] and Southern [12], and is based on the following Ising Hamiltonian H=
- ~JjSiS,-h,~Si, i
with -S
+S.
i (1)
The exchange bonds Jlj are treated as independent random variables, and the effects of disorder are incorporated through various approximation schemes for the moments of the effective field distribution. One such approximation scheme [12]
yields a Gaussian distribution of internal fields, and the following set of coupled equations for the magnetization m = ((S,),), and order parameter 4=
CCsi>‘,>J:
X
epa2j2 da,
[
S2B,2 fiS( .&m + yql”oL
X
eea2/’ da,
+
ha)]
(2)
where & and .? are related to the mean value and variance of the exchange bond distribution. The model does not employ the replica method and consequently avoids the spurious behaviour associated with the solutions of the SK model. Hence, while the coupled equations (2) and the zero field magnetic phase diagram reduce to those of the SK model when S = l/2, the corresponding expressions for the free energy and entropy are different. The thermodynamic properties are well-behaved at low temperatures and, in particular, the third law is not violated. Thus, within the limitations of this effective field approach, the solutions of the coupled equations (2) are stable everywhere and the reentrant phase boundary is well defined. These features make the model uniquely suited to the present calculation. (The absence of instabilities such as those exhibited by the SK solution, is a direct consequence of treating the thermal averages in the Weiss molecular field approximation [12]; the implications of an AT-like instability line for the present effective field calculations will be discussed in a later section.) The coupled equations (2) were differentiated with respect to h to obtain an expression for the susceptibility x = am/ah, and then solved numerically using Newton’s method. The integrals over (Y were performed by replacing the infinite integration limits with finite limits of +lO, dividing the domain of integration into 50 equal intervals, and applying a ten-point Gauss-Legendre quadrature over each interval.
325
K. Komik et al. / Identifying a reentrant phase transition
3. Results T>Tc
All of the numerical simulations presented here were performed for spin S = 5/2 and for a single value of - --7/J0 = 0.9 within the reentrant regime (0.8 I J/J, I 1.0). The reentrant F-SG transition temperature was determined numerically by establishing the temperature at which the spontaneous ferromagnetic magnetization vanished, and this procedure yielded a value of T,,/T, = 0.269285, where T, = (S(S + 1)/3)&/k, is the ferromagnetic critical temperature. The temperature dependence of the model susceptibility generated in a sequence of fixed reduced fields h = h,/k,T, exhibits features which are superficially identical to those obtained from the spin-l/2 SK model [13] and, as pointed out previously, bears a remarkable qualitative resemblance to the experimental frequency-dependent susceptibility of reentrant systems measured in finite static biasing fields (in zero applied field, the measured susceptibility possesses an essentially temperature-independent plateau between T, and TSG related to domain wall dynamics, which the model is incapable of reproducing): the susceptibility consists of two peaks which are suppressed in amplitude and driven apart in temperature by increasing the applied field h, with the reentrant peak amplitude > the ferromagnetic critical peak amplitude in a fixed field h. The zero field model susceptibility diverges in the vicinity of both transitions as It *-1 1-y with y = 1, where t* = T/T, for the P-F transition and t* = T/T,, for the reentrant F-SG transition. As mentioned earlier the “pure” P-SG transition is characterized by an order parameter which couples to the nonlinear components of the susceptibility, and expansions [14] of the coupled equations (2) in the “pure” spin glass regime (0 I &/.? -C1.0) for small values of the argument of the Brillouin function (that is, in the neighbourhood of the P-SG phase boundary), confirm that critical singularities are indeed present only in the nonlinear terms and, furthermore, that they are symmetric with respect to TsG, as observed in the PdMn system [15] (there is some evidence for symmetric critical behaviour in AgMn as well [16]). The focus of the present investigation was to
J/&=0.9
II OO
1.0
h2(1c?0) Fig. 1. Numerical susceptibility isotherms in the paramagnetic phase above T, for the following reduced temperatures (in order of decreasing vertical intercept): t = T/T, = 1.0004, 1.0010, 1.0014, 1.0020, 1.0030, 1.0050. The crossing of the isotherms is a consequence of the peak in the temperature dependence of the susceptibility above T,.
identify analogous critical anomalies in the leading field dependence of the model susceptibility in the reentrant regime, particularly in the neighbourhood of the reentrant phase boundary, and to compare these with the experimental anomalies observed in (PdFe)Mn. In order toreplicate the experimental investigation, the model susceptibility was generated over a comparable domain of reduced fields h and reduced temperatures C.Although the ferromagnetic critical analysis is relatively standard, some of the specific predictions of the model regarding the amplitude and range of validity of the various critical terms are relevant to subsequent comparisons with the experimental (PdFe)Mn data and are not available from a gene&scaling formalism. Figs. 1 and 2 show some typical susceptibility isotherms in the neighbourhood of the ferromagnetic critical temperature T,; in each temperature regime (T >< T,), the numerical susceptibility data has been plotted as a function of its dominant (leading) field dependence in that regime as pre-
K. Kornik et al. / Identifying
326
I
3.2
f
T < T,
h (lo-5) Fig. 2. Numerical susceptibility isotherms in the ferromagnetic phase below T, for the following reduced temperatures (top to t = 0.9995, 0.9992, 0.9988, 0.9984, 0.9919, 0.9965, bottom): 0.9950.
dieted by expansion of the coupled equations, that is, quadratic ( h2) above T, and linear (h) below T, (the latter is a consequence of the broken symmetry induced by a nonvanishing spontaneous magnetization). As t + l*, both sets of isotherms develop increasingly pronounced curvature and the range of applied fields over which the leading term dominates the magnetic response progressively diminishes, while the zero field slope of the isotherms, which measures the temperature dependence of the leading term, increases dramatically. The form of this temperature dependence was established by constructing double logarithmic plots of the zero field slope above and below T, as a function of the reduced temperature 1t - 11, as shown in fig. 3; this analysis yielded a divergent power law of the form 1t - 11 -y’ in both regimes, with y’ = 4 for T > T, (this is the expected mean field exponent y’ = 3-r + 2j3 = 4) and y’ = 5/2 for T-c T,. The weak deviations from a strict power law dependence which gradually appear at higher reduced temperatures reflect the presence of nonsingular (noncritical) components in the susceptibility, which become increasingly significant far - from T,, particularly as J/J, + 1 [17].
a reentrant phase transition
In contrast to other mean field models which exhibit sequential transitions, usually in the form of instability boundaries, the present model yields detailed, quantitative predictions concerning the field and temperature dependence of the magnetic response at the reentrant phase boundary, and thus permits specific comparisons with experimental measurements. Expansions of the coupled equations are, however, no longer valid in this regime since the order parameter q is large, and consequently a numerical determination of the leading field dependence of the susceptibility above and below TSG was necessary. A detailed examination of the very low field behaviour of the model susceptibility established that the initial response was linear in h above TSG and quadratic in h below TsG, and figs. 4 and 5 show typical numerical susceptibility isotherms in each of these temperature regimes plotted as a function of the appropriate leading field dependence. These iso-
T c Tc
i'
Fig. 3. Double logarithmic plots of the zero field slope of the P-F susceptibility isotherms above and below T, as a function of reduced temperature 1t - 1I.
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K. Komik et al. / Ia’entifVing a reentrant phase transition
I -0
1.0
h (IO-~)
Fig. 4. Numerical susceptibility isotherms in the ferromagnetic phase above Ts, for the following reduced temperatures (top t* = T/T,, = 1.0015, 1.0027, 1.0041, 1.0060, to bottom): 1.0101. 1.0212.
therms clearly possess many features in common with those generated in the vicinity of the P-F phase boundary and also bear a striking similarity to their experimental counterparts in the (PdFe) Mn system, near the onset of the apparentreentrant phase. Figs. 4 and 5 show that the reentrant phase boundary is indeed characterized by anomalous behaviour in the initial slope of the model isotherms, similar to that associated with the ferromagnetic transition. Below TsG, the behaviour is typical of a “pure” P-SG transition in the sense that the anomaly occurs in the nonlinear susceptibility (actually, in the quadratic component) while, above TsG, the presence of a spontaneous magnetization leads to a singular behaviour which manifests itself initially in the linear term. (The experimental (PdFe)Mn data appears to be compatible with an h* analysis above, as well as below, Tso, Possibly because the ferromagnetic domain structure between T,, and T, conceals the
J/J,=
0.9
1 T
J/J,= 0.9
I
OO
1.0
Fig. 5. Numerical susceptibility isotherms in the spin glass phase below T,, for the following reduced temperatures (in order of decreasing vertical intercept): t* = 0.9971, 0.9952, 0.9934, 0.9915, 0.9878, 0.9841,0.9804, 0.9581.
Fig. 6. Double logarithmic plots of the zero field slope of the F-SG reentrant susceptibility isotherms above and below TsG as a function of reduced temperature 1t * - 1 I.
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K. Kornik et al. / Identifying a reentrant phase transition
existence of the spontaneous magnetization to a large extent, with the result that the linear term is suppressed. Of course, the model also yields a singular h2 term above TsG, but this is not the dominant term in this regime.) As before, the values of the critical exponent y’ were determined from the double logarithmic plots shown in fig. 6; the straight line below TSG yields an exponent y’ = 4 in the reentrant phase, comparable to the experimental (PdFe)Mn value of y’ = 3.6 + 0.6, while above Tz y’ = 5/2 (the temperature dependence of the experimental h2 coefficient in this regime was too weak to establish a critical exponent [9]). Comparison of fig. 6 with fig. 3 shows that the h2 coefficient below TSG is several orders of magnitude larger than its ferromagnetic counterpart above T,, in agreement with the experimental data analysis.
4. Discussion The numerical simulations presented here suggest a valuable criterion for experimentally identifying a sequence of genuine phase transitions from P to F to SG, and lend support to the contention that the anomalous behaviour observed in the (PdFe)Mn system is indeed a manifestation of critical fluctuations. The calculations were based on a mean field model with thermodynamically well-behaved solutions at low temperatures, and, a well-defined reentrant phase consequently, boundary. As pointed out previously, the model is characterized by a particular hierarchy of mean field approximations which essentially amounts to neglecting the Onsager reaction term in the local mean field equations; hence, with the exception of the generalization to arbitrary spin S, the coupled equations (2) are identical to those of the SK model, but avoid the spurious behaviour associated with replica symmetry breaking. Moreover, other considerations suggest that these instabilities may not play a significant role here and, furthermore, that the Parisi solution may not be the relevant one in the present context: (1) There is some evidence to suggest that, in spite of the AT instabilities, numerical calculations of the differential susceptibility and magneti-
zation based on the replica symmetric SK model, or the present effective field version, may well possess physical relevance as the theoretical equivalent of dynamic probes such as the frequency dependent (ac) susceptibility and the zero field cooled (ZFC) magnetization. The peaks observed in frequency dependent measurements of the susceptibility in the “pure” spin glass regime of PdMn below TSG display systematics as a functionof applied field and temperature which are replicated remarkably well by the model calculations [18], while the symmetry of the critical behaviour observed in the nonlinear components of the experimental SG susceptibility with respect to TSG in PdMn [15] is also a characteristic of these mean fia models. Further evidence for such critical symmetry is provided by a recent dynamical study at ultralow frequencies of the nonlinear susceptibility of a very dilute AgMn spin glass [16]: in a static field of 90 G anGust below TsG, the system appears to approach quasistatic thermodynamic equilibrium for applied frequencies less than about 10m3 Hz, with an effective nonlinear critical exponent y which is close to its value above T,, and hence consistent with the predictions of the effective field model, but not with the Parisi solution which does not yield singular behaviour below TSG. By contrast, neither the ac susceptibility nor the ZFC magnetization in the “pure” SG regime ever seem to exhibit the temperature independent plateau predicted by the Parisi solution, even in the ultralow frequency limit [16]; this feature appears to be uniquely characteristic of the field cooled magnetization, which measures the total (equilibrium?) response. Recent numerical calculations indicate that these correlations extend into the reentrant regime of the magnetic phase diagram as well: measurements of the ZFC magnetization of very dilute -PdFe and PdMn alloys [19,20] with concentrations intermediate between the “pure” ferromagnetic and spin glass phases have been fitted successfully with numerical magnetization curves generated in the vicinity of the tricritical point of the effective field- model, with reentrant values for the parameter J/JO. (2) Although the zero field AT instability line in the ferromagnetic regime of the magnetic phase
K. Komik et al. / Identifying a reentrant phase transition
diagram lies above the reentrant phase boundary in temperature, the curvature of the instability surface, plotted as a function of h and .?,,/.? as independent variables, is strongly concave up, particularly in the vicinity of the tricritical point (z/J= 1) where the slope aT,,/ah approaches infinity. This means that the instability temperature in the reentrant regime is rapidly depressed by the application of a field h so that, while the zero field boundary itself may not be visible, some of the manifestations of the transition in finite field, both above and below the reentrant transition temperature, will penetrate the surface and survive in the regime where the solutions are considered to be stable. (An analogous situation occurs in the pure SG regime, where the model susceptibility peaks in fixed field lie above the AT instability line, but below the zero field critical temperature TsG, and furnish a valuable anomaly for comparing the systematics of the numerical and experimental data [18].) Thus, while the actual singularity in the nonlinear susceptibility may be obscured close to the reentrant temperature TsG, evidence for critical behaviour in the nonlinear’ components will nevertheless still be identifiable farther from TsG. (3) The effective field model (and its replicasymmetric SK equivalent) is currently the only model which is capable of simulating the anomaly in the nonlinear susceptibility observed in the vicinity of the reentrant transition in the (PdFe)Mn system; not only are the model calculations able to imitate the systematics of the field and temperature dependence of xNL, such as the progressively diminishing range of applied fields over which the leading term dominates the nonlinear response as T + TsG, as well as the relative magnitudes of the h* coefficients in the spin glass (T < T,.) and paramagnetic (T > T,) phases, but, perhaps more importantly, they allow these comparisons to be pursued below TSG. This is particularly significant from an experimental point of view since the analysis of the upper side of the transition is normally complicated considerably by interference from domain wall dynamics, and hence only the low temperature side is available for critical analysis in practice. This limitation is clearly illustrated by the (PdFe)Mn investigation: -
329
while the anomaly in the nonlinear susceptibility is distinct, it is asymmetric in the sense that the variation in the temperature dependent coefficient of the quadratic term is considerably weaker above TSG, so that power law behaviour and a corresponding critical exponent can be identified only below TSG. It should be emphasized that, while instability lines in both Ising and Heisenberg models are also accompanied by singularities in the “staggered” susceptibility xso, the implications of these instabilities for the nonlinear susceptibility in finite fields h have thus far remained unexplored. In summary, we have presented the first clear evidence of model calculations which predict an anomaly in the nonlinear susceptibility in the vicinity of the reentrant F-SG transition, with systematics which are highly reminiscent of those observed experimentally in the canonical reentrant (PdFe)Mn system. This anomaly may provide a valuable criterion for identifying critical reentrant behaviour.
Acknowledgement This work has been supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.
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