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Critical behavior induced by quenched disorder
Physica A 194 (1993) North-Holland
72-76
Critical behavior induced by quenched disorder A. Nihat
Berker
Department of Physics, Massachusetts Institute of Technology,
Cambridge,
MA 02139. USA
Domain arguments and renormalization-group calculations indicate that all temperaturedriven symmetry-breaking first-order phase transitions are converted to second order by quenched bond randomness. This occurs for infinitesimal randomness in d s 2 or d s 4 respectively for discrete or continuous (n = 1 or n 22 component) microscopic degrees of freedom. Even strongly first-order transitions undergo this conversion to second order! Above these dimensions this conversion still occurs but requires a threshold bond randomness, presumably with an intervening new tricritical point. For example, q-state Potts transitions can be made second order for any q in any d, via bond randomness. Non-symmetry-breaking “temperature-driven” first-order transitions are eliminated under the above conditions. These quenched-fluctuation-induced second-order phase transitions suggest the possibility of new universality classes of criticality and tricriticality.
It was recently realized [l-4] that quenched bond randomness has a very dramatic effect on first-order phase transitions, transforming them into secondorder phase transitions and, in many cases (such as q-state Potts models in three dimensions), introducing the possibility of new universality classes of criticality and tricriticality. The argument leading to this very genera1 phenomenon is such a simple adaptation of the Imry-Ma argument [5] for random fields that it is surprising that it has been overlooked throughout the many years in which the Imry-Ma argument has been known. We first consider first-order phase transitions that involve ing. They are represented in fig. 1. In the absence of bond column),
the
high-temperature
phase
(top
figure)
has
symmetry breakrandomness (left
a symmetry
that
is
spontaneously broken at low temperatures. Thus, at low temperatures, the system is in one of p symmetry-broken phases that map onto each other under the broken symmetry. Macroscopic domains of these p different phases can coexist in the system, as shown at the bottom of the left column in fig. 1. If the phase transition is first order, at the phase transition, the p symmetry-broken phases stretching from low temperature coexist with the symmetric phase stretching from high temperature, as shown in the middle of the left column in fig. 1. Presenting the most genera1 argument, we define as “bond” any coupling that is invariant under the symmetry that is broken in the low-temperature 037%4371/93/506.00
0
1993 - Elsevier
Science
Publishers
B.V. All rights
reserved
A.N. Berker I Critical behavior from quenched disorder No
random bonds
73
With random bonds
Fig. 1. Macroscopic domains above (symmetric - S), at (P), and below (symmetry-broken - B) the phase transition temperature in a system without (at left) or with (at right) bond randomness. The unmarked domains represent macroscopic domains of the symmetric phase. The domains marked by letters represent macroscopic domains of the symmetry-broken phases. Each letter corresponds to the same symmetry being broken differently. The bond randomness necessary to effect the change pictured in these figures is infinitesimal for d G 2 discrete systems and d s 4 continuum systems; above these dimensions, a finite threshold of bond randomness is needed.
phases. (On the other hand, a “field” is any coupling that is not invariant under this symmetry.) We call temperature change any change of a dimensionless (i.e., energy divided by k,T) bond strength or of a combination of dimensionless bond strengths (including the proportional change of all dimensionless bond strengths, which is the ordinary temperature T change). For the purposes of the discussion in this article, we call the symmetry-broken and -unbroken sides of the phase transition low and high temperatures, respectively. Consider introducing quenched randomness into the local bond strengths of the system previously discussed. First, consider the system at the phase transition. Randomness in local bond strengths is equivalent to randomness in local temperatures (as defined in the previous paragraph). Accordingly, inside a would-be symmetric macroscopic domain at the phase transition, there will be regions that have a lower average local temperature. By the formation of a symmetry-broken island at such a region, the bulk free energy will be decreased proportionately to Ld’*, where L is the linear extent of the island, d is the spatial dimensionality, and the square root is from the central-limit theorem of adding random variables; this will be accompanied by an interface
74
A. N. Berker
I Critical behavior from quenched
disorder
free energy increase proportionate to Ld-’ for discrete (n = 1 component) microscopic degrees of freedom and Ld-* for continuous (n 3 2 component) microscopic degrees of freedom. The exact converse will occur inside a would-be symmetry-broken macroscopic domain, due to regions that have a higher average local temperature. Balancing the opposite free energy changes, it is seen that for d < 2 discrete systems and d < 4 continuum systems, domain-within-domain formation is not limited by a maximal length scale. The would-be symmetric domain thus becomes indistinguishable with each of the would-be symmetry-broken domains, which thereby become indistinguishable with each other. The possibility of coexistence of distinct phases, at the phase transition, disappears, as shown in the middle of the right column in fig. 1. Renormalization-group calculations [6] with the analogous (see below) random-field problem indicate that this also occurs at the borderline dimensions of d = 2 and d = 4. Now consider the phases at low temperature, with the introduction of bond randomness. Recall that bonds are defined as couplings that are invariant under the symmetry that maps the low-temperature phases onto each other. Thus, local variations of bond strengths do not favor one of the p symmetrybroken phases over another. No bulk free energy decrease would occur by the formation of an island of one symmetry-broken phase inside another symmetry-broken phase, but the interface free energy increase would still occur. Consequently, no such island formation is induced by bond randomness. The p symmetry-broken phases remain distinct and coexist, as shown at the bottom of the right column in fig. 1. This means that different low- and high-temperature phases continue to exist under random bonds, but the transition between them cannot be first order (no coexistence) for d s 2 discrete systems and d c 4 continuum systems. This indicates that a second-order phase transition is induced by bond randomness. The phase transition point corresponds to a statistically uniform system that is at criticality. The argument above holds even for infinitesimal amounts of randomness in bond strengths. The above has used the contrasting outcomes, at the phase transition and at low temperatures, of a simple adaptation of the Imry-Ma argument [5] for random fields. The crux of the argument is that random bonds locally favor or disfavor the symmetric phase versus any symmetry-broken phase, and therefore act as a random field between them, but do not distinguish between the phases with differently broken symmetry. To understand what happens with random bonds for d > 2 discrete systems and d > 4 continuum systems, we again draw
on work on random-field systems. First of all, in the discussion above, the disappearance of the first-order transition under bond randomness (which need be no more than infinitesimal) is just a reflection of the fact that under infinitesimal field randomness, symmetry breaking disappears for d Q 2 discrete
A. N. Berker I Critical behavior from quenched disorder
75
systems and d s 4 continuum systems. On the other hand, renormalizationgroup calculations [6] indicate that, above these dimensions, symmetry breaking disappears with the application of a finite amount of field randomness. Thus, by repeating the adaptation above, it can be seen that, for d > 2 discrete systems and d > 4 continuum systems, coexistence at the phase transition disappears, and the phase transition is converted from first order to second order, by the application of a finite amount of bond randomness. Presumably, a tricritical point intervenes between the first- and second-order transitions. The criticality and tricriticality induced by bond randomness suggest the possibility of new universality classes of critical and tricritical phenomena, since phase transitions with characteristic symmetries can be made, for the first time, higher order by bond randomness. For example, Potts models, with any number of states q and in any dimension d, can be made to have a secondorder phase transition by bond randomness. Finally, we turn to first-order phase transitions that do not involve a symmetry breaking. In that case, fig. 1 is modified in that the bottom figures must show a single phase that has no symmetry change from the top figures. When coexistence at the phase transition (middle figures) disappears upon introduction of randomness that creates local preferences of the top or bottom phases, the phase transition disappears. For d d 2 discrete systems and d s 4 continuum systems, this occurs for infinitesimal such randomness. Otherwise, this occurs with a threshold of finite randomness, having an isolated critical point at the threshold. The above conclusions were first reached, simultaneously and independently, by renormalization-group calculations on multicritical phase diagrams [l] and rigorous mathematical proof on latent heats [2]. For the renormalization-group version of the above arguments, the reader is referred to ref. [3]. These general conclusions agree with and provide a unified explanation for results on a variety of specific experimental [7] and model [g-11] systems, as discussed in ref. [3]. The implication for multicritical phenomena, such as the disappearance of tricritical points under infinitesimal bond randomness for d G 2 and d G 4 and under threshold bond randomness for d > 2 and d > 4, for discrete and continuum systems respectively, is obtained in ref. [l] with actual calculations. Thus, fig. 2 here, reproduced from ref. [l], contains the results of a renormalization-group calculation explicitly showing the disappearance of firstorder coexistence under bond randomness. Finally, most recent Monte Carlo [12] results confirm the changeover under bond randomness from first-order to second-order transitions and present an intriguing critical behavior. This research was supported by NSF Grant No. DMR-90-22933 and JSEP Contract No. DAAL 03-92-COOOl.
A.N.
76
Berker
I Critical behavior from quenched
1.0
0.6
0.6 0.4 Density,(sp)
0.2
disorder
0
Fig. 2. Isotherm for the Blume-Emery-Griftiths model in d = 2 with the random-bond discalcutribution P(J,,) = [6(J,, - J - a) + S(J,, - J + a)], o = J/4, from the renormalization-group lation in ref. [ 11. The system is in the ferromagnetic phase at high density and in the paramagnetic phase at low density, with the phase transition occurring at the solid circle. In the pure system (D = 0), this isotherm has a density discontinuity, at constant chemical potential A/J, from (sf) = 0.971 to 0.058, corresponding to the first-order transition.
References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [ll] [12]
K. Hui and A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507; 63 (1989) 2433 (E). M. Aizenman and J. Wehr, Phys. Rev. Lett. 62 (1989) 2503. A.N. Berker. J. Appl. Phys. 70 (1991) 5941. A.N. Berker and K. Hui, in: Science and Technology of Nanostructured Magnetic G.C. Hadjipanayis and G.A. Prinz, eds. (Plenum, New York, 1991) p. 411. Y. Imry and S.-k. Ma, Phys. Rev. Lett. 35 (1975) 1399. S.R. McKay and A.N. Berker, J. Appl. Phys. 64 (1988) 5785. U.J. Cox, A. Gibaud and R.A. Cowley, Phys. Rev. Lett. 61 (1988) 982. G.S. Grest and E.F. Gabl, Phys. Rev. Lett. 43 (1979) 1182. G.N. Murthy, Phys. Rev. B 36 (1987) 7166. W.G. Wilson and C.A. Vause, Phys. Lett. A 134 (1989) 360. W.G. Wilson, Phys. Lett. A 137 (1989) 398. D.P. Landau, Phys. Rev. Lett. (1992), in press.
Materials,
72-76
Critical behavior induced by quenched disorder A. Nihat
Berker
Department of Physics, Massachusetts Institute of Technology,
Cambridge,
MA 02139. USA
Domain arguments and renormalization-group calculations indicate that all temperaturedriven symmetry-breaking first-order phase transitions are converted to second order by quenched bond randomness. This occurs for infinitesimal randomness in d s 2 or d s 4 respectively for discrete or continuous (n = 1 or n 22 component) microscopic degrees of freedom. Even strongly first-order transitions undergo this conversion to second order! Above these dimensions this conversion still occurs but requires a threshold bond randomness, presumably with an intervening new tricritical point. For example, q-state Potts transitions can be made second order for any q in any d, via bond randomness. Non-symmetry-breaking “temperature-driven” first-order transitions are eliminated under the above conditions. These quenched-fluctuation-induced second-order phase transitions suggest the possibility of new universality classes of criticality and tricriticality.
It was recently realized [l-4] that quenched bond randomness has a very dramatic effect on first-order phase transitions, transforming them into secondorder phase transitions and, in many cases (such as q-state Potts models in three dimensions), introducing the possibility of new universality classes of criticality and tricriticality. The argument leading to this very genera1 phenomenon is such a simple adaptation of the Imry-Ma argument [5] for random fields that it is surprising that it has been overlooked throughout the many years in which the Imry-Ma argument has been known. We first consider first-order phase transitions that involve ing. They are represented in fig. 1. In the absence of bond column),
the
high-temperature
phase
(top
figure)
has
symmetry breakrandomness (left
a symmetry
that
is
spontaneously broken at low temperatures. Thus, at low temperatures, the system is in one of p symmetry-broken phases that map onto each other under the broken symmetry. Macroscopic domains of these p different phases can coexist in the system, as shown at the bottom of the left column in fig. 1. If the phase transition is first order, at the phase transition, the p symmetry-broken phases stretching from low temperature coexist with the symmetric phase stretching from high temperature, as shown in the middle of the left column in fig. 1. Presenting the most genera1 argument, we define as “bond” any coupling that is invariant under the symmetry that is broken in the low-temperature 037%4371/93/506.00
0
1993 - Elsevier
Science
Publishers
B.V. All rights
reserved
A.N. Berker I Critical behavior from quenched disorder No
random bonds
73
With random bonds
Fig. 1. Macroscopic domains above (symmetric - S), at (P), and below (symmetry-broken - B) the phase transition temperature in a system without (at left) or with (at right) bond randomness. The unmarked domains represent macroscopic domains of the symmetric phase. The domains marked by letters represent macroscopic domains of the symmetry-broken phases. Each letter corresponds to the same symmetry being broken differently. The bond randomness necessary to effect the change pictured in these figures is infinitesimal for d G 2 discrete systems and d s 4 continuum systems; above these dimensions, a finite threshold of bond randomness is needed.
phases. (On the other hand, a “field” is any coupling that is not invariant under this symmetry.) We call temperature change any change of a dimensionless (i.e., energy divided by k,T) bond strength or of a combination of dimensionless bond strengths (including the proportional change of all dimensionless bond strengths, which is the ordinary temperature T change). For the purposes of the discussion in this article, we call the symmetry-broken and -unbroken sides of the phase transition low and high temperatures, respectively. Consider introducing quenched randomness into the local bond strengths of the system previously discussed. First, consider the system at the phase transition. Randomness in local bond strengths is equivalent to randomness in local temperatures (as defined in the previous paragraph). Accordingly, inside a would-be symmetric macroscopic domain at the phase transition, there will be regions that have a lower average local temperature. By the formation of a symmetry-broken island at such a region, the bulk free energy will be decreased proportionately to Ld’*, where L is the linear extent of the island, d is the spatial dimensionality, and the square root is from the central-limit theorem of adding random variables; this will be accompanied by an interface
74
A. N. Berker
I Critical behavior from quenched
disorder
free energy increase proportionate to Ld-’ for discrete (n = 1 component) microscopic degrees of freedom and Ld-* for continuous (n 3 2 component) microscopic degrees of freedom. The exact converse will occur inside a would-be symmetry-broken macroscopic domain, due to regions that have a higher average local temperature. Balancing the opposite free energy changes, it is seen that for d < 2 discrete systems and d < 4 continuum systems, domain-within-domain formation is not limited by a maximal length scale. The would-be symmetric domain thus becomes indistinguishable with each of the would-be symmetry-broken domains, which thereby become indistinguishable with each other. The possibility of coexistence of distinct phases, at the phase transition, disappears, as shown in the middle of the right column in fig. 1. Renormalization-group calculations [6] with the analogous (see below) random-field problem indicate that this also occurs at the borderline dimensions of d = 2 and d = 4. Now consider the phases at low temperature, with the introduction of bond randomness. Recall that bonds are defined as couplings that are invariant under the symmetry that maps the low-temperature phases onto each other. Thus, local variations of bond strengths do not favor one of the p symmetrybroken phases over another. No bulk free energy decrease would occur by the formation of an island of one symmetry-broken phase inside another symmetry-broken phase, but the interface free energy increase would still occur. Consequently, no such island formation is induced by bond randomness. The p symmetry-broken phases remain distinct and coexist, as shown at the bottom of the right column in fig. 1. This means that different low- and high-temperature phases continue to exist under random bonds, but the transition between them cannot be first order (no coexistence) for d s 2 discrete systems and d c 4 continuum systems. This indicates that a second-order phase transition is induced by bond randomness. The phase transition point corresponds to a statistically uniform system that is at criticality. The argument above holds even for infinitesimal amounts of randomness in bond strengths. The above has used the contrasting outcomes, at the phase transition and at low temperatures, of a simple adaptation of the Imry-Ma argument [5] for random fields. The crux of the argument is that random bonds locally favor or disfavor the symmetric phase versus any symmetry-broken phase, and therefore act as a random field between them, but do not distinguish between the phases with differently broken symmetry. To understand what happens with random bonds for d > 2 discrete systems and d > 4 continuum systems, we again draw
on work on random-field systems. First of all, in the discussion above, the disappearance of the first-order transition under bond randomness (which need be no more than infinitesimal) is just a reflection of the fact that under infinitesimal field randomness, symmetry breaking disappears for d Q 2 discrete
A. N. Berker I Critical behavior from quenched disorder
75
systems and d s 4 continuum systems. On the other hand, renormalizationgroup calculations [6] indicate that, above these dimensions, symmetry breaking disappears with the application of a finite amount of field randomness. Thus, by repeating the adaptation above, it can be seen that, for d > 2 discrete systems and d > 4 continuum systems, coexistence at the phase transition disappears, and the phase transition is converted from first order to second order, by the application of a finite amount of bond randomness. Presumably, a tricritical point intervenes between the first- and second-order transitions. The criticality and tricriticality induced by bond randomness suggest the possibility of new universality classes of critical and tricritical phenomena, since phase transitions with characteristic symmetries can be made, for the first time, higher order by bond randomness. For example, Potts models, with any number of states q and in any dimension d, can be made to have a secondorder phase transition by bond randomness. Finally, we turn to first-order phase transitions that do not involve a symmetry breaking. In that case, fig. 1 is modified in that the bottom figures must show a single phase that has no symmetry change from the top figures. When coexistence at the phase transition (middle figures) disappears upon introduction of randomness that creates local preferences of the top or bottom phases, the phase transition disappears. For d d 2 discrete systems and d s 4 continuum systems, this occurs for infinitesimal such randomness. Otherwise, this occurs with a threshold of finite randomness, having an isolated critical point at the threshold. The above conclusions were first reached, simultaneously and independently, by renormalization-group calculations on multicritical phase diagrams [l] and rigorous mathematical proof on latent heats [2]. For the renormalization-group version of the above arguments, the reader is referred to ref. [3]. These general conclusions agree with and provide a unified explanation for results on a variety of specific experimental [7] and model [g-11] systems, as discussed in ref. [3]. The implication for multicritical phenomena, such as the disappearance of tricritical points under infinitesimal bond randomness for d G 2 and d G 4 and under threshold bond randomness for d > 2 and d > 4, for discrete and continuum systems respectively, is obtained in ref. [l] with actual calculations. Thus, fig. 2 here, reproduced from ref. [l], contains the results of a renormalization-group calculation explicitly showing the disappearance of firstorder coexistence under bond randomness. Finally, most recent Monte Carlo [12] results confirm the changeover under bond randomness from first-order to second-order transitions and present an intriguing critical behavior. This research was supported by NSF Grant No. DMR-90-22933 and JSEP Contract No. DAAL 03-92-COOOl.
A.N.
76
Berker
I Critical behavior from quenched
1.0
0.6
0.6 0.4 Density,(sp)
0.2
disorder
0
Fig. 2. Isotherm for the Blume-Emery-Griftiths model in d = 2 with the random-bond discalcutribution P(J,,) = [6(J,, - J - a) + S(J,, - J + a)], o = J/4, from the renormalization-group lation in ref. [ 11. The system is in the ferromagnetic phase at high density and in the paramagnetic phase at low density, with the phase transition occurring at the solid circle. In the pure system (D = 0), this isotherm has a density discontinuity, at constant chemical potential A/J, from (sf) = 0.971 to 0.058, corresponding to the first-order transition.
References [l] [2] [3] [4] [5] [6] [7] [S] [9] [lo] [ll] [12]
K. Hui and A.N. Berker, Phys. Rev. Lett. 62 (1989) 2507; 63 (1989) 2433 (E). M. Aizenman and J. Wehr, Phys. Rev. Lett. 62 (1989) 2503. A.N. Berker. J. Appl. Phys. 70 (1991) 5941. A.N. Berker and K. Hui, in: Science and Technology of Nanostructured Magnetic G.C. Hadjipanayis and G.A. Prinz, eds. (Plenum, New York, 1991) p. 411. Y. Imry and S.-k. Ma, Phys. Rev. Lett. 35 (1975) 1399. S.R. McKay and A.N. Berker, J. Appl. Phys. 64 (1988) 5785. U.J. Cox, A. Gibaud and R.A. Cowley, Phys. Rev. Lett. 61 (1988) 982. G.S. Grest and E.F. Gabl, Phys. Rev. Lett. 43 (1979) 1182. G.N. Murthy, Phys. Rev. B 36 (1987) 7166. W.G. Wilson and C.A. Vause, Phys. Lett. A 134 (1989) 360. W.G. Wilson, Phys. Lett. A 137 (1989) 398. D.P. Landau, Phys. Rev. Lett. (1992), in press.
Materials,