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Volumn 65, Issue 14, 2002, Pages 1-21

Critical exponents and equation of state of the three-dimensional Heisenberg universality class

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EID: 84891799250     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.65.144520     Document Type: Article
Times cited : (71)

References (144)
  • 1
    • 85038334236 scopus 로고    scopus 로고
    • The isotropic Heisenberg model is a simplified model, and neglects several interactions that are present in real materials. Among them, we should mention the cubic anisotropies due to the lattice structure and the dipolar interactions. Even if, in the renormalization-group language, these effects are relevant perturbations of the Heisenberg fixed point (see Refs. 115116117118), the critical exponents are so close to those of the Heisenberg universality class that the difference is experimentally very difficult to observe, see, e.g., Refs. 119, 118, 120, and 56, and references therein
    • The isotropic Heisenberg model is a simplified model, and neglects several interactions that are present in real materials. Among them, we should mention the cubic anisotropies due to the lattice structure and the dipolar interactions. Even if, in the renormalization-group language, these effects are relevant perturbations of the Heisenberg fixed point (see Refs. 115116117118), the critical exponents are so close to those of the Heisenberg universality class that the difference is experimentally very difficult to observe, see, e.g., Refs. 119, 118, 120, and 56, and references therein.
  • 15
    • 85038298384 scopus 로고    scopus 로고
    • For some dopings and some divalent cation A, a first-order transition has been observed. Moreover, in systems in which the transition appears to be of second order, mean-field critical exponents have been measured. For instance, for (formula presented), a mean-field value for β was observed in Refs. 121122123, while an estimate compatible with the Heisenberg value was found in Refs. 124125126127. For (formula presented) there also exists (Ref. 94) an estimate of the exponent α, (formula presented), in agreement with the Heisenberg value
    • For some dopings and some divalent cation A, a first-order transition has been observed. Moreover, in systems in which the transition appears to be of second order, mean-field critical exponents have been measured. For instance, for (formula presented), a mean-field value for β was observed in Refs. 121122123, while an estimate compatible with the Heisenberg value was found in Refs. 124125126127. For (formula presented) there also exists (Ref. 94) an estimate of the exponent α, (formula presented), in agreement with the Heisenberg value.
  • 16
    • 85038269722 scopus 로고    scopus 로고
    • order to observe the correct exponents, it is essential to consider corrections to scaling in the analysis of the experimental data (Ref. 24). All results reported in Table II, except those of Ref. 21, have been obtained by assuming scaling corrections of the form (formula presented), with (formula presented) and (formula presented). Two observations are in order here. First, the results of the present paper provide a precise estimate of (formula presented) (formula presented). Second, the renormalization group also predicts corrections of order (formula presented) (formula presented), etc. which are more relevant than the term (formula presented), and should therefore be taken into account in the analysis of the data
    • In order to observe the correct exponents, it is essential to consider corrections to scaling in the analysis of the experimental data (Ref. 24). All results reported in Table II, except those of Ref. 21, have been obtained by assuming scaling corrections of the form (formula presented), with (formula presented) and (formula presented). Two observations are in order here. First, the results of the present paper provide a precise estimate of (formula presented) (formula presented). Second, the renormalization group also predicts corrections of order (formula presented) (formula presented), etc. which are more relevant than the term (formula presented), and should therefore be taken into account in the analysis of the data.
  • 61
    • 85038334329 scopus 로고    scopus 로고
    • Recently, a model with competing superexchange and double-exchange interactions was studied (Ref. 128). A preliminary analysis for the paramagnetic-ferromagnetic transition gives (formula presented) and (formula presented). While (formula presented) is reasonable agreement with the Heisenberg value, (formula presented) is significantly higher, so that the identification of this transition as a Heisenberg one is in doubt
    • Recently, a model with competing superexchange and double-exchange interactions was studied (Ref. 128). A preliminary analysis for the paramagnetic-ferromagnetic transition gives (formula presented) and (formula presented). While (formula presented) is reasonable agreement with the Heisenberg value, (formula presented) is significantly higher, so that the identification of this transition as a Heisenberg one is in doubt.
  • 73
    • 85038334875 scopus 로고    scopus 로고
    • the so-called critical-point renormalization method (see Ref. 129, and references therein), given two series (formula presented) and (formula presented) that are singular at the same point (formula presented) (formula presented) and (formula presented), one constructs a series (formula presented). The function (formula presented) is singular at (formula presented), and for (formula presented) behaves as (formula presented), where (formula presented). Therefore, the difference (formula presented) can be obtained by analyzing the expansion of (formula presented) by means of biased approximants with a singularity at (formula presented)
    • In the so-called critical-point renormalization method (see Ref. 129, and references therein), given two series (formula presented) and (formula presented) that are singular at the same point (formula presented) (formula presented) and (formula presented), one constructs a series (formula presented). The function (formula presented) is singular at (formula presented), and for (formula presented) behaves as (formula presented), where (formula presented). Therefore, the difference (formula presented) can be obtained by analyzing the expansion of (formula presented) by means of biased approximants with a singularity at (formula presented).
  • 98
    • 85038332499 scopus 로고    scopus 로고
    • the HT phase (formula presented) (formula presented) where (formula presented) is analytic in the complex (formula presented) plane except possibly for two branch cuts on the imaginary axis, for (formula presented) (Yang-Lee theorem). Thus the convergence radius is at most (formula presented), where k is the normalization of z
    • In the HT phase (formula presented) (formula presented) where (formula presented) is analytic in the complex (formula presented) plane except possibly for two branch cuts on the imaginary axis, for (formula presented) (Yang-Lee theorem). Thus the convergence radius is at most (formula presented), where k is the normalization of z.
  • 109
    • 85038277121 scopus 로고    scopus 로고
    • The amplitudes B, (formula presented) and (formula presented) were determined in several experiments-see, e.g., Ref. 20-and thus an estimate of the ratio (formula presented) is possible. However, these amplitudes have large systematic uncertainties, and we can only obtain a rough estimate, (formula presented)
    • The amplitudes B, (formula presented) and (formula presented) were determined in several experiments-see, e.g., Ref. 20-and thus an estimate of the ratio (formula presented) is possible. However, these amplitudes have large systematic uncertainties, and we can only obtain a rough estimate, (formula presented).
  • 114
    • 85038322087 scopus 로고    scopus 로고
    • We have evaluated (formula presented) (the position of the maximum) from Fig. 4 of Ref. 6: (formula presented). The variable u defined in Sec. IV D is related to (formula presented) by (formula presented) Note that the (formula presented) can also be derived from a knowledge of the nonuniversal constants B, (formula presented) and (formula presented), and using the results of Sec. IV D. Indeed, (formula presented)Numerically, using the estimates reported in Sec. IV D, we have (formula presented) (formula presented), and (formula presented)
    • We have evaluated (formula presented) (the position of the maximum) from Fig. 4 of Ref. 6: (formula presented). The variable u defined in Sec. IV D is related to (formula presented) by (formula presented) Note that the (formula presented) can also be derived from a knowledge of the nonuniversal constants B, (formula presented) and (formula presented), and using the results of Sec. IV D. Indeed, (formula presented)Numerically, using the estimates reported in Sec. IV D, we have (formula presented) (formula presented), and (formula presented).
  • 119
    • 85104369179 scopus 로고
    • edited by C. Domb and M.S. Green, Academic Press, London
    • M. Wortis, in Phase Transitions and Critical Phenomena, edited by C. Domb and M.S. Green (Academic Press, London, 1974), Vol. 3.
    • (1974) Phase Transitions and Critical Phenomena , vol.3
    • Wortis, M.1
  • 122
    • 0003493230 scopus 로고
    • For a review see, e.g., edited by C. Domb and J. Lebowitz, Academic, New York
    • For a review see, e.g., A.J. Guttmann, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic, New York, 1989), Vol. 13.
    • (1989) Phase Transitions and Critical Phenomena , vol.13
    • Guttmann, A.J.1
  • 125
    • 0001348177 scopus 로고
    • edited by C. Domb and J. Lebowitz, Academic Press, New York
    • A. Aharony, in Phase Transitions and Critical Phenomena, edited by C. Domb and J. Lebowitz (Academic Press, New York, 1976), Vol. 6, p. 357.
    • (1976) Phase Transitions and Critical Phenomena , vol.6 , pp. 357
    • Aharony, A.1


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