Lifshitz points in ising systems

Hornreich, R. M.; Liebmann, R.; Schuster, H. G.; Selke, W.

doi:10.1007/bf01322086

Cited by 90 publications

(26 citation statements)

References 23 publications

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“…Some later papers include Refs. [19,20]. All of these dealt with physical temperature; the model has not, to our knowledge, been studied for complex temperature.…”

Section: Introductionmentioning

confidence: 99%

Complex-temperature phase diagrams of one-dimensional spin models with next-nearest-neighbor couplings

Shrock¹,

Tsai

1997

Phys. Rev. E

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We study the dependence of complex-temperature phase diagrams on details of the Hamiltonian, focusing on the effect of non-nearest-neighbor spin-spin couplings. For this purpose, we consider a simple exactly solvable model, the 1D Ising model with nearest-neighbor (NN) and next-to-nearest-neighbor (NNN) couplings. We work out the exact phase diagrams for various values of J nnn /J nn and compare these with the case of pure nearest-neighbor (NN) couplings. We also give some similar results for the 1D Potts model with NN and NNN couplings.

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“…Some later papers include Refs. [19,20]. All of these dealt with physical temperature; the model has not, to our knowledge, been studied for complex temperature.…”

Section: Introductionmentioning

confidence: 99%

Complex-temperature phase diagrams of one-dimensional spin models with next-nearest-neighbor couplings

Shrock¹,

Tsai

1997

Phys. Rev. E

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“…15 In particular, the current belief is that the upper and lower an AB block copolymer, but relax the constraint of equal molecular weight of the two homopolymers. critical dimensions for the isotropic Lifshitz point are 8 and 4, respectively.…”

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confidence: 99%

Design of bicontinuous polymeric microemulsions

Fredrickson

Bates

1997

J. Polym. Sci. B Polym. Phys.

124

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“…Moreover, the isotropic LP has an upper critical dimension d"= 8 [1], whereas claims for the lower critical dimension vary: d& = 1 [15], d~~2 [16], and dr = 4 [17]. Nevertheless, fiuctuation effects are generally suppressed in polymer melts (the Ginzburg parameter scales as N '~, N ', and N~for diblock copolymers, binary homopolymer mixtures, and the associated LP, respectively [10]) making the three-component system illustrated in Fig.…”

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confidence: 99%

Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends

et al. 1995

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A series of mixtures composed of a symmetric A-8 diblock copolymer and a symmetric blend of A and B homopolymers was investigated by small-angle neutron scattering. Mean-field theory predicts that a line of lamellar-disorder transitions with wave-vector instability q~0 will meet a line of critical points with q' = 0 in the three-component mixture at an isotropic Lifshitz point. Mean-field Lifshitz 1 behavior (y = 1 and t = 4) was observed in the disordered state at the anticipated composition to within 1 K of the phase transition.PACS numbers: 61.41.+e, 61.12.Ex, 64.60.Cn, 64.70.Kb Fundamental to the study of critical transitions, where a system orders as a field (i.e., temperature) is varied, is the categorization of distinct universality classes. Each class possesses a unique set of critical exponents that describes how the various measurable quantities scale as the transition is approached. Two decades ago Hornreich, Luban, and Shtrikman [1] addressed the critical behavior that occurs when a wave-vector instability q* in the ordered state evolves continuously from a fixed value qo, as an appropriate nonordering field is varied. Here, the locus of critical transitions, the A line, is divided into q = qo and q 4 qo branches by a special multicritical point denoted as the Lifshitz point (LP) with its own set of characteristic exponents. A general review of Lifshitz phenomena has been presented by Selke [2). LP behavior has been suggested for a variety of systems including liquid crystals [3,4], ferroelectrics [5], magnets [6], microemulsions [7], polyelectrolytes [8], and block copolymer-homopolymer mixtures [9,10]. To our knowledge an isotropic LP (m = d, where m represents the number of dimensions in which the wave-vector instability occurs and d is the space dimension) has never been realized experimentally. In this Letter, we report experiments on symmetric diblock copolymer-homopolymer blends that demonstrate rneanfield isotropic Lifshitz behavior. The Ginzburg-Landau free-energy density for a symmetric isotropic system described by a scalar (n = I) order parameter, P(r), can be represented by F(P) = a2$ + a4$ + a6t/t + +~(~O)'+ 2(~'0)'+ where the coefficients, az, a4, a6, . . . , c&, c2, . . . , are system (and temperature, pressure, etc. ) dependent [2]. At an ordinary critical point a2 = 0, with all remaining coefficients positive. Such a critical point separates a disordered state from one that is uniformly ordered with q = 0. Ferromagnets, binary liquid mixtures, and singlecornponent fIuids display ordinary critical points with wellestablished scaling exponents. At a Lifshitz (critical) point both a2 and c~vanish, signaling the incipient development of a nonuniform ordered state with finite wave vector q ) 0. Thus the LP is a special type of critical point that connects three distinct phases: disordered (az~0), uniformly ordered (az ( 0, cI~0), and periodically ordered (a2 ( O, c& ( 0). Actual physical realizations of these states include paramagnetic, ferromagnetic, and helical in certain magnetics ...

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Lifshitz points in ising systems

Cited by 90 publications

References 23 publications

Complex-temperature phase diagrams of one-dimensional spin models with next-nearest-neighbor couplings

Complex-temperature phase diagrams of one-dimensional spin models with next-nearest-neighbor couplings

Design of bicontinuous polymeric microemulsions

Isotropic Lifshitz Behavior in Block Copolymer-Homopolymer Blends

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