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Volumn 60, Issue 1, 1999, Pages 206-257

Smectic liquid crystals in random environments

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EID: 5844286064     PISSN: 10980121     EISSN: 1550235X     Source Type: Journal    
DOI: 10.1103/PhysRevB.60.206     Document Type: Article
Times cited : (136)

References (97)
  • 18
    • 85037886581 scopus 로고    scopus 로고
    • For a review of the experimental status also see (especially Bellini et al., Chap. 19, and Rappaport et al., Chap. 20 in) Liquid Crystals in Complex Geometries, edited by G. P. Crawford and S. Z̆umer (Taylor & Francis, London, 1996).
    • For a review of the experimental status also see (especially Bellini et al., Chap. 19, and Rappaport et al., Chap. 20 in) Liquid Crystals in Complex Geometries, edited by G. P. Crawford and S. Z̆umer (Taylor & Francis, London, 1996).
  • 34
    • 85037885947 scopus 로고    scopus 로고
    • We expect the disorder to be short-range correlated in space. One might expect that the continuity of the random aerogel strands should lead to correlated disorder, with a wave vector-dependent variance that possibly diverges as (Formula presented). However, at the NA transition the order parameter is the Fourier transform of the density at a large wave vector q far from (Formula presented). We therefore expect that the disorder that couples to such an order parameter will be short-range correlated in space.
    • We expect the disorder to be short-range correlated in space. One might expect that the continuity of the random aerogel strands should lead to correlated disorder, with a wave vector-dependent variance that possibly diverges as (Formula presented). However, at the NA transition the order parameter is the Fourier transform of the density at a large wave vector q far from (Formula presented). We therefore expect that the disorder that couples to such an order parameter will be short-range correlated in space.
  • 37
    • 85037904557 scopus 로고    scopus 로고
    • For example, gauge glasses. L. Radzihovsky and J. Toner (unpublished).
    • For example, gauge glasses. L. Radzihovsky and J. Toner (unpublished).
  • 41
    • 85037905429 scopus 로고    scopus 로고
    • Our conclusion about the spatial algebraic decay of the smectic order parameter correlations only follows from the logarithmic growth of the layer displacement after a “Debye-Waller factor” approximation which neglects higher order (beyond second) phonon cumulants.
    • Our conclusion about the spatial algebraic decay of the smectic order parameter correlations only follows from the logarithmic growth of the layer displacement after a “Debye-Waller factor” approximation which neglects higher order (beyond second) phonon cumulants.
  • 47
    • 85037881829 scopus 로고    scopus 로고
    • It appears that in most bulk (pure) smectics it is difficult to get into this mean-field regime, with the liquid crystal freezing before it is reached. We thank Carl Garland and Germano Iannacchione for pointing this out to us.
    • It appears that in most bulk (pure) smectics it is difficult to get into this mean-field regime, with the liquid crystal freezing before it is reached. We thank Carl Garland and Germano Iannacchione for pointing this out to us.
  • 48
    • 85037909177 scopus 로고    scopus 로고
    • (unpublished)
    • L. Radzihovsky and B. Ward (unpublished);B. Ward, Ph.D. thesis, University of Colorado (1999).
    • Radzihovsky, L.1    Ward, B.2
  • 54
    • 85037922106 scopus 로고    scopus 로고
    • We have taken the distribution of disorder to be Gaussian, which relies on the assumption that, at long length scales, universality will ensure that small deviations of the true distribution from a Gaussian are not important, as long as the distribution is short-ranged, which we expect on physical grounds.
    • We have taken the distribution of disorder to be Gaussian, which relies on the assumption that, at long length scales, universality will ensure that small deviations of the true distribution from a Gaussian are not important, as long as the distribution is short-ranged, which we expect on physical grounds.
  • 55
    • 85037887240 scopus 로고    scopus 로고
    • We use the expression “line up” to designate a fixed preferred orientation of the nematogen axis relative to the local aerogel strand axis (or more generally with the local tangent to the surface of the confining structure), which might vary from parallel to perpendicular (homeotropic) alignment, depending on the system. It is easy to show that for isotropic confining structures, with no orientational order, e.g., aerogel, our results are independent of these microscopic details. However, for smectics confined, instead, in orientationally ordered structures, studied in Ref. 67, parallel and homeotropic alignments lie in different universality classes.
    • We use the expression “line up” to designate a fixed preferred orientation of the nematogen axis relative to the local aerogel strand axis (or more generally with the local tangent to the surface of the confining structure), which might vary from parallel to perpendicular (homeotropic) alignment, depending on the system. It is easy to show that for isotropic confining structures, with no orientational order, e.g., aerogel, our results are independent of these microscopic details. However, for smectics confined, instead, in orientationally ordered structures, studied in Ref. 67, parallel and homeotropic alignments lie in different universality classes.
  • 58
    • 85037886617 scopus 로고    scopus 로고
    • The choice of the field-rescaling exponent (Formula presented) is of course arbitrary. If, instead of the convenient (Formula presented) choice, we left it arbitrary, then to return the Hamiltonian to the form it had before the RG transformation, we would also have to allow the smectic wave vector (Formula presented) to flow as (Formula presented). In the end the flow equations for the dimensionless coupling constants would involve (Formula presented) and would be independent of the choice of (Formula presented).
    • The choice of the field-rescaling exponent (Formula presented) is of course arbitrary. If, instead of the convenient (Formula presented) choice, we left it arbitrary, then to return the Hamiltonian to the form it had before the RG transformation, we would also have to allow the smectic wave vector (Formula presented) to flow as (Formula presented). In the end the flow equations for the dimensionless coupling constants would involve (Formula presented) and would be independent of the choice of (Formula presented).
  • 59
    • 85037887736 scopus 로고    scopus 로고
    • Gauge invariance of the nonlinearities in the de Gennes model requires that even when dislocation loops are included K does not have any diagrammatic renormalization. This strict nonrenormalization of K can also be seen by studying the transition from the low-temperature side, in terms of the dislocation-loop unbinding picture (Refs. 48, 16). In this formulation, K is the coefficient of an operator quadratic in u, that is nonanalytic in wave vector, while the nonlinearities are all analytic in wave vector. Since it is impossible for these analytic nonlinearities to generate terms that are nonanalytic in k, there is strictly no renormalization of K, even in the presence of dislocation loops that drive the bulk modulus B to zero.
    • Gauge invariance of the nonlinearities in the de Gennes model requires that even when dislocation loops are included K does not have any diagrammatic renormalization. This strict nonrenormalization of K can also be seen by studying the transition from the low-temperature side, in terms of the dislocation-loop unbinding picture (Refs. 4816). In this formulation, K is the coefficient of an operator quadratic in u, that is nonanalytic in wave vector, while the nonlinearities are all analytic in wave vector. Since it is impossible for these analytic nonlinearities to generate terms that are nonanalytic in k, there is strictly no renormalization of K, even in the presence of dislocation loops that drive the bulk modulus B to zero.
  • 73
    • 85037898007 scopus 로고    scopus 로고
    • This choice of having the nonsingular part of (Formula presented) minimize the elastic Hamiltonian is not necessary, but is convenient because for such a choice the dislocation degrees of freedom (Formula presented) decouple from the phonons u; see Appendix C.
    • This choice of having the nonsingular part of (Formula presented) minimize the elastic Hamiltonian is not necessary, but is convenient because for such a choice the dislocation degrees of freedom (Formula presented) decouple from the phonons u; see Appendix C.
  • 80
    • 85037915728 scopus 로고    scopus 로고
    • Analogous elastic anharmonicities are of course also present in ordinary solids and other periodic elastic media. However, because of the stiffer, quadratic (Formula presented) (as opposed to quartic (Formula presented) dispersion of these systems, the anharmonicities are irrelevant in the renormalization group sense.
    • Analogous elastic anharmonicities are of course also present in ordinary solids and other periodic elastic media. However, because of the stiffer, quadratic (Formula presented) (as opposed to quartic (Formula presented) dispersion of these systems, the anharmonicities are irrelevant in the renormalization group sense.
  • 83
    • 85037892952 scopus 로고    scopus 로고
    • All (Formula presented) are finite despite the vanishing of (Formula presented) as (Formula presented) and (Formula presented).
    • All (Formula presented) are finite despite the vanishing of (Formula presented) as (Formula presented) and (Formula presented).
  • 84
    • 85037913873 scopus 로고    scopus 로고
    • D. S. Fisher used a real space renormalization group transformation to demonstrate the stability of the three-dimensional random-field XY model to vortex unbinding (Ref. 11), thereby arguing for the stability of the vortex “Bragg” glass phase in weakly disordered superconductors (Ref. 9, 10
    • D. S. Fisher used a real space renormalization group transformation to demonstrate the stability of the three-dimensional random-field XY model to vortex unbinding (Ref. 11), thereby arguing for the stability of the vortex “Bragg” glass phase in weakly disordered superconductors (Ref. 910).
  • 85
    • 85037913062 scopus 로고    scopus 로고
    • In Sec. VIII, for a periodic-disorder-only model (strictly only valid for (Formula presented), we explicitly demonstrated that this conclusion remained valid up to logarithmic corrections even in the presence of nonlinear elasticity.
    • In Sec. VIII, for a periodic-disorder-only model (strictly only valid for (Formula presented), we explicitly demonstrated that this conclusion remained valid up to logarithmic corrections even in the presence of nonlinear elasticity.
  • 97
    • 85037881278 scopus 로고    scopus 로고
    • We thank Tom Lubensky for pointing out to us the applicability of our theory to this system.
    • We thank Tom Lubensky for pointing out to us the applicability of our theory to this system.


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