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    188宝金博页面版: [精品]Evidence for a topological transition in nematic-to-isotropic phase transition in two dimen

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    内容提示： arXiv:cond-mat/0301065v2 [cond-mat.stat-mech] 3 Feb 2003Evidence for a topological transition in nematic-to-isotropicphase transition in two dimensionsA.I. Fari? nas Sanchez a,b , R. Paredes V. a and B. Berche baCentro de F??sica, Instituto Venezolano de Investigaciones Cient??f i cas,Apartado 21827, Caracas 1020A, VenezuelabLaboratoire de Physique des Mat? eriaux, Universit? e Henri Poincar? e, Nancy 1,F-54506 Vand?uvre les Nancy Cedex, FranceFebruary 2, 2008AbstractThe nematic-to-isotropic ori...

    文档格式：PDF | 页数：6 | 浏览次数：30 | 上传日期：2015-03-20 14:55:23 | 文档星级：
    arXiv:cond-mat/0301065v2 [cond-mat.stat-mech] 3 Feb 2003Evidence for a topological transition in nematic-to-isotropicphase transition in two dimensionsA.I. Fari˜ nas Sanchez a,b , R. Paredes V. a and B. Berche baCentro de F´?sica, Instituto Venezolano de Investigaciones Cient´?f i cas,Apartado 21827, Caracas 1020A, VenezuelabLaboratoire de Physique des Mat´ eriaux, Universit´ e Henri Poincar´ e, Nancy 1,F-54506 Vandœuvre les Nancy Cedex, FranceFebruary 2, 2008AbstractThe nematic-to-isotropic orientational phase transition, or equivalently the RP 2 model, is consid-ered in two dimensions and the question of the nature of the phase transition is addressed. Usingpowerful conformal techniques adapted to the investigation of critical properties of two-dimensionalscale-invariant systems, we report strong evidences for a transition governed by topological defectsanalogous to the Berezinskii-Kosterlitz-Thouless transition in two-dimensional XY model.Liquid crystals may be seen as constituted ofmolecules essentially represented by long rigid rods.From maximization of entropy at high tempera-tures, all the molecule orientations are equally prob-able, independently of the neighbouring moleculedirections and the system exists in an isotropicphase. At low temperatures a preferential orien-tation is more favourable in order to minimize in-teraction terms, and an ordered structure emerges.When order occurs along one space dimension only,the system is said to be nematic. Still at lowertemperatures, other ordered phases can appear, e.g.smectic phases.In a lattice model, each molecule may be repre-sented by a unit vector σ w at site w of an hyper-cubic lattice Λ of linear extent L. The σ’s live ina three-dimensional space attached to each latticesite. In the nematic phase, the preferential direc-tion def i nes a unit vector, n, called the director,and one can measure the deviation of molecule σ wwith respect to the director by the scalar productσ w · n = cosθ w . Due to the local Z 2 symmetry(the rods are not oriented), one cannot distinguishbetween opposite directions θ w and θ w + π, andcosθ w vanishes on average while cos 2 θ w does not.In the disordered phase on the other hand, the an-gles are measured with respect to any arbitrary di-rection, and the thermal average of course leads tohcosθ w i = 0 and hcos 2 θ w i =13 , so that hcos2 θ w i− 13represents a convenient order parameter. In the lit-erature on liquid crystals, one usually def i nes thelocal order parameter by the second Legendre poly-nomial,m(w) = hP 2 (σ w · n)i = hP 2 (cosθ w )i. (1)This def i nition suggests to consider the followingHamiltonian to describe the nematic transition,−Hk B T=Jk B TXwXµP 2 (σ w · σ w+µ ), (2)where µ stands for the unit basis vectors of the lat-tice, σ w ·σ w+µ = cos(θ w −θ w+µ ) is the scalar prod-uct between neighbouring vectors distant from onelattice spacing, and the interaction term −JP 2 (σ w ·σ w+µ ) is reminiscent from a dipole-dipole interac-tion. This Hamiltonian was introduced by Leb-wohl and Lasher [1] as a lattice version of the meanf i eld theory of Maier and Saupe [2], and its successcame from its ability to reproduce the weak f i rst or-der phase transition observed experimentally in the1

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