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CU-Boulder PHYS 7450 - Phase Diagram of an Ising Model

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Phase diagram of an Ising model with long-range frustrating interactions: A theoretical analysisM. Grousson,1G. Tarjus,1and P. Viot1,21Laboratoire de Physique The´orique des Liquides, Universite´Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex 05, France2Laboratoire de Physique The´orique, Batiment 210, Universite´Paris–Sud, 91405 Orsay Cedex, France共Received 16 June 2000兲We present a theoretical study of the phase diagram of a frustrated Ising model with nearest-neighborferromagnetic interactions and long-range 共Coulombic兲 antiferromagnetic interactions. For nonzero frustration,long-range ferromagnetic order is forbidden, and the ground state of the system consists of phases character-ized by periodically modulated structures. At finite temperatures, the phase diagram is calculated within themean-field approximation. Below the transition line that separates the disordered and the ordered phases, thefrustration-temperature phase diagram displays an infinite number of ‘‘flowers,’’ each flower being made by aninfinite number of modulated phases generated by structure combination branching processes. The specificitiesintroduced by the long-range nature of the frustrating interaction and the limitation of the mean-field approachare finally discussed.PACS number共s兲: 05.50.⫹q, 05.70.Fh, 64.60.CnI. INTRODUCTIONThere are many physical examples in which a short-ranged tendency to order is opposed by a long-range frustrat-ing interaction. In diblock copolymers formed by two mutu-ally incompatible polymer chains attached to each other, therepulsive short-range forces between the two types ofcomponents tend to induce phase separation of the melt, buttotal segregation is forbidden by the covalent bonds that linkthe subchains together. A microphase separation transitionoccurs instead at low enough temperature, and the systemthen forms phases with a periodical modulation of structuresrich in one component or the other, such as lamellar,hexagonal, or cubic phases 关1,2兴. In a similar way, self-assembly in water-oil-surfactant mixtures results from thecompetition between the short-range tendency of water andoil to phase separate and the stoichiometric constraints gen-erated by the presence of surfactant molecules, constraintsthat act as the electroneutrality condition in a system ofcharged particles 关3–5兴. The same kind of physics also arisesin quite different fields. For instance, stripe formation indoped antiferromagnets like cuprates has been ascribed to afrustrated electronic phase separation, by which a strong lo-cal tendency of the holes to phase separate into a hole-rich‘‘metallic’’ phase and a hole-poor antiferromagnetic phase isprohibited by the long-range Coulombic repulsion betweenthe holes 关6,7兴. A last example is provided by the structuralor topological frustration in glass-forming liquids: the dra-matic slowing down of the relaxation that leads to the glassformation has been interpreted as resulting from the presenceof frustration-limited domains whose formation comes fromthe inability of the locally preferred arrangement of the mol-ecules in the liquid to tile space periodically 关8兴; topologicalfrustration may also lead to low-temperature defect-orderedphases, such as the Frank-Kasper 关9兴 phases in bimetallicsystems.A coarse-grained description of the above-mentionedsituations1involves lattice or continuum models with com-peting short-range and Coulombic interactions. The purposeof the present work is to study the phase diagram of such amodel, namely the Coulomb frustrated Ising ferromagnet inwhich Ising spins placed on a three-dimensional cubic latticeinteract via both nearest-neighbor ferromagnetic couplingsand long-range Coulomb-like antiferromagnetic terms. Themodel is introduced in more detail in Sec. II and its groundstate as a function of the frustration parameter, i.e., of theratio of the antiferromagnetic coupling strength over the fer-romagnetic one, is studied in Sec. III. In Sec. IV, we inves-tigate the finite-temperature phase diagram in the mean-fieldapproximation. Finally, the effect of the long-range nature ofthe frustrating forces 共when comparing to the phase behaviorof the prototypical model with competing, but short-rangedinteractions, the axial next-nearest-neighbor Ising 共ANNNI兲model 关10–12兴兲, as well as the limitations of the mean-fieldapproach are discussed in Sec. V.II. THE COULOMB FRUSTRATED ISINGFERROMAGNETThe model is described by the HamiltonianH⫽⫺J兺具ij典SiSj⫹Q2兺i⫽ jv 共rij兲SiSj, 共1兲where, J,Q⬎ 0 are the ferromagnetic and antiferromagneticcoupling strengths, Si⫽⫾1 are Ising spin variables placedon the sites of a three-dimensional cubic lattice,具ij典denotesa sum restricted to nearest neighbors, rijis the vector joiningsites i and j, andv(r) represents a Coulomb-like interactionterm withv(r)⬃1/兩r兩when兩r兩→⬁. 共Throughout the paper,the lattice spacing is taken as the unit length.兲 The above1Additional examples include cross-linked polymer mixtures, in-terpenetrating networks, and ultrathin films.PHYSICAL REVIEW E DECEMBER 2000VOLUME 62, NUMBER 6PRE 621063-651X/2000/62共6兲/7781共12兲/$15.00 7781 ©2000 The American Physical SocietyHamiltonian is essentially the ‘‘hard-spin’’ version of thecoarse-grained free-energy functional derived by Ohta andKawasaki 关13兴 for symmetric diblock copolymer systems,with Si⫽⫾1 characterizing whether the system is locallyrich in one type of monomer or another, J/kBT playing therole of the Flory-Huggins interaction parameter, and Q/Jbeing proportional to 1/N2, where N is the overall degree ofpolymerization of a copolymer.In addition to considering the true Coulombic term,v(r)⫽ 1/兩r兩, we have also studied, for mathematical conveniencein the analytical treatment, an expression ofv(r) in terms ofthe lattice Green’s function that satisfies the Poisson equa-tion on the three-dimensional cubic lattice. In the latter case,v(r) is then simply given, up to a multiplying factor of 4␲,as the inverse Fourier transform of the inverse lattice Laplac-ian,v 共r兲⫽4␲N兺kexp共⫺ ik•r兲2兺␣⫽ x,y,z关1⫺ cos共k␣兲兴, 共2兲where N is the number of lattice sites and the sum over k⫽ (kx,ky,kz) is restricted to the first Brillouin zone. Forlarge兩r兩the lattice Green’s function behaves as 1/(4␲兩r兩),so that the expression in Eq. 共2兲 has the proper


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