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Rutkevich SB, Diehl HW. Inverse-scattering-theory approach to the exact n→∞ solutions of O(n) ϕ⁴ models on films and semi-infinite systems bounded by free surfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062114. [PMID: 26172668 DOI: 10.1103/physreve.91.062114] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/03/2015] [Indexed: 06/04/2023]
Abstract
The O(n) ϕ(4) model on a strip bounded by a pair of planar free surfaces at separation L can be solved exactly in the large-n limit in terms of the eigenvalues and eigenfunctions of a self-consistent one-dimensional Schrödinger equation. The scaling limit of a continuum version of this model is considered. It is shown that the self-consistent potential can be eliminated in favor of scattering data by means of appropriately extended methods of inverse scattering theory. The scattering data (Jost function) associated with the self-consistent potential are determined for the L=∞ semi-infinite case in the scaling regime for all values of the temperature scaling field t=(T-T(c))/T(c) above and below the bulk critical temperature T(c). These results are used in conjunction with semiclassical and boundary-operator expansions and a trace formula to derive exact analytical results for a number of quantities such as two-point functions, universal amplitudes of two excess surface quantities, the universal amplitude difference associated with the thermal singularity of the surface free energy, and potential coefficients. The asymptotic behaviors of the scaled eigenenergies and eigenfunctions of the self-consistent Schrödinger equation as function of x=t(L/ξ(+))(1/ν) are determined for x→-∞. In addition, the asymptotic x→-∞ forms of the universal finite-size scaling functions Θ(x) and ϑ(x) of the residual free energy and the Casimir force are computed exactly to order 1/x, including their x(-1)ln|x| anomalies.
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Affiliation(s)
- Sergei B Rutkevich
- Fakultät für Physik, Universität Duisburg-Essen, D-47058 Duisburg, Germany
| | - H W Diehl
- Fakultät für Physik, Universität Duisburg-Essen, D-47058 Duisburg, Germany
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Borjan Z. Crossover aspects in Ising strips under the influence of variable surface fields and a grain boundary. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032121. [PMID: 25871068 DOI: 10.1103/physreve.91.032121] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2014] [Indexed: 06/04/2023]
Abstract
I use an exact variational formulation of Mikheev and Fisher to study the critical Ising strip with a grain boundary and confining surfaces characterized by arbitrary and different surface magnetic fields. Energy density profiles that serve as order parameters of the system within the used method show strong nonmonotonous behavior in the vicinity of confining surfaces. I consider short-distance expansion of energy density profiles. New universal amplitudes associated with distant-wall corrections exhibit nontrivial crossover behaviors from positive to negative values as the field's variables are continuously varied. Casimir amplitudes calculated in a self-contained manner are characterized by complex manifolds which may comprise two disconnected positive wings separated by an area of negative values. I define and determine the generalized de Gennes-Fisher amplitude, which strongly suggests that a stress tensor is present within one of the distant-wall corrections. This is an unanticipated result given that a similar discovery for standard extraordinary (E) and ordinary (O) surface universality classes was based on the conformal invariance symmetry, which is broken under the present boundary conditions. A grain boundary essentially influences both energy density profiles and Casimir amplitudes, besides surface fields with a number of accompanying features that are examined in detail. We present closed analytic forms of energy density profiles for standard (N) and (O) BCs: ɛNN(x,g),ɛOO(x,g),ɛNO(x,g) [(N) is the normal surface universality class, g is the grain boundary's strength, x is the relative distance x:=z/L with L as a film width], thus enabling us to deduce their important symmetry properties and to have a detailed insight into their general behaviors.
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Affiliation(s)
- Z Borjan
- Faculty of Physics, University of Belgrade, P.O. Box 44, 11001 Belgrade, Serbia
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Eisenriegler E. Anisotropic colloidal particles interacting with polymers in a good solvent. J Chem Phys 2006; 124:144912. [PMID: 16626250 DOI: 10.1063/1.2185644] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study the consequences of chain self-avoidance for the interaction between nonadsorbing polymers and colloidal particles of anisotropic shape, such as ellipsoids, lenses, and dumbbells. In the framework of a field theoretic operator expansion for small mesoscopic particles, we obtain exact results for self-avoiding polymers in d=2 spatial dimensions and we compare ideal and self-avoiding polymers in 2<d<4. Changing the orientation of a particle, with a given size and shape, near a boundary wall requires less free energy if the chains are self-avoiding. The distance dependence of the anisotropic interation, mediated by the polymers between particle and wall, is changed both qualitatively and quantitatively by self-avoidance.
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Affiliation(s)
- E Eisenriegler
- Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany.
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Calabrese P, Cardy J. Time dependence of correlation functions following a quantum quench. PHYSICAL REVIEW LETTERS 2006; 96:136801. [PMID: 16712015 DOI: 10.1103/physrevlett.96.136801] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/13/2006] [Indexed: 05/09/2023]
Abstract
We show that the time dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some Hamiltonian and then evolves without dissipation according to some other Hamiltonian, may be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.
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Affiliation(s)
- Pasquale Calabrese
- Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands
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Dantchev D, Diehl HW, Grüneberg D. Excess free energy and Casimir forces in systems with long-range interactions of van der Waals type: general considerations and exact spherical-model results. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:016131. [PMID: 16486240 DOI: 10.1103/physreve.73.016131] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/11/2005] [Indexed: 05/06/2023]
Abstract
We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L --> infinity and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx-(d+sigma) as x --> infinity, with 2<sigma<4 and 2<d+sigma< or =6, on the Casimir effect at and near the bulk critical temperature Tc,infinity, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged--i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force Fc. We generalize the phenomenological scaling Ansätze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form LdFc/kBt approximately Xi0(L/xi infinity) + g omegaL -omega Xi omega (L/Xi infinity) + g sigma L -omega sigma Xi sigma (L/Xi infinity). Here Xi0, Xi omega, and Xi sigma are universal scaling functions; g omega and g sigma are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; omega and omega sigma = sigma + eta - 2 are the associated correction-to-scaling exponents, where eta denotes the standard bulk correlation exponent of the system without long-range interactions; xi infinity is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution proportional variant g sigma decays for T not = Tc,infinity algebraically in L rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and L. We derive exact results for spherical and Gaussian models which confirm these findings. In the case d + sigma = 6, which includes that of nonretarded van der Waals interactions in d = 3 dimensions, the power laws of the corrections to scaling proportional to b of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy omega = omega sigma = 4 - d that occurs for the spherical model when d + sigma = 6, in conjunction with the b dependence of g omega.
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Affiliation(s)
- Daniel Dantchev
- Fachbereich Physik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany
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Zinn SY, Fisher ME. Scaling for interfacial tensions near critical endpoints. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:011601. [PMID: 15697612 DOI: 10.1103/physreve.71.011601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2004] [Indexed: 05/24/2023]
Abstract
Parametric scaling representations are obtained and studied for the asymptotic behavior of interfacial tensions in the full neighborhood of a fluid (or Ising-type) critical endpoint, i.e., as a function both of temperature and of density/order parameter or chemical potential/ordering field. Accurate nonclassical critical exponents and reliable estimates for the universal amplitude ratios are included naturally on the basis of the "extended de Gennes-Fisher" local-functional theory. Serious defects in previous scaling treatments are rectified and complete wetting behavior is represented; however, quantitatively small, but unphysical residual nonanalyticities on the wetting side of the critical isotherm are smoothed out "manually." Comparisons with the limited available observations are presented elsewhere but the theory invites new, searching experiments and simulations, e.g., for the vapor-liquid interfacial tension on the two sides of the critical endpoint isotherm for which an amplitude ratio -3.25+/-0.05 is predicted.
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Affiliation(s)
- Shun-Yong Zinn
- Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
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Chen XS, Dohm V. Nonuniversal finite-size scaling in anisotropic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 70:056136. [PMID: 15600721 DOI: 10.1103/physreve.70.056136] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2004] [Indexed: 05/24/2023]
Abstract
We study the bulk and finite-size critical behavior of the O(n) symmetric phi4 theory with spatially anisotropic interactions of noncubic symmetry in d<4 dimensions. In such systems of a given (d,n) universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.
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Affiliation(s)
- X S Chen
- Institute of Theoretical Physics, Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China
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Schlesener F, Hanke A, Klimpel R, Dietrich S. Polymer depletion interaction between two parallel repulsive walls. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041803. [PMID: 11308870 DOI: 10.1103/physreve.63.041803] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2000] [Indexed: 05/23/2023]
Abstract
The depletion interaction between two parallel repulsive walls confining a dilute solution of long and flexible polymer chains is studied by field-theoretic methods. Special attention is paid to self-avoidance between chain monomers relevant for polymers in a good solvent. Our direct approach avoids the mapping of the actual polymer chains on effective hard or soft spheres. We compare our results with recent Monte Carlo simulations [A. Milchev and K. Binder, Eur. Phys. J. B 3, 477 (1998)] and with experimental results for the depletion interaction between a spherical colloidal particle and a planar wall in a dilute solution of nonionic polymers [D. Rudhardt, C. Bechinger, and P. Leiderer, Phys. Rev. Lett. 81, 1330 (1998)].
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Affiliation(s)
- F Schlesener
- Fachbereich Physik, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
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Eisenriegler E. Small mesoscopic particles in dilute and semidilute solutions of nonadsorbing polymers. J Chem Phys 2000. [DOI: 10.1063/1.1289239] [Citation(s) in RCA: 60] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Eisenriegler E, Krech M, Dietrich S. Short-distance behavior of the energy density near surfaces of critical systems. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 53:14377-14398. [PMID: 9983236 DOI: 10.1103/physrevb.53.14377] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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