Critical exponents, hyperscaling, and universal amplitude ratios for two- and three-dimensional self-avoiding walks

We make a high-precision Monte Carlo study of two- and three-dimensional self-avoiding walks (SAWs) of length up to 80,000 steps, using the pivot algorithm and the Karp-Luby algorithm. We study the critical exponentsv and 2Δ4 −γ as well as several universal amplitude ratios; in particular, we make an extremely sensitive test of the hyperscaling relationdv = 2Δ4 −γ. In two dimensions, we confirm the predicted exponentv=3/4 and the hyperscaling relation; we estimate the universal ratios /=0.14026±0.00007, /=0.43961±0.00034, and Ψ*=0.66296±0.00043 (68% confidence limits). In three dimensions, we estimatev=0.5877±0.0006 with a correctionto-scaling exponentΔ1=0.56±0.03 (subjective 68% confidence limits). This value forv agrees excellently with the field-theoretic renormalization-group prediction, but there is some discrepancy forΔ1. Earlier Monte Carlo estimates ofv, which were ≈0.592, are now seen to be biased by corrections to scaling. We estimate the universal ratios /=0.1599±0.0002 and Ψ*=0.2471±0.0003; since Ψ*>0, hyperscaling holds. The approach to Ψ* is from above, contrary to the prediction of the two-parameter renormalization-group theory. We critically reexamine this theory, and explain where the error lies. In an appendix, we prove rigorously (modulo some standard scaling assumptions) the hyperscaling relationdv = 2Δ4 −γ for two-dimensional SAWs.

[1]  P. Gennes.  Scaling Concepts in Polymer Physics , 1979 .

[2]  G. Stell.  Free‐Energy Density near the Critical Point , 1969 .

[3]  R. Roskies.  Reconciliation of high-temperature series and renormalization-group results by suppressing confluent singularities , 1981 .

[4]  D. Cannell,et al.  Photometer for quasielastic and classical light scattering , 1983 .

[5]  J. Westwater.  On Edwards' model for polymer chains. III. Borel summability , 1982 .

[6]  J. Collins,et al.  New methods for the renormalization group , 1974 .

[7]  Saleur,et al.  Exact tricritical exponents for polymers at the FTHETA point in two dimensions. , 1987, Physical review letters.

[8]  Kurt Binder,et al.  Monte Carlo studies on the freely jointed polymer chain with excluded volume interaction , 1979 .

[9]  Sign of corrections to scaling amplitudes: Field-theoretic considerations and results for self-repelling walks. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[10]  M. Westwater.  On Edwards' model for long polymer chains , 1980 .

[11]  B. Duplantier.  Lagrangian tricritical theory of polymer chain solutions near the θ-point , 1982 .

[12]  J. Magnen,et al.  A renormalizable field theory: The massive Gross-Neveu model in two dimensions , 1986 .

[13]  S. Weinberg.  New approach to the renormalization group , 1973 .

[14]  On the critical properties of the Edwards and the self-avoiding walk model of polymer chains , 1984 .

[15]  R. Friesner,et al.  A hierarchical algorithm for polymer simulations , 1992 .

[16]  D. C. Rapaport,et al.  On three-dimensional self-avoiding walks , 1985 .

[17]  Ferreira,et al.  Extrapolating Monte Carlo simulations to infinite volume: Finite-size scaling at xi /L >> 1. , 1995, Physical review letters.

[18]  B. Nickel.  Confluent singularities in 3D continuum Φ4 theory: resolving critical point discrepancies , 1991 .

[19]  Pearson,et al.  Finite-size scaling in the three-dimensional Ising model. , 1985, Physical review. B, Condensed matter.

[20]  Jean Zinn-Justin,et al.  Critical exponents from field theory , 1980 .

[21]  G. A. Baker,et al.  The continuous-spin Ising model, g0∶φ4∶d field theory, and the renormalization group , 1981 .

[22]  Gordon Slade,et al.  THE LACE EXPANSION FOR SELF-AVOIDING WALK IN FIVE OR MORE DIMENSIONS , 1992 .

[23]  Anthony J. Guttmann,et al.  On the critical behaviour of self-avoiding walks , 1987 .

[24]  A. Sokal,et al.  Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory , 1991, hep-lat/9210032.

[25]  Naeem Jan,et al.  Polymer conformations through 'wiggling' , 1985 .

[26]  G. Slade,et al.  Mean-field critical behaviour for percolation in high dimensions , 1990 .

[27]  Y. Kato,et al.  Solution properties of high molecular weight polystyrene , 1974 .

[28]  K. Šolc,et al.  Second virial coefficient of polydisperse polymers , 1982 .

[29]  G. Felder.  Construction of a non-trivial planar field theory with ultraviolet stable fixed point , 1985 .

[30]  A. J. Barrett.  Second osmotic virial coefficient for linear excluded volume polymers in the Domb-Joyce model , 1985 .

[31]  N. Madras,et al.  THE SELF-AVOIDING WALK , 2006 .

[32]  M. Lal,et al.  ‘Monte Carlo’ computer simulation of chain molecules , 1969 .

[33]  K. Gawȩdzki,et al.  Non-trivial continuum limit of a π44 model with negative coupling constant , 1985 .

[34]  N. Snider.  Remarks Concerning the Scaling Relations for Critical Exponents , 1971 .

[35]  Hiromi Yamakawa,et al.  Modern Theory of Polymer Solutions , 1971 .

[36]  H. Fujita.  Some unsolved problems on dilute polymer solutions , 1988 .

[37]  P. G. de Gennes,et al.  Exponents for the excluded volume problem as derived by the Wilson method , 1972 .

[38]  Gordon Slade.  The diffusion of self-avoiding random walk in high dimensions , 1987 .

[39]  Jean Dayantis,et al.  Monte Carlo precise determination of the end-to-end distribution function of self-avoiding walks on the simple-cubic lattice , 1991 .

[40]  R. Xu,et al.  Prism laser light‐scattering spectrometer , 1988 .

[41]  H. Stanley,et al.  Introduction to Phase Transitions and Critical Phenomena , 1972 .

[42]  J. Cardy,et al.  Universal distance ratios for two-dimensional polymers , 1989 .

[43]  J. Cotton.  Polymer excluded volume exponent v : An experimental verification of the n vector model for n = 0 , 1980 .

[44]  Alan D. Sokal,et al.  How to beat critical slowing down: 1990 update , 1991 .

[45]  Gordon Slade,et al.  Self-avoiding walk in five or more dimensions I. The critical behaviour , 1992 .

[46]  Richard M. Karp,et al.  Monte-Carlo Approximation Algorithms for Enumeration Problems , 1989, J. Algorithms.

[47]  K. Gawȩdzki,et al.  Massless lattice φ44 theory: Rigorous control of a renormalizable asymptotically free model , 1985 .

[48]  Gordon Slade.  The Scaling Limit of Self-Avoiding Random Walk in High Dimensions , 1989 .

[49]  N. Kawashima,et al.  A Monte-Carlo Calculation of the Renormalized Coupling Constant for the Three Dimensional Ising Model Using Improved Estimators , 1995 .

[50]  Michael Aizenman,et al.  Geometric analysis of φ4 fields and Ising models. Parts I and II , 1982 .

[51]  M. Muthukumar,et al.  Expansion of a polymer chain with excluded volume interaction , 1987 .

[52]  Bernard Nienhuis,et al.  Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas , 1984 .

[53]  J. Adler,et al.  Unbiased map of the temperature plane and its consequences for thed=3Ising model , 1982 .

[54]  J. Hughes,et al.  ß-functions and the exact renormalization group , 1988 .

[55]  Alan D. Sokal,et al.  Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory , 1992 .

[56]  J. D. Cloizeaux,et al.  Polymers in Solution: Their Modelling and Structure , 2010 .

[57]  Monte Carlo Simulation of Tetrahedral Chains II: Properties of “First Self-avoiding Walks” and their Usability as Starting Configurations for Dynamic Relaxation Mechanisms , 1991 .

[58]  Baker,et al.  Renormalized coupling constant for the three-dimensional ising model. , 1995, Physical review letters.

[59]  Sergio Caracciolo,et al.  Polymers and g|φ|4 theory in four dimensions , 1983 .

[60]  K. Freed,et al.  Renormalization and the two-parameter theory. 2. Comparison with experiment and other two-parameter theories , 1985 .

[61]  A. Sokal,et al.  The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk , 1988 .

[62]  B. Widom,et al.  The critical point and scaling theory , 1974 .

[63]  Jean Zinn-Justin,et al.  Critical Exponents for the N Vector Model in Three-Dimensions from Field Theory , 1977 .

[64]  R. Schrader.  A possible constructive approach to φ44 , 1976 .

[65]  B. Nickel,et al.  On hyperscaling in the Ising model in three dimensions , 1979 .

[66]  Static scaling behavior of high-molecular-weight polymers in dilute solution: A reexamination. , 1994 .

[67]  Joseph Klafter,et al.  Self-avoiding walks on a simple cubic lattice , 1993 .

[68]  A. J. Barrett,et al.  Universality approach to the expansion factor of a polymer chain , 1976 .

[69]  R. Roskies.  Hyperscaling in the Ising model on the simple cubic lattice , 1981 .

[70]  M. Fisher,et al.  The validity of hyperscaling in three dimensions for scalar spin systems , 1985 .

[71]  C. Domb,et al.  Mean‐Square Intrachain Distances in a Self‐Avoiding Walk , 1969 .

[72]  S. Edwards.  The statistical mechanics of polymers with excluded volume , 1965 .

[73]  Duplantier.  Tricritical disorder transition of polymers in a cloudy solvent: Annealed randomness. , 1988, Physical review. A, General physics.

[74]  A. J. Barrett,et al.  Numerical study of self-avoiding walks on lattices and in the continuum , 1991 .

[75]  B. Appelt,et al.  Characterization of Polystyrenes of Extremely High Molecular Weights , 1980 .

[76]  M. Daoud,et al.  Solutions of Flexible Polymers. Neutron Experiments and Interpretation , 1975 .

[77]  Universal amplitude combinations for self-avoiding walks, polygons and trails , 1993, cond-mat/9303035.

[78]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[79]  Keizo Suzuki,et al.  The Excluded Volume Effect of Very-long-chain Molecules , 1968 .

[80]  W. Stockmayer,et al.  Osmotic second virial coefficient and two-parameter theories , 1987 .

[81]  J. Noolandi,et al.  Renormalization‐group scaling theory for flexible and wormlike polymer chains , 1992 .

[82]  J. Zinn-Justin,et al.  Accurate critical exponents from field theory , 1989 .

[83]  K. Binder,et al.  Finite-Size Tests of Hyperscaling , 1985 .

[84]  B. Widom,et al.  Surface Tension and Molecular Correlations near the Critical Point , 1965 .

[85]  J. Zinn-Justin.  Analysis of high temperature series of the spin S Ising model on the body-centred cubic lattice , 1981 .

[86]  George A. Baker,et al.  Analysis of hyperscaling in the Ising model by the high-temperature series method , 1977 .

[87]  K. E. Newman,et al.  Critical exponents by the scaling-field method: The isotropic N-vector model in three dimensions , 1984 .

[88]  Karl F. Freed,et al.  Renormalization Group Theory of Macromolecules , 1987 .

[89]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[90]  B. Nickel.  Hyperscaling and universality in 3 dimensions , 1981 .

[91]  M. Fisher,et al.  Unbiased Estimation of Corrections to Scaling by Partial Differential Approximants , 1982 .

[92]  G. A. Baker,et al.  A test for hyperscaling violation in the three-dimensional Ising model , 1982 .

[93]  K. Gawȩdzki,et al.  Gross-Neveu model through convergent perturbation expansions , 1985 .

[94]  K. Gawȩdzki,et al.  Non-Gaussian fixed points of the block spin transformation. Hierarchical model approximation , 1983 .

[95]  P. Flory.  Principles of polymer chemistry , 1953 .

[96]  K. Binder.  Monte Carlo and molecular dynamics simulations in polymer science , 1995 .

[97]  Vladimir Privman,et al.  Finite Size Scaling and Numerical Simulation of Statistical Systems , 1990 .

[98]  F. Wegner.  Corrections to scaling laws , 1972 .

[99]  J. D. Cloizeaux,et al.  The Lagrangian theory of polymer solutions at intermediate concentrations , 1975 .

[100]  V. Privman.  Tests of a method for estimating corrections to scaling from series , 1983 .

[101]  J. Rehr.  Confluent singularities and hyperscaling in the spin-1/2 Ising model , 1979 .

[102]  M. Muthukumar,et al.  Perturbation theory for a polymer chain with excluded volume interaction , 1984 .

[103]  S. Caracciolo,et al.  Universal distance ratios for two-dimensonal self-avoiding walks: corrected conformal-invariance predictions , 1990 .

[104]  C. Domb,et al.  Cluster expansion for a polymer chain , 1972 .

[105]  Convergence of self-avoiding random walk to Brownian motion in high dimensions , 1988 .

[106]  D. Rapaport.  Two-dimensional polymers: universality and correction to scaling , 1985 .

[107]  Joseph Polchinski,et al.  Renormalization and effective lagrangians , 1984 .

[108]  B. Duplantier,et al.  Geometry of polymer chains near the theta‐point and dimensional regularization , 1987 .

[109]  H. Yamakawa,et al.  More on the Analysis of Dilute Solution Data: Polystyrenes Prepared Anionically in Tetrahydrofuran , 1971 .

[110]  R. Schrader,et al.  Analytic and numerical evidence from quantum field theory for the hyperscaling relation dv = 2Δ−γ in the d=3 ising model , 1981 .

[111]  H. Fujita,et al.  Excluded-Volume Effects in Dilute Polymer Solutions. 7. Very High Molecular Weight Polystyrene in Benzene and Cyclohexane , 1978 .

[112]  V. J. Emery.  Critical properties of many-component systems , 1975 .

[113]  J. D. Cloizeaux.  Polymers in solutions : principles and applications of a direct renormalization method , 1981 .

[114]  D. Amit,et al.  On dangerous irrelevant operators , 1982 .

[115]  Alon Orlitsky,et al.  Monte Carlo generation of self-avoiding walks with fixed endpoints and fixed length , 1990 .

[116]  Dayantis,et al.  Scaling exponents of the self-avoiding-walk-problem in three dimensions. , 1994, Physical review. B, Condensed matter.

[117]  D. S. McKenzie,et al.  The second osmotic virial coefficient of athermal polymer solutions , 1967 .

[118]  W. J. Camp,et al.  Series analysis of corrections to scaling for the spin-pair correlations of the spin-s Ising model: Confluent singularities, universality, and hyperscaling , 1976 .

[119]  J. Zinn-Justin,et al.  Analysis of ising model critical exponents from high temperature series expansion , 1979 .

[120]  Mean-field critical behaviour for correlation length for percolation in high dimensions , 1990 .

[121]  Bagnuls,et al.  Field-theoretical approach to critical phenomena. , 1990, Physical review. B, Condensed matter.

[122]  M. Benhamou,et al.  Long polymers in good solvent : ε-expansion of the ratio of the radius of gyration to the end to end distance , 1985 .

[123]  A. Compagner,et al.  Smooth finite-size behaviour of the three-dimensional Ising model , 1985 .

[124]  C. Hall.  Scaling in the ideal Bose gas , 1975 .

[125]  Shang‐keng Ma.  Modern Theory of Critical Phenomena , 1976 .

[126]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

[127]  F. Wegner,et al.  Logarithmic Corrections to the Molecular-Field Behavior of Critical and Tricritical Systems , 1973 .

[128]  Shlomo Sternberg,et al.  Local Contractions and a Theorem of Poincare , 1957 .

[129]  B. Nickel.  One-parameter recursion model for flexible-chain polymers , 1991 .

[130]  K. Freed,et al.  Penetration function and second virial coefficient for linear and regular star polymers , 1984 .

[131]  Guttmann.  Validity of hyperscaling for the d=3 Ising model. , 1986, Physical review. B, Condensed matter.

[132]  M. Fisher.  The theory of equilibrium critical phenomena , 1967 .

[133]  Bernard Nienhuis,et al.  Exact Critical Point and Critical Exponents of O ( n ) Models in Two Dimensions , 1982 .

[134]  A. Fisher,et al.  The Theory of critical phenomena , 1992 .

[135]  A. Sokal,et al.  New Monte Carlo method for the self-avoiding walk , 1985 .

[136]  Continuum limit of the hierarchicalO(N) non-linear σ-model , 1986 .

[137]  Andrea J. Liu,et al.  On the corrections to scaling in three-dimensional Ising models , 1990 .

[138]  J. Cardy.  Finite-size scaling , 1988 .

[139]  R. Schrader.  New rigorous inequality for critical exponents in the Ising model , 1976 .

[140]  Michael E. Fisher,et al.  Scaling, universality and renormalization group theory , 1983 .

[141]  Sidney Redner,et al.  Position-space renormalisation group for isolated polymer chains? , 1981 .