- Access by Xinjiang University
Renormalization group theory: Its basis and formulation in statistical physics
Rev. Mod. Phys. 70, 653 – Published 1 April, 1998
DOI: https://doi.org/10.1103/RevModPhys.70.653
Abstract
The nature and origins of renormalization group ideas in statistical physics and condensed matter theory are recounted informally, emphasizing those features of prime importance in these areas of science in contradistinction to quantum field theory, in particular: critical exponents and scaling, relevance, irrelevance and marginality, universality, and Wilson’s crucial concept of flows and fixed points in a large space of Hamiltonians.
See Also
Critical Behavior of Magnets with Dipolar Interactions. I. Renormalization Group near Four Dimensions
Phys. Rev. B 8, 3323 (1973)
References (123)
- Abe, R., 1972, “Expansion of a Critical Exponent in Inverse Powers of Spin Dimensionality,” Prog. Theor. Phys. 48, 1414–1415.
- Abe, R., 1973, “Expansion of a Critical Exponent in Inverse Powers of Spin Dimensionality,” Prog. Theor. Phys. 49, 113–128.
- Aharony, A., 1973, “Critical Behavior of Magnets with Dipolar Interactions. V. Uniaxial Magnets in Dimensions,” Phys. Rev. B 8, 3363–3370; 1974, erratum 9, 3946.
- Aharony, A., 1976, “Dependence of Universal Critical Behavior on Symmetry and Range of Interaction,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 357–424.
- Ahlers, G., A. Kornblit, and H. J. Guggenheim, 1975, “Logarithmic Corrections to the Landau Specific Heat near the Curie Temperature of the Dipolar Ising Ferromagnet ,” Phys. Rev. Lett. 34, 1227–1230.
- Amit, D. J., 1978, Field Theory, the Renormalization Group and Critical Phenomena (McGraw-Hill Inc., London); see also 1993, the expanded Revised 2nd Edition (World Scientific, Singapore).
- Bagnuls, C., and C. Bervillier, 1997, “Field-Theoretic Techniques in the Study of Critical Phenomena,” J. Phys. Studies 1, 366–382.
- Baker, G. A., Jr., 1961, “Application of the Padé Approximant Method to the Investigation of some Magnetic Properties of the Ising Model,” Phys. Rev. 124, 768–774.
- Baker, G. A., Jr., 1990, Quantitative Theory of Critical Phenomena (Academic, San Diego).
- Baker, G. A., Jr., and N. Kawashima, 1995, “Renormalized Coupling Constant for the Three-Dimensional Ising Model,” Phys. Rev. Lett. 75, 994–997.
- Baker, G. A., Jr., and N. Kawashima, 1996, “Reply to Comment by J.-K. Kim,” Phys. Rev. Lett. 76, 2403.
- Baker, G. A., Jr., B. G. Nickel, M. S. Green, and D. I. Meiron, 1976, “Ising Model Critical Indices in Three Dimensions from the Callan-Symanzik Equation,” Phys. Rev. Lett. 36, 1351–1354.
- Baker, G. A., Jr., B. G. Nickel, and D. I. Meiron, 1978, “Critical Indices from Perturbation Analysis of the Callan-Symanzik Equation,” Phys. Rev. B 17, 1365–1374.
- Baxter, R. J., 1982, Exactly Solved Models in Statistical Mechanics (Academic, London).
- Benfatto, G., and G. Gallavotti, 1995, Renormalization Group, Physics Notes, edited by P. W. Anderson, A. S. Wightman, and S. B. Treiman (Princeton University, Princeton, NJ).
- Bogoliubov, N. N., and D. V. Shirkov, 1959, Introduction to the Theory of Quantized Fields (Interscience, New York), Chap. VIII.
- Brézin, E., J. C. Le Guillou, and J. Zinn-Justin, 1976, “Field Theoretical Approach to Critical Phenomena,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 125–247.
- Brézin, E., D. J. Wallace, and K. G. Wilson, 1972, “Feynman-Graph Expansion for the Equation of State near the Critical Point,” Phys. Rev. B 7, 232–239.
- Buckingham, M. J., and J. D. Gunton, 1969, “Correlations at the Critical Point of the Ising Model,” Phys. Rev. 178, 848–853.
- Burkhardt, T. W. and J. M. J. van Leeuwen, 1982, Eds., Real Space Renormalization (Springer, Berlin).
- Cao, T. Y., 1998, Ed., The Conceptual Foundations of Quantum Field Theory (Cambridge University Press, Cambridge).
- Chen, J.-H., M. E. Fisher, and B. G. Nickel, 1982, “Unbiased Estimation of Corrections to Scaling by Partial Differential Approximants,” Phys. Rev. Lett. 48, 630–634.
- Creswick, R. J., H. A. Farach, and C. P. Poole, Jr., 1992, Introduction to Renormalization Group Methods in Physics (John Wiley & Sons, Inc., New York).
- De Bruijn, N. G., 1958, Asymptotic Methods in Analysis (North Holland, Amsterdam).
- Di Castro, C., and G. Jona-Lasinio, 1969, “On the Microscopic Foundation of Scaling Laws,” Phys. Lett. 29A, 322–323.
- Di Castro, C., and G. Jona-Lasinio, 1976, “The Renormalization Group Approach to Critical Phenomena,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 507–558.
- Domb, C., 1960, “On the Theory of Cooperative Phenomena in Crystals,” Adv. Phys. (Philos. Mag. Suppl.) 9, 149–361.
- Domb, C., 1996, The Critical Point: A Historical Introduction to the Modern Theory of Critical Phenomena (Taylor and Francis, London).
- Domb, C., and D. L. Hunter, 1965, “On the Critical Behavior of Ferromagnets,” Proc. Phys. Soc. (London) 86, 1147–1151.
- Domb, C., and M. F. Sykes, 1962, “Effect of Change of Spin on the Critical Properties of the Ising and Heisenberg Models,” Phys. Rev. 128, 168–173.
- Essam, J. W., and M. E. Fisher, 1963, “Padé Approximant Studies of the Lattice Gas and Ising Ferromagnet below the Critical Point,” J. Chem. Phys. 38, 802–812.
- Fisher, D. S., and D. A. Huse, 1985, “Wetting Transitions: A Functional Renormalization Group Approach,” Phys. Rev. B 32, 247–256.
- Fisher, M. E., 1959, “The Susceptibility of the Plane Ising Model,” Physica (Amsterdam) 25, 521–524.
- Fisher, M. E., 1962, “On the Theory of Critical Point Density Fluctuations,” Physica (Amsterdam) 28, 172–180.
- Fisher, M. E., 1964, “Correlation Functions and the Critical Region of Simple Fluids,” J. Math. Phys. 5, 944–962.
- Fisher, M. E., 1965, “The Nature of Critical Points,” in Lectures in Theoretical Physics, Vol. VIIC (University of Colorado Press, Boulder), pp. 1–159.
- Fisher, M. E., 1966a, “Notes, Definitions and Formulas for Critical Point Singularities,” in M. S. Green and J. V. Sengers, 1966, Eds., cited below.
- Fisher, M. E., 1966b, “Shape of a Self-Avoiding Walk or Polymer Chain,” J. Chem. Phys. 44, 616–622.
- Fisher, M. E., 1967a, “The Theory of Condensation and the Critical Point,” Physics (Long Island City, NY) 3, 255–283.
- Fisher, M. E., 1967b, “The Theory of Equilibrium Critical Phenomena,” Rep. Prog. Phys. 30, 615–731.
- Fisher, M. E., 1967c, “Critical Temperatures of Anisotropic Ising Lattices II. General Upper Bounds,” Phys. Rev. 162, 480–485.
- Fisher, M. E., 1969, “Phase Transitions and Critical Phenomena,” in Contemporary Physics, Vol 1, Proc. International Symposium, Trieste, 7–28 June 1968 (International Atomic Energy Agency, Vienna), pp. 19–46.
- Fisher, M. E., 1971, “The Theory of Critical Point Singularities,” in Critical Phenomena, Proc. 1970 Enrico Fermi Internat. Sch. Phys., Course No. 51, Varenna, Italy, edited by M. S. Green (Academic, New York), pp. 1–99.
- Fisher, M. E., 1972, “Phase Transitions, Symmetry and Dimensionality,” Essays in Physics (Academic, London), Vol. 4, pp. 43–89.
- Fisher, M. E., 1974a, “General Scaling Theory for Critical Points,” Proceedings of the Nobel Symp. XXIV, Aspenäsgården, Lerum, Sweden, June 1973, in Collective Properties of Physical Systems, edited by B. Lundqvist and S. Lundqvist (Academic, New York), pp. 16–37.
- Fisher, M. E., 1974b, “The Renormalization Group in the Theory of Critical Phenomena,” Rev. Mod. Phys. 46, 597–616.
- Fisher, M. E., 1983, “Scaling, Universality and Renormalization Group Theory,” in Lecture Notes in Physics, Vol. 186, Critical Phenomena, edited by F. J. W. Hahne (Springer, Berlin), pp. 1–139.
- Fisher, M. E., and A. Aharony, 1973, “Dipolar Interactions at Ferromagnetic Critical Points,” Phys. Rev. Lett. 30, 559–562.
- Fisher, M. E., and R. J. Burford, 1967, “Theory of Critical Point Scattering and Correlations. I. The Ising Model,” Phys. Rev. 156, 583–622.
- Fisher, M. E., and J.-H. Chen, 1985, “The Validity of Hyperscaling in Three Dimensions for Scalar Spin Systems,” J. Phys. (Paris) 46, 1645–54.
- Fisher, M. E., and D. S. Gaunt, 1964, “Ising Model and Self-Avoiding Walks on Hypercubical Lattices and ‘High Density’ Expansions,” Phys. Rev. 133, A224–A239.
- Fisher, M. E., S.-K. Ma, and B. G. Nickel, 1972, “Critical Exponents for Long-Range Interactions,” Phys. Rev. Lett. 29, 917–920.
- Fisher, M. E., and P. Pfeuty, 1972, “Critical Behavior of the Anisotropic -Vector Model,” Phys. Rev. B 6, 1889–91.
- Fisher, M. E., and M. F. Sykes, 1959, “Excluded Volume Problem and the Ising Model of Ferromagnetism,” Phys. Rev. 114, 45–58.
- Gawȩdski, K., 1986, “Rigorous Renormalization Group at Work,” Physica 140A, 78–84.
- Gell-Mann, M., and F. E. Low, 1954, “Quantum Electrodynamics at Small Distances,” Phys. Rev. 95, 1300–1312.
- Gorishny, S. G., S. A. Larin, and F. V. Tkachov, 1984, “ε-Expansion for Critical Exponents: The Approximation,” Phys. Lett. 101A, 120–123.
- Green, M. S., and J. V. Sengers, 1966, Eds., Critical Phenomena: Proceedings of a Conference held in Washington, D.C. April 1965, N.B.S. Misc. Publ. 273 (U.S. Govt. Printing Off., Washington, 1 December 1966).
- Griffiths, R. B., 1965, “Thermodynamic Inequality near the Critical Point for Ferromagnets and Fluids,” Phys. Rev. Lett. 14, 623–624.
- Griffiths, R. B., 1972, “Rigorous Results and Theorems,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 1, pp. 7–109.
- Gross, D., and F. Wilczek, 1973, “Ultraviolet Behavior of Non-Abelian Gauge Theories,” Phys. Rev. Lett. 30, 1343–46.
- Guggenheim, E. A., 1949, Thermodynamics (North Holland, Amsterdam), Fig. 4.9.
- Gunton, J. D., and M. S. Green, 1974, Eds., Renormalization Group in Critical Phenomena and Quantum Field Theory: Proceedings of a Conference, held at Chestnut Hill, Pennsylvania, 29–31 May 1973 (Temple University, Philadelphia).
- Heller, P., and G. B. Benedek, 1962, “Nuclear Magnetic Resonance in near the Critical Point,” Phys. Rev. Lett. 8, 428–432.
- Hille, E., 1948, Functional Analysis and Semi-Groups, 1948 (American Mathematical Society, New York).
- Hohenberg, P. C., and B. I. Halperin, 1977, “Theory of Dynamic Critical Phenomena,” Rev. Mod. Phys. 49, 435–479.
- Itzykson, D., and J.-M. Drouffe, 1989, Statistical Field Theory (Cambridge University Press, Cambridge).
- Jeffreys, H., and B. S. Jeffreys, 1956, Methods of Mathematical Physics, 3rd ed. (Cambridge University, London).
- Kadanoff, L. P., 1966, “Scaling Laws for Ising Models near ,” Physics (Long Island City, NY) 2, 263–272.
- Kadanoff, L. P., 1976, “Scaling, Universality and Operator Algebras,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 5a, pp. 1–34.
- Kadanoff, L. P., et al., 1967, “Static Phenomena near Critical Points: Theory and Experiment,” Rev. Mod. Phys. 39, 395–431.
- Kadanoff, L. P., and F. J. Wegner, 1971, “Some Critical Properties of the Eight-Vertex Model,” Phys. Rev. B 4, 3989–3993.
- Kaufman, B., and L. Onsager, 1949, “Crystal Statistics II. Short-Range Order in a Binary Lattice,” Phys. Rev. 76, 1244–52.
- Kiang, C. S., and D. Stauffer, 1970, “Application of Fisher’s Droplet Model for the Liquid-Gas Transition near ,” Z. Phys. 235, 130–139.
- Kim, H. K., and M. H. W. Chan, 1984, “Experimental Determination of a Two-Dimensional Liquid-Vapor Critical-Point Exponent,” Phys. Rev. Lett. 53, 170–173.
- Koch, R. H., V. Foglietti, W. J. Gallagher, G. Koren, A. Gupta, and M. P. A. Fisher, 1989, “Experimental Evidence for Vortex-Glass Superconductivity in Y-Ba-Cu-O,” Phys. Rev. Lett. 63, 1511–1514.
- Kouvel, J. S., and M. E. Fisher, 1964, “Detailed Magnetic Behavior of Nickel near its Curie Point,” Phys. Rev. 136, A1626–A1632.
- Landau, L. D., and E. M. Lifshitz, 1958, Statistical Physics, Vol. 5 of Course of Theoretical Physics (Pergamon, London), Chap. XIV.
- Larkin, A. I., and D. E. Khmel’nitskii, 1969, “Phase Transition in Uniaxial Ferroelectrics,” Zh. Eksp. Teor. Fiz. 56, 2087–2098 [Sov. Phys. JETP 29, 1123–1128].
- Le Guillou, J. C., and J. Zinn-Justin, 1977, “Critical Exponents for the -Vector Model in Three Dimensions from Field Theory,” Phys. Rev. Lett. 39, 95–98.
- Ma, S.-K., 1976a, Modern Theory of Critical Phenomena (W. A. Benjamin, Inc., Reading, MA).
- Ma, S.-K., 1976b, “Renormalization Group by Monte Carlo Methods,” Phys. Rev. Lett. 37, 461–474.
- MacLane, S. and G. Birkhoff, 1967, Algebra (MacMillan, New York), Chap. II, Sec. 9, Chap. III, Secs. 1, 2.
- Nelson, D. R., 1983, “Defect-mediated Phase Transitions” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 7, pp. 1–99.
- Nelson, D. R., and M. E. Fisher, 1975, “Soluble Renormalization Groups and Scaling Fields for Low-Dimensional Spin Systems,” Ann. Phys. (N.Y.) 91, 226–274.
- Niemeijer, Th., and J. M. J. van Leeuwen, 1976, “Renormalization Theory for Ising-like Systems,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 425–505.
- Nickel, B. G., 1978, “Evaluation of Simple Feynman Graphs,” J. Math. Phys. 19, 542–548.
- Onsager, L., 1944, “Crystal Statistics I. A Two-dimensional Model with an Order-Disorder Transition,” Phys. Rev. 62, 117–149.
- Onsager, L., 1949, “Discussion remark following a paper by G. S. Rushbrooke at the International Conference on Statistical Mechanics in Florence,” Nuovo Cimento Suppl. (9) 6, 261.
- Parisi, G., 1973, “Perturbation Expansion of the Callan-Symanzik Equation,” in Gunton and Green (1974), cited above.
- Parisi, G., 1974, “Large-Momentum Behaviour and Analyticity in the Coupling Constant,” Nuovo Cimento A 21, 179–186.
- Patashinskii, A. Z., and V. L. Pokrovskii, 1966, “Behavior of ordering systems near the transition point,” Zh. Eksp. Teor. Fiz. 50, 439–447 [Sov. Phys. JETP 23, 292–297].
- Patashinskii, A. Z., and V. L. Pokrovskii, 1979, Fluctuation Theory of Phase Transitions (Pergamon, Oxford).
- Pawley, G. S., R. H. Swendsen, D. J. Wallace, and K. G. Wilson, 1984, “Monte Carlo Renormalization Group Calculations of Critical Behavior in the Simple-Cubic Ising Model,” Phys. Rev. B 29, 4030–40.
- Pfeuty, P., and G. Toulouse, 1975, Introduction au Group des Renormalisation et à ses Applications (Univ. de Grenoble, Grenoble); translation, Introduction to the Renormalization Group and to Critical Phenomena (Wiley and Sons, London, 1977).
- Politzer, H. D., 1973, “Reliable Perturbative Results for Strong Interactions?” Phys. Rev. Lett. 30, 1346–49.
- Riesz, F. and B. Sz.-Nagy, 1955, Functional Analysis (F. Ungar, New York) Chap. X, Secs. 141, 142.
- Rowlinson, J. S., 1959, Liquids and Liquid Mixtures (Butterworth Scientific, London).
- Rushbrooke, G. S., 1963, “On the Thermodynamics of the Critical Region for the Ising Problem,” J. Chem. Phys. 39, 842–843.
- Rushbrooke, G. S., and P. J. Wood, 1958, “On the Curie Points and High-Temperature Susceptibilities of Heisenberg Model Ferromagnetics,” Mol. Phys. 1, 257–283.
- Schultz, T. D., D. C. Mattis, and E. H. Lieb, 1964, “Two-Dimensional Ising Model as a Soluble Problem of Many Fermions,” Rev. Mod. Phys. 36, 856–871.
- Shankar, R., 1994, “Renormalization-Group Approach to Interacting Fermions,” Rev. Mod. Phys. 66, 129–192.
- Stanley, H. E., 1971, Introduction to Phase Transitions and Critical Phenomena (Oxford University, New York).
- Stell, G., 1968, “Extension of the Ornstein-Zernike Theory of the Critical Region,” Phys. Rev. Lett. 20, 533–536.
- Suzuki, M., 1972, “Critical Exponents and Scaling Relations for the Classical Vector Model with Long-Range Interactions,” Phys. Lett. 42A, 5–6.
- Symanzik, K., 1966, “Euclidean Quantum Field Theory. I. Equations for a Scalar Model,” J. Math. Phys. 7, 510–525.
- Tarko, H. B., and M. E. Fisher, 1975, “Theory of Critical Point Scattering and Correlations. III. The Ising Model below and in a Field,” Phys. Rev. B 11, 1217–1253.
- Wallace, D. J., 1976, “The ε-Expansion for Exponents and the Equation of State in Isotropic Systems,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 293–356.
- Wegner, F. J., 1972a, “Corrections to Scaling Laws,” Phys. Rev. B 5, 4529–4536.
- Wegner, F. J., 1972b, “Critical Exponents in Isotropic Spin Systems,” Phys. Rev. B 6, 1891–93.
- Wegner, F. J., 1976, “The Critical State, General Aspects,” in Phase Transitions and Critical Phenomena, edited by C. Domb and M. S. Green (Academic, London), Vol. 6, pp. 7–124.
- Widom, B., 1965a, “Surface Tension and Molecular Correlations near the Critical Point,” J. Chem. Phys. 43, 3892–97.
- Widom, B., 1965b, “Equation of State in the Neighborhood of the Critical Point,” J. Chem. Phys. 43, 3898–3905.
- Widom, B., and O. K. Rice, 1955, “Critical Isotherm and the Equation of State of Liquid-Vapor Systems,” J. Chem. Phys. 23, 1250–1255.
- Wilson, K. G., 1971a, “Renormalization Group and Critical Phenomena I. Renormalization Group and Kadanoff Scaling Picture,” Phys. Rev. B 4, 3174–83.
- Wilson, K. G., 1971b, “Renormalization Group and Critical Phenomena II. Phase-Space Cell Analysis of Critical Behavior,” Phys. Rev. B 4, 3184–3205.
- Wilson, K. G., 1972, “Feynman-Graph Expansion for Critical Exponents,” Phys. Rev. Lett. 28, 548–551.
- Wilson, K. G., 1975, “The Renormalization Group: Critical Phenomena and the Kondo Problem,” Rev. Mod. Phys. 47, 773–840.
- Wilson, K. G., 1983, “The Renormalization Group and Critical Phenomena” (1982, Nobel Prize Lecture) Rev. Mod. Phys. 55, 583–600.
- Wilson, K. G., and M. E. Fisher, 1972, “Critical Exponents in 3.99 Dimensions,” Phys. Rev. Lett. 28, 240–243.
- Wilson, K. G., and J. Kogut, 1974, “The Renormalization Group and the ε Expansion,” Phys. Rep. 12, 75–200.
- Zinn, S.-Y., and M. E. Fisher, 1996, “Universal Surface-Tension and Critical-Isotherm Amplitude Ratios in Three Dimensions,” Physica A 226, 168–180.
- Zinn, S.-Y., S.-N. Lai, and M. E. Fisher, 1996, “Renormalized Coupling Constants and Related Amplitude Ratios for Ising Systems,” Phys. Rev. E 54, 1176–1182.