Renormalization Group and Monte Carlo Approach to Critical

Renormalization Group and Monte Carlo Approach to Critical Phenomena: a talk in memory of Professors Shang-keng Ma (馬上庚) and Yu-Ming Shih (石育民) • Chin-Kun Hu (胡進錕) Institute of Physics, Academia Sinica, Taipei 11529, Taiwan and CYCU • E-mail: huck@phys. sinica. edu. tw • http: //proj 1. sinica. edu. tw/~statphys/ At Dept. Phys. , NTHU, Hsinchu, at 13: 30 on 23 September 2009

Chin-Kun Hu: Education • 1971. 6. BS, Department of Physics, National Tsing Hua University, Taiwan • 1973. 6. MS, Institute of Physics, National Tsing Hua University, Taiwan • 1976. 6. Ph. D. , Institute of Physics, National Tsing Hua University, Taiwan

Chin-Kun Hu: Position held • • 1977. 1. -1978. 5. Associate Professor, Chung Cheng Institute of Technology in Taiwan 1978. 6. -1978. 10. Associate Research Fellow, Precision Instrument Development Center of National Science Council in Taiwan 1978. 11. -1979. 7. Postgraduate Research Physicist, Inst. for Pure and Applied Physical Sciences, Univ. of California at San Diego, USA 1979. 7. -1981. 8. Research Associate, Dept. of Physics and Astronomy, University of Maine, Orono, Maine, USA 1981. 9. -1983. 8. Research Associate, Dept. of Chemistry, University of Toronto, Ontario, Canada 1983. 9. -1984. 8. Visiting Expert, Institute of Physics, Academia Sinica, Taipei, Taiwan 1984. 9. -Present: Research Fellow, Institute of Physics, Academia Sinica, Taipei, Taiwan 1991. 6. -1993. 6. Visiting Scholar, Department of Physics, Harvard Univ. , Cambridge, MA 02138, USA

Publication List - Refereed Journal Papers • • • 1. Y. -Y. Lee, C. -K. Hu, and W. -D. Chen. Some considerations on divergences of Weinberg's lepton model. Chin. J. Phys. (Taipei) 12, 109 -130 (1974). 2. Y. -Y. Lee and C. -K. Hu. Gauge model of leptons involving gauge field of lepton numbers. Proceedings of the National Science Council, ROC, No. 10, part 1, 101 -116 (1977). 3. Y. -Y. Lee and C. -K. Hu. Alternative heavy lepton model for -e events, Phys. Rev. D 17, 2261 -2265 (1978). 4. Y. -M. Shih, D. -C. Jou, C. -K. Pan, W. -S. Lee, W. -D. Chen, H. -M. Huang, C. K. Hu, K. -G. Chen and H. -C. Tseng 1977. On the scaling factor in the renormalization group, Chinese Journal of Physics (Taipei) 15, 236 -242 (1978). 5. Y. -M. Shih, D. -C. Jou, C. -K. Pan, W. -S. Lee, W. -D. Chen, H. -M. Huang, C. K. Hu, K. -G. Chen, and H. -C. Tseng. On the critical phenomena and renormalization group transformations, Chinese Journal of Physics (Taipei) 16, 181 -187 (in Chinese) (1978).

• 6. S. -K. Ma, C. Dasgupta, and C. -K. Hu. Random antiferromagnetic Chain, Phys. Rev. Lett. 43, 1434 -1437 (1979). • 7. Y. -M. Shih, D. -C. Jou, C. -K. Pan, W. -S. Lee, W. -D. Chen, H. -M. Huang, C. -K. Hu, K. -G. Chen, and H. -C. Tseng. Comment on "Variational approximation for renormalization group transformations", Phys. Rev. B 19, 529 -532 (1979). • 8. C. Dasgupta, S. -K. Ma, and C. -K. Hu. Dynamic properties of a spin glass model at low temperatures, Phys. Rev. B 20, 3837 -3849 (1979). • 9. Y. -M. Shih, D. -C. Jou, C. -K. Pan, W. -S. Lee, W. -D. Chen. H. -M. Huang, C. -K. Hu, K. -G. Chen, and H. -C. Tseng. Calculation of free energies for a three dimensional Ising Model by a modified Kadanoff's variational method, Phys. Rev. B 21, 299 -303 (1980). • 10. C. -K. Hu et. al. Lower bound approximation to the free energies of an Ising model of ferrimagnetism on bcc lattice, Proceedings of the National Science Council, ROC 4, No. 4, 359 -363 (1980).

• 11. C. -K. Hu and P. Kleban. Direct calculation of the derivatives of the free energy for Ising models by a modified Kadanoff variational method, J. Computational Phys. 43, 289 -314 (1981). • 12. C. -K. Hu et. al. Lower bound approximation to the free energies of an Ising Model on simple cubic lattice, Canadian J. Phys. 59, 1291 -1295 (1981). • 13. S. -P. Liu, D. -S. Wang, C. -K. Hu et. al. Application of modified Kadanoff's variational method to one dimensional Ising model, Proceeding of the National Science Council, ROC 5 B, No. 2, 119 -124 (1981). • 14. C. -K. Hu and P. Kleban 1982. Phase diagram of an Ising model with random sublatice vacancies, Phys. Rev. B 25, 6760 -6764 (1982). • 15. C. -K. Hu and C. Y. Huang. Cooperative effects in Raman scattering, Optics Comm. 43, 395 -400 (1982).

• 16. C. -K. Hu. Correlated percolation and a sublaticedilute Potts model at T->0, Physica 116 A, 265 -271 (1982). • 17. C. -K. Hu. Bond-correlated percolation model and unusual behaviour of supercooled water, J. Phys. A: Math. Gen. 16, L 321 -326 (1983). • 18. C. -K. Hu. Exact results for a correlated percolation model, Physica 119 A, 609 -614 (1983). • 19. C. -K. Hu. An exact renormalization group transformation and properties of positive symmetric matrices, Chin. J. Phys. (Taipei) 21, Nos. 3 and 4, 24 -30 (1983). • 20. C. -K. Hu. Percolation, clusters, and phase transitions in spin models, Phys. Rev. B 29, 5103 -5108 (1984). • 21. C. -K. Hu. Site-bond-correlated percolation and a sublattice-dilute Potts model at finite temperatures, Phys. Rev. B 29, 5109 -5116 (1984).

• Shang-keng Ma (馬上庚): 24 Sep. 1940 -24 Nov. 1983 • Yu-Ming Shih (石育民): 1942 -2005

Memories of Shang-keng Ma by Prof. F. Y. Wu on 15 Dec. 1983 • Shang-keng was born in 1940 in Chungking. • He obtained his B. S. degree at UC-Berkeley in 1962 with the honor of the "most promising senior, " and continued on to earn a Ph. D. in 1966 at Berkeley under the direction of Professor Kenneth Watson. • He did his postdoctoral work with Professor Keith Brueckner at U. C. San Diego, where he eventually became a full professor in 1975.

Sheng-Keng Ma: some papers during 1979 -1984 • HUANG HM, MA SK, SHIH YM. THE ENTROPY OF ENTANGLED POLYMER, Solid State Commu. 51, 147 (1984). • MA S. CALCULATION OF ENTROPY FROM DATA OF MOTION, J. Stat. Phys. 26, 221 (1981). • MA SK, PAYNE M. ENTROPY OF A SPIN-GLASS MODEL WITH LONGRANGE INTERACTION, Phys. Rev. B 24, 3984 (1981). • MA S. DYNAMICS OF A VECTOR SPIN-GLASS MODEL, Phys. Rev. B 22, 4484 (1980). • DASGUPTA C, MA S. LOW-TEMPERATURE PROPERTIES OF THE RANDOM HEISENBERG ANTI-FERROMAGNETIC CHAIN, Phys. Rev. B 22, 1305 (1980). • DASGUPTA C, MA SK, HU CK. DYNAMIC PROPERTIES OF A SPINGLASS MODEL AT LOW-TEMPERATURES, Phys. Rev. B 20, 3837 (1979). • MA SK, DASGUPTA C, HU CK. RANDOM ANTI-FERROMAGNETIC CHAIN, Phys. Rev. Lett. 43, 1434 (1979).

Other publications • 統計力學 / 馬上庚著 , 臺北市 : 環華出版 事業, 民 71[1982] • Statistical mechanics / Shang-Keng Ma ; translated by M. K. Fung, Philadelphia : World Scientific, c 1985 • Modern theory of critical phenomena / Shang-keng Ma, Reading, Mass. : W. A. Benjamin, Advanced Book Program, 1976

Author=(ma sk or ma s) Timespan=1900 -1984. Databases=SCI-EXPANDED, • Results found: 71 -12=59, Sum of the Times Cited: 7, 222, hindex: 32 -1=31 • 1. RANDOM-FIELD INSTABILITY OF ORDERED STATE OF CONTINUOUS SYMMETRY, IMRY Y, MA S, PRL 35, 1399 (1975) Times Cited: 1, 571 • 13. RANDOM ANTI-FERROMAGNETIC CHAIN, MA SK, DASGUPTA C, HU CK, PRL 43, 1434(1979) Times Cited: 193 • 21. DYNAMIC PROPERTIES OF A SPIN-GLASS MODEL AT LOW-TEMPERATURES, DASGUPTA C, MA SK, HU CK, 20, 3837 -3849(1979) Times Cited: 90.

石育民畢業於台灣師範大學物理系，隨後進入清華大學物理研 究所攻讀碩士及博士。其博士論文是「左右型SU(3)⊕SU(3)對稱 的非線性具像」，指導教授為物理學家李怡巖。 In 1972, Dr. Shih visited Chia-Wei Woo (吳家瑋) and began to work on statistical physics. From 1973 to 1977, they published 11 joined papers. SHIH YM, WOO CW, SURFACE STATES OF HE-3 IN DILUTE HE 3 -HE 4 SOLUTIONS, Phys. Rev. Lett. 30, 478 (1973) LEE FT, TAN HT, SHIH YM, WOO CW, PHASE-DIAGRAM FOR LIQUID-CRYSTALS, Phys. Rev. Lett. 31, 1117 (1973).

物理學系/系所簡介/歷來成員/歷任 系主任 • • • • 主管姓名任職起迄年月 蘇賢錫 1963. 06 -1967. 12 張建邦 1967. 12 -1968. 07 賴其鵬 1968. 08 -1970. 07 石育民 1970. 08 -1971. 07 陳惟堯 1971. 08 -1973. 07 芮涵芝 1973. 08 -1978. 07 石育民 1978. 08 -1980. 07 顏三和1980. 08 -1982. 07 陳偉正 1982. 08 -1988. 07 錢凡之1988. 08 -1994. 07 林震安1994. 08 -1998. 07 彭維鋒 1998. 08 -2002. 07 張經霖 2002. 08 -2006. 07 曾文哲 2006. 08 -迄今

General historical introduction to critical phenomena and lattice models

Guggenheim (1945) Exact solution of 2 D Ising model Onsager(1944)

Statistical physics approach to physical systems: e. g. Ising model for phase transitions Ferromagnet to paramagnet Binary alloy: Cu. Zn Ising model: spin up, +1; spin down, -1

Founders of Statistical Mechanics

Crtitical exponents

Percolation Models

Percolation Model

Books on Percolation • Deutscher, G. ; Zallen, R. ; and Adler, J. (Eds. ). Percolation Structures and Processes. Bristol: Adam Hilger, 1983. • Grimmett, G. Percolation. New York: Springer-Verlag, 1989. • Kesten, H. Percolation Theory for Mathematicians. Boston, MA: Birkhäuser, 1982. • Stauffer, D. and Aharony, A. Introduction to Percolation Theory, 2 nd ed. London: Taylor & Francis, 1992.

Renormalization Group Theory: Historial Background

=2

L. P. Kandanoff’s theory of scaling, Physics 2, 263 (1966) M. E. Fisher, Rev. Mod. Phys (1998)

K. G. Wilson’s RG theory, Phys. Rev. B. 4, 3174 and 3184 (1971) M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998)

From field theory and particle physics to statistical physics

Chin-Kun Hu’s work on particle physics • • April 1976: finished Ph. D. thesis June 1976: Oral examination (July 1976: began military service) (January or February 1977: Annual Meeting of PSROC in Chung Cheng Institute of Technology) • Y. -Y. Lee and C. -K. Hu. Gauge model of leptons involving gauge field of lepton numbers. Proceedings of the National Science Council, ROC, No. 10, part 1, 101116 (1977). • Y. -Y. Lee and C. -K. Hu. Alternative heavy lepton model for -e events, Phys. Rev. D 17, 2261 -2265 (1978).

The Nobel Prize in Physics 1976: "for their pioneering work in the discovery of a heavy elementary particle of a new kind" Burton Richter, b. 1931 Samuel Chao Chung Ting: b. 1936 Chinese article: C. -K. Hu. Particle, in Chinese Learning and Modern Civilization Series 13, Natural Science, pp. 68 -83, Culture University Press, Taipei (1981).

Professor Yu Ming Shih gave me a copy of this paper after annual meeting of the PSROC in early 1977.

The name of this group was proposed by Prof. Shang-Keng Ma

This paper was kept in PRB editorial office for a long time. Prof. Ma wrote a letter to PRB, this paper was then accepted. PRB did not accept Group name as the author, thus all names in the group were listed.

Some papers on RG approach to spin models drafted in 1978 • Y. -M. Shih, D. -C. Jou, C. -K. Pan, W. -S. Lee, W. -D. Chen. H. -M. Huang, C. -K. Hu, K. -G. Chen, and H. -C. Tseng. Calculation of free energies for a three dimensional Ising Model by a modified Kadanoff's variational method, Phys. Rev. B 21, 299 -303 (1980). • C. -K. Hu et. al. Lower bound approximation to the free energies of an Ising model of ferrimagnetism on bcc lattice, Proceedings of the National Science Council, ROC 4, No. 4, 359 -363 (1980). • S. -P. Liu, D. -S. Wang, C. -K. Hu et. al. Application of modified Kadanoff's variational method to one dimensional Ising model, Proceeding of the National Science Council, ROC 5 B, No. 2, 119 -124 (1981).

From Taiwan to U. C. San-Diego

This paper was drafted in Taiwan, then finished and submitted at UCSD.

Monte Carlo approach to dynamic properties of a spin glass at low temperatures On my first visit to Prof. Ma’s office at the end of October 1978, he told me the meanings of ``Monte Carlo method” and “spin glass”.

Metastable state: energy increases by flipping any spin.

Dynamic properties of a spin glass model at low temperatures, Chandan Dasgupta, Shang-keng Ma, Chin-Kun Hu Phys. Rev. B 20, 3837 -3849 (1979).

Random 1 d Antiferromagnetic Quantum Spin Chain

Computing entropy from success rate: unfinished work • • Free Ising spin in external magnetic field. One dimensional Ising model Two dimensional Ising models Three dimensional Ising model

p. p. 221 -240

From UC San-Diego to University of Maine at Orono

For bond-decorated percolation on square lattice: n_c=0. 5.

Other Works • C. -K. Hu and C. Y. Huang. Cooperative effects in Raman scattering, Optics Comm. 43, 395 -400 (1982). • C. -K. Hu and Peter Kleban. A New Criterion for Phenomenological Renormalization Group Transformation, Bull. Am. Phys. Soc. 26, 241 (1982).

From University of Maine to University of Toronto • C. -K. Hu. Correlated percolation and a sublatice-dilute Potts model at T=0, Physica A, 265 -271 (1982). • C. -K. Hu. Exact results for a correlated percolation model, Physica 119 A, 609 -614 (1983). • C. -K. Hu. Bond-correlated percolation model and unusual behaviour of supercooled water, J. Phys. A: Math. Gen. 16, , L 321 -326 (1983).

pp. 5103 -5108 pp. 5109 -5116

RG approach to one-dimensional Ising model • This is a good example to understand RG idea and calculation procedure, see two papers below. • Wei-Tou Ni, Chinese Journal of Physics (CJP) 14, no. 3, 29 (1976). • Chin-Kun Hu, CJP 21, no. 3 and 4, 24 (1983), received on 17 March 1983. This paper considers the contribution to the free energy from a term at high temperature fixed point and thus the convergence to final result is faster.

From University of Toronto to Academia Sinica, Taipei

• H. M. Huang and Y. M. Shih, AN INVESTIGATION OF THE ANTIFERROMAGNETIC CASE OF THE ISING-MODEL BY THE MODIFIED KADANOFF VARIATIONAL RENORMALIZATION METHOD, Phys. Lett. A 78, 93 (1980). • H. M. Huang, Y. M. Shih, SOME FEATURES OF THE 2 PARAMETER MODIFIED KADANOFF LOWER BOUND RENORMALIZATION-GROUP METHOD, Phys. Lett. A, 89, 257 (1982). • H. M. Huang, Y. M. Shih, A NUMERICAL-CALCULATION OF TWO-DIMENSIONAL ISING-MODEL BY MODIFIED KADANOFF RENORMALIZATION-GROUP TRANSFORMATION WITH 2 -PARAMETER GENERATING FUNCTION, NUOVO CIMENTO DELLA SOCIETA ITALIANA DI FISICA D, 2, 1305 (1983).

Finite-size scaling

Histogram Monte Carlo Simulation Method

Histogram MCRG Method

Universal Finite-size Scaling Functions

Potts model and Bond-correlated Percolation Model

RG Method for Potts Model Based on Potts Model < ->Bond Correlated Percolation Model

RG transformational for q-state Potts model

RG Transformation for One-dimensional Lattice

RG Transformation for Two-dimensional Lattice

For one-dimensional q-state Potts model

For two-dimensional q-state Potts model

Backbone of Percolation Cluster

Histogram Monte Carlo RG for q-state Potts model

Summary and Final Comments • Professors Sheng-keng Ma and Yu-Ming Shih made essential contributions to early developments of statistical physics in Taiwan. • Critical many-body systems have simple universal and scaling behavior. • Lattice models are useful for understanding MC, RG, MCRG, etc. • Other related works: analytic calculations of finitesize scaling functions and finite-size corrections for the Ising model and dimer model.