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://proj1.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, 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 5B, 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 sublatice-dilute Potts model at T->0, Physica 116A, 265-271 (1982).

• 17.C.-K. Hu. Bond-correlated percolation model and unusual behaviour of supercooled water, J. Phys. A: Math. Gen. 16, L321-326 (1983).

• 18.C.-K. Hu. Exact results for a correlated percolation model, Physica 119A, 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 19

62 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 LONG-RANGE 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 SPIN-GL

ASS MODEL AT LOW-TEMPERATURES, Phys. Rev. B 20, 3837 (1979).• MA SK, DASGUPTA C, HU CK. RANDOM ANTI-FERROMAGNETIC CH

AIN, 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, c1985

• Modern theory of critical phenomena / Shang-keng Ma, Reading, Mass. : W. A. Benjamin, Advanced Book Program, 1976

劉源俊 : 物理教育從名詞談起

• 馬上庚主張「科學中文化」，他在《統計力學》（民 71 年）的序言中寫道：「我深信基礎科學，如不用本國文字，一定說不清。」他又說「英文名詞用法不合理之處甚多，習慣亦多有不合用於中文的」，於是將 boson 名為「合群粒子」， fermion 名為「不合群粒子」；這應算是經典之作。

Author=(ma sk or ma s)Timespan=1900-1984. Databases=SCI-EXPANDED,

• Results found:71-12=59, Sum of the Times Cited:7,222, h-index: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.

石育民昨通過口試獲授理學博士學位 , 為我國第一位理學博士 （ 1

971-02-02/ 聯合報 /02 版）• 本報訊• 國立清華大學物理研究所博士班研究生石育民，昨

（一）日通過教育部的博士學位口試，成為我國第一位理學博士石育民的博士論文為「左右型 SU(3) SU⊕(3) 對稱的非線性具像」，論文指導教授為李怡嚴。

• 石育民博士今年廿九歲，台灣省南投縣人，師範大學理化系畢業、清華大學物理研究所碩士班五十五年畢業。

• 昨日石育民博士學位評定考試，是由吳大猷擔任主席，考試委員有戴運軌、黃振麟、林其文、沈君山、蔣亨進、李怡嚴等六位教授，經全票通過授予理學博士

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。

SHIH YM, WOO CW, SURFACE STATES OF HE-3 IN DILUTE HE3-HE4 SOLUTIONS, Phys. Rev. Lett. 30, 478 (1973)

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.

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 introductionto critical phenomena and lattice models

Guggenheim (1945)

Guggenheim (1945)

)(-~c

Exact solution of 2D Ising model Onsager(1944)

Exact solution of 2D Ising model Onsager(1944)

Statistical physics approach to physical systems:

e.g. Ising model for phase transitions

Binary alloy: CuZnFerromagnet to paramagnet

Ising model: spin up, +1; spin down, -1

Founders of Statistical Mechanics

Crtitical exponents

Percolation Models

Percolation ModelPercolation 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, 2nd 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)

M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998)

K. G. Wilson’s RG theory, Phys. Rev. B. 4, 3174 and 3184 (1971)

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 involvin

g gauge field of lepton numbers. Proceedings of the National Science Council, ROC, No.10, part 1, 101-116 (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

Dept. Phys., NTHU 博士班系友

• 石育民 張坤樹 朱時宜 魏元勳 周東川（無聯絡資料） 侯萬壽 林家綬（無聯絡資料） 陳文典 胡進錕 陳國鎮 李文忠 陳浚楚 王明建 溫增明 蕭憲彥 蘇懿賢 林其隆 江進福 林美智 林達觀 李進榮 黃文宏 曾玄哲

• 婁祥麟 陳明克 楊淳青 陳龍銘 曾祥器 曾錦清 黃定維 陳永忠 林宏基 湯茂竹 周建和 徐紹維 嚴祖強 戴明鳳 呂孝文 劉偉強 林泰生 謝澄銀 陳企寧 蔡秀芬 潘小晞 陳俊君 吳慶軍 洪雪行 黃建華 杜全成 黃茂在 蔡裕勝 郭西川 楊明峰 林明發 邱木榮 杜明憲 江啟文 方正明 周哲仲 葉賢基 林財鈺

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 5B, 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 temperaturesOn 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 HuPhys. Rev. B 20, 3837-3849 (1979).

Random 1d 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, 26

5-271 (1982).• C.-K. Hu. Exact results for a correlated percolation model, Physica 119A, 609-614 (198

3).• C.-K. Hu. Bond-correlated percolation model and unusual behaviour of supercooled wate

r, J. Phys. A: Math. Gen. 16,, L321-326 (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

• 1971至 1976 年，筆者在李怡嚴教授指導下在新竹市國立清華大學物理研究所研究基本粒子理論，畢業後至中正理工學院服預官役並開始與石育民、陳文典、曾玄哲等教授用重整群理論 (renormalzation group theory) 研究相變模型的臨界現象。服完役後，於 1978 年 10月底至加州大學聖地牙哥校區加入馬上庚教授的研究群，以蒙地卡羅模擬法研究自旋玻璃與易行模型 (Ising model)並以重整群法研究量子自旋模型。 1979 年夏天至 1981 年夏天在緬因州立大學用重整群法研究表面吸附原子之相變並開始研究展透 (percolation)模型。 1981 年 9月至 1983 年 8月在多倫多大學研究展透模型和超冷水的模型。 1983 年 9月至中央研究院物理所任職後研究的間題包括發展展透重整群算法、新的蒙地卡羅算法、相變模型之普適尺度函數、非線性動力學、同步現象等問題。 1998 年起開始用蒙地卡羅法研究蛋白質， 2002 年起開始用解析方法研究 DNA 和病毒演化模型。

• H. M. Huang and Y. M. Shih, AN INVESTIGATION OF THE ANTI-FERROMAGNETIC 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 onPotts 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 forq-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 finite-size scaling functions and finite-size corrections for the Ising model and dimer model.

感謝 • 國立清華大學教師 ,尤其是李怡嚴教授• 石育民教授 (1942-2005)

• 馬上庚教授 (24 Sep. 1940-24 Nov. 1983)• 共同研究者 • Grants: National Science Council, Academia Sinic

a.