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arXiv:1010.1838 · International Journal of Modern Physics B

PROFESSOR C. N. YANG AND STATISTICAL MECHANICS

Fa Yueh Wu

Published May 10, 2008• 14 citations
arXivHTMLPDF

Abstract

Professor Chen Ning Yang has made seminal and influential contributions in many different areas in theoretical physics. This talk focuses on his contributions in statistical mechanics, a field in which Professor Yang has held a continual interest for over sixty years. His Master's thesis was on the theory of binary alloys with multi-site interactions, some 30 years before others studied the problem. Likewise, his other works opened the door and led to subsequent developments in many areas of modern day statistical mechanics and mathematical physics. He made seminal contributions in a wide array of topics, ranging from the fundamental theory of phase transitions, the Ising model, Heisenberg spin chains, lattice models, and the Yang–Baxter equation, to the emergence of Yangian in quantum groups. These topics and their ramifications will be discussed in this talk.

DOI
10.1142/s0217979208039198

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Professor Chen Ning Yang has made seminal and influential contributions in many different areas in theoretical physics. This talk focuses on his contributions in statistical mechanics, a field in which Professor Yang has held a continual interest for over sixty years. His Master's thesis was on the theory of binary alloys with multi-site interactions, some 30 years before others studied the problem. Likewise, his other works opened the door and led to subsequent developments in many areas of modern day statistical mechanics and mathematical physics. He made seminal contributions in a wide array of topics, ranging from the fundamental theory of phase transitions, the Ising model, Heisenberg spin chains, lattice models, and the Yang–Baxter equation, to the emergence of Yangian in quantum groups. These topics and their ramifications will be discussed in this talk.
In a seminal paper of 1967, Professor Chen Ning Yang found the full solution of the one-dimensional Fermi gas with a repulsive delta function interaction by using the Bethe ansatz and group theory. This work with a brilliant discovery of the Yang–Baxter equation has been inspiring new developments in mathematical physics, statistical physics, and many-body physics. Based on experimental developments in simulating many-body physics of one-dimensional systems of ultracold atoms, during a period from 2009 to 2011, Prof. Yang published seven papers on the exact properties of the ground state of bosonic and fermionic atoms with the repulsive delta function interaction and a confined potential to one dimension. Here I would like to share my experience in doing research work fortunately under the direct supervision of Prof. Yang in that period.
In the 60s Professor Chen Ping Yang with Professor Chen Ning Yang published several seminal papers on the study of Bethe’s hypothesis for various problems of physics. The works on the lattice gas model, critical behavior in liquid–gas transition, the one-dimensional (1D) Heisenberg spin chain, and the thermodynamics of 1D delta-function interacting bosons are significantly important and influential in the fields of mathematical physics and statistical mechanics. In particular, the work on the 1D Heisenberg spin chain led to subsequent developments in many problems using Bethe’s hypothesis. The method which Yang and Yang proposed to treat the thermodynamics of the 1D system of bosons with a delta-function interaction leads to significant applications in a wide range of problems in quantum statistical mechanics. The Yang and Yang thermodynamics has found beautiful experimental verifications in recent years.
The Conference on Statistical Physics, High Energy, Condensed Matter and Mathematical Physics was held in honor of Professor Chen-Ning Yang's 85th birthday in Singapore in Oct–Nov 2007. The conference paid tribute to the breadth and depth of Professor Yang's achievements in physics and science education since he received his Nobel Prize in Physics fifty years ago.This notable birthday volume is a collection of the presentations made at the conference by many eminent scientists who had worked closely with him or who have been influenced to some extent by his work.
In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of nonlinear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts model and the use of Rogers–Ramanujan identities to generalize our concepts of Bose/Fermi statistics. Each of these advances has led to the further discovery of major unsolved problems of great mathematical and physical interest. I will here discuss the mathematical advances, the physical insights and extraordinary lack of visibility of this field of physics.
Yang and Yang in 1969 [J. Math. Phys. 10, 1115 (1969)] for the first time proposed a rigorous approach to the thermodynamics of the one-dimensional system of bosons with a delta-function interaction. This paper was a breakthrough in exact statistical mechanics, after Yang [Phys. Rev. Lett. 19, 1312 (1967)] published his seminal work on the discovery of the Yang–Baxter equation in 1967. Yang and Yang’s brilliant method yields significant applications in a wide range of fields of physics. In this paper, we briefly introduce the method of the Yang–Yang equilibrium statistical mechanics and demonstrate a fundamental application of the Yang–Yang method for the study of thermodynamics of the Lieb–Liniger model with strong and weak interactions in a whole temperature regime. We also consider the equivalence between the Yang–Yang’s thermodynamic Bethe ansatz equation and the thermodynamics of the ideal gas with the Haldane’s generalized exclusion statistics.
Including topics not traditionally covered in literature, such as (1+1)-dimensional QFT and classical 2D Coulomb gases, this book considers a wide range of models and demonstrates a number of situations to which they can be applied. Beginning with a treatise of nonrelativistic 1D continuum Fermi and Bose quantum gases of identical spinless particles, the book describes the quantum inverse scattering method and the analysis of the related Yang–Baxter equation and integrable quantum Heisenberg models. It also discusses systems within condensed matter physics, the complete solution of the sine-Gordon model and modern trends in the thermodynamic Bethe ansatz. Each chapter concludes with problems and solutions to help consolidate the reader's understanding of the theory and its applications. Basic knowledge of quantum mechanics and equilibrium statistical physics is assumed, making this book suitable for graduate students and researchers in statistical physics, quantum mechanics and mathematical and theoretical physics.