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arXiv:1904.02283 · International Journal of Modern Physics B

Professor Chen Ping Yang’s early significant contributions to mathematical physics

Xi-Wen Guan, Feng He

Published Mar 7, 2019• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1142/s0217979219300020

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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.
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.
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.
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.
As early as the 1950s, Prof. Yang and his collaborators realized that the most important interaction effects in a dilute quantum gas can be described by the s-wave scattering length between particles. This insight leads to universal descriptions of the interaction effects without the detailed knowledge of the interaction potential. They derived a formula expanding the energy density in terms of the gas parameter. This formula is later known as the Lee-Huang-Yang correction. However, it took forty years for the experimentalists to overcome several challenges and finally achieve degenerate quantum gases of atoms in 1995. The developments after 1995 have led to an exciting field known as “quantum gases” or “ultracold atomic gases”. The ultracold atom system has flexible tunability, allowing both the scattering length and the dimensionality to vary. The Lee-Huang-Yang corrections were observed from several experiments on ultracold atoms by increasing the scattering length. In addition, by reducing the dimensionality to one-dimension, several experiments on ultracold atoms have confirmed the Yang-Yang thermodynamics for one-dimensional bosons that Prof. Yang obtained in the 1960s and the large-N limit of one-dimensional fermions that Prof. Yang obtained around 2010. By increasing the dimensionality through using the idea of synthetic dimension, the experiment on ultracold atoms has also demonstrated the Yang monopole in the SU(2) non-abelian gauge field proposed by Prof. Yang in the 1970s. All of these experiments show the long-lasting impact of Prof. Yang’s theoretical work over several decades.
Using Bethe's hypothesis, C N Yang exactly solved the one-dimensional (1D) delta-function interacting spin-1/2 Fermi gas with an arbitrary spin-imbalance in 1967. At that time, using a different method, M Gaudin solved the problem of interacting fermions in a spin-balanced case. Later, the 1D delta-function interacting fermion problem was named as the Yang-Gaudin model. It has been in general agreed that a key discovery of C N Yang's work was the cubic matrix equation for the solvability conditions. % This equation was later independently found by R J Baxter for commuting transfer matrices of 2D exactly solvable vertex models. % The equation has since been referred to Yang-Baxter equation, being the master equation to integrability. % The Yang-Baxter equation has been used to solve a wide range of 1D many-body problems in physics, such as 1D Hubbard model, SU(N)SU(N)SU(N) Fermi gases, Kondo impurity problem and strongly correlated electronic systems etc. % In this paper, we will briefly discuss recent developments of the Yang-Gaudin model on several breakthroughs of many-body phenomena, ranging from the universal thermodynamics to the Luttigner liquid, the spin charge separation, the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like pairing state and the quantum criticality. % These developments demonstrate that the Yang-Gaudin model has laid out a profound legacy of the Yang-Baxter equation.
This article presents an elementary introduction on various aspects of the prototypical integrable model the Lieb–Liniger Bose gas ranging from the cooperative to the collective features of many-body phenomena. In 1963, Lieb and Liniger first solved this quantum field theory many-body problem using Bethe's hypothesis, i.e., a particular form of wavefunction introduced by Bethe in solving the one-dimensional Heisenberg model in 1931. Despite the Lieb–Liniger model is arguably the simplest exactly solvable model, it exhibits rich quantum many-body physics in terms of the aspects of mathematical integrability and physical universality. Moreover, the Yang–Yang grand canonical ensemble description for the model provides us with a deep understanding of quantum statistics, thermodynamics, and quantum critical phenomena at the many-body physical level. Recently, such fundamental physics of this exactly solved model has been attracting growing interest in experiments. Since 2004, there have been more than 20 experimental papers that reported novel observations of different physical aspects of the Lieb–Liniger model in the laboratory. So far the observed results are in excellent agreement with results obtained using the analysis of this simplest exactly solved model. Those experimental observations reveal the unique beauty of integrability.