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Modern Physics Letters A

New contributions to physics by Prof. C. N. Yang: 2009–2011

Zhong-Qi Ma

Published Jan 3, 2016• 0 citations
Publisher

Abstract

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.

DOI
10.1142/s0217732316300019

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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.
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.
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.
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.
The exact solution of C.N. Yang's one-dimensional many-body problem with repulsive delta-function interactions and R.J. Baxter's eight-vertex statistical model are brilliant achievements in many-body statistical physics. A nonlinear equation, now known as the Yang-Baxter equation, is the key to the solution of both problems. The Yang-Baxter equation has also come to play an important role in such diverse topics as completely integrable statistical models, conformal and topological field theories, knots and links, braid groups and quantum enveloping algebras. This pioneering textbook attempts to make accessible results in this rapidly-growing area of research. The author presents the mathematical fundamentals at the outset, then develops an intuitive understanding of Hopf algebras, quantisation of Lie bialgebras and quantum enveloping algebras. The historical derivation of the Yang-Baxter equation from statistical models is recounted, and the interpretation and solution of the equation are systematically discussed. Throughout, emphasis is placed on acquiring calculation skills through physical understanding rather than achieving mathematical rigour. Originating from the author's own research experience and lectures, this book will prove both an excellent graduate text and a useful work of reference
The remarkable exact solution of 1D interacting fermions heralded a new era in the study of quantum many-body problems. The mathematical structure of the underlying Yang-Baxter equation led to profound advances in other fields. Recently there has been a significant revival of interest in 1D quantum many-body problems due to the striking and ongoing experimental developments in the study of cold quantum matter. In particular, through the trapping and cooling of interacting systems of atomic bosons and fermions in low dimensions. There is, e.g., already rather spectacular agreement between theory and experiment for 1D interacting bosons. This includes an experimental test of Yang-Yang thermodynamics for bosons on a chip...