Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems
DOI: 10.1063/1.2117827
In any field of physics, the discovery of an exact solution often brings about deep insights and fundamentally new concepts and methods. For example, the Schwarzschild solution of Einstein’s equations revealed the existence of black holes, and the Onsager solution of the two-dimensional Ising model proved the importance of fluctuations close to the critical point. Yet many theorists—rather than viewing exact solutions as important tools in their kit—consider them too unwieldy and difficult to obtain. This attitude is quite surprising since the alternative, the use of approximate methods, is not necessarily easier. History shows that several of the many-body models that were exactly solved were also studied by other methods that in many instances were more complicated and usually uncontrollable.
Beautiful Models: 70 Years of Exactly Solved Quantum Many-Body Problems by Bill Sutherland is a beautifully written book that aims to unveil some of the mystery behind the exact solutions and allow any advanced student access to this vast and fertile field. Having made his own fundamental contributions, Sutherland has developed a deep understanding of the subject and the ability to explain the main ideas in clear physical terms. Although the book involves a good amount of sophisticated mathematics, the author leads the reader with a sure hand through the forest of details.
In the first two chapters of the book, Sutherland outlines the ideas underlying the solvability of many-body quantum Hamiltonians. He emphasizes the nondiffractive nature of the scattering of the particles in the system and relates the phenomenon to the presence of conserved quantities. Subsequently, in chapter 3, he details the construction of the eigenstates. These eigenstates typically take a simple form (the BetheAnsatz), a consequence of the constraints imposed on the dynamics by the conserved quantities.
The solution of the one-dimensional Heisenberg Hamiltonian is presented in great detail in chapter 6. It is the model in which Hans Bethe, in 1931, first introduced the Bethe Ansatz, which in turn led to a characterization of the ground state and excitations and—many years later, through the fundamental contributions of C. N. Yang, C. P. Yang, Minoru Takahashi, and Michel Gaudin—to the full determination of thermodynamics.
My favorite part of the book, chapters 8 through 10, deals with Calogero—Sutherland models describing particles that interact via long-range exchange interactions. These models, not as heavily studied as the Heisenberg or Hubbard model, are fascinating in their own right and require significant generalizations of the standard methods. The book provides a welcome introduction presented with verve and clarity.
My only regret and criticism is that too much is left out of this slim monograph. But perhaps the omissions reflect the author’s intent to make Beautiful Models accessible rather than exhaustive.
More about the Authors
Natan Andrei. Rutgers University, Piscataway, New Jersey, US .
