Quantum Spin Systems and Their Operator Algebras

Huzihiro Araki

Masahico Saito

David Pierre Ruelle

The language of operator algebras is of great help for the formulation of questions and answers in quantum statistical mechanics. In Chapter 1 we present a minimal mathematical introduction to operator algebras, with physical applications in mind. In Chapter 2 we study some questions related to the quantum statistical mechanics of spin systems, with particular attention to the time evolution of infinite systems. The basic reference for these two chapters is Bratteli-Robinson: Operator algebras and quantum statistical mechanics I, II. In Chapter 3 we discuss the nonequilibrium statistical mechanics of quantum spin systems, as it is currently being developped.

Xijia Miao

A general operator algebra formalism is proposed for describing the unitary time evolution of multilevel spin systems. The time-evolutional propagator of a multilevel spin system is decomposed completely into a product of a series of elementary propagators. Then the unitary time evolution of the system can be determined exactly through the decomposed propagator. This decomposition may be simplified with the help of the properties of the finite dimensional Liouville operator space and of its three operator subspaces, and the operator algebra structure of spin Hamiltonian of the system. The Liouville operator space contains the even-order multiple-quantum, the zero-quantum, and the longitudinal magnetization and spin order operator subspace, and moreover, each former subspace contains its following subspaces. The propagator can be decomposed readily and completely for a spin system whose Hamiltonian is a member of the longitudinal magnetization and spin order operator subspace. If the Hamiltonian of a spin system is a zero-quantum operator this decomposition may be implemented by making a zero-quantum unitary transformation on the Hamiltonian to convert it into the diagonalized Hamiltonian, while if the Hamiltonian is an even-order multiple-quantum operator the decomposition may be carried out by diagonalizing the Hamiltonian with an even-order multiple-quantum unitary transformation. When the Hamiltonian is a member of the Liouville operator space but not any element of its three subspaces the decomposition may be achieved first by making an odd-order multiple-quantum and then an even-order multiple-quantum unitary transformation to convert it into the diagonalized Hamiltonian. Parameter equations to determine the unknown parameters in the decomposed propagator are derived for the general case and approaches to solve the equations are proposed.

Gunter Markus Schutz

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Using the Bethe ansatz and similarity transformations this yields new exact results. In a complementary approach we generalize previous work and present a new description of these and other processes and the related quantum chains in terms of an operator algebra with quadratic relations. The full solution of the master equation of the process is thus turned into the problem of finding representations of this algebra. We find a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. We obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal boundary fields.

Hal Tasaki