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Science China Mathematics

Application of Product Operator Formalism to the Strongly Coupled Spin (I=1/2) Systems

Miao Xi, Jie Hu

Published Jan 1, 1993• 0 citations
OpenAlex

Abstract

Product operator formalism has been developed to evaluate, in closed analytical form, the time evolution for strongly coupled spin (I=1/2) systems. This formalismis based on two facts: (ⅰ) the Hamiltonian for a strongly coupled spin system is a zero-quantum operator and when an arbitrary zero-quantum operator acts on a p-quantum operator, what is yielded is still a p-quantum operator and can be expressed in terms of a linear combination of a complete p-quantum operator base set of the spin system; (ⅱ) the zeroquantum and unitary transformation leads the Hamiltonian to be only a linear combination of zero-quantum longitudinal magnetization and spin order operators. Thus the time evolution for the spin systems can be evaluated in closed analytical form. The formalism retains completely the original character of the product operator formalism and enlarges its applicability. It can deal with both strongly and weakly coupled spin (I=1/2) systems in united and closed analytical forms.

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Product operator formalism has been developed to evaluate, in closed analytical form, the time evolution for strongly coupled spin (I=1/2) systems. This formalismis based on two facts: (ⅰ) the Hamiltonian for a strongly coupled spin system is a zero-quantum operator and when an arbitrary zero-quantum operator acts on a p-quantum operator, what is yielded is still a p-quantum operator and can be expressed in terms of a linear combination of a complete p-quantum operator base set of the spin system; (ⅱ) the zeroquantum and unitary transformation leads the Hamiltonian to be only a linear combination of zero-quantum longitudinal magnetization and spin order operators. Thus the time evolution for the spin systems can be evaluated in closed analytical form. The formalism retains completely the original character of the product operator formalism and enlarges its applicability. It can deal with both strongly and weakly coupled spin (I=1/2) systems in united and closed analytical forms.
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.
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The state of quantum systems, their energetics, and their time evolution is modeled by abstract operators. How can one visualize such operators for coupled spin systems? A general approach is presented that consists of several shapes representing linear combinations of spherical harmonics. It is applicable to an arbitrary number of spins and can be interpreted as a generalization of Wigner functions. The corresponding visualization transforms naturally under nonselective spin rotations as well as spin permutations. Examples and applications are illustrated for the case of three spins 1/2.
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Unitary transformations play a fundamental role in many-body physics, and except for special cases, they are not expressible in closed form. We present closed-form expressions for unitary transformations generated by a single fermionic operator for Hermitian and anti-Hermitian generators. We demonstrate the usefulness of these expressions in formal analyses of unitary transformations and numerical applications to Hamiltonian downfolding in quantum computing and Heisenberg dynamics. This work paves the way for new analytical treatments of unitary transformations and numerical many-body methods for fermions.