In this talk, we shall briefly review some results on the strongly correlated electron systems, derived recently by applying Lieb's spin-reflection-positivity method. To explain the basic ideas of this method to a wide audience, we emphasize the important role played by Marshall's rule in studying the many-body systems.
To begin with, we consider an one-dimensional quantum mechanical system in details. By applying the variational principle and exploiting the connectivity of region ?, in which the particle moves, we show that the ground state wave function of this system is strictly positive in ? and hence, satisfies Marshall's sign rule. Then, we prove that the similar conclusion can also be established for the antiferromagnetic Heisenberg model after some proper unitary transformation.
Unfortunately, for an itinerant electron model, such as the Hubbard model, this simple sign rule breaks down. However, by applying Lieb's spin-reflection positivity method, one can show that, when the system is half-filled, the expansion coefficients of its ground state wave function can be organized into a positive definite matrix after a proper transformation. Therefore, the ground state wave function of the half-filled Hubbard model also satisfies Marshall's rule in a more subtle way.
Finally, we show that this sign rule leads to several interesting results, such as the ferromagnetic long-range order in the half-filled Hubbard model on a certain type of lattices and some general relations satisfied by the charged and the spin-excitation gaps.