# Lieb’s spin-reflection-positivity method, Lieb lattice and all that

### Guang-Shan Tian

Peking University, Beijing, China

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## Abstract

In the present chapter we review some rigorous results derived with Lieb’s spin-reflectionpositivity
method in a pedagogical manner. To begin with, we emphasize the important role
played by the sign rule satisfied by the ground-state wave function of a quantum many-body
system. For example, we study some *localized* spin systems, such as the antiferromagnetic
Heisenberg model, in detail. Then we argue why such a rule is difficult to establish for the
ground-state wave functions of *itinerant* electron models. By using the spin-reflection-positivity
method, we show that this problem can be partially remedied. A weaker form of the sign rule
is uncovered. It enables us to prove several inequalities satisfied by either an electron pairing
correlation function or a magnetic correlation function of the Hubbard model on a bipartite
lattice. These inequalities imply that, if a bipartite lattice has macroscopically unequal numbers
of sublattice sites, then the Hubbard model at half-filling is either a supersolid or a ferrimagnet.
Similar inequalities can also be proven for the periodic Anderson model and the Kondo lattice
model. Finally, by applying a generalized version of the spin-reflection-positivity method, we
further establish an inequality satisfied by the charged and spin gaps of these models. Based on
them, we make some audacious conjectures, and then we summarize this chapter.