Theory of spin-reflection positivity developed in recent years is reviewed. This theory makes use of symmetries in an electron system and theory of matrix to investigate the ground state properties. Existences of anti- and ferromagnetic long-range orders in itinerant electron systems, and of off-diagonal long-range order are two successful applications of the theory. In this article, the author attempt to summarize exact results proved by utilizing this theory and related topics. First a general theory and basic theorems are introduced. Second, based on the band structures of conduction electrons, existences of a singlet state with strongly antiferromagnetic correlation, a state with both anti- and ferromagnetic long-range orders, and a fully saturated ferromagnetic state are proved. The theory is applied to several of the main theoretical models for strongly correlated electron systems, such as the Heisenberg model, the Hubbard model, the Anderson model, the single- and multichannel Kondo model, and the generalized Hubbard model, and a series of rigorous results are found in these models. Third, it is proved that off-diagonal long-range order and charge-density wave exist in the ground states of the attractive Hubbard model and the generalized Hubbard model. A relation between pseudospin symmetry and the uniform density theorem is introduced. Fourth, the theory is applied successfully to explain experimental observations of oscillatory interlayer magnetic coupling in ultrathin magnetic films. Finally several unsolved problems are discussed. All results introduced in this article are mathematically exact.
The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction
U is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``
CuO2 lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``
d orbitals'' occupy the vertices of the squares, while ``
p orbitals'' lie halfway between two
d orbitals; both
d and
p orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of
U and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case,
Ud=Up, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all
U. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases
Up=0,Ud=U and
Up=U,Ud=0. We found that the different environments of
d and
p sites lead to a ferromagnetic insulator when
Ud=0; by contrast,
Up=0 leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each
d site.
We prove rigourously that the ground state of a class of interacting Majorana fermion models and its equivalent spinless fermion models is either unique or doubly degenerate if the lattice size N is even, and is always doubly degenerate if N is odd. Our proof is based on the reflection positivity of the system in the eigen-operator representation and holds in all dimensions with arbitrary lattice structures. Understanding of physical properties of strongly correlated quantum lattice models has long been a challenging problem. Because of the subtlety of interactions, it is generally difficult to solve these models. Rigorous results and exact solutions are clearly useful bench marks, but these are rare. In 1989, Lieb proved that the ground state of the Hubbard model is unique by utilizing a Perron-Frobenius-type argument and the spinreflection positivity of the system[1]. This extended the proof for the uniqueness of the ground state from an antiferromagnetic Heisenberg model[2] to an interacting fermion system and led to a more general application of reflection positivity in quantum flux phases[3], frustrated Heisenberg antiferromagnets[4], and other quantum lattice models[5]. The spin-reflection positivity is intimately connected with the sign rule of the groundstate wave function, which can be used to understand the minus sign problem caused by the fermion characteristics of itinerant electrons in quantum Monte Carlo simulations[5]. In this letter, we present a rigorous proof for two theorems on the degeneracy of ground states for a class of interacting quantum lattice models of two-species Majorana fermions. We will show that the ground state is doubly degenerate when the lattice size N is odd and at most doubly degenerate when N is even. Our proof utilizes a new kind of reflection positivity between the two-species Majorana fermions. These theorems can be applied to a number of interacting spinless fermion models which are equivalent to the Majorana fermion model. It sheds light on the understanding of topological degeneracy and symmetry-protected topological phases of Majorana fermions[6, 7], and the minus sign problem in an interacting spinless fermion model that was recently studied by a number of groups[8–10]. The model studied here is defined on a lattice with N sites,
In this paper, we study a revised definition c of the charged gap for the strongly correlated electron models on small clusters, proposed by Nishino, in a mathematically rigorous way. By applying a simplified version of Lieb's spin-reflection-positivity method, we show that this quantity is always positive for the half-filled Hubbard model, the periodic Anderson model and the Kondo lattice model. We also establish a model-dependent lower bound to the charged gap. Our results show explicitly the role played by electron repulsion in opening up a nonvanishing charged gap of a cluster.