In this article we discuss the reflection positivity in spin space for the Heisenberg model and the Kondo Hamiltonian. We show that the ground state is unique for J ≠0 where J is the strength of the exchange interaction in the Heisenberg model or the exchange coupling between the localized spins and conduction electrons in the Kondo model. We introduce the Coulomb interaction U >0 between the conduction lectrons for the Kondo Hamiltonian to show the uniqueness of the ground state. We write the Hamiltonian in terms of fermions to express the exchange interactions and impose constraints. We employ the Lagrange-multiplier method to deal with the constraints in this work.
In this article we discuss the reflection positivity in spin space for the Heisenberg model and the Kondo Hamiltonian. We show that the ground state is unique for J ≠0 where J is the strength of the exchange interaction in the Heisenberg model or the exchange coupling between the localized spins and conduction electrons in the Kondo model. We introduce the Coulomb interaction U >0 between the conduction lectrons for the Kondo Hamiltonian to show the uniqueness of the ground state. We write the Hamiltonian in terms of fermions to express the exchange interactions and impose constraints. We employ the Lagrange-multiplier method to deal with the constraints in this work.
We examine the spin-reflection positivity of the ground state of the Kondo lattice model at half-filling with the antiferromagnetic and ferromagnetic exchange couplings J=0. For every positive U>0, where U is the Coulomb interaction between the conduction electrons, we can show that the ground state is unique.
We investigate the ground state of the Kondo-Hubbard model employing the recently introduced idea of spin-reflection positivity. It is shown exactly that the Kondo-Hubbard model for J<0 and U>∣J∣/4 at half filling has a unique ground state which can be ferromagnetic or singlet according to the lattice structure. In particular, for the Kondo lattice with U>∣J∣/4, the ground state is a singlet. We also show a theorem for the Kondo-Hubbard model with the anisotropic antiferromagnetic coupling ${J}_{\ensuremath{\perp}}>0$ and ${J}_{z}\ensuremath{\le}0$ for U>∣Jz∣/4.
The density-matrix renormalization-group method is applied to the one-dimensional Kondo-lattice model with the Coulomb interaction between the conduction electrons. The spin and charge gaps are calculated as a function of the exchange constant J and the Coulomb interaction Uc. It is shown that both the spin and charge gaps increase with increasing J and Uc. The spin gap vanishes in the limit of $J\ensuremath{\rightarrow}0$ for any Uc with an exponential form, ${\ensuremath{\Delta}}_{s}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\frac{1}{\ensuremath{\alpha}({U}_{c})J\ensuremath{\rho}}]$. The exponent, $\ensuremath{\alpha}({U}_{c})$, is determined as a function of Uc. The charge gap is generally much larger than the spin gap. In the limit of $J\ensuremath{\rightarrow}0$, the charge gap vanishes as ${\ensuremath{\Delta}}_{c}=\frac{1}{2}J$ for Uc=0 but for a finite Uc it tends to a finite value, which is the charge gap of the Hubbard model.
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.
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.
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,