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arXiv:2202.02271 · EMS Press eBooks

Quantum number towers for the Hubbard and Holstein models

James K. Freericks

Published Jun 29, 2022• 1 citation
arXivHTMLPDF

Abstract

In 1989, Elliott Lieb published a Physical Review Letter proving two theorems about the Hubbard model. This paper used the concept of spin-reflection positivity to prove that the ground state of the attractive Hubbard model was always a nondegenerate spin singlet and to also prove that the ground state for the repulsive model on a bipartite lattice had spin $\big{|}|\Lambda\_{A}|-|\Lambda\_{B}|\big{|}/2$, corresponding to the difference in number of lattice sites for the two sublattices. In addition, this work relates to quantum number towers—where the minimal energy state with a given quantum number, such as spin, or pseudospin, is ordered, according to the spin or pseudospin values. It was followed up in 1995 by a second paper that extended some of these results to the Holstein model (and more general electron-phonon models). These works prove results about the quantum numbers of these many-body models in condensed matter physics and have been very influential. In this chapter, I will discuss the context for these proofs, what they mean, and the remaining open questions related to the original work. In addition, I will briefly discuss some of the additional work that this methodology inspired.

DOI
10.4171/90-1/16

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In 1989, Elliott Lieb published a Physical Review Letter proving two theorems about the Hubbard model. This paper used the concept of spin-reflection positivity to prove that the ground state of the attractive Hubbard model was always a nondegenerate spin singlet and to also prove that the ground state for the repulsive model on a bipartite lattice had spin ∣∣Λ_A∣−∣Λ_B∣∣/2\big{|}|\Lambda\_{A}|-|\Lambda\_{B}|\big{|}/2​∣Λ_A∣−∣Λ_B∣​/2, corresponding to the difference in number of lattice sites for the two sublattices. In addition, this work relates to quantum number towers—where the minimal energy state with a given quantum number, such as spin, or pseudospin, is ordered, according to the spin or pseudospin values. It was followed up in 1995 by a second paper that extended some of these results to the Holstein model (and more general electron-phonon models). These works prove results about the quantum numbers of these many-body models in condensed matter physics and have been very influential. In this chapter, I will discuss the context for these proofs, what they mean, and the remaining open questions related to the original work. In addition, I will briefly discuss some of the additional work that this methodology inspired.
In 1989, Lieb proved two theorems about the Hubbard model. One showed that the ground state of the attractive model was a spin singlet state (S = 0) was unique and was positive definite. The other showed that the ground state of the repulsive model on a bipartite lattice at half-filling has a total spin given by [Formula: see text](N[Formula: see text]NB)/2[Formula: see text], corresponding to the difference of the number of lattice sites on the two sublattices divided by two. In the mid to late 1990s, Shen extended these proofs to show that the pseudospin of the attractive model was minimal until the electron number equaled 2NA where it became fixed at J = [Formula: see text](N[Formula: see text]NB)/2[Formula: see text] until the filling became 2NB, where it became minimal again. In addition, Shen showed that a spin tower exists for the spin eigenstates for the half-filled case on a bipartite lattice. The spin tower says the minimal energy state with spin S is higher in energy than the minimal energy state with spin S − 1 until we reach the ground-state spin given above. One long-standing conjecture about this model remains, namely, does the attractive model have such a spin tower for all fillings, which would then imply that the repulsive model has minimal pseudopsin in its ground state. While we do not prove this last conjecture, we provide a quick review of this previous work, provide a constructive proof of the pseudospin of the attractive model ground state, and describe the challenges by proving the remaining open conjecture.
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.
In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.
The Hubbard model is one of the most important models of many-body physics. It is a simplified model for how electrons move and interact in a crystal. It consists of a hopping term, where electrons can move between neighboring sites (sometimes more distant neighbors are allowed). When two electrons (of opposite spin) occupy the same lattice site, they feel a Coulomb repulsion. This model can be exactly solved only in one dimension, in the limit of infinite dimensions, or for small systems via exact diagonalization. It has been studied by thousands of physicists over the years. While much is known about the model, even more remains unknown, so it is still a great frontier to examine further. In this chapter, we focus on the known solutions and provide concrete examples and rigorous theorems about the behavior of the model. This includes the symmetries of spin, particle number, and pseudospin, the Nagaoka theorem for ferromagnetism, spin-reflection positivity, and spin and pseudospin towers.
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.
Using a cone-theoretical method, we prove the uniqueness of the ground state for two Bose Hubbard models. The first model is the usual Bose Hubbard model with real hopping coefficients and attractive interactions. The second model is a two-component Bose Hubbard model. Under certain conditions, we show that the ground state in the subspace with particle number N=2nN=2nN=2n(nnn is a positive integer) is unique for both models. For the second model, we show that the ground state has spin along the z-axis Sz=0S^{z}=0Sz=0. When the hopping coefficients are real, it has zero spin quantum number, i.e., it is a singlet. Our proofs work equally well for any arbitrary finite-size lattice.