Ring exchange and the Heisenberg and Hubbard models

Abstract

We study ring-exchange or cyclic-permutation correlations in one-dimensional quantum spin-half systems. For the Heisenberg model we show numerically that these correlations decay as Rn approximately 1/ square root n, although we can deduce nothing about any possible important logarithmic corrections. As such, ring-exchange correlations are much longer range than the more commonly considered spin-spin correlation functions. By considering the relationship between solitonic excitations and cyclic permutations, we suggest a way to predict the value of J2/J1 at which the phase transition between a gapped and gapless phase occurs in the next-nearest-neighbour Heisenberg model, suggesting J2=4J1 as the exact transition point. For the Hubbard model with a spin-charge-separated solution, we show that the occupation number, nk, is a 'convolution' of the cyclic-permutation correlations of the spin ground state with the anyonic occupation number of the charge ground state, with the integration being over statistical phase. We deduce the one-eighth singularity previously found for the U= infinity Hubbard model using this new route. We show that for the limit where nearest-neighbour hopping dominates longer-range hopping in the U= infinity Hubbard model, the single-particle correlation function, nk, for an infinitesimal concentration of holes in a half-filled system, is identical to the Fourier transform of the cyclic-exchange correlations of the corresponding spin wavefunction. For the elementary t1-t2 model, we show a relationship between the singularity which occurs at the Fermi surface, the so-called Luttinger-liquid singularity, and the long-range Heisenberg-model cyclic-permutation correlations.

DMRG study of ferromagnetism in a one-dimensional Hubbard model

“…In the following we express all energies in units of t 1 = 1 and only consider negative values of t 2 . Because a definite order of the particles is no longer enforced when t 2 = 0, the Lieb-Mattis theorem does not apply and, indeed, ferromagnetism has analytically been shown to exist at U = ∞ in three different limits: For one hole in a half-filled band Nagaoka ferromagnetism has been found [4]; for |t 2 | → 0, it has been shown [5] that the model is ferromagnetic for all densities; and for |t 2 | > 0.25, where the band structure has two minima, Müller-Hartmann [6] has shown that the low density limit is ferromagnetic. These three limits are indicated in the schematic phase diagram shown in Fig.…”

Variational cluster approach to ferromagnetism in infinite dimensions and in one-dimensional chains

“…Including a next-nearest-neighbor hopping t 2 , however, ferromagnetism is proven to exist for U = ∞ in the limit t 2 → 0 (t 2 < 0) for all densities. [67][68][69] This limit has to be contrasted to the limit t 1 = 0, but finite t 2 (two-chain model) where the Lieb-Mattis theorem applies again. In the low density limit, the ground state as obtained by the dynamical impurity approximation (DIA, blue).…”

From Nagaoka's Ferromagnetism to Flat-Band Ferromagnetism and Beyond: An Introduction to Ferromagnetism in the Hubbard Model

“…, N s }, and its hopping amplitudes are defined by t x,x+1 = t x+1,x = −t, t x,x+2 = t x+2,x = t ′ for all x ∈ Λ and t x,y = 0 otherwise. From a first order perturbation theory, it was suggested [66,67] that the model with U = ∞ exhibits ferromagnetism if t ′ > 0. By considering a continuum limit theory, Müller-Hartmann [68] argued that the model exhibits metallic ferromagnetism when 4t ′ > |t| > 0, U = ∞, and the electron density is sufficiently low.…”

“…In a fermion system on a finite lattice, a formal perturbation series alway converges because the operators are finite dimensional. Then one might think that the first order perturbation theories of[66,67,70] imply weak rigorous results that a finite model exhibits ferromagnetism for sufficiently small t ′ /t (or t /t ⊥ ). However, this is not the case since there are no estimates of the energy difference between the ground state and the first excited state, and there remains a possibility that the higher order perturbations change the nature of the ground state for any small values of the expansion parameter.…”

“…One does not face this competition in first order perturbation theory[66,67] 65. Take Λ ′ in Appendix A as the set of even sites.…”