The need for suitable many- or infinite-fermion correlation functions to describe strongly correlated systems is discussed, and the question linked to the need for a correlated basis, in which the ground state may be positive definite for certain low-dimensional geometries. In seeking a positive-definite basis a particular trial basis is proposed, based on that for hard-core bosons in pure one-dimensional systems. Single-particle correlations in this basis are evaluated for the case of the ground state of a quasi-1D Hubbard model in the limit of extreme correlation. The model is a strip of the 2D square lattice wrapped around a cylinder, and is related to a ladder geometry with periodic boundary conditions along its edges. This is done using both a novel mean-field theory and exact diagonalization, and the basis is indeed found to be well suited for examining (quasi-) order in the model. The model has a paramagnetic region and a Nagaoka ferromagnetic region. In the numerical calculation the correlation function in the paramagnetic phase has a power-law decay and the charge motion is qualitatively hard-core bosonic. The mean field leads to an example of a BCS-type model with single-particle bosonic long-range order.
The need for suitable many- or infinite-fermion correlation functions to describe strongly correlated systems is discussed, and the question linked to the need for a correlated basis, in which the ground state may be positive definite for certain low-dimensional geometries. In seeking a positive-definite basis a particular trial basis is proposed, based on that for hard-core bosons in pure one-dimensional systems. Single-particle correlations in this basis are evaluated for the case of the ground state of a quasi-1D Hubbard model in the limit of extreme correlation. The model is a strip of the 2D square lattice wrapped around a cylinder, and is related to a ladder geometry with periodic boundary conditions along its edges. This is done using both a novel mean-field theory and exact diagonalization, and the basis is indeed found to be well suited for examining (quasi-) order in the model. The model has a paramagnetic region and a Nagaoka ferromagnetic region. In the numerical calculation the correlation function in the paramagnetic phase has a power-law decay and the charge motion is qualitatively hard-core bosonic. The mean field leads to an example of a BCS-type model with single-particle bosonic long-range order.
We use exact diagonalization in order to study the infinite-U limit of the two-dimensional Hubbard model. As well as looking at single-particle correlations, such as ${\mathit{n}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$=〈${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$${\mathit{c}}_{\mathit{k}\mathrm{\ensuremath{\sigma}}}$〉, we also study N-particle correlation functions, which compare the relative positions of all the particles in different models. In particular we study 16- and 18-site clusters and compare the charge correlations in the Hubbard model with those of spinless fermions and hard-core bosons. We find that although low densities of holes favor a ``locally ferromagnetic'' fermionic description, the correlations at larger densities resemble those of pure hard-core bosons surprisingly well.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
Quantum mechanical systems of strongly interacting particles in two dimensions comprise a realm of condensed matter physics for which there remain many unanswered theoretical questions. In particular, the most formidable challenges may lie in cases where the ground states show no signs of ordering, break no symmetries, and support many gapless excitations. Such systems are known to exhibit exotic, disordered ground states that are notoriously difficult to study analytically using traditional perturbation techniques or numerically using the most recent methods (e.g., tensor network states) due to the large amount of spatial entanglement. Slave particle descriptions provide a glimmer of hope in the attempt to capture the fundamental, low-energy physics of these highly non-trivial phases of matter. To this end, this dissertation describes the construction and implementation of trial wave functions for use with variational Monte Carlo techniques that can easily model slave particle states. While these methods are extremely computationally tractable in two dimensions, we have applied them here to quasi-one-dimensional systems so that the results of other numerical techniques, such as the density matrix renormalization group, can be directly compared to those determined by the trial wave functions and so that exclusively one-dimensional analytic approaches, namely bosonization, can be employed. While the focus here is on the use of variational Monte Carlo, the sum of these different numerical and analytical tools has yielded a remarkable amount of insight into several exotic quantum ground states. In particular, the results of research on the d-wave Bose liquid phase, an uncondensed state of strongly correlated hard-core bosons living on the square lattice whose wave function exhibits a d-wave sign structure, and the spin Bose-metal phase, a spin-1/2, SU(2) invariant spin liquid of strongly correlated spins living on the triangular lattice, will be presented. Both phases support gapless excitations along surfaces in momentum space in two spatial dimensions and at incommensurate wave vectors in quasi-one dimension, where we have studied them on three- and four-leg ladders. An extension of this work to the study of d-wave correlated itinerant electrons will be discussed.
We study a model for two-species hard-core bosons in one dimension. In this model, the same-species bosons have a hard-core condition at the same site, while different-species bosons are allowed to occupy the same site with a local interaction U . At half-filling, by Jordan-Wigner transformation, the model can be exactly mapped to a fermionic Hubbard model. Due to this correspondence, the phase transition from superfluid ( U = 0) to Mott insulator ( U > 0) can be explained by simple one-band theory at half-filling. By using an exact diagonalization method adopting a modified Lanczos method, we obtain the ground states as a function of U for the lattice size upto L = 16. We calculate directional current-current correlation functions in this model, which indicate that there are some remaining counter-flow in the Mott insulating region ( U > 0) and co-flow in the charge-density-wave region ( U <0) for the finite lattices.
In the limit of dominant nearest-neighbour hopping and infinitesimal secondand third-nearest-neighbour hopping, the Hubbard model is solvable in the sense that it separates into two models which have previously been analysed in the literature. The solutions are spin-charge separated and involve highly correlated motion of the electrons. Although in general the solutions are too complicated to analyse, there are two quite specific states which are amenable to direct analytical attack: the phase boundary between the quantum paramagnet and the Nagaoka ferromagnet, and the 'dimer' phase where all spins pair up into nearest-neighbour singlets. We have been able to evaluate both (n(k)) and (S -k .S k ) accurately enough to deduce the qualitative behaviour for the two types of state in question. The second of these correlation functions is directly relevant to neutron-scattering studies. Although the solutions are highly correlated and spin-charge separated, the correlations are reminiscent of the non-interacting 'spinfull' free-electron gas solution, the spin response exhibiting a divergence where the noninteracting susceptibility does, for example.
Abstract. We study the influence of on-site disorder on the magnetic properties of the ground state of the infinite-U one-dimensional (1D) Hubbard model. We find that the ground state is not ferromagnetic. This is analysed in terms of the algebraic structure of the spin dependence of the Hamiltonian. A simple explanation is derived for the 1=N periodicity in the persistent current for this model. The Hubbard model is the simplest model one can study to examine the effects of correlations between electrons in narrow energy bands. The Hamiltonian consists of a nearest neighbour hopping term and an electron–electron repulsion, U, which acts only when two electrons are on the same site. The Hubbard model is also the canonical model for the study of itinerant ferromagnetism. The strong coupling regime is of special importance for the study of ferromagnetism, since a theorem by Nagaoka [1] states that in theU D 1 limit, the ground state (GS) is ferromagnetic given some connectivity property of the lattice (which holds in most cases for d> 1). The model is solvable in one dimension and it was shown [2] that for open boundary conditions (BCs) the GS for finiteU is a singlet, i.e. there can be no ferromagnetism unless one postulates explicitly spin- or velocity-dependent forces. For infinite U, the GS of