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arXiv:0907.4228 · Physical Review B

Exact ground states and correlation functions of chain and ladder models of interacting hardcore bosons or spinless fermions

Siew Ann Cheong, Christopher L. Henley

Published Oct 21, 2009• 15 citations
arXivHTMLPDF

Abstract

By removing one empty site between two occupied sites, we map the ground states of chains of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion to ground states of chains of hardcore bosons and spinless fermions without nearest-neighbor repulsion, respectively, and ultimately in terms of the one-dimensional Fermi sea. We then introduce the intervening-particle expansion, where we write correlation functions in such ground states as a systematic sum over conditional expectations, each of which can be ultimately mapped to a one-dimensional Fermi-sea expectation. Various ground-state correlation functions are calculated for the bosonic and fermionic chains with infinite nearest-neighbor repulsion, as well as for a ladder model of spinless fermions with infinite nearest-neighbor repulsion and correlated hopping in three limiting cases. We find that the decays of these correlation functions are governed by surprising power-law exponents.

DOI
10.1103/physrevb.80.165124

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By removing one empty site between two occupied sites, we map the ground states of chains of hardcore bosons and spinless fermions with infinite nearest-neighbor repulsion to ground states of chains of hardcore bosons and spinless fermions without nearest-neighbor repulsion, respectively, and ultimately in terms of the one-dimensional Fermi sea. We then introduce the intervening-particle expansion, where we write correlation functions in such ground states as a systematic sum over conditional expectations, each of which can be ultimately mapped to a one-dimensional Fermi-sea expectation. Various ground-state correlation functions are calculated for the bosonic and fermionic chains with infinite nearest-neighbor repulsion, as well as for a ladder model of spinless fermions with infinite nearest-neighbor repulsion and correlated hopping in three limiting cases. We find that the decays of these correlation functions are governed by surprising power-law exponents.
We use the density-matrix renormalization group to study the single-particle and two-particle correlation functions of spinless fermions in the ground state of a quarter filled ladder. This ladder consists of two chains having an in-chain extended Coulomb interaction reaching to third neighbor and coupled by interchain hopping. Within our short numerical coherence lengths, typically reaching 10 to 20 sites, we find a strong renormalization of the interchain hopping and the existence of a dimensional crossover at smaller interactions. We also find power exponents for single-particle hopping and interchain polarization consistent with the single chain. The total occupation correlation function has a larger power exponent and shows signs of a crossover from incoherent fermion hopping to coherent particle-hole pair motion between chains. We believe the ladder is gapless.
We consider the one-dimensional spinless Falicov-Kimball model of itinerant fermionic particles (``spinless electrons''), which can hop between nearest-neighbour sites only, and of immobile particles (``classical ions''), with an on-site attraction. Extensive studies of the ground-state phase diagram of this system and its higher dimensional counterparts, carried out up to now, concentrated on determining ground-state arrangements of ions on the underlying lattice, while the properties of electrons were typically ignored. We report studies of short- and long-range correlations between electrons, and between ions and electrons, and of the spatial decay of electron correlations (decay of single-particle density matrix), in the ground state. The studies have been carried out analytically and by means of well-controlled numerical procedures. In the case of period 2 ground state, the single-particle density matrix has been expressed in terms of a hypergeometric function, and its spatial decay has been extracted. Numerical calculations have been done for open chains of various lengths (up to a few thousand sites), in order to control the chain-size dependence of correlations and to extrapolate the results to the limit of infinite chain. A part of the obtained results refers to tight-binding electrons subjected to a periodic external potential due to the ions, which constitute simple models of metals and insulators.
One parametric one-dimensional spinless fermion models is formulated and solved exactly by the Bethe ansatz method. The Hamiltonian contains the kinetic terms describing nearest-neighbor hopping of electrons with different hopping integrals for many particle configurations. The Luttinger liquid behavior is studied for the case repulsion effective interactions. The ground state energy, the critical exponent, describing the asymptotic behavior of the density-density correlation function at long distances, the Fermi velocity have been calculated numerically for an arbitrary density of electrons. The model displays the condensation of particles in the case of the attractive interaction.
We study the two-band model of spinless fermions in one dimension for weak repulsive interactions. In this case, the model is equivalent to the weakly interacting spinless two-leg ladder. We obtain nonuniversal analytic expressions for the power-law decays $\ensuremath{\propto}{x}^{\ensuremath{-}\ensuremath{\gamma}}$ of the charge-density and $\ensuremath{\propto}{x}^{\ensuremath{-}1/\ensuremath{\gamma}}$ of the superconducting pairing correlation functions. To leading order in the doping $\ensuremath{\delta}$ away from half filling and ${t}_{\ensuremath{\perp}}/t,$ we find $\ensuremath{\gamma}=1+({\ensuremath{\pi}}^{2}{/8)(t}_{\ensuremath{\perp}}{/t)}^{2}{\ensuremath{\delta}}^{2}$ (t(t(t and ${t}_{\ensuremath{\perp}}$ are the hopping terms along and between the chains), such that superconducting pairing correlations dominate. We furthermore show that the transition from the superconducting phase to the usual one-dimensional (Luttinger) metal occurs via a mixed phase, where superconducting pairs are formed in the bonding band only. We give the phase diagram as a function of temperature and doping.
We study a model for interacting spinless bosons in one dimension. The bosons are under a hard-core condition, which does not allow two or more bosons in the same site. However, nearestneighbor interactions between bosons ( V ) and hoppings to the nearest empty site ( t ) are allowed. As V increases from a large negative value, the system undergoes a quantum phase transition from a phase-separation (PS) phase to a superfluid (SF) phase because the hopping term overcomes the attractive energy. When V becomes positive and is increased more, the superfluid phase becomes a charge-density-wave (CDW) phase because the repulsive energy blocks the movements of bosons. Via exact diagonalizations, we calculated the ground-state energies, the correlation energies, and the kinetic energies to obtain signatures of the quantum phase transitions. We adopted a fast stateseeking algorithm that enabled us to calculate the ground states and the ground-state energies up to L = 32 more efficiently. Some results are compared with those of quantum Monte Carlo simulations by using stochastic series expansion for the Heisenberg point, and the momentum distribution functions for the three phases are discussed.