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Physical review. B, Condensed matter

Finite-bandwidth effects in one-dimensional Wigner lattices

Hans A. De Raedt, Ad Lagendijk

Published Jan 15, 1983• 4 citations
Publisher

Abstract

The Feynman path-integral formalism is used to calculate the thermodynamic properties of a one-dimensional (1D) spinless fermion model with nearest- and next-nearest-neighbor interaction. The mapping of this 1D quantum model on a two-dimensional classical model allows the use of a Monte Carlo method to calculate the properties of long chains. We present results for typical values of the interaction parameters, density, and temperature. We find that a finite bandwidth smears out the details of the classical ground-state configurations discussed by Hubbard. If the density $\ensuremath{\rho}\ensuremath{\ne}\frac{1}{2}$, we find that the static structure factor and the static susceptibility display a maximum at $2{k}_{F}$ only when the next-nearest-neighbor interaction is nonzero.

DOI
10.1103/physrevb.27.921

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The Feynman path-integral formalism is used to calculate the thermodynamic properties of a one-dimensional (1D) spinless fermion model with nearest- and next-nearest-neighbor interaction. The mapping of this 1D quantum model on a two-dimensional classical model allows the use of a Monte Carlo method to calculate the properties of long chains. We present results for typical values of the interaction parameters, density, and temperature. We find that a finite bandwidth smears out the details of the classical ground-state configurations discussed by Hubbard. If the density \ensuremathρ\ensuremath≠12\ensuremath{\rho}\ensuremath{\ne}\frac{1}{2}\ensuremathρ\ensuremath=21​, we find that the static structure factor and the static susceptibility display a maximum at 2kF2{k}_{F}2kF​ only when the next-nearest-neighbor interaction is nonzero.
Feynman path-integral quantum Monte Carlo (QMC) simulations and an analytic many-body approach are used to study the ground state properties of one-dimensional (1D) chains in the theoretical framework of model Hamiltonians of the Hubbard type. The QMC algorithm is employed to derive position-space quantities, while band structure properties are evaluated by combining QMC data with expressions derived in momentum (k) space. Bridging link between both representations is the quasi-chemical approximation (QCA). Electronic charge fluctuations <(Δn 2 i )> and the fluctuations of the magnetic local moments <(Δs 2 i )> are studied as a function of the on-site density <n i > and correlation strength, which is given by the ratio between two-electron interaction and kinetic hopping. Caused by the non-analytic behaviour of the chemical potential μ = ∂E/∂<n i > (with E denoting the electronic energy), strict 1D systems with an on-site density <n i > of 1·0 do not exhibit the properties of a conductor for any non-zero interaction beyond the mean-field approximation. The QMC simulations lead to straightforward access to the probabilities P i (n) of finding n = 0, 1, 2 electrons at the ith lattice site. The P i (n) elements allow to calculate the enhancement factors on the electron spin susceptibility χ, effective electronic mass m* and Knight shift κ. m* is enhanced by a bandwidth renormalization factor D -1 0, κ by an element ηK mapping the additional localization of the correlated electrons in the presence of an external magnetic field B and χ by the product D -1 0 ηK. Available experimental data are discussed in the light of the present theoretical findings.
The magnetic phase diagram of the quarter-filled generalized Wigner lattice with nearest-neighbor and next-nearest-neighbor hoppings, t1{t}_{1}t1​ and t2{t}_{2}t2​, is explored. We find a region at negative t2{t}_{2}t2​ with fully saturated ferromagnetic ground states that we attribute to kinetic exchange. Such interaction disfavors antiferromagnetism at t2<0{t}_{2}<0t2​<0 and stems from virtual excitations across the charge gap of the Wigner lattice, which is much smaller than the Mott-Hubbard gap $\ensuremath{\propto}U$. Remarkably, we find a strong dependence of the charge structure factor on magnetism even in the limit $U\ensuremath{\rightarrow}\ensuremath{\infty}$, in contrast to the expectation that charge ordering in the Wigner lattice regime should be well described by spinless fermions. Our results, obtained using the density-matrix renormalization group and exact diagonalization, can be transparently explained by means of an effective low-energy Hamiltonian.
We investigate the phase diagrams of a one-dimensional lattice model of fermions and of a spin chain with interactions extending up to next-nearest-neighbor range.In particular, we investigate the appearance of regions with dominant pairing physics in the presence of nearest-neighbor and next-nearest-neighbor interactions.Our analysis is based on analytical calculations in the classical limit, bosonization techniques and large-scale density-matrix renormalization group numerical simulations.The phase diagram, which is investigated in all relevant filling regimes, displays a remarkably rich collection of phases, including Luttinger liquids, phase separation, charge-density waves, bond-order phases, and exotic cluster Luttinger liquids with paired particles.In relation with recent studies, we show several emergent transition lines with a central charge c = 3/2 between the Luttinger liquid and the cluster Luttinger liquid phases.These results could be experimentally investigated using highly tunable quantum simulators.
One parametric one-dimensional spinless fermion model 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 interaction. 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.
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.