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Physical review. B, Condensed matter

Spin-charge decoupling and the Green’s function of one-dimensional Mott insulators

Alberto Parola, Sandro Sorella

Published Jun 1, 1992• 39 citations
Publisher

Abstract

An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.

DOI
10.1103/physrevb.45.13156

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An exact expression for the Green's function G(k,\ensuremath{\omega}) of one hole in the U\ensuremath{\rightarrow}\ensuremath{\infty} one-dimensional Hubbard model is derived from the known Bethe-ansatz solution of the model. Due to the spin-charge decoupling of the one-hole excitation spectrum and the Fermi-like statistics of the spin excitations, G shows branch-cut singularities at \ensuremath{\omega}=\ifmmode\pm\else\textpm\fi{}2 sink and a nontrivial dependence on the momentum of the hole. As a result, the hole can propagate although the quasiparticle weight vanishes.
Starting from the known Lieb and Wu solution of the one-dimension-Hubbard model in the U to infinity limit, the authors show how the spin-charge decoupling of the elementary excitations is responsible for several peculiar features in one-particle properties, such as momentum distribution, quasiparticle weight and the Green function. In particular they analyse in detail the structure of the one-hole Green function at half-filling, which has not been previously calculated by field theory methods due to the breakdown of conformal invariance. A rich structure is found with branch cut singularities at omega =+or-2 sin k but no simple poles. The non-trivial dependence on the momentum of the hole allows for hole propagation although the analytic structure of G(k, omega ) is quite different from that usually characterizing band insulators. These results provide a precise characterization of one-dimensional Mott insulators. The relationship between the branch cuts of the Green function and the finite-size scaling of the quasiparticle weight is also discussed together with its implications for the analysis of numerical data.
On the basis of the Bethe ansatz solution, the characteristic properties of charge excitations are investigated for the one-dimensional Hubbard model by inserting a charged hole into the Mott insulator. The density of states for the charged hole shows divergent behavior in the lower-energy region, whose connection is discussed analytically with the compressibility and magnitude of the Hubbard gap. Momentum-dependent susceptibility is also investigated within the framework of the quasiparticle formalism in the Bethe ansatz method. The logarithmic anomaly of the susceptibilty for the charged hole near 4kF{k}_{F}kF​ (kF{k}_{F}kF​ being the Fermi momentum in the noninteracting case) is considerably enhanced in the vicinity of the Mott insulating phase, while it is suppressed at the insulating phase due to the formation of the Hubbard gap.
The recent advances in angle-resolved photoemission techniques allowed the unambiguous experimental confirmation of spin-charge decoupling in quasi-one-dimensional (1D) Mott insulators. This opportunity stimulates a quantitative analysis of the spectral function $A(k,\ensuremath{\omega})$ of prototypical one-dimensional correlated models. Here we combine Bethe Ansatz results, Lanczos diagonalizations, and field theoretical approaches to obtain $A(k,\ensuremath{\omega})$ for the 1D Hubbard model as a function of the interaction strength. By introducing a single spinon approximation, an analytic expression is obtained, which shows the location of the singularities and allows, when supplemented by numerical calculations, to obtain an accurate estimate of the spectral weight distribution in the $(k,\ensuremath{\omega})$ plane. Several experimental puzzles on the observed intensities and line-shapes in quasi-1D compounds such as SrCuO2{\text{SrCuO}}_{2}SrCuO2​, find a natural explanation in this theoretical framework.
Critical properties in the dynamical charge correlation function for the one-dimensional Mott insulator are studied. By properly taking into account {\it the final-state interaction} between the charge and spin degrees of freedom, we find that the edge singularity in the charge correlation function is governed by massless spinon excitations, although it is naively expected that spinons do not directly contribute to the charge excitation over the Hubbard gap. We obtain the momentum-dependent anomalous critical exponent by applying the finite-size scaling analysis to the Bethe ansatz solution of the half-filled Hubbard model.
We obtain an exact result on the single-hole dynamics in one-dimensional Mott-Hubbard insulators through a study of the supersymmetric $t\ensuremath{-}J$ model with long-range interaction at half filling. We calculate the expression G1s1h(x,t){G}_{1s1h}(x,t)G1s1h​(x,t) for the one-spinon one-holon contribution to the hole propagator, which contributes 7575%75 of the total spectral weight. The matrix element is independent of the holon momentum. The spectral weight has an inverse square root singularity along the holon band. Comparison with the hole dynamics of the $U\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ Hubbard model is also made.
The one-dimensional Hubbard model is characterized by charge-spin separation, but nevertheless in the doped system \ensuremath{\sigma}(\ensuremath{\omega}>0) does not vanish due to a small remaining coupling between charge and spin degrees of freedom. We derive an effective Hamiltonian and give in leading order (\ensuremath{\sim}J2{\mathit{J}}^{2}J2) an analytic expression for \ensuremath{\sigma}(\ensuremath{\omega}) for the case of the Mott insulator doped with one hole. The complete frequency dependence in the strong-coupling limit of the Hubbard model and for the t-J model agrees with results from exact diagonalization. The limit \ensuremath{\sigma}(\ensuremath{\omega}\ensuremath{\rightarrow}0) at T=0 is found to be very different for the two models, i.e., \ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{3/2}$ in the strong-coupling limit and \ensuremath{\sim}${\mathrm{\ensuremath{\omega}}}^{\mathrm{\ensuremath{-}}1/2}$ for the t-J model.