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Physical review. B, Condensed matter

Charge and spin dynamics of the two-dimensionaltmodel in leading order of a 1/Nexpansion

Lew Gehlhoff, R. Zeyher

Published Aug 15, 1995• 24 citations
Publisher

Abstract

The density- and spin-correlation functions of the t model are calculated on a square lattice in leading order of a 1/N expansion using X operators and a Baym-Kadanoff expansion. While the spectra of both correlation functions are similar at low densities, the energy scale for density fluctuations becomes much higher than that of spin fluctuations at large densities, due to collective effects. Our results agree quite well with recent exact diagonalizations for the case N=2. Furthermore, we examine the idea that the charge correlations of strongly correlated electrons with spin can be described by those of uncorrelated spinless electrons in two dimensions. While the equal-time density correlation functions of both systems are similar, the dynamic ones behave quite differently at high densities.

DOI
10.1103/physrevb.52.4635

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The density- and spin-correlation functions of the t model are calculated on a square lattice in leading order of a 1/N expansion using X operators and a Baym-Kadanoff expansion. While the spectra of both correlation functions are similar at low densities, the energy scale for density fluctuations becomes much higher than that of spin fluctuations at large densities, due to collective effects. Our results agree quite well with recent exact diagonalizations for the case N=2. Furthermore, we examine the idea that the charge correlations of strongly correlated electrons with spin can be described by those of uncorrelated spinless electrons in two dimensions. While the equal-time density correlation functions of both systems are similar, the dynamic ones behave quite differently at high densities.
We investigate the dynamical spin and charge correlation functions, $S(\mathrm{Q}, \ensuremath{\omega})$ and $N(\mathrm{Q}, \ensuremath{\omega})$, for the two-dimensional (2D) $t\ensuremath{-}J$ model by using the exact diagonalization technique. Both quantities are compared with the 1D $t\ensuremath{-}J$ model. At high electron density (n>0.7n>0.7n>0.7), a remarkable feature emerges: $S(\mathrm{Q}, \ensuremath{\omega})$ shows a pronounced peak at low energy of order JJJ which is consistent with 2kF2{k}_{F}2kF​ low-energy excitations, whereas low-energy 2kF2{k}_{F}2kF​-backscattering processes are strongly suppressed in the charge-density response. $N(\mathrm{Q}, \ensuremath{\omega})$ consists of a dense and broad continuum, qualitatively different from the 1D charge response, implying strong interactions between charge and spin degrees of freedom.
The spin- and density-correlation functions of the two-dimensional Hubbard model at low electronic density 〈n〉 are calculated in the ground state by using the power method, and at finite temperatures by using the quantum Monte Carlo technique. Both approaches produce similar results, which are in close agreement with numerical and high-temperature-expansion results for the two-dimensional t-J model. Using perturbative approximations, we show that the examination of the density-correlation function alone is not enough to support recent claims in the literature that suggested spin and charge separation in the low electronic density regime of the t-J model.
We present an exact diagonalization study of the dynamical spin and density correlation functions in small clusters of the t-J model, focusing on the regime of intermediate and low electron densities, ${\mathrm{\ensuremath{\rho}}}_{\mathit{e}}$0.5. In two dimensions (2D) both correlation functions agree remarkably well with the convolution of the single-particle spectral function, i.e., the simplest estimate possible within a Fermi-liquid picture. Deviations from the convolution are shown to originate from symmetry-related selection rules, which are unaccounted for in the convolution estimate. For all fillngs under consideration, we show that the low-energy peaks originate from particle-hole excitations between the Fermi momenta, as expected for a Fermi liquid. We contrast this with the behavior in 1D, where spin and density correlation function show the differences characteristic of spin-charge separation and where neither correlation function is approximated well by the convolution.
We have calculated the high temperature expansion for the density correlation function N(q) of the two-dimensional t-J model. On extrapolation by Pad\'e approximants to low temperatures we find that N(q) has 2kFSF{\mathbf{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ as a characteristic wave vector. Previous studies have shown that nk{\mathit{n}}_{\mathbf{k}}nk​ has a steplike feature at kF{\mathbf{k}}_{\mathit{F}}kF​ and S(q) has 2kF{\mathbf{k}}_{\mathit{F}}kF​ as a characteristic wave vector. Here kF{\mathit{k}}_{\mathit{F}}kF​ and kFSF{\mathit{k}}_{\mathit{F}}^{\mathrm{SF}}kFSF​ are the Fermi wave vectors of the nearest-neighbor square lattice tight-binding and spinless fermion models, respectively. By comparison to known results for one dimension this suggests spin-charge separation in the two-dimensional t-J model.
We study the ``t-J'' model with strongly correlated electrons (infinite-U Hubbard-model kinetic energy plus superexchange J) by exact diagonalization on small two-dimensional clusters of six, eight, and twelve sites on a square lattice with periodic boundary conditions. We find an attractive pairing potential for two holes when J/t\ensuremath{\ge}(J/t)c{)}_{\mathit{c}})c​\ensuremath{\simeq}0.2, with t the hopping matrix element. Ground-state calculations show that extended s-wave and d-wave pairing susceptibilities and equal-time pairing correlations grow weakly with increasing J and system size at fixed filling n. Pairing correlations may be enhanced for (J/t)>(J/t)c{)}_{\mathit{c}})c​\ensuremath{\approxeq}0.2. We compare equal-time pairing correlations to a mean-field theory for J=0 and note potential difficulties with interpreting numerical results for such small systems. The widths of one- and two-hole energy bands in low-spin states are of order J; the one-hole effective mass is larger than the two-hole effective mass in such states. Spin-charge correlation functions for a single hole in the eight- and twelve-site lattices are consistent with the spin-charge decoupling scenario for resonating-valence-bond states.
Using high temperature series we calculate temperature derivatives of the spin-spin and density-density correlation functions to investigate the low energy spin and charge excitations of the two-dimensional t-J model. We find that the temperature derivatives indicate different momentum dependences for the low energy spin and charge excitations. By comparing short distance density-density correlation functions with those of spinless fermions and hard core bosons, we find that the t-J model results are intermediate between the two cases, being closer to those of hard core bosons. The implications of these results for superconductivity are discussed.