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arXiv:cond-mat/9607104

Low energy excitations of the t-J model in 1D and 2D

R. Eder, Y. C. Chen, H. Q. Lin, Yutaka Ohta, C. T. Shih, T. K. Lee

Published Jul 15, 1996• 0 citations
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Abstract

We present an exact diagonalization study of the low energy singlet and triplet states for both 1D and 2D t-J model. A scan of the parameter ratio J/t shows that for nearly all low energy states in both 1D and 2D the excitation energy takes the linear form E(t,J) = a t + b J. In 1D this is the natural consequence of the factorization of the low energy wave functions, i.e. spin-charge separation. Examination of the low energy eigenstates in 2D shows that in the singlet sector the lowest states are collective modes, which for larger J correspond to `dynamical hole columns', i.e. a periodic modulation of the hole density. This modulation is well reproduced by treating holes as hard-core Bosons with an attractive interaction.

DOI
10.48550/arxiv.cond-mat/9607104

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We present an exact diagonalization study of the low energy singlet and triplet states for both 1D and 2D t-J model. A scan of the parameter ratio J/t shows that for nearly all low energy states in both 1D and 2D the excitation energy takes the linear form E(t,J) = a t + b J. In 1D this is the natural consequence of the factorization of the low energy wave functions, i.e. spin-charge separation. Examination of the low energy eigenstates in 2D shows that in the singlet sector the lowest states are collective modes, which for larger J correspond to `dynamical hole columns', i.e. a periodic modulation of the hole density. This modulation is well reproduced by treating holes as hard-core Bosons with an attractive interaction.
We present an exact diagonalization study of the low-energy singlet and triplet states for both the one-dimensional (1D) and 2D t-J models. A scan of the parameter ratio J/t shows that for most low-energy states in both 1D and 2D the excitation energy takes the form E(t,J)=a\ensuremath{\cdot}t+b\ensuremath{\cdot}J. In 1D this is the natural consequence of the factorization of the low-energy wave functions, i.e., spin-charge separation. Examination of the low-energy eigenstates in 2D shows that most of these are collective modes, which for larger J correspond to a periodic modulation of the hole density. The modulation is well reproduced by treating holes as hard-core bosons with an attractive interaction.
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 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.
We present an exact diagonalization study of the dynamical spin and density correlation functions in small clusters of 2D $t\ensuremath{-}J$ model for hole dopings $\ensuremath{\le}25$%. Both correlation functions show a remarkably regular, but very different, scaling with both hole density ${\ensuremath{\rho}}_{h}$ and parameters ttt and JJJ. The density correlation function is most consistent with that of condensed bosons in a band of width $\ensuremath{\sim}t$, the spin correlation function with that of fermions in a band of width $\ensuremath{\sim}J$. We show that the familiar spin bag scenario explains these results in a natural way.
The spin and charge excitations of the t-J ladder are studied by exact diagonalization techniques for several electron densities. The various modes are classified according to their spin (singlet or triplet excitations) and their parity under a reflection with respect to the symmetry axis along the chains and a finite-size scaling of the related gaps is performed. At low doping, formation of hole pairs leads to a spin gap for all J/t ratios. This phase is characterized by (at least) one vanishing energy mode only in the charge bonding channel when Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\rightarrow}0 consistent with the existence of superconducting pairing correlations. At larger doping the spin gap disappears. Although the antibonding modes remain gapped, low-energy Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\sim}0 and Kx{\mathit{K}}_{\mathit{x}}Kx​\ensuremath{\sim}2kF{\mathit{k}}_{\mathit{F}}kF​ spin and charge bonding modes are found consistent with a single-band Luttinger scenario. At sufficient low electron density and above a critical value of J/t we also expect another phase of electron pairs with gapped spin excitations.
The hole-doped two-dimensional(2D) $t\ensuremath{-}J$ model is studied by exact diagonalization on a $5\ifmmode\times\else\texttimes\fi{}4$ cluster which, unlike the standard tilted-square clusters, can in principle accomodate an antiphase domain wall. For hole concentration 1010%10 and J/t> 0.5J/t>~0.5J/t> 0.5 the ground-state energy/site is lower than the conventional tilted-square $\sqrt{20}\ifmmode\times\else\texttimes\fi{}\sqrt{20}$ cluster. In the ground state two holes form a loosely bound pair pinned to an antiphase domain wall. The dynamical density correlation function shows sharp quasiparticlelike peaks, reminiscent of the ``holons'' in 1D chains, which suggest the existence of solitonlike propagating domain walls. The dynamical correlation function of the bond-singlet operator has a low-energy peak structure characteristic of columnar spin-Peierls order, the dynamical spin-correlation function shows an intense and isolated ``resonance peak'' near $(\ensuremath{\pi},\ensuremath{\pi}).$