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Journal of Physics A Mathematical and General

Central charge for the half-filled Hubbard chain

Arkady L. Kholodenko

Published Dec 1, 1987• 2 citations
Publisher

Abstract

Zero-temperature finite-size Bethe ansatz calculations of the central charge for the Hubbard chain are compared with the finite-temperature Bethe ansatz calculations of the same charge.
DOI
10.1088/0305-4470/20/17/046

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We use the Bethe-ansatz equations to calculate the total and zero-frequency spectral weight in the optical conductivity of the half-filled one-dimensional Hubbard model as a function of the lattice size L and the on-site repulsion U. The zero-frequency spectral weight \ensuremath{\pi}D scales as L1/2{\mathit{L}}^{1/2}L1/2exp(-L/\ensuremath{\xi}) as L\ensuremath{\rightarrow}\ensuremath{\infty}. Near U=0, \ensuremath{\xi} varies as the inverse of the Lieb-Wu charge gap. In the strongly correlated regime (U\ensuremath{\gg}t), ${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$=ln(U/t)-1.48. $D--- is negative when L is a multiple of 4, corresponding to a negative inductance. We give a physical explanation of our results in terms of a simple model of ring exchange. The finite-size corrections to the total spectral weight scale as ${\mathit{L}}^{\mathrm{\ensuremath{-}}2}$. We discuss the implications of our results for exact diagonalization calculations of the optical conductivity.
We study the current, the curvature of levels, and the finite temperature charge stiffness, D(T,L),D(T,L),D(T,L), in the strongly correlated limit $U\ensuremath{\gg}t$ for Hubbard rings of L sites, with U the on-site Coulomb repulsion and t the hopping integral. Our study is done for finite-size systems and any band filling. Up to order t, we derive our results following two independent approaches, namely, using the solution provided by the Bethe ansatz and the solution provided by an algebraic method, where the electronic operators are represented in a slave-fermion picture. We find that, in the $U=\ensuremath{\infty}$ case, the finite-temperature charge stiffness is finite for electronic densities n smaller than 1. These results are essentially those of spinless fermions in a lattice of size L, apart from small corrections coming from a statistical flux, due to the spin degrees of freedom. Up to order t, the Mott-Hubbard gap is ${\ensuremath{\Delta}}_{\mathrm{MH}}=U\ensuremath{-}4t,$ and we find that D(T)D(T)D(T) is finite for n<1,n<1,n<1, but is zero at half filling. This result comes from the effective flux felt by the holon excitations, which, due to the presence of doubly occupied sites, is renormalized to ${\ensuremath{\Phi}}^{\mathrm{eff}}=\ensuremath{\varphi}{(N}_{h}\ensuremath{-}{N}_{d}{)/(N}_{d}{+N}_{h}),$ and which is zero at half filling, with Nd{N}_{d}Nd​ and Nh{N}_{h}Nh​ being the number of doubly occupied and empty lattice sites, respectively. Further, for half filling, the current transported by any eigenstate of the system is zero and, therefore, D(T)D(T)D(T) is also zero.
The gap equation at zero temperature for the half-filled Hubbard chain obtained by Lieb and Wu is rederived with use of the formalism of Yang and Yang and then the results are extended to the case of finite temperatures. For the values of the coupling constant c in the range 0<c\ensuremath{\le}1.5 the gap \ensuremath{\Delta}(T) exhibits BCS-like behavior \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​)1/2{)}^{1/2})1/2 while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/Tc{T}_{c}Tc​) with ${\ensuremath{\Delta}}_{0}$ being the zero-temperature result of Lieb and Wu for the gap and Tc{T}_{c}Tc​ some critical transition temperature.