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arXiv:cond-mat/0010072 · Physical review. B, Condensed matter

Truncated Lanczos method: Application to hole-doped spin ladders

O. Chiappa, Sylvain Capponi, Didier Poilblanc

Published Mar 2, 2001• 4 citations
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Abstract

The truncated Lanczos method using a variational scheme based on Hilbert space reduction as well as a local basis change [Dagotto et al., Phys. Rev. B $58,$ 12 063 (1998)] is reexamined. The energy is extrapolated as a power-law function of the Hamiltonian variance. This systematic extrapolation procedure is tested quantitatively on the two-leg t-$J$ ladder doped with two holes. For this purpose, we have carried out calculations of the spin gap and of the pair dispersion up to size $2\ifmmode\times\else\texttimes\fi{}15.$

DOI
10.1103/physrevb.63.115120

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The truncated Lanczos method using a variational scheme based on Hilbert space reduction as well as a local basis change [Dagotto et al., Phys. Rev. B 58,58,58, 12 063 (1998)] is reexamined. The energy is extrapolated as a power-law function of the Hamiltonian variance. This systematic extrapolation procedure is tested quantitatively on the two-leg t-JJJ ladder doped with two holes. For this purpose, we have carried out calculations of the spin gap and of the pair dispersion up to size 2\ifmmode×\else\texttimes\fi15.2\ifmmode\times\else\texttimes\fi{}15.2\ifmmode×\else\texttimes\fi15.
A method is proposed to improve the accuracy of approximate techniques for strongly correlated electrons that use reduced Hilbert spaces. As a first step, the method involves a change of basis that incorporates exactly part of the short-distance interactions. The Hamiltonian is rewritten in new variables that better represent the physics of the problem under study. A Hilbert-space expansion performed in the new basis follows. The method is successfully tested using both the Heisenberg model and the $t\ensuremath{-}J$ model with holes on two-leg ladders and chains, including estimations for ground-state energies, static correlations, and spectra of excited states. An important feature of this technique is its ability to calculate dynamical responses on clusters larger than those that can be studied using exact diagonalization. The method is applied to the analysis of the dynamical spin structure factor $S(\mathbf{q},\ensuremath{\omega})$ on clusters with $2\ifmmode\times\else\texttimes\fi{}16$ sites and 0 and 2 holes. Our results confirm previous studies [M. Troyer, H. Tsunetsugu, and T. M. Rice, Phys. Rev. B 53, 251 (1996)] that suggested that the state of the lowest energy in the spin-1 2-hole subspace corresponds to the bound state of a hole pair and a spin triplet. Implications of this result for neutron-scattering experiments both on ladders and planes are discussed.
The gap function $\ensuremath{\phi}(k,\ensuremath{\omega})$, determined from a Lanczos calculation for a doped 2-leg $t\text{\ensuremath{-}}J$ ladder, is used to provide insight into the spatial and temporal structure of the pairing interaction. It implies that this interaction is a local near-neighbor coupling which is retarded. The onset frequency of the interaction is set by the energy of an S=1S=1S=1 magnon-hole pair and it is spread out over a frequency region of the order of the bandwidth.
In this paper, we have systematically studied the single-hole problem in two-leg Hubbard and $t\ensuremath{-}J$ ladders by large-scale density-matrix renormalization-group calculations. We found that the doped holes in both models behave similarly, while the three-site correlated hopping term is not important in determining the ground-state properties. For more insights, we have also calculated the elementary excitations, i.e., the energy gaps to the excited states of the system. In the strong-rung limit, we found that the doped hole behaves as a Bloch quasiparticle in both systems where the spin and charge of the doped hole are tightly bound together. In the isotropic limit, while the hole still behaves like a quasiparticle in the long-wavelength limit, our results show that its spin and charge components are only loosely bound together inside the quasiparticle, whose internal structure can lead to a visible residual effect which dramatically changes the local structure of the ground-state wave function.
We derive an approximate theory for Heisenberg spin ladders with two legs by mapping the spin dynamics onto the problem of hard-core ``bond bosons.'' The parameters of the bosonic Hamiltonian are obtained by matching anomalous Green's functions to Lanczos results and we find evidence for a strong renormalization due to quantum fluctuations. Various dynamical spin correlation functions are calculated and found to be in good agreement with Lanczos results. We then enlarge the effective Hamiltonian to describe the coupling of the bosonic spin fluctuations to a single hole injected into the system and treat the hole dynamics within the rainbow-diagram approximation of Schmitt-Rink et al. Theoretical predictions for the single-hole spectral function are obtained and found to be in good agreement with Lanczos results.
The evolution of the spin gap of a 2-leg ladder upon doping depends upon the nature of the lowest triplet excitations in a ladder with two holes. Here we study this evolution using various numerical techniques for a $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}J$ ladder as the next-near-neighbor hopping ${t}^{\ensuremath{'}}$ is varied. We find that depending on the value of ${t}^{\ensuremath{'}},$ the spin gap can evolve continuously or discontinuously and the lowest triplet state can correspond to a magnon, a bound magnon-hole-pair, or two separate quasiparticles. Previous experimental results on the superconducting two-leg ladder ${\mathrm{Sr}}_{14\ensuremath{-}x}{\mathrm{Ca}}_{x}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41}$ are discussed.
The two-leg $t\ensuremath{-}J$ ladder forms a spin liquid at half filling that evolves to a Luther-Emery liquid upon doping. Our aim is to obtain a complete phase diagram for isotropic coupling (i.e., rungs and legs equal) as a function of electron density n and the ratio J/t.J/t.J/t. Two known limiting cases are n<12,n<\frac{1}{2},n<21​, which is a single band Luttinger liquid, and small hole doping $\ensuremath{\delta}\ensuremath{\ll}1$ for $J/\stackrel{\ensuremath{\rightarrow}}{t}0,$ which is a Nagaoka ferromagnet. Using Lanczos techniques we examine the region between the Nagaoka and Luther-Emery phases for 1>n>12.1>n>\frac{1}{2}.1>n>21​. We find evidences for gapless behavior in both spin and charge channels for J/t<0.3,J/t<0.3,J/t<0.3, consistent with Luttinger liquids in both bonding and antibonding bands (i.e., C2S2).C2S2).C2S2). This proposal is based on the behavior of spin and charge correlation functions. For example, the hole-hole correlation function that displays hole pairing at larger J/tJ/tJ/t shows hole-hole repulsion in this region. As a further test, we examined the dependence of the energy on a relative phase shift between bonding and antibonding bands. For J/t<0.3J/t<0.3J/t<0.3 this is very weak, indicating a lack of pairing between these channels.