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arXiv:cond-mat/9707205 · Physical review. B, Condensed matter

Diagonalization in reduced Hilbert spaces using a systematically improved basis: Application to spin dynamics in lightly doped ladders

Elbio R. Dagotto, George Balster Martins, José A. Riera, André Luiz Malvezzi, C. J. Gazza

Published Nov 1, 1998• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.58.12063

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•Jan 1, 1995•1 citation
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−Jt\ensuremath{-}Jt\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(q,\ensuremathω)S(\mathbf{q},\ensuremath{\omega})S(q,\ensuremathω) on clusters with 2\ifmmode×\else\texttimes\fi162\ifmmode\times\else\texttimes\fi{}162\ifmmode×\else\texttimes\fi16 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 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 $2\ifmmode\times\else\texttimes\fi{}15.$
We employ the recently proposed plaquette basis to investigate static and dynamic properties of isotropic two-leg Heisenberg spin ladders. Simple noninteracting multiplaquette states provide a remarkably accurate picture of the energy/site and dynamic spin response of these systems. Insights afforded by this simple picture suggest a very efficient truncation scheme for more precise calculations. When the small truncation errors are accounted for using recently developed contractor renormalization techniques, very accurate results requiring a small fraction of the computational effort of exact calculations are obtained. These methods allow us to determine the energy/site, gap, and spin response of $2\ifmmode\times\else\texttimes\fi{}16$ ladders. The former two values are in good agreement with density-matrix renormalization-group results. The spin-response calculations show that nearly all the strength is concentrated in the lowest triplet level and that coherent many-body effects enhance the response/site by nearly a factor of 1.6 over that found for $2\ifmmode\times\else\texttimes\fi{}2$ systems.
We have developed strong-coupling series expansion methods to study the two-particle spectra in quantum lattice models. The properties of bound states and multiparticle excitations can reveal important information about the dynamics of a given model. At the heart of this method lies the calculation of an effective Hamiltonian in the two-particle subspace. We use an orthogonal transformation to perform this block diagonalising, and find that maintaining orthogonality is crucial for cases where the ground state and the two-particle subspace have identical quantum numbers. The two-particle Schrödinger equation is solved by using a finite lattice approach in coordinate space or an integral equation in momentum space. These methods allow us to determine precisely the low-lying excitation spectra and dispersion relations for the two-particle bound states. The method has been tested for the (1+1) D transverse Ising model, and applied to the two-leg spin-1/2 Heisenberg ladder. We study the coherence lengths of the bound states, and how they merge with the two-particle continuum. Finally, these techniques are applied to the frustrated alternating Heisenberg chain, which has been of considerable recent interest due to its relevance to spin-Peierls systems such as CuGeO 3. Starting from a limit corresponding to weakly-coupled dimers, we develop high-order series expansions for the effective Hamiltonian in the two-particle subspace. In the regime of strong dimerisation, various properties of the singlet and triplet bound states, and the quintet antibound states, can be accurately calculated. We also study the behaviour as the external bond alternation vanishes, and the way in which the bound states of triplet dimer excitations make the transition to a soliton-antisoliton continuum.
We have numerically investigated the doped t-J ladder using exact diagonalization. We have studied both the limit of strong interchain coupling and isotropic coupling. The ladder scales to the Luther-Emery liquid regime in the strong interchain coupling limit. In this strong coupling limit there is a simple picture of the excitation spectrum that can be continued to explain the behavior at isotropic coupling. At J=0 we have indications of a ferromagnetic ground state. At a large J/t the ladder is phase separated into holes and a Heisenberg ladder. At intermediate coupling the ground state shows hole pairing with a modified d-wave symmetry. The excitation spectrum separates into a limited number of quasiparticles which carry charge +\ensuremath{\Vert}e\ensuremath{\Vert} and spin 1/2 and a triplet magnon mode. At half filling the former vanish but the latter evolves continuously into the magnon band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong attraction but simultaneously the Fermi wave vector, as would be measured in photoemission, is large. The dynamical structure factors are calculated and are found to be very similar to calculations on two-dimensional clusters. \textcopyright{} 1996 The American Physical Society.
We determine the dynamical structure factors of two gapped correlated electron systems, namely the Ising model in a strong transverse field and the two-leg spin-1/2 Heisenberg ladder in the limit of strong rung coupling. We consider the low-temperature limit, employing a variety of analytical and numerical techniques. The coherent modes of single-particle excitations, which are delta functions at zero temperature, are shown to broaden asymmetrically in energy with increasing temperature. Firstly, we apply a low-temperature “resummation” inspired by the Dyson equation to a linked-cluster expansion of the two-leg Heisenberg ladder. We include matrix elements to second order in the interaction between states containing up to two particles. A low-frequency response similar to the “Villain mode” is also observed. Next, we apply a cumulant expansion technique to the transverse field Ising model. We resolve the issue of negative spectral weight caused by double pole in the leading self-energy diagram by including a resummation of terms obtained from the six-point function, demonstrating that the perturbation series in 2n-spin correlation functions can be extended to higher orders. The result generalises to higher dimensions and the analytic calculation is compared to a numerical Pade approximant. We outline the extension of this method to the strong coupling ladder. Finally, we compare the previous results to numerical data obtained by full diagonalisation of finite chains and numerical evaluation of the Pfaffian, a method specific to the transverse field Ising chain. The latter method is used for a phenomenological study of the asymmetric broadening as well as an evaluation of fitting functions for the broadened lineshapes.
We present a novel approach for a systematic large–spin expansion of the t-J Hamiltonian which enables us to work without the constraint of no double occupancy. In our scheme we can perform the large–spin limit ensuring that the low energy spin excitations are in exact correspondence with the physical excitations of the s = 1 2 Hilbert space. As a consequence, we expect a smooth dependence of the physical quantities on the expansion parameter 1/s. As a first application of the method we study the case of a single hole in a Néel background. A systematic expansion in fluctuations about this stable solution indicates that by increasing t/J the quasiparticle weight strongly depends on the momentum carried by the hole. Results, obtained on small lattice sizes, are found in excellent agreement with exact diagonalization data.