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arXiv:cond-mat/9707207 · Physical review. B, Condensed matter

Dynamic spin response for Heisenberg ladders

J. Piekarewicz, James R. Shepard

Published May 1, 1998• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.57.10260

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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×\else\texttimes\fi162\ifmmode\times\else\texttimes\fi{}162\ifmmode×\else\texttimes\fi16 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×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 systems.
We employ a plaquette basis---generated by coupling the four spins in a $2\ifmmode\times\else\texttimes\fi{}2$ lattice to a well-defined total angular momentum---for the study of Heisenberg ladders with antiferromagnetic coupling. Matrix elements of the Hamiltonian in this basis are evaluated using standard techniques in angular-momentum (Racah) algebra. We show by exact diagonalization of small $(2\ifmmode\times\else\texttimes\fi{}4$ and $2\ifmmode\times\else\texttimes\fi{}6)$ systems that in excess of 90% of the ground-state probability is contained in a very small number of basis states. These few basis states can be used to define a severely truncated basis which we use to approximate low-lying exact eigenstates. We show how, in this low-energy basis, the isotropic spin-1/21/21/2 Heisenberg ladder can be mapped onto an anisotropic spin-111 ladder for which the coupling along the rungs is much stronger than the coupling between the rungs. The mapping thereby generates two distinct energy scales which greatly facilitates understanding the dynamics of the original spin-1/21/21/2 ladder. Moreover, we use these insights to define an effective low-energy Hamiltonian in accordance to the contractor renormalization group (CORE) method. We show how a simple range-2 CORE approximation to the effective Hamiltonian to be used with our truncated basis reproduces the low-energy spectrum of the exact $2\ifmmode\times\else\texttimes\fi{}6$ theory at the $\ensuremath{\lesssim}1%$ level.
We compute bulk properties of Heisenberg spin-1/2 ladders using Rayleigh-Schr\"odinger perturbation theory in the rung and plaquette bases. We formulate a method to extract high-order perturbative coefficients in the bulk limit from solutions for relatively small finite clusters. For example, a perturbative calculation for an isotropic $2\ifmmode\times\else\texttimes\fi{}12$ ladder yields an eleventh-order estimate of the ground-state energy per site that is within 0.02% of the density-matrix-renormalization-group value. Moreover, the method also enables a reliable estimate of the radius of convergence of the perturbative expansion. We find that for the rung basis the radius of convergence is ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}0.8,$ with $\ensuremath{\lambda}$ defining the ratio between the coupling along the chain relative to the coupling across the chain. In contrast, for the plaquette basis we estimate a radius of convergence of ${\ensuremath{\lambda}}_{c}\ensuremath{\simeq}1.25.$ Thus, we conclude that the plaquette basis offers the best currently available perturbative approach which can provide a reliable treatment of the physically interesting case of isotropic $(\ensuremath{\lambda}=1)$ spin ladders. We illustrate our methods by computing perturbative coefficients for the ground-state energy per site, the gap, and the one-magnon dispersion relation.
We investigate the NNN-leg spin-SSS Heisenberg ladders by using the density matrix renormalization group method. We present estimates of the spin gap ${\ensuremath{\Delta}}_{s}$ and of the ground-state energy per site ${e}_{\ensuremath{\infty}}^{N}$ in the thermodynamic limit for ladders with widths up to six legs and spin $S\ensuremath{\le}\frac{5}{2}$. We also estimate the ground-state energy per site ${e}_{\ensuremath{\infty}}^{2\mathrm{D}}$ for the infinite two-dimensional spin-SSS Heisenberg model. Our results support that for ladders with semi-integer spins the spin excitation is gapless for NNN odd and gapped for NNN even, whereas for integer spin ladders the spin gap is nonzero, independent of the number of legs. Those results agree with the well-known conjectures of Haldane and S\'en\'echal-Sierra for chains and ladders, respectively. We also observe edge states for ladders with NNN odd, similar to what happens in spin chains.
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.
We study the ground state and low-lying excited states of the Kitaev-Heisenberg model on a two-leg ladder geometry using the density-matrix renormalization-group and Lanczos exact diagonalization methods. The Kitaev and Heisenberg interactions are parametrized as $K=sin\ensuremath{\phi}$ and $J=cos\ensuremath{\phi}$ with an angle parameter $\ensuremath{\phi}$. Based on the results for several types of order parameters, excitation gaps, string order parameter, and entanglement spectra, the $\ensuremath{\phi}$-dependent ground state phase diagram is determined. Remarkably, the phase diagram is quite similar to that of the Kitaev-Heisenberg model on a honeycomb lattice, exhibiting the same long-range-ordered states, namely rung singlet (analog to N\'eel in three dimensions), zigzag, ferromagnetic, and stripy, and the presence of gapped spin liquids around the exactly solvable Kitaev points $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. We also calculate the expectation value of a plaquette operator corresponding to a $\ensuremath{\pi}$-flux state in order to establish how the gapped Kitaev spin liquid extends away from the $\ensuremath{\phi}=\ifmmode\pm\else\textpm\fi{}\ensuremath{\pi}/2$. Furthermore, we determine the dynamical spin structure factor and discuss the effect of the Kitaev interaction on the spin-triplet dispersion.
The goal of this project is to study the ground-state properties of frustrated spin-1/2 two- and three-leg Heisenberg ladders, and to investigate possible effects of diagonal couplings on spin gaps. The ground-state and the first-excited-state energies of these two systems are calculated systematically by using the density matrix renormalization group method. The ground state phase diagrams for these two systems are obtained. For the frustrated two-leg ladder, we found that the spin gap is insensitive to the change of the FM diagonal coupling constant. This is in agreement with experimental suggestions. For the frustrated three-leg ladder, we conclusively proved that introducing AF diagonal couplings can not change the gapless property of the ground state.