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arXiv:cond-mat/9804261 · Physical review. B, Condensed matter

Perturbation theory for spin ladders using angular-momentum coupled bases

J. Piekarewicz, James R. Shepard

Published Oct 1, 1998• 9 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.58.9326

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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×\else\texttimes\fi122\ifmmode\times\else\texttimes\fi{}122\ifmmode×\else\texttimes\fi12 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 \ensuremathλc\ensuremath≃0.8,{\ensuremath{\lambda}}_{c}\ensuremath{\simeq}0.8,\ensuremathλc​\ensuremath≃0.8, with \ensuremathλ\ensuremath{\lambda}\ensuremathλ 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 \ensuremathλc\ensuremath≃1.25.{\ensuremath{\lambda}}_{c}\ensuremath{\simeq}1.25.\ensuremathλc​\ensuremath≃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 (\ensuremathλ=1)(\ensuremath{\lambda}=1)(\ensuremathλ=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 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 study properties of two-leg Heisenberg spin ladders in a mean-field approximation using a variety of angular-momentum-coupled bases. The mean-field theory proposed by Gopalan, Rice, and Sigrist, which uses a rung basis, assumes that the mean-field ground state consists of a condensate of spin singlets along the rungs of the ladder. We generalize this approach to larger angular-momentum-coupled bases that incorporate---by their mere definition---a substantial fraction of the important short-range structure of these materials. In these bases the mean-field ground state remains a condensate of spin singlet---but now with each involving a larger fraction of the spins in the ladder. As expected, the ``purity'' of the ground state, as judged by the condensate fraction, increases with the size of the elementary block defining the basis. Moreover, the coupling to quasiparticle excitations becomes weaker as the size of the elementary block increases. Thus, the weak-coupling limit of the theory becomes an accurate representation of the underlying mean-field dynamics. We illustrate the method by computing static and dynamic properties of two-leg ladders in the various angular-momentum-coupled bases.
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.
Using the symmetric-group-approach method [N. Flocke and J. Karwowski, Phys. Rev. B 55, 8287 (1997)], exact diagonalization studies on a set of spin-1/2 Heisenberg ladders containing three different couplings JJJ, ${J}^{\ensuremath{'}}$ (chain), and ${J}_{\ensuremath{\perp}}$ (rungs) were performed and the behavior of the singlet-triplet energy gap was analyzed as a function of ${J}^{\ensuremath{'}}$ and ${J}_{\ensuremath{\perp}}$ in units of JJJ. Exact diagonalizations for the ground and first-exited state were performed for up to $2\ifmmode\times\else\texttimes\fi{}L=30$ spins for the isotropic ladder ${(J=J}^{\ensuremath{'}}{=J}_{\ensuremath{\perp}})$ and for up to $2\ifmmode\times\else\texttimes\fi{}L=28$ spins for all the anisotropic ones. All calculations were done under periodic/M\"obius boundary conditions for LLL even/odd, respectively. I found some evidence, that for certain ${J}_{\ensuremath{\perp}},$ ${J}^{\ensuremath{'}}$ pairs the gap may vanish in the bulk limit. For the isotropic ladder I obtained for the gap $\ensuremath{\Delta}(\ensuremath{\infty})=0.49996$, giving further support for the conjecture that for this system the exact value of $\ensuremath{\Delta}(\ensuremath{\infty})$ is equal to 1/2.
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.
We present a manifestly rotational invariant formulation of the matrix product method valid for spin chains and ladders. We apply it to two-legged spin ladders with spins , 1 and and different magnetic structures labelled by the exchange coupling constants, which can be ferromagnetic or antiferromagnetic along the legs and the rungs of the ladder. We compute ground-state energy densities, correlation lengths and string order parameters. We present numerical evidence of the duality properties of the three different nonferromagnetic spin ladders. We show that the long-range topological order characteristic of isolated spin 1 chains is broken by the interchain coupling. The string order correlation function decays exponentially with a finite correlation length that we compute. A physical picture of the spin 1 ladder is given in terms of a collection of resonating spin 1 chains. Finally, for ladders with spin equal to or greater than we define a class of AKLT states whose matrix product coefficients are given by 9- j symbols.