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arXiv:cond-mat/9808310 · Physical review. B, Condensed matter

Mean-field theory for spin ladders using angular-momentum coupled bases

J. Piekarewicz, James R. Shepard

Published Oct 1, 1999• 8 citations
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Abstract

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.

DOI
10.1103/physrevb.60.9456

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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 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 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.
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.
Two-leg t-J ladders are investigated in the framework of a combination of the phass string formulation and bond-operator representation. We develope a mean-field theory in the strong rung interaction regime, i.e. J┴ ＞＞ J, t, which provides a unified description of the undoped insulating phase and the low doping spin gap phase. The ground state of the doped phase is intrinsically a superconductor with a d-wave symmetry gap driven by the t- term. The ground-state energy is in good agreement with numerical results. Phase separation is shown to occur beyond some critical value of J/t for given doping concentration. The local structure of hole pairs as well as the spectra of various spin and charge modes are also analyzed.
Based on the generalized valence bond picture, a Schwinger boson mean-field theory is applied to the symmetric SU(4) spin-orbital systems. For a two-leg SU(4) ladder, the ground state is a spin-orbital liquid with a finite energy gap, in good agreement with recent numerical calculations. In two-dimensional square and triangle lattices, the SU(4) Schwinger bosons condense at $(\ensuremath{\pi}/2,\ensuremath{\pi}/2)$ and $(\ensuremath{\pi}/3,\ensuremath{\pi}/3),$ respectively. Spin, orbital, and coupled spin-orbital static susceptibilities become singular at the wave vectors; the Bose condensation arises at twice this value. It is also demonstrated that there are spin, orbital, and coupled spin-orbital long-range orderings in the ground state.
Two-leg $t\ensuremath{-}J$ ladders are investigated in the framework of a combination of the phase string formulation and bond-operator representation. We develop a mean-field theory in the strong rung interaction regime, i.e., ${J}_{\ensuremath{\perp}}\ensuremath{\gg}J,t,$ which provides a unified description of the undoped insulating phase and the low-doping phase---the so-called C1S0C1S0C1S0 phase. Both of them are characterized by the resonating-valence-bond (RVB) order parameter, with gap opened up in all spin excitations. The ground state of the doped phase is intrinsically a superconductor with a d-wave symmetry, which is driven by the RVB correlations. The ground-state energy is in good agreement with numerical results. Phase separation is shown to occur beyond some critical value of J/tJ/tJ/t for given doping concentration. The local structure of hole pairs as well as the spectra of various spin and charge modes are analyzed in comparison with other approaches.