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arXiv:cond-mat/0411541 · Physical Review B

Renormalized excitonic method in terms of block excitations: Application to spin lattices

Mohamad Al Hajj, Jean‐Paul Malrieu, Nathalie Guihéry

Published Dec 9, 2005• 24 citations
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Abstract

Dividing the lattice into blocks with a singlet ground state and knowing the exact low-energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the one-dimensional (1D) frustrated chain and the $1∕5$-depleted square and the plaquette 2D lattices. The method correctly locates the phase transitions between gapped and nongapped phases.

DOI
10.1103/physrevb.72.224412

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Dividing the lattice into blocks with a singlet ground state and knowing the exact low-energy spectrum of the blocks and of dimers (or trimers) of blocks, it is possible to approach the lowest part of the lattice spectrum through an excitonic type effective model. The potentialities of the method are illustrated on the one-dimensional (1D) frustrated chain and the 1∕51∕51∕5-depleted square and the plaquette 2D lattices. The method correctly locates the phase transitions between gapped and nongapped phases.
The real-space renormalization group for the study of periodic systems goes through the reduction of the Hilbert space to products of lowest eigenstates of identical blocks. Instead of working with the Hamiltonian, it is possible to estimate effective interactions between blocks from the spectrum of dimers or trimers of blocks, according to the effective Hamiltonian theory of Bloch. Tests on a series of spin problems [cohesive energy of the one-dimensional (1D) chain and of the 2D square lattice, excitation energies of the dimerized 1D spin chain, and behavior of the frustrated chain in the low frustration regime] show the potentiality of the method.
A variant of White's density-matrix renormalization-group scheme, which is designed to compute low-lying energies of one-dimensional quantum lattice models with a large number of degrees of freedom per site is described. The method is tested on two exactly solvable models---the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg chain and a dimerized XY\mathrm{XY}XY spin chain. To illustrate the potential of the method, it is applied to a model of spins interacting with quantum phonons. It is shown that the method accurately resolves a number of energy gaps on periodic rings that are sufficiently large to afford an accurate investigation of critical properties via the use of finite-size scaling theory.
The Hubbard Hamiltonian with infinite on-site repulsion and negative transfer is diagonalized numerically on 3×3 frustrated lattices, such as the square lattice with next-nearest-neighbor transfer and the triangular lattice. The number of electrons is chosen to be less by just one electron than that in the exactly half-filled band. When the strength of the frustration is sufficiently large, a singlet state appears as the ground state. Then, the total spin of the lowest excited state is unity. There is an excitation gap for the square lattice but not for the triangular lattice.
We study the ground state of one- and two-dimensional (square-lattice) spin-1/2 quantum antiferromagnets using a numerical real-space renormalization-group (RG) approach. In our RG approach we consider blocks of various sizes but with an odd number of sites; we retain only the doublet ground state and we integrate out the higher-energy states by means of second-order quasidegenerate perturbation theory. That is, we assume that the role of the excited states of a block, in the RG iteration process, is to renormalize the effective coupling parameters between blocks. We compute the ground-state energy of a spin-1/2 linear chain for various block sizes and find close agreement with the Bethe-ansatz exact solution. In the case of the spin-1/2 square-lattice quantum antiferromagnet, the obtained ground-state energy is in reasonable agreement with the available numerical estimates.
A zero-temperature lattice renormalization-group calculation is presented and applied to the antiferromagnetic one-dimensional dimerized spin-\textonehalf{} Heisenberg chain. The calculation indicates the existence of a gap in the spectrum of the dimer system, which vanishes only in the limit of a uniform spin chain. To improve the accuracy of the numerical estimates of the ground-state energy and gap, an iterative extension of finite-chain techniques is also applied. These approximate techniques are checked against the dimerized XY\mathrm{XY}XY chain, for which an exact solution is found.