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arXiv:cond-mat/0312487 · Physical Review Letters

Quantum Spins and Quasiperiodicity: A Real Space Renormalization Group Approach

Anuradha Jagannathan

Published Jan 28, 2004• 54 citations
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Abstract

We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.

DOI
10.1103/physrevlett.92.047202

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We study the antiferromagnetic spin-1/2 Heisenberg model on a two-dimensional bipartite quasiperiodic structure, the octagonal tiling, the aperiodic equivalent of the square lattice for periodic systems. An approximate block spin renormalization scheme is described for this problem. The ground state energy and local staggered magnetizations for this system are calculated and compared with the results of a recent quantum Monte Carlo calculation for the tiling. It is conjectured that the ground state energy is exactly equal to that of the quantum antiferromagnet on the square lattice.
We consider the antiferromagnetic spin-1∕21∕21∕2 Heisenberg model on a two-dimensional bipartite quasiperiodic tiling. The broken symmetry ground state in this model is inhomogeneous, reflecting the fact that there are a variety of local environments in such a structure. An important symmetry of the quasicrystal, namely that of invariance under discrete scale transformations is used to define an approximate real space renormalization scheme for the octagonal tiling. We solve for some of the fixed point properties of this quasiperiodic antiferromagnet. The ground state energy and local order parameters are calculated, and the results compare favorably with numerical values obtained by quantum Monte Carlo calculation. Despite the novel features of the ground state in this type of antiferromagnet, there are some interesting similarities with the well-known square lattice antiferromagnet. The most striking of these is the proximity of the values of the ground state energies of these two paradigms for two very dissimilar classes of solids.
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 density matrix renormalization group (DMRG) approach is presented for quantum systems of two spatial dimensions. In particular, it is shown that it is possible to create a multichain-type two-dimensional (2D) DMRG approach that utilizes previously determined system and environment blocks at all points. One first builds up an effective quasi-1D system and environment blocks of width L and these quasi-1D blocks are then used to as the initial building blocks of the 2D infinite-lattice algorithm introduced here. This algorithm is found to be competitive with those results of previous 2D DMRG algorithms and also of the best of other approximate methods. An illustration of this is given for the spin-half and spin-one Heisenberg models on the square lattice. The best results for the ground-state energies per bond of the spin-half and spin-one square-lattice Heisenberg antiferromagnets for the $N=20\ifmmode\times\else\texttimes\fi{}20$ lattice using this treatment are given by ${E}_{g}{/N}_{B}=\ensuremath{-}0.3321$ and ${E}_{g}{/N}_{B}=\ensuremath{-}1.1525,$ respectively.
We study the problem of the existence of long-range staggered magnetic order in the antiferromagnetic Heisenberg model on the square lattice by a renormalization group analysis. We first map the Heisenberg model onto an approximate effective model, described by the Hamiltonian: H = ∑Lj=1S(j)S(j + 1) + ∑Lj=1( − 1)j‖J(j)‖S(0)S(j), where S(0) represents a single spin located in the center of the ring. We show that the Ji tend to non-zero values as one increases the block size by successive renormalizations, thus establishing the existence of long-range staggered magnetic order in the effective model. We discuss the implications of this result for the original Heisenberg model.
A real space renormalization group method is applied to study the properties of the anisotropic and alternating Heisenberg antiferromagnets with spin 1/2, 1 and 3/2. To improve the accuracy of the numerical estimates larger spin cluster is used. The ground state energy and the energy gap are obtained. For the case of S = 1 at Heisenberg point, the ground state energy per site is E (0)/N = -1.4021 ± 0.0005 and the energy gap is 0.4097 ± 0.0005, in excellent agreement with a recent Monte Carlo calculation. For the cases of spin 1/2 and 3/2 the energy gap is absent. Therefore our results are shown to support Haldane's conjecture. We briefly discuss the properties of the novel phase.