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Journal of Applied Physics

A renormalization group study of the 2D antiferromagnetic Heisenberg model

Hua Lin, David K Campbell, C. Y. Pan

Published May 15, 1993• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.352714

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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.
We perform a real-space renormalization-group analysis for the two-dimensional antiferromagnetic Heisenberg model. We first derive an effective model, described by the Hamiltonian: H=J Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​ S(j)\ensuremath{\cdot}S(j+1)+ Jj=1L{\mathcal{J}}_{\mathit{j}=1}^{\mathit{L}}Jj=1L​(-1)j{)}^{\mathit{j}})j\ensuremath{\Vert}J(j)\ensuremath{\Vert}S(0)\ensuremath{\cdot}S(j), where S(0) represents a single spin located in the center of the ring. We show that the renormalized coupling constants \ensuremath{\Vert}J(j)\ensuremath{\Vert} tend to nonzero values as one increases the block size by successive renormalization. Using analytic and numerical arguments, we establish the existence of antiferromagnetic long-range order in the effective model and hence, in the original Heisenberg model. We compare our results with those obtained from perturbation theory, spin-wave theory, and exact-diagonalization calculations.
Applying a renormalization-group (RG) procedure to the 2D antiferromagnetic Heisenberg (AFH) model, we derive an effective AFH model in which spins on a 1D ring experience antiferromagnetic nearest-neighbor interactions and iteract individually with an effective central spin. Using analytic and numerical arguments, we establish the existence of staggered long-range order in the effective model and hence, given our RG prescription, in the 2D AFH. Our RG calculations give a value of the magnetic moment consistent with Monte Carlo results for the 2D AFH.
We propose an efficient numerical method to obtain local order parameter in two-dimensional systems using spiral boundary conditions. As a benchmark, we first estimate the magnitude of staggered magnetization for the S=1/2S=1/2S=1/2 XXZ Heisenberg model on a square lattice in the whole range of the XXZ anisotropy by density-matrix renormalization group technique. The validity of our method is confirmed by comparing our results with the previous analytical and numerical studies. Then, as further demonstration, we apply our method to obtain the staggered magnetization for the higher-spin cases from S=1S=1S=1 to S=6S=6S=6. The accuracy of the obtained results is validated using the series expansion and the spin-wave theory.
We give a rigorous proof of the existence of long-range antiferromagnetic order in the ground state of several two-dimensional spin-1/2 Heisenberg systems.Three types of systems are considered.The first type consists of an even number, N , of coupled square lattices with antiferromagnetic nearest-neighbor Heisenberg interactions.In this case, we prove the existence of long-range order (LRO) in the ground state for an interplane to inplane coupling ratio r in the range 0.16 ≤ r ≤ 2.1 for N ≥ 4. Further, we prove that the antiferromagnetic bilayer with ferromagnetic next-nearest-neighbor (nnn) interplane couplings possesses LRO in the ground state for 0.21 ≤ r < 1/4, where r is the absolute value of the ratio of the ferromagnetic and antiferromagnetic couplings.The final example consists of two antiferromagnetic spin-1/2 square lattices that are coupled via an antiferromagnetic nnn interplane coupling r.For r = 1, this system is effectively a spin-1 square lattice and has an ordered ground state.We show that LRO exists in the region 0.85 ≤ r ≤ 1. * ) Note that E r q in the notation of KLS corresponds to 2 + r -E q in our notation.
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.
In the presence of a staggered magnetic field, the plaquette expansion of the Lanczos matrix elements are obtained for the antiferromagnetic 2D Heisenberg model up to order 1/Np (Np is the number of plaquettes on the lattice). The resulting approximate tri-diagonal form of the Hamiltonian is diagonalised for various values of the field strength in the -> 00 limit for the ground state energy density. From the behaviour of the ground-state energy density at weak fields, the staggered magnetisation at this order in the plaquette expansion is found to be 0·71 (in units where the Neel state staggered magnetisation is 1· 0).