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Physical Review Letters

Long-range order in the 2D antiferromagnetic Heisenberg model: A renormalization perspective

H. Q. Lin, David K Campbell

Published Oct 19, 1992• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.69.2415

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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 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.
A real-space, zero-temperature renormalization-group method is used to study a Heisenberg antiferromagnetic spin chain with long-range interactions varying as ${(\mathrm{distance})}^{\ensuremath{-}p}$. Simple renormalization-group equations are obtained and studied analytically, and the results are checked against a more accurate numerical calculation. The calculations provide evidence for a phase transition at $p\ensuremath{\approx}1.85$ characterized by a change in the behavior of zero-temperature correlation functions. For $p\ensuremath{\gtrsim}1.85$ the large-distance physics is that of the nearest-neighbor antiferromagnet, while there is a line of fixed points for $p\ensuremath{\lesssim}1.85$. I conjecture that $p\ensuremath{\approx}1.85$ also marks an order-disorder transition at finite temperature such as occurs in the Ising model with power-law interactions.
A real space renormalization group method is applied to study the spin-1/2 two-dimensional anisotropic Heisenberg antiferromagnets. We carry out the calculation on a 5×5 cluster by using a variational approach. The ground-state energy per site is estimated as e0=−0.6845±0.0005 which is in good agreement with other numerical estimates. We also calculate the different ground state energies when anisotropy changes. By comparing with 3×3 cluster calculation the resulting threshold for long range order drops and the curve of the renormalization group parameter vs. anisotropy shows no hint of discontinuity over a wide range of anisotropic parameters. Hence, our calculation suggests that long-range order exists for the ground-state of the spin-1/2 Heisenberg antiferromagnets in two dimensions.
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 quantum phase transitions in Heisenberg antiferromagnetic chains with a staggered power-law decaying long-range interactions. Employing the density-matrix renormalization group (DMRG) algorithm and the fidelity susceptibility as the criticality measure, we establish more accurate values of quantum critical points than the results obtained from the spin-wave approximation, quantum Monte Carlo, and DMRG in literatures. The deviation is especially evident for strong long-range interactions. We extend isotropic long-range interactions to the anisotropic cases and find that kaleidoscope of quantum phases emerge from the interplay of anisotropy of the long-range exchange interaction and symmetry breaking. We demonstrate nonfrustrating long-range interactions induce the true long-range order in Heisenberg antiferromagnetic chains with a continuous symmetry breaking, lifting the restrictions imposed by the Mermin-Wagner theorem.