Spin-Wave Theory on Finite Lattices: Application to the
 J
 1
 -
 J
 2
 Heisenberg Model

Using examples of the square-and triangular-lattice Heisenberg models we demonstrate that the density matrix renormalization group method (DMRG) can be effectively used to study magnetic ordering in twodimensional lattice spin models. We show that local quantities in DMRG calculations, such as the on-site magnetization M , should be extrapolated with the truncation error, not with its square root, as previously assumed. We also introduce convenient sequences of clusters, using cylindrical boundary conditions and pinning magnetic fields, which provide for rapidly converging finite-size scaling. This scaling behavior on our clusters is clarified using finite-size analysis of the effective σ-model and finite-size spin-wave theory. The resulting greatly improved extrapolations allow us to determine the thermodynamic limit of M for the square lattice with an error comparable to quantum Monte Carlo. For the triangular lattice, we verify the existence of three-sublattice magnetic order, and estimate the order parameter to be M = 0.205(15).PACS numbers: 74.45.+c,74.50.+r,71.10.Pm Two-dimensional (2D) quantum lattice systems studied in condensed matter physics can be divided into two types: those with a sign problem in quantum Monte Carlo (QMC), and those without one. This is because recent developments in QMC [1,2,3] have enabled remarkably accurate large-scale studies of the latter systems, such as the square-lattice Heisenberg model (SLHM) [4]. In contrast, the former systems, such as the triangular lattice Heisenberg model (TLHM) and other models with geometric frustration, are often the subject of controversy even regarding questions of what type of order, if any, is present. For the TLHM, it is only recently that the rough agreement between several theoretical[5] and numerical [6,7,8] methods has made a convincing case that the model has three-sublattice, non-collinear 120• order.The density matrix renormalization group[9] (DMRG) is not subject to the sign problem, it has an error which can be systematically decreased by keeping more states, and even with modest computational effort it is extremely accurate for one dimensional and ladder systems. For 2D systems, the computational effort grows exponentially with the width. Ameliorating this effect is the very systematic behavior of the DMRG results versus the number of states kept, enabling the use of extrapolations to improve the accuracy. The extrapolation of the energy versus the truncation error ε (also known as the discarded weight) to the limit ε → 0 often can improve the accuracy of the energy by nearly an order of magnitude. For observables other than the energy, extrapolation has been more problematic and is much less used.In this Letter we show that the difficulty in extrapolating local measurements A is due to the incorrect assumption that the error ∆A ∼ ε 1/2 . In fact, the simplest way to measure local quantities within DMRG makes ∆A analytic in ε. The resulting improved extrapolations greatly improve one's ability to measure order parameters in two ...

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“…2(b) by FSSWT from this work and from Ref. 16, and in Fig. 3(b) by QMC using standard correlation function methods, Ref.…”

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confidence: 68%

Neél Order in Square and Triangular Lattice Heisenberg Models

2007

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“…Only quantum fluctuations lift the degeneracy in 0 . In order to estimate the effects of quantum fluctuations on the ground state energy, we made use of lowest order spinwave theory (SWT) for finite-size chains [20] which is free from singularities even in 1D. By applying SWT to the dimerized Heisenberg ring, we obtain the OS expression for E H 0 :…”

Section: Infm and Dipartimento DI Fisica E Matematica Università Delmentioning

confidence: 99%

Magnetoelastic Instability in Molecular Antiferromagnetic Rings

Parola

2004

Phys. Rev. Lett.

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Lattice stability in a model of an antiferromagnetic ring coupled to adiabatic phonons is investigated for different values of the spin and numbers of magnetic sites. The magnetoelastic transition is shown to be heavily affected by the spin value, displaying a qualitative difference in the nature of the instability for spin one-half. Among the different synthesized materials, Cu8 seems to be the best candidate to observe lattice dimerization in these systems. Our analysis excludes stable lattice distortions in higher spin rings. The effects of thermal fluctuations are studied in the Cu8 model, where a characteristic crossover temperature is estimated.

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“…As in the case of C 12 the transition of the ground state between different spin sectors as a function of applied field can be tracked in the complex λ plane. However the trajectory of the zeros of B a.c. N (λ), the off diagonal element in the effective Hamiltonian in equation (10), is different from C 12 where the zeros move from the complex plane onto the real axis at λ = 1 and h = h c and then move away for h > h c as seen in figure 17. These values of λ are branch points of the energy function for the ground and first excited states.…”

Section: Analytic Structurementioning

confidence: 95%

“…We apply the same methods towards the solution as in the twelve-site system case. The elements of the 2 × 2 effective Hamiltonian matrix H ef f in equation (10) are now real for all applied magnetic fields.…”

Section: C20mentioning

confidence: 99%

“…Approximation and numerical techniques that have been used include diagonalization of small clusters [5,6], Monte Carlo techniques [7], cluster expansions [8], spin wave expansions [9][10][11] and the density matrix renormalization group [12]. The methods that consider the full Hilbert space of the problem are limited by the size of the system, since the number of states is exponentially dependent on it.…”

Section: Introductionmentioning

confidence: 99%

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Accurate results from perturbation theory for strongly frustratedS=12Heisenberg spin clusters

Konstantinidis

Coffey

2001

Phys. Rev. B

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We investigate the use of perturbation theory in finite sized frustrated spin systems by calculating the effect of quantum fluctuations on coherent states derived from the classical ground state. We first calculate the ground and first excited state wavefunctions as a function of applied field for a 12-site system and compare with the results of exact diagonalization. We then apply the technique to a 20-site system with the same three fold site coordination as the 12-site system. Frustration results in asymptotically convergent series for both systems which are summed with Padé approximants.We find that at zero magnetic field the different connectivity of the two systems leads to a triplet first excited state in the 12-site system and a singlet first excited state in the 20-site system, while the ground state is a singlet for both. We also show how the analytic structure of the Padé approximants at |λ| ≃ 1 evolves in the complex λ plane at the values of the applied field where the ground state switches between spin sectors and how this is connected with the non-trivial dependence of the < S z > number on the strength of quantum fluctuations. We discuss the origin of this difference in the energy spectra and in the analytic structures. We also characterize the ground and first excited states according to the values of the various spin correlation functions.

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Spin-Wave Theory on Finite Lattices: Application to the J ₁ - J ₂ Heisenberg Model

Cited by 46 publications

References 15 publications

Neél Order in Square and Triangular Lattice Heisenberg Models

Neél Order in Square and Triangular Lattice Heisenberg Models

Magnetoelastic Instability in Molecular Antiferromagnetic Rings

Accurate results from perturbation theory for strongly frustratedS=12Heisenberg spin clusters

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Spin-Wave Theory on Finite Lattices: Application to the J 1 - J 2 Heisenberg Model

Cited by 46 publications

References 15 publications

Neél Order in Square and Triangular Lattice Heisenberg Models

Neél Order in Square and Triangular Lattice Heisenberg Models

Magnetoelastic Instability in Molecular Antiferromagnetic Rings

Accurate results from perturbation theory for strongly frustratedS=12Heisenberg spin clusters

Contact Info

Product

Resources

About

Spin-Wave Theory on Finite Lattices: Application to the J ₁ - J ₂ Heisenberg Model