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arXiv:1004.2330

Independent Bond Fluctuation Approximation to the Ground State of Quantum Antiferromagnets

Christian Rischel

Published Apr 14, 2010• 0 citations
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Abstract

A simple approach to estimation of the ground state energy of quantum antiferromagnets is developed, based on the approximation that quantum fluctuations around different bonds are independent. The ground state energy estimates are as good as spin wave theory or slightly better. A canonical transformation of the spin operators to generate bond quantum fluctuations is devised and applied to the classical ground state of the $S=\frac{1}{2}$ Heisenberg model on the square lattice. This simple picture of quantum spin fluctuations might be useful in more complex models. The resulting nearest neighbor and next-nearest neighbor correlations can be used in an alternative derivation of spin waves in the Heisenberg model, giving an accurate dispersion as well as raising and lowering operators.

DOI
10.48550/arxiv.1004.2330

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A simple approach to estimation of the ground state energy of quantum antiferromagnets is developed, based on the approximation that quantum fluctuations around different bonds are independent. The ground state energy estimates are as good as spin wave theory or slightly better. A canonical transformation of the spin operators to generate bond quantum fluctuations is devised and applied to the classical ground state of the S=12S=\frac{1}{2}S=21​ Heisenberg model on the square lattice. This simple picture of quantum spin fluctuations might be useful in more complex models. The resulting nearest neighbor and next-nearest neighbor correlations can be used in an alternative derivation of spin waves in the Heisenberg model, giving an accurate dispersion as well as raising and lowering operators.
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
We consider a resonating-valence-bond state as a trial wave function for the ground state of the antiferromagnetic Heisenberg model on the square lattice. The energy and the staggered spin-spin correlation functions are calculated on a small lattice and are compared with the results of the exact diagonalization of Oitmaa and Betts. The trial energy is only about 4.7% different from the exact ground-state energy on a 4\ifmmode\times\else\texttimes\fi{}4 lattice. The importance of the singlet pairs between the next-nearest antiferromagnetic sites is inferred. An argument against the previous interpretation of the numerical data as being evidence of long-range order is presented.
We have developed a nonlocal functional of the exchange interaction for the ground-state energy of quantum spin chains described by the Heisenberg Hamiltonian. An alternating chain is used to obtain the correlation energy and a local unit-cell approximation is defined in the context of the density-functional theory. The agreement with our exact numerical data, for small chains, is significantly better than a previous formulation, even for chains with several ferromagnetic or antiferromagnetic bond defects. The results can be particularly relevant in the study of finite spin-1/2 Heisenberg chains, with exchange couplings changing, magnitude, or even sign, from bond-to-bond.
Two-dimensional spin 1/2 antiferromagnetic Heisenberg models are numerically studied using Entanglement Perturbation Theory, where the ground state wave function is described by a product of local matrices defined at every site, and each matrix is optimized variationally to minimize the energy. We first apply this method for the spin 1/2 antiferromagnetic Heisenberg model on the square lattices to reproduce correctly the known ground state energy and the spin structure factors. Then, we study the spin 1/2 antiferromagnetic Heisenberg model on the triangular lattice with spatially anisotropic nearest neighbor couplings, J and J ', to establish the ground state phase diagram as a function of J'/ J.
In this thesis we consider quantum fluctuations in the ground state energy of antiferromagnetic spin configurations, and phase transitions between these configurations. We use a Holstein-Primakoff transformation up to second order to transform our spin operators, and a Bogoliubov transformation to diagonalize the corresponding Hamiltonian. First we consider a Hamiltonian with Heisenberg-exchange interaction, a magnetic field and anisotropy, for which we investigated both the the antiferromagnetic phase and the spin-flop phase. For the antiferromagnetic phase we find that quantum fluctuations lower the ground state energy. For the spin-flop phase we find that quantum fluctuations lower the ground state energy even more, such that the phase transition point between these phases is shifted to a slightly lower strength of the magnetic field. Furthermore, this shifted phase transition is caused by ground state energies of the spin-flop phase that become complex, instead of an energy crossing. This energy takes on complex values due to imaginary eigenvalues for magnons, which correspond to exponentially increasing semi-classical spin waves.\n\nNext we considered the Dzyaloshinskii-Moriya interaction (DMI) in one dimension, with the spiral phase it induces. We focused on the magnetic frustration of this phase due to anisotropy and the magnetic field. We used a superposition of two plane waves to describe the deviation due to this frustration, and minimized the ground state energy for their amplitudes numerically. By assuming only spirals of integer length and periodic boundary conditions, we constructed a semi-classical ground state phase diagram between the three mentioned phases. This phase diagram suggests a triple point between these phases when the DMI is zero, at the semi-classical phase transition point between the spin-flop phase and antiferromagnetic phase.
The variational wave function proposed by Bartkowski to treat the ground state of a quantum antiferromagnet is reexamined. The fundamental equations of this method, for the energy and magnetization, are formulated in terms of a linked cluster expansion. It is found that the energy which results from the numerical minimization of the formula derived by Bartkowski is lower than originally reported. The energy is further improved by including higher order linked diagrams, which gives the energy of E0=−0.3338J per bond and magnetization of 0.356 for the square lattice. The same wave function is next used to treat the motion of a single hole in a quantum antiferromagnet. Its motion becomes possible due to the local coupling to quantum fluctuations. It is found that the spectrum is dominated by the incoherent processes, but a coherent propagating mode may be also identified. The quasiparticle dispersion is only weakly modified by quantum fluctuations present on the neighbouring bonds.