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arXiv:cond-mat/9507060 · Physical review. B, Condensed matter

Variational Hilbert-space-truncation approach to quantum Heisenberg antiferromagnets on frustrated clusters

Normand A. Modine, Efthimios Kaxiras

Published Feb 1, 1996• 23 citations
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Abstract

We study the spin-1/2 Heisenberg antiferromagnet on a series of finite-size clusters with features inspired by the fullerenes. Frustration due to the presence of pentagonal rings makes such structures challenging in the context of quantum Monte Carlo methods. We use an exact diagonalization approach combined with a truncation method in which only the most important basis states of the Hilbert space are retained. We describe an efficient variational method for finding an optimal truncation of a given size which minimizes the error in the ground-state energy. Ground-state energies and spin-spin correlations are obtained for clusters with up to 32 sites without the need to restrict the symmetry of the structures. The results are compared to full-space calculations and to unfrustrated structures based on the honeycomb lattice. \textcopyright{} 1996 The American Physical Society.

DOI
10.1103/physrevb.53.2546

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We study the spin-1/2 Heisenberg antiferromagnet on a series of finite-size clusters with features inspired by the fullerenes. Frustration due to the presence of pentagonal rings makes such structures challenging in the context of quantum Monte Carlo methods. We use an exact diagonalization approach combined with a truncation method in which only the most important basis states of the Hilbert space are retained. We describe an efficient variational method for finding an optimal truncation of a given size which minimizes the error in the ground-state energy. Ground-state energies and spin-spin correlations are obtained for clusters with up to 32 sites without the need to restrict the symmetry of the structures. The results are compared to full-space calculations and to unfrustrated structures based on the honeycomb lattice. \textcopyright{} 1996 The American Physical Society.
The antiferromagnetic S=1/2S=1/2S=1/2 Heisenberg model on the C60_{60}60​ fullerene geometry
We solve the quantum-mechanical antiferromagnetic Heisenberg model with spins positioned on vertices of the truncated icosahedron using the density-matrix renormalization group (DMRG). This describes magnetic properties of the undoped C _{60} 60 fullerene at half filling in the limit of strong on-site interaction U U . We calculate the ground state and correlation functions for all possible distances, the lowest singlet and triplet excited states, as well as thermodynamic properties, namely the specific heat and spin susceptibility. We find that unlike smaller C _{20} 20 or C _{32} 32 that are solvable by exact diagonalization, the lowest excited state is a triplet rather than a singlet, indicating a reduced frustration due to the presence of many hexagon faces and the separation of the pentagonal faces, similar to what is found for the truncated tetrahedron. This implies that frustration may be tuneable within the fullerenes by changing their size. The spin-spin correlations are much stronger along the hexagon bonds and exponentially decrease with distance, so that the molecule is large enough not to be correlated across its whole extent. The specific heat shows a high-temperature peak and a low-temperature shoulder reminiscent of the kagomé lattice, while the spin susceptibility shows a single broad peak and is very close to the one of C _{20} 20 .
We study a variational wave function for the ground state of the two-dimensional S=1∕2S=1∕2S=1∕2 Heisenberg antiferromagnet in the valence bond basis. The expansion coefficients are products of amplitudes h(x,y)h(x,y)h(x,y) for valence bonds connecting spins separated by (x,y)(x,y)(x,y) lattice spacings. In contrast to previous studies, in which a functional form for h(x,y)h(x,y)h(x,y) was assumed, we here optimize all the amplitudes for lattices with up to $32\ifmmode\times\else\texttimes\fi{}32$ spins. We use two different schemes for optimizing the amplitudes; a Newton conjugate-gradient method and a stochastic method which requires only the signs of the first derivatives of the energy. The latter method performs significantly better. The energy for large systems deviates by only $\ensuremath{\approx}0.06%$ from its exact value (calculated using unbiased quantum Monte Carlo simulations). The spin correlations are also well reproduced, falling $\ensuremath{\approx}2%$ below the exact ones at long distances (corresponding to an $\ensuremath{\approx}1%$ underestimation of the sublattice magnetization). The amplitudes h(r)h(r)h(r) for valence bonds of long length rrr decay as ${r}^{\ensuremath{-}3}$. We also discuss some results for small frustrated lattices.
The sign problem breaks the polynomial scaling of Quantum Monte Carlo methods. We alleviate it by rotation of the local basis of the Hilbert space, such that the phase of off-diagonal matrix elements of the rotated Hamiltonian is minimized. These minima coincide with optima of the average sign. We benchmark our method for frustrated Heisenberg antiferromagnets in one dimension and on the two-dimensional maple-leaf lattice. Our approach reveals more efficient bases with improved sign in a large part of the parameter space for all models considered, enabling us to reach lower temperatures, on par with state-of-the-art numerical linked cluster expansions which we directly compare to.
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 wave functions 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 threefold site coordination as the 12-site system. Frustration results in asymptotically convergent series for both systems which are summed with Pad\'e 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 Pad\'e approximants at $|\ensuremath{\lambda}|\ensuremath{\simeq}1$ evolves in the complex $\ensuremath{\lambda}$ plane at the values of the applied field where the ground state switches between spin sectors and how this is connected with the nontrivial dependence of the 〈Sz〉〈{S}^{z}〉〈Sz〉 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.
We introduce a parameterization of the Hilbert space of a spin-12\frac{1}{2}21​ Heisenberg antiferromagnet using an octapartite description of the square lattice. This provides a systematic way to model the ground-state wave function within a truncated basis. To prove its effectiveness we study systems with up to 64 spins using exact diagonalization. At significantly reduced computational cost, we get ground-state energies within 1% of the best published values. Its noniterative nature and lack of exploitation of spatial symmetries make this approach suitable for generalization to doped systems and for integration into iterative procedures.
The exact many-body states of a 10-site square cluster with periodic boundary conditions of an S=(1/2 antiferromagnetic Heisenberg lattice model are investigated. A method that takes full advantage of the symmetries of the system (the cluster space-group symmetry and the total spin symmetry) greatly reduces the size of the Hamiltonian matrix and gives an analytic solution. A general form of a resonating-valence-bond wave function, written in a fermion representation, is examined. It is shown that the optimum variational wave function has complex singlet bond distribution amplitudes of s-wave + imaginary d-wave symmetry. It has a very large overlap with the ground state (0.995), belongs to the same space-group representation, and its energy functional has a minimum very close to the ground-state energy. On the other hand, if the amplitudes are restricted to real numbers, the local minimum has slightly higher energy and is nondegenerate but has an infinite manifold of equivalent representations in terms of projected BCS states related by a local SU(2) gauge transformation.