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arXiv:0912.5400 · Physical Review B

Computational scheme for the spin- 12 Heisenberg antiferromagnet based on an octapartite description of the square lattice

Bayo Lau, Mona Berciu, G. A. Sawatzky

Published May 5, 2010• 2 citations
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Abstract

We introduce a parameterization of the Hilbert space of a spin-$\frac{1}{2}$ 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.

DOI
10.1103/physrevb.81.172401

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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 present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1
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The trial function including a few variational parameters introduced by the consideration of geometrical characters is proposed for the purpose of discussing the exact ground-state wave function of spin-1/2 Heisenberg antiferromagnet on a square lattice. In this wave function, the essential contribution of singlet pairings with larger separations between spin sites rather than the 1st nearest neighbor is taken into account. The obtained solution for spin number N ≤26 on the basis of this extended RVB (resonating-valence-bond) picture comes quite close to the exact solution, though we use only a few geometrical parameters for the systems constituted by many degrees of freedom.
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Exploiting inherent symmetries is a common and effective approach to speed up the simulation of quantum systems. However, efficiently accounting for non-Abelian symmetries, such as the global SU(2) spin symmetry, remains a major challenge. In fact, expressing total-spin eigenstates in terms of the computational basis can require an exponentially large number of coefficients. In this work, we introduce a novel formalism for designing quantum algorithms directly in an eigenbasis of the total-spin operator. Our strategy relies on the symmetric group approach in conjunction with a truncation scheme for the internal degrees of freedom of total-spin eigenstates. For the case of the antiferromagnetic Heisenberg model, we show that this formalism yields a hierarchy of spin-adapted Hamiltonians, for each truncation threshold, whose ground-state energy and wave function quickly converge to their exact counterparts, calculated on the full model. These truncated Hamiltonians can be encoded with sparse and local qubit Hamiltonians that are suitable for quantum simulations. We demonstrate this by developing a state-preparation schedule to construct shallow quantum-circuit approximations, expressed in a total-spin eigenbasis, for the ground states of the Heisenberg Hamiltonian in different symmetry sectors.