TheS=1/2 Heisenberg antiferromagnet on the triangular lattice: Exact results and spin-wave theory for finite cells

Abstract

We study the ground state properties of the S= 1 2 Heisenberg antiferromagnet (HAF) on the triangular lattice with nearest-neighbour (J) and next-nearest neighbour (αJ) couplings. Classically, this system is known to be ordered in a 120 • Néel type state for values −∞ < α ≤ 1/8 of the ratio α of these couplings and in a collinear state for 1/8 < α < 1. The order parameter M and the helicity χ of the 120 • structure are obtained by numerical diagonalisation of finite periodic systems of up to N = 30 sites and by applying the spin-wave (SW) approximation to the same finite systems. We find a surprisingly good agreement between the exact and the SW results in the entire region −∞ < α < 1/8. It appears that the SW theory is still valid for the simple triangular HAF (α = 0) although the sublattice magnetisation M is substantially reduced from its classical value by quantum fluctuations. Our numerical results for the order parameter N of the collinear order support the previous conjecture of a first order transition between the 120 • and the collinear order at α ≃ 1/8.

Phase diagram of thespin−12triangularJ1−J2Heisenberg model on a three-leg cylinder

“…As a result a nonchiral and planar system should acquire values close to zero for this triple product. Some studies [54,58] predict that the THM should be chiral in some circumstances (e.g., considering couplings higher than two-body exchange interactions), while the others [18,26,27,29] strongly suggest that the quantum fluctuations always select a planar spin arrangement, so there is no chiral symmetry breaking.…”

“…For several decades, there has been much discussion [18,26,27,29,58] on the possibility of chiral order in the 2D model. A proper chiral order parameter will detect breaking of P and T symmetry of the wave function while the system preserves PT symmetry.…”

“…The ladder model has clear connections to the 2D THM and also extrapolates smoothly to the Majumdar-Ghosh point of the zig-zag chain. The J 1 -J 2 THM in two dimensions has been previously studied using semiclassical spin-wave theories (SWTs) and exact diagonalization (ED) [18,19,[26][27][28][29], but these studies did not cover the physics of the whole phase diagram. Recently, a coupled cluster method (CCM) study [30] and quantum Monte Carlo (QMC) studies [31,32] have identified a phase in this model that is a candidate for a spin liquid.…”

“…Exact diagonalization methods [18,[26][27][28]34] suffer from the exponentially growing size of the Hilbert space, which is especially a problem in two or more dimensions, while QMC techniques [35] suffer from the sign problem for frustrated lattices, and projected entangled pair states (PEPSs) [36,37] for this model are complex and computationally costly, even though, in principle, PEPSs have good computational scaling properties in two dimensions. More recently some numerical methods have been developed that are especially useful for frustrated systems and applied to the THM, for example, the large-scale parallel tempering Monte Carlo [38] and some tensor networks methods including entangled-plaquette states [39] and the multiscale entanglement renormalization ansatz (MERA) [40].…”

Frustrated Quantum Magnets

“…Chubukov and Jolicoeur [43] and Korshunov [44] have then shown that quantum fluctuations (evaluated in a spin wave approach) could, like thermal ones, lift this degeneracy of the classical ground states and lead to a selection of the collinear state (see Fig. 3.1) [45]. The first study of the exact spectrum of Eq.…”