Variational Studies of Triangular Heisenberg Antiferromagnet in Magnetic Field arXiv:1001.2863v2 [cond-mat.str-el] 20 Jul 2010 I. INTRODUCTION Tiamhock Tay and Olexei I. Motrunich Department of Physics, California Institute of Technology, Pasadena, CA 91125 (Dated: October 29, 2018) We present a variational study of the Heisenberg antiferromagnet on the spatially anisotropic triangular lattice in magnetic field. First we construct a simple yet accurate wavefunction for the 1/3-magnetization plateau uud phase on the isotropic lattice. Beginning with this state, we obtain natural extensions to nearby commensurate coplanar phases on either side of the plateau. The latter occur also for low lattice anisotropy, while the uud state extends to much larger anisotropy. Far away from the 1/3 plateau and for significant anisotropy, incommensurate states have better energetics, and we address competition between coplanar and non-coplanar states in the high field regime. For very strong anisotropy, our study is dominated by quasi-1d physics. The variational study is supplemented by exact diagonalization calculations which provide a reference for testing the energetics of our trial wavefunctions as well as helping to identify candidate phases. PACS numbers: The spin-1/2 Heisenberg antiferromagnet on a spa- tially anisotropic 2d triangular lattice is a deceptively simple spin system which nevertheless possesses very rich physics. Despite having attracted much attention, a complete understanding of the model has not been achieved. This can be attributed to the enhanced quan- tum fluctuations arising from a combination of low di- mensionality, small spin, geometrical frustration and spa- tial anisotropy, thus leading to a rich phase diagram. At zero field, studies have suggested that the anisotropic system may possibly remain disordered even at zero tem- perature. In experimental realizations of the triangular antiferromagnet, a 1/3-magnetization plateau was found for the approximately isotropic material Cs$_{2}$CuBr$_{4}$ but not for the more anisotropic Cs$_{2}$CuCl$_{4}$. 1–6 A recent ex- perimental study of Cs$_{2}$CuBr$_{4}$ further revealed a cascade of phases in the fields above the 1/3 plateau, which are still not understood. 7 Analytical studies on the model have been done on specific regions of the phase diagram, for instance low anisotropy near the 1/3-magnetization plateau, 8,9 large anisotropy limit, 10 and high field limit. 11,12 Sev- eral numerical studies using exact diagonalization, 13–16 series expansion, 17,18 coupled-cluster method, 19 den- sity matrix renormalization group, 20,21 and variational approaches 22–25 have also been used to analyze the model. Motivated by the experimental and theoretical works, we perform a variational study using simple yet powerful wavefunctions, attempting to cover a large por- tion of the phase diagram. We consider the Heisenberg model in external mag- netic field h , where ⃗ S$_{r}$ is the spin operator on site r and J$_{rr}$ ′ are the nearest neighbour exchange couplings. Throughout, we FIG. 1: Triangular antiferromagnet with coupling constants J and J ′ between nearest neighbours along horizontal and oblique directions respectively. Three sublattices A , B , and C important in the isotropic case are also labelled. FIG. 2: Spin orderings on the isotropic triangular lattice in the field, where three arrows refer to three sublattices in- dicated in Fig. 1: a) spiral (non-coplanar umbrella); b,c,d) coplanar Y , uud , and V . ˆ H = ∑ 〈 rr ′ 〉 J$_{rr}$ ′ ⃗ S$_{r}$ · ⃗ S$_{r}$ ′ − h ∑ r S z r , (1) J’ J’ J A B C V (d) Y (b) uud (c) Spiral (a) extensively use hard-core boson picture, mapping S + r = b$_{r}$ and S z r = 1 2 − n$_{r}$ : ˆ H = − ∑ 〈 rr ′ 〉 t$_{rr}$ ′ ( b † $_{r}$b$_{r}$ ′ + h . c . ) + ∑ 〈 rr ′ 〉 J$_{rr}$ ′ n$_{r}$n$_{r}$ ′ (2) − µ r n$_{r}$ + const , (3) t$_{rr}$ ′ = − 1 2 J$_{rr}$ ′ , µ = − h + 1 2 ∑ r ′ ∈ r J$_{rr}$ ′ . (4) The boson hopping amplitudes are negative and there- fore frustrated on the triangular lattice, making this a challenging interacting problem. Figure 2 depicts spin ordered phases considered in our